# Changing light beam direction

I'm using a rotating mirror to reflect a beam of light like that:

where A is the beam source, B is the projection of the reflected beam on a surface at a distance y from the mirror, x is the distance between the emitted and the projected ray, and α is the angle of incidence + the angle of reflection.
The angle α can be determined by the equation $$\alpha = \Pi -\arccos (-\frac{y}{\sqrt{x^{2}+y^{2}}})$$ By rotating the mirror, the distance x changes exponentially. Is there some kind of optical device (some kind of lens, perhaps) which I can put after the mirror to make x linearly dependent on α for some range? $$x\in (x_{0}, x_{1})$$

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The series expansion of cosine suggests not, unless the argument is small ($\theta<1$). Why do you need $x(\alpha)$ to be linear? – Kyle Kanos Oct 26 '14 at 19:08
I think it would be easier to use a not flat mirror which is much easier to make than a lens, or easier still, two additional flat mirrors in the path that are linked to the main mirror so the final angular direction is the one you want (in this case, one that makes x grow linearly with x. – Wolphram jonny Oct 26 '14 at 19:13
@Kyle Kanos I'm measuring the reflected beam digitally and if the measurment time is not linear, the bandwidth increases dramatically. – slaviber Oct 26 '14 at 20:37
@julian fernandez If I add more mirrors, the effective range at which the system works becomes very low, or the mirrors - very big. I'll try experimenting with a non flat mirror. – slaviber Oct 26 '14 at 20:52
In addition to using a non-flat mirror, changing the angular velocity $\omega$ of the mirror could be used to linearize the change in $x$. This might complicate the setup more than it is worth, though. – Dave Coffman Oct 27 '14 at 0:03