Is integer arithmetic fundamental to mental processing?: the mind's secret arithmetic.

Unlike the ability to acquire our native language, we struggle to learn multiplication and division. It may then come as a surprise that the mental machinery for performing lightning-fast integer arithmetic calculations could be within us all even though it cannot be readily accessed, nor do we have any idea of its primary function. We are led to this provocative hypothesis by analysing the extraordinary skills of autistic savants. In our view such individuals have privileged access to lower levels of information not normally available through introspection.

Beam splitting on weak illumination.

We demonstrate, in both two and three dimensions, how a self-guided beam in a non-Kerr medium is split into two beams on weak illumination. We also provide an elegant physical explanation that predicts the universal character of the observed phenomenon. Possible applications of our findings to guiding light with light are also discussed.

Autistic artists give clues to cognition.

Certain autistic children whose linguistic ability is virtually nonexistent cart draw natural scenes from memory with astonishing accuracy. In particular their drawings display convincing perspective. In contrast, normal children of the same preschool age group and even untrained adults draw primitive schematics or symbols of objects which they can verbally identify. These are usually conceptual outlines devoid of detail. It is argued that the difference between autistic child artists and normal individuals is that autistic artists make no assumptions about what is to be seen in their environment. They have not formed mental representations of what is significant and consequently perceive all details as equally important. Equivalently, they do not impose visual or linguistic schema-a process necessary for rapid conceptualisation in a dynamic existence, especially when the information presented to the eye is incomplete.

Mighty morphing spatial solitons and bullets.

We give what we believe to be the first closed-form exact expression for the dynamic evolution of nonstationary beams of arbitrary intensity and width propagating in a uniform nonlinear medium and in both two and three dimensions. This shows that periodic and quasi-periodic (nonradiating) beams can exist in a non-Kerr nonlinear medium. The Schrödinger equation is solved for Gaussian beams in a saturable medium. For one critical (initial) beam width, the Gaussian is a stable stationary soliton or bullet, independent of its intensity; otherwise, it breathes. New quasi-periodic beams (mighty morphing solitons) and bullets (mighty morphs) of elliptical cross section also exist whose ellipticity changes with propagation.

Spatial solitons of Maxwell's equations.

Spatial solitons of Maxwell's equations propagating in an isotropic Kerr material differ significantly from the classical soliton of the nonlinear Schrödinger equation unless the electric field is linearly polarized along a geometric axis of the soliton intensity pattern. In general the polarization state changes continuously as the beam propagates, with a period of millimeters for highly nonlinear materials. This effect is due to the form birefringence of the soliton-induced waveguide. Equivalently, a soliton of Maxwell's equations is composed of both the TE and TM modes of the axially uniform waveguide it induces. Modal beating leads to the polarization dynamics.

Interacting gray solitons.

We present what is to our knowledge the first general theory for describing interacting well-separated gray spatial solitons, propagating in an arbitrary self-defocusing nonlinearity. It reveals why two dark solitons will in general repel, becoming less dark as they propagate. The results are illustrated with analytic expressions for the Kerr and the saturating nonlinearities.

Spatial solitons of the power-law nonlinearity.

Simple, exact analytical solutions of Maxwell's equations are given for the TE-type self-guided modes of a medium that has a power-law dependence on intensity I(q) for the continuum values of q. An analytical criterion shows that such spatial solitons are stable for q < 2 only. Our derivation is novel in that solitons are borrowed from the known modes of the sech(2) profile (linear) waveguide, rather than by solving the nonlinear wave equation. The results reveal the change in soliton propagation as the nonlinear medium itself changes.

Collisions, steering, and guidance with spatial solitons.

By colliding two self-guided beams (solitons), we generate one, two, three, four, or possibly more stable solitons for a broad class of nonlinear material that spans threshold to power-law nonlinearities. This reproduction is shown to depend critically on beam stability, on the standard waveguide parameter V, and on a scaled angular beam separation theta/theta(c). The beams can also annihilate one another or create a stable beam from two unstable beams. Beam steering is possible by differentially changing soliton power or phase. Also, the colliding solitons induce versatile optical devices for the switching and steering of small-signal beams.

Self-tapered beams.

By introducing material gain within a bulk nonlinear medium, it is possible to taper adiabatically a self-guided beam and to preserve its planar wave front. The amount of tapering can be controlled externally by varying the gain through the intensity of a pump beam. This self-guided beam induces a tapered (linear) waveguide, which in turn can guide a low-power (signal) beam. Two counterpropagating self-guided beams induce tapered, mismatched (linear) couplers that allow for all-optical, tunable, adiabatic power splitting without waveguide fabrication.

Stable black self-guided beams of circular symmetry in a bulk Kerr medium.

We demonstrate the existence of a stable black self-guided beam of circular symmetry in a bulk self-defocusing Kerr medium. There are no gray or higher-order self-guided beams of circular symmetry. The stability of black beams of circular symmetry is in sharp contrast to analogous bright beams in a self-focusing medium, which are unstable. Black self-guided beams are ideal for all-optical devices in bulk material as they can induce a single-mode optical fiber to guide a (weak) signal beam.

Self-induced optical fibers: spatial solitary waves.

To be self-guided, a beam must exactly equal the mode of the linear-optical fiber that it induces. From this elementary consistency condition we can borrow solutions and their associated physics directly from the familiar literature of linear-optical waveguides. By considering a nonlinear medium characterized by ideal saturation, we present what is to our knowledge the first exact analytical solution of a two-dimensional self-guided beam. This beam is the familiar fundamental mode of a step-profile fiber. The stability of the beam is also determined.

Grating-assisted couplers.

A simple closed-form description of grating-assisted coupling between twin cores is presented. It is exact in the limit of weakly coupled cores and weak grating perturbations.

Texture segregation with luminance and chromatic contrast.

Preattentive texture discrimination was investigated using low spatial frequency texture elements. The contrast between the texture elements and the background was either purely luminance or purely chromatic, or some combination of both these types of contrast. The threshold to discriminate correctly the location of a different textured region was obtained from each subject, as was each subject's threshold to detect the elements of the texture. Using the ratio of texture to element detection as a measure of the effectiveness of texture discrimination, little difference could be found between the perception of luminance or chromatic texture. However, there were large and significant variations among subjects with otherwise normal colour vision.

Unification of nonlinear-optical fiber devices.

Nonlinear coupling can be described in a universal form applicable to a general class of optical fiber devices involving the interaction of two modes. By knowing the solution to one type of device, we then have the solution to all other types. For example, the descriptions of birefringent fibers used as switches and amplifiers have their direct counterparts in directional (twin-core) couplers, found by merely redefining the meaning of subscripts in equations or on the phase-space portrait. The key to this unification follows, not by actually solving any one problem but rather by interrelating the coupling of normal modes with the coupling of submodes, e.g., the individual core modes of directional couplers.

Mismatched directional couplers.

Various twin-core couplers composed of mismatched optical fibers are capable of complete power transfer over lengths only moderately longer than their matched analogs. Examples include tapered and piecewise-continuous mismatched couplers. These linear couplers also provide insight into the mechanism of nonlinear coupling because nonlinearity induces periodic mismatching betweén twin cores that are axially uniform.

Directional couplers composed of non-Kerr-law material.

The effects on propagation in directional couplers made of non-Kerr-law material are examined. Many unstable modes can exist at one input power level, in contrast to only one for Kerr-law material. Special design features are possible by tailoring the dependence of the refractive index on intensity.

Stiles-Crawford effect in garter snake.

The optical geometry of the eye of the American garter snake suggests a highly pronounced Stiles-Crawford effect occurs at the level of its cone photoreceptors. However, it is demonstrated here that inner segments of the cones in this species have a high density of microdroplets, up to 0.1 microns in diameter, which increases the refractive index of the ellipsoid. Theoretical considerations involving geometrical optics show that these droplets, like conventional oil droplets, may be a specific adaptation of the garter snake to increase the sensitivity of its cones to oblique rays, thus reducing the potentially large Stiles-Crawford effect predicted for this vertebrate.

Four-photon parametric mixing in optical fibers: effect of pump depletion.

The analytical solutions to differential equations governing nonlinear four-photon mixing in optical fibers are presented for the general case of depleted pump power. The expressions reduce to those derived earlier with the assumption of nondepleted pump power. We find that the maximum power available for conversion from the pump waves to the signal waves resulting from a third-order nonlinearity depends on the index mismatch and that the index mismatch required for achieving maximum possible power transfer is a function of the frequency shift.