20. Fresnel equations 20. Fresnel equations - Hanyangoptics.hanyang.ac.kr/~shsong/20-Fresnel equations.pdf · 20. Fresnel equations 20. Fresnel equations • EM Waves at an Interface

20. Fresnel equations 20. Fresnel equations

• EM Waves at an Interface

• Fresnel Equations: Reflection and Transmission Coefficients

• Brewster’s Angle

• Total Internal Reflection• Evanescent Waves

• The Complex Refractive Index• Reflection from Metals

2 2

2 2

2 2 2

2 2 2

cos sin:cos sin

cos sin:c

:

os sin

r

r

E nTE rE n

E

r reflection coefficient

n nTM rE n n

θ θ

θ θ

θ θ

θ θ

− −= =

+ −

− −= =

+ −

2 2

2 2 2

2cos:cos sin

2 cos:cos sin

:

t

t

t transE

TE tE nE nTM tE

mission coeffi e

n

ci nt

n

θ

θ θθ

θ θ

= =+ −

= =+ −

n2

θE

Et

Erθr

θt

n1

We will derive the Fresnel equations We will derive the Fresnel equations

1

2

nnn ≡

EM Waves at an InterfaceEM Waves at an Interface

r

TE mode


( )( )( )

exp

exp

:

:

:

( ), ,

exp

i oi i i

r or r r

t ot t t

E E i k r t

E E i k r t

E E

Incident beam

Reflected beam

Transmitted beam

At the boundary between the two media the x y plane

i k r t

all waves must exist simultaneouslya

ω

ω

ω

⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤− ⎦

−

= ⋅⎣

rr r r

rr r r

rr r r

( ) ( ) ( )

,

exp exp exp

oi i i or r r ot t t

i r t

nd .Therefore, for all t and for all r on the inter

the tangential component must be equal on both sides of the interface

n E n E n E

face

n E i k r t n E i k r t n E i k r tω ω ω⎡ ⎤ ⎡ ⎤ ⎡× ⋅ − + × ⋅ − = × ⋅ −⎣ ⎦ ⎣

×

⎦

× + = ×

r

r r rr r rr r r) )

)

)

r r r) )

( ) ( ) ( ) : Phase matching at the boundary!

,:

i i r r t t

the only way that this can be true over the entire interfAssuming that the wave amplitudes are cons

k r

ace and for al

t k r t k r

tl t is i

t

antf

ω ω ω⇒ ⋅ − = ⋅ − = ⋅ −

⎤⎣ ⎦

r r rr r r


( ) ( ) ( )

(Frequency does not change at the boundar

0,

y!)

(Phases on the boundary does not chan

0

)

,

ge!

i r t

i t

t

r

i r

i i r r t t

At r this results int t t

At t this results in

k r k r k

S

r

k r t k r t k r t

ω ω ωω ω ω

ω ω ω

== =

⇒

=

= =

⋅ ⋅ = ⋅⇒ =

⋅ − = ⋅ − = ⋅ −

r r rr

r

r r

r r rr r r

( ) ( ) ( )

,

0

.

,

i r i t t r

i i

i r t

ubtracting any pair of these factors results in

k k r k k r k k r

This equation k r constant is the equation for a plane perpendicular to k rConsequently the above relation implies t

k k a

a

nd k

h t

ar

− ⋅ = − ⋅ = − ⋅ =

⋅ = ⋅

r r

r r r r r rr r

r

r

r

rr r

. e coplanar in the plane of incidence


co

0,

,

sin si

ns

n

,

sin

ta

s

n

n

t

i

i r i i r r

i

ii r

t

r i ri

r

At t

Considering the relation for the incident and reflected beams

k r k r k r k r

Since the incident and reflected beams are in t

k r k r

he same mediumnk k

k r

c

θ θ

ω θ θ θ θ

=

⋅ = ⋅ ⇒ =

= = ⇒ = ⇒

⋅ = ⋅ =

=

⋅ =

r rr r

r r rr r r

: law of reflection

sin sin : law of

,

sin s n

re

i

,

i t i i t t

i ti ii tt t

Considering the relation for the incident and transmitted beams

k r k r k r k r

But the incident and transmitted

n

beams are in different median nk nk

c c

θ

θ

θ

ω ω θ =

⋅ = ⋅ ⇒ =

= = ⇒

r rr r

fraction

Development of the Fresnel EquationsDevelopment of the Fresnel Equations

cos co

' ,

s co

:

si r t

i i r r t t

E E EB B B

From Maxwell s EM field theorywe have the boundary conditions at the interface

Th tangential

components of both E and B are equal onboth sides o

e above co

f the i

nditions imply that th

for the T

e

E case

θ θ θ+ =

− =

r r

0

cos cos

.,

c

:

os

.

i i r r t t

i t

i r t

We have alsoassumed that as is true formost dielectric materia

nterface

E E EB B B

For the TM mode

lsμ μ μ

θ θ θ− + = −+ =

≅ ≅

TE-case

TM-case


1

1 1

1

2

2

1

:

cos cos

:

cos cos c

v

s

c

o

os

i i r

i r t

i

r t t

i

i r r t t

cRecall that E B Bn

Let n refractive index of incident mediumn refractive index of refracting me

For the TM m

diu

ode

E

For the TE mode

E E En E n E

E En

n

E n

n

c

m

E

EB

θ θ θ

θ θ θ

=⎛ ⎞= = ⇒⎜

−

⎟⎝ ⎠

==

=

=

=

−

−

+

+

+

2r tE n E=

TE-case

TM-case

n1

n2

n1

n2


2

1

cos cos:cos cos

cos cos:co

:

sin sin

cos 1

s cos

i tr

i i t

i tr

i

t

i

t

t

i

t

Eliminating E

nnn

n

n n

nETE case rE n

nETM case rE

from each set of equationsand solving for the reflection coefficient we obtain

where

We know that

n

θ θθ θ

θ θθ θ

θ θ

θ

−= =

+

−=

=

=

=

+

=

−2

2 2 22

sinsin 1 sinit in nnθθ θ= − = −

TE-case

TM-case

n1

n2

n1

n2

Now we have derived the Fresnel EquationsNow we have derived the Fresnel Equations

TE-case

TM-case

n1

n2

n1

n2

2 2

2 2

2 2 2

2 2 2

cos sin:

cos sin

cos sin:

cos

:

:

s

in

i ir

i i i

i ir

i i i

transmission coefficient t

TE

reflection coefficienSubst

c

ts r

nETE case rE n

n nETM case rE n

as

ituting we obtain the Fresnel equations for

For the

e

n

θ θ

θ θ

θ θ

θ θ

− −= =

+ −

− −= =

+ −

2 2

2 2 2

2cos:cos sin

2 co

:

s:

1: 1

cos sin

t i

i i i

t i

i i i

EtE n

E nTM case tE n

TE t rT nt r

n

M

θ

θ θ

θ

θ θ

=

= =+ −

= =+

+= +

−

1

2

nnn ≡

These mean just the boundary conditions

Power : Reflectance(R) and Transmittance(T)Power : Reflectance(R) and Transmittance(T).

1

,,

:

:

tr

i i

i r t

R and T are the ratios of reflected and transmitThe quantitiesThe ratiosrespectively to

ted powers

PPR TP P

R T

r and t are ratios of electric field amplitudes

From conservation of ener

the incident power

P P P

We can

gy

= =

= += + ⇒

21 0

20 00

:

cos cos

cos cos cos

1

cos

1 c22

i i i r r r t t t

i i r r

i i r r t

i i r r t

i

t

t

t t

iexpress the power in each of the fie

n terms of the product of an irradiance and areaP I A P I A P I A

I

lds

I A I A I A

But n c

I I

I n c

I A I I A

E

A

E

θ θ θ

ε

θ θ θ

ε

=⇒=+

=+

⇒

= = =

+

= 2 21 0 0 2 0 0

2 2 2 20 2 0 0

2 220 0

2 20 0

02 2 2 20 1 0 0 0

1 1os cos cos2 2

cos cos 1cos

cos coscos

co

s

s

co

i

r t t t

i i

r r t t

r t t r t t

i i i i

i

i i

i

n cE n cE

E n E E E

E ER r T n

n R TE

E

n E E

nE

E

θ ε θ ε θ

θ θθ

θ θθ θ

θ⎛ ⎞ ⎛ ⎞

=

= +

⎛ ⎞⇒ = + = + = +⎜ ⎟

⎝ ⎠

= = =⎜ ⎟ ⎜⎝ ⎝

⇒⎠ ⎠

2t⎟ 2

2

coscos*

coscos

*

tnttnT

rrrR

i

t

i

t⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

==

θθ

θθ

20-2. External and Internal Reflection20-2. External and Internal Reflection

1 2

1

2

1

1 22

:

1:

1

Internal reflection

External reflect

n n

nnn

i n

n

on

n

n

n

< = >

⇒ =>

⇒

<

2 2

,

sin i

For internal reflection

n may be an im total inteaginary rnal refnum lecber .tionθ− ⇒

( ) 1( ) :

0 tan p

pp

Note for polarizing angBrewster's angle for the Tle

r wh

M se

nen

ca

θ

θ

θ −=⇒ =

External reflection

r

transmission

t

θi

n=1.50

critical angle : θi = θc

1

2sinnnnc =≡θ

No reflection of TM mode

Derivation of Brewster’s AngleDerivation of Brewster’s Angle

( )

( )

4

2 2 2

2 2 2

2 2 2

4 2 2 2

2 22 22

2 2

2

cos sin0

c

( ) :

cos sin

cos sin 0

1 1 4 1 sin sin1 1 4cos sin2cos 2cos

1 1 4si

os si

n 4si

n

p

p p

p p

p p

p pp

p

p

p

p

p

p

p

Brewster's angle for polarizing angle for the TM case

n n

n n

n

n nr

n n

θ

θ θ

θ θ

θ θ

θ θθ

θ θ

θ θ

θ θ

θ

⇒ = −

− + =

± − −± −⇒ = =

± − +=

− −= =

+ −

( )42

1

2

2

22

2

1 1 2sinn2cos 2cos

sin s

1.5

cin

tos

ancos

0, 56.31

pp

p

pp

p

p

p

p

p

For n

n n n

θθ

θ θ

θθ

θθ

θ

θ −=

± −=

= ⇒ ⇒

= =

=

°

Exist both in external and internal

externalreflectioninternal

reflection

θp θp

θc

R

Total Internal Reflection (TIR) Total Internal Reflection (TIR)

2

1

1 1

total internal refl

sin , (TE and TM) .

, ,

* 1

:

ect

.

i

1

)

on(c

c

nFor internal reflectio

For n for both cases

called

TE ca

r is a complex numberR rr

r

For

n

s

n

R

n

TI

θ θ

θ θ

− == =

⇒⇒ =

= <

=

>

2 2

2 2

2 2 2

2 2 2

cos sin:

cos sin

cos sin:

cos sin

i ir

i i i

i ir

i i i

nEe rE n

n nETM cas

i

i

i

ie r

E n n

θ θ

θ θ

θ θ

θ θ

− −= =

+ −

− −= =

+ −

internalreflection

R

r

TIR

θc

20-3. Phase changes on External Reflection20-3. Phase changes on External Reflection

( ) ( )( )0

0

, ,,

.

:

exp

ex

180 ( ) 00

p

0

xp

0

e

r i

i

i

the phase shift is for r

For

When r is a real number as it always is for external reflectionthen the phase shift is fora

r

nd

E r E

r E i k r t

r E i k r t

r

i

π

π

π

ω

ω

° = <

<

= −

⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤= ⋅

°

⎣ ⎦

>

+−

r r

r r

externalreflection

TE TMπ π

r

수직입사인경우TE 경우만π-phase 변화,즉, 위상반전이일어나는가?

(Ex) Why two r’s are opposite in their signs at θi 0 ?(Ex) Why two r’s are opposite in their signs at θi 0 ?

Transmission coeff.

Reflection coeff.

Brewster angleonly for TM pol.

rTM > 0

rTE < 0

The opposite signs in r’s correspond completelyto the phase shift of π after reflection in both cases.

E3 is opposite to E1

E3 is opposite to E1

r

t

TE

TM

Phase Shifts for Internal ReflectionPhase Shifts for Internal Reflection

2

. :

0 0 18

cos sin sit

( )

ancos si

0

n

0

ii i

c

c

i

TIR case r is comWhen then r is a real number and the phase shift will be andWhen then and for both the TE and TM cases has the form

a ib i er e ea

p

ib

for r for rx

e

le

i

αα φ

α

θθ θ

α α αα α

θ

−− −

+

− −= = = = = ⇒

< ° > <

=+ +

≥°

2 2

2 2

2 2

2

ncos

cos sin:

cos si

( ).

n

cos sin

tan tan

i ir

i i i

i i

is the phase shift for total internal

ba

i nETE case rE i n

reflection TI

a b n

R

αα

θ θ

θ θ

θ

α

φ α

φ

θ

=

− −= =

⇒

=

+ −

= = −

=

2 21

2 21

2

2 2sin2 cos

sin

2 tancos

sin2 tan

c

:

os

iTE

iTM

i n

A similar analysis for the TM case gives

n

nn

θ

θφ

θ

θ

θ

φθ

φ

−

−

⎛ ⎞−⎜ ⎟=⎜ ⎟⎝ ⎠

⎛ ⎞−

−

⎜ ⎟=⎜ ⎟⎝

⎛ =⎟⎝ ⎠

⎠

⎞⎜ r

Internal reflection

TIR


π π

TM TE

2 2 2 21 1

2

sin sin2 tan 2 tan

cos cosi i

TM TE

n nn

θ θφ φ

θ θ− −⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


'p

'p c

2 21

c2

c

2 21

180 ( ) <0 <

sin2 tan <

cos

0 <

sin2 tan

cos

TM

i

TE i

nn

n

π θ θφ θ θ θ

θθ θ

θ

θ θ

φ θθ

−

−

⎧⎪⎪⎪⎪= θ θ

⎧⎪⎪ ⎞⎨ ⎟⎪ ⎟⎪ ⎠⎩

c

c

0 <:

0 >TM TEθ θ

φ φθ θ

⎧=− ⎨

>⎩

o

o

45 53 1.5TM TE iNote near when nφ φ θ− = = =o o

Phase Shifts for Internal Reflection:The Fresnel Rhomb

Phase Shifts for Internal Reflection:The Fresnel Rhomb

45 53 1.5TM TE iNote at when nφ φ θ− = = =o o

Linearly polarized light (45o)

CircularlyPolarized

light

20-5. Evanescent Waves at an Interface20-5. Evanescent Waves at an Interface

( )( )( )

( )

: exp

: exp

: exp

exp

si

:

n

i oi i i

r or r r

t ot t t

t ot t t

t t t

Incident beam E E i k r t

Reflected beam E E i k r t

Transmitted beam E E

For the transmitted

i k

bea

r t

E E i k r

k

m

t

r k

ω

ω

ω

ω

θ

⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤= ⋅ −⎣ ⎦

⎡ ⎤= ⋅ −⎣ ⎦

⋅ =

rr r r

rr r r

rr r r

r r

r r )( ) ( )( )

2

22

cos

sin cos

sin, cos 1 sin 1

, :

sin

sinc

( )

o s 1it

t t

t t t

it t

iWhen n

x k y x x yy

k x y

Butn

th

i a purely imaginary numbe

total internal reflection

r

en

n

θ

θ

θθ

θ

θθ

θ

θ

+ ⋅ +

= +

=

= −

>

−

⇒

− =

) ) )

Evanescent Waves at an InterfaceEvanescent Waves at an Interface( )

2

2 2

sin

sin 1

,:

sin

:

sin sin21 1

:

tt t

it

i

i

t

t

iwith an TIR condition n

i y

For the transmitted beamwe can write the phase factor as

k r k xn

Defining the coefficient

kn n

We can write the transmitted wa s

n

v

E

e a

θ

α

θ θπαλ

θ

θ⎛ ⎞⋅ = +⎜ ⎟

⎜ ⎟⎝ ⎠

=

−

− −

=

=

>

r r

( )0sin ep

.

xx pe t tt y

amplitude will decay rapidThe evanescent waveas it penetrates into the lower refractive ind

ly

k xE i t

ex med

n

ium

θ ω α⎡ ⎤⎛ ⎞− −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Note that the incident and reflection wavesform a standing wave in x direction

n2

n1

n1 > n2 h

( )0sinexp expt tt t

k xE E i t yn

θ ω α⎡ ⎤⎛ ⎞= − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

x

y

2

2

1 1sin2 1

t ot

i

E Ee

h

n

λα θπ

⎛ ⎞= ⇒ = =⎟⎝ ⎠

−⎜Penetration depth:

Frustrated TIRFrustrated TIR

Tp = fraction of intensity transmitted across gap

Zhu et al., “Variable Transmission Output Coupler and Tuner for Ring Laser Systems,”Appl. Opt. 24, 3611-3614 (1985).

d

n1=n2=1.5171.65

d/λ

Frustrated Total Internal ReflectanceFrustrated Total Internal Reflectance

Zhu et al., “Variable Transmission Output Coupler and Tuner for Ring Laser Systems,”Appl. Opt. 24, 3611-3614 (1985).

Pellin-Broca prism

20-6. Complex Refractive Index20-6. Complex Refractive Index

0

2 2 2

0

( )

1 2

1

I I

R I

R R

For a material with conductivity

n i n n i n

n i n

n

n i σε ω

σ

σε ω

⎛ ⎞= + = +

⎛ ⎞= + = − +⎜ ⎟

⎝

⎝ ⎠

⎠

⎜ ⎟

%

%

2 2

0 02

2

02

22 4 2

0 0

2

02

1 1 42

2

:

1 22

1 02 2

:

1 1 42

2 I

R I R I RI

I I II

I

Solving for the real and imaginary components we obtain

n n n n nn

n n nn

From the quadratic solution we obta n

n

i

n

σ σε ω ε ω

σ σε ω ε ω

σε ω

σε ω

− = = ⇒ =

⎛ ⎞ ⎛ ⎞⇒

⎛ ⎞+ +

− = ⇒ − − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞± + ⎜ ⎟

⎝⎜⎝=⇒

⎟⎠⎠=

.IWe need to take the positive root because n is a real number

Complex Refractive IndexComplex Refractive Index

2

0

0 0 00

, ,

2 2

,

2 2I I R I

With this approximation the complex refractive index for m

For most metals with light in the microwave region

n n

etals becomes

n

Substituting our expression for the complex refractive index

nσε ω

σ εω

σ σ σε ω ε ω ε ω

>>

= = =≅ ⇒

( ) ( ) ( )

( ) ( )

0 0

0

êx

êx êxp

p

.

exp

p R k

k

I

R I

k

back intoour expression for the electric field we obtain

E E i k r t E i

ni u r tc

n i n u r tc

The first exponential term is oscillatory The

n uc

E

E r

ω

ω ω

ω ω⎧ ⎫⎡ ⎤⎡ ⎤= ⋅ − = + ⋅ −⎨ ⎬⎢ ⎥

⎧ ⎫⎡ ⎤⋅ −⎨ ⎬⎢ ⎥⎣⎡ ⎤− ⋅⎢ ⎥⎣ ⎦

⎣ ⎦ ⎣ ⎦⎩

⎩

⎭

=⎦⎭

rr r rr r

r r r

/ .RThe second exponent

M wave propagates with aial has a real argument

velocity of(abso

n crbed).

Complex Refractive IndexComplex Refractive Index

( )* *0 0

0

..

ˆ2exp

ˆ2exp

I k

I k

The second term leads to absorption of the beam in metals due to inducinga current in the medium This causes the irradiance to decrease as the wavepropagates through the medium

n u rI EE E E

c

n uI I

ω

ω

⋅⎡ ⎤≡ = −⎢ ⎥

⎣ ⎦

= −

rr r r r

( ) ( )0 êxp

2 4:

k

I I

rI u r

c

The absorption coefficient is defined n ncω πα

α

λ

⋅⎡ ⎤= − ⋅⎡

= =

⎤⎢ ⎥ ⎣ ⎦⎣ ⎦

rr

( ) ( )0 êx êxpp Ik kRn uni u r

crt

cE E ω ω⎧ ⎫⎡ ⎤⋅ −⎨ ⎬⎢ ⎥

⎡ ⎤− ⋅=⎣ ⎦⎩ ⎢ ⎥⎣ ⎦⎭

rr r r

20-7. Reflection from Metals20-7. Reflection from Metals

2 2

2 2

2 2 2

2 2 2

cos sin:

cos sin

cos sin:

cos sin

:

i ir

i i i

i ir

i i i

Reflection from metals is analyzed

ETE case rE

nETM cas

by substituting the complex refractive index n in the Fresnel equation

e rE n

Sub

n

n

n

n

sti u

s

t t

θ θ

θ θ

θ θ

θ θ

− −= =

+ −

− −= =

+ −

% %

%

%

%

%

%

( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

2 2 2

2 2 2

2 2 2 2 2

2 2 2 2 2

:

cos sin 2:

cos sin 2

2 cos sin 2:

2 cos sin 2

i R I i R Ir

i i R I i R I

R I R I i R I i R Ir

i R I R I i R I i R I

R Iing we obtain

n n i n nErE n n i n n

n n i n n n n i n nErE n n i n n

TE case

T

n n

n n iM case

n n

i n

θ θ

θ θ

θ θ

θ θ

− − − += =

+ − − +

⎡ ⎤− + − − − +⎣ ⎦= =⎡ ⎤− + + − − +

= +

⎣ ⎦

%

Reflection from Metals at normal incidence (θi=0)Reflection from Metals at normal incidence (θi=0)

( ) ( )( ) ( )

( ) ( )( ) ( )

( )( )

( )( )

2 2 2

2 2 2

22 2

22 2

, 0 :

cos sin 2

cos sin 2

1 2 1

1 2 1

:

&

1

1

i

i R I i R Ir

i i R

R I

R I

I i R I

R I R I R I

R I R I R I

At normal incidence

n n i n nErE n n

TE TM cases

n i

i n n

n n i n n n i n

n n i n n n i n

power ref

nr

n i

T ehe

n

l c

θ

θ θ

θ θ

= °

− − − += =

+ − − +

− − + − −= =

+ − +

=−

+ −

− −+

∴

( )( )

( )( )

( )( )

2 2

2 2*

2 2

2 2

1 1 1 21 1 1 2

11

R I

R

R I R I R R I

R I

I

R I R R I

is given by

n i n n i n n n nR r rn i n n i n

tance R

n nR

n n

n n n⎡ ⎤ ⎡

− +=

⎤− − − + ⎛ ⎞− + += = =⎢ ⎥ ⎢ ⎥ ⎜ ⎟+ − + + + + +⎝ ⎠⎣ ⎦ ⎣

+

⎦

+

Reflection from MetalsReflection from Metals

At normal incidence(from Hecht, page 113)

Reflectance

θiλ

visible

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20. Fresnel equations 20. Fresnel equations - Hanyangoptics.hanyang.ac.kr/~shsong/20-Fresnel equations.pdf · 20. Fresnel equations 20. Fresnel equations • EM Waves at an Interface