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, and so on [1–6].

R. A. Gonsalves [6] proposed an odd–even function decomposition method that can be used to provide evidence for the solution to the phase retrieval problem. Based on this idea, Min Li et al. 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 was used in the calculation process. But it was impractical due to the point-to-point calculations. Its calculative quantity was huge and it was too sensitive to noise [7].

Generally, a phase can be accurately represented by a number of orthogonal bases such as Zernike polynomials. With the Hartmann–Shack wavefront sensor [8], there is a linear relationship between the Zernike polynomials and the measure parameters; that is, a=R·m·a is the vector of Zernike polynomials, R is the linear matrix, and m is the vector of the measure parameter. This linear phase reconstruction is favorable for actual application in wavefront sensing.

This paper will prove that the method offered by Min Li et al. can also realize the Zernike modal phase retrieval with a single far-field image. Though some of the literature [9] presupposes a linear relationship between the phase and the difference of far-field intensities and has validated this relation by numerical emulations and experiments, this paper will first test the linear relation in theory. The principle to deduce some equations about this linear phase retrieval method is introduced in Section 2. A numerical simulation is performed to test the performance of the linear phase retrieval method on atmosphere-disturbed aberrations with and without measurement noise. The numerical results are discussed in Section 3.

2. Theory

The principle of linear phase retrieval is shown in Fig. 1 [7]. It is performed on a conventional imaging system. An ideal beam source is used first to calibrate inherent aberration B of the imaging system itself. This calibrated far-field image intensity P serves as the standard.

 

Fig. 1. Principal diagram of the linear phase retrieval method.

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

where W_e and W_o are the even and odd parts of W, respectively. It is based on the theory of odd–even function decompositions. Any function f(x,y) can be decomposed uniquely into the sum of an even and an odd function:

where $f_o\left(x,y\right)=\genfrac{}{}{0.1ex}{}{f\left(x,y\right)-f(-x,-y)}{2},f_e\left(x,y\right)=\genfrac{}{}{0.1ex}{}{f\left(x,y\right)+f(-x,-y)}{2}.$.

In the same way, the vector–matrix description of an image f(x,y) can be decomposed as

in which f, f_o, and f_e are the vector formats of functions f(x, y), f_o(x, y), and f_e(x, y); and J_xy is the transform matrix to make the x-direction and the y-direction flip over a matrix.

We also break inherent phase aberration B into two parts: B=B _e+B_o. It is proved in [7] that if B_e≫B_o or B_o=0 and W and B are small, the odd part of disturbed phase W is proportional to the difference of odd part of intensities P_Bo and P_o:

The even part of an aberration is approximately proportional to the difference of the even part of intensities P_Be and P_e,

in which P is the calibrated far-field image intensity, P=P_e+P_o, and P_B is the far-field intensity with inherent system aberration and measuring aberrations P_B=P_Be+P_Bo.

The relationships can be described in a vector–matrix format:

ŵ_o and ŵ_e are the vector formats of Ŵ_o and Ŵ_e. R _o is the response matrix between phase aberration ŵ_o and intensity Δ _o ^o, which is the vector format of ΔI ^o; ΔI ^o=P_Bo-P_o; and R_e is the response matrix between phase aberration ŵ_e and intensity Δ_e, which is the vector format of ΔI_e, ΔI_e=P_Be-P_e.

Then the phase aberration can be retrieved as

in which R is the linear retrieval matrix between phase aberration ŵ and image difference Δ, Δ=Δ_o+Δ_e. Therefore, there is an inverse linear relationship between them:

in which R ⁺ is the pseudo inverse of matrix R and describes the linear matrix between them as well.

We use the Zernike circle polynomials, which were defined by Noll [9], as the orthonormal basis functions. So, the phase can be described as the sum of several Zernike modes :

in which P is the Zernike number, c_i is the coefficient, and M_i(x, y) is a description of the Zernike polynomials. Or, in matrix format,

in which D is the response matrix between w and Zernike polynomial c, and c is the vector format of c_i. Equation (9) becomes

It is proved that there is a linear relationship between the change of Zernike coefficients and the changes of intensities, too. The matrix Z represents the relationship between them. Matrix Z can be obtained formally according to the parameters of the imaging system. Then we can reconstruct the Zernike coefficients by

in which Z ⁺ is the pseudo inverse of matrix Z. The wavefront also can be reconstructed.

From Eqs. (12) and (13), a linear relationship has been established between the intensities and the Zernike coefficients of the aberration phase, and the aberration phase can be retrieved from intensities by using a linear calculation algorithm. Equations (12) and (13) are the principle of linear phase retrieval in the vector–matrix format, which is a kind of linear calculation and is suitable for real-time calculation.

3. Numerical Simulations

Numerical simulations are employed to analyze the performance of this linear phase retrieval algorithm. First, the inherent system phase aberration characteristic is analyzed. Second, the dynamic range of this method is tested without noise to judge how small the aberration must be to satisfy the method. Last, the performance of the method is tested with different levels of measurement noise added to the far-field images.

A series of phase aberrations are generated to test the performance of this linear 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 are used [10]. The entrance pupil is a 64×64 grid and the far-field image plane is in 128×128 pixels. The far-field image is calculated using a fast Fourier transform (FFT) program. During the FFT process, the far field is expanded 7 times experientially, and only 128×128 pixels in the center area are used.

In order to compare the difference between the unknown phase W(x, y) and the estimated phase Ŵ(x, y), which are expressed by 65 Zernike polynomials, we define the error wavefront as

An error coefficient η, which is the ratio between the rms of the error wavefront E(x, y) and the unknown wavefront W(x, y), is used as one criterion to determine the validity of the 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 intensity of the Airy spot. If the SR_e is closer to 1, the performance of this method is better.

The performance of this linear phase retrieval algorithm depends on the inherent phase aberration of the imaging system. It is proved in [7] that most of the inherent phase aberration of the imaging system must be an even type, such as defocus, to keep the method working well. Here we have done some simulations to test the tolerance of the inherent system phase aberration. The results are shown in Fig. 2. The atmosphere-disturbed phase aberration is set as σ=0.4 rad and the main type of system aberration is defocus.

 

Fig. 2. Relationship between the error coefficient η and the proportion of coma to the total system aberration; the defocus is set as σ=4 rad.

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From Fig. 2, if there is some other odd aberration such as coma in the system, the conclusion is that when the proportion of coma to the total system aberration is smaller than 0.1, it has faint effect on the retrieval result.

Next we have analyzed the retrieval effect under different atmosphere-disturbed phase aberration levels. For every situation, 100 frame simulations have been performed. The system aberration is defocus and its rms is σ=4 rad. The results are shown in Fig. 3.

 

Fig. 3. Numerical simulation results of the linear phase retrieval method proposed in this paper without noise. (a) Relationship between the average Strehl ratio $\overline{\mathrm{SR}}$ of aberration and the average RMS of atmosphere disturbed aberration σ̄ ; (b) relationship between the average error coefficient η̄ and the average rms of atmosphere-disturbed aberration σ̄.

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It can be seen from the results that with the increase of phase aberration, the retrieval effects become worse. When σ̄≥1 rad, the average residual Strehl ratio $\overline{{\mathrm{SR}}_e}<0.85$ and the average error coefficient η̄>0.6. 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 SNR is defined as

where P_I 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 σ̄=0.4 rad whose initial average Strehl ratio is 0.84, different levels of noise are added to the far-field image, and 100 frame simulations are performed. The system aberration is defocus and its rms is σ=4 rad. The retrieval results are presented in Table 1.

Table 1. Results of phase retrieval with different SNRs

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When the SNR changes from infinity to 3, the ${\overline{\mathrm{SR}}}_e$ decreases from 0.985 to less than 0.97 and the η̄ increases from 0.331 to more than 0.5. It shows that the Zernike modal phase retrieval method is not sensitive to noise. It is still effective when SNR≥3.

Comparing the results of literature [7], which is valid when the rms of disturbed phase aberration σ<0.7 rad and the SNR>100, it is obvious that the range of applicability is extended by a Zernike modal-based approach. The anti-noise ability especially improved greatly. It is due to the modal method, which is more stable on the border of the aperture, and the singular value of the matrix is smaller. So, it is not sensitive to noise. Certainly the performance of this Zernike modal-based approach relates to the order of Zernike polynomials. With the increase of the order, the retrieval result becomes more exact. But by aiming at the 128×128 pixels image plane, 65 Zernike polynomials are enough to retrieve the phase exactly.

4. Conclusions

A linear phase retrieval method to retrieve small-phase aberrations from a single far-field image by a Zernike modal-based approach is proposed. It is a vector–matrix format, which is a kind of linear calculation and is suitable for real-time calculation. The difference between a single measured image with aberration and a calibrated image with inherent aberration is used in the calculation process. The results of numerical simulations on atmosphere-disturbed phase aberrations show that the magnitude of the inherent phase aberration of an imaging system must be in a certain range, and it allows a fraction of system aberration to be an odd type. The proposed linear phase retrieval method works well when the rms of the phase error is less than 1 rad. It is also shown that the method is valid in a noise condition when the SNR≥3. Comparing the method discussed in literature [7], the Zernike modal-based approach enhances the performance, especially the tolerance to noise. So it is more realistic to do some actual experiments by using the linear phase retrieval approach.

In the future we will provide evidence of the performance of this method in detailed experiments. Because of its limited measuring range and linear calculation, this linear phase retrieval method will be more suitable for closed-loop adaptive optics systems.