Factoring in the hyperelliptic Torelli group

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. The authors and Putman proved that this group is generated by Dehn twists about separating curves fixed by the hyperelliptic involution. In this paper, we introduce an algorithmic approach to factoring a wide class of elements of the hyperelliptic Torelli group into such Dehn twists, and apply our methods to several basic elements.Comment: 9 pages, 7 figure

Recent Developments In Monolithic Phase-Locked Semiconductor Laser Arrays

Coherent combination of the power of several semiconductor lasers fabricated on the same substrate has been the subject of an intense research effort in recent years, the main motivation being to obtain higher power levels than those available from a single laser in a stable radiation pattern. Best results reported so far include 2.6 Watts cw emitted power and less than 10 far-field angle (in the array plane) in arrays where all the lasers are electrically connected in parallel. A different type of coherent array, where each element has a separate contact, has been recently demonstrated. While requiring the more complex two-level metallization technology, applying a separate contact to each laser provides an additional degree of freedom in the design and the operation of monolithic arrays. The separate contacts can be employed to tailor the near-field and far-field distributions and to compensate for device-to-device nonuniformities. Furthermore, the control of the currents of the array elements allows the performance of a variety of other functions, such as beam scanning, spectral mode control, wavelength tuning and control of the mutual coherence between array elements

Longitudinal-mode control in integrated semiconductor laser phased arrays by phase velocity matching

The spectrum of semiconductor laser arrays with separate contacts is investigated. It is demonstrated that the individual laser currents can be selected such that the array operates in a single longitudinal mode in contrast to the multimode nature of its individual constituents. Moreover, it is possible to tune the lasing frequency by varying the laser currents. Wavelength tuning range of ~50 Å, with tuning rate of ~5 Å/mA, is demonstrated. It is suggested that these spectral features, characteristic of lasers which are coupled in parallel, result from the strong frequency dependence of their spatial mode pattern near the phase-matching frequency of their coupled waveguides

Controlled fundamental supermode operation of phase-locked arrays of gain-guided diode lasers

Uniform semiconductor laser arrays tend to oscillate in a superposition of their supermodes, thus leading to large beam divergence and spectral spread. Discrimination among the supermodes in phase-locked arrays is discussed theoretically. It is shown that supermode discrimination in gain-guided arrays, in favor of the fundamental supermode, is made possible by the near-field interference patterns which result from the complex optical fields of the gain-guided lasers. A fundamental supermode operation is demonstrated, for the first time, in GaAlAs/GaAs gain-guided laser arrays. This is achieved by control of the current (gain) profile across the array by means of individual laser contacts

Commensurations of the Johnson kernel

Let K be the subgroup of the extended mapping class group, Mod(S), generated by Dehn twists about separating curves. Assuming that S is a closed, orientable surface of genus at least 4, we confirm a conjecture of Farb that Comm(K), Aut(K) and Mod(S) are all isomorphic. More generally, we show that any injection of a finite index subgroup of K into the Torelli group I of S is induced by a homeomorphism. In particular, this proves that K is co-Hopfian and is characteristic in I. Further, we recover the result of Farb and Ivanov that any injection of a finite index subgroup of I into I is induced by a homeomorphism. Our method is to reformulate these group theoretic statements in terms of maps of curve complexes.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper37.abs.htm

Chirped arrays of diode lasers for supermode control

We propose nonuniform structures of phase-locked diode lasers, which make it possible to discriminate efficiently against all the higher order array supermodes (lateral modes). In these nonuniform arrays, the effective mode index in each channel varies across the array. Consequently, the envelopes of the various supermodes, including the highest order one, differ significantly from each other. Thus, by proper tailoring of the gain distribution across the array, one can conveniently select the fundamental supermode. Such fundamental supermode oscillation is essential in order to obtain single lobe, diffraction limited beams and minimal spectral spread from phase-locked laser arrays

Coupling mechanism of gain-guided integrated semiconductor laser arrays

It is shown that a gain-guided laser array couples via propagating fields rather than the evanescent mode coupling typically responsible for directional coupling in passive (directional couplers) and active (laser array) devices. We show that these phase-locked modes exhibit an interference pattern, in the junction plane, which arises from the curvature of the phase fronts of optical fields of the interacting lasers. The experimental results are interpreted with the aid of a simple theoretical model, and the effect of the observed mode pattern on the coupling of gain-guided lasers is discussed

Control of mutual phase locking of monolithically integrated semiconductor lasers

The mutual coherence of two coupled semiconductor lasers is investigated experimentally. It is demonstrated that by varying the gain in the overlap region, the degree of phase coherence can be continuously controlled. The quantitative characterization of the degree of phase coherence by fringe visibility is demonstrated

Single contact tailored gain phased array of semiconductor lasers

We demonstrate a single contact tailored gain-guided array in which the gain profile across the array is made strongly asymmetric by varying the width of the contact stripes. A proton isolated array of six (GaAl)As lasers with 5-µm separations and widths varying linearly between 3 and 8 µm had a single lobed far field 2° wide, close to the diffraction limit for a single supermode. Fabrication of this device is simple, and suited to large-scale processing techniques. We also show that in such an asymmetric gain-guided array the fundamental mode is favored over higher order modes, and that higher order modes can have single lobed far-field patterns differing only slightly from that of the fundamental