Dynamical determination of the strength of cross-polarization coupling in a whispering-gallery microresonator

Cross-polarization coupling (CPC) between coresonant orthogonally polarized modes in a single whispering-gallery microresonator can lead to induced transparency effects. Depending on the CPC strength, coupled-mode-induced transparency (CMIT) or Autler-Townes splitting (ATS) can be observed. Determining the CPC strength is crucial in locating the exceptional point, where CMIT evolves into ATS and an enhancement in sensitivity to perturbations is expected. The CPC strength can be calculated theoretically based on the transverse structure of whispering-gallery modes. Experimentally, however, CPC strength has been inferred only indirectly by fitting the experimental throughput spectrum to a steady-state model. In this work, we propose and demonstrate a direct determination of the CPC strength based on the response of the throughput to a sinusoidal amplitude modulation of the input light. Our experimental results agree with numerical fitting values and are in the same range as predicted by theoretical calculations.

Figure 1

Ring cavity representing tapered-fiber coupling to a microresonator with intracavity cross-polarization coupling. This figure is reproduced from Ref. [22].

Figure 2

Experimental setup for dynamical determination of the CPC strength in a single microresonator. This figure is reproduced from Ref. [22].

Figure 3

A hollow-bottle resonator (HBR) obtained by manual compression of air inside the capillary. Initial diameter of the capillary is about 350 μm and the diameter at the bulge is about 500 μm. The wall thickness is about 5 μm after etching.

Figure 4

Modulation response on the fast detector channel (upper, blue trace) and on the slow detector channel (lower, yellow trace) at a modulation frequency of 1 MHz. As we change the modulation frequency, there is an obvious minimum in the amplitude on resonance (at the central peak) in the fast detector response while the slow detector response (output polarization orthogonal to input) is unaffected.

Figure 5

CMIT (TE input) with 200-μm-radius HBR. (a) Model throughput spectrum (blue solid line) fit to the experimental throughput (gray trace), giving $T_c=1.99\times {10}^{–8}$. This figure [panel (a)] is reproduced from Ref. [21]. (b) Model-predicted resonant throughput modulation (blue, amplitude of 0.40) relative to input modulation (gray, amplitude of 1.0), corresponding to the fitted $T_c$ in (a). Mode parameters: $Q_1=9.6\times {10}^6, Q_2=5.2\times {10}^7$; $M_1=0.70$ (undercoupled), $M_2=0.65$ (undercoupled); detuning = 2.3 MHz. An experimental minimum relative modulation amplitude of 0.40 was found (green dashed line) at 4.0 MHz using the AOM, giving $T_c=2.04\times {10}^{–8}$.

Figure 6

ATS (TE input) with 180-μm-radius HBR. (a) Model throughput spectrum (blue solid line) fit to the experimental throughput (gray trace), giving $T_c=3.98\times {10}^{–8}$. (b) Model-predicted resonant throughput modulation (blue, amplitude of 0.19) relative to input modulation (gray, amplitude of 1.0), corresponding to the fitted $T_c$ in (a). Mode parameters: $Q_1=8.0\times {10}^6, Q_2=2.3\times {10}^7$; $M_1=0.93$ (undercoupled), $M_2=0.72$ (undercoupled); detuning = $–2.0\mathrm{MHz}$. An experimental minimum relative modulation amplitude of 0.15 was found (green dashed line) at 6.7 MHz using the AOM, giving $T_c=4.65\times {10}^{–8}$.

Figure 7

Comparison of two methods of measuring $T_c$; data from Table 1. If the two methods gave exactly the same value, the point for each case would fall on the solid line. A point lying between the dashed lines indicates agreement of the two methods to within experimental uncertainty. Agreement is also indicated by the intersection of a point's error ellipse with the solid line.