*Article* **A Monolithically Integrated Laser-Photodetector Chip for On-Chip Photonic and Microwave Signal Generation**

#### **Hefei Qi 1,2,3, Guangcan Chen 1,2,3, Dan Lu 1,2,3,\* and Lingjuan Zhao 1,2,3**


Received: 29 August 2019; Accepted: 29 September 2019; Published: 30 September 2019

**Abstract:** An Indium-phosphide-based monolithically integrated photonic chip comprising of an amplified feedback laser (AFL) and a photodetector was designed and fabricated for on-chip photonic and microwave generation. Various waveforms including single tone, multi-tone, and chaotic signal generation were demonstrated by simply adjusting the injection currents applied to the controlling electrodes. The evolution dynamics of the photonic chip was characterized. Photonic microwave with frequency separation tunable from 26.3 GHz to 34 GHz, chaotic signal with standard bandwidth of 12 GHz were obtained. An optoelectronic oscillator (OEO) based on the integrated photonic chip was demonstrated without using any external electrical filter and photodetector. Tunable microwave outputs ranging from 25.5 to 26.4 GHz with single sideband (SSB) phase noise less than −90 dBc/Hz at a 10-kHz offset from the carrier frequency were realized.

**Keywords:** photonic integrated circuit; microwave generation; laser dynamics; optoelectronics oscillator

#### **1. Introduction**

Photonic microwave technologies have important applications in the field of radio over fiber system, radar, lidar, unmanned driving, etc. The development of photonic microwave technologies has received much attention. The generation of photonic microwave signals is generally based on discrete devices, including multiple active and passive devices, which are bulky, costly and lossy. With the development of photonic integration technology, photonic integrated chips are showing their potential in photonic microwave generation and processing [1,2], with the possibility to greatly reduce the system complexity, footprint, performance, and the cost.

To generate photonic microwave, at least two laser modes or sidebands are required, so that the heterodyning signal after photodetection will produce a microwave signal corresponding to the mode separation between the modes. Among various types of photonic integrated microwave generators, dual-mode semiconductor lasers are typical ones. Many dual-mode structures have been proposed, including the integration of two semiconductor lasers in series [3,4] or parallel [5], integrated feedback cavity lasers [6,7], etc. By controlling the mode separation between the two modes, photonic microwave with frequency ranging from GHz to THz can be obtained [3,7]. However, due to the lack of the necessary phase correlation between the laser modes, the heterodyning signal usually has a linewidth on the order of several or tens of MHz, which limits their potential applications in many fields. To address the problem, many techniques such as optical injection locking [8,9], electrical

modulation [10], or optoelectronic oscillation [11] have been proposed, which have greatly improved the signal quality of the integrated photonic microwave source to a level comparable or even superior to that obtained from electrical devices.

Another type of photonic microwave is the chaotic signal, which can be generated by using semiconductor lasers under optical injection [12,13], optical feedback [14,15] or optoelectric feedback [16,17]. Photonic integration technology provides a solution to combine the laser cavity and the feedback cavity or injection sources into a single chip [7,18–20] so that the needs for free-space or fiber-based feedback/injection are eliminated. Chaotic signals with bandwidth over tens of GHz have been demonstrated [21,22].

In the above-mentioned structure, however, external photodetectors were required to generate the microwave signal. In order to further include more functionality, the photodetectors have been integrated on-chip [23,24]. In [23,24], the photodetectors were integrated with two distributed feedback (DFB) lasers which were combined by a multimode interference (MMI) coupler. The tuning range of the on-chip-generated microwave signal could reach several tens of GHz. However, the active and passive integration requires additional regrowth process. Besides, the use of the MMI coupler resulted in a long device length of several mm.

In this paper, we present a simple Indium-phosphide(InP)-based monolithically integrated photonic microwave generator comprising of an amplified feedback laser (AFL) and a photodetector for on-chip photonic and microwave generation. The integrated photonic chip shares the same active material, no additional regrowth is required. The total length of the chip was only about 1.17 mm. By adjusting the injection currents applied on the controlling electrodes, microwaves with various waveforms including single tone, multitone and chaotic signal could be realized. Tunable microwave ranging from 26.3 GHz to 34 GHz, chaotic signal with standard bandwidth of 12 GHz were obtained. Furthermore, an optoelectronic oscillator (OEO) was constructed using the integrated photonic chip. Thanks to the multifunctionality of the integrated chip, there was no need for external lasers source, external microwave filter or external photodetector in the OEO system. Tunable microwave outputs ranging from 25.5 GHz to 26.4 GHz, with single sideband (SSB) phase noise of less than −90 dBc/Hz at 10 kHz offset from the carrier frequency were demonstrated.

#### **2. Device Structure and Fabrication Process**

The integrated laser-photodetector-chip comprises an amplifier feedback laser and a photodetector, as shown in Figure 1a. The AFL consists of a DFB section, a phase section, and an amplifier section. The DFB section functions as a laser source, while the phase section and the amplifier section forms an integrated feedback cavity, allowing the adjustment of the feedback phase and the feedback strength through current injection. The AFL can work in single-mode (S), period one (P1) state, period two (P2), chaos (C) state and dual-mode (D) state by controlling the bias currents. Normally, simple control of the amplifier's bias current will suffice to go through all of the states [7]. The on-chip integrated photodetector directly converts the various dynamic states from the optical domain into the electrical domain. The lengths of the DFB section, the phase section, the amplifier section, and the photodetector section are 300 μm, 240 μm, 510 μm, and 30 μm, respectively. Each adjacent section was electronically isolated by a 20-μm-long isolation region to prevent the electric crosstalk between adjacent sections. The AFL section and the photodetector section shared the same multiple quantum wells (MQWs) structure, which was grown on an S-doped n-type InP substrate by using metal-organic chemical vapor deposition (MOCVD). The schematic illustration of the monolithically integrated laser-photodetector chip is shown in Figure 1b. The epitaxial structure consists of six pairs of compressively strained InGaAsP MQWs sandwiched between two 120-nm-thick InGaAsP separated confinement heterostructure (SCH) layers. A gain-coupled Bragg grating was defined holographically on the upper-SCH layer of the DFB section. Then a p-InP cladding and a p-InGaAs contact layer were regrown by MOCVD. A 3-μm-wide ridge waveguide was fabricated by wet etching. The electrical isolation region between two adjacent sections was formed by etching the p-InGaAs contact layer

off, followed by He+ ion implanting, which provided a ~6 kΩ electrical resistance. A Ti-Au metal layer was sputtered on the p-type InGaAs contact layer to form a p-contact. Then the substrate is thinned, and Au-Ge-Ni/Au was evaporated on the backside. Finally, n-contact was formed after rapid thermal annealing.

**Figure 1.** (**a**) Microscopic picture; (**b**) schematic diagram; and (**c**) test system diagram of the integrated laser-photodetector chip. EA: electrical amplifier; EC: electrical coupler; OSA: optical Spectrum analyzer; OSC: oscilloscope; ESA: electrical spectrum analyzer).

#### **3. Experimental Setup and Results**

#### *3.1. Dynamic States*

The integrated laser-photodetector chip was mounted on a ground-signal-ground (GSG) subcarrier with the S-electrode connected to the p-contact of the photodetector to extract the on-chip electrical signal, as shown in Figure 1c. An electrical amplifier (EA) with 27-dB gain was used to boost the electrical signal. After passing through a DC block, the amplified electrical signal was split into two parts by a 50:50 electrical coupler to monitor the temporal waveforms and the RF spectra by using a real-time oscilloscope (OSC) (Tektronix DPO70000SX, 70-GHz bandwidth, Tektronix, Inc. Beaverton, OR, USA) and an electrical spectrum analyzer (ESA) (Agilent PXA N9030 A, 50-GHz bandwidth, Agilent Technologies Inc. Santa Clara, CA, USA), respectively. The optical spectra were measured by coupling the emission light from the photodetector-side using an optical signal analyzer (OSA) (Advantest Q8384, 0.01-nm resolution, Advantest Corporation, Tokyo, Japan). During the measurement, the working temperature was maintained at 20 ◦C by a thermo-electric cooler (TEC). Under −2.5 V bias condition, the on-chip photodetector had a −3 dB bandwidth of approximately 13 GHz (−10 dB bandwidth of ~26.5 GHz), which was measured by a 50-GHz vector network analyzer (VNA) (HP 8510c, Hewlett-Packard, Palo Alto, CA, USA ).

When characterizing the dynamic states of the chip, the injection currents of the DFB section and phase section were fixed at (IDFB, IPhase) = (78, 3) mA, and the dynamic state was controlled by tuning the amplifier section's injection currents. Figure 2 shows the optical spectra, radio frequency (RF) spectra, temporal waveforms and phase portrait of the device outputs under different amplifier currents. When IA = 0 mA, the chip works at the single-mode (S) state (Figure 2a) with side-mode suppression ratio (SMSR) > 55 dB. The corresponding temporal waveform shows a constant level. Accordingly, the phase portrait is a small spot. The RF spectrum reveals the characteristic relaxation-oscillation frequency of the DFB laser is around ~9 GHz, as shown in Figure 2(a-ii). When IA increases to 9 mA, as shown in Figure 2(b-i), the chip enters into the period-one (P1) state, and the temporal waveform shows a single period oscillation trace. The P1 state is also confirmed from the RF spectrum and the phase portrait as well, where a fundamental frequency appears around 5.5 GHz and the trajectories of phase portrait show clear limit cycle feature. Further increasing IA to 18.5 mA, the chip is driven into the chaos (C) state, which can be confirmed from Figure 2c. The optical spectrum

has been considerably broadened and the corresponding power spectrum covers a broad frequency range. Besides, the temporal waveform fluctuates dramatically, and the phase portrait shows a widely scattered distribution in a large area. As shown in Figure 2(c-ii), the chaotic electrical signal obtained from the on-chip photodetector has a standard bandwidth up to 12 GHz, where the standard bandwidth is defined as the span between the DC and the frequency where 80% of the energy is contained with the power spectrum. Further increasing IA from 18.5 mA to 24.5 mA, the output gradually evolves out of the C state and into a dual-mode (D) state. A typical optical spectrum of the dual-mode emission is presented in Figure 2(d-i). The RF spectrum in Figure 2(d-ii) shows an oscillation peak at ~26.3 GHz, corresponding to the dual-mode spacing. Due to the amplitude imbalance and the lack of coherence of the two laser modes, the temporal waveform does not show a well-defined sinusoidal shape, and the corresponding phase portrait also shows a limit cycle feature, but the traces are broadened compared to the P1 state.

**Figure 2.** Various dynamic states of the output of the integrated laser-photodetector chip at VPD = −2.5 V, IDFB = 78 mA, IPhase = 3 mA when IA varies from top to bottoms as (**a**) 0 mA (S state); (**b**) 9 mA (P1 state); (**c**) 18.5 mA (C state); (**d**) 24.5 mA (D state). (**a-i**)–(**d-i**): optical spectra, (**a-ii**)–(**d-ii**): RF spectra, (**a-iii**)–(**d-iii**): temporal waveforms, and (**a-iv**)–(**d-iv**): phase portraits of various dynamic states, respectively.

In our previous work [7,25,26], we have theoretically and experimentally demonstrated that in the dual-mode state, the beating frequency of the AFL's emission increases with the increase of the feedback strength as IA increases. The relationship between the frequency of the on-chip generated microwave signal and IA of the integrated laser-photodetector chip was investigated by increasing IA from 24.5 mA to 50.5 mA with a 2-mA step. When IA was fixed at 42.5 mA, the RF spectra were shown in Figure 3a, and the 3 dB linewidth of the beating RF signal was 7.6 MHz. As shown in Figure 3b, the on-chip generated microwave signal can be tuned from 26.3 GHz to 34 GHz.

**Figure 3.** When VPD, IDFB, and Iphase were fixed at −2.5 V, 78 mA, and 3 mA, respectively, (**a**) the RF spectra with IA = 42.5 mA, Inset: 100-MHz zoom-in view; and (**b**) the beating RF frequency when IA varied from 24.5 mA to 50.5 mA.

#### *3.2. High-Quality Microwave Signal Generation*

The dual-mode AFL can function as an active microwave photonic filter (MPF) and a pump source to start the optoelectronic oscillation in an optoelectronic oscillator (OEO) [11,27]. However, a discrete photodetector is still needed to achieve O/E conversion. By using the integrated laser-photodetector chip, a frequency tunable OEO with on-chip microwave generation capability was constructed.

The system diagram of the proposed OEO is shown in Figure 4, which contains an optical feedback loop (O-Loop) and an optoelectronic oscillation loop (OE-Loop). When working, the dual-mode signal travels through a circulator (Cir), a 99:1 optical coupler (OC), a 0.3-km-long single-mode fiber, a Mach-Zehnder modulator (MZM) driven by the amplified beating signal, which was extracted from the on-chip integrated photodetector and amplified by two electrical amplifiers (EA) with a total gain of ~60 dB (OE-Loop). Then, the modulated dual-mode light was sent to a 1-km-long single-mode fiber and injected back to chip through port 1 of the Cir to accomplish the O-Loop and OE-Loop. The dual-loop configuration performs a fine mode selection, which helps to suppress the unwanted cavity modes and improve the side-mode suppression ratio (SMSR) of the oscillation modes. With the assistance of O-Loop, two kinds of injection locking will happen inside the integrated laser-photodetector chip simultaneously. One is the delayed self-injection locking of an individual mode by its fiber-delayed replica. The delayed self-injection will considerably reduce the laser linewidth [9,28]. Besides, the equivalent bandwidth of the MPF will also be narrowed due to the narrowed beating signals. The other one is the mutual-injection locking between laser modes and the modulation sidebands. As a result, two laser modes will be synchronized at a fixed mode spacing and phase difference. Accordingly, high-quality beating signal originated from the dual-mode will be generated. The oscillation frequency was determined by the dual-mode spacing, which can be tuned with the injection current of IA. In the O-Loop, the polarization controllers (PCs) were used to match the polarization state between the feedback signal to the MZM and the integrated laser-photodetector, and the variable optical attenuator (VOA) was used control the feedback strength, respectively. The output signal was monitored through an OSA and an ESA.

**Figure 4.** Schematic diagram of the optoelectronic oscillator (OEO) based on the integrated laser-photodetector chip. EA, electrical amplifier; OC, optical coupler; EC, electrical coupler; Cir, circulator; MZM, Mach-Zehnder modulator; PC, polarization controller; VOA, variable optical attenuator; OSA, optical spectrum analyzer; ESA, electrical spectrum analyzer.

When IDFB, IPhase, and IA were biased at 75 mA, 1.5 mA, 42 mA, respectively, the integrated laser-photodetector chip exhibited a dual-mode emission with a mode spacing of 26.05 GHz. The dual-mode optical signal was converted to the electrical domain by the on-chip photodetector and served as the oscillation seeding to start the optoelectronic oscillation. At an optical feedback power of −12 dBm (measured at port 1 of the Cir), the OEO started to oscillate at the beating frequency. Figure 5a shows the generated microwave signal. The inset shows the zoom-in view of this signal in a 1-MHz span. The SSB phase noise spectrum of the obtained microwave signal was measured by the build-in phase noise module of the ESA. As shown in Figure 5b, the SSB phase noise of the 26.05 GHz signal is −92.2 dBc/Hz at a 10-kHz frequency offset of the carrier frequency. Other spurious modes have a maximal phase noise of <−73 dBc/Hz, indicating a good spectral purity of the OEO.

**Figure 5.** (**a**) The RF output from the OEO, Inset: zoom-in view in a frequency range of 1 MHz; (**b**) single sideband (SSB) phase noise spectrum of the generated 26.05 GHz microwave signal.

By tuning the injection current of IA from 40 mA to 46 mA, the output frequency of the OEO can be continuously tunable from 25.5 GHz to 26.4 GHz, as shown in Figure 6a. The SSB phase noise of the generated microwave signals was all below −90 dBc/Hz at a 10-kHz frequency offset over the whole frequency tuning range, as shown in Figure 6b. Due to the limited bandwidth of the photodetector, a further increase of the frequency tuning range was not attained. A widely frequency-tunable OEO can be expected if the bandwidth of the integrated photodetector can be further optimized.

**Figure 6.** The OEO's (**a**) RF spectrum; (**b**) SSB phase noise at the 10 kHz offset from the carrier frequency, with IA varied from 40 mA to 46 mA, when VPD, IDFB, and IPhase were fixed at −2.5 V, 75 mA, and 1.5 mA.

#### **4. Discussion**

Compared with [24], the integrated photonic chip is smaller, which only includes an AFL and a photodetector. The dual-mode light is generated by the AFL, which oscillates in the same resonant cavity. So, the phase correlation between the two modes is usually better than [24] in terms of optical linewidth and heterodyning microwave signal. With this integrated chip, both optical and electronic chaotic signal can be directly generated. The proposed frequency tunable OEO based on the integrated laser-photodetector chip shows a further step toward highly integrated on-chip OEO system. The small frequency adjustment range of the system can be improved by optimizing the material structure of the detector, adopting PIN photodiodes (PIN-PD) or uni-traveling-carrier photodiodes (UTC-PD) type detector structure using the butt-joint growth technique to increase the bandwidth of the detector.

#### **5. Conclusions**

In conclusion, we demonstrated an InP-based monolithically integrated photonic chip including an AFL and a detector for on-chip photonic and microwave generation. The device shows rich dynamic states, including single tone, multi-tone, and chaotic signal. The output optical signal can be directly converted into electrical signals by the on-chip photodetector. Single-tone photonic microwave signal with a frequency tunable from 26.3 GHz to 34 GHz and chaotic signal with a standard bandwidth of 12 GHz was obtained. An OEO based on the integrated photonic chip was built. High-quality microwave signal tunable 25.5 GHz to 26.4 GHz were obtained without the using of external electrical filters and photodetectors. SSB phase noise less than −90 dBc/Hz at a 10-kHz offset from the carrier frequency over the entire frequency range was realized.

**Author Contributions:** Conceptualization, L.Z., D.L., G.C.; methodology, G.C., H.Q.; formal analysis, G.C., H.Q.; investigation, H.Q.; writing—original draft preparation, H.Q.; writing—review and editing, D.L.; visualization, H.Q., G.C.; supervision, D.L., L.Z.; project administration, D.L.; funding acquisition, L.Z., D.L.

**Funding:** This research was funded by National Key Research & Development (R&D) Plan, grant number 2016YFB0402301 and the National Natural Science Foundation of China, grant number: 61975197.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Optical Feedback Sensitivity of a Semiconductor Ring Laser with Tunable Directionality**

#### **Guy Verscha**ff**elt 1,\*, Mulham Khoder <sup>2</sup> and Guy Van der Sande <sup>1</sup>**


Received: 6 September 2019; Accepted: 26 October 2019; Published: 28 October 2019

**Abstract:** We discuss the sensitivity to optical feedback of a semiconductor ring laser that is made to emit in a single-longitudinal mode by applying on-chip filtered optical feedback in one of the directional modes. The device is fabricated on a generic photonics integration platform using standard components. By varying the filtered feedback strength, we can tune the wavelength and directionality of the laser. Beside this, filtered optical feedback results in a limited reduction of the sensitivity for optical feedback from an off-chip optical reflection when the laser is operating in the unidirectional regime.

**Keywords:** semiconductor ring laser; optical feedback; laser stability

#### **1. Introduction**

Many studies have shown that semiconductor lasers are very sensitive to optical feedback, i.e., to part of the laser light being reflected back into the laser cavity with a delay [1–6]. Such coherent optical feedback (COF) is often difficult to avoid in practical systems, as it can be caused, for example, by reflections from a fiber tip or from other boundaries between materials with different refractive indices in the optical system to which the laser beam is coupled. COF can lead to linewidth narrowing for very weak feedback [2], but for larger feedback strengths it will typically introduce unwanted instabilities in the laser output [3]. For example, it has been shown that COF can lead to linewidth broadening [4], chaotic intensity fluctuations [5] and coherence collapse [6].

In order to avoid or suppress the COF-induced instabilities, several approaches have been investigated [7–9]. The most straightforward way to avoid them is to place an optical isolator with a large isolation ratio at the output of the laser. This works well to avoid COF-induced dynamics, but is an expensive approach as the isolator needs magneto-optic materials that—for technological reasons—cannot easily be integrated on the laser chip. Moreover, the optical isolator needs to be accurately aligned with the laser chip to avoid propagation losses of the emitted beam. Because of the high cost of such external isolators, there is considerable interest in other approaches to achieve the goal of suppressing the COF-induced dynamics in a semiconductor laser.

A laser with a ring-shaped cavity is inherently interesting for the purpose of suppressing feedback dynamics, as any externally reflected light will be re-injected in the cavity in the direction opposite to that of the initially emitted beam: imagine such a ring laser to emit in the clockwise (CW) directional mode, optical feedback will then result in part of this beam being coupled into the counterclockwise (CCW) directional mode. In [7], a weak optical isolator is integrated in the laser cavity in order to make one of the directional modes dominant, such that the COF is injected in the directional mode that is switched-off, hence reducing its destabilizing effect. But this approach requires complex components in the laser cavity to achieve the required weak optical isolation, making the laser system difficult to control. Another ring-laser based device was studied in [9], where the fabrication process of the semiconductor ring laser (SRL) is optimized to such a degree that coupling between the directional modes through backscattering is very low. This results in unidirectional operation (i.e., the laser emits in one of the directional modes) of the fabricated SRLs, which leads to a strong suppression of feedback-induced dynamics [8] as compared to a Fabry–Perot laser fabricated on the same chip. However, when using generic integration platforms—which are not optimized for one specific purpose—the backscattering will typically be much higher, resulting in bidirectional operation (i.e., the power in the two directional modes being roughly equal) of fabricated SRLs [10,11].

In this paper, we investigate the feedback sensitivity of an SRL that we designed and fabricated using the generic JeppiX fabrication platform [12]. Because of a substantial amount of backscattering between the directional modes, the SRL itself will typically emit bidirectionally. In this design, we included on-chip filtered optical feedback (FOF) paths that have been shown [11] to make the SRL emit in a single-longitudinal mode. Controlling the FOF also allows us to tune the emitted wavelength of the SRL. Moreover, as we will discuss in the next sections, the FOF in this SRL has as a side effect that it makes the emission (somewhat) unidirectional. Based on the above mentioned work in [8] on unidirectional SRLs, we thus expect our SRL design to be less sensitive to optical feedback from off-chip reflections. In order to check the effectiveness of this approach, we experimentally and numerically study in this paper the sensitivity of our SRL design to undesired external optical feedback.

The remainder of the paper is structured as follows. In Section 2 we describe the layout of the SRL and we detail the experimental setup. The results of the experiments and numerical simulations are shown in Section 3, whereas Section 4 is devoted to the discussions of the results. Finally, we end the paper with conclusions in Section 5.

#### **2. Materials and Methods**

#### *2.1. Layout of the SRL*

The layout of the device is illustrated by the picture shown in Figure 1. It has been fabricated using the standard building blocks from the Oclaro foundry, and a detailed description of the layout is given in [11]. As can be seen in Figure 1, the SRL has a racetrack-shaped geometry and optical gain is provided by two semiconductor optical amplifier (SOA) sections that are electrically interconnected. The laser cavity also contains two 2 × 2 multi-mode interference (MMI) couplers, which each couple 50% of the light out of the cavity. The outputs of the top MMI are coupled to the edges of the laser chip such that the CW and CCW modes can be measured. The bottom MMI in Figure 1 couples to two FOF branches. Each of these branches consists of a phase shifter (PS), an SOA and a distributed Bragg reflector (DBR). These components can be electrically tuned by adapting the current injected in the attached contact pad, such that we have control over the center wavelength (by changing the DBR current), the strength (by changing the SOA current) and the phase (by changing the PS current) of the FOF. Feedback arms 1 and 2 are used to control the FOF into the CW and CCW directions, respectively.

**Figure 1.** Image of the semiconductor ring laser with filtered feedback, in which the different laser and feedback components are indicated.

#### *2.2. Experimental Setup*

To measure the static and dynamic characteristics of the SRL, we used the setup that is schematically depicted in Figure 2. The SRL was mounted on a temperature-controlled heat sink, with which we stabilized the temperature of the laser chip at 21 ◦C. In principle, each of the contact pads visible in Figure 1 can be connected to a current source using electrical contact probes, but for the work presented in this paper only the laser pad and the SOA pad in feedback arm 2 were contacted. This allowed us to change the laser's injection current Ilaser and the current ISOA1 that controls the strength of the FOF of arm 2. It should be noted that we have obtained similar results when using FOF from feedback arm 1, with the difference being that the roles of the CW and the CCW modes are then reversed. Light emitted in the CW and in the CCW direction was collected outside the laser chip using lensed fibers. Light emitted in the CCW direction was sent through a feedback loop, and was coupled back with a time delay of about 50 ns into the CW directional mode. The COF feedback loop consisted of a circulator, an external SOA, an optical bandpass filter, a 2 × 2 single-mode splitter and a polarization controller. The circulator directed the CCW light from the laser towards the external SOA. The current ISOA2 injected in this external SOA was used to control the COF strength. Next, the amplified light was sent through a tunable bandpass filter with a bandwidth of 0.3 nm of which the center wavelength was tuned to the SRL's wavelength. This tunable filter was needed to remove the amplified spontaneous emission noise—introduced by the external SOA—from the feedback signal. The polarization controller was used to adjust the polarization of the re-injected light such that it matched the emitted polarization direction. Light was re-injected into the SRL chip using the third port of the circulator. The splitter coupled 50% of the light out of the feedback loop such that we could measure its temporal and spectral properties. The optical spectrum was measured with a scanning spectrum analyzer set at a resolution of 0.02 nm. Time traces of the intensity fluctuations were measured using a 12 GHz photo-detector coupled to a fast oscilloscope of which the input bandwidth was set at 13 GHz in the experiments discussed in Section 3.

**Figure 2.** Schematic of the experimental setup. LF, lensed fiber; SOA2, semiconductor optical amplifier used to tune the coherent optical feedback (COF) strength; TF, tunable optical bandpass filter; Det, fast opto-electronic detector; PC, polarization controller.

#### *2.3. Rate-Equation Model*

The behavior of the SRL under the effect of FOF and/or COF can be simulated using different models [13,14]. In this work, we used a two-directional mode rate equation model of the SRL [15], extended with Lang–Kobayashi terms, to take into account the optical feedbacks [16]. The equations of this models are:

$$\dot{E}^{\rm cw} = \kappa (1 + ia) [\mathcal{M}^{\rm cw} - 1] E^{\rm cw} - (k\_d + i \, k\_c) E^{\rm cw} + \eta\_1 E^{\rm cw} (t - \tau\_1) + \sqrt{D} \xi^{\rm cw}, \tag{1}$$

$$\dot{E}^{\rm czw} = \kappa (1 + ia) [\mathcal{M}\mathcal{C}^{\rm cw} - 1] E^{\rm czw} - (k\_d + i \, k\_c) E^{\rm cw} + \eta\_2 \, E^{\rm cw} (t - \tau\_2) + \sqrt{D} \xi^{\rm czw}, \tag{2}$$

$$\frac{1}{N}\dot{N} = \mu - N - N\left(\mathbf{G}^{\text{cw}} \left| \mathbf{E}^{\text{cw}} \right|^2 + \mathbf{G}^{\text{cw}} \left| \mathbf{E}^{\text{cw}} \right|^2\right). \tag{3}$$

Equations (1) and (2) describe the evolution of the slowly varying complex electric fields *Ecw* and *Eccw* of the CW and CCW directions, respectively. The number of carriers, *N*, is described by Equation (3). We have limited ourselves to one longitudinal mode (LM). The values of the different parameters are as follows: κ = 200 ns−<sup>1</sup> is the field decay rate, α = 3.5 is the linewidth enhancement factor, μ = 1.2 is the normalized injection current, γ = 0.4 ns−<sup>1</sup> is the carrier inversion decay rate. The effect of the backscattering is taken into account using the dissipative backscattering parameter kd = 0.2 ns−<sup>1</sup> and the conservative backscattering parameter *kc* = 0.88 ns−<sup>1</sup> which have been used for both of the two directional modes. The differential gain functions are given by:

$$G^{cw} = 1 - s \left| E^{cw} \right|^2 - c \left| E^{cw} \right|^2,\tag{4}$$

$$G^{\rm cuv} = 1 - s \left| E^{\rm cuv} \right|^2 - c \left| E^{\rm cuv} \right|^2,\tag{5}$$

where *s* = 0.005 is the self-saturation and *c* = 0.01 is the cross-saturation between the two directions of the same LM. η<sup>1</sup> represents the strength of the COF. τ<sup>1</sup> is the delay time of the COF which is measured in our setup to be 50 ns. η<sup>2</sup> represents the strength of the FOF. As the FOF couples the CW mode back into the CCW mode, we only include an FOF term in Equation (2). The bandwidth of the filter in the feedback loop is adiabatically eliminated from Equation (2) as this filter bandwidth is much larger than the bandwidth of the fluctuations in Ecw and Eccw. τ<sup>2</sup> is the propagation time in the FOF section which is integrated on the chip and is very small. Therefore, we take τ<sup>2</sup> equal to zero in the simulations. Here it is important to mention that the feedback scheme in this study is different from the feedback scheme which has been discussed in [17,18], where self-feedback has been investigated. The last terms in Equations (1) and (2) represent the effect of spontaneous emission noise coupled to the CW/CCW modes [18,19]. *D* represents the noise strength expressed as *D* = D0(N + G0N0/κ), where D0 is the spontaneous emission factor, G0 <sup>=</sup> <sup>10</sup>−<sup>12</sup> m3s−<sup>1</sup> is the gain parameter, N0 <sup>=</sup> 1.4 <sup>×</sup> 1024 <sup>m</sup>−<sup>3</sup> is the transparency carrier density. ξ*i* (t)(i = cw, ccw) are two independent complex Gaussian white noises with zero mean and correlation ξ*i*(*t*)ξ<sup>∗</sup> *j* . *t* <sup>=</sup> <sup>δ</sup>*ij t* − . *t* . Time is rescaled to photon lifetime τ*ph* = 5 ps.

#### **3. Results**

#### *3.1. Experimental Results*

Using the setup of Figure 2, we first measured the static characteristics of the studied SRL. The output power of the two directional modes is shown in Figure 3 as a function of the laser injection current (without pumping the SOAs in the FOF arms). The threshold current of this device was 34 mA. For all currents not too far above threshold, the power in the two directional modes was roughly equal, showing that this SRL always operates in the bidirectional regime [13], which indicates that there was a substantial amount of backscattering in SRLs fabricated on the used platform. For some laser bias currents, the SRL emitted in a single longitudinal mode, but for most values of the laser injection current, the laser emitted multiple longitudinal modes. The longitudinal mode spacing was measured to be 0.2 nm. The DBRs in the FOF arms have a peak intensity reflection of 0.58 and a reflection bandwidth of 2 nm. In [11] we have shown that a sufficiently large amount of feedback in either of the FOF channels resulted in single longitudinal mode operation, that the wavelength of the emitted mode could be changed by changing the DBR center reflection wavelength, and that this wavelength could be fine-tuned using the phase shifters in the FOF arms.

**Figure 3.** Output power of the two directional modes versus laser injection current while the current in the filtered optical feedback (FOF) section is equal to 0 mA.

If we only applied FOF in one of the arms, the FOF had an additional effect that made the SRL somewhat unidirectional. This is illustrated by the measurement shown in Figure 4, where we plot the power in the two directional modes as a function of the current ISOA1 injected in the SOA of FOF arm 2 in Figure 1. The laser current Ilaser was kept constant, as shown in Figure 4, at a value of 60 mA. For low values of ISOA1, most power was emitted in the CW direction. But as ISOA1 was ramped up, the power in the CCW direction gradually increased at the expense of the power in the CW direction. This is to be expected from the feedback configuration used in this experiment as the FOF in feedback arm 2 coupled light from the CW direction into the CCW direction. The power distribution over the two directional modes is further detailed at the right-hand side of Figure 4, where we plot the ratio between the power in the CCW direction and the power in the CW direction. This so-called directional mode suppression ratio (DMSR) increased most strongly when ISOA1 increased from 0 to 11 mA, and then continued to increase at a slower pace for still higher values of ISOA1.

**Figure 4.** Output power of the two directional modes versus current injected in the semiconductor optical amplifier (SOA) in the FOF arm 2 (left) and directional mode suppression ratio as a function of the SOA current in the FOF arm (right) at a laser injection current of 60 mA.

Based on Figure 4, we identified three interesting bias points (indicated by the black arrows) at which we wanted to investigate the sensitivity to COF. The first bias point, BP1, corresponds to ISOA1 = 0 mA, as in that case there was no FOF and we measured the feedback sensitivity of the SRL itself. The second bias point, BP2, that we would further investigate corresponds to ISOA1 = 11 mA, as in this case the FOF clearly favored the CCW directional mode. Finally, the third selected bias condition, BP3, corresponds to ISOA1 = 30 mA and in this case the directional mode suppression ratio was greatest. For BP2 and BP3, the SRL emitted a single longitudinal mode whose wavelength of 1551.555 nm was determined by the reflection spectrum of the DBR in feedback arm 2. For BP1, the output of the SRL was also single-mode but the emission wavelength of 1538.405 nm was determined by the gain maximum.

Next, we measured time traces of the intensity in the CCW direction for different values of the current ISOA2 injected in the external SOA. We first calibrated the amplification of the external amplifier by measuring the power transmitted through the external SOA as a function of its bias current (while keeping the laser current Ilaser and the FOF current ISOA1 constant). For small values of ISOA2, the CW intensity was rather constant with some noise-induced fluctuations around the steady state. This is illustrated in Figure 5 (left) at a setting (Ilaser, ISOA1, ISOA2) = (60 mA, 11 mA, 500 mA). Increasing ISOA2 eventually led to undamping of the relaxation oscillations as illustrated in Figure 5 (middle) for (Ilaser, ISOA1, ISOA2) = (60 mA, 11 mA, 600 mA). This marks the onset of the COF-induced dynamics. For larger values of the COF strength, the feedback-induced dynamical fluctuations became stronger and more complex as illustrated in Figure 5 (right) for (Ilaser, ISOA1, ISOA2) = (60 mA, 11 mA, 700 mA).

In order to quantify the strength of the feedback-induced dynamics in a simple way, we used the following metric: we extracted the rescaled STD as the ratio between the standard deviation of the laser intensity fluctuations σlaser and the mean value of the detector signal. Calculating this ratio is equivalent to rescaling the time traces such that the average value of the detector signal is equal to one. We performed this rescaling of the STD to make the extracted values independent of the average power coupled to the read-out fiber. The noise of the oscilloscope and the photo-detector are compensated for when extracting the value of σlaser from the time traces by assuming that the noise of these sources is Gaussian and is independent from the fluctuations in the laser's intensity. To perform this compensation, we measured a time trace of the detector signal (using the same oscilloscope settings as when measuring the laser's intensity) without optical input to the detector. From this time-trace, we determined the standard deviation σdet of the detector and oscilloscope noise (the mean value of the detector and oscilloscope noise was measured to be close to zero). Using the standard deviation σtimetrace extracted from the intensity time trace, we estimate the standard deviation of the intensity fluctuations σlaser to be σ*laser* = ! σ2 *timetrace* <sup>−</sup> <sup>σ</sup><sup>2</sup> *noise*.

In Figure 6 we plot the value of the rescaled STD for the three bias conditions BP1, BP2 and BP3 mentioned above. The COF signal strength, plotted on the horizontal axis of Figure 6, was changed by changing ISOA2 and was obtained by measuring the optical power after the splitter in Figure 2 when the feedback loop was open. For each of the bias conditions, the STD was small for small values of the COF strength, as there were not yet any feedback-induced dynamics in the time traces. When increasing the COF strength, we can see in Figure 6 that the onset of the feedback-induced dynamics was lowest for bias condition BP1, i.e., without FOF to stabilize the laser. When FOF was applied (see measurements for BP2 and BP3 in Figure 6), the onset of the COF dynamics was shifted to larger values of the feedback strength, but this shift was not large for BP2 and BP3: the shift in the onset when comparing BP1 to BP2 was roughly a factor of 2 and was thus rather modest as compared to the suppression of feedback dynamics in strongly unidirectional SRLs [7,8]. Moreover, when increasing the FOF strength from BP2 to BP3, we actually observed a slight drop in the onset of the COF dynamics. The experiments thus show only a limited effectiveness of the proposed FOF scheme to suppress these dynamical fluctuations, and this effectiveness is furthermore dependent on the exact value of the applied FOF strength. The reason behind these observations will be clarified based on numerical simulations of the system in Section 3.2.

**Figure 5.** Time traces of the laser's output as measured by the detector in the setup of Figure 2 at a laser injection current of 60 mA and an SOA current in the FOF path of 11 mA for different strengths of the COF by changing the current injected in SOA2 in the COF path: ISOA2 = 500 mA (left), ISOA2 = 600 mA (middle) and ISOA2 = 700 mA (right).

**Figure 6.** Rescaled STD from time traces of the laser's output as a function of the COF signal strength as measured after the splitter in Figure 2 for different values of the ISOA1 (which controls the FOF strength).

#### *3.2. Results from Numerical Simulations*

Using the rate-equations that have been introduced in Section 2.3, we performed a series of numerical simulations that mimic the experiments described above. In these simulations we set the normalized injection current to 1.2 and we selected particular values for the FOF and COF strengths in order to simulate time-traces of the directional powers. We remark here that we have obtained similar behavior for other values of the pump strength. From these time traces, we then extracted the STD of the intensity fluctuations in a similar manner to that used in the experiments represented in Figure 6. We show in Figure 7 (left) the simulated time traces when the strength of the COF was η<sup>1</sup> = 0.4 ns–1 (as this is a good setting to show the effect of the FOF on the onset of the laser dynamics). In the red time trace of Figure 7 (left), FOF was not used whereas the FOF strength was set to 2 ns–1 in the blue time trace of Figure 7 (left). Using FOF, the intensity fluctuations in the time trace became smaller as compared to the case without FOF. We also notice that the average intensity in the CCW direction increased due to the FOF, as it enhances the CCW mode (see also Figure 4). As a result, the rescaled STD was smaller for the trace in Figure 7 (left) corresponding to η<sup>2</sup> = 2 ns–1.

The rescaled STD of the time traces was measured in the experiments to be 0.02. We used this value to estimate D0 to be 2 <sup>×</sup> 10−<sup>6</sup> ns−<sup>1</sup> in order to find the same rescaled STD in the simulations without COF. Similarly to the experiments, we started by calculating the mean value and the STD of the time traces without FOF (η<sup>2</sup> = 0 ns−1). We increased the strength of the COF by increasing η<sup>1</sup> from 0 to 1.0 ns−<sup>1</sup> in steps of 0.05 ns−<sup>1</sup> while the rest of the parameters were fixed (η<sup>2</sup> = 0 ns<sup>−</sup>1). Next, we repeated the calculations of the mean value and the STD of the time traces, but this time with FOF by setting η<sup>2</sup> to 3 ns<sup>−</sup>1, 5 ns−<sup>1</sup> and 8 ns<sup>−</sup>1, while the rest of parameters were kept unchanged. We plot the rescaled STD from the simulations in Figure 7 (right) as a function of the COF strength η1. At low values of the COF strength, the STD is relatively small and remains approximately constant when changing the COF strength. The onset of COF-induced dynamics is visible in these curves as the point at which the STD starts to rapidly increase with increasing COF strength. Similarly to the experiments, the onset happened first for the laser without FOF around η<sup>1</sup> = 0.2 ns<sup>−</sup>1. When FOF was applied, the

onset first shifted to larger COF strengths, but this shift is albeit rather limited. When further increasing the FOF strength, the onset of the dynamics shifted erratically and we did not observe a continuous increase in the onset. These numerical results thus agree qualitatively with our experimental trends and observations discussed in Section 3.1, and show that the FOF scheme presented in Figure 1 does not really help to reduce the COF-induced dynamics.

**Figure 7.** Simulated time traces in the CCW direction when the COF strength is η<sup>1</sup> = 0.4 ns–1 without FOF in red and with FOF using η<sup>2</sup> = 2 ns–1 in blue (left). Standard deviation of the simulated time traces of the laser's output as a function of the COF strength η<sup>1</sup> for different values of the FOF strength η<sup>2</sup> (right).

To further elucidate the stabilizing effect of the FOF on the SRL's dynamical behavior, we computed and analyzed the so-called Lyapunov exponents, λi, from the model described in Equations (1)–(3) without noise (setting *D* = 0). By studying the Lyapunov spectrum, we tried to understand how FOF influences both the stability and complexity of the chaotic dynamics that might have arisen. For the computation of the Lyapunov exponents, we applied the ideas of Farmer [20] to our case. Specifically, we integrated the corresponding delay differential equations with an Euler method. This converts the original delay differential equations in a map. We computed the Lyapunov exponents of this map. Only a finite portion of the infinite set of λ<sup>i</sup> can be determined by such a numerical analysis. In Figure 8, we present the five largest Lyapunov exponents vs. the COF strength η1. Due to the field nature of the equations, one exponent will always be zero. If only the maximal exponent is zero, the SRL will be emitting in a continuous wave. If two exponents are zero, while the others are all negative, the laser output will be periodic. If more exponents are zero, the dynamics can correspond to either periodic or quasi-periodic behavior. Once the maximal Lyapunov exponent becomes positive, the SRL will be operating chaotically. From Figure 7 (right) and Figure 8 (left), in the case of no filtered feedback, the increase of the STD around η<sup>1</sup> = 0.1 to 0.4 ns–1 can be attributed to a bifurcation from continuous wave emission to periodic oscillations. It is only later, after a regime of quasi-periodic behavior, that the laser became chaotic (around η<sup>1</sup> = 0.8 ns–1). With FOF ( η<sup>2</sup> = 3.0 ns–1), in Figure 8 (middle), below η<sup>1</sup> = 0.7 ns–1, the SRL with filtered feedback was continuously lasing except for some very small windows of periodic behavior. While this seems to indicate that the SRL would be more stably lasing, the negative Lyapunov exponents were now much smaller in amplitude. This indicates that the SRL would be much easier to destabilize due to noise, for example. The bifurcation to chaotic behavior hardly moved and still appeared at feedback strengths around η<sup>1</sup> = 0.8 ns–1. However, its accompanying positive Lyapunov exponents were increased significantly, indicating a more complex and less damped dynamical chaotic behavior. For η<sup>2</sup> = 8.0 ns–1 (Figure 8 (right)), it is clear that the large region of chaos shifted to lower values of <sup>η</sup><sup>1</sup> (η1<sup>≈</sup> 0.4 ns–1). Around <sup>η</sup><sup>1</sup> = 0.2 ns–1, the laser was first destabilized as a small window of mildly chaotic behavior appeared (i.e., only one of the Lyapunov exponents was positive). This onset of chaotic oscillations corresponds to the abrupt change in the rescaled STD observed numerically in Figure 7 (right) and experimentally in Figure 6 for ISOA1 = 30 mA. To conclude, with filtered feedback, the dynamical behavior of the SRL was altered considerably. For some values of the filtered feedback this led to a larger but less stable continuous

wave regime and chaos which was more complex. Because of the larger continuous wave regime, the feedback sensitivity was somewhat reduced as compared to the device without FOF.

**Figure 8.** The five largest Lyapunov exponents: without FOF (i.e., η<sup>2</sup> = 0 ns–1) (left), with FOF (i.e., η<sup>2</sup> = 3 ns–1) (middle) and with large FOF strength (i.e., η<sup>2</sup> = 8 ns–1) (right).

#### **4. Discussion**

The above results show that the filtered feedback has only a marginal beneficial effect regarding feedback sensitivity of the SRL. Even more, in several cases the filtered feedback leads to a further destabilization of the laser dynamics. One reason that comes to mind as to why the addition of the filtered feedback does not deliver the desired outcome, is the fact that the SRL is not operating in an ideal unidirectional emission regime, i.e., the CW mode in which the COF signal is reinjected is not fully turned off. To investigate if this might be the issue, we have considered an ideal SRL with no backscattering between the two counter-propagating modes (i.e., kd = kc = 0) in the numerical simulations. In this case, the SRL without any feedback operates in a unidirectional regime with the full output power concentrated either in the CW or CCW mode. In Figure 9, we show the results from a numerical analysis of the Equations (1)–(3) for kd = kc = 0. The left-hand side of Figure 9 shows rescaled STDs obtained from time traces using the procedure described above. For all cases, we find that the STD increases for low COF strengths, which are even lower than in Figure 7. The right-hand side of Figure 9 shows the five largest Lyapunov values describing the noiseless dynamics of the SRL in the case of filtered feedback. Again, at a very low feedback strength (η<sup>1</sup> > 0.05 ns–1), the SRL becomes chaotic. It is clear that even in the case of no backscattering, the filtered feedback actually destabilizes the SRL. This indicates that—for the device layout studied here—a feedback signal in the quiescent directional mode is coupled (through the FOF branch) sufficiently strongly to the dominant directional mode in order to invoke delay-induced dynamical fluctuations.

**Figure 9.** Standard deviation of the simulated time traces of the laser's output as a function of the COF strength (left) and the five largest Lyapunov exponents (right) with FOF (η<sup>2</sup> = 3 ns–1) when the backscattering is set to zero.

#### **5. Conclusions**

In this paper we studied—both experimentally and numerically—an SRL in which on-chip filtered optical feedback is used to tune the wavelength, to enforce single-longitudinal mode operation and to enhance the directionality of the laser. More particularly, we focused on the sensitivity to coherent optical feedback from a longer off-chip delay path, and we initially speculated that the FOF might result in a higher tolerance to COF. However, our experiments and modeling show that the FOF does not result in a substantial shift of the COF-induced dynamics towards higher COF strengths. We attribute this to the fact that the COF signal after reinjection into the SRL is coupled back into the lasing mode via the filtered feedback. Even when the backscattering would be reduced strongly, our simulations show that this will not result in a beneficial effect for the studied SRL with FOF configuration.

**Author Contributions:** Conceptualization, G.V., M.K. and G.V.d.S.; methodology, G.V., M.K. and G.V.d.S.; experiments, G.V.; numerical simulations, M.K. and G.V.d.S.; formal analysis, G.V., M.K. and G.V.d.S.; writing—original draft preparation, G.V., M.K. and G.V.d.S.; writing—review and editing, G.V., M.K. and G.V.d.S.; funding acquisition, G.V. and G.V.d.S.

**Funding:** This research was funded by Research Foundation Flanders (FWO) a.o. under grant numbers G028618N and G029519N, the Hercules Foundation under grant "High-speed real-time characterization of photonic components" and the Research Council of the Vrije Universiteit Brussel. M.K. wishes to acknowledge partial financial support by FWO by ways of EOS Project No. G0F6218N (EOS ID 30467715).

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Development of an Interference Filter-Stabilized External-Cavity Diode Laser for Space Applications**

**Linbo Zhang 1,2,3, Tao Liu 1,\*, Long Chen 1,3, Guanjun Xu 1, Chenhui Jiang 1,3 and Jun Liu <sup>1</sup> and Shougang Zhang <sup>1</sup>**


Received: 13 November 2019; Accepted: 16 January 2020; Published: 18 January 2020

**Abstract:** The National Time Service Center of China is developing a compact, highly stable, 698 nm external-cavity diode laser (ECDL) for dedicated use in a space strontium optical clock. This article presents the optical design, structural design, and preliminary performance of this ECDL. The ECDL uses a narrow-bandwidth interference filter for spectral selection and a cat's-eye reflector for light feedback. To ensure long-term stable laser operation suitable for space applications, the connections among all the components are rigid and the design avoids any spring-loaded adjustment. The frequency of the first lateral rocking eigenmode is 2316 Hz. The ECDL operates near 698.45 nm, and it has a current-controlled tuning range over 40 GHz and a PZT-controlled tuning range of 3 GHz. The linewidth measured by the heterodyne beating between the ECDL and an ultra-stable laser with 1 Hz linewidth is about 180 kHz. At present, the ECDL has been applied to the principle prototype of the space ultra-stable laser system.

**Keywords:** external-cavity diode laser; interference filter; laser diode; laser stabilization; space optical clock

#### **1. Introduction**

Compared to most other types of laser, diode lasers are cheap and simple to use; they also have a high power and cover a large wavelength range. They have therefore become attractive light sources with versatile applications in many fields of optical technology and experimental physics, such as optical atomic clocks, precision measurement, astrophysics, and quantum communication [1–5]. With this wide spectrum of applications, it is not surprising that lasers are used in very different environments, with one of the most demanding being space [6].

In 1980, Lang and Kobayashi [7] applied external-cavity feedback technology to diode lasers. The increased external cavity can narrow the laser line width, and The wavelength can be tuned by changing the external cavity length. In The following years, Soviet scientists, for The first time, used a diffraction grating to feed back part of the output light of the diode laser to the active region, narrowing the linewidth of the laser to 1.5 MHz [8]. Common external-cavity diode laser (ECDL) designs such as the Littrow [9,10] and Littman–Metcalf configurations [11,12] use diffraction gratings for wavelength selection. Those lasers require precise alignment and are therefore sensitive to acoustic and mechanical disturbances, particularly when a spring-loaded kinematic mount is used to align the grating or feedback optics [13]. Another design uses a narrowband interference filter (IF) placed in a linear cavity as a frequency-selective component [14–16]. Because The wavelength of the transmission

maximum depends on the angle of incidence, where the angle is the angle of incidence and the maximum wavelength is the transmitted wavelength at normal incidence, the wavelength can be tuned by turning the filter. This leads to a sensitivity of *dλ* ! *dθ* = 0.017 nm/mrad, which is 60 times better than that of the Littrow configuration. Thus, the laser design is in principle less sensitive to mechanical vibrations and disturbances. With these advantages, the interference-filter configuration was chosen for the PHARAO [17,18] and SOC2 [19] projects for space laser systems.

As the atomic clock with the highest performance index in the world, the measurement accuracy of a strontium atomic optical clock has entered the order of 10−<sup>19</sup> [20]. In the microgravity environment of space, the performance of optical clocks is expected to be further improved [21]. The National Time Service Center (NTSC) of China is conducting research on the space Sr atomic optical clock. In the strontium atomic optical clock system, the wavelength of the clock transition <sup>1</sup>*S*<sup>0</sup> →<sup>3</sup> *<sup>P</sup>*<sup>0</sup> is 698 nm, and The natural linewidth is only 1 mHz. The linewidth of the detection light must reach the order of Hz or even sub-Hz. We use an ultra-narrow linewidth laser to detect the clock transition line. The ultra-narrow linewidth laser (also called clock laser) was obtained by locking the laser frequency to a high-finesse optical reference cavity by means of the Pound–Drever–Hall (PDH) technique [22]. The ECDL developed in this paper is used as the light source of an ultra-narrow linewidth laser system which is aimed to has a free-run linewidth at the level of 500 kHz or less. At present, commercial semiconductor lasers at 698 nm have wavelength tuning capabilities up to 10 nm. However, due to the use of an elastically loaded adjustment device, the structure is not very stable and usually needs to be readjusted every month to ensure good optical feedback and correct wavelength. Although The laser developed in this paper does not have a wide range of tuning capabilities, it has a stable structure and is one of the best choices to meet our special applications.

The objective of the present study is to develop and characterize a prototype of a 698 nm interference-filter external-cavity diode laser (IF-ECDL). The developed ECDL is compact and robust, and it will be planned for use in China's space Sr atomic light clock system in the future.

#### **2. Working Principle of the IF-ECDL**

In the IF-ECDL, the interference filter provides wavelength selection, and a partially reflective mirror provides optical feedback, as shown schematically in Figure 1. The interference filter is composed of alternating layers of dielectric material that can transmit a narrow frequency band while reflecting the light of other wavelengths. The narrowband interference filter is actually a thin Fabry–Perot etalon with only one transmission peak in the visible range, also known as a line filter. The bandpass section of an interference filter is made by the repetitive vacuum deposition of thin layers of partially reflecting dielectric compounds on a glass substrate. Dielectric layers are arranged to form reflective cavities. The spacer region is designed to be *λ*0/2 thick, where *λ*<sup>0</sup> is the central wavelength of the filter. This allows light that meets the reflection boundary conditions to be reinforced and transmitted by the cavity. The rejected light is reflected by the layers of dielectric material. The laser light generated by the semiconductor laser is collimated into parallel light by a lens, and then incident on the interference filter at a certain angle *θ*. Using the multi-beam isotropic interference theory, the wavelength *λ* at the peak of the transmittance is

$$
\lambda = \lambda\_{\text{max}} \sqrt{1 - \frac{\sin^2 \theta}{n\_{IF}^2}} \tag{1}
$$

where *nIF* is the interference filter's effective index of refraction, and *λmax* is the transmission wavelength value of the narrow-band filter when the beam is normally incident, and is also the maximum limit wavelength value in the tuning range of the narrow-band filter.

**Figure 1.** Schematic of interference-filter external-cavity diode laser (IF-ECDL). LD: laser diode; CL: collimating lens; IF: interference filter; L1: cat's-eye lens; PZT: piezotube; OC: partially reflective out-couple mirror; L2: re-collimating lens.

Assuming that the outgoing light intensity of the bare laser diode is *I*, then the total light intensity transmitted through the interference filter is given by the Airy formula

$$I\_T = I\_0 \frac{1}{1 + F\_{IF} \sin^2(\theta/2)}\tag{2}$$

where *FIF* is the fineness of the IF. By rotating the angle of the interference filter placed in the ECDL resonator cavity with respect to the laser, the wavelength of the laser exiting the IF-ECDL is tuned. Using the IF for frequency selection, a single longitudinal mode laser can be obtained.

#### **3. Design of the IF-ECDL**

#### *3.1. Optical Design*

A semiconductor laser diode (LD) with a central wavelength of 698 nm is anti-reflection coated on its output facet and is combined with an external cavity, leading to a large tuning range for the wavelength. The cavity length of the LD is 750 μm, and The reflectance of the two surfaces are 1 for the back face and 3 × <sup>10</sup>−<sup>4</sup> for the AR coated face. The light coming from the LD is collimated by a collimating lens (CL) with a focal length of 4.02 mm and a numerical aperture of 0.6. To collect as much light from the diode as possible, the CL must have a large numerical aperture.

The optical feedback is provided by a combination of a cat's-eye lens (L1) with a focal length of 15.29 mm and a partially reflective out-couple mirror (OC) with 30% reflectivity in focal distance. This cat's-eye configuration is less sensitive to misalignments of the OC compared to the case of feedback with no such lens. A cat's-eye reflector decreases the sensitivity to optical misalignment and maximizes the feedback efficiency. The overall external-cavity length from the LD output facet to the OC front facet is 50 mm and corresponds to an axial mode spacing of

$$
\Delta\_{FSR} = \frac{c}{2L} = 3\,\text{GHz}.\tag{3}
$$

The optical length of the external cavity is tuned with a piezotube (PZT) of 9 mm in length and with internal and external diameters of 5 and 10 mm, respectively. Applying a voltage of 100 V to the PZT changes the length of the external cavity by 1.4 μm. For a given optical mode, a variation Δ*l* in the optical path *l* of the external cavity yields a relative frequency detuning of

$$\frac{\Delta\nu}{\nu} = -\frac{\Delta l}{l} \tag{4}$$

allowing the PZT to tune the laser frequency with a response of −120 MHz/V.

The final optical component in the optical path is a re-collimating lens (L2) with a focal length of 11 mm, which is used to re-collimate the out-coupled laser. To narrow the output beam, the L2 focal length is chosen to be smaller than the L1 focal length.

An IF is placed inside the external cavity between the CL and L1 and is used for coarse wavelength tuning. It is part of a resonator that forces the laser to maintain a stable single mode and reduces the linewidth. The IF is made of a substrate that is coated with many dielectric layers on one side and anti-refection coated on the other side [23]. It has a 0.48 nm super-narrow passband and a peak transmission of 96%, and its measured spectrum is shown in Figure 2. The wavelength of the transmitted light is changed by adjusting the angle of the IF. Compared with the Littrow and Littman–Metcalf configurations, the IF and cat's-eye reflector replace the grating used to select the laser wavelength and form an external cavity, making it relatively easily to adjust the laser frequency and optimize the optical feedback. Furthermore, because The IF and cat's-eye reflector are insensitive to the incident angle [15], the present design has a higher mechanical stability than those of the Littrow and Littman–Metcalf configurations.

**Figure 2.** Measured transmission spectrum of interference filter (IF) with 6◦ angle of incidence. Data provided by manufacturer (Alluxa, Santa Rosa, CA, USA). The IF has a peak transmittance of up to 96% and a bandwidth of ∼0.48 nm(294.2 GHz).

#### *3.2. Structural Design*

The aim of the structural design of the ECDL is to provide a good mechanical environment for the optical components installed inside. The design should be able to resist external mechanical inputs, thereby ensuring the reliability and stability of the laser. Structural factors such as structural stability, machining and assembly accuracy, mechanical robustness, weight, and size should be considered in the design process.

The mechanical structure of the ECDL is shown in Figure 3. The mechanical parts comprise a laser base, a mount for the LD and CL, a mount for L1 and the PZT, a mount for L2, and a mount for the IF. In The present design, most of the parts are made of aluminum alloy, which is relatively light and has a consistent rate of thermal expansion, thereby reducing the effects of thermal stress on the optical components. The choice of material in this design was made primarily for principle verification; other material options that satisfy environmental requirements include AlSiC and Ti.

**Figure 3.** Computer-aided design view of the ECDL shown schematically in Figure 1. A sectional view is presented here to reveal the internal structure.

To ensure long-term stable laser operation suitable for space applications, the connections among all the components are rigid, and The design avoids any spring-loaded adjustment. The LD is fixed rigidly to its bracket by a retaining ring. Except for the IF, the mounts of the other optical components are cylindrical structures with the same outer diameter. All the mounts are inserted into the laser base after the components are mounted and locked by M2.5 screws. After setting the required angle of the IF holder, it is fixed to the platform of the laser base by two M3 screws. All the lenses, including CL, L1, and L2, are adjustable along the optical axis only; they are adjusted to their required positions and then fixed in place using slow-setting glue.

The laser base is machined from a solid aluminum block to ensure that the laser is stable, robust, and insensitive to outside interference. Figure 4 shows a photograph of the laser, the outer envelope of which is 75 mm × 65 mm × 39 mm. Because The laser frequency depends on the length of the cavity, precise temperature control of the laser is necessary [9,24]. A small hole with a diameter of 3 mm and a depth of 5 mm is found at the end of the laser base close to the LD; in this hole is placed a negative-temperature-coefficient thermistor for detecting variations in temperature of the LD. A Peltier thermoelectric cooler with dimensions of 40 mm × 40 mm × 4 mm is attached to the surface of the laser base to stabilize the temperature of the cavity and LD. To ensure a laser output height of 20 mm from the optical table, this ensemble is fixed on the optical platform with four M4 Teflon screws. The optical platform acts as a heat sink, as illustrated in Figure 4.

**Figure 4.** Photograph of the ECDL, which has been applied to space narrow-linewidth-laser demonstration systems.

#### *3.3. Simulation of Eigenmodes and Stress Distribution*

The structural design of the laser should have a high fundamental frequency and good dynamic characteristics to prevent structural damage caused by resonance of the low-frequency coupling during launch. We used finite element modeling to verify that the ECDL design is satisfactory. To reduce the amount of data needed for analog operation, we simplified the laser model appropriately. The model retains the physical structure of the laser, the optics, and The mounting brackets, but omits details such as chamfers and fillets. The overall structural material of the laser is aluminum alloy, the optical lens material is fused silica, and The PZT material is PZT-5A; the properties of these materials are given in Table 1. The simulation results are shown in Figure 5, where the lateral rocking frequency of the first eigenmode is 2316 Hz. Because The first-order natural frequency of the module at the aerospace standard component level exceeds 70 Hz, the modal analysis shows that the design meets the requirements. However, the eigenfrequency is too high, indicating that the structural design of the laser requires further optimization.


**Table 1.** Material properties.

Another issue is that the gravitational environment differs between space and the laboratory. The lab-mounted laser undergoes a tiny deformation once in microgravity. To reduce this effect, we adopted an integrated external cavity structure to reduce the relative mechanical deformation as much as possible. In addition, no adjustable elastic mechanical structure was used. The cantilever length and mass are reduced as much as possible while maintaining the mechanical strength. Figure 6 shows the displacement of the ECDL under a vertical acceleration of 1 g. The maximum deformation clearly occurs at L2 but is only 0.01 μm. When the optical board with the laser mounted is placed face up and back up, the performance of the optical system remains the same, indicating that this small deformation has no effect on the performance of the ECDL. We also analyze the deformation of two major components, IF and OC, that affect ECDL performance under the action of gravity along the optical axis (*z* axis). Among them, the deformation of IF is 0.003 μm. This deformation is mainly a translation in the direction of light transmission and has no effect on the angle of the IF. The deformation of OC is 0.005 μm. This deformation will increase the length of the external cavity

and affect the frequency of the output laser. This slight shift results in a frequency change of only 0.4 GHz. This deviation can be corrected by adjusting the voltage of the PZT.

**Figure 5.** The frequency of the first lateral rocking eigenmode is 2316 Hz.

**Figure 6.** Displacement of the ECDL under action of gravitational field.

#### **4. Test Results**

After adding the external cavity to the diode laser, iterative focal adjustment and external-cavity alignment were implemented to optimize the optical feedback. Optimum alignment was accomplished when the threshold was reduced to a minimum [25]. The variation of output power with laser diode current as measured using an optical powermeter is shown in Figure 7 both with and without an external cavity. The threshold current of the ECDL is 30 mA, and The diode current shifts the laser output power by 0.91 mW/mA. As can be seen in Figure 7, the threshold current was reduced by approximately 10 mA compared to the bare tube, and The output optical power was increased by 15 mW at a laser current of 65 mA. The output surface of the laser diode we used was coated with an anti-reflection coating, and The reflectivity was only 10−<sup>4</sup> orders of magnitude. In principle, the main reason for increasing the output laser power after increasing the external cavity was to reduce the threshold current: (1) adding an external cavity is equivalent to an increase in cavity length; (2) introducing optical feedback to help increase the stimulated emission suppresses the spontaneous

radiation. These results indicate that the external-cavity semiconductor laser of the present design achieves strong optical feedback and completes good alignment [25].

**Figure 7.** Output power versus laser diode current as measured using an optical powermeter at a controlled laser temperature of 22.3 ◦C.

The effect of injection current and temperature on the output wavelength of the ECDL was investigated. It can be seen in Figure 8 that, like the laser diode, the ECDL also has a mode hopping interval. It is necessary to avoid this mode hopping interval when the laser is working. The wavelength dependence on the injection current at a fixed temperature of 22.3 ◦C is shown in Figure 8a. The current adjustment range corresponding to the ECDL's no mode hopping interval is about 9 mA, and the corresponding frequency varies by approximately 43 GHz. From this, the frequency tuning rate of the injection current can be calculated to be approximately 4.8 GHz/mA. Figure 8b shows the wavelength dependence on temperature at a fixed current of 64 mA. The coefficient of the frequency with temperature is 25 GHz/◦C, and The non-mode hopping interval is 1.7 ◦C. The laser frequency can be tuned over a wide range by changing the temperature, but this adjustment is very rough, and it takes a long time for the laser temperature to become completely stable. With increasing injection current or temperature, the wavelength increased (frequency decreased). This is because the temperature of the laser diode increases as the injection current increases. The effective refractive index increased, leading to an increase in the optical length of the internal cavity. When the wavelength increased and entered the edge of the range selected by the interference filter, the operating mode competed with neighboring modes and mode hopping occurred. The range of continuous non-hopping mode is mainly determined by the FSR of the internal cavity and the full width at half maximum (FWHM) of the interference filter.

**Figure 8.** (**a**) Relationship of the fine tuning current and the output wavelengths. The injection current adjustment step size is 1 mA. (**b**) Wavelength dependence on temperature at a fixed current of 64 mA. The temperature change step size is 0.01 ◦C.

To determine the long-term stability of the IF-ECDL, we used a wavelength meter (WS7; HighFinesse) to monitor the frequency fluctuations of the laser during free running. The results obtained over a period of 18 h indicated good passive long-term stability, and The maximum deviation in laser frequency was only 200 MHz. Figure 9 shows the frequency stability of the ECDL derived from the Allan deviation. The measurements were conducted in an air-conditioned laboratory in which the temperature fluctuated by roughly 1 ◦C about an average of roughly 22 ◦C.

**Figure 9.** Long-term frequency fluctuations of the free-running 698 nm ECDL.

To determine the linewidth of the ECDL, we performed an optical heterodyne beat experiment involving the IF-ECDL and a 698 nm ultra-stable laser with an ultra-narrow linewidth. The spectrum of the beat signal is shown in Figure 10b. The full width at half maximum of the Lorentzian fit is 180 kHz. The linewidth of the ultra-stable laser is only ∼1 Hz [26], and this was obtained by locking the laser frequency to a high-finesse optical reference cavity by means of the Pound–Drever–Hall method [22]. The linewidth of the beat signal can be considered to be the linewidth of the IF-ECDL because the latter is far wider than the linewidth of the ultra-stable laser. Therefore, the linewidth of the ECDL is roughly 180 kHz.

**Figure 10.** (**a**) Beat experiment involving the IF-ECDL and an ultra-stable laser with the linewidth of about 1 Hz. (**b**) Spectrum of beat signal between ECDL and a 698 nm laser with ultra-narrow linewidth. The injection current of the laser during the measurement is 64 mA, and The temperature is 22.3 ◦C. The resolution bandwidth of the spectrum analyzer is 10 kHz, and The sweep time is 1.29 s. The black line shows the power spectrum of the beat signal, and The red line indicates a fitted line.

#### **5. Conclusions**

In summary, this paper presents the design of a compact and robust ECDL for space applications. This ECDL was created without using any position adjusters, taking advantage of insensitivity to misalignment. As a wavelength-selective element, the laser uses an IF rather than a diffraction grating. The frequency of the first lateral rocking eigenmode is 2316 Hz. The ECDL emits 35 mW of laser power at a wavelength of 698 nm with a linewidth of around 180 kHz. In future work, we will conduct an adaptive test of the mechanical and thermal environment of the ECDL and optimize the design to make it more suitable for use in space.

**Author Contributions:** Conceptualization, T.L. and S.Z.; methodology, L.Z. and L.C.; validation, L.Z., L.C., G.X., and C.J.; writing—original draft preparation, L.Z.; writing—review and editing, L.Z., C.J., and J.L.; supervision, T.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Major Scientific Instruments and National Development Funding Projects of China (61127901, 91636101) and the Young Scientists Fund of the National Natural Science Foundation of China (11403031). The project was supported by the Open Research Fund of State Key Laboratory of Transient Optics and Photonics (SKLST201909).

**Conflicts of Interest:** The authors declare that there is no conflict of interest.

#### **References**


c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Equivalent Circuit Model of High-Performance VCSELs**

#### **Marwan Bou Sanayeh 1, Wissam Hamad 2,\* and Werner Hofmann <sup>2</sup>**


Received: 12 December 2019; Accepted: 15 January 2020; Published: 18 January 2020

**Abstract:** In this work, a general equivalent circuit model based on the carrier reservoir splitting approach in high-performance multi-mode vertical-cavity surface-emitting lasers (VCSELs) is presented. This model accurately describes the intrinsic dynamic behavior of these VCSELs for the case where the lasing modes do not share a common carrier reservoir. Moreover, this circuit model is derived from advanced multi-mode rate equations that take into account the effect of spatial hole-burning, gain compression, and inhomogeneity in the carrier distribution between the lasing mode ensembles. The validity of the model is confirmed through simulation of the intrinsic modulation response of these lasers.

**Keywords:** high-speed VCSELs; multi-mode VCSELs; intrinsic laser dynamics; equivalent circuit modeling; intrinsic modulation response

#### **1. Introduction**

Vertical-cavity surface-emitting lasers (VCSELs) offer an excellent solution for many high-speed data communication challenges. Moreover, VCSELs have special features such as high integration level, low electrical power consumption, low divergence angle, simple packaging, low fabrication cost, high modulation speed at low currents, and good beam quality. These features led to the growth of the VCSEL market for a wide variety of applications, which are not only limited to the field of communications but also extends to consumer applications such as laser printers and optical mice [1,2]. Nowadays, despite the intensive research conducted to understand the underlying physics behind the multi-mode (MM) behavior in oxide-confined MM VCSELs and their impact on the intrinsic laser dynamics, many ambiguities still exist concerning the nature of the abnormal multi-peak phenomenon and the notches occurring in the small-signal modulation response of these VCSELs. These multiple local maxima which appear in their intrinsic dynamic response deviate substantially from the standard single-mode (SM) model normally applied to characterize these MM devices. The measured total small-signal modulation response of a laser is the result of the superposition of the intrinsic and the extrinsic responses. The need to accurately de-embed and analyze the intrinsic laser dynamic behavior of VCSELs becomes indispensable to understand and study their extrinsic chip behavior. However, since the intrinsic response is attributed to the structure and geometry of the VCSEL intrinsic region and lasing cavity, the only way to isolate its effects is by accurately modeling it. Hence, sufficient modeling and accurate parameter extraction strategies are needed for a reliable de-embedding approach of each of the intrinsic and extrinsic responses from the overall system response. This detailed understanding of the VCSELs modulation response enables further optimization of these lasers for next generation high-speed devices. Furthermore, analyzing and modeling these lasers enable the enhancement and optimization of their design and performance.

Recently, an advanced and accurate MM small-signal model, which is based on the carrier reservoir splitting approach, was developed [3,4]. This rate equation-based model enables the extraction of reliable information from the intrinsic dynamics of high-speed MM VCSELs, as it takes into account the effect of spatial hole-burning (SHB), gain compression, and inhomogeneity in the carrier distribution between the modes. Using these MM rate equations also ensures deeper understanding of the device MM laser dynamics and gives a better access to the nonlinear modal competition behavior for the carrier density in the active region for such high-performance VCSELs.

Accurate modeling is important for both device engineers and circuit designers. Device engineers require a model that simulates complex physical phenomena, resulting in long and complex simulation times, and circuit designers need a simple and relatively accurate model that can be implanted in a circuit simulator and drivers with fast computational time. Hence, detailed analysis of VCSEL operating characteristics is crucial to the design of high-speed optical links. Traditionally, the intrinsic dynamics of a laser have been analyzed using a direct solution of the rate equations. This method gives accurate results; however, it has some disadvantages as numerical optimization techniques that minimizes the difference between measured and modeled data can vary depending upon the optimization method and starting values and as the device–circuit interaction cannot be easily taken into account [5]. An alternative approach to that of using the rate equations to model the VCSELs' intrinsic dynamics is to transform these equations to an equivalent circuit model, in which electrical components model the different physical effects that contribute to the overall system response [5–8]. This technique presents several advantages, including that the circuit model gives an intuitive idea of the physics of the device and the modulation response and can be easily interfaced to the VCSEL standard parasitic network [5,8].

Circuit modeling includes an electronic and an optical part and permits the optimization of the devices' dynamic characteristics including the device–circuit interaction, and performance can be obtained using a general circuit simulator. For example, to improve the *f* <sup>3</sup>*dB* intrinsic modulation bandwidth of VCSELs, an intrinsic equivalent circuit model can be employed to accurately simulate the dynamic behavior inside the laser cavity and to understand in depth the effect of each device physical element on this intrinsic 3-dB frequency. Thus, using this advanced model and bearing in mind the relation between the circuit elements and the real word physical device layout, various simulations can be conducted by altering the values of some circuit components and by tracking the change in the resulting intrinsic 3-dB bandwidth. It was noticed that, inside our latest generation of MM VCSELs with highest carrier and photon densities, the common carrier reservoir splits up as a result of numerous effects such as mode competition, carrier diffusion, and SHB. Besides the well-understood mechanisms which control the strength and the form of relaxation oscillation frequency (e.g., carrier diffusion, nonlinear gain suppression, and carrier transport effects), the contribution of codominant higher-order modes is still under discussion. In general, these VCSELs are fabricated with a small circular aperture diameter, allowing only few modes to rise under operation. Hence, most of these transverse lasing modes are spatially localized in two main regions and therefore can be confined either in the center of the active region or are localized more towards the peripheral boundary of the carrier reservoir. Constituently, these lasing modes can be grouped into two mode ensembles: the central mode ensemble and the peripheral mode ensemble. For SM VCSELs, the solution of the rate equations is straight forward and the fitting procedure for parameter extraction is simple. For MM VCSELS, however, and as shown in Reference [4], even for two-mode ensembles, the analysis becomes very complex and the parameter extraction and development of an analytical intrinsic modulation expression becomes rigorous.

In this work, a general compact and comprehensive equivalent circuit model for MM VCSELs, which is based on our latest novel MM rate equations model, i.e., the carrier reservoir splitting approach, is presented. This circuit model has all the advantages of simple and fast simulation procedures of circuit modeling and still incorporates advanced features of lasing modes interactions given by the advanced MM rate equations model. Most importantly, the proposed equivalent circuit model can

reproduce the delicate measured intrinsic modulation response. The validity of the model is confirmed through simulations and plots of the intrinsic modulation response of a two-mode ensembles VCSEL equivalent circuit model. These simulation results are later compared to the experimentally measured intrinsic modulation response of our high-performance VCSELs.

#### **2. Rate Equations**

Small-signal advanced MM rate equations for high-speed MM VCSELs, which are based on mode competition for carrier density in the active region, were recently developed [3,4]. In order to map these rate equations to the proposed equivalent circuit model, we quickly review the different derivation steps leading to the system's intrinsic modulation response and interaction matrices. We first linearize a system of differential equations that represent the rates of change in the carrier and photon reservoir densities and rewrite them to get the rate coefficients above lasing threshold, which are

$$
\mu\_{\rm NiNi} = \delta f\_{\rm th\_i} / \delta N\_i + \upsilon\_{\rm \mathcal{S}} a\_i \mathbb{S}\_i \tag{1}
$$

$$
\mu\_{NiSi} = \upsilon\_{\mathcal{g}} \varrho\_{th\_i} - \upsilon\_{\mathcal{g}} a\_{p\_i} S\_i \tag{2}
$$

$$
\mu\_{\text{SiNi}} = \Gamma\_i \upsilon\_{\text{jl}} a\_i \mathbf{S}\_i \tag{3}
$$

$$
\mu\_{\text{SiSi}} = \Gamma\_i \upsilon\_{\text{\textquotedblleft}p\_i} S\_i \tag{4}
$$

where *i* represents the *i*th mode in the corresponding carrier or photon reservoirs; *gthi* is the gain at threshold; *ai* and *api* are the differential gain and the negative gain derivatives, respectively; *Ni* is the carrier density; and *Si* is the photon density in the active region and the optical cavity. Moreover, Γ*<sup>i</sup>* is the confinement factor, *vg* is the group velocity, and *Jthi* is the carrier recombination density due to spontaneous emission or losses. The system's relaxation oscillation frequency ω*Ri* and the damping factor γ*<sup>i</sup>* in terms of the simplified rate coefficients can be introduced as

$$
\mu\_{Ri}^2 = \mu\_{NiNi} \mu\_{SiSi} + \mu\_{NiSi} \mu\_{SiNi} \tag{5}
$$

$$
\gamma\_i = \mu\_{\text{NiNi}} + \mu\_{\text{SiSi}} \tag{6}
$$

For SM VCSELs, the resulting rate coefficients can be expressed in a matrix form as

$$
\begin{pmatrix}
j\omega + \mu\_{\rm NN} & \mu\_{\rm NS} \\ -\mu\_{\rm SN} & j\omega + \mu\_{\rm SS}
\end{pmatrix}
\begin{pmatrix}
dN \\ dS \\ \end{pmatrix} = \begin{pmatrix}
df \\ 0 \\ \end{pmatrix} \tag{7}
$$

where *J* is the driving current density. For MM VCSELs, the matrix representation of the SM model can be expanded to include the various interactions between the different carrier and photon reservoirs. When expanded, the matrix representation for the case of two lasing modes ensembles, which is for most purposes sufficient to describe the intrinsic dynamics of the reservoir splitting in MM VCSELs, is expressed as

$$
\begin{pmatrix}
j\omega + \mu\_{N\_1 N\_1} & \mu\_{N\_1 N\_2} & \mu\_{N\_1 S\_1} & \mu\_{N\_1 S\_2} \\
\mu\_{N\_2 N\_1} & j\omega + \mu\_{N\_2 N\_2} & \mu\_{N\_2 S\_1} & \mu\_{N\_2 S\_2} \\
0 & -\mu\_{S\_2 N\_2} & 0 & j\omega + \mu\_{S\_2 S\_2}
\end{pmatrix}
\begin{pmatrix}
dN\_1 \\
dN\_2 \\
dS\_1 \\
dS\_2
\end{pmatrix} = 
\begin{pmatrix}
dI\_1 \\
dI\_2 \\
0 \\
0
\end{pmatrix} \tag{8}
$$

The interaction between the two carrier reservoir densities *N*<sup>1</sup> and *N*<sup>2</sup> can be written as shown in Equations (9) and (10), where *s*<sup>12</sup> and *s*<sup>21</sup> represent the spatial dependency of the two interacting carrier reservoirs.

$$
\sigma\_1 \mu\_{N1N2} = s\_{21} \cdot \upsilon\_{\%} a S\_2 \simeq s\_{21} \cdot \mu\_{N2N2} \tag{9}
$$

$$
\pi\_1 \mu\_{N2N1} = s\_{12} \cdot \upsilon\_\% a S\_1 \simeq s\_{12} \cdot \mu\_{N1N1} \tag{10}
$$

Similarly, the interaction coefficients representing cross reabsorption can be written as

$$
\mu\_{N1S2} = s\_{21} \cdot v\_{\\$} a g\_{\text{th}2} = s\_{21} \cdot \mu\_{N2S2} \tag{11}
$$

$$
\mu\_{\rm N2S1} = \text{s}\_{\rm 12} \cdot v\_{\rm 8} a\_{\rm 6l\' 1} = \text{s}\_{\rm 12} \cdot \mu\_{\rm N1S1} \tag{12}
$$

For a system having any number of mode ensembles (m-mode ensembles), the matrix in Equation (8) can be further expanded and generalized into the interaction matrix shown in Equation (13).

⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *j*ω + μ*N*1*N*<sup>1</sup> ··· μ*N*1*Nn* μ*N*1*S*<sup>1</sup> ··· μ*N*1*Sn* . . . ... . . . . . . ... . . . μ*NnN*<sup>1</sup> ··· *j*ω + μ*NnNn* μ*NnS*<sup>1</sup> ··· μ*NnSn* −μ*S*1*N*<sup>1</sup> ··· −μ*S*1*Nn j*ω + μ*S*1*S*<sup>1</sup> ··· −μ*S*1*Sn* . . . ... . . . . . . ... . . . −μ*SnN*<sup>1</sup> ··· −μ*SnNn* μ*SnS*<sup>1</sup> ··· *j*ω + μ*SnSn* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *dN*<sup>1</sup> . . . *dNn dS*<sup>1</sup> . . . *dSn* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *dJ*<sup>1</sup> . . . *dJn* 0 . . . 0 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (13)

From the interaction matrices shown in Equations (7), (8) and (13), the intrinsic modulation responses of SM and MM VCSELs can be obtained. These can be used to model the intrinsic dynamics of VCSELs, but for MM VCSELs, they can be quite complicated to solve analytically and require either complex numerical calculations or the neglection of some minor physical effects. Alternatively, equivalent circuit modeling, presented in Section 3, can be adopted.

#### **3. Equivalent Circuit Modeling**

#### *3.1. Review on the Single Mode Model*

The standard equivalent electrical circuit model of a SM (single-mode) VCSEL intrinsic dynamic operation is shown in Figure 1, which is well established and can be found in different literatures [5,8]. This model can be easily integrated into the small-signal cascaded network model of the VCSEL diode that includes the source, cables, submout parasitics, and laser chip parasitics that represent the extrinsic laser dynamics (e.g., Figure 1 in Reference [8]). The different components in this circuit represent different elements of the rate equations. For example, the capacitance C is the sum of the space-charge capacitance of the heterojunction and the charge storage in the active layer. The small-signal photon storage is modeled by the inductance L. The small-signal photon density is proportional to the current over L and thus can be used as the output variable representing the optical output intensity. Using the interaction matrix in Equation (7), the intrinsic modulation response for SM VCSELs, *HSM*(ω) is found as

$$H\_{\rm SM}(\omega) = \frac{h\nu}{e} \eta\_d \frac{\omega' \mathbb{R}^2}{\omega\_R^2 + j\omega \gamma - \alpha^2} \tag{14}$$

where η*<sup>d</sup>* is the differential quantum efficiency and ω- *<sup>R</sup>*<sup>2</sup> = *vggth*μ*SN*. The relaxation oscillation frequency ω*<sup>R</sup>* usually replaces ω- *<sup>R</sup>* for standard physical device parameters and is a common approximation for the SM modulation approach [9]. By comparing the rate equation-based transfer function in Equation (14) with the calculated electrical transfer function of the circuit model shown in Figure 1, the latter can be written as

$$H\_{\rm SM,clcc}(\omega) = \frac{I\_{\rm out}}{I\_{\rm in}} = \frac{1/\text{LC}}{1/\text{LC} + \,^{R\_{\rm lb}}/\text{LR}\_{\rm la}\text{C} + j\omega (1/\text{R}\_{\rm in}\text{C} + \,^{R\_{\rm lb}}/\text{L}) - \alpha^2} \tag{15}$$

Comparing the two transfer functions, the interaction matrix in Equation (7) can be rewritten in term of its electrical circuit model equivalent, and the equivalencies acquired can be used afterwards to develop the MM VCSEL equivalent circuit model. Equations (16a) and (16b) show the comparison between the rate equation-based matrix and its electrical circuit model equivalent:

$$
\begin{pmatrix}
j\omega+\mu\_{NN} & \mu\_{NS} \\
\end{pmatrix}
\begin{pmatrix}
dN \\
dS
\end{pmatrix} = \begin{pmatrix}
d\boldsymbol{J} \\
0
\end{pmatrix} \tag{1}
$$

$$
\mathfrak{J} \tag{16}
$$

$$
\begin{pmatrix}
j\omega+\boldsymbol{1}/\boldsymbol{\eta}\_{\rm Li}\mathbb{C} & \boldsymbol{1}/\boldsymbol{\omega} \\
\end{pmatrix}
\begin{pmatrix}
\boldsymbol{\varepsilon}\mathbf{c}\_{\boldsymbol{\circ}}/q \\
\boldsymbol{\varepsilon}\boldsymbol{\omega}/q
\end{pmatrix} = \begin{pmatrix}
\eta\boldsymbol{\mu}\boldsymbol{\ell}/\_{q}\boldsymbol{V}\_{\boldsymbol{\circ}} \\
0
\end{pmatrix} \tag{b}
$$

where η*<sup>I</sup>* is the electrical efficiency, *vc* is the voltage over the capacitance, and *iL* the current in the inductance. Moreover, η*IdI*/*Vo* is represented by the current source *Iso* in Figure 1. Solving the matrix in Equation (16b) leads to Equations (17) and (18):

$$j\omega \frac{\mathbf{C} \mathbf{v}\_c}{q} + \frac{\mathbf{v}\_c}{q \mathbf{R}\_{1a}} + \frac{i\_L}{q} = \frac{\eta\_I dI}{qV\_o} \tag{17}$$

$$i\_L = \frac{v\_c}{R\_{1b} + j\omega L} \tag{18}$$

Replacing *iL* in Equation (17), the node equation of the circuit shown in Figure 1 can be obtained as

$$v\_c ja\mathcal{C} + \frac{v\_c}{R\_{1a}} + \frac{v\_c}{R\_{1b} + j\omega L} - \underbrace{\frac{\eta\mu I}{V\_o}}\_{I\_{so}} = 0\tag{19}$$

This SM model resembles a simple second-order low-pass filter. Moreover, in most literatures, the adapted SM based equivalent circuit model can only reproduce a single resonance peak and thus fails to replicate the delicate small-signal data (abnormal multi-peaks and the notches) of modern high-speed MM VCSELs accurately.

**Figure 1.** Standard equivalent electrical circuit model of a SM VCSEL intrinsic dynamic operation [5,8].

#### *3.2. Two-Mode Model*

To analyze the intrinsic behavior of high-performance MM VCSELs, a suitable equivalent circuit model is developed. In this section, we derive this model for a MM VCSEL having two mode ensembles. This model will be later expanded to comprise a system of any number of mode ensembles (Section 3.3). Using the relations in Equations (9)–(12) and the equivalencies that were extracted from Equation (16), the interaction matrix shown in Equation (8) is converted to its circuit model equivalent, shown in Equation (20).

$$
\begin{pmatrix} j\omega + \frac{1}{R\_{1\emptyset}C\_1} & s\_{21} \cdot \frac{1}{R\_{2\emptyset}C\_2} & \frac{1}{L\_1} & s\_{21} \cdot \frac{1}{L\_2} \\\ s\_{12} \cdot \frac{1}{R\_{1\emptyset}C\_1} & j\omega + \frac{1}{R\_{2\emptyset}C\_2} & s\_{12} \cdot \frac{1}{L\_1} & \frac{1}{L\_2} \\\ -\frac{1}{C\_1} & 0 & j\omega + \frac{R\_{1k}}{L\_1} & 0 \\\ 0 & -\frac{1}{C\_2} & 0 & j\omega + \frac{R\_{2k}}{L\_2} \end{pmatrix} \begin{pmatrix} \frac{C\_1\nu\_1}{q} \\\ \frac{C\_2\nu\_2}{q} \\\ \frac{L\_1L\_1}{q} \\\ \frac{L\_2L\_2}{q} \end{pmatrix} = \begin{pmatrix} \frac{\eta lH\_1}{qV\_{p1}} \\\ \frac{\eta lH\_2}{qV\_{p2}} \\\ 0 \\\ 0 \end{pmatrix} \tag{20}
$$

Solving the matrix in Equation (20) leads to the following relations:

$$j\omega \cdot \frac{C\_1 v\_{c1}}{q} + \frac{v\_{c1}}{R\_{1a} q} + s\_{21} \cdot \frac{v\_{c2}}{R\_{2a} q} + \frac{i\_{L1}}{q} + s\_{21} \cdot \frac{i\_{L2}}{q} = \frac{\eta\_l d I\_1}{q V\_{o1}} \tag{21}$$

$$j\omega \cdot \frac{C\_{1}v\_{\varepsilon 1}}{q} + \frac{v\_{\varepsilon 1}}{R\_{1a}q} + s\_{21} \cdot \frac{v\_{\varepsilon 2}}{R\_{2a}q} + \frac{i\_{L1}}{q} + s\_{21} \cdot \frac{i\_{L2}}{q} = \frac{\eta\_{l}qI\_{1}}{qV\_{o1}}\tag{22}$$

$$\dot{a}\_{L1} = \frac{v\_{c1}}{R\_{1b} + j\omega L\_1} \tag{23}$$

$$\dot{q}\_{L2} = \frac{v\_{c2}}{R\_{2b} + j\omega L\_2} \tag{24}$$

Replacing *iL*<sup>1</sup> and *iL*<sup>2</sup> in Equations (21) and (22), the node equations for the two-mode VCSEL model can be obtained as shown in Equations (25) and (26).

$$\underbrace{s\_{21}\left(\frac{\upsilon\_{c2}}{R\_{2a}} + \frac{\upsilon\_{c2}}{R\_{2b} + j\omega L\_2}\right)}\_{I\_{12}} + \underbrace{\upsilon\_{c1}j\omega C\_1}\_{I\_{12}} + \frac{\upsilon\_{c1}}{R\_{1b}} + \underbrace{\frac{\upsilon\_{c1}}{R\_{1b} + j\omega L\_1} - \frac{\eta\_{l}dI\_1}{V\_{o1}}}\_{I\_{s1}} = 0.\tag{25}$$

$$s\_{12} \cdot \underbrace{\left(\frac{\upsilon\_{c1}}{R\_{1a}} + \frac{\upsilon\_{c1}}{R\_{1b} + j\omega L\_1}\right)}\_{I\_{21}} + \upsilon\_{c2}j\omega C\_2 + \frac{\upsilon\_{c2}}{R\_{2a}} + \frac{\upsilon\_{c2}}{R\_{2b} + j\omega L\_2} - \underbrace{\frac{\eta\_l dI\_2}{V\_{o2}}}\_{I\_{o2}} = 0 \tag{26}$$

Using the node equations in Equations (25) and (26), the two-mode VCSEL equivalent circuit model can be obtained, as shown in Figure 2. In this circuit, we can consider, just like in the SM model, that *Iso*<sup>1</sup> = η*IdI*/*Vo*<sup>1</sup> and *Iso*<sup>2</sup> = η*IdI*/*Vo*<sup>2</sup> and that *s*21*I*<sup>12</sup> and *s*12*I*<sup>21</sup> are dependent current sources, which represent the interaction of the two carrier reservoirs with each other. This is an important aspect to consider, as it has been recently shown in MM VCSELs that the split carrier reservoirs of the lasing mode ensembles overlap and impact each other [3,4].

**Figure 2.** Equivalent electrical circuit model of a two-mode multi-mode VCSEL intrinsic dynamic operation.

#### *3.3. M-Mode Model*

Similar to the two-mode model equivalent circuit analysis, the electrical circuit equivalent matrix, shown in Equation (27), for a MM VCSEL with m-mode ensembles can be derived. From this matrix representation, the set of node equations (grouped in Equation (28)) can be obtained, following the same derivation procedure of the two-mode model case. Using these node equations, the m-mode equivalent electrical circuit model, depicted in Figure 3, can be developed.

⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *<sup>j</sup>*<sup>ω</sup> <sup>+</sup> <sup>1</sup> *<sup>R</sup>*1*aC*<sup>1</sup> ··· *sm*<sup>1</sup> · <sup>1</sup> *RmaCn* 1 *<sup>L</sup>*<sup>1</sup> ··· *sm*<sup>1</sup> · <sup>1</sup> *L*1 . . . ... . . . . . . ... . . . *<sup>s</sup>*1*<sup>m</sup>* · <sup>1</sup> *<sup>R</sup>*1*aC*<sup>1</sup> ··· *<sup>j</sup>*<sup>ω</sup> <sup>+</sup> <sup>1</sup> *RmaCm <sup>s</sup>*1*<sup>m</sup>* · <sup>1</sup> *<sup>L</sup>*<sup>1</sup> ··· <sup>1</sup> *Lm* − 1 *<sup>C</sup>*<sup>1</sup> ··· <sup>0</sup> *<sup>j</sup>*<sup>ω</sup> <sup>+</sup> *<sup>R</sup>*1*<sup>b</sup> <sup>L</sup>*<sup>1</sup> ··· 0 . . . ... . . . . . . ... . . . <sup>0</sup> ··· − <sup>1</sup> *Cm* <sup>0</sup> ··· *<sup>j</sup>*<sup>ω</sup> <sup>+</sup> *Rmb Ln* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *C*1*vc*<sup>1</sup> *q* . . . *Cmvcm q L*1*iL*<sup>1</sup> *q* . . . *LmiLm q* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ η*idI*<sup>1</sup> *qVo*<sup>1</sup> . . . η*idIm qVom* 0 . . . 0 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (27) *m i*=2 *si*<sup>1</sup> · *vci Ria* <sup>+</sup> *vci Rib*+*j*ω*Li* ⎭⎪⎪ ⎬⎪⎪ ⎫ *I*1*i* + *vc*<sup>1</sup> *<sup>j</sup>*ω*C*<sup>1</sup> + *vc*<sup>1</sup> *<sup>R</sup>*1*<sup>a</sup>* <sup>+</sup> *vc*<sup>1</sup> *<sup>R</sup>*1*b*+*j*ω*L*<sup>1</sup> <sup>−</sup> <sup>η</sup>*IdI*<sup>1</sup> *Vo*<sup>1</sup> " *Iso*<sup>1</sup> = 0 . . . *m*−1 *i*=1 *sim* · *vci Ria* <sup>+</sup> *vci Rib*+*j*ω*Li* ⎭⎪⎪ ⎬⎪⎪ ⎫ *Imi* + *vcm j*ω*Cm* + *vcm Rma* <sup>+</sup> *vcm Rmb*+*j*ω*Lm* <sup>−</sup> <sup>η</sup>*IdIm Vom* " *Isom* = 0 (28)

**Figure 3.** Equivalent electrical circuit model for an m-mode multi-mode VCSEL intrinsic dynamic operation.

#### **4. Circuit Simulation Results**

In order to derive the total small-signal modulation response *HTOT*(ω) of a MM VCSEL, its intrinsic transfer function *Hint*(ω) is multiplied by the extrinsic transfer function of its parasitic network *Hpar*(ω). This extrinsic response was recently developed for high-performance MM VCSELs [10]. In physical real-world devices, the intrinsic dynamic behavior is usually embedded in such a cascaded network that includes different parasitic elements, such as the submount and laser chip parasitics. The laser chip parasitics, also called the extrinsic response, play one of the most critical roles in limiting the intrinsic modulation speed, as their low-pass filter characteristics shunts the modulation current outside the active region at high frequencies and since the extrinsic response is attributed to the structure and geometry of the VCSEL chip; the only way to isolate its effects is by modeling it with an electrical equivalent circuit, of which electrical components represent the different physical effects that contribute to the overall system response. Having an equivalent circuit for the intrinsic response, as shown in this work, enables the combination of both the extrinsic and intrinsic modulation responses in the overall cascaded network of the entire link. Figure 4 shows the calibrated total small-signal modulation response of a 980-nm MM oxide-confined VCSEL with an aperture diameter of ~7 μm measured by a

40-GHz vector network analyzer (VNA-HP8722C). The curves describe the measured total relative modulation response data S21 for various driving currents at room temperature. The modulation current is increased gradually up to 14 mA. Thermal rollover is reached at around 17 mA. The maximum total 3-dB bandwidth of the device including chip parasitics is found to exceed 32 GHz at 14 mA.

**Figure 4.** Calibrated total (intrinsic and extrinsic) small-signal modulation response of a 980-nm MM oxide-confined VCSEL with an aperture diameter of ~7 μm: The curves depict the measured relative response data (S21) for various driving currents at room temperature.

In order to de-embed the pure intrinsic modulation response *Hint*(ω) from the total modulation response, either the direct rate equations solution or the proposed equivalent circuit model derived in this work can be used. As shown in Reference [4], even though it is very accurate, the calculated *Hint*(ω) is very complex to implement and an advanced fitting procedure is required to determine its physical parameters. Alternatively, the equivalent circuit model presented in this work has fewer fitting parameters compared to the rate equation model on one side, and secondly, it can be easily integrated in the overall system cascaded network.

To validate the proposed MM equivalent circuit model, MATLAB Simulink® was used to compute the intrinsic modulation response of the two-mode ensembles VCSEL circuit model shown in Figure 2. Results are presented in Figure 5 for three different driving currents. The curves represent the pure intrinsic small-signal modulation response of a two-mode ensembles VCSEL. The values of the circuit parameters used in this simulation are shown in the inset of Figure 5. These are a set of possible mathematical solutions that were extracted from fitting the intrinsic modulation response of the circuit model into its measured counterpart depicted in Figure 4. The parameter n shown in the inset of Figure 5 represents the injection current inhomogeneity factor, i.e., *n* and 1 − *n* are the fractions of the injection carrier densities in each carrier reservoir, and values have been experimentally determined in Reference [4] for different currents. In the model shown in Figure 2, this parameter will distribute the total current on *Iso*<sup>1</sup> and *Iso*<sup>2</sup> accordingly. This represents the inhomogeneity in the injection current distribution between the lasing mode ensembles. For the first two mode ensembles (*LP*<sup>01</sup> and *LP*11), *s*<sup>12</sup> = 0.67 and *s*<sup>21</sup> = 0.94 and were adopted from Reference [11]. At this point, it is important to mention that the chip-parasitics *Hpar*(ω) need to be de-embedded from the total measured small-signal modulation response before comparing it to the simulated pure intrinsic response shown in Figure 5. As shown in Figure 5, the intrinsic small-signal modulation response replicates a typical MM VCSEL intrinsic response with the multi-peaks and notches in the curves at low frequencies. It is worth noting that the advanced circuit for a two-mode ensembles VCSEL depicted in Figure 2 shows a much more

realistic modulation response of MM VCSELs compared to using the SM VCSEL circuit in Figure 1, which was traditionally used in various literatures. Using the SM intrinsic modulation response model (Equation (14)) is acceptable as an approximation for low-speed MM VCSELs sharing the same carrier reservoir. However, in high-speed and high-performance VCSELs, using this simple model gives rise to a lot of discrepancies when modeling the intrinsic performance of these MM VCSELs [3,4].

**Figure 5.** Simulation results of the relative intrinsic modulation response for three different driving currents using the two-mode VCSEL equivalent circuit model shown in Figure 2. Inset: testing parameters used in the two-mode VCSEL equivalent circuit model simulation.

#### **5. Conclusions**

In this study, a general, compact, and comprehensive equivalent circuit model based on the carrier reservoir splitting approach and that accurately describes the intrinsic dynamic behavior of MM VCSELs was presented. The model includes the case where the lasing modes do not share a common carrier reservoir and was derived from advanced MM rate equations that take into account the effect of spatial hole-burning, gain compression, and inhomogeneity in the carrier distribution into the different lasing modes. The validity of the model was confirmed through simulations of the intrinsic modulation response of a VCSEL having two lasing mode ensembles at different driving currents compared to measured data. This model can be expanded to include any number of mode ensembles. Moreover, this equivalent circuit model can be easily integrated in the overall system cascaded network that represents the extrinsic and intrinsic dynamics of MM VCSELs.

**Author Contributions:** Supervision, Writing—review & editing: M.B.S., W.H. (Wissam Hamad) and W.H. (Werner Hofmann). All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported in part by the Erasmus+ programme and the German Research Foundation (DFG) within the Collaborative Research Center "Semiconductor Nanophotonics" (CRC 787) and the Open Access Publication Fund of TU Berlin.

**Acknowledgments:** The authors would like to thank (in alphabetical order) Bassel Aboul-Hosn, Oliver Daou, Elio Nakhle, and Rami Yehia for their valuable support within this research.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

1. Larsson, A. Advances in VCSELs for Communication and Sensing. *IEEE J. Sel. Top. Quantum Electron.* **2011**, *17*, 1552–1567. [CrossRef]


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Parity–Time Symmetry in Bidirectionally Coupled Semiconductor Lasers**

#### **Andrew Wilkey, Joseph Suelzer, Yogesh Joglekar and Gautam Vemuri \***

IUPUI Department of Physics, 402 N. Blackford Street, Indianapolis, IN 46202, USA; arwilkey@iu.edu (A.W.); suelzer.joseph@gmail.com (J.S.); yojoglek@iupui.edu (Y.J.)

**\*** Correspondence: gvemuri@iupui.edu

Received: 5 September 2019; Accepted: 19 November 2019; Published: 27 November 2019

**Abstract:** We report on the numerical analysis of intensity dynamics of a pair of mutually coupled, single-mode semiconductor lasers that are operated in a configuration that leads to features reminiscent of parity–time symmetry. Starting from the rate equations for the intracavity electric fields of the two lasers and the rate equations for carrier inversions, we show how these equations reduce to a simple 2 × 2 effective Hamiltonian that is identical to that of a typical parity–time (PT)-symmetric dimer. After establishing that a pair of coupled semiconductor lasers could be PT-symmetric, we solve the full set of rate equations and show that despite complicating factors like gain saturation and nonlinearities, the rate equation model predicts intensity dynamics that are akin to those in a PT-symmetric system. The article describes some of the advantages of using semiconductor lasers to realize a PT-symmetric system and concludes with some possible directions for future work on this system.

**Keywords:** parity–time symmetry; semiconductor laser; intensity dynamics

#### **1. Introduction**

Semiconductor lasers (SCLs) with optical injection and feedback, as well as coupled SCLs, have been basic paradigms for investigating nonlinear dynamics for the last several years [1]. The dynamical response of these SCL systems has been shown to include low frequency fluctuations (LFFs), periodic doubling routes to chaos, and the occurrence of unstable attractors, and the dynamics have been exploited for chaotic encryption, random number generation, linewidth reduction, and optical waveform production [2]. Independent of these studies on SCLs, there has been enormous interest in systems that are described by non-hermitian Hamiltonians that arise in open systems, i.e., systems that are coupled to the environment [3–11]. Typically, the Hamiltonian in quantum mechanics is hermitian because one deals with closed systems, and the hermiticity leads to real eigenvalues, orthogonal eigenfunctions, unitary evolution, and conservation of probability. As soon as one deals with realistic systems, by including, say, dissipation, one has to work with non-hermitian Hamiltonians, and the varying dynamics that result in systems that are described by such Hamiltonians have attracted much attention in recent years. Part of this interest is driven by the fundamental physics inherent in such systems and part of of it by their predicted applications. The optics community has been particularly interested in one type of non-hermitian Hamiltonians called the parity–time (PT) symmetric Hamiltonians, which are a class of Hamiltonians that are symmetric under combined operations of parity (P) and time-reversal (T). The pioneering work of Bender and co-workers, and others [6–11], demonstrating that a non-hermitian Hamiltonian may have a real energy spectrum provided it is parity (P) and time-reversal (T) symmetric, has led to tremendous interest in experimental realizations of PT-symmetric laboratory systems [12–19]. Many experimental realizations have been in the

optical domain, largely because PT symmetry requires systems with balanced gain and loss, which are ubiquitous in optics. Thus, much effort has been put into developing integrated structures with appropriate gain and loss properties. The typical PT-symmetric dimer [12] consists of two coupled oscillators wherein the gain in one oscillator is exactly equal to the loss in the other. The resulting 2 × 2 Hamiltonian matrix that describes this system then has complex diagonal elements, which are complex conjugates of each other and represent gain and loss in each oscillator, and the off-diagonal elements are real and equal and represent the coupling between the oscillators.

In a typical PT-symmetric system, say a pair of evanescently coupled waveguides in which one waveguide has gain and the other an equal amount of loss [12], one finds that as the gain/loss parameter is varied, there is a critical value, called the PT threshold, at which the eigenvalues of the Hamiltonian transition from being real to complex. In the regime where the eigenvalues are real, the norm of the wavefunction is bounded, and once the eigenvalues are complex, the norm grows abruptly. In our work, we use this abrupt transition as a metric for the PT threshold.

One outcome of the studies on PT symmetry is that many of the features of these systems are a result of the exceptional point (EP) behaviors of the underlying Hamiltonian [20–23]. Coupled lasers are especially attractive for the experimental realization of PT-symmetric models and exceptional point (EP) behaviors, and a few recent experiments have fabricated synthetic microcavity lasers on an integrated chip and reported the PT-symmetric properties of the system [24]. The laser configuration is typically designed to exploit the balance between the gain and loss of the lasers in order to extract unexpected behaviors that arise when the system undergoes an abrupt PT phase transition or, more generally, approaches an EP. It is anticipated that the outcomes of our work will be important for systems described by non-hermitian rate equations, local and nonlocal, and their laboratory implementations.

In this paper, we report a realization of a time-delayed, non-hermitian system in a bulk optical system that is comprised of two optically coupled semiconductor lasers (SCLs), and a numerical investigation of the properties of this system. In particular, we show that the rate equation model that is typically used to describe these coupled lasers [25] can, under certain conditions, lead to an effective non-hermitian Hamiltonian that is strongly reminiscent of the Hamiltonians that arise in the study of conventional PT-symmetric systems. Our work demonstrates that the coupled SCL system possesses many of the features that PT-symmetric systems do. We note that our system is completely classical, and yet it has features of PT symmetry because many aspects of PT symmetry are a result of the characteristics of exceptional points in the governing Hamiltonian. The predictions of our numerical work can be implemented in commercially available, off-the-shelf SCLs, since it does not require any specially fabricated components with tailored properties. Furthermore, as we will show, the important PT parameters can be easily controlled in the laboratory, making coupled SCLs very useful for studying PT symmetry.

Among the key features of our system are the fact that unlike other PT-symmetric systems, which rely on coupling a system with gain to an identical one with loss, our configuration couples two lasers in which the frequency detuning between the two lasers and the coupling strength between them, respectively, are the relevant parameters. The advantage is that in contrast to other systems where a precise balance between gain and loss has to be engineered, our system always has the frequency detuning of one laser exactly equal and opposite in sign to the frequency detuning of the second laser, thereby guaranteeing that the diagonal elements of the effective PT Hamiltonian are equal and opposite in sign. PT-symmetric systems are of interest for making materials with unidirectional optical propagation [26], single mode lasing action [27], and the spontaneous generation of photons in a PT-symmetric medium by a vacuum field [28]. Due to the miniature size of SCLs and well established fabrication methods for incorporating several lasers and associated components on chips, our work may lead to PT-symmetric photonics on a chip.

#### **2. Numerical Model**

Our system is described by a rate equation model that is based on the Lang–Kobayashi model [25] wherein we assume that the two lasers are nearly identical in all of their characteristics, single-mode, and operate at slightly different frequencies, *ω*<sup>1</sup> and *ω*2. We write the rate equations in a frame that is rotating at the average frequency *θ* of the two lasers, i.e., *θ* = (*ω*<sup>1</sup> + *ω*2)/2 [29]. The rate equations describing the normalized complex electric fields, *E*1,2(*t*), and the normalized excess carrier densities, *N*1,2(*t*), may be written as follows [29]:

$$\frac{dE\_1}{dt} = (1 + ia)N\_1(t)E\_1((t) + i\Lambda\omega\nu E\_1(t) + \kappa \exp(-i\theta\tau)E\_2(t-\tau),\tag{1}$$

$$\frac{d\mathbb{E}\_2}{dt} = (1 + ia)N\_2(t)E\_2((t) - i\Lambda\omega\nu\mathbb{E}\_2(t) + \kappa\exp(-i\theta\tau)E\_1(t-\tau),\tag{2}$$

$$T\frac{dN\_1}{dt} = f\_1 - N\_1(t) - (1 + 2N\_1(t))|E\_1(t)|^2,\tag{3}$$

$$T\frac{dN\_2}{dt} = f\_2 - N\_2(t) - (1 + 2N\_2(t))|E\_2(t)|^2,\tag{4}$$

where *α* is the linewidth enhancement factor [28], *τ* is the time delay in coupling due to physical separation between the lasers, *J*1,2 is the injection current above threshold, and *T* is the ratio of the carrier lifetime to the photon lifetime. The model used in Equations (1)–(4) is a phenomenological model [30,31] that has been quite accurate in modeling the dynamical response of semiconductor lasers subject to optical injection and in reproducing the intensity response of mutually coupled SCLs. A detailed and rigorous model has been described in Ref. [29] for bidirectionally coupled SCLs, where the authors start from Maxwell's equations, apply appropriate boundary conditions, and obtain the time evolution of electric field amplitudes in each laser cavity. Equations for the time evolution of the carrier inversion in each laser are also obtained. Ref. [29] has shown that under the assumptions of (i) weak coupling between the lasers, (ii) both lasers operating at nearly identical optical frequencies, (iii) both lasers having equal gain coefficients despite a slight detuning between them, and (iv) neglecting multiple feedbacks, the rigorous model reduces to the phenomenological model.

The important and relevant PT parameters for our work are *κ* and Δ*ω*, which describe the coupling coefficient and the frequency detuning between the lasers, respectively. Note that in Equation (2), the coupling term accounts for the mutual coupling between the two lasers, and a phase accumulation term has been added to account for the time taken for the light to travel from one laser to the other. In our system, Δ*ω* physically represents the frequency pulling that is typical of coupled lasers operating at slightly different frequencies, and *κ* produces amplification of light in each laser.

To motivate the connection to non-hermitian Hamiltonians in general, and PT-symmetry in particular, the rest of this paper will focus on the zero-delay case. The effects of time-delay are profound and will be discussed in a future article. When the SCLs are operating in steady state, above threshold, the inversion above transparency is zero, i.e., *N*1,2 = 0 [32]. Therefore, Equations (1) and (2) reduce to

$$
\begin{bmatrix}
\dot{E}\_1\\\dot{E}\_2
\end{bmatrix} = \begin{bmatrix}
i\Delta\omega & \kappa\\\kappa & -i\Delta\omega
\end{bmatrix} \begin{bmatrix}
E\_1\\E\_2
\end{bmatrix} \tag{5}
$$

where the 2 × 2 effective Hamiltonian is isomorphic to typical PT-symmetric Hamiltonians under a *π*/2 rotation about Pauli matrix *σy*, with the difference being that the diagonal elements of the matrix that normally represent gain/loss terms [12] are replaced in Equation (5) by frequency detuning between the two lasers. The SCL model is a rate equation model, in contrast to typical PT systems that are studied

by invoking the Schroedinger equation. Thus, the complex i that occurs in the Schroedinger equation is missing in our model (such systems are referred to as anti-PT systems). In our system, the diagonal elements, instead of contributing to amplification or attenuation of light, now give rise to temporal oscillations in the field. The off-diagonal elements, instead of determining the frequency of exchange between the two oscillators, now contribute to laser intensity growth.

The eigenvalues, *λ*, of the effective 2 × 2 Hamiltonian above are given by *λ* = ± *<sup>κ</sup>*<sup>2</sup> <sup>−</sup> <sup>Δ</sup>*ω*2. For values of |Δ*ω*| < *κ*, the eigenvalues are real, and for |Δ*ω*| > *κ*, the eigenvalues are complex. Thus, the point at which |Δ*ω*| = *κ* marks the PT threshold.

The reduction of the rate equations to the simplified 2 × 2 effective Hamiltonian answers the question of why one might expect PT-symmetric behavior in coupled SCLs. The question still remains as to whether the full rate equation model also exhibits PT-symmetric features. We will show below that despite the simplifying assumptions made to get Equation (5) and the differences in the conventional PT model and our system, the coupled SCL system does behave like a PT-symmetric system. In fact, our work to date indicates that the coupled SCL system is a very robust PT system and that the signatures of PT symmetry persist even without some of the simplifying assumptions.

#### **3. Results**

Having motivated the existence of PT-symmetric behavior in a pair of coupled SCLs, we now investigate whether the system retains any features of PT symmetry when the full set of laser rate equations is solved numerically. We restrict our discussion to the zero time-delay case, i.e., *τ* = 0, to focus on the PT symmetry aspects of the system. The key signature we look for is whether there is an abrupt change in the intensities of the lasers at the PT threshold, i.e., when |Δ*ω*| = *κ*. In Figure 1a are shown the real parts of the two eigenvalues of Equation (5) vs. Δ*ω* when *τ* = 0, as well as the imaginary parts of the eigenvalues vs. Δ*ω*. It is seen that at |Δ*ω*| = *κ*, there is an abrupt change in the real eigenvalues to non-zeros values. At the same time, the imaginary parts of the eigenvalues transition from non-zero to zero values at |Δ*ω*| = *κ*. It is clear from the behavior of the eigenvalues that as in all PT-symmetric systems, there is a threshold at which the eigenvalues transition from purely imaginary to purely real. The solutions for Equation (5) have the form *exp*(*λt*), and so the real parts of the eigenvalues lead to amplification or decay of the laser intensities, depending on whether the real parts of the eigenvalues are positive or negative, respectively. Since the real parts take both positive and negative values (see Figure 1a, for example), the linear model gives physical results only if the real part of the eigenvalues is negative. For positive values of the real part of the eigenvalues, the solutions would diverge, and this unphysical result is a consequence of neglecting gain saturation. In a realistic laser system, gain saturation will prevent the laser intensities from growing to unphysical values, as shown later in Figure 2a.

Since the eigenvalues of the effective Hamiltonian in Equation (5) are given by <sup>√</sup> *κ*<sup>2</sup> − Δ*ω*2, the eigenvalues can be swept from real to complex by sweeping *κ* and holding Δ*ω* constant. In Figure 1b are shown the real and imaginary parts of the eigenvalues of the 2 × 2 effective Hamiltonian as a function of *κ* for a constant Δ*ω* = 0.2. Once again, it is seen that at the PT threshold, i.e., *κ* = Δ*ω*, the eigenvalues undergo a transition from real to imaginary. Thus, a pair of coupled SCLs provide multiple methods by which the PT threshold can be accessed, either by sweeping *κ* or by sweeping the relative detuning through either injection current modulation or temperature variation. The observations in Figure 1a,b are the characteristic behaviors for the eigenvalues of a PT-symmetric system. The regime where the time delay is non-zero leads to more complex behavior since the effective Hamiltonian now becomes infinite-dimensional instead of a simple 2 × 2 matrix, and this will be the subject of another article. As one illustration of the effect of time-delay, we show the real and imaginary parts of the eigenvalues of the effective Hamiltonian for a time delay *τ* = 85 in Figure 1c, when *κ* is swept and Δ*ω* is fixed at 0.2. Since one cannot show all the eigenvalues of an infinite dimensional system, we show the behavior of the dominant eigenvalue, i.e., eigenvalue with the largest real part, since the real part leads to laser intensity growth. It is observed that the real part of the eigenvalues shows a growth at Δ*ω* = *κ*, but there are also multiple other transitions for *κ* < Δ*ω*. The real part of the eigenvalues changes sign at all these transitions, and so the picture is quite different from Figure 1b.

**Figure 1.** Real and imaginary parts of eigenvalues of the effective Hamiltonian (**a**) vs. Δ*ω* for *τ* = 0, (**b**) vs. *κ* for *τ* = 0, and (**c**) vs. *κ* for *τ* = 85. The parity–time (PT) threshold is Δ*ω* = *κ* = 0.2 for all three plots.

The results in Figure 1 are obtained with simplifying assumptions, including the neglect of population dynamics and gain nonlinearities. We next investigate whether the features of PT symmetry persist if the full set of rate equations is numerically solved, which then includes gain saturation and population dynamics. In the simulations, all time scales are in units of the photon lifetime, taken to be 10 ps. For all simulations, we take *α* = 4, but note that the results are insensitive to the value of *α*. We also take the initial values for the intracavity electric fields to be the same for both lasers, chosen such that the lasers are operating at about 3%–5% above the lasing threshold. In Figure 2a are the intensities of the two lasers for a coupling strength *κ* = 0.2 and *τ* = 0. The relative detuning, Δ*ω*, is scanned, and we observe that for |Δ*ω*| > *κ*, the intensities of both lasers remain bounded, and this is the regime in which the eigenvalues of the effective Hamiltonian are complex. At the PT threshold, |Δ*ω*| = *κ*, there is an abrupt increase in the intensities of both lasers, consistent with the simplified 2 × 2 model. This observation is an indicator of the robustness of the PT-symmetric behavior of this system since the PT features persist in the presence of nonlinearities and population dynamics. It is surprising and remarkable that the predictions of the rate equation model match those of the 2 × 2 effective Hamiltonian so well since not only does the rate equation model include gain saturation and associated nonlinearities, and population dynamics, but it also assumes each laser is operating on a single longitudinal mode. However, in practice, it is unlikely that for the coupling strengths used here, the two lasers would still be single-mode.

To ensure that the abrupt change in the lasers' intensities is not an artifact of our simulations, we varied the relative detuning between the lasers by scanning the injection current to one of the SCLs since sweeping the pump changes the optical frequency of these lasers. Of course, varying the injection current also changes the output intensity of the laser, and so both lasers cannot be set to the same initial intensities. The dependence of the intensity and optical frequency of the lasers is given by

$$
\omega(\Delta \!\!\!/) = \omega\_\circ - k\Delta \!\!\!/ ,\tag{6}
$$

$$I(\Delta f) = I\_{thr} + \eta \Delta f,\tag{7}$$

where *ω<sup>o</sup>* is the optical frequency at the lasing threshold, *Ithr* is the lasing threshold intensity, and Δ*J* is the injection current with the threshold injection current subtracted. The slopes are intrinsic characteristics of the SCLs and were taken to be *k* = 1.84 GHz/mA and *η* = 0.55 mW/mA.

In Figure 2b, we show a case for *κ* = 0.0027, where the two lasers are operated at different initial output intensities, one at 2% above threshold and the other at 30% above threshold. The injection current to this latter, higher intensity laser is varied linearly, and it is seen that at the PT threshold, i.e., when Δ*ω* = *κ*, there is an abrupt increase in the intensity of the other SCL. This, once again, is a clear feature of the PT-symmetric properties of this system.

**Figure 2.** Intensities of the two lasers from numerical simulations of Equations (1)–(4). The vertical, dashed line indicates the PT threshold. (**a**) As a function of Δ*ω* for *κ* = 0.2, *τ* = 0. The intensities of the two lasers are indistinguishable from each other since we assume identical lasers and operating conditions. The red line is the intensity averaged over 10 ns to account for detector bandwidth; (**b**) as a function of Δ*ω* for *κ* = 0.0027 when injection current to one laser (shown in green) is swept to vary its optical frequency; intensities are averaged over 10 ns; (**c**) as a function of *κ* for a Δ*ω* = 0.2, *τ* = 85, intensities averaged over 10 ns.

Finally, to gain some insights into the limits of this numerical model for investigating PT symmetry, we show one example of the outcome of the numerical simulations for a non-zero time delay, when the coupling, *κ*, is strong so that the rate equation model is not valid. In Figure 2c, *τ* = 85, Δ*ω* = 0.2, and *κ* is swept, and we note that the intensities of both lasers increase as the PT threshold is crossed. However, one does not observe the sharp transition that is characteristic of PT-symmetric systems, and also, there are slow oscillations in the intensities of both lasers. This behavior, for *κ* > Δ*ω*, is not predicted by the eigenvalues picture obtained from the 2 × 2 effective Hamiltonian. The principal causes for this are that the infinite dimensional nature of the system, which is not captured by the 2 × 2 effective Hamiltonian, and that the rate equation model assumes weak coupling, which starts to break down for the strong couplings used here. We note that the time delay can cause the intensities of the two lasers to become chaotic. However, we are only interested in the global, average behavior of the intensities and not in the dynamical regimes, and the averaging over 10 ns hides the chaotic behaviors. This average behavior is more representative of the predictions of Equation (5).

In order to gain some further insight into the properties of this system, we re-write the rate equations for the field in terms of equations for the time evolution of the intensity and phase of the two lasers. The complex electric fields are written in terms of a real amplitude and a real phase modulation term as *E*1,2(*t*) = *A*1,2(*t*)*eiφ*1,2(*t*). Inserting this into the rate equation model and separating the real and imaginary components, the time evolution of the phases is given by

$$\dot{\phi}\_{1,2}(t) = aN\_{1,2}(t) \pm \Lambda \omega + \kappa \frac{A\_{2,1}(t-\tau)}{A\_{1,2}(t)} \sin(\phi\_{2,1}(t-\tau) - \phi\_{1,2}(t) - \theta \tau). \tag{8}$$

The time evolution of the phase difference, Δ*φ* = *φ*<sup>1</sup> − *φ*2, is given by

$$\begin{split} \dot{\Delta\dot{\phi}}(t) = \mathbf{a}(N\_1 - N\_2) + 2\Delta\omega + \kappa(\frac{A\_2(t-\tau)}{A\_1(t)}\sin(\phi\_2(t-\tau) - \phi\_1(t) - \theta\tau) \\ \qquad - \kappa(\frac{A\_1(t-\tau)}{A\_2(t)}\sin(\phi\_1(t-\tau) - \phi\_2(t) - \theta\tau). \end{split} \tag{9}$$

To establish the connection to PT symmetry in the SCL system, we follow an approach similar to the one above and assume that carrier inversion is negligible, i.e., *N*1,2 = 0, and that the time delay is zero, i.e., *τ* = 0. For identical lasers, with *A*<sup>1</sup> = *A*2, the time evolution of the phase difference simplifies to the Adler equation [33],

$$
\dot{\Delta\phi}(t) = 2\Delta\omega - 2\kappa \sin(\Delta\phi(t)).\tag{10}
$$

The above equation suggests that the phase locking condition is given by |Δ*ω*| < *κ*, which is the exact same condition that governs the PT threshold for a system with zero time-delay. This analysis establishes that the regime where the phase locking condition is satisfied is also the regime wherein the eigenvalues of the effective Hamiltonian are real, while the regime in which the phases are unlocked is the regime where the eigenvalues of the effective Hamiltonian are complex. This analysis, within the assumptions of neglecting population dynamics and zero time-delay, establishes the equivalence of the PT threshold and the phase locking condition [34].

#### **4. Discussion**

We have shown in this work that a pair of mutually coupled semiconductor lasers can serve as a template for investigating parity–time symmetry. The advantage of this system is that one does not need to fabricate a system in which the gain and loss are exactly balanced. As shown in this paper, the gain/loss terms are now replaced by the relative detuning between the lasers, which are identically equal and opposite in sign for a pair of SCLs. Since the optical frequency of an SCL is easily controlled via temperature or injection current, and as the coupling between the two lasers can also be easily controlled and measured, the coupled SCL system offers advantages over other PT-symmetric systems where to alter the coupling, one needs to fabricate a new system. Our work has shown how the rate equation model that describes the coupled SCL system can be reduced to a simple 2 × 2 effective Hamiltonian, which is identical to the typical PT-symmetric dimer. We then showed that the predictions of this simple effective Hamiltonian are reproduced by the full rate equation model, despite the latter having additional complexities such as population dynamics and gain nonlinearities. The model has some limitations, such as the assumption that both lasers operate on a single longitudinal model and that the coupling is weak. However, our results indicate that the features of PT symmetry are very robust and still evident in the rate equation model.

Among possibilities for further exploration, the SCL model for PT symmetry allows the inclusion of quantum noise due to spontaneous emission, as well shot noise in the carrier inversion. In the context of SCL dynamics, there are instances where noise plays a critical role, and it is, a priori, difficult to know when it will be important. Typically, it is only during a comparison of experiments and numerical simulations that the influence of noise is revealed. The effect of noise can be studied by augmenting Equation (2) with appropriate Langevin terms to account for spontaneous emission and shot noise in the inversion. The PT-symmetric SCL system is different from other PT systems since population dynamics are inextricably intertwined with the intensity dynamics. To get a handle on the properties of the system, it is instructive

to look at the populations and how they influence the intensities of the lasers. Our system allows us to make the coupling term purely real, complex, or purely imaginary, thereby offering further richness in the parameter space for probing the properties of the system. In summary, the use of commercially available, low-cost SCLs means that our PT system is flexible enough to add additional oscillators, whereas other systems do not offer this simplicity of extension since each additional oscillator requires fabrication of the entire array from scratch.

**Author Contributions:** Y.J. and G.V. conceptualized the project. A.W. and J.S. carried out the numerical computations and data analysis. All authors discussed the results and commented on the manuscript, which was written by G.V.

**Funding:** Y.J. was supported by NSF grant DMR-1054020.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**



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