# Focusing EM wave and the Linear Superposition Principle

1. Jul 20, 2012

### htg

Consider a lens of diameter d and a focal length f >> d.
Let the cross-section area of the lens be S = N*λ^2.
Let a plane wave be focused from S to an area S1=λ^2.
Then, by the linear superposition principle the electric field intensity
in the area S1 will be N times greater than it was in the original plane wave.
This means EM wave intensity N^2 times greater in the area S1,
which means N-fold increase of the power of the EM wave due to focusing.
Is focusing a process that violates energy conservation?

2. Jul 20, 2012

### Born2bwire

Focusing merely redirects the energy. If you find an area of increased energy density then you will find another region with an associated decrease in energy density.

3. Jul 20, 2012

### htg

We can consider a cylindrical beam which becomes focused, and according to the linear superposition principle its power would increase N times because of focusing, and saying that somewhere else there will be a decrease of EM wave power does not solve the problem - everywhere else you can have complete darkness and still there would be a violation of energy conservation.
Do you think linear superposition principle is true?

4. Jul 20, 2012

### Born2bwire

How does that violate energy conservation? The region where the wave is focused experiences an increase in energy density. The regions that the wave is redirected out of experiences an appropriate decrease in energy density. The total energy does not change; the density is simply shifted around.

5. Jul 20, 2012

### htg

Intensity of EM wave is N^2 times bigger and the area N times smaller
which gives N times bigger power, if the linear superposition principle is true.

6. Jul 20, 2012

### Drakkith

Staff Emeritus
What is your point? If you have 100 joules of EM radiation falling on a square cm of something, that's 1 j/mm2and stick a lens in that focuses that same 100 j on a square mm, the energy has simply been redirected. You have 100 times more energy falling on that square mm than you did before, but the other 99 square mm of surface area has 0 energy. No violation, the total energy is still 100 j. The total power would be 100 watts before and after.

7. Jul 21, 2012

### Staff: Mentor

If you focus the beam to 1/N of the initial width, the area is reduced to 1/N^2 of the initial area, as it is an area (two-dimensional).
The fields increase by a factor of N, and the intensity in the focus increases by a factor of N^2, therefore both changes cancel each other and you have conserved energy.

8. Jul 21, 2012

### Staff: Mentor

If the area is N times smaller then the power density would be N times larger, as mfb pointed out.

You are misapplying the principle of superposition. It says that if you have two solutions to Maxwells equations then the sum of those two solutions is also a solution. You still have to solve Maxwells equations to get the original two solutions. Here, I don't see a way, even in principle, to apply superposition.

9. Jul 21, 2012

### htg

When there are N "sources" (parts of the surface of the lens, each of area = λ^2 ) and the rays converge in the focal area, by the linear superposition principle we should have N times greater value of the E field (and of H field). In case you want to have no doubts that geometric optics applies, you can consider a "lens" made of N parts, each of which has cross-section area = A >> λ^2 , and which focuses N rays into area = A. It is important to assume that the focal length f >> d = diameter of the "lens".
The whole point is to look at it from the viewpoint of linear superposition. If you assume energy conservation, you refuse to look at the problem, as mfb did.

Last edited: Jul 21, 2012
10. Jul 21, 2012

### Staff: Mentor

OK, I can see how you could apply superposition. But then the question you need to ask yourself is "what is the field from 1 of those sources?" That is what you need to be adding.

Energy conservation is not an assumption, it can be derived directly from Maxwells equations. If your situation violates energy conservation, then it is not a solution to Maxwells equations. In this case, the problem is the wrong form of the field for 1 "source".

http://farside.ph.utexas.edu/teaching/em/lectures/node89.html

11. Jul 21, 2012

### htg

There are phenomena which clearly violate energy conservation - for example consider two tapering optic fibers with tips of diameter d << λ/4 and placed close to each other, at a distance d1 << λ/4. Let the tips emit light in antiphase with respect to each other. Then at a distance d2 > λ you get very small intensity of EM wave - you have almost completely destructive interference everywhere on the whole sphere of diameter d2.

12. Jul 21, 2012

### Staff: Mentor

So then I take it that your question from the OP has been completely resolved? You now understand that a lens conserves energy?

I recommend starting another thread for this new topic, but again, if it doesn't conserve energy then it isn't a solution to Maxwells equations. See the link I posted above.

13. Jul 21, 2012

### htg

Think about the destructive interference example I gave above. Each tip emits light according to Maxwell's equations, but due to destructive interference the energy of the sum of two waves is much smaller than the energy of any of the waves emitted by a single tip.
Somebody talked you into believing that solutions of Maxwell's equations cannot violate energy conservation. I presented a very clear counterexample.
Notice that if sin(wt+kx) is a solution and -sin(wt+kx) is a solution of a linear equation, then also their linear combination, in particular the sum of the two solutions which is equal to 0, also is a solution of this linear equation. This applies to Maxwell's equations.
But I still do not know if focusing a wave by a lens conserves energy.

Last edited: Jul 21, 2012
14. Jul 21, 2012

### Staff: Mentor

I have thought about it. It, and related examples, are very common questions on these forums. Generally someone makes the same mistake you are making on a weekly basis. But it is off topic from your lens question, so if you want to discuss it then you should make a new thread.

OK, then let's continue on the lens question. I have shown you a rigorous derivation of energy conservation from Maxwell's equations, proving your conclusion wrong. I have also pointed out the logical mistake that you are making in your analysis, showing why your analysis led you to a wrong conclusion. So, in your mind, what question remains?

Last edited: Jul 21, 2012
15. Jul 22, 2012

### htg

My assumption is that the area decreases from N*λ^2 to λ^2, not the width.