Abstract

An approximate analytical expression is derived for the two-dimensional incoherent optical transfer function (OTF) of an imaging system invariant to second-order aberrations. The system broadband behavior resulting from a third-order phase mask in its pupil plane is analyzed by using the two-dimensional stationary phase method. This approach does not require mathematical separability of the pupil function and can be applied to any pupil shape. The OTF is found to be a well-defined and smooth function at all nonzero spatial frequencies when the phase mask function includes third-order mixed terms in the pupil coordinates.

© 2009 Optical Society of America

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References

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    [Crossref] [PubMed]
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  17. R. Wong, Asymptotic Approximations of Integrals (Academic, 1989).
  18. H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London, Ser. A 231, 91-103 (1955).
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  20. F. R. Gantmacher, The Theory of Matrices (Chelsea, 1959), Vol. 1.
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  22. M. Frigo and S. G. Johnson, “FFTW,” http://www.fftw.org.

2008 (1)

2007 (1)

2006 (3)

2005 (1)

2004 (3)

2003 (1)

2002 (1)

1997 (1)

1996 (1)

1995 (1)

1994 (1)

D. Kaminski, “Exponentially improved stationary phase approximations for double integrals,” Methods Appl. Anal. 1, 44-56 (1994).

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London, Ser. A 231, 91-103 (1955).

Bagheri, S.

Bradburn, S.

Castro, A.

Cathey, W. T.

Christensen, M. P.

Deaver, D. M.

Dowski, E. R.

Farias, D. P.

Fedoruk, M.

M. Fedoruk, Analysis I. Asymptotic Methods in Analysis, Vol. 13 of Encyclopaedia of Mathematical Science (Springer, 1989).

Fedoruk, M. V.

M. V. Fedoruk, Saddle Point Method (Nauka, 1977) (in Russian).

Feng, H.

Frigo, M.

M. Frigo and S. G. Johnson, “FFTW,” http://www.fftw.org.

Gantmacher, F. R.

F. R. Gantmacher, The Theory of Matrices (Chelsea, 1959), Vol. 1.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).

Harvey, A. R.

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London, Ser. A 231, 91-103 (1955).

Johnson, S. G.

M. Frigo and S. G. Johnson, “FFTW,” http://www.fftw.org.

Kaminski, D.

D. Kaminski, “Exponentially improved stationary phase approximations for double integrals,” Methods Appl. Anal. 1, 44-56 (1994).

Lei, H.

Mezouari, S.

Muyo, G.

Ojeda-Castaneda, J.

Pauca, V. P.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High-resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14, 67-74 (2004).
[Crossref]

Plemmons, R. J.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High-resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14, 67-74 (2004).
[Crossref]

Prasad, S.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High-resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14, 67-74 (2004).
[Crossref]

Sherif, S. S.

Silveria, P. E. X.

Somayaji, M.

Tao, X.

Taylor, M. G.

Torgersen, T. C.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High-resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14, 67-74 (2004).
[Crossref]

van der Gracht, J.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High-resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14, 67-74 (2004).
[Crossref]

J. van der Gracht, E. R. Dowski, Jr., M. G. Taylor, and D. M. Deaver, “Broadband behavior of an optical-digital focus-invariant system,” Opt. Lett. 21, 919-921 (1996).
[Crossref] [PubMed]

Wong, R.

R. Wong, Asymptotic Approximations of Integrals (Academic, 1989).

Xu, Zh.

Appl. Opt. (8)

Int. J. Imaging Syst. Technol. (1)

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High-resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14, 67-74 (2004).
[Crossref]

J. Opt. Soc. Am. A (2)

Methods Appl. Anal. (1)

D. Kaminski, “Exponentially improved stationary phase approximations for double integrals,” Methods Appl. Anal. 1, 44-56 (1994).

Opt. Lett. (3)

Proc. R. Soc. London, Ser. A (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London, Ser. A 231, 91-103 (1955).

Other (6)

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).

F. R. Gantmacher, The Theory of Matrices (Chelsea, 1959), Vol. 1.

M. Frigo and S. G. Johnson, “FFTW,” http://www.fftw.org.

M. Fedoruk, Analysis I. Asymptotic Methods in Analysis, Vol. 13 of Encyclopaedia of Mathematical Science (Springer, 1989).

M. V. Fedoruk, Saddle Point Method (Nauka, 1977) (in Russian).

R. Wong, Asymptotic Approximations of Integrals (Academic, 1989).

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Figures (1)

Fig. 1
Fig. 1 MTF of the defocused optical system with the generalized cubic mask. Solid curves, MTF curves obtained with the approximate expression, Eq. (31); scatter plots represent the MTFs calculated numerically using the pupil function, as given by Eq. (36), for varying degrees of defocus w 0 .

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Equations (60)

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W ( x , y ) = w 0 ( a x 2 + b x y + c y 2 ) ,
H ( ω x , ω y , w 0 ) = 1 Ω ∫ ∫ − ∞ ∞ P ( x + ω x ∕ 2 , y + ω y ∕ 2 ) P * ( x − ω x ∕ 2 , y − ω y ∕ 2 ) d x d y ,
P ( x , y ) = f 0 ( x , y ) exp [ i θ ( x , y ) + i w 0 ( a x 2 + b x y + c y 2 ) ] .
f 0 ( x , y ) = { 1 ( x , y ) ∊ Ω 0 otherwise } .
H ( ω x , ω y , w 0 ) = 1 Ω ∫ ∫ − ∞ ∞ p 0 ( x , y , ω x , ω y ) exp [ i F ( x , y , ω x , ω y ) + i w 0 ( 2 a x ω x + b x ω y + b y ω x + 2 c y ω y ) ] d x d y ,
p 0 ( x , y , ω x , ω y ) = f 0 ( x + ω x ∕ 2 , y + ω y ∕ 2 ) f 0 ( x − ω x ∕ 2 , y − ω y ∕ 2 )
F ( x , y , ω x , ω y ) = θ ( x + ω x ∕ 2 , y + ω y ∕ 2 ) − θ ( x − ω x ∕ 2 , y − ω y ∕ 2 )
υ x = w 0 ( a ω x + b ω y ∕ 2 ) ∕ π , υ y = w 0 ( c ω y + b ω x ∕ 2 ) ∕ π ,
H ̃ ( ω x , ω y , υ x , υ y ) = 1 Ω ∫ ∫ − ∞ ∞ p 0 ( x , y , ω x , ω y ) exp [ i F ( x , y , ω x , ω y ) + i 2 π ( x υ x + y υ y ) ] d x d y ,
H ̃ ( ω x , ω y , υ x , υ y ) ≅ 2 π Ω ∣ Δ ( x c , y c ) ∣ − 1 ∕ 2 exp [ i F ( x c , y c , ω x , ω y ) + i 2 π ( x c υ x + y c υ y ) + i π δ ( x c , y c ) 4 ] ,
∂ F ∂ x ( x c , y c ) + 2 π υ x = 0 ,
∂ F ∂ y ( x c , y c ) + 2 π υ y = 0 .
M = [ ∂ 2 F ∂ x 2 ( x c , y c ) ∂ 2 F ∂ x ∂ y ( x c , y c ) ∂ 2 F ∂ y ∂ x ( x c , y c ) ∂ 2 F ∂ y 2 ( x c , y c ) ]
Δ ( x c , y c ) = det M = [ ∂ 2 F ∂ x 2 ∂ 2 F ∂ y 2 − ∂ 2 F ∂ x ∂ y ∂ 2 F ∂ y ∂ x ] ( x c , y c ) ,
∂ ∂ υ x Δ ( x c , y c ) = 0 ,
∂ ∂ υ y Δ ( x c , y c ) = 0 .
x = x c ( υ x , υ y ) ,
y = y c ( υ x , υ y ) .
∂ 2 F ∂ x ∂ υ x ( x c , y c ) + 2 π = 0 ,
∂ 2 F ∂ x ∂ υ y ( x c , y c ) = 0 ,
∂ 2 F ∂ y ∂ υ x ( x c , y c ) = 0 ,
∂ 2 F ∂ y ∂ υ y ( x c , y c ) + 2 π = 0 .
∂ 2 F ∂ x 2 ∂ x ∂ υ x + ∂ 2 F ∂ x ∂ y ∂ y ∂ υ x + 2 π = 0 ,
∂ 2 F ∂ x 2 ∂ x ∂ υ y + ∂ 2 F ∂ x ∂ y ∂ y ∂ υ y = 0 ,
∂ 2 F ∂ y ∂ x ∂ x ∂ υ x + ∂ 2 F ∂ y 2 ∂ y ∂ υ x = 0 ,
∂ 2 F ∂ y ∂ x ∂ x ∂ υ y + ∂ 2 F ∂ y 2 ∂ y ∂ υ y + 2 π = 0 .
Δ = 4 π 2 ∂ x ∂ υ x ∂ y ∂ υ y − ∂ x ∂ υ y ∂ y ∂ υ x .
∣ Δ ∣ − 1 ∕ 2 = 1 2 π ∣ ∂ x ∂ υ x ∂ y ∂ υ y − ∂ x ∂ υ y ∂ y ∂ υ x ∣ 1 ∕ 2 .
x c ( υ x , υ y ) = C 1 υ x + C 2 υ y ,
y c ( υ x , υ y ) = C 3 υ x + C 4 υ y ,
θ ( x , y ) = α ( β 1 x 2 y + β 2 y 2 x + γ 1 x 3 + γ 2 y 3 ) ,
F ( x , y , ω x , ω y ) = α ( β 1 x 2 ω y + β 2 y 2 ω x + 2 β 1 x y ω x + 2 β 2 x y ω y + 3 γ 1 x 2 ω x + 3 γ 2 y 2 ω y ) + α ( β 1 ω x 2 ω y + β 2 ω y 2 ω x + γ 1 ω x 3 + γ 2 ω y 3 ) ∕ 4 .
( β 1 ω y + 3 γ 1 ω x ) x c + ( β 1 ω x + β 2 ω y ) y c = − π υ x ∕ α ,
( β 1 ω x + β 2 ω y ) x c + ( β 2 ω x + 3 γ 2 ω y ) y c = − π υ y ∕ α ,
x c = − π α υ y ( β 1 ω x + β 2 ω y ) − υ x ( β 2 ω x + 3 γ 2 ω y ) ( β 1 ω x + β 2 ω y ) 2 − ( β 1 ω y + 3 γ 1 ω x ) ( β 2 ω x + 3 γ 2 ω y ) ,
y c = − π α υ x ( β 1 ω x + β 2 ω y ) − υ y ( β 1 ω y + 3 γ 1 ω x ) ( β 1 ω x + β 2 ω y ) 2 − ( β 1 ω y + 3 γ 1 ω x ) ( β 2 ω x + 3 γ 2 ω y ) .
∣ Δ ∣ − 1 ∕ 2 = 1 2 α ∣ ( β 1 ω x + β 2 ω y ) 2 − ( β 1 ω y + 3 γ 1 ω x ) ( β 2 ω x + 3 γ 2 ω y ) ∣ − 1 ∕ 2 .
27 ( γ 1 γ 2 ) 2 − 18 γ 1 γ 2 β 1 β 2 − ( β 1 β 2 ) 2 + 4 γ 1 β 2 3 + 4 γ 2 β 1 3 < 0 .
M = [ ∂ 2 F ∂ x 2 ( x c , y c ) ∂ 2 F ∂ x ∂ y ( x c , y c ) ∂ 2 F ∂ y ∂ x ( x c , y c ) ∂ 2 F ∂ y 2 ( x c , y c ) ] = 2 α [ ( β 1 ω y + 3 γ 1 ω x ) ( β 1 ω x + β 2 ω y ) ( β 1 ω x + β 2 ω y ) ( β 2 ω x + 3 γ 2 ω y ) ] .
λ 2 − λ ( β 1 ω y + β 2 ω x + 3 γ 1 ω x + 3 γ 2 ω y ) + ( β 1 ω y + 3 γ 1 ω x ) ( β 2 ω x + 3 γ 2 ω y ) − ( β 1 ω x + β 2 ω y ) 2 = 0 .
H ( ω x , ω y , w 0 ) ≅ π exp { i α 4 ( β 1 ω x 2 ω y + β 2 ω y 2 ω x + γ 1 ω x 3 + γ 2 ω y 3 ) + i π 4 δ } Ω α ∣ ( β 1 ω x + β 2 ω y ) 2 − ( β 1 ω y + 3 γ 1 ω x ) ( β 2 ω x + 3 γ 2 ω y ) ∣ 1 ∕ 2 .
H ( ω x , ω y , w 0 ) ≅ { π exp { i α 4 ( β 1 ω x 2 ω y + β 2 ω y 2 ω x + γ 1 ω x 3 + γ 2 ω y 3 ) + i π 4 δ } Ω α ∣ ( β 1 ω x + β 2 ω y ) 2 − ( β 1 ω y + 3 γ 1 ω x ) ( β 2 ω x + 3 γ 2 ω y ) ∣ 1 ∕ 2 ω x 2 + ω y 2 ≠ 0 1 ω x 2 + ω y 2 = 0 } .
H ( ω x , ω y , w 0 ) ≅ { π exp { i α 4 [ β 1 ω x 2 ω y + β 2 ω y 2 ω x ] } Ω α ∣ β 1 2 ω x 2 + β 2 2 ω y 2 + β 1 β 2 ω x ω y ∣ 1 ∕ 2 ω x 2 + ω y 2 ≠ 0 1 ω x 2 + ω y 2 = 0 } .
H ( ω x , ω y , w 0 ) ≅ π exp { i α 4 ( ω x 3 + ω y 3 ) + i π 4 [ sgn ( ω x ) + sgn ( ω y ) ] } 3 Ω α ∣ ω x ω y ∣ 1 ∕ 2 ,
H ( 0 , ω y , w 0 ) ≅ π 3 Ω α ∣ ω y ∣ exp { i α 4 ω y 3 + i π 4 sgn ( ω y ) } ,
H ( ω x , 0 , w 0 ) ≅ π 3 Ω α ∣ ω x ∣ exp { i α 4 ω x 3 + i π 4 sgn ( ω x ) } .
H ( ω x , ω y , w 0 ) = 2 π Ω F ̂ − 1 { ∣ F ̂ [ P ( x , y ) ] ∣ 2 } ,
∂ ∂ υ x [ ∂ 2 F ∂ x 2 ∂ 2 F ∂ y 2 − ∂ 2 F ∂ x ∂ y ∂ 2 F ∂ y ∂ x ] = 0 ,
∂ ∂ υ y [ ∂ 2 F ∂ x 2 ∂ 2 F ∂ y 2 − ∂ 2 F ∂ x ∂ y ∂ 2 F ∂ y ∂ x ] = 0 ,
∂ x c ∂ υ x [ ∂ 3 F ∂ x 3 ∂ 2 F ∂ y 2 + ∂ 2 F ∂ x 2 ∂ 3 F ∂ y 2 ∂ x − 2 ∂ 2 F ∂ y ∂ x ∂ 3 F ∂ x 2 ∂ y ] + ∂ y c ∂ υ x [ ∂ 3 F ∂ x 2 ∂ y ∂ 2 F ∂ y 2 + ∂ 2 F ∂ x 2 ∂ 3 F ∂ y 3 − 2 ∂ 2 F ∂ x ∂ y ∂ 3 F ∂ x ∂ y 2 ] = 0 ,
∂ x c ∂ υ y [ ∂ 3 F ∂ x 3 ∂ 2 F ∂ y 2 + ∂ 2 F ∂ x 2 ∂ 3 F ∂ y 2 ∂ x − 2 ∂ 2 F ∂ y ∂ x ∂ 3 F ∂ x 2 ∂ y ] + ∂ y c ∂ υ y [ ∂ 3 F ∂ x 2 ∂ y ∂ 2 F ∂ y 2 + ∂ 2 F ∂ x 2 ∂ 3 F ∂ y 3 − 2 ∂ 2 F ∂ x ∂ y ∂ 3 F ∂ x ∂ y 2 ] = 0 .
F ( x , y , ω x , ω y ) = ∑ n + m ⩽ 2 a n m ( ω x , ω y ) x n y m ,
F ( x , y , ω x , ω y ) = ω x ∂ θ ( x , y ) ∂ x + ω y ∂ θ ( x , y ) ∂ y + 2 3 ! ( ω x 2 ∂ ∂ x + ω y 2 ∂ ∂ y ) 3 θ ( x , y ) + ⋯ .
θ ( x , y ) = ∑ n + m ⩽ 3 b n m x n y m ,
a 00 = b 12 ω x ω y 2 + b 21 ω x 2 ω y ∕ 4 + b 01 ω y + b 10 ω x ,
a 01 = b 11 ω x ,
a 10 = b 11 ω y ,
a 11 = 2 b 12 ω y + 2 b 21 ω x ,
a 20 = 2 b 21 ω y ,
a 02 = 2 b 12 ω x .

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