# Characterization of Chirality in Diffractive Metasurfaces by Photothermal Deflection Technique
## 3 Results
### Heat Sources Localization from PTD Data
These last two quantities are calculated for the three wavelengths and are shown in Table 1. Note that the calculated internal angles are in good agreement with the theoretical prediction for the 2D photonic crystals given by \(\theta(\lambda)=\arcsin\left(\frac{2}{\sqrt{3}}\frac{\lambda/n}{a}\right)\) (where \(a=522\) nm is the period of the hexagonal 2D close packed structure and \(n=1.5\) is the glass refractive index). A different situation is found in case where the pump beam is incidental to the sample at angle \(\theta_{air}=20\) deg with respect to the norm of the metasurface, in the x-direction (transverse to the probe beam path). In this case, the incident beam is represented by wavevector \(k_{x,air}=-\frac{2\pi}{\lambda}n_{air}\sin\theta_{air}\) (with \(n_{air}\approx 1\)) along the k-vector surface plane, as shown in Figure 4a. In 2D photonic crystals, the k-vectors of the six diffracted beams can be calculated by adding wavenumber \(\frac{4\pi}{a\sqrt{3}}\), due to the metasurface hexagonal periodical pattern (where \(a=522\) nm is the period of the hexagonal, 2D, close-packed structure). \[\left\{ \begin{array}{l}k_{x,m}=-\frac{2\pi}{\lambda}n_{air}\sin\theta_{air}+ \frac{4\pi}{a\sqrt{3}}\cos\left(m\frac{\pi}{3}\right)\\ k_{y,m}=+\frac{4\pi}{a\sqrt{3}}\sin\left(m\frac{\pi}{3}\right)\end{array} \right. \tag{3}\]  
where \(m=1,2,\ldots,6\) is the integer corresponding to the \(m\)-th diffracted beam. The in-plane direction of each wavevector is identified by angle \(\beta_{m}\), and may be calculated with respect to the \(k_{x}\) direction, according to the relationship (see Figure 5a). \[\beta_{m}=\arctan\left(\frac{k_{y,m}}{k_{x,m}}\right)=\arctan\left(\frac{\sin \left(m\frac{\pi}{3}\right)}{\cos\left(m\frac{\pi}{3}\right)-\frac{\sqrt{3}}{ 3}\frac{\pi}{a}n_{air}\sin\theta_{air}}\right) \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Wavelength, nm** & \(R\)**, mm** & \(\theta\)**, deg** & \multicolumn{3}{c}{**Location of the Zeroes of the Lateral Deflection, mm**} \\  & & & \(d_{1}\) & \(d_{2}\) & \(d_{3}\) & \(d_{4}\) & \(d_{5}\) & \(d_{6}\) \\ \hline
514 & 2.23 & 48 & +0.60 & \(-1.50\) & \(-2.20\) & \(-0.65\) & +1.75 & +2.15 \\
496 & 2.07 & 46 & +0.55 & \(-1.35\) & \(-2.00\) & \(-0.60\) & +1.60 & +2.00 \\
458 & 1.76 & 42 & +0.50 & \(-1.20\) & \(-1.70\) & \(-0.50\) & +1.30 & +1.70 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Normal incidence: geometrical parameters of the heat sources produced by the absorption of the diffracted beams, as calculated from PDT data. Figure 4: Scheme of the photothermal deflection measurements of the diffracted orders. (**a**) Top view. Example of zeroes of the lateral deflection signal when the probe beam is passing above the center of the heat source produced by the diffracted orders; (**b**) typical antisymmetric behavior of the lateral beam deflection; (**c**) lateral view. Propagation of one diffracted order and the formation of the surface heat source. From Equation (4), the angles are calculated and reported in Table 2. It is worth noting that the incident angle of \(\theta_{air}=20\) deg already causes a relevant displacement from the symmetric case \(\theta_{air}=0\) deg, where \(\beta_{m}=m\cdot\pi/3\).