Q5Antenna And Wave Propagation

(a) Three-Element Yagi-Uda Antenna Design at 570 MHz

Operating wavelength: at f=570 MHz, the free-space wavelength is:

Driven element: the driven element is a half-wave dipole, fed directly by the transmission line, with a length close to (but slightly less than, due to the end-effect/velocity-factor correction typically around 0.95) a half wavelength:

Reflector: positioned behind the driven element (on the side opposite the desired direction of maximum radiation), the reflector is a parasitic (non-driven) element made electrically longer than the driven element (typically by about 5%, giving it an inductive reactance that causes it to re-radiate with a phase relationship that reinforces radiation toward the director side while cancelling radiation toward its own side), with a typical reflector length and spacing:

Director: positioned in front of the driven element (on the side of desired maximum radiation), the director is a parasitic element made electrically shorter than the driven element (typically by about 5%, giving it a capacitive reactance that causes it to re-radiate with a phase relationship reinforcing radiation further in its own forward direction), with a typical director length and spacing:

Design principle: the reflector and director are parasitic elements (not directly connected to the feed line), and function purely through mutual coupling with the driven element — the currents induced on each parasitic element (by the driven element's own radiated field) combine with the driven element's radiation such that the overall array pattern reinforces in the forward direction (toward the director) and cancels toward the reflector's side, giving the characteristic unidirectional, moderately high-gain end-fire pattern of the Yagi-Uda antenna, with a three-element design typically achieving a gain of approximately 7-9 dBi, a useful, low-cost improvement over a simple half-wave dipole's 2.15 dBi gain, widely used for television reception and amateur radio applications in the VHF/UHF bands.

(b) Categories of Lens Antennas and Dielectric Lens Operation

Lens antennas use a shaped, wave-refracting (or, for some types, wave-delaying/advancing) structure placed in front of a primary feed source (typically a horn or dipole) to convert the feed's naturally diverging, spherical wavefront into a collimated, plane wavefront, achieving high directivity/gain similar in principle to a parabolic reflector antenna, but operating through refraction (bending of the wave path through the lens material) rather than reflection.

Categories of lens antennas: dielectric lenses (using a solid dielectric material, such as polystyrene or similar low-loss material, shaped to introduce the required path-length-dependent phase delay through refraction, analogous in principle to an optical glass lens); E-plane and H-plane metal-plate lenses (using an array of parallel conducting plates spaced to act as a parallel-plate waveguide medium with an effective refractive index different from free space, achieving the required phase correction with lower weight and material cost than a solid dielectric lens, but generally narrower bandwidth); and zoned (stepped/Fresnel) lenses (in which the lens thickness is periodically reduced by integer multiples of a wavelength at certain radial zones, since a phase delay differing by a full 360-degree/2*pi cycle is electrically indistinguishable from zero additional delay, allowing the lens to be made significantly thinner and lighter than an unzoned lens providing the same phase-correction function, at the cost of somewhat reduced bandwidth and efficiency due to the abrupt thickness discontinuities at each zone boundary).

Basic principle of dielectric lens operation: a feed source (such as a horn) positioned at the lens's focal point radiates a diverging spherical wavefront toward the lens — rays traveling closer to the lens's central axis pass through a greater thickness of the (electrically denser, higher-refractive-index) dielectric material than rays traveling toward the lens's outer edge (since the lens is shaped, typically as a plano-convex or similar curved profile, to be thickest at its center and thinner toward its edge), and since electromagnetic waves travel more slowly through the dielectric material than through free space (by a factor equal to the material's refractive index n=sqrt(epsilon_r)), the central rays experience a correspondingly greater propagation delay through the thicker central lens material — by carefully shaping the lens's thickness profile so that this greater delay for the (geometrically shorter) central path exactly compensates for the naturally longer propagation time already required for the (geometrically longer) diverging outer rays to reach the same final wavefront plane, all rays emerge from the lens's exit face with equal total phase delay from the feed, producing a resulting wavefront that is planar (collimated) rather than spherical, exactly analogous to how an optical convex lens focuses/collimates light rays.

Derivation of the Lens Equation

Consider a feed source at the focal point F, with a ray traveling along the lens axis through the full lens thickness T (refractive index n), reaching the reference exit plane; and a second, off-axis ray traveling a distance r from the feed (in free space) to strike the lens surface at radial distance y from the axis, then continuing through the lens's now-shorter thickness at that radial position.

For the on-axis ray, the total optical path length (in terms of phase delay) is: r0+nT, where r0 is the axial distance from feed to lens front surface and T is the lens's maximum (axial) thickness. For an off-axis ray reaching the lens surface at radial position y, if the local lens thickness at that point is t(y) (necessarily less than T for a converging lens shape), the ray travels a free-space distance r (from feed to the lens surface at that point) plus a distance nt(y) through the reduced lens thickness there, plus any remaining free-space distance beyond the lens to reach the same final reference plane. Equating the total phase path length of the off-axis ray to that of the axial ray (the fundamental condition for a truly planar output wavefront) yields the lens equation, which for a hyperbolic-profile plano-convex dielectric lens (feed at the focal point, flat exit face) takes the standard form:

where f is the lens's focal length, n is the dielectric material's refractive index, and theta is the angle between the axis and the ray from the feed to a given point on the lens's curved (feed-facing) surface — this equation defines the required hyperbolic curvature profile of the lens's feed-facing surface that ensures all rays emerging from the flat exit face share exactly the same total phase path length from the feed, producing the desired collimated, planar output wavefront essential to achieving the lens antenna's high-directivity, narrow-beamwidth radiation performance.