## Solving conical diffraction grating problems with integral equations

JOSA A, Vol. 27, Issue 3, pp. 585-597 (2010)

http://dx.doi.org/10.1364/JOSAA.27.000585

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### Abstract

Off-plane scattering of time-harmonic plane waves by a plane diffraction grating with arbitrary conductivity and general surface profile is considered in a rigorous electromagnetic formulation. Integral equations for conical diffraction are obtained involving, besides the boundary integrals of the single and double layer potentials, singular integrals, the tangential derivative of single-layer potentials. We derive an explicit formula for the calculation of the absorption in conical diffraction. Some rules that are expedient for the numerical implementation of the theory are presented. The efficiencies and polarization angles compared with those obtained by Lifeng Li for transmission and reflection gratings are in a good agreement. The code developed and tested is found to be accurate and efficient for solving off-plane diffraction problems including high-conductive gratings, surfaces with edges, real profiles, and gratings working at short wavelengths.

© 2010 Optical Society of America

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(050.1960) Diffraction and gratings : Diffraction theory

(260.1960) Physical optics : Diffraction theory

(260.2110) Physical optics : Electromagnetic optics

(260.5430) Physical optics : Polarization

(290.5820) Scattering : Scattering measurements

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: October 5, 2009

Manuscript Accepted: December 25, 2009

Published: February 25, 2010

**Citation**

Leonid I. Goray and Gunther Schmidt, "Solving conical diffraction grating problems with integral equations," J. Opt. Soc. Am. A **27**, 585-597 (2010)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-27-3-585

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### References

- P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R.Petit, ed. (Springer, 1980), pp. 101-121. [CrossRef]
- M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2002).
- E. Popov and L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,” J. Opt. 17, 175-180 (1986). [CrossRef]
- S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393-6400 (1991). [CrossRef]
- L. Li, “Multilayer coated diffraction gratings: differential method of Chandezon revisited,” J. Opt. Soc. Am. A 11, 2816-2828 (1994). [CrossRef]
- J. P. Plumey, G. Granet, and J. Chandezon, “Differential covariant formalism for solving Maxwell's equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835-842 (1995). [CrossRef]
- L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581-2591 (1993). [CrossRef]
- F. Zolla and R. Petit, “Method of fictitious sources as applied to the electromagnetic diffraction of a plane wave by a grating in conical diffraction mounts,” J. Opt. Soc. Am. A 13, 796-802 (1996). [CrossRef]
- Ch. Hafner, Post-modern Electromagnetics: Using Intelligent Maxwell Solvers (Wiley, 1999).
- J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139-156 (2002). [CrossRef]
- R. Köhle, “Rigorous simulation study of mask gratings at conical illumination,” Proc. SPIE 6607, 66072Z (2007). [CrossRef]
- L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetics Waves: Numerical Simulations (Wiley, 2001), pp. 61-110.
- D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34-43 (1997). [CrossRef]
- J. M. Bendickson, E. N. Glytsis, and T. K. Gaylord, “Modeling considerations for rigorous boundary element method analysis of diffractive optical elements,” J. Opt. Soc. Am. A 18, 1495-1507 (2001). [CrossRef]
- E. G. Loewen and E. Popov, Diffraction Gratings and Applications, Vol. 58 of Optical Engineering Series (Marcel Dekker, 1997), pp. 367-399.
- D. Maystre, M. Neviere, and R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R.Petit, ed. (Springer, 1980), pp. 159-225. [CrossRef]
- L. I. Goray, I. G. Kuznetsov, S. Yu. Sadov, and D. A. Content, “Multilayer resonant subwavelength gratings: effects of waveguide modes and real groove profiles,” J. Opt. Soc. Am. A 23, 155-165 (2006). [CrossRef]
- B. H. Kleemann and J. Erxmeyer, “Independent electromagnetic optimization of the two coating thicknesses of a dielectric layer on the facets of an echelle grating in Littrow mount,” J. Mod. Opt. 51, 2093-2110 (2004). [CrossRef]
- M. Saillard and D. Maystre, “Scattering from metallic and dielectric surfaces,” J. Opt. Soc. Am. A 7, 982-990 (1990). [CrossRef]
- B. H. Kleemann, J. Gatzke, Ch. Jung, and B. Nelles, “Design and efficiency characterization of diffraction gratings for applications in synchrotron monochromators by electromagnetic methods and its comparison with measurement,” Proc. SPIE 3150, 137-147 (1997). [CrossRef]
- L. I. Goray and J. F. Seely, “Efficiencies of master replica, and multilayer gratings for the soft-x-ray-extreme-ultraviolet range: modeling based on the modified integral method and comparisons with measurements,” Appl. Opt. 41, 1434-1445 (2002). [CrossRef] [PubMed]
- A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).
- D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R.Petit, ed. (Springer, 1980), pp. 53-100. [CrossRef]
- A. Pomp, “The integral method for coated gratings: computational cost,” J. Mod. Opt. 38, 109-120 (1991). [CrossRef]
- B. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile. Theory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996). [CrossRef]
- E. Popov, B. Bozhkov, D. Maystre, and J. Hoose, “Integral method for echelles covered with lossless or absorbing thin dielectric layers,” Appl. Opt. 38, 47-55 (1999). [CrossRef]
- L. I. Goray, “Modified integral method for weak convergence problems of light scattering on relief grating,” Proc. SPIE 4291, 1-12 (2001). [CrossRef]
- L. I. Goray and S. Yu. Sadov, “Numerical modeling of coated gratings in sensitive cases,” in Diffractive Optics and Micro-Optics, R.Magnusson, ed., Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), 365-379.
- L. I. Goray, J. F. Seely, and S. Yu. Sadov, “Spectral separation of the efficiencies of the inside and outside orders of soft-x-ray-extreme ultraviolet gratings at near normal incidence,” J. Appl. Phys. 100, 094901 (2006). [CrossRef]
- L. I. Goray, “Specular and diffuse scattering from random asperities of any profile using the rigorous method for x-rays and neutrons,” Proc. SPIE 7390-30, 73900V (2009). [CrossRef]
- J. Elschner, R. Hinder, F. Penzel, and G. Schmidt, “Existence, uniqueness and regularity for solutions of the conical diffraction problem,” Math. Models Meth. Appl. Sci. 10, 317-341 (2000).
- G. Schmidt, “Boundary integral methods for periodic scattering problems,” in Around the Research of Vladimir Maz'ya II. Partial Differential Equations (Springer, 2010), pp. 337-364.
- L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553-573 (1993). [CrossRef]
- M. Mansuripur, L. Li, and W.-H. Yeh, “Diffraction gratings: part 2,” Opt. Photonics News 10, August 1, 1999, pp. 44-48. [CrossRef]
- L. Li and J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247-2255 (1996). [CrossRef]
- E. Popov, B. Chernov, M. Neviere, and N. Bonod, “Differential theory: application to highly conducting gratings,” J. Opt. Soc. Am. A 21, 199-206 (2004). [CrossRef]
- “IXO International X-ray observatory,” Goddard Space Flight Center, http://ixo.gsfc.nasa.gov/technology/xgs.html.
- “X-ray interactions with matter,” http://henke.lbl.gov/optical_constants/.
- J. F. Seely, L. I. Goray, B. Kjornrattanawanich, J. M. Laming, G. E. Holland, K. A. Flanagan, R. K. Heilmann, C.-H. Chang, M. L. Schattenburg, and A. P. Rasmussen, “Efficiency of a grazing-incidence off-plane grating in the soft-x-ray region,” Appl. Opt. 45, 1680-1687 (2006). [CrossRef] [PubMed]

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