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**Chapter 23: Fresnel equations Chapter 23: Fresnel equations**

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**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

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

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**E and B are harmonic Also, at any specified point in time and space,**

where c is the velocity of the propagating wave,

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**and the change in the phase upon reflection**

We’ll also determine the fraction of the light reflected vs. transmitted external reflection, R 1.0 .5 T qi R T 0° ° ° ° Incidence angle, qi and the change in the phase upon reflection

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**Let’s start with polarization… light is a 3-D vector field**

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

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**…and consider it relative to a plane interface**

Plane of incidence: formed by and k and the normal of the interface plane k normal

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**Polarization modes (= confusing nomenclature!)**

always relative to plane of incidence TE: transverse electric s: senkrecht polarized (E-field sticks in and out of the plane) TM: transverse magnetic p: plane polarized (E-field in the plane) M E E M perpendicular ( ), horizontal parallel ( || ), vertical E

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**Plane waves with k along z direction**

oscillating electric field y y x x Any polarization state can be described as linear combination of these two: “complex amplitude” contains all polarization info

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**Derivation of laws of reflection and refraction**

using diagram from Pedrotti3 boundary point

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**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

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**Reflection focus on first two terms:**

incident and reflected beams travel in same medium; same l hence we arrive at the law of reflection:

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**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:

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**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:

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**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

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**Summary of boundary conditions**

TE waves TM waves n1 n2 amplitudes are related:

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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 :

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

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**Fresnel equations, graphically**

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**External and internal reflections**

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**External and internal reflections**

occur when external reflection: internal reflection:

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**External reflections (i.e. air-glass)**

n = n2/n1 = 1.5 RTM = 0 (here, reflected light TE polarized; 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

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Glare

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**Internal reflections (i.e. glass-air)**

n = n2/n1 = 1.5 total internal reflection - incident angle where RTM = 0 is: both and reach values of unity before q=90° total internal reflection

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**Internal reflections (i.e. glass-air)**

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**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

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**Summary: Reflectance and Transmittance for an Air-to-Glass Interface**

Perpendicular polarization Incidence angle, qi 1.0 .5 0° ° ° ° R T Parallel polarization Incidence angle, qi 1.0 .5 0° ° ° ° R T

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**Summary: Reflectance and Transmittance for a Glass-to-Air Interface**

Perpendicular polarization Incidence angle, qi 1.0 .5 0° ° ° ° R T Parallel polarization Incidence angle, qi 1.0 .5 0° ° ° ° R T

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Back to reflections

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**or the polarizing angle**

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

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**Brewster’s angle for internal and external reflections**

at qp, TM is perfectly transmitted with no reflection at Brewster’s angle, “s skips and p plunges” s-polarized light (TE) skips off the surface; p-polarized light (TM) plunges in

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**Brewster’s angle Punky Brewster Sir David Brewster (1781-1868)**

( )

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**Brewster’s other angles: the kaleidoscope**

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**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…

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**Phase changes upon reflection: internal**

in the region , r is complex reflection coefficients in polar form: f phase shift on reflection

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**Phase changes upon reflection: internal**

depending on angle of incidence, -p < f < p

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**Exploiting the phase difference**

circular polarization - 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 each reflection produces a π/4 phase delay

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**Summary of phase shifts on reflection**

air glass external reflection TE mode TM mode air glass internal reflection TE mode TM mode

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A lovely example

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**How do we quantify beauty?**

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**Case study for reflection and refraction**

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Exercises You are encouraged to solve all problems in the textbook (Pedrotti3). The following may be covered in the werkcollege on 21 September 2011: Chapter 23: 1, 2, 3, 5, 12, 16, 20

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