Abstract

Polarization properties of Gaussian laser beams are analyzed in a manner consistent with the Maxwell equations, and expressions are developed for all components of the electric and magnetic field vectors in the beam. It is shown that the transverse nature of the free electromagnetic field demands a nonzero transverse cross-polarization component in addition to the well-known component of the field vectors along the beam axis. The strength of these components in relation to the strength of the principal polarization component is established. It is further shown that the integrated strengths of these components over a transverse plane are invariants of the propagation process. It is suggested that cross-polarization measurement using a null detector can serve as a new method for accurate determination of the center of Gaussian laser beams.

© 1987 Optical Society of America

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References

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  1. For a review of the early developments see, for example, H. Kogelnik, T. Li, “Laser Beams and Resonators,” Proc. IEEE 54, 1312 (1966).
    [CrossRef]
  2. H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).
  3. For a generalization of the abcd law to Gaussian Schell-model beams see R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized Rays in First Order Optics: Transformation Properties of Gaussian Schell-Model Fields,” Phys. Rev. A 29, 3273 (1984).
    [CrossRef]
  4. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-Model Beams: Passage Through Optical Systems and Associated Invariants,” Phys. Rev. A 31, 2419 (1985).
    [CrossRef] [PubMed]
  5. R. Simon, “Anisotropic Gaussian Beams,” Opt. Commun. 46, 265 (1983).
    [CrossRef]
  6. R. Simon, “A New Class of Anisotropic Gaussian Beams,” Opt. Commun. 55, 381 (1985).
    [CrossRef]
  7. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Wave Optics,” Phys. Rev. A 11, 1365 (1975).
    [CrossRef]
  8. R. C. Jones, “A New Calculus for the Treatment of Optical Systems. I. Description and Discussion of the Calculus,” J. Opt. Soc. Am. 31, 488 (1941). This and the other papers of this series along with several important contributions to polarization optics have been reprinted in W. Swindell, Ed., Polarized Light (Dowden Hutchinson & Ross, Stroudsburg, PA, 1975).
    [CrossRef]
  9. See, for example, S. Pancharatnam, “Generalized Theory of Interference and Its Applications,” Proc. Indian Acad. Sci. Sect. A 44, 247, 398 (1956); Proc. Indian Acad. Sci. Sect. A 45, 402 (1957); Proc. Indian Acad. Sci. Sect. A 46, 1 (1957).
  10. E. Wolf, “Coherence Properties of Partially Polarized Electromagnetic Radiation,” Nuovo Cimento 13, 1165 (1959).
    [CrossRef]
  11. G. G. Stokes, “On the Composition and Resolution of Streams of Polarized Light from Different Sources,” Trans. Cambridge Philos. Soc. 9, 399 (1852); H. Mueller “The Foundations of Optics,” J. Opt. Soc. Am. 38, 661 (1948).
  12. For the connection between these methods see R. Simon, “The Connection Between Mueller and Jones Matrices of Polarization Optics,” Opt. Commun. 42, 293 (1982).
    [CrossRef]
  13. N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Wave Optics and Relativistic Front Description II: The Vector Theory,” Phys. Rev. A 28, 2933 (1983).
    [CrossRef]
  14. N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier Optics for the Maxwell Field: Formalism and Applications,” J. Opt. Soc. Am. A 2, 416 (1985).
    [CrossRef]
  15. N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Maxwell Beams: Transformation by General Linear Optical Systems,” J. Opt. Soc. Am. A 2, 1291 (1985).
    [CrossRef]
  16. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian-Maxwell Beams,” J. Opt. Soc. Am. A 3, 536 (1986).
    [CrossRef]
  17. Y. Fainman, J. Shamir, “Polarization of Nonplanar Wave Fronts,” Appl. Opt. 23, 3188 (1984).
    [CrossRef] [PubMed]
  18. H. Bacry, M. Cadilhac, “Metaplectic Group and Fourier Optics,” Phys. Rev. A 23, 2533 (1981).
    [CrossRef]
  19. The intensity of the Ey component is smaller than that of the principal component Ex by a factor 1/(4k4 σ04) in the original beam. Yet, this is the leading component in the transmitted beam. Hence, with a sufficiently sensitive null detector the two orthogonal lines along which the intensity of the transmitted beam is zero can be fixed without difficulty; the intersection of these two lines then gives the beam center.

1986 (1)

1985 (4)

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier Optics for the Maxwell Field: Formalism and Applications,” J. Opt. Soc. Am. A 2, 416 (1985).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Maxwell Beams: Transformation by General Linear Optical Systems,” J. Opt. Soc. Am. A 2, 1291 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-Model Beams: Passage Through Optical Systems and Associated Invariants,” Phys. Rev. A 31, 2419 (1985).
[CrossRef] [PubMed]

R. Simon, “A New Class of Anisotropic Gaussian Beams,” Opt. Commun. 55, 381 (1985).
[CrossRef]

1984 (2)

For a generalization of the abcd law to Gaussian Schell-model beams see R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized Rays in First Order Optics: Transformation Properties of Gaussian Schell-Model Fields,” Phys. Rev. A 29, 3273 (1984).
[CrossRef]

Y. Fainman, J. Shamir, “Polarization of Nonplanar Wave Fronts,” Appl. Opt. 23, 3188 (1984).
[CrossRef] [PubMed]

1983 (2)

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Wave Optics and Relativistic Front Description II: The Vector Theory,” Phys. Rev. A 28, 2933 (1983).
[CrossRef]

R. Simon, “Anisotropic Gaussian Beams,” Opt. Commun. 46, 265 (1983).
[CrossRef]

1982 (1)

For the connection between these methods see R. Simon, “The Connection Between Mueller and Jones Matrices of Polarization Optics,” Opt. Commun. 42, 293 (1982).
[CrossRef]

1981 (1)

H. Bacry, M. Cadilhac, “Metaplectic Group and Fourier Optics,” Phys. Rev. A 23, 2533 (1981).
[CrossRef]

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Wave Optics,” Phys. Rev. A 11, 1365 (1975).
[CrossRef]

1966 (1)

For a review of the early developments see, for example, H. Kogelnik, T. Li, “Laser Beams and Resonators,” Proc. IEEE 54, 1312 (1966).
[CrossRef]

1965 (1)

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).

1959 (1)

E. Wolf, “Coherence Properties of Partially Polarized Electromagnetic Radiation,” Nuovo Cimento 13, 1165 (1959).
[CrossRef]

1956 (1)

See, for example, S. Pancharatnam, “Generalized Theory of Interference and Its Applications,” Proc. Indian Acad. Sci. Sect. A 44, 247, 398 (1956); Proc. Indian Acad. Sci. Sect. A 45, 402 (1957); Proc. Indian Acad. Sci. Sect. A 46, 1 (1957).

1941 (1)

1852 (1)

G. G. Stokes, “On the Composition and Resolution of Streams of Polarized Light from Different Sources,” Trans. Cambridge Philos. Soc. 9, 399 (1852); H. Mueller “The Foundations of Optics,” J. Opt. Soc. Am. 38, 661 (1948).

Bacry, H.

H. Bacry, M. Cadilhac, “Metaplectic Group and Fourier Optics,” Phys. Rev. A 23, 2533 (1981).
[CrossRef]

Cadilhac, M.

H. Bacry, M. Cadilhac, “Metaplectic Group and Fourier Optics,” Phys. Rev. A 23, 2533 (1981).
[CrossRef]

Fainman, Y.

Jones, R. C.

Kogelnik, H.

For a review of the early developments see, for example, H. Kogelnik, T. Li, “Laser Beams and Resonators,” Proc. IEEE 54, 1312 (1966).
[CrossRef]

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Wave Optics,” Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Li, T.

For a review of the early developments see, for example, H. Kogelnik, T. Li, “Laser Beams and Resonators,” Proc. IEEE 54, 1312 (1966).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Wave Optics,” Phys. Rev. A 11, 1365 (1975).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Wave Optics,” Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Mukunda, N.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian-Maxwell Beams,” J. Opt. Soc. Am. A 3, 536 (1986).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier Optics for the Maxwell Field: Formalism and Applications,” J. Opt. Soc. Am. A 2, 416 (1985).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Maxwell Beams: Transformation by General Linear Optical Systems,” J. Opt. Soc. Am. A 2, 1291 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-Model Beams: Passage Through Optical Systems and Associated Invariants,” Phys. Rev. A 31, 2419 (1985).
[CrossRef] [PubMed]

For a generalization of the abcd law to Gaussian Schell-model beams see R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized Rays in First Order Optics: Transformation Properties of Gaussian Schell-Model Fields,” Phys. Rev. A 29, 3273 (1984).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Wave Optics and Relativistic Front Description II: The Vector Theory,” Phys. Rev. A 28, 2933 (1983).
[CrossRef]

Pancharatnam, S.

See, for example, S. Pancharatnam, “Generalized Theory of Interference and Its Applications,” Proc. Indian Acad. Sci. Sect. A 44, 247, 398 (1956); Proc. Indian Acad. Sci. Sect. A 45, 402 (1957); Proc. Indian Acad. Sci. Sect. A 46, 1 (1957).

Shamir, J.

Simon, R.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian-Maxwell Beams,” J. Opt. Soc. Am. A 3, 536 (1986).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Maxwell Beams: Transformation by General Linear Optical Systems,” J. Opt. Soc. Am. A 2, 1291 (1985).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier Optics for the Maxwell Field: Formalism and Applications,” J. Opt. Soc. Am. A 2, 416 (1985).
[CrossRef]

R. Simon, “A New Class of Anisotropic Gaussian Beams,” Opt. Commun. 55, 381 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-Model Beams: Passage Through Optical Systems and Associated Invariants,” Phys. Rev. A 31, 2419 (1985).
[CrossRef] [PubMed]

For a generalization of the abcd law to Gaussian Schell-model beams see R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized Rays in First Order Optics: Transformation Properties of Gaussian Schell-Model Fields,” Phys. Rev. A 29, 3273 (1984).
[CrossRef]

R. Simon, “Anisotropic Gaussian Beams,” Opt. Commun. 46, 265 (1983).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Wave Optics and Relativistic Front Description II: The Vector Theory,” Phys. Rev. A 28, 2933 (1983).
[CrossRef]

For the connection between these methods see R. Simon, “The Connection Between Mueller and Jones Matrices of Polarization Optics,” Opt. Commun. 42, 293 (1982).
[CrossRef]

Stokes, G. G.

G. G. Stokes, “On the Composition and Resolution of Streams of Polarized Light from Different Sources,” Trans. Cambridge Philos. Soc. 9, 399 (1852); H. Mueller “The Foundations of Optics,” J. Opt. Soc. Am. 38, 661 (1948).

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian-Maxwell Beams,” J. Opt. Soc. Am. A 3, 536 (1986).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier Optics for the Maxwell Field: Formalism and Applications,” J. Opt. Soc. Am. A 2, 416 (1985).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Maxwell Beams: Transformation by General Linear Optical Systems,” J. Opt. Soc. Am. A 2, 1291 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-Model Beams: Passage Through Optical Systems and Associated Invariants,” Phys. Rev. A 31, 2419 (1985).
[CrossRef] [PubMed]

For a generalization of the abcd law to Gaussian Schell-model beams see R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized Rays in First Order Optics: Transformation Properties of Gaussian Schell-Model Fields,” Phys. Rev. A 29, 3273 (1984).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Wave Optics and Relativistic Front Description II: The Vector Theory,” Phys. Rev. A 28, 2933 (1983).
[CrossRef]

Wolf, E.

E. Wolf, “Coherence Properties of Partially Polarized Electromagnetic Radiation,” Nuovo Cimento 13, 1165 (1959).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Nuovo Cimento (1)

E. Wolf, “Coherence Properties of Partially Polarized Electromagnetic Radiation,” Nuovo Cimento 13, 1165 (1959).
[CrossRef]

Opt. Commun. (3)

R. Simon, “Anisotropic Gaussian Beams,” Opt. Commun. 46, 265 (1983).
[CrossRef]

R. Simon, “A New Class of Anisotropic Gaussian Beams,” Opt. Commun. 55, 381 (1985).
[CrossRef]

For the connection between these methods see R. Simon, “The Connection Between Mueller and Jones Matrices of Polarization Optics,” Opt. Commun. 42, 293 (1982).
[CrossRef]

Phys. Rev. A (5)

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Wave Optics and Relativistic Front Description II: The Vector Theory,” Phys. Rev. A 28, 2933 (1983).
[CrossRef]

H. Bacry, M. Cadilhac, “Metaplectic Group and Fourier Optics,” Phys. Rev. A 23, 2533 (1981).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Wave Optics,” Phys. Rev. A 11, 1365 (1975).
[CrossRef]

For a generalization of the abcd law to Gaussian Schell-model beams see R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized Rays in First Order Optics: Transformation Properties of Gaussian Schell-Model Fields,” Phys. Rev. A 29, 3273 (1984).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-Model Beams: Passage Through Optical Systems and Associated Invariants,” Phys. Rev. A 31, 2419 (1985).
[CrossRef] [PubMed]

Proc. IEEE (1)

For a review of the early developments see, for example, H. Kogelnik, T. Li, “Laser Beams and Resonators,” Proc. IEEE 54, 1312 (1966).
[CrossRef]

Proc. Indian Acad. Sci. Sect. A (1)

See, for example, S. Pancharatnam, “Generalized Theory of Interference and Its Applications,” Proc. Indian Acad. Sci. Sect. A 44, 247, 398 (1956); Proc. Indian Acad. Sci. Sect. A 45, 402 (1957); Proc. Indian Acad. Sci. Sect. A 46, 1 (1957).

Trans. Cambridge Philos. Soc. (1)

G. G. Stokes, “On the Composition and Resolution of Streams of Polarized Light from Different Sources,” Trans. Cambridge Philos. Soc. 9, 399 (1852); H. Mueller “The Foundations of Optics,” J. Opt. Soc. Am. 38, 661 (1948).

Other (1)

The intensity of the Ey component is smaller than that of the principal component Ex by a factor 1/(4k4 σ04) in the original beam. Yet, this is the leading component in the transmitted beam. Hence, with a sufficiently sensitive null detector the two orthogonal lines along which the intensity of the transmitted beam is zero can be fixed without difficulty; the intersection of these two lines then gives the beam center.

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Equations (24)

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ψ ( x , y ) = ψ ˜ ( k x , k y ) exp [ i ( k x x + k y x ) ] d k x d k y .
F ( x , y ) = ( E x ( x , y ) E y ( x , y ) E z ( x , y ) B x ( x , y ) B y ( x , y ) B z ( x , y ) ) ,
F ( x , y ; k x , k y ) = exp { i [ k x ( x + 1 k G x ) + k y ( y + 1 k G y ) ] } ( a 1 ( k x , k y ) a 2 ( k x , k y ) 0 - a 2 ( k x , k y ) a 1 ( k x , k y ) 0 )
G x = ½ ( - S 2 S 1 - S 1 - S 2 ) ,             G 2 = ½ ( S 1 S 2 - S 2 S 1 ) , S 2 = ( 0 0 0 0 0 - i 0 i 0 ) ,             S 2 = ( 0 0 i 0 0 0 - i 0 0 ) .
F ( x , y ) = F ( x , y ; k x , k y ) d k x d k y = exp { i [ k x ( x + 1 k G x ) ] + k y ( y + 1 k G y ) } × ( a 1 ( k x , k y ) a 2 ( k x , k y ) 0 - a 2 ( k x , k y ) a 1 ( k x , k y ) 0 ) d k x d k y .
[ G x , G y ] = 0 ,             G x 2 + G y 2 = 0.
a 1 ( k x , k y ) = ( 2 π ) - 3 / 2 · σ 0 · exp [ - σ 0 2 ( k x 2 + k y 2 ) / 4 ] , a 2 ( k x , k y ) = 0 ,
F ( x , y , z = 0 ) = ( 2 π ) 1 / 2 1 σ 0 exp ( - [ ( x + 1 k G x ) 2 + ( y + 1 k G y ) 2 ] / σ 0 2 ) ( 1 0 0 0 1 0 ) .
F ( x , y , z ) = exp [ - i z 2 k ( P x 2 + P y 2 ) ] F ( x , y , 0 ) ,
P x = - i x ,             P y = - i y .
F ( x , y , z ) = ( 2 π ) 1 / 2 1 σ ( z ) exp { i k 2 q ( z ) [ ( x + 1 k G x ) 2 + ( y + 1 k G y ) 2 ] } ( 1 0 0 0 1 0 ) .
1 q ( z ) = 1 R ( z ) + i 2 k σ ( z ) 2 ,             σ ( z ) = σ 0 [ 1 + ( 2 z k σ 0 2 ) 2 ] 1 / 2 ,             R ( z ) = z [ 1 + ( k σ 0 2 2 z ) 2 ] .
F ( x , y , z ) = ( E x ( x , y , z ) E y ( x , y , z ) E z ( x , y , z ) B x ( x , y , z ) B y ( x , y , z ) B z ( x , y , z ) ) = ( 2 π ) 1 / 2 1 σ ( z ) exp [ i k 2 q ( z ) ( x 2 + y 2 ) ] M ( 1 0 0 0 1 0 ) M = ( 1 + y 2 - x 2 8 q 2 - x y 4 q 2 x 2 q x y 4 q 2 y 2 - x 2 8 q 2 - y 2 q - x y 4 q 2 1 + x 2 - y 2 8 q 2 y 2 q y 2 - x 2 8 q 2 - x y 4 q 2 x 2 q x 2 q - y 2 q 1 y 2 q - x 2 q 0 - x y 4 q 2 x 2 - y 2 8 q 2 y 2 q 1 + y 2 - x 2 8 q 2 - x y 4 q 2 x 2 q x 2 - y 2 8 q 2 x y 4 q 2 - x 2 q - x y 4 q 2 1 + x 2 - y 2 8 q 2 y 2 q - y 2 q x 2 q 0 - x 2 q - y 2 q 1 )
E x ( x , y , z ) = ψ ( x , y , z ) [ 1 + y 2 - x 2 4 q ( z ) 2 ] ,
E y ( x , y , z ) = ψ ( x , y , z ) [ - x y 2 q ( z ) 2 ] ,
E z ( x , y , z ) = ψ ( x , y , z ) [ - x q ( z ) ] ,
ψ ( x , y , z ) = ( 2 π ) 1 / 2 1 σ ( z ) exp [ i k 2 q ( z ) ( x 2 + y 2 ) ]
ψ * ψ d x d y = 1.
x ( z ) = ± y ( z ) , x ( z ) 2 + y ( z ) 2 = σ ( z ) 2 / 2 ,
I ( z ) = E * ( r ) · E ( r ) d x d y = ( E x 2 + E y 2 + E z 2 ) d x d y I x ( z ) + I y ( z ) + I z ( z ) ,
ψ * ψ = 2 π 1 σ ( z ) 2 exp [ - 2 ( x 2 + y 2 ) / σ ( z ) 2 ] .
I x ( z ) = 1 + ¼ ( σ ( z ) 2 4 q ( z ) 2 ) 2 , I y ( z ) = ¼ [ σ ( z ) 2 4 q ( z ) 2 ] 2 , I z ( z ) = σ ( z ) 2 4 q ( z ) 2 .
σ ( z ) 2 / ( 4 q ( z ) 2 ) = ( k σ 0 ) - 2 .
I x ( z ) = 1 + ¼ ( k σ 0 ) - 4 , I y ( z ) = ¼ ( k σ 0 ) - 4 , I z ( z ) = ( k σ 0 ) - 2 .

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