The electro-optic matrix for lithium niobate, LiNbO₃, has the form given in Table 3.3, with four independent constants. Write down the index ellipsoid matrix η̃ for an electric field along the z-direction (i.e., the x₃ direction, which is the same as the c-axis).

(a) For a wave propagating in the x-direction, what are the natural polarizations and what are the indices of refraction felt by each?

(b) Figure 3.23 shows a typical electro-optic modulator geometry. Electro-optic devices are often characterized by their half-wave voltage, that is, the voltage needed to produce a phase shift of π between the two different polarizations. Show that for a crystal with length d in the x-direction, and thickness l in the z-direction across which the voltage is applied, the half-wave voltage of a LiNbO₃ crystal is given by

where λ₀ is the vacuum wavelength of the light and r is the appropriate electro-optic constant. If r = 30 × 10⁻¹² m/V, and l and d are both 2 mm, what is the half-wave voltage?

Table 3.3 Electro-optic matrix for C3v symmetry k r11 -11 0 0 142 0 E O 0 0 0 r13 13 r33 0 0 -11 0 142 0 2