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Cordierite-based optical resonators with extremely low thermal expansion

Nico Wagner, Thomas Legero, Stefanie Kroker

Abstract

Applications for ultra-stable lasers outside controlled laboratory environments require compact and robust optical resonators with reduced sensitivity to temperature fluctuations. The low thermal expansion coefficient (CTE) and the high stiffness make cordierite-based ceramics, such as NEXCERA, attractive for vibration insensitive room-temperature resonators. We revisit the effective CTE of resonators with spacers and mirrors made of different materials and use finite element simulations to analyze the impact of a CTE mismatch in a cordierite-based resonator with mirrors made of ultra-low expansion (ULE) glass or fused silica (FS). This enabled us to determine the CTE of a cordierite spacer from the measured effective CTE of a resonator. We confirm a six-fold larger CTE slope of cordierite around the zero-crossing temperature than in ULE glass. The steep CTE slope, in combination with the large stiffness, makes cordierite-based resonators far less sensitive to CTE mismatch with FS mirrors, thereby eliminating the need for additional compensation rings. We further consider the so far neglected case, where the CTE of the spacer is larger than that of the mirror, and propose resonator designs in which the thermal length change of the spacer is fully or partially compensated by the deflection of the mirrors. This results in a cordierite-based resonator with ULE mirrors whose effective CTE can be close to zero over a temperature range of several tens of Kelvin. We are extending our concept to resonators based on crystalline materials with high stiffness and low isothermal length change, such as silicon, enabling compact and robust room-temperature resonators for terrestrial and space-born applications.

Cordierite-based optical resonators with extremely low thermal expansion

Abstract

Applications for ultra-stable lasers outside controlled laboratory environments require compact and robust optical resonators with reduced sensitivity to temperature fluctuations. The low thermal expansion coefficient (CTE) and the high stiffness make cordierite-based ceramics, such as NEXCERA, attractive for vibration insensitive room-temperature resonators. We revisit the effective CTE of resonators with spacers and mirrors made of different materials and use finite element simulations to analyze the impact of a CTE mismatch in a cordierite-based resonator with mirrors made of ultra-low expansion (ULE) glass or fused silica (FS). This enabled us to determine the CTE of a cordierite spacer from the measured effective CTE of a resonator. We confirm a six-fold larger CTE slope of cordierite around the zero-crossing temperature than in ULE glass. The steep CTE slope, in combination with the large stiffness, makes cordierite-based resonators far less sensitive to CTE mismatch with FS mirrors, thereby eliminating the need for additional compensation rings. We further consider the so far neglected case, where the CTE of the spacer is larger than that of the mirror, and propose resonator designs in which the thermal length change of the spacer is fully or partially compensated by the deflection of the mirrors. This results in a cordierite-based resonator with ULE mirrors whose effective CTE can be close to zero over a temperature range of several tens of Kelvin. We are extending our concept to resonators based on crystalline materials with high stiffness and low isothermal length change, such as silicon, enabling compact and robust room-temperature resonators for terrestrial and space-born applications.
Paper Structure (9 sections, 7 equations, 5 figures, 2 tables)

This paper contains 9 sections, 7 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Thermal expansion of optical resonators
  3. Cordierite resonators with fused silica mirrors
  4. Experimental determination of the CTE of cordierite
  5. Resonators with extremely low CTE
  6. Zero-CTE resonators
  7. Resonators with mirror-determined CTE
  8. Silicon resonators with fused silica mirrors at room temperature
  9. Conclusion

Figures (5)

  • Figure 1: (a) Simulated coupling coefficient $\delta$ for a resonators with FS mirrors based on a ULE or cordierite spacer. The spacer length is 105.5mm, with outer and inner bore diameters of 32mm and 11mm, respectively. The mirrors have a diameter of 25.4mm and a thickness of 6.3mm. (b) and (c) show a zoom-in of the elastic deformation at the mirror–spacer interface caused by a temperature increase of 20, for the same resonator geometry with a ULE spacer and a cordierite spacer, respectively. The color scale represents the normal stress distribution within the resonator. The higher stiffness of cordierite reduces the coupling coefficient because the mirror undergoes less bending under thermally induced stresses. As a result, the coupling coefficient is up to 20% lower compared to ULE-based resonators. The marked points correspond to Fig. \ref{['fig:delta_over_L']}, where they appear as data points in the length-dependent coupling coefficient plot.
  • Figure 2: Influence of the spacer length $L$ on the coupling coefficient $\delta$ for spacer configurations with $D = 25.4mm$, $D = 52mm$, and $D = 104mm$, over a length range from 1mm to 200mm. With shorter spacers, the coupling coefficient is strongly dependent on the length. The results show that the coupling coefficient becomes constant once the spacer length is larger than its diameter. The marked points are related to Fig. \ref{['fig:delta_over_D']}.
  • Figure 3: (a) Relative length change of the resonator during a stepwise temperature ramp. The 4 temperature steps are separated by a thermalization time of 4 hours. The blue and red dots mark the achieved length change and temperature at the end of each step. (b) Temperature-dependent length changes. The black dots are the measurement points from (a). A least squares fit of Eq. (\ref{['eq:length_change']}) is shown as red line. (c) The residuals of the fit are well below 0.1%.
  • Figure 4: Calculated effective CTEs using Eq. (\ref{['eq:cte-model']}) and the linear and quadratic coefficients from Tab. \ref{['tab:cte_results']}. Shown is a resonator with a cordierite (Co) spacer and ULE mirrors for ZCT differences of 0.1 and 1 between spacer and mirror materials, with a correction factor of $k=1.19$. For comparison, the CTE of an all-ULE resonator is included. For a 0.1 ZCT difference, the resonator’s CTE exhibits a slope of only 7.9e-11, which is 18 times lower than that of ULE.
  • Figure 5: Effective CTE of a silicon--ULE resonator for $k=1$ and for small deviations from this value. For $k=1$, the effective CTE equals that of ULE, even though silicon is used as the spacer. Small variations in the correction factor shift the ZCT, while the slope remains nearly identical to that of ULE.