# Chapter 23: Fresnel equations 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: 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

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

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

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

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

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

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

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

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

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

Reflection focus on first two terms:
incident and reflected beams travel in same medium; same l hence we arrive at the law of reflection:

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:

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:

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

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

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 :

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

Fresnel equations, graphically

External and internal reflections

External and internal reflections
occur when external reflection: internal reflection:

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

Glare

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

Internal reflections (i.e. glass-air)

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

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

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

Back to reflections

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

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

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

Brewster’s 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 shift external reflection: all angles for TE and at for TM internal reflection: more complex…

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

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

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

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

A lovely example

How do we quantify beauty?

Case study for reflection and refraction

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