Fig. 1. (a) A single gain (n″ < 0) or lossy (n″ > 0) sphere in air impinged by a plane wave propagating along z direction. The corresponding optical torques and rotation speeds of the sphere as a function of ka are plotted in panels (b) and (c), respectively. Here k is the wavenumber and a is the radius of the sphere. For illustration purpose, Γ_z and ω denoted by green lines are multiplied by 0.1 in panels (b) and (c). (d) Absolute value of optical torque versus | n″ | is shown for a gain sphere with ka = 5.1 pointed out by the black arrow in panel (b). The dashed vertical green line is inserted to highlight the paradoxical symmetry between amplification and absorption at gain threshold.

Fig. 2. (a) A double-layer cluster made of a gain (n_gain = 2 – 0.005i) and a lossy (n_lossy = 2+0.005i) spheres with the separation between spheres D = 2a+0.002 μm. Optical torques for the upper gain (lossy) sphere and the lower lossy (gain) sphere are shown in panels (b) [(c)], respectively.

Fig. 3. (a) The configuration of a double-layer cluster. (b) The structure of each layer in the double-layer cluster or a single layer. The separation between adjacent spheres is D = 2a+0.002 μm. (c) Optical torques exerted on the PT-symmetric double-layer structure with the refractive indexes of the lower lossy and upper gain layers are n = 2± 0.005i, respectively. The insert figure displays the opposite torques of the two layers at resonance. (d) Optical torques exerted on a single lossy layer (n_lossy = 2+0.005i), the lower lossless layer (n_lossless = 2) in a lossless–lossless double-layer cluster, and a partial view of the black line in panel (c) are shown for comparison. Optical torques obtained by the upper lossless layer (n_lossless = 2) in a lossless–lossless double-layer cluster, and a partial view of the red line in panel (c) are also shown in panel (e) for comparison. Black dotted vertical lines in panels (d) and (e) are guides of eyes to highlight the optical twist.

Fig. 4. Structures are the same as Figs. 3(a) and 3(b) except the separation between adjacent spheres D = 2a+0.0008 μm. The refractive index of Ag is modeled by Drude model,^[6,40,41] and n_gain = 2 – 0.005i. (a) Optical torques exerted on the upper Ag layer and the lower gain layer in a gain–Ag double-layer cluster. (b) Optical torques exerted on a single Ag layer cluster and that on the upper Ag layer in a Ag–Ag double-layer cluster are plotted compared to the partial view of the black line in panel (a).

Fig. 5. (a) The configuration of a double-layer cluster. (b) The structure of each layer in the double-layer cluster or a single layer. The separation between adjacent spheres is D = 2a+0.002 μm. Optical torques of the upper glass layer (n_glass = 1.33) and the lower gain layer (n_gain = 2 – 0.005i) in the gain–glass double-layer cluster are shown in panel (c), and the inset exhibits optical twist at resonances. Optical torques exerted on a single glass layer and that on the upper glass layer in a Ag–glass double-layer cluster are shown in panel (d) compared to the partial view of the black line in panel (c).

Fig. 6. Structures are the same as Figs. 3(a) and 3(b), and the refractive index of gain spheres is n_gain = 2 – 0.005i. (a) Optical torques exerted on the gain–gain double-layer cluster (red and black lines) and on a single gain structure (green line). The inserts show the same rotation direction of the two layers in gain–gain double-layer cluster at resonances. The partial view of torques in panel (a) is shown in panel (b).