Abstract

The spectral function can be introduced into the transverse wave-number representation of TE modes in a dielectric slab guide. The modified representation provides a connection to the theory of singular ordinary differential equations. By making use of this theory, some exact solutions to novel dielectric slab guides can be found. The Gel’fand–Levitan integral equation is used to construct an example that has the interesting feature that Bragg reflection is used to confine the field.

© 1989 Optical Society of America

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References

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  1. A. J. Fox, “The grating guide—a component for integrated optics,” Proc. IEEE 62, 644–645 (1974).
    [CrossRef]
  2. P. Yeh, A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
    [CrossRef]
  3. E. C. Titchmarsh, Eigenfunction Expansions, Part I (Oxford U. Press, Oxford, 1962).
  4. E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).
  5. E. Hille, Lectures on Ordinary Differential Equations (Addison-Wesley, Reading, Mass., 1969).
  6. R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1982).
  7. K. Chadan, P. C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, Berlin, 1977).
    [CrossRef]
  8. S. Yukon, B. Bendow, “Design of waveguides with prescribed propagation constants,”J. Opt. Soc. Am. 70, 172–179 (1980).
    [CrossRef]
  9. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982).
  10. V. V. Shevchenko, Continuous Transitions in Open Waveguides (Golem, Boulder, Colo., 1971).
  11. I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation from its spectral function,” Izv. Akad. Nauk SSSR. Ser. Mat. 15, 309–360 (1951) [Am. Math. Soc. Transl. Ser. 2 1, 253–304 (1962)].
  12. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960).
  13. J. B. Bednar, R. Redner, E. Robinson, A. Weglein, eds., in Conference on Inverse Scattering: Theory and Application (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1983).
  14. G. M. L. Gladwell, S. R. A. Dods, S. K. Chaudhuri, “Nonuniform transmission-line synthesis using inverse eigenvalue analysis,”IEEE Trans. Circuits Syst. CS-35, 659–666 (1988).
    [CrossRef]
  15. R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962).
  16. A. Y. Cho, A. Yariv, P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471 (1977).
    [CrossRef]

1988 (1)

G. M. L. Gladwell, S. R. A. Dods, S. K. Chaudhuri, “Nonuniform transmission-line synthesis using inverse eigenvalue analysis,”IEEE Trans. Circuits Syst. CS-35, 659–666 (1988).
[CrossRef]

1980 (1)

1977 (1)

A. Y. Cho, A. Yariv, P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

1976 (1)

P. Yeh, A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

1974 (1)

A. J. Fox, “The grating guide—a component for integrated optics,” Proc. IEEE 62, 644–645 (1974).
[CrossRef]

1951 (1)

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation from its spectral function,” Izv. Akad. Nauk SSSR. Ser. Mat. 15, 309–360 (1951) [Am. Math. Soc. Transl. Ser. 2 1, 253–304 (1962)].

Bendow, B.

Chadan, K.

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

Chaudhuri, S. K.

G. M. L. Gladwell, S. R. A. Dods, S. K. Chaudhuri, “Nonuniform transmission-line synthesis using inverse eigenvalue analysis,”IEEE Trans. Circuits Syst. CS-35, 659–666 (1988).
[CrossRef]

Cho, A. Y.

A. Y. Cho, A. Yariv, P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

Coddington, E. A.

E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960).

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962).

Dods, S. R. A.

G. M. L. Gladwell, S. R. A. Dods, S. K. Chaudhuri, “Nonuniform transmission-line synthesis using inverse eigenvalue analysis,”IEEE Trans. Circuits Syst. CS-35, 659–666 (1988).
[CrossRef]

Fox, A. J.

A. J. Fox, “The grating guide—a component for integrated optics,” Proc. IEEE 62, 644–645 (1974).
[CrossRef]

Gel’fand, I. M.

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation from its spectral function,” Izv. Akad. Nauk SSSR. Ser. Mat. 15, 309–360 (1951) [Am. Math. Soc. Transl. Ser. 2 1, 253–304 (1962)].

Gladwell, G. M. L.

G. M. L. Gladwell, S. R. A. Dods, S. K. Chaudhuri, “Nonuniform transmission-line synthesis using inverse eigenvalue analysis,”IEEE Trans. Circuits Syst. CS-35, 659–666 (1988).
[CrossRef]

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962).

Hille, E.

E. Hille, Lectures on Ordinary Differential Equations (Addison-Wesley, Reading, Mass., 1969).

Levinson, N.

E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).

Levitan, B. M.

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation from its spectral function,” Izv. Akad. Nauk SSSR. Ser. Mat. 15, 309–360 (1951) [Am. Math. Soc. Transl. Ser. 2 1, 253–304 (1962)].

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982).

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1982).

Sabatier, P. C.

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

Shevchenko, V. V.

V. V. Shevchenko, Continuous Transitions in Open Waveguides (Golem, Boulder, Colo., 1971).

Titchmarsh, E. C.

E. C. Titchmarsh, Eigenfunction Expansions, Part I (Oxford U. Press, Oxford, 1962).

Yariv, A.

A. Y. Cho, A. Yariv, P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

P. Yeh, A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Yeh, P.

A. Y. Cho, A. Yariv, P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

P. Yeh, A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Yukon, S.

Appl. Phys. Lett. (1)

A. Y. Cho, A. Yariv, P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

G. M. L. Gladwell, S. R. A. Dods, S. K. Chaudhuri, “Nonuniform transmission-line synthesis using inverse eigenvalue analysis,”IEEE Trans. Circuits Syst. CS-35, 659–666 (1988).
[CrossRef]

Izv. Akad. Nauk SSSR. Ser. (1)

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation from its spectral function,” Izv. Akad. Nauk SSSR. Ser. Mat. 15, 309–360 (1951) [Am. Math. Soc. Transl. Ser. 2 1, 253–304 (1962)].

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

P. Yeh, A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Proc. IEEE (1)

A. J. Fox, “The grating guide—a component for integrated optics,” Proc. IEEE 62, 644–645 (1974).
[CrossRef]

Other (10)

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962).

E. C. Titchmarsh, Eigenfunction Expansions, Part I (Oxford U. Press, Oxford, 1962).

E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).

E. Hille, Lectures on Ordinary Differential Equations (Addison-Wesley, Reading, Mass., 1969).

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1982).

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

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982).

V. V. Shevchenko, Continuous Transitions in Open Waveguides (Golem, Boulder, Colo., 1971).

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960).

J. B. Bednar, R. Redner, E. Robinson, A. Weglein, eds., in Conference on Inverse Scattering: Theory and Application (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1983).

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Figures (8)

Fig. 1
Fig. 1

Dielectric slab guide over a reactive surface.

Fig. 2
Fig. 2

Possible values of the longitudinal wave number β.

Fig. 3
Fig. 3

Possible values of the transverse wave number κ.

Fig. 4
Fig. 4

Typical spectral functions.

Fig. 5
Fig. 5

(a) Potential V(x) with E0 = −6.4 × 1013 m−2 and c0 = 1.0 × 106 m−1. (b) Potential V(x) with E0 = 2.35 × 1014 m−2 and c0 = 5.0 × 105 m−1.

Fig. 6
Fig. 6

Example of a Bragg reflection waveguide: λ = 1.15 μm, ns = 3.35 (Al0.2Ga0.8As), na = 1.00; β0 = 1.00 × 107 m−1, ka = 5.46364 × 106 m−1; ks = 1.83032 × 107 m−1, κ0 = 1.53299 × 107 m−1; kg = 1.82895 × 107 m−1, c0 = 5.0 × 105 m−1; ng = 3.3475, xg = 0.855415 μm.

Fig. 7
Fig. 7

Magnitude of the reflection coefficient versus the normalized transverse wave number κ/ks.

Fig. 8
Fig. 8

Total normalized impedance Z ¯ t versus the longitudinal wave number β.

Equations (108)

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2 E y x 2 + 2 E y z 2 + k s 2 r ( x ) E y = 0.
k s = ω ( μ 0 s ) 1 / 2 ,
r ( x ) = ( x ) s
E y = ψ ( x ) exp ( - j β z ) .
ψ - β 2 ψ + k s 2 r ( x ) ψ = 0
ψ ( 0 ) + h ψ ( 0 ) = 0 ,             h = - j ω μ 0 Z s .
κ 2 = k s 2 - β 2 .
E = κ 2 .
ψ + [ E - V ( x ) ] ψ ( κ , x ) = 0 ,
V ( x ) = k s 2 [ 1 - r ( x ) ] .
E y ( x , z ) = m = 1 M a m ψ ( κ m , x ) exp ( - j β m z ) + 0 q ( κ ) ψ ( κ , x ) exp ( - j β z ) d κ ,
0 ψ 2 d x < .
ψ 2 < C x [ 0 , ) .
q ˜ ( κ ) = 0 ψ ( κ , x ) h a ( x ) d x .
h a ( x ) = - ψ ( κ , x ) q ˜ ( κ ) d ρ ( E ) .
E y ( x , z ) = - q ˜ ( κ ) ψ ( κ , x ) exp ( - j β z ) d ρ ( E ) .
0 b ψ ( κ n , x ) ψ ( κ m , x ) d x = ρ n δ n m .
q ˜ n = q ˜ ( κ n ) = 0 b ψ ( κ n , x ) h a ( x ) d x ,
E y ( x , y ) = n = 1 q ˜ ( κ n ) ψ ( κ n , x ) ρ n exp ( - j β n z ) .
ρ b ( E ) = E n < E 1 ρ n .
E y ( x , z ) = - q ˜ ( κ ) ψ ( κ , x ) exp [ - j β ( κ ) z ] d ρ b ( E ) .
ϕ aux + E ϕ aux = 0 , ϕ aux ( κ , x ) = cos κ x , ρ aux ( E ) = { ( 2 / π ) E 1 / 2 for E > 0 0 for E < 0 .
y ( κ , x ) = cos κ x + 0 x K ( x , s ) cos κ s d s .
K ( x , t ) + G ( x , t ) + 0 x K ( x , s ) G ( s , t ) d s = 0 ,
G ( x , t ) = - cos κ x cos κ t d σ ( E ) .
σ ( E ) = ρ ( E ) - ρ aux ( E ) .
y + [ E - V ( x ) ] y ( κ , x ) = 0
y ( 0 ) + h y ( 0 ) = 0 ,             h = - K ( 0 , 0 )
V ( x ) = 2 d K ( x , x ) d x .
ρ ( E ) = ρ aux ( E ) + c 0 U ( E - E 0 ) ,
K ( x , t ) = - c 0 cos ( E 0 1 / 2 x ) cos ( E 0 1 / 2 t ) 1 + c 0 0 x cos 2 ( E 0 1 / 2 s ) d s .
y ( κ , 0 ) + c 0 y ( κ , 0 ) = 0 ,
Z s = - j ω μ 0 c 0
0 y 2 ( κ 0 , x ) d x = 1 c 0 .
V ( x ) = - 2 d d x ( c 0 cosh 2 γ 0 x 1 + c 0 0 x cosh 2 γ 0 s d s )
β 0 = ( k s 2 - E 0 ) . 1 / 2
ρ ( E ) = ρ aux ( E ) + i = 1 N c i U ( E - E i )
G ( x , t ) = i = 1 N c i cos κ i x cos κ i t
K ( x , t ) = i = 1 N f i ( x ) cos κ i t
V ( x ) = - 2 d d x ( c 0 cos 2 κ 0 x 1 + c 0 0 x cos 2 κ 0 s d s ) .
Z + ( κ ) + Z - ( κ ) = 0
Z ¯ ( κ ) = 1 + Γ ( κ ) 1 - Γ ( κ ) .
ϕ ( κ , x ) = exp ( - j κ x ) + Γ ( κ ) exp ( j κ x )
ϕ ( κ , 0 ) - j κ ϕ ( κ , 0 ) = - 2 j κ .
ϕ ( κ , x ) = - g ( κ , κ ) y ( κ , x ) d ρ ( E ) .
F ( ϕ ) = 0 [ ( ϕ ) 2 + V ϕ 2 ] d x + j κ ϕ 2 ( κ , 0 ) - 4 j κ ϕ ( κ , 0 ) - E 0 ϕ 2 d x .
ρ ( E ) = ( 2 / π ) E 1 / 2 U ( E ) + c 0 U ( E - E 0 )
y ( κ , x ) + [ E - V ( x ) ] y ( κ , x ) = 0 ,             y ( κ , 0 ) = 1 , y ( κ , 0 ) = K ( 0 , 0 ) = - h = - c 0 .
y ( κ , b ) + H y ( κ , b ) = 0 ,
H b [ u , v ] = 0 b u ( x ) v ( x ) d x ,
D b [ u , v ] = 0 b ( u v + V u v ) d x - h u ( 0 ) v ( 0 ) + H u ( b ) v ( b ) .
H b [ y ( κ n , x ) , y ( κ m , x ) ] = ρ n δ n m , D b [ y ( κ n , x ) x , y ( κ m , x ) ] = E n ρ n δ n m .
χ ( κ , κ ) = 0 y ( κ , x ) y ( κ , x ) d x .
θ ( x ) = - y ( κ , x ) Θ ( κ ) d ρ ( E ) , Θ ( κ ) = 0 y ( κ , x ) θ ( x ) d x .
I = - χ ( κ , κ ) Θ ( κ ) d ρ ( E ) ;
I = 0 y ( κ , x ) 0 - y ( κ , x ) y ( κ , x ) d ρ ( E ) θ ( x ) d x d x .
I = 0 y ( κ , x ) 0 δ ( x - x ) θ ( x ) d x d x = 0 y ( κ , x ) θ ( x ) d x = Θ ( κ ) .
χ ( κ , κ ) = δ [ ρ ( E ) - ρ ( E ) ] ;
η ( κ , κ ) = lim b D b [ y ( κ , x ) , y ( κ , x ) ] .
- D b [ y ( κ n , x ) , y ( κ , x ) ] Θ ( κ ) d ρ b ( E ) = - E n H b [ y ( κ n , x ) , y ( κ , x ) ] Θ ( κ ) d ρ b ( E ) .
- η ( κ n , κ ) Θ ( κ ) d ρ ( E ) .
E n - χ ( κ n , κ ) Θ ( κ ) d ρ ( E ) = E n Θ ( κ n ) ;
η ( κ , κ ) = E χ ( κ , κ ) .
F ( ϕ ) = D [ ϕ , ϕ ] + ( j κ + h ) ϕ 2 ( κ , 0 ) - 4 j κ ϕ ( κ , 0 ) - E H [ ϕ , ϕ ] .
F ( ϕ ) = - - g ( κ , κ ) g ( κ , κ ) D [ y ( κ , x ) , y ( k , x ) ] × d ρ ( E ) d ρ ( E ) - E - - g ( κ , κ ) g ( κ , κ ) × H [ y ( κ , x ) , y ( κ , x ) ] d ρ ( E ) d ρ ( E ) + ( j κ + h ) ϕ 2 ( κ , 0 ) - 4 j κ ϕ ( κ , 0 ) = - ( E - E ) g 2 ( κ - κ ) d ρ ( E ) + ( j κ + h ) ϕ 2 ( κ , 0 ) - 4 j κ ϕ ( κ , 0 ) .
( 1 / 2 ) δ F = - ( E - E ) g ( κ , κ ) δ g ( κ , κ ) d ρ ( E ) + ( j κ + h ) ϕ ( κ , 0 ) × - δ g ( κ , κ ) y ( κ , 0 ) d ρ ( E ) - 2 j κ - δ g ( κ , κ ) y ( κ , 0 ) d ρ ( E ) .
g ( κ , κ ) = ( j κ + h ) ϕ ( κ , 0 ) - 2 j κ E - E .
ϕ ( κ , 0 ) = [ 2 j κ - ( h + j κ ) ϕ ( κ , 0 ) ] - d ρ ( E ) E - E .
m ( κ ) = - d ρ ( E ) E - E .
ϕ ( κ , 0 ) = 2 j κ m ( κ ) 1 + ( h + j κ ) m ( κ ) .
g ( κ , κ ) = - 2 j κ 1 + ( h + j κ ) m ( κ ) 1 E - E .
Γ ( κ ) = ( j κ - h ) m ( κ ) - 1 ( j κ + h ) m ( κ ) + 1 ,
Z ¯ + ( κ ) = j κ m ( κ ) h m ( κ ) + 1 .
m ( κ ) = c 0 κ 0 2 - κ 2 + 2 π 0 d κ κ 2 - κ 2 = c 0 κ 0 2 - κ 2 + 1 π - d κ ( κ - κ ) ( κ + κ ) .
m ( κ ) = c 0 κ 0 2 - κ 2 - j κ .
h = - K ( 0 , 0 ) = c 0 .
Γ ( κ ) = - c 0 κ 0 2 - j κ c 0 2 c 0 ( κ 0 2 - 2 κ 2 ) + j κ c 0 2 + 2 j κ ( κ 0 2 - κ 2 ) ,
Z ¯ + ( κ ) = j κ κ 0 2 - κ 2 + j κ c 0 j κ c 0 2 + ( c 0 + j κ ) ( κ 0 2 - κ 2 ) .
Γ ( κ 0 ) = j κ 0 - c 0 j κ 0 + c 0 ,
lim x ϕ ( κ , x ) = T ( κ ) exp ( - j κ x ) .
ϕ ( κ , x ) = u ( κ , x ) + 0 x K ( x , s ) u ( κ , s ) d s ,
u ( κ , x ) = - g ( κ , κ ) cos κ x d ρ ( E ) .
I = 2 π 0 cos κ x κ 2 - κ 2 d κ .
I = j κ exp ( - j κ x ) .
ϕ ( κ , x ) = 2 j κ 1 + ( c 0 + j κ ) m ( κ ) { - j κ exp ( - j κ x ) + c 0 cos κ 0 x 1 + c 0 0 x cos 2 κ 0 s d s [ 1 κ 0 2 - κ 2 + j κ 0 x exp ( - j κ s ) cos κ 0 s d s ] }
lim x ϕ ( κ , x ) = 2 exp ( - j κ x ) 2 + c 0 2 + j c 0 κ κ 0 2 - κ 2 - j c 0 κ .
Γ 2 + T 2 = 1.
Z + = ω μ 0 κ Z ¯ + = j ω μ 0 κ 0 2 - κ 2 + j κ c 0 j κ c 0 2 + ( c 0 + j κ ) ( κ 0 2 - κ 2 ) ,
Z - = ω μ 0 κ ,
2 κ 3 - 2 j c 0 κ 2 - ( c 0 2 + 2 κ 0 2 ) κ + j c 0 κ 0 2 = 0.
2 γ 3 + 2 c 0 γ 2 + ( c 0 2 + 2 κ 0 2 ) γ + c 0 κ 0 2 = 0.
V ( 0 ) = 2 c 0 2 ,
k g = ( k s 2 - 2 c 0 2 ) 1 / 2 .
Z ¯ s = κ g κ a ,
κ g = ( k g 2 - β 2 ) 1 / 2 ,             κ a = ( k a 2 - β 2 ) 1 / 2 .
Z ¯ - ( κ ) = Z ¯ s + j tan κ g x g 1 + j Z ¯ s tan κ g x g .
Z ¯ + ( κ ) = Z + ( κ ) ( ω μ 0 / κ g ) = j κ g κ 0 2 - κ 2 + j κ c 0 j κ c 0 2 + ( c 0 + j κ ) ( κ 0 2 - κ 2 ) .
Z ¯ + ( κ 0 ) = j κ g 0 c 0 ,
Z ¯ t ( κ ) = Z ¯ + ( κ ) + Z ¯ - ( κ ) .
tan κ g 0 x g = κ g 0 c 0 + j κ a 0 κ g 0 2 - j c 0 κ a 0 ,
lim x ϕ ( κ , x ) = 0.
x g = x g + n π κ g .
Z ¯ t ( κ ) = Z ¯ + ( κ ) + Z ¯ - ( κ ) ,
Z ¯ t ( κ ) = - j κ g j ( κ 0 2 - κ 2 ) - c 0 κ c 0 2 κ + ( κ - j c 0 ) ( κ 0 2 - κ 2 ) + Z ¯ s cos κ g x g + j sin κ g x g cos κ g x g + j Z ¯ s sin κ g x g ,
0 = κ g ( κ 0 2 - κ 2 ) cos κ g x g + j κ g c 0 κ cos κ g x g + j κ g Z ¯ s ( κ 0 2 - κ 2 ) sin κ g x g + Z ¯ s c 0 2 κ cos κ g x g + Z ¯ s κ ( κ 0 2 - κ 2 ) cos κ g x g - j c 0 Z ¯ s ( κ 0 2 - κ 2 ) cos κ g x g + j c 0 2 κ sin κ g x g + j κ ( κ 0 2 - κ 2 ) sin κ g x g + c 0 ( κ 0 2 - κ 2 ) sin κ g x g .
κ = - j γ ,             κ g = - j γ g ,             κ a = - j γ a ,
γ = ( β 2 - k s 2 ) 1 / 2 , γ g = ( β 2 - k g 2 ) 1 / 2 , γ a = ( β 2 - k a 2 ) 1 / 2 .
0 = - j γ g ( κ 0 2 + γ 2 ) cosh γ g x g - j γ g γ c 0 cosh γ g x g - j γ g Z ¯ s ( κ 0 2 + γ 2 ) sinh γ g x g - j Z ¯ s c 0 2 γ cosh γ g x g - j Z ¯ s γ ( κ 0 2 + γ 2 ) cosh γ g x g - j c 0 Z ¯ s ( κ 0 2 + γ 2 ) cosh γ g x g - j c 0 2 γ sinh γ g x g - j γ ( κ 0 2 + γ 2 ) sinh γ g x g - j c 0 ( κ 0 2 + γ 2 ) sinh γ g x g .

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