ELECTROMAGNETICS AND APPLICATIONS Lecture 22 Aperture Antennas Diffraction Luca Daniel

L21-2 Review of Fundamental Electromagnetic Laws Electromagnetic Waves in Media and Interfaces Digital & Analog Communications Wireless Communications oRadiation Fundamentals oTransmitting Antennas, Gain oReceiving Antennas; Effective Area oAperture Antennas; Diffraction  Examples of aperture antennas  Far field radiation from aperture antenna  Connection with Fourier Transform  Ex. rectangular aperture antenna  Gain and effective area of an aperture antenna  Slit diffraction Acoustic waves and Acoustic antennas e.g. speakers, musical instruments, voice Outline Today

Course Outline and Motivations Electromagnetics: –How to analyze and design antennas Applications –wire antennas (e.g. inside your iPhone, or wireless router) –aperture antennas (e.g. satellite, radar, parabola)

Aperture Antennas Dense phased arrays If each element is cancelled by another a distance D/2 away, the total is a null. => First null at  ≈ /D Aperture antennas: Derivation of far-field aperture radiation: Parabolic reflector Aperture screen Phased array /2  null cancel D 12- element array 1) Find equivalent current J s that would produce the same field in the aperture 2) Find the far fields radiated by each infinitesimal element of J s 3) Add up the contributions from all equivalent currents in the aperture 4) Simplify the expressions by using small-angle approximations => Fourier relationship between aperture and far fields ≈ apertures D 

Far-Field Radiation from Aperture Antennas Equivalent surface current Corresponding radiated far-field: JsJs z y x z y  aperture A r qp q p JsJs top view p k from Magnetic field boundary conditions: k

Far Field is the Fourier Transform of Aperture Field Radiated far field: Fourier relationship: ( time  frequency) Fraunhofer approximation (else, Fresnel) Fourier relationship: (space  k angle) z x q rqrq roro r qp x z y aperture A xx yy q p sin   1 xx roro side view p r qp p p F F 

Example: Rectangular Aperture For a distant observer close to the z axis: LxLx x’ ≈  x d d xx Null≈( /L x )d The Fourier transform of a box is a sinc Area rule: Area in angle domain

Aperture Antenna Gain and Effective Area Rectangular Aperture: G(  x,  y ) = I(  x,  y ) / (P t /4  r 2 ) where I(  x,  z ) = |E ff | 2 /2  o P t = L x L y |E o | 2 /2  o G(  x,  z ) = L x L y (4  / 2 )sinc 2 (  x L x / )sinc 2 (  y L y / ) G max = A (4  / 2 ) where A = L x L y [note: sin(    1 as   0] Hence, in theory A e =A i.e. for an aperture the effective area is the geometrical area But in practice, A e  0.65 A (i.e. “aperture efficiency”  65% x z y LxLx LyLy aperture xx yy xx yy Null at  y = /L y Null at  x = /L x First sidelobe G(  x,  y ) |E ff | 2 =|E o | 2 /( r) 2 ( L x L y ) 2 sinc 2 (  x L x / )sinc 2 (  y L y / )

Example: Slit Diffraction Pair of rectangular slots: Uniform illumination * = Fourier transform Spatial domain D L = Angular domain /D 0 /L x Convolution Multiplication  (x,z-D/2) z=-D/2 D z z=+D/2