Abstract

A general numerical technique for analyzing diffraction gratings of arbitrary groove shape as electromagnetic boundary-value problems is suggested. The method yields the induced surface current, from which the energies in various radiating orders can be conveniently obtained. Computed results are compared to the theoretical as well as the experimental results available in the literature. The accuracy and versatility of the technique are demonstrated together with its economy and convenience.

© 1971 Optical Society of America

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References

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  1. R. A. Sawyer, Experimental Spectroscopy (Prentice–Hall, New York, 1951).
  2. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965).
  3. J. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958).
  4. R. W. Wood, Phil. Mag. 4, 396 (1902).
  5. Lord Rayleigh, Phil. Mag. 14, 60 (1907).
  6. V. Twersky, IRE Trans. AP-4, 330 (1956).
  7. R. W. Wood, Phys. Rev. 48, 928 (1935).
    [Crossref]
  8. C. H. Palmer, J. Opt. Soc. Am. 42, 269 (1952).
    [Crossref]
  9. C. H. Palmer and F. W. Phelps, J. Opt. Soc. Am. 58, 1184 (1968).
    [Crossref]
  10. C. H. Palmer, J. Opt. Soc. Am. 59, 812 (1969).
    [Crossref]
  11. A. Wirgin and R. Deleuil, J. Opt. Soc. Am. 59, 1348 (1969).
    [Crossref]
  12. R. Deleuil, Opt. Acta 16, 23 (1969).
    [Crossref]
  13. T. Itoh and R. Mittra, IEEE Trans. MTT-6, 319 (1969).
  14. R. Petit, Rev. Opt. 42, 263 (1963).
  15. W. C. Meecham, J. Appl. Phys. 27, 361 (1956).
    [Crossref]
  16. R. Petit, Rev. Opt. 45, 249 (1966).
  17. A. R. Neureuther and K. Zaki, Alta Frequenza 38, 283 (1969).
  18. R. F. Harrington, Field Computation by Moment Method (Macmillan, New York, 1968).

1969 (5)

C. H. Palmer, J. Opt. Soc. Am. 59, 812 (1969).
[Crossref]

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

R. Deleuil, Opt. Acta 16, 23 (1969).
[Crossref]

T. Itoh and R. Mittra, IEEE Trans. MTT-6, 319 (1969).

A. R. Neureuther and K. Zaki, Alta Frequenza 38, 283 (1969).

1968 (1)

1966 (1)

R. Petit, Rev. Opt. 45, 249 (1966).

1963 (1)

R. Petit, Rev. Opt. 42, 263 (1963).

1956 (2)

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

V. Twersky, IRE Trans. AP-4, 330 (1956).

1952 (1)

1935 (1)

R. W. Wood, Phys. Rev. 48, 928 (1935).
[Crossref]

1907 (1)

Lord Rayleigh, Phil. Mag. 14, 60 (1907).

1902 (1)

R. W. Wood, Phil. Mag. 4, 396 (1902).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Deleuil, R.

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Method (Macmillan, New York, 1968).

Itoh, T.

T. Itoh and R. Mittra, IEEE Trans. MTT-6, 319 (1969).

Meecham, W. C.

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

Mittra, R.

T. Itoh and R. Mittra, IEEE Trans. MTT-6, 319 (1969).

Neureuther, A. R.

A. R. Neureuther and K. Zaki, Alta Frequenza 38, 283 (1969).

Palmer, C. H.

Petit, R.

R. Petit, Rev. Opt. 45, 249 (1966).

R. Petit, Rev. Opt. 42, 263 (1963).

Phelps, F. W.

Rayleigh, Lord

Lord Rayleigh, Phil. Mag. 14, 60 (1907).

Sawyer, R. A.

R. A. Sawyer, Experimental Spectroscopy (Prentice–Hall, New York, 1951).

Strong, J.

J. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958).

Twersky, V.

V. Twersky, IRE Trans. AP-4, 330 (1956).

Wirgin, A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Wood, R. W.

R. W. Wood, Phys. Rev. 48, 928 (1935).
[Crossref]

R. W. Wood, Phil. Mag. 4, 396 (1902).

Zaki, K.

A. R. Neureuther and K. Zaki, Alta Frequenza 38, 283 (1969).

Alta Frequenza (1)

A. R. Neureuther and K. Zaki, Alta Frequenza 38, 283 (1969).

IEEE Trans. (1)

T. Itoh and R. Mittra, IEEE Trans. MTT-6, 319 (1969).

IRE Trans. (1)

V. Twersky, IRE Trans. AP-4, 330 (1956).

J. Appl. Phys. (1)

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

J. Opt. Soc. Am. (4)

Opt. Acta (1)

R. Deleuil, Opt. Acta 16, 23 (1969).
[Crossref]

Phil. Mag. (2)

R. W. Wood, Phil. Mag. 4, 396 (1902).

Lord Rayleigh, Phil. Mag. 14, 60 (1907).

Phys. Rev. (1)

R. W. Wood, Phys. Rev. 48, 928 (1935).
[Crossref]

Rev. Opt. (2)

R. Petit, Rev. Opt. 42, 263 (1963).

R. Petit, Rev. Opt. 45, 249 (1966).

Other (4)

R. F. Harrington, Field Computation by Moment Method (Macmillan, New York, 1968).

R. A. Sawyer, Experimental Spectroscopy (Prentice–Hall, New York, 1951).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965).

J. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958).

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Figures (9)

F. 1
F. 1

Geometry of the echelette grating of period d and side angles β and α, with incidence at angle θi.

F. 2
F. 2

Source distribution for the determination of the periodic Green’s function.

F. 3
F. 3

Geometry of the reference cell, showing the surfaces of integration and the normal directions.

F. 4
F. 4

Shallow grating: d = 1.25 μm, λ = 0.546 μm, tanβ = tanα = 0.15, θi = 0, E polarized. Real and imaginary currents by the numerical technique (solid and dashed lines with circles, respectively), real and imaginary currents by physical optics (solid and dashed lines with triangles, respectively) vs distance along the grating.

F. 5
F. 5

Deep grating:d = 1.25 μm, λ = 0.546 μm, tanβ = tanα = 1.0, θi = 0, E polarized. Real and imaginary currents by the numerical technique (solid and dashed lines with circles, respectively), real and imaginary currents by physical optics (solid and dashed lines with triangles, respectively) vs distance along the grating.

F. 6
F. 6

Echelette grating: d = 28 μm, β = 66°, α = 20°, ψθi = 8.9°, E polarized. First-order energy by Deleuil (curve 1), Madden and Strong (curve 2), scalar theory (curve 3), Petit (dots), and the numerical technique (crosses) vs normalized wavelength.

F. 7
F. 7

Echelette grating: d = 28 μm, β = 66°, α = 20°, ψθi = 9°50′, H polarized. First-order energy by Deleuil (curve 1), Madden and Strong (curve 2), scalar theory (curve 3), Petit (dots), and the numerical technique (crosses) vs normalized wavelength.

F. 8
F. 8

Echelette grating: d = 15.8 μm, β=37°, α = 49°, ψθi = 8.9°, E polarized. First-order energy by Deleuil (curve 1), Madden and Strong (curve 2), scalar theory (curve 3), and the numerical technique (crosses) vs normalized wavelength.

F. 9
F. 9

Echelette grating: d = 15.8 μm, β = 37°, α = 49°, ψθi = 8.9°, H polarized. First-order energy by Deleuil (curve 1), Madden and Strong (curve 2), scalar theory (curve 3), and the numerical technique (crosses) vs normalized wavelength.

Tables (2)

Tables Icon

Table I Space-harmonic coefficients for the E-polarized case for a grating of d = 1.25 μm at λ = 0.546 μm.

Tables Icon

Table II Space-harmonic coefficients for the H-polarized case for a grating of d = 1.25 μm at λ = 0.546 μm.

Equations (9)

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( 2 + k 2 ) ϕ t ( x , z ) = 0 ,
ϕ t = 0 for the E polarize case ϕ t / n ̂ = 0 for the H polarized case .
( 2 + k 2 ) G P ( x , z , x 0 , z 0 ) = n = exp ( j β 0 x ) δ ( x x 0 n d ) δ ( z z 0 ) .
G p ( x , z , x 0 , z 0 ) = exp ( j β 0 x ) 2 j d n = 1 γ n exp j [ 2 n π d ( x x 0 ) γ n | z z 0 | ] ,
γ n = k 2 ( 2 n π / d + β 0 ) 2 .
G p ( x , z , x 0 , z 0 ) = j 4 n = H 0 2 { K [ ( x x 0 n d ) 2 + ( z z 0 ) 2 ] 1 2 } exp j β 0 ( x 0 + n d ) .
ϕ t ( x , z ) = exp ( j β 0 x ) S 1 + S 2 + S 3 + S 4 [ G p ( x , z , x 0 , z 0 ) ϕ t n ̂ 0 ϕ t G p ( x , z , x 0 , z 0 ) n ̂ 0 ] d s 0 ,
S 1 E t n ̂ ( x , z ) G p ( x , z , x 0 , z 0 ) d s = E 0 exp ( j γ 0 z 0 ) .
H t ( x 0 , z 0 ) exp ( j β 0 x 0 ) = S 1 H Z ( x , z ) G p ( x , z , x 0 , z 0 ) n ̂ d s + H 0 exp ( j γ 0 z 0 ) .