## 1 Introduction  
Aviation turbine engines must withstand higher temperatures to improve the aircraft's thrust-to-weight ratio [1, 2]. Because of their excellent high-temperature properties, nickel-based single crystal superalloys are frequently used in turbine blades; however, the temperature in the combustion chamber exceeds the melting point of the material [3]. As a result, thermal barrier coatings (TBCs) must be applied to the blade surface to provide insulation [4], implying that the properties of the TBCs have a direct impact on the service life of the turbine blade. The properties of TBCs are generally related to two aspects. One aspect is the bonding performance of the coating to the substrate. [PERSON] et al. prepared substrates with different surface roughness by surface sandblasting on nickel-based alloy DD3, investigated the effect of surface roughness on the adhesion state of the coating, and found that the coating performance on different roughness surfaces differed by thermal cycling experiments [5]. The other aspect is the growth morphology of the coating. [PERSON] et al. systematically analyzed the growth mechanism of the micro-convexity on the substrate surface leading to the columnar structure of the deposit, and the shape and depth of the micro-convexity played a decisive role in addition to the droplet size [6]. Therefore, all previous studies have shown that the properties of coatings are highly dependent on the micro-nano structure of the substrate surface [7, 8, 9, 10]. Experiments to obtain different surface topography by polishing and sandblasting are simple, but the topography is rather homogeneous, and the preparation of stable and controllable surface roughness on the surface of turbine blades with complex shapes is difficult. Several techniques have been employed over the past decades to prepare micro- and nano-structures on metal surfaces [11, 12, 13]. Among these, the use of chemical reactions for the preparation of micro-structures allows for more complex surface topography, but the surface area of turbine blades is generally large, and the controllability and integrity of the surface texture are limited. Furthermore, external micro-machining methods are frequently used in preparing metal surface micro/nano-structures, with laser processing being widely used because of their several applications such as in wettability, friction resistance, and trace level explosive sensing [14; 15]. In recent years, femtosecond lasers have been rapidly developed as a new technology primarily for coating material surfaces, where femtosecond laser-induced micro-structures can provide other desired functions on the inherent properties of the material, such as surface wettability, friction and wear properties, and particle filtration [16; 17; 18; 19; 20; 21]. Different shapes of micro/nano-structures serve different purposes, with the difference between the peak and trough of the micro-structure directly determining the surface roughness of the material [22]. More importantly, the femtosecond laser-induced micro-structures are almost entirely periodic, and the surface morphology is more refined. As a result, using a femtosecond laser, it is possible to induce periodic micro-structures with varying roughness on the substrate and then perform coating properties tests, with better engineering significance for turbine blade applications. However, femtosecond laser-induced surface micro-structure is a complicated process. [PERSON] et al. investigated the effect of laser fluence, gas environment, and other factors on the wave-peak-valley difference of micro/nanostructures [23]. [PERSON] et al. observed the evolution of micro-structure depth, width, and other properties by varying the number of pulses and scans [24]. However, only qualitative relationships between the evolution of micro-structures and parameters can be obtained experimentally, making precise control difficult. Using a two-temperature model, [PERSON] et al.