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A waveguide is a structure that guides waves, such as
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
s or
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities decrease according to the
inverse square law In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understoo ...
as they expand into three-dimensional space. There are different types of waveguides for different types of waves. The original and most common meaningInstitute of Electrical and Electronics Engineers, “The IEEE standard dictionary of electrical and electronics terms”; 6th ed. New York, N.Y., Institute of Electrical and Electronics Engineers, c1997. IEEE Std 100-1996. d. Standards Coordinating Committee 10, Terms and Definitions; Jane Radatz, (chair)/ref> is a hollow conductive metal pipe used to carry high frequency
radio wave Radio waves are a type of electromagnetic radiation with the longest wavelengths in the electromagnetic spectrum, typically with frequencies of 300 gigahertz (GHz) and below. At 300 GHz, the corresponding wavelength is 1 mm (short ...
s, particularly
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ran ...
s.
Dielectric waveguide An optical waveguide is a physical structure that guides electromagnetic waves in the optical spectrum. Common types of optical waveguides include optical fiber waveguides, transparent dielectric waveguides made of plastic and glass, liquid light g ...
s are used at higher radio frequencies, and transparent dielectric waveguides and
optical fiber An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a means to ...
s serve as waveguides for light. In
acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
, air ducts and
horn Horn most often refers to: *Horn (acoustic), a conical or bell shaped aperture used to guide sound ** Horn (instrument), collective name for tube-shaped wind musical instruments *Horn (anatomy), a pointed, bony projection on the head of various ...
s are used as waveguides for sound in musical instruments and
loudspeaker A loudspeaker (commonly referred to as a speaker or speaker driver) is an electroacoustic transducer that converts an electrical audio signal into a corresponding sound. A ''speaker system'', also often simply referred to as a "speaker" or " ...
s, and specially-shaped metal rods conduct ultrasonic waves in
ultrasonic machining Ultrasonic machining is a subtractive manufacturing process that removes material from the surface of a part through high frequency, low amplitude vibrations of a tool against the material surface in the presence of fine abrasive particles. The t ...
. The geometry of a waveguide reflects its function; in addition to more common types that channel the wave in one dimension, there are two-dimensional slab waveguides which confine waves to two dimensions. The frequency of the transmitted wave also dictates the size of a waveguide: each waveguide has a
cutoff wavelength In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced ( attenuated or reflected) rather than ...
determined by its size and will not conduct waves of greater wavelength; an
optical fiber An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a means to ...
that guides
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
will not transmit
microwaves Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency rang ...
which have a much larger wavelength. Some naturally occurring structures can also act as waveguides. The
SOFAR channel The SOFAR channel (short for sound fixing and ranging channel), or deep sound channel (DSC), is a horizontal layer of water in the ocean at which depth the speed of sound is at its minimum. The SOFAR channel acts as a waveguide for sound, and low ...
layer in the ocean can guide the sound of
whale song Whales use a variety of sounds for animal communication, communication and sensation. The mechanisms used to produce sound vary from one family of cetaceans to another. Marine mammals, including whales, dolphins, and porpoises, are much more d ...
across enormous distances. Any shape of cross section of waveguide can support EM waves. Irregular shapes are difficult to analyse. Commonly used waveguides are rectangular and circular in shape.


Uses

The uses of waveguides for transmitting signals were known even before the term was coined. The phenomenon of sound waves guided through a taut wire have been known for a long time, as well as sound through a hollow pipe such as a
cave A cave or cavern is a natural void in the ground, specifically a space large enough for a human to enter. Caves often form by the weathering of rock and often extend deep underground. The word ''cave'' can refer to smaller openings such as sea ...
or medical
stethoscope The stethoscope is a medical device for auscultation, or listening to internal sounds of an animal or human body. It typically has a small disc-shaped resonator that is placed against the skin, and one or two tubes connected to two earpieces. ...
. Other uses of waveguides are in transmitting power between the components of a system such as radio, radar or optical devices. Waveguides are the fundamental principle of
guided wave testing Guided wave testing (GWT) is a non-destructive evaluation method. The method employs acoustic waves that propagate along an elongated structure while guided by its boundaries. This allows the waves to travel a long distance with little loss in ...
(GWT), one of the many methods of
non-destructive evaluation Nondestructive testing (NDT) is any of a wide group of analysis techniques used in science and technology industry to evaluate the properties of a material, component or system without causing damage. The terms nondestructive examination (NDE), n ...
. Specific examples: * Optical fibers transmit light and signals for long distances with low attenuation and a wide usable range of wavelengths. * In a
microwave oven A microwave oven (commonly referred to as a microwave) is an electric oven that heats and cooks food by exposing it to electromagnetic radiation in the microwave frequency range. This induces polar molecules in the food to rotate and produce t ...
a waveguide transfers power from the
magnetron The cavity magnetron is a high-power vacuum tube used in early radar systems and currently in microwave ovens and linear particle accelerators. It generates microwaves using the interaction of a stream of electrons with a magnetic field while ...
, where waves are formed, to the cooking chamber. * In a radar, a waveguide transfers radio frequency energy to and from the antenna, where the impedance needs to be matched for efficient power transmission (see below). * Rectangular and circular waveguides are commonly used to connect feeds of parabolic dishes to their electronics, either low-noise receivers or power amplifier/transmitters. * Waveguides are used in scientific instruments to measure optical, acoustic and elastic properties of materials and objects. The waveguide can be put in contact with the specimen (as in a
medical ultrasonography Medical ultrasound includes diagnostic techniques (mainly medical imaging, imaging techniques) using ultrasound, as well as therapeutic ultrasound, therapeutic applications of ultrasound. In diagnosis, it is used to create an image of internal ...
), in which case the waveguide ensures that the power of the testing wave is conserved, or the specimen may be put inside the waveguide (as in a dielectric constant measurement), so that smaller objects can be tested and the accuracy is better. * A
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
is a commonly used specific type of waveguide.


History

The first structure for guiding waves was proposed by
J. J. Thomson Sir Joseph John Thomson (18 December 1856 – 30 August 1940) was a British physicist and Nobel Laureate in Physics, credited with the discovery of the electron, the first subatomic particle to be discovered. In 1897, Thomson showed that c ...
in 1893, and was first experimentally tested by
Oliver Lodge Sir Oliver Joseph Lodge, (12 June 1851 – 22 August 1940) was a British physicist and writer involved in the development of, and holder of key patents for, radio. He identified electromagnetic radiation independent of Heinrich Rudolf Hertz, H ...
in 1894. The first mathematical analysis of electromagnetic waves in a metal cylinder was performed by
Lord Rayleigh John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Amo ...
in 1897. For sound waves, Lord Rayleigh published a full mathematical analysis of
propagation mode Radio propagation is the behavior of radio waves as they travel, or are propagated, from one point to another in vacuum, or into various parts of the atmosphere. As a form of electromagnetic radiation, like light waves, radio waves are affected ...
s in his seminal work, “The Theory of Sound”.The Theory of Sound
by J. W. S. Rayleigh, (1894)
Jagadish Chandra Bose Sir Jagadish Chandra Bose (;, ; 30 November 1858 – 23 November 1937) was a biologist, physicist, Botany, botanist and an early writer of science fiction. He was a pioneer in the investigation of radio microwave optics, made significant contr ...
researched
millimetre wave file:EM Spectrum Properties edit.svg, 330px, Different lengths as in respect to the electromagnetic spectrum, measured by the metre and its derived scales. The microwave is between 1 meter to 1 millimeter. The millimetre (American and British Eng ...
lengths using waveguides, and in 1897 described to the Royal Institution in London his research carried out in Kolkata. reprinted in Igor Grigorov, Ed.,
Antentop
', Vol. 2, No.3, pp. 87–96.
The study of dielectric waveguides (such as optical fibers, see below) began as early as the 1920s, by several people, most famous of which are Rayleigh,
Sommerfeld Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
and
Debye The debye (symbol: D) (; ) is a CGS unit (a non- SI metric unit) of electric dipole momentTwo equal and opposite charges separated by some distance constitute an electric dipole. This dipole possesses an electric dipole moment whose value is give ...
.Advanced Engineering Electromagnetics
, by C. A. Balanis, John Wiley & Sons (1989).
Optical fiber began to receive special attention in the 1960s due to its importance to the communications industry. The development of radio communication initially occurred at the lower frequencies because these could be more easily propagated over large distances. The long wavelengths made these frequencies unsuitable for use in hollow metal waveguides because of the impractically large diameter tubes required. Consequently, research into hollow metal waveguides stalled and the work of Lord Rayleigh was forgotten for a time and had to be rediscovered by others. Practical investigations resumed in the 1930s by
George C. Southworth George Clark Southworth (August 24, 1890 – July 6, 1972), who published as G. C. Southworth, was a prominent American radio engineer best known for his role in the development of waveguides in the early 1930s. Biography Southworth was born in ...
at
Bell Labs Nokia Bell Labs, originally named Bell Telephone Laboratories (1925–1984), then AT&T Bell Laboratories (1984–1996) and Bell Labs Innovations (1996–2007), is an American industrial research and scientific development company owned by mult ...
and
Wilmer L. Barrow Wilmer Lanier Barrow (July 26, 1903 – August 29, 1975) was an American electrical engineer, inventor, teacher, industrial manager, and a counselor to government agencies. He obtained a BSEE degree in 1926 from Louisiana State University, and ...
at
MIT The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the mo ...
. Southworth at first took the theory from papers on waves in dielectric rods because the work of Lord Rayleigh was unknown to him. This misled him somewhat; some of his experiments failed because he was not aware of the phenomenon of
waveguide cutoff frequency In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced ( attenuated or reflected) rather than ...
already found in Lord Rayleigh's work. Serious theoretical work was taken up by
John R. Carson John Renshaw Carson (June 28, 1886 – October 31, 1940) was an American transmission theorist for early communications systems. He invented single-sideband modulation and developed the Carson bandwidth rule for estimating frequency modulation (FM) ...
and Sallie P. Mead. This work led to the discovery that for the TE01 mode in circular waveguide losses go down with frequency and at one time this was a serious contender for the format for long-distance telecommunications. The importance of
radar Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...
in
World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposin ...
gave a great impetus to waveguide research, at least on the
Allied An alliance is a relationship among people, groups, or states that have joined together for mutual benefit or to achieve some common purpose, whether or not explicit agreement has been worked out among them. Members of an alliance are called ...
side. The
magnetron The cavity magnetron is a high-power vacuum tube used in early radar systems and currently in microwave ovens and linear particle accelerators. It generates microwaves using the interaction of a stream of electrons with a magnetic field while ...
, developed in 1940 by John Randall and
Harry Boot Henry Albert Howard Boot (29 July 1917 – 8 February 1983) was an English physicist who with Sir John Randall and James Sayers developed the cavity magnetron, which was one of the keys to the Allied victory in the Second World War. Biography ...
at the University of Birmingham in the United Kingdom, provided a good power source and made microwave radar feasible. The most important centre of US research was at the
Radiation Laboratory The Radiation Laboratory, commonly called the Rad Lab, was a microwave and radar research laboratory located at the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts. It was first created in October 1940 and operated until 31 ...
(Rad Lab) at
MIT The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the mo ...
but many others took part in the US, and in the UK such as the
Telecommunications Research Establishment The Telecommunications Research Establishment (TRE) was the main United Kingdom research and development organization for radio navigation, radar, infra-red detection for heat seeking missiles, and related work for the Royal Air Force (RAF) d ...
. The head of the Fundamental Development Group at Rad Lab was
Edward Mills Purcell Edward Mills Purcell (August 30, 1912 – March 7, 1997) was an American physicist who shared the 1952 Nobel Prize for Physics for his independent discovery (published 1946) of nuclear magnetic resonance in liquids and in solids. Nuclear magne ...
. His researchers included
Julian Schwinger Julian Seymour Schwinger (; February 12, 1918 – July 16, 1994) was a Nobel Prize winning American theoretical physicist. He is best known for his work on quantum electrodynamics (QED), in particular for developing a relativistically invariant ...
,
Nathan Marcuvitz Nathan Marcuvitz (1913 - 2010) was an American electrical engineer, physicist, and educator who worked in the fields of microwave and electromagnetic field theory. He was head of the experimental group of the Radiation Laboratory (MIT). He was a ...
, Carol Gray Montgomery, and
Robert H. Dicke Robert Henry Dicke (; May 6, 1916 – March 4, 1997) was an American astronomer and physicist who made important contributions to the fields of astrophysics, atomic physics, physical cosmology, cosmology and gravity. He was the Albert Einstein ...
. Much of the Rad Lab work concentrated on finding
lumped element model The lumped-element model (also called lumped-parameter model, or lumped-component model) simplifies the description of the behaviour of spatially distributed physical systems, such as electrical circuits, into a topology consisting of discrete e ...
s of waveguide structures so that components in waveguide could be analysed with standard circuit theory.
Hans Bethe Hans Albrecht Bethe (; July 2, 1906 – March 6, 2005) was a German-American theoretical physicist who made major contributions to nuclear physics, astrophysics, quantum electrodynamics, and solid-state physics, and who won the 1967 Nobel Prize ...
was also briefly at Rad Lab, but while there he produced his small aperture theory which proved important for waveguide cavity filters, first developed at Rad Lab. The German side, on the other hand, largely ignored the potential of waveguides in radar until very late in the war. So much so that when radar parts from a downed British plane were sent to
Siemens & Halske Siemens & Halske AG (or Siemens-Halske) was a German electrical engineering company that later became part of Siemens. It was founded on 12 October 1847 as ''Telegraphen-Bauanstalt von Siemens & Halske'' by Werner von Siemens and Johann Geo ...
for analysis, even though they were recognised as microwave components, their purpose could not be identified. German academics were even allowed to continue publicly publishing their research in this field because it was not felt to be important. Immediately after World War II waveguide was the technology of choice in the microwave field. However, it has some problems; it is bulky, expensive to produce, and the cutoff frequency effect makes it difficult to produce wideband devices. Ridged waveguide can increase bandwidth beyond an octave, but a better solution is to use a technology working in
TEM mode A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern of the radiation in the plane perpendicular (i.e., transverse) to the radiation's propagation direction. Transverse modes occur in radio waves and microwav ...
(that is, non-waveguide) such as
coaxial In geometry, coaxial means that several three-dimensional linear or planar forms share a common axis. The two-dimensional analog is ''concentric''. Common examples: A coaxial cable is a three-dimensional linear structure. It has a wire conduc ...
conductors since TEM does not have a cutoff frequency. A shielded rectangular conductor can also be used and this has certain manufacturing advantages over coax and can be seen as the forerunner of the planar technologies (
stripline Stripline is a transverse electromagnetic (TEM) transmission line medium invented by Robert M. Barrett of the Air Force Cambridge Research Centre in the 1950s. Stripline is the earliest form of planar transmission line. Description A strip ...
and
microstrip Microstrip is a type of electrical transmission line which can be fabricated with any technology where a conductor is separated from a ground plane by a dielectric layer known as the substrate. Microstrip lines are used to convey microwave-frequ ...
). However, planar technologies really started to take off when printed circuits were introduced. These methods are significantly cheaper than waveguide and have largely taken its place in most bands. However, waveguide is still favoured in the higher microwave bands from around
Ku band The Ku band () is the portion of the electromagnetic spectrum in the microwave range of frequencies from 12 to 18 gigahertz (GHz). The symbol is short for "K-under" (originally german: Kurz-unten), because it is the lower part of the ori ...
upwards.


Properties


Propagation modes and cutoff frequencies

A
propagation mode Radio propagation is the behavior of radio waves as they travel, or are propagated, from one point to another in vacuum, or into various parts of the atmosphere. As a form of electromagnetic radiation, like light waves, radio waves are affected ...
in a waveguide is one solution of the wave equations, or, in other words, the form of the wave. Due to the constraints of the
boundary conditions In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
, there are only limited frequencies and forms for the wave function which can propagate in the waveguide. The lowest frequency in which a certain mode can propagate is the
cutoff frequency In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced ( attenuated or reflected) rather than ...
of that mode. The mode with the lowest cutoff frequency is the fundamental mode of the waveguide, and its cutoff frequency is the waveguide cutoff frequency. Propagation modes are computed by solving the
Helmholtz equation In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplacian"), is the eigenv ...
alongside a set of boundary conditions depending on the geometrical shape and materials bounding the region. The usual assumption for infinitely long uniform waveguides allows us to assume a propagating form for the wave, i.e. stating that every field component has a known dependency on the propagation direction (i.e. z). More specifically, the common approach is to first replace all unknown time-varying unknown fields u(x,y,z,t) (assuming for simplicity to describe the fields in
cartesian Cartesian means of or relating to the French philosopher René Descartes—from his Latinized name ''Cartesius''. It may refer to: Mathematics *Cartesian closed category, a closed category in category theory *Cartesian coordinate system, modern ...
components) with their complex
phasor In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to ...
s representation U(x,y,z), sufficient to fully describe any infinitely long single-tone signal at frequency f, (angular frequency \omega=2\pi f), and rewrite the Helmholtz equation and boundary conditions accordingly. Then, every unknown field is forced to have a form like U(x,y,z)=\hat(x,y)e^, where the \gamma term represents the propagation constant (still unknown) along the direction along which the waveguide extends to infinity. The Helmholtz equation can be rewritten to accommodate such form and the resulting equality needs to be solved for \gamma and \hat(x,y), yielding in the end an eigenvalue equation for \gamma and a corresponding eigenfunction \hat(x,y)_\gammafor each solution of the former. The propagation constant \gamma of the guided wave is complex, in general. For a lossless case, the propagation constant might be found to take on either real or imaginary values, depending on the chosen solution of the eigenvalue equation and on the angular frequency \omega. When \gamma is purely real, the mode is said to be "below cutoff", since the amplitude of the field phasors tends to exponentially decrease with propagation; an imaginary \gamma, instead, represents modes said to be "in propagation" or "above cutoff", as the complex amplitude of the phasors does not change with z.


Impedance matching

In
circuit theory Circuit may refer to: Science and technology Electrical engineering * Electrical circuit, a complete electrical network with a closed-loop giving a return path for current ** Analog circuit, uses continuous signal levels ** Balanced circui ...
, the impedance is a generalization of
electrical resistance The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallels ...
in the case of
alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...
, and is measured in
ohm Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm. Ohm or OHM may also refer to: People * Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm'' * Germán Ohm (born 1936), Mexican boxer * Jörg Ohm (b ...
s (\Omega). A waveguide in circuit theory is described by a
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
having a length and characteristic impedance. In other words, the impedance indicates the ratio of voltage to current of the circuit component (in this case a waveguide) during propagation of the wave. This description of the waveguide was originally intended for alternating current, but is also suitable for electromagnetic and sound waves, once the wave and material properties (such as
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
,
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
,
dielectric constant The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insulat ...
) are properly converted into electrical terms (
current Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stre ...
and impedance for example). Impedance matching is important when components of an electric circuit are connected (waveguide to antenna for example): The impedance ratio determines how much of the wave is transmitted forward and how much is reflected. In connecting a waveguide to an antenna a complete transmission is usually required, so an effort is made to match their impedances. The reflection coefficient can be calculated using: \Gamma=\frac, where \Gamma (Gamma) is the reflection coefficient (0 denotes full transmission, 1 full reflection, and 0.5 is a reflection of half the incoming voltage), Z_1 and Z_2 are the impedance of the first component (from which the wave enters) and the second component, respectively. An impedance mismatch creates a reflected wave, which added to the incoming waves creates a standing wave. An impedance mismatch can be also quantified with the
standing wave ratio In radio engineering and telecommunications, standing wave ratio (SWR) is a measure of impedance matching of loads to the characteristic impedance of a transmission line or waveguide. Impedance mismatches result in standing waves along the transmi ...
(SWR or VSWR for voltage), which is connected to the impedance ratio and reflection coefficient by: \mathrm=\frac=\frac, where \left, V\_ are the minimum and maximum values of the voltage
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
, and the VSWR is the voltage standing wave ratio, which value of 1 denotes full transmission, without reflection and thus no standing wave, while very large values mean high reflection and standing wave pattern.


Electromagnetic waveguides


Radio-frequency waveguides

Waveguides can be constructed to carry waves over a wide portion of the
electromagnetic spectrum The electromagnetic spectrum is the range of frequencies (the spectrum) of electromagnetic radiation and their respective wavelengths and photon energies. The electromagnetic spectrum covers electromagnetic waves with frequencies ranging from ...
, but are especially useful in the
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ran ...
and
optical Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
frequency ranges. Depending on the frequency, they can be constructed from either
conductive In physics and electrical engineering, a conductor is an object or type of material that allows the flow of charge (electric current) in one or more directions. Materials made of metal are common electrical conductors. Electric current is gener ...
or
dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
materials. Waveguides are used for transferring both
power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...
and communication signals.


Optical waveguides

Waveguides used at optical frequencies are typically dielectric waveguides, structures in which a
dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
material with high
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
, and thus high
index of refraction In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
, is surrounded by a material with lower permittivity. The structure guides optical waves by
total internal reflection Total internal reflection (TIR) is the optical phenomenon in which waves arriving at the interface (boundary) from one medium to another (e.g., from water to air) are not refracted into the second ("external") medium, but completely reflected b ...
. An example of an optical waveguide is
optical fiber An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a means to ...
, Other types of optical waveguide are also used, including
photonic-crystal fiber Photonic-crystal fiber (PCF) is a class of optical fiber based on the properties of photonic crystals. It was first explored in 1996 at University of Bath, UK. Because of its ability to confine light in hollow cores or with confinement charact ...
, which guides waves by any of several distinct mechanisms. Guides in the form of a hollow tube with a highly reflective inner surface have also been used as
light pipe Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
s for illumination applications. The inner surfaces may be polished metal, or may be covered with a multilayer film that guides light by
Bragg reflection In physics and chemistry , Bragg's law, Wulff–Bragg's condition or Laue–Bragg interference, a special case of Laue diffraction, gives the angles for coherent scattering of waves from a crystal lattice. It encompasses the superposition of wave ...
(this is a special case of a photonic-crystal fiber). One can also use small
prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...
s around the pipe which reflect light via total internal reflectio

mdash;such confinement is necessarily imperfect, however, since total internal reflection can never truly guide light within a ''lower''-index core (in the prism case, some light leaks out at the prism corners).


Acoustic waveguides

An ''acoustic waveguide'' is a physical structure for guiding sound waves. Sound in an acoustic waveguide behaves like electromagnetic waves on a
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
. Waves on a string, like the ones in a
tin can telephone A tin can phone is a type of acoustic (non-electrical) speech-transmitting device made up of two tin cans, paper cups or similarly shaped items attached to either end of a taut string or wire. It is a particular case of mechanical telephony, wh ...
, are a simple example of an acoustic waveguide. Another example are pressure waves in the pipes of an
organ Organ may refer to: Biology * Organ (biology), a part of an organism Musical instruments * Organ (music), a family of keyboard musical instruments characterized by sustained tone ** Electronic organ, an electronic keyboard instrument ** Hammond ...
. The term ''acoustic waveguide'' is also used to describe elastic waves guided in micro-scale devices, like those employed in piezoelectric delay lines and in
stimulated Brillouin scattering Brillouin scattering (also known as Brillouin light scattering or BLS), named after Léon Brillouin, refers to the interaction of light with the material waves in a medium (e.g. electrostriction and magnetostriction). It is mediated by the refractiv ...
.


Mathematical waveguides

Waveguides are interesting objects of study from a strictly mathematical perspective. A waveguide (or tube) is defined as type of boundary condition on the wave equation such that the wave function must be equal to zero on the boundary and that the allowed region is finite in all dimensions but one (an infinitely long cylinder is an example.) A large number of interesting results can be proven from these general conditions. It turns out that any tube with a bulge (where the width of the tube increases) admits at least one bound state that exist inside the mode gaps. The frequencies of all the bound states can be identified by using a pulse short in time. This can be shown using the variational principles. An interesting result by
Jeffrey Goldstone Jeffrey Goldstone (born 3 September 1933) is a British theoretical physicist and an ''emeritus'' physics faculty member at the MIT Center for Theoretical Physics. He worked at the University of Cambridge until 1977. He is famous for the discove ...
and
Robert Jaffe Robert Loren Jaffe (born May 23, 1946) is an American physicist and the Jane and Otto Morningstar Professor of Physics at the Massachusetts Institute of Technology (MIT). He was formerly director of the MIT Center for Theoretical Physics. Biograp ...
is that any tube of constant width with a twist, admits a bound state.


Sound synthesis

Sound synthesis uses
digital delay line A digital delay line is a discrete element in digital filter theory, which allows a Signalling (telecommunications), signal to be delayed by a number of sample (signal), samples. If the delay is an integer multiple of samples, digital delay lines ar ...
s as computational elements to simulate
wave propagation Wave propagation is any of the ways in which waves travel. Single wave propagation can be calculated by 2nd order wave equation ( standing wavefield) or 1st order one-way wave equation. With respect to the direction of the oscillation relative to ...
in tubes of
wind instruments A wind instrument is a musical instrument that contains some type of resonator (usually a tube) in which a column of air is set into vibration by the player blowing into (or over) a mouthpiece set at or near the end of the resonator. The pitc ...
and the
vibrating strings A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch. If the length or tension of the string is correctly adjusted, the sound produced is a musical tone. Vibrating ...
of
string instruments String instruments, stringed instruments, or chordophones are musical instruments that produce sound from vibrating strings when a performer plays or sounds the strings in some manner. Musicians play some string instruments by plucking the Str ...
.


See also

*
Circular polarization In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to t ...
*
Linear polarization In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation. The term ''linear polarizati ...
*
Orthomode transducer An orthomode transducer (OMT) is a waveguide component that is commonly referred to as a ''polarisation duplexer''. ''Orthomode'' is a contraction of ''orthoganal mode''. Orthomode transducers serve either to combine or to separate two orthog ...
* Polarization *
Earth–ionosphere waveguide The Earth–ionosphere waveguide refers to the phenomenon in which certain radio waves can propagate in the space between the ground and the boundary of the ionosphere. Because the ionosphere contains charged particles, it can behave as a conducto ...


References

* Han, C C; Hwang, Y, "Satellite antennas", in, Lo, Y T; Lee, SW, ''Antenna Handbook: Volume III Applications'', chapter 21, Springer, 1993 . * Levy, R; Cohn, S B
"A History of microwave filter research, design, and development"
''IEEE Transactions: Microwave Theory and Techniques'', pages 1055–1067, volume 32, issue 9, 1984. * Oliner, Arthur A, "The evolution of electromagnetic waveguides: from hollow metallic guides to microwave integrated circuits", chapter 16 in, Sarkar ''et al.'', ''History of Wireless'', Wiley, 2006 .


External links

{{Commons category, Waveguides
Electromagnetic Waves and Antennas: Waveguides
Sophocles J. Orfanidis, Department of Electrical and Computer Engineering, Rutgers University Applied and interdisciplinary physics Electrical components Telecommunications equipment British inventions Electromagnetic radiation