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.
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.
== History ==
During the 1890s theorists did the first analyses
of electromagnetic waves in ducts. Around
1893 J. J. Thomson derived the electromagnetic
modes inside a cylindrical metal cavity. In
1897 Lord Rayleigh did a definitive analysis
of waveguides; he solved the boundary-value
problem of electromagnetic waves propagating
through both conducting tubes and dielectric
rods of arbitrary shape. He showed that the
waves could travel without attenuation only
in specific normal modes with either the electric
field (TE modes) or magnetic field (TM modes),
or both, perpendicular to the direction of
propagation. He also showed each mode had
a cutoff frequency below which waves would
not propagate. Since the cutoff wavelength
for a given tube was of the same order as
its width, it was clear that a hollow conducting
tube could not carry radio wavelengths much
larger than its diameter. In 1902 R. H. Weber
observed that electromagnetic waves travel
at a slower speed in tubes than in free space,
and deduced the reason; that the waves travel
in a "zigzag" path as they reflect from the
walls.Prior to the 1920s, practical work on
radio waves concentrated on the low frequency
end of the radio spectrum, as these frequencies
were better for long-range communication.
These were far below the frequencies that
could propagate in even large waveguides,
so there was little experimental work on waveguides
during this period, although a few experiments
were done. In a June 1, 1894 lecture, "The
work of Hertz", before the Royal Society,
Oliver Lodge demonstrated the transmission
of 3 inch radio waves from a spark gap through
a short cylindrical copper duct. In his pioneering
1894-1900 research on microwaves, Jagadish
Chandra Bose used short lengths of pipe to
conduct the waves, so some sources credit
him with inventing the waveguide. However,
after this, the concept of radio waves being
carried by a tube or duct passed out of engineering
knowledge.During the 1920s the first continuous
sources of high frequency radio waves were
developed: the Barkhausen-Kurz tube, the first
oscillator which could produce power at UHF
frequencies; and the split-anode magnetron
which by the 1930s had generated radio waves
at up to 10 GHz. These made possible the first
systematic research on microwaves in the 1930s.
It was discovered that transmission lines
used to carry lower frequency radio waves,
parallel line and coaxial cable, had excessive
power losses at microwave frequencies, creating
a need for a new transmission method.The waveguide
was developed independently between 1932 and
1936 by George C. Southworth at Bell Telephone
Laboratories and Wilmer L. Barrow at the Massachusetts
Institute of Technology, who worked without
knowledge of one another. Southworth's interest
was sparked during his 1920s doctoral work
in which he measured the dielectric constant
of water with a radio frequency Lecher line
in a long tank of water. He found that if
he removed the Lecher line, the tank of water
still showed resonance peaks, indicating it
was acting as a dielectric waveguide. At Bell
Labs in 1931 he resumed work in dielectric
waveguides. By March 1932 he observed waves
in water-filled copper pipes. Rayleigh's previous
work had been forgotten, and Sergei A. Schelkunoff,
a Bell Labs mathematician, did theoretical
analyses of waveguides and rediscovered waveguide
modes. In December 1933 it was realized that
with a metal sheath the dielectric is superfluous
and attention shifted to metal waveguides.
Barrow had become interested in high frequencies
in 1930 studying under Arnold Sommerfeld in
Germany. At MIT beginning in 1932 he worked
on high frequency antennas to generate narrow
beams of radio waves to locate aircraft in
fog. He invented a horn antenna and hit on
the idea of using a hollow pipe as a feedline
to feed radio waves to the antenna. By March
1936 he had derived the propagation modes
and cutoff frequency in a rectangular waveguide.
The source he was using had a large wavelength
of 40 cm, so for his first successful waveguide
experiments he used a 16-foot section of air
duct, 18 inches in diameter.Barrow and Southworth
became aware of each other's work a few weeks
before both were scheduled to present papers
on waveguides to a combined meeting of the
American Physical Society and the Institute
of Radio Engineers in May 1936. They amicably
worked out credit sharing and patent division
arrangements.
The development of centimeter radar during
World War 2 and the first high power microwave
tubes, the klystron (1938) and cavity magnetron
(1940), resulted in the first widespread use
of waveguide. Standard waveguide "plumbing"
components were manufactured, with flanges
on the end which could be bolted together.
After the war in the 1950s and 60s waveguides
became common in commercial microwave systems,
such as airport radar and microwave relay
networks which were built to transmit telephone
calls and television programs between cities.
== Principle of operation ==
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,
∇
2
Φ
=
0
{\displaystyle \nabla ^{2}\Phi =0}
with boundary conditions guaranteeing only
the trivial solution). This contrasts with
two-conductor transmission lines used at lower
frequencies; coaxial cable, parallel wire
line and stripline, in which TEM mode is possible.
Additionally, the propagating modes (i.e.
TE and TM) inside the waveguide can be mathematically
expressed as the superposition of TEM waves.The
mode with the lowest cutoff frequency is termed
the dominant mode of the guide. It is common
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.
Rectangular waveguide TE-1,0 mode
Rectangular waveguide TE-0,1 mode
== 
Hollow metallic waveguides ==
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 serves as a feed path, and each
slot is a separate radiator, thus forming
an antenna. This structure has the capability
of generating a radiation pattern to launch
an electromagnetic wave in a specific relatively
narrow and controllable direction.
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
may 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 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 relation between the waveguide
dimensions and the lowest frequency is simple:
if
W
{\displaystyle \scriptstyle W}
is the greater of its two dimensions, then
the longest wavelength that will propagate
is
λ
=
2
W
{\displaystyle \scriptstyle \lambda \;=\;2W}
and the lowest frequency is thus
f
=
c
/
λ
=
c
/
2
W
{\displaystyle \scriptstyle f\;=\;c/\lambda
\;=\;c/2W}
With circular waveguides, the highest possible
bandwidth allowing only a single mode to propagate
is only 1.3601:1.Because rectangular waveguides
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
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
widthThe 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 standard waveguides. 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.
* 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"
(2.0 mm), WG11A–WG15: 0.064" (1.6 mm), WG16–WG17:
0.05" (1.3 mm), WG18–WG28: 0.04" (1.0 mm))For
the frequencies in the table above, the main
advantage of waveguides over coaxial 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. 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. 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.
== See also
