Abstract

It is shown that, for rectangular-core directional couplers, Marcatili's widely used analysis [Bell Syst. Tech. J. 48, 2071 (1969)] predicts almost the same value of coupling length as that obtained by the simple slab-waveguide approximation. A more accurate analysis, which takes the effect of corner regions into account through first-order perturbation theory, shows that Marcatili's analysis may lead to inaccurate results for small values of depths of channel waveguides, which are the most practical directional couplers involving single-mode channel waveguides.

© 1985 Optical Society of America

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References

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  1. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
  2. P. Yeh, H. F. Taylor, Appl. Opt. 19, 2848 (1980).
    [Crossref] [PubMed]
  3. A. Sharma, P. K. Mishra, A. K. Ghatak, presented at the Second European Conference on Integrated Optics, Italy, October 17–18, 1983.
  4. A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart and Winston, New York, 1976), p. 391.
  5. A. Kumar, K. Thyagarajan, A. K. Ghatak, Opt. Lett. 8, 63 (1983).
    [Crossref] [PubMed]
  6. M. Kuznetsov, Opt. Lett. 8, 499 (1983).
    [Crossref] [PubMed]

1983 (2)

1980 (1)

1969 (1)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

Ghatak, A. K.

A. Kumar, K. Thyagarajan, A. K. Ghatak, Opt. Lett. 8, 63 (1983).
[Crossref] [PubMed]

A. Sharma, P. K. Mishra, A. K. Ghatak, presented at the Second European Conference on Integrated Optics, Italy, October 17–18, 1983.

Kumar, A.

Kuznetsov, M.

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

Mishra, P. K.

A. Sharma, P. K. Mishra, A. K. Ghatak, presented at the Second European Conference on Integrated Optics, Italy, October 17–18, 1983.

Sharma, A.

A. Sharma, P. K. Mishra, A. K. Ghatak, presented at the Second European Conference on Integrated Optics, Italy, October 17–18, 1983.

Taylor, H. F.

Thyagarajan, K.

Yariv, A.

A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart and Winston, New York, 1976), p. 391.

Yeh, P.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

Opt. Lett. (2)

Other (2)

A. Sharma, P. K. Mishra, A. K. Ghatak, presented at the Second European Conference on Integrated Optics, Italy, October 17–18, 1983.

A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart and Winston, New York, 1976), p. 391.

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Figures (3)

Fig. 1
Fig. 1

(a) Cross section of (b) the rectangular-core directional coupler to be studied as described by Eqs. (1)(3).

Fig. 2
Fig. 2

Variation of coupling length Lc as a function of wavelength λ for a rectangular-core directional coupler with 2a = 2b = 2d = 2.0 μm.

Fig. 3
Fig. 3

Variation of coupling length Lc as a function of depth b for a rectangular-core directional coupling with 2a = 2d = 2.0 μm and λ = 0.60μm.

Equations (17)

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n 2 ( x , y ) = n 2 ( x ) + n 2 ( y ) n 1 2 ,
n 2 ( x , y ) = n s 2 , 0 < | x | < d , = n 1 2 , d < | x | < d + 2 a , = n s 2 , | x | > d + 2 a ,
n 2 ( y ) = n s 2 , y < b , = n 1 2 , b < y < b , = n 2 2 , y > b .
[ 2 x 2 + 2 y 2 + ( k 0 2 n 2 ( x , y ) β 2 ) ] ψ = 0
[ d 2 d x 2 + k 0 2 n 2 ( x ) β x 2 ] X ( x ) = 0 ,
[ d 2 d y 2 + k 0 2 n 2 ( y ) β y 2 ] Y ( y ) = 0 ,
β 2 = β x 2 + β y 2 k 0 2 n 1 2 .
ν 1 ν 2 tanh ( ν 1 d ) = [ tan ( 2 ν 2 a ) ν 1 ν 2 ] [ 1 + ν 1 ν 2 tan ( 2 ν 2 a ) ] .
ν 1 ν 2 coth ( ν 1 d ) = [ tan ( 2 ν a a ) ν 1 ν 2 ] [ 1 + ν 1 ν 2 tan ( 2 ν 2 a ) ] ,
tan ( 2 μ 2 b ) = n 1 2 μ 2 ( n 2 2 μ 3 + n s 2 μ 1 ) ( n 2 2 n s 2 μ 2 2 n 1 4 μ 1 μ 3 ) ,
ν 1 2 = ( β x 2 k 0 2 n s 2 ) , ν 2 2 = ( k 0 2 n 1 2 β x 2 ) ,
μ 1 2 = ( β y 2 k 0 2 n 2 2 ) , μ 2 2 = ( k 0 2 n 1 2 β y 2 ) , μ 3 2 = ( β y 2 k 0 2 n s 2 ) .
κ = ( β s β a ) / 2 .
κ = h 2 p exp ( 2 p d ) β a ( h 2 + p 2 ) ( 1 + 1 p a ) ,
h 2 = ( k 0 2 n 1 2 β x 2 ) , p 2 = ( β x 2 k 0 2 n s 2 ) .
β p 2 = β 2 + k 0 2 ( n 1 2 n s 2 ) Γ ,
Γ = b 0 d | ψ ( x , y ) | 2 d x d y + b ( d + 2 a ) | ψ ( x , y ) | 2 d x d y 0 0 | ψ ( x , y ) | 2 d x d y

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