## 3 Experimental Investigation  
Figure 1 illustrates the schematic of the Mach-Zehnder off-axis digital holographic setup used to measure temperature variations of a candle flame as a test transparent medium. The experiment was carried out at the temperature of 15 \({}^{\circ}\mathrm{C}\) and 1 atm pressure. In this experiment, we used Thorlab's CPS532-C2 collimated laser-diode-pumped DPSS laser module with a wavelength of 531.8 nm and an output power of 0.9 mW as the light source. The laser beam is split into two parts using a cubic beam splitter (BS). In this setup, a portion of the laser beam is designated as the reference beam, while another portion of the beam functions as the object beam. The object beam is diverged by a negative lens and passes through the test object (candle flame) and another cubic beam splitter before reaching the digital camera (FLIR-BFS-U3-120S4M-CS Monochrome Camera). The negative lens is used to illuminate a larger area of the flame. The reference beam is directed towards the camera at a small angle after passing through an ND filter, reflecting off a mirror, and passing through the cubic beam splitter. The mirror can be adjusted to provide a suitable small angle between the reference and object beams. The object and reference beams interfere at the camera sensor, and a complex hologram image is captured. The camera features a Sony IMX226 imaging sensor composed of 4000 \(\times\) 3000 pixels, with each pixel measuring 1.85 \(\upmu\)m \(\times\) 1.85 \(\upmu\)m. The camera's sensing area is 7.4 mm \(\times\) 5.55 mm with 31 fps. First, a complex holographic image was captured without a candle flame (representing the air) and kept as a reference image for lateral calibration issues. Then, multiple images were captured corresponding to different states of the air's refractive index due to the candle flame and variations in the flame's temperature. Image reconstruction and relevant post-processing algorithms were implemented using MATLAB R2021b software. The first step of the reconstruction procedure involves removing the zero-order term from the recorded hologram. We apply a method described in [37]. A spatial frequency mask is applied in the Fourier domain to select only the carrier frequencies that contain the required hologram information. The mask's size and shape are chosen to cover only one of the carrier frequencies of the twin lobes in the spectral domain (Figure 2), effectively filtering out all frequencies except the desired ones. The chosen frequencies are then cantered and zero-padded in the Fourier domain. The complex image \(\tilde{U}_{o}(x,y)\) is retrieved using the inverse fast Fourier transform. The phase distribution of the reconstructed images can be calculated using Equation (5). However, in addition to the dynamically varying phase changes caused by temperature variations, some pseudo phase can be introduced in the derived phase information. This pseudo phase change may originate from wave front and optical element aberrations, improper selection of the Fourier mask for carrier frequencies, or the residual effect of the reference wave tilt angle [35; 43]. However, in this setup, the phase changes between two states of the object will carry the same pseudo phase due to the shared source of aberrations. Therefore, using Equation (6) to calculate the phase changes cancels out the existing aberrations, and the final measured phase changes remain unaffected. Utilizing the reference image is beneficial, as it eliminates the need for precise optomechanical adjustments and accurate definition of the frequency carrier mask. However, there are always time-variant  
Figure 1: Schematic of the experimental setup Mach–Zehnder interferometer off-axis digital holography (**right**) and the test object (**left**). BS and ND represent beam splitter and neutral density filter, respectively. The two rectangles on the candle flame indicate the area where temperature measurements were taken within the \((X,Y)\) plane, perpendicular to the optical path axis \(Z\).