Strongly Localized Mode at the Intersection of the Phase Slips in a Photonic Crystal without Band Gap

A two-dimensional photonic crystal with a rectangular symmetry and low contrast ($<1$) of the dielectric constant is considered. We demonstrate that, despite the absence of a band gap, strong localization of a photon can be achieved for certain “magic” geometries of a unit cell by introducing two $\pi /2$ phase slips along the major axes. The long-living photon mode is bound to the intersection of the phase slips. We calculate analytically the lifetime of this mode for the simplest geometry—a square lattice of cylinders of a radius, $r$. We find the magic radius $r_c$ of a cylinder to be 43.10% of the lattice constant. For this value of $r$, the quality factor of the bound mode exceeds ${10}^6$. A small ($\sim 1\%$) deviation of $r$ from $r_c$ results in a drastic damping of the bound mode.

Figure 1

(a) Schematic illustration of two intersecting phase slips in a 2D structure with rectangular symmetry. (b) Omni-directional decay of the envelope of the mode localized at the intersection.

Figure 2

$\mathbf{k}$ plane for a periodic structure with low contrast of dielectric constant. Solid line is a contour of constant $|k|={ϵ}_0^{1/2}\omega /c$. Stop bands near the Bragg conditions are indicated with long-dashed lines. Thick solid lines: (1) second-order process responsible for the decay of the bound mode; (2) fourth-order process responsible for the decay of the bound mode. Virtual transition $\mathbf{2}$ is followed by one of the virtual transitions ${\mathbf{2}}_1$ or ${\mathbf{2}}_2$ shown with dashed lines.