## New formulation of the Fourier modal method for crossed surface-relief gratings

JOSA A, Vol. 14, Issue 10, pp. 2758-2767 (1997)

http://dx.doi.org/10.1364/JOSAA.14.002758

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### Abstract

A new formulation of the Fourier modal method (FMM) that applies the correct rules of Fourier factorization for crossed surface-relief gratings is presented. The new formulation adopts a general nonrectangular Cartesian coordinate system, which gives the FMM greater generality and in some cases the ability to save computer memory and computation time. By numerical examples, the new FMM is shown to converge much faster than the old FMM. In particular, the FMM is used to produce well-converged numerical results for metallic crossed gratings. In addition, two matrix truncation schemes, the parallelogramic truncation and a new circular truncation, are considered. Numerical experiments show that the former is superior.

© 1997 Optical Society of America

**History**

Original Manuscript: February 24, 1997

Manuscript Accepted: March 31, 1997

Published: October 1, 1997

**Citation**

Lifeng Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A **14**, 2758-2767 (1997)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-14-10-2758

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### References

- P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978). [CrossRef]
- D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978). [CrossRef]
- S. T. Han, Y.-L. Tsao, R. M. Walser, M. F. Becker, “Electromagnetic scattering of two-dimensional surface-relief dielectricgratings,” Appl. Opt. 31, 2343–2352 (1992). [CrossRef] [PubMed]
- G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979). [CrossRef]
- R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982). [CrossRef]
- G. Granet, “Diffraction par des surfaces bipériodiques: résolutionen coordonnées non-orthogonales,” Pure Appl. Opt. 4, 777–793 (1995). [CrossRef]
- J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996). [CrossRef]
- D. C. Dobson, J. A. Cox, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Application and Theory of Periodic Structures , J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 106–113 (1991). [CrossRef]
- R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993). [CrossRef]
- E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elementswith three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994). [CrossRef]
- J.-J. Greffet, C. Baylard, P. Versaevel, “Diffraction of electromagnetic waves by crossed gratings: a seriessolution,” Opt. Lett. 17, 1740–1742 (1992). [CrossRef] [PubMed]
- O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation ofboundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993). [CrossRef]
- O. P. Bruno, F. Reitich, “Calculation of electromagnetic scattering via boundary variations andanalytic continuation,” Appl. Computat. Electromagn. Soc. J. 11, 17–31 (1996).
- L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [CrossRef]
- P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996). [CrossRef]
- G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellargratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996). [CrossRef]
- See, for example, R. C. Wrede, Introduction to Vector and Tensor Analysis (Dover, New York, 1972), Chap. 1.
- We ignore the electromagnetic edge effect at the vertices of the zigzag contour. In the far field, this artificially introduced edge effect should be negligible.
- At the time of the writing, I have not mathematically proven the validity or invalidity of the hypothesis. However, the numerical examples given in Section 7 seem to support this hypothesis. The sum clearly corresponds to the Fourier coefficient of ∊E2 with respect to x1, albeit in a complicated way. The latter, as indicated in Eq. (26b), is continuous with respect to x2.
- L. Li, “Formulation and comparison of two recursive matrix algorithms for modelinglayered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996). [CrossRef]
- S. Zohar, “Toeplitz matrix inversion: the algorithm of W. F. Trench,” J. Assoc. Comput. Mach. 16, 592–601 (1969). [CrossRef]
- L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffractiongratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993). [CrossRef]

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