# Waveguide: Wikis

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# Encyclopedia

(animation) Electric field inside an x-band hollow metal waveguide. The waveguide is cut along its side to allow a view of the field inside.

A waveguide is a structure which guides waves, such as electromagnetic waves or sound waves. There are different types of waveguide for each type of wave. The original and most common[1] meaning is a hollow metal pipe used for this purpose.

Waveguides differ in their geometry which can confine energy in one dimension such as in slab waveguides or two dimensions as in fiber or channel waveguides. In addition, different waveguides are needed to guide different frequencies: an optical fiber guiding laser (high frequency) will not guide microwaves (which have a much lower frequency). As a rule of thumb, the width of a waveguide needs to be of the same order of magnitude as the wavelength of the guided wave.

There are structures in nature which act as waveguides: for example, a layer in the ocean can guide whale song to enormous distances.[2].

## Principle of operation

Waves in open space propagate in all directions, as spherical waves. In this way they lose their power proportionally to the square of the distance: i.e., at a distance R from the source the power is the source power divided by R2. The waveguide confines the wave to propagation in one dimension, so that it doesn't lose (in ideal conditions) power while propagating.

Waves are confined inside the waveguide due to complete reflection from the waveguide wall, so that the propagation inside the waveguide can be described approximately as a "zigzag" between the walls. This description is exact for electromagnetic waves in a rectangular or circular hollow metal tube.

## History

A 19th century drawing of women speaking over a Tin can telephone, using the tight cord as a sound waveguide.

The first structure for guideing waves was proposed by J. J. Thomson in 1893, and was first experimentally tested by O. J. Lodge in 1894. The first mathematical analysis of electromagnetic waves in a metal cylinder was performed by Lord Rayleigh in 1897[3]. For sound waves, lord Rayleigh published a full mathematical analysis of propagation modes in his seminal work, “The Theory of Sound”[4].

The study of dielectric waveguides (such as optical fibers, see below) began as early as the 1920's, by several people, most famous of which are Rayleigh, Sommerfeld and Debye[5]. The optical fiber started receiving special attention since the 1960's due to its importance in the communications area.

## 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 or medical stethoscope. Other uses of waveguides are in transmitting power between the components of a system such as radio, radar or optical devices.

• Optical fibers transmit light and signals for long distances and with a high signal rate.
• In a microwave oven a waveguide leads power from the magnetron where waves are formed to the cooking chamber.
• In a radar, a waveguide leads waves to the antenna, where their impedance needs to be matched for efficient power transmission (see below).
• A waveguide called stripline can be created on a printed circuit board, and is used to transmit microwave signals on the board. This type of waveguide is very cheap to manufacture and has small dimensions which fit inside printed circuit boards.
• 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), 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[6]), so that smaller objects can be tested and the accuracy is better.

## Theoretical analysis

The wave propagation along the waveguide axis is described by the wave equation, where the wavelength depends on the waveguide structure as well as on the frequency. Along the width of the waveguide the wave is confined in a standing wave pattern. The equation that describes the transverse wave form is more complicated, and is derived in the case of electromagnetic waves from maxwell's equations and in the case of sound waves from the equations of linear elasticity, along with boundary conditions that depend on the shape of the waveguide and the materials from which it is made. These equations have multiple solutions, called propagation modes (see below). Each mode propagates along the waveguide in a different form and velocity.

The band of frequencies that a waveguide can guide depends on its width. As a rule of thumb, the longer the wavelength, the wider the waveguide has to be. Since the wavelength is inversely proportional to the frequency, at higher frequencies the waveguide is narrower, and vice versa. A notable exception to this rule is a plane wave mode in certain waveguides (such as a coaxial wire for electromagnetic waves or a hollow tube for sound waves). A plane wave mode has a very large bandwidth and can have a wavelength many orders of magnitude larger than the width of the waveguide.

Closing the two ends of the waveguide produces a resonator, in which case only certain frequencies, called the normal modes of the resonator, can exist for long periods.

### Propagation modes and cutoff frequencies[5]

A Propagation mode 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, 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 basic mode of the waveguide, and its cutoff frequency is the waveguide cutoff frequency.

A particular waveguide mode is a plane wave mode. A plane wave exists in open space and its wave front is plane. A plane mode can propagate in a large band of frequencies, and in an ideal waveguide (where the walls are completely reflecting) its cutoff frequency approaches 0. A plane wave cannot propagate in all types of waveguides: For example, a coaxial cable supports a plane electromagnetic wave, while a hollow tube does not.[5]

The speed of the wave along the waveguide (z axis) is: $\ v_z=\frac{2\pi f}{k_z}$, where $\ f$ is the frequency, $\ \vec v$ is the wave speed in open space, and $\ \vec k$ is the wave number, which is a vector whose magnitude is $\ k=\frac{2\pi f}{v}$.

The connection between the wave number magnitude and its components is: $\ k^2=k_x^2+k_y^2+k_z^2$, where kx and ky are the transverse wave numbers, and depend on the structure of the waveguide and on the mode, but not on the wave frequency. The cutoff of the wave means that the wave does not propagate and therefore the longitudinal wave number kz equals zero. From this we get the cutoff wavenumber: $k_c=k=\sqrt{k_x^2+k_y^2}$, and therefore the cutoff frequency: $f_c=\frac{k_c v}{2\pi}=\frac{v}{2\pi}\sqrt{k_x^2+k_y^2}$.

### Impedance matching[5]

In circuit theory, the Impedance is a generalization of electrical resistivity in the case of alternate current, and is measured in Ohms (Ω). A waveguide in circuit theory is described by a transmission line having a length and self impedance. In other words the impedance is the resistance of the circuit component (in this case a waveguide) to the propagation of the wave. This description of the waveguide was originally intended for alternate currents, 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 that their impedances are matched.

The reflection coefficient can be calculated using: $\Gamma=\frac{Z_2/Z_1-1}{Z_2/Z_1+1}$, where Γ is the reflection coefficient (0 denotes full transmission, 1 full reflection, and 0.5 is a reflection of half the incoming voltage), Z1 and Z2 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: $VSWR=\frac{|V|_{max}}{|V|_{min}}=\frac{1+|\Gamma|}{1-|\Gamma|}$, where $\left|V\right|_{min, max}$ are the minimum and maximum values of the voltage absolute value, and the VSWR is the standing wave ratio, which value of 1 denotes full transmission and no standing wave, and very large values mean high reflection and standing wave pattern.

## Electromagnetic 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.

## 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. The most common 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 [1]—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.

## 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.

## References

1. ^ Institute 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. ISBN 1-55937-833-6 [ed. Standards Coordinating Committee 10, Terms and Definitions; Jane Radatz, (chair)]
2. ^ ORIENTATION BY MEANS OF LONG RANGE ACOUSTIC SIGNALING IN BALEEN WHALES, R. Payne, D. Webb, in Annals NY Acad. Sci., 188:110-41 (1971)
3. ^ N. W. McLachlan, Theory and Applications of Mathieu Functions, p. 8 (1947) (reprinted by Dover: New York, 1964).
4. ^ The Theory of Sound, by J. W. S. Rayleigh, (1894)
5. ^ a b c d Advanced Engineering Electromagnetics, by C. A. Balanis, John Wiley & Sons (1989).
6. ^ J. R. Baker-Jarvis, "Transmission / reflection and short-circuit line permittivity measurements", NIST tech. note 1341, July 1990