Ultra-stable narrow-linewidth lasers are essential in many applications, such as the optical atomic clock, high-resolution laser spectroscopy, tests of fundamental physics, and gravitational wave detection^{[13" target="_self" style="display: inline;">–3]}. Most of those lasers are achieved by actively stabilizing the laser frequency to the resonance of a Fabry–Perot (F-P) cavity using the Pound-Drever-Hall (PDH) technique^[4]. State-of-the-art laser systems have been developed with a linewidth below 1 Hz and fractional frequency instabilities at the 10−16 level or better at a 1 s averaging time^{[5–8" target="_self" style="display: inline;">–8]}.

In those cavity-stabilized laser systems, the laser frequency stability is mostly limited by the length stability of the reference cavities. A special design of optical cavities as well as their supporting configurations has been performed to achieve a low sensitivity to thermal fluctuation and vibration^{[810" target="_self" style="display: inline;">–10]}, approaching a thermal-noise-limited performance^[11].

In a cavity-stabilized laser system, the power of light incident onto a cavity is stabilized since the power fluctuation of cavity resonant light induces temperature fluctuation and thus the cavity length fluctuation is due to thermal expansion (TE) and the thermo-refractive (TR) effect. Fractional power instability of 1×10−4 can be achieved after stabilization^[12]. Usually, the power-dependent frequency shift of a cavity-stabilized laser is measured to be a few tens of hertz/microwatt (Hz/μW)^[12], corresponding to a cavity length sensitivity at the 10−14m/μW level. For a 10-cm-long cavity made of ultra-low expansion (ULE) glass, the light power fluctuation-induced fractional length instability is nearly 10−16, while the thermal-noise-limited length instability of the cavity at room temperature is approximately 8×10−16. In the pursuit of a laser frequency instability of 10−17 level or better, it is critical to reduce the cavity length instability resulting from light power fluctuation. In this Letter, we study the sensitivity of the cavity length to light power fluctuation (S), providing a method to evaluate the sensitivity S for cavity-stabilized laser systems, as well as providing a way to reduce S by designing an optical cavity with a particular size and material.

We use finite element analysis (FEA) to analyze the length deviation of an F-P cavity when the power of incident light varies. A portion of incident light is absorbed by mirror coatings, resulting in a temperature change. By performing a thermal-mechanical analysis, the displacement between the cavity mirrors due to TE (ΔLTE) is obtained. Meanwhile, the effective optical length change due to the TR effect (ΔLTR) is considered into the total cavity length change by evaluating the temperature variation based on thermal analysis. The length sensitivity of a 7.75-cm-long football cavity to the power fluctuation of incident light is simulated to be S=5×10−14/μW. It is in agreement with the experimental results, which are obtained by measuring the frequency change of a cavity-stabilized laser when the cavity incident light power is changed. Based on the FEA, we find that a long cavity made of materials with high thermal conductivity and a small coefficient of TE (CTE) has a low sensitivity S.

Consider an F-P cavity with two highly reflective mirrors optically contacted to a spacer. To achieve a reflectivity of 99.99%, the mirror substrates are usually covered by dielectric coatings, such as multiple layers of SiO2 and Ta2O5. Usually the coatings have a CTE of at the 10−5/K level. Compared with mirror substrates and a cavity spacer that made of fused silica (FS) or ULE glass whose CTE is on the order of 1×10−7/K or less, the deformation of the coatings is unneglectable. Therefore, mirror coatings must be taken into account in the simulation. However, we simplify the N alternating sequences of quarter-wavelength layers into one layer with thickness (d), an averaging density (ρ), Young’s modulus (E), and Poisson’s ratio (σ), which are listed in Table 1.

According to Refs. [19,22], the effective CTE (α), capacity (C), and thermal conductivity (κ) of the coating with N layers each of thickness dj are αceff=∑j=1Nαjdjd1+σs1−σj[1+σj1+σs+(1−2σs)EjEs],(1)Cceff=∑j=1NCjdjd,(2)κceff=(∑j=1N1κjdjd)−1,(3)where the subscripts j, s, and c stand for the coating layer j, the substrate, and the coating. The number of layer pairs of dielectric mirror coating for a reflectivity of 99.995% is typically N=19, while the number for a crystalline coating is typically N=40.5^[21]. For the dielectric coating at room temperature, Cceff=566J·(kg·K)−1, and κceff=2.27W·(m·K)−1, while at the cryogenic, Cceff=2J·(kg·K)−1, and κceff=0.1W·(m·K)−1. The αceff used throughout this Letter are listed in Table 2.

When a single frequency laser light is on resonance of a F-P cavity, the total loss is Pl=Pin−Pr−Pt,(4)where Pin, Pr, and Pt are the cavity input and the reflected and transmitted light power. Since the input laser light is phase modulated for frequency stabilization^[4], we assume the modulation index is nearly one, and thus 60% of Pin remains in the carrier and couples into the cavity, while the sidebands are nearly reflected from the cavity. Usually 90% of the light can be coupled to the TEM00 mode of the cavity, and what is left is coupled to higher order modes. Moreover, we assume the ratio of mirror absorption to scattering is about 1∶4^[22]. Hence each mirror absorbs a total power of 0.054×Pl.

In the simulation, on each mirror, a heat flow of 0.054×Pl is applied to an area in the coating center. We apply a step function (six steps within 1.52×ω0, where ω0 is the light beam radius) to simulate the Gaussian distribution of the light power, as shown in Fig. 1(a). The error due to the non-Gaussian distribution of the light power is within 5%. To obtain a steady solution, thermal radiation to the surroundings is imposed at the outside surface of the cavity with thermal emissivity (ε). The emissivity of the surroundings is assumed to be 0.1 for all the following simulations. Taking advantage of the geometrical symmetry of the cavities, the nodes in the symmetrical plane of the cavity are constrained within the symmetrical plane. A thermal-mechanical analysis is performed to investigate the distance displacement between the probe points along the cavity’s optical axis, and a thermal analysis is performed to obtain the temperature change of the probe points when the incident light power changes. The probe points are distributed within ω0. The cavity length change is obtained by averaging those probe points.

Fig. 1. (a) Diagram of a high-reflective mirror. It is coated at the center with a diameter of 10 mm. The red spot at the coating center represents a light spot to which an absorbed light power is applied. Schematic diagram of (b) a football cavity and (c) a cylinder cavity.

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We use this method to simulate the sensitivity S of our football cavity and compare it with the measurement.

The cavity structure is symmetrically wider in the middle and tapered at the ends, as shown in Fig. 1(b). The cavity spacer is 7.75-cm-long, and two mirrors are optically connected to the cavity spacer. Both the cavity spacer and the mirror substrates are made of ULE glass. The mirrors are coated with high-reflective films of SiO2 and Ta2O5 at 1064 nm.

Experimentally, we measured the sensitivity S by monitoring the frequency change of the beat note between two cavity-stabilized laser systems when the incident light power of one cavity was changed. The experimental diagram is shown in Fig. 2. Using the PDH technique, the frequency of each laser is stabilized to the resonance of its cavity by feeding back the error signal to a piezo transducer (PZT) for frequency adjustment. When the incident light power of Laser #1 was changed from 6.5 to 8.5 μW, the frequency change of the beat note monitored by a counter with a gate time of 1 s is 50 Hz, as shown in the inset of Fig. 2. It takes about 2 s for the cavity length to resettle to the new input light power. At the time of ∼130s, the incident light power of Laser #1 was changed back to 6.5 μW in order to cancel the residual drift of the cavity resonant frequencies on the measurement. The sensitivity S of this cavity to the power change of the incident light is measured to be 8.8×10−14/μW.

Fig. 2. Experimental setup for the measurement of the cavity sensitivity S. EOM, electro-optic modulator; PBS, polarization beam splitter; λ∕4, quarter-wave plate; PD, photo detector; RF, radio frequency; DBM, double balance mixer.

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We measured the cavity loss by measuring Pr and Pt to be 11.7 and 3.7 μW, accordingly, on the photo detectors (PD1 and PD2) when light with power Pin=17.7μW is incident and on resonance of the cavity. Thus, the fractional loss of the cavity is 13%×Pin.

In the simulation, d=6μm, and ω0=250μm. When Pin is changed from 6.5 to 8.5 μW, the cavity deforms due to the change of light power absorption. Figure 3(a) shows the cavity deformations due to TE. As we can see, only the points where heat flow is applied have an observable displacement. By averaging the probe points within ω0, ΔLTE=7.9×10−15m. It is worth noting that in the simulation ΔLTE includes the deformation of both cavity spacer and mirrors, while in the measurement the deformation of the cavity spacer is not measured since it takes a few hours for the cavity spacer to heat up and expand due to its large mass. By simulation, we calculated the length change of the cavity spacer to be on the order of 1×10−17m, which is negligible.

Fig. 3. (a) The thermally induced deformations of the cavity. (b) The temperature change of the probe points inside ω0 (blue open squares) and those on the mirror contacting surface (pink dots) over time.

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For another, when Pin is changed from 6.5 to 8.5 μW, the temperature of the coating also changed accordingly, which is ΔT=6μK based on the steady-static thermal analysis of the FEA. Figure 3(b) shows the temperature change of the probe points inside ω0 (blue open squares) based on the transient thermal analysis. It takes 2 s for the cavity to almost adjust to the new incident light power, which agrees with the temporal behavior shown in the inset of Fig. 2. On the contrary, it takes a longer time for the probe points on the mirror contacting surface of the cavity spacer due to its bigger mass.

The effective TR coefficient of the coating is^[22]βceff=BH+BL4(nH2−nL2),(5)where Bx=βx+αxnx(1+σx)/(1−σx) and subscript x=H, L stands for the high and low index material of the coating. βx and nx are the coefficients of thermo-refraction and index of the coating layer x. The βceff of various coatings on different substrates are listed in Table 2. Therefore, ΔLTR=−2βceffλ×ΔT,(6)where λ is the light wavelength. For the football cavity, ΔLTR=6×10−17m, which is two orders of magnitude smaller than ΔLTE.

As a result, the total cavity length change is ΔL=7.9×10−15m, corresponding to a fractional cavity length sensitivity S=ΔL/(L×ΔPin)=5×10−14/μW. The discrepancy between the measurement and the simulation may arise from an inaccurate estimate of the mirror absorption loss and the CTE of the coating. For example, if the mirror absorption loss equals the mirror scattering, the sensitivity S will be 1.3×10−13/μW.

Next, we study the sensitivity S by varying the cavity size and material when Pin changes from 10 to 11 μW. To simplify the cavity structure, in the following simulations the cavity is in the shape of a cylinder with a length of L=10cm, a diameter of D=10cm, and an axial hole of 10 mm for light access, as shown in Fig. 1(c). Two mirrors with a diameter of 25.4 mm and a thickness of 6.35 mm, made of ULE glass and coated with SiO2/Ta2O5, are bonded to the ULE spacer. Pl=13%×Pin, d=6μm, and ω0=250μm. For this cavity, S0 is simulated to be 3.2×10−14/μW. Then, the cavity size, CTE, Young’s modulus, thermal conductivity, and Poisson’s ratio of the cavity (including spacer and mirror substrate) and coating double separately. Based on the FEA, we obtain S in each case, where ΔLTR is also much smaller than ΔLTE. Figure 4(a) shows the relative sensitivity S/S0.

Fig. 4. Simulation results of S/S0 (a) when the parameters of the cavity and coating are doubled and (b) when the CTE (blue open squares, upper axis) or the thermal conductivity (black filled triangles, lower axis) of the cavity and mirror substrate changes.

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The sensitivity S is inversely proportional to L. Since the deformations of the cavity mirrors are almost the same, while increasing L, the fractional length and S reduces accordingly. The sensitivity S has a relatively weak correlation to D.

The CTE of the mirror coating has a great effect on S. S nearly doubles as αceff changes from −6.5×10−5 to −1.3×10−4/K. Therefore, it is important to use accurate CTE values of the coating. However, to date the CTE of Ta2O5 varies, i.e., 3.6×10−6 or −4.4×10−5/K^[14,15], and its CTE at a cryogenic temperature was deduced to be 5.8×10−7/K by only one group^[20]. The uncertainty of the coating CTE gives the biggest contribution to the simulation error. On the contrary, the CTE of the cavity has less of an effect on S. However, S increases nonlinearly when increasing the CTE of the cavity, as shown with blue open triangles in Fig. 4(b). Interestingly, S reduces when increasing E of the cavity, while it increases when increasing that of the coating. This situation happens when changing σ. It is due to the negative αceff. The thermal conductivity of the cavity also has a great effect on S, while that of the coatings has little effect on S. However, when κ of the cavity spacer and mirror substrate is larger than 50, S is less sensitive to it, as shown with black filled triangles in Fig. 4(b). When increasing κ, the temperature becomes more uniform, which makes less strain inside the cavity and coating, resulting in less of a change of the cavity length.

Next, we change the materials of the cavity, mirror substrate, and coating to explore the sensitivity S based on the FEA. The simulation results are listed in Table 3. The FS and ULE are usually used in room temperature cavities. While reducing the temperature of the cavities for less thermal noise, silicon and sapphire are employed since they have a relatively small CTE at low temperatures^[7,17,23,24]. The crystalline coating of GaAs/Al0.92Ga0.08As is also used for lower cavity thermal noise due to its lower mechanical loss compared to the dielectric coating of SiO2/Ta2O5^[21,22]. In the table, the corresponding fractional cavity length instabilities are listed when the incident light power is 10 μW with a fractional instability of 1×10−4.

As we can see in the table, for a dielectric coating at room temperature, ΔLTR is much smaller than ΔLTE. The sensitivity of an all-ULE cavity is smaller than that of an all-FS cavity since the CTE of ULE is three orders of magnitude smaller than that of FS. The sensitivity of a cavity with a ULE spacer and FS mirror substrates is about the same as that of an all-ULE cavity. The sensitivity S of the silicon cavity at 1.6 K is 1×10−17/μW, about the same as that at 124 K due to the smaller κ and CTE. For a 25-cm-long silicon cavity at 1.6 K, the sensitivity S is measured to be 3×10−14/μW in Ref. [15], while in our simulation for the same cavity, it is 4×10−18/μW. The biggest difference may arise from the CTE value of Ta2O5 at a cryogenic temperature. If we assume the CTE of Ta2O5 at 1.6 K is −2×10−4/K, the simulated sensitivity of the silicon cavity agrees with the measured result. The sensitivity for the sapphire cavity at 5 K is slightly smaller than the silicon cavity at 1.6 K due to its bigger E and κ. When the dielectric coating is replaced with the crystalline coating of GaAs/Al0.92Ga0.08As, the sensitivity S is reduced to 6×10−15/μW, which results from the smaller αceff and bigger κ of the crystalline coating.

In this Letter, we use FEA to simulate the sensitivity of cavity length to light power fluctuation. The simulated sensitivity of a 7.75-cm-long football cavity is 5×10−14/μW, which is in agreement with the experimental results. Based on the simulation, a cavity with a longer length is found to have less sensitivity. When the cavity is made of materials with higher thermal conductivity and Young’s modulus or a smaller CTE and Poisson’s ratio, it is less sensitive to light power fluctuation.