chapter2 - Chapter 2 Classical Electromagnetism and Optics...

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 à ! ³ ´ ∂E ∂P E μ 0 ∂t j + ± 0 ∂t + ∂t = ∂t × M + · E (2.3) and hence µ à ! 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)
Image of page 2
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 (
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern