Abstract

A general numerical technique for analyzing diffraction gratings of arbitrary groove shape as electromagnetic boundary-value problems is suggested. The method yields the scattered far-field modal amplitude without the necessity of initially computing the induced surface currents, which makes this technique very simple and convenient for analyzing the performance of diffraction gratings under normal and anomalous conditions. Some available computed results for rectangular gratings for the E-polarized case are compared with similar results in the literature and the adequacy and accuracy of the technique are demonstrated.

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References

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  1. H. A. Kalhor and A. R. Neureuther, J. Opt. Soc. Am. 61, 43 (1971).
    [CrossRef]
  2. Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. H, p. 171.
  3. W. C. Meecham, J. Appl. Phys. 27, 361 (1956).
    [CrossRef]
  4. R. Petit and M. Cadilhac, C.R. Acad. Sci. (Paris) 262, 468 (1966).
  5. R. F. Millar, Proc. Camb. Philos. Soc. 65, 773 (1969).
    [CrossRef]
  6. S. O. Rice, Theory of Electromagnetic Waves (Wiley-Interscience; Dover, New York, 1951).
  7. M. L. Burrows, Electron. Lett. 5, 277 (1969).
    [CrossRef]
  8. J. R. Wait, Radio Sci. 6, 387 (1971).
    [CrossRef]
  9. D. J. Fang, IEEE Trans. Antennas Propag. 20, 388 (1972).
    [CrossRef]
  10. V. Jamnejad-Dailami, R. Mittra, and T. Itoh, IEEE Trans. Antennas Propag. 20, 390 (1972).
    [CrossRef]
  11. R. Petit., Rev. Opt. Theor. Instrum. 45, 249 (1966).
  12. H. Kalhor, Ph.D. thesis, Univ. of California, Berkeley (1970) (University Microfilms, Ann Arbor, Mich., Order No. 71-15 809).
  13. K. A. Zaki and A. R. Neureuther, IEEE Trans. Antennas Propag. 19, 208 (1971).
    [CrossRef]
  14. A. Wirgin and R. Deleuil, J. Opt. Soc. Am. 59, 1348 (1969).
    [CrossRef]

1972

D. J. Fang, IEEE Trans. Antennas Propag. 20, 388 (1972).
[CrossRef]

V. Jamnejad-Dailami, R. Mittra, and T. Itoh, IEEE Trans. Antennas Propag. 20, 390 (1972).
[CrossRef]

1971

J. R. Wait, Radio Sci. 6, 387 (1971).
[CrossRef]

H. A. Kalhor and A. R. Neureuther, J. Opt. Soc. Am. 61, 43 (1971).
[CrossRef]

K. A. Zaki and A. R. Neureuther, IEEE Trans. Antennas Propag. 19, 208 (1971).
[CrossRef]

1970

H. Kalhor, Ph.D. thesis, Univ. of California, Berkeley (1970) (University Microfilms, Ann Arbor, Mich., Order No. 71-15 809).

1969

A. Wirgin and R. Deleuil, J. Opt. Soc. Am. 59, 1348 (1969).
[CrossRef]

R. F. Millar, Proc. Camb. Philos. Soc. 65, 773 (1969).
[CrossRef]

M. L. Burrows, Electron. Lett. 5, 277 (1969).
[CrossRef]

1966

R. Petit and M. Cadilhac, C.R. Acad. Sci. (Paris) 262, 468 (1966).

R. Petit., Rev. Opt. Theor. Instrum. 45, 249 (1966).

1956

W. C. Meecham, J. Appl. Phys. 27, 361 (1956).
[CrossRef]

Burrows, M. L.

M. L. Burrows, Electron. Lett. 5, 277 (1969).
[CrossRef]

Cadilhac, M.

R. Petit and M. Cadilhac, C.R. Acad. Sci. (Paris) 262, 468 (1966).

Deleuil, R.

A. Wirgin and R. Deleuil, J. Opt. Soc. Am. 59, 1348 (1969).
[CrossRef]

Fang, D. J.

D. J. Fang, IEEE Trans. Antennas Propag. 20, 388 (1972).
[CrossRef]

Itoh, T.

V. Jamnejad-Dailami, R. Mittra, and T. Itoh, IEEE Trans. Antennas Propag. 20, 390 (1972).
[CrossRef]

Jamnejad-Dailami, V.

V. Jamnejad-Dailami, R. Mittra, and T. Itoh, IEEE Trans. Antennas Propag. 20, 390 (1972).
[CrossRef]

Kalhor, H.

H. Kalhor, Ph.D. thesis, Univ. of California, Berkeley (1970) (University Microfilms, Ann Arbor, Mich., Order No. 71-15 809).

Kalhor, H. A.

H. A. Kalhor and A. R. Neureuther, J. Opt. Soc. Am. 61, 43 (1971).
[CrossRef]

Meecham, W. C.

W. C. Meecham, J. Appl. Phys. 27, 361 (1956).
[CrossRef]

Millar, R. F.

R. F. Millar, Proc. Camb. Philos. Soc. 65, 773 (1969).
[CrossRef]

Mittra, R.

V. Jamnejad-Dailami, R. Mittra, and T. Itoh, IEEE Trans. Antennas Propag. 20, 390 (1972).
[CrossRef]

Neureuther, A. R.

K. A. Zaki and A. R. Neureuther, IEEE Trans. Antennas Propag. 19, 208 (1971).
[CrossRef]

H. A. Kalhor and A. R. Neureuther, J. Opt. Soc. Am. 61, 43 (1971).
[CrossRef]

Petit, R.

R. Petit and M. Cadilhac, C.R. Acad. Sci. (Paris) 262, 468 (1966).

R. Petit., Rev. Opt. Theor. Instrum. 45, 249 (1966).

Rayleigh, Lord

Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. H, p. 171.

Rice, S. O.

S. O. Rice, Theory of Electromagnetic Waves (Wiley-Interscience; Dover, New York, 1951).

Wait, J. R.

J. R. Wait, Radio Sci. 6, 387 (1971).
[CrossRef]

Wirgin, A.

A. Wirgin and R. Deleuil, J. Opt. Soc. Am. 59, 1348 (1969).
[CrossRef]

Zaki, K. A.

K. A. Zaki and A. R. Neureuther, IEEE Trans. Antennas Propag. 19, 208 (1971).
[CrossRef]

Other

H. A. Kalhor and A. R. Neureuther, J. Opt. Soc. Am. 61, 43 (1971).
[CrossRef]

Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. H, p. 171.

W. C. Meecham, J. Appl. Phys. 27, 361 (1956).
[CrossRef]

R. Petit and M. Cadilhac, C.R. Acad. Sci. (Paris) 262, 468 (1966).

R. F. Millar, Proc. Camb. Philos. Soc. 65, 773 (1969).
[CrossRef]

S. O. Rice, Theory of Electromagnetic Waves (Wiley-Interscience; Dover, New York, 1951).

M. L. Burrows, Electron. Lett. 5, 277 (1969).
[CrossRef]

J. R. Wait, Radio Sci. 6, 387 (1971).
[CrossRef]

D. J. Fang, IEEE Trans. Antennas Propag. 20, 388 (1972).
[CrossRef]

V. Jamnejad-Dailami, R. Mittra, and T. Itoh, IEEE Trans. Antennas Propag. 20, 390 (1972).
[CrossRef]

R. Petit., Rev. Opt. Theor. Instrum. 45, 249 (1966).

H. Kalhor, Ph.D. thesis, Univ. of California, Berkeley (1970) (University Microfilms, Ann Arbor, Mich., Order No. 71-15 809).

K. A. Zaki and A. R. Neureuther, IEEE Trans. Antennas Propag. 19, 208 (1971).
[CrossRef]

A. Wirgin and R. Deleuil, J. Opt. Soc. Am. 59, 1348 (1969).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Geometry of the rectangular grating of period d, groove length a, and groove height b.

Fig. 2
Fig. 2

Arrangement of division of the groove area into cells and generation of nodes, showing p divisions in groove length and q divisions in its height.

Fig. 3
Fig. 3

Enlarged diagram of arrangement of nodes entering each finite-difference equation.

Fig. 4
Fig. 4

Rectangular grating: groove length = 0.5d, groove height = 0.25d, E-polarized θi = 30°. Normalized energy of three scattering modes by Wirgin and Deleuil (solid line), Kalhor (crosses), and the new numerical technique (dots) against normalized grating period (d/λ).

Fig. 5
Fig. 5

Rectangular grating: d = 17.2 mm, a = 6 mm, b = 4.3 mm, and λ = 8.58 mm. Normalized zero-order energy by Wirgin and Deleuil (solid line), by Kalhor (crosses), and the new numerical technique (dots) against angle of incidence.

Equations (19)

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( 2 + k 2 ) ϕ t ( x , z ) = 0 ,
ϕ t = 0 for the E - polarized case , d ϕ t d n ˆ = 0 for the H - polarized case .
E ¯ i = y ˆ E 0 exp ( j β 0 x ) exp ( j γ 0 z ) ,
β 0 = k sin θ i , γ 0 = k cos θ i ,
E ¯ s = Y ˆ n = A n exp ( j β n x ) · exp ( j γ n z ) ,
β n = β 0 + 2 π n d = k sin θ i + 2 π n d ,
γ n = ( k n 2 β n 2 ) 1 2 for propagating modes , γ n = j ( β n 2 k 2 ) 1 2 for evanescent modes .
E ¯ t = y ˆ E 0 exp ( j β 0 x ) · exp ( j γ 0 z ) + y ˆ n = A n exp ( j β n x ) exp ( j γ n z ) .
E ¯ t = y ˆ E 0 exp ( j β 0 x ) · exp ( j γ 0 z ) + y ˆ n = N N A n exp ( j β n x ) · exp ( j γ n z ) .
( p + 1 ) ( q + 1 ) { 2 q + ( p 1 ) } = ( p 1 ) q
( p 1 ) q ( p 1 ) = ( p 1 ) ( q 1 )
ϕ i + 1 , j t 2 ϕ i , j t + ϕ i 1 , j ( Δ x ) 2 + ϕ i , j + 1 t 2 ϕ i , j t + ϕ i , j 1 t ( Δ z ) 2 + k 2 ϕ i , j t = 0.
E 0 exp ( j β 0 x ) + n = N N A n exp ( j β n x ) = ϕ i , q + 1 t { μ ( x i ) μ ( x i 1 ) } for 0 < x < a , = 0 for a < x < d ,
i = 1 , 2 , 3 ,..., ( p + 1 ) ,
A n = 1 d 0 a ϕ i , q + 1 t { μ ( x i ) μ ( x i 1 ) } exp ( j β n x ) d x 1 d 0 a E 0 exp ( j β 0 x ) · exp ( j β n x ) d x .
A n = 1 d i = 2 p + 1 ϕ i , q + 1 t exp ( j β n x i ) exp ( j β n x i 1 ) j β n
A 0 = 1 d i = 2 p + 1 ϕ i , q + 1 t exp ( j β 0 x i ) exp ( j β 0 x i 1 ) j β 0 E 0 a d .
j γ 0 E 0 exp ( j β 0 x i ) + n = N N j γ n A m exp ( j β n x i ) = ϕ i , q + 1 t ϕ i , q t Δ z , i = 2 , 3 ,..., p .
( 2 N + 1 ) + ( p 1 ) q