Chapter.2.F08.commented - Chapter 2 Waves in Media 2.1...

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Chapter 2 Waves in Media 2.1 Introduction The propagation of waves in a medium depends on the magnetic permeability and electric permittivity functions µ ( ω ) and ε ( ω ) for the medium. We will generally deal with systems in which µ ( ω ) can be approximated as having the value 1 . A general form often used to approximate the electric permittivity is the Sellmeier equation ε ( ω )=1+ X m A m ω 2 m m ω ω 2 . (2.122) This equation is based on the assumption that the system behaves as a set of resonances. This will be observed in the two systems we will examine in this chapter. First we will consider electromagnetic waves in ionic crystals. The discussion will be restricted to frequencies below that of the electronic interband transitions. In this frequency range the dielectric function is determined by the ionic displacements and the polarizabilities of the ions. The second system will be electrical conductors, either metals or plasmas. In these systems the frequency dependent conductivity determines the dielectric function. The analysis of the models will be followed by a discussion of wave packets in dispersive materials. This discussion will use the permittivities obtained for the two models. The propagation of electromagnetic waves in a medium involves the index of refraction of the medium. From the deduced analytic properties of the complex index of refraction we will obtain the Kramer ­ Kronig relationship between the real and imaginary parts of the index of refraction. We will close the chapter with a discussion of re ectance and transmittance of electromagnetic waves at the interface between media. 2.2 Dielectric constant for ionic crystals Ionic crystals such as the alkali ­ halides are cubic crystals. Note that electromagnetic f elds only interact with the ‘optic modes’ for the crystal. These are the modes in which the positive and negative ions move in opposite directions. The dielectric function of cubic crystals is independent of the direction of the electric f eld and can be characterized by three parameters, the static dielectric constant ε s , the optical dielectric constant ε opt , and a characteristic resonant frequency ω T . For reference typical resonant frequencies are in the range 0.02eV ­ 0.04eV, optical frequencies are in the range 2eV ­ 3eV, and the electronic interband transitions begin around 4 or 5eV. An empirical dielectric function for these crystals is ε ( ω )= ε opt + ω 2 T ε opt ε s ω 2 + iγω ω 2 T (2.123) We will f nd that ω T can be identi f ed as the transverse optical phonon frequency for the crystal. A damping factor has been included to indicate what a real dielectric function might look like. This is an approximation to the Sellmeier equation, given in Eq.2.122, in a frequency range near one resonant term and far from any other resonance. Figure 2. shows a typical dielectric function (KBr 1 : ε s =4 . 90 opt =2 . 34 , and ω T . 26 × 10 13 rad/s, 14 . 3 meV ) with no damping.
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This note was uploaded on 12/19/2010 for the course PHYS 423 taught by Professor G. during the Spring '10 term at Missouri S&T.

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Chapter.2.F08.commented - Chapter 2 Waves in Media 2.1...

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