WO2018173032A9

WO2018173032A9 - System and method for mitigating optical dispersion

Info

Publication number: WO2018173032A9
Application number: PCT/IL2017/050369
Authority: WO
Inventors: Erel Granot; Shalom BLOCH; Shmuel Sternklar
Original assignee: Ariel Scientific Innovations Ltd.
Priority date: 2017-03-24
Filing date: 2017-03-24
Publication date: 2018-11-22
Also published as: WO2018173032A1

Abstract

A method of mitigating dispersion in an optical signal modulated relative to a DC component, comprises detecting the signal by a non-coherent optical detector generating an electrical signal; and applying a linear finite impulse response (FIR) filter to the electrical signal, thereby mitigating dispersion in the optical signal.

Description

SYSTEM AND METHOD FOR MITIGATING OPTICAL DISPERSION

FIELD AND BACKGROUND OF THE INVENTION

The present invention, in some embodiments thereof, relates to optics and, more particularly, but not exclusively, to a system and a method for mitigating optical dispersion.

Dispersion is a known constraint in optical communication systems. In particular, dispersion is problematic at high bit rates (e.g., in excess of 10 Gb/s) in fiber optic systems. Conventional dispersion compensation techniques include fiber Bragg gratings, dispersion compensating fibers, and digital dispersion compensation in coherent detection.

In fiber Bragg grating systems, the dispersed light is directed to a fiber Bragg grating. The short wavelengths have a higher group velocity, therefore they are ahead of the other wavelengths, and longer wavelengths are trailing. The pitch and the chirp in the pitch of the Bragg grating is selected so that the shorter wavelengths are delayed with respect to the longer wavelengths by just the right amount to compensate the dispersion. Dispersion compensating fiber systems include a spool of fiber with an inverse dispersion profile to counteract dispersion over a specific length of fiber, which length is typically not longer than a few tens of kilometers. In digital compensation techniques, the detection is coherent, and therefore the detected currents are proportional to the electromagnetic field, and the dispersion can thus be mitigated by applying a digital Finite Impulse Response (FIR) filter on the detected electric signals.

Another solution to the dispersion constraint is a duo-binary modulation scheme that has a narrow spectrum of an optical signal and thereby less influenced by the dispersion of an optical fiber. Additional methods, include Nyquist Sine pulses, and Orthogonal Frequency Division Multiplexing (OFDM).

SUMMARY OF THE INVENTION

According to an aspect of some embodiments of the present invention there is provided a method of mitigating dispersion in an optical signal modulated relative to a DC component. The method comprises: detecting the signal by a non-coherent optical detector generating an electrical signal; and applying a linear finite impulse response

(FIR) filter to the electrical signal, thereby mitigating dispersion in the optical signal.

According to some embodiments of the invention an AC component of the modulated optical signal is less than half the DC component. According to some embodiments of the invention an AC component of the modulated optical signal is less than a third of the DC component. According to some embodiments of the invention an

AC component of the modulated optical signal is less than a fifth of the DC component.

According to some embodiments of the invention an AC component of the modulated optical signal is less than a tenth of the DC component.

According to some embodiments of the invention a modulation frequency of the optical signal is up-shifted, and the method comprises down- shifting a frequency of the electrical signal, following the application of the FIR filter.

According to some embodiments of the invention the method comprises applying a band pass filter to the electrical signal, prior to the application of the FIR filter, to remove harmonics generated by the optical detector.

According to an aspect of some embodiments of the present invention there is provided a method of transmitting a signal. The method comprises: obtaining an optical signal and an electrical signal selected for modulating the optical signal to carry a data stream; up-shifting a frequency of an AC component of the electrical signal; modulating the optical signal relative to a DC component using the electrical signal; and coupling the optical signal, inclusive of both the AC and the DC components, into a single mode optical fiber.

According to some embodiments of the invention the method comprises applying a linear FIR filter to the electrical signal, prior to the modulation.

According to an aspect of some embodiments of the present invention there is provided a method of transmitting an optical signal modulated to carry a data stream, the method comprises: executing the method as described above; detecting the signal at an exit end of the optical fiber by a non-coherent optical detector generating an electrical signal; applying a band pass filter to the electrical signal to remove harmonics generated by said optical detector; and applying a FIR filter to the electrical signal, thereby mitigating dispersion in the optical signal. According to an aspect of some embodiments of the present invention there is provided a system for mitigating dispersion in an optical signal modulated relative to a DC component, the system comprises: a non-coherent optical detector detecting the signal to generate an electrical signal; electronic circuitry configured for applying a linear FIR filter to the band-pass filtered electrical signal, thereby mitigating dispersion in the optical signal.

According to some embodiments of the invention a modulation frequency of the optical signal is up-shifted, and the electronic circuitry is configured for down-shifting a frequency of an electrical signal output by the FIR filter circuit.

According to some embodiments of the invention the electronic circuitry configured for applying a band pass filter to the electrical signal to remove harmonics generated by the optical detector.

According to some embodiments of the invention the optical signal is transmitted over an optical fiber, wherein the non-coherent optical detector is coupled to receive the optical signal from the optical fiber.

According to some embodiments of the invention the optical signal is transmitted over a single mode optical fiber, wherein the non-coherent optical detector is coupled to receive the optical signal from the optical fiber.

According to an aspect of some embodiments of the present invention there is provided a system of transmitting a signal. The system comprises: electronic circuitry configured for receiving an electrical signal and up- shifting a frequency of an AC component of the electrical signal; an optical modulator configured for receiving an optical signal and the up-shifted electrical signal, and modulating the optical signal using the up-shifted electrical signal to carry the data stream; and an optical coupler configured for coupling the modulated optical signal into a single mode optical fiber.

According to some embodiments of the invention the electronic circuitry is configured for applying a linear FIR filter to the electrical signal, prior to the modulation.

According to an aspect of some embodiments of the present invention there is provided a system of transmitting an optical signal modulated to carry a data stream. The system comprises: a single mode optical fiber having a transmitting end and a receiving end; the system according to any of claim 40-44 coupled to the transmitting end; a non-coherent optical detector coupled to the receiving end and configured for detecting the signal to generate an electrical signal; and electronic circuitry configured for applying a band pass filter to the electrical signal to remove harmonics generated by the optical detector, and for applying a FIR filter to the band-pass filtered electrical signal, thereby mitigating dispersion in the optical signal.

According to some embodiments of the invention the FIR filter is a digital filter operating in a time domain.

According to some embodiments of the invention the FIR filter is characterized by a cutoff parameter ε, which is higher than 10% of a modulation depth of the optical signal.

According to some embodiments of the invention the FIR filter is characterized by a cutoff parameter ε, which is less than 50% of the modulation depth.

According to some embodiments of the invention the cutoff parameter ε equals about one third of the modulation depth.

According to some embodiments of the invention the wherein the FIR filter is a digital filter, operating in a time domain, and having a plurality of terms wherein a number of the terms is about 2N+1, and wherein N is a nearest integer above or below β₂ζ 2Δί , where β₂ is a dispersion parameter, optionally and preferably a second-order dispersion parameter), z is a distance propagated by the signal, and At is a sampling period characterizing a digitization of the signal.

According to some embodiments of the invention the optical signal is transmitted over an optical fiber. According to some embodiments of the invention a length of the optical fiber is at least 1 km. According to some embodiments of the invention a length of the optical fiber is at least 10 km. According to some embodiments of the invention a length of the optical fiber is at least 100 km. According to some embodiments of the invention the optical signal is transmitted over a single mode optical fiber.

According to some embodiments of the invention the optical signal carries a data stream at a bit rate of at least 20 Gb per second.

According to some embodiments of the invention the optical signal comprises a plurality of pulses having a predetermined shape, and wherein the FIR filter is an inverse of a filter whose convolution with a single pulse of the plurality of provides a dispersion distortion which is characteristic to the singe pulse. According to some embodiments of the invention the optical signal is selected from the group consisting of a pulse amplitude modulation (PAM) signal, an on-off keying (OOK) signal, a quadrature amplitude modulation (QAM) signal, a phase shift keying (PSK) signal, a binary PSK (BPSK) signal, a quadrature PSK (QPSK) signal, an octal PSK (OPSK) signal, a differential PSK (DPSK) signal, an amplitude shift keying (ASK) signal, a frequency shift keying (FSK) signal, a radio over fiber (RoF) signal, a frequency-division multiplexing (FDM) signal and an orthogonal FDM (OFDM) signal.

Unless otherwise defined, all technical and/or scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the invention pertains. Although methods and materials similar or equivalent to those described herein can be used in the practice or testing of embodiments of the invention, exemplary methods and/or materials are described below. In case of conflict, the patent specification, including definitions, will control. In addition, the materials, methods, and examples are illustrative only and are not intended to be necessarily limiting.

Implementation of the method and/or system of embodiments of the invention can involve performing or completing selected tasks manually, automatically, or a combination thereof. Moreover, according to actual instrumentation and equipment of embodiments of the method and/or system of the invention, several selected tasks could be implemented by hardware, by software or by firmware or by a combination thereof using an operating system.

For example, hardware for performing selected tasks according to embodiments of the invention could be implemented as a chip or a circuit. As software, selected tasks according to embodiments of the invention could be implemented as a plurality of software instructions being executed by a computer using any suitable operating system. In an exemplary embodiment of the invention, one or more tasks according to exemplary embodiments of method and/or system as described herein are performed by a data processor, such as a computing platform for executing a plurality of instructions. Optionally, the data processor includes a volatile memory for storing instructions and/or data and/or a non-volatile storage, for example, a magnetic hard-disk and/or removable media, for storing instructions and/or data. Optionally, a network connection is provided as well. A display and/or a user input device such as a keyboard or mouse are optionally provided as well. BRIEF DESCRIPTION OF SEVERAL VIEWS OF THE DRAWINGS

Some embodiments of the invention are herein described, by way of example only, with reference to the accompanying drawings. With specific reference now to the drawings in detail, it is stressed that the particulars shown are by way of example and for purposes of illustrative discussion of embodiments of the invention. In this regard, the description taken with the drawings makes apparent to those skilled in the art how embodiments of the invention may be practiced.

In the drawings:

FIG. 1 is a flowchart diagram describing a method suitable for mitigating dispersion in an optical signal propagating in a dispersive medium, such as, but not limited to, an optical fiber, according to some embodiments of the present invention;

FIG. 2 is a flowchart diagram describing a method suitable for transmitting an optical signal, according to some embodiments of the present invention;

FIG. 3 is a schematic illustration of a system for transmitting an optical signal and a system for mitigating dispersion in an optical signal, according to some embodiments of the present invention;

FIG. 4 is a schematic illustration of a system studied in experiments performed according to some embodiments of the present invention;

FIG. 5 shows a ratio between a second harmonic and a first harmonic, as obtained in experiments performed according to some embodiments of the present invention;

FIG. 6 is a graph showing shows a value of a parameter for which an error has a minimum as a function of a modulation depth, obtained in experiments performed according to some embodiments of the present invention;

FIGs. 7 A and 7B show a signal as a function of the time, for a first dispersion factor, as obtained in experiments performed according to some embodiments of the present invention;

FIGs. 8 A and 8B show a signal as a function of the time, for a second dispersion factor, as obtained in experiments performed according to some embodiments of the present invention; FIGs. 9A-F show eye patterns for different values of a dispersion factor, as obtained in experiments performed according to some embodiments of the present invention;

FIG. 10 shows normalized eye-openings, obtained in experiments performed according to some embodiments of the present invention;

FIG. 11 shows the bit error rate of FIG. 10;

FIGs. 12A-C show SNR penalty, obtained in experiments performed according to some embodiments of the present invention;

FIG. 13 is a schematic illustration of a suitable system for transmitting a signal over RFoF link, according to some embodiments of the present invention;

FIGs. 14A-D show power spectra, obtained in experiments performed according to some embodiments of the present invention;

FIGs. 15A-C show logarithmic plots of power spectra of a signal for weak modulation, as obtained in experiments performed according to some embodiments of the present invention;

FIGs. 16A-D show various spectra and spectrum differences as obtained in experiments performed according to some embodiments of the present invention;

FIGs. 17A-C show comparison between spectra of initial transmitted signal (FIG. 17A), a non-compensated received signal (FIG. 17B) and a reconstructed signal (FIG. 17C), as obtained in experiments performed according to some embodiments of the present invention;

FIGs. 18A and 18B show a comparison between an original signal and a detected signal after propagating in a first medium, as obtained in experiments performed according to some embodiments of the present invention;

FIGs. 19A and 19B show a comparison between an original signal and a detected signal after propagating in a second medium, as obtained in experiments performed according to some embodiments of the present invention; and

FIGs. 20A-C show a relative error function as a function of a normalized dispersion coefficient, as obtained in experiments performed according to some embodiments of the present invention. DESCRIPTION OF SPECIFIC EMBODIMENTS OF THE INVENTION

Before explaining at least one embodiment of the invention in detail, it is to be understood that the invention is not necessarily limited in its application to the details of construction and the arrangement of the components and/or methods set forth in the following description and/or illustrated in the drawings and/or the Examples. The invention is capable of other embodiments or of being practiced or carried out in various ways.

It is to be understood that, unless otherwise defined, the operations described hereinbelow can be executed either contemporaneously or sequentially in many combinations or orders of execution. Specifically, the ordering of the flowchart diagrams is not to be considered as limiting. For example, two or more operations, appearing in the following description or in the flowchart diagrams in a particular order, can be executed in a different order (e.g., a reverse order) or substantially contemporaneously. Additionally, several operations described below are optional and may not be executed.

FIG. 1 is a flowchart diagram describing a method suitable for mitigating dispersion in an optical signal propagating in a dispersive medium, such as, but not limited to, an optical fiber, according to some embodiments of the present invention. The method is particularly useful when the optical signal includes a non-zero DC component and is modulated relative to the DC component.

The method begins at 10 and continues to 11 at which the optical signal is detected, optionally and preferably by a non-coherent optical detector, which can be placed, for example, at the exit of the dispersive medium, e.g., optical fiber, carrying the optical signal. The optical detector generates an electrical signal.

As used herein a "non-coherent optical detector" refers to an optical detector that generates an electrical signal that is proportional to the optical intensity of the optical signal.

The received optical signal is typically a dispersed optical signal, wherein an optical pulse is widened during the propagation of the signal in the dispersive medium, e.g., optical fiber. It is recognized that while dispersion is a linear optical process, wherein the dispersion distortion can be expressed as a convolution between the electromagnetic field and the impulse response of the dispersive medium, the dispersion is not a linear process with respect to the electrical signal generated by the non-coherent optical detector providing signals indicative of the optical intensity. This is because the optical intensity is proportional to the square of the optical electromagnetic field, so that the optical detector destroys the linearity by generating an electrical signal which is nonlinear with respect to the optical electromagnetic field.

As demonstrated in the Examples section that follows, the technique of the present embodiments successfully mitigates the dispersion exhibited by the optical signal, even though the dispersion is non-linear with respect to the generated electrical signal.

The optical signal is optionally and preferably transmitted over an optical fiber having a length from about 100 m to more than 500 km. In some embodiments of the present invention the optical fiber has a length of at least 10 km or at least 20 km or at least 30 km or at least 40 km or at least 50 km or at least 60 km or at least 70 km or at least 100 km at least 200 km or at least 300 km or at least 400 km or more.

The optical fiber is preferably other than a dispersion compensating fiber. For example, the optical fiber can be a single mode optical fiber. The optical signal can optionally and preferably carry a data stream. The bit rate of the data stream can be at least 20 Gb per second or at least 50 Gb per second or at least 100 Gb per second or at least 200 Gb per second or at least 400 Gb per second or at least 800 Gb per second or more. The optical signal can alternatively carry an analog data stream.

In some embodiments of the present invention the duplication of the length of the optical fiber by the square of the bitrate is at least 40,000 m(Gb) .

The method optionally and preferably continues to 12 at which a band pass filter is applied to the electrical signal so as to remove harmonics generated by the optical detector. The cutoff frequencies of the band pass filter are selected so that the modulation frequency of the optical signal is maintained and the harmonics generated by the optical detector are filtered out. This operation is particularly useful when a modulation frequency of the optical signal is up-shifted. In these embodiments, the cutoff frequencies of the band pass filter are selected so that the up-shifted modulation frequency of the optical signal is maintained, and the harmonics generated by the optical detector are filtered out.

The method preferably continues to 14 at which a linear finite impulse response (FIR) filter is applied to the electrical signal, optionally and preferably the band-pass filtered electrical signal.

As used herein, a FIR filter is a signal processing circuit, which processes a signal to provide an output signal composed of a summation of successive samples weighted by individual coefficients, wherein the samples are not fed by a feedback loop.

A FIR filter circuit is typically described by a mathematical filtering function, but the circuit is a physical circuit, and its output is a physical signalThe filtering function optionally and preferably provides the individual coefficients and the number of samples. The FIR filter of the present embodiments is linear in the sense that in the time domain, it includes only teams that are linearly proportional to the respective sample, and, in some embodiments, also constant terms. Representative examples of filtering functions suitable for the present embodiments are provided below and in the Examples section that follows

The FIR filter circuit optionally and preferably operates on a sampled version of the signal. The sampling of the signal can be performed downstream FIR filter circuit, at the FIR filter circuit, or upstream of the FIR filter circuit, as desired.

It was found by the present Inventors that such a filter mitigates the dispersion in the optical signal. In some embodiments of the present invention the FIR filter is a digital filter operating in the time domain.

In some embodiments of the present invention the filtering function that describes the FIR filter is characterized by a cutoff parameter ε. The cutoff parameter ε is preferably dimensionless.

When the FIR filter operates in the frequency domain, the cutoff parameter ε can be an imaginary part of a complex argument of a function (e.g., a secant) in the complex frequency plane. When the FIR filter operates in the time domain, the cutoff parameter ε can correspond to a number of terms in a series expansion of the FIR filter. For example, the number of terms in the series expansion can be about l/ε, e.g., the nearest integer of l/ε or the floor of l/ε or the ceiling of 1/ε. When the FIR filter is a digital filter the number of the terms of the filter is optionally and preferably about 2N+1, wherein N is a nearest integer above or below β₂ζ 2Δί , where β₂ is a dispersion parameter, z is a distance propagated by the signal (e.g., a length of the optical fiber carrying the signal), and At is a sampling period characterizing a digitization of the signal. The dispersion parameter β₂ can be defined as the derivative of the reciprocal of the characteristic group velocity of the dispersive medium with respect to the frequency ω of the signal. When the dispersive medium is an optical fiber characterized by fiber mode propagation constant β, β₂ can be defined as the second derivative of β with respect to the frequency ω of the propagated signal.

In various exemplary embodiments of the invention the cutoff parameter ε is higher than 10% or higher than 15% or higher than 20% or higher than 25% or higher than 30% or of a modulation depth of the optical signal. In various exemplary embodiments of the invention the cutoff parameter ε is less than 50% or less than 45% or less than 40% or less than 35% of the modulation depth. Preferably, the cutoff parameter ε is from about 15% to about 45% or from about 20% to about 40% or from about 25% to about 35%, e.g., about one third of the modulation depth. In some embodiments, the cutoff parameter ε is less than 0.1 or less than 0.05 or less than 0.01 for any value of the modulation depth.

A representative example of a filtering function G(co) suitable in any of the embodiments in which the FIR filter operates in the frequency domain is:

where c is a normalization factor (e.g., c=l) ω is the frequency of the optical signal, ai and a₂ are constants, and z is the length of the optical fiber carrying the signal. In preferred embodiments of the invention, ai equals or approximately equals half of the characteristic group velocity dispersion parameter, β₂, of characteristic the dispersive medium. β₂ is typically defined as the derivative of the reciprocal of the characteristic group velocity of the dispersive medium with respect to co. When the dispersive medium is an optical fiber characterized by fiber mode propagation constant β, β₂ can be defined as the second derivative of β with respect to co. The constant a₂ can be set to zero. In cases in which the optical signal is modulated by a modulation that is accompanied by phase chirp with a linewidth enhancement factor a, constant a₂ optionally and preferably correlates with a. For example, a₂ can be set to be arctan(a).

A representative example of a filtering function g(t) suitable in any of the embodiments of the present invention, is:

where d is a normalization factor (e.g., ean satisfy d =1/π), and the notation [1/ε] represents the nearest integer of 1/ε.

A representative example of a filtering function g(nAt), when a digital FIR filter is employed is:

and g(nAt)=0 otherwise, so that the digital filter g(nAt) in this representative example has 2N+1 terms. Preferably

where the brackets correspond to the nearest integer value. In EQ. 3, qmax is a predetermined parameter. A preferred value of qmax is from about [2/m] to about [5/m], e.g., about [3/m], m being the modulation depth.

While EQs. 2 and 3 above are suitable, it is to be understood that other forms of the functions g(t) and g(nAt) are contemplated. For example, when some information regarding the signal are known in advance, such as, but not limited to, in the case in which the signal is provided in the form of a train of pulses (e.g., a train of rectangular pulses, a train of Gaussian pulses, a train of Nyquist-sinc pulses, a train of super Gaussian pulses, etc), the dispersion can be mitigated by a function that is simpler than EQ. 2 in the sense that the values of the filter's coefficients decays faster than in the case presented by EQ. 2 and then they can be implemented by filters having a length that us shorter than 2N+1, and may require lower sampling rate to achieve the same accuracy. That is, if using the filter in EQs. 2 and 3 with a sampling frequency fsl (or equivalently sampling period Atl) results in some accuracy, then a signal for which some information is known in advance (e.g., a train of pulses), can be processed with a shorter filter at a sampling frequency fs2 < fsl (or equivalently with a longer sampling period At2>Atl) to achieve the same accuracy. This is particularly useful when it is desired to fix low values of dispersion. In this case a filter shorter than the filter exemplified in EQ. 3 can be used with a sampling period which is approximately equal to the duration of a single bit, and it is not necessary to employ shorter sampling periods.

As a representative example, consider an optical signal that consists of a plurality of pulses, whose shape is known in advance and therefore the distortion due to dispersion of each of the pulses is also known in advance. In this case, the FIR filter can be an inverse of a filter whose convolution with a single pulse provides the dispersion distortion. Such an inverse filter can be obtained, by deconvolution as known in the art.

The method can optionally and preferably continue to 15 at which the modulation frequency is down-shifted. This embodiment is particularly useful when the modulation frequency of the optical signal is up-shifted prior to its transmission. In these embodiments, the down-shift is preferably by the same amount as the up-shift.

The method ends at 16.

FIG. 2 is a flowchart diagram describing a method suitable for transmitting an optical signal, according to some embodiments of the present invention.

The method begins at 20 and optionally and preferably continues to 21 at which an electrical signal suitable for modulating an optical signal to carry a data stream is received. Preferably the electrical signal is an RF signal. The method can optionally and preferably continue to 22 at which a frequency of an AC component of the electrical signal is up-shifted. This embodiment is particular useful, for example, when the electrical signal is used to generate a modulated optical signal that is to be transmitted over a dispersive medium (e.g., optical fiber) to a remote location, where it is detected, for example, by a non-coherent optical detector. The up-shifting is preferably excited such that the frequency after the up-shifting is at least 1.1 times or at least 1.2 times or at least 1.3 times or at least 1.4 times or at least 1.5 times the frequency before the up- shifting.

In some embodiments of the present invention the method applies 23 a linear FIR filter to the electrical signal. The FIR filter is applied after 21. Optionally, but not necessarily, The FIR filter is applied after 22. The FIR filter can be the linear FIR filter as further detailed hereinabove. In some embodiments of the present invention, operation 23 is skipped and not executed. These embodiments are useful when the signal is detected and processed by method 10, in which case it is sufficient to apply the

FIR filter at the receiving side.

At 24 the electrical signal is used for modulating an optical signal to carry the data stream. The modulation can be either a direct modulation or an external modulation, and of any type known in the art, including, without limitation, pulse amplitude modulation (PAM), on-off keying, quadrature amplitude modulation, phase shift keying (PSK), binary PSK (BPSK), quadrature PSK (QPSK), octal PSK (OPSK), differential PSK (DPSK), amplitude shift keying (ASK), frequency shift keying (FSK) signal, radio over fiber (RoF) modulation, and any combination thereof. The signal can also be multiplexed, according to any multiplexing scheme, including, without limitation, frequency-division multiplexing (FDM) and orthogonal FDM (OFDM).

It is expected that during the life of a patent maturing from this application many relevant optical modulation techniques will be developed and the scope of the term

"modulating" is intended to include all such new technologies a priori.

The modulation depth is optionally and preferably selected based on the length of the dispersive medium along which the signal is to be transmitted, wherein for longer lengths a weaker modulation depth is employed.

The modulation 24 can be done by an optical modulator. The working modulation point of the optical modulator is preferably selected such that the modulated signal includes a DC component. The working modulation point can be set by the electrical signal that drives the optical modulator. Thus, In some embodiments of the present invention the driving signal is such that the modulated signal includes a DC component.

At 25 the optical signal, is coupled into an optical fiber. When operation 22 is executed, both the AC and up-shifted DC components are coupled into the optical fiber. The optical fiber can be of any of the types and lengths described above.

The method ends at 26.

FIG. 3 is a schematic illustration of a system 30 for transmitting an optical signal and a system 50 for mitigating dispersion in an optical signal, according to some embodiments of the present invention. In the illustrated embodiment, the signal transmitted by system 30 is received by system 50, but the present embodiments also contemplate a configuration in which only system 30 is employed and the signal is received by another system, and a configuration in which only system 50 is employed and the signal is transmitted by another system.

In some embodiments of the present invention system 30 comprises an optical modulator 32 configured for modulating an optical signal 34 to carry a data stream. Optical modulator 32 can be a direct or indirect modulator, and be configured to apply any type of optical modulation as further detailed hereinabove. In the representative illustration of FIG. 3, which is not to be considered as limiting, optical modulator 32 applies a direct modulation to an optical field generated by a light source 36. Modulator 32 preferably provide a modulated optical signal that includes a DC component.

System 30 can also comprise electronic circuitry 38 that up-shifts a frequency of an AC component of the optical signal, as further detailed hereinabove. In some embodiments, system 30 comprises an optical coupler 40 that couples the modulated optical signal 34 into an optical fiber 42, as further detailed hereinabove. The optical fiber 42 can be of any of the types and lengths described above. In some embodiments of the present invention electronic circuitry 38 applies a linear FIR filter according to a filtering function, as further detailed hereinabove. The FIR filter can be applied before or after the modulation. When modulator 32 applies a direct modulation, the FIR filter preferably is applied to the electrical signal generated by modulator 32 before the modulation.

System 50 is optionally and preferably positioned at the receiving end of fiber

42. System 50 can comprise an optical detector 52 that detects the optical signal 34, for example, at the exit of fiber 42, and generate an electrical signal 54. Optical detector can be of any type known in the art, preferably of a non-coherent type. Representative examples of optical detectors suitable for use as detector 52 include, without limitation, a photodiode (e.g., a pin photodiode, an avalanche photodiode), a photomultiplier, a photocell, a light-sensitive integrated circuit (e.g., CCD, CMOS) and the like.

In some embodiments, an optical coupler (not shown) is provided between detector 52 and the exit of fiber 42 so as to out-couple signal 42 from fiber 42 and in- couple signal 42 into detector 52. System 50 can further comprise electronic circuitry 56 that applies a band pass filter to electrical signal 54 to remove harmonics generated by optical detector 52, as further detailed hereinabove. In embodiments in which the FIR filter is applied after the modulation, electronic circuitry 56 can apply the linear FIR filter, as further detailed hereinabove, to the band-pass filtered electrical signal. In some embodiments of the present invention electronic circuitry 56 transformed signal 54, optionally and preferably after the application of the band-pass filter, to the frequency domain, as further detailed hereinabove.

In preferred embodiments, the FIR filter of the present embodiments is applied at the receiving end of fiber 42 (by system 50 in the illustration of FIG. 3). This is advantageous since the FIR filter can also block parasitic noises.

As used herein the term "about" refers to ± 10 %.

The word "exemplary" is used herein to mean "serving as an example, instance or illustration." Any embodiment described as "exemplary" is not necessarily to be construed as preferred or advantageous over other embodiments and/or to exclude the incorporation of features from other embodiments.

The word "optionally" is used herein to mean "is provided in some embodiments and not provided in other embodiments." Any particular embodiment of the invention may include a plurality of "optional" features unless such features conflict.

The terms "comprises", "comprising", "includes", "including", "having" and their conjugates mean "including but not limited to".

The term "consisting of means "including and limited to".

The term "consisting essentially of" means that the composition, method or structure may include additional ingredients, steps and/or parts, but only if the additional ingredients, steps and/or parts do not materially alter the basic and novel characteristics of the claimed composition, method or structure.

As used herein, the singular form "a", "an" and "the" include plural references unless the context clearly dictates otherwise. For example, the term "a compound" or "at least one compound" may include a plurality of compounds, including mixtures thereof.

Throughout this application, various embodiments of this invention may be presented in a range format. It should be understood that the description in range format is merely for convenience and brevity and should not be construed as an inflexible limitation on the scope of the invention. Accordingly, the description of a range should be considered to have specifically disclosed all the possible subranges as well as individual numerical values within that range. For example, description of a range such as from 1 to 6 should be considered to have specifically disclosed subranges such as from 1 to 3, from 1 to 4, from 1 to 5, from 2 to 4, from 2 to 6, from 3 to 6 etc., as well as individual numbers within that range, for example, 1, 2, 3, 4, 5, and 6. This applies regardless of the breadth of the range.

Whenever a numerical range is indicated herein, it is meant to include any cited numeral (fractional or integral) within the indicated range. The phrases "ranging/ranges between" a first indicate number and a second indicate number and "ranging/ranges from" a first indicate number "to" a second indicate number are used herein interchangeably and are meant to include the first and second indicated numbers and all the fractional and integral numerals therebetween.

It is appreciated that certain features of the invention, which are, for clarity, described in the context of separate embodiments, may also be provided in combination in a single embodiment. Conversely, various features of the invention, which are, for brevity, described in the context of a single embodiment, may also be provided separately or in any suitable subcombination or as suitable in any other described embodiment of the invention. Certain features described in the context of various embodiments are not to be considered essential features of those embodiments, unless the embodiment is inoperative without those elements.

Various embodiments and aspects of the present invention as delineated hereinabove and as claimed in the claims section below find experimental support in the following examples.

EXAMPLES

Reference is now made to the following examples, which together with the above descriptions illustrate some embodiments of the invention in a non limiting fashion.

Example 1

Dispersion is a linear process that operates in the electromagnetic (EM) domain. A schematic illustration of a system studied in the present example is illustrated in FIG. 4. The system includes a laser, a dispersive fiber a detector and a filter. The laser can be directly modulated (excluding the need for external modulation), and the electronic filter can be placed at the receiver end (A) or transmitter end (B).

For an EM pulse envelope A(0, t) at the entrance to the fiber (z = 0), after a distance z, it transforms to

with the dispersion kernel

Since a non-coherent optical detector measures the power p(z, t) of the signal, where

the linearity of the dispersion is destroyed, and if the deformations exceeds a certain level, data reconstruction is not possible.

In networks in which the laser is directly modulated the data is encapsulated in the power of the signal, rather than in the field. Hence, the relation between the input power signal p(0, t) and the detected signal p(z, t') is

In the region of weak modulation, where p is the DC level of the optical power, and 5p(t) is the relative change:

where 5p(t')« l, the linearity is partially restored. As opposed to digital communications, where the loss of linearity is not necessarily a problem, in analog communications signal distortion causes loss of information. To avoid nonlinear distortions (especially third order) the optical channel is tuned to work in the weak modulation regime. For example, weak modulation is employed in RF-over-Fiber (RFoF) technology. In the weak modulation regime there a linear correspondence between the input and output RF signals. Weak modulation is beneficial since there are fewer modulation distortions. In this case, expanding EQ. (1.2) up to the first order in 5p(t) yields

After expanding the square and keeping only the first order in 5p(t),

where stands for the real part.

Thus, in the weak modulation regime, the linearity is restored. The relation between the input 5pi_n(t) and the output

is a convolution,

where the kernel in this case is

and the corresponding transfer function is

where F^~ stands for the inverse Fourier transform. Therefore, the dispersion effect can be reduced or eliminated according to some embodiments of the present invention b applying the inverse spectral filter

which corresponds to the intensity kernel

so that

or

The transmitted and received electronic signals are denoted by Vi_n(t) and v_out(t), respectively. When vi_n(t) and v_out(t) are proportional to 5pi_n(t) and 5p_out(t),

In case of modulation accompanied by phase chirp with a linewidth enhancement factor a, the kernel of EQ. (1.8) is optionally and preferably be replaced with

with similar modifications to all the corresponding formulas. An imaginary term is optionally and preferably introduced to the function's argument, namely,

where ε is a cutoff parameter. Correspondingly

EQ. (1.16) can be reduced to the finite summation, which is also the impulse response of the filter:

where [l/ε] corresponds to the value of l/ε rounded to the nearest integer. The justification for this reduction is that higher terms are exponentially smaller.

It was found that the cutoff parameter ε determines the number of terms that describe the filter. The present inventors found that the optimum value of ε is approximately mG, where m is the modulation depth. If the time period between adjacent measurements is At, then, the dispersion can be mitigated with the following FIR filter:

where and the brackets correspond to the nearest integer value. However,

unlike conventional approaches EQ. (1.17) can operate directly on the detected signal, which is proportional to the intensity, without the need for coherent detection. The advantage of the cutoff parameter ε can be understood with the following simple example.

Let

where vo corresponds to the modulation amplitude.

In the case of modulation depth m, the optical power satisfies

where po corresponds to the ower DC level, and the field can be described as:

Therefore, at the other end of the fiber,

and the power (p(t, z) oc IE(t,z)l ) approximately satisfies

Under the approximation after applying the transfer function of

the filter of EQ. (1.15) on EQ. (1.21), the reconstructed signal consists of at least two harmonics (the higher ones are ignored due to the weak modulation depth):

where

and

If the value of ε is large, ΙΗ1(Ω, ε)Ι « m for a wide spectral range near β₂ζΩ 2 = π(1/2+η) for any integer n. On the other hand, If the value of ε is small the second harmonic is large (about 2β₂ζΩ = π(1/2+η) for any integer n.

For the frequencies where the real part of the denominator of the second harmonic vanishes (for the worst case, see FIG. 5),

the ratio between the second and the first harmonics is

which demonstrates that a preferred value for the cutoff parameter ε is larger than 0.125m.

Simulation

To examine the benefits of the present embodiments, the filter was implemented with an OOK protocol. At z=0 an infinite pseudorandom sequence of rectangular intensity pulses was launched:

where x_n is a binary sequence (either -1 or 1), T is the time slot of a single bit,

is the rectangular function, and ξ is the pulse duty cycle, where ξ =1 stands for 100% duty cycle [the so called nonreturn- to-zero (NRZ) protocol], and ξ < 1 stands for return to zero (RZ) protocol. Since NRZ is less susceptible to dispersion, the simulation focus on this protocol.

Equation (1.26) is equivalent to launching the EM field signal:

where

is the modulation depth in the field domain.

To take account of the fact that the pulses are not exactly rectangular, but instead have a finite rise/fall time τ=Δ^_1, a Gaussian low-pass filter was added:

so that EQ. (1.27) becomes

where

is a smooth rectangular function with rise/fall time τ=Δ^~ , and erfc is the complementary error function.

After a finite dispersive medium, e.g., a fiber with spectral response

the input, EQ. (1.29) propagates to a form having an exact analytical solution,

Therefore, the detected signal, which is proportional to the detected optical power (after the reduction of the DC) obeys

On this signal, the reconstructing filter, EQ. (1.16), can be applied to yield the reconstructed signal

For laser bandwidth is considerably narrower than modulation bandwidth, EQs. (1.31) and (1.32) show agreement with the experimental results. Since the modulation bandwidth is larger than 10 GHz, and since there is no coherent detection, the laser bandwidth can be from less than 1 MHz to about 10 MHz without affecting the analysis. To evaluate the optimum value for ε, the error Ε(ε) can be evaluated as a function of ε:

where the ori inal signal is

In each one of the simulations, a 2 - 1 pseudorandom bit sequence (PRBS), was used for the bits' value x_n.

In a first simulation, the optimum so was evaluated. For nine values of the modulation depth m, from 0.01 to 1 (in logarithmic scale), the error of EQ. (1.35) was calculated for many values of ε [where v_rec,e(t) is the reconstructed signal, EQ. (1.34)]. FIG. 6 shows value of ε for which the error has a minimum, as a function of the modulation depth.

As shown, in FIG. 6, the optimum so can be fitted to the line:

This relation was used in the simulation that followed.

In FIGs. 7-9, the output-reconstructed signals are plotted for different values of the normalized distance

which is a measure of the dispersion effect in the communication link.

In these simulations, a computer program was used to generate the PRBS, use EQ. (1.33) to generate the nonreconstructed signal from the PRBS, and then to apply the filter of the present embodiments, EQs. (1.15) and (1.16) to reconstruct it. To emulate digital reconstruction, EQ. (1.17) can be used, but in the present Example the signal was corrected in the spectral domain using EQ. (1.15). In each of FIGs. 7A-B, 8A-B and 9A-F the upper panel (FIGs. 7A, 8A and 9A- C) represents the non-reconstructed signal, and the lower panel (FIGs. 7B, 8B and 9D-F) represents the reconstructed signal.

All the distortions in this Example are due to dispersion, and no artificial noises were added.

All three cases (FIGs. 7A-B, 8A-B and 9), show improvement. For example, for ζ =0.4 (FIGs. 8A and 8B) the data cannot be retrieved from the detected signal without the filter optionally and preferably, while with the filter of the present embodiments the signal distortion is very small.

FIGs. 9A-F illustrate the eye pattern of the signals for three different values of ζ. It is recognized that

is an upper limit for OOK. The results in FIGs. 9A-C are consistent with this limit, wherein the eye-opening is closed beyond ζ = l/π. However, with the filter of the present embodiments (FIGs. 9D-F), the eye pattern is wide open for much longer distances. Moreover, with this filter the degradation in the eye quality from ζ =0.3 to ζ=6 is minor. For bit-rate B = 50 Gb/s, ζ=0.3 and ζ=6 correspond to 6 km and 120 km of smf28, respectively.

To quantify the eye-opening, it is noted that the signal value at the center of the p-bit (for NRZ) is equal to

Therefore the values of the "1" and levels are

(the minimum of all the "l"s bits) and

(the maximum of all the bits), respectively. The ori inal signal's e e-opening is therefore

A similar process can be applied to the reconstructed signal to obtain the eye opening of the reconstructed signal

The improvement in the eye-opening is emphasized, when the Gaussian optical filter is replaced with a smooth rectangular filter (like a super- Gaussian in wavelength division multiplexing (WDM) networks) with a spectral width of about 2/Γ. The eye- opening (normalized to vo, the amplitude at z=0) as a function of the normalized parameter ζ is plotted in FIG. 10 for m=0.1 and a rectangular filter. As shown, while the eye-opening of the nonfiltered signal vanishes at about ζ = 0.3, the eye pattern of the signal once filtered using the filter of the present embodiments remains widely open.

The reconstructed signal's eye-opening wins for ζ > 0.15, which corresponds to 3 km of smf28 for 50 Gb/s and 75 km for 10 Gb/s, and the eye-opening degradation is small even for distances of several hundreds of kilometers.

Signal-To-Noise Consideration

In RFoF and low cost digital networks, there are primarily three types of noise sources: thermal noise, shot noise, and relative intensity noise (RIN). Thermal noise is independent of the detected signal, and both shot noise and RIN depend on the DC power of the transmitted signal, and therefore, none of the noise sources is affected by the eye-opening decrease.

The bit error rate (BER) evaluation counterpart is defined as

where R is the load resistance and N is the power noise consists of both thermal and shot-noise terms. The BER of FIG. 10 is plotted in FIG. 11 for 0 dBm transmit power source at 50 Gb/s, for the case in which the detection is by a high-quality detector whose sensitivity is bounded only by thermal noise and shot noise. As shown, even at that high bit-rate, with only 0 dBm, the presented method yields superior performances beyond ζ > 0.3 (6 km), and the forward-error-correction (FEC) limit (BER ~ 10 ) is reached only at ζ of about 1.4 (28 km). In FIG. 11, the non-compensated performance is illustrated by the vertical dashed line, and the FEC limit is marked by the horizontal dotted line.

Since the detected electrical RF-power is proportional to the square of the optical power modulation depth, this method begins with an SNR penalty of about 20 logio(m) (in dBm). However, since the eye pattern of OOK shrinks considerably for long fibers, the addition of the filter of the present embodiments moderates this decrease considerably (see FIG. 10). The technique of the present embodiments is therefore advantageous since the effect caused by SNR penalty decreases with any increase of the fiber's length.

In particular, since in ordinary OOK, the eye pattern is completely closed beyond the fundamental limit = Ι/π, then the technique of the present embodiments is advantageous for ζ > C^^.

A measure of performances for low-cost detectors is the SNR penalty. In FIGs. 12A-C, the SNR penalty (compared to the back-to-back value) is plotted as a function of the normalized fiber length ζ for three scenarios: without reconstruction (dotted line), with a the filter of the present embodiments and an additional rectangular low-pass electrical filter with spectral width 2 T for modulation depths m=0.25 (solid line) and m=0.1 (dashed line).

As shown, the SNR penalty of the non-reconstructed signal is practically infinite for ζ > 0.3. However, with the aid of the filter of the present embodiments, the SNR penalty remains finite for any distance. FIGs. 12A-C show that the penalty for ζ = 6 (equivalent to 120 km for 50 Gb/s and to 3000 km for 10 Gb/s) remains about 25 dB for both modulation depths (albeit the m=0.25 case fluctuates more with the distance). Nevertheless, in the region ζ = 0.5, there is a 6 dB advantage to the m=0.25 scenario. Therefore, for short distances, deep modulation is preferred, while for long distances, weak modulations (but preferably no less than m=0.06) are preferred.

For example, a traditional low cost 25 Gb/s OOK link (maximum transmit power of 4 dBm and detector sensitivity of -24 dBm) cannot operate when distances longer than 20 km (ζ = 0.25). However, when modulated with m=0.25, the link can operate for distances in excess of 40 and even 50 km, even with additional link losses of less than 10 dB. Example 2

In this Example, a dispersion compensation for RFoF links is exemplified. While analog and digital links share the same basic components, there are some fundamental differences between them. Two unique properties of RFoF links can be exploited to compensate dispersion-based distortions with an affordable approach. Firstly, unlike digital optical transmission, analog data is usually transmitted with weak modulation, since only weak modulation can insure linear operation. Secondly, RFoF data is carried by a high frequency RF carrier, which typically not the case in digital data links (in both cases there is an optical carrier but in the analog case there is an additional carrier in the RF domain).

Without loss of generality, an RF signal Vi_n(t) with a spectral bandwidth Δω, carried by a high carrier frequency ωο>Δω, is considered, since this is the case in most RF applications. After modulating the laser in the linear domain, the optical power entering the fiber is

where IApi_n(t)kp_Dc and the power amplitude is proportional to the signal:

The effect of the dispersion can be formulated as a linear operation on the electromagnetic field amplitude

where Ai_n(t) and A_out(t) are the field amplitudes at the two ends of the fiber and the kernel is expressed as

Since the signal is proportional to the optical power and the optical power is proportional to the square of the field, the linearity is destroyed

Using E . (2.1) and expanding gives

After opening the square and retaining the terms up to the second order

gives

where

stands for the real part. The followin notations will be used:

so that

If the bandwidth of (and of r|i_n(t)) is Δω then the spectrum of the first term

is bounded in the spectral band as

the second term is bounded in

and the third term is bounded in two bands 0<ω<Δω and 2coo- (in the following the positive frequencies will be referred to, the

ordinarily skilled person will appreciate that there is a negative counterpart as well). Therefore, if only the fundamental harmonic band is transmitted (coo- the original, undistorted signal, is restored.

Since the transmitting power is proportional to the input RF signal, the power equations can be replaced with voltage (or current) equations. For example, when the input RF signal is Vi_n(t) which is carried over a bias DC voltage VDC, and the output voltage is v_out(tX then EQ. (2.8) can be re laced with

Since

and its Fourier transform is

then when applying the spectral filter on EQ. (2.9),

where

is the rectangular function, and ε is the cutoff parameter, the original signal (multiplied by the carrier) can be restored r|i_n(t)cos(coot). In the present Example, ε can be substantially smaller than a third of the modulation index. The output signal can then be multiplied by a local oscillator cos(coot), and after a low-pass filter the original signal r|i_n(t) is restored. A schematic illustration of a suitable system for transmitting a signal over RFoF link is illustrated in FIG. 13. In the transmitter, a local oscillator is used to shift the input signal to a higher spectral band. At the receiver end the RF mixer uses the same oscillator signal to shift it back to the base-band.

Nonlinear Analysis

s erformance can be characterized by transmitting two harmonics

where in this Example the modulation depth (interchangeable referred to as modulation index) is denoted a.

Neglecting harmonics higher than a third order in a, the transmitted electromagnetic field is approximately

All power spectral components, whose contribution to the relevant spectral band are of the order of a⁴ or higher, were neglected as well (such as

3Ω).

Due to dispersion, each one of the harmonics accrues a different phase. Therefore, the electromagnetic field at the end of the fiber is

where

The power at the end of the fiber in the spectral sub-band

is, therefore:

so that

where the coefficients of the first and third harmonics are

and

with the notations ζ≡β₂ζΩ 12 and ρ≡ωο/Ω. Thus, only with the first and third harmonic terms remain, while the second harmonic terms disappeared.

A weak dispersion can be defined as

or less), in which case or less). For weak dispersion:

which demonstrates that this component is very weak.

When p is very large (namely when the two harmonics in EQ. (15) are close, e.g., within 10%, to the central frequency coo), EQ. (2.23) can be rewritten as:

where

For p»l, EQ. (2.24) reduces to:

After applying the filter as defined in EQ. (2.13) on EQ. (2.20) the

resultant signal is

where

Thus, in the weak dispersion case (ζ«1), the filter of the present embodiments has no correcting effect on the third harmonic, and therefore the filter only compensates the first harmonic with ne ligible additional nonlinear contribution:

p»l

In a typical RFoF channel, both the modulation depth and dispersion are weak, so that both a«l and ζ«1. Thus, the filter of the present embodiments successfully reduces the distortions in a typical RFoF channel to a negligible level in comparison to the original signal.

When the modulation of the optical signal is external (e.g., by an optical modulator such as, but not limited to, a Mach-Zehnder) the relation between the modulated power p_out(t), which up do a DC level is proportional to the detected signal, and the modulating signal's voltage Vi_n(t) is:

where T MZ is the modulator losses, o is the lasers' power and ν_π is the voltage required to change the Mach-Zehnder output power from its minimum value to its maximum value. Therefore, in this case, the noting the signal's amplitude by VQ:

and the nonlinear third harmonic coefficient is

That is, the remaining nonlinear term of EQ. (2.29) is smaller than the back-to- back nonlinearity of EQ. (2.34) caused by the modulator itself, so that this contribution can be neglected as well. Furthermore, expressions of the form of EQ. 2.34 can be canceled or at least suppressed using, for example, a non-linear amplifier.

Prior to filtering, the largest second harmonic coefficients are:

Therefore, the second harmonic coefficients of the fre uencies are

which for weak dispersion reduces to

Therefore, ¾₊<¾_. In fact H2_/H₃₊>8. When the frequency shifting of the present embodiments is used, wherein p increases from pi to p₂, the second harmonic power is reduced, so that for weak dispersion this technique advantageous provides

For large dispersion (ζ>1), the effect of an additional p increment is smaller so that the frequency shifting of the present embodiments advantageously reduces the second harmonic.

Results of such an analysis are illustrated in FIGs. 14A-D. A signal consisting of two frequencies is transmitted through the dispersive fiber. FIGs. 14A-D show the power spectra

is the Fourier transform of r|_out(t)- As shown in FIGs. 14A-D, without dispersion compensation the spurious harmonics are distributed throughout the spectral bandwidth ΙωΙ<Δω/2 much like white noise. When the signal is transmitted with low modulation and with the dispersion compensating filter, most of the noise is reduced substantially. Even the second harmonic component is reduced, albeit it does not disappear. On the other hand, when the signal is frequency- shifted, so that the data is carried over a high-frequency carrier and then filtered, the second harmonic can be substantially reduced and preferably eliminated with the cost of increasing the third harmonic power. In this case, the third harmonic component is considerably lower than the second harmonic, and therefore the nonlinearities are lower.

Simulation

To illustrate the performance of this method for a random analog sequence of spectral width Δω, which random analog sequence is not limited to two harmonics, the following signal is considered

where x_n is a random sequence with uniform probability in the range [-0.5,0.5], N=2 -1 is the number of random symbols, whose allocation time is T = 2π/Δω, in the sequence, coo is the carrier frequency and vo is the signal's amplitude.

This sequence is spectrally bound within

To simulate an infinite random sequence by a finite sequence, every symbol was multiplied by the periodic function N sin whose period equal to the

length of the sequence NT. This function converges to the well-known "sine" function in the limit N→∞.

To simulate sending the signal into a fiber, the spectral counterpart of (2.3) was applied on the EM field. Thereafter, the signal was detected. A typical scenario is illustrated in FIGs. 15A-C. In the upper panel, the power spectrum of the original spectrally-bounded signal is plotted in logarithmic scale. The center panel shows the spectrum of the received signal. This is the case when no FS was introduced, therefore, in addition to considerable spectral broadening nonlinear distortions arise, even within the original spectral band. However, FS (lower panel) splits the harmonic distortions to different spectral bands. At the fundamental harmonic, the original signal is located around the carrier frequency coo = 2Δω with minimal distortion (their shape remains rectangular), while the second harmonic components are located around 2coo = 4Δω and are twice as wide. The spectral filter (2.13) can easily filter them out.

In most RFoF channels the modulation is low and the carrier is of a high frequency. Therefore, the compensating filter is sufficient for compensating the dispersion distortions. Numerical simulations have been performed to demonstrate the performance of the compensating filter.

The effect of the compensating filter on a signal consisting of two harmonics (2.15) has been analyzed.

FIG. 16C (bottom-left) shows the non-compensated received spectrum, with the parameters: carrier ωο/Δω = 1.1892, modulation frequency Ω/Δω=0.0305, modulation depth m-0.25, and dispersion coefficient

FIG. 16D (bottom-right) shows the difference between the original spectrum and the non-compensated received spectrum. FIG. 16A (top-left) shows the spectrum of the received signal with the compensating filter, and FIG. 16B (top-right) shows the difference between the compensated spectrum and the original spectrum. Note that the frequencies are normalized to the spectral band, and therefore were subtracted in the plots. As shown in FIGs. 16A-D, the non-compensated signal is completely distorted: while the third harmonic is clearly seen, the main distortion is due to different amplitudes of the original harmonic spectral lines. The compensating filter did not eliminate the third harmonic power but it almost completely eliminates the original harmonic distortion.

The pseudo-random analog sequence (2.39) was also simulated. In this case the spectra are bounded in a spectral bandwidth Δω, and therefore the transmitted signal's spectrum has a rectangular profile.

FIGs. 17A-C show comparison between the spectra of the initial transmitted signal (FIG. 17A), the non-compensated received signal (FIG. 17B) and the reconstructed signal (FIG. 17C). The simulation parameters are ζ=0.3, m=0.1, and ωο/Δω=2. The spectrum of the detected signal is distorted due to dispersion and the compensating filter reconstructs the spectrum to its initial form.

The compensating effect is illustrated in the time domain in FIGs. 18A-B and 19A-B. FIG. 18A shows a comparison between the original signal (dashed line) and the detected signal (solid line) after a medium with β₂ζΔω =0.01. The distortion reaches a high level of about 50%., although the original signal can still be recognized. FIG. 18B shows the same comparison as in FIG. 18A except that for a signal transmitted with a modulation index m = 0.1, and carried on a carrier wave having carrier frequency ωο=2Δω. The curves for the original and detected signals substantially overlap, indicating. FIG. 19A shows a comparison between the original signal (dashed line) and the detected signal (solid line) after a medium with β₂ζΔω =0.3. The signal is totally distorted. FIG. 19B shows the same comparison as in FIG. 19A except that for a signal transmitted with a modulation index m = 0.1, and carried on a carrier wave having carrier frequency ωο=2Δω . The curves for the original and detected signals substantially overlap. FIGs. 18B and 19B demonstrate that the compensating filter reconstructs the signal with great accuracy.

To quantify the distortion of the signal, a relative error function is define as follows

Plots of the relative error function as a function of the normalized dispersion coefficient

is presented in FIGs. 20A-C as a function of modulation depth. In all these plots there is a clear transition. For small ζ the curve is smooth and the value of ε can be arbitrarily small. However, when the dispersion coefficient reaches the point where the compensatin filter diverges, namely

or in terms of ζ,

then the curve starts to fluctuate. For smaller the fluctuations are larger. The value of ε can be considerably smaller than the values in Example 1, and therefore it is advantageous to use this filter to mitigate dispersion distortions. For ζ<ζ₀ the error decreases by 10 orders of magnitudes for low modulation (FIG. 20C), but even for larger modulations index (FIGs. 20B-C) it decreases by 4-5 orders of magnitudes. For ζ>ζ₀ ώε error decreases by at least 3 orders of magnitudes. Moreover, the filter can compensate arbitrarily large distortions, even in cases where it is impossible to trace the original signal (Ε(ζ)≡1). For example, even for modulation index m=0.5 (FIG. 20C), a system with severe dispersion distortion ζ » 100 performs as a non-compensated system with ζ=0.003. With lower modulation index the advantage is even higher. This example described a technique for dispersion compensation in RFoF channels. The technique is based on two properties that are usually present in these channels: the modulation depth is lower than in digital links (e.g., lower than 1 to mitigate nonlinear effects), and the data's spectral bandwidth is narrower than the signals' carrier frequency. The first property ensures that distortions due to nonlinearities appear only in terms that are of high order of the modulation depth, and therefore can be neglected in the low modulation regime. The second property ensures the elimination of the second order nonlinear distortions. As a consequence, when both of these conditions are fulfilled the nonlinear distortions are of the third order, and in many cases can be neglected in comparison to the back- to-back third order distortions. Thus, in practice the system can be regarded as a linear one, and therefore the linear distortions can be compensated by a low-cost linear filter.

Although the invention has been described in conjunction with specific embodiments thereof, it is evident that many alternatives, modifications and variations will be apparent to those skilled in the art. Accordingly, it is intended to embrace all such alternatives, modifications and variations that fall within the spirit and broad scope of the appended claims.

All publications, patents and patent applications mentioned in this specification are herein incorporated in their entirety by reference into the specification, to the same extent as if each individual publication, patent or patent application was specifically and individually indicated to be incorporated herein by reference. In addition, citation or identification of any reference in this application shall not be construed as an admission that such reference is available as prior art to the present invention. To the extent that section headings are used, they should not be construed as necessarily limiting.

REFERENCES

[I] S. J. Savory, "Digital filters for coherent optical receivers," Opt. Express 16, 804-817 (2008).

[2] S. J. Savory, "Compensation of fibre impairments in digital coherent systems," in

Proceeding of IEEE European Conference on Optical Communication, Brussels,

Belgium, 2008, paper Mo.3.D. l.

[3] T. Xu, G. Jacobsen, S. Popov, J. Li, E. Vanin, K. Wang, A. T. Friberg, and Y.

Zhang, "Chromatic dispersion compensation in coherent transmission system using digital filters," Opt. Express 18, 16243-16257 (2010).

[4] X. Gu and L. C. Blank, "10 Gbit/s unrepeated three-level optical transmission over 100 km of standard fibre," Electron. Lett. 29, 2209-2211 (1993).

[5] K. Yonenaga, S. Kuwanto, S. Norimatsu, and N. Shibata, "Optical duobinary transmission system with no receiver sensitivity degradation," Electron. Lett. 31,

302-304 (1995).

[6] D. Penninckx, M. Chbat, L. Pierre, and J. -P. Thiery, "The phaseshaped binary transmission (PSBT): a new technique to transmit far beyond the chromatic dispersion limit," Photon. Technol. Lett. 9, 259-261 (1997).

[7] H. Bissessur, G. Charlet, C. Simonneau, S. Borne, L. Pierre, C. De Barros, P.

Tran, W. Idler, and R. Dischler, "3.2 Tb/s 80 x 40 Gb/s C-band transmission over 3 x 100 km with 0.8 bit/s/Hz efficiency," in Proceedings of European Conference on Optical Communication, Amsterdam, The Netherlands, 2001, Vol. 6, p. 22.

[8] A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).

[9] T. E. Gureyev, "Composite techniques for phase retrieval in the Fresnel region," Opt. Commun. 220, 49-58 (2003).

[10] T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).

[I I] T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, "Optical phase retrieval by use of first Born- and Rytov-type approximations," Appl. Opt. 43, 2418-2430 (2004).

[12] G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, 2002). [13] C. H. Cox III, Analog Optical Links, Theory and Practice (Cambridge University, 2004).

[14] S. Haykin and M. Moher, Introduction to Analog and Digital Communications (Wiley, 2007).

[15] E. Granot and A. Marchewka, "Emergence of currents as a transient quantum effect in nonequilibrium systems," Phys. Rev. A 84, 032110 (2011).

[16] E. Granot, E. Luz, and A. Marchewka, "Generic pattern formation of sharp- boundaries pulses propagation in dispersive media," J. Opt. Soc. Am. B 29, 763-768 (2012).

[17] M. Abramowitz and A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

[18] E. Granot, "Fundamental dispersion limit for spectrally bounded On- Off-Keying communication channels and its implications to quantum mechanics and the paraxial approximation," Europhys. Lett. 100, 44004 (2012).

[19] S. Marciano, S. Ben-Ezra, and E. Granot, "Eavesdropping and network analyzing using network dispersion," Appl. Phys. Res. 7(2), 27 (2015).

[36] A. M. Boers, B. Afshar, I. Vassiliou, S. Sarkar, S. T. Nicolson, E. Adabi, Bevin G. Perumana, T. Chalvatzis, S. Kavvadias, P. Sen, W. L. Chan, A. H.- T. Yu, A. Parsa, M. Nariman, S. Yoon, A. G. Besoli, C. A. Kyriazidou, G. Zochios, J. A. Castaneda, T. Sowlati, M. Rofougaran, and A. Rofougaran, " 16TX/16RX 60 GHz 802. Had Chipset With Single Coaxial Interface and Polarization Diversity," J. Solid-State Circuits, vol. 49, pp. 3031-3045, 2014.

[37] Y. Ma, Q. Yang, Y. Tang, S. Chen, and William Shieh, " 1-Tb/s singlechannel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth access," Opt. Exp., vol. 17,p. 9421, 2009.

[38] J. Hoversten, S. Schafer, M. Roberg, M. Norris, D. Maksimovic, and Zoya Popovic, "Codesign of PA, Supply, and Signal Processing for Linear Supply- Modulated RF Transmitters," IEEE Transactions on Microwave Theory and Techniques, vol. 60, pp. 2010-2020, 2012.

[39] H. Al-Raweshidy and Shozo Komaki, "Radio Over Fiber Technologies for Mobile Communications Networks," Artech House, 2002.

Claims

WHAT IS CLAIMED IS:

1. A method of mitigating dispersion in an optical signal modulated relative to a DC component, the method comprising:

detecting the signal by a non-coherent optical detector generating an electrical signal; and

applying a linear finite impulse response (FIR) filter to said electrical signal, thereby mitigating dispersion in the optical signal.

2. The method according to claim 1, wherein an AC component of the modulated optical signal is less than half the DC component.

3. The method according to claim 1, wherein an AC component of the modulated optical signal is less than a third of the DC component.

4. The method according to claim 1, wherein an AC component of the modulated optical signal is less than a fifth of the DC component.

5. The method according to claim 1, wherein an AC component of the modulated optical signal is less than a tenth of the DC component.

6. The method according to any of claims 1-5, wherein a modulation frequency of the optical signal is up-shifted, and the method comprises down-shifting a frequency of said electrical signal, following said application of said FIR filter.

7. The method according to any of claims 1-6 further comprising applying a band pass filter to said electrical signal, prior to said application of said FIR filter, to remove harmonics generated by said optical detector.

8. The method according to any of claims 1-7, wherein said FIR filter is a digital filter operating in a time domain.

9. The method according to any of claims 1-6, wherein said FIR filter is characterized by a cutoff parameter ε, which is higher than 10% of a modulation depth of said optical signal.

10. The method according to any of claims 1-9, wherein said FIR filter is characterized by a cutoff parameter ε, which is less than 50% of said modulation depth.

11. The method according to claim 10, wherein said cutoff parameter ε, is higher than 10% of a modulation depth of said optical signal.

12. The method according to any of claims 9 and 10, wherein said cutoff parameter ε equals about one third of said modulation depth.

13. The method according to any of claims 1-12, wherein said FIR filter is a digital filter, operating in a time domain, and having a plurality of terms wherein a number of said terms is about 2N+1, and wherein N is a nearest integer above or below β₂ζ 2Δί , where β₂ is a dispersion parameter, z is a distance propagated by the signal, and At is a sampling period characterizing a digitization of said signal.

14. The method according to any of claims 1-13, wherein the optical signal is transmitted over an optical fiber having a length of at least 100 km.

15. The method according to any of claims 1-14, wherein the optical signal is transmitted over a single mode optical fiber.

16. The method according to any of claims 1-15, wherein the optical signal carries a data stream at a bit rate of at least 20 Gb per second.

17. The method according to any of claims 1-16, wherein the optical signal comprises a plurality of pulses having a predetermined shape, and wherein said FIR filter is an inverse of a filter whose convolution with a single pulse of said plurality of provides a dispersion distortion which is characteristic to said singe pulse.

18. A method of transmitting a signal, the method comprising: obtaining an optical signal and an electrical signal selected for modulating said optical signal to carry a data stream;

up-shifting a frequency of an AC component of said electrical signal;

modulating said optical signal relative to a DC component using said electrical signal; and

coupling said optical signal, inclusive of both said AC and said DC components, into a single mode optical fiber.

19. The method according to claim 18, wherein a length of said optical fiber having is at least 1 km.

20. The method according to claim 18, wherein a length of said optical fiber having is at least 10 km.

21. The method according to claim 18, wherein a length of said optical fiber having is at least 100 km.

22. The method according to any of claims 18 and 19, wherein the data stream is characterized by a bit rate of at least 20 Gb per second.

23. The method according to any of claims 18-22, further comprising applying a linear finite impulse response (FIR) filter to said electrical signal, prior to said modulation.

24. The method according to claim 23, wherein said FIR filter is a digital filter operating in a time domain.

25. A method of transmitting an optical signal modulated to carry a data stream, the method comprising:

executing the method according to any of claim 18-22; detecting the signal at an exit end of said optical fiber by a non-coherent optical detector generating an electrical signal;

applying a band pass filter to said electrical signal to remove harmonics generated by said optical detector; and

26. The method according to claim 25, wherein the optical signal comprises a plurality of pulses having a predetermined shape, and wherein said FIR filter is an inverse of a filter whose convolution with a single pulse of said plurality of provides a dispersion distortion which is characteristic to said singe pulse.

27. A system for mitigating dispersion in an optical signal modulated relative to a DC component, the system comprising:

a non-coherent optical detector detecting the signal to generate an electrical signal;

electronic circuitry configured for applying a linear finite impulse response (FIR) filter to said band-pass filtered electrical signal, thereby mitigating dispersion in the optical signal.

28. The system according to claim 27, wherein a modulation frequency of the optical signal is up-shifted, and the electronic circuitry is configured for down-shifting a frequency of an electrical signal output by said FIR filter circuit.

29. The system according to any of claims 27 and 28, wherein said electronic circuitry configured for applying a band pass filter to said electrical signal to remove harmonics generated by said optical detector.

30. The system according to any of claims 27-29, wherein said FIR filter is a digital filter operating in a time domain.

31. The system according to any of claims 27-30, wherein said FIR filter is characterized by a cutoff parameter ε, which is higher than 10% of a modulation depth of said optical signal.

32. The system according to any of claims 27-31, wherein said FIR filter is characterized by a cutoff parameter ε, which is less than 50% of said modulation depth.

33. The system according to claim 32, wherein said cutoff parameter ε, is higher than 10% of a modulation depth of said optical signal.

34. The system according to any of claims 31 and 32, wherein said cutoff parameter ε equals about one third of said modulation depth.

35. The system according to any of claims 31-34, wherein said FIR filter is a digital filter, operating in a time domain, and having a plurality of terms wherein a number of said terms is about 2N+1, and wherein N is a nearest integer above or below β₂ζ 2Δί , where β₂ is a dispersion parameter, z is a distance propagated by the signal, and At is a sampling period characterizing a digitization of said signal.

36. The system according to any of claims 27-35, wherein the optical signal is transmitted over an optical fiber having a length of at least 100 km, and wherein said non-coherent optical detector is coupled to receive the optical signal from said optical fiber.

37. The system according to any of claims 27-36, wherein the optical signal is transmitted over a single mode optical fiber, and wherein said non-coherent optical detector is coupled to receive the optical signal from said optical fiber.

38. The system according to any of claims 27-37, wherein the data stream is characterized by a bit rate of at least 20 Gb per second.

39. The system according to any of claims 27-38, wherein the optical signal comprises a plurality of pulses having a predetermined shape, and wherein said FIR filter is an inverse of a filter whose convolution with a single pulse of said plurality of provides a dispersion distortion which is characteristic to said singe pulse.

40. A system of transmitting a signal, the system comprising:

electronic circuitry configured for receiving an electrical signal and up-shifting a frequency of an AC component of said electrical signal;

an optical modulator configured for receiving an optical signal and said up- shifted electrical signal, and modulating said optical signal using said up-shifted electrical signal to carry the data stream; and

an optical coupler configured for coupling said modulated optical signal into a single mode optical fiber.

41. The system according to claim 40, wherein a length of said optical fiber is at least 1 km.

42. The system according to claim 40, wherein a length of said optical fiber is at least 10 km.

43. The system according to claim 40, wherein a length of said optical fiber is at least 100 km.

44. The system according to any of claims 40 and 41, wherein the data stream is characterized by a bit rate of at least 20 Gb per second.

45. The system according to any of claims 40-44, wherein said electronic circuitry is configured for applying a linear finite impulse response (FIR) filter to said electrical signal, prior to said modulation.

46. The system according to claim 45, wherein said FIR filter is a digital filter operating in a time domain.

47. The system according to any of claims 45 and 46, wherein the optical signal comprises a plurality of pulses having a predetermined shape, and wherein said FIR filter is an inverse of a filter whose convolution with a single pulse of said plurality of provides a dispersion distortion which is characteristic to said singe pulse.

48. A system of transmitting an optical signal modulated to carry a data stream, the system comprising:

a single mode optical fiber having a transmitting end and a receiving end;

the system according to any of claim 40-44 coupled to said transmitting end; a non-coherent optical detector coupled to said receiving end and configured for detecting the signal to generate an electrical signal; and

electronic circuitry configured for applying a band pass filter to said electrical signal to remove harmonics generated by said optical detector, and for applying a linear finite impulse response (FIR) filter to said band-pass filtered electrical signal, thereby mitigating dispersion in the optical signal.

49. The system according to claim 48, wherein the optical signal comprises a plurality of pulses having a predetermined shape, and wherein said FIR filter is an inverse of a filter whose convolution with a single pulse of said plurality of provides a dispersion distortion which is characteristic to said singe pulse.

50. The method or system according to any of claims 1-49, wherein the optical signal is selected from the group consisting of a pulse amplitude modulation (PAM) signal, an on-off keying (OOK) signal, a quadrature amplitude modulation (QAM) signal, a phase shift keying (PSK) signal, a binary PSK (BPSK) signal, a quadrature PSK (QPSK) signal, an octal PSK (OPSK) signal, a differential PSK (DPSK) signal, an amplitude shift keying (ASK) signal, a frequency shift keying (FSK) signal, a radio over fiber (RoF) signal, a frequency-division multiplexing (FDM) signal and an orthogonal FDM (OFDM) signal.

51. The method according to claim 1, wherein a modulation frequency of the optical signal is up-shifted, and the method comprises down-shifting a frequency of said electrical signal, following said application of said FIR filter.

52. The method according to claim 1, further comprising applying a band pass filter to said electrical signal, prior to said application of said FIR filter, to remove harmonics generated by said optical detector.

53. The method according to claim 1, wherein said FIR filter is a digital filter operating in a time domain.

54. The method according to claim 1, wherein said FIR filter is characterized by a cutoff parameter ε, which is higher than 10% of a modulation depth of said optical signal.

55. The method according to claim 1, wherein said FIR filter is characterized by a cutoff parameter ε, which is less than 50% of said modulation depth.

56. The method according to claim 1, wherein said cutoff parameter ε, is higher than 10% of a modulation depth of said optical signal.

57. The method according to claim 1, wherein said cutoff parameter ε equals about one third of said modulation depth.

58. The method according to claim 1, wherein said FIR filter is a digital filter, operating in a time domain, and having a plurality of terms wherein a number of said terms is about 2N+1, and wherein N is a nearest integer above or below β₂ζ 2Δί , where β₂ is a dispersion parameter, z is a distance propagated by the signal, and At is a sampling period characterizing a digitization of said signal.

59. The method according to claim 1, wherein the optical signal is transmitted over an optical fiber having a length of at least 100 km.

60. The method according to claim 1, wherein the optical signal is transmitted over a single mode optical fiber.

61. The method according to claim 1, wherein the optical signal carries a data stream at a bit rate of at least 20 Gb per second.

62. The method according to claim 1, wherein the optical signal comprises a plurality of pulses having a predetermined shape, and wherein said FIR filter is an inverse of a filter whose convolution with a single pulse of said plurality of provides a dispersion distortion which is characteristic to said singe pulse.