- #1

LmdL

- 73

- 1

I'm considering the "beats" phenomena. I have two plane waves in some medium with a refractive index n(ω), one propagates in a z direction and second in a direction making an angle θ with z axis. Waves have frequencies ω

_{1}, ω

_{2}(not necessarily equal) and k-vectors k

_{1}, k

_{2}(not necessarily equal):

Now, I'm interested in interference (in the area where both waves overlap) along the z axis direction. I can write the initial fields contributions along the z axis as

So their sum is

First cosine is the envelope modulation and second is the carrier wave. Now, my question is regarding the

**modulation speed**. As far as I understand, the above describes the beats phenomena. Or, alternatively, an interference pattern with a period of 2π/Δk that moves in time. If I'm right, the modulation speed reads

And here is the question: this speed can exceed c (speed of light), once I change the θ. For some θ I can even get an infinite v

_{m}(when denominator vanishes). On the other hand, modulation speed should be limited by the speed of light. What I'm missing here?

The possibilities that I thought of:

1. This modulation speed is a sinusoidal and cannot transfer any information, thus isn't limited by c.

2. The expression for v

_{m}is correct, but for values of θ that lead to v

_{m}>c it should be constrained v

_{m}=c .

3. The expression for v

_{m}is an expression for phase velocity and I should use d(Δω)/d(Δk) instead (group velocity). However, in this case I also get velocities that exceed c (checked numerically).

Any advice or thoughts are appreciated.

Thank you.