volume | PIER Journals

Abstract

The Floquet-Bloch theorem allows waves in infinite, lossless periodic media to be expressed as a sum of discrete Floquet-Bloch modes, but its validity is challenged under the realistic constraints of loss and finite extent. In this work, we mathematically reveal the existence of Floquet-Bloch modes in the electromagnetic fields sustained by lossy, finite periodic layered media using Maxwell's equations alone without invoking the Floquet-Bloch theorem. Starting with a transfer-matrix representation of the electromagnetic field in a generic layered medium, we apply Fourier transformation and a series of mathematical manipulations to isolate a term explicitly dependent on Floquet-Bloch modes. Fourierdomain representation of the electromagnetic field can be reduced into a product of the Floquet-Bloch term and two other matrix factors: one governed by reflections from the medium boundaries and another dependent on layer composition. Electromagnetic fields in any finite, lossy, layered structure can now be interpreted in the Fourier-domain by separable factors dependent on distinct physical features of the structure. The developed theory enables new methods for analyzing and communicating the electromagnetic properties of layered metamaterials.

1. Brillouin, L., Wave Propagation in Periodic Structures, Dover, New York, 1953.

2. Rytov, S. M., "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP, Vol. 2, 466-475, 1956.

3. Sigelmann, R. A., "Radiation from periodic structures excited by an aperiodic source," IEEE Trans. Antennas Propag., Vol. 13, 354-364, 1965.
doi:10.1109/TAP.1965.1138437

4. Berreman, D. W., "Optics in stratified and anisotropic media: 4 × 4-matrix formulation," J. Opt. Soc. Am., Vol. 62, 502-510, 1972.
doi:10.1364/JOSA.62.000502

5. Yeh, P., A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am., Vol. 67, 423-438, 1977.
doi:10.1364/JOSA.67.000423

6. Yariv, A. and P. Yeh, "Electromagnetic propagation in periodic stratified media. II. Birefringence, phase-matching, and X-ray lasers," J. Opt. Soc. Am., Vol. 67, 438-448, 1977.
doi:10.1364/JOSA.67.000438

7. Yeh, P., "Electromagnetic propagation in birefringent layered media," J. Opt. Soc. Am., Vol. 69, 742-756, 1979.
doi:10.1364/JOSA.69.000742

8. Yeh, P., Optical Waves in Layered Media, Wiley, New York, 1988.

9. Macleod, H. A., Thin-film Optical Filters, 4th Edition, CRC Press, Boca Raton, 2010.

10. Sihvola, A., Electromagnetic Mixing Formulas and Applications, IEEE, London, 1999.
doi:10.1049/PBEW047E

11. Born, M. and E. Wolf, Principles of Optics, 4th Edition, Pergamon Press, Oxford, 1970.

12. Nicholson, A. M. and G. F. Ross, "Measurement of the intrinsic properties of materials by timedomain techniques," IEEE Trans. Instrum. Meas., Vol. 19, 377-382, 1970.
doi:10.1109/TIM.1970.4313932

13. Weir, W. B., "Automatic measurement of complex dielectric constant and permeability at microwave frequencies," Proc. IEEE, Vol. 62, 33-36, 1974.
doi:10.1109/PROC.1974.9382

14. Smith, D. R., D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. B, Vol. 71, 036617, 2005.
doi:10.1103/PhysRevE.71.036617

15. Pozar, D. M., Microwave Engineering, 3rd Edition, 174–189, Wiley, New York, 2005.

16. Mortensen, N. A., M. Yan, O. Sigmund, and O. Breinbjerg, "On the unamibiguous determination of effective optical properties of periodic metamaterials: A one-dimensional case study," J. Europ. Opt. Soc. Rap. Public., Vol. 5, 10010, 2010.
doi:10.2971/jeos.2010.10010

17. Clausen, N. C. J., S. Arslanagic, and O. Breinbjerg, "Comparison of spatial harmonics in infinite and finite Bragg stacks for metamaterial homogenization," Photon. Nanostruct.: Fundam. Appl., 2014, http://dx.doi.org/10.1016/j.photonics.2014.06.006.

18. Arslanagic, S., T. V. Hansen, N. A. Mortensen, A. H. Gregersen, O. Sigmund, R. W. Ziolkowski, and O. Breinbjerg, "A review of the scattering-parameter extraction method with clarification of ambiguity issues in relation to metamaterial homogenization," IEEE Antennas Propag. Mag., Vol. 55, 91-106, 2013.
doi:10.1109/MAP.2013.6529320

19. Smith, D. R. and J. B. Pendry, "Homogenization of metamaterials by field averaging," J. Opt. Soc. Am. B, Vol. 23, 391-403, 2006.
doi:10.1364/JOSAB.23.000391

20. Chau, K. J., "Homogenization of waveguide-based metamaterials by energy averaging," Phys. Rev. B, Vol. 85, 125101, 2012.
doi:10.1103/PhysRevB.85.125101

21. Floquet, G., "Sur les equations differentielles linearies a coefficients periodique," Ann. Ecole Norm. Sup., Vol. 12, 47-88, 1883.

22. Bloch, F., "Uber die quantenmachanick der electronen in kristallgittern," Z. Phys., Vol. 52, 555-600, 1928.
doi:10.1007/BF01339455

23. Ramo, S., J. R. Whinnery, and T. van Duzer, Fields and Waves in Communication Electronics, 474-479, Wiley, New York, 1965.

24. Chu, R.-S. and J. A. Kong, "Modal theory of spatially periodic media," IEEE Trans. Microw. Theory Techn., Vol. 25, 18-24, 1977.
doi:10.1109/TMTT.1977.1129025

25. Gralak, B., S. Enoch, and G. Tayeb, "Anomalous refractive properties of photonic crystals," J. Opt. Soc. Am. A, Vol. 17, 1012-1020, 2000.
doi:10.1364/JOSAA.17.001012

26. Lombardet, B., L. A. Dunbar, R. Ferrini, and R. Houdre, "Fourier analysis of Bloch wave propagation in photonic crystals," J. Opt. Soc. Am. B, Vol. 22, 1179-1190, 2005.
doi:10.1364/JOSAB.22.001179

27. Lombardet, B., L. A. Dunbar, R. Ferrini, and R. Houdr, "Bloch wave propagation in twodimensional photonic crystals: Influence of the polarization," Opt. Quant. Electron., Vol. 37, 293-307, 2005.
doi:10.1007/s11082-005-1186-4

28. Eshrah, I. A. and A. A. Kishk, "A periodically loaded transmission line excited by an aperiodic source --- A Green’s function approach," IEEE Trans. Microw. Theory Techn., Vol. 55, 1118-1123, 2007.
doi:10.1109/TMTT.2007.897667

29. Valerio, G., P. Baccarelli, P. Burghignoli, A. Galli, R. Rodriguez-Berral, and F. Mesa, "Analysis of periodic shielded microstrip lines excited by nonperiodic sources through the array scanning method," Radio Sci., Vol. 43, RS1009, 2008.
doi:10.1029/2007RS003697

30. Sjoberg, D., C. Engstr¨om, G. Kristensson, D. J. N. Wall, and N. Wellander, "A Floquet-Bloch decomposition of Maxwell’s equations applied to homogenization," Multiscale Model. Simul., Vol. 4, 149-171, 2006.
doi:10.1137/040607034

31. Tsukerman, I., "Negative refraction and the minimum lattice cell size," J. Opt. Soc. Am. B, Vol. 25, 927-936, 2008.
doi:10.1364/JOSAB.25.000927

32. Cabuz, A. I., D. Felbacq, and D. Cassagne, "Spatial dispersion in negative-index composite metamaterials," Phys. Rev. A, Vol. 77, 013807, 2008.
doi:10.1103/PhysRevA.77.013807

33. Rockstuhl, C., T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. P. Meyrath, and H. Giessen, "Transition from thin-film to bulk properties of metamaterials," Phys. Rev. B, Vol. 77, 035126, 2008.
doi:10.1103/PhysRevB.77.035126

34. Alu, A., "First-principles homogenization theory for periodic metamaterials," Phys. Rev. B, Vol. 84, 075153, 2011.
doi:10.1103/PhysRevB.84.075153

35. Andryieuski, A., S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Labrinenko, "Bloch-mode analysis for retrieving effective parameters of metamaterials," Phys. Rev. B, Vol. 86, 035127, 2012.
doi:10.1103/PhysRevB.86.035127

36. Fan, S., P. R. Villeneuve, and J. D. Joannopuolos, "Large omnidirectional band gaps in metallodielectric photonic crystals," Phys. Rev. B, Vol. 54, 11245-11251, 1996.
doi:10.1103/PhysRevB.54.11245

37. Huang, K. C., E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B, Vol. 69, 195111, 2004.
doi:10.1103/PhysRevB.69.195111

38. Parisi, G., P. Zilio, and F. Romanato, "Complex Bloch-modes calculation of plasmonic crystal slabs by means of finite elements method," Opt. Express, Vol. 20, 16690-16703, 2012.
doi:10.1364/OE.20.016690

39. Kong, J. A., Electromagnetic Wave Theory, 6th Edition, EMW Publishing, Cambridge, 2005.

40. Depine, R. A. and A. Lakhtakia, "A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity," Microw. Opt. Technol. Lett., Vol. 41, 315-316, 2004.
doi:10.1002/mop.20127

41. Al Shakhs, M. H., P. Ott, and K. J. Chau, "Band diagrams of layered plasmonic metamaterials," J. App. Phys., Vol. 116, 173101, 2014.
doi:10.1063/1.4900532

42. Johnson, P. B. and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B, Vol. 4, 4370-4379, 1972.
doi:10.1103/PhysRevB.6.4370

43. Xu, T., M. Abashin, A. Agrawal, K. J. Chau, and H. J. Lezec, "All-angle negative refraction and active flat lensing of ultraviolet light," Nature, Vol. 497, 470-474, 2013.
doi:10.1038/nature12158

44. Verhagen, E., R. de Waele, L. Kuipers, and A. Polman, "Three-dimensional negative index of refraction at optical frequencies by coupling plasmonic waveguides," Phys. Rev. Lett., Vol. 105, 223901, 2010.
doi:10.1103/PhysRevLett.105.223901

45. Ott, P., M. H. Al Shakhs, H. J. Lezec, and K. J. Chau, "Flat lens criterion by small-angle phase," Opt. Express, Vol. 22, 29340-29355, 2014.
doi:10.1364/OE.22.029340

Dr. Kenneth J. Chau

Dr. Mohammed H. Al Shakhs

Prof. Peter Ott