Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles

It is a well-known fact that a single nanoparticle placed in the evanescent field of an optical microdisk leads to coherent backscattering of light between counterpropagating whispering-gallery modes. This backscattering lifts the spectral degeneracy giving rise to a doublet of standing-wave modes. Here, we show that the evanescent coupling of two or more nanoparticles leads in general to asymmetric backscattering (i.e., the strength of the scattering of light from the clockwise to the counterclockwise propagation direction is different than the other way around). Even if the strength of the backscattering is weak its asymmetry can have a dramatic impact on the mode structure. In the regime of overlapping resonances the modes do not have a standing-wave character. Instead nonorthogonal pairs of mainly copropagating modes are formed. In the extreme case the pair of optical modes coalesce at a so-called exceptional point. We derive an effective Hamiltonian within a two-mode approximation which reveals that this unexpected behavior is due to interference of the scattered waves which can be destructive or constructive depending on the propagation direction of the light.

Figure 1

Microdisk of refractive index $n$ and radius $R$ with two nanoparticles of refractive index $n_j$ at distance $d_j$ from the microdisk. The azimuthal position of the second nanoparticle is specified by the angle $\beta$.

Figure 2

Intensity ${\left|\psi \right|}^2$ of a nearly degenerate pair of modes (mode pair A) in the disk perturbed by two nanoparticles. The azimuthal position of the second nanoparticle is given by $\beta =1$; cf. Fig. 1. The other parameters are specified in the text. (a) Dimensionless frequency ${\Omega }_{+}=9.87196-i0.00451$ and quality factor $Q_{+}\approx 1100$. (b) ${\Omega }_{-}=9.88362-i0.00019$ and $Q_{-}\approx 26000$.

Figure 3

Intensity ${\left|\psi \right|}^2$ of mode pair B with $\beta =1.072$ in the disk perturbed by two nanoparticles in the regime of overlapping resonances. (a) ${\Omega }_{+}=9.87722-i0.00243$ and $Q_{+}\approx 2050$. (b) ${\Omega }_{-}=9.87869-i0.00266$ and $Q_{-}\approx 1850$.

Figure 4

Intensity ${\left|\psi \right|}^2$ of mode pair C in the regime of strongly overlapping resonances close to an exceptional point; $r_2/R=0.048325$ and $\beta =1.08468125$. (a) ${\Omega }_{+}=9.87814-i0.00236$ and $Q_{+}\approx 2100$. (b) ${\Omega }_{-}=9.87801-i0.00248$ and $Q_{-}\approx 2000$.

Figure 5

Variation of light intensity along the boundary of the microdisk for the mode pair A (top panel), B (middle), and C (bottom); solid and dashed curves correspond to (a) and (b) in Figs. 2, 3, 4, respectively.

Figure 6

Angular momentum distributions ${\alpha }_m^{\left(1\right)}$ (circles and black solid lines) and ${\alpha }_m^{\left(2\right)}$ (crosses and green dashed lines) of the modes of pair B in Fig. 3 (real part normalized to 1 at maximum): (a) absolute value squared, (b) real and (c) imaginary part, (d) superpositions ${\alpha }_m^{\mathrm{CCW}}=({\alpha }_m^{\left(1\right)}+{\alpha }_m^{\left(2\right)})/2$ (squares and blue solid lines) and ${\alpha }_m^{\mathrm{CW}}=({\alpha }_m^{\left(1\right)}-{\alpha }_m^{\left(2\right)})/2$ (diamonds and red dashed lines magnified by a factor of 3.5). The lines are guides to the eye.

Figure 7

Angular momentum distributions ${\alpha }_m^{\left(1\right)}$ (circles and black solid lines) and ${\alpha }_m^{\left(2\right)}$ (crosses and green dashed lines) of the modes of pair C in Fig. 4 (real part normalized to 1 at maximum): (a) absolute value squared, (b) real (the inset magnifies the region around $m=16$) and (c) imaginary part, (d) superpositions ${\alpha }_m^{\mathrm{CW}}=({\alpha }_m^{\left(1\right)}+{\alpha }_m^{\left(2\right)})/2$ (squares and blue solid lines) and ${\alpha }_m^{\mathrm{CCW}}=({\alpha }_m^{\left(1\right)}-{\alpha }_m^{\left(2\right)})/2$ (diamonds and red dashed lines multiplied by a factor of 100 to be clearly visible). The lines are guides to the eye.

Figure 8

(top) Dimensionless frequency splitting $\Delta \Omega =|\text{Re}\left({\Omega }_{+}\right)-\text{Re}\left({\Omega }_{-}\right)|$ as a function of the azimuthal position of the second nanoparticle. (bottom) Individual dimensionless decay rates $-\text{Im}\left({\Omega }_{\pm }\right)$ of the two modes. The solid curves are the full results of Maxwell's equations; the dashed curves are the predictions of the two-mode approximation introduced in Sec. 4. Solid circles mark the mode pairs A and B (cf. Figs. 2 and 3). The insets magnify the region $\beta \in [1.06,1.1]$.

Figure 9

The top panel shows the chirality $\alpha$ as function of $\beta$. The solid curves (two curves are on top of each other) are the full results of Maxwell's equations using Eq. (3), the dashed curve is the prediction of the two-mode approximation in Eq. (19). In the shaded regions the CW component is smaller than the CCW component. (bottom) Mode-splitting quality $Q_{\mathrm{sp}}$ computed from Maxwell's equations versus $\beta$. Overlapping resonances are located below the dotted line $Q_{\mathrm{sp}}=1$.

Figure 10

Complex-square-root topology with a branch point singularity at the EP. Shown is the real part of $\Omega$ (i.e., the dimensionless frequency) versus the parameters $r_2/R$ and $\beta$. Dark (light) areas indicate values of the chirality close to unity (0.8). The area is darkest at the EP.

Figure 11

Complex-square-root topology with a branch point singularity at the EP (cf. Fig. 10). Shown is the negative of the imaginary part of $\Omega$ (i.e., the dimensionless decay rate) versus the parameters $r_2/R$ and $\beta$. Dark (light) areas indicate values of the chirality close to unity (0.8).

Figure 12

Real (solid curve) and imaginary part (dashed) of the deviation of the mean dimensionless frequencies versus the azimuthal position of the second scatterer $\beta$. (Inset) Logarithmic plot of the mode intensity with $\beta =0.751$. The double-sided arrow indicates the interaction of nanoparticles via scattering across the microdisk.

Figure 13

Intensity ${\left|\psi \right|}^2$ of a nearly degenerate pair of modes in a microdisk perturbed by eight nanoparticles. (a) ${\Omega }_{+}=9.86672-i0.00412$ and $Q_{+}\approx 1200$. (b) ${\Omega }_{-}=9.85770-i0.00889$ and $Q_{-}\approx 550$.