Abstract

An efficient, simple numerical method is employed in, conjuntion with an optimum perturbation scheme to solve for the radiating and guided waves of a periodic corrugated dielectric waveguide. Results for sinusoidal, triangular, and rectangular grating profiles on waveguides and waveguide-lasers are obtained and compared with approximate analytic expressions.

© 1978 Optical Society of America

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References

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  1. H. Kogelnik and C. V. Shank, “Coupled mode theory of distributed feedback lasers,” J. Appl. Phys.,  43, 2327 (1972).
    [Crossref]
  2. D. R. Scifres, R. D. Burnham, and W. Streifer, “Output coupling and distributed feedback utilizing substrate corrugations in double-heterostructure GaAs lasers,” Appl. Phys. Lett,  27, 295 (1975).
    [Crossref]
  3. R. D. Burnham, D. R. Scifres, and W. Streifer, “Low divergence beam from grating coupled composite guide heterostructure GaAlAs diode lasers,” Appl. Phys. Lett,  26, 644 (1975).
    [Crossref]
  4. W. Ng and A. Yariv, “Highly collimated broadside emission from room-temperature GaAs distributed Bragg reflector lasers,” Appl. Phys. Lett.,  31, 613 (1977).
    [Crossref]
  5. V. A. Kiselev, “Diffraction coupling of radiation into a thin-film waveguide,” Sov. J. Quantum Electron.,  4, 872 (1975).
    [Crossref]
  6. A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhal’skii, and V. A. Sychugov, “Emission of surface light wave from a corrugated part of a thin film waveguide,” Sov. J. Quantum Electron.,  4, 839 (1975).
    [Crossref]
  7. 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 (1976).
    [Crossref]
  8. W. W. Rigrod and D. Marcuse, “Radiation loss coefficients of asymmetric dielectric waveguides with shallow sinusoidal corrugations,” IEEE J. of Quantum Electron. QE-12, 673 (1976).
    [Crossref]
  9. S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. MTT-23, 123 (1975).
  10. K. Handa, S. T. Peng, and T. Tamir, “Improved perturbation analysis of dielectric grating,” Appl. Phys. 5, 325 (1975).
    [Crossref]
  11. S. T. Peng and T. Tamir, “TM-Mode perturbation analysis of dielectric gratings,” Appl. Phys. 7, 35 (1975).
    [Crossref]
  12. 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 (1976).
    [Crossref]
  13. W. Streifer, R. D. Burnham, and D. R. Scifres, “Analysis of grating coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quant. Electron, QE-12, 494 (1976).
    [Crossref]
  14. C. Lanczos, “Trigonometric interpolation of empirical and analytical Functions,” J. Math. Phys., 17, 123 (1938).
  15. C. Lanczos, Applied Analysis (Prentice Hall, Englewood Cliffs, N.J., 1956), Chap 7.
  16. Y. L. Luke, Mathematical Functions and their Approximations (Academic, New York, 1975), p. 495.

1977 (1)

W. Ng and A. Yariv, “Highly collimated broadside emission from room-temperature GaAs distributed Bragg reflector lasers,” Appl. Phys. Lett.,  31, 613 (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 (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, 673 (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 (1976).
[Crossref]

W. Streifer, R. D. Burnham, and D. R. Scifres, “Analysis of grating coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quant. Electron, QE-12, 494 (1976).
[Crossref]

1975 (7)

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. MTT-23, 123 (1975).

K. Handa, S. T. Peng, and T. Tamir, “Improved perturbation analysis of dielectric grating,” Appl. Phys. 5, 325 (1975).
[Crossref]

S. T. Peng and T. Tamir, “TM-Mode perturbation analysis of dielectric gratings,” Appl. Phys. 7, 35 (1975).
[Crossref]

V. A. Kiselev, “Diffraction coupling of radiation into a thin-film waveguide,” Sov. J. Quantum Electron.,  4, 872 (1975).
[Crossref]

A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhal’skii, and V. A. Sychugov, “Emission of surface light wave from a corrugated part of a thin film waveguide,” Sov. J. Quantum Electron.,  4, 839 (1975).
[Crossref]

D. R. Scifres, R. D. Burnham, and W. Streifer, “Output coupling and distributed feedback utilizing substrate corrugations in double-heterostructure GaAs lasers,” Appl. Phys. Lett,  27, 295 (1975).
[Crossref]

R. D. Burnham, D. R. Scifres, and W. Streifer, “Low divergence beam from grating coupled composite guide heterostructure GaAlAs diode lasers,” Appl. Phys. Lett,  26, 644 (1975).
[Crossref]

1972 (1)

H. Kogelnik and C. V. Shank, “Coupled mode theory of distributed feedback lasers,” J. Appl. Phys.,  43, 2327 (1972).
[Crossref]

1938 (1)

C. Lanczos, “Trigonometric interpolation of empirical and analytical Functions,” J. Math. Phys., 17, 123 (1938).

Bertoni, H. L.

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. MTT-23, 123 (1975).

Burnham, R. D.

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 (1976).
[Crossref]

W. Streifer, R. D. Burnham, and D. R. Scifres, “Analysis of grating coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quant. Electron, QE-12, 494 (1976).
[Crossref]

D. R. Scifres, R. D. Burnham, and W. Streifer, “Output coupling and distributed feedback utilizing substrate corrugations in double-heterostructure GaAs lasers,” Appl. Phys. Lett,  27, 295 (1975).
[Crossref]

R. D. Burnham, D. R. Scifres, and W. Streifer, “Low divergence beam from grating coupled composite guide heterostructure GaAlAs diode lasers,” Appl. Phys. Lett,  26, 644 (1975).
[Crossref]

Handa, K.

K. Handa, S. T. Peng, and T. Tamir, “Improved perturbation analysis of dielectric grating,” Appl. Phys. 5, 325 (1975).
[Crossref]

Kiselev, V. A.

A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhal’skii, and V. A. Sychugov, “Emission of surface light wave from a corrugated part of a thin film waveguide,” Sov. J. Quantum Electron.,  4, 839 (1975).
[Crossref]

V. A. Kiselev, “Diffraction coupling of radiation into a thin-film waveguide,” Sov. J. Quantum Electron.,  4, 872 (1975).
[Crossref]

Kogelnik, H.

H. Kogelnik and C. V. Shank, “Coupled mode theory of distributed feedback lasers,” J. Appl. Phys.,  43, 2327 (1972).
[Crossref]

Lanczos, C.

C. Lanczos, “Trigonometric interpolation of empirical and analytical Functions,” J. Math. Phys., 17, 123 (1938).

C. Lanczos, Applied Analysis (Prentice Hall, Englewood Cliffs, N.J., 1956), Chap 7.

Luke, Y. L.

Y. L. Luke, Mathematical Functions and their Approximations (Academic, New York, 1975), p. 495.

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, 673 (1976).
[Crossref]

Ng, W.

W. Ng and A. Yariv, “Highly collimated broadside emission from room-temperature GaAs distributed Bragg reflector lasers,” Appl. Phys. Lett.,  31, 613 (1977).
[Crossref]

Peng, S. T.

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. MTT-23, 123 (1975).

K. Handa, S. T. Peng, and T. Tamir, “Improved perturbation analysis of dielectric grating,” Appl. Phys. 5, 325 (1975).
[Crossref]

S. T. Peng and T. Tamir, “TM-Mode perturbation analysis of dielectric gratings,” Appl. Phys. 7, 35 (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 (1976).
[Crossref]

A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhal’skii, and V. A. Sychugov, “Emission of surface light wave from a corrugated part of a thin film waveguide,” Sov. J. Quantum Electron.,  4, 839 (1975).
[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, 673 (1976).
[Crossref]

Scifres, D. R.

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 (1976).
[Crossref]

W. Streifer, R. D. Burnham, and D. R. Scifres, “Analysis of grating coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quant. Electron, QE-12, 494 (1976).
[Crossref]

D. R. Scifres, R. D. Burnham, and W. Streifer, “Output coupling and distributed feedback utilizing substrate corrugations in double-heterostructure GaAs lasers,” Appl. Phys. Lett,  27, 295 (1975).
[Crossref]

R. D. Burnham, D. R. Scifres, and W. Streifer, “Low divergence beam from grating coupled composite guide heterostructure GaAlAs diode lasers,” Appl. Phys. Lett,  26, 644 (1975).
[Crossref]

Shank, C. V.

H. Kogelnik and C. V. Shank, “Coupled mode theory of distributed feedback lasers,” J. Appl. Phys.,  43, 2327 (1972).
[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 (1976).
[Crossref]

A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhal’skii, and V. A. Sychugov, “Emission of surface light wave from a corrugated part of a thin film waveguide,” Sov. J. Quantum Electron.,  4, 839 (1975).
[Crossref]

Streifer, W.

W. Streifer, R. D. Burnham, and D. R. Scifres, “Analysis of grating coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quant. Electron, QE-12, 494 (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 (1976).
[Crossref]

D. R. Scifres, R. D. Burnham, and W. Streifer, “Output coupling and distributed feedback utilizing substrate corrugations in double-heterostructure GaAs lasers,” Appl. Phys. Lett,  27, 295 (1975).
[Crossref]

R. D. Burnham, D. R. Scifres, and W. Streifer, “Low divergence beam from grating coupled composite guide heterostructure GaAlAs diode lasers,” Appl. Phys. Lett,  26, 644 (1975).
[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 (1976).
[Crossref]

A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhal’skii, and V. A. Sychugov, “Emission of surface light wave from a corrugated part of a thin film waveguide,” Sov. J. Quantum Electron.,  4, 839 (1975).
[Crossref]

Tamir, T.

S. T. Peng and T. Tamir, “TM-Mode perturbation analysis of dielectric gratings,” Appl. Phys. 7, 35 (1975).
[Crossref]

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. MTT-23, 123 (1975).

K. Handa, S. T. Peng, and T. Tamir, “Improved perturbation analysis of dielectric grating,” Appl. Phys. 5, 325 (1975).
[Crossref]

Yariv, A.

W. Ng and A. Yariv, “Highly collimated broadside emission from room-temperature GaAs distributed Bragg reflector lasers,” Appl. Phys. Lett.,  31, 613 (1977).
[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 (1976).
[Crossref]

A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhal’skii, and V. A. Sychugov, “Emission of surface light wave from a corrugated part of a thin film waveguide,” Sov. J. Quantum Electron.,  4, 839 (1975).
[Crossref]

Appl. Phys. (2)

K. Handa, S. T. Peng, and T. Tamir, “Improved perturbation analysis of dielectric grating,” Appl. Phys. 5, 325 (1975).
[Crossref]

S. T. Peng and T. Tamir, “TM-Mode perturbation analysis of dielectric gratings,” Appl. Phys. 7, 35 (1975).
[Crossref]

Appl. Phys. Lett (2)

D. R. Scifres, R. D. Burnham, and W. Streifer, “Output coupling and distributed feedback utilizing substrate corrugations in double-heterostructure GaAs lasers,” Appl. Phys. Lett,  27, 295 (1975).
[Crossref]

R. D. Burnham, D. R. Scifres, and W. Streifer, “Low divergence beam from grating coupled composite guide heterostructure GaAlAs diode lasers,” Appl. Phys. Lett,  26, 644 (1975).
[Crossref]

Appl. Phys. Lett. (1)

W. Ng and A. Yariv, “Highly collimated broadside emission from room-temperature GaAs distributed Bragg reflector lasers,” Appl. Phys. Lett.,  31, 613 (1977).
[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, 673 (1976).
[Crossref]

IEEE J. Quant. Electron, (1)

W. Streifer, R. D. Burnham, and D. R. Scifres, “Analysis of grating coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quant. Electron, QE-12, 494 (1976).
[Crossref]

IEEE J. Quantum Electron. (1)

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 (1976).
[Crossref]

IEEE Trans. (1)

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. MTT-23, 123 (1975).

J. Appl. Phys. (1)

H. Kogelnik and C. V. Shank, “Coupled mode theory of distributed feedback lasers,” J. Appl. Phys.,  43, 2327 (1972).
[Crossref]

J. Math. Phys., (1)

C. Lanczos, “Trigonometric interpolation of empirical and analytical Functions,” J. Math. Phys., 17, 123 (1938).

Sov. J. Quantum Electron. (3)

V. A. Kiselev, “Diffraction coupling of radiation into a thin-film waveguide,” Sov. J. Quantum Electron.,  4, 872 (1975).
[Crossref]

A. A. Zlenko, V. A. Kiselev, A. M. Prokhorov, A. A. Spikhal’skii, and V. A. Sychugov, “Emission of surface light wave from a corrugated part of a thin film waveguide,” Sov. J. Quantum Electron.,  4, 839 (1975).
[Crossref]

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 (1976).
[Crossref]

Other (2)

C. Lanczos, Applied Analysis (Prentice Hall, Englewood Cliffs, N.J., 1956), Chap 7.

Y. L. Luke, Mathematical Functions and their Approximations (Academic, New York, 1975), p. 495.

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Figures (6)

FIG. 1
FIG. 1

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

FIG. 2
FIG. 2

The TE mode patterns and their gradients in corrugated waveguides with sinusoidal profile (solid curves) and rectangular profile (broken curves). The waveguide parameters are: n1 = n3 = 3.4, n2 = 3.6, t = 1.0 m,g = 0.4 μm and λ0 = 0.88 μm.

FIG. 3
FIG. 3

Total radiated power α(cm−1) calculated by using approximation analytical results (Ref. 6) (broken curves) is compared with those calculated by the perturbation method (solid curve). The specific points (x) are numerically exact solutions obtained by the iteration procedure outlined in Ref. 12.

FIG. 4
FIG. 4

Total radiated power, α(cm−1), vs grating thickness g for three TE modes. Discrete points are iteration results. (a) rectangular corrugation, (b) sinusoidal corrugation, (c) triangular corrugation.

FIG. 5
FIG. 5

TE mode radiated power, (solid curves) and radiation angles (broken curves) vs grating period for three partial waves.

FIG. 6
FIG. 6

Total radiated power, vs grating thickness, of an output coupler with n = 1, n2 = 3.6, n3 = 3.4, t = 0.5 μm and λ0 = 0.88 μm for three different grating profiles (R = rectangular, S = sinusoidal, T = triangular). The grating period is related to β by /β, similar to those use in Bragg reflectors.

Equations (28)

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E y ( x , z ) = m = - E m ( x ) e i β m z ,
β m = β 0 + 2 π m / Λ
n g 2 ( x , z ) = q = - A q ( x ) e i 2 π q z / Λ ,             0 < x < g ,
A q ( x ) = [ ( n 2 2 - n 1 2 ) / π q ] × sin [ π q w ( x ) / Λ ] e i 2 π q z 0 ( x ) / Λ ,             q 0
A 0 ( x ) = ( n 1 2 + n 2 2 ) / 2 + ( n 2 2 - n 1 2 ) [ w ( x ) / Λ - 1 / 2 ] ,
( 1 ) rectangular :             w ( x ) = w , ( 2 ) triangular :             w ( x ) = ( x / g ) Λ , ( 3 ) sinusoidal :             w ( x ) = ( Λ / π ) cos - 1 ( 1 - 2 x / g ) .
d 2 E m ( x ) d x 2 + [ k 0 2 n 2 ( x ) - β m 2 ] E m ( x ) = - k 0 2 q = - 1 m A m - q ( x ) E q ( x ) m ,
n 2 ( x ) = [ n 1 2 , x < 0 A 0 ( x ) , 0 < x < g n 2 2 , g < x < t n 3 2 , t < x
P m = ( 1 / 2 ω μ ) { E m ( 0 ) 2 Re [ k 0 2 n 1 2 - β m 2 ] 1 / 2 + E m ( t ) 2 Re [ k 0 2 n 3 2 - β m 2 ] 1 / 2 } .
α = m P m / ( β 0 / 2 ω μ 0 ) - E 0 ( x ) 2 d x ,
β 0 = β r + i β i ,
θ j = cos - 1 [ β m / ( 2 π n j / λ o ) ] ,             j = 1 , 3 ,
d 2 E ( x ) d x 2 + a ( x ) E ( x ) = b ( x ) ,
E ( x ) = p = 0 P - 1 C p T p * ( x ) ,             0 x 1 ,
d 2 E d x 2 + a ( x ) E ( x ) = b ( x ) + T P * ( x ) ( τ 1 + τ 2 x ) ,
p = 0 P - 1 C p ( d 2 T p * ( x ) d x 2 + a ( x ) T p * ( x ) ) = b ( x ) + T P * ( x ) ( τ 1 + τ 2 x ) .
Λ = M π / β 0 ,
d 2 E m ( x ) d x 2 + ( k 0 2 n j 2 - β m 2 ) E m ( x ) = 0 ,             j = 1 , 2 , 3 ,
β m = β 0 + 2 π m / Λ .
d 2 E m ( x ) d x 2 + { k 0 2 ( n 1 2 - n 2 2 ) / 2 + k 0 2 ( n 2 2 - n 1 2 ) [ w ( x ) / Λ - 1 / 2 ] - β m 2 } E m ( x ) = - k 0 2 q = q m A m - q ( x ) E q ( x ) ,
A q ( x ) = [ ( n 2 2 - n 1 2 ) π q ] sin [ π q w ( x ) / Λ ] .
E m ( x ) = { C m 1 exp ( - i k m 1 x ) , x 0 , C m 1 f h m ( x ) + f p m ( x ) , 0 < x g , C m 3 [ cos k m 2 ( t - x ) - ( i k m 3 / k m 2 ) sin k m 2 ( t - x ) ] , g < x < t , C m 3 exp [ - i k m 3 ( x - t ) ] , x > t ,
k m j = k 0 2 n j 2 - β m 2 ,             j = 1 , 2 , 3 ,
f h m ( 0 ) = 1 ,             f h m ( 0 ) = - i k m 1 , and f p m ( 0 ) ,             f p m ( 0 ) = 0.
C 03 U 0 = f h 0 ( g ) and k 02 C 03 V 0 = f h 0 ( g ) ,
U m = cos [ k m 2 ( t - g ) ] - ( i k m 3 / k m 2 ) sin [ k m 2 ( t - g ) ] , and V m = sin [ k m 2 ( t - g ) ] + ( k m 3 / k m 2 ) cos [ k m 2 ( t - g ) ] .
Q ( β ) = f h 0 ( g ) U 0 - k 02 f h 0 ( g ) V 0 = 0.
C m 1 f h m ( g ) - C m 3 U m = - f p m ( g ) , C m 1 f h m ( g ) - k m 2 C m 3 V m = - f p m ( g ) .