Waveguides with Absorbing Boundaries: Nonlinearity Controlled by an Exceptional Point and Solitons

We reveal the existence of continuous families of guided single-mode solitons in planar waveguides with weakly nonlinear active core and absorbing boundaries. Stable propagation of TE and TM-polarized solitons is accompanied by attenuation of all other modes, i.e., the waveguide features properties of conservative and dissipative systems. If the linear spectrum of the waveguide possesses exceptional points, which occurs in the case of TM polarization, an originally focusing (defocusing) material nonlinearity may become effectively defocusing (focusing). This occurs due to the geometric phase of the carried eigenmode when the surface impedance encircles the exceptional point. In its turn, the change of the effective nonlinearity ensures the existence of dark (bright) solitons in spite of focusing (defocusing) Kerr nonlinearity of the core. The existence of an exceptional point can also result in anomalous enhancement of the effective nonlinearity. In terms of practical applications, the nonlinearity of the reported waveguide can be manipulated by controlling the properties of the absorbing cladding.

Figure 1

(a) Real part vs imaginary part of the propagation constants for the waveguide with ${\eta }^{\mathrm{TE}}=-1.25-3.16i$, $\epsilon =1.5-0.062i$, $|{\chi }_{\mathrm{NL}}|=0.025$ and $k_0\ell =2\pi /3.1$. Squares and triangles represent the $\cos (Qy)$ and $\sin (Qy)$ modes, respectively. The guided and weakest decaying modes are indicated by red and blue color, respectively; their transverse profiles are shown in the inset. (b) Dynamics of the single bright TE-soliton input $A=2\tilde{A}=0.5\text{sech}(\sqrt{g/8}x)$ with the amplitude profiles shown in ($b^{\prime }$) (see also Ref. [29]). (c) Dynamics of the in-phase two-soliton input: $A=2\tilde{A}=0.5\{\text{sech}[\sqrt{g/8}(x+5)]+\text{sech}[\sqrt{g/8}(x-5)]\}$. The effective nonlinearities are $g=1.24$, $\tilde{g}=1.02+0.02i$, $g_1=1.34+0.07i$, ${\tilde{g}}_1=2.4+0.002i$. Simulations were carried out on the window $-200<x<200$ and $0<z<1000$. The inputs were perturbed by noise of order 5% of the amplitude.

Figure 2

Central panels: Imaginary (c) and real (d) parts of the eigenvalue $Q$ corresponding to a TM mode in the vicinity of EP, ${\eta }^{\mathrm{EP}}=1.6506+2.05998i$, vs real and imaginary parts of the impedance, for $0<r<0.04$ and $0\le \mathrm{\Theta }\le 2\pi$. The mode is chosen to be guided at two values of the total phase $\phi$: at black discs $\mathrm{\Theta }=0$, $\phi =0$, and $g>0$, while white discs represent the same points after change of the impedance resulting in $\phi =\pi$ and $g<0$. The positions of the phase changes are shown at $(r,\mathrm{\Theta })=(0.01,1.25\pi ),(0.02,0.91\pi )$, and $(0.03,0.722\pi )$, when $\chi =0.12+1.29i,0.61+1.14i$, and $0.87+0.95i$, respectively. Different $\chi$ for different radii ensure that $g$ is real at both white and black points. Upper panels: (a) Propagation constants, and profiles of the guided and weakly decaying TM modes (in the inset), and (b) dynamics of a bright TM soliton with $A=2\tilde{A}=0.5\text{sech}(\sqrt{g/8}x)$ at the input, for $g=1.42$ at $r=0.03$ [black discs in (c) and (d)] observed for ${\eta }^{\mathrm{TM}}=1.681+2.060i$, $\epsilon =1.5-0.986i$, $q=1.63$, and $\tilde{q}=1.897+0.386i$. Lower panels: (e) Propagation constants, and the guided and weakly decaying modes (in the inset) and (f) dynamics of a dark TM soliton excited by the $A=2\tilde{A}=0.5\tanh (\sqrt{|g|/8}x)$ input for $g=-0.073$ at $r=0.03$ [white discs in (c) and (d)] obtained for ${\eta }^{\mathrm{TM}}=1.631+2.083i$, $\epsilon =1.5-0.999i$, $q=1.85$, and $\tilde{q}=1.609+0.409i$. In all panels $k_0\ell =2\pi /3.1$. Simulations were carried out on the window $-200<x<200$ and $0<z<1000$. Inputs were perturbed by noise of order 5% of the amplitude.