## 4 Result and Discussion  
A reference hologram of air without a candle flame was recorded and is shown in Figure 3. The interference pattern includes the information of both the reference beam and the object beam passing through the air. The direction of the interference fringes can indicate the tilt angle between the reference beam and the object beam. This direction is also related to the direction of the twin side lobes in the Fourier domain, as can be seen in the spectrum of the recorded hologram in Figure 2. The hologram was reconstructed using the numerical reconstruction method described in the previous section. The phase distribution map of the reference hologram was then extracted using Equation (5), revealing the wrapped phase representation shown in Figure 4 (left). This wrapped phase map demonstrates a periodicity of \(2\pi\) modulo (due to the nature of the arctan function). Another hologram was recorded from the candle flame from the area corresponding to the rectangle indicated at the bottom of Figure 1. Using the same reconstruction procedure, the hologram of the candle flame was reconstructed, and the phase distribution was calculated (Figure 4 (left)). The phase difference between the holograms with and without the flame was then computed using Equation (6) and is shown in Figure 5. It is important to note that the frequency carrier mask should remain unchanged for the entire hologram reconstruction procedure. Figure 3: The captured reference hologram displays the interference pattern, which is visible in the scaled region. As observed, the phase change map wraps quickly on the left side and gradually becomes slower from left to right. The hologram quality is higher in the interval from pixel number 500 to 3500 on the horizontal axis, where the results are shown and discussed. So, a cross-sectional profile plot of the phase difference map at pixel row number 1500 (red line) was chosen and plotted in Figure 5 (left). The line profile of the wrapped phase indeed reaches minima at positions that correspond to dark fringes. In Figure 6 (left), the unwrapped phase changes of the chosen line are plotted. The unwrapped phase initially decreases and then sharply increases until the end. The slope of the increment decreases as it moves towards the right. Using Equation (9), the refractive index differences are also plotted and shown in Figure 6. Similar to the phase changes, the refractive index first decreases, changes slope, and then sharply increases until it reaches the end, becoming smoother. The change in the refractive index is proportional to the phase difference with a proportionality coefficient factor of \(2\pi L/\lambda\). By using the relation introduced in Equation (12), the temperature variation is calculated and plotted in Figure 6, left. Contrary to the refractive index and phase changes, as we move away from the center towards the outer region of the flame, the temperature first increases from \(575\ ^{\circ}\)C to almost \(650\ ^{\circ}\)C and then gradually decreases until the end. Figure 4: Phase distribution for the reference hologram (**left**) and for the measurement hologram with the flame (**right**), corresponding to the rectangular area at the bottom of the flame (Figure 1, (right)). Figure 5: Phase change map (**left**) and a cross-sectional profile plot at pixel row number 1500 (**right**). The red line indicates the pixels chosen for the cross-sectional profile. Figures 4-6 depict the progressive implementation of the proposed method outlined in the theoretical section, as elucidated by Equations (5)-(15). These figures demonstrate the step-by-step flow of the methodology, visually illustrating the application of the theoretical principles discussed. By sequentially presenting the results in Figures 4-6, we provide a comprehensive and tangible representation of the proposed approach. Another hologram was recorded from the rectangular area indicated at the top of the flame, as shown in Figure 1 (right).