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# Lecture9 - Lecture 9 Notes 95.658 Spring 2012...

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Lecture 9 Notes, 95.658, Spring 2012, Electromagnetic Theory II Dr. Christopher S. Baird, UMass Lowell 1. Scattering Review - At long wavelengths, we can assume that dipole scattering dominates and that the incident wave, induced dipoles and scattered wave all oscillate in step. In addition, we are typically concerned only with the far-field waves. The effects can be approximated to be instantaneous so that the induced dipoles are the same as is found in electrostatics. - The differential scattering cross section (normalized scattered power) with these approximations in place becomes: d σ d Ω = k 4 ( 4 πϵ 0 E 0 ) 2 ϵ * p + 1 c ( ̂ k ×ϵ * )⋅ m 2 where p is the electric dipole moment, m is the magnetic dipole moment, and ε is the polarization unit vector of the polarization being measured in the scattered wave. - The extent to which the scattered wave is polarized is called the average polarization and is defined by: = d unpol, H d d unpol, V d d unpol, unpol d 2. Scattering by a Small Perfectly Conducting Sphere - Consider a perfectly conducting sphere of radius a . If a is much smaller than the wavelength of the incident light, we can use the long-wavelength approximation and all of the results that this approximation leads to. - The electric dipole induced by a uniform incident field was found previously to be: p = 4  0 a 3 E inc 0 - The magnetic moment of the sphere is found to be: m =− 2 a 3 H inc 0 - In the far-field, these dipoles produce scattered radiation according to: d d = k 4 4  0 E 0 2 * p 1 c k × * ⋅ m 2 d d = k 4 4  0 E 0 2 * ⋅ 4  0 a 3 E inc 0  1 c k × * ⋅− 2 a 3 H inc 0  2

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d d = k 4 4  0 E 0 2 * ⋅ 4  0 a 3 0 E 0 1 c k × * ⋅ 2 a 3 1 0 k k 0 × 0 E 0 2 d σ d Ω = k 4 a 6 ϵ * ⋅ϵ 0 1 2 ( ̂ k ×ϵ * )⋅( ̂ k 0 ×ϵ 0 ) 2 - Let us expand this explicitly into the different polarization cases: d HH d = k 4 a 6 1 1 2 cos 2 d HV d = 0 d VH d = 0 d VV d = k 4 a 6 cos − 1 2 2 - If unpolarized light is incident: d unpol, H d = k 4 a 6 2 1 1 2 cos 2 d unpol, V d = k 4 a 6 2 cos − 1 2 2 - If unpolarized light is incident and all scattered polarizations are measured:
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Lecture9 - Lecture 9 Notes 95.658 Spring 2012...

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