Constructive interference and energy conservation

[Wminus]

Hi. Let's say two monochromatic laser beams superimpose in a single point in space in such a way that there's constructive interference. Because there's constructive interference there, the total intensity at that point will be larger than the sum of the separate intensities.

Will this mean that the area illuminated by the lasers in the point of superimposition will shrink such that energy is conserved?

thanks :)

 

[Dale]

Essentially, yes. What happens is that if you have regions of constructive interference then you also have regions of destructive interference also. The energy density is higher where there is constructive interference and lower where there is destructive interference, for overall energy conservation.

 

[Philip Wood]

What happens when two {small) wave sources are placed much less than half a wavelength apart? Destructive interference everywhere, presumably. What then happens to the energy? I think I know the answer to this paradox, but I post it here as a brain teaser. [If a mentor considers it an inappropriate posting, I'd be happy for it to be removed,]

 

[nasu]

Why would be destructive interference everywhere?

 

[tech99]

Hi. Let's say two monochromatic laser beams superimpose in a single point in space in such a way that there's constructive interference. Because there's constructive interference there, the total intensity at that point will be larger than the sum of the separate intensities.

Will this mean that the area illuminated by the lasers in the point of superimposition will shrink such that energy is conserved?

thanks :)

The total power over the combined spot will be the sum of the power of the two beams. There will be a fringe pattern, and the bright fringes represent the coherent addition of the two amplitudes, so they have four times the intensity (W/m^2) of a single beam. The additional power to provide these bright fringes comes from the dark ones, where cancellation is occurring. For the special case where the lasers are side by side and the measurement is made at great distance, in the radiation far field, the round spot of a single laser will become narrower, forming an ellipse, with a peak intensity four times that of one laser. The additional intensity comes from the smaller spot area, in the way you describe.

 

[Philip Wood]

My mistake, and too late to edit! Should say 'constructive everywhere' (because difference in path distances from sources to any point P must also be much less than half a wavelength). So at every point P amplitude is double what it would be from a single source, so energy of oscillation four times as much. Where has the extra energy come from?

 

[Dale]

I think that you are forgetting about diffraction. In the single source situation the light diffracts over a very wide angle. In the two source situation the diffraction is less and the light is much more concentrated. It has a higher energy density in the middle, but lower energy density at large angles. The overall energy is still conserved.

 

[Wminus]

Thanks for the answers guys! :)

Essentially, yes. What happens is that if you have regions of constructive interference then you also have regions of destructive interference also. The energy density is higher where there is constructive interference and lower where there is destructive interference, for overall energy conservation.

Is it really impossible to construct a case where the interference is solely positive? I mean in my example the two rays cross in a single point of constructive interference, so were will the destructive interference take place?

 

[BruceW]

In that 'single point' there will be places of constructive interference and destructive interference. The total radiated power must be the same, regardless of interference between the two beams.

edit: to be able to talk about interference, we need to think of the rays as having some wave-like properties, so we can think of the region of interference between the two rays, we get constructive and destructive interference. If we shrink this region of interference, the smaller region will still have constructive and destructive interference. It is not possible to keep only constructive interference by making the region of interference smaller.

 

[Dale]

Thanks for the answers guys! :)
Is it really impossible to construct a case where the interference is solely positive? I mean in my example the two rays cross in a single point of constructive interference, so were will the destructive interference take place?

Yes, it is impossible. Similarly to Philip, you seem to be forgetting about diffraction. It prevents waves that would cross at a single point. You will always have a finite volume of crossing.

 

[USeptim]

What happens when two {small) wave sources are placed much less than half a wavelength apart? Destructive interference everywhere, presumably. What then happens to the energy? I think I know the answer to this paradox, but I post it here as a brain teaser. [If a mentor considers it an inappropriate posting, I'd be happy for it to be removed,]

For the poynting theorem the energy of the laser beams should be conserved no matter how they interfere. A deeper analysis would show the intensity is distributed.