Analysis of Stability of Generation in Quantum Well Lasers

Abstract

A stability analysis of two modes of generation in semiconductor quantum well lasers is performed. These modes correspond to two solutions of the rate equations obtained by taking into account the internal optical loss that depends on the density of charge carriers injected into the laser waveguide region and, hence, on the injection current. It is shown that, in contrast to the first (“conventional”) mode of generation, which is always stable and hence observable, the second (“additional”) mode, which is entirely due to the internal loss that depends on the carrier density, is unstable and hence cannot be observed under the steady-state conditions in the laser structure considered in this work.

1 INTRODUCTION

Semiconductor injection lasers based on heterostructures emerged more than 50 years ago. They were invented in 1970, when a laser based on an AlGa-As–GaAs heterostructure was pioneered at the A.F. Ioffe Institute of Physics and Technology, operating in continuous wave mode at room temperature [1].

Modern life is almost impossible to imagine without the use of heterostructure lasers. Such lasers have been widely applied, in particular, in information technology, industry, medicine and everyday life. Recently, much attention has also been paid to the development of lidars (“light radars”) based on semiconductor lasers for use in unmanned aerial vehicles and underwater vehicles, as well as in cars.

Over the years, heterostructure lasers, especially lasers with a low-dimensional active region, have been the subject of extensive research (see, for example, [2–29]). Back in the 2000s, it was predicted in works [30–33] that in such lasers there should be a decrease in optical power with an increase in pump current (the so-called rollover of the light-current characteristic (LCC) – dependence of the output optical power on the pump current) for the “conventional” (first) mode of generation of stimulated emission and the existence of an “additional” (second) mode of generation is also possible. Both the LCC rollover for the first mode of generation and the very existence of the second mode are entirely due to (i) the non-instantaneous capture of charge carriers from the bulk waveguide region (optical confinement layer (OCL)) of the laser into the low-dimensional active region and (ii) internal optical loss that depends on the carrier concentration in the waveguide region. (There is also internal optical loss in the lowdimensional active region, but it is significantly lower than the loss in the bulk waveguide region). The characteristics of lasers, in particular the LCC, are radically different for the two modes of generation and, thus, are degenerate (have two branches) (Fig. 1).

Fig. 1.

Steady-state light-current characteristic of a single QW laser with a broadened waveguide and a lasing wavelength of 0.98 μm. 1 – stable first (“conventional”) mode of generation, 2 – unstable second (“additional”) mode of generation.

The criterion for the existence of the second mode of generation (and, accordingly, the second branch in laser characteristics) can be presented in the following form:

$$\frac{{{{\sigma }_{{\operatorname{int} }}}{{{v}}_{{{\text{capt}}{\text{,0}}}}}}}{{2b{{B}_{{3{\text{D}}}}}{{g}^{{\max }}}}} > 1,$$

(1)

where σ_int—cross section of internal optical loss in the waveguide region of the laser structure; ${{{v}}_{{{\text{capt}},0}}}$—capture velocity of charge carriers from the bulk (three-dimensional) waveguide region into an unfilled low-dimensional active region, which is a single or multiple quantum wells or layer(s) with quantum dots, b— width of the waveguide region; B_3D—coefficient of spontaneous radiative recombination in the waveguide region [11, 19, 34]; g^max—maximum modal gain coefficient due to stimulated radiative transitions in the low-dimensional active region of the laser [11, 15, 18, 34].

The dependence of internal optical loss on the concentration of carriers in the waveguide region is taken into account by the non-zero value of the cross section σ_int (in units of cm²) of internal loss in the waveguide region (see below expression (5) for the coefficient of internal optical loss).

The non-instantaneous capture of carriers from the waveguide region into the low-dimensional active region is taken into account by the finite (not infinitely high) value of the capture velocity ${{{v}}_{{{\text{capt}},0}}}$ (in units of cm/s) [34–38] (see below equations (2) and (3)).

The non-instantaneous capture, in combination with spontaneous radiative and (or) Auger recombinations of carriers in the waveguide region, leads to intrinsic sublinearity of the LCC in lasers with a low-dimensional active region [34–38]. However, this non-instantaneousness in itself cannot lead to a LCC rollover. Rollover of LCC does not occur even in a situation where charge carriers are first captured from the waveguide region to the nonemitting upper level of size quantization in the active region, and then relax to the emitting lower (ground) level. In this case, the non-instantaneousness of relaxation, combined with the non-instantaneousness of capture, further enhances the sublinearity of the LCC is the output optical power saturates (asymptotically approaches its maximum possible value) with increasing pump current [39, 40].

Thus, it is the combination of internal optical loss, which depends on the concentration of carriers in the waveguide region, with the non-instantaneousness of capture from the waveguide region that leads to the LCC rollover for the “conventional” (first) mode of generation. And it is the combination of these factors that leads to the emergence of the “additional” (second) mode of generation.

Both the LCC rollover for the conventional mode of generation and the emergence of an additional mode of generation can significantly affect the output optical power and other important characteristics of lasers with a low-dimensional active region. In connection with this, it seems relevant to conduct a detailed study of such lasers, properly taking into account the above factors. In works [30–33, 41–45], as a first step, both modes of generation (in particular, both branches of the LCC) were theoretically studied under steady-state conditions.

The next step, which naturally follows from the first, is a theoretical study of the stability of the two modes of generation, which is the focus of this article in the context of quantum well (QW) lasers. The goal of further work could be a direct experimental study of this issue, which, as far as we know, is currently missing.

2 THEORETICAL MODEL: RATE EQUATIONS

In this work, we carry out a linear analysis of the stability of two modes of generation and, accordingly, two branches of laser characteristics. We use a model based on the following three rate equations (for charge carriers in the waveguide region and in the QW, as well as for photons in the cavity):

$$\begin{gathered} \frac{{\partial {{n}^{{{\text{OCL}}}}}}}{{\partial t}} = \frac{j}{{eb}} + {{N}_{{{\text{QW}}}}}\frac{1}{b}\frac{{{{n}^{{{\text{QW}}}}}}}{{{{\tau }_{{{\text{esc}}}}}}} - {{N}_{{{\text{QW}}}}}\frac{{{{{v}}_{{{\text{capt}},0}}}}}{b} \\ \times \;(1 - {{f}_{n}}){{n}^{{{\text{OCL}}}}} - {{B}_{{3{\text{D}}}}}{{({{n}^{{{\text{OCL}}}}})}^{2}}, \\ \end{gathered} $$

(2)

$$\begin{gathered} \frac{{\partial {{n}^{{{\text{QW}}}}}}}{{\partial t}} = {{{v}}_{{{\text{capt}},0}}}(1 - {{f}_{n}}){{n}^{{{\text{OCL}}}}} - \frac{{{{n}^{{{\text{QW}}}}}}}{{{{\tau }_{{{\text{esc}}}}}}} - {{B}_{{2{\text{D}}}}}{{({{n}^{{{\text{QW}}}}})}^{2}} \\ - \;{{c}_{g}}{{g}^{{\max }}}(2{{f}_{n}} - 1){{n}^{{{\text{ph}}}}}, \\ \end{gathered} $$

(3)

$$\frac{{\partial {{n}^{{{\text{ph}}}}}}}{{\partial t}} = {{N}_{{{\text{QW}}}}}{{c}_{g}}{{g}^{{\max }}}(2{{f}_{n}} - 1){{n}^{{{\text{ph}}}}} - {{c}_{g}}(\beta + {{\alpha }_{{\operatorname{int} }}}){{n}^{{{\text{ph}}}}}.$$

(4)

The model of three rate equations assumes electron-hole symmetry (i.e., the sameness of all parameters and quantities related to electrons and holes). Taking electron-hole asymmetry into account would require the use of a system of five rate equations (for electrons and holes in the waveguide region and in the QW, as well as for photons in the cavity) and would introduce additional complications without significantly affecting the physical picture as a whole.

The physical quantities included in equations (2)–(4) and not introduced earlier are: n^OCL—concentration of free charge carriers in the waveguide region, j – pump (injection) current density, e—electron charge, n^QW—two-dimensional concentration of charge carriers in each quantum well (QW), N_QW—number of QWs, B_2D—coefficient of spontaneous radiative recombination in a two-dimensional QW material [19, 34], c_g—group speed of light, n_ph—two-dimensional concentration of photons (number of photons per unit area of the stripe contact) of stimulated radiation in the laser cavity, β = (1/L) ln(1/R)—cavity loss associated with the radiation yield through the mirror(s), L—cavity length, R—mirror reflectivity.

The internal optical loss coefficient α_int, which depends on the concentration of charge carriers n^OCL in the waveguide region and is included in equation (4), is given in the form

$${{\alpha }_{{\operatorname{int} }}} = {{\alpha }_{0}} + {{\sigma }_{{\operatorname{int} }}}{{n}^{{{\text{OCL}}}}},$$

(5)

where α₀—constant component that includes “built-in” (independent of n^OCL) loss. The non-instantaneousness of carrier capture from the waveguide region into the low-dimensional active region leads to the dependence of the concentration n^OCL on the injection current [34–38, 41–45] and, thus, to the dependence of the internal loss α_int on the injection current.

The occupancy (filling factor) f_n of the lower edge of the size quantization subband in the QW by charge carriers is not an independent quantity and is expressed through the two-dimensional concentration of charge carriers n^QW in the QW as follows [23, 34]:

$${{f}_{n}} = 1 - \exp \left( { - \frac{{{{n}^{{{\text{QW}}}}}}}{{N_{c}^{{2{\text{D}}}}}}} \right).$$

(6)

The thermal escape time τ_esc of charge carriers from the QW into the waveguide region is also not an independent quantity and is expressed through the velocity of capture ${{{v}}_{{{\text{capt}}{\text{,0}}}}}$ of charge carriers from the waveguide region into the unfilled QW as follows:

$${{\tau }_{{{\text{esc}}}}} = \frac{1}{{{{{v}}_{{{\text{capt}}{\text{,0}}}}}(1 - {{f}_{n}})}}\frac{{N_{c}^{{2{\text{D}}}}}}{{{{n}_{1}}}}.$$

(7)

In expressions (6) and (7), the quantity $N_{c}^{{2{\text{D}}}}$ = $m_{c}^{{{\text{QW}}}}T{\text{/}}(\pi {{\hbar }^{2}})$—two-dimensional effective density of states in the QW, $m_{c}^{{{\text{QW}}}}$—the effective mass of charge carriers in the QW, T—temperature in energy units.

The quantity n₁ in expression (7) is determined as follows [34]:

$${{n}_{1}} = N_{c}^{{3{\text{D}}}}\exp \left( { - \frac{{\Delta {{E}_{c}} - \varepsilon _{n}^{{{\text{QW}}}}}}{T}} \right),$$

(8)

where $N_{c}^{{3{\text{D}}}}$ = 2[($m_{c}^{{{\text{OCL}}}}T$)/($2\pi {{\hbar }^{2}}$)]^3/2—three-dimensional effective density of states in the waveguide region, $m_{c}^{{{\text{OCL}}}}$—effective mass of charge carriers in the waveguide region, ΔE_c—QW depth for charge carriers (band offset at the heterointerface between the waveguide region and the QW), $\varepsilon _{n}^{{{\text{QW}}}}$—energy of the lower edge of the subband of size quantization of charge carriers in a QW.

3 ANALYSIS OF THE STABILITY OF MODES OF GENERATION

Within the framework of linear stability analysis, the solutions of rate equations (2)–(4) (time-dependent concentrations of charge carriers in the waveguide region (n^OCL) and in the QW (n^QW), as well as the concentration of photons (n^ph) in the resonator) are presented in the following form:

$${{n}^{{{\text{OCL}}}}}(t) = n_{0}^{{{\text{OCL}}}} + (\delta n_{m}^{{{\text{OCL}}}})\exp (\Lambda t),$$

(9)

$${{n}^{{{\text{QW}}}}}(t) = n_{0}^{{{\text{QW}}}} + (\delta n_{m}^{{{\text{QW}}}})\exp (\Lambda t),$$

(10)

$${{n}^{{{\text{ph}}}}}(t) = n_{0}^{{{\text{ph}}}} + (\delta n_{m}^{{{\text{ph}}}})\exp (\Lambda t),$$

(11)

where $n_{0}^{{{\text{OCL}}}}$, $n_{0}^{{{\text{QW}}}}$ and $n_{0}^{{{\text{ph}}}}$ – solutions of the steady state rate equations, and $\delta n_{m}^{{{\text{OCL}}}}$, $\delta n_{m}^{{{\text{QW}}}}$, and $\delta n_{m}^{{{\text{ph}}}}$— small amplitudes of time-dependent components.

3.1 Steady-State Light-Current Characteristic

Solving the rate equations (2)–(4) at $\partial {\text{/}}\partial t$ = 0 gives the steady-state values of the concentrations of free charge carriers in the waveguide region $n_{0}^{{{\text{OCL}}}}$, charge carriers localized in the QW, $n_{0}^{{{\text{QW}}}}$, as well as stimulated emission photons $n_{0}^{{{\text{ph}}}}$ as functions of the pump current density j and parameters of the laser structure. Next, the steady-state output optical power of the laser P₀ = $\hbar \omega {{c}_{g}}\beta n_{0}^{{{\text{ph}}}}S$ is calculated, where $\hbar \omega$—photon energy, S = WL—stripe contact area, W—stripe contact width.

Here, as in works [41–45], an edge-emitting laser structure with a broadened waveguide is considered. The active region of the laser presents a single stressed QW. The materials of the QW, waveguide region and emitters (cladding layers) are In_0.31Ga_0.69As, GaAs and Al_0.1Ga_0.9As, respectively. The widths of the QW and waveguide region are 30 Å and 0.9 μm, respectively. The QW is not located in the center of the waveguide region, but closer to the p-emitter (at a distance of 0.1 μm from it). The concentrations of the majority charge carriers in the emitters n_clad = p_clad = 5 × 10¹⁸ cm^–3. The cross section of internal optical loss in the waveguide region σ_int = 3 × 10^–18 cm²; the velocity of capture of charge carriers from the waveguide region into an empty QW ${{{v}}_{{{\text{capt}}}}}{{_{{\text{,}}}}_{{\text{0}}}}$ = 10⁶ cm/s; length of the Fabry–Perot resonant cavity L = 1.5 mm, reflectivities of both mirrors R = 0.32, width of the stripe contact W = 100 μm, temperature T = 300 K. The calculated maximum modal gain in a single QW g^max = 30.73 cm^–1. The lasing wavelength is 0.98 μm.

For the structure under consideration, the quantity included in the left side of criterion (1) is equal to 2.65 and, thus, the rate equations have two solutions, which correspond to two modes of generation. The threshold current densities for the first and second modes j_{th, 1} = 31.6 A/cm² and j_{th, 2} = 34 kA/cm². The maximum pump current density, above which laser generation is disrupted, is j^max = 48.06 kA/cm² (at this injection current density, the first and second branches of the LCC merge together—see Fig. 1).

Figure 1 shows the dependence of the output optical power on the injection current density (LCC) for the structure under consideration. As can be seen from the figure, the LCC consists of two branches. Above the first threshold, the first (“conventional”) mode of generation is switched on, which corresponds to the first branch of the LCC (branch 1 in Fig. 1), above the second threshold, the second (“additional”) mode of generation emerges, which corresponds to the second branch of LCC (branch 2 in Fig. 1). The generation in both branches is disrupted at the pump current density j = j^max.

3.2 Characteristic Equation

For the existence of non-zero solutions for small amplitudes $\delta n_{m}^{{{\text{OCL}}}}$, $\delta n_{m}^{{{\text{QW}}}}$ and $\delta n_{m}^{{{\text{ph}}}}$ included in (9)–(11), the following characteristic equation (obtained by linearizing equations (2)–(4) using the expres-sions (9)–(11)) should be satisfied to find the eigenvalues Λ:

$${{\Lambda }^{3}} + {{A}_{2}}{{\Lambda }^{2}} + {{A}_{1}}\Lambda + {{A}_{0}} = 0,$$

(12)

where the expressions for the coefficients A₀, A₁ and A₂ are:

$$\begin{gathered} {{A}_{0}} = {{c}_{g}}{{g}^{{\max }}}(2{{f}_{{n0}}} - 1) \\ \times \;\left\{ {2{{g}^{{\max }}}\left[ {{{N}_{{{\text{QW}}}}}\frac{{{{{v}}_{{{\text{capt}}{\text{,0}}}}}}}{b}(1 - {{f}_{{n0}}}) + 2{{B}_{{{\text{3D}}}}}n_{0}^{{{\text{OCL}}}}} \right]} \right. \\ \left. { - \;\frac{{{{{v}}_{{{\text{capt}}{\text{,0}}}}}}}{b}({{n}_{1}} + n_{0}^{{{\text{OCL}}}}){{\sigma }_{{\operatorname{int} }}}} \right\}{{N}_{{{\text{QW}}}}}\frac{{1 - {{f}_{{n0}}}}}{{N_{c}^{{2{\text{D}}}}}}{{c}_{g}}n_{0}^{{{\text{ph}}}}, \\ \end{gathered} $$

(13)

$$\begin{gathered} {{A}_{1}} = {{N}_{{{\text{QW}}}}}\frac{{{{{v}}_{{{\text{capt}}{\text{,0}}}}}}}{b}(1 - {{f}_{{n0}}}) \\ \times \;\left( {2{{B}_{{2{\text{D}}}}}n_{0}^{{{\text{QW}}}} + 2{{c}_{g}}{{g}^{{\max }}}\frac{{1 - {{f}_{{n0}}}}}{{N_{c}^{{2{\text{D}}}}}}n_{0}^{{{\text{ph}}}}} \right) \\ \end{gathered} $$

$$ + \;2{{B}_{{{\text{3D}}}}}n_{0}^{{{\text{OCL}}}}\left[ {{{{v}}_{{{\text{capt}}{\text{,0}}}}}(1 - {{f}_{{n0}}})\frac{{{{n}_{1}} + n_{0}^{{{\text{OCL}}}}}}{{N_{c}^{{2{\text{D}}}}}}} \right.$$

(14)

$$\begin{gathered} \left. { + \;2{{B}_{{{\text{2D}}}}}n_{0}^{{{\text{QW}}}} + 2{{c}_{g}}{{g}^{{\max }}}\frac{{1 - {{f}_{{n0}}}}}{{N_{c}^{{2{\text{D}}}}}}n_{0}^{{{\text{ph}}}}} \right] \\ + \;{{c}_{g}}{{g}^{{\max }}}(2{{f}_{{n0}}} - 1)\left[ {2{{N}_{{{\text{QW}}}}}{{c}_{g}}{{g}^{{\max }}}\frac{{1 - {{f}_{{n0}}}}}{{N_{c}^{{2{\text{D}}}}}}n_{0}^{{{\text{ph}}}}} \right], \\ \end{gathered} $$

$$\begin{gathered} {{A}_{2}} = {{{v}}_{{{\text{capt}}{\text{,0}}}}}(1 - {{f}_{{n0}}})\left( {\frac{{{{N}_{{{\text{QW}}}}}}}{b} + \frac{{{{n}_{1}} + n_{0}^{{0{\text{CL}}}}}}{{N_{c}^{{2{\text{D}}}}}}} \right) \\ + \;2{{B}_{{{\text{3D}}}}}n_{0}^{{0{\text{CL}}}} + 2{{B}_{{{\text{2D}}}}}n_{0}^{{{\text{QW}}}} + 2{{c}_{g}}{{g}^{{\max }}}\frac{{1 - {{f}_{{n0}}}}}{{N_{c}^{{2{\text{D}}}}}}n_{0}^{{{\text{ph}}}}. \\ \end{gathered} $$

(15)

The characteristic equation (12) is cubic since our model is based on three rate equations (see equations (2)–(4)).

Expressions (13)–(15) for the coefficients A₀, A₁ and A₂ include the steady-state values of the concentrations of charge carriers and photons $n_{0}^{{{\text{OCL}}}}$, $n_{0}^{{{\text{QW}}}}$ and $n_{0}^{{{\text{ph}}}}$, which are the solutions to the system of rate equations (2)–(4) at $\partial {\text{/}}\partial t$ = 0 and depend on the parameters of the laser structure and pump current density j. Therefore, the coefficients A₀, A₁ and A₂ also depend on j.

Since for the laser structure under consideration the rate equations have two solutions, each of the two modes of generation corresponds to its own values of the coefficients A₀, A₁ and A₂ and its eigenvalues Λ.

Within the framework of linear analysis, in order for the considered mode of generation to be stable, the exponential components in the expressions (9)–(11) should be evanescent (solutions (9)–(11) of the system of rate equations (2)–(4) should return over time to their steady-state values $n_{0}^{{{\text{OCL}}}}$, $n_{0}^{{{\text{QW}}}}$ and $n_{0}^{{{\text{ph}}}}$), i.e. the real parts of all three eigenvalues Λ should be negative:

$$\operatorname{Re} (\Lambda ) < 0.$$

(16)

A positive sign of the real part of even one of the three eigenvalues Λ would mean an exponential increase in solutions (9)–(11), i.e., their departure from steady-state values and the instability of the mode of generation under consideration.

Thus, to determine the stability of modes of generation, it would suffice to determine the signs of the real parts of the roots of the cubic equation (12). Since the coefficients A₀, A₁ and A₂ in equation (12) are real quantities, there are the following two possibilities for the roots of equation (12) (eigenvalues Λ): (i) one real and a pair of complex conjugate roots; (ii) three real roots.

If all three coefficients A₀, A₁ and A₂ were always (at any pump current) positive, then, as can be seen from (12), the real eigenvalue(s) Λ would necessarily be negative, and the corresponding solutions would be stable. Since f_n0 < 1 and all quantities included in expressions (14) and (15) for A₁ and A₂ are positive, the coefficients A₁ and A₂ are indeed always positive. However, unlike A₁ and A₂, as can be seen from expression (13) for A₀, it is impossible a priori to draw any conclusion about the sign of the coefficient A₀. Thus, to even determine the stability of the solutions corresponding to the real eigenvalue(s), an analysis of at least the dependence of the sign of the coefficient A₀ on the injection current j is required. If it turns out that the coefficient A₀ is negative, then this will also not automatically lead to any conclusion about the sign of the real eigenvalue(s) Λ—since the coefficient A₂ is always positive, the second (quadratic) term on the left side of the equation (12) is always positive and formally the real eigenvalue(s) Λ can be both positive and negative. Thus, in the case of a negative coefficient A₀, to determine the sign of the real eigenvalue(s) Λ, the solution of equation (12) will be required.

It should be noted that since 2f_n0 – 1 > 0 (the condition for the positivity of the steady state population inversion and the gain coefficient in the active region—see [11]), as can be seen from (13), in the absence of internal optical loss that depends on the concentration of charge carriers in the waveguide region (i.e., at σ_int = 0), the coefficient A₀ is also always positive. Thus, in this case the real eigenvalue(s) Λ are negative. This result reflects the fact that in the absence of internal loss, which depends on the concentration of charge carriers in the waveguide region, there is only one solution to the system of rate equations and, accordingly, only one (“conventional”) mode of generation, which is always stable.

In Subsection 3.2.1 we analyze the coefficients A₀, A₁ and A₂. Since no conclusion can be made about the sign of the real part of the complex-conjugate eigenvalues even in the case of all positive coefficients A₀, A₁ and A₂, and it turns out that the coefficient A₀ is negative for the second mode of generation, then further (in Section 3.2.2) we solve equation (12) to find the eigenvalues Λ.

3.2.1. Coefficients A₀, A₁ and A₂ of the characteristic equation as functions of the pump current density. Having calculated the steady state concentrations $n_{0}^{{{\text{OCL}}}}$, $n_{0}^{{{\text{QW}}}}$ and $n_{0}^{{{\text{ph}}}}$ for each of the modes of generation and using further formulas (13)–(15), we determined the coefficients A₀, A₁ and A₂ for two branches. Figure 2 shows these coefficients as functions of the pump current density. As can be seen from the figure, the coefficients A₀, A₁ and A₂ for the first mode of generation (curves 1) are always positive. For the second mode of generation, the coefficients A₁ and A₂ (curves 2) are also always positive, but the coefficient A₀ for the laser structure under study is always negative. At the laser emission cut-off point (j = j^max), the coefficients A₀, A₁ and A₂ for the two modes (as well as all other characteristics for them) coincide.

Fig. 2.

Coefficients A₀, A₁ and A₂ of the characteristic equation (12) against pump current density. 1 – first mode of generation, 2 – second mode of generation.

3.2.2. Roots Λ₁, Λ₂ and Λ₃ of the characteristic equation as functions of the pump current density. Having calculated the coefficients A₀, A₁ and A₂, we then find the roots Λ of the cubic equation (12) for each of the two modes of generation (Figs. 3–6).

Fig. 3.

Real eigenvalue Λ₁ (a) and real parts of eigenvalues Λ₂ and Λ₃ (b) for the first (“conventional”) mode of generation against pump current density.

Fig. 4.

Imaginary parts of the eigenvalues Λ₂ and Λ₃ for the first (“conventional”) mode of generation against pump current density.

Fig. 5.

Real eigenvalues Λ for the second (“additional”) mode of generation against pump current density: (a) positive root Λ₁; (b) negative roots Λ₂ and Λ₃.

Fig. 6.

Real eigenvalue Λ₁ (a) and real parts of the eigenvalues Λ₂ (b) and Λ₃ (c) against pump current density. 1 – first (“conventional”) mode of generation, 2 – second (“additional”) mode of generation.

Let us start with an analysis of the roots for the first (“conventional”) mode of generation. As can be seen from Fig. 3a, one of the eigenvalues (Λ₁) for the first mode of generation is always (for all values of the pump current, i.e., for j_{th, 1} < j < j^max) a real negative quantity. It turned out that the other two roots (Λ₂ and Λ₃) for the first mode of generation are complex conjugate quantities at pump current densities in the range j_{th, 1} < j < 9.6 kA/cm². The real part of these roots is also negative (Fig. 3b). The imaginary parts of the roots Λ₂ and Λ₃, which are equal in magnitude but opposite in sign, are shown in Fig, 4. Solutions in the form (9)–(11), corresponding to the roots Λ₂ and Λ₃ at pump currents in the range j_{th, 1} < j < 9.6 kA/cm², describe underdamped relaxation oscillations in the laser—the absolute value of the imaginary parts of the roots Λ₂ and Λ₃ is the angular frequency of these oscillations (see [46, 47]).

As can be seen from Fig. 4, at j = 9.64 kA/cm², the imaginary parts of the roots Λ₂ and Λ₃ (i.e., the angular frequency of relaxation oscillations) go to zero (compare with Fig. 2 in [46, 47]). (In Fig. 4 we should also note a peculiarity in the dependence of Im(Λ₂) and Im (Λ₃) on the pump current density at values of j slightly exceeding j_{th, 1}.)

As can be seen from Fig. 3b, at j > 9.64 kA/cm², the roots Λ₂ and Λ₃ become purely real (while differing from each other) negative quantities. Solutions in the form (9)–(11) at pump currents j > 9.64 kA/cm² describe overdamping, i.e. purely exponentially evanescent (without oscillations) relaxation of the concentrations of charge carriers and photons to their steady state values.

Thus, as can be seen from our calculations, for the first (“conventional”) mode of generation, the real parts of the eigenvalues (or real eigenvalues) of all three roots Λ are negative in the entire range of operating currents in the laser j_th,1 < j < j^max (Fig. 3). This proves the stability of the first mode of generation: the second terms on the right side of expressions (9)–(11) decay exponentially in time, and, thus, the concentrations of charge carriers and photons return to their steady state values $n_{0}^{{{\text{OCL}}}}$, $n_{0}^{{{\text{QW}}}}$ and $n_{0}^{{{\text{ph}}}}$.

Let us proceed to the analysis of the roots of the characteristic equation (12) for the second (“additional”) mode of generation as functions of the pump current density. These roots are shown in Fig. 5. It turned out that all three roots are real quantities: one root (Λ₁) is always (for all values of the pump current for the second mode of generation, i.e., for j_{th, 2} < j < j^max) positive (Fig. 5a), and the other two (Λ₂ and Λ₃) are always negative (Fig. 5b).

Figures 6a–6c shows the real parts of the roots Λ₁, Λ₂ and Λ₃ for two modes of generation (“conventional” and “additional”) as functions of the pump current density. It can be seen from the figure that at the lasing cut-off point (at j = j^max) the corresponding eigenvalues for the two modes coincide. This, in particular, means that the roots Λ₁ for two modes of generation (which have opposite signs for j < j^max) should necessarily vanish at this point (Fig. 6a).

Thus, our calculations show that for the second mode of generation the eigenvalue Λ₁ is positive over the entire range of pump currents (j_{th, 2} < j < j^max). This proves the instability of the second mode of generation in the considered laser structure: the second terms on the right side of expressions (9)–(11), corresponding to this root Λ₁, increase exponentially in time and, thus, the concentrations of charge carriers and photons move away from their steady state values $n_{0}^{{{\text{OCL}}}}$, $n_{0}^{{{\text{QW}}}}$ and $n_{0}^{{{\text{ph}}}}$.

4 CONCLUSIONS

A linear analysis of the stability of two modes of generation in semiconductor quantum well lasers has been carried out. Two modes of generation correspond to two solutions to the system of rate equations, obtained taking into account the internal optical loss that depends on the concentration of charge carriers in the waveguide region of the laser. It is shown that, in contrast to the always stable and, therefore, observable first (“conventional”) mode of generation, the second (“additional”) mode, which is entirely due to the carrier-concentration-dependent internal loss, is unstable and, therefore, cannot be observed under steady state conditions in the considered laser structure.

Despite the fact that the second mode of generation is unstable and cannot be observed under steady state conditions, i.e., with a constant pump current (continuous wave operation), its existence can manifest itself under pulsed pumping at current densities j > j_{th, 2}. Even possible short-term switches from the stable first (“conventional”) mode to the unstable second should be accompanied by a strong increase in the intensity of radiation associated with the spontaneous recombination of charge carriers (in the waveguide region in the first place, but also in the quantum well) and, accordingly, a drop in the useful power of stimulated radiation from the quantum well (see Fig. 1). Thus, fluctuations of spontaneous and stimulated emissions under pulsed pumping can serve as indicators of the existence of an unstable second mode of generation.