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# Ray tracing (physics)

In physics, ray tracing is a method for calculating the path of waves or particles through a system with regions of varying propagation velocity, absorption characteristics, and reflecting surfaces. Under these circumstances, wavefronts may bend, change direction, or reflect off surfaces, complicating analysis. Ray tracing solves the problem by repeatedly advancing idealized narrow beams called rays through the medium by discrete amounts. Simple problems can be analyzed by propagating a few rays using simple mathematics. More detailed analyses can be performed by using a computer to propagate many rays.

When applied to problems of electromagnetic radiation, ray tracing often relies on approximate solutions to Maxwell's equations that are valid as long as the light waves propagate through and around objects whose dimensions are much greater than the light's wavelength. Ray theory does not describe phenomena such as interference and diffraction, which require wave theory (involving the phase of the wave).

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 distance, possibly very small, 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 adjustments to the ray's direction if a collision is found. Other properties of the ray may be altered as the simulation advances as well, such as intensity, wavelength, or polarization. The process is repeated with as many rays as are necessary to understand the behavior of the system.

Uses

Radio signals

See also: Radio propagation

Radio signals 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.

See also: Computation of radiowave attenuation in the atmosphere

Ocean acoustics

See also: Underwater 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 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, underwater acoustic communication, and acoustic thermometry.

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

See also: Optical lens 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

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. 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 most frequently used algorithm could be found in Spencer and Murty's fundamental paper: "General ray tracing Procedure".[1]

Seismology

This ray tracing of seismic waves through the interior of the Earth shows that paths can be quite complicated, and reveals telling information about the structure of our planet

In seismology, geophysicists use ray tracing to aid in earthquake location and tomographic reconstruction of the Earth's interior.[2][3] Seismic wave velocity varies within and beneath Earth's crust, causing these waves to bend and reflect. Ray tracing may be used to compute paths through a geophysical model, following them back to their source, such as an earthquake, or deducing the properties of the intervening material.[4] In particular, the discovery of the seismic shadow zone (illustrated at right) allowed scientists to deduce the presence of Earth's molten core.

Plasma Physics

Energy transport and the propagation of waves plays an important role in the wave heating of plasmas. Power-flow trajectories of electromagnetic waves through a spatially nonuniform plasma can be computed using direct solutions of Maxwell’s equations. Another way of computing the propagation of waves in the plasma medium is by using Ray tracing method. Studies of wave propagation in plasmas using ray tracing method can be found in.[5]

See also

Atmospheric refraction

Ocean acoustic tomography

Ray transfer matrix analysis

Gradient index optics

Ray tracing (graphics)

List of ray tracing software

References

G. H. Spencer and M. V. R.K. Murty (1962). "General ray tracing Procedure" (PDF). J. Opt. Soc. Am. 52 (6): 672–678. doi:10.1364/JOSA.52.000672.

Rawlinson, N., Hauser, J. and Sambridge, M., 2007. Seismic ray tracing and wavefront tracking in laterally heterogeneous media. Advances in Geophysics, 49. 203-267.

Cerveny, V. (2001). Seismic Ray Theory. ISBN 0-521-36671-2.

Purdue University

Bhaskar Chaudhury and Shashank Chaturvedi (2006). "Comparison of wave propagation studies in plasmas using three-dimensional finite-difference time-domain and ray-tracing methods". Physics of Plasmas 13 (12): 123302. Bibcode:2006PhPl...13l3302C. doi:10.1063/1.2397582.

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