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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:

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Presentation Transcript
slide1

Chapter 23: Fresnel equations

Chapter 23: Fresnel equations

slide2

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

slide4

E and B are harmonic

Also, at any specified point in time and space,

where c is the velocity of the propagating wave,

slide5

We’ll also determine the fraction of the light reflected vs. transmitted

external reflection, 

1.0

.5

0

R

T

qi

R

T

0° 30° 60° 90°

Incidence angle, qi

and the change in the phase upon reflection

slide6

Let’s start with polarization…

y

light is a 3-D vector field

circular polarization

linear polarization

z

x

slide7

k

…and consider it relative to a plane interface

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

normal

slide8

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

slide9

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

slide10

Derivation of laws of reflection and refraction

using diagram from Pedrotti3

boundary point

slide11

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

slide12

Reflection

focus on first two terms:

incident and reflected beams travel in same medium; same l

hence we arrive at the law of reflection:

slide13

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:

slide14

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:

slide15

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

slide16

Summary of boundary conditions

TM waves

TE waves

n1

n2

amplitudes are related:

slide17

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 :

slide18

Fresnel equations

TE waves

TM waves

For transmission: eliminate Er , separate Ei and Et , take ratio…

And together:

slide21

External and internal reflections

occur when

external reflection:

internal reflection:

slide22

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

slide23

Glare

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

slide24

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
slide29

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

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

Summary:Reflectance and Transmittance for anAir-to-Glass Interface
summary reflectance and transmittance for a glass to air interface

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

Summary:Reflectance and Transmittance for aGlass-to-Air Interface
slide33

Brewster’s angle

or the polarizing angle

is the angle qp, at which RTM = 0:

slide34

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

slide35

Brewster’s angle

Punky Brewster

Sir David Brewster

(1984-1986)

(1781-1868)

slide36

Brewster’s other angles: the kaleidoscope

http://www.brewstersociety.com/cbs_sundaymorning_09.html

slide37

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…
slide38

Phase changes upon reflection: internal

in the region , r is complex

reflection coefficients in polar form:

f  phase shift on reflection

slide39

Phase changes upon reflection: internal

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

slide40

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

http://www.halbo.com/fr_rhmb.htm

slide41

Summary of phase shifts on reflection

external reflection

TE mode

TM mode

air

glass

internal reflection

TE mode

TM mode

air

glass

slide45

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