The point spread function (PSF) of wavefront coding system with a rectangular pupil has been theoretically analyzed and numerically simulated by our proposed method based on the stationary phase method [Opt. Express 15, 1543 (2007)]. This method is extended to a cubic phase wavefront coding system with a circular pupil, which has rarely been studied in space domain. The approximated analytical representation of the PSF is deduced and boundaries of the focused PSF are proved to form an isosceles right triangle. The analysis indicates that the PSF is affected by the absolute value but not the sign of the defocus aberration. Defocus leads to the alteration of PSF in four aspects including position shift, boundary expansion, boundary deformation and oscillation frequency. Defocus also influenced the decoded image and caused position shift and image blurring. However, the influences introduced by defocus can be ignored when the defocus is very small compared to the cubic parameter. The similarities and differences of the PSF between the rectangular pupil system and the circular pupil system are discussed. The present method is helpful to analyze and design wavefront coding systems with a circular pupil.
© 2012 OSA
Wavefront coding  is a hybrid optical-digital technology, consisting of optical encoding and digital decoding. Digital decoding, which makes up for the deficiencies introduced by optical design, leads to the increase of the degrees of freedom in optical design. Consequently, this technology provides new opportunities for optical instruments, which cannot be achieved based on traditional optics design methods.
The depth of field extending is one of classic examples based on wavefront coding. With a special phase mask added on the pupil, the system is insensitive to defocus. The defocus invariant PSF (point spread function) is used as the deconvolution kernel in digital decoding to obtain sharp images. It is apparent that choosing an optimal phase mask is a key process for the design of wavefront coding system. Although there are many kinds of phase masks [1–3], cubic phase mask is one of most widely used types.
Most analysis of wavefront coding has been carried out within the systems with a rectangular pupil [1, 4, 5] due to the simplicity of the analysis in either frequency or space domain. The rectangular pupil assumption allows the double integral involved in the analysis to be written as the product of two separate 1D integrals. However, most of pupils in optical systems are circular instead of rectangular. Thereby, those analysis results cannot accurately reflect the imaging characteristics of most wavefront coding systems. Bagheri et al  investigated the modulation transfer function (MTF) of the circular pupil system. However, they just focused on the axial properties of MTF and did not give the analysis of the entire frequency plane. In our previous paper , the stationary phase method was proposed to analyze the PSF of the wavefront coding system with a rectangular pupil and the obtained results were agreed well with those obtained by Fast Fourier Transform approach. In this paper, the PSF analysis method is extended to the circular pupil in space domain. The approximated representation of the PSF in a circular pupil wavefront coding system is deducted based on the stationary phase method. The defocus influence on PSF and decoded images are discussed in detail. The similarities and differences of the PSFs between the rectangular and circular pupil systems are given.
2. Approximated PSF with a circular pupil
2.1 Defocused PSF
The defocused PSF of an incoherent wavefront coding system can be written as
Figure 1(a) displays the PSF representation when and, which is a piecewise function because of different stationary points exist in different PSF regions. Please refer to Appendix A for details. Figure 1(b) gives the grey-scale map of the approximated PSF. Sampling frequency used here is . Logarithm function of the PSF is used in Fig. 1(b) for the clarification. The black represents zero, while white represents the maximum value of the PSF. The discontinuities of the PSF at the boundaries are due to the use of the stationary method and will not affect the analysis of imaging characteristics . The boundaries are derived from Eq. (4). Notice that the defocused PSF does not depend on the sign but on the absolute value of the defocus aberration.
2.2 Focused PSF
Consider focused PSF, i.e. . Then Eq. (3) is simplified asEq. (7), three boundaries are written as
They form an isosceles right triangle shown in the red lines in Fig. 2 . The right-angle sides are located on and axes, while the hypotenuse is located at the first quadrant. The length of right-angle sides are .
3. PSF analysis
According to Eq. (3) and Eq. (7), the defocused and the focused PSFs can be divided into six and two regions, respectively. The boundaries of defocused PSF are described in Eq. (4) when equalities hold. Black lines in Fig. 2 represent boundaries of the defocused PSF. Notice that the outer boundaries are determined by Eq. (4a) and Eq. (4e), as shown in black bold lines in Fig. 2. Considering Eq. (4e) and Eq. (5), we obtain
Equation (9) can be further simplified as , where
The differences between the defocused PSF and the focused PSF can be found by comparing Eq. (8) and Eq. (12). Firstly, defocus leads to position shift in both and directions. The offset is proportional to the square of the defocus aberration, and inversely proportional to the cubic parameter. Secondly, the PSF’s boundary expands as increases. The increase in length of right angle side is about . Thirdly, the defocus bends the hypotenuse of the boundaries and introduces boundary deformation.
Finally, the oscillations occur in most regions of the PSF, as shown in Eq. (3). The oscillation frequencies and depend on and , respectively. Combining Eq. (3), Eq. (5) and Eq. (6), oscillation frequencies can be written as
Thus, the frequency of oscillation increases as increases.
4. Defocus influences on decoded image
Defocus can also influence decoded images. The imaging process is usually described as the convolution of object and the PSF . Notice that convolution in space domain is the equivalent to multiplication in frequency domain. Therefore, the imaging process in frequency domain can be described as Eq. (14), where represents image, and represents Fourier transform
Here we use as the deconvolution kernel in digital decoding. The solution of Eq. (14) is
Using the translation property of Fourier transform , we obtain
Using the translation property of Fourier transform again, the decoded image is
Therefore, the defocus introduces position shift. The offset is related to the defocus aberrations used in optical encoding and digital decoding, and inversely proportional to the cubic parameter.
Furthermore, the defocused decoded PSF is a disc of confusion instead of an impulse, which leads to image blurring. This is attributed to the differences between the two PSFs used in encoding and decoding, such as boundary and oscillation frequency alteration introduced by defocus. Figure 3 gives the decoded PSFs by using cubic parameter and different defocus aberrations and . Here, Wiener filter  is used as the decoding algorithm and the focused PSF as the deconvolution kernel. It is apparent that the defocus results in position shift and image blurring, which is consistent with the analysis above.
However, all of the influences can be ignored if the defocus aberration is very small compared to the cubic parameter. Then Eq. (3) is simplified as Eq. (7) when . The details are displayed in Appendix B. In this way, the PSF is defocus invariant, i.e. the system is insensitive to defocus. One deconvolution kernel is efficient and effective in digital decoding to obtain sharp images.
5. Circular pupil and rectangular pupil
We further focus on the rectangular pupil system and the circular pupil system. The approximated PSF for the rectangular pupil system  is described as
Take and . The schematic diagram of the defocused PSF is shown in Fig. 4(a) and the grey-scale map of the defocused PSF is shown in Fig. 4(b). Similarly, the logarithm function of the PSF is used here for the clarity. Sampling frequency used here is . Notice that the boundaries of the PSF are determined by Eq. (22) and shown in Fig. 4(c). The black lines represent the boundaries of the defocused PSF. The black bold lines form the closed outer boundary, i.e.
The boundaries of the focused PSF are also given as a contrast, as shown in red lines in Fig. 4(c).
Three similarities between rectangular pupil system and circular pupil system can be clearly observed. Firstly, the defocus leads to position shift along and directions. The offset is . Secondly, the defocus leads to the boundary expansion and increase in the length of side. Thirdly, the oscillation frequencies are the same as described in Eq. (13). This can be explained that both oscillation frequencies of the PSFs depend on and , as shown in Eq. (3) and Eq. (21). There are also three differences between rectangular pupil system and circular pupil system. The first one is the shapes of the PSF’s boundaries. The rectangular pupil leads to a rectangular boundary, while the circular pupil does an isosceles right triangle boundary, which is obvious when comparing Fig. 2 and Fig. 4. The second one is the piece number of the PSF function. We compare Eq. (3) and Eq. (21) and find that the latter has six pieces while the former has five ones because of no region containing three stationary points in the former. Thirdly, the PSF boundaries remain rectangular even if in the rectangular pupil system. In contrast, the defocus aberration causes the boundary deformation in the circular pupil system.
A method based on the stationary phase method is proposed to analyze the point spread function (PSF) of a circular pupil wavefront coding system and an approximated analytical expression of the defocused PSF was obtained. The absolute value of the defocus aberration affects the PSF and leads to the alteration of PSF in four aspects including position shift, boundary expansion, boundary deformation and oscillation frequency. The defocus also influenced the decoded image and caused position shift and image blurring. However, the influences on either PSF or decoded image can be omitted when the defocus aberration is much smaller than the cubic parameter. Three similarities (including position shift, boundary expansion and oscillation frequency) and three differences (including boundary shape, pieces number and boundary deformation) of PSF between the rectangular pupil system and the circular pupil system were detailed analyzed. In this way, the image characteristics of a cubic phase wavefront coding system with a circular pupil have been revealed clearly. It is helpful to analyze and design circular pupil wavefront coding systems.
Appendix A: Deduction of the defocused PSF in a circular pupil wavefront coding system
The PSF function described in Eq. (1) is rewritten as
where . The first and the second derivations of are
According to the stationary phase method, stationary points exist if and only if Eq. (A3) is satisfied.
Thus , and . Possible solutions of Eq. (A3a) are , , and , where
Figure A1(a) uses color to distinguish regions containing different stationary points in plane when . All of the nonzero regions are located at the first quadrant because . Consider positive defocus aberration, i.e. first. The red region, , has 4 stationary points, , , and . The yellow region, , has 3 stationary points, , and . The rose region, , has 2 stationary points, and . The gray region, , has 2 stationary points, and . The cyan region, , has only 1 stationary point, . If the defocus aberration is negative, i.e. , then the subscripts, “01” and “02”, change places with each other in the text above. With Eq. (A4) substituted in Eq. (A6), the distribution of the stationary points in plane is obtained and shown in Fig. A1(b).
where are the stationary points existing in the area shown in Fig. A1(a).
Substitute stationary points’ value in Eq. (A7) and the PSF is described as
where, , , .
Consider and rewrite Eq. (5) as
It is apparent that can be omitted when . Therefore, , , and can be approximated to be one set , i.e.
is described as
In other word, boundaries of the defocused PSF are simplified as Eq. (8) when .
This material is based upon work funded by Zhejiang Provincial Natural Science Foundation of China under Grant No. Y1110455, Scientific Research Fund of Zhejiang Provincial Education Department under Grant No. Y200909691, and Science Foundation of Zhejiang Sci-Tech University (ZSTU) under Grant No. 0913849-Y.
References and links
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