Photo-Thermal Transfer Function Indicator

• Transfer function-type indicators quantify how modulated optical power absorption in dielectric coatings produces effective mirror displacement via thermo-elastic and thermo-refractive effects.
• They utilize multilayer thermal diffusion models to detect sign-switching phenomena, enabling micron-scale determination of absorption depth within complex coatings.
• Applications include precision optical metrology, noise prediction in gravitational-wave detectors, and optomechanical system stabilization.

The transfer function-type indicator, specifically the photo-thermal transfer function, quantifies the relationship between modulated absorbed optical power at the surface of a dielectric mirror and the effective mirror displacement as perceived by an interferometric read-out. Formally, for an incident intensity fluctuation $j(\omega)$ [W/m²] at angular frequency $\omega$, the transfer function $H(\omega)$ relates this to a displacement $\delta z(\omega)$ such that $H(\omega) \equiv \delta z(\omega)/j(\omega)$. This multidimensional quantity integrates the periodic thermal response of a multilayer dielectric mirror and substrates, encompassing thermo-elastic expansion and thermo-refractive optical phase shifts. The functional form and frequency dependence of $H(\omega)$ serve as a sensitive diagnostic, notably providing a robust indicator of the absorption depth within complex coatings, and hence play a pivotal role in both precision optical metrology and optomechanical system design (Ballmer, 2014).

1. Mathematical Formulation and Multilayer Thermal Response

The derivation of the photo-thermal transfer function begins with the thermal diffusion equation in each of the $k=1\dots N$ dielectric layers and the semi-infinite substrate $k=s$. In the Fourier domain, the coupled equations involve layer-specific density $\rho_k$, specific heat $C_k$, and thermal conductivity $\kappa_k$: $$\frac{\partial j_k(z,\omega)}{\partial z} + i\omega \rho_k C_k T_k(z,\omega) = 0,$$

$$j_k(z,\omega) = -\kappa_k \frac{\partial T_k(z,\omega)}{\partial z},$$

where $T_k(z,\omega)$ is the complex temperature perturbation, and $j_k(z,\omega)$ is the (one-dimensional) heat flux. The spatial temperature profile obeys

$$\xi_k^2 T_k(z) = \frac{\partial^2 T_k(z)}{\partial z^2}, \qquad \xi_k = \sqrt{\frac{i\omega \rho_k C_k}{\kappa_k}},$$

yielding general solutions within each layer: $$T_k(z) = A_k e^{+\xi_k(z-z_k)} + B_k e^{-\xi_k(z-z_k)},$$ with $A_k$ and $B_k$ determined by interface continuity conditions and boundary constraints.

The layer-averaged temperature $\bar T_k(\omega)$ for expansion and refractive-index coupling is

$$\bar T_k(\omega)d_k = \int_{z_k-d_k/2}^{z_k+d_k/2} T_k(z,\omega)\,dz = \frac{2}{\xi_k}\sinh\left(\frac{\xi_k d_k}{2}\right)(A_k + B_k).$$

For the substrate,

$$\bar T_s(\omega)d_s = \frac{T_{R,s}(\omega)}{\xi_s}.$$

2. Conversion to Effective Displacement: Thermo-Elastic and Thermo-Optic Coupling

Thermal fluctuations in each layer produce mechanical expansion with a constrained expansion coefficient $\bar\alpha_k$ and modify the coating's optical phase due to refractive-index changes ($\beta_k = dn_k/dT$). The round-trip optical phase change is: $$\delta\phi_k = \frac{4\pi}{\lambda_0} \int_k (\,\beta_k + \bar\alpha_k n_k)\,T_k(z)\,dz,$$ with $\lambda_0$ the probe wavelength, $n_k$ the refractive index. The constrained thermal expansion is defined by elastic moduli and Poisson ratios of each layer and the substrate. The overall reflected phase shift $\delta\phi_c(\omega)$ is obtained by

$$\delta\phi_c(\omega) = \sum_{k=1}^N \frac{4\pi}{\lambda_0}\Big[\frac{\partial\phi_c}{\partial\phi_k}(\beta_k+\bar\alpha_k n_k)+\bar\alpha_k\Big]\bar T_k(\omega)d_k,$$

and the equivalent displacement as read out by the Gaussian beam: $$\delta z(\omega) = \frac{\lambda_0}{4\pi} \delta\phi_c(\omega).$$ Thus,

$$H(\omega) = \sum_{k=1}^N \Big[\frac{\partial\phi_c}{\partial\phi_k}(\beta_k+\bar\alpha_k n_k)+\bar\alpha_k\Big] \frac{2}{\xi_k}\sinh\left(\frac{\xi_k d_k}{2}\right)\frac{A_k+B_k}{j(\omega)},$$

with all parameters defined as above.

3. Frequency Response and Sign-Switching Phenomena

The frequency dependence of $H(\omega)$ is driven by the relative scale of the thermal diffusion length $d_\mathrm{diff} = \sqrt{\kappa/(\rho C \omega)}$ and coating thickness $d_\mathrm{coat}$. At low frequencies $d_\mathrm{diff} \gg d_\mathrm{coat}$, the system approaches the single-layer (substrate) limit,

$$H_\mathrm{sub}(\omega) = \frac{\bar\alpha}{i\omega\rho C},$$

characterized by $1/\omega$ amplitude roll-off and $-90^\circ$ phase lag. At intermediate frequencies ($d_\mathrm{diff}\sim d_\mathrm{coat}$), the top layers dominate, leading to a bump in $|H(\omega)|$ and a phase rotation across $+90^\circ$, marking a sign change in $H(\omega)$. At very high frequencies ($d_\mathrm{diff}\ll d_\mathrm{coat}$), only nanometric-scale layers contribute, with $H(\omega)$ rolling off as $1/\sqrt{\omega}$ and phase tending to $+90^\circ$. For Advanced LIGO Ta$_2$O$_5$:SiO$_2$ coatings ($d_\mathrm{coat}\sim 5\,\mu$m), the critical frequency is $f_\mathrm{coat} \sim 10^5$ Hz.

4. Transfer Function as an Absorption-Depth Indicator

The magnitude and zero-crossing behavior of $H(\omega)$ at high frequencies uniquely diagnose the localization of optical absorption. If absorption is concentrated at the front surface, the negative $\frac{\partial\phi_c}{\partial\phi_k}$ of the first quarter-wave layer yields a distinctive positive amplitude bump and a sharp sign flip near $f_\mathrm{coat}$. If the absorption is distributed deeper (e.g., at an interface), the bump diminishes or vanishes, and the sign-crossing shifts upward or disappears. When absorption follows the optical power penetration profile (indicative of bulk absorption), the high-frequency features interpolate between these extremes. Experimentally, a modulated heating beam (e.g., acousto-optic modulator) and interferometric read-out enable measurement of $\delta z(\omega)$, and fitting the observed $H(\omega)$ yields the effective absorption depth with micron-scale precision.

5. Implementation and Experimental Requirements

Precise realization of the transfer function-based absorption-depth indicator involves:

• A modulation source for heating at frequencies up to $2f_\mathrm{coat}$ ($\sim 200$ kHz for LIGO coatings, $1$ MHz for AlGaAs).
• An interferometric read-out of $\delta z$ with sub-fm/$\sqrt{\text{Hz}}$ sensitivity from DC to MHz.
• Calibrated incident heating intensity $j(\omega)$ via controlled modulator drive.
• A comprehensive multilayer thermal model for $H(\omega)$ using known dielectric layer design and material constants.
• Parameter fitting of measured $|H|$ and arg$(H)$ to model families (surface, interface, and bulk absorption) to extract absorption depth.

6. Applications in Precision Measurement, Noise Prediction, and Optomechanics

These transfer function indicators underpin several advanced applications:

• In optomechanical stabilization ("optical spring"), the $1/\omega$ substrate effect adds a $-90^\circ$ phase lag, causing destabilization above resonance. If the resonance frequency is high enough to cross $+90^\circ$ phase (sign flip), photothermal feedback shifts to positive damping, permitting passive cavity self-locking.
• Thermo-optic noise predictions in coatings require the full $H(\omega)$ formalism above $\sim$10 kHz; using coating-averaged models for $\omega \ll \omega_\mathrm{coat}$ is insufficient, and high-frequency corrections can reach $10$–$100\%$ above 100 kHz.
• Al$_x$Ga$_{1-x}$As crystalline coatings (with high $\kappa$ and tailored layer counts) leverage near-total cancellation between thermo-elastic and thermo-refractive noise, but high-frequency deviations dominate residual noise, as predicted by the detailed $H(\omega)$ heat flow model.

7. Summary and Significance

The photo-thermal transfer function $H(\omega)$ acts as both a predictive model for photothermal noise and a quantitative probe of optical absorption location in precision dielectric coatings. Its high-frequency amplitude and sign dynamics provide a robust, micron-scale indicator for distinguishing absorption sources (coating-internal vs. surface contamination). This utility extends to critical tasks in gravitational-wave detector mirror characterization, optomechanical system stabilization, and the design of next-generation, low-noise crystalline coatings (Ballmer, 2014).