CN104331857A - Phase position difference iteration compensation method in light intensity transmission equation phase retrieval - Google Patents

Abstract

The invention discloses a phase position difference iteration compensation method in light intensity transmission equation phase retrieval. The method comprises the steps of solving a light intensity transmission equation through a known light intensity axial differential and a light intensity distribution on a focusing surface by using a quick FFT (Fast Fourier Transform), so as to obtain a non exact solution; substituting the non-exact solution into the right end of the light path transmission equation to obtain a manually calculated light intensity axial differential, and taking an experimental data difference obtained by the manually calculated light intensity axial differential and actual measurement as an error function of the light intensity axial differential; taking the error function of the light intensity axial differential and the light intensity distribution on the focusing surface as an input so as to resolve the light intensity transmission equation by reusing the quick FFT, so as to obtain a phase error compensation item; and finally, adding the phase error compensation item and an original imprecision solution to finish a turn of iteration compensation, and performing the iteration to meet a final condition. According to the phase position difference iteration compensation method provided by the invention, the recovery precision of the light intensity transmission equation can be improved.

Description

Phase difference iteration compensation method in light intensity transmission equation phase recovery

Technical field

The invention belongs to the phase recovery in optical measurement and quantitative phase imaging technique, the phase difference iteration compensation method particularly in a kind of light intensity transmission equation phase recovery.

Background technology

Phase recovery is an important topic of optical measurement and imaging technique, and no matter in biomedical or field of industry detection, phase imaging technology is all playing an important role.Make a general survey of the progress of optical measurement nearly half a century, the most classical Method for Phase Difference Measurement should not belong to by non-interfering mensuration.But the shortcoming of interferometry is also fairly obvious: interferometry generally needs the light source (as laser) of high coherence, thus need comparatively complicated interference device; The introducing of extra reference path causes the requirement for measurement environment to become very harsh; The speckle coherent noise of the light source introducing of high coherence limits spatial resolution and the measuring accuracy of imaging system.

Difference and interferometry, another kind of very important phase measurement does not need by interference, and they are referred to as phase recovery.Because the PHASE DISTRIBUTION directly measuring light wave fields is very difficult, and the amplitude/intensity measuring light wave fields is very easy.Therefore, can be thought of as being recovered by intensity distributions this process of (estimation) phase place one mathematical " inverse problem ", i.e. phase retrieval problem.Phase recovery method also can be subdivided into process of iteration and direct method.Light intensity transmission equation method is the typical direct method of one in phase recovery method.Light intensity transmission equation is a Some Second Order Elliptic partial differential equation, that illustrates the quantitative relationship of the phase place of light wave in the variable quantity of light intensity is vertical with optical axis on optical axis direction plane.When the axial differential of light intensity and light distribution known, directly can obtain phase information by numerical solution light intensity transmission equation.Compare and interferometric method and iterative phase restoring method, its major advantage comprises: (1) non-interfering, only by measurement object plane light intensity direct solution phase information, does not need to introduce additional reference light; (2) non-iterative, obtains phase place by the direct solution differential equation; (3) can well white-light illuminating be applied to, as in traditional light field microscope kohler's illumination ( ); (4) without the need to Phase-un-wrapping, directly obtain the absolute profile of phase place, there are not 2 π phase place parcel problems in general interferometry; (5) optical system that need not be complicated, does not have harsh requirement for experimental situation, vibrates insensitive.

For solving of light intensity transmission equation, existing many methods propose at present: as Green Function Method (M.Reed Teague, " Deterministic phase retrieval:a Green's function solution, " J.Opt.Soc.Am.73, 1434-1441 (1983) .), the zernike polynomial method of development (T.E.Gureyev and K.A.Nugent, " Phase retrieval with the transport-of-intensity equation.II.Orthogonal series solution for nonuniform illumination, " J. Opt.Soc.Am.A 13, 1670-1682 (1996) .), Fast Fourier Transform (FFT) method (L.J.Allen and M.P.Oxley, " Phase retrieval from series of images obtained by defocus variation, " Opt Commun 199, 65-75 (2001)).But, solve in the algorithm of light intensity transmission equation at these, often need to introduce Teague auxiliary function, thus Poisson equation simplification light intensity transmission equation being converted to two standards solves.But Teague auxiliary function implies a stronger hypothesis, namely light intensity can flow field be laterally a conservation law.Generally this hypothesis being false, so Teague auxiliary function inherently causes solve error (being called phase difference), cause traditional method for solving can not provide the exact solution (J.A.Schmalz of light intensity transmission equation, T.E.Gureyev, D.M.Paganin, and K.M.Pavlov, " Phase retrieval using radiation and matter-wave fields:Validity of Teague's method for solution of the transport-of-intensity equation, " Phys Rev A 84, 023808 (2011)).So far still do not propose for the effective solution of this problem.If do not adopted an effective measure for this problem, greatly will affect the accuracy that light intensity transmission equation method solves phase place, make it be difficult to be applied in high-precision phase measurement and quantitative phase imaging field.

Comparatively detailed background introduction is carried out below by for this problem.

Consider that a monochrome propagated along z-axis is concerned with paraxial light wave fields, its complex amplitude expression formula U (r) is wherein j is imaginary unit, the PHASE DISTRIBUTION of φ (r) for recovering.Light intensity transmission equation can be expressed as

$-k\frac{\∂I\left(r\right)}{\∂z}=\▿\·[I\left(r\right)\▿\mathrm{\φ}\left(r\right)]---\left(1\right)$

Wherein k is wave number 2 π/λ, λ is adopted optical wavelength, and r is lateral attitude coordinate (x, y).I (r), for focusing on light distribution, thinks that it is positioned at the plane of z=0 without loss of generality.▽ is the Hamiltonian operator acting on r plane, for the axial differential signal of light intensity.

Teague supposes existence auxiliary function ψ, and it meets

I(r)▽φ(r)＝▽ψ(r) (2)

Such light intensity transmission equation just can be reduced to Poisson equation (the M.Reed Teague of following two standards, " Deterministic phase retrieval:a Green's function solution; " J.Opt.Soc.Am.73,1434-1441 (1983))

$\frac{\∂I\left(r\right)}{\∂z}=-\frac{1}{k}{\▿}^2\mathrm{\ψ}\left(r\right)---\left(3\right)$

With

▽·[I(r) ^-1▽ψ(r)]＝▽ ²φ(r) (4)

Thus the Solve problems of light intensity transmission equation can be converted into the problem solving above two Poisson equations.

The essence of Teague hypothesis thinks that I (r) ▽ φ (r) is a conservation law, it can be expressed as the gradient of ψ (r), but due to I (r) ▽ φ (r) just common two-dimentional scalar field itself, obviously its differ be decided to be conservative.Can be obtained by Helmholtz's decomposition theorem, any one bivector field can be decomposed into the gradient of a scalar potential ψ and the curl of a vector potential η

I(r)▽φ(r)＝▽ψ(r)+▽×η(r) (5)

Compared to formula (2) can find out curl item ▽ × η (r) by Teague suppose have ignored, this adds an implicit hypothesis virtually, and namely I (r) ▽ φ (r) is irrotationality.This hypothesis is obviously not necessarily set up.By getting divergence or curl respectively to formula both sides, two Poisson equations about scalar potential ψ and vector potential η can be obtained

▽ ²ψ(r)＝▽[I(r)▽φ(r)] (6)

With

▽ ²η(r)＝I(r)×▽φ(r) (7)

Here character ▽ × (I ▽ φ)=▽ I × ▽ φ is applied.

The impact that uncared-for curl item brings Phase Build Out is discussed below.I (r) ▽ φ (r) (formula (5)) after decomposing is substituted in light intensity transmission equation (formula (1)), one and formula (3) identical Poisson equation can be obtained.This is because the divergence of curl item ▽ × η (r) own is 0.So scalar potential ψ itself can be solved out accurately.But when getting divergence to both sides again after the I (r) of formula (5) being moved on to right side, second following Poisson equation can be obtained

▽ ²φ(r)＝▽·[I(r) ^-1▽ψ(r)]+▽I(r) ^-1×▽η(r) (8)

Here make use of character ▽ (I ^-1▽ × η)=▽ I ^-1× ▽ η.Obvious formula (8) is different from formula (2).Note under suitable boundary condition, the solution of these two Poisson equations is all unique (or unique to any additivity constant).When formula (8) that this just means that and if only if is completely the same with the right term of formula (2), their Xie Caihui is identical.Therefore can obtain adopting Teague hypothesis can obtain the necessary and sufficient condition of exact solution:

▽I(r) ^-1×▽η(r)＝0 (9)

From formula (7), function η (r) depends on the value of I (r) and φ (r), and it can be expressed as

η(r)＝▽ ^-2[I(r)×▽φ(r)] (10)

Substituted in formula (9), the necessary and sufficient condition obtaining exact solution can be re-expressed as

▽I(r) ^-1×▽ ^-2{▽·[▽I(r)×▽φ(r)]}＝0 (11)

The some special cases being rich in clear and definite physical significance making above formula set up being discussed: first two situations the simplest comprise uniform intensity distribution (now no longer needing Teague to suppose) once below, is a constant (physically nonsensical) with phase place.Another interesting special case is when ▽ I (r) × ▽ φ (r)=0, and this is also the sufficient and necessary condition (because η (r) itself is defined by formula (10)) that Teague supposes to set up.This means that ▽ I (r) must be parallel with ▽ φ (r) everywhere, namely for arbitrary scalar function c (r), have ▽ φ (r)=c (r) ▽ I (r) to set up.The form of further hypothesis c (r) is γ I ^-1r (), thus ▽ φ (r)=γ ▽ lnI (r), can obtain φ (r)=γ lnI (r) like this.A kind of expression-form of this inherently lambert Bill absorption law, which show light intensity transmitance and becomes logarithm attenuation relation with Media density or light path.So for the transparent objects of a typical single dielectric material, due to the correlativity of light intensity and phasetophase itself, can be similar to and think that ▽ I (r) and ▽ φ (r) are almost parallel.But must emphasize, for common situation (or reflecting object), given one arbitrary light intensity and PHASE DISTRIBUTION, the condition expressed by formula (11) is generally ungratified.So employing Teague approximate solution light intensity transmission equation itself is the exact solution that can not get light intensity transmission equation.Claim this error for " phase difference " here, and make it equal true phase distribution and the difference adopting Teague to suppose the out of true solution of trying to achieve.If do not adopted an effective measure for this " phase difference ", greatly will affect the accuracy that light intensity transmission equation method solves phase place, make it be difficult to be applied in high-precision phase measurement and quantitative phase imaging field.

Summary of the invention

The object of the present invention is to provide the phase difference iteration compensation method in a kind of light intensity transmission equation phase recovery, improve the precision of light intensity transmission equation phase recovery.

The technical solution realizing the object of the invention is: the phase difference iteration compensation method in a kind of light intensity transmission equation phase recovery, and step is as follows:

The first step, by the axial differential of known light intensity, with the light distribution on focusing surface, utilizes Fast Fourier Transform (FFT) method to solve light intensity transmission equation, obtains an inaccurate, namely by the axial differential of known light intensity with light distribution I (r) on focusing surface, by formula (12), adopting Fast Fourier Transform (FFT) to solve light intensity transmission equation, to obtain phase place be an inaccurate φ _nr (), now n=0, represent iterations:

$\mathrm{\φ}(x,y)=-k{\▿}^{-2}\▿\·[I^{-1}(x,y)\▿{\▿}^{-2}\frac{\∂I(x,y)}{\∂z}]---\left(12\right)$

▽ in formula ^-2be inverse Laplace's operation symbol, ▽ is gradient operator, and be vector dot, k is wave number, ▽ and ▽ ^-2operational symbol is all realized by Fourier transform, namely

${\▿}^{-2}\{\·\}=F^{-1}\left\{F\right\{\·\left\}\frac{1}{-{4\mathrm{\π}}^2(u^2+v^2)}\right\}---\left(13\right)$

▽{·}＝F ^-1{i2πuF{·},i2πvF{·}} (14)

Wherein F represents Fourier transform, and (u, v) is the frequency domain coordinates corresponding with volume coordinate (x, y), and i is imaginary unit;

Second step, by inaccurate φ _nr () substitutes into the right-hand member of light intensity transmission equation again, obtain the axial differential of the artificial light intensity calculated, and the experimental data that axial for the light intensity artificially calculated differential and actual measurement obtain is made the error function of difference as the axial differential of light intensity;

3rd step, by the error function of axial for light intensity differential, with the light distribution on focusing surface, re-uses Fast Fourier Transform (FFT) method as input and solves light intensity transmission equation, obtain phase error compensation item Δ φ _n(r);

4th step, by phase error compensation item Δ φ _n(r) and last inaccurate φ _nr () is added, this completes the iterative compensation of one bout, and the second to the four step is incited somebody to action always ceaselessly iteration and performed, until till meeting end condition.

The present invention compared with prior art, its remarkable advantage: the phase difference that the present invention is directed to caused by Teague hypothesis proposes simply effective iteration compensation method, solve the difficult problem that light intensity transmission equation is difficult to obtain exact solution, it is by the precision for improving light intensity transmission equation phase recovery, and it plays positive role in the application in quantitative phase imaging, the field such as micro-.In addition iteration converges of the present invention is very fast, generally only need 2-5 iteration just to can converge to exact solution, and only need to adopt fast discrete cosine transform to carry out Numerical Implementation, calculate simple, memory space is little, is very suitable for the application scenario high to requirement of real-time.

Below in conjunction with attached figurethe present invention is described in further detail.

Accompanying drawing explanation

figure1 is the flow process of phase difference iteration compensation method in light intensity transmission equation phase recovery of the present invention figure.

figure2 (a) is the emulation light intensity function defined by formula (21).

figure2 (b) is the simulation bit function defined by formula (22).

figure3 (a) is that fast fourier transform algorithm solves the original phase error (RMSE 9.53%) obtained.

figure3 (b) is through the distribution of the error after an iterative compensation (RMSE 0.99%).

figure3 (c) is through the distribution of the error after three iterative compensations (RMSE 0.016%).

figure3 (d) is the relation curve of RMSE and iterations.

figure4 (a) is that fast fourier transform algorithm solves the original phase error (RMSE10.64%) obtained under there is noise situations.

figure4 (b) is through the distribution of the error after an iterative compensation (RMSE 1.90%).

Embodiment

Phase difference iteration compensation method in light intensity transmission equation phase recovery of the present invention, performing step is as follows:

The first step, by the axial differential of known light intensity, with the light distribution on focusing surface, utilizes Fast Fourier Transform (FFT) method to solve light intensity transmission equation, obtains an inaccurate, namely by the axial differential of known light intensity with light distribution I (r) on focusing surface, by formula (12), adopt Fast Fourier Transform (FFT) to solve light intensity transmission equation, be similar to, so obtaining phase place is an inaccurate φ owing to being employed herein Teague _n(r) φ ₁r (), now n=0, represent iterations.

$\mathrm{\φ}(x,y)=-k{\▿}^{-2}\▿\·[I^{-1}(x,y)\▿{\▿}^{-2}\frac{\∂I(x,y)}{\∂z}]---\left(12\right)$

▽ in formula ^-2be inverse Laplace's operation symbol, ▽ is gradient operator, and be vector dot, k is wave number, ▽ and ▽ ^-2operational symbol is all realized by Fourier transform, namely

${\▿}^{-2}\{\·\}=F^{-1}\left\{F\right\{\·\left\}\frac{1}{-{4\mathrm{\π}}^2(u^2+v^2)}\right\}---\left(13\right)$

▽{·}＝F ^-1{i2πuF{·},i2πvF{·}} (14)

Wherein F represents Fourier transform, and (u, v) is the frequency domain coordinates corresponding with volume coordinate (x, y), and i is imaginary unit.

Second step, substitutes into the right-hand member of light intensity transmission equation again by inaccurate, obtain the axial differential of the artificial light intensity calculated, by inaccurate φ _nr () substitutes into the right-hand member of light intensity transmission equation (formula (15)) again

$\frac{\∂I\left(r\right)}{\∂z}=-\frac{1}{k}\▿\·[I\left(r\right)\▿\mathrm{\φ}\left(r\right)]---\left(15\right)$

Wherein k is wave number 2 π/λ, λ is adopted optical wavelength, by the axial differential of the light intensity of the left end calculated be referred to as the axial differential of the artificial light intensity calculated here, namely.

${\left(\frac{\∂I\left(r\right)}{\∂z}\right)}_n=-\frac{1}{k}\▿\·[I\left(r\right)\▿{\mathrm{\φ}}_n\left(r\right)]---\left(16\right)$

3rd step, the existence of the phase difference caused by Teague hypothesis, the axial differential of the artificial light intensity calculated the axial differential of the light intensity obtained with actual measurement result also certainly exists difference, the two is made the error function of difference as the axial differential of light intensity

$\mathrm{\Δ}{\left(\frac{\∂I\left(r\right)}{\∂z}\right)}_n={\left(\frac{\∂I\left(r\right)}{\∂z}\right)}_n-\frac{\∂I\left(r\right)}{\∂z}---\left(17\right)$

4th step, by the error function of axial for light intensity differential with light distribution I (r) on focusing surface, as inputting again through type (18), adopting Fast Fourier Transform (FFT) to solve light intensity transmission equation, obtaining phase error compensation item Δ φ _n(r)

$\mathrm{\Δ}{\mathrm{\φ}}_n\left(r\right)=-k{\▿}^{-2}\▿\·[I^{-1}\left(r\right)\▿{\▿}^{-2}\mathrm{\Δ}{\left(\frac{\∂I\left(r\right)}{\∂z}\right)}_n]---\left(18\right)$

▽ in formula ^-2be inverse Laplace's operation symbol, ▽ is gradient operator, and be vector dot, k is wave number, ▽ and ▽ ^-2operational symbol is all realized by Fourier transform, namely

${\▿}^{-2}\{\·\}=F^{-1}\left\{F\right\{\·\left\}\frac{1}{-{4\mathrm{\π}}^2(u^2+v^2)}\right\}---\left(19\right)$

▽{·}＝F ^-1{i2πuF{·},i2πvF{·}} (20)

Wherein F represents Fourier transform, and (u, v) is the frequency domain coordinates corresponding with volume coordinate (x, y), and i is imaginary unit.

5th step, phase error compensation item Δ φ _nthe inaccurate φ of (r) and previous step _nr () is added

φ _n+1(r)=φ _n(r)+Δ φ _nr () obtains the estimated value of a new round, this completes the iterative compensation of one bout, and make n ← n+1.This completes the iterative compensation of one bout.The second to the five step is incited somebody to action always ceaselessly iteration and is performed, until till meeting end condition.The end condition of iteration comprises, the error function of the axial differential of light intensity (recommended value is the axial differential of light intensity to be less than a given threshold value 1% of mean square value), phase error compensation item Δ φ _nr () is less than a given threshold value (recommended value is φ _n1% of mean square value), and exceed maximum iteration time (recommended value is 20 times).Three any one meet just can termination of iterations.Finally obtain the exact solution of light intensity transmission equation, PHASE DISTRIBUTION φ (r) namely to be asked.

Adopt a simulation example to verify the validity of the phase difference iteration compensation method in bright light intensity transmission equation phase recovery proposed by the invention below, in the simulation, light wave fields complex amplitude to be measured is defined within the pixel grid of 256 × 256, Pixel Dimensions is 0.025 μm × 0.025 μm, and wavelength is 632.8nm.As figure2 (a), figureshown in 2 (b), phase place and light intensity function be predefined into

I(x,y)＝exp(-a ₀x ²-b ₀y ²) (21)

With

φ(x,y)＝a ₀x ²-b ₀y ²-a ₁x ⁸+b ₁y ⁸(22)

Wherein a ₁=b ₂=0.25 ^-2× 10 ^-8μm ^-8, a ₁=b ₂=0.25 ^-2× 10 ^-8μm ^-8.

figure2 (a) is the light intensity function defined by formula (21). figure2 (b) is the phase function defined by formula (22).

Relative root-mean-square error (relative root-mean-square error, RMSE) is adopted to go to weigh objectively the precision of phase recovery.Before calculating RMSE, all phase places of trying to achieve all first deduct their piston item. figure3 (a) extremely figure3 (d) shows the simulation result of iterative compensation algorithm.

figurethe error that fast fourier transform algorithm solves the original phase obtained is shown in 3 (a), because the necessary and sufficient condition formula (11) that can obtain exact solution all cannot meet in most of position, so obviously can find out very significantly phase difference (RMSE 9.53%), this error level is unacceptable for quantitative phase is recovered.Through this section propose algorithm iterative compensation after error distribution as figureshown in 3 (b), its residual error obviously declines, and RMSE drops to 0.99% by 9.53%.After three iteration, residual error has been reduced to a negligible level (RMSE0.016%), as figureshown in 3 (c).? figurethe relation curve of RMSE and iterations is given in 3 (d).The speed of convergence that this shows the phase difference iteration compensation method in light intensity transmission equation phase recovery proposed by the invention is quickly.

In the measuring system based on light intensity transmission equation of reality, the intensity collected figurepicture is all containing noise.In order to test method proposed by the invention performance under noise, emulation is obtained adopting figurepicture with the addition of Gaussian reflectivity mirrors artificially, and the signal to noise ratio (S/N ratio) of the axial differential signal of the noisy light intensity obtained is about 10dB, and phase error not only derives from Teague hypothesis in the case, and the impact of noise on it also be can not ignore, as figureshown in 4 (a): fast fourier transform algorithm solves the original phase error (RMSE 10.64%) obtained.After phase difference iteration compensation method three iterative compensations in light intensity transmission equation phase recovery proposed by the invention, most phase difference is removed, its residual error as figureshown in 4 (b): error distribution (RMSE 1.90%) after an iterative compensation.Can know and find out that the present invention successfully eliminates significantly " arch " error because phase difference causes, only leave nebulous random noise, this is caused by added intensity noise.The validity of backoff algorithm under noise that proposes of above result verification.

Claims (6)

1. the phase difference iteration compensation method in light intensity transmission equation phase recovery, is characterized in that step is as follows:

The first step, by the axial differential of known light intensity, with the light distribution on focusing surface, utilizes Fast Fourier Transform (FFT) method to solve light intensity transmission equation, obtains an inaccurate, namely by the axial differential of known light intensity with light distribution I (r) on focusing surface, by formula (12), adopting Fast Fourier Transform (FFT) to solve light intensity transmission equation, to obtain phase place be an inaccurate φ _nr (), now n=0, represent iterations:

$\mathrm{\φ}(x,y)=-k{\▿}^{-2}\▿\·[I^{-1}(x,y)\▿{\▿}^{-2}\frac{\∂I(x,y)}{\∂z}]---\left(12\right)$

In formula inverse Laplace's operation symbol, for gradient operator, be vector dot, k is wave number, with operational symbol is all realized by Fourier transform, namely

${\▿}^{-2}\{\·\}=F^{-1}\left\{\begin{array}{cc}F& \{\·\}\frac{1}{-4{\mathrm{\π}}^2(u^2+v^2)}\end{array}\right\}---\left(13\right)$

$\▿\{\·\}=F^{-1}\left\{\begin{array}{ccc}i2\mathrm{\πuF}& \{\·\},i2\mathrm{\πvF}& \{\·\}\end{array}\right\}---\left(14\right)$

Wherein F represents Fourier transform, and (u, v) is the frequency domain coordinates corresponding with volume coordinate (x, y), and i is imaginary unit;

Second step, by inaccurate φ _nr () substitutes into the right-hand member of light intensity transmission equation again, obtain the axial differential of the artificial light intensity calculated, and the experimental data that axial for the light intensity artificially calculated differential and actual measurement obtain is made the error function of difference as the axial differential of light intensity;

3rd step, by the error function of axial for light intensity differential, with the light distribution on focusing surface, re-uses Fast Fourier Transform (FFT) method as input and solves light intensity transmission equation, obtain phase error compensation item Δ φ _n(r);

4th step, by phase error compensation item Δ φ _n(r) and last inaccurate φ _nr () is added, this completes the iterative compensation of one bout, and the second to the four step is incited somebody to action always ceaselessly iteration and performed, until till meeting end condition.

2. the phase difference iteration compensation method in light intensity transmission equation phase recovery according to claim 1, is characterized in that in second step, by inaccurate φ _nr () substitutes into light intensity transmission equation again, i.e. the right-hand member of formula (15)

$\frac{\∂I\left(r\right)}{\∂z}=-\frac{1}{k}\▿\·[I\left(r\right)\▿\mathrm{\φ}\left(r\right)]---\left(15\right)$

Wherein k is wave number 2 π/λ, λ is adopted optical wavelength, by the axial differential of the light intensity of the left end calculated be referred to as the axial differential of the artificial light intensity calculated, namely

${\left(\frac{\∂I\left(r\right)}{\∂z}\right)}_n=-\frac{1}{k}\▿\·[I\left(r\right)\▿{\mathrm{\φ}}_n\left(r\right)].---\left(16\right)$

3. the phase difference iteration compensation method in light intensity transmission equation phase recovery according to claim 1, is characterized in that in second step, the existence of the phase difference caused by Teague hypothesis, the axial differential of the artificial light intensity calculated the axial differential of the light intensity obtained with actual measurement result also certainly exists difference, the two is made the error function of difference as the axial differential of light intensity

$\mathrm{\Δ}{\left(\frac{\∂I\left(r\right)}{\∂z}\right)}_n={\left(\frac{\∂I\left(r\right)}{\∂z}\right)}_n-\frac{\∂I\left(r\right)}{\∂z}---\left(17\right)$

4. the phase difference iteration compensation method in light intensity transmission equation phase recovery according to claim 1, is characterized in that in the third step, by the error function of axial for light intensity differential with light distribution I (r) on focusing surface, as inputting again through type (18), adopting Fast Fourier Transform (FFT) to solve light intensity transmission equation, obtaining phase error compensation item Δ φ _n(r), namely

$\mathrm{\Δ}{\mathrm{\φ}}_n\left(r\right)=-k{\▿}^{-2}\▿\·[I^{-1}\left(r\right)\▿{\▿}^{-2}\mathrm{\Δ}{\left(\frac{\∂I\left(r\right)}{\∂z}\right)}_n]---\left(18\right)$

In formula inverse Laplace's operation symbol, for gradient operator, be vector dot, k is wave number, with operational symbol is all realized by Fourier transform, namely

${\▿}^{-2}\{\·\}=F^{-1}\left\{\begin{array}{cc}F& \{\·\}\frac{1}{-4{\mathrm{\π}}^2(u^2+v^2)}\end{array}\right\}---\left(19\right)$

$\▿\{\·\}=F^{-1}\left\{\begin{array}{ccc}i2\mathrm{\πuF}& \{\·\},i2\mathrm{\πvF}& \{\·\}\end{array}\right\}---\left(20\right)$

Wherein F represents Fourier transform, and (uv) is the frequency domain coordinates corresponding with volume coordinate (xy), and i is imaginary unit.

5. the phase difference iteration compensation method in light intensity transmission equation phase recovery according to claim 1, is characterized in that in the 4th step, phase error compensation item Δ φ _n(r) and inaccurate φ _nr () is added φ _n+1(r)=φ _n(r)+Δ φ _nr () obtains the estimated value of a new round, this completes the iterative compensation of one bout, and make n ← n+1, and the second to the four step is incited somebody to action always ceaselessly iteration and performed, until till meeting the end condition of iteration.

6. the phase difference iteration compensation method in light intensity transmission equation phase recovery according to claim 5, is characterized in that the end condition of iteration comprises the error function of the axial differential of light intensity be less than a given threshold value, phase error compensation item Δ φ _nr () is less than a given threshold value, and exceed maximum iteration time, three any one meet just can termination of iterations, finally obtain the exact solution of light intensity transmission equation, PHASE DISTRIBUTION φ (r) namely to be asked.