Airy Resonances in Photonic Crystal Superpotentials

Abstract

Airy wavefunctions are associated with one of the simplest scenarios in wave mechanics: a quantum bouncing ball. In other words, they are the eigenstates of the time-independent Schrodinger equation with a linear potential. In the domain of optics, laser beams that are spatially shaped as Airy functions (`Airy beams') have been shown to exhibit a prominent lobe that follows a curved path, rather than propagating in a straight line, and which has self-healing properties in the presence of obstacles. Here, we observe the presence of Airy resonances in two-dimensional photonic crystals composed of a lattice of holes in a silicon slab. Analogously to electrons in a linear potential, these Airy resonances arise due to a linear spatial variation in the lattice constant of the holes. We map the electromagnetic description of the photonic crystal onto a 2D non-Hermitian Schrodinger equation with a linear potential, which we call a `superpotential'. The non-Hermiticity appears in the form of a complex effective mass due to out-of-plane radiation and fundamentally alters the collective optical response of the Airy resonances.

Figures (16)

• Figure 1: Photonic crystal slab superpotential. (a) The in-plane geometry of a periodic photonic crystal slab (upper panel) and its simulated band structure (lower panel). (b) The photonic crystal structure (upper panel) after a linear superpotential ($\kappa=0.311a^{-1}$) has been introduced. The lattice constant in the $x$ direction $a_x$ is slowly varied as a function of $x$. The corresponding superpotential is shown in the lower panel. (c) The simulated band structure of the structure in (b). The first $n=1$ Airy state is marked with a red arrow. (d) The eigenstate profiles ($\left|H_z\right|^2$ at the center of the slab) of the structure in (b) at $k_y=0$. The $n=1$, $n=2$, $n=3$ and $n=9$ Airy states are shown from top to bottom. The corresponding theoretical Airy function solutions are also shown in the lower panels.
• Figure 2: Observation of Airy resonances. (a) An SEM image of the sample with $\kappa=0.237a^{-1}$. (b) The isolated quadratic band in the control sample with no superpotential. (c) Several Airy bands observed in an Airy sample with $\kappa = 0.152a^{-1}$ (d) The frequency separation of Airy states vs. superpotential strength. The solid lines show the theoretical frequency separation between Airy states, and the dots with error bars show the measured values. (e) The linewidth of the first $n=1$ Airy state vs. superpotential strength. The solid line shows the linewidth predicted from superpotential formalism, and the dots with error bars show the measured values. The experimental linewidths are consistently $\sim 10^{-4}\ (2\pi ca^{-1})$ larger than the theoretical linewidths. This discrepancy is likely caused by some additional loss in our system not associated with the superpotential such as scattering from fabrication-induced roughness or absorption.
• Figure 3: Spatial profiles of Airy resonances. (a) The intensity distribution of light at the surface of an Airy sample with $\kappa = 0.108a^{-1}$ at two different frequencies. The main feature of the resonance - several parallel "lobes" - moves smoothly across the sample as the frequency of the probe beam is swept. (b) The spatial profile of an Airy resonance at $\omega = 0.416\ (2\pi ca^{-1})$. The red dots show the experimentally measured spatial profile. The black line shows $\left|\alpha_{17}(x)\right|^2$, which is predicted in the theory. The blue line shows the modified Airy envelope which is approximated by adding an exponential decay factor on top of the Airy function. (c) The intensity distribution of light at the surface of the Airy sample with $\kappa = 0.108a^{-1}$ averaged in the $y$ direction at different probe beam frequencies. (d) The position of the brightest lobe in the Airy resonance as a function of probe beam frequency for nine samples with different potential strengths. The solid lines and dots show the theoretical predictions and experimental measurements, respectively.
• Figure 4: Accelerating Airy beam generated from Airy resonances. (a) The intensity of light averaged in the $y$-direction as the distance above the sample ($z$) is swept. The dashed red curves are the quadratic fits for the bending of the lobes. For this measurement, the Airy sample with $\kappa = 0.108a^{-1}$ was used and the frequency of the probe beam was $\omega = 0.411\ (2\pi c a^{-1})$. (b) The beam acceleration as a function of superpotential strength. The solid line and dots with error bars represent the theory and experimental values of the beam acceleration respectively.
• Figure S1: The periodic photonic structure and the definition of the local potential. (a) The unit cell (in the $x-y$ plane, and $z$ is the out-of-plane direction) of our photonic crystal slab. The structure contains a circular air hole ($\varepsilon=1.0$) with $r_0=0.34a$ in a rectangular lattice, where $a_x$ and $a_y$ are the lattice constant in the $x$ and $y$ directions. The slab is made of silicon ($\varepsilon=12.11$) with a thickness of $h=0.35a$ and sits on top of a silica substrate ($\varepsilon=2.25$). (b) The simulated band structure of $a_x=a_y=a$ along the $k_x=0$ line. Only the transverse electric (TE)-like modes are shown. (c) The simulated band structure of $a_x=1.15a$ and $a_y=a$ along the $k_x=0$ line. (d) The tip energy ($E=\left(\omega/c\right)^2$) of the quadratic band as a function of $a_x$ when the lattice constant in the $y$ direction is fixed at $a_y=a$.