1. Outline Index of Refraction Introduction Classical Model
Jing Li
Outline
Introduction
Classical Model
Typical measurement methods
Application
Reference

2. Definition of Index of Refraction
In uniform isotropic linear media, the wave equation is:
They are satisfied by plane wave
y=A e i(k r- wt)
y can be any Cartesian components of E and H
The phase velocity of plane wave travels in the direction of k is

3. Definition of Index of Refraction
We can define the index of refraction as
Most media are nonmagnetic and have a magnetic permeability m=m0, in this case
In most media, n is a function of frequency.

4. Classical Electron Model ( Lorentz Model)+ - w0 X E
Let the electric field of optical wave in an atom be
E=E0e-iwt
the electron obeys the following equation of motion
X is the position of the electron relative to the atom
m is the mass of the electron
w0 is the resonant frequency of the electron motion
g is the damping coefficient

5. Classical Electron Model ( Lorentz Model)
The solution is
The induced dipole moment is
a 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 a E = e0 c E

6. Classical Electron Model ( Lorentz Model)
The dielectric constant of the medium is given by
e = e0 (1+c) = e0 (1+Na/ e0)
If the medium is nonmagnetic, the index of refraction is
n= (e /e0)1/2 = (1+Na/ e0 )1/2
If the second term is small enough then

7. Classical Electron Model ( Lorentz Model)
The complex refractive index is
at w ~w0 ,
Normalized plot of n-1 and k versus w-w0

8. For more than one resonance frequencies for each atom,
Classical Electron Model ( Drude model)
If we set w0=0, the Lorentz model become Drude model. This model can be used in free electron metals

9. Relation Between Dielectric Constant and Refractive Index
By definition,
We can easily get:

10. An Example to Calculate Optical Constants
Aleksandra B. Djuris¡ic and E. H. Li, JOURNAL OF APPLIED PHYSICS, 85(1999)2848
The symbols in these two graphs are not clear to me. I assume they are the real part and the imaginary part of the dielectric constant on the left, but I do not know what is n and k– the real part only?
Real and imaginary part of the index of refraction of GaN vs. energy;

11. Kramers-Kronig Relation
The real part and imaginary part of the complex dielectric function e (w) 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
J. Appl. Phys. 75 (1994)4176

12. A Method Based on Reflection
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 (q); (8) PbS detector(2q)
Alibert et al. J. Appl. Phys., Vol. 69(1991)3208

13. Calculation In this case, n1=1, and n2=nr+i n i z E1'q E1
E1'
n1=1 z x
n2=nr+i n i
Snell Law become:
Reflection coefficient:
Explain the fitting procedure: how many unknowns and typically how many angles are used
Reflectance:
R(q1, l, nr, n i)=|r p|2
From this measurement, they got R, q for each wavelength l, Fitting the experimental curve, they can get nr and n i .
Reflection of p-polarized light

14. Results Based on Reflection Measurement
Single effective oscillator model
(Eq. 1)
(Eq. 2)
Need to explain the figures– Eq.(1), (2) quoted in the figure, can you explain them? Refractive index is the real part only?
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 energy
Ed: dispersion energy
EG: lowest direct band gap energy

15. Use AFM to Determine the Refractive Index Profiles of Optical Fibers
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
S. T. Huntington, J. Appl. Phys., 82(1997) say more about the fiber and the measurement method with AFM and the principle—how the AFM has anything to do with the dielectric constant
Modern communication cables are optical waveguides in the form of glass fibers. The basic configuration consists of a hair like, cylindrical, dielectric region surrounded by a concentric layer of somewhat lower refractive index( cladding). An additional layer of optically lossy material (jacket) may also be present.
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.

16. AFM
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.

17. Result

18. A Method Based on Transmission
For q=0, input wave function a e if ,
tm=aTT’R’2m-1 e i(f-(2m-1)d ) (m=1,2…)
d=2pdn/l
The transmission wave
function is superposed by all tm
a T = a T T’ e if S m(R’2m-1 e-i(2m-1)d )
=(1-R2)a e i(f-d) /(1-R2e-i2d)
(TT’=1-R2 ; R’=-R)
If R<<1, then
a T =a e i(f-d)
maximum condition is 2d=2pm= 4pdn/l
n(lm)=m lm/2d d q q1 t1 t2 t3 r1 r2 r3 n

19. Result Based on Transmission Measurement

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

21. Thickness Measurement
n(lm)=m lm/2d
Steps to calculate thickness
Get peak position lm
d=(lm lm-1)/2/[lm-1 n(lm) - lm n(lm-1)]
Average d
get m min= n(l max)*2d/ l max
Calculate d : d=m lm/2/n(lm) (from m min for each peak)
Average d again
Limit
Minimum thickness:~500/n
Error<l/2n

22. Reference
Pochi Yeh, "Optical Wave in Layered Media", 1988, John Wiley & Sons Inc
E. E. Kriezis, D. P. Chrissoulidis & A. G. Papafiannakis, Electromagnetics and Optics, 1992, World Scientific Publishing Co.,
Aleksandra B. Djurisic and E. H. Li, J. OF Appl. Phys., 85 (1999) 2848 (mode for GaN)
C. Alibert, M. Skouri, A. Joullie, M. Benounab andS. Sadiq , J. Appl. Phys., 69(1991)3208 (Reflection)
Kun Liu, J. H. Chu, and D. Y. Tang, J. Appl. Phys. 75 (1994)4176 (KK relation)
G. Yu, G. Wang, H. Ishikawa, M. Umeno, T. Soga, T. Egawa, J. Watanabe, and T. Jimbo, Appl. Phys. Lett. 70 (1997) 3209
Jagat, (AFM)
Authors needed to be listed in the references– before you conclude, can you say anything about the pro and con of the methods mentioned?
J. Appl. Phys., Vol. 73 (1993)1732
S. T. Huntington, J. Appl. Phys., 82(1997)2730