Dimensionality constraints of light induced rotation
László Oroszi, András Búzás, Péter Galajda, Lóránd Kelemen, Anna Mathesz, Tamás Vicsek, Gaszton Vizsnyiczai, Pál Ormos
DDimensionality constraints of light-induced rotation
L´aszl´o Oroszi , Andr´as B´uz´as , P´eter Galajda , L´or´and Kelemen ,Anna Mathesz , Tam´as Vicsek , Gaszton Vizsnyiczai and P´al Ormos Institute of Biophysics, Biological Research Centre,Hungarian Academy of Sciences,H-6726 Szeged, Hungary. Department of Biological Physics, E¨otv¨os University,Statistical and Biological Physics Research Group of the Hungarian Academy of Sciences,H-1117 Budapest, Hungary. (Dated: October 12, 2018)We have studied the conditions of rotation induced by collimated light carrying no angular mo-mentum. Objects of different shapes and optical properties were examined in the nontrivial casewhere the rotation axis is perpendicular to the direction of light propagation. This geometry offersimportant advantages for application as it fundamentally broadens the possible practical arrange-ments to be realised. We found that collimated light cannot drive permanent rotation of 2D orprism-like 3D objects (i.e. fixed cross-sectional profile along the rotation axis) in the case of fullyreflective or fully transparent materials. Based on both geometrical optics simulations and theo-retical analysis, we derived a general condition for rotation induced by collimated light carrying noangular momentum valid for any arrangement: Permanent rotation is not possible if the scatteringinteraction is two-dimensional and lossless. In contrast, light induced rotation can be sustainedif partial absorption is present or the object has specific true 3D geometry. We designed, simu-lated, fabricated, and experimentally tested a microscopic rotor capable of rotation around an axisperpendicular to the illuminating light.
Light-induced rotation has been an intensively studiedeffect ever since the availability of optical micromanipula-tion, partly because of the possible technical applicationsin micromechanics, at the same time the phenomenon isattractive for pure intellectual reasons, too [1]. Differentrotation mechanisms have been introduced and discussedwhich can be classified into two main categories. In one,light itself carries the angular momentum needed for therotation (e.g. circularly polarized light [2–4] LaguerreGaussian beam [4, 5], etc.). In the other scheme, lightwith no angular momentum is scattered by an object in ahelical manner [6–10] consequently, angular momentumis transferred to the object resulting in rotation (a pro-cess analogous to the rotation of a windmill). Here wediscuss a yet not considered aspect of this latter case.In the above mentioned cases rotation usually tookplace around an axis parallel to the optical axis. Thereis a distinctly different case, where the axis of rotation isperpendicular to the direction of illumination. This ar-rangement may be important for practical purposes: ro-tors could be arranged to roll in arbitrary directions ona plane illuminated perpendicularly by a single sharedlight beam (e.g. particles moving in independent di-rections on a flat surface could be supplied with en-ergy by a single light source similar to the non-rotatingwedge shaped quasy-autonomous robots introduced re-cently [11]). Intuitively, one might think that this type ofrotation should be possible simply by using properly de-signed asymmetric rotors. Indeed, water mill type light-driven rotors have been introduced earlier [12–14], how-ever, in these cases rotation was achieved by partial illu-mination of the wheel (in analogy with a water mill par- tially submerged in water). In contrast, here we discussfully illuminated rotors. According to our results, rotorgeometries capable of sustained rotation are not trivial toconstruct. We conducted a systematic modelling studyand theoretical analysis to determine the conditions forrotation and then experimentally verified our findings.In the modelling we studied objects of homogeneousmaterial, either transparent or reflective (primarily non-absorbing, but absorption was also considered in specificcases). The illumination is a collimated unpolarised orlinearly polarised light beam of homogeneous intensitydistribution, fully covering the objects. In a geometri-cal optics approach, we calculated the forces acting onthe rotors resulting from the linear momentum trans-fer between light and matter. Continuous illuminationis modelled by the introduction of a high density arrayof light rays. Each ray is associated initially with unitlight intensity. During the simulations these intensitiesare modified according to the Fresnel equations. Whena light ray hits the interface of two regions having dif-ferent refractive indices, then the reflected and refractedrays are created dynamically and new coordinates andintensities are calculated based on the law of reflection,Snell-Descartes law and the Fresnel equations in a re-cursive manner. The polarization state of the light is afreely adjustable parameter. The change in the angularmomentum of light (with respect to some point of origin),thus the torque upon the object is a result of elementarylinear momentum transfer processes integrated over allray-object intersection points. We restrict ourselves tothe following case (if not otherwise stated): (a) rotationsare investigated around a fixed axis perpendicular to the a r X i v : . [ phy s i c s . op ti c s ] D ec illumination, (b) the illumination is modelled by a largenumber of homogeneously distributed, parallel, linearlypolarized light rays, (c) mechanical interaction is basedexclusively on transfer of linear momentum of light, (d)the motion is over-damped: the angular velocity is pro-portional to the torque exerted by light (e.g. microscopicparticles in fluid).Two basic types of rotor shapes were investigated.First, we analysed prism like shapes, i.e. 3D objects thatare created by the extrusion of a 2D footprint, and subse-quently, we considered true 3D objects having a varyingstructure along the rotation axis.In the case of prism-like rotors the problem is es- (a)(b) T o r que ( a . u . ) Angle (°) T o r que ( a . u . ) Angle (°)
FIG. 1. Simulation of prism shape rotors in homogeneouscollimated illumination perpendicular to the rotation axis.Acharacteristic rotor shape with reflective (a) and transparent(b) material and the respective torque exerted by light as afunction of the rotation angle. In the images the reflected andrefracted light rays are also presented. Illumination comesfrom the top, the refractive index is 1.2. sentially two-dimensional: light scattering occurs in theplane perpendicular to the rotation axis. We investigateddifferent shapes that one would intuitively regard as can-didates for rotation. Fig. 1 shows a rotor with charac-teristic shape made of reflective and transparent materialinteracting with collimated light. We have tried a largenumber of different shapes and we found that in the loss-less case (no absorption) regardless of the shape of theobject, either refractive or reflective, sustained rotationwould not be driven by momentum transfer: Although incertain orientations light exerted torque upon the object,the integral of the torque for a full rotation cycle was al-ways exactly zero, and in addition the object eventuallyended up in an angular trap.The statement that prism shaped objects can not berotated by collimated light can be proven with the follow-ing quantitative theoretical argumentation. The integralof torque for a full revolution of the object is equal to the torque experienced by the object at rest illuminated bylight coming from all directions in the plane (i.e. isotropicillumination). This equivalent case is easier to anal-yse because the angular dependence has been removed.First, let us consider the case of the reflecting body.Any light beam with a given radiance ( L = d PdA proj · d Ω :power/projected area/steric angle) hitting the object willeventually leave it after a certain number of reflections,propagating away from the object in the plane of illumi-nation. The radiance is known to be invariant for transferin a lossless system: While during interaction with an op-tical surface parameters of the particular light beam maychange, the radiance is conserved, and the outgoing beamwill leave the object with unchanged radiance, indepen-dently of the distortions during particular refractions orreflections on surfaces. (We note that in our ray op-tics simulations this invariance is automatically fulfilled.)Since the illumination is isotropic in the plane, the beamleaving the object will have its counterpart coming to-wards the object with exactly the same radiance (this ischaracteristic to the isotropic radiation field). It followsthat the radiation field after the reflections will remainunchanged, consequently no net momentum change oc-curs, and therefore light will not have a mechanical effectupon the object. Thus we have also proven that for illu-mination from a single direction the integral of the torquefor a full rotation is zero.The case of transparent objects can be understood us-ing a similar argumentation. Let us consider an incidentlight beam of infinitesimally small cross section and radi-ance L . When hitting the surface of the object the beamis split into reflected and refracted beams typically sev-eral times before leaving the object, resulting in multiplelight paths. Traveling along a particular one of thesepaths the radiance of the incident light beam decreasesafter each transmission or reflection as determined by theFresnel equations, giving an output radiance of c i · L . Theoriginal radiance is distributed among all possible lightpaths: (cid:80) c i =1. Since the illumination is isotropic, lightwill also enter each path from the opposite direction withradiance L . Due to the direction invariance of the Fresnelcoefficients the total attenuation of radiance will be thesame in the two directions (i.e. counterpropagating lightwill leave the path with output radiance c i · L as well).Now, if we take into account all light paths into whicha single incident light beam is divided, we can see thatdue to the direction invariance of the attenuation, theradiance of counterpropagating light will add up exactlyto L , the original input radiance. This results again inan unchanged radiation field around the object, and thusno transferred torque.We have shown that in the two-dimensional case col-limated light cannot induce sustained rotation of non-absorbing objects (either reflective or transparent). In-terestingly, on the other hand, if losses are present, ro-tation becomes possible for certain rotor shapes. Wepresent simulations of this case on rotors of the shapeshown in Fig. 1. In order to find the characteristic fea-tures of the phenomenon, we performed the simulationsfor different rotor geometries including varying number ofblades (both odd and even) and calculated the torque ex-erted by light for the whole absorption range, as shown inFig. 2. Assuming negligible inertia (e.g. microscopic ob-jects in fluid) sustained rotation occurs only if the torqueacting on the rotor does not change sign during a com-plete revolution (otherwise the object will be trapped inan angular trap). The condition of sustained rotationis satisfied only in certain absorption ranges specific tothe actual rotor geometry. Note e.g. that in Fig. 2. inthe case of the 5 blade rotor there is no absorptance re-gion where the torque does not change sign, consequently,in this particular case continuous rotation does not takeplace at all. In other words, sustained rotation is possi-ble only for specific combinations of absorption and rotorshape. Unfortunately, it is not possible to test this phe-nomenon experimentally, since in the case of significantabsorption the sample would heat up to a level prevent-ing the experiment.The observation that permanent rotation is feasi- T o r que ( a . u . ) Absorptance n=4 T o r que ( a . u . ) Absorptance n=5 T o r que ( a . u . ) Absorptance n=6 T o r que ( a . u . ) Absorptance n=7
FIG. 2. Torque exerted by light on rotors of the shape shownin Fig. 1. with partially reflecting and partially absorbingsurfaces. n is the order of the rotational symmetry of theobject, i.e. the number of blades of the rotor. Average, mini-mum and maximum of the torque (evaluated over one revolu-tion) is plotted as a function of absorptance. Steady rotationis possible only in certain regions of the absorptance wherethe torque acting on the rotor does not change sign during acomplete revolution (marked by blue), specific for the rotorgeometry. An arrow is positioned at the absorptance valuewhere the average torque has its maximum. ble if light is extracted (due to losses) from the two-dimensional plane where scattering occurs, suggests anadditional possibility for rotation: deflection of light intothe third dimension. To demonstrate this latter concept, (a)(b) T o r que ( a . u . ) Torsion (°)continuous rotationtrapping selected i n c i den t li gh t r o t a t i o n a x i s t o r s i o n FIG. 3. Three-dimensional reflective rotor capable of sus-tained rotation around an axis perpendicular to a collimatedlight beam. (a) Geometry of the rotor and illustration of theworking principle. Torque results from the asymmetric mo-mentum change of light on opposite sides of the rotation axis.The rotor has a simple geometry with one parameter, thetorsion angle, and provides a smooth profile to enable rollingon a planar surface. (b) The average and the extreme val-ues of the torque exerted by light as a function of the torsionangle. Sustained rotation is possible when the torque doesnot change sign during rotation (assuming negligible inertia,i.e. low Reynolds number regime). ”Selected” indicates thetorsion angle chosen for the realization of the rotor. we designed a true three-dimensional object where alsothe scattering process is three-dimensional. We chose astructure with four-fold symmetry, generated by the tor-sional deformation of a cross-based prism, twisted sym-metrically in opposite directions along the two halves ofthe rotation axis (see Fig.3.a.). The rotor fits into a cylin-der, having identical height and diameter. With theseconstraints a single torsion angle parameter fully deter-mines the geometry and consequently the torque exertedby light reflected form the surfaces. Fig.3.b shows theresults of the simulations, the capability of permanentrotation (i.e. minimum, maximum and average torqueduring a complete revolution) as a function of the tor-sion angle. The rotor can exhibit sustained rotation in afairly wide range of torsion, thus making it more robustto potential fabrication errors. To experimentally con-firm the results of the simulations we built a rotor with (b)(a)
FIG. 4. The constructed rotor. (a) Electron microscopic im-age of the photopolymerized structure, the scale bar corre-sponds to 10 µ m. (b) Optical image of the microrotors lyingon the microscope slide. the essential features of the simulated object.The rotors were produced by the two-photon pho-topolymerization technique [8, 15, 16] from SU-8 pho-toresist (Michrochem, USA). After photopolymerizationthe surface of the rotors was made reflective at the wave-length of 1070 nm by covering them with a 200 nm thickAu metal mirror laye, on top of a 5 nm Cr layer for stabil-ity, both deposited on the surface by sputtering (EmitechK975X, UK). Fig. 4 shows the fabricated microstructure.In the experiments we realized rotation as rolling therotor on a horizontal surface (microscope slide). Thedifferences between the concept in Fig. 3. and the real-ization in Fig. 4. are due to practical necessities: Tworings were added at the edge of the structure to improvemechanical rigidity as well as to facilitate smooth rolling.Extruding tips were added to the rotation axes to ensureproper orientation of the rotors after sedimentation.The experiments were carried out in an inverted mi-croscope (Zeiss Axio Observer A1). The sample com-partment was formed by a microscope slide and coverslip separated by a 200 µ m spacer, and filled with watercontaining the rotors. After sedimentation, the rotorswere rolling on the horizontal glass surface when illu-minated vertically by an infrared fiber laser (IPG YLS-1070, I=10 W, λ =1070 nm). In addition to the illumi-nation from the top we also applied illumination fromthe bottom with slightly less intensity: this scheme pre-vented the light pressing the rotor against the surfacethat could impede easy rolling of the rotors. To achievethis the linearly polarised light from the laser was passedthrough a λ /2 plate and then split into two beams by apolarising cube. One beam was directed to the samplecompartment from below, one from above. The inten-sities of the two beams could be finely tuned by rotat-ing the λ intensity corresponds to a velocity of 1.56 µ m/s.Regular rolling of the rotors on the surface is demon-strated by the distribution of relative angles betweentravelling direction and rotor symmetry plane (Fig. 4b, also see video supplement).We experimentally demonstrated that true 3D objects (a)(b)
250 300 350 400 450 500 5500246810121416 S peed ( � m / s ) Light intensity (W/cm ) -180 -135 -90 -45 0 45 90 135 1800.00.20.40.60.81.0 N o r m a li z ed f r equen cy Orientation mismatch (degrees)
FIG. 5. Characteristics of microrotor motion. (a) Rotor speedas a function of the illumination intensity. The velocity val-ues and the corresponding errors are determined by imageanalysis based on video recordings. The solid line representsa linear fit to the data points resulting in a slope of 0.0156( µ m/s)/(W/cm ). (b) Histogram of the relative angle be-tween the direction of rotor velocity and the symmetry planeof the rotor (based on the data set used in a). can be rotated by light in the discussed geometry. Thecrucial condition is that light is extracted from the scat-tering plane normal to the rotation axis by deflection oflight into the third dimension (where the rotation axispoints). To stress the peculiar nature of this character-istics, consider the cup anemometer that is rotated bywind. If it is reflective, it will also be rotated by light.If we take a two-dimensional version (projection) of it,simulations show that it will not rotate, in accordancewith our findings concerning the dimensionality of thelight scattering process.Finally we note that in all of the other arrangementsfor rotation discussed in previous works [1–10], where therotation axis is parallel to (or has a component in) thedirection of light propagation (propeller, windmill, etc.),the scattering process is three-dimensional. Taking allobservations into account, we reach a general conclusion:Collimated light carrying no angular momentum can-not drive permanent rotation of any object (reflective ortransparent) if the scattering process is two-dimensionaland lossless.This work was supported by the grants OTKANN102624, NN114692, COST MP1205 and TAMOP-4.2.2.A-11/1/KONV-2012-0060 (to LO, AB, LK, AM,GV, PO) and EU ERC COLLMOT-227878 (to TV). [1] M. Padgett and R. Bowman, Nature Photonics , 343-348 (2011)[2] M.E.J. Friese, T.A. Nieminen, N.R. Heckenberg and H.Rubinsztein-Dunlop, Nature , 348 (1998)[3] A.I. Bishop, T.A. Nieminen, N.R. Heckenberg and H.Rubinsztein-Dunlop, Physical Review A , 033802(2003)[4] N.B. Simpson, K. Dholakia, L. Allen and M.J. Padgett,Optics Letters , 52-54, (1997)[5] L. Paterson, M.P. MacDonald, J. Arlt, W. Sibbett, P.E. Bryant and K. Dholakia, Science , 912-914 (2001)[6] A. Yamamoto and I. Yamaguchi, Jpn. J. Appl. Phys. ,3104-3108 (1995)[7] R.C. Gauthier, Appl. Phys. Lett. , 2269-2271 (1995)[8] P. Galajda and P. Ormos, Appl. Phys. Lett. , 249-251(2001)[9] T. Asavei, V.L.Y. Loke, M. Barbieri, T.A. Nieminen,N.R. Heckenberg and H. Rubinsztein-Dunlop, New Jour-nal of Physics , (2009) 093021[10] R. Di Leonardo, A. B´uz´as, L. Kelemen, G. Vizsnyiczai,L. Oroszi and P. Ormos, Phys. Rev. Lett. , 034104(2012)[11] A. B´uz´as, L. Kelemen, A. Mathesz, L. Oroszi, G. Vizs-nyiczai, T. Vicsek and P. Ormos, Appl. Phys. Lett. ,041111 (2012)[12] E. Higurashi, R. Sawada and T. Ito, Appl. Phys. Lett. , 2951 (1995)[13] R.C. Gauthier, R.N. Tait, H. Mende and C. Pawlowicz,Applied Optics , 930-937 (2001)[14] L. Kelemen, S. Valkai and P. Ormos, Applied Optics ,2777-2780 (2006)[15] S. Mauro, O. Nakamura and S. Kawata, Optics Lett.15