Nonlinear optical beam propagation and solitons in photorefractive media

Conclusions  The last few years have witnessed a great renewal of interest in the photorefractive effect thanks to the realization of the possibility of propagating stable 2D spatial solitons at very low optical powers in PR crystals. This result, besides possessing a great scientific relevance in itself, associated with the first experimental demonstration of a non-diffracting two-dimensional pencil of light, has also an interesting applicative potential thanks to the guidance properties of the waveguide associated with the soliton, which survives in the dark also after the soliton has been turned off. In this paper, we have tried to present a self-contained approach to the theory of self-trapped nonlinear propagation, which is far from being definitive, together with the most important experimental demonstrations obtained in the last few years. The novelty of the field, and the fact that it is still undergoing a rapid growth, has made our task not an easy one and we apologize to the readers for the many inevitable omissions both in the subjects we have chosen to emphasize and in the references. We cannot conclude this review without mentioning self-trapping of planar optical beams by the use of the PR effect in a semiconductor (InP:Fe) [94, 95], the main advantages over standard PR materials being associated with its sensitivity in the range of the near-infrared wavelengths and faster response time.     

Partially Coherent Waves in Nonlinear Periodic Lattices

We study the propagation of partially coherent (random-phase) waves in nonlinear periodic lattices. The dynamics in these systems is governed by the threefold interplay between the nonlinearity, the lattice properties, and the statistical (coherence) properties of the waves. Such dynamic interplay is reflected in the characteristic properties of nonlinear wave phenomena (e.g., solitons) in these systems. While the propagation of partially coherent waves in nonlinear periodic systems is a universal problem, we analyze it in the context of nonlinear photonic lattices, where recent experiments have proven their existence.     

On completely reducible solvable subgroups of GL(n, Δ)

Let Δ be a finite field and denote by GL(n, Δ) the group ofn×n nonsingular matrices defined over Δ. LetR⊆GL(n, Δ) be a solvable, completely reducible subgroup of maximal order. For |Δ|≧2, |Δ|≠3 we give bounds for |R| which improve previous ones. Moreover for |Δ|=3 or |Δ|>13 we determine the structure ofR, in particular we show thatR is unique, up to conjugacy. This work is part of a Ph.D. thesis done at the Hebrew University under the supervision of Professor A. Mann.     

Spatial solitons in centrosymmetric photorefractive media

We predict spatial solitons in photorefractive centrosymmetric media driven by the dc Kerr effect.     

Primarily isotropic nature of photorefractive screening solitons and the interactions between them

We present experimental evidence demonstrating that the photorefractive-index change responsible for the formation of photorefractive spatial screening solitons and coherent collisions between them is primarily isotropic in nature, even though the photorefractive medium is inherently anisotropic.     

Optically induced photovoltaic self-defocusing-to-self-focusing transition

We show theoretically and experimentally that the photovoltaic nonlinearity that gives rise to spatial solitons can be switched from self-defocusing to self-focusing (or vice versa) by use of background illumination. This raises the possibility of bright photovoltaic solitons in LiNbO(3) .     

Dynamics of incoherent bright and dark self-trapped beams and their coherence properties in photorefractive crystals

Using the coherent density approach, we study the propagation dynamics of incoherent bright and dark beams in biased photorefractive crystals. We show that, under appropriate initial conditions, bright as well as darklike incoherent quasi-solitons can be established in this material system. Our numerical simulations demonstrate that the coherence properties of these beams can be significantly affected by the self-trapping process.     

Apartments and the cayley-algebra module for G2(k)

The 7-dimensional Cayley-algebra module for the group G₂(k) over a field k has a presentation by 1-subspaces corresponding to the points of the associated hexagon, subject to relations determined by the lines, and one further relation on the neighbors of a point.The proof is geometric, based on spanning by apartments of the building.     

A Correction of an Inconsistency in my Paper “Cauchy's Theorem on Manifolds”

The Cauchy postulates are required for the formulation and proof of Cauchy's theorem for the existence of stress. The generalized postulates and theorem in the geometric setting of differentiable manifolds was considered in a previous paper. This note presents an inconsistency in one of the proposed postulates, the boundedness postulate, and corrects it by specifying a weaker requirement.