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.

© 2012 OSA

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References

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2010

2008

2007

2005

1995

Bagheri, S.

Cathey, W. T.

Chen, Y.

de Farias, D. P.

Dowski, E. R.

Harvey, A. R.

Li, Y.

Muyo, G.

Silveira, P. E.

Ye, Z.

Yu, F.

Zhang, W.

Zhao, H.

Zhao, T.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Other

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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Figures (5)

Fig. 1
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=30π and W 20 =5λ .

Fig. 2
Fig. 2

Boundaries of the focused PSF (in red) and the defocused PSF (in black). Here α=30π , the defocus aberration W 20 =5λ .

Fig. 3
Fig. 3

Decoded PSFs based on Wiener filter using focused PSF as the deconvolution kernel when α=30π and (a) W 20 =0 ; (b) W 20 =10λ .

Fig. 4
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=30π , the defocus aberration W 20 =5λ .

Fig. A1
Fig. A1

Color maps to distinguish regions containing different stationary points in (a): AB plane and (b): xy plane. Here W 20 =5λ,α=30π .

Equations (34)

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h( x,y, W 20 )= | u 2 + v 2 1 q( u,v )exp[ jk W 20 ( u 2 + v 2 )j2π( xu+yv ) ] | 2 ,
q( u,v )={ 1 2 exp[ jα( u 3 + v 3 ) ], u 2 + v 2 1 0,otherwise .
h( x,y, W 20 )={ 1 4 κ,( A,B )( Ω ¯ 1 Ω ¯ 2 Ω ¯ 3 Ω 4 ) 1 2 κ( 1+sinη ),( A,B )( Ω ¯ 1 Ω 2 Ω ¯ 3 Ω 4 ) 1 2 κ( 1+sinξ ),( A,B )( Ω ¯ 1 Ω ¯ 2 Ω 3 Ω 4 ) 1 4 κ[ ( 1+sinη )( 1+sinξ )+2+2cosηcosξ ],( A,B )( Ω ¯ 1 Ω 2 Ω 3 Ω 4 ) κ( 1+sinη )( 1+sinξ ),( A,B )( Ω 1 Ω 2 Ω 3 Ω 4 ) 0,( A,B )( Ω ¯ 1 Ω ¯ 2 Ω ¯ 3 Ω ¯ 4 )orx,y< ( k W 20 ) 2 6πα ,
{ Ω 0 ={ ( x,y )|x ( k W 20 ) 2 6πα ,y ( k W 20 ) 2 6πα }(a) Ω 1 ={ ( x,y )| ( A+| k W 20 | ) 2 + ( B+| k W 20 | ) 2 9 α 2 }(b) Ω 2 ={ ( x,y )| ( A+| k W 20 | ) 2 + ( B| k W 20 | ) 2 9 α 2 }(c) Ω 3 ={ ( x,y )| ( A| k W 20 | ) 2 + ( B+| k W 20 | ) 2 9 α 2 }(d) Ω 4 ={ ( x,y )| ( A| k W 20 | ) 2 + ( B| k W 20 | ) 2 9 α 2 }(e) ,
A= ( k W 20 ) 2 +6παx ,B= ( k W 20 ) 2 +6παy
κ= π 2 / ( AB ) ,η= 4 A 3 / ( 27 α 2 ) ,ξ= 4 B 3 / ( 27 α 2 ) .
h( x,y )={ 0,x+y> 3α 2π orx<0ory<0 κ'( 1+sinη' )( 1+sinξ' ),otherwise ,
x=0,y=0,x+y= 3α 2π .
[ 6παk W 20 ( xy ) 6πα( x+y )9 α 2 +4 ( k W 20 ) 2 ] 2 + [ 6πα( x+y )9 α 2 4k W 20 ] 2 = 9 2 α 2 .
cosθ= 2 2 π| k W 20 |( xy ) 6πα( x+y )9 α 2 +4 ( k W 20 ) 2 ,sinθ= 2 2 π( x+y )3 2 α 4| k W 20 | .
x= ( k W 20 ) 2 6πα ,y= 9 α 2 +2| k W 20 | 9 α 2 ( k W 20 ) 2 ( k W 20 ) 2 6πα .
x= ( k W 20 ) 2 6πα ,y= ( k W 20 ) 2 6πα ,x+y= 2 | k W 20 |sinθ π + 3α 2π ( 2 2 sinθ1 ).
f x = 6πα ( 27 α 2 π 2 ) 2 3 ( k W 20 ) 2 , f y = 6πα ( 27 α 2 π 2 ) 2 3 ( k W 20 ) 2 .
I'( x,y, W 20 )= F 1 { F[ I( x,y ) ]F[ h( x,y, W 20 ) ] F[ h( x,y, W 20 ' ) ] },
I'( x,y )= F 1 { F[ I( x,y ) ]exp[ j2π( f x ( k W 20 ) 2 ( k W 20 ' ) 2 6πα + f y ( k W 20 ) 2 ( k W 20 ' ) 2 6πα ) ] } .
h( x,y,ψ )={ 1 4 κ, T 2 <x,y T 3 1 2 κ( 1+sinη ), T 1 <x T 2 , T 2 <y T 3 1 2 κ( 1+sinξ ), T 2 <x T 3 , T 1 <y T 2 κ( 1+sinη )( 1+sinξ ), T 1 <x,y T 2 0,x,y> T 3 orx,y T 1 ,
T 1 = ( k W 20 ) 2 6πα , T 2 = 3α2| k W 20 | 2π , T 3 = 3α+2| k W 20 | 2π .
x= ( k W 20 ) 2 6πα ,y= ( k W 20 ) 2 6πα ,x= 3α+2| k W 20 | 2π ,y= 3α+2| k W 20 | 2π .
x=0,y=0,x= 3α 2π ,y= 3α 2π .
h( x,y, W 20 )= 1 2 | u 2 + v 2 1 exp[ jϕ( u,v ) ]dudv | 2 ,
{ ϕ u =3α u 2 +2k W 20 u2πx ϕ v =3α v 2 +2k W 20 v2πy and{ 2 ϕ u 2 =6αu+2k W 20 2 ϕ v 2 =6αv+2k W 20 2 ϕ uv =0 .
{ ϕ u =0and ϕ v =0(a) ( 2 ϕ u 2 )( 2 ϕ v 2 ) ( 2 ϕ uv ) 2 0(b) u 2 + v 2 1(c) .
A= ( k W 20 ) 2 +6παx ,B= ( k W 20 ) 2 +6παy .
{ u 01 = A+k W 20 3α , u 02 = Ak W 20 3α v 01 = B+k W 20 3α , v 02 = Bk W 20 3α .
{ Ω 1 ={ ( A,B )| ( A+| k W 20 | ) 2 + ( B+| k W 20 | ) 2 9 α 2 } Ω 2 ={ ( A,B )| ( A+| k W 20 | ) 2 + ( B| k W 20 | ) 2 9 α 2 } Ω 3 ={ ( A,B )| ( A| k W 20 | ) 2 + ( B+| k W 20 | ) 2 9 α 2 } Ω 4 ={ ( A,B )| ( A| k W 20 | ) 2 + ( B| k W 20 | ) 2 9 α 2 } .
h( x,y, W 20 ) 1 4 | 2πj σ m,n | 2 ϕ u 2 ( u m , v n ) 2 ϕ v 2 ( u m , v n ) | 1 2 exp[ jϕ( u m , v n ) ] | 2 ,
σ m,n ={ +1, 2 ϕ u 2 ( u m , v n )>0and 2 ϕ v 2 ( u m , v n )>0 1, 2 ϕ u 2 ( u m , v n )<0and 2 ϕ v 2 ( u m , v n )<0 j,[ 2 ϕ u 2 ( u m , v n ) ][ 2 ϕ v 2 ( u m , v n ) ]<0 .
h( x,y, W 20 )={ 1 4 κ,( A,B )( Ω ¯ 1 Ω ¯ 2 Ω ¯ 3 Ω 4 ) 1 2 κ( 1+sinη ),( A,B )( Ω ¯ 1 Ω 2 Ω ¯ 3 Ω 4 ) 1 2 κ( 1+sinξ ),( A,B )( Ω ¯ 1 Ω ¯ 2 Ω 3 Ω 4 ) 1 4 κ[ ( 1+sinη )( 1+sinξ )+2+2cosηcosξ ],( A,B )( Ω ¯ 1 Ω 2 Ω 3 Ω 4 ) κ( 1+sinη )( 1+sinξ ),( A,B )( Ω 1 Ω 2 Ω 3 Ω 4 ) 0,( A,B )( Ω ¯ 1 Ω ¯ 2 Ω ¯ 3 Ω ¯ 4 )orx,y< ( k W 20 ) 2 6πα ,
A= 3α ( k W 20 ) 2 3α +2πx 6παx ,B= 3α ( k W 20 ) 2 3α +2πy 6παy .
κ= π 6α xy ,η= 4 ( 6παx ) 3/2 27 α 2 ,ξ= 4 ( 6παy ) 3/2 27 α 2 .
Ω 0 ={ ( x,y )|x0,y0 }.
{ Ω 1 ={ ( x,y )| ( 2πx 3α + | k W 20 | 3α ) 2 + ( 2πy 3α + | k W 20 | 3α ) 2 1 } Ω 2 ={ ( x,y )| ( 2πx 3α + | k W 20 | 3α ) 2 + ( 2πy 3α | k W 20 | 3α ) 2 1 } Ω 3 ={ ( x,y )| ( 2πx 3α | k W 20 | 3α ) 2 + ( 2πy 3α + | k W 20 | 3α ) 2 1 } Ω 4 ={ ( x,y )| ( 2πx 3α | k W 20 | 3α ) 2 + ( 2πy 3α | k W 20 | 3α ) 2 1 } .
{ Ω ¯ 1 Ω ¯ 2 Ω ¯ 3 Ω 4 =0 Ω ¯ 1 Ω 2 Ω ¯ 3 Ω 4 =0 Ω ¯ 1 Ω ¯ 2 Ω 3 Ω 4 =0 Ω ¯ 1 Ω 2 Ω 3 Ω 4 =0 .
Ω ' 1 ={ ( x,y )|x+y 3α 2π }.

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