## Abstract

A method to retrieve small-phase aberrations from a single far-field image is proposed. It is found that in a small-phase condition, the odd and even parts of a phase aberration can be obtained with a simple linear calculation method. The difference between a single measured image with aberration and a calibrated image with inherent aberration is used in the calculation process. It is proved that most of the inherent phase aberration of the imaging system must be of an even type, such as defocus, astigmatism, etc., to keep the method working. The results of numerical simulations on atmosphere-disturbed phase aberrations show that the proposed small-phase retrieval method works well when the RMS phase error is less than 1 rad. It is also shown that the method is valid in a noise condition when the SNR>100.

© 2008 Optical Society of America

## 1. Introduction

The purpose of phase retrieval is to obtain the unknown phase from far-field image data. It has been proposed for a variety of applications and has received much attention over many years. Solutions to the problem include measuring the modulus in one or two conjugate domains, using a single intensity measurement plus a non-negativity constraint, using two defocused images, etc. [1–5]. There is an analytic method of solving the phase retrieval problem under the condition of small aberration proposed by R. A. Gonsalves [6]. The method requires capturing two far-field images simultaneously with one in focus and another out of focus. The disturbed phase can be found by using a method that characterizes odd and even components of the disturbed phase by decomposing the original intensity profile.

It is useful in a real-time system if the phase aberration can be obtained from only one far-field image instead of two or more simultaneous images. In this paper, we provide a novel solution to the small-phase retrieval problem using only one far-field image. The feasibility of this method and the uniqueness of the solution are analyzed. The method used here is compared to that proposed by R. A. Gonsalves [6].

The basic principal is introduced in Section 2 to deduce some equations about this small-phase retrieval method and the type of inherent system aberration is discussed. A numerical simulation is performed to test the performance of the method on atmosphere-disturbed aberrations with and without measurement noise. The numerical results are discussed in Section 3.

## 2. Principal of the small-phase retrieval method

The small-phase retrieval is performed on a conventional imaging system. A CCD camera is used to measure the far-field intensity profile of incoming phase aberration. There is always inherent aberration in the imaging system. An ideal beam source is used first to calibrate inherent aberration of the imaging system itself. This calibrated far-field image serves as the standard. When the same imaging system is used to measure a beam with small-phase aberration, the far-field image with aberration is obtained. With the help of odd- and even-function decompositions, the odd and even parts of phase aberration are obtained, respectively. The principal of the small-phase retrieval method is shown in Fig. 1.

It is possible that any function *f*(*x*,*y*) can be decomposed uniquely into the sum of an even and an odd function:

where ${f}_{o}(x,y)=\frac{f(x,y)-f(-x,-y)}{2}$ , ${f}_{e}(x,y)=\frac{f(x,y)+f(-x,-y)}{2}$ .

Assuming there is an inherent aberration in the imaging system

where A is an aperture function, which is a real even function in the circle aperture, A=1 in the aperture, and A=0 out of the aperture. B is the inherent phase aberration of the imaging system and assumed to be a small real function, which satisfies the approximate relationship exp(*iB*)≈1+*iB*. Then *S*≈*A*(1+*iB*).

If we break B into two parts:

where *B _{e}* and

*B*are the even and odd parts of

_{o}*B*, respectively, then

*S*=

*A*+

*iA*·

*B*+

_{e}*iA*·

*B*.

_{o}Let *Z*=*A*·*B _{e}*,

*T*=

*A*·

*B*, then

_{o}According to the principal of imaging systems, the complex optical field in the focal plane is *s*=*a*+*iz*+*it*, where *s*, *a*, *z*, and *t* are the Fourier transforms of *S*, *A*, *Z*, and *T*, respectively. Based on the Fourier transform theory, *a* and *z* are real and even, and *t* is imaginary and odd. We define *x*=*it* such that *x* is real and odd. Then *s* becomes *s*=*a*+*iz*+*x*. So the modulus squared of *s* is

where *P* is the far-field intensity with inherent system aberration. The even and odd parts of *P* are

After getting the calibrated image, the same imaging system is used to measure the beam with small disturbed phase W. We also break W into two parts:

where *W _{e}* and

*W*are the even and odd parts of

_{o}*W*, respectively. Assuming the RMS phase error of the phase aberration W is small, the term B+W is small, too. Then the complex optical field with aberration W and B may be approximated by

$$\phantom{\rule{.2em}{0ex}}=A+iA\xb7\left({B}_{e}+{B}_{o}+{W}_{e}+{W}_{o}\right)=A+iZ+iT+iV+iQ,$$

where *V*=*A*·*W _{e}*,

*Q*=

*A*·

*W*,

_{o}*V*is real and even, and

*Q*is real and odd.

A complex optical field in the focal plane is

where *a*,*z*,*t*,*v*, and *q* are the Fourier transforms of *A*,*Z*,*T*,*V*, and *Q*, respectively. From Fourier-transform theory, we know that *a*,*z*, and *v* are real and even, respectively, and *t* and *q* are imaginary and odd. We define *y*=*iq* such that *y* is real and odd. Then *h _{B}* becomes

The modulus squared of *h _{B}* is

$$\phantom{\rule{.2em}{0ex}}={a}^{2}+{x}^{2}+{y}^{2}+{z}^{2}+{v}^{2}+2\mathrm{xy}+2vz+2\mathrm{ax}+2\mathrm{ay},$$

where *P _{B}* is the far-field intensity with aberration. The even and odd parts of

*P*are

_{B}*W _{o}* should be calculated first. From Eq. (14),

Since *y* is real and odd, *Y*, the inverse Fourier-transform of *y*, is imaginary and odd. Using the previous definitions: *y*=*iq*, *Q*=*AW _{o}*, the odd part of the estimated aberration

*Ŵ*is obtained by

*W _{e}* is calculated secondly. Equation (13) becomes:

Supposing the unknown phase aberration *W* is small, and *v* and *y* are the Fourier-transforms of the unknown aberration *W*’s even part *W _{e}* and odd part

*W*, respectively, so

_{o}*v*and

*y*are small, too.

If most of the inherent phase aberration of the imaging system is odd, that is *B _{e}*≪

*B*,

_{o}*z*≪

*x*, we can solve this equation for

*v*by neglecting

*z*and the quadratic terms of

*y*, then Eq. (17) becomes:

It is not possible to get the unique *v* from Eq. (18).

If there is no phase aberration in the imaging aberration, that is *B*=0, *z*=0, *x*=0, then Eq. (17) becomes:

It is not possible to get the unique *v* from Eq. (19) either.

If most of the inherent phase aberration of the imaging system is even, that is, *B _{e}*≫

*B*,

_{o}*z*≫

*x*, we can solve this equation for

*v*by neglecting

*x*and the quadratic terms of

*v*and

*y*, then

Since *v* is real and even, then *V*, the inverse Fourier-transform of *v*, is also real and even. Based on the definition *V*=*A*·*W _{e}*, then the even part of the estimated aberration is obtained:

Then the estimated phase *Ŵ* can be obtained by *Ŵ*=*Ŵ _{e}*+

*Ŵ*.

_{o}In Eqs. (15) and (20), to avoid the zeros of *a* and *z*, we have replaced 1/*a* with *a*/(*a*
^{2}+*e*) and 1/*z* with *z*/(*z*
^{2}+*e*), respectively, where *e* is an appropriately small constant. The divisor A in Eqs. (16) and (21) is replaced by *A*/(*A*
^{2}+*e*) to avoid the zero of A, where *e* is an appropriate small constant. In each equation, the e may be equivalent or not.

From the above deduction, it is clear that only when most of the inherent phase aberration of the imaging system is even and the disturbed phase aberration is small, the disturbed phase aberration can be retrieved by intensity *P _{B}* and

*P*. Here

*P*is fixed and calibrated in advance. It only needs to measure intensity

*P*in real time, and then the estimated phase aberration can be obtained directly.

_{B}Equations (15) and (16), which derive the method for estimating the odd part of the disturbed phase aberration, are almost the same when compared with Gonsalves’ work. However, for the even part of the disturbed phase aberration, the method is different. The basic assumption of the even part underlying Gonsalves’ method is to have two image measurements simultaneously, one in focus and one out of focus. For this novel method, though, it only needs one far-field image with aberration and a known calibrated image with inherent aberration to formulate the even part of the phase aberration.

## 3. Numerical simulations

Computer simulations are employed to analyze the performance of this phase retrieval algorithm. First, the dynamic range of this method is discussed without noise to judge how small it must be to satisfy the method. Second, the performance of the method is tested with different levels of measurement noise added to far-field images.

A series of phase aberrations is generated to test the performance of this phase retrieval method. The method proposed by N. Roddier is used to generate the phase screen with the Kolmogorov atmosphere spectrum, and a total of 65 Zernike polynomials were used [7]. The entrance pupil is a 64×64 grid and the far-field imaging plane is 128×128 pixels. The far-field imaging is calculated using a fast Fourier transform (FFT) program. During the process of FFT, the far field has been expanded 7 times experientially, and only 128×128 pixels in the center area have been used. Because defocus is an even and familiar aberration, the inherent aberration of the imaging system is set as defocus with RMS *σ*=0.7 rad in this simulation. If the imaging system is not changed, the defocus is fixed and calibrated once to get its intensity profile P. In the simulation, S and *H _{B}* are calculated from

*S*=

*A*exp(

*iB*) and

*H*=

_{B}*A*exp[

*i*(

*B*+

*W*)], respectively.

In order to compare the difference between the unknown phase *W*(*x*,*y*) and the estimated phase *Ŵ*(*x*,*y*), we define the error wavefront as

An error coefficient *η*, the ratio between the RMSs of the error wavefront *E*(*x*,*y*) and the unknown wavefront *W*(*x*,*y*), is used as one criterion to determine the validity of phase retrieval method:

If *η*<1, the retrieval effect is valid.

Another metric is the residual Strehl ratio (*SR _{e}*), which is the ratio between the peak intensity of the far-field image produced by the error wavefront

*E*(

*x*,

*y*) and the maximum of intensity of the Airy spot. If the

*SR*is closer to 1, the performance of this method is better.

_{e}When the wavefront and the far-field intensity profile are reversed and puckered during the decomposition into an even part and an odd part, their brims produce invalid information. It is caused by the limited grid number. We neglect one pixel of the skirt of the wavefront by setting the value to 0. We have analyzed the retrieval effect under different atmosphere-disturbed phase aberration levels. For every situation, 100 frame simulations have been performed. The results are shown in Fig. 2.

It can be seen from the results that with the increase of phase aberration, the retrieval effects become worse. When *$\overline{\sigma}$*≥1 rad, the average residual Strehl ratio
$\overline{{\mathrm{SR}}_{e}}<0.78$ and the average error coefficient *$\overline{\eta}$*>0.65. So we can conclude under the conditions of this paper that the valid dynamic range of this method is approximately *σ*<1 rad.

To examine the applicability of this method in a noise condition, we investigate the sensitivity of the method to noise. Random noise, which satisfies the Gaussian distribution, is added to the imaging plane. All the values of noise are positive. The signal-to-noise ratio (SNR) is defined as:

where *P* is the peak value of the far-field image without noise, and *σ _{n}* is the RMS value of noise.

Because the mean value of noise is nonzero, a threshold has to be subtracted from the noisy image during data processing. In this paper, the threshold is denoted as:

After subtraction, if the intensity value of a pixel is negative, it is set to zero.

For an average wavefront aberration *σ*=0.38 rad, whose initial average Strehl ratio is 0.86, different levels of noise are added to the far-field image, and 100 frame simulations are performed. The retrieval results are presented in Table 1.

When the *SNR* changes from infinity to 100,
$\overline{{\mathrm{SR}}_{e}}$
decreases from 0.972 to less than 0.94, and *$\overline{\eta}$* increases from 0.468 to more than 0.75. It shows that noise influences the retrieval effect of the sensor as long as it exists. But the phase retrieval method in this paper is effective enough when SNR>100.

The detailed results for a random aberration of the last frame of 100 whose *σ*=0.467 rad and the initial Strehl ratio of aberration is 0.812 under the condition of SNR=∞ and SNR=120, respectively, are shown in Fig. 3: (a) is the initial disturbed wavefront; (b) and (c) are the even and odd parts of the initial disturbed wavefront; (d) is the retrieved wavefront with SNR=∞; (e) and (f) are the even and odd parts of the retrieved wavefront with SNR=∞; (g) is the retrieved wavefront with SNR=120; (h) and (i) are the even and odd parts of the retrieved wavefront with SNR=120. When SNR=∞, the residual Strehl ratio *SR _{e}* is 0.963, and the error coefficient

*η*is 0.471; when SNR=120, the residual Strehl ratio

*SR*is 0.945 and the error coefficient

_{e}*η*is 0.656.

## 4. Conclusions

A new method of phase retrieval is proposed to solve the small-phase retrieval problem from a single far-field image with aberration and a calibrated image with inherent aberration. A model is founded to obtain the odd and even parts of the random phase aberration, respectively, from the difference between the measured image with aberration and the calibrated image with inherent aberration. It is proved that most of the inherent phase aberration of an imaging system must be an even type, such as defocus, astigmatism, etc., to allow the method to work. Results of numerical simulations on atmosphere-disturbed phase aberrations imply that the novel phase retrieval method works well when the RMS<1 rad. It is also shown that the method can be done in a noise condition when the SNR>100.

Here we retrieve the phases one by one. In the future, we will do the phase retrieval with the Zernike polynomials mode. We hope the modal algorithm can enhance the performance of this method to some degree. We will analyze the dynamic range and the anti-noise ability of the modal method and study the type, magnitude, and tolerance of the inherent system phase aberration. Also, we will do some actual experiments to test its performance.

## Acknowledgments

This project is funded by the National Natural Science Foundation of China with contract 60408005. The authors thank Hao Xian, Changhui Rao, Shen Feng and Bincheng Li for their helpful comments and suggestions.

## References and links

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