## 2 Theory
So, \(\Delta\varphi(x,y)\) can be written as [20,38]  
\[\Delta\varphi(x,y)=\frac{2\pi}{\lambda}\Delta OPL. \tag{7}\]  
However, \(\Delta OPL\) originates from the change of refractive index and can be calculated as [20]  
\[\Delta OPL=\int_{L_{1}}^{L_{2}}\Delta n(x,y,z)dz,\]  
where \(\Delta n(x,y,z)\) denotes the change of the refractive index in the measured volume. For the temperature change \(\Delta T=T_{t_{2}}-T_{t_{1}}\), the relation between \(\Delta\varphi\) and the refractive index variation \(\Delta n=n_{2}-n_{1}\) can be derived by altering the \(\Delta\)OPL with the index variation of the transparent media in Equation (7). So, Equation (7) can be expressed as [20]  
\[\Delta\varphi(x,y)=\frac{2\pi}{\lambda}\int_{L_{1}}^{L_{2}}\Delta n(x,y,z)dz. \tag{8}\]  
In our analysis, we work with a three-dimensional representation using X, Y, and Z axes. The optical axis aligns with the Z-axis, representing the direction of the optical path. To simplify the study and for the case of a 2D temperature field, we focus on the 2D plane (X, Y) perpendicular to the optical axis. This choice allows us to assume that the temperature variation occurs only within this plane. So, a constant temperature is assumed in the direction of the optical axis, while the refractive index changes in the X and Y directions but remains constant along the optical axis. Consequently, Equation (8) simplifies then to  
\[\Delta\varphi(x,y)=\frac{2\pi}{\lambda}L\Delta n, \tag{9}\]  
where \(L=L_{2}-L_{1}\) denotes the differential distance that the beam passes through the medium (the total length of the media). Equation (9) can be written as  
\[\frac{\Delta\varphi(x,y)}{\Delta T}=\frac{2\pi}{\lambda}L\frac{\Delta n}{ \Delta T} \tag{10}\]So, the temperature variations can be related to \(\frac{dn}{dT}\) as  
\[\Delta T=\frac{\Delta\varphi(x,y)\lambda}{2\pi L\frac{dn}{dT}} \tag{11}\]  
where \(\frac{dn}{dT}\) is thermal coefficients of the air refractive index as a ratio of changes of the refractive index to the changes of temperature [39; 40; 41]. So, the temperature can be found as:  
\[T_{t_{2}}=T_{t_{1}}+\frac{\Delta\varphi(x,y)\lambda}{2\pi L\frac{dn}{dT}} \tag{12}\]  
The refractive index of air for the wavelength range 230 nm \(\leq\lambda\leq\) 1630 nm and the temperature of T with the pressure of P can be calculated with good accuracy [42] with  
\[n_{air}(\lambda,T,P)=1+\frac{P}{P_{0}}\times\frac{n_{air}(\lambda,T=15\ ^{\circ} \mathrm{C},P_{0})-1}{1+3.478\times 10^{-3}\frac{1}{\mathrm{C}}\times(T-15\ ^{\circ}\mathrm{C})} \tag{13}\]  
where \(n_{air}(\lambda,T=15\ ^{\circ}\mathrm{C},P_{0})\) is the reference refractive index of the air at the temperature of \(T=15\ ^{\circ}\mathrm{C}\) with a pressure of 1 atmosphere, \(P_{0}=101325\ \mathrm{N/m^{2}}\), which for the wavelength \(\lambda\) can be calculated as [42]  
\[n_{air}(\lambda,15\ ^{\circ}\mathrm{C},P_{0})=1+10^{-8}\times\left[6432.8+\frac{2 949810\frac{1}{\mu m^{2}}\lambda^{2}}{146\frac{1}{\mu m^{2}}\lambda^{2}-1}+ \frac{25540\frac{1}{\mu m^{2}}\lambda^{2}}{41\frac{1}{\mu m^{2}}\lambda^{2}-1}\right] \tag{14}\]  
Using Equations (13) and (14), the thermal coefficients of the air refractive index can be expressed as  
\[\frac{dn_{air}(\lambda,T,P)}{dT}=-0.00367\frac{n_{air}(\lambda,T,P)-1}{1+0.003 67\frac{1}{\mathrm{C}}T} \tag{15}\]  
So, by knowing the wavelength of the laser source, which was \(\lambda=531.8\) nm in our experiment, \(n_{air}(531.8\ \mathrm{nm},15\ ^{\circ}\mathrm{C},P_{0})\) becomes 1.00027259905. Further, the thermal coefficient of the refractive index for the \(\lambda=531.8\) nm at \(T=15^{\circ}\mathrm{C}\) is then determined as  
\[\Delta OPL=\int_{L_{1}}^{L_{2}}\Delta n(x,y,z)dz,\]0