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We investigate the spectral properties of violet 405 nm (Al,In)GaN laser diodes (LDs). Depending on the substrate the LDs are grown on, the lasing spectra show significant differences. LDs grown on low dislocation GaN substrate have a broad spectrum with several longitudinal modes, while LDs grown on SiC substrate are lasing on a single longitudinal mode. With increasing current, the laser emission of LDs grown on SiC substrate jumps from one longitudinal mode to another (mode hopping), whereas GaN substrate LDs show a smooth but asymmetric mode comb. The different envelopes of these spectra can be understood by assuming a variation of the gain for each individual longitudinal mode. With a high spectral resolution setup, we measure the gain of each longitudinal mode, employing the Hakki–Paoli method. Measurements show a slightly fluctuating gain for the modes of GaN substrate LDs, but much larger fluctuations for LDs on SiC substrate. We carry out simulations of the longitudinal mode spectrum of (Al,In)GaN laser diodes using a rate equation model with nonlinear gain (self saturation, symmetric and asymmetric cross saturation) and including gain fluctuations. With a set of parameters which is largely identical for LDs on either substrate, the simulated spectra truly resemble those typical for LDs on GaN or SiC substrate.
©2008 Optical Society of America
1. Introduction
Nitride based semiconductor laser diodes (LDs) have already found their way into today’s consumer electronics and optical storage devices. In spite of their current application, (Al,In)GaN LDs have many properties which are not yet investigated, but play an important role both for future applications and for the basic understanding of nitride based optoelectronic devices. Issues of current research focus are, among others, the extension of the accessible wavelength range towards deep UV and green [1, 2], (Al,In)GaN VCSELs [3], optical gain [4], waveguide properties [5], and nonpolar and semipolar (Al,In)GaN LDs [6, 7]. While long living (Al,In)GaN LDs are on the marked, the understanding of degradation mechanisms is still an issue of current research [8].
In this paper, we will study and simulate the spectral properties of LDs and concentrate on the influence of the substrate thereon. While LDs on GaN substrate are generally lasing on many longitudinal modes, those on a SiC substrate lase on a single or on a few longitudinal modes. We trace this characteristic behavior to slight static fluctuations of the modal gain of each individual longitudinal mode. Hakki–Paoli gain spectroscopy [9, 10] shows that these fluctuations are larger by roughly one order of magnitude for LDs on SiC substrate when compared to those on GaN substrate. The link between the gain spectra below threshold and the lasing spectra is made by a standard rate–equation model [11, 12, 13] which truly reproduces the envelope of the longitudinal mode comb in both cases. The asymmetric envelope in the case of LDs on GaN substrate allows us to estimate the effect of symmetric and asymmetric cross saturation [14, 15] for (Al,In)GaN LDs in the near UV (≈405nm) spectral range. Thus, from the longitudinal mode spectra above threshold and from gain spectra we provide a full set of parameters to describe the temporal and spectral behavior of this class of LDs on a rate–equation basis.
2. Current dependency of laser spectra
We investigate two types of 405 nm (Al,In)GaN laser diodes from OSRAM Opto Semiconductors. One of them was grown on low dislocation density GaN substrate, the other on SiC substrate. GaN substrates offer several advantages in comparison to SiC substrate, such as lower internal losses and suppression of degradation, but they are more expensive than SiC substrates. The laser diodes are standard test structures with a ridge waveguide, as described in detail e.g. in [16].
Above threshold spectra of investigated lasers on GaN substrate show a characteristic shape, as seen in Fig. 1(a). Several longitudinal modes are lasing, some of them having a higher amplitude than the others. Figure 2 shows the current dependency of the spectrum of the GaN substrate LD. At threshold, some longitudinal modes at approximately 405.9 nm start lasing. With increasing current, the modes shift slightly to the red side of the spectrum, because the effective refractive index and the resonator length increase with temperature. Additionally, the envelope of the spectrum shifts to longer wavelengths, as the band gap and thus the transition energy of the quantum wells decrease with increasing temperature [17], which is affected by the injection current.
Fig. 1. (a) The spectrum of a LD on GaN substrate at I=1.3×I th. (b) The spectrum of a LD on SiC substrate at I=1.3×I th.
SiC substrate LDs exhibit a rather different spectrum. Figure 1(b) shows the spectrum at 1.3 times the threshold current I th, where only one single longitudinal mode is lasing. This behavior can be seen over a wide range of the injection current I. Some mA above threshold, a single longitudinal mode starts lasing (see Fig. 3). As the current increases, the laser emission jumps to another longitudinal mode with greater wavelength. This phenomenon is known as mode hopping and can result in an increase of the intensity noise. At higher currents, the emission shows several jumps to longer wavelengths, which is already known from SiC substrate LDs under pulsed operation [18]. In some of the investigated SiC substrate LDs, the number of emitting modes at high currents increases, but the spectra never have such a regular envelope as GaN substrate LD spectra at comparable currents.
The jumps of the emission only occur between longitudinal modes of the same mode comb, i.e. the LD is always emitting on the same lateral and transverse mode and no other lateral or transverse modes are emitting. If other lateral or transverse modes were emitting, another mode comb with a different mode spacing would appear. We observe a second mode comb only for laser diodes with broader ridges [19, 20].
3. Gain fluctuations
The laser spectrum of the GaN substrate LD, as shown in Fig. 2, does not have such a completely smooth, Gaussian envelope. Some longitudinal modes have a higher intensity than others. Consequently, the modal gain for these modes has to be higher. For the SiC substrate LDs, the same considerations apply, but the higher differences between the intensity of longitudinal modes imply even higher differences in the gain.
Fig. 2. The current dependency of the spectrum of a LD on GaN substrate. In this density plot color represents the intensity of the longitudinal modes (white: zero; blue: low; red: high intensity).
Fig. 3. The current dependency of the spectrum of a LD on SiC substrate. Density plot in analogy to Fig. 2.
Figures 4(a) and (b) show electroluminescence (EL) spectra belowthreshold for GaN and SiC substrate LDs, respectively. In contrast to the spectra above threshold, many longitudinal modes are visible. The spectra have an envelope which is approximately parabolic, due to the shape of the gain spectrum in the immediate vicinity of the gain maximum [21, 22]. The difference between SiC and GaN substrate LDs can be seen in these spectra below threshold, but is not as pronounced as above threshold. Figure 4(b), the SiC substrate LD spectrum, shows variations in the amplitude of the modes. They are also present in the GaN substrate spectrum, however less prominent. This leads to the assumption that gain spectra of LDs on SiC substrate fluctuate more than those for LDs on GaN substrate. These fluctuations are time independent and reproducible for individual LDs.
Fig. 4. The spectrum of a GaN substrate LD (a) and a SiC substrate LD (b) slightly below threshold. Intensities are given in arbitrary units. (c) Magnification of a single longitudinal mode from (a).
The Hakki–Paoli method provides the possibility to estimate the gain from EL spectra below threshold [9, 10], but the narrow spacing of the longitudinal modes for (Al,In)GaN LDs request a very high spectral resolution setup [23]. With a double Czerny–Turner monochromator operated in second order, we achieve a resolution of approximately 1.5 pm. This allows us to resolve the maxima and minima of the longitudinal mode comb with very high accuracy, which is mandatory for the Hakki–Paoli method. Figure 4(c) shows a single longitudinal mode, the maximum and minimum of this mode is clearly resolved.
As explained before, we expect higher variations in the gain spectra for SiC substrate LDs than for GaN substrate LDs. Figure 5 shows the gain spectra determined from the two EL spectra shown in Fig. 4(a) and (b), each point represents the gain of a longitudinal mode. Both the GaN substrate and SiC substrate LD gain spectra have a parabolic shape, but show slight deviations of the ideal form, herein after called gain fluctuations. These gain fluctuations exist for both the GaN and SiC substrate diodes, but it can be clearly seen that the gain fluctuations are higher for the SiC substrate diode than for the GaN substrate LD.
The question arises whether the different gain fluctuations for GaN and SiC substrate LDs lead to the individual spectra above threshold, as shown in Fig. 2 and 3. In the next section, we will discuss the possible origins of these fluctuations and will afterwards present a theoretical model that reproduces the longitudinal mode spectra above threshold.
4. Possible origins of the gain fluctuations
The fluctuations of the gain of the single longitudinal modes can be either due to a different gain or loss observed by each mode, i.e. a variation of the imaginary part of the refractive index, or by a modulation of the real part of the refractive index along the waveguide. Peters and Cassidy attribute irregular longitudinal mode spectra to interference caused by scattering from randomly distributed scattering centers along the ridge [24]. This mechanism could also be active in (Al,In)GaN LDs, where an inhomogeneous distribution of threading dislocations may cause inhomogeneous strain and local scattering centers. For LDs on GaN and SiC substrates the defect density and therefore the strain and scattering center distribution are different. Corbett and McDonald employ this effect to force a 1.3µm Fabry–Perot LD to single mode operation by etchings slots acting as scatterers at certain positions along the ridge waveguide [25, 26]. The optimum position of these intentionally introduced scatterers can be determined in an perturbative inverse scattering approach [27]. For a few strong scatterers the LD is in the coupled cavity regime. For an (Al,In)GaN LD with a single crack in the waveguide we observed a strong periodic modulation of the longitudinal mode gain together with a side–peak in the Fourier transform of the longitudinal mode spectrum [28]. The small and irregular fluctuation of the gain for the present LDs (see Fig. 5) is distinctly different from that regime of one or a few strong scattering centers. Another mechanism causing a second feedback path are substrate modes. They cause a periodic modulation of the gain spectra with a much longer period of approximately 2nm in similar (Al,In)GaN laser diodes [29]. In the present LDs the n–side cladding layer thickness was increased to suppress substrate modes [30]. We cannot rule out that the roughness of the etched ridge waveguide is affected by the defect density. This could be a mechanism causing different losses and/or feedback to individual longitudinal modes, adding to the differences between LDs on SiC and GaN substrate.
Fig. 5. Gain spectra for a LD on GaN and SiC substrate, respectively. Each point is the gain of an individual longitudinal mode.
There is manifold evidence of gain fluctuations on several length scales for (Al,In)GaN LDs. Therefore we favor gain fluctuations as origin for the irregular longitudinal mode spectrum, while we cannot exclude the mechanism acting through feedback from irregularly spaced scatterers. There are several mechanisms that can cause an increase or decrease of the gain for specific longitudinal modes. Spatial fluctuations of the quantum well depth have been proposed as origin [31]. The gain of an individual longitudinal mode is then given by the overlap of the mode intensity distribution and the gain profile along the waveguide. Therefore gain fluctuation is bound to the individual longitudinal modes and not to the wavelength. With increasing current, the temperature of the waveguide changes, leading to a change of the effective refractive index, resonator length, and wavelength shift of each longitudinal mode. However, the positions of the oscillation maxima and nodes of the longitudinal modes remain on the same locations in the waveguide, as well as the gain and loss centers. Thus every longitudinal mode experiences the same gain and loss centers across the full injection current range. This was confirmed by experiments where we measured the gain for different temperatures.
Quantum well thickness fluctuations on a sub–micrometer scale are clearly visible by transmission electron microscopy [32]. As carrier density and optical gain depends on the position of the band edge relative to the Fermi energy level, the optical gain will fluctuate on this lateral scale [4]. Yet one would expect that the effect of gain fluctuations on a sub–micrometer length scale will be averaged out in the modal gain of a 600µm long cavity. From spatially resolved photoluminescence spectra of InGaN quantum wells we know that photoluminescence intensity and peak wavelength also fluctuate on a length scale of several micrometers [33]. The origin of these fluctuations may be a variation in the nonradiative recombination rate or indium composition due to different growth modes triggered by larger areas of high defect density [34]. This could be due to the higher threading dislocation density which can affect the growth of the quantum wells and interfaces [35, 36]. Furthermore, it is reported that threading dislocations advantage the indium incorporation [37] and affect the quantum well transition energy [38, 39]. Rossetti et al. observed several dark regions in micro–electroluminescence spectroscopy along the 600µm long ridge of a degraded (Al,In)GaN LD and attributed those tentatively to locally decreased carrier injection [8]. Those dark spots were spatially correlated with defect arrays. While the prominent decrease in electroluminescence was only observed after degradation, it is reasonable to assume that carrier injection and gain are inhomogeneous from the very beginning.
We tried unsuccessfully to estimate the number and size of the centers causing additional gain or loss from the Hakki–Paoli gain spectra. The number of points along the gain curve is given by the number of longitudinal modes and thus fixed for a given resonator length. While this discrete spectrum does not appear to be completely random, as would be the case for a large (>100) number of scatterers, the Fourier spectrum does not show distinct peaks, as expected for a few (<5) scatterers.
In the next section, we will present a rate equation model which includes these gain fluctuations.
5. Multimode rate equation model with gain fluctuations
We use a multimode rate equation model with self– and cross–saturation for semiconductor lasers (see e.g. [11, 13]) as described by Yamada et al. [12, 40, 41, 42]. In this model, we include gain fluctuations and other modifications.
The dynamics is described in the rate equations for the electron number N and the photon numbers Sp for the longitudinal modes p.
The current injected into the laser diode is denoted as I inj, q e is the elementary charge. η inj describes the injection efficiency into the quantum wells, as not all electrons and holes will recombine in the wells. τ s is the spontaneous carrier lifetime, which is given by τ -1 s=τ -1 r+τ -1 nr, where τ r and τ nr are the time constant belonging to the radiative and nonradiative recombination, respectively.
Every longitudinal mode is denoted with a mode number p=0,±1,±2,… and has the wavelength λp. The central mode with p=0, i.e. the mode where the laser emission is highest when reaching laser threshold, has the wavelength λ 0, modes with p>0 lie on the long-wavelength side of the spectrum. g̃ th is the threshold gain and is given by
c vac stands for the speed of light in vacuum and n gr is the group refractive index of the waveguide. α int are the internal losses of the waveguide, L is the length of the resonator, R 1 and R 2 are the mirror reflectivities.
To describe spectral properties of a laser diode with rate equations, the gain must be considered individually for every longitudinal mode. The modal gain g̃p is given by
where Ap is the linear gain coefficient. B describes the self-saturation and Dpq and Hpq describe the cross–saturation effects between longitudinal modes [42]. The factor ζp accounts for the gain fluctuations and will be explained later. Differently from Yamada et al. we include the modal gain g̃p from Eq. 4 in the last term of Eq. 1. The gain coefficients are given by the following expressions:
Here, a is the slope coefficient characterizing the linear gain and was measured for the discussed LDs by the Hakki–Paoli method. Γ is the confinement factor into the quantum wells. Ng denotes the transparency electron number and is given by
V is the volume of the quantum wells. The parameter b describing the dispersion of the gain was derived from the curvature at gain maximum of measured gain spectra. In summary, the linear gain coefficient Ap describes the linear dependence of the gain on the carrier number N and the spectral shape of the gain as a parabola around the gain maximum located at λg.
The other gain coefficients contain the vacuum permittivity ε 0, the intraband relaxation time τ in, the dipole moment of the quantum wells Rcv, the antiguiding or Henry factor α, and the carrier number N s characterizing the saturation coefficient B.
The saturation coefficients play an important role for the shape of the laser spectrum. B leads to a saturation of the gain at high photon numbers and is caused by spectral hole burning. Dpq is called the symmetrical cross–saturation coefficient and influences modes q on the higher and shorter wavelength side of the mode p in the same manner. In contrast, the asymmetrical cross–saturation coefficient Hpq enhances the gain on the longerwavelength side and suppresses the modes on the shorter wavelength side. Thus, it leads to an asymmetrical envelope of the spectrum.
The rate equation for the photon numbers include the spontaneous emission factor βp, which characterizes the fraction of spontaneously generated light that is coupled into the longitudinal mode p [11, 13]. We take into account the Lorentzian shape of the spontaneous emission.
K en is a numerical factor that depends on the waveguiding of the laser (i.e. ridge waveguide or gain guided LD), n ref is the refractive index of the waveguide, δλ sp is the half–width at half–maximum of the spontaneous emission and λ sp is the wavelength of the peak of the spontaneous emission.
Additionally, the shift of both the peak gain and the longitudinal modes has to be considered. Without a current dependent shift, the spectrum would remain nearly unchanged over a wide range of the injection current. As approximation, the shift of the longitudinal modes can be regarded linearly with the injection current I inj:
λ 0,th is the wavelength of the maximum laser emission at the threshold current I th. Δλ is the mode spacing and dλ g/dI inj is a constant factor given by the current dependent shift of the modes which is estimated from longitudinal mode spectra above threshold. Similarly, the shift of the peak gain is given by
The constant factor dλ g/dI inj leads to the shift of the peak gain and thus the spectrum’s envelope with increasing current.
A factor yet unmentioned is the gain fluctuations factor ζp. It is included in both the rate equations for electrons and for photons and modifies the linear gain coefficient Ap. For our simulations, we use a gain fluctuations factor given by
where rp is a random number given by the standard normal distribution. ζp is a random number which increases or decreases the linear gain Ap independently for every longitudinal mode and is not affected by the injection current. High values for the amplitude δζ lead to large gain fluctuations and a value of δζ=0 would produce a smooth gain spectrum without any fluctuations.
Ropars et al. also observe fluctuations of the gain and an asymmetric envelope of the longitudinal mode spectrum of (Al,In)GaN LDs [43]. They used gain guided LDs and explained the asymmetric envelope in terms of defocussing which was supported by far-field measurements. As we observe a similar asymmetric envelope for our index guided LDs, where defocussing should play a minor role, we are the opinion that symmetric and asymmetric cross saturation are the origin of the observed asymmetry. The ridge widths of our index guided LDs are between 1.5 µm and 2.5 µm which is narrow enough to guarantee lateral ground mode operation.
6. Simulation of (Al,In)GaN laser spectra with gain fluctuations
We use the multimode rate equation model with gain fluctuations to simulate the current dependency of the spectra of LDs. The parameters used for these simulations are given in tables 1 and 2.
Figure 6 shows a simulated spectrum with GaN substrate parameters. Compared with the measured spectrum (see Fig. 2), the simulation looks quite similar: The longitudinal modes of the simulated spectrum shift to the long wavelength side as well as the envelope of the spectrum, which is due to Eq. 11 and 12. Furthermore, the increasing current broadens the envelope of the spectrum slightly. Without gain fluctuations (i.e. δζ=0), the envelope of the spectrum would have a smooth, slightly asymmetric shape. Here, we use a value for the gain fluctuations of δζ=5×10-4 to describe the gain fluctuations. With this value, the simulated spectra are in
Table 1. Common parameters for GaN and SiC substrate LDs
best agreement with the measured GaN substrate LD spectra. It is also possible to calculate the amplitude of the modal gain fluctuations from Eq. 4 and 5. For the GaN substrate LD, δζ=5×10-4 leads to a fluctuation of the modal gain of δg=0.020 cm-1. This is in good agreement with the value obtained from the high resolution gain spectrum (see Fig. 5), which is approximately 0.027 cm-1.
A simulation of a SiC substrate LD is shown in Fig. 7. All characteristic features of the measured spectrum (Fig. 3) are present here, too: The laser emission occurs mostly on only one longitudinal mode and the emission is hopping towards longer wavelengths with increasing current. At high injection currents, some additional modes start lasing, which can be observed at some SiC substrate LDs. In comparison with the GaN substrate LD simulations, only a small subset of the parameters was changed for the SiC substrate LDs (see table 2). The most influential parameter is δζ, which is 10 times higher in the SiC substrate LD simulations. This value of δζ=5×10-3 is equivalent to a modal gain fluctuations amplitude of δg=0.35 cm-1, compared to a value of 0.17 cm-1 from the measured gain spectrum (see Fig. 5).
In summary, the gain fluctuations of the SiC substrate LD are significantly higher than the
Table 2. Parameters for GaN and SiC substrate LDs, respectively
ones for the GaN substrate LD, both in the simulations and the experimentally acquired gain spectra. Furthermore, the values of the modal gain fluctuations used in the simulations are in good agreement with the ones obtained from the measured gain spectra, both for the GaN and SiC substrate LDs. This leads to the assumption that the gain fluctuations are the key to describe the characteristic shape of the longitudinal mode spectra of laser diodes on GaN and SiC substrate.
Fig. 6. The simulated current dependency of the spectrum of a LD on GaN substrate. Density plot in analogy to Fig. 2.
Fig. 7. The simulated current dependency of the spectrum of a LD on SiC substrate. Density plot in analogy to Fig. 2.
7. Conclusion
We measured above threshold spectra of 405 nm (Al,In)GaN laser diodes grown on SiC and GaN substrates, respectively. While the GaN substrate LD shows a broad spectrum with several longitudinal modes, the SiC substrate LD is lasing on single longitudinal modes with jumps of the laser emission with increasing current. This implies differences in the gain spectra between the LDs, which are demonstrated by high resolution Hakki–Paoli gain measurements, where the SiC substrate LD shows a gain spectrum with larger modeȓtoȓmode fluctuations than the GaN substrate LD.
We propose that fluctuations of the modal gain on a several micrometer length scale for LDs on SiC substrate are the origin of the spectral gain fluctuations, as these spatial gain and loss centers in the laser resonator amplify certain longitudinal modes stronger than others.
A multimode rate equation model was modified to include gain fluctuations and simulations were carried out for laser diodes on GaN and SiC substrate. The simulated spectra resemble the measured ones very well, reproducing characteristic features of both GaN and SiC substrate LDs. We obtained values for the modal gain fluctuations from the simulations which are in good agreement with values from measured gain spectra.
Furthermore, we provide a parameter set for (Al,In)GaN based laser diodes which can be used with the multimode rate equation model and other types of simulations. In particular the symmetric and asymmetric cross saturation parameters are determined. This allows to describe the asymmetric longitudinal mode spectrum for (Al,In)GaN LDs on GaN substrate. The majority of commercial (Al,In)GaN LDs are grown on GaN substrate, so that the parameter set should provide valuable input for advanced simulations of (Al,In)GaN LDs for various applications ranging from single mode over fast modulation to high output power operation.
References and links
1. M. Kneissl, Z. Yang, M. Teepe, C. Knollenberg, O. Schmidt, P. Kiesel, N. M. Johnson, S. Schujman, and L. J. Schowalter, “Ultraviolet semiconductor laser diodes on bulk AlN,” Appl. Phys. Lett. 101, 123103 (2007).
2. T. Miyoshi, T. Yanamoto, T. Kozaki, S. Nagahama, Y. Narukawa, M. Sano, T. Yamada, and T. Mukai, “Recent status of white LEDs and nitride LDs,” Proc. SPIE 6894, 689414 (2008). [CrossRef]
3. T.-C. Lu, C.-C. Kao, H.-C. Kuo, G.-S. Huang, and S.-C. Wang, “CW lasing of current injection blue GaN-based vertical cavity surface emitting laser,” Appl. Phys. Lett. 92, 141102 (2008). [CrossRef]
4. K. Kojima, U. T. Schwarz, M. Funato, Y. Kawakami, S. Nagahama, and T. Mukai, “Optical gain spectra for near UV to aquamarine (Al,In)GaN laser diodes,” Opt. Express 15, 7730 (2007). [CrossRef] [PubMed]
5. D. F. Feezell, M. C. Schmidt, R. M. Farrell, K.-C. Kim, M. Saito, K. Fujito, D. A. Cohen, J. S. Speck, S. P. Den-Baars, and S. Nakamura, “AlGaN-cladding-free nonpolar InGaN/GaN laser diodes,” Jpn. J. Appl. Phys. 46, L284 (2007). [CrossRef]
6. R. M. Farrell, D. F. Feezell, M. C. Schmidt, D. A. Haeger, K. M. Kelchner, K. Iso, H. Yamada, M. Saito, K. Fujito, D. A. Cohen, J. S. Speck, S. P. DenBaars, and S. Nakamura, “Continuous-wave operation of AlGaN-cladding-free nonpolar m-plane InGaN/GaN laser diodes,” Jpn. J. Appl. Phys.46, L761 (2007). [CrossRef]
7. K. Kojima, M. Funato, Y. Kawakami, S. Masui, S. Nagahama, and T. Mukai, “Stimulated emission at 474nm from an InGaN laser diode structure grown on a (112̄2) GaN substrate,” Appl. Phys. Lett. 91, 251107 (2007). [CrossRef]
8. M. Rosetti, R. M. Smeeton, W.-S. Tan, M. Kauer, S. E. Hooper, J. Heffernan, H. Xiu, and C. J. Humphreys, “Degradation of InGaN/GaN laser diodes analyzed by microphotoluminescence and microelectroluminescence mappings,” Appl. Phys. Lett. 92, 151110 (2008). [CrossRef]
9. B. W. Hakki and T. L. Paoli, “cw degradation at 300 K of GaAs double-heterostructure junction lasers. II. Electronic gain,” J. Appl. Phys. 44, 4113 (1973). [CrossRef]
10. B. W. Hakki and T. L. Paoli, “Gain spectra in GaAs double-heterostructure injection lasers,” J. Appl. Phys. 46, 1299 (1974). [CrossRef]
11. G. P. Agrawal and N. K. Dutta, Semiconductor lasers, 2nd ed. (Kluwer Academic Publishers, 1993).
12. M. Yamada, “Advanced theory of semiconductor lasers,” in Handbook of semiconductor lasers and photonic integrated circuits, Y. Suematsu and A. R. Adams , eds. (Chapman & Hall, 1994).
13. L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, Wiley series in microwave and optical engineering (Wiley, 1995).
14. M. Yamada and Y. Suematsu, “Analysis of gain suppression in undoped injection lasers,” J. Appl. Phys. 52, 2653 (1981). [CrossRef]
15. M. Yamada, “Theoretical analysis of nonlinear optical phenomena taking into account the beating vibration of the electron density in semidonductor lasers,” J. Appl. Phys. 66, 81 (1989). [CrossRef]
16. B. Witzigmann, V. Laino, M. Luisier, F. Roemer, G. Feicht, and U. T. Schwarz, “Simulation and design of optical gain in In(Al)GaN/GaN short wavelength lasers,” Proc. SPIE 6184, 61840E (2006). [CrossRef]
17. Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica 34, 149 (1967). [CrossRef]
18. C. Eichler, S.-S. Schad, M. Seyboth, F. Habel, M. Scherer, S. Miller, A. Weimar, A. Lell, V. Härle, and D. Hofstetter, “Time resolved study of laser diode characteristics during pulsed operation,” Phys. Status Solidi C 0, 2283 (2003). [CrossRef]
19. H. Braun, H.-M. Solowan, D. Scholz, T. Meyer, U. T. Schwarz, S. Brüninghoff, A. Lell, and U. Strauß, “Lateral and longitudinal mode pattern of broad ridge 405 nm (Al,In)GaN laser diodes,” J. Appl. Phys. 103, 073102 (2008). [CrossRef]
20. H. Braun, T. Meyer, D. Scholz, U. T. Schwarz, S. Brüninghoff, A. Lell, and U. Strauß, “Spectral and time resolved scanning near-field microscopy of broad area 405 nm InGaN laser diode dynamics,” Phys. Status Solidi C (to be published).
21. B. Witzigmann, V. Laino, M. Luisier, U. T. Schwarz, G. Feicht, W. Wegscheider, K. Engl, M. Furitsch, A. Leber, A. Lell, and V. Härle, “Microscopic analysis of optical gain in InGaN/GaN quantum wells,” Appl. Phys. Lett. 88, 021104 (2006). [CrossRef]
22. U. T. Schwarz, H. Braun, K. Kojima, M. Funato, Y. Kawakami, S. Nagahama, and T. Mukai, “Investigation and comparison of optical gain spectra of (Al,In)GaN laser diodes emitting in the 375 nm to 470 nm spectral range,” Proc. SPIE 6485, 648506 (2007). [CrossRef]
23. U. T. Schwarz, E. Sturm, W. Wegscheider, V. Kümmler, A. Lell, and V. Härle, “Optical gain, carrier-induced phase shift, and linewidth enhancement factor in InGaN quantum well lasers,” Appl. Phys. Lett. 83, 4095 (2003). [CrossRef]
24. F. H. Peters and D. T. Cassidy, “Strain and scattering related spectral output of 1.3-µm InGaAsP semiconductor diode lasers,” Appl. Optics 30, 1036 (1991). [CrossRef]
25. B. Corbett and D. McDonald, “Single longitudinal mode ridge waveguide 1.3 µm Fabry-Perot laser by modal perturbation,” Electron. Lett. 31, 2181 (1995). [CrossRef]
26. B. Roycroft, P. Lambkin, S. Riesner, B. Corbett, and J. F. Donegan, “Transition from perturbed to coupled-cavity behavior with asymmetric spectral emission in ridge lasers emitting at 1.55µm,” IEEE Photon. Technol. Lett. 19, 58 (2007). [CrossRef]
27. S. O’Brien, A. Amann, R. Fehse, S. Osborne, and E. P. O’Reilly, “Spectral manipulation in Fabry-Perot lasers: perturbative inverse scattering approach,” J. Opt. Soc. Am. B 23, 1046 (2006). [CrossRef]
28. U. T. Schwarz, W. Wegscheider, A. Lell, and V. Härle, “Nitride-based in-plane laser diodes with vertical current path,” Proc. SPIE 5365, 267 (2004). [CrossRef]
29. V. Laino, F. Roemer, B. Witzigmann, C. Lauterbach, U. T. Schwarz, C. Rumbolz, M. O. Schillgalies, M. Furitsch, A. Lell, and V. Härle, “Substrate Modes of (Al,In)GaN Semiconductor Laser Diodes on SiC and GaN Substrates,” IEEE J. Quantum Electron. 43, 16 (2007). [CrossRef]
30. U. Strauss, C. Eichler, C. Rumbolz, A. Lell, S. Lutgen, S. Tautz, M. O. Schillgalies, and S. Brüninghoff, “Beam quality of blue InGaN lasers for projection,” Phys. Status Solidi C (to be published).
31. C. Eichler, S.-S. Schad, F. Scholz, D. Hofstetter, S. Miller, A. Weimar, A. Lell, and V. Härle, “Observation of temperature-independent longitudinal-mode patterns in violet-blue InGaN-based laser diodes,” IEEE Photon. Technol. Lett. 17, 1782 (2005). [CrossRef]
32. N. K. van der Laak, R. A. Oliver, M. J. Kappers, and C. J. Humphreys, “Role of gross well-width fluctuations in bright, green-emitting single InGaN/GaN quantum well structures,” Appl. Phys. Lett. 90, 121911 (2007). [CrossRef]
33. U. Strauß, S. Brüninghoff, M. Schillgalies, C. Vierheilig, N. Gmeinwieser, V. Kümmler, G. Brüderl, S. Lutgen, A. Avramescu, D. Dini, C. Eichler, A. Lell, and U. T. Schwarz. “True blue InGaN laser for pico size projectors,” Proc. SPIE 6894, 689417 (2008). [CrossRef]
34. R. A. Oliver, M. J. Kappers, C. J. Humphreys, and G. A. D. Briggs, “Growth modes in heteroepitaxy of InGaN on GaN,” J. Appl. Phys. 97, 013707 (2005). [CrossRef]
35. H. Sato, T. Sugahara, Y. Naoi, and S. Sakai, “Compositional inhomogeneity of InGaN grown on sapphire and bulk GaN substrates by metalorganic chemical vapor deposition,” Jpn. J. Appl. Phys. 37, 2013 (1998). [CrossRef]
36. K. Akita, T. Kyono, Y. Yoshizumi, H. Kitabayashi, and K. Katayama, “Improvements of external quantum efficiency of InGaN-based blue light-emitting diodes at high current density using GaN substrates,” J. Appl. Phys. 101, 033104 (2007). [CrossRef]
37. N. Duxbury, U. Bangert, P. Dawson, E. J. Thrush, W. Van der Stricht, K. Jacobs, and I. Moerman, “Indium segregation in InGaN quantum-well structures,” Appl. Phys. Lett. 76, 1600 (2000). [CrossRef]
38. A. Hangleiter, F. Hitzel, C. Netzel, D. Fuhrmann, U. Rossow, G. Ade, and P. Hinze, “Suppression of nonradiative recombination by V-shaped pits in GaInN/GaN quantum wells produces a large increase in the light emission efficiency,” Phys. Rev. Lett. 95 , 127, 402 (2005).
39. C. Netzel, H. Bremers, L. Hoffmann, D. Fuhrmann, U. Rossow, and A. Hangleiter, “Emission and recombination characteristics of Ga1-xInxN/GaN quantum well structures with nonradiative recombination suppression by V-shaped pits,” Phys. Rev. B 76, 155322 (2007). [CrossRef]
40. M. Ahmed, M. Yamada, and M. Saito, “Numerical modeling of intensity and phase noise in semiconductor lasers,” IEEE J. Quantum Electron. 37, 1600 (2001). [CrossRef]
41. M. Ahmed, M. Yamada, and S. Abdulrahmann, “A multimode simulation model of mode-competition low-frequency noise in semiconductor lasers,” Fluctuation and Noise Letters 1, L163 (2001). [CrossRef]
42. M. Ahmed and M. Yamada, “Influence of instantaneous mode competition on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 38, 682 (2002). [CrossRef]
43. G. Ropars, A. Le Floch, and G. P. Agrawal, “Spectral and spatial dynamics in InGaN blue-violet lasers,” Appl. Phys. Lett. 89, 241128 (2006). [CrossRef]
