Abstract

Perturbation analysis is applied to calculate the radiation loss from corrugated waveguides for TM polarized guided waves. It is shown that the radiated power is strongly dependent upon radiation angle. A significant drop in output radiation occurs for radiation normal to the waveguide. Numerical results for waveguide structures having rectangular and sinusoidal corrugation profiles are included.

© 1979 Optical Society of America

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References

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  1. A. A. Zlenko, A. M. Prokhorov, A. A. Spikhal’skii, and V. A. Sychugov, “Emission of E Waves from a Corrugated Section of a Waveguide,” Sov. J. Quantum Electron. 6, 565–567 (1976).
    [Crossref]
  2. E. M. Zolotov, V. A. Kiselev, V. M. Pelekhatyi, A. M. Prokhov, and R. A. Shcherbakov, “Diffraction Emission and Excitation of E and H Waves in Diffused Optical Waveguides,” Sov. J. Quantum Electron. 7, 806–809 (1977).
    [Crossref]
  3. S. T. Peng and T. Tamir, “TM-Mode Perturbation Analysis of Dielectric Gratings,” Appl. Phys. 7, 35–38 (1975).
    [Crossref]
  4. W. W. Rigrod and D. Marcuse, “Radiation Loss Coefficients of Asymmetric Dielectric Waveguides with Shallow Sinusoidal Corrugations,” IEEE J. of Quantum Electron. QE-12, 679–685 (1976).
  5. Y. Yamamoto, T. Kamiya, and H. Yanai, “Improved Coupled Mode Analysis of Corrugated Waveguides and Lasers—II: TM Mode,” IEEE J. Quantum Electron. QE-14, 620–624 (1978).
    [Crossref]
  6. W. H. Lee and W. Streifer, “Radiation Loss Calculations for Corrugated Dielectric Waveguides,” J. Opt. Soc. Am. 68, 1701–1707 (1978).
    [Crossref]
  7. W. Streifer, D. R. Scifres, and R. D. Burnham, “TM Coupling Coefficients in Guide-Wave Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
    [Crossref]
  8. H. Kogelnik and T. P. Sosnowski, “Holographic Thin Film Couplers,” Bell Syst. Tech. J. 49, 1602–1608 (1970).
    [Crossref]
  9. W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of Grating Coupled Radiation in GaAs:GaAlAs Lasers and Waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
    [Crossref]

1978 (2)

Y. Yamamoto, T. Kamiya, and H. Yanai, “Improved Coupled Mode Analysis of Corrugated Waveguides and Lasers—II: TM Mode,” IEEE J. Quantum Electron. QE-14, 620–624 (1978).
[Crossref]

W. H. Lee and W. Streifer, “Radiation Loss Calculations for Corrugated Dielectric Waveguides,” J. Opt. Soc. Am. 68, 1701–1707 (1978).
[Crossref]

1977 (1)

E. M. Zolotov, V. A. Kiselev, V. M. Pelekhatyi, A. M. Prokhov, and R. A. Shcherbakov, “Diffraction Emission and Excitation of E and H Waves in Diffused Optical Waveguides,” Sov. J. Quantum Electron. 7, 806–809 (1977).
[Crossref]

1976 (4)

A. A. Zlenko, A. M. Prokhorov, A. A. Spikhal’skii, and V. A. Sychugov, “Emission of E Waves from a Corrugated Section of a Waveguide,” Sov. J. Quantum Electron. 6, 565–567 (1976).
[Crossref]

W. W. Rigrod and D. Marcuse, “Radiation Loss Coefficients of Asymmetric Dielectric Waveguides with Shallow Sinusoidal Corrugations,” IEEE J. of Quantum Electron. QE-12, 679–685 (1976).

W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of Grating Coupled Radiation in GaAs:GaAlAs Lasers and Waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[Crossref]

W. Streifer, D. R. Scifres, and R. D. Burnham, “TM Coupling Coefficients in Guide-Wave Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
[Crossref]

1975 (1)

S. T. Peng and T. Tamir, “TM-Mode Perturbation Analysis of Dielectric Gratings,” Appl. Phys. 7, 35–38 (1975).
[Crossref]

1970 (1)

H. Kogelnik and T. P. Sosnowski, “Holographic Thin Film Couplers,” Bell Syst. Tech. J. 49, 1602–1608 (1970).
[Crossref]

Burnham, R. D.

W. Streifer, D. R. Scifres, and R. D. Burnham, “TM Coupling Coefficients in Guide-Wave Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
[Crossref]

W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of Grating Coupled Radiation in GaAs:GaAlAs Lasers and Waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[Crossref]

Kamiya, T.

Y. Yamamoto, T. Kamiya, and H. Yanai, “Improved Coupled Mode Analysis of Corrugated Waveguides and Lasers—II: TM Mode,” IEEE J. Quantum Electron. QE-14, 620–624 (1978).
[Crossref]

Kiselev, V. A.

E. M. Zolotov, V. A. Kiselev, V. M. Pelekhatyi, A. M. Prokhov, and R. A. Shcherbakov, “Diffraction Emission and Excitation of E and H Waves in Diffused Optical Waveguides,” Sov. J. Quantum Electron. 7, 806–809 (1977).
[Crossref]

Kogelnik, H.

H. Kogelnik and T. P. Sosnowski, “Holographic Thin Film Couplers,” Bell Syst. Tech. J. 49, 1602–1608 (1970).
[Crossref]

Lee, W. H.

Marcuse, D.

W. W. Rigrod and D. Marcuse, “Radiation Loss Coefficients of Asymmetric Dielectric Waveguides with Shallow Sinusoidal Corrugations,” IEEE J. of Quantum Electron. QE-12, 679–685 (1976).

Pelekhatyi, V. M.

E. M. Zolotov, V. A. Kiselev, V. M. Pelekhatyi, A. M. Prokhov, and R. A. Shcherbakov, “Diffraction Emission and Excitation of E and H Waves in Diffused Optical Waveguides,” Sov. J. Quantum Electron. 7, 806–809 (1977).
[Crossref]

Peng, S. T.

S. T. Peng and T. Tamir, “TM-Mode Perturbation Analysis of Dielectric Gratings,” Appl. Phys. 7, 35–38 (1975).
[Crossref]

Prokhorov, A. M.

A. A. Zlenko, A. M. Prokhorov, A. A. Spikhal’skii, and V. A. Sychugov, “Emission of E Waves from a Corrugated Section of a Waveguide,” Sov. J. Quantum Electron. 6, 565–567 (1976).
[Crossref]

Prokhov, A. M.

E. M. Zolotov, V. A. Kiselev, V. M. Pelekhatyi, A. M. Prokhov, and R. A. Shcherbakov, “Diffraction Emission and Excitation of E and H Waves in Diffused Optical Waveguides,” Sov. J. Quantum Electron. 7, 806–809 (1977).
[Crossref]

Rigrod, W. W.

W. W. Rigrod and D. Marcuse, “Radiation Loss Coefficients of Asymmetric Dielectric Waveguides with Shallow Sinusoidal Corrugations,” IEEE J. of Quantum Electron. QE-12, 679–685 (1976).

Scifres, D. R.

W. Streifer, D. R. Scifres, and R. D. Burnham, “TM Coupling Coefficients in Guide-Wave Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
[Crossref]

W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of Grating Coupled Radiation in GaAs:GaAlAs Lasers and Waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[Crossref]

Shcherbakov, R. A.

E. M. Zolotov, V. A. Kiselev, V. M. Pelekhatyi, A. M. Prokhov, and R. A. Shcherbakov, “Diffraction Emission and Excitation of E and H Waves in Diffused Optical Waveguides,” Sov. J. Quantum Electron. 7, 806–809 (1977).
[Crossref]

Sosnowski, T. P.

H. Kogelnik and T. P. Sosnowski, “Holographic Thin Film Couplers,” Bell Syst. Tech. J. 49, 1602–1608 (1970).
[Crossref]

Spikhal’skii, A. A.

A. A. Zlenko, A. M. Prokhorov, A. A. Spikhal’skii, and V. A. Sychugov, “Emission of E Waves from a Corrugated Section of a Waveguide,” Sov. J. Quantum Electron. 6, 565–567 (1976).
[Crossref]

Streifer, W.

W. H. Lee and W. Streifer, “Radiation Loss Calculations for Corrugated Dielectric Waveguides,” J. Opt. Soc. Am. 68, 1701–1707 (1978).
[Crossref]

W. Streifer, D. R. Scifres, and R. D. Burnham, “TM Coupling Coefficients in Guide-Wave Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
[Crossref]

W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of Grating Coupled Radiation in GaAs:GaAlAs Lasers and Waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[Crossref]

Sychugov, V. A.

A. A. Zlenko, A. M. Prokhorov, A. A. Spikhal’skii, and V. A. Sychugov, “Emission of E Waves from a Corrugated Section of a Waveguide,” Sov. J. Quantum Electron. 6, 565–567 (1976).
[Crossref]

Tamir, T.

S. T. Peng and T. Tamir, “TM-Mode Perturbation Analysis of Dielectric Gratings,” Appl. Phys. 7, 35–38 (1975).
[Crossref]

Yamamoto, Y.

Y. Yamamoto, T. Kamiya, and H. Yanai, “Improved Coupled Mode Analysis of Corrugated Waveguides and Lasers—II: TM Mode,” IEEE J. Quantum Electron. QE-14, 620–624 (1978).
[Crossref]

Yanai, H.

Y. Yamamoto, T. Kamiya, and H. Yanai, “Improved Coupled Mode Analysis of Corrugated Waveguides and Lasers—II: TM Mode,” IEEE J. Quantum Electron. QE-14, 620–624 (1978).
[Crossref]

Zlenko, A. A.

A. A. Zlenko, A. M. Prokhorov, A. A. Spikhal’skii, and V. A. Sychugov, “Emission of E Waves from a Corrugated Section of a Waveguide,” Sov. J. Quantum Electron. 6, 565–567 (1976).
[Crossref]

Zolotov, E. M.

E. M. Zolotov, V. A. Kiselev, V. M. Pelekhatyi, A. M. Prokhov, and R. A. Shcherbakov, “Diffraction Emission and Excitation of E and H Waves in Diffused Optical Waveguides,” Sov. J. Quantum Electron. 7, 806–809 (1977).
[Crossref]

Appl. Phys. (1)

S. T. Peng and T. Tamir, “TM-Mode Perturbation Analysis of Dielectric Gratings,” Appl. Phys. 7, 35–38 (1975).
[Crossref]

Bell Syst. Tech. J. (1)

H. Kogelnik and T. P. Sosnowski, “Holographic Thin Film Couplers,” Bell Syst. Tech. J. 49, 1602–1608 (1970).
[Crossref]

IEEE J. of Quantum Electron. (1)

W. W. Rigrod and D. Marcuse, “Radiation Loss Coefficients of Asymmetric Dielectric Waveguides with Shallow Sinusoidal Corrugations,” IEEE J. of Quantum Electron. QE-12, 679–685 (1976).

IEEE J. Quantum Electron. (3)

Y. Yamamoto, T. Kamiya, and H. Yanai, “Improved Coupled Mode Analysis of Corrugated Waveguides and Lasers—II: TM Mode,” IEEE J. Quantum Electron. QE-14, 620–624 (1978).
[Crossref]

W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of Grating Coupled Radiation in GaAs:GaAlAs Lasers and Waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[Crossref]

W. Streifer, D. R. Scifres, and R. D. Burnham, “TM Coupling Coefficients in Guide-Wave Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-12, 74–78 (1976).
[Crossref]

J. Opt. Soc. Am. (1)

Sov. J. Quantum Electron. (2)

A. A. Zlenko, A. M. Prokhorov, A. A. Spikhal’skii, and V. A. Sychugov, “Emission of E Waves from a Corrugated Section of a Waveguide,” Sov. J. Quantum Electron. 6, 565–567 (1976).
[Crossref]

E. M. Zolotov, V. A. Kiselev, V. M. Pelekhatyi, A. M. Prokhov, and R. A. Shcherbakov, “Diffraction Emission and Excitation of E and H Waves in Diffused Optical Waveguides,” Sov. J. Quantum Electron. 7, 806–809 (1977).
[Crossref]

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Figures (5)

FIG. 1
FIG. 1

(a) Waveguide geometry and grating profile; (b) variation in square refractive index.

FIG. 2
FIG. 2

TM mode patterns and their gradients in corrugated waveguides with (a) rectangular and (b) sinusoidal profile. The broken curves show the function ν 0 ( x ) H 0 ( x ). The waveguide parameters are: n1 = n3 = 3.4, n2 = 3.6, t = 1.0 μm, g = 0.4 μm, and λ = 0.88 μm.

FIG. 3
FIG. 3

Total radiated power α (cm−1) versus grating thickness g for three TM modes. (a) Rectangular corrugation; (b) sinusoidal corrugation.

FIG. 4
FIG. 4

Total radiated power α (cm−1) versus grating thickness g for three TM modes in a corrugated waveguide with a rectangular profile. The radiation angle is approximately normal to the waveguide.

FIG. 5
FIG. 5

TM modes radiated powers (solid curves) and radiation angles (broken curves) for three partial waves. (a) Rectangular corrugation profile; (b) sinusoidal corrugation profile.

Equations (28)

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z [ ν ( x , z ) H y ( x , z ) z ] + x [ ν ( x , z ) H y ( x , z ) x ] + k 0 2 H y ( x , z ) = 0 ,
H y ( x , z ) = m = H m ( x ) e i β m z ,
β m = β 0 + 2 π m / Λ .
ν ( x , z ) = B 0 ( x ) + q = q 0 B q ( x ) e i 2 π q z / Λ .
B q ( x ) = [ ( 1 n 2 2 1 n 1 2 ) l π q ] sin [ π q w ( x ) / Λ ] ,
B 0 ( x ) = 1 n 1 2 + ( 1 n 2 2 1 n 1 2 ) ω ( x ) ,
d d x [ ν 0 ( x ) d H m ( x ) d x ] + [ k 0 2 β m 2 ν 0 ( x ) ] H m ( x ) = q m q = [ β q β m B m q ( x ) H q ( x ) d d x ( B m q ( x ) d H q ( x ) d x ) ] ,
ν 0 ( x ) = { 1 / n 1 2 , x < 0 , B 0 ( x ) , 0 < x < g , 1 / n 2 2 , g < x < t , 1 / n 3 2 , t < x .
Λ = π M / β 0 , M = 1 , 2 , ,
P m = ( 1 2 ω 0 ) [ | H m ( 0 ) | 2 Re ( k 0 2 β m 2 ) 1 / 2 / n 1 2 + | H m ( t ) | 2 Re ( k 0 2 β m 2 ) 1 / 2 / n 3 2 ]
α = m P m [ ( β 2 ω 0 ) | H 0 ( x ) | 2 ν 0 ( x ) d x ] 1 .
β 0 β m B m ( x ) H 0 ( x ) d d x ( B m d H 0 d x ) ,
k 0 2 A m ( x ) E 0 ( x ) .
d d x [ 1 n i 2 ( x ) d H 0 d x ] + [ k 0 2 β 0 2 n i 2 ( x ) ] H 0 = 0 , i = 1 , 2 , 3 , 4
1 n 4 2 ( x ) = B 0 ( x ) .
d d x [ 1 n i 2 ( x ) d H m d x ] + [ k 0 2 β m 2 n i 2 ( x ) ] H m = β 0 β m B m ( x ) H 0 ( x ) d d x [ B m ( x ) d H 0 d x ] ,
d 2 H 0 ( x ) d x 2 ( k 0 2 n i 2 β 0 2 ) H 0 ( x ) = 0 , i = 1 , 2 , 3 , 4 ,
1 n 4 2 = 1 2 ( 1 n 1 2 + 1 n 2 2 ) + ( 1 n 2 2 1 n 1 2 ) ( w Λ 1 2 ) ,
H 0 ( x ) = { e i k 1 x , x < 0 cos ( k 4 x ) ( i k ˆ 1 / k ˆ 4 ) sin ( k 4 x ) , 0 < x < g C 3 { cos [ k 2 ( t x ) ] ( i k ˆ 3 / k ˆ 2 ) sin [ k 2 ( t x ) ] } , q < c < t C 3 e i k 3 ( x t ) , x > t .
d d x ( 1 n i 2 d H m d x ) + ( k 0 2 β m 2 n i 2 ) H m = β 0 β m B m ( x ) H 0 ( x ) d d x ( B m d H 0 d x )
B m ( x ) = { ( 1 / n 2 2 1 / n 1 2 ) sin ( π m w / Λ ) / π m 0 < x < g , 0 x < 0 , g < x .
H m ( x ) = { C m 1 e i k m 1 x , x < 0 , C m 4 cos ( k m 4 x ) + C ¯ m 4 sin ( k m 4 x ) + T m ( x ) , 0 x g , C m 3 { cos [ k m 2 ( t x ) ] ( i k ˆ m 3 / k ˆ m 2 ) sin [ k m 2 ( t x ) ] } , q x t , C m 3 e i k m 3 ( x t ) , t x ,
T m ( x ) = 1 k ˆ m 4 0 x β 0 β m B m H 0 ( ξ ) sin [ k m 4 ( x ξ ) ] d ξ n 4 2 0 x B m d H 0 d ξ ( ξ ) cos [ k m 4 ( x ξ ) ] d ξ .
d T m d x = n 4 2 0 x β 0 β m B m H 0 ( ξ ) × cos [ k m 4 ( x ξ ) ] d ξ n 4 2 B m d H 0 d x + n 4 2 k m 4 0 x B m d H 0 d ξ sin [ k m 4 ( x ξ ) ] d ξ ,
d 2 T m d x 2 = k m 4 2 T m + n 4 2 [ β 0 β m B m H 0 ( x ) d d x ( B m d H 0 d x ) ] .
i k ˆ m 1 C m 1 = k ˆ m 4 C ¯ m 4 B m d H 0 / d x | 0 + ,
C m 3 { cos [ k m 2 ( t g ) ] ( i k ˆ m 3 / k ˆ m 2 ) sin [ k m 2 ( t g ) ] } = C m 1 cos ( k m 4 g ) + C ¯ m 4 sin ( k m 4 g ) + T m ( g )
C m 3 { k ˆ m 2 sin [ k m 2 ( t g ) ] + i k ˆ m 3 cos [ k m 2 ( t g ) ] } = k ˆ m 4 C m 1 sin ( k m 4 g ) + k ˆ m 4 C ¯ m 4 cos ( k m 4 g ) + T m 1 ( g ) / n 4 2 ,