# Characterization of Chirality in Diffractive Metasurfaces by Photothermal Deflection Technique
## 3 Results
### Determination of the Thermal Parameters of the Metasurface  
Among the various techniques, photothermal deflection (mirage effect) [27, 28, 39] has been established as a noncontact method for optical and thermal property characterization. In particular, from the measurements of lateral deflection \(\Phi_{x}\) vs. the pump-probe distance, it is possible to directly determine the thermal diffusivity by using, for example, the phase method [39, 47, 48, 49]. According to such a method, the phase lag of the lateral deflection is \(\varphi(x)=-x/\mu+\varphi_{o}=-x\sqrt{\pi f/D}+\varphi_{o}\), where sample thermal diffusivity \(D\) is calculated simply from the linear slope of the x scan (\(\mu=\sqrt{D/\pi f}\) is the thermal diffusion length). Unfortunately, this method can be applied to material thermal diffusivities larger than the air thermal diffusivity of \(D_{air}\) = 20 mm\({}^{2}\)/s, with severe limitations for materials with a low thermal diffusivity, such as glasses (with \(D\) = 0.5 mm\({}^{2}\)/s). To extend the method to these materials, a strong reduction in the probe beam height above the sample is recommended (z = 10-20 \(\upmu\)m), and/or the bouncing configuration should be used, sending the probe beam to the sample surface at a glazing angle (typically 1 degree) [50, 51]. For these cases, the optothermal parameters can be worked out by fitting the lateral deflection (both amplitude and phase) with the complete theoretical model given by the following expression [50, 51, 52, 53, 54]:  
\[\Phi_{x}=-\frac{1}{n_{air}}\bigg{(}\frac{dn_{air}}{dT}\bigg{)}\frac{\alpha P} {\pi}\int\limits_{o}^{+\infty}\frac{\delta\exp\left[-\frac{(\delta\varphi)^{2 }}{8}\right]\exp(-\beta_{air}z)\sin(\delta x)}{(\beta_{s}+\alpha)(k_{air} \beta_{air}+k_{s}\beta_{s})}d\delta \tag{9}\]  
where \(P\) is the optical power, \(\alpha\) is the absorption coefficient, w is the pump beam spot size, \(\beta_{air}=\sqrt{\imath\omega/D_{air}}\), \(\beta_{s}=\sqrt{\imath\omega/D}\), and \(k_{air}\) and \(k_{s}\) are the air and sample thermal conductivities, respectively.  
By analyzing both the PTD amplitude and the phase signals in the proximity of each heat source (<0.3 mm), the effective material thermal diffusion length, the material thermal diffusivity, and the effective spot size, can be calculated using Equation (9). These values  
Figure 7: Distribution of the absorbed power in the unit cell when the metasurface is excited with LCP or RCP wavelength at 514 nm; absorption for orders 5 and 6 shown in the top and bottom panels, respectively, while the color scale is kept constant.  
are reported in Table 4. The effective thermal diffusivity has been estimated to be about \(D=0.6\) mm\({}^{2}\)/s for all diffracted orders and light polarizations. This value is increased with respect to the typical glass diffusivity value of \(D=0.5\) mm\({}^{2}\)/s because of the silver and polystyrene surface layer. This demonstrates that, at a low modulation frequency of 36 Hz, the effective thermal diffusivity is only weakly affected by the submicron metasurface which has a potentially higher thermally diffusive process, which could be specifically detected in the MHz range, but by using other experimental techniques [55, 56, 57]. Concerning the values of the effective spot sizes, one may notice that the values found for diffracted beams are larger with respect to the 0.09 mm found for the incident pump beam, in agreement with the elliptical enlargement shown in Figure 4 (the spot size is defined as the distance at which the Gaussian light intensity drops to \(1/\mathrm{e}^{2}\) of its maximum value). As an example, a good agreement between lateral deflection experimental data (amplitude and phase) and the theoretical expectations from Equation (9) is demonstrated in Figure 8. The best fit between the PDT data (symbols) and theory (continuous line) is shown for both amplitude and phase, and for \(m=0\) and \(m=2\). The probe beam height has been kept at z = 30 \(\upmu\)m.