Abstract

The recently developed fast Fourier factorization method, which has greatly improved the application range of the differential theory of gratings, suffers from numerical instability when applied to metallic gratings with very low losses. This occurs when the real part of the refractive index is small, in particular, smaller than 0.1–0.2, for example, when silver and gold gratings are analyzed in the infrared region. This failure can be attributed to Li’s “inverse rule” [L. Li, J. Opt. Soc. Am. A 13, 1870 (1996)] as shown by studying the condition number of matrices that have to be inverted. Two ways of overcoming the difficulty are explored: first, an additional truncation of the matrices containing the coefficients of the differential system, which reduces the numerical problems in some cases, and second, an introduction of lossier material inside the bulk, thus leaving only a thin layer of the highly conducting metal. If the layer is sufficiently thick, this does not change the optical properties of the system but significantly improves the convergence of the differential theory, including the rigorous coupled-wave method, for various types of grating profiles.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
    [CrossRef]
  2. M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
    [CrossRef]
  3. M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
    [CrossRef]
  4. M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
  5. 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]
  6. M. Nevière, E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2003).
  7. 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]
  8. 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 17, 1773 (2001).
    [CrossRef]
  9. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  10. 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]
  11. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
    [CrossRef]
  12. J. R. Andrewartha, G. H. Derrick, R. C. McPhedran, “A general modal theory for reflection gratings,” Opt. Acta 28, 1501–1516 (1981).
    [CrossRef]
  13. S. T. Peng, T. Tamir, H. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  14. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  15. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  16. M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmission matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  17. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

2001 (1)

2000 (1)

1996 (2)

1995 (1)

1982 (1)

1981 (4)

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

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]

C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

J. R. Andrewartha, G. H. Derrick, R. C. McPhedran, “A general modal theory for reflection gratings,” Opt. Acta 28, 1501–1516 (1981).
[CrossRef]

1975 (1)

S. T. Peng, T. Tamir, H. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

1974 (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

1973 (2)

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

1971 (1)

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
[CrossRef]

Adams, J. 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]

C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Andrewartha, J. R.

C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

J. R. Andrewartha, G. H. Derrick, R. C. McPhedran, “A general modal theory for reflection gratings,” Opt. Acta 28, 1501–1516 (1981).
[CrossRef]

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]

Bertoni, H.

S. T. Peng, T. Tamir, H. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Botten, C.

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]

C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Cadilhac, M.

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
[CrossRef]

Cerutti-Maori, G.

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
[CrossRef]

Craig, M. S.

C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

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]

Derrick, G. H.

J. R. Andrewartha, G. H. Derrick, R. C. McPhedran, “A general modal theory for reflection gratings,” Opt. Acta 28, 1501–1516 (1981).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Gaylord, T. K.

Grann, E. B.

Li, L.

McPhedran, R. C.

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]

J. R. Andrewartha, G. H. Derrick, R. C. McPhedran, “A general modal theory for reflection gratings,” Opt. Acta 28, 1501–1516 (1981).
[CrossRef]

C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Moharam, M. G.

Nevière, M.

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 17, 1773 (2001).
[CrossRef]

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]

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
[CrossRef]

M. Nevière, E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2003).

Peng, S. T.

S. T. Peng, T. Tamir, H. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Petit, R.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Pommet, D. A.

Popov, E.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Tamir, T.

S. T. Peng, T. Tamir, H. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Vincent, P.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. T. Peng, T. Tamir, H. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Nouv. Rev. Opt. (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

Opt. Acta (3)

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]

C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

J. R. Andrewartha, G. H. Derrick, R. C. McPhedran, “A general modal theory for reflection gratings,” Opt. Acta 28, 1501–1516 (1981).
[CrossRef]

Opt. Commun. (3)

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par un réseau infiniment conducteur,” Opt. Commun. 3, 48–52 (1971).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

Other (2)

M. Nevière, E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, New York, 2003).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (17)

Fig. 1
Fig. 1

Convergence with respect to the truncation parameter N of the -1st-order diffraction efficiency of a metallic sinusoidal grating with period d equal to the groove depth h=0.5 µm in TM polarized light with wavelength 0.6328 µm, incident from air at an angle of 45°. Calculations were made for three different complex refractive indices of the substrate with values shown in the figure.

Fig. 2
Fig. 2

Schematical representation of a lamellar grating.

Fig. 3
Fig. 3

-1st-order efficiency of a lamellar grating with optical index of the material n1=0+i10 as function of the groove width c for N=15. d=h=0.5 µm, λ=0.6328 µm, θ=30 deg. (a) RCW method, (b) rigorous modal method.

Fig. 4
Fig. 4

Condition number of 〚〛 as a function of the groove width c. All the parameters are the same as in Fig. 3.

Fig. 5
Fig. 5

Matrix truncation parameters M, Δ, and N.

Fig. 6
Fig. 6

Two-step truncation method applied to the multiplication of two matrices.

Fig. 7
Fig. 7

Two-step truncation method applied to the multiplication of three matrices.

Fig. 8
Fig. 8

Effect of the truncation type on the reconstruction of a continuous function obtained by the product of two discontinuous ones. In this example, N=50 and M=150.

Fig. 9
Fig. 9

Schematic representation of a slanted grating illuminated in conical mounting.

Fig. 10
Fig. 10

Improvement of convergence as a function of truncation parameter with the use of the FFF method compared with the conventional formulation. Shown is the effect of the truncation parameter Δ on the fluctuations with the FFF method.

Fig. 11
Fig. 11

Influence of the Δ parameter on the elimination of the fluctuations when the truncation number N is increased.

Fig. 12
Fig. 12

Performance of the two-step truncation method when it is applied to the eigenvalue problems stated by Eqs. (3) and (4).

Fig. 13
Fig. 13

Performance of the two-step truncation method as a function of the fill-in-ratio grating parameter.

Fig. 14
Fig. 14

Schematical representation of the lamellar grating with artificially changed bulk material.

Fig. 15
Fig. 15

(a) same as in Fig. 3(a), but with N=20 instead of 15; (b) same as (a) but with a bulk material with losses.

Fig. 16
Fig. 16

Convergence with respect to N of the total diffracted energy (sum of efficiencies) for the grating presented in Figs. 3 and 15 with c/d=0.5. (a) nm=n1=0+i10; (b) n1=0+i10 and nm=2+i10; (c) nm=n1=2+i10; (d) n1=0+i10 and nm=2+i0.

Fig. 17
Fig. 17

Convergence of the -1st-order efficiency of a sinusoidal grating when the truncation parameter N increases. The grating is the same as in Fig. 1. The bulk region with an index nm is covered with a 40-nm-thick layer with index n1. Thick curve, nm=3+i10 and n1=0+i10; thin curve, nm=n1=0+i10 (the same as for the dashed curve in Fig. 1).

Equations (12)

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

dFdy=MF,
M=i0αω-1α-ωμ-ω1-10.
F(y)=Vφ(y)V-1F(0).
det(M-RI)=0,
M1,2M2,1-R2I=0.
U=ω2μ1-1I-αω2μ-1α.
M˜i,j=Mi,j+L,
f(x)=a,|x|<π2a2,π2<|x|π,
hn(M)=m=-MMfn-mgm.
h(MM)(x)=n=-MMhn(M) exp(inx),
h(MN)(x)=n=-NNhn(M) exp(inx),
h(NN)(x)=n=-NNhn(N) exp(inx).

Metrics