Abstract

Corrections to classical radiometry are considered for the case of a uniform medium. The corrections represent wave effects of physical optics and of the effects of partial coherence. New results include integral transforms and infinite series connecting the real and the imaginary parts of Walther’s distribution function and the Wigner function, infinite series representing all the corrections to the evolution of different distributions along rays, and estimates and comparisons of the degree to which different distribution functions are conserved along rays. In addition, a new distribution function is presented, which is exactly conserved along rays.

© 1993 Optical Society of America

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References

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  1. E. Wolf, “Coherence and radiometry,”J. Opt. Soc. Am. 68, 6–17 (1978).
    [CrossRef]
  2. L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk. 142, 689–711 (1984) [Sov. Phys. Usp. 27, 301–313 (1984)].
    [CrossRef]
  3. A. Walther, “Radiometry and coherence,”J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  4. A. Walther, “Radiometry and coherence,”J. Opt. Soc. Am. 63, 1622–1623 (1973).
    [CrossRef]
  5. E. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  6. H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1–46 (1927).
    [CrossRef]
  7. J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
    [CrossRef]
  8. T. Jannson, “Radiance transfer function,”J. Opt. Soc. Am. 70, 1544–1549 (1980).
    [CrossRef]
  9. T. Jannson, R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik (Stuttgart) 56, 429–441 (1980).
  10. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  11. S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986).
  12. K.-J. Kim, “Brightness, coherence and propagation characteristics of synchrotron radiation,” Nucl. Instrum. Methods 246, 71–76 (1986).
    [CrossRef]
  13. G. S. Agarwal, E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics,” Phys. Rev. D 2, 2161–2225 (1970).
    [CrossRef]
  14. N. L. Balazs, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase space,” Phys. Rep. 104, 347–391 (1984).
    [CrossRef]
  15. M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
    [CrossRef]
  16. S. W. McDonald, “Phase-space representations of wave equations with applications to the eikonal approximation for short-wavelength waves,” Phys. Rep. 158, 337–416 (1988).
    [CrossRef]
  17. R. G. Littlejohn, “Semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
    [CrossRef]
  18. W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,”J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  19. W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, New York, 1989).
  20. K. Kim, E. Wolf, “Propagation law for Walther’s first generalized radiance function and its short-wavelength limit with quasi-homogeneous sources,” J. Opt. Soc. Am. A 4, 1233–1236 (1987).
    [CrossRef]
  21. A. Walther, “Propagation of the generalized radiance through lenses,”J. Opt. Soc. Am. 68, 1606–1610 (1978).
    [CrossRef]
  22. 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]
  23. R. Winston, X. Ning, “Constructing a conserved flux from plane waves,” J. Opt. Soc. Am. A 3, 1629–1631 (1986).
    [CrossRef]

1988

S. W. McDonald, “Phase-space representations of wave equations with applications to the eikonal approximation for short-wavelength waves,” Phys. Rep. 158, 337–416 (1988).
[CrossRef]

1987

1986

R. Winston, X. Ning, “Constructing a conserved flux from plane waves,” J. Opt. Soc. Am. A 3, 1629–1631 (1986).
[CrossRef]

R. G. Littlejohn, “Semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
[CrossRef]

K.-J. Kim, “Brightness, coherence and propagation characteristics of synchrotron radiation,” Nucl. Instrum. Methods 246, 71–76 (1986).
[CrossRef]

1984

N. L. Balazs, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase space,” Phys. Rep. 104, 347–391 (1984).
[CrossRef]

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk. 142, 689–711 (1984) [Sov. Phys. Usp. 27, 301–313 (1984)].
[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]

1980

T. Jannson, R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik (Stuttgart) 56, 429–441 (1980).

T. Jannson, “Radiance transfer function,”J. Opt. Soc. Am. 70, 1544–1549 (1980).
[CrossRef]

1978

1977

1973

1970

G. S. Agarwal, E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics,” Phys. Rev. D 2, 2161–2225 (1970).
[CrossRef]

1968

1949

J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
[CrossRef]

1932

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

1927

H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1–46 (1927).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics,” Phys. Rev. D 2, 2161–2225 (1970).
[CrossRef]

Apresyan, L. A.

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk. 142, 689–711 (1984) [Sov. Phys. Usp. 27, 301–313 (1984)].
[CrossRef]

Balazs, N. L.

N. L. Balazs, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase space,” Phys. Rep. 104, 347–391 (1984).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Carter, W. H.

Crosignani, B.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986).

Di Porto, P.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986).

Hillery, M.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Janicki, R.

T. Jannson, R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik (Stuttgart) 56, 429–441 (1980).

Jannson, T.

T. Jannson, R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik (Stuttgart) 56, 429–441 (1980).

T. Jannson, “Radiance transfer function,”J. Opt. Soc. Am. 70, 1544–1549 (1980).
[CrossRef]

Jennings, B. K.

N. L. Balazs, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase space,” Phys. Rep. 104, 347–391 (1984).
[CrossRef]

Kim, K.

Kim, K.-J.

K.-J. Kim, “Brightness, coherence and propagation characteristics of synchrotron radiation,” Nucl. Instrum. Methods 246, 71–76 (1986).
[CrossRef]

Kravtsov, Yu. A.

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk. 142, 689–711 (1984) [Sov. Phys. Usp. 27, 301–313 (1984)].
[CrossRef]

Littlejohn, R. G.

R. G. Littlejohn, “Semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
[CrossRef]

McDonald, S. W.

S. W. McDonald, “Phase-space representations of wave equations with applications to the eikonal approximation for short-wavelength waves,” Phys. Rep. 158, 337–416 (1988).
[CrossRef]

Moyal, J. E.

J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
[CrossRef]

Mukunda, N.

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]

Ning, X.

O’Connell, R. F.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Scully, M. O.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Simon, R.

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]

Solimeno, S.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986).

Sudarshan, E. C. G.

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]

Walther, A.

Welford, W. T.

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, New York, 1989).

Weyl, H.

H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1–46 (1927).
[CrossRef]

Wigner, E.

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

Wigner, E. P.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

Winston, R.

R. Winston, X. Ning, “Constructing a conserved flux from plane waves,” J. Opt. Soc. Am. A 3, 1629–1631 (1986).
[CrossRef]

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, New York, 1989).

Wolf, E.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nucl. Instrum. Methods

K.-J. Kim, “Brightness, coherence and propagation characteristics of synchrotron radiation,” Nucl. Instrum. Methods 246, 71–76 (1986).
[CrossRef]

Optik (Stuttgart)

T. Jannson, R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik (Stuttgart) 56, 429–441 (1980).

Phys. Rep.

N. L. Balazs, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase space,” Phys. Rep. 104, 347–391 (1984).
[CrossRef]

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984).
[CrossRef]

S. W. McDonald, “Phase-space representations of wave equations with applications to the eikonal approximation for short-wavelength waves,” Phys. Rep. 158, 337–416 (1988).
[CrossRef]

R. G. Littlejohn, “Semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
[CrossRef]

Phys. Rev.

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

Phys. Rev. A

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]

Phys. Rev. D

G. S. Agarwal, E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics,” Phys. Rev. D 2, 2161–2225 (1970).
[CrossRef]

Proc. Cambridge Philos. Soc.

J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
[CrossRef]

Usp. Fiz. Nauk.

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk. 142, 689–711 (1984) [Sov. Phys. Usp. 27, 301–313 (1984)].
[CrossRef]

Z. Phys.

H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1–46 (1927).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986).

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, New York, 1989).

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Equations (157)

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2 ψ + k 0 2 ψ = 0 ,
J = n c k 0 Im ψ * ψ .
ψ ( r , z ) = r ψ ( z )
ψ ˜ ( k , z ) = d 2 r 2 π exp ( - i k · r ) ψ ( r , z ) ,
ψ ( r , z ) = d 2 k 2 π exp ( + i k · r ) ψ ˜ ( k , z ) ,
ψ ˜ ( k , z ) = k ψ ( z ) .
d 2 r ψ ( r , z ) 2 = d 2 k ψ ˜ ( k , z ) 2 .
ψ ˜ ( k , z ) = exp ( i k z z ) ψ ˜ 0 ( k ) ,
k z = { ( k 0 2 - k 2 ) 1 / 2 if k k 0 , i ( k 2 - k 0 2 ) 1 / 2 if k k 0 .
ψ ( r , z ) = d 2 k 2 π exp ( i k · r ) ψ ˜ 0 ( k ) .
k · r = k · r + k z z ,
ψ ( r , z ) = d 2 r K ( r - r , z ) ψ 0 ( r ) ,
K ( r , z ) = d 2 k ( 2 π ) 2 exp ( i k · r ) .
r K ^ ( z ) r = K ( r - r , z ) ,
k K ^ ( z ) k = exp ( i k z z ) δ ( k - k ) .
ψ ( z ) = K ^ ( z ) ψ ( 0 ) .
G ( r ) = - exp ( i k 0 r ) 4 π r = - i 2 d 2 k ( 2 π ) 2 exp ( i k · r ) k z = - d 3 k ( 2 π ) 3 exp ( i k · r ) k 2 - k 0 2 .
2 G + k 0 2 G = δ ( r ) .
K ( r , z ) = 2 G ( r ) z = z exp ( i k 0 r ) 2 π ( 1 r 3 - i k 0 r 2 ) .
d d z d 2 r ψ ( r , z ) 2 = d d z d 2 k ψ ( k , z ) 2 = d d z k > k 0 d 2 k exp [ - 2 z ( k 2 - k 0 2 ) 1 / 2 ] ψ 0 ( k ) 2 0 .
i z ψ ( z ) = H ^ ψ ( z ) ,
k H ^ k = - k z δ ( k - k ) ,
r H ^ r = - exp ( i k 0 r ) 2 π ( 1 ρ 3 - i k 0 ρ 2 ) ,
K ^ ( z ) = exp ( - i H ^ z ) .
H ^ = - ( k 0 2 - k ^ 2 ) 1 / 2 = - ( k 0 2 + 2 ) 1 / 2 ,
s x = sin θ cos ϕ , s y = sin θ sin ϕ , s z = cos θ ,
d 2 k = k 0 2 d 2 s = s z k 0 2 d Ω ,
ψ ( r , z ) = d Ω a ˜ ( s ) exp ( i k · r ) ,
a ˜ ( s ) = k 0 2 2 π s z ψ ˜ 0 ( k ) .
α = 1 N c α = 1.
Γ ( r , r ) = α c α ψ α ( r ) ψ α * ( r ) = ψ ( r ) ψ * ( r ) ¯ ,
Γ ( r , r ; z ) = r Γ ^ ( z ) r = ψ ( r , z ) ψ * ( r , z ) ¯ ,
Γ ^ ( z ) = α ψ α ( z ) c α ψ α ( z ) .
Γ ˜ ( k , k ; z ) = k Γ ^ ( z ) k
Γ ^ ( z ) = K ^ ( z ) Γ ^ ( 0 ) K ^ ( z ) ,
Γ ˜ ( k , k ; z ) = exp [ i ( k z - k z * ) z ] Γ ˜ 0 ( k , k ) ,
i d Γ ^ ( z ) d z = H ^ Γ ^ ( z ) - Γ ^ ( z ) H ^ .
A w ( r , k ) = d 2 a exp ( i k · a ) × r + ½ a A ^ r - ½ a = d 2 q exp ( + i q · r ) × k + ½ q A ^ k - ½ q .
Tr ( A ^ B ^ ) = d 2 r d 2 k ( 2 π ) 2 A w ( r , k ) * B w ( r , k ) .
Tr ( A ^ ) = d 2 r d 2 k ( 2 π ) 2 A w ( r , k ) .
C w ( r , k ) = A w ( r , k ) × exp [ i 2 ( r · k - k · r ) ] B w ( r , k ) .
C w = A w B w + i 2 { A w , B w } + ,
{ A w , B w } = A w r · B w k - A w k · B w r .
C w = 2 i A w sin [ 1 2 ( r · k - k · r ) ] B w .
1 1.
r ^ r ,             k ^ k .
f ( r ^ ) f ( r ) ,             g ( k ^ ) g ( k ) .
r 0 r 0 δ ( r - r 0 ) ,
k 0 k 0 δ ( k - k 0 ) .
Γ ^ ( z ) W ( r , k ; z ) .
W ( r , k ; z ) = d 2 a exp ( - i k · a ) × Γ ( r + ½ a , r - ½ a ; z ) .
W ( r , k ; z ) = d 2 q exp { i [ q · r + ( κ + - κ - * ) z } × Γ ˜ 0 ( k + ½ q , k - ½ q ) ,
κ ± = [ k 0 2 - ( k ± ½ q ) 2 ] 1 / 2
W ( r , k ; z ) = d 2 k d 2 k δ ( k - k + k 2 ) × exp [ i ( k - k * ) · r ] Γ ˜ 0 ( k , k ) ,
d 2 k ( 2 π ) 2 W ( r , k ; z ) = Γ ( r , r , z ) = ψ ( r , z ) 2 ¯ = I ( r ) ¯ .
d 2 r ( 2 π ) 2 W ( r , k ; z ) = Γ ˜ ( k , k ; z ) = ψ ˜ ( k , z ) 2 ¯ .
d 2 r d 2 k ( 2 π ) 2 W ( r , k ; z ) = d 2 r ψ ( r , z ) 2 ¯ ,
d J = ( n · s ) B ( r , s ) d A d Ω ,
d N = n 2 ( 2 π ) 2 ω W ( r , k ; z ) d 3 r d 2 k
d J = n c s z ( 2 π ) 2 W ( r , k ; z ) d A d 2 k ,
B ( r , s ) = c n s z ( k 0 2 π ) 2 W ( r , k ; z ) .
n c B ( r , s ) d Ω = n 2 d 2 k ( 2 π ) 2 W ( r , k ; z ) = n 2 ψ ( r ) 2 ¯ ,
J ( r ) = n c k 0 Im ψ ( r ) ψ ( r ) * ¯ = n c k 0 Re d 2 k d 2 k ( 2 π ) 2 k × exp [ i ( k - k * ) · r ] Γ ˜ 0 ( k , k ) ,
Γ ˜ 0 ( k , k ) * = Γ ˜ 0 ( k , k ) ,
J ( r ) = n c k 0 d 2 k d 2 k ( 2 π ) 2 ( k + k 2 ) × exp [ i ( k - k * ) · r ] Γ ˜ 0 ( k , k ) .
J ( r ) = n c k 0 d 2 k ( 2 π ) 2 k W ( r , k ; z ) = s B ( r , s ) d Ω .
k 0 n c J ( r 0 , z ) = Im ψ ( r 0 , z ) ψ ( r 0 , z ) * ¯ = Re r 0 k ^ ψ ψ r 0 ¯ = Re r 0 k ^ Γ ^ ( z ) r 0 = Re Tr [ r 0 r 0 k ^ Γ ^ ( z ) ] = ½ Tr [ ( r 0 r 0 k ^ + k ^ r 0 r 0 ) Γ ^ ( z ) ] .
r 0 r 0 k ^ k ^ δ ( r - r 0 ) + i 2 δ ( r - r 0 ) .
½ ( r 0 r 0 k ^ + k ^ r 0 r 0 ) k ^ δ ( r - r 0 ) .
J ( r 0 , z ) = n c k 0 d 2 r d 2 k ( 2 π ) 2 k δ ( r - r 0 ) W ( r , k ; z ) ,
J z ( r ) = n c k 0 Im ψ ( r ) z ψ ( r ) * ¯ = n c k 0 d 2 k d 2 k ( 2 π ) 2 ( k z + k z * 2 ) × exp [ i ( k - k * ) · r ] Γ ˜ 0 ( k , k ) .
s z B ( r , s ) d Ω = n c k 0 d 2 k ( 2 π ) 2 k z W ( r , k ; z ) = n c k 0 d 2 k d 2 k ( 2 π ) 2 [ k 0 2 - ( k + k 2 ) 2 ] 1 / 2 × exp [ i ( k - k * ) · r ] Γ ˜ 0 ( k , k ) ,
1 2 [ ( k 0 2 - k 2 ) 1 / 2 + ( k 0 2 - k 2 ) 1 / 2 ] [ k 0 2 - ( k + k 2 ) ] 1 / 2 .
Γ ˜ 0 ( k , k ) F ˜ ( k ) δ ( k - k ) ,
J z ( r ) = n c k 0 d 2 k ( 2 π ) 2 k z F ˜ ( k ) .
W ( r + s k k 0 , k ; z + s k z k 0 ) = d 2 q exp { i [ q · r + ( κ + - κ - ) z ] } × Γ ˜ 0 ( k + ½ q , k - ½ q ) × exp { i s [ q · k + ( κ + - κ - ) k z ] / k 0 } ,
q · k + ( κ + - κ - ) k z = - 1 8 k z 4 [ k z 2 ( k · q ) q 2 + ( k · q ) 3 ] + O ( q 5 ) .
i d Γ ˜ ( z ) d z = [ H ^ , Γ ^ ( z ) ] .
H = - ( k 0 2 - k 2 ) 1 / 2 .
W ( r , k ; z ) z = 2 ( k 0 2 - k 2 ) 1 / 2 × sin [ 1 2 ( k · r ) ] W ( r , k ; z ) .
- 1 ( k 0 2 - k 2 ) 1 / 2 k · W ,
d W d z = W d z + 1 k z k · W = m = 1 ( - 1 ) m 2 2 m ( 2 m + 1 ) ! 2 m + 1 k z k 2 m + 1 · 2 m + 1 W r 2 m + 1 .
- 1 2 2 3 ! i j l 3 k z k i k j k l 3 W r i r j r l .
d W d s ~ k z k 0 k k z 3 W L 3 ~ θ λ ( λ L ) 3 W ,
d A d s ~ 1 λ ( λ L ) 2 A .
d W d s ~ k z k 0 ( k k z 3 + k 3 k z 5 ) W L 3 ~ 1 λ ( λ L ) 3 W ,
d W d s ~ k z k 0 k 3 k z 5 W L 3 ~ 1 λ α 4 ( λ L ) 3 W ,
Γ ( r , r ) = F ( r + r 2 , r - r ) ,
W ( r , k ) = d 2 a exp ( i k · a ) F ( r , a ) ,
ψ 2 ¯ = Γ ( r , r ) = F ( r , 0 ) .
λ l L .
H - k 0 + k 2 2 k 0 ,
d W d z = W z + { W , H } = 0.
k 0 n c J z ( r 0 , z ) = Im ψ ( r 0 , z ) z ψ ( r 0 , z ) * ¯ = - Re r 0 H ^ ψ ψ r 0 ¯ = - Re r 0 H ^ Γ ^ ( z ) r 0 = - Re Tr [ r 0 r 0 H ^ Γ ^ ( z ) ] = - ½ Tr [ ( r 0 r 0 H ^ + H ^ r 0 r 0 ) Γ ^ ( z ) ] .
1 2 ( r 0 r 0 H ^ + H ^ ( r 0 r 0 ) δ ( r - r 0 ) cos [ 1 2 ( r · k ) ] H .
J z ( r 0 , z ) = n c k 0 d 2 r d 2 k ( 2 π ) 2 W ( r , k ; z ) × { δ ( r - r 0 ) cos [ 1 2 ( r · k ) ] ( k 0 2 - k 2 ) 1 / 2 } .
J z ( r ) = n c k 0 d 2 k ( 2 π ) 2 k z W ( r , k ; z ) + n c k 0 m = 1 ( - 1 ) m 2 2 m ( 2 m ) ! d 2 k ( 2 π ) 2 2 m W r 2 m · 2 m k z k 2 m .
A ( r , k ; z ) = ( 2 π ) exp ( i k · r ) ψ ˜ ( k , z ) ψ ( r , z ) * ¯ = d 2 k exp [ i ( k - k ) · r ] × Γ ˜ ( k , k ; z ) = d 2 k exp [ i ( k - k * ) · r ] Γ ˜ 0 ( k , k ) = d 2 r exp [ i k · ( r - r ) ] Γ ( r , r ; z ) .
B ( r , s ) = c n s z ( k 0 2 π ) 2 Re A ( r , k ; z ) ,
d 2 k ( 2 π ) 2 A ( r , k ; z ) = Γ ( r , r ; z ) = I ( r ) ¯ ,
d 2 r ( 2 π ) 2 A ( r , k ; z ) = Γ ˜ ( k , k ; z ) ,
d 2 r Im A ( r , k ; z ) = d 2 k Im A ( r , k ; z ) = 0.
J ( r ) = n c k 0 Re d 2 k ( 2 π ) 2 k A ( r , k ; z ) ,
J = n c k 0 d 2 k ( 2 π ) 2 k Re A ,
W ( r , k ; z ) = d 2 r d 2 k π 2 × exp [ - 2 i ( r - r ) · ( k - k ) ] A ( r , k ; z ) ,
A ( r , k ; z ) = d 2 r d 2 k π 2 × exp [ + 2 i ( r - r ) · ( k - k ) ] W ( r , k ; z ) .
A ( r , k ) = d 2 a d 2 q π 2 exp ( 2 i a · q ) × W ( r + a , k + q ) = d 2 a d 2 q π 2 exp ( 2 i a · q ) × m = 0 1 m ! ( q · k ) m W ( r + a , k ) .
A ( r , k ) = d 2 a d 2 q π 2 m = 0 1 m ! ( 2 i ) m × [ exp ( 2 i a · q ) ( a · k ) m × W ( r + a , k ) ] = d 2 a m = 0 1 m ! ( 2 i ) m × [ δ ( a ) ( a · k ) m W ( r + a , k ) ] = m = 0 1 m ! ( i 2 ) m ( 2 r · k ) m W ( r , k ) .
D ^ = 2 r · k ,
A ( r , k ) = exp [ ( i / 2 ) D ^ ] W ( r , k ) .
W ( r , k ) = exp [ - ( i / 2 ) D ^ ] A ( r , k ) .
Re A ( r , k ) = cos ( ½ D ^ ) W ( r , k ) ,
Im A ( r , k ) = sin ( ½ D ^ ) W ( r , k ) = tan ( ½ D ^ ) Re A ( r , k ) ,
W ( r , k ) = sec ( ½ D ^ ) Re A ( r , k ) .
D ^ ~ l / L ,
Γ ( x , x ) = I 0 L 2 π exp [ - 1 2 L 2 ( x + x 2 ) 2 - ( x - x ) 2 2 l 2 ] ,
Γ ( x , x ) d x = I 0 .
ψ ( x ) = ( I 0 L 2 π ) 1 / 2 exp ( - x 2 / 4 L 2 ) ;
W ( x , k ) = I 0 l L exp [ - x 2 2 L 2 - l 2 k 2 2 ] ,
A ( x , k ) = 2 I 0 l 4 L 2 + l 2 exp [ 2 ( - x 2 - L 2 l 2 k 2 + i l 2 x k ) 4 L 2 + l 2 ] .
d m d z m exp ( - z 2 ) = ( - 1 ) m H m ( z ) exp ( - z 2 ) ,
( 2 x k ) m W ( x , k ) = ( l 2 L ) m H m ( x L 2 ) × H m ( l k 2 ) W ( x , k ) ,
exp [ ( i / 2 ) D ^ ] W ( x , k ) = m = 0 1 m ! ( i l 4 L ) m H m ( x L 2 ) × H m ( l k 2 ) W ( x , k ) .
H m ( z ) = 2 m π - + ( z + i t ) m exp ( - t 2 ) d t
exp [ ( i / 2 ) D ^ ] W ( x , k ) = l π L d s d t exp [ - x 2 2 L 2 - l 2 k 2 2 + i l L ( x L 2 + i s ) ( l k 2 + i t ) - s 2 - t 2 ] .
A z = exp [ ( i / 2 ) D ^ ] W z ,
S ^ n = n k z k n · n r n ,
W z = m = 0 ( - 1 ) m 2 2 m ( 2 m + 1 ) ! S ^ 2 m + 1 W .
A z = m = 0 ( - 1 ) m 2 2 m ( 2 m + 1 ) ! exp [ ( i / 2 ) D ^ ] S ^ 2 m + 1 exp [ - ( i / 2 ) D ^ ] A .
e A ^ B ^ e - A ^ = B ^ + [ A ^ , B ^ ] + 1 2 ! [ A ^ , [ A ^ , B ^ ] ] + ,
[ D ^ , S ^ n ] = S ^ n + 1 ,             [ D ^ , [ D ^ , S ^ n ] ] = S ^ n + 2 ,
A z = m = 0 ( - 1 ) m 2 2 m ( 2 m + 1 ) ! p = 0 1 p ! ( i 2 ) p S ^ 2 m + p + 1 A .
E ^ = m = 0 p = 0 ( - 1 ) m + p 2 2 m + 2 p ( 2 m + 1 ) ! ( 2 p ) ! S ^ 2 m + 2 p + 1 ,
O ^ = m = 0 p = 0 ( - 1 ) m + p 2 2 m + 2 p + 1 ( 2 m + 1 ) ! ( 2 p + 1 ) ! S ^ 2 m + 2 p + 2 .
E ^ = m = 0 ( - 1 ) m S ^ 2 m + 1 2 2 m p = 0 m 1 ( 2 m - 2 p + 1 ) ! ( 2 p ) ! .
E ^ = m = 0 ( - 1 ) m ( 2 m + 1 ) ! S ^ 2 m + 1 .
O ^ = m = 0 ( - 1 ) m ( 2 m + 2 ) ! S ^ 2 m + 2 .
A z = ( E ^ + i O ^ ) A = m = 0 ( - 1 ) m [ S ^ 2 m + 1 ( 2 m + 1 ) ! + i S ^ 2 m + 2 ( 2 m + 2 ) ! ] A .
d A d z i 2 S ^ 2 A = i 2 2 k z k 2 · 2 A r 2 ~ A k z L 2 ~ 1 λ ( λ L ) 2 A .
Re A z = E ^ ( Re A ) - O ^ ( Im A ) = [ E ^ - O ^ tan ( ½ D ^ ) ] ( Re A ) ,
d Re A d z - 1 4 S ^ 2 D ^ ( Re A ) = - 1 4 2 k z k 2 · 4 ( Re A ) r 2 ( r · k ) ~ 1 k z l L 3 ( Re A ) ~ 1 λ ( l L ) ( λ L ) 3 ( Re A ) .
J ( r ) = n c k 0 d 2 k d 2 k ( 2 π ) 2 ( k + k 2 ) × exp [ i ( k - k ) · r ] Γ ˜ 0 ( k , k ) ,
q = k - k ,
k = k + k 2 D ,
D = [ 1 2 ( 1 + k · k k 0 2 ) ] 1 / 2 = k + k 2 k 0 .
k 2 = k 0 2 ,
q · k = 0.
q z = - k · q k z ,
D = [ 1 - q 2 / 4 k 0 2 ] 1 / 2 ,
k = D k + q / 2 , k = D k - q / 2.
| ( k , k ) ( k , q ) | = k z k z k z 2 .
J ( r ) = n c k 0 d 2 k ( 2 π ) 2 k R ( r , k ) ,
R ( r , k ) = d 2 q k z k z D k z 2 exp ( i q · r ) Γ ˜ 0 ( k , k ) .
R ( r + s k / k 0 , k ) = R ( r , k ) ,
ψ ( r ) = I 0 2 π exp ( i K · r ) ,
Γ ˜ 0 ( k , k ) = I 0 δ ( k - K ) δ ( k - K ) = I 0 δ ( q ) δ ( k - K ) .
R ( r , k ) = I 0 δ ( k - K ) .

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