Antennas radiate spherical waves that propagate in the radial direction for a coordinatesystem centered on the antenna. At large distances, spherical waves can be approximatedby plane waves. Plane waves are useful because they simplify the problem.They are not physical, however, because they require infinite power.

The Poynting vector describes both the direction of propagation and the powerdensity of the electromagnetic wave. It is found from the vector cross product of theelectric and magnetic fields and is denoted S:

S = E × H∗W/m2

Root mean square (RMS) values are used to express the magnitude of the fields. H∗ isthe complex conjugate of the magnetic field phasor. The magnetic field is proportionalto the electric field in the far field. The constant of proportion is η, the impedance offree space (η = 376.73Ω):

|S| = S = W/m2 (1)

Because the Poynting vector is the vector product of the two fields, it is orthogonal toboth fields and the triplet defines a right-handed coordinate system: (E, H, S).

Consider a pair of concentric spheres centered on the antenna. The fields around theantenna decrease as 1/R, 1/R², 1/R³, and so on. Constant-order terms would requirethat the power radiated grow with distance and power would not be conserved. Forfield terms proportional to 1/R², 1/R³, and higher, the power density decreases withdistance faster than the area increases. The energy on the inner sphere is larger than thaton the outer sphere. The energies are not radiated but are instead concentrated aroundthe antenna; they are near-field terms. Only the 1/R² term of the Poynting vector(1/R field terms) represents radiated power because the sphere area grows as R² andgives a constant product. All the radiated power flowing through the inner sphere willpropagate to the outer sphere. The sign of the input reactance depends on the near-fieldpredominance of field type: electric (capacitive) or magnetic (inductive). At resonance(zero reactance) the stored energies due to the near fields are equal. Increasing thestored fields increases the circuit Q and narrows the impedance bandwidth.

Far from the antenna we consider only the radiated fields and power density. Thepower flow is the same through concentric spheres:

4πR₁²S_1,avg = 4πR₂²S_2,avg

The average power density is proportional to 1/R². Consider differential areas on thetwo spheres at the same coordinate angles. The antenna radiates only in the radialdirection; therefore, no power may travel in the θ or φ direction. Power travels in fluxtubes between areas, and it follows that not only the average Poynting vector but alsoevery part of the power density is proportional to 1/R²:

S₁R₁² sinθ dθ dφ = S₂R₂² sinθ dθ dφ

Since in a radiated wave S is proportional to 1/R², E is proportional to 1/R. It isconvenient to define radiation intensity to remove the 1/R2 dependence:

U(θ,φ) = S(R, θ, φ)R²W/solid angle

Radiation intensity depends only on the direction of radiation and remains the sameat all distances. A probe antenna measures the relative radiation intensity (pattern)by moving in a circle (constant R) around the antenna. Often, of course, the antennarotates and the probe is stationary.

Some patterns have established names. Patterns along constant angles of the sphericalcoordinates are called either conical (constant θ) or great circle (constant φ). Thegreat circle cuts when φ = 0â—¦ or φ = 90â—¦ are the principal plane patterns. Other namedcuts are also used, but their names depend on the particular measurement positioner,and it is necessary to annotate these patterns carefully to avoid confusion betweenpeople measuring patterns on different positioners. Patterns are measured by usingthree scales: (1) linear (power), (2) square root (field intensity), and (3) decibels (dB).The dB scale is used the most because it reveals more of the low-level responses(sidelobes).

Figure 1 demonstrates many characteristics of patterns. The half-power beamwidthis sometimes called just the beamwidth. The tenth-power and null beamwidths are usedin some applications. This pattern comes from a parabolic reflector whose feed is movedoff the axis. The vestigial lobe occurs when the first sidelobe becomes joined to themain beam and forms a shoulder. For a feed located on the axis of the parabola, thefirst sidelobes are equal.