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Augustin Cauchy August 21, 1789 – May 23, 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 -- Cauchys Dispersion Equation

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Cauchys Dispersion Equation Simple Works well in the visible spectrum (400750nm) for transparent material 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 SiO2: A = 1.451, B = , C = 0

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An Application of the Cauchy Equation 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 Å and if off more than a few angstroms the device may not work correctly Assumptions for Example Initial medium is air (n 0 = 1) Transparent film (k=0) Normal incident light source

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Measurement Sequence

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Spectral Reflectometry Measurement

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Measurement Data 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 films 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)

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Model and Fit 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.

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Recursive Fit

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Thickness 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

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References

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