Transportable all-optical atomic clocks represent the next-generation devices for precision time keeping, ushering a new era in encompassing a wide range of PNT (Positioning, Navigation and Timing) applications in the civil and strategic sectors. Their performance relies on ultra-stable, narrow-linewidth lasers, frequency stabilized to a compact portable optical cavity. Among various designs, the cubic spacer-based ultra-stable cavity is particularly well-suited for transportable applications due to its low sensitivity to vibrations, owing to its symmetric geometry and robust mounting structure. While longer cavities offer a lower fundamental thermal noise floor, one needs to strike a balance between transportability and size. In this aspect, the 7.5 cm dual-axis cubic cavity offers a lower fundamental thermal noise floor in comparison to smaller counterparts, while still retaining a reasonable SWaP (Size, Weight and Power) for terrestrial and aerial PNT applications. Its dual-axis design also enables multi-wavelength laser stabilization, making it a promising candidate for future transportable clock applications. This work presents a detailed study of the 7.5 cm dual-axis cubic cavity using FEM (Finite Element Method) to evaluate its mechanical and thermal stability. We analyze the impact of various geometric factors, mounting forces, and machining imperfections, while also modelling thermal effects such as conduction, radiation, and mirror heating within a vacuum chamber and thermally shielded environment. Our findings provide design insights for developing robust dual-axis optical reference cavities, advancing the deployment of portable atomic clocks for next-generation applications in PNT, geodesy, VLBI (Very Long Baseline Interferometry) and deep space missions.
The pursuit of ever more precise time and frequency standards has transformed modern science and technology, underpinning applications from satellite navigation and telecommunications to tests of the fundamental laws of physics [1][2][3][4][5][6]. Optical atomic clocks have emerged as the most accurate and stable timekeepers, with fractional frequency instabilities reaching the 10 -18 level [7]. Such performance is made possible by interrogating ultranarrow line-width dipole forbidden atomic transitions whose frequencies are stabilized to ultra-stable high-finesse optical resonators. In this context, the frequency stability of the laser and thus of the clock itself is directly determined by the dimension stability of the reference cavity to which it is referenced. The fractional frequency stability of the stabilized laser is directly proportional to the fractional length stability of the cavity. In state-of-the-art laboratory systems, ultra-stable cavities are generally longer operated in vibration-isolated, well temperature-controlled environments, pushing laser instabilities into the low 10 -16 range over 1 s averaging times [8]. However, future applications increasingly demand transportable and even space-qualified clock systems, where such environmental isolation is not always feasible. Transportable optical clocks are being developed for inter-laboratory comparisons, precision time transfer in remote locations, spaceborne fundamental physics missions, and navigation systems that extend beyond the limits of present PNT-based systems [9]. In these scenarios, the reference cavity must maintain exceptional stability despite exposure to mechanical shocks, varying orientations, and fluctuating thermal conditions.
A key challenge in transportable cavities [9] lies in minimizing acceleration sensitivity without relying solely on external vibration isolation. Among the various approaches to achieving robust frequency references outside controlled laboratory settings, symmetry-based mounting strategies have proven particularly effective in suppressing sensitivity to inertial forces. One of the notable examples is the force insensitive cubic cavity [10,11], where a cubic spacer is truncated at the vertices and supported at the four vertices in a tetrahedral arrangement. In this geometry, each support force is applied toward the geometric centre of the cube, constraining all translational and rotational degrees of freedom while maintaining force symmetry. This configuration effectively cancels first-order cavity length changes induced by both static preload forces and linear or rotational accelerations. With careful optimization of the vertex truncation depth, such cavities have demonstrated passive acceleration sensitivities as low as 10 -11 /g [10,12] making them strong candidates for use in transportable and spaceborne optical clock systems [13]. Building on this foundation, the dual-axis cubic cavity [14,15] extends the concept by incorporating two orthogonal optical axes within the same cubic spacer. This enables simultaneous stabilization of multiple lasers [16] for example, a narrow linewidth clock laser and other cooling or trapping lasers within a single mechanically robust platform.
Many research groups have demonstrated cubic spacer cavities [10,[12][13][14][15][16], among which the 5 cm cubic cavity (patented by NPL, UK [10]) has become a prominent choice. The compactness and portability of this 5 cm cubic cavity make it particularly suitable for transportable optical clock applications, positioning it as a leading candidate for future space missions [9,17,18]. On the other hand, some groups have demonstrated 10 cm cubic cavities [19][20][21], which benefit from a longer cavity length that significantly reduces the thermal noise floor. This makes them a better candidate for transportable systems which require high frequency stability. However, the larger size limits their portability, reducing suitability for transportable or space applications and increasing costs due to greater complexity in thermal management and vibration isolation.
Therefore, to strike a balance on size (which determines portability) and noise performance, an analysis comparing different sizes of cubic cavities is essential. We perform a comparative study of 5 cm, 7.5 cm and 10 cm dual-axis cubic cavities, analysing their vibration sensitivity and thermal noise floor. This analysis is conducted for terrestrial applications (with supporting forces of 50 N) and space applications (with supporting forces of 500 N). The sensitivity is determined through the FEA simulation in COMSOL, with the results presented in Figure 1. The thermal noise floor for the three spacer sizes is calculated analytically following the procedure from [17]. Both the simulated sensitivities and the analytical noise floor values are summarized in Table I.
From Table I, it is evident that both sensitivity and thermal noise performance improve as the size of
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