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
Fig. 1. Principal diagram of the linear phase retrieval method.
It is possible that any function f(x,y) can be decomposed uniquely into the sum of an even and an odd function:
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 Be and Bo are the even and odd parts of B, respectively, then S=A+iA·Be+iA·Bo.
Let Z=A·Be, T=A·Bo, then
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 We and Wo are the even and odd parts of 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
where V=A·We, Q=A·Wo, 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 hB becomes
The modulus squared of hB is
where PB is the far-field intensity with aberration. The even and odd parts of PB are
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=AWo, the odd part of the estimated aberration Ŵ is obtained by
is calculated secondly. Equation (13
Supposing the unknown phase aberration W is small, and v and y are the Fourier-transforms of the unknown aberration W’s even part We and odd part Wo, respectively, so v and y are small, too.
If most of the inherent phase aberration of the imaging system is odd, that is Be
, we can solve this equation for v
by neglecting z
and the quadratic terms of y
, then Eq. (17
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, then Eq. (17
It is not possible to get the unique v
from Eq. (19
If most of the inherent phase aberration of the imaging system is even, that is, Be≫Bo, 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·We, then the even part of the estimated aberration is obtained:
Then the estimated phase Ŵ can be obtained by Ŵ=Ŵe+Ŵo.
In Eqs. (15
) and (20
), to avoid the zeros of a
, we have replaced 1/a
) and 1/z
), respectively, where e
is an appropriately small constant. The divisor A in Eqs. (16
) and (21
) is replaced by A
) 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 PB and P. Here P is fixed and calibrated in advance. It only needs to measure intensity PB in real time, and then the estimated phase aberration can be obtained directly.
) 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
7. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990). [CrossRef]
]. 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 HB
are calculated from S
) and HB
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 (SRe), 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 SRe is closer to 1, the performance of this method is better.
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
Fig. 2. Numerical simulation results of the phase retrieval method proposed in this paper without noise. (a) Relationship between the average Strehl ratio SR of aberration and the average RMS of atmosphere disturbed aberration σ; (b) relationship between the average error coefficient η and the average RMS of atmosphere disturbed aberration σ.
It can be seen from the results that with the increase of phase aberration, the retrieval effects become worse. When ≥1 rad, the average residual Strehl ratio
and the average error coefficient >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 σn 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
Table 1. Results of phase retrieval with different SNRs
When the SNR changes from infinity to 100,
decreases from 0.972 to less than 0.94, and 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 SRe
is 0.963, and the error coefficient η
is 0.471; when SNR=120, the residual Strehl ratio SRe
is 0.945 and the error coefficient η
Fig. 3. Detailed retrieved wavefront comparison under the conditions of with and without noise. (a) Initial disturbed wavefront; (b) (c) even and odd parts of the initial disturbed wavefront; (d) retrieved wavefront with SNR=∞; (e) (f) even and odd parts of the retrieved wavefront with SNR=∞; (g) retrieved wavefront with SNR=120; (h) (i) even and odd parts of the retrieved wavefront with SNR=120.