Antenna Temperature ( ) is a parameter that describes how much noise an antenna produces in a given environment. This temperature is not the physical temperature of the antenna. Moreover, an antenna does not have an intrinsic "antenna temperature" associated with it; rather the temperature depends on its gain pattern and the thermal environment that it is placed in.

To define the environment, we'll introduce a temperature distribution - this is the temperature in every direction away from the antenna in spherical coordinates. For instance, the night sky is roughly 4 Kelvin; the value of the temperature pattern in the direction of the Earth's ground is the physical temperature of the Earth's ground. This temperature distribution will be written as . Hence, an antenna's temperature will vary depending on whether it is directional and pointed into space or staring into the sun.

For an antenna with a radiation pattern given by , the noise temperature is mathematically defined as:

This states that the temperature surrounding the antenna is integrated over the entire sphere, and weighted by the antenna's radiation pattern. Hence, an isotropic antenna would have a noise temperature that is the average of all temperatures around the antenna; for a perfectly directional antenna (with a pencil beam), the antenna temperature will only depend on the temperature in which the antenna is "looking".

The noise power received from an antenna at temperature can be expressed in terms of the bandwidth (B) the antenna (and its receiver) are operating over:

In the above, K is Boltzmann's constant (1.38 * 10^-23 [Joules/Kelvin = J/K]). The receiver also has a temperature associated with it ( ), and the total system temperature (antenna plus receiver) has a combined temperature given by

. This temperature can be used in the above equation to find the total noise power of the system. These concepts begin to illustrate how antenna engineers must understand receivers and the associated electronics, because the resulting systems very much depend on each other.

A parameter often encountered in specification sheets for antennas that operate in certain environments is the ratio of gain of the antenna divided by the antenna temperature (or system temperature if a receiver is specified). This parameter is written as G/T, and has units of dB/Kelvin [dB/K].

Antenna Impedance:An antenna's impedance relates the voltage to the current at the input to the antenna. This is extremely important as we will see.

Let's say an antenna has an impedance of 50 ohms. This means that if a sinusoidal voltage is input at the antenna terminals with amplitude 1 Volt, the current will have an amplitude of 1/50 = 0.02 Amps. Since the impedance is a real number, the voltage is in-phase with the current.

Let's say the impedance is given as Z=50 + j*50 ohms (where j is the square root of -1). Then the impedance has a magnitude of

and a phase given by

This means the phase of the current will lag the voltage by 45 degrees. To spell it out, if the voltage (with frequency f) at the antenna terminals is given by

then the current will be given by

So impedance is a simple concept, which relates the voltage and current at the input to the antenna. The real part of an antenna's impedance represents power that is either

radiated away or absorbed within the antenna. The imaginary part of the impedance represents power that is stored in the near field of the antenna (non-radiated power). An antenna with a real input impedance (zero imaginary part) is said to be resonant. Note that an antenna's impedance will vary with frequency.

While simple, we will now explain why this is important, considering both the low frequency and high frequency cases.

Low Frequency

When we are dealing with low frequencies, the transmission line that connects the transmitter or receiver to the antenna is short. Short in antenna theory always means "relative to a wavelength". Hence, 5 meters could be short or very long, depending on what frequency we are operating at. At 60 Hz, the wavelength is about 3100 miles, so the transmission line can almost always be neglected. However, at 2 GHz, the wavelength is 15 cm, so the little length of line within your cell phone can often be considered a 'long line'. Basically, if the line length is less than a tenth of a wavelength, it is reasonably considered a short line.

Consider an antenna (which is represented as an impedance given by ZA) hooked up to a voltage source (of magnitude V) with source impedance given by ZS. The equivalent circuit of this is shown in Figure 1.

Figure 1. Circuit model of an antenna hooked to a source.The power that is delivered to the antenna can be easily found to be (recall your circuit theory, and that P=I*V):

If ZA is much smaller in magnitude than ZS, then no power will be delivered to the antenna and it won't transmit or receive energy. If ZA is much larger in magnitude than ZS, then no power will be delivered as well.

For maximum power to be transferred from the generator to the antenna, the ideal value for the antenna impedance is given by:

The * in the above equation represents complex conjugate. So if ZS=30+j*30 ohms, then for maximum power transfer the antenna should impedance ZA=30-j*30 ohms. Typically, the source impedance is real (imaginary part equals zero), in which case maximum power transfer occurs when ZA=ZS. Hence, we now know that for an antenna to work properly, its impedance must not be too large or too small. It turns out that this is one of the fundamental design parameters for an antenna, and it isn't always easy to design an antenna with the right impedance.

High Frequency

This section will be a little more advanced. In low-frequency circuit theory, the wires that connect things don't matter. Once the wires become a significant fraction of a wavelength, they make things very different. For instance, a short circuit has an impedance of zero ohms. However, if the impedance is measured at the end of a quarter wavelength transmission line, the impedance appears to be infinite, even though there is a dc conduction path.

In general, the transmission line will transform the impedance of an antenna, making it very difficult to deliver power, unless the antenna is matched to the transmission line. Consider the situation shown in Figure 2. The impedance is to be measured at the end of a transmission line (with characteristic impedance Z0) and Length L. The end of the transmission line is hooked to an antenna with impedance ZA.

Figure 2. High Frequency Example.

It turns out (after studying transmission line theory for a while), that the input impedance Zin is given by:

This is a little formidable for an equation to understand at a glance. However, the happy thing is:

If the antenna is matched to the transmission line (ZA=ZO), then the input impedance does not depend on the length of the transmission line.

This makes things much simpler. If the antenna is not matched, the input impedance will vary widely with the length of the transmission line. And if the input impedance isn't well matched to the source impedance, not very much power will be delivered to the antenna. This power ends up being reflected back to the generator, which can be a problem in itself (especially if high power is transmitted). Hence, we see that having a tuned impedance for an antenna is extremely important. For more information on transmission lines, see the transmission line tutorial.

Dipole

The short dipole antenna is the simplest of all antennas. It is simply an open-circuited wire, fed at its center as shown in Figure 1.

Figure 1. Short dipole antenna of length L.

The words "short" or "small" in antenna engineering always imply "relative to a wavelength". So the absolute size of the above dipole does not matter, only the size of the wire relative to the wavelength of the frequency of operation. Typically, a dipole is short if its length is less than a tenth of a wavelength:

If the antenna is oriented along the z-axis with the center of the dipole at z=0, then the current distribution on a thin, short dipole is given by:

The current distribution is plotted in Figure 2. Note that this is the amplitude of the current distribution; it is oscillating in time sinusoidally at frequency f.

Figure 2. Current distribution along a short dipole.

The fields radiated from this antenna in the far field are given by:

The above equations can be broken down and understood somewhat intuitively. First,

note that in the far-field, only the and fields are nonzero. Further, these fields

are orthogonal and in-phase. Further, the fields are perpendicular to the direction of

propagation, which is always in the direction (away from the antenna). Also, the

ratio of the E-field to the H-field is given by (the intrinsic impedance of free space). This indicates that in the far-field region the fields are propagating like a plane-wave.

Second, the fields die off as 1/r, which indicates the power falls of as

Third, the fields are proportional to L, indicated a longer dipole will radiate more power. This is true as long as increasing the length does not cause the short dipole assumption to become invalid. Also, the fields are proportional to the current

amplitude , which should make sense (more current, more power).

The exponential term:

describes the phase-variation of the wave versus distance. Note also that the fields are oscillating in time at a frequency f in addition to the above spatial variation.

Finally, the spatial variation of the fields as a function of direction from the antenna

are given by . For a vertical antenna oriented along the z-axis, the radiation will be maximum in the x-y plane. Theoretically, there is no radiation along the z-axis far from the antenna.

In this section, the dipole antenna with a very thin radius is considered. The dipole is similar to the short dipole except it is not required to be small compared to the wavelength (of the frequency the antenna is operating at).

For a dipole antenna of length L oriented along the z-axis and centered at z=0, the current flows in the z-direction with amplitude which closely follows the following function:

Note that this current is also oscillating in time sinusoidally at frequency f. The current distributions for a quarter-wavelength (left) and full-wavelength (right)

dipoles are given in Figure 1. Note that the peak value of the current is not reached along the dipole unless the length is greater than half a wavelength.

Figure 1. Current distributions on finite-length dipoles.

Before examining the fields radiated by a dipole, consider the input impedance of a dipole as a function of its length, plotted in Figure 2 below. Note that the input impedance is specified as Z=R + jX, where R is the resistance and X is the reactance.

Figure 2. Input impedance as a function of the length (L) of a dipole antenna.

Note that for very small dipoles, the input impedance is capacitive, which means the impedance is dominated by a negative reactance value (and a relatively small real impedance or resistance). As the dipole gets larger, the input resistance increases,

along with the reactance. At slightly less than 0.5 the antenna has zero imaginary component to the impedance (reactance X=0), and the antenna is said to be resonant.

If the dipole antenna's length becomes close to one wavelength, the input impedance becomes infinite. This wild change in input impedance can be understood by studying high frequency transmission line theory. As a simpler explanation, consider the one wavelength dipole shown in Figure 1. If a voltage is applied to the terminals on the right antenna in Figure 1, the current distribution will be as shown. Since the current at the terminals is zero, the input impedance (given by Z=V/I) will necessarily be infinite. Consequently, infinite impedance occurs whenever the dipole is an integer multiple of a wavelength.

The half-wave dipole antenna is just a special case of the dipole antenna, but its important enough that it will have its own section.

The half-wave dipole antenna is as you may expect, a simple half-wavelength wire fed at the center as shown in Figure 1:

Figure 1. Current on a half-wave dipole.

The input impedance is given by Zin = 73 + j42.5 Ohms. The fields from the dipole are given by:

The directivity of a half-wave dipole antenna is 1.64 (2.15 dB). The HPBW is 78 degrees.

In viewing the impedance as a function of the dipole length in the section on dipole antennas, it can be noted that by reducing the length slightly the antenna can become

resonant. If the dipole's length is reduced to 0.48 , the input impedance of the antenna becomes Zin = 70 Ohms, with no reactive component. This is a desirable property, and hence is often done in practice. The radiation pattern remains virtually the same.

The above length is valid if the dipole is very thin. In practice, dipoles are often made with fatter or thicker material, which tends to increase the bandwidth of the antenna. When this is the case, the resonant length reduces slightly depending on the thickness

of the dipole, but will often be close to 0.47 .

A standard rule of thumb in antenna design is: an antenna can be made more broadband by increasing the volume it occupies. Hence, a dipole antenna can be made more broadband by increasing the radius A of the dipole.

As an example, method of moment simulations will be performed on dipoles of length 1.5 meters. At this length, the dipole is a half-wavelength long at 100 MHz. Three cases are considered:

A=0.001 m = (1/3000th) of a wavelength at 100 MHz A=0.015 m = (1/100th) of a wavelength at 100 MHz A=0.05 m = (1/30th) of a wavelength at 100 MHz

The resulting S11 for each of these three cases is plotted versus frequency in Figure 1 (assuming matched to a 50 Ohm load).

Figure 1. Magnitude of S11 for Dipoles of Varying Radii.

The first thing apparent from Figure 1 is that the fatter the dipole is made, the larger the bandwidth becomes. For instance, if the bandwidth is measured as the frequency range over which |S11|<-9 dB, then the bandwidths are 6.5 MHz, 14 MHz, and 24 MHz, for the blue, green and red curves, respectively.

Secondly, the fatter the dipole gets the lower the resonant frequency becomes. In other words, if an antenna is to resonate at 100 MHz, the resonant length decreases as the dipole gets fatter.

Reflector AntennaThe most well-known reflector antenna is the parabolic reflector antenna, commonly known as a satellite dish antenna. Examples of this dish antenna are shown in the following Figures.

Figure 1. The "big dish" of Stanford University.

Figure 2. A random "direcTV dish" on a roof.

Parabolic reflectors typically have a very high gain (30-40 dB is common) and low cross polarization. They also have a reasonable bandwidth, with the fractional bandwidth being at least 5% on commercially available models, and can be very wideband in the case of huge dishes (like the Stanford "big dish" above, which can operate from 150 MHz to 1.5 GHz).

The smaller dish antennas typically operate somewhere between 2 and 28 GHz. The large dishes can operate in the VHF region (30-300 MHz), but typically need to be extremely large at this operating band.

The basic structure of a parabolic dish antenna is shown in Figure 3. It consists of a feed antenna pointed towards a parabolic reflector. The feed antenna is often a horn antenna with a circular aperture.

Figure 3. Components of a dish antenna.

Unlike resonant antennas like the dipole antenna which are typically approximately a half-wavelength long at the frequency of operation, the reflecting dish must be much larger than a wavelength in size. The dish is at least several wavelengths in diameter, but the diameter can be on the order of 100 wavelengths for very high gain dishes (>50 dB gain). The distance between the feed antenna and the reflector is typically several wavelenghts as well. This is in contrast to the corner reflector, where the antenna is roughly a half-wavelength from the reflector.

Helical AntennaHelix antennas have a very distinctive shape, as can be seen in the following picture.

Photo courtesy of Dr. Lee Boyce.

The most popular helical antenna (often called a 'helix') is a travelling wave antenna in the shape of a corkscrew that produces radiation along the axis of the helix. These helixes are referred to as axial-mode helical antennas. The benefits of this antenna is it has a wide bandwidth, is easily constructed, has a real input impedance, and can produce circularly polarized fields. The basic geometry is shown in Figure 1.

Figure 1. Geometry of Helical Antenna.

The parameters are defined below. D - Diameter of a turn on the helix. C - Circumference of a turn on the helix (C=pi*D). S - Vertical separation between turns. - pitch angle, which controls how far the antenna grows in the z-direction per

turn, and is given by N - Number of turns on the helix. H - Total height of helix, H=NS.

The antenna in Figure 1 is a left handed helix, because if you curl your fingers on your left hand around the helix your thumb would point up (also, the waves emitted from the antenna are Left Hand Circularly Polarized). If the helix was wound the other way, it would be a right handed helical antenna.

The pattern will be maximum in the +z direction (along the helical axis in Figure 1). The design of helical antennas is primarily based on empirical results, and the fundamental equations will be presented here.

Helices of at least 3 turns will have close to circular polarization in the +z direction when the circumference C is close to a wavelength:

Once the circumference C is chosen, the inequalites above roughly determine the operating bandwidth of the helix. For instance, if C=19.68 inches (0.5 meters), then the highest frequency of operation will be given by the smallest wavelength that fits

into the above equation, or =0.75C=0.375 meters, which corresponds to a frequency of 800 MHz. The lowest frequency of operation will be given by the largest

wavelength that fits into the above equation, or =1.333C=0.667 meters, which corresponds to a frequency of 450 MHz. Hence, the fractional BW is 56%, which is true of axial helices in general.

The helix is a travelling wave antenna, which means the current travels along the antenna and the phase varies continuously. In addition, the input impedance is primarly real and can be approximated in Ohms by:

The helix functions well for pitch angles ( ) between 12 and 14 degrees. Typically, the pitch angle is taken as 13 degrees.

The normalized radiation pattern for the E-field components are given by:

For circular polarization, the orthogonal components of the E-field must be 90 degrees out of phase. This occurs in directions near the axis (z-axis in Figure 1) of the helix. The axial ratio for helix antennas decreases as the number of loops N is added, and can be approximated by:

The gain of the helix can be approximated by:

In the above, c is the speed of light. Note that for a given helix geometry (specified in terms of C, S, N), the gain increases with frequency. For an N=10 turn helix, that has a 0.5 meter circumference as above, and an pitch angle of 13 degrees (giving S=0.13 meters), the gain is 8.3 (9.2 dB).

For the same example helix, the pattern is shown in Figure 2.

Figure 2. Normalized radiation pattern for helical antenna (dB).

The Half-Power Beamwidth for helical antennas can be approximated (in degrees) by:

Yagi-Uda

The Yagi-Uda antenna or Yagi is one of the most brilliant antenna designs. It is simple to construct and has a high gain, typically greater than 10 dB. These antennas typically operate in the HF to UHF bands (about 3 MHz to 3 GHz), although their bandwidth is typically small, on the order of a few percent of the center frequency. You are probably familiar with this antenna, as they sit on top of roofs everywhere. An example of a Yagi-Uda antenna is shown below.

The Yagi antenna was invented in Japan, with results first published in 1926. The work was originally done by Shintaro Uda, but published in Japanese. The work was presented for the first time in English by Yagi (who was either Uda's professor or colleague, my sources are conflicting), who went to America and gave the first English talks on the antenna, which led to its widespread use. Hence, even though the antenna is often called a Yagi antenna, Uda probably invented it. A picture of Professor Yagi with a Yagi-Uda antenna is shown below.

The basic geometry of a Yagi-Uda antenna is shown in Figure 1.

Figure 1. Geometry of Yagi-Uda antenna

The antenna consists of a single 'feed' or 'driven' element, typically a dipole or a folded dipole antenna. This is the only member of the above structure that is actually excited (a source voltage or current applied). The rest of the elements are parasitic - they reflect or help to transmit the energy in a particular direction. The length of the feed element is given in Figure 1 as F. The feed antenna is almost always the second from the end, as shown in Figure 1. This feed antenna is often altered in size to make

it resonant in the presence of the parasitic elements (typically, 0.45-0.48 wavelengths long for a dipole antenna).

The element to the left of the feed element in Figure 1 is the reflector. The length of this element is given as R and the distance between the feed and the reflector is SR. The reflector element is typically slightly longer than the feed element. There is typically only one reflector; adding more reflectors improves performance very slightly. This element is important in determining the front-to-back ratio of the antenna.

Having the reflector slightly longer than resonant serves two purposes. The first is that the larger the element is, the better of a physical reflector it becomes. Secondly, if the reflector is longer than its resonant length, the impedance of the reflector will be inductive. Hence, the current on the reflector lags the voltage induced on the reflector. The director elements (those to the right of the feed in Figure 1) will be shorter than resonant, making them capacitive, so that the current leads the voltage. This will cause a phase distribution to occur across the elements, simulating the phase progression of a plane wave across the array of elements. This leads to the array being designated as a travelling wave antenna. By choosing the lengths in this manner, the Yagi-Uda antenna becomes an end-fire array - the radiation is along the +y-axis as shown in Figure 1.

The rest of the elements (those to the right of the feed antenna as shown in Figure 1) are known as director elements. There can be any number of directors N, which is typically anywhere from N=1 to N=20 directors. Each element is of length Di, and separated from the adjacent director by a length SDi. As alluded to in the previous paragraph, the lengths of the directors are typically less than the resonant length, which encourages wave propagation in the direction of the directors.

The above description is the basic idea of what is going on. Yagi antenna design is done most often via measurements, and sometimes computer simulations. For instance, lets look at a two-element Yagi antenna (1 reflector, 1 feed element, 0 directors). The feed element is a half-wavelength dipole, shortened to be resonant (gain = 2.15 dB). The gain as a function of the separation is shown in Figure 2.

Figure 2. Gain versus separation for 2-element Yagi antenna.

The above graph shows that the gain is increases by about 2.5 dB if the separation SD is between 0.15 and 0.3 wavelengths. Similarly, the gain can be plotted as a function of director spacings, or as a function of the number of directors used. Typically, the first director will add approximately 3 dB of overall gain (if designed well), the second will add about 2 dB, the third about 1.5 dB. Adding an additional director always increases the gain; however, the gain in directivity decreases as the number of elements gets larger. For instance, if there are 8 directors, and another director is added, the increases in gain will be less than 0.5 dB.

The design of a Yagi-Uda antenna is actually quite simple. Because Yagi antennas have been extensively analyzed and experimentally tested, the process basically follows this outline: Look up a table of design parameters for Yagi antennas Build it (or model it numerically), and tweak it till the performance is acceptable

As an example, consider the table published in "Yagi Antenna Design" by P Viezbicke from the National Bureau of Standards, 1968, given in Table I. Note that the "boom" is the long element that the directors, reflectors and feed elements

are physically attached to, and dictates the lenght of the antenna.

Table I. Optimal Lengths for Yagi-Uda Elements, for Distinct Boom Lengths

d=0.0085

SR=0.2Boom Length of Yagi-Uda Array (in )

0.4 0.8 1.2 2.2 3.2 4.2

R 0.482 0.482 0.482 0.482 0.482 0.475

D1 0.442 0.428 0.428 0.432 0.428 0.424

D2 0.424 0.420 0.415 0.420 0.424

D3 0.428 0.420 0.407 0.407 0.420

D4 0.428 0.398 0.398 0.407

D5 0.390 0.394 0.403

D6 0.390 0.390 0.398

D7 0.390 0.386 0.394

D8 0.390 0.386 0.390

D9 0.398 0.386 0.390

D10 0.407 0.386 0.390

D11 0.386 0.390

D12 0.386 0.390

D13 0.386 0.390

D14 0.386

D15 0.386

Spacing between directors,

(SD/ )

0.20 0.20 0.25 0.20 0.20 0.308

Gain (dB) 9.25 11.35 12.35 14.40 15.55 16.35

There's no real rocket science going on in the above table. I believe the authors of the above document did experimental measurements until they found an optimized set of spacings and published it. The spacing between the directors is uniform and

given in the second-to-last row of the table. The diameter of the elements is given

by d=0.0085 . The above table gives a good starting point to estimate the required length of the antenna (the boom length), and a set of lengths and spacings that achieves the specified gain. In general, all the spacings, lengths, diamters (including the boom diameter) are design variables and can be continuously optimized to alter performance. There are thousands of tables that further give results, such as how the diamter of the boom affects the results, and the optimal diamters of the elements. As an example of Yagi-antenna radiation patterns, a 6-element Yagi antenna (with axis along the +x-axis) is simulated in FEKO (1 reflector, 1 driven half-wavelength dipole, 4 directors). The resulting antenna has a 12.1 dBi gain, and the plots are given in Figures 1-3.

Figure 1. E-plane gain of Yagi antenna.

Figure 2. H-Plane gain of Yagi antenna.

Figure 3. 3-D Radiation Pattern of Yagi antenna.

The above plots are just an example to give an idea of what the radiation pattern of

the Yagi-Uda antenna resembles. The gain can be increased (and the pattern made more directional) by adding more directors or optimizing spacing (or rarely, adding another refelctor). The front-to-back ratio is approximately 19 dB for this antenna, and this can also be optimized if desired.

Horn Antenna

Horn antennas are very popular at UHF (300 MHz-3 GHz) and higher frequencies (I've heard of horns operating as high as 140 GHz). They often have a directional radiation pattern with a high gain , which can range up to 25 dB in some cases, with 10-20 dB being typical. Horns have a wide impedance bandwidth, implying that the input impedance is slowly varying over a wide frequency range (which also implies low values for S11 or VSWR). The bandwidth for practical horn antennas can be on the order of 20:1 (for instance, operating from 1 GHz-20 GHz), with a 10:1 bandwidth not being uncommon.

The gain often increases (and the beamwidth decreases) as the frequency of operation is increased. Horns have very little loss, so the directivity of a horn is roughly equal to its gain.

Horn antennas are somewhat intuitive and not relatively simple to manufacture. In addition, acoustic horns also used in transmitting sound waves (for example, with a megaphone). Horn antennas are also often used to feed a dish antenna, or as a "standard gain" antenna in measurements. Popular versions of the horn antenna include the E-plane horn, shown in Figure 1. This horn is flared in the E-plane, giving the name. The horizontal dimension is constant at w.

Figure 1. E-plane horn.

Another example of a horn is the H-plane horn, shown in Figure 2. This horn is flared in the H-plane, with a constant height for the waveguide and horn of h.

Figure 2. H-Plane horn.

The most popular horn is flared in both planes as shown in Figure 3. This is a pyramidal horn, and has width B and height A at the end of the horn.

Figure 3. Pyramidal horn.

Horns are typically fed by a section of a waveguide, as shown in Figure 4. The waveguide itself is often fed with a short dipole, which is shown in red in Figure 4. A waveguide is simply a hollow, metal cavity. Waveguides are used to guide electromagnetic energy from one place to another. The waveguide in Figure 4 is a rectangular waveguide of width b and height a, with b>a. The E-field distribution for the dominant mode is shown in the lower part of Figure 1.

Figure 4. Waveguide used as a feed to horn antennas.

Antenna texts typically derive very complicated functions for the radiation pattern. To do this, first the E-field across the aperture of the horn is assumed, and the far-field radiation pattern is calculated using the radiation equations. While this is conceptually straight forward, the resulting field functions end up being extremely complex, and personally I don't feel add a whole lot of value. If you would like to see these derivations, pick up any antenna textbook that has a section on horn antennas. (Also, as a practicing antenna engineer, I can assure you that we never use radiation integrals to estimate patterns. We always go on previous experience, computer simulations and measurements.)

Instead of the traditional academic derivation approach, I'll state some results for the horn and show some typical radiation patterns, and attempt to provide a feel for the design parameters of horn antennas. Since the pyramidal horn is the most popular, we'll analyze that. The E-field distribution across the aperture of the horn is what is responsible for the radiation.

The radiation pattern of a horn will depend on B and A (the dimensions of the horn at the opening) and R (the length of the horn, which also affects the flare angles of the horn), along with b and a (the dimensions of the waveguide). These parameters are optimized in order to taylor the performance of the antenna, and are illustrated in the following Figures.

Figure 1. Cross section of waveguide, cut in the H-plane.

Figure 2. Cross section of waveguide, cut in the E-plane.

Observe that the flare angles ( and ) depend on the height, width and length of the horn. Given the coordinate system of Figure 2 (which is centered at the opening of the horn), the radiation will be maximum in the +z-direction (out of the screen).

Figure 2. Coordinate system used, centered on the horn opening.

The E-field distribution across the opening of the horn can be approximated by:

The E-field in the far-field will be linearly polarized, and the magnitude will be given by:

The above equation states that the far-fields of the horn antenna are the Fourier Transform of the fields at the opening of the horn. Many textbooks evaluate this integral, and end up with supremely complicated functions, that I don't feel sheds a whole lot of light on the patterns.