Why do we say diffraction is the resolution limit if resolution comes from interference patterns?

[pipnina]

I probably misunderstand this somewhere, but I have been reading recently and notice that since the property of optical resolution comes from the differences in length of light paths from the source to lens to focal plane producing a complicated fringe interference pattern. This would seem to me to be a separate effect to diffraction yet when we talk about perfect optics we describe them as "diffraction limited". Similarly for obstructed designs we call the spiked stars "diffraction spikes" but seemingly these emerge not from light waves bending around the support structure, but from "holes" in the aperture, removing the fringe patterns that would form the final image without diffraction spikes? But this doesn't sound like diffraction at all.

Is diffraction involved in a way I don't understand in this scenario or is it simply another way of thinking about the same optical effect?

[globular]

1 hour ago, pipnina said:

for obstructed designs we call the spiked stars "diffraction spikes" but seemingly these emerge not from light waves bending around the support structure, but from "holes" in the aperture

An SCT has a big hole in the aperture but no spiked stars. I thought this was because it has a glass corrector plate holding the secondary mirror, rather than solid vanes like in Newts, for example, which do have spiked stars.  i.e. it seems to me more like the vanes causing the spikes rather than the secondary mirror / hole in the aperture.
But I feel like the blind leading the blind, so I'll exit stage left and wait for someone to post the real position.

[pipnina]

8 minutes ago, globular said:

An SCT has a big hole in the aperture but no spiked stars. I thought this was because it has a glass corrector plate holding the secondary mirror, rather than solid vanes like in Newts, for example, which do have spiked stars.  i.e. it seems to me more like the vanes causing the spikes rather than the secondary mirror / hole in the aperture.
But I feel like the blind leading the blind, so I'll exit stage left and wait for someone to post the real position.

The distinction there is the shape of the obstruction. Removing a circular profile from the middle reduces contrast but won't affect star shape. Meanwhile the spiders cross the whole aperture in a straight line so a whole section of the horizontal point spread function is removed. It's a small difference and only affects a small contrast level, but it notices on very bright stars.

[DaveS]

I *think* I know from my "A" Level physics but this is a question for @vlaiv.

[saac]

The different path lengths taken by the light which in turn create the interference fringes are created by the process of diffraction. The single ray of light becomes two (multiple) rays which in turn create the interference pattern.  If you want to look further you could take a look at interference by division of amplitude or wavelength. 

Hyper Physics provides a nice description of diffraction limitation criteria in terms of (Rayleigh Criterion) interference fringes.

Diffraction Limited - Rayleigh Criterion 

Jim 

Edited August 21 by saac

[vlaiv]

In sense both are correct, or rather diffraction is the effect that is the same as "different paths".

There are several ways you can "describe" what is going on - and while there is only "one correct" way - they are all really the same thing - we just look at it from a different perspective.

One way of looking at diffraction is from perspective of Heisenberg's uncertainty principle. We might think of a photon being particle that travels thru the space but we have this relationship that tells us that we can't know both exact position and exact momentum of a particle.

When a particle is close to the edge - we in essence know where it is - but uncertainty in momentum must then be larger - and since "speed" of light is fixed - angle of momentum then changes and it "scatters" in direction perpendicular to the edge.

You get the same exact result if you observe something called Feynman's path integral and in that sense - that would be correct interpretation.

What it states is that you should take any path that light takes with a "little clock" (actually a phase of a wave) and this little clock is ticking as particle moves along the path and at a certain point all paths taken by such particles are summed. But they are summed as complex numbers - meaning that phase is taken into account - or if two paths arrive at the same spot - one at 12 o'clock and other at 6 o'clock - those two cancel out - but if both arrive at 12 o'clock - they add up.

Now this is very simplified application of Feynman's path integral - but it is true none the less, and if you think about it - it is the same as letting the EM wave propagate all over the place and letting it do constructive and destructive interference with itself. This is actually what is happening according to QM (or rather QFT - particularly QED) except that wave is not EM wave in classical sense - it is a wave in quantum field - a wave function and it represents the state of quantum field and it's amplitude represents likelihood of finding a photon (particle of said field) at any particular place.

Actual limit to resolution comes from mathematics related to above concepts. Above is description of how nature behaves - we have some quantum fields and there are particles that are excitations of said fields and they propagate as wave-function in said field. If we look at the math behind that we see that Fourier transformation is involved and that aperture placed in such field acts as low pass filter - meaning it effectively cuts off higher frequencies (here we think of spatial higher frequencies - or ability to distinguish direction of incoming photon based on its position, or rather position of detection).

Hope that makes some sense (happy to clarify or expand if needed).

 

 

[vlaiv]

Just to add up - what we consider diffraction in classical physics / wave mechanics - can also be easily explained with above path integral.

Some other interesting phenomena can also be explained with path integral - like fact that glass reflects light at certain frequency (depending on thickness of the glass) - effect that is responsible for seeing rainbow color on puddle of water with oil in it (oil form thin film) - or maybe if we look at CD - we see the same effect.

There is also something called diffraction mirror - which should not work according to classical wave theory - but is nicely explained with path integral.

 

[vlaiv]

Here is nice video explaining how mirror works using path integral:

 

[Ags]

I always thought diffraction and interference were the sane thing? As in the diffraction ring around a star is an interference pattern...

[vlaiv]

2 minutes ago, Ags said:

I always thought diffraction and interference were the sane thing? As in the diffraction ring around a star is an interference pattern...

Technically interference is wave phenomena which results in either cancellation or enhancement of the wave (positive / negative interference), while diffraction is term that describes how wave can travel around obstacles.

To quote wiki:
 

Quote

Diffraction is the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture.

If we think of light as particles of light / photons traveling in straight line - then it looks like photons bend their trajectory near edges (like example with Heisenberg's uncertainty principle as explanationI gave above).

But all of them - including for example refraction can be explained with adding of little clocks along all possible paths.

[CraigT82]

17 minutes ago, vlaiv said:

Here is nice video explaining how mirror works using path integral:

 

She lost me at 4.09, how did she determine the direction of each of those vector arrows beneath the graph? 

[vlaiv]

3 minutes ago, CraigT82 said:

She lost me at 4.09, how did she determine the direction of each of those vector arrows beneath the graph? 

Ok, so she is just writing them down, but actual way to "calculate" - would be to have a little clock run for the duration of travel along the trajectory - you see a graph plotted above those arrows - for each trajectory?

That graph represents "time" it takes to travel particular path - or how much little clock will advance along said path. Trick is that clocks advance until they reach 12 where they "reset" or start another turn. Direction that is recorded is direction of the hand on the clock on the last "day" or last turn of the clock hand.

Say that we have a speed of 1 hour for 1mm and we need to record 27mm of distance and we start at 12 o'clock - so first 12 mm hand will complete one circle, next 12mm (from 12 to 24) hand will complete another circle and last 3mm hand will move to 3 o'clock position. So resulting arrow will point to the right and be level (like hand pointing when it is 3 am/pm).

By the way - speed at which this clock turns is related to frequency or wavelength of the light, so for blue light it will run faster than for red light. That is the reason why CD splits colors of rainbow (and why diffraction gratings work) - each different frequency will have different ending hand orientation as clocks will spin at different rates along the trajectory. If you are close in frequency - difference will be small and we always split rainbow so that close frequencies end up next to each other as total difference in final arrow orientation is small.

[vlaiv]

Btw, for those that wonder what little clock has to do with photons and waves and all of that ...

[CraigT82]

1 minute ago, vlaiv said:

Btw, for those that wonder what little clock has to do with photons and waves and all of that ...

But that clock is running backwards.  Even I know you can’t travel back in time! 😉

[globular]

Vlaiv is in your screen looking out, so from his perspective it's clockwise. 😉

[CraigT82]

On 21/08/2024 at 18:16, globular said:

Vlaiv is in your screen looking out, so from his perspective it's clockwise. 😉

I wouldn’t mind a Vlaiv personal assistant in my phone… he’s a lot more useful than Siri! 

[saac]

2 hours ago, Ags said:

I always thought diffraction and interference were the sane thing? As in the diffraction ring around a star is an interference pattern...

I think the best way to think about it is that diffraction is the process (ability of wave to bend around an obstacle or gap) which in turn produces conditions (multiple coherent waves) required for interference. Hence we use a diffraction grating to produce an interference pattern.  In this case, diffraction created by the grating creates multiple (and importantly) coherent rays which in turn are now able to exhibit interference. 

Jim 

Edited August 21 by saac

[bosun21]

2 hours ago, vlaiv said:

Technically interference is wave phenomena which results in either cancellation or enhancement of the wave (positive / negative interference), while diffraction is term that describes how wave

That's the way I was taught also about interference for my physics A level. Constructive and destructive interference. I can clearly recall doing the experiments and demonstrating the effects in a wave tank.

[saac]

37 minutes ago, bosun21 said:

That's the way I was taught also about interference for my physics A level. Constructive and destructive interference. I can clearly recall doing the experiments and demonstrating the effects in a wave tank.

Indeed, interference is the phenomena used to test for wave nature - constructive and destructive interference relying on the interaction of coherent waves. On the other hand particle nature is exhibited by the photoelectric effect. Producing coherent waves, while fairly straightforward (diffraction gratings, lasers, Young's slit etc) does not generally arise from everyday interaction with light. Hence we don't see interference patterns when we look around the room or outside - none of the multiple waves reflecting of the multiple (often extended) surfaces are coherent. However, we can see interference in everyday life in some special cases which do produce coherent waves via diffraction/path difference . For example - the colours in a CD reflection,  the colour (iridescence) in a beetle wing case or peacock feather,  or the colour in a thin film of petrol on a wet road are all produced by interference.

Jim 

Edited August 21 by saac

[johannes1]

"Interference" is a very broad term. I struggle to think of a scenario of diffraction, where one could not argue that somehow there is some form of interference causing the effect. But that does not work the other way - I can easily think of examples of interference that would not normally be described as diffraction. For example a Mach-Zehnder interferometer. Or - in this context very interesting - a Hanbury Brown and Twiss interferometer.

Most cases of interference arise from the phase of the wavefunction of single particle/photon, and this is also underlying the diffraction limit. This simplification of considering the image to be built from individual single photons arriving independently of each other is justified in most cases. However, there can also be correlations between multiple photons. A Hanbury Brown and Twiss interferometer measures a 2-photon correlation. It has been used to measure star diameters, which are obviously much smaller than the diffraction limit of our telescopes. A lot of approaches to beat the diffraction limit rely on some form of multi-photon correlation.

[Gfamily]

On 21/08/2024 at 15:38, pipnina said:

I probably misunderstand this somewhere, but I have been reading recently and notice that since the property of optical resolution comes from the differences in length of light paths from the source to lens to focal plane 

Fundamentally, resolution depends not simply on the differences in length of light paths, but on using the differences in path length to equalise the time it takes for the light to reach the sensor.

In refractors, because light travels slower through 'glass', it means that light that has gone through the centre of the lens is relatively delayed, and the light from the edges that goes 'further' only goes through less glass, so it is delayed less. 

Ultimately, the lens design is to ensure that all the light arriving at a point on the sensor was emitted by the corresponding point on the source at the same time.

For reflectors, the curve of the mirror is chosen to make sure the light hitting the mirror away from the centre line is reflected earlier, so that the physical light path length from source to sensor is identical across the full diameter of the mirror. 

This also explains why turbulence in the atmosphere can affect the seeing, as air density can affect the speed of light through the air. 

Being able to equalise the time of flight for all the signal to reach the sensor is part of the way that interferometry works, so that signals captured at different telescopes  can be brought together into a coherent image.

 

[Captain Scarlet]

On 22/08/2024 at 22:46, johannes1 said:

... I struggle to think of a scenario of diffraction, where one could not argue that somehow there is some form of interference causing the effect. …

I live near the coast at an elevated position overlooking Baltimore Harbour (Ireland) which is a roughly circular area of water about a mile in diameter with a narrow entrance to the open sea (the North Atlantic). I can regularly see diffracted waves arising from the outside swell and the narrow aperture. Mesmerising sometimes, actually.

Magnus