Abstract

The odd-symmetric quadratic (OSQ) phase mask is examined as a candidate for reduction of working distance and enhancement of light collection in multiplex imaging systems. The knowledge gained from the exact mathematical representation of the optical transfer function of the OSQ phase mask imager is exploited to explain the limits of system performance and quantify the upper bound on the magnitude of defocus within which this wavefront coding imager can successfully operate. The sensitivity of this imaging system to defocus about the special imaging condition that yields an enhanced dynamic range is examined, and it is shown that the modulation transfer function (MTF) degradation when the magnitude of misfocus is increased past this condition is much more gradual than the degradation of a conventional imager past a zero-misfocus state. The condition required for the spatial frequency and angular resolution of this OSQ phase mask imager to exceed that of its counterpart scaled imager is established, and results of simulated imaging under a reduced working distance configuration are presented.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
  9. E. R. Dowski, Jr. and G. E. Johnson, "Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems," Proc. SPIE 3779, 137-145 (1999).
    [CrossRef]
  10. E. R. Dowski, Jr., R. H. Cormack, and S. D. Sarama, "Wavefront coding: jointly optimized optical and digital imaging systems," Proc. SPIE 4041, 114-120 (2000).
    [CrossRef]
  11. R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," Proc. SPIE 5299, 163-174 (2004).
    [CrossRef]
  12. M. Somayaji and M. P. Christensen, "Frequency analysis of the wavefront coding odd-symmetric quadratic phase mask," Appl. Opt. 46, 216-226 (2007).
    [CrossRef] [PubMed]
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    [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2007 (1)

2006 (2)

2004 (4)

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes--different concepts and their application to ultra flat image acquisition sensors," Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

A. Sauceda and J. Ojeda-Castañeda, "High focal depth with fractional-power wavefronts," Opt. Lett. 29, 560-562 (2004).
[CrossRef] [PubMed]

A. Castro and J. Ojeda-Castañeda, "Asymmetric phase masks for extended depth of field," Appl. Opt. 43, 3474-3479 (2004).
[CrossRef] [PubMed]

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," Proc. SPIE 5299, 163-174 (2004).
[CrossRef]

2002 (1)

2001 (1)

2000 (2)

E. R. Dowski, Jr., R. H. Cormack, and S. D. Sarama, "Wavefront coding: jointly optimized optical and digital imaging systems," Proc. SPIE 4041, 114-120 (2000).
[CrossRef]

J. Liu, V. Boopathi, T. C. Chong, and Y. Wu, "Space-invariant patches in diffraction-limited imaging," Opt. Eng. 39, 396-400 (2000).
[CrossRef]

1999 (1)

E. R. Dowski, Jr. and G. E. Johnson, "Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems," Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

1998 (2)

H. B. Wach, E. R. Dowski, Jr., and W. T. Cathey, "Control of chromatic focal shift through wavefront coding," Appl. Opt. 37, 5359-5367 (1998).
[CrossRef]

J. Nagy and D. O'Leary, "Restoring images degraded by spatially-variant blur," SIAM J. Sci. Comput. 19, 1063-1082 (1998).
[CrossRef]

1996 (1)

1995 (1)

1984 (1)

1978 (1)

H. J. Trussell and B. R. Hunt, "Image restoration of space-variant blurs by sectional methods," IEEE Trans. Acoust. Speech Signal Process. 26, 608-609 (1978).
[CrossRef]

1974 (1)

1971 (1)

Baron, A. E.

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," Proc. SPIE 5299, 163-174 (2004).
[CrossRef]

Bartelt, H.

Boden, A.

Boopathi, V.

J. Liu, V. Boopathi, T. C. Chong, and Y. Wu, "Space-invariant patches in diffraction-limited imaging," Opt. Eng. 39, 396-400 (2000).
[CrossRef]

Born, M.

M. Born and E. Wolf, "The diffraction theory of aberrations," in Principles of Optics-Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1989), pp. 459-490.
[PubMed]

Bräuer, A.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes--different concepts and their application to ultra flat image acquisition sensors," Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Castro, A.

Cathey, W. T.

Chong, T. C.

J. Liu, V. Boopathi, T. C. Chong, and Y. Wu, "Space-invariant patches in diffraction-limited imaging," Opt. Eng. 39, 396-400 (2000).
[CrossRef]

Christensen, M. P.

Chumachenko, V.

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," Proc. SPIE 5299, 163-174 (2004).
[CrossRef]

Cormack, R. H.

E. R. Dowski, Jr., R. H. Cormack, and S. D. Sarama, "Wavefront coding: jointly optimized optical and digital imaging systems," Proc. SPIE 4041, 114-120 (2000).
[CrossRef]

Dannberg, P.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes--different concepts and their application to ultra flat image acquisition sensors," Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Dowski, E. R.

W. T. Cathey and E. R. Dowski, "New paradigm for imaging systems," Appl. Opt. 41, 6080-6092 (2002).
[CrossRef] [PubMed]

E. R. Dowski, Jr., R. H. Cormack, and S. D. Sarama, "Wavefront coding: jointly optimized optical and digital imaging systems," Proc. SPIE 4041, 114-120 (2000).
[CrossRef]

E. R. Dowski, Jr. and G. E. Johnson, "Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems," Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

H. B. Wach, E. R. Dowski, Jr., and W. T. Cathey, "Control of chromatic focal shift through wavefront coding," Appl. Opt. 37, 5359-5367 (1998).
[CrossRef]

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

Duparré, J.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes--different concepts and their application to ultra flat image acquisition sensors," Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Enrique, E. S.

Goodman, J. W.

F. D. Russell and J. W. Goodman, "Nonredundant arrays and postdetection processing for aberration compensation in incoherent imaging," J. Opt. Soc. Am. 61, 182-191 (1971).
[CrossRef]

J. W. Goodman, "Frequency analysis of optical imaging systems," in Introduction to Fourier Optics, 2nd ed., L. Cox and J. M. Morriss, eds. (McGraw-Hill, 1996), pp. 126-171.

Greengard, A.

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," Proc. SPIE 5299, 163-174 (2004).
[CrossRef]

Haney, M. W.

Hanisch, R.

Herzig, H.-P.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes--different concepts and their application to ultra flat image acquisition sensors," Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Hunt, B. R.

H. J. Trussell and B. R. Hunt, "Image restoration of space-variant blurs by sectional methods," IEEE Trans. Acoust. Speech Signal Process. 26, 608-609 (1978).
[CrossRef]

Johnson, G. E.

E. R. Dowski, Jr. and G. E. Johnson, "Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems," Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

Kumagai, T.

Liu, J.

J. Liu, V. Boopathi, T. C. Chong, and Y. Wu, "Space-invariant patches in diffraction-limited imaging," Opt. Eng. 39, 396-400 (2000).
[CrossRef]

Miyatake, S.

Mo, J.

Nagy, J.

J. Nagy and D. O'Leary, "Restoring images degraded by spatially-variant blur," SIAM J. Sci. Comput. 19, 1063-1082 (1998).
[CrossRef]

Narayanswamy, R.

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," Proc. SPIE 5299, 163-174 (2004).
[CrossRef]

Ojeda-Castañeda, J.

O'Leary, D.

J. Nagy and D. O'Leary, "Restoring images degraded by spatially-variant blur," SIAM J. Sci. Comput. 19, 1063-1082 (1998).
[CrossRef]

Pelli, P.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes--different concepts and their application to ultra flat image acquisition sensors," Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Redding, D.

Russell, F. D.

Sarama, S. D.

E. R. Dowski, Jr., R. H. Cormack, and S. D. Sarama, "Wavefront coding: jointly optimized optical and digital imaging systems," Proc. SPIE 4041, 114-120 (2000).
[CrossRef]

Sauceda, A.

Sawchuk, A. A.

Scharf, T.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes--different concepts and their application to ultra flat image acquisition sensors," Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Schreiber, P.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes--different concepts and their application to ultra flat image acquisition sensors," Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Somayaji, M.

Tanida, J.

Trussell, H. J.

H. J. Trussell and B. R. Hunt, "Image restoration of space-variant blurs by sectional methods," IEEE Trans. Acoust. Speech Signal Process. 26, 608-609 (1978).
[CrossRef]

Völkel, R.

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes--different concepts and their application to ultra flat image acquisition sensors," Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

Wach, H. B.

Wolf, E.

M. Born and E. Wolf, "The diffraction theory of aberrations," in Principles of Optics-Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1989), pp. 459-490.
[PubMed]

Wu, Y.

J. Liu, V. Boopathi, T. C. Chong, and Y. Wu, "Space-invariant patches in diffraction-limited imaging," Opt. Eng. 39, 396-400 (2000).
[CrossRef]

Yamada, K.

Appl. Opt. (9)

IEEE Trans. Acoust. Speech Signal Process. (1)

H. J. Trussell and B. R. Hunt, "Image restoration of space-variant blurs by sectional methods," IEEE Trans. Acoust. Speech Signal Process. 26, 608-609 (1978).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Eng. (1)

J. Liu, V. Boopathi, T. C. Chong, and Y. Wu, "Space-invariant patches in diffraction-limited imaging," Opt. Eng. 39, 396-400 (2000).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (4)

J. Duparré, P. Schreiber, P. Dannberg, T. Scharf, P. Pelli, R. Völkel, H.-P. Herzig, and A. Bräuer, "Artificial compound eyes--different concepts and their application to ultra flat image acquisition sensors," Proc. SPIE 5346, 89-100 (2004).
[CrossRef]

E. R. Dowski, Jr. and G. E. Johnson, "Wavefront coding: a modern method of achieving high performance and/or low cost imaging systems," Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

E. R. Dowski, Jr., R. H. Cormack, and S. D. Sarama, "Wavefront coding: jointly optimized optical and digital imaging systems," Proc. SPIE 4041, 114-120 (2000).
[CrossRef]

R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, "Applications of wavefront coded imaging," Proc. SPIE 5299, 163-174 (2004).
[CrossRef]

SIAM J. Sci. Comput. (1)

J. Nagy and D. O'Leary, "Restoring images degraded by spatially-variant blur," SIAM J. Sci. Comput. 19, 1063-1082 (1998).
[CrossRef]

Other (2)

M. Born and E. Wolf, "The diffraction theory of aberrations," in Principles of Optics-Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1989), pp. 459-490.
[PubMed]

J. W. Goodman, "Frequency analysis of optical imaging systems," in Introduction to Fourier Optics, 2nd ed., L. Cox and J. M. Morriss, eds. (McGraw-Hill, 1996), pp. 126-171.

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Figures (7)

Fig. 1
Fig. 1

Setup for form factor enhancement. (a) An OSQ phase mask is introduced at the pupil of a conventional imager, and the detector is brought forward to obtain (b) a flat form factor wavefront coding system. The performance of the wavefront coding imager is compared with (c) that of a scaled imager with the same working distance as the former. The configuration in (b) collects more light that that in (c) by a factor of ( L 2 / L 1 ) 2 .

Fig. 2
Fig. 2

MTF and AF magnitude plots of an OSQ phase mask imager. (a) Normalized 1-D MTF as a function of the normalized spatial frequency u. (b) Magnitude of the ambiguity function, where an absence of nulls inside the passband of the MTF marks the operating region of this imager. The MTFs at the boundaries of this region display a higher dynamic range than those for other misfocus values within the area where imaging is possible, when | ψ | = α . The value of α is set at 30 π .

Fig. 3
Fig. 3

Simulated flat form factor imaging by an OSQ phase mask system. Left column, intermediate images formed at the image capture plane, translated to offset the lateral shifts due to the phase of the OTF. Right column, final images after digital reconstruction. The rows from top to bottom represent images captured by a 10-bit detector at d c = 0.8 f , d c = 0.641 f ( ψ = α ) , d c = 0.6281 f ( ψ = α 4 π ) , and d c = 0.5 f , respectively. A 2× reduction in working distance ( d c = 0.5 f ) resulted in severe noise in the reconstructed image due to a greatly lowered MTF. Image scales have been normalized to enable comparison of similar object features in the scene. The value of α was set at 70 π .

Fig. 4
Fig. 4

Design range of the OSQ phase mask imager. (a) MTF when ψ = α 4 π , along with the magnitude of I 3 ( u , ψ ) , which touches down onto the spatial frequency axis at u z = ±¼. (b) The corresponding misfocus radial line on the AF magnitude plot demarcates the boundary of the region within which imaging is possible. The value of α was taken to be 30 π .

Fig. 5
Fig. 5

Bandwidth versus working distance for an OSQ phase mask imager. The normalized spatial frequency bandwidth is shown plotted against the working distance for various configurations of the wavefront coding system. The bandwidth curves are shown for working distances such that | ψ | α .

Fig. 6
Fig. 6

Angular resolution versus working distance of an OSQ phase mask imager. The plot shows various configurations of the wavefront coding imager wherein the angular resolution worsens as the working distance is reduced. The solid curves represent the angular resolution of conventional imaging systems with specified F/# values. Each curve of the wavefront coding system extends up to the point where the magnitude of misfocus equals the strength of the phase mask.

Fig. 7
Fig. 7

Spatial variation of PSFs due to aberrations. (a) schematic showing the distance AB on the optical axis between the image capture plane and the diffraction-limited imaging surface, which is different from the off-axis distance CD, thereby resulting in variations in the PSFs at these two locations. (b) Schematic representing the partitioning of the image into isoplanatic regions, within each of which the PSFs are considered to be linear and shift invariant. All misfocus values due to this variation across the image fall between ψ min and ψ max as seen in (c) the AF magnitude plots.

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

η ( x ) = α   sign ( x ) | x | 2 ; 1 x 1 ; α > 0 ,
H ( u , ψ ) = { ( 1 2 | u | ) sinc [ 4 u π ( ψ α ) ( 1 2 | u | ) ] exp [ j 2 u ( ψ α ) ] + ( π 8 α ) 1 / 2  exp [ j 2 α u 2 ( 1 ψ 2 α 2 ) sign ( u ) ] × 1 2 { [ C ( b C ( u ) ) C ( a C ( u ) ) ] + j   sign ( u ) [ S ( b C ( u ) ) S ( a C ( u ) ) ] } + ( 1 2 | u | ) × sinc [ 4 u π ( ψ + α ) ( 1 2 | u | ) ] exp [ j 2 u ( ψ + α ) ] = H C ( u , ψ ) 0 | u | 1 2 ( π 8 α ) 1 / 2 exp [ j 2 α u 2 ( 1 ψ 2 α 2 ) sign ( u ) ] 1 2 { [ C ( b T ( u ) ) C ( a T ( u ) ) ] + j   sign ( u ) × [ S ( b T ( u ) ) S ( a T ( u ) ) ] } = H T ( u , ψ ) 1 2 | u | 1 0 otherwise ,
ψ = π L 2 4 λ ( 1 f 1 d o 1 d c ) ,
a T ( u ) = ( 4 α π ) 1 / 2 [ | u | ( 1 + ψ α ) 1 ] ,
b T ( u ) = ( 4 α π ) 1 / 2 [ 1 | u | ( 1 ψ α ) ] ,
a C ( u ) = ( 4 α π ) 1 / 2 | u | ( ψ α 1 ) ,
b C ( u ) = ( 4 α π ) 1 / 2 | u | ( ψ α + 1 ) .
I 1 ( u , ψ ) = ( 1 2 | u | ) sinc [ 4 u π ( ψ α ) ( 1 2 | u | ) ] × exp [ j 2 u ( ψ α ) ] ,
I 2 ( u , ψ ) = ( π 8 α ) 1 / 2 exp [ j 2 α u 2 ( 1 ψ 2 α 2 ) sign ( u ) ] × 1 2 { [ C ( b C ( u ) ) C ( a C ( u ) ) ] + j   sign ( u ) × [ S ( b C ( u ) ) S ( a C ( u ) ) ] } ,
I 3 ( u , ψ ) = ( 1 2 | u | ) sinc [ 4 u π ( ψ + α ) ( 1 2 | u | ) ] × exp [ j 2 u ( ψ + α ) ] .
u c = α α + | ψ | , | ψ | α , α > 0 .
Δ l 3 = ( 1 u c ) ( λ f 2 L 2 ) .
θ 3 = ( 1 u c ) ( λ f 2 L 2 ) ( 1 f 1 ) .
α π L 2 4 λ [ ( f 1 / L 1 ) ( f 2 / L 2 ) ] ( f 2 f 1 1 ) , ( f 1 / L 1 ) > ( f 2 / L 2 ) .
H ( u , ψ ) = { ( 1 | u | ) sinc [ 4 ψ π u ( 1 | u | ) ] 1 u 1 0 otherwise .
sin [ 4 u ψ ( 1 | u | ) ] = 0 , 0 < | u | < 1.
4 ψ u ( 1 u ) = ±m π , m = 0 , 1 , 2 ,   …   .
4 | ψ | u ( 1 u ) = m π , m = 1 , 2 ,   …   .
u 2 u + m π 4 | ψ | = 0 , m = 1 , 2 ,   …   ,
u z = 1 2 ± 1 2 1 m π | ψ | , m = 1 , 2 ,   …   .
| ψ | m π , m = 1 , 2 ,   …   .
sin [ 4 u ( ψ + α ) ( 1 2 | u | ) ] = 0 , 0 < | u | < 1.
4 | α + ψ | u ( 1 2 u ) = m π , m = 1 , 2 ,   …   .
u 2 u 2 + m π 4 | α + ψ | = 0 , m = 1 , 2 ,   …   .
u z = 1 4 ± 1 2 1 4 m π | α + ψ | , m = 1 , 2 ,   …   .
| ψ | α + 4 m π , | ψ | α | ψ | α 4 m π , | ψ | < α ,
| ψ | α + 4 π .
p π L 2 π L 2 + 4 λ ( F / # ) ( α + 4 π ) .
L 2 4 λ ( F / # ) π ( p 1 p ) ( α + 4 π ) .
F / # π L 2 4 λ ( α + 4 π ) ( 1 p p ) .

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