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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 17, Iss. 10 — Oct. 1, 2000
  • pp: 1773–1784

Grating theory: new equations in Fourier space leading to fast converging results for TM polarization

Evgeni Popov and Michel Nevière  »View Author Affiliations

JOSA A, Vol. 17, Issue 10, pp. 1773-1784 (2000)

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Using theorems of Fourier factorization, a recent paper [J. Opt. Soc. Am. A 13, 1870 (1996)] has shown that the truncated Fourier series of products of discontinuous functions that were used in the differential theory of gratings during the past 30 years are not converging everywhere in TM polarization. They turn out to be converging everywhere only at the limit of infinitely low modulated gratings. We derive new truncated equations and implement them numerically. The computed efficiencies turn out to converge about as fast as in the TE-polarization case with respect to the number of Fourier harmonics used to represent the field. The fast convergence is observed on both metallic and dielectric gratings with sinusoidal, triangular, and lamellar profiles as well as with cylindrical and rectangular rods, and examples are shown on gratings with 100% modulation. The new formulation opens a new wide range of applications of the method, concerning not only gratings used in TM polarization but also conical diffraction, crossed gratings, three-dimensional problems, nonperiodic objects, rough surfaces, photonic band gaps, nonlinear optics, etc. The formulation also concerns the TE polarization case for a grating ruled on a magnetic material as well as gratings ruled on anisotropic materials. The method developed is applicable to any theory that requires the Fourier analysis of continuous products of discontinuous periodic functions; we propose to call it the fast Fourier factorization method.

© 2000 Optical Society of America

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(050.1940) Diffraction and gratings : Diffraction
(050.1970) Diffraction and gratings : Diffractive optics
(050.2770) Diffraction and gratings : Gratings
(260.2110) Physical optics : Electromagnetic optics

Evgeni Popov and Michel 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)

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  1. M. Nevière, R. Petit, and M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
  2. M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
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  5. M. Nevière and J. Flamand, “Electromagnetic theory as it applies to X-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
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  17. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
  18. F. Montiel, M. Nevière, and P. Peyrot, “Waveguide confinement of Čerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1988).
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  27. E. Popov and M. Nevière, “Differential theory for diffraction gratings: new formulation for TM polarization with rapid convergence,” Opt. Lett. 25, 598–600 (2000).
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  29. D. Maystre and M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
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  31. M. Nevière, E. Popov, R. Reinisch, and G. Vitrant, Electromagnetic Resonances in Nonlinear Optics (Gordon & Breach, London, UK) (to be published).
  32. P. L. Knight, ed., Special issue on Photonic Band Structures, J. Mod. Opt. 41(2) (1994).
  33. M. Nevière and E. Popov, “Grating electromagnetic theory user guide,” J. Imaging Sci. Technol. 41, 315–323 (1997).

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