Mode Competition Between Longitudinal and Lateral Modes in III-Nitride Broad-Ridge Laser Diodes

Mode competition in semiconductor laser diodes typically causes a continuously repeating spectral modulation by periodic mode hopping through longitudinal modes. High-power laser diodes include a broad ridge waveguide that supports multiple lateral mode operation, interacting with the dynamically changing spatial distribution of charge carriers. We investigate the time-dependent behavior of longitudinal-lateral mode competition in III-nitride-based broad-ridge laser diodes, involving several longitudinal mode combs, each corresponding to a different lateral mode. For this, we perform lateral scans of streak camera measurements, take single shot images, and develop a rate equation model for numerical simulation of the process. We observe mode competition that involves all active lateral modes, driven by asymmetric mode coupling. In particular, the small wavelength spacing between adjacent lateral modes and spatial lateral hole burning are influencing mode competition dynamics in broad-ridge devices. Furthermore, we investigate the role of mode clustering on the dynamical behavior and how long-time periodicity depends on the laser diode properties.

SECTION I. Introduction

Broad-ridge laser diodes are among the main concepts for effective power scaling of semiconductor lasers, using a single emitter [1], [2], [3], [4], [5]. Other options, such as bars with multiple emitters or stacks of bars offer higher total optical power at the expense of beam quality [6], [7], [8]. Photonic crystal surface emitting lasers (PCSELs) show the potential for high power and brightness [9], [10], [11], while their wall plug efficiency is currently lower than for edge-emitting lasers. High-power laser diodes are demanded for medical applications, industrial metal processing [6], optical pumping sources e.g. for other lasers [12], [13], [14], sensing, especially LiDAR (light detection and ranging) [15], and many more applications. Especially blue lasers have advantages for lighting [16], [17] and the machining of copper and gold [18]. Understanding the time-dependent behavior of a laser diode in terms of power, spectrum, and beam shape (i.e. lateral mode dynamics) is required for precise short pulse operation in the nanosecond to microsecond regime [19]. However, also for continuous wave (cw) operation, a modulation can arise from dynamic effects. As an example, a time-periodic wavelength variation in combination with a wavelength-sensitive optical element like a waveplate results in an intensity modulation.

Compared with narrow-ridge edge-emitting laser diodes, broad-ridge devices reduce gain saturation and avoid catastrophic optical damage (COD) at a given output power by scaling up the active volume and the facet area [20], [21]. The ridge waveguide is usually between 5 µm and 100 µm wide, which leads to multiple lateral mode operation [22], [23], [24]. The interaction between the lateral modes and the spatial charge carrier distribution has been studied in different material systems and gives rise to various dynamic effects, such as mode switching [25], [26], [27], filamentation [28], [29], [30], [31], oscillations [25], [32], or spatiotemporal chaos [33], [34], [35], [36], [37].

Longitudinal mode competition was mostly studied in conventional narrow-ridge laser diodes. It appears as continuously repeating process, where the active longitudinal mode is switching to the adjacent mode towards longer wavelength and this continues through the whole laser spectrum [19], [38], [39], [40], [41]. When the long-wavelength side of the spectrum is reached, the gain is too low to continue this process further and mode competition starts again from the gain maximum. This mode rolling process arises from asymmetric coupling of two modes that are close in wavelength, where the gain of the longer-wavelength mode increases and the shorter-wavelength mode receives less gain [42]. This interaction is due to the beating of both optical fields at the frequency difference $\Omega =\omega _{2}-\omega _{1}$, which modulates the charge carrier population with $\Omega$ [43]. With the carrier population, also the refractive index and gain are modulated, forming dynamical gratings that affect both modes again in terms of refractive index and gain [44], [45], [46]. The gain variation is classified into a symmetric and an asymmetric contribution, regarding its wavelength dependency. Particularly the asymmetric term is responsible for mode rolling, whereas the symmetric term accounts for spectral hole burning.

Longitudinal mode competition is well understood in narrow-ridge edge-emitting lasers, originally investigated in infrared and red lasers and later in the blue and green wavelength range in III-nitride-based devices. However, there are few reports on mode competition in broad-ridge laser diodes [47], [48]. The main challenges are to include the effect of multiple lateral modes, i.e. multiple longitudinal mode combs, as well as the role of lateral carrier dynamics.

In the following, we present our experimental investigations on 442 nm broad-ridge laser diodes by high-resolution spectroscopy and streak camera measurements, both covering the lateral dimension of the device using a scanning setup. Furthermore, we develop a rate equation based simulation model to complement our experimental observations and to understand the coupled longitudinal-lateral mode competition. We also assess the role of mode cluster formation on the dynamic effects and briefly investigate the stability of the mode competition process under parameter variation based on our simulations.

SECTION II. Spectral-Spatial Distribution of Longitudinal and Lateral Modes

To experimentally observe the static and dynamical behavior of broad-ridge laser diodes over the whole lateral range of the ridge waveguide ($w=40\,$µm), we use a scanning technique that has been detailedly presented in [49]. We employ a 10 µm wide vertical slit (perpendicular to the lateral direction), where the magnified near field of the laser diode is imaged onto (see Fig. 1). In this way, every point in the lateral near field can be observed one after another, by moving the imaging lens with the help of a motorized linear stage. The two lenses form a Gaussian telescope with 20 x magnification for accurate imaging of the lateral modes [50]. The light passing through the slit is then guided either to a high-resolution double spectrometer or to the streak camera setup including a grating monochromator. From the laterally scanning streak camera measurements, we get spatial-spectral-temporal resolved data, which will be analyzed in section III.

The high-resolution spectra taken at each position during a lateral near-field scan form the time-averaged lateral-spectral intensity distribution, as shown in Fig. 2. The longitudinal modes appear as horizontal lines at specific wavelengths with their corresponding individual lateral intensity profile. The spectrum contains several longitudinal mode clusters, i.e. a spectral modulation of mode intensity. Between the clusters, a few modes are less active but not completely suppressed, which indicates a weak modulation only. Due to the regular structure with a period of $\Delta \lambda _\text{cluster}=0.15$ nm, we attribute the clustering effect to a weak interaction with substrate modes, as previously reported in [51]. This can be understood as an effective spectral modulation of the mirror losses. Other possible mechanisms behind the formation of mode clusters include structural imperfections along the cavity, such as defects [52] or variations of the quantum well thickness [53], as well as dynamic mode cross-saturation due to spectral hole burning [40].

Fig. 1.

Schematic view of the measurement principle. With the help of the Gaussian telescope, the laser diode near field is imaged onto the vertical slit. By moving the achromatic lens, specific lateral positions in the image are selected to study the spectral-temporal characteristics with the streak camera or high-resolution spectra using the double monochromator.

Fig. 2.

Longitudinal-lateral mode configuration measured by lateral resolved high-resolution spectroscopy. The laser diode is driven in 30 ns long pulses using a current of 2 $I_\text{th}$. Here we use a nonlinear color function to display the high dynamic range of the measurement.

The longitudinal mode spacing of one mode comb is $\Delta \lambda =26\,\text{pm}$, corresponding to a cavity length of 1200 µm and a group refractive index of 3.15. With a spectral resolution of 7 pm, it is possible to distinguish multiple longitudinal mode combs that have a small wavelength offset against each other. Each mode comb corresponds to a different lateral mode (or supermode, see [54]), due to their slightly different effective group velocity. Note that adjacent modes of different combs do not need to belong to the same longitudinal mode order. In other words, the offset between different combs (different lateral mode) can become larger than the longitudinal mode spacing, especially for a long cavity and a high number of active lateral modes. In red or infrared laser diodes, mode groups involving all lateral modes are observed to form a parabolic shape [36], [55], [56], [57], [58]. In our case, these mode groups are usually overlapping, thus the wavelength spacing between some modes can become very small [37], [56], [59]. In Ref. [60], we approached this question using a Fabry-Pérot interferometer in combination with the double spectrometer. Up to 11 separate mode combs and wavelength spacings as small as 1 pm could be observed in the same device that we investigate in this article. In the measurement in Fig. 2, not all longitudinal mode combs can be resolved, however such an overview over the whole spectrum and lateral scans are not possible in the highest-resolution measurement with the Fabry-Pérot interferometer.

The measurement in Fig. 2 shows not only different lateral intensity distributions between the longitudinal combs, but also noticeable differences from one spectral cluster to the other. This means, the gain for each lateral mode depends on their position in the spectrum, or in other words, on the longitudinal mode they belong to. Their individual strength depends on the combined longitudinal-lateral gain dynamics, which involves the effects of lateral carrier dynamics e.g. spectral and lateral spatial hole burning and mode competition.

SECTION III. Mode Competition Involving Multiple Mode Clusters

In order to approach the lateral-spectral-temporal dynamics of mode competition in this system, we perform analog integrated streak camera measurements during a stepwise scan through the lateral near field image. Here we investigate the time range of 100 ns long pulses, which are repeated at 10 kHz, corresponding to a low duty cycle of 0.1%. Despite the continuous nature of the mode competition process, which is also present in cw operation, we investigate this effect in pulsed conditions for two main reasons: (i) to reduce self-heating and separate thermal effects from the carrier and optical mode dynamics; (ii) to restart the mode competition process with each pulse to selectively observe the deterministic dynamics, reducing chaotic influences. The time window of 100 ns is sufficient to study mode competition in detail and still allows for integrated measurements without washing out the signal.

The streak camera images in Fig. 3(a), (b) clearly show mode competition that involves several mode clusters. In each cluster, the intensity is modulated, and during the high-intensity periods, the emission wavelength undergoes a continuous red-shift, as typical for mode competition. The modulation frequency of one cluster lies between 60–100 MHz for the current of 384 mA, but is generally faster for higher currents, as reported for usual longitudinal mode competition [40]. Over the measured time range of 100 ns after pulse onset, the frequency decreases slightly, which we attribute to onset effects regarding carrier dynamics [19] or self-heating [61] that influence the first few ten nanoseconds. The mode competition frequency is shared between all involved mode clusters and we observe a fixed phase relation between the intensity modulation in adjacent mode clusters. This indicates coupled dynamics and suggests a mode competition process that extends across multiple mode clusters.

Fig. 3.

Integrated streak camera images of the spectral-temporal dynamics at two positions in the near field (a, b). From the same measurement, the spectral-lateral crosssections of intensity (c) and modulation amplitude (d) are shown, where the two x-positions of (a) and (b) are marked with vertical dashed lines. The horizontal dashed lines indicate the five strongest mode clusters, which also appear in (d) at the same wavelengths. The device is driven at the current 1.75 $I_\text{th}$.

Fig. 3(c) shows the intensity distribution over the spectral-lateral crosssection, similar to the high-resolution measurement in Fig. 2. For this, the data is averaged over time from 50 ns to 100 ns to exclude onset effects. During the same time span, the modulation amplitude is evaluated for each data point from the same wavelength-position space, as shown in Fig. 3(d). Here we use the standard deviation of the intensity over time to measure how strongly the signal is modulated due to mode competition. This is the most robust method, and more complex approaches like peak detection or Fourier transformation did not produce more reliable or clearer results. The modulation amplitude is mostly correlated to the intensity, which means that all the active modes are generally participating in mode competition - not only the strongest lateral mode or only few clusters. Some small deviations might arise from weakly active modes that are quickly overcome by their neighboring modes or from some modes with a small spatial overlap to their spectrally surrounding modes, which limits their coupling strength. The noisy character of Fig. 3(d) arises from the laser dynamics itself. While the spectral-lateral intensity distribution is stable, the visibility and the frequency of mode competition dynamics varies noticeably between two measurements, even without changing the lateral position. To further investigate the unstable behavior of mode competition, single shot measurements are considered in the next section. Furthermore, from these measurements alone, it is not obvious whether mode competition processes happen mostly separately in each cluster with only little coupling between the clusters, or if the same process covers all mode clusters and the active modes jump from cluster to cluster. In the following, we use single shot measurements to clarify this.

Fig. 4 shows four single shot images taken under the same conditions, illustrating the variations in individual laser pulses. Instability of the mode competition process is observed in terms of which modes or mode clusters are dominantly active, their sequence, and repetition frequency. The regular mode rolling, where one mode after the other becomes active, only occurs in some instances. However, switching between different modes at a characteristic frequency is observed in each frame. The differences between single pulses are most likely explained by the influence of noise sources [38], [39], [62] (photon shot noise and current injection) and by thermal fluctuations, which affect many properties of the laser diode. But even in the absence of any noise, stable and periodic mode competition requires a certain combination of device parameters, especially when multiple lateral modes are involved, as we can learn from numerical simulation, which is presented later. A larger number of single shot images are included in the supplementary material in the form of a movie.

Fig. 4.

Single shot measurements taken at one position under constant conditions at a current of 1.5 $I_\text{th}$.

Apart from the instability, the single shot measurements give insights about the mechanism of mode competition involving several lateral modes and mode clusters. At most points in time, only one mode cluster is dominantly active and after 10 ns to 20 ns, the intensity is shifted to a different mode cluster. While one mode cluster is active, mode rolling occurs inside this cluster, which we observe as small red-shifts in the integrated measurements (Fig. 3(a) and (b)). After the completed red-shift, a different mode cluster is favored due to the asymmetric gain component, but this behavior is not sequential and sometimes multiple clusters are amplified at the same time, indicating a chaotic component in this process.

SECTION IV. Mechanism of Mode Competition Involving Multiple Lateral Modes

A. Model for Longitudinal-Lateral Mode Competition

In the following, we set up a rate equation based model to describe the time-dependent dynamics of the laterally distributed charge carrier density $N(x,t)$ and the photon density $S_{kl}(t)$ in the combined mode $(k,l)$, where $k$ is the longitudinal mode number and $l$ is the order of the lateral mode. The first step is to calculate the spatial intensity distribution $M_{l}(x)$ of all lateral modes by solving the wave equation. The complex effective refractive index profile is given by: \begin{align*} \widetilde{n}(x)=& n_\text{slab}+\Gamma \frac{\mathrm{d}n}{\mathrm{d}N}\left[ N(x)-N_\text{tr}\right]-\frac{i}{2k_{0}} \frac{\mathrm{d}g}{\mathrm{d}N}\left[ N(x)-N_\text{tr}\right] \\ & +\left\lbrace{\begin{array}{ll}\Delta n_\text{ridge} &, |x|\leq w/2 \\ 0 &, |x|> w/2 \end{array}}\right. \\ =& n_\text{slab}-\frac{R}{2 k_{0}} a \left[ N(x)-N_\text{tr}\right]-\frac{i}{2k_{0}} a\left[ N(x)-N_\text{tr}\right] \\ & +\left\lbrace{\begin{array}{ll}\Delta n_\text{ridge} &, |x|\leq w/2 \\ 0 &, |x|> w/2 \end{array}}\right.\tag{1} \end{align*} This includes the effective index of the slab waveguide $n_\text{slab}$ and the refractive index step due to the ridge $\Delta n_\text{ridge}$, as well as the influence of the carrier density $N(x)$ on the index and the gain. As we consider the effective refractive index of a laser mode, we use the confinement factor $\Gamma$ and $g=\Gamma G$ denotes the modal gain, whereas $G$ is the material gain. Here, $k_{0}$ is the vacuum wavenumber and $N_\text{tr}$ stands for the transparency carrier density. The antiguiding factor is defined as $R=-2 \Gamma k_{0} \frac{\mathrm{d}n/\mathrm{d}N}{\mathrm{d}g/\mathrm{d}N}$ [63], [64] and we use the small signal, linear gain approximation with the coefficient $a=\mathrm{d}g/\mathrm{d}N$.

We can now use the squared effective index to solve the one-dimensional wave equation: \begin{equation*} \left[ \frac{\partial ^{2}}{\partial x^{2}}+k_{0}^{2} \widetilde{n}^{2}(x)-\beta _{l}^{2}\right] \psi _{l}(x)=0 \tag{2} \end{equation*} As result, we obtain the propagation constants $\beta _{l}$ and wave functions $\psi _{l}(x)$ of each lateral mode, which give the mode intensity profiles $M_{l}(x)=|\psi _{l}(x)| ^{2}$.

The equation for the carrier density involves current injection, spontaneous and stimulated recombination, and lateral diffusion: \begin{align*} & \frac{\partial }{\partial t}N(x,t) = \frac{\eta _\text{inj} j(x)}{q d_\text{QW}} - \frac{N(x,t)}{\tau _\text{sp}(N(x,t))} \\ & - v_\text{gr} \sum _{k, l} \left[ a(N(x,t)-N_\text{tr})-\frac{b}{2}\left(\lambda _{kl}-\lambda _{0}[N_{l}(t)]\right) ^{2}\right] \\ & \times S_{kl}(t) M_{l}(x) + D \frac{\partial ^{2}}{\partial x^{2}} N(x,t)\tag{3} \end{align*} We describe the lateral current density profile $j(x)$ by a step function, which is equal to the 2D current density $j_{0}$ in the region of the ridge and zero beside the ridge. For numerically solving the rate equation system, we use a slightly rounded profile to avoid discontinuities. Note that $j(x)$ is a current density per area, but we use a carrier density per volume. The carrier injection rate is further described by the injection efficiency $\eta _\text{inj}$, the elementary charge $q$, and the total thickness $d_\text{QW}$ of the (multi) quantum well. The spontaneous carrier lifetime is expressed with the ABC model, including Shockley-Read-Hall- ($A$), radiative ($B$), and Auger-Meitner-recombination ($C$): \begin{equation*} \tau _\text{sp}(N) = \left(A +B N+C N^{2}\right) ^{-1} \tag{4} \end{equation*} The stimulated emission rate is summed over all modes and includes group velocity $v_\text{gr}=c/n_\text{gr}$, where $c$ is the speed of light and $n_\text{gr}$ is the group refractive index, as well as curvature $b$ of the gain spectrum, wavelength of each mode $\lambda _{kl}$ and the carrier density-dependent wavelength of the gain peak $\lambda _{0}$. We approximate this by a linear relationship $\lambda _{0}(N)=\lambda _{0}+\frac{\mathrm{d}\lambda _{0}}{\mathrm{d}N}(N-N_\text{th})$ that is valid near the threshold carrier density $N_\text{th}$. For lateral carrier diffusion, the diffusion constant $D$ is used. The mode-specific carrier density $N_{l}$ corresponds to the weighted average of $N(x)$ regarding the lateral mode function $M_{l}(x)$.

The rate equation for the photon density $S_{kl}(t)$ includes most importantly the modal gain $g_{kl}$ as well as the optical losses $g_\text{th}=\alpha _\text{int}+\alpha _\text{mir}$, namely the internal and mirror losses respectively. \begin{equation*} \frac{\partial }{\partial t}S_{kl}(t) = v_\text{gr} \left(g_{kl}-g_\text{th} \right) S_{kl}(t) + \beta _\text{sp} B N_{l}^{2}(t) \tag{5} \end{equation*} The last term describes the spontaneous emission into each mode using the factor $\beta _\text{sp}$. The modal gain contains the following contributions: \begin{align*} g_{kl} =& a\left(N_{l}-N_\text{tr} \right) -\frac{b}{2}\left[ \lambda _{kl}-\lambda _{0}(N_{l})\right] ^{2} -B_\text{sat} S_{kl}\\ &-\sum _{k^{\prime }, l^{\prime }} \left(D_{k k^{\prime } l l^{\prime }}+H_{k k^{\prime } l l^{\prime }} \right) U_{l l^{\prime }} S_{k^{\prime } l^{\prime }}\tag{6} \end{align*} The first two terms stand for the linear gain dependency on the carrier density and the parabolic spectral distribution. Self-saturation of each mode is described with the coefficient $B_\text{sat}$ and for cross-saturation effects the coefficients $D_{k k^{\prime } l l^{\prime }}$ and $H_{k k^{\prime } l l^{\prime }}$ are used, referring to interaction between the modes $(k, l)$ and $(k^{\prime }, l^{\prime })$. These expressions for self-saturation and symmetric ($D_{k k^{\prime } l l^{\prime }}$) and asymmetric ($H_{k k^{\prime } l l^{\prime }}$) mode coupling have been developed to model mode competition in narrow-ridge infrared laser diodes [38], [45], [46]. The spatial overlap integral $U_{ll^{\prime }}$ reduces the interaction strength between different lateral modes $l$ and $l^{\prime }$ that only overlap partially [65]. The mode interaction coefficients are defined as follows, where we insert the wavelengths $\lambda _{kl}$ of our longitudinal-lateral modes: \begin{align*} B_\text{sat}&=\frac{9}{2} \frac{\pi c}{\epsilon _{0} n_\text{gr}^{2} \hbar \lambda _{0}}\Gamma \tau _\text{in}^{2} a \left| R_{cv}\right| ^{2} \left(N-N_\text{sat} \right) \tag{7}\\ D_{k k^{\prime } l l^{\prime }}&=\frac{4}{3} \frac{B_\text{sat}}{\left(2 \pi c \tau _\text{in} /\lambda _{kl}^{2} \right)^{2} \left(\lambda _{k^{\prime }l^{\prime }}-\lambda _{kl} \right)^{2} +1} \tag{8}\\ H_{k k^{\prime } l l^{\prime }}&=\frac{3 \lambda _{kl}^{2} R a^{2}}{8 \pi n_\text{gr}} \frac{N-N_\text{tr}}{\lambda _{k^{\prime }l^{\prime }}-\lambda _{kl}} \tag{9} \end{align*} Here, we use the dielectric constant $\epsilon _{0}$, the reduced Planck constant $\hbar$, the transition dipole moment $R_{cv}$, the intraband relaxation time $\tau _\text{in}$, and the saturation carrier density $N_\text{sat}$. The wavelength dependency of the symmetric mode interaction $D_{k k^{\prime } l l^{\prime }}$ corresponds to a Lorentzian with a full width at half maximum of 8.6 nm (using the parameters from Table I), so it is not varying strongly over the width of the laser spectrum of below 1 nm. However, the asymmetric coupling coefficient $H_{k k^{\prime } l l^{\prime }}$ depends strongly on the wavelength difference between two modes as it is proportional to $1/(\lambda _{k^{\prime }l^{\prime }}-\lambda _{kl})$. Therefore, the smallest wavelength spacings between two different mode combs have a dominant effect on mode competition.

TABLE I Parameter Set for Numerical Solution of the Rate-Equation Model

An important aspect of our model is the definition of the wavelengths of all modes. The longitudinal mode combs are characterized by the typical mode spacing \begin{equation*} \Delta \lambda =\frac{\lambda _{0}^{2}}{2 n_\text{gr} L}, \tag{10} \end{equation*} where $L$ is the cavity length. For higher lateral modes, a constant offset is added to the whole mode comb due to the different effective refractive index. Using a simple box model for the cavity (see e.g. [56]), the offset between the lateral mode combs can be estimated. However, these values are very sensitive to uncertainties in ridge width or refractive index and in many cases the wavelength offset is larger than the mode spacing $\Delta \lambda$. For this reason, we distribute the lateral mode combs evenly across one longitudinal mode spacing, i.e. the offsets range between 0 and $\Delta \lambda$. Small differences in the spectral location of the mode combs or a different spectral ordering of the lateral modes are unlikely to cause largely different dynamics. If such values were known precisely, they could easily be used in this model.

B. The Role of Multiple Lateral Modes for Mode Competition

In the following simulations, we first approach the case of four (or six) lateral modes and a ridge width of 10 µm, to develop a fundamental understanding of the fast mode switching process and the long-time periodic behavior. These insights can then be generalized to lasers with a broader ridge waveguide and more involved lateral modes, such as the experimentally investigated 40 µm ridge laser diode with up to 11 mode combs. We solve the rate (3) and (5) numerically using the parameter set presented in the methods section, Table I. The resulting mode competition behavior is plotted in Fig. 5(a), showing the time-dependent intensity of each longitudinal-lateral mode at their respective wavelength. Here, the mode competition process appears unstable and only shows short-time periodicity in the sense of repeated hopping towards the next closest mode in the direction of longer wavelengths. However, no long-time periodicity is observed, which means that each cycle through the spectrum shows a different spectral-temporal behavior. We consider this as an analogy to the short range order in amorphous materials, where no long-range periodicity exists. This result reflects the unstable dynamics that we observe experimentally, showing a repetitive process with substantial deviations in each cycle. Also the mode competition frequency is in the range of 50-80 MHz, similar to the measurements, although this is additionally influenced by the true number of involved lateral modes and the current. Fig. 5(b) illustrates the mode hopping process that repeats regularly on short time scales of a few ten nanoseconds. Every few nanoseconds, a new mode on the longer-wavelength side of the currently active modes is switching on and modes with shorter wavelengths lose their intensity again. This involves all the mode combs, i.e. all the lateral modes, in the same way. Thus, longitudinal mode rolling also causes intensity variations between the lateral modes.

Fig. 5.

Simulated spectral-temporal dynamics involving four lateral modes, using realistic parameters ($b=74$ µm$^{-3}$, $\tau _\text{in}=25\,\text{fs}$, $R=5.9$) and a current of 2 $I_\text{th}$. (a) Mode competition dynamics over 500 ns and the whole wavelength range. (b) A small section of (a), as indicated by the dashed, white rectangle. Here, the contributions of the four lateral modes are marked in different colors.

However, to better analyze how mode competition works involving multiple lateral modes, we investigate the stable, periodic case. For this, we change three parameters that specifically influence the mode dynamics, namely the gain spectrum curvature $b$, the intraband relaxation time $\tau _\text{in}$, and the antiguiding factor $R$. Using the adjusted parameter set ($b=236$ µm$^{-3}$, $\tau _\text{in}=100\,\text{fs}$, $R=3$), stable mode competition is observed involving e.g. four or six lateral modes, see Fig. 6(a) and (c), respectively. The first important factor for longitudinal-lateral mode competition is the number of active lateral modes. With more active lateral modes, their wavelength spacing is smaller and the additional gain from mode coupling becomes stronger (see (9)), which results in the higher mode competition frequency we find in Fig. 6(c).

Fig. 6.

Simulated spectral-temporal behavior at 2 $I_\text{th}$. Systems of (a) four and (c) six lateral modes are compared using the adjusted parameters ($b=236$ µm$^{-3}$, $\tau _\text{in}=100\,\text{fs}$, $R=3$). For the case of one lateral mode (b), the mode spacing is artificially decreased to 50% $\Delta \lambda$. This is comparable to the system of two lateral modes (d), where lateral carrier dynamics are the major difference. In (b) and (d), the parameters ($b=236$ µm$^{-3}$, $\tau _\text{in}=50\,\text{fs}$, $R=3$) are used for stability reasons.

Apart from the major influence of the mode spacing, lateral carrier dynamics is the second important factor. To assess this effect, we compare the dynamics of a single mode comb with only half of the usual mode spacing (see Fig. 6(b)) with the case of two lateral modes (see Fig. 6(d)), where adjacent modes of different combs have the same wavelength distance as in the first case. The system of two lateral modes exhibits faster mode competition, even though the asymmetric and symmetric mode interactions are weaker due to the spatial overlap of both modes that is smaller than unity. However, the lateral carrier distribution varies dynamically according to the switching between both lateral modes, whereas it stays practically constant in the case of a single lateral mode. Here, we clearly identify lateral spatial hole burning as an additional driving force leading to faster mode competition if more than one lateral mode is involved. The case of two lateral modes (Fig. 6(d)) is used as an example to illustrate the equal contribution of both modes in mode rolling in Supplementary Figure S1.

To investigate the mode rolling dynamics more closely, let us focus on the simulation shown in Fig. 6(a) using four lateral modes. After a build-up time of about 100 ns, typical mode rolling towards longer wavelengths starts, repeating in a nearly periodic way. At each point in time, all four lateral modes are generally active, but only one longitudinal mode of each comb is strong. This means, there are four active modes, belonging to the four lateral modes, with minimal wavelength spacing. In Fig. 7, as an example we focus on a later part of the simulation from 400 ns to 450 ns. Fig. 7(a) illustrates how switching to new modes at longer wavelength is accompanied by overshooting intensity of the respective lateral mode. The successive switching of the dominant lateral mode is also reflected in the lateral intensity distribution over time, as shown in Fig. 7(b). Such repeating variations should be observable in the experiment as well, but in our case the mode competition process is too irregular and such variations in the near field cannot be distinguished from noise. According to the intensity distribution, also the spatial distribution of charge carriers varies over time, as depicted in Fig. 7(c). This mainly shows the effect of spatial hole burning, where the carriers are depleted faster in the regions of high intensity, whereas they accumulate in low-intensity areas. Lateral carrier diffusion works against this effect, so a high diffusion coefficient leads to smaller variations in carrier density. Such an inhomogeneous carrier distribution again affects the optical modes, because the gain of high-intensity lateral modes is reduced but low intensity modes receive additional gain after the carrier density has locally risen over the threshold value $N_\text{th}$.

Fig. 7.

Analysis of simulated longitudinal-lateral mode dynamics using four lateral modes, see Fig. 6(a). The time-dependent photon density in each lateral mode (a) is shown during switching through the spectrum, which also influences the near-field intensity profile (b) and the spatial carrier distribution (c) over time. The gain spectrum for all four mode combs at 424 ns is shown as an overview image (d), including the normalized photon density in each mode. The switching process from dominating mode 2 to mode 1 is shown at three points in time (e-g), covering a smaller wavelength range. The three time steps are marked in (a), (b) and (c) as gray lines.

If one mode profits not only from an increased local carrier density, but also from asymmetric mode coupling, i.e. if it follows on the long-wavelength side of the currently dominating mode, the excess gain above threshold causes a fast rise in intensity. This process is depicted in Fig. 7(d)–​(g) as gain spectra that contain all four mode combs. Fig. 7(d) shows the parabolic basic shape as well as some gain offset between the lateral modes and asymmetric mode interaction near the lasing modes. The gain offset is caused by the spatial overlap of each mode with the lateral distribution of charge carriers and thus strongly active modes are suppressed by spatial hole burning, whereas weak modes are favored. This promotes faster switching from mode to mode. Especially if carrier diffusion is slow, this effect is even more pronounced. The asymmetric effect of mode competition is strongest around the high-intensity modes, according to (9). Both contributions to the total gain are on the same order of magnitude, around 0.2 cm$^{-1}$ to 0.4 cm$^{-1}$, corresponding to a rise time of about 1.2 ns to 2.5 ns. The detailed switching process is shown as an example in Fig. 7(e)–​(g), where initially mode 2 predominates and the modes 1 and 4 are weaker. Both modes 1 and 4 receive additional gain due to a locally high carrier density, but the asymmetric contribution is mainly responsible for raising the gain for the next mode from comb 1. While this mode rises in intensity, the initially high carrier density $N_{1}$ accessible to mode 1 decreases to below the corresponding values for the other modes, which is indicated by the vertical shift of the gain spectra. At the same time, the previously active mode from comb 1 becomes diminished, especially due to the asymmetric mode coupling that further decreases the gain. On the long-wavelength side, the rising mode is overshooting the other active modes, because asymmetric mode interaction increases its gain beyond the usual level. In Fig. 7(g), the gain of the next mode (lateral mode 4) is already high above $g_\text{th}$ and it will now start to rise in intensity (see also Fig. 7(a), around 427 ns), continuing the mode competition process. An animation of this process is available in the supplementary material.

C. Influence of Mode Clustering on Dynamic Effects

To model the formation of mode clusters, we introduce a small modulation of the optical losses that enter into $g_\text{th}$ in (5). This is written as: \begin{equation*} \widetilde{g}_\text{th}(\lambda)=g_\text{th}+\Delta g_\text{mod} \cos \left[ 2\pi \nu (\lambda -\lambda _{0})\right] \tag{11} \end{equation*} From our measurements, we estimate the spectral frequency to be $\nu =(\Delta \lambda _\text{cluster})^{-1}=(\text{0.15}\,\text{nm})^{-1}$ and the modulation depth is chosen as $\Delta g_\text{mod}=0.1\,\text{cm}^{-1}$. We use four lateral modes and the same parameters as in Fig. 6(a) to simulate mode competition under the influence of mode clustering, as presented in Fig. 8. Three mode clusters participate in mode rolling and we observe similar dynamical behavior as in the measurements, see Fig. 3(a), (b). In each cluster, mode competition leads to mode switching towards longer wavelength and when the wavelength-dependent losses become too high, the next mode cluster is favored and mode competition continues there. The frequency of mode competition is higher than in the unperturbed case in Fig. 6(a), because the mode hopping between adjacent clusters happens without delay and thus the end of the active spectrum is reached faster, before the process starts over from the shorter-wavelength side. Furthermore, the shorter-wavelength cluster starts already before the active modes on the long-wavelength side have vanished, which contributes to faster mode rolling too. The transition between mode competition with and without clustering is presented in Supplementary Figure S2. This shows that already a small gain modulation of $\Delta g_\text{mod}=0.05\,\text{cm}^{-1}$ is sufficient to form mode clusters.

Fig. 8.

Simulated mode competition with spectral modulation of the optical losses, leading to mode clustering. The same conditions are used as in Fig. 6(a).

For some parameter sets, the introduction of mode clusters through modulation of the optical losses can cause instabilities in the mode competition process, where the behavior is stable and periodic if there is no cluster formation. An example of this is given in Supplementary Figure S3.

SECTION V. Stability of Mode Competition

As mentioned earlier, some modification of the simulation parameters is necessary to obtain a stable and long-time periodic mode competition process. We noticed this especially for a higher number of active lateral modes. In the following, we give an overview over the regions of stability and instability in the parameter space, where we investigate the influence of the gain curvature $b$, intraband relaxation time $\tau _\text{in}$, and antiguiding factor $R$. These parameters selectively determine the three main aspects of gain dynamics, which are the parabolic gain dispersion, i.e. the built-in gain difference between two modes, symmetric and asymmetric cross-saturation. We calculated several scenarios for different values of $b$ and $\tau _\text{in}$ using either $R=3$ or $R=5.9$. For each parameter set, the mode competition dynamics is subjectively classified into a fast chaotic behavior, stable mode rolling, or slow wavelength shifts over a large wavelength range. The corresponding parameter maps are shown in Fig. 9. For all of these calculations, four lateral modes and a current of 2 $I_\text{th}$ were used.

Fig. 9.

Parameter maps indicating the stability of mode competition for several combinations of $b$ and $\tau _\text{in}$ and two different values for $R$. All simulations include four lateral modes and the current is set to 2 $I_\text{th}$. The example for periodic mode competition from Fig. 6(a) is marked with “A” and the unstable case from Fig. 5 is marked with “B” in the parameter space map.

The strongest influence in the investigated parameter space comes from $\tau _\text{in}$ and therefore from the symmetric mode coupling term. Small values of $\tau _\text{in}$ lead to fast, unordered mode dynamics and large values practically suppress any strong gain differences. The latter causes very slow mode rolling over a large wavelength range. If $\tau _\text{in}$ is small and symmetric cross-saturation is weak, the asymmetric mode interaction influences the dynamics more strongly. A similar effect is achieved by increasing the antiguiding factor, which causes the corresponding red area in Fig. 9 to become larger. In such a case, multiple longitudinal modes tend to become active at the same time due to the larger influence that weaker modes have on the gain distribution. This disrupts the periodicity of regular mode rolling cycles, where usually only one longitudinal mode per comb is active simultaneously.

The gain spectrum curvature $b$ determines the spectral confinement of mode competition. For strong saturation (large $\tau _\text{in}$), a large curvature can still cause regular mode rolling by reasonably limiting the active wavelength range. On the other hand, if the saturation effect is low and the impact of asymmetric interaction is higher, too strong curvature can also disrupt periodic mode rolling. In such a case the spectral confinement is so tight that after a completed cycle, asymmetric interaction still suppresses these modes that would normally start the next mode rolling cycle.

Simulation results for several parameter combinations from the stability map in Fig. 9 are shown in Supplementary Figure S4.

Apart from these three parameters, also many other values influence the mode dynamics. A higher number of lateral modes (or e.g. a longer cavity) reduces the wavelength spacing between adjacent modes, which strongly increases the asymmetric coupling and has a similar effect as a large antiguiding factor. A higher current increases the photon density and leads to stronger self- and cross-saturation. As a consequence, faster mode competition is usually observed at higher current [40]. The differential gain influences all the contributions to the modal gain as well as the threshold and transparency carrier densities, therefore its role is more complex.

SECTION VI. Conclusion

In this paper we investigate the dynamics of longitudinal-lateral mode competition in broad-ridge laser diodes as well as the influence of mode clustering on this effect. Streak camera measurements show characteristic mode rolling that involves all active lateral modes. We describe mode competition with a set of rate equations involving the lateral carrier distribution and multiple longitudinal mode combs. The sequential mode hopping involves all the active lateral modes in a periodic way, so that all lateral modes are mostly simultaneously active, but there is only one high-intensity mode per mode comb at a time. In contrast to single-lateral-mode devices, asymmetric mode coupling - the main driving force of mode rolling - is strongly enhanced by the small wavelength distances between modes of different mode combs. Additionally, the sequential switching between different lateral modes is further assisted by lateral carrier dynamics and the effect of spatial hole burning. In this way, the longitudinal-lateral mode competition and spatial carrier dynamics are directly coupled.

However, this stable behavior is only observed on short time scales, but no long-time periodicity is seen in our investigated devices. Single shot images reveal substantial deviations from pulse to pulse in the form of jitter, incomplete mode rolling, and wavelength jumps. This is because the properties of the measured devices allow for no long-time periodic mode rolling, as we confirm by numerical simulation. Only the basic short-time periodic behavior of sequential mode hopping between the active lateral modes is present in this situation. Additionally, in reality there is also the influence of noise that causes random variations between each single pulse and that substantially affects the long-time periodicity of mode rolling. The inclusion of noise contributions in the rate equation model through Langevin terms is beyond the scope of this paper, but it seems interesting for future work to investigate the transition between regular and chaotic behavior.

Broad-ridge laser diodes are often considered as static systems regarding their output power, spectrum, and lateral mode configuration, or on the other hand their chaotic dynamics are studied on short time scales. We show that in fact a steady state is never reached, even without considering thermal effects. With this paper, we contribute to bridging the gap between static and chaotic treatment of broad-ridge lasers by describing their deterministic dynamics.

For many applications of high-power broad-ridge laser diodes it might be actually beneficial to have no periodic spectral effects, but rather an intensity and spectrum that stay constant over the whole pulse duration. This can be obtained by averaging over a number of pulses, if the device does not show periodic mode competition. The artifical introduction of mode clustering could help to obtain non-periodic dynamics. This can be done e.g. by weak coupling to a substrate mode or by periodic structures along the cavity, similar to a distributed feedback (DFB) laser. However, for single shot operation, a consistent and predictable behavior would be preferred. Another straightforward way to control the mode coupling strength is to vary the cavity length or the ridge width to change the longitudinal mode spacing or the number of lateral modes, respectively.

SECTION VII. Methods

A. Experimental Setup

We use commercially available broad-ridge laser diodes with a wavelength around 442 nm, a threshold current of 220 mA, a cavity length of 1200 µm, and a 40 µm wide ridge. It is mounted in an active temperature-controlled copper heatsink and is driven using a pulse generator (PicoLAS LDP-V 03-100 V3.3 with PicoLAS PLCS-21). The Gaussian telescope for near-field imaging is formed by an aspheric lens (focal length $f_{1}=8\,\text{mm}$, numerical aperture $N\!A=0.5$) and an achromatic lens ($f_{2}=150\,\text{mm}$) and thus the magnification is $f_{2}/f_{1}=18.75$. The first lens is 8 mm apart from the laser diode and distance between both lenses is $f_{1}+f_{2}$. In the image plane, 150 mm after the second lens, a 10 µm wide fixed vertical slit is mounted to select a fraction of the near field image. The transmitted beam is again collimated using an achromatic lens ($f_{3}=200\,\text{mm}$) and passes a beamsplitter, where 90% of the power is reflected sideways towards the high-resolution spectrometer. With an achromatic lens ($f_{4}=75\,\text{mm}$), the transmitted part is focused into the entrance slit of the monochromator (Princeton Instruments Acton SP-2300 with 2400 l/mm grating) that is connected to the streak camera (Hamamatsu C10910).

The reflected beam from the beamsplitter is coupled into an optical fiber (multimode, 105 µm diameter, $N\!A=0.22$), with the help of an aspheric lens ($f_{5}=30\,\text{mm}$). After outcoupling at the spectrometer setup, the beam is collimated again with an achromatic lens ($f_{6}=18\,\text{mm}$) and finally focused into the high-resolution double spectrometer (SPEX 1404) using another achromatic lens ($f_{7}=100\,\text{mm}$). The spectrometer is operated in second order and spectral resolution has been measured to be 7 pm.

For scanning the lateral near field, only the second lens in the Gaussian telescope is moved, so that the beam path from the vertical slit to both streak camera and high-resolution spectrometer stays constant. The lens is mounted on a horizontal linear stage and can be moved with a stepper motor. Considering the slit width of 10 µm, the stage is moved 5 µm between two measurements to obtain partially overlapping data points.

The laser diode is operated in pulses of 30 ns or 100 ns length at a moderate repetition rate of 10 kHz. The streak camera uses a CMOS integration time of 100 ms and usually 100 images are averaged, except for single shot measurements, where the repetition rate is set to 10 Hz (one laser pulse per image). For the high-resolution spectroscopy, a long integration time of 200 ms helps to reduce noise.

B. Numerical Simulation

The rate equation system is numerically solved with a fourth/fifth-order Runge-Kutta algorithm using the Dormand-Prince method [66] with an adaptive time step size of maximum 30 ps. After each time step, the shape of the lateral modes $M_{l}(x)$ is calculated again to account for the influence of the lateral carrier density distribution on the refractive index.

The parameters for the simulation model are presented in Table I. We use different values for the ridge width $w$ according to the number of involved lateral modes to prevent unrealistic inefficient pumping (e.g. only one mode in a 20 µm wide ridge). For 1 or 2 lateral modes, we use $w=5$ µm, as well as $w=10$ µm for 4 modes and $w=20$ µm for 6 modes. In a narrower ridge, not all of these lateral modes would start lasing due to carrier diffusion, even though they might still be guided. We measured several of those parameters, such as the threshold current density $j_\text{th}$, group index $n_\text{gr}$, differential gain $a$, gain curvature $b$, and antiguiding factor $R$ for the blue 445 nm laser diode. Parameters like the ABC-coefficients [67], refractive index step of the ridge waveguide $\Delta n_\text{ridge}$ [31], transition dipole moment $R_\text{cv}$ [40], intraband relaxation time $\tau _\text{in}$ [68], saturation carrier density $N_\text{sat}$ [38], [40], and diffusion constant $D$ [69], [70], [71], [72] are taken from the literature, although there might be some uncertainty. For other parameters that are not directly accessible, we use typical values, e.g. for the quantum well thickness $d_\text{QW}$ (assuming three quantum wells), confinement factor $\Gamma$, slab refractive index $n_\text{slab}$, spontaneous emission factor $\beta _\text{sp}$, and the injection efficiency $\eta _\text{inj}$.