Gradient-index Medium Research Articles

Fractional Fourier transform and the elliptic gradient-index medium

We demonstrate that the propagation of a light wave in an elliptic gradient-index medium can be regarded as a joint fractional Fourier transform. The distinctive features of the Wigner distributon function and the recurrence of the original function are analyzed qualitatively.

Finite element ray tracing: a new method for ray tracing in gradient-index media

I apply the principle of finite elements, known in optics for calculating the beam propagation in waveguide structures, for calculation of meridional rays in inhomogeneous media. The plane is divided into finite elements that have a constant refractive index. The ray trajectory is calculated by a simple algorithm. Contrary to the existing methods, the model I propose in this research does not require an explicit formula for the index distribution. Only the numerical representation is sufficient, which can be a major advantage for calculation of the light propagation in real problems, such as the thermal lensing effect.

Numerical evaluation of the Luneburg integral and ray tracing

The Luneburg integral has many applications in optics and optoelectronics, among which is determination of the refractive-index profile of a Luneburg lens with a full or nonfull aperture. Consequently, computationally efficient and accurate methods for evaluating this integral represent an important challenge. An alternative approach to numerical evaluation of the Luneburg integral that is five times faster than existing methods is described. Several improvements in the ray-tracing procedure in gradient-index media are also presented. A combination of these methods increases the speed of ray tracing through the generalized Luneburg lens by as many as 2 orders of magnitude compared with earlier algorithms. The precision of our method can be easily controlled.

Sulfonated polystyrene as a new gradient-index medium for light-focusing elements

We have discovered that the sulfonation of polystyrene can form a gradient-index medium useful for lightfocusing purposes. We found that the refractive index of sulfonated polystyrene varies with the degree of sulfonation and that the refractive index of the fully sulfonated polystyrene decreased by approximately 0.06 at 0.633 microm from that of pure polystyrene.

Seventh-order aberrations of gradient-index media with axial symmetry

The theory of the seventh-order aberrations of gradient-index media with axial symmetry is presented. This theory is based on Hamilton’s equations. Formulas for seventh-order aberrations are derived. On the basis of the derived formulas the explicit algebraic expressions for all seventh-order aberration coefficients for gradient-index media with cylindrical symmetry are evaluated. The correctness of the obtained coefficients is verified for secant hyperbolic exact and helical exact gradient-index distributions. It is also shown that, for meridional rays, some of the aberration coefficients vanish for other index distributions besides the secant hyperbolic exact gradient-index distribution.

Beam modes in graded-index media and topological phases

The change in the polarization plane of light vector beams retaining their shape (beam modes) under propagation through gradient-index media is evaluated as a topological phase acquired by cyclic and noncyclic evolutions of these beams on their projective Hilbert space (momentum sphere). The polarization changes are evaluated by means of the characteristic parameters of the light beam selected.

Optical propagator in a graded-index medium with a hyperbolic secant refractive-index profile

Light transmission through a gradient-index medium, whose refractive-index profile can be represented by a one-dimensional hyperbolic secant function, is analyzed by a wave-optics approach. An asymptotic expression of the corresponding optical propagator (the Green's function) in this medium is evaluated in a such a way that nonparaxial propagation can be studied. Thus optical transformations such as focusing and collimation are obtained, and the pupil effect (diffractive effect) is shown.

Numerical determination of ray tracing: a new method

A practical method for tracing rays in a gradient-index medium (electron rays in a potential field) is described. First, the Euler equation is transformed into a convenient form of four first-order differential equations. Next, a standard numerical technique is used for solving the resulting equations that contain at most two first-order spatial derivatives of a refractive index. A comparison of this method with previous methods is made. Some special distributions of the refractive index and the potential of a spherical field capacitor have been used as a verification of the proposed method.

Three-dimensional focal curves generated by one-dimensional gradient-index tapers

Three-dimensional focal curves generated by one-dimensional gradient-index taper have been obtained when the taper is illuminated by an off-axis plane wave. The intensity distribution along the focal curve was evaluated by taking into account the particular form of the exit pupil produced by this gradient-index medium.

Three-dimensional focal curves generated by one-dimensional gradient-index tapers

Three-dimensional focal curves generated by one-dimensional gradient-index taper have been obtained when the taper is illuminated by an off-axis plane wave. The intensity distribution along the focal curve was evaluated by taking into account the particular form of the exit pupil produced by this gradient-index medium.

Geometrical model for the design of gradient-index-rod lens coupling devices

Gaussian beam propagation in a gradient-radial-index medium is presented and discussed. A simple geometrical method for modeling the Gaussian beam inside the gradient-index medium has been developed. The results with this method and those predicted by wave optics have been compared in a number of practical cases. Results show the validity of the ray approach. This is also confirmed by experimental tests on a high-power laser-fiber coupler.

Light rays in gradient index media: A laboratory exercise

A simple laboratory experiment is described for tracing light rays in a medium with spatially varying index of refraction. The analysis is at a level appropriate to an introductory optics course.

Fresnel Diffraction of Gaussian Beams by a Circular Aperture in Gradient-index Media

Abstract This work presents a method of evaluating the intensity distribution of a Gaussian beam in a weakly inhomogeneous medium when it has been truncated by a circular aperture. An analytical expression for the diffracted field is obtained in terms of Bessel functions.

The Nature of the Paraxial Approximation II. Systems Containing Media of Radial and of Axial Non-uniform Refractive Index: Invariants

Abstract The methods described in Part I are applied to systems containing media of non-uniform refractive index. For both radial and axial gradients, the parametric equations X = X(Z) and Y = Y(Z), and the values of the direction cosines of the ray-tangent at Z, of any ray are represented by power series in the initial object plane and entrance pupil variables (σ, τ, x y), where the coefficients are functions of the axial coordinate Z. The coefficients of the linear terms in (σ, τ; x y) are denoted by X 1(Z) and Y 1(Z), and paraxial refraction and transfer equations for the rays then appear as algebraic relations between the individual coefficients of the variables (σ, τ; x y) in X 1(Z) and Y 1(Z), and in the linear parts of the direction cosines of the ray-tangent at Z. Explicit forms are given, for both focusing and defocusing, radial and axial gradient-index media.

Ray tracing through gradient-index media: recent improvements

A general ray tracing algorithm for gradient-index media that treats various refractive index distributions in a unified manner has been developed by implementing the concept of isoindicial surfaces, position variables, and cubic splines. A novel and simpler algorithm for optical path length calculations is also presented.

On-axis and off-axis propagation of Gaussian beams in gradient index media

The object-image formula and an expression for the lateral magnification are obtained for a gradient radial index medium used with Gaussian beams. Angular acceptance conditions for on-axis and off-axis incidence are discussed. Meridional and sagittal sections of a Gaussian beam are analyzed, and numerical methods are used to find the evolution of the beam. The eccentricity parameter of the elliptical spot and the astigmatism of the beam are obtained as a function of propagation distance.

Optical path integrals in gradient-index media

A path integral has been used to evaluate, in paraxial approximation, the complex amplitude transmitted through gradient-index media. This formulation is useful for examining propagation in gradient-index media.

Ray tracing in gradient-index lenses: computation of ray–surface intersection

We present a simple, yet accurate, method to obtain the intersection point of a ray in a gradient-index medium and the surface following the medium. The method does not involve additional ray tracing, unlike the commonly used iterative technique, and the coordinates of the intersection point are obtained analytically.

Gradient-index objective lens for the compact disk system

A radial gradient-index aplanatic singlet pickup objective lens for the compact disk (CD) system has been designed, manufactured, and evaluated. A plano-convex aplanatic singlet was designed by optimizing the refractive-index coefficients of the gradient-index medium, lens thickness, and one curvature. The necessary ion exchange technology and surface fabrication technology have been developed to successfully fabricate the objective lens. The gradient-index singlet was mounted in a CD player, and tests confirmed that the lens meets the system requirements.

Design of three-dimensional nonimaging concentrators with inhomogeneous media

A three-dimensional nonimaging concentrator is an optical system that transforms a given four-parametric manifold of rays reaching a surface (entry aperture) into another four-parametric manifold of rays reaching the receiver. A procedure of design of such concentrators is developed. In general, the concentrators use mirrors and inhomogeneous media (i.e., gradient-index media). Using this method, we obtain an axisymmetrical concentrator with flat-entry aperture and receiver, which collects every ray that forms, at the entry aperture, an angle lower than a given value with the normal to this surface. The concentrator has the maximum concentration allowed by the theorem of conservation of phase-space volume. This is the first known concentrator with such properties. The Welford-Winston edge-ray principle in 3-D geometry is proven under several assumptions. The linear compound parabolic concentrator is derived as a particular case of the procedure of design.