Telescopes have angular diffraction limit depending on the observed wavelength and aperture diameter.
I've read that it's possible to go beyond the limit for microscopes. But is it possible to do the same for telescopes too?
[Physics] Is it possible to surpass the diffraction limit for telescopes
diffractionopticstelescopes
Related Solutions
To understand this explanation, you need to understand Fourier decomposition of the electromagnetic field.
In any homogeneous medium, any electromagnetic field can be thought of as a linear superposition of plane waves, all in different directions. Because they run in different directions, the phase delays they undergo in propagating from, say, your aperture to another, parallel plane are all different. Therefore the wavefront gets "scrambled" owing to these direction-dependent phase delays. This interference between the different plane wave components of the electromagnetic field is what we commonly call "diffraction". I further explain this idea, as well as draw some diagrams in this answer here as well as this one here.
So with this introduction in mind, let's look at your paragraph. For simplicity, assume only one transverse direction and one axial (in the direction of propagation) direction. Let's also assume scalar optics, i.e. that the electromagnetic field is well represented by the behaviour of one of its Cartesian components, so that we can do Fourier optics on scalar field.
So we have a uniformly lit aperture of width $a$. Its transverse profile is therefore the function ${\rm rect}(2 x/a)$ where ${\rm rect}(x) = 1;\,|x| \leq 1$ and ${\rm rect}(x) = 0;\,|x| > 1$. We take a Fourier transform to find the superposition weights of each plane wave component, because each such component has a transverse variation $\exp(i\,k_x\,x)$ where $k_x$ is the Fourier transform variable with units of reciprocal length. The Fresnel distance is, as the paragraph says, simply the axial distance needed for this spread to double the beam width. So it is a rough measure of how quickly the light spreads.
So this is how the "divergence" arises from diffraction, i.e. the interference between an optical field's plane wave components as they propagate. Also $\sin\theta = k_x/k$ where $k = 2\pi/\lambda$ defines the angle that this plane wave component makes with the axial direction. We take the Fourier transform, we find that there is a spread of $k_x$ values such that the plane wave components most skewed to the axial direction make an angle without direction of roughly $\lambda/a$. So, owing to these skewed components, the field's energy spreads out.
The beam width diverges slowly at first and then, after an axial distance of several Fresnel distances, the divergence speeds up so that the propagation becomes well modelled by the cone of rays diverging from the centre of the aperture. Indeed if you plot contours of constant intensity, they are hyperbolas which begin at right angles to the aperture but bend so that their asymptotes are the cone defined by ray theory. The Fresnel distance defines how far the "knee" of the hyperbol is from the aperture.
For your question:
Is it not that the validity holds when all objects are comfortably larger, and not smaller, than the wavelength of light?
This is in general right, but it breaks down near focuses and in situations like this where we are near and aperture and if the aperture is comparable to the light wavelength. In this case you should be able to understand from the Fourier analysis the reciprocal relationship between the aperture width and the angular spread.
Best Answer
Short answer - no, it wouldn't be possible to beat the diffraction limit with a telescope.
Longer answer - The way that microscopes get around the diffraction limit is by getting really close to what they're looking at, or into the near-field of the object. For visible light, this only exists within a couple of nanometres of the surface. The near-field contains information at all spatial frequencies (i.e. arbitrarily high resolution) but it decays away exponentially (like light in the cladding of an optical fibre or the wavefunction in the wall of a quantum well). Near-field microscopes beat the diffraction limit by converting the near-field into propagating light - into the far-field - where it can be detected and measured. The various forms of SNOM are probably the most obvious examples of this.
As telescopes can never get close to what they're looking at, the near-field will always be unavailable to them - in other words, they will only ever be using the far-field. This means that they can never gain information beyond the diffraction limit - they are fundamentally incapable of retrieving the lost information. This is also the reason that superlenses built with metamaterials won't work with telescopes - they work by amplifying the near-field.
The only ways I'm aware of for getting around the diffraction limit in the far-field is to pre-arm yourself with more information - either by cleverly structuring the light you use to take the picture, or to modify surface you're looking at with fluorescing dyes, neither of which are possible with a telescope. Maybe someone knows more though?
Fourier optics is the bit of physics that covers all the near-field/far-field information loss if you want to look in more detail.