Supersymmetry Laser Arrays with High‐Order Exceptional Point

Supersymmetry (SUSY) laser array with superpartner structure can suppress excess modes to achieve high‐intensity and high‐coherent radiation. Compared with complex superpartner, using parity time (PT) symmetry broken to manipulate SUSY laser arrays is a more flexible approach. Herein, based on ultrathin perovskite single crystal, a SUSY laser array is constructed without superpartner, but with auxiliary gain–loss structure. PT symmetry broken with high‐order exceptional point (EP) is realized in the visible spectral range. It intrinsically avoids superpartner field mismatch and improves side mode suppression ratio 8.8–12 dB for single‐mode lasing. Furthermore, variations in EP‐order enable the transition from single‐mode to dual‐mode or tri‐mode radiation and retain amplified radiation characteristics. In addition, these SUSY laser arrays with the high‐order EP have the potential to be small‐volume optical sensor devices. The new design combining SUSY and PT symmetry broken presents its potential in micro‐nanophotonic devices with the benefit of small size, flexibility in spectral control, expandability, simplicity, and multifunctionality feature.


Introduction
Integrated semiconductor laser arrays are made by a large number of unit emitters, which present high-intensity and high-beam quality. They are the laser sources for the next generation applications. [1][2][3] Phase locking and coherent coupling within a laser array system [4,5] are indispensable to scale up the radiance and improve beam quality. However, due to adjacent cavity coupling, the degenerate longitudinal modes would split, [6] which indicates that the laser array inevitably operates on closely packed multiple transverse-mode oscillations. The mode competition often results in lasing pulsation and filamentation that degrade the spectral, spatial, and temporal properties. Therefore, eliminating the complex transverse mode caused by the coupling is a daunting task. Fortunately, novel symmetry concepts, such as supersymmetry (SUSY) and parity time (PT) symmetry, [7,8] have provided a rich and flexible means for spectral engineering.
SUSY describes the transformation between a fermion and its boson superpartner which originated from quantum field theory. [9,10] Interestingly, the isomorphism of Schrödinger and scalar Helmholtz equations enables manipulating the localization, [11] scattering, [12] and propagation properties [13] of light with the concept of SUSY. In physics, SUSY transformations construct two isospectral Hamiltonian optical systems except the ground state, [14,15] as shown in Figure 1a. The superpartner can couple with the original optical structure to control the optical properties. Therefore, new applications in photonics are emerging such as SUSY mode converter [13] and SUSY single-mode laser array, [16][17][18] etc. However, the existence of superpartner also limits the optical system: 1) SUSY transformations, such as orthogonal-triangular QR matrix decomposition, [16] result in complex optical structures. In addition, the superpartner inevitably occupies the optical device volume; 2) The increase in the number of structural units degrades the mode performance due to accumulated field mismatch between superpartner and main array [19] ; and 3) Once the superpartner structure is determined, the system is inflexible.
To manipulate the spectrum while eliminating the negative effects of superpartners, the introduction of PT-symmetry is a novel approach. In a non-Hermitian system with PT symmetry, the gain and loss can be controlled electrically or optically, [20,21] which brings convenient and flexible way to construct spectrum. However, to generate high-order exceptional points on demand DOI: 10.1002/adpr.202300143 Supersymmetry (SUSY) laser array with superpartner structure can suppress excess modes to achieve high-intensity and high-coherent radiation. Compared with complex superpartner, using parity time (PT) symmetry broken to manipulate SUSY laser arrays is a more flexible approach. Herein, based on ultrathin perovskite single crystal, a SUSY laser array is constructed without superpartner, but with auxiliary gain-loss structure. PT symmetry broken with high-order exceptional point (EP) is realized in the visible spectral range. It intrinsically avoids superpartner field mismatch and improves side mode suppression ratio 8.8-12 dB for single-mode lasing. Furthermore, variations in EP-order enable the transition from single-mode to dual-mode or tri-mode radiation and retain amplified radiation characteristics. In addition, these SUSY laser arrays with the high-order EP have the potential to be small-volume optical sensor devices. The new design combining SUSY and PT symmetry broken presents its potential in micro-nanophotonic devices with the benefit of small size, flexibility in spectral control, expandability, simplicity, and multifunctionality feature.
in the system, fully constituted desired energy levels ( Figure 1b) is crucial. [22,23] Based on the requirement, we look at combining SUSY and PT symmetry to design novel devices with small size, flexibility in spectral control, expandability, simplicity, and multifunctionality.
In this article, ultrathin single-crystal perovskite films are processed by a focused ion beam (FIB) to produce laser arrays. [24] Here, the SUSY transformation of adding target energy levels, instead of the previous elimination of energy levels, construct a SUSY laser array with uniformly spaced optical frequency. The structure units are uniform in size. Simple auxiliary structures are designed to introduce gain/loss in the system. No requirement on superpartner simplifies the structural complexity. Then, PT symmetry is used to control the energy level of SUSY laser arrays to achieve single-mode laser in the visible wavelength, rather than the previous infrared band. [16][17][18] Compared to the SUSY device with superpartner, the side-mode suppression ratio and quality factor are improved because there is no field mismatch from the coupling between the superpartner and the main array. [19] In addition, the change on auxiliary structure effectively adjusts the order of the EPs, which makes laser radiation modes reconfigurable (single-mode radiation can be converted to dual-mode or tri-mode radiation with scale-up intensity). Finally, the system has the capability of high-order exceptional point sensing. [25][26][27] The device has a small area of 12 μm Â2.5 μm. The research reveals that novel design by combining SUSY and non-Hermitian physics can open a new route for photonic devices.

Material
Perovskite has gained 450 cm À1 in visible wavelengths greater than other semiconductors. [28] Its gain and absorption properties allow the non-Hermitian symmetry breaking to occur effectively at very small scales, making it a new material for studying non-Hermitian photonics. [29,30] The non-Hermitian laser array in this article is realized by patterned perovskite. Material synthesis and FIB structure processing detail are in the Section SI, Supporting Information.

The Energy Level System Constructed by the Supersymmetry
SUSY laser array is built here with the purpose to construct equally spaced energy levels, facilitating the further introduction of higher-order EPs arising from PT symmetry. [23] The Hamiltonian for N units system is obtained by SUSY transfor- where N is the number of energy levels, a m (a þ m ) is the annihilation (producing) operator, J is the coupling between units, and w 0 is the resonant frequency.
In the experiment, the laser arrays based on SUSY transformation were realized on an ultrathin single-crystal perovskite film. Setting N ¼ 5 in Equation (1), its structural Hamiltonian is The main array is composed by coupling five identical waveguide cavities, as shown in Figure 2a. Each waveguide unit is a single longitudinal mode laser. The coupling strength J is controlled by the gap between waveguides, which is designed by simulation in Section SIII, Supporting Information. The designed array structure details are shown in Section SIV, Supporting Information, which has low complexity due to high symmetry and uniform waveguide size. In this laser array, the resonant modes couple and split into a cluster of five frequencies, which are called supermodes of the active array (Figure 2b). In the experiment, the pump light source (wavelength 450 nm, repeat at 1 kHz, pulse width 20 ps) was focused onto the laser array. When the pump intensity was low, the emission spectrum showed a broad peak centered at 538 nm with a full width at half-maximum (FWHM) of %21 nm. When the pump power www.advancedsciencenews.com www.adpr-journal.com exceeds the threshold of 1 mJ cm À2 , equally spaced laser peaks appear, corresponding to the designed Hamiltonian. As shown in Figure 2b, the spacing of the peaks is almost equal, with the coupling efficiency of 2J ¼ 3 AE 0.2 THz, extracted from the experimental spectra. The red box in Figure 2b indicates a nearly invisible peak at the gain band edge. The central peak wavelength is 545.6 nm, with half-maximum width of about 0.7 nm. The pump-emission correlations are shown in Figure 2c, illustrating the intensity amplification properties of the laser array toward the unit radiator. The radiation intensity of the array is %6 times to a single emitter.

Spectral Degeneracy under PT Symmetry Manipulation
Laser arrays with intensity amplification were implemented in the previous section, but no single-mode oscillations. The superpartner structure is a common mean for filtering out redundant modes in laser arrays, which suffers from high complexity and degrade mode performance. Instead, the PT symmetry is used here to degenerate the eigenlevels constructed. N-order EP can be realized by the following Hamiltonian where Equation (1) introduces a non-Hermitian term, AEg, which represents the gain and loss.
Setting N ¼ 5 in Equation (3), the real part and imaginary part of the eigenenergy changes as gained with the numerical calculation, as shown in Figure 3a,b. Concurrently, the PT symmetry breaking can effectively suppress the appearance of side modes and achieve a well-performing single-mode laser. In the simulation, the degenerate resonance appears after the EP with PT symmetry breaking, which implies the possibility of a singlefrequency radiation in this mode-coupled, multilevel system.
In the experiment, the PT symmetry breaking can be realized by strong uneven gain-loss configuration, which may  www.advancedsciencenews.com www.adpr-journal.com be performed electronically or optically, etc. In our study, to introduce PT symmetry into the laser array, triangular nonresonant auxiliary structures with 150 nm gap from the array ports act as energy addition and loss auxiliary device, as shown in Figure 3c. When the triangular structure is pumped, the waveguide neighboring to it gets more gain. When no pump present, it presents absorption and works as an additional loss. In the experiment, the realization of PT-symmetry breaking is that the pump spot is shifted slightly to the red circle in Figure 3c. When the pump intensity was low, the emission spectrum showed a broad peak centered at 538 nm. When the pump intensity is over the threshold, the single-mode laser FWHM is 1.22 nm at 5.0 mJ cm À2 due to incomplete degeneracy of the eigenenergy. With higher pump intensity of 6.7 mJ cm À2 , the FWHM reduces to 0.5 nm because the appearance of EP (see Figure 3d), which indicates the occurrence of the PT phase transition. The experimental results in Figure 3d show good lasing single mode with a side mode suppression ratio of 12 dB. The output power and FWHM of the device vary with the pump intensity, as shown in Figure 3e. The radiation intensity is approximately 3 times to a single waveguide at a near threshold. As a comparison, we made devices with superpartner to implement single-mode lasers under the same conditions (details in Section SV, Supporting Information). The structure of the superpartner is more complex and larger area than the auxiliary one. The experimentally generated single-mode spectra with superpartner mode suppression are shown in Figure 3f. The FWHM is about 0.8 nm (quality factor %675, compared to 1,200 with auxiliary structure), and side-mode suppression ratio is about 8.8 dB. Compared with the SUSY laser arrays of PT symmetry control, the quality factor is 0.6 times, and the side mode suppression ratio is 3.2 dB lower. Experiments illustrate that PT symmetry combined with SUSY to implement single-mode laser arrays can avoid complex redundant structures and degraded optical performance due to mode field mismatch and with better mode suppression.

Controllable Spectrum under EP Order Manipulation
The auxiliary structure in Figure 3c degenerates the 5-order EP in Figure 3a,b. Reconfiguring the gain/loss distribution could lead to other high-order EP, such as 3-or 2-order, which significantly changes the radiation spectrum of the SUSY laser array. This is a beneficial flexible feature when PT symmetry replaces the superpartner. It could be proposed that such uneven gain/loss can be achieved not only by auxiliary structure in this article, but also by patterned optical excitation, or by electronics in electric pumped lasing system with patterned current injection, etc.
On the premise of maintaining PT symmetry, we remove some auxiliary to construct a new Hamiltonian form.
The eigenenergy as a function of gain is shown in Figure 4a. Whether the eigenenergy can generate laser radiation is determined by the numerically solved imaginary part of the Hamiltonian, which represents the gain of the mode (Section SVI, Supporting Information). A third-order EP exists in the system, leading to symmetry breaking. Beyond EP, the intermediate mode of the system is a high-gain mode, while the other two modes are low-gain modes (see Figure 4a). The SEM image of the engineered optical structure is shown in Figure 4b. In Figure 4c, one high-intensity laser peak (Q = 885) and two low-intensity laser peaks (Q = 540) appear at 5.6 mJ cm À2 above the threshold. The two low-intensity laser peaks are at wavelengths of 542.2 and 548.6 nm, which are symmetrical with respect to the middle peak at 545.4 nm. Spectra of SUSY laser arrays match the predicted theoretical design in Figure 4a. The pump-emission intensity curve in Figure 4d shows that the radiation intensity of the SUSY laser array is about 8 times that of the unit waveguide. The experimental results favorably demonstrate the mode control of the reconfigurable EP for the SUSY laser array, while preserving the high radiation intensity of the array relative to the unit waveguide.
Likewise, the 2-order EPs are achievable by Hamiltonian The real part of eigenenergy as a function of gain is shown in Figure 5a, where two 2-order EPs exist in the system. The imaginary part in Section SVI, Supporting Information, determines whether the mode can form laser radiation. The intermediate www.advancedsciencenews.com www.adpr-journal.com mode of the system is a low-gain mode and the other two modes are high-gain modes (see Figure 5a). The SEM image of the optical structure is shown in Figure 5b. In experiment, when the pump power exceeds the threshold, dual-mode lasing peaks are presented with wavelength 543.4 and 546.7 nm, retaining a very low-intensity lasing peak sandwiched between them (the low-gain mode), as shown in Figure 5c. The FWHM is about 0.65 nm, corresponding quality factor %840. The pump-emission intensity curve in Figure 5d shows that the radiation intensity of the SUSY laser array is about 5 times that of the unit waveguide. The experimental results show not only that the SUSY combined with PT symmetry realizes high-intensity single-mode lasing, but also the novel design of manipulating spectral mode while scale up radiation.

Small Volume Optical Sensing
Besides the benefit of lasing, the presence of N-order EP in SUSY laser array can provide high-sensing capability for weak perturbations. [25][26][27]31,32] When the system is perturbed, the degenerate eigenenergy will split. The splitting energy spacing is proportional to 1/N power of the perturbation. Here, the EP system in Figure 3a is used as the demonstration of the study. The numerically solved eigenenergy splitting as a function of the perturbation intensity is shown in Figure 6a. The detailed experimental description is presented in Section SVII, Supporting Information. Figure 6b shows a typical splitting spectrum under perturbation in experiment, where single peak in Figure 3d splitting into multiple peaks. Figure 6c illustrates that the spectral splitting in the experiment at EP is proportional to the 1/5 power of the perturbation strength. In addition, the fifth-order EP sensor constructed here only occupies an on-chip volume of 12 μmÂ2.5 μm, which is very miniaturized compared with the previous second-order [25] (diameters 80 μm) and third-order EP [26] (60 μm Â 20 μm) sensing systems. In addition, it has the potential ability to reconstruct the EP order as mentioned above.
Our experiments provide a forward-looking approach to flexibly using SUSY and PT symmetry to design energy levels and achieve reconfigurable sensing.

Conclusion
In conclusion, we provide a new approach to designing photonic devices by combining SUSY and non-Hermitian physics. This combination presents benefits for designing new photonic devices, such as SUSY at the visible range, flexibility in spectral control, small size, expandability, simplicity, and multifunctionality. This type of device has an equal energy interval which forms SUSY laser array. With working on PT symmetry, single-mode laser is realized, avoiding the poor side-mode suppression ratio caused by coupling between the main array and superpartner. The laser FWHM is compressed from 0.8 to 0.5 nm, corresponding to an increase in quality factor to 1.6 times. The side-mode suppression ratio is also increased to 12 dB. It works at visible range, indicating its high-mode selectivity. It has variable EP order, which provides spectral control. Dual-mode and tri-mode lasers can achieve and maintain radiant intensity amplification in laser arrays (5 and 8 times, respectively). It has significantly smaller size, which presents effective mode coupling and potential expendability compared to other larger SUSY structures. Finally, it is multifunctional. Compact high-order EP pointsensing laser properties are experimentally demonstrated, which provides a method to realize arbitrary-order EP sensing. The eigenenergy as a function of gain for EP 2 . b) SEM image of a SUSY laser array with 2-order EP. Scale bar: 2 μm. c) Spectral mode under second-order EP control at 6.9 mJ cm À2 . d) Optical output intensity as a function of input intensity for EP 2 SUSY arrays and unit waveguides. Figure 6. Mode sensing. a) The eigenenergy as a function of perturbation for EP 5 . b) The splitting spectrum due to the perturbation. c) Experimentally measured sensing as a function of perturbation. The red line is the 1/5th power fitting curve. ε is perturbation calculated by simulation.