Open Access
Add to BibSonomyAbstract
We investigate the spectral linewidth of a monolithic photonic crystal nanocavity laser. The nanocavity laser is based on a buried heterostructure cavity in which an ultra-small InGaAsP active region is embedded in an InP photonic crystal. Although it was difficult to achieve narrow linewidth operation in previously reported photonic crystal nanocavity lasers, we have successfully demonstrated a linewidth of 143.5 MHz, which is far narrower than the cold cavity linewidth and the narrowest value yet reported for nanolasers and photonic crystal lasers. The narrow linewidth is accompanied by a low power consumption and an ultrasmall footprint, thus making this particular laser especially suitable for use as an integrated multi-purpose sensor.
©2012 Optical Society of America
1. Introduction
Optical micro/nano cavities with a high quality (Q) factor have recently been extensively studied for efficient and fast optical light sources [1–3]. Among the various micro/nano cavities, photonic crystal (PhC) nanocavities are especially attractive because they have a high Q factor with a very small effective cavity mode volume (Veff) around a cubic wavelength. It has been well established that Q/Veff, which represents the degree of light-matter interaction, is very large for PhC nanocavities. Since this property is advantageous when constructing small lasers, there have been a number of studies of PhC-nanocavity-based lasers that have demonstrated tiny size and ultralow threshold power [4,5].
One of the important properties of laser is the spectral purity of the lasing mode, namely, linewidth. Linewidth of semiconductor laser is affected by spontaneous emission and also by coupling between the optical gain and the refractive index [6]. Narrow linewidth lasers are strongly demanded in the area of optical sensor, frequency metrology, and coherent optical communication. Tremendous efforts have been made to achieve a narrow linewidth from semiconductor-based lasers for such purposes [7–12]. In terms of applications, it is very important to integrate a number of narrow linewidth lasers in a single chip. For example, if we are able to achieve the dense integration of a vast number of ultrasmall narrow linewidth lasers with different lasing wavelengths in a single chip with an appropriate “laboratory on a chip” setup, we can realize a very efficient multi-purpose sensor. In fact, PhC cavity based platforms become potential “laboratory on a chip” devices for robust biomedical and chemical sensors owing to their compactness with integration capabilities [13–16]. The sensitivity of the sensors relies on a refractive index change induced by an analyte, and this index change is determined by a resonance wavelength shift. The resolution of the sensors is directly related to the spectral linewidth since a narrow linewidth enables the sensors to detect a very small shift in the resonance wavelength. So PhC based nanolasers with a narrow linewidth should be realized for ultra-compact and high-resolution sensor applications.
However, conventional narrow linewidth lasers have substantially long cavities and require fairly high power operation [12]. Since, as explained later, the laser linewidth is strongly related to the cavity size and operating power, it will be particularly challenging to realize narrow linewidth from small size lasers such as micro and nanolasers, although large a Q/Veff ratio for PhC nanocavities is still very advantageous in this context.
In addition, there is another difficulty as regards achieving a narrow linewidth in PhC nanolasers. Although various PhC nanolasers have been developed and demonstrated, continuous wave lasing is still difficult to achieve and its performance is very limited [4, 17, 18]. These lasers suffer severely from heating, and therefore it is only possible to lase at very close to the lasing threshold. This is a serious headache because lasers should operate far above the threshold in order to achieve a narrow linewidth. The reason for this is related to the conventional structure of PhC nanocavities. PhC lasers were created by making vertical holes in a thin membrane of a gain medium, e.g., InGaAsP. The gain medium has poor thermal conductivity and moreover is suspended in air. The distinctive structures adopted to realize a high quality factor have left us with a thermal issue. PhC lasers experience a heating just after the threshold and eventually reach a thermal rollover not far from the threshold. The weak output power of PhC lasers hindered by the near threshold thermal rollover is obviously a fundamental obstacle with respect to realizing a narrow linewidth. The weak output power also leads to ambiguity as regards the PhC laser’s linewidth because it constrains linewidth measurement methods. Linewidth measurement methods such as optical heterodyne or homodyne methods provide extremely high resolution but require a certain optical power level due to the finite responsivity of the photodiodes in the setup. Another issue with the PhC laser is that there are no built-in structures to isolate generated carriers and photons. Optical pumping has normally been achieved by focusing light on the top of nanocavities, and the beam areas are much wider than the nanocavity area. The generated carrier and photons outside the nanocavities can couple into the nanocavities and this phenomenon eventually adds additional phase and amplitude noise to the lased nanocavity mode.
As far as we know, there have been no quantitative studies of the linewidth of PhC nanolasers with a mode volume of approximately 1 (λ/n)3 except near the threshold. In the previous studies, the laser linewidth near the threshold was comparable to the cold cavity linewidth (several GHz). Linewdith measurement has been reported for a relatively large PhC cavity laser bonded on a sapphire substrate, which serves as an effective heatsink but the observed width is 1.5 GHz, which is not so different from the above value [18]. Their result seems to be limited by the relatively high lasing threshold.
In this paper, we aim to achieve a sub-GHz linewidth from a PhC nanocavity laser with the goal of realizing PhC based high resolution sensors. To achieve this goal, we adopt PhC lasers based on a buried heterostructure (BH) cavity [19]. The structure is shown schematically in Fig. 1 . In this design, a tiny gain material (InGaAsP quantum wells (QWs)) is embedded in a line defect of an InP PhC slab. A high-Q cavity mode is formed by the refractive index modulation of the PhC line defect by the presence of InGaAsP [20]. Owing to the large Q/Veff ratio and strong carrier confinement in the gain region, we can achieve an extremely low lasing threshold below 10 μW. In addition, the heating problem can be substantially reduced because InP has 17 times the heat conductivity of InGaAsP, which means that lasers can operate far above the threshold. The strong carrier confinement is also effective for a narrow linewidth because it reduces the extra phase noise of the nanocavity mode caused by carrier interaction between inside and outside the nanocavity. Therefore, the problems of obtaining a narrow linewidth with previous PhC lasers can be mostly solved in this particular laser. The detailed design of the BH PhC laser including the PhC cavity and active gain medium is discussed in the next section.
Fig. 1 A schematic diagram and simulated magnetic field profile for the BH PhC laser. An ultra-small active region is buried in the line defect waveguide, and the quality factor is controlled by controlling the offset distance from the edge of the output waveguide
2. Design of PhC nanocavity laser
A BH PhC nanolaser was designed and fabricated for the experiment. The designed lattice constant, a, is 438nm and the air-hole radius is 100 nm. The line-defect input waveguide is located along the Γ-K direction with a width of 0.98 W0 where W0 is a. The ultra-small active region (3.5 × 0.3 × 0.16 μm3) consisting of InGaAs/InGaAsP three QWs is buried inside the line-defect input waveguide. The photoluminescence peaks of the InGaAs QWs and InGaAsP barriers are around 1.55 and 1.35 μm, respectively.
The local and small refractive index modulation introduced by the buried active region provides 3D light confinement and a high Q nanocavity with a small effective mode volume. The calculated effective mode index (neq = 2.73) in the active region is slightly larger than that of the surrounding InP layer (neq = 2.509). A line-defect output waveguide is created along the Γ-K direction by removing some of the fourth row air holes from the line defect input waveguide. The output waveguide is designed to overlap with the nanocavity mode extending in the Γ-M direction. The output waveguide has a width of 1.1 W0 to couple the cavity modes into the output waveguide as a result of a band edge shift of the mode gap. The BH PhC laser is shown schematically in Fig. 1.
The Q factor of the BH PhC laser is controlled by the coupling strength between an optical nanocavity mode and the output waveguide. The quality factor of the designed BH PhC laser without the output waveguide is estimated to be around 1 × 106 using a finite-difference time-domain (FDTD) simulation. The calculated effective modal volume is 1.09 (λ/n)3 and the corresponding TE-like mode is shown in Fig. 1.
3. Experiment
The fabricated BH PhC laser is optically pumped with 1.3 μm light through an input waveguide to achieve continuous wave lasing at room temperature. No antireflection coating was applied to the input facet while the output facet had an antireflection coating. For the input and output waveguide sides, the overall losses from the fiber to the cavity were experimentally estimated to be 10 and 8.5 dB, respectively. Continuous wave lasing around 1.55 μm was achieved from the nanocavity laser. Some of the total output power of the lased nanocavity mode was coupled to the output waveguide and used for measurements. Light coupling into or from the facet of the BH PhC laser was achieved through polarization maintained fiber collimators.
The BH PhC laser was optically excited with different pump powers, and the corresponding linewidth of the fundamental mode was measured in terms of the output power and input power. The spectral linewidth at far above the threshold was measured using a delayed self-homodyne method [21]. An erbium-doped fiber amplifier (EDFA) was used to amplify the signal because the output power collected from the BH PhC laser was too low for the linewidth to be measured. The noise figure of the EDFA was less than 4.3 dB for a −20 dBm input signal. A tunable bandpass filter was used to suppress amplified spontaneous emission beat noise, and the full-width at half-maximum of the filter was 0.3nm. The amplified signal after the band pass filter was split into an unbalanced fiber interferometer and then combined. The interference of the combined two fields was detected at a photodetector with a 3.5 GHz bandwidth. For incoherent interference between the signal and the delayed signal, a 10 km long optical fiber was used in the delayed arm of the interferometer. The photocurrent spectrum that resulted from the beating between the two fields was analyzed using an RF spectrum analyzer with a resolution bandwidth of 100 kHz. Since linewidth of the BH PhC laser was measured by the self-homodyne method, the photocurrent power spectrum was broadened by the frequency jitter or 1/f noise of the BH PhC laser [22]. The optical delayed self-homodyne measurement setup is shown in Fig. 2 .
Fig. 2 Optical delayed self-homodyne measurement setup. VOA: variable optical attenuator, I: isolator, BPF: band pass filter.
Light-in light-out (L-L) curve near the threshold is shown in Fig. 3(a) . The optical powers in the L-L curve are the estimated powers in the input and output waveguides. The lasing behavior of the BH PhC laser is clearly confirmed by the threshold shown in the L-L curve. The estimated threshold power in the input waveguide is 2.5 μW and the optical spectrum at the threshold is shown in the inset, which exhibits a clear resonance peak at 1560.2 nm. The linewidth measured at the threshold using an optical spectrum analyzer is 0.04 nm and this corresponds to a frequency of 4.9 GHz. Note that this value is not limited by the spectral resolution since the resolution of the spectrometer is 0.01 nm. Figure 3(b) shows the linewidth versus the input power around the threshold. These results are close to our previously reported results for a BH-PhC laser [19]. From this measurement, we can estimate the cold cavity quality factor (Q) at the transparency condition to be around 40,000.
Fig. 3 (a) “L-L curve” near threshold. The inset shows the optical spectrum at the threshold. (b) Spectral linewidth versus input power. The input and output powers are estimated powers in the input and output waveguides (WG), respectively.
Next, we investigated the linewidth far above the threshold. The L-L curve far above the threshold is shown in Fig. 4(a) . The optical spectra in Fig. 4(b) were obtained for input powers of over 100 μW, which is 40 times above the threshold. Under such a condition, the BH PhC laser achieved lasing via a fundamental resonant mode at 1560.8 nm, and higher order modes (1542 nm and 1554 nm) were suppressed over 30 dB even though the pump power was close to the saturation power. Note that this highly pumped condition was hard to achieve in previous nanolasers except for BH-PhC lasers. As the pumping power increased, the linewidth became narrower than our spectral resolution. Therefore, we employed the optical delayed self-homodyne method for the linewidth measurement. Figure 4(c) shows the measured linewidth as a function of the input power. Apparently, the linewidth of the fundamental mode decreased as the input power increased. It should be emphasized that the BH PhC nanolaser achieved a linewidth of ~150 MHz with an extremely small input power of 66 μW. Figure 4(d) shows the result for the narrowest linewidth we observed. For a given input power of 81 μW, we achieved a linewidth of 143.5 MHz with a collected output power of 6.5 µW. To the best of our knowledge, this linewidth is the narrowest ever achieved in nanocavity lasers and PhC-based lasers.
Fig. 4 (a) “L-L curve” far above the threshold. (b) Optical spectrum when the input power was over 100 μW in the input waveguide. The inset shows the optical spectrum of the fundamental mode. (c) Spectral linewidth vs. the input power in the input waveguide. (d) Photocurrent spectrum of the narrowest linewidth. The same color of square is used to denote the same input power in “L-L curve” far above threshold, “linewidth vs. input power”, and “photocurrent power spectrum”.
It is well known that the laser linewidth is described by the Schawlow-Townes formula as shown in Eq. (1),
where Δνc is the cold cavity bandwidth, N1 and N2 are the carrier populations for the ground and excited levels, respectively, and Pe is the total power emitted by the stimulated emission [23]. In semiconductor gain materials, the laser linewidth is normally (1 + α2) times wider than this formula, where α is the linewidth enhancement factor. The α value of BH PhC laser is estimated around 3.9 using an injection locking method [24,25]. Since the last term of Eq. (1) is almost constant except under a very high pumping condition, the laser linewidth is inversely proportional to the output power, which means that the linewidth-power product is constant. Figure 5 shows the linewidth versus the inverse of the output power. We clearly observed the trend predicted by Eq. (1) and found that this product was 348 MHz μW in our case. Under a high pumping condition, the linewidth is saturated around 150 MHz. In a simple two level system, under a high pumping condition, N2 becomes almost proportional to Pe, although N2 - N1 is constant, which means that Δν approaches a certain residual value independent of Pe as the pumping power is increased. This residual linewidth corresponds to 72.2 MHz in our case. Note that the linewidth of our laser is saturated at a much larger linewidth of around 150 MHz.Similar linewidth saturation has been encountered for various semiconductor lasers [6,26]. Several mechanisms have been suggested to explain the origin of the saturation. They are spatial-hole burning, side-mode interaction, cross saturation, and 1/f-noise [27–30]. Since the BH PhC nanocavity laser has an ultra-small active volume and a side mode suppression of over 30 dB, the linewidth saturation observed in our laser must be related to the cross saturation and the 1/f-type FM noise rather than the spatial-hole burning and side-mode interaction. The 1/f FM noise can be induced by the carrier mobility fluctuation and the nonradiative recombination current at the BH interface [31,32]. The cross saturation of the side mode suppressed even by 20 dB can enhance the low frequency noise of the main mode. Therefore, single frequency BH PhC lasers with a reduced nonradiative recombination current would be potential candidates for realizing a sub-100 MHz linewidth.
Here we compare our laser with vertical-cavity surface-emitting lasers (VCSELs) in terms of linewidth, input power, and mode volume. The comparison is shown in Table 1 . Although VCSELs are much smaller and may be possible to integrate, the linewidth is marginal. If we look at Eq. (1), it is clear that if we want to achieve a narrower linewidth, we should increase the photon number and the cavity photon lifetime. In other words, we should increase the pumping power and the cavity size. This means that it is challenging to reduce the linewidth while simultaneously maintaing a small device size and low power consumption. In fact, our laser is advantageous in this context since it can achieve a fairly narrow linewidth in spite of its extremely small footprint and power consumption. We list the input power and cavity mode volume for a BH PhC laser and micro lasers in Table 1, which shows that the volume and power of BH PhC lasers are orders of magnitude smaller than those of other types of lasers. To clarify this point, we introduce two new products using input power and mode volume. The two products are “input power (Pin) × minimum linewidth (Δνmin)” and “mode volume (Vmode) × minimum linewidth (Δνmin)”. Readers may be puzzled by this definition because the product of “output power (Pout) × linewidth” is conventionally used to evaluate a laser’s linewidth. This conventional definition leads to 0.347 MHz mW for our laser, which is extremely small. However, this value is misleading because the observed output power is only a fraction of the total lasing power due to the small external quantum efficiency (this is generally the case for most nanolasers). In the present nanolaser case, the input power is a more appropriate value for the comparison than the output power. This is easily seen by modifying Eq. (1) as follows,
where Pin is the input power, Pth is the threshold power, and η is the internal quantum efficiency of the laser. Note that we can expect an internal quantum efficiency of around unity and the last term does not depend on the external quantum efficiency. This justifies the above definition of the new product. In Table 1, the input power is electrical or optical power applied to the lasers. The mode volume of the VCSELs is estimated based on their cavity length and aperture diameter [33]. Since there is a large ambiguity in estimating the electrical power and the mode size for lasers in the literature, this comparison is far from rigorous, but we still regard it as valuable for discussing orders-of-magnitude differences.
Table 1. Linewidth and Related Parameters of Monolithic Lasers Emitting at 1.5 μm
The minimum linewidth of the VCSELs are one order of magnitude smaller than that of the BH PhC laser. But it is clear from “Pin × Δνmin” that the VCSELs required an input power two or three orders of magnitude higher than that of the BH PhC laser for a one order of magnitude smaller linewidth. The “Vmode × Δνmin” values of the VCSELs are one or two orders of magnitude larger than that of the BH PhC lasers. Hence, BH-PhC lasers are superior to VCSELs in terms of micro narrow linewidth lasers.
Finally, in Table 1, the BH PhC laser is also compared with distributed feedback (DFB) lasers which can exhibit very narrow linewidth of around several kHz even though they are not suited for the integrated sensor applications because the footprint is fairly large (typically longer than 1mm). This comparison is intended to clarify the general performance of the BH PhC laser among narrow-linewidth monolithic lasers besides integrated applications. Although we cannot determine an appropriate Pin value for DFB lasers, the product of Pin × Δνmin for DFB lasers is likely to be smaller than that for BH-PhC lasers since Pin should be an order of 100 mW. Thus, in terms of the power efficiency needed for generating a narrow linewidth, DFB lasers are still better than BH-PhC lasers. This is mostly due to the fact that nanolasers are operating at a condition close to linewidth saturation, although large DFB lasers are operating far from this condition. Of course, the dimensions of DFB lasers prevent large-scale integration such as integrated sensors, as we noted before.
Consequently, BH PhC lasers are the most advantageous lasers for the large-scale integration of narrow linewidth lasers, which should be useful as high performance sensors with a small power consumption.
4. Summary
We have undertaken a series of studies on the linewidth of the BH-PhC nanocavity laser. Such studies had been difficult to conduct on previous nanocavity lasers because they are hard to operate under a highly pumped condition (which is required for obtaining a narrow linewidth) due to the heating problem. The unique design of the BH-PhC laser enabled us to observe a narrow linewidth under a highly pumped condition far above the threshold. The minimum linewidth was as narrow as 143.5 MHz, which is the narrowest linewidth reported for any nanocavity lasers and PhC lasers. Since our laser is small and has a small power consumption, it is suitable for use as an integrated narrow linewidth laser. Although VCSELs have been regarded the most suitable devices for such purposes, we have clarified that BH-PhC lasers are more advantageous in comparison with VCSELs by examining two representative products (Pin × Δνmin and Vmode × Δνmin) for narrow linewidth lasers.
Acknowledgment
Part of this work was supported by the New Energy and Industrial Technology Development Organization (NEDO).
References and links
1. K. Inoue and K. Ohtaka, eds., Photonic Crystal: Physics, Fabrication and Applications (Springer, 2004).
2. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432(7014), 200–203 (2004). [CrossRef] [PubMed]
3. K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003). [CrossRef] [PubMed]
4. K. Nozaki, S. Kita, and T. Baba, “Room temperature continuous wave operation and controlled spontaneous emission in ultrasmall photonic crystal nanolaser,” Opt. Express 15(12), 7506–7514 (2007). [CrossRef] [PubMed]
5. T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Top. Quantum Electron. 3(3), 808–830 (1997). [CrossRef]
6. G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993), Chap. 6.
7. P. Signoret, F. Marin, S. Viciani, G. Belleville, M. Myara, J. P. Tourrenc, B. Orsal, A. Plais, F. Gaborit, and J. Jacquet, “3.6-MHz linewidth 1.55- μm monomode vertical-cavity surface-emitting laser,” IEEE Photon. Technol. Lett. 13(4), 269–271 (2001). [CrossRef]
8. R. Shau, H. Halbritter, F. Riemenschneider, M. Ortsiefer, J. Rosskopf, G. Böhm, M. Maute, P. Meissner, and M.-C. Amann, “Linewidth of InP-based 1.55 μm VCSELs with buried tunnel junction,” Electron. Lett. 39(24), 1728–1729 (2003). [CrossRef]
9. N. M. Margalit, J. Piprek, S. Zhang, D. I. Babic, K. Streubel, R. P. Mirin, J. R. Wesselmann, J. E. Bowers, and E. L. Hu, “64 °C continuous-wave operation of 1.5 μm vertical-cavity laser,” IEEE J. Sel. Top. Quantum Electron. 3(2), 359–365 (1997). [CrossRef]
10. M. Okai, M. Suzuki, and T. Taniwatari, “Strained multiquantum-well corrugation-pitch-modulated distributed feeback laser with ultranarrow (3.6 kHz) spectral linewidth,” Electron. Lett. 29(19), 1696–1697 (1993). [CrossRef]
11. H. Bissessur, C. Starck, J.-Y. Emery, F. Pommereau, C. Duchemin, J.-G. Provost, J.-L. Beylat, and B. Fernier, “Very narrow-linewdith (70 kHz) 1.55 μm strained MQW DFB lasers,” Electron. Lett. 28(11), 998–999 (1992). [CrossRef]
12. W. Loh, F. J. O’Donnell, J. J. Plant, M. A. Brattain, L. J. Missaggia, and P. W. Juodawlkis, “Packaged, high-power, narrow-linewidth slab-coupled optical waveguide external cavity laser (SCOWECL),” IEEE Photon. Technol. Lett. 23(14), 974–976 (2011). [CrossRef]
13. E. Chow, A. Grot, L. W. Mirkarimi, M. Sigalas, and G. Girolami, “Ultracompact biochemical sensor built with two-dimensional photonic crystal microcavity,” Opt. Lett. 29(10), 1093–1095 (2004). [CrossRef] [PubMed]
14. D. Dorfner, T. Zabel, T. Hürlimann, N. Hauke, L. Frandsen, U. Rant, G. Abstreiter, and J. Finley, “Photonic crystal nanostructures for optical biosensing applications,” Biosens. Bioelectron. 24(12), 3688–3692 (2009). [CrossRef] [PubMed]
15. M. Lončar, A. Scherer, and Y. Qiu, “Photonic crystal laser sources for chemical detection,” Appl. Phys. Lett. 82(26), 4648–4650 (2003). [CrossRef]
16. M. L. Adams, M. Lončar, A. Scherer, and Y. Qiu, “Microfluidic integration of porous photonic crystal nanolasers for chemical sensing,” IEEE J. Sel. Areas Comm. 23(7), 1348–1354 (2005). [CrossRef]
17. M. Nomura, S. Iwamoto, K. Watanabe, N. Kumagai, Y. Nakata, S. Ishida, and Y. Arakawa, “Room temperature continuous-wave lasing in photonic crystal nanocavity,” Opt. Express 14(13), 6308–6315 (2006). [CrossRef] [PubMed]
18. M. Bagheri, M. H. Shih, Z.-J. Wei, S. J. Choi, J. D. O’Brien, P. D. Dapkus, and W. K. Marshall, “Linewidth and modulation response of two-dimensional microcavity photonic crystal lattice defect lasers,” IEEE Photon. Technol. Lett. 18(10), 1161–1163 (2006). [CrossRef]
19. S. Matsuo, A. Shinya, T. Kakitsuka, K. Nozaki, T. Segawa, T. Sato, Y. Kawaguchi, and M. Notomi, “High-speed ultracompact buried heterostructure photonic-crystal laser with 13 fJ of energy consumed per bit transmitted,” Nat. Photonics 4(9), 648–654 (2010). [CrossRef]
20. M. Notomi and H. Taniyama, “On-demand ultrahigh-Q cavity formation and photon pinning via dynamic waveguide tuning,” Opt. Express 16(23), 18657–18666 (2008). [CrossRef] [PubMed]
21. T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980). [CrossRef]
22. J.-P. Tourrenc, P. Signoret, M. Myara, M. Bellon, J.-P. Perez, J.-M. Gosalbes, R. Alabedra, and B. Orsal, “Low-frequency FM-noise-induced lineshape: a theoretical and experimental approach,” IEEE J. Quantum Electron. 41(4), 549–553 (2005). [CrossRef]
23. A. Yariv, Quantum Electronics, 3rd ed. (John Wiley & Sons Inc., New York, 1989), Chap. 9.
24. G. Liu, X. Jin, and S. L. Chuang, “Measurement of linewidth enhancement factor of semiconductor lasers using an injection-locking technique,” IEEE Photon. Technol. Lett. 13(5), 430–432 (2001). [CrossRef]
25. J. Kim and P. J. Delfyett, “Above threshold spectral dependence of linewidth enhancement factor, optical duration and linear chirp of quantum dot lasers,” Opt. Express 17(25), 22566–22570 (2009). [CrossRef] [PubMed]
26. Y. Yamamoto, ed., Coherence, Amplification, and Quantum Effects in Semiconductor Lasers (John Wiley & Sons, Inc., 1991), Chap. 2.
27. H. Yasaka, M. Fukuda, and T. Ikegami, “Current tailoring for lowering linewidth floor,” Electron. Lett. 24(12), 760–761 (1988). [CrossRef]
28. U. Krüger and K. Petermann, “Dependence of the linewidth of a semiconductor laser on the mode distribution,” IEEE J. Quantum Electron. 26(12), 2058–2064 (1990). [CrossRef]
29. G. R. Gray and G. P. Agrawal, “Effect of cross saturation on frequency fluctuations in a nearly single-mode semiconductor laser,” IEEE Photon. Technol. Lett. 3(3), 204–206 (1991). [CrossRef]
30. K. Kikuchi, “Effect of 1/f-Type FM noise on semiconductor-laser linewidth residual in high-power limit,” IEEE J. Quantum Electron. 25(4), 684–688 (1989). [CrossRef]
31. M. Fukuda, T. Hirono, T. Kurosaki, and F. Kano, “1/f noise behavior in semiconductor laser degradation,” IEEE Photon. Technol. Lett. 5(10), 1165–1167 (1993). [CrossRef]
32. F. N. Hooge, “1/f noise sources,” IEEE Trans. Electron. Dev. 41(11), 1926–1935 (1994). [CrossRef]
33. A. Mutig, High Speed VCSELs for Optical Interconnects (Springer Thesis, 2011), Chap. 2.
34. J. Shimizu, H. Yamada, S. Murata, A. Tomita, M. Kitamura, and A. Suzuki, “Optical-confinement-factor dependencies of the K factor, differential gain, and nonlinear gain coefficient for 1.55 µm InGaAs/InGaAsP MQW and strained-MQW lasers,” IEEE Photon. Technol. Lett. 3(9), 773–776 (1991). [CrossRef]
