Changes

where <math> n _1 </math> is the index of refraction for air and <math> n _2 </math> is the index of refraction for diamond. We can look these up and calculate R:

where <math> n _1 </math> is the index of refraction for air and <math> n _2 </math> is the index of refraction for diamond. We can look these up and calculate R:

<math>R = 0.17189</math>

<math>R = 0.17189\,</math>

We also need the coefficient of transmission T. However, because <math> R + T = 1 </math>, calculation is easy.

We also need the coefficient of transmission T. However, because <math> R + T = 1 </math>, calculation is easy.

<math>T = 0.82811</math>

<math>T = 0.82811\,</math>

This tells us that about 83% of the laser will be transmitted through the diamond at each reflection. This tells us <math> C _1 </math>:

This tells us that about 83% of the laser will be transmitted through the diamond at each reflection. This tells us <math> C _1 </math>:

<math> C _1 = R = 0.17189 </math>

<math> C _1 = R = 0.17189 \,</math>

For <math> C _2 </math>, we must take into account two transmissions and one reflection. The calculation is easy:

For <math> C _2 </math>, we must take into account two transmissions and one reflection. The calculation is easy:

<math> C _2 = R T^2 = 0.117876 </math>

<math> C _2 = R T^2 = 0.117876 \,</math>

We can continue and calculate <math> C _3</math>, <math> C _4</math>, and <math> C _5</math>.

We can continue and calculate <math> C _3</math>, <math> C _4</math>, and <math> C _5</math>.

<math> C _3 = R^3 T^2 = 0.003483 </math>

<math> C _3 = R^3 T^2 = 0.003483 \,</math>

<math> C _4 = R^5 T^2 = 0.000103 </math>

<math> C _4 = R^5 T^2 = 0.000103 \,</math>

<math> C _5 = R^7 T^2 = 0.000003 </math>

<math> C _5 = R^7 T^2 = 0.000003 \,</math>

Generally, for <math> C _n </math> where <math> n > 2 </math>:

Generally, for <math> C _n </math> where <math> n > 2 </math>:

<math> C _n = T^2 R^{2(n-1)-1} </math>

<math> C _n = T^2 R^{2(n-1)-1} \,</math>

Using just the five waves <math> C _1 </math> to <math> C _5 </math> to begin with, we can recalculate our shape term.

Using just the five waves <math> C _1 </math> to <math> C _5 </math> to begin with, we can recalculate our shape term.

<math>\frac{A^2 _{12345}}{A ^2} = C^2 _1 + C^2 _2  + C^2 _3 + C^2 _4 + C^2 _5 + 2 C _1 C _2 \cos ( d _2 - d _1 ) + 2 C _1 C _3 \cos ( d _3 - d _1 ) + 2 C _1 C _4 \cos ( d _4 - d _1 )</math>

<math>\frac{A^2 _{12345}}{A ^2} = C^2 _1 + C^2 _2  + C^2 _3 + C^2 _4 + C^2 _5 + 2 C _1 C _2 \cos ( d _2 - d _1 ) </math>

<math> + 2 C _1 C _5 \cos ( d _5 - d _1 ) + 2 C _2 C _3 \cos ( d _3 - d _2 ) + 2 C _2 C _4 \cos ( d _4 - d _2 ) + 2 C _2 C _5 \cos ( d _5 - d _2 ) + 2 C _3 C _4 \cos ( d _4 - d _3 )</math>

::: <math> + 2 C _1 C _3 \cos ( d _3 - d _1 ) + 2 C _1 C _4 \cos ( d _4 - d _1 ) + 2 C _1 C _5 \cos ( d _5 - d _1 ) \,</math>

<math> + 2 C _3 C _5 \cos ( d _5 - d _3 )+ 2 C _4 C _5 \cos ( d _5 - d _4 ) </math>

::: <math> + 2 C _2 C _3 \cos ( d _3 - d _2 ) + 2 C _2 C _4 \cos ( d _4 - d _2 ) + 2 C _2 C _5 \cos ( d _5 - d _2 ) \,</math>

::: <math> + 2 C _3 C _4 \cos ( d _4 - d _3 ) + 2 C _3 C _5 \cos ( d _5 - d _3 )+ 2 C _4 C _5 \cos ( d _5 - d _4 ) \,</math>

This is an unnerving equation. However, we only have eleven C-terms that need to be calculated. We can begin with the sum of C-terms and compare it to the idealized version.

This is an unnerving equation. However, we only have eleven C-terms that need to be calculated. We can begin with the sum of C-terms and compare it to the idealized version.

<math> C^2 _1 + C^2 _2  + C^2 _3 + C^2 _4 + C^2 _5 = 0.043453</math>

<math> C^2 _1 + C^2 _2  + C^2 _3 + C^2 _4 + C^2 _5 = 0.043453\,</math>

<math> C^2 _1 + C^2 _2  = 0.043441</math>

<math> C^2 _1 + C^2 _2  = 0.043441\,</math>

Because the difference between these terms is three orders of magnitude less than the measurement, we can count it as error and do not need to compensate for it.

Because the difference between these terms is three orders of magnitude less than the measurement, we can count it as error and do not need to compensate for it.

Next, we need to calculate all of the inner products.

Next, we need to calculate all of the inner products.

<math>2 C _1 C _2 = 0.0405</math>

<math>2 C _1 C _2 = 0.0405\,</math>

<math>2 C _1 C _3 = 0.0012</math>

<math>2 C _1 C _3 = 0.0012\,</math>

<math>2 C _1 C _4 = 0.0000</math>

<math>2 C _1 C _4 = 0.0000\,</math>

<math>2 C _1 C _5 = 0.0000</math>

<math>2 C _1 C _5 = 0.0000\,</math>

<math>2 C _2 C _3 = 0.0008</math>

<math>2 C _2 C _3 = 0.0008\,</math>

<math>2 C _2 C _4 = 0.0000</math>

<math>2 C _2 C _4 = 0.0000\,</math>

<math>2 C _2 C _5 = 0.0000</math>

<math>2 C _2 C _5 = 0.0000\,</math>

<math>2 C _3 C _4 = 0.0000</math>

<math>2 C _3 C _4 = 0.0000\,</math>

<math>2 C _3 C _5 = 0.0000</math>

<math>2 C _3 C _5 = 0.0000\,</math>

<math>2 C _4 C _5 = 0.0000</math>

<math>2 C _4 C _5 = 0.0000\,</math>

Once again, the terms become very small very quickly. Because even the largest internal-reflection induced term (<math> 2 C _2 C _3 </math>) is more than an order of magnitude smaller than the needed terms, we can treat all internal reflection as error and ignore it.

Once again, the terms become very small very quickly. Because even the largest internal-reflection induced term (<math> 2 C _2 C _3 </math>) is more than an order of magnitude smaller than the needed terms, we can treat all internal reflection as error and ignore it.