Lecture 22 Aperture Antennas Diffraction

1. Lecture 22Aperture AntennasDiffraction 6.013 ELECTROMAGNETICS AND APPLICATIONS Luca Daniel

2. Outline • Review of Fundamental Electromagnetic Laws • Electromagnetic Waves in Media and Interfaces • Digital & Analog Communications • Wireless Communications • Radiation Fundamentals • Transmitting Antennas, Gain • Receiving Antennas; Effective Area • Aperture 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 Today

3. 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)

4. Aperture Antennas Parabolic reflector Phased array Dense phased arrays ≈ apertures cancel If each element is cancelled by another a distance D/2 away, the total is a null. => First null at q ≈ l/D 12- element array qnull D l/2 D>>l Aperture antennas: Aperture screen Derivation of far-field aperture radiation: 1) Find equivalent current Js that would produce the same field in the aperture 2) Find the far fields radiated by each infinitesimal element of Js 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

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

6. rqp rqp Far Field is the Fourier Transform of Aperture Field Radiated far field: aperture A side view x x p p q Fraunhofer approximation(else, Fresnel) q p ro  ro rq p ax y ax z z ay sin  1 F Fourier relationship: (space  k angle) F Fourier relationship: ( time  frequency)

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

8. Aperture Antenna Gain and Effective Area Rectangular Aperture: ax aperture x Null at ax = l/Lx First sidelobe G(ax,ay) Lx ax y z ay ay Null at ay = l/Ly Ly G(ax,ay) = I(ax,ay) / (Pt/4pr2) where I(ax,az) = |Eff|2/2ho Pt = LxLy |Eo|2/2ho G(ax,az) = LxLy(4p/l2)sinc2(axLx/l)sinc2(ayLy/l) Gmax = A (4/2) where A = LxLy [note: sin(q)/(q)  1 as q  0] Hence, in theory Ae=A i.e. for an aperture the effective area is the geometrical area But in practice, Ae  0.65 A (i.e. “aperture efficiency”  65% |Eff|2=|Eo|2/(lr)2 (LxLy)2sinc2(axLx/l)sinc2(ayLy/l)

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