# How can I simulate michelson's interferometer circular fringes?

I wanto to simulate the circular fringes on a screen for the michelson interferometer. I know how to identify if there will be a destructive or constructive interference, but what about the radius of the circles? how is the intensity of the fringes when optical path diffference is not an $$\lambda/2$$ or $$\lambda / 4$$ and is like $$0.35 \lambda$$ I want to do something like this. https://demonstrations.wolfram.com/MichelsonInterferometerAndHaidingerFringes/ And also i'm getting confused about the focal distance parameter.

Consider two parallel rays reflected from the two mirrors (separated by distance $$d$$), both with angle $$\theta$$ deviating from the optical axis.

They have a path difference $$\Delta s = 2d\cos\theta \tag{1}$$ and hence a phase difference of $$\Delta\phi = \frac{2\pi}{\lambda}\Delta \tag{2}s$$

So you have two parallel rays with the same amplitude, but different phases \begin{align} A_1(\theta,t)&=A_0\sin(\omega t) \\ A_2(\theta,t)&=A_0\sin(\omega t + \Delta\phi) \end{align} \tag{3}

Superposing these two parallel rays with the lens on the same spot of the screen, you get \begin{align} A(\theta,t) &= A_1(\theta,t)+A_2(\theta,t) \\ &= A_0\sin(\omega t) + A_0\sin(\omega t+\Delta\phi) \\ &= 2A_0 \sin\left(\omega t+\frac 12 \Delta\phi\right) \cos\left(\frac 12 \Delta\phi \right) \end{align} \tag{4}

Calculating the intensity (by $$I=A^2$$), taking the average over time $$t$$, and inserting (2) and (1) you get \begin{align} I(\theta) &= \overline{I(\theta,t)} = \overline{A(\theta,t)^2} = 2A_0^2\cos^2\left(\frac 12\Delta\phi \right) \\ &= 2A_0^2\cos^2\left(\frac{\pi}{\lambda}\Delta s\right) = 2A_0^2\cos^2\left(\frac{2\pi d}{\lambda}\cos\theta\right) \end{align} \tag{5}

When you now draw this $$I(\theta)$$ versus $$\theta$$ (for a fixed $$d$$ and $$\lambda$$), then you get images like those generated with Wolfram Demonstrations Project - Michelson Interferometer and Haidinger Fringes:

In simulations, I feel that the aim should be to try if we can to make the phenomenon we want to simulate emerge indirectly from simpler mechanisms... If we simulate the wave inteference equations directly, putting numbers into the equation will of course give us a wave interference pattern...

Today I was looking exactly to accomplish what you want to do. After some time struggling to try and make the simple output plot with Fourier optics to become the bidimensional Haidinger fringes, I gave up. Then I found this wonderful library by Ian Cooper which contains MATLAB scripts for this, the one you are looking for is 'opMichC.m'. I am definitely going to do some digging in these other scripts later.