Abstract

A rigorous design rule of corrugated waveguide filters is developed by employing the Gel’fand–Levitan–Marchenko inverse-scattering method for the two-component coupled-wave equations of the Zakharov–Shabat type. In the course of developing the design method, the coupled Gel’fand–Levitan–Marchenko integral equations for the Zakharov–Shabat system having no discrete spectrum are shown to be reducible to a set of linear simultaneous equations amenable to simple numerical calculations when the reflection coefficient is a rational function.

© 1985 Optical Society of America

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  1. K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer, New York, 1977).
    [CrossRef]
  2. M. J. Ablowitz, “Lectures on the inverse scattering transform,” Stud. Appl. Math. 58, 17–94 (1978).
  3. G. L. Lamb, Elements of SolitonTheory (Wiley, New York, 1980).
  4. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, 2nd ed., T. Tamir, ed. (Springer, New York, 1979).
  5. M. Matsuhara, K. O. Hill, A. Watanabe, “Optical waveguide filters; synthesis,” J. Opt. Soc. Am. 65, 804–809 (1975).
    [CrossRef]
  6. H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).
    [CrossRef]
  7. P. C. Cross, H. Kogelnik, “Sidelobe suppression in corrugated waveguide filters,” Opt. Lett.1, 43–45 (1977).
  8. L. Weinberg, Network Analysis and Synthesis (McGraw-Hill, New York, 1962), p. 495.
  9. I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. XIII, 371–393 (1960).
    [CrossRef]
  10. H. H. Szu, C. E. Caroll, C. C. Yang, S. Ahn, “A new functional equation in the plasma inverse problem and its analytic properties,” J. Math. Phys. 7, 1236–1247 (1976).
    [CrossRef]
  11. H. Mathews, Surface Wave Filters (Wiley, New York, 1977), pp. 381–442.
  12. C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276–278 (1977).
    [CrossRef]
  13. B.-G. Kim, S.-Y. Shin, “An asymptotic approximation of linear-chirped grating filter response,” Opt. Commun. 44, 371–376 (1983).
    [CrossRef]
  14. For the −q* case, the physical interpretations are well described in H. A. Haus, “Physical interpretation of inverse scattering formalism applied to self-induced transparency,” Rev. Mod. Phys. 51, 331–339 (1979).
    [CrossRef]
  15. We shall call the (z, τ) plane the time domain, since that τ provides the time base to the timelike evolution of waves according to Eqs. (13).
  16. R. P. Boas, Entire Functions (Academic, New York, 1954).
  17. K. R. Pechenik, J. M. Cohen, “Inverse scattering—exact solution of the Gel’fand–Levitan equation,” J. Math. Phys. 22, 1513–1516 (1981).
    [CrossRef]
  18. H. E. Moses, “An example of the effect of the rescaling of the reflection coefficient on the scattering potential for the one-dimensional Schrödinger equation,” Stud. Appl. Math. 60, 177–181 (1979).
  19. P. Deift, E. Trubowitz, “Inverse scattering on the line,” Commun. Pure Appl. Math. XXXII, 121–151 (1979).
    [CrossRef]
  20. G.-H. Song, S.-Y. Shin, “Inverse scattering problem for the coupled-wave equations when the reflection coefficient is a rational function,” Proc. IEEE 71, 266–268 (1983).
    [CrossRef]

1983

B.-G. Kim, S.-Y. Shin, “An asymptotic approximation of linear-chirped grating filter response,” Opt. Commun. 44, 371–376 (1983).
[CrossRef]

G.-H. Song, S.-Y. Shin, “Inverse scattering problem for the coupled-wave equations when the reflection coefficient is a rational function,” Proc. IEEE 71, 266–268 (1983).
[CrossRef]

1981

K. R. Pechenik, J. M. Cohen, “Inverse scattering—exact solution of the Gel’fand–Levitan equation,” J. Math. Phys. 22, 1513–1516 (1981).
[CrossRef]

1979

H. E. Moses, “An example of the effect of the rescaling of the reflection coefficient on the scattering potential for the one-dimensional Schrödinger equation,” Stud. Appl. Math. 60, 177–181 (1979).

P. Deift, E. Trubowitz, “Inverse scattering on the line,” Commun. Pure Appl. Math. XXXII, 121–151 (1979).
[CrossRef]

For the −q* case, the physical interpretations are well described in H. A. Haus, “Physical interpretation of inverse scattering formalism applied to self-induced transparency,” Rev. Mod. Phys. 51, 331–339 (1979).
[CrossRef]

1978

M. J. Ablowitz, “Lectures on the inverse scattering transform,” Stud. Appl. Math. 58, 17–94 (1978).

1977

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276–278 (1977).
[CrossRef]

P. C. Cross, H. Kogelnik, “Sidelobe suppression in corrugated waveguide filters,” Opt. Lett.1, 43–45 (1977).

1976

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).
[CrossRef]

H. H. Szu, C. E. Caroll, C. C. Yang, S. Ahn, “A new functional equation in the plasma inverse problem and its analytic properties,” J. Math. Phys. 7, 1236–1247 (1976).
[CrossRef]

1975

1960

I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. XIII, 371–393 (1960).
[CrossRef]

Ablowitz, M. J.

M. J. Ablowitz, “Lectures on the inverse scattering transform,” Stud. Appl. Math. 58, 17–94 (1978).

Ahn, S.

H. H. Szu, C. E. Caroll, C. C. Yang, S. Ahn, “A new functional equation in the plasma inverse problem and its analytic properties,” J. Math. Phys. 7, 1236–1247 (1976).
[CrossRef]

Boas, R. P.

R. P. Boas, Entire Functions (Academic, New York, 1954).

Caroll, C. E.

H. H. Szu, C. E. Caroll, C. C. Yang, S. Ahn, “A new functional equation in the plasma inverse problem and its analytic properties,” J. Math. Phys. 7, 1236–1247 (1976).
[CrossRef]

Chadan, K.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer, New York, 1977).
[CrossRef]

Cohen, J. M.

K. R. Pechenik, J. M. Cohen, “Inverse scattering—exact solution of the Gel’fand–Levitan equation,” J. Math. Phys. 22, 1513–1516 (1981).
[CrossRef]

Cross, P. C.

P. C. Cross, H. Kogelnik, “Sidelobe suppression in corrugated waveguide filters,” Opt. Lett.1, 43–45 (1977).

Deift, P.

P. Deift, E. Trubowitz, “Inverse scattering on the line,” Commun. Pure Appl. Math. XXXII, 121–151 (1979).
[CrossRef]

Haus, H. A.

For the −q* case, the physical interpretations are well described in H. A. Haus, “Physical interpretation of inverse scattering formalism applied to self-induced transparency,” Rev. Mod. Phys. 51, 331–339 (1979).
[CrossRef]

Hill, K. O.

Hong, C. S.

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276–278 (1977).
[CrossRef]

Katzir, A.

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276–278 (1977).
[CrossRef]

Kay, I.

I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. XIII, 371–393 (1960).
[CrossRef]

Kim, B.-G.

B.-G. Kim, S.-Y. Shin, “An asymptotic approximation of linear-chirped grating filter response,” Opt. Commun. 44, 371–376 (1983).
[CrossRef]

Kogelnik, H.

P. C. Cross, H. Kogelnik, “Sidelobe suppression in corrugated waveguide filters,” Opt. Lett.1, 43–45 (1977).

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).
[CrossRef]

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, 2nd ed., T. Tamir, ed. (Springer, New York, 1979).

Lamb, G. L.

G. L. Lamb, Elements of SolitonTheory (Wiley, New York, 1980).

Livanos, A. C.

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276–278 (1977).
[CrossRef]

Mathews, H.

H. Mathews, Surface Wave Filters (Wiley, New York, 1977), pp. 381–442.

Matsuhara, M.

Moses, H. E.

H. E. Moses, “An example of the effect of the rescaling of the reflection coefficient on the scattering potential for the one-dimensional Schrödinger equation,” Stud. Appl. Math. 60, 177–181 (1979).

Pechenik, K. R.

K. R. Pechenik, J. M. Cohen, “Inverse scattering—exact solution of the Gel’fand–Levitan equation,” J. Math. Phys. 22, 1513–1516 (1981).
[CrossRef]

Sabatier, P. C.

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer, New York, 1977).
[CrossRef]

Shellan, J. B.

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276–278 (1977).
[CrossRef]

Shin, S.-Y.

G.-H. Song, S.-Y. Shin, “Inverse scattering problem for the coupled-wave equations when the reflection coefficient is a rational function,” Proc. IEEE 71, 266–268 (1983).
[CrossRef]

B.-G. Kim, S.-Y. Shin, “An asymptotic approximation of linear-chirped grating filter response,” Opt. Commun. 44, 371–376 (1983).
[CrossRef]

Song, G.-H.

G.-H. Song, S.-Y. Shin, “Inverse scattering problem for the coupled-wave equations when the reflection coefficient is a rational function,” Proc. IEEE 71, 266–268 (1983).
[CrossRef]

Szu, H. H.

H. H. Szu, C. E. Caroll, C. C. Yang, S. Ahn, “A new functional equation in the plasma inverse problem and its analytic properties,” J. Math. Phys. 7, 1236–1247 (1976).
[CrossRef]

Trubowitz, E.

P. Deift, E. Trubowitz, “Inverse scattering on the line,” Commun. Pure Appl. Math. XXXII, 121–151 (1979).
[CrossRef]

Watanabe, A.

Weinberg, L.

L. Weinberg, Network Analysis and Synthesis (McGraw-Hill, New York, 1962), p. 495.

Yang, C. C.

H. H. Szu, C. E. Caroll, C. C. Yang, S. Ahn, “A new functional equation in the plasma inverse problem and its analytic properties,” J. Math. Phys. 7, 1236–1247 (1976).
[CrossRef]

Yariv, A.

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276–278 (1977).
[CrossRef]

Appl. Phys. Lett.

C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, A. Katzir, “Broad-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 31, 276–278 (1977).
[CrossRef]

Bell Syst. Tech. J.

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).
[CrossRef]

Commun. Pure Appl. Math.

I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. XIII, 371–393 (1960).
[CrossRef]

P. Deift, E. Trubowitz, “Inverse scattering on the line,” Commun. Pure Appl. Math. XXXII, 121–151 (1979).
[CrossRef]

J. Math. Phys.

K. R. Pechenik, J. M. Cohen, “Inverse scattering—exact solution of the Gel’fand–Levitan equation,” J. Math. Phys. 22, 1513–1516 (1981).
[CrossRef]

H. H. Szu, C. E. Caroll, C. C. Yang, S. Ahn, “A new functional equation in the plasma inverse problem and its analytic properties,” J. Math. Phys. 7, 1236–1247 (1976).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

B.-G. Kim, S.-Y. Shin, “An asymptotic approximation of linear-chirped grating filter response,” Opt. Commun. 44, 371–376 (1983).
[CrossRef]

Opt. Lett.

P. C. Cross, H. Kogelnik, “Sidelobe suppression in corrugated waveguide filters,” Opt. Lett.1, 43–45 (1977).

Proc. IEEE

G.-H. Song, S.-Y. Shin, “Inverse scattering problem for the coupled-wave equations when the reflection coefficient is a rational function,” Proc. IEEE 71, 266–268 (1983).
[CrossRef]

Rev. Mod. Phys.

For the −q* case, the physical interpretations are well described in H. A. Haus, “Physical interpretation of inverse scattering formalism applied to self-induced transparency,” Rev. Mod. Phys. 51, 331–339 (1979).
[CrossRef]

Stud. Appl. Math.

H. E. Moses, “An example of the effect of the rescaling of the reflection coefficient on the scattering potential for the one-dimensional Schrödinger equation,” Stud. Appl. Math. 60, 177–181 (1979).

M. J. Ablowitz, “Lectures on the inverse scattering transform,” Stud. Appl. Math. 58, 17–94 (1978).

Other

G. L. Lamb, Elements of SolitonTheory (Wiley, New York, 1980).

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, 2nd ed., T. Tamir, ed. (Springer, New York, 1979).

K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer, New York, 1977).
[CrossRef]

We shall call the (z, τ) plane the time domain, since that τ provides the time base to the timelike evolution of waves according to Eqs. (13).

R. P. Boas, Entire Functions (Academic, New York, 1954).

L. Weinberg, Network Analysis and Synthesis (McGraw-Hill, New York, 1962), p. 495.

H. Mathews, Surface Wave Filters (Wiley, New York, 1977), pp. 381–442.

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Figures (8)

Fig. 1
Fig. 1

Pictorial representations for support of the solutions Φ(z, τ) and U(z, τ) in the (z, τ) plane, (a) Φ(z, τ). (b) U(z, τ). The hatching direction indicates the characteristic lines in the region of the respective supports.

Fig. 2
Fig. 2

Coupling potentials of the Butterworth filters of various orders with r0 = −1 in common: (a) the first-order, (b) the second-order, (c) the third-order, (d) the fourth-order, and (e) the fifth-order Butterworth filters.

Fig. 3
Fig. 3

(a) Poles and zeros of the reflection coefficient r(is) of the uniformly corrugated waveguide filter with q0L = 1.5. The poles and zeros are denoted by × and ○, respectively, (b) The zeros of the characteristic polynomial D(s)D*(−s*) − N(s)N*(−s*) of the sixth-order approximation for the uniformly corrugated waveguide filter with q0L = 1.5. The poles and zeros of N(s)N*(−s*)/D(s)D*(−s*) are denoted by ×’s and ○’s, respectively, and the zeros of 1 − N(s)N*(−s*)/D(s)D*(−s*) are denoted by ⋄’s.

Fig. 4
Fig. 4

Comparison of the reflectivities; the original and its approximations. (a) |r(ξ)|2, the original reflectivity, (b) |ra(ξ)|2 for the reflection coefficient ra(ξ) approximated with 16 poles and 14 zeros. (c) That with six poles and four zeros, (d) That with four poles and two zeros.

Fig. 5
Fig. 5

Reconstruction of q(z) from rational functions to which the original reflection coefficient is approximated with different numbers of poles and zeros, (a) The original, (b) q(z) reconstructed with two zeros and four poles, (c) That with four zeros and six poles, (d) That with 14 zeros and 16 poles.

Fig. 6
Fig. 6

The coupling coefficients of the third-order Butterworth filters with various values of r0: (a) r0 = −1.0, (b) r0 = −0.999, (c) r0 = −0.99, (d) r0 = −0.95, (e) r0 = −0.8.

Fig. 7
Fig. 7

The coupling coefficients of the Butterworth waveguide filters of various orders with r0 = −0.95 in common: (a) the first-order, (b) the second-order, (c) the third-order, (d) the sixth-order, and (e) the tenth-order filters.

Fig. 8
Fig. 8

The reflectivities |r(ξ)|2 of the tenth-order Butterworth waveguide filters with truncation of q(z) at different zero-crossing points: (a) at the first, (b) at the second, and (c) at the sixth zero-crossing points.

Tables (1)

Tables Icon

Table 1 Last Two Poles of ra(is) in the s/q0 Plane

Equations (79)

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E t ( z , t ) = d ω 2 π exp ( i ω t ) { υ 1 ( z , ξ ( ω ) ) exp ( i π z / Λ ) + υ 2 ( z , ξ ( ω ) ) exp ( i π z / Λ ) } ,
Δ h ( z ) = α ( z ) sin [ 2 π z / Λ + θ ( z ) ] .
d υ 1 ( z , ξ ) / d z + i ξ υ 1 ( z , ξ ) = q ( z ) υ 2 ( z , ξ ) ,
d υ 2 ( z , ξ ) / d z i ξ υ 2 ( z , ξ ) = q * ( z ) υ 1 ( z , ξ ) ,
q ( z ) = A α ( z ) exp [ i θ ( z ) ] ,
ρ ( z , ξ ) = υ 1 ( z , ξ ) / υ 2 ( z , ξ ) ,
d ρ ( z , ξ ) / d z = 2 i ξ ρ ( z , ξ ) + q ( z ) q * ( z ) ρ 2 ( z , ξ ) ,
r ( ξ ) = lim z ρ ( z , ξ ) exp ( i 2 ξ z ) .
r ( ξ ) = sinh { q 0 L [ 1 ( ξ / q 0 ) 2 ] 1 / 2 } / [ 1 ( ξ / q 0 ) 2 ] 1 / 2 i ξ q 0 sinh { g 0 L [ 1 ( ξ / q 0 ) 2 ] 1 / 2 } [ 1 ( ξ / q 0 ) 2 ] 1 / 2 + cosh { q 0 L [ 1 ( ξ / q 0 ) 2 ] 1 / 2 } .
q c ( z ) = ξ c q 1 ( ξ c z ) .
q ( z ) = 2 A 1 ( z , z ) 2 R ( 2 z ) = r ( ξ ) exp ( i 2 ξ z ) d ξ π .
V j ( z , τ ) = υ j ( z , ξ ) exp ( i ξ τ ) d ξ 2 π , j = 1 , 2 ,
q ( z ) = 0 , z < 0 .
( / z / τ ) V 1 ( z , τ ) = q ( z ) V 2 ( z , τ ) ,
( / z + / τ ) V 2 ( z , τ ) = ± q * ( z ) V 1 ( z , τ ) .
A ( z , τ ) = 0 , z < | τ | ,
U ( z , τ ) = C [ ϕ ¯ ( z , k ) + b ( k ) a ( k ) ϕ ( z , k ) ] exp ( i k τ ) d k 2 π .
U ( z , τ ) = [ 0 δ ( z τ ) ] + A ( z , τ ) + [ F ( z + τ ) 0 ] + z i z [ A 2 * ( z , y ) ± A 1 * ( z , y ) ] F ( y + τ ) d y ,
Y ( z , τ ) = U ( z , τ ) [ 0 δ ( z τ ) ] .
y 1 ( μ , ν ) / ν = q ( μ + ν ) [ y 2 ( μ , ν ) + δ ( 2 ν ) ] ,
y 2 ( μ , ν ) / μ = ± q * ( μ + ν ) y 1 ( μ , ν ) .
y 1 ( μ , 0 ) = Y 1 ( z , z + ) = q ( z ) / 2 .
y 2 ( μ , 0 ) = Y 2 ( z , z + ) = ½ 0 z | q ( y ) | 2 d y .
( z , s ) = [ 1 ( z , s ) 2 ( z , s ) ] = [ V 1 ( z , τ ) V 2 ( z , τ ) ] exp ( s τ ) d τ ,
G ( s ) = 0 R ( y ) exp ( s y ) d y
| G ( s ) | < | 0 T R ( y ) exp ( s y ) d y | + T | R ( y ) | d y , T | R ( y ) | d y < / 2
| ĵ ( z , s ) | C j exp ( σ z ) / σ , σ = Re s > 0 , j = 1 , 2
| Ã j ( z , s ) | D j sinh ( z σ ) / σ , σ = Re s > 0 , j = 1 , 2
Ũ ( z , s ) = [ 0 exp ( s z ) ] + Ã ( z , s ) + G ( s ) { [ exp ( s z ) 0 ] + [ Ã 2 * ( z , s * ) ± Ã 1 * ( z , s * ) ] } ,
( z , s ) = Ã ( z , s ) + G ( s ) { [ exp ( s z ) 0 ] + [ Ã 2 * ( z , s * ) ± Ã 1 * ( z , s * ) ] } .
[ q ( z ) ± 0 z | q ( z ) | 2 d z ] = 2 lim s s ( z , s ) exp ( s z ) = 2 lim s s à ( z , s ) exp ( s z ) .
G ( s ) = N ( s ) D ( s ) , N ( s ) = r 0 s m n = m + 1 M ( 1 s μ n ) , D ( s ) = n = 1 N ( 1 s λ n ) , 0 m M , N M + 1 , Re λ n < 0 .
D ( s ) Ã ( z , s ) + N ( s ) { [ exp ( s z ) 0 ] + [ Ã 2 * ( z , s * ) ± Ã 1 * ( z , s * ) ] } = D ( s ) ( z , s ) .
D ( s ) ( z , s ) = P ( z , s ) exp ( s z ) ,
D ( s ) Ã ( z , s ) + N ( s ) [ Ã 2 * ( z , s * ) ± Ã 1 * ( z , s * ) ] = N ( s ) [ exp ( s z ) 0 ] + P ( z , s ) exp ( s z ) .
D * ( s * ) [ Ã 2 * ( z , s * ) ± Ã 1 * ( z , s * ) ] ± N * ( s * ) Ã ( z , s ) = N * ( s * ) [ 0 exp ( s z ) ] + [ P 2 * ( z , s * ) ± P 1 * ( z , s * ) ] exp ( s z ) .
à ( z , s ) = 1 Δ ( s ) ( N ( s ) { D * ( s * ) ( 1 0 ) + [ P 2 * ( z , s * ) ± P 1 * ( z , s * ) ] } exp ( s z ) + [ P ( z , s ) D * ( s * ) ± N ( s ) N * ( s * ) ( 0 1 ) ] exp ( s z ) ) ,
Δ ( s ) = D ( s ) D * ( s * ) N ( s ) N * ( s * ) .
κ 1 , κ 2 , , κ N ; κ 1 * , κ 2 * , , κ N * .
( z , s ) = [ G ( s ) exp ( s z ) 0 ] , z < 0 .
à ( z , s ) = 2 n = 1 N { [ d 1 , n ( z ) d 2 , n ( z ) ] sinh [ z ( s κ n ) ] s κ n + [ f 1 , n ( z ) f 2 , n ( z ) ] sinh [ z ( s + κ n * ) ] s + κ n * } .
f 1 , n ( z ) / d 2 , n * ( z ) = ± f 2 , n ( z ) / d 1 , n * ( z ) = G ( κ n * ) = 1 / G * ( κ n ) .
G ( s ) { ( 1 0 ) + n = 1 N [ d 2 , n * ( z ) ± d 1 , n * ( z ) ] exp ( κ n * z ) s + κ n * n = 1 N 1 G ( κ n ) [ d 1 , n ( z ) d 2 , n ( z ) ] exp ( κ n z ) s κ n } + n = 1 N [ d 1 , n ( z ) d 2 , n ( z ) ] exp ( κ n z ) s κ n n = 1 N G ( κ n * ) [ d 2 , n * ( z ) ± d 1 , n * ( z ) ] exp ( κ n * z ) s + κ n * .
m = 1 N { 1 G ( κ m ) exp ( κ m z ) λ n κ m [ d 1 , m ( z ) d 2 , m ( z ) ] exp ( κ m * z ) λ n + κ m * [ d 2 , m * ( z ) ± d 1 , m * ( z ) ] } = ( 1 0 )
[ q ( z ) 0 z | q ( y ) | 2 d y ] = 2 n = 1 N { [ d 1 , n ( z ) ± d 2 , n ( z ) ] exp ( κ n z ) G ( κ n * ) [ d 2 , n * ( z ) d 1 , n * ( z ) ] exp ( κ n * z ) } .
m = M = 0 , λ n = ξ c exp [ i π 2 ( 1 + 2 n 1 N ) ] , n = 1 , , N ,
G ( s ) = r 0 / n = 0 N c n ( s / ξ c ) n , | r 0 | = 1 .
Δ ( s ) = D ( s ) D ( s ) N ( s ) N ( s ) .
D ( s ) D * ( s * ) N ( s ) N * ( s * ) = s 2 N + ( 1 | r 0 | 2 ) .
à ( z , s ) = 1 s 2 N n = 0 2 N 1 s n { [ g 1 , n ( z ) g 2 , n ( z ) ] exp ( s z ) + [ h 1 , n ( z ) h 2 , n ( z ) ] exp ( s z ) } .
q ( z ) = 2 h 1 , 2 N 1 ( z ) , 0 z | q ( z ) | 2 d z = 2 h 2 , 2 N 1 ( z ) .
( s 2 N 0 ) + n = 0 2 N 1 { ( s ) n [ h 2 , n * ( z ) h 1 , n * ( z ) ] + s n G ( s ) [ g 1 , n ( z ) g 2 , n ( z ) ] } = 0 .
g 1 , N ( z ) = r 0 / c N , g 2 , N ( z ) = 0 ; g 1 , n ( z ) = g 2 , n ( z ) = 0 , n N + 1 .
exp ( 2 s z ) n = 0 N [ g 1 , n ( z ) g 2 , n ( z ) ] s n + n = 0 2 N 1 [ h 1 , n ( z ) h 2 , n ( z ) ] s n
n = m N 1 c N + m n [ g 1 , n ( z ) g 2 , n ( z ) ] + ( 1 ) N + m r 0 [ h 2 , N + m * ( z ) h 1 , N + m * ( z ) ] = ( r 0 c m / c N 0 ) ,
n = 0 N 1 w N + m n [ g 1 , n ( z ) g 2 , n ( z ) ] + [ h 1 , N + m ( z ) h 2 , N + m ( z ) ] = ( r 0 w m / c N 0 )
[ h 1 , 2 N 1 ( z ) h 2 , 2 N 1 ( z ) ] = [ r 0 c N g 2 , N 1 * ( z ) r 0 c N g 1 , N 1 * ( z ) c N 1 / c N ] .
n = m N 1 c N + m n [ g 1 , n ( z ) g 2 , n ( z ) ] ( 1 ) N + m r 0 n = 0 N 1 w N + m n [ g 2 , n * ( z ) g 1 , n * ( z ) ] = [ r 0 c m / c N ( 1 ) N + m 1 w m / c N ]
q ( z ) = 2 r 0 c N g 2 , N 1 * ( z ) 0 z | q ( z ) | 2 d z = 2 c N 1 c N 2 r 0 c N g 1 , N 1 * ( z ) .
q ( z ) = { 1 , 0 z 1.5 0 , otherwise .
N ( s ) = tanh ( 1.5 ) n = 1 N 2 ( 1 s μ n ) , D ( s ) = n = 1 N ( 1 s λ n ) .
κ n = ( 1 | r 0 | 2 ) 1 / 2 N × exp [ i π 2 ( 1 + 2 n 1 N ) ] , n = 1 , , N .
0 30 q 0 | r a ( ξ ) r ( ξ ) | 2 d ξ
± [ υ 1 ( z , k ) υ 2 ( z , k ) ] = k [ υ 1 ( z , k ) υ 2 ( z , k ) ] , ± = ( i d / d z i q ( z ) ± i q * ( z ) i d / d z ) ,
lim z ψ ( z , k ) exp ( i k z ) = lim z ϕ ¯ ( z , k ) exp ( i k z ) = ( 0 1 ) ,
lim z ψ ¯ ( z , k ) exp ( i k z ) = lim z ϕ ( z , k ) exp ( i k z ) = ( 1 0 ) .
ψ ( z , k ) = a ( k ) ϕ ¯ ( z , k ) + b ( k ) ϕ ( z , k ) .
a ( k ) a * ( k * ) b ( k ) b * ( k * ) = 1 .
ϕ ( z , k ) = a ( k ) ψ ¯ ( z , k ) b * ( k * ) ψ ( z , k ) .
ϕ ¯ ( z , k ) = ( 0 1 ) exp ( i k z ) + z A ( z , τ ) exp ( i k τ ) d τ ,
[ A 1 ( z , τ ) A 2 ( z , τ ) ] + ( 1 0 ) F ( z + τ ) + z [ A 2 * ( z , y ) ± A 1 * ( z , y ) ] F ( y + τ ) d y = 0 , z > τ ,
F ( z ) = C b ( k ) a ( k ) exp ( i k z ) d k 2 π .
R ( z ) = b ( ξ ) a ( ξ ) exp ( i ξ z ) d ξ 2 π .
A 1 ( z , z ) = q ( z ) / 2 , A 2 ( z , z ) = ± ½ z | q ( y ) | 2 d y ,
2 exp ( s z ) n = 1 N { [ d 1 , n ( z ) d 2 , n ( z ) ] exp ( κ n z ) s κ n + [ f 1 , n ( z ) f 2 , n ( z ) ] exp ( κ n * z ) s + κ n * } 2 exp ( s z ) n = 1 N { [ d 1 , n ( z ) d 2 , n ( z ) ] exp ( κ n z ) s κ n + [ f 1 , n ( z ) f 2 , n ( z ) ] exp ( κ n * z ) s + κ n * } .
2 [ d 1 , n ( z ) d 2 , n ( z ) ] = N ( κ n ) { D * ( κ n * ) ( 1 0 ) + [ P 2 * ( z , κ n * ) ± P 1 * ( z , κ n * ) ] } [ d Δ ( s ) / d s ] s = κ n exp ( κ n z ) = P ( z , κ n ) D * ( κ n * ) ± N ( κ n ) N * ( κ n * ) ( 0 1 ) [ d Δ ( s ) / d s ] s = κ n exp ( κ n z ) ,
2 [ f 1 , n ( z ) f 2 , n ( z ) ] = N ( κ n * ) { D * ( κ n ) ( 1 0 ) + [ P 2 * ( z , κ n ) ± P 1 * ( z , κ n ) ] } [ d Δ ( s ) / d s ] s = κ n * exp ( κ n * z ) = P ( z , κ n * ) D * ( κ n ) ± N ( κ n * ) N * ( κ n ) ( 0 1 ) [ d Δ ( s ) / d s ] s = κ n * exp ( κ n * z ) .
f 1 , n ( z ) d 2 , n * ( z ) = P 1 ( z , κ n * ) D * ( κ n ) exp ( κ n * z ) [ d Δ ( s ) / d s ] s = κ n * ± P 1 ( z , κ n * ) N * ( κ n ) exp ( κ n * z ) [ d Δ ( s ) / d s | s = κ n ] * = 1 G * ( κ n ) = G ( κ n * ) ,
f 2 , n ( z ) d 1 , n * ( z ) = N ( κ n * ) D ( κ n * ) = G ( κ n * ) = 1 G * ( κ n ) .

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