# All-Optical Method of Determining Laser Power from the Photomechanical Effect
## 3 Results
The transmittance remained constant over time while illuminated by the pump, and optical hole burning effects were not observed at the maximum pump power over the experimental illumination period of \(\sim\)10 s. Data from the linear CCD array was collected at 100 ms intervals. The data stream from the linear CCD array was immediately recorded after the external shutter in front of the red probe laser was opened. The green pump laser's external shutter was opened \(\sim\)1 s after data recording began. Absorption of the pump laser resulted in photothermal heating of the dye-doped PDMS which resulted in thermal expansion. This thermal expansion is observed from the time-dependent decrease in the diffracted beam's angle as measured with the linear CCD array. Figure 3a shows a contour plot of the intensity as a function of time and position on the linear CCD array. Note that the maximum values of 1 shown in the color-scale of Figure 3 are from normalization and not from any saturation of the CCD array. The CCD array only detects the probe beam, where the pump beam is filtered below the detection threshold of the linear CCD array. The thermal expansion of PDMS is a reversible process where Figure 3b shows the grating period being restored after closing the shutter  
Figure 2: (**a**) The solid lines show the intensity profile of the \(n=+1\) diffracted beam incident on the surface of the linear CCD array. Gaussian fits to both experimental intensity profiles are given by the dashed lines. (**b**) A straight line was fitted to the linear strain as a function of temperature. The strain was determined from the change in the diffraction beam’s angle shown in the lower left inset. The upper right inset shows the peak position of the Gaussian fits as a function of temperature used to determine the angle of the diffraction beam. in front of the pump beam. The insets of Figure 3 show the peak vertical position of the diffracted beam incident on the array as determined from Gaussian fits at each time step. The linear strain as a function of time can be determined from the peak positions of the time-dependent intensity profile shown in the example illustrated in Figure 3. The contour plots show normalized profiles of the diffracted probe beam incident on the CCD array. The probe beam intensity was tuned to an intensity level that resulted in a peak intensity that was midway between the CCD threshold and saturation intensity limits. The diffraction angle as a function of time is first determined from the peak position and the experimental dimensions. The grating period is calculated using Equation (3) and the strain follows as \(\Delta L/L_{0}\). A straightforward model connecting the time dependence of the local strain to the thermalization of the absorbed pump power through the local PDMS temperature is afforded by the heat kernel. For a localized heat impulse \(Q\) being delivered at time \(t\) and at \(x=0\), we expect the temperature field  
\[T(x,t)=\beta^{\prime}Qt^{-\gamma}e^{-\frac{x^{2}}{4 Dt}} \tag{5}\]  
where \(\gamma\) is a positive exponent that depends on the effective (spatial) dimensionality of the heat flow field and \(\beta^{\prime}\) and \(D\) are constants depending on the assumed isotropic thermal conductivity and heat capacity. Linearity of the heat flow equations allows us to build up the effect of a continuously constant supplied heating over a two-dimensional variegated spatial distribution by adding the individual effects up in the Green's function methodology, to arrive at  
\[\bar{T}(t)=\beta^{\prime}\int\mathrm{d}^{2}x^{\prime}\mathrm{d}^{2}x\int_{0}^ {t}\mathrm{d}t^{\prime}\,P(t)\,(t-t^{\prime})^{-\gamma}e^{-\frac{(x-t^{\prime })^{2}}{4D(t-t^{\prime})}}Y(x,x^{\prime})\,. \tag{6}\]  
here \(\bar{T}(t)\) represents a probe spot-averaged temperature and \(Y(x,x^{\prime})\) is a function that represents the spatial overlap between the pump and probe.