# Characterization of Chirality in Diffractive Metasurfaces by Photothermal Deflection Technique
## 3 Results
### Heat Sources Localization from PTD Data  
In this section we present a simple geometrical model to allow the clear interpretation of the PTD experimental data, giving a quick quantitative estimate of the main parameters of the pump beam diffraction patterns. First, we analyzed the case of a normal incidence. The Ar pump beam produced six diffracted beams, which were reflected by the back mirror, and absorbed by the metasurface, eventually forming a regular hexagonal pattern of heat sources, as shown in Figure 3a. The He-Ne probe beam travels in air, skimming the metasurface along the y-axis at a minimum height (z) from the sample surface (typically 20-50 \(\upmu\)m), so as to probe the heat sources induced by the pump beam (Figure 4b). The theory of PDT in a transverse configuration states that the probe beam is deflected from its original trajectory due to the air's thermal gradient, produced by the heat source (mirage effect). The lateral deflection angle, \(\Phi_{x}\), is accumulated along the probe beam path, and is eventually detected by a photodiode position sensor. Quantity \(\Phi_{x}\) is given by the well-known expression [27; 39]:  
\[\Phi_{x}= \begin{array}{cc}\frac{1}{n}\frac{dn}{dT}\int\limits_{y}\frac{dT}{ dx}\ dy\end{array}  \tag{1}\]where, in air, the refractive index is \(n\approx 1\) and the optothermal coefficient in air at room temperature is \(dn/dT\approx-10^{-6}\) K\({}^{-1}\). It is worth noting that each heat source exhibits a regular elliptical shape (Figure 4a), reaching the maximum temperature in the center of the ellipse. By scanning the probe along the x-axis, the deflection angle will show an anti-symmetrical behavior with \(x\), becoming zero where the probe passes along the heat source center, corresponding to the maximum temperature increase (see Figure 4b). We were able to measure the absorption-induced PDT for all seven heat sources (incident + six diffracted orders), at the same time, by properly misaligning the probe beam and the symmetry lines of the hexagonal diffraction pattern (see Figure 4a) and then scanning the probe beam from \(-2.5\) mm to \(+2.5\) mm using a translation stage. The position of each heat source center was calculated from the condition \(\Phi_{x}\)= 0, and is reported in Table 1 for each diffracted order m, as well as for three different wavelengths. From simple geometrical considerations, taken from Figure 3a, the position of the heat sources are linked to each other by the following relationship:  
\[\left\{ \begin{array}{l}d_{1}=R\sin(\frac{\pi}{6}-\alpha)\\ d_{5}=R\sin(\frac{\pi}{6}+\alpha)\\ d_{6}=R\cos(\alpha)\end{array} \right. \tag{2}\]  
where the misalignment angle can be calculated as \(\alpha=\arctan(\frac{1}{\sqrt{3}}\frac{d_{5}-d_{1}}{d_{6}})\approx 15^{\circ}\) averaged for all wavelengths. By combining the equations, one may calculate the side \(R\) of the hexagon pattern as \(R=d_{6}/\cos\alpha\), while the internal angle of the diffracted beams is obtained by \(\theta=\arctan(R/2L)\), where \(L\) = 1 mm is the thickness of the glass substrate  
Figure 3: Amplitude of the photothermal deflection vs. pump–probe offset on Ag-SNSA sample excited with an Ar pump beam under different conditions: (**a**) normal incidence, wavelength \(\lambda\) = 458 nm; modulation frequency 36 Hz; (**b**) oblique incidence at 20 deg, wavelength \(\lambda\) = 458 nm; modulation frequency 16 Hz; (**c**) normal incidence, wavelength \(\lambda\) = 514 nm; modulation frequency 36 Hz; (**d**) oblique incidence at 20 deg, wavelength \(\lambda\) = 514 nm; modulation frequency 16 Hz. Red and blue lines indicate RCP and LCP responses, respectively. (see Figure 4c).