## 1 Introduction  
Thermal power generation using gas turbines is expected to expand in the long term as a clean and economical power generation method. Gas turbines are a type of internal combustion engine. They are thermal engines that obtain power from the combustion of gases by expanding high-temperature gas obtained by burning petroleum, natural gas, or other fuels to rotate the turbine. At present, the improvement in the thermal efficiency of various high-temperature equipment in power plants is required to further improve performance. In order to achieve high efficiency, it is essential to increase the temperature of the combustion gas and the turbine inlet temperature. It is therefore necessary to design the turbine blades for high temperatures. A cooling passage can be formed inside the turbine blade, which is then cooled by passing a cooling medium, such as air or steam, through this cooling passage, thereby ensuring the sufficient heat resistance of the blade. This technology contributes the most to such requirements for high temperature resistance. A typical blade cooling structure is the pin fin cooling system [1,2]. This system performs convection cooling, impingement cooling, and film cooling inside the moving blade, and enables the cooling air to flow out of the blade. In addition, the mainstream method of lowering the surface temperature of the turbine blade is applied in thermal barrier coatings (TBCs), which not only lowers the surface temperature but also improves thermal fatigue characteristics. Therefore, ceramics are coated on the gas turbine surface using the electron-beam physical vapor deposition (EB-PVD) method proposed by [PERSON] et al. [3]. However, there are concerns about damage and deterioration due to an increase in the thermal load applied to the TBC, and the improvement of durability is an issue. Therefore, in recent years, in addition to TBC, transpiration cooling of the turbine has been considered as an effective method for improving the durability of the turbine blades by using a porous material to form them. [PERSON] [4] developed a porous ceramic coating (P-TBC) through which cooling air could pass. P-TBC has a thermal conductivity as low as 50% of a conventional TBC, and it has been shown that its adhesion strength is almost the same. During the operation of the power plant, the turbine blades are subjected to high centrifugal loads due to the rotation of the rotor, flexural loads due to the working fluid, and vibration loads. These occur under the severe condition of a high temperature, hence cracks may develop in the cooling passage. Because of this reason, power plants regularly conduct non-destructive inspections, such as ultrasonic inspection tests and radiation transmission tests based on Japanese industrial standards [5] to detect turbine defects. Additionally, the cooling structure is checked, and the remaining life of the turbine is determined. [PERSON] et al. [6] proposed a prediction technique based on metallographic image analysis and a life assessment technique using damage trend analysis. These were based on statistical parameters of information from regular inspections of gas turbines where surface damage and material deterioration were prominent. In order to extend the inspection interval and increase the operation rate by accurately diagnosing the remaining life of the turbine, it is necessary to improve the precision of the inspection method to detect cracks generated in turbine blades in power plants. To meet this requirement, one of the authors has proposed a new flaw detection method using digital holographic microscopy (DHM). This method can quickly obtain a nanometer-order height distribution over a wide area of 1 mm\({}^{2}\) on the material surface and in an atmospheric environment [7; 8; 9]. Digital holographic microscopes can acquire 3D images of a sample surface in real time with a high resolution by using holograms. The vertical resolution of these is excellent at 0.2 nm, and the fine behavior of the material surface can be captured.