Changes

<math>A^2 _{12} / A^2 = C _1 + C _2 + 2 C _1 C _2 \cos ( \frac{4 \pi \tau}{\lambda} ) </math>

<math>A^2 _{12} / A^2 = C _1 + C _2 + 2 C _1 C _2 \cos ( \frac{4 \pi \tau}{\lambda} ) </math>

== Calculating the Shape ==

== Ideal Shape Calculation ==

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

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

Although this equation looks very complicated, we know that

Although this equation looks very complicated, we know that

<math>( d _2 - d _1 ) = \frac{4 \pi \tau}{\gamma}</math>

<math>( d _2 - d _1 ) = \frac{4 \pi \tau}{\lambda}</math>

so

so

<math>d _2  = \frac{4 \pi \tau}{\gamma} + d _1</math>

<math>d _2  = \frac{4 \pi \tau}{\lambda} + d _1</math>

To further simplify, we need a "shape term". This term will represent the difference in the distance traveled by waves 1 and 3. If the phase-shift between 1 and 3 is constant, the diamond is flat; otherwise, it is deformed by some distance expressed by s.

To further simplify, we need a "shape term". This term will represent the difference in the distance traveled by waves 0 and 1. If the phase-shift between 0 and 1 is constant, the diamond is flat; otherwise, it is deformed by some distance expressed by s.

<math>s = d _1 \frac{v _a}{2}</math>

<math>s = \frac{d _1 \lambda}{4 \pi}</math>

where <math> v _a </math> is the speed of light in air. We can rewrite this in terms of <math> d _1 </math>:

We can rewrite this in terms of <math> d _1 </math>:

<math>d _1 = 2 s / v _a</math>

<math>d _1 = \frac{4 \pi s}{\lambda}</math>

Simplifying our initial equation, we find that  

Simplifying our initial equation, we find that  

<math>\frac{A^2 _{123}}{A^2} = C^2 _1 + C^2 _2 + C^2 _3 + 2 C _1 C _2 \cos ( 2 \tau / V _d ) + 2 C _1 \cos ( - 2 s / v _a ) + 2 C _2 \cos ( - 2 \tau / v _d - 2 s / v _a ) </math>

<math>\frac{A^2 _{012}}{A^2} = C^2 _0 + C^2 _1 + C^2 _2 + 2 C _1 C _2 \cos ( \frac{4 \pi \tau}{\lambda} ) + 2 C _0 C _1 \cos ( - \frac{4 \pi s}{\lambda} ) + 2 C _0 C _2 \cos ( - \frac{4 \pi (\tau + s)}{\lambda} ) </math>

Because the cosine is an even function, we can slightly simplify this further.

Because the cosine is an even function, we can slightly simplify this further.

<math>\frac{A^2 _{123}}{A^2} = C^2 _1 + C^2 _2 + C^2 _3 + 2 C _1 C _2 \cos ( 2 \tau / V _d ) + 2 C _1 \cos ( 2 s / v _a ) + 2 C _2 \cos ( 2 \tau / v _d + 2 s / v _a )</math>

<math>\frac{A^2 _{012}}{A^2} = C^2 _0 + C^2 _1 + C^2 _2 + 2 C _1 C _2 \cos ( \frac{4 \pi \tau}{\lambda} ) + 2 C _0 C _1 \cos ( \frac{4 \pi s}{\lambda} ) + 2 C _0 C _2 \cos ( \frac{4 \pi (\tau + s)}{\lambda} ) </math>

This equation, athough complicated, is completely solvable, as the only unknown is the s-term.

This equation, athough complicated, is completely solvable, as the only unknown is the s-term.