1. Introduction

Parity-time (PT) symmetry is originally discovered in quantum mechanics for non-Hermitian Hamiltonians [1]. Due to the similarity between Schrodinger and Maxwell equations, optics have become an ideal platform to investigate non-Hermitian systems, where the non-Hermiticity is denoted by optical gain and loss. Several novel behaviors have been theoretically and experimentally discovered in the PT symmetry optics, such as mode discrimination and non-reciprocity in optical waveguides [2–9]. Particularly, the mode discrimination in PT-symmetric waveguides has been employed to realize single mode PT symmetric lasers [10,11]. Such devices are mostly limited to optical injection [12,13], and they are mainly focused on laser devices. However, laser devices exhibit gain clamping after lasing and limit the further exploration of physics in PT-symmetric devices. In this context, we report a novel application of PT symmetry in the electrically driven broadband superluminescent light emitting diodes (SLEDs) where the mode discrimination based on PT symmetry is utilized in the coupled waveguides to achieve single-transverse-mode operation in tunable gain/loss configuration. In the SLED devices, both the gain and loss can be tuned and the broadband emission provide another challenge to investigate the frequency dependence of PT-symmetric conditions, which is studied theoretically in this work.

As an important category of the optoelectronic devices, SLED devices have found applications in several areas. For example, owing to the recent development of optical coherence tomography (OCT) systems for biomedical imaging, broadband SLEDs in the telecommunication wavelength regime have drawn great interest [14,15], as the key components in OCT. The inhomogeneous gain spectrum broadening of InAs quantum dot (QD) materials has made it a supreme candidate for broadband light sources [16–19]. Despite being considered as a promising candidate, it remains challenging to achieve high-performance, highly focused beam and high power, broadband QD SLEDs for OCT applications [20–25]. There are two major reasons hindering the further development of QD SLEDs, non-uniform gain spectrum and smaller gain compared with their quantum well (QW) counterparts. The former issue has been largely investigated and several unique structures and techniques have been employed to obtain a Gaussian-like broadband gain spectrum [26]. To improve the device output power, it is natural to use broad-area waveguide design, while mode filters, such as tapered waveguide design, has to be employed to maintain single-transverse-mode output for efficient fiber coupling and diffraction limit focus [27]. However, the device performance is largely scarified, since the mode filter techniques often introduce large loss to the fundamental modes. In this context, the reported PT-symmetric SLEDs provide a unique pathway to achieve single-transverse-mode operation in broad-area coupled waveguide configurations. In addition to the device applications, the presented PT-symmetric SLED devices also provide a platform to investigate PT symmetry with both tunable gain and loss.

2. PT-symmetric waveguide design

The optical response of any active optical system, including lasers and LEDs, can be related to the interplay between the active (gain) material that generates photons and the encompassing waveguide or resonator structure that provides spectral and spatial feedback to the gain medium. In this work the active region is incorporated in the coupled gain/loss planar waveguides running parallel to each other. A schematic of the waveguide structure is illustrated in Fig. 1. The waveguide core and cladding layers consist of GaAs and Al_0.4Ga_0.6As with the thickness of 300 nm and 1.5 µm, respectively. The coupled waveguides have identical geometry and conjugate refractive index, representing gain and loss in the core region and total width of 60 µm.

 

Fig. 1 Schematic of the gain/loss coupled waveguide structure, where ε^’ and ε^” represents the real and imaginary part of the permittivity in the waveguide region, respectively and κ is the coupling coefficient; time dependence $\propto \exp i\omega t$ is assumed

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Monochromatic electromagnetic radiation propagating in the waveguide structure can be represented as a linear combination of waveguide modes, with each mode characterized by its spatial profile, as well as its overall gain/loss coefficient. The overall modal gain coefficient can be related to the imaginary part of permittivity of the (active) waveguide core. The parity (P) and time (T) symmetries of a particular mode can be related to its behavior under reflection of the geometry and gain/loss conjugation, respectively.

The compound nature of the waveguide in our SLED implies that modes of the compound waveguide can be represented as a combination of the modes of its components. Analytically, the propagation constant of the compound mode can be related to the properties of the modes in the components via coupled-mode theory. Alternatively, the properties of the modes of the compound waveguide can be calculated numerically.

3. Coupled mode theory

In two coupled waveguides with conjugate imaginary index, the modal amplitudes $a_m$ and $b_m$ of the $m_{th}$ modes in these two guides is described through the coupled mode theory as follow;

Even though PT-symmetric breaking can be achieved in conjugate gain/loss coupled waveguide configurations, it is almost impossible to maintain the conditions of equal gain and loss in practical devices. Thus, it is preferable to operate the devices with varied gain while the loss is fixed. It has been shown by Christodoulides et. al. and Li et. al. that, in unbalanced gain/loss conditions, the PT symmetry conditions can significantly deviate from their exact PT balance [28,29]. The PT symmetry breaking threshold becomes $k_m=\frac{\left|g_m\right|+\left|{\alpha }_m\right|}{2}$, where κm, gm, and αm are the coupling coefficient, gain and loss of the m_th order mode, respectively. With a fixed loss of αm, the thresholdless PT symmetry breaking condition becomes αm = 2 κ_m.

4. Waveguide simulation

The PT-symmetric waveguides described above are investigated with a 2-dimensional (2D) model by commercial Maxwell equation solver, COMSOL Multiphysics, to simulate the mode profile and effective mode index. Wavelength of 1.3 µm is considered in the simulation. Transverse electric (TE) modes are calculated in the simulations, since only the TE modes are supported in Fabry-Perot waveguides with QD active materials [30]. Figure 2(a) shows the imaginary part of the mode propagation constant vs. g/α, gain/loss of the waveguide. It is calculated that, in the balanced gain/loss waveguide configuration, the PT symmetry exceptional point (EP) of TE₀ mode is at α₀ = g₀ = 6.2 cm⁻¹, and, by further increasing the gain/loss in the waveguide, higher order TE₁ supermodes subsequently reach their EP and break their PT symmetry. The coupled mode theory therefore implies that the effective coupling coefficient of TE₀ mode, κ₀ = g₀ = 6.2 cm⁻¹, is determined. In addition, we further investigate the PT-symmetric breaking conditions in coupled waveguide containing fixed loss and varied gain. In typical InAs quantum dots, the modal loss can be adjusted in the range of 0 to 50 cm⁻¹ by changing the biasing conditions in the loss waveguide. Figures 2(b)-2(d) illustrate the simulated imaginary part of the mode propagation constant vs. gain of the gain waveguide, where the loss in the loss waveguide is fixed at 8.2 cm⁻¹, 20 cm⁻¹, and 37 cm⁻¹, respectively. Our simulation results have shown that PT-symmetric breaking, and similar behaviors can be obtained in the fixed loss and varied gain configurations as well, and the exceptional points of the supermodes in the coupled waveguides is largely depending on the fixed loss. Figure 2(e) illustrates the EP of TE₀ and TE₁ modes as a function loss, where it is shown that the EP is decreasing at increased loss. When the loss is at α₀ = = 2*κ₀ = 11.4 cm⁻¹, the coupled PT symmetric waveguides exhibit thresholdless PT-symmetric broken of TE₀ mode, EP = 0 cm⁻¹. The thresholdless PT-symmetric broken condition agrees well with the coupled model theory.

 

Fig. 2 (a) Imaginary part of the mode propagation constant vs. g/α, gain/loss of the waveguide. (b)-(d) the simulated imaginary part of the mode propagation constant vs. gain of the gain waveguide in the coupled waveguide configurations, where the loss in the loss waveguide is fixed at 8.2 cm⁻¹, 20 cm⁻¹, and 37 cm⁻¹, respectively (e) EPs of TE₀ and TE₁ modes as a function loss. (f) Required gain vs. fixed loss for single-transverse-mode operation.

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In addition, we discuss the mode selections for single-transverse-mode operation in the couple waveguide cavity. Single-transverse-mode operation is achieved when there is only one pair of supermode break their PT symmetry and exhibit net gain. The gray areas in Figs. 2(b)-2(d) illustrate the window meeting the aforementioned conditions. It is worth noting that, as shown in Fig. 2(b), since the loss is small, the single-transverse-mode operation is prohibited before TE₁ supermodes reach their EP, since the non-broken supermodes start to exhibit net gain. Figure 2(f) shows the required gain vs. different fixed loss in gain and loss coupled waveguide for single-transverse-mode output. It is found that the single transverse mode operation exhibits largest window if the loss in the loss waveguide is tuned to ~10 cm⁻¹.

Since the broadband SLEDs exhibit emissions across a broad-spectrum range, it is essential to investigate the frequency dependence of EPs and single-transverse-mode operation conditions. Figure 3 illustrates the EP of TE₀ (square) and lower (dot) and upper (triangle) boundaries of the single mode operation windows vs. wavelength, where the loss of 20 cm⁻¹ is fixed in the loss waveguide. It is found that the single mode operation window, shaded area in Fig. 3, exhibits weakly dependence on the wavelength, which is favorable for PT symmetric devices with broadband emissions.

 

Fig. 3 EP of TE₀ (square) and lower (dot) and upper (triangle) boundaries of the single-mode-operation windows vs. wavelength.

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5. SLED fabrication process

Shown in Fig. 4(a), the QD SLED heterostructures is grown by a Veeco Gen-II molecular beam epitaxy (MBE) system. The top and bottom GaAs and Al_0.4Ga_0.6As layers are doped with beryllium and silicon, respectively, to provide the electrical contacts and carrier injections. To achieve broadband output emission spectrum, chirped QD active region with a varied In_xAl_1-xAs strain reducing layers is used. This technique has been previously demonstrated by us to effectively improve the QD SLED emission bandwidth and reduce the spectrum dip [26]. The coupled waveguide PT symmetric SLEDs are fabricated by standard photolithography, wet chemical etching and metallization. As shown in Fig. 4(b), the coupled waveguides with a total width of 60 µm are obtained and two p-type Ohmic contacts are defined on top of the coupled waveguides to provide independent control of the gain and loss in the waveguides, where the electrical isolation between them are achieved by three-step deep H⁺ ion implantation process. The ion implanted region has the width of 3 µm and depth of 1.5 µm. In addition, the top 200 nm p⁺ GaAs contact region is removed by wet chemical etching to assure a good electrical isolation. The oblique view of the fabricated device is shown in scanning electron microscope (SEM) image in Fig. 4(c). The facets are roughened on purpose to increase the mirror loss of the SLED cavity and suppress gain clamping.

 

Fig. 4 Heterostructures (a), schematic (b) and SEM image (c) of the InAs QD PT symmetric coupled waveguide SLEDs

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6. Experiment results and discussions

To minimize the heating effect, the coupled waveguide PT-symmetric SLEDs are characterized under pulse current injection conditions with the pulse width and duty cycle of 1 µs and 1%, respectively, at room temperature. To characterize the cavity modes and output beam profile, near-field and far-field patterns are measured. Figures 5(a) and 5(b) show the near- and far-field patterns of the SLED under gain and loss bias current of 600 mA and 60 mA, respectively. It is found that near single-lobe far-field pattern is obtained from the PT-symmetric SLEDs, and the SLED emission is from the gain waveguide, which implies that the coupled waveguide is operating in the PT-symmetric broken regime. Finally, it is also measured that the SLED far-field pattern remains single-lobe with the gain bias current varied from 300 to 800 mA.

 

Fig. 5 PT-symmetric SLED near-field (a) and far-field (b) patterns with the basing condition of I_gain = 600 mA and I_loss = 60 mA.

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Finally, Fig. 6 shows the emission spectrum of the QD SLEDs, where the 3-dB bandwidth > 100 nm is obtained. More importantly, the emission spectrum shows no obvious spectrum dip with a near Gaussian-like shape.

 

Fig. 6 EL spectra of the PT symmetric SLED at I_gain = 400, 600 and 800 mA and I_loss = 60 mA.

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7. Summary

In summary, in this work, we have numerically investigated the PT-symmetric breaking conditions in gain/loss coupled waveguides, where the supermode EPs in varied both gain and fixed loss configurations are studied. The single-transverse-mode operation window is determined numerically as well. In addition, we have experimentally demonstrated a single-transverse-mode broadband QD SLED based on PT symmetry. By introducing the gain and loss in the coupled waveguide and concept of PT symmetry, single transverse mode can be achieved in the broad-area waveguides, which has the potential to significantly improve the SLED output power while still maintaining the preferred mode profile. Further work is underway to optimize the device efficiencies and reduce the device Joule heating. Nevertheless, this work opens a pathway for practical device applications of PT symmetry in optics. It is argued that the PT symmetric QD SLEDs can play a significant role in biomedical imaging applications after further optimizing the cavity design for the broadband.