Four-wave mixing analysis on injection-locked quantum dot semiconductor lasers

Chih-Hao Lin; Fan-Yi Lin

doi:10.1364/OE.21.021242

1. Introduction

The unique behaviors and nonlinear dynamics of quantum dot (QD) semiconductor lasers have attracted much attentions in recent years [1–3]. To faithfully predict the dynamics of a particular device, the value of each intrinsic laser parameter used in the theoretical model has to be carefully measured. Among those parameters, the linewidth enhancement factor α has been extensively studied that it determines the chirp when a laser is under modulation [4,5]. For QD devices, however, different values of α ranging from 0 to 60 have been reported [6, 7]. Moreover, it is found to vary when operated in different bias currents and under different external feedback conditions [8–10].

To measure the α, methods utilizing the FM/AM response ratio under small signal current modulation [10, 11], the linewidth measurement [12], and the injection locking [12–15] are commonly adopted. Among these methods, however, some can only extract the α but not other intrinsic laser parameters needed in the theoretical models. To extract the intrinsic parameters of QD lasers all at the same time, a method based on the four-wave mixing (FWM) analysis has been developed and demonstrated [1, 16]. By fitting the experimentally obtained FWM spectra of the regenerative and the amplitude modulation signals with the respective theoretical curves, the α and other intrinsic laser parameters used in the rate equations including the carrier decay rates in the quantum dots γ_d, the photon decay rate in the laser cavity γ_s, the differential gain g₀, the interaction cross section of the carriers in the dots with the electric field ς, and the gain saturation coefficient ε can all be simultaneously obtained.

Since the intrinsic parameters of a laser can be influenced by external perturbations, in this paper, we develop a simplified rate equation model for FWM analysis on QD lasers subject to optical injection. The effects of each intrinsic and injection parameters on the FWM spectra are theoretically studied. In experiments, the α and other intrinsic parameters of a commercial single-mode QD laser under different injection strengths and detuning frequencies are extracted. A reduction of α to below its free-running value with optical injection is demonstrated.

2. Theoretical model

For the FWM analysis, a single-mode DFB QD laser is optically-injected by a probe signal with an effective complex amplitude E_p and a detuning frequency Δ for the injection field. The time evolution of the complex amplitude of the electric field E, the occupancy probability of the energy level in quantum dots ρ, and the carrier density in the surrounding quantum wells N_W can be expressed as: [17–20]

\frac{d E}{d t} = \frac{1}{2} υ_{g} g_{0} (\frac{2 ρ - 1}{1 + ε {| E |}^{2}} - \frac{γ_{s}}{υ_{g} g_{0}}) (1 - i α) E + γ_{s} E_{p} e^{- i Δ t}

\frac{d ρ}{d t} = - γ_{d} ρ + C N_{W} (1 - ρ) - υ_{g} ς (\frac{2 ρ - 1}{1 + ε {| E |}^{2}}) {| E |}^{2}

\frac{d N_{W}}{d t} = - γ_{N} N_{W} + \frac{J}{q} - 2 C N_{W} (1 - ρ),

where γ_s is the photon decay rate in the laser cavity, γ_N and γ_d are the carrier decay rates in the quantum wells and the quantum dots, respectively, C is the capture rate from the quantum wells into the quantum dots, J is the bias current per quantum dot, ς is the interaction cross section of the carriers in the dots with the electric field, α is the linewidth enhancement factor, υ_g is the group velocity, g₀ is the differential gain, and ε is the gain saturation coefficient, respectively.

While these rate equations have already been simplified [21, 22], some of the intrinsic parameters included are found to be insensitive in the FWM analysis [1]. Therefore, similar to the process shown in [19] and [20], the parameters including N_W, γ_N, C, υ_g, and ε are further eliminated and the rate equations are rewritten as follows:

\frac{d E}{d t} = γ_{s} {\frac{1}{2} (1 - i α) [g (2 ρ - 1) - 1] E + E_{inj} + E_{p} e^{- i Δ t}} + i Δ_{inj} E

\frac{d ρ}{d t} = γ_{d} [- ρ + \tilde{J} - (2 ρ - 1) {| E |}^{2}],

where J̃ = J/(2γ_dq) is the pump and g is the gain coefficient. To study the effect of optical injection on QD lasers, an additional injection term is added to the complex electric field in Eq. (4) where E_inj and Δ_inj are the respective field amplitude and frequency of the injected light. When injection-locked, the frequency of the QD laser is locked to the frequency of the injected light at Δ_inj. (Note that, while we focus on the effects of the optical injection in this paper, similar process can be done to derive the simplified rate equation model for FWM analysis on QD lasers subject to external perturbations such as optical feedback or optoelectronic feedback.)

To derive the analytical solution, steady-state electric field and carrier density of the injection-locked QD laser are calculated before injecting the probe light (i.e. E_p = 0). In the degenerate FWM states, the electric field can be expanded approximately to the first order from the steady-state point. Thus the electric field can be expressed as the composition of the free-oscillating signal, the regenerative amplification signal, and the FWM signal:

E (t) = E_{0} + E_{r} e^{- i Δ t} + E_{f} e^{i Δ t},

where E₀, E_r, and E_f are the steady-state field amplitude of the free-oscillating signal, the complex amplitude of the regenerative amplification signal, and the complex amplitude of the FWM signal, respectively. Note that the Δ is now the frequency detuning of the probe signal from the injected light at Δ_inj.

Since the optical modulation from the beating of the fields also alters the carrier populations, the occupancy probability of the quantum dots ρ will also oscillate at the frequencies of the beat signals. To the first order, the occupancy probability can be described as

ρ (t) = ρ_{0} + ρ_{1} e^{- i Δ t} + ρ_{1}^{*} e^{i Δ t},

where ρ₀ is the steady-state occupancy probability of the quantum dots without the perturbation from the probe signal and ρ₁ is the amplitudes of the carrier fluctuation.

For the amplitude modulation σ of the complex amplitude, it is generally much smaller than the steady-state field amplitude E₀. Therefore the higher order terms can be ignored, which gives

{| E |}^{2} \approx {| E_{0} |}^{2} (1 + σ e^{- i Δ t} + σ^{*} e^{i Δ t})

By solving the steady-state solutions and substituting Eqs. (6)–(8) into Eqs. (4) and (5), the FWM spectra including the complex amplitudes of the regenerative field E_r, the FWM field E_f, and the amplitude modulation σ at different detuning frequencies Δ can be obtained:

\frac{E_{r}}{E_{0}} = - \frac{γ_{s} [(1 - i α) g ρ_{1} + E_{p} / E_{0}]}{i (Δ_{inj} + Δ) + (1 - i α) G}

\frac{E_{f}}{E_{0}} = - \frac{γ_{s} (1 - i α) g ρ_{1}^{*}}{i (Δ_{inj} - Δ) + (1 - i α) G}

σ = \frac{E_{r}}{E_{0}} + {(\frac{E_{f}}{E_{0}})}^{*},

where

ρ_{1} = \frac{E_{p} / E_{0}}{Z + W}, G = \frac{γ_{s}}{2} [g (2 ρ_{0} - 1) - 1], W = \frac{2 g [α Δ_{inj} - i Δ - (1 + α^{2}) G]}{i (Δ_{inj} - Δ) - (1 + i α) G}

Z = \frac{[2 γ_{d} (\tilde{J} - ρ_{0}) + (γ_{d} - i Δ) (2 ρ_{0} - 1)] [i (Δ_{inj} + Δ) + (1 - i α) G]}{γ_{s} γ_{d} (2 ρ_{0} - 1) (\tilde{J} - ρ_{0})}

Calculated with Eqs. (9)–(11), the FWM spectra of the regenerative signal, the FWM signal, and the amplitude modulation as a function of the detuning frequency Δ between the probe signal and the frequency of the injection-locked QD laser are plotted in Fig. 1 (blue curves). To validate the theoretical model derived, the FWM spectra obtained numerically from Eqs. (4) and (5) are also plotted for comparison (green dots). Figures 1(a), 1(c), and 1(e) and Figs. 1(b), 1(d), and 1(f) show the magnitudes of the FWM spectra for the regenerative signals, the FWM signals, and the amplitude modulation of a QD laser without and with injection, respectively. The laser parameters used are α = 1.37, γ_s = 34.8 (ns⁻¹), γ_d = 0.71 (ns⁻¹), g = 29.8, and the injection parameters used are k_inj = 1.0 (k_inj = E_inj/E₀ is the normalized injection strength) and Δ_inj = 0, respectively. (Similar results are obtained with the parameters previously used in [1].)

Fig. 1 (a)–(c) FWM spectra of regenerative, FWM, and amplitude modulation signals without optical injection, respectively. (d)–(f) FWM spectra of regenerative, FWM, and amplitude modulation signals with optical injection, respectively. The green dots are from the numerical simulations and the blue curves are from the analytical solutions.

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As can be seen, the results derived from the analytical model match well with those obtained from the numerical simulations for both the cases with or without injection. As the result, the analytical model derived can be used to obtain the intrinsic laser parameters of an optically-injected QD laser through curve fitting the FWM spectra.

3. Characteristics of the FWM spectra of a QD laser subject to optical injection

To investigate the effects of different intrinsic parameters on a QD laser subject to optical injection, the FWM spectra obtained from Eqs. (9)–(11) with different α, γ_s, γ_d, and g are shown in Fig. 2. Since the regenerative and the amplitude modulation signals contain all the information of the FWM signal [16], only the spectra of the regenerative and the amplitude modulation signals are shown. The blue curves in each plot are obtained with the same parameters as those used in Fig. 1 to serve as the benchmark. The values of the variable parameters used in each plot are arbitrary chosen to show the effect of each parameters on the FWM spectra.

Fig. 2 Calculated spectra of the (a)–(d) regenerative and (e)–(h) amplitude modulation signals of the QD laser with different α, γ_s, γ_d, and g, respectively.

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To investigate the influences of each parameter on the FWM spectra qualitatively, we focus on the magnitudes and the detuning frequencies of the characteristic peaks and valleys (labeled with green arrows in the following plots) when the parameters are varied. As can be seen in Figs. 2(a) and 2(e), while in a solitary QD laser the α only alters the depth of the valley in the regenerative spectrum [1], the α alters both the depths of the valleys in the regenerative and amplitude modulation spectra for a QD laser subject to optical injection. When γ_s increases, as shown in Figs. 2(b) and 2(f), the valleys in the spectra become deeper and shift toward higher detuning frequency. Meanwhile, the peaks in the spectra of the amplitude modulation are shifted away from zero detuning. Figures 2(c) and 2(g) show the spectra when γ_d increases. As can be seen, the valleys and the peaks are shifted away from zero detuning while the valleys become shallower and the peaks reduce in their magnitudes. For the effect of the gain coefficient g, similar behaviors are observed for the spectra of the regenerative and amplitude modulation signals shown in Figs. 2(d) and 2(h) as those shown in Figs. 2(b) and 2(g), respectively. Moreover, the peaks in the spectra of the amplitude modulation become more symmetric as the g increases.

Figure 3 shows the effects of the optical injection on the spectra of the regenerative and amplitude modulation signals. As can be seen in Fig. 3(a), the valleys in the spectra of the regenerative signals become shallower and shift toward higher detuning frequency as the injection strength k₀ increases. At the same time, the peaks in the spectra of the amplitude modulation shown in Fig. 3(c) become more asymmetric. On the other hand, increasing the detuning frequency of the injected light Δ_inj makes the valleys deeper in both the spectra of the regenerative and amplitude modulation signals as shown in Figs. 3(b) and 3(d), respectively. Also, the peaks in the spectra of the amplitude modulation become more asymmetric. With the distinct features of each parameter on the characteristic peaks and valleys shown in Figs. 2 and 3, the parameters of a QD laser subject to optical injection are expected to be extracted effectively by curve fitting with the analytical model.

Fig. 3 Calculated spectra of the (a)–(b) regenerative (c)–(d) and amplitude modulation signals of the QD laser with different k_inj and Δ_inj, respectively.

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4. Experimental setup

Figure 4 shows the schematic setup of the FWM analysis to extract the parameters of a QD laser subject to optical injection. A commercial single-mode DFB QD laser (LD)(QDLaser QLD 1334) with a threshold current of 8.75 mA and a wavelength of about 1296 nm is used for laser parameter characterization. When biasing at 20 mA, the output power is about 1.6 mW. Two light sources are used to optically inject the QD laser, where one (TL1)(NTT Electronics NLK1352SGC-L) is used as the master laser for injection locking and the other (TL2)(Yenista Tunics T100S-O) is used as the probe for the FWM analysis. The injections to the QD laser is through a free-space optical circulator formed by a polarizing beam splitter (PBS), a half-wave plate (HW2), and a Faraday rotator (FR). The amplitude modulation of the QD laser is detected by a photodiode (PD) with 12 GHz frequency response (NewFocus 1554-A) and resolved with a 26.5 GHz spectrum analyzer (Agilent E4407B). The regenerative signal is measured by heterodyning the QD laser output with part of the TL2 output at the PD, where the TL2 output is shifted by about 100 MHz with an acousto-optic modulator (AOM)(IntraAction ACM-1002AA1) to improve the signal to noise ratio.

Fig. 4 Schematic setup of the FWM analysis. TL: tunable laser; LD: QD laser diode; FR: Faraday rotator; HW: half-wave plate; PBS: polarizing beamsplitter; PD: photodiode; SA: spectrum analyzer; ATT: variable attenuator; AOM: acousto-optic modulator; PC: polarization controller; FC: 50/50 fiber coupler.

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5. Experimental results

Figures 5(a)–5(d) and 5(e)–5(h) show the spectra of the regenerative and amplitude modulation signals of the injection-locked QD laser (red dots) with injection power of P_inj = 0 mW, 0.1 mW, 0.2 mW, and 0.3 mW at a detuning frequency Δ_inj= 0, respectively. Here the injection power P_inj is measured before the injection light being coupled into the QD laser. Figures 6(a)–6(d) and 6(e)–6(h) show the respective spectra at detuning frequencies of Δ_inj = −1.22, −0.67, 0, and 0.67 GHz with an injection power of P_inj = 0.15 mW. By the least squares fitting with the analytically derived curves from Eqs. (9)–(11) (blue curves), the laser parameters including the linewidth enhancement factor α, the photon decay rate γ_s, the carrier decay rates in the quantum dots γ_d, the gain coefficient g, and the normalized injection strength within the laser cavity k_inj are extracted and listed in Table 1. Note that, from the previous study of the FWM analysis on QD lasers, the error ranges of the α extracted with this method is expected to be less than 5% while the γ_s, γ_d, and g are expected to be less than 15%, respectively [1].

Fig. 5 Experimentally obtained spectra of (a)–(d) regenerative and (e)–(h) amplitude modulation signals of the injection-locked QD laser with different injection powers (red dots). Blue curves are the least square fitting of the experimental data.

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Fig. 6 Experimentally obtained spectra of (a)–(d) regenerative and (e)–(h) amplitude modulation signals of the injection-locked QD laser with different detuning frequencies (red dots). Blue curves are the least square fitting of the experimental data.

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Table 1. The extracted parameters of the QD laser for different injection powers P_inj and detuning frequencies Δ_inj.

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When increasing the injection power, as shown in Fig. 5, the characteristic valleys in the spectra of the regenerative signal become shallower and the peaks in the spectra of the amplitude modulation become more asymmetric. When increasing the detuning frequency, as shown in Fig. 6, the valleys in the spectra of the regenerative signal become deeper and the peaks in the spectra of the amplitude modulation become more asymmetric. These trends of the FWM spectra shown in Figs. 5 and 6 are qualitatively the same as those predicted in the theoretical investigation as shown in Fig. 3.

From Table 1, it is apparent that the α decreases as the injection power increases. Moreover, it has a minimum value at around Δ_inj = 0. Figure 7(a) shows the α of the injection-locked QD laser for different injection powers with Δ_inj = 0. As can be seen, the α decreases distinctly as the injection power increases. A minimum α of 0.84 is observed at an injection power of 0.9 mW, which is reduced by 39% from its free-running value. (Note that similar behavior of the reduction of the α in optically-injected Fabry-Pérot quantum dash lasers at zero detuning has also been reported [23, 24].) To investigate the reduction of the α under different detuning frequencies, Fig. 7(b) shows the α of the QD laser with different injection conditions in the stable locking region (bounded by the green curves). As can be seen, while the α in general decreases as the injection power increases, local minima occur around zero detuning under different injection powers. To the best of our knowledge, this variations of the α for the single-mode DFB QD laser under different injection powers and detuning frequencies are experimentally demonstrated the first time. The behaviors shown in Fig. 7 qualitatively agree with those previously reported in the theoretical study [2].

Fig. 7 Extracted linewidth enhancement factor α of the injection-locked QD laser with (a) different injection powers at Δ_inj = 0 and (b) different injection condition within the locking boundaries.

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6. Conclusion

We have successfully applied the FWM analysis on an optically-injected QD laser for laser parameter extraction. A theoretical model for the FWM analysis is derived, which is verified with the numerical simulations for its validity. By curve fitting the experimentally obtained FWM spectra with the analytic ones, the laser parameters including the linewidth enhancement factor α, the photon decay rate γ_s, the carrier decay rates in the quantum dots γ_d, the gain coefficient g, and the normalized injection strength k_inj are successfully extracted. To the best of our knowledge, the variations of the α in an injection-locked single-mode DFB QD laser at different injection powers and detuning frequencies are experimentally demonstrated the first time. A reduction of α up to 39% from its free-running value is shown, which is expected to reduce the chirp in optical communications for long distance transmission. While in this paper we focus on the effects of the optical injection, similar process can be done to derive the simplified rate equation model for FWM analysis on QD lasers subject to external perturbations such as optical feedback or optoelectronic feedback. How the external feedback affects the laser parameters will be next studied.

Acknowledgments

This study was funded by the National Science Council of Taiwan under contract NSC 100-2112-M-007-012-MY3 and by the National Tsing Hua University under grant 102N2081E1.

References and links

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P_inj (mW)	Δ_inj (GHz)	k_inj	α	γ_s (ns⁻¹)	γ_d (ns⁻¹)	g
0	0	0	1.37	34.8	0.71	29.8
0.1	0	0.79	1.09	41.0	0.75	31.2
0.2	0	1.17	1.09	43.3	0.65	30.6
0.3	0	1.43	1.06	46.2	0.56	33.7
0.15	0.67	1.01	1.30	31.4	1.34	16.8
0.15	0	0.97	1.09	39.8	0.73	28.5
0.15	−0.67	0.95	1.22	45.3	0.56	33.8
0.15	−1.22	0.96	1.24	47.5	0.41	43.7

Four-wave mixing analysis on injection-locked quantum dot semiconductor lasers

Abstract

1. Introduction

2. Theoretical model

3. Characteristics of the FWM spectra of a QD laser subject to optical injection

4. Experimental setup

5. Experimental results

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (13)

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