Optical resonators with ten million finesse (2024)

We report a significant improvement of the finesse and quality factor of a calcium fluoride whispering gallery mode optical resonator achieved with iterative thermal annealing. The initial and final values of the resonator finesse are ℱ_i ≈ 5 × 10⁵ and ℱ_f ≥ 10⁷ respectively. The measurements are conducted at room temperature in the laboratory atmosphere.

1. Introduction

Resonators are usually characterized with two partially dependent parameters – finesse (ℱ) and quality factor (Q). The finesse of an empty Fabry-Perot (FP) resonator is defined solely by the reflectivity of its mirrors and is calculated as ℱ = π√R/(1 - R). The maximum reported value of reflectivity R≃ 1 - 1.6 × 10^-6 is achieved with dielectric mirrors [1,2]. A FP resonator made with the mirrors has finesse ℱ = 1.9 × 10⁶. Further practical increase of the finesse of FP resonators is problematic because of the absorption and scattering of light in the mirror material [3, 4], though fundamental limit on the reflection losses given by the internal material losses and by thermodynamic density fluctuations is of the order of parts in 10⁹ [3]. Quality factor of a resonator depends on both its finesse and its geometrical size. The one-dimensional FP resonator has Q = 2ℱL/λ, where L is the distance between the mirrors, and λ is the wavelength. It is easy to see that the quality factor of the resonator is essentially unlimited because L is unlimited.

ℱ and Q are typically equally important in the majority of applications. In some cases, though, finesse is technically more valuable than the quality factor. For instance, the buildup of optical power inside the resonator and the Purcell factor [5] are proportional to finesse. Sometimes quality factor is more valuable. For example, the inverse threshold power of intracavity hyperparametric oscillation is proportional to Q² [6], and efficiency of parametric frequency mixing is proportional to Q³ [7]. Therefore, it is important to know both the maximally achievable finesse and quality factor values of a resonator.

Whispering gallery mode (WGM) resonators allow achieving larger finesse compared with FP resonators. For instance, fused silica resonators with finesse 2.3 × 10⁶ [8] and 2.8 × 10⁶ [9] have been demonstrated. Crystalline WGM resonators reveal even larger finesse values, ℱ = 6.3 × 10⁶ [10], because of low attenuation of light in the transparent optical crystals. The large values of ℱ and Q result in the enhancement of various nonlinear processes. Low threshold Raman lasing [11, 12], opto-mechanical oscillations [13], frequency doubling [7], and hyper-parametric oscillations [14,15] based on these resonators have been recently demonstrated. Theory predicts the possibility of achieving nearly 10¹⁴ for Q-factors of optical crystalline WGM resonators at room temperature [7,10], which correspond to a finesse level higher than 10⁹. Experiments, thus far, have shown numbers that are a thousand times lower. The difference between the theoretical prediction and the experimental values is due to media imperfections. To bridge this gap, a technique to substantially reduce the optical losses caused by the imperfections is reported in this paper. We utilize specific multi-step asymptotic processing of the resonator. The technique has been initially developed to reduce microwave absorption in dielectric resonators [16]. One step of the process consists of mechanical polishing performed after high temperature annealing. Several repeated subsequent steps lead to a significant reduction of the optical attenuation and, as a result, to the increase of Q-factor as well as finesse of the resonator. We demonstrate a CaF₂ WGM resonator with ℱ > 10⁷ and Q > 10¹¹.

2. Measurement technique

There are several practical issues related to the study of properties of ultra-high-Q resonators. The study is challenging because the spectral width of the corresponding WGM is expected to be less than a kHz. Direct observation of such a narrow line could be done with a narrowband tunable laser with a high level of frequency stability. Such lasers are not available in many laboratories. Another problem is related to the parasitic contributions from nonlinear effects. Because the threshold of stimulated Raman scattering (SRS) is very low in high-Q resonators [12] the direct observation of the intrinsic spectral width of a WGM with a continuous wave laser radiation is a problem. Thermal oscillations and drifts of the position of the optical resonance also make a time-averaged observation problematic. The recently demonstrated nonlinear ring-down technique [17] helps us avoid these limitations. We sweep the laser frequency rapidly across the resonance line. A part of the light is accumulated in the mode during the sweep and is reemitted in the direction of the detector. The reemitted radiation interferes with the laser light, and beatnote signals are subsequently observed. The power of the laser light is generally much larger than the power of the reemitted light. Hence, the envelop of the decaying oscillations follows the decay of the amplitude of the reemitted light. This decay is twice as long as the intensity decay. Optical Q-factor of the cavity is expressed as Q = ωτ/2.

3. Annealing procedure

Fig. 1. Left: Ring-down signal after one annealing step. Right: Ring-down signal after three annealing steps. The exponential fit (red solid curve) is nonlinear in the logarithmic scale because the exponents have constant offsets. Keeping in mind that Q = ωτ/2, and ω = 2πc/λ, λ = 1.55 μm, we find that the values of the quality factors after the first and third annealing steps are Q = 1.2 × 10¹⁰ and Q = 6.7 × 10¹⁰ respectively. It is important to note that the initial value of the Q-factor corresponds to the earlier observations [10], while the final value is the apparent improvement at the given wavelength.

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We selected a fluorite WGM resonator with optical loss limited by the material attenuation, not by the surface scattering [18]. The resonator had 4.5 mm in diameter, 0.5 mm in thickness, and 32 μm in diameter of sidewall curvature. The resonator was placed into the center of a 3-feet long air-filled transparent tube made of annealed fused silica. The tube was installed into a 20 cm long horizontal tube furnace. The heated furnace core had approximately one inch in diameter and three inches in length. We increased the temperature of the furnace core from room temperature to 650 C during 3 hours, kept the temperature stabilized for one day, cooled the core back during 3 hours, and repolished the fluorite resonator. We repeated iterations three times keeping the same annealing duration but gradually decreasing the size of the grain of diamond slurry we used for polishing. We obtained a significant increase of the ringdown time at the end of the process.

4. Results

The measured ringdown spectrum did not change substantially after the first annealing stage. However, the ringdown time increased significantly after the third stage. The measured ring-down signals are shown in Fig. 1. A five-fold increase of the optical ringdown time is clearly observed. It is also worth to note that the measured quality factor is several times larger compared with the quality factor of calcium fluoride resonators observed previously at 1.55 μm (see [18, 10]).

The power in the mode was clearly above the threshold of the nonlinear loss related to SRS [17]. The measurement method we used has a low dynamic range, so we were unable to see the nonlinear decay looking on the high-amplitude signals. For that reason we have measured the ringdown time using a smaller optical power. In such a measurement two major factors were taken into account: (i) the laser wavelength should interact with only one mode, which means that no WGM doublets are allowed within the investigated frequency span. (ii) the sweep of the frequency of the laser should be controlled and the local frequency of the beatnote signal should never increase; such an increase can occur if the laser is swept back to the starting frequency rapidly. We insure the single excitation of a single WGM in this way.

The best ringdown signal measured with all the precautions is shown in Fig. 2(left). To ensure that the measured line is not a result of several consecutive excitations of the optical mode because of dithering of the laser carrier frequency we have evaluated the oscillation period of the beatnote (Fig. 2(right)). This shows that the carrier frequency indeed moves gradually from the resonance and the WGM has not been excited twice.

Fig. 2. Left: Ringdown signal after the third annealing step taken with low-power laser radiation (blue solid line). The theoretical fit of the signal is shown by the red dotted line. Right: Evaluated change of the period of the signal. The solid line stands for the linear fit of the time dependence of the period of the ringdown signal.

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Let us discuss the fitting procedure to the theoretical curve in more detail. To evaluate the oscillation period we have determined the positions of the maxima and minima of the beatnote peaks as well as the zero crossings of the beatnote signal. We subtracted the time coordinate of each peak from the coordinate of the adjacent peak and obtained the averaged time for the two adjacent peaks. The same procedure has been repeated for the zero-crossing points. In this way we deduced the time dependence of the period of the beat note signal, shown in picture Fig. 2(right). The dependence is linear at the tail of the curve. The initial period has a different time dependence because the frequency of the WGM changes much faster immediately after the frequency of the pump laser is tuned away from the WGM. The change is determined by multiple nonlinear processes, e.g. WGM frequency shift due to after-interaction cooling of the resonator.

Using the linear approximation for the beatnote period we found the period of the waveform to fit the experimental data. We solved the equation ϕ(t + Period(t)/2) - ϕ(t) = π approximating it by a linear differential equation ϕ(t) = 2π/Period(t). using the experimental results we obtain Period(t) = ξ - ζt, where ξ =71.1 μs, and ζ = 0.11 (time t is measured in microseconds). As a result, we have the expression

(1) $ϕ (t) = Const - (\frac{2 π}{ζ}) \ln (1 - \frac{ζt}{ξ})$

for the phase. The final oscillation waveform presented in Fig. 2(right) is given by cos[ϕ(t)].

The time dependence of the beatnote amplitude has been extracted in the following way. We subtracted the amplitude of a minimum of the oscillating beatnote signal from the adjacent maximum. The value shows the relative oscillation amplitude. We related this value to the moment of time equally separated from the maximum and the minimum time coordinates. The result of the evaluation is shown in Fig. 3. Using the first three points of the dependence we have found initial decay time τ₁ = 130 μs. Using the part of the signal in the interval between 0.15 ms and 0.35 ms we have found the final decay time, τ₂ = 510 μs. Finally, we have used the expression for the nonlinear decay rate discussed in [19] to fit the amplitude decay rate. The resultant fit of the beatnote waveform is shown in Fig. 2(left). We can conclude that the intrinsic linear quality factor of the CaF₂ WGM resonator approaches 3 × 10¹¹ at 1.55 μm (twenty fold improvement compared with the initial quality factor).

Fig. 3. Change of the signal amplitude shown in Fig. 2(left) with time. Initial and final quality factors are Q = (7.9±0.5) × 10¹⁰ and Q = (3±1) × 10¹¹ respectively. Solid line stands for the theoretical fit of the ringdown signal found using formalism presented in [19]. A small oscillation of the decaying signal may result from the residual scattering in the material becoming observable at the given value of quality factor.

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Let us find the finesse of such a resonator. The expression for the finesse as well as the quality factor of WGM resonators naturally includes index of refraction of the resonator host material, unlike to those of an empty FP resonator: Q = 2πaℱn₀/λ, and ℱ = c/[2an₀(γ₀ + γ_c)], where a is the radius of the resonator, n₀ is the index of refraction of the resonator host material, γ₀ and γ_c are the intrinsic and coupling amplitude decay rates, respectively. The values of the light intensity inside I_in and outside I₀ the resonator are related as I_in/I₀ = 2γcℱ[π(γ₀ + γ_c)]. Using our experimental data (2a = 0.45 cm, n₀ = 1.42) we find that the finesse of the resonator is ℱ = (2.1 ± 0.6) × 10⁷.

5. Discussion

The annealing process discussed above improves the transparency of the material because an increased temperature results in the enhancement of the mobility of defects induced by the fabrication process, and also reduces any residual stress birefringence [20]. The increased mobility leads to the recombination of defects and their migration to the surface [16]. It is worth noting that our annealing technique is similar to the previously developed procedures [21, 22].

The significant improvement of the Q-factor and finesse of a fluorite WGM resonator demonstrated here does not reach the fundamental limit. The straightforward annealing of a WGM resonator leads to Q > 10¹¹ at 1.55 μm while our earlier theoretical prediction gives Q ≃ 10¹³ at this wavelength [18]. To improve the quality of the annealing we suggest using a larger furnace with much lower thermal gradients. Thermal gradients within a sample play the leading role in the defect redistribution [20]. The amplitude of the gradient can be reduced not only by increasing the volume of the oven but also by increasing thermal conductivity and optimizing shape of the container the resonator is placed into during the annealing procedure. A fluorite container is the best choice for annealing fluorite WGM resonators.

6. Conclusion

We have demonstrated that proper thermal annealing procedure combined with mechanical polishing allows fabricating whispering gallery mode resonators with vastly improved (record) finesse and quality factors. As the result, we demonstrate a CaF₂ WGM resonator with ℱ > 10⁷ and Q > 10¹¹ at 1.55 μm. We expect that described here technique will allow improving quality factors and finesses of WGM resonators made of other transparent crystalline materials.

Acknowledgement

The research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration and sponsorship from DARPA. A. Matsko appreciates useful discussions with D. Strekalov.

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