CN104976962B - Method for measuring plane mirror absolute surface shape based on conjugate difference method - Google Patents

Abstract

The invention discloses a method for measuring plane mirror absolute surface shape based on a conjugate difference method. The method comprises the following steps: a plane mirror to be measured and a standard plane mirror forming interference images on a CCD of an interferometer respectively, judging the sizes of light spots of the plane mirror to be measured and the standard plane mirror formed on the CCD of the interferometer and enabling the light spot of the standard plane mirror to be larger than the light spot of the plane mirror to be measured; determining translation amount delta of the plane mirror to be measured corresponding to one pixel in the CCD of the interferometer, delta＞0; defining the direction of optical axis of the interferometer as the z axis and carrying four-step test in the positive and negative directions of the x axis and y axis respectively and recording wavefront data phi (x+delta, y), phi (x-delta, y), phi (x, y+delta) and phi (x, y-delta) respectively; obtaining surface shape gradient of the plane mirror to be measured according to the measured wavefront data; and recovering the surface shape w of the plane mirror to be measured according to the surface shape gradient of the plane mirror to be measured by utilizing a wave surface recovering method. The method improves precision of differential approximation and prevents coupling between step-by-step tests, thereby improving stability of the test; and the method can also be used for measuring the absolute surface shape of a cylindrical mirror, a conical mirror and a spherical mirror.

Description

Method of Measuring Absolute Surface Shape of Plane Mirror Based on Conjugate Difference Method

technical field

The invention belongs to the technical field of absolute detection of plane mirrors, in particular to a method for measuring the absolute surface shape of a plane mirror based on a conjugate difference method.

Background technique

At present, the research on the absolute detection of plane mirrors mainly includes the three-plane mutual detection method, the liquid level reference method and the difference method, that is, the pseudo-shear method. The traditional three-plane mutual inspection method cannot obtain the two-dimensional surface shape of the entire plane of the surface to be tested, but can only obtain the situation of a line. The improved method can obtain the surface shape of the entire surface, but the testing process is very complicated; The surface reference method uses a liquid flat crystal as the standard surface, but the liquid surface reference is difficult to manufacture, requires high environmental conditions and is not easy to move, and the systematic error in the test results cannot be eliminated; The three positions have a coupling relationship with each other. It is necessary to maintain the stability of the environment and other factors in the three tests to ensure the constant static error, but the external environment and other factors are difficult to control, and it is difficult to maintain its stability, so it is difficult to ensure the constant static error, which affects the difference method. detection accuracy.

Contents of the invention

The purpose of the present invention is to provide a method for measuring the absolute surface shape of a plane mirror based on the conjugate difference method with high measurement accuracy and convenient operation.

The technical solution to realize the object of the present invention is: a method for measuring the absolute surface shape of a plane mirror based on the conjugate difference method, comprising the following steps:

Step 1: The plane mirror to be tested and the standard plane mirror form an interference image on the CCD of the interferometer, and the size of the spot formed by the standard plane mirror and the plane mirror to be tested on the CCD of the interferometer is judged, so that the spot of the standard plane mirror is larger than the spot of the plane mirror to be tested ;

The second step is to determine the translation amount δ of the plane mirror to be measured corresponding to one pixel in the CCD of the interferometer, δ>0;

Step 3: Set the direction of the optical axis of the interferometer as the z-axis, perform four-step tests in the positive and negative directions of the x-axis and y-axis, and record the wavefront data Φ(x+δ,y), Φ(x-δ ,y), Φ(x,y+δ), Φ(x,y-δ), the data size is N×N, and N is a positive integer;

Step 4, according to the measured wavefront data Φ(x+δ,y), Φ(x-δ,y), Φ(x,y+δ), Φ(x,y-δ), obtain the Measure the surface gradient dx(x,y) and dy(x,y) of the plane mirror;

The fifth step is to recover the surface shape w of the plane mirror to be tested according to the surface gradient dx(x, y) and dy(x, y) of the plane mirror to be tested by using the wave surface restoration method.

Compared with the prior art, the present invention has the following remarkable advantages: (1) the absolute detection of the surface shape of the plane mirror to be measured can be realized by the interferometer, the standard mirror, and the mirror to be tested, and the structure is simple and the operation is convenient; (2) the use of common The yoke difference can omit the test process of the initial position, which improves the accuracy of the differential approximation; (3) The four-step test process is independent in pairs, that is, the test processes in the two orthogonal directions are independent of each other and do not affect each other, which further simplifies the experimental operation and improves the accuracy. Inspection accuracy; (4) This method can also measure the absolute surface shape of cylindrical mirror, conical mirror and spherical mirror.

Description of drawings

Fig. 1 is a schematic diagram of the hardware structure of the method for measuring the absolute surface shape of a plane mirror based on the conjugate difference method of the present invention.

Fig. 2 is a schematic diagram of a four-step test process for measuring the absolute surface shape of a plane mirror based on the conjugate difference method in the present invention.

detailed description

The present invention will be described in further detail below in conjunction with the accompanying drawings and specific embodiments.

In conjunction with Fig. 1～2, the method for measuring the absolute surface shape of a plane mirror based on the conjugate difference method of the present invention comprises the following steps:

Step 1: The plane mirror to be tested and the standard plane mirror form an interference image on the CCD of the interferometer, and the size of the spot formed by the standard plane mirror and the plane mirror to be tested on the CCD of the interferometer is judged, so that the spot of the standard plane mirror is larger than the spot of the plane mirror to be tested ;

As shown in Figure 1, the interferometer is a Fizeau interferometer, and the inclination between the standard plane mirror and the plane mirror to be measured is less than 0.02 times of the laser wavelength of the interferometer, because the inclination term is a first-order term in the wave aberration, and after the gradient is calculated, it is is a constant term, which cannot be recovered by using the wave surface restoration method, so it is necessary to make this term as small as possible. If the light spot of the plane mirror to be tested is larger than that of the standard plane mirror, then make a square diaphragm and place it on the plane mirror to be tested so that the center of the plane mirror to be tested coincides with the center of the square diaphragm, so that the spot size of the standard plane mirror is larger than that of the plane mirror to be tested.

The second step is to determine the translation amount δ of the plane mirror to be measured corresponding to one pixel in the CCD of the interferometer, δ>0;

Using the ratio method, the translation distance of the plane mirror to be measured is compared with the moving number of pixels in the CCD of the corresponding interferometer, and the translation amount δ of the plane mirror to be measured corresponding to one pixel is obtained by averaging multiple measurements.

Step 3: Set the direction of the optical axis of the interferometer as the z-axis, perform four-step tests in the positive and negative directions of the x-axis and y-axis, and record the wavefront data Φ(x+δ,y), Φ(x-δ ,y), Φ(x,y+δ), Φ(x,y-δ), the data size is N×N, and N is a positive integer; combined with Figure 2, the specific process of the four-step test is as follows:

The plane mirror to be tested moves 1 pixel in the negative direction of the x-axis, and records the wavefront data Φ(x-δ,y); the plane mirror to be tested moves 2 pixels in the positive direction of the x-axis, and records the wavefront data Φ(x+δ ,y); the plane mirror to be tested moves 1 pixel to the negative direction of the x-axis, and returns to the initial position; the plane mirror to be tested moves 1 pixel to the positive direction of the y-axis, and records the wavefront data Φ(x,y+δ); Finally, the plane mirror to be tested moves 2 pixels in the negative direction of the y-axis, and records the wavefront data Φ(x,y-δ).

Step 4, according to the measured wavefront data Φ(x+δ,y), Φ(x-δ,y), Φ(x,y+δ), Φ(x,y-δ), obtain the Measure the surface shape gradient dx(x,y) and dy(x,y) of the plane mirror, the formula is as follows:

The fifth step is to recover the surface shape w of the plane mirror to be tested according to the surface gradient dx(x, y) and dy(x, y) of the plane mirror to be tested by using the wave surface restoration method. Described wave surface recovery method is Fourier transform method, path integral method or least squares method, and each method is specifically as follows:

(1) Fourier transform method

The spectrum Dx(u,v) and Dy(u,v) are respectively obtained by Fourier transform of dx(x,y) and dy(x,y), and the surface spectrum is obtained from this or Perform inverse Fourier transform on W to obtain surface shape w;

(2) Path integral method

Take any point on the restoration surface as the starting point P ₀ (x ₀ ,y ₀ ), let the surface value of this point be w(x ₀ ,y ₀ )=0, then the point P ₁ (x ₀ +△x,y ₀ ) Surface value is The surface value of point P ₂ (x ₀ ,y ₀ +△y) is By analogy, the entire surface shape w is obtained, where △x=n ₁ δ, △ _y =n ₂ δ, n ₁ and n ₂ are integers;

(3) Least square method

The least squares method is divided into Zernike polynomial fitting and Hudgin wavefront sensing:

a. Zernike polynomial fitting

Depend on Find the Zernike coefficient C _k , where n is the number of Zernike polynomial items used, Z _k x(x,y), Z _k y(x,y) are the Zernike polynomial Z _k (x,y) to the independent variable x , the derivative of y, then the surface shape of the plane mirror to be tested is

b. Hudgin wavefront sensing

A[q+(p-1)(N-1),q+(p-1)N]=-1

A[q+(p-1)(N-1),1+q+(p-1)N]=1(p=1,...,N q=1,...,N-1)

A[r+N(N-1),r]=-1

A[r+N(N-1),r+N]=-1(r=1,...,N(N-1))

Then w=(A ^T A) ^-1 A ^T S, where S is a vector of 2N×(N-1) data containing surface gradients in the x-axis and y-axis directions, and A is the 2N×(N-1) data of the Hudgin wavefront sensor A matrix with (N-1) rows and N ² columns.

Example 1

Taking the Fizeau interferometer as an example, the absolute surface shape of the plane mirror to be measured is measured according to the method of the present invention.

1. The measuring device is shown in Figure 1, adjust the standard lens and the frame knob:

(1) Open the Fizeau interferometer application;

(2) Install the standard flat mirror: put the two probe angles on the lens frame into the groove of the interferometer bayonet bracket, rotate the lens clockwise, and then tighten it;

(3) Adjust the standard plane mirror: Press the "Adjustment/Test" button on the remote control to switch the display screen to the "Adjustment" state, and a crosshair image appears on the plane. Adjust the knob of the interferometer bayonet bracket until the bright spot on the screen is in the middle of the crosshairs;

(4) Put the plane mirror under test on the five-dimensional adjustment frame, and adjust the knob of the adjustment frame until the bright spot of the plane mirror under test on the screen is also in the middle of the crosshairs.

2. Determine the size of the light spot formed by the standard plane mirror and the plane mirror to be tested. If the light spot of the plane mirror to be tested is larger than the light spot of the standard plane mirror, make a square diaphragm and place it on the plane mirror to be tested so that the center of the plane mirror to be tested coincides with the center of the square diaphragm , so as to meet the requirement that the spot size of the standard mirror is larger than that of the plane mirror to be tested.

3. Taking the optical axis direction as the z-axis direction, the x-axis and y-axis directions are also determined. At the same time, we need to determine the translation of the DUT corresponding to 1 pixel of the interferometer used in the experiment, and use the ratio method to compare the length of a distance in the experiment with the pixel size, and obtain 1 by averaging multiple measurements. The translation amount δ of the DUT corresponding to the pixel.

4. First adjust the positions of the standard mirror and the mirror to be tested so that the tilt term is small enough, less than 0.02 wavelength. The laser wavelength of the Fizeau interferometer used in the experiment is 632.8nm. As shown in Figure 2, move 1 to the negative direction of the x-axis pixels, record the wavefront data Φ(x-δ,y); move 2 pixels to the positive direction of the x-axis, and record the wavefront data Φ(x+δ,y); move 1 pixel to the negative direction of the x-axis , return to the initial position; then move 1 pixel to the positive direction of the y-axis, and record the wavefront data Φ(x,y+δ); finally move 2 pixels to the negative direction of the y-axis, and record the wavefront data Φ(x ,y-δ). The size of the data is N×N, that is, the size of the Mask in the MetroPro software that sets the interferometer in the experiment is N×N, and N is a positive integer.

Read the test data measured in the experiment into the dat2txt.exe software, press the "Enter" key, and get the txt files corresponding to the test data: fringe.txt, mask.txt, wave.txt. The wave file is the data we want to use for wave surface restoration.

5. According to the measured phase: Φ(x, y) = φ _{to be measured flat mirror} (x, y) - Ψ _{reference flat mirror} (x, y), so that after translation in the x-axis and y-axis directions, Similar four phase equations are obtained:

Φ(x-δ, y)=φThe flat mirror to _{be measured} (x-δ,y)-Ψ _{reference flat mirror} (x,y)

Φ(x+δ,y)=φThe flat mirror to _{be measured} (x+δ,y)-Ψ _{reference flat mirror} (x,y)

Φ(x,y+δ)=φThe flat mirror to _{be tested} (x,y+δ)-Ψ _{reference flat mirror} (x,y)

Φ(x, y-δ)=φThe flat mirror to _{be measured} (x,y-δ)-Ψ _{reference flat mirror} (x,y)

Subtracting the conjugated two equations yields:

Φ(x+δ,y)-Φ(x-δ,y)= _{φThe flat mirror to be tested} (x+δ,y)-φThe _{flat mirror to be tested} (x-δ,y)

Φ(x,y+δ)-Φ(x,y-δ)= _{φThe flat mirror to be tested} (x,y+δ)-φThe _{flat mirror to be tested} (x,y-δ)

Thus, the surface shape gradient of the plane mirror to be measured is obtained:

6. Using the wave surface restoration method, the surface shape w of the plane mirror to be tested is restored according to the surface shape gradient dx(x, y) and dy(x, y) of the plane mirror to be tested. Described wave surface recovery method is Fourier transform method, path integral method or least squares method, and each method is specifically as follows:

(1) Fourier transform method

The spectrum Dx(u,v) and Dy(u,v) are respectively obtained by Fourier transform of dx(x,y) and dy(x,y), and the surface spectrum is obtained from this or Perform inverse Fourier transform on W to obtain surface shape w;

(2) Path integral method

Take any point on the restoration surface as the starting point P ₀ (x ₀ ,y ₀ ), let the surface value of this point be w(x ₀ ,y ₀ )=0, then the point P ₁ (x ₀ +△x,y ₀ ) Surface value is The surface value of point P ₂ (x ₀ ,y ₀ +△y) is By analogy, the entire surface shape w is obtained, where △x=n ₁ δ, △y=n ₂ δ, n ₁ and n ₂ are integers;

(3) Least square method

The least squares method is divided into Zernike polynomial fitting and Hudgin wavefront sensing:

a. Zernike polynomial fitting

Depend on Find the Zernike coefficient C _k , where n is the number of Zernike polynomial items used, Z _k x(x,y), Z _k y(x,y) are the Zernike polynomial Z _k (x,y) to the independent variable x , the derivative of y, then the surface shape of the plane mirror to be tested is

b. Hudgin wavefront sensing

A[q+(p-1)(N-1),q+(p-1)N]=-1

A[q+(p-1)(N-1),1+q+(p-1)N]=1(p=1,...,N q=1,...,N-1)

A[r+N(N-1),r]=-1

A[r+N(N-1),r+N]=-1(r=1,...,N(N-1))

Then w=(A ^T A) ^-1 A ^T S, where S is a vector of 2N×(N-1) data containing surface gradients in the x-axis and y-axis directions, and A is the 2N×(N-1) data of the Hudgin wavefront sensor A matrix with (N-1) rows and N ² columns.

In summary, the present invention is a method for measuring the absolute surface shape of a plane mirror based on the conjugate difference method. The conjugate difference can not only improve the accuracy of the differential approximation, but also avoid the coupling between step-by-step tests, thereby reducing the impact of environmental factors. Influencing and improving the stability of the test, this method can also be applied to the situation of measuring the surface shape of cylindrical mirror, conical mirror and spherical mirror.

Claims (1)

1. a method for measuring the absolute surface shape of a plane mirror based on the conjugate difference method, is characterized in that, comprises the following steps: Step 1: The plane mirror to be tested and the standard plane mirror form an interference image on the CCD of the interferometer, and the size of the spot formed by the standard plane mirror and the plane mirror to be tested on the CCD of the interferometer is judged, so that the spot of the standard plane mirror is larger than the spot of the plane mirror to be tested ; The second step is to determine the translation amount δ of the plane mirror to be measured corresponding to one pixel in the CCD of the interferometer, and δ>0; Step 3: Set the direction of the optical axis of the interferometer as the z-axis, perform four-step tests in the positive and negative directions of the x-axis and y-axis, and record the wavefront data Φ(x+δ,y), Φ(x-δ ,y), Φ(x,y+δ), Φ(x,y-δ), the data size is N×N, and N is a positive integer; Step 4, according to the measured wavefront data Φ(x+δ,y), Φ(x-δ,y), Φ(x,y+δ), Φ(x,y-δ), obtain the Measure the surface shape gradient dx(x,y) and dy(x,y) of the plane mirror; Step 5, using the wave surface restoration method to recover the surface shape w of the plane mirror to be measured according to the surface gradient dx(x, y) and dy(x, y) of the plane mirror to be measured; The interferometer described in the first step above is a Fizeau interferometer, and the inclination between the standard plane mirror and the plane mirror to be measured is less than 0.02 times the laser wavelength of the interferometer; Judging the size of the light spot formed by the standard plane mirror and the plane mirror to be measured on the interferometer CCD as described in the first step above, so that the light spot of the standard plane mirror is larger than the light spot of the plane mirror to be tested, specifically: if the light spot of the plane mirror to be measured is larger than that of the standard plane mirror spot, then make a square diaphragm and place it on the plane mirror to be tested so that the center of the plane mirror to be tested coincides with the center of the square diaphragm, so that the spot size of the standard plane mirror is larger than that of the plane mirror to be tested; As described in the second step above, determine the translation amount δ of the plane mirror to be measured corresponding to one pixel in the CCD of the interferometer. The number of moves is used as a ratio, and the translation amount δ of the plane mirror to be measured corresponding to one pixel is obtained by averaging multiple measurements; Carry out four-step tests in the positive and negative directions of the x-axis and y-axis as described in the third step above, and record the wavefront data Φ(x+δ,y), Φ(x-δ,y), Φ(x, y+δ), Φ(x,y-δ), specifically: the plane mirror to be measured moves 1 pixel to the negative direction of the x-axis, and the wavefront data Φ(x-δ,y) is recorded; the plane mirror to be measured moves to the x-axis Move 2 pixels in the positive direction, and record the wavefront data Φ(x+δ,y); move the plane mirror to be tested by 1 pixel in the negative direction of the x-axis, and return to the initial position; move the plane mirror to be tested in the positive direction of the y-axis by 1 pixels, record the wavefront data Φ(x, y+δ); finally, the plane mirror to be measured moves 2 pixels to the negative direction of the y-axis, and record the wavefront data Φ(x, y-δ); According to the measured wavefront data Φ(x+δ,y), Φ(x-δ,y), Φ(x,y+δ), Φ(x,y-δ), as described in the fourth step above, Obtain the surface gradient dx(x,y) and dy(x,y) of the plane mirror to be tested, the formula is as follows: $\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&Phi;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&delta;</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Phi;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&delta;</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&delta;</span>}}$ $\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&Phi;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&delta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Phi;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&delta;</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&delta;</span>}}$ The wave surface restoration method described in the above step 5 is the Fourier transform method, the path integral method or the least square method, each method is as follows: (1) Fourier transform method The spectrum Dx(u,v) and Dy(u,v) are respectively obtained by Fourier transform of dx(x,y) and dy(x,y), and the surface spectrum is obtained from this or Perform inverse Fourier transform on W to obtain surface shape w; (2) Path integral method Take any point on the recovery surface as the starting point P ₀ (x ₀ ,y ₀ ), let the surface shape value of this point be w(x ₀ ,y ₀ )=0, then the surface of point P ₁ (x ₀ +Δx,y ₀ ) shape value The surface shape value of point P ₂ (x ₀ ,y ₀ +Δy) is By analogy, the entire surface shape w is obtained, where Δx=n ₁ δ, Δy=n ₂ δ, n ₁ and n ₂ are integers; (3) Least square method The least squares method is divided into Zernike polynomial fitting and Hudgin wavefront sensing: a. Zernike polynomial fitting Depend on Find the Zernike coefficient C _k , where n is the number of Zernike polynomial items used, Z _k x(x,y), Z _k y(x,y) are the Zernike polynomial Z _k (x,y) to the independent variable x , the derivative of y, then the surface shape of the plane mirror to be tested is b. Hudgin wavefront sensing A[q+(p-1)(N-1),q+(p-1)N]=-1 A[q+(p-1)(N-1),1+q+(p-1)N]=1(p=1,...,N q=1,...,N-1) A[r+N(N-1),r]=-1 A[r+N(N-1),r+N]=-1(r=1,...,N(N-1)) Then w=(A ^T A) ^-1 A ^T S, where S is a vector of 2N×(N-1) pieces of data containing surface gradients in the x-axis and y-axis directions, and A is the 2N×(N-1) data vector of the Hudgin wavefront sensor A matrix with (N-1) rows and N ² columns.