1 optical properties of materials … reflection … refraction (snell’s law) … index of...

1

Optical Properties of Materials

0

1

1

0

0

1

1

0

00

sin

sin

sin

sin

v

vv

cn

n

n

… reflection

… refraction (Snell’s law)

… index of refraction

Index of refraction

Absorption

2

Maxwell’s Equations

HμμHμB

EεεEεD

Ej

DDr

t

BEE

r

r

r

0

0

div

rot

… electric field … magnetic field … electric displacement field … magnetic induction … current density … electric charge density … electrical conductivity … permittivity … permeability

Materials equations

3

Maxwell’s Equations

0div0

divdiv

rot

0

0

E

EεD

t

HμE

r

r

0div

rot 0

H

Et

EεH r

… no free charge

t

Eσ

t

EεεμμE

r

E

r

EEE

t

E

t

Eε

t

H

t

HμE

rr

r

r

2

2

002

2

2

2

0

2

2

0

0

rotrot

divgradrotrot

rot

rotrotrot

… wave equation

The Wave Equation

4

rrμεv

cn

2

22

t

Eσ

t

EεεμμE

r

Err

2

2

002

2

Et

E

Eit

E

22

2

Ekr

E

22

2

00

1

c

trkiEE

exp0

2

2220

2

ck

EσiωωεεμEk rr

2

002

ωε

σμinkk

ωε

σμin

c

ωk

ωσμiμωc

nk

r

r

r

0

220

2

0

22

22

02

2

22

ωε

σiεεεn

k

kμr

0

220

2~~~1

5

Refraction and Absorption

k … wave vector

… angular frequency

c … velocity of light

n … index of refraction

… electrical conductivity

Complex permittivity:permittivity and losses

Complex index of refraction:refraction and absorption

nn

inin

in

ininn

inn

i

2;

2

~~

2~

~

~

02

221

01

22

212

2222

0

6

Amplitude and Intensity of the Propagating Wave

rkrkntirkntiEI

EEEI

rkrkntiEE

inrktiEE

inkk

rktiEE

000

2

0

2

absorption

0

wavegpropagatin

00

00

0

0

2expexpexp

expexp

exp

exp

7

Relationship between Dielectric and Optical Constants

210

12

2222

~~

2~

iin

inninn

00

02

221

42

2

nn

n

n

122

212

11

2

0

21

2

122

212

11

2

0

21

2

2

1

2

1

n

* dielectric constant = permittivity

Insulator

8

nn

nn2

200 0;024

0 … non-conducting material

… no absorption, no losses

… the index of refraction is a real quantity

9

Penetration Depth

zc

IzkII

zknzktinzktiEI

EEEI

zknzktiEE

2exp2exp

2expexpexp

expexp

000

000

2

0

2

absorption

0

wavegpropagatin

00

4421

2

12exp

1:

00

0

cczz

c

Ie

zc

II

Ie

Iz

ee

e

… dependent on frequency (wavelength) and absorption

0

04

cnz

n

e

10

Penetration Depth and Absorption(Examples)

* absorption = damping

11

Reflection and Transmission

𝜃i 𝜃r

𝜃t

1

2

vexp

2exp

exp

0

0

0

rstiEE

rstiEE

rktiEE

111

211

sinsinsin

sin

v

θ

v

θ

v

θ

θs

v

s

v

s

v

s

tri

x

(t)x

(r)x

(i)x

121

2

11

22

2

1

sin

sin

sinsin

nn

n

με

με

v

v

θ

θ

θθ

t

i

ri

Reflection:

Transmission:(Snell’s law)

Same amplitude and phase of wave at the point “0”

12

Electric and Magnetic Field

The original wave:

11

)(

1)(

1||)(

1)(

||)()(

||)(

v

cossin

v

sincos

sincos

iii

i

ii

iz

iiy

ii

ix

ii

iz

iiy

ii

ix

zxt

rst

eAHeAHeAH

eAEeAEeAE

iii

iii

𝐼 𝑅

𝑇

EsH

The vectors of the electric and magnetic fields are perpendicular to the propagation direction of the wave.

𝜃i𝜃r

EsHE

𝒔𝑯

𝑬

13

Electric and Magnetic FieldThe transmitted wave:

22

222

cossin

sincos

sincos

v

θzθxtω

v

rstωτ

eεθTHeεTHeεθTH

eθTEeTEeθTE

tt(t)

t

iτt

(t)z

iτ||

(t)y

iτt

(t)x

iτt||

(t)z

iτ(t)y

iτt||

(t)x

ttt

ttt

The reflected wave:

11

)(

1)(

1||)(

1)(

||)()(

||)(

v

cossin

v

sincos

sincos

rrr

r

ir

rz

iry

ir

rx

ir

rz

iry

ir

rx

zxt

rst

eRHeRHeRH

eREeREeRE

rrr

rrr

14

Fresnel Equations… are obtained from the boundary conditions: Tangential components of

and have to be continuous at the interface (surface).

)t(y

)r(y

)i(y

)t(x

)r(x

)i(x

)t(y

)r(y

)i(y

)t(x

)r(x

)i(x

HHHHHH

EEEEEE

TRA

TRA

TRA

TRA

2||||1

t2i1

t||i||||

coscos

coscos

15

Fresnel Coefficients

Ann

nnR

Ann

nnR

Ann

nT

Ann

nT

ti

ti

ti

ti

ti

i

ti

i

coscos

coscos

coscos

coscos

coscos

cos2

coscos

cos2

21

21

||12

12||

21

1

||12

1||

ti

ti

ti

ti

ti

i

ti

i

nn

nnr

nn

nnr

nn

nt

nn

nt

coscos

coscos

coscos

coscos

coscos

cos2

coscos

cos2

21

21

12

12||

21

1

12

1||

it

ti

nnn

nn

221

22

2

21

sin1

cos

sinsin

Snell

ii

ii

inn

i

inn

i

ii

i

inn

i

i

nnn

nnnr

nnn

nnnr

nnn

nt

nnn

nt

221

221

221

221

221

222

221

222

||

221

221

1

221

222

1||

sincos

sincos

sincos

sincos

sincos

cos2

sincos

cos2

2

1

2

1

2

1

16

Index of Refraction(Experimental Examples)

17

Materials with different refractive indices are very important for complex optical systems

18

Transmission and Reflection

0

2

0

2

0

2

||

t

r

tII

rII

EEEI

R

T

R

Brewster angle – complete polarization of reflected electromagnetic wave (polarization of light)

Vacuum Glass (n=1,5)

RR

RRP

RRR

||

||

||21

Vacuum Glass: n=1.5

19

Transmission and Reflection

Vacuum Germanium (n=5,3)

Vacuum Germanium: n=5,3

20

Optical Reflection

Glass (n=1,5) Vacuum

Total internal reflection

Glass (n=1,5) Vacuum

21

Total Internal Reflection

1

2

2

1

2

1

21

21

arcsin

1sinsin

1sinsin

sinsin

n

n

n

n

n

n

nn

nn

c

tc

ti

ti

n1

n2

c

Glass (n = 1,5): c = 41,8°

Water (n = 2): c = 30°

22

Transmission and Reflectionwith Complex Index of Refraction

23

Transmission and Reflectionwith an Incident Angle of 0°

ii

ii

inn

i

inn

i

ii

i

inn

i

i

nnn

nnnr

nnn

nnnr

nnn

nt

nnn

nt

221

221

221

221

221

222

221

222

||

221

221

1

221

222

1||

sincos

sincos

sincos

sincos

sincos

cos2

sincos

cos2

2

1

2

1

2

1

21

21

12

12||

21

1

12

1||

22

0sin1cos0

nn

nnr

nn

nnr

nn

nt

nn

nt

iii

rr

tt

||

||

2

21

21

nn

nnR

2

1

1

n

nRInterface material - vacuum:

24

Table 11.2Refractive index and absorption index of some materials with nm

n

4

… absorption index… absorption coefficient… index of refraction… wavelength

25

Transmission and Reflection with Complex Index of Refraction

Copper

n = 0.14

k = 3.35

R = 95.6 %

Vacuum Copper (n=0.14-3.35i)

26

Transmission and Reflection with Complex Index of Refraction

Sodium

n = 0.048

k = 1.86

R = 95.8 %

Vacuum Sodium (n=0.048-1.86i)

27

Transmission and Reflection with Complex Index of Refraction

Gallium

n = 3.69

k = 5.43

R = 71.3 %

Vacuum Gallium (n=3.69-5.43i)

28

Transmission and Reflectionwith Complex Index of Refraction

Cobalt

n = 2.0

k = 4.0

R = 68.0 %

Vacuum Cobalt (n=2.0-4.0i)

29

30

Reflection with Complex Index of Refraction

22

222

1

1

11

11

1

1

n

n

inin

inin

n

nR

Influence of absorption (weakening, damping)

on the reflection

31

Reflection with Complex Index of Refraction

Total external reflection vanishes

32

Reflectivity as function of Refractive Index and Absorption

Reflectivity increases with increasing index of refraction and

an increasing absorption index

Fig. 11.2Reflectivity as function of absorption and refractive index

33

Refractive Index as function of Wavelength

Color of Materials

(Sphalerite)

(Rutile)

Material

Fig. 11.5Refractive index as function of absorption index and absorption coefficient as function of wavelength for Si (a), KCl (b) and Cu (c).

34

Reflection and Transmissionof a Thin Film

3322

332223

3322

2223

2211

221112

2211

1112

coscos

coscos

coscos

cos2

coscos

coscos

coscos

cos2

nn

nnr

nn

nt

nn

nnr

nn

nt

Fresnel coefficients at the interfaces:

22

11

33

22312

22312

22312

2312

cos

cos

11

rtn

n

err

errr

err

ettt

i

i

i

i

RT

cos

2cos0 nttnkk Phase shift:

35

Reflection and Transmissionof a Thin Film

Constant wavelength (monochromatic radiation)

Thickness of the film is ten times of the wavelength

Reflection

Vacuum Glass (n = 1.5, t = 6 μm) Vacuum, λ = 600 nm

Angle of incidence (degree)

Inte

nsi

ty (

%)

36

Reflection and Transmissionof a Thin Film

Constant wavelength (monochromatic radiation)

Thickness of the film is two times of the wavelength

Reflection

Vacuum Glass (n = 1.5, t = 1.2 μm) Vacuum, λ = 600 nm

Angle of incidence (degree)

Inte

nsi

ty (

%)

37

Reflection and Transmissionof a Thin Film

Constant wavelength (monochromatic radiation)

Thickness of the film is 40 times of the wavelength

Reflection

Vacuum Glass (n = 1.5, t = 24 μm) Vacuum, λ = 600 nm

Angle of incidence (degree)

Inte

nsi

ty (

%)

38

Reflection and Transmissionof a Thin Film

Different wavelengths (polychromatic radiation)

Thickness of film is 1.2 m

Different “Colors” are reflected and transmitted differently.

Vacuum Glass (n = 1.5, t = 1.2 μm) Vacuum, λ = 300-600 nm

Angle of incidence (degree)

Inte

nsi

ty (

%)