Theoretical Modelling of Single-Mode Lasing in Microcavity Lasers via Optical Interference Injection

: The effective manipulation of mode oscillation and competition is of fundamental importance for controlling light emission in semiconductor lasers. Here we develop a rate equation model which considers the spatially modulated gain and spontaneous emission, which are inherently governed by the ripple of the vacuum electromagnetic field in a Fabry-PØrot (FP) microcavity. By manipulating the interplay between the spatial oscillation of the vacuum field and external optical injection via dual-beam laser interference, single longitudinal mode operation is observed in a FP-type microcavity with a side mode suppression ratio exceeding 40 dB. An exploration of this extended rate equation model bridges the gap between the classical model of multimode competition in semiconductor lasers and a quantum-optics understanding of radiative processes in microcavities.


Introduction
In the rapidly developing field of photonic integrated circuits and photonic signal processing, there is a general demand for small-size and high-efficiency light sources to enable dense integration [1,2]. Usually, a smaller laser cavity results in a larger free spectral range (FSR) and therefore a larger mode separation. However, in this case, an inhomogeneously broadened gain spectrum [3] can lead to the mode competition and effective mode manipulation can be a demanding task [4,5]. The technical path leading to single mode operation has so far relied on the spatial modulation of the real and imaginary parts of the refractive index. A periodically modulated active region acts as an optical grating to enable wavelength selection via distributed optical feedback. For example, index-coupled distributed feedback (DFB) lasers [6], distributed Bragg reflector lasers [7], and photonic crystal lasers [8] operate on a single longitudinal mode based on a modulation of the real-part of the refractive index, whilst gain/loss modulation is successfully employed in gain-coupled DFB lasers [9][10][11], lasers with periodic metal structures [4] and lasers based on a pair of cavities with broken parity-time (PT) symmetry [5].
Very recently, optical interference pumping has emerged as an alternative approach to demonstrate single mode lasing in micro-sphere lasers [12,13]. While the underlying mechanism for single mode operation is in general based on gain/loss modulation, it fundamentally differs from the previous cases, such as gain-coupled DFB lasers. In the case of gain-coupled DFB lasers, the lasing wavelength is primarily dominated by the optical grating itself, i.e. the spatial variation of the semiconductor structure, whilst the variation of the imaginary part in the refractive index breaks the PT symmetry and hence enables single mode operation [4,5] . In the case of optical interference pumping discussed in this work, the spatial variation of the imaginary part of the refractive index determines both the lasing wavelength and the single mode operation. In addition, carrier injection is uniformly applied where k 0 is the propagation constant corresponding to the longitudinal mode of wavelength . Combining formula (1) and (2), the overlap between the interference pattern and the vacuum field of standing wave can be manipulated by altering the incident angles and the wavelength of interference beams.

Travelling-wave rate equation
A travelling wave rate equation is employed to analyse the lasing properties of the spatially pumped FP cavity. The electric field in the FP cavity can be expressed as where ω is the angular frequency, F(t,z) and R(t,z) represent the forward and backward optical fields in the FP cavity, respectively. The travelling-wave equation is derived from the timedependent coupled wave equations by neglecting the coupling between the forward and backward optical fields. The fields F and R are periodically modulated using interference pumping [16][17][18][19][20], where v g is the group velocity, is the confinement coefficient factor, g is the optical gain, P is the spatial factor governed by the vacuum field of standing wave, F P is the Purcell factor, is the internal loss, F sp and R sp represent the forward and reverse field of the spontaneous emission coupled into the lasing mode, and L is the length of active region. By applying the rotation wave approximation, the spatial factor of gain modulation is simplified to that of the following expression: Because the periodicity of the optical interference is at the same scale as the lasing wavelength, the variation in the Purcell factor due to the spatial oscillation of the vacuum field in the FP-type microcavity should not be neglected. According to the theoretical description of the Einstein coefficients and the experiment confirmation [21][22][23][24], we take the spatially-varied Purcell enhancement into account in both the spontaneous emission and stimulated emission [25,26]. The vacuum zero-point energy, which exists ubiquitously due to the quantization of electromagnetic fields in the resonant cavity, has the same distribution as the field of standing wave in the longitudinal direction of the cavity. Governed by Fermi's golden rule, the transition rate is proportional to the density of final states which are the vacuum states [27,28], resulting in spatially modulated spontaneous and stimulated emission. Single-mode operation, depending on the spatial overlap between the distribution of the vacuum field and optical interference pumping, can be analysed using the above rate equations. The general mechanism for single mode operation is sketched in Fig. 1b. The Purcell factor in the discussion is defined as: where Q is the quality factor and V is the cavity mode volume [29]. A parabolic approximation of the optical gain is used [30] ( ) where g N is the differential gain, N 0 is the transparent carrier density, G 0 is the parabolic gain fitting factor, and is the gain compression factor [31] resulted from spectral hole burning and carrier heating at high photon density. Photon density N P is calculated by where 0 is the vacuum dielectric constant, 0 |E| 2 is the total energy of the optical field at the given frequency, . Considering the relationship between optical field and photon density, the forward and reverse field of the spontaneous emission coupled into the lasing mode holds as where is the spontaneous emission factor, N is the carrier density, sp is the spontaneous emission lifetime. The boundary conditions are given by: where r 1 and r 2 are the mirror reflectivity of the FP resonator's left and right facets where r 1 = r 2 . Reducing the mirror loss increases the Q factor of the microcavity, which further increases the Purcell factor and enhances single mode oscillation via optical interference pumping.

Carrier rate equation
By taking into account the optical interference pumping and the Purcell enhancement of the spontaneous and stimulated emission, the time-dependent rate equation for the carrier densities in the active region are described by [32] ( ) where i is the internal efficiency, L in is the injected light power, hv is the photon energy, V is the volume of active region, nr is the non-radiative lifetime, and D is carrier diffusion coefficient [33,34]. The value of S is 1 while using the uniform optical pumping, and S is governed by formula (1) with optical interference pumping.

Multi-mode rate equations
The longitudinal mode competition is governed by the following multimode rate equations [35,36] here, i represents the different longitudinal modes. To keep it simple, only three main modes near to the central wavelength of the gain spectrum are simulated.  Table 1 shows the parameters used in the simulation. The length of the FP microresonator is 20 m which is divided into 1200 sections in the simulation. We have assumed that a quantum well active region is used in the simulation. We consider three lasing modes, including 1 = 1.243 m with a longitudinal node number of m = 104, 2 = 1.255 m with m = 103 and 3 = 1.267 m with m = 102. The FSR between the three modes is nearly 12 nm. In the proposed experimental system, a continuous wave laser source with = 976 nm is selected for interference. The incident angle is set to 51 and the required angular resolution is therefore around 1 . The spatial distribution of the vacuum field and hence the optical gain  When the interference pumping effectively overlaps with the field distribution of the central mode (CEN lines in Fig. 4), single-mode lasing is realized with a SMSR = 43.39 dB to the left mode and 40.55 dB to the right mode. Comparing with the 1.41 dB difference between modes at 1 and 2 under uniform pumping, the central mode is very effectively selected by optical interference pumping. The modified distribution of the photon density in the interference pumping case is a consequence of spatially enhanced stimulated and spontaneous emission.

Simulation results and discussion
Following the m 103 interference pumping, we further change the number of nodes in the interference pattern to effectively overlap with the field of different modes in the longitudinal direction of the cavity. As shown in POS lines in Fig. 4, the output powers of the three modes have been altered under the m 102 spatial pumping. The SMSR at the main lasing mode 3 1.267 m is 42.53 dB to 1 and 37.42 dB to 2 . The situation can be a little different under m 104 interference pumping since the mode 1 is the weakest mode in these three lasing modes. The SMSR at 1 is 27.60 dB to 2 and 32.20 dB to 3 , respectively (NEG lines in Fig. 4). A carrier diffusion coefficient of 2 cm 2 s -1 is used in the simulation, which is a typical value taken from the literature for InGaAsP microcavity lasers [34]. In the optical interference injection system, the carrier diffusion coefficient has a non-negligible impact on mode selection. To investigate the influence, we observe that the SMSR at the central mode 2 = 1.255 m reduces to 35 dB to 1 and 28 dB to 3 when the carrier diffusion coefficient increases to 10 cm 2 s -1 . This degradation is attributed to reduced contrast in the spatial distribution of carriers due to the interference injection. Comparing the characteristic light-light curves of laser cavities with uniform (Fig. 3a) and interference pumping (Fig. 5a), a clear transition from multimodal to single mode lasing is observed. A detailed comparison of the output intensity in the desired mode ( 2 ) (Fig. 5b) and non-desired modes ( 1 and 3 ) show the superior mode selection when the longitudinal profile of the pump light is effectively overlapped with the desired lasing mode. The normalized intensity of the desired mode reaches almost unity when the spatial interference pumping is above threshold. The spatially pumped cavity displays a significant growth in the value of SMSR beyond the threshold power, as shown in Fig. 5c. The value of SMSR increases with higher input power, indicating that the SMSR can be precisely controlled by changing the input. A smaller size cavity expands the FSR, revealing a higher SMSR (Fig.  5d). Applying spatial gain modulation to realise single mode lasing is suitable for microcavities with different sizes. Both SMSR and laser performance, such as threshold pump power are related to the size of the microcavity. A smaller laser cavity expands the FSR and results in a larger mode separation, therefore a higher SMSR can be obtained. In the simulation, we have kept the same pumping density for uniformly and nonuniformly pumped microcavities. This proves that the SMSR is improved under nonuniform injection compared to a cavity with uniform injection. Fig 6 shows the time evolution of the carrier density (Fig. 6a) and photon density of the central mode (Fig. 6b)  delay is an important parameter for semiconductor lasers and extensive studies have indicated that the delay time is principally determined by the carrier dynamics [37,38]. The simulation results indicate that there is a shorter turn-on delay for the spatially pumped cavity, in which the interference injection interacts with the vacuum field distribution. This affects the carrier dynamics and hence increases the non-radiative electron scattering rates. However, the decrease of the turn-on delay time in our system is relatively small. This is due to the slight differences in the threshold carrier density under uniform and spatial optical pumping, which directly affect the carrier recombination rate. Moreover, the damping rate of the turn-on process is modified by changes to the spontaneous emission process caused by spatial pumping Fig. 6 Turn-on delay of uniformly and spatially pumped cavities with different pumping power. a. Turn-on delay dynamics of carriers. b. Turn-on delay dynamics of photon density of the central mode.
We have calculated the small signal response [39][40][41] of both the uniformly and spatially pumped microcavity lasers based on the extended rate equation model. The results are shown in Fig. 7. Clear relaxation resonances are observed for injected light powers ranging from 3 mW to 48 mW. The corresponding 3 dB bandwidth varies from 4.7 to 15.1 GHz for the uniformly pumped cavity (Fig. 7a solid lines) and 6.2 to 21.3 GHz for the spatially pumped cavity (Fig. 7a dotted lines), respectively. Focusing on the change of modulation response amplitude, the damping factor is decreased in the spatially pumped case. We plot the 3 dB linewidth versus the square root of light power minus threshold light power (Fig. 7b). The proportional constant is 2.23 GHz mW -0.5 for the uniformly pumped cavity, but the modulation efficiency increased to 3.70 GHz mW -0.5 for the spatially pumped cavity. The higher-speed performance is achieved due to an immediate interaction between non-uniform carrier injection with optical interference pumping and the vacuum field of the standing wave. function of the light power (L-L th ) 1/2 , the modulation efficiency of spatial pumping is increased comparing to the uniform pumping.

Conclusions
In conclusion, we have proposed an extended rate equation model which considers the interplay between the vacuum electromagnetic field of standing waves and external optical injection via laser interference. Simulation results show that the lasing mode can be effectively selected (with a SMSR over 40 dB) when the longitudinal profile of the pumping light effectively overlaps that of the field distribution of the desired lasing mode. This mode selection mechanism shows that the value of SMSRs can be precisely controlled by altering the input and that the model can be applied to micro-resonators with different sizes and shapes, such as FP and whispering-gallery-mode resonators [42]. Moreover, a shorter turn-on delay and higher-speed modulation can be possible using optical interference pumping compared to uniformly pumped lasers. This model has been used to explore the performance of micro-and nano-resonators through a combination of a classical model of multimode competition in semiconductor lasers and a quantum-optics understanding of radiative processes in microcavities.