Abstract

For higher-order moment matrices of a light distribution, combinations of moment matrices and the antisymmetric J matrix can be found for each order such that these combinations satisfy a similarity transformation at propagation through first-order optical systems. The similarity transformations for higher-order moment matrices of a light distribution are identified, and some properties of their invariant eigenvalues are discussed.

© 1994 Optical Society of America

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References

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  1. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  2. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  3. J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
    [CrossRef]
  4. R. Martinez-Herrero, P. M. Mejias, J. L. H. Neira, M. Sanchez, “Propagation invariance of laser beam parameters through optical systems,” in 8th International Symposium on Gas Flow and Chemical Lasers (Sept. 1990, Madrid), C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1397, 627–630 (1991).
  5. G. Piquero, P. M. Mejias, R. Martinez-Herrero, “On the kurtosis parameter of laser beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martínez-Herrero, A. González-Urena, eds. (SEDO, Madrid, 1993), pp. 141–148.
  6. M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).
  7. M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).
  8. D. Onciul, “Invariance properties of general astigmatic beams through first-order optical systems,” J. Opt. Soc. Am. A 10, 295–299 (1993).
    [CrossRef]
  9. J. A. Arnaud, H. Kogelnik, “Gaussian light beams with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969).
    [CrossRef] [PubMed]
  10. T. A. C. M. Claasen, W. F. G. Meklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part I. Continuous-time signals,” Philips J. Res. 35, 217–250 (1980).
  11. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  12. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
    [CrossRef]
  13. S. Barnett, Matrices: Methods and Applications (Clarendon, Oxford, 1990).
  14. V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).

1993 (1)

1992 (1)

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
[CrossRef]

1991 (1)

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).

1989 (1)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

1986 (1)

1980 (1)

T. A. C. M. Claasen, W. F. G. Meklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part I. Continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

1978 (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

1969 (1)

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Arnaud, J. A.

Barnett, S.

S. Barnett, Matrices: Methods and Applications (Clarendon, Oxford, 1990).

Bastiaans, M. J.

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Claasen, T. A. C. M.

T. A. C. M. Claasen, W. F. G. Meklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part I. Continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

Guillemin, V.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).

Kogelnik, H.

Martinez-Herrero, R.

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, J. L. H. Neira, M. Sanchez, “Propagation invariance of laser beam parameters through optical systems,” in 8th International Symposium on Gas Flow and Chemical Lasers (Sept. 1990, Madrid), C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1397, 627–630 (1991).

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “On the kurtosis parameter of laser beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martínez-Herrero, A. González-Urena, eds. (SEDO, Madrid, 1993), pp. 141–148.

Mejias, P. M.

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, J. L. H. Neira, M. Sanchez, “Propagation invariance of laser beam parameters through optical systems,” in 8th International Symposium on Gas Flow and Chemical Lasers (Sept. 1990, Madrid), C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1397, 627–630 (1991).

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “On the kurtosis parameter of laser beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martínez-Herrero, A. González-Urena, eds. (SEDO, Madrid, 1993), pp. 141–148.

Meklenbrauker, W. F. G.

T. A. C. M. Claasen, W. F. G. Meklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part I. Continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

Neira, J. L. H.

R. Martinez-Herrero, P. M. Mejias, J. L. H. Neira, M. Sanchez, “Propagation invariance of laser beam parameters through optical systems,” in 8th International Symposium on Gas Flow and Chemical Lasers (Sept. 1990, Madrid), C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1397, 627–630 (1991).

Onciul, D.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Piquero, G.

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “On the kurtosis parameter of laser beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martínez-Herrero, A. González-Urena, eds. (SEDO, Madrid, 1993), pp. 141–148.

Sanchez, M.

R. Martinez-Herrero, P. M. Mejias, J. L. H. Neira, M. Sanchez, “Propagation invariance of laser beam parameters through optical systems,” in 8th International Symposium on Gas Flow and Chemical Lasers (Sept. 1990, Madrid), C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1397, 627–630 (1991).

Serna, J.

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
[CrossRef]

Sternberg, S.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Opt. Quantum Electron. (1)

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
[CrossRef]

Optik (2)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).

Philips J. Res. (1)

T. A. C. M. Claasen, W. F. G. Meklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part I. Continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Other (5)

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

S. Barnett, Matrices: Methods and Applications (Clarendon, Oxford, 1990).

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).

R. Martinez-Herrero, P. M. Mejias, J. L. H. Neira, M. Sanchez, “Propagation invariance of laser beam parameters through optical systems,” in 8th International Symposium on Gas Flow and Chemical Lasers (Sept. 1990, Madrid), C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1397, 627–630 (1991).

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “On the kurtosis parameter of laser beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martínez-Herrero, A. González-Urena, eds. (SEDO, Madrid, 1993), pp. 141–148.

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Equations (41)

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J = [ O - I I O ] ,
J T = J - 1 = - J ,
S = [ A B C D ] ,
( q p ) ,
( q p ) o = S ( q p ) i .
ξ ¯ = 1 / 4 π 2 ξ W ( q , p ) d q d p 1 / 4 π 2 W ( q , p ) d q d p ,
q i p j ¯ = 1 / 4 π 2 ( q - q ¯ ) i ( p - p ¯ ) j W ( q , p ) d q d p 1 / 4 π 2 W ( q , p ) d q d p ,
W ( q , p ) = φ ( q + q 2 ) φ * ( q - q 2 ) exp ( i k 0 q p ) d q ,
W ( q , p ) = Γ ( q + q 2 , q - q 2 ) exp ( i k 0 q p ) d q ,
M j = ( q T p T ) ( q p ) ( q T p T ) . ¯ j times
M 1 o = S M 1 i ,
M 2 o = S M 2 i S T ,
M 3 o = S M 3 i ( S S ) T ,
M 4 o = ( S S ) M 4 i ( S S ) T .
M j o = ( S S ) [ j / 2 ] times M j i ( S S ) T ( j - [ j / 2 ] ) times ,
( A A ) j times
S T JS = J .
( S ) j T ( J ) j ( S ) j = ( J ) j .
M 2 k o ( J ) k = ( S ) k M 2 k i ( J ) k ( S ) k - 1 .
K o = T K i T - 1 ,
M 2 k + 1 o ( J ) k + 1 M 2 k + 1 o T ( J ) k = ( S ) k M 2 k + 1 i × ( J ) k + 1 M 2 k + 1 T ( J ) k ( S ) k - 1 ,             k 0 ,
M 2 k + 1 o T ( J ) k M 2 k + 1 o ( J ) k + 1 = ( S ) k + 1 M 2 k + 1 i T × ( J ) k M 2 k + 1 i ( J ) k + 1 ( S ) k + 1 - 1 ,             k > 0 ,
K = M 2 k + 1 ( J ) k + 1 M 2 k + 1 T ( J ) k , T = ( S ) k , K = M 2 k + 1 T ( J ) K M 2 k + 1 ( J ) k + 1 , T = ( S ) k + 1 .
K = P Λ P - 1 ,
Λ o = Λ i ,
P o = T P i ,
M 2 k ( J ) k = P Λ P - 1 ,
M 2 k = M 2 k T = ( - 1 ) k ( J ) k T P - T Λ P T ,
M 2 k = ( - 1 ) k ( J ) k - 1 P - T Λ P T ,
M 2 k ( J ) k = ( - 1 ) k [ P T ( J ) k ] - 1 Λ [ P T ( J ) k ] .
M 2 k + 1 ( J ) k + 1 M 2 k + 1 T ( J ) k = P Λ P - 1 ,
M 2 k + 1 ( J ) k + 1 M 2 k + 1 T ( J ) k = ( - 1 ) [ P T ( J ) k ] - 1 × Λ [ P T ( J ) k ]
M 2 k ( J ) k p = λ p ,
[ ( J ) k p ] + M 2 k [ ( J ) k p ] = λ p + ( J ) k + p = ( - 1 ) k λ p + ( J ) k p ,
M 2 k + 1 ( J ) k + 1 M 2 k + 1 T ( J ) k p = λ p ,
M 3 = ( q T p T ) ( q p ) ( q T p T ) ¯ = [ q 3 ¯ q 2 p ¯ q 2 p ¯ q p 2 ¯ q 2 p ¯ q p 2 ¯ q p 2 ¯ p 3 ¯ ] , M 3 ( J J ) M 3 T = [ 2 q 3 ¯ q p 2 ¯ - 2 ( q 2 p ¯ ) 2 q 3 ¯ p 3 ¯ - q 2 p ¯ q p 2 ¯ q 3 ¯ p 3 ¯ - q p 2 ¯ q 2 p ¯ 2 p 3 ¯ q 2 p ¯ - 2 ( q p 2 ¯ ) 2 ] .
M 2 k = m k ( [ G - 1 G - 1 H H G - 1 G + H G - 1 H ] ) k .
M 2 k ( J ) k M 2 k T ( J ) k = m 2 k ( N ) k ( J ) k × ( N ) k T ( J ) k = m 2 k I ,
[ M 2 k ( J ) k ] 2 = m 2 k I .
M 2 k o ( J ) k M 2 k o T ( J ) k = ( S ) k M 2 k i ( S ) k T × ( J ) k ( S ) k M 2 k i T ( S ) k T ( J ) k .
M 2 k o ( J ) k M 2 k o T ( J ) k = ( S ) k M 2 k i × ( J ) k M 2 k i T ( J ) k ( S ) k - 1 .

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