CN103884681B - A kind of phase microscope formation method based on SHWS - Google Patents

Abstract

The invention discloses a kind of phase microscope formation method based on SHWS, comprising: (1) gathers Hartmanngram; (2) barycenter distribution is calculated; (3) wavefront slope is calculated; (4) Wave-front phase is rebuild; (5) reconstruction of three-dimensional tomography refractive index.The present invention effectively utilize SHWS to improve complexity built by hardware that the phase microscope based on interferometric method exists, reconstruction procedures is loaded down with trivial details, is easily subject to vibration effect thus introduces the problem of much noise; Compare with the experiment of existing method for reconstructing and show, enormously simplify its hardware, also simplify reconstruction procedures, and avoid being subject to vibration effect.

Description

A kind of phase microscope formation method based on SHWS

Technical field

The invention belongs to microscope imaging technical field, be specifically related to a kind of phase microscope formation method based on SHWS (Shack-HartmannWavefrontSensor, picogram Hartman wavefront detector).

Background technology

In recent years, along with the progress of science and technology, optical microscope is with higher spatial resolution, and larger imaging depth and the feasibility to active somatic cell lossless detection, become the focus of biologist and clinical research personnel's research.It has become requisite instrument in biology and medical research, is the important foundation that cytology and cell biology are set up and developed.At present in cell biology, except ordinary optical microscope, also have the multiple microscopes such as fluorescent microscope, laser confocal microscope, phase microscope.

Wherein, phase microscopy comprises phasecontrast microscope, Differential interference contrast microscope and quantitative phase microscope etc., and the phase delay that they are caused by measurement sample obtains visual biological structure.In biological cell, the phase delay that different index distribution will cause light wave different, the spatial diversity of refractive index is converted into the contrast of image by these microtechnics.And the refractive index contrast source that to be object intrinsic, be also an important biochemical parameter simultaneously, be directly proportional to the concentration of molecule.The absorption under visible light of most of biological cell can be ignored, but there is the difference of refractive index between its inner different organelle.Therefore, in biological study, refractive index has become a specific absorption rate better interior raw contrast source.But under normal conditions, the refractive index difference of active somatic cell is very faint, this just needs to carry out modulation to improve image contrast to light signal.

Usually, phasecontrast microscope adopts one to make the phase board of zero-frequency decay simultaneous phase shifting 90 ° carry out spatial filtering, by the light distribution in the phase structure conversion imaging plane of object.Differential interference contrast microscope adopts two wollaston prisms to make two-beam in the adjacent regions of different time through sample, because the thickness of sample is different with refractive index, cause two-beam generation optical path difference, thus the minute differences in sample is changed into the light and shade difference of image.These two kinds of microtechnics utilize interference technique to improve picture contrast, and the phase place change of the transmitted light caused by refractive index difference in cell converts intensity distributions to.But what these technology provided is not quantitative phase place variation diagram.Quantitative phase microscope, by modulating reference beam, makes to produce phase shift between sample beam and reference beam, then interferes two light beams and measure the small refractive index difference of sample.And the phase microscopy of development in recent years can be quantitative the phase delay that caused by sample of record.But phase delay is proportional to the product of refractive index and path, it is more generally the convolution of refractive index and optical system point spread function.Therefore, these technology only can provide the mean refractive index parameter of cell or the thickness of cell, and do not have the detailed three-dimensional structure of cell.

Recent years, on the basis of quantitative phase microtechnic, a lot of research group has developed the various new microscopy can measuring the distribution of biological cell three-dimensional refractive index, as tomography phase microscope, STPM (syntheticaperturetomographicphasemicroscopy, synthetic aperture chromatography phase microscope), DHM (digitalholographicmicroscopy, digital holographic microscope), OSH (opticalscanningholography, optical scanning holographic microscope) etc.The common policy of these microscopies is the modes adopting computed tomographic scanner multi-angle to gather absorption coefficient data for projection, the phase projection data of record different angles, thus the three-dimensional refractive index distribution reconstructing biological cell.These novel microscopies have the precision higher than conventional optical microscope and spatial resolution, and can obtain the 3-D quantitative structural information of sample.

But, traditional 3-D quantitative phase microscope system is built based on heterodyne Mach-Zehnder interferometer, temporal frequency modulation method is adopted to make reference light produce a fixing difference on the frequency relative to sample light wave, then from time dependent interference field, obtain the quantitative phase images of sample, thus calculate the refractive index of each position, then obtain the structural information of cell interior.Such method, makes computation process very complicated, in addition, vibrates the interference brought, can introduce huge noise, thus make measurement result inaccurate to measurement result.

Summary of the invention

For the above-mentioned technical matters existing for prior art, the invention provides a kind of phase microscope formation method based on SHWS, solve hardware construction complexity, process of reconstruction is loaded down with trivial details, and the problems such as noise are introduced in vibration.

Based on a phase microscope formation method of SHWS, comprise the steps:

(1) for not setting-out product and setting-out product two kinds of situations, utilize SHWS carry out detecting obtain two kinds of situations under Hartmanngram (picogram Hartmann figure) corresponding to each angle light;

(2) the barycenter distribution of each lens corresponding detection window in the Hartmanngram that each angle light is corresponding in SHWS front lens square formation is in above-mentioned two situations calculated according to described Hartmanngram;

(3) each angle light is calculated by the wavefront slope after each lens according to the distribution of described barycenter;

(4) utilize region process of iteration to estimate the Wave-front phase that each lens is introduced each angular light line imaging according to described wavefront slope, and then obtain the Wave-front phase square formation of corresponding each angle light;

(5) utilize FBP method (FilteredBackProjection, Inverse Projection after filtering) to carry out the reconstruction of three-dimension disclocation refractive index to sample according to described Wave-front phase square formation, obtain sample and respectively to cut into slices corresponding refractive index square formation.

In described step (2), calculate barycenter distribution according to following formula:

$\begin{array}{cc}{X}_{m,n,\mathrm{\θ}}=\frac{\underset{i,j}{\overset{M,N}{\Σ}}{x}_{m,n,\mathrm{\θ}}(i,j)I_{m,n,\mathrm{\θ}}(i,j)}{\underset{i,j}{\overset{M,N}{\Σ}}{I}_{m,n,\mathrm{\θ}}(i,j)}& Y_{m,n,\mathrm{\θ}}=\frac{\underset{i,j}{\overset{M,N}{\Σ}}{y}_{m,n,\mathrm{\θ}}(i,j)I_{m,n,\mathrm{\θ}}(i,j)}{\underset{i,j}{\overset{M,N}{\Σ}}{I}_{m,n,\mathrm{\θ}}(i,j)}\end{array}$

Wherein: X _{m, n, θ}and Y _{m, n, θ}be respectively barycenter distribution cross stream component and the barycenter distribution longitudinal component of the capable n-th row lens of m in lens square formation corresponding detection window in the Hartmanngram that θ ° of light is corresponding, x _{m, n, θ}(i, j) and y _{m, n, θ}(i, j) is respectively the capable n-th row lens of m in lens the square formation horizontal ordinate of the i-th row jth row picture dot and ordinate in corresponding detection window in the Hartmanngram that θ ° of light is corresponding, I _{m, n, θ}(i, j) be the capable n-th row lens of m in lens square formation signal intensity of the i-th row jth row picture dot in corresponding detection window in the Hartmanngram that θ ° of light is corresponding, i, j, m and n are natural number and 1≤i≤M, 1≤j≤N, 1≤m≤L, 1≤n≤L, M and N is respectively line number and the columns of the corresponding detection window in the Hartmanngram that θ ° of light is corresponding of kth lens in lens square formation, L is the dimension of lens square formation, and θ is angle.

In described step (3), calculate wavefront slope according to following formula:

$\begin{array}{cc}{\mathrm{\βx}}_{m,n,\mathrm{\θ}}=\frac{1}{f}(X_{m,n,\mathrm{\θ}}^c-X_{m,n,\mathrm{\θ}}^r)& {\mathrm{\βy}}_{m,n,\mathrm{\θ}}=\frac{1}{f}(Y_{m,n,\mathrm{\θ}}^c-Y_{m,n,\mathrm{\θ}}^r)\end{array}$

Wherein: β x _{m, n, θ}with β y _{m, n, θ}be respectively cross stream component and the longitudinal component of the wavefront slope in θ ° of rays pass through lens square formation after the capable n-th row lens of m, with to be respectively in not setting-out product situation barycenter distribution cross stream component and the barycenter distribution longitudinal component of the corresponding detection window in the Hartmanngram that θ ° of light is corresponding of capable n-th row lens of m in lens square formation, with to be respectively in setting-out product situation barycenter distribution cross stream component and the barycenter distribution longitudinal component of the corresponding detection window in the Hartmanngram that θ ° of light is corresponding of capable n-th row lens of m in lens square formation, f is the focal length of SHWS and lens square formation, m and n is natural number and 1≤m≤L, 1≤n≤L, L is the dimension of lens square formation, and θ is angle.

In described step (4), region process of iteration is utilized to estimate Wave-front phase based on following formula:

$\begin{array}{c}{\mathrm{\ω}}_{m,n,\mathrm{\θ}}=\frac{1}{g_{m,n,\mathrm{\θ}}}\[{\mathrm{\ω}}_{m+1,n,\mathrm{\θ}}^{*}+{\mathrm{\ω}}_{m-1,n,\mathrm{\θ}}^{*}+{\mathrm{\ω}}_{m,n-1,\mathrm{\θ}}^{*}+{\mathrm{\ω}}_{m,n+1,\mathrm{\θ}}^{*}\\ +\frac{1}{2h}({\mathrm{\βy}}_{m,n+1,\mathrm{\θ}}-{\mathrm{\βy}}_{m,n-1,\mathrm{\θ}}+{\mathrm{\βx}}_{m+1,n,\mathrm{\θ}}-{\mathrm{\βx}}_{m-1,n,\mathrm{\θ}})\]\end{array}$

Wherein: ω _{m, n, θ}for the Wave-front phase that the capable n-th row lens of m in current iteration lens square formation are introduced θ ° of image formation by rays, for the Wave-front phase that the capable n-th row lens of m+1 in a front iteration lens square formation are introduced θ ° of image formation by rays, for the Wave-front phase that the capable n-th row lens of m-1 in a front iteration lens square formation are introduced θ ° of image formation by rays, for the Wave-front phase that the capable (n-1)th row lens of m in a front iteration lens square formation are introduced θ ° of image formation by rays, for the Wave-front phase that the capable (n+1)th row lens of m in a front iteration lens square formation are introduced θ ° of image formation by rays, β y _{m, n+1, θ}for the longitudinal component of the wavefront slope after the capable (n+1)th row lens of m in θ ° of rays pass through lens square formation, β y _{m, n-1, θ}for the longitudinal component of the wavefront slope after the capable (n-1)th row lens of m in θ ° of rays pass through lens square formation, β x _{m+1, n, θ}for the cross stream component of the wavefront slope after the capable n-th row lens of m+1 in θ ° of rays pass through lens square formation, β x _{m-1, n, θ}for the cross stream component of the wavefront slope after the capable n-th row lens of m-1 in θ ° of rays pass through lens square formation, h is the diameter of lens, m and n is natural number and 1≤m≤L, 1≤n≤L, L is the dimension of lens square formation, and θ is angle.

The present invention effectively utilize SHWS to improve complexity built by hardware that the phase microscope based on interferometric method exists, reconstruction procedures is loaded down with trivial details, is easily subject to vibration effect thus introduces the problem of much noise; Compare with the experiment of existing method for reconstructing and show, enormously simplify its hardware, also simplify reconstruction procedures, and avoid being subject to vibration effect.

Accompanying drawing explanation

Fig. 1 is the steps flow chart schematic diagram of the inventive method.

Fig. 2 is the optical system schematic diagram based on the imaging of SHWS phase microscope.

Fig. 3 (a) is for adopting the microscopical reconstructed results of conventional phase.

Fig. 3 (b) the present invention is based on the reconstructed results of SHWS phase microscope for adopting.

Embodiment

In order to more specifically describe the present invention, below in conjunction with the drawings and the specific embodiments, technical scheme of the present invention is described in detail.

As shown in Figure 1, a kind of phase microscope formation method based on SHWS, comprises the steps:

(1) for not setting-out product and setting-out product two kinds of situations, utilize SHWS carry out detecting obtain two kinds of situations under Hartmanngram corresponding to each angle light;

As shown in Figure 2, helium neon laser beam He-Ne (λ=633nm) first incides on scanning galvanometer GM.Then, successively through lens L1 (f=200mm), after oil immersion collector lens CL (Nikon, NA=1.4), be irradiated on sample.4f optical system is formed between GM and sample.Sample is immersed in the solution of certain refractive index, and is positioned between two cover glasses, is separated by and seals between two cover glasses with plastic spacer.Light beam through enter after sample a high-NA oil immersion objective OL (Nikon, 100 ×, NA=1.4) in, then image in SHWS via tube lens L2 (f=200mm).

(2) distribute based on the barycenter of each lens corresponding detection window in the Hartmanngram that each angle light is corresponding in following formulae discovery in above-mentioned two situations SHWS front lens square formation according to Hartmanngram:

$\begin{array}{cc}{X}_{m,n,\mathrm{\θ}}=\frac{\underset{i,j}{\overset{M,N}{\Σ}}{x}_{m,n,\mathrm{\θ}}(i,j)I_{m,n,\mathrm{\θ}}(i,j)}{\underset{i,j}{\overset{M,N}{\Σ}}{I}_{m,n,\mathrm{\θ}}(i,j)}& Y_{m,n,\mathrm{\θ}}=\frac{\underset{i,j}{\overset{M,N}{\Σ}}{y}_{m,n,\mathrm{\θ}}(i,j)I_{m,n,\mathrm{\θ}}(i,j)}{\underset{i,j}{\overset{M,N}{\Σ}}{I}_{m,n,\mathrm{\θ}}(i,j)}\end{array}$

Wherein: X _{m, n, θ}and Y _{m, n, θ}be respectively barycenter distribution cross stream component and the barycenter distribution longitudinal component of the capable n-th row lens of m in lens square formation corresponding detection window in the Hartmanngram that θ ° of light is corresponding, x _{m, n, θ}(i, j) and y _{m, n, θ}(i, j) is respectively the capable n-th row lens of m in lens the square formation horizontal ordinate of the i-th row jth row picture dot and ordinate in corresponding detection window in the Hartmanngram that θ ° of light is corresponding, I _{m, n, θ}(i, j) be the capable n-th row lens of m in lens square formation signal intensity of the i-th row jth row picture dot in corresponding detection window in the Hartmanngram that θ ° of light is corresponding, i, j, m and n are natural number and 1≤i≤M, 1≤j≤N, 1≤m≤L, 1≤n≤L, M and N is respectively line number and the columns of the corresponding detection window in the Hartmanngram that θ ° of light is corresponding of kth lens in lens square formation, L is the dimension of lens square formation, and θ is angle.

(3) each angle light is gone out by the wavefront slope after each lens according to barycenter distribution based on following formulae discovery:

In described step (3), calculate wavefront slope according to following formula:

$\begin{array}{cc}{\mathrm{\βx}}_{m,n,\mathrm{\θ}}=\frac{1}{f}(X_{m,n,\mathrm{\θ}}^c-X_{m,n,\mathrm{\θ}}^r)& {\mathrm{\βy}}_{m,n,\mathrm{\θ}}=\frac{1}{f}(Y_{m,n,\mathrm{\θ}}^c-Y_{m,n,\mathrm{\θ}}^r)\end{array}$

Wherein: β x _{m, n, θ}with β y _{m, n, θ}be respectively cross stream component and the longitudinal component of the wavefront slope in θ ° of rays pass through lens square formation after the capable n-th row lens of m, with to be respectively in not setting-out product situation barycenter distribution cross stream component and the barycenter distribution longitudinal component of the corresponding detection window in the Hartmanngram that θ ° of light is corresponding of capable n-th row lens of m in lens square formation, with to be respectively in setting-out product situation barycenter distribution cross stream component and the barycenter distribution longitudinal component of the corresponding detection window in the Hartmanngram that θ ° of light is corresponding of capable n-th row lens of m in lens square formation, f is the focal length of SHWS and lens square formation.

(4) utilize region process of iteration to estimate based on following formula the Wave-front phase that each lens is introduced each angular light line imaging according to wavefront slope, and then obtain the Wave-front phase square formation of corresponding each angle light:

$\begin{array}{c}{\mathrm{\ω}}_{m,n,\mathrm{\θ}}=\frac{1}{g_{m,n,\mathrm{\θ}}}\[{\mathrm{\ω}}_{m+1,n,\mathrm{\θ}}^{*}+{\mathrm{\ω}}_{m-1,n,\mathrm{\θ}}^{*}+{\mathrm{\ω}}_{m,n-1,\mathrm{\θ}}^{*}+{\mathrm{\ω}}_{m,n+1,\mathrm{\θ}}^{*}\\ +\frac{1}{2h}({\mathrm{\βy}}_{m,n+1,\mathrm{\θ}}-{\mathrm{\βy}}_{m,n-1,\mathrm{\θ}}+{\mathrm{\βx}}_{m+1,n,\mathrm{\θ}}-{\mathrm{\βx}}_{m-1,n,\mathrm{\θ}})\]\end{array}$

Wherein: ω _{m, n, θ}for the Wave-front phase that the capable n-th row lens of m in current iteration lens square formation are introduced θ ° of image formation by rays, for the Wave-front phase that the capable n-th row lens of m+1 in a front iteration lens square formation are introduced θ ° of image formation by rays, for the Wave-front phase that the capable n-th row lens of m-1 in a front iteration lens square formation are introduced θ ° of image formation by rays, for the Wave-front phase that the capable (n-1)th row lens of m in a front iteration lens square formation are introduced θ ° of image formation by rays, for the Wave-front phase that the capable (n+1)th row lens of m in a front iteration lens square formation are introduced θ ° of image formation by rays, β y _{m, n+1, θ}for the longitudinal component of the wavefront slope after the capable (n+1)th row lens of m in θ ° of rays pass through lens square formation, β y _{m, n-1, θ}for the longitudinal component of the wavefront slope after the capable (n-1)th row lens of m in θ ° of rays pass through lens square formation, β x _{m+1, n, θ}for the cross stream component of the wavefront slope after the capable n-th row lens of m+1 in θ ° of rays pass through lens square formation, β x _{m-1, n, θ}for the cross stream component of the wavefront slope after the capable n-th row lens of m-1 in θ ° of rays pass through lens square formation, h is the diameter of lens.

(5) utilize FBP method to carry out the reconstruction of three-dimension disclocation refractive index to sample according to Wave-front phase square formation, obtain sample and respectively to cut into slices corresponding refractive index square formation; Three-dimensional refractive index is rebuild based on following system of equations:

$f(x,y)={\∫}_0^{\mathrm{\π}}q(s,\mathrm{\θ}){|}_{s=x\cos \mathrm{\θ}+ysin\mathrm{\θ}}d\mathrm{\θ}$

Wherein: q (s, θ) is for subpoint is in the value of this polar coordinate position, and f (x, y) is the index distribution after rebuilding.

In order to evaluate the image quality of chromatography phase microscope, we adopt real polystyrene microbeads to test as sample below.Be 4.5m by diameter, refractive index be 1.588 polystyrene microbeads to be immersed in refractive index be in the immersion oil of 1.518.Microballon center is placed on the focal plane of microcobjective.Incident beam scans between 67 to 67, gathers piece image, collect 200 width interference images altogether every 0.67.Select the reconstruction of the good 180 amplitude phase diagram pictures of wherein image quality for successive image, the change of its incident angle is roughly between 60 to 60.

Fig. 3 (a) is conventional phase microscope reconstructed results, Fig. 3 (b) is for present embodiment is based on the phase microscope reconstructed results of picogram Hartmann wave front sensor, and to the phase microscope of picogram Hartmann wave front sensor, we make use of calculation visualization technology and show.Can find out that the microscope after improvement not removing only background artifact and eliminates cell edges noise, resolution rises to about 0.01um by original about 0.5um.Extraordinaryly improve image quality.

Claims (4)

1., based on a phase microscope formation method of SHWS, comprise the steps:

(1) for not setting-out product and setting-out product two kinds of situations, utilize SHWS carry out detecting obtain two kinds of situations under each angle light corresponding Hartmanngram figure;

(2) the barycenter distribution of each lens corresponding detection window in the Hartmanngram figure that each angle light is corresponding in SHWS front lens square formation is in above-mentioned two situations calculated according to described Hartmanngram figure;

(3) each angle light is calculated by the wavefront slope after each lens according to the distribution of described barycenter;

(4) utilize region process of iteration to estimate the Wave-front phase that each lens is introduced each angular light line imaging according to described wavefront slope, and then obtain the Wave-front phase square formation of corresponding each angle light;

(5) utilize FBP method to carry out the reconstruction of three-dimension disclocation refractive index to sample according to described Wave-front phase square formation, obtain sample and respectively to cut into slices corresponding refractive index square formation.

2. phase microscope formation method according to claim 1, is characterized in that: in described step (2), calculates barycenter distribution according to following formula:

$\begin{array}{cc}{X}_{m,n,\mathrm{\θ}}=\frac{\underset{i,j}{\overset{M,N}{\Σ}}{x}_{m,n,\mathrm{\θ}}(i,j)I_{m,n,\mathrm{\θ}}(i,j)}{\underset{i,j}{\overset{M,N}{\Σ}}{I}_{m,n,\mathrm{\θ}}(i,j)}& Y_{m,n,\mathrm{\θ}}=\frac{\underset{i,j}{\overset{M,N}{\Σ}}{y}_{m,n,\mathrm{\θ}}(i,j)I_{m,n,\mathrm{\θ}}(i,j)}{\underset{i,j}{\overset{M,N}{\Σ}}{I}_{m,n,\mathrm{\θ}}(i,j)}\end{array}$

Wherein: X _{m, n, θ}and Y _{m, n, θ}be respectively barycenter distribution cross stream component and the barycenter distribution longitudinal component of the capable n-th row lens of m in lens square formation corresponding detection window in the Hartmanngram figure that θ ° of light is corresponding, x _{m, n, θ}(i, j) and y _{m, n, θ}(i, j) is respectively the capable n-th row lens of m in lens the square formation horizontal ordinate of the i-th row jth row picture dot and ordinate in corresponding detection window in the Hartmanngram figure that θ ° of light is corresponding, I _{m, n, θ}(i, j) be the capable n-th row lens of m in lens square formation signal intensity of the i-th row jth row picture dot in corresponding detection window in the Hartmanngram figure that θ ° of light is corresponding, i, j, m and n are natural number and 1≤i≤M, 1≤j≤N, 1≤m≤L, 1≤n≤L, M and N is respectively line number and the columns of the corresponding detection window in the Hartmanngram figure that θ ° of light is corresponding of kth lens in lens square formation, L is the dimension of lens square formation, and θ is angle.

3. phase microscope formation method according to claim 1, is characterized in that: in described step (3), calculates wavefront slope according to following formula:

$\begin{array}{cc}{\mathrm{\βx}}_{m,n,\mathrm{\θ}}=\frac{1}{f}(X_{m,n,\mathrm{\θ}}^c-X_{m,n,\mathrm{\θ}}^r)& {\mathrm{\βy}}_{m,n,\mathrm{\θ}}=\frac{1}{f}(Y_{m,n,\mathrm{\θ}}^c-Y_{m,n,\mathrm{\θ}}^r)\end{array}$

Wherein: β x _{m, n, θ}with β y _{m, n, θ}be respectively cross stream component and the longitudinal component of the wavefront slope in θ ° of rays pass through lens square formation after the capable n-th row lens of m, with to be respectively in not setting-out product situation barycenter distribution cross stream component and the barycenter distribution longitudinal component of the corresponding detection window in the Hartmanngram figure that θ ° of light is corresponding of capable n-th row lens of m in lens square formation, with to be respectively in setting-out product situation barycenter distribution cross stream component and the barycenter distribution longitudinal component of the corresponding detection window in the Hartmanngram figure that θ ° of light is corresponding of capable n-th row lens of m in lens square formation, f is the focal length of SHWS and lens square formation, m and n is natural number and 1≤m≤L, 1≤n≤L, L is the dimension of lens square formation, and θ is angle.

4. phase microscope formation method according to claim 1, is characterized in that: in described step (4), utilizes region process of iteration to estimate Wave-front phase based on following formula:

$\begin{array}{c}{\mathrm{\ω}}_{m,n,\mathrm{\θ}}=\frac{1}{g_{m,n,\mathrm{\θ}}}\[{\mathrm{\ω}}_{m+1,n,\mathrm{\θ}}^{*}+{\mathrm{\ω}}_{m-1,n,\mathrm{\θ}}^{*}+{\mathrm{\ω}}_{m,n-1,\mathrm{\θ}}^{*}+{\mathrm{\ω}}_{m,n+1,\mathrm{\θ}}^{*}\\ +\frac{1}{2h}\left({\mathrm{\βy}}_{m,n+1,\mathrm{\θ}}-{\mathrm{\βy}}_{m,n-1,\mathrm{\θ}}+{\mathrm{\βx}}_{m+1,n,\mathrm{\θ}}-{\mathrm{\βx}}_{m-1,n,\mathrm{\θ}}\right)\]\end{array}$

Wherein: ω _{m, n, θ}for the Wave-front phase that the capable n-th row lens of m in current iteration lens square formation are introduced θ ° of image formation by rays, for the Wave-front phase that the capable n-th row lens of m+1 in a front iteration lens square formation are introduced θ ° of image formation by rays, for the Wave-front phase that the capable n-th row lens of m-1 in a front iteration lens square formation are introduced θ ° of image formation by rays, for the Wave-front phase that the capable (n-1)th row lens of m in a front iteration lens square formation are introduced θ ° of image formation by rays, for the Wave-front phase that the capable (n+1)th row lens of m in a front iteration lens square formation are introduced θ ° of image formation by rays, β y _{m, n+1, θ}for the longitudinal component of the wavefront slope after the capable (n+1)th row lens of m in θ ° of rays pass through lens square formation, β y _{m, n-1, θ}for the longitudinal component of the wavefront slope after the capable (n-1)th row lens of m in θ ° of rays pass through lens square formation, β x _{m+1, n, θ}for the cross stream component of the wavefront slope after the capable n-th row lens of m+1 in θ ° of rays pass through lens square formation, β x _{m-1, n, θ}for the cross stream component of the wavefront slope after the capable n-th row lens of m-1 in θ ° of rays pass through lens square formation, h is the diameter of lens, m and n is natural number and 1≤m≤L, 1≤n≤L, L is the dimension of lens square formation, and θ is angle.