Abstract

The optical transfer function of a cubic phase mask wavefront coding imaging system is experimentally measured across the entire range of defocus values encompassing the system’s functional limits. The results are compared against mathematical expressions describing the spatial frequency response of these computational imagers. Experimental data shows that the observed modulation and phase transfer functions, available spatial frequency bandwidth and design range of this imaging system strongly agree with previously published mathematical analyses. An imaging system characterization application is also presented wherein it is shown that the phase transfer function is more robust than the modulation transfer function in estimating the strength of the cubic phase mask.

© 2012 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  27. M. R. Arnison, C. J. Cogswell, C. J. R. Sheppard, and P. Török, “Wavefront coding fluorescence microscopy using high aperture lenses,” in Optical Imaging and Microscopy: Techniques and Advanced Systems, P. Török and F.-J. Kao, eds. (Springer-Verlag, Berlin, 2003), pp. 143–165.
  28. S. Bradburn, W. T. Cathey, and E. R. Dowski, “Realizations of focus invariance in optical-digital systems with wave-front coding,” Appl. Opt. 36(35), 9157–9166 (1997).
    [CrossRef] [PubMed]
  29. R. Narayanswamy, A. E. Baron, V. Chumachenko, and A. Greengard, “Applications of wavefront coded imaging,” Proc. SPIE 5299, 163–174 (2004).
    [CrossRef]
  30. E. R. Dowski 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]
  31. M. Somayaji, V. R. Bhakta, and M. P. Christensen, “Experimental validation of exact optical transfer function of cubic phase mask wavefront coding imaging systems,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2010), paper FThT7.
  32. S. Bradburn, W. T. Cathey, and E. R. Dowski, “Realizations of focus invariance in optical-digital systems with wave-front coding,” Appl. Opt. 36(35), 9157–9166 (1997).
    [CrossRef] [PubMed]
  33. K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44(5), 323–326 (1983).
    [CrossRef]
  34. Q. Kim, G. Yang, C. J. Wrigley, T. J. Cunningham, and B. Pain, “Modulation transfer function of active pixel focal plane arrays,” Proc. SPIE 3950, 49–56 (2000).
    [CrossRef]
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    [CrossRef] [PubMed]
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2011 (2)

2010 (2)

2009 (2)

M. Demenikov, E. Findlay, and A. R. Harvey, “Miniaturization of zoom lenses with a single moving element,” Opt. Express 17(8), 6118–6127 (2009).
[CrossRef] [PubMed]

G. Muyo and A. R. Harvey, “The effect of detector sampling in wavefront-coded imaging systems,” J. Opt. A, Pure Appl. Opt. 11(5), 054002 (2009).
[CrossRef]

2008 (2)

2007 (2)

2006 (2)

2005 (3)

2004 (3)

A. Sauceda and J. Ojeda-Castañeda, “High focal depth with fractional-power wave fronts,” Opt. Lett. 29(6), 560–562 (2004).
[CrossRef] [PubMed]

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[CrossRef]

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

2003 (1)

2001 (1)

2000 (3)

G. E. Johnson, E. R. Dowski, and W. T. Cathey, “Passive ranging through wave-front coding: information and application,” Appl. Opt. 39(11), 1700–1710 (2000).
[CrossRef] [PubMed]

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

Q. Kim, G. Yang, C. J. Wrigley, T. J. Cunningham, and B. Pain, “Modulation transfer function of active pixel focal plane arrays,” Proc. SPIE 3950, 49–56 (2000).
[CrossRef]

1999 (2)

D. L. Marks, R. A. Stack, D. J. Brady, and J. van der Gracht, “Three-dimensional tomography using a cubic-phase plate extended depth-of-field system,” Opt. Lett. 24(4), 253–255 (1999).
[CrossRef] [PubMed]

E. R. Dowski 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 (1)

1997 (2)

1996 (1)

1995 (1)

1983 (1)

K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44(5), 323–326 (1983).
[CrossRef]

Bagheri, S.

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]

Barwick, S.

Bhakta, V. R.

Bradburn, S.

Brady, D. J.

Brenner, K.-H.

K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44(5), 323–326 (1983).
[CrossRef]

Cathey, W. T.

Chang, H.

Chen, S.

Chi, W.

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, R. H. Cormack, and S. D. Sarama, “Wavefront coding: jointly optimized optical and digital imaging systems,” Proc. SPIE 4041, 114–120 (2000).
[CrossRef]

Cunningham, T. J.

Q. Kim, G. Yang, C. J. Wrigley, T. J. Cunningham, and B. Pain, “Modulation transfer function of active pixel focal plane arrays,” Proc. SPIE 3950, 49–56 (2000).
[CrossRef]

de Farias, D. P.

Deaver, D. M.

Demenikov, M.

Dowski, E.

Dowski, E. R.

Escamilla, H. M.

Fan, Z.

Findlay, E.

George, N.

Greengard, A.

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

Harvey, A. R.

Johnson, G. E.

Kim, Q.

Q. Kim, G. Yang, C. J. Wrigley, T. J. Cunningham, and B. Pain, “Modulation transfer function of active pixel focal plane arrays,” Proc. SPIE 3950, 49–56 (2000).
[CrossRef]

Kubala, K.

Landgrave, J. E. A.

Lee, S.-H.

Lohmann, A.

K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44(5), 323–326 (1983).
[CrossRef]

Marks, D. L.

Muyo, G.

G. Muyo and A. R. Harvey, “The effect of detector sampling in wavefront-coded imaging systems,” J. Opt. A, Pure Appl. Opt. 11(5), 054002 (2009).
[CrossRef]

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]

Narayanswamy, R.

Ojeda-Castaneda, J.

Ojeda-Castañeda, J.

A. Sauceda and J. Ojeda-Castañeda, “High focal depth with fractional-power wave fronts,” Opt. Lett. 29(6), 560–562 (2004).
[CrossRef] [PubMed]

K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44(5), 323–326 (1983).
[CrossRef]

Pain, B.

Q. Kim, G. Yang, C. J. Wrigley, T. J. Cunningham, and B. Pain, “Modulation transfer function of active pixel focal plane arrays,” Proc. SPIE 3950, 49–56 (2000).
[CrossRef]

Park, N.-C.

Park, Y.-P.

Pauca, V. P.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[CrossRef]

Plemmons, R. J.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[CrossRef]

Prasad, S.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[CrossRef]

Sarama, S. D.

E. R. Dowski, 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.

Silveira, P. E. X.

Somayaji, M.

Stack, R. A.

Taylor, M. G.

Torgersen, T. C.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[CrossRef]

van der Gracht, J.

Wach, H. B.

Wrigley, C. J.

Q. Kim, G. Yang, C. J. Wrigley, T. J. Cunningham, and B. Pain, “Modulation transfer function of active pixel focal plane arrays,” Proc. SPIE 3950, 49–56 (2000).
[CrossRef]

Xu, Z.

Yang, G.

Q. Kim, G. Yang, C. J. Wrigley, T. J. Cunningham, and B. Pain, “Modulation transfer function of active pixel focal plane arrays,” Proc. SPIE 3950, 49–56 (2000).
[CrossRef]

Appl. Opt. (12)

S. Bradburn, W. T. Cathey, and E. R. Dowski, “Realizations of focus invariance in optical-digital systems with wave-front coding,” Appl. Opt. 36(35), 9157–9166 (1997).
[CrossRef] [PubMed]

S. Bradburn, W. T. Cathey, and E. R. Dowski, “Realizations of focus invariance in optical-digital systems with wave-front coding,” Appl. Opt. 36(35), 9157–9166 (1997).
[CrossRef] [PubMed]

H. B. Wach, E. R. Dowski, and W. T. Cathey, “Control of chromatic focal shift through wave-front coding,” Appl. Opt. 37(23), 5359–5367 (1998).
[CrossRef] [PubMed]

G. E. Johnson, E. R. Dowski, and W. T. Cathey, “Passive ranging through wave-front coding: information and application,” Appl. Opt. 39(11), 1700–1710 (2000).
[CrossRef] [PubMed]

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

M. Somayaji and M. P. Christensen, “Enhancing form factor and light collection of multiplex imaging systems by using a cubic phase mask,” Appl. Opt. 45(13), 2911–2923 (2006).
[CrossRef] [PubMed]

P. E. X. Silveira and R. Narayanswamy, “Signal-to-noise analysis of task-based imaging systems with defocus,” Appl. Opt. 45(13), 2924–2934 (2006).
[CrossRef] [PubMed]

M. Somayaji and M. P. Christensen, “Frequency analysis of the wavefront-coding odd-symmetric quadratic phase mask,” Appl. Opt. 46(2), 216–226 (2007).
[CrossRef] [PubMed]

M. Somayaji and M. P. Christensen, “Improving photon count and flat profiles of multiplex imaging systems with the odd-symmetric quadratic phase modulation mask,” Appl. Opt. 46(18), 3754–3765 (2007).
[CrossRef] [PubMed]

R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, “Extending the imaging volume for biometric iris recognition,” Appl. Opt. 44(5), 701–712 (2005).
[CrossRef] [PubMed]

S. Barwick, “Catastrophes in wavefront-coding spatial-domain design,” Appl. Opt. 49(36), 6893–6902 (2010).
[CrossRef] [PubMed]

S. Chen, Z. Fan, H. Chang, and Z. Xu, “Nonaxial Strehl ratio of wavefront coding systems with a cubic phase mask,” Appl. Opt. 50(19), 3337–3345 (2011).
[CrossRef] [PubMed]

J. Opt. A, Pure Appl. Opt. (1)

G. Muyo and A. R. Harvey, “The effect of detector sampling in wavefront-coded imaging systems,” J. Opt. A, Pure Appl. Opt. 11(5), 054002 (2009).
[CrossRef]

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

Opt. Commun. (1)

K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44(5), 323–326 (1983).
[CrossRef]

Opt. Express (5)

Opt. Lett. (6)

Proc. SPIE (5)

Q. Kim, G. Yang, C. J. Wrigley, T. J. Cunningham, and B. Pain, “Modulation transfer function of active pixel focal plane arrays,” Proc. SPIE 3950, 49–56 (2000).
[CrossRef]

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

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

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[CrossRef]

Other (5)

M. Somayaji and M. P. Christensen, “Form factor enhancement of imaging systems using a cubic phase mask,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, Technical Digest (Optical Society of America, 2005), paper CMB4.

M. Somayaji and M. P. Christensen, “Form factor enhancement of imaging systems using a cubic phase mask,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2005), paper FThU5.

M. R. Arnison, C. J. Cogswell, C. J. R. Sheppard, and P. Török, “Wavefront coding fluorescence microscopy using high aperture lenses,” in Optical Imaging and Microscopy: Techniques and Advanced Systems, P. Török and F.-J. Kao, eds. (Springer-Verlag, Berlin, 2003), pp. 143–165.

M. Somayaji, V. R. Bhakta, and M. P. Christensen, “Experimental validation of exact optical transfer function of cubic phase mask wavefront coding imaging systems,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2010), paper FThT7.

V. R. Bhakta, M. Somayaji, and M. P. Christensen, “Phase transfer function of sampled imaging systems,” in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuB1.

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Figures (7)

Fig. 1
Fig. 1

Experimental setup used to measure the spatial frequency response of a cubic phase mask wavefront coding imaging system across a large range of defocus values.

Fig. 2
Fig. 2

(a) MTF and (b) PTF at the in-focus plane (ψ = 0) of the cubic phase mask imaging system shown in Fig. 1. Ma and Θa are the approximate MTF and PTF from Eq. (4); Mt and Θt are the exact theoretical MTF and PTF given by Eq. (7) and Eq. (11) respectively; and Mm and Θm are the measured MTF and PTF. Mp is the pixel MTF as per Eq. (12) and Θc is a linear phase correction applied to the measured PTF. Inset in (a): Measured PSF at the in-focus plane, shown in pseudo-color to highlight its features.

Fig. 3
Fig. 3

MTF versus normalized spatial frequency u and distance from best focus ziza in mm, for a cubic phase mask wavefront coding imaging system. (a) Theoretical approximate MTF Ma(u,ψ) lowered by the pixel MTF Mp(u); (b) theoretical exact MTF Mt(u,ψ) lowered by the pixel MTF and; (c) measured system MTF Mm(u,ψ).

Fig. 4
Fig. 4

Theoretical versus experimentally observed values of the cutoff spatial frequency uc marking the available bandwidth, as a function of the distance from best focus ziza.

Fig. 5
Fig. 5

AF magnitude plots derived from (a) theoretical data Mt × Mp and (b) measured data Mm. Brighter colors denote regions of higher power. Slices of these plots along the green radial lines yield the MTFs at ±|ψm| when projected onto the horizontal u axis. The AF plot obtained from Mm extends only up to the range spanning –4.937α < ψ < 4.685α for which this data was collected.

Fig. 6
Fig. 6

PTF in multiples of π radians, versus normalized spatial frequency u and distance from best focus ziza for a cubic phase mask wavefront coding imaging system. (a) Theoretical PTF Θt(u,ψ)/π; and (b) measured PTF Θm(u,ψ)/π to which a linear phase correction Θc(u,ψ)/π has been applied.

Fig. 7
Fig. 7

Estimations of the cubic phase mask strength α at various defocus values from theoretical and experimental data sources, using a polynomial curve-fitting approach. The dashed black line in each figure marks the true value of α.

Tables (1)

Tables Icon

Table 1 Mean values of estimates of the cubic phase mask strength α across a defocus range of –20 ≤ ψ ≤ 20, based on MTF and PTF data corresponding to a design value of α = 38.822

Equations (14)

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

P( x )={ 1 2 exp[ j( ψ x 2 + α x 3 ) ] | x |1, α> α min 0 otherwise .
α= 2πξ λ ,
ψ= πL 2 4λ ( 1 f 1 z o 1 z a ),
H a ( u,ψ )={ ( π 24α| u | ) 1/2 exp[ 2j( α u 3 ψ 2 u 3α ) ] 0<| u |1 1 u=0 ,
H t ( u,ψ )= ( π 24α| u | ) 1/2 exp[ 2j( α u 3 ψ 2 u 3α ) ] × 1 2 { C( b( u ) )C( a( u ) )+jS( b( u ) )jS( a( u ) ) }, 0| u |1.
a( u )= ( 12αu π ) 1/2 ( ψ 3α ( 1| u | ) ), b( u )= ( 12αu π ) 1/2 ( ψ 3α +( 1| u | ) ).
M t ( u,ψ )= ( π 24α| u | ) 1/2 × 1 2 { [ C( b( u ) )C( a( u ) ) ] 2 + [ S( b( u ) )S( a( u ) ) ] 2 } 1/2 .
u c =( 1 | ψ | 3α ).
H( u,ψ )=A( u, 2uψ /π ).
| ψ m |=3α [ ( 9αt 4 ) 2 ] 1/3 .
Θ t ( u,ψ )=2α u 3 2 u 3α + tan 1 { S( b( u ) )S( a( u ) ) C( b( u ) )C( a( u ) ) }.
M p ( u )=sinc( u η o p ),
αu= π 24 M 2 .
2α u 3 2 3α u=Θ.

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