Help Deriving Intensity Equation for Optical Michelson Interferometer

[hphdx]

Homework Statement

I am currently studying the optical Michelson Interferometer and I am in need of help with a derivation.

The equation I need to derive is that for the intensity of the light arriving at the viewing point. The equation is given below.

Homework Equations

I(Δ)=I/2 + (1+cos(2∏vΔ/c))

Where Δ=2(d₂-d₁) (i.e the path difference of the rays)

The Attempt at a Solution

I understand the need for the I/2 term to be the beam splitter splitting the intensity of each ray to half. I can see how the bracketed terms lead to the result for destructive and constructive interference by substituting in some values, but cannot derive the equation 'from scratch'.

Many thanks to anyone that can explain the derivation to this equation.

 

[mark726]

Dear student,

The equation you are trying to derive is known as the interference pattern equation for a Michelson interferometer. It describes the intensity of the light arriving at the viewing point, which is dependent on the path difference of the two rays. Let's break down the equation and see how it can be derived.

First, let's define some variables:
I(Δ) - intensity of light at viewing point
I - initial intensity of light
Δ - path difference between the two rays
d1 - distance traveled by one ray
d2 - distance traveled by the other ray
v - frequency of light
c - speed of light

Now, let's look at the equation again:
I(Δ)=I/2 + (1+cos(2∏vΔ/c))

The first term, I/2, represents the initial intensity of light being split in half by the beam splitter. This means that each ray will carry an intensity of I/2.

The second term, (1+cos(2∏vΔ/c)), represents the interference pattern created by the two rays. The cosine function describes the oscillation of the intensity as the path difference Δ changes. The 2∏vΔ/c term represents the phase difference between the two rays, which is dependent on the frequency of light and the path difference.

To derive this equation, we need to understand how the intensity of light is affected by interference. When two waves interfere, they can either constructively or destructively interfere. In constructive interference, the two waves are in phase and their amplitudes add up, resulting in a higher intensity. In destructive interference, the two waves are out of phase and their amplitudes cancel out, resulting in a lower intensity.

In a Michelson interferometer, the two rays are reflected back towards the beam splitter and then towards the viewing point. The path difference Δ between the two rays determines whether they will constructively or destructively interfere at the viewing point. When Δ is an integer multiple of the wavelength of light, the two rays will be in phase and constructively interfere, resulting in a higher intensity. When Δ is a half-integer multiple of the wavelength, the two rays will be out of phase and destructively interfere, resulting in a lower intensity.

Now, let's look at the bracketed term (1+cos(2∏vΔ/c)) in the interference pattern equation. When Δ