Abstract

The space–bandwidth product (SW) is fundamental for judging the performance of an optical system. Often the SW of a system is defined only as a pure number that counts the degrees of freedom of the system. We claim that a quasi-geometrical representation of the SW in the Wigner domain is more useful. We also represent the input signal as a SW in the Wigner domain. For perfect signal processing it is necessary that the system SW fully embrace the signal SW.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. See O. Lummer, F. Reiche, Die Lehre von der Bildentstehung im Mikroskop von E. Abbe (Vieweg, Braunschweig, 1910).
  2. M. von Laue, Ann. Phys. (Leipzig) 44, 1197 (1914).
  3. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1996).
    [CrossRef]
  4. A. W. Lohmann, “The space–bandwidth product, applied to spatial filtering and holography,” Research Paper RJ-438 (IBM San Jose Research Laboratory, San Jose, Calif., 1967), pp. 1–23.
  5. A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992), pp. 10, 50.
  6. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  7. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]

1996

1993

1979

1914

M. von Laue, Ann. Phys. (Leipzig) 44, 1197 (1914).

Bastiaans, M. J.

Lohmann, A. W.

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

A. W. Lohmann, “The space–bandwidth product, applied to spatial filtering and holography,” Research Paper RJ-438 (IBM San Jose Research Laboratory, San Jose, Calif., 1967), pp. 1–23.

Lukosz, W.

Lummer, O.

See O. Lummer, F. Reiche, Die Lehre von der Bildentstehung im Mikroskop von E. Abbe (Vieweg, Braunschweig, 1910).

Reiche, F.

See O. Lummer, F. Reiche, Die Lehre von der Bildentstehung im Mikroskop von E. Abbe (Vieweg, Braunschweig, 1910).

VanderLugt, A.

A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992), pp. 10, 50.

von Laue, M.

M. von Laue, Ann. Phys. (Leipzig) 44, 1197 (1914).

Ann. Phys. (Leipzig)

M. von Laue, Ann. Phys. (Leipzig) 44, 1197 (1914).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

See O. Lummer, F. Reiche, Die Lehre von der Bildentstehung im Mikroskop von E. Abbe (Vieweg, Braunschweig, 1910).

A. W. Lohmann, “The space–bandwidth product, applied to spatial filtering and holography,” Research Paper RJ-438 (IBM San Jose Research Laboratory, San Jose, Calif., 1967), pp. 1–23.

A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992), pp. 10, 50.

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 (3)

Fig. 1
Fig. 1

(a) Simple space–bandwidth product SWI(x, ν) in the Wigner domain, (b) SWI of (a) after a Fourier transform, (c) SWI of (a) after a Fresnel transform, (d) SWI of (a) after passage through a lens, (e) SWI of (a) after a fractional Fourier transform.

Fig. 2
Fig. 2

SWY of a system with space-variant bandwidth.

Fig. 3
Fig. 3

SWI of a signal, multiplied by a grating.

Equations (16)

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

u ˜ ( ν ) = u ( x ) exp ( - 2 π i ν x ) d x = 0             outside of ν Δ ν / 2.
u ( x ) = 0             outside of x < Δ x / 2.
N I = Δ x / δ x = Δ x Δ ν .
x Δ x / 2 ,             sin α λ Δ ν / 2.
N Y = Δ x Δ ν .
W ( x , ν ) = u ( x + x / 2 ) u * ( x - x / 2 ) exp ( - 2 π i ν x ) d x .
W ( x / 2 , ν ) exp ( 2 π i ν x ) d ν = u ( x ) u * ( 0 ) ,
u ( 0 ) 2 = W ( 0 , ν ) d ν .
W ( x , ν ) = u ˜ ( ν + ν / 2 ) u ˜ * ( ν - ν / 2 ) exp ( + 2 π i ν x ) d ν .
S W I ( x , ν ) S W I ( A x + B ν , C x + D ν ) .
necessary :             N I I Y             ( not sufficient ) ,
sufficient :             S W I ( x , ν ) S W Y ( x , ν ) .
G ( x ) = exp [ i ϕ ( x ) ] = A m exp ( 2 π i m ν 0 x )
v ( x ) = u ( x ) G ( x ) .
W u ( x , ν ) = u ( x + x / 2 ) u * ( x - x / 2 ) exp ( - 2 π i ν x ) d x ,
W v ( x , ν ) = A m A n exp [ 2 π i ν 0 x ( m - n ) ] × W u [ x , ν - ν 0 ( m + n ) / 2 ] .

Metrics