Single Transverse-Mode Semiconductor Laser Arrays by Parity-Time Symmetry Breaking

Electrically pumped semiconductor laser array operating in the fundamental mode is of vital importance. In general, the competition of multiple transverse modes in the laser array leads to spatiotemporally unstable and degradation of beam quality. In this letter, we propose and experimentally demonstrate electrically pumped single transverse-mode semiconductor laser arrays by exploiting the concept of parity-time symmetry, which is insensitive to fabrication and frequency detuning. Laser arrays composed of four, six and ten-ridge gain lattice have been fabricated, and single transverse-mode operations are observed from the spectral response and far-field radiation. Insensitivity to the fabrication have also been experimentally demonstrated by varying the gap from <inline-formula> <tex-math notation="LaTeX">$1.06 \mu \text{m}$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$1.42 \mu \text{m}$ </tex-math></inline-formula>. The emission power of PT-symmetric laser arrays composed of a four-ridge gain lattice is about five times higher than that of the single-ridge laser with the same current density.


I. INTRODUCTION
E LECTRICALLY pumped semiconductor lasers with high power output play a crucial role in a wide range of applications such as light detection and ranging systems. An integrated array of lasers can enlarge the active region and allow higher energies within the cavities, while avoiding the impact of optical nonlinearities [1]. Unfortunately, merely enlarging the active region inevitably leads to deterioration of beam quality. In general, a laser array supports multiple transverse modes (supermodes) of the same longitudinal order, and the spatial characteristic of the radiation degrades due to the non-linear modal competition [2]. A laser array design strategy based on supersymmetry (SUSY) is proposed to enforce fundamental mode operation in the inherently multi-mode laser array [3], [4], [5]. The propagation eigenvalues of the superpartner are carefully designed to match all higher-order supermodes except the fundamental mode associated with the main array. However, as the number of ridge waveguides in the array increases, more higherorder modes need to be carefully considered, and the design difficulty increases sharply. Fabrication error can also lead to Manuscript  propagation eigenvalues mismatch, resulting in deterioration of beam quality.
Recently, non-Hermitian parity-time (PT) symmetric systems have raised considerable attention. Investigations of PT-symmetric photonics have introduced many intriguing phenomena, such as coherent perfect absorption, double refraction, nonreciprocal light propagation, and sensitivity enhancement at an exceptional point [6], [7], [8], [9]. Periodic transverse PT-symmetric optical waveguides also have been reported [10], [11]. Explorations of the PT symmetry in photonics also provide new powerful methods for device design. Several experimental results demonstrate that PT symmetry can be employed for mode control in lasers [12], [13], [14], [15], [16], [17]. However, for an electrically pumped PT-symmetric laser, frequency detuning is inevitable because of carrier dispersion and thermal effects, which can be detrimental, especially if the coupling between the lasers is weak. For the wide-area coupled waveguide laser, increasing the area of the active region allows higher energies within the cavities but greatly reduces the coupling coefficient, making it sensitive to frequency detuning.
In this letter, we propose and experimentally demonstrate electrically pumped single transverse-mode semiconductor laser arrays based on parity-time symmetry, which is insensitive to fabrication and frequency detuning. By adopting a laser array structure, high coupling coefficients are obtained while increase the active region, thereby ensuring insensitivity to frequency detuning. The large difference in the coupling coefficient between the fundamental mode (TE 0 ) and firstorder (TE 1 ) mode also ensure fine fabrication tolerance. This design method can be easily generalized to the case of a large number of arrays for that only TE 0 and TE 1 modes need to be carefully engineered regardless of the number of ridge waveguides in the array. Single transverse-mode PT-symmetric laser arrays composed of four, six and ten-ridge gain lattice have been experimentally demonstrated. Insensitivity to the fabrication has also been experimentally demonstrated. Single transverse-mode operations are experimentally obtained in three PT-symmetric laser arrays composed of a four-ridge gain lattice with a gap of 1.06 µm, 1.25 µm and 1.42 µm, respectively. The emission power of PT-symmetric laser arrays composed of four ridges is about five times higher than that of the single-ridge laser with the same current density.  We assume that the gain array on the left and the loss array on the right. The ridge waveguides of the two arrays are identical, with the same ridge width W and gap g. For ease of fabrication, the gap between the left and right arrays is also chosen as G = g. Two isolated P-side electrodes are employed, and the PT-symmetry can be established or broken by independently tuning of gain-loss coefficients in the two laser arrays.

II. PRINCIPLE AND DESIGN
According to the coupled mode theory, the eigenfrequencies ω (1,2) m of the PT-symmetric laser array can be given as [12]: where ω m is the angular frequency of the m th transverse modes in each laser array, κ m is the coupling coefficient between the two laser arrays, γ ave = (γ am + γ bm )/2 and γ diff = (γ am − γ bm )/2, in which γ am and γ bm are the modal gain and loss of the two laser arrays, respectively. According to (1), as soon as the gain-loss contrast exceeds the coupling coefficient γ diff > κ m , the PT symmetry is spontaneously broken, a conjugate pair of lasing and decaying modes emerges. In our PT symmetric laser array, when the first-order mode (TE 1 ) is PT-symmetric, all other higherorder modes (m > 1) will remain in the PT-symmetric phase because that the coupling coefficient increases with mode order. Therefore, regardless of the number of ridge waveguides in the array, only the two transverse modes of the lowest order (TE 0 and TE 1 ) need to be carefully designed to achieve single transverse-mode operation. In specific, with appropriate gain-loss contrast and coupling coefficient κ 0 < γ diff < κ 1 , only the fundamental mode can reach the PT symmetry breaking threshold and single transverse-mode operation is achieved, as shown in Fig. 1(b). The largest gain contrast ( g = κ 2 1 − κ 2 0 ) between the fundamental mode and first order mode can also be obtained while γ diff = κ 1 .
However, considering the carrier dispersion effect and thermal effect, frequency detuning occurs in electrically laser even if two coupled resonators are structurally identical when the injection currents are different in two resonators. The frequency detuning must be considered and the eigenfrequencies ω of the PT-symmetric laser can be given as where ω diff = (ω am − ω bm )/2. In this way, PT symmetry breaking is thresholdless due to the frequency detuning, and the gain difference between TE 0 and TE 1 decreases sharply while the coupling between the lasers is weak. Without sufficient gain difference, multimode competition occurs and the beam quality deteriorates.
To solve this problem, a laser array structure is adopted for its larger coupling coefficients compared to the widearea coupled laser [16]. As shown in Table I, two different configurations with the same active region width of 12 µm are compared: a PT-symmetric laser array composed of a fourridge gain lattice and a wide-area coupled laser. In the PTsymmetric laser array, ridge width of W = 2.1 µm and gap g = 1.2 µm are adopted, while ridge width of W = 12 µm and gap G = 1.2 µm are adopted in the wide-area coupled laser. The PT-symmetric laser array has a weaker constraint on the optical field, and the coupling coefficients of the fundamental mode and first-order mode are both larger than those of the wide-area coupled waveguide laser. In this way, the largest gain contrast of the PT-symmetric laser array is much larger than that of the wide-area coupled waveguide laser. More importantly, the PT-symmetric laser array is insensitive to frequency detuning due to its larger coupling coefficients. With the same frequency detuning, the PT-symmetric laser array can still maintain high largest gain contrast, while the largest gain contrast of the broad area-coupled waveguide laser decrease to 3.3 GHz.
Single transverse-mode operation can be achieved when the TE 0 reaches the PT symmetry breaking threshold and TE 1 keep in the PT-symmetric phase, while all other higher-order modes (m > 1) will automatically remain in the PT-symmetric phase. The large coupling coefficient difference between the TE 0 and TE 1 also ensures large fabrication tolerance of the PT-symmetric laser array. The fabrication tolerance of the PTsymmetric laser array is further investigated by varying the gap from 800 to 1600 nm with a step of 40 nm, and the simulated results are shown in Fig. 1(d). The coupling coefficients of both TE 0 and TE 1 decrease with the increase of the gap, but even if such gap changes by 800 nm, the two coupling coefficients still maintain a large difference, and the single transverse-mode lasing condition κ 0 < γ diff < κ 1 still holds within a large range of gain-loss contrast, as depicted in the blue region in Fig. 1(d).

III. EXPERIMENTAL CHARACTERIZATION
Several PT-symmetric laser arrays, a conventional laser array and a single-ridge laser are fabricated, with the same epitaxial structure (InAlGaAs based multi-quantum well) and similar device fabrication process in [17]. The scanning electron microscope (SEM) image of the PT-symmetric laser array composed of a four-ridge gain lattice is shown in Fig. 2(a). Each ridge is 200 µm in length, 1.96 µm in width, 1.96 µm in height, and each gap is 1.25 µm. The front and back facets are coated with optical films to enhance the output light power with the reflectivity of 70% and 92% respectively.
The light-current characteristics corresponding to three arrangements are shown in Fig. 2(b). The emission power of three PT-symmetric laser arrays are about five times higher than that of the single-ridge laser with current density of 3000 A/cm 2 . The thresholds and slope efficiencies of the PT-symmetric laser arrays are similar to the conventional laser array, because that the fundamental mode is mainly distributed in the gain array and loss array only plays the role of suppressing undesired higher-order modes.
The single-ridge laser and the conventional laser array are electrically pumped with a single electrode while the PT-symmetric laser array is electrically pumped with two independent electrodes. The gain-loss contrast can be effectively adjusted with different pump currents in the gain array and the loss array, and both the PT-symmetric phase and the PT-broken phase of the PT-symmetric laser array are observed. As shown in Fig. 3, the spectral response and far-field radiation profiles of four different configurations are compared. The single-ridge laser with an injection current of 36 mA lases in a single transverse mode with a few longitudinal modes. In contrast, when the conventional quadridge laser array is electrically pumped with an injection current of 36 mA, the optical spectrum of each longitudinal mode shows multiple peaks, implying that several supermode resonances are involved. For the PT-symmetric laser array in the PT-symmetric phase, an emission spectrum with multiple peaks in each longitudinal mode is observed when the gain and loss arrays are electrically even pumped with the same currents of 36 mA, as shown in Fig. 3(c). However, single transversemode lasing occurs once the PT symmetry is broken by tuning the injection currents of two arrays, with the injection currents of 36 mA and 0 mA. As shown in Fig. 3(d), the PT-symmetric laser array demonstrates the similar optical spectrum with single-ridge laser that only one peak in each longitudinal mode.
To validate the single transverse-mode response of the PTsymmetric laser array, the far-field radiation profiles are collected. These experimental results are shown in Figs. 3(e)-3(h). The far-field radiation profile of the PT-symmetric laser array in the broken phase is approximately single-lobe distribution, which is similar to that of the single-ridge laser. It indicates that the PT-symmetric laser array operates in single transverse mode. In addition, the far-field distribution along the slow axis of the laser array in the broken phase become narrower than the single ridge laser for its larger width. In contrast, when the bias of loss array is increased to 36 mA and the PTsymmetric phase is reconstructed, a multilobe profile similar to that of the conventional quad-ridge laser array appears, indicating the mode competition among supermodes. The farfield distribution along the slow axis of the laser array in the PT-symmetric phase is broader than that of the conventional quad-ridge laser array, because more higher order modes appear. To further investigate the fabrication tolerance of the PTsymmetric laser array, we altered the gap between ridges to 1.06 µm and 1.42 µm while hold the same active region width. For both of the PT-symmetric laser arrays under broken phase, the optical spectrums of each longitudinal mode show only one peak, and near single-lobe far-field radiation profiles are obtained with the bias currents of 37 mA and −5 mA, 45 mA and −5 mA respectively, indicating single transversemode operations. These PT-symmetric laser arrays at the broken phase have slightly different bias currents due to differences in their structure and performance.
To demonstrate the scalability of the PT-symmetric laser array, we design and experimentally demonstrate another two laser arrays composed of six and ten-ridge gain lattice with the same design method and fabrication process. For the arrangement of the six ridges, the ridge is 200 µm in length, 1.3 µm in width, 1.96 µm in height, and each gap is 1.29 µm. For the arrangement of the ten ridges, the ridge is 400 µm in length, 1.4 µm in width, 2.2 µm in height and each gap is 1.1 µm. In addition, to verify the applicability of this design method in C band, the laser arrays composed of ten-ridge gain lattice is fabricated on a wafer with a gain peak near 1530 nm. The spectral responses and far-field radiation profiles of the single transverse-mode operating characteristics are both obtained with the bias currents of 44 mA and −10 mA, 90 mA and −10 mA respectively, as shown in Fig. 4. The far-field diffraction pattern of the laser arrays composed of ten-ridge gain lattice becomes elliptical because the divergence angle of the slow axis becomes narrower and the divergence angle of the fast axis remains roughly the same.

IV. CONCLUSION
In conclusion, by exploiting the concept of parity-time symmetry, we provide the experimental realization of electrically pumped single transverse-mode semiconductor laser arrays. The propagation eigenvalues matching for all higher-order supermodes are not required. Single transverse-mode operation can be achieved when the TE 0 reaches the PT symmetry breaking threshold and TE 1 keep in the PT-symmetric phase, regardless of the number of ridge waveguides in the array. Our experimental results indicate that the PT-symmetric laser array is insensitive to fabrication. Single transverse-mode operations are still obtained with the gap varying from 1.06 µm to 1.42 µm. The emission power of PT-symmetric laser arrays composed of a four-ridge gain lattice is about five times higher than that of the single-ridge laser with the same current density. This design approach can be easily extended to the case of a large number of ridge waveguides in the array, and a single transverse-mode PT-symmetric laser array composed of a ten-ridge gain lattice is experimentally demonstrated. Our findings may have practical implications for designing single transverse-mode, wide-area and high-efficiency lasers.