Abstract

Coupled-mode theory for parallel waveguides is extended to include systems with multimode guides. The basic set of coupled differential equations is similar to that for single-mode guides, but the matrices are more broadly defined to allow for coupling among all the modes. Cases of anisotropic dielectric waveguides and/or an anisotropic embedding medium are also considered.

© 1986 Optical Society of America

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References

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  1. A. Hardy, W. Streifer, Opt. Lett. 10, 335 (1985).
    [CrossRef] [PubMed]
  2. A. Hardy, W. Streifer, IEEE J. Lightwave Technol. LT-3, 1135 (1985).
    [CrossRef]
  3. A. Hardy, W. Streifer, IEEE J. Lightwave Technol. LT-4, 90 (1986).
    [CrossRef]
  4. A. Hardy, W. Streifer, IEEE J. Quantum Electron. QE-22, 528 (1986).
    [CrossRef]
  5. H. Kogelnik, in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1975), Chap. 2.
  6. E. A. J. Marcatili, IEEE J. Quantum Electron. QE-22, 988 (1986).
    [CrossRef]
  7. E. A. J. Marcatili, “Experimental verification of the improved coupled-mode equations for dielectric guides,” Appl. Phys. Lett. (to be published).
  8. H. A. Haus, W. P. Huang, S. Kawakami, N. A. Whitaker, “Coupled mode theory of optical waveguides,” IEEE J. Lightwave Technol. (to be published).
  9. S. L. Chuang, “New coupled mode formulation by reciprocity and a variational principle,” IEEE J. Lightwave Technol. (to be published).
  10. S. L. Chuang, “A coupled mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” IEEE J. Lightwave Technol. (to be published).
  11. T. Tamir, F. Y. Kou, IEEE J. Quantum Electron. QE-22, 544 (1986).
    [CrossRef]
  12. M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), Chap. 2.
  13. W. Streifer, M. Osiński, A. Hardy, “Reformulation of the coupled mode theory of multiwaveguide systems,” J. Lightwave Technol. (to be published).

1986 (4)

A. Hardy, W. Streifer, IEEE J. Lightwave Technol. LT-4, 90 (1986).
[CrossRef]

A. Hardy, W. Streifer, IEEE J. Quantum Electron. QE-22, 528 (1986).
[CrossRef]

E. A. J. Marcatili, IEEE J. Quantum Electron. QE-22, 988 (1986).
[CrossRef]

T. Tamir, F. Y. Kou, IEEE J. Quantum Electron. QE-22, 544 (1986).
[CrossRef]

1985 (2)

A. Hardy, W. Streifer, Opt. Lett. 10, 335 (1985).
[CrossRef] [PubMed]

A. Hardy, W. Streifer, IEEE J. Lightwave Technol. LT-3, 1135 (1985).
[CrossRef]

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), Chap. 2.

Chuang, S. L.

S. L. Chuang, “New coupled mode formulation by reciprocity and a variational principle,” IEEE J. Lightwave Technol. (to be published).

S. L. Chuang, “A coupled mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” IEEE J. Lightwave Technol. (to be published).

Hardy, A.

A. Hardy, W. Streifer, IEEE J. Lightwave Technol. LT-4, 90 (1986).
[CrossRef]

A. Hardy, W. Streifer, IEEE J. Quantum Electron. QE-22, 528 (1986).
[CrossRef]

A. Hardy, W. Streifer, IEEE J. Lightwave Technol. LT-3, 1135 (1985).
[CrossRef]

A. Hardy, W. Streifer, Opt. Lett. 10, 335 (1985).
[CrossRef] [PubMed]

W. Streifer, M. Osiński, A. Hardy, “Reformulation of the coupled mode theory of multiwaveguide systems,” J. Lightwave Technol. (to be published).

Haus, H. A.

H. A. Haus, W. P. Huang, S. Kawakami, N. A. Whitaker, “Coupled mode theory of optical waveguides,” IEEE J. Lightwave Technol. (to be published).

Huang, W. P.

H. A. Haus, W. P. Huang, S. Kawakami, N. A. Whitaker, “Coupled mode theory of optical waveguides,” IEEE J. Lightwave Technol. (to be published).

Kawakami, S.

H. A. Haus, W. P. Huang, S. Kawakami, N. A. Whitaker, “Coupled mode theory of optical waveguides,” IEEE J. Lightwave Technol. (to be published).

Kogelnik, H.

H. Kogelnik, in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1975), Chap. 2.

Kou, F. Y.

T. Tamir, F. Y. Kou, IEEE J. Quantum Electron. QE-22, 544 (1986).
[CrossRef]

Marcatili, E. A. J.

E. A. J. Marcatili, IEEE J. Quantum Electron. QE-22, 988 (1986).
[CrossRef]

E. A. J. Marcatili, “Experimental verification of the improved coupled-mode equations for dielectric guides,” Appl. Phys. Lett. (to be published).

Osinski, M.

W. Streifer, M. Osiński, A. Hardy, “Reformulation of the coupled mode theory of multiwaveguide systems,” J. Lightwave Technol. (to be published).

Streifer, W.

A. Hardy, W. Streifer, IEEE J. Quantum Electron. QE-22, 528 (1986).
[CrossRef]

A. Hardy, W. Streifer, IEEE J. Lightwave Technol. LT-4, 90 (1986).
[CrossRef]

A. Hardy, W. Streifer, IEEE J. Lightwave Technol. LT-3, 1135 (1985).
[CrossRef]

A. Hardy, W. Streifer, Opt. Lett. 10, 335 (1985).
[CrossRef] [PubMed]

W. Streifer, M. Osiński, A. Hardy, “Reformulation of the coupled mode theory of multiwaveguide systems,” J. Lightwave Technol. (to be published).

Tamir, T.

T. Tamir, F. Y. Kou, IEEE J. Quantum Electron. QE-22, 544 (1986).
[CrossRef]

Whitaker, N. A.

H. A. Haus, W. P. Huang, S. Kawakami, N. A. Whitaker, “Coupled mode theory of optical waveguides,” IEEE J. Lightwave Technol. (to be published).

IEEE J. Lightwave Technol. (2)

A. Hardy, W. Streifer, IEEE J. Lightwave Technol. LT-3, 1135 (1985).
[CrossRef]

A. Hardy, W. Streifer, IEEE J. Lightwave Technol. LT-4, 90 (1986).
[CrossRef]

IEEE J. Quantum Electron. (3)

A. Hardy, W. Streifer, IEEE J. Quantum Electron. QE-22, 528 (1986).
[CrossRef]

E. A. J. Marcatili, IEEE J. Quantum Electron. QE-22, 988 (1986).
[CrossRef]

T. Tamir, F. Y. Kou, IEEE J. Quantum Electron. QE-22, 544 (1986).
[CrossRef]

Opt. Lett. (1)

Other (7)

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), Chap. 2.

W. Streifer, M. Osiński, A. Hardy, “Reformulation of the coupled mode theory of multiwaveguide systems,” J. Lightwave Technol. (to be published).

E. A. J. Marcatili, “Experimental verification of the improved coupled-mode equations for dielectric guides,” Appl. Phys. Lett. (to be published).

H. A. Haus, W. P. Huang, S. Kawakami, N. A. Whitaker, “Coupled mode theory of optical waveguides,” IEEE J. Lightwave Technol. (to be published).

S. L. Chuang, “New coupled mode formulation by reciprocity and a variational principle,” IEEE J. Lightwave Technol. (to be published).

S. L. Chuang, “A coupled mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” IEEE J. Lightwave Technol. (to be published).

H. Kogelnik, in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1975), Chap. 2.

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Figures (2)

Fig. 1
Fig. 1

Cross section of two misoriented rectangular waveguides.

Fig. 2
Fig. 2

Schematic illustration of refractive-index squared and modal propagation constants in one-dimensional array of N parallel waveguides.

Equations (19)

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Δ ( m ) ( x , y ) = ( x , y ) - ( m ) ( x , y ) ,             m = 1 , , N
E t ( x , y , z ) = p = 1 N m = 1 n p u m ( p ) ( z ) E t m ( p ) ( x , y ) ,
E t m ( p ) ( x , y ) = ν = 1 C ν , m ( q , p ) E t ν ( q ) ( x , y ) ,
H t m ( p ) ( x , y ) = ν = 1 D ν , m ( q , p ) H t ν ( q ) ( x , y ) , q , p = 1 , , N ,             m = 1 , , n p .
2 z ^ · - [ E t μ ( p ) × H t ν ( p ) ] d x d y = δ μ ν ,             p = 1 , , N ,
C ν , m ( q , p ) = 2 z ^ · - [ E t m ( p ) × H t ν ( q ) ] d x d y ,
D ν , m ( q , p ) = C m , ν ( p , q ) , p , q = 1 , , N ,             m = 1 , , n p ,             ν = 1 , , .
P ν , m ( q , p ) = 1 2 [ C ν , m ( q , p ) + D ν , m ( q , p ) ] ,
K ˜ l , m ( q , p ) = ω - Δ ( q ) [ E t l ( q ) E t m ( p ) - ( p ) 0 n 2 E z l ( q ) E z m ( p ) ] d x d y , q , p = 1 , , N ,             l = 1 , , n q ,             m = 1 , , n p ,
B l , m ( q , p ) = β m ( p ) δ q p δ l m ,
d U d z = i [ P - 1 B P + P - 1 K ˜ ] U ,
U ( z ) = [ u 1 ( 1 ) , , u n 1 ( 1 ) , u 1 ( 2 ) , , u n 2 ( 2 ) , ] ,
P = [ P ( 1 , 1 ) P ( 1 , 2 ) P ( 1 , N ) P ( 2 , 1 ) P ( 2 , 2 ) P ( 2 , N ) P ( N , 1 ) P ( N , 2 ) P ( N , N ) ] ,
K ˜ = [ K ˜ ( 1 , 1 ) K ˜ ( 1 , 2 ) K ˜ ( 1 , N ) K ˜ ( 2 , 1 ) K ˜ ( 2 , 2 ) K ˜ ( 2 , N ) K ˜ ( N , 1 ) K ˜ ( N , 2 ) K ˜ ( N , N ) ] ,
B = [ B ( 1 , 1 ) 0 0 0 B ( 2 , 2 ) 0 0 0 B ( N , N ) ] .
B P + K ˜ = P B + K ,
d U d z = i [ B + P - 1 K ] U .
( x , y ) = [ 11 ( x , y ) 12 ( x , y ) 0 21 ( x , y ) 22 ( x , y ) 0 0 0 33 ( x , y ) ] .
K ˜ l m ( q , p ) = ω - { E t l ( q ) [ Δ ( q ) E t m ( p ) - E z l ( q ) [ Δ ( q ) - 1 ( p ) E z m ( p ) ] } d x d y = ω - E t l ( q ) [ Δ ( q ) E t m ( p ) ] d x d y - ω - [ Δ ( q ) ] 33 [ ( p ) ] 33 [ ( x , y ) ] 33 E z l ( q ) E z m ( p ) d x d y ,

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