Abstract

Gaussian-beam-type solutions to the Maxwell equations are constructed by using results from relativistic front analysis, and the propagation characteristics of these beams are analyzed. The rays of geometrical optics are shown to be the trajectories of energy flow, as given by the Poynting vector. The longitudinal components of the field vectors in the direction of the beam axis, though small, are shown to be essential for a consistent description.

© 1986 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. See, for example, H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966), where further references can be found.
    [CrossRef]
  2. For a critical analysis of this assumption, see M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  3. N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (1985).
    [CrossRef]
  4. N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Maxwell beams: transformation by general linear optical systems,” J. Opt. Soc. Am. A 2, 1291–1296 (1985).
    [CrossRef]
  5. The front form of dynamics was first introduced in P. A. M. Dirac, “Forms of relativistic dynamics,” Rev. Mod. Phys. 21, 392–399 (1949). It has been used for a systematic analysis of paraxial wave optics, both the scalar case and the vector case, in Refs. 6 and 7.
    [CrossRef]
  6. E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (1983).
    [CrossRef]
  7. N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial wave optics and relativistic front description. II. The vector theory,” Phys. Rev. A 28, 2933–2942 (1983).
    [CrossRef]
  8. H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981); D. Gloge, D. Marcuse, “Formal quantum theory of light rays,”J. Opt. Soc. Am. 59, 1629–1631 (1969).
    [CrossRef]
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 60.
  10. For a proof, see E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985). Compare this assertion with the comments to the contrary in O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 288, 294. See also S. Cornbleet, “Geometrical optics revisited: a new light on an old subject,” Proc. IEEE 71, 471–502 (1983).
    [CrossRef]
  11. M. Nazarathy, J. Shamir, “First order optics—a canonical operator representation: lossless sytems,”J. Opt. Soc. Am. 72, 356–364 (1982).
    [CrossRef]
  12. For an analysis of this rule, see H. Bacry, “Group theory and paraxial optics,” invited paper presented at the XIII International Colloquium on Group Theoretical Methods in Physics, University of Maryland, May, 1984, where it has been called the MSS postulate. It was postulated in Ref. 7 and subsequently proved in Ref. 3.
  13. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), p. 307.
  14. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell. Syst. Tech. J. 44, 455–494 (1965).
  15. 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, 3273–3279 (1984).
    [CrossRef]
  16. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980); A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982); F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
    [CrossRef]
  17. L. Fishman, J. J. McCoy, “Factorization, path integral representations, and the construction of direct and inverse wave propagation theories,”IEEE Trans. Geosci. Rem. Sen. G-22, 682–692 (1948).
  18. J. Corones, B. DeFacio, R. J. Krueger, “Parabolic approximations to the time-independent elastic wave equation,”J. Math. Phys. 23, 577–586 (1982).
    [CrossRef]
  19. R. J. Hill, “A stochastic parabolic wave equation and field moment equations for random media having spatial variation of mean refractive index,”J. Accoust. Soc. Am. 77, 1742–1753 (1985).
    [CrossRef]

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, 416–426 (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, 1291–1296 (1985).
[CrossRef]

For a proof, see E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985). Compare this assertion with the comments to the contrary in O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 288, 294. See also S. Cornbleet, “Geometrical optics revisited: a new light on an old subject,” Proc. IEEE 71, 471–502 (1983).
[CrossRef]

R. J. Hill, “A stochastic parabolic wave equation and field moment equations for random media having spatial variation of mean refractive index,”J. Accoust. Soc. Am. 77, 1742–1753 (1985).
[CrossRef]

1984 (1)

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, 3273–3279 (1984).
[CrossRef]

1983 (2)

E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (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, 2933–2942 (1983).
[CrossRef]

1982 (2)

M. Nazarathy, J. Shamir, “First order optics—a canonical operator representation: lossless sytems,”J. Opt. Soc. Am. 72, 356–364 (1982).
[CrossRef]

J. Corones, B. DeFacio, R. J. Krueger, “Parabolic approximations to the time-independent elastic wave equation,”J. Math. Phys. 23, 577–586 (1982).
[CrossRef]

1981 (1)

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981); D. Gloge, D. Marcuse, “Formal quantum theory of light rays,”J. Opt. Soc. Am. 59, 1629–1631 (1969).
[CrossRef]

1980 (1)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980); A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982); F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

1975 (1)

For a critical analysis of this assumption, see M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1966 (1)

See, for example, H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966), where further references can be found.
[CrossRef]

1965 (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell. Syst. Tech. J. 44, 455–494 (1965).

1949 (1)

The front form of dynamics was first introduced in P. A. M. Dirac, “Forms of relativistic dynamics,” Rev. Mod. Phys. 21, 392–399 (1949). It has been used for a systematic analysis of paraxial wave optics, both the scalar case and the vector case, in Refs. 6 and 7.
[CrossRef]

1948 (1)

L. Fishman, J. J. McCoy, “Factorization, path integral representations, and the construction of direct and inverse wave propagation theories,”IEEE Trans. Geosci. Rem. Sen. G-22, 682–692 (1948).

Bacry, H.

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981); D. Gloge, D. Marcuse, “Formal quantum theory of light rays,”J. Opt. Soc. Am. 59, 1629–1631 (1969).
[CrossRef]

For an analysis of this rule, see H. Bacry, “Group theory and paraxial optics,” invited paper presented at the XIII International Colloquium on Group Theoretical Methods in Physics, University of Maryland, May, 1984, where it has been called the MSS postulate. It was postulated in Ref. 7 and subsequently proved in Ref. 3.

Cadilhac, M.

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981); D. Gloge, D. Marcuse, “Formal quantum theory of light rays,”J. Opt. Soc. Am. 59, 1629–1631 (1969).
[CrossRef]

Corones, J.

J. Corones, B. DeFacio, R. J. Krueger, “Parabolic approximations to the time-independent elastic wave equation,”J. Math. Phys. 23, 577–586 (1982).
[CrossRef]

DeFacio, B.

J. Corones, B. DeFacio, R. J. Krueger, “Parabolic approximations to the time-independent elastic wave equation,”J. Math. Phys. 23, 577–586 (1982).
[CrossRef]

Dirac, P. A. M.

The front form of dynamics was first introduced in P. A. M. Dirac, “Forms of relativistic dynamics,” Rev. Mod. Phys. 21, 392–399 (1949). It has been used for a systematic analysis of paraxial wave optics, both the scalar case and the vector case, in Refs. 6 and 7.
[CrossRef]

Fishman, L.

L. Fishman, J. J. McCoy, “Factorization, path integral representations, and the construction of direct and inverse wave propagation theories,”IEEE Trans. Geosci. Rem. Sen. G-22, 682–692 (1948).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 60.

Gori, F.

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980); A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982); F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

Hill, R. J.

R. J. Hill, “A stochastic parabolic wave equation and field moment equations for random media having spatial variation of mean refractive index,”J. Accoust. Soc. Am. 77, 1742–1753 (1985).
[CrossRef]

Kogelnik, H.

See, for example, H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966), where further references can be found.
[CrossRef]

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell. Syst. Tech. J. 44, 455–494 (1965).

Krueger, R. J.

J. Corones, B. DeFacio, R. J. Krueger, “Parabolic approximations to the time-independent elastic wave equation,”J. Math. Phys. 23, 577–586 (1982).
[CrossRef]

Lax, M.

For a critical analysis of this assumption, see M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Li, T.

See, for example, H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966), where further references can be found.
[CrossRef]

Louisell, W. H.

For a critical analysis of this assumption, see M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

McCoy, J. J.

L. Fishman, J. J. McCoy, “Factorization, path integral representations, and the construction of direct and inverse wave propagation theories,”IEEE Trans. Geosci. Rem. Sen. G-22, 682–692 (1948).

McKnight, W. B.

For a critical analysis of this assumption, see M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mukunda, N.

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (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, 1291–1296 (1985).
[CrossRef]

For a proof, see E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985). Compare this assertion with the comments to the contrary in O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 288, 294. See also S. Cornbleet, “Geometrical optics revisited: a new light on an old subject,” Proc. IEEE 71, 471–502 (1983).
[CrossRef]

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, 3273–3279 (1984).
[CrossRef]

E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (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, 2933–2942 (1983).
[CrossRef]

Nazarathy, M.

M. Nazarathy, J. Shamir, “First order optics—a canonical operator representation: lossless sytems,”J. Opt. Soc. Am. 72, 356–364 (1982).
[CrossRef]

Shamir, J.

M. Nazarathy, J. Shamir, “First order optics—a canonical operator representation: lossless sytems,”J. Opt. Soc. Am. 72, 356–364 (1982).
[CrossRef]

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), p. 307.

Simon, R.

For a proof, see E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985). Compare this assertion with the comments to the contrary in O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 288, 294. See also S. Cornbleet, “Geometrical optics revisited: a new light on an old subject,” Proc. IEEE 71, 471–502 (1983).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Maxwell beams: transformation by general linear optical systems,” J. Opt. Soc. Am. A 2, 1291–1296 (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, 416–426 (1985).
[CrossRef]

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, 3273–3279 (1984).
[CrossRef]

E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (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, 2933–2942 (1983).
[CrossRef]

Sudarshan, E. C. G.

For a proof, see E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985). Compare this assertion with the comments to the contrary in O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 288, 294. See also S. Cornbleet, “Geometrical optics revisited: a new light on an old subject,” Proc. IEEE 71, 471–502 (1983).
[CrossRef]

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (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, 1291–1296 (1985).
[CrossRef]

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, 3273–3279 (1984).
[CrossRef]

E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (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, 2933–2942 (1983).
[CrossRef]

Bell. Syst. Tech. J. (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell. Syst. Tech. J. 44, 455–494 (1965).

IEEE Trans. Geosci. Rem. Sen. (1)

L. Fishman, J. J. McCoy, “Factorization, path integral representations, and the construction of direct and inverse wave propagation theories,”IEEE Trans. Geosci. Rem. Sen. G-22, 682–692 (1948).

J. Math. Phys. (1)

J. Corones, B. DeFacio, R. J. Krueger, “Parabolic approximations to the time-independent elastic wave equation,”J. Math. Phys. 23, 577–586 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

M. Nazarathy, J. Shamir, “First order optics—a canonical operator representation: lossless sytems,”J. Opt. Soc. Am. 72, 356–364 (1982).
[CrossRef]

J. Accoust. Soc. Am. (1)

R. J. Hill, “A stochastic parabolic wave equation and field moment equations for random media having spatial variation of mean refractive index,”J. Accoust. Soc. Am. 77, 1742–1753 (1985).
[CrossRef]

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

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Paraxial Maxwell beams: transformation by general linear optical systems,” J. Opt. Soc. Am. A 2, 1291–1296 (1985).
[CrossRef]

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

N. Mukunda, R. Simon, E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416–426 (1985).
[CrossRef]

Opt. Acta (1)

For a proof, see E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985). Compare this assertion with the comments to the contrary in O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), pp. 288, 294. See also S. Cornbleet, “Geometrical optics revisited: a new light on an old subject,” Proc. IEEE 71, 471–502 (1983).
[CrossRef]

Opt. Commun. (1)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980); A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982); F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

Phys. Rev. A (5)

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, 3273–3279 (1984).
[CrossRef]

For a critical analysis of this assumption, see M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

E. C. G. Sudarshan, R. Simon, N. Mukunda, “Paraxial wave optics and relativistic front description. I. The scalar theory,” Phys. Rev. A 28, 2921–2932 (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, 2933–2942 (1983).
[CrossRef]

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981); D. Gloge, D. Marcuse, “Formal quantum theory of light rays,”J. Opt. Soc. Am. 59, 1629–1631 (1969).
[CrossRef]

Proc. IEEE (1)

See, for example, H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966), where further references can be found.
[CrossRef]

Rev. Mod. Phys. (1)

The front form of dynamics was first introduced in P. A. M. Dirac, “Forms of relativistic dynamics,” Rev. Mod. Phys. 21, 392–399 (1949). It has been used for a systematic analysis of paraxial wave optics, both the scalar case and the vector case, in Refs. 6 and 7.
[CrossRef]

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 60.

For an analysis of this rule, see H. Bacry, “Group theory and paraxial optics,” invited paper presented at the XIII International Colloquium on Group Theoretical Methods in Physics, University of Maryland, May, 1984, where it has been called the MSS postulate. It was postulated in Ref. 7 and subsequently proved in Ref. 3.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), p. 307.

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.


Equations (22)

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

U out ( x ) = Ω ( x , P ) U in ( x ) , P a = - i x a ,             a = 1 , 2.
Ω f ( x ) = exp ( - i k 2 f x a x a ) ,
Ω d ( P ) = exp ( - i d 2 k P a P a ) .
F ( x ) = ( E 1 ( x ) E 2 ( x ) E 3 ( x ) B 1 ( x ) B 2 ( x ) B 3 ( x ) ) .
Ω ( x , P ) Ω ( x + 1 k G , P ) ; G 1 = 1 2 ( - S 2 S 1 - S 1 - S 2 ) ,             G 2 = 1 2 ( S 1 S 2 - S 2 S 1 ) , S 1 = ( 0 0 0 0 0 - i 0 i 0 ) ,             S 2 = ( 0 0 i 0 0 0 - i 0 0 ) .
U ( x ; z = 0 ) = ( 2 / π ) 1 / 2 1 σ 0 exp ( - x 2 / σ 0 2 ) .
U ( x ; z ) = exp ( - i z P a P a / 2 k ) U ( x ; 0 ) = ( 2 / π ) 1 / 2 1 σ ( z ) exp [ - x 2 / σ ( z ) 2 ] × exp [ i k x 2 / 2 R ( z ) ] , 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 ] .
1 q ( z ) = 1 R ( z ) + 2 i k σ ( z ) 2 ,
U ( x ; z ) = ( 2 / π ) 1 / 2 1 σ ( z ) exp ( i k x 2 / 2 q ) .
Ω ( x ) = ( 2 / π ) 1 / 2 1 σ 0 exp ( - x 2 / σ 0 2 ) .
F ( x ; 0 ) = ( 2 / π ) 1 / 2 1 σ 0 exp [ - ( x + 1 k G ) 2 / σ 0 2 ] ( 1 0 0 0 1 0 ) .
F ( x ; z ) = ( 2 / π ) 1 / 2 1 σ ( z ) exp [ ( i k / 2 q ) ( x + 1 k G ) 2 ] ( 1 0 0 0 1 0 ) .
G 1 G 2 = G 2 G 1 , G a G a = 0 , G a G b G c = 0.
F ( x ; z ) = ( 2 / π ) 1 / 2 1 σ ( z ) exp ( i k x 2 / 2 q ) × x [ 1 + i q x · G - 1 q 2 ( x · G ) 2 ] ( 1 0 0 0 1 0 ) .
E 1 ( x ; z ) = B 2 ( x ; z ) = ( 2 / π ) 1 / 2 1 σ ( z ) exp ( i k x 2 / 2 q ) , E 2 ( x ; z ) = B 1 ( x ; z ) = 0 , E 3 ( x ; z ) = - x q E 1 ( x ; z ) , B 3 ( x ; z ) = - y q B 2 ( x ; z ) .
E 3 ( x ; 0 ) = - ( 2 / π ) 1 / 2 1 σ 0 2 i x k σ 0 2 exp ( - x 2 / σ 0 2 ) , B 3 ( x ; 0 ) = - ( 2 / π ) 1 / 2 1 σ 0 2 i y k σ 0 2 exp ( - x 2 / σ 0 2 ) .
σ ( z ) σ 0 ,             R ( z ) 1 z ( k σ 0 2 / 2 ) 2 ,
1 q 2 i k σ 0 2 .
q z ,
S = Re E Λ * B = 2 π 1 σ ( z ) 2 [ x R ( z ) , 1 ] .
x = ( x ) 0 [ 1 + ( 2 z k σ 0 2 ) 2 ] 1 / 2 ,
d x d z = x R ( z ) .

Metrics