Fresnel Reflection Lecture13

7/31/2019 Fresnel Reflection Lecture13

1/19

13. Fresnel's Equations for Reflection

and RefractionIncident, transmitted, and reflected beams

Boundary conditions: tangential fields are continuous

Reflection andtransmission

coefficients

The "Fresnel Equations"

Brewster's Angle

Total internal reflection

Power reflectanceand transmittance

Augustin Fresnel1788-1827


2/19

Posing the problem

What happens when light, propagating in auniform medium, encounters a smooth interfacewhich is the boundary of another medium with adifferent refractive index?

k-vector of the

incident light

boundarynincident

ntransmitted

This is really two problems, not just one.


3/19

Definitions: Planes of Incidence, the

interface and S and P polarizationsS polarization is the perpendicular polarization, and it sticks upout of the plane of incidence

Plane of the interface (y=0, the xzplane) (perpendicular to screen)

P polarization is the parallel polarization, and it lies parallelto the plane of incidence.

Plane of incidence (z=0)is the plane thatcontains the incident

and reflected k-vectors.

x

y

z

I R

T


4/19

ni

nt

ik

rk

tk

i r

t

EiBi

Er

Br

Et

Bt

Interface

Fresnel EquationsWe would like to compute the fraction of a light wave reflected andtransmitted by a flat interface between two media with different refrac-tive indices. Fresnel was the first to do this calculation (early 1800s).

x

y

zBeam geometry forlight with its electric

field sticking up out ofthe plane of incidence(i.e., out of the page)

We treat the case of S polarization first:

the xz plane (y = 0)


5/19

ni

nt

ik

rk

tk

i r

t

EiBi

Er

Br

Et

Bt

Interface

Boundary Condition for the Electric

Field at an Interface: s polarizationx

y

z

In other words,

The Tangential Electric Field is Continuous

So: Ei(y = 0) +Er(y = 0) = Et(y = 0)

The component ofthe E-field that lies inthe xz plane iscontinuous as you

move across theplane of the interface.

Here, all E-fields arein the z-direction,

which is in the planeof the interface.

(Were not explicitly writingthe x, z, and t dependence,

but it is still there.)


6/19

Boundary Condition for the Magnetic

Field at an Interface: s polarization

ni

nt

ik

rk

tk

i r

t

Ei

Bi

ErBr

Et

Bt

Interface

x

y

z

ii

*It's really the tangentialB/, but we're using i t0

Bi(y = 0) cosi +Br(y = 0) cosr = Bt(y = 0) cost

The Tangential Magnetic Field* is Continuous

In other words,

The total B-field in theplane of the interface iscontinuous.

Here, all B-fields are inthe xy-plane, so we takethe x-components:


7/19

Reflection and Transmission for

Perpendicularly Polarized Light

Ignoring the rapidly varying parts of the light wave and keeping

only the complex amplitudes:

0 0 0

0 0 0

cos( ) cos( ) cos( )

i r t

i i r r t t

E E E

B B B

0 0 0 0

0 0 0 0

:

( ) cos( ) ( ) cos( )

Substituting for using

t i r t

i r i i t r i t

E E E E

n E E n E E

0 0 0( ) cos( ) cos( ) i r i i t t t n E E n E

0 0/( / ) / .

But and

i rB E c n nE c Substituting into the second equation:


8/19

Reflection & Transmission Coefficients

for Perpendicularly Polarized Light

0 0 0 0

0 0

( ) cos( ) ( ) cos( ) :

cos( ) cos( ) cos( ) cos( )

i r i i t r i t

r i i t t i i i t t

n E E n E E

E n n E n n

Rearranging yields

0 0/ 2 cos( ) / cos( ) cos( )t i i i i i t t t E E n n n

0 0/ , ist iE EAnalogously, the transmission coefficient,

0 0/ cos( ) cos( ) / cos( ) cos( )r i i i t t i i t t r E E n n n n

0 0/r iE ESolving for yields the reflection coefficient :

These equations are called the Fresnel Equations for

perpendicularly polarized (s-polarized) light.


9/19

ni

nt

ik

rk

tk

i r

t

EiBi Er

Br

EtBt

Interface

Fresnel EquationsParallel electric field

x

y

z

Beam geometryfor light with itselectric fieldparallel to theplane of incidence(i.e., in the page)

Note that the reflected magnetic field must point into the screen toachieve for the reflected wave. The x with a circle

around it means into the screen.

E B k

Note that Hechtuses a differentnotation for the

reflected field,which is confusing!

Ours is better!

This leads to adifference in the

signs of someequations...

Now, the case of P polarization:


10/19

Reflection & Transmission Coefficients

for Parallel Polarized Light

These equations are called the Fresnel Equations for

parallel polarized (p-polarized) light.

|| 0 0/ cos( ) cos( ) / cos( ) cos( )r i i t t i i t t ir E E n n n n

|| 0 0/ 2 cos( ) / cos( ) cos( )t i i i i t t it E E n n n

Solving forE0r/ E0i yields the reflection coefficient, r||:

Analogously, the transmission coefficient, t|| = E0t/ E0i, is

For parallel polarized light, B0i B0r = B0t

and E0icos(i) +E0rcos(r) = E0tcos(t)


11/19

To summarize

||

cos( ) cos( )

cos( ) cos( )

i t t i

i t t i

n nr

n n

||

2 cos( )

cos( ) cos( )

i i

i t t i

nt

n n

2 cos( )

cos( ) cos( )

i i

i i t t

nt

n n

cos( ) cos( )

cos( ) cos( )

i i t t

i i t t

n nr

n n

s-polarized light: p-polarized light:

And, for both polarizations: sin( ) sin( )i i t t n n

planeofincidenceincidentwave

transmitted wave

interface

planeofincidenceincidentwave

transmitted wave

interface

E-field vectors are red.k vectors are black.


12/19

Reflection Coefficients for an

Air-to-Glass Interface

Incidence angle, i

Reflection

coefficient,

r

1.0

.5

0

-.5

-1.0

r||

r

0 30 60 90

The two polarizations areindistinguishable at = 0

Total reflection at = 90for both polarizations.

nair 1 < nglass 1.5

Brewsters angle

r||=0!Zero reflection for parallel

polarization at 56.3

Brewster's angleThe value of this angledepends on the value ofthe ratio ni/nt:

Brewster = tan-1(nt/ni)

Sir David Brewster1781 - 1868


13/19

Incidence angle, i

Reflection

coefficient,

r

1.0

.5

0

-.5

-1.0

r||

r

0 30 60 90

Brewstersangle

Total internalreflection

Critical

angle

Criticalangle

Total internal reflection

above the "critical angle"

crit

sin-1

(nt/ni) 41.8for glass-to-air

nglass > nair

(The sine in Snell's Law

can't be greater than one!)

Reflection Coefficients for a

Glass-to-Air Interface


14/19

Reflectance (R)

R Reflected Power / Incident Power r r

i i

I A

I A

Because the angle of incidence = the angle of reflection,the beam area doesnt change on reflection.

Also, n is the same for both incident and reflected beams.

A = Area

20 0

02

c

I n E

iwi nint

r wi

2R rSo: since

2

0 2

2

0

r

i

Er

E


15/19

Transmittance (T)

t t

i i

I A

I A A = Area

20 0

02

cI n E

cos( )cos( )

t t t

i i i

A wA w

t

iwi

wt

nint

If the beamhas width wi:

20 020

0 2

220 0 0

0

2

2

t tt t tt t t t t

i i i i ii i ii i

cn E

n E wI A w n wT t

cI A w n wn E w

n E

The beam expands (or contracts) in one dimension on refraction.

since

2

0 2

2

0

t

i

Et

E

2cos

cos

t t

i i

nT t

n

T Transmitted Power / Incident Power


16/19

Reflectance and Transmittance for an

Air-to-Glass Interface

Note that it is NOT true that: r+ t= 1.

But, it is ALWAYS true that: R + T = 1

Perpendicular polarization

Incidence angle, i

1.0

.5

00 30 60 90

R

T

Parallel polarization

Incidence angle, i

1.0

.5

00 30 60 90

R

T

Brewstersangle


17/19

Perpendicular polarization

Incidence angle, i

1.0

.5

00 30 60 90

R

T

Reflectance and Transmittance for a

Glass-to-Air Interface

Parallel polarization

Incidence angle, i

1.0

.5

00 30 60 90

R

T

Note that the critical angle is the same for both polarizations.

And still, R + T = 1


18/19

Reflection at normal incidence,i= 0

2

t i

t i

n nR

n n

2

4t i

t i

n nT

n n

When i = 0, the Fresnelequations reduce to:

For an air-glass interface (ni = 1 and nt= 1.5),

R = 4% and T= 96%

The values are the same, whicheverdirection the light travels, from air toglass or from glass to air.

This 4% value has big implicationsfor photography.

lens flare


19/19

Windows look like mirrors at night (when youre in the brightly lit room)

One-way mirrors (used by police to interrogate bad guys) are just

partial reflectors (actually, with a very thin aluminum coating).

Disneyland puts ghouls next to you in the haunted house using partialreflectors (also aluminum-coated).

Optical fibers use total internal reflection.

Where youve seen Fresnels Equations in action

R = 100% R = 90%Laser medium

0% reflection!

0% reflection!

Some lasers use Brewsters angle components to avoid reflective losses:

Fresnel Reflection Lecture13

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