Lens-like Medium Research Articles

Equivalence Relations in a Class of Distributed Optical Systems–Lenslike Media

Equivalence properties of lenslike media with parabolic index profiles are analyzed on the basis of geometrical optics. By equivalent media we mean here those with the same ray-transfer matrix representation. An equivalence theorem is presented that provides a simple method of equivalent transformation of parabolic media. The practical importance of the equivalent transformation is that it leads to flexibility in the design of distributed optical structures, such as gas lenses and inhomogeneous optical fibers.

Equivalence of Sturm-Liouville-Type Distributed-Parameter Systems

This paper analyzes equivalence properties of linear distributed-parameter systems characterized by Sturm-Liouville-type differential equations. The approach taken here may be regarded as an application of the method of independent-variable transformation to those fundamental equations. In accord with the meaning of equivalence used for lumped-parameter systems, distributed-parameter systems are considered here as equivalent when they possess the same terminal behavior. The principal result is an equivalence theorem which establishes a general method of equivalent transformation of Sturm-Liouville systems. The fact that we can generate any number of equivalent parameter distributions from a given one is not only of theoretical interest but also of practical importance, because it offers a flexibility in the design of distributed-parameter structures. The derivation of the transformation function which plays a central role in the equivalent transformation is briefly described. Discussion is also given to the quantities which are invariant under the equivalent transformation. In the later part of the paper applications of the general theory to several problems of practical interest are considered, which include guiding systems for electromagnetic and space-charge waves such as nonuniform transmission lines, lens-like media, and nonuniform drift regions. The physical meanings of the transformation function, the parameters introduced, and various transformation invariants are discussed in detail. A simple example of equivalent systems is also illustrated in the case of nonuniform transmission lines.

On the ray transfer matrix of a tapered lenslike medium

In this letter are analyzed interrelationships among the four ray transfer matrix parameters of a tapered lenslike medium with a transversely square-law index profile. It is shown that knowing any one of them is sufficient to determine the other three parameters.

Propagation of light beams in continuously focusing lens-like media

A method is presented which allows a wave optical calculation of the Gaussian light beam in continuously focusing lenslike media. It is based on a formal quantum theory of physical optics. An example is worked out.

Mode Conversion due to Fluctuations in a Lens-like Medium

Mode conversion due to fluctuations caused by the dielectric-constant inhomogeneities of gaussian autocorrelation in a lens-like medium is discussed. Mean-square value of the mode-conversion coefficient, i.e., the power occupation of the higher mode in the total converted power, is obtained by means of the perturbation technique with the assumption of the fluctuations being smaller than square-law variation of the dielectric constant. The results show that the mode conversion increases in an undulating manner along the length of the gaseous medium, and is proportional to the longitudinal correlation length, whereas it decreases as the transverse correlation length becomes larger.

Influence of longitudinal temperature gradient on light propagation in lens-like medium

It is shown that the combined longitudinal and transverse temperature gradient can be used to obtain new types of focusing properties in lens-like media.

The theory of the thermohydrodynamic gas lens

We find an analytical solution for the equation of the beam trajectory for arbitrary lens-like media of the type n=n(y). We examine the optical properties of certain specific lens-type media.

Kinetics of an Optical Wave Packet in a Lens-like Medium

The motion of a wave packet in a nonuniform medium is discussed. It is found that the wave-packet motion, obtained by WKB method, is the same as one obtained by geometrical optics. The wave packet, in general, undulates around the axis. As the direction of the beam and the local refractive index change as the packet travels, the axial velocity component of the beam is modulated. This seems to be in contradiction to the constancy of the group velocity. This apparent contradiction is eliminated by examining the shape of a wave packet composed of a single transverse mode, which turns out to be warped. As an example, a medium is assumed in which the dielectric constant decreases quadratically, away from some spatial axis. The group velocity of several undulating packets in the medium is very close to that of a straight ray. Numerical calculation is presented.

HERMITE-GAUSSIAN MODE PATTERNS IN GaAs JUNCTION LASERS

Far-field radiation patterns with Hermite-Gaussian symmetry in the junction plane have been observed in both p+nn+ and p+n+ GaAs lasers specially constructed with a stripe geometry contact. The Hermite-Gaussian symmetry implies the presence of a lens-like medium between the cleavage plane mirrors. Compared with conventional junction lasers, improved mode control is obtained.

Off-Axis Wave-Optics Transmission in a Lens-Like Medium with Aberration

Normal modes and their propagation constants have been found for a two-dimensional lens-like medium in which the transverse refractive index varies essentially with quadratic law but has perturbing terms of higher order. In such a realistic guiding medium with aberrations, those modes are used to find the field configuration of a Gaussian beam of half width W entering off-axis. Close to the input the beam oscillates periodically with amplitude x i as if the medium were aberration-free, but slowly the beam cross-section changes shape, breaking up in several maxima, and increases size, reaching a maximum approximately equal to 2(W + x i ). Afterwards the beam slowly shrinks back to the starting field configuration and the process repeats again. These results are applicable to a sequence of lenses with aberrations and become important when the lenses are closely spaced. If redirectors are to be used to compensate for lens misalignments, the corrections must be made before a large break-up of the beam occurs.

Light Rays in Lens-Like Media

In this paper, the geometrical light rays in lens-like media are discussed. The eikonal equation curlK = 0, written in terms of the wave vector, is used and the characteristic differential equations of the light ray, h1du/ku = h2dυ/kυ = h3dw/kw, are obtained. The quantities which are conserved along the ray are also discussed and the equations of the rays in idealized lens-like media whose refractive indexes vary along one direction, are obtained to integrate the differential equations. For a practical example, the trajectory of the ray in a lens-like medium with square-law index variation is discussed.

A Light Beam Waveguide Using Hyperbolic-Type Gas Lenses

This paper is concerned with the optimum design of a light beam waveguide constructed with a proposed lens-like medium, namely, a hyperbolic-type gas lens which has a hyperbolic temperature distribution on its transverse cross section. In such a medium, the temperature distribution is ideally quadratic in the transverse directions, and so the mode conversion due to the higher order variation of the dielectric constant can be minimized. Moreover, this guide has the following merits; the design procedure is simple, the mode matching at the input part is easy, and it is possible to construct a waveguide using ordinary air as the lens medium. And this consideration can be easily extended to the guide with curved configuration. It was shown that the experimental convergency of this gas lens was in agreement with the theoretical one.

Imaging of Optical Modes - Resonators with Internal Lenses

This paper discusses the modes of optical resonators, and optical modes of propagation or Gaussian beams of light. The passage of Gaussian beams through lenses, telescopes, sequences of lenses, and lenslike media is studied. Mode matching formulae are derived. A complex beam parameter is introduced for which the law of transformation by any given optical structure can be written in the simple form of a bilinear transformation (ABCD law). Resonators with internal optical elements and their transmission line duals are also investigated. Effective Fresnel numbers and curvature parameters are determined which allow one to infer the diffraction losses, the resonant conditions, and the mode patterns of the various systems. Results are obtained for resonators with internal lenses, sequences of lenses with irises inserted between the lenses, resonators with internal lenslike media, transmission lines consisting of a lenslike medium with periodically spaced irises, and resonators with one very large mirror.