Abstract

Based on the exact scalar wave equations of bent slab waveguides, locally leaky-mode attenuation constants for both TE and TM modes are derived by using the WKB tunneling theory. Changes in propagation constant due to waveguide bending are considered in our calculation. Under some assumptions, our results are found to be identical to other results obtained using entirely different methods.

© 1989 Optical Society of America

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References

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  1. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1982), p. 406.
  2. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).
  3. E. Kuester, D. C. Chang, IEEE J. Quantum Electron. QE-11, 903 (1975).
    [CrossRef]
  4. Y. Takuma, M. Miyagi, S. Kawakami, Appl. Opt. 20, 2291 (1981).
    [CrossRef] [PubMed]
  5. Y. H. Cheng, W. G. Lin, Microwave Opt. Technol. Lett. 1, 329 (1988).
    [CrossRef]
  6. Y. H. Cheng, W. G. Lin, Microwave Opt. Technol. Lett. 2, 285 (1989).
    [CrossRef]
  7. J. D. Love, C. Winkler, J. Opt. Soc. Am. 67, 1627 (1977).
    [CrossRef]
  8. That is, the conditions βe/[n2k0(1 + a/Rc)] > 1 and βe/[n1k0(1 − a/Rc)] < 1 are both required for the propagation constant of a locally tunneling leaky mode in bent dielectric optical waveguides.

1989 (1)

Y. H. Cheng, W. G. Lin, Microwave Opt. Technol. Lett. 2, 285 (1989).
[CrossRef]

1988 (1)

Y. H. Cheng, W. G. Lin, Microwave Opt. Technol. Lett. 1, 329 (1988).
[CrossRef]

1981 (1)

1977 (1)

1975 (1)

E. Kuester, D. C. Chang, IEEE J. Quantum Electron. QE-11, 903 (1975).
[CrossRef]

1969 (1)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

Chang, D. C.

E. Kuester, D. C. Chang, IEEE J. Quantum Electron. QE-11, 903 (1975).
[CrossRef]

Cheng, Y. H.

Y. H. Cheng, W. G. Lin, Microwave Opt. Technol. Lett. 2, 285 (1989).
[CrossRef]

Y. H. Cheng, W. G. Lin, Microwave Opt. Technol. Lett. 1, 329 (1988).
[CrossRef]

Kawakami, S.

Kuester, E.

E. Kuester, D. C. Chang, IEEE J. Quantum Electron. QE-11, 903 (1975).
[CrossRef]

Lin, W. G.

Y. H. Cheng, W. G. Lin, Microwave Opt. Technol. Lett. 2, 285 (1989).
[CrossRef]

Y. H. Cheng, W. G. Lin, Microwave Opt. Technol. Lett. 1, 329 (1988).
[CrossRef]

Love, J. D.

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1982), p. 406.

Miyagi, M.

Takuma, Y.

Winkler, C.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2103 (1969).

IEEE J. Quantum Electron. (1)

E. Kuester, D. C. Chang, IEEE J. Quantum Electron. QE-11, 903 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

Microwave Opt. Technol. Lett. (2)

Y. H. Cheng, W. G. Lin, Microwave Opt. Technol. Lett. 1, 329 (1988).
[CrossRef]

Y. H. Cheng, W. G. Lin, Microwave Opt. Technol. Lett. 2, 285 (1989).
[CrossRef]

Other (2)

That is, the conditions βe/[n2k0(1 + a/Rc)] > 1 and βe/[n1k0(1 − a/Rc)] < 1 are both required for the propagation constant of a locally tunneling leaky mode in bent dielectric optical waveguides.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1982), p. 406.

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Figures (1)

Fig. 1
Fig. 1

(a) Coordinate system used to describe the radiation loss of a bent-slab waveguide, (b) Illustration of locally leaky-mode characteristics due to waveguide bending. The hatched area indicates the region where the modes experience radiation loss.

Equations (18)

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d 2 F d x 2 + [ k 0 2 n 2 ( x ) β e 2 / h s 2 ] F = 0 ,
e x ( x ) = h s 3 / 2 F ( x ) exp ( i β s ) for TM modes ,
e y ( x ) = h s 1 / 2 F ( x ) exp ( i β s ) for TM modes ,
α = 2 β i = T f exp [ 2 a x c w e ( x ) / ( h s a ) d x ] 2 β a 0 a [ u e ( x ) h s ] 1 d x ,
T f = { 4 u e ( a ) w e ( a ) u e 2 ( a ) + w e 2 ( a ) for TE modes 4 u e ( a ) w e ( a ) n 1 2 n 2 2 n 2 4 u e 2 ( a ) + n 1 4 w e 2 ( a ) for TM modes .
α = K exp ( S R c ) ,
K = u e ( a ) 2 β a 2 T f ,
S = 2 β { t h 1 [ w e ( a ) β e a ] w e ( a ) β e a } ,
t h 1 [ w e ( a ) β e a ] w e ( a ) β e a t h 1 ( w 1 β e a ) w 1 β e ( 1 a + 1 R c ) ,
α K exp { 2 β 0 [ t h 1 ( w ˜ 1 β 0 a ) w ˜ 1 β 0 a ] R c } exp ( 2 w ˜ 1 ) ,
S 2 3 w e 3 ( a ) / ( β e 2 a 3 ) .
S 1 2 n 1 k 0 [ w ˜ 1 2 / ( n 2 k 0 a ) 2 2 a / R c ] 3 / 2 ,
t h 1 ( w 1 β e a ) w 1 β e a t h 1 ( w ˜ 1 β 0 a ) w ˜ 1 β 0 a δ β 1 β 0 2 w ˜ 1 a 1 R c 1 3 w ˜ 1 3 ( β 0 a ) 3 + δ β 1 β 0 2 w ˜ 1 a R c .
S 2 3 w ˜ 1 3 β 0 2 a 3 + w ˜ 1 a δ β 1 β 0 2 R c 2 w ˜ 1 R c .
δ β 1 = β 0 a ( 1 / w ˜ 1 1 / w ˜ 2 ) / 2 .
α K exp ( 2 3 w ˜ 1 3 β 0 2 a 3 R c ) exp [ 2 w ˜ 1 ( 1 w ˜ 1 / w ˜ 2 ) ] ,
S = 2 3 [ ( β k 0 n 2 ) ( β k 0 n 2 ) 2 β 2 a / R c β 2 ] 3 / 2 ( 1 + a / R c ) 3 ,
S 2 3 n 1 k 0 { 2 [ β ( 1 a / R c ) n 2 k 0 1 ] } 3 / 2 ,

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