Exceptional points of degeneracy with indirect band gap induced by mixing forward and backward propagating waves

We demonstrate that exceptional points of degeneracy (EPDs) are obtained in two coupled waveguides without resorting to gain and loss. We show the general concept that modes resulting from a proper coupling of forward and backward waves exhibit EPDs of order two and that there the group velocity vanishes. We verify our insight by using coupled-mode theory and also by full wave numerical simulations of light in a dielectric slab coupled to a grating, when one supports a forward wave, whereas the other (the grating) supports a backward wave. We also demonstrate how to realize a photonic indirect band gap in guiding systems supporting a backward and a forward wave, show its relations to the occurrence of EPDs, and offer a design procedure.

Figure 1

(a) Coupling between two electromagnetic waves whose complex amplitudes are $A$ and $B$. Conditions that lead to EPDs are obtained by introducing proper coupling between (b) two waveguides with $\mathcal{PT}$ symmetry where the two media have gain and loss supporting exponentially growing and attenuating waves and (c) two waveguides with forward and backward propagating waves, without resorting to $\mathcal{PT}$ symmetry (i.e., in this case the waveguides do not have gain and loss). The waves with black arrows represent the directions of phase propagation. The blue and red arrows represent the directions of power flow.

Figure 2

Two wave numbers of the guiding system versus the coupling parameter $\kappa$, showing the existence of an EPD. Two cases are examined: (a) coalescence of modes in $\mathcal{PT}$-symmetrical waveguides and (b) coalescence of modes obtained by coupling a forward wave (phase and group velocities have the same direction) and a backward wave (phase and group velocities have opposite directions). Both cases exhibit an EPD, represented by the bifurcation point.

Figure 3

Example of modal EPD of order 2 between a forward wave (phase and group velocities have the same direction) and backward wave (phase and group velocities have opposite directions). (a) Two coupled Si layers where the top one supports a forward wave (in red) while the bottom one is periodically corrugated to support a backward wave (positive phase velocity and negative group velocity, in blue). The red and blue arrows represent the direction of power flow. (b) Dispersion relation showing the propagating eigenmodes when the two waveguides are uncoupled (dashed) and when they are coupled (solid). The dispersion of modes in the coupled waveguides show the existence of two EPDs. The blue and red colors of the curves are related to the power flow directions. Note also that because two EPDs are found, an indirect band gap is present between the upper and lower branches that can be designed ad hoc. In the shown case we have $\mathrm{\Delta }{k}_e\equiv k_{e2}-k_{e1}<0$.

Figure 4

Schematic of a dispersion diagram showing the indirect band gap that results from two EPDs based on contradirectional coupling. The red-dashed line represents the wave number of the forward wave ${\beta }_a^{\prime }$, whereas the blue-dashed line the one of the backward wave ${\beta }_b^{\prime }$, when the waveguides are uncoupled. The coupling yields the two curves with two EPDs that are labeled as EPD1 and EPD2. The indirect band gap width is $\mathrm{\Delta }{\omega }_{IB}$. In the shown case $\mathrm{\Delta }{k}_e\equiv k_{e2}-k_{e1}>0$.