Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities

Recently, it has been shown that spiral-shaped microdisk cavities support highly nonorthogonal pairs of copropagating modes with a preferred sense of rotation (spatial chirality) [J. Wiersig et al., Phys. Rev. A 78, 053809 (2008)]. Here, we provide numerical evidence which indicates that such pairs are a common feature of deformed microdisk cavities which lack mirror symmetries. In particular, we demonstrate that discontinuities of the cavity boundary such as the notch in the spiral cavity are not needed. We find a quantitative relation between the nonorthogonality and the chirality of the modes which agrees well with the predictions from an effective non-Hermitian Hamiltonian. A comparison to ray-tracing simulations is given.

Figure 1

The left panel shows the boundary parametrization $\rho \left(\phi \right)$ in Eq. (2) with ${\varepsilon }_1=0.1$ and ${\varepsilon }_2=0.075$, for different values of $\delta$. The curves are shifted vertically for better comparison. The right panel shows a top view of the cavity with $\delta =\pi \frac{\sqrt{5}-1}{2}$.

Figure 2

Position of dimensionless complex resonance frequencies with TM polarization for the cavity in Eq. (2) with refractive index $n=3.3$ and shape parameters ${\varepsilon }_1=0.1$, ${\varepsilon }_2=0.075$, and $\delta =\pi \frac{\sqrt{5}-1}{2}$. Open circles (crosses) mark the slightly higher-$Q$ (lower-$Q$) mode of a given pair of modes.

Figure 3

Intensity ${\left|\psi \right|}^2$ of the nearly degenerate pair of modes in the asymmetric limaçon with (a) ${\Omega }_1=12.319807-i0.00089$ and (b) ${\Omega }_2=12.319851-i0.0009$; cf. Fig. 2. The same color map has been used in panels (a) and (b). Panels (c) and (d) show the corresponding exterior mode pattern at some small distance away from the cavity (white region). The far-field patterns are shown in Fig. 4.

Figure 4

Far-field patterns of the modes in Fig. 3. The solid and dashed lines correspond to the modes in Figs. 3a and 3b, respectively. For the definition of the far-field angle $\phi$ see Fig. 1.

Figure 5

Angular momentum distributions ${\alpha }_m^{\left(1\right)}$ (black solid line) and ${\alpha }_m^{\left(2\right)}$ (green dashed) of the modes in Fig. 3 (real part normalized to 1 at maximum): (a) absolute value squared, (b) real and (c) imaginary parts, and (d) superpositions ${\alpha }_m^{+}=({\alpha }_m^{\left(1\right)}+{\alpha }_m^{\left(2\right)})/2$ (black solid) and ${\alpha }_m^{-}=({\alpha }_m^{\left(1\right)}-{\alpha }_m^{\left(2\right)})/2$ (red dashed, multiplied by a factor of 6).

Figure 6

Time evolution of CCW (solid lines) and CW (dashed) components in semilogarithmic scale. The traveling waves are superpositions of the modes with frequencies ${\Omega }_1$ and ${\Omega }_2$ depicted in Fig. 3. The upper (lower) panel contains the dynamics starting with a pure CCW (CW) traveling wave. Time is measured in units of $T_1=2\pi /\text{Re}{\Omega }_1$.

Figure 7

Upper panel: individual decay rates $-\text{Im}\left({\Omega }_i\right)$ (solid and dashed line) and the level splitting $\Delta \Omega =|\text{Re}\left({\Omega }_1\right)-\text{Re}\left({\Omega }_2\right)|$ (dotted line, scaled by a factor of 10) of the pair of modes in Fig. 3 vs shape parameter $\delta$. Lower panel: corresponding chirality $\alpha$ as a function of $\delta$. Note that two curves are on top of each other.

Figure 8

Intensity ${\left|\psi \right|}^2$ of the nearly degenerate pair of modes in the asymmetric limaçon with (a) ${\Omega }_1=12.73070292-i1.83\times {10}^{-6}$ and (b) ${\Omega }_2=12.73070286-i1.88\times {10}^{-6}$ (cf. Fig. 2).

Figure 9

Angular momentum distributions ${\alpha }_m^{\left(1\right)}$ (black solid line) and ${\alpha }_m^{\left(2\right)}$ (green dashed) of the modes in Fig. 8 normalized to 1 at maximum: (a) absolute value squared, (b) real and (c) imaginary parts, and (d) superpositions ${\alpha }_m^{+}=({\alpha }_m^{\left(1\right)}+{\alpha }_m^{\left(2\right)})/2$ (black solid) and ${\alpha }_m^{-}=({\alpha }_m^{\left(1\right)}-{\alpha }_m^{\left(2\right)})/2$ (red dashed, multiplied by a factor of 8).

Figure 10

Chirality $\alpha$ vs spatial overlap $S$ of pairs of almost degenerate modes in the asymmetric limaçon. Open circles (crosses) mark the slightly higher-$Q$ (lower-$Q$) mode of a given pair of modes computed numerically from Maxwell's equations (cf. Fig. 2). The solid line is the analytical prediction of the theoretical model, Eq. (15).

Figure 11

Complex-square-root topology with a branch point singularity at the exceptional point of the Hamiltonian (6).

Figure 12

Time evolution of amplitude (defined as square root of intensity) in semilogarithmic scale corresponding to CCW (solid lines) and CW (dashed) propagating light rays in the asymmetric limaçon. The upper (lower) panel shows the dynamics starting with a set of pure CCW (CW) propagating rays in analogy to the wave dynamical considerations in Fig. 6. Time is proportional to the geometric length of ray trajectories.

Figure 13

Intensity of long-lived rays vs angle of incidence $\chi$ in the asymmetric limaçon. Positive (negative) $\chi$ correspond to CCW (CW) propagation direction.

Figure 14

Billiard boundary of constant width as defined in Eq. (28) with $z\left(0\right)=(1/4-i)R$, $a_0=R$, $a_3=iR/8$, $a_5=(1+i)R/4$, and $a_{2k+1}=0$ for $k>2$. The straight lines show four possible ways (out of infinitely many) to measure the width $W$. The value of $W$ is always $2R$.

Figure 15

Intensity ${\left|\psi \right|}^2$ of the nearly degenerate pair of modes in Gutkin's billiard of constant width with (a) ${\Omega }_1=12.386847-i0.00561$ and (b) ${\Omega }_2=12.386995-i0.005695$.

Figure 16

Angular momentum distributions ${\alpha }_m^{\left(1\right)}$ (black solid line) and ${\alpha }_m^{\left(2\right)}$ (green dashed) of the modes in Fig. 15 normalized to 1 at maximum: (a) absolute value squared, (b) real and (c) imaginary part, and (d) superpositions ${\alpha }_m^{+}=({\alpha }_m^{\left(1\right)}+{\alpha }_m^{\left(2\right)})/2$ (black solid) and ${\alpha }_m^{-}=({\alpha }_m^{\left(1\right)}-{\alpha }_m^{\left(2\right)})/2$ (red dashed, multiplied by a factor of 3.5).