# Electrically Tunable and Reconfigurable Topological Edge State Laser

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Model and Simulation

#### 2.1. Non-Hermitian Topological Laser Chain

_{1}and C

_{2}represent the intradimer and interdimer coupling coefficients, respectively, and C

_{1}> C

_{2}. The intradimer coupling denotes field coupling between the loss (blue) waveguides and the gain (orange) waveguides, and interdimer coupling denotes coupling between gain (orange) waveguides and next loss (blue) waveguides. The trivial and nontrivial topological states were achieved by the strong and weak coupling in the dimers consisting of a gain and loss, respectively. A topological defect in the chain was introduced by removing the loss waveguide in the middle dimer (fourth pair). As a result, the designed active SSH chain consisted of 13 coupled ridge waveguide lasers with a width of 3 µm. The laser active region was composed of seven-layer InAs quantum dots (QDs) sandwiched between Al

_{0.4}Ga

_{0.6}As cladding layers. Figure 1b shows the heterostructures of the InAs QD lasers grown by the molecular beam epitaxy (MBE) system. In the laser chain, the top p

^{+}-GaAs contact layer and p-Al

_{0.4}Ga

_{0.6}As cladding layer were etched for electrical isolation between each waveguide. The coupling strengths, C

_{1}and C

_{2}, in the dimers were controlled by the width of the isolation trenches, 1 µm and 2 µm, respectively. All the gain and loss waveguides were interconnected, allowing the simultaneous and fully electrical control of all the gain and loss waveguides. A one-dimensional SSH model was first employed to investigate the passive Hermitian dimer chain without a gain and loss [35]. The SSH model studied two sites per unit cell with different coupling coefficients: intradimer coupling C

_{1}and interdimer coupling C

_{2}. It is well understood that, in the SSH mode, the topological zero states depend on the configuration of the unit cell. The topological state exists only when the smaller coupling, C

_{2}, is located at the edges [35,36]. In contrast, there is no interface state if the chain is terminated by a dimer with a larger coupling coefficient, C

_{1}. These two different scenarios could be identified by the winding number in the Brillouin zone (BZ) SSH model [37]:

_{ZAK}) divided by π [38]. It has been shown that when W

_{h}= 0, the system exhibits a trivial topological phase; in contrast, if W

_{h}= 1, the system transits into a nontrivial topological phase [39]. In the explored topological configuration, as shown in Figure 1a, all sites in the left side of the center site could be treated as subsystem A, which had a winding number of 0, and the rest of the sites, including the center site and all the other sites on the right side of the center site, could be treated as subsystem B, which had a winding number of 1 due to the different bounded edge with the coupling coefficient C

_{1}in the left and C

_{2}in the right. To obtain the coupling coefficients ${\mathrm{C}}_{1}$ and ${\mathrm{C}}_{2}$, the mode coupling between the waveguides was studied by the commercial software COMSOL. ${\mathrm{C}}_{1}$= 10.05 cm

^{−1}and ${\mathrm{C}}_{2}$ = 5.32 cm

^{−1}were obtained from COMSOL simulations. A topological transition would occur from configuration A to B, where the winding number changed from 0 to 1; as a result, the defect waveguide, the fourth pair dimer with a gain waveguide only, was located at the interface. Therefore, there had to be a topological interface state residing at the zero energy band, located at the defect waveguide. Szameit et al. already investigated such a topological waveguide configuration containing passive components, and showed the topological mode appearing in the center defect while preserving the PT symmetry [40,41,42]. In this work, we employed the dimer chain containing a gain and loss and an active gain defect to achieve a topological laser. In this context, a modified non-Hermitian SSH model was employed to study the density of state (DOS) of the SSH laser chain. The DOS was calculated by evaluating the number of eigenstates at the corresponding eigenvalues in momentum space (see Supplementary Materials for details). As shown in Figure 2a, the DOS was calculated with a passive defect in the center and all modes were grouped into two sets and separated by the middle gap. Different from defect-less SSH chains, topologically induced states manifested in the middle of the gap, which resulted from the PT symmetry breaking in the system. By introducing a gain into the center defect, as shown in Figure 2b, topologically induced states in the midgap were amplified, while states in the upper and lower groups were suppressed. It is worth noting that the DOS enhancement could be obtained not only by introducing the active defect, but also by introducing an absorptive lossy defect as well due to the time symmetry of the Hamiltonian.

#### 2.2. Phase Transition of Non-Hermitian Topological Laser Chain

^{−1}. Figure 3 shows the eigenvalues and corresponding dispersion relation at different γ = g/C

_{1}, a dimensionless variable, where g was the gain applied to the FP waveguides. It was found that the topological zero-mode exhibited complex eigenvalues with any nonzero gain, as shown in the red circles in Figure 3a–c. As shown in the phase diagram in Figure 3d, in phase I, only the zero mode, represented by the yellow curve, exhibited a complex eigenvalue with a gain, and the SSH laser chain lased at the single topological mode. At a small value of γ = 0.1, the optical field was predominately located in the defect gain waveguide in the SSH system, as shown in the Figure 3d inset. The topological mode profile presented the exponential localization on the odd site n, where the wavefunction ${\left|{\psi}_{n}\right|}^{2}~{\left(\frac{{c}_{1}}{{c}_{2}}\right)}^{\left|n-7\right|}$ for odd n, and ${\left|{\psi}_{n}\right|}^{2}~0$ for even n. This was in agreement with the analytical solutions without a gain/loss on the SSH chain. With the increase in γ, other bulk modes showed complex eigenvalues, as shown in Figure 3b,c. In phase II, two bulk modes, represented by the pink curve, were bifurcated with complex conjugate eigenvalues, which suggested the bulk modes may have started to lase alongside the topological mode. With a further increase in γ, more bulk modes showed complex eigenvalues and the corresponding gain of these bulk modes was comparable with the topological mode. The green curve, in phase III, was one representative mode of those modes. The bifurcation of bulk modes resulted from the spontaneously PT symmetry breaking once the gain applied to the waveguide was nonzero. It is worthwhile to notice that, different from bulk modes, which only presented complex eigenvalues with a certain transition point γ > 0.6, the nontrivial topological zero mode would maintain complex eigenvalues with nonzero γ, as shown by the yellow curve.

#### 2.3. Complex Eigenvalue Diagrams of Non-Hermitian Topological Laser Chain

## 3. Experimental Results

#### 3.1. Experimental Demonstration of Non-Hermitian Topological Laser Chain

_{1}and C

_{2}, in the SSH was determined by the trench between each waveguide (see the Materials and Methods the section). The trench width of 1 and 3 µm was employed to define the coupling strength between waveguides C

_{1},and C

_{2}. Figure 5a shows the top-view scanning electron microscope (SEM) image of the fabricated laser device, where the finger patterns provided independently controlled gain and loss laser waveguides. The isolation trenches not only controlled the coupling strength, but also provided electrical isolation between each waveguide. The gain/loss SSH chain was operated by electrically biasing the gain and loss waveguides at different levels. The bias current of the loss waveguides was maintained at 0 mA, which corresponded to a loss of ~ 35 cm

^{−1}in the loss waveguide, and the gain waveguide bias current varied from 0 to 1000 mA to electrically tune the gain in the gain waveguides. Figure 5b shows the light–current (L–I) characteristic of the SSH-coupled waveguide laser chain. It was observed that the lasing threshold current was ~600 mA. The inset shows the electroluminescence (EL) spectrum of the laser chain. The lasing wavelength was at ~1.33 µm, and the relative broader laser linewidth was due to the lasing from multiple longitudinal modes in the Fabry–Perot laser cavities.

#### 3.2. Reconfiguration of Non-Hermitian Topological Laser Chain

#### 3.3. Topological Laser Chain Fabrication

_{0.4}Ga

_{0.6}As, with a thickness of 300 nm and 1.5 µm, respectively. In the active region, there were seven layers of InAs quantum dot gain materials with a height of ~5 nm. The QD layers were separated by a 30 nm GaAs spacer layer. The QD laser waveguide was designed at ~1.3 µm.

_{0.4}Ga

_{0.6}As cladding layer, 200 nm above the top InAs QD layer in the active region. The trench isolation was precisely controlled to provide electrical isolation while yielding coupling strength, C

_{1}, and C

_{2}, in the dimers. Finally, two finger patterns on the gain and loss waveguides were deposited through the SiN

_{x}passivation layer and via holes. The finger patterns provided the simultaneous control of all the gain and loss waveguides electrically, indicating a fast tuning/reconfiguration speed and programmable response. The schematic of the fabricated device is shown in the Supplementary Materials. In total, 24 laser arrays were fabricated on ~ 1x1 cm

^{2}epitaxial wafers, and over 90% of the laser bars exhibited lasing. Among all the lasing devices, similar topological phase transitions were observed.

#### 3.4. Near-Field Measurement

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic of the proposed non-Hermitian SSH dimer chain, with staggered gain (orange) and loss (blue) waveguides, except the center single defected gain waveguide. ${\mathrm{C}}_{1}$ and ${\mathrm{C}}_{2}$ represent the coupling coefficients for intradimer and interdimer, respectively, where ${\mathrm{C}}_{1}$ > ${\mathrm{C}}_{2}$. (

**b**) Heterostructures of the InAs QD lasers grown by molecular beam epitaxy.

**Figure 2.**The density of states (DOS) of SSH chain without (

**a**) and with (

**b**) active defect, where the energy band was presented as normalized energy E/${\mathrm{C}}_{1}$. In passive system (

**a**), with zero gain, DOS of zero-energy states presented as less dominant compared to other bulk modes; on the contrary, in the active system (

**b**), the DOS of the zero-energy mode was largely enhanced as a dominant mode by including an active defect.

**Figure 3.**Complex eigenvalue diagrams of the SSH laser chain system in phase I (

**a**), II (

**b**), and III (

**c**), and red circles denote imaginary parts of the topological zero state. (

**d**) Phase diagram of the complex SSH chain, the yellow curve (color online) represents the topological mode, pink and green curves show the representative bulk modes in the system. The inset shows the topological mode profile calculated at γ = 0.1, along with the theoretical profile ∝(t

_{1}/t

_{2})

^{−|n|}(red curve).

**Figure 4.**(

**a**) Band diagram of SSH chain with neither gain nor loss: a bandgap presents clearly between two gray dashed lines, where the midgap states (red) sit in between. (

**b**) Evolution of complex eigenvalues with the γ varied from 0.1 (blue dots) to 0.3 (red dots): the gain increment of the topological zero-mode was ~10 times of the ones of trivial modes (with nonzero real eigenvalue) as the gain level increased.

**Figure 5.**Top-view SEM image (

**a**) and light–current (L–I) characteristic (

**b**) of the SSH laser chain. It was observed that the lasing threshold current was ~600 mA. Inset: electroluminescence (EL) spectrum of the laser chain. The lasing wavelength was at 1.33 µm.

**Figure 6.**Near-field patterns measured at varied gain bias currents: (

**a**–

**c**) bias current was 630 mA, 700 mA, and 730 mA. (

**d**) The corresponding intensity line scans to the different bias currents.

**Figure 7.**(

**a**) The optical field intensity distribution of bulk mode and topological mode in configuration with lossy defect, where normalized lossy value was given as |γ| = g/C

_{1}= 1.2; (

**b**) near-field characteristics of the gain and lossy defected topological lasers.

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## Share and Cite

**MDPI and ACS Style**

Li, H.; Yao, R.; Zheng, B.; An, S.; Haerinia, M.; Ding, J.; Lee, C.-S.; Zhang, H.; Guo, W. Electrically Tunable and Reconfigurable Topological Edge State Laser. *Optics* **2022**, *3*, 107-116.
https://doi.org/10.3390/opt3020013

**AMA Style**

Li H, Yao R, Zheng B, An S, Haerinia M, Ding J, Lee C-S, Zhang H, Guo W. Electrically Tunable and Reconfigurable Topological Edge State Laser. *Optics*. 2022; 3(2):107-116.
https://doi.org/10.3390/opt3020013

**Chicago/Turabian Style**

Li, Hang, Ruizhe Yao, Bowen Zheng, Sensong An, Mohammad Haerinia, Jun Ding, Chi-Sen Lee, Hualiang Zhang, and Wei Guo. 2022. "Electrically Tunable and Reconfigurable Topological Edge State Laser" *Optics* 3, no. 2: 107-116.
https://doi.org/10.3390/opt3020013