Abstract

Most of the previous imaging characteristics analysis of the wavefront coding system has been carried out within the frequency domain. In this paper, the stationary phase method is employed to perform the analysis within the spatial domain. The approximate expression of point spread function (PSF) in the presence of defocus aberration is derived for the system with a cubic phase mask, which shows a good agreement with the Fast Fourier Transform (FFT) approach. Based on this, the PSF characteristics are analyzed in terms of the boundaries, oscillations and sensitivities to defocus, astigmatism and coma.

© 2007 Optical Society of America

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References

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  1. E. R. Dowski and W. T. Cathey, "Extended depth of field through wavefront coding," Appl. Opt. 34, 1859-1866 (1995).
    [CrossRef] [PubMed]
  2. S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," Proc. SPIE 5249, 238-248 (2004).
    [CrossRef]
  3. A. Castro and J. O. Castañeda, "Increased depth of field with phase-only filters: ambiguity function," Proc. SPIE 5827, 1-11 (2005).
    [CrossRef]
  4. S. S. Sherif, E. R. Dowski, and W. T. Cathey, "A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems," Proc. SPIE 4471, 272-280 (2001).
    [CrossRef]
  5. S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," Proc. SPIE 4768, 21-31 (2002).
    [CrossRef]
  6. N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A 5, 157-163 (2003).
    [CrossRef]
  7. S. Mezouari, G. Muyo, and A. R. Harvey, "Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism," J. Opt. Soc. Am. A 23, 1058-1062 (2006).
    [CrossRef]
  8. G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, "Analysis tools for computational imaging systems," Proc. SPIE 5817, 34-44 (2005).
    [CrossRef]
  9. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1985).
  10. D. L. Marks, R. A. Stack, and D. J. Brady, "Three-dimensional tomography using a cubic-phase plate extended depth-of-field system," Opt. Lett. 24, 253-255 (1999).
    [CrossRef]
  11. J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, 1996), Chap. 6.
  12. M. R. Arnison, "Phase control and measurement in digital microscopy" (Sydney Digital Theses, Physics, 2006), http://hdl.handle.net/2123/569>.
  13. M. Somayji and M. P. Christensen, "Enhancing form factor and light collection of multiplex imaging systems by using cubic phase mask," Appl. Opt. 45, 2911-2923 (2006).
    [CrossRef]
  14. T. Hellmuth, A. Bich, R. Börret, and A. Kelm, "Variable phaseplates for focus invariant optical systems," in Optical Design and Engineering II, L. Mazuray, R. Wartmann, eds., Proc. SPIE 5962, 596215 (2005).
    [CrossRef]

2006 (2)

2005 (2)

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, "Analysis tools for computational imaging systems," Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

A. Castro and J. O. Castañeda, "Increased depth of field with phase-only filters: ambiguity function," Proc. SPIE 5827, 1-11 (2005).
[CrossRef]

2004 (1)

S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," Proc. SPIE 5249, 238-248 (2004).
[CrossRef]

2003 (1)

N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A 5, 157-163 (2003).
[CrossRef]

2002 (1)

S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," Proc. SPIE 4768, 21-31 (2002).
[CrossRef]

2001 (1)

S. S. Sherif, E. R. Dowski, and W. T. Cathey, "A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems," Proc. SPIE 4471, 272-280 (2001).
[CrossRef]

1999 (1)

1995 (1)

Brady, D. J.

Castañeda, J. O.

A. Castro and J. O. Castañeda, "Increased depth of field with phase-only filters: ambiguity function," Proc. SPIE 5827, 1-11 (2005).
[CrossRef]

Castro, A.

A. Castro and J. O. Castañeda, "Increased depth of field with phase-only filters: ambiguity function," Proc. SPIE 5827, 1-11 (2005).
[CrossRef]

Cathey, W. T.

S. S. Sherif, E. R. Dowski, and W. T. Cathey, "A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems," Proc. SPIE 4471, 272-280 (2001).
[CrossRef]

E. R. Dowski and W. T. Cathey, "Extended depth of field through wavefront coding," Appl. Opt. 34, 1859-1866 (1995).
[CrossRef] [PubMed]

Chi, W.

N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A 5, 157-163 (2003).
[CrossRef]

Christensen, M. P.

Dowski, E. R.

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, "Analysis tools for computational imaging systems," Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

S. S. Sherif, E. R. Dowski, and W. T. Cathey, "A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems," Proc. SPIE 4471, 272-280 (2001).
[CrossRef]

E. R. Dowski and W. T. Cathey, "Extended depth of field through wavefront coding," Appl. Opt. 34, 1859-1866 (1995).
[CrossRef] [PubMed]

George, N.

N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A 5, 157-163 (2003).
[CrossRef]

Harvey, A. R.

S. Mezouari, G. Muyo, and A. R. Harvey, "Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism," J. Opt. Soc. Am. A 23, 1058-1062 (2006).
[CrossRef]

S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," Proc. SPIE 5249, 238-248 (2004).
[CrossRef]

S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," Proc. SPIE 4768, 21-31 (2002).
[CrossRef]

Johnson, G. E.

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, "Analysis tools for computational imaging systems," Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

Marks, D. L.

Mezouari, S.

S. Mezouari, G. Muyo, and A. R. Harvey, "Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism," J. Opt. Soc. Am. A 23, 1058-1062 (2006).
[CrossRef]

S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," Proc. SPIE 5249, 238-248 (2004).
[CrossRef]

S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," Proc. SPIE 4768, 21-31 (2002).
[CrossRef]

Muyo, G.

S. Mezouari, G. Muyo, and A. R. Harvey, "Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism," J. Opt. Soc. Am. A 23, 1058-1062 (2006).
[CrossRef]

S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," Proc. SPIE 5249, 238-248 (2004).
[CrossRef]

Sherif, S. S.

S. S. Sherif, E. R. Dowski, and W. T. Cathey, "A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems," Proc. SPIE 4471, 272-280 (2001).
[CrossRef]

Silveira, P. E. X.

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, "Analysis tools for computational imaging systems," Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

Somayji, M.

Stack, R. A.

Appl. Opt. (2)

J. Opt. A (1)

N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A 5, 157-163 (2003).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Proc. SPIE (5)

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, "Analysis tools for computational imaging systems," Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

S. Mezouari, G. Muyo, and A. R. Harvey, "Amplitude and phase filters for mitigation of defocus and third-order aberrations," Proc. SPIE 5249, 238-248 (2004).
[CrossRef]

A. Castro and J. O. Castañeda, "Increased depth of field with phase-only filters: ambiguity function," Proc. SPIE 5827, 1-11 (2005).
[CrossRef]

S. S. Sherif, E. R. Dowski, and W. T. Cathey, "A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems," Proc. SPIE 4471, 272-280 (2001).
[CrossRef]

S. Mezouari and A. R. Harvey, "Primary aberrations alleviated with phase pupil filters," Proc. SPIE 4768, 21-31 (2002).
[CrossRef]

Other (4)

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

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

M. R. Arnison, "Phase control and measurement in digital microscopy" (Sydney Digital Theses, Physics, 2006), http://hdl.handle.net/2123/569>.

T. Hellmuth, A. Bich, R. Börret, and A. Kelm, "Variable phaseplates for focus invariant optical systems," in Optical Design and Engineering II, L. Mazuray, R. Wartmann, eds., Proc. SPIE 5962, 596215 (2005).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Comparison of the FFT and approximate PSFs in the presence of defocus aberration for α=30π. The defocus values in (a), (b) and (c) are W20 =0, 2.5λ and 5λ, respectively.

Fig. 2.
Fig. 2.

PSF of an unbounded aperture for α=30π and W20 =2.5λ.

Fig. 3.
Fig. 3.

PSFs of a 2D system for α=30π and W20 =2.5λ. The left image is the FFT PSF, and the right image is the approximate PSF.

Fig. 4.
Fig. 4.

PSFs calculated from Eq. (4) for α=30π and W20 =0, 1λ, 2λ, 3λ.

Fig. 5.
Fig. 5.

PSFs calculated from Eq. (4) for α=30π and W31 =0, 1λ, 2λ, 3λ.

Fig. 6.
Fig. 6.

FFT PSFs of the system with a quartic phase mask for α=30π and W31 =0, 2λ, 4λ, 6λ.

Equations (32)

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h x W 20 = + q ( u ) exp ( j k W 20 u 2 j 2 πxu ) d u 2 ,
q ( u ) = { 1 2 exp ( j α u 3 ) u 1 , 0 u > 1
h x W 20 = 1 2 1 + 1 exp ( j α u 3 + j k W 20 u 2 j 2 πxu ) d u 2 .
h x W 20 = { 0 x ( k W 20 ) 2 6 πα π ( k W 20 ) 2 + 6 π α x { 1 + sin [ 4 ( ( k W 20 ) 2 + 6 π α x ) 3 2 27 α 2 ] } ( k W 20 ) 2 6 π α < x 3 α 2 k W 20 2 π π 2 ( k W 20 ) 2 + 6 π α x 3 α 2 k W 20 2 π < x 3 α + 2 k W 20 2 π 0 x > 3 α + 2 k W 20 2 π
h x W 20 = 1 2 + exp ( j α u 3 + j k W 20 u 2 j 2 π x u ) d u 2
u = t k W 20 3 α
= 1 2 + exp [ j α t 3 j 2 π ( x + ( k W 20 ) 2 6 π α ) t ] dt 2 ,
= 2 π 2 ( 3 α ) 2 3 A i 2 [ 2 π ( 3 α ) 1 3 ( x + ( k W 20 ) 2 6 π α ) ]
D = 3 α + 2 k W 20 2 π [ ( k W 20 ) 2 6 πα ] = ( 3 α + k W 20 ) 2 3 π α f 0 3 α π f 0 ( for small W 20 ) .
f ( x ) = d [ 4 ( ( k W 20 ) 2 + 6 π α x ) 3 2 27 α 2 1 2 π ] d x = 2 3 α ( k W 20 ) 2 + 6 π α x , ( k W 20 ) 2 6 π α < x 3 α 2 k W 20 2 π .
h x W 20 = 1 2 + exp ( j α u 3 + j k W 20 u 2 j 2 π x u ) du 2
u = t k W 20 3 α
= 1 2 1 + k W 20 3 α + 1 + k W 20 3 α exp [ j α ( t k W 20 3 α ) 3 + j ( k W 20 ) ( t k W 20 3 α ) 2 j 2 πx ( t k W 20 3 α ) ] d t 2 ] .
= 1 2 1 + k W 20 3 α + 1 + k W 20 3 α exp [ j α t 3 j 2 π ( x + ( k W 20 ) 2 6 πα ) t ] d t 2
h x W 31 = 1 2 1 + 1 exp ( j α u 3 + j k W 31 u 3 j 2 π x u ) d u 2
= 1 2 1 + 1 exp [ j ( α + k W 31 ) u 3 j 2 π x u ) d u 2
h x W 31 = 1 2 1 + 1 exp ( j α u 4 + j β u 3 j 2 π x u ) d u 2
u = t β 4 α
= 1 2 1 + β 4 α + 1 + β 4 α exp [ j α ( t β 4 α ) 4 + j β ( t β 4 α ) 3 j 2 π x ( t β 4 α ) ] d t 2 ,
= 1 2 1 + β 4 α + 1 + β 4 α exp { j [ α t 4 3 β 2 8 α t 2 2 π ( x β 3 16 π α 2 ) t ] } d t 2
PSF c ( x ) = 1 + 1 exp ( j α u 3 + j ψ u 2 j 2 πxu ) du ,
μ ( u ) = 3 α u 2 + 2 ψ u 2 π x ,
μ ′′ ( u ) = 6 α u + 2 ψ .
{ μ ( u 0 ) = 3 α u 0 2 + 2 ψ u 0 2 π x = 0 μ ′′ ( u 0 ) = 6 α u 0 + 2 ψ 0 . u 0 1
{ u 01 = 2 ψ + 4 ψ 2 + 24 π α x 6 α = ψ + ψ 2 + 6 π α x 3 α u 02 = 2 ψ 4 ψ 2 + 24 π α x 6 α = ψ ψ 2 + 6 π α x 3 α .
PSF c ( x ) 2 π μ ′′ ( u 01 ) exp { jsign [ μ ′′ ( u 01 ) ] π 4 } exp [ j μ ( u 01 ) ] +
2 π μ ′′ ( u 02 ) exp { jsign [ μ ′′ ( u 02 ) ] π 4 } exp [ j μ ( u 02 ) ] ,
sign ( x ) = { 1 x > 0 0 x = 0 · 1 x < 0
PSF c ( x ) 2 π μ ′′ ( u 01 ) exp { jsign [ μ ′′ ( u 01 ) ] π 4 } exp [ j μ ( u 01 ) ] .
PSF c ( x ) 2 π μ ′′ ( u 02 ) exp { jsign [ μ ′′ ( u 02 ) ] π 4 } exp [ j μ ( u 02 ) ] .
{ μ ′′ ( u 01 ) = μ ′′ ( u 02 ) = 2 ψ 2 + 6 π α x μ ( u 01 ) μ ( u 02 ) = 4 27 α 2 ( ψ 2 + 6 π α x ) 3 2 .
PSE c ( x ) { 0 2 π μ ( u 01 ) { 2 + 2 cos [ π 2 + μ ( u 01 ) μ ( u 02 ) ] } 2 π μ ( u 01 ) 0 x ψ 2 6 πα ψ 2 6 πα < x 3 α 2 ψ 2 π 3 α 2 ψ 2 π < x 3 α + 2 ψ 2 π x > 3 α + 2 ψ 2 π

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