Persistence and Lifelong Fidelity of Phase Singularities in Optical Random Waves

Phase singularities are locations where light is twisted like a corkscrew, with positive or negative topological charge depending on the twisting direction. Among the multitude of singularities arising in random wave fields, some can be found at the same location, but only when they exhibit opposite topological charge, which results in their mutual annihilation. New pairs can be created as well. With near-field experiments supported by theory and numerical simulations, we study the persistence and pairing statistics of phase singularities in random optical fields as a function of the excitation wavelength. We demonstrate how such entities can encrypt fundamental properties of the random fields in which they arise.

Figure 1

Near-field measurements of the optical field in the chaotic cavity. (a) Optical micrograph of the chaotic cavity; the dark area is a photonic crystal that confines light inside the cavity. (b) A $17\times 17\mu \mathrm{m}$ near-field map of the amplitude of the $x$ component of the electric field in the cavity. (c)–(f) Zoomed-in image of panel (b) for different input wavelengths $\lambda$. (g)–(j) False-color map of the measured phase for different input wavelengths $\lambda$; in such maps we pinpoint phase singularities with positive (dark spots) or negative (light spots) topological charge. The black circles highlight annihilation and creation events.

Figure 2

A 3D representation of the trajectories of singularities that propagate through a $2\mu \mathrm{m}\times 2\mu \mathrm{m}\times 0.64\mathrm{nm}$ observation volume. The red and green trajectories are of faithful and unfaithful singularities, respectively. In both cases, bright and dark colors differentiate oppositely charged singularities. Please note that the parts of the trajectories that continue outside of the observation volume are not shown. The semitransparent trajectories are of singularities that propagate outside of the total wavelength range; these are not taken into account in our statistics.

Figure 3

Overview of the statistics for the persistence and fidelity of singularities in random waves, in experiment [(a),(c)] and finite-element method (FEM) simulation [(b),(d)]. In the upper panels the dashes indicate the number of singularities $\mathcal{N}$ which persisted in the field for a given wavelength shift $\mathrm{\Delta }\lambda$. The main figures refer to experiment or simulation where the wavelength was swept over a range $\mathcal{L}=1.2\mathrm{nm}$ with a step $\delta =0.02\mathrm{nm}$, whereas the insets show the results of wider scans ($\mathcal{L}=8\mathrm{nm}$, $\delta =0.1\mathrm{nm}$). The gray lines are representative of the prediction for the persistence histogram given by our model. In the same panels, the boxes are still a persistence histogram, but in which only the ${\mathcal{N}}_f$ singularities that were faithful to each other are taken into account. In the lower panels we report the fidelity fraction $\mathcal{F}$ of phase singularities, again as a function of their persistence in the field [$\mathcal{F}(\mathrm{\Delta }\lambda )={\mathcal{N}}_f(\mathrm{\Delta }\lambda )/\mathcal{N}(\mathrm{\Delta }\lambda )$].

Figure 4

Correlation coefficient $\rho ({\lambda }_1,{\lambda }_2)$ of the measured (upper left) and FEM-simulated (lower right) optical random field, at wavelengths ${\lambda }_1$ and ${\lambda }_2$ (${\lambda }_0=1550\mathrm{nm}$).

Figure 5

Persistence histograms for singularities arising in numerically generated random fields in which two families of eigenstates with different spectral widths $\gamma$ and ${\gamma }^{\prime }$ coexist [31]. The lines are biexponential fits to the data points.