Abstract

Two theoretical formalisms, the most widely used version of coupled-mode theory and a first-order boundary-perturbation theory, are used to analyze the interaction between optical guided waves and surface roughness on an asymmetric planar waveguide. Each formalism is applied to two distinct problems for both TE and TM guided waves: (1) the determination of the far-field radiation patterns from randomly rough surfaces (with explicit results for the case of a sinusoidal roughness); and (2) the determination of guided-wave reflectivities from waveguide diffraction gratings. In all cases, the theoretical predictions agree for the TE polarization but disagree significantly for the TM polarization.

© 1984 Optical Society of America

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References

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  1. D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).
  2. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  3. H. Kogelnik, “Theory of Dielectric Waveguides,” in Integrated Optics, T. Tamir, Ed. (Springer, New York, 1979).
  4. Y. Suematsu, K. Furuya, M. Hakuta, K. Chiba, Electron. Commun. Jpn. 56-C, 62 (1973).
  5. D. G. Hall, Appl. Opt. 19, 1732 (1980).
    [Crossref] [PubMed]
  6. G. I. Stegeman, D. Sarid, J. J. Burke, D. G. Hall, J. Opt. Soc. Am. 71, 1497 (1981).
    [Crossref]
  7. T. Tsai, H. S. Tuan, IEEE J. Quantum Electron. QE-10, 326 (1974).
    [Crossref]
  8. D. Marcuse, Bell Syst. Tech. J. 48, 3233 (1969).
  9. Y. M. Chen, J. Math. Phys. 9, 439 (1968).
    [Crossref]
  10. E. Bradley, D. G. Hall, Opt. Lett. 7, 235 (1982).
    [Crossref] [PubMed]

1982 (1)

1981 (1)

1980 (1)

1974 (1)

T. Tsai, H. S. Tuan, IEEE J. Quantum Electron. QE-10, 326 (1974).
[Crossref]

1973 (1)

Y. Suematsu, K. Furuya, M. Hakuta, K. Chiba, Electron. Commun. Jpn. 56-C, 62 (1973).

1969 (2)

D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).

D. Marcuse, Bell Syst. Tech. J. 48, 3233 (1969).

1968 (1)

Y. M. Chen, J. Math. Phys. 9, 439 (1968).
[Crossref]

Bradley, E.

Burke, J. J.

Chen, Y. M.

Y. M. Chen, J. Math. Phys. 9, 439 (1968).
[Crossref]

Chiba, K.

Y. Suematsu, K. Furuya, M. Hakuta, K. Chiba, Electron. Commun. Jpn. 56-C, 62 (1973).

Furuya, K.

Y. Suematsu, K. Furuya, M. Hakuta, K. Chiba, Electron. Commun. Jpn. 56-C, 62 (1973).

Hakuta, M.

Y. Suematsu, K. Furuya, M. Hakuta, K. Chiba, Electron. Commun. Jpn. 56-C, 62 (1973).

Hall, D. G.

Kogelnik, H.

H. Kogelnik, “Theory of Dielectric Waveguides,” in Integrated Optics, T. Tamir, Ed. (Springer, New York, 1979).

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).

D. Marcuse, Bell Syst. Tech. J. 48, 3233 (1969).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Sarid, D.

Stegeman, G. I.

Suematsu, Y.

Y. Suematsu, K. Furuya, M. Hakuta, K. Chiba, Electron. Commun. Jpn. 56-C, 62 (1973).

Tsai, T.

T. Tsai, H. S. Tuan, IEEE J. Quantum Electron. QE-10, 326 (1974).
[Crossref]

Tuan, H. S.

T. Tsai, H. S. Tuan, IEEE J. Quantum Electron. QE-10, 326 (1974).
[Crossref]

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).

D. Marcuse, Bell Syst. Tech. J. 48, 3233 (1969).

Electron. Commun. Jpn. (1)

Y. Suematsu, K. Furuya, M. Hakuta, K. Chiba, Electron. Commun. Jpn. 56-C, 62 (1973).

IEEE J. Quantum Electron. (1)

T. Tsai, H. S. Tuan, IEEE J. Quantum Electron. QE-10, 326 (1974).
[Crossref]

J. Math. Phys. (1)

Y. M. Chen, J. Math. Phys. 9, 439 (1968).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Other (2)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

H. Kogelnik, “Theory of Dielectric Waveguides,” in Integrated Optics, T. Tamir, Ed. (Springer, New York, 1979).

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Figures (2)

Fig. 1
Fig. 1

Geometry for the calculation.

Fig. 2
Fig. 2

Normalized TM angular prefactors A(α) predicted by IM-CMT and BVT for an asymmetric waveguide with the parameters nc = 1, nf = 1.56, ns = 1.47, h = 0.35 μm, and λ = 0.84 μm.

Equations (57)

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E y = n = 0 C n ( z ) ξ n ( x , z ) + Σ 0 g ( z , Δ ) ξ ( x , z , Δ ) d Δ
H y = n = 0 b n ( z ) h n ( x , z ) + Σ 0 j ( z , Δ ) h ( x , z . Δ ) d Δ ,
ξ n ( x , z ) = exp [ i ( β n z - ω t ) ] { A exp ( - γ c x ) , x 0 A cos ( k x ) + B sin ( k x ) , 0 x - h [ A cos ( k h ) - B sin ( k h ) ] exp [ γ s ( x + h ) ] , x - h ,
ξ ( x , z , Δ ) = C r exp [ i ( β z - ω t ) ] { cos ( Δ x ) + ( σ / Δ ) F i sin ( Δ x ) , x 0 cos ( σ x ) + F i sin ( σ x ) , 0 x - h [ cos ( σ h ) - F i sin ( σ h ) ] cos [ ρ ( x + h ) ] + σ / ρ [ sin σ h ) + F i cos ( σ h ) ] sin [ ρ ( x + h ) ] , x - h
h n ( x , z ) = β n β n exp [ i ( β n z - ω t ) ] { C exp ( - γ c x ) , x 0 C cos ( k x ) + D sin ( k x ) , 0 x - h [ C cos ( k h ) - D sin ( k h ) ] exp [ γ s ( x + h ) ] , x - h
h ( x , z , Δ ) = β β S r exp [ i ( β z - ω t ) ] { cos ( Δ x ) + ( n c / n f ) 2 ( σ / Δ ) R i sin ( Δ x ) , x 0 cos ( σ x ) + R i sin ( σ x ) , 0 x - h { [ cos ( σ h ) - R i sin ( σ h ) ] cos [ ρ ( x + h ) ] + ( n s / h f ) 2 ( σ / ρ ) [ sin ( σ h ) + R i cos ( σ h ) ] · sin [ ρ ( x + h ) ] } , x - h
n ( x ) = { n c for x 0 , n f for 0 x - h , n s for x - h .
[ 2 x 2 + 2 z 2 + { n 2 ( x ) + δ n 2 } k 0 2 ] E y = 0 ,
[ 2 x 2 + 2 z 2 + { n 2 ( x ) + δ n 2 } k 0 2 ] H y = 0 ,
2 g ( z , Δ ) z 2 + 2 i β g ( z , Δ ) z = G ( z , Δ ) ,
2 j ( z , Δ ) z 2 + 2 i β j ( z , Δ ) z = J ( z , Δ ) ,
G ( z , Δ ) = - β k 0 2 2 ω μ P ¯ [ n = 0 c n ( z ) - d x ξ n ( x , z ) δ n 2 ξ * ( x , z , Δ ) + 0 d Δ g ( z , Δ ) - d x ξ ( x , z , Δ ) δ n 2 ξ * ( x , z , Δ ) ] ,
J ( z , Δ ) = - β k 0 2 2 ω 0 P ¯ [ n = 0 b n ( z ) - d x h n ( x , z ) δ n 2 h * ( x , z , Δ ) n 2 ( x ) + 0 d Δ j ( z , Δ ) - d x h ( x , z , Δ ) δ n 2 h * ( x , z , Δ ) n 2 ( x ) ] .
g ( z , Δ ) = ( 2 i β ) - 1 0 z d z G ( z , Δ ) ,
j ( z , Δ ) = ( 2 i β ) - 1 0 z d z J ( z , Δ ) .
G ( z , Δ ) = - β k 0 2 2 ω μ P ¯ - d x ξ 0 ( x , z ) δ n 2 ξ * ( x , z , Δ ) ,
J ( z , Δ ) = - β k 0 2 2 ω 0 P ¯ - d x h 0 ( x , z ) δ n 2 h * ( x , z , Δ ) n 2 ( x ) .
g ( L , Δ ) = k 0 2 ( n f 2 - n c 2 ) A C r 4 i ω μ P ¯ ϕ ( β 0 - β ) ,
j ( L , Δ ) = k 0 2 ( n f 2 - n c 2 ) C S r 4 i ω 0 P ¯ ϕ ( β 0 - β ) ,
ϕ ( β 0 - β ) = 0 L d z f ( z ) exp [ i ( β 0 - β ) z ] ,
ϕ ( β 0 - β ) = 0 L d z f ( z ) n 2 ( x , z ) exp [ i ( β 0 - β ) z ] .
E y = A k 0 2 ( n f 2 - n c 2 ) 8 i ω μ P ¯ 0 d Δ [ C r 1 2 ( 1 - i σ Δ F 1 ) + C r 2 2 ( 1 - i σ Δ F 2 ) ] ϕ ( β 0 - β ) exp [ i ( β z + Δ x - ω t ) ] ,
H y = C k 0 2 ( n f 2 - n c 2 ) 8 i ω 0 P ¯ 0 d Δ [ S r 1 2 { 1 - i ( n c n f ) 2 ( σ Δ ) R 1 } + S r 2 2 { 1 - i ( n c n f ) 2 ( σ Δ ) R 2 } ] ϕ ( β 0 - β ) exp [ i ( β z + Δ x - ω t ) ] .
[ C r 1 2 ( 1 - i σ Δ F 1 ) + C r 2 2 ( 1 - i σ Δ F 2 ) ] 0 d Δ e i ( β z + Δ x - ω t ) ,
[ S r 1 2 { 1 - i ( n c n f ) 2 ( σ Δ ) R 1 } + S r 2 2 { 1 - i ( n c n f ) 2 ( σ Δ ) R 2 } ] × 0 d Δ e i ( β z + Δ x - ω t ) ,
0 d Δ e i ( β z + Δ x ) = 1 r 2 π k 0 n c ( cos ( α ) ) × ( e - i π 4 ) ( e i n c k 0 [ x sin ( α ) + z cos ( α ) ] ) ,
E y 2 = A 2 k 0 5 n c ( n f 2 - n c 2 ) 2 sin 2 α 2 π r | cos ( σ h ) - i ( ρ / σ ) sin ( σ h ) ( Δ + ρ ) cos ( σ h ) { 1 - i ( σ 2 + Δ ρ ) tan ( σ h ) σ Δ + σ ρ } | 2 · | 0 L d z f ( z ) exp [ i ( β 0 - β ) z ] | 2 ,
H y 2 = C 2 k 0 5 n c ( n f 2 - n c 2 ) sin 2 α 2 π r | cos ( σ h ) - i ( n f / n s ) 2 ( ρ / σ ) sin ( σ h ) ( Δ n c 2 + ρ n s 2 ) cos ( σ h ) { 1 - i ( σ 2 n f 2 + Δ ρ n c 2 n s 2 ) tan ( σ h ) σ Δ n 2 f 2 n c 2 + σ ρ n f 2 n s 2 } | 2 · | 0 L d z f ( z ) n 2 ( x , z ) exp [ i ( β 0 - β ) z ] | .
[ n 2 ( x ) + δ n 2 ] E z = n 2 ( x ) n = 0 C n ξ n z ,
E y 2 = A 2 k 0 5 n c ( n f 2 - n c 2 ) 2 sin 2 ( α ) 2 π r | cos ( σ h ) - i ( ρ / σ ) sin ( σ h ) ( Δ + ρ ) cos ( σ h ) { 1 - i ( σ 2 + Δ ρ ) tan ( σ h ) σ Δ + σ ρ } | · | 0 L d z f ( z ) exp [ i ( β 0 - β ) z ] | 2 ,
H y 2 = C 2 k 0 ( n f 2 - n c 2 ) 2 sin 2 α 2 π r n c 4 n f 4 · | ( β β 0 + i ρ γ c [ n f / n s ] 2 ) cos ( σ h ) + ( γ c σ - i ( ρ / σ ) β β 0 [ n f / n s ] 2 ) sin ( σ h ) ( Δ n c 2 + ρ n s 2 ) cos ( σ h ) { 1 - i ( σ 2 n f 4 + Δ ρ n c 2 n s 2 ) tan ( σ h ) Δ σ n c 2 n f 2 + ρ σ n s 2 n f 2 } | 2 · | 0 L d z f ( z ) exp [ i ( β 0 - β ) z ] | 2 .
0 L d z f ( z ) exp [ i ( β 0 - β ) z ] = - a L 2 i δ β 0 , β + 2 π / Λ
d z f ( z ) n 2 ( x , z ) exp [ i ( β 0 - β ) z ] = - a L 4 i ( 1 n f 2 + 1 n c 2 ) δ β 0 , β + 2 π / Λ
k = π λ a h eff n f 2 - N 2 N ,             N = β / k 0 , and h eff = h + 1 γ c + 1 γ s .
k = π 2 λ a h eff ( n f 2 - N 2 N 2 ) ( n f 2 n c 2 + n c 2 n f 2 ) ( N 2 n f 2 - N 2 n c 2 + 1 ) ( N 2 n f 2 + N 2 n c 2 - 1 ) ,
k = π λ a h eff ( n f 2 - N 2 h eff ) ( N 2 n f 2 - N 2 N c 2 + 1 ) ( N 2 n f 2 + N 2 n c 2 - 1 ) .
1 2 ( n f 2 n c 2 + n c 2 n f 2 ) ,
C r 2 = 4 ω μ P ¯ π β [ ( cos ( σ h ) - F i sin ( σ h ) ) 2 + ( σ ρ ) 2 · ( sin ( σ h ) + F i cos ( σ h ) ) 2 + ( 1 + ( σ Δ ) 2 F i 2 ) Δ ρ ] - 1 ,
S r 2 = 4 π 0 P ¯ π β [ ( 1 n s ) 2 ( cos ( σ h ) - R i sin ( σ h ) ) 2 + ( n s n f 2 ) 2 · ( σ ρ ) 2 · ( sin ( σ h ) + R i cos ( σ h ) ) 2 + ( 1 n c 2 + n c 2 n f 4 ( σ Δ ) 2 · R i 2 ) Δ ρ - 1 ] ,
F 1 , 2 = [ ( σ 2 - ρ 2 ) sin ( 2 σ h ) ] - 1 [ ( σ 2 - ρ 2 ) cos ( 2 σ h ) + ρ Δ ( σ 2 - Δ 2 ) ± { ( σ 2 - ρ 2 ) 2 + 2 ρ Δ ( σ 2 - ρ 2 ) ( σ 2 - Δ 2 ) cos ( 2 σ h ) + ( ρ Δ ) 2 · ( σ 2 - Δ 2 ) 2 } 1 / 2 ] ,
R 1 , 2 = [ ( n s 4 σ 2 - n f 4 ρ 2 ) sin ( 2 σ h ) ] - 1 [ ( n s 4 σ 2 - n f 4 ρ 2 ) cos ( 2 σ h ) + ( n s n c ) 2 · ρ Δ · ( n c 4 σ 2 - n f 4 Δ 2 ) ± { ( n s 4 σ 2 - n f 4 ρ 2 ) 2 + ( n s n c ) 4 · ( ρ Δ ) 2 · ( n c 4 σ 2 - n f 4 Δ 2 ) 2 + 2 ( n s n c ) 2 · ( ρ Δ ) · ( n s 4 σ 2 - n f 4 ρ 2 ) ( n c 4 σ 2 - n f 4 Δ 2 ) cos ( 2 σ h ) } 1 / 2 ] ,
β β P ¯ δ ( Δ - Δ ) = β 2 μ ω - d x ξ ( x , z , Δ ) ξ * ( x , z , Δ ) ,
β β P ¯ δ ( Δ - Δ ) = β 2 0 ω - d x h ( x , z , Δ ) h * ( x , z , Δ ) n 2 ( x ) ,
4 ω μ P ¯ π β { Δ 2 ρ 2 - i σ Δ ρ 2 F 1 Δ 2 [ F 1 2 ( B cos 2 ( σ h ) + ( ρ / Δ ) A ) + 2 B F 1 sin ( σ h ) cos ( σ h ) + ( A - B cos 2 ( σ h ) ) ] + Δ 2 ρ 2 - i σ Δ ρ 2 F 2 Δ 2 [ F 2 2 ( B cos 2 ( σ h ) + ( ρ / Δ ) A ) + 2 B F 2 sin ( σ h ) cos ( σ h ) + ( A - B cos 2 ( σ h ) ) ] } ,
numerator = Δ ρ { Δ ρ ( F 1 2 + F 2 2 ) ( Δ 2 B cos 2 ( σ h ) + Δ ρ A ) + 2 Δ 3 ρ B ( F 1 + F 2 ) sin ( σ h ) cos ( σ h ) + 2 Δ 3 ρ ( A - B cos 2 ( σ h ) ) + i σ ρ [ ( F 1 + F 2 ) ( Δ 2 B cos 2 ( σ h ) + ρ Δ A - Δ 2 A + Δ 2 B cos 2 ( σ h ) ) + 4 Δ 2 B sin ( σ h ) cos ( σ s ) ] } , and
denominator = Δ 2 { ( ρ 2 + Δ 2 ) A 2 + 2 Δ ρ B C cos 2 ( σ h ) - 6 Δ 2 B 2 sin 2 ( σ h ) cos 2 ( σ h ) + 2 Δ B ( F 1 + F 2 ) sin ( σ h ) cos ( σ h ) [ A ( Δ - ρ ) - 2 Δ B cos 2 ( σ h ) ] + Δ ( F 1 2 + F 2 2 ) [ ρ A 2 - ρ B C cos 2 ( σ h ) + Δ B 2 sin 2 ( σ h ) cos 2 ( σ h ) ] } .
F 1 + F 2 = 2 Δ B cos 2 ( σ h ) - A ( Δ - ρ ) Δ B sin ( σ h ) cos ( σ h ) , F 1 2 + F 2 2 = A 2 ( Δ - ρ ) 2 + 2 Δ B cos 2 ( σ h ) [ 2 ρ C - Δ B sin 2 ( σ h ) ] Δ 2 B 2 sin 2 ( σ h ) cos 2 ( σ h ) ,
numerator = Δ ρ B 2 sin 2 ( σ h ) cos 2 ( σ h ) × [ ρ 2 A + Δ ρ B cos 2 ( σ h ) + i σ ρ B sin ( σ h ) cos ( σ h ) ] · [ A 2 ( Δ - ρ ) 2 + 4 Δ ρ B C cos 2 σ h ] , and
denominator = Δ ρ B 2 sin 2 ( σ h ) cos 2 ( σ h ) [ A 2 - B cos 2 ( σ h ) ] · [ A 2 ( Δ - ρ ) 2 + 4 Δ ρ B C cos 2 ( σ h ) ] .
4 ω μ P ¯ ρ π β cos ( σ h ) - i ( ρ / σ ) sin ( σ h ) ( Δ + ρ ) cos ( σ h ) [ 1 - i ( σ 2 + Δ ρ ) tan ( σ h ) σ Δ + σ ρ ] .
4 ω 0 P ¯ ρ π β [ cos ( σ h ) - i ( n f n s ) 2 ( ρ σ ) sin ( σ h ) ( Δ n c 2 + ρ n s 2 ) cos ( σ h ) { 1 - i ( σ 2 n f 4 + Δ ρ n c 2 n s 2 ) tan ( σ h ) δ Δ n f 2 n c 2 + Δ ρ n f 2 n s 2 } ]
f ( z ) [ 1 n f 2 + W ( z ) ] exp [ i ( β 0 - β ) z ] ,
W ( z ) = ( 1 n c 2 - 1 n f 2 ) n = - sin ( π n 2 ) π n exp ( - i π n 2 ) exp ( i 2 π n z Λ ) .
0 L d z f ( z ) n 2 ( x , z ) exp [ i ( β 0 - β ) z ]
a ( 1 n c 2 - 1 n f 2 ) n = - sin ( π n 2 ) π n exp ( - i π n 2 ) 0 L d z sin ( 2 π z Λ ) × exp [ i ( β 0 - β + 2 π n Λ ) z ] + a n f 2 0 L d z sin ( 2 π z Λ ) exp [ i ( β 0 - β ) z ] ,
f ( z ) = a sin ( 2 π z Λ ) .
0 L d z f ( z ) n 2 ( x , z ) exp [ i ( β 0 - β ) z ] = - a L 4 i ( 1 n f 2 + 1 n c 2 ) δ β 0 , β + 2 π / Λ .

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