Thermal analysis of optical reference cavities for low sensitivity to environmental temperature fluctuations

Xiaojiao Dai; Yanyi Jiang; Chao Hang; Zhiyi Bi; Longsheng Ma

doi:10.1364/OE.23.005134

1. Introduction

Optical reference cavities with finesse of >100,000 are widely used to stabilize the frequency of lasers in optical atomic clocks, precision metrology, high resolution spectroscopy, tests of fundamental physics and deep space navigation [1–7], which rely on the high frequency stability of lasers. When a laser is tightly frequency-stabilized to a reference cavity using the Pound-Drever-Hall technique [8], the laser frequency stability is dominated by the effective length stability of the reference cavity. To minimize the length fluctuation, reference cavities are often made of ultra-low expansion (ULE) glass or single crystal silicon, and are installed in vacuum chambers with high stability temperature stabilization for less sensitivity to environmental temperature fluctuations via thermal expansion [9–12]. Special geometry and mounting configurations for reference cavities have been developed for less sensitivity to environmental vibration [13–17]. When stabilized to those carefully designed reference cavities, lasers with a frequency instability at the 10⁻¹⁶ level at 1–10 s averaging time have been achieved [11, 12, 14, 18, 19], approaching the thermal noise limit of reference cavities [20].

However, for the applications of ultra-stable laser systems, both the short-term stability on a few seconds’ timescale as well as the long-term stability over a day are essential. In most of those ultra-stable laser systems, the laser frequency drifts linearly at tens of mHz/s to several Hz/s on a relatively short timescale, and drifts nonlinearly on hundreds of seconds’ timescale which is hardly compensated by applying a feed forward frequency correction. The aging of the cavity material and optical contact between cavity mirrors and spacers leads to a fractional length change on the order of 10⁻¹⁷/s to 10⁻¹⁶/s [21]. Therefore, temperature fluctuation of reference cavities is the key cause of laser frequency drift. Well-designed temperature controllers can achieve sub-mK stability over several days [10,22]. When a reference cavity made of ULE glass is temperature-stabilized at the zero-crossing thermal expansion temperature (T_CTE=0) with accuracy of 0.1 °C, it has a coefficient of thermal expansion (CTE) of less than 2 × 10⁻¹⁰ /K [18]. To achieve laser frequency instability of 10⁻¹⁶ or even lower, careful considerations of passive thermal shielding and supporting configuration for reference cavities should be made. For systems that can not be temperature stabilized close to T_CTE=0 or that have a lower temperature stability, it is also possible to realize high length stability of reference cavities by carefully designing their thermal shields [23].

This paper gives two models on how reference cavities respond to (1) a step environmental temperature change and (2) a periodic temperature fluctuation through heat transfer of thermal conduction and thermal radiation separately. The analysis shows that passive thermal shielding configurations of reference cavities with a larger thermal time constant will be less sensitive to environmental temperature fluctuations. All the thermal shields discussed in this paper are passive ones without active temperature control. With additional numerical simulation results based on finite element analysis, we will discuss the details on the design of cavity thermal shielding configurations to make it insensitive to environmental temperature perturbations, including material, layer numbers and aperture size of thermal shields. The number of thermal shields has a great effect on the time constant of reference cavities. A two-layer thermal shield will help to reduce the temperature fluctuation-induced length instability of a reference cavity below 1×10⁻¹⁵ on a day timescale when the cavity has a CTE of 1×10⁻¹⁰ /K and is enclosed in a vacuum chamber with temperature fluctuation amplitude of 1 mK and period of 24 hours.

2. Time constant

2.1. Thermal conduction

Consider body M₁ that has a length of L₁, cross-section area of A₁ and heat conductivity of k₁, as shown in Fig. 1(a). Another body M₂ with a mass of m₂ and heat capacity of C₂ is on the top of M₁. Assume that M₂ is in good thermal contact with M₁, and both are initially in thermal equilibrium.

Fig. 1 Thermal conduction. (a) One-layer thermal conduction. (b) Two-layer thermal conduction. (c) The temperature variation of different layers over time when the temperature of M₁ changes from T_i = 300 K to T_f = 350 K at t = 0.

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At time t = 0, the temperature at the bottom of M₁ changes from T_i to T_f (ΔT₁ = T_f − T_i). If ignoring the heat absorbed by M₁ itself when M₁ has a much smaller mass and thermal capacity compared with those of M₂, the heat flow rate transferred between M₁ and M₂ is given by

\frac{d Q}{d t} = \frac{k_{1} A_{1}}{L_{1}} (T_{f} - T_{2}),

where Q is transferred heat and T₂ is the temperature of M₂. M₂ absorbs the heat that flows from M₁, resulting in a temperature change

d T_{2} = \frac{d Q}{C_{2} m_{2}} .

By solving the above equations, we obtain the temperature change of M₂ over time

T_{2} (t) = T_{f} - Δ T_{1} \cdot e^{- \frac{k_{1} A_{1}}{C_{2} m_{2} L_{1}} t} = T_{f} - Δ T_{1} \cdot e^{- \frac{t}{τ_{c 1}}} .

It takes M₂ time of

τ_{c 1} = \frac{C_{2} m_{2} L_{1}}{k_{1} A_{1}}

(time constant of thermal conduction) to change its temperature by

(1 - \frac{1}{e}) Δ T_{1}

.

For the case of thermal conduction in a typical supporting configuration of reference cavities, M₁ could be supporting rods of reference cavities or supporting rods of thermal shields. To enlarge the time constant, those supporting rods might have small thermal conductivity and cross-section area, while thermal shields on the top of the supporting rods have a large thermal capacity and mass. The red dashed line in Fig. 1(c) shows an illustration of the temperature change of M₂ (made of copper with m₂ = 20 kg) assuming the temperature of M₁ (Teflon cylinders with L₁ = 3 cm and total cross-section area A₁ = 10⁻³ m²) changes from 300 K to 350 K. Here the time constant of thermal conduction is τ_c1 ≈ 256.7 h. Parameters for materials used throughout this paper can be found in Table 1.

Table 1. Parameters of materials.

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Suppose there are other layers M₃ and M₄ on the top of M₂, as shown in Fig. 1(b). The temperature change of M₄ is determined by

\frac{d T_{4}}{d t} = \frac{k_{3} A_{3}}{C_{4} m_{4} L_{3}} (T_{f} - Δ T_{1} \cdot e^{- \frac{t}{τ_{c 1}}} - T_{4}) .

Thus the temperature change of M₄ which responds to the temperature variation of M₁ is

T_{4} (t) = T_{f} - \frac{Δ T_{1}}{τ_{c 1} - τ_{c 2}} (τ_{c 2} \cdot e^{- \frac{t}{τ_{c 1}}} - τ_{c 2} \cdot e^{- \frac{t}{τ_{c 2}}}),

where

τ_{c 1} = \frac{C_{2} m_{2} L_{1}}{k_{1} A_{1}}

and

τ_{c 2} = \frac{C_{4} m_{4} L_{3}}{k_{3} A_{3}}

. If M₃ is also Teflon rods with L₃ = 3 cm and A₃ = 10⁻³ m², and M₄ is a piece of thermal shield made of copper with m₄ = 15 kg, the time constants are τ_c1 ≈ 256.7 h and τ_c2 ≈ 192.5 h. The blue short dashed line in Fig. 1(c) shows how M₄ responds to the step temperature change of M₁. It takes M₄ about 480.0 h to change its temperature by

(1 - \frac{1}{e}) Δ T_{1}

. As it shows, in the two-layer thermal conduction the time constant is almost twice of that in the single-layer thermal conduction.

2.2. Thermal radiation

Thermal radiation is a dominant way of heat transfer for optical reference cavities since those cavities are usually put in vacuum chambers. Suppose there is an outer layer P₁ and an inner layer P₂, as shown in Fig. 2(a). Layer P₁ could be a vacuum chamber, and P₂ could be a layer of thermal shield. Initially, both of them are in thermal equilibrium. At t = 0, the temperature of P₁ changes from T_i to T_f (ΔT₁ = T_f − T_i). According to ref. [25], when ignoring the geometry of radiation bodies and distances between radiation surfaces, and supposing the view factor of radiation surfaces is unity (Indeed, the value of view factor is less than 1, which is related to radiation angle and relative spacing of radiation surfaces.), the rate of radiated energy absorbed by P₂ is

\frac{d Q}{d t} = \frac{σ (T_{f}^{4} - T_{2}^{4})}{\frac{1}{A_{1}} \cdot (\frac{1}{ε_{1}} - 1) + \frac{1}{A_{2}} \cdot \frac{1}{ε_{2}}},

where A₁ and A₂ are inner surface area of P₁ and outer surface area of P₂ respectively, and σ = 5.67 × 10⁻⁸ W/(m²·K⁴) is the Stefan-Boltzmann constant. ε₁ and ε₂ are the emissivity of the corresponding surfaces of P₁ and P₂ respectively. Since the emissivity of a material varies as a function of temperature and surface finish, the exact emissivity of a material should be determined with absolute measurements. All the emissivity values concerning in this paper refer to Table 1, which are used just for reference. The temperature of P₂ changes by

\frac{d T_{2}}{d t} = \frac{σ (T_{f}^{4} - T_{2}^{4})}{C_{2} m_{2} [\frac{1}{A_{1}} \cdot (\frac{1}{ε_{1}} - 1) + \frac{1}{A_{2}} \cdot \frac{1}{ε_{2}}]} = β_{12} (T_{f}^{4} - T_{2}^{4}) .

Using the first order approximation when ΔT₁ is much smaller than T_f and T_i, the temperature variation of P₂ over time is given by

T_{2} (t) \approx T_{f} - Δ T_{1} \cdot e^{- 4 β_{12} T_{f}^{3} \cdot t} = T_{f} - Δ T_{1} \cdot e^{- \frac{t}{τ_{r 1}}},

where

τ_{r 1} = \frac{1}{4 β_{12} T_{f}^{3}}

is the time constant of thermal radiation for P₂ to respond to the temperature change of P₁. Note that in Eq. (8) τ_r1 depends on T_f instead of T_i due to the approximation(ΔT₁ ≪ T_i, T_f), while in fact τ_r1 has a weak dependence on T_i. This approximation gives an error of less than 5% for τ_r1 when ΔT₁ < 10 K, T_f and T_i ≈ 300 K. The red dashed line in Fig. 2(c) shows an illustration of the temperature change of P₂ over time when the temperature of P₁ changes from 300 K to 310 K, assuming P₁ is made of aluminum with ε₁ = 0.2 and A₁ = 1 m², and P₂ is made of copper with m₂ = 20 kg, A₂ = 0.6 m² and ε₂ = 0.1. The time constant is τ_r1 ≈ 6.5 h.

Fig. 2 Thermal radiation. (a) One-layer thermal radiation. (b) Two-layer thermal radiation. (c) The temperature change of different layers over time when the temperature of P₁ changes from T_i = 300 K to T_f = 310 K at t = 0.

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If there is another layer P₃, which could be a cavity, inside P₂, as shown in Fig. 2(b), the temperature change of P₃ is determined by

\frac{d T_{3}}{d t} = β_{23} [{(T_{f} - Δ T_{1} \cdot e^{- \frac{t}{τ_{r 1}}})}^{4} - T_{3}^{4}],

where

β_{23} = \frac{σ}{C_{3} m_{3} [\frac{1}{A_{2}} (\frac{1}{ε_{2}} - 1) + \frac{1}{A_{3}} \frac{1}{ε_{3}}]}

. By solving Eq. (9) to the first order approximation, we have the temperature change of P₃, which is

T_{3} (t) \approx T_{f} - \frac{Δ T_{1}}{τ_{r 1} - τ_{r 2}} \cdot [τ_{r 1} \cdot e^{- \frac{t}{τ_{r 1}}} \cdot τ_{r 2} \cdot e^{- \frac{t}{τ_{r 2}}}],

where

τ_{r 1} = \frac{1}{4 β_{12} T_{f}^{3}}

and

τ_{r 2} = \frac{1}{4 β_{23} T_{f}^{3}}

. If P₃ is made of ULE glass with A₃ = 0.2 m², m₃ = 10 kg and ε₃ = 0.85, τ_r2 ≈ 6.6 h. The blue short dashed line in Fig. 2(c) shows the response of P₃ to the step temperature change of P₁ of 10 K. It takes P₃ about 14.2 h to change its temperature by

(1 - \frac{1}{e}) Δ T_{1}

. The green dash-dot line shows the response of P₃ to the step temperature change of P₁ if the intermediate layer P₂ is missing (τ_r ≈ 3.1 h). As shown, the intermediate layer P₂ helps to enlarge the time constant of P₃ by more than four times.

3. Thermal sensitivity

Next we will discuss the situation in which the environmental temperature fluctuates periodically, we will see how reference cavities respond to the temperature variations.

3.1. Thermal conduction

In Fig. 1(a), suppose the temperature at the bottom of M₁ fluctuates as $T_{1} (t) = \bar{T} + Δ T_{1} \cdot sin (\frac{2 π t}{ξ})$ , in which T̄ is the mean temperature, ΔT₁ and ξ are the amplitude and period of the temperature fluctuation respectively. Then the temperature of M₂ changes periodically due to the existence of the heat transfer between M₁ and M₂ via thermal conduction. We use similar equations as Eqs. (1) and (2), and obtain the temperature of M₂ as

\frac{d T_{2}}{d t} = \frac{1}{τ_{c 1}} [\bar{T} + Δ T_{1} \cdot sin (\frac{2 π t}{ξ}) - T_{2}] .

By solving the above equation, we have

T_{2} (t) = \bar{T} + \frac{ξ}{\sqrt{ξ^{2} + 4 π^{2} τ_{c 1}^{2}}} \cdot Δ T_{1} \cdot sin (\frac{2 π t}{ξ} - ϕ) + \frac{2 π ξ τ_{c 1}}{ξ^{2} + 4 π^{2} τ_{c 1}^{2}} \cdot Δ T_{1} \cdot e^{- \frac{t}{τ_{c 1}}} .

Figure 3(a) shows the temperature change of M₂ when the temperature at the bottom of M₁ fluctuates as

T_{1} (t) = 300 + 10 \cdot sin (\frac{2 π t}{5 \times 10^{5}})

. The large fluctuation period (ξ = 5 × 10⁵ s) used here is to make the temperature fluctuation of M₂ large enough to be clearly seen in the figure. As shown, the temperature of M₂ fluctuates with the same period as that of M₁, but it has a phase lag

ϕ = arcsin (\frac{2 π τ_{c 1}}{\sqrt{ξ^{2} + 4 π^{2} τ_{c 1}^{2}}})

. And the mean value of T₂(t) does not equal to T̄ due to the last term of Eq. (12). The temperature fluctuation amplitude of M₂ is

Δ T_{2} = \frac{ξ Δ T_{1}}{\sqrt{ξ^{2} + 4 π^{2} τ_{c 1}^{2}}}

.

Fig. 3 Thermal sensitivity. (a) The temperature change of M₂ when the temperature of M₁ fluctuates as $T_{1} (t) = 300 + 10 \cdot sin [\frac{2 π t}{5 \times 10^{5}}]$ . (b) The sensitivity of thermal conduction as a function of temperature fluctuation period. S (S₂) is the sensitivity of one (two)-layer thermal conduction. (c) The sensitivity of thermal radiation as a function of temperature fluctuation period. S (S₂) is the sensitivity of one (two)-layer thermal radiation.

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The normalized sensitivity of M₂ to the temperature fluctuation of M₁ can be written

S = \frac{Δ T_{2}}{Δ T_{1}} = \frac{ξ}{\sqrt{ξ^{2} + 4 π^{2} τ_{c 1}^{2}}} .

As shown in Eq. (13), if the period of the temperature fluctuation is much smaller than the time constant (ξ ≪ τ_c1),

S \approx \frac{ξ}{2 π τ_{c 1}}

. In the other limit, if ξ ≫ τ_c1, S ≈ 1.

If there are more layers on the top of M₂, then the thermal sensitivity of M_n is

S_{n} = \prod_{i = 1}^{n} \frac{ξ}{\sqrt{ξ^{2} + 4 π^{2} τ_{c_{i}}^{2}}} .

Figure 3(b) shows the thermal sensitivity of conduction when using τ_c1 ≈ 256.7 h for S (the blue solid line) and τ_c1 ≈ 256.7 h and τ_c2 ≈ 192.5 h for two-layer sensitivity S₂ (the red dashed line). As we can see in the figure, multiple-layer structure helps to make reference cavities less sensitive to temperature fluctuations with short fluctuation periods. For example, as seen in Fig. 3(b), when the temperature perturbations have fluctuation periods of 1 s to 10⁶ s, the two-layer thermal sensitivity is S₂ ≈ S².

3.2. Thermal radiation

For the model shown in Fig. 2(a), if the temperature of P₁ fluctuates as $T_{1} (t) = \bar{T} + Δ T_{1} \cdot sin (\frac{2 π t}{ξ})$ , due to thermal radiation the temperature of the inner layer P₂ varies as

\frac{d T_{2}}{d t} = β_{12} [{(\bar{T} + Δ T_{1} \cdot sin (\frac{2 π t}{ξ}))}^{4} - T_{2}^{4}] .

When ΔT₁ is small, the temperature variation of P₂ is obtained to the first order approximation as

T_{2} (t) \approx \bar{T} + \frac{ξ}{\sqrt{ξ^{2} + 4 π^{2} τ_{r 1}^{2}}} \cdot Δ T_{1} \cdot sin (\frac{2 π t}{ξ} - ψ) + \frac{2 π ξ τ_{r 1}}{ξ^{2} + 4 π^{2} τ_{r 1}^{2}} \cdot Δ T_{1} \cdot e^{- \frac{t}{τ_{r 1}}},

where

ψ = arcsin (\frac{2 π τ_{r 1}}{\sqrt{ξ^{2} + 4 π^{2} τ_{r 1}^{2}}})

. As we can see from the above equation, the temperature of P₂ also fluctuates with the same period as that of P₁. And the normalized sensitivity of P₂ to the temperature fluctuation of P₁ is given by

S = \frac{ξ}{\sqrt{ξ^{2} + 4 π^{2} τ_{r 1}^{2}}} .

When the outer temperature fluctuation period ξ is much smaller compared to the thermal time constant of radiation τ_r1,

S \approx \frac{ξ}{2 π τ_{r 1}}

. In the opposite limit, if ξ ≪ τ_r1, S ≈ 1.

If there are other layers inside P₂, the thermal sensitivity of the most inside layer P_n to the temperature fluctuation of P₁ is given by

S_{n} = \prod_{i = 1}^{n} \frac{ξ}{\sqrt{ξ^{2} + 4 π^{2} τ_{r i}^{2}}} .

Figure 3(c) shows the thermal sensitivity of P₂ when using τ_r1 ≈ 6.5 h (the blue solid line) and the thermal sensitivity of P₃ with P₂ as an intermediate layer using τ_r1 ≈ 6.5 h and τ_r2 ≈ 6.6 h (the red dashed line). The green dash-dot line is the thermal sensitivity of P₃ without P₂ using τ_r1 ≈ 3.1 h. As we can see that multiple-layer thermal radiation also helps to make reference cavities less sensitive to temperature fluctuations with short fluctuation period.

4. Simulation results for less thermal sensitivity

In the above analysis, whenever heat transfers via thermal conduction or thermal radiation, cavity thermal shielding configurations with a large time constant are insensitive to temperature fluctuations. Thereby, it is significant to enlarge the time constant of reference cavities.

Considering thermal conduction in an apparatus of a reference cavity, its thermal time constant (τ_c) is proportional to thermal capacity and mass of thermal shields and reference cavities, and inversely proportional to thermal conductivity and cross section area of supporting rods. To enlarge τ_c, material with small thermal conductivity should be chosen for supporting rods, such as vacuum compatible ceramic or Teflon. The geometry of those supporting rods is usually designed to be small in radius and large in length as long as it is stable enough to support.

For thermal radiation in an apparatus of a reference cavity, its thermal time constant τ_r is also proportional to thermal capacity and mass of thermal shields and reference cavities. For those reasons, the thermal shields for reference cavities are usually have a large mass. In the case of limited weight, aluminum is a good choice since its density is nearly one-third of that of copper while its heat capacity is three times bigger if without considering emissivity. To reduce thermal radiation between thermal shields and reference cavities, the emissivity of thermal shields should be small. Even for a specified material, the emissivity varies according to its surface treatment. Usually highly polished surfaces have lower emissivity, though it is technically difficult [24].

From the estimations of time constant listed above, the time constant of thermal radiation is more than an order of magnitude smaller than that of thermal conduction. Therefore thermal conduction could be neglected when calculating the combined time constant of reference cavities. It is also true when reference cavities are designed to be even longer in length for lower thermal noise [18, 19, 26], which makes the geometry of thermal shields even bigger. Since the mass as well as the radiation areas of thermal shields increase, the time constant of thermal conduction increases more than that of thermal radiation.

In the above analysis, both time constant and thermal sensitivity of thermal radiation are given to the first order approximation. Moreover, the geometry of radiation bodies and the distance between radiation surfaces are not taken into consideration, which affect the view factor of radiation and thus time constant of reference cavities as well. Therefore, in the following section, finite element analysis is performed in order to obtain a clearer and more accurate answer on the design of cavity shielding configurations, aiming at making reference cavities less sensitive to environmental temperature perturbations.

4.1. Material for thermal shields

We use ANSYS software to numerically simulate the temperature change of a reference cavity over time. The simulation model is shown in Fig. 4. A reference cavity and thermal shields are in the shape of cylinder. The cavity is made of ULE glass, and the outer vacuum chamber is made of aluminum. Only one layer of thermal shield, layer B as shown in the figure, is used to study the effect of different materials of thermal shield on the thermal time constant. The initial temperature of the whole setup is 25 °C. At t = 0, the temperature of the outer vacuum chamber raises to 30 °C.

Fig. 4 The simulation model. The outer layer is a vacuum chamber made of aluminum, while the most inner layer is a reference cavity made of ULE glass. There are three layers of thermal shields inserted between the vacuum chamber and the reference cavity, labeled as A, B and C. All are in the shape of cylinder.

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The simulation results are shown in Fig. 5(a) and listed in Table 2. The thermal time constant of the reference cavity is 26.3 h, 23.8 h and 11.0 h when the thermal shield with the same size is made of gold, copper (roughly polished) and aluminum (roughly polished), respectively. For aluminum thermal shield, if radiation surfaces are highly polished, the emissivity could be smaller, e.g. 0.07, resulting in a thermal time constant of 26.0 h for the reference cavity. For the copper thermal shield, if it is plated by gold, the time constant increases to 32.5 h.

Table 2. The thermal time constants when the material of thermal shield varies.

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When using Eq. (10), the calculated time constants are also listed in the table, which are slightly smaller than the results based on finite element analysis since finite element analysis takes the geometry of each part, radiation angle and relative spacing of radiation surfaces into consideration. While in the calculation the thermal radiation is more efficient since the view factor is assumed to be unity. However, the calculation gives an estimation.

If there are three supporting rods made of Teflon with a diameter of 20 mm and length of 50 mm inserted between layers, a similar simulation is performed when considering both thermal radiation and thermal conduction. The combined thermal time constant of the reference cavity is 31.9 h when the thermal shield is made of gold-plated copper, as shown in Fig. 5(a). Comparing with the time constant of 32.5 h in the case that without supporting rods, it indicates that thermal conductivity plays a less significant role.

Fig. 5 The simulation results of temperature change for the reference cavity over time when the temperature of outer vacuum chamber is changed from 25 °C to 30 °C if (a) the material of one-layer thermal shield varies and (b) the layer number of thermal shields varies. (c) The simulated and calculated thermal sensitivity of a reference cavity to outer temperature fluctuation when it is enclosed in zero to two layers thermal shields. (d) The simulated time constant of a reference cavity when the aperture size of the three-layer thermal shield varies.

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4.2. Layers of thermal shield

To study the effect of multiple-layer thermal shields, we numerically simulated the temperature variation of the reference cavity when the temperature of the outer vacuum chamber changes from 25 °C to 30 °C using the same model with different layer numbers of thermal shield made of gold-plated copper. As shown in Fig. 5(b), the time constant of the reference cavity is 1.5 h, 32.5 h, 87.2 h and 153.8 h if there are zero to three layers of thermal shields, respectively. This indicates that when adding one more layer, the thermal time constant of the reference cavity increases significantly.

The thermal sensitivities of the reference cavity as a function of temperature fluctuation period are also simulated when it is enclosed in different layers of thermal shields, as shown in Fig. 5(c). For each data in the figure, the temperature of the outer vacuum chamber fluctuates at a certain period with different amplitude ΔT_out. By simulation, the temperature change of the inner reference cavity ΔT_cav is obtained separately and thus the thermal sensitivity S = ΔT_cav/ΔT_out is obtained. In the figure, the simulation results of the sensitivities for reference cavities with no thermal shield, one-layer thermal shield and two-layer thermal shield are shown with dots, squares and triangles respectively, while the corresponding calculations results based on Eq. (10) are shown with dashed lines, which agree with the simulation results quite well.

From the above analysis, we can see that when using multiple-layer thermal shield the thermal time constant of reference cavities gets larger and reference cavities become less sensitive to environmental temperature fluctuations. However, it is not necessary to use multiple-layer thermal shields in each case. In Table 3, the temperature fluctuation-induced fractional length instability of reference cavities (ΔL/L) are estimated according to achievable outer temperature instability and the CTE of reference cavities. For example, if an outer vacuum chamber is temperature-stabilized with instability within 1 mK (24 hour fluctuation period) near T_CTE=0, which might have a CTE of 1× 10⁻¹⁰/K, then the reference cavity can have a length instability at the 10⁻¹⁶ level on hundreds of seconds’ timescale even without any thermal shield, assuming the thermal noise limit of the reference cavity is below 1 × 10⁻¹⁶. If a reference cavity can not be conveniently controlled near T_CTE=0, for example, which has a CTE of 5 × 10⁻⁹/K (for ULE glass, about 30 °C above T_CTE=0), in order to achieve 10⁻¹⁶ length instability in an averaging time above 100 s, it should be enclosed in thermal shields. If the temperature variation rate of the outer vacuum chamber is large, the need for multiple-layer thermal shield becomes significant. Moreover, multiple-layer thermal shield is also important to achieve a high frequency stability on the timescale over a day. For example, as shown in Table 3, if the environmental temperature fluctuates with amplitude of 1 mK and period of 24 hours, a reference cavity with a CTE of 1 × 10⁻¹⁰/K is enclosed in a two-layer thermal shield, the temperature-induced length instability of the reference cavity can be at the 10⁻¹⁶ level over one day.

Table 3. Estimation of temperature fluctuation-induced fractional length instability based on outer temperature variation rate, the CTE and the sensitivity of a reference cavity.

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4.3. Aperture size of thermal shield

In the previous simulations, the thermal shields do not have any holes to let laser light access the reference cavities. However, all designs are going to need an aperture. To study the effect of the aperture size in the thermal shield on the thermal time constant, we numerically analyze the thermal time constants of reference cavities when the thermal shields having optical apertures with different diameters. In the case of one-layer thermal shield, layer B, the time constant decreases from 31.6 h to 30.0 h (changes by 5%) if the diameter of aperture in the thermal shield increases from 5 mm to 20 mm. If the reference cavity is enclosed in three-layer gold-plated copper thermal shields that have optical apertures d of 5 mm to 20 mm, the time constant decreases from 151 h to 135 h (changes by 12%), as shown in Fig. 5(d). This implies that the aperture size of thermal shields has a relatively weak effect on the thermal time constant compared to the material and layer numbers of thermal shields.

5. Conclusion

In this paper, thermal time constant and temperature sensitivity of reference cavities are studied in order to describe how reference cavities respond to environmental temperature fluctuations via heat transfer of thermal conduction and thermal radiation. The thermal sensitivity of reference cavities is proportional to fluctuation periods of temperature perturbations, and inversely proportional to the time constant of reference cavities. Both calculations based on equations and numerical simulations indicate that thermal radiation plays a significant role in enlarging the thermal time constant of reference cavities. To make reference cavities less sensitive to temperature perturbations, they might be enclosed in multiple layers of thermal shields with large mass, high thermal capacity and small emissivity. One choice for the material of thermal shields is copper with low emissivity. The aperture size of thermal shields has a weak effect on the thermal time constant of reference cavities. The layer number of thermal shields can be chosen according to experimentally achievable temperature stability and the CTE of reference cavities. By using multiple-layer thermal shield, the temperature fluctuation-induced fractional length instability at the 10⁻¹⁶ level on a day timescale could be achieved.

Acknowledgments

The authors would like to thank Chad Hoyt and Wei Zhang for helpful discussions. This work was supported by the National Natural Science Foundation of China (grant Nos. 11374102, 11104077, 11334002, 11127405 and 11475063) and the National Basic Research Program of China (grant No. 2012CB821302).

References and links

1. J. L. Hall, “Nobel lecture: defining and measuring optical frequencies,” Rev. Mod. Phys. 78(4), 1279–1295 (2006). [CrossRef]

2. T. W. Hansch, “Nobel lecture: passion for precision,” Rev. Mod. Phys. 78(4), 1297–1309 (2006). [CrossRef]

3. N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, “An atomic clock with 10⁻¹⁸ instability,” Science 341(6151), 1215–1218 (2013). [CrossRef] [PubMed]

4. B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10⁻¹⁸ level,” Nature 506(7486), 71–75 (2014). [CrossRef] [PubMed]

5. T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011). [CrossRef]

6. T. M. Fortier, N. Ashby, J. C. Bergquist, M. J. Delaney, S. A. Diddams, T. P. Heavner, L. Hollberg, W. M. Itano, S. R. Jefferts, K. Kim, F. Levi, L. Lorini, W. H. Oskay, T. E. Parker, J. Shirley, and J. E. Stalnaker, “Precision atomic spectroscopy for improved limits on variation of the fine structure constant and local position invariance,” Phys. Rev. Lett. 98(7), 070801 (2007). [CrossRef] [PubMed]

7. S. G. Turyshev, “Experimental tests of general relativity: recent progress and future directions,” Phys. Uspekhi 52(1), 1–27 (2009). [CrossRef]

8. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983). [CrossRef]

9. A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye, “Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1 × 10⁻¹⁵,” Opt. Lett. 32(6), 641–643 (2007). [CrossRef] [PubMed]

10. H. Chen, Y. Jiang, S. Fang, Z. Bi, and L. Ma, “Frequency stabilization of Nd:YAG lasers with a most probable linewidth of 0.6 Hz,” J. Opt. Soc. Am. B 30(6), 1546–1550 (2013). [CrossRef]

11. B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82(19), 3799–3802 (1999). [CrossRef]

12. T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6(10), 687–692 (2012). [CrossRef]

13. L. Chen, J. L. Hall, J. Ye, T. Yang, E. Zang, and T. Li, “Vibration-induced elastic deformation of Fabry-Perot cavities,” Phys. Rev. A 74(5), 053801 (2006). [CrossRef]

14. H. Stoehr, F. Mensing, J. Helmcke, and U. Sterr, “Diode laser with 1 Hz linewidth,” Opt. Lett. 31(6), 736–738 (2006). [CrossRef] [PubMed]

15. D. R. Leibrandt, M. J. Thorpe, M. Notcutt, R. E. Drullinger, T. Rosenband, and J. C. Bergquist, “Spherical reference cavities for frequency stabilization of lasers in non-laboratory environments,” Opt. Express 19(4), 3471–3482 (2011). [CrossRef] [PubMed]

16. Y. N. Zhao, J. Zhang, A. Stejskal, T. Liu, V. Elman, Z. H. Lu, and L. J. Wang, “A vibration-insensitive optical cavity and absolute determination of its ultrahigh stability,” Opt. Express 17(11), 8970–8982 (2009). [CrossRef] [PubMed]

17. B. Argence, E. Prevost, T. Leveque, R. Le Goff, S. Bize, P. Lemonde, and G. Santarelli, “Prototype of an ultra-stable optical cavity for space applications,” Opt. Express 20(23), 25409–25420 (2012) [CrossRef] [PubMed]

18. Y. Y. Jiang, A. D. Ludlow, N. D. Lemke, R. W. Fox, J. A. Sherman, L. S. Ma, and C. W. Oates, “Making optical atomic clocks more stable with 10⁻¹⁶-level laser stabilization,” Nat. Photonics 5(3), 158–161 (2011). [CrossRef]

19. T. L. Nicholson, M. J. Martin, J. R. Williams, B. J. Bloom, M. Bishof, M. D. Swallows, S. L. Campbell, and J. Ye, “Comparison of two independent Sr optical clocks with 1 × 10⁻¹⁷ stability at 10³ s,” Phys. Rev. Lett. 109(23), 230801 (2012). [CrossRef]

20. K. Numata, A. Kemery, and J. Camp, “Thermal-noise limit in the frequency stabilization of lasers with rigid cavities,” Phys. Rev. Lett. 93, 250602 (2004). [CrossRef]

21. P. Dube, A. A. Madej, J. E. Bernard, L. Marmet, and A. D. Shiner, “A narrow linewidth and frequency-stable probe laser source for the ⁸⁸Sr⁺ single ion optical frequency standard,” Appl. Phys. B 95(1), 43–54 (2009). [CrossRef]

22. J. Alnis, A. Matveev, N. Kolachevsky, Th. Udem, and T. W. Hansch, “Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry-Perot cavities,” Phys. Rev. A 77, 053809 (2008). [CrossRef]

23. C. Hagemann, C. Grebing, C. Lisdat, S. Falke, T. Legero, U. Sterr, F. Riehle, M. J. Martin, and J. Ye, “Ultrastable laser with average fractional frequency drift rate below 5 × 10⁻¹⁹/s,” Opt. Lett. 39(17), 5102–5105 (2014). [CrossRef] [PubMed]

24. “Table of total emissivity,” www.omega.com/temperature/Z/pdf/z088-089.pdf.

25. J. G. Knudsen, H. C. Hottel, A. F. Sarofim, P. C. Wankat, and K. S. Knaebel, “Heat and mass transfer,” in Perry’s Chemical Engineers’ Handbook, R. H. Perry and D. W. Green, eds. 8th ed. (Academic, 2007), pp. 25–32.

26. M. Notcutt, L. S. Ma, A. D. Ludlow, S. M. Foreman, J. Ye, and J. L. Hall, “Contribution of thermal noise to frequency stability of rigid optical cavity via hertz-linewidth lasers,” Phys. Rev. A 73, 031804(R) (2006). [CrossRef]

Material	ρ (kg/m³)	k (W/(m·K))	C (J/(kg·K))	ε^a
ULE glass	2210	1.31	767	0.85
Copper	8930	398	385	0.1
Aluminum	2700	210	900	0.2
Teflon	2200	0.25	1400	0.85
Gold	19320	301	128	0.07

Material of thermal shield	Time constant (hour)
Material of thermal shield	Simulation	Calculation
Gold	26.3	19.9 (τ_r1 = 15.8 h and τ_r2 = 3.6 h)
Copper	23.8	19.2 (τ_r1 = 16.1 h and τ_r2 = 2.8 h)
Aluminum	11.0	8.7 (τ_r1 = 6.5 h and τ_r2 = 1.9 h)

ΔT_out/ξ	dT_out/dt (K/s)	CTE (1/K)	$S = \frac{Δ T_{cav}}{Δ T_{out}}$	ΔL/L
ΔT_out/ξ	dT_out/dt (K/s)	CTE (1/K)	$S = \frac{Δ T_{cav}}{Δ T_{out}}$	1 s	100 s	1 day
1 mK/24 hour	7.3 × 10⁻⁸	1 × 10⁻¹⁰	0.8 ^a	6 × 10⁻¹⁸	6 × 10⁻¹⁶	8 × 10⁻¹⁴
			0.07 ^b	5 × 10⁻¹⁹	5 × 10⁻¹⁷	7 × 10⁻¹⁵
			0.006 ^c	4 × 10⁻²⁰	4 × 10⁻¹⁸	6 × 10⁻¹⁶
1 mK/24 hour	7.3 × 10⁻⁸	5 × 10⁻⁹	0.8 ^a	3 × 10⁻¹⁶	3 × 10⁻¹⁴	4 × 10⁻¹²
			0.07 ^b	3 × 10⁻¹⁷	3 × 10⁻¹⁵	4 × 10⁻¹³
			0.006 ^c	2 × 10⁻¹⁸	2 × 10⁻¹⁶	3 × 10⁻¹⁴
1 mK/1 hour	1.7 × 10⁻⁶	1 × 10⁻¹⁰	6 × 10⁻² ^a	1 × 10⁻¹⁷	1 × 10⁻¹⁵	6 × 10⁻¹⁵
			3 × 10⁻⁴ ^b	5 × 10⁻²⁰	5 × 10⁻¹⁸	3 × 10⁻¹⁷
			3 × 10⁻⁶ ^c	5 × 10⁻²²	5 × 10⁻²⁰	3 × 10⁻¹⁹
1 mK/1 hour	1.7 × 10⁻⁶	5 × 10⁻⁹	6 × 10⁻² ^a	5 × 10⁻¹⁶	5 × 10⁻¹⁴	3 × 10⁻¹³
			3 × 10⁻⁴ ^b	3 × 10⁻¹⁸	3 × 10⁻¹⁶	2 × 10⁻¹⁵
			3 × 10⁻⁶ ^c	3 × 10⁻²⁰	3 × 10⁻¹⁸	2 × 10⁻¹⁷

Material	ρ (kg/m³)	k (W/(m·K))	C (J/(kg·K))	ε^a
ULE glass	2210	1.31	767	0.85
Copper	8930	398	385	0.1
Aluminum	2700	210	900	0.2
Teflon	2200	0.25	1400	0.85
Gold	19320	301	128	0.07

Material of thermal shield	Time constant (hour)
Material of thermal shield	Simulation	Calculation
Gold	26.3	19.9 (τ_r1 = 15.8 h and τ_r2 = 3.6 h)
Copper	23.8	19.2 (τ_r1 = 16.1 h and τ_r2 = 2.8 h)
Aluminum	11.0	8.7 (τ_r1 = 6.5 h and τ_r2 = 1.9 h)

Thermal analysis of optical reference cavities for low sensitivity to environmental temperature fluctuations

Abstract

1. Introduction

2. Time constant

2.1. Thermal conduction

2.2. Thermal radiation

3. Thermal sensitivity

3.1. Thermal conduction

3.2. Thermal radiation

4. Simulation results for less thermal sensitivity

4.1. Material for thermal shields

4.2. Layers of thermal shield

4.3. Aperture size of thermal shield

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Tables (3)

Equations (18)

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