Abstract

Analytical expressions for the width, far-field divergence, crossed moment, and generalized beam-quality parameter of a coherent, lowest-order Gaussian beam propagating through hard-edged slits are given in terms of the size of the opening and of the position of the aperture plane with respect to the beam waist. Some optimization criteria of these beam-shape parameters are also inferred.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Lavi, R. Prochaska, E. Keren, Appl. Opt. 27, 3696 (1988).
    [CrossRef] [PubMed]
  2. R. Simon, N. Mukunda, E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
    [CrossRef]
  3. A. E. Siegman, Proc. Soc. Photo-Opt. Instrum. Eng. 1224, 2 (1990).
  4. M. J. Bastiaans, Optik 88, 163 (1991).
  5. J. Serna, R. Martínez-Herrero, P. M. Mejías, J. Opt. Soc. Am. A 8, 1094 (1991).
    [CrossRef]
  6. H. Weber, Opt. Quantum Electron. 24, 1027 (1992).
    [CrossRef]
  7. R. Martínez-Herrero, P. M. Mejías, Opt. Lett. 18, 1669 (1993).
    [CrossRef] [PubMed]
  8. R. Martínez-Herrero, P. M. Mejías, in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), p. 197.
  9. P. A. Belanger, Y. Champagne, C. Paré, in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), p. 173.
  10. B. R. Frieden, in Progress in Optics IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), p. 311.
    [CrossRef]

1993

1992

H. Weber, Opt. Quantum Electron. 24, 1027 (1992).
[CrossRef]

1991

1990

A. E. Siegman, Proc. Soc. Photo-Opt. Instrum. Eng. 1224, 2 (1990).

1988

R. Simon, N. Mukunda, E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

S. Lavi, R. Prochaska, E. Keren, Appl. Opt. 27, 3696 (1988).
[CrossRef] [PubMed]

Bastiaans, M. J.

M. J. Bastiaans, Optik 88, 163 (1991).

Belanger, P. A.

P. A. Belanger, Y. Champagne, C. Paré, in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), p. 173.

Champagne, Y.

P. A. Belanger, Y. Champagne, C. Paré, in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), p. 173.

Frieden, B. R.

B. R. Frieden, in Progress in Optics IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), p. 311.
[CrossRef]

Keren, E.

Lavi, S.

Martínez-Herrero, R.

R. Martínez-Herrero, P. M. Mejías, Opt. Lett. 18, 1669 (1993).
[CrossRef] [PubMed]

J. Serna, R. Martínez-Herrero, P. M. Mejías, J. Opt. Soc. Am. A 8, 1094 (1991).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), p. 197.

Mejías, P. M.

R. Martínez-Herrero, P. M. Mejías, Opt. Lett. 18, 1669 (1993).
[CrossRef] [PubMed]

J. Serna, R. Martínez-Herrero, P. M. Mejías, J. Opt. Soc. Am. A 8, 1094 (1991).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), p. 197.

Mukunda, N.

R. Simon, N. Mukunda, E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

Paré, C.

P. A. Belanger, Y. Champagne, C. Paré, in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), p. 173.

Prochaska, R.

Serna, J.

Siegman, A. E.

A. E. Siegman, Proc. Soc. Photo-Opt. Instrum. Eng. 1224, 2 (1990).

Simon, R.

R. Simon, N. Mukunda, E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

Weber, H.

H. Weber, Opt. Quantum Electron. 24, 1027 (1992).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

R. Simon, N. Mukunda, E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

H. Weber, Opt. Quantum Electron. 24, 1027 (1992).
[CrossRef]

Optik

M. J. Bastiaans, Optik 88, 163 (1991).

Proc. Soc. Photo-Opt. Instrum. Eng.

A. E. Siegman, Proc. Soc. Photo-Opt. Instrum. Eng. 1224, 2 (1990).

Other

R. Martínez-Herrero, P. M. Mejías, in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), p. 197.

P. A. Belanger, Y. Champagne, C. Paré, in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Española de Optica, Madrid, 1993), p. 173.

B. R. Frieden, in Progress in Optics IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), p. 311.
[CrossRef]

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 (5)

Fig. 1
Fig. 1

Schematic of a coherent lowest-order Gaussian beam propagating through a centered slit. PA, plane of the aperture; Pw0, plane where the incident beam reaches its waist (plane Z = 0); Z, distance (in units of the Rayleigh length zR) between PA and Pw0. Notice that Z can be positive or negative depending on whether the incident beam waist is placed before or after the aperture, respectively (here Z < 0).

Fig. 2
Fig. 2

Squared beam width 〈x2〉 versus Z and L at the aperture plane. Ordinates are given in units of w02.

Fig. 3
Fig. 3

Plot of the product k2w02u2〉 in terms of Z and L.

Fig. 4
Fig. 4

Diffracted beam waist position d0 versus Z and L.

Fig. 5
Fig. 5

Generalized beam-quality product k2QG in terms of Z and L. The planar surface at the bottom of the figure corresponds to the value QG = 1/4k2 (as if no opening were present).

Equations (17)

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

x 2 = 1 I 0 - D + D     x 2 f 2 d x ,
u 2 = 1 k 2 I 0 - D + D     f 2 d x + 8 f ( D ) 2 k 2 I 0 D ,
x u = 1 2 i k I 0 - D + D { x [ f ( x ) ] * f ( x ) - x f ( x ) f * ( x ) } d x ,
I 0 = - D + D f 2 d x
x 2 s = A 2 x 2 o + B 2 u 2 o + 2 A B x u o ,
u 2 s = C 2 x 2 o + D 2 u 2 o + 2 C D x u o ,
x u s = A C x 2 o + B D u 2 o + ( A D + B C ) x u o ,
M = [ x 2 x u x u u 2 ] ,
M s = P M 0 P t ,
Q G = det M = x 2 u 2 - x u 2 ,
f ( x , Z ) = A 0 ( Z 2 + 1 ) - 1 / 2 exp { - x 2 w 0 2 ( Z 2 + 1 ) + i [ Z x 2 w 0 2 ( Z 2 + 1 ) + k 2 w 0 2 2 Z - arctan ( Z ) ] } ,
q ( Z , L ) - π - 1 / 2 s [ erf ( s ) ] - 1 exp ( - s 2 ) ,
x 2 = w 0 2 2 ( 1 / 2 - q ) ( Z 2 + 1 ) .
u 2 = ( k w 0 ) - 2 [ 1 - 2 q ( 1 - 4 L 2 ) ] .
x u = ( 1 2 - q ) Z k + d k ( 1 2 - q + 4 q L 2 ) ,
d 0 = - Z [ 1 + 8 q L 2 ( 1 - 2 q ) ] - 1 ,
k 2 Q G = ( 1 2 - q ) 2 + 4 L 2 [ ( Z 2 + 1 ) ( 1 / 2 - q ) q ] .

Metrics