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

$\Psi _{FrontOfDiamond}=\Psi _{1}=C_{1}A\sin(\omega t+d_{1})$

$\Psi _{BackOfDiamond}=\Psi _{2}=C_{2}A\sin(\omega t+d_{2})$

$\Psi _{Mirror}=\Psi _{3}=C_{3}A\sin(\omega t)$

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

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. To remove the third wave, which reflects from the mirror, we can simply obscure the mirror with something that absorbs light, like a black cloth.

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

$A_{12}^{2}=C_{1}A^{2}+C_{2}A^{2}+2C_{1}C_{2}A^{2}\cos(d_{2}-d_{1})$

$A_{12}^{2}/A^{2}=C_{1}+C_{2}+2C_{1}C_{2}\cos(d_{2}-d_{1})$

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 $\tau$.

$(d_{2}-d_{1})V_{d}/2=\tau$

Therefore, we can rewrite the earlier equation in terms of $\tau$:

$A_{12}^{2}/A^{2}=C_{1}+C_{2}+2C_{1}C_{2}\cos(2\tau /V_{d})$

Calculating the Shape

Of course, thickness is not the only thing we need. After calculating $\tau$ 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

$A_{123}^{2}=C_{3}^{2}A^{2}+C_{1}^{2}A^{2}+C_{2}^{2}A^{2}+2C_{1}C_{2}A^{2}\cos(d_{2}-d_{1})+2C_{1}A^{2}\cos(-d_{1})+2C_{2}A^{2}\cos(-d_{2})$

Although this equation looks very complicated, we know that

$(d_{2}-d_{1})=2\tau /V_{d}$

so

$d_{2}=2\tau /v_{d}+d_{1}$

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.

$s=d_{1}v_{a}/2\,$

where $v_{a}$ is the speed of light in air. We can rewrite this in terms of $d_{1}$:

$d_{1}=2s/v_{a}$

Simplifying our initial equation, we find that

$A_{123}^{2}/A^{2}=C_{1}^{2}+C_{2}^{2}+C_{3}^{2}+2C_{1}C_{2}\cos(2\tau /V_{d})+2C_{1}\cos(-2s/v_{a})+2C_{2}\cos(-2\tau /v_{d}-2s/v_{a})$

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

$A_{123}^{2}/A^{2}=C_{1}^{2}+C_{2}^{2}+C_{3}^{2}+2C_{1}C_{2}\cos(2\tau /V_{d})+2C_{1}\cos(2s/v_{a})+2C_{2}\cos(2\tau /v_{d}+2s/v_{a})$

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

Color of the Laser

The laser used must have a wavelength greater than the maximum expected thickness of the diamond. Because the diamond should be, at its thickest, roughly 50 angstroms (50,000 nm) thick, we need light with a wavelength of more than 50,000 nm. Since the wavelength of visible light peaks at less than 800 nm, we will need to use extreme infrared light.

Future Updates

• Add diagrams to this page
• Compensate for interference and internal reflection