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We know for EM waves undergoing destructive interference, there's constructive interference somewhere else that compensates for the loss in energy during destructive interference. However, I tried constructing a case (below) where such arguments seemingly don't apply.

Statement: Laser A, placed at the origin, produces a monochromatic EM wave with E field $\vec{E}_a(x,t)=\vec{E}_0e^{i(k_x x-\omega t)}$. Laser B, placed at $x=-\frac{\pi}{k_x}$, produces a monochromatic wave $\vec{E}_b(x,t)=\vec{E}_0e^{i(k_xx-\omega t+\pi)}$. At $t=0$, both lasers are fired for one cycle. Let's say it takes an energy $U_0$ for each laser to produce this wave (since they produce the same wave, they obviously take the same amount of energy). Notice that after a long time, the two waves propagate through space with the EM waves canceling for half a cycle in the region of overlap -- we effectively have one cycle of the same wave, transporting $U_0$, not $2U_0$ (as one would expect). Where did the energy go?

Three things to note with the construction of the problem:

  1. The resulting wave is still a monochromatic plane wave (locally), so the usual calculations (mechanical properties) apply: energy density is still uniform along the waves
  2. I've avoided unrealistic waves that extend out to infinity by restricting the waves to one period. (if you find this troubling, the same problem arises with any plane wave)
  3. The $(\vec{k},\vec{E},\vec{B})$ triad are all oriented in the same direction, so the EM waves do indeed cancel in both electric and magnetic components in the region of overlap (ie. there are no fields in the region of interference)

It seems like a problem that can be solved in classical EM, but I just can't figure out where the energy is going... With waves in media, you can always argue that the energy goes into the medium through which the wave propagates (ex.potential energy in the particles in a string) but this doesn't apply to EM waves in vacuum. Any help is appreciated.

Huchaney
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1 Answers1

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Where did the energy go?

The energy went into source A.

Let's say it takes an energy U0 for each laser to produce this wave (since they produce the same wave, they obviously take the same amount of energy)

This actually is not the case. Although they are producing the same wave, their EM environments are different. They are not symmetrical.

Source B has no external field while generating the pulse. Source A is generating half of its pulse in the presence of an external field that is out of phase with A. This results in energy going into A. So the process is as follows:

Both A and B produce a half-pulse with energy $U_0/2$. A’s half pulse continues on, while B’s is absorbed by A and B produces another half-pulse with energy $U_0/2$.

A output a total of $U_0/2$. B output a total of $U_0$, the first half of which was absorbed by A and the second half continued on after A stopped absorbing it.

Dale
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