Ellipsometry … a method with which we can measure the complex index of refraction of a material or the thickness of thin films by performing a reflection experiment. Definition of the “plane of incidence”: Plane of incidence = the plane that contains incoming ray, reflected ray, refracted ray and surface normal. Definition of “p” and “s” polarization in reflection: p‐polarized parallel to plane of incidence s‐polarized senkrecht to plane of incidence (“senkrecht” = German for “perpendicular”) p s Fresnel Equations: cos
cos
cos
cos
cos
cos
cos
cos
,
complex reflection
coefficient for amplitude for s and p polarized light complex index of refraction complex index of refraction of medium with incident wave
complex index of refraction of medium with transmitted wave
,
incident and transmitted wave angles
with respect to surface normal Use Snell’s Law to Eliminate and
When Ni , Nt are complex, then Snells Law: sin
sin
cos
sin
1
sin
1
Substituting: cos
1
cos
1
2
2
2
2
cos
1
cos
1
2
2
2
2
If you know ,
, and
, you can calculate and
. Absorbing Media Definitions : n = Index of Refraction k = Extinction Coefficient (both are real numbers) When the electric field enters the medium its amplitude diminishes as a function of depth of penetration: Electric Field: ,
where Intensity: Reflection from Absorbing Media for the p and s polarized Assume we have incident amplitudes of components and they are in‐phase (linearly polarized light). The angle of polarization is shown in the picture below (looking against the direction of light propagation, for example): direction of light propagation We will now write the complex reflection coefficients as “ Amplitude*eiPhase “ : | |
and
The reflected electric field amplitudes are: | |
and
There also will be a phase shift between the two electric fields Ψ
phase shift φ
direction of light propagation tan Ψ
If we choose | |
° ) then (
tan
| |
The phase shift  is expressed as the difference in the phase shifts for s and p polarized light: φ
‐ | |
tan ψ
, where
tan
| |
and
φ
‐
Before and after reflection – looking along the axis of the laser beam | | 1 Θ
Before reflection 45°
After reflection 1
Determining the complex index of refraction of the reflecting medium:  Measure and φ using an ellipsometer ; Know Ni  Calculate Nt , the complex index of refraction of the reflecting medium. cos
1
cos
1
cos
1
cos
1
cos
1
cos
1
Assuming tan
cos
1
cos
1
45° ,
Solve for , Ψ,
… .. sin
1
1
1
tan
tan
tan
sin
1
1
1
tan
tan
tan
This can also be written as: sin
1
1
1
tan
, where ρ tan
Predicting the polarization angle after reflection and the phase shift from known complex indices of refraction and
: cos
1
cos
1
tan
cos
1
cos
1
Calculate this complex number and write it in the complex number form tan
Ψ
arctan
Performing an ellipsometry experiment L A S E R Polarizer P1 at 45 Esi Epi Babinet‐Soleil Compensator Photodiode Sample Polarizer Epr BSC at 0 Esr  P2 DET 110 Esr  = ‐ 





Epr Rotate P2 to minimize the intensity I measured by the DET110 detector. Change the micrometer setting on the BSC to adjust the phaseshift to minimize intensity I. Go back to readjust P2, then readjust phase shift on the BSC, then P2, etc….. Once you are done with the minimizing (nulling), read off: x (micrometer screw on BSC)  (orientation of P2) From x, using the BSC calibration, determine the phase shift  needed to compensate for the phase shift at the glass plate. Determine , the orientation of the reflected light. Measurement of the thickness of thin dielectric films Outside: No Film: N1 d1
Substrate: N2 1
cos
cos
1
cos
1
cos
1
cos
1
cos
1
cos
1
cos
1
Snells Law: sin
sin
sin
The two interfaces yield a combined reflection coefficient which can be obtained using the Fabry‐Perot equations: 1
1
Where and d1=thickness of layer 1
1
tan
One can obtain values for and Δ as a function of thickness d1 by calculating and making a graph as follows:  -  curves for silicon dioxide on silicon. The layer thickness increases counter
clock wise from 0 (square marker on the left) in steps of 10 nm (black diamonds)
and in steps of 100 nm(squares). The incident angle of the He/Ne laser beam
(632.8 nm) is 70 degrees