# All-Optical Method of Determining Laser Power from the Photomechanical Effect
## 4 Discussion  
The coefficient of thermal expansion was determined by transmitting a 650 nm laser beam through a dye-doped PDMS film with an embossed diffraction grating on the back. The strain depends linearly on the temperature of the sample which allows for the linear coefficient of thermal expansion to be determined within a \(<\)5% uncertainty. Although this measurement was used to determine the \(\beta\) parameter for the proposed thermal model of a transparent sample doped with oil red O, the results indicate that any transparent elastomer cured on a form with a surface relief grating, or a thermoplastic embossed with a grating, could have its linear coefficient of thermal expansion measured with the reported method. Furthermore, the method could easily be expanded to opaque materials by placing the grating on the same side as the incident probe beam to create a reflective surface grating. Furthermore, placing the probe beam on the opposite side of the pump beam would allow for a more compact experiment or enclosed device design. In such a geometry, an inorganic dye could be doped in PDMS, or another material with a large photomechanical strain, to make a CW laser power sensor that is both (i) based on the photomechanical effect and (ii) resistant to photobleaching. For transmission-based power sensors with the probe and pump beam on the same side, self-healing chromophore/polymer pairs could be implemented to increase longevity [104; 105; 106; 107; 108]. The versatility of this method for determining a linear coefficient of thermal expansion and the ability for micron sized samples that have grating periods in the hundreds of nanometers makes it useful for determining coefficients of low volume materials that are either high in cost to procure or low in yield to produce. A well-characterized sample could also be used as a thermometer, similar to the PDMS-clad fiber Bragg grating temperature sensor developed by [PERSON] et al. [109]. This proposed model for a pump laser power sensor works satisfactorily because the strain was linearly correlated with the pump power referenced by a photodiode detector. A single, passive rate parameter \(a^{2}/4D\) was found to fit the data for all experimental pump powers. The model does suffer from deviations with experimental values at short time-scales immediately after the shutter was actuated. In fact, pump flash measurements of the shutter times in the each automated experiment confirm that the physical open and close times of the shutter relative to the beginning of each data run are at \((1.00\pm 0.05)\) s and \((11.00\pm 0.05)\) s. These experimental values are about 300-400 ms greater than the values for \(t_{\rm on}\) and \(t_{\rm off}\) fitted by our model. The reasons for the deviations in shutter times follow from the assumptions that (1) the heat source is a point source in a two-dimensional plane with the temperature measured at an adjacent point on the plane point near it (i.e., \(f(t)\) is not negligible at small time scales), (2) the absorption of the pump is uniform through the thin film while also absorbing most of the intensity, and (3) the two-dimensional plane is perfectly insulated from the three-dimensional environment. The second assumption  
Figure 5: The dimensionless amplitude of the model \(\alpha_{L}\beta P_{0}\) shows a linear dependence with the incident pump power referenced by a calibrated Ophir PD300 photodiode detector. is only true for low absorption films, where high absorption films will inherently suffer from slow diffusion constants when surface absorption dominates. The first assumption requires a well-focused beam to make a better approximation while the last assumption admits a Green's function for a two-dimensional heat equation that does not include loss to the environment. The two-dimensional heat equation can obviously be reformulated to contain a loss term that is linearly dependent on the local temperature, which would follow a Robin-type boundary condition for a disk in three dimensions.