Abstract

Analytical expressions describing the propagation of surface waves for both transverse-electric and transverse-magnetic fields in semi-infinite superlattice structures are obtained. The modes are characterized by electric fields that decay exponentially in the region of lower refractive index while being oscillatory in the region of greater refractive index, and both are modulated by an envelope function that decays exponentially away from the end of the truncated superlattice. A criterion for the existence of the surface modes is developed. Numerical results are presented for several illustrative cases.

© 1991 Optical Society of America

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References

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  1. For a recent review, see J. W. Y. Lit, Y.-F. Li, D. W. Hewak, “Guiding properties of multilayer dielectric planar waveguides,” Can. J. Phys. 66, 914–940 (1988).
    [CrossRef]
  2. 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]
  3. J. F. Revelli, “Mode analysis and prism coupling for multilayered optical waveguides,” Appl. Opt. 20, 3158–3167 (1981).
    [CrossRef] [PubMed]
  4. Y.-F. Li, J. W. Y. Lit, “Guided even and odd modes in symmetric periodic stratified dielectric waveguides,” J. Opt. Soc. Am. A 5, 1050–1057 (1988).
    [CrossRef]
  5. 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]
  6. W. Ng, P. Yeh, P. C. Chen, A. Yariv, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 370–374 (1978).
    [CrossRef]
  7. P. Yeh, “Resonant tunneling of electromagnetic radiation in superlattice structures,” J. Opt. Soc. Am. A 2, 568–571 (1985).
    [CrossRef]

1988 (2)

For a recent review, see J. W. Y. Lit, Y.-F. Li, D. W. Hewak, “Guiding properties of multilayer dielectric planar waveguides,” Can. J. Phys. 66, 914–940 (1988).
[CrossRef]

Y.-F. Li, J. W. Y. Lit, “Guided even and odd modes in symmetric periodic stratified dielectric waveguides,” J. Opt. Soc. Am. A 5, 1050–1057 (1988).
[CrossRef]

1985 (2)

1981 (1)

1978 (1)

W. Ng, P. Yeh, P. C. Chen, A. Yariv, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 370–374 (1978).
[CrossRef]

1977 (1)

Chen, P. C.

W. Ng, P. Yeh, P. C. Chen, A. Yariv, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 370–374 (1978).
[CrossRef]

Hewak, D. W.

For a recent review, see J. W. Y. Lit, Y.-F. Li, D. W. Hewak, “Guiding properties of multilayer dielectric planar waveguides,” Can. J. Phys. 66, 914–940 (1988).
[CrossRef]

Hong, C.-S.

Li, Y.-F.

For a recent review, see J. W. Y. Lit, Y.-F. Li, D. W. Hewak, “Guiding properties of multilayer dielectric planar waveguides,” Can. J. Phys. 66, 914–940 (1988).
[CrossRef]

Y.-F. Li, J. W. Y. Lit, “Guided even and odd modes in symmetric periodic stratified dielectric waveguides,” J. Opt. Soc. Am. A 5, 1050–1057 (1988).
[CrossRef]

Lit, J. W. Y.

Y.-F. Li, J. W. Y. Lit, “Guided even and odd modes in symmetric periodic stratified dielectric waveguides,” J. Opt. Soc. Am. A 5, 1050–1057 (1988).
[CrossRef]

For a recent review, see J. W. Y. Lit, Y.-F. Li, D. W. Hewak, “Guiding properties of multilayer dielectric planar waveguides,” Can. J. Phys. 66, 914–940 (1988).
[CrossRef]

Ng, W.

W. Ng, P. Yeh, P. C. Chen, A. Yariv, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 370–374 (1978).
[CrossRef]

Revelli, J. F.

Walpita, L. M.

Yariv, A.

W. Ng, P. Yeh, P. C. Chen, A. Yariv, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 370–374 (1978).
[CrossRef]

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]

Yeh, P.

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Figures (6)

Fig. 1
Fig. 1

TE dispersion relation for the surface modes (dotted curves) of the semi-infinite superlattice and those of the bulk modes (hatched areas).

Fig. 2
Fig. 2

TM dispersion relation using the same parameters as in Fig. 1, except that Nc = 2.

Fig. 3
Fig. 3

Transverse field distribution for the fundamental mode guided by a periodic superlattice. The inset shows the refractive-index profile.

Fig. 4
Fig. 4

Transverse field distribution for a typical surface mode guided by a semi-infinite periodic stratified medium.

Fig. 5
Fig. 5

TE-frequency modes for the surface modes (dotted curves) of the semi-infinite superlattice and those of the bulk modes (hatched areas) as a function of the thickness of the low refractive-index layer for δ = 1.

Fig. 6
Fig. 6

TM-frequency modes using the same parameters as in Fig. 3, except that Nc = 2.

Equations (38)

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E y = E y ( x ) exp ( i ω t - i β z ) ,
E y ( x ) = exp ( i k z ) E k ( x ) ,
E k ( x + n d ) = E k ( x ) .
d 2 E y ( x ) d x 2 + [ N 2 ( x ) ( ω / c ) 2 - β 2 ] E y ( x ) = 0 ,
N ( x ) = { N a n d < x < n d + a N b n d + a < x < ( n + 1 ) d ,
E k ( x ) = exp [ - i k ( x - n d ) ] { A exp [ i α a ( x - n d ) ] + B exp [ - i α a ( x - n d ) ] }             n d < x < n d + a
E y ( x ) = exp ( i k n d ) { A exp [ i α a ( x - n d ) ] + B exp [ - i α a ( x - n d ) ] } .
E y ( x ) = exp ( i k n d ) { C exp [ α b ( x - n d - a ) ] + D exp [ - α b ( x - n d - a ) ] } ,
α a = [ N a 2 ( ω / c ) 2 - β 2 ] 1 / 2
α b = [ β 2 - N b 2 ( ω / c ) 2 ] 1 / 2 .
A exp ( i α a a ) + B exp ( - i α a a ) = C + D
i α a [ A exp ( i α a a ) - B exp ( - i α a a ) ] = α b ( C - D ) .
A + B = exp ( - i k d ) [ C exp ( α b b ) + D exp ( - α b b ) ] ,
i α a ( A - B ) = α b exp ( - i k d ) [ C exp ( α b b ) - D exp ( - α b b ) ] .
α b 2 - α a 2 2 α b α a sinh α b b sin α a a + cosh α b b cos α a a = cosh d .
E z ( x ) = α a ω a exp ( i k d ) { A exp [ i α a ( x - n d ) ] - B exp [ - i α a ( x - n d ) ] }             n d < x < n d + a
E z ( x ) = - i α b ω b exp ( i k d ) { C exp [ α b ( x - n d - a ) ] - D exp [ - α b ( x - n d - a ) ] }             n a < x < ( n + 1 ) d ,
( a 2 α a 2 - b 2 α b 2 ) sinh α b b sin α a a + 2 a b α a α b ( cos α a a cosh α b b - cosh d ) = 0.
E y ( x ) = exp ( - λ n d ) { A exp [ i α a ( x - n d ) ] + B exp [ - i α a ( x - n d ) ] }             n d < x < n d + a ,
E y ( x ) = exp ( - λ n d ) { C exp [ α b ( x - n d - a ) ] + D exp [ - α b ( x - n d - a ) ] } .
E y ( x ) = E c exp ( α c x ) ,
α c 2 = β 2 - N c 2 ( ω / c ) 2 .
( 1 + i α a α b ) [ exp ( i α a a ) - exp ( - α b b - λ d ) ] A + ( 1 - i α a α b ) [ exp ( - i α a a ) - exp ( - α b b - λ d ) ] B = 0
( 1 - i α a α b ) [ exp ( i α a a ) - exp ( α b b - λ d ) ] A + ( 1 + i α a α b ) [ exp ( - i α a a ) - exp ( α b b - λ d ) ] B = 0.
cosh λ d = α b 2 - α a 2 2 α a α b sin α a a sinh α b b + cos α a a cosh α b b ,
E c = A + B
α c E c = i α a ( A - B ) ,
( 1 - i α a α c ) A + ( 1 + i α a α c ) B = 0.
exp ( - λ d ) = exp ( - α b b ) ( cos α a a + T - sin α a a ) ,
T - = α a 2 + α b α c α a ( α b - α c ) ,
exp ( - λ d ) = exp ( α b b ) ( cos α a a + T + sin α a a ) ,
T + = - α a 2 - α b α c α a ( α b + α c ) .
2 sinh α b b cos α a a + [ T + exp ( α b b ) - T - exp ( - α b b ) ] × sin α a a = 0.
exp ( - γ d ) = exp ( - α b b ) ( cos α a a - P - sin α a a ) ,
P - = α a 2 b c + α b α c a 2 α a a ( b α c - α b c )
exp ( γ d ) = exp ( α b b ) ( cos α a a - P + sin α a a )
P + = α a 2 b c - α b α c a 2 α a a ( b α c + α b c ) ,
2 sinh α b b cos α a a + [ P - exp ( - α b b ) - P + exp ( α b b ) ] × sin α a a = 0.

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