Elctromagntic scattring Graduat Cours Elctrical Enginring (Communications) 1 st Smstr, 1388-1389 Sharif Univrsity of Tchnology
Contnts of lctur 8 Contnts of lctur 8: Scattring from small dilctric objcts (Rayligh scattring) Volum intgral quation for a dilctric particl Small particl (uniform fild) approximation Th dpolariation fild and factors for llipsoids Solution of th problm Th scattrd fild Exampls Scattring from sphrical objcts
Introduction W considr th problm of scattring by a dilctric objct with a constant prmittivity, but assum th objct to hav dimnsions much smallr than th wavlngth (insid th objct) k i To xprss th actual fild distribution insid th particl, lt us rturn to th volum intgral quation E ( r ) E ( r ) k G ( r, r) E ( r) dv Total fild i V Incidnt fild Scattring from sphrical objcts 3
Introduction Rmmbr that 1 1 E( r) E( r) P( r) Elctric polariation Considr now th fild gnratd by this polariation insid th dilctric particl k 1 V V G ( r, r) P ( r) dv k G ( r, r) P ( r) G ( r, r) P ( r) dv Scattring from sphrical objcts 4
Introduction To gain mor insight, w compar th ratio of th two trms insid th intgral. W hav G ( r, r) RR 1 I 1 jk R k R 3 jk R 3 k G R k R ( r, r) R r r, R R If th particl is small compard to wavlngth, thn k R and this ratio can b hug. Thrfor, th first trm in th intgral is ngligibl. 1 Scattring from sphrical objcts 5
Th lctrostatic approximation Now, considr th nd trm G ( r, r) RR I 1 jk R k R 3 j k R 3 R RR 1 1 I 3 3 R 4 R 4 R xp 3 4 R jk R This is th lctrostatic approximation 1 E ( r ) E i ( r ) P ( r) dv 4 V R Scattring from sphrical objcts 6
Th lctrostatic approximation Lt us rwrit this fild as P ( r) nˆ P ( r) E ( r ) E i ( r ) ds 4 R 4 R S V dv Now, insid a small particl th incidnt fild is almost a constant: E r E k r i ( ) i xp j i E xp jk r i i ki r Scattring from sphrical objcts 7
Th lctrostatic approximation W hav to solv th lctrostatic problm of a dilctric in a uniform, xtrnally applid lctric fild For a dilctric particl of llipsoid shap, it is known that th solution is a uniform (constant) polariation vctor It gnrats a uniform dpolariation fild E dp ˆ 1 P n ds N P 4 R S Dpolariation tnsor Scattring from sphrical objcts 8
To show this, considr a 3D llipsoid with its principal axs chosn along th x,y, and axs of th coordinat systm Lt us dcompos th constant polariation vctor also into its x,y, and componnts. Th total dpolariation fild with b th suprposition of th filds gnratd by ach componnt E E E E x y dp dp dp dp ˆ ˆ Edp P υ n ds, x, y, 4 R υˆ xˆ, υˆ yˆ, υˆ ˆ x y S P y x Scattring from sphrical objcts 9
Th cas of a gnral llipsoid is too complicatd. Thus, w analy th problm for th particular cas of an llipsoid of rvolution. To obtain such an objct, on can draw an llips on th x- plan and rotat it around th -axis. W 1 st us cylindrical coordinats in which th surfac of th llipsoid is givn by th quation b 1 a b x a y Scattring from sphrical objcts 1
Th problm w ar trying to solv is th lctrostatic problm of th lctric fild inducd by th surfac charg υˆ s P So w may solv th Laplac quation insid and outsid th llipsoid, and match th solutions on th surfac nˆ In cylindrical coordinats th Laplac quation rads 1 1 Scattring from sphrical objcts 11
But this coordinat systm is inappropriat to solv th problm. Instad considr th paramtriation (>) sinh sin, cosh cos, Obviously, coordinat surfacs corrsponding to = =constant ar half llipsoids with axs lngth s cosh sinh sinh cosh 1 Scattring from sphrical objcts 1
To b abl to simplify th problm, th surfac of our llipsoid should coincid with on of ths surfacs: a sinh b cosh b a This is possibl if. Thn th corrsponding is a arctanh b Not that w hav assumd b > a. If that is not th cas w intrchang sinh and cosh in th formulation. Scattring from sphrical objcts 13
Th surfacs corrsponding to =constant ar hyprboloids 1 cos sin For corrct paramtriation w dmandd that and In this way all th spac with > will b covrd Scattring from sphrical objcts 14
Th Laplac quation in this nw coordinat systm is 1 1 sinh sin sinh sin sinh sin sinh sin Th outward normal drivativ on th surfac of th llipsoid is givn by nˆ nˆ 1 sinh sin ρˆ cosh sin ˆ sinh cos sinh sin nˆ Scattring from sphrical objcts 15
Now w solv th Laplac quation both insid and outsid th llipsoid and match th solutions using,,,, s nˆ ˆ n υˆ nˆ s P nˆ P υˆ ρˆ cosh sin ˆ sinh cos Scattring from sphrical objcts 16
1 st cas: polariation along -axis,,,, P sinh cos nˆ In both rgions try solution of th typ 1 d df cos f sinh f sinh d d Scattring from sphrical objcts 17
Solving th quation: d f df cosh 1 s s s f ds ds Gnral solution: s s 1 f ( s) As B 1 ln s 1 cosh cosh 1 f ( ) Acosh B 1 ln cosh 1 Scattring from sphrical objcts 18
Scattring from sphrical objcts 19 Insid th llipsoid (to nsur finitnss at =) ( ) cosh f A Outsid th llipsoid (to nsur finitnss at =) cosh cosh 1 ( ) 1 ln cosh 1 f B Matching: cosh cosh 1 cosh 1 ln cosh 1 A B cosh 1 cosh 1 ln cosh 1 sinh P B A
P cosh cosh 1 A sinh 1 ln cosh 1 P B sinh cosh Insid th llipsoid P P cos f N cos cosh N N E cosh cosh 1 sinh 1 ln cosh 1 N P ˆ dp a arctanh b Scattring from sphrical objcts
nd cas: polariation along x-axis, th boundary conditions ar,,,, Px cosh sin cos In both rgions try solution of th typ sin cos f Scattring from sphrical objcts 1
From th Laplac quation w gt 1 1 sinh sin sinh sin sinh sin sinh sin 1 d df 1 sinh f sinh d d sinh d f df s 1 s s s f ds ds s 1 cosh 1 s 1 s 1 f A s 1 B s 1ln s 1 s 1 Scattring from sphrical objcts
Dpolariation factors Insid th llipsoid only th first trm is allowd. It thn follows that th lctric fild is constant, and is in th x-dirction. Summariing th rsults, it follows that th lctric fild insid a uniformly polarid llipsoid is givn by th constant fild E 1 dp N N P x N N y N x N y N 1 N In cas of an llipsoid of rvolution w hav du to symmtry: 1 N N 1 N x y Scattring from sphrical objcts 3
Dpolariation factors Exampls: som limiting cass of llipsoids sphr N x N y N 1 3 y x Infinitly thin disk N 1 N N x y Infinitly long rod 1 N x N y N Scattring from sphrical objcts 4
W can now asily find th (uniform) lctric fild insid th dilctric llipsoid: 1 E E i xp jki r N P P E 1 r i xp j i E I N E k r r 1 Scattring from sphrical objcts 5
Th scattrd fild k xp jkr ( ) ˆ ˆ E s r k s k s r xp jk s r E ( r) dv 4 r V Using prvious rsults, & assuming th particl to b small: E s ( r ) Vk r xp j k s ki r 4 r jkr 1 ˆ s s r i kˆ k I N E Scattring from sphrical objcts 6