Thermorefringent noise in crystalline optical materials

Crystalline materials are increasingly employed to construct precision optical instruments because of their reduced mechanical dissipation and consequent reduction of thermal Brownian noise. However, the anisotropy of the crystalline state implies a fundamental source of thermal noise; depolarization induced by thermal fluctuations of its birefringence. We establish the theory of this effect, which is a generalization of prior treatments of thermo-optic noises in amorphous materials. This theory—in conjunction with poorly understood anisotropic thermal stress coefficients of crystalline coatings—predict that thermo-refringent noise in crystalline mirror coatings may be lurking within an order of magnitude of Brownian noise (below 100 Hz). Thus, in order to appreciate the full promise of crystalline optical materials, a more precise understanding of their anisotropic material constants is necessary. Barring that, we elucidate measurement techniques that can affect partial coherent cancellation of thermorefringent noise. In passing, our general formalism also predicts the existence of thermal beam-pointing noise.

Figure 1

Thermorefringent noise for cryogenic silicon (left) and room-temperature lithium niobate (right), using Eqs. (3.10)–(3.12). In both cases the wavelength is 1550 nm, the beam size is $r_0=100\mathrm{\mu }\mathrm{m}$, and the material length is $\ell =1\mathrm{cm}$.

Figure 2

Schematic picture of the mirror layers that shows the elements that correspond to propagation matrices and direction of relevant field modes.

Figure 3

Thermorefringent noise [Eqs. (3.40) and (3.42)] for an AlGaAs/GaAs high reflector of 50 quarter-wave layers, with 1500 nm light with radius $r_0=4\mathrm{cm}$. The comparison with the Brownian noise of the coating is also shown.