Energy emitted by EM sources under constructive interference

[IcedCoffee]

TL;DR Summary

How much energy is needed to emit EM wave when there are arrays of same emitters that interferes constructively?

I'm trying to wrap my head around the energy increment under constructive interference. In short, why does energy increase quadratically when each source emit EM wave that interferes constructively?

Suppose we have an array of identical and equidistant sources, each of which span the entire x-y plane and emits plane EM wave of the same frequency and amplitude. They emit EM wave in both positive and negative z directions, and the phase of each source is such that the EM waves interfere constructively in +z direction.

I get that the net field amplitude increases linearly as the wave propagates, and hence the energy increases quadratically. However, wouldn't each source emit the same amount of energy? Suppose the first source emits energy E in +z direction. For energy to increase quadratically, the second source must emit energy 3E, the third source must emit energy 5E, the fourth source must emit energy 7E, and so on.

The only difference seems to be that for each of the "identical" sources, there is pre-existing EM field of different amplitude. Does this make the difference? In other words, does the amount of energy required to generate EM field of a certain magnitude increases (linearly?) with the magnitude of pre-existing EM field?

 

[Paul Colby]

Perhaps you are neglecting the coupling between elements? Each element will be immersed in the field generated by all the other array elements. The power source (transmitter) will have to work against this local field if it's to establish a current identical to an isolated element you are using in your power estimates. This extra work is what you are not taking into account.

 

[Drakkith]

the phase of each source is such that the EM waves interfere constructively in +z direction.

I don't think this would happen in your setup. The sources should interfere with each other, both constructively and destructively, in all sorts of directions.

I get that the net field amplitude increases linearly as the wave propagates, and hence the energy increases quadratically. However, wouldn't each source emit the same amount of energy? Suppose the first source emits energy E in +z direction. For energy to increase quadratically, the second source must emit energy 3E, the third source must emit energy 5E, the fourth source must emit energy 7E, and so on.

Your setup makes no sense. The emitters can't interact with each other at all if all of the energy is being emitted in the z direction, as they are scattered about the xy plane. Perhaps simplify your example and just use two sources?

Note that the energy increases or decreases only where the waves interfere. A line passing through a region occupied by two propagating waves can potentially pass through many regions of both destructive and constructive interference. The sources have little to do with this unless you're talking about how the sources couple with the local field.

 

[Dale]

Summary:: How much energy is needed to emit EM wave when there are arrays of same emitters that interferes constructively?

there is pre-existing EM field of different amplitude. Does this make the difference?

Yes. Clearly. Poynting’s theorem says that the energy requires includes a term $\vec J \cdot \vec E$. The external field increases the $\vec E$ and therefore it takes more energy to produce the same $\vec J$

 

[IcedCoffee]

Yes. Clearly. Poynting’s theorem says that the energy requires includes a term $\vec J \cdot \vec E$. The external field increases the $\vec E$ and therefore it takes more energy to produce the same $\vec J$

Thanks a lot! And I guess that explains why the required energy increases linearly with external field.

 

[IcedCoffee]

Perhaps you are neglecting the coupling between elements? Each element will be immersed in the field generated by all the other array elements. The power source (transmitter) will have to work against this local field if it's to establish a current identical to an isolated element you are using in your power estimates. This extra work is what you are not taking into account.

Thank you! So when a transmitter has a phase so that the wave it emits interferes destructively with the external field, it would rather absorb the external field I believe?

 

[Drakkith]

Thank you! So when a transmitter has a phase so that the wave it emits interferes destructively with the external field, it would rather absorb the external field I believe?

The external field will be trying to induce a current/voltage change in the transmitter, which the transmitter has to expend energy to overcome in addition to the energy it uses to emit its own EM wave.

 

[tech99]

Regarding the variation of field with distance, then for large distances of course it reduces inversely with distance.
For an ordinary array, the power is shared among the elements. However, the received field in the desired direction is the sum of the individual fields - in other words it is "voltage addition", not power addition. So for instance if we have two elements, they will each radiate half the power but the received field will be twice that for one element. So overall we have twice the received power. We see a directive gain about equal to the number of elements.
When we arrange the phasing of two sources so there is some destructive interference, we create a super directive array, which can provide a stronger field at the receiver than for "correct" phasing. This is because the coupling between the elements induces an opposing voltage on a particular driven element, and this means that the driving resistance is reduced. To obtain the same radiated field from that element, the current must remain the same, but we can now provide this current with the expenditure of less power, a consequence of the reduced driving resistance, so the directive gain of the array is increased.
An ordinary array has a directive gain approx equal to the number of elements. A super directive array can exceed this figure, with the proviso that Ohmic losses will be increased and the bandwidth will be drastically reduced.