## Alvarez lenses and other strangely shaped optical elements

In typical microscopes, lenses or mirrors are moved forth and back to change the position of their focus. Tunable lenses like the electro-tunable lens or the TAG lens, on the other hand, are deformed by an external force and thereby change their focal length. One interesting concept that I had not noticed until recently is the idea of the Alvarez lens, named after its inventor (described in this 1964 patent). I came across it in a 2017 paper from the lab of Monika Ritsch-Marte from Innsbruck/Austria. The following picture adapted from their paper very nicely illustrates the effect:

By lateral displacement of the two lens elements against each other one can focus or de-focus the beam. In two papers from this lab (paper 1, paper 2), the authors used a method that sort of replaces this lateral (slow) movement with a (fast) rotation of a galvo mirror by using a creative optical configuration (check out the paper for the details, it is a pleasure to read).

There a couple of things to notice: The Alvarez lens is a bit more complex in 2D (the above schematic illustrates a (de-)focusing system for 1D only). The authors use diffractive instead of refractive Alvarez lenses. They use only visible light (no near-infrared light, which I would prefer). And they mention some other shortcomings of their approach.

Still, I find the principle very interesting and inspiring, and I hope that somebody will invest his or her time to put a system together that is not only a proof-of-principle, but an optimized system that reaches the best possible performance. This would probably also be a nice playground for a study of optical modeling and optimization: to find out which shape of the lens could perform much better than the Alvarez lens (like this study, but a bit more systematic with respect to possible lens surfaces).

Overall, this is a fascinating piece of optics, and I got interested also because I had always been intrigued by optical scanning methods where a simple movement of the beam is translated to a complex scanning scheme by an optical element (see for example this blog post on entirely passive scanning at MHz rates). For a long time, I hoped that a method similar to an Alvarez lens and based on a strangely shaped mirror (or lens) surface could be used to transform a linearly scanned pattern into something more complex (like a spiral scan, or a 2D raster scan). In theory, this is possible, but in practice the finite beam diameter would create a lot of problems. In addition, constructing an arbitrarily shaped mirror with good surface flatness and broadband reflective coatings would be quite costly.

One field where I long thought that such an approach could be applied, because it would be applicable for many microscopes, is the un-distortion of the non-linear angular scanning trajectory of resonant scanners (described in detail in a previous blog post). The idea would be that an optical element (the ‘black box lens’ in the schematic below) placed after the resonant scanner would somehow convert the angular dependency $\sim \sin(\omega t)$ into a relationship that is rather linear in time, $\sim t$. Such that at time points close to the turnaround of the sine (blue time point below), the ‘black box lens’ would increase the angular deflection angle, eventually inversing the sine function:

I have the suspicion that this problem is practically not solvable due to the finite beam diameter, but it would be interesting to know whether there is a solution for this problem at least for the assumption of infinitely small scanning beams using geometric optics. This could be done by a lens whose diffractive power increases with distance $x$ from the center of the lens.

Let’s assume a scan angle $\alpha = \sin( \omega t)$. The scanned beam hits the black box lens at a position $x(t) = \tan(\alpha) \cdot d$ with the distance $d$ between the resonant scanner and the lens. The refractive power $f(x)$ of the lens must therefore change with $x$ such that the outgoing beam is linear in time. In approximative ABCD optics:

$\left( \begin{array}{cc} 1 & 0 \\ -\frac{1}{f(x(t))} & 1 \end{array} \right) \cdot \left( \begin{array}{cc} x(t) \\ \alpha (t) \end{array} \right) \stackrel{!}{=} \left( \begin{array}{cc} x(t) \\ t \end{array} \right)$

This results in the following expression for the local radius of the lens depending on the location $x$:

$f(x) = \frac{x}{\arctan(x/d) - \arcsin(\arctan(x/d))}$

You can find the equations and some plots also in a Jupyter notebook on Github. For small absolute values of $x$, $f(x)$ diverges, indicating an infinite curvature, i.e., a lens that simply transmits light without deflection. With increasing/decreasing $x$, $f(x)$ tends to zero, indicating increasing refractory power and a stronger local curvature of the lens surface.

Such a lens would only work optimally at one single zoom setting, which is probably one of the many reasons why nobody ever has tried this out. But it’s still interesting to think about it.

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