Abstract

For plane electromagnetic wave incidence on a perfectly conducting periodic surface with a rectangular groove profile, the coefficients of the matrices governing the scattered mode amplitudes are given. These matrices decompose when the Bragg condition is satisfied, facilitating computation. Numerical results converge to those of the comb grating as the groove width approaches the period. Data for perfect blazing to the n = −1 spectral order (i.e., all the power is confined to this order) for TE, TM, and arbitrary polarization are given. Perfect blazing with arbitrary polarization for near-grazing incidence is shown to be possible in principle with deep wide grooves. Numerical and experimental results are shown also for narrow groove widths, effective for TM polarization only.

© 1978 Optical Society of America

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References

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  1. A. Wirgin, “On the theory of scattering from rough lamellar surfaces,” Alta. Freq. 38, 327–331 (1969).
  2. A. Wirgin and R. Deleuil, “Theoretical and experimental investigation of a new type of blazed grating,” J. Opt. Soc. Am. 59, 1348–1357 (1969).
    [Crossref]
  3. A. Hessel, J. Shmoys, and D. Y. Tseng, “Bragg-angle blazing of diffraction gratings,” J. Opt. Soc. Am. 65, 380–384 (1975).
    [Crossref]
  4. J. L. Roumiguieres, D. Maystre, and R. Petit, “On the efficiencies of rectangular-groove gratings,” J. Opt. Soc. Am. 66, 772–775 (1976).
    [Crossref]
  5. E. V. Jull, J. W. Heath, and G. R. Ebbeson, “Gratings that diffract all incident energy,” J. Opt. Soc. Am. 67, 557–560 (1977).
    [Crossref]
  6. J. A. DeSanto, “Scattering from a periodic corrugated structure: Thin comb with soft boundaries,” J. Math. Phys. 12, 1913–1923 (1971); “Scattering from a periodic corrugated structure: Thin comb with hard boundaries,” J. Math. Phys. 13, 336–341 (1972).
    [Crossref]
  7. G. R. Ebbeson, “TM polarized electromagnetic scattering from fin-corrugated periodic surfaces,” J. Opt. Soc. Am. 66, 1363–1367 (1976).
    [Crossref]
  8. J. W. Heath, “Scattering by a conducting periodic surface with a rectangular groove profile,” M.A.Sc. Thesis, University of British Columbia (1977).
  9. E. V. Jull and G. R. Ebbeson, “The reduction of interference from large reflecting surfaces,” IEEE Trans. Antennas Prop. AP-25, 565–570 (1977).
    [Crossref]
  10. P. Facq, “Application des matrices de Toeplitz a la théorie de la diffraction par les structures cylindriques périodiques limitées,” D.Sc. Thesis, University of Limoges (1977).

1977 (2)

E. V. Jull and G. R. Ebbeson, “The reduction of interference from large reflecting surfaces,” IEEE Trans. Antennas Prop. AP-25, 565–570 (1977).
[Crossref]

E. V. Jull, J. W. Heath, and G. R. Ebbeson, “Gratings that diffract all incident energy,” J. Opt. Soc. Am. 67, 557–560 (1977).
[Crossref]

1976 (2)

1975 (1)

1971 (1)

J. A. DeSanto, “Scattering from a periodic corrugated structure: Thin comb with soft boundaries,” J. Math. Phys. 12, 1913–1923 (1971); “Scattering from a periodic corrugated structure: Thin comb with hard boundaries,” J. Math. Phys. 13, 336–341 (1972).
[Crossref]

1969 (2)

A. Wirgin, “On the theory of scattering from rough lamellar surfaces,” Alta. Freq. 38, 327–331 (1969).

A. Wirgin and R. Deleuil, “Theoretical and experimental investigation of a new type of blazed grating,” J. Opt. Soc. Am. 59, 1348–1357 (1969).
[Crossref]

Deleuil, R.

DeSanto, J. A.

J. A. DeSanto, “Scattering from a periodic corrugated structure: Thin comb with soft boundaries,” J. Math. Phys. 12, 1913–1923 (1971); “Scattering from a periodic corrugated structure: Thin comb with hard boundaries,” J. Math. Phys. 13, 336–341 (1972).
[Crossref]

Ebbeson, G. R.

Facq, P.

P. Facq, “Application des matrices de Toeplitz a la théorie de la diffraction par les structures cylindriques périodiques limitées,” D.Sc. Thesis, University of Limoges (1977).

Heath, J. W.

E. V. Jull, J. W. Heath, and G. R. Ebbeson, “Gratings that diffract all incident energy,” J. Opt. Soc. Am. 67, 557–560 (1977).
[Crossref]

J. W. Heath, “Scattering by a conducting periodic surface with a rectangular groove profile,” M.A.Sc. Thesis, University of British Columbia (1977).

Hessel, A.

Jull, E. V.

E. V. Jull and G. R. Ebbeson, “The reduction of interference from large reflecting surfaces,” IEEE Trans. Antennas Prop. AP-25, 565–570 (1977).
[Crossref]

E. V. Jull, J. W. Heath, and G. R. Ebbeson, “Gratings that diffract all incident energy,” J. Opt. Soc. Am. 67, 557–560 (1977).
[Crossref]

Maystre, D.

Petit, R.

Roumiguieres, J. L.

Shmoys, J.

Tseng, D. Y.

Wirgin, A.

A. Wirgin, “On the theory of scattering from rough lamellar surfaces,” Alta. Freq. 38, 327–331 (1969).

A. Wirgin and R. Deleuil, “Theoretical and experimental investigation of a new type of blazed grating,” J. Opt. Soc. Am. 59, 1348–1357 (1969).
[Crossref]

Alta. Freq. (1)

A. Wirgin, “On the theory of scattering from rough lamellar surfaces,” Alta. Freq. 38, 327–331 (1969).

IEEE Trans. Antennas Prop. (1)

E. V. Jull and G. R. Ebbeson, “The reduction of interference from large reflecting surfaces,” IEEE Trans. Antennas Prop. AP-25, 565–570 (1977).
[Crossref]

J. Math. Phys. (1)

J. A. DeSanto, “Scattering from a periodic corrugated structure: Thin comb with soft boundaries,” J. Math. Phys. 12, 1913–1923 (1971); “Scattering from a periodic corrugated structure: Thin comb with hard boundaries,” J. Math. Phys. 13, 336–341 (1972).
[Crossref]

J. Opt. Soc. Am. (5)

Other (2)

J. W. Heath, “Scattering by a conducting periodic surface with a rectangular groove profile,” M.A.Sc. Thesis, University of British Columbia (1977).

P. Facq, “Application des matrices de Toeplitz a la théorie de la diffraction par les structures cylindriques périodiques limitées,” D.Sc. Thesis, University of Limoges (1977).

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Figures (8)

FIG. 1
FIG. 1

A plane wave incident on a periodic surface with rectangular groove profile.

FIG. 2
FIG. 2

TM reflected power for large aspect ratios: θi = 81.25°, d = λ/2 sinθi; - - -, a/d = 1.0 (from Ref. 7); —, a/d = 0.9999.

FIG. 3
FIG. 3

TM reflected power for small aspect ratios: θi = 55°, d = λ/(2 sinθi).

FIG. 4
FIG. 4

Blazing depth curves for large aspect ratios a/d, d = λ/2 sinθi; - - -, a/d = 0.99, — - - —, a/d = 0.995; —, a/d = 0.9999; — - —, a/d = 1.0 from [7].

FIG. 5
FIG. 5

Blazing depth curves for small aspect ratios a/d, d = λ/2 sinθi.

FIG. 6
FIG. 6

Blazing depth curves for a/d = 0.99, d = λ/2 sinθi; —, lowest order; - - -, higher order. Simultaneous blazing occurs at: θi; = 40.5° (h/λ = 1.088), θi = 58.5° (h/λ = 0.556), and θi = 73.1° (h/λ = 1.049).

FIG. 7
FIG. 7

Measured and calculated TM reflected power for a grating with 56 grooves, λ = 8.57 mm, d = 4.75 mm, a = 1.20 mm, and h = 1.78 mm.

FIG. 8
FIG. 8

Measured and calculated reflected power for a grating with: 51 grooves, λ = 8.57 mm, d = 5.23 mm, a = 4.97 mm, and h = 4.80 mm. (a) TM polarization, (b) TE polarization. Experimental results (—○—) from Ref. 5, Fig. 5.

Tables (1)

Tables Icon

TABLE I Dimensions of dual-blazed surfaces for shallow rectangular grooves.

Equations (15)

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H y ( x , z ) = A i e - j k ( x sin θ i - z cos θ i ) + n = - A n e - j k ( x sin θ n + z cos θ n ) ,             n = 0 , ± 1 , ± 2 ,
sin θ n = sin θ i + n λ / d ,
cos θ n = ( 1 - sin 2 θ n ) 1 / 2             sin θ n 1 = - j ( sin 2 θ n - 1 ) 1 / 2             sin θ n > 1.
H y = m = 0 C m cos [ m π a ( x + a 2 ) ] cos [ k m ( z + h ) ] , E y = m = 0 D m sin [ m π a ( x + a 2 ) ] sin [ k m ( z + h ) ] ,             m = 0 , 1 , 2 ,
k m = [ k 2 - ( m π a ) 2 ] 1 / 2 , m π a k = - j [ ( m π a ) 2 - k 2 ] 1 / 2 , m π a > k .
m = 0 C m k m tan ( k m h ) j k cos θ n b n m M = A n
m = 1 D m b n m E = B n ,
b n m M = ( a d ) 1 / 2 sin [ k sin θ n ( a / 2 ) ] 2 k sin θ n ( k sin θ n ) 2 - ( m π / a ) 2 ,             m = 0 , 2 , 4 , j ( 2 a d ) 1 / 2 cos [ k sin θ n ( a / 2 ) ] 2 k sin θ n ( k sin θ n ) 2 - ( m π / a ) 2 ,             m = 1 , 3 , 5 , ½ ( a d ) 1 / 2 ( j ) m ,             k sin θ n = ± m π a , ( a d ) 1 / 2 ,             k sin θ n = m π a = 0 , b n m E = - ( 2 a d ) 1 / 2 cos [ k sin θ n ( a / 2 ) ] 2 m π / a ( k sin θ n ) 2 - ( m π / a ) 2 ,             m = 1 , 3 , 5 , j ( 2 a d ) 1 / 2 sin [ k sin θ 0 ( a / 2 ) ] 2 m π / a ( k sin θ n ) 2 - ( m π / a ) 2 ,             m = 2 , 4 , 6 , j ( a 2 d ) 1 / 2 ( j ) m ,             k sin θ n = ± m π a , A 0 = A ref - A i , B 0 = B ref - B i
= { 1 , m = 0 2 , m 0.
m = 0 [ C m tan ( k m h ) ] ( n = - { c m n M b n m M } k m j k cos θ n - δ m m cot ( k m h ) ) = - 2 A i c m 0 M ,
m = 0 D m ( n = - { c m n E b n m E } j k cos θ n k m - δ m m cot ( k m h ) ) = 2 B i c m 0 E ,
c m n M = ( - 1 ) m b n m M , c m n E = ( - 1 ) m + 1 b n m E ,
k d sin θ i = p π ,             p = 1 , 2 , 3 ,
a m m = 2 n = - ( p - 1 ) c m n b n m ,             p = 1 , 3 , 5 , = c m ( - p / 2 ) b ( - p / 2 ) m + 2 n = - ( p / 2 - 1 ) c m n b n m ,             p = 2 , 4 , 6 ,
c m n = c m n M or c m n E , b n m = b n m M or b n m E ,