Definitions

In the field of antenna design the term radiation pattern most commonly refers to the directional (angular) dependence of radiation from the antenna or other source (synonyms: antenna pattern, far-field pattern).

Particularly in the fields of fiber optics, lasers, and integrated optics, the term radiation pattern, or near-field radiation pattern, may also be used as a synonym for the near-field pattern or Fresnel pattern. This refers to the positional dependence of the electromagnetic field in the near-field, or Fresnel region of the source. The near-field pattern is most commonly defined over a plane placed in front of the source, or over a cylindrical or spherical surface enclosing it.

The far-field pattern of an antenna may be determined experimentally at an antenna range, or alternatively, the near-field pattern may be found using a near-field scanner, and the radiation pattern deduced from it by computation.

The far field radiation pattern may be represented graphically as a plot of one of a number of related variables, including; the field strength at a constant (large) radius (an amplitude pattern or field pattern), the power per unit solid angle (power pattern) and the gain or directive gain. Very often, only the relative amplitude is plotted, normalized either to the amplitude on the antenna boresight, or to the total radiated power. The plotted quantity may be shown on a linear scale, or in dB. The plot is typically represented as a three dimensional graph (as at right), or as separate graphs in the vertical plane and horizontal plane.

## Reciprocity

### Reciprocity applied to antennas

It is a fundamental property of antennas that the receiving pattern (sensitivity as a function of direction) is identical to the far-field radiation pattern. This is a consequence of the reciprocity theorem.

### Proof

For a complete proof, see the reciprocity (electromagnetism) article. Here, we present a common simple proof limited to the approximation of two antennas separated by a large distance compared to the size of the antenna, in a homogeneous medium. The first antenna is the test antenna whose patterns are to be investigated; this antenna is free to point in any direction. The second antenna is a reference antenna, which points rigidly at the first antenna.

Each antenna is alternately connected to a transmitter having a particular source impedance, and a receiver having the same input impedance (the impedance may differ between the two antennas).

It will be assumed that the two antennas are sufficiently far apart that the properties of the transmitting antenna are not affected by the load placed upon it by the receiving antenna. Consequently, the amount of power transferred from the transmitter to the receiver can be expressed as the product of two independent factors; one depending on the directional properties of the transmitting antenna, and the other depending on the directional properties of the receiving antenna.

For the transmitting antenna, by the definition of gain, $G$, the radiation power density at a distance $r$ from the antenna (i.e. the power passing through unit area) is

$mathrm\left\{W\right\}\left(theta,Phi\right) = frac\left\{mathrm\left\{G\right\}\left(theta,Phi\right)\right\}\left\{4 pi r^\left\{2\right\}\right\} P_\left\{t\right\}$.

Here, the arguments $theta$ and $Phi$ indicate a dependence on direction from the antenna, and $P_\left\{t\right\}$ stands for the power the transmitter would deliver into a matched load. The gain $G$ may be broken down into three factors; the directive gain (the directional redistribution of the power), the radiation efficiency (accounting for ohmic losses in the antenna), and lastly the loss due to mismatch between the antenna and transmitter. Strictly, to include the mismatch, it should be called the realized gain, but this is not common usage.

For the receiving antenna, the power delivered to the receiver is

$P_\left\{r\right\} = mathrm\left\{A\right\}\left(theta,Phi\right) W,$.

Here $W$ is the power density of the incident radiation, and $A$ is the effective area or effective aperture of the antenna (the area the antenna would need to occupy in order to intercept the observed captured power). The directional arguments are now relative to the receiving antenna, and again $A$ is taken to include ohmic and mismatch losses.

Putting these expressions together, the power transferred from transmitter to receiver is

$P_\left\{r\right\} = A frac\left\{G\right\}\left\{4 pi r^\left\{2\right\}\right\} P_\left\{t\right\}$,

where $G$ and $A$ are directionally dependent properties of the transmitting and receiving antennas respectively. For transmission from the reference antenna (2), to the test antenna (1), that is

$P_\left\{1r\right\} = mathrm\left\{A_\left\{1\right\}\right\}\left(theta,Phi\right) frac\left\{G_\left\{2\right\}\right\}\left\{4 pi r^\left\{2\right\}\right\} P_\left\{2t\right\}$,

and for transmission in the opposite direction

$P_\left\{2r\right\} = A_\left\{2\right\} frac\left\{mathrm\left\{G_\left\{1\right\}\right\}\left(theta,Phi\right)\right\}\left\{4 pi r^\left\{2\right\}\right\} P_\left\{1t\right\}$.

Here, the gain $G_\left\{2\right\}$ and effective area $A_\left\{2\right\}$ of antenna 2 are fixed, because the orientation of this antenna is fixed with respect to the first.

Now for a given disposition of the antennas, the reciprocity theorem requires that the power transfer is equally effective in each direction, i.e.

$frac\left\{P_\left\{1r\right\}\right\}\left\{P_\left\{2t\right\}\right\} = frac\left\{P_\left\{2r\right\}\right\}\left\{P_\left\{1t\right\}\right\}$,

whence

$frac\left\{mathrm\left\{A_\left\{1\right\}\right\}\left(theta,Phi\right)\right\}\left\{mathrm\left\{G_\left\{1\right\}\right\}\left(theta,Phi\right)\right\} = frac\left\{A_\left\{2\right\}\right\}\left\{G_\left\{2\right\}\right\}$.

But the right hand side of this equation is fixed (because the orientation of antenna 2 is fixed), and so

$frac\left\{mathrm\left\{A_\left\{1\right\}\right\}\left(theta,Phi\right)\right\}\left\{mathrm\left\{G_\left\{1\right\}\right\}\left(theta,Phi\right)\right\} = mathrm\left\{constant\right\}$,

i.e. the directional dependence of the (receiving) effective aperture and the (transmitting) gain are identical (QED). Furthermore, the constant of proportionality is the same irrespective of the nature of the antenna, and so must be the same for all antennas. Analysis of a particular antenna (such as a Hertzian dipole), shows that this constant is $frac\left\{lambda^\left\{2\right\}\right\}\left\{4pi\right\}$, where $lambda$ is the free-space wavelength. Hence, for any antenna the gain and the effective aperture are related by

$mathrm\left\{A\right\}\left(theta,Phi\right) = frac\left\{lambda^\left\{2\right\} mathrm\left\{G\right\}\left(theta,Phi\right)\right\}\left\{4 pi\right\}$.

Even for a receiving antenna, it is more usual to state the gain than to specify the effective aperture. The power delivered to the receiver is therefore more usually written as

$P_\left\{r\right\} = frac\left\{lambda^\left\{2\right\} G_\left\{r\right\} G_\left\{t\right\}\right\}\left\{\left(4 pi r\right)^\left\{2\right\}\right\} P_\left\{t\right\}$

(see link budget). The effective aperture is however of interest for comparison with the actual physical size of the antenna.

### Practical consequences

• When determining the pattern of a receiving antenna by computer simulation, it is not necessary to perform a calculation for every possible angle of incidence. Instead, the radiation pattern of the antenna is determined by a single simulation, and the receiving pattern inferred by reciprocity.
• When determining the pattern of an antenna by measurement, the antenna may be either receiving or transmitting, whichever is more convenient.