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Chapter 23: Fresnel equations

Chapter 23: Fresnel equations. Chapter 23: Fresnel equations. Recall basic laws of optics. normal. Law of reflection:. q i. q r. n 1. n 2. Law of refraction “Snell’s Law”:. q t. Incident, reflected, refracted, and normal in same plane. Easy to derive on the basis of:

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Chapter 23: Fresnel equations

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  1. Chapter 23: Fresnel equations Chapter 23: Fresnel equations

  2. Recall basic laws of optics normal Law of reflection: qi qr n1 n2 Law of refraction “Snell’s Law”: qt Incident, reflected, refracted, and normal in same plane 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

  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… y light is a 3-D vector field circular polarization linear polarization z x

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

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

  8. y y x x 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

  9. Derivation of laws of reflection and refraction using diagram from Pedrotti3 boundary point

  10. 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

  11. Reflection focus on first two terms: incident and reflected beams travel in same medium; same l. Since k = 2p/l, hence we arrive at the law of reflection:

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

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

  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 TE waves magnetic fields: continuity requires: same analysis can be performed for TM waves

  15. Summary of boundary conditions TM waves TE waves n1 n2 amplitudes are related:

  16. Fresnel equations TE waves TM waves Get all in terms of E and apply law of reflection (qi = qr): For reflection: eliminate Et, separate Ei and Er, and take ratio: Apply law of refraction and let :

  17. Fresnel equations TE waves TM waves For transmission: eliminate Er, separate Ei and Et, take ratio… And together:

  18. External and internal reflections

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

  20. External reflections (i.e. air-glass) n= n2/n1 = 1.5 here, reflected light TE polarized; RTM = 0 RTE = 15% at normal : 4% normal grazing • - 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 = Note:R+T = 1

  21. When is a window a mirror? at night (when you’re in a brightly lit room) Iin >> Iout Indoors Outdoors Window R = 8% T = 92%

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

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

  24. Internal reflections (i.e. glass-air) n= n2/n1= 1/1.5 total internal reflection • - incident angle where RTM = 0 is: • both and reach values of unity before q=90° •  total internal reflection

  25. Perpendicular polarization Parallel polarization 1.0 .5 0 1.0 .5 0 T T R R 0° 30° 60° 90° 0° 30° 60° 90° Incidence angle, qi Incidence angle, qi Reflectance and Transmittance for anAir-to-Glass Interface

  26. Perpendicular polarization Parallel polarization 1.0 .5 0 1.0 .5 0 R R T T 0° 30° 60° 90° 0° 30° 60° 90° Incidence angle, qi Incidence angle, qi Reflectance and Transmittance for aGlass-to-Air Interface

  27. Conservation of energy it’s always true that and in terms of irradiance (I, W/m2) using laws of reflection and refraction, you can deduce and

  28. Back to reflections

  29. Brewster’s angle or the polarizing angle is the angle qp, at which RTM = 0:

  30. Brewster’s angle for internal and external reflections at qp, TM is perfectly transmitted with no reflection

  31. 0% reflection! R = 100% R = 90% Laser medium 0% reflection! Brewster’s angle

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

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

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

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

  36. Phase changes upon reflection: internal depending on angle of incidence, -p < f < p

  37. Summary of phase shifts on reflection external reflection TE mode TM mode air glass internal reflection TE mode TM mode air glass

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

  39. A lovely example

  40. How do we quantify beauty?

  41. Case study for reflection and refraction

  42. Exercises You are encouraged to solve all problems in the textbook (Pedrotti3). The following may be covered in the werkcollege on 15 September 2010: Chapter 23: 1, 2, 3, 5, 12, 16, 20 http://sites.google.com/site/sciencecafeenschede/vooruitblik-3/-beam-me-up-scotty-50-jaar-laserstraal

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