Question

I need help with this Electromagnetic Waves homework

(VIII.2) Michelson interferometer Consider a Michelson interferometer as shown in the figure. Two mirrors, MI and M2, are located at distances L and L2 L away from the center of the beam splitter. M has a slight misalignment angle 0 from the horizontal direction. A plane wave is incident from the left. (a) Calculate the spacing of the interference fringes on the screen. M1 M2 Source Beam splitter (b) If M2 is moving away from the mirror at a slow speed, , the interference fringes will move as well Find the direction and speed of the pattern motion Screen

Answer 1

Solution:

Let the origin of coordinates be at the center of M2. Waves are reflected normally by M2 (wave 1) and at an angle θ by M1 (wave 2). The resulting plane waves are:

$\small E_1= E_0 exp\, i [k(z+2\Delta d)-\omega t]\\ E_2= E _0 exp\, i [kzcos\Theta + kx'sin\Theta -\omega t]$

Now the resultant electric field is:

E_total =E₁+E₂

$\small E_{total}=E_0 e^{-i\omega t}exp\, i [k(z+2\Delta d)-\omega t]+exp\, i [kzcos\Theta + kx'sin\Theta]$

The intensity can now be calculated using:

$\small I_{total}=E_{total}\times E^*_{total}$

$\small I_{total}=4I_0cos^2k\left ( \Delta d-\frac{x'}{2} sin\Theta +zsin^2\frac{\Theta}{2} \right )$

for fixed Δd, we have fringes at intervals Δx'

$\small \Delta x'=\frac{2\lambda }{sin\Theta }$

When mirror M2 is moved in a direction normal to its face, the fringe pattern moves as Δd changes. The number, n, of the fringes that cross the center of the screen when M1 is moved a distance Δd is given by:

$\small n=\frac{2\Delta d }{\lambda }$