Unidirectional emission from circular dielectric microresonators with a point scatterer

Circular microresonators are micron-sized dielectric disks embedded in material of lower refractive index. They possess modes of extremely high $Q$-factors (low-lasing thresholds), which makes them ideal candidates for the realization of miniature laser sources. They have, however, the disadvantage of isotropic light emission caused by the rotational symmetry of the system. In order to obtain high directivity of the emission while retaining high $Q$-factors, we consider a microdisk with a pointlike scatterer placed off-center inside of the disk. We calculate the resulting resonant modes and show that some of them possess both of the desired characteristics. The emission is predominantly in the direction opposite to the scatterer. We show that classical ray optics is a useful guide to optimizing the design parameters of this system. We further find that exceptional points in the resonance spectrum influence how complex resonance wave numbers change if system parameters are varied.

Figure 1

(Color online) Level dynamics (black curves) of the TM resonances in the complex wave number plane for a dielectric disk with $n=3.0$ and $R=1.0\mu \text{m}$ and a point scatterer of varying coupling parameter $a$. The solid circles mark the unperturbed resonances with azimuthal and radial modal indices $\left(m,q\right)$. For the upper panel the scatterer is placed at distance $d=0.9\mu \text{m}$ from the center, for the lower panel $d=0.5\mu \text{m}$. The color code (grayscale) in the $kR$ plane indicates the directivity $D$ of the emission. The directivity increases from blue (lower dark gray regions) through green and yellow (light gray) to red (upper dark regions).

Figure 2

(Color online) Analog of Fig. 1 for TE resonances. The parameters are the same as in Fig. 1.

Figure 3

The far-field intensities of the unperturbed TM resonance with modal indices (21,4) (thin curve), and of the highly directional TM resonance (thick curve) that is obtained from the unperturbed mode by perturbing a dielectric disk with $n=3.0$ and $R=1.0\mu \text{m}$ with a point scatterer of strength $a={10}^{-6}$ placed at distance $d=0.5\mu \text{m}$ from the disk center.

Figure 4

The field intensity in the far-field of the unperturbed TE resonance with modal indices (21,4) (thin curve), and of the highly directional TE resonance (thick curve) that is obtained from the unperturbed mode by perturbing a dielectric disk with $n=3.0$ and $R=1.0\mu \text{m}$ with a point scatterer of strength $a={10}^{-6}$ placed at distance $d=0.5\mu \text{m}$ from the disk center.

Figure 5

(Color online) Level dynamics of the TM resonances in the complex wave number plane for a dielectric disk with $n=1.4$ and $R=10.51\text{mm}$ and a point scatterer of varying coupling parameter $a$. The solid circles mark the unperturbed resonances with azimuthal and radial modal indices $\left(m,q\right)$. For the upper panel the scatterer is placed at distance $d=9.585\text{mm}$ from the center, for the lower panel $d=9.385\text{mm}$. As before, the directivity increases from blue (dark gray) through green and yellow (light gray).

Figure 6

(Color online) Analog of Fig. 5 for TE resonances. The parameters are the same as in Fig. 5. The label $\left(23,s\right)$ indicates a shape resonance with azimuthal model index $m=23$.

Figure 7

The far-field intensities $\left(r=5000\text{mm}\right)$ of the unperturbed TM resonance with $kR=37.599462-i0.488553$, $m=31$, $q=5$ (thin curve), and of the highly directional perturbed TM resonance with $kR=37.621007-i0.523645$, $d=9.385\text{mm}$, $a\cong 0.07$ (thick curve) in a dielectric disk with $n=1.4$ and $R=10.51\text{mm}$.

Figure 8

The far-field intensities $\left(r=5000\text{mm}\right)$ of the unperturbed TE resonance with $kR=37.129055-i0.000177$, $m=46$, $q=1$ (thin curve), and of the directional perturbed TE resonance with $kR=37.142373-i0.001832$, $d=9.585\text{mm}$, $a\cong 4\cdot {10}^{-5}$ (thick curve) in a dielectric disk with $n=1.4$ and $R=10.51\text{mm}$.

Figure 9

The average directivity, $D_{\text{av}}$, of TM (thick curves) and TE (thin curves) polarized light in the range $12<\text{Re}\left(kR\right)<13$, $-0.1<\text{Im}\left(kR\right)<0.0$ for each position, $d$, of a point scatterer in microdisks of radius $R=1\mu \text{m}$ and various effective refractive indices as shown.

Figure 10

The average directivity, $D_{\text{av}}$, of TM (thick curves) and TE (thin curves) polarized light in the range $37<\text{Re}\left(kR\right)<38$, $-1<\text{Im}\left(kR\right)<0.0$ for each position, $d$, of a point scatterer in microdisks of radius $R=10.51\mu \text{m}$ and various effective refractive indices as shown.

Figure 11

(Color online) Level dynamics of the TM resonances in the complex wave number plane for a dielectric disk with $n=1.4$ and $R=10.51\text{mm}$ and a point scatterer of varying coupling parameter $a$ for several positions $d$ of the scatterer. All start from the unperturbed resonance (34,4). The solid circles mark the unperturbed resonances with modal indices $\left(m,q\right)$. The color code (grayscale) indicates the directivity $D$ of the emission, where dark blue (dark gray) marks small values of $D$ and light blue (light gray) marks high values of $D$.

Figure 12

(Color online) Level dynamics of the TM resonances in the complex wave number plane for a dielectric disk with $n=1.4$ and $R=10.51\text{mm}$ and a point scatterer of varying coupling parameter $a$ for two positions $d$ of the scatterer. The solid circles mark the unperturbed resonances with azimuthal and radial modal indices $\left(m,q\right)$. As the position of the scatterer changes from $d=9.409$ to $d=9.411$ the connections of the unperturbed resonances change.

Figure 13

(Color online) Two almost degenerate TM resonance states for a dielectric disk with $n=1.4$ and $R=10.51\text{mm}$. The corresponding wave numbers are shown by crosses in Fig. 12.

Figure 14

Far field of the two resonance states shown by crosses in Fig. 12.