Abstract

The Fourier modal method (FMM), often also referred to as rigorous coupled-wave analysis (RCWA), is known to suffer from numerical instabilities when applied to low-loss metallic gratings under TM incidence. This problem has so far been attributed to the imperfect conditioning of the matrices to be diagonalized. The present analysis based on a modal vision reveals that the so-called instabilities are true features of the solution of the mathematical problem of a binary metal grating dealt with by truncated Fourier representation of Maxwell’s equations. The extreme sensitivity of this solution to the optogeometrical parameters is the result of the excitation, propagation, coupling, interference, and resonance of a finite number of very slow propagating spurious modes. An astute management of these modes permits a complete and safe removal of the numerical instabilities at the price of an arbitrarily small and controllable reduction in accuracy as compared with the referenced true-mode method.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. P. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
    [Crossref]
  2. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [Crossref]
  3. G. Granet, and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
    [Crossref]
  4. M. Nevière and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).
  5. E. Popov and 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]
  6. E. Popov, B. Chernov, M. Nevière, and N. Bonod, "Differential theory: application to highly conducting gratings," J. Opt. Soc. Am. A 21, 199-206 (2004).
    [Crossref]
  7. K. Watanabe, "Study of the differential theory of lamellar gratings made of highly conducting materials," J. Opt. Soc. Am. A 23, 69-72 (2006).
    [Crossref]
  8. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
    [Crossref]
  9. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
    [Crossref]
  10. M. Foresti, L. Menez, and A. Tishchenko, "Modal method in deep metal-dielectric gratings: the decisive role of hidden modes," J. Opt. Soc. Am. A 23, 2501-2509 (2006).
    [Crossref]
  11. J. W. Gibbs, "Fourier series," Nature 59, 200 (1898).
    [Crossref]
  12. J. W. Gibbs, "Fourier series," Nature 59, 606 (1899).
    [Crossref]
  13. J. Chandezon, M. T. Dupuis, and G. Cornet, "Multicoated gratings: a differential formalism applicable in the entire optical region," J. Opt. Soc. Am. 72, 839-846 (1982).
    [Crossref]
  14. T. Vallius, "Comparing the Fourier modal method with the C method: analysis of conducting multilevel gratings in TM polarization," J. Opt. Soc. Am. A 19, 1555-1561 (2002).
    [Crossref]
  15. M. Foresti, "Etude et développement de systèmes nanostructurés pour verres optiquement fonctionnels," Ph.D. thesis (Université Jean Monnet, Saint-Etienne, 2007).

2006 (2)

2004 (1)

2002 (1)

2000 (1)

1996 (3)

1982 (1)

1981 (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[Crossref]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[Crossref]

1899 (1)

J. W. Gibbs, "Fourier series," Nature 59, 606 (1899).
[Crossref]

1898 (1)

J. W. Gibbs, "Fourier series," Nature 59, 200 (1898).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

Nature (2)

J. W. Gibbs, "Fourier series," Nature 59, 200 (1898).
[Crossref]

J. W. Gibbs, "Fourier series," Nature 59, 606 (1899).
[Crossref]

Opt. Acta (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[Crossref]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[Crossref]

Other (2)

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

M. Foresti, "Etude et développement de systèmes nanostructurés pour verres optiquement fonctionnels," Ph.D. thesis (Université Jean Monnet, Saint-Etienne, 2007).

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 (12)

Fig. 1
Fig. 1

Groove width dependence of the minus-first-order diffraction efficiency with λ = 632.8 nm and Λ = 500 nm .

Fig. 2
Fig. 2

Example of FMM modal H field of mode N 1 (multi-peaked curve, red online) and regular plasmon mode 0 (lower curve, black) for the truncation number 31. The groove is located between x = 0 and 93.518 nm .

Fig. 3
Fig. 3

Effective index spurious modes dependence versus groove width.

Fig. 4
Fig. 4

Resonance effect of spurious Fourier mode N 3 self-interference in the grating layer.

Fig. 5
Fig. 5

Set of resonances under a fine scan of the groove width.

Fig. 6
Fig. 6

Field enhancement in the grating region under spurious mode resonance. The air groove is located between 0 and 0.187 of the X period axis.

Fig. 7
Fig. 7

Diffraction efficiency of the minus first order calculated by the FMM with the described spurious modes filtering.

Fig. 8
Fig. 8

Difference of diffraction efficiencies calculated by TMM and FMM for 31 (upper curve, black) and 61 modes (lower curve, red online).

Fig. 9
Fig. 9

H field distribution in the grating region calculated by the FMM with mode filtering.

Fig. 10
Fig. 10

H field distribution in the grating region calculated by the reference TMM.

Fig. 11
Fig. 11

Diffraction efficiency of the minus first order of a 20 nm thin lossy metal grating without (noisy curve, black) and with (smooth curve, red online) spurious modes filtering.

Fig. 12
Fig. 12

Diffraction efficiency of the minus first order of a sinusoidal grating calculated by the C method (horizontal line) and for a sliced grating calculated by the FMM without (middle curve, red online) and with (lower curve, blue online) spurious modes filtering and calculated by the TMM (upper curve, green online).

Tables (2)

Tables Icon

Table 1 Complex Effective Index of a Few First- and Last-Order TMM and FMM Modes with Truncation Number N = 31 a

Tables Icon

Table 2 Evolution of the Effective Index of the Spurious FMM Modes in a Lossless Metal Grating with the Truncation Number N a

Metrics