Microscopic origin of the gain-saturation parameters governing longitudinal mode competition in (Al,In)GaN laser diodes

Multimode rate equations with self-saturation $B$ , symmetric cross-saturation $D_{pq}$ and asymmetric cross-saturation $H_{pq}$ quantitatively reproduce the 10-150 MHz longitudinal-mode competition and mode clustering observed in blue-green (Al,In)GaN Fabry-Perot laser diodes, but the model's key parameters enter as fitted quantities: the intraband relaxation time $\tau_{\mathrm{in}}$ (19 fs in a blue, 113 fs in a green device) and the effective antiguiding factor $\alpha_{\mathrm{ag}}=9$ , the latter far above independent measurements ( $4.1\pm0.5$ ). The open problem is the microscopic origin and quantitative, device-specific mapping of these phenomenological gain-saturation parameters: which carrier and polarization processes in (Al,In)GaN quantum wells determine $\tau_{\mathrm{in}}$ and how alloy disorder, indium-composition fluctuations and the quantum-confined Stark effect modify it; which mechanisms set the effective $\alpha_{\mathrm{ag}}$ entering the asymmetric term and why the simulation value exceeds measured antiguiding factors; and how many-body effects and ultrafast spectral hole burning enter $B$ , $D_{pq}$ and $H_{pq}$ . Later work derived the effective mode-interaction terms from Coulomb scattering in an idealized InGaN quantum well (Kuhn and Thranhardt, Phys. Rev. B 108, 115304 (2023)), but a validated device-specific mapping including alloy disorder, phonon scattering and a microscopic reconciliation of the antiguiding discrepancy remains open.

Background

Blue and green semiconductor lasers based on III-nitride alloys - (Al,In)GaN quantum-well laser diodes grown on GaN - are the light sources of laser projection displays and, increasingly, of AR/VR and visible-light-communication systems. In an edge-emitting Fabry-Perot laser diode (a resonator formed by two cleaved crystal facets), the cavity supports a comb of longitudinal modes: discrete resonance wavelengths separated by roughly 0.05 nm in these devices. Above threshold several modes lase at once, and single-shot streak-camera spectrochronograms (time- and wavelength-resolved single-shot intensity traces) show that blue-green (Al,In)GaN laser diodes exhibit pronounced longitudinal mode competition - the lasing power periodically rolls from one mode to the next across the gain spectrum at 10-150 MHz - as well as mode clustering, in which a few modes separated by many mode spacings coexist stably.

Weig et al. (2014) showed that a multimode rate-equation model reproduces both phenomena quantitatively. Rate equations are ordinary differential equations for the photon density $S_p$ of each longitudinal mode $p$ and for the carrier density $N$ in the quantum well; the essential nonlinearity is gain saturation, introduced through a self-saturation coefficient $B$ (a mode saturates its own gain), a symmetric cross-saturation coefficient $D_{pq}$ (mode $p$ saturates the gain seen by mode $q$ symmetrically under $p\leftrightarrow q$ ) and an asymmetric cross-saturation coefficient $H_{pq}$ (the interaction differs under exchange of $p$ and $q$ , which drives the rolling). This structure originates in Yamada's theory of mode competition in injection lasers (1986). There, $D_{pq}$ has a Cauchy (Lorentzian) shape in the mode spacing with full width at half maximum $\lambda_p^2/(\pi c_0 \tau_{\mathrm{in}})$ , governed by the intraband relaxation time $\tau_{\mathrm{in}}$ : the characteristic time on which carrier-carrier scattering restores the equilibrium carrier distribution within their bands after the optical field burns a hole in it (spectral hole burning). The asymmetric term is proportional to the antiguiding factor $\alpha_{\mathrm{ag}}$ (also called the linewidth-enhancement or Henry factor): the ratio of the carrier-induced refractive-index change to the carrier-induced gain change, which converts intensity beating between two modes into a differential gain shift. In GaN-based ridge lasers this same factor quantifies index antiguiding - the weakening of the effective lateral waveguide by the carrier-induced index depression inside the ridge.

Fitting the model to measured spectrochronograms, Weig et al. obtained $\tau_{\mathrm{in}}=19$ fs for a blue (446 nm) and 113 fs for a green (513 nm) device, and $\alpha_{\mathrm{ag}}=9$ for both. Each value is in tension with independent knowledge. Non-Markovian many-body gain calculations for wurtzite InGaN/GaN quantum-well lasers (Park et al., 2005) give $\tau_{\mathrm{in}}\approx 24$ fs at the subband edge, nearly independent of carrier density - close to the blue-device fit but almost five times smaller than the green-device fit. Measurements of the antiguiding factor of GaN-based laser diodes from the violet to the green (Scheibenzuber et al., 2010, extending earlier gain and carrier-induced phase-shift measurements by Schwarz et al., 2003) yield $\alpha=4.1\pm0.5$ , far below the value of 9 needed to reproduce strong mode competition. The nitride-specific physics that could modify these parameters includes alloy disorder and indium-composition fluctuations (the random In/Ga occupation of the ternary InGaN well, which causes inhomogeneous broadening and partial carrier localization, a mechanism absent in binary III-V lasers) and the quantum-confined Stark effect (QCSE: the tilting of the quantum-well potential by spontaneous and piezoelectric polarization fields, which separates electrons and holes, red-shifts the emission and reduces the optical transition strength). The open question is thus the microscopic origin and the quantitative mapping between this microscopic physics and the phenomenological gain-saturation parameters of the validated modal model.

Problem Statement

The problem, as posed by Weig et al. (2014) on the basis of their validated multimode rate-equation description of longitudinal mode competition and mode clustering in (Al,In)GaN laser diodes, is to establish the microscopic origin and the quantitative, device-specific mapping of the phenomenological gain-saturation parameters of that model: the self-saturation coefficient $B$ (a mode saturating its own gain), the symmetric cross-saturation coefficient $D_{pq}$ (mode $p$ saturating the gain seen by mode $q$ , symmetric under exchange of $p$ and $q$ ), the asymmetric cross-saturation coefficient $H_{pq}$ (the interaction differing under exchange of $p$ and $q$ , which drives the mode rolling), the intraband relaxation time $\tau_{\mathrm{in}}$ (the characteristic time on which carrier-carrier scattering restores the equilibrium carrier distribution within the energy bands after the lasing field burns a hole in it), and the effective antiguiding factor $\alpha_{\mathrm{ag}}$ (the ratio of the carrier-induced refractive-index change to the carrier-induced gain change):

(i) Which microscopic carrier and polarization processes in (Al,In)GaN quantum wells determine the intraband relaxation time $\tau_{\mathrm{in}}$ that controls the width and amplitude of the symmetric cross-saturation $D_{pq}$ , and how do alloy disorder, indium-composition fluctuations and the quantum-confined Stark effect (the tilting of the quantum-well potential by spontaneous and piezoelectric polarization fields) modify the effective $\tau_{\mathrm{in}}$ in practice - in particular, what reconciles the fitted values (19 fs for a blue and 113 fs for a green device) with many-body calculations of about 24 fs? The required target here is the general, device-specific microscopic mapping across the (Al,In)GaN device family; the two fitted values quoted here are instances that this mapping must account for, not the whole of its object, and the mapping must treat each of the three named mechanisms - alloy disorder, indium-composition fluctuations and the quantum-confined Stark effect.

(ii) Which microscopic mechanisms set the effective antiguiding factor $\alpha_{\mathrm{ag}}$ that governs the asymmetric cross-saturation $H_{pq}$ , and why does the value required in simulations to obtain strong mode competition ( $\alpha_{\mathrm{ag}}=9$ ) differ from independent antiguiding measurements ( $4.1\pm0.5$ )?

(iii) How do many-body effects and spectral hole burning (the depletion by the optical field of the carrier distribution at the lasing transition energies) on ultrafast time scales quantitatively enter the gain-saturation terms $B$ , $D_{pq}$ and $H_{pq}$ for (Al,In)GaN laser diodes?

A complete answer must be microscopically grounded - it must establish the microscopic origin of these parameters and the quantitative, device-specific mapping, not a re-fit of the phenomenological quantities - and must demonstrate consistency with the measured modal dynamics (10-150 MHz rolling mode competition, clustering of up to about two dozen modes) that the phenomenological model reproduces. All three parts (i)-(iii) are required for a complete answer; they are separately resolvable, so an answer that fully resolves one part while leaving the others open counts as partial progress rather than a complete resolution, and the idealized-quantum-well derivation of the mode-interaction terms cited above does not by itself settle any of (i)-(iii).

References

T. Weig, T. Hager, G. Brüderl, U. Strauss, U. T. Schwarz, 'Longitudinal mode competition and mode clustering in (Al,In)GaN laser diodes', Optics Express 22(22), 27489 (2014). DOI: 10.1364/oe.22.027489, https://doi.org/10.1364/oe.22.027489
M. Yamada, 'Theory of mode competition noise in semiconductor injection lasers', IEEE Journal of Quantum Electronics 22(7), 1052-1059 (1986). DOI: 10.1109/JQE.1986.1073087, https://doi.org/10.1109/JQE.1986.1073087
U. T. Schwarz, E. Sturm, W. Wegscheider, V. Kümmler, A. Lell, V. Härle, 'Optical gain, carrier-induced phase shift, and linewidth enhancement factor in InGaN quantum well lasers', Applied Physics Letters 83(20), 4095-4097 (2003). DOI: 10.1063/1.1628825, https://doi.org/10.1063/1.1628825
S.-H. Park, D. Ahn, E. H. Park, T. K. Yoo, Y.-T. Lee, 'Many-body optical gain and intraband relaxation time of wurtzite InGaN/GaN quantum-well lasers and comparison with experiment', Applied Physics Letters 87(4), 044103 (2005). DOI: 10.1063/1.2001744, https://doi.org/10.1063/1.2001744
W. G. Scheibenzuber, U. T. Schwarz, T. Lermer, S. Lutgen, U. Strauss, 'Antiguiding factor of GaN-based laser diodes from UV to green', Applied Physics Letters 97(2), 021102 (2010). DOI: 10.1063/1.3464172, https://doi.org/10.1063/1.3464172
L. Redaelli, H. Wenzel, J. Piprek, T. Weig, S. Einfeldt, M. Martens, G. Lukens, U. T. Schwarz, M. Kneissl, 'Index-Antiguiding in Narrow-Ridge GaN-Based Laser Diodes Investigated by Measurements of the Current-Dependent Gain and Index Spectra and by Self-Consistent Simulation', IEEE Journal of Quantum Electronics 51(8), 1-6 (2015). DOI: 10.1109/JQE.2015.2444662, https://doi.org/10.1109/JQE.2015.2444662
A. Thränhardt, E. Kuhn, L. Uhlig, M. Wachs, U. T. Schwarz, 'Modelling of the Laser Dynamics of an (Al,In)GaN Laser Diode', 2018 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), 107-108 (2018). DOI: 10.1109/NUSOD.2018.8570238, https://doi.org/10.1109/NUSOD.2018.8570238
E. Kuhn, A. Thränhardt, 'Modeling mode competition in laser diodes', Optical and Quantum Electronics 51(6), 206 (2019). DOI: 10.1007/s11082-019-1916-7, https://doi.org/10.1007/s11082-019-1916-7
E. Kuhn, L. Uhlig, M. Wachs, U. T. Schwarz, A. Thränhardt, 'Mode rolling effects in nitride laser diodes', Engineering Research Express 2(3), 035036 (2020). DOI: 10.1088/2631-8695/abb6c8, https://doi.org/10.1088/2631-8695/abb6c8
L. Uhlig, M. Wachs, D. J. Kunzmann, U. T. Schwarz, 'Spectral-temporal dynamics of (Al,In)GaN laser diodes', Optics Express 28(2), 1771 (2020). DOI: 10.1364/oe.382257, https://doi.org/10.1364/oe.382257
S. M. S. Imran, H. M. A. Tanmay, R. Sultana, 'Mode Clustering and Mode Hopping Phenomena in Multi-Quantum Well InGaN Blue Laser Diodes', Physics of Wave Phenomena 29(4), 299-306 (2021). DOI: 10.3103/S1541308X21040075, https://doi.org/10.3103/S1541308X21040075
E. Kuhn, A. Thränhardt, 'Influence of scattering effects on the interaction between longitudinal modes in laser diodes', Physical Review B 108, 115304 (2023). DOI: 10.1103/PhysRevB.108.115304, https://doi.org/10.1103/PhysRevB.108.115304 (full text also available as WIAS Preprint No. 3038, DOI: 10.20347/WIAS.PREPRINT.3038, https://doi.org/10.20347/WIAS.PREPRINT.3038)
L. Uhlig, D. J. Kunzmann, U. T. Schwarz, 'Characterization of Lateral and Longitudinal Mode Competition in Blue InGaN Broad-Ridge Laser Diodes', physica status solidi (a) 220(16), 2200751 (2023). DOI: 10.1002/pssa.202200751, https://doi.org/10.1002/pssa.202200751
E. Kuhn, 'Simulation of the Mode Dynamics in Broad-Ridge Laser Diodes', IEEE Photonics Journal 16(2), 1-8 (2024). DOI: 10.1109/JPHOT.2024.3374448, https://doi.org/10.1109/JPHOT.2024.3374448
L. Uhlig, D. J. Kunzmann, U. T. Schwarz, 'Mode Competition Between Longitudinal and Lateral Modes in III-Nitride Broad-Ridge Laser Diodes', IEEE Journal of Selected Topics in Quantum Electronics 31(2), 1-12 (2025). DOI: 10.1109/JSTQE.2024.3440179, https://doi.org/10.1109/JSTQE.2024.3440179