# Michelson interferometer average power derivation

1. Sep 21, 2013

### leroyjenkens

1. The problem statement, all variables and given/known data
Using the trig product identity, $cosαcosβ=\frac{1}{2}[cos(α+β)+cos(α-β)]$, show that the time-average power at the detector can be written as Pavg = 1+cos(δ)

That = is supposed to be a proportional symbol.

2. Relevant equations
Other than the ones given in the problem statement, there are a few:

E1=E0cos(wt)
E2=E0cos(wt+δ)

$$δ=\frac{2∏(2x)}{λ}$$
Etot=E1+E2

P = Etot2

That last = is supposed to be a proportional symbol.

3. The attempt at a solution
Well, I started off by trying to square Etot, which gives me a long expression:
E02cos2(wt)+E02[cos(2wt+δ)+cos(δ)]+E02cos2(wt+δ)

I'm not sure I did that right. I used the trig product rule.

From here, I can factor out an E02, but I still have a bunch of cosine terms that I don't know what to do with. How in the world could I turn those into 1+cos(δ)?

Thanks.

2. Sep 21, 2013

### haruspex

You want average power, so don't you need to integrate wrt t over one cycle?

3. Sep 21, 2013

### leroyjenkens

I don't know. Actually, the question is asking to derive the relationship Pavg= cos(δ)

So to derive that expression, I need to integrate? Do I integrate Etot2?

Thanks

4. Sep 23, 2013

### leroyjenkens

Anyone with any idea how to do this?

5. Sep 23, 2013

### haruspex

Yes, I think it gives the desired answer. Integrate over one period (0 to 2pi/w) and divide by the length of the period.