Electrodynamics, chap10

Electrodynamics, chap10 - Chapter 10: Scattering and...

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scatterer inc inc E H sc sc E H , p m θ 0 n n d λ ± 0 00 0 Consider a plane wave Assume free space. (10.1) Z/ incident onto an object of di ik inc inc inc Ee Z με ⎡⎤ = ⎢⎥ ⎣⎦ nx Differential Scattering Cross Section : E ε Hn E 0 1 0 2 mension , where can be real (linearly polarized) or complex [e.g. for circularly polarized wave, ( )]. and will induce multipoles on the object, which in turn generate xy inc inc d i ± ε εε ε EH ± () [] 2 0 4 0 scattered radiation ( , ). For , only the induced and are important. From (9.19) and (9.36), we have [in far zone] (10.2) Henc ikr sc sc ke sc r sc sc d c Z πε × × pm En p n n m E ² e, to find and , we need to find the induced and sc sc p m . 10.1 Scattering at Long Wavelength Chapter 10: Scattering and Diffraction
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d Ω / θ r n 0 n For scattering problems, a useful figure of merit is the scattered power ralative to incident power. Furthermore, it is often important to know the polarization state of the scattered radiation. () 0 2 00 radiated power in -direction with -polarization unit solid angle incident power in -direction with -pol 0 Thus we define a differential scattering cross section (with dimension ) as ,; , d d m σ Ω n n ε ε n ε n ε 0 0 2 2 1 2 2 1 0 2 arization unit area * The meaning of will (10.3) become clear in (10.11). * : (i) For a circularly polarized state, an be written sc Z inc Z r Note ⎡⎤ = ⎢⎥ ⎣⎦ ε E ε E ε 1 12 2 0 0 0 ( ), where . (ii) and * ; and * ; * 1; * 1 (iii) is not necessarily the direction of . * gives the -component o sc sc i ± ⊥⊥ = = εε ε ε ε n n ε ε ε E ε E ε f . sc E 10.1 Scattering at Long Wavelength ( continued )
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d Ω / θ r n 0 n () [] 0 0 2 0 0 4 2 00 2 2 1 2 2 1 0 2 4 2 4 * Rewrite (10.3): , ; , (10.3) * Sub. and into (10.3) , * * ikr sc Z inc Z ik ke inc sc r d d d d E k r Ee c c πε σ Ω Ω = == × × × =⋅ + × nx ε E n ε n ε ε E E ε En p n n m n ε n εε pn ε m (10.4) 10.1 Scattering at Long Wavelength ( continued ) ( ) [ ] 0 * * *( * ) ( ) * ×× = ⋅− ε np n ε n p ε p ε nn p ε p ±²³
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() 0 0 3 0 1 2 0 / (relative permitivity) From (4.56), we obtain the electric dipole moment induced on the scatterer by 4 (4.56 r r r inc inc a ε μ εε πε + =⇒ = = = m p E pE 4 2 00 2 4 2 2 46 0 1 2 ) & (10.5) Sub. (10.5) into * * (10.4) , ; , * (10.6) r r d k d E d d c ka σ Ω Ω + =⋅ + × ε pn ε m n ε n ε total electric field inc E 0 / = 0 = r a 10.1 Scattering at Long Wavelength ( continued ) Examples: Example 1 : Scattering by a small ( a << λ ), uniform dielectric sphere with μ = μ 0 and arbitrary ε Question : (4.56) is derived for a dielectric sphere in a static field. Why is it valid for the time-dependent field here? 0 ik Ee = nx ε 0 by assumption =
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0 direction of incident wave n 0 φ (1) ε (2) ε 0 n 0 ε x y z n θ 2 π + 00 (2) 0 We define the - plane as the scattering plane. Let be along the -axis and lie on the - plane. The orientations ( , ) of unit vectors , , and are specified accordingly as fol zx z θφ nn n n εε ε 2 2 22 lows polarization of , incident wave polarization state , 0 of scattered wave to scattering plane polarization state of scattered wave to scat () ππ ⎡⎤ = ⎢⎥ ⎣⎦ =+ = ε ε ε & (2)
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This note was uploaded on 05/14/2010 for the course EAD 234 taught by Professor Ncl during the Spring '10 term at École Normale Supérieure.

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Electrodynamics, chap10 - Chapter 10: Scattering and...

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