Abstract

We describe the use of a Wigner distribution function approach for exploring the problem of extending the depth of field in a hybrid imaging system. The Wigner distribution function, in connection with the phase–space curve that formulates a joint phase–space description of an optical field, is employed as a tool to display and characterize the evolving behavior of the amplitude point spread function as a wave propagating along the optical axis. It provides a comprehensive exhibition of the characteristics for the hybrid imaging system in extending the depth of field from both wave optics and geometrical optics. We use it to analyze several well-known optical designs in extending the depth of field from a new viewpoint. The relationships between this approach and the earlier ambiguity function approach are also briefly investigated.

© 2006 Optical Society of America

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    [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. T. Alieva and M. J. Bastiaans, "Phase-space distributions in quasi-polar coordinates and the fractional Fourier transform," J. Opt. Soc. Am. A 17, 2324-2329 (2000).
    [CrossRef]
  21. K. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323-326 (1983).
    [CrossRef]
  22. C. J. R. Sheppard and K. G. Larkin, "Wigner function and ambiguity function for nonparaxial wavefields," in Optical Processing and Computing: a Tribute to Adolf Lohmann, D.P.Casasent, H.J.Caulfield, W.J.Dallas, and H.H.Szu, eds., Proc. SPIE 4392,99-103 (2001).
  23. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2001), pp. 883-891.
  24. N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, 1986), pp. 252-320.

2005

2004

2003

2002

2000

1998

1995

D. Zalvidea, M. Lehman, S. Granieri, and E. E. Sicre, "Analysis of the Strehl ratio using the Wigner distribution function," Opt. Commun. 118, 207-214 (1995).
[CrossRef]

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

1994

1990

1988

1986

1985

1983

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

1979

1978

M. J. Bastianns, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
[CrossRef]

1971

Alieva, T.

Alonso, M. A.

Andres, P.

Athale, R.

Bastiaans, M. J.

T. Alieva and M. J. Bastiaans, "Phase-space distributions in quasi-polar coordinates and the fractional Fourier transform," J. Opt. Soc. Am. A 17, 2324-2329 (2000).
[CrossRef]

M. J. Bastiaans, "Wigner distribution function and its application to first-order optics," J. Opt. Soc. Am. 69, 1710-1716 (1979).
[CrossRef]

M. J. Bastiaans, "Application of the Wigner distribution function in optics," in The Wigner Distribution; Theory and Applications in Signal Processing, W.Mechlenbräuker and F.Hlawatsch, eds. (Elsevier, 1997), pp. 375-426.
[PubMed]

Bastianns, M. J.

M. J. Bastianns, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
[CrossRef]

Berriel-Valdos, L. R.

Bleistein, N.

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, 1986), pp. 252-320.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2001), pp. 883-891.

Brenner, K.

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

Castro, A.

Cathey, W. T.

Colautti, C.

Diaz, A.

Dowski, E. R.

Forbes, G. W.

Granieri, S.

D. Zalvidea, M. Lehman, S. Granieri, and E. E. Sicre, "Analysis of the Strehl ratio using the Wigner distribution function," Opt. Commun. 118, 207-214 (1995).
[CrossRef]

Handelsman, R. A.

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, 1986), pp. 252-320.

Lang, H.

Larkin, K. G.

C. J. R. Sheppard and K. G. Larkin, "Wigner function and ambiguity function for nonparaxial wavefields," in Optical Processing and Computing: a Tribute to Adolf Lohmann, D.P.Casasent, H.J.Caulfield, W.J.Dallas, and H.H.Szu, eds., Proc. SPIE 4392,99-103 (2001).

Lehman, M.

D. Zalvidea, M. Lehman, S. Granieri, and E. E. Sicre, "Analysis of the Strehl ratio using the Wigner distribution function," Opt. Commun. 118, 207-214 (1995).
[CrossRef]

Liu, L.

Lohmann, A. W.

A. W. Lohmann and B. H. Soffer, "Relationships between the Radon-Wigner and fractional Fourier transforms," J. Opt. Soc. Am. A 11, 1798-1801 (1994).
[CrossRef]

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

Mait, J. N.

Mino, M.

Montes, E. L.

Ojeda-Castañeda, J.

Okano, Y.

Sheppard, C. J. R.

C. J. R. Sheppard and K. G. Larkin, "Wigner function and ambiguity function for nonparaxial wavefields," in Optical Processing and Computing: a Tribute to Adolf Lohmann, D.P.Casasent, H.J.Caulfield, W.J.Dallas, and H.H.Szu, eds., Proc. SPIE 4392,99-103 (2001).

Sherif, S. S.

Sicre, E. E.

Soffer, B. H.

Tepichin, E.

van der Gracht, J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2001), pp. 883-891.

Yang, Q.

Zalvidea, D.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

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

D. Zalvidea, M. Lehman, S. Granieri, and E. E. Sicre, "Analysis of the Strehl ratio using the Wigner distribution function," Opt. Commun. 118, 207-214 (1995).
[CrossRef]

M. J. Bastianns, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26-30 (1978).
[CrossRef]

Opt. Express

Opt. Lett.

Other

M. J. Bastiaans, "Application of the Wigner distribution function in optics," in The Wigner Distribution; Theory and Applications in Signal Processing, W.Mechlenbräuker and F.Hlawatsch, eds. (Elsevier, 1997), pp. 375-426.
[PubMed]

C. J. R. Sheppard and K. G. Larkin, "Wigner function and ambiguity function for nonparaxial wavefields," in Optical Processing and Computing: a Tribute to Adolf Lohmann, D.P.Casasent, H.J.Caulfield, W.J.Dallas, and H.H.Szu, eds., Proc. SPIE 4392,99-103 (2001).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2001), pp. 883-891.

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, 1986), pp. 252-320.

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Figures (19)

Fig. 1
Fig. 1

Geometry of a modified optical system.

Fig. 2
Fig. 2

Equivalence of the two approaches to compute the incoherent out-of-focus PSF from the projection of the WDF.

Fig. 3
Fig. 3

(a) Gray-scale picture of the WDF of the rectangularly clear pupil function: (a) original and (b) sheared version of (a).

Fig. 4
Fig. 4

Rays of an ordinary optical imaging system.

Fig. 5
Fig. 5

Incoherent PSF of an ordinary optical system at the (a) in-focus plane and (b) out-of-focus plane.

Fig. 6
Fig. 6

(a) Gray-scale picture of the WDF of the pupil function with a cubic ( n = 3 ) phase part:(a) original and (b) sheared version of (a).

Fig. 7
Fig. 7

Incoherent PSF of an optical system modified by a cubic ( n = 3 ) phase plate at (a) in-focus plane and (b) out-of-focus plane.

Fig. 8
Fig. 8

(a) Gray-scale picture of the WDF of the pupil function with a quartic ( n = 4 ) phase part:(a) original and (b) sheared version of (a).

Fig. 9
Fig. 9

(a) Gray-scale picture of the WDF of the pupil function with a quintic ( n = 5 ) phase part:(a) original and (b) sheared version of (a).

Fig. 10
Fig. 10

Incoherent PSF of an optical system modified by a quartic ( n = 4 ) phase plate at the (a) in-focus plane and (b) out-of-focus plane.

Fig. 11
Fig. 11

Incoherent PSF of an optical system modified by a quintic ( n = 5 ) phase plate at the (a) in-focus plane and (b) out-of-focus plane.

Fig. 12
Fig. 12

(a) Gray-scale picture of the WDF of the pupil function with a logarithmic phase part:(a) original and (b) sheared version of (a).

Fig. 13
Fig. 13

Incoherent PSF of an optical system modified by a logarithmic phase plate at the (a) in-focus plane and (b) out-of-focus plane.

Fig. 14
Fig. 14

Plot of an Airy function.

Fig. 15
Fig. 15

Gray-scale picture of the WDF of a field with an odd-symmetric phase part; the PSC is shown as a dashed curve.

Fig. 16
Fig. 16

Geometric interpretation of the stationary phase equation given by Eq. (A2).

Fig. 17
Fig. 17

Half-scale replica of the PSC described by ( u ¯ , v ¯ ) when u 1 is fixed and u 2 varies, or u 2 is fixed and u 1 varies.

Fig. 18
Fig. 18

Superposition of a half-scale replica of the PSC, for which the PSC (shown as a thick curve) is an envelope, and the phase–space-area element (shown as a shadow) enclosed by two pairs of neighboring replicas is proportional to the reciprocal square of the amplitude factor of the WDF in Eq. (A3) at this region.

Fig. 19
Fig. 19

Exaggerated picture of the rays of an optical imaging system modified by a phase plate with an odd-symmetric form.

Equations (24)

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p ( ξ / a ) = { 1 2 exp [ j 2 π φ ( ξ / a ) / λ ] for | ξ | a 0 for | ξ | > a ,
| h ( x ¯ , z ) | 2 = | p ( u ) exp [ j 2 π z a 2 u 2 2 λ f ( f + z ) j 2 π a x ¯ u λ ( f + z ) ] d u | 2 ,
| h ( x ¯ , z ) | 2 = p ( u + u 2 ) p * ( u u 2 ) × exp [ j 2 π a 2 z u a f x ¯ λ f ( f + z ) u ] d u d u .
W p ( u , v ) = p ( u + u 2 ) p * ( u u 2 ) × exp ( j 2 π v u ) d u .
| h ( x , z ) | 2 = W p [ u , x 2 ω ( z ) u ] d u ,
| h ( x , z ) | 2 = W h ( u , x ) d u .
tan [ ϕ ( u ) ] = d φ ( u ) d u .
φ ( u ) = α sgn ( u ) | u | n ,
v = n α | u | n 1 ,
v = n α | u | n 1 + 2 ω ( z ) u ,
φ ( u ) = sgn ( u ) α u 2 ( ln | u | + β ) + γ u ,
v = 2 α | u | ln | u | + α ( 2 β + 1 ) | u | + γ ,
v = 2 α | u | ln | u | + [ sgn ( u ) α ( 2 β + 1 ) + ω ( z ) ] u + γ .
| h ( x , z ) | 2 = sin θ W p ( u cos θ v sin θ , u sin θ + v cos θ ) d v = W p ( u , cot θ + u sin θ ) d u ,
| h ( x , z ) | 2 = sin θ | FRT p ( u , θ ) | 2 .
H ( u , z ) = sin θ | FRT p ( x sin θ , θ ) | 2 × exp ( j 2 π u x ) d x .
H ( u , z ) = A p ( R , ϑ ) ,
W p ( u , v ) = 4 a 2 2 ( 1 | u | ) 2 ( 1 | u | ) exp { j 2 π [ φ ( u + u 2 ) φ ( u u 2 ) v u ] } d u ,
v 1 2 [ φ ( u + 1 2 u s ) + φ ( u 1 2 u s ) ] = 0.
W ˜ p f ( u , v ) = 4 a 2 cos ( 2 π Φ + σ π / 4 ) | φ ( u + u s / 2 ) φ ( u u s / 2 ) | 1 / 2 ,
δ 1 8 φ‴ ( u ) u s 2 0,
Φ δ u s + 1 24 φ‴ ( u ) u s 3 ,
φ ( u + 1 2 u s ) φ ( u 1 2 u s ) φ‴ ( u ) u s .
W ˜ p n [ u , φ ( u ) + δ ] = a 2 | 32 π 2 φ‴ ( u ) | 1 / 3 Ai { | 32 π 2 φ‴ ( u ) | 1 / 3 × sgn [ φ‴ ( u ) ] δ } .

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