Photonic-crystal surface-emitting lasers (PCSELs) recently reached 50 W of purely single-mode continuous-wave (CW) output from a 3-mm-diameter resonator (Yoshida et al., Nature 2023), but the authors explicitly left open a general theory of single-mode stability: no general quantitative criterion specifies how large the threshold gain margin between the fundamental and higher-order modes must be to guarantee single-mode oscillation in the presence of the non-uniform in-plane refractive-index distributions (thermal, carrier-induced, fabrication-related) that inevitably arise in large devices, and the route to 10-mm, 1-kW-class CW single-mode operation with near-diffraction-limited beam quality was unproven. Later work has narrowed the theory: Inoue et al. (IEEE JSTQE 2024/2025) used perturbation analysis to derive an analytical condition for maintaining single-mode lasing under band-edge-frequency non-uniformity, showing that the modal frequency separation matters alongside the gain margin, and Inoue et al. (IEEE JSTQE 2026) simulated carrier-photon mode redistribution in 3-10 mm devices; independent theoretical frameworks and scaling laws for the modal threshold difference have also appeared. However, a criterion that holds for arbitrary in-plane non-uniformity profiles of arbitrary magnitude and spatial form — not only the gradual, random or perturbative classes studied so far — and that is stated as an explicit general relation, together with its validation at 10-mm kW-class CW single-mode operation with M2M^{2} approaching unity, remain unestablished.

Background

A photonic-crystal surface-emitting laser (PCSEL) is a semiconductor laser whose resonator is a two-dimensional photonic crystal — a square lattice of air holes etched into the waveguide core — operating at a photonic band edge, where four degenerate in-plane waves couple through Bragg diffraction and emit vertically, so that the emission area (and hence output power) can be scaled while keeping a narrow circular beam. As the resonator diameter LL grows to thousands of wavelengths, the in-plane losses (light leaking from the resonator periphery) of the fundamental and higher-order transverse modes both vanish, so mode discrimination must instead come from vertical radiation loss. In the three-dimensional coupled-wave theory of square-lattice PCSELs, the near-band-edge modes are described by Hermitian in-plane coupling coefficients RR (real part) and II (imaginary part), which exchange the four fundamental waves without radiation, and a non-Hermitian coupling coefficient μ\mu that accompanies vertical radiation; the radiation constant αv\alpha_{\mathrm{v}} of each mode (its vertical radiation-loss rate) sets the modal threshold gain, so the threshold gain margin Δαv\Delta\alpha_{\mathrm{v}} — the difference in radiation constants (equivalently, in threshold gain) between the first higher-order mode and the fundamental mode — is the quantity that must be kept large for robust single-mode oscillation. Double-lattice PCSELs, whose unit cell contains two air holes placed so that the waves diffracted at 180° and 90° interfere destructively, weaken the Hermitian coupling and enlarge Δαv\Delta\alpha_{\mathrm{v}} (Yoshida et al., Nature Materials 2019 demonstrated 7 W CW from 800-μm devices). A design guideline by Inoue et al. (Nature Communications 2022) showed theoretically that simultaneously reducing RR and μ\mu to increase the curvature of the radiation-constant dispersion should enable 100-W-to-1-kW single-mode operation at diameters of 3 mm and beyond; an independent route based on open-Dirac singularities (Dirac-type degeneracies of the photonic band structure that couple radiatively to the continuum and pin the lasing mode) was proposed by Contractor et al. (Nature 2022). Under CW current injection the device heats non-uniformly (centre hotter than edge), and the thermo-optic effect converts the temperature distribution into a convex spatial variation of the band-edge frequency, which destroys single-mode oscillation; Yoshida et al. (Nature 2023) countered this by pre-installing a lattice-constant distribution Δa(x,y)\Delta a(x,y) chosen to cancel the predicted band-edge distortion, achieving 50 W CW single-mode lasing at 3 mm with 0.05° divergence and a beam-quality factor M2≈2.36M^{2}\approx 2.36 (M2M^{2} measures how many times the beam divergence times waist exceeds the diffraction limit of a Gaussian; M2=1M^{2}=1 is a diffraction-limited Gaussian beam). That paper left several issues explicitly unresolved, recorded in its Methods as a note on the threshold gain margin: 'It is difficult to specify a general value of Δαv\Delta\alpha_{\mathrm{v}} sufficient for single-mode oscillation in PCSELs', even though increasing Δαv\Delta\alpha_{\mathrm{v}} by simultaneously reducing RR and μ\mu was found empirically to preserve single-mode oscillation in the presence of a non-uniform in-plane refractive-index distribution 'borne by various physical phenomena', with those findings to be reported separately. In addition, the M2≈2.36M^{2}\approx 2.36 beam quality was attributed to the super-Gaussian field profile (a flat-topped rather than Gaussian near-field intensity distribution) caused predominantly by uniform current injection; a residual far-field side lobe (about 1/e21/e^{2} of the main peak, larger in the yy direction) remained to be removed by further refining the pre-installed lattice-constant distribution; and the expected extension to 10-mm-diameter, 1-kW-class single-mode CW operation raised unresolved thermal-management and current-uniformity challenges.

Problem Statement

Establish a general quantitative criterion that guarantees purely single-mode oscillation of an ultra-large-area photonic-crystal surface-emitting laser (PCSEL) in the presence of the non-uniform in-plane refractive-index (band-edge-frequency) distributions that arise during operation from various physical phenomena — thermal gradients under CW injection, carrier-induced index changes, and fabrication imperfections. The criterion should specify how large the threshold gain margin Δαv\Delta\alpha_{\mathrm{v}} (the difference in threshold gain, set by the radiation constants, between the first higher-order mode and the fundamental mode) must be — together with any companion modal quantities the analysis shows to be necessary, such as the frequency separation between the fundamental and higher-order modes, the Hermitian and non-Hermitian coupling coefficients RR, II and μ\mu, and the resonator diameter — as an explicit general relation in terms of the magnitude and spatial profile of the non-uniformity, rather than a criterion that can only be evaluated numerically for one profile at a time, going beyond the perturbative treatment of gradual and random band-edge frequency fluctuations. Validate the criterion by extending the demonstrated 3-mm, 50-W CW single-mode architecture: (a) determine whether the super-Gaussian field profile that currently bounds the beam-quality factor at M2≈2.36M^{2}\approx 2.36 can be converted into a near-Gaussian profile so that M2M^{2} is brought within 10% of the diffraction limit (M2≤1.1M^{2}\le 1.1); (b) eliminate the residual far-field side lobe (about 1/e21/e^{2} of the main peak, larger in the yy direction), suppressing it to at most 1% of the main peak in every transverse direction; and (c) scale the device to 10 mm±0.5 mm10\,\mathrm{mm}\pm 0.5\,\mathrm{mm} diameter with the thermal-load and current-uniformity management required for 1 kW±10%1\,\mathrm{kW}\pm 10\% output power with high beam quality in purely single-mode CW operation, at a side-mode-suppression ratio of at least 30 dB. All requested results serve the single overarching objective: a priori-guaranteed, robust single-mode, high-brightness CW operation of scalable PCSELs, with the general criterion providing the theory that the device demonstrations validate. The general quantitative criterion and its validation through the device demonstrations (a)–(c) are jointly posed; each demonstration must report the threshold gain margin Δαv\Delta\alpha_{\mathrm{v}} and the modal frequency separation of the demonstrated device and show them to satisfy the criterion's single-mode condition.

References

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