Non-Hermitian arrangement for stable semiconductor laser arrays

We propose and explore a physical mechanism for the stabilization of the complex spatiotemporal dynamics in arrays (bars) of broad area laser diodes taking advantage of the symmetry breaking in non-Hermitian potentials. We show that such stabilization can be achieved by specific pump and index profiles leading to a PT-symmetric coupling between nearest neighboring lasers within the semiconductor bar. A numerical analysis is performed using a complete (2+1)-dimensional space-temporal model, including transverse and longitudinal spatial degrees of freedom and temporal evolution of the electric field and carriers. We show regimes of temporal stabilization and light emission spatial redistribution and enhancement. We also consider a simplified (1+1)-dimensional model for an array of lasers holding the proposed non-Hermitian coupling with a global axisymmetric geometry. We numerically demonstrate a two-fold benefit: the control over the temporal dynamics over the EELs bar and the field concentration on the central lasers leading to a brighter output beam, facilitating a direct coupling to an optical fiber.

Recently, the interplay between gain and index modulation has emerged as a fruitful new research area in photonics. Initially introduced as a curiosity in quantum mechanics [21], parity-time (PT-) symmetry, found experimental realizations in the field of photonics in artificial materials with spatial distributions of real and complex permittivities, showing the ability of molding the flow of light [22,23,24,25,26]. The attentions to those systems that while being non-conservative could still hold real energy eigenvalues, derives from the unusual, even counter intuitive properties they hold arising from an asymmetric coupling of modes. Beyond the particular class of open non-conservative systems holding PT-symmetry however, there is a larger class of non-conservative as non-Hermitian Hamiltonians [27]. Indeed complex Non-Hermitian photonics has led to technologically accessible novel effects, from transparency and invisibility [28,29] to light transport [30] including various applications in laser science [31,32]. In particular, the new concepts of non-Hermitian photonics have successfully been applied to the control of the dynamics of broad semiconductor lasers [33,34], and arrays of vertical emitting semiconductor lasers or a ring array of semiconductor lasers [35,36].
Our proposal is intended to obtain a stable emission from an array of EELs and the improvement of its beam quality and energy distribution within the laser array. Altogether allows the direct coupling to fiber or optical guide without any optical component that should strongly enhance the coupling efficiency. The light generated in every single semiconductor laser is expected to be spatially redistributed and temporally stabilized via non-Hermitian coupling between neighboring lasers induced by a particular gain (pump) and index modulation (stripes) of the structure. The system is described by a complete (2+1)dimensional space-temporal model, including transverse and longitudinal spatial directions and temporal evolution of the electric field and carriers.
We first identify the onset of spatiotemporal instabilities for a single laser source, the regime of temporally stable and monomode emission severely restricts the power of the laser source. However, splitting a broad EEL source in an array of stable, thinner lasers with stable emission parameters, is not a solution, since new temporal and synchronization instabilities arise from the coupling between neighboring lasers leading again to irregular spatiotemporal behaviors. Thus, we propose a non-Hermitian asymmetric coupling between EELs within the array for stabilization and redistribution of the light emission. This emission improvement is first demonstrated for the simple two coupled laser system. We determine the stabilization performance as a function of the shift between the pumped region and laser stripes and the distance between the two lasers. Next, we analyze a system formed by three lasers holding a global mirror symmetry to induce an inward coupling in the laser array. In the following, a simplified (1+1)-dimensional model is used to extend the study to a full EEL bar formed by an array of many lasers. The simulations show both temporal stabilization and simultaneous spatial redistribution, i.e. localization, of the generated light.

Model for semiconductor laser arrays
In order to model the spatial redistribution and temporal stabilization of coupled EEL sources, we use a well-established model including the spatiotemporal evolution of the electromagnetic field and carrier density inside the cavity [20]. EELs are usually described either by stationary models [37] or dynamical models of the mean field [38]. Here, the complete dynamical model is used for the forward and backward fields propagating within the cavity and the carrier density. It was recently used to demonstrate spatial filtering of broad EEL sources [17]. Since the round-trip time of the cavity (on the order of ps) is small compared to the carrier's relaxation time (on the order of ns), the temporal evolution of the field in one roundtrip may be calculated by its propagation along the cavity assuming constant carriers. Applying the slowly varying envelope approximation, the forward and backward envelopes of the electric field, A ± are integrated along the EEL followed by the second step, the temporal integration of carriers considering a constant field. We neglect the frequency dependence of material gain, spatial hole burning of carriers, and heating-induced changes of model parameters since we assume we do not reach high powers [39,40]. Overall, this results in the following non-linear system of three coupled equations: where k 0 is the wavevector, n is the effective refractive index, σ is a parameter inversely proportional to the light matter interaction length, h is the Henry factor or linewidth enhancement factor of the semiconductor, α corresponds to losses, p 0 is the pump, D is the carrier diffusion coefficient and γ is the inverse of carriers' relaxation time, τ nr . In our calculations the transverse and longitudinal spatial coordinates are in units of the wavelength, time is normalized to the roundtrip time; N is normalized to N 0 (the carrier's density to achieve transparency) and the electric field envelope is normalized to τ ω  nr a being a the gain parameter, ω the angular frequency of light. Polarization of the material is eliminated in eq.
(1) as the semiconductor laser is considered a class B laser and the fine longitudinal interference between the forward and backward fields, considered to be blurred by the carrier diffusion, is disregarded. Finally, the transverse modulations of the refractive index Δn(x) account for the individual laser stripes; and the pump, Δp(x-s), where s is a spatial shift the spatial profile of the electrodes. Both modulations, Δn(x) and Δp(x-s), induce the real and imaginary parts of the non-Hermitian potential, which, properly designed, may lead to an asymmetric field coupling. See Fig. 1. (a) for a schematic representation of the laser architecture. In order to avoid discontinuities in the derivatives of these modulations, the two spatial transverse profiles are mathematically described as consecutive sharp sigmoids.
In the proposed scheme, the two profiles, Δn(x) and Δp(x), can be slightly spatially shifted a distance s, one with respect to another, and it is precisely this interplay between index and gain profiles that is expected to induce a non-Hermitian potential and asymmetric coupling between neighboring lasers. The boundary conditions are straightforwardly determined by the Fabry-Perot cavity mirrors located at z = 0 and z = L are where L is the length of the laser and r 0/L are the corresponding reflection of the edge mirrors at z = 0/L, respectively.
First, we numerically study the spatiotemporal behavior of a single EEL source through the system model in eq. (1) to determine its dynamics for different working conditions. As it is well known, decreasing the laser width acts as a mode selection mechanism when light is confined and a the broad and strongly multimode semiconductor emission turns into a monomode emission regime. However, to achieve a brighter source it is not enough to split a broad EEL source into an array of spatially stable thin EELs by patterning longitudinal separation slits between them. It is also necessary to engineer the coupling between lasers within the array to obtain stability and improve the quality of the emission, see Fig. 1(a). Numerical simulations supporting this idea are provided in Fig 1. (b), showing the total emitted output power of a single EEL for widths, w, ranging from 2.5 µm to 50 µm, and analyzing the field profile within the laser while decreasing width seeking for the onset of the monomode emission. The maximum width for a monomode emission determined by max 2 w n λ = Fig.1) is in go quality factor a Gaussian be w is the near provided in th Fig. 1. (

Symmetric and asymmetric coupling
Once the main dynamics and parameters of a single laser are determined we proceed to analyze the effect of the coupling between lasers on the dynamics. Such coupling depends on the distance between neighboring lasers and can be further engineered by introducing a displacement between the laser profile and the pump, as schematically shown in Fig. 1. (a). We first analyze the coupling between two identical lasers with the same intrinsic parameters and where the index and the pump profiles perfectly coincide, and are, therefore, symmetrically coupled. While the two standing alone lasers may have a spatially and temporally stable emission, as the distance between them decreases -the coupling strength increases-and keeping the rest of parameters, both lasers become temporally unstable. Spatial asymmetries are evident in every snapshot of the numerically calculated intensity distribution of two close EEL sources, see Fig. 2. (a). Besides, the temporal evolution of the transverse profile of the intensity at any position of the cavity length is aperiodic, see Fig. 2. (b). Next, we slightly shift the index profile of both lasers with respect to the gain profiles to induce a mirror-symmetric coupling. As expected, the light generated in one laser is partially transferred to the other one, see Fig. 2. (c). As an important consequence, when the energy is redistributed due to the asymmetric coupling, both lasers become temporally stable, as shown in the temporal evolution of the intensity transverse profile in Fig. 2. (d). This temporal stabilization tendency is in agreement with the general behavior of coupled nonlinear oscillators generally showing less complex dynamics for unidirectional than for bidirectional couplings.
The performance of the proposed asymmetric coupling is assessed by the asymmetric energy enhancement and temporal stability of the attained regimes. We calculate the enhancement as the relative intensity of the laser to which the energy is accumulated, i.e. as the ratio of the temporally averaged intensity of the enhanced laser, for s ≠ 0, versus the unshifted case, s = 0. We explore the parameter space of the distance between lasers, d, and asymmetry shift parameter, s (spatial shift between the pump and the refractive index profiles) for a fixed value of the pump. The results are summarized in Figs. 2. (e). While a larger enhancement could be expected by increasing the shift parameter s. However, we observe that the emission decreases for a maximum around a given shift value, namely s ≈ 0.25 µm. This decrease may be attributed to the asymmetric configuration of both lasers which also induces an asymmetric leaking of energy opposite to the direction of the other laser. Such leaked energy is therefore lost, and the net gain for the whole system is reduced; being the largest relative intensity of the enhanced factor around about 2. Moreover, such enhancement increases for a smaller distance, d. In addition, the temporal stability of the emission may be evaluated by mapping the amplitude of the temporal oscillations of the enhanced laser also in the distance-shift, (d,s), parameter space. Temporal instabilities arise for small d and s values, i.e. when the distance between lasers or coupling asymmetry decrease. For center-to-center distances close to the laser width the emission is found to be unstable for all values of s. On the contrary, stability is found either increasing the coupling asymmetry for a given laser distance, as also, trivially, at larger distances between lasers. Interestingly, inspecting Figs. 2. (e) and (f), we observe that there is a range of parameters around the maximum relative intensity region which coincides with a temporally stable behavior. profile Δp (red cu sity enhancement f d spatial shift, s. ations (normalized Fig. 1, and p where, / are complex numbers standing for the coupling parameter from the lasers at positions j+1 and j-1, respectively, and the spatial and temporal coordinates and all parameters are normalized as in eq. (1). In the simplest case, a periodic non-Hermitic potential in one-dimension may be approximated by a complex harmonic form: where d is the distance between lasers and and the amplitudes of the real and imaginary part of the non-Hermitian potential. Therefore, from eq. (2) we may express the coupling between neighboring laser as deriving from this simple harmonic complex potential as For Φ = ± π/2, the coupling is perfectly symmetric while for Φ = 0, the coupling becomes PT-symmetric. In turn, and for simplicity we assume m r = m i = 1, the PTsymmetry breaking point, which entails no restriction but assuming the maximally asymmetric situation. Despite the relationship between parameters of spatial modulation in both models is nontrivial, we can infer a logarithmic relationship between m m and the laser distance d as far as the coupling strength should decrease exponentially with the laser separation given by (d-w). This relationship is verified by direct comparison of the temporal instability in both models shown in Fig.5(b). The phase shift between the real and imaginary components of the coupling, π/2-Φ is proportional to the spatial shift s. The other parameters:σ , h, α, γ and p 0 are the same as in eq. (1).
The system of eq. (2) is integrated with boundary conditions given by the two mirrors of the Fabry-Perot cavity as for the complete model described by eq. (1). In order to compare the results of this simplified model to the complete model, we find equivalences between parameters comparing behaviors. We first determine the pump threshold p th and the onset of temporal instability, Hopf bifurcation p H , for a single laser to locate equivalent pump values, see Fig. 5. (a). The temporal instability onset corresponds to smooth oscillations of small amplitude, just for a pump interval above the bifurcation, abruptly changing to a pulsed regime with short and bright pulses, starting from almost zero constant output power, as observed with the complete model in Fig. 1. (c). In order to provide a comparison between the coupling parameters of the simplified model, namely just m m and Φ, coupling strength and phase shift, with the parameters of the full model, d and s, distance between lasers and shift between the gain and index profiles, we consider two coupled lasers and map the amplitude of the temporal oscillations of the field amplitude of the laser to which the energy is transferred. Inspecting the comparison in Fig. 5. (b) we conclude that for a symmetric coupling, hence Φ = π/2, instabilities arise for values of the coupling strength between lasers, m m , above 10 -6 equivalent to no asymmetric shift, s = 0 and distances d larger than 5.0 μm in the full model. Moreover, for coupling strengths of 10 -4 and a symmetric coupling the simplified model shows behaviors analogous to a distance between lasers d = 3.0-3.5 μm, and no asymmetric shift. As expected, the laser is temporally stable at the symmetry breaking point, hence for Φ = 0, although the transition to stabilization is reached for much larger values of Φ, about Φ ≈ π/3, which comparing the two models, turns out to be analogous to an asymmetric shift about s ≈ 0.3 µm. Thus, both models agree that it is not necessary to reach totally asymmetric unidirectional to attain stab intended to l generated fro elements and    for the considered parameters, in good agreement with numeric simulations.

Conclusions
In summary, we propose a physical mechanism for the temporal stabilization and localization of the emission of a coupled array of EELs or an EEL bar. The scheme is based on a non-Hermitian coupling between neighboring lasers with a global mirror-symmetric geometry. While the monomode emission of a single laser is assured by reducing its width, spatiotemporal instabilities may still arise from the coupling between lasers in an array. Such temporal instabilities are molded by a non-Hermitian coupling that may be simply introduced by a lateral shift between the pump and index profile, technically by a spatial shift between the individual laser stripe and corresponding electrode. Such asymmetric coupling, while temporally stabilizing the emission also redistributes and localizes energy close to the central symmetry axis.
The proposed stabilization scheme is analyzed by a complete spatiotemporal model, including transverse and longitudinal spatial degrees of freedom and the temporal evolution of the electric fields and carriers. We perform a comprehensive numerical analysis in terms of the design parameters, namely the distance between lasers and non-Hermitian shift observing regimes of simultaneous temporal stabilizations and light localization. In turn, the validity of the proposal is also demonstrated for an array with a large number of lasers using a simplified model where the transverse space is accounted by the coupling between neighboring laser cavities.