Abstract

One-dimensional propagation of transient electromagnetic waves in periodic media is studied. The media are periodic in the direction of propagation and can be of finite or infinite length. Wave propagators that map a transient field from one point in the medium to another are introduced. A number of useful relations for the propagators are presented. Some of these relations are used in the determination of explicit expressions for the short time behavior of a transient wave as it propagates in a periodic medium. The theory is exemplified by several numerical examples.

© 1995 Optical Society of America

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References

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  1. F. Gilbert, G. E. Backus, “Propagator matrices in elastic wave and vibration problems,” Geophysics 31, 326–332 (1966).
    [CrossRef]
  2. L. Fishman, “Exact and operator rational approximate solutions of the Helmholtz, Weyl composition equation in underwater acoustics—the quadratic profile,” J. Math. Phys. 33, 1887–1914 (1992).
    [CrossRef]
  3. J. P. Corones, M. E. Davison, R. J. Krueger, “Direct and inverse scattering in the time domain via invariant imbedding equations,” J. Acoust. Soc. Am. 74, 1535–1541 (1983).
    [CrossRef]
  4. R. J. Krueger, R. L. Ochs, “A Green’s function approach to the determination of internal fields,” Wave Motion 11, 525–543 (1989).
    [CrossRef]
  5. V. H. Weston, “Invariant imbedding for the wave equation in three dimensions and the applications to the direct and inverse problems,” Inverse Probl. 6, 1075–1105 (1990).
    [CrossRef]
  6. C. Elachi, “Waves in active and passive periodic structures: a review,” Proc. IEEE 64, 1666–1698 (1976).
    [CrossRef]
  7. S. He, A. Karlsson, “Time domain Green function technique for a point source over a dissipative stratified half-space,” Radio Sci. 28, 513–526 (1993).
    [CrossRef]
  8. G. Kristensson, R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. Part 3. Scattering operators in the presence of a phase velocity mismatch,” J. Math. Phys. 28, 360–370 (1987).
    [CrossRef]
  9. G. Kristensson, R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. Part I. Scattering operators,” J. Math. Phys. 27, 1667–1682 (1986).
    [CrossRef]
  10. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  11. K. E. Oughstun, G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
    [CrossRef]
  12. A. Karlsson, H. Otterheim, R. Stewart, “Transient wave propagation in composite media: Green’s function approach,” J. Opt. Soc. Am. A 10, 886–895 (1993).
    [CrossRef]
  13. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

1993 (2)

S. He, A. Karlsson, “Time domain Green function technique for a point source over a dissipative stratified half-space,” Radio Sci. 28, 513–526 (1993).
[CrossRef]

A. Karlsson, H. Otterheim, R. Stewart, “Transient wave propagation in composite media: Green’s function approach,” J. Opt. Soc. Am. A 10, 886–895 (1993).
[CrossRef]

1992 (1)

L. Fishman, “Exact and operator rational approximate solutions of the Helmholtz, Weyl composition equation in underwater acoustics—the quadratic profile,” J. Math. Phys. 33, 1887–1914 (1992).
[CrossRef]

1990 (1)

V. H. Weston, “Invariant imbedding for the wave equation in three dimensions and the applications to the direct and inverse problems,” Inverse Probl. 6, 1075–1105 (1990).
[CrossRef]

1989 (1)

R. J. Krueger, R. L. Ochs, “A Green’s function approach to the determination of internal fields,” Wave Motion 11, 525–543 (1989).
[CrossRef]

1988 (1)

1987 (1)

G. Kristensson, R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. Part 3. Scattering operators in the presence of a phase velocity mismatch,” J. Math. Phys. 28, 360–370 (1987).
[CrossRef]

1986 (1)

G. Kristensson, R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. Part I. Scattering operators,” J. Math. Phys. 27, 1667–1682 (1986).
[CrossRef]

1983 (1)

J. P. Corones, M. E. Davison, R. J. Krueger, “Direct and inverse scattering in the time domain via invariant imbedding equations,” J. Acoust. Soc. Am. 74, 1535–1541 (1983).
[CrossRef]

1976 (1)

C. Elachi, “Waves in active and passive periodic structures: a review,” Proc. IEEE 64, 1666–1698 (1976).
[CrossRef]

1966 (1)

F. Gilbert, G. E. Backus, “Propagator matrices in elastic wave and vibration problems,” Geophysics 31, 326–332 (1966).
[CrossRef]

Backus, G. E.

F. Gilbert, G. E. Backus, “Propagator matrices in elastic wave and vibration problems,” Geophysics 31, 326–332 (1966).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Corones, J. P.

J. P. Corones, M. E. Davison, R. J. Krueger, “Direct and inverse scattering in the time domain via invariant imbedding equations,” J. Acoust. Soc. Am. 74, 1535–1541 (1983).
[CrossRef]

Davison, M. E.

J. P. Corones, M. E. Davison, R. J. Krueger, “Direct and inverse scattering in the time domain via invariant imbedding equations,” J. Acoust. Soc. Am. 74, 1535–1541 (1983).
[CrossRef]

Elachi, C.

C. Elachi, “Waves in active and passive periodic structures: a review,” Proc. IEEE 64, 1666–1698 (1976).
[CrossRef]

Fishman, L.

L. Fishman, “Exact and operator rational approximate solutions of the Helmholtz, Weyl composition equation in underwater acoustics—the quadratic profile,” J. Math. Phys. 33, 1887–1914 (1992).
[CrossRef]

Gilbert, F.

F. Gilbert, G. E. Backus, “Propagator matrices in elastic wave and vibration problems,” Geophysics 31, 326–332 (1966).
[CrossRef]

He, S.

S. He, A. Karlsson, “Time domain Green function technique for a point source over a dissipative stratified half-space,” Radio Sci. 28, 513–526 (1993).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Karlsson, A.

A. Karlsson, H. Otterheim, R. Stewart, “Transient wave propagation in composite media: Green’s function approach,” J. Opt. Soc. Am. A 10, 886–895 (1993).
[CrossRef]

S. He, A. Karlsson, “Time domain Green function technique for a point source over a dissipative stratified half-space,” Radio Sci. 28, 513–526 (1993).
[CrossRef]

Kristensson, G.

G. Kristensson, R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. Part 3. Scattering operators in the presence of a phase velocity mismatch,” J. Math. Phys. 28, 360–370 (1987).
[CrossRef]

G. Kristensson, R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. Part I. Scattering operators,” J. Math. Phys. 27, 1667–1682 (1986).
[CrossRef]

Krueger, R. J.

R. J. Krueger, R. L. Ochs, “A Green’s function approach to the determination of internal fields,” Wave Motion 11, 525–543 (1989).
[CrossRef]

G. Kristensson, R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. Part 3. Scattering operators in the presence of a phase velocity mismatch,” J. Math. Phys. 28, 360–370 (1987).
[CrossRef]

G. Kristensson, R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. Part I. Scattering operators,” J. Math. Phys. 27, 1667–1682 (1986).
[CrossRef]

J. P. Corones, M. E. Davison, R. J. Krueger, “Direct and inverse scattering in the time domain via invariant imbedding equations,” J. Acoust. Soc. Am. 74, 1535–1541 (1983).
[CrossRef]

Ochs, R. L.

R. J. Krueger, R. L. Ochs, “A Green’s function approach to the determination of internal fields,” Wave Motion 11, 525–543 (1989).
[CrossRef]

Otterheim, H.

Oughstun, K. E.

Sherman, G. C.

Stewart, R.

Weston, V. H.

V. H. Weston, “Invariant imbedding for the wave equation in three dimensions and the applications to the direct and inverse problems,” Inverse Probl. 6, 1075–1105 (1990).
[CrossRef]

Geophysics (1)

F. Gilbert, G. E. Backus, “Propagator matrices in elastic wave and vibration problems,” Geophysics 31, 326–332 (1966).
[CrossRef]

Inverse Probl. (1)

V. H. Weston, “Invariant imbedding for the wave equation in three dimensions and the applications to the direct and inverse problems,” Inverse Probl. 6, 1075–1105 (1990).
[CrossRef]

J. Acoust. Soc. Am. (1)

J. P. Corones, M. E. Davison, R. J. Krueger, “Direct and inverse scattering in the time domain via invariant imbedding equations,” J. Acoust. Soc. Am. 74, 1535–1541 (1983).
[CrossRef]

J. Math. Phys. (3)

L. Fishman, “Exact and operator rational approximate solutions of the Helmholtz, Weyl composition equation in underwater acoustics—the quadratic profile,” J. Math. Phys. 33, 1887–1914 (1992).
[CrossRef]

G. Kristensson, R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. Part 3. Scattering operators in the presence of a phase velocity mismatch,” J. Math. Phys. 28, 360–370 (1987).
[CrossRef]

G. Kristensson, R. J. Krueger, “Direct and inverse scattering in the time domain for a dissipative wave equation. Part I. Scattering operators,” J. Math. Phys. 27, 1667–1682 (1986).
[CrossRef]

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

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

Proc. IEEE (1)

C. Elachi, “Waves in active and passive periodic structures: a review,” Proc. IEEE 64, 1666–1698 (1976).
[CrossRef]

Radio Sci. (1)

S. He, A. Karlsson, “Time domain Green function technique for a point source over a dissipative stratified half-space,” Radio Sci. 28, 513–526 (1993).
[CrossRef]

Wave Motion (1)

R. J. Krueger, R. L. Ochs, “A Green’s function approach to the determination of internal fields,” Wave Motion 11, 525–543 (1989).
[CrossRef]

Other (2)

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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Figures (7)

Fig. 1
Fig. 1

Permittivity (z) = 1 + sin2πz for the periodic dielectric medium.

Fig. 2
Fig. 2

Propagator kernel G+(0, z, t) for the dielectric medium at z = 1 m (solid curve), z = 4 m (dashed curve), and z = 8 m (dotted curve).

Fig. 3
Fig. 3

Propagator kernel G+(0, z, t) for the dielectric medium at z = 64 m (solid curve) and z = 128 m (dashed curve).

Fig. 4
Fig. 4

Propagator kernel G+(0, z, t) for the dielectric medium at z = 1024 m (solid curve) and the asymptotic value of G+ given by relation (6.1) at the same z value (dashed curve).

Fig. 5
Fig. 5

Reflection kernel for the Lorentz medium given by Eqs. (8.1) and (8.2).

Fig. 6
Fig. 6

Transmission kernel (solid curve) and the asymptotic Green’s kernel given by relation (6.1) (dashed curve) for the Lorentz medium given by Eqs. (8.1) and (8.2).

Fig. 7
Fig. 7

Propagator kernel G+ and the asymptotic kernel in relation (6.1) at z = 256 m for the Lorentz medium given by Eqs. (8.1) and (8.2).

Equations (83)

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z 2 E ( z , t ) - 1 c 0 2 { ( z ) t 2 E ( z , t ) + F [ E ] } = 0.
F [ E ] = - t χ ( z , t - τ ) τ 2 E ( z , τ ) d τ = χ ( z , 0 ) t E ( z , t ) + [ t χ ( z , 0 ) ] E ( z , t ) + - t t 2 χ ( z , t - τ ) E ( τ ) d τ .
D ( z , t ) = 0 ( z ) E ( z , t ) + 0 - t χ ( z , t - τ ) E ( τ ) d τ .
z [ E z E ] = [ 0 1 1 c 0 2 { ( z ) t 2 + F [ · ] } 0 ] [ E z E ] = A [ E z E ] .
[ E + E - ] = 1 2 [ 1 - c ( z ) t - 1 1 c ( z ) t - 1 ] [ E z E ] = P [ E z E ] ,
t - 1 E ( z , t ) = - t E ( z , τ ) d τ .
z [ E + E - ] = ( z P ) P - 1 [ E + E - ] + P A P - 1 [ E + E - ] ,
P - 1 = [ 1 1 - 1 c ( z ) t 1 c ( z ) t ] .
z [ E + E - ] = [ α β γ δ ] [ E + E - ] ,
α = - 1 c ( z ) t - c ( z ) 2 c 0 2 t - 1 F [ · ] + 1 2 c ( z ) z c ( z ) ,
β = - c ( z ) 2 c 0 2 t - 1 F [ · ] - 1 2 c ( z ) z c ( z ) ,
γ = c ( z ) 2 c 0 2 t - 1 F [ · ] - 1 2 c ( z ) z c ( z ) ,
δ = 1 c ( z ) t + c ( z ) 2 c 0 2 t - 1 F [ · ] + 1 2 c ( z ) z c ( z ) .
t - 1 F [ E ] = - t χ ( z , t - τ ) τ E ( z , τ ) d τ .
E ( z , t ) = E + ( z , t ) + E - ( z , t ) .
E + [ z 1 , t + l ( z 0 , z 1 ) ] = G + ( z 0 , z 1 ) E + ( z 0 , t ) ,
E - [ z 1 , t + l ( z 0 , z 1 ) ] = G - ( z 0 , z 1 ) E + ( z 0 , t ) ,
l ( z 0 , z 1 ) = z 0 z 1 1 c ( z ) d z .
E + [ z 1 , t + l ( z 0 , z 1 ) ] = a ( z 0 , z 1 ) E + ( z 0 , t ) + [ G + ( z 0 , z 1 , · ) * E + ( z 0 , · ) ] ( t ) ,
E - [ z 1 , t + l ( z 0 , z 1 ) ] = [ G - ( z 0 , z 1 , · ) * E + ( z 0 , · ) ] ( t ) .
[ G + ( z 0 , z 1 , · ) * E + ( z 0 , · ) ] ( t ) = 0 t G + ( z 0 , z 1 , t - τ ) × E + ( z , τ ) d τ ,
a ( z 0 , z 1 ) = c ( z 1 ) c ( z 0 ) exp [ - 1 2 c 0 2 z 0 z 1 c ( z ) χ ( z , 0 ) d z ] .
G ± ( z , z ) = G ± ( z , z ) G + ( z , z ) .
G + ( z , z , t ) = a ( z , z ) G + ( z , z , t ) + a ( z , z ) G + ( z , z , t ) + [ G + ( z , z , · ) * G + ( z , z , · ) ] ( t ) ,
G - ( z , z , t ) = a ( z , z ) G - ( z , z , t ) + [ G - ( z , z , · ) * G + ( z , z , · ) ] ( t ) .
a ( z , z ) G + ( z , z , t ) + a ( z , z ) G + ( z , z , t ) + [ G + ( z , z , · ) * G + ( z , z , · ) ] ( t ) = 0.
G ± ( 0 , z , t ) = the Green ' s kernels used in the Green ' s function technique ,
G + ( z , L , t ) = T ( z , t ) , the transmission kernel for the imbedded subslab [ z , L ] ,
G - ( z , z , t ) = R ( z , t ) , the reflection kernel for the imbedded subslab [ z , L ] ,
G + ( 0 , L , t ) = T ( t ) , the transmission kernel for the slab ,
G - ( 0 , 0 , t ) = R ( t ) , the reflection kernel for the slab .
G - ( z , z ) = R ( z , t ) + [ R ( z , · ) * G + ( z , z , · ) ] ( t ) .
ζ G + ( η , ζ , t ) = c ζ ( ζ ) 2 c ( ζ ) [ G + ( η , ζ , t ) - G - ( η , ζ , t ) ] - c ( ζ ) 2 c 0 2 ( a ( η , ζ ) χ t ( ζ , t ) + χ ( ζ , 0 ) × [ G + ( η , ζ , t ) + G - ( η , ζ , t ) ] + { χ t ( ζ , · ) * [ G + ( η , ζ , · ) + G - ( η , ζ , · ) ] } ( t ) ) ,
ζ G - ( η , ζ , t ) - 2 c ( ζ ) t G - ( η , ζ , t ) = c ζ ( ζ ) 2 c ( ζ ) [ G - ( η , ζ , t ) - G + ( η , ζ , t ) ] + c ( ζ ) 2 c 0 2 ( a ( η , ζ ) χ t ( ζ , t ) + χ ( ζ , 0 ) × [ G + ( η , ζ , t ) + G - ( η , ζ , t ) ] + { χ t ( ζ , · ) * [ G + ( η , ζ , · ) + G - ( η , ζ , · ) ] } ( t ) ) .
G - ( η , ζ , 0 ) = 1 4 a ( η , ζ ) { c ζ ( ζ ) - [ c ( ζ ) c 0 ] 2 χ ( ζ , 0 ) } .
η G + ( η , ζ , t ) = c η ( η ) 2 c ( η ) { a ( η , ζ ) R ( η , t ) - G + ( η , ζ , t ) + [ G + ( η , ζ , · ) * R ( η , · ) ] ( t ) } + c ( η ) 2 c 0 2 ( χ ( η , 0 ) { a ( η , ζ ) R ( η , t ) + G + ( η , ζ , t ) + [ G + ( η , ζ , · ) * R ( η , · ) ] ( t ) } + a ( η , ζ ) χ t ( η , t ) + { χ t ( η , · ) * [ a ( η , ζ ) × R ( η , · ) + G + ( η , ζ , · ) ] } ( t ) + [ χ t ( η , · ) * G + ( η , ζ , · ) * R ( η , · ) ] ( t ) ) ,
η G - ( η , ζ , t ) = c η ( η ) 2 c ( η ) { [ G - ( η , ζ , · ) * R ( η , · ) ] ( t ) - G - ( η , ζ , t ) } + c ( η ) 2 c 0 2 [ χ ( η , 0 ) × { [ G - ( η , ζ , · ) * R ( η , · ) ] ( t ) + G - ( η , ζ , t ) } + ( χ t ( η , · ) * { G - ( η , ζ , · ) + [ G - ( η , ζ , · ) * R ( η , · ) ] ( · ) } ) ( t ) ] .
2 R η ( η , t ) - 4 c ( η ) R t ( η , t ) = c η ( η ) c ( η ) [ R ( η , · ) * R ( η , · ) ] ( t ) + c ( η ) c 0 2 { χ t ( η , t ) + 2 χ ( η , 0 ) × R ( η , t ) + χ ( η , 0 ) [ R ( η , · ) * R ( η , · ) ] ( t ) + 2 [ χ t * R ] ( η , t ) + [ χ t * R * R ] ( η , t ) } .
R ( η , 0 ) = ¼ { c η ( η ) - [ c ( η ) / c 0 ] 2 χ ( η , 0 ) } .
G + ( 0 , z j ) = [ G + ( 0 , z 1 ) ] j ,
G + ( 0 , z ) = G + ( z j , z ) G + ( 0 , z j ) , G - ( 0 , z ) = G - ( z , z ) G + ( 0 , z ) .
R ( q , t ) = R ( 0 , t ) ,
G + ( q , q ) = 0.
E + [ z j , t + l ( z i , z j ) ] = G + ( z i , z j ) E + ( z i , t ) , E - [ z j , t + l ( z i , z j ) ] = G + ( z i , z j ) G - ( z j , z j ) E + ( z i , t ) , E - ( z i , t ) = G - ( z i , z i ) E + ( z i , t ) .
E - [ z j , t + l ( z i , z j ) ] = G + ( z i , z j ) G - ( z i , z i ) E + ( z i , t ) = G + ( z i , z j ) E - ( z i , t ) , E [ z j , t + l ( z i , z j ) ] = G + ( z i , z j ) E ( z i , t ) .
G + ( 0 , z ) = [ G + ( 0 , d z ) ] N .
G + ( 0 , z ) { [ a ( 0 , d z ) δ ( t ) + G + ( 0 , d z , 0 ) ] * } ( N - 1 ) × [ a ( 0 , d z ) δ ( t ) + G + ( 0 , d z , 0 ) ] = { [ a ( 0 , d z ) δ ( t ) + d z G z + ( 0 , 0 , 0 ) ] * } ( N - 1 ) × [ a ( 0 , d z ) δ ( t ) + d z G z + ( 0 , 0 , 0 ) ] ,
[ G z + ( 0 , 0 , 0 ) * ] k G z + ( 0 , 0 , 0 ) = G z + ( 0 , 0 , 0 ) [ G z + ( 0 , 0 , 0 ) t ] k k !
lim N ( N k ) ( d z ) k = lim N ( N d z ) k k ! = z k k !
G + ( 0 , z ) a ( 0 , z ) { δ ( t ) + z G z + ( 0 , 0 , 0 ) × k = 0 [ t z G z + ( 0 , 0 , 0 ) ] k ( k + 1 ) ! k ! } = a ( 0 , z ) { δ ( t ) + z G z + ( 0 , 0 , 0 ) × J 1 [ 2 - t z G z + ( 0 , 0 , 0 ) ] - t z G z + ( 0 , 0 , 0 ) } ,
G z ( 0 , 0 , 0 ) = c ( 0 ) 8 c 0 2 { χ ( 0 ) 2 [ c ( 0 ) c 0 ] 2 - 4 χ t ( 0 ) } .
G + ( 0 , N q ) { [ a ( 0 , q ) δ ( t ) + G + ( 0 , q , 0 ) ] * } ( N - 1 ) × [ a ( 0 , q ) δ ( t ) + G + ( 0 , q , 0 ) ] = a ( 0 , N q ) { δ ( t ) + G + ( 0 , q , 0 ) a ( 0 , q ) × k = 1 N ( N k ) [ a - 1 ( 0 , q ) G + ( 0 , q , 0 ) t ] k - 1 ( k - 1 ) ! } .
G + ( 0 , N q ) a ( 0 , N q ) { δ ( t ) + N G + ( 0 , q , 0 ) × J 1 [ 2 - a - 1 ( 0 , q ) t N G + ( 0 , q , 0 ) ] - a ( 0 , q ) t N G + ( 0 , q , 0 ) } .
G + ( 0 , q , 0 ) = a ( 0 , q ) 8 0 q 1 c ( z ) { [ χ ( z , 0 ) ] 2 [ c ( z ) c 0 ] 2 - 4 [ c ( z ) c 0 ] 2 χ t ( z , 0 ) - [ c z ( z ) ] 2 } d z .
E + ( z i + 1 , t + l ) = T 0 E + ( z i , t ) = a E + ( z i , t ) + [ T 0 ( · ) * E + ( z i , · ) ] ( t ) , E - ( z i , t ) = R 0 + E + ( z i , t ) = [ R 0 + ( · ) * E + ( z i , · ) ] ( t ) .
E - ( z i , t + l ) = T 0 E - ( z i + 1 , t ) = a E - ( z i + 1 , t ) + [ T 0 ( · ) * E - ( z i + 1 , · ) ] ( t ) , E + ( z i + 1 , t ) = R 0 - E - ( z i + 1 , t ) = [ R 0 - ( · ) * E - ( z i + 1 , · ) ] ( t ) .
G 0 + ( 0 , 0 , t ) = 0 , G 0 - ( 0 , q , t ) = 0 ,
G 0 + ( q , q , t ) = 0 , R 0 + ( q , t ) = 0.
E + ( z i + 1 ) = G + ( z i , z i + 1 ) E + ( z i ) , E - ( z i + 1 ) = G - ( z i , z i + 1 ) E + ( z i ) .
G + ( z i , z i + 1 ) E + ( z i ) = T 0 E + ( z i ) + R 0 - E - ( z i + 1 ) ,
E - ( z i + 1 ) = G - ( z i + 1 , z i + 1 ) E + ( z i + 1 ) = G - ( z i + 1 , z i + 1 ) × [ T 0 E + ( z i ) + R 0 - E - ( z i + 1 ) ] .
E - ( z i + 1 ) = [ 1 - G - ( z i + 1 , z i + 1 ) R 0 - ] - 1 G - ( z i + 1 , z i + 1 ) × T 0 E + ( z i )
G + ( z i , z i + 1 ) = T 0 + R 0 - [ 1 - G - ( z i + 1 , z i + 1 ) R 0 - ] - 1 × G - ( z i + 1 , z i + 1 ) T 0 ,
G - ( z i , z i + 1 ) = [ 1 - G - ( z i + 1 , z i + 1 ) R 0 - ] - 1 G - ( z i + 1 , z i + 1 ) T 0 .
[ 1 - G - ( z i + 1 , z i + 1 ) R 0 - ] - 1 E ( t ) = E ( t ) + [ K i + 1 ( · ) * E ( · ) ] ( t ) .
K i ( t ) - [ R ( z i , · ) * R 0 - ( · ) ] ( t ) - [ R ( z i , · ) * R 0 - ( · ) * K i ( · ) ] ( t ) = 0.
G + ( z i , z i + 1 , t ) = T 0 ( t ) + a { [ R 0 - ( · ) * R ( z i + 1 , · ) ] ( t ) + [ R 0 - ( · ) * K i + 1 ( · ) * R ( z i + 1 , · ) ] ( t ) } + [ R 0 - ( · ) * R ( z i + 1 , · ) * T 0 ( · ) ] ( t ) + [ R 0 - ( · ) * K i + 1 ( · ) * R ( z i + 1 , · ) * T 0 ( · ) ] ( t ) ,
G - ( z i , z i + 1 , t ) = a { R ( z i + 1 , t ) + [ R ( z i + 1 , · ) * K i + 1 ( · ) ] ( t ) } + [ R ( z i + 1 , · ) * T 0 ( · ) ] ( t ) + [ R ( z i + 1 , · ) * K i + 1 ( · ) * T 0 ( · ) ] ( t ) .
G + ( z N - 1 , z N , t ) = T 0 ( t ) , R ( z N - 1 , t ) = R 0 ( t ) .
E - ( z N - 2 ) = R ( z N - 2 ) E + ( z N - 2 ) = T 0 E - ( z N - 1 ) + R 0 E + ( z N - 2 ) ,
R ( z N - 2 , t ) = R 0 ( t ) + a 2 R ( z N - 1 , t ) + a 2 [ K N - 1 ( · ) * R ( z N - 1 , · ) ] ( t ) + 2 a [ T 0 ( · ) * R ( z N - 1 , · ) ] ( t ) + [ T 0 ( · ) * R ( z N - 1 , · ) * R ( z N - 1 , · ) ] ( t ) + 2 a [ R ( z N - 1 , · ) * K N - 1 * T 0 ( · ) ] ( t ) + [ T 0 ( · ) * K N - 1 ( · ) * R ( z n - 1 , · ) * T 0 ( · ) ] ( t ) .
( z ) = 1 + sin 2 π z ,
χ ( z , t ) = ω p 2 sin ω 0 ( z ) t ω 0 ( z ) H ( t ) ,
ω 0 = ½ ω m [ cos ( 2 π z ) + 2 ] ,
d d ζ { a ( η , ζ ) E + ( η , t ) + [ G + ( η , ζ , · ) * E + ( η , · ) ] ( t ) } = a ζ ( η , ζ ) E + ( η , t ) + [ G ζ + ( η , ζ , · ) * E + ( η , · ) ] ( t ) .
d d ζ E + [ ζ , t + l ( η , ζ ) ] = ( ζ + 1 c ( ζ ) t ) E + [ ζ , t + l ( η , ζ ) ] = [ α ( ζ ) + 1 c ( ζ ) t ] E + ( ζ , t + l ) + β ( ζ ) E + ( ζ , t + l ) ,
c ζ ( ζ ) 2 c ( ζ ) { a ( η , ζ ) E + ( η , t ) + [ G + ( η , ζ , · ) * E + ( η , · ) ] ( t ) - [ G - ( η , ζ , · ) * E + ( η , · ) ] ( t ) } = a ζ ( η , ζ ) E + ( η , t ) + [ G ζ + ( η , ζ , · ) * E + ( η , · ) ] ( t ) .
a ζ ( η , ζ ) = [ c ζ ( ζ ) 2 c ( ζ ) - c ( ζ ) 2 c 0 2 χ ( ζ , 0 ) ] a ( η , ζ ) ,
G ζ + ( η , ζ , t ) = c ζ ( ζ ) 2 c ( ζ ) [ G + ( η , ζ , t ) - G - ( η , ζ , t ) ] .
γ ( ζ ) { a ( η , ζ ) E + ( η , t ) + [ G + ( η , ζ , · ) * E + ( η , · ) ] ( t ) } + δ ( ζ ) × [ G - ( η , ζ , · ) * E + ( η , · ) ] ( t ) + 1 c ( ζ ) { G - ( η , ζ , 0 ) E + ( η , t ) + [ G t - ( η , ζ , · ) * E + ( η , · ) ] ( t ) } = [ G ζ - ( η , ζ , · ) * E + ( η , · ) ] ( t ) .
- 1 c ( η ) { a ( η , ζ ) E t + ( η , t ) + G + ( η , ζ , 0 ) E + ( η , t ) + [ G t + ( η , ζ , · ) * E + ( η , · ) ] ( t ) } = a η ( η , ζ ) E + ( η , t ) + a ( η , ζ ) [ a ( η ) E + ( η , t ) + β ( η ) E - ( η , t ) ] + [ G η + ( η , ζ , · ) * E + ( η , · ) ] ( t ) + { G + * [ α ( η ) E + ( η , t ) + β ( η ) E - ( η , t ) ] } ( t ) .
E - ( η , t ) = [ R ( η , · ) * E + ( η , · ) ] ( t )
- 1 c ( η ) { G - ( η , ζ , 0 ) E + ( η , t ) + [ G t - ( η , ζ , · ) * E + ( η , · ) ] ( t ) } = [ G η - ( η , ζ , · ) * E + ( η , · ) ] ( t ) + { G - ( η , ζ , · ) * [ α ( η ) E + ( η , · ) + β ( ζ ) E - ( η , · ) ] } ( t ) .

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