Waveguide

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]

In circuit theory, the impedance is a generalization of electrical resistance 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: ${\displaystyle \Gamma ={\frac {Z_{2}-Z_{1}}{Z_{2}+Z_{1}}}}$, where ${\displaystyle \Gamma }$ (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: ${\displaystyle \mathrm {VSWR} ={\frac {|V|_{\rm {max}}}{|V|_{\rm {min}}}}={\frac {1+|\Gamma |}{1-|\Gamma |}}}$, where ${\displaystyle \left|V\right|_{\rm {min/max}}}$ 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.

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.

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,[13]

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).