135
edits
Changes
Jump to navigation
Jump to search
Line 35:
Line 35: − + − + Line 45:
Line 45: − + − + − + − Because we have values for A and <math> A _t </math>, we can almost find <math>A _{recieved} </math>. Unfortunately, we lack values for the new phase-shift term, which is a function of the phase-shifts of <math> \Psi _1 </math> and <math> \Psi _2 </math>. This can be easily expressed as+ − <math>\tan { d_f } = \frac{ \sin d_1 + \sin d_2 }{ \cos d_1 + \cos d_2 } </math> − − Unfortunately, this term is not simple to calculate. Worse yet, the shape is represented by − − <math> d _1 c / 2 </math> − − As long as this term, which represents the phase-shift between the mirror and the front of the diamond, remains constant, the diamond is a constant shape. Any variation in the diamond's shape (and therefore distance from the apparatus) will cause a greater phase-shift. − − We can use small-angle approximations to estimate <math> d _f </math>, but that approach seems invalid when all numbers are extremely small. A solution to this problem is necessary to proceed. Line 70:
Line 61: − +
Target Diamond Structural Analysis (view source)
Revision as of 14:13, 19 March 2009
, 14:13, 19 March 2009no edit summary
The combined wave equation is unimportant, since we only record its amplitude, which is
The combined wave equation is unimportant, since we only record its amplitude, which is
<math>A^2 _{total} = C _1 A^2 + C _2 A^2 + 2 C _1 C _2 A^2 \cos ( d _2 - d _1 ) </math>
<math>A^2 _{12} = C _1 A^2 + C _2 A^2 + 2 C _1 C _2 A^2 \cos ( d _2 - d _1 ) </math>
<math>A^2 _{total} / A^2 = C _1 + C _2 + 2 C _1 C _2 \cos ( d _2 - d _1 ) </math>
<math>A^2 _{12} / A^2 = C _1 + C _2 + 2 C _1 C _2 \cos ( d _2 - d _1 ) </math>
Because the wave reflecting off the back of the diamond travels through the diamond twice, the term <math> d _2 - d _1 </math> is twice the thickness of the diamond, in seconds. Because this measurement is in unhelpful units, we can multiply it by the speed of light in a diamond and divide by two for the thickness <math> \tau </math>.
Because the wave reflecting off the back of the diamond travels through the diamond twice, the term <math> d _2 - d _1 </math> is twice the thickness of the diamond, in seconds. Because this measurement is in unhelpful units, we can multiply it by the speed of light in a diamond and divide by two for the thickness <math> \tau </math>.
Therefore, we can rewrite the earlier equation in terms of <math> \tau </math>:
Therefore, we can rewrite the earlier equation in terms of <math> \tau </math>:
<math>A^2 _{total} / A^2 = C _1 + C _2 + 2 C _1 C _2 \cos ( 2 \tau / V _d ) </math>
<math>A^2 _{12} / A^2 = C _1 + C _2 + 2 C _1 C _2 \cos ( 2 \tau / V _d ) </math>
== Calculating the Shape ==
== Calculating the Shape ==
Of course, thickness is not the only thing we need. We will also detect a third laser, which reflects off the mirror. We can calculate that the amplitude of this new combined wave will be
Of course, thickness is not the only thing we need. After calculating <math> \tau </math> and uncovering the mirror, we will also detect the third laser reflection. We can calculate that the amplitude of this new combined wave will be
<math>A^2 _{recieved} = A^2 + A^2 _t + 2 A A _t \cos ( d _t ) </math>
<math>A^2 _{123} = A^2 + C^2 _1 A^2 + C^2 _2 A^2 + 2 C _1 C _2 \cos ( d _2 - d _1 ) + 2 C _1 \cos ( - d _1 ) + 2 C _2 \cos ( - d _2 ) </math>
== Future Updates ==
== Future Updates ==
* Improve the mathematical syntax on this page
* Improve the mathematical syntax on this page
* Calculate precise shape terms- solve the <math> d _f </math> problem
* Calculate precise shape terms- solve the <math> d _f </math> problem
* Compensate for interference and internal reflection</math>
* Compensate for interference and internal reflection