Augustin Cauchy
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August 21, 1789 – May 23, 1857
1810 - Graduated in civil engineering and went to work as a
junior engineer where Napoleon planned to build a naval base
1812 – (age 23) Lost interest in engineering, being more
attracted to abstract mathematics
Cauchy had many major accomplishments in both
mathematics and science in areas such as complex functions,
group theory, astronomy, hydrodynamics, and optics
Cauchy made 789 contributions to scientific journals
One of his most significant accomplishments involved
determining when an infinite series will converge on a
solution
In wave theory, he defined an empirical relationship between
the refractive index and wavelength of light for transparent
materials -- Cauchy’s Dispersion Equation
Cauchy’s Dispersion Equation
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Simple
Works well in the visible
spectrum (400→750nm) for
transparent material
SiO2: A = 1.451, B = 317410, C = 0
n – refractive index
λ – wavelength (um)
A,B,C - coefficients that can be determined for a material by fitting the equation to
measured refractive indices at known wavelengths
An Application of the Cauchy Equation
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The Cauchy Dispersion Equation is
used in semiconductor manufacturing
when monitoring film thickness
Films less than a few hundred
angstroms in thickness are required in
semiconductor manufacturing (1um =
10,000 angstroms)
A gate oxide on a transistor might be
between 50-100 Å and if off more than
a few angstroms the device may not
work correctly
Assumptions for Example
Initial medium is air (n0 = 1)
Transparent film (k=0)
Normal incident light source
Measurement Sequence
Spectral Reflectometry Measurement
Measurement Data
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As light strikes the surface of a film, it is
either transmitted or reflected
Light that is transmitted hits the bottom
surface and again is either transmitted or
reflected
The light reflected from the upper and
lower surfaces will interfere
The amplitude and periodicity of the
reflectance of a thin film is determined by
the film’s thickness and optical constants.
The reflections add together constructively
or destructively, due to the wavelike nature
of light and the phase relationship
determined by the difference in optical
path lengths of two reflections.
Fresnel Equations
(normal incidence)
Model and Fit
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To obtain the best fit between the
theoretical and measured spectra, the
dispersion for the measured material is
needed
A material dispersion is typically
represented mathematically by an
approximation model that has a limited
number of parameters. One commonly
used model is the Cauchy model.
Best fit is determined through a regression
algorithm, varying the values of the
thickness and selected dispersion model
parameters in the equation until the best
correlation is obtained between theoretical
and measured spectra.
Recursive Fit
Thickness
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The two parallel beams leaving the film at
A and C can be brought together by a
converging lens
The wavelength of light n in a medium of
refractive index n is given by n = 0 /n,
where 0 is the wavelength in air
The optical path difference (OPD) for
normal incidence is (AB+BC) times the
refractive index of the film.
(AB+BC) is approximately equal to twice
the thickness, so OPD = n(2t)
Reflections are in-phase and therefore
add constructively when the light path is
equal to one integral multiple of the
wavelength of light.
Reflectance of thin films will vary
periodically with 1/ n
References
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http://utopia.cord.org/step_online/st1-4/st14eiii3.htm
http://en.wikipedia.org/wiki/Fresnel_equations
http://en.wikipedia.org/wiki/Thin-film_interference
http://en.wikipedia.org/wiki/Ellipsometry
http://en.wikibooks.org/wiki/Waves/Thin_Films
www.chem.agilent.com/Library/applications/uv90.pdf
http://www.jawoollam.com/resources.html