Abstract

Based on the solution of a simplified canonical problem, the possible margin of error on the predicted coupling length according to the coupled-mode theory as applied to fiber and integrated optical guides is inferred. It was found that coupling length obtained according to the coupled-mode theory is usually accurate to within 20% of the actual value provided that the frequency of operation is above the cutoff frequency of the antisymmetric mode of the coupled structure.

© 1978 Optical Society of America

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References

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  1. D. Marcuse, Light Transmission Optics, (Van Nostrand–Reinhold, New York, 1972, pp. 417–421.
  2. D. B. Hall, “Frequency Selective Coupling between Planar Waveguides,” NELC Technical Note TN-2583 (1974), Naval Electronics Laboratory Center, San Diego, Calif. 92152.
  3. S. E. Miller, Bell Syst. Tech. J. 33, 661 (1954).
    [Crossref]

1954 (1)

S. E. Miller, Bell Syst. Tech. J. 33, 661 (1954).
[Crossref]

Hall, D. B.

D. B. Hall, “Frequency Selective Coupling between Planar Waveguides,” NELC Technical Note TN-2583 (1974), Naval Electronics Laboratory Center, San Diego, Calif. 92152.

Marcuse, D.

D. Marcuse, Light Transmission Optics, (Van Nostrand–Reinhold, New York, 1972, pp. 417–421.

Miller, S. E.

S. E. Miller, Bell Syst. Tech. J. 33, 661 (1954).
[Crossref]

Bell Syst. Tech. J. (1)

S. E. Miller, Bell Syst. Tech. J. 33, 661 (1954).
[Crossref]

Other (2)

D. Marcuse, Light Transmission Optics, (Van Nostrand–Reinhold, New York, 1972, pp. 417–421.

D. B. Hall, “Frequency Selective Coupling between Planar Waveguides,” NELC Technical Note TN-2583 (1974), Naval Electronics Laboratory Center, San Diego, Calif. 92152.

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Figures (9)

FIG. 1
FIG. 1

Geometry of the canonical problem.

FIG. 2
FIG. 2

Normalized propagation constants as a function of normalized frequencies. The lowest-order mode n = 1 has zero cutoff frequency.

FIG. 3
FIG. 3

Normalized propagation constants as a function of normalized frequencies. All antisymmetric modes have cutoff frequencies.

FIG. 4
FIG. 4

Normalized power in the core region of guides A and B as a function of the normalized longitudinal distance. The coupling length (defined as the length for which complete exchange of power in the cores of guide A and guide B occurs) is longer for less tightly bounded fields.

FIG. 5
FIG. 5

Normalized power in the core region of guides A and B as a function of the normalized longitudinal distance. PC is the power in the core region of guide A or guide B as appropriate. PT is the total guided power.

FIG. 6
FIG. 6

Maximum normalized power in the core region of guide A as a function of separation distance of the two guides.

FIG. 7
FIG. 7

Transverse electric field distribution across the two coupled guides.

FIG. 8
FIG. 8

Normalized coupling length as a function of the normalized separation distance. Note that the coupling length ceases to exist for k0d < 7.0 when k0a = 1.0 according to the normal-mode theory.

FIG. 9
FIG. 9

Percent coupling length differences between normal-mode theory and coupled-mode theory as a function of normalized separation distance.

Equations (28)

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E x = - 1 j ω H y z ,
E z = 1 j ω H y x .
2 H y x 2 + 2 H y z 2 + k i 2 H y = 0 ,
Ê x = ( β / ω ) Ĥ y ,
Ê z = 1 j ω d Ĥ y d x ,
d 2 Ĥ y d x 2 + ( k i 2 - β 2 ) Ĥ y = 0
- x - a - d / z :             Ĥ y n ( 1 ) ( x ) = A n exp [ v n ( x + a + d / 2 ) ] ;
- a - d / 2 x - d / 2 :             Ĥ y n ( 2 ) ( x ) = B n sin u n ( x + d / 2 ) + C n cos u n ( x + d / 2 ) ;
- d / 2 x d / 2 :             Ĥ y n ( 3 ) ( x ) = D n sinh v n x + E n cosh v n x ;
d / 2 x a + d / 2 :             Ĥ y n ( 4 ) ( x ) = F n sin u n ( x - d / 2 ) + G n cos u n ( x - d / 2 ) ;
a + d / 2 x :             Ĥ y n ( 5 ) ( x ) = H n exp [ - v n ( x - a - d / 2 ) ] ;
[ tan u n a - 2 q u n v n q 2 u n 2 - v n 2 + ( q 2 u n 2 + v n 2 ) exp ( - v n d ) ] × [ tan u n a - 2 q u n v n q 2 u n 2 - v n 2 - ( q 2 u n 2 + v n 2 + v n 2 ) exp ( - v n d ) ] = 0
b n e = f n e = ( 1 / Δ n e ) ( v n / 2 ) cosh ( v n d / 2 ) ;
c n e = - g n e = - ( 1 / Δ n e ) ( u n / 1 ) sinh ( v n d / 2 ) ;
d n e = 1 Δ n e u n 1 ,             e n e = 0 ,             h n e = - 1 ;
Δ n e = ( - v n / 2 ) cosh ( v n d / 2 ) sin u n a - ( u n / 1 ) sinh ( v n d / 2 ) cos u n a .
b n o = - f n o = ( 1 / n o ) ( v n / 2 ) sinh ( v n d / 2 ) ;
c n o = g n o = - ( 1 / Δ n o ) ( u n / 1 ) cosh ( v n d / 2 ) ;
d n o = 0 ,             e n o = - 1 Δ n o u n 1 ,             h n o = 1 ;
Δ n o = - ( v n / 2 ) sinh ( v n d / 2 ) sin u n a - ( u n / 1 ) cosh ( v n d / 2 ) cos u n a .
E 0 [ U ( x - d / 2 ) - U ( x - a - d / 2 ) ] = n - β n ω ( x ) Ĥ y n ( x ) + ( radiated field ) x = 0 ,
- 1 ( x ) Ĥ y ( x ; β ) Ĥ y * ( x ; β ) d x = 0             ( β β ) ,
- ω β n E 0 d / 2 a + d / 2 Ĥ y n ( 4 ) * ( x ) d x = 1 2 - - a - d / 2 Ĥ y n ( 1 ) ( x ) 2 d x + 1 1 - a - d / 2 - d / 2 Ĥ y n ( 2 ) ( x ) 2 d x + 1 2 - d / 2 d / 2 Ĥ y n ( 3 ) ( x ) 2 d x + 1 1 d / 2 a + d / 2 Ĥ y n ( 4 ) ( x ) 2 d x + 1 2 a + d / 2 Ĥ y n ( 5 ) ( x ) 2 d x + ( radiated fields ) .
A n = - ω E o u n β n [ f n * ( 1 - cos u n a ) + g n * sin u n a ] × { 1 v n 2 + 1 u n 1 [ g n 2 ( u n a + ½ sin 2 u n a ) + f n 2 ( u n a - ½ sin 2 u n a ) + ( f n g n * + f n * g n ) sin 2 u n a ] + u n 2 2 Δ n 2 v n 1 2 2 ( sinh v n d ± v n d ) } - 1
( 2 a / q d ) < u a tan u a < ,
P z A ( x , z ) = 1 2 ω Re { 1 1 [ β e Ĥ y e ( 4 ) ( x ) 2 + β o Ĥ y o ( 4 ) ( x ) 2 + β e Ĥ y e ( 4 ) ( x ) Ĥ y o ( 4 ) * ( x ) e - j ( β e - β o * ) z + β o Ĥ y o ( 4 ) ( x ) Ĥ y e ( 4 ) * ( x ) e j ( β e * - β o ) z ] } .
P z B ( x , z ) = 1 2 ω Re { 1 1 [ β e Ĥ y e ( 2 ) ( x ) 2 + β o Ĥ y o ( 2 ) ( x ) 2 + β e Ĥ y e ( 2 ) ( x ) Ĥ y e ( 2 ) * ( x ) e - j ( β e - β o * ) z + β o Ĥ y o ( 2 ) ( x ) Ĥ y e ( 2 ) * ( x ) e - j ( β e * - β o ) z ] } .
P A ( z ) = d / 2 a + d / 2 P z A ( x , z ) d x , P B ( z ) = - a - d / 2 - d / 2 P z B ( x , z ) d x .