Need help reducing exponential rotated plane wave

[mmpstudent]

I have an EM problem (michelson interferometerish) where I have a term that I need to reduce. It is

$$|1+e^{ik \Delta cos\theta}|^{2}+| e^{ik \Delta sin\theta}|^{2}$$

I have foiled it and squared the last term but is there something that I am missing. I am multiplying it by a large matrix and hope that it reduces to something simple.

Thanks in advance

 

[cepheid]

Are 'k' and 'Δcos(θ)' always real and positive? If so, it simplifies a fair bit. First of all, the second term is easy: it's just 1. The magnitude of a complex exponential is unity.

Similarly, the first term can be simplified a lot by applying the Euler relation for complex exponentials.

 

[mmpstudent]

k I'll review that material. I should have said that k is a 4 vectore driven at a frequency in the z direction

 

[cepheid]

Really? k is a 4-vector? Like, in the relativistic sense?

I would have thought that k would be the wavevector (an ordinary 3-vector) and k would be its magnitude, which is the wavenumber 2pi/lambda.

 

[mmpstudent]

i understand that the second term is 1

but my attempt at reducing the first term

$$|1+e^{ik \Delta cos\theta}|=|(1+cos(k \Delta cos\theta))+i sin(k \Delta cos\theta)|$$
$$\sqrt{(1+cos(k \Delta cos\theta))^{2}+sin^{2}(k \Delta cos\theta)}$$
$$=\sqrt{2+2cos(k \Delta cos\theta)}$$

in which I'm stuck

 

[cepheid]

i understand that the second term is 1

but my attempt at reducing the first term

$$|1+e^{ik \Delta cos\theta}|=|(1+cos(k \Delta cos\theta))+i sin(k \Delta cos\theta)|$$
$$\sqrt{(1+cos(k \Delta cos\theta))^{2}+sin^{2}(k \Delta cos\theta)}$$
$$=\sqrt{2+2cos(k \Delta cos\theta)}$$

in which I'm stuck

It's squared. So you can get rid of the square root. Then you can add 1 to that. Isn't that much simpler?

 

[mmpstudent]

yeah I guess, but i was hoping to be able to take it back into a form that would have exponentials again. I wonder if me dropping the complex conjugate terms in the prior steps be the reason why its not simplifying the way I want it to. I will try with them included i guess

 

[cepheid]

yeah I guess, but i was hoping to be able to take it back into a form that would have exponentials again.

You mean complex exponentials? I don't see why you want this, since the magnitude of those two complex numbers obviously gives you a real number, which would seem more desirable. BUT, if you really insist, I suppose you could always express the cosine term in terms of complex exponentials. You know how to do that, right?

I wonder if me dropping the complex conjugate terms in the prior steps be the reason why its not simplifying the way I want it to. I will try with them included i guess

What complex conjugate terms?

 

[mmpstudent]

I forgot I didn't post the whole problem in the beginning. Disregard the complex conjugate part. This might be helpful need to chug thru this now. Thanks