Waveguide

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A section of flexible waveguide with a pressurizable flange WaveguideJ-Band.png
A section of flexible waveguide with a pressurizable flange
Electric field Ex component of the TE31 mode inside an x-band hollow metal waveguide. Waveguide x EM rect TE31.gif
Electric field Ex component of the TE31 mode inside an x-band hollow metal waveguide.

A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave amplitudes decrease according to the inverse square law as they expand into three dimensional space.

Contents

There are different types of waveguides for different types of waves. The original and most common [1] meaning is a hollow conductive metal pipe used to carry high frequency radio waves, particularly microwaves. Dielectric waveguides are used at higher radio frequencies, and transparent dielectric waveguides and optical fibers serve as waveguides for light. In acoustics, air ducts and horns are used as waveguides for sound in musical instruments and loudspeakers, and specially-shaped metal rods conduct ultrasonic waves in ultrasonic machining.

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 determined by its size and will not conduct waves of greater wavelength; an optical fiber that guides light will not transmit microwaves which have a much larger wavelength. Some naturally occurring structures can also act as waveguides. The SOFAR channel layer in the ocean can guide the sound of whale song across enormous distances. [2]

Principle

Example of waveguides and a diplexer in an air traffic control radar Diplexer1.jpg
Example of waveguides and a diplexer in an air traffic control radar

Waves propagate in all directions in open space as spherical waves. The power of the wave falls with the distance R from the source as the square of the distance (inverse square law). A waveguide confines the wave to propagate in one dimension, so that, under ideal conditions, the wave loses no power while propagating. Due to total reflection at the walls, waves are confined to the interior of a waveguide.

Uses

Waveguide supplying power for the Argonne National Laboratory Advanced Photon Source. Waveguide.aps.anl.gov.jpg
Waveguide supplying power for the Argonne National Laboratory Advanced Photon Source.

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 or medical stethoscope. 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 (GWT), one of the many methods of non-destructive evaluation.

Specific examples:

History

The first structure for guiding waves was proposed by J. J. Thomson in 1893, and was first experimentally tested by Oliver Lodge in 1894. The first mathematical analysis of electromagnetic waves in a metal cylinder was performed by Lord Rayleigh in 1897. [4] For sound waves, Lord Rayleigh published a full mathematical analysis of propagation modes in his seminal work, “The Theory of Sound”. [5] Jagadish Chandra Bose researched millimetre wavelengths using waveguides, and in 1897 described to the Royal Institution in London his research carried out in Kolkata. [6]

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 and Debye. [7] 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 at Bell Labs and Wilmer L. Barrow at MIT. 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 already found in Lord Rayleigh's work. Serious theoretical work was taken up by John R. Carson 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. [8]

The importance of radar in World War II gave a great impetus to waveguide research, at least on the Allied side. The magnetron, developed in 1940 by John Randall and Harry Boot at the University of Birmingham in the United Kingdom, provided a good power source and made microwave radars feasible. The most important centre of US research was at the Radiation Laboratory (Rad Lab) at MIT but many others took part in the US, and in the UK such as the Telecommunications Research Establishment. The head of the Fundamental Development Group at Rad Lab was Edward Mills Purcell. His researchers included Julian Schwinger, Nathan Marcuvitz, Carol Gray Montgomery, and Robert H. Dicke. Much of the Rad Lab work concentrated on finding lumped element models of waveguide structures so that components in waveguide could be analysed with standard circuit theory. Hans Bethe 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 for analysis, even though they were recognised as microwave components, their purpose could not be identified.

At that time, microwave techniques were badly neglected in Germany. It was generally believed that it was of no use for electronic warfare, and those who wanted to do research work in this field were not allowed to do so.

H. Mayer, wartime vice-president of Siemens & Halske

German academics were even allowed to continue publicly publishing their research in this field because it was not felt to be important. [9]

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 (that is, non-waveguide) such as coaxial 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 and microstrip). 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 upwards. [10]

Properties

Propagation modes and cutoff frequencies

A propagation mode in a waveguide is one solution of the wave equations, or, in other words, the form of the wave. [7] Due to the constraints of the boundary conditions, 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 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 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. ). More specifically, the common approach is to first replace all unknown time-varying unknown fields (assuming for simplicity to describe the fields in cartesian components) with their complex phasors representation , sufficient to fully describe any infinitely long single-tone signal at frequency , (angular frequency ), and rewrite the Helmholtz equation and boundary conditions accordingly. Then, every unknown field is forced to have a form like , where the 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 and , yielding in the end an eigenvalue equation for and a corresponding eigenfunction for each solution of the former. [11]

The propagation constant 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 . When 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 , instead, represents modes said to be "in propagation" or "above cutoff", as the complex amplitude of the phasors does not change with . [12]

Impedance matching

In circuit theory, the impedance is a generalization of electrical resistance in the case of alternating current, and is measured in ohms (). [7] A waveguide in circuit theory is described by a transmission line 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, density, dielectric constant) are properly converted into electrical terms (current 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: , where is the reflection coefficient (0 denotes full transmission, 1 full reflection, and 0.5 is a reflection of half the incoming voltage), and 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 (SWR or VSWR for voltage), which is connected to the impedance ratio and reflection coefficient by: , where are the minimum and maximum values of the voltage absolute value, 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, but are especially useful in the microwave and optical frequency ranges. Depending on the frequency, they can be constructed from either conductive or dielectric materials. Waveguides are used for transferring both power and communication signals.

In this military radar, microwave radiation is transmitted between the source and the reflector by a waveguide. The figure suggests that microwaves leave the box in a circularly symmetric mode (allowing the antenna to rotate), then they are converted to a linear mode, and pass through a flexible stage. Their polarisation is then rotated in a twisted stage and finally they irradiate the parabolic antenna. Radar waveguide.JPG
In this military radar, microwave radiation is transmitted between the source and the reflector by a waveguide. The figure suggests that microwaves leave the box in a circularly symmetric mode (allowing the antenna to rotate), then they are converted to a linear mode, and pass through a flexible stage. Their polarisation is then rotated in a twisted stage and finally they irradiate the parabolic antenna.

Optical waveguides

Waveguides used at optical frequencies are typically dielectric waveguides, structures in which a dielectric material with high permittivity, and thus high index of refraction, is surrounded by a material with lower permittivity. The structure guides optical waves by total internal reflection. An example of an optical waveguide is optical fiber .

Other types of optical waveguide are also used, including photonic-crystal fiber, 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 pipes for illumination applications. The inner surfaces may be polished metal, or may be covered with a multilayer film that guides light by Bragg reflection (this is a special case of a photonic-crystal fiber). One can also use small prisms around the pipe which reflect light via total internal reflection 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. A duct for sound propagation also behaves like a transmission line. The duct contains some medium, such as air, that supports sound propagation.

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. This can be shown using the variational principles. An interesting result by Jeffrey Goldstone and Robert Jaffe [13] is that any tube of constant width with a twist, admits a bound state.

Sound synthesis

Sound synthesis uses digital delay lines as computational elements to simulate wave propagation in tubes of wind instruments and the vibrating strings of string instruments.

See also

Related Research Articles

Cutoff frequency frequency response boundary

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 rather than passing through.

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

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 transmission line, and SWR is defined as the ratio of the partial standing wave's amplitude at an antinode (maximum) to the amplitude at a node (minimum) along the line.

Transmission line specialized cable or other structure designed to carry alternating current of radio frequency

In radio-frequency engineering, a transmission line is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

Transmission medium material substance that can propagate energy waves

A transmission medium is something that can mediate the propagation of signals for the purposes of telecommunication.

The wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields. For a transverse-electric-magnetic (TEM) plane wave traveling through a homogeneous medium, the wave impedance is everywhere equal to the intrinsic impedance of the medium. In particular, for a plane wave travelling through empty space, the wave impedance is equal to the impedance of free space. The symbol Z is used to represent it and it is expressed in units of ohms. The symbol η (eta) may be used instead of Z for wave impedance to avoid confusion with electrical impedance.

In electromagnetics, an evanescent field, or evanescent wave, is an oscillating electric and/or magnetic field that does not propagate as an electromagnetic wave but whose energy is spatially concentrated in the vicinity of the source. Even when there is a propagating electromagnetic wave produced, one can still identify as an evanescent field the component of the electric or magnetic field that cannot be attributed to the propagating wave observed at a distance of many wavelengths.

Impedance matching practice in electronics

In electronics, impedance matching is the practice of designing the input impedance of an electrical load or the output impedance of its corresponding signal source to maximize the power transfer or minimize signal reflection from the load. A source of electric power such as a generator, amplifier or radio transmitter has a source impedance which is equivalent to an electrical resistance in series with a reactance. An electrical load, such as a light bulb, transmission line or antenna similarly has an impedance which is equivalent to a resistance in series with a reactance. The maximum power theorem says that maximum power is transferred from source to load when the load resistance equals the source resistance and the load reactance equals the negative of the source reactance. Another way of saying this is that the load impedance must equal the complex conjugate of the source impedance. If this condition is met the two parts of the circuit are said to be impedance matched.

A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern of the radiation in the plane perpendicular to the radiation's propagation direction. Transverse modes occur in radio waves and microwaves confined to a waveguide, and also in light waves in an optical fiber and in a laser's optical resonator.

Microstrip electrical transmission line for microwave-frequency signals on printed circuit board

Microstrip is a type of electrical transmission line which can be fabricated using printed circuit board technology, and is used to convey microwave-frequency signals. It consists of a conducting strip separated from a ground plane by a dielectric layer known as the substrate. Microwave components such as antennas, couplers, filters, power dividers etc. can be formed from microstrip, with the entire device existing as the pattern of metallization on the substrate. Microstrip is thus much less expensive than traditional waveguide technology, as well as being far lighter and more compact. Microstrip was developed by ITT laboratories as a competitor to stripline.

Waveguide (radio frequency) waveguide for the transmission of electromagnetic waves; linear structure that conveys electromagnetic waves between its endpoints

In radio-frequency engineering and communications engineering, waveguide is a hollow metal pipe used to carry radio waves. This type of waveguide is used as a transmission line mostly at microwave frequencies, for such purposes as connecting microwave transmitters and receivers to their antennas, in equipment such as microwave ovens, radar sets, satellite communications, and microwave radio links.

An optical waveguide is a physical structure that guides electromagnetic waves in the optical spectrum. Common types of optical waveguides include optical fiber and transparent dielectric waveguides made of plastic and glass.

Stripline transverse electromagnetic (TEM) transmission line

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.

Eigenmode expansion (EME) is a computational electrodynamics modelling technique. It is also referred to as the mode matching technique or the bidirectional eigenmode propagation method. Eigenmode expansion is a linear frequency-domain method.

Reflections of signals on conducting lines

A signal travelling along an electrical transmission line will be partly, or wholly, reflected back in the opposite direction when the travelling signal encounters a discontinuity in the characteristic impedance of the line, or if the far end of the line is not terminated in its characteristic impedance. This can happen, for instance, if two lengths of dissimilar transmission lines are joined together.

Metal-mesh optical filters are optical filters made from stacks of metal meshes and dielectric. They are used as part of an optical path to filter the incoming light to allow frequencies of interest to pass while reflecting other frequencies of light.

Waveguide filter electronic filter that is constructed with waveguide technology

A waveguide filter is an electronic filter that is constructed with waveguide technology. Waveguides are hollow metal tubes inside which an electromagnetic wave may be transmitted. Filters are devices used to allow signals at some frequencies to pass, while others are rejected. Filters are a basic component of electronic engineering designs and have numerous applications. These include selection of signals and limitation of noise. Waveguide filters are most useful in the microwave band of frequencies, where they are a convenient size and have low loss. Examples of microwave filter use are found in satellite communications, telephone networks, and television broadcasting.

Substrate integrated waveguide

A Substrate integrated waveguide (SIW) is a synthetic rectangular electromagnetic waveguide formed in a dielectric substrate by densely arraying metallized posts or via-holes which connect the upper and lower metal plates of the substrate. The waveguide can be easily fabricated with low-cost mass-production using through-hole techniques where the post walls consists of via fences. SIW is known to have similar guided wave and mode characteristics to conventional rectangular waveguide with equivalent guide wavelength.

Planar transmission line Transmission lines with flat ribbon-like conducting or dielectric lines

Planar transmission lines are transmission lines with conductors, or in some cases dielectric (insulating) strips, that are flat, ribbon-shaped lines. They are used to interconnect components on printed circuits and integrated circuits working at microwave frequencies because the planar type fits in well with the manufacturing methods for these components. Transmission lines are more than simply interconnections. With simple interconnections, the propagation of the electromagnetic wave along the wire is fast enough to be considered instantaneous, and the voltages at each end of the wire can be considered identical. If the wire is longer than a large fraction of a wavelength, these assumptions are no longer true and transmission line theory must be used instead. With transmission lines, the geometry of the line is precisely controlled so that its electrical behaviour is highly predictable. At lower frequencies, these considerations are only necessary for the cables connecting different pieces of equipment, but at microwave frequencies the distance at which transmission line theory becomes necessary is measured in millimetres. Hence, transmission lines are needed within circuits.

Frequency selective surface any thin, repetitive surface designed to reflect, transmit or absorb electromagnetic fields based on the frequency of the field

A frequency-selective surface (FSS) is any thin, repetitive surface designed to reflect, transmit or absorb electromagnetic fields based on the frequency of the field. In this sense, an FSS is a type of optical filter or metal-mesh optical filters in which the filtering is accomplished by virtue of the regular, periodic pattern on the surface of the FSS. Though not explicitly mentioned in the name, FSS's also have properties which vary with incidence angle and polarization as well - these are unavoidable consequences of the way in which FSS's are constructed. Frequency-selective surfaces have been most commonly used in the radio frequency region of the electromagnetic spectrum and find use in applications as diverse as the aforementioned microwave oven, antenna radomes and modern metamaterials. Sometimes frequency selective surfaces are referred to simply as periodic surfaces and are a 2-dimensional analog of the new periodic volumes known as photonic crystals.

References

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