Fig. 1. Ray trajectories, 2D refractive index profiles, and 3D nanohole distributions of a CS limaçon TC and the corresponding homogeneous limaçon cavity. (a) Well-ordered ray trajectory in the CS limaçon TC exhibiting similar caustic to a uniform disk. Curved blue lines are transformed images of the Cartesian grid lines in the original virtual space, and the red line is the boundary of a CS limaçon TC. (b) Two-dimensional CS limaçon TC and its refractive index profile. (c) Top view of 3D CS limaçon TC implemented by nanohole distribution. (d) Chaotic ray trajectory in the homogeneous limaçon cavity. The ray is launched with the same incident angle as in (a). Straight blue lines are grid of the Cartesian coordinates. (e) Homogeneous limaçon-shaped cavity having the same shape as the CS limaçon TC and its refractive index. (f) Top view of 3D homogeneous limaçon cavity implemented by the nanohole distribution.

Fig. 2. Resonant mode patterns of the CS limaçon TC, the homogeneous limaçon cavity, and the homogeneous disk cavity. The free-space wavelengths of these resonant modes for all three cavities with 500 nm thickness are near 1550 nm (k∼4.05×106), which is the gain center of the InGaAsP-based WQM wafer. (a) Resonant mode (azimuthal mode number m=19 and radial mode number l=1) of the CS limaçon TC and its Q-factor (1.36×104). (b) Corresponding resonant mode of the homogeneous limaçon cavity and its Q-factor (3.24×103). In contrast to the CS limaçon TC, the resonant mode pattern is distorted. (c) Resonant mode (m=19,l=1) of the homogeneous disk cavity and its Q-factor (1.50×104).

Fig. 3. (a) SEM image of the fabricated CS limaçon TC laser. (b) Top view image of the nanoholes structure. (c) Continuous 2D refractive index profile of the CS limaçon TC is overlapped to the mask design for visual aid. (d) and (f) SEM images of the fabricated homogeneous limaçon cavity and homogeneous disk cavity lasers. (e) and (g) Mask designs for the homogeneous limaçon cavity and homogeneous disk cavity, respectively; for visual aid, the green color represents the mean refractive index of the CS limaçon TC. (h) Vertical cross section of the nanohole structure. Protection carbon is deposited on the cavity to prevent structural collapse when vertical milling is carried out for visualization. (i) Vertical profile of the epitaxial heterostructure wafer. (j) Steady-state room-temperature photoluminescence spectrum of the epitaxial heterostructure wafer.

Fig. 4. Measurement setup and lasing results for the three cavity lasers. (a) Schematic of experimental setup to measure lasing characteristics of the CS limaçon TC laser, the homogeneous limaçon cavity laser, and the homogeneous disk laser. OSA and BS mean optical spectrum analyzer and beam splitter, respectively. (b) Measured lasing curve of the CS limaçon TC laser. (c) CCD images of the TC laser in action just before and after the lasing threshold. (d) Lasing spectra of the CS limaçon TC laser at pumping power 92 μW and 108 μW. FWHMs at these pumping powers are 0.215 nm and 0.418 nm, respectively. An inset image depicts a spectrum at 123 μW pumping power, which shows single-mode lasing operation. (e) Lasing curves of the three cavity lasers for comparison. (f) The mean lasing thresholds and their standard deviations measured from every eight samples of the CS limaçon TC laser, the homogeneous disk cavity laser, and the homogeneous limaçon cavity laser are 88.3±6.1  μW, 77.3±8.7  μW, and 256.6±8.5  μW, respectively.

Fig. 5. Effect of center-shift Möbius mapping on refractive index distribution. (a) A grid line and refractive index distribution of the unit disk with uniform index. (b) Using a conformal mapping z1 to the unit disk. (c) Using the subgroup of Möbius mapping z2 to the unit disk. (d) Using center-shift limaçon mapping z=z1∘z2 to the unit disk.

Fig. 6. Two-dimensional effective refractive index is obtained from a 60 nm diameter nanohole in a material with a refractive index of 3.4 and a thickness of 500 nm.

Fig. 7. Q factors and mode patterns of disk cavities versus Din/Dout.

Fig. 8. FDTD simulation results with active medium. (a) Lasing spectrum of a CS limaçon TC laser and (b) its near-field pattern.

Fig. 9. Various-shaped TCs and their implementation. (a) Refractive index distribution of a rounded-hexagon cavity. (b) Refractive index distribution of a stadium cavity. (c) Refractive index distribution of a rounded-isosceles triangle cavity. (d)–(f) Their design using a fixed size deep-sub-wavelength-scale hole for implementation.