Constructive interference - energy non-conservation?

[htg]

Consider two identical optic fibers with deeply subwavelength tips. Let their tips be placed next to each other, at a distance much smaller than the wavelength. Let light of equal intensity come out of them in phase. Then, by the principle of linear superposition, at some distance much bigger than the distance between the tips, the intensity of light is 4 times greater than it would be if we had light coming only from one tip. If each tip sends power P, then out there we have power 4P.

 

[Petr Mugver]

I don't understand much of what you have said, but if in some region the I = power/m^2 of light is big, then in other regions it is small, and when you make the flux of I on a closed surface that surrounds the light source, you get exactly P.

 

[Dr Lots-o'watts]

... the intensity of light is 4 times greater than it would be if we had light coming only from one tip. If each tip sends power P, then out there we have power 4P...

No it is not. Is is 2P.

 

[htg]

By linear superposition, E and H will be 2 times stronger, so the power will be 4P.

 

[Dale]

Don't forget that E and B (squared) are proportional to energy density. A high energy density does not by itself imply non-conservation of energy.

 

[pallidin]

Consider two identical optic fibers with deeply subwavelength tips. Let their tips be placed next to each other, at a distance much smaller than the wavelength. Let light of equal intensity come out of them in phase. Then, by the principle of linear superposition, at some distance much bigger than the distance between the tips, the intensity of light is 4 times greater than it would be if we had light coming only from one tip. If each tip sends power P, then out there we have power 4P.

I see nothing particularly significant with this. And, some errors.
2p is possible, but is momentary, like a "pulse"
Even 4p(8, 16, etc) is possible, but the pulse separations are spread even further apart.

Thus, though it might SEEM like a huge power gain, this is not the ACTUAL case at all. The TOTAL pulse power equals or is commonly LESS than the total input.

Even still, there are certainly many notable applications of pulse power.

 

[Dr Lots-o'watts]

By linear superposition, E and H will be 2 times stronger, so the power will be 4P.

Beams that come out of fibers are not plane waves. They are more often approximated as beams in which E is a function of radius. This gives rise to a pattern of bright and dark rings enclosed within each other on the far out screen. For 2 initial beams E=E(r), the bright rings have an irradiance up to I=4kE^2 W/m^2, and the dark rings, down to I=0, but the total power, integrated over the entire set of bright and dark rings, will be 2P.

 

[htg]

...This gives rise to a pattern of bright and dark rings...

The tips of the optic fibers are at a distance much smaller than the wavelength. So there are no dark fringes - constructive interference takes place at every point.

 

[Dr Lots-o'watts]

What is the diameter of each fiber?

 

[Dr Lots-o'watts]

Oh that's right, subwavelength. In any case, I don't see what's backing your statement.

 

[Dale]

The tips of the optic fibers are at a distance much smaller than the wavelength. So there are no dark fringes - constructive interference takes place at every point.

Don't forget the magnetic field.

 

[htg]

Diameter of fibers is not that important. They have to have conical tips, whose diameter at the ends must be small compared to wavelength.
There are scanning optical microscopes which use such tapering fibers.

 

[Dale]

The E-field, by itself, is also not important for energy conservation. I have tried to get you to think about these issues, but not successfully it seems.

The E- and B-fields are proportional to the square of energy density, not energy. You can have an arbitrarily high energy density in a sufficiently small volume and still have conservation of energy.

Conservation of energy is related to the energy within a volume (integral of the square of the E- and B- fields over the volume), the flux of energy across the surface enclosing the volume (integral of the cross product of the E- and B- fields over the surface), and the work done on matter within the volume (dot product of the current and the E field). See equation 1034 at http://farside.ph.utexas.edu/teaching/em/lectures/node89.html

So, with those comments, what about the B-field in the region between the fibers? How does that affect your considerations about energy conservation?

 

[Dr Lots-o'watts]

Diameter of fibers is not that important. They have to have conical tips, whose diameter at the ends must be small compared to wavelength.
There are scanning optical microscopes which use such tapering fibers.

So you're saying that if the ends of optical fibers are conical, then energy is created, right?

 

[htg]

At some distance from the tips, say at least 1 wavelength, both the B field and the E field will be about 2 times stronger than in the case of light coming only from one tip. So the power will be about 4 times the power in the case of a single tip radiating.

 

[Dr Lots-o'watts]

You know, I'm interested in these systems, but I don't understand if you're making an announcement, asking a question, wanting attention or something else. Just let me know if I can help.

 

[Dale]

And if the energy density in some region is higher what does that imply about the energy density in other regions?

 

[Dadface]

Diameter of fibers is not that important. They have to have conical tips, whose diameter at the ends must be small compared to wavelength.
There are scanning optical microscopes which use such tapering fibers.

The smaller the diameter the greater the diffraction and the wider the emergent beams.Each diffraction pattern on its own is an interference pattern with bright and dark regions.If the emergent beams are not coherent there will be overlapping diffraction patterns and if they are coherent there will be Youngs type fringes modulated by the diffraction envelopes.

 

[pallidin]

Would it be wrong to suggest that constructive and destructive interference is immutably intertwined?
That is, one can not exist without the other.

 

[Dale]

In free space that is correct. In the presence of matter it is possible to have completely destructive or completely constructive interference, in which case there will be energy transferred to or from the matter.

 

[Dr Lots-o'watts]

In free space that is correct. In the presence of matter it is possible to have completely destructive or completely constructive interference, in which case there will be energy transferred to or from the matter.

Isn't that what we call absorption and amplification?