Explicit Formulae for the Medium Parameters of Optically-Active Molecules and Crystals from the Microscopic Theory

Abstract

Dyadic formulae for the medium parameters of the electromagnetic point-localized constitutive relations from the microscopic theory of optically-active molecules and crystals have been newly found, which clears out long-held controversy on reciprocity and equivalence.2006 Optical Society of America OCIS codes: 160.1190, 260.1180. It is quite extraordinary that, although the finding of optical activity in one of the optically-active crystals, α-quartz, was made almost two hundred years ago, 1 some significant controversy in its understanding around reciprocity characteristics still exists. 2, 3 The work in the chemistry community 4 focused on molecular structures and the calculation of rotatory power, while the work in the physics community focused on bianisotropic characteristics of the crystalline medium after accepting certain electromagnetic constitutive relations connecting the E, D, B, H fields. We believe that the afore-mentioned controversy has been largely due to the missing connection between the two viewpoints and to so-far nonexistent explicit microscopic expressions for the relevant medium tensors for the constitutive relations. In this study, we give such a theoretical connection between the two branches of study and attempt to develop a general theory by finding the explicit general expressions for the involved dyadic coefficients of the constitutive relations by showing the microscopic connection between the coefficient dyadics to the atomic dipoles, for the first time to our knowledge, thus demonstrating directly the reciprocity of the medium as a resulting macroscopic system. This ab-initio theory will then generalize the available classical model, 5, 6 mostly developed for dissymmetric optically active molecules in a liquid.In a molecule with dissymmetric bonds such as in most sugar molecules or in optically-active liquid crystals, external electric field polarizes the individual molecules to generate a nonvanishing macroscopic magnetization field 5, 6 M • (r, t) ≡ [n mol /2] N i = 1 [r i − r] × dπ i (t)/dt as well as the polarization field P(r, t) ≡ n mol N i = 1 π i (t) summing up all of N modes of electric dipole moments π i (t) ≡ π i π i (t), along the fixed π i direction, located at position r i within a single molecule, and finally taking the statistical expectation or even randomization over n mol molecules in unit volume. In the above formula, r is chosen near the center of a given molecule with "canonical magnetic dipoles" as the ones illustrated in Figs. 1(c)-(d). For a molecule with such a well-defined cylindrical symmetry, we suppose that N dipole resonance modes are well localized all in individual atoms. Then, there are N coupled-mode equations such asfor the jth dipole with charge q e , reduced mass m j , damping constant Γ j , resonance frequency Ω j , and the well-known coupling coefficients from dipole-dipole interaction, 5, 6 Fig. 1. The phase, direction, and magnitude of the oscillating dipoles attached to atoms in a molecule: Each of (a) and (b) gives a non-canonical magneti...