1. Introduction

Nonlinear dynamics of semiconductor lasers and their applications [1–5] have been investigated extensively in recent years. For the quantum dot (QD) lasers, the dynamical behaviors and modulation characteristics are significantly influenced by the intrinsic laser parameters especially the linewidth enhancement factor α [6]. The linewidth enhancement factor of QD lasers can be measured with several methods under different operation conditions. For the material α below the threshold, it is usually measured with the amplified spontaneous emission (ASE) [7]. For the device α above the threshold, techniques such as the FM/AM response ratio under small signal current modulation [8, 9], the linewidth measurement [10], and the injection locking [10–13] are commonly used.

In these techniques, however, the FM/AM method is limited by the electric parasitic effects where careful calibrations of the laser and the photodetector responses are required. In the injection locking technique, one can either measure the variations of the output power [11] or the junction voltage [12] under different detunings to extract the value of α. However, the variations are typically small and the value of α is difficult to be precisely determined. While the α can also be measured from the slope ratio of the upper and the lower injection locking boundaries [13], strong injections are needed to obtain the accurate locking bandwidths. When the laser under test is biased at a higher bias level, injection locking of the laser becomes difficult to achieve.

In this paper, we study the four-wave mixing (FWM) analysis [14] to measure the α of a QD laser. By fitting the experimentally obtained regenerative signals and the power spectra of the FWM states at different detuning frequencies to the respective theoretical curves, intrinsic laser parameters such as the linewidth enhancement factor α, the relaxation resonance frequency ν_r, the carrier decay rates in the quantum dots γ_d, the differential gain g₀, the photon decay rate γ_s, the interaction cross section of the carriers ς, the gain saturation coefficient ɛ, the capture rate from the quantum wells into the dots C, and the carrier decay rates in the quantum wells γ_N can all be extracted simultaneously. Moreover, unlike the injection locking technique, FWM states can be easily obtained with just weak injections. As the results, α of the QD lasers at very high bias levels can still be measured.

2. Model and method

The dynamics of QD lasers with optical injection can be described by the rate equations for the complex amplitude of electric field E, the occupancy probability of the quantum dots ρ, and the carrier density in the surrounding quantum wells N_W [2]

By deriving the equations of E, ρ, and N_W, the steady-state solutions of the rate equations at the FWM states can be obtained. In the degenerate FWM states, the E-field of the QD laser is composed of the free-oscillating signal, the regenerated amplification signal, and the FWM signal. Therefore, the output field can be expressed as

N_W is nearly constant (≃ N₀) based on the simulation results with large capture rates, where N₀ is the steady-state solution of N_W without perturbation. Therefore, we can set Eq. (3) equals to zero to get the steady-state solution, which gives

To simplify the calculation, some approximations are made based on the simulation results. First, the complex amplitude of the amplitude modulation (σ) is much smaller than the steady-state field amplitude (E₀), which gives

Since the capture rate from the quantum wells into the dots (C) is generally much larger than the carrier decay rates in the quantum wells (γ_N) while the occupancy probability of the quantum dots (ρ) is not close to 1, Eq. (6) can be reduced to

To validate this analytically derived model, the regenerative signals, the FWM signals, and the power spectra obtained from Eqs. (10)–(12) are plotted in Figs. 1(a)–1(c) and compared with the numerical simulation results obtained from the original rate equations Eqs. (1)–(3). The parameters used are listed in Table 1 with 2J_th, which are adopted from those used in Ref. [19]. As shown in Figs. 1, except some minor discrepancies around the dips in the regenerative and FWM signals, the curves derived from the analytical model (blue curves) match well with the simulation results (green dots). Thus, by fitting the experimentally obtained regenerative signals, FWM signals, and power spectra with the respective derived curves using the analytically model shown in Eqs. (10)–(12), the intrinsic laser parameters can be extracted. In fact, since the regenerative signal and the power spectrum contain all the information of the FWM signal as can be seen in Eqs. (10)–(12), only the regenerative signals and the power spectra are needed in determining the parameters.

 

Fig. 1 (a) Regenerated signals, (b) four-wave mixing signals, and (c) power spectra from the numerical simulation (green dots) and the analytical solutions based on the simplified model (blue curves), respectively.

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Table 1. Parameters of the Quantum Dot Laser used in Fig. 1

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3. Experimental setup

Figure 2 shows the schematic setup of the FWM analysis. A commercial QD laser diode (LD)(QDLaser QLD 1334) with a threshold current J_th = 8.7 mA is used as a sample for intrinsic laser parameter characterization, which has a wavelength of about 1296 nm and an output power of about 1.6 mW when biased at 20 mA. The QD laser is optically injected by a tunable laser (TL)(Yenista Tunics T100S-O) through a free space optical circulator formed by a polarizing beam splitter (PBS 2), a half-wave plate (HW 2), and a Faraday rotator (FR). The injected power is less than 1 μW to prevent the QD laser from injection-locking or any instability. The power spectrum of the QD laser is detected by a photodiode with 12 GHz frequency response (NewFocus 1554-A) and resolved with a 26.5 GHz spectrum analyzer (Agilent E4407B). The regenerative signal of the QD laser is measured by heterodyning the QD laser output with the TL output at the photodiode (when beam block (B) is removed), where an acousto-optic modulator (IntraAction ACM-1002AA1) is used to shift the beat signal from the DC to about 100 MHz for a better signal to noise ratio.

 

Fig. 2 Schematic setup of the four-wave mixing analysis. TL: tunable laser; LD: QD laser diode; FR: Faraday rotator; HW: half-wave plate; PBS: polarizing beamsplitter; PD: photodiode; SA: spectrum analyzer; VA: variable attenuator; AOM: acousto-optic modulator; B: beam block; FC: 50/50 fiber coupler.

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4. Result and discussion

Figures 3(a)–3(d) and 3(e)–3(h) show the magnitudes of the regenerative signals and the power spectra of the QD laser (red dots) with different detuning frequencies between the TL and the QD laser at bias currents of 1.5J_th, 1.75J_th, 2J_th, and 2.25J_th, respectively. By the least squares curve fitting with the analytically derived curves from Eqs. (10)–(12) (blue curves), the intrinsic parameters including the linewidth enhancement factor α, the carrier decay rates in the quantum dots γ_d, the differential gain g₀, the photon decay rate γ_s, the interaction cross section of the carriers ς, the gain saturation coefficient ɛ, the capture rate from the quantum wells into the dots C, and the carrier decay rates in the quantum wells γ_N of the QD laser are obtained and shown in Table 2. The parameters used in Ref. [19] are also listed for reference.

 

Fig. 3 Experimentally obtained (a)–(d) regenerated signals and (e)–(h) power spectra of the QD laser under the FWM states at different bias currents (red dots). Blue curves are the least square fitting calculated from the analytical model.

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Table 2. The Extracted Intrinsic Parameters of the QD Laser and Their Error Ranges

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To show the sensitivity and accuracy of the extracted parameters with FWM, a normalized error range (listed in the parentheses of Table 2) measuring a 10% increase in the standard deviation (σ) from the best-fitted parameter (that has a least standard deviation σ_opt) is calculated. As can be seen in Table 2, the FWM method is particularly sensitive in determining the linewidth enhancement factor α where the error range is less than 5% (which means that changing the α from its best-fitted value by 5% will result in an increase of the standard deviation by 10%). The increasing trend of α as the bias current increases can be clearly determined [19,20]. The linewidth enhancement factor α of the very same QD laser is also measured with the injection locking method [13] to verify the FWM result. At the same bias currents of 1.5J_th, 1.75J_th, 2J_th, and 2.25J_th, α of 0.98(0.11), 0.96(0.08), 0.97(0.04), and 0.98(0.02) are obtained respectively (The values in the parentheses are the standard deviations for various measurements under different injection levels). As can be seen, similar values of α are obtained and confirmed the feasibility of the FWM method. Note that, a relatively strong injection is needed to locked the QD laser being examined in the injection locking method. At higher bias levels, locking the laser becomes more difficult and determining α becomes not possible. On the contrary, only a weak injection is needed to generate the FWM state in the QD laser at any bias levels. Moreover, except α, other intrinsic laser parameters can be extracted simultaneously under the same operating conditions.

As can be seen from Table 2, the FWM method is also good in extracting the γ_d, g₀, and γ_s that have the error ranges within about 15%. Nonetheless, other parameters such as ς, ɛ, C, and γ_N are insensitive to the regenerative signals and the power spectra where the values of these parameters are not able to extract accurately through the FWM method (The capture rate from the quantum wells into the dots (C) and the carrier decay rates in the quantum wells (γ_N) barely affect the fitting results and thus are set with the same values used in Ref. [19]). Note that, the shapes of the regenerative signals shown in Figs. 3(a)–3(d) fitted with the experimental data are very different from the one shown in Fig. 1(a). As can be seen in Table 2, this is mainly due to the relatively small γ_s of the QD laser evaluated.

To show the effects and characteristics of each parameter, Fig. 4 shows the regenerative signals and power spectra obtained from Eqs. (10)–(12) with different α, γ_d, g₀, and γ_s, respectively. Using the curves from the best-fitted values as the references (blue solid curves), curves calculated with larger (red dashed curves) and smaller (red dotted curves) parameters are presented for comparisons. The values of the parameters are arbitrary chosen to exaggerate the effect. Except the parameters that are being discussed, other parameters are fixed at those parameters shown in Table 2 with 2J_th. As can be seen in Fig. 4(a), the α alters the depth of the dip on the positive detuning significantly, while making almost no change on the negative detuning side. As has been discussed in Table 2, only 5% change in the α will increase the standard deviation in the fitting to 110% from its minimum σ_opt. Compared with other reported methods [21], the FWM method is relatively sensitive and precise in determining the α. As can be seen in Eq. (12), the power spectrum is independent of α. Therefore, no change in the power spectra is shown in Fig. 4(b) as expected. When increasing the carrier decay rate in the dots γ_d, as shown in Fig. 4(c), the dips in the regenerative signal become shallower and shifts toward larger detuning. The shoulder on the negative detuning becomes more smooth and almost disappears for γ_d greater than 0.2 ns⁻¹. As expected, the resonance peaks in the power spectra shown in Fig. 4(d) also shifts toward the larger detuning and becomes lower as γ_d increases. Similar results are obtained and shown in Figs. 4(e) and 4(f) when the differential gain g₀ is varied. As the result, from the position and the depth of the dip in the regenerative signal and the position and the height of the resonance peaks, the γ_d and g₀ can be successfully determined. Another sensitive parameter is the photon decay rate γ_s. Compared to the previous parameters, γ_s seems to only alter the regenerative signal close to the dip but not those away from the dip as shown in Fig. 4(g). From the power spectrum in Fig. 4(h), the magnitudes of the spectra at lower detuning frequencies vary significantly when varying the γ_s compared to the other parameters. With each parameter has its distinct features in the regenerative signal and the power spectrum, the laser parameters α, γ_d, g₀, and γ_s can be effectively extracted.

 

Fig. 4 Calculated regenerative signals (left column) and power spectra (right column) of the QD laser with different values of (a)(b) α, (c)(d) γ_d, (e)(f) g₀, and (g)(h) γ_s, respectively.

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5. Conclusion

In conclusion, we apply the FWM analysis to a QD laser for simultaneously extracting the linewidth enhancement factor and other intrinsic parameters. A model for the FWM analysis of the QD lasers is derived and validated. The linewidth enhancement factors of 0.93, 0.94, 0.95, and 1.03 at bias currents of 1.5J_th, 1.75J_th, 2J_th, and 2.25J_th are obtained with the error ranges of less than 5%, where similar values are obtained with the injection locking method using the very same QD laser. Other parameters such as γ_d, g₀, and γ_s are also effectively measured, which have the error ranges of about 15%. Unlike the injection locking technique, the parameters at higher bias levels can still be extracted with the FWM method where only weak injections are needed to generate the FWM states. While different models have to be derived for semiconductor lasers with different structures, the advantages of the FWM method shown here are generally applicable to any types of semiconductor lasers.

While the FWM method is shown to successfully extract the intrinsic parameters of the QD laser, minor discrepancies are still observed between the experimentally measured regenerative signals and power spectra from the respective fitting curves calculated with the derived model. A more complete and complex model including the effects of the carrier dynamics in the excited states, the nonlinear effects of the α, and the phonon bottleneck may reduce the discrepancies, which will be investigated in the future.