Supersymmetry-inspired non-Hermitian optical couplers.

Supersymmetry has been shown to provide a systematic and effective framework for generating classes of isospectral optical structures featuring perfectly-phase-matched modes, with the exception of one (fundamental) mode which can be removed. More recently, this approach has been extended to non-Hermitian scenarios characterized by spatially-modulated distributions of optical loss and gain, in order to allow the removal of higher-order modes as well. In this paper, we apply this approach to the design of non-Hermitian optical couplers with higher-order mode-selection functionalities, with potential applications to mode-division multiplexing in optical links. In particular, we highlight the critical role of the coupling between non-Hermitian optical waveguides, which generally induces a phase transition to a complex eigenspectrum, thereby hindering the targeted mode-selection functionality. With the specific example of an optical coupler that selects the second-order mode of a given waveguide, we illustrate the aforementioned limitations and propose possible strategies to overcome them, bearing in mind the practical feasibility of the gain levels required.

denote the propagation constants along the x-direction, and B 0 , A 1 , B 1 , A 2 , B 2 , A 3 are unknown expansion coefficients. From Eq. (S2), we can calculate the tangential magnetic field via the appropriate Maxwell's curl equation, viz., with η 0 denoting the vacuum characteristic impedance. By enforcing the electric and magnetic tangential-field continuity at the three interfaces x = ±w and x = 0, we obtain a 6 × 6 homogeneous system of linear equations in the unknown expansion coefficients, from which the dispersion relationship follows by zeroing the determinant, viz., with τ 1 = tan (γ 1 w) .
Equation (S6) needs to be generally solved (with respect to γ b or, equivalently, β 1 ) in the complex plane. However, for the assumed parameter configurations that maintain the PT -symmetric waveguide in the unbroken phase, real-valued solutions are expected.

Beat-length estimate
The numerical results from the supermode study in the balanced gain/loss scenario can also be utilized to obtain some rough estimates of relevant design parameters, such as the beat-length L B . Such parameter, defined as the propagation distance for which the power associated with a given mode of the original waveguide has been completely transferred to the partner waveguide, dictates the (minimum) length of the coupler. Our approximate semi-analytical approach is based on projecting the transverse profile U with β (m) denoting the numerically-computed supermode propagation constants. From Eq. (S8), we can estimate the beat-length by finding the positions of the nulls (or minima) in the field intensity peak distribution along z. Although, in view of the lack of exact orthogonality between the supermodes, this is not strictly rigorous, it still provides a rough estimate in qualitatively good agreement with the finite-element results.

Spontaneous symmetry breaking
Results in Table 2 and Fig. 5 indicate that the supermodes of order m = 1 and m = 2 pertaining to the compound structure in Eq. (15) are complex conjugate. This implies that the desired modeselection functionality (cf. Fig. 1) cannot be attained, as the exponential amplification of the m = 1 supermode would eventually dominate over the coupling effects. Figure S1 shows the effects of a slight loss unbalance iν (with ν > 0) introduced in the SUSY-partner profile ∆ε 2 , as an attempt to overcome the above limitation. It can be observed that the imaginary part of the propagation constant pertaining to the m = 1 supermode (and, hence, its exponential amplification) actually decreases, but this is accompanied by the appearance of positive imaginary parts (and, hence, exponential attenuation) in the propagation constants of the remaining modes. For this configuration, our approximate model in Eq. (S8) predicts that the power associated with the fundamental mode (n = 1) of the original waveguide is never significantly transferred to the partner waveguide, and hence a beat-length cannot be meaningfully defined.
To better understand these effects, Fig. S2 shows a finite-element-computed intensity field map, and three representative transverse cuts, pertaining to the compound structure in Eq. (15) (assuming w = 2λ 0 , D = 1.02, |Im (∆ε 1 ) | = 0.015, and ν = 0.008) excited with a linear combination of the three modes of the original waveguide (cf. Fig. 2c), with coefficients chosen so that the total power density is equally distributed among the modes. As expected, the m = 1 supermode profile dominates, resulting in output profiles (at two representative distances) that are quite different from the targeted one (i.e., the n = 2 mode of the original waveguide, shown as dashed curves in the transverse cuts).
SUSY * -based configuration Figure S3 and Table S1 show the numerically-computed intensity profiles and propagation constants, respectively, pertaining to the alternative SUSY * -based compound profile in Eq. (18). As it can be observed, the mode profiles are very similar to the SUSY case (compare with Fig. 4), but the propagation constants are markedly different (compare with Table 2). Figure S4 shows the effect of a slight loss unbalance in the SUSY * partner profile. We observe that, by properly tailoring this unbalance, it is now possible to drive all undesired supermodes in the exponentially-decaying regime, while maintaining the desired one (m = 3) in the exponentially-growing (or unattenuated) regime.    Table   S1) of the SUSY * -based compound profile in Eq. (18), for w = 2λ 0 and D = 1.02. The traces are vertically offset for clarity, and different colors are used to facilitate the association with the modes of the isolated waveguides (cf. Figs. 2c and 3c). Note that the profiles closely resemble those in