I Energy of Electromagnetic Waves in Destructive Interference

[Quarker]

Energy is not 'transferred'. The Airy Disc is just a diffraction pattern which is formed by contributions from all parts of what would be the image. We start by considering simple 'interference' between ideal point sources but the image of a star (for example) is due to contributions from all over the objective lens. The resulting Airy pattern is given by an Integral of all contributions. The amplitudes in all directions all scale according to the brightness of the source (star). The energy is spread over an infinite range (in theory) but in practice it's confined to the usual disc image.

And how long you are prepared to wait for an exposure. The outer rings can in fact coincide with another very faint star and that can determine the resolution between to stars.

Assuming an Airy disc with one ring, does the dark ring around the central disc have the same area as the disc and bright ring? Do the bright and dark areas balance out exactly? Is that how energy is conserved?

 

[Charles Link]

Assuming an Airy disc with one ring, does the dark ring around the central disc have the same area as the disc and bright ring? Do the bright and dark areas balance out exactly? Is that how energy is conserved?

The energy (total power) incident onto the aperture is the same as the energy (total power) in the diffraction pattern.

 

[Dale]

Is that how energy is conserved?

No. The preceding two questions have nothing to do with the conservation of energy.

 

[sophiecentaur]

Do the bright and dark areas balance out exactly? Is that how energy is conserved?

What would they 'balance out' to except the total energy? 'Balancing out' implies cancelling out to a zero; that's not what happens. Conservation of energy is a 'law' which you can use to check on your results when you calculate or measure the diffraction pattern. You would expect the Integral of the energy all over the (two dimensional) pattern to equal the total energy entering the aperture / lens. Put a large area thermal detector to measure the energy and it would (should) give the same answer whatever distance the detector is placed and whatever area is covered by the visible pattern (a small dot or a wide disc + rings).

 

[tech99]

It is interesting that where we have an interference pattern, such as from a knife edge or large aperture, individual bright zones can have greater intensity than the average. But due to the smaller area of the bright zone, energy is conserved.

 

[sophiecentaur]

But due to the smaller area of the bright zone, energy is conserved.

I'd put it the other way round, rather. Conservation of energy forces the bright zones to be narrower than you'd maybe expect. There's nowhere else for energy in the pattern to come from or to go.

 

[Quarker]

It is interesting that where we have an interference pattern, such as from a knife edge or large aperture, individual bright zones can have greater intensity than the average. But due to the smaller area of the bright zone, energy is conserved.

Could the central disc of an Airy disc be “brighter” than the surrounding ring? Or could the central disc itself have a higher intensity in the very center of the disc?

 

[Dale]

Could the central disc of an Airy disc be “brighter” than the surrounding ring? Or could the central disc itself have a higher intensity in the very center of the disc?

Certainly. But this has nothing whatsoever to do with conservation of energy. You could also arrange lenses so that the center had a low or zero brightness. Again, with nothing to do with conservation of energy.

The energy flows from the light source to the disks and from the source to the rings, not from the disk to the rings.

 

[Charles Link]

The airy disc is very similar to a single slit diffraction pattern in two dimensions with a circular aperture being very similar to the square that results from the two-dimensional single slit. See also: https://www.physicsforums.com/threa...erfect-focusing-comments.907511/#post-5749159

The peak of the pattern appears in the center.

There is Poisson's bright spot that can get generated when the center of the aperture is blocked, but there is no Poisson's dark spot for an open aperture.

Edit: I need to study this further because, ignoring any obliquity factor, the phasor diagram for a circular aperture is a circle, (that returns to the starting point=zero intensity), if I recall correctly from previous calculations. That would indicate you can get something of a dark spot in the center of the pattern from an open aperture if you have just the right number (=even number) of Fresnel zones.

and I think I just answered this question: For the far field pattern which appears in the focal plane of the lens, the entire aperture all has zero phase angle at the center of the diffraction pattern. The result is you always get the bright spot in the center if you sample in the focal plane of the lens.

 

[sophiecentaur]

But this has nothing whatsoever to do with conservation of energy.

I wouldn't say that; it's more a matter of cause and effect. COE always applies but the actual patterns are more. obviously due to vector additions of a multiplicity of fields. Energy, being a scalar, is difficult (inconvenient) to use as a way of predicting the energy distribution in a diffraction pattern.

Both descriptions of the process will apply and measurement of the accuracy of a pattern can be verified / checked by integrating what you get over a sphere (or whatever). With radio antennae, power can be 'lost' in directions that the designer ignored. Summing the measured powers around the direction of interest could give, say half the expected main beam power and COE would instantly lead you to smell a rat.

 

[Dale]

Could the central disc of an Airy disc be “brighter” than the surrounding ring?

But this has nothing whatsoever to do with conservation of energy.

I wouldn't say that

The reason I said it is that energy is conserved if you arrange for a bright center or if you arrange for a dark center. Energy flows from the source to the disk and from the source to the rings. It doesn’t flow from the rings to the disk or vice versa. So it doesn’t matter which is darker or brighter

 

[sophiecentaur]

energy is conserved if you arrange for a bright center or if you arrange for a dark center.

Yes. Whatever diffraction pattern you manage to produce with your source (and you can design a source to produce any pattern you want - even a hologram), the total energy (integral) over the whole pattern (the hemisphere) will be the energy from the source.