Aperture Antennas

Theory — Circular Apertures

z y x O a P(r,θ,φ) θ φ r = distance θ = polar angle φ = azimuthal angle E_θ E_φ Circular Aperture in Spherical Coordinates
Figure: Circular aperture (broken blue) of radius a in the xy-plane, with observation point P defined by spherical coordinates (r, θ, φ).
The electric field components E_θ and E_φ are shown at point P.

A circular aperture antenna radiates electromagnetic waves. At a far-field observation point, the electric field components in spherical coordinates are:

\[ E_r = 0 \] \[ E_\theta = j \frac{k a^2 E_0 e^{-j k r}}{r} \left\{ \sin \phi \left[ \frac{J_1 (k a \sin \theta)}{k a \sin \theta} \right] \right\} \] \[ E_\phi = j \frac{k a^2 E_0 e^{-j k r}}{r} \left\{ \cos \theta \cos \phi \left[ \frac{J_1 (k a \sin \theta)}{k a \sin \theta} \right] \right\} \]

Here: - \(a\) is the radius of the circular aperture (shown in diagram), - \(r\) is the distance to the observation point, - \(k = 2 \pi / \lambda\) is the wave number, - \(E_0\) is the constant field over the aperture, - \(J_1\) is the first-order Bessel function of the first kind, - \(\theta\) is the polar angle from the z-axis (normal to aperture), - \(\phi\) is the azimuthal angle in the xy-plane.

The polar pattern depends strongly on the radius \(a\) relative to the wavelength \(\lambda\): increasing the radius narrows the main lobe and increases directivity.

Radiation Pattern (Polar)

Value: 1 λ
Main lobe always visible. Larger radius → narrower beam.

Radiation Pattern (Rectangular)

Shows E-field amplitude along x (blue, φ=0) and y (red, φ=π/2). Both main lobe and sidelobes are visible for any radius.