A closed system is an ideal system that is independent and has no interaction with the external environment, ensuring that the Hamiltonian is a Hermitian operator. However, when the system becomes open, introducing gain and loss, the Hamiltonian of the system will become non-Hermitian [1–3]. This introduces many novel physical phenomena and properties distinct from Hermitian systems [4–12]. Parity-time (PT) symmetry is a novel physical phenomenon based on the non-Hermitian system, exhibiting the exceptional point where the system undergoes a phase transition [13,14]. Beyond the exceptional point, the originally PT-symmetric system experiences a spontaneous symmetry breaking. Research on PT symmetry initially stemmed from studies of electrons in quantum systems [15–19]. However, due to challenges, such as the presence of decoherence in non-Hermitian PT systems, severe effects of many-body interactions, and difficulties in achieving non-Hermiticity, research on PT symmetry gradually shifted towards the photonics domain [2,20–25].

The laser provides a fabulous platform to manipulate PT symmetry because of its characteristics of convenience to achieve gain and loss [2]. Since 2009, an increasing number of studies have applied PT symmetry to models of semiconductor waveguides [26,27]. From the perspective of laser model structures, the main structures include stripe double-ridge lasers [28–31], micro-ring lasers [32–34], VCSELs [35], distributed feedback lasers [36,37], and microcavity structures [38–41]. It has been demonstrated in existing research that the application of PT symmetry in semiconductor lasers leads to improvements in the far-field divergence angle, near-field beam size, and spectral characteristics (e.g., under specific manipulation, it is possible to achieve single-mode output and improve lateral mode characteristics) [32,35,41]. For a dual-element coupled system, the critical parameters determining the aforementioned performance are the relative phase and the ratio of field amplitudes between the two elements, both of which are determined by the system’s mode characteristics. In semiconductor lasers, the temporal coupled-mode theory (TCMT) and rate equation (RE) can, respectively, describe the temporal characteristics of the carrier density and electric field in the time domain for each individual unit, providing a foundational framework for studies on coupled laser arrays [31,42–47]. However, research that links PT symmetry with actual semiconductor laser models to analyze their temporal characteristics and compare them with experimental results is relatively scarce. This scarcity can be attributed to the presence of many complex nonlinear phenomena inherent in semiconductor lasers, particularly under electrical pumping conditions [34].

In this work, we theoretically investigate the relative phase, the ratio of field amplitudes, and the dynamic process in PT-symmetric system using TCMT and RE. Experimentally, we design stripe double-ridge lasers by electrical injection and test their horizontal far-field, near-field, and spectrum characteristics, including the relative phase and the ratio of field amplitude information. We introduce the concept of the phase transition region in the phase transition process of electrically injected PT-symmetric lasers, which was conventionally perceived as an exceptional point in previous studies. In this specific region, multimode oscillation is observed, while stable single-mode outputs are present both before and after this region. The relative phase and amplitude ratio within this region also exhibit anomalies. By varying the coupling strength and the pump intensity in the loss waveguide within a certain range, the position of the phase transition region can be altered. Additionally, the pump intensity in the loss waveguide can also change the range of the phase transition region. These conclusions have been validated through near-field and far-field measurements in different structures. This work demonstrates a distinct phase transition process of PT-symmetric lasers under the manipulation of two dimensions (gain contrast and total frequency detuning), contrasting with previous studies. The model accurately describes the process of symmetry breaking and is consistent with experimental results. In practical applications, this study provides a method for manipulating the single-mode output range and avoiding multimode outputs in PT-symmetric electrically injected lasers, which also establishes the theoretical foundation for artificially tuning multimode oscillation pulses in this system.

In our dual-element coupled model depicted in Fig. 1(a), the temporal evolution characteristics of carrier densities N1 (in the gain waveguide) and N2 (in the loss waveguide) can be investigated by the RE of the laser, while TCMT can be utilized to analyze the temporal evolution of electric fields E1 and E2 in the corresponding waveguides. Due to the PT-symmetric modulation, the gain contrast between the principal mode and the next-strongest mode in the Fabry–Perot (FP) cavity is enhanced [32] (as evidenced by the spectra in our experiments). Therefore, we only consider the coupling of the main modes in the two waveguides. They can be described by dN1,2dt=ηiI1,2qV−N1,2τ−vgg1,2Fele_pho|E1,2|2,(1)d|E1,2|dt=vgG1,2|E1,2|∓vg|E2,1|κsinΔϕ,(2)dΔϕdt=αvg(G1−G2)+κ(|E1||E2|−|E2||E1|)vgcosΔϕ−Δω.(3)

Here, |E1,2| and I1,2 are the field amplitudes and current in gain and loss waveguides, while Δϕ=ϕ2−ϕ1 and Δω=ω2−ω1 are the relative phase and the detuning frequency between them, respectively. ηi is the injection efficiency, V is the volume of the active region, τ is the carrier lifetime, κ is the real coupling coefficient between two waveguides, vg is the group velocity, and α is the linewidth enhancement factor describing the changes in the refractive index affected by the carrier density in the laser. In this model, the frequency detuning Δω is primarily affected by the redshift induced by the injection current I, represented by Δω=rΔI, where r denotes the frequency detuning factor. In order to investigate the characteristics of the electric field, the photon density Np in the original rate equations is transformed into field intensity, and their relationship is Np=Fele−pho|E|2=|E|2ε0ngnm/2ℏω. The optical gain g1,2 determined by N1,2 is in spatial domain, respectively, and we define the net gain G1,2=(Γg1,2−(αi+αm))/2, where the confinement factor is Γ and αi and αm are the internal loss and the mirror loss, respectively [47].

Linked with PT theory, Eqs. (2) and (3) of the model have formal similarities with the Schrödinger equation [1,2]. The expression for the Hamiltonian of the system is obtained as $$\begin{gathered} H^=vg[iG1+(αG1+ω1/vg)κκiG2+(αG2+ω2/vg)] \\ =vg(κσx+(igdiff+γdiff)σz+(igsam+γsam)σ0), \end{gathered}$$(4)where gdiff=(G1−G2)/2, gsam=(G1+G2)/2, γdiff=((G1−G2)α+(ω1−ω2)/vg)/2, and γsam=((G1+G2)α+(ω1+ω2)/vg)/2. The complex propagation constants β (in spatial domain) of the supermodes in this system are the eigenvalues of Eq. (4) as β=(igsam+γsam)±κ2+(igdiff+γdiff)2.(5)

To explore the properties of the PT symmetry, we employ the Runge-Kutta method of order (4,5) to numerically solve Eqs. (1)–(3). Under specific conditions, |E1,2|, N1,2, and Δϕ are solved when time t is big enough for the steady states, and as a time-averaged value over an extended duration for the unsteady states. Therefore, the β can be solved from the Eq. (5), as g1,2 is determined by N1,2.

Utilizing the approach mentioned above, we conduct the following analysis (see details in Appendix A). The magnitude of the coupling coefficient κ (relative to the magnitude of gain and loss G1,2) determines the system’s phase transition process and the modal properties at each stage. To elucidate the physical properties of PT symmetry, the model is idealized, i.e., frequency detuning and the linewidth enhancement factor are ignored. First, we investigate the weak coupling case in the PT-symmetric model. In this model, we assume coupling coefficient κ=1.57cm−1, current in loss waveguide I2=35mA, and other parameters with standard values in semiconductor laser research. The intrinsic propagation constant of a single FP resonator is defined as β0. The real part and the imaginary part of the complex propagation constants (β−β0) are calculated for different currents in gain waveguide I1 in Fig. 2(a). The difference of the real parts represents the frequency splitting between modes in physics (when transformed into the time domain), while the imaginary part represents gain and loss. We find the exceptional point in Fig. 2(a), corresponding to findings reported in previous studies. The spontaneous symmetry breaking happens at the exceptional point. The relative phase Δϕ and field amplitude ratio |E1|/|E2| are shown in Fig. 2(b). Before the exceptional point, the field amplitude ratio is pinned to 1, and the relative phases are in the range of (0, π/2) and (π/2, π), corresponding to in-phase mode and out-of-phase mode, respectively. In the structures investigated in this study, the out-of-phase mode is preferentially stimulated due to experiencing lower loss. After the exceptional point, the two supermodes evolve into the gain mode and the loss mode. Relative phases are both locked at π/2, and the field amplitude ratios are no longer 1. Figure 2(c) depicts the temporal evolution of |E1,2|, N1,2, and Δϕ, under I1=42mA (before the exceptional point) and I1=250mA (after the exceptional point), respectively, all in the form of steady-state solutions. We also analyze the strong coupling case and find some novel phenomena, e.g., the emergence of unsteady state and the existence of two exceptional points during the phase transition process [48].

However, in practical semiconductor lasers by electric injection, thermal effects, gain saturation, and variations in carrier density induce changes in the refractive index, deviating from ideal models [31,35]. We consider a part of these factors to refine our model, aiming for improved concordance with practical phenomena. Based on this, we modify the previous ideal PT-symmetric laser model to analyze the practical PT-symmetric laser model here. In the practical model, beyond the impact of gain contrast between the two waveguides, the frequency detuning and linewidth enhancement factor contribute to the system in an additional dimension. This can be demonstrated through Eq. (5), where they collectively influence the parameter γdiff. Here we set linewidth enhancement factor α=2.5 and frequency detuning factor r=7.6×1011A−1·s−1, obtained from the measurement in a single FP laser with the same structure. We utilize a method similar to that shown in Fig. 2 for analyzation, resulting in the findings depicted in Fig. 3 (see details in Appendix A). There exhibits a phase transition region, corresponding to the exceptional point during the process of PT symmetry breaking. However, here it is not merely a point but rather a range. The real part and the imaginary part of the complex propagation constants (β−β0) as a function of current I1 are given in Figs. 3(a) and 3(b), where the main difference from the ideal laser model is that the real part after the phase transition and the imaginary part before the phase transition are no longer degenerate. This results in one mode being suppressed both before and after the phase transition region, leading to single-mode output. The kink in the curve of total output power in Fig. 3(c) indicates the instability of the mode in this region. Figure 3(d) depicts the field amplitude |E1| and |E2| under different currents I1.

In the following sections, we will analyze the relative phase and amplitude ratio data obtained from experimental results in both near-field and far-field measurements. Subsequently, we will compare them with the relative phase and amplitude ratio calculated theoretically in the practical model. Finally, we will conduct a comprehensive analysis of PT-symmetric lasers under different structures, revealing the underlying physical principles.

To investigate the PT symmetry in the semiconductor laser, we construct the dual-element coupled PT stripe laser experimentally. The dual waveguides are obtained by the photolithography process and inductively coupled plasma (ICP) etching. The coupling strength is controlled by the distance between two waveguides. Based on this, we design waveguides with two structures: one with the distance of 2 μm (corresponding to κ=6.5cm−1) and the other with the distance of 4 μm (corresponding to κ=3.9cm−1). The epitaxial structure is described in Ref. [49], where the active region consists of double quantum wells. The laser operates around 960 nm, and the optical field is guided by the refractive index of the ridge waveguide. The cavity length is designed to be 0.6 mm. The thin-film coatings on the front and rear cavity facets have reflectivity of 0.49 and 0.99, respectively. The scanning electron microscope (SEM) image of the device is shown in Fig. 1(b). The different gain between two waveguides is implemented via patterned electrodes, which alter the pump strength between them by different currents I1 and I2. The injected current exhibits lateral diffusion. Through theoretical and experimental measurements of the diode current and voltage characteristics, we find that the current leaking into the other waveguide is at the order of one-twentieth. Therefore, this factor has minimal impact on the model (see details in Appendix B).

We first discuss the situation when the distance between two waveguides is 2 μm and the current in loss waveguide I2 is maintained at 0. The light-current-voltage (LIV) curves are given in Fig. 4(a). The kink in the LI curve indicates the phase transition as the increase of gain in the gain waveguide, and this also occurs in our theoretical model. The threshold current of the dual-element stripe laser is around 30 mA. As the pumping current I1 in the gain waveguide increases, a kink appears around 180 mA, indicating the occurrence of mode hopping. The phenomenon is reflected in the spectra of Fig. 4(b). From the spectrum in Fig. 4(b), we observe the presence of multimode oscillation in this region. When the current is lower than this range, due to weaker gain, the laser operates in single longitudinal mode. However, when the current exceeds this range, spontaneous symmetry breaking occurs, hence maintaining single-mode operation. The side-mode suppression ratios of the spectra at 120 mA and 200 mA are 31.6 dB and 35.0 dB, respectively. A detailed analysis will be provided in the subsequent sections.

To investigate the phase transition process in this PT-symmetric laser, the near-field and horizontal far-field are measured under different currents I1. The experimental platform maintains a constant temperature of 25°C using the thermoelectric cooler (TEC), while the current source operates in continuous wave (CW) mode. The horizontal far-field images are acquired by scanning the output beam spot along a circular arc at distances beyond the Fraunhofer distance as shown in Fig. 5(a). The near-field images are obtained using the 4f imaging system as shown in Fig. 5(b). In this test system, the optical spot is directed through a 100× objective lens with a working distance of 12 mm, followed by Fourier transformation, and ultimately converged onto a CCD camera via a collector lens with a focal length of 150 mm. Before and after the phase transition region, both the near-field and far-field patterns transition from the double-lobed to the single-lobed, indicating a drastic change in both field amplitude ratio |E1|/|E2| and relative phase Δϕ.

For a deeper understanding of the phase transition and symmetry breaking in the practical PT-symmetric laser in our system, we calculate the amplitude ratio |E1|/|E2| and relative phase Δϕ as follows. We derive the field intensity distribution of the FP cavity with the same structure through simulation. This distribution is assigned to one waveguide, while the field in another waveguide is multiplied by the relative phase and amplitude ratio. By superimposing them, we obtain the near-field distribution, and through diffraction transformations, the far-field distribution is obtained. These distributions are then compared with experimentally obtained near-field and far-field distributions to determine the corresponding relative phase and amplitude ratio. The amplitude ratio and relative phase under different currents I1 are depicted in Fig. 6(b). It can be observed that there exists a phase transition region here, as described before, rather than a single transition point (the exceptional point). We observe steady-state behavior in both modes before and after the phase transition region. By performing fast Fourier transform (FFT) on the output electric field, we obtain single-mode outputs in both cases. However, within the phase transition region, the output electric field oscillates in the time domain, and its FFT results in a multimode behavior, indicating the occurrence of multimode oscillation. The relative phase in this region cannot be locked, thus exhibiting instability temporally (see details in Appendix C). Therefore, the relative phase in this region depends on the initial settings (such as initial carrier density and initial relative phase), while stable regions are unaffected by the initial settings within a large range of variation. Thus, in this region for Figs. 7(c, i) and 7(f, i), we obtain the average value of the relative phase over a sufficiently long period of time as the relative phase under the specific current I1. As for Figs. 6(b, i), 6(c, i), 7(b, i), and 7(e, i), we fix a specific initial value for consistency in our analysis.

In the experiment, we derived the relative phase and amplitude ratio information of the electric field from the near-field and far-field data in Fig. 5, as depicted in Fig. 6(b, ii). When the pump current I1 exceeds 170 mA, the precision of the CCD camera cannot measure the field distribution in the loss waveguide. This suggests that the amplitude ratio is at a high value, depicted by the black arrow in the figure. Additionally, this implies that the relative phase cannot be accurately determined at this case.

Based on the model established above, we conduct theoretical analyses and experiments on several different structures. This allows us to investigate the influence of coupling strength and pumping intensity in the loss waveguide on the phase transition process in electrically injected PT-symmetric lasers. First, we manipulate the separation between the two waveguides to adjust the magnitude of the coupling strength in Fig. 6(a). The structures are all pumped by a single side, and the spacing is 2 μm in Fig. 6(b) and 4 μm in Fig. 6(c). The amplitude ratios both decrease towards 1 before the phase transition region, approach unity within the phase transition region, and sharply increase afterwards. The relative phases both experience a sharp decline within the phase transition region. The impact of coupling strength on the system is primarily characterized by its ability to shift the position of the phase transition region when coupling coefficient k is below about 7cm−1. Increasing the coupling strength within a certain range can lead to the forward displacement of the phase transition region. Furthermore, by comparing the results from Figs. 6(b) and 6(c), the weakening of coupling strength results in a reduction of the power coupled from the gain waveguide to the loss waveguide, leading to a total upward shift in the field amplitude ratio.

Second, the influence of the pump intensity in the loss waveguide on the phase transition process is investigated by manipulating the value of I2 as shown in Fig. 7. We calculate the relative phase and field amplitude ratio under different I1 and I2 for two different coupling structures. The spacing is 2 μm in the structure depicted in Fig. 7(a), and 4 μm in the structure depicted in Fig. 7(d). A total of 15 mA [in Figs. 7(b) and 7(e)] and 35 mA [in Figs. 7(c) and 7(f)] of current is pumped into the loss waveguides of these two structures, respectively. Increasing the pump intensity in the loss waveguide within a certain range can also lead to the forward displacement of the phase transition region. When the current I2 in the loss waveguide increases to about 30 mA, the phase transition process becomes different compared to the previous situation. Specifically, when I2=35mA, the phase transition region becomes significantly broader compared to 15 mA, with almost no steady-state region before this phase transition region, and the system returns to steady-state output after symmetry breaking. Here in Figs. 7(c, i) and 7(f, i), the relative phase and amplitude ratio under different initial settings are scanned in the phase transition region, and we obtain the density distribution maps. At 35 mA, the amplitude ratio and the relative phase within the phase transition region exhibit a trend of increasing first and then decreasing, which has been experimentally verified. Conversely, within the corresponding range of I1, the trend observed at I2=15mA is a gradual decrease in the amplitude ratio.

In summary, we propose the model of an electrically injected PT-symmetric laser and analyze its phase transition and symmetry breaking process, and we demonstrate it experimentally. TCMT and RE are applied to our model to analyze the entire process, from which we can obtain the relative phase and amplitude ratio of fields in the two waveguides, as well as the dynamic characteristics of field intensity and relative phase in the time domain. In experiments, the relative phase and amplitude ratio can be obtained by testing near-field and far-field. We find that there exists a dynamic phase transition region in the phase transition process for the electrically injected PT-symmetric laser, rather than a single transition point (the exceptional point) in the ideal PT-symmetric laser. In this region, states are temporally unsteady, and multimode oscillation occurs. The phase transition region can also be characterized by the relative phase and amplitude ratio. Specifically, when the current I2 in the loss waveguide is less than 30 mA, there is a sharp decrease in the relative phase within the phase transition region. When I2 is greater than 30 mA, both the relative phase and the amplitude ratio within the phase transition region exhibit an increase followed by a decrease. Based on this model, we investigate the effects of coupling strength and pumping intensity in the loss waveguide on the phase transition process and validate them experimentally. This study theoretically analyzes and predicts the dynamic processes of the electrically injected PT-symmetric laser, demonstrating the dynamic phase transition region in the phase transition process and establishing a foundational framework for transitioning from theory to practical applications of PT symmetry in optics.