# Concave magnifier lens Two questions in one [closed]

Consider a support made from a wide (1m), short (30cm) (approximate sizes just for reference), rigid, empty, open ended cylinder. Cover the ends with a perfectly flexible transparent sheet. Suck the air out of the contraption. It should look somewhat like a concave lens. The sketch below provides a section view of the "lens".

Question:

1 - Would this "lens" concentrate light ?

• I am assuming it would act like a lens because of the difference in density between the inside medium and the outside medium. I am suspecting it would concentrate light because the medium inside is less dense than the outside so it should have the opposite effect of a solid glass concave lens.

2 - What shape will the membranes take under pressure? Will it take the shape that's necessary for a lens?

Temporary Edit to address the "off topic" issue:

I did a search in physics SE for "What shape" and found lots of questions asking what shape would x material take under x circumstances. Why is mine different? Just because this physics concept would be used in some engineering project later on?

## closed as off-topic by Aaron Stevens, Thomas Fritsch, Jon Custer, GiorgioP, Kyle KanosJun 19 at 13:46

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question appears to be about engineering, which is the application of scientific knowledge to construct a solution to solve a specific problem. As such, it is off topic for this site, which deals with the science, whether theoretical or experimental, of how the natural world works. For more information, see this meta post." – Aaron Stevens, Thomas Fritsch, Kyle Kanos
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• Why do you think it should (or should not) act as a lens? I presume you want the light to be propagating vertically? – Jon Custer Jun 18 at 18:36
• @JonCuster That is correct. (As in sunlight). I added "light" to the image. Thanks for the suggestion. – Alex Doe Jun 18 at 19:00
• @aaronstevens and the other 4 close votes- Which part of my question is off topic? Should I just start guessing? The votes to close a question should be accompanied by a decent explanation imho – Alex Doe Jun 19 at 17:59
• @JonCuster - Which part of my question is off topic? – Alex Doe Jun 21 at 2:47
• Well, you asked two questions with no particular evidence that you did any background work for either of the loosely coupled problems. That makes it, to me, a bad question that requires a lot of clarification on what you are having difficulty with. Others appear to think (also rightly in my opinion) that your request for the shape the membranes would take is an engineering question. Now, having taken more than the few minutes overall to address problems with the question I’m done... – Jon Custer Jun 21 at 3:10

According to this reference https://byjus.com/physics/the-lens-makers-formula/

the lens maker's formula for a lens made of one substance immersed in a different medium is:

In most cases, the surrounding medium is air, and $$n_2$$ is quite close to $$1$$ ($$1.00029$$ for dry air at standard temperature and pressure) $$n_1$$ is of the order of $$1.4$$ for many lens materials

In this case, $$n_2$$ is still very close to $$1$$, while $$n_1$$ is even closer to $$1$$ approaching exactly $$1$$ as the pressure inside your air lens reaches zero.

This appears to be a tiny difference, However, it is enough to change the value of the refractive index ratio from greater than $$1$$ to less than $$1$$. Thus subtracting $$1$$ from the ratio produces a negative result rather than a positive result for the first factor, and changes a diverging lens to a verrrry slightly converging lens.

• So the answer to question 1 is basically yes (verrry slightly :) Does the shape of the surface of a lens have to be part of a sphere? Or of parabolic shape? Would it be very difficult to approximate the focal distance of such a lens? – Alex Doe Jun 18 at 19:42
• Spherical surfaces are the simplest to work with, both theoretically and in manufacturing. Aspherical lenses offer optical advantages and much higher prices... – DJohnM Jun 18 at 20:00
• The surface of the lens should be a hyperbolic cosine revolved around its vertex. The problem is very similar to the classical chain hanged between two points. But I suppose it can be approximated as a paraboloid pretty well. That said, I think the effect of the polycarbonate sheet has to be taken into account when considering the net effect. It depends on the thickness of the sheets, but it can potentially hide the small convergence due to the vacuum. – GRB Jun 18 at 23:34
• @GRB Does the air pressure in question result in vertical forces (like the chain example)? Or will they be perpendicular to the surface? – Alex Doe Jun 19 at 0:24
• @AlexDoe You probably spot a mistake in my reasoning. Pressure should exert a force perpendicular to the surface. So the shape is not described by cosh(). Sorry for the mistake. – GRB Jun 19 at 8:35

Your system acts rather like two planar sheets and less like a lens. The fundamental physics of the lens is given by Snell's law, $$n_1 \sin(\alpha_1) = n_2 \sin(\alpha_2)$$. By using thin plastic sheets you mainly get a shift in the position, but only a tiny change in the direction. Instead of vacuum you should fill it with some oil.

Sidemark: Your lens is concave. Therefore it would not focus, but defocus an incoming plane wave.

• Plus it isn’t a constant curvature as well. The slight difference in refractive index won’t help either. – Jon Custer Jun 18 at 19:02
• Does the fact that the medium inside is less dense than the outside not account for anything? Or is it too little of a difference? – Alex Doe Jun 18 at 19:12
• The light changes direction only at the surface of the lens. The change of direction is due to Snell's law (see above). Hence, it only depends on the refractive indices and the angles. The "width" of the lens is rather unimportant, as the light propagates in a straight line. Of course, if we are interested in an exact description, the last statement is not correct. Therefore, in linear optics we use the ABCD-formalism to obtain a more accurate description of lenses. – Semoi Jun 18 at 19:21
• @JonCuster - Can you elaborate on the curvature not being constant please? – Alex Doe Jun 18 at 19:43
• @AlexDoe - the deformation will be a fairly complicated problem depending on factors such as the shape of the membrane, boundary conditions, linearity of the deformation, etc. If the plastic sheets are thin, isotropic, round, and have a very small deflection, the result might be a constant curvature over much of the area. In particular, the assumption of isotropy is wrong for any plastic sheet you can buy, and the deformations are not small for the example you sketched. – Jon Custer Jun 18 at 23:25