Abstract

We present analytical expressions for the effective indices of a three-waveguide directional coupler, using the modified scalar coupled-mode theory. Our analysis takes into account all perturbation correction terms, all coupling coefficients, and all overlap integrals and leads to a simple analytical condition for maximum power transfer between the two outside guides.

© 1995 Optical Society of America

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References

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  1. H. A. Haus, C. G. Fonstad, IEEE J. Quantum Electron. QE-17, 2321 (1981).
    [CrossRef]
  2. J. P. Donnelly, N. L. DeMeo, G. A. Ferrante, J. Lightwave Technol. LT-1, 417 (1983).
    [CrossRef]
  3. A. Hardy, W. Streifer, J. Lightwave Technol. LT-4, 90 (1986).
    [CrossRef]
  4. E. A. J. Marcatilli, IEEE J. Quantum Electron. QE-22, 988 (1986).
    [CrossRef]
  5. R. G. Peall, R. R. Syms, J. Lightwave Technol. 7, 540 (1989).
    [CrossRef]
  6. R. G. Peall, R. R. Syms, IEEE J. Quantum Electron. 25, 729 (1989).
    [CrossRef]
  7. J. P. Donnelly, H. A. Haus, N. Whitaker, IEEE J. Quantum Electron. QE-23, 401 (1987).
    [CrossRef]
  8. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York), Chap. 3, p. 17.

1989

R. G. Peall, R. R. Syms, J. Lightwave Technol. 7, 540 (1989).
[CrossRef]

R. G. Peall, R. R. Syms, IEEE J. Quantum Electron. 25, 729 (1989).
[CrossRef]

1987

J. P. Donnelly, H. A. Haus, N. Whitaker, IEEE J. Quantum Electron. QE-23, 401 (1987).
[CrossRef]

1986

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

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

1983

J. P. Donnelly, N. L. DeMeo, G. A. Ferrante, J. Lightwave Technol. LT-1, 417 (1983).
[CrossRef]

1981

H. A. Haus, C. G. Fonstad, IEEE J. Quantum Electron. QE-17, 2321 (1981).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York), Chap. 3, p. 17.

DeMeo, N. L.

J. P. Donnelly, N. L. DeMeo, G. A. Ferrante, J. Lightwave Technol. LT-1, 417 (1983).
[CrossRef]

Donnelly, J. P.

J. P. Donnelly, H. A. Haus, N. Whitaker, IEEE J. Quantum Electron. QE-23, 401 (1987).
[CrossRef]

J. P. Donnelly, N. L. DeMeo, G. A. Ferrante, J. Lightwave Technol. LT-1, 417 (1983).
[CrossRef]

Ferrante, G. A.

J. P. Donnelly, N. L. DeMeo, G. A. Ferrante, J. Lightwave Technol. LT-1, 417 (1983).
[CrossRef]

Fonstad, C. G.

H. A. Haus, C. G. Fonstad, IEEE J. Quantum Electron. QE-17, 2321 (1981).
[CrossRef]

Hardy, A.

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

Haus, H. A.

J. P. Donnelly, H. A. Haus, N. Whitaker, IEEE J. Quantum Electron. QE-23, 401 (1987).
[CrossRef]

H. A. Haus, C. G. Fonstad, IEEE J. Quantum Electron. QE-17, 2321 (1981).
[CrossRef]

Marcatilli, E. A. J.

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

Peall, R. G.

R. G. Peall, R. R. Syms, J. Lightwave Technol. 7, 540 (1989).
[CrossRef]

R. G. Peall, R. R. Syms, IEEE J. Quantum Electron. 25, 729 (1989).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York), Chap. 3, p. 17.

Streifer, W.

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

Syms, R. R.

R. G. Peall, R. R. Syms, J. Lightwave Technol. 7, 540 (1989).
[CrossRef]

R. G. Peall, R. R. Syms, IEEE J. Quantum Electron. 25, 729 (1989).
[CrossRef]

Whitaker, N.

J. P. Donnelly, H. A. Haus, N. Whitaker, IEEE J. Quantum Electron. QE-23, 401 (1987).
[CrossRef]

IEEE J. Quantum Electron.

H. A. Haus, C. G. Fonstad, IEEE J. Quantum Electron. QE-17, 2321 (1981).
[CrossRef]

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

R. G. Peall, R. R. Syms, IEEE J. Quantum Electron. 25, 729 (1989).
[CrossRef]

J. P. Donnelly, H. A. Haus, N. Whitaker, IEEE J. Quantum Electron. QE-23, 401 (1987).
[CrossRef]

J. Lightwave Technol.

R. G. Peall, R. R. Syms, J. Lightwave Technol. 7, 540 (1989).
[CrossRef]

J. P. Donnelly, N. L. DeMeo, G. A. Ferrante, J. Lightwave Technol. LT-1, 417 (1983).
[CrossRef]

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

Other

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York), Chap. 3, p. 17.

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Figures (4)

Fig. 1
Fig. 1

(a) Three coupled waveguides, WG1, WG2, and WG3, of widths d1, d2, and d3 as a seven-layered structure with waveguide separations s1 and s2. (b) Transverse refractive-index variations nI(x), nII(x), nIII(x), and n(x) of WG1, WG2, and WG3 and the coupled structure, respectively.

Fig. 2
Fig. 2

(a) Variation of effective index with normalized separation s/d for three identical coupled waveguides (d = 1.2 μm, n0 = 3.4406, ns = 3.4145). (b) Effect of various approximations on effective index (d = 1.2 μm, n0 = 3.4406, ns = 3.4145).

Fig. 3
Fig. 3

Modal field distribution for three identical coupled waveguides (d = 1.2 μm, s = 0.4 μm, n0 = 3.4406, ns = 3.4145).

Fig. 4
Fig. 4

Variation of Δ1 = neaneb, Δ2 = nebnec, and θ with Δn, where Δn is the change in refractive index of WG2, i.e., n4 = n0 + Δn (d = 1.2 μm, s = 0.6 μm, n2 = n6 = n0 = 3.4406, ns = 3.4145).

Equations (24)

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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 = I , II , III ,             j = 1 , 2 , 3.
ψ ( x , z ) = j = 1 3 A j ( z ) exp ( - i n e j k 0 z ) ψ j ( x ) ,
k = 1 3 e j k n e k d a k d z = - i k = 1 3 e j k n e k 2 a k k 0 - i k = 1 3 n e k κ j k a k k 0 ,
e j k = ψ j ψ k d x ,             e k j = e j k ,             e j j = 1 ,
κ j k = ( n 2 - n K 2 ( x ) ) ψ j ψ k d x 2 n e k ,             K = I , II , III .
a j = a j 0 exp ( - i k 0 n e z ) ,
k = 1 3 ( e j k n e - κ j k ) n e k a k 0 = 0 ,             j = 1 , 2 , 3 ,
n e j = n e j + κ j j ,             κ j k = n e k e j k + κ j k .
M j k = ( e j k n e - κ j k ) n e k ,             j , k = 1 , 2 , 3 ,
A n e 3 - B n e 2 + C n e + D = 0 ,
A = 1 - ( e 12 2 + e 23 2 + e 31 2 ) + 2 e 12 e 23 e 31 ,
B = n e 1 ( 1 - e 23 2 ) + n e 2 ( 1 - e 31 2 ) + n e 3 ( 1 - e 12 2 ) + 2 κ ˜ 12 ( e 23 e 31 - e 12 ) + 2 κ ˜ 23 ( e 31 e 12 - e 23 ) + 2 κ ˜ 31 ( e 12 e 23 - e 31 ) ,
C = n e 1 n e 2 + n e 2 n e 3 + n e 3 n e 1 - ( κ 12 κ 21 + κ 23 κ 32 + κ 31 κ 13 ) + e 12 ( κ 23 κ 31 + κ 32 κ 13 ) + e 23 ( κ 31 κ 12 + κ 21 κ 13 ) + e 31 ( κ 23 κ 12 + κ 32 κ 21 ) - 2 ( n e 1 e 23 κ ˜ 23 + n e 2 e 31 κ ˜ 31 + n e 3 e 12 κ ˜ 12 ) ,
D = - n e 1 n e 2 n e 3 - ( κ 12 κ 23 κ 31 + κ 21 κ 32 κ 13 ) + ( n e 1 κ 32 κ 23 + n e 2 κ 13 κ 31 + n e 3 κ 21 κ 12 ) ,
κ ˜ j k = κ j k + κ k j 2 .
n e a = 2 q cos θ 3 + B 3 A ,
n e b , c = B 3 A - q cos θ 3 ± 3 q sin θ 3 .
q = - ( C 3 A - B 2 9 A 2 ) ,
θ = tan - 1 ( p r ) ,
p = q 3 - r 2 ,
r = | ( B 3 A ) 3 - B C 6 A 2 - D 2 A | .
Δ 1 = 3 q cos θ 3 - 3 q sin θ 3 ,
Δ 2 = 2 3 q sin θ 3 .
tan θ 3 = 1 3             or             θ = π 2 .

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