1,359
edits
Changes
Jump to navigation
Jump to search
Mapping diamond surfaces using interference (view source)
Revision as of 14:14, 6 July 2007
, 14:14, 6 July 2007→Links to Other Wiki Pages
This page represents a ongoing project dealing with using interference patterns to map the surface of a diamond wafer. Since this is my first page, you'll have to excuse any blatant errors that I do not pick up on immediately. Currently this page will represent my work with Dr. Richard Jones on an model of the beam splitter featured in the Michelson interferometer. I will start by giving a brief introduction to electromagnetic radiation, then move on to the model itself (including graphs,etc.).
==Interference and Light Waves==
Light (electromagnetic radiation) [[Image:emwave.jpg|thumb|An Electromagnetic Wave! (courtesy of [http://www.mtholyoke.edu/~mlyount/MySites/ForensicSpectroscopy/Vocab.html])]] is comprised of an temporally and spatially oscillating electric and magnetic field components. These components are orthogonal to each other as well as to the direction of propagation given by the Poynting vector. Wave equations for the electric and magnetic fields can be derived using [[Maxwell's Equations]].
{|width="50%"
|align="right"|
<math>\nabla^2 \boldsymbol{E} = \frac{1}{c^2} \frac{\partial^2 \boldsymbol{E}}{\partial t^2}</math>
|align="center" width="80"|(1)
|}
{|width="50%"
|align="right"|
<math>\nabla^2 \boldsymbol{B} = \frac{1}{c^2} \frac{\partial^2 \boldsymbol{B}}{\partial t^2}</math>
|align="center" width="80"|(2)
|}
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.
{|width="50%"
|align="right"|
<math>\boldsymbol{E} = \boldsymbol{E_0} e^{i\left(\boldsymbol{k \cdot x} - \omega t\right)}</math>
|align="center" width="80"|(3)
|}
Electromagnetic waves also have the property that they superimpose. [[Image:MichIntInt.jpg|thumb|Interference from a Michelson Interferometer (courtesy of [http://www.arikah.com/encyclopedia/Interference])]] Like mechanical waves, EM waves can interfere destructively or constructively, but, unlike mechanical waves, there is an additional condition for light wave interference. In order for this to occur, the light waves must be traveling in the same direction, be of the same wavelength and have a constant phase with respect to each other. The reason that light waves must be travelling in the same direction to interfere is that in order for interference to occur, both the electric and magnetic field components must "line up". A picture of EM wave interference is shown at right.
==Interferometry and the Michelson Interferometer==
Interferometry is the splitting of light beams into two or more paths and the recombining of those different beams to measure difference in optical path length and refractive index [[#References|[1]]] via interference fringes that form as a result of the recombined beams.
The Michelson interferometer [[Image:MIint.gif|thumb|A Michelson Interferometer! (courtesy of [http://scienceworld.wolfram.com/physics/MichelsonInterferometer.html])]] was invented by Albert Michelson in 1882 “to detect a change in the velocity of light due to the motion of the [ether]” [[#References|[2]]]. The findings of Michelson's experiment eventually went on to support Einstein's theory of relativity.
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 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 [[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.
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.
{|width="80%"
|align="right"|
<math>M = \begin{bmatrix}
-1 & 1 & 1 & 0\\
Z_1^{-1} & Z_2^{-1} & -Z_2^{-1} & 0\\
0 & e^{ik_2a} & e^{-ik_2a} & -e^{-ik_1a}\\
0 & Z_2^{-1} e^{ik_2a} & Z_2^{-1} e^{-ik_2a} & -Z_1^{-1} e^{-ik_1a}
\end{bmatrix}</math>
|align="center" width="80"|(4)
|}
{|width="80%"
|align="right"|
<math> v = \begin{bmatrix} E_r\\ E_f\\ E_b\\ E_t\end{bmatrix}</math>
|align="center" width="80"|(5)
|}
{|width="80%"
|align="right"|
<math>b = \begin{bmatrix} 1\\ Z_1^{-1}\\ 0\\ 0\end{bmatrix}</math>
|align="center" width="80"|(6)
|}
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.
{|align=center
|[[Image:Thickness.jpg|thumb|Transmission probability through the silvered mirror versus thickness of the silver coating.]]
|[[Image:phaseshift.jpg|thumb|Phase shift of the transmitted (linear) and reflected (upper) waves versus thickness of the mirror coating.]]
|}
<br style="clear:both;"/>
==References==
<div class="references-small">
1. C. Candler, ''Modern Interferometers'', Hilger & Watts Ltd., 1951, p.9.
2. ibid, p.110.
</div>
==Links to Other Wiki Pages==
[[Surface Resolution]]
[[Using Wavelet Transforms to Analyze Interference Patterns]]
[[Using a Genetic Algorithm to Analyze Interference Patterns]]
==Interference and Light Waves==
Light (electromagnetic radiation) [[Image:emwave.jpg|thumb|An Electromagnetic Wave! (courtesy of [http://www.mtholyoke.edu/~mlyount/MySites/ForensicSpectroscopy/Vocab.html])]] is comprised of an temporally and spatially oscillating electric and magnetic field components. These components are orthogonal to each other as well as to the direction of propagation given by the Poynting vector. Wave equations for the electric and magnetic fields can be derived using [[Maxwell's Equations]].
{|width="50%"
|align="right"|
<math>\nabla^2 \boldsymbol{E} = \frac{1}{c^2} \frac{\partial^2 \boldsymbol{E}}{\partial t^2}</math>
|align="center" width="80"|(1)
|}
{|width="50%"
|align="right"|
<math>\nabla^2 \boldsymbol{B} = \frac{1}{c^2} \frac{\partial^2 \boldsymbol{B}}{\partial t^2}</math>
|align="center" width="80"|(2)
|}
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.
{|width="50%"
|align="right"|
<math>\boldsymbol{E} = \boldsymbol{E_0} e^{i\left(\boldsymbol{k \cdot x} - \omega t\right)}</math>
|align="center" width="80"|(3)
|}
Electromagnetic waves also have the property that they superimpose. [[Image:MichIntInt.jpg|thumb|Interference from a Michelson Interferometer (courtesy of [http://www.arikah.com/encyclopedia/Interference])]] Like mechanical waves, EM waves can interfere destructively or constructively, but, unlike mechanical waves, there is an additional condition for light wave interference. In order for this to occur, the light waves must be traveling in the same direction, be of the same wavelength and have a constant phase with respect to each other. The reason that light waves must be travelling in the same direction to interfere is that in order for interference to occur, both the electric and magnetic field components must "line up". A picture of EM wave interference is shown at right.
==Interferometry and the Michelson Interferometer==
Interferometry is the splitting of light beams into two or more paths and the recombining of those different beams to measure difference in optical path length and refractive index [[#References|[1]]] via interference fringes that form as a result of the recombined beams.
The Michelson interferometer [[Image:MIint.gif|thumb|A Michelson Interferometer! (courtesy of [http://scienceworld.wolfram.com/physics/MichelsonInterferometer.html])]] was invented by Albert Michelson in 1882 “to detect a change in the velocity of light due to the motion of the [ether]” [[#References|[2]]]. The findings of Michelson's experiment eventually went on to support Einstein's theory of relativity.
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 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 [[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.
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.
{|width="80%"
|align="right"|
<math>M = \begin{bmatrix}
-1 & 1 & 1 & 0\\
Z_1^{-1} & Z_2^{-1} & -Z_2^{-1} & 0\\
0 & e^{ik_2a} & e^{-ik_2a} & -e^{-ik_1a}\\
0 & Z_2^{-1} e^{ik_2a} & Z_2^{-1} e^{-ik_2a} & -Z_1^{-1} e^{-ik_1a}
\end{bmatrix}</math>
|align="center" width="80"|(4)
|}
{|width="80%"
|align="right"|
<math> v = \begin{bmatrix} E_r\\ E_f\\ E_b\\ E_t\end{bmatrix}</math>
|align="center" width="80"|(5)
|}
{|width="80%"
|align="right"|
<math>b = \begin{bmatrix} 1\\ Z_1^{-1}\\ 0\\ 0\end{bmatrix}</math>
|align="center" width="80"|(6)
|}
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.
{|align=center
|[[Image:Thickness.jpg|thumb|Transmission probability through the silvered mirror versus thickness of the silver coating.]]
|[[Image:phaseshift.jpg|thumb|Phase shift of the transmitted (linear) and reflected (upper) waves versus thickness of the mirror coating.]]
|}
<br style="clear:both;"/>
==References==
<div class="references-small">
1. C. Candler, ''Modern Interferometers'', Hilger & Watts Ltd., 1951, p.9.
2. ibid, p.110.
</div>
==Links to Other Wiki Pages==
[[Surface Resolution]]
[[Using Wavelet Transforms to Analyze Interference Patterns]]
[[Using a Genetic Algorithm to Analyze Interference Patterns]]