1ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityLecture 35Diffraction and Aperture AntennasIn this lecture you will learn:• Diffraction of electromagnetic radiation• Gain and radiation pattern of aperture antennasECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityDiffraction and Aperture Antennas“Aperture antenna” usually refers to a (metallic) sheet with a hole (or an aperture) of some shape through which radiation comes outThe natural spreading of electromagnetic waves in free space when emanating from a source is called “diffraction”Questions• What happens on this side?• How does the radiation coming out of the aperture looks like when the dimensions of the hole are of the order of the wavelength?• What is the radiation pattern?zxIncoming radiation2ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityAperture Antennas in Practice: Rectangular Waveguides How does radiation coming out of a rectangular waveguide looks like?∞=σ∞=σPRadiation coming out of a rectangular apertureSome fraction of the incident power is reflected from the open end and some is radiated outMetal rectangular waveguideDiffractionECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityAperture Antennas in Practice: Dielectric WaveguidesOptical fiberRadiation coming out of a circular apertureIntegrated Photonics (dielectric waveguides on a chip)Radiation coming out of a integrated dielectric waveguide (e.g. a semiconductor laser)DiffractionDiffraction3ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityzxIncident radiationAssumption and GoalAssumption: Assume that we know the field for all time right at the aperture() ,,0, tzyxE =ryThis we could know for example from our knowledge of the incident (and reflected) fields behind the aperture Goal: To find the field for y > 0()? ,,, =tzyxErECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityH-field and Surface Current Density Boundary Condition ()KHHnrrr=−×12ˆ1Hr2HrKrFirst recall the surface current boundary condition for the H-field (now in vector form):For a left-right symmetric problem:KHnrr=×22ˆ1Hr2HrKr21HHrr−=nˆnˆ4ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityPrinciple of Equivalence (Huygens Principle)Principle of equivalence says that if one knows the radiation E- and H-fields at every point on an imaginary closed surface, then the radiation outside the closed surface can be described as the radiation generated from a surface current density that flows on the closed surface()rEsrr()rHsrr()()rHnrKsrrr2ˆ×=()()rHrEsosrrrrη=Principle of Equivalence is a mathematical statement of the old Huygens Principlethat said that every point on a wave-front can be considered a source of radiation???Equivalent problemHsEsKECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityAperture Antenna and the Equivalent ProblemzxyAssumption: Knowing the E-field and H-field phasors at the aperture allows us to consider the equivalent problem of radiation by a current sheet densityEH()() ,ˆ,0, zxEzzyxEa−==r()()oazxExzyxHη,ˆ,0, −==rzxy()()()oaazxEzzxHyzxKη,2ˆ ,2ˆ,=×=rr()()()yzxEzzyxJoaδη,2ˆ,, =⇒r5ECE 303 – Fall 2005 – Farhan Rana – Cornell Universityzxy()()()yzxEzzyxJoaδη,2ˆ,, =r()()''4' 'dverrrJrArrkjo∫∫∫−=−−rrrrrrrrπµKnowing the current density, use the superposition integral for the vector potential to calculate the fields: Make the far-field (or the Fraunhoffer) apprximation:() ()''4 '.ˆdverJerrArrkjrkjoff∫∫∫=⇒−rrrrrπµCompute the E-field in the far-field approximation:Aperture Antennas: Analysis() ()()()[]()[]''ˆˆ4 ˆˆ '.ˆfield far in2dverJrrerkjrArrjrAjcrErrkjrkjoffffff∫∫∫××=××≈×∇×∇=−rrrrrrrrrπηωω'.ˆ' rrrrrrrr−≈−Note that in the far-field: ()()()rArjkrAkjrAffffffrrrrrrr×−=×−≈×∇ˆECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityRectangular Apertures: General Casezxy()()()⎩⎨⎧=0,2ˆ,,yzxEzzyxJoaδηrfor22xxLxL≤≤−&22zzLzL≤≤−otherwise() ()[]''ˆˆ4 '.ˆdverJrrerkjrErrkjrkjoff∫∫∫××=−rrrrrπηUse the formulas:[]()θθsinˆˆˆˆ=×× zrrTo get:()() ( )''','sin2ˆ 2222'.dzdxezxEerkjrEzzxxLLLLrkjarkjff∫∫=−−−rrrrθπθ()() ( )''','sin2ˆ '2222'dzdxeezxEerkjrEzkjLLLLxkjarkjffzzzxxx∫∫=−−−θπθrrOr:Far-field is proportional to the 2D Fourier transform of the field at the aperturezLxLzkykxkkzyxˆˆˆ++=rzzxxrˆ'ˆ''+=r6ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityRectangular Apertures with Uniform Field at the Aperturezxy()()⎩⎨⎧=02ˆ,,yEzzyxJoaδηrfor22xxLxL≤≤−&22zzLzL≤≤−otherwise()()''sin2ˆ '2222'dzdxeeeErkjrEzkjLLLLxkjrkjaffzzzxxx∫∫=−−−θπθrrFar-field is proportional to the 2D Fourier transform of the shape of the aperturezLxLOr:()()''sin2ˆ 22'22'dxedzeeErkjrExxxzzzLLxkjLLzkjrkjaff∫∫=−−−θπθrrECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityFourier Transforms and the Rectangular Aperture Far-Fieldx2xL−2xLConsider the 1D box function FT of the 1D box functionkx1FT:()()dxexfkFxkjxx∫=∞∞−IFT:()()π2xxkjxdkekFxfx−∞∞−∫=()xf()xkF()22sinxxxxxxLkLkLkF⎟⎠⎞⎜⎝⎛=()()()()()()()()22sin22sinsin2ˆ ''sin2ˆ 2222''zzzzxxxxzxrkjaLLLLzkjxkjrkjaffLkLkLkLkLLeErkjdzedxeeErkjrExxzzzx−−−−=∫∫=θπθθπθrrThe far-field E-field is proportional to the 2D FT of the aperture shapexxLkπ2=xxLkπ2−=Width of main lobe in k-space = xLπ47ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityRectangular Aperture: Far-FieldyzxzLxL()()()()()22sin22sinsin2ˆ zzzzxxxxzxrkjaffLkLkLkLkLLeErkjrE−=θπθrrEHE-field amplitude on a plane perpendicular to the y-axis is plottedxzLL =()()φθcossinkkx=()θcoskkz=ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityzxzLxLEHxzLL =Angular width of main lobe in vertical direction is governed by the function:()()()()2'sin2'sinsin22sinθθzzzzzzLkLkLkLk=ynull'θRectangular Aperture: Angular Widths of the Main LobeThe angular half-width is determined by when the term inside the sine function becomes ±π()zznullLkLλπθ±=±=2'sin()()'sincosθθkkkz==For:'2θπθ−=8ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityzxzLxLEHxzLL =ynull'φRectangular Aperture: Angular Widths of the Main LobeAngular width of main lobe in horizontal direction is governed by the function:()()()()2'sin2'sinsin22sinφφxxxxxxLkLkLkLk=The angular half-width is determined by when the term inside the sine function