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Mapping diamond surfaces using interference (view source)
Revision as of 13:47, 14 March 2007
, 13:47, 14 March 2007no edit summary
<math>\nabla^2 \boldsymbol{E} = \frac{1}{c^2} \frac{\partial^2 \boldsymbol{E}}{\partial t^2}</math>
<math>\nabla^2 \boldsymbol{E} = \frac{1}{c^2} \frac{\partial^2 \boldsymbol{E}}{\partial t^2}</math>
<math>\nabla^2 \boldsymbol{B} = \frac{1}{c^2} \frac{\partial^2 \boldsymbol{B}}{\partial t^2}</math>
<math>\nabla^2 \boldsymbol{B} = \frac{1}{c^2} \frac{\partial^2 \boldsymbol{B}}{\partial t^2}</math>
Something about plane wave solutions here.
Something about plane wave solutions here.
<math>\boldsymbol{E} = \boldsymbol{E_0} e^{\boldsymbol{k \cdot x} - \omega t}</math>
<math>\boldsymbol{E} = \boldsymbol{E_0} e^{i\left(\boldsymbol{k \cdot x} - \omega t\right)}</math>
Electromagnetic waves also have the property that they interfere. [[Image:MichIntInt.jpg|thumb|Interference from a Michelson Interferometer (courtesy of [http://www.arikah.com/encyclopedia/Interference])]] Like interfering mechanical waves, EM waves can interfere destructively or constructively depending on the phase difference between the two waves. 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” (Serway and Jewett 1177). A picture of EM wave interference is shown at right.
Electromagnetic waves also have the property that they interfere. [[Image:MichIntInt.jpg|thumb|Interference from a Michelson Interferometer (courtesy of [http://www.arikah.com/encyclopedia/Interference])]] Like interfering mechanical waves, EM waves can interfere destructively or constructively depending on the phase difference between the two waves. 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” (Serway and Jewett 1177). A picture of EM wave interference is shown at right.
==INTERFEREOMETRY AND THE MICHELSON INTERFEROMETER==
==INTERFEREOMETRY 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” (Candler 9) via interference fringes that form as a result of the recombined beams.
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” (Candler 9) 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]” (110). The findings of Michelson's experiment eventually went on to support Einstein's theory of relativity.
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]” (110). The findings of Michelson's experiment eventually went on to support Einstein's theory of relativity. Add something here describing the parts of the Michelson interferometer.
For our experiment we will be utilizing the fringes of the Michelson interferometer to gather information about the topology of synthetic diamond wafers.
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 approximation 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 two sides of a thin piece of optical glass. When a beam of light is directed towards the beam splitter, half of the light travels through to the other side of the splitter and the other half is reflected.
Using our knowledge of electric and magnetic fields in conductors and [[Maxwell's Equations]], we can create a simple approximation of the beam splitter with a light wave at normal incidence. We are interested in finding two main quantities in this approximation: the width of the conducting film and the phase shift that occurs in the conducting film. Need more here about the calculations we did. Using programming power of Matlab, we can solve our system of equations
<math>\left(
\begin{array}{c c c c}
-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{array}
\right)\\
</math>
[[Image:fig2.jpg|thumb|A Graph of the Phase shift]]
[[Image:fig1.jpg|thumb|A Graph of Amplitude versus Width]]
[[Maxwell's Equations]]
[[Maxwell's Equations]]
[[Maxwell's Equations]]