Ray Tracing In Inhomogeneous Media Research Articles

General scheme for paraxial ray tracing and aberration calculation in inhomogeneous media

A set of general formulae for paraxial ray tracing in inhomogeneous media are derived, which can be used together with the aberration formulae by Sands for the first- and third-order analyses of inhomogeneous lenses of all kinds of gradient function. The choice of an optimum step size for paraxial ray tracing and aberration integration is discussed, and it is shown that results with good accuracy can be obtained with a relatively large step size and very economic computing effort, even for complex gradient functions such as the tapered hyperbolic secant.

General scheme for paraxial ray tracing and aberration calculation in inhomogeneous media

Abstract A set of general formulae for paraxial ray tracing in inhomogeneous media are derived, which can be used together with the aberration formulae by Sands for the first- and third-order analyses of inhomogeneous lenses of all kinds of gradient function. The choice of an optimum step size for paraxial ray tracing and aberration integration is discussed, and it is shown that results with good accuracy can be obtained with a relatively large step size and very economic computing effort, even for complex gradient functions such as the tapered hyperbolic secant.

Differential ray tracing in inhomogeneous media

Differential ray tracing determines an optical system's first-order properties by finding the first-order changes in the configuration of an exiting ray in terms of changes in that ray's initial configuration. When one or more of the elements of a system is inhomogeneous, the only established procedure for carrying out a first-order analysis of a general ray uses relatively inefficient finite differences. To trace a ray through an inhomogeneous medium, one must, in general, numerically integrate an ordinary differential equation, and Runge–Kutta schemes are well suited to this application. We present an extension of standard Runge–Kutta schemes that gives exact derivatives of the numerically approximated rays.

Optimal interpolants for Runge–Kutta ray tracing in inhomogeneous media

Runge–Kutta integration schemes are well suited to the determination of ray trajectories in inhomogeneous media. There is a fundamental difference, however, between Runge–Kutta schemes and many other schemes for numerically integrating ordinary differential equations: Runge–Kutta schemes are not based on approximating the continuous trajectory by a polynomial. Consequently, these schemes do not implicitly provide a continuous trajectory; they yield only a series of points through which the ray passes, together with the ray direction at those points. A supplementary method must be devised when a continuous trajectory is required. The accuracy of a continuous trajectory for Runge–Kutta schemes is limited by the error introduced in a single iteration of the integrator. A trajectory that attains this limit is referred to here as optimal. The existing method of calculating trajectories for a widely used Runge–Kutta scheme is, in fact, not optimal. Accordingly, an efficient method of determining optimal intermediate trajectories is presented. This new technique is shown to be superior to the existing method for locating ray–surface intersections and allows accuracy doubling (a recently proposed method for accelerating the analysis of systems with inhomogeneous elements) to be used to full advantage.

A Direct Method for Ray Tracing in Gradient-index Fibres

Abstract A direct method for ray-tracing in inhomogeneous media is presented. In the case of gradient-index fibres, it is shown that satisfactory results can be obtained over distances typical of the discrete elements in fibre links. Illustrative examples are presented for the circularly symmetric fibre and the Y-divider.

Ray tracing in gradient-index media

The problem of ray tracing in inhomogeneous media has been examined in an attempt to find an accurate yet computationally efficient method. Several numerical techniques have been compared. It is shown that Montagnino’s Taylor-series-expansion method [ J. Opt. Soc. Am.58, 1667 ( 1968)] is superior to Euler’s method, which is a standard numerical method for solving the differential equation. It is also shown that, by a transformation of the independent coordinate from the physical path length to the optical path length, more-accurate optical path calculations may be obtained.

Ray Tracing in Inhomogeneous Media

Get PDF Email Share Share with Facebook Tweet This Post on reddit Share with LinkedIn Add to CiteULike Add to Mendeley Add to BibSonomy Get Citation Copy Citation Text Lucian Montagnino, "Ray Tracing in Inhomogeneous Media," J. Opt. Soc. Am. 58, 1667-1668 (1968) Export Citation BibTex Endnote (RIS) HTML Plain Text Citation alert Save article

Approximate Ray Tracing in Inhomogeneous Media

A method of approximate ray tracing is developed that can be applied to media that are inhomogeneous in two or three dimensions. The medium is subdivided into regions in each of which the velocity is approximated by a linear function of the space coordinates. The approximation results in a constant-velocity gradient in each region and a continuous velocity across the boundary between any two regions. A ray trace through the regions consists of a sequence of circular arcs that are tangent to one another at each boundary between regions. The technique may be applied to study the effects of inhomogeneity in more than one dimension in the construction of soundfield models, using geometric acoustics.