Nonlinear Lens Research Articles

Self-induced Mode as a Building Element of Transversal Pattern Formation

Abstract Optical patterns, forming in transversely extended nonlinear feedback systems, sometimes consist of ordered or disordered arrangements of many bright circular spots. We derive an approximate analytical description for a localized individual spot for the case of a diffusive Kerr nonlinearity. Each spot represents a Gaussian mode, which induces its own spherical subresonator by supporting a nonlinear lens in the Kerr medium. The properties of these self-induced modes are related to system parameters and their multistable behaviour is investigated. Interaction between two or more spots is estimated, leading to an easy explanation of hexagonal pattern formation.

Application of group theory on the nonlinear transport system following ion source

The transfer group theory is used to study the nonlinear transport system that follows the ion source. The properties of the transfer group and the method of computing the phase-space contours, distribution functions, and moments of ion beams are studied. It is shown that the nonlinear element can be used to compensate for the aberration. As an example, a method of improving the quality of the beam extracted from an ion source by means of a nonlinear space-charge lens is discussed.

Nonlinear propagation and transformation of arbitrary laser beams by means of the generalized ABCD formalism

The evolution of non-Gaussian and nonspherical high-power laser beams in cubic nonlinear media is described by means of their mean or gross parameters: width, mean curvature radius, and quality factor. The influence of the beam over its own propagation is contained in a new mean parameter that measures the ability of a beam to build its own waveguide. Beam quality and threshold power for self-focusing are connected. The ABCD and invariance laws for modified complex beam parameter and quality factor allow one to transform in one step the mean beam parameters through a sequence of nonlinear propagations, lenses, mirrors, and nonlinear quadratic graded index.

Formulation of a constrained automatic lens design problem by means of the augmented Lagrangian function

It is formulated a constrained nonlinear automatic lens design problem by means of the augmented Lagrangian function. By introducing transformation matrices, the solution is transformed into the normal equation of a linear least squares problem.

Formulation of a nonlinear automatic lens design problem with boundary conditions by means of the Kuhn-Tacker's optimality condition

Presents a constrained nonlinear automatic lens design problem formulated with the Kuhn-Tucker's optimality conditions. At first, the first order solutions are given, then the second order solutions are presented together with their formal linearization process.

Nelineinye linzy i ikh primeneniya

Nelineinye linzy i ikh primeneniya

Imaging properties of Gaussian beams with a nonlinear graded-index rod lens

The imaging properties of Gaussian beams of light passing through a nonlinear graded-index rod are studied for the first time to our knowledge. The beam-width parameter, the distance of the focal plane, and the linear magnification in terms of the beam power, diffraction, and the rod dimensions are derived by using the variational approach and the ABCD law of Gaussian beam propagation. A comparison of these results with those derived from the ray equation and the linear case shows that the effects of the beam power on the imaging characteristics are significant.

Some Optical Properties of the Elaidicum Acid in Mesophase

Abstract The liquid crystal (smectic C* type) properties of the elaidicum acid, from 28°C to 51°C have been studied and utilised in this paper. The electrical potential, the capacitance and the electric current intensity versus applied voltage on a membrane of liquid crystal were measured. The experimental results of these measurements and the microstructural aspects show that this acid has the dielectric properties of a liquid crystal and is ferroelectric. The dependence of the current on time, when an electric field is applied and removed, were investigated, too, the results confirming the previous conclusions. Taking into account these findings, different liquid crystal cells were realised and utilised for the “nonlinear lens” effect. This is the external self-focusing of laser light in the liquid crystal film. A He-Ne laser (Λ = 6328 A, P = 16-20 mW) was utilised for the experimental setup. The f NL change with the sample thickness and with the temperature was determined, the optical nonlinearity ϵ2 v...

A formal linearization procedure of constrained nonlinear automatic lens design problems. II. Use of slack variables and Lagrange multipliers

For pt. I, see J. Opt. (Paris), vol.21, no.6, p.241 (1990). Presents a constrained nonlinear automatic lens design problem formulated by means of Lagrange multipliers and slack variables. The first order and the second order solutions are presented. Formal linearization of the second order solutions is also derived.

A formal linearization procedure of constrained nonlinear automatic lens design problems. I. Formulation by means of penalty function method

Presents a process to linearize formally a constrained nonlinear automatic lens design problem, which is formulated by means of the penalty function method. Also is given the solutions derived by the QR factorization method.

Linearization of nonlinear automatic lens design problems

A simple method to linearize formally a nonlinear automatic lens design problem is presented with two different solvers.

Root-mean-square emittance increase in a nonlinear lens

An exact derivation of the root-mean-square emittance increase when a beam of arbitrary distribution passes through a nonlinear lens system is presented. It is found that the emittance increase depends on the spherical aberration coefficient of the lens and characteristics of the beam distribution.

Experimental studies of electron beam transport in the maryland periodic solenoid channel

In the University of Maryland electron beam transport experiment, a 5 kV, 0.2 A electron beam from a thermionic electron source (cathode radius r c = 1.27 cm and cathode temperature kT = 0.12 eV) is injected into a periodic focusing channel consisting of 38 solenoid lenses with period length S = 13.6 cm. The magnetic focusing fields of the channel, and hence the phase advance without space charge, can be varied over a wide range. We report emittance measurements and results of the effects of beam mismatch, misalignments and nonlinear lens forces on beam transport and emittance growth.

Formation of arcs by nearly circular gravitational lenses.

An extended source located near the axis of a nearly circular, sufficiently nonlinear lens can produce extended-arc-like images. Examples of these may have been found near three distant clusters of galaxies which seem to be acting as gravitational lenses. We propose a simple geometrical method for locating images of objects using the Einstein ring and the corresponding caustic. We also give an approximate theory of lenses for which deviations from circularity are small perturbations. We discuss noncircular perturbations of practical interest for gravitational lensing and properties of the arcs that are thus formed.

Dynamics of the self-interaction of picosecond light pulses in semiconducting cadmium suifide

The dynamics of the self-interaction of picosecond laser pulses in semiconducting crystals was investigated for the first time. A nonlinear lens method was analyzed theoretically allowing for diffraction and then this method was used in direct measurements of the nonlinearity of the anisotropy of the refractive index of CdS single crystals excited with picosecond light pulses.

Coherent light oscillator by self-induced holographic gratings in SBN:Ce crystals

The coherent light oscillation in linear and ring cavities with the gain medium on holographic induced gratings in Ce doped Sr1-xBaxNb2O6 crystals has been obtained. Oscillation appeared without any apparent pumping power threshold at the excitation levels down to 75 μW on 488 nm Ar laser line. The angular dependence of gain reaching 50cm-1 has been measured. The strong selffocusing was observed with the nonlinear lens power reaching 60 diopters in a 1 mm thick sample for the first time in ferroelectrics. The positive lens aris;es mainly due to heating of crystal by absorbed radiation and its optical power is determined by the peak values of the thermooptical constants in the vicinity of the phase transition.

Effects of space charge and lens aberrations in the focusing of an electron beam by a solenoid lens

The effects of space charge and lens aberrations on a 5-keV, 80–200-mA electron beam focused by a solenoid lens are investigated theoretically and experimentally. A third-order radial trajectory equation is derived with space charge added in the form of discrete annular current elements. This equation is numerically solved for the trajectories of focused electrons in a short solenoid lens. Trajectories launched near the axis are ‘‘reflected’’ from the axis at the waist due to space charge, and trajectories launched farther from axis, cross the axis. The resulting discontinuity is best illustrated in a transverse phase-space plot where it may be described as a ‘‘tearing’’ of the electron distribution. Due to the combined action of the nonlinear lens forces and the space charge, the initially uniform radial current density distribution becomes markedly nonuniform (hollow or peaked depending on distance) downstream from the lens. Detailed experimental measurements of the density profiles at different positions are found to be in very good agreement with computations.

Investigation of Stochastic Beam Growth with a Non-Linear Lens

The problem of the onset of stochastic motion in a non-linear system is of considerable importance for the understanding of the long-time stability of beams in hadron storage rings in the presence of strong high-order non-linear resonances. A non-linear lens has been installed in the SPS in order to simulate resonances driven by the beam-beam effect or by magnetic field imperfections in superconducting storage rings. In this paper, a criterion for the onset of stochastic motion is derived and compared with the results of computer simulations and of experiments performed on the machine.

A Nonlinear Lens System to Smooth the Intensity Distribution of a Gaussian Beam

The intensity distribution of a typical ion beam is quite nonuniform with a bright center and a diffuse edge; in many cases, the actual distribution is adequately approximated by a Gaussian distribution. However, many uses for such beams require uniform illumination from the center of the target to its edges. It is shown here that a system of controlled third-order aberrations produced by a pair of octupole magnets is able to sustantially smooth the intensity distribution by spreading the center and folding in the edges of the beam such that the intensity is nearly uniform over the entire target.

Amplitude growth of (anti)proton colliding beams

The connection between beam loss in a p-p (or p- p ) storage ring and multiple Coulomb scattering, related to head-on collisions, is considered. It is demonstrated that transversal heating can push particles into the unstable region of a nonlinear resonance, whilst stable area bounds are shrunk by the beam-beam tune-shift and by the establishment of a beam-beam nonlinear lens. Beam lifetimes are evaluated and compared with experimental results. The agreement looks quite encouraging.