Telescope Resolving Power Limits

In summary, the conversation discusses the possibility of a telescope being able to resolve images beyond the diffraction limit by using a birefringent crystal and interference patterns. The design is similar to a system that combines the output from several telescopes into a single image without the spatial coherence issues. However, there are factors such as clear air turbulence and system noise that may affect the resolution limit. Despite this, the idea is worth exploring for potential improvements in space telescope technology.
  • #1
kmartin
10
0
Is it possible for a telescope to resolve images beyond the diffraction limit? In other words, is the information in the light entering a telescope insufficient to resolve beyond the diffraction limit, or is information lost when the light is focused onto a screen?

Attached to this post is a diagram of my thought experiment telescope design. The collimated output from a telescope is split into two beams. Each beam is then passed through a birefringent crystal, and then the two beams are focused down onto a single screen.

If, for example, the telescope is looking at two far away point sources (Point A and Point B), then the collimated beam will consist of two overlapping plane waves, traveling in slightly different directions. Because of the small difference in propagation direction, they will experience slightly different refractive indices when going through the birefringent crystals. This causes a difference in path lengths for light from point A compared to light from point B. If the path lengths are arranged so that light from point A constructively interferes at the focus point, and the correct length of birefringent crystals is chosen, then light from point B will interfere destructively. This allows light from point A to be separated from light from point B, even if this would be impossible conventionally due to the resolving power limit. By my calculations, if the collimated beam is 5mm in diameter, the birefringent crystals are made of calcite and 1.5cm long, and the light wavelength is around 500nm, then the resolving power of the telescope would be doubled. Using thicker crystals, or a more exotic crystal (e.g. calomel) could produce a much higher increase. Also the beam could be split multiple times to improve the resolving power in both dimensions, and to reduce the effects of interference patterns.

Note that the light experiences a 'walk off' as it enters and exits the crystal, as shown in the diagram, however this shouldn't affect the operation of the telescope. Also the phase difference from the change in refractive index is proportional to the angle between the propagation directions of the light from the two sources. This is much more significant than the geometric path difference, which is proportional to the square of the angle.

My telescope design can be thought of as similar to a system that combines the output from several telescopes into a single image. However in my design the alignment is much simpler, and there would be no spatial coherence issues, allowing for better nulling.

So would my telescope design work?
 

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  • #2
The resolution power is usually given by the size of the first, big mirror*. Even with perfect analysis of the light reflected by that mirror, you won't get beyond the resolution limit.

*without atmosphere or with adaptive optics
 
  • #3
What you are describing is very similar to an imaging technique in microscopy- differential interference contrast. There are telescopes that use interference:

http://en.wikipedia.org/wiki/Astronomical_interferometer

I saw the Palomar testbed during construction- very impressive technological feat. One thing you are forgetting is that the incoming wave is not a plane wave- clear air turbulence means the incoming wavefront has a stochastic component. Keeping the two 'arms' of the interferometer a constant length is another trick.
 
  • #4
Thanks for the replies, but I don't think they really answer my main question.

To put it another way, if I could measure the phase of the light at every point of the focused image of a telescope, as well as the intensity, wouldn't that give me more information about the light source than a conventional telescope that just measures intensity?

Also I would be grateful if someone could give the fundamental reason why my thought experiment telescope design in the opening post doesn't work.
 
  • #5
The 'resolution limit' is not a hard and fast figure. The intensity / angle figures of two images which is just at the so-called Rayleigh criterion shows a dip to half power between the two maxima. This is a good rule of thumb as a limit for human eyballing (I guess??) but if you move the images much closer and if the system noise level is low enough (and if you know the detailed spatial impulse response of the optics) you can still resolve the two. The limit is, theoretically, only imposed by the system noise (and your processing power). Whilst it is clearly better if you know the phase profile of the waves, the intensity at the detector can still give you more information than is suggested by the criteria that have been used historically. This is the same process they use for 'image enhancing' photos of car number plates and faces which, at first sight, apprear illegible.
 
  • #6
kmartin said:
<snip>

Also I would be grateful if someone could give the fundamental reason why my thought experiment telescope design in the opening post doesn't work.

Why do you think it would not work? As I mentioned, there are existing systems that use interferometry. You may not appreciate the need to account for atmospheric turbulence, because your incoming wavefront is not a plane wave.

Edit: you may appreciate David Saint-Jacques dissertation "astronomical seeing in space and time"

http://www.google.com/url?sa=t&rct=...sg=AFQjCNEDyip2nSJBPT7FxI9scBgPQB5MIg&cad=rja
 
  • #7
Andy Resnick said:
Why do you think it would not work? As I mentioned, there are existing systems that use interferometry.

If it worked then it could potentially allow the construction of a small relatively cheap space telescope that has a resolving power much greater than any current telescope*. Also it could make conventional astronomical interferometry obsolete, as only one big telescope is required. So its probably not going to work, but its worth asking for other peoples opinions in case there is something in the idea.

*Admittedly there might be a bit of a brightness problem.
 
  • #10
kmartin said:
That design has two big mirrors. The design in the opening post just has one, so it is very different.

I must say, I couldn't see how the design in the OP gave any increase in actual Aperture - which is the essence of the improvement in resolution for an interferometer. It has that great lens and then messes about with it and loses most of its energy gathering power (another factor in telescopes) - not to mention the energy loss in the polarisers.
 
  • #11
I think it boils down to the following.
Can you split the light after it enters the aperture and then recombine it later to give you a better resolution?
 
  • #12
Drakkith said:
I think it boils down to the following.
Can you split the light after it enters the aperture and then recombine it later to give you a better resolution?

Would it give you a bigger effective aperture? I don't think so. Interferometers are based on increasing the baseline to extend over a much bigger distance than the aperture of just one telescope. This system doesn't do that.
 
  • #13
Drakkith said:
I think it boils down to the following.
Can you split the light after it enters the aperture and then recombine it later to give you a better resolution?

I think the more important question is:

Can you use a birefringent crystal to adjust the phase in order to mimic the effect of translating a telescope laterally?
 
  • #14
Here's a (hopefully) better explanation.


Consider two identical telescopes a few metres apart, both pointing at the same far away light source. The images formed by the two telescopes are indistinguishable because the source is far way. However if the images are combined they can create a higher resolution image. How is this possible if the images are identical? The answer is that there is a difference in the phase of the light. For example, consider the attached diagram, where two telescopes are looking at two far away point sources (coloured blue and green for illustrative purposes). The light entering telescope 1 is momentarily in phase, where as the light entering telescope 2 is out of phase. The only difference between the light entering telescope 1 and the light entering telescope 2, is that in telescope 2 the light from the 'green' source has been delayed by half a wavelength. This difference is what allows the light from the two telescopes to be combined together to form a high resolution image. Is it possible to modify the light from telescope 1 to make it identical to the light from telescope 2? Possibly yes, if the light is passed through a birefringent crystal where light traveling in different directions experiences different refractive indices. The light from the green source would experience a higher refractive index, giving it a longer optical path length, causing it to lag behind the light from the blue source. So now we can take light from telescope 1, split it with a beam splitter, pass half of it through a birefringent crystal, then recombine the light to create the same image we would have got from combining the light from telescopes 1 and 2.

This technique, if it works, would allow the construction of a small relatively cheap single mirror space telescope that has a resolving power much greater than any current telescope.
 

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  • #15
How do you think a single aperture (defined by the dimensions of the single telescope) can be transformed by a wider aperture, merely by jiggling about with two different polarities?

What you are saying, implies that a single slit can be made to have the same interference pattern as two similar slits, separated by a significant distance. Can you demonstrate this to be true, with some calculations and geometry? Where do you get your angle dependent path differences?
 
  • #16
sophiecentaur said:
Where do you get your angle dependent path differences?

The angular dependent path difference comes from the angular dependence of the birefringent crystal. Small changes in propagation direction cause small changes in the effective refractive index, which cause small changes in the optical path length.
 
  • #17
The way an interferometer works is that the wide baseline gives a useful phase change as the angle of arrival changes by a small amount. I understand, now, what you are saying but can you explain how the path length through a crystal can change by what must be at least one wavelength for a change in angle of a few arc seconds? I looked for some information but couldn't find more than general stuff and nice pictures.
 
  • #18
I guess one practical problem with this would be to get that thickness of calcite crystals and to ensure good optical quality.
 
  • #19
sophiecentaur said:
The way an interferometer works is that the wide baseline gives a useful phase change as the angle of arrival changes by a small amount. I understand, now, what you are saying but can you explain how the path length through a crystal can change by what must be at least one wavelength for a change in angle of a few arc seconds? I looked for some information but couldn't find more than general stuff and nice pictures.

The effective refractive index is given by [itex]n(\vartheta) [/itex] where

[itex]\frac{1}{n(\vartheta)^2} = (\frac{cos(\vartheta)}{n_o})^2+(\frac{sin( \vartheta)}{n_e})^2[/itex]
http://www.radiantzemax.com/kb-en/KnowledgebaseArticle50260.aspx

For calcite
[itex]n_o=1.658[/itex]
[itex]n_e=1.486
[/itex]
http://en.wikipedia.org/wiki/Birefringence

We want to align the crystal at an angle [itex]\vartheta_m[/itex], where the rate of change of refractive index is maximum, and the response is most linear. This gives a maximum rate of change of refractive index as

[itex]\frac{dn(\vartheta_m)}{d\vartheta}≈ 0.17[/itex]

So a 1.5cm length of calcite would give a path length change of 2.5μm per milliradian.

However the act of focusing the light and collimating it amplifies the angle. So for a telescope with an aperature of say 50cm, which collimates the incoming light to a diameter of 0.5cm, this gives 250μm per milliradian, or 0.25μm per microradian.

However I am far from being an expert on birefringence, so if someone could confirm my thinking it would be good.
 
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  • #20
Sounds more convincing now. Thanks for the figures.
 
  • #21
sophiecentaur said:
Sounds more convincing now. Thanks for the figures.

Its nice to know I'm not going completely mad. Thanks :smile:
 
  • #22
This has set me thinking and I think I have found snags. If your source is unpolarised then what will be the relationship between the two linearly polarised signals that emerge from your polarisers?
The only time that the interference between the two components will produce the required pattern will be for the contributions that are exactly in the appropriate plane. I mean that the two phasors will only cancel properly when the two waves which are polarised at 45degrees to the polariser planes so that the two vectors have equal magnitudes. You would need to pre-polarise the incoming light to make it linearly polarised in this plane.
Also, the resultant interference pattern on the screen will be modified by geometry of the 'two slit' arrangement where the two beams recombine. I'm not sure how relevant this is as it may be producing a very fine set of fringes as the 'throw' is so short.
 
  • #23
sophiecentaur said:
This has set me thinking and I think I have found snags. If your source is unpolarised then what will be the relationship between the two linearly polarised signals that emerge from your polarisers?

The light has to be polarised as the angular dependent refractive index only works when the polarisation is in the same plane as the crystal's optic axis and the propagation direction. However it would be possible to split the light into its two polarisations and process them separately to prevent any significant brightness loss.
 
  • #24
Yes, I understand about the function of the two different paths for the two components. Your original diagram doesn't have a polariser at the input and I assumed that the your system would just deal with light straight from an object. How would you propose to split and recombine the light so that the two input polarisations could be treated completely separately? I think there would be other interference problems if the two images were to be allowed to combine on a single screen.

I have no experience of optics but I do know about RF antenna arrays. Their beamwidths are normally 'aperture limited' but it is possible to make 'super gain' (narrow beam) arrays by suitable weighting of array elements. This gives a narrower beam than normal but it is very difficult to go very far without producing large sidelobes, for instance. Now, with RF, one has the chance of controlling phase and amplitude much easier than with optics and it is still pretty well impossible to obtain a big improvement. It worries me that you are, in effect, trying to do the same sort of thing (a phase slope on half of an RF array would possibly achieve a similar thing) but with a necessarily more crude arrangement. I don't have a specific objection because I can't exactly put my finger on it but, somewhere, I feel there has to be a fundamental problem. (Maverick Cop: "There's something wrong, sergeant - it just doesn't feel right") Could it be to do with the assumed performance of the extraordinary ray's path through the birefractor? Diffraction will also be at work there and be limited by the aperture. It may cast doubt on your simple assertion that the path length is only affected by ray angle. We are not actually dealing with 'rays' in a diffraction limited system.
This just might be spoiling your day. If so, sorry.
 

1. What is telescope resolving power?

Telescope resolving power, also known as angular resolution, is the ability of a telescope to distinguish between two nearby objects in the sky that appear close together. It is a measure of the smallest angle between two objects that can be resolved by the telescope.

2. How is telescope resolving power determined?

Telescope resolving power is determined by the diameter of the telescope's primary lens or mirror. The larger the diameter, the better the resolving power. Other factors that can affect resolving power include atmospheric conditions, optical quality, and design of the telescope.

3. What is the theoretical limit to telescope resolving power?

The theoretical limit to telescope resolving power is known as the diffraction limit. This is determined by the wavelength of light and the diameter of the telescope's primary lens or mirror. It is calculated using the formula: θ = 1.22λ/D, where θ is the smallest angle between two objects that can be resolved, λ is the wavelength of light, and D is the diameter of the telescope's primary lens or mirror.

4. Can telescope resolving power be improved?

Yes, telescope resolving power can be improved by increasing the diameter of the telescope's primary lens or mirror, using higher quality optics, and reducing atmospheric distortion through techniques such as adaptive optics.

5. Are there any limits to improving telescope resolving power?

Yes, there are limits to improving telescope resolving power. The diffraction limit sets a maximum theoretical resolution that cannot be surpassed, and atmospheric conditions can also limit the resolving power of a telescope. Additionally, larger telescopes can be more difficult and expensive to build and maintain, making it challenging to continually improve resolving power beyond a certain point.

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