Abstract

We present a two-dimensional function that graphically illustrates the effects of defocus on the optical transfer function (OTF) associated with a circularly symmetric pupil function. We call it the defocus transfer function (DTF). The function is similar in application to the ambiguity function, which can be used to display the OTF associated with a defocused rectangularly separable pupil function. The properties of the DTF make it useful for analyzing optical systems with circularly symmetric pupils when one is interested in the OTF as a function of defocus. In addition to presenting these properties, we give examples of the DTF for systems with clear, bifocal, and annular pupil functions.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Brenner, A. Lohmann, J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
    [CrossRef]
  2. E. R. Dowski, W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. 34, 1859–1866 (1995).
    [CrossRef] [PubMed]
  3. E. R. Dowski, W. T. Cathey, “Single-lens, single-image, incoherent passive ranging systems,” Appl. Opt. 33, 6762–6773 (1994).
    [CrossRef] [PubMed]
  4. A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, New York, 1969).
  5. C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967).
  6. A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
    [CrossRef]
  7. H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. Phys. Soc. 231, 91–103 (1955).
  8. M. Larsson, C. Beckman, A. Nystrom, S. Hard, J. Sjostrand, “Optical properties of diffractive, bifocal, intraocular lenses,” Appl. Opt. 31, 2377–2384 (1992).
    [CrossRef] [PubMed]
  9. W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt. Soc. Am. 50, 749–753 (1960).
    [CrossRef]
  10. H. H. Hopkins, “The aberration permissable in optical systems,” Proc. Phys. Soc. B 70, 449–470 (1955).
    [CrossRef]
  11. H. Bartelt, J. Ojeda-Castañeda, E. S. Enrique, “Misfocus tolerance seen by simple inspection of the ambiguity function,” Appl. Opt. 23, 2693–2696 (1984).
    [CrossRef] [PubMed]
  12. A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, New York, 1969), p. 120.
  13. J. Ojeda-Castañeda, R. Ramos, A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. 27, 2583–2586 (1988).
    [CrossRef] [PubMed]
  14. T.-C. Poon, M. Motamedi, “Optical/digital incoherent image processing for extended depth of field,” Appl. Opt. 26, 4612–4615 (1987).
    [CrossRef] [PubMed]

1995 (1)

1994 (1)

1992 (1)

1988 (1)

1987 (1)

1984 (1)

1983 (1)

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

1974 (1)

1960 (1)

1955 (2)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. Phys. Soc. 231, 91–103 (1955).

H. H. Hopkins, “The aberration permissable in optical systems,” Proc. Phys. Soc. B 70, 449–470 (1955).
[CrossRef]

Bartelt, H.

Beckman, C.

Bernfeld, M.

C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967).

Brenner, K.

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

Cathey, W. T.

Cook, C. E.

C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967).

Dowski, E. R.

Enrique, E. S.

Hard, S.

Hopkins, H. H.

H. H. Hopkins, “The aberration permissable in optical systems,” Proc. Phys. Soc. B 70, 449–470 (1955).
[CrossRef]

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. Phys. Soc. 231, 91–103 (1955).

Larsson, M.

Lohmann, A.

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

Motamedi, M.

Noyola-Isgleas, A.

Nystrom, A.

Ojeda-Castañeda, J.

Papoulis, A.

Poon, T.-C.

Ramos, R.

Rihaczek, A. W.

A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, New York, 1969), p. 120.

A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, New York, 1969).

Sjostrand, J.

Welford, W. T.

Appl. Opt. (6)

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

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

Proc. Phys. Soc. (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. Phys. Soc. 231, 91–103 (1955).

Proc. Phys. Soc. B (1)

H. H. Hopkins, “The aberration permissable in optical systems,” Proc. Phys. Soc. B 70, 449–470 (1955).
[CrossRef]

Other (3)

A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, New York, 1969), p. 120.

A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, New York, 1969).

C. E. Cook, M. Bernfeld, Radar Signals (Academic, New York, 1967).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Measuring defocus.

Fig. 2
Fig. 2

The MTF for a particular value of defocus is found from the AF by taking a slice through the origin with slope proportional to the defocus. Top, the magnitude of the AF for a clear one-dimensional pupil and three slices for defocus values of (a) W20 = 0, (b) W20 = 0.25λ, (c)W20 = 0.5λ. Bottom, the corresponding MTF’s.

Fig. 3
Fig. 3

Magnitude of the DTF for a clear circular pupil.

Fig. 4
Fig. 4

Top, magnitude of the DTF for a clear circular pupil. Bottom, MTF’s corresponding to (a) W20 = 0, (b) W20 = (1/π)λ, (c)W20 = (2/π)λ.

Fig. 5
Fig. 5

Pupil function and DTF of a bifocal lens for α = 8. Top, the phase of the pupil function. Center and bottom, the magnitude of the DTF. The two bright ridges correspond to the two foci of the lens.

Fig. 6
Fig. 6

Plots of the magnitude of the DTF for an annular pupil with ∊ = 4/5. The dotted triangle shows the DOF according to Hopkins’criterion (W20max = ±(5/9)λ).

Fig. 7
Fig. 7

Magnitude of the DTF for a clear pupil. The dotted triangle shows the DOF according to Hopkins’ criterion (W20max = ±(1/5)λ).

Fig. 8
Fig. 8

Magnitude of the DTF for a clear pupil compressed by a = 5/3. The dotted triangle shows the DOF according to Hopkins’ criterion.

Fig. 9
Fig. 9

Extended DTF, where ρ is allowed to be positive and negative.

Fig. 10
Fig. 10

Volume distributions in ρ and y for a clear pupil.

Fig. 11
Fig. 11

Volume distributions in ρ and y for an annular pupil, ∊ = 4/5.

Equations (38)

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

W20 = θ22 δz=12D2di2δz,
u = 2x/D.
Qu =Puexpj 2πλ W20 u2
Hu=-Qv+ u/2Q*v - u/2dv=-Pv + u/2P*v - u/2 ×expj2π 2W20λ uvdv.
Au, y =-Pv + u/2P*v - u/2expj2πvydv.
Hu =Au, 2W20λ u.
Pu ={1u10otherwise.
Au1, u2,y1, y2 = -Pv1 + u1/2,v2 + u2/2×P*v1 - u1/2,v2 - u2/2 × expj2π v1 y1 + v2 y2dv1dv2.
-Pu1, u22du1du2= 1.
Hu1,u2 = Au1,u2, 2W20λu1, 2W20λu2.
Hu1,u2 = HRρ.
HRρ= Hρ, 0 =Aρ, 0, 2W20λ ρ, 0.
Acsρ,y = Aρ, 0, y,0.
Acsρ,y = - Pv1 +ρ/2, v2P*v1 - ρ/2,v2 ×expj2πv1 ydv1dv2.
HRρ= Acsρ, 2W20λρ.
PRρ={1ρ  10otherwise.
PRρ={expjπαρ2 mod 2π2ρ10otherwise,
PRρ={1ρ10otherwise.
Gρ,v1 = -Pv1 + ρ/2,v2 P*v1 -ρ/2, v2dv2.
Acsρ,y = -Gρ, v1expj2πv1 ydv1.
Pu1,u2Acsρ,y,
a2Pau1,au2Acsaρ,y/a.
Pu1,u2expjπα u12 + u22Acsρ, y +αρ.
Acsρ, y2 Acs20,0= 1.
A˜csρ˜, y˜ =-Pv1 +ρ/2, v2P*v1 - ρ/2,v2 × expj2πv1 y ×exp-j2πρρ˜ + yy˜dv1dv2dρdy,
A˜csρ˜, y˜ =2Acs2y˜, -2ρ˜.
-0Acsρ, ydρdy = 12 -Acsρ, ydρdy= Acs0,0 = 1.
-Acsρ, ydy = 2 -Acs0, -2yexpj2πρydy,
0Acsρ, ydρ= -Acs2ρ, 0expj2πyρdρ.
volρ= 2 -Acs0, -2yexpj2πρydy = 2 -P v1, v22exp-j4πyv1dv1dv2 ×expj2πρydy = 2 -P v1, v22δρ - 2v1dv1dv2 =-Pρ/2, v22dv2.
Acs2ρ,0 = HR2ρ= H2ρ,0.
Hu1,u2 = - hx1, x2× exp-j2πx1 v1 + x2v2dx1dx2.
voly= -hx1,x2exp-j4π x1ρ×dx1dx2 expj2πyρdρ.
voly= -hx1,x2δy -2x1dx1dx2=12-hy2,x2dx2.
A˜csρ˜, y˜ = -Pv1 + ρ/2,v2 P*v1 -ρ/2, v2 ×exp-j2πρρ˜δv1 - y˜dv1dv2dρ
A˜csρ, y = - Py˜ + ρ/2,v2 P*y˜ -ρ/2, v2 ×exp-j2πρρ˜dρdv2.
A˜csρ˜,y˜ =2-Pv^1 + y˜,v2P* v^1- y˜, v2 ×expj2πv^1-2ρ˜dv^1 dv2.
A˜csρ˜,y˜ =2Acs2y˜, -2ρ˜.

Metrics