Modulation of high quality factors in rolled-up microcavities

We systematically investigate the evolution of resonant modes in a rolled-up microcavity as the overlap length between structural notches increases, which presents a modulation behavior for high $Q$ factors. The resonant modes in the rolled-up microcavity display a deterministic mode chirality, which is well correlated to the $Q$ factor. We derive a two-mode non-Hermitian Hamiltonian to clarify these unusual findings. It reveals that strong resonant interactions of scattered waves between the structural notches are responsible for the high mode chirality (thus high $Q$ factor) and its modulation behavior in rolled-up microcavities.

Figure 1

Schematic of a rolled-up microcavity. A U-shaped strained nanomembrane is defined at the beginning. Then, the self-rolling mechanism starts from the starting edge, and a tubular structure is formed over the full width of the strained nanomembrane. After the rolling distance $L_z$, only the side pieces of the structure continue to roll. As a result, the middle piece of the structure is separated from the substrate (not shown here) so that scattering loss of resonant modes caused by the substrate is avoided. In the end, a rolled-up microcavity is formed in the middle piece of the self-rolling tubular structure with a well-defined overlap length $L$, as shown in the inset.

Figure 2

Evolution of (a) resonant wavelength, (b) $Q$ factor, (c) mode chirality, and (d) $Q_{\text{sp}}$ of nearly degenerate mode pairs as the overlap length $L$ increases. Crossing of the resonant wavelengths occurs in (a) when anticrossing of their $Q$ factors does in (b). Roman numerals in (b) mark the first ten local maximums of the $Q$ factors. In the shaded region in (b), the $Q$ factor is larger than 10 000. Modulation behavior of the locally high $Q$ factors is clearly visible in (b). The same modulation behavior is reproduced for the mode chirality in (c). The inset in (c) gives the angular momentum distribution $|{\alpha }_m{|}^2$ for two successive local maximums of the $Q$ factors, showing that the CCW traveling-wave component is dominant for resonant modes in the rolled-up microcavity. The nearly degenerate mode pairs could be resolved in experiments if $Q_{\text{sp}}>1$, as shown in the shaded region in (d).

Figure 3

Electric field intensity ${\left|\psi \right|}^2$ of resonant modes at the first ten local maximums of $Q$ factors marked by Roman numerals in Fig. 2. As can be seen, the Roman numerals also indicate approximately the number of antinodes in the overlap area. These modes are classified into two groups in (a) and (b) according to the parity of the number of antinodes in the overlap area. The black arrows show the total internal reflection of resonant light, while the white arrows display its refractive escape.

Figure 4

Internal emerging Husimi function at the outer boundary of the rolled-up microcavity for peaks (a) iii, (b) iv, and (c) vi in Fig. 2. A different color map is applied for the leaky region bounded by the critical lines $\text{sin}\left(\chi \right)=\pm 1/n$. Black arrows in (c) are guides to the eye.

Figure 5

(a) Left (right): schematic of interactions between two particles (inner and outer notches) via scattered waves propagating across the solid core of two-particle-microdisk systems (through the subwavelength-thin wall of rolled-up microcavities). (b) Scattering of a waveguide mode at the overlap segment formed by two identical waveguides in an add-drop-like configuration. (c) Evolution of mode chirality as a function of the overlap length $L$ calculated with and without considering interactions between notches.