Abstract

It is shown, contrary to the assertions of van den Berg,1 that (1) the validity of the Rayleigh hypothesis has not been confirmed numerically, (2) the Rayleigh–Fourier (RF) method gives useful far-field results for sinusoidal gratings with height/period ratios far exceeding 0.3, (3) the Rayleigh–Yasuura method is not generally convergent in a numerical sense and is much less useful than the RF method, (4) the integrated-square error of the boundary condition is not a more reliable check of a numerical solution than the energy-balance error.

© 1982 Optical Society of America

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References

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  1. P. M. van den Berg, “Reflection by a grating: Rayleigh methods,” J. Opt. Soc. Am. 71, 1224–1229 (1981).
    [Crossref]
  2. A. Wirgin, “Sur la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. B 288, 179–182 (1979).
  3. A. Wirgin, “Sur trois variantes de la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. A 289, 259–262 (1979).
  4. A. Wirgin, “Aspects numériques du problème de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. B 289, 273–276 (1979).
  5. A. Wirgin, “Reflection from a corrugated surface,” J. Acoust. Soc. Am. 68, 692–699 (1980).
    [Crossref]
  6. A. Wirgin, “On Rayleigh’s theory of sinusoidal diffraction gratings,” Opt. Acta 27, 1671–1692 (1980).
    [Crossref]
  7. A. Wirgin, “On Rayleigh’s theory of partially reflecting gratings,” Opt. Acta 28, 1377–1404 (1981).
    [Crossref]
  8. J. P. Hugonin, R. Petit, and M. Cadilhac, “Plane wave expansions used to describe the field diffracted by a grating,” J. Opt. Soc. Am. 71, 593–597 (1981).
    [Crossref]
  9. R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. B 262, 468–471 (1966).
  10. R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
    [Crossref]
  11. R. F. Millar, “Singularities of two-dimensional exterior solutions of the Helmholtz equation,” Proc. Cambridge Philos. Soc. 69, 175–188 (1971).
    [Crossref]
  12. R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
    [Crossref]
  13. G. R. Jiracek, “Numerical comparisons of a modified Rayleigh approach with other rough surface EM scattering solutions,” IEEE Trans. Antennas Propag. AP-21, 393–396 (1973).
    [Crossref]
  14. N. Garcia and N. Cabrera, “New method for solving the scattering of waves from a periodic hard surface: solutions and numerical comparisons with the various formalisms,” Phys. Rev. B 18, 576–589 (1978).
    [Crossref]
  15. R. Petit, “Contribution à l’étude de la diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 42, 263–281 (1963).
  16. W. C. Meecham, “Variational method for the calculation of the distribution of energy reflected from a periodic surface,” J. Appl. Phys. 27, 361–367 (1956).
    [Crossref]
  17. H. Ikuno and K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
    [Crossref]
  18. P. M. van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971).
  19. P. M. van den Berg, Ph.D. Thesis (Delft University of Technology, Delft, The Netherlands, 1971).
  20. P. M. van den Berg and J. C. M. Borbough, “Dispersion of surface plasmons in InSb-gratings,” Appl. Phys. 3, 55–60 (1974).
    [Crossref]
  21. J. T. Fokkema and P. M. van den Berg, “Elastodynamic diffraction by a periodic rough surface (stress-free boundary),” J. Accoust. Soc. Am. 62, 1095–1101 (1977).
    [Crossref]
  22. R. Petit, “Contribution à l’étude de la diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 45, 249–276; 353–370 (1966).
  23. H. P. Bonzel and N. A. Gjostein, “Diffraction theory of sinusoidal gratings and application to in situ surface self-diffusion measurements,” J. Appl. Phys. 39, 3480–3491 (1968).
    [Crossref]
  24. D. Maystre, Ph.D. Thesis (Faculté des Sciences de Marseille, 1974).
  25. J. Pavageau and J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta. 17, 469–478 (1970).
    [Crossref]
  26. K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile: TE polarization,” IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
    [Crossref]
  27. G. Whitman and F. Schwering, “Scattering by periodic metal surfaces with sinusoidal height profiles—a theoretical approach,” IEEE Trans. Antennas Propag. AP-25, 869–876 (1977).
    [Crossref]
  28. K. Yasuura, “A view of numerical methods in diffraction problems,” in Progress in Radio Science 1966–1969, W. V. Tilston and M. Sauzade, eds. (International Union of Radio Science, Brussels, Belgium, 1971), Vol. 3.
  29. P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
    [Crossref]
  30. J. C. Bolomey and A. Wirgin, “Numerical comparison of the Green’s function and the Waterman and Rayleigh theories of scattering from a cylinder with arbitrary cross section,” Proc. IEE 121, 794–804 (1974).

1981 (3)

1980 (2)

A. Wirgin, “Reflection from a corrugated surface,” J. Acoust. Soc. Am. 68, 692–699 (1980).
[Crossref]

A. Wirgin, “On Rayleigh’s theory of sinusoidal diffraction gratings,” Opt. Acta 27, 1671–1692 (1980).
[Crossref]

1979 (3)

A. Wirgin, “Sur la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. B 288, 179–182 (1979).

A. Wirgin, “Sur trois variantes de la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. A 289, 259–262 (1979).

A. Wirgin, “Aspects numériques du problème de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. B 289, 273–276 (1979).

1978 (1)

N. Garcia and N. Cabrera, “New method for solving the scattering of waves from a periodic hard surface: solutions and numerical comparisons with the various formalisms,” Phys. Rev. B 18, 576–589 (1978).
[Crossref]

1977 (2)

J. T. Fokkema and P. M. van den Berg, “Elastodynamic diffraction by a periodic rough surface (stress-free boundary),” J. Accoust. Soc. Am. 62, 1095–1101 (1977).
[Crossref]

G. Whitman and F. Schwering, “Scattering by periodic metal surfaces with sinusoidal height profiles—a theoretical approach,” IEEE Trans. Antennas Propag. AP-25, 869–876 (1977).
[Crossref]

1975 (2)

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[Crossref]

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
[Crossref]

1974 (2)

J. C. Bolomey and A. Wirgin, “Numerical comparison of the Green’s function and the Waterman and Rayleigh theories of scattering from a cylinder with arbitrary cross section,” Proc. IEE 121, 794–804 (1974).

P. M. van den Berg and J. C. M. Borbough, “Dispersion of surface plasmons in InSb-gratings,” Appl. Phys. 3, 55–60 (1974).
[Crossref]

1973 (2)

G. R. Jiracek, “Numerical comparisons of a modified Rayleigh approach with other rough surface EM scattering solutions,” IEEE Trans. Antennas Propag. AP-21, 393–396 (1973).
[Crossref]

H. Ikuno and K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[Crossref]

1971 (4)

P. M. van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971).

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
[Crossref]

R. F. Millar, “Singularities of two-dimensional exterior solutions of the Helmholtz equation,” Proc. Cambridge Philos. Soc. 69, 175–188 (1971).
[Crossref]

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile: TE polarization,” IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[Crossref]

1970 (1)

J. Pavageau and J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta. 17, 469–478 (1970).
[Crossref]

1968 (1)

H. P. Bonzel and N. A. Gjostein, “Diffraction theory of sinusoidal gratings and application to in situ surface self-diffusion measurements,” J. Appl. Phys. 39, 3480–3491 (1968).
[Crossref]

1966 (2)

R. Petit, “Contribution à l’étude de la diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 45, 249–276; 353–370 (1966).

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. B 262, 468–471 (1966).

1963 (1)

R. Petit, “Contribution à l’étude de la diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 42, 263–281 (1963).

1956 (1)

W. C. Meecham, “Variational method for the calculation of the distribution of energy reflected from a periodic surface,” J. Appl. Phys. 27, 361–367 (1956).
[Crossref]

Bolomey, J. C.

J. C. Bolomey and A. Wirgin, “Numerical comparison of the Green’s function and the Waterman and Rayleigh theories of scattering from a cylinder with arbitrary cross section,” Proc. IEE 121, 794–804 (1974).

Bonzel, H. P.

H. P. Bonzel and N. A. Gjostein, “Diffraction theory of sinusoidal gratings and application to in situ surface self-diffusion measurements,” J. Appl. Phys. 39, 3480–3491 (1968).
[Crossref]

Borbough, J. C. M.

P. M. van den Berg and J. C. M. Borbough, “Dispersion of surface plasmons in InSb-gratings,” Appl. Phys. 3, 55–60 (1974).
[Crossref]

Bousquet, J.

J. Pavageau and J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta. 17, 469–478 (1970).
[Crossref]

Cabrera, N.

N. Garcia and N. Cabrera, “New method for solving the scattering of waves from a periodic hard surface: solutions and numerical comparisons with the various formalisms,” Phys. Rev. B 18, 576–589 (1978).
[Crossref]

Cadilhac, M.

J. P. Hugonin, R. Petit, and M. Cadilhac, “Plane wave expansions used to describe the field diffracted by a grating,” J. Opt. Soc. Am. 71, 593–597 (1981).
[Crossref]

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. B 262, 468–471 (1966).

Fokkema, J. T.

J. T. Fokkema and P. M. van den Berg, “Elastodynamic diffraction by a periodic rough surface (stress-free boundary),” J. Accoust. Soc. Am. 62, 1095–1101 (1977).
[Crossref]

Garcia, N.

N. Garcia and N. Cabrera, “New method for solving the scattering of waves from a periodic hard surface: solutions and numerical comparisons with the various formalisms,” Phys. Rev. B 18, 576–589 (1978).
[Crossref]

Gjostein, N. A.

H. P. Bonzel and N. A. Gjostein, “Diffraction theory of sinusoidal gratings and application to in situ surface self-diffusion measurements,” J. Appl. Phys. 39, 3480–3491 (1968).
[Crossref]

Hugonin, J. P.

Ikuno, H.

H. Ikuno and K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[Crossref]

Jiracek, G. R.

G. R. Jiracek, “Numerical comparisons of a modified Rayleigh approach with other rough surface EM scattering solutions,” IEEE Trans. Antennas Propag. AP-21, 393–396 (1973).
[Crossref]

Maystre, D.

D. Maystre, Ph.D. Thesis (Faculté des Sciences de Marseille, 1974).

Meecham, W. C.

W. C. Meecham, “Variational method for the calculation of the distribution of energy reflected from a periodic surface,” J. Appl. Phys. 27, 361–367 (1956).
[Crossref]

Millar, R. F.

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
[Crossref]

R. F. Millar, “Singularities of two-dimensional exterior solutions of the Helmholtz equation,” Proc. Cambridge Philos. Soc. 69, 175–188 (1971).
[Crossref]

Neureuther, A. R.

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile: TE polarization,” IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[Crossref]

Pavageau, J.

J. Pavageau and J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta. 17, 469–478 (1970).
[Crossref]

Petit, R.

J. P. Hugonin, R. Petit, and M. Cadilhac, “Plane wave expansions used to describe the field diffracted by a grating,” J. Opt. Soc. Am. 71, 593–597 (1981).
[Crossref]

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
[Crossref]

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. B 262, 468–471 (1966).

R. Petit, “Contribution à l’étude de la diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 45, 249–276; 353–370 (1966).

R. Petit, “Contribution à l’étude de la diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 42, 263–281 (1963).

Schwering, F.

G. Whitman and F. Schwering, “Scattering by periodic metal surfaces with sinusoidal height profiles—a theoretical approach,” IEEE Trans. Antennas Propag. AP-25, 869–876 (1977).
[Crossref]

van den Berg, P. M.

P. M. van den Berg, “Reflection by a grating: Rayleigh methods,” J. Opt. Soc. Am. 71, 1224–1229 (1981).
[Crossref]

J. T. Fokkema and P. M. van den Berg, “Elastodynamic diffraction by a periodic rough surface (stress-free boundary),” J. Accoust. Soc. Am. 62, 1095–1101 (1977).
[Crossref]

P. M. van den Berg and J. C. M. Borbough, “Dispersion of surface plasmons in InSb-gratings,” Appl. Phys. 3, 55–60 (1974).
[Crossref]

P. M. van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971).

P. M. van den Berg, Ph.D. Thesis (Delft University of Technology, Delft, The Netherlands, 1971).

Waterman, P. C.

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[Crossref]

Whitman, G.

G. Whitman and F. Schwering, “Scattering by periodic metal surfaces with sinusoidal height profiles—a theoretical approach,” IEEE Trans. Antennas Propag. AP-25, 869–876 (1977).
[Crossref]

Wirgin, A.

A. Wirgin, “On Rayleigh’s theory of partially reflecting gratings,” Opt. Acta 28, 1377–1404 (1981).
[Crossref]

A. Wirgin, “Reflection from a corrugated surface,” J. Acoust. Soc. Am. 68, 692–699 (1980).
[Crossref]

A. Wirgin, “On Rayleigh’s theory of sinusoidal diffraction gratings,” Opt. Acta 27, 1671–1692 (1980).
[Crossref]

A. Wirgin, “Sur la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. B 288, 179–182 (1979).

A. Wirgin, “Sur trois variantes de la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. A 289, 259–262 (1979).

A. Wirgin, “Aspects numériques du problème de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. B 289, 273–276 (1979).

J. C. Bolomey and A. Wirgin, “Numerical comparison of the Green’s function and the Waterman and Rayleigh theories of scattering from a cylinder with arbitrary cross section,” Proc. IEE 121, 794–804 (1974).

Yasuura, K.

H. Ikuno and K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[Crossref]

K. Yasuura, “A view of numerical methods in diffraction problems,” in Progress in Radio Science 1966–1969, W. V. Tilston and M. Sauzade, eds. (International Union of Radio Science, Brussels, Belgium, 1971), Vol. 3.

Zaki, K. A.

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile: TE polarization,” IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[Crossref]

Appl. Phys. (1)

P. M. van den Berg and J. C. M. Borbough, “Dispersion of surface plasmons in InSb-gratings,” Appl. Phys. 3, 55–60 (1974).
[Crossref]

Appl. Sci. Res. (1)

P. M. van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971).

C. R. Acad. Sci. A (1)

A. Wirgin, “Sur trois variantes de la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. A 289, 259–262 (1979).

C. R. Acad. Sci. B (3)

A. Wirgin, “Aspects numériques du problème de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. B 289, 273–276 (1979).

A. Wirgin, “Sur la théorie de Rayleigh de la diffraction d’une onde par une surface sinusoidale,” C. R. Acad. Sci. B 288, 179–182 (1979).

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. B 262, 468–471 (1966).

IEEE Trans. Antennas Propag. (4)

H. Ikuno and K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[Crossref]

G. R. Jiracek, “Numerical comparisons of a modified Rayleigh approach with other rough surface EM scattering solutions,” IEEE Trans. Antennas Propag. AP-21, 393–396 (1973).
[Crossref]

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile: TE polarization,” IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[Crossref]

G. Whitman and F. Schwering, “Scattering by periodic metal surfaces with sinusoidal height profiles—a theoretical approach,” IEEE Trans. Antennas Propag. AP-25, 869–876 (1977).
[Crossref]

J. Accoust. Soc. Am. (1)

J. T. Fokkema and P. M. van den Berg, “Elastodynamic diffraction by a periodic rough surface (stress-free boundary),” J. Accoust. Soc. Am. 62, 1095–1101 (1977).
[Crossref]

J. Acoust. Soc. Am. (2)

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[Crossref]

A. Wirgin, “Reflection from a corrugated surface,” J. Acoust. Soc. Am. 68, 692–699 (1980).
[Crossref]

J. Appl. Phys. (2)

W. C. Meecham, “Variational method for the calculation of the distribution of energy reflected from a periodic surface,” J. Appl. Phys. 27, 361–367 (1956).
[Crossref]

H. P. Bonzel and N. A. Gjostein, “Diffraction theory of sinusoidal gratings and application to in situ surface self-diffusion measurements,” J. Appl. Phys. 39, 3480–3491 (1968).
[Crossref]

J. Opt. Soc. Am. (2)

Nouv. Rev. Opt. (1)

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
[Crossref]

Opt. Acta (2)

A. Wirgin, “On Rayleigh’s theory of sinusoidal diffraction gratings,” Opt. Acta 27, 1671–1692 (1980).
[Crossref]

A. Wirgin, “On Rayleigh’s theory of partially reflecting gratings,” Opt. Acta 28, 1377–1404 (1981).
[Crossref]

Opt. Acta. (1)

J. Pavageau and J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta. 17, 469–478 (1970).
[Crossref]

Phys. Rev. B (1)

N. Garcia and N. Cabrera, “New method for solving the scattering of waves from a periodic hard surface: solutions and numerical comparisons with the various formalisms,” Phys. Rev. B 18, 576–589 (1978).
[Crossref]

Proc. Cambridge Philos. Soc. (2)

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
[Crossref]

R. F. Millar, “Singularities of two-dimensional exterior solutions of the Helmholtz equation,” Proc. Cambridge Philos. Soc. 69, 175–188 (1971).
[Crossref]

Proc. IEE (1)

J. C. Bolomey and A. Wirgin, “Numerical comparison of the Green’s function and the Waterman and Rayleigh theories of scattering from a cylinder with arbitrary cross section,” Proc. IEE 121, 794–804 (1974).

Rev. Opt. (2)

R. Petit, “Contribution à l’étude de la diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 45, 249–276; 353–370 (1966).

R. Petit, “Contribution à l’étude de la diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 42, 263–281 (1963).

Other (3)

P. M. van den Berg, Ph.D. Thesis (Delft University of Technology, Delft, The Netherlands, 1971).

D. Maystre, Ph.D. Thesis (Faculté des Sciences de Marseille, 1974).

K. Yasuura, “A view of numerical methods in diffraction problems,” in Progress in Radio Science 1966–1969, W. V. Tilston and M. Sauzade, eds. (International Union of Radio Science, Brussels, Belgium, 1971), Vol. 3.

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Tables (5)

Tables Icon

Table 1 Approximation Sequences of the Diffraction Efficiences ξn and Total Energy ξ (EBE = 1 − ξ) Relative to a Sinusoidal Perfectly Conducting Grating of Height (h) = 0.375 μm = 0.3D and Period (D) = 1.25 μm, Illuminated by a Normally Incident E-Polarized or H-Polarized Plane Wave of Wavelength (λ) = 0.546 μma

Tables Icon

Table 2 Same as Table 1 for h = 0.5 μm = 0.4D

Tables Icon

Table 3 Same as Table 1 for h = 0.7 μm = 0.56D

Tables Icon

Table 4 Same as Table 1 for h = 0.875 μm = 0.7D

Tables Icon

Table 5 Approximation Sequences of the Diffraction Coefficients An+ Relative to a Sinusoidal Perfectly Conducting Grating of Height h = 0.375 μm = 0.3D and Period D = 1.25 μm, Illuminated by an E-Polarized Normally Incident Plane Wave of Wavelength λ = 0.546 μm. Comparison of RY and RF Results in Evanescent Orders

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

ϕ m , e n + = exp i [ α n x m + γ n h 2 ( 1 + cos 2 π x m D ) ] ,
ϕ m , e n + ~ [ exp i α n x m ] exp - π h n D ( 1 + cos 2 π x m D ) .