## 1 Introduction  
Temperature measurement plays a crucial role in various fields, including heat transfer, non-isothermal flow, thermal boundary layers, and combustion processes. Accurate temperature measurement plays a vital role in various aspects, including optimizing system design, comprehending physical phenomena, and ensuring the safety of engineering systems. These systems encompass a wide range of applications, such as heat exchangers, aerospace engines, scramjet combustors, and solid propellant plumes. Additionally, temperature measurement is crucial in areas such as turbine operations, the cooling of electronic devices, hot water pipelines, and the food industry. Furthermore, it extends its significance to the advancement of emerging technologies, including sustainable energy, biotechnology, information technology, and nanotechnology [1, 2, 3, 4, 5, 6]. Traditional contact methods have limitations in providing full-field temperature data and can influence the measured phenomena, as it is necessary for them to be situated within the experimental domain. Thermocouples, thermometers, resistance temperature detectors, thermistors, and a capacity or change in electrical resistance are typically the predominant types of sensors employed for this purpose. As a result, non-contact methods have gained significant attention because of their non-invasive characteristics and capacity to capture temperature distributions over large areas [7, 8]. Non-contact methods for temperature measurement offer several advantages over contact methods, making them desirable in many applications. Firstly, non-contact methods eliminate the need for physical contact with the object or medium being measured, which is particularly advantageous when dealing with delicate or sensitive materials. Itprevents any interference or disruption of the measurement caused by the presence of a sensor or probe. Secondly, non-contact methods provide full-field measurements, allowing temperature mapping over a larger area or volume. This capability is especially beneficial in applications where temperature distribution and gradients are of interest, such as in fluid dynamics, combustion processes, or heat transfer studies. Non-contact methods enable the visualization and analysis of temperature variations across the entire field of view, providing valuable insights into complex thermal phenomena. Furthermore, non-contact techniques offer faster measurement capabilities, enabling real-time or high-speed monitoring of the temperature changes. This is particularly crucial in dynamic processes where the temperature fluctuations occur rapidly or when precise temporal resolution is required. Additionally, certain non-contact techniques might have constraints in terms of distance, field of view, or environmental factors. For example, thermal imaging methods might be affected by factors such as ambient temperature, air flow, or optical obstructions, leading to potential measurement errors or limitations. Other non-contact techniques for the measurement of the temperature are the spectroscopic method [1], Raman and Rayleigh scattering [9; 10], Schlieren photography [11], laser speckle shearing interferometry and photography [12; 13], moire deflectometry [14], and interferometric methods, such as holography [15; 16], shearing [12; 13; 14; 17], Talbot [18], and Lau phase [19]. In transparent media, temperature changes induce corresponding refractive index variations, which subsequently affect the phase distribution and variation of a laser beam's wave front passing through the medium [20]. Digital holography provides an advantageous solution for measuring these changes accurately and non-invasively with its differential measurement capabilities. Digital holography allows for the direct collection of the complex amplitude distribution of the wave front in multiple states by dynamically recording a sequence of holograms of the object field at distinct time points [20].