Abstract

We present a complete analysis of power exchange in multiple coupled waveguides, using the modified scalar coupled-mode theory and give generalized expressions for power exchange between any two waveguides of a multiple-waveguide system. We thus obtain a reciprocity principle applicable to power exchange between any two waveguides. Completely generalized and accurate analytical expressions for all coupling coefficients, all perturbation correction terms, and all nonorthogonality terms involved in the analysis are presented.

© 1996 Optical Society of America

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  1. A. Hardy, W. Streifer, “Coupled modes of multi-waveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90–99 (1986).
    [CrossRef]
  2. A. Hardy, W. Streifer, “Coupled-mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. QE-22, 528–534 (1986).
    [CrossRef]
  3. S. L. Chuang, “Application of the strongly coupled mode theory to integrated optical devices,” IEEE J. Quantum Electron. QE-23, 499–509 (1987).
    [CrossRef]
  4. E. A. J. Marcatilli, “Improved coupled mode equations for dielectric guides,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
    [CrossRef]
  5. R. G. Peall, R. R. Syms, “Comparison between strongly coupling theory and experiment for three arm directional couplers in Ti:LiNbO3,” J. Lightwave Technol. 7, 540–554 (1989).
    [CrossRef]
  6. R. G. Peall, R. R. Syms, “Further evidence of coupling effects in three arm Ti:LiNbO3directional couplers,” IEEE J. Quantum Electron. 25, 729–735 (1989).
    [CrossRef]
  7. N. Gupta, E. K. Sharma, “Scalar coupled mode analysis: a comparison of various approximations,” J. Opt. Commun. 14, 28–31 (1993).
  8. D. W. Mills, L. S. Tamil, “Coupling in multilayer optical waveguides: an approach based on scattering data,” J. Lightwave Technol. 12, 1560–1568 (1994).
    [CrossRef]
  9. N. Gupta, E. K. Sharma, “Modified variational analysis for coupled optical waveguides,” J. Opt. Soc. Am. A 10, 1549–1552 (1993).
    [CrossRef]
  10. W. H. Press, S. A. Teuklosky, W. T. Vellertig, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press), pp. 38, 39, and 41.
  11. J. P. Donnelly, H. A. Haus, N. Whitaker, “Symmetric three guide coupler with nonidentical center and outside guides,” IEEE J. Quantum Electron. QE-23, 401–406 (1987).
    [CrossRef]
  12. E. K. Sharma, M. P. Singh, “Multilayer waveguide devices with absorbing layers: an exact analysis,” J. Opt. Commun. 14, 134–136 (1993).
  13. S. Srivastava, E. K. Sharma, “Analytical expressions for effective indices in a three-waveguide coupler,” Opt. Lett. 20, 1005–1007 (1995).
    [CrossRef] [PubMed]

1995 (1)

1994 (1)

D. W. Mills, L. S. Tamil, “Coupling in multilayer optical waveguides: an approach based on scattering data,” J. Lightwave Technol. 12, 1560–1568 (1994).
[CrossRef]

1993 (3)

N. Gupta, E. K. Sharma, “Scalar coupled mode analysis: a comparison of various approximations,” J. Opt. Commun. 14, 28–31 (1993).

E. K. Sharma, M. P. Singh, “Multilayer waveguide devices with absorbing layers: an exact analysis,” J. Opt. Commun. 14, 134–136 (1993).

N. Gupta, E. K. Sharma, “Modified variational analysis for coupled optical waveguides,” J. Opt. Soc. Am. A 10, 1549–1552 (1993).
[CrossRef]

1989 (2)

R. G. Peall, R. R. Syms, “Comparison between strongly coupling theory and experiment for three arm directional couplers in Ti:LiNbO3,” J. Lightwave Technol. 7, 540–554 (1989).
[CrossRef]

R. G. Peall, R. R. Syms, “Further evidence of coupling effects in three arm Ti:LiNbO3directional couplers,” IEEE J. Quantum Electron. 25, 729–735 (1989).
[CrossRef]

1987 (2)

J. P. Donnelly, H. A. Haus, N. Whitaker, “Symmetric three guide coupler with nonidentical center and outside guides,” IEEE J. Quantum Electron. QE-23, 401–406 (1987).
[CrossRef]

S. L. Chuang, “Application of the strongly coupled mode theory to integrated optical devices,” IEEE J. Quantum Electron. QE-23, 499–509 (1987).
[CrossRef]

1986 (3)

E. A. J. Marcatilli, “Improved coupled mode equations for dielectric guides,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
[CrossRef]

A. Hardy, W. Streifer, “Coupled modes of multi-waveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

A. Hardy, W. Streifer, “Coupled-mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. QE-22, 528–534 (1986).
[CrossRef]

Chuang, S. L.

S. L. Chuang, “Application of the strongly coupled mode theory to integrated optical devices,” IEEE J. Quantum Electron. QE-23, 499–509 (1987).
[CrossRef]

Donnelly, J. P.

J. P. Donnelly, H. A. Haus, N. Whitaker, “Symmetric three guide coupler with nonidentical center and outside guides,” IEEE J. Quantum Electron. QE-23, 401–406 (1987).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teuklosky, W. T. Vellertig, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press), pp. 38, 39, and 41.

Gupta, N.

N. Gupta, E. K. Sharma, “Modified variational analysis for coupled optical waveguides,” J. Opt. Soc. Am. A 10, 1549–1552 (1993).
[CrossRef]

N. Gupta, E. K. Sharma, “Scalar coupled mode analysis: a comparison of various approximations,” J. Opt. Commun. 14, 28–31 (1993).

Hardy, A.

A. Hardy, W. Streifer, “Coupled-mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. QE-22, 528–534 (1986).
[CrossRef]

A. Hardy, W. Streifer, “Coupled modes of multi-waveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

Haus, H. A.

J. P. Donnelly, H. A. Haus, N. Whitaker, “Symmetric three guide coupler with nonidentical center and outside guides,” IEEE J. Quantum Electron. QE-23, 401–406 (1987).
[CrossRef]

Marcatilli, E. A. J.

E. A. J. Marcatilli, “Improved coupled mode equations for dielectric guides,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
[CrossRef]

Mills, D. W.

D. W. Mills, L. S. Tamil, “Coupling in multilayer optical waveguides: an approach based on scattering data,” J. Lightwave Technol. 12, 1560–1568 (1994).
[CrossRef]

Peall, R. G.

R. G. Peall, R. R. Syms, “Comparison between strongly coupling theory and experiment for three arm directional couplers in Ti:LiNbO3,” J. Lightwave Technol. 7, 540–554 (1989).
[CrossRef]

R. G. Peall, R. R. Syms, “Further evidence of coupling effects in three arm Ti:LiNbO3directional couplers,” IEEE J. Quantum Electron. 25, 729–735 (1989).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teuklosky, W. T. Vellertig, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press), pp. 38, 39, and 41.

Sharma, E. K.

S. Srivastava, E. K. Sharma, “Analytical expressions for effective indices in a three-waveguide coupler,” Opt. Lett. 20, 1005–1007 (1995).
[CrossRef] [PubMed]

N. Gupta, E. K. Sharma, “Scalar coupled mode analysis: a comparison of various approximations,” J. Opt. Commun. 14, 28–31 (1993).

E. K. Sharma, M. P. Singh, “Multilayer waveguide devices with absorbing layers: an exact analysis,” J. Opt. Commun. 14, 134–136 (1993).

N. Gupta, E. K. Sharma, “Modified variational analysis for coupled optical waveguides,” J. Opt. Soc. Am. A 10, 1549–1552 (1993).
[CrossRef]

Singh, M. P.

E. K. Sharma, M. P. Singh, “Multilayer waveguide devices with absorbing layers: an exact analysis,” J. Opt. Commun. 14, 134–136 (1993).

Srivastava, S.

Streifer, W.

A. Hardy, W. Streifer, “Coupled modes of multi-waveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

A. Hardy, W. Streifer, “Coupled-mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. QE-22, 528–534 (1986).
[CrossRef]

Syms, R. R.

R. G. Peall, R. R. Syms, “Further evidence of coupling effects in three arm Ti:LiNbO3directional couplers,” IEEE J. Quantum Electron. 25, 729–735 (1989).
[CrossRef]

R. G. Peall, R. R. Syms, “Comparison between strongly coupling theory and experiment for three arm directional couplers in Ti:LiNbO3,” J. Lightwave Technol. 7, 540–554 (1989).
[CrossRef]

Tamil, L. S.

D. W. Mills, L. S. Tamil, “Coupling in multilayer optical waveguides: an approach based on scattering data,” J. Lightwave Technol. 12, 1560–1568 (1994).
[CrossRef]

Teuklosky, S. A.

W. H. Press, S. A. Teuklosky, W. T. Vellertig, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press), pp. 38, 39, and 41.

Vellertig, W. T.

W. H. Press, S. A. Teuklosky, W. T. Vellertig, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press), pp. 38, 39, and 41.

Whitaker, N.

J. P. Donnelly, H. A. Haus, N. Whitaker, “Symmetric three guide coupler with nonidentical center and outside guides,” IEEE J. Quantum Electron. QE-23, 401–406 (1987).
[CrossRef]

IEEE J. Quantum Electron. (5)

A. Hardy, W. Streifer, “Coupled-mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. QE-22, 528–534 (1986).
[CrossRef]

S. L. Chuang, “Application of the strongly coupled mode theory to integrated optical devices,” IEEE J. Quantum Electron. QE-23, 499–509 (1987).
[CrossRef]

E. A. J. Marcatilli, “Improved coupled mode equations for dielectric guides,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
[CrossRef]

R. G. Peall, R. R. Syms, “Further evidence of coupling effects in three arm Ti:LiNbO3directional couplers,” IEEE J. Quantum Electron. 25, 729–735 (1989).
[CrossRef]

J. P. Donnelly, H. A. Haus, N. Whitaker, “Symmetric three guide coupler with nonidentical center and outside guides,” IEEE J. Quantum Electron. QE-23, 401–406 (1987).
[CrossRef]

J. Lightwave Technol. (3)

A. Hardy, W. Streifer, “Coupled modes of multi-waveguide systems and phased arrays,” J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

D. W. Mills, L. S. Tamil, “Coupling in multilayer optical waveguides: an approach based on scattering data,” J. Lightwave Technol. 12, 1560–1568 (1994).
[CrossRef]

R. G. Peall, R. R. Syms, “Comparison between strongly coupling theory and experiment for three arm directional couplers in Ti:LiNbO3,” J. Lightwave Technol. 7, 540–554 (1989).
[CrossRef]

J. Opt. Commun. (2)

N. Gupta, E. K. Sharma, “Scalar coupled mode analysis: a comparison of various approximations,” J. Opt. Commun. 14, 28–31 (1993).

E. K. Sharma, M. P. Singh, “Multilayer waveguide devices with absorbing layers: an exact analysis,” J. Opt. Commun. 14, 134–136 (1993).

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Other (1)

W. H. Press, S. A. Teuklosky, W. T. Vellertig, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press), pp. 38, 39, and 41.

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Figures (6)

Fig. 1
Fig. 1

(a) N coupled waveguides, WGj of width dj with waveguide separation sj centered at x = δj. (b) Transverse refractive-index variation nj(x) and n(x) of WGi and of the coupled structure, respectively.

Fig. 2
Fig. 2

Variation of the effective index with normalized separation s/d for five identical coupled waveguides (d = 1.2 μm, nc = 3.4406, n0 = 3.4145).

Fig. 3
Fig. 3

Effect of various approximations on the outcoupled power for five identical coupled waveguides (d = 1.2 μm, s = 0.5 μm, nc = 3.4406, n0 = 3.4145).

Fig. 4
Fig. 4

Variation of power outcoupled versus device length l for a symmetric five-waveguide coupler (nc1 = nc3 = nc5 = 3.4406, nc2 = nc4 = 3.4405, n0 = 3.4145, d1 = d5 = 1.14 μm, d2 = d4 = 1.24 μm, d3 = 1.25 μm, s1 = s4 = 0.82 μm, and s2 = s3 = 0.67 μm).

Fig. 5
Fig. 5

Variation of power outcoupled versus device length l for a symmetric three-waveguide coupler (nc1 = nc3 = 3.4406, nc2 = 3.4419, n0 = 3.4145, d1 = d2 = d3 = 1.2 μm, s1 = s2 = 0.6 μm).

Fig. 6
Fig. 6

Variation of power outcoupled versus device length l for a three-waveguide coupler (nc1 = 3.4333, nc2 = 3.4406, nc3 = 3.4478, n0 = 3.4145, d1 = d2 = d3 = 1.2 μm, s1 = s2 = 0.6 μm).

Equations (50)

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d 2 ψ j d x 2 + k 0 2 ( n j 2 n e j 2 ) ψ j = 0 , j = 1 , , N ,
ψ j = B j exp γ j ( x δ j + d j / 2 ) , x < δ j d j / 2 , = B j cos α j ( x δ j ) / cos ( α j d j / 2 ) , δ j d j / 2 < x < δ j + d j / 2 , = B j exp γ j ( x δ j d j / 2 ) , x > δ j + d j / 2 ,
B j = ( d j / 2 + 1 / γ j ) 1 / 2 cos ( α j d j 2 ) ,
α j = k 0 n c j 2 n e j 2 ,
γ j = k 0 n e j 2 n 0 2 ,
δ j = d j 2 + k = 1 j 1 ( d k 1 + s k 1 ) for j > 1 δ 1 = d 1 / 2 .
ϕ ( x , z ) = j = 1 N A j ( z ) exp ( i n e j k 0 z ) ψ j ( x ) ,
k = 1 N e j k n e k d a ˜ k d z = i k = 1 N e j k n e k 2 a ˜ k k 0 i k = 1 N n e k κ j k a ˜ k k 0 ,
κ j k = [ n 2 n k 2 ( x ) ] ψ j ψ k d x 2 n e k ,
e j k = ψ j ψ k d x , e k j = e j k , e j j = 1 .
κ j k = u j k + ν j k , i k ,
κ j j = ν j j , j = k ,
u j k = B j B k exp ( r j k ) [ n c j 2 n 0 2 n e k ( γ k 2 + α j 2 ) ] [ γ k sinh ( γ k d j 2 ) + γ j cosh ( γ k d j 2 ) ]
ν j k = B j B k 2 n e k i j , k N d i ( n c i 2 n 0 2 ) exp ( r i j + r i k ) × sinch ( g i j + g i k 2 d i ) ,
r j k = ( d k / 2 | δ j δ k | ) γ k , g j k = ( k j ) | k j | γ k ,
Sinch ( x ) = sinh x x [ as x 0 , sinch ( x ) 1 ] .
e j k = exp ( r j k ) ( r j k / γ k + d j / 2 ) B j B k × exp ( γ k d j / 2 ) + w j k + w k j ,
w j k = exp [ γ k ( | δ j δ k | + d k d j 2 ) ] 2 γ k B k B j + 2 u j k n e k n c j 2 n 0 2 .
e j k = 2 k 0 2 ( n e k 2 n e j 2 ) ( w j k w k j ) ,
w j k = exp ( r k j ) [ γ k cosh ( γ j d k / 2 ) + γ j sinh ( γ j d k / 2 ) ] B j B k + ( n e k 2 n e j 2 ) k 0 2 n c j 2 n 0 2 u j k n e k .
a ˜ j = a j exp ( i k 0 n e z ) .
k = 1 N ( e j k n e κ j k ) n e k a k = 0 , j = 1 , , N .
κ j k = n e k e j k + κ j k ,
[ M ] [ a ] = 0 ,
m j k = ( e j k n e κ j k ) n e k , j , k = 1 , , N .
m j k = e j k n e 2 κ j k , j , k = 1 , , N ,
κ j k = n e k ( 2 κ j k + e j k n e k ) .
ϕ trial = j = 1 N a j ψ j ( x ) ,
| M | = 0.
[ M ˜ ] [ b ] = [ M ˜ 1 ] ,
M ˜ = [ m 12 m 13 m 1 N m 22 m 23 m 2 N m n 2 m n N ] , where n = N 1
b = [ b 2 b 3 b N ] , M ˜ 1 = [ m 11 m 21 m n 1 ] .
ϕ l ( x ) = a 1 l ( b 1 l ψ 1 + b 2 l ψ 2 + b N l ψ N ) × exp ( i n e l k 0 z ) ; b 1 l = 1 ,
a 1 l = 1 j , k = 1 N b j l b k l e j k
| ϕ l | 2 d x = 1.
ψ i ( x ) = l = 1 N C i l ϕ l ( x ) ,
C i l = 1 j , k 1 N b j l b k l e j k k = 1 N b k l e i k .
ψ T ( x , z ) = l = 1 N C i l ϕ l ( x ) exp ( i n e l k 0 z ) .
P i j = | ψ j ψ T | 2 d x ,
p i j = l , m = 1 N X l X m cos ( n e l n e m ) k 0 z ,
X l = C i l C j l .
P i j ( z ) = P j i ( z ) .
P 11 ( z = 0 ) = ( l = 1 N C 1 l 2 ) 2 ,
P 1 N ( z = 0 ) = ( l = 1 N C 1 l C N l ) 2 .
C N l = C 1 l , for the symmetric modes ( l = 1,3 , ) ,
C N l = C 1 l , for the antisymmetric modes ( l = 2,4 , ) ,
P 1 N ( z = 0 ) = [ l = 1 N ( 1 ) l + 1 C 1 l 2 ] 2 .
n e 1 n e 2 = n e 2 n e 3 = = n e ( N 1 ) n e N = Δ n e ,
P 11 ( z = l c ) = [ l = 1 N ( 1 ) l + 1 C 1 l 2 ] 2 = P 1 N ( z = 0 ) ,
P 1 N ( z = l c ) = ( l = 1 N C 1 l 2 ) 2 = P 11 ( z = 0 ) .

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