In physics, ray tracing is a method for calculating the path of particles or waves through a medium of varying propagation velocity and absorbtion characteristics. Under these circumstances, wavefronts may bend, change direction, or reflect off of surfaces, complicating analysis. Ray tracing solves the problem by repeatedly advancing idealized narrow beams through the medium by very small amounts.

Technique

Ray tracing of a beam of light passing through a medium with changing refractive index. The ray is advanced by a small amount, and then the direction is re-calculated.

Ray tracing works by assuming that the particle or wave can be modeled as a large number of very narrow beams (rays), and that there exists some small distance over which such a ray is locally straight. The ray tracer will advance the ray over this distance, and then use a local derivative of the medium to calculate the ray's new direction. From this location, a new ray is sent out and the process is repeated until a complete path is generated. If the simulation includes solid objects, the ray may be tested for intersection with them at each step, making special adjustments to the ray's direction if a collision is found.

Uses

Radio Signals

Radio signals (rays) traced from the transmitter at the left to the receiver at the right (triangles on the base of the 3D grid).

One particular form of ray tracing is radio signal ray tracing, which traces radio signals, modeled as rays, through the ionosphere where they are refracted and/or reflected back to the Earth. This form of ray tracing involves the integration of differential equations that describe the propagation of electromagnetic waves through dispersive and anisotropic media such as the ionosphere. An example of physics-based radio signal ray tracing is shown to the right. Radio communicators use ray tracing to help determine the precise behavior of radio signals as they propagate through the ionosphere.

The image at the right illustrates the complexity of the situation. Unlike optical ray tracing where the medium between objects typically has a constant refractive index, signal ray tracing must deal with the complexities of a spatially varying refractive index, where changes in ionospheric electron densities influence the refractive index and hence, ray trajectories. Two sets of signals are broadcast at two different elevation angles. When the main signal penetrates into the ionosphere, the magnetic field splits the signal into two component waves which are separately ray traced through the ionosphere. The ordinary wave (red) component follows a path completely independent of the extraordinary wave (green) component.

Ocean Acoustics

Sound velocity in the ocean varies with depth due to changes in density and temperature, reaching a local minimum near a depth of 800-1000 meters. This local minimum, called the SOFAR channel, acts as a waveguide, as sound tends to bend towards it. Ray tracing may be used to calculate the path of sound through the ocean up to very large distances, incorporating the effects of the SOFAR channel, as well as reflections and refractions off of the ocean surface and bottom. From this, locations of high and low signal intensity may be computed, which are useful in the fields of ocean acoustics and underwater acoustic communication.

A ray tracing of acoustic wavefronts propagating through the varying density of the ocean. The path can be seen to oscillate about the SOFAR channel

Optical Design

Ray tracing may be used in the design of lenses and optical systems, such as in cameras, microscopes, telescopes, and binoculars, and its application in this field dates back to the 1900s. Geometric ray tracing is used to describe the propagation of light rays through a lens system or optical instrument, allowing the image-forming properties of the system to be modeled. The following effects can be integrated into a ray tracer in a straightforward fashion:

Dispersion leads to chromatic aberration
Gradient index optics
Polarization
- Crystal optics
- Fresnel equations
Laser light effects
Thin film interference (optical coating, soap bubble) can be used to calculate the reflectivity of a surface.

For the application of lens design, two special cases of wave interference are important to account for. In a focal point, rays from a point light source meet again and may constructively or destructively interfere with each other. Within a very small region near this point, incoming light may be approximated by plane waves which inherit their direction from the rays. The optical path length from the light source is used to compute the phase. The derivative of the position of the ray in the focal region on the source position is used to obtain the width of the ray, and from that the amplitude of the plane wave. The result is the point spread function, whose Fourier transform is the optical transfer function. From this, the Strehl ratio can also be calculated.

The other special case to consider is that of the interference of wavefronts, which, as stated before, are approximated as planes. When the rays come close together or even cross, however, the wavefront approximation breaks down. Interference of spherical waves is usually not combined with ray tracing, thus diffraction at an aperture cannot be calculated.

These techniques are used to optimize the design of the instrument by minimizing aberrations, for photography, and for longer wavelength applications such as designing microwave or even radio systems, and for shorter wavelengths, such as ultraviolet and X-ray optics.

Before the advent of the computer, ray tracing calculations were performed by hand using trigonometry and logarithmic tables. The optical formulas of many classic photographic lenses were optimized by roomfuls of people, each of whom handled a small part of the large calculation. Now they are worked out in optical design software such as OSLO or TracePro from Lambda Research, Code-V or Zemax. A simple version of ray tracing known as ray transfer matrix analysis is often used in the design of optical resonators used in lasers. The basic principles of the mostly used algorithm could be found in Spencer and Murty's fundamental paper: "General ray tracing Procedure".^[1]

References

^ G. H. Spencer and M. V. R.K. Murty (1962). "General ray tracing Procedure" (PDF). J. Opt. Soc. Am. 52 (6): 672–678.

[1] G. H. Spencer and M. V. R.K. Murty (1962). "General ray tracing Procedure" (PDF). J. Opt. Soc. Am. 52 (6): 672–678.

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User:Timrb/Ray Tracing (physics)

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