According to the widely accepted Rayleigh criterion, in order to resolve two distant point sources, their angular separation must be larger than 1.22 (the wavelength of light divided by the telescope’s aperture diameter. To achieve higher resolutions, telescope systems with ever larger primary mirrors are therefore being designed, making them extremely expensive and technologically challenging to build.
By calculating the Fisher information for the case of conventional image-plane photon counting, it has been shown that the information drops to zero as the separation of the point sources goes to zero – a phenomenon that has been dubbed Rayleigh’s curse. Nevertheless, by changing the measurement scheme, it is possible to have a non-zero Fisher information for every source separation and rendering the Rayleigh criterion moot. Limitations on our ability to resolve two incoherent point sources is therefore not a fundamental constraint, but rather a consequence of the most commonly applied measurement technique – wherein a pixelated detector is used to measure the light intensity distribution in the image plane of the system.
Stephen Felming and colleagues has an improved measurement scheme uses spatial light demultiplexing, in which the electric field in the image plane is decomposed into a set of spatially independent modes – for example HermiteGaussian (also TEM) modes. By measuring the corresponding signal in a number of lower order modes, it is then possible to extract information about the separation and brightness of the two point sources, even for separations below the conventional resolution limit. By decomposing the image field into a symmetric and an antisymmetric mode, the method has already been experimentally tested and proven to work for two equally bright incoherent point sources.
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A long LONG time ago (in today’s terms), oh say in the 1990s to 2000s, I worked out ‘ab initio’ what the Raleigh’s Criterion was about. Turns out, it is pretty geometrically simple.
FIRST, let’s say that a focal spot of ‘a lens or mirror’ focusses to an absolutely perfect spot, for every photon it focusses. Naturally, since photons are statistical phenomena, the ‘spot’ is the point X, Y at the focal plane where the photon(s) image out to be, but … necessarily, also an airy circle of about 1.2 times the wavelength of the light involved. Wavelength of the photons. Why? Well, because at the very least, the ‘spot’ for each photon of a given wavelength and angular focual origin can only illuminate the ‘spot’ with a diamter not just of the wavelength of the photon, but in a gaussian distribution around the center of it. And, moreover, when overlapping Gaussians (in the 2D axis) are closer than ¹⁄₁.22² of a diameter, they become smeared by their normal-distribution nature into a single homogeneous spot. Not even very oval. Just a wee bit.
Sources with angular separations closer even than the Raleigh limit (1.22 λ / D) in fact cannot be separated at all.
So… there’s that.
But from another point of view, even with synthetic sparse-coverage mirrors (imagine a large super-mirror, in space, composed of 196 ‘segments’, all extraordinarily well figured as parts of the parabola or hyperbola of the intended synthetic aperture mirror … but separated by dozens of diameters from each other) that are many, and extraordinary, and ridiculously well spaced in real-time, that same synthetic mirror might occasionally focus well collimated photons to within the theoretical limit of the super-mirror’s’ synthetic diamter … but hundreds, if not millions more, much blurred out … because of the interference fringes of all the sparse composite mirrors.
EVEN WITH THE WEBB, it being a truly awesome ‘synthesis mirror’ composed of a couple of dozen mirror-Léts, but close-packed (without appreciable mirror-to-mirror gaps) well, and figured to within hundredth’s of a wavelength of light … even with this absolutely mind-boggling monstrosity, there are enormous hexagonal and unavoidable interference fringes where each of the hexagonal mirrors a-bûtt. Nothing can be done about it, except post-processing (mathematical) enterprise, to synthetically subtract out the hexagonal fringes with a basically Poisson’s distribution error range in the mathematics.
Nice but not perfect.
Imagining a truely sparse mirror, 100× the diameter of Webb, with perhaps only 3× to 5× the imagining area of mirror-Léts … and the ‘hexagonal spike’ problem multiplies exponentially.
This is what we face.
Mathematics certainly helps.
But ultimately, we want really giant, really edge-less mirrors.
And that’s hard.
⋅-⋅-⋅ Just saying, ⋅-⋅-⋅
⋅-=≡ GoatGuy ✓ ≡=-⋅
” that same synthetic mirror might occasionally focus well collimated photons to within the theoretical limit of the super-mirror’s’ synthetic diameter”
And almost equally often, focus them to what looks like that theoretical limit, only somewhere else… With the distribution of those ghost images being… Gaussian! So, how do we distinguish these super-sharp “ghost” images from the one true evanescent image? I don’t know that we can.
If I’m reading the above right, and admittedly high end optics isn’t my gig, they’re taking the incredibly small current pulse generated by the individual photons, and instead of just saying, “Yup, a photon struck there”, (Which quantum magically sums up according to the wave nature of light.) they’re decomposing it into its various modes, and measuring them individually, thus getting a lot more information out of the photon than just, “Yup, I was here!”
This allows them to, essentially, look at the shape of the Gaussian distribution each individual photon represents, and decide where its center was. Instead of blurring them all together. Getting at the information that gross summing throws way, thus computing a sharper image.
Looking at individual photons in this manner would have been practically unimaginable to Raleigh, except maybe as a thought experiment.
Nice. And yes.
I think your observation (minus the pun that could be made… ahem) of the photons not exactly helpfully carrying little flags that identify whether they’re the esquisitely lucky ones to be focussed ‘right’ compared to all the imposters that aren’t … is actually at the root of the problem that even your penultimate paragraph doesn’t quite answer.
I estimate that the gaussian blending of such photons seriously upsets the hoped-for quantum entanglement that might be elucidated from them individually. Indeed, even imagining a quantum per-photon-with-detection-of-flags-and-eigenstates detector … for billions or trillions of quanta per second … whew!!!
⋅-=≡ GoatGuy ✓ ≡=-⋅
I’ve thought about ways to make giant mirrors for space. I have no idea if this would work but…
The problem is trying to make a large mirror in one big piece. So make a hut that melts one piece/segment of a mirror. You can spin mercury to make mirrors. Same idea but you spin single segments to make them. What’s the difference between spinning the whole thing and a piece? Nothing I suspect? In this case you put the hut on the end of a long strut with a wheeled drive and drive/spin it around in something like an aircraft hangar. Large reasonably flat surface. You melt the glass then cool it with it still rotating. Change the length of the strut to change where the mirror is located. I suspect with an active suspension on the hut where each segment is melted you could get a high degree of accuracy on any flat floor area you spun the hut around while each mirror segment is made. Another cost saving measure is do not use borosilicate glass. More expensive. Use regular glass from pop bottles. Once the mirror is in space behind sunscreens, the temperature could be stable so you don’t need special glass that takes a lot of energy to melt. Make the mirror as thin as you can to cut cost and save space. Along with the “thin” mirror, you cast octagonal struts to hold them in place. Assemble in space with micro adjusters to do final alignment. The goal is not to make something perfect, but to lower cost by adjusting in low gravity and constant temperature. I know there are waves in mercury mirrors from spinning. Maybe you rotate the mirror in space to average those out. This saves you from spending a fortune on grinding these out.
Does anyone know if you even need a large space for this? Normal mercury spinning mirror, the outside spins faster than the inside. Could you make the circle it follows the same for each mirror, but just vary the speed of rotation to get the curvature desired???? I’m not sure about this because radius would be smaller, and it might change the curvature. If it could be done in a single radius circle by varying the speed it would vastly reduce the cost.
Well, that’s interesting.
The Rayleigh limit is, IIRC, based on the wave properties of light, essentially treated as a continuous phenomenon. Which through quantum weirdness, it actually does act that way, despite being individual photons, if you look at the cumulative count.
This new approach, I take it, instead of analyzing the sum of the photons, does a detailed analysis of the properties of each individual photon. Obviously, the individual photons carry a lot of information that doesn’t end up in the count.
This sounds very promising. I’m surprised. I thought the Rayleigh limit was general theorem. So if they can improve 100x just by changing the sensor, that’s great.
And it could be combined with another improvement. This hasn’t been done, as far as I know, but it seems at least theoretically possible. If you want to create a giant telescope in space, you don’t have to use a giant mirror. You could use a giant lens, instead. But it doesn’t have to be a literal lens. It could be a thin sheet of aluminized mylar, with a dense pattern of small holes in the aluminum. That could act as a hologram, to behave like a lens. And it could be built kilometers across. It could rotate, to keep it taut, and to average out any imperfections over time.
I suspect you’d need to build it to work at only a single wavelength, which could still be useful for some purposes. Though maybe you could get it to focus each wavelength at a different focal point, and have multiple detectors, one for each wavelength.