27. Optical properties of materials

(1)

27. Optical properties of materials

Polarization of a dielectric medium : P

: collective dipole moment per unit volume P = − Ner

JG G

E

e

The equation of motion of the oscillating electron,

d r d r

K r m e E m

dt dt

γ

− − − =

²

2

G G

G JJG

(2)

When the applied -field is static,

: static polarization E

K r e E P Ne E

− − = 0 ⇒ = K²

JJG

G JJG JG JJG

&

: time-dependent polarization

i t i t

o o

When E E e r r e

d r Ne

r P E

dt m im K

ω ω

ω ω ωγ

− −

= =

 

= − ⇒ =  

− − +

 

2 2

2

2 2

JJG JJG G G

G G JG JJG

2 o

Since the actual field at the dipole is a superposition of and the field due to other dipoles,

Defining , (

o

E E E P

Ne P

P E

m im K

K Ne

RESONANCE FREQ

m m

ε

ω ωγ ε

ω ε

→ +

 

=   + 

− − +

  

≡ −

2 2

2

3

JJG JJG JJG JG

JG JJG JG

( )

) /

o

UENCY

Ne Ne m

P E P E

ω ω

i

ωγ ω ω ω γ

 

=  − −  ⇒ = − +

2 2

2 2 2

2 2 2 2

JG JJG JG JJG

(3)

Propagation of light waves in a dielectric

•

• -

o

o o o

E B E B

t B J E

t ρ

ε

µ µ ε

∇ =

∇ × = ∂

∂

∇ × = + ∂

∂ 0

-

o

E P

B E B

t

E P

c B

t t

ε

∇ ⋅ = −∇ ⋅

∇ ⋅ =

∇ × = ∂

∂

∂ ∂

∇ × = +

∂ ∂

2

0

1 P J P

t ρ = −∇ ⋅

= ∂

∂

( - )

( )

:

, since .

( )

o o o

i kz t o

o o

E P Ne E

c E

t t m i t

For a harmonic wave E E e

Ne f

k n k n

c v c

c m i

ω

ε ε ω ω ωγ

ω ω π π ω

ε ω ω ωγ λ

 

∂ ∂ ∂

∇ = + =  + 

∂ ∂  − −  ∂

=

     

=  + − −    =   = = =   

2 2 2 2

2 2

2 2 2 2 2

2 2 2

2 2

2 2 2

1 1

2 2

1

(4)

Complex refractive index

( )

:

: (the fraction of dipoles having the resonance frequency )

j

R I

o j j j

j

j j

Ne f

n n in

m i

resonance frequencies f oscillator strength

ε ω ω ωγ

ω

= + = +

− −

2

∑

2 2

1

( )

&

( ) ( )

R I R I

R I

o o

o

R I R I

o o o o

k ik n in

c

n in Ne

m i

Ne Ne

n n n n

m m

ω

ε ω ω ωγ

ω ω γω

ε ω ω γ ω ε ω ω γ ω

+ =     +

 

+ =  + 

− −

 

 

 −   

− = +   =  

− + − +

   

   

2 2

2 2 2 2 2 2 2 2 2 2

1

1 2

(5)

Dispersion equation : n=n(ω)

( ,

_o

)

o o

n Ne Assume

m γ ω ω

ε ω ω

 

= +   =

 − 

 

2 2

1 1 0

-1

- =

o o o o o o

o o o o

n Ne

m

B C n A

ω ω ω

ω ω ω ω ω ω ω

ω ω

ε ω ω ω

λ λ

   

=       + + +  

−    

 

= +    + + +   

= + + +

2 2 4

2 2 2 2 2 2 4

2 2 4

2

2 2 4

2 4

1 1 1

1 1

"

(6)

Conduction current in a matal

The equation of motion of a free electron (not bound to a particular nucleus; ),

The conduction current density .

K

d r d r d v

K r m e E m m m v e E

dt dt dt

J Nev

d J Ne

J E

dt m

γ γ

γ

=

− − − = ⇒ + = −

= −

 

+ =  

 

2 2

2

G G G 0

G JJG G JJG

JG G

JG JG JJG

For a harmonic wave , the current desity varies at the same rate

( )

= ( )

/

i t i t

o o

E E e J J e

i J Ne E

m

J E Ne conductivity

i m

ω ω

ω γ

σ σ

ω γ γ

− −

= =

 

− + =  

 

 

=  − 

2

1

JJG JJG JG JG

JG JJG

(7)

Propagation of light waves in a metal

•

• -

o

o o o

E B E B

t B J E

t ρ

ε

µ µ ε

∇ =

∇ × = ∂

∂

∇ × = + ∂

∂ 0

-

o

E B E B

t

E J

c B

t ε

∇ ⋅ =

∇ × = ∂

∂

∇ × = ∂ +

∂

2

0 0

no net free-charge ρ = 0

( )

/

, , ,

since .

/

o

i kz t o

o

o o

E E

E c t c i t

For plane harmonic wave E E e

k i c

c i

ω

σ ε ω γ

ω σωµ

ω γ ε µ

−

 ∂    ∂

∇ =  ∂ +  −  ∂

=

 

= +  −  =

2 2

2 2 2

2

2 2

2

1 1

1

1 1

(8)

Skin depth

/

/ / /

: low frequency with a large conductivity,

( )

, &

o

R I

o

R I

o

R R I I

o o

Special case

k i

k i k ik

k k

c c c

n k n k

σωµ σωµ

σωµ

σµ σ σ

ω ω ωε ω ωε

=

 

= +   ≡ +

 

= =  

     

= =   =   = ==  

2

1 2

1 2 1 2 1 2

2

1 2

2

2 2 2

( )

, exp( ) exp( ).

The skin depth where the field amplitude drops to / ,

i kz t

o o I R

I o

For harmonic wave E E e E k z ik z i t e

k

ω

δ σµ ω

= − = − −

≡ =

1

1 2

(9)

Plasma frequency

( / )

( )

is a resonance frequency for the free oscillation of the electrons in a metal.

o o

p

R I

p o o

i c c

n c k

i i

n n in

i The PLASMA FREQUENCY

Ne Ne

c c

m

σ µ σ µ γ

ω ω ω γ ω ωγ

ω ω ωγ

ω σ µ γ µ γ

γ

 

=   = + − = − +

= + = −

+

 

≡ =   =

 

2 2 2

2

2 2

2

2 2

1 1

1

m

ε

o

by neglecting , valid for high frequency ( ).

For , is complex and radiation is attenuated.

For , is real and radiation is not attenuated(transparent).

p

p p

n

n n

ω ω

γ ω γ

ω ω ω ω

= −

<

>

2 2

1 2

(10)

If the electrons in a plasma are displaced from a uniform background of ions, electric fields will be built up in such a direction as to restore the neutrality of the plasma by pulling the electrons back to their original positions.

Because of their inertia, the electrons will overshoot and oscillate around their equilibrium positions with a characteristic frequency known as the plasma frequency.

Plasma frequency

/ ( ) / : electrostatic field by small charge separation exp( ) : small-amplitude oscillation

27. Optical properties of materials