Jing LiJing Li

Outline

•Introduction•Classical Model•Typical measurement methods•Application•Reference

Index of Refraction

In uniform isotropic linear media, the wave equation is:

02

22

tEE

02

22

tHH

They are satisfied by plane wave

=A e i(k r- t)

can be any Cartesian components of E and H

The phase velocity of plane wave travels in the direction of k is

1

kωv

ωk k

Definition of Index of Refraction

We can define the index of refraction as

00

cvn

Most media are nonmagnetic and have a magnetic permeability , in this case

0n

In most media, n is a function of frequency.

Definition of Index of Refraction

Let the electric field of optical wave in an atom be

E=E0e-it

the electron obeys the following equation of motion

EXXX emdtdmdt

dm 202

2

X is the position of the electron relative to the atom

m is the mass of the electron

0 is the resonant frequency of the electron motion

is the damping coefficient

Classical Electron Model ( Lorentz Model)

+ -

0

X

E

The solution is tieim

e

)( 220

0EX

The induced dipole moment is

EEXp

)( 22

0

2

imee

)( 220

2

ime

is atomic polarizability

The dielectric constant of a medium depends on the manner in which the atoms are assembled. Let N be the number of atoms per unit volume. Then the polarization can be written approximately as

P = N p = N E = 0E

Classical Electron Model ( Lorentz Model)

)(1

2200

2

0

2

imNen

)(21

2200

2

imNen

If the second term is small enough then

The dielectric constant of the medium is given by

= 0 (1+) = 0 (1+N/ 0)

If the medium is nonmagnetic, the index of refraction is

n= ( /0)1/2 = (1+N/ 0 )1/2

Classical Electron Model ( Lorentz Model)

])[(2])[(2

)(1

2222200

2

2222200

220

2

m

Nei

m

Neinnn ir

The complex refractive index is

Normalized plot of n-1 and k versus

])2/()[(8])2/()[(4

)(1 22

000

2

22000

02

m

Nei

m

Neinn ir

at ~0 ,

Classical Electron Model ( Lorentz Model)

j jj

j

i

f

mNen

)(1

220

22

For more than one resonance frequencies for each atom,

Zfj

j

Classical Electron Model ( Drude model)

If we set 0=0, the Lorentz model become Drude model. This model can be used in free electron metals

)(1 2

0

22

imNen

0

2

n

ir innn

21 i

02/1

12/12

22

1 /]})[(21{ rn

02/1

12/12

22

1 /]})[(21{ in

By definition,

We can easily get:

Relation Between Dielectric Constant and Refractive Index

Real and imaginary part of the index of refraction of GaN vs. energy;

An Example to Calculate Optical Constants

The real part and imaginary part of the complex dielectric function ) are not independent. they can connected by Kramers-Kronig relations:

P indicates that the integral is a principal value integral.

K-K relation can also be written in other form, like

dP

0 222

01

)(2)(

dP0 22

012

)(2)(

Kramers-Kronig Relation

dPn0 2)/(1

)(1)(

Typical experimental setup

( 1) halogen lamp;

(2) mono-chromator; (3) chopper; (4) filter;

(5) polarizer (get p-polarized light); (6) hole diaphragm;

(7) sample on rotating support (); (8) PbS detector(2)

A Method Based on Reflection

xirx

xirx

xx

xxp kinnk

kinnk

knkn

knknr

12

2

12

2

12

222

1

12

222

1

)(

)(

Snell Law become:

Reflection of p-polarized light

sin2

21 zz kk

2/1221 ])2[(

xk

2/12222 ])()2[(

irx innk

Reflection coefficient:

In this case, n1=1, and n2=nr+i n i

Reflectance:R(1, , nr, n i)=|r p|2

From this measurement, they got R, for each wavelength Fitting the experimental curve, they can get nr and n i .

Calculation

E1

E1'

n1=1

z

x

n2=nr+i n i

Results Based on Reflection Measurement

220

02 1EE

EEn d

r

)ln(122

24

30

2

0

2

EE

EEE

E

EEEE

n fddr

FIG. 2. Measured refractive indices at 300 K vs. photon energy of AlSb and AlxGa1-xAsySb1-y layers lattice matched to GaSb (y~0.085 x).

Dashed lines: calculated curves from Eq. ( 1); Dotted lines: calculated curves from Eq. (2)

E0: oscillator energyEd: dispersion energyE: lowest direct band gap energy

220

2 2 EEE f

)(2 220

30

EEE

Ed

Single effective oscillator model

(Eq. 1)

(Eq. 2)

Use AFM to Determine the Refractive Index Profiles of Optical Fibers

Fiber samples were• Cleaved and mounted in holder• Etched with 5% HF solution• Measured with AFM

There is no way for AFM to measure refractive index directly.People found fiber material with different refractive index have different etch rate in special solution.

The basic configuration of optical fiber consists of a hair like, cylindrical, dielectric region (core) surrounded by a concentric layer of somewhat lower refractive index( cladding).

n1 n2

2a

•The optical lever operates by reflecting a laser beam off the cantilever. Angular deflection of the cantilever causes a twofold larger angular deflection of the laser beam.

• The reflected laser beam strikes a position-sensitive photodetector consisting of two side-by-side photodiodes.

• The difference between the two photodiode signals indicates the position of the laser spot on the detector and thus the angular deflection of the cantilever.

• Because the cantilever-to-detector distance generally measures thousands of times the length of the cantilever, the optical lever greatly magnifies motions of the tip.

AFM

Result

For =0, input wave function a e itm=aTT’R’2m-1 e i(-(2m-1) ) (m=1,2…)=2dn/The transmission wavefunction is superposed by all tm

a T = a T T’ e i m(R’2m-1 e-i(2m-1) )=(1-R2)a e i() /(1-R2e-i2)

(TT’=1-R2 ; R’=-R)If R<<1, then

a T =a e i()

maximum condition is 2=2m= 4dn/

n(m=m m/2d

d

t1 t2 t3

r1 r2 r3

n

A Method Based on Transmission

294

2 304.7 2.27n 222 2

Result Based on Transmission Measurement

ApplicationIn our lab., we have a simple system to measure the thickness of epitaxial GaN layer.

n(m=m m/2d

Thickness Measurement

Steps to calculate thickness

• Get peak position m

• d=(m m-1)/2/[m-1 n(mmn(m-1

• Average d

• get m min= n(max*2d/ max

• Calculate d : d=m m/2/n(mfrom m min for each peak)

• Average d again

LimitMinimum thickness:~500/nError/2n

1. Pochi Yeh, "Optical Wave in Layered Media", 1988, John Wiley & Sons Inc

2. E. E. Kriezis, D. P. Chrissoulidis & A. G. Papafiannakis, Electromagnetics and Optics, 1992, World Scientific Publishing Co.,

3. Aleksandra B. Djurisic and E. H. Li, J. OF Appl. Phys., 85 (1999) 2848 (mode for GaN)

4. C. Alibert, M. Skouri, A. Joullie, M. Benounab andS. Sadiq , J. Appl. Phys., 69(1991)3208 (Reflection)

5. Kun Liu, J. H. Chu, and D. Y. Tang, J. Appl. Phys. 75 (1994)4176 (KK relation)

6. G. Yu, G. Wang, H. Ishikawa, M. Umeno, T. Soga, T. Egawa, J. Watanabe, and T. Jimbo, Appl. Phys. Lett. 70 (1997) 3209

7. Jagat, http://www.phys.ksu.edu/~jagat/afm.ppt (AFM)

Reference