Ray tracing (physics)
Encyclopedia
In physics, ray tracing is a method for calculating the path of wave
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

s or particle
Subatomic particle
In physics or chemistry, subatomic particles are the smaller particles composing nucleons and atoms. There are two types of subatomic particles: elementary particles, which are not made of other particles, and composite particles...

s through a system with regions of varying propagation velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

, absorption characteristics, and reflecting surfaces. Under these circumstances, wavefront
Wavefront
In physics, a wavefront is the locus of points having the same phase. Since infrared, optical, x-ray and gamma-ray frequencies are so high, the temporal component of electromagnetic waves is usually ignored at these wavelengths, and it is only the phase of the spatial oscillation that is described...

s may bend, change direction, or reflect
Reflection (physics)
Reflection is the change in direction of a wavefront at an interface between two differentmedia so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves...

off surfaces, complicating analysis. Ray tracing solves the problem by repeatedly advancing idealized narrow beams called rays
Ray (optics)
In optics, a ray is an idealized narrow beam of light. Rays are used to model the propagation of light through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of ray tracing. This allows even very...

through the medium
Medium (optics)
An optical medium is material through which electromagnetic waves propagate. It is a form of transmission medium. The permittivity and permeability of the medium define how electromagnetic waves propagate in it...

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
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

, ray tracing often relies on approximate solutions to Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.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
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

. Ray theory does not describe phenomena such as interference and diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

, which require wave theory (involving the phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

of the wave).

Technique

Ray tracing works by assuming that the particle or wave can be modeled as a large number of very narrow beams (rays
Ray (optics)
In optics, a ray is an idealized narrow beam of light. Rays are used to model the propagation of light through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of ray tracing. This allows even very...

), 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
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

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
Intensity (physics)
In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...

, wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

, or polarization. The process is repeated with as many rays as are necessary to understand the behavior of the system.

One particular form of ray tracing is radio
Radio is the transmission of signals through free space by modulation of electromagnetic waves with frequencies below those of visible light. Electromagnetic radiation travels by means of oscillating electromagnetic fields that pass through the air and the vacuum of space...

signal ray tracing, which traces radio signals, modeled as rays, through the ionosphere
Ionosphere
The ionosphere is a part of the upper atmosphere, comprising portions of the mesosphere, thermosphere and exosphere, distinguished because it is ionized by solar radiation. It plays an important part in atmospheric electricity and forms the inner edge of the magnetosphere...

where they are refracted and/or reflected back to the Earth. This form of ray tracing involves the integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

of differential equations that describe the propagation of electromagnetic waves
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

through dispersive and anisotropic
Anisotropy
Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions. It can be defined as a difference, when measured along different axes, in a material's physical or mechanical properties An example of anisotropy is the light...

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
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

, signal ray tracing must deal with the complexities of a spatially varying refractive index, where changes in ionospheric electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

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
Birefringence
Birefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientist Rasmus Bartholin in 1669, who saw it in calcite...

(red) component follows a path completely independent of the extraordinary wave
Birefringence
Birefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientist Rasmus Bartholin in 1669, who saw it in calcite...

(green) component.

Ocean acoustics

Sound velocity in the ocean
Ocean
An ocean is a major body of saline water, and a principal component of the hydrosphere. Approximately 71% of the Earth's surface is covered by ocean, a continuous body of water that is customarily divided into several principal oceans and smaller seas.More than half of this area is over 3,000...

varies with depth due to changes in density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

and temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

, reaching a local minimum near a depth of 800–1000 meters. This local minimum, called the SOFAR channel
Sofar channel
The SOFAR channel , or deep sound channel , is a horizontal layer of water in the ocean at which depth the speed of sound is minimal. The SOFAR channel acts as a waveguide for sound, and low frequency sound waves within the channel may travel thousands of miles before dissipating...

, acts as a waveguide
Waveguide
A waveguide is a structure which guides waves, such as electromagnetic waves or sound waves. There are different types of waveguides for each type of wave...

, 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
Reflection (physics)
Reflection is the change in direction of a wavefront at an interface between two differentmedia so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves...

and refraction
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...

s 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
Acoustical oceanography
Acoustical oceanography is the use of underwater sound to study the sea, its boundaries and its contents.-History:The earliest and most widespread use of sound and sonar technology to study the properties of the sea is the use of an echo sounder to measure water depth...

, underwater acoustic communication
Underwater acoustic communication
Underwater acoustic communication is a technique of sending and receiving message below water . There are several ways of employing such communication but the most common is using hydrophones...

, and acoustic thermometry.

Optical design

Ray tracing may be used in the design of lenses
Lens (optics)
A lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...

and optical systems, such as in camera
Camera
A camera is a device that records and stores images. These images may be still photographs or moving images such as videos or movies. The term camera comes from the camera obscura , an early mechanism for projecting images...

s, microscope
Microscope
A microscope is an instrument used to see objects that are too small for the naked eye. The science of investigating small objects using such an instrument is called microscopy...

s, telescopes
Optical telescope
An optical telescope is a telescope which is used to gather and focus light mainly from the visible part of the electromagnetic spectrum for directly viewing a magnified image for making a photograph, or collecting data through electronic image sensors....

, and binoculars
Binoculars
Binoculars, field glasses or binocular telescopes are a pair of identical or mirror-symmetrical telescopes mounted side-by-side and aligned to point accurately in the same direction, allowing the viewer to use both eyes when viewing distant objects...

, 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
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

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
Dispersion (optics)
In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...

Chromatic aberration
In optics, chromatic aberration is a type of distortion in which there is a failure of a lens to focus all colors to the same convergence point. It occurs because lenses have a different refractive index for different wavelengths of light...

• Polarization
• Crystal optics
Crystal optics
Crystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media in which light behaves differently depending on which direction the light is propagating. The index of refraction depends on both composition and crystal structure and can be...

• Fresnel equations
Fresnel equations
The Fresnel equations , deduced by Augustin-Jean Fresnel , describe the behaviour of light when moving between media of differing refractive indices...

• Laser light effects
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

• Thin film interference
Thin-film optics
Thin-film optics is the branch of optics that deals with very thin structured layers of different materials. In order to exhibit thin-film optics, the thickness of the layers of material must be on the order of the wavelengths of visible light...

(optical coating
Optical coating
An optical coating is one or more thin layers of material deposited on an optical component such as a lens or mirror, which alters the way in which the optic reflects and transmits light. One type of optical coating is an antireflection coating, which reduces unwanted reflections from surfaces, and...

, soap bubble
Soap bubble
A soap bubble is a thin film of soapy water enclosing air, that forms a hollow sphere with an iridescent surface. Soap bubbles usually last for only a few seconds before bursting, either on their own or on contact with another object. They are often used for children's enjoyment, but they are also...

) 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
Focus (optics)
In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...

, 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
Optical path length
In optics, optical path length or optical distance is the product of the geometric length of the path light follows through the system, and the index of refraction of the medium through which it propagates. A difference in optical path length between two paths is often called the optical path...

from the light source is used to compute the phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

. The derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

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
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

of the plane wave. The result is the point spread function
The point spread function describes the response of an imaging system to a point source or point object. A more general term for the PSF is a system's impulse response, the PSF being the impulse response of a focused optical system. The PSF in many contexts can be thought of as the extended blob...

, whose Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

is the optical transfer function
Optical transfer function
The optical transfer function of an imaging system is the true measure of resolution that the system is capable of...

. From this, the Strehl ratio
Strehl ratio
The Strehl ratio, named after the German physicist and mathematician Karl Strehl , is a measure for the optical quality of telescopes and other imaging instruments...

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
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

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
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

applications such as designing microwave or even radio systems, and for shorter wavelengths, such as ultraviolet
Ultraviolet
Ultraviolet light is electromagnetic radiation with a wavelength shorter than that of visible light, but longer than X-rays, in the range 10 nm to 400 nm, and energies from 3 eV to 124 eV...

and X-ray
X-ray
X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...

optics.

Before the advent of the computer
Computer
A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...

, ray tracing calculations were performed by hand using trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

and logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

ic tables. The optical formulas of many classic photographic
Photography
Photography is the art, science and practice of creating durable images by recording light or other electromagnetic radiation, either electronically by means of an image sensor or chemically by means of a light-sensitive material such as photographic film...

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
Optical lens design
Optical lens design refers to the calculation of lens construction parameters that will meet a set of performance requirements and constraints, including cost and schedule limitations....

. A simple version of ray tracing known as ray transfer matrix analysis
Ray transfer matrix analysis
Ray transfer matrix analysis is a type of ray tracing technique used in the design of some optical systems, particularly lasers...

is often used in the design of optical resonators
Optical cavity
An optical cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light. They are also used in optical parametric...

used in laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

s. The basic principles of the mostly used algorithm could be found in Spencer and Murty's fundamental paper: "General ray tracing Procedure".

Seismology

In seismology
Seismology
Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies. The field also includes studies of earthquake effects, such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic,...

, geophysicists use ray tracing to aid in earthquake location and tomographic
Tomography
Tomography refers to imaging by sections or sectioning, through the use of any kind of penetrating wave. A device used in tomography is called a tomograph, while the image produced is a tomogram. The method is used in radiology, archaeology, biology, geophysics, oceanography, materials science,...

reconstruction of the Earth's interior
Structure of the Earth
The interior structure of the Earth, similar to the outer, is layered. These layers can be defined by either their chemical or their rheological properties. The Earth has an outer silicate solid crust, a highly viscous mantle, a liquid outer core that is much less viscous than the mantle, and a...

. Seismic wave
Seismic wave
Seismic waves are waves of energy that travel through the earth, and are a result of an earthquake, explosion, or a volcano that imparts low-frequency acoustic energy. Many other natural and anthropogenic sources create low amplitude waves commonly referred to as ambient vibrations. Seismic waves...

velocity varies within and beneath Earth's crust
Crust (geology)
In geology, the crust is the outermost solid shell of a rocky planet or natural satellite, which is chemically distinct from the underlying mantle...

, 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 . 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 .

• Ocean acoustic tomography
Ocean acoustic tomography
Ocean Acoustic Tomography is a technique used to measure temperatures and currents over large regions of the ocean. On ocean basin scales, this technique is also known as acoustic thermometry. The technique relies on precisely measuring the time it takes sound signals to travel between two...

• Ray transfer matrix analysis
Ray transfer matrix analysis
Ray transfer matrix analysis is a type of ray tracing technique used in the design of some optical systems, particularly lasers...