1. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-171 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

2. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-172

3. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-173

4. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-174 Analysis of Maxwell’s equations without sources -- continued:

5. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-175 Analysis of Maxwell’s equations without sources -- continued: Both E and B fields are solutions to a wave equation:

6. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-176 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).

7. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-177 Analysis of Maxwell’s equations without sources -- continued: E0E0 B0B0 k

8. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-178 Reflection and refraction of plane electromagnetic waves at a plane interface between dielectrics (assumed to be lossless)  ’  ’  k’ kiki kRkR iR 

9. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-179 Reflection and refraction -- continued  ’  ’  k’ kiki kRkR iR 

10. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1710 Reflection and refraction -- continued  ’  ’  k’ kiki kRkR iR 

11. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1711 Reflection and refraction -- continued  ’  ’  k’ kiki kRkR iR 

12. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1712 Reflection and refraction -- continued  ’  ’  k’ kiki kRkR iR 

13. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1713 Reflection and refraction -- continued  ’  ’  k’ kiki kRkR iR  s-polarization – E field “polarized” perpendicular to plane of incidence

14. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1714 Reflection and refraction -- continued  ’  ’  k’ kiki kRkR iR  p-polarization – E field “polarized” parallel to plane of incidence

15. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1715 Reflection and refraction -- continued  ’  ’  k’ kiki kRkR iR 

16. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1716 For s-polarization For p-polarization

17. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1717 Special case: normal incidence (i=0,  =0)

18. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1718 Multilayer dielectrics (Problem #7.2) n1n1 n2n2 n3n3 kiki kRkR ktkt kbkb kaka d

19. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1719 Extension of analysis to anisotropic media --

20. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1720 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:

21. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1721 Inside the anisotropic medium, Maxwell’s equations are: After some algebra, the equation for E is: From E, H can be determined from

22. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1722 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.

23. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1723 Solution for s-polarization

24. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1724 Solution for p-polarization

25. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1725 Extension of analysis to complex dielectric functions

26. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1726 Paul Karl Ludwig Drude 1863-1906 http://photos.aip.org/history/Thumbnails/drude_paul_a1.jpg

27. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1727 http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png Drude model: Vibrations of charged particles near equilibrium: rr

28. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1728 Drude model: Vibration of particle of charge q and mass m near equilibrium: rr http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png Note that:   > 0 represents dissipation of energy.   0 represents the natural frequency of the vibration;  0 =0 would represent a free (unbound) particle

29. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1729 Drude model: Vibration of particle of charge q and mass m near equilibrium: rr http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png

30. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1730 Drude model: Vibration of particle of charge q and mass m near equilibrium: rr http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png

31. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1731 Drude model: Vibration of particle of charge q and mass m near equilibrium: rr http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png

32. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1732 Drude model dielectric function:

33. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1733 Drude model dielectric function:

34. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1734 Drude model dielectric function – some analytic properties:

35. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1735 Drude model dielectric function – some analytic properties:

36. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1736 Analytic properties of the dielectric function (in the Drude model or from “first principles” -- Kramers-Kronig transform Re(z) Im(z) 

37. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1737 Kramers-Kronig transform -- continued Re(z) Im(z)  =0

38. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1738 Kramers-Kronig transform -- continued

39. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1739 Kramers-Kronig transform -- continued

40. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1740 a u s b u

41. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1741

42. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1742 Analysis for Drude model dielectric function:

43. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1743 Analysis for Drude model dielectric function – continued -- Analytic properties:

44. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1744 Kramers-Kronig transform – for dielectric function:

45. 02/21/2014PHY 712 Spring 2014 -- Lecture 16-1745 Further comments on analytic behavior of dielectric function