We derive a simplified rate equation model for the four-wave mixing (FWM) analysis on quantum dot (QD) semiconductor lasers subject to optical injection. The regenerative and the amplitude modulation spectra of the FWM signals with different intrinsic laser parameters and external injection conditions are investigated. By curve fitting the regenerative and the amplitude modulation spectra obtained experimentally, the intrinsic parameters of a commercial single-mode QD laser under different injection conditions are extracted. The linewidth enhancement factor α at different injection levels and detunings are shown, where a reduction of up to 39% from its free-running value is demonstrated. By increasing the injection strength, the α can be further reduced to minimized the chirp in optical communications.
© 2013 OSA
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 , 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 γd, the photon decay rate in the laser cavity γs, 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 Ep 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 NW can be expressed as: [17–20]
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 . Therefore, similar to the process shown in  and , the parameters including NW, γN, C, υg, and ε are further eliminated and the rate equations are rewritten as follows:Eq. (4) where Einj and Δinj 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 Δinj. (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. Ep = 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:
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
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
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 Er, the FWM field Ef, and the amplitude modulation σ at different detuning frequencies Δ can be obtained:
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, γs = 34.8 (ns−1), γd = 0.71 (ns−1), g = 29.8, and the injection parameters used are kinj = 1.0 (kinj = Einj/E0 is the normalized injection strength) and Δinj = 0, respectively. (Similar results are obtained with the parameters previously used in .)
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, γs, γd, and g are shown in Fig. 2. Since the regenerative and the amplitude modulation signals contain all the information of the FWM signal , 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.
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 , 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 γs 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 γd 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 Δinj 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.
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.
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 Pinj = 0 mW, 0.1 mW, 0.2 mW, and 0.3 mW at a detuning frequency Δinj= 0, respectively. Here the injection power Pinj 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 Δinj = −1.22, −0.67, 0, and 0.67 GHz with an injection power of Pinj = 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 γs, the carrier decay rates in the quantum dots γd, the gain coefficient g, and the normalized injection strength within the laser cavity kinj 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 γs, γd, and g are expected to be less than 15%, respectively .
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 Δinj = 0. Figure 7(a) shows the α of the injection-locked QD laser for different injection powers with Δinj = 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 .
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 γs, the carrier decay rates in the quantum dots γd, the gain coefficient g, and the normalized injection strength kinj 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.
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
1. C. H. Lin, H. H. Lin, and F. Y. Lin, “Four-wave mixing analysis of quantum dot semiconductor lasers for linewidth enhancement factor extraction,” Opt. Express 20, 101–110 (2012). [CrossRef] [PubMed]
2. B. Lingnau, K. Lüdge, W. W. Chow, and E. Schöll, “Failure of the α factor in describing dynamical instabilities and chaos in quantum-dot lasers,” Phys. Rev. E 86, 065201 (2012). [CrossRef]
4. S. K. Hwang and J. M. Liu, “Dynamical characteristics of an optically injected semiconductor laser,” Opt. Commun. 183, 195–205 (2000). [CrossRef]
5. Y. Okajima, S. K. Hwang, and J. M. Liu, “Experimental observation of chirp reduction in bandwidth-enhanced semiconductor lasers subject to strong optical injection,” Opt. Commun. 219, 357–364 (2003). [CrossRef]
6. B. Dagens, A. Markus, J. X. Chen, J. G. Provost, D. Make, O. L. Gouezigou, J. Landreau, A. Fiore, and B. Thedrez, “Giant linewidth enhancement factor and purely frequency modulated emission from quantum dot laser,” Electron. Lett. 41, 323–324 (2005). [CrossRef]
7. T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, and L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photon. Technol. Lett. 11, 1527–1529 (1999). [CrossRef]
8. K. Kechaou, F. Grillot, J. G. Provost, B. Thedrez, and D. Erasme, “Self-injected semiconductor distributed feedback lasers for frequency chirp stabilization,” Opt. Express 20, 26062–26074 (2012). [CrossRef] [PubMed]
9. F. Grillot, B. Dagens, J. G. Provost, H. Su, and L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-μ m InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 44, 946–951 (2008). [CrossRef]
10. J. G. Provost and F. Grillot, “Measuring the chirp and the linewidth enhancement factor of optoelectronic devices with a Mach-Zehnder interferometer,” IEEE Photon. J. 3, 476–488 (2011). [CrossRef]
11. S. Gerhard, C. Schilling, F. Gerschutz, M. Fischer, J. Koeth, I. Krestnikov, A. Kovsh, M. Kamp, S. Hofling, and A. Forchel, “Frequency-dependent linewidth enhancement factor of quantum-dot lasers,” IEEE Photon. Technol. Lett. 20, 1736–1738 (2008). [CrossRef]
12. T. Fordell and A. M. Lindberg, “Experiments on the linewidth-enhancement factor of a vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron. 43, 6–15 (2007). [CrossRef]
13. K. Iiyama, K. Hayashi, and Y. Ida, “Simple method for measuring the linewidth enhancement factor of semiconductor lasers by optical injection locking,” Opt. Lett. 17, 1128–1130 (1992). [CrossRef] [PubMed]
14. R. Hui, A. Mecozzi, A. D’ottavi, and P. Spano, “Novel measurement technique of alpha factor in DFB semiconductor lasers by injection locking,” Electron. Lett. 26, 997–998 (1990). [CrossRef]
15. I. Petitbon, P. Gallion, G. Debarge, and C. Chabran, “Locking bandwidth and relaxation oscillations of an injection-locked semiconductor laser,” IEEE J. Quantum Electron. 24, 148–154 (1988). [CrossRef]
16. J. M. Liu and T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron. 30, 957–965 (1994). [CrossRef]
17. D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett. 98, 153903 (2007). [CrossRef] [PubMed]
18. D. O’Brien, S. P. Hegarty, G. Huyet, and A. V. Uskov, “Sensitivity of quantum-dot semiconductor lasers to optical feedback,” Opt. Lett. 29, 1072–1074 (2004). [CrossRef]
20. B. Kelleher, D. Goulding, S. P. Hegarty, G. Huyet, E. A. Viktorov, and T. Erneux, “Optically injected single-mode quantum dot lasers,” in Quantum Dot Devices,Zhiming M. Wang, eds. (Springer, 2012), pp. 1–22. [CrossRef]
21. M. Sugawara, N. Hatori, H. Ebe, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Modeling room-temperature lasing spectra of 1.3-μ m self-assembled InAs/GaAs quantum-dot lasers: Homogeneous broadening of optical gain under current injection,” J. Appl. Phys. 97, 043523 (2005). [CrossRef]
22. M. Gioannini, A. Sevega, and I. Montrosset, “Simulations of differential gain and linewidth enhancement factor of quantum dot semiconductor lasers,” Opt. Quantum Electron. 38, 381–394 (2006). [CrossRef]
23. L. F. Lester, F. Grillot, N. A. Naderi, and V. Kovanis, “Differential gain enhancement in a quantum dash laser using strong optical injection,” Proc. SPIE 8619, 861907, (2013). [CrossRef]
24. N. A. Naderi, F. Grillot, V. Kovanis, and L. F. Lester, “Simultaneous low linewidth enhancement factor and high bandwidth quantum-dash injection-locked laser,” in Proceedings of IEEE Photonics Conference (IEEE, 2011), pp. 115–116.