Abstract

The basic formalism for first-order paraxial Fourier optics of the Maxwell field is established. Analysis of the relativistic symmetry of Maxwell’s equations leads to specific mathematical quantities by which the passage from scalar to vector Fourier optics becomes unambiguous. The inconsistency of viewing scalar Fourier optics as the description of plane-polarized Maxwell waves is recognized. System operators giving the actions of lenses, magnifiers, Fourier transformers, and general first-order systems on Maxwellian waves are constructed. Both field-strength and vector-potential treatments are given.

© 1985 Optical Society of America

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  1. E. C. G. Sudarshan, “Quantum theory of partial coherence,”J. Math. Phys. Sci. 3, 121–175 (1969).
  2. E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96A, 315–320 (1979).
  3. See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968); H. J. Butterweck, “General theory of linear, coherent optical data-processing systems,” J. Opt. Soc. Am. 67, 60–72 (1977).
    [Crossref]
  4. For a critical discussion of this assumption in the context of Gaussian beams see M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [Crossref]
  5. This follows from the divergence-free nature of the electric and magnetic vectors in free space.
  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. The relativistic-front form of dynamics was first elaborated by P. A. M. Dirac, “Forms of relativistic dynamics,” Rev. Mod. Phys. 21, 392–399 (1949).
    [Crossref]
  9. E. C. G. Sudarshan, N. Mukunda, Classical Dynamics—A Modern Perspective (Wiley, New York, 1974), Chap. 20. See, also, any standard text on electromagnetism or field theory, such as W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962); J. D. Bjorken, S. D. Drell, Relativistic Quantum Fields (McGraw-Hill, New York, 1965).
  10. See, in this context, L. Susskind, “Models of self-induced strong interactions,” Phys. Rev. 165, 1535–1546 (1968); J. B. Kogut, D. E. Soper, “Quantum electrodynamics in the infinite-momentum frame,” Phys. Rev. D 1, 2901–2914 (1970); L. C. Biedenharn, H. van Dam, “Galilean subdynamics and the dual resonance model,” Phys. Rev. D 9, 471–486 (1974).
    [Crossref]
  11. H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1931), Sec. 14.
  12. C2 as a first-order system corresponds to rotation about the system axis, namely, the x3axis.
  13. See, for instance, M. Nazarathy, J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am.72, 356–364 (1982), in which further references can be found.
    [Crossref]
  14. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.
  15. A detailed analysis of the front-form radiation gauge has been presented in Ref. 7.
  16. Explicit expressions for these generators can be found in Ref. 7.
  17. E. C. G. Sudarshan, “Pencils of rays in wave optics,” Phys. Lett. 73A, 269–272 (1979); “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981). These generalized rays in the context of paraxial-wave optics have been studied in Refs. 6 and 7 and also in R. Simon, “Generalized pencils of rays in statistical wave optics,” Pramana 20, 105–124 (1983).
    [Crossref]
  18. See A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell model beams,” Opt. Commun. 41, 383–387 (1982), in which further references can be found.
    [Crossref]
  19. E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” preprint (Department of Physics, Center for Particle Theory, University of Texas at Austin, Austin, Texas 78712, 1983).
  20. H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
    [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 (1)

See A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell model beams,” Opt. Commun. 41, 383–387 (1982), in which further references can be found.
[Crossref]

1981 (1)

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[Crossref]

1979 (2)

E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96A, 315–320 (1979).

E. C. G. Sudarshan, “Pencils of rays in wave optics,” Phys. Lett. 73A, 269–272 (1979); “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981). These generalized rays in the context of paraxial-wave optics have been studied in Refs. 6 and 7 and also in R. Simon, “Generalized pencils of rays in statistical wave optics,” Pramana 20, 105–124 (1983).
[Crossref]

1975 (1)

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

1969 (1)

E. C. G. Sudarshan, “Quantum theory of partial coherence,”J. Math. Phys. Sci. 3, 121–175 (1969).

1968 (1)

See, in this context, L. Susskind, “Models of self-induced strong interactions,” Phys. Rev. 165, 1535–1546 (1968); J. B. Kogut, D. E. Soper, “Quantum electrodynamics in the infinite-momentum frame,” Phys. Rev. D 1, 2901–2914 (1970); L. C. Biedenharn, H. van Dam, “Galilean subdynamics and the dual resonance model,” Phys. Rev. D 9, 471–486 (1974).
[Crossref]

1949 (1)

The relativistic-front form of dynamics was first elaborated by P. A. M. Dirac, “Forms of relativistic dynamics,” Rev. Mod. Phys. 21, 392–399 (1949).
[Crossref]

Bacry, H.

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[Crossref]

Cadilhac, M.

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[Crossref]

Dirac, P. A. M.

The relativistic-front form of dynamics was first elaborated by P. A. M. Dirac, “Forms of relativistic dynamics,” Rev. Mod. Phys. 21, 392–399 (1949).
[Crossref]

Friberg, A. T.

See A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell model beams,” Opt. Commun. 41, 383–387 (1982), in which further references can be found.
[Crossref]

Goodman, J. W.

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968); H. J. Butterweck, “General theory of linear, coherent optical data-processing systems,” J. Opt. Soc. Am. 67, 60–72 (1977).
[Crossref]

Lax, M.

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

Louisell, W. H.

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

McKnight, W. B.

For a critical discussion of this assumption in the context of Gaussian beams 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.

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]

E. C. G. Sudarshan, N. Mukunda, Classical Dynamics—A Modern Perspective (Wiley, New York, 1974), Chap. 20. See, also, any standard text on electromagnetism or field theory, such as W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962); J. D. Bjorken, S. D. Drell, Relativistic Quantum Fields (McGraw-Hill, New York, 1965).

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” preprint (Department of Physics, Center for Particle Theory, University of Texas at Austin, Austin, Texas 78712, 1983).

Nazarathy, M.

See, for instance, M. Nazarathy, J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am.72, 356–364 (1982), in which further references can be found.
[Crossref]

Shamir, J.

See, for instance, M. Nazarathy, J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am.72, 356–364 (1982), in which further references can be found.
[Crossref]

Simon, R.

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]

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]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” preprint (Department of Physics, Center for Particle Theory, University of Texas at Austin, Austin, Texas 78712, 1983).

Sudarshan, E. C. G.

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]

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]

E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96A, 315–320 (1979).

E. C. G. Sudarshan, “Pencils of rays in wave optics,” Phys. Lett. 73A, 269–272 (1979); “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981). These generalized rays in the context of paraxial-wave optics have been studied in Refs. 6 and 7 and also in R. Simon, “Generalized pencils of rays in statistical wave optics,” Pramana 20, 105–124 (1983).
[Crossref]

E. C. G. Sudarshan, “Quantum theory of partial coherence,”J. Math. Phys. Sci. 3, 121–175 (1969).

E. C. G. Sudarshan, N. Mukunda, Classical Dynamics—A Modern Perspective (Wiley, New York, 1974), Chap. 20. See, also, any standard text on electromagnetism or field theory, such as W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962); J. D. Bjorken, S. D. Drell, Relativistic Quantum Fields (McGraw-Hill, New York, 1965).

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” preprint (Department of Physics, Center for Particle Theory, University of Texas at Austin, Austin, Texas 78712, 1983).

Sudol, R. J.

See A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell model beams,” Opt. Commun. 41, 383–387 (1982), in which further references can be found.
[Crossref]

Susskind, L.

See, in this context, L. Susskind, “Models of self-induced strong interactions,” Phys. Rev. 165, 1535–1546 (1968); J. B. Kogut, D. E. Soper, “Quantum electrodynamics in the infinite-momentum frame,” Phys. Rev. D 1, 2901–2914 (1970); L. C. Biedenharn, H. van Dam, “Galilean subdynamics and the dual resonance model,” Phys. Rev. D 9, 471–486 (1974).
[Crossref]

Weyl, H.

H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1931), Sec. 14.

Wolf, K. B.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.

J. Math. Phys. Sci. (1)

E. C. G. Sudarshan, “Quantum theory of partial coherence,”J. Math. Phys. Sci. 3, 121–175 (1969).

Opt. Commun. (1)

See A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell model beams,” Opt. Commun. 41, 383–387 (1982), in which further references can be found.
[Crossref]

Phys. Lett. (1)

E. C. G. Sudarshan, “Pencils of rays in wave optics,” Phys. Lett. 73A, 269–272 (1979); “Quantum theory of radiative transfer,” Phys. Rev. A 23, 2802–2809 (1981). These generalized rays in the context of paraxial-wave optics have been studied in Refs. 6 and 7 and also in R. Simon, “Generalized pencils of rays in statistical wave optics,” Pramana 20, 105–124 (1983).
[Crossref]

Phys. Rev. (1)

See, in this context, L. Susskind, “Models of self-induced strong interactions,” Phys. Rev. 165, 1535–1546 (1968); J. B. Kogut, D. E. Soper, “Quantum electrodynamics in the infinite-momentum frame,” Phys. Rev. D 1, 2901–2914 (1970); L. C. Biedenharn, H. van Dam, “Galilean subdynamics and the dual resonance model,” Phys. Rev. D 9, 471–486 (1974).
[Crossref]

Phys. Rev. A (4)

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[Crossref]

For a critical discussion of this assumption in the context of Gaussian beams 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]

Physica (1)

E. C. G. Sudarshan, “Quantum electrodynamics and light rays,” Physica 96A, 315–320 (1979).

Rev. Mod. Phys. (1)

The relativistic-front form of dynamics was first elaborated by P. A. M. Dirac, “Forms of relativistic dynamics,” Rev. Mod. Phys. 21, 392–399 (1949).
[Crossref]

Other (10)

E. C. G. Sudarshan, N. Mukunda, Classical Dynamics—A Modern Perspective (Wiley, New York, 1974), Chap. 20. See, also, any standard text on electromagnetism or field theory, such as W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962); J. D. Bjorken, S. D. Drell, Relativistic Quantum Fields (McGraw-Hill, New York, 1965).

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968); H. J. Butterweck, “General theory of linear, coherent optical data-processing systems,” J. Opt. Soc. Am. 67, 60–72 (1977).
[Crossref]

This follows from the divergence-free nature of the electric and magnetic vectors in free space.

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” preprint (Department of Physics, Center for Particle Theory, University of Texas at Austin, Austin, Texas 78712, 1983).

H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1931), Sec. 14.

C2 as a first-order system corresponds to rotation about the system axis, namely, the x3axis.

See, for instance, M. Nazarathy, J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am.72, 356–364 (1982), in which further references can be found.
[Crossref]

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.

A detailed analysis of the front-form radiation gauge has been presented in Ref. 7.

Explicit expressions for these generators can be found in Ref. 7.

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Equations (93)

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U out ( x ) = t ( x ) U in ( x ) .
0 E - B = 0 , 0 B + E = 0 , · E = 0 , · B = 0
F j 0 = E j ,             F j k = j k l B l ,
x μ x μ = x μ + ω μ ν x ν + a μ , F μ ν ( x ) F μ ν ( x ) = F μ ν ( x ) + ω μ λ F λ ν ( x ) - ω ν λ F λ μ ( x ) .
δ F μ ν ( x ) = F μ ν ( x ) - F μ ν ( x ) = - ( ω α β x β + a α ) α F μ ν ( x ) + ω μ λ F λ ν ( x ) - ω ν λ F λ μ ( x ) .
G = ½ ω μ ν M μ ν - a μ P μ = ω 0 j K j + ω 23 J 1 + ω 31 J 2 + ω 12 J 3 - a μ P μ .
F ( x ) = ( E 1 ( x ) E 2 ( x ) E 3 ( x ) B 1 ( x ) B 2 ( x ) B 3 ( x ) ) .
P μ = - i μ , J = - i x + ( S 0 0 S ) , K = i ( ct + x 0 ) + ( 0 S - S 0 ) .
( S j ) k l = - i j k l .
M 0 j = K j ,             M j k = j k l J l .
σ = x 0 - x 3 , τ = ½ ( x 0 + x 3 )             x a , a = 1 , 2.
P a = - i / x a , M = ½ ( P 0 + P 3 ) = i / σ , E = P 0 - P 3 = i / τ , G a = ½ ( K a - a b J b ) = M x a - τ P a + G a ( spin ) , H a = K a + a b J b = E x a - σ P a + H a ( spin ) , J 3 = a b x a P b + J 3 ( spin ) , K 3 = σ M - τ E + K 3 ( spin ) ,
G a ( spin ) = ½ ( - a b S b S a - S a - a b S b ) , H a ( spin ) = ( a b S b S a - S a a b S b ) , J 3 ( spin ) = ( S 3 0 0 S 3 ) , K 3 ( spin ) = ( 0 S 3 - S 3 0 ) .
[ G a , P b ] = i M δ a b , [ J 3 , G a or P a ] = i a b ( G b or P b ) .
Q a = M - 1 G a
[ Q a , Q b ] = [ P a , P b ] = 0 , [ Q a , P b ] = i δ a b ,
[ E , P a or M or J 3 ] = 0 , [ E , G a ] = - i P a .
C 1 = 2 M E - P a P a , C 2 = M J 3 - a b G a P b
( 2 σ τ - ½ 2 ) F ( x ) = 0 ,
C 1 = 0.
C 2 = M J 3 ( spin ) + a b P a G b ( spin ) .
x a Q a = x a + ( 1 / k ) G a ( spin ) .
U ( x ) = d 2 s exp ( i k s · x ) U ˜ ( s ) .
U out ( x ) = exp ( i a · x ) U in ( x ) .
F out ( x ) = exp ( i a · Q ) F in ( x ) .
F ˜ T = ( E ˜ 1 E ˜ 2 0 - E ˜ 2 E ˜ 1 0 ) .
exp ( i k s · Q ) F ˜ T .
F ( x ) = d 2 s exp ( i k s · Q ) F ˜ T ( s ) , F ˜ T ( s ) = [ E ˜ 1 ( s ) E ˜ 2 ( s ) 0 - E ˜ 2 ( s ) E ˜ 1 ( s ) 0 ] ,
F ( x ) e - i ω t [ E 1 ( x ) E 2 ( x ) ( i / k ) a E a ( x ) - E 2 ( x ) E 1 ( x ) - ( i / k ) a b a E b ( x ) ] .
F ( x ) = [ E 1 ( x ) E 2 ( x ) ( i / k ) a E a ( x ) - E 2 ( x ) E 1 ( x ) ( - i / k ) a b a E b ( x ) = ] d 2 s exp ( i k s · x ) F ˜ ( s ) , F ˜ ( s ) = [ E ˜ 1 ( s ) E ˜ 2 ( s ) - s a E ˜ a ( s ) - E ˜ 2 ( s ) E ˜ 1 ( s ) a b s a E ˜ b ( s ) ] , F ( x ) = d 2 s exp [ i k s · x + i k [ 1 - ( 1 / 2 ) s 2 ) x 3 ] F ˜ ( s ) .
F ( x ) exp [ ( i / k ) G a ( spin ) P a ] F T ( x ) , F T ( x ) = [ E 1 ( x ) E 2 ( x ) 0 - E 2 ( x ) E 1 ( x ) 0 ] .
F ˜ ( s ) = exp [ i G a ( spin ) s a ] F ˜ T ( s ) ,
F T ( x ) = d 2 s exp ( i k s · x ) F ˜ T ( s ) .
F in ( x ) = exp [ ( i / k ) G a ( spin ) P a ] F T , in ( x ) ,
F out ( x ) = exp [ ( i / k ) G a ( spin ) P a ] t ( x ) F T , in ( x ) = exp [ ( i / k ) G a ( spin ) P a ] t ( x ) × exp [ ( - i / k ) G b ( spin ) P b ] F in ( x ) .
F out ( x ) = t ( Q ) F in ( x ) .
t ( x ) = exp [ ( - i k / 2 f ) x 2 ]
t ( Q ) = Ω f ( x ) = exp [ ( - i k / 2 f ) x 2 ] exp [ ( - i / f ) G a ( spin ) x a ] = t ( x ) { 1 - ( i / f ) x · G ( spin ) - ( 1 / 2 f 2 ) × [ x · G ( spin ) ] 2 }
[ G a ( spin ) , G b ( spin ) ] = 0 G a ( spin ) G a ( spin ) = 0 G a ( spin ) G b ( spin ) G c ( spin ) = 0.
Ω f ( x ) t ( x ) × [ 1 + ( 1 / 2 f ) ( 0 0 - x 0 0 y 0 0 - y 0 0 - x x y 0 - y x 0 0 0 - y 0 0 - x 0 0 x 0 0 - y y - x 0 x y 0 ) ] .
F out ( x ) = Ω f ( x ) F in ( x ) .
U out ( x ) = g M U in ( x ) = ( 1 / M ) U in ( x M ) .
F in ( x ) = exp [ ( i / k ) G a ( spin ) P a ] F T ( x ) , F out ( x ) = Ω M F in ( x ) = exp [ ( i / k ) G a ( spin ) P a ] g M F T ( x ) = exp [ ( i / k ) G a ( spin ) P a ] ( 1 / M ) × exp [ ( - i / k ) M G b ( spin ) P b ] F in ( x M ) = exp [ ( i / k ) ( 1 - M ) G a ( spin ) P a ] g M F in ( x ) ,
Ω M = exp [ ( i / k ) ( 1 - M ) G a ( spin ) P a ] g M .
U out ( x ) = g F U in ( x ) = d 2 x × exp ( - i k 2 x · x ) U in ( x ) = ( 2 π k ) 2 U ˜ in ( k x ) ,
F in ( x ) = exp [ ( i / k ) G a ( spin ) P a ] F T ( x ) , F out ( x ) = Ω F F in ( x ) = exp [ ( i / k ) G a ( spin ) P a ] g F F T ( x ) = exp [ ( i / k ) G a ( spin ) P a ] d 2 x × exp ( - i k 2 x · x ) F T ( x ) = ( 2 π k ) 2 exp [ ( i / k ) G a ( spin ) P a ] F ˜ T ( k x ) = ( 2 π k ) 2 exp [ ( i / k ) G a ( spin ) P a ] × exp [ ( - i k ) G b ( spin ) x b ] F ˜ in ( k x ) , Ω F = exp [ ( i / k ) G a ( spin ) P a ] exp [ ( - i k ) G b ( spin ) x b ] g F .
e A e B = exp { A + B + ½ [ A , B ] }
Ω F = exp [ i G a ( spin ) ( P a / k - k x a ) ] g F .
[ J 3 ( spin ) ,             G a ( spin ) ] = i a b G b ( spin ) ,
C 2 = exp [ ( i / k ) G a ( spin ) P a ] k J 3 ( spin ) exp [ ( - i / k ) G b ( spin ) P b ] .
C 2 F ( x ) = C 2 exp [ ( i / k ) G a ( spin ) P a ] F T ( x ) = exp [ ( i / k ) G a ( spin ) P a ] k J 3 ( spin ) F T ( x ) = exp [ ( i / k ) G a ( spin ) P a ] ( - i k ) ( E 2 ( x ) - E 1 ( x ) 0 E 1 ( x ) E 2 ( x ) 0 ) .
S = ( a b c d ) ,             a d - b c = 1 , ( x p ) out = ( a b c d ) ( x p ) in
g s ( x : x ) = { - i 2 π b exp [ i 2 b ( d x 2 - 2 x · x + a x 2 ) ] ,             b 0 exp ( i c 2 a x 2 ) 1 a δ ( 2 ) ( x a - x ) ,             b = 0
U out ( x ) = g s U in ( x ) = d 2 x g s ( x ; x ) U in ( x ) .
F in ( x ) = exp [ ( i / k ) G a ( spin ) P a ] F T ( x ) F out ( x ) = Ω s F in ( x ) = exp [ ( i / k ) G a ( spin ) P a ] g s F T ( x ) = exp [ ( 1 / k ) G ( spin ) · ] d 2 x g s ( x ; x ) F T ( x ) , Ω s F in ( x ) = exp [ ( 1 / k ) G ( spin ) · ] d 2 x g s ( x ; x ) exp [ ( - 1 / k ) G ( spin ) · ] F in ( x ) .
Ω s F in ( x ) = exp [ ( 1 / k ) G ( spin ) · ] d 2 x { exp [ ( 1 / k ) G ( spin ) · ] g s ( x ; x ) } F in ( x ) .
exp [ ( 1 / k ) G ( spin ) · ] exp ( α / 2 x 2 + β · x = { 1 + ( 1 / k ) G ( spin ) · + ( 1 / 2 k 2 ) [ G ( spin ) · ] 2 } × exp [ ( α / 2 ) x 2 + β · x ] = [ 1 + ( 1 / k ) G ( spin ) · ( α x + β ) + ( 1 / 2 k 2 ) G ( spin ) · G ( spin ) · ( α x + β ) ] · exp [ ( α / 2 ) x 2 + β · x ] = exp [ ( 1 / k ) G ( spin ) · ( α x + β ) ] exp [ ( α / 2 ) x 2 + β · x ] .
exp [ ( 1 / k ) G ( spin ) · ] g s ( x ; x ) = exp [ ( i / k b ) G ( spin ) · ( d x - x ) ] g s ( x ; x ) ,
exp [ ( 1 / k ) G ( spin ) · ] g s ( x ; x ) = exp [ ( i / k b ) G ( spin ) · ( a x - x ) ] g s ( x ; x ) .
Ω s F in ( x ) = exp [ ( 1 / k ) G ( spin ) · ] d 2 x { exp [ i / k b G ( spin ) · ( a x - x ) ] g s ( x ; x ) } F in ( x ) = exp [ ( 1 / k ) G ( spin ) · ] exp [ ( - i / k ) b G ( spin ) · x ] d 2 x { exp [ ( i a d / k b ) G ( spin ) · x ] x × exp [ ( - a / k ) G ( spin ) · ] g s ( x ; x ) } F in ( x ) = exp [ ( 1 / k ) G ( spin ) · ] × exp { i [ ( a d - 1 ) / k b ] G ( spin ) · x } × exp [ ( - a / k ) G ( spin ) · ] g s F in ( x ) .
Ω s = exp ( ( 1 / k ) G ( spin ) · { ( 1 - a ) + i [ ( a d - 1 ) / b ] x } ) g s .
Ω s = exp { ( i / k ) G ( spin ) · [ ( 1 - a ) P + c x ] } g s .
S M = ( M 0 0 1 M ) ,
S F = ( 0 1 / k 2 - k 2 0 ) .
A σ ( x ) = ½ [ A 0 ( x ) - A 3 ( x ) ] = 0 , A 0 ( x ) = A 3 ( x ) = ½ A τ ( x ) , a A a ( x ) = σ A τ ( x ) .
½ ( E a + a b B b ) = - σ A a , E 3 = - σ A τ = - a A a , E a - a b B b = a A τ - τ A a , B 3 = a b a A b = 1 A 2 - 2 A 1 .
A ( x ) = ( A 1 ( x ) A 2 ( x ) A τ ( x ) ) .
P a = - i a , M = ½ ( P 0 + P 3 ) = i σ , E = P 0 - P 3 = i τ , J 3 = a b x a P b + S 3 , G a = M x a - τ P a + G ˜ a = M Q ˜ a - τ P a ,
G ˜ 1 = ( 0 0 0 0 0 0 i 0 0 ) ,             G ˜ 2 = ( 0 0 0 0 0 0 0 i 0 ) .
G ˜ a G ˜ b = 0.
A ( x ) ( - i / k ) ( E 1 ( x ) E 2 ( x ) E 3 ( x ) ) ( - i / k ) ( E 1 ( x ) E 2 ( x ) ( i / k ) a E a ( x ) ) .
A ( x ) ( A 1 ( x ) A 2 ( x ) ( i / k ) a A a ( x ) ) .
A ( x ) = exp [ ( i / k ) G ˜ a P a ] A T ( x ) , A T ( x ) = ( A 1 ( x ) A 2 ( x ) 0 ) .
A ( x ) = d 2 s exp ( i k s · x ) A ˜ ( s ) , A T ( x ) = d 2 s exp ( i k s · x ) A ˜ T ( s ) , A ˜ ( s ) = exp ( G ˜ a s a ) A ˜ T ( s ) , A ˜ T ( s ) = - i k ( E ˜ 1 ( s ) E ˜ 2 ( s ) 0 ) .
C 2 = M S 3 + a b P a G ˜ b .
C 2 = exp [ ( i / k ) G ˜ a p a ] k S 3 exp [ ( - i / k ) G ˜ b P b ] .
A in ( x ) = exp [ ( i / k ) G ˜ a P a ] A T ( x ) A out ( x ) = exp [ ( i / k ) G ˜ a P a ] t ( x ) A T ( x ) = { exp [ ( i / k ) G ˜ a P a ] t ( x ) exp [ - ( i / k ) G ˜ b P b ] } A in ( x ) = t ( Q ˜ ) A in ( x ) .
( x ) a ( Q ˜ ) a = ( x ) a + G ˜ a k .
Ω ˜ f ( x ) = t ( Q ˜ ) = exp { - ( i k / 2 f ) } [ x + ( 1 / k ) G ˜ ] 2 } = t ( x ) exp [ - ( i / f ) x · G ˜ ] = exp [ - ( i k / 2 f ) x 2 ] [ 1 + ( 1 / f ) ( 0 0 0 0 0 0 x y 0 ) ] .
A a , out ( x ) = exp [ ( - i k / 2 f ) x 2 ] A a , i n ( x ) , A τ , out ( x ) = exp [ ( - i k / 2 f ) x 2 ] [ A τ , in ( x ) + ( x a / f ) A a , in ( x ) ] .
Γ α β ( σ 1 , x ; σ 2 , y ; τ ) = [ A α ( σ 1 , x , τ ) ] * A β ( σ 2 , y , τ ) ,
A α ( σ , x , τ ) = exp ( - i 0 σ ) A α ( x , τ ) ,
Γ α β ( 0 ) ( x ; y ; τ ) = [ A α ( x , τ ) ] * A β ( y , τ )
W α β ( x ; p ; τ ) = ( 2 π ) - 2 d 2 ξ exp ( i p · ξ ) × Γ α β ( 0 ) ( x + ½ ξ ; x - ½ ξ ; τ ) .
W a τ ( x ; p ; τ ) = i k ( 1 2 x b + i p b ) W a b ( x ; p ; τ )
Γ ( x + ½ ξ ; x - ½ ξ ) = I exp ( - x 2 / 2 σ 2 I ) exp ( - ξ I 2 / 2 σ g 2 ) exp [ ( - i k / 2 R ) x · ξ ] ,
i τ A α = P a P a 2 k A α ,
W α β ( x ; p ; τ ) = W α β ( x - p τ k ; p ; 0 ) .
A α ( x ; 0 ) = exp ( - i k 2 f x 2 ) T α β ( x ) A β ( x ; 0 ) ,
T α β ( x ) = δ α β + δ α τ δ β b x b f .
W a b ( x ; p ; 0 ) = W a b ( x ; p + k f x ; 0 ) , W a τ ( x ; p ; 0 ) = W a τ ( x ; p + k f x ; 0 ) + 1 f ( x b + ½ p b ) W a b ( x ; p + k f x ; 0 ) .
U ( x ) t ( x , p ) U ( x ) ,
F ( x ) t ( Q , p ) F ( x ) , A ( x ) t ( Q ˜ , p ) A ( x ) .

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