Abstract

An exact scalar wave analysis of a directional coupler obtained by coupling two identical graded-index waveguides having exponential profiles is given. This analysis has been used to study the validity of the widely used coupled-mode theory for studying such problems. The study shows that coupled-mode theory indeed leads to very good results for weak coupling but may lead to quite inaccurate results for sufficiently strong coupling.

© 1981 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. W. J. Tomlinson, H. P. Weber, C. A. Pryde, E. A. Chandross, Appl. Phys. Letts. 26, 303 (1975).
    [CrossRef]
  3. T. Suhara, Y. Handa, Nishihara, J. Koyama, J. Opt. Soc. Am. 69, 807 (1979).
    [CrossRef]
  4. M. Papuchon, Y. Cobemale, X. Mathieu, D. B. Ostrowsky, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).
    [CrossRef]
  5. M. Kobayashi, H. Terui, Appl. Opt. 17, 3253 (1978).
    [CrossRef] [PubMed]
  6. D. Marcuse, Bell. Syst. Tech. J. 50, 1791 (1971).
  7. A. Sharma, E. Khular, K. Thyagarajan, A. K. Ghatak, Opt. Commun. 30, 166 (1979).
    [CrossRef]
  8. R. C. Alferness, R. V. Schmidt, E. H. Turner, Appl. Opt. 18, 4012 (1979).
    [CrossRef] [PubMed]
  9. H. P. Hsu, A. F. Milton, IEEE J. Quantum Electron. QE-13, 224 (1977).
    [CrossRef]
  10. Y. Suematsu, K. Kishino, Radio Sci. 12, 587 (1977).
    [CrossRef]
  11. H. P. Hsu, W. S. C. Chang, M. S. Chang, Wave Electron, 1, 191 (1974/1975).
  12. G. A. Teh, V. C. Y. So, G. I. Stegman, IEEE Trans. Microwave Theory Tech. MTT-27, 265 (1979).
    [CrossRef]
  13. D. G. Dalgoutte, R. B. Smith, G. Achutaramayya, J. H. Harris, Appl. Opt. 14, 1860 (1975).
    [CrossRef] [PubMed]
  14. J. N. Fields, Appl. Opt. 18, 3533 (1979).
    [CrossRef] [PubMed]
  15. P. K. Tien, Rev. Mod. Phys. 49, 361 (1977).
    [CrossRef]
  16. A. Sharma, E. Sharma, I. C. Goyal, A. K. Ghatak, Opt. Commun. 34, 39 (1980).
    [CrossRef]
  17. E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
    [CrossRef]
  18. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
  19. A. Kumar, K. Thyagarajan, A. K. Ghatak, IEEE J. Quantum Electron. QE-10, 902 (1974).
    [CrossRef]

1980 (1)

A. Sharma, E. Sharma, I. C. Goyal, A. K. Ghatak, Opt. Commun. 34, 39 (1980).
[CrossRef]

1979 (5)

J. N. Fields, Appl. Opt. 18, 3533 (1979).
[CrossRef] [PubMed]

G. A. Teh, V. C. Y. So, G. I. Stegman, IEEE Trans. Microwave Theory Tech. MTT-27, 265 (1979).
[CrossRef]

T. Suhara, Y. Handa, Nishihara, J. Koyama, J. Opt. Soc. Am. 69, 807 (1979).
[CrossRef]

A. Sharma, E. Khular, K. Thyagarajan, A. K. Ghatak, Opt. Commun. 30, 166 (1979).
[CrossRef]

R. C. Alferness, R. V. Schmidt, E. H. Turner, Appl. Opt. 18, 4012 (1979).
[CrossRef] [PubMed]

1978 (1)

1977 (3)

P. K. Tien, Rev. Mod. Phys. 49, 361 (1977).
[CrossRef]

H. P. Hsu, A. F. Milton, IEEE J. Quantum Electron. QE-13, 224 (1977).
[CrossRef]

Y. Suematsu, K. Kishino, Radio Sci. 12, 587 (1977).
[CrossRef]

1975 (3)

M. Papuchon, Y. Cobemale, X. Mathieu, D. B. Ostrowsky, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).
[CrossRef]

W. J. Tomlinson, H. P. Weber, C. A. Pryde, E. A. Chandross, Appl. Phys. Letts. 26, 303 (1975).
[CrossRef]

D. G. Dalgoutte, R. B. Smith, G. Achutaramayya, J. H. Harris, Appl. Opt. 14, 1860 (1975).
[CrossRef] [PubMed]

1974 (1)

A. Kumar, K. Thyagarajan, A. K. Ghatak, IEEE J. Quantum Electron. QE-10, 902 (1974).
[CrossRef]

1973 (1)

E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
[CrossRef]

1971 (1)

D. Marcuse, Bell. Syst. Tech. J. 50, 1791 (1971).

1969 (1)

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

Achutaramayya, G.

Alferness, R. C.

Chandross, E. A.

W. J. Tomlinson, H. P. Weber, C. A. Pryde, E. A. Chandross, Appl. Phys. Letts. 26, 303 (1975).
[CrossRef]

Chang, M. S.

H. P. Hsu, W. S. C. Chang, M. S. Chang, Wave Electron, 1, 191 (1974/1975).

Chang, W. S. C.

H. P. Hsu, W. S. C. Chang, M. S. Chang, Wave Electron, 1, 191 (1974/1975).

Cobemale, Y.

M. Papuchon, Y. Cobemale, X. Mathieu, D. B. Ostrowsky, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).
[CrossRef]

Conwell, E. M.

E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
[CrossRef]

Dalgoutte, D. G.

Fields, J. N.

Ghatak, A. K.

A. Sharma, E. Sharma, I. C. Goyal, A. K. Ghatak, Opt. Commun. 34, 39 (1980).
[CrossRef]

A. Sharma, E. Khular, K. Thyagarajan, A. K. Ghatak, Opt. Commun. 30, 166 (1979).
[CrossRef]

A. Kumar, K. Thyagarajan, A. K. Ghatak, IEEE J. Quantum Electron. QE-10, 902 (1974).
[CrossRef]

Goyal, I. C.

A. Sharma, E. Sharma, I. C. Goyal, A. K. Ghatak, Opt. Commun. 34, 39 (1980).
[CrossRef]

Handa, Y.

Harris, J. H.

Hsu, H. P.

H. P. Hsu, A. F. Milton, IEEE J. Quantum Electron. QE-13, 224 (1977).
[CrossRef]

H. P. Hsu, W. S. C. Chang, M. S. Chang, Wave Electron, 1, 191 (1974/1975).

Khular, E.

A. Sharma, E. Khular, K. Thyagarajan, A. K. Ghatak, Opt. Commun. 30, 166 (1979).
[CrossRef]

Kishino, K.

Y. Suematsu, K. Kishino, Radio Sci. 12, 587 (1977).
[CrossRef]

Kobayashi, M.

Koyama, J.

Kumar, A.

A. Kumar, K. Thyagarajan, A. K. Ghatak, IEEE J. Quantum Electron. QE-10, 902 (1974).
[CrossRef]

Marcatili, E. A. J.

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

Marcuse, D.

D. Marcuse, Bell. Syst. Tech. J. 50, 1791 (1971).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

Mathieu, X.

M. Papuchon, Y. Cobemale, X. Mathieu, D. B. Ostrowsky, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).
[CrossRef]

Milton, A. F.

H. P. Hsu, A. F. Milton, IEEE J. Quantum Electron. QE-13, 224 (1977).
[CrossRef]

Nishihara,

Ostrowsky, D. B.

M. Papuchon, Y. Cobemale, X. Mathieu, D. B. Ostrowsky, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).
[CrossRef]

Papuchon, M.

M. Papuchon, Y. Cobemale, X. Mathieu, D. B. Ostrowsky, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).
[CrossRef]

Pryde, C. A.

W. J. Tomlinson, H. P. Weber, C. A. Pryde, E. A. Chandross, Appl. Phys. Letts. 26, 303 (1975).
[CrossRef]

Reiber, L.

M. Papuchon, Y. Cobemale, X. Mathieu, D. B. Ostrowsky, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).
[CrossRef]

Roy, A. M.

M. Papuchon, Y. Cobemale, X. Mathieu, D. B. Ostrowsky, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).
[CrossRef]

Schmidt, R. V.

Sejourne, B.

M. Papuchon, Y. Cobemale, X. Mathieu, D. B. Ostrowsky, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).
[CrossRef]

Sharma, A.

A. Sharma, E. Sharma, I. C. Goyal, A. K. Ghatak, Opt. Commun. 34, 39 (1980).
[CrossRef]

A. Sharma, E. Khular, K. Thyagarajan, A. K. Ghatak, Opt. Commun. 30, 166 (1979).
[CrossRef]

Sharma, E.

A. Sharma, E. Sharma, I. C. Goyal, A. K. Ghatak, Opt. Commun. 34, 39 (1980).
[CrossRef]

Smith, R. B.

So, V. C. Y.

G. A. Teh, V. C. Y. So, G. I. Stegman, IEEE Trans. Microwave Theory Tech. MTT-27, 265 (1979).
[CrossRef]

Stegman, G. I.

G. A. Teh, V. C. Y. So, G. I. Stegman, IEEE Trans. Microwave Theory Tech. MTT-27, 265 (1979).
[CrossRef]

Suematsu, Y.

Y. Suematsu, K. Kishino, Radio Sci. 12, 587 (1977).
[CrossRef]

Suhara, T.

Teh, G. A.

G. A. Teh, V. C. Y. So, G. I. Stegman, IEEE Trans. Microwave Theory Tech. MTT-27, 265 (1979).
[CrossRef]

Terui, H.

Thyagarajan, K.

A. Sharma, E. Khular, K. Thyagarajan, A. K. Ghatak, Opt. Commun. 30, 166 (1979).
[CrossRef]

A. Kumar, K. Thyagarajan, A. K. Ghatak, IEEE J. Quantum Electron. QE-10, 902 (1974).
[CrossRef]

Tien, P. K.

P. K. Tien, Rev. Mod. Phys. 49, 361 (1977).
[CrossRef]

Tomlinson, W. J.

W. J. Tomlinson, H. P. Weber, C. A. Pryde, E. A. Chandross, Appl. Phys. Letts. 26, 303 (1975).
[CrossRef]

Turner, E. H.

Weber, H. P.

W. J. Tomlinson, H. P. Weber, C. A. Pryde, E. A. Chandross, Appl. Phys. Letts. 26, 303 (1975).
[CrossRef]

Werner, M.

M. Papuchon, Y. Cobemale, X. Mathieu, D. B. Ostrowsky, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (2)

E. M. Conwell, Appl. Phys. Lett. 23, 328 (1973).
[CrossRef]

M. Papuchon, Y. Cobemale, X. Mathieu, D. B. Ostrowsky, L. Reiber, A. M. Roy, B. Sejourne, M. Werner, Appl. Phys. Lett. 27, 289 (1975).
[CrossRef]

Appl. Phys. Letts. (1)

W. J. Tomlinson, H. P. Weber, C. A. Pryde, E. A. Chandross, Appl. Phys. Letts. 26, 303 (1975).
[CrossRef]

Bell. Syst. Tech. J. (2)

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

D. Marcuse, Bell. Syst. Tech. J. 50, 1791 (1971).

IEEE J. Quantum Electron. (2)

H. P. Hsu, A. F. Milton, IEEE J. Quantum Electron. QE-13, 224 (1977).
[CrossRef]

A. Kumar, K. Thyagarajan, A. K. Ghatak, IEEE J. Quantum Electron. QE-10, 902 (1974).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

G. A. Teh, V. C. Y. So, G. I. Stegman, IEEE Trans. Microwave Theory Tech. MTT-27, 265 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

A. Sharma, E. Sharma, I. C. Goyal, A. K. Ghatak, Opt. Commun. 34, 39 (1980).
[CrossRef]

A. Sharma, E. Khular, K. Thyagarajan, A. K. Ghatak, Opt. Commun. 30, 166 (1979).
[CrossRef]

Radio Sci. (1)

Y. Suematsu, K. Kishino, Radio Sci. 12, 587 (1977).
[CrossRef]

Rev. Mod. Phys. (1)

P. K. Tien, Rev. Mod. Phys. 49, 361 (1977).
[CrossRef]

Wave Electron (1)

H. P. Hsu, W. S. C. Chang, M. S. Chang, Wave Electron, 1, 191 (1974/1975).

Other (1)

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

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Figures (3)

Fig. 1
Fig. 1

Directional coupler formed by two identical graded-index waveguides having an exponential variation in refractive index and separated by a distance 2d [see Eq. (1)].

Fig. 2
Fig. 2

Variation of normalized propagation constants of the lowest order symmetric and antisymmetric modes of the directional coupler as a function of the normalized separation d/a between the two waveguides.

Fig. 3
Fig. 3

Variation of the normalized coupling coefficient κ ˜ as a function of normalized separation d/a between the two waveguides using the exact analysis (solid curve) and the coupled-mode analysis (dashed curve) for K0 = 2.25, K1 = 1.0, KN = 15.0 and (a) V = 2.6 and (b) V = 2.0.

Equations (23)

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K ( x ) = { K 0 + Δ K exp [ ( x d ) a ] x > d K 1 d > x > d K 0 + Δ K exp [ ( x + d ) a ] x < d ,
E y = F ( x ) exp [ i ( ω t β z ) ] ,
d 2 F d x 2 + [ K ( x ) ω 2 c 2 β 2 ] F = 0
F ( x ) = { A J 2 a p 0 ( ξ ) x > d B 1 exp ( p 1 x ) + B 2 exp ( p 1 x ) d < x < d C J 2 a p 0 ( η ) x < d ,
ξ = 2 V exp [ ( x d ) 2 a ] , η = 2 V exp [ ( x + d ) 2 a ] ,
V = a ω c Δ K ,
p 0 = β 2 ω 2 c 2 K 0 , p 1 = β 2 ω 2 c 2 K 1 ,
tanh p 1 d = V a p 1 J ( 2 V ) J 2 a p 0 ( 2 V ) ,
coth p 1 d = V a p 1 J ( 2 V ) J 2 a p 0 ( 2 V ) ,
L c = π β s β a ,
κ = π 2 L c = β s β a 2 .
K N = K 0 K 1 Δ K , d ˜ = d a ,
p ˜ 0 = p 0 a , p ˜ 1 = p 1 a ,
p ˜ 1 2 = p ˜ 0 2 + V 2 K N .
V p ˜ 1 J 2 p ˜ 0 ( 2 V ) J 2 p ˜ 0 ( 2 V ) = { tanh p ˜ 1 d ˜ symmetric coth p ˜ 1 d ˜ antisymmetric
κ = ω 0 4 P d [ K ( x ) K 1 ] E * y 1 E y 2 d x ,
p = β 0 2 ω μ 0 | E y | 2 d n ,
E y 1 = A J 2 p 0 ( ξ ) x > d = A J 2 p 0 ( 2 V ) exp [ p 2 ( x d ) a ] x < d ,
E y 2 = A J 2 p 0 ( η ) x < d = A J 2 p 0 ( 2 V ) exp [ p 2 ( x + d ) a ] x > d ,
p 0 = a β 0 2 ω 2 c 2 K 0 , p 2 = a β 0 2 ω 2 c 2 K 1 .
κ = κ 0 exp [ 2 p 2 d ˜ ] ,
κ 0 = k 0 2 Δ K 2 β 0 ( 2 V ) 2 p 2 J 2 p 0 ( 2 V ) [ K n 0 2 V J 2 p 0 ( ξ ) ξ 2 p 2 1 d ξ + ( 2 V ) 2 0 2 V J 2 p 0 ( ξ ) ξ 2 p 2 + 1 d ξ ] + [ J 2 2 p 0 ( 2 V ) ( p 0 + p 2 ) 4 p 0 p 2 + 1 p 0 0 2 V J 2 p 0 ( ξ ) J 2 p 0 + 1 ( ξ ) d ξ ] .
β ˜ ( = a β 2 ω 2 c 2 K 0 )

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