# Waveguide

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A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting expansion to one dimension or two. There is a similar effect in water waves constrained within a canal, or guns that have barrels which restrict hot gas expansion to maximize energy transfer to their bullets. Without the physical constraint of a waveguide, wave amplitudes decrease according to the inverse square law as they expand into three dimensional space.

In physics, sound is a vibration that typically propagates as an audible wave of pressure, through a transmission medium such as a gas, liquid or solid.

## Contents

There are different types of waveguides for each type of wave. The original and most common [1] meaning is a hollow conductive metal pipe used to carry high frequency radio waves, particularly microwaves.

Radio waves are a type of electromagnetic radiation with wavelengths in the electromagnetic spectrum longer than infrared light. Radio waves have frequencies as high as 300 gigahertz (GHz) to as low as 30 hertz (Hz). At 300 GHz, the corresponding wavelength is 1 mm, and at 30 Hz is 10,000 km. Like all other electromagnetic waves, radio waves travel at the speed of light in vacuum. They are generated by electric charges undergoing acceleration, such as time varying electric currents. Naturally occurring radio waves are emitted by lightning and astronomical objects.

Microwaves are a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter; with frequencies between 300 MHz (1 m) and 300 GHz (1 mm). Different sources define different frequency ranges as microwaves; the above broad definition includes both UHF and EHF bands. A more common definition in radio engineering is the range between 1 and 100 GHz. In all cases, microwaves include the entire SHF band at minimum. Frequencies in the microwave range are often referred to by their IEEE radar band designations: S, C, X, Ku, K, or Ka band, or by similar NATO or EU designations.

The geometry of a waveguide reflects its function. Slab waveguides confine energy in one dimension, fiber or channel waveguides in two dimensions. The frequency of the transmitted wave also dictates the shape of a waveguide: an optical fiber guiding high-frequency light will not guide microwaves of a much lower frequency.

An optical fiber 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 transmit light between the two ends of the fiber and find wide usage in fiber-optic communications, where they permit transmission over longer distances and at higher bandwidths than electrical cables. Fibers are used instead of metal wires because signals travel along them with less loss; in addition, fibers are immune to electromagnetic interference, a problem from which metal wires suffer excessively. Fibers are also used for illumination and imaging, and are often wrapped in bundles so they may be used to carry light into, or images out of confined spaces, as in the case of a fiberscope. Specially designed fibers are also used for a variety of other applications, some of them being fiber optic sensors and fiber lasers.

Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.

Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum. The word usually refers to visible light, which is the visible spectrum that is visible to the human eye and is responsible for the sense of sight. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), or 4.00 × 10−7 to 7.00 × 10−7 m, between the infrared and the ultraviolet. This wavelength means a frequency range of roughly 430–750 terahertz (THz).

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]

The SOFAR 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 frequency sound waves within the channel may travel thousands of miles before dissipating. This phenomenon is an important factor in submarine warfare. The deep sound channel was discovered and described independently by Maurice Ewing, Stanley Wong and Leonid Brekhovskikh in the 1940s.

## Principle

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.

Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. The law of reflection says that for specular reflection the angle at which the wave is incident on the surface equals the angle at which it is reflected. Mirrors exhibit specular reflection.

## 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. Waveguides are the fundamental principle of guided wave testing (GWT), one of the many methods of non-destructive evaluation.

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 also refer to much smaller openings such as sea caves, rock shelters, and grottos, though strictly speaking a cave is exogene, meaning it is deeper than its opening is wide, and a rock shelter is endogene.

The stethoscope is an acoustic medical device for auscultation, or listening to the 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. A stethoscope can be used to listen to the sounds made by the heart, lungs or intestines, as well as blood flow in arteries and veins. In combination with a manual sphygmomanometer, it is commonly used when measuring blood pressure.

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 energy. Nowadays, GWT is widely used to inspect and screen many engineering structures, particularly for the inspection of metallic pipelines around the world. In some cases, hundreds of meters can be inspected from a single location. There are also some applications for inspecting rail tracks, rods and metal plate structures.

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 waveguide transfers power from the magnetron, 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), 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 [3] ), so that smaller objects can be tested and the accuracy is better.
• Transmission line s are a specific type of waveguide, very commonly used.

## 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]

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. ${\displaystyle z}$). More specifically, the common approach is to first replace all unknown time-varying unknown fields ${\displaystyle u(x,y,z,t)}$ (assuming for simplicity to describe the fields in cartesian components) with their complex phasors representation ${\displaystyle U(x,y,z)}$, sufficient to fully describe any infinitely long single-tone signal at frequency ${\displaystyle f}$, (angular frequency ${\displaystyle \omega =2\pi f}$), and rewrite the Helmholtz equation and boundary conditions accordingly. Then, every unknown field is forced to have a form like ${\displaystyle U(x,y,z)={\hat {U}}(x,y)e^{-\gamma z}}$, where the ${\displaystyle \gamma }$ 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 ${\displaystyle \gamma }$ and ${\displaystyle {\hat {U}}(x,y)}$, yielding in the end an eigenvalue equation for ${\displaystyle \gamma }$ and a corresponding eigenfunction ${\displaystyle {\hat {U}}(x,y)_{\gamma }}$for each solution of the former. [11]

The propagation constant ${\displaystyle \gamma }$ 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 ${\displaystyle \omega }$. When ${\displaystyle \gamma }$ 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 ${\displaystyle \gamma }$, instead, represents modes said to be "in propagation" or "above cutoff", as the complex amplitude of the phasors does not change with ${\displaystyle z}$. [12]

### Impedance matching

In circuit theory, the impedance is a generalization of electrical resistivity in the case of alternating current, and is measured in ohms (${\displaystyle \Omega }$). [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: ${\displaystyle \Gamma ={\frac {Z_{2}/Z_{1}-1}{Z_{2}/Z_{1}+1}}}$, where ${\displaystyle \Gamma }$ is the reflection coefficient (0 denotes full transmission, 1 full reflection, and 0.5 is a reflection of half the incoming voltage), ${\displaystyle Z_{1}}$ and ${\displaystyle Z_{2}}$ 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 (SWR or VSWR for voltage), which is connected to the impedance ratio and reflection coefficient by: ${\displaystyle VSWR={\frac {|V|_{max}}{|V|_{min}}}={\frac {1+|\Gamma |}{1-|\Gamma |}}}$, where ${\displaystyle \left|V\right|_{min,max}}$ 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

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. 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.

## Related Research Articles

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.

Polarization is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave. A simple example of a polarized transverse wave is vibrations traveling along a taut string (see image); for example, in a musical instrument like a guitar string. Depending on how the string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string. In contrast, in longitudinal waves, such as sound waves in a liquid or gas, the displacement of the particles in the oscillation is always in the direction of propagation, so these waves do not exhibit polarization. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves, gravitational waves, and transverse sound waves in solids.

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 physics, a surface wave is a 90 degree wave that propagates along the interface between differing media. A common example is gravity waves along the surface of liquids, such as ocean waves. Gravity waves can also occur within liquids, at the interface between two fluids with different densities. Elastic surface waves can travel along the surface of solids, such as Rayleigh or Love waves. Electromagnetic waves can also propagate as "surface waves" in that they can be guided along a refractive index gradient or along an interface between two media having different dielectric constants. In radio transmission, a ground wave is a guided wave that propagates close to the surface of the Earth.

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.

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.

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 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.

In optics, an ultrashort pulse of light is an electromagnetic pulse whose time duration is of the order of a picosecond or less. Such pulses have a broadband optical spectrum, and can be created by mode-locked oscillators. They are commonly referred to as ultrafast events. Amplification of ultrashort pulses almost always requires the technique of chirped pulse amplification, in order to avoid damage to the gain medium of the amplifier.

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.

In electromagnetics and communications engineering, the term waveguide may refer to any linear structure that conveys electromagnetic waves between its endpoints. However, the original and most common meaning 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 rectangular waveguides.

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.

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.

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.

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.

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.

Marcatili’s method is an approximate analytical method that describes how light propagates through rectangular dielectric optical waveguides. It was published by Enrique Marcatili in 1969.

## 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. J. R. Baker-Jarvis, "Transmission / reflection and short-circuit line permittivity measurements", NIST tech. note 1341, July 1990
4. N. W. McLachlan, Theory and Applications of Mathieu Functions, p. 8 (1947) (reprinted by Dover: New York, 1964).
5. The Theory of Sound, by J. W. S. Rayleigh, (1894)
6. Emerson, D. T. (1997). The work of Jagadis Chandra Bose: 100 years of MM-wave research. IEEE Transactions on Microwave Theory and Research. 45. pp. 2267–2273. Bibcode:1997imsd.conf..553E. CiteSeerX  . doi:10.1109/MWSYM.1997.602853. ISBN   9780986488511. reprinted in Igor Grigorov, Ed., Antentop , Vol. 2, No.3, pp. 87–96.
7. Advanced Engineering Electromagnetics Archived 2009-05-14 at the Wayback Machine , by C. A. Balanis, John Wiley & Sons (1989).
8. Oliner, pp. 544-548
9. Oliner, pp. 548-554
• Levy & Cohn, pp. 1055, 1057
10. Oliner, pp. 556-557
• Han & Hwang, pp. 21-7, 21-50
11. D. Pozar, "Microwave Engineering", Third Edition, John Wiley and Sons, 2005, Chapter 3.
12. Ramo, Simon; Whinnery, John R.; Van Duzer, Theodore (1994). Fields and Waves in Communication Electronics. New York: Joh Wiley and Sons. pp. 321–324. ISBN   978-0-471-58551-0.