Abstract

An electromagnetic study of the staircase approximation of arbitrary shaped gratings is conducted with three different grating theories. Numerical results on a deep aluminum sinusoidal grating show that the staircase approximation introduces sharp maxima in the local field map close to the edges of the profile. These maxima are especially pronounced in TM polarization and do not exist with the original sinusoidal profile. Their existence is not an algorithmic artifact, since they are found with different grating theories and numerical implementations. Since the number of the maxima increases with the number of the slices, a greater number of Fourier components is required to correctly represent the electromagnetic field, and thus a worsening of the convergence rate is observed. The study of the local field map provides an understanding of why methods that do not use the staircase approximation (e.g., the differential theory) converge faster than methods that use it. As a consequence, a 1% accuracy in the efficiencies of a deep sinusoidal metallic grating is obtained 30 times faster when the differential theory is used in comparison with the use of the rigorous coupled-wave theory. A theoretical analysis is proposed in the limit when the number of slices tends to infinity, which shows that even in that case the staircase approximation is not well suited to describe the real profile.

© 2002 Optical Society of America

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References

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  1. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
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    [CrossRef]
  3. D. M. Pai, K. A. Awada, “Analysis of dielectric gratingsof arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
    [CrossRef]
  4. M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
    [CrossRef]
  5. D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
    [CrossRef]
  6. J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom-molecular reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
    [CrossRef]
  7. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  8. F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
    [CrossRef]
  9. P. Lalanne, G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–783 (1996).
    [CrossRef]
  10. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  11. E. Popov, M. Nevière, “Differential theory for diffraction gratings: a new formulation for TM polarization with rapid convergence,” Opt. Lett. 25, 598–600 (2000).
    [CrossRef]
  12. E. Popov, M. Nevière, “Grating theory: New equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
    [CrossRef]
  13. J. M. Elson, P. Tran, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A 12, 1765–1771 (1995).
    [CrossRef]
  14. G. Tayeb, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique: comment,” J. Opt. Soc. Am. A 13, 1766–1767 (1996).
    [CrossRef]
  15. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
    [CrossRef]
  16. G. Tayeb, R. Petit, “On the numerical study of deep conducting lamellar diffraction gratings,” Opt. Acta 31, 1361–1365 (1984).
    [CrossRef]
  17. S. E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: An exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
    [CrossRef]
  18. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
    [CrossRef]
  19. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., (Springer-Verlag, Berlin, 1980), Chap. 3.
  20. G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” special issue on “Generalized Multipole Techniques (GMT)” of Appl. Comput. Electromagn. Soc. J. 9, 90–100 (1994).
  21. E. Popov, M. Nevière, “Maxwell equations in Fourier space: fast converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001).
    [CrossRef]
  22. P. Lalanne, “Convergence performance of the coupled-wave and the differential method for thin gratings,” J. Opt. Soc. Am. A 14, 1583–1591 (1997).
    [CrossRef]

2001 (1)

2000 (2)

1998 (1)

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

1997 (1)

1996 (4)

1995 (2)

1994 (2)

G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” special issue on “Generalized Multipole Techniques (GMT)” of Appl. Comput. Electromagn. Soc. J. 9, 90–100 (1994).

M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
[CrossRef]

1993 (1)

S. E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: An exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

1991 (1)

1984 (1)

G. Tayeb, R. Petit, “On the numerical study of deep conducting lamellar diffraction gratings,” Opt. Acta 31, 1361–1365 (1984).
[CrossRef]

1982 (1)

1981 (2)

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

1976 (2)

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom-molecular reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Awada, K. A.

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Elson, J. M.

Gaylord, T. K.

Lalanne, P.

Li, L.

Light, J. C.

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom-molecular reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

Maystre, D.

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., (Springer-Verlag, Berlin, 1980), Chap. 3.

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Moharam, M. G.

Montiel, F.

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
[CrossRef]

Morf, R. H.

Morris, G.

Nevière, M.

Pai, D. M.

Petit, R.

S. E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: An exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

G. Tayeb, R. Petit, “On the numerical study of deep conducting lamellar diffraction gratings,” Opt. Acta 31, 1361–1365 (1984).
[CrossRef]

Peyrot, P.

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

Popov, E.

Sandström, S. E.

S. E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: An exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

Tayeb, G.

G. Tayeb, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique: comment,” J. Opt. Soc. Am. A 13, 1766–1767 (1996).
[CrossRef]

G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” special issue on “Generalized Multipole Techniques (GMT)” of Appl. Comput. Electromagn. Soc. J. 9, 90–100 (1994).

S. E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: An exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

G. Tayeb, R. Petit, “On the numerical study of deep conducting lamellar diffraction gratings,” Opt. Acta 31, 1361–1365 (1984).
[CrossRef]

Tran, P.

Walker, R. B.

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom-molecular reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

Zvijac, D. J.

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

Appl. Comput. Electromagn. Soc. J. (1)

G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” special issue on “Generalized Multipole Techniques (GMT)” of Appl. Comput. Electromagn. Soc. J. 9, 90–100 (1994).

Chem. Phys. (1)

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

J. Chem. Phys. (1)

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom-molecular reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

J. Electromagn. Waves Appl. (1)

S. E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: An exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

J. Mod. Opt. (1)

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (10)

E. Popov, M. Nevière, “Grating theory: New equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

E. Popov, M. Nevière, “Maxwell equations in Fourier space: fast converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001).
[CrossRef]

P. Lalanne, “Convergence performance of the coupled-wave and the differential method for thin gratings,” J. Opt. Soc. Am. A 14, 1583–1591 (1997).
[CrossRef]

D. M. Pai, K. A. Awada, “Analysis of dielectric gratingsof arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
[CrossRef]

P. Lalanne, G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–783 (1996).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

G. Tayeb, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique: comment,” J. Opt. Soc. Am. A 13, 1766–1767 (1996).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
[CrossRef]

J. M. Elson, P. Tran, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A 12, 1765–1771 (1995).
[CrossRef]

Opt. Acta (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

G. Tayeb, R. Petit, “On the numerical study of deep conducting lamellar diffraction gratings,” Opt. Acta 31, 1361–1365 (1984).
[CrossRef]

Opt. Commun. (1)

M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
[CrossRef]

Opt. Lett. (1)

Other (1)

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., (Springer-Verlag, Berlin, 1980), Chap. 3.

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Figures (12)

Fig. 1
Fig. 1

Schematic representation of a sinusoidal profile and a five-stair approximation of the same profile, together with some notations used in the text.

Fig. 2
Fig. 2

Mean error in the reflected diffraction orders as a function of the number of slices in the staircase approximation (Fig. 1) of an aluminum grating with a sinusoidal groove profile, period d=0.5 μm, groove depth h=0.2 μm, aluminum refractive index nAl=1.3+i7.6, illuminated at 40° incidence with TE polarized light with wavelength λ=0.6328 μm.

Fig. 3
Fig. 3

Convergence of the mean error for the grating with parameters given in Fig. 2 as a function of the truncation parameter N, the number of the Fourier components or modes being equal to 2N+1. The RCW method (heavy solid curve) and the modal method (thin curve) are used with M=200 slices, the differential method (curve “diff.”) is applied to the smooth sinusoidal profile. TE polarization.

Fig. 4
Fig. 4

Convergence of the -1st order efficiency in TM polarization of the RCW and modal methods (indicated in the figure) (a) for a single-step (M=1) lamellar grating, (b) for a staircase grating with M=20, (c) M=200, compared with the convergence of the differential method for a smooth sinusoidal profile (curve “diff.”). The grating parameters are given in Fig. 2.

Fig. 5
Fig. 5

Three-dimensional view of the distribution of the x component of the field squared inside the groove for a five-step staircase approximation in TM polarization.

Fig. 6
Fig. 6

Field map inside the groove of a five-step staircase grating in TM polarization. (a) |Ex|2, (b) |Ey|2.

Fig. 7
Fig. 7

Spatial field distribution in the vicinity of several steps inside the groove of a 20-step staircase profile, used to approximate the sinusoidal grating under study. TM polarization, (a) |Ex|2, (b) |Ey|2.

Fig. 8
Fig. 8

Spatial distribution of |Ex|2 calculated for a nondiscretized sinusoidal profile in TM polarization. (a) The entire groove region as in Fig. 5, (b) the same region as presented in Fig. 7.

Fig. 9
Fig. 9

Spatial field distribution inside the groove region of a five-step staircase grating, presented in Fig. 1, in TE polarization. (a) |Hx|2, (b) |Hy|2.

Fig. 10
Fig. 10

Distribution of |Hx|2 inside the groove for a 20-step staircase approximation of the sinusoidal grating in TE polarization. Same groove region as in Fig. 7(a).

Fig. 11
Fig. 11

Same as in Fig. 7, except for the wavelength λ=13 μm.

Fig. 12
Fig. 12

Convergence rates of the RCW method compared when using the three different sets of equations as marked on the figure [Eqs. (1)–(3), (5) and (6), and (7) and (8)], compared with the differential method used for the sinusoidal profile. (a) λ=0.6328 μm, M=20, and M=200; (b) λ=13 μm and M=20. The convergence of the differential method for a sinusoidal profile is shown by a solid curve marked “diff.”

Equations (8)

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

d[Hz]dy=-iωNy2+1-1Nx2[Ex]- -1-1NxNy[Ey],
d[Ex]dy=-iωμ0[Hz]+iα[Ey],
[Ey]=Nx2+1-1Ny2-1αω [Hz]+-1-1NxNy[Ex].
fmn=[f]m-n.
d[Hz]dy=-iω1-1[Ex],
d[Ex]dy=-iωμ0[Hz]+iα-1αω [Hz].
d[Hz]dy=-iω[Ex],
d[Ex]dy=-iωμ0[Hz]+iα1αω [Hz].

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