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The light waves that we used in our model of the Michelson interferometer are what are called plane waves.  Plane waves are waves that have planes of constant phase, which are orthogonal to the wave vector ''k''. Mathematically, plane waves look as follows:

The simplest solution to these differential equations are ''plane waves'', given by Eq. (1) below.  They are called plane waves because the fields appear as planar surfaces of constant amplitude and direction which advance along the direction of ''k'' at the speed of light.

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The Michelson interferometer consists of a source, a beam splitter, a reference mirror, a target and a detector (see schematic at right).  Light leaves the source and moves towards the beam splitter.  The beam splitter allows half of the incident light to be transmitted and the other half to be reflected.  The reflected light travels a known distance to the reference mirror, while the transmitted light travels towards the target.  Both beams reflect off of their respective targets and travel back towards the beam splitter, where a similar half-half selection process occurs again.  Only this time, the beams that travel back to the source are removed through an optical device, and the beams that travel to the detector "survive".

The Michelson interferometer consists of a source, a beam splitter, a reference mirror, a target and a detector (see schematic at right).  Light leaves the source and moves towards the beam splitter.  The beam splitter allows half of the incident light to be transmitted and the other half to be reflected.  The reflected light travels a known distance to the reference mirror, while the transmitted light travels towards the target.  Both beams reflect off of their respective targets and travel back towards the beam splitter, where a similar half-half selection process occurs again.  Only this time, the beams that travel back to the source are removed through an optical device, and the beams that travel to the detector "survive".

For our experiment we will be utilizing the fringes of the Michelson interferometer to gather information about the topology of synthetic diamond wafers.  In order to be able to utilize a computer program to analyze the data gathered from the Michelson interferometer, we start with an model of the beam splitter present at the center of the interferometer.  We know that the beam splitter is comprised of a thin layer of a conducting substance present on one side of a thin piece of optical glass.  When a beam of light is incident on the beam splitter, a fraction of the photons travel through to the other side of the splitter and the remaining photons reflected.  Here we consider wavelengths at which the fraction of the beam absorbed by the mirror is negligible.

For our experiment we will be utilizing the fringes of the Michelson interferometer to gather information about the topology of synthetic diamond wafers.  In a Michelson interferometer, an incoming beam of light is ''split'' by a partially silvered mirror.  This mirror is comprised of a thin layer of a metallic conductor plated onto one side of a thin piece of optical glass.  When a beam of light is incident on the beam splitter, a fraction of the photons travel through to the other side of the splitter and the remaining photons are reflected or absorbed.  An optical-quality beam splitter is designed to have neglibible absorption at the wavelengths at which it is designed to operate.

Using our knowledge of electric and magnetic fields in conductors and [[Maxwell's Equations]], we can create a simple model of the beam splitter with a light wave at normal incidence.  We are interested in finding two main quantities in this model: the thickness of the conducting film and the phase shift that occurs at the mirror surface.   

Using [[Maxwell's Equations]] and the properties of conductors, a simple model of the beam splitter was created with a light wave at normal incidence.  There are two main quantities to be determined by this model: the thickness of the conducting film corresponding to a given transmission fraction, and the phase shift of the transmitted and reflected waves that occurs at the mirror surface.   

Using the programming power of Matlab, we can solve our system of equations Mv=b, where M, v, and b are given below.

Ignoring the glass-air interface on the back side of the mirror, there are three regions in which Maxwell's equations must hold: the incident region (air, treated as vacuum), the interior of the metallic layer (silver, treated as an imperfect conductor), and the transmission region (air, treated as vacuum).  The complex amplitude of the waves in these three regions are related by boundary-matching conditions that can be expressed as a linear equation '''M'''''v=b'', where '''M''', ''v'', and ''b'' are given by Eq. (4) below.

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Using Matlab we were able to extract information about the desired depth of the conducting film and the phase shift that occurred when the light wave traveled through the the film.  Our graphs dislpaying phase shift versus depth and amplitude versus depth are shown below.

These equations were solved using Matlab.  The results for the transmitted fraction and the phase shifts of the transmitted and reflected waves are shown in the figures at the right.   

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