Abstract

With a new mathematical formulation expressed in terms of the direct product of matrices, the propagation laws for the first-, second-, third-, and fourth-order moments of a general astigmatic beam are derived. It is emphasized that the possibility of finding invariants for propagation through linear optical systems results from the symplectic structure of optical phase space. Invariant quantities for the propagation of general astigmatic beams are defined.

© 1993 Optical Society of America

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References

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  1. J. A. Arnaud, H. Kogelnik, “Gaussian light beams with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969).
    [CrossRef] [PubMed]
  2. 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, Proc. Soc. Photo-Opt. Instrum. Eng. 1397, 627–630 (1991).
  3. J. Serna, R. Martinez-Herrero, P. M. Mejias, “Spatial characterization of high-power multimode laser beams,” in 8th International Symposium on Gas Flow and Chemical Lasers, Proc. Soc. Photo-Opt. Intrum. Eng. 1397, 631–634 (1991).
  4. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).
  5. J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1973), Vol. XI.
    [CrossRef]
  6. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  7. M. J. Bastiaans, “Propagation law for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).
  8. S. Barnett, Matrices: Methods and Applications (Clarendon, Oxford, 1990).
  9. J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagating through ABCDoptical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [CrossRef]
  10. M. J. Bastiaans, “Wigner distribution function and its applications to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  11. V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).
  12. M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).
  13. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), p. 70.
  14. P. M. Cohn, Algebra (Wiley, New York, 1978), Vol. 1, pp. 208–210.
  15. J. Dieudonné, Treatise on Analysis, Vol. 10-III of Pure and Applied Mathematics (Academic, New York, 1970), p. 370.

1991 (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, Proc. Soc. Photo-Opt. Instrum. Eng. 1397, 627–630 (1991).

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Spatial characterization of high-power multimode laser beams,” in 8th International Symposium on Gas Flow and Chemical Lasers, Proc. Soc. Photo-Opt. Intrum. Eng. 1397, 631–634 (1991).

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

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

1989 (1)

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

1979 (1)

1969 (1)

Arnaud, J. A.

J. A. Arnaud, H. Kogelnik, “Gaussian light beams with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969).
[CrossRef] [PubMed]

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), p. 70.

J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1973), Vol. XI.
[CrossRef]

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 law for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

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

Cohn, P. M.

P. M. Cohn, Algebra (Wiley, New York, 1978), Vol. 1, pp. 208–210.

Dieudonné, J.

J. Dieudonné, Treatise on Analysis, Vol. 10-III of Pure and Applied Mathematics (Academic, New York, 1970), p. 370.

Guillemin, V.

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

Kogelnik, H.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).

Martinez-Herrero, R.

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Spatial characterization of high-power multimode laser beams,” in 8th International Symposium on Gas Flow and Chemical Lasers, Proc. Soc. Photo-Opt. Intrum. Eng. 1397, 631–634 (1991).

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, Proc. Soc. Photo-Opt. Instrum. Eng. 1397, 627–630 (1991).

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

Mejias, P. M.

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagating through ABCDoptical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[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, Proc. Soc. Photo-Opt. Instrum. Eng. 1397, 627–630 (1991).

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Spatial characterization of high-power multimode laser beams,” in 8th International Symposium on Gas Flow and Chemical Lasers, Proc. Soc. Photo-Opt. Intrum. Eng. 1397, 631–634 (1991).

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, Proc. Soc. Photo-Opt. Instrum. Eng. 1397, 627–630 (1991).

Papoulis, A.

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

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, Proc. Soc. Photo-Opt. Instrum. Eng. 1397, 627–630 (1991).

Serna, J.

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Spatial characterization of high-power multimode laser beams,” in 8th International Symposium on Gas Flow and Chemical Lasers, Proc. Soc. Photo-Opt. Intrum. Eng. 1397, 631–634 (1991).

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

Sternberg, S.

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

8th International Symposium on Gas Flow and Chemical Lasers (2)

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, Proc. Soc. Photo-Opt. Instrum. Eng. 1397, 627–630 (1991).

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Spatial characterization of high-power multimode laser beams,” in 8th International Symposium on Gas Flow and Chemical Lasers, Proc. Soc. Photo-Opt. Intrum. Eng. 1397, 631–634 (1991).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Optik (2)

M. J. Bastiaans, “Propagation law 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).

Other (8)

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), p. 70.

P. M. Cohn, Algebra (Wiley, New York, 1978), Vol. 1, pp. 208–210.

J. Dieudonné, Treatise on Analysis, Vol. 10-III of Pure and Applied Mathematics (Academic, New York, 1970), p. 370.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).

J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1973), Vol. XI.
[CrossRef]

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).

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Equations (22)

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[ x y n cos ϑ x n cos ϑ y ] .
[ x y p x p y ] o = [ A B C D ] · [ x y p x p y ] i = S · [ x y p x p y ] i ,
ψ ( x , y ; z ) = A ( z ) exp { i k 2 [ x 2 q 1 ( z ) + y 2 q 2 ( z ) + 2 x y q ( z ) ] } ,
ψ ¯ ( p x , p y ; z ) = ψ ( x , y ; z ) exp ( i k p x x + i k p y y ) d x d y .
W ( ζ , ρ ) = ψ ( ζ + 1 2 ζ ) · ψ * ( ζ 1 2 ζ ) exp ( i k ρ T ζ ) d ζ = 1 ( 2 π ) 2 ψ ¯ ( ρ + 1 2 ρ ) · ψ ¯ * ( ρ 1 2 ρ ) exp ( i k ζ T ρ ) d ρ ,
ζ = [ x y ] , ρ = [ p x p y ] ,
ξ ¯ = 1 ( 2 π ) 2 ξ · W ( ζ , ρ ) d ζ d ρ 1 ( 2 π ) 2 W ( ζ , ρ ) d ζ d ρ ,
x i y j p x k p y l ¯ = 1 ( 2 π ) 2 ( x x ¯ ) i ( y y ¯ ) j ( p x p ¯ x ) k ( p y p ¯ y ) l W ( ζ , ρ ) d ζ d ρ 1 ( 2 π ) 2 W ( ζ , ρ ) d ζ d ρ ,
W i ( ζ , ρ ) = W o ( A · ζ + B · ρ , C · ζ + D · ρ ) .
[ ζ ρ ] o ¯ = [ A B C D ] · [ ζ ρ ] i ¯ = S · [ ζ ρ ] i ¯ ,
M 2 = [ ζ ρ ] [ ζ T ρ T ] ¯ .
M 2 o = [ ζ ρ ] o [ ζ T ρ T ] o ¯ = ( S · [ ζ ρ ] i ) ( [ ζ T ρ T ] i · S T ) ¯ = S · ( [ ζ ρ ] i [ ζ T ρ T ] i ) S T ¯ = S · M 2 i · S T ,
M 3 = [ ζ T ρ T ] [ ζ ρ ] [ ζ T ρ T ] ¯ , M 3 o = ( [ ζ T ρ T ] i · S T ) ( S · [ ζ ρ ] i ) ( [ ζ T ρ T ] i · S T ) ¯ = S · ( [ ζ T ρ T ] i [ ζ ρ ] i [ ζ T ρ T ] i ) ¯ ( S T S T ) = S · M 3 i · T T ,
M 3 T = [ ζ ρ ] [ ζ T ρ T ] [ ζ ρ ] ¯ , M 3 o T = ( S · [ ζ ρ ] i ) ( [ ζ T ρ T ] i · S T ) ( S · [ ζ ρ ] i ) ¯ = ( S S ) · ( [ ζ ρ ] i [ ζ T ρ T ] i [ ζ ρ ] i ) ¯ · S T = T · M 3 i T · S T ,
T = S S .
M 4 = [ ζ ρ ] [ ζ T ρ T ] [ ζ ρ ] [ ζ T ρ T ] ¯ , M 4 o = ( S · [ ζ ρ ] i ) ( [ ζ T ρ T ] i · S T ) ( S · [ ζ ρ ] i ) ( [ ζ T ρ T ] i · S T ) ¯ = ( S S ) · ( [ ζ ρ ] i [ ζ T ρ T ] i [ ζ ρ ] i [ ζ T ρ T ] i ) ¯ · ( S T S T ) = T · M 4 i · T T .
ω ( ζ , ρ , ζ , ρ ) = [ ζ T ρ T ] · J [ ζ ρ ] , J = [ 0 I I 0 ] ,
[ ζ T ρ T ] [ ζ T ρ T ]
S T · J · S = J .
I 2 l = ω 21 ( ζ , ρ , ζ , ρ ) · W ( ζ , ρ ) · W ( ζ , ρ ) d ζ d ρ d ζ d ρ W ( ζ , ρ ) · W ( ζ , ρ ) · d ζ d ρ d ζ d ρ ,
det S = 1 ,
det ( M 3 T · J · M 3 ) , det [ M 3 · M 4 1 · M 3 T · ( M 2 1 ) T ] , tr [ M 3 · M 4 1 · M 3 T · ( M 2 1 ) T ] .

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