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I recently read (sorry but I don’t have a reference) that interference is not only about destructive and constructive interference but moving energy from destructive to constructive regions according to conservation of energy. The place where this really bothers me is when light is being transmitted through glass. As light moves through the glass there is forward scattering of light where there is constructive interference between the primary and secondary waves; so light propagates forward. On the other hand, the light that back-scatters (or side scatters as well) destructively interferes with the primary waves (I am assuming that the light has already passed the surface atoms so reflection has already been accounted for). Said another way, I can set up a thin film such that the reflectance is zero (R = 0) and transmittance is one (T = 1), don’t I need energy in the waves to cause destructive interference to begin with? So how does interference account for this energy exchange?

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The basic surprise here is that energy is always conserved when waves superimpose. This seems like it shouldn't be true, since superposition causes amplitudes to add, whereas energy is proportional to the square of the amplitude. This nonlinear relationship makes it seem like energy should not be additive, and therefore can't be conserved.

This is a generic issue for waves, not just light waves, so let's consider it first in the case of waves on a string.

For example, if you take a light string and a heavy string and join them together end to end, waves that hit the boundary will be partially reflected and partially transmitted. One thing we can predict immediately is that energy is conserved, since the interaction between neighboring bits of the strings are governed by the ordinary laws of mechanics. Since energy is conserved at all points in space and time during the evolution of the waves, it must be conserved over all.

Another helpful example is a single uniform string, with oppositely propagating sine waves of the same wavelength encountering one another and superposing. There are moments when the waves cancel and the string is flat, but the motion recovers after such a moment -- was energy destroyed and then created again? No, because the flatness of the string only implied zero potential energy. The string had kinetic energy at the moment of flatness. If each wave separately had 1 unit of KE and 1 unit of PE, for a total of 4 units, then at the moment of flatness, we have 4 units of KE (due to doubled velocity) and 0 units of PE.

So in general, the idea is that conservation of energy has to be proved from the equations of motion, which are a differential equation holding at every point in space and time. For light, these equations of motion are Maxwell's equations, and one can indeed prove conservation of energy from them.

It's a nice exercise to verify conservation of energy for the case of oppositely propagating light waves, as in the second example with the rope above.

I can set up a thin film such that the reflectance is zero (R = 0) and transmittance is one (T = 1), don’t I need energy in the waves to cause destructive interference to begin with?

(I corrected T = 0 to T = 1 in this quote.)

Here we can analyze the reflected wave as a superposition of waves reflected from the front and back surfaces. Amplitudes add, not energies. Therefore these two waves, each of which individually would have had, say, 1 unit of energy, together do not have to have 1+1=2 units of energy. Their amplitudes add to 0, so together they have 0 units of energy.

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  • $\begingroup$ I’ve now read the last paragraph about 20 times and I don’t get it, and I really want to understand this. I don’t understand the sentence “Therefore these two waves, each of which individually would have had, say, 1 unit of energy, together do not have to have 1+1=2 units of energy.” Could you please explain or expand your answer with a different approach? $\endgroup$ – Carlos Aug 21 '13 at 15:59
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Helen, this is likely not a full answer to your question, but I'm pretty sure it's relevant to it: https://physics.stackexchange.com/a/23953/7670

My point in that answer was that what really happens when you look at destructive interference over time, rather than as a sort of given, it always resolves into some variation of reflection or diffraction. The energy is not so much transferred as it is directed as the event evolves.

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    $\begingroup$ thanks for the above link and your answer is fantastic. However, I do not understand and need further explanation. In your analogy, the friend has to put energy into the skip rope so that she does not get hit by the incoming wave; i.e. she is inputting energy. If I apply this to the OP’s question, atoms are radiating energy backwards in order to destructively interfere; i.e. atoms are radiating energy into the system. How is this “energy” being account for in the interference process? This energy is not going forward! $\endgroup$ – Carlos Aug 1 '13 at 4:00
  • $\begingroup$ I agree with Carlos that with waves in glass, energy is radiated backwards so I still see the dilemma. $\endgroup$ – Helen Aug 1 '13 at 14:11
  • $\begingroup$ Helen, and Carlos, I'll read your question more carefully and get back to this, hopefully today. Helen, you likely are already familiar with it, but just in case you have not read it, I would strongly recommend Richard Feynman's paperback "QED: The Strange Theory of Light and Matter". It's a delightful book that talks in detail without using math (remarkable, that) about just how peculiar the reflection of light from matter is. $\endgroup$ – Terry Bollinger Aug 1 '13 at 19:22
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Ben Crowell's answer is good; let me just add a useful way to think about it, in a sort of quantum-mechanical way while retaining the theme of the question: the fully developed superposition of waves (including the reflected, in your thin-film example) says where the energy actually can be transferred.

The places where the waves add up to 0, there the energy can't be deposited. So in your example, the reflected waves from the two surfaces of the thin film will perfectly destructively interfere, hence no energy transfer takes place into the reflected direction.

It's kind of misleading to see the waves that "bounce back and forth" as to actually carrying energy around individually. Energy is one thing, the waves are another. The waves add together interferingly, but energy here, like Ben explains, isn't something with plus and minus parts that can cancel out. An atom doesn't expend energy to re-send a wave that subsequently is perfectly cancelled out.

If you want to think about it at a fairly low level, you can see the resulting wave pattern as shaping where a photon (and hence the energy) sent into the system is most likely to end up.

Also I agree with Terry: Feynman's paperback QED is a must read :)

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Since this came up again I will add my two cents, which consists in delving down to the quantum mechanical level.

Current day physics accepts that the fundamental framework of nature is quantum mechanical. Classical mechanics, classical electromagnetism are emergent theories from the quantum mechanical foundations, in an analogous way that thermodynamics is an emergent theory on the substratum of statistical mechanics. (If one is theoretically inclined here is a link which explains how photons build up the electromagnetic wave).

Conservation of energy is a law in the quantum mechanical framework and keeps on being a law in all emergent theories/frameworks, due to Noether's theorem.

In quantum mechanics the electromagnetic wave, which does not need a medium to propagate or interfere with if there are more than one waves, consists of an enormous number of photons which add up coherently to build up the classical wave with the frequency given by the energy of the photon E=h*nu, where h is the Planck constant and nu the frequency that the emergent classical electromagnetic wave displays.

The simplest way, in my opinion, to see that energy is conserved in a wavefront, whether there is interference or not is to look at the photon framework, since the energy of the wave is contained in the set of photons that create it macroscopically. Where due to interference in the classical wave there is no "light" it means that the photons were deflected to the positions of the bright strips. The path of the photons changed but the collective energy is conserved. This deflection can be clearly seen in the double slit experiments one photon at a time , which in the end collectively display an interference pattern.

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