## 1 Introduction  
The high-finesse Fabry-Perot (FP) cavity is one of the most vital components of ultra-stable lasers (USLs), which have been widely used in several space missions such as LISA (Laser Interferometer Space Antenna) [1], Taiji Program in Space [2], DECIGO (DECI-Hertz Interferometer Gravitational wave Observatory) [3], Post-GRACE (Gravity Recovery and Climate Experiment) [4], etc., as well as in other high-precision fields [5; 6; 7; 8; 9; 10].  
The Pound-Drever-Hall (PDH) method, by which the continuous-wave lasers are locked to the resonance frequency of the FP cavity via high-speed and wide-band electronic control system, is extensively employed in USLs [11; 12]. The instability of a laser's fractional frequency is entirely defined by the length instability of the FP cavity, and perturbations of this length must be minimized to ensure spectral purity [13]. Therefore, the designation of the FP cavity must meet the demand of ultra-high precision coupling in multiple physics fields (optical/thermal/mechanical), including its shape (cylindrical, cubic, spherical, multiple-bore, midplane, etc.), fixing strategy (as small a deformation as possible), and vacuum temperature control (getter and ion pumps, three or more layers of heat shielding, etc.) [14; 15; 16; 17; 18].  
There have been numerous studies conducted on laser systems, including nonlinear optical materials [19; 20] and the design of optical cavities with different shapes [21; 22; 23; 24]. A cubic FP cavity has the advantage of being force-insensitive, making it appropriate for space missions [25]. Based on this idea, follow-up studies have developed several ground-transportable USLs [26]. A typical USL system based on a cubic FP cavity is shown in Figure 1[27]. An FP cavity made of ultra-low-expansion (ULE) glass is attached to a titanium bracket by four titanium screws, which support the truncated vertices symmetrically. Gaskets made of PEEK are placed between the screws and the spacer for heat and vibration insulation. In the center of the top and bottom surfaces, two mirrors (one planar and one concave) are optically attached. Subsequently, the bracket is affixed to the inner heat shield. Three heat shields employed in all, as depicted in Figure 1b, between which PEEK gaskets are inserted to prevent heat transfer. Finally, the heat shields are secured to the vacuum chamber using titanium screws. The ion and getter pumps maintain a high vacuum level below \(2\times 10^{-6}\) Pa.  
As evident from the system composition above, The FP cavity's performance is significantly affected by the shape parameters of the cavity and the temperature fluctuation. Mechanical simulations have been conducted on cubic FP cavities in many studies, primarily focusing on the static deformation under fixing forces [25] and the variation in performance of different materials [28]. However, as far as we know, there is currently no research that specifically focuses on optimizing the design parameters of the FP cavity in consideration of multi-physics field coupling, which this paper precisely addresses.  
The rest of this paper is organized as follows. In Section 2, the multi-physics coupling theory and the finite element method are introduced for problem modelling. The performance results of the cubic FP cavity are thoroughly discussed in Section 3. Section 4 consists of the mechanical optimization process, through which the optimal combination of key design parameters is obtained. Finally, Section 5 presents concluding remarks.