1. Chapter 23: Fresnel equations

2. Recall basic laws of optics Law of reflection: ii normal n1n1 n2n2 rr tt Law of refraction “Snell’s Law”: Easy to derive on the basis of: Huygens’ principle:every point on a wavefront may be regarded as a secondary source of wavelets Fermat’s principle:the path a beam of light takes between two points is the one which is traversed in the least time Incident, reflected, refracted, and normal in same plane

3. Today, we’ll show how they can be derived when we consider light to be an electromagnetic wave.

4. E and B are harmonic Also, at any specified point in time and space, where c is the velocity of the propagating wave,

5. Let’s start with polarization… light is a 3-D vector field linear polarization circular polarization y x z

6. Plane of incidence: formed by and k and the normal of the interface plane …and consider it relative to a plane interface k normal

8. TE:Transverse electric s:senkrecht polarized (E-field sticks in and out of the plane) Polarization modes (= confusing nomenclature!) TM:Transverse magnetic p:plane polarized (E-field in the plane) E M M EEE perpendicular, horizontal parallel, vertical always relative to plane of incidence

9. y x x y Plane waves with k along z direction oscillating electric field Any polarization state can be described as linear combination of these two: “complex amplitude” contains all polarization info

10. Derivation of laws of reflection and refraction boundary point using diagram from Pedrotti 3

11. At the boundary point: phases of the three waves must be equal: true for any boundary point and time, so let’s take or hence, the frequencies are equal and if we now consider which means all three propagation vectors lie in the same plane

12. focus on first two terms: incident and reflected beams travel in same medium; same  Since k = 2  hence we arrive at the law of reflection: Reflection

13. now the last two terms: reflected and transmitted beams travel in different media (same frequencies; different wavelengths!): which leads to the law of refraction: Refraction

14. Boundary conditions from Maxwell’s eqns for both electric and magnetic fields, components parallel to boundary plane must be continuous as boundary is passed complex field amplitudes electric fields: TE waves continuity requires: parallel to boundary plane

15. Boundary conditions from Maxwell’s eqns for both electric and magnetic fields, components parallel to boundary plane must be continuous as boundary is passed magnetic fields: continuity requires: same analysis can be performed for TM waves TE waves

16. TM waves n2n2 Summary of boundary conditions n1n1 amplitudes are related:

17. TE wavesTM waves For reflection: eliminate E t, separate E i and E r, and take ratio: Get all in terms of E and apply law of reflection (  i =  r ): Apply law of refraction and let : Fresnel equations

18. For transmission: eliminate E r, separate E i and E t, take ratio… And together: TE wavesTM waves Fresnel equations

19. External and internal reflections

20. internal reflection: External and internal reflections external reflection: occur when n is called the relative refractive index ( ) reflection coefficient: r TM Reflectance:R transmission coefficient: t TM Transmittance:T characterize by as a function of angle of incidence

21. n = n 2 / n 1 = 1.5 External reflections (i.e. air-glass) normalgrazing - at normal and grazing incidence, coefficients have same magnitude - negative values of r indicate phase change - fraction of power in reflected wave = reflectance = - fraction of power transmitted wave = transmittance = here, reflected light TE polarized; at normal : 4% Note: R + T = 1 R TM = 0 R TE = 15%

22. at night (when you’re in a brightly lit room) IndoorsOutdoors Window I in >> I out R = 8% T = 92% When is a window a mirror?

23. when viewing a police lineup When is a mirror a window?

24. http://www.ray-ban.com/clarity/index.html?lang=uk Glare

25. - incident angle where R TM = 0 is: - both and reach values of unity before  =90°  total internal reflection Internal reflections (i.e. glass-air) n = n 2 / n 1 = 1/1.5 total internal reflection

28. Reflectance and Transmittance for an Air-to-Glass Interface Perpendicular polarization Incidence angle,  i 1.0.5 0 0° 30° 60° 90° R T Parallel polarization Incidence angle,  i 1.0.5 0 0° 30° 60° 90° R T

29. Reflectance and Transmittance for a Glass-to-Air Interface Perpendicular polarization Incidence angle,  i 1.0.5 0 0° 30° 60° 90° R T Parallel polarization Incidence angle,  i 1.0.5 0 0° 30° 60° 90° R T

30. Conservation of energy it’s always true that and in terms of irradiance ( I, W/m 2 ) using laws of reflection and refraction, you can deduce and

32. Brewster’s angle or the polarizing angle is the angle  p, at which R TM = 0:

33. at  p, TM is perfectly transmitted with no reflection Brewster’s angle for internal and external reflections

34. Brewster’s angle R = 100% R = 90% Laser medium 0% reflection!

35. Brewster’s angle Punky Brewster Sir David Brewster by Calum Colvin, 2008 (1781-1868) (1984-1986)

36. http://www.youtube.com/watch?v=-zksq0gVZvI Brewster’s other angles: the kaleidoscope

37. Phase changes upon reflection -recall the negative reflection coefficients -indicates that sometimes electric field vector reverses direction upon reflection: -  phase shift external reflection: all angles for TE and at for TM internal reflection: more complex…

38. Phase changes upon reflection: internal in the region, r is complex reflection coefficients in polar form:  phase shift on reflection

39. Phase changes upon reflection: internal depending on angle of incidence, 

40. Summary of phase shifts on reflection TE modeTM mode air glass external reflection TE modeTM mode air glass internal reflection

41. Exploiting the phase difference circular polarization -consists of equal amplitude components of TE and TM linear polarized light, with phases that differ by ±  /2 -can be created by internal reflections in a Fresnel rhomb each reflection produces a π/4 phase delay

42. A lovely example

43. How do we quantify beauty?

44. Case study for reflection and refraction

45. You are encouraged to solve all problems in the textbook (Pedrotti 3 ). The following may be covered in the werkcollege on 15 September 2010: Chapter 23: 1, 2, 3, 5, 12, 16, 20 Exercises http://sites.google.com/site/sciencecafeenschede/vooruitblik-3/-beam-me-up-scotty-50-jaar-laserstraal