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→Huygen's Principle
== Surface Resolution Approximation ==
== Surface Resolution Approximation ==
An important part of topological interferometry is that the surface be ''retained'' in the reflected image. Plane wave solutions are most effective when the surface gradually curves and no sharp peaks are present.
An important part of topological interferometry is that the surface profile is imprinted on the phase of the wave reflected from the surface. Plane wave solutions can be used when the height of surface features is much smaller than the features' transverse size. Huygen's principle can be used to estimate the distance the reflected wave propagates before there is significant smearing due to transverse diffusion of the phase gradient.
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<math>\boldsymbol{E} = \boldsymbol{E_0} e^{i \left( \boldsymbol{k \cdot x} - \omega t \right)}</math>
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== Huygen's Principle ==
== Huygen's Principle ==
Each point on the surface can be approximated by an outgoing spherical wave (away from the surface). These waves have solutions such as <math>\boldsymbol{E} = \boldsymbol{E_0} e^{i \left( \boldsymbol{k \cdot x} - \omega t \right)}</math>
Each point on the surface can be approximated by an outgoing spherical wave. [[Image:huygensprinciple.gif|right|Huygen's Principle Illustrated (courtesy of [http://www.mathpages.com/home/kmath242/kmath242.htm]]] The shape of the surface is imprinted on the phase of these outgoing spherical waves, but will gradually diffuse as the wave propagates forward. For nearly flat surfaces this diffusion will only occur gradually, and approximate plane wave solutions can be used as long as the wave has traveled significantly less than some diffusion length scale.
[[Image:ResDiagram1.jpg|thumb|Surface Schematic]]
[[Image:huygensprinciple.gif|thumb|Huygen's Principle Illustrated (courtesy of http://www.mathpages.com/home/kmath242/kmath242.htm]]
In our model of the diamond surface, Huygen's principle can be used to determine the forward distance from the surface when the reflected light will no longer contain an undistorted image of the surface. Using the diagram at right, [[Image:DIAGRAM1.jpeg|thumb|Surface Schematic]] and some knowledge of the experimental setup, an estimate for the forward distance ''L'' can be calculated using a small angle approximation for the angle. Assuming that the diamond surface is about 5 ·10<sup>-3</sup> m in transverse dimensions and 5 ·10<sup>-6</sup> m in profile height, the forward distance for image loss can then be deduced to be 5m. Since this is much longer than the distance that the beams travel inside the Michelson interferometer we are using (typical scale 10 cm), the plane-wave approximation is appropriate.