Unidirectional emission from a $\mathcal{P}\mathcal{T}$-symmetric annular microcavity

The introduction of a scattering center to microdisk lasers has been considered as an effective way to tailor their far-field emissions. However, this approach is practically limited by its nearby modes with higher quality ($Q$) factors and worse directionalities. In this research, by applying the simple concept of parity-time ($\mathcal{P}\mathcal{T}$) symmetry, we demonstrate that the resonances with well-designed directionalities can be selectively excited in optical microcavities. Taking the annular microcavities as examples, the unidirectional laser emissions and single-mode operations have been realized simultaneously by breaking the $\mathcal{P}\mathcal{T}$ symmetry. More interestingly, we find that the $\mathcal{P}\mathcal{T}$-symmetric modulations on the imaginary parts of the refractive index in a circular microdisk can affect the internal ray dynamics and thus generate highly directional laser emission. This research can boost both the fundamental studies and the practical applications of $\mathcal{P}\mathcal{T}$-symmetric systems.

Figure 1

Mode properties of the annular microdisk. (a) The schematic picture of the annular microcavity. (b) The $Q$ factors of resonances in annular microdisk with $r=0.22R$ and $d=0.389R$. (c) The top and bottom panels show the far-field patterns of resonant modes 1 and 2, which are marked in (b). The insets are their field distributions. (d) The angular momentum distributions of the resonance WGM (17, 1) with unidirectional emission. (e) The near-field and far-field patterns of two nearly degenerated WGMs with the same azimuthal number. They correspond to the resonant modes 3 marked in (b). (f) The calculated $U({\theta }_1,{\theta }_2)$ factors and the percentage of components ${\alpha }_m$ at $m=13$ of high-$Q$ resonances in (b).

Figure 2

Mode properties of the $\mathcal{P}\mathcal{T}$-symmetric annular microdisk. (a) The schematic of the $\mathcal{P}\mathcal{T}$-symmetric microdisk. (b) The real (top panel) and imaginary (bottom panel) wave numbers of designed WGM with $m=17$ as a function of $n^{″}$. (c) The dependence of real (top panel) and imaginary (bottom panel) wave numbers of undesigned WGM with $m=18$ on the modulation $n^{″}$. (d)The far-field patterns of odd and even modes with $m=17$ at $n^{″}=0.02$.

Figure 3

Resonances in the microdisk with a point scatter. (a) The schematic picture of the microdisk with a point defect. (b) The resonant modes inside the microdisk with a point defect. (c) The far-field angular distributions of WGM (20, 4) (top panel) and WGM (32, 1) (bottom panel). (d) The $U({\theta }_1,{\theta }_2)$ factor of the resonances in (b). Here $U({\theta }_1,{\theta }_2)$ follows the same definition as in Fig. 1.

Figure 4

Resonances in the $\mathcal{P}\mathcal{T}$-symmetric microdisk without any deformation. (a) Schematic picture of the undeformed $\mathcal{P}\mathcal{T}$-symmetric microdisk. (b) The upward and downward triangles are the real and imaginary parts of resonant frequencies as a function of $n^{″}$. The undersigned modes are plotted as left- and right-pointing triangles. (c) The far-field pattern of the designed mode with $m=18$ at different $n^{″}$. (d) The near-field and far-field patterns of the WGM with $m=18$ in the circular cavity without $\mathcal{P}\mathcal{T}$ symmetry. Here the $kR$ of the mode is 7.695. (e) The near-field and far-field patterns of the WGM with $m=18$ in the annular cavity with an air hole inside. Here the $kR$ of the mode is 7.26. (f) The dependence of the $U$ (–10°, 10°) factor on the modulation $n^{″}$.

Figure 5

The comparisons between the annular cavity and the scatter cavity. (a) $Q$ factor and unidirectionality of fundamental modes in the annular cavity. (b) $Q$ factor and unidirectionality of the high-order modes in the scatter cavity. (c) $Q$ factor and unidirectionality of the fundamental modes in the scatter cavity. (d) The corresponding field distributions of the modes marked in (a,b,c), respectively.

Figure 6

Resonances in the $\mathcal{P}\mathcal{T}$-symmetric microdisk with a point scatter. (a) The schematic picture of the $\mathcal{P}\mathcal{T}$-symmetric microdisk. (b) The real (top panel) and imaginary (bottom panel) wave numbers of the designed WGM with $m=20$ as a function of $n^{″}$. (c) The dependence of real (top panel) and imaginary (bottom panel) wave numbers of the undesigned WGM with $m=32$ on the modulation $n^{″}$. (d) The far-field pattern of the odd mode with $m=20$ at $n^{″}=0.02$.

Figure 7

Unidirectional emission of the low-refractive-index $\mathcal{P}\mathcal{T}$-symmetric microdisk. (a) Schematic picture of the low-refractive-index $\mathcal{P}\mathcal{T}$-symmetric microdisk. The period number of loss modulation is 35. (b) The far-field pattern of the designed odd WGM (35,1). Inset is the corresponding field distributions.