1. Introduction

Quantum dot (QD) semiconductor lasers have already demonstrated many interesting properties such as temperature insensitive low threshold current densities [1], high modulation band-widths [2, 3], and a strong resistance to optical feedback [4], therefore showing great prospects towards realizing uncooled, isolator-free, directly modulated semiconductor lasers. All of these features originate from the quantum confinement that usually characterizes atoms or molecules in contrast to semiconductor materials. Indeed bulk and quantum well semiconductor materials have a large density of states at high energy and, as a result, the maximum gain suffers a blue shift with increasing carrier density.

To quantify the effect of the change of refractive index with carrier density on the dynamical properties of semiconductor lasers, Henry introduced the linewidth enhancement factor α [5],

where n is the refractive index, g is the gain per unit length, λ is the wavelength and N is the carrier density. The introduction of the α-factor in rate equations not only explained the experimentally observed broad linewidth and large chirp, but also the onset of coherence collapse induced by optical feedback and filamentation in broad area lasers and amplifiers.

Since its definition, researchers have developed various experimental techniques to measure the α-factor. The most commonly used techniques are based on the analysis of the amplified spontaneous emission (ASE) [6], on the FM/AM response ratio under small signal current modulation [7], linewidth measurements [8, 9, 10], and pump-probe experiments [11].

Simple analytical calculations based on rate equations for bulk and quantum well semiconductor lasers show that all of these measurements should theoretically give the same value for the α-factor. For this reason, experimentalists have used these techniques to measure the α-factor of bulk, quantum well and, more recently, of QD semiconductor lasers.

For QD lasers, simple models predict a small if not zero α-factor due to the discrete density of states. However, experiments have reported a great variety of values for the α-factor ranging from 0 [2, 6], to values of the order of those commonly observed in bulk and quantum well materials (1.5 - 3) [12, 13]. Larger values for the a-factor than those published for quantum well devices have also been reported (4 - 11) [14], and a value for α as high as 60 has been published [15]. It is also worth noticing that while the values for α reported from FM/AM measurements are much larger than those obtained from the analysis of the ASE, both techniques have shown an increase in α with injection [12, 14, 15]. The non-zero α-factor could be explained as a direct consequence of the plasma effect [16], or dispersion of the dot sizes [17]. However, the large variance in the values for α suggests that the unique carrier dynamics needs to be taken into account to explain these experimental results.

The aim of this paper is to compare three techniques commonly used to measure the α-factor of QD lasers, namely, analysis of ASE, small signal FM/AM response and linewidth measurements. Our analysis is based on QD rate equations incorporating both the finite capture time of carriers into the dots and plasma effects [16]. It is found that each of the three techniques gives a different value for the measured α-factor, which increases with injection current due to the plasma effect. These results are in qualitative agreement with recent experimental results and suggest that the α-factor might not be a well-defined parameter to characterise QD devices.

2. Quantum dot rate equation model

In QD semiconductor devices, the carriers are first injected into a wetting layer before being captured into a dot at a capture rate that depends strongly on the dot population. Thus, rate equations that commonly describe carrier dynamics of QD materials read [18, 19],

where N is the number of carriers in the wetting layer per dot and ρ is the occupation probability of a dot in the laser pumped with the current per dot J. The parameters γ_n and γ_d are the non-radiative decay rates, C is the capture rate from the wetting layer into an empty dot, v_g is the group velocity and σ is the cross section of interaction of the carriers in a dot with the electric field. The constant q is the elementary charge and E is the complex amplitude of the electric field in the cavity, normalized to the photon density S: |E|² = S.

The gain g ₀ (2ρ-1) is defined by the dot population. Experimental [20], and theoretical analyses [21, 16], have shown the refractive index of quantum dot materials can depend on the ground state population ρ and on the surrounding carriers N. Thus, the temporal evolution for the electric field reads

where γ_s is the photon decay rate in the cavity and g0 is the differential gain. The coefficients α_d and α_nr describe the change of refractive index with the change of dot carrier population and with non-resonant population respectively. In real QD materials the non-resonant carriers are described by more than one quantity, as carriers can be present in the wetting layers or in various excited states of the dots. The size dispersion of the dots and thermalisation through a wetting layer also lead to a non-zero α-factor which is here described by the parameter α_d . Our analysis could also be carried out using a model describing the evolution for each dot size as in [22]. However, it would include several other unknown parameters and make the derivation of analytical results much more difficult. Here, for simplicity, only a single non-resonant carrier density N and a single dot population ρ are taken into account.

 

Fig. 1. Values of the α-factor calculated by the following techniques: solid line is the ASE method, dashed line is the FM/AM method, dotted line is the linewidth method. (a) Laser operates for non-saturated dots (ρ_th = 0.6). (b) Laser operates close to full inversion (ρ_th = 0.9). For the non-saturated operating point (a) the α-factor does not show strong dependence on the injection current. This is not the case for operation close to full inversion (b).

Download Full Size | PDF

3. Analysis of the amplified spontaneous emission

The first technique commonly used to measure the α-factor is based on the analysis of the ASE of a laser biased below threshold. In this regime, a change of the injection current δJ induces changes in the carrier densities δN and δr. This in turns leads to changes in the gain δg = 2g ₀ δρ and laser frequency δω= v_gg ₀(α_dδρ + α_nrδN). From (1), it follows that $\alpha =\frac{2}{v_g}\frac{\mathit{\delta \omega }}{\mathit{\delta g}}$ and

This shows that the α-factor depends on the non-resonant carrier density N and that this contribution becomes more important near full inversion as shown in Fig. 1. Close to full inversion the dot occupation probability saturates to 1, but the carrier density in the wetting layer continues to increase with current, leading to a dramatic increase in α. From Eqs. (2) and (3) it follows that δN/δρ = γ_d /(C(1 - ρ_ss )²), where ρ_ss is a steady state value, which increases with injection current up to the threshold current J_th , and then clamps at a value ρ_th .

In [20], it was demonstrated that the non-resonant carriers contribute approximately half the total index change. For this reason α_d and α_nr were chosen to be 1.2 and 130.6 respectively. In this case the first and the second terms in Eq. (5) are equal to 60% and 40% of total α = 2 when ρ_ss = 0.6. The following values for other parameters were used in the calculations: γ_n = γ_d = 10^-3 (ps)^-1, v_g = 0.3 (ps)^-1, v_g = 167 ∙ 10⁶ m/s and C = 1ps. Values of g ₀ = 90(cm)^-1, σ = 7.5 ∙ 10^-7(μm)² are implied for ρ_th = 0.6; g ₀ = 36(cm)^-1, σ = 3 ∙ 10^-7(μm)² are implied for ρ_th = 0.75 and correspondingly g ₀ = 22.5(cm)^-1, σ = 1.88 ∙ 10^-7(μm)² are implied for ρ_th = 0.9.

This technique can indeed give a low α-factor if the device operates with a low carrier density. However, if the carrier density is increased and the dots are almost fully occupied then the a-factor increases [12].

 

Fig. 2. (a) FM/AM response under small signal current modulation as defined by Eq. (6). The minimum of this function corresponds to α_FM /AM as defined in Eq. (7). Dotted line is the value of α calculated just below threshold by the ASE method. Solid line is the FM/AM response just above threshold. Dashed line is the FM/AM response at twice threshold. Here ρ_th = 0.6. (b) Linewidth due to α_lw . Solid line is for non-zero values of α_d and α_nr . Dashed line is for α_nr = 0. Here the current ranges from threshold to 8 times threshold and α_th = 0.75.

Download Full Size | PDF

4. Small signal response

The second technique [7], commonly used to measure the α-factor is based on the FM/AM response of a laser biased above threshold to small modulation of the injection current J = J_dc + δJ cos(ωt). In this case, the electric field can be written as E = √Se^iφ with S = S_dc + ΔS _cos(ωt + ϕ ₁) and φ= φ_dc + δφcos(ωt + ϕ ₂). The values of δS and δφ can be calculated by solving the small signal response. The FM/AM response is defined by

A typical curve of the FM/AM response calculated for Eqs. (2), (3), (4) is depicted in Fig. 2 (a). This function first decreases, reaches a minimum at ω_min = √2S_dcγ^gσ and increases linearly at high frequency. This situation is very different from conventional rate equations where the FM/AM response converges toward the α-factor at high frequencies. In this technique, α is defined as F(ω_min )

It increases linearly with the injection current J_dc and, at threshold current J_th , is equal to the α-factor as measured from the ASE for the same injection current. Notice that the slope of F(ω) at high frequency is given by dF/dω ≃ α_nrτ_cap and thus is directly proportional to the plasma contribution to the linewidth enhancement factor and to the effective dot carrier capture time τ_cap = 1/(C(1 - ρ_th )). When this technique is applied to QD devices to extract an a-factor, extremely large values for the α above the lasing threshold can occur, as experimentally observed in reference [15].

5. Linewidth measurements

Since the α-factor was originally introduced to explain the broad linewidth of semiconductor lasers, it is worthwhile to calculate the value of this parameter based on a measurement of the laser linewidth. This can be performed by adding a white noise term to Eq. (4) and calculating the root mean square of the phase fluctuations [10]. Such a calculation gives

This yields a different value for α than those calculated using any of the previous two techniques as shown in Fig. 1. This means that a prediction of the laser linewidth using the α-factor measured by either the ASE or FM/AM technique is not straightforward. However, α_lw increases linearly with injection current as N_ss ∝ J_dc , as in the case for α_FM/AM . As a result, the linewidth, which varies as Δv ∝ (1 + ${\mathrm{\alpha }}_{\mathit{\text{lw}}}^2$)/S_dc , increases linearly at high injection currents as shown in Fig. 2 (b). This is confirmed experimentally in [23].

6. Conclusion

In conclusion, we have shown that various experimental techniques commonly used to measure the α-factor of semiconductor lasers lead to different values when applied to QD semiconductor lasers, depending on device design. For devices operating at low carrier densities or if α_nr = 0, the various measurement techniques will give similar values for the α-factor. However, for saturated dots and if α_nr ≠ 0, the various measurements may give great discrepancies since the plasma effect plays an important role in this regime. Indeed, the ASE measurements, carried out at lower injection current, will give small values for the α-factor, but the other two techniques, carried out well above threshold, will give extremely large values for α. For example, Fig. 1 shows that analysis of the ASE can give α = 1.5 at J = 0.4J_th , while the FM/AM technique carried out on the same device could give α = 28 at J = 2J_th . These results are consistent with experimental results for the ASE method [12] and the FM/AM technique [15].

This analysis also suggests that one should deal carefully with the α-factor of QD materials and devices. For some applications, such as direct modulation, it is clear that the FM/AM technique may give the best indication about device performance but for others, such as filamentation in broad area devices or feedback sensitivity, the ASE technique may also be a good indicator of the device behaviour. Since such behaviour is directly linked to the finite capture time and the plasma effect, which are characteristic of QD materials, it might be useful to compute these parameters from experimental measurements.