Abstract

General formulas describing field distributions and eigenvalue equations for both transverse-electric and transverse-magnetic modes are used to analyze symmetric periodic stratified media. The half-phase shifts are discussed, and the eigenvalue equations for even and odd modes are obtained. Numerical results for some special cases are also given.

© 1988 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. P. Yeh, A. Yariv, C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,”J. Opt. Soc. Am. 67, 423–438 (1977).
    [CrossRef]
  4. J. F. Revelli, “Mode analysis and prism coupling for multilayered optical waveguides,” Appl. Opt. 20, 3158–3167 (1981).
    [CrossRef] [PubMed]
  5. S. Ruschin, E. Marom, “Coupling effects in symmetrical three-guide structures,” J. Opt. Soc. Am. A 1, 1120–1128 (1984).
    [CrossRef]
  6. L. M. Walpita, “Solutions for planar waveguide equations by selecting zero elements in a characteristic matrix,” J. Opt. Soc. Am. A 2, 595–602 (1985).
    [CrossRef]
  7. Y.-F. Li, J. W. Y. Lit, “General formulas for the guiding properties of a multilayer slab waveguide,” J. Opt. Soc. Am. A 4, 671–677 (1987).
    [CrossRef]
  8. Ref. 3, Eq. (45).
  9. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  10. Y.-F. Li, J. W. Y. Lit, “Contribution of low-index layers to mode number in multilayer slab waveguides,” J. Opt. Soc. Am. A 4, 2233–2239 (1987).
    [CrossRef]

1987 (2)

1985 (1)

1984 (1)

1981 (1)

1977 (1)

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]

Fox, A. J.

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

Hong, C. S.

Li, Y.-F.

Lit, J. W. Y.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Marom, E.

Revelli, J. F.

Ruschin, S.

Walpita, L. M.

Yariv, A.

Yeh, P.

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Figures (12)

Fig. 1
Fig. 1

Geometry of an L-layer planar waveguide with L = l + m + 1.

Fig. 2
Fig. 2

Field distribution in the ith layer, showing the location of the maximum or minimum point, xim [see Eq. (17)]. (a) and (b) Real points. (c) and (d) Virtual points.

Fig. 3
Fig. 3

Field distribution in the ith layer, showing the point xit at which the sign of the field changes [see Eq. (22)]. (a) and (b) Real points. (c) and (d) Virtual points.

Fig. 4
Fig. 4

Refractive-index profile of a symmetric periodic stratified planar waveguide with 2m + 1 layers. −1, 0, 1, …, 2m − 1 are the layer order numbers.

Fig. 5
Fig. 5

Effective refractive index N of TE modes as a function of d/λ for symmetric three-layer waveguide (m = 1), where n1 = 3.36, n2 = 3.62, d = d1 = d2, and λ is the wavelength of the guided wave in the vacuum.

Fig. 6
Fig. 6

Effective refractive index N of TE modes as a function of d/λ for symmetric five-layer periodic stratified waveguide (m = 2). The data are the same as those given in Fig. 5.

Fig. 7
Fig. 7

Effective refractive index N of TE modes as a function of d/λ for symmetric seven-layer periodic stratified waveguide (m = 3). The data are the same as those given in Fig. 5.

Fig. 8
Fig. 8

Transverse field distributions for the TE mode of a symmetric three-layer waveguide (m = 1).

Fig. 9
Fig. 9

Transverse field distributions for the TE mode in the first band of a symmetric five-layer waveguide (m = 2).

Fig. 10
Fig. 10

Transverse field distributions for the TE mode in the second band of a symmetric five-layer waveguide (m = 2).

Fig. 11
Fig. 11

Transverse field distributions for the TE mode in the first band of a symmetric seven-layer waveguide (m = 3).

Fig. 12
Fig. 12

Transverse field distributions for the TE mode in the second band of a symmetric seven-layer waveguide (m = 3).

Equations (77)

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t + 0 t 0 = d 0 ,
t ± i = t ± 0 ± k = 1 i d ± k ( + : i = 1 , , l 1 ; : i = 1 , , m 1 ) ,
F 0 ( x ) = F + 0 cos [ h 0 ( x t + 0 ) + ϕ + 0 ] cos ϕ + 0 = F 0 cos [ h 0 ( x t 0 ) ϕ 0 ] cos ϕ 0 ( t 0 x t + 0 ) .
F ± i ( x ) = F ± i cosh [ p ± i ( x t ± ( i 1 ) ) ψ ± i ] cosh ψ ± i ( + : i = 1 , 2 , , l , t + ( i 1 ) x t + i , t + l = + ; : i = 1 , 2 , , m , t i x t ( i 1 ) , t m = ) .
F + 0 = F 0 cos ( h 0 d 0 ϕ 0 ) cos ϕ 0 ,
F ± 1 = F ± 0 ,
F ± i = F ± 0 k = 1 i 1 cosh ( p ± k d ± k ψ ± k ) cosh ψ ± k .
h i 2 = k 0 2 n i 2 β 2 ,
p i = j h i ( i = m , , 1 , 0 , + 1 , , + l ) ,
tanh ψ + l = tanh ψ m = 1.
h 0 d 0 = ϕ + 0 + ϕ 0 + q 0 π ( q 0 = 0 , 1 , ) ,
ϕ ± 0 = tan 1 ( η 0 , ± 1 p ± 1 h 0 tanh ψ ± 1 ) ,
ψ ± i = p ± i d ± i + tanh 1 [ η ± i , ± ( i + 1 ) p ± ( i + 1 ) p ± i tanh ψ ± ( i + 1 ) ] ( + : i = 1 , 2 , , l 1 ; : i = 1 , 2 , , m 1 ) .
η r , s = { 1 , TE modes ( n r / n s ) 2 , TM modes .
ϕ i = j ψ i ,
ϕ ± i = h ± i d ± i tan 1 [ η ± i , ± ( i + 1 ) p ± ( i + 1 ) h ± i tanh ψ ± ( i + 1 ) ] q ± i π ( q ± i = 0 , 1 , ) ,
tanh ( p i d i ψ i ) = η i , i + 1 p i + 1 p i tanh ψ i + 1 .
tan ( h i d i ) = tan ( ϕ i + ϕ i )
tan ϕ i = η i , i + 1 p i + 1 h i tanh ψ i + 1 .
tan ϕ i = { tan ( h i d i ϕ i ) ( n i > β / k 0 ) j tanh ( p i d i ψ i ) ( n i < β / k 0 ) .
tanh ψ i = tanh ( p i d i ψ i )
tan ϕ i = tan ( h i d i ϕ i )
x = x ± i m = t ± ( i 1 ) ± ψ ± i p ± i
tanh ( ψ ± i ) = tanh ( p ± i d ± i ) + η ± i , ± ( i + 1 ) p ± ( i + 1 ) p ± i tanh ψ ± ( i + 1 ) 1 + η ± i , ± ( i + 1 ) p ± ( i + 1 ) p ± i tanh ( p ± i d ± i ) tanh ψ ± ( i + 1 ) ( + : i = 1 , 2 , , l 1 ; : i = 1 , 2 , , m 1 ) .
coth ( χ ± i ) = tanh ( p ± i d ± i ) + η ± i , ± ( i + 1 ) p ± ( i + 1 ) p ± i tanh ψ ± ( i + 1 ) 1 + η ± i , ± ( i + 1 ) p ± ( i + 1 ) p ± i tanh ( p ± i d ± i ) tanh ψ ± ( i + 1 ) .
tanh ( p i d i ) = tanh ( χ i + χ i ) ,
F ± i ( x ) = F ± i sinh [ p ± i ( x t ± ( i 1 ) ) χ ± i ] sinh χ ± i ( + : i = 2 , 3 , , l , t + ( i 1 ) x t + i ; : i = 2 , 3 , , m , t i x t ( i 1 ) ) .
x = x ± i t = t ± ( i 1 ) ± χ ± i p ± i
n ( x ) = { n 2 ( 0 < x < d 2 ) n 1 ( d 2 < x < Λ ) ,
n 1 < n 2 ,
n ( x + Λ ) = n ( x ) ,
Λ = d 1 + d 2 ,
A = exp ( p 1 d 1 ) [ cos ( h 2 d 2 ) 1 2 ( ζ 12 ζ 21 ) sin ( h 2 d 2 ) ] ,
B = exp ( p 1 d 1 ) [ 1 2 ( ζ 12 + ζ 21 ) sin ( h 2 d 2 ) ] ,
C = exp ( p 1 d 1 ) [ 1 2 ( ζ 12 + ζ 21 ) sin ( h 2 d 2 ) ] ,
D = exp ( p 1 d 1 ) [ cos ( h 2 d 2 ) + 1 2 ( ζ 12 ζ 21 ) sin ( h 2 d 2 ) ] ,
U i = sin ( i + 1 ) K Λ / sin K Λ ,
cos K Λ = 1 2 ( A + D ) .
tan ϕ 0 = tan ϕ 0 = ζ 21
tanh ψ 2 i + 1 = ζ 12 tan ( h 2 d 2 ϕ 2 i ) ( i = 0 , 1 , , m 1 ) ,
tanh ϕ 2 i = ζ 21 tanh ( p 1 d 1 ψ 2 i 1 ) ( i = 1 , , m 1 ) ,
tanh ψ 2 m 1 = 1 ,
ζ 21 = η 21 p 1 h 2
ζ 12 = 1 ζ 21 .
tanh ( p 1 d 1 ψ 2 i 1 ) = ( A + B ) U i 1 U i 2 ( A B ) U i 1 U i 2 .
A U m 1 U m 2 = 0.
tanh ( p 1 d 1 ψ 2 i 1 ) = ( A C ) U m i 1 U m i 2 ( A + C ) U m i 1 + U m i 2 ,
tanh ψ 2 i 1 = X 2 i 1 ,
tanh ψ 2 i 1 = tanh ( p 1 d 1 ) X 2 i 1 1 tanh ( p 1 d 1 ) X 2 i 1 ,
tan ϕ 2 i = tan ( h 2 d 2 ) ζ 21 X 2 i 1 1 + ζ 21 tan ( h 2 d 2 ) X 2 i 1 ,
tan ϕ 2 i = ζ 21 X 2 i 1 .
F 1 ( x ) = F 1 exp ( p 1 x ) ( < x < 0 ) .
F 2 k ( x ) = F 2 k cos [ h 2 ( x k Λ ) ϕ 2 k ] cos ϕ 2 k ( k = 0 , 1 , , m 1 , k Λ < x < k Λ + d 2 ) .
F 2 k + 1 ( x ) = F 2 k + 1 cosh [ p 1 ( x k Λ d 2 ) ψ 2 k + 1 ] cosh ψ 2 k + 1 [ k = 0 , 1 , , m 2 , k Λ + d 2 < x < ( k + 1 ) Λ ] .
F 2 m 1 ( x ) = F 2 m 1 exp [ p 1 ( x m Λ + d 1 ) ] ( m Λ d 1 < x < + ) ,
F 2 i = F 1 [ ( A B ) U i 1 U i 2 ] ,
F 2 i + 1 = F 1 { [ cos ( h 2 d 2 ) + ζ 21 sin ( h 2 d 2 ) ] U i exp ( p 1 d 1 ) U i 1 } .
R = X m 1 ,
R = { tanh ( p 1 d 1 / 2 ) ( even modes ) coth ( p 1 d 1 / 2 ) ( odd modes ) .
S = ζ 21 X m 2 ,
S = { tan ( h 2 d 2 / 2 ) ( even modes ) cot ( h 2 d 2 / 2 ) ( odd modes ) .
F 1 ( x ) = F 1 exp ( p 1 x ) ( < x < 0 ) ,
F 0 ( x ) = F 1 cos ( h 2 x ϕ 0 ) cos ϕ 0 ( 0 < x < d 2 ) ,
F 1 ( x ) = F 1 exp [ p 1 ( x d 2 ) ] ( d 2 < x < + ) ,
F 1 = F 1 [ cos ( h 2 d 2 ) + ζ 21 sin ( h 2 d 2 ) ] .
S = ζ 21 X 1 = ζ 21 ,
F 1 ( x ) = F 1 exp ( p 1 x ) ( < x < 0 ) ,
F 0 ( x ) = F 1 cos ( h 2 x ϕ 0 ) cos ϕ 0 ( 0 < x < d 2 ) ,
F 1 ( x ) = F 1 cosh [ p 1 ( x d 2 ) ψ 1 ] cosh ψ 1 ( d 2 < x < Λ ) ,
F 2 ( x ) = F 2 cos [ h 2 ( x Λ ) ϕ 2 ] cosh ϕ 2 ( Λ < x < Λ + d 2 ) ,
F 3 ( x ) = F 3 exp [ p 1 ( x Λ d 2 ) ] ( Λ + d 2 < x < + ) .
F 1 = F 1 [ cos ( h 2 d 2 ) + ζ 21 sin ( h 2 d 2 ) ] ,
F 2 = F 1 ( A B ) ,
F 3 = F 1 { [ cos ( h 2 d 2 ) + ζ 21 sin ( h 2 d 2 ) ] ( A + D ) e p 1 d 1 } .
R = X 1 = A + B A B .
R = 1 ζ 12 tan ( h 2 d 2 ) 1 + ζ 21 tan ( h 2 d 2 ) .
S = ζ 21 X 1 = ζ 21 [ 1 + tanh ( p 1 d 1 ) ] + [ ζ 21 tanh ( p 1 d 1 ) ζ 12 ] tan ( h 2 d 2 ) [ 1 + tanh ( p 1 d 1 ) ] + [ ζ 21 ζ 12 tanh ( p 1 d 1 ) ] tan ( h 2 d 2 ) .

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