Copyright 2006 Regents of University of California1EE 210 Applied EM Fall 2006, Neureuther Lecture #23 Ver 11/15/06EE243 Advanced Electromagnetic TheoryLec # 23 Scattering and Diffraction• Scalar Diffraction Theory• Vector Diffraction Theory• Babinet and Other Principles• Optical TheoremReading: Jackson Chapter 10.5-10.9, 10.10-10.11 liteCopyright 2006 Regents of University of California2EE 210 Applied EM Fall 2006, Neureuther Lecture #23 Ver 11/15/06OverviewObjects large compared to a wavelength are generally treated by approximate integrals over the assumed fields on their surfaces.• In many cases (where the polarization is not important) scalar diffraction can be used.• Where polarization effects are important a vector formulation is needed.• The two key factors in the approximation– The assumed fields on the surfaces or apertures– The source free Green’s function used in the integralCopyright 2006 Regents of University of California3EE 210 Applied EM Fall 2006, Neureuther Lecture #23 Ver 11/15/06Scalar Integral Representation for Far Field• General Representation for solution to scalar wave equation• Choose scalar Green’s function (R to simplifies notation)• Integral that closes surface at infinity goes to zero – radiation condition–f(θ,φ) is the radiation pattern()()()()()[]()()()⎟⎠⎞⎜⎝⎛−→∂∂⇒→′⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛++∇′⋅′−=′−==′′′∇′⋅′′−′∇′⋅′′=∫∫rikrrefadRRkRiiknRexxxRRexxGadxnxxGxxGnxxikrSikRikRS114,14414,,,ψψπφθψψψππψπψψψJackson 10.5Copyright 2006 Regents of University of California4EE 210 Applied EM Fall 2006, Neureuther Lecture #23 Ver 11/15/06Kirchhoff Approximation Representation• Apply to Screen with aperture• Assumptions– ψ and its normal derivative vanish except on opening– ψ and its derivative are equal to the those incident on aperture with no screen• Inherent inconsistencies– Since scattered field is zero everywhere on screen it is zero everywhere– Integral does not yield the assumed values on the openings• Enforcing either Dirichlet or Neuman Boundary Conditions results in a consistent formulation()() ()adxRRnkRiReixadRRkRiiknRexSikRDSikRGEN′′⋅′⎟⎠⎞⎜⎝⎛+−=′⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛++∇′⋅′−=∫∫11121141ψπψψψπψnJackson 10.5Copyright 2006 Regents of University of California5EE 210 Applied EM Fall 2006, Neureuther Lecture #23 Ver 11/15/06Kirchhoff Approximation: Green’s FunctionExample for a point source on one side of Screen• Approximating ψ, δψ/δn or keeping both (Kirchhoff) gives the same integral except for the obliquity factorϑ(θ,θ’) that weights the rays by the cos of the arrival or takeoff angle.()() ()()()()()θθθθϑθθθϑθθθϑθθϑπψψψπψ′+=′′=′=′′′′−=′⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛++∇′⋅′−=∫∫′coscos21,cos,cos,,214111adrereikxadRRkRiiknRexSrikikrSikRGENnJackson 10.5apertureCopyright 2006 Regents of University of California6EE 210 Applied EM Fall 2006, Neureuther Lecture #23 Ver 11/15/06Vector Integral Representation for Far Field• Start with x in volume and interaction integral•Treat x as singular point plus rest of volume• Apply divergence theorem• Use free space Green Function• Integral on surface at infinity goes to zero• Rewrite in transvere only components of E and B on surface()()[]()()[]()()()()()[]adEnkeBneeikkFekkFrexEereGadGEnGEnGBnixEadEnGGnExEssSxkiikrsxnikrikSS′×′×⋅+×′⋅=⋅→′′→′∇′⋅′+∇′××′+×′=′∇′⋅′−∇′⋅′=∫∫∫⋅⋅′′**0*0ˆˆ)(ˆ4,ˆ,4)()()(1ωππωJackson 10.7Copyright 2006 Regents of University of California7EE 210 Applied EM Fall 2006, Neureuther Lecture #23 Ver 11/15/06Diffraction by Screen with Aperture• B is given by integrating B values on the screen geometry.• E is given by integrating E fields on the apertures• This suggest that dual problems are related (Babinet’s principle)()()()()()()adReEnxEadReBnxBadReBnxAikRaperturesdiffikRscreenikRscreen′××∇=′××∇=′′×=∫∫∫ˆ21ˆ21ˆ21πππnscreenapertureCopyright 2006 Regents of University of California8EE 210 Applied EM Fall 2006, Neureuther Lecture #23 Ver 11/15/06Vector Theorems and Concepts• Equivalence theorem: Contributions from sources outside of a volume can be found from tangential E and H on the surface of the volume. • Reaction integral: Integral of tangential fields on the surface is same as calculating E dot J and H dot M throughout the volume.• Green’s Function choice: The region outside the volume could be filled with p.e.c. material to cancel E and double effect of H or magnetic material to cancel H and double the effect of E• Babinet’s Principle: For perfectly conducting thin screen and its complement the electric and magnetic fields for complementary problems are given by the same integral. For example in the case of a slot the magnetic field in a an aperture is used and the complementary case is a metallic bar (screen) and the electric field over the bar is used.Jackson 10.8Copyright 2006 Regents of University of California9EE 210 Applied EM Fall 2006, Neureuther Lecture #23 Ver 11/15/06Diffraction by a Circular Aperture Far Field• Approximate kR by Taylor series• Use Scalar or• Use Vector for A plus B curl A, E curl B()[]() ()[]() ()()adexEnkriexEadxnkixnerexxnrrkxnkkRkRxkiSikrVECTORSxkiikrSCALAR′′××=′′⋅+′∇′⋅−=+′⋅−′+′⋅−=′⋅′⋅−∫∫11ˆ2ˆ4)(...ˆ2ˆ22πψψπψnplanarscreenapertureJackson 10.9xx’RCopyright 2006 Regents of University of California10EE 210 Applied EM Fall 2006, Neureuther Lecture #23 Ver 11/15/06Diffraction by a Circular Aperture Far Field• Plane wave in x-z plane incident from below–ETANreduced by cos α; linear phase in x direction• Find field in direction k – linear phase in x and y directions– Combine all phases; recognize azimuthal integral as J0; integrate in ρ => J1•Result is J1(v)/v with weighting for tangential components of arrival and scattering()()[]()()()()()()()()απξξθφθπαξξαρξβπφαθαθξβρρπαπβρξπβφθβαρcos22sincoscos4coscos21cossinsin2sinsin2cos20202122221202020cos212220cossincossin00aZEPkakaJkaPddPkakaJekEariexEkJededdrEiexEiiikrikikaikr⎟⎟⎠⎞⎜⎜⎝⎛=+=Ω×==′−+==∫∫∫′−−−zaxkk0yαφθEincBincin y dirCopyright 2006 Regents of