cos sin

20. Fresnel Equations

2 2

2 2

2 2 2

2 2 2

:

cos sin

:

cos sin

cos sin

:

cos sin

r

r

r reflection coefficient

E n

TE r

E n

E n n

TM r

E n n

θ θ

θ θ

θ θ

θ θ

− −

= =

+ −

− −

= =

+ −

2 2

2 2 2

:

2 cos :

cos sin

2 cos :

cos sin

t

t

t transmission coefficient TE t E

E n

E n

TM t

E n n

θ

θ θ

θ

θ θ

= =

+ −

= =

+ −

n

E θ

E

E

θ

θ

n

Phase matching at the boundary

: exp{ ( )}

: exp{ ( )}

: exp{ ( )}

o

r or r r

t ot t t

incident light E E i k r t

reflected light E E i k r t transmitted light E E i k r t

ω ω

ω

= −

= −

= −

i i

i

, :

( ) ( ) ( )

int 0,

r r t t

r t

In the boundary the three phases must themselves be equal

k r t k r t k r t

At the boundary po r

all fr

ω ω ω

ω ω ω

− = − = −

=

= = ⇒

i i i

=

.

0 ,

sin sin :

r t

r r r

equencies are equal At t in the boundary plane

k r k r k r

kr θ k r θ θ θ law of reflection

=

• = • = •

⇒ = →

⇒ k r_r sinθ_r = k r_t sinθ_t → n_r sinθ_r=n_t sinθ_t : law of refraction E

E_t

E_r

x z

y

k r

θ

Boundary conditions

cos cos cos

r t

r t t

E E E

B θ B θ B θ

+ =

− + = −

For TE-polarization

E

E_t

E_r

x

B k θ

E

E_t

E_r

x B k

cos cos cos

r t

r t t

B B B

E θ E θ E θ

+ =

− =

For TM-polarization

1 1 2

1 1 2

cos cos cos

cos cos cos

r t

r t t

r t

r t t

E E E c

E vB B

n E n E n E n

n E n E n E

E E E

θ θ θ

θ θ θ

+ =

  = =  

 − =    

 

+ =

 

 − = 

 

TE: since

TM:

2 1

2 2 2

cos cos

/ )

cos cos

cos cos cos cos

sin sin , cos 1 sin sin ,

t r

t r t

t

t t t

E n

r n n n

E n

n r E

E n

n n n n

θ θ

θ θ

θ θ

θ θ

θ θ θ θ θ

= = − ≡

+

= = −

+

= = − = −

TE: (

TM:

Since

Reflection coefficients : r = E _r /E

2 2

2 1

2 2

2 2 2

2 2 2

cos sin

/ )

cos sin

cos sin

cos sin

r

t r

E n

r n n n

E n

E n n

r E n n

θ θ

θ θ

θ θ

θ θ

− −

= = ≡

+ −

− −

= =

+ −

TE: (

TM:

2 2

2 2 2

2 cos

cos sin

2 cos

cos sin

t

t

t E

E n

E n

t E n n

θ

θ θ

θ

θ θ

= =

+ −

= =

+ −

TE:

TM:

Transmission coefficients : t = E _t /E

1 1 t r

nt r

= +

= + TE:

TM:

i r t

r i t i

P P P

R P

P T P

P

= +

=

=

: power conservation

: Reflectance

: Transmittance

Conservation of energy

1= +R T ⇒ I A_i( cos

θ

θ

θ

2

2 2 2

2

2 2

2 cos

cos

1

o

t t t

oi or ot

i i i

t t t i i i

i t o

i i i t t t

I v E

E E v E

v

v v v v

n v

v v v v

ε

ε θ

ε θ

ε µ

µ µ µ

ε µ εµ

 

=  

 

  

= +   

  

= = = = = =

Using the relation of

where, and

Reflectance and Transmittance

2 2 2

2 2

2 2

2

cos cos 1 cos

cos

cos cos

t

oi or ot

i

t i

or

r r

i i oi

t t

i i

E E n E

r n t

E

P I

R r

P I E

T n t P

P θ

θ

θ θ

θ θ

 

= +  

 

 

⇒ = +  

 

   

 

=  = = =   

 

 

 

   

=    = 

   

: Reflectance

: Transmittance

2

1 2

1 2

1 2

1

1

1

n n n n

n

n n n n

n

< = >

> = <

external reflection: or

internal reflection: or

External and internal reflections

n

E θ

E

E

θ

θ

n

See

Fig.20-3 : t, r with π-phase change

Fig.20-4 : Brewster angle (R_TM=0 when θ=tan^-1n) Critical angle (θ=sin^-1n)

Phase changes on internal reflection

2 2

2 2

2 2 2

2 2 2

cos sin

cos sin

cos sin

cos sin

r

r

E i n

r E i n

E n i n

r E n i n

θ θ

θ θ

θ θ

θ θ

− −

= =

+ −

− −

= =

+ −

TE:

TM:

When n < 1 (internal reflection) and

θ θ

(2 ) i

i i

i

r Ae e e r

Ae

α α φ

α

φ

− − −

= + = ≡ ( : the phase of )

2 2

2 2

2

tan sin

2 cos

tan sin

2 cos

n

n n

φ θ

θ

φ θ

θ

  = −

  

  = −

   TE:

TM:

See

Fig.20-6 : phase shift Fig.20-7 : Fresnel rhomb

Phase changes on internal reflection

2 2

1

'

'

2 tan sin

cos

2 tan

c

c

p

p c

n

θ θ

φ θ θ θ

θ

θ θ

φ θ θ θ

−

 < 

 

   

  −  > 

   

   

 

<

<

o

TE

o

o TM

0 ;

= ;

180 ;

= 0 ; <

2 2

1

2

sin

cos ^c

n n

θ θ θ

θ

−

 

 

 

 

 

 

 

 

  −  > 

   

   

 

;

See Fig.20-8

Evanescent waves

:

exp{ (

)}

transmitted wave E = E i k r i − ω t

E

E_t

E_r

x

y

k

r k_t

θ

2 2

2

(sin , cos , 0) ( , , 0)

( sin cos )

cos 1 sin 1 sin

t t t t

t t t

t t

k r k x y

k x y

n

θ θ

θ θ

θ θ θ

=

= +

= − = −

i i

For angles such that

sin θ > n ( TIR : total int ernal reflection )

[ ]

2 2

2 2

2 2

cos sin 1

sin sin

1

exp ( sin / ) exp , sin 1

t

t t t

t ot t t

i n

k r k x ik y

n n

E E i k x n t y k

n θ θ

θ θ

θ ω α α θ

= −

∴ = + −

=   −   − ≡ −

i

Reflection from metals

1 / 2

2 2

2 2

2 2 2

2 2 2

cos sin

1

cos sin

cos sin

cos sin

r

R I

t o

r

E n

r n i n in

E n

E n n

r E

n n

θ θ σ

θ θ ε ω

θ θ

θ θ

  

− −

= = =  +   ≡ +

  

 

+ −

− −

= =

+ −

TE:

TM:

( )

/

, --- ( )

/

, , ,

since .

/

2 2

2 2 2

2 1 2

2 2

2

1 1

27 27 52

1

1 1

1

o

i kz t o

o

o o o

E E

From Chapter E

i t

c t c

For plane harmonic wave E E e

k i c k i

i c c

c

ω

σ ε ω γ

ω σωµ ω σ ω

ω γ ε µ ε ω

−

∂   ∂

∇ =  ∂ +  −  ∂ −

=

  

 

= +  −  = ∴ =  +   = ( ) - - - (n 27 53- )