Phased-Array Radar in one common type of radar installation, a rotating antenna sweeps a radio beam around the sky. But in a phased-array radar system, the antennas remain stationary and the beam is swept electronically. To see how this is done, consider an array of N antennas that are arranged along the horizontal x-axis at x = 0, ± d, ± 2d, ... , ± (N – l)d/2. (The number N is odd.) Each antenna emits radiation uniformly in all directions in the horizontal xy-plane. The antennas all emit radiation coherently, with the same amplitude E0 and the same wavelength λ. The relative phase δ of the emission from adjacent antennas can be varied, however. If the antenna at x = 0 emits a signal that is given by E0cos wt, as measured at a point next to the antenna, the antenna at x = d emits a signal given by E0cos (wt + δ), as measured at a point next to that antenna. The corresponding quantity for the antenna at x = – d is E0 cos (wt – δ); for the antennas at x = ±2d, it is E0cos (wt ± 2δ); and so on.
(a) If δ = 0, the interference pattern at a distance from the antennas is large compared to d and has a principal maximum at θ = 0 (that is, in the +y-direction, perpendicular to the line of the antennas). Show that if d < λ, this is the only principal interference maximum in the angular range -90o < θ < 90o. Hence this principal maximum describes a beam emitted in the direction θ = O. As described in Section 36.4, if N is large, the beam will have a large intensity and be quite narrow.
(b) If δ ≠ 0, show that the principal intensity maximum described in part (a) is located at θ = arc sin (δλ/2πd) where δ is measured in radians. Thus, by varying δ from positive to negative values and back again, which can easily be done electronically, the beam can be made to sweep back and forth around δ = 0.
(c) A weather radar unit to be installed on an airplane emits radio waves at 8800MHz. The unit uses 15 antennas in an array 28.0 cm long (from the antenna at one end of the array to the antenna at the other end). What must the maximum and minimum values of δ be (that is, the most positive and most negative values) if the radar beam is to sweep 45° to the left or right of the airplane's direction of flight? Give your answer in radians.