Abstract

The scattering properties of dielectric gratings having asymmetric triangular profiles are determined by a simple approach that views the grating structure as a double set of parallel planes. The energy of an incident surface wave is then scattered in preferential directions that are consistent with the grating periodicity and with a Bragg condition. As a result, the triangular shape can be chosen so that scattering occurs mostly into the region above the grating or into that below it, thus making it possible to maximize the efficiency of beam couplers or other devices. This desirable blazing property can be achieved by satisfying simple design criteria that are obtained from the scattering approach presented here. The range of grating parameters for strong blazing is derived, and gratings with trapezoidal profiles are also discussed.

© 1980 Optical Society of America

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References

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  1. S. T. Peng, T. Tamir, IEEE Trans. Microwave Theory Tech. MTT-23, 123 (1975).
    [CrossRef]
  2. N. Nevière, R. Petit, M. Cadilhac, Opt. Commun. 8, 113 (1973).
    [CrossRef]
  3. D. Marcuse, Bell. Syst. Tech. J. 55, 1295 (1976).
  4. K. C. Chang, “Surface-Wave Scattering by Dielectric Gratings with Arbitrary Profiles,” Ph.D. Dissertation, Polytechnic Institute of New York (June1979).
  5. W. H. Lee, W. Streifer, J. Opt. Soc. Am. 68, 1701 (1978).
    [CrossRef]
  6. W. Streifer, R. D. Burnham, R. D. Scifres, IEEE J. Quantum Electron. QE-12, 422, 494 (1976).
    [CrossRef]
  7. T. Tamir, S. T. Peng, Appl. Phys. 14, 235 (1977).
    [CrossRef]
  8. K. C. Chang, T. Tamir, Opt. Commun. 26, 327 (1978).
    [CrossRef]
  9. T. Tamir, “Beam and Waveguide Couplers,” in Integrated Optics, T. Tamir, Ed. (Springer, Berlin, 1975), Chap. 3, p. 84.
  10. S. T. Peng, T. Tamir, Opt. Commun. 11, 405 (1974).
    [CrossRef]
  11. T. Aoyagi, Y. Aoyagi, S. Namba, Appl. Phys. Lett. 29, 303 (1976).
    [CrossRef]
  12. L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953), p. 136.
  13. H. Kogelnik, T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).
  14. Ref. 12, p. 172.
  15. T. Tamir, H. C. Wang, A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-13, 297 (1965).
  16. T. Tamir, H. C. Wang, J. Res. Natl. Bur. Stand. 698, 101 (1965).

1978 (2)

K. C. Chang, T. Tamir, Opt. Commun. 26, 327 (1978).
[CrossRef]

W. H. Lee, W. Streifer, J. Opt. Soc. Am. 68, 1701 (1978).
[CrossRef]

1977 (1)

T. Tamir, S. T. Peng, Appl. Phys. 14, 235 (1977).
[CrossRef]

1976 (3)

W. Streifer, R. D. Burnham, R. D. Scifres, IEEE J. Quantum Electron. QE-12, 422, 494 (1976).
[CrossRef]

T. Aoyagi, Y. Aoyagi, S. Namba, Appl. Phys. Lett. 29, 303 (1976).
[CrossRef]

D. Marcuse, Bell. Syst. Tech. J. 55, 1295 (1976).

1975 (1)

S. T. Peng, T. Tamir, IEEE Trans. Microwave Theory Tech. MTT-23, 123 (1975).
[CrossRef]

1974 (1)

S. T. Peng, T. Tamir, Opt. Commun. 11, 405 (1974).
[CrossRef]

1973 (1)

N. Nevière, R. Petit, M. Cadilhac, Opt. Commun. 8, 113 (1973).
[CrossRef]

1970 (1)

H. Kogelnik, T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

1965 (2)

T. Tamir, H. C. Wang, A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-13, 297 (1965).

T. Tamir, H. C. Wang, J. Res. Natl. Bur. Stand. 698, 101 (1965).

Aoyagi, T.

T. Aoyagi, Y. Aoyagi, S. Namba, Appl. Phys. Lett. 29, 303 (1976).
[CrossRef]

Aoyagi, Y.

T. Aoyagi, Y. Aoyagi, S. Namba, Appl. Phys. Lett. 29, 303 (1976).
[CrossRef]

Brillouin, L.

L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953), p. 136.

Burnham, R. D.

W. Streifer, R. D. Burnham, R. D. Scifres, IEEE J. Quantum Electron. QE-12, 422, 494 (1976).
[CrossRef]

Cadilhac, M.

N. Nevière, R. Petit, M. Cadilhac, Opt. Commun. 8, 113 (1973).
[CrossRef]

Chang, K. C.

K. C. Chang, T. Tamir, Opt. Commun. 26, 327 (1978).
[CrossRef]

K. C. Chang, “Surface-Wave Scattering by Dielectric Gratings with Arbitrary Profiles,” Ph.D. Dissertation, Polytechnic Institute of New York (June1979).

Kogelnik, H.

H. Kogelnik, T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

Lee, W. H.

Marcuse, D.

D. Marcuse, Bell. Syst. Tech. J. 55, 1295 (1976).

Namba, S.

T. Aoyagi, Y. Aoyagi, S. Namba, Appl. Phys. Lett. 29, 303 (1976).
[CrossRef]

Nevière, N.

N. Nevière, R. Petit, M. Cadilhac, Opt. Commun. 8, 113 (1973).
[CrossRef]

Oliner, A. A.

T. Tamir, H. C. Wang, A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-13, 297 (1965).

Peng, S. T.

T. Tamir, S. T. Peng, Appl. Phys. 14, 235 (1977).
[CrossRef]

S. T. Peng, T. Tamir, IEEE Trans. Microwave Theory Tech. MTT-23, 123 (1975).
[CrossRef]

S. T. Peng, T. Tamir, Opt. Commun. 11, 405 (1974).
[CrossRef]

Petit, R.

N. Nevière, R. Petit, M. Cadilhac, Opt. Commun. 8, 113 (1973).
[CrossRef]

Scifres, R. D.

W. Streifer, R. D. Burnham, R. D. Scifres, IEEE J. Quantum Electron. QE-12, 422, 494 (1976).
[CrossRef]

Sosnowski, T. P.

H. Kogelnik, T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

Streifer, W.

W. H. Lee, W. Streifer, J. Opt. Soc. Am. 68, 1701 (1978).
[CrossRef]

W. Streifer, R. D. Burnham, R. D. Scifres, IEEE J. Quantum Electron. QE-12, 422, 494 (1976).
[CrossRef]

Tamir, T.

K. C. Chang, T. Tamir, Opt. Commun. 26, 327 (1978).
[CrossRef]

T. Tamir, S. T. Peng, Appl. Phys. 14, 235 (1977).
[CrossRef]

S. T. Peng, T. Tamir, IEEE Trans. Microwave Theory Tech. MTT-23, 123 (1975).
[CrossRef]

S. T. Peng, T. Tamir, Opt. Commun. 11, 405 (1974).
[CrossRef]

T. Tamir, H. C. Wang, J. Res. Natl. Bur. Stand. 698, 101 (1965).

T. Tamir, H. C. Wang, A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-13, 297 (1965).

T. Tamir, “Beam and Waveguide Couplers,” in Integrated Optics, T. Tamir, Ed. (Springer, Berlin, 1975), Chap. 3, p. 84.

Wang, H. C.

T. Tamir, H. C. Wang, A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-13, 297 (1965).

T. Tamir, H. C. Wang, J. Res. Natl. Bur. Stand. 698, 101 (1965).

Appl. Phys. (1)

T. Tamir, S. T. Peng, Appl. Phys. 14, 235 (1977).
[CrossRef]

Appl. Phys. Lett. (1)

T. Aoyagi, Y. Aoyagi, S. Namba, Appl. Phys. Lett. 29, 303 (1976).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, T. P. Sosnowski, Bell Syst. Tech. J. 49, 1602 (1970).

Bell. Syst. Tech. J. (1)

D. Marcuse, Bell. Syst. Tech. J. 55, 1295 (1976).

IEEE J. Quantum Electron. (1)

W. Streifer, R. D. Burnham, R. D. Scifres, IEEE J. Quantum Electron. QE-12, 422, 494 (1976).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

S. T. Peng, T. Tamir, IEEE Trans. Microwave Theory Tech. MTT-23, 123 (1975).
[CrossRef]

T. Tamir, H. C. Wang, A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-13, 297 (1965).

J. Opt. Soc. Am. (1)

J. Res. Natl. Bur. Stand. (1)

T. Tamir, H. C. Wang, J. Res. Natl. Bur. Stand. 698, 101 (1965).

Opt. Commun. (3)

S. T. Peng, T. Tamir, Opt. Commun. 11, 405 (1974).
[CrossRef]

K. C. Chang, T. Tamir, Opt. Commun. 26, 327 (1978).
[CrossRef]

N. Nevière, R. Petit, M. Cadilhac, Opt. Commun. 8, 113 (1973).
[CrossRef]

Other (4)

K. C. Chang, “Surface-Wave Scattering by Dielectric Gratings with Arbitrary Profiles,” Ph.D. Dissertation, Polytechnic Institute of New York (June1979).

T. Tamir, “Beam and Waveguide Couplers,” in Integrated Optics, T. Tamir, Ed. (Springer, Berlin, 1975), Chap. 3, p. 84.

Ref. 12, p. 172.

L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953), p. 136.

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Figures (9)

Fig. 1
Fig. 1

Scattering of a surface wave incident on a triangular grating: (a) geometry of the scattering configuration; (h) Bragg-reflection mechanism for wave scattering inside the grating region.

Fig. 2
Fig. 2

Circle diagram for analyzing scattering by the facets of triangular gratings. As shown, the angle γ1 satisfies the Bragg condition (8).

Fig. 3
Fig. 3

Circle diagram for assessing scattering by the grating represented in Fig. 1. Here γ1 is adjusted to satisfy a Bragg condition, as in Fig. 2, but γ2 is assumed variable, with g 2 ( 0 ) - g 2 ( 5 ) representing various possible situations.

Fig. 4
Fig. 4

Efficiency ηa and normalized leakage αλ for a grating having a variable triangular profile. Here tg = tgB satisfies a Bragg condition if either γ1 or γ2 is zero.

Fig. 5
Fig. 5

Efficiency ηa and leakage α for various heights tg in a dielectric grating, for which the Bragg condition (12) is satisfied at tg = 0.25 μm, as indicated by the arrow.

Fig. 6
Fig. 6

Efficiency ηa and normalized leakage αλ for varying height tg in a dielectric grating, for which the Bragg condition (12) is satisfied at tg/λ = 0.688, as indicated by the arrow.

Fig. 7
Fig. 7

Efficiency ηa and normalized leakage αλ for the grating in Fig. 6, but with the teeth missing their upper portion, to show the effect of horizontal cuts in the grating profile.

Fig. 8
Fig. 8

Efficiency ηa and normalized leakage αλ for the grating in Fig. 6, but with the teeth missing their right-hand portion, to show the effect of vertical cuts in the grating profile.

Fig. 9
Fig. 9

Efficiency ηa and normalized leakage αλ for a trapezoidal grating with variable horizontal portions d0. The Bragg condition (12) is now satisfied at d0 = 0.

Equations (27)

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k x n = β 0 + 2 n π d + i α ,             ( n = 0 , ± 1 , ± 2 ) .
α β 0 β s w ,
sin θ a = ( 2 π / d ) - β 0 k 0 n a ( λ / n a d ) - ( N / n a ) ,
β - 1 = β 0 - ( 2 π / d )
η a + η s = 1 ,
n g = ( n r 2 + n a 2 2 ) 1 / 2 .
β s w · A B ¯ + k 0 n g · B C ¯ = β s w d + k 0 n g d cos 2 γ 1 = 2 π ,
cos 2 γ 1 = - β - 1 k 0 n g = 1 n g ( λ d - N ) .
n a sin θ a = n g sin θ g ,
g 1 = - ( n g - N + λ d ) ( x ^ 0 - z ^ 0 tan γ 1 ) ,
g 2 = - ( n g - N + λ d ) ( x ^ 0 + z ^ 0 tan γ 2 ) ,
t g B d = cot γ B = [ n g - N + ( λ / d ) n g + N - ( λ / d ) ] 1 / 2 .
( x ) = n g 2 ( 1 + M cos 2 π x d ) ,
d = d cos γ 1 ,
M = M ( z ) = 2 π · n r 2 - n a 2 n g 2 sin π z t g .
M = n r 2 - n a 2 n g 2 A ,
E ( x , z ) = F ( x ) exp ( i k 0 n g z sin γ 1 ) .
d 2 F ( d x ) 2 + ( p - 2 q cos 2 π x d ) F = 0 ,
p = [ ( 2 n g d / λ ) cos γ 1 ] 2 ,
q = ( M / 2 ) ( 2 n g d / λ ) 2 .
1 - q p 1 + q .
Δ p = ± q .
Δ γ = ± A 4 · n r 2 - n a 2 n g 2 sin 2 γ B .
n g 2 = n g 2 B + Δ n g 2 ,
± Δ n g 2 = n r 2 - n a 2 2 ( 1 - t g t g B ) ,
M = n r 2 - n a 2 n g 2 · t g t g B A
t g / t g B = ( 1 + A cos 2 γ B ) - 1 ,

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