Abstract

Spatial-domain design for wavefront-coding systems frequently simplifies the defining oscillatory integral of the point spread function (PSF) by means of the stationary phase approximation (SPA). Although the SPA applies over much of the support of the PSF, it tends to break down at or near the regions of highest intensity. A branch of mathematics known as catastrophe theory is shown to provide tools that can ferret out important design information precisely at the points where the SPA is unphysical.

© 2010 Optical Society of America

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References

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    [CrossRef]
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2010 (2)

2009 (3)

2008 (1)

2007 (3)

2006 (2)

2005 (1)

G. E. Johnson, P. Silveira, and E. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE 5817, 34–44 (2005).
[CrossRef]

2004 (2)

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

S. Sherif, W. Cathey, and E. Dowski, “Phase plate to extend depth of field of incoherent hybrid imaging systems,” Appl. Opt. 43, 2709–2721 (2004).
[CrossRef] [PubMed]

2003 (1)

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003).
[CrossRef]

1995 (1)

1988 (1)

1982 (1)

F. Wright, G. Dangelmayer, and D. Lang, “Singular coordinate sections of the conic umbilic catastrophes,” J. Phys. A 15, 3057–3071 (1982).
[CrossRef]

Andersson, M.

Barwick, S.

Bertsekas, D.

D. Bertsekas, Nonlinear Programming (Athena Scientific, 1999).

Born, M.

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

Bustin, N.

T. Vettenburg, N. Bustin, and A. Harvey, “Fidelity optimization for aberration tolerant hybrid imaging systems,” Opt. Express 18, 9220–9228 (2010).
[CrossRef] [PubMed]

T. Vettenburg, A. Wood, N. Bustin, and A. Harvey, “Optimality of pupil-phase profiles for increasing the defocus tolerance of hybrid digital-optical imaging systems,” Proc. SPIE 7429, 742903 (2009).
[CrossRef]

Casteñeda, J. O.

Cathey, W.

Chen, Y.

Christodoulides, D.

Dangelmayer, G.

F. Wright, G. Dangelmayer, and D. Lang, “Singular coordinate sections of the conic umbilic catastrophes,” J. Phys. A 15, 3057–3071 (1982).
[CrossRef]

Dowski, E.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Harvey, A.

T. Vettenburg, N. Bustin, and A. Harvey, “Fidelity optimization for aberration tolerant hybrid imaging systems,” Opt. Express 18, 9220–9228 (2010).
[CrossRef] [PubMed]

T. Vettenburg, A. Wood, N. Bustin, and A. Harvey, “Optimality of pupil-phase profiles for increasing the defocus tolerance of hybrid digital-optical imaging systems,” Proc. SPIE 7429, 742903 (2009).
[CrossRef]

Harvey, A. R.

Huckridge, D.

Johnson, G. E.

G. E. Johnson, P. Silveira, and E. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE 5817, 34–44 (2005).
[CrossRef]

Lang, D.

F. Wright, G. Dangelmayer, and D. Lang, “Singular coordinate sections of the conic umbilic catastrophes,” J. Phys. A 15, 3057–3071 (1982).
[CrossRef]

Li, Y.

Liu, L.

Lu, W.

Montes, E.

Muyo, G.

Nye, J.

J. Nye, “Evolution of the hyberbolic umbilic diffraction pattern from Airy rings,” J. Opt. A Pure Appl. Opt. 8, 304–314 (2006).
[CrossRef]

Pauca, V.

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003).
[CrossRef]

Pauca, V. P.

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

Plemmons, R.

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

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003).
[CrossRef]

Poston, T.

T. Poston and I. Stewart, Catastrophe Theory and Its Application (Pitman, 1978).

Prasad, S.

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

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003).
[CrossRef]

Sherif, S.

Silveira, P.

G. E. Johnson, P. Silveira, and E. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE 5817, 34–44 (2005).
[CrossRef]

Singh, A.

Siviloglou, G.

Stewart, I.

T. Poston and I. Stewart, Catastrophe Theory and Its Application (Pitman, 1978).

Sun, J.

Thom, R.

R. Thom, Structural Stability and Morphogenesis(Benjamin, 1972).

Torgersen, T.

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

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003).
[CrossRef]

Valdos, L. R. B.

van der Gracht, J.

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

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003).
[CrossRef]

Vettenburg, T.

T. Vettenburg, N. Bustin, and A. Harvey, “Fidelity optimization for aberration tolerant hybrid imaging systems,” Opt. Express 18, 9220–9228 (2010).
[CrossRef] [PubMed]

T. Vettenburg, A. Wood, N. Bustin, and A. Harvey, “Optimality of pupil-phase profiles for increasing the defocus tolerance of hybrid digital-optical imaging systems,” Proc. SPIE 7429, 742903 (2009).
[CrossRef]

Wolf, E.

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

Wood, A.

T. Vettenburg, A. Wood, N. Bustin, and A. Harvey, “Optimality of pupil-phase profiles for increasing the defocus tolerance of hybrid digital-optical imaging systems,” Proc. SPIE 7429, 742903 (2009).
[CrossRef]

G. Muyo, A. Singh, M. Andersson, D. Huckridge, A. Wood, and A. R. Harvey, “Infrared imaging with a wavefront-coded singlet lens,” Opt. Express 17, 21118–21123 (2009).
[CrossRef] [PubMed]

Wright, F.

F. Wright, G. Dangelmayer, and D. Lang, “Singular coordinate sections of the conic umbilic catastrophes,” J. Phys. A 15, 3057–3071 (1982).
[CrossRef]

Yang, Q.

Ye, Z.

Yu, F.

Zhang, W.

Zhao, H.

Zhao, T.

Zhu, Y.

Appl. Opt. (6)

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

J. Nye, “Evolution of the hyberbolic umbilic diffraction pattern from Airy rings,” J. Opt. A Pure Appl. Opt. 8, 304–314 (2006).
[CrossRef]

J. Phys. A (1)

F. Wright, G. Dangelmayer, and D. Lang, “Singular coordinate sections of the conic umbilic catastrophes,” J. Phys. A 15, 3057–3071 (1982).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Proc. SPIE (4)

G. E. Johnson, P. Silveira, and E. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE 5817, 34–44 (2005).
[CrossRef]

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

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003).
[CrossRef]

T. Vettenburg, A. Wood, N. Bustin, and A. Harvey, “Optimality of pupil-phase profiles for increasing the defocus tolerance of hybrid digital-optical imaging systems,” Proc. SPIE 7429, 742903 (2009).
[CrossRef]

Other (5)

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

R. Thom, Structural Stability and Morphogenesis(Benjamin, 1972).

T. Poston and I. Stewart, Catastrophe Theory and Its Application (Pitman, 1978).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

D. Bertsekas, Nonlinear Programming (Athena Scientific, 1999).

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Figures (14)

Fig. 1
Fig. 1

(a) B at z = 0 for the 1-D cubic with w = 5 (black solid), 10 (blue dotted), and 15 (green dashed). (b) T for the same parameters.

Fig. 2
Fig. 2

Plots of x 3 + p x for (a) p < 0 , (b) p = 0 , and (c) p > 0 .

Fig. 3
Fig. 3

B projected onto an image plane for a typical HU at (a) z = 0 and (b) z > 0 .

Fig. 4
Fig. 4

Plots of D E (left, blue dashed) and C mtf at k z = 2 (right, red solid) versus r for ϕ in Eq. (3).

Fig. 5
Fig. 5

FI (left, blue dashed) with respect to y-astig and C mtf (right, red solid) are plotted versus r. Both FI and C mtf were calculated for k q = 1 assuming no defocus.

Fig. 6
Fig. 6

(a) Position of singularities in the t y plane that give rise to the vertical branch of B for r = 0 in Eq. (3) are plotted for z = 0 (black solid) and 20 (green dashed). (b) The corresponding Bs are shown at those values of z. Red thick s are associated with two degenerate CPs in the pupil.

Fig. 7
Fig. 7

(a) B of a cubic EU for r = 3 in Eq. (3) is shown for z > 0 . (b) B for r 1 = 5 and r 2 = 10 in Eq. (8) is shown for z > 0 .

Fig. 8
Fig. 8

“Angle” that the normal of the tangent-space “plane” makes with the x 2 , y 2 -plane for masks described by Eq. (3) plotted versus r.

Fig. 9
Fig. 9

(a) Surface plot of the ratio of C mtf to defocus FI at k z = 2 versus r 1 and r 2 is shown for normalized masks described by Eq. (8). (b) The corresponding gray-scale contour plot is shown with the data in (a) interpolated for easier viewing.

Fig. 10
Fig. 10

Ratio of the defocus FI to C mtf is plotted versus k z for the (black solid) fifth-order mask r e , as well as the cubic masks with (green dashed) r = 0 and (red dotted) r = 3 in Eq. (3).

Fig. 11
Fig. 11

C mtf is plotted versus k q for the masks (black solid) ϕ s p , (red dotted) ϕ c , and (green dashed) ϕ p . Note: ϕ c and ϕ p had been optimized based on MTF alone.

Fig. 12
Fig. 12

Position of critical points in the s x plane is plotted at z = 0 for (black solid) ϕ s with normalized modulation w = 1.75 , (blue dotted) ϕ sopt with c = 0.52 and the same w, and ϕ sopt with renormalized w. Note: s and w are in waves.

Fig. 13
Fig. 13

(a) Defocus FI at k z = 2 is plotted versus the position of the degeneracy at ± x d 1 as c is increased in Eq. (11) and w is normalized. (b) Same FI is plotted versus the minimum of s at ± x d 2 for z = 0 as c is increased.

Fig. 14
Fig. 14

The ratio of the defocus FI to C mtf is plotted versus k z for (black solid) ϕ sopt with c = 0.52 , (red-dashed) ϕ s , and the cubic masks with (green dotted) r = 0 and (blue dashed–dotted) r = 3 in Eq. (3). A square pupil of width 2 was assumed.

Equations (12)

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h ( s , t , z ) = | Σ exp ( i k F ( x , y , s , t , z ) ) d x d y | 2 ,
d F d x = 3 w x 2 2 z x s = 0 , d 2 F d x 2 = 6 w x 2 z = 0 .
ϕ ( x , y ) = w ( x 3 + y 3 + r ( x 2 y + x y 2 ) ) ,
y = ( r 1 ) ± ( r 1 ) 2 4 2 .
ϕ x = ϕ x = 3 w x 2 + w r y 2 + 2 w r x y , ϕ y = ϕ y = w r x 2 + 3 w y 2 + 2 w r x y .
x = z o 3 w , s = z o 2 3 w , t = 3 w y 2 2 z o y ,
x 2 + y 2 = z 2 18 w 2 .
ϕ ( x , y ) = w ( x 5 + y 5 + r 1 ( x 4 y + x y 4 ) + r 2 ( x 3 y 2 + x 2 y 3 ) ) ,
x 2 + y 2 = ( z 2 200 w 2 ) 1 / 3 .
ϕ s ( x ) w ( β x 5 β 3 3 ! x 7 + β 5 5 ! x 9 β 7 7 ! x 11 ) .
ϕ sopt ( x ) = w ( c x 3 + β x 5 β 3 3 ! x 7 + β 5 5 ! x 9 β 7 7 ! x 11 ) .
s = w ( 3 c x 2 + 5 β x 4 7 β 3 3 ! x 6 + 9 β 5 5 ! x 8 11 β 7 7 ! x 10 ) .

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