1. Introduction

Spontaneous build-up of coherences and phase transition to macroscopic quantum state of matter has always been a fascinating subject for scientific and engineering communities. Numerous application domains have benefited from fundamental research in macroscopic quantum coherent phenomena. One example is the Bose-Einstein condensation (BEC) of atoms which stimulated works on laser-cooled atomic clocks with unprecedented fractional frequency stability. Another example is the wide spreading use of superconducting magnets in which coil undergoes BCS (Bardeen, Cooper, and Schrieffer) transition. The search for spontaneous build-up of macroscopic coherences in optically-pumped semiconductor microcavities has resulted in intensive research towards the polariton laser.

Interestingly, photonic microcavities implemented in group-III nitrides are capable of reaching macroscopic coherences at room temperature conditions unlike their counterparts implemented in conventional III–V alloys (e.g. AlGaAs). Larger reduced exciton mass in InGaN/GaN quantum well (QW) results in larger exciton binding energy (∼ 30 meV), smaller exciton Bohr radius and higher Mott transition density of exciton dissociation. Due to large oscillator strength, the exciton-polariton Rabi splitting in nitride-based microcavities may significantly exceed the thermal energy k_BT. Thanks to these remarkable features of group-III nitrides, room-temperature polariton lasing in InGaN microcavities has been demonstrated [1]. The high critical temperature of exciton-polariton BEC is conditioned by their low effective mass (in the plane of QWs), so the critical BEC density can be achieved below the excitonic Mott transition density. At the same time, although such 2D BEC is a transient dynamic state, it can be analyzed in the framework of stationary Schrödinger equation [2].

Superradiance (SR) in semiconductor edge-emitting laser diode structures can be considered as another example of spontaneous macroscopic coherence in a solid state system [3]. In distinguishing from BEC of cavity polaritons, SR occurs at non-equilibrium carrier densities significantly exceeding the excitonic Mott density: in the electron-hole (e-h) plasma phase [4]. In order to reach it, a high carrier injection is used instead of optical pumping of microcavities. The associated second-order phase transition to a coherent state can be regarded as occurring in a one-dimensional system [5].

The cooperative radiative recombination in an ensemble of quantum oscillators (e.g. atoms or molecules) has been predicted before the invention of lasers [6]. Since then, it has been extensively studied both theoretically and experimentally [7, 8]. The characteristic features of SR emission are the temporal and spatial coherences, highly anisotropic emission pattern, quadratic dependence of pulse intensity on the number of excited atoms I ∝ n² and afterpulse ringing oscillations [6–13]. The SR pulse duration decreases with the number of emitters with relationship τ_c ∝ 1/n, while the pulse energy is proportional to the ensemble population (Iτ_c ∝ n) in agreement with the energy conservation considerations. On the other hand, the spontaneous nature of the transition to the transient macroscopically coherent state is responsible for large fluctuations in the shape, duration and amplitude of SR pulses [9, 10].

Many different approaches have been proposed to treat SR in atomic or molecular gases in the framework of semiclassical or quantum models and distinguishing samples of elongated shape, point-like shape or disk-like shape [11, 12]. All of them have predicted a quadratic dependence of the pulse intensity on the number of excited atoms I ∝ n². The analytical expression for conditions of SR, cooperation time and threshold density have been almost directly applied to semiconductor laser heterostructures [13, 14] while the numerical models based on semi-classical Maxwell-Bloch equations were either too complex [13] or lack some of the essential details [15]. As a consequence, the existing set of conditions for reaching SR in semiconductors is quite confusing.

In semiconductors, unlike atomic or molecular gases, the energy distribution of carriers is very broad, thus the interpretation in the framework of Dicke SR theory for atomic and molecular gases cannot be perfectly justified. A hypothesis has been drawn that SR is assisted by the formation of a transient coupled electron-hole pair state mediated by photons [16, 17]. According to this hypothesis, during SR pulse emission, the e-h system undergoes a second order phase transition to coherent BSC-like state. From the practical point of view, SR has been considered to be one of the promising approaches to generate femtosecond optical pulses whose peak power exceeds the output powers of semiconductor laser devices in lasing or amplified spontaneous emission regimes by a few orders of magnitude [18–20]. Several phenomenological interpretations have been given [21,22] but none of them have attempted to demonstrate the Ginzburg-Landau equation for BCS transition.

In this paper, inspired by the remarkable features of the group-III nitrides, we studied numerically the Dicke superradiance in an edge-emitting ridge-waveguide multiple-section cavity with InGaN/GaN QWs (see Fig. 1). We provide comparison between lasing and SR regimes in such a cavity. Our numerical model is based on a semiclassical description of macroscopic coherences (polarization) in the ensemble of electrons and holes in the QWs. Contrasting from previous numerical model developed for the bulk and QW GaAs-based materials [15], our numerical model correctly takes into account the carrier populations and coherences as a function of the pump current and absorber bias.

 

Fig. 1 Schematic of two-section laser structure for SR emission generation.

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We show that the traveling wave Maxwell-Bloch equations governing the SR emission can be converted to the Ginzburg-Landau equation for the BCS-like (second-order) phase transition. This enables us to define the critical density in the system, the order parameter and the coherence time (and length). Unlike previous analytical treatment, only one requirement is imposed for achieving the SR regime, namely the condition to exceed the critical carrier density.

2. Numeric model

As a model system we utilize a tandem edge-emitting cavity consisting of two separately contacted sections (Fig. 1). The shorter, negatively biased section is used as a saturable absorber. The longer section is pumped with short current pulses and provides the optical gain [16].

The numerical simulations are performed assuming InGaN/GaN double QW separate confinement heterostructure with composition of epitaxial layers and optical gain parameters typical for an edge emitting laser [23, 24]. (The detailed description of the epitaxial layers does not enter the model simulations directly.) Apart from one important exception, we use the same set of approximations and the same set of model parameters as usually used in the traveling-wave rate equation model of a semiconductor laser (Table 1). Unlike lasing dynamics of a semiconductor laser, SR emission is a very fast process. The adiabatic approximation, where the medium polarization follows the electromagnetic field instantaneously, is not valid. The parameters of our model accounting for polarization decoherence time T₂ are summarized in Table 1.

Table 1. Model parameters for SR emission in wide-bandgap semiconductors, using data from Refs. [24–26]

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Following the semiclassical Maxwell-Bloch equations [27], the forward (A₊) and backward (A₋) traveling waves in the cavity are excited by macroscopic medium polarization. The evolution of the wave amplitude also accounts for the intrinsic material loss (including free carrier absorption) :

Equation (1) has one and the same form for the gain and absorber cavity sections. We use single-mode approximation, which has been proven in analysis of semiconductor laser dynamics [28]. In Eq. (1), we distinguish the medium’s polarization associated with the forward and backward traveling waves. The medium’s polarization or coherences induced by electrons and holes (the ensemble average for the off-diagonal elements of the density matrix [27]) are normalized in the following way

The usual boundary conditions for the wave amplitudes are applied at left (z=0) and right (z=L) cavity facets

The evolution of carrier population n and coherences P_± in the gain and absorber sections is treated separately. The model equations below take into account that (i) the carrier lifetime in saturable absorber τ_a is much shorter as compared to the gain section (τ_n); (ii) the differential absorption is higher by a factor of σ with respect to differential gain [28]; (iii) the gain section is electrically pumped; (iv) the modal absorption coefficient in absorber section can be altered via Quantum Confined Stark Effect (QCSE) by applying a negative bias to the absorber. The dynamics of carrier populations and coherences in the gain and absorber section (index a) are described by the following [29]:

In the limit of slow radiation dynamics (spontaneous emission and lasing regimes), the system of Eqs. (1)–(5) can be transformed to a traveling-wave rate equation model of semiconductor laser (see Annex A). Figure 2 shows the difference between lasing (a) and SR emission (b) regimes in a cavity of overall length L=600μm and relative absorber length of 20% (of the overall cavity length). The evolution of the carrier density n, medium polarization P_± (coherences) and output power at the gain and absorber section facets is obtained by numerically integrating the system of Eqs. (1)–(5).

 

Fig. 2 Radiation dynamics in InGaN/GaN tandem cavity laser in the self Q-switching regime (a) and superradiance regime (b). Normalized carrier densities n/n_t (left panels, left axis), normalized macroscopic polarizations |P_±|/n_t (middle panels) and radiated output power (right panels) are displayed for the gain (red curves) and absorber (blue curves) sections. The overall cavity length is 600 μm including the 120 μm long absorber section. Left panels also indicate normalized threshold carrier density n_th/n_t (left axis) estimated from Eq. (6) and evolution of the applied pump current normalized at its steady-state value (green curve, right axis). In (a), the pump current of 80 mA is adiabatically turned on, the time constant is 6 ns, the absorber bias is Va=0. In (b), large current of 3.7 A is switched abruptly while absorber is kept at large negative bias (Va=−5).

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In both cases, the carriers are initially depleted. If the pump current of small amplitude (80 mA) is adiabatically switched-on at zero absorber bias [Fig. 2(a)], the emission appears almost instantaneously after the carrier density reaches the lasing threshold (red solid curve and red dotted line, respectively). In the steady-state operation, such tandem cavity structure exhibits self Q-switching regime, emitting pulses of 0.3 W peak power and 30 ps width at repetition rate of 1 GHz. After the onset of emission, the carrier density in the gain section oscillates in the vicinity of threshold.

However if the pump current and absorber bias are adjusted in such a way that large non-equilibrium carrier density can be accumulated prior to emission of the first pulse, the radiation dynamics changes drastically. In the example shown in Fig. 2(b), this is achieved by applying a large constant current (3.7 A) and large absorber bias (normalized bias parameter V_a=−5.2). (These values can be reduced by tailoring the cavity length, relative lengths of the gain and absorber sections and changing the shape of the current pulse.) Under these operating conditions, the structure emits sub-picosecond pulses of peak power in the range 100–200 W. In this particular example, the output pulses associated with repetitive accumulation of carriers have significantly smaller amplitude and follow the main SR pulses after 50 ps.

The major differences between the two emission regimes depicted in Fig. 2 are the magnitude and evolution of the medium polarization P_± (middle panels). In the lasing regime [Fig. 2(a)], no substantial medium polarization is induced, |P_±| ≪ n. The opposite situation is revealed in Fig. 2(b). The macroscopic coherences P_± and carrier density n exhibit variations of the same scale, indicating that during SR pulse emission, the magnitude of the Bloch vector is conserved. The system thus preserves macroscopic coherence, which is possible only if the effective decoherence time $T_2^{\text{eff}}$ exceeds the SR pulse width τ_c. An interested reader can find other criteria to distinguish superradiance from amplified spontaneous emission and lasing in Ref. [11].

As shown in Fig. 2(b), the key requirement for SR emission is the large initial carrier density. In the rest of the paper we assume that such initial carrier density has been created in one way or another. Furthermore, we focus on solitary SR pulse generation which corresponds better to the experimental conditions used so far [16, 30, 31]. The question of how such a non-equilibrium system can be prepared, which is in fact a question about cavity optimization, is left for another study. It is not of such fundamental importance here since experimental evidences are available that SR in semiconductor lasers can occur below as well as above the lasing threshold [20]. In practice, this is achieved by optimizing the length of the cavity sections, the shape of the current pulse and applying pump current of sufficiently high amplitude so that the SR pulse is emitted before the onset of lasing.

The spatiotemporal dynamics of SR emission in our model cavity system is detailed in Fig. 3. The dynamics of the carrier population [Fig. 3(a)], coherences [Figs. 3(b) and 3(c)] and optical field [Figs. 3(d) and 3(e)] is quite complicated. (The absorber section is located at the left cavity facet (at z=0). The dynamical variables n and P_± are normalized on transparency carrier density n_t and A_± is normalized on $\sqrt{n_t}.$) Initially, a high reversed bias is applied to the absorber leading to depletion of carriers in the QWs [blue color in the scale of Fig. 3(a)]. It prevents the structure from lasing and enables accumulation of carriers in the gain section. The gain section QWs are pumped to a high level. In this particular case, the carrier population is about 14 times above the transparency n_t [red color in Fig. 3(a)].

 

Fig. 3 Spatiotemporal dynamics of the carrier population (a), macroscopic polarization in InGaN/GaN QWs associated with the forward (b) and backward (c) waves; field amplitudes of the forward (d) and backward (e) waves. The overall cavity length is 600 μm (z axis) including the absorber section of 120 μm long located at the coordinate origin. (Other parameters can be found in Table 1.) The time axis shows the interval of two cavity roundtrip times 2T_cav. The dynamic variables n, P_± and A_± are normalized at the carrier density at transparency n_t and $\sqrt{n_t}$. The initial carrier density in the gain section n₀ is 2.3·10²⁰ cm⁻³. The output SR pulses emitted from the two cavity facets gain are shown in Fig. 4(b).

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The process starts from spontaneous polarization noise. We use a noise source Λ_± with the normal distribution and diffusion coefficient [32] of 2D_PP = ∫ du ∫ dτ 〈Λ_±(z,t)Λ_±(z + u,t + τ)〉 = 1.6 · 10⁴⁴cm⁻⁶s⁻¹. (In the units of the axis scale in Fig. 3, the diffusion coefficient is very small, $2D_{PP}=8.7\cdot {10}^{-6}{n}_t^2/T_{\mathit{cav}}$). The corresponding rate of spontaneous polarization noise into the cavity mode is defined by the term $\sim \frac{1}{2}\mathrm{\Gamma }\sqrt{g_0{T}_2}{\mathrm{\Lambda }}_{\pm }$. It reaches 27 photons per cavity round trip (T_cav = 14 ps). This background spontaneous emission of 0.92 μW average power is clearly visible if the output SR pulse is plotted in logarithmic scale (Fig. 4(b), right panel).

 

Fig. 4 Output SR pulses at the gain section facet (red curve) and absorber section facet (blue curve) for the diffusion coefficient of spontaneous polarization noise 2D_PP of (a) $8.7\cdot {10}^{-7}{n}_t^2/T_{\mathit{cav}}$, (b) $8.7\cdot {10}^{-6}{n}_t^2/T_{\mathit{cav}}$ [the same as in Fig. 3] and (c) $8.7\cdot {10}^{-4}{n}_t^2/T_{\mathit{cav}}$. The corresponding spontaneous emission background of 0.09 μW, 0.92 μW and 92 μW can be seen in logarithmic power scale (right panels). Other parameters are the same as in Fig. 3.

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Interestingly, the noise sources Λ_±(z,t) are homogeneously distributed along the cavity. However, the macroscopic coherences P_± build up at the edges of the gain section after about half of the cavity roundtrip time [at t ∼7ps in Figs. 3(b) and 3(c)]. Because of the asymmetric cavity and random spontaneous polarization noise of the medium, the shape and emission time of the output SR pulses from the left and right cavity facets are different [Fig. 4(b)]. The emitted SR pulses are not related by reflections at the cavity facets.

The build-up of the macroscopic polarization is followed by ringing oscillations at a frequency dependent on the SR pulse intensity (and hence of the initial carrier density) [33]. They are clearly seen in the wave amplitudes [Figs. 3(d) and 3(e)]. The carrier density [Fig. 3(a)] drops abruptly at the same time when the polarization rises [Figs. 3(b) and 3(c)], indicating that almost all carriers contribute to the field. When the backward traveling pulse hits the absorber section at z=0.2L, the pulse intensity is sufficiently high to saturate the absorber [the blue step in the region 0<z<120μm and t∼7ps in Fig. 3(a)]. From this time, there is no absorption and gain in the cavity. The backward traveling SR pulse reflected off of the left cavity facet can now freely travel through the structure as seen in Fig. 3(d), yielding emission of the secondary pulse at the right cavity facet. The secondary pulse of ∼1 W peak power can be seen at t∼14ps in Fig. 4(b) (red curve).

In Fig. 4, the output SR pulses obtained at three different rates of spontaneous polarization noise Λ_±(z,t) are shown in linear (left panels) and logarithmic (right panels) power scale. The spontaneous emission background can be examined by observing the lowest bound, which can be seen at t = 0 and in between the main and secondary pulses. There are 3 orders of magnitude variations between Figs. 4(a) and 4(b). However these have no impact on the peak power and width of the emitted SR pulses. The spontaneous polarization noise is thus important only at the initial stage, triggering the process of SR pulse build up. As soon as the growing pulse amplitude exceeds the spontaneous emission background, the system dynamics follows the evolution predicted by Eqs. (1)–(5) for macroscopic dynamical variables. The magnitude of spontaneous polarization noise with average 〈Λ_±〉 = 0 impacts only the dispersion of the SR pulses between individual realizations but not the statistically average pulse parameters. This observation brings us to the idea about a master equation for statistically average realization of SR pulse, which is presented in the next section.

The build-up of macroscopic polarization has a stochastic nature, which results in differences between individual realizations of SR pulses. The peak power and pulsewidth of SR pulses vary within the range of 50–200 W and 0.5–5 ps, respectively, depending on the driving conditions (Fig. 6). Varying the overall cavity length and the relative length of absorber, we find that the intensity ratio and the width of the forward and backward SR pulses can be effectively altered. The common feature of all realizations is that the polarization P_± and the carrier excess above transparency (n – n_t) vanish after the build up of SR pulse. From this moment, the optical pulses travel freely through resonantly transparent medium without experiencing gain or absorption. (However, the non-resonant losses α_i are always present in the cavity.)

 

Fig. 5 Output SR pulses in a single-section cavity of 480 μm length without absorber (a) [α = 0] and without facet reflections (b) [R = 0 in Eq. (4)]. Blue and red curves are pulses emanating from the two cavity facets. The cavity length and initial carrier density are the same as in the gain section of Figs. 3 and 4.

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Fig. 6 Simulated peak power (a) and pulse FWHM width (b) of output SR pulses at the gain section facet (red curve) and absorber section facet (blue curve) as a function of the initial carrier density n₀. Black dashed curves indicate theoretically predicted behavior Eq. (15).

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In Figs. 3(b) and 3(c), the density of the coherence excitations P_± reaches the maximum of 9·10¹⁹ cm⁻³, which is smaller by only a factor of 2.5 than the initial density of uniformly distributed carriers n₀=2.3·10²⁰ cm⁻³. However, the associated photon density in the cavity is, at first glance, much lower, at 1.2·10¹⁹ cm⁻³ [Figs. 3(d) and 3(e)]. In fact, the dynamical variables in Eq. (1) account for the optical confinement factor Γ due to the difference between the cavity mode size and the QW width. The effective peak photon density reduced to the QW width is 6.1·10²⁰ cm⁻³, which is spectacularly large compared to the initial density n₀. A hypothesis has been previously stated [16, 17] that the SR in a semiconductor is mediated by a formation of the transient BCS-like state of e-h pairs.

In the next section, we derive master equation for statistically average SR pulse. It has canonical form of the master equation for BCS-like phase transition.

3. Analytic model

In a laser, a resonant cavity is used to reduce the photon density of states, sending spontaneous or stimulated recombination photons into just a few state of electromagnetic field. This allows the modal gain to become high enough to overcome the optical losses in the cavity. The phase transition from amplified spontaneous emission to lasing is associated with the change of the effective lifetime of photon in the cavity. At well below threshold, the lifetime of spontaneously emitted photon is determined mainly by the cavity loss. At threshold, where the optical gain equals the losses, the effective lifetime of a photon is infinite. Once electromagnetic field oscillations occur in the cavity, they will continue to oscillate for an infinitely long time. Note that the coherence time of the laser is much longer than the lifetime of photons in the cold cavity.

Unlike lasing operation, experimental observations of SR in atomic and molecular gases indicate that there is no need to introduce a resonant cavity [7]. In semiconductors, cooperative recombination of almost all nonequilibrium carriers with emission of photons into the rising SR pulse is achieved faster than the cavity mode structure of that field sets in. This is confirmed by our numerical simulations in Fig. 3, revealing the SR pulse formation after half of the cavity roundtrip time. However, due to inevitable Fresnel reflections of the cavity facets, optical feedback cannot be avoided, leading to a lasing operation. Therefore in reported experimental realizations of SR emission in semiconductor laser diode structures [14, 30, 31] (see also the review [3]), multiple section cavity with an integrated saturable absorber has been employed. The absorber section increases the lasing threshold, resulting in high initial carrier density. For example, in a tandem cavity, the threshold carrier density can be made very high, by increasing the relative length of absorber section α and applying a large negative bias V_a [24]:

The saturable absorber itself does not play a vital role in the SR pulse formation. Indeed, in Fig. 4, the SR pulse emitted at the absorber facet (blue curve) is just delayed by a propagation time in the absorber, evidencing that the SR pulse is formed in the gain section.

Figure 5(a) shows the numerical simulation results in a single-section cavity of 480 μm length. The cavity length and initial carrier density correspond to the gain section in Figs. 3 and 4. In Fig. 5(b), zero-reflection facets are assumed as well. A comparison between Figs. 4 and 5 indicates that the main parameters of the highest SR peak are all essentially the same. Inadvertent variations between individual SR pulse realizations are within a range of the dispersion caused by spontaneous polarization noise. Cavity facet reflections and saturable absorber affect only the afterpulse ringing oscillations, which are of minor importance here. Therefore, we confine our analytical model to the gain section, ignoring possible reflections at the cavity facets. Such interpretation is consistent with the treatment of SR in atomic or molecular ensembles [6].

In semiconductors, unlike atomic and molecular gases, the initial energy distribution of carriers is very broad. Furthermore, Fig. 2 indicates that the carrier density variable may take negative values during SR pulse build-up. [See red curve in Fig. 2(b) at t ∼ 0.05 ns.] An attempt to interpret such state in terms of Fermi-Dirac distribution of carriers results in a paradox: The quasi-Fermi level of holes in the valance band is above the Fermi level of electrons in the conduction band. As such, the analytic model must assume a possibility of BCS-like condensation.

To analyze SR emission in terms of BCS-like transition in semiconductor gain medium, we have to transform Eqs. (1) and (5) into Ginzburg-Landau equation (GLE) for the second-order phase transition. However GLE is a stationary equation while SR is a transient process taking place in an open system. To overcome difficulties arising from the non-stationary nature of SR, we change variables to the internal coordinates of the pulse. For concreteness, we focus our analysis on the forward traveling pulse and introduce in Eqs. (1) and (5) a new coordinate system with independent variables ζ =t – z/v_g and z :

The numerical simulations in Fig. 3 indicate that large macroscopic polarization P_± and associated field amplitudes A_± are correlated in time and space. For example, in Figs. 3(b) and 3(d), the macroscopic coherence domain builds up at the nearby right cavity facet at z ∼ 600μm and t ∼ 7ps. Within this domain, a simple proportionality relationship A₊∝P₊ holds. The same relationship A₋∝P₋ is maintained for the backward propagating SR pulse at the left boundary of the gain section [blue colored regions in Figs. 3(c) and 3(e) at z ∼ 120μm and t ∼ 7ps]. The spatial derivative in the leftmost Eq. (7) accounts for the change of the pulse amplitude. For a laser, the amplitude change would be described by a factor ∝ exp(z/L^*) with 1/L^* being the net modal gain. This brings us to the idea of the substitution 1/L^*→∂/∂z:

We shall distinguish two cases. In short samples, the coherence length of the SR pulse exceeds the sample length. The size of the domain L^* with large macroscopic coherences is defined by the sample length. In long samples, the sample length is larger than the coherence length. Different domains of the sample contribute to the SR pulse incoherently and L^* is set by the coherence length. These assumptions have been confirmed in the study [5] by comparing the numerical and analytic model predictions for the short and long samples.

The ansatz Eq. (8) allows us to exclude the wave amplitude from the two last equations (7), yielding

We have examined the Lyapunov stability of spontaneous decay by substituting $P_{+}^2={\delta }_{P^2} \text{exp} (s\zeta )$ and n=n₀ exp(−ζ/τ_n)+δ_n exp (sζ) in Eq. (9) and linearizing these equations with respect to small variations δ_P² and δ_n. For ζ ≪ τ_n, the linearized system has two independent solutions (normal modes). One of them is always stable, yielding just the usual spontaneous decay n=(n₀+δ_n) exp(−ζ/τ_n) and $P_{+}^2=0$. The second solution has the following form

For high initial carrier density n₀ > n_cr, the macroscopic coherences P₊ build up, triggered by fluctuations δ_P². The characteristic time constant that emerges from Eq. (11),

It follows that n_cr in Eq. (10) is the critical (threshold) density for macroscopic polarization build-up and a transition to the superradiance emission regime. In order to show that this is BCS-like phase transition, we shall convert Eq. (9) into the Ginzburg-Landau equation.

At first, we notice that the carrier relaxation, spontaneous decay (τ_n ∼ 1ns) and pumping term (n_t e_qd/J ∼ 100ps) can be neglected during the SR pulse, at the time scale set by the lifetime of photons (1/v_gα_i=3ps) and decohernece time T₂. This is also consistent with the observation that the normal mode of Eq. (9) associated with the usual spontaneous decay does not exhibit phase-transition changes at n_cr.

Neglecting the terms n/τ_n end J/e_qd in the last Eq. (9) allows us to recover the Bloch vector conservation (n(ζ)−n_cr)²+P₊(ζ)²=(n₀−n_cr)² with n₀ = n(−∞) being the initial carrier density in the system. Taking the second derivative in the first Eq. (9), one obtains after some trivial algebra the following master equation

It is interesting to compare the predictions of our simplified analytical model with the results of numerical calculations (Fig. 6). The critical carrier density of n_cr=2.52n_t is achieved at 4.2 kA/cm² pump current density (n=τ_nJ/e_qd). For n>n_cr, the condensation condition is fulfilled. However at n_cr2=2.57n_t (at J=4.3kA/cm²), the coherence length reduces down to the size of the cavity. Thus, only in the narrow range n_cr<n₀<n_cr2 does the SR pulse peak power exhibits quadratic growth [Eq. (14)]. At higher injection levels, it shows less steep growth characterized by asymptotic expression Eq. (15). This last feature is perfectly reproduced by the numerical model in Fig. 6. In conditions of Fig. 3, where the initial carrier density n₀ = 14.3n_t, the FWHM of the SR pulse estimated from the analytic model 1.76τ_p is 600fs, in agreement with the prediction of the numerical model.

Large stochastic variations of the SR pulse FWHM width might be partially caused by the pulse ringing effect. Thus if one of the sub-pulses reaches the half of maximum of the main pulse [as in Fig. 4, blue curves], the FWHM as plotted in Fig. 6(b) is enlarged by the time delay of the sub-pulse. However, fundamentally the output power and pulse width curves in Fig. 6 exhibit large amplitude and timing instabilities. Note that such macroscopically large fluctuations are one of the characteristic features of a condensed state [34].

The proposed consideration of SR emission is different from all previous treatments (see Sec. 1). We have shown that SR in a semiconductor follows the GLE (13) for BCS-like phase transition, while the medium polarization P_± is the order parameter of the system. The unique requirement for reaching the SR regime is that the excess of initial non-equilibrium carrier density must be above the critical density Eq. (10). We find that the cooperative time Eq. (12) defining the growth rate of the SR pulse is exactly half of the pulse width Eq. (14), which seems to be a logically self-consistent conclusion.

The original GLE describes phenomenologically the behavior seen in supercondcutors. It has been elaborated prior to the microscopic BCS theory of superconductivity. Our Eq. (13) is phenomenological as well: we have not made microscopic consideration of interactions between composite bosons composed of e-h pairs and photons. Since phenomenological GLE have been proven to describe BCS transition in superconductors, we may expect that the GLE form of the master equation for SR should assume that there is a BCS-like condensation transition. Moreover, the fact of condensation has been evoked in experimental observations [17].

One question that might arise is on the critical temperature of condensation. The GLE for SR (13) does not involve the system temperature explicitly. The studies clarifying this feature are ongoing [5, 35]. It appears that the effective mass of condensing quasiparticles is so small (∼ 10⁻¹¹m₀) that corrections due to the thermal energy k_BT are negligible. We may reference these quasi particles (polarization waves [12]) as longitudinal polaritons.

It is useful to highlight several important differences between eventual BCS-like condensation of longitudinal polaritons during SR emission and BEC of microcavity exciton-polaritons. For one, the condensate in Eq. (13) is one dimensional while BEC of microcavity polaritons is two dimensional [2]. Also, the typical carrier density for SR regime assumes that the inter-particle distance is much smaller than the exciton Bohr radius; in our model simulations, the inter-particle spacing $1/\sqrt{\pi n_{0}d}$ is 0.48 nm while the 2D exciton Bohr radius in InGaN/GaN QWs is a_B =3.5 nm [1]. The excitons cannot exist at such density. SR occurs in the e-h plasma. An excellent overview of what will happen at various carrier densities, including a discussion on BEC-BCS crossover in driven polariton condensates, can be found in Ref. [4].

4. Conclusion

In summary, we presented the numerical model of Dicke SR in semiconductor laser cavities and showed that it corresponds to the analytical Ginzburg-Landau equation for BCS phase transition. On example of InGaN/GaN QW laser heterostructures, the theoretically predicted performance of SR pulses exceeds substantially that of the mode-locked or Q-switched lasers.

Gain nonlinearities, such as the spectral hole burning (SHB) effect, are important at a high lasing power. Unlike high-power cw operating lasers, where e-h pais recombine individually, SR regime forces simultaneous cooperative recombination of most of the carriers. Therefore, the eventual intraband dynamics of carriers (e.g., SHB) can be taken into account as a perturbation in Eq. (13). Future work on the model will be to incorporate the impact of finite intraband relaxation time of carriers. Preliminary results indicate that at very high carrier density, the minimum pulse width in Eq. (15) will be clamped at 4ξ/3g₀ with ξ being the gain compression coefficient and g₀ being the differential gain. Respectively, the growth of peak power with the carrier density will become linear.