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11.26 Gratings: Theory and Numeric Applications, 2012 of (11.40)-(11.41), we consider the case of square inclusions of sidelength a = d/2, where d is the pitch of a bi-periodic grating. The eigenfunctions are ψnm(y) = 2sin(nπy1)sin(nπy2) in (11.41) and the corresponding eigenvalues are k 2 nm = π 2 (n 2 + m 2 ). The right-hand side in the homogenized equation (11.40) can then be interpreted in terms of effective magnetism: µhom(k) = 1 + 64a2 π 4 ∑ (n,m)odd k2 n2m2 (k2 nm/a2 − k2 . (11.44) ) This function can be computed numerically for instance with Matlab and demonstrates that negative values can be achieved for µhom near resonances, see Fig. 11.15(b). This allows for superlensing via negative refraction, as shown in Fig. 11.15(a). Finally, we would like to point out that high-order homogenization techniques [67] suggest that most gratings display some artificial magnetism and chirality when the wavelength is no longer much larger than the periodicity [68]. We hope we have convinced the reader that there is a whole new range of physical effects in gratings which classical, high-frequency and high-contrast homogenization theories can capture.
S. Guenneau et al.: Homogenization Techniques for Periodic Structures 11.27 References: [1] Petit, R., 1980. Electromagnetic theory of gratings, Topics in current physics, Springer- Verlag, Berlin. [2] Bakhvalov, N. S.,1975. Averaging of partial differential equations with rapidly oscillating coefficients. Dokl. Akad. Nauk SSSR 221, 516–519. English translation in Soviet Math. Dokl. 16, 1975. [3] De Giorgi, E., Spagnolo, S., 1973. Sulla convergenza degli integrali dell’energia per operatori ellitici del secondo ordine. Boll. Unione Mat. Ital., Ser 8, 391–411. [4] Bensoussan, A., Lions, J.L., Papanicolaou, G., 1978. Asymptotic analysis for periodic structures, North-Holland, Amsterdam [5] Marchenko, V. A., Khruslov, E. Ja., 1964. Boundary-value problems with fine-grained boundary. (Russian) Mat. Sb. (N.S.) 65 (107) 458–472. [6] Tartar, L., 1974. Problème de côntrole des coefficients dans des équations aux dérivées partielles. Lecture Notes in Economics and Mathematical Systems 107, 420–426. [7] Murat, F., 1978. Compacité par compensation. (French) Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5(3), 489507. [8] Nguetseng, G., 1989. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (3), 608–623. [9] Cioranescu, D., Damlamian, A., Griso, G., 2002. Periodic unfolding and homogenization, C. R. Math. Acad. Sci. Paris, 335, 99-104. [10] Bakhvalov, N. S., Panasenko, G. P., 1984. Homogenization: Averaging Processes in Periodic Media. Nauka, Moscow (in Russian). English translation in: Mathematics and its Applications (Soviet Series) 36, Kluwer Academic Publishers. [11] Jikov, V. V., Kozlov, S. M., Oleinik, O. A., 1994. Homogenization of Differential Operators and Integral Functionals. Springer, Berlin. [12] Sanchez-Palencia, E., 1980 Nonhomogeneous media and Vibration Theory. Lecture Notes in Physics 127, Springer, Berlin. [13] Chechkin, G. A., Piatnitski, A. L., Shamaev, A. S., 2007. Homogenization: Methods and Applications. AMS Translations of Mathematical Monographs 234. [14] Kozlov, S. M., 1979. The averaging of random operators. Mat. Sb. (N.S.), 109(151)(2),188–202.
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