Observation of Photon Droplets and Their Dynamics

We present experimental evidence of photon droplets in an attractive (focusing) nonlocal nonlinear medium. Photon droplets are self-bound, finite-sized states of light that are robust to size and shape perturbations due to a balance of competing attractive and repulsive forces. It has recently been shown theoretically, via a multipole expansion of the nonlocal nonlinearity, that the self-bound state arises due to competition between the $s$-wave and $d$-wave nonlinear terms, together with diffraction. The theoretical photon droplet framework encompasses both a solitonlike stationary ground state and the nonsolitonlike dynamics that ensue when the system is displaced from equilibrium, i.e., driven into an excited state. We present numerics and experiments supporting the existence of these photon droplet states and measurements of the dynamical evolution of the photon droplet orbital angular momentum.

Figure 1

(a) Pseudoenergy landscape $H_{*}(\delta ,\xi )$ for $P_0=0.8\mathrm{W}$ and $E_0(r)\propto r{e}^{-r^2/{\xi }_0^2}$. (b),(c) $320\mu \mathrm{m}\times 320\mu \mathrm{m}$ square images showing the transverse intensity profile $I(x,y)$ and nonlinear potential $\mathrm{\Delta }n(x,y)$, respectively, for the ground state. The minimum pseudoenergy occurs for $|\delta |=1$ and $\xi \sim 100\mu \mathrm{m}$, and there is an instability at $H_{*}(\delta =0)$ in contrast to the SM model.

Figure 2

Photon droplet dynamics from numerics. Plot of $|\delta |$ versus $⟨r⟩\propto \xi$ over 80 cm of propagation for $|{\delta }_0|=0.2$, ${\xi }_0=210\mu \mathrm{m}$ ($⟨r{⟩}_0=220\mu \mathrm{m}$), and varying input beam powers $P_0$. See Supplemental Material [35] for details of the spatial evolution.

Figure 3

Experimental setup. Lenses $L1$ and $L2$ expand the 532-nm laser beam. An SLM creates a diffraction grating with variable combinations of OAM $\ell =+1$ and $\ell =-1$. The first-order diffraction is weakly focused ($f\sim 2\mathrm{m}$) onto the entrance facet ($P0$) of a set of four $10\text{−}\mathrm{cm}\times 4\text{−}\mathrm{cm}\times 0.5\text{−}\mathrm{cm}$ glass slabs. HWP2 is used to vary the SLM diffraction efficiency. HWP3 and PBSC provide additional control of $P_0$. The beam intensity profile is imaged at the entrance facet $P0$ and at the exit facet of each glass plate $P1–P4$. The $M=6$ imaging system is fixed with respect to the camera and translated as a single entity. The camera can be replaced with an SLM and single-mode fiber to perform an OAM decomposition.

Figure 4

$1\text{−}\mathrm{mm}\times 1\text{−}\mathrm{mm}$ square images of the intensity profile for input conditions $|{\delta }_0|=0.7$ and ${\xi }_0=160\mu \mathrm{m}$. (a) Experiment: Effectively linear propagation for reference $P_0=0.03$ W. (b) Experiment: Photon droplet propagation for $P_0=0.86\mathrm{W}$. (c) Numerics: Propagation for $P_0=0.86\mathrm{W}$. (d) Numerics: Plots of rotation angle $\theta$ versus $z$.

Figure 5

Photon droplet dynamics with ${\xi }_0=160\mu \mathrm{m}$. (a) Representative experimental OAM spectrum, after 20 cm of propagation for $|{\delta }_0|=0.7$. (b) Plot of ${\mathrm{\Delta }}_{+}-{\mathrm{\Delta }}_{-}$ versus $z$, where ${\mathrm{\Delta }}_{\pm }=|A_{\mathrm{HP},\pm }{|}^2-|A_{\mathrm{LP},\pm }{|}^2$. HP (LP) denotes $P_0=0.86\mathrm{W}$ ($P_0=0.03\mathrm{W}$). Dashed black line: Numerics for $|{\delta }_0|=0.7$. Solid blue line: Numerics for $|{\delta }_0|=1.4$. Black squares: Experiment with $|{\delta }_0|=0.7$. Blue diamonds: Experiment with $|{\delta }_0|=1.4$. (c) Plot of $|\delta |$ versus $⟨r⟩$ from numerics with $|{\delta }_0|=0.7$.