# Lecture5 - 7 Propagation of light and interaction with...

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Lecture 5 7.1.Interaction of light with matter 7.2.Scattering 7.3.Huygens principle 7.4.Reﬂection and Refraction 7.5.Illustration using Huygens principle 7.6.Fermat Principle 7.7. Electromagnetic Approach (Fresnel equation) Part 1 7. Propagation of light and interaction with matter 7.1nteraction of light with matter • As an EM wave hits a material the electric ﬁeld tries to move the charges within the material. It induces a polarization • Let’s assume a single bound electron (Lorentz model) The Lorentz-model We describe the atom as a classical oscillator the equation of motion can be written down easily x electron equilibrium position E Harmonic oscillator q, m e

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Equation of motion • Depending on the binding potential (“due to the spring force”) the oscillator will have an eigenfrequency of ! o • Due to the displacement of the charge from its equilibrium position we will have a dipole moment: • Polarization density: d 2 d 2 t + " o 2 # \$ % ( r x = q m e r E r R , t ( ) r p = q r x r P = N r p = Nq r x = 0 # r E P depends on E r E depends on r P we have to solve this self consistently electric susceptibility Plane waves We use plane waves: E x t , z ( )= E o " cos t \$ kz + % ( ) k 2 = 2 c 2 d 2 d 2 t + o 2 # \$ % ( r x = q m e E o ) cos t * kz + + ( ) r p = e r q m e E o o 2 \$ 2 % ( ) * cos t \$ kz ( ) = = q 2 m e o 2 \$ 2 r E o cos t \$ kz ( )= again: not a constant and not always a scalar r x = r # q m e E o \$ o 2 % 2 ( ) * + cos t % kz ( ) • This equation is fulﬁlled for E electronic polarizability Forced oscillation Self-consistent solution We assume to have identical oscillator with a density N in the medium the electronic polarization of the medium can be expressed as: r P = N r p = N "# ( ) v E = q r N q 2 m e o 2 % 2 ( ) * + E o cos t % kz ( ) • but we have to consider the wave equation as well " 2 v E # 1 c vac 2 2 r E 2 t = 1 o c vac 2 2 r P 2 t = N ( ) o c vac 2 2 r E 2 t k 2 2 2 k 2 " 2 c vac 2 = 2 c vac 2 N \$# ( ) o k 2 = 2 c vac 2 1 + N #" ( ) o % ( ) * Refractive index in the Lorentz model • in vacuum we have: • in a medium:
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Lecture5 - 7 Propagation of light and interaction with...

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