## Introduction

It is well known that two identical elastic bodies must exchange energy at the same rate as long as the conservative collision process is reciprocal. This situation naturally arises in Hermitian Hamiltonian systems where two parties can exchange energy in a fully symmetric fashion. Interestingly, more than two decades ago, Hatano and Nelson predicted the emergence of delocalization in a special class of exponentiated non-Hermitian random quantum Hamiltonians in order to address some outstanding problems in classical statistical mechanics1,2,3. Such Hamiltonians are quite ubiquitous in nature, describing a wide span of phenomena ranging from various non-equilibrium processes4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25 to asymmetric XXZ spin chains and spatial inhomogeneities in biological networks26,27, to name a few. Recent studies also suggest that such lattices provide a route for realizing a new category of topologically non-trivial states in non-Hermitian systems4,6,28.

The Hatano-Nelson model features a lattice with asymmetric coupling terms induced by an external field that operates like an imaginary vector potential. A schematic of a 1D Hatano-Nelson array along with a conceptual realization in the optical domain are shown in Fig. 1a. The Hamiltonian of this lattice is described by:

$$\hat H = - t/2\mathop {\sum }\limits_n \left( {{{{\mathrm{exp}}}}\left( { - g} \right)\hat a_{n + 1}^{\dagger} \hat a_n + {{{\mathrm{exp}}}}\left( g \right)\hat a_{n - 1}^{\dagger} \hat a_n} \right)$$
(1)

where $$\hat a_n^{\dagger}$$ and $$\hat a_n$$ are the bosonic creation and annihilation operators at sites n; t is the hopping strength, and $$g \in {\mathbb R}$$ represents a “non-Hermitian external field” (Fig. 1b). In 2D, the Hatano–Nelson model reduces to a quantum Hall system for spatially varying imaginary values of g. When subject to periodic boundary conditions, the Hatano-Nelson lattice supports a set of delocalized states with pair-wise complex eigenvalues (see Fig. 1c). On the other hand, under open boundary conditions the eigenvalues are entirely real, whereas the eigenmodes exhibit non-Hermitian skin effects and the energy distribution in the array tilts towards one of the two ends (see Fig. 1d). The directionality and strength of the power imbalance across the array is dictated by the sign and value of the field parameter g. Recently, this type of lattice has been demonstrated in acoustic systems10, mechanical metamaterials29,30 and electrical circuits31,32, and also proposed in elastic media33 and cold atoms20. In optics, such lattices have been only implemented in synthetic dimensions, leading to the observation of light funneling with interface localization12, non-Hermitian bands with arbitrary winding numbers13, and complex-energy braiding14. However, the realization of such non-reciprocal coupling processes in real space have so far remained difficult if not elusive34,35,36.

In this work, we utilize non-symmetric exchange interactions arising in judiciously coupled active resonators in order to tailor the response of a corresponding Hatano–Nelson phased locked laser array. The building block of our system is shown in Fig. 2a, where two III–V compound semiconductor active microring resonators are coupled through a pair of asymmetrically terminated link waveguides. These connecting sections provide an asymmetric delayed mutual coupling by reflecting the field at one end (sharply terminated), while gradually dissipating the energy at the tapered side. This coupling scheme affects the spontaneous emission by modifying the density of states, thus promoting an energy circulation in the rings only in one direction37 (here counter-clockwise) as shown in Fig. 2a. The mathematical description of this behavior is provided in the Supplementary Section 1. The coupling from resonator is given by $$\kappa _R = {\rm{i}}\gamma _u{\rm{e}}^{{\rm{i}}\beta L}$$, whereas that from is expressed by $$\kappa _L = {\rm{i}}\gamma _l{\rm{e}}^{{\rm{i}}\beta L}$$, where β is the propagation constant of the TE0 waveguide mode and L is the length of the links. The right-left coupling strengths between the two rings are determined by the gain/loss coefficients, $$\gamma _u$$ and $$\gamma _l$$, that are in turn controlled by the pumping profiles of the upper and lower links, respectively. It should be noted that the resulting asymmetric energy exchange between the resonators is enabled in part by the unidirectional circulation of power in the rings16,17,18, something that is impossible to achieve in passive structures, even if the links endure varying levels of loss37.

## Results

While the ratio between $$\gamma _u$$ and $$\gamma _l$$ determines the degree of asymmetry between the coupling coefficients ($$g = 1/2\ln (\gamma _l/\gamma _u)$$), the value of βL plays an important role in governing the lasing properties of the array. When βL=, where m is an integer, both coupling coefficients are imaginary, resulting in a splitting in the imaginary parts of the corresponding eigenfrequencies. The two modes of the system will then occur at the same wavelength albeit with different quality factors. Clearly, in this case, the mode with the higher quality factor is poised to lase. This situation is favorable for phase locking as it results in single mode lasing operation. Furthermore, depending on whether the integer m is even or odd, this mode can be in-phase or π-out-of-phase. It should be noted that the situation described here, where the asymmetric coupling is complex, is unique to our experimental system, and thus represents a generalization of the original Hatano–Nelson model in which under open boundary conditions all eigenvalues are real. On the other hand, when $$\beta L = (m + 1/2)\pi$$, the coupling coefficients are real, and therefore two lasing modes with identical quality factors will emerge. For any values in between, the system is expected to support two modes at two different frequencies and with varying quality factors. In practice, however, as long as βL remains in the vicinity of , the two eigenvalues are complex with a substantial difference between their imaginary components, and therefore single mode lasing is expected to prevail.

The response of the asymmetrically coupled microring lasers is experimentally characterized by examining their emission properties. The aforementioned coupled resonant systems are fabricated on an InP semiconductor wafer, covered by 6 quantum wells of InGaAsP with an overall thickness of 200 nm. The fabrication procedure is outlined in Supplementary Section 2. The ring resonators have a radius of 5 µm. All waveguiding sections feature a high-contrast core ($$n_{{{{\mathrm{core}}}}} = 3.4$$) with a width of 500 nm and a height of 200 nm that is embedded in a silicon dioxide film ($$n_{{{{\mathrm{SiO}}}}_2} = 1.45$$) and is exposed to air on top. These structures are designed to support the TE0 mode with an effective index of $$n_{{{{\mathrm{eff}}}}} = 2.24$$. To promote lasing in a single mode, βL of the upper and lower links are designed to be close to at the operating wavelength (see Supplementary Section 3 for more detail). The fabricated samples are then tested in a μ-photoluminescence setup at room temperature with a pulsed pump laser (wavelength: 1064 nm, pulse width: 15 ns, repetition rate: 290 kHz). To establish the asymmetric coupling, the pump profile is shaped using a combination of knife edges before being imaged on the sample plane, where it is partially blocked from the upper or lower links depending on the sign of asymmetry. In order to confirm the light direction of circulation, each ring is accompanied by a bus waveguide that is terminated at two grating couplers. In the experiments throughout this study, all resonators are uniformly pumped. For more information about the measurement station and methodology see Supplementary Section 4.

Figure 2b, c show the measured emission profiles and spectra of the asymmetrically coupled two-resonator system with $$\beta L \cong m\pi$$. This phase condition is verified by 16 samples with varying length L (see Supplementary Section 3 for more detail). For the purpose of visual comparison, the layout of the structure is inserted in the background, and the pumped area is specified by a bright rectangle. In this configuration, pumping the upper link leads to g < 0 (g = −1.66 in Fig. 2b), while g > 0 when the lower link is illuminated (g = 1.66 in Fig. 2c). Examining the emission intensity from the gratings (which is expected to be linearly proportional to that in the two counter propagating directions) confirms that indeed the energy circulates in a counter-clockwise direction in the rings. In addition, a significant intensity imbalance is observed between the two resonators (attributed to the coupling asymmetry), leading to an energy shift in the array towards the left or right ring as a result of pumping the lower or upper link, respectively. Furthermore, in both cases the emission spectra are single-moded, indicating that the proper coupling phase conditions are experimentally established. Notice that, in Fig. 2b, the bus waveguides are partially pumped in order to reduce the loss in the path towards the gratings, which causes a small power residual appears at the clockwise output. On the other hand, in Fig. 2c, the lower links are not fully pumped, which reduce the spontaneous emission in the bus-waveguides and maintain a relatively high visibility. The light-light curve displayed in Fig. 2d and the spectrum evolution presented in Fig. 2e further confirm that the array indeed operates as a phased-locked laser system.

Next, we examine the emission properties of a 5-element Hatano–Nelson laser array, having the above asymmetrically coupled two-resonator system as a building block. Figure 3a depicts the microscope and scanning electron micrograph (SEM) images of the fabricated array. Similar to a two-level system, the coupling ratios can be adjusted by spatially varying the pump profile overlap with the links. Figure 3b, e, and h show the emission intensity of the array when the non-Hermitian field parameter changes from g = −1.78 to g = 0 to g = 1.78. Consequently, the corresponding mode profiles are reported in Fig. 3c, f, and i, where the peak of the lasing mode shifts from one end of the array to the other. As expected, at g = 0 the array supports a fully symmetric mode. By gradually shifting the pump profile from covering the upper to the lower link, one can steer the beam in the near field from one end of the array to the other. This situation is recorded in Supplementary Section 5 and Supplementary Videos 1 and 2. In all these cases, the system operates in a single longitudinal mode (Fig. 3d, g, and j) dictated by the mode discrimination afforded by the proper link length choice ($$\beta L \approx m\pi$$). Similar results are observed in larger lattices of 11 elements (see Supplementary Section 6).

Finally, so far we focused on structures with link lengths satisfying $$\beta L \cong m\pi$$. However, the non-Hermitian skin effect still persists in asymmetrically coupled lasing arrays even when βL significatly deviates from this condition (Fig. 4a). Figure 4b, c, f, and g display the intensity profiles and spectra of a two-level system under such phase locking conditions (see Supplementary Sections 7 and 8 for detailed analysis). Even though in these cases the emission spectra involve two lasing modes, the intensity profile across the array nevertheless follows the same trend observed in their single mode counterparts—a result of incoherent superposition of the fields (intensity superposition) associated with various modes in each site. Similar behavior is also observed in a 5-element lattice (Fig. 4d, e, h, and i) (see Supplementary Section 9 for detailed analysis). In these cases, the eigen-frequencies are in general complex due to the coupling phase, despite the fact that the array does not feature periodic boundary conditions.

## Discussion

In this work, we focus on the simplest case of 1D Hatano–Nelson model. More complicated arrangements, for example, the ones that expand the demonstrated array to 2D and utilize high order hopping such as next nearest neighbor couplings, can pave the path towards large-scale phase-locked laser arrays with controllable intensity profiles and phase profiles. Theoretical works that involve similar concepts have proposed pulse-shortening in synthetic dimensions38.

In conclusion, we have reported the first realization of Hatano–Nelson lattices in a laser setting by judiciously introducing a non-symmetric coupling between active resonators. The skin effect observed in such lattices may provide a new approach for near-field beam steering where the energy distribution throughout the array is globally controlled by modulating the gain/loss levels in the link areas. We also explored the effect of non-Hermitian delayed mutual coupling on laser phase locking. Our work may provide new avenues for near-field beam steering of phase locked laser arrays while shedding light on the intriguing physics of non-symmetrically coupled systems.

## Materials and methods

The details of sample fabrication and experimental setup are provided in Supplementary Information Sections 2 and 4, respectively.