AAn optical n -body gravitational lens analogy Markus Selmke *Fraunhofer Institute for Applied Optics and Precision Engineering IOF, Albert-Einstein-Str. 7, 07745 Jena, Germany ∗ (Dated: November 12, 2019)Raised menisci around small discs positioned to pull up a water-air interface provide a well con-trollable experimental setup capable of reproducing much of the rich phenomenology of gravitationallensing (or microlensing events) by n -body clusters. Results are shown for single, binary and triplemass lenses. The scheme represents a versatile testbench for the (astro)physics of general relativity’sgravitational lens effects, including high multiplicity imaging of extended sources. I. INTRODUCTION
Gravitational lenses make for a fascinating and inspir-ing excursion in any optics class. They also confront stu-dents of a dedicated class on Einstein’s general theory ofrelativity with a rich spectrum of associated phenomena.Accordingly, several introductions to the topic are avail-able, just as there are comprehensive books on the fullspectrum of observations (for a good selection of articlesand books the reader is referred to a recent Resource Let-ter in this Journal ). Instead of attempting any furtherintroduction to the topic, which the author would cer-tainly not be qualified to do in the first place, the articleproposes a new optical analogy in this field.Gravitational lenses found their optical refraction anal-ogy at least as early as 1969. Over the following 50years, several variants of logarithmic axicon lenses orother glass/plastic lenses with similar profiles (the sim-plest one being a wine glass stem) have been used tosimulate and tangibly convey the effects of gravitationallensing in class rooms, physics labs or museums around the globe. All of these simulators are for singlemasses or mass distributions only , ruling out for instanceaccess to the growing field of microlensing (an importanttool in the search for exoplanets) and exotic gravi-tational lenses. Quoting the pioneers of the theory ofbinary lens systems, one may call attention to[. . . ] the fact that most stars are members ofa binary (or multiple) system and that galax-ies appear in pairs, groups or clusters [. . . ].Inspired by earlier studies on the caustics of floatingbubbles, the herein proposed model system thus usesliquid menisci around fixed solid discs to extend pastphysical models to multiple component ( n -body) grav-itational lenses . Although a given component’s liquidmeniscus shape (exponential in its large-distance fall-off)does not provide a perfect approximation to the logarith-mic profile needed to generate a proper optical analogyto a point mass, nor to other commonly assumed massdistributions (though resembling some models of ex-ponential spiral galaxy disks ), the phenomenology isagain very similar. Using thin (compared to the cap-illary length) vertical rods instead can even provide alogarithmic profile accurately mimicking point masses atthe cost of a more difficult observation due to the smaller camera screen (a) (b) acrylic pool(water-filled) sensor LED FIG. 1. A suitably mounted acrylic water basin allows forviews through menisci-cluster lenses. Drawing pins are heldabove the unperturbed water level by carbon fiber rods (diam-eter ∅ = 0 . ∅ = 10 mm). The rods are attachedadjustable to the basin using M3-screws and two plastic wash-ers. Variants: (a) to study caustics, (b) to study images (byeye or camera). The distances were: D l = 1 . D sl = 1 . s = 1 . f -stop ≥ size of such an installation, which is why larger discswere used in this work. While it appears as if constel-lations of two glass lens models may have been used inthe past to mimic binary lenses , the present model pro-vides a perfectly smooth and connected single adjustable refracting interface geometry (in a single plane) by theaction of surface tension, also enabling the simulation ofdynamic microlensing events. To produce such strong gravitational n -body lensing or microlensing analogies one may attach n soliddiscs to support rods, e.g. via drawing pins (tacks) su-perglued to thin carbon rods, and position them in sucha way that they pull up the liquid around them, as shownschematically in Fig. 1 (menisci sketched in blue, tacks inorange). The liquid (e.g. water) should be held in a flat-based basin and mounted such as to allow views throughit (loc. Fig. 16(a) of Ref. [ 17] shows an example of awooden mount). Alternatively, separated bubbles (e.g. inthe process of merger due to their mutual attraction )also produce in a dynamic way the same interface geom-etry, much in contrast to typical floating objects depress-ing water around them . While the bubble scenario hasthe advantage of an unobstructed geometry (no rods),the more controlled setup using positioned discs lends it-self naturally for the proposed task and is considered in a r X i v : . [ phy s i c s . e d - ph ] N ov this article.More mathematically, in the gentle slope approxima-tion the liquid surface perturbation z = f ( x, y ) is gov-erned by the linearized Young-Laplace equation ∇ f = a − f, (1)with a being the capillary length of the liquid (water: a =2 .
73 mm), and solved with proper boundary conditions.Light is then deflected via refraction at the interface f by an angle (direction from incident to deflected ray) (cid:126)α = ( n − ∇ f, (2)where ∇ is the gradient operator acting on the xy -plane (see Appendix VII). Both of these expressions re-semble the case of gravitational lensing where the de-flecting gravitational potential Φ is governed by Poisson’sequation ∇ Φ = 4 πGρ and (cid:126)α = − (2 /c ) (cid:82) OS ∇ ⊥ Φ d z (fi-nal tangent vector at observer O minus initial tangentvector at source S . Typically, the deflection is definedvice versa.), where ∇ ⊥ (gradient perpendicular to thelight path) can be approximated as the gradient opera-tor ∇ acting on the plane of the projected thin lens φ = (cid:82) Φd z . The difference in signs in the deflectionformulas is compensated for by the inverted topology of φ vs. f . Loosely then, the analogy is − f ↔ φ and h ↔ ρ .In contrast to the glass lens model this system,through the differential equation determining f (also anelliptic PDE, a Helmholtz equation with imaginary wavenumber), provides an interactive analogy to the mass-warped space-time fabric as well, somewhat like the rub-ber membrane visualization (with all its limitations ). II. EXPERIMENTAL SETUP
The experiment uses illumination by a distant conven-tional whitish LED light source. Though small in size,it still actually represents an extended source. However,for convenience, it is referred to as a ”point light source”mostly in the manuscript as it is small enough to allowcounting of non-overlapping images. Now, the experi-ment can be performed in at least two modi:(a) observation of the caustics associated with a given n -body realization, see Fig. 1(a), or(b) observation of the images associated with a given n -body realization, see Fig. 1(b).Experiments of type (a) are conveniently carried outfirst, i.e. prior to any effort to recreate a given astro-physical lensing image scenario as the caustic patternsare commonly used for classification of relevant configu-rations. They are typically shown in journal articles andcan thus be used as a starting point to recreate a givenanalog configuration. Experiments of this type can alsobe used to recreate and study light-curves, that is time series of intensities for a given astronomical microlensingevent as it would be observed for the integrated (unre-solved) signal recorded with a telescope: To this end,one would photograph a certain caustic pattern and takea line profile through it in some direction. Alternatively,a dynamic caustic may be created (via a moving tack)with a fixed detector placed below the setup taking atime-series of intensities. The sequence of peaks appear-ing is characteristic of a certain n -body scenario.Experiments of type (b) in contrast reveal the phe-nomenology of source images, akin to astronomical back-ground objects lensed by heavy foreground objects ashave for instance been catalogued in many surveys, including several exotic examples. The multiplicity forvarious lensing configurations can be explored by count-ing the number of images observed for the light source asrecorded by a camera or seen by the unaided eye whenviewed through the setup. Here, the LED can be replacedby a printed image of any extended model source as wellto study the complex distortion patterns.
A fixed number of images corresponds to a given regionof the caustic bound by folds and cusps in which the cam-era (or the eye of the observer) is placed. Crossing foldscauses a change of the number of images by at least ± ± While admittedly the formercorrelation between caustic regions and image multiplic-ity is hard to make in the simplified setup considered here(cf. Fig. 1), the more controlled though light-sacrificingsetup variant using a pinhole screen could be adopted forthis purpose. Effectively, setting the camera lenses’ f -stop to high values functions as a pinhole in the presentsetup, though projecting on a smaller screen (the camera-chip). III. THREE SCENARIOS
The complexity of pure n -point mass lensing scenar-ios (no added external shear) grows rapidly with in-creasing number n of lenses. A thorough analysis andcomprehensive description has been done for two-masslenses. This situation is particularly useful for thesearch of exoplanets.
For three-mass lenses, exten-sive studies exist as well, although the system is alreadyhardly tractable.
Thus, only scenarios up to n = 3have been considered experimentally here.For a single point mass ( n = 1), there are exactly 2 im-ages of a point source. For an extended lensing mass dis-tribution (a non-singular bounded transparent lens, e.g.an elliptical lens) the number of images is odd , andthe observable image multiplicity depends on whetherthe central lensing mass is transparent or outshining acentral image (whereby typically either 2 or 4 imageshave been observed). The number of images for n > n + 1 to 5 n − i.e. for binary systems ( n = 2) from 3 to 5, andfor triplet systems ( n = 3) from 4 to 10. For non-singular(extended) multi-component lenses, the odd-number im- FIG. 2. Images for a single lens using the setup of type (b).The top series was taken looking vertically through the lenstowards the LED, whereas the bottom row corresponds tolooking towards the LED under an angle of about 40 ◦ to thevertical. In the top row, the image positions qualitativelyagree with the series for gravitational lensing of an extendedsource by a point mass (cf. loc. Fig. 8.20(a)-(c) in [28], in-cluding an Einstein ring). The bottom row image positionscorrespond to gravitational lensing of an elliptical mass dis-tribution (or a point mass with external shear, cf. loc. Fig.8.20(d)-(f) in [28], including asymmetric Einstein cross ). age theorem holds as well . Again, apparent dis-crepancies between observations and theory can typicallybe explained by unseen ”ghost images” hidden by thedeflectors or strongly demagnified and thereby weaksub-images. A. Single mass gravitational lenses ( n = 1 ) The best known example of strong gravitational lens-ing is that of an extended source by a single mass in themost simple scenario: A configuration in which the ob-server, the lensing mass (at least axis-symmetric) as wellas the lensed source are lying on a single axis: in thiscase, an
Einstein ring is observed (e.g. SDP.81,LRG 3-757, B1938+666). For the case of a single tack ofradius R , setting f ( r ) | r = R = h , the angular Einstein ringradius θ E ( D l , D sl ) may be computed from the approxi-mate profile f = h exp( − ( r − R ) /a ) and the experimentalconfiguration, yielding an expression indeed resemblingthe gravitational lens case (see Appendix VII. In the ex-periments, θ E ∼ ◦ ). Now, when either the lens or thesource are displaced away from the axis, a splitting intwo elongated arcs is seen for an extended source, or asplitting into two points for a point source (one inside, abrighter one outside the Einstein ring radius). For larger displacements, one of the images becomes significantlyfainter while the other becomes unlensed. The experi-mental recreation using a single positioned tack is shownin the central row of Fig. 2.When the single lensing mass is elliptical (as projectedin the lens plane), the symmetry is reduced and theEinstein-ring for the perfect alignment scenario breaksup into the so-called Einstein cross (e.g. QSO2237+0305, J2211-0350). For any displacement of eitherthe source or the lens, two of the images making up thecross move towards one of the remaining two images be-fore finally merging to yield then only two images. Uponfurther displacement, again one one of the images be-comes fainter while the other becomes the unlensed one.The experimental recreation is shown in the bottom rowof Fig. 2, where an inclined view through the setup givesan elliptical lens as projected perpendicular to the view-ing axis. Note, that a central image is obstructed by theopaque disc, such that the observed numbers of images(2 and 4) are consistent with the theoretical expecta-tions for an extended asymmetric mass distribution. Atransparent acrylic disc instead of a tack could possiblyovercome this shortcoming of the model.The same phenomenology also occurs for a point sourcewith external shear (a Chang-Refsdal lens), mimick-ing an asymmetric environment (say galaxies or clustersnear the lenses, or structures along the ray path). Exter-nal shear may be thought of as an extremely asymmetrictwo-mass lens system, yielding 3 and 5 images, withthe central ones unobserved (effectively totalling then the2 and 4 images of the Chang-Refsdal lens ). B. Two mass gravitational lenses ( n = 2 ) For two point masses, the phenomenology becomesricher. Three qualitatively different configurations canbe identified for gravitational (micro)lensing by binarymass systems: wide, intermediate, and close, each havingqualitatively different caustics.
The categorizationof a given system is determined by two parameters only,the mass ratio (in this analogy: the ratio of meniscusheights h ) and the distance s (cf. Fig. 1(b)) of the twoinvolved masses (or tacks). The resulting caustics of twopoint masses entail only cusps and folds.An experimental recreation of the no-shear system(vertical view) began by adjusting the two tacks to yieldapproximately the characteristic intermediate-distancecaustic (type (a) experiment, see Fig. 3). The result-ing images for the corresponding type (b) experimentsare shown in Fig. 4, yielding from 3 up to 5 separate im-ages of the single point source when including the centralweak image (which has a high probability to be missingin astronomical observations). The image configurationsattainable may be compared to astronomical examplessuch as CLASS B1608+656 or SL2SJ1405+5502. It must be noted that the ”close” configuration could not be reproduced at all, see Fig. 3. Also, on closeinspection it was found that for the two extended discsthe 6-cusped intermediate caustic expected for pointsources was easily perturbed by small misalignments: thetwo cusps on the symmetry axis then evolved into tiny butterfly caustics . This caustic structure resembled theone of two point masses with added external shear, asshown in loc. Fig. 11(a) of [ 32] (rather than e.g. loc.Fig. 2 ( X = 0 .
5) of [16]). Indeed, theory tells that withadded external shear (effectively then corresponding toat least three lensing masses), or again by perturbing thespherical symmetry by an extended mass distribution (orelliptical ones, i.e. considering an inclined view throughthe setup), higher-order caustics can appear: the swal-lowtail and the butterfly caustics, with a correspondinglyincreased (odd) number of images .It should be a worthwhile advanced laboratory courseexercise to try and access the higher multiplicitiesthrough highly inclined view experiments of type (b) cor-responding to the caustics shown in the third column ofFig. 3 (symmetry axis connecting the two tacks in theplane of inclined incidence). (a) (b) (c) (d) * FIG. 3. Caustic metamorphoses (top to bottom) below twotacks for decreasing distances s ∼ (2 . −
1) cm (cf. Fig. 1, s = 2 R = 1 cm ≡ contact) and four different illuminationscenarios: (a) Light source directly above, (b) inclined (Γ ∼ ◦ ) incidence perpendicular, (c) parallel and (d) diagonal tothe symmetry axis. A configuration similar to the one markedwith an asterisk (*) was chosen for the images in Fig. 4. C. Three mass gravitational lenses ( n = 3 ) For three point masses, the phenomenology becomeseven richer . Already, there are five parameters re-quired to fully describe the situation: two mass ratios(menisci height ratios) along with three relative posi-tions defining the two-dimensional configuration of the
FIG. 4. Images for a double (binary) lens using the setup oftype (b). The image locations qualitatively agree with thoseshown in loc. Fig. 6 in [16] for a gravitational two-point-masslens for an intermediate distance case (corresponding causticin loc. Fig. 2, likewise observed here (not shown)). masses (tacks). Many gravitational triple lens scenarioshave been observed, although almost all have been veryasymmetrical in nature: either binary stars with a sin-gle planet or single stars with two planets . Thesesituations can be viewed as perturbed single mass or bi-nary systems. To the best of the author’s knowledge,the most prominent triple mass system where all threemasses contributed roughly equally to the lensing hasbeen identified in 2001: CLASS B1359+154 is a groupof three compact galaxies lensing a radio source (and itshost galaxy), resulting in 6 images (likely a scenario in-volving extended lens components yielding 9 images, 6 ofwhich are observable) .An experimental recreation using three positionedtacks (roughly attempting to recreate the asymmetric 3-mass configuration of the astronomical counterpart) isshown in Fig. 5. A fair match was found rather quickly,where the closeness was somewhat surprising and is cer-tainly to some extend accidental, given that no optimiza-tion of the menisci heights, their diameters or their de-tailed positions was undertaken and given that the anal-ogy is rather to a system of point masses. The match-ing image was one of several found image configurations,where up to 10 images were observed (in line with theexpectation for a three point mass system).It should be an interesting advanced laboratory courseexercise to try to recreate the zoo of caustics reported forvarious different parameter triple lens systems . IV. ALTERNATIVE: FREE-FORMS OPTICS?
The present model’s strengths can also be appreci-ated from the difficulty one encounters when trying tocreate a freeform multi-component lens mimicking the
B1359+154
Credit: NRAO/AUI/NSF causticsLED images (octuplet) (decuplet)(sextuplet)
FIG. 5. Images for a triple lens using the setup of type (b).The top and center rows show a slightly asymmetric config-uration of the lenses, and the image locations qualitativelyagree with those observed for CLASS B1359+154 . The bot-tom row shows a symmetric configuration in which the largestmultiplicity which could be observed was 10. phenomenology of the triple lens by hand. Such an at-tempt is shown in Fig. 6, which was done starting from a5 cm × × . n -component lensing geometry which isautomatically decaying for large distances and interpolat-ing in-between components to yield all images expectedfrom the astrophysical counterparts. V. CONCLUSION
Using the simple experimental scheme proposed in thisarticle its potential has been demonstrated using thethree cases of single, double and triple mass systems. Ineach case, the phenomenology of the associated causticsas well as the image multiplicity and image configurations defocusedcausticsbulges
FIG. 6. (Failed) attempt of a hand-crafted freeform acrylictriple lens. 7 images are clearly visible (defocused brightspots, black crosses), while indications of 6 more can onlybe guessed (weak and deformed spots, yellow crosses). The 3images due to the central bulges (locations marked by orangediscs) are absent in the tack menisci model (cf. Fig. 5). could be reproduced faithfully for selected configurations.Admittedly, the analogy has not been probed exhaus-tively, i.e. the immense parameter spaces for n = 2 , n point masses. Although the underlying details of theimaging are different, the analogy is close enough to af-ford a good match of the imaging characteristics. Thesetup should thus be a valuable advanced demonstrationpiece, supplementing the simpler single mass (distribu-tion) glass simulators, and maybe also a tool allowingto gain insight into more complex gravitational lensingconfigurations by hinting at an analogy to the remotefield of optical caustics of swimming objects. VI. ACKNOWLEDGEMENTS
I thank K.-H. Lotze for fruitful discussions on glasslens experiments, which prompted the attempted free-form triple lens.
VII. APPENDIX
Since the differential equation for the surface, ∇ f = a − f , is linear, solutions for the individual discs maybe superimposed, just like in the gravitational lensingcase. In cylindrical coordinates, and for a single disconly, the differential equation has the solution f ( r ) = hK ( r/a ) /K ( R/a ), with K being the modified Besselfunction of the second kind of zeroth order, and R beingthe radius of the disc.From Fig. 7 the the bending angle vector can be seento be (cid:126)α = ( n − ∇ f (where ∇ f ∝ − ˆ ρ for raised meniscipoints towards the optical axis), where in the gentle slopeapproximation ∇ = ˆ ρ∂/∂ρ + . . . in cylindrical coordinatesmay be taken as the gradient operator ∇ = ˆx ∂/∂x + ˆy ∂/∂y acting in the xy -plane.For the parameters used in the experiments( h ∼ r E /D l + r E /D sl = α ( r E ) (3)to find θ E ∼ . ◦ (angular diameter of ∼ . ◦ ). This an-gular diameter is indeed small compared to the (lateral)angle of view of 22 . ◦ of the used macro camera lens andcorresponds nicely to the observed ca. 7 . f ( r, φ ) = h exp( − ( r − R ) /a ) for asingle lens, yields the following analytical expression forthe radius r E of the Einstein ring: r E ≈ a · W (cid:18) ( n − h exp( R/a ) a D l D sl D s (cid:19) (4)where D s = D l + D sl was used (an identity which isnot true on cosmological scales for gravitational lens-ing) and W ( x ) is the Lambert W-Function (ProductLog-function), i.e. the inverse of x exp( x ). The correspondingEinstein angular radius is θ E = r E /D l .Again using the parameters of the experiments, theresult is θ E ∼ . ◦ (an angular diameter of ∼ . ◦ ). Expression (4) is similar to the gravitational Einsteinradius expression, which for α = 4 GM/rc solves to r E =(4 GM/c ) × D l D sl /D s , and θ E = r E /D l . VIII. THE CHASHIRE CAT CHALLENGE
The ”Cheshire Cat” system should be a worthwhile ex-ample of a beautiful and well-known wide binary lens ,resembling the face of a smiling cat. It is a complexsystem consisting of multiple arcs on two different Ein-stein radii, foreground and lensing galaxies. Luckily, thehard work of figuring out the system’s likely configura-tion has already been done using spectroscopy and gravi-tational lens modeling : At least 7 images can be clearly FIG. 7. Calculation of the deflection angle α = | (cid:126)α | (blue) byconsidering the inverse situation (light from below): Snell’slaw for small angles (red) reads nθ r = θ i , also θ r = − f (cid:48) (green) such that α = θ i − θ r ≈ − f (cid:48) ( n − r E then follows (see right part): r E /D l + r E /D sl = α ( r E ), and the angular diameter is θ E ≈ r E /D l . identified and belong to two sources in different planes ( D s, (cid:54) = D s, ) (take two LEDs at different distances),whereas the two lensing galaxies lie roughly in the sameplane at D l (use two tacks). A third foreground galaxynot partaking in lensing forms the nose of the cat (adda third LED between the observer and the lenses). Bytrial and error, given the information derived by the as-tronomers, a recreation of the Cheshire Cat should bepossible for an ambitious experimentalist or within anadvanced lab course. The cat’s eyes could be superposedimages of the tacks (or use phosphorescent paint on thetwo tacks). ∗ [email protected]; http://photonicsdesign.jimdo.com T. Treu, P.J. Marshall, D. Clowe, “Resource Letter GL-1: Gravitational Lensing,” Am. J. Phys. (9), 753–763(2012). S. Liebes, “Gravitational Lens Simulator,” Am. J. Phys. (1), 103–104 (1969). V. Icke, “Construction of a gravitational lens,” Am. J.Phys. (10), 883–886 (1980). J. Higbie, “Gravitational lens,” Am. J. Phys. (7), 652–655 (1981). J. Higbie, “Galactic lens,” Am. J. Phys. (9), 860–861(1983). J. Surdej, S. Refsdal, and A. Pospieszalska-Surdej, “TheOptical Gravitational Lens Experiment,” in GravitationalLenses in the Universe, Proceedings of the 31st Liege Inter-national Astrophysical Colloquium (LIAC 93), held June R.J. Adler, W.C. Barber, and M.E. Radar, “Gravitationallenses and plastic simulators,”Am. J. Phys. (6), 536–541(1995). M.J. Nandor, T.M. Helliwell , “Fermats principle and mul-tiple imaging by gravitational lenses,”Am. J. Phys. (1),45–49 (1996). M. Falbo-Kenkel, J. Lohre, “Simple gravitational lensdemonstrations,” Phys. Teacher (9), 555–557 (1996). P. Huwe, S. Field, “Modern Gravitational Lens Cosmologyfor Introductory Physics and Astronomy Students,” Phys.Teacher (5), 266–270 (2015). Brown University, Physics Labs, Lecture Demonstrations,Astronomy, 8C20.40 Gravitational Lens, Retrieval date:11/10/2019, https://wiki.brown.edu/confluence/display/PhysicsLabs/8C20.40+Gravitational+Lens American Museum of Natural History, Cullman Hall ofthe Universe, Universality of Physical Laws, Gravita-tional Lenses, Retrieval date: 11/10/2019, E. Giannini, J.I. Lunine, “Microlensing detection of extra-solar planets,” Rep. Prog. Phys. , 056901 (2013). B.S. Gaudi, “Microlensing Surveys for Exoplanets,” Ann.Rev. Astron. Astrophys. (1), 411–453 (2012) G.O. de Xivry, P. Marshall, “An atlas of predicted exoticgravitational lenses,” Mon. Not. R. Astron. Soc. (1),2–20 (2009). P. Schneider, A. Weiß, “The two-point-mass lens: detailedinvestigation of a special asymmetric gravitational lens,”Astron. Astrophys. (2), 237–259 (1986). M. Selmke, “Bubble optics,” Appl. Optics, accepted(2019). K.C. Freeman, “On the disks of Spiral and S0 galaxies,”ApJ , 811–830 (1970). MOA collaboration homepage (Microlensing Observa-tions in Astrophysics), ”Demonstrations”, Retrievaldate: 10/16/2019, T. Treu, “Strong Lensing by Galaxies,” Annu. Rev. As-tron. Astrophys. (1), 87–125 (2010). D. Vella and L. Mahadevana, “The ”Cheerios effect””, Am.J. Phys. (9), 817–825 (2005). C.L. Adler, J.A. Lock, “Caustics due to complex watermenisci,” Appl. Optics (4), B207–B221 (2015). A. O. Petters, H. Levine, J. Wambsganss,
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