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# Destructive Interference and the Conservation of Energy

05/24/2010 8:54 PM

This is a question concerning the wave nature of light and the conservation of energy. Consider a prism that combines two sources of monochromatic, coherent (laser) light into one beam. The sources are adjusted so that the laser beams combine in phase resulting in constructive interference. So, the resultant beam has the same wavelength as the parent beams but it's electric and magnetic vectors have twice the amplitude. Now consider the same setup but with the one parent beam adjusted so that the beams combine 180 degrees out of phase, resulting in 100 percent destructive interference. The electric field vector of the first beam exactly cancels the electric field vector of the second beam and vice versa; the same is true for the magnetic field vector. It seems that at this point, with the two light beams perfectly combined 180 degrees out of phase, that the resultant beam of light is effectively obliterated. If so, where does the energy go? How is the law of conservation of energy maintained when considering this system?

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#1

### Re: Destructive Interference and the Conservation of Energy

05/24/2010 10:02 PM

Well first, not even light from a laser is truly monochromatic. First there's the slight energy separation that happens because of the different electron spin variation that the two electrons occupying each quantum energy state. Next the laser light is not perfectly coherent, I don't remember the reason for this right now but I believe that it is true. So this scenario is more of a thought experiment than an obtainable one. But I think point is still the same. In the thought experiment to obtain two perfectly synchronous beams to converge into one perfectly cancelling beam requires knowing too many things to an absolute precision. Remember that another puzzling aspect of quantum mechanics is that you cannot know more than one parameter absolutely, the Uncertainty Principle.

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#2

### Re: Destructive Interference and the Conservation of Energy

05/24/2010 10:18 PM

The purpose of the question was indeed a thought experiment. So, assuming that we did have the precision to combine two perfectly synchronous laser beams such that each cancels exactly, what is the consequence for the universe? If the waves do indeed combine then they cannot be 'separated' again with some other apparatus. What then accounts for the energy lost to the universe?

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#3

### Re: Destructive Interference and the Conservation of Energy

05/24/2010 11:19 PM

My point of bringing up the uncertainty principle is not that this is difficult to obtain, this is impossible to obtain. This is less possible to obtain as spontaneously making matter appear out of nothing or to momentarily at a location have matter and energy seem to disappear.

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#4

### Re: Destructive Interference and the Conservation of Energy

05/24/2010 11:21 PM

I don't know, but I suspect it may be geometrically impossible to combine two nonaligned beams of the same λ into a single beam.

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#5

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 12:25 AM

I think you're right. No beam of light has zero divergence, not even laser light. If the two beams were successfully combined such that there were perfect cancellation (180° out of phase) at beam center, anywhere off center will not be perfect cancellation, and at some angle, there will be a cone of perfect addition (in phase).

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#11

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 1:32 PM

This is quite interesting, could you elaborate on it a bit further?

If no beam of light has zero divergence, and our two "perfect" beams have exactly equal angles of divergence, then why would the wave fronts go out of phase (or back in phase) away from the central axis?

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#13

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 2:07 PM

Since all beams have divergence, only one geometrical line (such as the center of the beam) can enter your magic prism at the correct angle to interfere completely with one geometrical line of the other beam. The majority of the energy will be entering at at least slightly different angles, and therefore have less than complete destructive interference.

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#6

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 12:38 AM

Heat.

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#7

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 2:52 AM

Good question!

How does this theoretical prism work?

A light beam must always travel in the same direction (follow the same path), even if its sense (forward/backward) is reversed.

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#9

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 11:35 AM

It could be a Glan-Taylor Prism, for example. One of the input beams would be polarized in one axis, the other beam polarized in the orthogonal axis. The two beams could then combine to provide constructive or destructive interference.

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#8

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 9:10 AM

In any kind of optical interference (constructive or destructive) the photons are interacting with a layer (or layers) of electrons in the material. Whether the interference is like the scenario you describe, or the kind of destructive interference that occurs in a reflection-reducing coating on a filter or lens, the photons must interact with something. The photons that are 'destroyed' are absorbed by the electrons they are interacting with. Once absorbed, the photons may show up as heat, or may show up as stray light emitted at some other angles (perhaps as a glow at the edges of the optical component).

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#10

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 1:26 PM

First lets assume all this is occurring in perfect vacuum so we can eliminate scattering from objects in the path of the beam.

Indeed, some scattering, heat generation, etc. will occur in the theoretical prism. However it should be equal for both light sources. My concern is what comes out of the prism. If a device (photo voltaic or something else) is positioned to absorb the emerging beam, energy can be captured in the first case but not in the second. I'm interested in what happens to the energy in the second case. A related question is, for two out-of-phase beams traveling together perfectly on the same trajectory, do the beams exist at all? And if they do, (even if they could not be separated by physical means), is that a manifestation of the particle nature of light? In other words, are the photons still there even if their wave nature is now (perhaps permanently) undetectable?

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#12

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 1:44 PM

Again, all light beams have divergence. The perfect destructive interference can only occur along a line of zero thickness, such as the center of the beam(s). At any other angle, no matter how small, the phase difference will be something less than 180°, so the cancellation will not be complete, at least according to wave theory.

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#14

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 5:02 PM

Okay, so you've got a magical prism that combines two laser beams in a way that causes complete destructive interference as the two beams 'emerge' from one of the prism faces. So nothing comes out of the prism, yet your magic prism manages to absorb none of this energy. (Whereas in a real prism the energy is absorbed and either converted to heat or re-radiated at a different angle.)

And now you want someone else to explain how your magic prism works. Perhaps you could send a note to J K Rowling. Maybe she has an answer.

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#15

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 5:38 PM

"Indeed, some scattering, heat generation, etc. will occur in the theoretical prism. However it should be equal for both light sources."

Two points:

1) I'm not concerned with how the theoretical prism works (it's not possible in the real world and irrelevant to the question asked). Remember that this is just a thought experiment.

2) "energy is absorbed and either converted to heat or re-radiated at a different angle." - right, but shouldn't this amount of energy be equal? I see no reason why one of two identical beams should have more energy absorbed or re-radiated than another.

I've tried to attach a picture of the set up to clarify.

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#16

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 6:26 PM

This is precisely the trap of thought experiments. One can think of things that cannot exist and then wonder what might happen. Once one accepts one impossibility, any following possibility can happen.

But let me ask you what will happen with the light from source A in your diagram. The light from Source A must have some width dimension. Because of this the light in your prism from A will no longer be precisely entering the crystal at the same phase angle across the entire width of the beam width. On top of that the transition to a different media itself will cause some added divergence.

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#18

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 8:29 PM

"I see no reason why one of two identical beams should have more energy absorbed or re-radiated than another."

It doesn't. The same energy is involved in both situations.

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#17

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 7:28 PM

You are correct that electromagnetic radiation can combine destructively to create nulls. This is most commonly seen with radio transmisions where 2 or more signal of similar strengths at the same frequency cancel each other out (simulcast distortion) creating nulls and beating patterns. If you look at the situation of signals from two different sources at the same frequency and measure the energy in a large area around them you find that the areas with nulls (destructive interference) and the areas with peaks (constructive interference) are such that the total energy is conserved. In your example I suspect that while you will have a null along a certain path from the prism there will be other paths around/near this where there will be constructive interference such that the total energy is conserved. So in your thought experiment your theoretical prism will need to have dimensions suitable to the frequency of light that is being combined, ie, it cannot be infinitely small, therefore there will be a range of paths that the light can take through the prism since the light itself cannot follow an infinitely narrow beam (once again related to its frequency, heisenberg uncertainty etc) so there will be many paths but only one of which allows perfect cancellation. The end result is that it is not possible (even in a thought experiment unless you ignore fundamental physical laws) to completely cancel two light beams by destructive interference such that the energy disapears everywhere. It is only possible to form localised nulls along certain paths.

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#19

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 10:12 PM

This ties in with what dkwarner said and is probably the best answer I heard yet. But I'm left wondering if a single photon can exhibit divergence?

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#20

### Re: Destructive Interference and the Conservation of Energy

05/26/2010 10:32 PM

Yes, a single photon can have divergence. A single electron can diverge and interfere with itself. For those of you who think that multiple electrons must be present to interact with each other, this experiment worked with about 1000 electrons per second (about 10^-16 amperes). A single electron diverges and passes through two slits to then interfere with itself.

Now don't forget, the wavelength of an electron is much smaller than a photon.

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#22

### Re: Destructive Interference and the Conservation of Energy

05/27/2010 9:32 PM

I think it can be misleading to mix up terms from the wave-like nature of light and the particle like nature. Talking about divergence of photons is such a case. If we are now looking at a "particle nature of light" interpretation of the constructive and destructive interference then it is more accurate to say that the probability of a photon taking a particular path is higher or lower depending on how the photons interact. In this interpretation photons can be considered as being particles with energy dependent on their frequency and position/movement determined by a probability function. This probability function interacts with both the environment (as with the double slit experiment where the probability function of a single photon can interact with both slits) and with other photon's probability functions. When photons from two or more sources are combined they do not cancel each other or combine to form photons of higher intensity, instead the probability of any photon taking a particular path is modified so that there is very little or no probability of taking some paths (nulls) and high probability of taking others (peaks). The total number of photons (and therefore the total energy) remains the same, its just the paths that they take that are altered by the interaction.

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#23

### Re: Destructive Interference and the Conservation of Energy

05/27/2010 10:05 PM

I agree that it is confusing, but I would not call it misleading. Quantum mechanics is just plain weird. Everything acts both as a particle and a wave. The best approach I find when working in the energies and dimensions that quantum mechanics rule is to try to forget our macroscopic world of classical mechanics. You can never precisely know anything more than one parameter at a time. Things rarely make smooth transitions. Things cannot be blocked by a forbidden region acting as a wall. The wave property just has to have enough energy on the other side of the forbidden region to cross the region without ever residing in the forbidden region.

On that last apparent quantum mechanics paradox, are there particles that can tunnel across the event horizon of a black hole?

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#24

### Re: Destructive Interference and the Conservation of Energy

05/28/2010 1:14 AM

I totally agree with you that QM is weird and that is the reason that I consider the use of divergence for a single photon as misleading. Looking at where divergence has been used in this thread it seems clear to me that most people are talking about the standard definition of divergence as the angular measure of the increase in beam diameter with increasing distance. As far as I am aware quantum mechanics does not have a generaly accepted explanation for what happens to photons or electrons when they travel through a double slit as per your reference (do they spread out and travel through both slits or stay the same "size" while their probabilistic wavefunction travels through both and decides which path they take). I think it is misleading to apply divergence to the necessarily QM motion of a single photon as it assumes that we know what is happening to the photon while it is in motion.

Re your question about tunnelling across the event horizon of a black hole - I think this is called Hawking radiation. Particle/antiparticle pairs are created at the event horizon and some particles tunnel out of it while the anti-particles fall into the black hole with the end result that black holes emit radiation and if they are small enough (so that the amount they emit is larger than what falls into them) they lose mass and completely disapear.

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#25

### Re: Destructive Interference and the Conservation of Energy

05/28/2010 1:18 PM

My point of posting the electron scattering paper is that one and only one electron was in flight as it interacted with the slits and the region after the slits. It could only have interfered with itself. So applying the natural classical mechanics idea of the electron (particle) passing through one slit or the other is wrong. The wave like probability distribution for each particle passed through both slits by dispersion and deposited their energy in the expected interference pattern. Now you can validly say that single entity items cannot disperse, only waves can disperse. But then you need a new word for when a single particle acts like a wave.

Lastly, I believe that Hawking radiation originates from particles already inside the event horizon. I'm proposing a step up from this idea of a particle with a trajectory to intersect the black hole's event horizon that never resides inside the event horizon.

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#21

### Re: Destructive Interference and the Conservation of Energy

05/27/2010 1:17 AM

I think you said it more clearly than I did! GA!

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