# chapter 23: fresnel equations

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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: - PowerPoint PPT Presentation

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• Recall basic laws of opticsLaw of reflection:qinormaln1n2qrqtLaw of refractionSnells Law:Easy to derive on the basis of:Huygens principle:every point on a wavefront may be regarded as a secondary source of wavelets

Fermats principle:the path a beam of light takes between two points is the one which is traversed in the least timeIncident, reflected, refracted, and normal in same plane

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

• E and B are harmonicAlso, at any specified point in time and space,where c is the velocity of the propagating wave,

Plane of incidence: formed by and k and the normal of the interface planeand consider it relative to a plane interfacenormal

TE:Transverse electrics:senkrecht polarized(E-field sticks in and out of the plane)

Polarization modes (= confusing nomenclature!)TM:Transverse magneticp:plane polarized(E-field in the plane)

EMMEperpendicular, horizontalparallel, verticalalways relative to plane of incidence

Plane waves with k along z directionoscillating electric fieldAny polarization state can be described as linear combination of these two:complex amplitude contains all polarization info

• Derivation of laws of reflection and refractionboundary pointusing diagram from Pedrotti3

• At the boundary point:phases of the three waves must be equal:true for any boundary point and time, so lets take orhence, the frequencies are equaland if we now considerwhich means all three propagation vectors lie in the same plane

• 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:Reflection

• 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

• Boundary conditions from Maxwells eqnsfor both electric and magnetic fields, components parallel to boundary plane must be continuous as boundary is passedcomplex field amplitudeselectric fields:TE wavesparallel to boundary plane

• Boundary conditions from Maxwells eqnsfor both electric and magnetic fields, components parallel to boundary plane must be continuous as boundary is passedmagnetic fields: same analysis can be performed for TM wavesTE waves

TE wavesTM wavesn2Summary of boundary conditionsn1amplitudes are related:

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

For transmission: eliminate Er, separate Ei and Et, take ratioAnd together:TE wavesTM wavesFresnel equations

• External and internal reflections

• internal reflection: External and internal reflectionsexternal reflection: occur whenn is called the relative refractive index( )reflection coefficient: rTM Reflectance:Rtransmission coefficient: tTM Transmittance:Tcharacterize byas a function of angle of incidence

• n = n2/n1 = 1.5External reflections (i.e. air-glass)- 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 = 1RTM = 0 RTE = 15%

• at night (when youre in a brightly lit room)IndoorsOutdoorsWindowIin >> IoutR = 8% T = 92%When is a window a mirror?

• when viewing a police lineupWhen is a mirror a window?

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

• - incident angle where RTM = 0 is:

both and reach values of unity before q=90 total internal reflection

Internal reflections (i.e. glass-air)n = n2/n1 = 1/1.5total internal reflection

• Reflectance and Transmittance for anAir-to-Glass Interface

• Reflectance and Transmittance for aGlass-to-Air Interface

• Conservation of energyits always true that andin terms of irradiance (I, W/m2)using laws of reflection and refraction, you can deduceand

• Brewsters angleor the polarizing angleis the angle qp, at which RTM = 0:

• at qp, TM is perfectly transmitted with no reflectionBrewsters angle for internal and external reflections

• Brewsters angle

• Brewsters anglePunky Brewster

(1984-1986)

• http://www.youtube.com/watch?v=-zksq0gVZvI Brewsters other angles: the kaleidoscope

• Phase changes upon reflection-recall the negative reflection coefficients

indicates that sometimes electric field vector reverses direction upon reflection:

-p phase shiftexternal reflection: all angles for TE and at for TM internal reflection: more complex

• Phase changes upon reflection: internalin the region , r is complex

reflection coefficients in polar form:

f phase shift on reflection

• Phase changes upon reflection: internaldepending on angle of incidence, -p < f < p

• Summary of phase shifts on reflectionTE modeTM modeTE modeTM mode

• 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 rhombeach reflection produces a /4 phase delay

• A lovely example

• How do we quantify beauty?

• Case study for reflection and refraction

• 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