UKH

My 6yo just asked me how many of reflections there are in a hall of mirrors (two mirrors facing each other, child in between, you know..)

so for the geeks amongst us.. answers on a post (card) please.

It's infinite. 

What's that short story (Guy de Maupassant?) where somebody spots another figure in one of the receding reflections? When he looks again each time the figure is coming one frame ever closer .... until! Can't remember how it ends. Might scare your kid into stopping asking awkward questions.

can you find the story?  Sounds as much fun as MR James!

Its not because mirrors aren't perfect reflectors. And other reasons as well I'm sure!

42.

Get him to set up 2 mirrors and count them 😉

It was so long ago. I often recall the gist of a book but the author is rarely something I can remember. I'm fairly sure it's a well known author but then I did used to wade through some right old tosh at times. Especially sci-fi.

Even in the case of perfect reflectors, there's the issue of the finite speed of light, a matter of interest to the greatest of all fictional savants, de Selby (in Flann O'Brien's "The Third Policeman") ...

"But de Selby, even loath to leave well enough alone, insists on reflecting the first reflection in a further mirror and professing to detect minor changes in the second image.  Ultimately, he constructed the familiar arrangement of parallel mirrors, each reflecting diminishing images of an interposed object indefinitely.  The interposed object in this case was de Selby's own face and this he claims to have studied backwards through an infinity of reflections by means of 'a powerful glass'.  What he states to have seen through his glass is astonishing.  He claims to have noticed a growing youthfulness in the reflections of his face according as they receded, the most distant of them - too tiny to be visible to the naked eye - being the face of a beardless boy of twelve, and, to use his own words, 'a countenance of singular beauty and nobility'.  He did not succeed in pursuing the matter back to the cradle 'owing to the curvature of the earth and the limitations of the telescope'."

In terms of discernible (by-eye) reflection of the images, between 10^1 and 10^2.

In terms of photon reflections, order of 10^20 per second.

Other answer will fall within those bounds.

Next question:

A lens orbits a point like star with an orbital radius of one focal length.  What direction is the force on the lens from photon pressure?

If it was, the mirrors would explode from the infinite photon pressure.  Which would be counteracted by the whole experiment collapsing in to a black hole.

Optics - messing with forces you can’t control!

There is an attractive force between the mirrors, though - the Casimir force.  Even though the photon energy density between the mirrors is infinite it's less than it would be if the mirrors weren't there, because the boundary conditions limit the number of possible standing wave modes.  (I know this sounds very dodgy but that's field theory for you).

That applies to the zero point of the field, but I’m not clear how you can apply it to additional excitations only introduced inside the cavity…?  

I’d have said the limiting factor is the finite length of time for which illumination can be introduced and the speed of light, so that cavity pressure would only become infinite after an infinite amount of time had passed.  

As a fellow past-dabbler in optical tweezers, I feel obliged to take the bait here

You're right, I'm thinking of equilibrium (though the Casimir argument works at finite temperature too). I expect you have to take into account the energy dissipation in real mirrors or it all gets pathological.

My go at this:

To see any chain of reflection, there has to be an offset between the observing eye and the object being reflected and/or the mirrors have to be non-parallel (in the real world, they can't be perfectly parallel anyway). The practical limit to the number of reflections is set by how big the offset and non-parallel angle are and how big the mirrors are - sooner or later the reflections go off the edges. 

The popular view seems to be that when we eventually make a black hole by mistake that it’ll be the particle physicists and their colliders.  My money is on some quiet optical coatings breakthrough and someone getting carried away breaking the record for Q…

Good question that.

Although if concave...

Good interview question, this. You can tell a theoretical physicist from an experimenter quite easily. One will start a long monologue about diminishing reflections and quantisation and photon pressure. The other will say your head's in the way, so you can't be looking perpendicular, so how wide is your mirror?

In the absence of any info on spectrum and material you'd have to assume it's absorbing most of what hits it.

You need more 'light inextensible frictionless' words in there before I'll even start thinking about the momentum argument. And specify whether the star is even emitting anything or whether you're fishing for way more complex arguments.

Not quite: if the mirrors are infinite and perfectly reflecting then it's infinite (someone will be along to explain they need to be parallel as well but that's not actually a condition that is required and it doesn't hold unless your observer is infinity small or your observed object is infinity far away).

If you assume (reasonable for mirrors in a lift) the reflectivity is ~95% and you can detect (see) the reflection with 1% of the original brightness then its around 40 complete (double) reflections. These assumptions are fairly generous so expect the practical limit to be a little lower.

With fancy mirrors you can go much higher.

Not a great believer in the power of metamaterials then?

In reply to  Longsufferingropeholder:

Convex lens collimating the light from the star, no absorbance.  

I don’t know about the photon pressure, but your point like star just made another black hole which destroyed the lens, merged with the one from the mirrors and caused an almost indistinguishable phase difference in a separate hall of mirrors several billion light years away. Some physicists got excited.

Define light.

edit: lens falls towards star due to photon pressure, but as it’s a black hole, light can’t escape, so is also pulled into the black hole. So then you’re actually asking about Hawking radiation I think.. 

So you could go into a long winded momentum argument involving lots of diagrams, but it's succinctly just a rocket nozzle for photons, right? What have I missed?

As you well know these aren't practical on the scale of a hall of mirrors, don't work in the visible (at present) and aren't broadband enough to work for the entire visible range. Plus practical implementations would _probably_ be much lossier than their classical counterparts...

We can’t have a monochromatic hall of mirrors in the Ku band then?  The replies to this thread are proving a great Rorschach test.  Years since I’ve read anything on superlenses and their journey to shorter wavelengths.

In reply to Longsufferingropeholder:

You’ve not missed anything, it’s a simple question with a simple solution, but one that surprises a lot of people (not optical tweezer types as jonny taylor’s reply shows…)

For extra credit, what happens when you use an extended spherical star, not a point source? Does the phenomenology change or is it just detail?

...monochromatic hall of mirrors in the Ku band then?

If you must, but surely this lies outside the scope of the original question?

😉 ... outed!

I only did it because I couldn’t identify your waterfall..

I was taking my lead away from reality into theory land from your implied suggestion the diffraction limit would have any relevance to the original question…  

It’s a great question, far more to it than first meets the eye.  

So Flann O’Brien invented cavity ring down (well, ok, up.. but CPT invariance and all..) spectroscopy 25 yrs before the laser was invented...?!

If it meets the eye, surely, that's the end of the reflection path...

Considering the star as a collection of independent point sources, with the forces from each source superposing at the lens, the lens always moves towards the light.  As the angular size of the source increases, the force  reduces until it becomes zero for an infinity large source.  

He says having sketched on a piece of paper and done no maths…

What about the surface curvature of the star?

ps. How ‘f-ing’ big is your lens?

Unless you’re close to the star and it’s occluding some rays, what about it? The lens doesn’t need to be specifically one focal length from the source; threw that in for misdirection…

Edit to your edit; are you suggesting a larger diameter for the star than for the lens means this does not work?  That was my first thought, but considering it as an array of point sources I don’t think it matters. 

For your infinite planar star, I agree.. because luminance on the lens falls as 1/d^2 from the source, but the star is not infinite (residual force) but part of the source is further away than in the planar case, (and occlusion at the edges but ignore that)

The only way I see distance having an effect is that for sources more than 2f from the lens the direction of the force reverses.  So probably best to bound f to be larger than the diameter of the star; stops it from being vaporised as well…

nice. someone should tell Elon Musk sooner rather than later

The force is not perfectly opposed for off axis sources so you should consider the projection of the vector of the central ray leaving the lens onto the vector of the incident ray. This decreases the contribution for off axis sources I think.. so I suspect there is a critical source size even for a planar source?

Agreed.  Assuming symmetric off axis sources the further off axis they go, the weaker their axial force vector.

Not I think for a paraxial lens an an array of point sources; but perhaps that just goes to show how misleading paraxial lenses and point sources are…