Awaveguide is a structure that guides waves by restricting the transmission of energy to one direction. Common types of waveguides includeacoustic waveguides which directsound,optical waveguides which directlight, andradio-frequency waveguides which directelectromagnetic waves other than light likeradio waves.
Without the physical constraint of a waveguide, waves would expand into three-dimensional space and their intensities would decrease according to theinverse square law.
There are different types of waveguides for different types of waves. The original and most common meaning is a hollow conductive metal pipe used to carry high frequencyradio waves, particularlymicrowaves.[1]Dielectric waveguides are used at higher radio frequencies, and transparent dielectric waveguides andoptical fibers serve as waveguides for light. Inacoustics, air ducts andhorns are used as waveguides for sound in musical instruments andloudspeakers, and specially-shaped metal rods conduct ultrasonic waves inultrasonic 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 acutoff wavelength determined by its size and will not conduct waves of greater wavelength; anoptical fiber that guideslight will not transmitmicrowaves which have a much larger wavelength. Some naturally occurring structures can also act as waveguides. TheSOFAR channel layer in the ocean can guide the sound ofwhale song across enormous distances.[2]Any shape of waveguide can support EM waves, however irregular shapes are difficult to analyse. Commonly used waveguides are rectangular or circular in cross-section.
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 acave or medicalstethoscope. 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 ofguided wave testing (GWT), one of the many methods ofnon-destructive evaluation.[3]
Specific examples:
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The first structure for guiding waves was proposed byJ. J. Thomson in 1893, and was first experimentally tested byOliver Lodge in 1894. The first mathematical analysis of electromagnetic waves in a metal cylinder was performed byLord Rayleigh in 1897.[6]: 8 For sound waves, Lord Rayleigh published a full mathematical analysis ofpropagation modes in his seminal work, "The Theory of Sound".[7]Jagadish Chandra Bose researchedmillimeter wavelengths using waveguides, and in 1897 described to the Royal Institution in London his research carried out in Kolkata.[8][9]
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 andDebye.[10] 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 byGeorge C. Southworth atBell Labs andWilmer L. Barrow atMIT. 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 ofwaveguide cutoff frequency already found in Lord Rayleigh's work. Serious theoretical work was taken up byJohn R. Carson andSallie 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.[11]: 544–548
The importance ofradar inWorld War II gave a great impetus to waveguide research, at least on theAllied side. Themagnetron, developed in 1940 byJohn Randall andHarry Boot 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 theRadiation Laboratory (Rad Lab) atMIT but many others took part in the US, and in the UK such as theTelecommunications Research Establishment. The head of the Fundamental Development Group at Rad Lab wasEdward Mills Purcell. His researchers includedJulian Schwinger,Nathan Marcuvitz, Carol Gray Montgomery, andRobert H. Dicke. Much of the Rad Lab work concentrated on findinglumped 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 forwaveguide 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 toSiemens & 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.[12]: 548–554 [13]: 1055, 1057
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 inTEM mode (that is, non-waveguide) such ascoaxial 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 andmicrostrip). 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 aroundKu band upwards.[12]: 556–557 [14]: 21–27, 21–50
A propagation mode in a waveguide is one solution of the wave equations, or, in other words, the form of the wave.[10] Due to the constraints of theboundary 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 thecutoff 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.[15]: 38
Propagation modes are computed by solving theHelmholtz 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 fields (assuming for simplicity to describe the fields incartesian components) with their complexphasors 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 eigenfunctionfor each solution of the former.[16]
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.[17]
Incircuit theory, theimpedance is a generalization ofelectrical resistance in the case ofalternating current, and is measured inohms ().[10] A waveguide in circuit theory is described by atransmission line having a length andcharacteristic impedance.[18]: 2–3, 6–12 [19]: 14 [20] 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 aspressure,density,dielectric constant) are properly converted into electrical terms (current and impedance for example).[21]: 14
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.[20]
Thereflection coefficient can be calculated using:, where (Gamma) 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.[22]
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 thestanding 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 voltageabsolute 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.[20]
Waveguides can be constructed to carry waves over a wide portion of theelectromagnetic spectrum, but are especially useful in themicrowave andoptical frequency ranges. Depending on the frequency, they can be constructed from eitherconductive ordielectric materials. Waveguides are used for transferring bothpower and communication signals.[15]: 1–3 [23]: xiii–xiv
Waveguides used at optical frequencies are typically dielectric waveguides, structures in which adielectric material with highpermittivity, and thus highindex of refraction, is surrounded by a material with lower permittivity. The structure guides optical waves bytotal internal reflection. An example of an optical waveguide isoptical fiber.[24]
Other types of optical waveguide are also used, includingphotonic-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 aslight pipes for illumination applications. The inner surfaces may be polished metal, or may be covered with a multilayer film that guides light byBragg reflection (this is a special case of a photonic-crystal fiber). One can also use smallprisms around the pipe which reflect light via total internal reflection[25] —such confinement is necessarily imperfect, however, since total internal reflection can never truly guide light within alower-index core (in the prism case, some light leaks out at the prism corners).[26]
Anacoustic waveguide is a physical structure for guiding sound waves. Sound in an acoustic waveguide behaves like electromagnetic waves on atransmission line. Waves on a string, like the ones in atin can telephone, are a simple example of an acoustic waveguide. Another example are pressure waves in the pipes of anorgan. The termacoustic waveguide is also used to describe elastic waves guided in micro-scale devices, like those employed in piezoelectric delay lines and instimulated Brillouin scattering.
A waveguide (or tube) impose a 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 but one dimension. An infinitely long cylinder is an example. Mathematically, any tube with a bulge, where the width of the tube increases, admits at least one non-propagating bound state. Using the variational principles,Jeffrey Goldstone andRobert Jaffe have shown a tube of constant width with a twist admits a bound state.[27]
Sound synthesis usesdigital delay lines as computational elements to simulatewave propagation in tubes ofwind instruments and thevibrating strings ofstring instruments.[28]
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