• A critical aspect of any remote sensor is the coupling from the circuit (or transmission line) medium of the sensor to the external medium in which the target is embedded (usually free space), and/or the coupling of the external medium to the sensor• The component that carries out this coupling is traditionally called the “antenna”. • RF sensor antennas generally fall into one of two categories: (1) wire antennas, and (2) aperture antennas. • At the low end of the RF spectrum, roughly up to 10 GHz, the wire antennas take on the form of dipoles, spirals, helices and other simple shapes. The aperture antennas usually take on the form of parabolic or elliptical dishes.• At the high end of the RF spectrum, the wire antenas usually occur on substrates in the form of patches, slots, or other “printed-circuit” antennas. The aperture antennas usually have the form of feedhorns or small dishes.Coupling of THz Radiation to Free Space: Antennas**Good reference on antennas:R.S. Elliott, “Antenna Theory and Design,” (Prentice Hall, Englewood Cliffs, 1981).• All antennas are also classified by their electromagnetic properties (i.e. radiation or beam“pattern” in the external medium) and their circuit properties (i.e., the impedance) in theinternal medium. Because the antenna is a reciprocal passive element, the propertiesin reception are related to those in transmission, but the transmit case is easier to do firstElectromagnetic Properties in Transmit(1) The radiated electric field at a far distance from the antenna will tend to displaya modified spherical-wave of the formreFrEjkr−∝ ),(),,(φθφθwhere k is the free-space propagation constant (= ω/c = 2π/λ) and F is the (normalized) intensity pattern function, F ≡ |S(r,θ,φ)|/Smaxwith S being the Poyntingvector and Smaxis its maximum magnitude, wherever in space that occurs.• All antennas display a limited direction in space where F(θ,φ) is large and other regions where it is negligible, in contrast to isotropic (point) sources. Therefore, a useful metric is the directivity, D.BdFD Ω≡⎟⎟⎠⎞⎜⎜⎝⎛Ω=−∫∫/4),(414πφθππwhere ΩBis the beam solid angle. Conceptually D defines how much greater the intensity is at the peak of F compared to the isotropic radiator emittingthe same total power, for which ΩB= 4π and D = 1.83Notes #4, ECE594I, Fall 2009, E.R. BrownIn this case it is useful to approximate F(θ,φ) by an equivalent spherical cone or sector having a symmetry axis along θP, φP, and polar angular width (or widths) equal to the full-widths at the half-maximum points β(φ) of the real major lobe. • Throughout the cone or sector, F(θ,φ) =1.0• If the pattern has perfect conical symmetry (generally true for parabolic dishes and lenses, and often the design goal for feedhorns), then one finds)]2/cos(1[2sin),(2/0204βπθθφφθβππ−=⋅≈Ω≡Ω∫∫∫∫dddFB)2/cos(12β−≈DandIn the limit of a narrow “pencil” beam where β is small (<< 1 rad), one can Taylor expand the denominator, yielding216β≈DNote that in most books make the simpler approximation ΩB≈β2, so that D ≈ 4π/β2 -a less precise expression but one easier to remember.The characterization of the parabolic dish then reduces to knowing the-3-dB full-width the main lobe, β• An important aspect of all antennas is their degree of non-ideal behavior relatedto radiation in “sidelobes”. These are peaks of radiation in addition to the main lobe that arise from the phenomenon of diffraction.• Diffraction is most easily explained for aperture antennas because they arewider than a wavelength, so that scalar diffraction theory applies:Non-Ideal Behavior of the Radiation Pattern: Diffraction• All RF antennas are generally designed to have a pattern function that displays a predominant, symmetric or quasi-symmetric peak (i.e, “major lobe”) in a single direction of space θP, φPNotes #4, ECE594I, Fall 2008, E.R. Brown84212)(|),(|⎥⎦⎤⎢⎣⎡∝θθϕθkakaJF222/)/sin(/)/sin(|),(|⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡∝rkbyrkbyrkaxrkaxFϕθFor a uniformly illuminated circular aperture, one getswhere J1is the ordinary Bessel function of 1storder, a is the radius, and θ is the angle of the measurement point relative to the optical axis• A key issue in using this integral is the geometric shape of the aperture andthe amplitude distribution of the point sources inside the aperture. In the special case of uniform illumination, scalar diffraction predicts a far-field pattern for a rectangular aperture that goes as:• Both of these functions oscillate with respect to polar angle, θ , reaching relative minimaand relative maxima in between. Each portion of the radiation pattern between the relativeminima is called the “sidelobe”. The totality of oscillating behavior of the radiation pattern is called “diffraction.”xyθMain-lobePatternSide-lobeSide-lobexzWhere k=2π/λ, a is the half-width of the rectangle along x,b is the half-width along y, and r is the range to the measurement point• Scalar diffraction theory provides an approximate solution to the vector EMwave (Helmholtz) equation for radiation passing through the aperture.• The scalar formalism results in the famous Kirchoff-Fresnel integral which, in essence, approximates the radiation pattern as the superposition of point sources filling the aperture, each point source radiating a spherical wave .Notes #4, ECE594I, Fall 2008, E.R. Brown85-0.4-0.3-0.2-0.100.10.20.30.40.50.60246810XJ1(x)0.00010.0010.010.110246810X[J1(x)/x]^2(a)(b)-0.4-0.3-0.2-0.100.10.20.30.40.50.60246810XJ1(x)0.00010.0010.010.110246810X[J1(x)/x]^2(a)(b)•J1(x)/x peaks at x = 0. But its peak value is 0.25, not 1.0 as for the sinc(x). • The first null occurs at the first zero of the J1function, x = 3.835 or θ = 3.835 λ/(2πa) =0.610 λ/a. • A secondary peak of magnitude 0.00437 occurs at approximately x = 5.14, correspondingto θ = 0.818 λ/a. Note that this secondary peak (first sidelobe) has a value of 0.0175 or –17.6 dB relative to the main lobe (this is to be contrasted to the more familiar value of–13.2 dB for the first sidelobe for a square aperture of uniform illuminationExample: the uniformly illuminated circular aperture• From this case, the –3-dB point is at x = 1.616, so the beam full-width is given by β = 2θ = 3.232 λ/(2πa).  Substitution into directivity expressions yields (for β