## Abstract

A simple and sensitive photothermal technique—photothermal detuning, in which the spectral shift of an optical coating caused by absorption-induced temperature rise is used to measure the photothermal signal—and its application for the absorption measurement of coated optical components are developed theoretically and experimentally in detail for the first time to the best of our knowledge. The theoretical description of the photothermal detuning signal with a continuous-wave modulated laser beam excitation is presented.
Experiments are conducted with a highly reflective coating used at
$532\text{\hspace{0.17em} nm}$ to measure the photothermal detuning signal and to evaluate the absorption at
$532\text{\hspace{0.17em} nm}$ by detecting the spectral shift with a probe beam at a wavelength of
$632.8\text{\hspace{0.17em} nm}$. By optimizing the incident angle of the probe beam, the amplitude of the photothermal detuning signal is maximized. Good agreement is obtained between the experimental results and the theoretical predictions.

© 2008 Optical Society of America

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### Equations (11)

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(1)
$${{{\displaystyle n}}_{f}}^{T}={{{\displaystyle n}}_{f}}^{0}+\left\{{\left(\frac{\mathrm{d}n}{\mathrm{d}T}\right)}_{f}\Delta T+\left[1-{{{\displaystyle n}}_{f}}^{0}-{\left(\frac{\mathrm{d}n}{\mathrm{d}T}\right)}_{f}\Delta T\right]\times \left(\frac{{A}_{f}\Delta T}{1+\left(3{\alpha}_{f}+{A}_{f}\right)\Delta T}\right)\right\}\text{,}$$
(2)
$${{d}_{f}}^{T}={{{\displaystyle d}}_{f}}^{0}\{1+({\alpha}_{f}-{B}_{f})\mathrm{\Delta}T\}\text{,}$$
(3)
$${A}_{f}=\frac{2\left(1-2{v}_{f}\right)}{\left(1-{v}_{f}\right)}\left({\alpha}_{s}-{\alpha}_{f}\right)\text{,}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}{B}_{f}=\frac{2{v}_{f}}{\left(1-{v}_{f}\right)}\left({\alpha}_{s}-{\alpha}_{f}\right)\text{,}$$
(4)
$$\left[{M}_{j}\right]=\left[\begin{array}{cc}\text{cos \hspace{0.17em}}{\delta}_{j}& \frac{i}{{\eta}_{j}}\text{\hspace{0.17em} sin \hspace{0.17em}}{\delta}_{j}\\ i{\eta}_{j}\text{\hspace{0.17em} sin \hspace{0.17em}}{\delta}_{j}& \text{cos \hspace{0.17em}}{\delta}_{j}\end{array}\right]\text{,}$$
(5)
$${\delta}_{j}=\frac{2\pi}{\lambda}\text{\hspace{0.17em}}{{{\displaystyle n}}_{f}}^{T}{{{\displaystyle d}}_{f}}^{T}\text{\hspace{0.17em} cos \hspace{0.17em}}{\phi}_{f}\text{.}$$
(6)
$$\text{}M=\left[\begin{array}{c}B\\ C\end{array}\right]=\left[\begin{array}{cc}\text{cos \hspace{0.17em}}{\delta}_{1}& \frac{i}{{\eta}_{1}}\text{\hspace{0.17em} sin \hspace{0.17em}}{\delta}_{\text{1}}\\ i{\eta}_{\text{1}}\text{\hspace{0.17em} sin \hspace{0.17em}}{\delta}_{\text{1}}& \text{cos \hspace{0.17em}}{\delta}_{1}\end{array}\right]\times \left[\begin{array}{cc}\text{cos \hspace{0.17em}}{\delta}_{2}& \frac{i}{{\eta}_{2}}\text{\hspace{0.17em} sin \hspace{0.17em}}{\delta}_{\text{2}}\\ i{\eta}_{\text{2}}\text{\hspace{0.17em} sin \hspace{0.17em}}{\delta}_{\text{2}}& \text{cos \hspace{0.17em}}{\delta}_{2}\end{array}\right]\cdots \times \left[\begin{array}{cc}\text{cos \hspace{0.17em}}{\delta}_{j}& \frac{i}{{\eta}_{j}}\text{\hspace{0.17em} sin \hspace{0.17em}}{\delta}_{j}\\ i{\eta}_{j}\text{\hspace{0.17em} sin \hspace{0.17em}}{\delta}_{j}& \text{cos \hspace{0.17em}}{\delta}_{j}\end{array}\right]\left[\begin{array}{c}1\\ {N}_{b}\end{array}\right]\text{.}$$
(7)
$$R\left(\lambda ,T\right)=\left(\frac{{N}_{0}B-C}{{N}_{0}B+C}\right)\left(\frac{{N}_{0}B-C}{{N}_{0}B+C}\right){}^{*}\text{,}$$
(8)
$$\frac{\mathrm{d}R}{\mathrm{d}T}=|{\frac{\partial R\left(\lambda ,T\right)}{\partial T}|}_{T={T}_{0}}\text{.}$$
(9)
$$S\left(r\right)=\frac{1}{{R}_{0}}\text{\hspace{0.17em}}\frac{\mathrm{d}R}{\mathrm{d}T}\text{\hspace{0.17em}}\Delta T\left(r\right)\text{,}$$
(10)
$$\Delta T\left(r\right)=\frac{{A}_{0}P}{2\pi {K}_{th}}\text{\hspace{0.17em}}{\displaystyle {\int}_{0}^{\infty}\frac{{J}_{0}\left(\delta r\right)}{\beta}}\text{\hspace{0.17em} exp}\left(-\frac{{a}^{2}{\delta}^{2}}{4}\right)\delta \mathrm{d}\delta $$
(11)
$${\beta}^{2}={\delta}^{2}+i\left(\omega /D\right)\text{,}$$