Abstract

The theory of second-order coherence in connection with wave propagation through a stratified N-layer (SNL) medium is developed. Especially, the influence of the SNL medium on the propagation of the coherence generated by a given state of coherence at the entrance plane of the medium is considered. The generalization of the van Cittert–Zernike theorem is obtained, and the propagation of the second-order coherence from a quasi-homogeneous surface distribution or a rough surface is calculated. Furthermore, the influence of SNL media on the coherence properties of a pulse is calculated.

© 2005 Optical Society of America

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  1. H. Wolter, Optik dünner Schichten, S. Flügge, ed., Encyclopedia of Physics (Springer, Berlin, 1956), Vol. 24, pp. 461–554.
  2. P. Yeh, Optical Waves in Layered Media, Wiley Series in Pure and Applied Optics (Wiley, Chichester, UK, 1988).
  3. M. W. Ewing, W. S. Jardetzky, F. Press, Elastic Waves in Layered Media (McGraw-Hill, London, 1957).
  4. M. Schoenberg, “Wave propagation in alternating solid and fluid layers,” Wave Motion 6, 303–320 (1984).
    [CrossRef]
  5. A. T. de Hoop, “Acoustic radiation from an impulsive point source in a continuously layered fluid—an analysis based on the Cagniard method,” J. Acoust. Soc. Am. 5, 2376–2388 (1990).
  6. C. H. Chapman, “Generalized ray theory for an inhomogeneous medium,” Geophys. J. R. Astron. Soc. 36, 673–704 (1974).
    [CrossRef]
  7. C. H. Chapman, “Exact and approximate generalized ray theory in vertically inhomogeneous media,” Geophys. J. R. Astron. Soc. 46, 201–233 (1976).
    [CrossRef]
  8. C. H. Chapman, “A new method for computing synthetic seismograms,” Geophys. J. R. Astron. Soc. 36, 673–704 (1974).
  9. S. Barkeshli, “On the dyadic Green’s function for a planar multilayered dielectric/magnetic media,” IEEE Trans. Microwave Theory Tech. 40, 128–141 (1992).
    [CrossRef]
  10. M. S. Tomaš, “Green function for multilayers: light scattering in planar cavities,” Phys. Rev. A 51, 2545–2559 (1995).
    [CrossRef] [PubMed]
  11. C. E. Reed, J. Giergiel, J. C. Hemminger, S. Ushioda, “Dipole radiation in a multilayer geometry,” Phys. Rev. B 36, 4990–5000 (1987).
    [CrossRef]
  12. E. Wolf, “Spatial coherence of resonant modes in a maser interferometer,” Phys. Lett. 3, 166–168 (1963).
    [CrossRef]
  13. W. Streifer, “Spatial coherence in periodic systems,” J. Opt. Soc. Am. 56, 1481–1489 (1966).
    [CrossRef]
  14. P. Yeh, A. Yariv, C.-S. Hong, “Electromagnetic propagation in periodic stratified media. 1. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
    [CrossRef]
  15. E. Hilb, “Zur Theorie der Linearen funktionalen Differentialgleichungen,” Math. Ann. 78, 137–170 (1916).
  16. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, (Cambridge U. Press, New York, 1995), Chap. 1–9.
    [CrossRef]
  17. G. V. Rozhnov, “Electromagnetic scattering from statistically rough sourfaces,” Sov. Phys. JETP 67, 240–247 (1988).
  18. G. V. Rozhnov, “Electromagnetic-wave diffraction by multilayer media with rough interfaces,” Sov. Phys. JETP 96, 646–651 (1989).

1995

M. S. Tomaš, “Green function for multilayers: light scattering in planar cavities,” Phys. Rev. A 51, 2545–2559 (1995).
[CrossRef] [PubMed]

1992

S. Barkeshli, “On the dyadic Green’s function for a planar multilayered dielectric/magnetic media,” IEEE Trans. Microwave Theory Tech. 40, 128–141 (1992).
[CrossRef]

1990

A. T. de Hoop, “Acoustic radiation from an impulsive point source in a continuously layered fluid—an analysis based on the Cagniard method,” J. Acoust. Soc. Am. 5, 2376–2388 (1990).

1989

G. V. Rozhnov, “Electromagnetic-wave diffraction by multilayer media with rough interfaces,” Sov. Phys. JETP 96, 646–651 (1989).

1988

G. V. Rozhnov, “Electromagnetic scattering from statistically rough sourfaces,” Sov. Phys. JETP 67, 240–247 (1988).

1987

C. E. Reed, J. Giergiel, J. C. Hemminger, S. Ushioda, “Dipole radiation in a multilayer geometry,” Phys. Rev. B 36, 4990–5000 (1987).
[CrossRef]

1984

M. Schoenberg, “Wave propagation in alternating solid and fluid layers,” Wave Motion 6, 303–320 (1984).
[CrossRef]

1977

P. Yeh, A. Yariv, C.-S. Hong, “Electromagnetic propagation in periodic stratified media. 1. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
[CrossRef]

1976

C. H. Chapman, “Exact and approximate generalized ray theory in vertically inhomogeneous media,” Geophys. J. R. Astron. Soc. 46, 201–233 (1976).
[CrossRef]

1974

C. H. Chapman, “A new method for computing synthetic seismograms,” Geophys. J. R. Astron. Soc. 36, 673–704 (1974).

C. H. Chapman, “Generalized ray theory for an inhomogeneous medium,” Geophys. J. R. Astron. Soc. 36, 673–704 (1974).
[CrossRef]

1966

W. Streifer, “Spatial coherence in periodic systems,” J. Opt. Soc. Am. 56, 1481–1489 (1966).
[CrossRef]

1963

E. Wolf, “Spatial coherence of resonant modes in a maser interferometer,” Phys. Lett. 3, 166–168 (1963).
[CrossRef]

1916

E. Hilb, “Zur Theorie der Linearen funktionalen Differentialgleichungen,” Math. Ann. 78, 137–170 (1916).

Barkeshli, S.

S. Barkeshli, “On the dyadic Green’s function for a planar multilayered dielectric/magnetic media,” IEEE Trans. Microwave Theory Tech. 40, 128–141 (1992).
[CrossRef]

Chapman, C. H.

C. H. Chapman, “Exact and approximate generalized ray theory in vertically inhomogeneous media,” Geophys. J. R. Astron. Soc. 46, 201–233 (1976).
[CrossRef]

C. H. Chapman, “A new method for computing synthetic seismograms,” Geophys. J. R. Astron. Soc. 36, 673–704 (1974).

C. H. Chapman, “Generalized ray theory for an inhomogeneous medium,” Geophys. J. R. Astron. Soc. 36, 673–704 (1974).
[CrossRef]

de Hoop, A. T.

A. T. de Hoop, “Acoustic radiation from an impulsive point source in a continuously layered fluid—an analysis based on the Cagniard method,” J. Acoust. Soc. Am. 5, 2376–2388 (1990).

Ewing, M. W.

M. W. Ewing, W. S. Jardetzky, F. Press, Elastic Waves in Layered Media (McGraw-Hill, London, 1957).

Giergiel, J.

C. E. Reed, J. Giergiel, J. C. Hemminger, S. Ushioda, “Dipole radiation in a multilayer geometry,” Phys. Rev. B 36, 4990–5000 (1987).
[CrossRef]

Hemminger, J. C.

C. E. Reed, J. Giergiel, J. C. Hemminger, S. Ushioda, “Dipole radiation in a multilayer geometry,” Phys. Rev. B 36, 4990–5000 (1987).
[CrossRef]

Hilb, E.

E. Hilb, “Zur Theorie der Linearen funktionalen Differentialgleichungen,” Math. Ann. 78, 137–170 (1916).

Hong, C.-S.

P. Yeh, A. Yariv, C.-S. Hong, “Electromagnetic propagation in periodic stratified media. 1. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
[CrossRef]

Jardetzky, W. S.

M. W. Ewing, W. S. Jardetzky, F. Press, Elastic Waves in Layered Media (McGraw-Hill, London, 1957).

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, (Cambridge U. Press, New York, 1995), Chap. 1–9.
[CrossRef]

Press, F.

M. W. Ewing, W. S. Jardetzky, F. Press, Elastic Waves in Layered Media (McGraw-Hill, London, 1957).

Reed, C. E.

C. E. Reed, J. Giergiel, J. C. Hemminger, S. Ushioda, “Dipole radiation in a multilayer geometry,” Phys. Rev. B 36, 4990–5000 (1987).
[CrossRef]

Rozhnov, G. V.

G. V. Rozhnov, “Electromagnetic-wave diffraction by multilayer media with rough interfaces,” Sov. Phys. JETP 96, 646–651 (1989).

G. V. Rozhnov, “Electromagnetic scattering from statistically rough sourfaces,” Sov. Phys. JETP 67, 240–247 (1988).

Schoenberg, M.

M. Schoenberg, “Wave propagation in alternating solid and fluid layers,” Wave Motion 6, 303–320 (1984).
[CrossRef]

Streifer, W.

W. Streifer, “Spatial coherence in periodic systems,” J. Opt. Soc. Am. 56, 1481–1489 (1966).
[CrossRef]

Tomaš, M. S.

M. S. Tomaš, “Green function for multilayers: light scattering in planar cavities,” Phys. Rev. A 51, 2545–2559 (1995).
[CrossRef] [PubMed]

Ushioda, S.

C. E. Reed, J. Giergiel, J. C. Hemminger, S. Ushioda, “Dipole radiation in a multilayer geometry,” Phys. Rev. B 36, 4990–5000 (1987).
[CrossRef]

Wolf, E.

E. Wolf, “Spatial coherence of resonant modes in a maser interferometer,” Phys. Lett. 3, 166–168 (1963).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, (Cambridge U. Press, New York, 1995), Chap. 1–9.
[CrossRef]

Wolter, H.

H. Wolter, Optik dünner Schichten, S. Flügge, ed., Encyclopedia of Physics (Springer, Berlin, 1956), Vol. 24, pp. 461–554.

Yariv, A.

P. Yeh, A. Yariv, C.-S. Hong, “Electromagnetic propagation in periodic stratified media. 1. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
[CrossRef]

Yeh, P.

P. Yeh, A. Yariv, C.-S. Hong, “Electromagnetic propagation in periodic stratified media. 1. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
[CrossRef]

P. Yeh, Optical Waves in Layered Media, Wiley Series in Pure and Applied Optics (Wiley, Chichester, UK, 1988).

Geophys. J. R. Astron. Soc.

C. H. Chapman, “Generalized ray theory for an inhomogeneous medium,” Geophys. J. R. Astron. Soc. 36, 673–704 (1974).
[CrossRef]

C. H. Chapman, “Exact and approximate generalized ray theory in vertically inhomogeneous media,” Geophys. J. R. Astron. Soc. 46, 201–233 (1976).
[CrossRef]

C. H. Chapman, “A new method for computing synthetic seismograms,” Geophys. J. R. Astron. Soc. 36, 673–704 (1974).

IEEE Trans. Microwave Theory Tech.

S. Barkeshli, “On the dyadic Green’s function for a planar multilayered dielectric/magnetic media,” IEEE Trans. Microwave Theory Tech. 40, 128–141 (1992).
[CrossRef]

J. Acoust. Soc. Am.

A. T. de Hoop, “Acoustic radiation from an impulsive point source in a continuously layered fluid—an analysis based on the Cagniard method,” J. Acoust. Soc. Am. 5, 2376–2388 (1990).

J. Opt. Soc. Am.

W. Streifer, “Spatial coherence in periodic systems,” J. Opt. Soc. Am. 56, 1481–1489 (1966).
[CrossRef]

P. Yeh, A. Yariv, C.-S. Hong, “Electromagnetic propagation in periodic stratified media. 1. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
[CrossRef]

Math. Ann.

E. Hilb, “Zur Theorie der Linearen funktionalen Differentialgleichungen,” Math. Ann. 78, 137–170 (1916).

Phys. Lett.

E. Wolf, “Spatial coherence of resonant modes in a maser interferometer,” Phys. Lett. 3, 166–168 (1963).
[CrossRef]

Phys. Rev. A

M. S. Tomaš, “Green function for multilayers: light scattering in planar cavities,” Phys. Rev. A 51, 2545–2559 (1995).
[CrossRef] [PubMed]

Phys. Rev. B

C. E. Reed, J. Giergiel, J. C. Hemminger, S. Ushioda, “Dipole radiation in a multilayer geometry,” Phys. Rev. B 36, 4990–5000 (1987).
[CrossRef]

Sov. Phys. JETP

G. V. Rozhnov, “Electromagnetic scattering from statistically rough sourfaces,” Sov. Phys. JETP 67, 240–247 (1988).

G. V. Rozhnov, “Electromagnetic-wave diffraction by multilayer media with rough interfaces,” Sov. Phys. JETP 96, 646–651 (1989).

Wave Motion

M. Schoenberg, “Wave propagation in alternating solid and fluid layers,” Wave Motion 6, 303–320 (1984).
[CrossRef]

Other

H. Wolter, Optik dünner Schichten, S. Flügge, ed., Encyclopedia of Physics (Springer, Berlin, 1956), Vol. 24, pp. 461–554.

P. Yeh, Optical Waves in Layered Media, Wiley Series in Pure and Applied Optics (Wiley, Chichester, UK, 1988).

M. W. Ewing, W. S. Jardetzky, F. Press, Elastic Waves in Layered Media (McGraw-Hill, London, 1957).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, (Cambridge U. Press, New York, 1995), Chap. 1–9.
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Geometry of the N-layer medium.

Equations (76)

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[ 2 + k 0 2 n 2 ( r ) ] E z ( r ) = 0 , r = ( x , y ) ,
[ 2 + η ( r ) + k 0 2 n 2 ( r ) ] H z ( r ) = 0 , η ( r ) = ϵ 1 ( r ) ,
E z ( r ; k ) = ψ ( r ; k ) = { a n ( α ) exp [ i k α x ( x n Λ ) ] + b n ( α ) exp [ i k α x ( x n Λ ) ] } exp ( i k α y y ) , α = 1 , 2 ,
k = ( k α x , k α y ) , k α x = ± [ ( ω n α c ) 2 k α y 2 ] 1 2 .
( a n 1 b n 1 ) = [ A B C D ] ( a n b n ) ,
A = exp ( i k 1 x a ) [ cos ( k 2 x b ) i 2 ( k 2 x k 1 x + k 1 x k 2 x ) sin ( k 2 x b ) ] ,
B = exp ( i k 1 x a ) [ i 2 ( k 2 x k 1 x k 1 x k 2 x ) sin ( k 2 x b ) ] ,
C = exp ( i k 1 x a ) [ i 2 ( k 2 x k 1 x k 1 x k 2 x ) sin ( k 2 x b ) ] ,
D = exp ( i k 1 x a ) [ cos ( k 2 x b ) + i 2 ( k 2 x k 1 x + k 1 x k 2 x ) sin ( k 2 x b ) ] .
A = exp ( i k 1 x a ) [ cos ( k 2 x b ) i 2 ( n 2 2 k 1 x n 1 2 k 2 x + n 1 2 k 2 x n 2 2 k 1 x ) sin ( k 2 x b ) ] ,
B = exp ( i k 1 x a ) [ i 2 ( n 2 2 k 1 x n 1 2 k 2 x n 1 2 k 2 x n 2 2 k 1 x ) sin ( k 2 x b ) ] ,
C = exp ( i k 1 x a ) [ i 2 ( n 2 2 k 1 x n 1 2 k 2 x n 1 2 k 2 x n 2 2 k 1 x ) sin ( k 2 x b ) ] ,
D = exp ( i k 1 x a ) [ cos ( k 2 x b ) + i 2 ( n 2 2 k 1 x n 1 2 k 2 x + n 1 2 k 2 x n 2 2 k 1 x ) sin ( k 2 x b ) ] .
( a 0 b 0 ) = [ A B C D ] n ( a n b n ) .
[ A B C D ] n
b ( n ) = 1 a b [ ( C ̃ A C A ̃ ) ( 1 b n + 1 1 a n + 1 ) + C ̃ ( 1 a n 1 b n ) ] ,
a ( n ) = 1 a b [ ( A ̃ D B C ̃ ) ( 1 a n + 1 1 b n + 1 ) + A ̃ ( 1 a n 1 b n ) ] ,
C ̃ = C a ( 0 ) + D b ( 0 ) , A ̃ = A a ( 0 ) + B b ( 0 ) .
a , b = A + D 2 ± [ ( A + D 2 ) 2 1 ] 1 2
[ A B C D ]
R N ( b 0 a 0 ) b N = 0 = C ( a N b N ) a N + 1 b N + 1 + D ( b N a N ) ,
T N ( a N a 0 ) b N = 0 = 1 a b [ a N + 1 b N + 1 + ( A + B R N ) ( b N a N ) ] .
ψ ( r , t ) = + A ( k ) exp ( i k r i ω t ) d k + + B ( k ) exp ( i k r + i ω t ) d k ,
A ( k ) + B ( k ) = f ̃ ( k ) ,
i ω A ( k ) + i ω B ( k ) = g ̃ ( k ) ,
f ̃ ( k ) = + f ( r ) exp ( i k r ) d r , g ̃ ( k ) = + g ( r ) exp ( i k r ) d r .
ψ = 1 2 [ 1 c x c t x + c t g ( x ) d x + f ( x + c t ) + f ( x c t ) ] ,
ψ x + ψ c t = 0 .
ψ ( x , t ) = + A ( k ) exp ( i k r i ω t ) d k = f ( x c t ) .
g ̃ ( k ) = i ω f ̃ ( k ) .
a 0 = A , b 0 = R N a 0 , b N = T N a 0
ψ ( r , t ) = + A ( k ) E ( r ; k ) exp ( i ω t ) d k ,
E ( r ; k ) = a n exp [ i k x ( x N Λ ) ] + b n exp [ i k x ( x N Λ ) ] exp ( i k y y ) .
exp ( ± i k 2 x b ) , exp ( ± i k 1 x a ) ,
Γ ( r 1 , r 2 ; t 1 , t 2 ) ψ ( r 1 , t 1 ) ψ * ( r 2 , t 2 ) ,
γ ( r 1 , r 2 ; t 1 , t 2 ) = Γ ( r 1 , r 2 ; t 1 , t 2 ) [ Γ ( r 1 , r 1 ; t 1 ) ] 1 2 [ Γ ( r 2 , r 2 ; t 1 ) ] 1 2 ,
0 γ ( r 1 , r 2 ; t 1 , t 2 ) 1 .
ϵ ( r ) = ϵ 1 θ [ x h ( r σ ) ] + ϵ 2 θ [ h ( r σ ) x ] ,
Δ ϵ ( r ) ϵ ( r ) ϵ 1 θ ( x ) ϵ 2 θ ( x ) ,
Δ ϵ ( r ) = [ h ( y ) δ ( x ) 1 2 h 2 ( y ) d d x δ ( x ) + ] ,
[ h ̃ ( k y 1 ) h ̃ * ( k y 2 ) = S ( k y 1 ) δ ( k y 1 k y 2 ) .
[ Δ 2 + k 0 2 n 2 ( x ) G ( r , r ; k 0 2 ) = δ ( r r ) ,
G ( r , r ; k 0 2 ) = exp i k y ( y y ) g ( x , x ; k y ) d k y ,
d 2 g ( x , x ; k y ) d x 2 + [ k 0 2 n 2 ( x ) k y 2 ] g ( x , x ; k y ) = δ ( x x ) ,
g ( x , x ; k y ) = E 1 ( x < , x ; k y ) E 2 ( x > , x ; k y ) W ( E 1 , E 2 ) ,
ψ ( r ) = A exp ( i k 0 r ) + τ k 0 2 Δ ϵ ( r ) G ( r , r ) ψ ( r ) d r ,
ψ ( r ) = A exp ( i k 0 r ) + A k 0 2 exp ( i k y y ) h ̃ ( k y k 0 y ) g ( x , x = x 0 ; k y ) d k y ,
Γ ( r 1 , r 2 ; τ ) = exp ( i k y 1 y 1 + i k y 2 y 2 ) h ̃ ( k y 1 k 0 y ) h ̃ * ( k y 2 k 0 y ) g ( x 1 , x = x 0 ; k y 1 ) g * ( x 2 , x = x 0 ; k y 2 ) d k y 1 k y 2 .
Γ ( r 1 , r 2 ; τ ) = exp [ i k y ( y 1 y 2 ) ] S ( k y k 0 y ) g ( x 1 , x = x 0 ; k y ) g * ( x 2 , x = x 0 ; k y ) d k y .
Γ ( r 1 , r 2 ; t 1 , t 2 ) = + A ( k 1 ) E ( r 1 ; k 1 ) exp ( i ω 1 t 1 ) d k 1 + A * ( k 2 ) E * ( r 2 ; k ) exp ( i ω 2 t 2 ) d k 2 , ω 1 , 2 = c n k 1 , 2 .
+ + d k 1 d k 2 T N ( k 1 ) T N * ( k 2 ) a 0 ( k 1 ) a 0 * ( k 2 ) × exp ( i k 1 r 1 i k 2 r 2 ) exp ( i ω 2 t 2 i ω 1 t 1 ) d k 1 d k 2 , ω 1 , 2 = c n k 1 , 2 .
exp ( ± i k 2 x b ) , exp ( ± i k 1 x a ) ,
Γ ( y 1 , y 2 ; ν ̃ ) ψ ( y 1 ) ψ * ( y 2 ) = S 0 [ 1 2 ( y 1 + y 2 ) ; ν ̃ ] g 0 ( y 1 y 2 ; ν ̃ ) .
Γ ( k y 1 , k y 2 ; ν ̃ ) = a 0 ( k y 1 ) a 0 * ( k y 2 ) = S ̃ 0 [ 1 2 ( k y 1 + k y 2 ) ; ν ̃ ] g ̃ 0 ( k y 1 k y 2 ; ν ̃ ) .
Γ ( y 1 , y 2 ; ν ̃ ) = I ( y 1 ) δ ( y 1 y 2 ) ,
Γ ( k y 1 , k y 2 ; ν ̃ ) = a 0 ( k y 1 ) a 0 * ( k y 2 ) = I ̃ ( k y 1 k y 2 ) .
Γ ( y 1 , y 2 , ν ̃ ) = ψ ( y 1 ) ψ * ( y 2 ) ,
Γ ( s 1 , s 2 ; ν ̃ ) = σ σ Γ ( y 1 , y 2 , ν ̃ ) exp [ i k 0 ( y 1 s y 1 y 2 s y 2 ) ] d y 1 d y 2 ,
I ̃ [ k 0 ( s y 1 s y 2 ) ] T [ k 0 ( s y 1 s y 2 ) ] T * [ k 0 ( s y 1 s y 2 ) ] .
T n = c l , m exp [ i ( l k 1 x a + m k 2 x b ) ] ,
ψ b l ( x , t = 0 ) = + sin ( Δ k ) k exp ( i x ) d k .
ψ b l ( x , t ) = + T N ( k ) sin ( Δ k ) k exp ( i k x i ω t ) d k .
ψ ( r , ω ) = ψ ( inc ) ( r , ω ) + τ G ( r , r ; ω ) k 0 2 [ 1 n 2 ( x ) ] ψ ( inc ) ( r , ω ) d r ,
ψ ( r , ω ) = ψ ( inc ) ( r , ω ) + τ R ( r , r ; ω ) ψ ( inc ) ( r , ω ) d r ,
R ( r , r ; ω ) = G ( r , r ; ω ) k 0 2 [ 1 n 2 ( x ) ] + τ G ( r , r ; ω ) R ( r , r ; ω ) k 0 2 [ 1 n 2 ( x ) ] d r .
[ 2 + k 0 2 n 2 ( x ) ] R ( r , r ; ω ) = [ 1 n 2 ( x ) ] δ ( r r )
R ( r , r ; ω ) = exp [ i k y ( y y ) ] γ ( x , x ; k y ) d k y ,
d 2 γ ( x , x ; k y ) d x 2 + [ k 0 2 n 2 ( x ) k y 2 ] γ ( x , x ; k y ) = δ ( x x ) [ 1 n 2 ( x ) ] ,
γ ( x , x ; k y ) = E 1 ( x < ; k y ) E 2 ( x > ; k y ) W ( E 1 , E 2 ) [ 1 n 2 ( x ) ] ,
ψ ( r , t ) = ψ ( inc ) ( r , t ) + 0 + d t τ R ( r , r , t t ) ψ ( inc ) ( r , t ) d r ,
R ( r , r , t ) = + exp ( i ω t ) R ( r , r , ω ) d ω ;
Γ ( r 1 , r 2 ; t 1 , t 2 ) = [ ψ ( inc ) ( r 1 , t 1 ) + 0 + d t τ R ( r 1 , r , t 1 t ) ψ ( inc ) ( r , t ) d r ] [ ψ ( inc ) ( r 2 , t 2 ) + 0 + d t τ R ( r 2 , r , t 2 t ) ψ ( inc ) ( r , t ) d r ] * .
exp ( i k y y ) γ ( x , x ; k y )
c l , m exp [ i ( l k 1 x a + m k 2 x b ) ] ,
exp [ i ( l k 1 x a + m k 2 x b ) ] exp [ i k 1 , 2 ( x ± x ) ] exp ( i ω t ) .
R ( r , r , t ) = c l , m { U [ n ( x ) x + n ( x ) x + n 1 l a + n 2 m b ] } x ,

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