Optomechanical probes of resonances in amplifying microresonators

We investigate whether the force and torque exerted by light pressure on an irregularly shaped dielectric resonator allow to detect resonant frequencies, delivering information complemental to the scattering cross section by mechanical means. The peak-to-valley ratio in the torque signal can be many times larger than in the scattering cross section, and, furthermore, depends on the structure of the resonance wave pattern. The far-field emission pattern of the associated quasibound states can be tested by the angular dependence of the mechanical mechanical probes at finite amplification rate. We relate the force and torque to the scattering matrix and present numerical results for an annularly shaped dielectric resonator.

Figure 1

The annular resonator used in the numerical investigations of this paper is composed of two circles with radii $R$ and $R^{\prime }=0.6R$, and eccentricity $\delta =0.22R$. The refractive index is $n=1$ outside the resonator, $n=1.8$ in the annular region between the circles, and $n=3.3$ inside the interior circle.

Figure 2

(Color) The top panels show four quantities (in arbitrary units) as a function of $k$ that probe for resonances in the annular resonator: The Wigner delay time ${\tau }_W$, the angle-of-incidence averaged scattering cross-section ${\sigma }_0$, the angle-of-incidence averaged force in forward direction $F_0$, and the variance of the torque $\mathrm{var}N$. The lower panel shows the complex resonance wave numbers $k_c$ of quasibound states, classified by their wave patterns as described in the text: $W_{\mathrm{int}}$ (엯), $W_{\mathrm{ext}}$ (▵), $A$ (◻), $C$ (◇). Open symbols indicate modes of even parity, full symbols are modes of odd parity. The complex resonances are also indicated by the spikes in the top panels, located at $k=\mathrm{Re}{k}_c$ with height proportional to the life time $1∕(-2c\mathrm{Im}{k}_c)$. Additionally, the resonances of Fig. 3a, 3b, 3c, 3d, 3e are indicated by the arrows at the very top.

Figure 3

(Color) The panels (a–e) show results for various combinations of $k$ and $\mathrm{Im}n$ tuned very close to resonance with the quasibound states of odd parity shown on the right, with (a) $k_c=6.009-i0.008$ (class $W_{\mathrm{int}}$), (b) $k_c=6.228-i0.116$ (class $C$), (c) $k_c=7.554-i0.063$ (class $W_{\mathrm{ext}}$), (d) $k_c=7.847-i0.145$ (class $C$), (e) $k_c=9.026-i0.022$ (class $A$). In the top graph of each panel, the angular dependence of the far field $I_c$ (red) on the radiation direction $\varphi$ is compared to the angular dependence on the illumination direction ${\varphi }_0$ (note Ref. 28) of the scattering cross-section $\sigma$ (green), the weighted delay time $\tau$ (blue), and the $x$ component of the force, $F_x$ (purple)—the four lines are almost indistinguishable. In the bottom graph of each panel, the (squared) torque $N^2$ (red) is compared to the interference term $I_c^{\prime 2}$ (green), defined in Eq. (25). The blue curve is $N^2$ evaluated at real $n$. All quantities are in arbitrary units.