## 2 Theory  
Consider the object beam as \(U_{o}(x,y)=A_{o}(x,y)exp(i\varphi_{o}(x,y))\) and the reference beam in the detector plane as \(U_{r}(x,y)=A_{r}(x,y)exp(i\varphi_{r}(x,y))\), respectively, where \(A\) representsthe beam's amplitude and \(\phi\) denotes the phase of the respective components. Consequently, the resulting image intensity recorded, denoted as \(I(x,y)\), can be expressed as  
\[I(x,y)=A_{o}^{2}(x,y)+A_{r}^{2}(x,y)+A_{r}A_{o}((x,y)exp(i\Delta\varphi)+A_{r}A_{ o}(x,y)exp(-i\Delta\varphi) \tag{1}\]  
In Equation (1), we assume \(A_{r}(x,y)=A_{r}\) to be a constant value, and \(\Delta\varphi=\varphi_{o}(x,y)-\varphi_{r}(x,y)\) represents the phase difference between the object beam \(U_{o}(x,y)\) and the reference beam \(U_{r}(x,y)\). Equation (1)'s first two intensity components relate to the zero order and gradual variations across the detector. The remaining two components correspond to the interference terms, which are highly responsive to changes in \(\Delta\varphi\). Modifying the size of the aperture allows for the adjustment of the spatial frequency content of the interference terms. Further, adjusting the incidence angle allows the interference component to be positioned at various positions in the Fourier domain. In our analysis, we focus on the third term in Equation (1), denoted as \(J(x,y)=U_{o}(x,y)U_{r}^{*}(x,y)=A_{r}A_{o}exp(i\Delta\varphi)\). We assume that the reference beam \(U_{r}(x,y)\) primarily captures the variations in phase \(\varphi\) caused by the tilt angle and curvature of the field across the detector. For digitally recorded holograms, we utilize \(\tilde{U}_{nd}(x,y)\) as a simplified approximation of the actual experimental reference beam \(U_{r}(x,y)\) during the reconstruction process [35]. \(\tilde{U}_{rd}(x,y)\) can be presented as  
\[\tilde{U}_{rd}(x,y)=\big{|}\tilde{U}_{rd}(x,y)\big{|}exp\bigg{[}i\frac{2\pi}{ \lambda}\Big{(}ax+by+c(x^{2}+y^{2})\Big{)}\bigg{]}. \tag{2}\]  
In this context, \(\lambda\) represents the wavelength of the laser beam used. It is important to note that the symbol \(\  
The phase information may be wrapped within the interval of \([-\pi,+\pi]\) radians due to the principles of the arctan function. However, it is important to note that the actual phase may vary over the range of \(2\pi\). When a laser beam propagates through transparent media (e.g., a flame) in a parallel manner, the phase of the object wave front changes, due to the change of temperature and consequently change of refractive index inside the media [20]. The phase differences can be reconstructed numerically from the retrieved digital hologram \(\widetilde{U}_{o}(x,y)\), as is stated in Equation (5). In the digital holography set up, at least two digital holograms should be captured in order to measure the temperature variations. The first hologram corresponds to the reference or initial state of the temperature, and the second hologram is recorded when the temperature field has been changed. If the two holograms are recorded separately and their phase information is calculated, the phase difference \(\Delta\varphi(x,y)\) by a modulo of \(2\pi\) subtraction provides information on the refractive index change occurring between the two states, which relates to the temperature field variations. \[\Delta\varphi(x,y)=\left\{ \begin{array}{c}\varphi_{1}-\varphi_{2}\text{ if }\varphi_{1}\geq\varphi_{2}\\ \varphi_{1}-\varphi_{2}+2\pi\text{ if }\varphi_{1}<\varphi_{2}\end{array} ,\right. \tag{6}\]  
where \(\varphi_{1}\) and \(\varphi_{2}\) are the phase distribution of the holograms at the reference state and measurement state, respectively. Let us assume the first hologram (reference) is recorded at the time \(t_{1}\), with refractive index \(n_{1}\), and the second hologram at the time \(t_{2}\) with \(n_{2}\). The phase change \(\Delta\varphi\) between two states of the hologram originates from the change in optical path length (OPL), \(\Delta\)OPL.