Fig. 1. Schematic for generation of Fano spectrum and laser frequency locking. RayA and RayB interfere to produce the Fano spectrum. The left inset shows a schematic of the coupling between the prism and the crystalline cavity. A light ray with a wave vector kB satisfies kB·cos θ=kC, which can excite the WGM with a wave vector kC. The right inset depicts the Fano spectrum used as the error signal for laser frequency locking. Frequency fluctuations caused by external disturbances (horizontal direction) will be converted into intensity fluctuations (vertical direction). The conversion ratio depends on the frequency discrimination accuracy, which is related to the Q-factor of the WGM cavity and the photoelectric conversion efficiency.

Fig. 2. Fano spectra for different values of R and Δϕpath. (a)–(d) correspond to Δϕpath=0, 0.5π, π, 1.5π, respectively. The Fano spectra in (b) and (d) have transmission intensities linearly dependent on the frequency, which can be used as the error signal for laser frequency locking.

Fig. 3. (a) Experimental setup for Fano laser frequency locking. (b) Self-heterodyne frequency noise measurement system. (c) Prism coupled crystalline cavity system utilizes integrated fiber focusing systems for both inputting and receiving the laser beams. (d)–(f) Pictures of the MgF2 crystalline WGM cavity and the calculated mode field inside the cavity. The scale bars are 1 mm, 50 μm, and 5 μm, respectively. (g) Picture of the packaged system for the cavity coupled by the prism.

Fig. 4. (a) Lorentzian lineshape of the transmission spectrum of a WGM in MgF2 crystalline cavity. (b) Corresponding Fano transmission spectrum used for laser frequency locking.

Fig. 5. Power spectral density of the laser frequency noise.

Fig. 6. Contribution of different noise sources to the laser frequency noise after locking.

Fig. 7. Laser frequency noise floor at 10 Hz Fourier frequency, limited by the TRN of the crystalline cavity, is related to the WGM mode field area. The inset shows the field distributions of WGM modes of different polar orders for 1, 2, and 5.

Fig. 8. Field distributions for WGMs of the crystalline cavity. The red and blue curves represent the light intensity distribution along the radial and azimuthal directions, respectively, and the horizontal dashed lines represent the 1/e2 reference lines. (a)–(f) correspond to the quantum numbers (p,l)=(1,1), (1, 2), (2, 1), (3, 1), (4, 1), (5, 1) of the WGMs; p and l indicate the radial and azimuthal quantum numbers, respectively. All modes have an azimuthal quantum number of m=13,985.