Surface Resolution

Surface Resolution Approximation

An important part of topological interferometry is that the surface be retained in the reflected image. Plane wave solutions can be used when the surface gradually curves and no sharp peaks are present. Using Huygen's principle and knowledge of the surface, the forward distance for image loss can be calculated.

${\boldsymbol {E}}={\boldsymbol {E_{0}}}e^{i\left({\boldsymbol {k\cdot x}}-\omega t\right)}$

(1)

Huygen's Principle

Each point on the surface can be approximated by an outgoing spherical wave.

Huygen's Principle Illustrated (courtesy of http://www.mathpages.com/home/kmath242/kmath242.htm

Depending on the nature of the surface topology, the shape will be contained in these outgoing spherical waves, but will diffuse over some distance.

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 a valid image of the surface. Using the diagram at right,

Surface Schematic

and some knowledge of the experimental setup, a rought estimate for the forward distance L can be calculated using a small angle approximation for the angle. Using the the diamond surface is about Failed to parse (unknown function "\cross"): {\displaystyle 5 \cross 10^{-3} m} and the thickness is Failed to parse (unknown function "\cross"): {\displaystyle 5 \cross 10^{-6} m} , the forward distance for image loss can then be deduced to be 5m. Since this is much longer than the feature length. Our experimental setup can be considered effective.