Thermal Noise in Cubic Optical Cavities

Abstract

Thermal noise in optical cavities sets a fundamental limit to the frequency instability of ultra-stable lasers. Numata et al. derived three equations based on strain energy and the fluctuation–dissipation theorem to estimate the thermal noise contributions of the spacer, substrates, and coating. These equations work well for cylindrical cavities. Extending from that, an expression for the thermal noise for a cubic spacer based on the fluctuation–dissipation theorem is derived, and the thermal noise in cubic optical cavities is investigated in detail by theoretical analysis and finite element simulation. The result shows that the thermal noise of the analytic estimate fits well with that of finite element analysis. Meanwhile, the influence of the compressive force Fp on the thermal noise in cubic optical cavities is analyzed for the first time. For a 50 mm long ultra-low expansion cubic cavity with fused silica substrates and GaAs/AlGaAs crystalline coating, the displacement noise contributed from every Fp of 100 N is about three times more than that of the substrate and coating.

Laser frequency stabilisation for the LISA mission using a cubic cavity

“…Following removal of the cavity isothermal drift, the Allan deviation (Figure 11) is independent of measurement time over timescales between 1 and ~20 s, indicating flicker noise. This is consistent with the plot in Figure 10 which shows that on these timescales, the frequency noise PSD reduces as 1/ f. Although well within the LISA specification on these timescales we note that the observed flicker noise could be further reduced to the thermal noise limit of ~2 x 10 -15 [4,5,6] . In our setup for example, the lock error signal to noise ratio could be improved by increasing the modulation index but we chose not to do this since an additional amplifier would increase the overall device heat load.…”

“…In this section, we discuss the choice of material for the cavity spacer and mirror substrate with focus on the frequency stability specification (see Section 1). The thermal noise limit of a 5-cm cubic cavity with ULE substrate mirrors [4,5,6] in terms of a fractional Allan deviation is typically 2 ×10 -15 over timescales of a few seconds once the isothermal cavity drift is removed. Lower thermal noise limits are observed with fused silica substrates with the lowest limits observed with crystalline mirror coatings [7] .…”

“…For this measurement, the beat frequency between our laser and the NPL reference was recorded once every second. These beat frequency results were then converted to an effective temperature using the results from section 4.1 which gives the relationship as: ( 6 ) In the above equation, = 0.158 MHz/K 2 ; this relates to the thermal expansion for bore #1. The cavity set-point temperature can then be fitted to an exponential of the form: Fitting (as shown in Figure 9) yields = 18.4 hours which compares extremely well with the modelled time constant of 17.8 hours (see Section 2.3).…”

Design of a transportable miniaturized optical reference cavity with flexibly tunable thermal expansion properties

“…As for the thermal noise induced by the compensation structure that can be negligible for the 10 -15 level estimation since a finite cylinder produces lower noise than the infinite half spacer, mirror substrates with smaller radius and increased thickness will produce lower thermal noise [36]. In the condition that brown noise contributes low enough, thermo-refractive noise and thermooptical noise should be taken into consideration for precise estimation [37,38], and the underestimated thermal noise of the spacer induced by additional strain energy stored near the mirror should not be ignored [26,39].…”