Method for detecting wavefront aberration for optical imaging system based on grating shearing interferometer

Abstract

Method for detecting wavefront aberration for optical imaging system based on grating shearing interferometer, the grating shearing interferometer system 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 and two-dimensional diffraction grating plates are respectively placed on the object plane and the image plane of the optical imaging system to be tested. By collecting interferograms with phase-shifting amounts of 0, π/2, π, 3π/2 and N sets of α, π-α, 2π-α (where, $N=2\left(\mathrm{fix}\left(\frac{\mathrm{ceil}\left(1\text{/}s\right)}{2}\right)+1\right),$
s is the shear ratio of the grating shearing interferometer system), 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 detection accuracy of wavefront aberration for the imaging system to be tested is improved.

Claims

1. A method for detecting wavefront aberration for optical imaging system based on grating shearing interferometer, comprising (1) placing an optical imaging system to be tested (3) in a grating shearing interferometer, a light source and illumination system (8) and a one-dimensional diffraction grating plate (1) are located on an object side of the optical imaging system to be tested (3), and a two-dimensional diffraction grating plate (4) is located on an image side of the optical imaging system to be tested (3); a first three-dimensional stage (2) is adjusted so that a one-dimensional diffraction grating plate (1) is located on an object plane of the imaging system to be tested (3); a 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); (2) determining the phase-shifting amounts according to a shear ratio s of the grating shearing interferometer by firstly determining a maximum diffraction order as $m=\mathrm{ceil}\left(\frac{1}{s}\right)-1,$ and a diffraction order of the grating shearing interferometer system is in turn as follows: ±1, ±3, . . . , ±(2n−1) , wherein n is a total amount of positive high-order or a negative high-order in the grating shearing interferometer system, $n=\mathrm{fix}\left(\frac{m+1}{2}\right),$ function ceil(X) returns a smallest integer greater than or equal to X, and function fix(X) returns a biggest integer less than or equal to X; then, according to n, a movement of the two-dimensional diffraction grating plate (4) when the interferogram is acquired is respectively determined as 0, ¼, ½, ¾ and $\frac{{\alpha }_i}{2\pi },\frac{\pi -{\alpha }_i}{2\pi },\frac{2\pi -{\alpha }_i}{2\pi }$ checkerboard grating period, wherein $i=2,3\mathrm{.Math.}n,{\alpha }_i<\pi \mathrm{and}{\alpha }_i\ne \frac{\pi }{2};$ (3) moving the first three-dimensional stage (2) so that a first grating (101) along an Y axis direction of a grating line on the one-dimensional grating diffraction plate (1) is moved to a position in a field of the object side of the optical imaging system to be tested (3); moving the second three-dimensional stage (5) so that a checkerboard grating on the two-dimensional diffraction plate (4) is moved to a position in a field of the image side 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; (4) moving the second three-dimensional stage (5) along the X axis direction according to the 0, ¼, ½, ¾ and $\frac{{\alpha }_i}{2\pi },\frac{\pi -{\alpha }_i}{2\pi },\frac{2\pi -{\alpha }_i}{2\pi }$ checkerboard grating period, wherein $i=2,3\mathrm{.Math.}n,{\alpha }_i<\pi \mathrm{and}{\alpha }_i\ne \frac{\pi }{2};$ after each movement, having a two-dimensional photoelectric sensor (6) to acquire a interferogram I.sub.α.sub.i, transmitting the interferogram I.sub.α.sub.i to a data processing unit (7), and obtaining a total of 3n+1 interferograms, wherein phase is calculated as follows: $\begin{array}{cc}\left[\begin{array}{cccc}1& 1& \mathrm{.Math.}& 1\\ \cos {\alpha }_2& \cos 3{\alpha }_2& \mathrm{.Math.}& \cos \left(2n-1\right){\alpha }_2\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ \cos {\alpha }_n& \cos 3{\alpha }_n& \mathrm{.Math.}& \cos \left(2n-1\right){\alpha }_n\end{array}\right].Math.\hspace{1em}\left[\begin{array}{c}2a_1\cos \phi \\ 2a_3\left(\cos {\phi }_3+\cos {\phi }_{-3}\right)\\ \mathrm{.Math.}\\ 2a_{2n-1}\left(\cos {\phi }_{2n-1}+\cos {\phi }_{-\left(2n-1\right)}\right)\end{array}\right]=\left[\begin{array}{c}{I}_0-I_{\pi }\\ I_{{\alpha }_2}-I_{\pi -{\alpha }_2}\\ \mathrm{.Math.}\\ I_{{\alpha }_n}-I_{\pi -{\alpha }_n}\end{array}\right]& \left(1\right)\\ \left[\begin{array}{cccc}\sin \frac{\pi }{2}& \sin \frac{3\pi }{2}& \mathrm{.Math.}& \sin \frac{\left(2n-1\right)\pi }{2}\\ \sin {\alpha }_2& \sin 3{\alpha }_2& \mathrm{.Math.}& \sin \left(2n-1\right){\alpha }_2\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ \sin {\alpha }_n& \sin 3{\alpha }_n& \mathrm{.Math.}& \sin \left(2n-1\right){\alpha }_n\end{array}\right].Math.\hspace{1em}\left[\begin{array}{c}2a_1\sin \phi \\ 2a_3\left(\sin {\phi }_3-\sin {\phi }_{-3}\right)\\ \mathrm{.Math.}\\ 2a_{2n-1}\left(\sin {\phi }_{2n-1}+\sin {\phi }_{-\left(2n-1\right)}\right)\end{array}\right]=\left[\begin{array}{c}{I}_{\frac{\pi }{2}}-I_{\frac{3\pi }{2}}\\ I_{{\alpha }_2}-I_{2\pi -{\alpha }_2}\\ \mathrm{.Math.}\\ I_{{\alpha }_n}-I_{2\pi -{\alpha }_n}\end{array}\right]& \left(2\right)\end{array}$ obtaining 2a.sub.1 cos φ and 2a.sub.1 sin φ by solving linear equations (1) and (2), respectively, and obtaining a gradient phase of the X axis direction φ.sub.x as follows: $\begin{array}{cc}{\phi }_x=\arctan \left(\frac{2a_1\sin \phi }{2a_1\cos \phi }\right);& \left(3\right)\end{array}$ (5) moving the first three-dimensional stage (2) so that a second grating (102) along the X axis direction of a grating line on the one-dimensional grating diffraction plate (1) is moved to a position in a field of the object side of the optical imaging system to be tested (3); performing same periodic movement of the second three-dimensional stage (5) along the Y axis direction; after each movement, having the two-dimensional photoelectric sensor (6) to acquire a interferogram I.sub.α.sub.i, transmitting the shearing interferogram I.sub.α.sub.i to the data processing unit (7), and obtaining a total of 3n+1 interferograms; calculating 2a.sub.1 cos φ and 2a.sub.1 sin φ according to formula (1) and formula (2) to obtain a gradient phase of the Y axis direction φ.sub.y as follows: $\begin{array}{cc}{\phi }_y=\arctan \left(\frac{2a_1\sin \phi }{2a_1\cos \phi }\right);& \left(4\right)\end{array}$ and (6) unwrapping the gradient phase φ.sub.x of the X axis direction and the gradient phase φ.sub.y of the Y axis direction, and obtaining a differential wavefront ΔW.sub.x of the X axis direction and a differential wavefront ΔW.sub.y of the Y axis direction, respectively; using a wavefront reconstruction algorithm of shearing interference to obtain a wavefront aberration W of the optical imaging system to be tested (3), wherein the grating shearing interferometer comprises the light source and illumination system (8), the one-dimensional diffraction grating plate (1), the first three-dimensional stage (2), the two-dimensional diffraction grating plate (4), the second three-dimensional stage (5), the two-dimensional photoelectric sensor (6), and the computing unit (7), wherein the light source and illumination system (8) outputs 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), the one-dimensional diffraction grating plate (1) comprises two sets of linear gratings with 50% duty-cycle, the two-dimensional diffraction grating plate (4) comprises a set of checkerboard gratings, an output of the two-dimensional photoelectric sensor (6) is connected with the computing unit (7); an xyz coordinate is established as follows: Z axis direction of the xyz coordinate is along an optical axis direction of the grating shearing interferometer, the X axis direction of the xyz 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 xyz coordinate is along the grating line direction of the linear grating (101) on the one-dimensional diffraction grating plate (1), and set motion axes of the first three-dimensional stage (2) and the second three-dimensional stage (5) as the X axis, the Y axis, and the Z axis, respectively, and the period of the one-dimensional grating on the one-dimensional diffraction grating plate (1) is equal to the period of the checkerboard grating on the two-dimensional diffraction grating plate (4) times a magnification of the optical imaging system to be tested (3).

2. The method of claim 1, wherein in the phase-shifting amounts, 0, ¼, ½, ¾ and $\frac{{\alpha }_i}{2\pi },\frac{\pi -{\alpha }_i}{2\pi },\frac{2\pi -{\alpha }_i}{2\pi }$ times 2π, i=2, 3 . . . n, ${\alpha }_i<\pi \mathrm{and}{\alpha }_i\ne \frac{\pi }{2},$ and α.sub.i is an equally spaced distribution or a non-equal interval distribution within [0 π).

3. The method of claim 1, wherein in the phase-shifting amounts, 0, ¼, ½, ¾ and $\frac{{\alpha }_i}{2\pi },\frac{\pi -{\alpha }_i}{2\pi },\frac{2\pi -{\alpha }_i}{2\pi }$ times 2π, i=2, 3 . . . n, ${\alpha }_i<\pi \mathrm{and}{\alpha }_i\ne \frac{\pi }{2},$ when α.sub.i is an equally spaced distribution within [0 π), the phase-shifting amounts $\frac{{\alpha }_i}{2\pi }\mathrm{and}\frac{\pi -{\alpha }_i}{2\pi }$ coincide, and an actual total phase-shifting step number is 2(n+1).

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a schematic drawing of the detection device of the wavefront aberration for grating shearing interferometer of the present invention.

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

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

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

(5) Reference numbers used in the drawings are as follows: 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 embodiments hereunder, the present invention is described in detail, while the protection scope of the present invention is not limited to the figures and the embodiment.

(7) As shown in FIG. 1, the grating shearing interferometer system used by the detection method of wavefront aberration for optical imaging system based on grating shearing interferometer disclosed in the present invention 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) In the system of the present invention, 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, and set the 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.

(9) The first three-dimensional stage 2 is used to move the first grating 101 and the second grating 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) As shown in FIG. 2, the one-dimensional diffraction grating plate 1 in the present invention includes 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) In the present invention, the first grating 101 and the second grating 102 are phase gratings or amplitude gratings.

(15) As shown in FIG. 3, the two-dimensional diffraction grating plate 4 of the present invention is a checkerboard grating with a period of P2 and a 50% duty-cycle. The checkerboard grating is composed of square grids, 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 checkerboard grating P2 satisfy the following formula:
P1=M.Math.P2  (1),
wherein M is a magnification of the imaging optical imaging system to be tested 3.

(18) As shown in FIG. 4, 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 are as follows. The shear ratio s is defined as the ratio of the diffraction angle to the angle with full aperture:

(19) $\begin{array}{cc}s=\frac{\beta }{\alpha }=\frac{\arcsin \left(\frac{\lambda }{P2}\right)}{2\mathrm{ars}\sin \left(\mathrm{NA}\right)}& \left(2\right)\end{array}$

(20) Where, β is the diffraction angle of the 1st-order diffraction beam, and α is the angle with full aperture of the beam.

(21) 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 linear grating 101 or 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:

(22) $I\left(x,y\right)=A_0^2+\underset{m=-\left(2n+1\right)}{\overset{m=2n+1}{.Math.}}{A}_n^2+\mathrm{.Math.}+2\underset{m=-\left(2n+1\right)}{\overset{m=2n+1}{.Math.}}{A}_0{A}_m{\gamma }_m\cos \left[\varphi \left(x,y\right)-\varphi \left(x-m\Delta ,y\right)-{\alpha }_m\right],(n=0,1,2,\mathrm{.Math.})$
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 0th-order beam, ϕ.sub.(x, y) is the wavefront to be measured, γ.sub.m is the degree of coherence between the m-order diffraction beam and the 0th-order beam, m is the diffraction order, and Δ is the offset of the 1st-order diffraction beam relative to the 0th-order beam.

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

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

(25) For small shear conditions, the phase between ±1th 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)

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

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

(28) Where,

(29) 0$I0=A_0^2+\underset{m=-\left(2n+1\right)}{\overset{m=2n+1}{.Math.}}{A}_n^2,a_1=4A_0{A}_1{\gamma }_1,a_m=2A_0{A}_m{r}_m.$
When a phase shift δ is introduced, the above equation can be described as:

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

(31) The detection method of wavefront aberration for optical imaging system using the above described grating shearing interference includes the following steps:

(32) (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 to be tested 3; 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.

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

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

(35) $n=\mathrm{fix}\left(\frac{m+1}{2}\right),$
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, according to n, a movement of the two-dimensional diffraction grating plate (4) when the interferogram is acquired is respectively determined as 0, ¼, ½, ¾ and

(36) $\frac{{\alpha }_i}{2\pi },\frac{\pi -{\alpha }_i}{2\pi },\frac{2\pi -{\alpha }_i}{2\pi }$
checkerboard grating period (where

(37) $i=2,3\mathrm{.Math.}n,{\alpha }_i<\pi \mathrm{and}{\alpha }_i\ne \frac{\pi }{2}),$
and has a total of 3n+1 step number.

(38) (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 into 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.

(39) (4) Moving the second three-dimensional stage 5 along the X axis direction according to the above mentioned 0, ¼, ½, ¾ and

(40) $\frac{{\alpha }_i}{2\pi },\frac{\pi -{\alpha }_i}{2\pi },\frac{2\pi -{\alpha }_i}{2\pi }$
checkerboard grating period (where

(41) $i=2,3\mathrm{.Math.}n,{\alpha }_i<\pi \mathrm{and}{\alpha }_i\ne \frac{\pi }{2}).$
After each movement, the two-dimensional photoelectric sensor 6 acquires a interferogram I.sub.α.sub.i, transmits it to the data processing unit 7, and obtains a total of 3n+1 interferograms. Calculating phase as follows:

(42) $\begin{array}{cc}\left[\begin{array}{cccc}1& 1& \mathrm{.Math.}& 1\\ \cos {\alpha }_2& \cos 3{\alpha }_2& \mathrm{.Math.}& \cos \left(2n-1\right){\alpha }_2\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ \cos {\alpha }_n& \cos 3{\alpha }_n& \mathrm{.Math.}& \cos \left(2n-1\right){\alpha }_n\end{array}\right].Math.\hspace{1em}\left[\begin{array}{c}2a_1\cos \phi \\ 2a_3\left(\cos {\phi }_3+\cos {\phi }_{-3}\right)\\ \mathrm{.Math.}\\ 2a_{2n-1}\left(\cos {\phi }_{2n-1}+\cos {\phi }_{-\left(2n-1\right)}\right)\end{array}\right]=\left[\begin{array}{c}{I}_0-I_{\pi }\\ I_{{\alpha }_2}-I_{\pi -{\alpha }_2}\\ \mathrm{.Math.}\\ I_{{\alpha }_n}-I_{\pi -{\alpha }_n}\end{array}\right]& \left(7\right)\\ \left[\begin{array}{cccc}\sin \frac{\pi }{2}& \sin \frac{3\pi }{2}& \mathrm{.Math.}& \sin \frac{\left(2n-1\right)\pi }{2}\\ \sin {\alpha }_2& \sin 3{\alpha }_2& \mathrm{.Math.}& \sin \left(2n-1\right){\alpha }_2\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ \sin {\alpha }_n& \sin 3{\alpha }_n& \mathrm{.Math.}& \sin \left(2n-1\right){\alpha }_n\end{array}\right].Math.\hspace{1em}\left[\begin{array}{c}2a_1\sin \phi \\ 2a_3\left(\sin {\phi }_3-\sin {\phi }_{-3}\right)\\ \mathrm{.Math.}\\ 2a_{2n-1}\left(\sin {\phi }_{2n-1}+\sin {\phi }_{-\left(2n-1\right)}\right)\end{array}\right]=\left[\begin{array}{c}{I}_{\frac{\pi }{2}}-I_{\frac{3\pi }{2}}\\ I_{a_2}-I_{2\pi -{\alpha }_2}\\ \mathrm{.Math.}\\ I_{a_n}-I_{2\pi -{\alpha }_n}\end{array}\right]& \left(8\right)\end{array}$

(43) By solving linear equations (1) and (2), 2a.sub.1 cos φ and 2a.sub.1 sin φ are obtained, respectively, and the gradient phase of the X axis direction is obtained:

(44) $\begin{array}{cc}{\phi }_x=\arctan \left(\frac{2a_1\sin \phi }{2a_1\cos \phi }\right).& \left(9\right)\end{array}$

(45) (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 I.sub.α.sub.i, transmits it to the data processing unit 7, and obtains a total of 3n+1 interferograms. Also calculate 2a.sub.1 cos φ and 2a.sub.1 sin φ according to formula (7) and formula (8) to obtain the gradient phase of the Y axis direction:

(46) 0$\begin{array}{cc}{\phi }_y=\arctan \left(\frac{2a_1\sin \phi }{2a_1\cos \phi }\right).& \left(10\right)\end{array}$

(47) (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.

(48) The phase-shifting amounts 0, ¼, ½, ¾ and

(49) $\frac{{\alpha }_i}{2\pi },\frac{\pi -{\alpha }_i}{2\pi },\frac{2\pi -{\alpha }_i}{2\pi }$
times 2π, where

(50) $i=2,3\mathrm{.Math.}n,{\alpha }_i<\pi \mathrm{and}{\alpha }_i\ne \frac{\pi }{2},{\alpha }_i$
is an equally spaced distribution or a non-equal interval distribution within [0 π).

(51) The phase-shifting amounts 0, ¼, ½, ¾ and

(52) $\frac{{\alpha }_i}{2\pi },\frac{\pi -{\alpha }_i}{2\pi },\frac{2\pi -{\alpha }_i}{2\pi }$
times 2π, where

(53) $i=2,3\mathrm{.Math.}n,{\alpha }_i<\pi \mathrm{and}{\alpha }_i\ne \frac{\pi }{2},$
when α.sub.i is an equally spaced distribution within

(54) $\left[0\pi \right),\frac{{\alpha }_i}{2\pi }\mathrm{and}\frac{\pi -{\alpha }_i}{2\pi }$
overlap, and the actual phase-shifting step number is simplified to 2 (n+1).

(55) An embodiment of the phase-shifting amounts setting of the present invention is given below.

(56) Assuming that the shear ratio s of the grating shearing interferometer system is 2%, there are high diffraction orders such as ±3th-, ±5th- . . . and ±49th-order beams in the interference field.

(57) When α.sub.i is a non-equal interval distribution within [0 π), the exact solution of the equation requires a 76-step phase shift, and the phase-shifting amounts can be set to:

(58) $0,\frac{n}{M}\frac{\pi }{2},\frac{\pi }{2},\pi -\frac{n}{M}\frac{\pi }{2},2\pi -\frac{n}{M}\frac{\pi }{2},$
where M=25, n=1,2 . . . M−1; and there can be other forms of phase-shifting distribution.

(59) In particular, when α.sub.i is an equally spaced distribution within [0 π), the phase shifts in the first quadrant and the second quadrant coincide, the actual phase-shifting step number is reduced by 24, the exact solution of the equation requires a 52-step phase shift, and the phase-shifting distribution is:

(60) $0,\frac{N}{M}\frac{\pi }{2},\pi /2,\pi -\frac{N}{M}\frac{\pi }{2},\pi ,\pi +\frac{N}{M}\frac{\pi }{2},3\pi /2,2\pi -\frac{N}{M}\frac{\pi }{2},$
where M=13, N=1, 2, 3 . . . M−1. That is, the phase-shifting amounts is equally spaced within [0 2π) period, the interval of the phase shift is

(61) $\frac{\pi }{26},$
corresponding to the checkerboard grating of the image plane moves 52 times in a period, and the moving interval of each move is

(62) $\frac{1}{52}$
checkerboard grating period.

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