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Abstract
(Al,In)GaN laser diodes have various relevant applications, especially in projection systems for virtual and augmented reality devices and in optical communication, all requiring fast modulation. This corresponds to pulses in the nanosecond to microsecond range, where a rich longitudinal mode dynamics occurs. We investigate this spectral-temporal dynamics experimentally with a streak camera system and simulate it using a longitudinal multi-mode rate equation model. We observe an interplay of effects, which have been observed selectively, such as relaxation oscillations, mode competition and inhomogeneous pumping of multiple quantum wells. A mechanism is included in the simulations to model the red-shift of the gain spectrum due to the carrier density in the quantum wells exceeding threshold density, which is amplified by inhomogeneous pumping. Mode competition leads to spectral cycles of the active mode with a noticeable jitter, which is observed in single pulse measurements in comparison to multi pulse averaged measurements where blurring occurs. Here, some statistical behavior as well as repeating patterns are investigated at the same effect. Also thermal effects as laser diode self-heating are discussed and have been measured over six orders of magnitude in time.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Augmented reality (AR), virtual reality (VR), and mixed reality (MR) glasses are recent applications for near-eye displays. A promising approach is the use of waveguide optical display systems, which has been evaluated in [1] by D. Gallagher et al. as head-up displays for e.g. scuba divers that require physically small components and low power consumption. Also commercial AR/VR/MR glasses as Microsoft HoloLens 2, Google Glass or Epson Moverio are based on waveguide optical display systems. Currently available devices mostly use a micro pixel display as image source with the emitted light being coupled into a waveguide, usually with the help of diffraction gratings. Inside the waveguide, the light is conducted through multiple internal total reflections until it hits another diffraction grating, which is responsible for outcoupling. The resulting light beam enters the eye and produces a virtual image. In the case of AR glasses, the observer will see the real scenery through the transparent waveguide superimposed with the projected image.
Complex applications require high resolution, a large field of view, and high frame rates of the projection light source, but commercial and professional usage demands also miniaturized devices. So the common micro displays are restricted by their miniaturization limits, especially for high resolutions. Scanning systems using digital micromirror devices can be a solution, but very fast modulated light sources are necessary for this, such as laser diodes [2]. The light source must generate all pixels of an image, e. g. 1920 $\cdot$ 1080 pixels for Full HD and repeat this with 60 Hz frame rate, which yields a required modulation with 124.4 MHz that corresponds to about 8 ns per pixel. The periods for single lines and frames of an image are in the 10 to 20 microsecond and 10 to 20 millisecond range, respectively, so these time ranges must also be investigated. Common light sources, such as LEDs, are too slow and cannot be modulated at these frequencies. Neither can a LED be efficiently coupled to the waveguide of the optical system. So the application of laser diodes in display systems is promising and for this, the spectral-temporal dynamics in short pulses and in multiple consecutive pulses must be understood [3].
Green InGaN laser diodes have undergone rapid development and improvement in recent years [4–8] and exhibit many desired properties such as compact size at high emission powers, direct electrical modulation and various spectral ranges and widths. So several novel applications are emerging, not only in imaging systems, but in optical communication too [9,10], also in the combination with lighting [11], or optical clocks [12]. In all these applications a fast modulation and/or spectral stability are important. Even when the laser is used for excitation, like in spectroscopy or projection, the spectral-temporal dynamics is relevant, because it transforms into an intensity modulation when the laser beam passes wavelength-sensitive optics like gratings or wave plates plus polarizers.
The basic effects in laser diode dynamics [13], including turn-on delay [8] and relaxation oscillations [14] are well known and can be calculated using a simple rate equation model considering only one longitudinal mode. Here, we use a longitudinal multi-mode simulation model in order to reproduce multi-mode effects, such as mode competition occurring in longer pulses, as well as the fast red-shift at pulse onset.
Furthermore, we investigate the time-dependent self-heating of laser diodes. This is important for applications where a variation of the threshold and slope with temperature impairs the intensity modulation pattern e.g. in image projection. We use the wavelength shift of the longitudinal modes as a precise thermometer for the active region [15,16].
2. Description of experiment and samples
Experimentally we observe full laser diode dynamics with a Hamamatsu C10910 streak camera setup that is able to measure the spectral and temporal resolved intensity with high resolution in wavelength and in time [17–19]. The setup is schematically shown in Fig. 1. The current pulses for driving the laser diode are generated using a Stanford Research Systems delay generator DG645 and a constant bias can be added with the help of a bias T. The delay generator is computer-controlled and also triggers the streak camera synchronously with the laser diode. A simple circuit with 50 $\Omega$ striplines enables time-resolved current monitoring using a iC Haus iC227 sampling oscilloscope. The laser diode is mounted in a passive copper heatsink and is aligned together with the imaging optics that include a collimating aspherical lens, a ND filter wheel and a focusing lens for coupling the beam into the Princeton Instruments Acton SpectraPro SP-2300 monochromator. There, a 600 l/mm or 2400 l/mm grating is employed to split up the incident light horizontally by wavelength, and temporal resolution is achieved in the streak camera by photoelectron generation and vertical deflection by a time-varying field inside an evacuated tube. A typical streak camera measurement of a 10 ns long pulse is shown in Fig. 2, where both gratings are compared. In Fig. 2(a), the grating with 600 l/mm is used, which offers lower spectral resolution and the longitudinal modes cannot be distinguished, but it enables measuring the whole spectral range of the laser diode at once. The inset curve represents the intensity integrated over the whole spectrum, from which the relaxation oscillation frequency can be determined and the decay after the end of the current pulse can be evaluated regarding the spontaneous recombination coefficients. The 2400 l/mm grating used in Fig. 2(b) allows to resolve the individual longitudinal modes, but a smaller spectral range can be captured and the sensitivity across the spectrum may be slightly uneven. Dust particles in the streak camera entrance slit form irregularities that appear as supressed longitudinal modes.
Fig. 2. Measurement of 10 ns long pulses with $I=2\:I_{\mathrm {th}}$ using (a) the 600 l/mm grating and (b) the 2400 l/mm grating. The inset in (a) shows the spectrally integrated intensity over time. The threshold current is $I_{\mathrm {th}}=57.6\,{\textrm {mA}}$.
The investigated (Al,In)GaN laser diodes are generic commercial devices. These narrow ridge Fabry-Pérot type laser diodes operate in the fundamental mode in lateral and transverse directions. They show however a complex longitudinal mode dynamics. All effects described here are universal in a sense that we have observed them consistently in a variety of green and blue (Al,In)GaN laser diodes from different suppliers. Different design of the individual laser diodes causes only a gradual variation of the dynamical behavior. The parameter set needed as input to the multi longitudinal mode rate equation model consequently depends on the individual laser diode type. In principle, this parameter set can be derived from the epitaxial structure and geometry of the laser diode. The uncertainty in the calculation of the optical gain spectra, recombination coefficients, internal loss, and other parameters is still significant. Our approach is therefore to adjust these parameters by a calibration of the rate equation model with the basic characteristics of each type of laser diode. The derived parameter set used throughout this article is listed in Table 1. In order to allow a reproduction and verification of our model, we next describe the basic characteristics of the investigated individual type of green laser diode. Corresponding gain spectra are shown further down in section 5. on the initial red-shift.
Table 1. Simulation parameters for the investigated laser diode.
The static laser characteristics showing the relations between current, voltage, and optical output power are depicted in Fig. 3, also including a logarithmic plot of the optical power to illustrate the sub-threshold behavior, from which the spontaneous emission factor $\beta$ can be determined. These cw measurements indicate a threshold current of $I_{\mathrm {th}}=57.6\,{\textrm {mA}}$ and a slope efficiency of $\mathrm {d}P_{\mathrm {opt}}/\mathrm {d}I=0.3\,{\textrm {W}}/A$.
Fig. 3. Static laser diode characteristics, (a) voltage, optical power and current on a linear scale, (b) logarithmic plot of optical power vs. current.
The relaxation oscillation frequency $f_{\mathrm {r}}$ can be evaluated as a function of the current $I$ (see Fig. 4) and is a practical way for determining fundamental properties such as the differential gain from dynamic processes. From theoretical analysis, the relation between $I$ and $f_{\mathrm {r}}$ in Eq. (1) can be derived [13], including the proportionality $f_{\mathrm {r}}^2\sim I$.
Fig. 4. Measurement of relaxation oscillation frequency $f_{\mathrm {r}}(I)$ and fit of the linear relationship between $f_{\mathrm {r}}^2$ and current $I$.
Here, $c_{\mathrm {vac}}$ and $q_{\mathrm {e}}$ stand for the speed of light in vacuum and the elementary charge respectively, and with the known group refractive index $n_{\mathrm {gr}}$, the differential gain $\mathrm {d}g/\mathrm {d}N$ (with respect to the total carrier number $N$) can be calculated that is $7.62\cdot 10^{-8}\:\mathrm {cm}^{-1}$ for this particular laser diode.
3. Introduction to multi-mode rate equations
For simulating the dynamics of the whole laser diode system and to enable a direct comparison with the streak camera images, the rate equation model that has been described e.g. in [13] by L. A. Coldren et al. needs to be expanded. In this paper, different aspects of the new model are presented in subsequent sections to deliver a comprehensible connection to the various measurements, yet for all calculations the complete model is used.
The expanded model must consider the individual longitudinal modes that will be described with the index $p$ and the goal is to determine the photon number $S_{\mathrm {p}}$ in each mode time-dependently. Also the modal gain $g_{\mathrm {p}}$ is different for each longitudinal mode and time-dependent as well. The wavelengths $\lambda _{\mathrm {p}}$ that correspond to the modes are fixed in this model:
because thermal effects that can lead to a wavelength shift of the modes [15,20] are neglected for the calculation of the photon density of each mode. The mode spacing $\Delta \lambda$ equals and is determined by the cavity length $L$ and group refractive index $n_{\mathrm {gr}}=c_{\mathrm {vac}}/v_{\mathrm {gr}}$, where $v_{\mathrm {gr}}$ is the group velocity in the material, and $\Delta \lambda$ can be considered constant near the central wavelength $\lambda _0$.(Al,In)GaN laser diodes have usually two or three quantum wells (QWs) [21]. The optimization of the number of quantum wells is an empirical process and part of the development process [22]. Single quantum well laser diodes are less efficient than double or triple quantum well laser diodes, caused by a lower confinement factor and by the inferior epitaxial quality of the first quantum well. A larger number of quantum wells is again leading to higher threshold current density, probably due to the effect that the carrier injection into the multiple quantum wells is not equal. In particular, the hole density is largest for the p-side quantum well and decreasing for the subsequent quantum wells towards n-side, due to low hole mobility. This effect of unequal pumping is already observed in double quantum wells and is the cause for a distinct hysteresis of the current-optical power curve near threshold [23,24]. We include this effect by accounting for two different carrier densities in two quantum wells and an unequal pumping parameter $\chi$ describing the imbalanced carrier injection. This model can easily be expanded to three quantum wells. However, with $\chi$ as free parameter, we are not able to distinguish a double or triple quantum well laser diode regarding unequal pumping [23,24]. Here we use the model with two quantum wells that are described with the two different carrier numbers $N_1$ and $N_2$. So the threshold ($N_{\mathrm {th}}$), transparency ($N_{\mathrm {tr}}$) and saturation ($N_{\mathrm {sat}}$) carrier numbers are half of the values for the one QW model, because in the case of two QWs the active area has been divided into two equal sized parts that are now considered separately.
The expanded rate equation system contains about ninety equations for the photon number time derivative $\mathrm {d}S_{\mathrm {p}}/ {\mathrm {d}}t$ and two equations for the carrier number derivative $\mathrm {d}N_{1,2}/ \mathrm {d}t$, as well as the starting conditions with no photons in each mode and no carriers in each QW. The simple rate equations are now modified as follows [25,26]:
After the injection terms in Eqs. (4) and (5), the terms for spontaneous recombination and stimulated emission follow. The recombination rate $R$ of the carriers in one quantum well contains nonradiative Shockley-Read-Hall (SRH) recombination ($\sim N$), radiative recombination ($\sim N^2$) as well as Auger processes ($\sim N^3$), and is given by:
Here we neglect the impact of Fermi statistics on the dependency of the recombinations terms as function of carrier density which would lead to a radiative recombination proportional to $N$ and Auger recombination proportional to $N^2$ in the high carrier density limit [27,28]. Furthermore, screening of the piezoelectric field with increasing carrier density changes the spontaneous recombination rates and optical gain. Including these effects would modify the turn-on delay and also have an impact on the derived parameters. However, in our rate equation model we use coefficients $A_{\mathrm {SRH}}$, $B_{\mathrm {rad}}$, and $C_{\mathrm {Auger}}$ which are independent of carrier density.The stimulated emission rate is computed as product of modal gain and photon number, summed up over all longitudinal modes. For including unequal pumping, the modal gain is divided into the two parts resulting from the different QWs. This means $g_{\mathrm {p,1}}$ describes the amplification of the p-th longitudinal mode in the first QW and $g_{\mathrm {p,2}}$ is the contribution of the second QW to the p-th mode. Usually, the gain has the same unit as absorption (length$^{-1}$) but when using the number notation the unit time$^{-1}$ is required, which leads to multiplication with the group velocity $v_{\mathrm {gr}}$ in the rate equations.
In Eq. (6), the photon number in each mode increases through stimulated emission and radiative spontaneous emission in both QWs. The losses reducing the photon numbers equal $-\sum _{p} v_{\mathrm {gr}} g_{\mathrm {th}}S_{\mathrm {p}}$ and threshold gain $g_{\mathrm {th}}$ corresponds to the sum of internal losses $\alpha _{\mathrm {int}}$ and mirror losses $\alpha _{\mathrm {mir}}$, which are connected to the cavity length and the mirror reflectivities $R_1$ and $R_2$ of the facets forming the Fabry-Pérot cavity:
The spontaneous emission factor $\beta$ is used to describe the fraction of the spontaneously emitted photons that contribute to the laser mode of the cavity [13] and is computed by Eq. (10). Here, the phase refractive index $n_{\mathrm {ph}}$, the active volume $V$, the spontaneous emission bandwidth $\Delta \lambda _{\mathrm {sp}}$, and the the confinement factor $\Gamma$, which denotes the fraction of the laser mode overlapping with the active volume are used.The coefficients $A_{\mathrm {p}}$, $B$, $D_{\mathrm {pq}}$ and $H_{\mathrm {pq}}$ depend on the carrier number and several parameters [25,29]:
The self saturation parameter $B$ depends on the intraband relaxation time constant $\tau _{\mathrm {in}}$, the dipole transition matrix element $R_{\mathrm {cv}}$, and the saturation carrier number $N_{\mathrm {sat}}$ and includes the vacuum permittivity $\varepsilon _0$ and the reduced Planck constant $\hbar$. It describes the influence of the already present photon number in one longitudinal mode on the modal gain in the same mode. $B$ depends on the carrier number in the considered quantum well and contains a saturation effect regarding $N_{\mathrm {sat}}$. Also, $B$ is the same for all longitudinal modes, only the two quantum wells have a different impact, depending on $N_1$ and $N_2$.
The term $D_{\mathrm {pq}}$ describes the symmetric part of the mode cross-saturation, that means each mode suppresses the gain in the neighboring modes and the impact scales with the photon number in the considered mode. This is also referred to as spectral hole burning. When inserting $B$ from Eq. (13) into Eq. (14), it becomes clearer visible that $D_{\mathrm {pq}}$ describes basically a Lorentz-distribution in the wavelength spectrum, so the influence of this interaction decreases with larger wavelength differences between the two coupling modes. The intraband relaxation time $\tau _{\mathrm {in}}$ corresponds to the homogeneous linewidth and is the most characteristic parameter influencing symmetric mode coupling [26].
The asymmetric part of the mode coupling effect is characterized by the term $H_{\mathrm {pq}}$ where the asymmetric character is modeled with $(\lambda _{\mathrm {q}}-\lambda _{\mathrm {p}})^{-1}$, that means the gain of the neighboring modes in longer wavelength direction is increased and the neighboring modes with lower wavelength are decreased in gain. This models the mechanism of mode competition where the lasing mode moves through the spectrum towards longer wavelengths [25]. The strength of the influence is mainly determined by the antiguiding factor $\alpha _{\mathrm {ant}}$ that describes the ratio between carrier-induced change of refractive index and change of gain [31]:
Fig. 5. Simulated gain spectrum showing the four components, resulting from multi-mode simulation. The dots indicate the individual longitudinal modes. This static image represents the gain at one point in time and is a snapshot of the dynamic behavior.
The multi-mode rate equation system consists of equations for the photon number in each mode and for the carrier number in each quantum well. The expressions for the modal gain are evaluated for the two QWs using the parameters in Table 1 and the differential equation system is numerically solved yielding the photon numbers $S_{\mathrm {p}}(t)$ and carrier numbers $N_{\mathrm {1,2}}(t)$ time-dependently.
The simulation results correspond to the single-mode results by adding all longitudinal mode photon numbers to get the total $S$ and the carrier number in the two quantum wells to total $N$. So the single-mode effects such as turn-on delay and relaxation oscillations can also be observed, of course in the sums of photons and carriers but in the multi-modal results as well. Also the $P_{\mathrm {opt}}(I)$-characteristics are computed from the multi-mode model. This integrated intensity is equal to the single-mode simulation result.
In this article, all experimental measurements are performed on one specific commercial green InGaN/GaN laser diode and the simulation parameters are determined to model this laser diode as accurate as possible (see Table 1). The resulting images for $I=2\: I_{\mathrm {th}}$ and $\chi =0.7$ on time ranges of 10 ns, 100 ns and 1 µs are shown in Fig. 6. In these images, the simulated mode dynamics can be seen from the onset dynamics in Fig. 6(a) up to an almost steady-state mode competition in Fig. 6(c). Note that the colors show the intensity, separately normalized in each image, and that the wavelength axis is the same in Figs. 6(a) and 6(b) but covers a smaller range in Fig. 6(c).
Fig. 6. Simulated streak images at $I=2\: I_{\mathrm {th}}$ with the time ranges (a) 10 ns; (b) 100 ns and (c) 1 µs.
The current pulse starts at $t=0\:\mathrm {ns}$ and is constant over time. The optical pulse begins after the turn-on delay which equals about 1.7 ns in this case. Then in Fig. 6(a) the relaxation oscillations are clearly visible and at the same time the active part of the laser spectrum becomes narrower and the central wavelength shifts about 2 nm during the first 5 ns. This corresponds very well to the measurement results that are shown in Fig. 2 and later throughout the article. In Fig. 6(b) the transition from the fast initial red-shift to the mode competition wavelength shift can be seen. In the first about 15 ns, multiple modes emit light at the same time and therefore the intensity of each mode is noticeably lower, compared to later times when nearly only one mode is active at once. This corresponds to the spectrally integrated intensity being constant after the first few nanoseconds, which is clearly visible in Fig. 7. So the mode coupling effects start to dominate after about 15 ns, where the symmetric cross saturation leads to preferred single-mode behavior and the asymmetric cross saturation causes the lasing mode to move in the longer wavelength direction until it starts over from the shorter wavelength side. Figure 6(c) shows the mode competition behavior on a longer time scale, where the active part of the spectrum is now less than 1 nm wide. Until about 0.5 µs, the mode cycles contain some irregularities in terms of frequency and intensity. There are both weaker peaks in the sequence of the rolling modes and also some peaks extend into and through the region between two cycles. However, in the second half of the first microsecond, mode competition stays mostly stable and this continues also at later times.
Fig. 7. Calculated carrier numbers $N_{1,2}(t)$ and total photon number $S(t)$ for $I=2\:I_{\mathrm {th}}$ and $\chi =0.7$ .
Besides the rate equation based model, microscopical simulations [32] also show an improving accordance to the measurements [33]. Although the direct connection of this approach to the rate equation model and the measurements is challenging, this promises new insights on microscopic processes.
4. Turn-on delay
The turn-on delay is the time difference between the onsets of the current pulse and the optical pulse, which is a major limitation to the pulse repetition rate in the case of modulated laser operation. During the turn-on delay, carriers are injected into the system and spontaneous recombination processes start immediately. But the modal absorption is still strong and prevents a noticeable contribution of stimulated emission, until the carrier density approaches the threshold. After reaching the threshold, stimulated emission becomes dominant and the photon density rises exponentially in the first approximation.
We define the turn-on delay between the time of the current reaching 50% of the bias-pulse current step and the optical power reaching 50% of its steady-state value. By this, we avoid errors arising from imperfect electrical pulse forms and from difficulties in both exactly measuring and simulating relaxation oscillation amplitudes.
The turn-on delay of a laser diode is characterized by the spontaneous radiative and nonradiative recombination coefficients that determine the threshold current, injection efficiency, differential gain, pulse current and a possibly present bias current. For a given laser diode, the turn-on delay can be tuned with pulse and bias current, which is interesting for many kinds of applications. A higher pulse current over threshold corresponds to faster carrier injection and thus a decreased turn-on delay. In the case of a bias current close below threshold, turn-on delay is also greatly reduced in comparison with lower or no prebias, because the carrier number is already close to threshold and the remaining part is substantially decreased.
We measure the turn-on delay using the streak camera for the time-resolved optical power and a sampling oscilloscope for the current signal. Both signals are synchronized with respect to the trailing edge of the pulse, assuming that current modulation above threshold translates directly to the optical output power. Typical measurements in dependence on bias and pulse current are shown in Fig. 8. The measured data sets are compared to rate equation-based simulations, where the turn-on delay is defined the same way as for the measurements. For these calculations, even a simpler model with only one optical mode can be used, which gives the same values as the previously presented multi-mode rate equation model.
Fig. 8. Measurement results (dots) and simulation (continuous lines) of the turn-on delay at different bias and pulse currents. This is measured on a different yet comparable green laser diode with a threshold current of 43 mA.
The simulation results show good accordance with the measured values, except in the case of no bias, where the actual turn-on delay is underestimated by the simulation model. This extra delay is caused by the capacity of the p-n junction, as extra carriers are needed to fill the depletion region before charging the quantum wells. Because the rate equation model does not cover the electronic properties of the p-n junction, this extra delay is obviously not included. If a bias is applied, the depletion region is already charged and this difference vanishes.
5. Initial red-shift
The fast initial red-shift takes place in the first nanoseconds of the optical pulse and can be well distinguished from the wavelength shift that arises from mode coupling. In the case of mode coupling, mostly one single active mode is observed that changes from one longitudinal mode to the neighboring one towards higher wavelengths. But the initial red-shift moves the central wavelength of a bunch of modes in the direction of longer wavelengths and these two effects also happen on a significantly different time scale, as visible in Fig. 6.
The fast initial red-shift originates from the blue-shift of the peak of the gain spectrum with increasing charge carrier density. While the carrier density is clamped at longer times, it is deviating from this equilibrium value during the dynamics around pulse onset. The blue-shift is caused both by band filling and screening of the internal electric field. The quantum confined Stark effect (QCSE) describes the reduced electron and hole wavefunction overlap and a larger effective bandgap due to the internal electric field and band tilting in the QWs. A high carrier density leads to stronger screening of the internal electric field and thus weakening the QCSE. So as the carrier density drops after reaching its maximum at the onset of the optical pulse (see Fig. 4), the effective bandgap is lowered, contributing to this fast red-shift.
In contrast to the mode coupling effects, the fast initial red-shift was not yet included in the model of T. Weig et al. [26]. So the gain model is modified to contain the dependence of the central wavelength $\lambda _{\mathrm {c}}$ in the parabolic gain spectrum ($A_{\mathrm {p}}$) on the carrier number in the quantum wells:
The shift of $\lambda _{\mathrm {c}}(I)$ with the current $I$ can be well described with a linear relation in the range of the measurement data and it is reasonable to extrapolate this linear model also in the simulation for the currents above threshold. So the slope $\mathrm {d}\lambda _{\mathrm {c}}/\mathrm {d}I$ is evaluated from the measurements:
Connection between the current and carrier number in steady state below threshold is obtained by a simple rate equation approach, which assumes equal pumping and yields the relation $I(N)$. Differentiation to $N$ and inserting threshold carrier number $N_{\mathrm {th}}$ for $N$ leads to the following approximation:Unequal pumping of the two quantum wells has also an important influence here [24,35,36]. An unequal pumping coefficient of $\chi =0.7$ yields the $N$- and $S$-curves shown in Fig. 7 for a 10 ns long pulse. In this case, a noticeable rise of the carrier number $N_1$ over $N_{\mathrm {th}}$ is visible, which is stronger than in the case of equal pumping with $\chi =0.5$ (not shown). Note that $N_{\mathrm {th}}$ equals now the threshold value for every single one of the two quantum wells i. e. in the case of equal pumping. The carrier number exceeding the threshold (equilibrium) number leads in combination with the gain shift from Eq. (20) to a red-shifted gain maximum in the first nanoseconds. When the carrier numbers approach their equilibrium values, the gain maximum shifts back to the cw-emission wavelength. This corresponds to the initial emission of shorter wavelength light and the following red-shift towards longer wavelength. Because of the photon lifetime and the photon induced amplification, the shift of the emission wavelength happens slower than the shift of the gain curve. This leads to an observable red-shift in the first about 5 ns and not only in the time range of the very first relaxation oscillation, although the shift is strongest in this first range. So the strength of the red-shift is enhanced by stronger unequal pumping of multiple quantum wells, but this influence is moderate and the red-shift also occurs at equal pumping, as shown in Fig. 10 with a comparison of equal ($\chi =0.5$) and unequal ($\chi =0.7$, $\chi =0.8$) pumping. Considering the measurements that are shown e.g. in Fig. 2, a value of $\chi =0.7$ is found to be the most suitable for the investigated laser diode.
Fig. 10. Comparison of the initial red-shift for unequal (a, b) and equal pumping (c) at $I=2\:I_{\mathrm {th}}$.
6. Mode competition
The mode rolling behavior resulting from the interaction between neighboring longitudinal modes has been shown in (Al,In)GaN laser diodes [25,37] and is described by the symmetric and asymmetric cross saturation of the modal gain in the rate equation model, as introduced by M. Yamada [29]. The general phenomenon of mode competition occurs in most types of semiconductor lasers [38–40] and its cause has been investigated by H. Ishikawa et al. [41]. While the symmetric gain saturation can be explained using density matrix analysis, the asymmetric mode interaction was found not to be caused by spatial hole burning but by pulsations of the electron population, that have been described by A. Bogatov et al. [42]. These pulsations are induced by the beating of neighboring longitudinal modes at mode spacing frequency which equals about 80 GHz in our investigated green InGaN laser diode. In this case, the beating frequency does not affect the interband recombination, because this recombination happens too slow, but the fast intraband dynamics of the electrons inside the quantum well correspond to spectral hole burning and yield a modulation of the intraband population. The resulting population pulsations lead to a modulation of both refractive index and gain causing the asymmetric component in the gain spectrum. The theoretical framework for these calculations is four-wave mixing, that has been used by G. P. Agraval [43] to explain asymmetric mode coupling resulting from high beating frequencies of the longitudinal modes in the gigahertz range, which is the case in InGaN laser diodes.
The measurements yield similar results to the simulations in Fig. 6. The streak camera images covering the same time ranges are shown in Fig. 11 but for these measurements, a pulse current of $I=1.2\:I_{\mathrm {th}}$ is used. This causes a lower mode rolling frequency, which is a function of the injected current [25]. Apart from this frequency, the same behavior and especially the transition from the initial effects to the later mode competition dominated state is observed in good accordance to the simulations. In Fig. 11(a), not the complete emission spectrum is captured due to using the high resolution 2400 l/mm grating in the monochromator.
Fig. 11. Measured streak images with the pulse length (a) 10 ns, (b) 100 ns, and (c) 0.9 µs at $I=1.2\:I_{\mathrm {th}}$.
An important feature in this context is the temporal stability of the mode cycles. In the measurement of the 1 µs long pulse in Fig. 11(c), the first mode cycles are clearly resolved but the later ones towards the end of the pulse are blurred. This is due to jitter in the mode cycle repetition frequency, resulting from noise processes [44], which leads to such results when integrating 200 images that contain about 1000 pulses each. In the simulation in Fig. 6(c), also some irregularities in the mode cycles and variations in the repetition frequency occur, even when assuming no gain noise and no present photons at $t=0\:\mathrm {ns}$. Little variations in the real system in terms of initial photons, some gain noise or shot noise in spontaneous emission can then lead to these temporal instabilities.
To experimentally investigate this behavior, single shot measurements are performed, for which the pulse repetition rate is greatly reduced until exactly one pulse is captured per image and the images are not integrated. Some of the measurement results are shown in Fig. 12. The images contain more noise due to the missing integration over multiple pictures, but the mode dynamics is still well visible. The differences and the deviations from the regular behavior are noticeably, also it seems to occur that few active modes move through the spectrum at once. The measurements indicate that mode competition processes are rather unstable but also that mode rolling persists in longer pulses and in cw-operation and does not cease as Fig. 11(c) might suggest. Stabilization of the mode cycles after about 0.5 µs as in the simulation shown in Fig. 6(c) is not observed in the single shot measurements.
To illustrate how the blurring in Fig. 11(c) is caused by the temporal irregularities between the pulses and the integration, 50 single shot images are superimposed after the measurement. The resulting image is shown in Fig. 13 and it resembles well the measurement in Fig. 11(c).
Fig. 13. Superimposition of 50 single shot measurements of a 1 µs long pulse at $I=1.2\:I_{\mathrm {th}}$.
7. Thermal effects
Self-heating of laser diodes impacts the optical and spectral emission properties on various time scales from the nanosecond to millisecond range. The static effects like the influence of temperature on the threshold current, center wavelength, slope efficiency, and recombination rates are well investigated [45,46]. But beneath these, the dynamic behavior of these mechanisms in pulsed laser operation must also be understood looking towards applications. Two different self-heating effects have been distinguished by W. Scheibenzuber et al. [16] that correspond to heating of the crystal lattice and the charge carrier plasma. While heating the crystal lattice changes the modal refractive index and thus the position of the longitudinal modes [15], heating of the charge carrier plasma influences the band gap and population inversion (described by the Fermi-Dirac distibution) resulting in a red-shift of the gain spectrum.
Both the red-shift of the mode positions and the central wavelength can be observed experimentally in the streak camera measurements. Especially the mode shift can be used to track the thermal behavior of the laser diode over up to six orders of magnitude in time, which gives information about the different time scales at which the separate parts of the laser diode heat up. In order to relate the mode shift with a temperature difference, the mode spectrum in cw operation has been measured with the laser diode mounted in an actively temperature-controlled heatsink [16]. For this, the steady-state temperature difference between the heatsink and the active region is assumed to be constant, so that external temperature changes directly translate to the active material. So the mode shift can be measured with a streak camera setup and evaluated to receive the quantitative temperature change over time.
In Fig. 14, the self-heating behavior is shown for green and blue InGaN laser diodes and a red AlGaInP laser diode for comparison. The three devices exhibit different heating processes, which take place on distinct time scales. They can be characterized by their different slopes which correspond to the thermal capacity of the separate laser diode parts. After the fast heating of the charge carrier plasma which can not be investigated by tracing the mode shift [16], the crystal lattice in the active region heats up. This is followed by the whole chip, the substrate, the metal stem, the TO can and finally the passive heatsink. Measurement of the self-heating processes allows to attribute the different slopes and time scales to the corresponding parts. Comparing the experimental results to a rather simple thermal model, that takes the various thermal capacities and thermal resistances into account, enables estimating these quantities. In contrast to such a qualitative approach, quantitative results could be expected from finite element modeling, but because this is unrelated to the rate equations, it is outside of the scope of this article.
Fig. 14. Self-heating behavior $\Delta T (t)$ of commercial red, green, and blue laser diodes, derived from the longitudinal mode shift in streak camera measurements on five different time scales (indicated by different colors). The absolute heating rate depends mostly on the current density, which is for green 6.7 kA cm$^{-2}$, for blue 3.5 kA cm$^{-2}$, and for red 2.5 kA cm$^{-2}$.
In pulsed laser operation, also cooling effects play an important role, determining the initial temperature for the following pulse in dependence on the spacing time between the pulses. Amongst other methods [47], this can be investigated by measuring a short probe pulse using the streak camera, which succeeds after a defined long heating pulse. The cooling time between the two pulses can be varied and the longitudinal mode position from the end of the heating pulse is compared to the beginning of the probe pulse. This allows quantifying the temperature difference from the mode shift as a function of time.
In order to include the thermal effects directly in the rate equation model, the temperature dependence of essential parameters has to be included into the rate equations. This means that at least recombination rates, optical gain, and the internal losses have to be characterized and described as temperature dependent quantities.
8. Conclusion
We present an extensive description of the static properties and dynamic processes in InGaN/GaN laser diodes in cw operation down to few nanosecond long pulses. A streak camera setup allows the experimental investigation of the full longitudinal mode dynamics in high spectral and temporal resolution. Our expanded multi-mode rate-equation model is a powerful tool to describe, understand, and simulate the spectral and temporal behavior including turn-on delay, relaxation oscillations and the fast initial red-shift at pulse onset as well as mode competition. It is straightforward to further include effects like noise, fluctuations, and photon statistics into the model [43]. Not included is the time regime of pulses which are short with respect to the cavity round trip time, like in the case of mode locking.
These results are relevant regarding recent applications that involve fast laser diode modulation, e.g. in projection, AR/VR/MR or optical communication. The spectral-temporal onset dynamics strongly determine the time-integrated spectral width in the case of repeated pulses, which can thus be tuned by changing e.g. pulse length or current. Interaction effects between subsequent pulses also play an important role in this context [48], so the time between the pulses and bias current influence the resulting spectral characteristics as well. Also static laser diode operation is discussed in terms of spectral properties and it is important to note that in general also during cw operation, mode competition effects take place if the device is not actively stabilized.
Another recent topic is spatial dynamics, both in the near field and the far field of laser diodes. Next to the longitudinal mode dynamics discussed in this article, also lateral dynamics and formation of filaments is of particular interest for broad ridge laser diodes that are typically used in high-power applications, and which will thus be subject to future research.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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