Fractal frequency spectrum in laser resonators and three-dimensional geometric topology of optical coherent waves

Abstract

We theoretically verify that the symmetry breaking in spherical resonators can result in a fractal frequency spectrum that is full of numerous new accidental degeneracies to cluster around the unperturbed degenerate cavity. We further experimentally discover that the fractal frequency spectrum excellently reflects the intimate connection between the emission power and the degenerate mode numbers. It is observed that the wave distributions of lasing modes at the accidental degeneracies are strongly concentrated on three-dimensional (3D) geometric topology. Considering the overlapping effect, the wave representation of the coherent states is analytically derived to manifest the observed 3D geometric surfaces.

Characterizing classical periodic orbits from quantum Green's functions in two-dimensional integrable systems: Harmonic oscillators and quantum billiards

“…As shown in Fig. 1(a), the wave patterns corresponding to the Lissajous periodic orbits of two-dimensional (2D) commensurate harmonic oscillators have been confirmed to be the predominant modes in the selectively pumped large-Fresnel-number spherical cavity with astigmatism [31][32][33][34][35]. Furthermore, as shown in Fig.…”

“…In contrast, the emergence of classical features in 2D square billiards is found to arise from the superposition of nearly degenerate eigenstates induced by the tiny damping effect. The present model can provide deep insight into mesoscopic physics [24][25][26][27] and transverse pattern formations in laser resonators [31][32][33][35][36][37][38][39].…”

“…We further verify that the quantum Green's function with an appropriate damping factor can be decomposed to correspond to the individual classical periodic orbit. Since the quantum-classical or ray-wave correspondence has been ubiquitously observed in numerous systems [22][23][24][25][26][27][31][32][33][34][35][36][37][38][39][40], the present analysis can offer important insights into quantum physics, mesoscopic physics, and laser physics.…”

“…Quantum wave functions correlated with classical periodic orbits have been verified to play an important role in explaining the quantum phenomena such as shell effects in nuclei and metallic clusters [22,23], conductance fluctuations in mesoscopic quantum transports [24,25], and oscillations in photodetachment cross sections [26,27]. On the other hand, thanks to Hamilton's ingenious optico-mechanical theory [28][29][30], modern laser resonators have been widely exploited to analogously explore the formation of quantum coherent waves in the mesoscopic regime [31][32][33][34][35][36][37][38][39][40]. As shown in Fig.…”

“…(a) Lasing modes generally observed in the selectively pumped large-Fresnel-number spherical cavity with astigmatism[31][32][33][34][35]. (b) Lasing modes generally observed in the large-aperture VCSELs with large detuning[36][37][38][39][40].…”