# Michelson interferometer average power derivation

• leroyjenkens
In summary, the trig product identity cosαcosβ=\frac{1}{2}[cos(α+β)+cos(α-β)] can be used to show that the time-average power at the detector, given by Pavg = 1+cos(δ), can be derived by integrating over one period (0 to 2pi/w) and dividing by the length of the period. This is done by first squaring Etot and then factoring out E02, resulting in a long expression with multiple cosine terms. By integrating over one period and dividing by the length of the period, the final expression of 1+cos(δ) is obtained.

## Homework Statement

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.

## Homework 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.

## 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.

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

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

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

Anyone with any idea how to do this?

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