Target Diamond Structural Analysis

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The Target Diamond
Section in progress

Probing the Diamond's Structure
We must be able to determine the precise thickness and shape of the diamond chip. Because it is tiny (on the order of 50 microns thick), conventional measurements are impossible. Instead, we will use a modified Michelson interferometer. In our design, we will replace one of the mirrors with the target diamond chip. In this modified design, the plane wave returned to the detector will be a combination of three waves (neglecting internal reflection): one reflected off the front of the diamond, one reflected off the back of the diamond, and one reflected off the remaining mirror. However, all that the detector can record is the wave's amplitude. From this, we need to extract thickness and shape.

Thickness Calculation
Both the front and back planes of the diamond are two-dimensional surfaces in three-dimensional space. The recorded amplitudes will form a two-dimensional graph and record amplitude at points across the diamond's surface. Basically, the light wave can be treated as a massive grid of one-dimensional waves normal to the diamond. All of the following calculations are applied to the recorded amplitude of one of these waves, which is the amplitude at one specific point on the diamond.

Light is a wave, and can be expressed as

$$\Psi = A \sin ( \omega t + d ) $$

where A is the amplitude, $$\omega$$ is the frequency, t is time, and d is the phase-shift.

We have a sum of three waves, which can be expressed as

$$\Psi _{Front Of Diamond} = A \sin ( \omega t + d _1 ) $$ $$\Psi _{Back Of Diamond} = A \sin ( \omega t + d _2 ) $$ $$\Psi _{Mirror} = A \sin ( \omega t) $$

(For simplicity, we will say that the wave leaving the mirror has not been phase-shifted.)

Because all three waves are reflections of the same original wave, they all have the same amplitude and frequency.

To find the thickness of the diamond, we only need the first two waves. Although later all three waves will be inexorably tied together, we can begin with only two.

The combined wave equation is unimportant, since we only record its amplitude, which is

$$A^2 _{total} = A^2 _1 + A^2 _2 + 2 A _1 A _2 \cos ( d _2 - d _1 ) $$

=

$$A^2 _{total} = 2 A^2 + 2 A^2 \cos ( d _2 - d _1 ) $$

=

$$A^2 _{total} = 2 A^2 (1 + \cos ( d _2 - d _1 ) ) $$

Because the wave reflecting off the back of the diamond travels through the diamond twice, the term $$ d _2 - d _1 $$ 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 $$ \tau $$.

$$ ( d _2 - d _1 ) V / 2 = \tau $$

We can calculate A by returning the mirror, removing the diamond, and measuring the amplitude from the mirror alone.

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

$$A^2 _{recieved} = A^2 + A^2 _t + 2 A A _t \cos ( d _t ) $$

Because we have values for A and $$ A _t $$, we can almost find $$A _{recieved} $$. Unfortunately, we lack values for the new phase-shift term, which is a function of the phase-shifts of $$ \Psi _1 $$ and $$ \Psi _2 $$. This can be easily expressed as

$$\tan { d_f } = \frac{ \sin d_1 + \sin d_2 }{ \cos d_1 + \cos d_2 } $$

Unfortunately, this term is not simple to calculate. Worse yet, the shape is represented by

$$ d _1 c / 2 $$

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 $$ d _f $$, but that approach seems invalid when all numbers are extremely small. A solution to this problem is necessary to proceed.

Future Updates

 * Add diagrams to this page
 * Improve the mathematical syntax on this page
 * Calculate precise shape terms- solve the $$ d _f $$ problem
 * Compensate for interference and internal reflection