# rotating mirror system for shifting a beam?

I am looking for some mirror setup where rotation one or more elements in the mirror setup will allow to shift the beam along its axis.

Two mirrors in a galvanometer scanner type setup work well for controlling the angle of the beam.

Rotating risley prisms can be used to only offset the beam without changing its angle but the possible positions are limited.

Dove prisms are great for rotating the beam along itself or rotate in a circular fashion if prism center is shifted.

But as for offsetting (shifting) a beam along its axis from a (-1,-1) position to (1,1) without changing the angle of the beam, I can't find any setup which involves a rotating mirror rather than shifting mirror. But I feel there may be something missing from my limited optics knowledge.

4 rotating galvo scanner mirrors may do this but this may be an overkill and the size of the mirrors on the last two will increase to impractical sizes for decent 45 degree FOVs or more. So I'm crossing this one out.

In case not clear, https://i.imgur.com/3qkvRpS.jpg

You could use a slab of glass or other transparent medium with a different refractive index then air. The slab should have two surfaces which are as parallel as possible and be mounted on a two axis gimbal. Namely the more you tilt the slab in any direction relative to axis of the beam the more the beam will shift in that direction, while maintaining the beam direction, as shown below:

However you will get more losses due to reflections the further you tilt the slab. And depending on how large of a displacement or the beam you want you might also need a quite thick slab. In order to get your hand on a thick slab of transparent material you could also use a liquid filled container.

The lateral displacement of the beam $l$ can be expressed as follows

$$l = d\,\sin\theta\left(1 - \frac{\cos\theta}{\sqrt{n^2 - \sin^2\theta}}\right)$$

where $d$ is the thickness of the slab, $\theta$ the angle between the beam and the normal of the slab and $n$ the refractive index of the slab.

• How about acrylic? Those are way cheaper and lighter than glass. What will the calculation be for determining how much offset I will get given a slab refractive index, slab thickness, slab angle relative to beam variables? – Mark Legault Mar 30 '18 at 7:19
• @MarkLegault I have added the calculation to my answer. And yes, acrylic should work as well, from what I have looked up it seems to have a refractive index very similar to that of glass. – fibonatic Mar 30 '18 at 9:26
• I think the same effect can also be achieved with two bounce mirrors attached together and rotating around a center point, yes? With an acrylic first surface mirrors would weigh a lot less – Mark Legault Mar 30 '18 at 10:23
• @MarkLegault That should work as well. However your image gave me the impression that the outgoing beam should be close the axis of the incoming beam. This would be impossible with a two mirror setup. – fibonatic Mar 30 '18 at 10:52
• not at all, as long as it can shift from a hypothetical -1 to 1 not relative to source beam position it should work – Mark Legault Mar 30 '18 at 11:00