A novel phase locked array design, based on direct reflection feedback among adjacent cavities using an
external grating, is analyzed and proposed. As a result of both longitudinal and transverse wavenumber
selection, caused respectively by the narrow grating reflection bandwidth and by the array geometry, only
one among the free running cavity eigenmodes can couple into a phase-locked, collective array eigenmode.
The coupled array mode is experiencing the high reflectivity of the grating and surpasses the low gain of the
free running modes, experiencing only a much lower reflectivity from the cavity edge mirror (anti-reflective
coating). Thus phase locking and single mode operation can be concurrently achieved.
An important feature of coupled laser arrays is that the gain in one cavity is modulated by the radiation
from another cavity. This is a generic effect under many forms of coupling, including fringe field interactions
in closely packed arrays and reflection feedback from external mirrors. Cross-cavity gain depletion can
occur in many types of laser arrays, including edge emitting semiconductor lasers, VCSELs and fiber laser
bundles. The interplay between frequency pulling and the characteristic cavity oscillations creates a rich
behavior, setting the properties of active photonic lattices apart from the well known passive photonic
lattices involving radiation interference due to real index variations. The case of planar VCSEL arrays
is chosen as generic example for studying the physics of active photonic lattices. Results of theoretical
calculations and numerical simulations are presented, addressing the following issues: (a) Non-linear phase-locked
Bloch eigenmodes and boundary layer formation for finite arrays (b) Lattice defects, including sites
that fail to lase, and defect tolerance (c) Excitation of stable, slow-light, lattice waves and photonic sound
propagation (d) Unstable lattice behavior at high coupling strengths, with self-excited array oscillations and
chaotic transitions (e) Phase locking in realistic arrays, with random variations in the cold-cavity parameters
(manufacturing tolerances), via self-regulated frequency pulling (f) Existence and properties of randomly
phase-locked arrays with "fuzzy" eigenmodes.
The longitudinal component of the polarization field inherent in polar materials, combined with constrained carrier motion along the quantum wells, causes formation of equilibrium plasma nano-sheaths at intersections of quantum wells. The induced short range (nm) potentials of peak voltages much larger than the thermal carrier energy cause wavefunction localization, which further reduces the dimensionality of the carrier behavior. The associated energy band-bending causes enhanced carrier accumulation at quantum wedges and quantum tips formed by intersecting quantum wells. In addition, the total carrier number over the QW length increases, manifesting spontaneous intrinsic pumping due to polarization. As a result, the spontaneous emission is localized at quantum wedges, and the total emission exceeds that from a flat quantum well of similar parameters, as experimentally observed. The sheath potentials are sufficiently high for 1-D or 0-D carrier localization at quantum wedges and quantum tips.
An active photonic lattice is characterized by cross-cavity gain modulation (hole-burning) due to carrier depletion from adjacent cavity interactions. A VCSEL array offers a generic example for studying the physics of these, inherently nonlinear, active photonic lattices. The interplay between the non-linear frequency pulling, which allows phase locking over the array, and the coupling among the individual slow cavity oscillations, creates a rich behavior involving both steady-state and dynamic effects.
A novel theoretical approach combining scattering theory with supercritical angle transmission is introduced for treating light incidence on nanotextured surfaces. The theory is used to evaluate enhanced light extraction from interfaces with sub-wavelength feature sizes, where the ray tracing approach breaks down. A unified analytic formula covering the transition from periodic to random surface texturing is obtained. The results will be compared with experimental enhanced light extraction results from GaN textured interfaces. The extraction efficiency is studied as a function of the average feature size and the rms deviation from the average values. It is argued that enhanced extraction occurs due to both supercritical transmission for single wave incidence, and the quick randomization of the incident wave-vector directions via internal scattering.
This work addresses the effects of finite boundaries in coupled arrays as well as those from "defects" from individual bias failures, of interest for practical applications. Analysis and numerical simulations based on the tight-binding approximation show that phase-locking persists in finite arrays. Self-regulation of the edge cavity density and power generate boundary layers of differentiated cavity operation values. The inter-cavity phase shift remains nearly uniform, with a small superimposed linear slope caused by cross-cavity reflection interference from DBRs. Phase locking is robust against partial or complete failure to lase for individual cavity sites, or even entire rows, due to biasing errors. The inherent gain dependence on carrier depletion and on the lateral cavity interactions is shown to be important.
The use of micro-structured interfaces has been shown to increase the extracted light portion far beyond the total reflection cone that limits extraction from flat surfaces. However, earlier theoretical treatments based on ray tracing, break down for nano-structured features smaller than the material wavelength. We apply a new analytic method converting the boundary conditions at the interface into surface currents following from the surface discontinuity in the perpendicular electric and parallel displacement vectors. These currents serve as source terms driving the transmitted radiation, computed by applying the full wave propagator. Analytic formulae yield the transmitted fraction for quasi-periodic surface features involving random variation of the feature size and period. A single incidence of an incoming plane wavefront suffices for transmission, while, in the ray tracing approach, many bounces with an opposing surface are required until the randomly changing angle of incidence falls within the extraction cone.
Closely packed VCSEL arrays phase lock due to active lateral interactions from cross gain and cross hole-burning. Breaking the lattice symmetry either by geometry, or by staggered application of two biases IA, IB creates a "two cavity species" photonic lattice. The relative phase among locked cavities is controlled by the applied bias ratio IA/IB. While uniform arrays allow only in- or out- of phase locking, two-species lattices allows a small but continuous range of phase shifts for the Bragg steering condition, without movable parts.
Lateral interactions due to fringe field overlap or "stray" reflections from optical elements in VCSEL array based interconnects are analyzed. Interacting cavity pairs act as coupled oscillators. The cavity that happens to switch on first, determined by the bit sequence between neighbors, acts as a master oscillator that affects the switch on jitter for the next cite. Earlier analytic results for the BER rate are extended to include the influence of the cavity coupling strength on the switch on jitter. Numerical examples, including pre-biasing cases, demonstrate the potential for a large degradation in BER rate at small coupling strengths. Extreme cases result into complete pulse suppression (bit skipping).
A novel, fast simulation tool for transient response is developed to study jitter and noise caused by lateral cavity interactions in VCSEL arrays. The cavity mode profiles, obtained from a paraxial eigenmode analysis, are used to derive fast 1-D rate equations that implement gain confinement, edge clipping, wide angle scattering and diffraction (self-interference) losses. These equations are augmented by lateral coupling terms describing the interactions among nearest neighbor cavities. Slow time scale coupling describes interactions of phase-shifted cavities via mutually induced electric polarization, cross-hole burning and cross-cavity gain due to optical fringe-field interactions. The tool is used to study cavity cross-talk, lateral bit pattern error effects, and the possibility of excitation of long range modulations over the array. Conclusions relating VCSEL packing density to BER, bit suppression by neighboring cavities, and array phase locking are given.
Since VCSEL cavities posses a natural oscillation frequency, 1-D or 2-D periodic VCSEL arrays behave as oscillator lattices coupled through near neighbor interactions. Modulation of an
individual VCSEL will excite modulation waves through the lattice. A dispersion relation for the lattice waves is obtained using optical interaction among nearest VCSELs. It is shown that
the excited frequency bandwidth is given by the product of the VCSEL natural frequency times the coupling strength; the latter is parameterized by the ratios a/w and b/a, with a the VCSEL active radius, w the cavity mode waist and b the array period. A connection is made between the lattice wave decay constant
and the channel cross-talk for VCSEL array-based interconnects / switch fabrics.
It is demonstrated numerically that photoactive layers placed in a VCSEL cavity can eliminate the power modulation during laser switch on. As well known, the transient response is generated by the cavity relaxation oscillations (omega) R, stemming from accelerated carrier depletion with increasing laser power. The presence of photoactive layers with appropriately chosen parameters reverses the sign of the depletion rate. The relaxation frequency then becomes pure imaginary, and the laser cavity behaves as an overdamped oscillator that asymptotes to the final steady-state without power modulation or 'spiking.' Superior bandwidth is anticipated by frequency response analysis.
On-axis channeling through the use of photoactive layers in VCSEL cavities is proposed to counteract hole burning and mode switching. The photoactive layers act as variable resistivity screens whose radial `aperture' is controlled by the light itself. It is numerically demonstrated that absorption of a small fraction of the light intensity suffices for significant on axis current peaking and single mode operation at currents many times threshold, with minimum efficiency loss and optical mode distortion. Fabrication is implemented during the molecular beam epitaxy phase without wafer post processing, as for oxide apertures.
The recirculating hub charge causes nonlinear feedback between the input and the output in Crossed-Field Amplifiers. By varying some control parameters, such as the secondary emission coefficient or the recirculating charge fraction, the operation exhibits transitions from steady state, to limit cycles and finally to unpredictable (chaotic) behavior. The amplifier dynamics is described by a truncated system of fluid equations following guiding center trajectories1. Numerical integration of these equations yields very good agreement with experimental results, while reducing the computation time by three orders of magnitude compared to particle simulations. A pass-to-pass map is employed to investigate whether the chaotic transitions in CFAs follow the 'universal' rout to chaos for nonlinear-dissipative systems.
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