Consider an \(n\) -element broadside linear array. Increasing the number of elements has the effect of narrowing the main beam. Demonstrate this by evaluating the separation in \(\phi\) between the zeros on either side of the principal maximum at \(\phi=90^{\circ} .\) Show that for large \(n\) this separation is approximated by \(\Delta \phi \doteq 2 \lambda / L\), where \(L \doteq n d\) is the overall length of the array.

The array factor (AF) is a function that describes the amplitude and phase distribution of an antenna array, shedding light on how it radiates in space. For a linear array antenna, AF is crucial for determining its radiation pattern, beam direction, and sidelobe levels, among other characteristics.

Specifically, AF is determined by the relative spacing and phase of each antenna element in the array. As the number of elements in a broadside linear array increases, AF becomes narrowly distributed around the main beam direction, which is perpendicular to the line of the array. In mathematical terms, AF is expressed as a summation of the contributions from each element, with each influenced by the signal's wavenumber, the element's position, and the angle of interest.

Beam narrowing refers to the phenomenon where the main lobe of the antenna's radiation pattern becomes narrower as the number of elements within the array is increased. This feature is crucial for achieving better directivity and reducing the width of the main beam, which in turn enhances pointing accuracy and mitigates interference from unwanted directions.

With more elements, the overall array aperture, symbolically the length 'L' of the array, increases. This increase in aperture directly translates to a narrower beamwidth, as can be seen with the inverse relationship between \(\Delta\phi\) and 'L' in our given exercise. The larger the aperture, the smaller the beamwidth, given by the approximation \(\Delta\phi \doteq \frac{2\lambda}{L}\), which showcases how a longer array leads to a more focused beam.

In electromagnetics, geometric series often arise when analyzing periodic structures like antenna arrays. Such series allow for simplifying the mathematical expression of the array factor AF.

When applying the geometric series to simplify the AF, we transition from what would be a complex summation of exponential terms to a form that can be more easily analyzed and interpreted. In the case of a broadside linear array, this simplification is possible because the phase difference between adjacent elements adds up in a regular, repeating pattern, which is the essence of a geometric progression. This greatly aids in predicting the behavior of the array and understanding how the number of elements and their spacing affect the antenna's performance, such as beamwidth and the angle at which nulls in the radiation pattern occur.

The angle of array zeros is where the radiation pattern of the array exhibits minimum or no radiation, commonly referred to as nulls. In a linear array antenna, angles of zeros are important in determining the performance and directivity of the antenna.

The separation between these null angles is instrumental for characterizing the antenna's beamwidth, and as the exercise demonstrates, this separation can be approximated when the number of array elements is large. Specifically, by solving the equation \(AF(\phi) = 0\), we find the angles \(\phi_1\) and \(\phi_2\) corresponding to consecutive nulls and deduce their separation \(\Delta\phi\). With increased number of elements, these zeros become more narrowly spaced, leading to the beam narrowing effect that enhances the array's directivity. This information is vital for antenna design, particularly in applications like radar and telecommunications where precise beam control is essential.