fresnel equations

7/23/2019 Fresnel Equations

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23. Fresnel Equations

• EM Waves at boundaries

• Fresnel Equations:

Reflection and Transmission Coefficients

• Brewster’s Angle

• Total Internal Reflection (TIR)

• 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

TE

rTM

E n r

E n

E n n r

r reflection coef

E n n

ficient

θ θ

θ θ

θ θ

θ θ

= =

= =

2 2

2 2 2

2cos

cos sin

2 cos

cos s

:

in

tTE

tTM

t transmission coefficie

E t

E n

E n t

E

n

n

t

n

θ

θ θ

θ

θ θ

= =

= =

n

θ

E

E

t

E

r

θ

r

θ

t

n

We will derive the Fresnel equations

1

2

n

n

n

nn

incident

d transmitte =≡



EM Waves at an Interface

( )

( )

( )

: exp

: exp

: exp

oi

or

i i i

r r

t t

r

t ot

k

k

k

Incident beam E i r t

Reflected beam E i r t

Tran

E

E

smitted beam E i t E r

ω

ω

ω

⎡ ⎤= ⋅ −⎣ ⎦

⎡ ⎤= ⋅ −⎣ ⎦

⎡ ⎤= ⋅ −⎣ ⎦

rr r r

r r

r

r

r

r

r

r

1 0

1 0

2 0

i

r

t

k n k

k n k

k n k

=

=

=

r

r

r

r

TE mode

n1

n2

iθ

n1

n2

TM mode

Note the definition of the positive E-field directions in both cases.

n

n

i E r

r E r

t E r

ik r

r k r

t k r

oi E r

or E

r

ot E r




( ), ,

points

,

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

At the boundary between the two media the x y plane all waves must exist simultaneously

and .

Therefore for all time t and fo

f

all

ac

boundary r on th te

e

r e in

−

r,rface

( )

( )

( )

exp

exp

exp

:

:

:

i oi i

r or

i

r

t ot t

r

t

E E i r t

E E i r

Incident beam

Reflected beam

Transmitted bea

k

k

k m

t

E E i r t

ω

ω

ω

⎡ ⎤= ⋅ −⎣ ⎦

⎡ ⎤= ⋅ −⎣ ⎦

⎡ ⎤= ⋅ −⎣ ⎦

r r r

r r r

r 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 interfac

k r t k r t k r t

e and for all t is i

Assuming that the wave amplitudes are constan

f

t

ω ω ω ⇒ ⋅ − = ⋅ − = ⋅ −r r rr r r

( ) ( ) ( )exp exp expoi i i or r r ot t

i r t

t n E i k r t n E i k r t n E i k

n E n E n

r t

E

ω ω ω ⎡ ⎤ ⎡ ⎤ ⎡ ⎤× ⋅ − + ×

× +

⋅ − = × ⋅ −⎣ ⎦ ⎣ ⎦ ⎣ ⎦

× = ×r r rr r r

r r r ) )

r r r ) ) )

)n

n

i E r

r E r

t E r

ik

r

r k

r

t k r

oi E r

or E r

ot E r

n̂

r r




Normal

x

ik r

r k r

t k r

1 0n k

2 0n k

r r

1 0

1 0

2 0

i

r

t

k n k

k n k

k n k

=

=

=

r

r

r

( ) ( ) ( )Phase matching condition:

i i r r t t k r t k r t k r t ω ω ω ⋅ − = ⋅ − = ⋅ −r r rr r r

(Frequency does not change at the boundary!)

0,

t

i t

i r

r

At r this results in

t t t ω ω ω

ω ω ω

=

= =

⇒ = =

r

, ,

, ,

,

and .

, .i r

i r t

i r

t

t

k r constant

the equation for a plane perpendicular to k

k k and k are coplanar in the plane of incidence

r ⇒ ⋅

⇒

=→

r r

r

r

r

r r

(Phases on the boundary does not change

,

!)

0

i r t

At t this

k r

results i

k k

n

r r ⋅ ==

=⇒ ⋅ ⋅r r rr r r

n

n

i E r

r E r

t E

r

ik r

r k r

t k

r

r r




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 r i

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 medium

nk k

k r

c

θ θ

ω θ θ θ θ

=

⋅ = ⋅ ⇒ =

= = ⇒ = ⇒

⋅ = ⋅ =

=

⋅ =

r rr r

r r rr r r

: law of reflection

,

sin sin

sin sin : law of refractio

,

n

i t i

i i t

i t t

i t t t i

Considering the relation for the incident and transmitted beams

k r k r k r k r

But the incident and transmitted beams are in different media

n nk k

c c

n n

θ θ

ω θ

ω θ =

⋅ = ⋅ ⇒ =

= = ⇒

r rr r

iθ

r θ

t θ

Normal

x

ik r

r k r

t k r

1 0n k

2 0n k

r r

n

θ

ι

θ

r

θ

t

n

x

i E r

r E

r

t E

r

ik r

r k r

t k r



Development of the Fresnel Equations

cos co

' ,

s co

:

s

i r t

i i r r t t

E E E

B B B

From Maxwell s EM field theory

we have the boundary conditions at the interface

Th tangential

components of both E and B are equal on

both 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 also

assumed that as is true for

most dielectric materia

nterface

E E E

B

For the TM mod

B B

e

ls

θ θ θ

μ μ μ

+ =

− + = −

≅ ≅

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

c Recall that E B B

n

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

For the TM m

diu

For the TE mod

nE B

c

ode

E E

e

E E E

n E n E E

n

m

n

E

n E E

θ θ

θ θ θ

θ

=⎛ ⎞

= = ⇒⎜ ⎟⎝ ⎠

==

−

+

+

=

+

−

=

=

2r t n E = −

TE-case

TM-case

n1

n2

n1

n2




2

1

sin sin

cos cos:cos cos

cos cos:

cos co

:

s

cos

i t r T

t

i t

E

i i t

i t r TM

i i t

from each set of e

n E TE case r E n

n E TM ca

quations

and solving for the refle

Eliminating

ction coefficient we obtain

wher

E

nne

We know t

se r E

n

a

n

t

n

n

h

θ

θ θ

θ

θ

θ θ

θ θ

θ

θ −= =+

− += =

=

=

+

22 2 2

2

sin1 sin 1 sini

t t in n n

n

θ θ θ = − = − = −

TE-case

TM-case

n1

n2

n1

n2



TE-case

TM-case

n1

n2

n1

n2

Now we have derived the Fresnel Equations

2 2

2 2

2 2 2

2 2 2

cos sin

cos si

:

:

:

:

n

cos sin

cos sin

i ir TE

i i i

i ir TM

i i i

Substituting we obtain t reflection coefficiehe Fresnel equa nts r

n E r

E n

n n E r E n

transmission c

tions for

TE case

TM case

For the

T

oeffici n t

n

e t

θ θ

θ θ

θ θ

θ θ

− −= =

+ −

− + −= =+ −

2 2

2 2 2

2coscos sin

2 cos

: 1

:

cos sin

1

:

:

TE TE

TM T

t iTE

i i i

t iTM

i

M

i i

E case

TM ca

E t E

TE t r

TM n

n

E nt

E n

t

nse

r

θ θ θ

θ

θ θ

= =+ −

= =

+

= −

+ −

=

1

2n

nn ≡

These just mean the boundary conditions.

:

:

i r t

i r t

For the TE case E E E

For the TM mode B B B

+ =

− + = −



Power : Reflectance (R) and Transmittance (T)

.

1

,

,

:

:

t r

i i

i r t

R and T are the ratios of reflected and transmit

The quantities

The ratios

respectively to

ted powers

PP R T

P 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

= =

= += + ⇒

2

1 0

2

0 00

:

cos cos

cos cos cos

1

cos

1c

22

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

i

express the power in each of the fie

n terms of the product of an irradiance and area

P 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 2

1 0 0 2 0 0

2 2 2 2

0 2 0 0

2 22

0 02 2

0 0

0

2 2 2 2

0 1 0 0 0

1 1os cos cos

2 2

cos cos 1

cos

cos cos

cos

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 E R r T n

n R T E

E

n E E

n E

E

θ ε θ ε θ

θ θ

θ

θ θ

θ θ

θ

⎛ ⎞ ⎛ ⎞

=

= +

⎛ ⎞⇒ = + = + = +⎜ ⎟

⎝ ⎠

= = =⎜ ⎟ ⎜⎝ ⎝ ⇒ ⎠ ⎠

2

t ⎟

2

2

cos

cos*cos

cos

*

t ntt nT

r rr R

i

t

i

t ⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

=⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

=

==

θ

θ

θ

θ

2

2

coscos _

cos cos

out out out out out

in in in in in

n E I Power ratio

I n E

θ θ

θ θ

⎛ ⎞= = ⎜ ⎟

⎜ ⎟⎝ ⎠

A



23-2. External and Internal Reflection

( )

2 1

2 2

2 1

,

/ 1 sin 0 are always real

TE TM

n n n nr

n n

θ

⇒

⇒

>

= > ⇒ − ≥

⇒

rTE

t

n=1.50

Brewster’s angle (or, polarizing angle)(No reflection of TM mode)( ) 1

:

tan0TM p p

f No or tht e TM case

n

e

r whenθ θ θ −=⇒ = =

External Reflection

, 0TE TM r >

, 0TE TM r <

, 0TE TM

t >2 2

2

2 2

2

2

2 2 2

cos sin

c

cos sin

co os sins sin

i ii

TM

i

i

TE

i ii

n nr

n

nr

nn

θ θ

θ

θ θ

θ θ θ

− + −=

+ −

− −=

+ −

, If 0 then there are no phase changes after reflection.TE TM r ⇒ >

,

, , ,

If 0 then there are always ( 180 ) phase changes.

TE TM

TE TM TE TE M

i

TM T

r

er r r π

π ⇒ < =→ = =−

o

[ ]2 1/ 1n n n= >

rTM



( )2 2

,

,

, ,

, If sin 0, =1,

=1

(- ~ ) phase change may occur after reflect

ion

!TE T TE TM

TE TM

i i

TE TM TE

M

TM

BUT r are compn r

r

r r e e

lex

φ φ

θ

φ π π

⇒ − <

→

→ = =

→ +

( )1tan

: 0

p

TM

No Brewster's angle n

for the TM

e

case

t

r

θ −=

=

critical angle

Internal Reflection

2 2

2

2 2

2

2

2 2 2

cos sin

c

cos sin

co os sins sin

i ii

TM

i

i

TE

i ii

n nr

n

nr

nn

θ θ

θ

θ θ

θ θ θ

− + −=

+ −

− −=

+ −

, 0TE TM

r >

, 0TE TM

r <

Total internal reflection (TIR) when θ > θ c

TIR region

( ) ( )21

2 2

2 1

2 2

/ 1

sin 0, , sin 0

n n n

n o

n n

r nθ θ

⇒ = <

⇒ − > − <

>

( )2 2

,

2 1

If sin 0, =1

sin ( / )

TE TM

c

n r

n n n

θ

θ

⇒ − =

→ = =

( )2 2

,

,

,

If sin 0, are always real

If 0 then there are no phase changes after reflection.

If 0 then there are ( 180 ) phase changes.

TE TM

TE TM

TE TM

n r

r

r

θ

π

⇒ − >

→ >

→ < = o

[ ]2 1/ 1n n n= <



Derivation of Brewster’s Angle

( )

4 2 2 2

2 2 2

4 2 2 2

1

2 2 2

2 2 2 2

c

( ) :

cos sin

cos s

os sin0

cos sin

1.50, 56.

in

( 1) os si

tan

n 0

31

p

p p

p p

TM p

p p

p

p

p p

p p

Brewster's angle for polarizing ang

n nr

n n

For n

le

n n

n n

n n c

n

θ

θ θ

θ θ

θ θ θ

θ

θ

θ

θ

θ

θ

−

⇒ = −

− +

− + −= =

⎡ ⎤= − − =⎣

+

= =

=

⎦

−

⇒

°

externalreflectioninternal

reflection

θ p θ p

θ c

R

External & Internal reflections, but TM-polarization only

1: sin nnc <=θ

or nnn p 11 : tan <>=θ

TE & TM polarizations, but Internal reflection only

Brewster ‘s angle :

Critical angle :

TETM



Total Internal Reflection (TIR)

2

1

1 , ,

and for both (TE and TM) cases.

sin t

: 1

1

otal internal r

* 1

eflection( )

cos

r TE

i

c

n Internal reflection n

n

r R r

r is a complex

call

numb

eFor n TI d

e

r

E r E

r

Rθ θ −

⇒

= <

= = =

≥

=

=

⇒

=

2 2

2 2

2 2 2

2 2 2

sincos sin

cos sin

cos sin

i i

i i

i ir TM

i i i

i nn

n n E r

E n

i

i n

i

θ θ θ θ

θ θ

θ θ

− −+ −

− + −= =

+ −

internalreflection

R

r

θ c

Complex value

R = 1



rTEn=1.50

, 0TE TM

r >

, 0TE TM r <

, 0TE TM

t >

23-3. Phase changes on reflection

,

,

,

18

,

,0

) 0

0

.0 (

TE T

TE TM

TE TM

M the phase

r is always a real numb

the

er for ex

phase sh

shift is fo

ift is

ternal reflection

then

an for r

r

d

r

π

° >

° = <

TE TM

Phase shift after External Reflection

For TE case, π phase shift for all incident angles For TM case, π phase shift for θ < θp

No phase shift for θ > θp

External Reflection External Reflection

External reflection

rTM



cθ θ >:

Complex value

In TIR region

Phase shift after Internal Reflection Internal reflection

1

0 for sin is complex in TIR region where

TE TE

TE c

TE c

i i

TE TE

r nr

r r e eφ φ

θ θ θ θ

−

⇒ > < =⇒ >

→ = =

For TM case, no phase shift for θ < θp

π phase shift for θp < θ < θc

TM(θ) phase shift for θ > θc

For TE case, no phase shift for θ < θ

c

TE(θ) phase shift for θ > θc

TIR TIR

1

0 for tan 0 for

TM

TM p

TM p c

i

TM TM TM TM

r nr

r r e r φ

θ θ θ θ θ

φ π

−

⇒ > < =⇒ < < <

→ = − = → =

is complex in TIR region where

TM TM

TM c

i i

TM TM

r

r r e eφ φ

θ θ ⇒ >

→ = =



2

:

cos sin sintan

cos sin2

( )

s

co

ii i

i

c TI When then and for both the TE and TM cases has the form

a ib i e br e e

a ib i

R case r is complex

e a

α α φ

α

α α α φ α α

α

θ θ

α α

−−

+

− −= = = =

≥

== = −= ⇒+ +

2 2

2 2

2 2

2

1

2 2

2

cos sin:

cos sin

cos sin

sintan t

( ).

sin2tan

co

an2 cos

s

i ir TE

i i i

i i

iT

i

E

TE

i

is the phase shift on total internal reflection TIR

n

i n E TE case r

E i n

a b n

n

A simi

φ

θ θ

θ θ

θ θ

θ φ

θ

θ φ α

θ

−

− −= =

+ −

= = −−⎛ ⎞

⎛ ⎞−⎜ ⎟= −⎜ ⎟

⇒ = − =⎜⎝ ⎠

⎝ ⎠

⎟

2 2

1

2

sin2tan

cos

:

i

TM

i

lar analysis for the TM case giv

n

es

nθ φ π

θ

− ⎛ ⎞−⎜ ⎟= −⎜ ⎟

⎝ ⎠

Internal reflection

: i cθ θ >

: i cθ θ >

TIR(Complex r )

Phase shifts on total Internal Reflection for both TE- and TM-cases



2 2

1

2

2 2

1

sin2tan

cos

sin2tan

cos

i

TM

i

TE

n

n

n

θ φ π

θ

θ φ

θ

−

−

⎛ ⎞−⎜ ⎟= −⎜ ⎟

⎝ ⎠⎛ ⎞−⎜ ⎟= −⎜ ⎟⎝ ⎠

Internal reflection

Complex value

Therefore, after TIR is ………..,TE TM r

,TE TM r

,TE TM φ

2 2

2 2

2 2 2

2 2 2

cos sin

cos sin

cos sin

cos sin

i ir TE

i i i

i ir TM

i i i

n E r

E n

n n

i

i

in n

i

E r

E

θ θ

θ θ

θ θ

θ θ

− −= =+ −

− + −= =

+ −

( )cincident θ θ >caseTIR For



Internal reflectionTIR(Complex r )

Summary of Phase Shifts on Internal Reflection

'

p

'

p c

2 2

1c2

0 <

( 180 ) <

sin2 tan <cos

TM

i nn

θ θ

φ π θ θ θ

θ π θ θ θ

−

⎧⎪⎪⎪⎪

= = <⎨⎪

⎛ ⎞−⎪ ⎜ ⎟−⎪ ⎜ ⎟

⎪ ⎝ ⎠⎩

o

o

p

p c

c

0 <

<

0

TM TE

θ θ

φ φ φ π θ θ θ

θ θ

⎧=⎪Δ = − = <⎨⎪

> <⎩

o

o

c

2 2

1c

0 <

sin2 tan >cos

TE i n

θ θ

φ θ θ θ θ

−

⎧⎪⎪ ⎛ ⎞=

−⎨ ⎜ ⎟−⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

o

φ Δ

TM φ

TE φ



Fresnel Rhomb

Linearly polarized light (45o)

Circularly

Polarizedlight

3 53 1.5

4

After two consequentive TIRs,

3

2

2

TM TE i

TM TE

TM TE

Note near when n

Quarter wave retarder

π φ φ θ

π

φ φ π

φ φ φ

− = = =

→

→ − =

→ Δ = − =

→ −

o

φ ΔTM φ

TE φ



Linearlypolarized light

(45o)

CircularlyPolarized

light

69 ???2

2

TM TE i

TM TE

Note near when n

Quarter wave retarder

π φ φ θ

π φ φ φ

− = = =

→ Δ = − =

→ −

oφ ΔTM φ

TE φ

Quarter-wave retardation after TIR

n



23-5. 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

( )

os 1it

t t

t t t

it t

iWhen n

x k z x x zz

k x z

But n

th

i a purely imaginary numbe

total internal reflection

r

en

n

θ

θ

θ θ

θ

θ θ

θ

θ

+ ⋅ +

= +

=

= −

− −

⇒

=

>

) ) )



Evanescent Waves at an Interface

( )

2

2 2

sin

sin1

,

:

sin

:

sin sin21 1

:

t t t

it

i

i

t

t

iwith an TIR condition n

i z

For the transmitted beam

we can write the phase factor as

k r k x

n

Defining the coefficient

k n n

We can write the transmitted wa s

n

v

E

e a

θ

α

θ θ π α

λ

θ

θ ⎛ ⎞⋅ = +⎜ ⎟

⎜ ⎟⎝ ⎠

=

−

− −

=

=

>

r r

( )0

sine p

.

xx pe t t t z

amplitude will decay rapid The evanescent wave

as it penetrates into the lower refractive ind

ly

k x E i t

ex med

n

ium

θ ω α

⎡ ⎤⎛ ⎞− −⎜ ⎟⎢ ⎥

⎝ ⎠⎣ ⎦

Note that the incident and reflection waves

form a standing wave in x direction

n2

n1

n1 > n2 h

( )0

sinexp expt t

t t

k x E E i t z

n

θ ω α

⎡ ⎤⎛ ⎞= − −⎜ ⎟⎢ ⎥

⎝ ⎠⎣ ⎦

x

z

2

2

1

1

sin2 1

t ot

i

E E

e

h

n

λ

α θ π

⎛ ⎞= ⇒ = =⎟

⎝ ⎠ −

⎜Penetration depth:



Frustrated TIR

Tp = fraction of intensitytransmitted across gap

Zhu et al., “Variable Transmission OutputCoupler and Tuner for Ring Laser Systems,” Appl. Opt. 24, 3610-3614 (1985).

d

n1=n2=1.517

1.65

d/λ



Frustrated Total Internal Reflectance

Zhu et al., “Variable Transmission OutputCoupler and Tuner for Ring Laser Systems,” Appl. Opt. 24, 3610-3614 (1985).

Pellin-Broca prism

d = 1 ~ λ: changing the reflectanceRotation: changing the wavelength resonant at θB

d



23-6. Complex Refractive Index

2 2 2

0

0

( ) :

1

1

2 R I R I

R I For a material with conductivity

n i n n i n

n i n i n

n

σ

ε ω σ

σ

ε ω

⎛

⎛ ⎞

= + = −

⎞= + = +⎜ ⎟

⎝ ⎠

+⎜ ⎟⎝ ⎠%

%

2

0

2

0

2 2

2 4 2

0 0

2

0

2

022

:

1 2

1 02 2

:

1 1

2

1 1 42

42

2 2

R

I

I

R I R I

I I I

I

I

Solving for the real and imaginary components we obtain

n n n n

n n nn

From the quadratic solution we obt

n

n

n

ain

n

σ

ε ω

σ σ

ε ω ε ω

σ

ε

σ

ε

σ

ω

ω

ε ω

− = = ⇒

⎛ ⎞ ⎛ ⎞⇒ − = ⇒ − − =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞± +

=

⎛ ⎞+ + ⎜ ⎟

⎝ ⎜⎝

=⇒

⎟ ⎠

=

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



Complex Refractive Index

( )

( ) ( )

( )

0

0

0

exp

ˆexp

eˆ xexp p Rk

R k

I

I

Substituting our expression for the complex refractive index back into

our expression for the electric field we obtain

E E i k r t

E i n i n u r t c

E n

i u r t c

n

cω

ω

ω

ω

ω

⎡ ⎤= ⋅ −⎣

⎧ ⎫⎡ ⎤⋅ − −

⎦

⎧ ⎫⎡ ⎤= + ⋅ −⎨ ⎬⎢ ⎥

⎨ ⎬⎢ ⎥⎣ ⎦

⎣ ⎦⎭

⎩

⎩

=⎭

rr

r

r r

r r

r

( )

.

.

ˆ

/

k

R

The first exponential term is oscillatory

The EM wave propagates w

u r

The second exponential has a r

ith a ve

eal arg

locity of

ument (absor .

n c

bed)

⎡ ⎤⋅⎢ ⎥⎣ ⎦

r



Complex Refractive Index

( )* *

0 0

0

.

.

ˆ2exp

ˆ2exp

I k

I k

The second term leads to absorption of the beam in metals due to inducing

a current in the medium This causes the irradiance to decrease as the wave

propagates through the medium

n u r I EE E E c

n u I I

ω

ω

⋅⎡ ⎤≡ = −⎢ ⎥⎣ ⎦

= −

r

r r r r

( )( )0

êxp

2 4:

k

I I

r I u r

c

The absorption coefficient is defined n n

c

ω π α

α

λ

⋅⎡ ⎤= − ⋅⎡

= =

⎤⎢ ⎥ ⎣ ⎦⎣ ⎦

rr

( ) ( )0êx êxp p I

k k R n

un

i u r c

r t c

E E ω ω ⎧ ⎫⎡ ⎤

⋅ −⎨ ⎬⎢ ⎥⎡ ⎤

− ⋅=⎣ ⎦⎩

⎢ ⎥⎣ ⎦⎭

rr r r



23-7. Reflection from Metals

2 2

2 2

2 2 2

2 2 2

cos sin:

cos sin

cos sin:

cos sin

:

i ir TE

i i i

i ir TM

i i i

Reflection from metals is analyzed

E TE case r

E

n E TM cas

by substituting the complex refractive index n i

e r E n

S

n the Fresnel

n

n

equa

ub

ti

n

n

n

o s

s

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 I r

i i R I i R I

R I R I i R I i R I r

i R I R I i R

R I

I i R I

TE case

TM case

tituting we obtain

n n i n n E r

E n n i n n

n n i n n n n i n n E r

E n n i n n n n

n n i

i n n

n

θ θ

θ θ

θ θ

θ θ

− − − += =

+ − − +

⎡ ⎤− − + + − − +⎣ ⎦= =⎡ ⎤− + + − +

=

⎣

+

−⎦

%

Reflectance

θ i



Reflection from Metals at normal incidence (θi=0)

( )

( )

( )( )

( )( )

( )

( )

2 2

2 2

*

2 2

2 2

1 1 1 2

1 1

1

1

1

1

1

2

R I

R I R I R R I

R I R I R R

R I

R I

I

R I

The is given by

R r r

n i n n i n n n n

n i n n i n n n n

power refl

n i n

r n i

ectance R

n

n n

R n n

∴

=⎡ ⎤ ⎡ ⎤− − − + ⎛ ⎞− + +

= =⎢ ⎥ ⎢ ⎥ ⎜ ⎟+ − + + + + +⎝ ⎠

− +

= +

− −

= + −

⎦

+

⎣ ⎣ ⎦

2 2

2 2

2 2 2

2 2 2

cos sin 1

1cos sin

cos sin 1

1cos sin

i i

TE

i i

i i

TM

i i

nr

n

n n

n

n

nr

nn n

θ θ

θ θ

θ θ

θ θ

− − −= =

++ −

− + − −= =

++ −

%

%

% % %

%

%

% %

%

, 0 :i At normal incidence θ = ° At normal incidence(from Hecht, page 113)

visible

fresnel equations

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