The propagation of light in an anisotropic impedance-matched metamaterial is studied in the frame of geometrical optics. We prove that directions of fields D, B and v (ray velocity) are a triad of conjugate directions with respect to the inverse relative dielectric permittivity tensor and constitutes a local basis, whose reciprocal one is formed by directions of E, H fields and wave-vector k. Consequently, both dual bases are intrinsically related to the physical properties of medium. We have identified these bases with direct and reciprocal bases of a curvilinear coordinates system, showing that physics defines geometry. This identification provides a powerful tool to solve two kinds of problems (direct and inverse ones) that currently arise: In direct problems, medium properties are given and it suffices to know ε = μ tensor at every point, to obtain the wave structure. In inverse problems, medium properties must be found for the rays to propagate along prescribed trajectories. The procedure is applied to an illustrating example.
2. Pendry, J. B., D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science, Vol. 312, 1780-1782, 2006. doi:10.1126/science.1125907
3. Chen, H., C. T. Chan, and P. Sheng, "Transformation optics and metamaterials," Nature Mater., Vol. 9, 387-396, 2010. doi:10.1038/nmat2743
4. Leonhardt, U. and T. G. Philbin, Geometry and Light: The Science of Invisibility, Dover, Mineola, 2010.
5. Bellver-Cebreros, C. and M. Rodriguez-Danta, "Conjugacy and reciprocal bases in anisotropic dielectric media," Optica Pura y Aplicada, Vol. 44, No. 3, 571-580, 2011.
6. Bellver-Cebreros, C. and M. Rodriguez-Danta, "Analogy between the torque-free motion of a rigid body about a fixed point and light propagation in anisotropic media," European Journal of Physics, Vol. 30, 285-294, 2009. doi:10.1088/0143-0807/30/2/007
7. Fung, Y. C., Foundations of Solid Mechanics, Prentice Hall, Englewood Cliffs, New Jersey, 1965.
8. Salmon, G., Treatise on Conic Sections, AMS Chelsea Publishing, Providence, Rhode Island, 1960.
9. Zwikker, C., The Advanced Geometry of Plane Curves and Their Applications, Dover, Mineola, 2005.
10. Melrose, D. B. and R. C. McPhedran, Electromagnetic Processes in Dispersive Media, Cambridge University Press, Cambridge, 1991.
11. Born, M. and E. Wolf, Principles of Optics, Pergamon Press, New York, 1999.
12. Bellver-Cebreros, C. and M. Rodriguez-Danta, "Refraccion conica externa en medios biaxicos a partir de la construccion de Mohr," Optica Pura y Aplicada, Vol. 36, 33-37, 2003.
13. Jackson, J. D., Classical Electrodynamics, 3th Edition, Wiley and Sons, Oxford, 2009.
14. Joets, A. and R. Ribotta, "A geometrical model for the propagation of rays in an anisotropic inhomogeneous medium," Optics Communications, Vol. 107, 200-204, 1994. doi:10.1016/0030-4018(94)90020-5
15. Bellver-Cebreros, C. and M. Rodriguez-Danta, "Eikonal equation from continuum mechanics and analogy between equilibrium of a string and geometrical light rays," American Journal of Physics, Vol. 69, 360-367, 2001. doi:10.1119/1.1317560