Low-noise, narrow-linewidth lasers are highly valued for their wide range of applications, including high-precision measurement, atomic clocks, coherent optical communication, and so on [1–3]. To achieve ultra-low laser noise, a stable and high-precision frequency reference is often required. Common methods include using ultra-high-finesse Fabry–Perot (FP) cavities [4–6] and fine absorption lines of atoms or molecules [7–9]. The state of the art for such low-noise lasers has demonstrated laser linewidths as low as 10 mHz and frequency instabilities down to 4×10−17 [10]. However, these precise frequency reference systems require extremely stable operating conditions and vacuum cryogenic environments, which limit their application outside laboratory settings. Some low-cost frequency references, such as fiber interferometers, are also used to achieve low-noise lasers [11,12]. However, due to their long fiber delay lines being sensitive to environmental disturbances, complex and large volume packaging systems are necessary to meet the practical requirements for low laser noise.

Whispering-gallery-mode (WGM) cavities have demonstrated significant advantages in achieving low-noise lasers with their high Q-factor, small volume, and monolithic structure [13]. These benefits make WGM cavities an excellent choice for frequency references, enabling compact and simplified low-noise laser systems [14–19]. WGM cavities made from fluoride crystals, such as MgF2 and CaF2, exhibit even higher Q-factors (>109), lower material thermal noise, and a wider transparency range (350–7000 nm) [20–22]. These characteristics provide a promising solution for achieving movable, miniaturized, low-noise laser systems [23,24]. On the other hand, laser frequency locking systems often require active electronic devices to convert optical signals into electrical error signals that can be used for laser locking. The Pound-Drever-Hall (PDH) technique is the most commonly used method [25]. It can “differentiate” the Lorentzian transmission spectrum of the WGM cavity, thereby obtaining a transmission that is linearly dependent on the frequency at resonance. The PDH technique is a well-established method for laser frequency stabilization, which involves phase modulation, demodulation, and mixing to generate an error signal for feedback control. However, the PDH technique not only introduces electronic noise from the modulators and demodulator electronics, but also increases the overall power consumption and complexity, which contradicts the motivation of using WGM cavities to achieve a compact low-noise laser.

Recently, “all-optical” error signals generated through the interference between optical modes have been proposed for laser frequency locking, eliminating the need for modulation [26,27]. Xu et al. showed the modulation-free laser frequency locking based on the Fano resonance arising from the interference between radiative modes and WGMs within a cylindrical microcavity [26]. However, radiative mode generation efficiency is generally low, requiring ultra-thin tapered fibers for excitation. This setup not only makes the tapered fibers fragile and sensitive to environmental disturbances but also places the WGM in an over-coupled state, reducing the Q-factor [28]. In another work [27], Idjadi et al. demonstrated modulation-free laser frequency locking on an on-chip silicon-on-insulator (SOI) waveguide by utilizing interference between the light from a Mach-Zehnder interferometer and a microring resonator to generate frequency linearly dependent transmission for laser locking. This method simplifies the locking system and provides a promising approach for achieving fully on-chip low-noise lasers. While monolithic integration on a chip and reducing the optical path length can mitigate environmental impacts, these solutions inevitably increase inherent thermal noise. Furthermore, due to the structural and material limitations of the WGM cavity in the above two works, it is not possible to suppress the laser noise to below 1Hz2/Hz. In particular, the thermal noise of the microcavity significantly increases the low-frequency noise (<103Hz) of the laser.

In this paper, while ensuring the small size and compactness of the WGM cavity, we utilize a MgF2 crystalline WGM cavity with excellent thermal properties to achieve a modulation-free laser frequency locking system. We generate Fano resonance modes through the interference between the background light totally internally reflected by a prism and the high-Q WGMs from the crystalline cavity. This method can produce a high-slope, easily controllable error signal. Our experimental results demonstrate laser frequency noise below 1Hz2/Hz in the Fourier frequency range of 103–105Hz, with a minimum noise level of 0.2Hz2/Hz at 104Hz. This performance is more than two orders of magnitude lower than previous results. Typically, modulation-free laser frequency locking schemes require balanced detection to eliminate the impact of laser intensity fluctuations on the error signal. However, our results indicate that using WGMs with higher Q-factors helps mitigate this effect, thereby further simplifying the system. The performance of this locking method is comparable to that of traditional PDH techniques, but it is more compact and has lower power consumption.

As shown in Fig. 1, the incident light is focused by a lens onto the interface of a prism, where total internal reflection occurs. The focused beam can be considered as a composition of rays propagating in various directions (with different wave vectors) indicated by the arrows in the left inset. Only the rays whose wave vectors match the cavity modes can effectively excite the whispering-gallery modes (WGMs) within the cavity. We assume that this matching ray is RayB, with its wave vector kB approximately satisfying the relation kB·cosθ=kC, where kC represents the wave vector of the cavity mode, and θ denotes the incident angle of RayB. The remaining rays can collectively be referred to as RayA, which are directly subjected to total internal reflection by the prism. These two types of rays, RayA and RayB, will interfere in the far field, such as at a photodetector.

From the perspective of Fano resonance, RayB represents the WGM modes with discrete energy spectra (Lorentzian transmission spectrum), and RayA represents the continuous energy spectra. The interference between these two spectra will produce a Fano transmission spectrum. As shown in the right-hand inset in Fig. 1, the line shape of the Fano transmission spectrum is linearly dependent on the frequency near the cavity resonance. If external noise perturbs the relative frequency between the laser and the cavity resonance, the Fano effect can convert this disturbance directly into transmission intensity fluctuations. Thus, it can serve as an error signal similar to that in classical PDH locking techniques. Furthermore, for high-Q WGM cavities, the resulting Fano transmission spectrum offers high frequency discrimination accuracy, enabling precise laser frequency locking.

The interference intensity of the fields of RayA and RayB in the far field can be expressed as I=|EA|2+|Γ|2|EB|2+2|EA||Γ||EB|cosΔφ.(1)

Here, EA and EB represent the amplitudes of RayA and RayB, respectively. Δϕ=Δϕpath−ϕΓ represents the phase difference between the two interfering rays. For this interference process, it is crucial to account not only for the phase difference arising from their different optical paths Δϕpath, but also for an additional phase shift ϕΓ induced by the cavity for RayB. Similar to the add-drop coupling system between a WGM cavity and a waveguide, the complex reflectivity of the WGM cavity for RayB can be expressed as Γ(ω)=r−aeiϕ(ω)1−r·aeiϕ(ω),(2)where r is the coupling coefficient of RayB to the crystal mode, which can be controlled experimentally by adjusting the relative position of the prism and the WGM cavity, as well as the incident beam spot. ϕ(ω) and 1−a represent the roundtrip phase shift and loss of the cavity mode, respectively. The phase shift ϕ(ω)=ω·neff·L/c, where ω is the laser frequency, neff is the effective refractive index of the cavity mode, and L is the roundtrip length. Therefore, the phase angle of the complex reflectivity Γ(ω) is the phase shift: ϕΓ=angle[Γ(ω)]. Define the amplitude ratio of RayA to RayB as R≡|EA/EB|, and the normalized laser frequency as ωN≡ω/ΔωFSR=ϕΓ/2π, where ΔωFSR=(2πc/neff)·L represents the frequency range of one free spectral range (FSR) of the cavity. Thus, the intensity in Eq. (1) can be normalized to the incident light intensity as follows: IN=R2+|Γ(ω)|2+2R|Γ(ω)|cos(Δϕpath−2πωN)1+R2.(3)

According to Eqs. (2) and (3), we calculate the Fano spectrum for different values of R and Δϕpath, as shown in Fig. 2. Figures 2(a)–2(d) show the Fano spectra for Δϕpath=0, 0.5π, π, 1.5π, respectively, corresponding to the q parameter of 0, −1, ∞, and 1 in the classical Fano resonance theory. Clearly, the Fano spectra in Figs. 2(b) and 2(d) can be used as error signals for laser frequency locking. Taking Fig. 2(b) as an example, as R increases, the amplitude of the Fano spectrum decreases, which in turn reduces the slope of the error signal at the locking point. This decrease in slope implies lower frequency discrimination accuracy for the laser, which is undesirable.

On the other hand, when the R value is relatively small (less than one), it is easy to find another point with the same intensity near the locking point, as shown by the two blue points on the curve for R=0.5 in Fig. 2(b). When the laser is locked, we use intensity as the sole criterion for laser frequency error. In this situation, it is easy for the locking point to switch between these two points, resulting in frequent loss of locking. Therefore, there exists an optimal parameter range for R and Δϕpath that achieves the desired Fano spectrum for laser frequency locking. In experiments, we aim to achieve a Fano spectrum in Fig. 2(b) where the R value is in the range of two to three.

In the experiment, a WGM cavity made of MgF2 crystal was used, as shown in Fig. 3. The diameter of the cavity is approximately 5 mm, and it was fabricated through mechanical cutting and precision optical polishing. The crystalline WGM cavity has a surface roughness on the nanometer scale and features a narrowed side profile on the rim, which ensures an extremely high Q-factor. The curvature radius of the narrowed rim is about 100 μm, which, compared to other types of WGM cavities, can support a large mode area for the cavity modes. Figure 3(f) shows the calculated field distribution of the fundamental mode using the finite element method. The mode area is estimated to be ∼200μm2. The large mode area of the WGM, combined with the excellent thermal properties of the MgF2 crystal, allows this cavity to achieve low inherent thermal noise. This ensures that the laser, once locked, can exhibit an ultra-low frequency noise.

Figure 3(a) depicts the experimental setup used for achieving Fano laser frequency locking. The output from a commercial external cavity diode laser (Toptica, CTL1550) is split into two parts by a fiber coupler. One part is coupled into the crystalline cavity through a prism to obtain the Fano spectrum, which serves as the error signal. This signal is fed back to the original laser via the PID servo circuits to control the laser frequency. We employ a dual servo control loop: one loop adjusts the laser current to achieve rapid laser frequency tuning, while the other loop uses a piezoelectric transducer (PZT) to control the laser cavity length, eliminating long-term laser frequency drift.

For the coupling between the prism and the crystalline cavity used in our experiment [Fig. 3(c)], the laser output from a single-mode fiber (SMF) with a quartz end cap is focused onto the total internal reflection interface of the prism by a graded-index (GRIN) lens. The fiber end cap and GRIN lens are encased together in a glass tube. By adjusting the distance between the end cap and the GRIN lens, the focus size at the coupling point of the crystalline cavity can be controlled. This configuration eliminates beam misalignment caused by environmental disturbances to the optical elements in the path, providing excellent stability. Consequently, it could ensure the reliability of subsequent laser frequency locking experiments. The light collection system also employs the same configuration. To improve collection efficiency, the receiving fiber can be replaced with a high numerical aperture multimode fiber (MMF). In this case, the interference effect described in Eq. (3) will occur at the photodetector following the MMF. The analysis of the impact of the MMF on laser noise is discussed in detail in the appendices.

In the experiment, we first excited only the WGMs of the crystalline cavity. By adjusting the incident angle of the laser beam and the air gap between the prism and the cavity, we can control the WGM to approach the critical coupling condition. At this point, the measured mode linewidth was approximately 0.24 MHz, as shown in Fig. 4(a), corresponding to a Q-factor of ∼0.8×109 at a wavelength of 1550 nm. Next, by finely adjusting the positions and angles of the GRIN lenses at both the incident and receiving fiber ends, we transformed the transmission spectrum with a Lorentzian profile into a Fano resonance spectrum, as shown in Fig. 4(b). In our experiments, the Fano resonance arises from the interference between the WGM and the background light in continuum at the end face of the receiving fiber (here we use an SMF to collect the laser beam). By adjusting the position and angle of the GRIN lens, we effectively adjust R and Δϕpath in Eq. (3) to obtain the Fano transmission spectrum. The intensity in Fig. 4(b) has been normalized to the incident laser intensity. By performing a linear fit to the spectrum near the resonance (locking point), we can determine the normalized frequency discrimination accuracy, KD=1.23×10−6Hz−1. It should be pointed out that the maximum amplitude of the Fano spectrum obtained experimentally is around 0.3, which is smaller than the results obtained from numerical calculations [see Fig. 2(b)]. This is because, in the experiment, we cannot independently adjust R and Δϕpath. However, the existing low-noise, high-gain photodetectors are sufficient to convert this signal into an adequate voltage amplitude to meet the requirements of the electronic frequency locking circuit.

We used a self-heterodyne method based on an unbalanced Michelson interferometer to measure the frequency noise of the laser after locking, as shown in Fig. 3(b). One arm of the interferometer includes a 100 m optical fiber delay line, while the other arm is connected to a fiber stretcher to apply a carrier modulation of several hundred Hz to the beat signal. Both arms are terminated with Faraday rotators to eliminate the impact of polarization fluctuations on the interference signal. The entire interferometer is also well isolated from vibrations and temperature fluctuations, and we use low-noise photodetectors and data acquisition cards to minimize the influence of electronic noise on the measurement system.

To enhance the stability of the coupling system between the prism and the crystalline cavity, we encapsulated the system as shown in Fig. 3(g). Each optical component is fixed onto a specially designed aluminum substrate using low thermal expansion glue. This structure allows precise micro-adjustment of the coupling gap between the prism and the crystalline cavity. The entire system is enclosed in an aluminum box and employs a thermoelectric cooler (TEC) for precise temperature control, effectively isolating it from environmental mechanical and thermal fluctuations. Figure 5 shows the measured power spectral density (PSD) of the laser frequency noise based on Fano laser locking. Using the packaged crystalline cavity as the frequency reference, the laser frequency was locked using both the Fano resonance spectrum and the classical PDH technique. The resulting frequency noise of the locked laser is shown by the blue and red curves in Fig. 5, respectively. Both locking schemes achieved laser noise levels close to the thermorefractive noise (TRN) limit of the crystalline cavity (red dashed line), demonstrating the capacity of the Fano resonance laser locking scheme proposed in this paper. Compared to the free-running laser noise (black curve), the frequency noise was reduced by more than 50 dB at the maximum, with the lowest frequency noise of approximately 0.2Hz2/Hz at an offset frequency of 10 kHz. Multiple spurious peaks in the low-frequency range (<1kHz) are due to environmental vibrations and electronic noise in the locking system. In the high-frequency range from 1 kHz to1 MHz, the noise floor of the laser is limited by the electronic noise in the PID servo control loop. Using the β-line method, we estimated the integrated linewidth of the laser before and after locking to be 62.4 kHz and 89 Hz (0.1 s integration time), respectively, achieving a reduction of over 700 times.

The key of our Fano laser locking technique is the generation of an error signal analogous to that in the classical PDH technique through optical interference. This “all-optical” error signal eliminates certain electronic noise sources, such as residual modulation noise introduced by electro-optic modulators. However, it is also essential to discuss the unique optical noise inherent in this laser locking system. As shown in Fig. 6, we present the contributions of various optical noise sources to the frequency noise of the locked laser. The mathematical derivations of these noise contributions are provided in Appendix C.

First, we find that, the laser frequency noise is still limited by the inherent TRN of the crystalline cavity. This implies that employing larger mode areas and larger-sized cavities could help further reduce laser noise. Theoretically, the WGM mode field area of the crystalline cavity increases with its polar quantum number. However, considering the Q-factor of the WGM and the coupling conditions, we can only excite a few low-order WGMs in the experiment. Figure 7 shows that the laser frequency noise at a 10 Hz Fourier frequency decreases with the increase in the WGM mode field area. It also presents the field distributions and corresponding frequency noise of several low-order WGMs that can be excited in the experiment. The second source of frequency noise is due to laser intensity fluctuations. Since the error signal in the Fano locking method is derived directly from the detected transmission intensity, intensity noise of the laser will directly translate into fluctuations in the error signal. According to Eq. (D1) in Appendix D, we can reduce this type of noise by increasing the frequency discrimination accuracy KD. In our laser locking system, the contribution of intensity noise is still an order of magnitude lower than the experimentally measured frequency noise. Even if it surpasses the thermal noise of the crystalline cavity, we can mitigate the impact of intensity fluctuations on the error signal through balanced detection techniques. In the experiment, we used an MMF to achieve higher efficiency in light collection, as shown by the red fiber in the encapsulation structure in Fig. 3(g). In this configuration, the interference between the WGM and the background light in the continuum occurs at the photodetector at the end of the fiber. Consequently, the difference in propagation constants between these two modes within the multimode fiber leads to a Δϕpath that is influenced by the noise of the fiber itself. Numerical estimates indicate that this type of noise contributes much less to the laser frequency noise compared to other types and can therefore be neglected. The final type of noise, arising from the thermal expansion of the MgF2 crystal, is also negligible.

In conclusion, we utilized a coupling system between a prism and a crystalline WGM cavity to obtain a Fano transmission spectrum, which served as the error signal for laser frequency locking. This approach successfully reduced the laser frequency noise to a level close to the thermal noise limit of the crystalline cavity. Due to the high Q-factor and excellent thermo-mechanical properties of the crystalline cavity, it allows us to reduce the laser frequency noise to below 1Hz2/Hz within the Fourier frequency range of 103 to 105Hz, achieving a minimum noise level of 0.2Hz2/Hz. To evaluate the potential performance of this method, we also discussed the contributions of unique optical noise sources in the Fano locking technique to the frequency noise of the locked laser. The experimental results revealed that the primary limiting factor for the locked laser noise is still the inherent thermal noise of the crystalline cavity. It demonstrates that this method achieves the same level of laser noise suppression as the classical PDH technique, with noise levels more than two orders of magnitude lower than previously reported modulation-free locking methods, which is shown in Table 1. Therefore, the proposed Fano locking technique has significant potential to simplify laser locking systems, enhance stability, and reduce overall power consumption and cost.Table 1.

Comparison of Laser Locking System Based on WGM Cavities