Abstract

Numerical solution of integral equations for the surface current on perfectly conducting echelette-type diffraction gratings is used for comparison with the results of five approximate-analysis techniques. The edge singularity of the current for the P(E) polarization and the relatively large current in the shadow region for the S(H) polarization give some insight into the rather restricted limits of validity of the perturbation, scalar diffraction, and physical-optics techniques. Two more-general approaches, based on the Rayleigh hypothesis of using only outward-diffracted orders, while considerably more versatile, are found to be at best of marginal use in studying the physical phenomena associated with typical echelette gratings.

© 1972 Optical Society of America

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References

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  1. J. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958).
  2. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).
  3. J. Pavageaus, Rev. Opt. 42, 451 (1963).
  4. W. C. Meecham, J. Appl. Phys. 27, 361 (1955).
    [Crossref]
  5. R. Petit, Compt. Rend. 257, 2018 (1962).
  6. H. A. Kalhor and A. R. Neureuther, J. Opt. Soc. Am. 61, 43 (1971).
    [Crossref]
  7. Lord Rayleigh, Proc. Roy. Soc., (London) A79, 399 (1907).
  8. S. Sesnic, Ph.D. thesis, University of California, Berkeley (1966).

1971 (1)

1963 (1)

J. Pavageaus, Rev. Opt. 42, 451 (1963).

1962 (1)

R. Petit, Compt. Rend. 257, 2018 (1962).

1955 (1)

W. C. Meecham, J. Appl. Phys. 27, 361 (1955).
[Crossref]

1907 (1)

Lord Rayleigh, Proc. Roy. Soc., (London) A79, 399 (1907).

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).

Kalhor, H. A.

Meecham, W. C.

W. C. Meecham, J. Appl. Phys. 27, 361 (1955).
[Crossref]

Neureuther, A. R.

Pavageaus, J.

J. Pavageaus, Rev. Opt. 42, 451 (1963).

Petit, R.

R. Petit, Compt. Rend. 257, 2018 (1962).

Rayleigh, Lord

Lord Rayleigh, Proc. Roy. Soc., (London) A79, 399 (1907).

Sesnic, S.

S. Sesnic, Ph.D. thesis, University of California, Berkeley (1966).

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).

Strong, J.

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

Compt. Rend. (1)

R. Petit, Compt. Rend. 257, 2018 (1962).

J. Appl. Phys. (1)

W. C. Meecham, J. Appl. Phys. 27, 361 (1955).
[Crossref]

J. Opt. Soc. Am. (1)

Proc. Roy. Soc., (London) (1)

Lord Rayleigh, Proc. Roy. Soc., (London) A79, 399 (1907).

Rev. Opt. (1)

J. Pavageaus, Rev. Opt. 42, 451 (1963).

Other (3)

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

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).

S. Sesnic, Ph.D. thesis, University of California, Berkeley (1966).

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Figures (12)

F. 1
F. 1

Surface currents on echelette grating: d = 1.224λ, β = 25.5°, α = 64.5°, θi = 34.5°, H polarization. Real and imaginary parts of the numerical solution (solid and dashed lines with data points, respectively) and real and imaginary parts of the approximate physical-optics solution (solid and dashed lines, respectively).

F. 2
F. 2

Surface current for second harmonic on echelette grating: d = 2.449λ, β = 25.5°, α = 64.5°, θi = 34.5°, H polarization. Real and imaginary parts of the numerical solution (solid and dashed lines with data points, respectively) and real and imaginary parts of the approximate physical-optics solution (solid and dashed lines, respectively).

F. 3
F. 3

Surface currents on echelette grating: d = 1.224λ, β = 25.5°, α = 64.5°, θi = 34.5°, E polarization. Real and imaginary parts of the numerical solution (solid and dashed lines with data points, respectively) and real and imaginary parts of the approximate physical-optics solution (solid and dashed lines, respectively).

F. 4
F. 4

Surface current for second harmonic on echelette grating: d = 2.449λ, β = 25.5°, α = 64.5°, θi = 34.5°, E polarization. Real and imaginary parts of the numerical solution (solid and dashed lines with data points, respectively) and real and imaginary parts of the approximate physical-optics solution (solid and dashed lines, respectively).

F. 5
F. 5

Fraction of incident energy scattered into zero and first orders as a function of wavelength: β = 25.5°, α = 64.5°, mλ/d = sinθi + sin(θi − 20°). Scalar theory (— – — –) is the same for both polarizations. Physical-optics result is given by solid and dashed curves for H and E polarizations, respectively. Numerical solution is shown by solid and dashed curves with data points for H and E polarizations, respectively.

F. 6
F. 6

Fraction of second-harmonic energy scattered into second order. β = 25.5°, α = 64.5°, 2λ2/d = sinθi + sin(θi−20°). Scalar theory (— – — –) is the same for both polarizations. Physical-optics result is given (solid and dashed) curves for H and E polarizations, respectively. Numerical solution is shown by solid and dashed curves with data points for H and E polarizations, respectively.

F. 7
F. 7

Energy in various spectral orders as a function of slope for a symmetric echelette grating: β = α, d/λ = 2.29, normal incidence, H polarization. Physical-optics and numerical results shown by dots and crosses, respectively.

F. 8
F. 8

Energy in various spectral orders as a function of slope for a symmetric echelette grating: β = α, d/λ = 2.29, normal incidence, E polarization. Physical-optics and numerical results shown by dots and crosses, respectively.

F. 9
F. 9

Percentage error in conservation of energy as a function of slope for a symmetric echelette grating. β = α, d/λ = 2.29, H polarization. Curves 1 and 2 for physical optics with θi = 0° and 10°, respectively. Curves 3 and 4 for physical optics with θi = 0° and 10°, respectively.

F. 10
F. 10

Percentage error in various orders and conservation of energy for a symmetric echelette. β = α, d/λ = 2.29, θi = 0°, H polarization. Data sets: 1, Petit’s approximate method; 2 and 4, physical optics; 3 and 5, perturbation. Sets 1, 2, and 3 are for tanα = 0.15 and 4 and 5 are for tanα = 1.0.

F. 11
F. 11

Percentage error in various orders and conservation of energy for a symmetric echelette. β = α, d/λ = 2.29, θi = 0°, E polarization. Data sets: 1 and 4, Petit’s approximate method; 2 and 4, physical optics; 3 and 6, perturbation. Sets 1, 2, and 3 are for tanα = 0.15 and 4, 5, and 6 are for tan α = 1.0.

F. 12
F. 12

Percentage error in various orders and conservation of energy for an echelette grating. β = 80°, α = 10°, d/λ = 1.115, θ = 0°, E polarization. Data sets: 1, Meecham’s variational method; 2, physical optics; 3, scalar theory; and 4, perturbation.

Equations (1)

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m λ m / d = sin θ i + sin ( θ i 20 ° ) ,