Surface Resolution - Revision history

Jonesrt: /* Huygen's Principle */

2007-09-12T19:54:22Z

Huygen's Principle

Jonesrt: /* Huygen's Principle */

2007-09-12T19:53:21Z

Huygen's Principle

DemasMa at 13:52, 18 April 2007

2007-04-18T13:52:37Z

New page

== Surface Resolution Approximation ==
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.
{|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"|(1)
|}

== Huygen's Principle ==
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.

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. Using the the diamond surface is about 5 ·10<sup>-3</sup> m and the thickness is 5 ·10<sup>-6</sup> m, 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.

DemasMa at 15:25, 11 April 2007

2007-04-11T15:25:28Z

DemasMa at 15:24, 11 April 2007

2007-04-11T15:24:36Z

DemasMa at 15:21, 11 April 2007

2007-04-11T15:21:20Z

DemasMa at 13:57, 11 April 2007

2007-04-11T13:57:10Z

DemasMa at 13:54, 11 April 2007

2007-04-11T13:54:53Z

DemasMa: /* Huygen's Principle */

2007-04-11T13:54:11Z

Huygen's Principle

DemasMa at 13:51, 11 April 2007

2007-04-11T13:51:46Z

← Older revision		Revision as of 19:54, 12 September 2007
Line 11:		Line 11:


−	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 ~~profile~~ is about 5 ·10<sup>-3</sup> m ~~high~~ and ~~the thickness is~~ 5 ·10<sup>-6</sup> m, 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.	+	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.

← Older revision		Revision as of 19:53, 12 September 2007
Line 11:		Line 11:


−	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. ~~Using the~~ the diamond surface is about 5 ·10<sup>-3</sup> m and the thickness is 5 ·10<sup>-6</sup> m, 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.	+	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 profile is about 5 ·10<sup>-3</sup> m high and the thickness is 5 ·10<sup>-6</sup> m, 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.

@@ Line 9: / Line 9: @@
 == Huygen's Principle ==
-Each point on the surface can be approximated by an outgoing spherical wave. [[Image:huygensprinciple.gif|thumb|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 as the inverse of the distance traveled.  For nearly flat surfaces this diffusion will not be appreciable for a distance depending on the shape of the surface and plane wave solutions can be used to mirror reflected light.
+Each point on the surface can be approximated by an outgoing spherical wave. [[Image:huygensprinciple.gif|thumb|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 as the inverse of the distance traveled.  For nearly flat surfaces this diffusion will not be appreciable for a distance depending on the shape of the surface, and plane wave solutions can be used to mirror reflected light.
 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, [[Image:ResDiagram1.jpg|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.  Using the the diamond surface is about <math>5 \times 10^{-3} m</math> and the thickness is <math>5 \times 10^{-6} m</math>, 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.

@@ Line 9: / Line 9: @@
 == Huygen's Principle ==
-Each point on the surface can be approximated by an outgoing spherical wave. [[Image:huygensprinciple.gif|thumb|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.
+Each point on the surface can be approximated by an outgoing spherical wave. [[Image:huygensprinciple.gif|thumb|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 as the inverse of the distance traveled.  For nearly flat surfaces this diffusion will not be appreciable for a distance depending on the shape of the surface and plane wave solutions can be used to mirror reflected light.
 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, [[Image:ResDiagram1.jpg|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.  Using the the diamond surface is about <math>5 \times 10^{-3} m</math> and the thickness is <math>5 \times 10^{-6} m</math>, 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.

@@ Line 9: / Line 9: @@
 == Huygen's Principle ==
-Each point on the surface can be approximated by an outgoing spherical wave. [[Image:huygensprinciple.gif|thumb|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.
+Each point on the surface can be approximated by an outgoing spherical wave. [[Image:huygensprinciple.gif|thumb|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, [[Image:ResDiagram1.jpg|thumb|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 <math>5 \times 10^{-3} m</math> and the thickness is <math>5 \times 10^{-6} m</math>, 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.
+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, [[Image:ResDiagram1.jpg|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.  Using the the diamond surface is about <math>5 \times 10^{-3} m</math> and the thickness is <math>5 \times 10^{-6} m</math>, 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.

← Older revision		Revision as of 13:57, 11 April 2007
Line 12:		Line 12:


−	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, [[Image:ResDiagram1.jpg\|thumb\|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 <math>~~\small{~~5 \times 10^{-3} m}</math> and the thickness is <math>~~\small{~~5 \times 10^{-6} m}</math>, 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.	+	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, [[Image:ResDiagram1.jpg\|thumb\|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 <math>5 \times 10^{-3} m</math> and the thickness is <math>5 \times 10^{-6} m</math>, 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.

← Older revision		Revision as of 13:54, 11 April 2007
Line 12:		Line 12:


−	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, [[Image:ResDiagram1.jpg\|thumb\|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 <math>\~~smaller~~{5 \times 10^{-3} m}</math> and the thickness is <math>\~~smaller~~{5 \times 10^{-6} m}</math>, 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.	+	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, [[Image:ResDiagram1.jpg\|thumb\|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 <math>\small{5 \times 10^{-3} m}</math> and the thickness is <math>\small{5 \times 10^{-6} m}</math>, 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.

← Older revision		Revision as of 13:51, 11 April 2007
Line 12:		Line 12:


−	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, [[Image:ResDiagram1.jpg\|thumb\|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 <math>5 \~~cross~~ 10^{-3} m</math> and the thickness is <math>5 \~~cross~~ 10^{-6} m</math>, 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.	+	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, [[Image:ResDiagram1.jpg\|thumb\|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 <math>5 \times 10^{-3} m</math> and the thickness is <math>5 \times 10^{-6} m</math>, 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.