Abstract

Transformation laws for the Wigner distribution function, the radiant intensity, the radiant emittance, and the first- and second-order moments of the Wigner distribution function through an inhomogeneous, Kerr-type medium have been derived as well as for the beam quality factor and the kurtosis parameter. It is shown that the inhomogeneous Kerr-type medium can be approximated from the Wigner-distribution-function transformation-law point of view with a symplectic ABCD matrix with elements depending on the field distribution.

© 1996 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, “Wigner distribution function and its applications to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  3. M. J. Bastiaans, “Applications of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1237 (1986).
    [CrossRef]
  4. D. Onciul, “Invariance properties of general astigmatic beams through first-order optical systems,” J. Opt. Soc. Am. A 10, 295–298 (1993).
    [CrossRef]
  5. D. Dragoman, “Higher-order moments of the Wigner distribution function in first-order optical systems,” J. Opt. Soc. Am. A 11, 2643–2646 (1994).
    [CrossRef]
  6. J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [CrossRef]
  7. R. Martinez-Herrero, G. Piquero, P. M. Mejias, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
    [CrossRef]
  8. N. Marcuvitz, “Quasiparticle view of wave propagation,” Proc. IEEE 68, 1380–1395 (1980).
    [CrossRef]
  9. V. Magni, G. Cerullo, S. De Silvestri, “ABCD matrix analysis of propagation of Gaussian beams through Kerr media,” Opt. Commun. 96, 348–355 (1993).
    [CrossRef]
  10. R. Martinez-Herrero, P. M. Mejias, “Beam characterization through active media,” Opt. Commun. 85, 162–166 (1991).
    [CrossRef]
  11. M. A. Porras, J. Alda, E. Bernabeu, “Nonlinear propagation and transformation of arbitrary laser beams by means of the generalized ABCD formalism,” Appl. Opt. 32, 5885–5892 (1993).
    [CrossRef] [PubMed]
  12. C. Pare, P. A. Belanger, “Beam propagation in a linear or nonlinear lens-like medium using ABCD ray matrices: the method of moments,” Opt. Quantum Electron. 24, 1051–1070 (1992).
    [CrossRef]
  13. Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991), Chap. 3.
  14. M. J. Bastiaans, “Transport equation for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979).
    [CrossRef]
  15. A. Ankiewicz, C. Pask, “Geometric optics approach to light acceptance and propagation in graded index fibers,” Opt. Quantum Electron. 9, 87–109 (1977).
    [CrossRef]
  16. L. D. Faddeev, L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).
  17. V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).
  18. D. Dragoman, “Wigner distribution function for Gaussian–Schell beams in complex matrix optical systems,” Appl. Opt. 34, 3352–3357 (1995).
    [CrossRef] [PubMed]
  19. A. Yariv, Optical Electronics (CBS College Publications, New York, 1985), Chap. 2.
  20. K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1 D signals,” Opt. Commun. 42, 310–314 (1982).
    [CrossRef]
  21. H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [CrossRef]
  22. H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
  23. N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space analyzer,” Opt. Commun. 24, 927–949 (1992).

1995 (2)

R. Martinez-Herrero, G. Piquero, P. M. Mejias, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

D. Dragoman, “Wigner distribution function for Gaussian–Schell beams in complex matrix optical systems,” Appl. Opt. 34, 3352–3357 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (3)

1992 (3)

C. Pare, P. A. Belanger, “Beam propagation in a linear or nonlinear lens-like medium using ABCD ray matrices: the method of moments,” Opt. Quantum Electron. 24, 1051–1070 (1992).
[CrossRef]

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space analyzer,” Opt. Commun. 24, 927–949 (1992).

1991 (2)

1986 (1)

1982 (1)

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1 D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

1980 (2)

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

N. Marcuvitz, “Quasiparticle view of wave propagation,” Proc. IEEE 68, 1380–1395 (1980).
[CrossRef]

1979 (2)

M. J. Bastiaans, “Wigner distribution function and its applications to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

M. J. Bastiaans, “Transport equation for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979).
[CrossRef]

1977 (1)

A. Ankiewicz, C. Pask, “Geometric optics approach to light acceptance and propagation in graded index fibers,” Opt. Quantum Electron. 9, 87–109 (1977).
[CrossRef]

1932 (1)

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

Alda, J.

Ankiewicz, A.

A. Ankiewicz, C. Pask, “Geometric optics approach to light acceptance and propagation in graded index fibers,” Opt. Quantum Electron. 9, 87–109 (1977).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Bastiaans, M. J.

Belanger, P. A.

C. Pare, P. A. Belanger, “Beam propagation in a linear or nonlinear lens-like medium using ABCD ray matrices: the method of moments,” Opt. Quantum Electron. 24, 1051–1070 (1992).
[CrossRef]

Bernabeu, E.

Brenner, K. H.

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1 D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Cerullo, G.

V. Magni, G. Cerullo, S. De Silvestri, “ABCD matrix analysis of propagation of Gaussian beams through Kerr media,” Opt. Commun. 96, 348–355 (1993).
[CrossRef]

De Silvestri, S.

V. Magni, G. Cerullo, S. De Silvestri, “ABCD matrix analysis of propagation of Gaussian beams through Kerr media,” Opt. Commun. 96, 348–355 (1993).
[CrossRef]

Dragoman, D.

Faddeev, L. D.

L. D. Faddeev, L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).

Guillemin, V.

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

Haase, T.

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space analyzer,” Opt. Commun. 24, 927–949 (1992).

Hodgson, N.

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space analyzer,” Opt. Commun. 24, 927–949 (1992).

Kim, Y. S.

Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991), Chap. 3.

Kostka, R.

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space analyzer,” Opt. Commun. 24, 927–949 (1992).

Lohmann, A. W.

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1 D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Magni, V.

V. Magni, G. Cerullo, S. De Silvestri, “ABCD matrix analysis of propagation of Gaussian beams through Kerr media,” Opt. Commun. 96, 348–355 (1993).
[CrossRef]

Marcuvitz, N.

N. Marcuvitz, “Quasiparticle view of wave propagation,” Proc. IEEE 68, 1380–1395 (1980).
[CrossRef]

Martinez-Herrero, R.

R. Martinez-Herrero, G. Piquero, P. M. Mejias, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, “Beam characterization through active media,” Opt. Commun. 85, 162–166 (1991).
[CrossRef]

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

Mejias, P. M.

R. Martinez-Herrero, G. Piquero, P. M. Mejias, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, “Beam characterization through active media,” Opt. Commun. 85, 162–166 (1991).
[CrossRef]

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

Noz, M. E.

Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991), Chap. 3.

Onciul, D.

Pare, C.

C. Pare, P. A. Belanger, “Beam propagation in a linear or nonlinear lens-like medium using ABCD ray matrices: the method of moments,” Opt. Quantum Electron. 24, 1051–1070 (1992).
[CrossRef]

Pask, C.

A. Ankiewicz, C. Pask, “Geometric optics approach to light acceptance and propagation in graded index fibers,” Opt. Quantum Electron. 9, 87–109 (1977).
[CrossRef]

Piquero, G.

R. Martinez-Herrero, G. Piquero, P. M. Mejias, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

Porras, M. A.

Serna, J.

Sternberg, S.

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

Takhtajan, L. A.

L. D. Faddeev, L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).

Weber, H.

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space analyzer,” Opt. Commun. 24, 927–949 (1992).

Wigner, E.

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

Yariv, A.

A. Yariv, Optical Electronics (CBS College Publications, New York, 1985), Chap. 2.

Appl. Opt. (2)

J. Mod. Opt. (1)

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Acta (1)

M. J. Bastiaans, “Transport equation for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979).
[CrossRef]

Opt. Commun. (6)

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1 D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

R. Martinez-Herrero, G. Piquero, P. M. Mejias, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

V. Magni, G. Cerullo, S. De Silvestri, “ABCD matrix analysis of propagation of Gaussian beams through Kerr media,” Opt. Commun. 96, 348–355 (1993).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, “Beam characterization through active media,” Opt. Commun. 85, 162–166 (1991).
[CrossRef]

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space analyzer,” Opt. Commun. 24, 927–949 (1992).

Opt. Quantum Electron. (2)

C. Pare, P. A. Belanger, “Beam propagation in a linear or nonlinear lens-like medium using ABCD ray matrices: the method of moments,” Opt. Quantum Electron. 24, 1051–1070 (1992).
[CrossRef]

A. Ankiewicz, C. Pask, “Geometric optics approach to light acceptance and propagation in graded index fibers,” Opt. Quantum Electron. 9, 87–109 (1977).
[CrossRef]

Phys. Rev. (1)

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

Proc. IEEE (1)

N. Marcuvitz, “Quasiparticle view of wave propagation,” Proc. IEEE 68, 1380–1395 (1980).
[CrossRef]

Other (4)

Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991), Chap. 3.

L. D. Faddeev, L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).

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

A. Yariv, Optical Electronics (CBS College Publications, New York, 1985), Chap. 2.

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Equations (32)

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W ( x , p ; z ) = φ ( x + x 2 , z ) φ* ( x x 2 , z ) exp ( i x p ) d x ,
2 i k φ z = 2 φ x 2 + 2 k 2 ( n 1 n 0 | φ | 2 n 2 n 0 x 2 ) φ,
i φ z = α 2 φ x 2 + β | φ | 2 φ γ x 2 φ .
i W z = α [ 2 x 2 φ ( x + x 2 , z ) φ* ( x x 2 , z ) φ ( x + x 2 , z ) 2 x 2 φ* ( x x 2 , z ) ] exp ( i p x ) d x + β [ | φ ( x + x 2 , z ) | 2 | φ ( x x 2 , z ) | 2 ] × φ ( x + x 2 , z ) φ* ( x x 2 , z ) exp ( i p x ) d x 2 γ x x φ ( x + x 2 , z ) φ* ( x x 2 , z ) exp ( i p x ) d x
f ( x + x 2 ) f ( x x 2 ) = n = 0 2 ( 2 n + 1 ) ! ( x 2 ) 2 n + 1 ( x ) 2 n + 1 f ( x ) , | φ ( x , z ) | 2 = 1 2 π W ( x , p ; z ) d p .
W z = 2 p α W x 2 γ x W p + β 2 π n = 0 1 ( 2 n + 1 ) ! ( i 2 ) 2 n ( p ) 2 n + 1 × W ( x , p ; z ) ( x ) 2 n + 1 W ( x , q ; z ) d q .
i φ z = [ L ( x , i x ; z ) + N L ( | φ | 2 ; z ) ] φ .
i z W ( x , q ; z ) exp ( i q x ) d q = L ( x + x 2 , q i 2 x ; z ) L * ( x x 2 , q i 2 x ; z ) + NL [ | φ ( x + x 2 ; z ) | 2 ] NL * [ | φ ( x x 2 ; z ) | 2 ] } × W ( x , q ; z ) exp ( i q x ) d q .
i W z = [ L ( x + i 2 p , p i 2 x ; z ) L * ( x + i 2 p , p i 2 x ; z ) ] W + { NL [ n = 0 1 n ! ( i 2 ) n ( x ) n W ( x , q ; z ) d q ( p ) n ] NL* [ n = 0 1 n ! ( i 2 ) n ( x ) n W ( x , q ; z ) d q ( p ) n ] } W .
i W z = O ( x + i 2 p , p i 2 x , n = 0 1 n ! ( i 2 ) n × ( x ) n W ( x , q ; z ) d q ( p ) n W O * ( x + i 2 p , p i 2 x , n = 0 1 n ! ( i 2 ) n × ( x ) n W ( x , q ; z ) d q ( p ) n W .
E / z = 0 .
J z = 2 γ p x W ( x , p ; z ) d x β 2 π n = 0 ( i 2 ) 2 n 1 ( 2 n + 1 ) ! × W ( x , q ; z ) ( 2 x p ) 2 n + 1 W ( x , p ; z ) d x d q .
R z = 2 α x p W ( x , p ; z ) d p + β 2 π n = 0 1 ( 2 n + 1 ) ! ( i 2 ) 2 n ( p ) 2 n × W ( x , p ; z ) ( x ) 2 n + 1 W ( x , q ; z ) d q .
x i p j ¯ = x i p j W ( x , p ; z ) d x d p / W ( x , p ; z ) d x d p ,
x ¯ / z = 2 α p ¯ ,
p ¯ / z = 2 γ x ¯ ,
x 2 ¯ / z = 4 α x p ¯ ,
p 2 ¯ / z = 4 γ x p ¯ ( β / 2 α ) I / z ,
x p ¯ / z = 2 α p 2 ¯ + 2 γ x 2 ¯ + β I / 2 ,
I = ( 1 / 2 π ) W ( x , p ; z ) W ( x , q ; z ) d x d p d q / W ( x , p ; z ) d x d p = | φ ( x , z ) | 4 d x / | φ ( x , z ) | 2 d x .
M 2 = x 2 ¯ p 2 ¯ ( x p ¯ ) 2 .
M 2 / z = ( β / 4 α I ) ( x 2 ¯ I 2 ) / z ,
z ( M 2 β x 2 ¯ I / 2 α ) = β I x p ¯ .
I z = 2 α ( p q ) W ( x , q ; z ) W ( x , p ; z ) x d x d p d q .
K = x 4 ¯ / ( x 2 ) ¯ 2 .
x 4 ¯ / z = 8 α x 3 ¯ p ,
K / z = 8 α ( x 2 ¯ x 3 p ¯ x 4 ¯ x p ¯ ) / x 2 ¯ 3 .
W o ( x , p ) = W i ( D x B p , C x + A p ) ,
( x p ) o = ( A B C D ) ( x p ) i .
W z = 2 α p W x ( 2 γ x β | φ | 2 x ) W p .
A = D = cos [ a ( x ) z ] , B = K 1 ( x ) sin [ a ( x ) z ] , C = K 2 ( x ) sin [ a ( x ) z ] ,
a ( x ) = [ 2 α ( 2 γ β | φ | 2 / x ) ] 1 / 2 , K 1 ( x ) = 2 α / a ( x ) , K 2 ( x ) = ( 2 γ + β | φ | 2 / x ) / a ( x ) .

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