⇦Chinese Physics Letters, 2020, Vol. 37, No. 4, Article code 044207 Mode Control of Quasi-PT Symmetry in Laterally Multi-Mode Double Ridge Semiconductor Laser * Ting Fu (傅廷)1,3, Yu-Fei Wang (王宇飞)1,2, Xue-You Wang (王学友)1,3, Xu-Yan Zhou (周旭彦)1, Wan-Hua Zheng (郑婉华)1,2,3,4** Affiliations 1Laboratory of Solid State Optoelectronics Information technology, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083 2College of Future Technology, University of Chinese Academy of Sciences, Beijing 101408 3Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049 4State Key Laboratory on Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083 Received 30 December 2019, online 24 March 2020 *Supported by the National Key R&D Program of China (Grant Nos. 2016YFB0401804 and 2016YFA0301102), and the National Natural Science Foundation of China (Grant Nos. 91850206, 61535013, and 11981260014).
**Corresponding author. Email: whzheng@semi.ac.cn Citation Text: Fu T, Wang Y F, Wang X Y, Zhou X Y and Zheng W H et al 2020 Chin. Phys. Lett. 37 044207    Abstract In traditional semiconductor lasers, it is usual to obtain single lateral mode operation by narrowing the ridge of waveguide, which is sensitive to fabrication inaccuracies. To overcome this shortcoming, a quasi-PT (parity-time) symmetric double ridge semiconductor laser is proposed to reach single lateral mode operation for an intrinsic multi-mode stripe laser. The coupled mode theory is used to analyze the non-Hermitian modulation of the gain (or loss) of the PT symmetric double ridge laser to obtain the coupling coefficient between the two ridge waveguides. Finally, the mode field distributions of the quasi-PT symmetric double ridge laser are simulated before and after the spontaneous PT symmetry breaking, which keep the laser operating in single lateral mode. DOI:10.1088/0256-307X/37/4/044207 PACS:42.55.Px, 42.60.Fc, 11.30.Er © 2020 Chinese Physics Society Article Text Hermitian quantum systems possess an entirely real energy spectrum, which has been analyzed in detail in traditional quantum mechanics. However, it is shown that a non-Hermitian quantum system can also have an entirely real energy spectrum as long as the Hamiltonian of the system and the PT (parity-time) operator share the same eigenfunctions,[1–3] which requires that they commute with each other. If the Hamiltonian of the non-Hermitian system is PT symmetric, then the complex potential needs to follow $V(x)=V^{\ast} (- x) $.[4] Although the PT symmetry concept originates from the quantum mechanics, it also applies to the optical field because of the similarity between quantum mechanics and paraxial optics,[3–6] which can result in a mathematically identical solution. Similarly, the Hamiltonian of the optical system can be PT symmetric when the complex refractive index of the system satisfies the relation $n(x)=n^{\ast} (-x$). Furthermore, compared with the quantum system, the optical system is an intriguing platform to explore the novel properties of PT symmetric systems and many new phenomena have been studied extensively, such as exceptional point,[7–10] PT symmetric laser,[11–19] and unidirectional invisibility.[20–23] In the area of PT symmetric laser, single longitudinal mode micro-ring lasers[18,19] based on PT symmetry have been achieved. The mode degeneracy after spontaneous PT symmetry breaking[16] has also been demonstrated. Single lateral mode micro-ring lasers[15] based on PT symmetry have also been obtained due to the lower coupling coefficient of fundamental lateral mode compared with that of higher order modes. However, to date, lateral coupled PT symmetric stripe semiconductor lasers have not been investigated, so a quasi-PT symmetric double ridge semiconductor laser is proposed to achieve a single lateral mode in this work and it may pave the way for designing PT symmetric stripe laser arrays.[7,14,24–26]
cpl-37-4-044207-fig1.png
Fig. 1. Schematic of PT symmetric double ridge semiconductor laser. The original point is at the geometry center of the GaAs layer in the front facet of the device.
Coupled mode theory is a common and effective way to analyze PT symmetric laser.[18,19] The coupling coefficient between the two ridge waveguides can be acquired based on the theory, among which one has net gain $\gamma_{1}$ while the other has equal loss $\gamma_{2}\,=\,-\gamma_{1}$ as shown in Fig. 1. In Fig. 1, the two ridges have the same width of $w\,=\,4$ µm, the distance between the two ridge waveguides is $d\,=\,4$ µm, the etched thickness is $h_{2}\,=\,0.8$ µm, the p-type AlGaAs cladding layer's thickness is $h_{1}\,=\,0.82$ µm and GaAs core layer's thickness is $h_{0}\,=\,0.064$ µm. In addition, the refractive index of the AlGaAs layer is $n_{1}\,=\,3.401$ and that of the GaAs layer is $n_{0}\,=\,3.6$. Here the wavelength of propagation wave is set as $\lambda_{0}\,=\,0.895$ µm (in vacuum). According to the coupled mode theory,[18,19,27] a simple set of equations can be expressed as $$\begin{align} &\frac{da_{1} }{dz}=-j\beta_{1} a_{1} -j\kappa a_{2} {\rm }, \\ &\frac{da_{2} }{dz}=-j\beta_{2} a_{2} -j\kappa a_{1} {\rm },~~ \tag {1} \end{align} $$ where $a_{1}$ and $a_{2}$ are the mode amplitudes of gain and loss waveguide, respectively, $\beta_{1}$ and $\beta_{2}$ are the complex propagation constants of gain and loss waveguide, respectively, and $\kappa$ is the coupling coefficient between the two waveguides. Solving Eq. (1), the propagation constants of the obtained supermodes[7,27] can be given as follows: $$ \beta_{\pm } =\beta_{\rm r} +j\frac{\gamma_{1} +\gamma_{2} }{2}\pm \sqrt {\kappa^{2}-\Big({\frac{\gamma_{1} -\gamma_{2} }{2}} \Big)^{2}},~~ \tag {2} $$ where $\beta_{\rm r}$ is the real part of $\beta_{1}$ or $\beta_{2}$, and $\gamma_{1}$ and $\gamma_{2}$ are the imaginary part of $\beta_{1}$ and $\beta_{2}$, respectively, which represent the gain when the amplitude gain coefficient is larger than zero and the loss when smaller than zero. In this case, the laser structure is PT symmetric in the lateral direction ($y$ direction), which requires that the complex refractive index satisfies $n(y)=n^{\ast} (-y$), i.e., Re[$n(y)$] = Re[$n(-y)$] (real part) and Im[$n(y)$] = $-$Im[$n(-y)$] (imaginary part), $0 \le y \le 8\,µ$m. Because the amplitude gain coefficients of the two waveguides, $\gamma_{1}$ and $\gamma_{2}$, are determined by the imaginary parts of the complex refractive indices, we have $\gamma_{1}\,=\,-\gamma_{2}$ and Eq. (2) can be reduced to $$ \beta_{\pm } =\beta_{\rm r} \pm \sqrt {\kappa^{2}-\gamma_{1}^{2} }.~~ \tag {3} $$ From Eq. (3), $\kappa$ equals $\gamma_{1}$ at the exceptional point,[18,28] where not only the eigenvalues of the non-Hermitian system, $\beta_{+}$ and $\beta_{-}$, degenerate but also the mode field distributions of the two supermodes degenerate. Thus, the coupling coefficient can be obtained by finding out the amplitude gain coefficient $\gamma_{1}$ at the spontaneous PT symmetry breaking point. Using the finite element method and varying the imaginary part of the complex refractive index, we can plot the changes of simulated complex propagation constant as shown in Fig. 2. From this figure, it is obvious that the spontaneous PT symmetry breaking point takes place at Im[$n(y)$] =  $-$Im[$n(-y)$] = $7.605\times 10^{-4}$ and the corresponding amplitude gain coefficient of the gain ridge waveguide is $\gamma_{1}\,=\,7.03$ cm$^{-1}$. Then, according to Eq. (3), the coupling coefficient is 7.03 cm$^{-1}$. Based on this method, the coupling coefficients of lasers with different ridge widths and waveguide distances are calculated as shown in Fig. 3, which shows that the coupling coefficient becomes greater when the ridge width is smaller or the waveguide distance is smaller. Therefore, the ridge widths and waveguide distances should be carefully chosen in order to obtain an appropriate coupling coefficient. For example, $w\,=\,10$ µm and $d\,=\,6$ µm are inappropriate because the corresponding coefficient $\kappa\,=\, 0.424$ cm$^{-1}$ is too small.
cpl-37-4-044207-fig2.png
Fig. 2. Real part (a) and imaginary part (b) of the complex propagation constants versus the imaginary part of the complex refractive index.
cpl-37-4-044207-fig3.png
Fig. 3. Coupling coefficient versus ridge width and waveguide distance.
Although this analysis is useful to acquire the coupling coefficient of PT symmetric double ridge semiconductor laser, the loss of a laser is usually fixed and the gain can vary as a function of injection current on the contrary. For example, in our case, the injection current is added to the gain waveguide to change its gain, while the loss waveguide has no injection current so it possesses fixed loss. Fortunately, a non-Hermitian system with fixed loss can also have a spontaneous PT symmetry breaking point,[10,16,29] which occurs at a different point compared with that of the PT symmetric one, and in this case, the system is quasi-PT symmetric, which has gain-loss contrast, too. Therefore, the imaginary part of the complex refractive index of the loss ridge waveguide is fixed to $7.373\times 10^{-4}$ and the corresponding amplitude gain coefficient is $\gamma_{2}\,=\,-6.815$ cm$^{-1}$, the magnitude is half the total loss of laser, $\alpha_{\rm total }=\alpha_{i }+ \alpha_{m}$, where $\alpha_{i}$ is the internal loss equal to 2.239 cm$^{-1}$ and $\alpha_{m}$ is the mirror loss equal to 11.394 cm$^{-1}$.
cpl-37-4-044207-fig4.png
Fig. 4. Complex propagation constants of supermodes versus the amplitude gain coefficient of the gain ridge waveguide (a) the real part, (b) the imaginary part. Here, red color stands for the amplified mode and blue color stands for the lossy mode; square and circle points are obtained using the finite element method and the solid and dotted lines are calculated using the coupled mode theory.
Similarly, according to the finite element method, the changes of the complex propagation constant of the supermodes versus the amplitude gain coefficient of the gain ridge waveguide are plotted in Fig. 4, as shown by the point data. Before the spontaneous PT symmetry breaking point, the real parts of the complex propagation constants bifurcate with each other but their imaginary parts remain degenerate, and the laser stays in the PT symmetry phase; at the spontaneous breaking point, both the real parts and the imaginary parts become degenerate, which means the occurrence of an exception point; after the spontaneous breaking point, the real parts remain degenerate but the imaginary parts bifurcate, which shows that the laser gets into the broken PT symmetry phase. In the broken PT symmetry phase, one supermode gets more gain when the imaginary part of the complex refractive index becomes greater while the other one gets less gain and becomes lossy lastly. Thus, to be clarified, the former is called the amplified mode and the latter is called the lossy mode. Furthermore, by bringing the acquired coupling coefficient $\kappa$, the fixed amplitude gain coefficient of the loss ridge waveguide $\gamma_{2}$ and numerical calculated real propagation constant $\beta_{\rm r}$ into Eq. (2), the results of coupled mode theory are plotted with solid and dotted lines in Fig. 4, which are in good agreement with those of the finite element method. This conclusion can also be justified by the mode field distributions of the supermodes shown in Fig. 5. Obviously, both the amplified mode and the lossy mode uniformly distribute in the two ridge waveguides before reaching the spontaneous PT symmetry breaking point, so both of them get small gains. However, after reaching the spontaneous PT symmetry breaking point, the amplified mode will focus on the gain waveguide and get more gain, while the lossy mode will focus on the loss waveguide and get less gain and even experience loss. Therefore, only the amplified mode will lase finally as the additional stimulation increases.
cpl-37-4-044207-fig5.png
Fig. 5. Simulated mode field distributions of the fundamental supermodes using the finite element method. Here (a) and (c) are the mode profiles of the amplified mode before and after spontaneous PT symmetry breaking respectively; (b) and (d) are the mode profiles of the lossy mode before and after spontaneous PT symmetry breaking respectively. In (a) and (b), Im[$n(-y)$] = $-7.7\times 10^{-4}$; in (c) and (d), Im[$n(-y)$] = $-9.0\times 10^{-4}$.
These above analyses are concentrated on the fundamental supermodes, which originate from the coupling between fundamental lateral modes of the two ridge waveguides. The method can also apply to higher order supermodes resulting from the coupling between higher order lateral modes of the two ridge waveguides. However, the coupling coefficient of fundamental lateral mode is smaller than that of the higher order modes,[15,28] so the spontaneous PT symmetry breaking point of the former can be reached more easily than that of the latter. This can be proved by the mode field distributions of the first order supermodes with the same imaginary part of the complex refractive index of the gain ridge waveguide, as shown in Fig. 6. In this figure, the mode field distributions of the two first order supermodes barely change but uniformly distribute in the two ridge waveguides all the time, so the two first order supermodes will feel gain and loss at the same time and not lase eventually as long as there is only fundamental modes staying in broken PT symmetry phase. As shown in Fig. 7, the amplitude gain coefficient of the fundamental amplified supermode is far larger than that of the first order and the second order amplified supermodes before the imaginary part of the complex refractive index reaches 0.0072, which corresponds to the spontaneous PT symmetry breaking point of the first order supermode. In short, according to the selective breaking of PT symmetry of the fundamental lateral supermodes,[15] the quasi-PT symmetric double ridge laser can achieve a single lateral model lasing.
cpl-37-4-044207-fig6.png
Fig. 6. Simulated mode field distributions of the first order supermodes using the finite element method. In (a) and (b), Im[$n(-y)$] = $-7.7\times 10^{-4}$; in (c) and (d), Im[$n(-y)$] = $-9.0\times 10^{-4}$. Here (a) and (c) are the mode profiles of the amplified mode before and after the spontaneous PT symmetry breaking of the fundamental supermodes respectively; (b) and (d) are the mode profiles of the lossy mode before and after the spontaneous PT symmetry breaking of the fundamental supermodes respectively.
cpl-37-4-044207-fig7.png
Fig. 7. Amplitude gain coefficients of amplified supermodes versus the imaginary part of the complex refractive index. The blue line represents the fundamental supermode, the red line represents the first order supermode and the yellow line represents the second order supermode.
In summary, we have obtained the coupling coefficient between two ridge waveguides according to the properties of the PT symmetric double ridge laser at the spontaneous PT symmetry breaking point of fundamental supermodes, and we have used it to analyze the quasi-PT symmetric double ridge laser. It is found that the results of the finite element method match well with these of the coupled mode theory, which demonstrates the validity of the way of calculating coupling coefficient. Lastly, the mode field distributions of different order supermodes are simulated before and after the spontaneous PT symmetry breaking of fundamental supermodes, which shows that the quasi-PT symmetric double ridge laser can operate in a single lateral mode as long as the fundamental supermode is a selective breaking of the PT symmetry. Therefore, compared with traditional narrow ridge semiconductor lasers, this type of stripe laser is more tolerant of fabrication inaccuracies[15] because it can stay in single-mode operation for a wider ridge, although its output power will be reduced by the introduction of the additional loss waveguide. We believe that our work will pave the way for further research of the nonlinearity properties[25,30] of PT symmetric ridge laser and PT symmetric stripe laser arrays. References Real Spectra in Non-Hermitian Hamiltonians Having P T SymmetryGeneralized PT symmetry and real spectraMaking sense of non-Hermitian HamiltoniansBeam Dynamics in P T Symmetric Optical LatticesObservation of parity–time symmetry in opticsTheory of coupled optical PT-symmetric structuresNon-Hermiticity and exceptional points in coherently coupled vertical cavity laser diode arraysEnhanced sensitivity at higher-order exceptional pointsParity-time-symmetric coupled microring lasers operating around an exceptional pointReversing the pump dependence of a laser at an exceptional pointHigh-power edge-emitting laser based on a parity-time-structured Bragg reflection waveguideA Single-Mode Laser Based on Parity-Time-Symmetry Structured Vertical Bragg Reflection WaveguideAn integrated parity-time symmetric wavelength-tunable single-mode microring laserParity-time symmetry in coherently coupled vertical cavity laser arraysSingle mode lasing in transversely multi-moded PT-symmetric microring resonatorsExperimental demonstration of PT-symmetric stripe lasersThe combination of directional outputs and single-mode operation in circular microdisk with broken PT symmetryParity-time-symmetric microring lasersSingle-mode laser by parity-time symmetry breakingParity–time-symmetric circular Bragg lasers: a proposal and analysisNonreciprocal waveguide Bragg gratingsUnidirectional Invisibility Induced by P T -Symmetric Periodic StructuresModal Gain Analysis of Parity-Time-Symmetric Distributed Feedback LasersBreakup and Recovery of Topological Zero Modes in Finite Non-Hermitian Optical LatticesRate equation analysis and non-Hermiticity in coupled semiconductor laser arraysLocalized Single Frequency Lasing States in a Finite Parity-Time Symmetric Resonator ChainLarge area single-mode parity–time-symmetric laser amplifiersVisualization of Branch Points in P T -Symmetric WaveguidesNonlinear reversal of the PT -symmetric phase transition in a system of coupled semiconductor microring resonators
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