# All-Optical Method of Determining Laser Power from the Photomechanical Effect
## 3 Results  
The sample's grating period \(L\) increases with temperature due to thermal expansion. An increase in the grating period reduces the angle of the diffracted probe beam. The absolute angle was measured by the linear CCD array, where a Gaussian function was fitted to the intensity profile at different steady-state temperatures. The Gaussian function is given by  
\[G(y)=\frac{A}{\sigma\sqrt{2\tau}}e^{-(y-y_{0})^{2}/2\sigma^{2}}\,, \tag{1}\]  
where \(A\) is the area of the Gaussian, \(y_{0}\) is the location of the peak, and \(\sigma\) is the standard deviation. The position of the Gaussian peak was used to determine the angle \(\theta\) of the diffracted beam at each elevated temperature. Figure 2a shows the normalized intensity profile incident on the linear CCD array for two different elevated temperatures along with the Gaussian fits to those profiles. The grating period was determined by the equation  
\[L\sin\theta_{n}-L\sin\theta_{\rm pr}=n\lambda\,, \tag{2}\]  
where \(\theta_{\rm pr}\) is the incident angle of the probe beam, \(\theta_{n}\) is the angle of the diffracted beam, \(n=+1\) for the first-order diffraction beam, \(\lambda=650\,\)nm for the probe diode laser. When the incident angle of the probe beam in the experimental apparatus is negligibly small, \(\theta_{\rm pr}\approx 0\), Equation (2) simplifies to the form  
\[L\sin\theta_{n}=n\lambda\,. \tag{3}\]  
The grating period was assumed to changed solely from thermal expansion of the PDMS. To first-order approximation, the change in the grating period, \(\Delta L=L-L_{0}\) with \(L_{0}\) being the initial grating period at an initial temperature \(T_{0}\), depends linearly on the change  
Figure 1: (**a**) Illustrated steps for fabricating the dye-doped PDMS sample including (i) mixing and centrifuging under vacuum, (ii) pouring between guides and blade coating, and (iii) placing in an oven to cure. (**b**) The optical experiment with a probe beam for measuring the coefficient of thermal expansion as well as the pump laser power. (**c**) Illustration of an elastomer expanding via photothermal heating, where the stretched diffraction grating causes the diffracted probe beam’s angle to decrease. (**d**) An AFM image of the grating structure on the surface of the PDMS sample. (**e**) The beam profiles incident at the sample plane for the pump and probe lasers. in temperature \(\Delta T=T-T_{0}\). The linear thermal expansion equation relating the linear strain \(\Delta L/L_{0}\) to the change in temperature is given by  
\[\frac{\Delta L}{L_{0}}=\alpha_{L}\,\Delta T\,, \tag{4}\]  
where \(\alpha_{L}\) is the linear coefficient of thermal expansion. The linear strain as a function of temperature change is shown in Figure 2b. From Equation (4), the slope of the graph gives the linear coefficient of thermal expansion. The value of \(340\,\mathrm{ppm}/^{\circ}\mathrm{C}\) quoted by the technical data sheet [103] is within the experimental value of \((343\pm 16)\,\mathrm{ppm}/^{\circ}\mathrm{C}\) determined by linear regression with the line being constrained through the origin. The grating period shifted \(<\)2 nm, but the method has a high degree of sensitivity owing to the diffracting beam acting as an optical lever. A polymer with a typical coefficient of thermal expansion of an order of magnitude less than PDMS can also be measured with the same precision by simply increasing the arm length of the optical lever (propagation length of the diffracted probe beam) by an order of magnitude. A diode pumped, frequency doubled, Nd-doped garnet laser with CW operation was focused on the 650 nm probe beam after letting the enclosed sample reach the ambient temperature in the laboratory. After filtering the infrared light from the green laser beam, the transmittance of the film was measured to be \(\sim\)0.8%.