Can any mirror arrangement give an optical advantage?

Question

I'm trying to see tiny movements in a membrane using a laser spot that shines on it, and looking at the reflected spot. I was wondering if it is possible for any plane mirror arrangement to give me some optical advantage.

Moving the screen further away is an option, but not a good one mainly because my laser is diverging, and the spot size also enlarges with distance, rendering the displacement as a fraction of spot size the same. (Also, due to practical limitations).

Is it possible to have an optical advantage such that if the input angle is $\theta_{\text{in}}$,

the output light angle $\theta_{\text{out}} = f(m\theta_{\text{in}})$ where $m>1$

I tried a two angled mirror setup, with calculation below. the output is still linear in $\theta$. Even with multiple reflections between the wedges,

$$\theta_{\text{out}} = \theta_{\text{in}}+n\alpha$$ where alpha is the angle between the mirrors.

EDIT

Thanks @Gilbert and @Andrew Steane I hadnt considered refraction as an option.

So it seems a prism would do the job. the deviation $\delta$ is a non-linear function of angle of incidence $i$, and for a glass prism, there is a small window of $\approx 5^o$ just after the critical angle point where we can get an optical advantage.

Here's my analysis : https://www.desmos.com/calculator/ts3lsacstn

still have to figure out how to collimate the LASER beam. As I move away from the LASER, the spot size decreases initially and then starts increasing... it should be a focusing issue.

Answer 1

An optical device which magnifies small changes in direction of light is the telescope. A telescope can be made with a pair of lenses of different focal length, or by using a curved mirror and lens, or two curved mirrors.

However, any such device will amplify both the direction change you are looking for, and also the spreading of the laser beam itself. Therefore it won't necessarily help you.

If it is possible to reduce the spreading of the laser beam, you should try to do that. This will depend on what has caused it. It will be a combination of diffraction effects and whatever lensing is in the beam path. To reduce diffraction you have to make the laser beam width larger at the place where it hits the membrane, which might not be practical for you.

Another option is to try to use an interference effect. For example you may have noticed the laser speckle, which is a complicated pattern in the beam intensity as a function of position. If your detector is small enough then it might detect the shift in the laser beam through the movement of the speckle pattern.

Answer 2

You've correctly observed that you can't get any advantage from flat optics. You might get some advantage, however, through curved optics (lenses and/or focusing mirrors). For example, a telescope (two confocal lenses with different focal lengths) is designed to magnify an image. Equivalently, it can magnify a beam displacement. But of course, the whole beam will be magnified, so you may or may not actually gain from this depending on your detection scheme.

You just have to remember that a lens converts angle information into displacement information so you don't get confused. If you're currently measuring angle, collimate the beam with a lens (choose the focal length wisely!) to switch to displacement, and then you can amplify it with a telescope. If you want a large amplification, you'll need large and high-power optics.