Abstract

Diffraction of electromagnetic waves by longitudinally periodic media (reflection gratings) is rigorously analyzed using an exact coupled-wave approach. The analysis is formulated in a simple matrix form easily implemented on a digital computer. The intensities of the two waves, the diffracted (reflected) and transmitted waves, are calculated for a wide range of parameters. These exact results are then compared to results obtained with approximate two-wave modal, and with multiwave and two-wave coupled-wave analyses. The applicability of these approximate theories is discussed. The case of other than first-order Bragg incidence is also included.

© 1981 Optical Society of America

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References

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  1. T. Tamir, H. C. Wang, A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-12, 323 (1964).
    [CrossRef]
  2. T. Tamir, H. C. Wang, Can. J. Phys. 44, 2073 (1966).
    [CrossRef]
  3. T. Tamir, Can. J. Phys. 44, 2461 (1966).
    [CrossRef]
  4. C. B. Burckhardt, J. Opt. Soc. Am. 56, 1502 (1966).
    [CrossRef]
  5. L. Bergstein, D. Kermisch, Proc. Symp. Modern Opt. 17, 655 (1967).
  6. R. S. Chu, T. Tamir, IEEE Trans. Microwave Theory Tech. MTT-18, 486 (1970).
  7. R. S. Chu, T. Tamir, Proc. IEE 119, 797 (1972).
  8. F. G. Kaspar, J. Opt. Soc. Am. 63, 37 (1973).
    [CrossRef]
  9. R.S. Chu, J. A. Kong, IEEE Trans. Microwave Theory Tech. MTT-25, 18 (1977).
  10. S. T. Peng, E. S. Cassedy, Proc. Symp. Modem Opt. 17, 299 (1967).
  11. P. Phariseau, Proc. Indian Acad. Sci. Sect. A. 44, 165 (1965).
  12. W. R. Klein, B. D. Cook, IEEE Trans. Sonic Ultrason. SU-14, 123 (1967).
    [CrossRef]
  13. H. Iogelnik, Bell Sys. Tech. J. 48, 2909 (1969).
  14. R. Magnusson, T. K. Gaylord, J. Opt. Soc. Am. 67, 1165 (1977).
    [CrossRef]
  15. J. A. Kong, J. Opt. Soc. Am. 67, 825 (1977).
    [CrossRef]
  16. M. G. Moharam, T. K. Gaylord, J. Opt. Soc. Am., (1981) submitted.
  17. M. G. Moharam, T. K. Gaylord, R. Magnusson, Opt. Commun. 32, 14 (1980).
    [CrossRef]

1980 (1)

M. G. Moharam, T. K. Gaylord, R. Magnusson, Opt. Commun. 32, 14 (1980).
[CrossRef]

1977 (3)

1973 (1)

1972 (1)

R. S. Chu, T. Tamir, Proc. IEE 119, 797 (1972).

1970 (1)

R. S. Chu, T. Tamir, IEEE Trans. Microwave Theory Tech. MTT-18, 486 (1970).

1969 (1)

H. Iogelnik, Bell Sys. Tech. J. 48, 2909 (1969).

1967 (3)

W. R. Klein, B. D. Cook, IEEE Trans. Sonic Ultrason. SU-14, 123 (1967).
[CrossRef]

S. T. Peng, E. S. Cassedy, Proc. Symp. Modem Opt. 17, 299 (1967).

L. Bergstein, D. Kermisch, Proc. Symp. Modern Opt. 17, 655 (1967).

1966 (3)

T. Tamir, H. C. Wang, Can. J. Phys. 44, 2073 (1966).
[CrossRef]

T. Tamir, Can. J. Phys. 44, 2461 (1966).
[CrossRef]

C. B. Burckhardt, J. Opt. Soc. Am. 56, 1502 (1966).
[CrossRef]

1965 (1)

P. Phariseau, Proc. Indian Acad. Sci. Sect. A. 44, 165 (1965).

1964 (1)

T. Tamir, H. C. Wang, A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-12, 323 (1964).
[CrossRef]

Bergstein, L.

L. Bergstein, D. Kermisch, Proc. Symp. Modern Opt. 17, 655 (1967).

Burckhardt, C. B.

Cassedy, E. S.

S. T. Peng, E. S. Cassedy, Proc. Symp. Modem Opt. 17, 299 (1967).

Chu, R. S.

R. S. Chu, T. Tamir, Proc. IEE 119, 797 (1972).

R. S. Chu, T. Tamir, IEEE Trans. Microwave Theory Tech. MTT-18, 486 (1970).

Chu, R.S.

R.S. Chu, J. A. Kong, IEEE Trans. Microwave Theory Tech. MTT-25, 18 (1977).

Cook, B. D.

W. R. Klein, B. D. Cook, IEEE Trans. Sonic Ultrason. SU-14, 123 (1967).
[CrossRef]

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, R. Magnusson, Opt. Commun. 32, 14 (1980).
[CrossRef]

R. Magnusson, T. K. Gaylord, J. Opt. Soc. Am. 67, 1165 (1977).
[CrossRef]

M. G. Moharam, T. K. Gaylord, J. Opt. Soc. Am., (1981) submitted.

Iogelnik, H.

H. Iogelnik, Bell Sys. Tech. J. 48, 2909 (1969).

Kaspar, F. G.

Kermisch, D.

L. Bergstein, D. Kermisch, Proc. Symp. Modern Opt. 17, 655 (1967).

Klein, W. R.

W. R. Klein, B. D. Cook, IEEE Trans. Sonic Ultrason. SU-14, 123 (1967).
[CrossRef]

Kong, J. A.

R.S. Chu, J. A. Kong, IEEE Trans. Microwave Theory Tech. MTT-25, 18 (1977).

J. A. Kong, J. Opt. Soc. Am. 67, 825 (1977).
[CrossRef]

Magnusson, R.

M. G. Moharam, T. K. Gaylord, R. Magnusson, Opt. Commun. 32, 14 (1980).
[CrossRef]

R. Magnusson, T. K. Gaylord, J. Opt. Soc. Am. 67, 1165 (1977).
[CrossRef]

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, R. Magnusson, Opt. Commun. 32, 14 (1980).
[CrossRef]

M. G. Moharam, T. K. Gaylord, J. Opt. Soc. Am., (1981) submitted.

Oliner, A. A.

T. Tamir, H. C. Wang, A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-12, 323 (1964).
[CrossRef]

Peng, S. T.

S. T. Peng, E. S. Cassedy, Proc. Symp. Modem Opt. 17, 299 (1967).

Phariseau, P.

P. Phariseau, Proc. Indian Acad. Sci. Sect. A. 44, 165 (1965).

Tamir, T.

R. S. Chu, T. Tamir, Proc. IEE 119, 797 (1972).

R. S. Chu, T. Tamir, IEEE Trans. Microwave Theory Tech. MTT-18, 486 (1970).

T. Tamir, H. C. Wang, Can. J. Phys. 44, 2073 (1966).
[CrossRef]

T. Tamir, Can. J. Phys. 44, 2461 (1966).
[CrossRef]

T. Tamir, H. C. Wang, A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-12, 323 (1964).
[CrossRef]

Wang, H. C.

T. Tamir, H. C. Wang, Can. J. Phys. 44, 2073 (1966).
[CrossRef]

T. Tamir, H. C. Wang, A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-12, 323 (1964).
[CrossRef]

Bell Sys. Tech. J. (1)

H. Iogelnik, Bell Sys. Tech. J. 48, 2909 (1969).

Can. J. Phys. (2)

T. Tamir, H. C. Wang, Can. J. Phys. 44, 2073 (1966).
[CrossRef]

T. Tamir, Can. J. Phys. 44, 2461 (1966).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

R. S. Chu, T. Tamir, IEEE Trans. Microwave Theory Tech. MTT-18, 486 (1970).

R.S. Chu, J. A. Kong, IEEE Trans. Microwave Theory Tech. MTT-25, 18 (1977).

T. Tamir, H. C. Wang, A. A. Oliner, IEEE Trans. Microwave Theory Tech. MTT-12, 323 (1964).
[CrossRef]

IEEE Trans. Sonic Ultrason. (1)

W. R. Klein, B. D. Cook, IEEE Trans. Sonic Ultrason. SU-14, 123 (1967).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt. Commun. (1)

M. G. Moharam, T. K. Gaylord, R. Magnusson, Opt. Commun. 32, 14 (1980).
[CrossRef]

Proc. IEE (1)

R. S. Chu, T. Tamir, Proc. IEE 119, 797 (1972).

Proc. Indian Acad. Sci. Sect. A. (1)

P. Phariseau, Proc. Indian Acad. Sci. Sect. A. 44, 165 (1965).

Proc. Symp. Modem Opt. (1)

S. T. Peng, E. S. Cassedy, Proc. Symp. Modem Opt. 17, 299 (1967).

Proc. Symp. Modern Opt. (1)

L. Bergstein, D. Kermisch, Proc. Symp. Modern Opt. 17, 655 (1967).

Other (1)

M. G. Moharam, T. K. Gaylord, J. Opt. Soc. Am., (1981) submitted.

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Figures (4)

Fig. 1
Fig. 1

Geometry for diffraction of an electromagnetic wave by a planar reflection grating.

Fig. 2
Fig. 2

Calculated intensity of the diffracted (reflected) wave for incidence at the first Bragg condition (B = 1).

Fig. 3
Fig. 3

The absolute error in the diffraction efficiency of the diffracted (reflected) wave defined as the difference between the diffraction efficiency calculated using the present exact analysis and the diffraction efficiency calculated using various approximate theories: (a) Kogelnik’s two-wave coupled-wave analysis; (b) Magnusson and Gaylord’s multiwave coupled-wave analysis; and (c) Bergstein and Kermisch’s two-wave modal analysis.

Fig. 4
Fig. 4

Calculated diffraction efficiency of the diffracted (reflected) wave for incidence at the second Bragg condition (B = 2).

Equations (17)

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( x , z ) = 2 + Δ cos [ K ( x sin ϕ - z cos ϕ ) ] ,
E 1 = exp [ - j ( β 0 x + ξ 10 z ) ] + i R i exp [ - j ( β i x - ξ 1 i z ) ] .
E 3 = i T i exp { - j [ β i x + ξ 3 i ( z - d ) ] } .
E 2 = i S ^ i ( z ) exp [ - j ( β i x + ξ 2 i z ) ] ,
2 E 2 + ( 2 π / λ ) 2 ( x , z ) E 2 = 0 ,
1 ρ B 2 d 2 S i ( v ) d v 2 = [ 1 - ( 2 i / B ) ] d S i ( v ) d v - ρ i ( i - B ) S i ( v ) + S i + 1 ( v ) + S i - 1 ( v ) ,
[ S S ] = [ b r s ] [ S S ] ,
S i ( v ) = m C m w i m exp ( q m v ) ,
2 δ i 0 ξ 1 i d = m C m w i m ( ξ 2 i d + ξ 1 i d - q m γ ) ,
0 = m C m w i m ( ξ 2 i d - ξ 3 i d - q m γ ) exp ( j q m γ ) ,
δ i 0 + R i = m C m w i m ,
T i = m C m w i m exp [ j ( q m γ - ξ 2 i d ) ] ,
S 0 ( v ) = cos h ( γ + j v ) / cos h γ ,
S 1 ( v ) = - j sin h ( γ + j v ) / cos h γ ,
S i ( v ) = 0 ,             i 0 , 1.
I r = R 1 R 1 * = S 1 ( 0 ) S 1 * ( 0 ) = tan h 2 γ .
I r = i > 0 ( 2 i - 1 ) S i ( 0 ) S i * ( 0 ) .

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