Geometry-Driven Resonance and Localization of Light in Fractal Phase Spaces

Abstract

Geometry can fundamentally govern the propagation of light, independent of material constraints. Here, we demonstrate that a fractal phase space, endowed with a non-Euclidean, scale-dependent geometry, can intrinsically induce resonance quantization, spatial confinement, and tunable damping without the need for material boundaries or external potentials. Employing a fractional formalism with a fixed scaling exponent, we reveal how closed-loop geodesics enforce constructive interference, leading to discrete resonance modes that arise purely from geometric considerations. This mechanism enables light to localize and dissipate in a controllable fashion within free space, with geometry acting as an effective quantizing and confining agent. Numerical simulations confirm these predictions, establishing geometry itself as a powerful architect of wave dynamics. Our findings open a conceptually new and experimentally accessible paradigm for material-free control in photonic systems, highlighting the profound role of geometry in shaping fundamental aspects of light propagation.

Figures (6)

• Figure 1: Geometry as a confining agent for light (schematic). Left (Euclidean, $\alpha=1$): straight geodesics yield open rays and delocalized modes. Right (fractal, $\alpha=1.71$): geometry generates closed geodesic loops obeying the quantization condition $\oint \mathbf{p}\!\cdot d\mathbf{x}=n h$, leading to resonance and localization without material boundaries. This figure is conceptual; quantitative comparisons with the conventional case appear in subsequent figures.
• Figure 2: Fractal metric modulation as a function of $\alpha$. Effective fractal metric profile $\chi_\alpha(x) \sim |x|^{\alpha-1}$ for $\alpha = 1.0, 1.3, 1.5, 1.71, 1.9$. While $\alpha = 1$ shows no curvature, increasing $\alpha$ introduces geometric curvature, modifying the effective propagation environment without material boundaries.
• Figure 3: Geometry-induced localization: Euclidean baseline versus fractal metric. Normalized mode intensity ( $|\psi(x,y)|^2$ ) on the same spatial domain for (a) $\alpha=1$ (Euclidean) and (b) $\alpha\approx 1.71$ (fractal). The Euclidean case shows extended interference patterns (low inverse participation ratio, large $L_{\mathrm{loc}}$), whereas the fractal metric generates concentric resonance islands and geometry-induced spatial confinement (high inverse participation ratio, small $L_{\mathrm{loc}}$) without external potentials. For visualization, intensities are normalized independently to $[0,1]$ in each panel.
• Figure 4: Normalized resonance energy spacing versus mode index $n$. Shown are the cases $\alpha = 1.30,\,1.50,\,1.71$, computed under identical domain, resolution, and boundary conditions (see Sec. III). Across the plotted range the spacing decreases smoothly and monotonically with $n$; at fixed $n$ the ordering $\text{spacing}(\alpha{=}1.71) > \text{spacing}(1.50) > \text{spacing}(1.30)$ holds, indicating geometry-dependent spectral separation.
• Figure 5: Geometry-tuned damping of the EM mode amplitude $A(t)$. Three traces correspond to $\alpha=1.30$ (blue), $\alpha=1.50$ (green), and $\alpha=1.71$ (red); all runs use identical domain, resolution, and boundary conditions (see Sec. III). The vertical dashed line marks the resonance threshold $t_c$. Across the window shown, $A(t)$ decreases smoothly and monotonically. For early times ($t \lesssim t_c$) the ordering is $A_{1.71}(t) \ge A_{1.50}(t) \ge A_{1.30}(t)$; for late times ($t \gtrsim t_c$) it reverses to $A_{1.71}(t) < A_{1.50}(t) < A_{1.30}(t)$, indicating systematically faster late-time decay for larger $\alpha$.