Parametric amplification in quasi-PT symmetric coupled waveguide structures

The concept of non-Hermitian parametric amplification was recently proposed as a means to achieve an efficient energy conversion throughout the process of nonlinear three wave mixing in the absence of phase matching. Here we investigate this effect in a waveguide coupler arrangement whose characteristics are tailored to introduce passive PT symmetry only for the idler component. By means of analytical solutions and numerical analysis, we demonstrate the utility of these novel schemes and obtain the optimal design conditions for these devices.

Inspired by this progress, we have recently proposed a novel application of non-Hermitian photonic to control the nonlinear interactions inside optical parametric amplifiers (OPAs) [21]. For OPAs to operate properly, a phase matching condition between the interacting components must be satisfied. In the absence of this condition, the amplification process does not take place and energy oscillates back and forth between the different wave components. This stringent constraint has long posed a problem for utilizing the power of certain nonlinear material systems such as semiconductors [22].
In our recent publications, we have shown that by carefully engineering the dissipative spectral features of the nonlinear optical system under considerations, it is possible for the process of parametric amplification to take place even in the absence of Hermitian phase matching [21]. This is done by introducing optical losses to the idler component.
In this work, we investigate in detail a particular implementation of non-Hermitian parametric amplifiers based on quasi-PT symmetric coupled waveguide structures (a system where the idler component experiences passive PT symmetry as will be explained in details later). As shown in figure 1, the system consists of two coupled asymmetric waveguides. The nonlinear interaction takes place in one of them (say the left one) which we will call the nonlinear guide. The second channel is designed to support a mode that matches only the idler component. Thus by introducing optical loss to this second waveguide, one can provide a dissipative coupling to the idler wave. We will call this second waveguide a reservoir channel. Thus effectively, the nonlinear waveguide together with the reservoir channel constitute a passive PT symmetric structure [5] for the idler component.
Here we refer to this frequency selective passive PT symmetry as quasi-PT symmetry.

Analysis and results
Within the context of linear coupled mode theory between two waveguides and the nonlinear coupled wave analysis between the different components of the three wave mixing process, the system can be described by [23][24][25]: where E S,I,P are the signal, idler and pump field amplitudes respectively, while E D is the field amplitude of the dissipative optical mode; k ¢ S,I,P are the nonlinear coupling coefficients, which are functions of the second order nonlinearity as well as the optical frequencies w S,I,P , whereas k D is the linear coupling between the idler and the dissipative modes.
I is the propagation constant mismatch between the three different beams [24], g D is the linear loss coefficient associated with the dissipative mode. In equation (1), z is the propagation distance, and asterisks denote complex conjugation.
We first consider the undepleted pump approximation (i.e. E P can be taken as a constant). Under this condition, and by using the substitutions  The two channels are asymmetric with the pump, signal and idler waves propagating in the first waveguide (the main one) and only the idler components can travel in the second auxiliary channel. Thus by introducing loss to the second channel, the structure as a passive modal PT coupler at the idler frequency. By tuning the coupling between the two waveguides and the optical loss on the auxiliary channel, the dynamics of the wave mixing process can be engineered and amplification can be achieved even in the absence of the Hermitian phase matching condition.
Interestingly, as we can see in figure 2(a), in the case when g = 0 DN , one can still find a regime where amplification can occur even when b . This can be understood by thinking in terms of supermodes of the two asymmetric waveguides at the idler frequency, i.e.
. As a result of this hybridization, these supermodes exhibit shifted propagation constants that can satisfy the phase matching condition and eventually lead to amplification. We note however that this occurs only at a finite region bounded by two asymptotic lines. Outside this region, the dynamics exhibit only energy oscillations between the modes without any change in average powers. The detailed mathematical treatment of this case is presented in the appendix.
We now focus on the situation where losses are involved. As an example, we consider the case of g = 1 DN as shown in figure 2(b). Here we note that amplification is not pinned to a finite region in the parameter space but rather occur for all points in the b k D ( ) , can be chosen to achieve maximum amplification. In practice, this can be done by carefully engineering the separation between the two waveguides.  values are within the region discussed in figure 2(a). Outside this regime, amplification can be only attained by introducing optical dissipation to the auxiliary waveguide.
In order to verify our predictions based on the eigenstate analysis, we perform numerical integration of equation (1) for several different scenarios under the appropriate initial conditions. In doing so, we first rescale equation (1)  , and k = 1.3 D or 4.5, the system is outside the amplification regime and the optical power associated with signal, idler and auxiliary beams oscillate without any change in average powers as shown in figures 3(a) and (d). By changing the above values of k D to 1.45 or 4.35, we observe that the different components grow exponentially as a function of the propagation distance, as shown in figures 3(b), (c). Interestingly we observe some fast and small oscillatory behavior in figure 3. This can be attributed to the beating effects between the different eigenvectors associated with equation (2). Figure 4 depicts the dynamical behavior in the presence of optical loss in the dissipative mode where for completeness we also present the oscillatory behavior of the system in the absence of any linear coupling ( figure 4(a)). Figures 4(b)-(e) show the dynamics when g = 1 D for different value of k D . Evidently, in this regime, as the coupling is first increased from 0.5 to 1.5, the energy conversion occurs at a faster rate. As the coupling is further increased to 6, the amplification rate becomes less pronounced-in perfect agreement with the eigenvalue analysis. In all cases, the initial conditions were taken to be We now treat the general case where the undepleted pump approximation no longer holds. Figure 5 depicts the numerical results for this scenario for different design parameters. In our simulations, we assumed that k = 0.1 and x = = ( ) a 0 10 P . In the phase-matching condition ( b D = 0), oscillatory power transfer between the pump and both signal and idler beams is observed in the absence of linear coupling and loss, as shown in figure 5(a). When the dissipative mode is introduced without losses (g = 0 D ), the phase-matching condition will be changed. Choosing a proper linear coupling k D can still obtain parametric amplification, as shown in figure 5(d)   . Evidently, adding loss changes the dynamics drastically by allowing unidirectional energy conversion from the pump beam to the signal component. that adding a dissipative mode can force the system to undergo a unidirectional energy conversion from the pump beam to the signal component-a task that could not be otherwise achieved in Hermitian systems.

Discussion and concluding remarks
In summary, we have investigated the dynamical behavior of non-Hermitian parametric amplifiers based on quasi-PT symmetric coupled waveguide channels. In these configurations, the two waveguides are asymmetric and their modal structure is assumed to be engineered in order to induce crosstalk only between the idler components. Our theoretical and analytical study revealed the existence of different regimes of operation depending on the interplay between the phase mismatch of the different frequency components from one side and the optical losses and coupling coefficient between the two channels on the other. In particular, we have shown in a certain regime of operation, introducing the auxiliary channel was enough to achieve phase matching even without introducing any loss. This is rather a form of dispersion engineering. On the other hand, outside this domain, amplification takes place only if optical loss is introduced to the idler component. These results might provide an alternative solution for utilizing the high nonlinearity of semiconductor materials which have been so far hindered by complex dispersion engineering requirements [22].
We note that in this work we have not discussed the details of how the optical loss can be introduced into the optical mode of the auxiliary waveguide. One possibility is to use metal strips on the waveguide surface as has been done in [5]. Another option is to use dopants that exhibit frequency selective absorption spectrum in the bandwidth of interest. These two strategies are material dependent and might not provide enough flexibility to operate at any desired frequency. Another attractive alternative is thus to employ structures whose effective complex index can be tailored at will by means of geometric designs. These include plasmonics, metamaterial and nano-antennas that, if designed properly, can function as meta-absorbers [26]. Another advantage of using meta-absorbers is also the possibility of engineering the spatial profile of the optical absorption by controlling the spatial distribution of the density and geometric design of these configurations. We carry out this study in future work.
Finally we note that other recent works that studied wave mixing processes in the presence of gain and loss can be found in [27,28]. Additionally, we mention in passing that the similarity between PT phase transition and the crossover between the oscillatory and amplifying behavior in parametric down conversion in Hermitian systems was noted in [21] and soon after investigated in details in [29,30].  We now seek solutions of equation (2) that imply amplification, i.e. solutions that exhibit imaginary parts of λ. We do so by noting that the solutions of equation  Asymptotically, for large x, the above equations reduce to the linear relation -=  y x 2 . The simplicity of this last expression begs for an intuitive explanation.
In order to gain more insight into this behevaior, we rewrite equation (1)