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Abstract
Electrically injected Parity-time (PT)-symmetric double ridge stripe semiconductor lasers lasing at 980 nm range are designed and measured. The spontaneous PT-symmetric breaking point or exceptional point (EP) of the laser is tuned below or above the lasing threshold by means of varying the coupling constant or the mirror loss. The linewidth of the optical spectrum of the PT-symmetric laser is narrowed, compared with that of traditional single ridge (SR) laser and double ridge (DR) laser. Furthermore, the far field pattern of the PT-symmetric laser with EP below the lasing threshold is compared with that of the PT-symmetric laser with EP above the lasing threshold experimentally. It is found that when the laser start to lase, the former is single-lobed while the latter is double-lobed. when the current continues to increase, the former develops into double lobe directly while the latter first develops into single lobe and then double lobe again.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
PT symmetry is a fascinating physical terminology, which was firstly applied to non-Hermitian quantum system by Bender in order to get real spectra for such a complex system [1]. Interestingly, it is found that non-Hermitian systems can possess real spectra as long as their Hamiltonians share the same eigen functions with the PT operator [1–3], which require that they can commute with each other, i.e., HPT = PTH. After some simplifications, it requires the quantum potentials should obey the relation [4,5], V(-x) = V*(x). Because of the similarity between the quantum mechanical Schrödinger equation and the optical wave equation, optical systems can also respect PT symmetry if their optical complex indices satisfy the equation [6–12], n(-x) = n*(x). So far, many optical applications based on PT symmetry have been proposed and accomplished, such as highly sensitive sensors resulting from EP [13–18], optically pumped PT-symmetric lasers [19–25], unidirectional invisibility based on PT-symmetric complex refractive index gratings [26–30] and so on. Besides, electrically injected PT-symmetric lasers have been simulated and demonstrated recently [31–36], which paves the way for the further study of electrically driven systems with PT symmetry. However, there are few researches to investigate the actual location of EP in laser systems. Therefore, in this article, electrically injected PT-symmetric lasers lasing at 980 nm range are also demonstrated by comparing their optical spectra and far field distributions with those of ordinary double ridge lasers. Furthermore, the approaches to tuning the EP are depicted in detail and, at the same time, the lasing characteristics of PT-symmetric laser with EP below the lasing threshold are experimentally compared with those of PT-symmetric laser with EP above the lasing threshold.
2. Theoretical designs
The coupled mode theory (CMT) is a useful tool for designing laser arrays [20,37,38], based on which PT-symmetric double ridge stripe lasers are designed and fabricated with different ridge widths and cavity lengths to tune the spontaneous PT-symmetric breaking point (or EP) of the systems. Figure 1(a) shows the geometrical structure of the PT-symmetric laser, the epitaxial structure of which is an asymmetric super large optical cavity (ASLOC) consisting of contact layer, P-cladding layer, P-waveguiding layer, double quantum wells (QWs), N-waveguiding layer, N-cladding layer and substrate [39]. Figure 1(b) is the corresponding scanning electron microscope (SEM) of the cross section of the PT-symmetric laser with ridge width w of 5.0 μm. The lasers are designed to lase around 980.0 nm with ridge depth t of 1.0 μm and distance between waveguides d of 2.0 μm. In order to produce a PT-symmetric optical complex potential, only the left waveguide has electrically injected window to get a variable gain with respect to the injected current while the right waveguide is not injected to keep the intrinsic loss constant all the time [6,22]. Besides, the geometric dimension of the two waveguides is exactly the same, as shown in Fig. 1(a).
Fig. 1. Structure of the PT-symmetric lasers. (a) The schematic of the PT-symmetric laser. The active region is composed of a set of DQWs. (b) SEM of the cross section of the PT-symmetric laser with ridge width of 5 μm and waveguide spacing of 2 μm. The actual ridge depth is 944.3 nm, which may lead to the small shift of the EP.
The internal loss of the laser is αi = 0.58 cm-1 and the mirror loss for the uncoated laser with cavity length L of 2.0 mm is αm = 6.03 cm-1 [39]. Therefore, the total loss of the laser is αt = αm + αi = 6.61 cm-1. According to the relation between optical power and electrical amplitude, the intrinsic mode loss of the laser is γ0 = -3.30 cm-1. Based on these parameters, the EP can be adjusted by either changing the coupling constant or the mirror loss. Firstly, for the laser with a fixed cavity length of 2.0 mm, the EP can be tuned above the lasing threshold by adapting a narrower ridge waveguide which corresponds to a larger coupling constant. For the laser with ridge width w of 5 μm, the coupling constant is calculated to be κ = 5.57 cm-1 by the properties of EP [40]. According to the CMT [37], the complex propagation constants of the supermodes of the PT-symmetric laser can be expressed as
Fig. 2. Mode gains of the supermodes of the PT-symmetric lasers versus the mode gain of the gain waveguide. (a) w = 5.0 μm, d = 2 μm, t = 1μm and L = 2 mm. The points A, B, C point out the locations where near field distributions and far field distributions are calculated in Fig. 3. In this case, the EP is above the lasing threshold indicated by the dashed green line. (b) w = 14.0 μm, d = 2 μm, t = 1μm and L = 2 mm. In this case, the EP is below the lasing threshold. The lines are the results of CMT and the circles are the results of the numerical calculation.
Secondly, in order to observe the optical spectrum more easily and keep the location of the EP fixed relative to the lasing threshold, the cavity length has to be shortened but the mirror loss must keep constant. Here, the cavity length is chosen to be 0.6 mm and the rest of the geometric parameters are kept same as those depicted in Fig. 2. Because the expression for mirror loss is
where R1 and R2 are the reflectivity of front facet and rear facet of the laser [37], respectively, the mirror loss can be constant if the product of R1R2 is increased when the cavity length is shortened. For a cavity length of 0.6 mm and mirror loss of 6.03 cm-1, R1 can equal 0.49 and R2 can equal 0.99. Then, mode spacing between longitudinal modes is increased from 0.05 nm to 0.18 nm, which is convenient for spectrum observation.At last, the near field distributions and far field distributions of the PT-symmetric laser with ridge width of 5 μm are calculated numerically. Figure 3(a, b) shows the near field distributions of the amplified mode and the lossy mode at the point A of Fig. 2(a), respectively. In this case, the PT-symmetric laser works at the PT-symmetric phase, so both amplified mode and the lossy mode uniformly distribute in the two waveguides. At the point B, the PT-symmetric laser still works at the PT-symmetric phase, which leads to the similar near field distributions, as shown in Fig. 3(c, d). However, at the point C, the PT-symmetric laser works at the broken PT-symmetric phase, where the amplified mode mainly locates in the gain waveguide while the lossy mode mainly locates in the lossy waveguide, as shown in the Fig. 3 (e, f). Next, after performing Fourier transformation, the horizontal far field distributions at the three points are calculated and shown in Fig. 3(g). For points A and B, both the lossy mode and the amplified mode possess comparative gain, so they contribute to lasing equally. In this case, the lossy mode would lead to double-lobed far field patterns, as indicated by the red and blue solid lines in Fig. 3(g). While for point C, only the amplified mode possesses positive gain and contributes to the lasing. The far field distribution of the amplified mode at point C is shown with green solid line in Fig. 3(g), which shows that the PT-symmetric breaking of specified modes can really suppress the side lobe of far field pattern. Similarly, the numerical analysis on the far field profiles of the PT-symmetric laser with ridge width of 14.0 μm can be performed. However, the main difference is that the PT-symmetric laser with ridge width of 14.0 μm always works at the broken PT-symmetric phase after the threshold, so the far field distribution of the laser is single-lobed after lasing.
Fig. 3. Numerically calculated near field patterns and far field patterns of the PT-symmetric laser with w = 5 μm. (a, b) Near field distributions of the amplified mode and lossy mode, respectively, when γ1 = 3.76 cm-1 (Point A in Fig. 2(a)). (c, d) Near field distributions of the amplified mode and lossy mode, respectively, when γ1 = 6.31 cm-1 (Point B). (e, f) Near field distributions of the amplified mode and lossy mode, respectively, when γ1 = 10.26 cm-1 (Point C). (g) Numerically calculated far field distributions of the laser under different mode gain of the gain waveguide (Points A, B and C).
3. Materials and methods
Based on the above discussions, four PT-symmetric lasers were fabricated and tested. Two of them have cavity length of 2.0 mm with the cavity facets uncoated, the ridge widths of which are 14 μm and 5 μm, respectively. The other two of them have cavity length of 0.6 mm with the cavity facets coated, the ridge widths of which are 14 μm and 5 μm, respectively. The actual reflectivity of the front facet and rear facet of the laser with cavity length of 0.6 mm was tested to be 44.5% and 98.5%, respectively. It is due to the growth error of electron beam evaporation, which had small influence on the location of the EP. The epitaxy of the laser based on ASLOC structure was grown by metal organic chemical vapor deposition (MOCVD) [39]. Following it, waveguide patterns were transferred into the wafer by standard photolithography process and inductively coupled plasma (ICP) etching. Then, Ti/Pt/Au metal stacks were sputtered on the wafer surface by magnetron sputtering to form P-side electrode. After the wafer was thinned to 150 μm, AuGeNi/Au metal stacks were deposited on the wafer backside by magnetron sputtering to form N-side electrode. Then, the wafer was cleaved and packaged into single diode chips. At last, the optical spectra, near field patterns and the far field distributions were tested and analyzed. The optical spectra were measured by the optical spectrum analyzer of Yokogawa, 70D, with resolution of 0.02 nm. The near fields were measured by the near infrared charge coupled device (CCD) camera and the far fields were measured by the COS Tester of Raybow Opto. Figure 1(b) shows the SEM of the cross section of the laser with ridge width of 5 μm and waveguide spacing of 2 μm. From this figure, it can be found that the actual ridge depth is 944.3 nm, which may lead to the small shift of the EP.
4. Results
For the lasers with cavity length of 2.0 mm, the horizontal far field patterns of them were tested, which could reveal the lateral mode characteristics of the lasers. Figure 4 shows the LIV (Light-current-voltage) curves of these lasers, from which it can be known threshold current is around 0.1 A. In Fig. 5(a), the horizontal far field distributions of the PT-symmetric laser with ridge width of 5.0 μm under different injected currents are displayed on the left side of this figure. From this figure, the far field pattern is double-lobed when the injected current is 0.2 A, as shown in the lower panel of Fig. 5(a). In this case, the EP is above the lasing threshold of the laser. Even though the laser can lase at this current point, the mode gain of gain waveguide, γ1, is in the range from 3.3 cm-1 to 9.4 cm-1, as shown in Fig. 2(a), where both the amplified mode and the lossy mode have positive mode gain and therefore can lase, which leads to the double-lobed far field. With the current increased to 0.6 A, γ1 becomes larger than 9.4 cm-1, where the amplified mode has larger mode gain but the lossy mode has negative mode gain, so only the amplified mode can lase and the far field becomes single-lobed, as shown in the middle panel of Fig. 5(a). The evolution of the far field distribution of the PT-symmetric laser with ridge width of 5 μm agrees well with the result of numerical calculation in Fig. 3(g). With further increasing the current to 1.0 A, the far field becomes double-lobed again, as shown in the upper panel of Fig. 5(a), which means the laser enters the nonlinear unbroken phase resulting from the nonlinear effects [41]. Both the Joule heating of the active layer and the spatial hole burning can increase the effective index in the center of gain waveguide [42], which not only breaks the condition of PT symmetry but also concentrates high-order modes into the waveguide. However, for the DR laser with the same ridge width as the PT-symmetric laser, the evolution of its far field distribution with current shows characteristics of ordinary laser arrays [43,44], as shown on the right side of Fig. 5(a), which is mainly determined by the relative size of the confinement factor of the in-phase supermode and that of out-of-phase supermode. Both ridges of the DR laser are injected with current at the same time. For the PT-symmetric laser with ridge width of 14.0 μm, its horizontal far field distribution is single-lobed at small injected currents but multi-lobed with the increasing of injected currents, as shown on the left side of Fig. 5(b). In this case, the EP is below the lasing threshold of the laser, as shown in Fig. 2(b), so only the amplified mode will possess positive mode gain while the lossy mode will possess negative mode gain when the laser works above the lasing threshold. Therefore, only the amplified mode can lase at small injected currents, leading to the single-lobed far field pattern. With further increasing the injected currents, the laser will also enter the nonlinear unbroken phase, resulting in multi-lobed far fields [41]. Similarly, the far field distribution of the corresponding DR laser shows the characteristics of ordinary laser array, as shown on the right side of Fig. 5(b).
Fig. 4. LIV (Light-current-voltage) curves for the lasers with L = 2.0 mm. (a) LIV curves for the PT-symmetric laser and DR laser with w = 5.0 μm. (b) LIV curves for the PT-symmetric laser and DR laser with w = 14.0 μm.
Fig. 5. Horizontal far field patterns of the lasers with cavity length of 2.0 mm versus injected currents. (a) w = 5.0 μm. The left side of the figure is the far field distributions of the PT-symmetric laser and the right side of the figure is the far field distributions of the corresponding DR laser. (b) w = 14.0 μm. The left side of the figure is the far field distributions of the PT-symmetric laser and the right side of the figure is the far field distributions of the corresponding DR laser.
Figure 6 shows the LIV curves of the PT-symmetric lasers with cavity length of 0.6 mm, from which it can be known threshold current is around 40.0 mA. For the lasers with cavity length of 0.6 mm and ridge width of 5.0 μm, the spectra of them were tested under current of 89.0 mA and shown in Fig. 7(a). The spectrum of the PT-symmetric laser is narrower than those of the single ridge (SR) laser and the DR laser. The side mode suppression ratio (SMSR) of the PT-symmetric laser is 37. 0 dB, which is because the spontaneous PT-symmetric breaking of the system can not only suppress the lasing of higher order lateral modes but also reduce the number of longitudinal modes [20,35,38]. Besides, as shown on the left side of Fig. 7(b), the far field distribution of this PT-symmetric laser also becomes single-lobed with increasing the injected current, which confirms that the location of the EP is nearly the same as that of the PT-symmetric laser with same ridge width but cavity length of 2.0 mm. The insets in Fig. 7(b) show the scattered field of this PT-symmetric laser near the cavity mirror from the top view, which was collected by the near infrared CCD camera. Obviously, the near field distributes in the two ridge waveguides under small injected currents. when the injected current increases, the near field exclusively distributes in the gain waveguide, which results from the mode field separation after the spontaneous PT-symmetric breaking of the system and proves that the EP is above the lasing threshold [19,20,22,38]. For the PT-symmetric laser with ridge width of 14.0 μm, the evolution of its far field is also the same as that of the corresponding PT-symmetric laser with cavity length of 2.0 mm, as shown on the right side of Fig. 7(b). Therefore, tuning the mirror loss can surely keep or change the location of the EP.
Fig. 6. LIV curves for the PT-symmetric lasers with L = 0.6 mm. Red solid lines are the LIV of the PT-symmetric laser with w = 5.0 μm. Blue solid lines are the LIV of the PT-symmetric laser with w = 14.0 μm.
Fig. 7. L = 0.6 mm. (a) w = 5.0 μm. The spectra of SR, DR and PT lasers tested under injected current of 89.0 mA. (b) The left side of the figure is the horizontal far field distributions of the PT-symmetric laser with ridge width of 5.0 μm and the right side of the figure is the horizontal far field distributions of the PT-symmetric laser with ridge width of 14.0 μm. The insets in Fig. 7(b) are the scattered fields of the PT-symmetric laser with ridge width of 5.0 μm near cavity mirror from the top view.
5. Conclusions
PT-symmetric double ridge stripe lasers with different cavity lengths and ridge widths are designed, fabricated and tested. The far field distributions of these devices are compared with each other and those of the DR lasers. It is found that tuning the coupling constant between two ridge waveguides can truly change the location of the EP and the evolution of the far field distribution with injected current is different between the two types of PT-symmetric lasers. What’s more, the spectra of PT-symmetric laser with cavity length of 0.6 mm and ridge width of 5.0 μm demonstrate that the spontaneous PT-symmetric breaking of the system can narrow the linewidth of the spectrum. Furthermore, it is demonstrated that the evolutions of the far field distributions of the PT-symmetric lasers with cavity length of 0.6 mm are the same as those of the corresponding PT-symmetric lasers with cavity length of 2.0 mm, which shows that tunning the mirror loss can also keep or change the location of EP. Therefore, our methods of tuning the location of the EP can be very useful to design PT-symmetric lasers in the order of micrometers, which are very compatible with the traditional ultraviolet photolithography technology and the process of growth of thin films. If the PT-symmetric laser is designed to work at the neighborhood of threshold current, the EP can be tuned below the threshold to get a single-lobed far field pattern. However, if the PT-symmetric laser works at higher current according to the application, the EP can be tuned above the threshold to get a single-lobed far field pattern as well. The single-lobed far field pattern can increase the coupling efficiency between the laser and other optical element. Lastly, we found that the small disturbances in size of device and reflectivity of mirror had little effect on the location of the EP, increasing the reliability of the device. However, further study based on the coupled rate equation is still needed to include the nonlinear effects of laser [41,45], which we will investigate in our future work.
Funding
National Key Research and Development Program of China (2016YFA0301102, 2016YFB0401804); National Natural Science Foundation of China (62075213, 91850206).
Acknowledgements
Ting Fu thanks Dr. Jeremy Sanders for his free software, Veusz, for graphing.
Disclosures
The authors declare no conflict of interest.
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