Fano Combs in the Directional Mie Scattering of a Water Droplet

When light scatters off a sphere, it produces a rich Mie spectrum full of overlapping resonances. Single resonances can be explained with a quantum analogy and result in Fano profiles. However, the full spectrum is so complex that recognizable patterns have not been found, and is only understood by comparing to numerical simulations. Here we show the directional Mie spectrum of evaporating water droplets arranged in consecutive Fano Combs. We then fully explain it by expanding the quantum analogy. This turns the droplet into an “optical atom" with angular momentum, tunneling, and excited states.

Figure 1

Counterpropagating trap used to record directional Mie spectra of evaporating water droplets. The green arrows mark the laser’s polarization. The $I_{\theta }$ polarization was focused onto a position sensitive detector to measure the scattering intensity. $I_{\varphi }$ was projected onto a transparent screen to observe the interference pattern and calculate the droplet’s radius.

Figure 2

Fano Comb structure of the directional Mie scattering. (a) Directional scattering intensity $I_{\theta }$ as the droplet shrinks. Highlighted are the first three combs composed of equidistant, evolving Fano resonances. (b) Evolution of the resonances from Lorenztians to Fano through five selected segments of the first comb. A fit to the Fano equation, Eq. (1), is shown in green. (c) Mie simulation showing the interference between two resonances at the intersection of two combs (below) producing a sharp Fano profile (above). (d)–(f) Fano parameters obtained from fits of Eq. (1) on the second comb. The separation between resonances inside a comb is constant (d), the phase shift $\delta$ starts at zero and decreases linearly until $0.3\pi$ (e), and the resonance width $\gamma$ decreases exponentially (f).

Figure 3

The shape of the wedged well potential defines the profile of the resonance. (a) $V_{\mathrm{eff}}(r)$ for $\ell =11$, 20, 31, and 35. The energy levels inside the wedge are plotted with green (first comb) and orange (second comb). The magenta lines mark the potential barriers for $\ell =11$ and 20. The energy level $a_{31}^1$ is dashed to mark that it does not resonate with the laser. See the Supplemental Material [36] for a video of the energy levels in a growing sphere. (b) All the calculated energy levels up to $4.5\mathrm{\mu }\mathrm{m}$, showing the basis for the comb structure. (c) The directional scattering at three selected size ranges of the droplet (blue) and a fit using Eq. (1).