Modal identification at the folded Gamma point and the quantitative accuracy of effective two-dimensional band models in triangular-lattice photonic-crystal surface-emitting lasers

The 1999 demonstration of coherent two-dimensional lasing in a wafer-fused InGaAsP/InP device containing a triangular-lattice photonic crystal (Imada et al., Appl. Phys. Lett. 75, 316) left two intertwined questions open: which of the closely spaced band edges of the folded (second-order) Gamma point actually lases, and how gain, two-dimensional coupling among the six equivalent Gamma-X directions, and radiative loss select that mode; and how accurately the effective two-dimensional band model used to predict the Gamma-point band edges (infinite rods with a reduced dielectric contrast standing in for the finite-depth rods, the vertical multilayer stack, and out-of-plane radiation) represents the real three-dimensional device. Later work assigned lasing band edges by comparing measured polarization and far-field patterns with computed band-edge field patterns, developed fully three-dimensional FDTD and coupled-wave models, and made band-edge mode selection a designable quantity in modern photonic-crystal surface-emitting lasers (PCSELs). However, a systematic characterization of the quantitative accuracy and validity limits of effective-index models of PCSEL band-edge frequencies, mode splittings, and radiation constants relative to rigorous three-dimensional methods and experiment remains open.

Background

A photonic-crystal surface-emitting laser (PCSEL) is a semiconductor laser in which the optical feedback that sustains lasing comes not from cleaved facets or a vertical stack of mirror layers but from a two-dimensional (2D) photonic crystal: a periodic array of air holes etched into a semiconductor layer with a period comparable to the optical wavelength, whose folded photonic band structure controls which in-plane wave vectors can resonate. In the originating 1999 experiment (Imada et al., Applied Physics Letters 75, 316), a triangular lattice of air holes (lattice constant $a=0.462~\mu\mathrm{m}$, hole depth $0.1~\mu\mathrm{m}$, filling factor, the fraction of the unit-cell area occupied by air, about 0.2) was etched into $n$-InP and wafer-fused, meaning joined by heat and pressure in a hydrogen atmosphere, to an InGaAsP/InP multiple-quantum-well (MQW) gain layer; a quantum well is an ultrathin semiconductor sandwich whose confined electron and hole states provide optical gain, and this one was designed to emit near $1.3~\mu\mathrm{m}$. Light propagating along the six symmetry-equivalent $\Gamma$-$X$ directions of the lattice ($\Gamma$ is the center of the hexagonal Brillouin zone, where the in-plane wave vector vanishes, and $X$ is a zone-boundary point) is Bragg-coupled to the other five directions and, by second-order diffraction, deflected out of the plane, so the device both resonates and emits through its surface. At a band edge the group velocity, i.e. the slope of the frequency-versus-wave-vector dispersion, vanishes, which enhances the density of states and favors distributed-feedback (DFB) lasing, in which the optical feedback comes from Bragg scattering off a periodic refractive-index modulation rather than from mirrors, the mechanism of conventional one-dimensional DFB lasers; the relevant band edges here are the several closely spaced edges of the folded, second-order $\Gamma$ point. The device showed coherent 2D lasing with a threshold current density of $3.2~\mathrm{kA/cm^2}$, a near-field filling the whole patterned area, a far-field divergence below $1.8^{\circ}$, and a two-mode lasing spectrum whose $28~\text{\AA}$ splitting was close to the computed spacing between two $\Gamma$-point band edges (labeled A and B), suggesting that the spectrum reflects a 2D stop band, the frequency interval in which in-plane propagation along $\Gamma$-$X$ is forbidden.

The band structure had been computed by plane-wave expansion (PWE), a standard method that expands the electromagnetic eigenmodes in Fourier components of the periodic dielectric function, applied to an idealized 2D problem: the finite hole depth, the vertical multilayer stack, and out-of-plane radiation were replaced by a single effective dielectric contrast $\Delta\epsilon=0.12$ derived from a computed field-confinement factor $\Gamma_{g}=1.26\%$ (the fraction of modal energy overlapping the patterned region), and the air rods were treated as infinitely long. This approximation reproduced a separate one-dimensional DFB stop-band measurement but was never validated against the 2D device itself. Two limitations were therefore acknowledged: the data did not uniquely determine which of the multiple, closely spaced $\Gamma$-point band edges actually lased, nor the detailed mechanism by which gain and 2D coupling select that mode; and the quantitative accuracy of the effective 2D model for the real, three-dimensional (3D) device geometry, including finite rod depth, the vertical multilayer structure, and radiative coupling, was unsettled.

The tools now standard for this question were introduced by later work in the same lineage: the finite-difference time-domain (FDTD) method, which integrates Maxwell's equations numerically on a space-time grid and can treat the full 3D device; coupled-wave theory, which reduces the band-edge resonance to a small set of coupled wave amplitudes propagating along the equivalent lattice directions and yields analytic relations between resonant frequencies and coupling constants; and rigorous grating solvers such as rigorous coupled-wave analysis (RCWA) and the Fourier modal method (FMM). In current square-lattice PCSELs the $\Gamma$-point band-edge resonances are routinely labeled by symmetry (for example modes A, B, and higher) and their thresholds, radiation constants, and far fields are computed as design quantities.

Problem Statement

Connect the measured lasing behavior of a two-dimensional-photonic-crystal surface-emitting laser to its computed photonic band structure with quantified, mode-resolved accuracy. Concretely, for the triangular-lattice wafer-fused device of Imada et al. (1999) and its successors: (i) determine which of the multiple, closely spaced band edges of the folded (second-order) $\Gamma$ point the lasing state corresponds to, by confronting the measured lasing spectrum, near- and far-field patterns, and polarization with the computed field patterns and symmetry of the individual $\Gamma$-point band-edge modes and reporting the residual ambiguity of the assignment, and determine the mechanism by which gain, two-dimensional coupling among the six equivalent $\Gamma$-$X$ directions, finite device size, and radiative loss select that mode over its competitors; and (ii) establish quantitatively how accurately an effective two-dimensional band model, in which infinite rods carry a reduced dielectric contrast in place of the finite rod depth, the vertical multilayer stack, and out-of-plane radiation, predicts the $\Gamma$-point band-edge frequencies and the band-edge splittings and stop-band width of the real three-dimensional device, and how the band edges it identifies compare with the radiative properties of the real device obtained from the coupled-wave and rigorous three-dimensional treatments; that is, determine the accuracy and validity limits of effective-index models of photonic-crystal surface-emitting lasers relative to fully three-dimensional electromagnetic computation and to experiment, including for triangular-lattice geometries. Existing band-edge assignments for successor devices of this family rest on effective-index band models of exactly the kind whose accuracy part (ii) questions, and no published three-dimensional re-analysis of the exact 1999 device exists. Both parts, (i) and (ii), are required and serve one objective: a quantitative, mode-resolved account of the $\Gamma$-point band-edge resonance that both explains which band edge lases and certifies the model used to predict it, so that measured spectral and spatial features can be assigned to specific computed band-edge modes with stated confidence. Within each part, the listed selection factors and accuracy metrics characterize the resonance and the accuracy of the model used to predict it: for each accuracy metric, the quantitative deviation between the effective model and the rigorous three-dimensional computation or measurement, together with the geometry and parameter range over which the comparison is established.

References

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