Analytical study of birefringent cavities for axion-like dark matter search

TL;DR

This work develops a nonperturbative Jones-calculus framework to quantify mirror-birefringence effects in ring cavities used for axion-like particle (ALP) searches. It derives how birefringence splits resonance peaks and modifies intracavity polarization, and extends the model to include ALP-induced sidebands, providing expressions for signal power and SNR. The study finds that birefringence degrades low-mass sensitivity but can be mitigated by postselection, while ALP-induced resonances can enhance high-mass sensitivity; it also proposes hardware strategies, including a 3D cavity design, to suppress birefringence. Overall, the results guide the design and operation of high-finesse cavities for ALP detection, balancing polarization control with practical mitigation techniques.

Abstract

Light polarization plays a crucial role in optical-cavity experiments; however, mirror birefringence presents a significant challenge that must be addressed carefully. In this study, a rigorous, nonperturbative framework is developed to quantify birefringence effects by incorporating variations in reflectance and polarization misalignment. We analyze the impact of this framework on the sensitivity of axion-like particle (ALP) dark-matter searches. The results show that both birefringence and misalignment contribute to sensitivity degradation in the low-mass regime; however, the adverse effects of misalignment can be mitigated by selecting a postselection angle greater than the misalignment angle. Furthermore, birefringence produces an additional resonance peak in the high-mass region, which remains largely unaffected by misalignment and postselection variations. This rigorous framework underscores the importance of considering birefringence in high-precision optical-cavity experiments for ALP detection.

Figures (8)

• Figure 1: (a) Schematic of the experimental setup considered in this work. $\omega_0$, angular frequency of laser light; $P$, laser power; $|H\rangle$, horizontal linear polarization; $R_{1,2}~(T_{1,2})$, Jones matrices of reflection (transmission) representing the two successive mirrors; $L$, cavity length; $\epsilon$, rotation angle of linear polarization; PBS, polarizing beam splitter; HWP, half-wave plate; PD, photodetector. (b) Schematic of the theoretical setup considered in this work. The two successive mirrors at the shorter sides of the ring cavity are regarded as a single birefringent mirror. (c) Model of a birefringent mirror. $r_{1,2}$, (geometric) average of reflectances along the fast and slow axes of the waveplate. (d) Waveplates characterized by retardation $\alpha_{1,2}$, and angle between the $|H\rangle$ polarization and its fast axes $\theta_{1,2}$.
• Figure 2: Population transfer with the real retardation $\alpha_1$ and angle of birefringence axis $\theta_1$. $P_f~(P'_f)$ is the laser power at the detection (reference) port, in which the polarization is orthogonal (parallel) to the polarization of light just before entering the cavity. The minimum value $\sin^2{\epsilon}\simeq\epsilon^2$ is due to the postselection. Common parameters are $\mathcal{F}=10^5~(r_1=r_2)$, $L=10.64~\mathrm{m}$, $\lambda=1064~\mathrm{nm}$, $\alpha_2=\alpha_1$, $\theta_2=\theta_1$, and $\epsilon=0.01$. The resonance condition in (\ref{['eq: resonance condition theta_1,2=0']}) holds.
• Figure 3: Reflectance of the cavity with the real retardation $\alpha_1$ and angle of birefringence axis $\theta_1$. $P$ is the laser power just before entering the cavity. $P_f~(P'_f)$ is the laser power at the detection (reference) port, in which the polarization is orthogonal (parallel) to the polarization of light before entering the cavity. Common parameters are $\mathcal{F}=10^5~(r_1=r_2)$, $L=10.64~\mathrm{m}$, $\lambda=1064~\mathrm{nm}$, $\alpha_2=\alpha_1$, $\theta_2=\theta_1$, and $\epsilon=0.01$. The resonance condition in (\ref{['eq: resonance condition theta_1,2=0']}) holds.
• Figure 4: Sensitivity curves with (a) real and (b) imaginary retardation. The black lines are the cases without birefringence. We set $\theta_1=\theta_2=0$ so that the fast and slow axes of the waveplate coincide with polarizations of the carrier and signal light, respectively. The laser frequency is fixed at the resonance frequency of the carrier light, which means that the birefringence with real retardation effectively shifts the resonance curves for the sideband light. Degradations in (a) occur in the low-mass region because the resonance frequency is shifted from on-resonance to off-resonance by the birefringence. Dips in (a) are located at $m_aL=\mathrm{Re}[\alpha_1]$, where both carrier and signal light resonate in the cavity. In (b), the imaginary part of $\alpha_1$ is limited by (\ref{['eq: Im alpha limitation approx']}) because the actual reflectance $r_1e^{\pm\mathrm{Im}[\alpha_1]}$ is smaller than $1$. A nonzero $\mathrm{Im}[\alpha_1]$ splits reflectance, and, hence, the finesse is also split; a positive (negative) $\mathrm{Im}[\alpha_1]$ enlarges the finesse of carrier (signal) light as in (\ref{['eq: actual finesses']}). Common parameters are $\mathcal{F}=10^5~(r_1=r_2)$, $L=10.64~\mathrm{m}$, $\lambda=1064~\mathrm{nm}$, $P=1~\mathrm{W}$, $T_\mathrm{obs}=1~\mathrm{yr}$, $\alpha_2=\alpha_1$, $\theta_1=\theta_2=0$, and $\epsilon=0.01~\mathrm{rad}$. The resonance condition in (\ref{['eq: resonance condition theta_1,2=0']}) holds.
• Figure 5: Sensitivity curves with finite values of $\theta_2$. The light green line is the same line as that shown in Fig. \ref{['fig: sensitivity optimistic']}(a) in which birefringence is relatively small and the sidebands are almost on resonance in the low-mass region. Degradation is enhanced when the ALP mass is lower than $\pi/2\mathcal{F}L$, where the resonance of signal light in the cavity is relevant. For finite $\theta_1$ and/or $\theta_2$, birefringence modifies linear polarizations to slightly elliptic ones. Roughly speaking, a finite $\theta_{1,2}$ makes the carrier and signal light flow to the detection and reference ports, respectively. Common parameters are $\mathcal{F}=10^5~(r_1=r_2)$, $L=10.64~\mathrm{m}$, $\lambda=1064~\mathrm{nm}$, $P=1~\mathrm{W}$, $T_\mathrm{obs}=1~\mathrm{yr}$, $\alpha_1=\alpha_2=10^{-5}~\mathrm{rad}$, and $\theta_1=0$. The resonance condition in (\ref{['eq: resonance condition theta_1,2=0']}) holds.