Device and method for detecting wavefront error by modal-based optimization phase retrieval using extended Nijboer-Zernike theory

Abstract

The disclosure provides a device for detecting a wavefront error by modal-based optimization phase retrieval using an extended Nijboer-Zernike (ENZ) theory. The detection device includes a point light source (1), a half mirror (2), a lens (3) to be tested, a plane mirror (4) and an image sensor (5). The wavefront error of the component under test is characterized by using a Zernike polynomial, and a Zernike polynomial coefficient is solved based on an ENZ diffraction theory. The present disclosure realizes the one-time full-aperture measurement on the wavefront error of a large-aperture optical component, and can use a partially overexposed image to achieve accurate wavefront error retrieval. Meanwhile, the present disclosure overcomes the contradiction between underexposure and high signal-to-noise ratio (SNR) caused by a limited dynamic range when the image sensor (5) acquires an image. The detection device is simple and does not have high requirements for the experimental environment.

Claims

1. A method for detecting a wavefront error by modal-based optimization phase retrieval using an extended Nijboer-Zernike (ENZ) theory, wherein the method is implemented by using a detection device, comprising a point light source (1), a half mirror (2), a lens (3) to be tested, a plane mirror (4) and an image sensor (5), wherein the half mirror (2) is located behind the point light source (1) at an angle of 45 degrees; the lens (3) to be tested and the plane mirror (4) are sequentially arranged behind the point light source (1), and the point light source (1), the half mirror (2), the lens (3) to be tested and the plane mirror (4) share an optical axis; a front focus of the lens (3) to be tested is located at the point light source (1); the image sensor (5) is located on an optical path of reflected light of the half mirror (2), and is located at a defocus position of the lens (3) to be tested; the image sensor (5) shares an optical axis with the reflected light of the half mirror (2); and wherein the method comprises the following steps: S1: emitting, by the point light source (1), a spherical wave; collimating, by the lens (3) to be tested, the spherical wave; reflecting, by the plane mirror (4) and the half mirror (2), the spherical wave; acquiring, by the image sensor (5), a defocus intensity image with a wavefront error of the lens (3) to be tested; and S2: performing phase retrieval on the defocus intensity image acquired by step S1 by modal-based optimization phase retrieval using the ENZ theory to obtain the wavefront error of the tested lens; wherein step S2 specifically comprises: S2.1: using defocus intensity image I.sub.m′ obtained in step S1 as an initial defocus intensity image for a subsequent iteration in an ENZ mode; setting an initial defocus distance of the image sensor (5) as f.sub.0; setting a total number K of iterations to remove a cross term, an initial number k=1 of iterations to remove the cross term, a total number L of iterations for a defocus position, an initial number l=1 of iterations for the defocus position, an aperture of the lens (3) to be tested and a number of terms in a Zernike polynomial; S2.2: calculating each element of a modal gradient matrix of a system of equations for solving a coefficient of the Zernike polynomial according to a defocus distance, to obtain a modal gradient matrix V of the system of equations for solving a coefficient of the Zernike polynomial; S2.3: removing an overexposed pixel of initial defocus intensity image I.sub.m′ to obtain a defocus intensity image with the overexposed pixel removed, and removing an element in a row corresponding to the overexposed pixel from modal gradient matrix V to obtain a modal gradient matrix with the overexposed pixel removed; S2.4: calculating a Zernike coefficient matrix A of the Zernike polynomial by using a least squares method according to the defocus intensity image and the modal gradient matrix with the overexposed pixel removed; S2.5: calculating cross term I.sub.c of an ENZ diffraction mode according to the Zernike coefficient matrix A; S2.6: removing cross term I.sub.c from initial defocus intensity image I.sub.m′ to obtain a defocus intensity image with the cross term removed; S2.7: determining whether the number k of iterations to remove the cross term is greater than K; if not, letting k=k+1, using the defocus intensity image with the cross term removed as an initial defocus intensity image, and returning to step S2.3; if yes, proceeding to the next step; S2.8: determining whether the number l of iterations for the defocus position is greater than L; if not, calculating a correction value Δf of the defocus distance according to $\Delta f=\frac{{\beta }_{\mathrm{power}}\lambda }{\pi \left(1-\sqrt{1-N{A}^2}\right)},$ subtracting the correction value of the defocus distance from a current value thereof as a defocus distance for a subsequent iteration, taking the defocus intensity image obtained in step S1 as an initial defocus intensity image, letting the number of iterations for the defocus position be l=l+1, letting the number of iterations to remove the cross term be k=1, and returning to step S2.2, wherein, β.sub.power is a coefficient of a fourth term of the Zernike polynomial, λ represents a wavelength of incident light, and N represents the number of pixels in a lateral direction of the defocus intensity image; if yes, using the obtained Zernike coefficient matrix to fit the wavefront phase error of the lens (3) to be tested, and calculating a true defocus position U of the image sensor (5) by $U=\exp \left(i\times \underset{n,m}{.Math.}{\beta }_n^m\times Z_n^m\right)$ according to the Zernike coefficient matrix, wherein, β.sub.n.sup.m represents a Zernike coefficient in the Zernike coefficient matrix, Z.sub.n.sup.m represents the Zernike polynomial, i is an imaginary number, (m,n) represents a pixel of the defocus intensity image, m=1, 2, . . . , M, n=1, 2, . . . , N; M represents the number of pixels in a vertical direction of the defocus intensity image.

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) In order to illustrate the embodiments of the present disclosure or the technical solutions of the prior art, the accompanying drawing to be used will be described briefly below. Notably, the following accompanying drawing merely illustrates some embodiments of the present disclosure, but other accompanying drawings can also be obtained those of ordinary skill in the art based on the accompanying drawing without any creative efforts.

(2) FIG. 1 is a schematic diagram of a wavefront error detection device according to the present disclosure.

(3) FIG. 2 is a flowchart of detecting a wavefront error by modal-based optimization phase retrieval using an extended Nijboer-Zernike (ENZ) theory.

(4) Reference Numerals: 1. point light source; 2. half mirror; 3. lens to be tested; 4. plane mirror; and 5. image sensor.

DETAILED DESCRIPTION

(5) The following clearly and completely describes the technical solutions in the embodiments of the present disclosure with reference to accompanying drawings in the embodiments of the present disclosure. Apparently, the described embodiments are merely a part rather than all of the embodiments of the present disclosure. All other embodiments obtained by a person of ordinary skill in the art based on the embodiments of the present disclosure without creative efforts should fall within the protection scope of the present disclosure.

(6) In order to overcome the shortcomings of the prior art, an objective of the present disclosure is to provide a device and method for detecting a wavefront error by modal-based optimization phase retrieval using an extended Nijboer-Zernike (ENZ) theory. The present disclosure solves the problems of the conventional iterative phase retrieval method in measuring the wavefront error of a large-aperture optical lens. The conventional iterative phase retrieval method uses a sub-aperture stitching method, which has a detection error, a large amount of calculation due to a high sampling number, and high difficulty to accurately determine the defocus position.

(7) To make the above objectives, features and advantages of the present disclosure clearer and more comprehensible, the present disclosure is further described in detail below with reference to the accompanying drawings and specific embodiments.

(8) As shown in FIG. 1, a device for detecting a wavefront error by modal-based optimization phase retrieval using an extended Nijboer-Zernike (ENZ) theory includes a point light source 1, a half mirror 2, a lens 3 to be tested, a plane mirror 4 and an image sensor 5. The half mirror 2 is located behind the point light source 1 at an angle of 45 degrees. The lens 3 to be tested and the plane mirror 4 are sequentially arranged behind the point light source 1, and the point light source 1, the half mirror 2, the lens 3 to be tested and the plane mirror 4 share an optical axis. A front focus of the lens 3 to be tested is located at the point light source 1. The image sensor 5 is located on an optical path of reflected light of the half mirror 2, and is located at a defocus position of the lens 3 to be tested. The image sensor 5 shares an optical axis with the reflected light of the half mirror 2. In the detection device, the half mirror 2 is used to split light to realize the simultaneous measurement of an interferometer and a phase retrieval method; the plane mirror 4 reflects an emitted light wave to realize the return of the light wave.

(9) In one of the embodiments, the point light source 1 is a spherical interferometer that emits a spherical wave with a wavelength of 632.8 nm.

(10) In one of the embodiments, the image sensor 5 is a charge coupled device (CCD) camera, which is used to acquire a light spot image.

(11) A method for detecting a wavefront error by modal-based optimization phase retrieval using an ENZ theory, implemented by using the above detection device, includes the following steps (as shown in FIG. 2):

(12) S1: Emit, by a point light source 1, a spherical wave; collimate, by a lens 3 to be tested, the spherical wave; reflect, by a plane mirror 4 and a half mirror 2, the spherical wave; acquire, by an image sensor 5, a defocus intensity image with a wavefront error of the lens 3 to be tested.

(13) S2: Perform phase retrieval on the defocus intensity image acquired by step S1 by modal-based optimization phase retrieval using an ENZ theory to obtain the wavefront error of the tested lens.

(14) S2.1: Use I obtained in step S1 as an initial value of the defocus intensity image for a subsequent iteration in an ENZ mode; set an initial defocus position z0 of the image sensor 5; set a total number K of iterations to remove a cross term and an initial number k=1 of iterations, a total number L of iterations for a defocus position and an initial number l=1 of iterations, an aperture of the lens 3 to be tested, an initial value of a cross term of an ENZ diffraction mode and a number of terms in a Zernike polynomial.

(15) S2.2: Calculate each element of a modal gradient matrix of a system of equations for solving a coefficient of the Zernike polynomial, to obtain a modal gradient matrix V of the system of equations for solving a coefficient of the Zernike polynomial:

(16) $V=.Math.\begin{array}{cccccc}4{.Math.V_0^0\left(1,1\right).Math.}^2& {\mathrm{Hc}}_1^1\left(1,1\right)& {\mathrm{Gc}}_1^1\left(1,1\right)& {\mathrm{Hs}}_1^1\left(1,1\right)& {\mathrm{Gs}}_1^1\left(1,1\right)& \mathrm{.Math.}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \ddots \\ 4{.Math.V_0^0\left(M,1\right).Math.}^2& {\mathrm{Hc}}_1^1\left(M,1\right)& {\mathrm{Gc}}_1^1\left(1,N\right)& {\mathrm{Hs}}_1^1\left(1,N\right)& {\mathrm{Gs}}_1^1\left(1,N\right)& \mathrm{.Math.}\\ 4{.Math.V_0^0\left(2,1\right).Math.}^2& {\mathrm{Hc}}_1^1\left(2,1\right)& {\mathrm{Gc}}_1^1\left(2,1\right)& {\mathrm{Hs}}_1^1\left(2,1\right)& {\mathrm{Gs}}_1^1\left(2,1\right)& \mathrm{.Math.}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \ddots \\ 4{.Math.V_0^0\left(M,N\right).Math.}^2& {\mathrm{Hc}}_1^1\left(M,N\right)& {\mathrm{Gc}}_1^1\left(M,N\right)& {\mathrm{Hs}}_1^1\left(M,N\right)& {\mathrm{Gs}}_1^1\left(M,N\right)& \mathrm{.Math.}\end{array}\begin{array}{ccccccccc}{\mathrm{Hc}}_n^m\left(1,1\right)& {\mathrm{Gc}}_n^m\left(1,1\right)& {\mathrm{Hs}}_n^m\left(1,1\right)& {\mathrm{Gs}}_n^m\left(1,1\right)& \mathrm{.Math.}& {\mathrm{Hc}}_N^M\left(1,1\right)& {\mathrm{Gc}}_N^M\left(1,1\right)& {\mathrm{Hs}}_N^M\left(1,1\right)& {\mathrm{Gs}}_N^M\left(1,1\right)\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \ddots & \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ {\mathrm{Hc}}_n^m\left(1,N\right)& {\mathrm{Gc}}_n^m\left(1,N\right)& {\mathrm{Hs}}_n^m\left(1,N\right)& {\mathrm{Gs}}_n^m\left(1,N\right)& \mathrm{.Math.}& {\mathrm{Hc}}_N^M\left(1,N\right)& {\mathrm{Gc}}_N^M\left(1,N\right)& {\mathrm{Hs}}_N^M\left(1,N\right)& {\mathrm{Gs}}_N^M\left(1,N\right)\\ {\mathrm{Hc}}_n^m\left(2,1\right)& {\mathrm{Gc}}_n^m\left(2,1\right)& {\mathrm{Hs}}_n^m\left(2,1\right)& {\mathrm{Gs}}_n^m\left(2,1\right)& \mathrm{.Math.}& {\mathrm{Hc}}_N^M\left(2,1\right)& {\mathrm{Gc}}_N^M\left(2,1\right)& {\mathrm{Hs}}_N^M\left(2,1\right)& {\mathrm{Gs}}_N^M\left(2,1\right)\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \ddots & \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ {\mathrm{Hc}}_n^m\left(M,N\right)& {\mathrm{Gc}}_n^m\left(M,N\right)& {\mathrm{Hs}}_n^m\left(M,N\right)& {\mathrm{Gs}}_n^m\left(M,N\right)& \mathrm{.Math.}& {\mathrm{Hc}}_N^M\left(M,N\right)& {\mathrm{Gc}}_N^M\left(M,N\right)& {\mathrm{Hs}}_N^M\left(M,N\right)& {\mathrm{Gs}}_N^M\left(M,N\right)\end{array}.Math.$$\mathrm{where}$${\mathrm{Hc}}_n^m\left(m,n\right)=8\mathrm{Re}\left[i^m{V}_n^m\left(x_{\left(m,n\right)},y_{\left(m,n\right)},f\right)V_0^{0*}\left(x_{\left(m,n\right)},y_{\left(m,n\right)},f\right)\right]\cos \left[m\phi \left(x_{\left(m,n\right)},y_{\left(m,n\right)}\right)\right],{\mathrm{Gc}}_n^m\left(m,n\right)=-8\mathrm{Im}\left[i^m{V}_n^m\left(x_{\left(m,n\right)},y_{\left(m,n\right)},f\right)V_0^{0*}\left(x_{\left(m,n\right)},y_{\left(m,n\right)},f\right)\right]\cos \left[m\phi \left(x_{\left(m,n\right)},y_{\left(m,n\right)}\right)\right],{\mathrm{Hs}}_n^m\left(m,n\right)=-8\mathrm{Re}\left[i^m{V}_n^m\left(x_{\left(m,n\right)},y_{\left(m,n\right)},f\right)V_0^{0*}\left(x_{\left(m,n\right)},y_{\left(m,n\right)},f\right)\right]\sin \left[m\phi \left(x_{\left(m,n\right)},y_{\left(m,n\right)}\right)\right],{\mathrm{Gs}}_n^m\left(m,n\right)=-8\mathrm{Im}\left[i^m{V}_n^m\left(x_{\left(m,n\right)},y_{\left(m,n\right)},f\right)V_0^{0*}\left(x_{\left(m,n\right)},y_{\left(m,n\right)},f\right)\right]\cos \left[m\phi \left(x_{\left(m,n\right)},y_{\left(m,n\right)}\right)\right],$

(17) where, (x.sub.(m,n), y.sub.(m,n)) represents coordinates of pixel (m,n) in an image plane; f represents a defocus distance; V.sub.n.sup.m(.Math.) represents a kernel of the ENZ theory; Hc.sub.n.sup.m(m,n), Gc.sub.n.sup.m(m,n), Hs.sub.n.sup.m(m,n) and Gs.sub.n.sup.m(m,n) are intermediate variables; i is an imaginary number; mφ(x.sub.(m,n), y.sub.(m,n)) is an azimuthal frequency of pixel (m,n) in the image plane; φ(x.sub.(m,n), y.sub.(m,n)) represents angles in polar coordinates of image coordinates (x.sub.(m,n), y.sub.(m,n)); Re represents taking a real part; Im represents taking an imaginary part; N and M represent a radial order and an azimuthal frequency of the Zernike polynomial, respectively.

(18) S2.3: Remove an overexposed pixel of initial defocus intensity image I.sub.m′ to obtain I.sup.(k), and remove an element in a corresponding row of modal gradient matrix V.sub.mm.

(19) S2.4: Calculate a coefficient matrix A of the Zernike polynomial by using a least squares method according to the defocus intensity image and the modal gradient matrix with the overexposed pixel removed.
I.sup.(k)=V×A

(20) S2.5: Calculate cross term I.sub.c of the ENZ diffraction mode according to the Zernike coefficient matrix A.

(21) $I_c\left(x,y\right)=4\underset{n1,m1;n2,m2}{.Math.}\mathrm{\text{'}\text{'}}\mathrm{Re}\left[{\beta }_{n_1}^{m_1}{\beta }_{n_2}^{m_{2^{*}}}{i}^{m_1-m_2}{V}_{n_1}^{m_1}{V}_{n_2}^{m_{2^{*}}}\right]\cos m_1\mathrm{\phi cos}{m}_2\phi \hspace{1em}\hspace{1em}4\underset{n1,m1;n2,m2}{.Math.}\mathrm{\text{'}\text{'}}\mathrm{Im}\left[{\beta }_{n_1}^{m_1}{\beta }_{n_2}^{m_{2^{*}}}{i}^{m_1-m_2}{V}_{n_1}^{m_1}{V}_{n_2}^{m_{2^{*}}}\right]\sin m_1\mathrm{\phi cos}{m}_2\phi \hspace{1em}4\underset{n1,m1;n2,m2}{.Math.}\mathrm{\text{'}\text{'}}\mathrm{Re}\left[{\beta }_{n_1}^{m_1}{\beta }_{n_2}^{m_{2^{*}}}{i}^{m_1-m_2}{V}_{n_1}^{m_1}{V}_{n_2}^{m_{2^{*}}}\right]\cos m_1\mathrm{\phi sin}{m}_2\phi \underset{n1,m1;n2,m2}{4.Math.}\mathrm{\text{'}\text{'}}\mathrm{Im}\left[{\beta }_{n_1}^{m_1}{\beta }_{n_2}^{m_{2^{*}}}{i}^{m_1-m_2}{V}_{n_1}^{m_1}{V}_{n_2}^{m_{2^{*}}}\right]\sin m_1{\phi \mathrm{sion}m}_2\phi$

(22) where, β.sub.n.sup.m represents an element of the Zernike coefficient matrix A.

(23) S2.6: Remove cross term I.sub.c from initial defocus intensity image I.sub.m′.
I=I.sub.m′−I.sub.c

(24) S2.7: Determine whether the number k of iterations is greater than K; if not, let k=k+1, use the defocus intensity image with the cross term removed as an initial defocus intensity image, and return to step S2.3; if yes, proceed to the next step.

(25) S2.8: Determine whether the number l of iterations is greater than L; if not, calculate a correction value Δf of the defocus distance according to

(26) $\Delta f=\frac{{\beta }_{\mathrm{power}}\lambda }{\pi \left(1-\sqrt{1-N{A}^2}\right)},$
subtract the correction value of the defocus distance from a current value thereof as a defocus distance for a subsequent iteration, take the defocus intensity image obtained in step S1 as an initial defocus intensity image, let the number of iterations for the defocus position be l=l+1, let the number of iterations to remove the cross term be k=1, and return to step S2.2, where, β.sub.power is a coefficient of a fourth term of the Zernike polynomial, λ represents a wavelength of incident light, and N represents the number of pixels in a lateral direction of the defocus intensity image; if yes, use the obtained Zernike coefficient matrix to fit the wavefront phase error of the lens (3) to be tested, and calculate a true defocus position U of the image sensor (5) by

(27) $U=\exp \left(i\times \underset{n,m}{.Math.}{\beta }_n^m\times Z_n^m\right)$
according to the Zernike coefficient matrix, where, β.sub.n.sup.m represents a Zernike coefficient in the Zernike coefficient matrix, Z.sub.n.sup.m represents the Zernike polynomial, and i is an imaginary number.

(28) The embodiments of the present disclosure are described above with reference to the accompanying drawings, but the present disclosure is not limited to the above specific embodiments. The above specific embodiments are merely illustrative and not restrictive. Those of ordinary skill in the art may make several modifications to the present disclosure without departing from the purpose of the present disclosure and the scope of protection of the claims, but these modifications should all fall within the protection of the present disclosure.