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02/21/2014PHY 712 Spring Lecture PHY 712 Electrodynamics 9-9:50 & 10-10:50 AM Olin 107 Plan for Lecture 16-17: Read Chapter 7 1.Plane polarized electromagnetic waves 2.Reflectance and transmittance of electromagnetic waves – extension to anisotropy and complexity 3.Frequency dependence of dielectric materials; Drude model 4.Kramers-Kronig relationships

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02/21/2014PHY 712 Spring Lecture

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02/21/2014PHY 712 Spring Lecture

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02/21/2014PHY 712 Spring Lecture Analysis of Maxwell’s equations without sources -- continued:

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02/21/2014PHY 712 Spring Lecture Analysis of Maxwell’s equations without sources -- continued: Both E and B fields are solutions to a wave equation:

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02/21/2014PHY 712 Spring Lecture Analysis of Maxwell’s equations without sources -- continued: Note: , n, k can all be complex; for the moment we will assume that they are all real (no dissipation).

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02/21/2014PHY 712 Spring Lecture Analysis of Maxwell’s equations without sources -- continued: E0E0 B0B0 k

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02/21/2014PHY 712 Spring Lecture Reflection and refraction of plane electromagnetic waves at a plane interface between dielectrics (assumed to be lossless) ’ ’ k’ kiki kRkR iR

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02/21/2014PHY 712 Spring Lecture Reflection and refraction -- continued ’ ’ k’ kiki kRkR iR

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02/21/2014PHY 712 Spring Lecture Reflection and refraction -- continued ’ ’ k’ kiki kRkR iR

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02/21/2014PHY 712 Spring Lecture Reflection and refraction -- continued ’ ’ k’ kiki kRkR iR

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02/21/2014PHY 712 Spring Lecture Reflection and refraction -- continued ’ ’ k’ kiki kRkR iR

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02/21/2014PHY 712 Spring Lecture Reflection and refraction -- continued ’ ’ k’ kiki kRkR iR s-polarization – E field “polarized” perpendicular to plane of incidence

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02/21/2014PHY 712 Spring Lecture Reflection and refraction -- continued ’ ’ k’ kiki kRkR iR p-polarization – E field “polarized” parallel to plane of incidence

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02/21/2014PHY 712 Spring Lecture Reflection and refraction -- continued ’ ’ k’ kiki kRkR iR

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02/21/2014PHY 712 Spring Lecture For s-polarization For p-polarization

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02/21/2014PHY 712 Spring Lecture Special case: normal incidence (i=0, =0)

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02/21/2014PHY 712 Spring Lecture Multilayer dielectrics (Problem #7.2) n1n1 n2n2 n3n3 kiki kRkR ktkt kbkb kaka d

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02/21/2014PHY 712 Spring Lecture Extension of analysis to anisotropic media --

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02/21/2014PHY 712 Spring Lecture Consider the problem of determining the reflectance from an anisotropic medium with isotropic permeability and anisotropic permittivity where: By assumption, the wave vector in the medium is confined to the x-y plane and will be denoted by The electric field inside the medium is given by:

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02/21/2014PHY 712 Spring Lecture Inside the anisotropic medium, Maxwell’s equations are: After some algebra, the equation for E is: From E, H can be determined from

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02/21/2014PHY 712 Spring Lecture The fields for the incident and reflected waves are the same as for the isotropic case. Note that, consistent with Snell’s law: Continuity conditions at the y=0 plane must be applied for the following fields: There will be two different solutions, depending of the polarization of the incident field.

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02/21/2014PHY 712 Spring Lecture Solution for s-polarization

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02/21/2014PHY 712 Spring Lecture Solution for p-polarization

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02/21/2014PHY 712 Spring Lecture Extension of analysis to complex dielectric functions

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02/21/2014PHY 712 Spring Lecture Paul Karl Ludwig Drude

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02/21/2014PHY 712 Spring Lecture Drude model: Vibrations of charged particles near equilibrium: rr

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02/21/2014PHY 712 Spring Lecture Drude model: Vibration of particle of charge q and mass m near equilibrium: rr Note that: > 0 represents dissipation of energy. 0 represents the natural frequency of the vibration; 0 =0 would represent a free (unbound) particle

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02/21/2014PHY 712 Spring Lecture Drude model: Vibration of particle of charge q and mass m near equilibrium: rr

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02/21/2014PHY 712 Spring Lecture Drude model: Vibration of particle of charge q and mass m near equilibrium: rr

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02/21/2014PHY 712 Spring Lecture Drude model: Vibration of particle of charge q and mass m near equilibrium: rr

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02/21/2014PHY 712 Spring Lecture Drude model dielectric function:

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02/21/2014PHY 712 Spring Lecture Drude model dielectric function:

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02/21/2014PHY 712 Spring Lecture Drude model dielectric function – some analytic properties:

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02/21/2014PHY 712 Spring Lecture Drude model dielectric function – some analytic properties:

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02/21/2014PHY 712 Spring Lecture Analytic properties of the dielectric function (in the Drude model or from “first principles” -- Kramers-Kronig transform Re(z) Im(z)

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02/21/2014PHY 712 Spring Lecture Kramers-Kronig transform -- continued Re(z) Im(z) =0

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02/21/2014PHY 712 Spring Lecture Kramers-Kronig transform -- continued

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02/21/2014PHY 712 Spring Lecture Kramers-Kronig transform -- continued

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02/21/2014PHY 712 Spring Lecture a u s b u

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02/21/2014PHY 712 Spring Lecture

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02/21/2014PHY 712 Spring Lecture Analysis for Drude model dielectric function:

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02/21/2014PHY 712 Spring Lecture Analysis for Drude model dielectric function – continued -- Analytic properties:

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02/21/2014PHY 712 Spring Lecture Kramers-Kronig transform – for dielectric function:

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02/21/2014PHY 712 Spring Lecture Further comments on analytic behavior of dielectric function

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