# Waveguide (electromagnetism)

In electromagnetics and communications engineering, the term waveguide may refer to any linear structure that conveys electromagnetic waves between its endpoints. However, the original[1] and most common[1] 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.

A dielectric waveguide employs a solid dielectric rod rather than a hollow pipe. An optical fibre is a dielectric guide designed to work at optical frequencies. Transmission lines such as microstrip, coplanar waveguide, stripline or coaxial cable may also be considered to be waveguides.

The electromagnetic waves in a (metal-pipe) waveguide may be imagined as travelling down the guide in a zig-zag path, being repeatedly reflected between opposite walls of the guide. For the particular case of rectangular waveguide, it is possible to base an exact analysis on this view. Propagation in a dielectric waveguide may be viewed in the same way, with the waves confined to the dielectric by total internal reflection at its surface. Some structures, such as non-radiative dielectric waveguides and the Goubau line, use both metal walls and dielectric surfaces to confine the wave.

 Short length of rectangular waveguide (WG17 with UBR120 connection-flanges) Section of flexible waveguide

## History

The first waveguide was proposed by J. J. Thomson in 1893 and experimentally verified by Oliver Lodge in 1894; the mathematical analysis of the propagating modes within a hollow metal cylinder was first performed by Lord Rayleigh in 1897. (McLachan, 1947)

## Principle of operation

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

Depending on the frequency, waveguides can be constructed from either conductive or dielectric materials. Generally, the lower the frequency to be passed the larger the waveguide is. For example the natural waveguide the earth forms given by the dimensions between the conductive ionosphere and the ground as well as the circumference at the median altitude of the Earth is resonant at 7.83 Hz. This is known as Schumann resonance. On the other hand, waveguides used in extremely high frequency (EHF) communications can be less than a millimeter in width.

## Analysis

Electromagnetic waveguides are analyzed by solving Maxwell's equations, or their reduced form, the electromagnetic wave equation, with boundary conditions determined by the properties of the materials and their interfaces. These equations have multiple solutions, or modes, which are eigenfunctions of the equation system. Each mode is characterized by a cutoff frequency below which the mode cannot exist in the guide.

Waveguide propagation modes depend on the operating wavelength and polarization and the shape and size of the guide. The longitudinal mode of a waveguide is a particular standing wave pattern formed by waves confined in the cavity. The transverse modes are classified into different types:

• TE modes (transverse electric) have no electric field in the direction of propagation.
• TM modes (transverse magnetic) have no magnetic field in the direction of propagation.
• TEM modes (transverse electromagnetic) have no electric nor magnetic field in the direction of propagation.
• Hybrid modes have both electric and magnetic field components in the direction of propagation.

In hollow waveguides (single conductor), TEM waves are not possible, since Maxwell's Equations will give that the electric field must then have zero divergence and zero curl and be equal to zero at boundaries, resulting in a zero field (or, equivalently, $\nabla ^2 \Phi=0$ with boundary conditions guaranteeing only the trivial solution). However, TEM waves can propagate in coaxial cables because there are two conductors.

The mode with the lowest cutoff frequency is termed the dominant mode of the guide. It is usual to choose the size of the guide such that only this one mode can exist in the frequency band of operation. In rectangular and circular (hollow pipe) waveguides, the dominant modes are designated the TE1,0 mode and TE1,1 modes respectively.

## Hollow metallic waveguides

Rectangular hollow Waveguide
Flexible waveguide from a J-Band radar

In the microwave region of the electromagnetic spectrum, a waveguide normally consists of a hollow metallic conductor. These waveguides can take the form of single conductors with or without a dielectric coating, e.g. the Goubau line and helical waveguides. Hollow waveguides must be one-half wavelength or more in diameter in order to support one or more transverse wave modes.

Waveguides may be filled with pressurized gas to inhibit arcing and prevent multipaction, allowing higher power transmission. Conversely, waveguides may be required to be evacuated as part of evacuated systems (e.g. electron beam systems).

A slotted waveguide is generally used for radar and other similar applications. The waveguide structure has the capability of confining and supporting the energy of an electromagnetic wave to a specific relatively narrow and controllable path.

A closed waveguide is an electromagnetic waveguide (a) that is tubular, usually with a circular or rectangular cross section, (b) that has electrically conducting walls, (c) that may be hollow or filled with a dielectric material, (d) that can support a large number of discrete propagating modes, though only a few may be practical, (e) in which each discrete mode defines the propagation constant for that mode, (f) in which the field at any point is describable in terms of the supported modes, (g) in which there is no radiation field, and (h) in which discontinuities and bends cause mode conversion but not radiation.

The dimensions of a hollow metallic waveguide determine which wavelengths it can support, and in which modes. Typically the waveguide is operated so that only a single mode is present. The lowest order mode possible is generally selected. Frequencies below the guide's cutoff frequency will not propagate. It is possible to operate waveguides at higher order modes, or with multiple modes present, but this is usually impractical.

Waveguides are almost exclusively made of metal and mostly rigid structures. There are certain types of "corrugated" waveguides that have the ability to flex and bend but only used where essential since they degrade propagation properties. Due to propagation of energy in mostly air or space within the waveguide, it is one of the lowest loss transmission line types and highly preferred for high frequency applications where most other types of transmission structures introduce large losses. Due to the skin effect at high frequencies, electric current along the walls penetrates typically only a few micrometers into the metal of the inner surface. Since this is where most of the resistive loss occurs, it is important that the conductivity of interior surface be kept as high as possible. For this reason, most waveguide interior surfaces are plated with copper, silver, or gold.

Voltage standing wave ratio (VSWR) measurements may be taken to ensure that a waveguide is contiguous and has no leaks or sharp bends. If such bends or holes in the waveguide surface are present, this may diminish the performance of both transmitter and receiver equipment connected at either end. Poor transmission through the waveguide may also occur as a result of moisture build up which corrodes and degrades conductivity of the inner surfaces, which is crucial for low loss propagation. For this reason, waveguides are nominally fitted with microwave windows at the outer end that will not interfere with propagation but keep the elements out. Moisture can also cause fungus build up or arcing in high power systems such as radio or radar transmitters. Moisture in waveguides can typically be prevented with silica gel, a desiccant, or slight pressurization of the waveguide cavities with dry nitrogen or argon. Desiccant silica gel canisters may be attached with screw-on nibs and higher power systems will have pressurized tanks for maintaining pressure including leakage monitors. Arcing may also occur if there is a hole, tear or bump in the conducting walls, if transmitting at high power (usually 200 watts or more). Waveguide plumbing [2] is crucial for proper waveguide performance. Voltage standing waves occur when impedance mismatches in the waveguide cause energy to reflect back in the opposite direction of propagation. In addition to limiting the effective transfer of energy, these reflections can cause higher voltages in the waveguide and damage equipment.

### Waveguide in practice

In practice, waveguides act as the equivalent of cables for super high frequency (SHF) systems. For such applications, it is desired to operate waveguides with only one mode propagating through the waveguide. With rectangular waveguides, it is possible to design the waveguide such that the frequency band over which only one mode propagates is as high as 2:1 (i.e. the ratio of the upper band edge to lower band edge is two). The relationship between the longest wavelength that will propagate through a rectangular waveguide is a simple one. Given that W is the greater of its two dimensions, and lambda is the wavelength, then lambda = 2W.

With circular waveguides, the highest possible bandwidth allowing only a single mode to propagate is only 1.3601:1.[3]

Because rectangular waveguide have a much larger bandwidth over which only a single mode can propagate, standards exist for rectangular waveguides, but not for circular waveguides. In general (but not always), standard waveguides are designed such that

• one band starts where another band ends, with another band that overlaps the two bands[4]
• the lower edge of the band is approximately 30% higher than the waveguide's cutoff frequency
• the upper edge of the band is approximately 5% lower than the cutoff frequency of the next higher order mode
• the waveguide height is half the waveguide width

The first condition is to allow for applications near band edges. The second condition limits dispersion, a phenomenon in which the velocity of propagation is a function of frequency. It also limits the loss per unit length. The third condition is to avoid evanescent-wave coupling via higher order modes. The fourth condition is that which allows a 2:1 operation bandwidth. Although it is possible to have a 2:1 operating bandwidth when the height is less than half the width, having the height exactly half the width maximizes the power that can propagate inside the waveguide before dielectric breakdown occurs.

Below is a table of waveguide standards. The waveguide name WR stands for waveguide rectangular, and the number is the inner dimension width of the waveguide in hundredths of an inch (0.01 inch = 0.254 mm) rounded to the nearest hundredth of an inch.

Standard sizes of rectangular waveguide
Waveguide name Frequency band name Recommended frequency band of operation (GHz) Cutoff frequency of lowest order mode (GHz) Cutoff frequency of next mode (GHz) Inner dimensions of waveguide opening (inch)
EIA RCSC * IEC
WR2300 WG0.0 R3 0.32 — 0.45 0.257 0.513 23.000 × 11.500
WR2100 WG0 R4 0.35 — 0.50 0.281 0.562 21.000 × 10.500
WR1800 WG1 R5 0.45 — 0.63 0.328 0.656 18.000 × 9.000
WR1500 WG2 R6 0.50 — 0.75 0.393 0.787 15.000 × 7.500
WR1150 WG3 R8 0.63 — 0.97 0.513 1.026 11.500 × 5.750
WR975 WG4 R9 0.75 — 1.15 0.605 1.211 9.750 × 4.875
WR770 WG5 R12 0.97 — 1.45 0.766 1.533 7.700 × 3.850
WR650 WG6 R14 L band (part) 1.15 — 1.72 0.908 1.816 6.500 × 3.250
WR510 WG7 R18 1.45 — 2.20 1.157 2.314 5.100 × 2.550
WR430 WG8 R22 1.72 — 2.60 1.372 2.745 4.300 × 2.150
WG9 ‡§ S band (part) 2.20 — 3.30 1.686 3.372 3.500 × 1.750
WR340 WG9A R26 S band (part) 2.20 — 3.30 1.736 3.471 3.400 × 1.700
WR284 WG10 R32 S band (part) 2.60 — 3.95 2.078 4.156 2.840 × 1.340
WG11 ‡§ C band (part) 3.30 — 4.90 2.488 4.976 2.372 × 1.122
WR229 WG11A R40 C band (part) 3.30 — 4.90 2.577 5.154 2.290 × 1.145
WR187 WG12 R48 C band (part) 3.95 — 5.85 3.153 6.305 1.872 × 0.872
WR159 WG13 R58 C band (part) 4.90 — 7.05 3.712 7.423 1.590 × 0.795
WR137 WG14 R70 C band (part) 5.85 — 8.20 4.301 8.603 1.372 × 0.622
WR112 WG15 R84 7.05 — 10.00 5.260 10.520 1.122 × 0.497
WR102  7.00 — 11.00 5.786 11.571 1.020 × 0.510
WR90 WG16 R100 X band 8.20 — 12.40 6.557 13.114 0.900 × 0.400
WR75 WG17 R120 10.00 — 15.00 7.869 15.737 0.750 × 0.375
WR62 WG18 R140 Ku band 12.40 — 18.00 9.488 18.976 0.622 × 0.311
WR51 WG19 R180 15.00 — 22.00 11.572 23.143 0.510 × 0.255
WR42 WG20 R220 K band 18.00 — 26.50 14.051 28.102 0.420 × 0.170
WR34 WG21 R260 22.00 — 33.00 17.357 34.715 0.340 × 0.170
WR28 WG22 R320 Ka band 26.50 — 40.00 21.077 42.154 0.280 × 0.140
WR22 WG23 R400 Q band 33.00 — 50.00 26.346 52.692 0.224 × 0.112
WR19 WG24 R500 U band 40.00 — 60.00 31.391 62.782 0.188 × 0.094
WR15 WG25 R620 V band 50.00 — 75.00 39.875 79.750 0.148 × 0.074
WR12 WG26 R740 E band 60.00 — 90.00 48.373 96.746 0.122 × 0.061
WR10 WG27 R900 W band 75.00 — 110.00 59.015 118.030 0.100 × 0.050
WR8 WG28 R1200 F band 90.00 — 140.00 73.768 147.536 0.080 × 0.040
WR6, WR7 WG29 R1400 D band 110.00 — 170.00 90.791 181.583 0.0650 × 0.0325
WR5 WG30 R1800 140.00 — 220.00 115.714 231.429 0.0510 × 0.0255
WR4 WG31 R2200 172.00 — 260.00 137.243 274.485 0.0430 × 0.0215
WR3 WG32 R2600 220.00 — 330.00 173.571 347.143 0.0340 × 0.0170
* Radio Components Standardization Committee
For historical reasons the outside rather than the inside dimensions of these waveguides are 2:1 (with wall thickness WG6–WG10: 0.08", WG11A–WG15: 0.064", WG16–WG17: 0.05": WG18–WG28: 0.04")[5]
Not an international standard
§ Obsolete

For the frequencies in the table above, the main advantage of waveguides over coax cables is that waveguides support propagation with lower loss. For lower frequencies, the waveguide dimensions become impractically large, and for higher frequencies the dimensions become impractically small (the manufacturing tolerance becomes a significant portion of the waveguide size).

## Dielectric waveguides

Dielectric rod and slab waveguides are used to conduct radio waves, mostly at millimeter wave frequencies and above.[6][7] These confine the radio waves by total internal reflection from the step in refractive index due to the change in dielectric constant at the material surface.[8] At millimeter wave frequencies and above, metal is not a good conductor, so metal waveguides can have increasing attenuation. At these wavelengths dielectric waveguides can have lower losses than metal waveguides. Optical fiber is a form of dielectric waveguide used at optical wavelengths.

One difference between dielectric and metal waveguides is that at a metal surface the electromagnetic waves are tightly confined; at high frequencies the electric and magnetic fields penetrate a very short distance into the metal. In contrast, the surface of the dielectric waveguide is an interface between two dielectrics, so the fields of the wave penetrate outside the dielectric in the form of an evanescent (non-propagating) wave.[8]

## References

This article is based in part on material from Federal Standard 1037C and from MIL-STD-188, and ATIS

1. ^ a b 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. ^ [1]
3. ^ For bandwidths lower than 2:1 it is more common to express them as a percentage of the center frequency, which in the case of 1.360:1 is 26.55%. For reference, a 2:1 bandwidth corresponds to a 66.67% bandwidth. The reason for expressing bandwidths as a ratio of upper to lower band edges for bandwidths greater than 66.67% is that in the limiting case that the lower edge goes to zero (or the upper edge goes to infinity), the bandwidth approaches 200%, which means that the entire range of 3:1 to infinity:1 map into the range 100% to 200%.
4. ^ Harvey, A. F. (July 1955). "Standard waveguides and couplings for microwave equipment". Proceedings of the IEE - Part B: Radio and Electronic Engineering 102 (4): 493–499.
5. ^ Baden Fuller, A. J. (1969). Microwaves (1 ed.). Pergamon Press. ISBN 0-08-006616-X.
6. ^ Lioubtchenko, Dmitri; Sergei Tretyakov, Sergey Dudorov (2003). Millimeter-Wave Waveguides. Springer. p. 149. ISBN 1402075316.
7. ^ Shevgaonkar, R. K. (2005). Electromagnetic Waves. Tata McGraw-Hill Education. p. 327. ISBN 0070591164.
8. ^ a b Rana, Farhan (Fall 2005). "Lecture 26: Dielectric slab waveguides". Class notes ECE 303: Electromagnetic Fields and Waves. Electrical Engineering Dept. Cornell Univ. Retrieved June 21, 2013. p. 2-3, 10
• J. J. Thomson, Recent Researches (1893).
• O. J. Lodge, Proc. Roy. Inst. 14, p. 321 (1894).
• Lord Rayleigh, Phil. Mag. 43, p. 125 (1897).
• N. W. McLachlan, Theory and Applications of Mathieu Functions, p. 8 (1947) (reprinted by Dover: New York, 1964).