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Lecture8 - Lecture 8 Notes 95.658 Spring 2012...

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Lecture 8 Notes, 95.658, Spring 2012, Electromagnetic Theory II Dr. Christopher S. Baird, UMass Lowell 1. Scattering Introduction - Consider a localized object that contains no net sources. Left by itself there are no net electric or magnetic fields present. - An electromagnetic wave propagates until it reaches the object. Part of the wave may pass through through unaffected. - The fields in the wave exert forces on the electrons in the object and induces oscillating charges. - The oscillating charges in turn radiate waves that propagate radially away, possibly in every direction. - The incident wave is said to be scattered by the object. - Scattering experiments – shooting a plane wave at an object and measuring the scattered waves in every direction – can be very useful in probing the structure of the object. 2. Scattering at Long Wavelengths - If the wavelength of the incident light is very long compared to the size of the scattering object, we can use the concepts we developed when describing radiation; the object can be described by the lowest multipole, which oscillates in step with the incident wave. - Such scattering is elastic: the incident and scattered waves have the same frequency. - This is often called Rayleigh scattering or long-wavelength scattering - Consider an object in free space sitting at the origin. - A plane monochromatic wave of polarization ε 0 traveling in the z direction is incident on the object: E inc = 0 E 0 e i k z − t - Assume these fields induce an electric dipole p and magnetic dipole m in the object - The scattered waves has fields that are those due to these dipoles in the far field: E sc =− k 2 4   0 [ k × k × p  1 c k × m ] e i k r − t r - The power scattered is proportional to the square of the electric field. - With this in mind, we can immediately see that the power scattered at long wavelengths is proportional to k 4 (or equivalently λ -4 ) independent of the particulars of the object. - This dependence is known as Rayleigh's Law and it has remarkably wide application. - The scattering effects are typically measured experimentally as a differential scattering cross section . This is defined as the power radiated in the direction n with polarization ε , per unit solid angle, per unit incident flux: d d = r 2 * E sc 2 0 * E inc 2 Polarization Specific Differential Scattering Cross Section - Let us plug in the incident and scattered fields:

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d d = r 2 * ⋅− k 2 4  0 [ k × k × p  1 c k × m ] e i k r − t r 2 E 0 2 d d = k 4 4  0 E 0 2 * [ k × k × p  1 c k × m ] 2 d d = k 4 4  0 E 0 2 * ⋅ k × k × p  1 c * ⋅ k × m 2
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