i

An optical reference cavity is commonly employed to reduce the frequency (or phase) noise of a laser, leading to a significant reduction in the integrated linewidth and a corresponding improvement in the laser coherence time (or length). The ultimate noise floor in such a laser system will be determined by the noise in the reference cavity itself. With sufficient isolation, technical sources of noise can be neglected, and the fundamental noise limit will be given by the thermomechanical fluctuations of the constituent materials of the cavity.^{[1]} These unavoidable perturbations in cavity length will be directly imparted onto the frequency noise of the locked laser system. State-of-the-art laser systems employing high-performance optical cavities are now capable of achieving line widths <10 mHz, with coherence times>15 s,^{[2, 3]} with the high-reflectivity optical coatings remaining the limiting aspect of the ultimate laser performance.^{[4]}

As a means to further improve such systems, in 2013, GaAs/AlGaAs-based crystalline coatings were demonstrated as a promising alternative to traditional amorphous coatings. These single-crystal structures yield a significant reduction in the coating Brownian noise contribution, stemming from the low elastic loss of these materials.^{[5]}

In this Demonstration, we calculate the fundamental noise contributions commonly considered for Fabry–Perot optical cavities. There are a variety of design options you can choose from, with inputs being cavity length, mirror curvature, laser wavelength, spacer radius (cavity bore radius is kept constant at 5 mm), substrate and spacer material, plus coating materials. For the latter, two options are included: i) traditional ion-beam sputtered SiO_{2}/Ta_{2}O_{5} amorphous coatings^{[4]} and ii) substrate-transferred GaAs/AlGaAs crystalline coatings.^{[5]} This allows you to compare the performance of various cavity geometries and the impact of the mirror coatings on the overall cavity stability. There are two design choices for the crystalline coating, a standard quarter-wave stack or a "TO-optimized" design, which illustrates the effect of using an aperiodic multilayer engineered for thermooptic noise minimization.^{[6]}

You can also choose to display the calculated cavity noise in a variety of different units (frequency, phase or displacement noise), as well as the resulting Allan deviation.

The following noise contributions, in units of m^{2}/Hz, are considered^{[6]}:

**Spacer Brownian noise**

${S}_{x}^{\mathrm{spBr}}=\frac{4{k}_{b}T}{\pi f}\frac{L{\varphi}_{\mathrm{sp}}}{2\pi {E}_{\mathrm{sp}}\left({R}^{2}-{r}^{2}\right)}$

**Substrate Brownian noise**

${S}_{x}^{\mathrm{sBr}}=\frac{2{k}_{b}T}{\pi \surd \pi f}\frac{\text{}\left(1\text{}-\text{}{\sigma}_{s}^{2}\right)}{{E}_{s}{\omega}_{i}}{\varphi}_{s}$

**Substrate thermoelastic noise**

${S}_{x}^{\mathrm{sTE}}=\frac{4{k}_{b}{T}^{2}}{\surd \pi}\frac{\text{}{\alpha}_{s}^{2}\left(1\text{}+\text{}{\sigma}_{s}\right){\omega}_{i}}{{\kappa}_{s}\text{}}J\left(f/{f}_{T}\right)$

**Coating Brownian noise**

${S}_{x}^{\mathrm{cBr}}=\frac{2{k}_{b}T}{{\pi}^{2}f}\frac{d}{{\omega}_{i}^{2}{E}_{s}^{2}}\frac{{\text{}}_{{\varphi}_{c}}}{{E}_{c}\left(1\text{}-\text{}{\sigma}_{c}^{2}\right)}\left[{E}_{c}^{2}\left(1+{\sigma}_{s}^{2}\right)\left(1-2{\sigma}_{s}^{2}\right)+{E}_{s}^{2}\left(1+{\sigma}_{c}^{2}\right)\left(1-2{\sigma}_{c}\right)\right]$

**Coating thermooptic noise**

${S}_{x}^{\mathrm{cTO}}=$ $\text{}$ ${S}_{T}\left(f\right){\Gamma}_{\mathrm{tc}}{\left(\stackrel{\_}{{\alpha}_{c}}d-\stackrel{\_}{\beta}\lambda -\stackrel{\_}{{\alpha}_{s}}d\frac{{C}_{c}}{{C}_{s}}\right)}^{2}$

where the spectrum for thermal fluctuations is described by

${S}_{T}=\frac{2\surd 2{k}_{b}{T}^{2}}{\pi {\kappa}_{s}{\omega}_{i}}K\left(f/{f}_{T}\right)$

${\Gamma}_{\mathrm{tc}}$ is a correction to account for the finite thickness of the coating (see [7] for more information). When the thickness of the coating becomes comparable to the thermal diffusion length in the coating, that is, $d~{r}_{T}$ , where ${r}_{T}=\sqrt{\frac{{\kappa}_{c}}{2\pi f{C}_{\mathrm{c.}}}}$ For a GaAs/AlGaAs crystalline coating, this diffusion length is on the order of ~100 µm at 1 kHz, thus it is important to take this into account.

$K\left(f/{f}_{T}\right)$ and $J\left(f/{f}_{T}\right)$ are special functions that consider the low frequency response, which has a low pass filter response at the characteristic frequency ${f}_{T}=\frac{{\kappa}_{s}}{\pi {\omega}_{i}^{2}{C}_{s}}$ . See [8] for further details.

It can be seen in the spectrum for coating thermooptic noise, that proper engineering of the average values $\stackrel{\_}{{\alpha}_{c}}$ $\text{,}$ $\stackrel{\_}{\beta}$ $\text{,}$ $d$ can lead to minimization of this noise contribution $\left({S}_{x}^{\mathrm{cTO}}\right)$ . When coating Brownian noise is greatly reduced, as in the case of crystalline coatings, thermooptic noise can dominate the noise spectrum at low frequencies. You can see how this noise contribution can be greatly reduced by selecting "coating design TO-optimized".

The physical constants and symbols in this Demonstration (s → substrate, sp → spacer, c → coating) include the following:

$\begin{array}{ccc}\mathrm{Symbol}& \mathrm{Units}& \mathrm{Description}\\ L& m& \mathrm{Cavity\; length}\\ E& \mathrm{Pa}& \mathrm{Young\text{'}s\; modulus}\\ \sigma & \u2013& \mathrm{Poisson\text{'}s\; ratio}\\ C& \frac{J}{K{m}^{3}}& \mathrm{Heat\; capacity\; per\; unit\; volume}\\ \kappa & \frac{W}{Km}& \mathrm{Thermal\; conductivity}\\ \alpha & \frac{1}{K}& \mathrm{Coefficient\; of\; thermal\; expansion}\\ \beta & \frac{1}{K}& \mathrm{Thermo-refractive\; coefficient}\\ \varphi & \u2014& \mathrm{Elastic\; loss\; angle}\\ {\omega}_{1,2}& m& \mathrm{Laser\; beam\; radius\; of\; mirror\; 1\; or\; 2,\; defined\; as}1/{e}^{2}\mathrm{power\; decay}\\ d& m& \mathrm{Coating\; thickness}\\ {k}_{b}& \frac{J}{K}& \mathrm{Boltzmann\; constant}\\ R& m& \mathrm{Spacer\; outer\; radius}\\ r& m& \mathrm{Spacer\; bore\; radius}\\ f& \mathrm{Hz}& \mathrm{Frequency}\\ \lambda & m& \mathrm{Laser\; wavelength}\end{array}$

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