Appid M Fa 6, Nuuth Lctu # V //6 43 Advancd ctomagntic Thoy Lc # Scatting and Diffaction Scatting Fom Sma Obcts Scatting by Sma Dictic and Mtaic Sphs Coction of Scatts Sphica Wav xpansions Scaa Vcto Rading: Jackson Chapt.,.3, it on both. and.4 Copyight 6 Rgnts of Univsity of Caifonia
Appid M Fa 6, Nuuth Lctu # V //6 Ovviw Scatting is simia to adiation but oftn quis simutanousy moding th cation of poaization and cunts fom stimuation by an xtna souc. Sma scatts a tatd by dipo momnts. Intmdiat scatts qui xpansion in many sphica hamonics. Lag scatts can b tatd by appoximation in vaious scaa and vctod diffaction intgas Copyight 6 Rgnts of Univsity of Caifonia
Appid M Fa 6, Nuuth Lctu # V //6 Scatting by Dipos Inducd in Sma Scatts n n Incidnt fid is in diction n and has poaization Thy induc ctic and magntic dipo momnts Scattd fid is in diction n and has poaization Not that fo th fa fid th a two choics fo ach of and but on choic ativ to th pan fomd by n and n Copyight 6 Rgnts of Univsity of Caifonia 3
Appid M Fa 6, Nuuth Lctu # V //6 Scatting by Dipos Inducd in Sma Scatts H p m H sc sc nˆ 4πε iknˆ ˆ n k sc Z x Z inducd _ ctic _ dipo inducd _ magntic _ dipo ik [( nˆ p) n n m / c] Jackson..A Incidnt fids induc ctic and magntic dipo momnts Fa fids fom a thn found fom ths momnts Copyight 6 Rgnts of Univsity of Caifonia 4
Appid M Fa 6, Nuuth Lctu # V //6 Diffntia Scatting Coss Sction dσ dω dσ dω ( nˆ, ˆ; nˆ, ˆ ) ( nˆ, ˆ; nˆ, ˆ ) Z ˆ Z k ( 4πε ) 4 ˆ ˆ sc p + ( nˆ ˆ ) m / c n is in obsvation diction with poaization, whi idnt fux is in diction n with poaization. Dfind as th outgoing pow adiatd p unit soid ang dividd by th idnt pow p unit aa. It is atd to th bistatic coss sction. Thn spciaiz to th cas of th ctic and magntic dipo momnts of sma scatts. Intgating ov both poaizations and a angs givs th ffctiv aa of th scatt Copyight 6 Rgnts of Univsity of Caifonia 5
Copyight 6 Rgnts of Univsity of Caifonia 6 Appid M Fa 6, Nuuth Lctu # V //6 Scatting fom a Sma Dictic Sph Dipo p is in th diction of th idnt fid and qua to th static poaization (sam wight facto and popotiona to voum). Radiation is popotiona th obsvation poaization diction dottd with th idnt poaization. This givs cosθ in on ang and constant in φ. Stngth is 6-th pow of siz (voum squad) and 4-th pow ativ to siz in wavngths. (This xpains th cation of th bu sky succss of hoizontay poaizd sun gasss). Stongst and qua in fowad and backwad dictions. ( ) 6 4 6 4 3 3 8 ˆ ˆ ˆ, ˆ ˆ; ˆ, 4 + Ω Ω + Ω + a k d d d a k n n d d a p ε ε π σ σ ε ε σ ε ε πε Jackson..B p p
Appid M Fa 6, Nuuth Lctu # V //6 Scatting fom a Sma p..c. Sph p m dσ dω 4πε a πa 3 3 ( ) ( ) ( ) 4 6 nˆ, ˆ; nˆ, ˆ k a ˆ ˆ nˆ ˆ nˆ ˆ H Jackson..C p and m Both xist A at ight angs Intf cohnty poduc a + b cosθ typ pattns ow fowad (/3) and high backwad (x) scatting Copyight 6 Rgnts of Univsity of Caifonia 7
Appid M Fa 6, Nuuth Lctu # V //6 dσ dω q F F ( q ) ( q ) ( nˆ, ˆ; nˆ, ˆ ) knˆ knˆ Coction of Scatts iq x i iq ( 4πε ) ( x x ) i k 4 [ p + ( n ) m c] ˆ ˆ ˆ / Assum p and m coctd fo bing insid mdia Sum ov a scatts uding ativ phas masud with spct to idnt diction n and scattd diction n F(q) is N (numb of scatts) in fowad diction and dops quicky to zo xcpt fo cysta stuctus with Bagg ffct. Can b usd to masu ang of intmocua focs that poduc dnsity fuxuations (citica opascnc). Copyight 6 Rgnts of Univsity of Caifonia iq x Jackson..D 8
Appid M Fa 6, Nuuth Lctu # V //6 Scaa Sphica Wav Rpsntation Ψ Ψ h h ( x, ω) f ( ) Y ( θ, φ) [ ] Y ( θ, φ) () () ( ) ( ( x, ω) A h + A h ) (), m, m i + [ ] ( ) ( h ) m m k ik m Soution to scaa wav quation Sphica Hamonics Y m (θ,φ) Radia vaiation dpnds ony on indx m Match bounday conditions on sufac(s) at fixd m Jackson..D Copyight 6 Rgnts of Univsity of Caifonia 9
Appid M Fa 6, Nuuth Lctu # V //6 Scaa Sphica Wav Rpsntation: xamps ik x ik x ik x x 4π x x ik i i ik J ( ) () k h ( k ) Y ( θ, φ ) Y ( θ, φ) Y ( θ, φ ) Y ( θ, φ) 4π J m < m ( + ) J Y ( γ ) Scaa Gn s Function Pan wav in two foms Rpac Numica Gid outsid obct (Mi Mthod) Tansation/otation in coodinat systms Addition Thom fo Sphica Hamonics Sph-Sph intactions > Copyight 6 Rgnts of Univsity of Caifonia m m m ( ) h Jackson 9.6,.3 m Numica Sphica Hamonic xpansion Outsid
Appid M Fa 6, Nuuth Lctu # V //6 L X H g f Vcto Sphica Wav Rpsntation i m ( ) ( θ, φ), m Z a, m A B ( + ) ( θ, φ) (, m) f X a (, m) a k (, m) f X + a (, m) () () ( ) ( ) h () () ( ) ( ) h LY + A + B m m h h g Opato L givs compact notation xpssd in tms of ctic and magntic mutipos Souc and Bounday conditions on sufac at fixd Radia and H componnt on a Sph a adquat Copyight 6 Rgnts of Univsity of Caifonia i k M m M X g Jackson.3 m X m
Appid M Fa 6, Nuuth Lctu # V //6 Vcto Sphica Wav Rpsntation: xamp Jackson.4 Tota fid sum of idnt and scattd xpand scattd fid Outgoing wavs (ony) outsid Both typs but no idint fid insid Rotationay symmtic m + and - (ony) Bounday conditions tan and H tan continuous on bounday of sph Tactab fo Conducting Sph (Mi) Dictic Sph? Usfu fo chcking numica simuatos Copyight 6 Rgnts of Univsity of Caifonia