Characterization of Lateral and Longitudinal Mode Competition in Blue InGaN Broad‐Ridge Laser Diodes

In blue GaN‐based laser diodes with a 40 μm broad‐ridge waveguide, lateral modes of different order are selectively observed by high‐resolution spectroscopy combined with lateral near‐field scanning. Longitudinal mode clusters contribute to different lateral modes, close above the threshold current. Longitudinal mode competition is observed in each longitudinal mode cluster as intensity modulation and wavelength shift, repeating itself with a frequency of around 60–80 MHz. Despite the spectral separation of the different clusters and different lateral mode order, the mode competition processes share the same frequency and show a stable phase relation, especially at low currents. This indicates coupling of separate mode clusters via the charge carrier density. Due to the lateral modes occurring at separate wavelengths, lateral–longitudinal mode competition is found, which not only causes spectral effects but also time‐dependent periodic variations in the lateral near field.


Introduction
Broad-ridge laser diodes are specialized for high optical output power, while preserving the typical advantages of laser diodes such as small size, high efficiency, wavelength flexibility, and direct electrical modulation at high speed. In recent years, great progress has been made in blue broad-ridge lasers based on the III-nitride material system. [1][2][3] Blue high-power laser diodes have unique advantages for white light generation, [4,5] optical pumping, [6] and industrial processing of metals like copper, [7,8] which has huge importance in the context of electromobility and energy storage.
The maximum optical power is limited by catastrophic optical damage [9,10] occuring at a critical power density at the facet, so broader ridge waveguides are used for power scaling in one emitter. If the ridge width is increased beyond 2 μm, lateral modes of higher order arise, [11][12][13] limiting the beam quality.
In earlier devices, filamentation dominated the lateral dynamics, which was shown to be thermally driven and influenced by material composition fluctuations. [14][15][16][17][18] In this work, we investigate the dynamic interplay between lateral modes and spectral effects, i.e., longitudinal mode competition. The lateral mode dynamics have been investigated in detail for infrared broad area laser diodes. [19][20][21][22][23] In singlelateral-mode laser diodes, the interaction between neighboring longitudinal modes is known to result in symmetric and asymmetric cross-saturation, where the latter gives rise to self-repeating mode rolling through the spectrum. [24][25][26][27][28][29][30][31] Additionally, the laser spectrum is influenced by the formation of longitudinal mode clusters, [29,[32][33][34][35] which is attributed to fluctuations in quantum well thickness, indium content, optical gain, as well as spectral hole burning or plasma oscillations. We experimentally investigate devices with a 40 μm wide ridge at currents close above the threshold, because in this case the presence of fewer, lower order modes allows better to trace their dynamics. For higher currents, more modes closely fill the gain volume spatially and spectrally, so that dynamics of individual modes cannot be disentangled. [36] Furthermore, we observe mode competition in simultaneously active longitudinal mode clusters, as well as coupling between different clusters.

Lateral-Longitudinal Mode Distribution
Few nanoseconds after onset of the electrical pulse, the laser diode reaches the threshold and starts lasing. During the following 10-20 ns, the active lateral and longitudinal modes undergo transitional dynamics, which are driven by carrier dynamics and mode interaction effects, such as spectral hole burning, lateral hole burning, and mode competition. As a result, a specific spectral-lateral mode pattern stabilizes over time, apart from mode rolling. This means, we observe an uneven spectral distribution of longitudinal modes, i.e., mode clustering, while at the same time different lateral modes predominate in different parts of the spectrum. Due to the involved nonlinear processes, this mode pattern depends on the injected current.
We investigate these dynamics by stepwise scanning through a magnified image of the near field and performing spectraltemporal measurements at each point using a streak camera. [36] DOI: 10.1002/pssa.202200751 In blue GaN-based laser diodes with a 40 μm broad-ridge waveguide, lateral modes of different order are selectively observed by high-resolution spectroscopy combined with lateral near-field scanning. Longitudinal mode clusters contribute to different lateral modes, close above the threshold current. Longitudinal mode competition is observed in each longitudinal mode cluster as intensity modulation and wavelength shift, repeating itself with a frequency of around 60-80 MHz. Despite the spectral separation of the different clusters and different lateral mode order, the mode competition processes share the same frequency and show a stable phase relation, especially at low currents. This indicates coupling of separate mode clusters via the charge carrier density. Due to the lateral modes occurring at separate wavelengths, lateral-longitudinal mode competition is found, which not only causes spectral effects but also timedependent periodic variations in the lateral near field.
Here, we average over the time range in which the laterallongitudinal mode pattern has mostly stabilized, which means to leave out the initial onset behavior and average over the mode rolling features. The resulting lateral-spectral intensity distribution is shown in Figure 1 for different currents close above threshold. Note that individual longitudinal modes can not be resolved with the monochromator used in combination with the streak camera due to their small spacing of 26 pm. In most cases, the observed peaks in the spectrum are actually clusters of about two to five longitudinal modes. This is illustrated in Figure 2, showing the lateral-longitudinal mode distribution measured at 1.1 I th using the high-resolution (HR) spectrometer that allows to spectrally resolve longitudinal and lateral modes. As this device offers no time resolution, the signal is averaged over the whole pulse length, which leads to different relative intensities of the modes in comparison to Figure 1d.
We identify four main mode clusters that are populated by different lateral modes of varying intensity (see Figure 1). Directly at threshold only fundamental lateral modes are present, which is remarkable in a 40 μm wide waveguide, but the effectively used area is smaller (about 20 μm), due to ineffective current spreading at low currents. At high current, the whole active area contributes to lasing, as shown in ref. [36]. For currents of 1.02 I th and above, a first-order lateral mode becomes  www.advancedsciencenews.com www.pss-a.com active, and the fundamental modes on the short-wavelength side decrease in intensity. At 1.2 I th , we observe a 4th order mode. In this current range I < 1.2 I th , measurements with the HR spectrometer indicate that each longitudinal mode cluster substantially contributes to one lateral mode only. Thus, the lateral modes can be clearly identified by the spatial intensity profile of a whole cluster, which agrees well with the theoretical expectations in a simple box model for the cavity.

Mode Competition in the Presence of Multiple Lateral Modes
In low power laser diodes with a ridge width of about 2 μm and stable single lateral mode operation, longitudinal mode competition appears as mode rolling. In this case, the active mode moves through the laser spectrum toward longer wavelength and if the gain becomes too low, this process starts again from the short-wavelength side and repeats itself. [28,29,37] But in the investigated broad-area laser diodes, longitudinal mode clusters are formed and in most cases different lateral modes appear in each cluster. The observed mode competition behavior is shown in Figure 3 for different currents. Here, all the involved mode clusters show mode competition as intensity modulation accompanied by a slight wavelength shift during each period. This indicates that mode competition primarily happens between the few longitudinal modes within each cluster. However, at this point it is unclear, whether each mode cluster is acting independently or if mode competition in different clusters is coupled.

Mode Competition Frequency
To approach this question, the frequency of intensity modulation f is investigated in each of the four main clusters separately. Mode competition is a quite unstable process and strongly affected by jitter that arises from various effects, such as current noise, thermal fluctuations, and shot noise. This leads to a random phase offset in mode competition from one streak camera image to another. For this reason, each position x is separately evaluated in terms of the frequency f, and the resulting distribution is given in Figure 4 as histogram for each cluster and all the investigated currents. The data points are only used if their intensity exceeds 15% of the maximum value at the respective current to avoid wrong frequencies from noisy data. When comparing the peaks in these histograms, for currents of up to 1.04 I th all active clusters share the same mode competition frequency for each current. In the case of 1.1 I th , only cluster (4) is active. At 1.2 I th , a 4th order lateral mode arises in cluster (2), whereas cluster (4) still occupies a first-order mode (see Figure 1e). Here we observe a higher mode competition frequency on the shorter wavelength side where the 4th order mode is found, compared to the longer wavelength clusters.  www.advancedsciencenews.com www.pss-a.com The dependence of mode competition frequency on the current is depicted in Figure 5 and shows a sublinear increase with current. Here the whole spectrum was taken into account. The same sublinear trend has also been reported in previous papers. [28,29] In the investigated current range, in each mode cluster, there is only one predominant lateral mode to which multiple longitudinal modes belong. Mode competition between these longitudinal modes follows exactly the same mechanism as mode competition in narrow-ridge laser diodes, where only the fundamental lateral mode is active. Beating between spectrally adjacent modes causes the modulation of the intraband relaxation of charge carriers into the quantum well. [38] This affects the refractive index and gain and leads to symmetric and asymmetric gain cross-saturation between two longitudinal modes that are spectrally close. [24][25][26][27][28][29][30][31] The asymmetric contribution is the reason behind mode rolling by increasing the gain for neighboring modes with longer wavelength and suppressing the gain for the shorter wavelength modes. The strength of this gain variation Δg asym p on mode p is given by [27] Δg asym p This equation includes the group refractive index n gr , differential gain dg dN , antiguiding factor α, carrier density N, transparency carrier density N tr , wavelength λ of both modes p and q, as well as the photon density S q in the mode q. At higher currents, the photon density rises and consequently the asymmetric mode coupling becomes stronger and mode rolling through the active longitudinal modes becomes faster. The sublinear dependency of mode competition frequency on the current is not straightforward to see from the equations, but numerical simulations of the spectral-temporal mode rolling behavior using the full set of rate equations [27,29,30] have been showing quantitative agreement with the measured behavior in narrow-ridge laser diodes. [28,29]

Phase Relation Between Two Clusters
In most cases, we find the same frequency of mode competition in all longitudinal mode clusters. This is required for coupling between different clusters. If we would have observed mode rolling in two separate clusters at different speed, no significant interaction is possible. In our case, this frequency condition is met and for checking if mode competition processes in different clusters interact with each other, we investigate the phase relation considering the intensity modulation.
Considering one longitudinal mode cluster, the strong jitter leads to random phase differences in mode competition between two consecutive streak camera measurements, as shown in    www.advancedsciencenews.com www.pss-a.com Figure 6a,b. If mode competition would occur in two separate mode clusters with no interaction between them, uncorrelated jitter would occur in both clusters, and the phase difference between both is again randomly distributed. In contrast, our measurements show a stable phase relation between the clusters (1) and (4) even though the phase of each cluster varies heavily (see Figure 6). In Figure 7, we compare the phase relation of the two strongest mode clusters for different currents. In general, the phase difference distribution becomes broader with increased current, which means the dynamics within a cluster becomes stronger than the dynamical coupling between clusters. In some cases, the phase difference seems to be slowly varying over the x-range (corresponding also to the duration of the measurement: about 20 min for one scan). This indicates some slow variation affecting the coupling mechanism.

Discussion and Conclusion
We measure the lateral-spectral-temporal dynamics in a blue broad-area InGaN laser diode and gain insights on the appearance of lateral and the corresponding longitudinal modes at currents close above the laser threshold. The spectrum shows pronounced longitudinal mode clustering and at these low currents (<1.2 I th ) only one lateral mode dominates in each cluster. The order of active lateral modes and their relative intensity depend on the current: from fundamental lateral mode operation (but multiple longitudinal modes) at 1.0 I th to first and 4th order contributions at 1.2 I th .
Mode competition happens between the few longitudinal modes that constitute each mode cluster. For currents close to threshold current (<1.1 I th ), all the active mode clusters show mode competition at the same current-dependent frequency and the phase relation between different clusters is stable. This indicates coupling between the parallel mode competition processes despite the spectral separation of the mode clusters. Direct interaction in terms of cross-saturation seems unlikely due to the wavelength difference, which is about 10Â larger than usual (between two adjacent longitudinal modes) and the fact that the coupling coefficient is inversely proportional to this wavelength spacing (see Equation (1)). In addition, for interaction between two lateral modes of different order, the coupling is further reduced by their spatial overlap integral that is smaller than one. For direct coupling between two mode clusters via the asymmetric cross-saturation, the contribution to modal gain would have to be at least of the same order of magnitude as the mode competition within one cluster. This is not the case, so direct asymmetric interaction between separated mode clusters can be ruled out as reason for the phase coupling of mode rolling.
We propose coupling of different mode clusters via the charge carrier density instead. The observed intensity modulation in one mode cluster is naturally associated with an opposite modulation of the carrier density, i.e., carriers are depleted faster at high intensity and build up at low intensity. Thus, for two mode clusters, antiphase intensity modulation is more favorable than being in-phase. This mechanism is independent of the wavelength separation between the involved modes. Assuming an idealized case, where two mode clusters show antiphased intensity oscillations, both the total intensity and the carrier density stay constant over time at their respective equilibrium values. However if these oscillations would be in phase, the total intensity varies strongly, and the carrier density is consequently decreased during each intensity peak and gets higher during low intensity. This effects leads to strong damping of any inphase intensity modulation, whereas (nearly) antiphased mode competition effects are not suppressed. If there are variations in the total intensity, in the first approximation, the damping effect via carrier density on any optical mode is of the same amount as the original modulation. Thus, we attribute the (a) (b) (c) Figure 6. Phase jitter in a,b) clusters (1) and (4). c) Consistent phase relation between these two clusters. Measured at 1.0 I th (230 mA). www.advancedsciencenews.com www.pss-a.com observed coupling of mode competition in separate clusters to interaction via the carrier density. Mode competition in broad-ridge laser diodes is not only affecting the spectrum but also the lateral intensity profile in a time-periodic way. In this coupled longitudinal-lateral mode competition, not only the longitudinal modes are switching cyclically but also the lateral modes do. This is an interesting analogy, which has practical implications on the time-dependent stability of the lateral beam shape of broad-ridge lasers.

Experimental Section
Lateral-Spectral-Temporal Characterization: We investigated commercially available broad-ridge laser diodes with a 40 μm wide ridge and emission wavelength of 442 nm. The laser was packaged in a TO 90 can and mounted in a passive copper heatsink. The device was operated in 100 ns long pulses at a duty cycle of 1:1000 using a PicoLAS LDP-V 03-100 driver. The near field of the laser diode was magnified with a Gaussian telescope (see Figure 8) consisting of an aspheric lens (L1, f 1 ¼ 8 mm) and an achromatic lens (L2, f 2 ¼ 150 mm), which yielded a 18.75Â magnification, while all the different lateral Gaussian modes were correctly imaged. [39] To achieve lateral resolution, a 10 μm wide slit was mounted in the image plane perpendicular to the slow axis, where the lateral waveguide modes appeared. The second lens L2 can be laterally moved by a stepper motor thus moving the near-field image over the fixed slit. In this way, the lateral near field was scanned stepwise. The light passing the slit was collimated and focused using two lenses L3 and L4 ( f 3 ¼ 200 mm, f 4 ¼ 75 mm). The beam entered the monochromator (Princeton Instruments Acton SP-2300 using grating with 2400 L mm À1 ) that was attached to the streak camera (Hamamatsu C10910). This allowed measuring the spectral-temporal dynamics at each position in the near field. The integration time of the streak camera was 100 ms for each frame and we integrated over 100 frames for each position.
Time-Integrated High-Resolution Spectroscopy: Using a SPEX 1404 HR double spectrometer, individual longitudinal modes can be well resolved, but with an integration time of 200 ms the signal was averaged over roughly 2000 pulses. To combine HR spectroscopy with the lateral near field scan, a beamsplitter was introduced between the lenses L3 and L4, and the reflected part of the beam (90%) was guided to the HR spectrometer using a multimode optical fiber.