Method for wavefront measurement of optical imaging system based on grating shearing interferometry

Abstract

A method for wavefront measurement of optical imaging system based on grating shearing interferometry, the grating shearing interferometer comprising: a light source and illumination system, an optical imaging system to be tested, a one-dimensional diffraction grating plate, a two-dimensional diffraction grating plate, a two-dimensional photoelectric sensor and a computing unit. The one-dimensional diffraction grating plate and the two-dimensional diffraction grating plate are respectively placed on the object side and the image side of the optical imaging system to be tested. By collecting N sets of interferograms with a $\frac{2\pi }{N}$
phase-shifting interval (where, $N=2\left(\mathrm{fix}\left(\frac{\mathrm{ceil}\left(1/s\right)}{2}\right)+1\right),$
s is the shear ratio of the grating shearing interferometer), combined with a certain phase retrieval algorithm, the influence of all high-order diffraction beams on the phase retrieval accuracy is eliminated, and finally the wavefront measurement accuracy for the optical imaging system is improved.

Claims

1. A method for wavefront measurement of an optical imaging system based on a grating shearing interferometry, comprising: (a) placing an optical imaging system (3) in a grating shearing interferometer system, the optical imaging system (3) is defined by an object side and an image side and a magnification, wherein an object plane is on the object side and perpendicular to an optical axis of the optical imaging system (3), and an image plane is on the image side and perpendicular to the optical axis of the optical imaging system (3), and the grating shearing interferometer system comprises an illumination system (8), wherein the illumination system (8) outputs spatial incoherent light in a direction of an optical axis of the grating shearing interferometer system, a one-dimensional diffraction grating plate (1) fixed on a first three-dimensional stage (2) and comprising a first set of linear gratings (101) and a second set of linear gratings (102), and duty-cycle of one-dimensional diffraction grating is 50%, the first three-dimensional stage (2) for moving the one-dimensional diffraction grating plate (1), a two-dimensional diffraction grating plate (4) fixed on a second three-dimensional stage (5) and comprising a set of checkerboard gratings, and each of the checkerboard grating is square-shaped with a diagonal line, the second three-dimensional stage (5) for moving the two-dimensional diffraction grating plate (4), a two-dimensional photoelectric sensor (6) being connected with and sending outputs to a data processing unit (7), and the data processing unit(7) being connected with and receiving outputs of the two-dimensional photoelectric sensor (6), establishing an xyz coordinate axis system of the grating shearing interferometer system by setting motion axes of the first three-dimensional stage (2) and the second three-dimensional stage (5) as X axis, Y axis, and Z axis, respectively, wherein direction of the Z axis is along the direction of the optical axis of the grating shearing interferometer system, direction of the X axis is along the direction of the grating line of the second set of linear gratings (102) on the one-dimensional diffraction grating plate (1), and direction of the Y axis is along the direction of the grating line of the first set of linear gratings (101) on the one-dimensional diffraction grating plate (1), wherein the illumination system (8) is located on the object side of the optical imaging system (3), and the two-dimensional diffraction grating plate (4) is located on the image side of the optical imaging system (3); adjusting the first three-dimensional stage (2) to locate the one-dimensional diffraction grating plate (1) on the object plane of the optical imaging system (3); adjusting the second three-dimensional stage (5) to locate the two-dimensional diffraction grating plate (4) on the image plane of the optical imaging system (3); (b) determining number of phase-shifting moving steps N of the two-dimensional diffraction grating plate (4) by the following steps of determining $m=\mathrm{ceil}\left(\frac{1}{s}\right)-1,$ maximum diffraction order m according to formula wherein s is shear ratio $\mathrm{ceil}\left(\frac{1}{s}\right)$ of the grating shearing interferometer system, and equals to (1/s) when (1/s) is an integer, $\mathrm{ceil}\left(\frac{1}{s}\right)$ or is an immediate next positive integer that is greater than (1/s) when (1/s) is not an integer, determining total number n of positive high-order and negative low-order diffraction beams in the grating shearing interferometer system according to formula $n=\mathrm{fix}\left(\frac{m+1}{2}\right),$ wherein n equals to [(m+1)/2] when [(m+1)/2] is an integer, or n is an immediate next positive integer that is less than [(m+1)/2] when [(m+1)/2] is not an integer, determining diffraction order based on the total number n of the positive high-order and negative low-order diffraction beams as follows: ±1, ±3, . . . , ±(2n−1), determining number N of phase-shifting moving steps of the two-dimensional diffraction grating plate (4) for acquiring interferograms according to formula N=2 (n+1), and determining period of movement of the two-dimensional diffraction grating plate (4) $\frac{i}{N}$ which is times the period of the set of checkerboard gratings, wherein i is an integer that is 0, 1, 2, . . . (N-1); (c) moving the first three-dimensional stage (2) so that the first set of linear gratings (101) on the one-dimensional grating diffraction plate (1) are moved along the Y axis direction to a position in a field of view on the object side of the optical imaging system (3); moving the second three-dimensional stage (5) so that the set of checkerboard gratings on the two-dimensional diffraction plate (4) are moved to a position in a field of view on the image side of the optical imaging system (3), and direction of the diagonal line in each square of the checkerboard gratings is along the X axis direction or Y axis direction; (d) moving the second three-dimensional stage (5) along direction of the X axis $\frac{i}{N}$ direction according to the period of movement which is times the period of the set of checkerboard gratings, wherein i is an integer that is 0, 1, 2, . . . (N-1) $I_x\left(\frac{i}{N}\right)$ acquiring an interferogram by the two-dimensional photoelectric sensor (6) after each movement and transmitting the interferograms to the data processing unit (7); obtaining a total number of N interferograms and performing Fourier transform on light intensity of each position corresponding to each of the N numbers of the interferograms in $I_x\left(\frac{i}{N}\right)$ as in formula (i): $\begin{array}{cc}{I}_x\left(w\right)=FFT\left(I_x\left(\frac{i}{N}\right)\right);& \left(i\right)\end{array}$ (i); and calculating argument of I.sub.x (w) component when diagonal frequency of frequency domain is 2π and obtaining a gradient phase φ.sub.x of the X axis direction according to formula (ii): φ.sub.x=angle (I.sub.x (w=2π)) (ii), wherein φ.sub.x is the phase angle function of complex element I.sub.x (w); (e) moving the first three-dimensional stage (2) so that the second set of linear gratings (102) on the one-dimensional grating diffraction plate (1) are moved along the X axis direction to a position in the field of view on the object side of the optical imaging system; moving the second three-dimensional stage (5) $\frac{i}{N}$ along the direction of the Y axis according to the period of movement which is times the period of the set of checkerboard gratings, wherein i is an integer that is 0, 1, 2, . . . (N-1), $I_y\left(\frac{i}{N}\right)$ acquiring an interferogram by the two-dimensional photoelectric sensor (6) after each movement, transmitting the interferograms to the data processing unit (7), and obtaining a total of N numbers of shearing interferograms; performing Fourier transform on a light intensity of each position corresponding to each $I_y\left(\frac{i}{N}\right)$ of the N numbers of the shearing interferograms in according to formula (iii): $\begin{array}{cc}{I}_y\left(w\right)=FFT\left(I_y\left(\frac{i}{N}\right)\right),& \left(\mathrm{iii}\right)\end{array}$ (iii), calculating argument of I.sub.y(w) component when diagonal frequency w of frequency domain is 2π, and obtaining a gradient phase φ.sub.y of the Y axis direction according to formula (iv): φ.sub.y=angle(I.sub.y (w=2π)) (iv), wherein φ.sub.y is the phase angle function of complex element I.sub.y(w); and (f) unwrapping the gradient phase φ.sub.x of the X axis direction to obtain differential wavefront ΔW.sub.x of the X axis direction, unwrapping the gradient phase φ.sub.y of the Y axis direction to obtain differential wavefront ΔW.sub.y of the Y axis direction, and obtaining wavefront aberration W of the optical imaging system (3) by wavefront reconstruction algorithm of the grating shearing interferometer system.

2. The method for wavefront measurement of the optical imaging system based on grating shearing interferometry of claim 1, wherein a period of the first set of linear gratings (101) and the second set of linear gratings (102) on the one-dimensional diffraction grating plate (1) is determined by multiplying the period of the set of checkerboard gratings on the two-dimensional diffraction grating plate (4) with the magnification of the optical imaging system (3).

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a schematic of the detection device of the wavefront measurement for grating shearing interferometer;

(2) FIG. 2 is a schematic of the one-dimensional diffraction grating plate.

(3) FIG. 3 is a schematic of the two-dimensional diffraction grating plate.

(4) FIG. 4 is a schematic showing the relationship between the shear ratio s of the grating shearing interferometer system and the numerical aperture N.A.

(5) Reference numbers in the figures refer to the following structures: 1-one-dimensional diffraction grating plate; 2-first three-dimensional stage; 3-the optical imaging system to be tested; 4-two-dimensional diffraction grating plate; 5-second three-dimensional stage; 6-two-dimensional photoelectric sensor; 7-computing unit; 8-light source and illumination system.

DETAILED DESCRIPTIONS OF THE INVENTION AND EMBODIMENTS

(6) In combination with the figures and the embodiment hereunder, the present invention will be described in detail, but the protection scope of the present invention is not limited to the figures and the embodiment described below.

(7) The grating shearing interferometer system used by the method for wavefront measurement of optical imaging system based on grating shearing interferometry disclosed in the present invention is shown in FIG. 1, and the system comprises a light source and illumination system 8, a one-dimensional diffraction grating plate 1, a first three-dimensional stage 2, a two-dimensional diffraction grating plate 4, a second three-dimensional stage 5, a two-dimensional photoelectric sensor 6, and a computing unit 7. The light source and the illumination system 8 output spatially incoherent light, the one-dimensional diffraction grating plate 1 is fixed on the first three-dimensional stage 2, the two-dimensional diffraction grating plate 4 is fixed on the second three-dimensional stage 5, and the output of the two-dimensional photoelectric sensor 6 is connected with the computing unit 7.

(8) The xyz coordinate is established as follows. The Z axis direction of the coordinate is along the optical axis direction of the system. The X axis direction of the coordinate is along the grating line direction of the linear grating 102 on the one-dimensional diffraction grating plate 1. The Y axis direction of the coordinate is along the grating line direction of the linear grating 101 on the one-dimensional diffraction grating plate 1. The motion axes of the first three-dimensional stage 2 and the second three-dimensional stage 5 are set as X axis, Y axis, and Z axis, respectively.

(9) The first three-dimensional stage 2 is used to move the two linear gratings 101 and 102 on the first diffraction plate 1 to the center of the object side view of the optical imaging system to be tested 3.

(10) The second three-dimensional stage 5 is used to move the checkerboard grating on the second diffraction plate 4 to the center of the image side view of the optical imaging system to be tested 3, and perform specific movement along X axis direction and Y axis direction of the two-dimensional diffraction grating plate 4.

(11) The two-dimensional photoelectric sensor 6 is a charge coupled device CCD or a CMOS sensor, and the detecting surface receives shearing interference fringes generated by diffraction orders of the checkerboard grating.

(12) The computing unit 7 is used to collect and store interferograms, and perform process and analyze of the interferograms.

(13) FIG. 2 is a schematic of the one-dimensional diffraction grating plate 1, including two linear diffraction gratings, which are the first grating 101 along Y axis direction of the grating line and the second grating 102 along X axis direction of the grating line, respectively. The period of the linear diffraction grating is P1, and the duty-cycle is 50%.

(14) The first grating 101 and the second grating 102 are phase gratings or amplitude gratings.

(15) FIG. 3 is a schematic of the two-dimensional diffraction grating plate 4, which is a checkerboard grating with a period of P2 and a 50% duty-cycle. The checkerboard grating is composed of a square grid, and the diagonal direction of the square is along the X axis direction or Y axis direction.

(16) The checkerboard grating is a phase grating or an amplitude grating.

(17) The period of the linear grating P1 and the period of the chenckerboard grating P2 satisfy the following formula:
P1=M.Math.P2  (1),
wherein M is a magnification of the optical imaging system to be tested 3.

(18) FIG. 4 is a schematic showing the relationship between the shear ratio s of the shearing interferometer system, the numerical aperture NA of the optical imaging system to be tested 3, and the period of the checkerboard grating P2. The shear ratio s is defined as the ratio of the diffraction angle to the angle with full aperture:

(19) $\begin{array}{cc}s=\frac{\beta }{\alpha }=\frac{\arcsin \left(\frac{\lambda }{P2}\right)}{2\mathrm{ars}\sin \left(\mathrm{NA}\right)},& \left(2\right)\end{array}$
wherein β is the diffraction angle of the 1st-order diffraction beam, and a is the angle with full aperture of the beam.

(20) Based on the grating shearing interference system described above, only high-order diffraction wavefront with odd-order is generated after passing through the checkerboard grating of the two-dimensional grating diffraction plate 4. Combining with the first linear grating 101 or the second linear grating 102 on the object side of the one-dimensional grating diffraction plate 1, the spatial coherence of the light field is modulated such that there is only interference between the 0th-order beam and other high diffraction orders in the light field, and there is no interference between these high diffraction orders. The interference field received by the two-dimensional photoelectric sensor 6 can then be described as:

(21) $\begin{array}{cc}I\left(x,y\right)=A_0^2+\underset{m=-\left(2n+1\right)}{\overset{m=2n+1}{.Math.}}{A}_n^2+\mathrm{.Math.}+2\underset{m=-\left(2n+1\right)}{\overset{m=2n+1}{.Math.}}{A}_0{A}_m{\gamma }_m\cos \left[\varphi \left(x,y\right)-\varphi \left(x-m\Delta ,y\right)-{\alpha }_m\right],(n=0,1,2,\mathrm{.Math.}),& \left(3\right)\end{array}$
Wherein A.sub.0 is the amplitude of the 0th-order beam, A.sub.m is the amplitude of the mth-order diffraction beam, α.sub.m is the optical path difference between the mth-order diffraction beam and the 0-order beam, φ(x,y) is the wavefront to be measured, γ.sub.m is the degree of coherence between the mth-order diffraction beam and the 0th-order beam, m is the diffraction order, and A is the offset of the 1st-order diffraction beam relative to the 0th-order beam.

(22) Assuming A.sub.0 is 1, the A.sub.m and γ.sub.m coefficients satisfy the following relationship:

(23) $\begin{array}{cc}{A}_m={\gamma }_m=\frac{2}{m\pi }& \left(4\right)\end{array}$

(24) For small shear conditions, the phase between ±1st-order beams and 0th-order beam is as follows:
cos[φ(x,y)−φ(x−Δ,y)]+cos[φ(x,y)−φ(x+Δ,y)]≈2 cos(dφ.Math.Δ)

(25) Let dφ.Math.Δ=φ, φ.sub.m=φ(x,y)−φ(x+mΔ, y), then formula (3) can be further simplified as:

(26) $\begin{array}{cc}I=I0+a_1\cos \phi +\underset{m=3}{\overset{m=2n-1}{.Math.}}{a}_m\left(\cos {\phi }_{-m}+\cos {\phi }_m\right),\left(n=2,3\mathrm{.Math.}\right)& \left(5\right)\end{array}$

(27) Where,

(28) $I0=A_0^2+\underset{m=-\left(2n+1\right)}{\overset{m=2n+1}{.Math.}}{A}_n^2,$
α.sub.1=4A.sub.0A.sub.1γ.sub.1, α.sub.m=2A.sub.0A.sub.mγ.sub.m. When a phase shift δ is introduced, the above equation can be described as:

(29) $\begin{array}{cc}I=I0+a_1\cos \left(\phi +\delta \right)+\underset{m=3}{\overset{m=2n-1}{.Math.}}{a}_m\left(\cos \left({\phi }_{-m}-m\delta \right)+\cos \left({\phi }_m+m\delta \right)\right)& \left(6\right)\end{array}$

(30) The method for wavefront measurement of optical imaging system using the aboved described grating shearing interferometer of the present invention includes the following steps:

(31) (1) the optical imaging system to be tested 3 is placed in the grating shearing interferometer, the light source and the illumination system 8 are located on the object side of the optical imaging system to be tested 3, and the two-dimensional diffraction grating plate 4 is located on the image side of the optical imaging system to be tested 3; the first three-dimensional stage 2 is adjusted so that the one-dimensional diffraction grating plate 1 is located on the object plane of the imaging system 3 to be tested; the second three-dimensional stage 5 is adjusted so that the two-dimensional diffraction grating plate 4 is located on the object plane of the imaging system to be tested 3;

(32) (2) the phase-shifting amounts are determined according to the shear ratio s of the grating shearing interferometer: first, the maximum diffraction order is determined as

(33) $m=\mathrm{ceil}\left(\frac{1}{s}\right)-1,$
and the diffraction order of the grating shearing interferometer system is in turn as follows: ±1, ±3, . . . , ±(2n−1), where n is the total number of positive high-order diffraction beams or the negative high-order diffraction beams in the shearing interferometer system,

(34) $n=\mathrm{fix}\left(\frac{m+1}{2}\right),$
the function ceil(X) returns the smallest integer greater than or equal to X, and the function fix(X) returns the biggest integer less than or equal to X; then, the number N of moving steps of the two-dimensional diffraction grating plate 4 when the interferogram is acquired is determined according to n, N=2(n+1), and the movement is determined as

(35) $\frac{i}{N}$
checkerboard grating period (i=0, 1, 2 . . . N−1);

(36) (3) the first three-dimensional stage 2 is moved so that the first grating 101 along the Y axis direction of the grating line on the one-dimensional grating diffraction plate 1 is moved into the position of object side field of the optical imaging system to be tested 3; the second three-dimensional stage 5 is moved so that the checkerboard grating on the two-dimensional diffraction plate 4 is moved to the position of image side field of the imaging system to be tested 3, and the diagonal direction of the square is along the X axis direction or Y axis direction;

(37) (4) moving the second three-dimensional stage 5 along the X axis direction according to the aboved mentioned

(38) 0$\frac{i}{N}$
checkerboard grating period (i=0, 1, 2 . . . N−1). After each movement, the two-dimensional photoelectric sensor 6 acquires a interferogram

(39) $I_x\left(\frac{i}{N}\right),$
transmits it to the data processing unit 7, and obtains a total of N shearing interferograms; performing Fourier transform on the light intensity of each position in

(40) $\begin{array}{cc}{I}_x\left(\frac{i}{N}\right)\text{:}{I}_x\left(w\right)=FFT\left(I_x\left(\frac{i}{N}\right)\right),& \left(7\right)\end{array}$

(41) calculating the argument of I.sub.x (w) component when diagonal frequency of frequency domain is 2π according to formula (8), and the gradient phase φ.sub.x of the X axis direction is obtained:
φ.sub.xangle(I.sub.x(w=2π))  (8),
wherein the function angle (X) returns the phase angle of a complex element X.

(42) (5) the first three-dimensional stage 2 is moved so that the second grating 102 along the X axis direction of the grating line on the one-dimensional grating diffraction plate 1 is moved to the position of object side field of the optical imaging system to be tested 3; performing the same movement of the second three-dimensional stage 5 along the Y axis direction; after each movement, the two-dimensional photoelectric sensor 6 acquires a interferogram,

(43) $I_y\left(\frac{i}{N}\right),$
transmits it to the data processing unit 7, and obtains a total of N interferograms; performing Fourier transform on the light intensity of each position in

(44) $\begin{array}{cc}{I}_y\left(\frac{i}{N}\right)\text{:}{I}_y\left(w\right)=FFT\left(I_y\left(\frac{i}{N}\right)\right),& \left(9\right)\end{array}$

(45) calculating the argument of I.sub.x(w) component when diagonal frequency of frequency domain is 2π, and the gradient phase φ.sub.y of the Y axis direction is obtained:
φ.sub.y=angle(I.sub.y(w=2π))  (10);

(46) (6) by unwrapping the gradient phase φ.sub.x of the X axis direction and the gradient phase φ.sub.y of the Y axis direction, the differential wavefront ΔW.sub.x of the X axis direction and the differential wavefront ΔW.sub.y of the Y axis direction are respectively obtained. The wavefront reconstruction algorithm of shearing interference is used to obtain the wavefront aberration of the optical imaging system to be tested 3.

(47) The method of the present invention has the advantages of high precision, large measurable range of numerical aperture, and adjustable shear rate of the grating interferometer.