The material will expand or contract under thermal stress when the temperature changes. The thermal expansion can be written as [23] d L = α L d T (1)

For a cavity length of L and a mirror radius of R, the instantaneous CTE of the ULE spacer and FS substrate are α _ULE and α _FS. The equivalent CTE α _equ can be described as α e q u = α U L E + 2 δ R L α F S − α U L E (2) where δ is the coupling coefficient of the mirror axial expansion. Since both ULE and FS are isotropic materials, the coupling coefficient δ is only related to the geometry of the cavity and the mechanical properties of the material. The coupling coefficient δ is a constant when a given cavity structure is in place. Therefore, we can use the coupling coefficient to calculate the zero-thermal-expansion temperature for a specific structure.

The instantaneous CTE of ULE glass around its zero-thermal-expansion temperature T ₀ can be approximated as quadratic temperature dependence α U L E T = a T − T 0 + b T − T 0 2 (3)

For a typical ULE, the quadratic coefficient b is much smaller than the linear coefficient. In the FEA, we assume that a = 2.4 × 10^–9/K² and b = 0. The CTE of FS is a constant value of 5 × 10^–7/K, which is consistent with the parameters assumed in the simulation in Ref. [23], and it is experimentally demonstrated that these parameters and the model can describe the CTE variation of the composite cavity very well.

Substituting Eq. 3 into Eq. 2, the equivalent CTE of the composite cavity can be described as a linearly temperature-dependent function around the zero-thermal-expansion temperature. α e q u = a T − T 0 + 2 δ R L α F S − a T − T 0 = 1 − 2 δ R L a T − T 0 + 2 δ R α F S / L 2 δ R / L − 1 a = a e q u T − T 0 + Δ T = a e q u T − T e q u (4)

Typically, the zero-thermal-expansion temperature T ₀ of ULE is in the range of 5°C–35°C ¹ , ΔT is the compensation temperature difference of the composite cavity, and T _equ is the zero-thermal-expansion temperature of the composite cavity. The aim is to optimize the structure so that the zero-thermal-expansion temperature of the composite cavity is slightly above the room temperature, around 28°C. Therefore, the tuning range of ΔT is expected to cover the range from -10 K to 23 K.

The 3-cm cavity is designed as a ULE cylindrical cavity, with a central bore hole punched in the center of both the end faces and an evacuating hole punched in the perpendicular direction with an aperture of 4 mm. The mirror is selected as a standard product with a diameter of 12.7 mm and a thickness of 6 mm, optically contact to both end faces of the spacer, and the cavity is symmetrical about three orthogonal planes. The O-ring is used to apply a force parallel to the direction of the optical axis to extrude the two end faces of the cavity to fix the cavity. As the radius of the spacer is designed to be 30 mm after comprehensive consideration, there exists a certain O-ring diameter for some certain structure, where the cavity is insensitive to the applied extrusion force. However, the diameter is closely related to the structure of the composite cavity, which will be described in detail in the next section.

The thermal properties of the cavity are analyzed with the help of the Comsol Multiphysics FEA platform. Simulations are performed to calculate the equivalent zero-thermal-expansion temperature of the corresponding structure. The compensation temperature difference ΔT can be derived by Eqs 1 and 4, where ΔL is the axial displacement of the mirror center for a positive temperature step of 1 K. Δ T = α F S Δ L − a L a Δ L − α F S L (5)

Cavity lengths at multiple temperatures are computed for a certain geometric parameter, and the zero-thermal-expansion temperature is determined by fitting the length-temperature curve. The compensation temperature difference obtained this way is used to confirm the result according to Eq. 5. Due to the high symmetry of the cavity structure, we only simulate the thermodynamics of 1/8 of the cavity to improve the speed of the simulation calculation. The symmetry is ensured not to be broken during the calculation.

First, we analyzed the effect of the change of the zero-thermal-expansion temperature on the variation of three structural parameters of the “sandwich” structure, where the ULE ring is connected to the back side of the FS mirror and may change [23]. The geometry and thermal deformation are presented in Figures 1A,B. And the simulation results are indicated in Figures 1C,D. It can be seen that the ring thickness has a small effect on the compensation temperature difference, and the increase in the bore-aperture and the inner diameter of the ring decreases the difference. It is due to the fact that the axial constraint of the mirror is weakened when the bore-aperture or ring inner diameter is increased, allowing the mirror to expand more freely in the axial direction under thermal stress. Although the compensation temperature difference can be increased to some extent by decreasing the bore-aperture and the inner diameter of the ring, they cannot be too small; otherwise, the assumption of an open cavity will be invalid and additional diffraction loss will be incurred when the laser interacts with the cavity, thus distorting the beam. As shown in Figures 1C, D, the compensation temperature difference regulation of the sandwich structure ranges from −30 K to −20 K, far away from our desired range.

We also analyzed a modified “re-entrant” structure in which the FS mirrors optically contact to the FS ring, followed by the FS ring optically contacted to the spacer so that the mirrors are embedded inside the spacer [24], as shown in Figures 2A,B. This allows the mirrors to expand in the opposite direction to the spacer, while the FS ring expands in the same direction, inhibiting the mirrors from expanding excessively inward to avoid excessive compensation temperature difference. As shown in Figure 2C, varying the ring thickness and ring outer diameter gives the structure a temperature difference tuning capability from 5 K to 23 K when the bore-aperture is 15 mm and the ring inner diameter is 5 mm. The reason for the trend is not monotonous, that is, the volume of the compensation ring and the contact area play different roles in the thermal expansion. The larger the volume, the more the compensation ring expands toward the outside of the cavity, making the compensation temperature difference lower. While larger contact area will restrain this effect. As shown in Figure 2D, further increasing the bore-aperture can achieve our desired temperature difference regulation capability from −10 K to 23 K. Nevertheless, this structure requires the mirrors to be embedded into the spacer, which increases the volume of the cavity, and in addition the compensation ring further increases the overall volume and increases the material consumption. Fixation is also problematic and will be described in the next section.

The proposed “lantern” structure is well-suited for the design of compensation rings for a miniaturized cavity, with two components: a cover ring and a side ring, shown in Figure 3. The bore-aperture and the inner diameter of the side ring are slightly larger than the diameter of the mirrors. First, the cover ring is optically contacted to the mirror. Then, the side ring is optically contacted to the cover ring, and finally the side ring is optically contacted to the spacer so that the mirror is in contact with the cover ring only. The cover rings are made of ULE, which inhibits expansion of the mirror with the spacer in the same direction. This structure allows the mirror to expand freely against the spacer. To avoid inward over expansion, which leads to high compensation temperature difference, the side ring is made of FS material, which can expand in the same direction as the spacer and compensate the inward over expansion of the mirror so that the tuning range meets our requirements. The FEA 3D model and thermal deformation are pictured in Figures 4A,B.

As shown in Figure 4C, the tuning range can be made to cover our desired range from −10 K to 23 K, by adjusting the width of the side ring and the thickness of the cover ring. Larger side ring width will lead to more expansion outward the cavity, resulting in the compensation temperature difference to be lower, while the cover rings restrain this effect. The thicker the cover ring, the more solid the restraint. The compensation temperature difference can also be further adjusted by changing the bore-aperture and the side ring gap as shown in Figures 4D,E, where the gray shadow shares the same structural parameters with that in Figure 4C. The tendency can be easily seen that increasing the bore-aperture will decrease the compensation temperature difference, while increasing the side ring gap will increase it.

The tuning ranges of the “sandwich” structure do not meet the requirements according to Section 2.2 and are therefore not discussed here. Only the “re-entrant” structure and the “lantern ring” structure are analyzed. For the “re-entrant” structure, see Figure 2C, the compensation temperature difference ΔT > 5°C when the dimensions of the compensation ring are changed at the bore-aperture of 15 mm, and the compensation temperature difference can be reduced only by increasing the bore-aperture. Therefore, we calculated the relative axial displacement of the mirror center for the FS compensation ring size fixed at od = 44 mm and tk = 10 mm. It can be seen from Figure 6 that there is no zero-crossing point in each curve, which means that there is no extrusion pressure insensitive O-ring diameter for this structure.

This indicates that although the “re-entrant” structure can be changed to meet the desired tuning range, it cannot necessarily find a suitable installation method to make it insensitive to the extrusion pressure.

For the “lantern ring” structure, as shown in Figure 4C, when the cover ring thickness is 2 mm and the side ring width is varied from 3 mm to 7 mm, the compensation temperature difference can cover our requirement of −10 K to 23 K. As shown in Figure 7, the corresponding O-ring diameters can be found in all the parameter ranges covered in Figure 4C, making the cavity insensitive to the extrusion force applied. The pressure sensitivity of the O-ring is about 2 × 10^–10/N/mm when the cover ring thickness is 2 mm. Therefore, the “lantern ring” structure is more suitable for the design of thermal compensation for miniaturized cavities.

In term of rigid fixation, the original design is fixed by directly extruding the two end faces of the spacer with three elastomer balls [25], which is not only difficult to align, but also the force perpendicular to the axial direction exists only by friction, which may shift under the influence of external vibration and then change its extrusion force sensitivity and vibration sensitivity, resulting in a risk of performance deterioration after long-term operation or transportation. To overcome the problem, we propose to dig small slots in the end faces of the cavity, shown in Figure 3. The FEA has determined the O-ring position so that the O-ring is plunged about 3/10 of 2 mm, the cross section’s diameter of O-ring. This can provide additional support forces in the direction perpendicular to the optical axis and facilitate the alignment during installation. As indicated by Figure 8, the presence or absence of a small slot has been found to have a negligible effect on the thermal and mechanical properties of the cavity. Moreover, in practice, the pressure pressing against the O-ring may not be evenly distributed around the circle. We increased the pressure on one semicircle by 10% and lowered the value for the other half by the same amount. The simulation shows little difference from that of pressure evenly distributed, which indicates that whether the pressure is evenly distributed or not does not affect it much.

We assume that the zero-thermal-expansion temperature of ULE handy is 20°C. Using the proposed design, we can obtain ΔT = 8 K for a cover ring thickness of 2 mm and a side ring width of 4.9 mm, which means the zero-thermal-expansion temperature of the composite cavity is around 28°C. As indicated in Figure 8, the optimal size of the O-ring diameter is about 54.75 mm. In order to verify its vibration insensitivity, we applied 1 g of gravity in x, y, and z directions, and the symmetry constraint on the mid-plane except perpendicular to the direction of gravity. The translation of the slot in the x direction was allowed, while the other directions were rejected. For the gravity in the x direction, fixed constraint should be applied to one of the slot. Then, the displacement of mirrors can be calculated by static analysis to obtain the theoretical vibration sensitivity of the cavity of 2 × 10^–12/g in the x direction and far more lower in the other directions. Therefore, this type of cavity is considered to have good vibration insensitivity.