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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi = A \sin ( \omega t + d ) }

where A is the amplitude, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi _{Front Of Diamond} = C _1 A \sin ( \omega t + d _1 ) } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi _{Back Of Diamond} = C _2 A \sin ( \omega t + d _2 ) } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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. 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^2 _{12} = C _1 A^2 + C _2 A^2 + 2 C _1 C _2 A^2 \cos ( d _2 - d _1 ) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^2 _{12} / A^2 = C _1 + C _2 + 2 C _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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau } .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( d _2 - d _1 ) V _d / 2 = \tau }

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^2 _{12} / A^2 = C _1 + C _2 + 2 C _1 C _2 \cos ( 2 \tau / V _d ) }

Calculating the Shape

Of course, thickness is not the only thing we need. After calculating Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 ) }

Future Updates

• Add diagrams to this page
• Improve the mathematical syntax on this page
• Calculate precise shape terms- solve the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d _f } problem
• Compensate for interference and internal reflection