Direct Kerr electro-optic effect in noncentrosymmetric materials

In materials lacking inversion symmetry, both Pockels and Kerr electro-optic effects are simultaneously present, with the former effect generally dominating the latter one. The theoretical findings of this article provide the crystal physics community with concrete tabulated evidence showing that it is possible in principle to selectively bypass contributions from the linear effect(s) and directly obtain information only about the genuine (Kerr-like) quadratic effects in $90\%$ of the noncentrosymmetric point groups. The general idea and treatment used for the electro-optic effect can be extended and adapted to other optical or non-optical (phenomenological) purely quadratic effects in media lacking inversion symmetry.

Figure 1

The two-dimensional geometry of the electro-optic effect. The probing light propagates normal (in)to the plane of the (article's) page. (a) The initial index ellipse of the unperturbed crystal; its intersection with the two Cartesian axes ${\mathrm{x̂}}_p,{\mathrm{x̂}}_q$ generates the two initial (eigen)indexes of refraction $n_{\mathit{pp}},n_{\mathit{qq}}$. (b) The new index ellipse induced by the application of an external electric field; its intersection points $n_{\mathit{pp}}^{\prime },n_{\mathit{qq}}^{\prime }$ with the principal Cartesian axes ${\mathrm{x̂}}_p,{\mathrm{x̂}}_q$ are related to $n_{\mathit{pp}},n_{\mathit{qq}}$ through the electro-optic effect [Eq. (1)]. (c) The new index ellipse in its proper system of orthogonal coordinates ${\mathrm{x̂}}_{\mathit{pq}+},{\mathrm{x̂}}_{\mathit{pq}-}$; its intersections generate the new (eigen)indexes of refraction $n_{\mathit{pq}+},n_{\mathit{pq}-}$. (d) The three (in-plane) geometrical changes that quantitatively characterize the existence of an electro-optic effect: $\Delta n_{\mathit{pp}}=n_{\mathit{pp}}^{\prime }-n_{\mathit{pp}}$, $\Delta n_{\mathit{qq}}=n_{\mathit{qq}}^{\prime }-n_{\mathit{qq}}$, and ${\theta }_{\mathit{pq}}$. If at least one of these three parameters is nonzero, then the particular configuration indicates a potentially detectable electro-optic effect.

Figure 2

Experimental schematic of the direct Kerr electro-optic effect in noncentrosymmetric crystals. The experimental setup combines a Mach-Zehnder interferometer (top left corner), a Michelson interferometer (bottom right corner), and an amplitude-modulated null polarimeter (top right corner) in a three-in-one compact design. BS1, BS2, BS3, and BS4 are similar antireflection-coated and nonpolarizing beamsplitters; D1, D2, D3 and CCD1, CCD2 are similar high-sensitivity power detectors and similar high-resolution CCD-CMOS cameras, respectively. L, S, and PBS are a low-power laser, the (crystal) sample, and a no-multiple-reflections and nonpolarizing pellicle beamsplitter, respectively. P and A are two similar high-extiction rate polarizers oriented at 90${}^{°}$ to each other in a null format.