Point spread function analysis of a cubic phase wavefront coding system with a circular pupil

doi:10.1364/OE.20.002408

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Point spread function analysis of a cubic phase wavefront coding system with a circular pupil

Tingyu Zhao and Feihong Yu »View Author Affiliations

Optics Express, Vol. 20, Issue 3, pp. 2408-2419 (2012)
http://dx.doi.org/10.1364/OE.20.002408

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Abstract

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.

1. Introduction

Wavefront coding [1

E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]

] 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

E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]

–3

H. Zhao and Y. Li, “Optimized logarithmic phase masks used to generate defocus invariant modulation transfer function for wavefront coding system,” Opt. Lett. 35(15), 2630–2632 (2010). [CrossRef] [PubMed]

], 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

E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]

, 4

G. Muyo and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. 30(20), 2715–2717 (2005). [CrossRef] [PubMed]

, 5

W. Zhang, Z. Ye, T. Zhao, Y. Chen, and F. Yu, “Point spread function characteristics analysis of the wavefront coding system,” Opt. Express 15(4), 1543–1552 (2007), http://www.opticsinfobase.org/abstract.cfm?&id=127217. [CrossRef] [PubMed]

] 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 [6

S. Bagheri, P. E. Silveira, and D. P. de Farias, “Analytical optimal solution of the extension of the depth of field using cubic-phase wavefront coding. part I. reduced-complexity approximate representation of the modulation transfer function,” J. Opt. Soc. Am. A 25(5), 1051–1063 (2008). [CrossRef] [PubMed]

] 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 [5

], 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

h (x, y, W_{20})

of an incoherent wavefront coding system can be written as

h (x, y, W_{20}) = {| \iint_{u^{2} + v^{2} \leq 1} q (u, v) \exp [j k W_{20} (u^{2} + v^{2}) - j 2 π (x u + y v)] |}^{2},

(1)

where

k

is the wavenumber;

j

is the square root of −1;

W_{20}

is the defocus aberration in unit of wavelength;

x

and

y

are coordinates in image plane;

u

and

v

are normalized pupil coordinates; and

q (u, v)

is the normalized pupil function with a cubic phase mask and can be expressed as

q (u, v) = {\begin{cases} \frac{1}{2} \exp [j α (u^{3} + v^{3})], u^{2} + v^{2} \leq 1 \\ 0, otherwise \end{cases} .

(2)

Here

α

is the cubic parameter of the phase mask. Take

α > 0

as an example. Stationary phase method [7

M. Born and E. Wolf, Principles of Optics (Pergamon, 1985).

] is used to obtain the approximated PSF as shown in Eq. (3) and the details can be seen in Appendix A.

h (x, y, W_{20}) = {\begin{cases} \frac{1}{4} κ, (A, B) \in ({\bar{Ω}}_{1} \cap {\bar{Ω}}_{2} \cap {\bar{Ω}}_{3} \cap Ω_{4}) \\ \frac{1}{2} κ (1 + \sin η), (A, B) \in ({\bar{Ω}}_{1} \cap Ω_{2} \cap {\bar{Ω}}_{3} \cap Ω_{4}) \\ \frac{1}{2} κ (1 + \sin ξ), (A, B) \in ({\bar{Ω}}_{1} \cap {\bar{Ω}}_{2} \cap Ω_{3} \cap Ω_{4}) \\ \frac{1}{4} κ [(1 + \sin η) (1 + \sin ξ) + 2 + 2 \cos η \cos ξ], (A, B) \in ({\bar{Ω}}_{1} \cap Ω_{2} \cap Ω_{3} \cap Ω_{4}) \\ κ (1 + \sin η) (1 + \sin ξ), (A, B) \in (Ω_{1} \cap Ω_{2} \cap Ω_{3} \cap Ω_{4}) \\ 0, (A, B) \in ({\bar{Ω}}_{1} \cap {\bar{Ω}}_{2} \cap {\bar{Ω}}_{3} \cap {\bar{Ω}}_{4}) or x, y < \frac{- {(k W_{20})}^{2}}{6 π α} \end{cases},

(3)

where

{\begin{cases} Ω_{0} = {(x, y) | x \geq - \frac{{(k W_{20})}^{2}}{6 π α}, y \geq - \frac{{(k W_{20})}^{2}}{6 π α}} (a) \\ Ω_{1} = {(x, y) | {(A + | k W_{20} |)}^{2} + {(B + | k W_{20} |)}^{2} \leq 9 α^{2}} (b) \\ Ω_{2} = {(x, y) | {(A + | k W_{20} |)}^{2} + {(B - | k W_{20} |)}^{2} \leq 9 α^{2}} (c) \\ Ω_{3} = {(x, y) | {(A - | k W_{20} |)}^{2} + {(B + | k W_{20} |)}^{2} \leq 9 α^{2}} (d) \\ Ω_{4} = {(x, y) | {(A - | k W_{20} |)}^{2} + {(B - | k W_{20} |)}^{2} \leq 9 α^{2}} (e) \end{cases},

(4)

A = \sqrt{{(k W_{20})}^{2} + 6 π α x}, B = \sqrt{{(k W_{20})}^{2} + 6 π α y}

(5)

κ = π^{2} / (A B), η = 4 A^{3} / (27 α^{2}), ξ = 4 B^{3} / (27 α^{2}) .

(6)

Figure 1(a) displays the PSF representation when

α = 30 π

and

W_{20} = 5 λ

, 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

500 \times 500

. Logarithm function of the PSF

\log (h)

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 [5

]. 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.

Fig. 1 Approximated PSF of the wavefront coding system with a circular pupil. (a) Schematic diagram of the piecewise PSF function; (b) Grey-scale map of the log-PSF. Here

α = 30 π

and

W_{20} = 5 λ

2.2 Focused PSF

Consider focused PSF, i.e.

W_{20} = 0

. Then Eq. (3) is simplified as

h (x, y) = {\begin{cases} 0, x + y > \frac{3 α}{2 π} or x < 0 or y < 0 \\ κ' (1 + \sin η') (1 + \sin ξ'), otherwise \end{cases},

(7)

where,

κ' = π / (6 α \sqrt{x y})

η' = [4 {(6 π α x)}^{3 / 2}] / (27 α^{2})

ξ' = [4 {(6 π α y)}^{3 / 2}] / (27 α^{2})

. From Eq. (7), three boundaries are written as

x = 0, y = 0, x + y = \frac{3 α}{2 π} .

(8)

They form an isosceles right triangle shown in the red lines in Fig. 2 . The right-angle sides are located on

x

and

y

axes, while the hypotenuse is located at the first quadrant. The length of right-angle sides are

3 α / (2 π)

Fig. 2 Boundaries of the focused PSF (in red) and the defocused PSF (in black). Here

α = 30 π

, the defocus aberration

W_{20} = 5 λ

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

{[\frac{6 π α k W_{20} (x - y)}{6 π α (x + y) - 9 α^{2} + 4 {(k W_{20})}^{2}}]}^{2} + {[\frac{6 π α (x + y) - 9 α^{2}}{4 k W_{20}}]}^{2} = \frac{9}{2} α^{2} .

(9)

Equation (9) can be further simplified as

\cos^{2} θ + \sin^{2} θ = 1

, where

\cos θ = \frac{2 \sqrt{2} π | k W_{20} | (x - y)}{6 π α (x + y) - 9 α^{2} + 4 {(k W_{20})}^{2}}, \sin θ = \frac{2 \sqrt{2} π (x + y) - 3 \sqrt{2} α}{4 | k W_{20} |} .

(10)

The variable

\sin θ

depends on

x

and

y

and reaches its maximum value of 1 when

y = x

\sin θ

takes the minimum when equalities hold in Eqs. (4d), (4e) or Eqs. (4c), (4e). Consider Eq. (4d) and Eq. (4e) first. We obtain

x = - \frac{{(k W_{20})}^{2}}{6 π α}, y = \frac{9 α^{2} + 2 | k W_{20} | \sqrt{9 α^{2} - {(k W_{20})}^{2}} - {(k W_{20})}^{2}}{6 π α} .

(11)

Substitute Eq. (11) in Eq. (10), then

\sin θ = \frac{\sqrt{2}}{2} (\sqrt{1 - {(k W_{20})}^{2} / (9 α^{2})} - | k W_{20} | / (3 α))

. Thereby,

\sin θ \approx \sqrt{2} / 2

| k W_{20} | < < 3 α

. Similarly,

\sin θ \approx \sqrt{2} / 2

when equalities hold in Eqs. (4c), (4e). Thereby, the boundaries of defocused PSF are written as

x = - \frac{{(k W_{20})}^{2}}{6 π α}, y = - \frac{{(k W_{20})}^{2}}{6 π α}, x + y = \frac{\sqrt{2} | k W_{20} | \sin θ}{π} + \frac{3 α}{2 π} (\frac{\sqrt{2}}{2} \leq \sin θ \leq 1) .

(12)

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

- x

and

- y

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

| W_{20} |

increases. The increase in length of right angle side is about

| k W_{20} | / π + {(k W_{20})}^{2} / (6 π α)

. 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

f_{x}

and

f_{y}

depend on

η

and

ξ

, respectively. Combining Eq. (3), Eq. (5) and Eq. (6), oscillation frequencies can be written as

f_{x} = \frac{6 π α}{{(\frac{27 α^{2} π}{2})}^{\frac{2}{3}} - {(k W_{20})}^{2}}, f_{y} = \frac{6 π α}{{(\frac{27 α^{2} π}{2})}^{\frac{2}{3}} - {(k W_{20})}^{2}} .

(13)

Thus, the frequency of oscillation increases as

| W_{20} |

increases.

4. Defocus influences on decoded image

Defocus can also influence decoded images. The imaging process is usually described as the convolution of object

I

and the PSF

h (x, y, W_{20})

. 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

J

represents image, and

F

represents Fourier transform

F [J (x, y, W_{20})] = F [I (x, y)] \cdot F [h (x, y, W_{20})] .

(14)

Here we use

h (x, y, W_{20}')

as the deconvolution kernel in digital decoding. The solution of Eq. (14) is

I' (x, y, W_{20}) = F^{- 1} {\frac{F [I (x, y)] \cdot F [h (x, y, W_{20})]}{F [h (x, y, W_{20}')]}},

(15)

where

I'

represents decoded image and

F^{- 1}

represents inverse Fourier transform. Notice that

h (x, y, W_{20})

is used in optical encoding, while

h (x, y, W_{20}')

is used in digital decoding. When consider the position shift and ignore other effects, the defocused PSF is described as

h (x, y, W_{20}) = h [x + \frac{{(k W_{20})}^{2}}{6 π α}, y + \frac{{(k W_{20})}^{2}}{6 π α}, 0] .

(16)

Using the translation property of Fourier transform [8

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 6.

], we obtain

F [h (x + \frac{{(k W_{20})}^{2}}{6 π α}, y + \frac{{(k W_{20})}^{2}}{6 π α}, 0)] = F [h (x, y, 0)] \exp [j 2 π (ν_{x} \frac{{(k W_{20})}^{2}}{6 π α} + ν_{y} \frac{{(k W_{20})}^{2}}{6 π α})],

(17)

where

ν_{x}

and

ν_{y}

are coordinates in frequency plane. Similarly,

F [h (x + \frac{{(k W_{20}')}^{2}}{6 π α}, y + \frac{{(k W_{20}')}^{2}}{6 π α}, 0)] = F [h (x, y, 0)] \exp [j 2 π (f_{x} \frac{{(k W_{20}')}^{2}}{6 π α} + f_{y} \frac{{(k W_{20}')}^{2}}{6 π α})] .

(18)

Substitute Eq. (17) and Eq. (18) in Eq. (15) and we obtain

I' (x, y) = F^{- 1} {F [I (x, y)] \cdot \exp [j 2 π (f_{x} \frac{{(k W_{20})}^{2} - {(k W_{20}')}^{2}}{6 π α} + f_{y} \frac{{(k W_{20})}^{2} - {(k W_{20}')}^{2}}{6 π α})]} .

(19)

Using the translation property of Fourier transform again, the decoded image is

I' (x, y) = I (x + \frac{{(k W_{20})}^{2} - {(k W_{20}')}^{2}}{6 π α}, y + \frac{{(k W_{20})}^{2} - {(k W_{20}')}^{2}}{6 π α}) .

(20)

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

α = 30 π

and different defocus aberrations

W_{20} = 0 λ

and

W_{20} = 10 λ

. Here, Wiener filter [9

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using Matlab (Prentice-Hall, 2004), Chap. 5.

] 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.

Fig. 3 Decoded PSFs based on Wiener filter using focused PSF as the deconvolution kernel when

α = 30 π

and (a)

W_{20} = 0

; (b)

W_{20} = 10 λ

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

| k W_{20} | < < 3 α

. 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 [5

] is described as

h (x, y, ψ) = {\begin{cases} \frac{1}{4} κ, T_{2} < x, y \leq T_{3} \\ \frac{1}{2} κ (1 + \sin η), T_{1} < x \leq T_{2}, T_{2} < y \leq T_{3} \\ \frac{1}{2} κ (1 + \sin ξ), T_{2} < x \leq T_{3}, T_{1} < y \leq T_{2} \\ κ (1 + \sin η) (1 + \sin ξ), T_{1} < x, y \leq T_{2} \\ 0, x, y > T_{3} or x, y \leq T_{1} \end{cases},

(21)

where

κ = π^{2} / (A B)

η = 4 A^{3} / (27 α^{2})

, and

ξ = 4 B^{3} / (27 α^{2})

T_{1} = \frac{- {(k W_{20})}^{2}}{6 π α}, T_{2} = \frac{3 α - 2 | k W_{20} |}{2 π}, T_{3} = \frac{3 α + 2 | k W_{20} |}{2 π} .

(22)

Take

W_{20} = 5 λ

and

α = 30 π

. 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

500 \times 500

. 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.

Fig. 4 Approximated PSF of the wavefront coding system with a rectangular pupil. (a) Schematic diagram of the piecewise PSF function; (b) Grey-scale map of the defocused log-PSF; (c) Boundaries of the focused PSF (in red) and the defocused PSF (in black). Here

α = 30 π

, the defocus aberration

W_{20} = 5 λ

x = \frac{- {(k W_{20})}^{2}}{6 π α}, y = \frac{- {(k W_{20})}^{2}}{6 π α}, x = \frac{3 α + 2 | k W_{20} |}{2 π}, y = \frac{3 α + 2 | k W_{20} |}{2 π} .

(23)

The boundaries of the focused PSF are also given as a contrast, as shown in red lines in Fig. 4(c).

x = 0, y = 0, x = \frac{3 α}{2 π}, y = \frac{3 α}{2 π} .

(24)

Three similarities between rectangular pupil system and circular pupil system can be clearly observed. Firstly, the defocus leads to position shift along

- x

and

- y

directions. The offset is

- {(k W_{20})}^{2} / (6 π α)

. Secondly, the defocus leads to the boundary expansion and

| k W_{20} | / π + {(k W_{20})}^{2} / (6 π α)

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

η

and

ξ

, 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

W_{20} \neq 0

in the rectangular pupil system. In contrast, the defocus aberration causes the boundary deformation in the circular pupil system.

6. Conclusion

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.

Appendices

Appendix A: Deduction of the defocused PSF in a circular pupil wavefront coding system

The PSF function described in Eq. (1) is rewritten as

h (x, y, W_{20}) = \frac{1}{2} {| \iint_{u^{2} + v^{2} \leq 1} \exp [j ϕ (u, v)] d u d v |}^{2},

(A1)

where

ϕ (u, v) = α (u^{3} + v^{3}) + k W_{20} (u^{2} + v^{2}) - 2 π (x u + y v)

. The first and the second derivations of

ϕ (u, v)

are

{\begin{cases} \frac{\partial ϕ}{\partial u} = 3 α u^{2} + 2 k W_{20} u - 2 π x \\ \frac{\partial ϕ}{\partial v} = 3 α v^{2} + 2 k W_{20} v - 2 π y \end{cases} and {\begin{cases} \frac{\partial^{2} ϕ}{\partial u^{2}} = 6 α u + 2 k W_{20} \\ \frac{\partial^{2} ϕ}{\partial v^{2}} = 6 α v + 2 k W_{20} \\ \frac{\partial^{2} ϕ}{\partial u \partial v} = 0 \end{cases} .

(A2)

According to the stationary phase method, stationary points exist if and only if Eq. (A3) is satisfied.

{\begin{cases} \frac{\partial ϕ}{\partial u} = 0 and \frac{\partial ϕ}{\partial v} = 0 (a) \\ (\frac{\partial^{2} ϕ}{\partial u^{2}}) (\frac{\partial^{2} ϕ}{\partial v^{2}}) - {(\frac{\partial^{2} ϕ}{\partial u \partial v})}^{2} \neq 0 (b) \\ u_{}^{2} + v_{}^{2} \leq 1 (c) \end{cases} .

(A3)

Let

A = \sqrt{{(k W_{20})}^{2} + 6 π α x}, B = \sqrt{{(k W_{20})}^{2} + 6 π α y} .

(A4)

Thus

x \geq - {(k W_{20})}^{2} / (6 π α)

, and

y \geq - {(k W_{20})}^{2} / (6 π α)

. Possible solutions of Eq. (A3a) are

(u_{01}, v_{01})

(u_{01}, v_{02})

(u_{02}, v_{01})

and

(u_{02}, v_{02})

, where

{\begin{cases} u_{01} = - \frac{A + k W_{20}}{3 α}, u_{02} = \frac{A - k W_{20}}{3 α} \\ v_{01} = - \frac{B + k W_{20}}{3 α}, v_{02} = \frac{B - k W_{20}}{3 α} \end{cases} .

(A5)

With Eq. (A4) substituted in Eq. (A3c), the boundaries of the PSF are:

{\begin{cases} Ω_{1} = {(A, B) | {(A + | k W_{20} |)}^{2} + {(B + | k W_{20} |)}^{2} \leq 9 α^{2}} \\ Ω_{2} = {(A, B) | {(A + | k W_{20} |)}^{2} + {(B - | k W_{20} |)}^{2} \leq 9 α^{2}} \\ Ω_{3} = {(A, B) | {(A - | k W_{20} |)}^{2} + {(B + | k W_{20} |)}^{2} \leq 9 α^{2}} \\ Ω_{4} = {(A, B) | {(A - | k W_{20} |)}^{2} + {(B - | k W_{20} |)}^{2} \leq 9 α^{2}} \end{cases} .

(A6)

Fig. A1 Color maps to distinguish regions containing different stationary points in (a):

A - B

plane and (b):

x - y

plane. Here

W_{20} = 5 λ, α = 30 π

Figure A1(a) uses color to distinguish regions containing different stationary points in

A - B

plane when

W_{20} = 5 λ, α = 30 π

. All of the nonzero regions are located at the first quadrant because

A, B \geq 0

. Consider positive defocus aberration, i.e.

W_{20} > 0

first. The red region,

(Ω_{1} \cap Ω_{2} \cap Ω_{3} \cap Ω_{4})

, has 4 stationary points,

(u_{01}, v_{01})

(u_{01}, v_{02})

(u_{02}, v_{01})

and

(u_{02}, v_{02})

. The yellow region,

({\bar{Ω}}_{1} \cap Ω_{2} \cap Ω_{3} \cap Ω_{4})

, has 3 stationary points,

(u_{01}, v_{02})

(u_{02}, v_{01})

and

(u_{02}, v_{02})

. The rose region,

({\bar{Ω}}_{1} \cap Ω_{2} \cap {\bar{Ω}}_{3} \cap Ω_{4})

, has 2 stationary points,

(u_{01}, v_{02})

and

(u_{02}, v_{02})

. The gray region,

({\bar{Ω}}_{1} \cap {\bar{Ω}}_{2} \cap Ω_{3} \cap Ω_{4})

, has 2 stationary points,

(u_{02}, v_{01})

and

(u_{02}, v_{02})

. The cyan region,

({\bar{Ω}}_{1} \cap {\bar{Ω}}_{2} \cap {\bar{Ω}}_{3} \cap Ω_{4})

, has only 1 stationary point,

(u_{02}, v_{02})

. If the defocus aberration is negative, i.e.

W_{20} < 0

, 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

x - y

plane is obtained and shown in Fig. A1(b).

Equation (A1) is approximated as the sum of the stationary phase approximations evaluated at the stationary points [7

M. Born and E. Wolf, Principles of Optics (Pergamon, 1985).

h (x, y, W_{20}) \approx \frac{1}{4} {| \sum \sum 2 π j σ_{m, n} {| \frac{\partial^{2} ϕ}{\partial u^{2}} (u_{m}, v_{n}) \frac{\partial^{2} ϕ}{\partial v^{2}} (u_{m}, v_{n}) |}^{- \frac{1}{2}} \exp [j ϕ (u_{m}, v_{n})] |}^{2},

(A7)

where

u_{m}, v_{n}

are the stationary points existing in the area shown in Fig. A1(a).

σ_{m, n} = {\begin{cases} + 1, \frac{\partial^{2} ϕ}{\partial u^{2}} (u_{m}, v_{n}) > 0 and \frac{\partial^{2} ϕ}{\partial v^{2}} (u_{m}, v_{n}) > 0 \\ - 1, \frac{\partial^{2} ϕ}{\partial u^{2}} (u_{m}, v_{n}) < 0 and \frac{\partial^{2} ϕ}{\partial v^{2}} (u_{m}, v_{n}) < 0 \\ - j, [\frac{\partial^{2} ϕ}{\partial u^{2}} (u_{m}, v_{n})] [\frac{\partial^{2} ϕ}{\partial v^{2}} (u_{m}, v_{n})] < 0 \end{cases} .

Substitute stationary points’ value in Eq. (A7) and the PSF is described as

h (x, y, W_{20}) = {\begin{cases} \frac{1}{4} κ, (A, B) \in ({\bar{Ω}}_{1} \cap {\bar{Ω}}_{2} \cap {\bar{Ω}}_{3} \cap Ω_{4}) \\ \frac{1}{2} κ (1 + \sin η), (A, B) \in ({\bar{Ω}}_{1} \cap Ω_{2} \cap {\bar{Ω}}_{3} \cap Ω_{4}) \\ \frac{1}{2} κ (1 + \sin ξ), (A, B) \in ({\bar{Ω}}_{1} \cap {\bar{Ω}}_{2} \cap Ω_{3} \cap Ω_{4}) \\ \frac{1}{4} κ [(1 + \sin η) (1 + \sin ξ) + 2 + 2 \cos η \cos ξ], (A, B) \in ({\bar{Ω}}_{1} \cap Ω_{2} \cap Ω_{3} \cap Ω_{4}) \\ κ (1 + \sin η) (1 + \sin ξ), (A, B) \in (Ω_{1} \cap Ω_{2} \cap Ω_{3} \cap Ω_{4}) \\ 0, (A, B) \in ({\bar{Ω}}_{1} \cap {\bar{Ω}}_{2} \cap {\bar{Ω}}_{3} \cap {\bar{Ω}}_{4}) or x, y < - \frac{{(k W_{20})}^{2}}{6 π α} \end{cases},

(A8)

where,

κ = π^{2} / (A B)

η = 4 A^{3} / (27 α^{2})

ξ = 4 B^{3} / (27 α^{2})

Appendix B: Proof of equivalence of Eq. (3) and Eq. (7) when $| k W_{20} | < < 3 α$

Consider

| k W_{20} | < < 3 α

and rewrite Eq. (5) as

A = \sqrt{3 α} \sqrt{\frac{{(k W_{20})}^{2}}{3 α} + 2 π x} \approx \sqrt{6 π α x}, B = \sqrt{3 α} \sqrt{\frac{{(k W_{20})}^{2}}{3 α} + 2 π y} \approx \sqrt{6 π α y} .

(B1)

Substitute Eq. (B1) in Eq. (6) and obtain

κ = \frac{π}{6 α \sqrt{x y}}, η = \frac{4 {(6 π α x)}^{3 / 2}}{27 α^{2}}, ξ = \frac{4 {(6 π α y)}^{3 / 2}}{27 α^{2}} .

(B2)

Consequently,

κ = κ', η = η', ξ = ξ' .

(B3)

Then consider boundaries of the defocused PSF described in Eq. (4). Notice that

| k W_{20} | < < 3 α

, and Eq. (4a) can be simplified as

Ω_{0} = {(x, y) | x \geq 0, y \geq - 0} .

(B4)

With Eq. (B1) substituted in Eq. (4b)–(4e),

{\begin{cases} Ω_{1} = {(x, y) | {(\sqrt{\frac{2 π x}{3 α}} + \frac{| k W_{20} |}{3 α})}^{2} + {(\sqrt{\frac{2 π y}{3 α}} + \frac{| k W_{20} |}{3 α})}^{2} \leq 1} \\ Ω_{2} = {(x, y) | {(\sqrt{\frac{2 π x}{3 α}} + \frac{| k W_{20} |}{3 α})}^{2} + {(\sqrt{\frac{2 π y}{3 α}} - \frac{| k W_{20} |}{3 α})}^{2} \leq 1} \\ Ω_{3} = {(x, y) | {(\sqrt{\frac{2 π x}{3 α}} - \frac{| k W_{20} |}{3 α})}^{2} + {(\sqrt{\frac{2 π y}{3 α}} + \frac{| k W_{20} |}{3 α})}^{2} \leq 1} \\ Ω_{4} = {(x, y) | {(\sqrt{\frac{2 π x}{3 α}} - \frac{| k W_{20} |}{3 α})}^{2} + {(\sqrt{\frac{2 π y}{3 α}} - \frac{| k W_{20} |}{3 α})}^{2} \leq 1} \end{cases} .

It is apparent that

| k W_{20} | / (3 α)

can be omitted when

| k W_{20} | < < 3 α

. Therefore,

Ω_{1}

Ω_{2}

Ω_{3}

and

Ω_{4}

can be approximated to be one set

Ω'_{1}

, i.e.

{\begin{cases} {\bar{Ω}}_{1} \cap {\bar{Ω}}_{2} \cap {\bar{Ω}}_{3} \cap Ω_{4} = 0 \\ {\bar{Ω}}_{1} \cap Ω_{2} \cap {\bar{Ω}}_{3} \cap Ω_{4} = 0 \\ {\bar{Ω}}_{1} \cap {\bar{Ω}}_{2} \cap Ω_{3} \cap Ω_{4} = 0 \\ {\bar{Ω}}_{1} \cap Ω_{2} \cap Ω_{3} \cap Ω_{4} = 0 \end{cases} .

Ω'_{1}

is described as

Ω'_{1} = {(x, y) | x + y \leq \frac{3 α}{2 π}} .

(B5)

In other word, boundaries of the defocused PSF are simplified as Eq. (8) when

| k W_{20} | < < 3 α

As a consequence, Eq. (3) can be simplified as Eq. (7). In other words, the defocused PSF is approximated as the focused PSF when

| k W_{20} | < < 3 α

Acknowledgments

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

1.	E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]
2.	H. Zhao and Y. Li, “Optimized sinusoidal phase mask to extend the depth of field of an incoherent imaging system,” Opt. Lett. 35(2), 267–269 (2010). [CrossRef] [PubMed]
3.	H. Zhao and Y. Li, “Optimized logarithmic phase masks used to generate defocus invariant modulation transfer function for wavefront coding system,” Opt. Lett. 35(15), 2630–2632 (2010). [CrossRef] [PubMed]
4.	G. Muyo and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. 30(20), 2715–2717 (2005). [CrossRef] [PubMed]
5.	W. Zhang, Z. Ye, T. Zhao, Y. Chen, and F. Yu, “Point spread function characteristics analysis of the wavefront coding system,” Opt. Express 15(4), 1543–1552 (2007), http://www.opticsinfobase.org/abstract.cfm?&id=127217. [CrossRef] [PubMed]
6.	S. Bagheri, P. E. Silveira, and D. P. de Farias, “Analytical optimal solution of the extension of the depth of field using cubic-phase wavefront coding. part I. reduced-complexity approximate representation of the modulation transfer function,” J. Opt. Soc. Am. A 25(5), 1051–1063 (2008). [CrossRef] [PubMed]
7.	M. Born and E. Wolf, Principles of Optics (Pergamon, 1985).
8.	J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 6.
9.	R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using Matlab (Prentice-Hall, 2004), Chap. 5.

OCIS Codes
(110.2960) Imaging systems : Image analysis
(110.7348) Imaging systems : Wavefront encoding

ToC Category:
Imaging Systems

History
Original Manuscript: September 28, 2011
Revised Manuscript: November 14, 2011
Manuscript Accepted: January 12, 2012
Published: January 19, 2012

Citation
Tingyu Zhao and Feihong Yu, "Point spread function analysis of a cubic phase wavefront coding system with a circular pupil," Opt. Express 20, 2408-2419 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2408

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References

E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt.34(11), 1859–1866 (1995). [CrossRef] [PubMed]
H. Zhao and Y. Li, “Optimized sinusoidal phase mask to extend the depth of field of an incoherent imaging system,” Opt. Lett.35(2), 267–269 (2010). [CrossRef] [PubMed]
H. Zhao and Y. Li, “Optimized logarithmic phase masks used to generate defocus invariant modulation transfer function for wavefront coding system,” Opt. Lett.35(15), 2630–2632 (2010). [CrossRef] [PubMed]
G. Muyo and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett.30(20), 2715–2717 (2005). [CrossRef] [PubMed]
W. Zhang, Z. Ye, T. Zhao, Y. Chen, and F. Yu, “Point spread function characteristics analysis of the wavefront coding system,” Opt. Express15(4), 1543–1552 (2007), http://www.opticsinfobase.org/abstract.cfm?&id=127217 . [CrossRef] [PubMed]
S. Bagheri, P. E. Silveira, and D. P. de Farias, “Analytical optimal solution of the extension of the depth of field using cubic-phase wavefront coding. part I. reduced-complexity approximate representation of the modulation transfer function,” J. Opt. Soc. Am. A25(5), 1051–1063 (2008). [CrossRef] [PubMed]
M. Born and E. Wolf, Principles of Optics (Pergamon, 1985).
J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 6.
R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using Matlab (Prentice-Hall, 2004), Chap. 5.

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Optics Express

Point spread function analysis of a cubic phase wavefront coding system with a circular pupil

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Abstract

1. Introduction

2. Approximated PSF with a circular pupil

2.1 Defocused PSF

2.2 Focused PSF

3. PSF analysis

4. Defocus influences on decoded image

5. Circular pupil and rectangular pupil

6. Conclusion

Appendices

Appendix A: Deduction of the defocused PSF in a circular pupil wavefront coding system

Appendix B: Proof of equivalence of Eq. (3) and Eq. (7) when $| k W_{20} | < < 3 α$

Acknowledgments

References and links

References

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Other

Cited By

Figures

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OpticsInfoBase

Optics Express

Point spread function analysis of a cubic phase wavefront coding system with a circular pupil

Article Tools

Article Navigation

Abstract

1. Introduction

2. Approximated PSF with a circular pupil

2.1 Defocused PSF

2.2 Focused PSF

3. PSF analysis

4. Defocus influences on decoded image

5. Circular pupil and rectangular pupil

6. Conclusion

Appendices

Appendix A: Deduction of the defocused PSF in a circular pupil wavefront coding system

Appendix B: Proof of equivalence of Eq. (3) and Eq. (7) when | k W 20|<<3α

Acknowledgments

References and links

References

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Other

Cited By

Figures

Appendix B: Proof of equivalence of Eq. (3) and Eq. (7) when $| k W_{20} | < < 3 α$