Abstract

The second-order theory of partial coherence for scalar and TE or TM fields is developed for weakly periodic media, and the van Cittert–Zernike theorem of classical coherence theory is generalized for such media. The coherence properties of a wave field, generated by a quasi-homogeneous source distribution at the entrance plane of a finite weakly periodic medium, are calculated both inside such a structure and in the far field. The second-order theory of partial coherence for pulse propagation through weakly periodic media is also developed.

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References

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  1. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).
  2. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
    [CrossRef]
  3. E. Wolf, “Spatial coherence of resonant modes in a maser interferometer,” Phys. Lett. 3, 166–168 (1963).
    [CrossRef]
  4. W. Streifer, “Spatial coherence in periodic systems,” J. Opt. Soc. Am. 56, 1481–1489 (1966).
    [CrossRef]
  5. R. Becker, Theorie der Elektriztät (B.G. Teubner, 1969), Vol. 3, Chap. A2, p. 35.
  6. E. C. Titchmarsh, Eigenfunction Expansions (Oxford U. Press, Oxford, 1958), Vol. 2.
  7. B. J. Hoenders, D. N. Pattanayak, “Interaction of a moving charged particle with a spatially dispersive medium. I. Structure of the electromagnetic field,” Phys. Rev. D 13, 282–290 (1976).
    [CrossRef]
  8. B. J. Hoenders, D. N. Pattanayak, “Interaction of a moving charged particle with a spatially dispersive medium. II. Cerenkov and transition radiation,” Phys. Rev. D 13, 291–298 (1976).
    [CrossRef]
  9. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Chaps. 1–9, pp. 1–436.
  10. P. H. van Cittert, “Die wahrscheinliche Schwingungsverteilung in einer von einer Lichtquelle direkt oder mittels einer Linse beleuchteten Ebene,” Physica (Amsterdam) 1, 201–210 (1934).
    [CrossRef]
  11. F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica (Amsterdam) 5, 785–795 (1938).
    [CrossRef]
  12. 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]
  13. P. Yeh, Optical Waves in Layered Media, Wiley Series in Pure and Applied Optics (Wiley, 1988).

1977 (1)

1976 (2)

B. J. Hoenders, D. N. Pattanayak, “Interaction of a moving charged particle with a spatially dispersive medium. I. Structure of the electromagnetic field,” Phys. Rev. D 13, 282–290 (1976).
[CrossRef]

B. J. Hoenders, D. N. Pattanayak, “Interaction of a moving charged particle with a spatially dispersive medium. II. Cerenkov and transition radiation,” Phys. Rev. D 13, 291–298 (1976).
[CrossRef]

1966 (1)

1963 (1)

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

1938 (1)

F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica (Amsterdam) 5, 785–795 (1938).
[CrossRef]

1934 (1)

P. H. van Cittert, “Die wahrscheinliche Schwingungsverteilung in einer von einer Lichtquelle direkt oder mittels einer Linse beleuchteten Ebene,” Physica (Amsterdam) 1, 201–210 (1934).
[CrossRef]

Becker, R.

R. Becker, Theorie der Elektriztät (B.G. Teubner, 1969), Vol. 3, Chap. A2, p. 35.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

Hoenders, B. J.

B. J. Hoenders, D. N. Pattanayak, “Interaction of a moving charged particle with a spatially dispersive medium. I. Structure of the electromagnetic field,” Phys. Rev. D 13, 282–290 (1976).
[CrossRef]

B. J. Hoenders, D. N. Pattanayak, “Interaction of a moving charged particle with a spatially dispersive medium. II. Cerenkov and transition radiation,” Phys. Rev. D 13, 291–298 (1976).
[CrossRef]

Hong, C.-S.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Chaps. 1–9, pp. 1–436.

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

Pattanayak, D. N.

B. J. Hoenders, D. N. Pattanayak, “Interaction of a moving charged particle with a spatially dispersive medium. II. Cerenkov and transition radiation,” Phys. Rev. D 13, 291–298 (1976).
[CrossRef]

B. J. Hoenders, D. N. Pattanayak, “Interaction of a moving charged particle with a spatially dispersive medium. I. Structure of the electromagnetic field,” Phys. Rev. D 13, 282–290 (1976).
[CrossRef]

Streifer, W.

Titchmarsh, E. C.

E. C. Titchmarsh, Eigenfunction Expansions (Oxford U. Press, Oxford, 1958), Vol. 2.

van Cittert, P. H.

P. H. van Cittert, “Die wahrscheinliche Schwingungsverteilung in einer von einer Lichtquelle direkt oder mittels einer Linse beleuchteten Ebene,” Physica (Amsterdam) 1, 201–210 (1934).
[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).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Chaps. 1–9, pp. 1–436.

Yariv, A.

Yeh, P.

Zernike, F.

F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica (Amsterdam) 5, 785–795 (1938).
[CrossRef]

J. Opt. Soc. Am. (2)

Phys. Lett. (1)

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

Phys. Rev. D (2)

B. J. Hoenders, D. N. Pattanayak, “Interaction of a moving charged particle with a spatially dispersive medium. I. Structure of the electromagnetic field,” Phys. Rev. D 13, 282–290 (1976).
[CrossRef]

B. J. Hoenders, D. N. Pattanayak, “Interaction of a moving charged particle with a spatially dispersive medium. II. Cerenkov and transition radiation,” Phys. Rev. D 13, 291–298 (1976).
[CrossRef]

Physica (Amsterdam) (2)

P. H. van Cittert, “Die wahrscheinliche Schwingungsverteilung in einer von einer Lichtquelle direkt oder mittels einer Linse beleuchteten Ebene,” Physica (Amsterdam) 1, 201–210 (1934).
[CrossRef]

F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica (Amsterdam) 5, 785–795 (1938).
[CrossRef]

Other (6)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Chaps. 1–9, pp. 1–436.

R. Becker, Theorie der Elektriztät (B.G. Teubner, 1969), Vol. 3, Chap. A2, p. 35.

E. C. Titchmarsh, Eigenfunction Expansions (Oxford U. Press, Oxford, 1958), Vol. 2.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

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

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

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Figures (2)

Fig. 1
Fig. 1

Geometry of the system.

Fig. 2
Fig. 2

Geometry and various wave amplitudes of the plane waves inside and outside the medium.

Equations (76)

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[ 2 + k 0 2 n 2 ( r ) ] E z ( r ) = 0
( 2 + η ( r ) + k 0 2 n 2 ( r ) ) H z ( r ) = 0 , η ( r ) = ϵ 1 ( r ) ,
( 2 + k 0 2 n 2 ( r ) ) ψ ( r ) = 0 ,
n 2 ( r ) = n a n exp ( i n r ) , a 0 = 1
n = ( n x π a , n y π b ) ,
n x = 0 , ± 1 , ± 2 , , n x = 0 , ± 1 , ± 2 , ,
ψ m ( r ) = A ( k ) exp ( i k r ) { 1 n 0 d n exp ( i n r ) } ,
k 2 = k 0 2 ,
ψ m ( r ) = A ( k ) exp ( i k r ) { 1 k 0 2 n 0 a n exp ( i n r ) k 0 2 ( k + n ) exp 2 } .
k 0 2 ( k + n ) 2 = 0 .
ψ ( r ) = k σ gap A ( k σ ) exp ( i k r ) { 1 k 0 2 n 0 a n exp ( i n r ) k 0 2 ( k + n ) 2 } d k σ ,
ψ ( r σ ) = k σ gap A ( k σ ) exp ( i k r σ ) { 1 k 0 2 n 0 a n exp ( i n r σ ) k 0 2 ( k + n ) 2 } d k σ ,
A ( k σ ) k 0 2 n 0 a n A ( k σ + n σ ) k 0 2 ( k + n σ + n ) 2 = ψ ̃ ( k σ ) ,
A ( k σ ) = ψ ̃ ( k σ ) + k 0 2 n 0 a n ψ ̃ ( k σ + n σ ) k 0 2 ( k + n σ + n ) 2 .
ψ ( r ) = ( ψ ̃ ( k σ ) + k 0 2 n 0 a n ψ ̃ ( k σ + n σ ) k 0 2 ( k + n σ + n ) 2 ) exp ( i k r ) { 1 k 0 2 n 0 a n exp ( i n r ) k 0 2 ( k + n σ + n ) 2 } d k σ .
ψ ( r ) = exp ( i k 0 R ) R σ ψ ( r σ ) exp ( i k 0 r σ s ) ( 1 + k 0 2 n 0 a n exp ( i n r σ ) k 0 2 ( k + n ) 2 k 0 2 m 0 a m exp ( i m r ) k 0 2 ( k + m ) 2 ) d r σ ,
k 2 = k 0 2 , k x = k 0 cos ( φ ) , k y = k 0 sin ( φ ) .
( ψ ̃ ( k σ ) exp ( i k r ) k 0 2 n 0 a n exp ( i n r ) k 0 2 ( k + n σ + n ) 2 )
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 .
Γ ( r 1 , r 2 ; ν ̃ ) R 2 Γ ( r 1 , r 2 ; ν ̃ ) = σ σ ψ ( r 1 σ ) ψ * ( r 2 σ ) exp ( i k 0 ( r 1 σ s 1 r 2 σ s 2 ) ) ( 1 + k 0 2 n 0 a n exp ( i n r 1 σ ) k 0 2 ( k + n ) 2 k 0 2 m 0 a m exp ( i m r 1 ) k 0 2 ( k + m ) 2 ) ( 1 + k 0 2 n 0 a n * exp ( i n r 2 σ ) k 0 2 ( k + n ) 2 k 0 2 m 0 a m * exp ( i m r 2 ) k 0 2 ( k + m ) 2 ) d r 1 σ d r 2 σ
ψ ( r 1 σ ) ψ * ( r 2 σ ) = I ( r 1 σ ) δ ( r 1 σ r 2 σ ) .
Γ ( r 1 , r 2 ; ν ̃ ) = σ I ( r σ ) exp ( i k 0 r σ ( s 1 s 2 ) ) ( 1 + k 0 2 n 0 a n exp ( i n r σ ) k 0 2 ( k + n ) 2 k 0 2 m 0 a m exp ( i m r 1 ) k 0 2 ( k + m ) 2 ) ( 1 + k 0 2 n 0 a n * exp ( i n r σ ) k 0 2 ( k + n ) 2 k 0 2 m 0 a m * exp ( i m r 2 ) k 0 2 ( k + m ) 2 ) d r σ
ψ ( r 1 σ ; ν ̃ ) ψ * ( r 2 σ ; ν ̃ ) = S ( 0 ) [ ( 1 2 ( r σ 1 + r σ 2 ) ; ν ̃ ) ] g ( 0 ) ( r σ 2 r σ 1 ; ν ̃ ) .
Γ ( r 1 , r 2 ; ν ̃ ) = S ̃ ( 0 ) [ k ( s 2 σ s 1 σ ; ν ̃ ) ] g ̃ ( 0 ) [ 1 2 k ( s 2 σ + s 1 σ ; ν ̃ ) ] ( 1 k 0 2 m 0 a m * exp ( i m r 2 ) k 0 2 ( k + m ) 2 k 0 2 m 0 a m exp ( i m r 1 ) k 0 2 ( k + m ) 2 ) + k 0 2 m 0 a m S ̃ ( 0 ) [ k ( s 2 σ s 1 σ m σ ; ν ̃ ) ] g ̃ ( 0 ) [ 1 2 k ( s 2 σ + s 1 σ + m σ ; ν ̃ ) ] k 0 2 ( k + m σ + m ) 2 + k 0 2 m 0 a m * S ̃ ( 0 ) [ k ( s 2 σ s 1 σ + m σ ; ν ̃ ) ] g ̃ ( 0 ) [ 1 2 k ( s 2 σ + s 1 σ + m σ ; ν ̃ ) ] k 0 2 ( k + m σ + m ) 2 ,
S ̃ ( 0 ) [ k ( s 2 σ s 1 σ ) ] g ̃ ( 0 ) [ 1 2 k ( s 2 σ + s 1 σ ) ] ,
A ( k ) exp ( i k σ r σ + i k x x ) { 1 k 0 2 n 0 a n exp ( i n σ r σ + i n x x ) k 0 2 ( k + n ) 2 }
B ( k ) exp ( i k σ r σ i k x x ) { 1 k 0 2 n 0 a n exp ( i n σ r σ + i n x x ) k 0 2 ( k + n ) 2 } .
A 0 ( k σ , r σ ) A ( k σ ) { 1 k 0 2 n 0 a n exp ( i n σ r σ ) k 0 2 ( k + n ) 2 } ,
B 0 ( k σ , r σ ) B ( k σ ) { 1 k 0 2 n 0 a n exp ( i n σ r σ ) k 0 2 ( k + n ) 2 } ,
A 1 ( k σ , r σ ) A ( k σ ) { 1 k 0 2 n 0 a n exp ( i n σ r σ + i n x a ) k 0 2 ( k + n ) 2 }
A 2 ( k σ , r σ ) A ( k σ ) { 1 k 0 2 n 0 a n exp ( i n σ r σ + i n x a ) i ( k x + n x ) k 0 2 ( k + n ) 2 }
B 2 ( k σ , r σ ) B ( k σ ) { 1 k 0 2 n 0 a n exp ( i n σ r σ + i n x a ) i ( n x k x ) k 0 2 ( k + n ) 2 } .
ψ ( r ) = ( A 0 ( k σ ( 1 ) , r σ ) exp ( i k σ ( 1 ) r σ + i k x ( 1 ) x ) + B 0 ( k σ ( 1 ) r σ ) exp ( i k σ ( 1 ) r σ i k x ( 1 ) x ) ) d k σ ( 1 ) .
ψ out ( r ) = C ( k σ ( 2 ) ) exp ( i k σ ( 2 ) r σ + i k x ( 2 ) x ) d k σ ( 2 ) ,
ψ ̃ ( k σ ( 1 ) ) = A ̃ 0 ( k σ ( 1 ) ) + B ̃ 0 ( k σ ( 1 ) )
A ̃ 1 ( k σ ( 1 ) ) exp ( i k x ( 1 ) a ) + B ̃ 1 ( k σ ( 1 ) ) exp ( i k x ( 1 ) a ) = C ( k σ ( 2 ) ) ,
A ̃ 2 ( k σ ( 1 ) ) exp ( i k x ( 1 ) a ) B ̃ 2 ( k σ ( 1 ) ) exp ( i k x ( 1 ) a ) = i k x ( 2 ) i k x ( 1 ) C ( k σ ( 2 ) ) .
A ( unp ) ( k σ ( 1 ) ) = ψ ̃ ( k σ ( 1 ) ) ( ν exp ( i k x ( 1 ) a ) exp ( i k x ( 1 ) a ) ) ( ν 1 ) exp ( i k x ( 1 ) a ) ( 1 + ν ) exp ( i k x ( 1 ) a ) ,
B ( unp ) ( k σ ( 1 ) ) = ψ ̃ ( k σ ( 1 ) ) ( 1 ν ) exp ( i k x ( 1 ) a ) ( ν 1 ) exp ( i k x ( 1 ) a ) ( 1 + ν ) exp ( i k x ( 1 ) a ) ,
C ( unp ) ( k σ ( 2 ) ) = 2 ψ ̃ ( k σ ( 1 ) ) ( ν 1 ) exp ( i k x ( 1 ) a ) ( 1 + ν ) exp ( i k x ( 1 ) a ) , with ν = i k x ( 2 ) i k x ( 1 ) .
C ( k σ ( 2 ) ) = 2 ψ ̃ ( k σ ( 1 ) ) ( ν 1 ) exp ( i k x ( 1 ) a ) ( 1 + ν ) exp ( i k x ( 1 ) a ) + k 0 2 ( n 0 a n A ( unp ) ( k σ ( 1 ) + n σ ) + exp ( i n x a ) k 0 2 ( k + n σ + n ) 2 ) × exp ( i k x ( 1 ) a ) + k 0 2 ( n 0 a n B ( unp ) ( k σ ( 1 ) + n σ ) + exp ( i n x a ) k 0 2 ( k + n σ + n ) 2 ) exp ( i k x ( 1 ) a ) .
Γ ( r 1 σ ( 2 ) , r 2 σ ( 2 ) ; x = a ) = ψ ( r 1 σ ( 2 ) ) ψ * ( r 2 σ ( 2 ) )
Γ ( s 1 ( 2 ) , s 2 ( 2 ) ; ν ̃ ) = σ σ ψ ( r 1 σ ( 2 ) ) ψ * ( r 2 σ ( 2 ) ) exp ( i k 0 ( r 1 σ ( 2 ) s 1 ( 2 ) r 2 σ ( 2 ) s 2 ( 2 ) ) ) d r 1 σ ( 2 ) d r 2 σ ( 2 ) ,
Γ ( r 1 σ ( 2 ) , r 2 σ ( 2 ) ; x = a ) = C ( k 1 σ ( 2 ) ) C * ( k 2 σ ( 2 ) ) ,
σ σ I ( r 1 σ ( 2 ) ) δ ( r 1 σ ( 2 ) r 2 σ ( 2 ) ) exp ( i k 0 ( r 1 σ ( 2 ) s 1 ( 2 ) r 2 σ ( 2 ) s 2 ( 2 ) ) ) d r 1 σ ( 2 ) d r 2 σ ( 2 ) = I ̃ ( k 1 σ ( 2 ) k 2 σ ( 2 ) ) ,
σ σ S ( 0 ) [ ( 1 2 ( r σ 1 ( 2 ) + r σ 2 ( 2 ) ) ; ν ̃ ) ] g ( 0 ) ( r σ 2 ( 2 ) r σ 1 ( 2 ) ; ν ̃ ) exp ( i k 0 ( r 1 σ ( 2 ) s 1 ( 2 ) r 2 σ ( 2 ) s 2 ( 2 ) ) ) d r 1 σ ( 2 ) d r 2 σ ( 2 ) = S ̃ ( 0 ) [ ( 1 2 ( s σ 1 ( 2 ) + s σ 2 ( 2 ) ) ; ν ̃ ) ] g ̃ ( 0 ) ( s σ 2 ( 2 ) s σ 1 ( 2 ) ; ν ̃ ) .
ψ ( r , t ) = + D ( k ) exp ( ( i k r i ω t ) ) d k + + E ( k ) exp ( ( i k r + i ω t ) ) d k ,
D ( k ) + E ( k ) = f ̃ ( k + n σ )
i ω D ( k ) + i ω E ( k ) = g ̃ ( k ) ,
f ̃ ( k ) = + f ( r ) exp ( i k r ) d r ,
g ̃ ( k ) = + g ( r ) exp ( i k r ) d r .
ψ x + ψ c t = 0 .
ψ ( x , t ) = + D ( k ) exp ( ( i k r i ω t ) ) d k = f ( x c t ) .
g ̃ ( k ) = i ω f ̃ ( k ) .
ψ ( ) ( r , t ) = ( D ( k ( 1 ) ) exp ( i k σ ( 1 ) r σ + i k x ( 1 ) x i ω t ) + E ( k ( 1 ) ) exp ( i k σ ( 1 ) r σ i k x ( 1 ) x i ω t ) ) d k σ ( 1 ) d ω ,
ω = c k ( 1 )
ψ ( cryst ) ( r , t ) = ( A 0 ( k ( 2 ) , r ) exp ( i k σ ( 2 ) r σ + i k x ( 2 ) x i ω t ) + B 0 ( k ( 2 ) , r ) exp ( i k σ ( 2 ) r σ i k x ( 2 ) x i ω t ) ) d k σ ( 2 ) d ω , ω = c k ( 2 ) ,
A 0 ( k ( 2 ) , r ) A ( k σ ( 2 ) ) { 1 k 0 2 n 0 a n exp i n r k 0 2 ( k ( 2 ) + n ) 2 } ,
B 0 ( k ( 2 ) , r ) B ( k σ ( 2 ) ) { 1 k 0 2 n 0 a n exp ( i n r ) k 0 2 ( k ( 2 ) + n ) 2 }
ψ ( out ) ( r , t ) = C ( k ( 3 ) ) exp ( i k σ ( 3 ) r σ + i k x ( 3 ) x i ω t ) d k σ ( 3 ) d ω , ω = c k ( 3 )
D ( k ( 1 ) ) + E ( k ( 1 ) ) = A ̃ 0 ( k ( 2 ) ) + B ̃ 0 ( k ( 2 ) ) , k σ ( 1 ) = k σ ( 2 ) ,
i k ( 1 ) ( D ( k ( 1 ) ) E ( k ( 1 ) ) ) = ( A ̃ 2 ( k ( 2 ) ) B ̃ 2 ( k ( 1 ) ) ) i k ( 2 ) .
A ̃ 1 ( k ( 2 ) ) exp ( i k x ( 2 ) a ) + B ̃ 1 ( k ( 2 ) ) exp ( i k x ( 2 ) a ) = C ( k ( 3 ) ) exp ( i k x ( 3 ) a ) ,
k σ ( 2 ) = k σ ( 3 ) ,
i k x ( 2 ) a A ̃ 2 ( k ( 2 ) ) exp ( i k x ( 2 ) a ) i k x ( 2 ) a B ̃ 2 ( k ( 2 ) ) exp ( i k x ( 2 ) a ) = i k x ( 3 ) C ( k ( 3 ) ) exp ( i k x ( 3 ) a ) .
1 2 [ exp ( i k x ( 2 ) a ) ( 1 + k x ( 1 ) k x ( 2 ) ) exp ( i k x ( 2 ) a ) ( 1 + k x ( 1 ) k x ( 2 ) ) exp ( i k x ( 2 ) a ) ( 1 k x ( 1 ) k x ( 2 ) ) exp ( i k x ( 2 ) a ) ( 1 k x ( 1 ) k x ( 2 ) ) ] ( E ( unp ) ( k ( 1 ) ) D ( unp ) ( k ( 1 ) ) ) = ( A ( unp ) ( k ( 2 ) ) B ( unp ) ( k ( 2 ) ) )
E ( unp ) ( k ( 1 ) ) = R D ( unp ) ( k ( 1 ) ) , C ( unp ) ( k ( 3 ) ) = T D ( unp ) ( k ( 3 ) ) ,
R = ( g 2 + g 1 ) ( g 3 g 2 ) exp ( i ρ 2 a ) + ( g 2 + g 3 ) ( g 2 g 1 ) exp ( i ρ 2 a ) ( g 2 + g 3 ) ( g 1 + g 2 ) exp ( i ρ 2 a ) + ( g 2 g 3 ) ( g 1 g 2 ) exp ( i ρ 2 a ) ,
T = 2 g 2 g 3 ( g 2 + g 3 ) ( g 1 + g 2 ) exp ( i ρ 2 a ) + ( g 2 g 3 ) ( g 1 g 2 ) exp ( i ρ 2 a ) ,
ρ 2 = k x ( 2 ) , g l = k x ( l ) μ l for TE ; g l = k x ( l ) μ l n l for TM , l = 1 , 2 , 3 .
C ( k ( 3 ) ) exp ( i k x ( 3 ) a ) = T D ( unp ) ( k ( 3 ) ) + k 0 2 ( n 0 a n A ( unp ) ( k σ ( 2 ) + n σ , k x ( 2 ) ) exp ( i n x a ) k 0 2 ( k ( 2 ) + n σ + n ) 2 ) exp ( i k x ( 2 ) a ) + k 0 2 ( n 0 a n B ( unp ) ( k σ ( 1 ) + n σ , k x ( 1 ) ) exp ( i n x a ) k 0 2 ( k ( 2 ) + n σ + n ) 2 ) exp ( i k x ( 2 ) a ) .
ψ ( r , t ) = + C ( k ( 3 ) ) exp ( i k σ ( 3 ) r σ + i k x ( 3 ) x i ω t ) d k ( 3 )
if ω = c k ( 3 ) .
Γ ( r 1 , r 2 ; t 1 , t 2 ) = + + C ( k ( 3 ) ) C * ( k ( 3 ) ) exp ( i k σ ( 3 ) r 1 σ + i k x ( 3 ) x 1 i ω t 1 i k σ ( 3 ) r 2 σ i k x ( 3 ) x 1 + i ω t 2 ) d k ( 3 ) d k ( 3 ) .

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