PT-symmetric coupler withχ(2)nonlinearity

Abstract

We introduce the notion of a PT -symmetric dimer with a χ (2) nonlinearity. Similarly to the Kerr case, we argue that such a nonlinearity should be accessible in a pair of optical waveguides with quadratic nonlinearity and gain and loss, respectively. An interesting feature of the problem is that because of the two harmonics, there exist in general two distinct gain/loss parameters, different values of which are considered herein. We find a number of traits that appear to be absent in the more standard cubic case. For instance, bifurcations of nonlinear modes from the linear solutions occur in two different ways depending on whether the first or the second harmonic amplitude is vanishing in the underlying linear eigenvector. Moreover, a host of interesting bifurcation phenomena appear to occur including saddle-center and pitchfork bifurcations which our parametric variations elucidate. The existence and stability analysis of the stationary solutions is corroborated by numerical timeevolution simulations exploring the evolution of the different configurations, when unstable.

CPsymmetry in optical systems

“…Two o f them w ere taken w ith a G aussian pulse in either com ponent, u(z -0 ,f) = ex p (-0 .0 5 0 /2 ) , v(z = 0 , 0 = 0, (32) u(z = 0 , 0 = 0, v(z = 0 ,f) = exp(-0.0 5 0 /2 ) , (33) and two other initial sets are given by Eqs. (19) and (20), and (23), w ith z = 0.…”

“…The nonlinearity gives rise to families of "PT-symmetric soli tons, which were investigated in detail in continuous and discrete systems [12,[22][23][24][25][26][27], including PT-symmetric dual core couplers [28][29][30]. Models combining the V T symmetry with quadratic nonlinearity in the dynamical equations (i.e., cubic terms in the respective Hamiltonians) were also elabo rated [31][32][33], As a subject of the quantum field theory, the C V T and CV symmetries mainly relate to elementary particles [34][35][36][37]. On the other hand, the above-mentioned works on the imple mentation of non-Hermitian PT-symmetric Hamiltonians in photonics suggest looking for a possibility to design optical settings that would realize non-Hermitian Hamiltonians fea turing the full C V T symmetry, as well as its CV reduction.…”

Nonlinear switching and solitons in PT‐symmetric photonic systems

“…Nonlinear modes in PT potentials can also be supported by quadratic (χ (2) ) optical nonlinearity [45][46][47]. Compared to the case of Kerr-type nonlinearity discussed above, quadratic nonlinear response involves parametric coupling of the fundamental and second-harmonic optical waves.…”

Solitons in a PT-symmetric χ^(2) coupler

“…iqz E j , where j = 1, 2 labels the waveguide, for the electric field envelopes E j of the FFs (u j ) and SHs (v j ), the dimensionless propagation distance z = Z/2L d and the transverse coordinate x = X/η, the linear coupling between harmonics in different arms κ i = K i L d and the mismatch of the propagation constants q = ∆kL d , assumed the same in both arms, where K 1,2 are the physical couplings of FF and SH (we consider them positive), The model (1) includes diffraction effects and generalizes the model of the PT -symmetric coupler considered in [19]. On the other hand inclusion of gain and losses represents a PT -symmetric generalization of the conservative coupler considered in [13].…”

“…[15] and references therein). In the case of χ (2) nonlinearity the previous studies were restricted to stationary (nondiffractive) propagation [19].…”