CN100590421C - System and method for phase measurement - Google Patents

Abstract

The preferred embodiments of the present invention are primarily directed to phase measurement systems that address the phase noise problem through a combination of several strategies including, but not limited to, common-path interferometry, phasing references, active stabilization, and differential measurement. Embodiments relate to optical devices for imaging small organisms with light. These embodiments can be applied, for example, in the fields of cell physiology and neuroscience. These preferred embodiments are based on the principles of phase measurement and imaging techniques. Scientific motivation for using phase measurement and imaging techniques stems, for example, from sub-micron levels of cell biology, which can include, without limitation, imaging sources of dysplasia, cellular communication, neuronal transmission, and genetic code execution. The structure and dynamics of subcellular components have not been universally studied in their natural state using existing methods and techniques (including, for example, X-ray and neutron scattering). In contrast, lightwave-based techniques with nanometer resolution allow the study of cellular structures in their native state. Thus, a preferred embodiment of the invention comprises a system based on the principle of interferometry and/or phase measurement and is used to study cell physiology. The system comprises the principle of Low Coherence Interferometry (LCI) using optical interferometry for measuring phase, or the principle of Light Scattering Spectroscopy (LSS) in which interference within cellular components is used, or in the alternative, the principles of LCI and LSS can be combined to obtain the system of the invention.

Description

Systems and methods for phase measurement

technical field

This application is a successor-in-part of U.S. Patent Application No. 10/823,389 (filed April 13, 2004), which is a successor-in-part of U.S. Patent Application No. 10/024,455 (filed December 2001 18), and claims the benefit of US Patent Provisional Application No. 60/479,732 (filed June 19, 2003).

The entire content of the above application is hereby incorporated by reference in its entirety.

Background technique

Phase-based optical interferometry techniques have been widely used for optical distance measurements requiring subwavelength distance sensitivity. Optical distance is defined as the product of refractive index and length. However, most such techniques are limited by problems well known in the field to 2π ambiguity or integer ambiguity which can be defined as describing the difficulty in axially scanning interferograms moving away from each other. Low-coherence interferometry (LCI) based on the unmodified harmonic phase can be used to determine the difference optical distance (n _λ2 -n _λ1 )L, where L is the actual distance and n _λ1 and n _λ2 are in the respective The refractive index at the wavelength λ ₁ , λ ₂ , if the optical distance is gradually increased so that the difference phase measured with LCI can be tracked by its 2π superposition. In order to determine (n _λ2 −n _λ1 ) of DNA in a solution, for example, the DNA concentration is gradually increased in a measuring tube. Although such a measurement works well in a controlled environment, it cannot be achieved when the sample is less manipulable. For example, this method does not work on fixed slab material that is forced to be preserved intact.

The problem is that unmodified LCI cannot account for the fact that the axially scanned interferograms are far from each other, described here as a 2π ambiguity problem. This is the problem that plagues most phase-based optical interferometry techniques. Therefore, these techniques cannot fully determine the optical distance. Therefore, most such techniques are used in applications such as computing the structure of continuous surfaces or detecting time-varying distance changes, where the phase unwrapping is achieved by the Comparisons are possible.

In many applications, it is important to quantitatively measure the phase of light waves transmitted through or reflected from a sample. Specifically, phased light waves transmitted through or reflected from biological samples can form effective probes of structure and function in living or non-living cells.

Interferometry is a widely used technique for measuring the phase of light waves. A common problem with quantitative interferometry is the sensitivity to phase noise due to external perturbations such as vibrations, air motion, and thermal drift. Therefore, there remains a need for a phase measurement system that addresses the phase noise problem.

Interferometry is one way to obtain phase information associated with a sample. Techniques such as phase contrast and Nomarski microscopy use optical phase only as a contrast element and do not provide quantitative information about its magnitude. Several techniques exist for measuring the phase of light waves transmitted through a nearly transparent sample. These include digital recording interferometric microscopy (DRIMAPS) and non-interferometric detection of phase profiles delivered via intensity equations.

Reflection interferometry can have a sensitivity much smaller than the wavelength of the light wave used. Measurements on the scale of fractions of a nanometer or smaller are common in metrology and microstructural characterization. However, little work has been done using nanoscale interferometry on weakly reflective samples such as biological cells and tissues. Optical coherence tomography (OCT) - an interferometry technique using biological samples - is primarily related to amplitude rather than phase from interference of reflected light waves, so is limited in resolution to the coherence length of the light waves used , usually 2-20 microns.

Phase-inducing reflection interferometry has been used to measure volume changes in monolayers of cells. The harmonic phase-based interferometer used requires two light sources, is relatively low (5 Hz), and has a phase sensitivity of about 20 mrad across the bandwidth. Therefore, there remains a need for efficient phase measurement systems that address the phase noise issue and aid in the development of different imaging applications.

Contents of the invention

Preferred embodiments of the invention relate to phase measurement systems that deal with problems such as phase noise, for example, using several strategies including, but not limited to, co-directional optical path interferometry, phase referencing, active stabilization, and differential measurement combination. Embodiments relate to optical devices for imaging tissue or small organisms with light waves. These embodiments can be applied, for example, to the fields of cell physiology and neuroscience. The preferred embodiment is based on the principles of phase measurement and imaging techniques. The scientific motivation to use phase measurement and imaging techniques arises, for example, from cell biology at the submicron level, which can include without limitation imaging sources of developmental abnormalities, cellular communication, neuronal transmission, and execution of programs using the genetic code. The structure and dynamics of subcellular components cannot currently be studied in their native state using existing methods and techniques (including, for example, X-ray and neutron scattering). Conversely, nanometer-resolution light-wave-based techniques enable cellular machinery to be studied in its natural state. Accordingly, preferred embodiments of the present invention include systems based on the principles of interferometry and/or phase measurement and used to study cell physiology. These systems include low-coherence interferometry (LCI), which uses optical interferometers to measure phase, or where the principle of interferometric light-wave scattering spectroscopy (LSS) is used within the cellular components themselves, or, in the alternative, LCI and LSS The principles of can be combined to lead to the system of the present invention.

Preferred embodiments of phase measurement and imaging systems include actively stabilized interferometers, isolation interferometers, co-directional optical path interferometers, and can include phase contrast microscopy using spatial light wave modulation.

In a preferred embodiment, the method of the present invention involves the technique of precise phase-based measurement of arbitrarily long optical distances, preferably with sub-nanometer precision. A preferred embodiment of the invention uses an interferometer with harmonically related sources (one continuous wave (CW) and a second source with low coherence (LC)), eg, a Michelson interferometer. Low coherence sources provide broad spectral bandwidth, preferably greater than 5 nm for a wavelength of 1 micrometer (μm), eg, the necessary bandwidth can vary with wavelength and application. By fine-tuning the central wavelength of the low-coherence light source between scans of the target sample, the phase relationship between the heterodyne signals of the CW and low-coherence light waves can be used to measure the separation between reflective interfaces with sub-nanometer precision. Because this method is completely free of the 2π ambiguity that plagues most phase-based techniques, it can be used to measure arbitrarily long optical distances without loss of accuracy. An application of a preferred embodiment of the method of the invention is the precise determination of the refractive index of a sample at a given wavelength of known actual thickness. Another application of the preferred embodiment of the method of the present invention is the precise determination of the actual thickness of a sample using a known index of refraction. A further application of the preferred embodiment of the method of the invention is the precise determination of the refractive index ratio at two given wavelengths.

In other possible preferred embodiments, the low-coherence light source provides light waves with a bandwidth sufficiently wide, preferably greater than 5 nm, to simultaneously provide a first low-coherence wavelength and a second low-coherence wavelength with respective center wavelengths separated from each other by more than about 2 nm. The spectrum of the low coherence wavelengths does not sufficiently overlap. Additional detectors and filters are arranged in the interferometer to transmit and detect two low coherence wavelengths.

The method of the preferred embodiment can be used to make precise optical distance measurements. Based on such measurement results, the optical properties of the target object can be accurately measured. By measuring the dispersion profile of a target, the structural and/or chemical properties of the target can be estimated. The dispersion profile plots the difference in refractive index at various wavelengths. In a biomedical context, preferred embodiments of the present invention provide precise determination of the dispersion properties of biological tissue in a non-contact and non-invasive manner. The dispersion measurement can be used on the cornea or aqueous fluid of the eye. The achieved sensitivity is sufficient to detect optical changes that are dependent on glucose concentration. In a preferred embodiment of the method of the invention, the blood glucose level can be determined by non-invasively measuring the dispersion profile of the aqueous fluid of the eye and/or the vitreous or cornea. Preferred embodiments of the present invention can be applied as a metrology technique in semiconductor manufacturing to measure small features formed during the manufacture of integrated circuits and/or optoelectronic components. Because preferred embodiments of the method are non-contact and non-destructive, the thickness of semiconductor structures or optical components can be monitored while they are being manufactured.

According to a preferred embodiment of the present invention using a Mach-Zender heterodyne interferometer, the method for measuring the phase of a light wave passing through a portion of a sample comprises the steps of: providing a first wavelength of the light wave; The optical path guides light waves of a first wavelength, the first optical path extends onto the sample medium to be measured and the second path undergoes a change in path length, and detects light waves from the sample medium and light waves from the second optical path for measurement The change in phase of a light wave passing through two separate points on a sample medium. The vehicle comprises biological tissue, eg, neurons. The method involves simultaneously imaging the phase of the sample at numerous locations using a photodiode array or a fiber optic bundle coupled to the photodiodes. The method further includes frequency shifting said light waves in a second optical path. The method includes providing a HeNe laser source ( ) or a low coherence light source emitting at a first wavelength.

According to another aspect of the invention, an actively stabilized interferometer is used in a method of measuring the phase of light waves passing through a portion of a sample, the method comprising the steps of: providing a first signal and a second signal generated by a first light source and a second light source, respectively For two signals, the second light source is a low coherence source. The method includes directing a first signal and a second signal along a first optical path and a second optical path; varying a path length difference between the first optical path and the second optical path; generating a first signal indicative of an optical path delay therebetween. an output signal summed with the second signal; modulating the output signal at an interferometer lock modulation frequency; and determining the phase of the sample by means of the time progression of the interferometer lock phase. The first and second signals are two low coherence signals. The method further includes demodulating the first signal with a mixer or a lock-in amplifier. The method includes electronically generating an interferometer-locked phase.

According to another aspect of the invention, a two-beam reflective interferometer is used in a system for measuring the phase of light waves passing through a portion of a sample. The system includes a first light source that generates a first signal; an interferometer that generates a second signal that is two pulses separated from the first signal by a time delay; a linked second optical path; and a detector system for measuring the first heterodyne signal based on the interference between the first and second signals respectively from the sample and the reference surface and light waves reflected from the sample and the reference surface. The system includes detecting a phase of a heterodyne signal indicative of a phase of a sample reflection relative to a reference surface reflection. The first signal is a low coherence signal. The first source of light waves can include without limitation one of a super light emitting diode or a multimode laser diode. The second path of the interferometer further includes the first path and the second path, and the second path has an acousto-optic modulator. The system includes an optical path including optical fibers. The system includes a seismically isolated heterodyne Michelson interferometer. The interferometer further includes mirrors attached to the translation stage to adjust the difference in optical path length. The detection system includes a first detector for detecting signals reflected from the sample and a second detector for detecting signals reflected from the reference surface.

According to another aspect, the invention provides a method of imaging a sample using phase contrast microscopy and spatial light wave modulation. In various embodiments, the method includes illuminating the sample, since light waves originating from illuminating the sample have a low frequency spatial component and a high frequency spatial component. The phases of the low frequency spatial components are shifted to provide at least three phase shifted low frequency spatial components. Preferably, the phase is shifted by, for example, increments of π/2 to produce low frequency spatial components phase shifted by π/2, π and 3π/2.

The unshifted low-frequency spatial component and the at least three phase-shifted low-frequency spatial components separately interfere with the high-frequency spatial components along the co-directional optical path, producing an intensity signal for each separately interfered. Then, for example, an image or a phase image of the sample is generated using at least four intensity signals.

According to another aspect, the present invention provides a method of non-contact optical measurement of a sample having a reflective surface, the method having the steps of: providing a first light source for generating a first signal; A second signal that separates the two pulses; providing a first optical path from the interferometer to the sample and a second optical path from the interferometer to the reference plane; based on the first and second signals from the sample and reference plane respectively and interferometrically measuring the first heterodyne signal between light waves reflected from the sample and the reference surface; and detecting a phase of the heterodyne signal indicative of a phase of the sample reflection relative to the reference surface reflection.

In a preferred embodiment, the first signal is a low coherence signal. The first light source may be a super light emitting diode or a multimode laser diode. The interferometer further includes a first path and a second path, the second path having an acousto-optic modulator. The method further includes an optical path comprising an optical fiber. The sample may be a portion of a nerve cell.

In a preferred embodiment, the interferometer comprises a vibration-isolated heterodyne Michelson interferometer. The interferometer further includes mirrors attached to the translation stage to controllably adjust the optical path length difference. A preferred embodiment includes a heterodyne low-coherence interferometer performing the first non-contact and first interferometric measurements of nerve expansion. The biophysical mechanism of nerve expansion can be imaged and analyzed using individual axons according to a preferred embodiment of the present invention. Two-beam low-coherence interferometers may have many other applications in measuring nanoscale motion in living cells. Other embodiments can include interferometer-based microscopy to detect the mechanical changes associated with action potentials in single neurons. A related interferometry method is also used to measure cell volume changes in cultured monolayers.

Another aspect of the present invention includes a fiber optic probe for optically imaging a sample, the fiber optic probe comprising a housing having a proximal end and a distal end; a fiber optic collimator coupled to a light source at the proximal end of the housing; and a A graded index lens having first and second surfaces, wherein the first surface is a reference surface, and the numerical aperture of the probe provides efficient collection of light waves from the diffuse surface of the sample. The probe further includes a fiber optic probe mounted on a translational stage for at least one of two-dimensional phase imaging and three-dimensional confocal phase imaging. The translating stage includes scanning piezoelectric transducers. The numerical aperture of the probe is in the range of about 0.4 to 0.5.

The above and other features and advantages of the described system and method for phase measurement are illustrated from the following figures of the system and method in which like reference characters denote the same parts everywhere in different views A more specific description of preferred embodiments will become apparent. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention.

Description of drawings

Figure 1 is a schematic diagram of a preferred embodiment of a system for measuring optical distances according to the present invention;

Figure 2 illustrates a low coherence heterodyne signal associated with a reflective interface, wherein tuning of the low coherence wavelength compresses or expands the heterodyne signal around the interface (depending on the direction of modulation of the center wavelength of the low coherence source), according to a preferred embodiment;

Figure 3 illustrates heterodyne signals associated with two reflective interfaces in a sample, wherein the heterodyne signal around the low coherence wavelength compressing interface is reduced, according to a preferred embodiment;

Figure 4 illustrates a scan of a sample with two interfaces, (a) a low-coherence heterodyne signal, (b) traces, in accordance with a preferred embodiment of the present invention, where the enlarged view shows phase fringes, each fringe is optically related to λ _CW Distance correspondence, (c) Traces at the two differences of Δ, where arrows indicate phase intersections, and the vertical axis is in radians;

Figure 5 illustrates a method for determining the correct estimate of (n _775nm L) measured by selecting a value that minimizes the error between S _Phase and S _frigE based estimates in accordance with a preferred embodiment of the present invention;

6A and 6B are flowcharts illustrating a method of measuring optical distance according to a preferred embodiment of the present invention;

Figure 7 is a schematic diagram of other possible preferred embodiments of the system for measuring optical distances according to the present invention;

8A and 8B are flowcharts illustrating alternative methods of measuring optical distance in accordance with a preferred embodiment of the present invention;

Figure 9 schematically illustrates a preferred embodiment of a fiber optic based system for measuring the thickness of an optically transparent material such as a glass plate, tissue sample or tissue layer;

Figure 10 illustrates a preferred embodiment of the system of the present invention for use in a vitreous and/or aqueous humoral glucose measurement system according to the present invention;

Figure 11 illustrates an actively stabilized Michelson interferometer where M is a mirror, MM is a moving mirror, BS is a beam splitter, PM is a phase modulator, D is a detector, and LO is a local oscillator according to a preferred embodiment of the invention source, MX is the mixer, S is the summing amplifier;

Figure 12 illustrates a stabilized interferometer for optical delay phase-sensitive low-coherence interferometry (LCI), wherein the DBS is a dichroic beamsplitter according to a preferred embodiment of the present invention;

Figure 13 illustrates sample demodulation interferograms of a pair of interfaces when the optical path length difference ΔL is varied according to a preferred embodiment of the present invention;

Figure 14A illustrates a system for stable phase-sensitive low-coherence interferometry (LCI), wherein LC1 and LC2 are low-coherence beams according to a preferred embodiment of the present invention;

Figure 14B illustrates an alternative embodiment of a system for actively stabilized phase-sensitive low-coherence interferometry (LCI) in accordance with the present invention using piezoelectric transducers to generate phase changes;

15A and 15B are demodulated fringe plots of LC1 and LC2, where the two peaks of the LC2 signal represent the coverslip reflection (large) and the reflection from the sample (small) according to a preferred embodiment of the invention;

Figure 16 illustrates an imaging system for stable phase-sensitive low-coherence interferometry in accordance with a preferred embodiment of the present invention;

Figure 17 illustrates an unwrapped optical design for two-dimensional phase imaging, wherein solid lines represent incident light rays and dashed lines represent backscattered light rays, in accordance with a preferred embodiment of the present invention;

Figure 18A illustrates a two-point Mach-Zender heterodyne interferometer according to a preferred embodiment of the present invention;

Figure 18B illustrates an imaging Mach-Zender heterodyne interferometer in accordance with a preferred embodiment of the present invention;

Figure 18C illustrates the heterodyne and gating signals associated with the embodiment described with reference to Figure 18B;

Figure 18D illustrates an imaging two-beam heterodyne interferometer in accordance with a preferred embodiment of the present invention;

Figure 19 illustrates an isolated two-beam heterodyne low-coherence interferometer in accordance with a preferred embodiment of the present invention;

Figure 20 illustrates a dual datum heterodyne low coherence interferometer according to a preferred embodiment of the present invention;

Figure 21 illustrates a preferred embodiment of an optical reference interferometer in accordance with a preferred embodiment of the present invention;

Figure 22 schematically illustrates the components of the measured phase due to the fact that the datum point as the sampling object is located on the same surface (glass) according to a preferred embodiment of the present invention;

23A and 23B diagrammatically illustrate the voltage and corresponding phase changes of a piezoelectric transducer (PZT), respectively, for the embodiment illustrated with reference to FIG. 21 , in accordance with a preferred embodiment of the present invention;

Figure 24 graphically illustrates noise performance in units of radians in accordance with the interferometer illustrated in Figure 21;

25A and 25B are schematic representations of calibration components for sample and datum signals in accordance with a preferred embodiment of the present invention;

Figure 26 schematically illustrates a preferred embodiment of an interferometer system according to a preferred embodiment of the present invention;

Figure 27 illustrates a schematic diagram of a system for measuring nerve displacement in accordance with a preferred embodiment of the present invention;

28A and 28B graphically illustrate neural displacement (nm) and potential (μV) with respect to time (ms), according to a preferred embodiment of the present invention;

Figure 29 schematically illustrates peak potentials (crosses) and displacements (circles) of a single nerve with variable stimulus current amplitudes in accordance with a preferred embodiment of the present invention;

Figure 30 illustrates the optical design of a scanning system for a two-beam interferometer in accordance with a preferred embodiment of the present invention;

Figure 31 illustrates galvanometer positions and phase data collected from empty coverslips using Lissajous scanning in accordance with a preferred embodiment of the present invention;

Figures 32A and 32B illustrate a color map and a retroreflected intensity image, respectively, of a phase image using the data represented and graphically illustrated in Figure 31, in accordance with a preferred embodiment of the present invention;

Figure 33 schematically illustrates the focus problem solved by means of the preferred embodiment of the present invention;

Figure 34 illustrates a design for a bifocal lens according to a preferred embodiment of the present invention;

Figure 35 illustrates an alternative design for a bifocal lens in accordance with a preferred embodiment of the present invention;

Figure 36 illustrates the calculation of the optimum distance between lenses f3 (bifocal lens) and f2 according to a preferred embodiment of the present invention;

Figure 37 illustrates the fabrication of a bifocal lens in accordance with a preferred embodiment of the present invention;

Figure 38 illustrates retroreflected intensity measured by an optical circulator as an objective lens is scanned towards a glass coverslip, in accordance with a preferred embodiment of the present invention;

Figure 39 illustrates retroreflection intensity as a function of focus position of an objective using bifocal lens f3, in accordance with a preferred embodiment of the present invention;

Figure 40 illustrates retroreflection intensity as a function of focus position of an objective using a bifocal lens for two coverslip reflections in accordance with a preferred embodiment of the present invention;

Fig. 41 illustrates, in accordance with a preferred embodiment of the present invention, when the distance between f2 and f3 is adjusted to match the gap between the front and rear glass surfaces, the retroreflection intensity varies with the use of two coverslip reflections. The focus position of the objective lens of the bifocal lens changes;

Figures 42A and 42B, collectively referred to as Figure 42, illustrate the formation of additional smaller peaks due to axial and marginal beam coupling in an optical system, in accordance with a preferred embodiment of the present invention;

Figure 43A illustrates a dual beam probe having a datum surface as a monolithic element in accordance with a preferred embodiment of the present invention;

Figure 43B is another preferred embodiment of a dual-beam interferometer probe;

Figure 43C is an image of two nerve fibers;

Figure 43D is a graph of heterodyne signal amplitude as a function of position;

Figure 43E is a reflection phase image of the same sample seen in Figure 43D;

Figure 44 is a schematic illustration of a dual-beam probe suitable for studying the geometry of displacement effects observed in nerves during action potentials, in accordance with a preferred embodiment of the present invention;

Figure 45 is a dual beam probe system for imaging by scanning the probe or sample in accordance with a preferred embodiment of the present invention;

46A-46C illustrate intensity, phase, and bright field images of light waves backscattered from dried human buccal epithelial cells using a bifocal two-beam microscope, in accordance with a preferred embodiment of the present invention;

Figures 46D-46G illustrate the profiling capabilities of the dual beam microscope exemplified in Figure 43, where Figure 46D is an intensity image of the central portion of the plano-convex lens system illustrated in Figure 46E, and Figure 46F is a phase map of reflected light waves , while Figure 46G is a cross-section of a phase image, phase unfolded by quadratic fitting according to a preferred embodiment of the present invention.

47A-47E illustrate a schematic diagram of a phase shifting interferometry system, phase stepping and barrel integration, respectively, in accordance with a preferred embodiment of the present invention;

48A-48C illustrate the principle of stroboscopic heterodyne interferometry system and barrel integration, respectively, according to a preferred embodiment of the present invention;

Figure 49A illustrates a two-beam stroboscopic heterodyne interferometer in accordance with a preferred embodiment of the present invention;

49B and 49C illustrate data representing the phase noise of a dual-beam probe focused on a stationary glass surface in accordance with a preferred embodiment of the present invention;

Figure 50A illustrates another preferred embodiment in which light waves from separate paths are directed towards a common path and are focused on different regions of the material being measured;

Figure 50B is a preferred embodiment of a dual beam system using the system shown in Figure 50A;

Figure 50C provides details about the polarization components within the system illustrated in Figure 50B;

Figures 51A-51D are schematic representations of various features of such image descriptions in accordance with a preferred embodiment of the present invention;

Figure 52 schematically illustrates various embodiments of microscope systems based on transmission geometry in accordance with the present invention;

Figure 53 schematically illustrates various embodiments of microscope systems based on reflection geometry in accordance with the present invention;

Figures 54A and 54B, collectively referred to as Figure 54, schematically illustrate one embodiment of integrating various embodiments of the present invention with an optical microscope;

Figure 55 schematically illustrates various embodiments of the systems and methods of the present invention utilizing the 4-f system;

Figure 56 schematically illustrates one embodiment of a phase contrast microscope system utilizing spatial lightwave modulation (SLM) in accordance with the present invention;

Figures 57A and 57B schematically illustrate the photoelectric effect on a pixel of an image in amplitude mode and phase mode, in accordance with a preferred embodiment of the present invention;

58A-58C are block diagrams of various embodiments of SLM modes of operation in accordance with a preferred embodiment of the present invention;

Figure 59 is an example of a calibration curve obtained for an instrument operating in amplitude mode in accordance with a preferred embodiment of the present invention;

60A-60D show images acquired at four different phase shifts by a system using reflection geometry in accordance with a preferred embodiment of the present invention;

Figure 61 schematically illustrates the relationship between the electromagnetic field vector E and the high frequency wave vector component E _H of the electromagnetic field and the low frequency wave vector component E _L of the electromagnetic field according to a preferred embodiment of the present invention;

Figure 62 is a Δφ image of a calibration sample generated using data such as illustrated in Figures 35A-35D and Equation 55 in accordance with a preferred embodiment of the present invention;

Figure 63 is a phase image of a calibration sample of Example 1 using the systems and methods in accordance with the present invention;

Figure 64 shows a phase image obtained using transmission geometry in accordance with a preferred embodiment of the present invention;

Figure 65 shows onion cell intensity images obtained in accordance with a preferred embodiment of the present invention;

Figure 66 shows an onion cell phase image obtained using transmission geometry in accordance with the present invention;

67 illustrates the experimental setup according to a preferred embodiment, wherein VPS is a virtual point source; CL is a correction lens; IP is an imaging plane; P is a polarizer; BS is a beam splitter; FL is a Fourier lens; Programmed phase modulator; CCD is charge-coupled device, PC is personal computer;

Figures 68A and 68B illustrate experimental results for polystyrene microspheres immersed in 100% glycerol obtained using a 10x microscope objective according to a preferred embodiment, wherein Figure 68A is an intensity image and Figure 68B is a quantitative phase image . Colored bars represent phase expressed in nm;

69A-69C illustrate LCPM images obtained using a 40× microscope objective according to a preferred embodiment, wherein FIG. 69A is a phase image of HE1A cancer cells undergoing mitosis, FIG. 69B is a phase image of an entire blood smear, and FIG. 69C is a phase image with Points lacking cells are associated with instantaneous phase shifts (standard deviation σ is indicated). Colored bars represent phase expressed in nm.

Detailed ways

Harmonic interferometry for distance measurement:

The present invention relates to a system and method for measuring optical distance based on phase intersection, which solves the integer or 2π ambiguity problem by introducing dispersion imbalance in the interferometer. A preferred embodiment of the method is capable of accurately measuring the relative height difference of two adjoining points on a surface. Furthermore, it has been found that the accuracy of the refractive index of the sample is only limited by the accuracy of the experimental measurement of the actual thickness of the sample.

Replacing one of the low coherence light sources with a continuous wave (CW) light source in harmonic phase based interferometry (HPI) allows the use of the associated CW heterodyne signal as a form of optical scale for measuring the low coherence heterodyne signal. Low coherence light sources provide spectral bandwidth, for example greater than 5 nm for 1 micron wavelength. One of the advantages of using the improved HPI is that the measured phase is now sensitive to the length scale nL instead of (n _λ2 −n _λ2 )L, where n is the refractive index at low coherence wavelengths. The quantity n is actually much more useful than the composite quantity (n _λ2 -n _λ2 ). By adjusting the low-coherence wavelength slightly, for example, around 2nm, one can find the value nL without 2π ambiguity but with sub-nanometer sensitivity. This method uses the CW heterodyne interference signal as a datum optical scale for measuring optical distance.

Systems for measuring interferometric optical distances using readily available low-coherence light sources have achieved resolutions on the order of tens of wavelengths. Although this technique is relatively insensitive, it does not have to resolve the 2π ambiguity problem. Preferred embodiments include low-coherence interferometry using phase to measure arbitrarily long optical distances with sub-nanometer precision. This method uses a low-coherence phase intersection technique to determine integer interference fringes and additional phase information from the measurements to accurately obtain fractional fringes. Among other things, it provides depth resolution and can be used for X-ray tomographic profiling of layered samples. Because the method can accurately measure long optical distances, it can be used to accurately determine the refractive index of numerous materials. Since this is a phase-based approach, the refractive index thus found is a phase refractive index rather than a population refractive index.

Figure 1 illustrates a preferred embodiment of a system 10 of the present invention comprising a modified Michelson interferometer. The input light wave 12 is a two-color composite beam consisting of a 150-fs mode-locked light wave (e.g. emitting at 775.0 nm) from a Ti:sapphire laser and a continuous wave (CW) 1550.0 nm light wave from a semiconductor laser, for example. In a preferred embodiment, the method calculates the optical distance based on the CW wavelength (1550.0 nm to be precise in this embodiment), and all optical distances are calculated on this basis. The composite beam is split in two at the beam splitter 14 . A portion of the signal is incident on the target sample 16 and another portion is incident on a reference mirror 32 which induces a Doppler shift in the reference beam 34, preferably moving at a speed of, for example, about 0.5 mm/sec. Doppler shift can be induced by other means, for example, by using electro-optic modulators. The back-reflected light beams are recombined at the beam splitter 14 , split into their wavelength components by means of the dichroic mirror 18 and measured separately by the photodetectors 20 , 22 . The resulting signal is digitized by means of an analog-to-digital converter (ADC) 24 (for example, a 16-bit 100 kHz A/D converter). A data processor, such as a personal computer (PC) 26, communicates with ADC 24 for further processing of the data. The resulting heterodyne signals have passbands around their respective central heterodyne frequencies at their respective Doppler-shifted frequencies and are Hilbert transformed to deduce the corresponding phases of the heterodyne signals, φ _CW and φ _LC . The subscripts CW and LC represent the continuous wave component of 1550.0 nm and the low coherence wavelength component of 775.0 nm, respectively.

Then, adjust the center wavelength of the low-coherence light wave by about 1-2nm, and measure the second set of φ _CW and φ _LC values. From these two sets of readouts, various interfaces in the sample of interest can be localized with sub-nanometer precision. Data processing for localization is described below.

Consider a sample consisting of a single interface at an unknown distance x ₁ from beamsplitter 14 . The distance x from beamsplitter 14 to reference mirror 32 is a known quantity at each point in time during the scan of the reference mirror.

An approximation of _x is found by scanning x in the recombined low-coherence beam and monitoring the resulting heterodyne signal. The peaking of the heterodyne signal amplitude is expected when x is approximately equal to _x1 . The accuracy of this method is limited by the coherence length L _C of the light source and the signal-to-noise quality of the heterodyne signal. Under realistic experimental conditions, the error in determining _x1 cannot be better than one-fifth of the coherence length.

Assume that the coherence length of a typical low-coherence light source is nominally about 10 μm. This means that the error of the method for determining the length is limited to approximately 2 μm.

When considering the phase of the heterodyne signal, the different components of the detected heterodyne signal can be expressed as:

${\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; heterodyne</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; ref</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&omega;t</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; sig</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&omega;t</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; .</span>$

$<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; ref</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; sig</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

Where _ErEf and E _sig are the reference electromagnetic field amplitude and the signal electromagnetic field amplitude respectively, k is the optical wavenumber, and ω is the optical frequency. A factor of 2 in the index means that the light wave travels the path twice, hitting the mirror/sample and returning to the beamsplitter.

Note that the heterodyne signal is expected to peak when x matches _x1 exactly. The two returning beams are in constructive interference. Therefore, this property is used to localize the interface. _x1 is found by finding the value of x where the two beams are in constructive interference. Since the phase can be accurately measured, this method gives a length sensitivity of about 5 nm. Unfortunately, this approach is computationally intensive because there are multiple values of x at which the heterodyne signal peaks; specifically, the heterodyne signal peaks when x satisfies:

x＝x ₁ +aλ/2 (2)

where a is an integer and λ is the optical wavelength. This is a manifestation of the 2π ambiguity problem.

Preferred embodiments include methods to identify the correct peak. Note that the heterodyne signal peaks when x is exactly equal to _x1 , regardless of the optical wavelength. On the other hand, as illustrated in Fig. 2, the later peaks depend on the wavelength. Figure 2 illustrates a low coherence heterodyne signal associated with a reflective interface 52 in a sample. So, by adjusting the low coherence wavelength, the heterodyne signal is compressed around the interface and the correct peak associated with the case that xx ₁ can be clearly distinguished. One should note that the heterodyne signal may be compressed or expanded around the interface, depending on the direction of accommodation. An intuitive way to observe localization with the naked eye is to draw _the heterodyned signal squeezed into or extended away from the fringes where x is exactly equal to _x1 .

A CW light source is required in the localized approach for two reasons. First, it is very difficult to identify this value with absolute accuracy in practical interferometers. The CW component of the interferometer allows very precise measurements of x while scanning the reference mirror. In a particularly preferred embodiment, in order to determine the distance between two interfaces in a sample, the positions occurring at x ₁ equal to the distance to the first interface shown in FIG. 1 and x ₂ (x ₂ =x ₁ + nL, where n is the refractive index of the sample) is the number of CW interference fringes between positions equal to the distance to the second interface. Figure 3 illustrates the heterodyne signal associated with two reflective interfaces in the sample. Adjusting the low coherence wavelength compresses 82, 84 the heterodyne signals 78, 80 around the interface.

Second, the previously described methods of interface localization may partially fail if there is a phase shift associated with the reflection process. For example, if the surface is metallic, then the phase shift is not trivial, and the phase of the heterodyne signal has some other value when x is exactly equal to _x1 . Although previous methods allow identification of correct interference fringes when x = x ₁ , subwavelength sensitivity may be a tradeoff. The presence of a CW heterodyne signal allows finding the difference phase by means of the HPI method. Knowledge of this value allows the interface to be localized with a high level of sensitivity.

The principle of the HPI method can be illustrated by an exemplary implementation of a sample of thickness L having a refractive index n _{775 nm} at a wavelength of 775 nm. The two interfaces of the sample are located at optical distances x ₁ and x ₂ from the beam splitter (where x ₂ =x ₁ +n _775nm L). Note that this method will only work if the optical distance separation is larger than the coherence length, eg typically between 1 µm and 100 µm for low-coherence sources. Otherwise, the heterodyned phase signals associated with the interface merge together and result in erroneous interface localization. For clarity of explanation, the incorporation of phase shifts associated with reflections is postponed until later.

Figure 4 is a scan illustrating the mathematical description. This scan is of a sample with two interfaces. Signal 100 is a low coherence heterodyne signal. Trace 102 is φ _CW (x). The enlarged view 104 shows phase fringes. Each fringe corresponds to an optical distance of λ _CW . The lower φ _D (x) traces are two different delta values. Arrows 106, 110 indicate phase intersections. The vertical axis is in radians. When scanning the reference mirror, the phase of the low-coherence heterodyne signal is given by:

${\mathrm{<span\; class="notranslate">\; \&psi;</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; arg</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; \&ap;</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

where R _LCj is the reflectivity of interface j at the low coherence wavelength, k is the optical wavenumber, a=4ln(2)/l _c , l _c is the coherence length, x is the distance from the reference mirror to the beamsplitter, h _c (x) is a piecewise continuous function, when |x|<2l _c , its value is 1, otherwise it is 0. The factor of 2 in the index is due to the fact that the optical path length is effectively doubled in the retroreflective geometry. Equation 3 reflects the fact that phase far beyond the coherence envelope cannot be measured due to noise. Although the modeled coherence envelope is Gaussian in profile, the same phase treatment is valid for the profile of any slowly varying envelope.

The phase of the CW heterodyne signal is given by:

${\mathrm{<span\; class="notranslate">\; \&psi;</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; arg</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="x"\; filenames="C2004800208380107H1.png,C2004800208380113H1.png"\; state="\{\{state\}\}">x</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; 1550</span>}\mathrm{<span\; class="notranslate">\; nm</span>}}\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; arg</span>}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

where R _CWj is the reflectance of interface j at CW wavelength, n _1550nm is the refractive index of the sample, R and X are the effective average reflectance and the effective average distance to the beamsplitter, respectively. If the center wavelengths of the two sources are chosen such that

k _LC =2k _CW +Δ, (5)

where Δ is a small intentionally added offset, then a difference phase φ _D of the form is obtained:

ψ _D (x) = ψ _LC (x)-2ψ _cw (x)

＝h _c (xx ₁ )mod _2π (4k _cw (xx ₁ )+2Δ(xx ₁ )) (6)

+h _c (xx ₂ )mod _2π (4k _cw (xx ₂ )+2Δ(xx ₂ ))

The above quantities provide both the approximate number of fringes in the interval (x ₂ -x ₁ ) and the fractional fringes that provide sub-wavelength precision.

When the parameter Δ is changed by a small amount (corresponding to a wavelength shift of about 1-2 nm), the slope of φ _D (x) rotates around the points x=x ₁ and x=x ₂ . In other words, the phase is swept where different values of delta intersect the point. The optical distance from x ₁ to x ₂ can be found by computing the fringes that φ _CW (x) passes between two phase intersection points. Twice the number so found is indicated by the non-integer S _fringe and corresponds to the number of fringes at low coherence wavelengths. For a single interface, if there are multiple phase intersections, the point corresponding to the interface position can be found by performing various scans with multiple additional Δ values. The interface location is the only location where φ _D (x) will intersect for all values of Δ.

The phase offset information is used to further localize the interfacial spacing. Specifically, the difference between the phase offsets at x= _x1 and x= _x2 is:

${\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; phase</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&psi;</span>}}_{\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&psi;</span>}}_{\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

This measures fractional fringes with high sensitivity.

The absolute optical spacing (x ₂ -x ₁ ) can be precisely determined from S _fringe and S _Phase by the following formula:

${<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>}_{\mathrm{<span\; class="notranslate">\; measured</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; 775</span>}\mathrm{<span\; class="notranslate">\; nm</span>}}\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; )</span>}_{\mathrm{<span\; class="notranslate">\; measured</span>}}<span\; class="notranslate">\; =</span>$

$\frac{{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; int</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; fringe</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&Delta;S</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Delta;S</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; phase</span>}}<span\; class="notranslate">\; )</span>$

Where ΔS=rEs(S _fringe )-S _Phase , U() is the unit step function. Here, int() and res() represent the integer part and the fractional part of the argument, respectively. The first term localizes the correct integer number of optical distances to fringes by minimizing the error between S _Phase and the fractional part of S _fringe . The optical distance measurement error is only limited by the S _Phase measurement error. In an embodiment, such an error translates to an error in (n _775nm L) _measured of about 0.5nm. The measurement error of S _Phase only needs to be less than half a fringe, so that the correct interference fringe can be established; this criterion is met, it does not enter the (n _775nm L) _measured error. The maximum measurable optical distance depends only on the system's ability to accurately count the number of fringes between two intersection points and the frequency stability of the light source.

The above formula is a concise expression of the method used to find the correct and fractional stripes. Operation can be exemplified by the following example and Figure 5 showing that the correct estimate is determined by choosing the value that minimizes the error based on the estimate between S _Phase and S _fringe . Assume that S _fringe and S _Phase are 26.7 and 0.111. From the measurement results of S _Phase , the value of the optical distance is:

(n _775nm L) _measured = λ _CW (a+0.111)/4 (9)

where a is an integer. Given the value of S _fringe , the possible (n _775nm L) _measured values are limited to the following three values: λ _CW (25.111)/4, λ _CW (26.111)/4 and λ _CW (27.111)/4. If the value of λ _CW (27.111)/4 is closest to λ _CW (S _fringe )/4, then it is the correct estimate of (n _775nm L) _measured .

In a preferred embodiment for light source-based interferometry measurements of harmonic relations, appropriately selected light source pairs and methods of inferring the differential phase allow minimizing and preferably eliminating the otherwise high precision optics in the interferometer. Distance measurement becomes impossible due to jitter effects. Dejittering also allows comparison of scans done at different times.

To demonstrate the capability of a preferred embodiment of the method, the system was used to probe the optical distance between the top and bottom surfaces of a fused silica coverslip of actual thickness L=237±3 μm. In this embodiment, there is a π phase shift associated with the reflection from the first interface, which marks a positive refractive index transition. Thus, in Equations 1 and 2 there are e ^-iπ terms associated with the factors R _LC,1 and R _cw,1 . This results in a correction factor of one-half for S _fringe and S _Phase . Figure 4 shows typical scan results at LC wavelengths of 773.0 nm and 777.0 nm. The results of the four sets of scans are summarized in Table 1 showing the measurements of (n _775nm L) on the quartz coverslip. The repeatability of the experimental data shows that the light source is sufficiently stable in terms of frequency.

Table 1

λ_CWS_fringe/4(μm) λ_CWS_Phase/4(μm) (n_775nmL)_measured(μm) Group 1 350.86±0.17 0.3496±0.0004 351.0371±0.0004 Group 2 351.08±0.17 0.3497±0.0004 351.0372±0.0004 Group 3 351.15±0.16 0.3502±0.0004 351.0377±0.0004 Group 4 351.04±0.18 0.3498±0.0004 351.0373±0.0004 average 351.0373±0.0004

The experimental data yield absolute optical distance measurements with sub-nanometer precision. The optical distances found are associated with low coherence light sources. The CW heterodyne signal acts as an optical scale. If the L of the quartz coverslip is known precisely, then the n _775nm of the quartz at a wavelength of 775.0nm can be determined with very high accuracy from (n _775nm L) _measured .

Alternatively, without knowing the exact value of L, the refractive index ratio at two different wavelengths can be determined by measuring the corresponding optical distances using low coherence lightwaves at these wavelengths and CW lightwaves at their respective harmonics. Using a range of low coherence wavelengths, the dispersion profile of a material can be accurately determined. A dispersion profile plots the difference in refractive index at various wavelengths. According to a preferred embodiment, these experimental results predict that an accuracy of about seven significant figures can be achieved with samples about 1 millimeter thick.

In another preferred embodiment, the light source of the system is changed to a low coherence super light emitting diode (SLD) emitting at 1550.0 nm and a CWTi:sapphire laser emitting at 775.0 nm. By adjusting the operating current through the SLD, the center wavelength is changed by about 2nm; this is suitable for achieving phase crossing. Using the described preferred embodiment of the system of the present invention, optical distance can be measured at 1550.0 nm. Using the ratio of said measurements to the previous measurements, the refractive index ratio n _775nm /n _1550nm of quartz can be determined. One should note that the refractive index ratios found are due to the harmonic relationship wavelengths for the light sources used in the preferred embodiment. Refractive index ratios at other wavelengths can be measured with appropriately selected other light sources. For comparison, the corresponding data for glass and acrylic plastic are listed in Table 2 as n _775nm /n _1550nm measurements for different materials.

Table 2

n_775nm/n_1550nm Quartz 1.002742±0.000003 Glass (German Borosilicate) 1.008755±0.000005 Acrylic plastic 1.061448±0.000005

Note that some of the formulas used when the low coherence wavelength is half the CW wavelength are slightly different from those previously presented here. For example:

${\mathrm{<span\; class="notranslate">\; \&psi;</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; arg</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; \&ap;</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; \&psi;</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; arg</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="x"\; filenames="C2004800208380107H1.png,C2004800208380113H1.png"\; state="\{\{state\}\}">x</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; 1550</span>}\mathrm{<span\; class="notranslate">\; nm</span>}}\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; arg</span>}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; cw</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 11</span>}<span\; class="notranslate">\; )</span>$

2k _LC =k _CW +Δ (12)

ψ _D (x)＝2ψ _LC (x)-ψ _cw (x)

＝h _c (xx ₁ )mod _2π (4k _LC (xx ₁ )+2Δ(xx ₁ )) (13)

+h _c (xx ₂ )mod _2π (4k _LC (xx ₂ )+2Δ(xx ₂ ))

${\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; phase</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&psi;</span>}}_{\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&psi;</span>}}_{\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="mod"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">mod</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 14</span>}<span\; class="notranslate">\; )</span>$

${<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>}_{\mathrm{<span\; class="notranslate">\; measured</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; 775</span>}\mathrm{<span\; class="notranslate">\; nm</span>}}\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; )</span>}_{\mathrm{<span\; class="notranslate">\; measured</span>}}$

$<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; LC</span>}}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; int</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; fringe</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&Delta;S</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Delta;S</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; phase</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 15</span>}<span\; class="notranslate">\; )</span>$

Preferred embodiments of the method to resolve 2π ambiguity are useful in applications such as high-precision depth ranging and high-precision refractive index determination of thin-film solid materials.

The use of the preferred method can be exemplified by considering a glass plate. There are some systems that can measure the distance from the system to the mean center of the glass plate very accurately. There are also systems that measure the roughness of glass surfaces very accurately. A preferred embodiment of the system of the present invention measures the thickness of the end face of a glass plate with nanometer sensitivity.

The steps for implementing a preferred embodiment of the method for determining optical distances are illustrated by flow diagram 124 in FIGS. 6A and 6B. Method 124 involves using two harmonically related light sources in a Michelson interferometer, one of which is a CW light source and the other is a low coherence light source. The sample whose optical distance between interfaces needs to be measured is used every step 126 as the end mirror of the signal arm of the interferometer. The reference mirror in the reference arm of the interferometer is scanned every step 128 . The method includes the step 130 of combining reflections from the signal arm and the reference arm and then separating them by wavelength. Further, at each step 132 a heterodyne oscillation of the combined light wave intensity is detected. Then, each step 134 finds the phase of the heterodyne signal at the two wavelengths by means of, for example, the Hilbert transform or any phase derivation alternative. Each step 136 estimates the difference phase for the entire scan by twice subtracting the longer wavelength phase from the shorter wavelength phase. Each step 137 repeats the scan with the slightly detuned optical wavelength. Then, steps 130-136 are repeated.

Then, each step 138 superimposes the two difference phases found from the two scans on a graph with the x-axis representing the displacement of the reference mirror. One should note that derivation of the differential phase can also be done with appropriate light sources or filters or software/hardware signal processing of a single scan.

Subsequent steps in method 124 include determining phase intersection points on the graph at each step 140 to mark the location of the sample interface. At each step 142, the optical distance between the interface is determined with an accuracy of approximately a fraction of the wavelength (e.g., approximately 0.2) by counting the number of times the heterodyne signal associated with the CW light wave overlaps with 2π between the two points of intersection. spacing. This spacing is further localized and/or refined to very small fractions of wavelength, eg, about 0.001, by measuring the difference phase at the intersection point.

In another preferred embodiment illustrated in Figure 7, which is used as a schematic diagram of a system for measuring optical distances, the low-coherence light source may be sufficiently wide in bandwidth, eg, exceeding 4 nm. At the end of detection, a third detector 174 is added to the two detectors 166,176. This results in the low coherence lightwave signal 168 being further split in two. The two beams pass through different filters 170, 172 before reaching the detector. These two filters transmit different parts of the spectrum. One passes spectral components with longer wavelengths, while the second passes spectral components with shorter wavelengths. It is preferred that the two transmitted beams are separated by more than 2 nm in their spectra.

The beams are then incident on the detectors and their heterodyned signals are processed in the manner discussed with reference to FIG. 1 . An advantage of the method according to other possible preferred embodiments is that the method eliminates repetitions of the procedure with adjusted low coherence wavelengths. These two signals were acquired in the same scan.

8A and 8B illustrate a flowchart 184 of an alternative method of measuring optical distance in accordance with a preferred embodiment of the present invention. Method 184 includes using two harmonically related light sources in an interferometer, one of which is a CW light source and the other is a low coherence light source. At each step 186, the sample whose optical distance is to be measured is used as the end mirror of the signal arm of the interferometer. At each step 188, the reference mirror in the reference arm of the interferometer is scanned. The method further includes combining the reflections from the signal arm and the reference arm at each step 190 and separating them by wavelength. At each step 192, filters are used to further separate the low coherence wavelengths. Method 184 includes a step 194 of detecting heterodyne oscillations with at least three detectors. The next step 196 involves detecting heterodyned vibrations of the combined light wave intensity. Then, each step 198 finds the phase of the heterodyne signal at the two wavelengths by means of, for example, the Hilbert transform or any phase derivation alternative. Then, at every step 200, the phase difference between each low-coherence signal and the CW signal is calculated.

Then, every step 202, the two difference phases are superimposed on each other on the x-axis of the graph representing the displacement of the reference mirror. The remaining steps 204, 206, 208 are similar to steps 140, 142, 144 discussed with reference to Fig. 6B.

Preferred embodiments of the method can absolutely be used to measure arbitrarily long optical distances with sub-nanometer precision. Preferred embodiments of the system can be free space based or fiber optic based. Figure 9 illustrates a preferred embodiment of an optical fiber based system for measuring optical distance.

The input light waves 256 include approximately harmonically low coherence light waves (wavelength λ ₁ ) and CW light beams (wavelength λ ₂ ) propagating in the optical fiber 251 . The composite beam is split in two, one part of the signal is incident on the objective lens 254 and the sample 256 and transmitted in the optical fiber 253 , while the other part of the signal is incident on the reference mirror 266 via the lens 268 and transmitted in the optical fiber 251 . Movement of the reference mirror introduces a Doppler shift in the reflected beam. The reflected beams are combined again and then, by means of dichroic mirror 258, separated into their constituent wavelength components. The wavelength components are measured separately by means of photodetectors 260 , 262 . These heterodyne signals generated at their respective Doppler shifted frequencies have passbands around their respective central heterodyne frequencies and are Hilbert transformed to deduce the corresponding phases φ _CW and φ _LC of the heterodyne signals .

The method of the preferred embodiment can be used to make precise optical distance measurements. From such measurements, the optical properties of the target object can be accurately determined. By measuring the dispersion profile of the target, the structural and/or chemical properties of the target can be calculated. In the biomedical field, preferred embodiments of the present invention can be used to accurately determine the dispersion properties of biological tissues in a non-contact and non-invasive manner. Such dispersion measurements can be used on the cornea or aqueous fluid of the eye. The achieved sensitivity is sufficient to detect optical changes that are dependent on glucose concentration. In a preferred embodiment of the method of the invention, the blood glucose level can be determined by non-invasive measurement of the dispersion profile of the aqueous or vitreous fluid of the eye or of the cornea.

As discussed above, phase-based interferometry can measure optical distance very sensitively. However, said measurements are limited in their application by problems well known in the field such as 2π ambiguity. The crux of the problem is that it is impossible to distinguish a length of 10.1 wavelengths from a length of 11.1 wavelengths. Preferred embodiments of the present invention overcome this limitation and allow absolute optical distance measurements with sub-nanometer accuracy.

There are many phase-based methods for measuring optical distance changes with sensitivities in the approximate nanometer range. As long as the change is small and gradual, the change can be continuously tracked. There are low-coherence methods of measuring absolute optical distance by tracking the delay to the detector for light waves reflected from different interfaces whose mirror sensitivity is on the order of microns. As discussed above, the simultaneous use of a CW source and a low coherence source in an interferometer allows for a method of measuring optical distance. The heterodyned phases of the signals associated with the two wavelengths are inherently correlated. By manipulating the phase of each preferred embodiment, motion noise is minimized and preferably eliminated from our measurements.

An application of the preferred embodiment is the determination of glucose levels using measurements of the refractive index of the vitreous and/or aqueous fluid of the eye. The sensitivity of this technique provides the ability to measure chemical concentrations with clinically appropriate sensitivity. One of the more obvious applications of the method of the preferred embodiment is the determination of blood glucose levels by measurements done on the eye. The glucose level of the fluid in the eye mirrors the glucose level of the blood with a clinically insignificant time delay.

The method of the preferred embodiment measures the optical path length of the vitreous humor and/or aqueous fluid layer in the eye using at least two separate sets of wavelengths as illustrated in FIG. 10 . This method measures the product of the index of refraction at low coherence wavelengths and the actual separation between two interfaces. The refractive index difference is measured at different wavelengths by varying the wavelength of the low coherence light source (and appropriately varying the CW wavelength for matching). For example, one set of measurements was done with a tunable 500nm low-coherence light source and a 1-micron CW light source to extract n _500nm L, where L is the actual thickness of the vitreous humor and/or aqueous fluid at the point of measurement. Another set of measurements was done with a tunable 1000nm low-coherence light source and a 1800nm CW light source to extract n _900nm L. By taking the ratio of these two measurements, the refractive index ratio n _500nm /n _900nm of the vitreous humor and/or aqueous fluid is deduced. Using existing sensitivities, e.g., 0.5 nm optical path sensitivity, a preferred embodiment of the system is capable of measuring a refractive index ratio n _{500 nm} /n with a sensitivity of 10 ⁻⁸ for materials of thickness and human vitreous humor and/or aqueous fluid _900nm . This provides the sensitivity to glucose level changes of approximately 0.25 mg/dl. Assuming typical blood glucose levels are about 100 mg/dl, preferred embodiments of the present invention are well suited for blood glucose assays. The choice of optical wavelength is flexible and the wavelengths used above are for illustration only. For highest sensitivity, the wavelength separation is preferably as large as possible. Preferred embodiments include spacings greater than 500nm.

In cases where such refractive index ratios are insufficient to determine absolute blood glucose levels due to the presence of other chemicals that are changing in the vitreous humor and/or aqueous fluid, a more complete set of optical path length measurements can be made at a range of other wavelengths. conduct. This more complete set of measurements allows determination of glucose levels and concentrations of other chemicals by fitting the measurements to known dispersion profiles of glucose and other chemicals.

The preferred embodiment of the invention can be applied as a measurement technique in the semiconductor manufacturing industry. Because the preferred embodiment of the method is non-contact and non-destructive, it can be used to monitor the thickness of semiconductor structures during fabrication. In addition, the composition of the semiconductor structure can be assayed in as many ways as discussed with respect to the characteristics of vitreous humor and/or aqueous fluid measurements.

Phase Measurement and Imaging Systems:

Other possible preferred embodiments of the invention involve imaging small organisms or features with light waves. These embodiments can be applied in many fields, for example, cell physiology and neuroscience. These preferred embodiments are based on the principles of phase measurement and imaging techniques. The scientific motivation to use phase measurement and imaging techniques arises, for example, from submicron level cell biology that can include without limitation developmental abnormalities, cellular communication, neuronal transmission, and imaging causes of genetic code execution. The structure and dynamics of subcellular components cannot currently be studied in their native state using existing methods and techniques including, for example, scattering of X-rays and neutrons. Conversely, lightwave-based techniques with nanoscale resolution enable cellular machinery to be studied in its natural state. Accordingly, preferred embodiments of the present invention include systems based on the principles of interferometry and/or phase measurement and used to study cell physiology. These systems include the principles of low-coherence interferometry (LCI), which uses optical interferometers to measure phase, or light-wave scattering spectroscopy (LSS), in which interference within the cellular components themselves is used, or, in the alternative, the combination of LCI and LSS. The principles can be combined in the system of the present invention.

Preferred embodiments for phase measurement and imaging systems include actively stabilized interferometers, isolation interferometers, codirectional optical path interferometers, and interferometers providing differential measurements. Embodiments involving differential measurement systems include two-point heterodyne interferometers and two-beam interferometers. Embodiments using co-directional interferometers can include phase contrast microscopy using spatial light wave modulation.

Optical low-coherence interferometry (LCI) has found many applications in the study of biological media. The most widely used LCI technique is optical coherence tomography (OCT) imaging the 2D or 3D backscatter profile of biological samples. Drexder, W. et al. have described the LCI technique in "In vivo ultrahigh-resolution optical Coherencetomography" (Optics Letters, Volume 24, No. 17, pages 1221-1223), the entire teachings of which are hereby incorporated by reference. OCT has depth sensitivity limited by the coherence length of the light source used. Ultra-broadband light sources have been able to resolve features on the order of 1 micron in size.

Phase-sensitive low-coherence interferometry is sensitive to subwavelength pathlength variations of the sample. The main difficulty of phase-sensitive LCI is the phase noise caused by the optical path variation in the arms of the interferometer. Laser beams of different wavelengths passing through nearly the same optical path can be used to measure the interferometer phase noise, which is then subtracted from a similarly noisy sample signal in order to extract the true sample phase shift. Other researchers have used laser polarization orthogonal along a common optical path to measure differential phase contrast or birefringence with high phase sensitivity. In both techniques, the reference arm path is scanned and computer calculations are required to deduce the phase from the resulting fringe data (via the Hilbert transform); in addition, phase unwrapping algorithms must be used to remove the 2π ambiguity. The fringe scanning and information processing routines substantially reduce measurement speed and may increase noise.

Systems including actively stabilized interferometers:

A preferred embodiment of the invention uses the LCI method, where active stabilization of the interferometer by means of a reference beam allows uninterrupted detection of very small phase shifts with high bandwidth and minimal computer processing. A reference beam locked at an arbitrary phase angle gives direct sample phase measurements without reference arm scanning. Preferred embodiments provide two-dimensional and three-dimensional phase imaging.

A preferred embodiment relies on active stabilization of the Michelson interferometer by means of a reference laser beam. A schematic diagram of a preferred embodiment of an actively stabilized interferometer 300 is shown in FIG. 11 . Actively stabilized Michelson interferometer system 300 includes mirror 306 ; moving mirror 310 ; beamsplitter 304 ; phase modulator 308 ; detector 318 ; local oscillator sources 320 , 322 ; The continuous wave laser beam split by beamsplitter 304 traverses the two interferometer arms and is recombined at detector 318 . One arm of the interferometer contains a phase modulating element 308, such as an optoelectronic modulator or a mirror mounted on a piezoelectric transducer. Large adjustments to the optical path difference can be accomplished by means of translating mirror 310 or any other variable optical delay line. The processing means (eg, computer 315) is in communication with electronics for providing feedback and processing the phase offset measurements. Electronic image display 317 is used to display the phase shift and associated images.

The phase difference between the two interferometer arms is modulated sinusoidally:

φ＝φ+φ _d sin(Ωt) (16)

where φ=k(L ₁ −L ₂ )=kΔL is the phase difference between the two arms, φ _d < 2π is the modulation depth, and Ω is the modulation frequency. The detected interferometer signal is given by the coherent addition of the beams from the two arms of the interferometer:

I＝I ₁ +I ₂ +2(I ₁ I ₂ ) ^1/2 cosφ (17)

The non-linear relationship between I and φ causes the detected signal to have components at many harmonic frequencies of the modulation frequency Ω. The first (I ^Ω ) and second (I ^2Ω ) harmonic terms are given by:

^IΩ ＝4J ₁ (φ _d )(I ₁ I ₂ ) ^1/2 sinφsin(Ωt) (18)

I ^2Ω ＝4J ₂ (φ _d )(I ₁ I ₂ ) ^1/2 cosφcos(2Ωt) (19)

The demodulation of I ^Ω and I ^2Ω into Ω and 2Ω respectively is done by means of a mixer 316 or a lock-in amplifier, and both signals are amplified as a function of φ to give equal amplitudes:

V ₁ =V ₀ sinφ (20)

V ₂ =V ₀ cosφ (21)

Using an analog or digital circuit, the linear combination _V0 is computed with a time-varying parameter θ:

V _θ ＝cosθ*V ₁ -sinθ*V ₂ ＝V ₀ sin(φ-θ) (22)

This signal is used as an error signal to lock the interferometer to any zero crossing with a positive slope. V _θ (t) is integrated, filtered and amplified before being fed back to the phase modulator (high frequency) and path length modulator (low frequency) to actively cancel the interferometer noise. The linear combination V _θ (t) is used as the error signal to allow locking to an arbitrary phase θ.

Stabilized interferometers can be combined with phase-sensitive low-coherence interferometry as described herein. The system components described here can all be realized by means of free-space optics or fiber optics. For clarity, the illustration shows the implementation of a free-space optical system.

A schematic diagram of a reference beam stabilized interferometer for optical delay phase-sensitive LCI is shown in FIG. 12 . Beam 353 from a low-coherence source travels the same path as lock-in beam 355 in a stabilized interferometer. By varying the (stable) path length difference between the two arms of the interferometer, the preparation consists of the sum of two "copies" of the LC beam modulated at the interferometer lock modulation frequency with a very stable continuously variable optical path delay between them output beam.

Sample 382 was placed on a coverslip that had been coated on the side in contact with the sample with an anti-reflection coating at the LC wavelength. The LC beam is focused through the coverslip and sample by means of microscope objective 380 . The backscattered light waves are collected using the same optical system and focused on a detector 366 . The detected signal is the autocorrelation of the backscattered scanning electromagnetic field with the time delay ΔL/c. It can be displayed to show the interferogram at zero delay and a delay corresponding to twice the optical path length between pairs of scattering or reflecting surfaces according to the sample arms illustrated in FIG. 13 . Specifically, the uncoated side of the coverslip is located approximately one coverslip thickness d away from the sample; the interference signal from the sample is seen with an optical path delay nd given the refractive index n of the glass.

Since ΔL=~2nd, the sample signal is demodulated at the modulation frequency by means of a mixer or lock-in amplifier, providing a continuous measurement of the sample phase. The interferometer lock phase θ can in turn be changed electronically in order to lock to zero crossings in the demodulated low-coherence signal. In the described manner, the time progression of the locked phase of the interferometer is used as a direct measure of the phase of the sample. The locking scheme has the advantage of being independent of the amplitude of the sample signal.

The system according to the preferred embodiment is similar to the two-beam optical computed tomography (OCT) technique in that the optical delay is prepared before the low-coherence light waves enter the sample, and the detected signal is sensitive to the distance between the sample and the interferometer. Changes are insensitive. In other possible preferred embodiments, a Mach-Zender interferometer configuration can also be used to prepare low coherence beams.

By introducing variable attenuators into one or both arms of the interferometer, the relative amplitudes of the two time-delayed electromagnetic fields can be modulated in order to optimize the interfering signal.

A schematic of a phase-sensitive LCI stabilized by a reference beam is shown in Figure 14A. The system components are similar to FIG. 11 , but with the sample on a cover glass 430 instead of one mirror of the interferometer. Two beams 422, 424 from two low coherence light sources (LC1 and LC2) are the incident beams on the input of the interferometer. The reference beam has a coherence length equal to or greater than the thickness of the coverslip (eg, approximately 150 microns). The coverslip reflection was used to lock the interferometer. The short coherence length of the reference beam prevents interferometer locking from being affected by spurious reflections from microscope objectives and other surfaces.

In order to distinguish between reflections from the sample surface and the rear surface, the signal beam has a coherence length several times smaller than the thickness of the cover glass. The reference arm length is adjusted to give the interferogram from the sample and, as previously described, the signal is demodulated to give the sample phase as illustrated in Figures 15A and 15B. Locking of the interferometer with respect to the uncoated side of the coverslip results in a phase measurement of the sample relative to this interface and virtually excludes all external interferometer noise.

Compared to the optical delay method described with reference to Figure 11, this preferred embodiment has the disadvantage of having both the signal beam and the reference beam as incident beams on the sample. For biological materials, especially living cells, this may limit the reference beam power that can be used, resulting in reduced locking quality. Scanning the reference mirror, on the other hand, allows more straightforward identification of reflections from the sample. Processing means such as a computer 453 is in communication with the electronics used to provide feedback and process the phase offset measurements. Electronic image display 455 is used to display the phase shift and associated images.

A preferred embodiment can phase image a sample over a region using two methods. In the preferred first method, the incident beam can be scanned across the sample along the X-Y direction as in most OCT devices. In implementations that include an LCI that is stabilized by a reference beam, it is important to maintain reference beam interferometer lock while the beam is scanning. According to the second method, a charge-coupled device (CCD) or photodiode array can be used to detect these signals without scanning. Figure 16 illustrates an imaging system 500 for a stable phase-sensitive LCI. This optical system is used to illuminate the extended area and image the scattered light on the detector. Figure 17 illustrates a simplified development of a system configuration to illustrate optical design in accordance with a preferred embodiment of the present invention. Solid lines represent incident rays, while dashed lines represent backscattered rays.

In the case of CCD imaging, the measurement of relative phase can be done by analyzing a sequence of 4 images, each different in phase from the previous one by π/2. Figure 14B illustrates an embodiment using CCD imaging in a system for actively stabilized phase-sensitive low-coherence interferometry using mirrors with piezoelectric transducers (PZT) 461 to produce phase shifts in accordance with the present invention. Circuit 469 is the electronics to generate the phase shift using the PZT and the electronics to detect the phase shift. A CCD is an array of pixels integrated into a compact electronic chip. A CCD controller 477, itself connected to a processing device such as a computer 478, communicates with the CCD. Image display 479 is used to display the phase shift and associated images.

For high-bandwidth phase imaging, the signal from the photodiode array can be individually demodulated in first and second harmonics; this allows unambiguous measurement of the phase of each pixel.

The higher sensitivity and bandwidth of preferred embodiments of the interferometry system of the present invention provide new possibilities for measuring small optical phase shifts in biological or non-biological media. For example, these preferred embodiments enable the study of cell membrane motion and fluctuations. Dual-wavelength phase-sensitive LCI has been used to observe cell volume regulation and membrane dynamics in human colonic cell cultures. Recently, low frequency oscillations of cell membranes have been observed after addition of sodium azide in culture. Preferred LCI implementations allow the study of membrane dynamics on smaller time scales, where thermally driven fluctuations and mechanical vibrations may be more important. The two-dimensional imaging method according to a preferred embodiment of the present invention allows the study of membrane fluctuations while collecting interacting cells. Oscillations and correlations can provide information about cellular signaling.

Preferred embodiments of the present invention can be used for the measurement of neuronal action potentials. There is great interest in neuroscience in improved optical methods for non-invasively monitoring electrical signals of neurons. Current approaches rely on calcium-sensitive or voltage-sensitive dyes, which suffer from a number of problems, including short lifetimes, phototoxicity, and slow response times.

It has been known for decades that action potentials are accompanied by optical changes in nerve fibers and cell bodies. In addition to this, nerves have been shown to exhibit transient volume increases during stimulation. These changes have been explained in terms of phase transitions in the cell membrane and exponential shifts due to the reorientation of dipoles in the cell membrane.

The phase-sensitive LCI method according to the preferred embodiment can be used to measure optical and mechanical changes associated with action potentials. The increased bandwidth allows sensitive measurement of phase on an action potential timescale of approximately 1 ms. Preferred embodiments of the invention can be used to provide non-invasive long-term measurements of neural signals and provide the ability to image many neurons simultaneously. These embodiments help analyze the spatiotemporal patterning of neural activity that is important for understanding the brain. The small ( ^≈10-4 rad) exponential shifts and membrane fluctuations known to accompany action potentials can be detected in preferred embodiments of the invention which provide high levels of sensitivity speed and high bandwidth (>1 kHz). These embodiments use methods of noise cancellation such as isolation to prevent noise ingress; stabilization to cancel noise using feedback elements; differential measurements to provide common path interferometry without feedback noise cancellation to minimize noise effects .

The embodiments described herein can be used in many medical applications. For example, cortical mapping can be accomplished during neurosurgery with improvements in speed and resolution compared to prior art electrode methods. Additionally, these preferred embodiments can be used to localize epileptic focus during neurosurgery. These embodiments are also capable of monitoring retinal neural activity in the eye. Other applications of preferred embodiments of the present invention include two-dimensional and three-dimensional scanning due to the high speed provided by these embodiments; high dynamic range and DC suppression provided by photodiode detectors; nanoscale imaging in cell biology; epithelial Characterization of tissues and detection of membrane vibrations, such as, but not limited to, auditory cells and blood vessels.

Systems including a dual-beam interferometer:

A preferred embodiment of the invention includes a fiber-based optical retardation phase-sensitive low-coherence interferometer integrated in a conventional lightwave microscope. Simultaneous electrical and optical measurements can be made in cultures of hippocampal neurons. Preferred embodiments include imaging systems comprising photodiode arrays or rapidly scanning beams. Methods for optically stimulating neurons combined with LCI measurements of action potentials could form extremely useful new tools for studying neural network dynamics, synaptic plasticity and other fundamental questions in neuroscience.

Another embodiment applies phase-sensitive imaging techniques to brain slices and even living neurons. Tracking and compensating for motion on the brain surface presents significant challenges. Optical scattering limits the depth at which neuronal signals may be extracted, but depths of approximately 100 micrometers may be possible.

The preferred embodiment of the actively stabilized interferometer described heretofore has included a two wavelength system, where the first wavelength is used for stabilization and the second wavelength is used for phase measurement. Figure 18A illustrates a schematic diagram of a two-point Mach-Zender heterodyne interferometer system in which one wavelength is used. This point-stabilized/reference interferometer system measures the phase difference of light waves passing through two points on a sample 586. The geometry of the nearly co-directional optical path reduces the phase noise of the interferometer.

A collimated laser beam or low coherence light source is split by a beam splitter 584 into a sample 586 path and a reference path. The sample beam passes through the sample 586 and lenses L ₁ (objective lens) 588 and L2 (tube lens) 590 in front of the final beam splitter 592 . Lenses _L1 588 and _L2 590 have focal lengths _f1 and _f2 respectively and constitute a microscope of magnification M= _f2 / _f1 . These lenses are aligned such that the distance between sample 586 and L ₁ 588 is f ₁ , the distance between L ₁ and L ₂ is f ₁ + f ₂ , and the imaging plane lies at a distance f ₂ from L ₂ Location.

The reference beam passes through two acousto-optic modulators 594, AOM ₁ and AOM ₂ , driven with radio frequency RF electromagnetic fields of frequencies ω ₁ and ω ₂ respectively. The variable diaphragm is used to select the +1st order diffracted beam from AOM ₁ and the -1st order diffracted beam from AOM ₂ . Therefore, a light wave incident on AOM ₁ at frequency ω ₀ exits the second pinhole at frequency ω _R =ω ₀ +Ω (where Ω=ω ₁ −ω ₂ ). This dual AOM configuration is used to obtain a relatively low heterodyne frequency Ω of about 100kHz. A low heterodyne frequency may be preferable for the use of high-sensitivity photodetectors and also facilitates optical alignment, since Ω may be set equal to zero for very small beam direction changes. If a higher heterodyne frequency is required, a single AOM can be used. Processing means such as computer 609 and image display 611 are in communication with the system.

The frequency of the frequency shifted reference beam is extended by lenses _L3 598 and _L4 600 separated by a distance equal to the sum of their focal lengths. The signal electromagnetic field and the reference electromagnetic field in the two imaging planes can be described in complex notation by the following formula:

E _S (x, y, t) = E _S ⁰ (x, y) exp[i(φ _s (x, y, t) + φ _{N, S} (x, y, t) - ωt)] (23)

E _R (x, y, t) = E _R ⁰ (x, y) exp[iφ _{N, R} (x, y, t)-(ω+Ω)t] (24)

Here x and y are the transverse coordinates along the optical path, φ _S (x, y, t) is the phase of the sample under study, φ _{N, S} (x, y, t) and φ _{N, R} (x, y , t) represent the interferometer noise in the sample and reference arms, while E _S ⁰ (x, y), E _R ⁰ (x, y) are the electromagnetic field amplitude profiles, which may be, for example, but not limited to, Gaussian of.

The sample phase φ _S (x, y, t) can be expressed by the refractive index distribution n _S (x, y, z, t) of the sample over time:

${\mathrm{<span\; class="notranslate">\; \&phi;</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; z</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="z"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">z</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dz</span>}$

where z is the axial coordinate and integration is done over the depth of the sample. Note the magnification factor M. The intensities at the two imaging planes are given by:

I _± ＝|E _S ±E _R | ² ＝|E _S ⁰ | ² +|E _R ⁰ | ²

(26)

±2|E _S ⁰ ||E _R ⁰ |cos[φ _S (x, y, t) + φ _{N, S} (x, y, t) - φ _{N, R} (x, y, t) + Ωt]

The heterodyne signal is detected with two photodiodes PD1 604 and PD2 606 positioned at positions (x ₁ , y ₁ ) and (x ₂ , y ₂ ). Light waves may be collected through optical fibers or pinholes. The AC component of the detected intensity is given by:

I ₁ (t)＝2|E _S ⁰ ||E _R ⁰ |cos[φ _S (x ₁ , y ₁ , t)+φ _{N, S} (x ₁ , y ₁ , t)-φ _{N, R} ( x ₁ ，y ₁ ，t)+Ωt] (27)

I ₂ (t)＝-2|E _S ⁰ ||E _R ⁰ |cos[φ _S (x ₂ , y _{2 , t} )+φ _{N, S} (x ₂ , y ₂ , t)-φ _{N, R} (x ₂ , y ₂ , t)+Ωt] (28)

The phase difference between the heterodyne signals I ₁ and -I ₂ is then measured with a lock-in amplifier or phase detector circuit 608 .

Φ ₁₂ (t) = [φ _S (x ₁ , y ₁ , t) + φ _{N, S} (x ₁ , y ₁ , t) - φ _{N, R} (x ₁ , y ₁ , t)] - [φ _S (x ₂ , y ₂ , t)+φ _{N, S} (x ₂ , y ₂ , t)-φ _{N, R} (x ₂ , y ₂ , t)]

= φ _S (x ₁ , y ₁ , t) - φ _S (x ₂ , y ₂ , t) + φ _{N, S} (x ₁ , y ₁ , t) - φ _{N, S} (x ₂ , y ₂ , t)-φ _{N, R} (x ₁ , y ₁ , t)+φ _{N, R} (x ₂ , y ₂ , t)

(29)

If it is now assumed that the interferometer noise is independent of lateral position, i.e.,

φ _{N, S} (x ₁ , y ₁ , t) = φ _{N, S} (x ₂ , y ₂ , t) (30a)

φ _{N, R} (x ₁ , y ₁ , t) = φ _{N, R} (x ₂ , y ₂ , t) (30b),

Then the measured phase difference is simply the difference in sample phase at the chosen point:

φ ₁₂ (t) = φ _S (x ₁ , y ₁ , t) - φ _S (x ₂ , y ₂ , t) (31)

The method according to the preferred embodiment of the present invention can be implemented with many photodetectors in the imaging plane subject only to physical constraints. Photodiode arrays or photodiode-coupled fiber bundles can be used to image the phases of many locations simultaneously. Any single detector can be chosen as a "reference" detector against which to measure the phase difference of all other points.

A schematic diagram of a heterodyne interferometer imaging Mach-Zender is shown in Figure 18B. Device 670 images the phase of light waves that have passed through sample 673 .

${\mathrm{<span\; class="notranslate">\; \&phi;</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; z</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="z"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">z</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dz</span>}$

The optical design is similar to the two-point Mach-Zeder heterodyne interferometer described in connection with FIG. 18A , but with two changes: (i) the imaging detector (e.g., a charge-coupled device (CCD) 682) is located in one of the imaging planes, and (ii) Use the photoelectric polarization modulator 672 and the polarizer 681 to perform stroboscopic detection. Quantitative phase images were obtained using phase-shift interferometry.

The time-varying intensity distribution at the CCD imaging plane is given by:

I_(x, y, t)＝|E _S ±E _R | ² ＝

(32a)

|E _S ⁰ | ² +|E _R ⁰ | ² -2|E _S ⁰ ||E _R ⁰ |cos[φ _S (x, y, t)+φ _{N, S} (x, y, t)-φ _N,R (x,y,t)+Ωt]

Stroboscopic phase interferometry is used to image this heterodyne interferogram in a phase-sensitive manner. This requires detection by "gated" CCDs and can be done in several ways. Enhanced CCDs can be gated by controlling the intensifier voltage. The large-aperture photoelectric element in front of the CCD can be used as a fast shutter. In the system illustrated in Figure 18B, an optoelectronic polarization switch is used to control the polarization of the input beam to the interferometer. The two polarization energies are labeled "in-plane" and "out-of-plane", corresponding to Figure 18B. A linear polarizer 681 is placed in front of the CCD imaging device 682 to detect only in-plane polarized light; out-of-plane polarized light is absorbed or reflected by the polarizer.

A photodiode array aligned to the first imaging plane (via fiber optics, if desired) is used to obtain the following signals in a two-point heterodyne interferometer:

I ₁ (t)＝2|E _S ⁰ ||E _R ⁰ |cos[φ _S (x ₁ , y ₁ , t)+φ _{N, S} (x ₁ , y ₁ , t)-φ _{N, R} ( x ₁ , y ₁ , t)+Ωt] (32b)

The gating signal is then derived from the heterodyne signal I ₁ as described below. The electronic comparator outputs "high level" when the heterodyne signal is positive and has a positive slope. This corresponds to the gating signal at phase 0. Similar signals at phase shifts of π/2, π and 3π/2 can be generated by triggering when the heterodyne signal is positive with a negative slope, negative with a negative slope, and negative with a positive slope, respectively. In accordance with a preferred embodiment of the present invention, heterodyne signal 687 and gating signals 688-691 are shown in Figure 18C.

The gating signal is then used to continuously gate the CCD detector. The sequence is under computer 685 control. Only when the gating signal is "high", light waves are allowed to fall on the CCD. Four exposures corresponding to four gating signals instead of an equal number of heterodyne cycles are captured by the CCD in order to obtain the corresponding intensities at the four exposures. The four measured fringe images are called I ₀ (x, y), I _π/2( x, y), I _π (x, y), I _3π/2 (x, y). Then, the relative sample phase can be calculated by:

${\mathrm{<span\; class="notranslate">\; \&phi;</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; the\; tan</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\frac{{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; \&pi;</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 32</span>}\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; )</span>$

Since the phase is shifted between each of the four frames. Other methods for changing the phase and calculating the phase can be used, for example, Creath, K. in "Phase-Measurement Interferometry Techniques", Progress in Optics, Vol.XXVI, E.Wolf, the entire teaching of which is hereby incorporated by reference , Ed., those described in Elsevier Science Publishers, Amsterdam, 1988, pp, 349-393. Furthermore, interferometer noise can be eliminated by means of the reference correlated noise heterodyne signal I ₁ (t) as long as it is constant in the image plane. Stroboscopic phase imaging can be viewed as a form of "barrel" integration where integration is done versus time from a common heterodyne reference signal.

According to a preferred embodiment of the present invention, stroboscopic phase imaging can also be accomplished with a two-beam heterodyne interferometer. This requires a low coherence wavelength, such as 850nm, that can be detected by the CCD. It also requires modification of the sample beam delivery system to the imaging system shown in FIG. 18D as compared to FIG. 19 described below. In this embodiment, the reference heterodyne signals used to generate the four gating signals are provided by optical reference signals. Masking of the probe signal may be done with polarizers by means of fiber optic switches or polarization modulators.

A preferred embodiment of the invention includes a two-beam reflective interferometer. A preferred embodiment of two-beam reflection interferometry involves isolating two-beam heterodyne LCI. A heterodyne two-beam interferometer 620 is shown in FIG. 19 . The interferometer is used to measure the phase change of reflected light waves from the sample relative to a partially reflecting surface located in front of the sample. For example, one might measure the phase of light waves reflected from a sample on a glass slide. As another example, measurements may be made relative to reflections from a fiber optic probe tip placed near the sample under study.

A low-coherence source 622 such as a superluminescent diode (SLD) or a multimode laser diode is coupled into a single-mode optical fiber that enters the vacuum chamber 640 through a vacuum feedthrough. Enclosed within this vacuum chamber is a shock-isolated free-space heterodyne Michelson interferometer. The low-coherence light beam is emitted from the fiber via a collimating lens, and then split by a beam splitter 626 . The two arms of the interferometer (called 1 (656) and 2 (658)) contain acousto-optic modulators (AOM1 628 and AOM2634) driven by radio frequency electromagnetic fields at frequencies _ω1 and _ω2 , respectively. In each arm, a positively offset first-order diffracted beam is selected by means of a pinhole. The light waves are focused by lenses 630 and 636 and then reflected by mirrors M1 632 and M2 638 back to the two AOMs. The lens is placed one focal length away from both the AOM and the mirror. This design allows the AOM retroreflective correction to continue to preserve the spectrum of low coherence (broad spectrum) light waves.

Since the AOM is operated in a dual optical path configuration, light waves incident at frequency ω ₀ are shifted to ω ₀ +2ω ₁ and ω ₀ +2ω ₂ after passing through arms 1 ( 656 ) and 2 ( 658 ), respectively. The frequency difference between the two beams passing arms 1 and 2 is Ω=2(ω ₁ −ω ₂ ).

One of the mirrors M1 (632) is attached to the translation stage in order to adjust the optical path length difference Δl=l ₁ -l ₂ between the two arms. After passing through both arms, the combined beam can be seen as a beam of two pulses separated by a time delay Δl/c. The reflections from the two interferometer arms are focused back into the fiber optic by means of collimator 660 and exit chamber 640 .

Optical circulators are used to separate the retroreflected beam from the incident beam. The light wave is emitted by another collimator 662 as a free-space beam and is focused on the sample 642 , thus first passing through a partially reflecting surface 664 . Backscattered light waves are collected by the same collimator and detected by a photodiode 650 after passing through another optical circulator. The optical retardation of the Michelson interferometer is adjusted to match the optical path difference Δs between the reflection from the sample S and the reflection from the reference surface. When the condition ΔL=Δs is maintained within the coherence length L of the light source, a heterodyne signal at frequency Ω is detected due to interference between light waves reflected from surfaces S642 and R664. The phase of the heterodyne signal represents a measure of the phase of the sample reflection relative to the reference reflection, measured relative to a local oscillator provided by the mixing and doubling of the electromagnetic fields driven by the two AOMs. To prevent heterodyning signals from single surface reflections, the length Δs must be substantially greater than the coherence length LC. It is assumed that the sample thickness is smaller than the glass thickness Δs, so that the signal relates to the glass surface and not to scattering from the sample.

A quantitative description of the interferometer follows. Consider first a monochromatic light source with wavenumber k ₀ . The electromagnetic field amplitude at the input of the Michelson interferometer can be described by:

E _i ＝A _i cos(k ₀ z-ω ₀ t) (33)

The electromagnetic field returning from the beamsplitter after passing through the AOM is given by the sum of the electromagnetic fields from the two arms of the interferometer:

E _m ＝E ₁ +E ₂ ＝A _i cos(2k ₁ l ₁ -(ω ₀ +2ω ₁ )t)+A _i cos(2k ₂ l ₂ -(ω ₀ +2ω ₂ )t)(34)

where k ₁ =k ₀ +2ω ₁ /c and k ₂ =k ₀ +2ω ₂ /c.

The double beam is now the incident beam on the sample. Let _s1 be the optical distance for the reference reflection and _s2 the optical distance for the sample reflection. If the reflectances of the reference and sample reflections are R ₁ and R ₂ , respectively, and multiple reflections are neglected, then the electromagnetic field reflected from the sample is given by:

${\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="l"\; filenames="C2004800208380107H1.png,C2004800208380113H1.png"\; state="\{\{state\}\}">l</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="l"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">l</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; ]</span>$

$<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="l"\; filenames="C2004800208380107H1.png,C2004800208380113H1.png"\; state="\{\{state\}\}">l</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="l"\; filenames="C2004800208380107H1.png,C2004800208380113H1.png"\; state="\{\{state\}\}">l</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 35</span>}<span\; class="notranslate">\; )</span>$

The detected intensity i _D is proportional to the square of the electromagnetic field amplitude:

${\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; \&Proportional;</span><span\; class="notranslate">\; <</span>{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ></span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="R"\; filenames="C2004800208380107H1.png,C2004800208380113H1.png"\; state="\{\{state\}\}">R</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="k"\; filenames="C2004800208380089H1.png,C2004800208380102H1.png"\; state="\{\{state\}\}">k</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;l</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Omega;t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>$

$\mathrm{<span\; class="notranslate">\; 2</span>}\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="k"\; filenames="C2004800208380089H1.png,C2004800208380102H1.png"\; state="\{\{state\}\}">k</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;s</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&Delta;l</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;s</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Omega;t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&Delta;l</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Delta;s</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Omega;t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 36</span>}<span\; class="notranslate">\; )</span>$

The optical frequency oscillation term has been neglected, and it is assumed that the wavenumber shift Ω/c caused by the frequency shift is negligible compared with the inverse of the path length difference Δs and Δl.

To model a low-coherence (broadband) source, it is assumed to have a Gaussian power spectral density with a central wavenumber k ₀ and a full-wavelength half-maximum (FWHM) spectral bandwidth of Δk.

$\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\sqrt{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="ln"\; filenames="C2004800208380104H1.png,C2004800208380108H1.png"\; state="\{\{state\}\}">ln</figure-callout></span>}\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; \&Delta;k</span>}\sqrt{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}}\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; [</span><span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="k"\; filenames="C2004800208380089H1.png,C2004800208380102H1.png"\; state="\{\{state\}\}">k</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}{\mathrm{<span\; class="notranslate">\; \&Delta;k</span>}<span\; class="notranslate">\; /</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\sqrt{\mathrm{<span\; class="notranslate">\; ln</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 37</span>}<span\; class="notranslate">\; )</span>$

The detected intensity for low coherence radiation is found by integrating the monochromatic result over the spectral distribution:

$\stackrel{<span\; class="notranslate">\; ~</span>}{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; D.</span>}}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dk</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="R"\; filenames="C2004800208380107H1.png,C2004800208380113H1.png"\; state="\{\{state\}\}">R</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&Delta;l</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="k"\; filenames="C2004800208380089H1.png,C2004800208380102H1.png"\; state="\{\{state\}\}">k</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;l</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Omega;t</span>}<span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&Delta;s</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="k"\; filenames="C2004800208380089H1.png,C2004800208380102H1.png"\; state="\{\{state\}\}">k</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;s</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&Delta;l</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;s</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&Delta;l</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;s</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Omega;t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 38</span>}<span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&Delta;l</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Delta;s</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&Delta;l</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Delta;s</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Omega;t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>$

in

$\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; [</span><span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; x</span>}}{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; /</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; ln</span>}\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 39</span>}<span\; class="notranslate">\; )</span>$

is the source coherence function of the selected spectral density. Here l _c is the coherence length

${\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; ln</span>}\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; \&Delta;k</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; ln</span>}\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; \&pi;\&Delta;\&lambda;</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 40</span>}<span\; class="notranslate">\; )</span>$

If the path length difference is chosen such that within the coherence length Δl=Δs, and Δl>>l _c , then the dominant time-dependent signal has the form:

$\stackrel{<span\; class="notranslate">\; ~</span>}{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; D.</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; AC</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&Delta;l</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Delta;s</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&Delta;l</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Delta;s</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Omega;t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 41</span>}<span\; class="notranslate">\; )</span>$

By measuring the phase of this signal relative to the local oscillator 652 LO=cos(Ωt), the change in Δs can be measured. Note that the isolation of the Michelson interferometer is essential in order to prevent phase noise from affecting the measurement results through changes in Δl.

Figure 20 illustrates an isolated dual reference heterodyne low coherence interferometer in accordance with a preferred embodiment of the present invention. An interferometer is used to measure the phase of light waves reflected from selected sample depths relative to scattering from different sample depths. This assembly is superior to simpler two-beam interferometers because no glass reflective surfaces are required. This system is ideal for in vivo measurements. Dual reference Michelson interferometers can be used to image neural activity in three-dimensional volumes in sufficiently thin or transparent samples. The system can be used to study the growth of neural networks.

Light waves from low coherence light source 702 are split by fiber coupler 706 into upper and lower paths. The upper path is similar to the two-beam interferometer described above in connection with Figure 19, with a two-beam AOM shifted by _ω1 replacing the sample now in the lower path of the interferometer. The two electromagnetic fields are recombined in another fiber coupler 742 . Photodiodes 746, 748 are arranged in a double-balanced pattern.

In the case of a monochromatic light source, the quantitative description of the interferometer is as follows. The electromagnetic field for the above path can be written as:

E ₁ ＝A _i cos(2k ₀ l ₁ -(ω ₀ +2ω ₁ -2ω ₃ )t)+A _i cos(2k ₀ l ₂ -(ω ₀ +2ω2-2ω ₃ )t)

(42)

And the following path is (again assuming the sample contains two reflections at positions _s1 and _s2 ):

${\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="k"\; filenames="C2004800208380089H1.png,C2004800208380102H1.png"\; state="\{\{state\}\}">k</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="k"\; filenames="C2004800208380089H1.png,C2004800208380102H1.png"\; state="\{\{state\}\}">k</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 43</span>}<span\; class="notranslate">\; )</span>$

It has been assumed that the path lengths of the fiber optic cables are equal between the two arms. The mirror 740 associated with the AOM 736 at frequency _ω3 can be translated so that the path lengths are equal.

The AC component of the photodetector signal is given by:

${\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; \&Proportional;</span>{<span\; class="notranslate">\; <</span>{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ></span>}_{\mathrm{<span\; class="notranslate">\; AC</span>}}<span\; class="notranslate">\; =</span>\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 13</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; twenty\; three</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 44</span>}<span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; +</span>\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 13</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; twenty\; three</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>$

Where Ω ₁₃ =2(ω ₁ −ω ₃ ), and Ω ₂₃ =2(ω ₂ −ω ₃ ). In terms of a Gaussian spectral distribution, the polychromatic case gives:

$\stackrel{<span\; class="notranslate">\; ~</span>}{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; D.</span>}}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; \&Integral;</span>{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dk</span>}<span\; class="notranslate">\; \&Proportional;</span>\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 13</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; twenty\; three</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>$

$<span\; class="notranslate">\; +</span>\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 13</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; twenty\; three</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 45</span>}<span\; class="notranslate">\; )</span>$

If within the coherence length, l ₁ ≈ s ₁ , l ₂ ≈ s ₂ , and Δl, Δs<<l _c , then the main term is

$\stackrel{<span\; class="notranslate">\; ~</span>}{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; D.</span>}}}<span\; class="notranslate">\; \&Proportional;</span>\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 13</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; twenty\; three</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 46</span>}<span\; class="notranslate">\; )</span>$

Next, these two frequency components are combined in a mixer, and a passband filter selects the difference frequency Ω ₁₂ =Ω ₁₃ -Ω ₂₃ =ω ₁ -ω ₂ :

$\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; =</span>\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 12</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 47</span>}<span\; class="notranslate">\; )</span>$

A phase-sensitive detector then measures the phase of the signal relative to a local oscillator at Ω generated by mixing and _doubling of the AOM-driven electromagnetic field. The measured phase is φ=2k ₀ (Δl−Δs).

The phase shifter is used to cancel interferometer noise which may have some influence on the phase measurement despite its differential properties. Phase of photodiode signal components

$\sqrt{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 13</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 48</span>}<span\; class="notranslate">\; )</span>$

is measured and used as an error signal, locking the reflection from _s1 to a constant phase with the aid of a phase shifter.

In an embodiment using a real sample, there will be no two reflections other than the scatter distribution. By setting the reference arm position, the interferometer measures the phase difference between light waves scattered from two different depths.

The embodiment described in connection with FIG. 19 measures the phase of an optical heterodyne signal relative to an electrical signal generated by mixing with an acousto-optic modulator (AOM) radio-frequency electromagnetic field. There are numerous sources of noise associated with the implementation. They include a slow drift of about one wavelength (1λ) on the order of minutes and phase noise at 60 Hz and 120 Hz perhaps due to AOM heating from radio frequency electromagnetic fields. In addition, amplitudes appear in the AOM tuning voltage and may be due to transmission line noise. Other sources of broadband amplitude and phase noise that change as the fiber moves outside the chamber are most likely due to polarization mode dispersion (PMD) in the optical circulator.

Preferred embodiments of the present invention minimize and preferably eliminate noise and include optical reference measurements, or AOM tuning voltages using precision voltage supplies or alternatively using polarization maintaining fiber optics to reduce drift and noise. Figure 21 illustrates a preferred embodiment of an optical reference interferometer that minimizes noise. The implementation addresses the drift and noise issues experienced by the system described in connection with FIG. 19 . The embodiment illustrated with reference to Figure 21 is a heterodyne two-beam interferometer. Interferometers are used to measure the phase change of reflected light waves from a sample using different transverse points that may serve as references on the sample object or individual elements. A low-coherence light source 762, such as an SLD, is coupled into a single-mode optical fiber that enters the vacuum chamber 782 through a vacuum feedthrough. The heterodyne Michelson interferometer operates as described for FIG. 19 . Reflections from the two interferometer arms are focused by collimators 766, 792 back into the two optical fibers and exit the chamber. Additional optical paths are provided as references for different lateral points on individual elements. The backscattered light waves are collected by two collimators 788, 790 and detected by two photodiodes 796, 800 after passing through optical circulators 794 and 798, respectively.

In a dual-reference interferometry implementation, transverse fiducial and reflection-reference phase measurements are combined. Ideally, both the fiducial and the sample object are on the same glass. This has the added benefit of eliminating any tilt, vibration and/or expansion effects in the coverslip. As shown in Figure 22, the measured phase is:

φ(t)＝(φ ₁ -φ ₁ ′)-(φ ₂ -φ ₂ ′) (49)

A preferred embodiment of a dual reference interferometer includes photodetectors with similar gain and frequency response to eliminate noise. In addition, polarization-maintaining components and fibers can be used to address polarization effects in fibers. Specifically, polarization mode dispersion in optical circulators produces a variable delay between two orthogonal polarizations that results in amplitude and phase noise that can be mitigated through the use of polarization-maintaining components. In a preferred embodiment, digital passband filters are used to process harmonics found in optical signals.

FIG. 23A schematically illustrates the voltages used to scale the piezoelectric transducer (PZT) of the system described in connection with FIG. 21 . The PZT calibration assembly is illustrated in Figure 25A.

The phase change corresponding to the displacement of mirror 888 is illustrated diagrammatically in Figure 23B. This phase change corresponds to a nominal distance change of 27nm.

FIG. 24 graphically illustrates, in radians, the noise performance associated with the interferometer illustrated in FIG. 21 over a total time period of 50 ms, where the two arms of the interferometer are equal.

25A and 25B are schematic representations of calibration components for a sample signal and a reference signal in accordance with a preferred embodiment of the present invention. The spacing between mirrors and beamsplitters is varied with PZT. The motion of the PZT is calibrated by monitoring the emission of the light source (He-Ne or Ti:sapphire).

A preferred embodiment of the invention involves a system comprising a two-beam low-coherence interferometer for non-contact measurement of subtle movements of weakly reflective surfaces such as nerve displacement movements during action potentials. Nerve fibers exhibit a rapid lateral displacement during the action potential. This "bulging" of axons, usually attributed to water influx, was first observed in crab nerves and later in many other experimental specimens of invertebrates and vertebrates. All observations of nerve expansion to date have relied on optical or piezoelectric sensors in physical contact with the nerve. Noncontact optical methods for measuring nerve displacement can eliminate contact-related artifacts and allow simultaneous imaging of the activity of multiple nerves in their native state.

Preferred embodiments of the invention include a two-beam heterodyne low-coherence interferometer and its use in measuring the expansion effect of lobster nerve bundles. Existing interferometric methods of observing nerve expansion have not been successful because of low sensitivity and the inability to detect any movement associated with action potentials in frog or lobster nerves. Recently, changes in the refractive index of nerves during action potentials have been successfully measured using transmitted light wave interferometry.

Measuring neural displacements on the order of nanometers in the millisecond time scale requires fast and stable interferometric measurement systems capable of recording from low reflectivity surfaces. According to a preferred embodiment, a dual-beam system consisting of single-mode fiber and free-space elements is shown in FIG. 26 . The light wave from the optical fiber coupled with the super light-emitting diode 922 (Optospeed SLD, FWHM has a central wavelength of 1550nm and a bandwidth of 40nm) collimates and then enters the optical fiber that is included in the dual optical path configuration and is composed of frequencies ω=110.1MHz and ω ₂ = 110 MHz RF electromagnetic field driven acousto-optic modulator (AOM) 946, 952 Michelson interferometer. Mirrors mounted on the translation stage allow control of the optical path difference ΔL to and fro between the two interferometer arms. Light waves enter and exit the Michelson interferometer through a fiber collimator.

The output from each of the two ports of the Michelson interferometer is a dual beam consisting of two low coherence electromagnetic fields with different frequency shifts and variable delays. One of the dual beams is the incident beam on the neural compartment assembly (detailed in Figure 27), while the other is the incident beam on the fiducial gap. The neural assembly and reference gap each contain two reflective surfaces separated by an adjustable distance, and are aligned to reflect incident light waves back to their respective optical fibers. One of these surfaces is almost the interface between air and uncoated glass. The second reflection in the sample comes from the interface between the air and the nerve surface.

ΔL _S and ΔL _R are the round-trip optical path differences between reflections from surfaces 1 and 2 of the sample and reference gap, respectively. The various components are adjusted so that the three path lengths ΔL, ΔL _S and ΔL _R are all equal to within the coherence length of the source. When the conditions are met, photodetectors 932, 962 (new focus 2011) record light waves due to (1) traversing arm 1 of the Michelson interferometer and reflected from surface 2 of the sample (or reference gap) and (2) traversing Michelson interference The interference between the arm 2 of the instrument and the reflected light waves from the surface 1 of the sample (or reference gap) results in a heterodyne signal of frequency Ω=2(ω ₁ −ω ₂ )=200 kHz. The phase difference (up to a multiple of 2π) between two heterodyne signals is φ(t)=k ₀ [(ΔL _S -ΔL)-(ΔL _R -ΔL)]=k ₀ (ΔL _S -ΔL _R ) , where _k0 is the central wavenumber of the source. The quantity most susceptible to phase noise, the Michelson path delay ΔL, is deleted in this differential measurement method. Polarization independent optical circulators 926, 930, 960 are used to maximize the detected power and prevent reflected light waves from re-entering the Michelson interferometer. A polarization controller (not shown) is used to minimize the effect of polarization mode dispersion in the fiber optic component.

To measure the phase difference φ(t), the output of the photodetector was digitized by a 12-bit A/D card (National Instruments PCI-6110) at a rate of five million samples per second. The instruction sequence in the computer calculates the phase difference between the two signals by means of the Hilbert transformation and expresses this phase shift as the relative surface displacement d(t)=φ(t)/2k ₀ .

To verify that the interferometer accomplished the displacement measurements, the neural component was replaced by a planar Fabry-Perot resonator that modulated the resonator spacing sinusoidally using a piezoelectric transducer at a frequency of 300 Hz and an amplitude of 27 nm. The two-beam interferometer's amplitude and frequency measurements agree well with those determined by monitoring the transmission of a 632.8-nm He-Ne laser beam when the cavity is scanned over several micrometers.

According to a preferred embodiment, an ambulatory nerve (~1 mm diameter, ~50 mm long) from the American lobster (L. americana) was dissected and placed on the nerve compartment machined from acrylic resin as shown in FIG. 27 . The chamber contained five containers filled with lobster saline solution, and the nerves between the containers were surrounded by a layer of petroleum jelly insulation to maximize electrical resistance between the containers. The stimulation isolator delivered current pulses of variable amplitude (0-10 mA, duration 1 ms) to stimulate the nerve through the stimulation electrodes. The current delivered by the stimulation isolator may be much greater than the true current through the nerve due to the parallel conductance through the saline. Compound action potentials generated in the nerve are detected with a pair of recording electrodes 998a, 998b and amplified with a gain of 104. In the central hole, the nerve is placed on a small glass stage so that it is not submerged in saline solution. Nerves were kept moist with saline during dissection and data collection.

Figures 28A and 28B illustrate the electrical potential and optically measured displacement of a nerve in an experiment according to a preferred embodiment of the present invention. The spike at time zero in the electrical signal is artificially stimulated. It is followed by a series of spikes describing the action potentials of multiple axons in the nerve tract. The optical signal exhibits a peak with a height of approximately 5 nm and a FWHM duration of approximately 10 ms, the direction of which corresponds to the upper surface displacement. The optical signal exhibits a single peak, rather than multiple spikes like the electrical signal; this may be due to the single-phase (single-symbol) nature of the displacement signal. The rms noise of the displacement measurement is about 0.25nm for a 1kHz bandwidth.

Shifts were observed in approximately half of the neural specimens and varied in amplitude from 0 nm to 8 nm for 5 mA stimulation. Large variability may reflect differences in the nerve itself or preparation procedures. Similar displacement amplitudes have also been reported using nerves from crabs and crayfish. About ~10 smaller displacements were observed in a recent study of lobster nerve expansion using optical levers, which may reflect technical artifacts.

To control for artifacts of nerve heating due to stimulation current, such as thermal expansion from ohmic heating, the peak electrical and displacement signals of a single nerve were measured while varying the stimulation current (as shown in Figure 29). The electrical and displacement signals exhibited nearly identical threshold currents (approximately 1.5 mA) and saturation currents (approximately 5 mA), suggesting that the observed displacements were associated with action potentials. In contrast, the Ohmic effect will feature a quadratic dependence on current independent of saturation. Accordingly, preferred embodiments in accordance with the present invention provide neural displacement studies capable of controlling artificial stimulation. A preferred embodiment includes a heterodyne low-coherence interferometer that performs the first non-contact and first interferometric nerve dilation measurements. According to a preferred embodiment of the present invention, the biophysical mechanisms of nerve expansion can be imaged and analyzed for individual axons. Two-beam low-coherence interferometers may have many other applications in measuring nanoscale motion in living cells. Other embodiments can include interferometer-based microscopy to detect mechanical changes in single neurons associated with action potentials. Correlative interferometry is also used to measure changes in cell volume in cultured monolayers.

Figure 30 illustrates the optical design of the scanning system of a two-beam interferometer, in accordance with a preferred embodiment of the present invention. Mirrors mounted on motorized galvanometers 1024, 1030 are placed in the Fourier plane of the imaging system and allow the beam to sweep across the sample at a constant angle. The mirror was scanned at approximately 30 Hz with 1-2 degree amplitude (50-100 nm on the sample). The galvanometer can be set to raster or scan according to the Lissajous diagram 1052 .

Figure 31 illustrates galvanometer positions and phase data collected from blank coverslips using Lissajous scanning in accordance with a preferred embodiment of the present invention. The total field of view is approximately 100 microns. The vertical axis is the measured phase. The noise profile is about 25 mrad at 1 kHz.

32A and 32B illustrate color maps of retroreflected phase and intensity (amplitude) images in accordance with a preferred embodiment of the present invention. Beam scan data were collected from blank coverslips. The noise profile is about 25 mrad at 1 kHz. Power is highest at the center due to misalignment and clipping that occurs when the beam moves off the optical axis. Black dots in the image correspond to pixels that are not accessed with the Lissajous map.

Figure 33 schematically illustrates the focus problem solved by an embodiment of the present invention. Cells 2002 are placed on a glass coverslip 2004 of a dual-beam microscope. Solid lines represent beams focused on the upper glass surface and cells. Dashed lines represent beams reflected from the bottom surface of the glass. A phase-reference interferometry system is a two-beam interferometry method according to a preferred embodiment of the present invention, and entails collecting not only light waves scattered from the sample of interest but also reflections from a fixed reference surface located in front of the sample. The different axial positions of the sample and fiducial should effectively collect the scattering from both the fiducial and the sample simultaneously, especially for high numerical aperture optics. Two solutions to the focus problem are provided by two systems in the preferred embodiment that allow efficient collection of sample and reference light waves at a numerical aperture of 0.50. First, the bifocal lens system brings the sample surface and the reference surface into focus simultaneously by splitting the numerical aperture into marginal and paraxial rays that pass through differently curved portions of the bifocal optic. Bifocal elements can be constructed by flattening the central region on the convex surface of a plano-convex lens. Placing the bifocal lens near the Fourier plane of the microscope allows scanning of the beam via the mirror galvanometer for sample imaging.

Figure 34 illustrates a bifocal lens design in accordance with a preferred embodiment of the present invention. It is easier to place the bifocal lens in front of the objective rather than behind it. Said embodiment facilitates translation.

Figure 35 illustrates another design of a bifocal lens system in accordance with a preferred embodiment of the present invention. Bifocal lens system 2050 allows beam scanning by placing the bifocal lens at or near the Fourier plane of the imaging system. Beam 2068 is a paraxial beam, while beam 2070 is a marginal beam. The optical properties are invariant to first order inclined beams.

Figure 36 illustrates the calculation of the optimal distance between lenses f3 (bifocal) and f2 such that the spacing between the two focal lengths behind the objective equals Δ=100 microns according to a preferred embodiment of the invention. The optical design is provided by ray tracing.

Figure 37 illustrates the fabrication of a bifocal lens in accordance with a preferred embodiment of the present invention. Bifocal lenses can be made by polishing the center portion of a plano-convex lens. Very small glass thicknesses (approximately 2-10 μm) are removed. Small changes result in path length differences between the signal and reference.

Figure 38 illustrates the measured retroreflection intensity through the optical circulator as the objective lens scans across the glass coverslip in accordance with a preferred embodiment of the present invention. Reflections from the back and front are separated by approximately 100 microns. There is no overlap and therefore no interference.

Figure 39 illustrates the relationship between retroreflection intensity and objective focal point position for a bifocal lens with a single reflection from a mirror in accordance with a preferred embodiment of the present invention. Using the bifocal lens f3, (as the objective lens position is scanned) the single peak from the mirror is split in two, corresponding to the paraxial and marginal beams focused at the mirror at different objective focus positions. The spacing between the peaks depends on the distance between lenses f2 and f3. In this embodiment, the pitch is about 100 microns.

Figure 40 illustrates the relationship between retroreflection intensity and objective focal point position when using a bifocal lens with double-sided coverslip reflection in accordance with a preferred embodiment of the present invention. This graph puts the two earlier graphs together to provide four major peaks and several smaller peaks.

Figure 41 illustrates the relationship between retroreflection intensity and objective focal point position when the distance between f2 and f3 is adjusted to match the spacing between the front and rear glass surfaces in accordance with a preferred embodiment of the present invention. The marginal rays focus on the posterior back, at the same position the paraxial rays focus on the anterior back. This provides the large peak that can be seen in the center.

Figure 42 illustrates an additional smaller peak due to the coupling of paraxial and marginal beams according to a preferred embodiment of the present invention. These extra peaks can be explained by the near-axis edge beam coupling occurring precisely halfway between the two peaks. The amplitude of the extra peaks is highly dependent on optical alignment and can be minimized and preferably eliminated by fine-tuning the optics.

Figure 43A illustrates a dual beam probe with reference surface as an integrated component in accordance with a preferred embodiment of the present invention. The probe consists of a fiber optic collimator 2382 and a graded index of refraction (GRIN) lens 2390 . The uncoated back surface of the GRIN lens provides the reference reflection. Since no separate reference surface is required, the probe is well suited for in vivo applications. For 2D phase imaging or 3D confocal phase imaging, the probe may be mounted on a fast scanning piezoelectric transducer. The described embodiments provide pre-aligned fiber optic probes that are easily used for displacement measurements. The probe has a high numerical aperture (NA), in the range of approximately 0.4-0.5, providing efficient focusing of light waves from scattering surfaces. Integral datum surfaces solve problems caused by depth of field. The probe of the preferred embodiment replaces an assembly of complex optics and is suitable for in vivo use.

FIG. 43B illustrates another preferred embodiment of the fiber-optic dual-beam interferometer probe 2381 . In this example, the reference reflection is provided by the end 2385 of the fiber interlayer. In this case, the end 2385 is polished at right angles to the fiber axis. The optical fiber 2383 is mounted in a glass ferrule 2389 inserted into the housing 2387. The light waves from fiber end 2385 are focused using a graded index of refraction (GRIN) lens, which in this example has a pitch of approximately 0.29 or 0.3. The straightened fiber has a magnification M approximately equal to 3.5 to match the numerical apertures of the fiber (NA=0.13) and the GRIN lens (NA=0.50). The beam is focused on the sample, which in this example is approximately 300 microns from the distal surface of the probe.

Figure 43C is a bright field microscope image of two nerve fibers (axons or dendrites) from a culture of rat hippocampus. Figure 43D is a representation of the heterodyne signal amplitude as a function of position measured using a bifocal two-beam microscope to scan the sample.

Figure 43E is a reflection phase image of the same sample seen in Figure 43D.

44 is a schematic diagram 2400 of a dual-beam probe used to study the geometry of displacement effects observed in nerves during action potentials in accordance with a preferred embodiment of the present invention. By changing the angle of the probe, displacements in different directions can be measured.

Figure 45 illustrates a dual beam probe system that can be used for imaging by scanning the probe or sample in accordance with a preferred embodiment of the present invention. To avoid introducing artifacts due to sample motion, scanning probes are preferred. The probe can be scanned very fast (about 1 kHz) due to its light weight (about 2-3 grams). Because the reference surface is integrated in the probe, three-dimensional confocal imaging (both intensity and phase) is possible. The dual beam probe systems listed can measure either transmission or reflection. Preferred embodiments include scanning dual-beam probe microscopes, so the objective-to-sample distance is highly stable.

Figures 46A-46C illustrate, respectively, images obtained from dried human cheek epithelial cells (or two overlapping cells) placed on anti-reflective coated glass coverslips using a bifocal two-beam microscope in accordance with a preferred embodiment of the present invention. Intensity images, measured phase images of backscattered light waves, and bright field images. These images are generated by scanning a microscope stage with a motorized translation mechanism. The images of Figures 46A and 46B (field of view 130 microns horizontal x 110 microns vertical, scan is 100 x 100 pixels) show the amplitude and phase of the heterodyne signal, respectively. Figure 46C is a bright field image of the same cheek cells (field of view approximately 60 microns x 40 microns). The phase image shows less than one wave of contrast which likely reflects the topography of the lower surface of the cell due to the nearly flat surface in contact with the glass substrate.

Figures 46D-46G illustrate the profilometry capabilities of the two-beam microscope illustrated in Figure 43. The concave surface of a plano-convex lens with a focal length of 25 mm is placed on a coverslip as shown in Figure 46E, and the top of the coverslip has an anti-1.5 Coatings reflective at micron wavelengths. Figure 46D illustrates an intensity image of the central portion of the lens. Figure 46F is a phase map of reflected light waves. Figure 46G illustrates a cross-section of a phase image, with the phase unwrapped by a quadratic fit. The coefficient of the second order term corresponds to a radius of curvature of 11.7 millimeters consistent with the known curvature of the lens surface. Outlying spots in the intensity and phase images can be caused by dust particles or dents on the lens.

47A-47E illustrate the principle of phase-shifting interferometry, including interferograms and collection of frames such as phase stepping (FIGS. 47B and 47C) and barrel integration (FIGS. 47D and 47E) in accordance with a preferred embodiment of the present invention. Schematic illustration of different methods. The method involves modulating the phase, recording the minimum of the three frames and calculating the optical phase shift.

48A-48C illustrate the principles of a stroboscopic heterodyne interferometry system in accordance with a preferred embodiment of the present invention. To give very low phase noise, this system includes acousto-optic modulation that results in a continuous phase ramp similar to barrel integration except that barrel switching involves a correlated noise-referenced heterodyne signal. In contrast, stroboscopic heterodyne interferometers provide continuous measurements because there are no pauses caused by mechanical mirror displacements.

Figure 49 illustrates a schematic diagram of a stroboscopic two-beam heterodyne interferometer 2570 in accordance with a preferred embodiment of the present invention. The light waves from the two-beam interferometer are collimated and then enter the optoelectronic polarization modulator 2594. The light waves reflect from beam splitter 2582, pass through two lenses 2580, 2576 (f1 and f0, objective) in a telescopic configuration, and then illuminate the sample as a collimated beam. Retroreflected light waves from the sample (e.g., cell 2573) and the rear surface of the coverslip are collected by the objective and f1 onto the CCD 2586. Lens f1 is adjusted to be one focal distance from the CCD and one focal distance from the Fourier plane (FP) 2578 . The objective f02576 is also one focal length from the Fourier plane and one focal length from the sample. The optoelectronic polarization modulator in combination with the polarizer 2584 in front of the CCD acts as a fast optical switch. The optoelectronic polarization modulator is gated according to the phase of the reference signal which is the heterodyne signal from the reference gap in the two-beam interferometer. An 820nm SLD is used in the preferred embodiment. The light waves on sample 2573 were collimated, so any focus problems were precluded.

Figures 49B and 49C illustrate data showing the phase noise caused by the dual beam probe shown in Figure 43 focusing on a stationary glass surface in accordance with a preferred embodiment of the present invention.

The system shown and described in connection with FIG. 26 presents a challenge to attempting to focus light waves on a reflective surface and sample that are axially separated by about 100 microns in a typical example. In the case of numerical aperture > 0.1, this is much larger than the depth of field. It is possible to change the bifocal optics or probe design to solve this problem, however, bifocal systems generally have low collection efficiencies that are worsened with high numerical aperture objectives, and diffraction from off-axis parts of the lenses can produce edge effects . The probe design is limited by the fact that the surface fiducial is separated from the sample and the NA is limited to 0.5 when using GRIN lenses.

Figure 50A illustrates a system 2700 in which light waves from two paths are combined along a co-directional optical path before being directed onto a sample. Light waves from path 1 are focused on surface 1, while light waves from path 2 are focused on surface 2.

An interferometer incorporating this feature is shown in the dual beam system 2800 of Figure 50B. The system 2800 has a light source 2801, two moving mirrors 2803, 2805 and a polarizing beamsplitter 2811 that directs the light waves from the two paths to the surface of the sample 2807 and the surface of the reference 2809. The orientation of the polarization components is clearly shown in Figure 50C. Each arm of the interferometer is directed onto a surface. Light waves are thus used more efficiently. The beams are focused separately without fringing effects from bifocal lenses. There is no resulting numerical aperture tradeoff, and the system transaction can be configured in free space requiring no fiber coupling, except optionally from the light source. This improves light wave collection efficiency.

Quantitative Phase Contrast Microscopy Using Spatial Lightwave Modulation:

In another aspect, the present invention provides microscopy systems and methods that combine phase contrast microscopy with phase shift interferometry. The systems and methods of the present invention can be applied to both transmission and reflection geometries. In various embodiments, the methods and systems use a co-directional optical path for waves of different spatial frequencies and shift the phase between waves of different spatial frequencies originating from the same point on the sample.

Phase in optics has been used for many years to provide the sub-wavelength accuracy required in many applications. For example, biological systems, which are inherently weak scatterers, have been made visible using the principles of phase contrast microscopy. Interferometry is one way to obtain phase information, so various interferometry techniques have been developed over the past few years with the aim of retrieving the phase associated with the sample. Techniques such as phase contrast and Nomarski microscopy, while very useful and popular methods, use only optical phase as a contrast tool and do not provide quantitative information about other sizes.

On the other hand, phase shifting techniques can quantitatively determine phase information, and various interferometry schemes have been proposed in the past decades. Differential phase contrast techniques based on polarization optics have been interfaced with conventional optical coherence tomography. Barrel integration techniques, as a specific case of phase-shifting interferometry, have also been used for 2D phase imaging. However, most such interferometers require the formation of two physically separated beams, making them susceptible to uncorrelated environmental noise. This problem often requires specific measures in order to actively cancel the noise. Phase locked loops have been used for this purpose. What is needed are microscopy systems and methods that reduce or eliminate uncorrelated noise from interferometric signals.

The systems and methods of the present invention use co-directional optical paths to coherent light waves of different spatial frequencies emanating from a sample. In various embodiments, the systems and methods presented herein provide phase images of a sample that are substantially free of uncorrelated ambient phase noise. In addition, in preferred embodiments, the method of the present invention is capable of obtaining phase images even in the presence of phase singularities when using low coherence illumination sources.

In various preferred embodiments, the present invention provides instruments that are insensitive to ambient phase noise and provide very accurate and stable phase information over an arbitrary range of exposure times. In various embodiments, the invention is based on image descriptions as interferograms. An example of such a description is the Abbe imaging theory. Each point in the image plane is seen as a superposition (interference) of waves propagating at different angles with respect to the optical axis. If we consider the zero-order scattering from the sample as the basis of the interferometer, then the image can be viewed as the interference between the zero-order electromagnetic field and the electromagnetic field propagating off the optical axis.

Figures 51A-5D are schematic representations of various features of such image descriptions. FIG. 51A is a schematic representation 1100 of an interferogram 1102 formed by high spatial frequency components 1104 and zero order components 1106 . FIG. 51B is a schematic representation 1110 of an interferogram 1112 formed by low frequency components 1114 and zero order components 1106 . FIG. 51C is a schematic representation 1120 of a diffraction spot 1122 formed by the overlapping of beams 1124 of broad spatial frequency at an image plane 1126 . FIG. 51D is a schematic representation 1130 of a wider diffraction spot 1132 at the image plane 1126 produced by a narrower spatial frequency spectrum 1134 . Besides, for example, zero-order components and higher-order components can be regarded as DC components and AC components.

The amplitude of the electron field in the imaging plane and the intensity in the image plane can be expressed as:

代表光波的相位，I_image代表光波在成像平面上某点的强度，而其中的下标0和1分别代表用于方程式50-56中的零阶组份和高阶组份，例如，单一的振幅已被考虑。方程式51举例说明对于在样品上与π相比较很小的相位变化

图像平面中的强度缓慢地改变，这等于说图像缺乏反差。然而，通过把零阶相位

偏移π/2，图像强度分布能被表示成：Where E _image represents the electric field amplitude of the light wave at a point on the imaging plane,

Represents the phase of the light wave, I _image represents the intensity of the light wave at a point on the imaging plane, and the subscripts 0 and 1 represent the zero-order components and high-order components used in equations 50-56, for example, a single Amplitude has been taken into account. Equation 51 illustrates that for small phase changes on the sample compared to π

The intensity in the image plane changes slowly, which is equivalent to saying that the image lacks contrast. However, by placing the zero-order phase

Offset by π/2, the image intensity distribution can be expressed as:

的周围是非常敏感的，相当于该图像即使对于纯粹地调整物体相位也呈现明显的反差。Equation 52 illustrates that the intensity of the image plane is now at the value

is very sensitive to the surroundings, which means that the image shows obvious contrast even for purely adjusting the phase of the object.

In addition to improving intensity contrast, shifting the phase of the zero-order light component can also provide quantitative information about the phase distribution of the object. For example, consider shifting the phase of the zero frequency component by an amount [delta] that can be controllably varied. The total electric field E(x, y) _image and the intensity I _image (x, y, δ) at any point (x, y) in the image plane can be expressed by the following formula, thus remembering that the zero-order electromagnetic field is not changing:

where _I0 is the intensity associated with the low frequency component and _I1 is the intensity associated with the high frequency component.

Here we refer generally to the order of the light waves from the sample. However, when using SLMs, it is very difficult in practice to controllably shift only the phase of the zeroth order component of the light wave from the sample. Therefore, in a preferred embodiment, we shift the phase of the low frequency spatial components comprising all zeroth order light waves. Thus, it will be appreciated that the systems and methods of the present invention can be practiced by only shifting the phase of the zeroth order component, and that shifting the phases of other orders is not necessary. By changing δ, one can get and expressions:

能使用方程式53和54获得，而这两个方程式本身使用，例如，总电场E(x，y)的相位表达式。与物体相关联的相位能用下式表示：phase associated with the sample

can be obtained using Equations 53 and 54, which themselves use, for example, the phase expression for the total electric field E(x,y). The phase associated with an object can be expressed by:

In Equation 56, β=I ₁ /I ₀ and represents the ratio of the intensities associated with the high and low spatial frequency components, respectively. The value of β can be obtained, for example, from I _image (x, y, δ) at four values δ.

In various embodiments, the systems and methods of the present invention are based on the transmission geometry of a microscope system. Figure 52 schematically illustrates one embodiment of a microscope system 1200 based on transmission geometry in accordance with the present invention. Referring to Figure 52, a pair of lenses, objective lens 1204 and tube lens 1206, image a sample 1210 at image plane _P2 1212 in transmission geometry. Lens L ₃ 1214 can be used to form a Fourier transform of an image on a spatial lightwave modulator (SLM) 1216 . The central region on the SLM 1216 can apply a controllable phase shift δ to the central region of the incident beam 1220 relative to the rest of the SLM and reflect the entire incident beam 1220 . The central region of the incident beam 1220 corresponds to low spatial frequency waves delineated by the beam inner boundary 1222 . The outer boundary 1224 illustrates the path of the high frequency beam component, magnifying the divergence of both the low frequency beam component and the high frequency beam component to be seen. Lens _L3 1214 can also act as the second lens of the 4-f system, using beam splitter BS 1232 to form the final image on a detector 1230 such as a Charge Coupled Device (CCD).

A wide variety of devices can be used to control the SLM and obtain images of the sample. For example, in various embodiments, computer 1250 controls modulation of SLM 1216 to increase delta by π/2 and preferably synchronizes image acquisition by detector 1230. The computation of Equation 55 can be done in real time; thus the speed at which the phase image can be displayed is limited only by the acquisition time of the detector 1230 and the refresh rate of the SLM 1216 in the preferred embodiment.

A wide variety of illumination modes and illumination sources can be used to provide illumination 1260 for the transmissive geometry of the present invention. This illumination can be done in bright field mode or dark field mode. Other than that, there are no special requirements on the coherent nature of the sources used. The systems and methods of the present invention can use laser light, partially coherent radiation, or "white" light (eg, from discharge lamps). However, the illumination source should have good spatial coherence.

As shown in Figure 52, the coherent low frequency and high frequency electromagnetic fields are components of the same light beam; and thus share a common optical path. Thus, low frequency components and high frequency components are similarly affected by phase noise, and various embodiments of the system of the present invention can thus be viewed as optically noise-free quantitative phase contrast microscopes. For example, in various embodiments, a phase sensitivity of λ/1000 over an arbitrary range of acquisition timescales is possible.

In various embodiments, the systems and methods of the present invention are based on reflection geometries suitable for microscope systems. The difference between transmission geometry and reflection geometry is the lighting geometry. Transmission geometry can be converted to reflection geometry.

Figure 53 schematically illustrates one embodiment of a reflection geometry based microscope system 1300 in accordance with the present invention. The path outer boundaries of both the low frequency component and the high frequency component are not illustrated in FIG. 53 for clarity. In addition, the divergence of the beam is exaggerated for visibility in FIG. 53 . In various embodiments, beamsplitter BS ₁ 1301 allows a second illumination source 1302 to be coupled into the system and provide illumination 1303 . In one embodiment, the second illumination source comprises a super light emitting diode (SLD). In order to avoid interference due to reflections from various interfaces in the optical path, a low coherence light source such as an SLD is desirable.

Referring to FIG. 53 , a pair of lenses, objective lens 1304 and tube lens 1306 , image sample 1310 at image plane P ₂ 1312 . Lens L ₃ 1314 can be used to cause the Fourier transform of the image to be formed on a spatial lightwave modulator (SLM) 1316 . The central region 1317 on the SLM 1316 can apply a controllable phase shift delta to the central region 1318 of the incident beam 1320 and reflect the entire incident beam 1320 relative to the rest of the SLM. The central region 1318 of the incident light beam 1320 corresponds to low spatial frequency waves. Lens _L3 1314 can also act as the second lens of the 4-f system forming the final image on a detector 1330 such as a CCD using beam splitter BS1332.

A wide variety of devices can be used to control the SLM and obtain images of the sample. For example, in various embodiments, computer 1350 controls modulation of SLM 1316 to increase delta by π/2 and preferably synchronizes image acquisition by detector 1330. The computation of Equation 55 can be done in real time; thus the speed at which the phase image can be displayed is limited only by the acquisition time of the detector 1330 and the refresh rate of the SLM 1316 in the preferred embodiment.

Reflective geometry according to preferred embodiments of the present invention can also include illumination 1360 such as is used in transmissive geometry. Suitable trans-illumination modes include, but are not limited to, bright field and dark field modes. As in the transmission geometry according to the invention, there are no special requirements regarding the coherence properties of the illumination source. The systems and methods of the present invention can use laser light, partially coherent radiation, or "white" light from sources such as discharge lamps. However, the illumination source should have good spatial coherence.

In reflective geometry according to the present invention, the coherent low-frequency and high-frequency electromagnetic fields are also components of the same beam and thus share a common optical path. Thus, low frequency components and high frequency components are similarly affected by phase noise, and various embodiments of the system of the present invention can be viewed as optically noise-free quantitative phase contrast microscopes. For example, in various embodiments, a phase sensitivity of λ/1000 is possible at any range of acquisition timescales.

In various embodiments, the present invention provides a phase contrast microscope system utilizing spatial lightwave modulation, the system comprising an imaging component and a phase imaging component. Imaging components and phase imaging components can, for example, be built independently, making them easier to use with existing optical microscopes.

Figures 54A and 54B, collectively referred to as Figure 54, schematically illustrate one embodiment 1400 of integrating the present invention with an optical microscope. The path outer boundaries of both the low and high frequency components are not illustrated in Figure 54 for clarity. In addition, the beam divergence is exaggerated for visibility in FIG. 54 .

Phase imaging head 1450 can interface with optical microscope 1410 using, for example, the video output of a microscope. Optical microscope 1410 includes a pair of lenses L ₁ 1412 , L ₂ 1414 and mirror 1416 capable of manipulating light waves from a sample 1420 to the microscope video output. Typically, a portion of the light waves are directed by beam splitter 1424 to eyepiece 1426 which focuses the light waves for viewing by human eye 1430 .

The phase imaging head 1450 includes a Fourier transformed lens L ₃ 1454 for forming an image on a spatial light modulator (SLM) 1456 . The central region of the SLM 1456 can apply a controllable phase shift delta to the central region of the incident beam 1460 relative to the rest of the SLM and reflect the entire incident beam 1460 . The central region of the incident beam 1460 corresponds to low spatial frequency waves. Lens _L3 1454 also acts as the second lens of the 4-f system forming the final image on a detector 1470 such as a CCD using a beam splitter BS 1472.

Control of the SLM and image acquisition of the sample can be accomplished, for example, using a computer 1480 that controls the modulation of the SLM 1456 to increase delta by π/2 and preferably synchronizes the image acquisition of the detector 1470. The computer may be a stand-alone computer, eg, equipped with a phase imaging head, or the "computer" can include instructions according to the present invention residing on a computer associated with the microscope. The computation of Equation 55 can be done in real time; therefore, the speed at which the phase image can be displayed is limited only by the acquisition time of the detector 1470 and the refresh rate of the SLM 1456 in the preferred embodiment.

In various embodiments, the lateral resolution of the system of the present invention can be improved by extension of the 4-f system. The 4-f system can be used in both transmission and reflection geometries. In addition, the 4-f system can be used in a system including a calibration system. The 4-f system makes it easy to exploit other Fourier operations performed on the image.

Figure 55 schematically illustrates one embodiment of a system 1500 and method of the present method utilizing a 4-f system. The 4-f system includes a pair of lenses L ₄ 1504 , L ₅ 1506 and can include a spatial filter F 1508 . Spatial filter F 1508 provides amplitude control of individual spatial frequencies. Combined with the phase control provided by the SLM according to the invention, said amplitude control facilitates, for example, the study of small organelles inside cells, since the enhancement of the high frequency component improves contrast. One can imagine other applications where the spatial filter F preferentially attenuates certain spatial frequencies.

The 4-f system can be added to various transmission and reflection geometry embodiments of the present invention. A reflective light source (eg, second illumination source 1302 in FIG. 53) can readily be added to the embodiment of FIG. 55 in accordance with the present invention by one of ordinary skill in the art using the disclosure provided herein.

Systems and methods for phase contrast microscopy utilizing spatial light wave modulation have a variety of applications. For example, these systems and methods can be used to image microscale and nanoscale structures. An important class of applications lies in the study of intercellular and intracellular organization, dynamics, and behavior. The stability provided by the use of co-directional optical paths for both low and high frequency components, and the ability to perform measurements in both transmission and backscatter modes make the various preferred embodiments of the present invention suitable for extended periods of time Study single cells and cell ensembles over a period of time (from hours to days). Thus, in various embodiments, the phase imaging provided by the preferred embodiments of the present invention is used to provide information about the slow dynamic processes of cells, for example, the size of living cells during their life cycle from mitosis to cell death and shape changes.

In various preferred embodiments, the methods and systems of the present invention are used to study the separation process of cells after division with nanometer precision and to provide information about the size, properties, or both of cell membranes. A phenomenon that has received particular attention recently is programmed cell death - apoptosis. Given that apoptosis can be controlled in the laboratory, in various preferred embodiments, the methods and systems of the present invention are used to study the cellular deformation induced in this process. In various preferred embodiments, the methods and systems of the present invention are used to study and detect differences in the life cycle of various types of cells (eg, cancer cells versus normal cells).

One would expect some degree of shared interaction of confluent layers of cells that leads to collective mechanical behavior. In various preferred embodiments, the methods and systems of the present invention are used to study such shared interactions, for example, by performing cross-correlations between different points of a phase image obtained in accordance with the present invention.

The phase imaging provided by the preferred embodiments of the present invention can also be used to provide information about fast dynamic processes in cells, eg, responses to stimuli. For example, processes such as cell volume adjustments are responses of living cells to biochemical stimuli. The timescale for this response may be from milliseconds to minutes and should be very accurately measurable using preferred embodiments of the systems and methods of the present invention. In various preferred embodiments, the methods and systems of the invention are used to study cellular responses to biochemical stimuli and to measure mechanical properties of cellular structures (eg, the cytoskeleton).

In various preferred embodiments, the methods and systems of the present invention are used to study cell structure information of great interest, such as understanding the transport phenomena of organelles in cells and creating artificial biomaterials. In various preferred embodiments, the methods and systems of the present invention are used to study cellular structures, e.g., by stimulating cell membranes with mechanical vibrations and measuring the amplitude of cell membrane oscillations, e.g., to correlate them with cell mechanical properties and cellular matter relation. Traditionally, magnetic or trapping beads have been used to stimulate this movement. In various preferred embodiments, the methods and systems of the present invention are used to study cellular structures using magnetic or trapped beads to stimulate mechanical vibrations, mechanical stimulation induced by photon pressure of femtosecond laser pulses, or both.

An important class of applications is the study of intracellular organization and organelle dynamics. In various preferred embodiments, the methods and systems of the present invention are used to study the transport of various particles within cells.

In addition to the diversity of biological research, preferred embodiments of the present invention are also suitable for industrial applications, for example, studying the nanostructure of semiconductors. The semiconductor industry lacks reliable rapid inspection of wafer quality during nanoscale processing. In various embodiments, the methods and systems of the present invention are used to provide nanoscale information about semiconductor structures, eg, in a quantitative manner. In a preferred embodiment, nanoscale information possesses about one second of measurement.

Figure 56 schematically illustrates one embodiment of a phase contrast microscope system 1600 utilizing spatial lightwave modulation (SLM) in accordance with the present invention. The system exemplified by 1600 can use both reflective and transmissive geometries. In addition, the system has a calibration subsystem.

Referring to FIG. 56 , a pair of lenses, objective lens L11607 and tube lens L21606 , image a sample 1610 using a mirror 1613 in a plane P1612 . Imaging can be accomplished using transmission geometry, for example, through a fiber coupler (FC) 1614 and a first fiber optic 1616 light wave from a first illumination source 1620 (exemplified here as a helium-neon (HENE) laser source) to the sample. 1610 coupling. Imaging can also be accomplished using reflection geometry, for example, by coupling light waves from a second illumination source 1624 (illustrated here as an SLD) through FC 1614 and second optical fiber 1622 and then using first beamsplitter 1626 to combine light waves from the second illumination source The light waves at 1624 are directed onto the sample 1610.

Lens L3 1630 is used to form the Fourier transform of the image on the first spatial light modulator (SLM) 1632 . The SLM 1632 is used to add a controllable phase shift δ to the central region of the incident beam 1634. In one embodiment, lens L ₃ acts as the second lens 1630 of the 4-f system to form the final image on the first detector 1636 (exemplified here as a CCD) using a second beamsplitter 1638 . In one embodiment, the system shown in FIG. 56 further comprises a second 4-f system, eg, schematically exemplified in FIG. 49 . The system of Figure 50 also includes a calibration subsystem for calibrating the SLM. The paths of the calibration lightwaves are schematically illustrated by dashed lines 1640 , while the paths of the illumination and imaging lightwaves are schematically illustrated by solid lines 1642 . The calibration subsystem uses a pair of lenses _L4 1652 and _L5 1654 (forming a beam expander) to collect a portion of the light wave from the first beamsplitter 1626 and pass it through a polarizer Pc1656 which can be used to switch the SLM operation between phase mode and amplitude mode Send that light wave. For calibration, the SLM1658 scans across phase shifts from 0 to 2π in amplitude mode, so that the resulting phase-shifted light wave is attenuated as it returns through the polarizer. The light waves are then collected by lens L ₆ 1660 and focused on detector 1664 .

Various devices and schemes can be used to control the system of Figure 50 and to calibrate the phase images. In one embodiment, a first control unit PC ₁ 1670 is used for image acquisition via the first detector 1636, while a second control unit PC ₂ 1672 is used to control the first and second SLM 1632, 1658 and to collect images via the oscilloscope 1674. Data from detector 1664. The control units PC ₁ , PC ₂ may be separate units or a single unit. For example, PC ₁ and PC ₂ may be separate computers or the same computer, and the control unit may include analog and/or digital circuits suitable for implementing the functions of the control unit.

In various embodiments, the systems and methods of the present invention include dynamic focus using microlenses, for example. In various embodiments, the systems and methods of the invention include parallel focal points to, for example, simultaneously image two or more points on a sample. In various embodiments, the systems and methods of the present invention include coherence functions adapted, for example, to access several points simultaneously in depth.

The phase contrast microscope system of the present invention utilizing spatial light wave modulation can operate in two modes. In the first mode (hereinafter referred to as "amplitude mode"), Fourier filtering is obtained and calibration is done. In the second mode (hereafter referred to as "phase mode"), the wavefront of the light wave is reconstructed and phase imaging is performed.

In various embodiments, in "phase mode", there is no polarizer in front of the SLM, and the light waves are aligned to the fast axis of the SLM. The incident light wave is phase shifted, for example, by the value addressed on the SLM.

In "amplitude mode", the polarizer is placed in front of the SLM. The incident light wave on the SLM is phase shifted (as, for example, in "phase mode"), and the polarization is rotated. When the light wave reflects from the SLM, it returns through the polarizer and the signal is attenuated. Therefore, there is a nominal reduction in amplitude based on the phase shift of the SLM.

57A and 57B schematically illustrate the photoelectric effect on a pixel of an image in amplitude mode 1700 and phase mode 1750, where E _i 1702, 1752 is the electric field emission of the incident wavefront and s-axis 1704, 1754 is the SLM's The slow axis, while the f-axis 1706, 1756 is the fast axis of the SLM.

是在标定中确定的)。RGB被转换成电压而且用来确定象素在SLM1804上的地址。该电压被加到，例如，SLM上的液晶上，把相位偏移给予入射光波1806。标定结果能在振幅模式中获得。图58B举例说明发生在标定中的转化。强度是作为灰度图像的函数使用探测器1852(例如，光电探测器)获得的。然后，作为灰度图像的函数的强度被作为灰度的函数1854转换成相位。图58C举例说明控制-相位模式而且举例说明在各种不同的实施方案中标定结果(灰度)是怎样变成SLM控制和SLM引起的实际相移之间关系的。从此，例如，人们能制订用于该仪器的标定查询表。然后作为灰度的函数的相位被用来产生灰度图像1856(例如，用于计算机上的显示器)并且与SLM所引起的相移1858相关联。58A-58C are block diagrams 1800, 1850, 1855 of various embodiments of SLM modes of operation. Figure 58A illustrates the normal mode of operation of the setup for phase imaging and depicts SLM operation. RGB 1802 is the gray value obtained by the control unit (e.g. computer) which controls the SLM (RGB to

is determined in the calibration). RGB is converted to voltages and used to address the pixels on the SLM1804. This voltage is applied to, for example, a liquid crystal on the SLM, imparting a phase shift 1806 to the incident light wave. Calibration results can be obtained in amplitude mode. Figure 58B illustrates the transformations that occur in calibration. The intensity is obtained as a function of the grayscale image using a detector 1852 (eg, a photodetector). The intensity as a function of the grayscale image is then converted 1854 to phase as a function of grayscale. Figure 58C illustrates the control-phase mode and illustrates calibration results in various implementations How (gray scale) becomes the relationship between the SLM control and the actual phase shift caused by the SLM. From this, for example, one can develop a calibration look-up table for the instrument. The phase as a function of grayscale is then used to generate a grayscale image 1856 (eg, for a display on a computer) and correlated to the phase shift 1858 caused by the SLM.

Figure 59 is an example of a calibration curve 1900 obtained for an instrument operating in the amplitude mode. Calibration curve 1900 is a curve of phase shift 1902 in radians versus grayscale 1904 in RGB values. The resultant curve 1906 obtained shows the relationship between the computer control of the SLM and the actual phase shift induced by the SLM in the format of a calibrated look-up table. Figure 59 can be used as a calibration lookup table. Separation point 1908 in curve 1906 is an overlap of phases.

Example

Some embodiments using transmissive geometry in accordance with the invention and some embodiments using reflective geometry in accordance with the invention are provided. The rotated expansion notation that appears, for example, in Figures 62-66B shows that the 2π ambiguity has been removed.

Example 1: Phase imaging of calibration samples

In this example, calibrated samples were studied and illustrated that the present invention can provide quantitative information at the nanometer (nm) scale. The samples consisted of metal deposits on a glass substrate, which were then etched. The metal deposit pattern is in the shape of a figure "8" and the thickness of the metal layer is about 140nm as measured by a microscopic light wave interferometer.

Figures 60A-60D show images acquired by the system using reflection geometry at four different phase shifts δ. Figure 60A is an image 2000 for δ=0; Figure 60B is an image 2200 for δ=π; Figure 60C is an image 2400 for δ=π/2; and Figure 60D is an image 2600 for δ=3π/2.

是物体的“真实”相位。Figure 61 schematically illustrates the relationship 2100 between the electric field vector E 2102 and the high frequency wave vector component E _H of the electric field and the low frequency wave vector component E _L of the electric field. As shown in Figure 61, the Y-axis 2110 and X-axis 2112 represent the CCD pixel size. phase

is the "true" phase of the object.

FIG. 62 is an image of a calibration sample 2200 generated using data (eg, the data illustrated in FIGS. 60A-60D ) and Equation 55. In FIG. 62, both the Y-axis 2202 and the X-axis 2204 are in units of pixels on the CCD. The scale 2206 on the right side of Figure 62 represents in radians

Figure 63 is a phase image of a sample 2300 calibrated using the systems and methods in accordance with the present invention. Figure 63 was generated using Equation 56 and data (eg, the data illustrated in Figure 62). Figure 63 can also be generated using Equations 55 and 56 and data such as those illustrated in Figures 60A-60D. Both the Y-axis 2302 and the X-axis 2304 are in units of CCD pixels and the vertical scale 2306 is in units of nm.

As shown in Figure 63, one can see that the height of the deposited metal pattern 2310 has been correctly restored to illustrate the ability of the system and method of the present invention to provide quantitative feature information. The noise present in the phase image 2300 is mainly caused by the low quality (8 bit) camera used for recording.

Example 2: Phase Imaging of Phase Grating

Figure 64 shows a phase image 2400 of a phase grating with trenches nominally 10 microns wide and nominally 266 nm deep obtained using transmission geometry. In FIG. 64, the unit of Z-axis 2402 is nm, and the unit of Y-axis 2404 and X-axis 2406 is CCD pixel. Vertical scale 2408 is also in nm and is provided to further aid in determining depth (Z-axis dimension) from phase image 2400 .

Example 3: Phase image of onion cells

In this example, onion cells were phase imaged using transmission geometry in accordance with the present invention. An intensity image 2500 of onion cells is shown in FIG. 65 for comparison with a phase image 2550 shown in FIG. 66 . In Figures 65 and 66, the y-axis 2502, 2552 and the x-axis 2504, 2554 are in units of CCD pixels. Scale bar 2556 in Figure 66 is in nm.

The intensity image (FIG. 65) represents the first frame taken without a phase shift δ=0 between the low and high frequency components. As shown by comparing Figure 65 and Figure 66, the conventional microscope (intensity) image has very low contrast relative to the phase image obtained in accordance with the present invention. As seen in Figure 66, the contrast in the phase image is greatly enhanced, and in this case much finer cellular structures can be distinguished. In addition, the information in the phase image is quantified to nanometer-level precision and can be scaled according to the optical path length of the electromagnetic field through the cell. This type of information represents a great improvement not only with conventional light microscopy but also with conventional phase contrast and Nomarski microscopy.

A preferred embodiment of the invention involves the use of coherence decomposition to decompose a low coherence optical image field into two distinct spatial components that can be controllably phase shifted relative to each other to develop a phase imaging instrument. This technique transforms a typical light microscope into a quantitative phase contrast microscope featuring high accuracy and a sensitivity of λ/5500. The results obtained on living biological cells suggest that the apparatus according to the preferred embodiments of the present invention has great potential for quantitatively studying biological structures and dynamics.

Phase contrast and differential interference contrast (DIC) microscopy provide high-contrast intensity images of transparent biological structures without the need for sample preparation. Structural information encoded in the phase of the light wave is retrieved through an interferometric process. However, while both techniques reveal sample structure in the transverse (x-y) plane, the information provided on the longitudinal axis (z) is largely qualitative.

As described earlier in this paper, phase-shifting interferometry has been used for a considerable time in the quantitative metrology of phase samples, and various interferometry techniques have been proposed. Phase noise, which naturally occurs in any interferometer due to air fluctuations and mechanical vibrations, makes the quantitative recovery of the phase associated with the optical domain a specific challenge in practice. Preferred embodiments include correlated wavelengths to overcome this obstacle.

Furthermore, non-interferometric techniques based on the irradiance transfer equation have been proposed for full-field phase imaging at the expense of time-consuming numerical calculations. Spatial lightwave modulation is used with laser radiation to obtain phase images with λ/30 sensitivity. Digital Recording Interferometry Microscopy with Automatic Phase Shifting (DRIMAPS) is a method that utilizes conventional interference microscopy to provide phase images of biological samples. Although no measures are taken in DRIMAPS to prevent phase noise that ultimately limits the sensitivity of any phase measurement technique, the potential of this instrument for cell biology has been demonstrated.

A preferred embodiment of the present invention includes a low coherence phase contrast microscope (LCPM) as a new instrument for biological research. This technique transforms a conventional light microscope into a quantitative phase-contrast microscope featuring very good accuracy and extremely low noise. The principle of this technique relies on coherent decomposition to decompose the electromagnetic field associated with an optical image into its spatially averaged and spatially varying electromagnetic fields, which can be controllably phase-shifted relative to each other. Let E(x,y) be the complex image field assumed to be stationary over the entire spatial domain. This field can be expressed as

E(x, y) = E ₀ +E ₁ (x, y) (57)

where E ₀ is the spatial average and E ₁ is the component in which E varies in space. Therefore, any image can be viewed as the result of an interference phenomenon between a plane wave (mean electromagnetic field) and a spatially varying electromagnetic field. One should note that E ₀ and E ₁ can be compared to the zero- and high-spatial-frequency components of the electromagnetic field E in each point of the image as a consequence of the central right theorem. Therefore, these two spatial components can be easily separated and independently phase modulated by performing Fourier decomposition.

能被测量。它能被展示作为重要数量与像场相关联的空间相位分布有如下的表达式。Experimental components are depicted in Figure 67. An inverted microscope (Axioert35, Zeiss Co.) was used to image the sample IP in the image plane. A low coherence electromagnetic field (central wavelength in the range 800-850, eg λ ₀ =824nm, bandwidth Δλ=21nm, or alternatively λ ₀ =809nn, Δλ=20nm) emitted by a superluminescent diode is used for the transmission method. To ensure full spatial coherence of the illuminating electromagnetic field, the light waves are coupled into single-mode fibers and then collimated with a fiber collimator. Light traces emerging from the image are shown with dotted and continuous lines representing unbiased light waves and high spatial frequency components corresponding to electromagnetic fields E ₀ and E ₁ , respectively. To decompose the image field into the components described in Equation 57, a Fourier lens FL (50 cm focal length) is placed at a focal distance from the image plane IP. One can see in Figure 67 that the microscope image formed at IP appears to be illuminated with a virtual point source (VPS) that is the microscope image of the fiber end used for illumination. Therefore, in order to obtain an accurate (phase and amplitude) Fourier transform of the image field at the back focal plane of FL, corrective lens CL is placed at plane IP. The focal length of the CL is such that the VSP is imaged at infinity; thus, the new image of the sample maintains its position and magnification, and it appears to be illuminated with a plane wave. In the Fourier plane of FL, the zero-spatial-frequency component _E0 is focused on the optical axis, while the high-frequency component _E1 is distributed off-axis. To control the phase delay between E ₀ and E ₁ , a programmable phase modulator (PPM) (Hamamatsu Co.) was placed in the Fourier plane. The PPM consists of an optically addressed two-dimensional liquid crystal array that, due to its birefringent properties, provides precise control over the phase range of the light waves reflected by its surface. The smallest addressable area on the PPM surface is 20 x 20 µm ² or alternatively 26 x 26 µm ² , while the dynamic range of the phase control is 8 bits over one wavelength or 2π. The PPM can modify the phase (phase mode of operation) or amplitude (amplitude mode) of the incident electromagnetic field in a spatially resolved manner, depending on the orientation of the polarizer P relative to the liquid crystal axis. The light waves reflected by the PPM travel back through the FL and are collected by the CCD placed at the conjugate position of the IP after being reflected on the beam splitter BS. Thus, in the absence of PPM modulation, an exact phase and amplitude replica of the image at the IP is recorded by the CCD. The phase of the high spatial frequency component is continuously increased in four π/2 increments, and the resulting irradiance distribution can be recorded with a CCD. The phase modulation in PPM and the acquisition rate of the CCD are synchronized by a computer PC using, for example, LabVIEW (National Instruments). Using standard 4-frame phase-shift interferometry, the phase difference between E ₁ and E ₀

can be measured. It can be shown as the spatial phase distribution of the important quantity associated with the image field with the following expression.

$\mathrm{<span\; class="notranslate">\; \&phi;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; the\; tan</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; [</span>\frac{\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; \&Delta;\&phi;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; \&Delta;\&phi;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 58</span>}<span\; class="notranslate">\; )</span>$

In Equation 58, the factor β represents the amplitude ratio of the two electromagnetic field components, β(x,y)=|E ₁ (x,y)|/|E ₂ |. The parameter β is measured so as to operate as a π/2 wave plane (amplitude mode) to selectively perform PPM filtering of the two spatial frequency components. Therefore, using Equation 58, the spatial phase distribution of a given transparent sample can be uniquely retrieved. The optimum value for the on-axis modulation area in the Fourier plane was found to be 160×160 μm ² , while the diffraction spots based on the FWHM intensity associated with the optical system in the same plane had a diameter of about 100 μm. Since the numerical computation of Equation 58 is virtually instantaneous, the speed at which the phase image is retrieved is limited only by the PPM refresh rate, which in one embodiment is 8 Hz. However, the overall technology speed can potentially be increased by using other spatial modulators such as ferroelectric liquid crystals.

To demonstrate its potential for quantitative phase imaging, the LCPM technique was applied to study various standard samples. Figures 68A and 68B show examples of such measurements obtained from polystyrene microsphere imaging. Particle diameter was 3±0.045 μm as provided by the manufacturer (Duke Scientific). To better mimic transparent biological samples, the spheres were submerged in 100% glycerol and sandwiched between two coverslips. The refractive index difference achieved between the particle and the surrounding medium is Δn=0.12. The typical transmitted intensity image shown in Figure 68A was obtained without modulation in PPM. The contrast one can see in this image is very unsatisfactory due to the transparency of the sample. Figure 68B shows an LCPM image obtained with the program outline described above. Here, the obtained contrast is relatively high in nature, while the third dimension (Z-axis) provides quantitative information about the thickness of the sample. Using the contouring of the center of the sphere shown in Figure 68B, the value obtained from the corresponding diameter was 2.97 ± 7.7%, which is in good agreement with the value indicated by the manufacturer. Existing errors could be caused by imperfect beam quality and potential impurities present in the solution.

LCPM tools are further used to form phase images of living biological cells. Figure 69A shows quantitative phase images of the final stages of mitosis in HeLa cancer cells. One should note that cells were surrounded by culture medium and lived under typical culture conditions prior to imaging without any additional preparation. It has been previously pointed out that the phase delay accumulated by electromagnetic fields propagating through biological cells is proportional to the non-aqueous mass of the cells. Quantitative phase images should therefore find important application in the automatic analysis of cell kinematics at various stages of cell physiology such as mitosis, cell growth and death.

A phase image of a whole blood smear is shown in Figure 69B. Samples were prepared by simply sandwiching a small drop of fresh whole blood from a healthy volunteer between two coverslips. One can see that the well-known oblate shape of red blood cells (RBCs) is restored. A simple analysis considering the refractive index of hemoglobin relative to plasma readily provides quantitative information on cell volume. This level of detail in RBC analysis is currently only available for electron and atomic force microscopy. Non-invasive optical techniques have the potential to provide a rapid screening procedure for pathology, as it is well known that RBC shape is often a good indicator of cell health. Among other things, the complex dynamic properties of the RBC cell membrane and surrounding proteins that this technique according to a preferred embodiment of the present invention can monitor are responsible for blood clotting.

To assess the stability of the instrument against phase noise and ultimately quantify its sensitivity, cell vessels containing only culture medium (no cells) were imaged at 15 second intervals over a 100 minute time period. Figure 69C shows an example of instantaneous phase fluctuations associated with a point contained in the field of view. Phase values are averaged over an ^area of 0.6 × 0.6 μm, which represents the lateral resolution limit of the microscope. The standard deviation of these fluctuations has a value of 0.15 nm, which is equivalent to λ/5500. The results demonstrate the extraordinary sensitivity of the LCPM instrument. The extremely low noise that characterizes the instrument can be explained by the fact that two coherent electromagnetic fields spatially overlap each other and propagate in an optical path affected by similar phase noise that eventually cancels out in the interference term. In contrast to laser radiation, the use of low-coherence electromagnetic fields favors the sensitivity of the method, since possible fringes formed by multiple reflections on the various components are eliminated.

Accordingly, preferred embodiments of the present invention include a low coherence phase contrast microscope characterized by high accuracy and sensitivity at the level of lambda/5500. Preliminary results on living cancer cells and erythrocytes suggest that the described devices and methods have the potential to become valuable tools for structural and dynamical studies of biological systems. By incorporating a conventional optical microscope into the system components, the instrument according to the preferred embodiment of the present invention is characterized by a high degree of versatility and exceptional ease of use.

The claims should not be falsely read as limited to the described order or elements unless stated otherwise to that effect. Therefore, all embodiments appearing within the scope and spirit of the claims and equivalents thereof are within the scope of the present invention.

Claims (24)

1. method that is used for measuring through the phase place of a part of vectorial light wave, this method comprises the steps:

Light wave from light emitted first wavelength;

Along first light path and second light path guide the light wave of first wavelength, first light path to comprise the first movable reflecting surface and by optical coupled to the media that will measure, second path has the second movable reflecting surface to be provided at the change of path aspect; And

Detection is from vectorial light wave with from the light wave of second light path, so that measure the phase change by the medium light wave.

2. according to the process of claim 1 wherein that described medium comprises the sample of biological tissue.

3. according to the method for claim 1, further comprise photodiode array is provided at least and with one of fibrous bundle of photodiode coupling so that the phase place of sample in a plurality of locations imaging simultaneously.

4. according to the method for claim 1, further comprise the step of the light wave in frequency displacement second light path.

5. according to the method for claim 1, further comprise the step that changes by at least two photodetection detecting phases.

6. according to the method for claim 1, further comprise the He-Ne Lasers light source that emission first wavelength is provided.

7. according to the method for claim 1, it is right further to comprise low-coherence light source and form light pulse.

8. according to the method for claim 1, further comprise first gap and second gap between second reference plane and the 3rd reference plane that are provided between reference plane and the sample.

9. according to the process of claim 1 wherein first reflecting surface experience displacement, second reflecting surface and low coherence light source optical coupled.

10. according to the method for claim 1, further comprise the step of using polarization discrimination sample gap signal and benchmark event signal.

11., further comprise sample arranged with respect to first reflecting surface and second reflecting surface according to the method for claim 1.

12. a twin-beam measuring system, comprising:

Light source;

Light wave from light source is divided into spectroscope at first component on first light path and second component on second light path;

Change the first movable reflecting surface of first optical path length;

Change the second movable reflecting surface of second optical path length; And

Guide the vectorial compositor that to measure into from the light wave of first light path and second light path.

13. according to the system of claim 12, wherein said compositor comprises the light beam spectroscope.

14. according to the system of claim 12, wherein said compositor is the reference field of guiding first reflecting surface that separates by the gap and second reflecting surface from the light wave of first light path and second light path into.

15. according to the system of claim 12, wherein the medium that will measure comprises the tissue that is placed between first reflecting surface and second reflecting surface.

16. according to the system of claim 15, wherein said tissue comprises nerve fiber.

17., comprise that further the light wave that makes from first light path focuses on vectorial first side and makes light wave from second light path focus on lens combination on vectorial second side according to the system of claim 12.

18., further comprise second compositor of light wave being guided into first polarization detector and second polarization detector according to the system of claim 1.

19., wherein be drawn towards the medium and the reference field that the second polarization component is arranged of the first polarization component from the light wave in first path according to the system of claim 1.

20., wherein be drawn towards and medium from the polarization of the light wave quadrature in first path from the light wave in second path according to the system of claim 19.

21., wherein guide the light wave of reference field and the light wave quadrature of guiding reference field from first path into into from second path according to the system of claim 20.

22. according to the system of claim 15, wherein said tissue comprises cancerous tissue.

23. according to the system of claim 12, wherein said light source comprises that low-coherence light source and optical system are right to form light pulse.

24., further comprise fiber coupler according to the system of claim 12.

Images