Question

In this problem, you will derive the equations used to explain the Michelson interferometer for incident light of a single frequency. a. Show that the expression \\[ A(t)=\frac{A_{0}}{\sqrt{2}}\left(1+e^{i \delta(t)}\right) \exp \left[i\left(k y_{D}-\omega t\right)\right] \\] represents the sum of two waves of the form \(A_{0} / \sqrt{2} \exp [i(k x-\omega t)],\) one of which is phase shifted by the amount \(\delta(t)\) evaluated at the position \(y_{D}\) b. Show using the definition \(I(t)=A(t) A^{*}(t)\) that \\[ I(t)=I_{0} /[2(1+\cos \delta(t))] \\] c. Expressing \(\delta(t)\) in terms of \(\Delta d(t),\) show that \\[ I(t)=\frac{I_{0}}{2}\left(1+\cos \frac{2 \pi \Delta d(t)}{\lambda}\right) \\] d. Expressing \(\Delta d(t)\) in terms of the mirror velocity \(v,\) show that \\[ I(t)=\frac{I_{0}}{2}\left(1+\cos \left[\frac{2 \mathrm{v}}{c} \omega t\right]\right) \\]

Short answer

In summary, we have derived the equations related to a Michelson Interferometer for incident light of a single frequency. We showed that the given expression represents the sum of two waves with a phase shift. We derived the intensity function, expressed the phase difference in terms of path difference, and finally, found the relationship between path difference and mirror velocity. The final equation for the intensity function is: \\[ I(t) = \frac{I_0}{2}\left(1 + \cos\left[\frac{2v}{c}\omega t\right]\right) \\]

Step-by-step solution

Write the Sum of the Two Waves

Let us first write the sum of the two given waves, one of which has a phase shift of \(\delta(t)\), as follows: \\[ A_1(t) = \frac{A_0}{\sqrt{2}}\exp[i(ky_D-\omega t)], \quad A_2(t) = \frac{A_0}{\sqrt{2}}\exp[i(ky_D-\omega t+\delta(t))] \\]

Add the Two Waves

Now, let's add these two waves: \\[ A(t) = A_1(t) + A_2(t) = \frac{A_0}{\sqrt{2}}\exp[i(ky_D-\omega t)]\left(1 + \exp[i\delta(t)]\right) \\] This expression matches the given expression for \(A(t)\), proving that it represents the sum of the two waves with the given phase shift. b. Deriving the Intensity function

Find the Complex Conjugate of A(t)

To find the intensity \(I(t)\), we need to find the product \(A(t)A^*(t)\). First, we find the complex conjugate \(A^*(t)\) of \(A(t)\): \\[ A^*(t) =\frac{A_0^*}{\sqrt{2}}\left(1+e^{-i \delta(t)}\right) \exp \left[-i\left(ky_{D}-\omega t\right)\right] \\]

Multiply A(t) and A(t)*

Now, we multiply \(A(t)\) and \(A^*(t)\): \\[ I(t) = A(t)A^*(t) = \frac{A_0A_0^*}{2}\left(1+e^{i \delta(t)}\right)\left(1+e^{-i \delta(t)}\right) \\]

Simplify the expression

Simplify the expression: \\[ I(t) = \frac{A_0A_0^*}{2}(1 + \cos\delta(t)) \\] Let \(I_0 = A_0A_0^*\), then the intensity function is: \\[ I(t) = \frac{I_0}{2}(1 + \cos\delta(t)) \\] c. Expressing phase difference in terms of path difference

Express delta(t) in terms of Delta d(t)

We can express \(\delta(t)\) in terms of path difference \(\Delta d(t)\) using the relationship: \\[ \delta(t) = \frac{2\pi \Delta d(t)}{\lambda} \\]

Substitute delta(t) into I(t)

Now, substitute this expression for \(\delta(t)\) into the intensity function \(I(t)\): \\[ I(t) = \frac{I_0}{2}\left(1 + \cos\frac{2\pi\Delta d(t)}{\lambda}\right) \\] d. Expressing path difference in terms of mirror velocity

Express Delta d(t) in terms of mirror velocity v

We can express the path difference \(\Delta d(t)\) in terms of the mirror velocity \(v\) using the relationship: \\[ \Delta d(t) = \frac{vt}{2} \\]

Substitute Delta d(t) into I(t)

Now, substitute this expression for \(\Delta d(t)\) into the intensity function \(I(t)\): \\[ I(t) = \frac{I_0}{2}\left(1 + \cos\frac{2\pi \frac{vt}{2}}{\lambda}\right) \\] Simplifying further, we obtain: \\[ I(t) = \frac{I_0}{2}\left(1 + \cos\left[\frac{2v}{c}\omega t\right]\right) \\] This is the desired equation for the intensity function with respect to time and mirror velocity in a Michelson Interferometer.

Key concepts

Interference of Light

Interference of light is a beautiful phenomenon that occurs when two or more light waves overlap, creating regions of constructive and destructive interference. Depending on the phase difference between the waves, the combined effect can lead to areas of increased brightness (where the waves amplify each other) or darkness (where the waves cancel each other out). In the context of the Michelson Interferometer, this principle is used to measure very small distances or changes in distance, by observing how the interference pattern shifts.

In practical terms, light from a single source is split into two paths. Each path reflects off mirrors and recombines, creating an interference pattern. The sensitivity of this setup allows for precision measurements at a scale where ordinary rulers are ineffective. The outcome depends on the exact wavelength of light, making it essential for scientific experiments requiring high accuracy.

Phase Shift

A phase shift in a light wave refers to a change in the phase of that wave. In the Michelson Interferometer, phase shifts are critical in determining the interference pattern. This shift can occur due to differences in the path lengths that the light travels.

In mathematical terms, the phase shift \(\delta(t)\) can be represented as an additional term in the wave's exponential function. This term alters the wave's phase angle relative to another wave, leading to different interference effects. For instance, if two waves are completely in-phase, they will interfere constructively. Conversely, if they have a phase difference of \pi\ (half a wavelength), they will interfere destructively.

Understanding phase shifts is key in controlling the resulting interference and accurately measuring the path differences or changes over time, especially in applications such as metrology and optical coherence tomography.

Intensity Function

The intensity function in an interferometer describes how the intensity of light varies as a function of time or position. For the Michelson Interferometer, this function can be expressed as \(I(t) = \frac{I_0}{2}(1 + \cos \delta(t))\).

This tells us that the observed intensity depends on the phase difference \(\delta(t)\) between the two light paths. The function uses trigonometric cosine, indicating the periodic nature of light interference. Constructive interference (when \(\cos \delta(t) = 1\)) results in maximum intensity, while destructive interference (when \(\cos \delta(t) = -1\)) results in minimum intensity.

By analyzing the variation of intensity, one can deduce changes in the optical path difference, which is fundamental in precision measurements. This equation is vital in determining the characteristics of the interference pattern created by the superimposition of light waves.

Path Difference

Path difference is the difference in the length of the paths traveled by two waves before they recombine. In the Michelson Interferometer, by adjusting the mirrors, you can change this path difference and observe the resultant shifts in the interference pattern.

The relationship between phase difference and path difference is given by \(\delta(t) = \frac{2 \pi \Delta d(t)}{\lambda}\), where \(\Delta d(t)\) is the path difference, and \(\lambda\) is the wavelength of light. This relationship shows how a change in one path length relative to the other affects the phase and thus the interference pattern.

By determining how changes in the path length affect the interference, one can derive precise measurements related to distances or motion, crucial in fields like physics and optical engineering.

Mirror Velocity

Mirror velocity refers to the speed at which one of the mirrors in the Michelson Interferometer is moving. This movement changes the path difference over time, affecting the observed interference pattern.

The path difference is expressed in terms of the mirror velocity \(v\) as \(\Delta d(t) = \frac{vt}{2}\). Substituting this into the intensity function allows us to see how variations in velocity influence interference. The equation becomes \(I(t) = \frac{I_0}{2}(1 + \cos\left[\frac{2v}{c}\omega t \right])\), showing that the intensity oscillates based on the product of time and velocity, where \(c\) is the speed of light.

Understanding how mirror velocity affects the interference allows researchers to measure velocity and other dynamic processes with high precision, pivotal in experimental physics and other technical fields.