Abstract

VanderLugt [Appl. Opt. 29, 3352 (1990)] presented sampling rates for the amplitude of Fresnel diffraction patterns. These apply to any plane in a coherent optical system. Although these sampling rates represent the amplitude of diffraction patterns accurately, they are not adequate for the retention of complete information in complex-valued Fresnel diffraction patterns. I show this by considering the ability to reconstruct the original input image through backward diffraction of the forward diffraction pattern of such an image. I then extend the VanderLugt sampling techniques such that reliable sampling of the phase of these Fresnel diffraction patterns can also be achieved. The analysis is restricted to lensless optical systems. The new sampling rates are tested with numerical computations of Fresnel diffraction patterns and rigorous scalar diffraction patterns in both forward and backward directions.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. G. Stremler, Introduction to Communication Systems, 3rd ed. (Addison-Wesley, Reading, Mass., 1990), Chap. 3, p. 129.
  2. A. VanderLugt, “Optimal sampling of Fresnel transforms,” Appl. Opt. 29, 3352–3361 (1990).
    [CrossRef] [PubMed]
  3. A. J. Devaney, G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
    [CrossRef]

1990 (1)

1973 (1)

A. J. Devaney, G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
[CrossRef]

Devaney, A. J.

A. J. Devaney, G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
[CrossRef]

Sherman, G. C.

A. J. Devaney, G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
[CrossRef]

Stremler, F. G.

F. G. Stremler, Introduction to Communication Systems, 3rd ed. (Addison-Wesley, Reading, Mass., 1990), Chap. 3, p. 129.

VanderLugt, A.

Appl. Opt. (1)

SIAM Rev. (1)

A. J. Devaney, G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
[CrossRef]

Other (1)

F. G. Stremler, Introduction to Communication Systems, 3rd ed. (Addison-Wesley, Reading, Mass., 1990), Chap. 3, p. 129.

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 (8)

Fig. 1
Fig. 1

Different diffraction regions behind the input plane.

Fig. 2
Fig. 2

Interference of two plane waves propagating at maximum opposite angles.

Fig. 3
Fig. 3

Profiles of the intensity and the phase sampling rates for different intersections of the regions behind the input plane. (S denotes the sampling rate of the input image.)

Fig. 4
Fig. 4

Intensity of the input image.

Fig. 5
Fig. 5

Phase of the input image.

Fig. 6
Fig. 6

Intensity of the Fresnel diffraction pattern.

Fig. 7
Fig. 7

Reconstructed input image.

Fig. 8
Fig. 8

Phase of the inadequately sampled Fresnel diffraction.

Equations (16)

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

S out = Δ x / λ z ,
T = z = L / S λ
L = λ z S + L ,
S = L / λ z .
T = L / S λ = λ z 2 S / L + z ,
f = exp ( j k u 1 ) + exp ( j k u 2 ) = 2 exp [ j k 2 ( u 1 + u 2 ) ] cos [ k 2 ( u 1 u 2 ) ]
u 1 = a x + ( 1 a 2 ) 1 / 2 z a x + ( 1 a 2 2 ) z ,
u 2 = b x + ( 1 b 2 ) 1 / 2 z b x + ( 1 b 2 2 ) z .
u 1 u 2 a x b x a 2 z 2 + b 2 z 2 = ( a b ) [ x ( a + b ) z 2 ] .
f = 2 exp [ j k 2 ( u 1 + u 2 ) ] cos [ k 2 ( a b ) ] [ x ( a + b ) 2 z ] .
α f = ( a b ) 2 λ .
S = L / λ z .
u 1 + u 2 a x + b x + ( 1 a 2 2 ) z + ( 1 b 2 2 ) z = ( a + b ) x + 2 z ( a 2 + b 2 2 ) z ,
α p = ( a + b ) / 2 λ ,
S = Δ x / λ z ,
S = L / λ z

Metrics