Raised menisci around small discs positioned to pull up a water-air interface provide a highly controllable experimental setup capable of reproducing much of the rich phenomenology of gravitational lensing (or microlensing) by *n*-body clusters. Results are shown for single, binary, and triple mass systems. The scheme represents a versatile testbench for the (astro)physics of general relativity's gravitational lens effects, including high multiplicity imaging of extended sources.

## I. INTRODUCTION

Gravitational lenses make for a fascinating and inspiring excursion in any optics class. They also confront students of Einstein's general theory of relativity with a rich set of associated phenomena. Accordingly, several introductions to the topic are available, just as there are comprehensive books and articles on the full spectrum of observations (for a good selection, the reader is referred to a recent Resource Letter in this Journal^{1}).

Gravitational lenses found their optical refraction analogy at least as early as 1969.^{2} Over the following 50 years, several variants of logarithmic axicon lenses or other glass/plastic lenses with similar profiles (the simplest one being a wine glass stem) have been used to simulate and tangibly convey the effects of gravitational lensing in class rooms,^{3–11} physics labs,^{12,13} or museums^{14–16} around the globe. All of these simulators are for *single masses or mass distributions only*, hence impeding access to the growing field of microlensing (an important tool in the search for exoplanets)^{17,18} and exotic gravitational lenses.^{19} Quoting the pioneers of the theory for binary lens systems,^{20} one may call attention to the fact

“[…] that most stars are members of a binary (or multiple) system and that galaxies appear in pairs, groups or clusters […].”

Indeed, it has been estimated^{21} that about 10% of all observed stellar gravitational microlensing events should exhibit signatures of a binary lens. Moreover,^{22}

“[…] in roughly a quarter of all strong lenses, the lensing potential is known to be more complex than a single lensing galaxy, with usually a pair or group of galaxies being implicated.”

This article therefore proposes a new interactive optical analogy in the field addressing this gap.

### A. A multi-component lens simulator

A natural extension is to use multiple glass or plastic lenses. While it appears as if indeed constellations of two glass lens models (in different planes) have been used in the past to mimic binary lenses,^{23} edge effects and unwanted images occurring in such setups likely discouraged past attempts in this direction. The only true multi-component optical simulator was proposed by Mertz in 1996:^{24} using a microscopic arrangement of salt-grains stochastically dispersed on a flat piece of glass, he used a thin layer of transparent, UV-curable cement to generate a menisci landscape. This ensemble of gravitational lenslets caused a chaotic caustic pattern reminiscent of the situation encountered, for instance, in quasar microlensing by a random distribution of compact lenses.^{25–27}

Inspired by recent studies on the caustics of floating bubbles,^{28} the model system proposed herein uses *liquid menisci* around objects with a circular cross section to complement past physical models with an *adjustable multi-component gravitational lens simulator*. The perturbed liquid, when held in a flat-based transparent basin and mounted such as to allow views through it, represents the *n*-components lens system and provides a highly controllable, *perfectly smooth,* and *connected single refracting interface geometry* (in a single plane) by the action of surface tension, thus effectively avoiding unwanted reflections and associated images by edge effects. The basic idea is sketched in Fig. 1.

For thin (compared to the capillary length) vertical rods, the surface topography around each component even becomes (as for the microscopic salt grains) logarithmic close to the rods^{29} and the corresponding deflections rather faithfully mimic point masses, while also providing access to more caustics patterns (more on this later). However, for a practical installation larger centimetric objects are beneficial. Although in this case, a given component's liquid meniscus shape (exponential in its large-distance fall-off) will not yield the logarithmic profile needed to generate a proper optical analogy to a point mass, nor to other commonly assumed mass distributions^{6,7} (though resembling some models of exponential spiral galaxy disks^{30}), the phenomenology is again very similar (exceptions noted later), much in the same way the wine glass stem provides a good analogy for single masses.

The macroscopic scale of the setup makes it very flexible and particularly suitable for lab courses as well as seminar and open science demonstrations.

### B. The analogy

Whereas for glass lens simulators, the lens profiles can be freely chosen and designed to match certain mass distributions, the liquid menisci lens system is determined by the object arrangement (and cross section) and the physics of surface tension. In the gentle slope approximation, the liquid surface height perturbation $z=f(x,y)$ is governed by the linearized Young–Laplace equation^{31}

with *a* being the capillary length of the liquid (for water $a=2.73\u2009mm$), and solved with proper boundary conditions (e.g., the liquid forming the equilibrium contact angle $\varphi c$ at a cylinder wall). A light ray is then deflected via refraction at the interface *f* by an angle (direction from incident to deflected ray)

where $\u2207r$ is the gradient operator acting in the *xy*-plane (see Appendix A). Both of these expressions resemble the case of gravitational lensing, where the deflecting gravitational potential $\Phi $ is governed by Poisson's equation $\u22072\Phi =4\pi G\rho $ (cf. Eq. (1)), with *G* being the gravitational constant and *ρ* being the mass density (for point masses *m _{i}*, $\rho =\u2211imi\delta (r\u2212ri)$), and the deflection angle is $\alpha \u2192=\u2212(2/c2)\u222bSO\u2207\u22a5\Phi \u2009dz$. (This is the final tangent vector at observer

*O*minus initial tangent vector at source

*S*; typically, the deflection is defined vice versa.) The operator $\u2207\u22a5$ is the gradient perpendicular to the light path and can be approximated by the gradient operator $\u2207r$ acting in the plane of the projected two-dimensional thin lens potential $\varphi =\u222b\Phi dz$, i.e., analogous to Eq. (2) the gravitational deflection reads $\alpha \u2192=\u2212(2/c2)\u2207r\varphi $.

^{27}The difference in signs in the deflection formulas is compensated for by the inverted topography of $\varphi $ vs

*f*. Loosely speaking then, the analogy is $\u2212f\u2194\varphi $ and $hi\u2194mi$, where

*h*are the menisci's rise heights at the

_{i}*i*th component. The analogy works because the resulting (inverted) surface topographies (and their derivatives) resemble gravitational potential landscapes, as sketched in Fig. 1.

In contrast to the glass lens model^{2–11} this system, through the differential equation determining *f* (also an elliptic PDE, a Helmholtz equation with imaginary wave number), provides an *interactive and readily visible analogy to the mass-warped space-time fabric* as well, somewhat like the rubber membrane visualization (with all its limitations^{32}).

## II. EXPERIMENTAL DESIGN

### A. Lenslet design

To produce liquid menisci gravitational lenslets, there are several options: First, one may *attach solid discs to support rods* and position them in such a way that they *pull up the liquid* around them to some height *h _{i}*, as shown schematically in Fig. 2 (menisci sketched in blue and discs in orange). Experimental realizations are discussed in Sec. II B.

Alternatively, though producing weaker lenses fixed in strength (rise height *h*), one may use *short cylinders* (cut from glass rods, say length $\u223c1\u2009cm$, diameter $\u2205\u223c5\u2009mm$, rubbed with an anti-fog cloth for better wetting/higher *h*) standing in the basin. In this scenario, a *shallow* filling level is required, for otherwise the cylinders themselves will obstruct the focused light. Board game playing pieces or Halma pawns may serve as practical substitutes for glass cylinders. Using transparent bevelled glass pieces such as *cabochons* instead allows, much in contrast to mounted discs or cylinders, to even observe additional central images,^{5} which are otherwise hidden by the deflectors (see Sec. III). Although in these glass lenslet scenarios further reflections occur, they can be well separated from the actual images due to their comparatively weak intensities (see the supplementary material).^{33–35}

*Thin rods* ($\u2205\u226aa=2.73\u2009mm$) glued to rods, or thin craft wire, represent another lenslet variant. However, straightforward image multiplicity studies are then troubled by the correspondingly small size of the resulting caustics, and the use of thin rods is not recommended. On the other hand, only this scenario allows for the realization of *all* gravitational lens scenarios of *n*-point masses (cf. Sec. III B).

An even further variation is given by separated bubbles (e.g., in the process of merger due to their mutual attraction^{36}) which produce, in a dynamic way, the same interface geometry, much in contrast to typical floating objects depressing water around them.^{31}

In conclusion, while the bubble and the standing glass piece scenarios each have the advantage of an unobstructed geometry (no rods), and while the thin rods give access to more caustics, only the positioned discs represent strong lenslets which are highly controllable and give large enough caustics for easy observations. Also, by the obstruction of component-centric images, they more closely resemble the typically observed astronomical situations (see Sec. III).

One may wonder why tacks and not larger discs are suggested for experimentation. Choosing a disc size too large (more than $\u223c\u2009cm$) has the effect of increasing beyond practicality the required height of the overall setup (cf. Sec. II B) before the caustics develop. The reason is that deflection angles *α* (for a given *h*) remain roughly constant, while the observer (or camera or screen) needs to be further away from the menisci to receive light from within the focal area (caustic). Also imperfections in the wetting along the perimeter of too large discs or rods cause defects in the menisci, which tend to create complex patterns reminiscent of leaves floating on water^{31} rather than of *n*-body clusters. Sizes of $\u2272\u2009cm$ were found to produce good results, and a corresponding experimental realization as reported in Sec. II B was used for the images shown in this article.

At least two complementary types of experiments can be performed for a given *n*-body realization:

In both cases, a conventional white LED can play the role of a compact cosmic object to be lensed. Though small in size, it still really represents an extended source. However, for convenience, it is usually referred to as a “point light source” in the manuscript since it is small enough to allow counting of non-overlapping (though distorted) images, much as for quasars. Dispersion, a co-phenomenon of all refractive optical simulators but absent in gravitational lensing (which is inherently achromatic^{27}), was not found to have much noticeable effects.

*Experiments of type (a)* are conveniently carried out first, i.e., prior to any effort to recreate a given astrophysical lensing image scenario, as the caustic patterns are commonly used for classification of relevant configurations. They are typically shown in journal articles and can thus be used as a starting point to recreate a given analog configuration. Experiments of this type can also be used to recreate and study light curves,^{27} that is, time series of intensities for a given astronomical microlensing event as it would be observed for the integrated (unresolved) signal recorded via a telescope. To this end, one would photograph a certain caustic pattern and take a line profile through it in some direction, or move a single pixel detector along the caustic. Alternatively, a dynamic caustic may be created by moving a tack with a fixed detector placed below the setup taking a time-series of intensities. (Although in this case, a careful experimental design realizing smooth movements would be required to avoid the excitement of capillary waves.) In both cases, the sequence and shape of peaks appearing is characteristic of a certain *n*-body scenario. Wallace *et al.*^{37} offered an accessible tutorial on this idea.

*Experiments of type (b)* in contrast reveal the phenomenology of source images, akin to astronomical background objects lensed by heavy foreground objects as have, for instance, been catalogued in many surveys,^{1,38} including exotic examples.^{19} The multiplicity for various lensing configurations can be explored by counting the number of images observed for the light source as recorded by a camera or seen by the unaided eye when viewed through the setup. A fixed number of images corresponds to a given region of the caustic bound by folds and cusps, in which the camera (or the eye of the observer) is placed. Crossing folds causes a change of the number of images by at least ±2 (the usual case) and by at most ±4 (at special points only).^{27,39,40} Admittedly, the former correlation between caustic regions and image multiplicity is hard to make in the simplified setup considered here. A mildly more complex setup variant using a pinhole screen could be adopted for this purpose.^{6} Effectively, setting the camera lenses' *f*-stop to high values already functions as a pinhole in the present setup, though it projects on a smaller screen (the camera-chip). Since individual *images become unresolved and merge if the aperture crosses/covers caustic boundaries* (or the lensed light source is too large),^{41} a larger overall caustic structure is beneficial, whereby again the benefit of using the strong macroscopic lenslets based on positioned tacks plays out.

In experiments of type (b), the LED can also be replaced by a printed image of any extended model source to study complex distortion patterns.^{3,25,26}

### B. Setup realizations

Realizations of a single setup *suitable for both types* of experiments as sketched in Fig. 2 are shown in Figs. 3 and 4. In practice, many variants of the setup are thinkable. A good starting point can be a commercial C-table (laptop stand) or console table with a glass top. Alternatively, a simple custom mount for a given basin can be constructed either from wooden planks or from aluminum strut profiles (cf. Fig. 4). The flat-based acrylic basin is best built on one's own from acrylic sheets (thickness $\u223c5\u2009mm$) using transparent aquarium silicone adhesive, since most commercial transparent basins tend to have non-flat bases from an injection moldling process. To adjust the light source (any small LED, e.g., a flashlight, with reflectors removed if present) vertically above a given setup, one may conveniently use a small mass suspended with a thread to act as a plumb line. Both setups cost approximately $150–$250 to realize, although less expensive variants are thinkable.

The actual pictures in Figs. 5–8 of this paper were taken with the custom wooden setup of Ref. 28 (cf. loc. Fig. 16(a), not reproduced here; conceptually equivalent to the one of Fig. 3, basin as in Fig. 4, though with a ceiling-mounted LED). The camera was a Fuji X-Pro 2 equipped with the XF60 mm F2.4R Macro lens, set to a large *f*-stop $\u226516$. Thumb tacks (drawing pins) were superglued to black carbon fiber rods (diameter $\u2205=0.5\u2009mm$, pins: $\u2205=10\u2009mm$), which in turn were attached to the basin using M3-screws and two plastic washers (using magnetic ball joints further improves controllability).^{62} This allowed to freely (within some limits and with some practice) position the pins above the water (filling level adjusted to be $h\u22485\u2009mm$ below them). A brief forced dipping of the elastic rods can then be used to pull up the menisci. Referring to the sketch in Fig. 2, the distances were: $Dl=1.0\u2009m,\u2009Dsl=1.3\u2009m$ (and $s=1.4\u2009cm$ for Sec. III B).

## III. THREE SCENARIOS

The complexity of pure (no external shear) *n*-point mass lensing scenarios grows rapidly with increasing *n*. A systematic analysis and comprehensive description has been done for two-mass lenses.^{20,42–45} This situation is particularly useful in the search for exoplanets.^{17,18} For three-mass lenses, extensive studies exist as well, although the system is already hardly tractable.^{46,47} Thus, only scenarios up to *n *=* *3 components have been considered experimentally here.

For a single point mass (*n *=* *1), there are exactly 2 images of a point source.^{27} For an arbitrary extended lensing mass distribution (a *non-singular* bounded transparent lens, e.g., an elliptical lens) the number of images is odd,^{48} and the observable image multiplicity depends on whether the central lensing mass is transparent or outshining a central image^{49} (whereby typically either 2 or 4 images have been observed). The number of images for *n *>* *1 point masses ranges from *n *+* *1 to $5n\u22125$ (formerly, $n2+1$ images were believed to be possible,^{50} which is not true) in increments of two,^{51,52} i.e., for binary systems (*n *=* *2) from 3 to 5, and for triplet systems (*n *=* *3) from 4 to 10. For non-singular (extended) multi-component lenses, the odd-number image theorem^{48} holds as well.^{53} Again, apparent discrepancies between observations and theory can typically be explained by unseen “ghost images” hidden by the deflectors^{39,54} or strongly demagnified, and thereby weak, sub-images.^{55}

### A. Single component lens (*n *=* *1)

The best known example of strong gravitational lensing is that of an extended source by a single mass (at least axisymmetric) all lying on a single axis: In this case, an *Einstein ring* is observed^{38,40,56} (e.g., SDP.81, LRG 3–757, and B1938 + 666). For the case of a single tack of radius *R*, setting $f(r)|r=R=h$, the angular Einstein ring radius $\theta E(Dl,Dsl)$ may be computed from an approximate meniscus profile and the experimental configuration, yielding an expression indeed resembling the gravitational lens case (see Appendix A). In the experiments, $\theta E\u223c1\xb0$, making longer focal length lenses beneficial for taking images with a camera. Now, when either the lens, the camera or the source are displaced away from the axis, a splitting in two elongated arcs is seen for an extended source, or a splitting into two point images for a point source (one inside, a brighter one outside the Einstein ring radius). For larger displacements, the image inside becomes significantly fainter, before eventually becoming unobservable due to an aperture effect (finite *D _{l}* and bounded

*α*), while the other becomes only weakly lensed (distorted). The experimental realization using a single positioned tack is shown in the central row of Fig. 5.

When the single lensing mass is elliptical (as projected in the lens plane), the symmetry is reduced and the Einstein-ring for the perfect alignment scenario breaks up into the so-called *Einstein cross*^{38,40,46} (e.g., QSO 2237 + 0305, J2211–0350). For any displacement of either the source or the lens, two of the images making up the cross move towards one of the remaining two images before finally merging to yield then only two images. Upon further displacement, again one of the images becomes fainter while the other becomes the weakly lensed one. The experimental realization is shown in the bottom row of Fig. 5, where an *inclined view* through the setup gives an elliptical lens as projected perpendicular to the viewing axis (using an elliptical lenslet and a vertical configuration yields the same results). Note, that a central image is obstructed by the opaque disc, such that the observed numbers of images (2 and 4) are consistent with the theoretical expectations for an extended asymmetric mass distribution. As mentioned in Sec. II A, a transparent cabochon lenslet instead of a tack yields the central image as well.

The same phenomenology also occurs for a point source with external shear (a Chang–Refsdal lens),^{55,56} mimicking an asymmetric environment (say galaxies or clusters near the lenses, or structures along the ray path). External shear may be thought of as an extremely asymmetric two-mass lens system,^{20,44} yielding 3 and 5 images, with the central ones unobserved (effectively matching then the 2 and 4 images of the Chang–Refsdal lens^{55}).

### B. Two component lens (*n *=* *2)

For two components, the phenomenology becomes richer. Three different configurations have been identified: *wide*, *intermediate/resonant*, and *close*, each having qualitatively different caustics.^{20,42–45} The categorization of a given binary system is determined by two parameters only, the mass ratio (in this analogy: the ratio of meniscus heights *h*) and the distance *s* (cf. Fig. 2(b)) of the two components. The resulting caustics entail cusps and folds only, and for gravitational lensing the image positions can be efficiently calculated numerically.^{57}

An experimental realization of the no-shear system (vertical view) began by adjusting the two tacks to yield approximately the characteristic intermediate-distance caustic (type (a) experiment, see Fig. 6). The resulting images for the corresponding type (b) experiments are shown in Fig. 7, yielding from 3 up to 5 separate images when including the central weak image (which has a high probability to be missing in astronomical observations). The image configurations attainable may be compared to astronomical examples such as CLASS B1608 + 656 (Ref. 58) or SL2SJ1405 + 5502.^{19}

It must be noted that the “close” configuration^{20,42–45} could *not* be reproduced using tacks, see Fig. 6. (However, using thin rod lenslets of diameter $\u2205=0.7\u2009mm\u226aa$, the configuration *did* become accessible for small *s*. A separate study, co-authored by Lock, analyzes these binary liquid meniscus caustics in detail.^{62}) Also, on close inspection, it was found that for the two extended discs the 6-cusped intermediate caustic was easily perturbed by small misalignments (from the vertical configuration) or tilts of the tacks: the two cusps on the symmetry axis then evolved into tiny *butterfly caustics*. This caustic structure resembled the one of two point masses with added external shear, as shown in Fig. 11(a) of Ref. 43 (rather than, e.g., Fig. 2 (X = 0.5) of Ref. 20). Indeed, theory tells that with added external shear (effectively then corresponding to at least three components), or again by perturbing the spherical symmetry by an extended mass distribution (or elliptical ones, i.e., considering an inclined view through the setup), higher-order caustics can appear: the swallowtail and the butterfly caustics, with a correspondingly increased (odd) number of images.^{43}

It should be a worthwhile advanced laboratory course exercise to try and access the higher image multiplicities through highly inclined view experiments of type (b) corresponding to the caustics shown in the third column of Fig. 6 (symmetry axis connecting the two tacks in the plane of inclined incidence).

### C. Three component lens (*n *=* *3)

For three components, the phenomenology becomes even richer. For triple point mass systems, as many as 32 different caustic structures have already been identified in a subset of configurations,^{46} while the full caustic structure cataloging has yet to be done.^{47} Already, there are five parameters required to fully describe the situation: two mass ratios (menisci height ratios) along with three parameters defining the two-dimensional configuration of the components. Many gravitational triple lens scenarios have been observed, although almost all have been very asymmetrical in nature: either binary stars with a single planet or single stars with two planets.^{18,46,47} These situations can be viewed as perturbed single mass or binary systems. To the best of the author's knowledge, the most prominent triple mass system where all three masses contributed roughly equally to the lensing has been identified in 2001: CLASS B1359 + 154 is a group of three compact galaxies lensing a radio source (and its host galaxy), resulting in 6 images (likely a scenario involving extended lens components yielding 9 images, 6 of which are observable).^{59}

The attempt of an experimental realization using three tacks is shown in Fig. 8. A fair match was found rather quickly, where the closeness was somewhat surprising and is likely accidental to some extent, given that no optimization of the menisci heights, their diameters, or their detailed positions was undertaken and given that the analogy is rather to a system of point masses. The matching image was one of several found image configurations, where up to 10 images were observed (in line with the expectation for a triple point mass system).

Similar to the binary point mass analogy, not all configurations could be reproduced using tacks. For instance, a “close” configuration counterpart in the triple lens system, denoted by TE:G_{2} (triangular equilateral, parameter space region G, caustic structure 2 with 18 cusps),^{46} is not accessible unless again thin rods are used.

It should be an interesting advanced laboratory course exercise to try to recreate further members of the fascinating zoo of caustics reported for various different parameter triple lens systems.^{46,47}

## IV. CONCLUSION

Using the simple experimental scheme proposed in this article, its potential has been demonstrated using the three scenarios of single, double, and triple mass systems. In each case, the phenomenology of the associated caustics as well as the image multiplicity and image configurations could be reproduced faithfully for selected configurations. Admittedly, the analogy has not been probed exhaustively, i.e., the immense parameter spaces for *n *=* *2, 3-body lenses have not been sampled wholly (including mass/menisci height ratios), and neither have the effects of distances nor transitions from weak to strong lensing, etc., been investigated. Still, the utility of the model was hopefully sufficiently motivated and leaves much room for experimentation to still be done (cf. also the “Cheshire Cat Challenge” in Appendix B). The present manuscript also leaves room for a deeper theoretical exploration of the optical analogy and its exact limitations.

In summary, the system affords optically smooth surfaces representing analog scenarios, which closely mimic ideal theoretical predictions for the strong gravitational lensing by *n* point masses. Although the underlying details of the imaging are different, the analogy is close enough to afford a good match of the imaging characteristics. The setup should thus be a valuable advanced demonstration piece, supplementing the simpler single mass (distribution) glass simulators,^{2–11} and may also be a tool for insight into more complex gravitational lensing configurations by hinting at an analogy to the remote field of optical caustics of perturbed liquid interfaces.

## ACKNOWLEDGMENTS

The author thanks the anonymous referees for helpful suggestions. Among them is the suggestion of further extending the setup to use horizontal rods or other shapes to simulate, for instance, (weak) gravitational filament lensing Ref. 61. The author also thanks K.-H. Lotze for fruitful discussions on glass lens experiments. Finally, the author thanks James A. Lock for his initiation of an inspiring collaboration further exploring lensing by multiple objects in water.

### APPENDIX A: ANALOGOUS EINSTEIN RADII

Since the differential equation (gentle slope approximation) for the surface, $\u22072f=a\u22122f$, is linear, solutions for the individual discs (or cylinders) may be superimposed, just as in the gravitational lensing case. In cylindrical coordinates, and for a single disc only, the differential equation has the solution

with *K*_{0} being the modified Bessel function of the second kind of zeroth order, and *R* being the radius of the disc.

From Fig. 9, the bending angle vector can be seen to be $\alpha \u2192=(n\u22121)\u2207f$ (where $\u2207f\u221d\u2212\rho \u0302$ for raised menisci points towards the optical axis), where in the gentle slope approximation $\u2207=\rho \u0302\u2202/\u2202\rho +\cdots $ in cylindrical coordinates may be taken as the gradient operator $\u2207r=x\u0302\u2202/\u2202x+y\u0302\u2202/\u2202y$ acting in the *xy*-plane (the lens plane).

For the parameters used in the experiments ($h\u223c5\u2009mm$), and using the geometry of the lensing situation as depicted in Fig. 9, the Einstein angle can be found numerically by solving

to find $\theta E\u223c0.7\xb0$ (angular diameter of $\u223c1.5\xb0$). This angular diameter is indeed small compared to the (lateral) angle of view of $22.3\xb0$ of the used macro camera lens and corresponds well to the observed $\u223c7.5%$ lateral filling fraction of the APS-C sensor (the images in the figures were cropped).

Using instead the *exponentially damped approximation,*^{31} $f(r,\varphi )=h\u2009exp\u2009(\u2212(r\u2212R)/a)$ for a single lens, yields the following *analytical* expression for the radius *r _{E}* of the Einstein ring:

where $Ds=Dl+Dsl$ was used (an identity which is not true on cosmological scales for gravitational lensing) and *W*(*x*) is the Lambert W-Function (ProductLog-function), i.e., the inverse of $x\u2009exp\u2009(x)$. The corresponding Einstein angular radius is $\theta E=rE/Dl$. Again using the parameters of the experiments, the result is $\theta E\u223c0.8\xb0$ (an angular diameter of $\u223c1.6\xb0$). Expression (5) is similar to the gravitational Einstein radius expression, which for $\alpha =4GM/rc2$ solves to $rE2=(4GM/c2)\xd7DlDsl/Ds$, and $\theta E=rE/Dl$.

### APPENDIX B: THE CHESHIRE CAT CHALLENGE

The “Cheshire Cat” system should be a worthwhile example of a beautiful and well-known *wide binary* lens,^{22} resembling the face of a smiling cat. It is a complex system consisting of multiple arcs on two different Einstein radii, foreground and lensing galaxies. Luckily, the hard work of figuring out the system's likely configuration has already been done using spectroscopy and gravitational lens modeling:^{60} At least 7 images can be clearly identified and belong to *two sources in different planes* ($Ds,1\u2260Ds,2$) (take two LEDs at different distances), whereas the two lensing galaxies lie roughly in the same plane at *D _{l}* (use two tacks). A third foreground galaxy, not partaking in lensing, forms the nose of the cat (add a third LED between the observer and the lenses). By trial and error, given the information derived by the astronomers, a realization of the Cheshire cat should be possible for an ambitious experimentalist or within an advanced lab course. The cat's eyes could be superposed images of the tacks (or phosphorescent paint on the two tacks).

## References

*n*= 2 (intermediate regime).