Go to space. Take a blob of viscous polymeric whatever. Squirt a little air inside. Blows up automagically. Let the goo cure. Now you have a ~10 kilometer bubble.
Inject some vaporized silver, using electrostatic deposition or whatever.
Cut the bubble in half. Now you've got 2 concave mirrors.
It's not close enough to just put a sensor out in front, but it's definitely useful. Lots of multi-mirror telescopes use several conic sections; if you needed maximum light you could probably build a telescope with a monstrous 10km ellipsoidal primary, and let that bring the light down to a <1km secondary, and then bring the light back to an aspheric or hyperbolic tertiary optical system built with conventional mechanisms to actually focus the light.
However, I don't think you need to cook your optics with that many square kilometers of background radiation if you don't mind a longer exposure. Instead, you'd be better off installing a few small subsets of that hypothetical single mirror 10s of km away from each other, tracking the distance of each satellite through interferometry, and then putting it back together in software.
the data volumes are probably too high to make this work well. interferometry requires ridiculous amounts of data (hundreds of terabytes per image) and doing that processing in space will be tough within the thermal requirements.
After making that suggestion, I realize that Earth's gravity would make perfectly parabolic lenses if the suggestion was viable. That people aren't already getting perfect(-enough) parabolic lenses from this technique suggests to me that it's probably not actually viable. Oh well.
Come to think of it, a film suspended from a ring would create a catenary[1] under constant gravity -- which is pretty darned close to a parabola but, I suspect, not good enough for astronomy.
I think the link you want is https://en.wikipedia.org/wiki/Liquid-mirror_telescope - "The liquid and its container are rotated at a constant speed around a vertical axis, which causes the surface of the liquid to assume a paraboloidal shape. This parabolic reflector can serve as the primary mirror of a reflecting telescope."
This method in this article uses the same principle: “It has long been known that rotating liquids that are aligned with the local gravitational axis will naturally form a paraboloid surface shape,...“
I'm familiar with liquid mirror telescopes, but I was responding to some theorizing about using an inflated film. Without gravity, the film would produce a sphere; suspended in gravity, it would produce a catenary.
Could a technique evolve someday where you don't even need the mirror but just the sensor and the collected data is then processed as if it was focused by a mirror?
Could a liquid be developed based on a molecule with near perfect reflectivity, a few liters of it is taken into space and then pulled out like a soap bubble at molecule-level thinness?
For your first thought - if we could capture the phase information of the incoming light, then yes. We actually already do this for radio waves- you may have heard of synthetic aperature radar, or AESA/PESA radars in a military context, or the event horizon telescope network (the thing that got us the 'first picture of a blackhole').
For your second thought - as literally described, such a thing would probably be badly impacted by solar wind.
Since there are no gravity loads in space, one could have humongous mirrors with this technique.
Not an expert but I imagine thermal load design becomes an issue too.
Current rocket fairings can be 5 m in diameter and 20 m in length. One could roll up a 20 m diameter mirror in there....
Micrometeorites and solar wind seem like they'd be pesky obstacles to a membrane. JWST experienced a strike almost immediately after it was deployed, which is quite tragic. The bigger the target, the easier it is to hit.
And if we get anything like the launch prices SpaceX has talked about for Starship, we could launch a lot of them.
There's still the cost of the telescope itself, but it's probably possible to save money on that when cheap launch lets you build heavier, and mass-produce the telescope.
Hmm if the mirror has a large hole in the center, you can have one cut and roll it along the tangent. The major radius can be big and the rolled mirror only be the length of the minor radius.
Ie if you have a 45 meter circumference and 5 m hole, the major radius is 22.5 meters but the minor radius is only 20 meters and it can roll into 20 meters. It will have one discontinuity though that would cause artefacts and might destroy any advantages from the additional size.
I think a Jupiter sized planet, 20 light years away, imaged at 500nm wavelength, would need a 826 meter diameter mirror just to make it 2 pixels instead of 1.
Fortunately a large mirror doesn't have to be a continuous surface, and it doesn't need all of that area either.
A constellation of optically linked telescopes each with small mirrors would be able to image the same exoplanet.
Controlling the optical linkages or equivalently the trajectories precise and fast enough is difficult, but as far as I know this is not a fundamental limitation.
Go to space. Take a blob of viscous polymeric whatever. Squirt a little air inside. Blows up automagically. Let the goo cure. Now you have a ~10 kilometer bubble.
Inject some vaporized silver, using electrostatic deposition or whatever.
Cut the bubble in half. Now you've got 2 concave mirrors.
How to make it parabolic? Hmm