# chapter2 - Chapter 2 Classical Electromagnetism and Optics...

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Chapter 2 Classical Electromagnetism and Optics The classical electromagnetic phenomena are completely described by Maxwell’s Equations. The simplest case we may consider is that of electrodynamics of isotropic media 2.1 Maxwell’s Equations of Isotropic Media Maxwell’s Equations are ∂D × H = ∂t + J, (2.1a) ∂B × E = ∂t , (2.1b) · D = ρ, (2.1c) · B = 0 . (2.1d) The material equations accompanying Maxwell’s equations are: D = ± 0 E + P, (2.2a) B = μ 0 H + M. (2.2b) Here, E and H are the electric and magnetic fi eld, D the dielectric ux, B the magnetic ux, J the current density of free chareges, ρ is the free charge density, P is the polarization, and M the magnetization. 13

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14 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS Note, it is Eqs.(2.2a) and (2.2b) which make electromagnetism an inter - esting and always a hot topic with never ending possibilities. All advances in engineering of arti fi cal materials or fi nding of new material properties, such as superconductivity, bring new life, meaning and possibilities into this fi eld. By taking the curl of Eq. (2.1b) and considering ³ ´ ³ ´ × × E = · E E, where is the Nabla operator and the Laplace operator, we obtain Ã ! ³ ´ ∂E ∂P E μ 0 ∂t j + ± 0 ∂t + ∂t = ∂t × M + · E (2.3) and hence µ Ã ! ³ ´ 1 2 j 2 c 2 0 ∂t 2 E = μ 0 ∂t + ∂t 2 P + ∂t × M + · E . (2.4) with the vacuum velocity of light s 1 c 0 = . (2.5) μ 0 ± 0 For dielectric non magnetic media, which we often encounter in optics, with no free charges and currents due to free charges, there is M = 0 , J = 0 , ρ = 0 , which greatly simpli fi es the wave equation to µ ³ ´ 1 2 2 c 2 ∂t 2 E = μ 0 ∂t 2 P + · E . (2.6) 0 2.1.1 Helmholtz Equation In general, the polarization in dielectric media may have a nonlinear and non local dependence on the fi eld. For linear media the polarizability of the medium is described by a dielectric susceptibility χ ( r, t ) Z Z P ( r, t ) = ± 0 dr 0 dt 0 χ ( r r 0 , t t 0 ) E ( r 0 , t 0 ) . (2.7)
15 2.1. MAXWELL’S EQUATIONS OF ISOTROPIC MEDIA The polarization in media with a local dielectric suszeptibility can be de - scribed by Z P ( r, t ) = ± 0 dt 0 χ ( r, t t 0 ) E ( r, t 0 ) . (2.8) This relationship further simpli fi es for homogeneous media, where the sus - ceptibility does not depend on location Z P ( r, t ) = ± 0 dt 0 χ ( t t 0 ) E ( r, t 0 ) . (2.9) which leads to a dielectric response function or permittivity ± ( t ) = ± 0 ( δ ( t ) + χ ( t )) (2.10) and with it to Z D ( r, t ) = dt 0 ± ( t t 0 ) E ( r, t 0 ) . (2.11) In such a linear homogeneous medium follows from eq.(2.1c) for the case of no free charges Z dt 0 ± ( t t 0 ) ( · E ( r, t 0 )) = 0 . (2.12) This is certainly ful fi lled for E = 0 , which simpli fi es the wave equation · (2.4) further µ 1 2 2 E = μ 0 P. (2.13) 2 c 0 ∂t 2 ∂t 2 This is the wave equation driven by the polarization of the medium. If the medium is linear and has only an induced polarization, completely described in the time domain χ ( t ) or in the frequency domain by its Fourier transform, the complex susceptibility χ ˜( ω ) = ˜ ± r ( ω ) 1 with the relative permittivity ˜ ± r (

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