1

Investigation of Optical Properties

n, k … index of refraction and damping

1, 2 … polarization and absorption

nkkn 2; 222

1

Problems:

The penetration depth of light is usually very low (metals) surface sensitive measurement

Influences on measured optical properties of the material:

State of oxidation, distortion, adsorption and etc.

2

Problem with Complex Parameters

22

222

1

1

11

11

1

1

n

n

inin

inin

n

nR

>>> Solely the measurement of reflectivity is not sufficient <<<

Solution:

1. The Kramers-Kronig analysis

2. Optical ellipsometry

3. Differential reflectometry

3

The Kramers-Kronig Analysis

Relation between the real and imaginary part of a complex quantity

0

0

022

21

0

lnln1

lnln

2

1

ln

2

2

lnlnln

exp

dd

Rd

dd

Rd

dR

iRiRr

iRrE

ER

022

022

2

2

d

d

i

Relation between the reflectivity (real part) and the phase shift (imaginary part)

cos21

sin2

cos21

1

RR

Rk

RR

Rn

4

The Kramers-Kronig Analysis

Polarizability

Ion resonance

Electron resonance

Frequency

Microwaveradiation

Infraredradiation

Ultraviolet radiation and x-rays

Val

idity

of M

axw

ell’s

equ

atio

ns

Dipole relaxation

Measurement of reflectivity in a wide spectrum of wavelengths(measured in a broad frequency range)

5

Color Table

c

hhE E (eV) 1 2 3 4 5 6 10 15 20

(nm) 1240 620 413 310 248 207 124 83 62

3.10 2.76 2.48 2.26 2.07 1.91 1.77 1.65 (eV)

6

SilverReflectivity and permittivity from the Kramers-Kronig analysis

The reflectivity for visible light is nearly one

White light is reflected as white light

Dispersion curve: Model of free and bound electrons can be applied

7

CopperReflectivity and permittivity from the Kramers-Kronig analysis

Reflectivity decreases for E > 2 eV ( < 620 nm)

IR light has the best reflectivity

For visible light the reflectivity is better for higher wavelengths „red“ color of copper

Dispersion curve as calculated from the model of free and bound electrons

8

AluminumReflectivity and permittivity from the Kramers-Kronig analysis

Reflectivity is nearly constant (≈ 90%) for the whole spectrum of visible light

White light is reflected as white light

Dispersion curve: Model of free and bound electrons

9

Optical Ellipsometry

Reflected light is always partially polarized

RR

RRP

RRR

||

||

||21

2

2||||

r

r

R

R

ti

ti

ti

ti

nn

nnr

nn

nnr

coscos

coscos

coscos

coscos

21

21

12

12||

Vacuum Germanium: n=5,3

10

Optical Ellipsometry

Incident light: a linear polarized wave

… angle of incidence

Reflected light: an elliptical polarizes wave

s

pe E

Etan

s

pr R

Rtan

11

Elliptical Polarization of a Propagating Wave

… phase shift of two linear polarized waves, which describes an elliptical polarized wave

An elliptical polarized wave can be decomposed into two linear polarized waves.

12

Experimental Arrangement for Optical Ellipsometry

222222222212

222222222212

sin4sin

sin4sin

bababak

bababan

r

r

r

ab

a

2tansin

2sincos1

2costansin

… index of refraction, … absorption, … incidence angle of primary beam,… phase shift of the linear polarized wave… azimuth of reflected wave

13

Measurement of and

Compensator: Measurement of phase shift . The thickness of the compensator is adjusted, so that the phase shift after the compensator is equal to zero creation of a linear polarized wave.

Analyzer: Measurement of the angel . Analyzer is rotated until the polarized light can’t pass.

14

Index of Refraction and Permittivity

222

2

2

2222222

1

cos2sin1

sintansin4sin2

cos2sin1

sin2sin2costan1sin

r

r

r

rr

nk

kn

… index of refraction

… absorption

… polarization

… absorption

15

Differential ReflectometryComparative measurement for similar materials

Investigation of differences in real structures (distortion, oxidation, composition, …)

16

Differential ReflectometryInvestigation of “critical points” in the band structure

… first derivative of

… third derivative of

17

Alloys

Cu-Zn alloy

Change in color

ED

18

Change of Optical Properties Caused by Corrosion

Oxidation of copper

CuO2 layer on the surface of copper

Change in color