Radiation Torque Exerted on a Spheroid: Analytical Solution

Transcription

1 Cleveland State University Physics Faculty Publications Physics Deartment Radiation Torque Exerted on a Sheroid: Analytical Solution Feng Xu James A. Lock Cleveland State University, j.lock@csuohio.edu Gérard Gouesbet Cameron Troea Follow this and additional works at: htts://engagedscholarshi.csuohio.edu/scihysics_facub Part of the Physics Commons How does access to this work benefit you? Let us know! Publisher's Statement Coyright 8 American Physical Society. Available on ublisher's site at htt://ra.as.org/ abstract/pra/v78/i1/e Original Citation Xu, Feng, James A. Lock, Gerard Gouesbet, and Cameron Troea. "Radiation Torque Exerted on a Sheroid: Analytical Solution." Physical Review A 78 (8): Reository Citation Xu, Feng; Lock, James A.; Gouesbet, Gérard; and Troea, Cameron, "Radiation Torque Exerted on a Sheroid: Analytical Solution" (8). Physics Faculty Publications. 51. htts://engagedscholarshi.csuohio.edu/scihysics_facub/51 This Article is brought to you for free and oen access by the Physics Deartment at EngagedScholarshi@CSU. It has been acceted for inclusion in Physics Faculty Publications by an authorized administrator of EngagedScholarshi@CSU. For more information, lease contact library.es@csuohio.edu.

2 PHYSICAL REVIEW A 78, Radiation torque exerted on a sheroid: Analytical solution Feng Xu* Fachgebiet Strömungslehre und Aerodynamik, Technische Universität Darmstadt, Petersenstrass 3, 6487 Darmstadt, Germany and Institute of Particle and Two-Phase Flow Measurement, University of Shanghai for Science and Technology, Shanghai 93, China James A. Lock Deartment of Physics, Cleveland State University, Cleveland, Ohio 44115, USA Gérard Gouesbet LESP, UMR 6614/CORIA CNRS, Université et INSA de Rouen, Site du Madrillet, Avenue de l Université, BP Saint Etienne du Rouvray, France Cameron Troea Fachgebiet Strömungslehre und Aerodynamik, Technische Universität Darmstadt, Petersenstrass 3, 6487 Darmstadt, Germany Received 1 March 8; ublished 9 July 8 As a comanion work to our revious study F. Xu, et al., Phys. Rev. E. 75, on radiation force rediction for a homogeneous sheroid, we rovide in this aer the analytical solution to the radiation torque exerted by an arbitrarily shaed beam on a sheroid, which can be rolate or oblate, transarent or absorbing. Calculations based uon this theoretical develoment are erformed for both linearly and circularly olarized incident beams, and the results are comared to those of a shere. Stable orientations of sheroids inside a linearly and a circularly olarized Gaussian beam are redicted. We analyze two hysical mechanisms, the olarization torque and the reaction force torque, which do not exist or have no contribution to the torque on a shere but cause rotation of a sheroid. As verification, the diole method is also develoed for the torque calculation for sheroids of size much less than the wavelength, and geometrical otics is develoed to qualitatively analyze the torque exerted on sheroids of large size. DOI: 1.113/PhysRevA PACS numbers: 4.5.Fx, 87.8.Cc I. INTRODUCTION *f3_xu@yahoo.com Transfer of momentum and angular momentum from a laser beam to a article are known to roduce radiation forces and torques, which can be emloyed to maniulate and characterize microscoic objects 1 8. For examle, viruses, bacteria, organelles, and even DNA strands can be traed by a highly focused laser beam inducing forces in iconewton range 6. The orientation of nonsherical articles can be controlled through the torque exerted by laser beams carrying sin angular momentum 7. By using two counterroagating laser beams stretching forces can be induced to measure the mechanical rigidity of a cell 8. Scientific design and otimization of otical instruments imlementing these measurements requires the relevant develoment of scattering theory describing the interaction between laser beams and articles. Though rigorous electromagnetic scattering theory with analytical solution to the radiation force and torque exerted on a homogeneous and radially inhomogeneous shere have been well established 9 15, much less work has been reorted for nonsherical articles, excet for the case of a cylinder with circular cross section 16. To deal with the comlex article shaes to which analytical solution is hard to obtain or does not exist, the numerical aroaches, e.g., the surface-based T-matrix method 17,18 and volume-based finite-difference timedomain method 19 have been develoed. To avoid the mathematical difficulties in establishing the rigorous theory, some aroximate aroaches, e.g., the geometrical otics aroximation 3 and Rayleigh theory 4,5 become alternatives when extreme recision is not of rimary imortance. For examle, to evaluate the force and torque exerted on many regularly shaed articles of large size, e.g., sheroid, cylinder, ring-shaed article, etc., geometrical otics can be emloyed 3. In the other limit, when the article size is considerably smaller than the wavelength, Rayleigh theory rovides an efficient choice 4,5. For some regularly shaed articles, e.g., sheres, cylinders, sheroids, etc., these two limiting cases can be unified by the rigorous theory called generalized Lorenz-Mie theory GLMT 6. Within the framework of GLMT, once the beam is exanded in relevant coordinates mirroring the geometry of the article, analytical solutions to all hysical quantities, e.g., scattered intensities, scattering and extinction cross sections, and radiation forces and torques are available. Obviously, the extension of GLMT from shere to some nonsherical articles is significant since most articles existing in nature or generated in industrial rocesses are not exactly sherical. Even for some soft sherical articles, e.g., drolets and biological cells, the larger they are, the easier it is to deform them, due to the decreasing efficiency of surface tension to maintain their shericity. Actually, when these articles are only slightly deformed from sherical shae they might be well modeled by a sheroid, e.g., otically or acoustically levitated drolets, atomized drolets, etc. For their characterization by otical means, a rigorous theory /8/781/ The American Physical Society

3 XU et al. with analytical solution describing their scattering characteristics is of great imortance. In the ast decade, GLMT for a sheroid has been develoed stewise. Analytical solutions to scattering amlitudes, extinction, and scattering cross sections, as well as radiation forces, were successively found 7,8. The resent contribution adds the torque to this list. Before introducing our work, we briefly look back at the hysical mechanisms of the torque exerted on a shere by a laser beam. First, we consider the simlest system of a shere illuminated by a lane wave. To generate torque on the article, Marston and Crichton found the lane wave must be circularly olarized and the shere must be absorbing 9. A circularly olarized lane wave carries sin angular momentum 9. When it is incident on an absorbing article, art of circular olarization angular momentum is lost by the incident beam and transorted to the shere. We call this the sin-absortion mechanism, which also occurs for the case of a shere illuminated by an on-axis shaed beam. Next, we consider the system of a shere illuminated by an off-axis shaed beam. As found by Chang and Lee 1 and Barton 14, a torque can be induced by a linearly olarized shaed beam. We call this the orbit-absortion mechanism since an off-axis beam has orbital angular momentum with resect to the shere s axis and this orbital angular momentum is transferred to the rotation of the shere through absortion. The third circumstance is that the shere is illuminated by an off-axis circularly olarized beam. The torque induced in this case was also redicted by Chang and Lee 1 and Barton 14. It can be attributed to both the sinabsortion and orbit-absortion mechanisms. All these mechanisms exlaining the torque exerted on a shere also occur for a sheroid. The majority of the results resented in this aer will be for the geometries where the torque on a shere is zero. Two tyes of torques which do not occur for sheres, i.e., the olarization torque and reaction force torque, are identified and analyzed. II. THEORY According to the conservation law of angular momentum, the radiation torque exerted on a article of arbitrary shae is equal to the average rate at which the angular momentum is conveyed to it. Mathematically, it can be exressed as a surface integral of the dot roduct of the outward normal unit vector nˆ and the seudotensor A J r over a surface enclosing the article designated by S 3,31, T = S nˆ A J rds, 1 where the symbol denotes time average and A J is the Maxwell stress tensor exressed by AJ = EE * + HH* 1 E + H I J, where E and H are the total electric and magnetic fields outside the article, I J is the unit tensor, and denote the u O B v ermittivity and ermeability of the medium, resectively, and the asterisk indicates the comlex conjugate. A. Descrition of the electromagnetic fields We consider a monochromatic arbitrarily oriented shaed beam whose electric field is linearly olarized in the u direction at the waist in its own Cartesian coordinates O B -uvw. The time-deendent art of its electromagnetic fields is ex it, with being the angular frequency. Such a beam is incident on a sheroid of semimajor axis length a and semiminor axis length b. As indicated by Fig. 1, through coordinate translation the beam center, O B can be moved to the article center O P to generate the system O P -uvw. Taking the axis of symmetry of the sheroid as the z axis, we set the x axis of the Cartesian coordinates of the article to lie in the lane formed by the w and z axes. This determines the article coordinate system O P -xyz. Then the two coordinate systems can be related by two beam angles. One is the incident angle bd indicating the beam roagation direction in O P -xyz, and the other is the olarization angle bd indicating the olarization direction of the beam relative to the O P -xz lane. Note that the signs of bd and bd are the same as those of cross roducts of unit vectors, e z e w and e up e u resectively, with u P denoting the rojection of u axis in the O P -xz lane. These two angles, in conjunction with the coordinates of beam center x,y,z in O P -xyz are used to evaluate the sheroidal beam shae coefficients or m m more briefly, sheroidal BSCs, G n,te and G n,tm, for an arbitrarily shaed beam 3. When the sheroidal BSCs are obtained, the incident fields E i,h i can be exanded in terms of sheroidal vector functions M mn,n mn in the sheroidal coordinates,,, E i = m= n=m,n H i = ik I I m + G n,tm N i mn C I ;,,, PHYSICAL REVIEW A 78, w y uí m= n=m,n Ф bd i n+1 ig m n,te M i mn C I ;,, i n+1 m G n,tm M i mn C I ;,, + ig m n,te N i mn C I ;,,, 4 where k I is the wave number of the surrounding medium ví x O P FIG. 1. Coordinate systems: O B -uvw is attached to the incident shaed beam and O P -xyz is attached to the sheroid. wí z

4 RADIATION TORQUE EXERTED ON A SPHEROID: denoted by the subscrit I and assumed to be linear, isotroic, and nonmagnetic I =1, and C is the size arameter calculated by multilying the wave number k I by the semifocal length of the sheroid f, namely C I =k I f =k I aa/b 1 1/. Likewise, the scattered fields E s,h s can be exressed in terms of vector wave functions in the following way: E s = m= n=m,n H s = ik I I + A m n N s mn C I ;,,, m= n=m,n + ib m n N s mn C I ;,,, i n+1 ib m n M s mn C I ;,, i n+1 A m n M s mn C I ;,, where the scattering coefficients, A n m and B n m, are determined via solving the equations established from the boundary conditions ensuring the continuity of tangential comonents of electric and magnetic vectors across the surface of a sheroid 7. A suerosition of the incident and scattered field gives the total external field for the torque calculation. In this aer we only resent the formulas for a rolate sheroid since all the equations and exressions ertaining to the rolate sheroidal coordinates can be converted to their counterarts in the oblate system through relacing the size arameter C I by ic I and the sheroidal radial coordinate by i. B. Analytical solution to the radiation torque To erform the integral indicated by Eq. 1, an arbitrary Gaussian surface surrounding the sheroid can be used. For the sake of mathematical convenience, the surface characterized by an infinite sheroidal radial coordinate is chosen. In this limit this surface is essentially sherical so that we have f r and cos. Then Eq. 1 becomes T = 1 Re I E r E * + I H r H * e I E r E * + I H r H * e r 3 sin dd, where E=E i +E s and H=H i +H s. Utilizing the rojections of the unit vectors e and e, we can decomose the torque into the following three comonents about the x, y, and z axes, resectively, T x = 1 Re I E r s E s* + E s r E i* + E i r E s* + E i r E i* + I H s r H s* + H s r H i* + H i r H s* + H i r H i* cos cos + I E r s E s* + E r i E s* + E r s E i* + E r i E i* + I H r s H s * H r i H s* + H r s H i* + H r i H i* sin r 3 sin dd, 8 T y = 1 Re I E r s E s* + E s r E i* + E i r E s* + E i r E i* + I H s r H s* + H s r H i* + H i r H s* + H i r H i* cos sin I E r s E s* + E r i E s* + E r s E i* + E r i E i* + I H r s H s * + H r i H s* + H r s H i* + H r i H i* cos r 3 sin dd, 9 T z = 1 Re I E r s E s* + I H r s H s* + I E r i E i * + I H r i H i* + I E r s E i* + I E r i E s* + I H r s H i * + I H r i H s* sin r 3 sin dd. 1 All the comonents of electric and magnetic fields contain the sheroidal radial function R mn and the angular function S mn as well as their derivatives 33. Invoking the orthogonality relationshi of the angular wave functions, using the 1 following asymtotic behavior of the radial functions R mn and R 3 mn concerning the incident and scattered fields, resectively: R 1 mn 1 in+1e ikr i n+1eikr kr +, kr R 3 mn i n+1eikr kr, 11 1 and neglecting the terms of order higher than 1/r, the analytical solution to the radiation torque can be found in series form after a great deal of algebra, + T x + it y = M 3 c k I T z = M 3 c k I + + = n= n= 1 A * 1 n G n,tm + A n 1 G * n,tm n J nn,1 + B * 1 n G n,te + B 1 n G * n,te +A * n A 1 n +B * n B 1 n, ReA n = n= n= G * n,tm PHYSICAL REVIEW A 78, n H nn,1 C I J nn, 13 C I H nn, + B n G * n,te + A * n A n + B * n B n. 14 In Eqs. 13 and 14, n is the searation constant of the differential equations for sheroidal radial and angular functions in sheroidal coordinates 33, M is the refractive index of the medium, c is the seed of light in vacuum, the minus lus sign corresonds to the rolate-oblate sheroid, and J nn,1, J nn,, H nn,1, and H nn, are the orthogonality relationshi of the angular functions and are given by

5 XU et al. PHYSICAL REVIEW A 78, J nn,1 S n 1S 1n cos ds 1n sin d =, = d n n = odd, = 1 + r +1! d n +r +1 r 1! r d 1n r 1, n n = even,, r=,1 = 1 + r! d n +r +1 r! r d 1n r+1, n n = even,, 15 r=,1 J nn, = cos J nn,1 = = + r r=,1 +r +1 S n 1S 1n cos ds 1n d r + r +1 + r 1! +r 1 r 1! sin cos d =, d n r d 1n r 1 + r r 1 + r + r 1 + r 1! d n r=4,3 +r 1 +r 3 +r +1 r 1! r d 1n r 3 + r +1r + + r + + r +3 r=,1 +r +1 +r +3 +r +5 r +1 + r=,1 +r +1 r + r +1 + r +1! +r +3 r +1! + r 1 + r 1! = r=,1 +r +1 +r 1 r 1! + r +1! r +1! d n r d 1n r+1 n n = odd, d,n r d r 1 1,n, n n = even,, d n r d 1n r rr 1 + r 1! d n r=,1 +r 3 +r 1 +r +1 r 1! r d 1n r r +r +1 r=,1 +r +1 +r +3 +r +5 r r +1! + +r +1 +r +3 r +1! r=,1 + r +1! r +1! d n r d 1n r+3 d n r d r+1 1n, n n = even,, 16 and H nn,1 = S n S n sin d =, n n = odd, 1 + r! d n r=,1 +r +1 r! r d n r, n n = even, 17 H nn, = cos H nn,1 = S n S n sin cos d =, = 1 + r r=,1 +r 1 +r +1 r=,3 n n = odd, + r 1! d n r 1! r d n r + r 1 + r + r 1! d n +r 3 +r 1 +r +1 r 1! r d n r + r +1 + r + + r +1! d n r=,1 +r +1 +r +3 +r +5 r +1! r d n r+ + 1 r +1 +r +3 +r +1 + r +1! d n r +1! r d r n, n n = even, 18 r=,

6 RADIATION TORQUE EXERTED ON A SPHEROID: PHYSICAL REVIEW A 78, where d n r s are the exansion coefficients of angular functions S mn in sheroidal coordinates 33, and the suerscrit rime sign in the sum symbol indicates a series summation starting from 1 over the even odd indices of r when n m is even odd. Equations 13 and 14 are also valid for a circularly or an ellitically olarized incident beam when the BSCs are reevaluated. For examle, to calculate the torque exerted by a circularly olarized beam, the relevant coefficients G,circular n,te and G,circular n,tm should be used. They can be obtained from the following suerosition of BSCs evaluated for linearly olarized beams: T z 4 x1 17 Edge of shere Shere Prolate a/b=1.5 Oblate a/b=1.5 Prolate a/b=1. Oblate a/b=1. G,circular n,te = 1 G n,te ig n,tm, 19 4 Edge of shere G,circular n,tm = 1 G n,tm ig n,te, where the suerscrit + of the BSCs denotes the two linear olarizations shifted in hase by / so that right left circular olarization is roduced. Note that the torque comonents exressed by Eqs. 13 and 14 are normalized in such a way that the beam intensity at the focal oint I, which can be related to the beam ower P and the beam shae confinement arameter s by I =P/ 1+s +1.5s 4 for a circular Gaussian beam 34, is omitted. It is needed for torque evaluation when a realistic beam is used. III. NUMERICAL CALCULATIONS 1 1 y (µm) FIG.. Torque about the z axis T z versus the location of the beam center along y axis y. A linearly olarized Gaussian beam of wavelength =.785 m and waist radius =1. m is incident on a olystyrene shere of radius r s =1. m, two rolate sheroids of axis ratios a/b=1.5 and 1., and two oblate ones of the same ratios. These articles have the identical volume V =4.19 m 3 and refractive index M= i. The incidence and olarization angles of the beam are 9 and, resectively bd =9 and bd =. Numerical rocedures have been develoed to imlement the theory resented in the receding section. As a check of the analytical solution, results have been comared to those obtained from direct numerical quadratures of Eqs. 8 1, and a comlete agreement was found. After this verification, numerical calculations were made for both linearly and circular olarized Gaussian beams. We assumed the beam is roduced by a cw Ti:sahire laser =.785 m and focused to a waist radius of =1. m. It is incident on a hard olystyrene article of refractive index M= i 35 surrounded by air M =1., or a single red blood cell RBC of refractive index M= i 36 embedded in a buffer solution with osmolarity adjusted to 13 milliosmoles so that its refractive index is M = All the numerical results resented in this section are for the beam ower P=1 W. The most imortant ste in GLMT calculations concerns the beam descrition and beam shae coefficients evaluation. For a loosely focused Gaussian beam, Davis first-order corrected exressions 37 are a highly accurate aroximation to a solution to Maxwell s equations, thereby sufficient for the beam descrition. In this case, the localized beam model 38,39 can rovide an accurate and efficient in BSC evaluation 4,41. When the beam is focused to the same order as the wavelength, our case, the first-order descrition becomes less accurate. In this case, Barton and Alexander s fifth-order beam descrition was used instead in our calculation 34. And we find the quadrature method 4 is more ractical to evaluate the BSCs for such a tightly focused beam. A. Linearly olarized beam: Off-axis incidence First, we exlore the influence the article s location inside the beam has on the torque it receives. The articles considered here are two rolate olystyrene sheroids of axis ratios a/b=1.5 and 1., two oblate ones of these ratios, and a shere of radius r s =1. m. They are in air and all have the same volume V= 4 3 r 3 s =4.19 m 3. The beam is assumed to be olarized in the O P -xz lane bd =. It illuminates these sheroids vertically, i.e., side on bd =9. The beam center moves along the y axis in the equatorial lane of the sheroid. In this case, the beam lays a role similar to a thin encil of light that carries momentum. When it hits a sheroid, a radiation force is exerted on the article along or oosite to the beam roagation direction. When the force is alied off center on the article, there is a leverarm associated with the force and the orbital angular momentum can be successfully transferred to the article through absortion, resulting in an orbit-absortion torque. Such a torque is about the z axis in the current case. It is observed in Fig. that the T z -y curves for a shere and the sheroids have a similar shae. As the beam center moves along the y axis, the lever-arm increases, and the torque corresondingly increases. Maxima of the torque aear at the vicinity of y =.8 m, near the edge of the article see the labels in Fig.. Further increase in y causes less beam light flux incident on the article and less angular momentum is trans

7 XU et al. PHYSICAL REVIEW A 78, T y 4 x a/b=1. (Shere) a/b=1.1 (Prolate sheroid) a/b=1.1 (Oblate sheroid) a/b=1.1 (Prolate sheroid) a/b=1.1 (Oblate sheroid) rotation is the oosite. Since during the rotation the incident angle gradually decreases for the rolate sheroid but increases for the oblate one, the article finds its stable balance orientation corresonding to end-on incidence of the beam on the rolate sheroid bd = and side-on incidence on the oblate one bd =9. Mathematically, the stable orientation s is determined by T y s =, T y. 1 bd bd = s FIG. 3. Torque about the y axis T y versus the incident angle bd when a linearly olarized Gaussian beam of waist radius =1. m and wavelength =.785 m is incident on a shere and two sheroids of axis ratios a/b=1.1 and 1.1. The sheroids have the same volume as the shere of radius r s =1. m. Their refractive index is M= i. The beam center coincides with that of the article and the beam olarization angle is bd =. Note that in this figure T y is zero for the shere T y =. ferred, so the torque corresondingly decreases. In addition, a symmetry relation of the torque with resect to the oint T z =,y = can be observed, as it should. For a given y, we did not find a monotonic relationshi between T z and a/b during the deformation of the shere to a rolate or oblate sheroid of axis ratio 1.. This is because, even for a shere, the absortion efficiency as a function of size arameter is a very raidly varying function see Fig. 7. on. 17 of Ref. 43. The situation for a sheroid would be similar when its size arameter defined by axis ratio and semimajor axis length, C I =k I aa/b 1 1/ varies as the axis ratio increases. Therefore looking for trends in data as a function of a/b or C I will be comlicated when the article s refractive index greatly differs from that of the surrounding medium. B. On-axis beam incidence: Effect of incident angle Second, we investigate the influence of the incident angle on the torque. The beam center is set to coincide with that of the article so that the beam is incident on axis. Torques are calculated for the incident angle increasing from 9 to 9 9 bd 9. The articles considered here are two rolate sheroids and two oblate ones of axis ratios 1.1 and 1.1. They still have the same volume as a shere of radius r s =1. m. Since there are no torque comonents about the x and z axes T x =T z = in this case, only T y is lotted in Fig. 3. It can be observed that at a ositive negative incident angle, there exists a ositive negative torque causing counterclockwise clockwise rotation of the rolate sheroid about the y axis. For an oblate sheroid, the direction of The torque found for these slightly absorbing sheroids is dominated by a reaction force which actually does not require the articles to be absorbing. To exlain its hysical mechanism, we develo in Aendix A the geometrical otics GO method in which the focused Gaussian beam illuminating a large sheroid is treated as a single ray. When it is refracted into the sheroid, the change of roagation direction roduces a change of momentum of the reflected and transmitted rays and brings in the reaction force of order =. When it is refracted out, the reaction force of order = 1 is contributed. Reaction forces of higher orders are from the rays exeriencing 1 internal reflections. As roved in Aendix A, the reaction force arising from both reflection and refraction is along or oosite to the outwardly directed normal vector to the surface of the article. For the secial case of a shere, this surface normal is along the direction of the lever-arm R so that the reaction force torques caused by both reflection R F R and refraction R F T are equal to zero for all orders. For a sheroid, however, the surface normal does not coincide with the lever arm in direction unless an on-axis sheroid is illuminated by an end-on or side-on incident beam here the term on-axis means the center of sheroid is on the beam axis. In the current case of an on-axis sheroid illuminated by an oblique incident beam, torques of orders = and 1 evaluated by R F R +F T and R 1 F R 1 +F T 1, resectively, are found to have the leading contribution to the torque. They have the same sign when the sheroid is not extremely deformed from shere, say 1.a/b. for ReM = Since the normal to an oblate sheroid lies on the oosite side of the incidence direction as does the normal for a rolate sheroid, an oosite direction of rotation is induced, as indicated by Fig. 3. In the addition to the dominating contribution from the reaction force induced torque, the orbit-absortion torque also aears on an absorbing sheroid even when it is onaxis located. To exlain the aearance of the orbitabsortion torque, we trace two rays contained in the Gaussian beam and symmetric about the beam axis. Since for an on-axis sheroid, the lever-arms R associated with hoton absortion induced forces d/dt along the otical aths L of these two rays are not equal, a residual torque evaluated by T= L1 dr 1 d 1 dt + L dr d dt is induced. For a shere, however, the numbers of absorbed hotons from these two rays are exactly identical and the lever arms are symmetric about the beam axis along the internal light ath so that the two induced torques counteract each other T =. To demonstrate the existence and effect of this orbit

8 RADIATION TORQUE EXERTED ON A SPHEROID: PHYSICAL REVIEW A 78, x1 8 4 M=1.1 M= i M= i M=1.1+1.i x1 15 M=1.573 (Prolate sheroid) M=1.573 (Oblate sheroid) M= i (Prolate sheroid) M= i (Oblate sheroid) M= i (Prolate sheroid) M= i (Oblate sheroid) T y T y 1.5 = 45 = FIG. 4. Torque about the y axis T y versus the incident angle bd when a linearly olarized Gaussian beam of waist radius =1. m and wavelength =.785 m is incident on a rolate sheroid of axis ratio a/b=1.1 and volume equal to that of a shere of radius r s =1. m. The real art of its refractive index is set to 1.1 and the imaginary art increases from,.1 3, 1.1, and then to 1.. The beam has its center coinciding with that of the article and its electric field E is olarized in the O P -xy lane. Note that for clarity the results for the sheroid of M =1.1+1.i has been divided by 6. absortion torque for an on-axis sheroid, we take a rolate sheroid of axis ratio 1.1 and with volume equal to that of a shere of radius r s =1. m as an examle. The real art of its refractive index is set to 1.1 so that the angular momentum change due to refraction and reflection is small and the contribution of the reaction force torque can be effectively reduced. Meanwhile, we let the imaginary art of the article s refractive index increase from to.1 3, 1. 1, and then to 1.. It can be observed from Fig. 4 that with the increase of article s absortion caacity, the torque first decreases and then flis its sign. Similar effects also occur for an oblate sheroid. This means there exists an orbit-absortion torque that rotates the sheroid in a direction oosite to that caused by reaction force induced torque. To comletely overcome the torque resulting from the reaction force, the absortion of the sheroid should reach a certain level so that the rotation is dominated by the orbitabsortion torque. The transition oint ImM t is found to be greatly deendent on the real art of the article s refractive index. For the current case of ReM=1.1, its value is found to be and for rolate and oblate sheroids, resectively. When ReM increases to 1.573, its value increases to and 9.41 for rolate and oblate sheroids, resectively. Note that when the imaginary art of the article s refractive index is increased to be sufficiently large so that the article tends to be totally reflecting, the reaction force dominates again the torque. And the rotation still becomes reverse since the torque caused by the reflected rays at the order = and evaluated by R F R has a direction oosite to that of the torque caused by the refracted rays F T in this case. But in our former FIG. 5. Torque about the y axis T y versus the incident angle bd when a linearly olarized Gaussian beam of waist radius =1. m and wavelength =.785 m is incident on olystyrene rolate and oblate sheroids of axis ratio a/ b=1.1 and volume equal to that of a shere of radius r s =5. m. The beam has its center coinciding with that of the article and its electric field E is olarized in the O P -xy lane. The articles are susended in air and have real refractive index while the imaginary art increases from to 1.1 3, and then to.1 3. case of the slightly absorbing sheroid see Fig. 3, F T is larger than F R and hence dominates the reaction force torque of order =. As a qualitative verification of GLMT for micrometer scale articles, the GO model develoed in Aendix A is used. Ideally, the GO model requires the size of sheroid to be much larger than the beam waist so that the curvature of the sheroidal surface in y direction can be neglected and the three dimensional roblem can be simlified to two dimensions. However, an extremely large size is beyond the caability of our current comutational rogram develoed for GLMT. Therefore to comare the results obtained by GO and GLMT, the torque is calculated for sheroids with volume equal to that of a shere of not very large radius r s =5. m. They are assumed to have an axis ratio a/b =1.1 and refractive indices M=1.573, i, and i. A Gaussian beam of waist radius =1. m and electric field E lying in the O P -xy lane is used in the GLMT calculation. In the GO calculation such a beam is aroximated by a single ray and the maximum order of ray tracing is taken to be max =. The results are lotted in Fig. 5 for GLMT and Fig. 6 for GO. Both calculations redict a ositive torque for a rolate oblate sheroid at ositive negative incident angles. The maximum torque aears at an incident angle smaller larger than 45 for a rolate oblate sheroid. Such an angle is found corresonding to the maximal momentum change of transmitted rays of orders = and =1, which contribute most to the final torque. More calculations show that further increasing the axis ratio will shift the maximal torque on a rolate and/or oblate sheroid toward even smaller larger incident angles. Figures 5 and 6 also exhibit the decrease of the

9 XU et al. PHYSICAL REVIEW A 78, x1 15 M=1.573 (Prolate sheroid) M=1.573 (Oblate sheroid) M= i (Prolate sheroid) M= i (Oblate sheroid) M= i (Prolate sheroid) M= i (Oblate sheroid) x a/b=1.1 (Prolate, Diole) a/b=1.1 (Oblate, Diole) a/b=1.1 (Prolate, Diole) a/b=1.1 (Oblate, Diole) a/b=1.1 (Prolate, GLMT) a/b=1.1 (Oblate, GLMT) a/b=1.1 (Prolate, GLMT) a/b=1.1 (Oblate, GLMT) T y (a.u.) 1 T y Θ bd Θ bd FIG. 6. Torque about the y axis T y versus the incident angle bd when a light ray of wavelength =.785 m is incident on the olystyrene rolate and oblate sheroids of axis ratio a/ b =1.1 and volume equal to that of a shere of radius r s =5. m. The light ray is directed toward the article center O P and it has its electric field E olarized in the O P -xy lane. The articles are susended in air and have real refractive index while the imaginary art increases from to 1.1 3, and then to.1 3. Torque in the figure has arbitrary units a.u.. torque due to increasing absortion. There are two reasons for this henomenon. First, increasing the imaginary art of a slightly absorbing article s refractive index means more intensity attenuated in the sheroid so that the rays of order =1 contribute less and the reaction force induced torque decreases accordingly. Second, the orbital momentum transferred from the absorbed hotons induces an orbit-absortion torque of oosite sign to that caused by the reaction force. Therefore the overall torque decreases further. The orbitabsortion torque is not included in our GO model, therefore the decrease of torque redicted by GO in Fig. 6 is less remarkable than that redicted by GLMT in Fig. 5. The stable orientations found in the receding examles for slightly absorbing sheroids only ertain to the articles of size aroaching or larger than the wavelength. It does not ertain to the nanometer scale articles of size much smaller than the wavelength. An extremely small sheroid embedded in an external electric field E olarizes with the induced diole moment P, but usually not along the direction of the external field unless the E field direction is along one of the article s symmetry axes. A olarization induced torque T =PE aears in this case and tends to orient the major axis of the sheroid in the direction of E to minimize the stored energy in the field 44. As an examle, we give in Fig. 7 the T y - bd curves calculated by GLMT for two rolate and two oblate sheroids with volume equal to that of a shere of radius r s =1 nm. According to the criterion exressed by Eq. 1, the stable orientations are found to be characterized by the incident angle bd =9 for a rolate sheroid and by bd = for an oblate one. As indicated by the dotted lines in Fig. 7, these calculations are confirmed by diole method whose theoretical develoment is described in FIG. 7. Torque about the y axis T y versus the incident angle bd when a linearly olarized lane wave of wavelength =.785 m is incident on the olystyrene sheroids of refractive index M= i and axis ratios a/b=1.1 and 1.1. These sheroids have volume equal to that of a shere of radius r s =1 nm. The lane wave has the olarization angle bd =. The incident electric field magnitude E is assumed to be unity E =1. detail in Aendix B. Numerical calculations show that the diole method has errors less than 1.% for the articles of size about times smaller than the incident wavelength. Since the comarison of Fig. 3 to Fig. 7 has indicated that sheroids of nanometer scale and micrometer scale sizes have different stable orientations, it is interesting to know the transition size r s,t. In our current case using a linearly olarized Gaussian beam of waist radius =1. m, an average value of r s,t for the rolate olystyrene sheroids of axis ratio 1.a/b. and refractive index M= i is found to be about. m. C. On-axis beam incidence: Effect of olarization angle To further exlore the effect of the olarization induced torque, we examine the influence of the olarization angle bd on the torque at the stable orientation of the sheroid. First we discuss the case of oblate sheroids of micrometer scale size. When bd is not equal to or 9, the induced diole P of the sheroid is not along the direction of E field. Using the oblate sheroids of the same arameters as those in Fig. 3, we give in Fig. 8 the torque about the x axis T x for the oblate sheroids of stable orientation characterized by bd =9 no torque comonents about y and z axes exist in this situation, namely T y =T z =. It can be found that T x rotates the articles to the stable orientation of the beam olarization angle bd =9. Mathematically, this stable olarization angle s is described by T x s =, T x. s bd = s Our rediction is consistent with the exerimental observation of Bayoudh et al

10 RADIATION TORQUE EXERTED ON A SPHEROID: T x 3 6 x a/b=1. (Shere) a/b=1.1 (Oblate sheroid) a/b=1.1 (Oblate sheroid) Φ bd FIG. 8. Torque about the x axis T x versus the olarization angle bd when a linearly olarized Gaussian beam is incident on a shere and two oblate sheroids. The arameters of the beam and the articles are the same as in Fig. 3. But the incident angle is now fixed to be 9 bd =9. Note that T x is zero for the shere at all olarization angles T x =. Next we show in Fig. 9 the T x - bd curves for two nanometer scale rolate sheroids of axis ratios a/b=1.1 and 1.1, resectively, and having a volume equal to that of a shere of radius r s =1 nm. They are vertically illuminated by the lane wave bd =9. According to Eq., the stable olarization angle is found to be characterized by bd = for a rolate sheroid. These calculations are confirmed by a diole method which gives the results lotted as dotted lines in Fig. 9. T x 1 3 x a/b=1.1 (Prolate, Diole) a/b=1.1 (Prolate, Diole) a/b=1.1 (Prolate, GLMT) a/b=1.1 (Prolate, GLMT) Φ bd FIG. 9. Torque about the x axis T x versus the olarization angle bd when a linearly olarized lane wave of wavelength =.785 m is incident on two rolate olystyrene sheroids of refractive index M= i and axis ratios a/b=1.1 and 1.1. The sheroids have volume equal to that of a shere of radius r s =1 nm. The incident angle of the beam is fixed to be 9 bd =9. PHYSICAL REVIEW A 78, Note that due to the rotational symmetry about the beam axis, olarization torque does not exist on a rolate sheroid of micrometer scale at the beam incident angle bd = or on an oblate sheroid of nanometer scale at bd =9. Thus we can end this art of the discussion by concluding that the torque exerted on an extremely small sheroid r s is dominated by the olarization induced torque which leads to the stable orientation bd =9, bd = and bd =, bd for a rolate sheroid and for an oblate one, resectively. When the size of the article increases to the geometrical otics limit and its absortion is slight, the reaction force induced torque becomes significant and leads to end-on side-on incidence of the beam on the rolate oblate sheroid. Finally, the stable orientation is characterized by bd =, bd and bd =9, bd =9 for a rolate sheroid and for an oblate one, resectively. Since comletely oosite rotation directions are involved in the two limits, there exists a trade-off size r s,t in the transition region. In addition, increasing article s absortion caacity to a certain level might cause the stable orientation of the sheroids of micrometer scale to be same as that found for the sheroids of nanometer scale. D. Circularly olarized beam incidence: Effect of axis ratio As discussed in the receding subsection, the torque exerted by a linearly olarized beam aligns a sheroid to a given orientation. There should be some oscillations of the sheroid back and forth about the stable orientation. After a certain eriod, the oscillations are damed out by friction and the article finally remains stationary inside the beam. However, when a circularly olarized beam is used the induced torque makes the sheroid rotate at the formerly stable orientations. As an examle, we simulated the otical stretcher emloying two coaxial but two counterroagating Gaussian beams to exert forces on a RBC. The beams are assumed right- and left-circularly olarized, resectively though they are linearly olarized in actual exeriments. The originally sherical RBC of radius r s =3. m is embedded in a buffer solution of refractive index M = It is traed where the beam cross sections are same and the local waist radius is 1% larger than that of the article =1.1r s 8. During the deformation its shae is aroximated by a rolate sheroid of axis ratio a/b growing from 1. to 1.5. The changes of torque about the z axis T z and absortion cross section C abs are illustrated in Figs. 1 and 11, resectively. With the refractive index of the RBC aroaching that of the medium M = i, a monotonic increase of the torque during the deformation can be observed. Since the magnitude of torque deends on the number of absorbed hotons carrying angular momentum, the torque curve exhibits the same shae as that of absortion cross section. Assuming the central beam intensity I to be unity, the torque exerted by a circularly olarized beam can be related to the absortion cross section for the case of on-axis beam incidence bd = and G n,circular = when 1 by

11 XU et al. 3.4 x1 19 C abs = + Re PHYSICAL REVIEW A 78, = n= n= nn A * n G n,tm.91 + B * n G n,te + A * n A n + B * n B n, 4 T z a/b FIG. 1. Torque about the z axis T z versus axis ratio a/b for a rolate sheroid. Two coaxial but counterroagating Gaussian beams, which are right- and left-circularly olarized, resectively, stretch an originally sherical RBC of refractive index M= i and radius r s =3. m. It is embedded in a buffer solution of refractive index M = The beams have the waist radius =1. m. The RBC is located where the beam cross sections are same and the radius of local beam waist is 1% larger than that of the article =1.1r s. During the deformation, its axis ratio is assumed to increase from 1. to 1.5. Note that in this figure, T z is the contribution of each beam. T z = 4M ck I 3 I C abs, 3 where + and corresond to the right- and left-circularly olarized beams, resectively, and the absortion cross section C abs can be obtained by relacing the BSCs in the following formula for a linearly olarized beam by those for a circularly olarized beam Eqs. 19 and : C abs (m ) x a/b FIG. 11. Absortion cross section of the sheroid during deformation. Parameters of the beam and sheroid are the same as in Fig. 1. where nn =, n n = odd, + r + r +1 + r! d n r=,1 +r +1 r! r d n r, n n = even. 5 E. Circularly olarized beam incidence: Effect of incident angle Next, we investigate the influence of the incident angle on the torque. The micrometer scale articles used here are the same as those used for Fig. 3 excet that the beam is now right-circularly olarized and the article has a urely real refractive index In this case, the reaction force induced torque still dominates the rotation about the y axis T y, Fig. 1b causing stability at end-on side-on incidence of the beam on the rolate oblate sheroid. But contrary to the case when a linearly olarized beam is used, an additional torque about the x axis can also be observed in Fig. 1a. This should be the olarization induced torque due to the temoral variation of the olarization direction of the electric field E. Though the variation is quite raid in time scale, a residual effect still occurs. This torque kees a transarent oblate sheroid rotating at its stable orientation characterized by the beam incident angle bd =9 but not for a rolate sheroid at its stable orientation with bd =. Its aearance has also been redicted by couled diole method develoed for microroeller driven by a circularly olarized beam 48. It is noteworthy that the torque about the z axis T z is zero for all incident angles. The reason is that when we decomose the local electric field E into comonents in the x and z directions, the sinning art about the z axis actually induces no torque because of the sheroid s symmetry about this axis and its nonabsortion of light. To generate a torque about the z axis, the article has to be absorbing so that the angular momentum of some incident hotons is transferred to the sheroid via sin-absortion mechanism. It can be seen in Fig. 13c that T z aears when the imaginary art of the refractive index of the sheroids in the receding examle is increased to Moreover, at the stable orientation of an oblate sheroid bd =9, comarison of the torque about the x axis for an absorbing article Fig. 13a to that for a transarent one Fig. 1a indicates that torques induced by olarization and sinabsortion rotate the article toward a same direction in site of their difference in magnitude. As to absorbing sheroids of nanometer scale, all torque comonents occur as well when the article is illuminated by an oblique incident beam. In Fig. 14 is illustrated an examle

12 RADIATION TORQUE EXERTED ON A SPHEROID: T x 1 x for olystyrene sheroids with volume equal to that of a shere of radius r s =1 nm. But in this case, the olarization induced torque tends to align a rolate oblate sheroid into the orientation characterized by bd =9 bd =. For a transarent rolate sheroid, a olarization induced torque will rotate the article about the beam axis at its stable orientation. For sheres of whatever size, however, there is no rotation about the y axis T y =. They have to be absorbing to induce the torque comonents T x and T z which are related by the following equation: T x,bd + T z,bd = T z,bd = a/b=1.1 (Prolate sheroid) a/b=1.1 (Oblate sheroid) a/b=1.1 (Prolate sheroid) a/b=1.1 (Oblate sheroid) (a) T y 4 x1 16 a/b=1.1 (Prolate sheroid) a/b=1.1 (Oblate sheroid) a/b=1.1 (Prolate sheroid) a/b=1.1 (Oblate sheroid) (b) FIG. 1. Torque comonents T x a and b T y versus the incident angle bd of the beam. The articles used here are same as those in Fig. 3, but their refractive indices are now urely real, M= And the beam is now right-circularly olarized. = T x,bd =9. 6 Equation 6 indicates an overall rotation of the shere about the beam axis at a constant angular velocity. PHYSICAL REVIEW A 78, IV. CONCLUSION Within the framework of generalized Lorenz-Mie theory, the analytical solution to the radiation torque exerted on a sheroid by an arbitrarily shaed beam is derived. Numerical results indicate two tyes of torques not found for a shere: the olarization induced torque and the reaction force induced torque. Their hysical mechanisms are analyzed using geometrical otics for large sheroids and the diole olarization method for small ones. Due to these torques, a olarized Gaussian beam obliquely incident on a sheroid causes its rotation into a stable orientation. This orientation deends on the article size relative to the incident wavelength as well as its refractive index. For both transarent and slightly absorbing sheroids, when its size aroaches or is larger than the wavelength e.g., micrometer scale sheroids the stable orientation dominated by the reaction force indcued torque is associated with end-on incidence of the beam on a rolate sheroid and side-on incidence on an oblate sheroid. While for a article of size much smaller than the wavelength e.g., nanometer scale sheroids, the stable orientation is decided by the olarization indueced torque and corresonds to side-on incidence of the beam on a rolate sheroid and end-on incidence on an oblate one. Moreover, the olarization induced torque aligns the equatorial lane of a micrometer scale nanometer scale oblate rolate sheroid arallel erendicular to the olarization direction of the E field at stable orientations. It should be noted that the stable orientation redicted for sheroids of micrometer scale might change to those found for sheroids of nanometer scale when articles increases its absortion to a certain level and the reaction force induced torque is dominated by an orbit-absortion torque rotating the on-axis sheroid in the oosite direction. The same effect can be found for a highly absorbing sheroid which totally reflects the incident light, bringing in a reaction force urely contributed by external reflection. When illuminated by a circularly olarized beam, the stable orientation is still related to the beam incident angle bd = bd =9 for a rolate oblate sheroid of micrometer scale and by bd =9 bd = for a rolate oblate sheroid of nanometer scale. And the sinabsortion induced torque dominates the rotation of an absorbing sheroid about the beam axis at the final stable orientation. However, the rotation is dominated by olarization torque for two tyes of transarent sheroids: rolate ones of nanometer scale and oblate ones of micrometer scale. ACKNOWLEDGMENT This research is suorted by Alexander von Humboldt Foundation Grant No. CHN/1177. APPENDIX A: REACTION FORCE INDUCED TORQUE ON A SPHEROID ANALYZED BY GEOMETRICAL OPTICS As shown in Fig. 15, a focused Gaussian beam roagating in the O P -xz lane illuminates a sheroid with z axis being its rotational symmetric axis. For simlicity in the

13 XU et al. PHYSICAL REVIEW A 78, x x a/b=1. (Shere) a/b=1.1 (Prolate sheroid) a/b=1.1 (Oblate sheroid) a/b=1.1 (Prolate sheroid) a/b=1.1 (Oblate sheroid).5 T x T y 1.5 a/b=1. (Shere) a/b=1.1 (Prolate sheroid) a/b=1.1 (Oblate sheroid) a/b=1.1 (Prolate sheroid) a/b=1.1 (Oblate sheroid) (a) (b) 8 x T z 4 a/b=1. (Shere) a/b=1.1 (Prolate sheroid) a/b=1.1 (Oblate sheroid) a/b=1.1 (Prolate sheroid) a/b=1.1 (Oblate sheroid) (c) FIG. 13. Torque comonents T x a, T y b, and T z c versus the incident angle bd of the beam. The articles used here are same as those used for Fig. 3 having a comlex refractive index while the beam is now right-circularly olarized. In b, T y is zero for a shere at all incident angles T y =. Note that in a, the T x - bd curves of a shere and the sheroids of axis ratio a/b=1.1 nearly coincide with each other. qualitative analysis of the torque about the y axis, this beam is aroximated by the ray A O. The olarization of the electric field of the beam and the equivalent ray is in the O P -xz lane. Assuming the sheroid is very large comared to the width of the focused beam, the curvature of the sheroidal surface in the y direction can be neglected and the rolate sheroid simlifies to an ellise in the O P -xz lane. Keeing the incident ray fixed and rotating the ellise 9 about the y axis gives the case for an oblate sheroid. The torque exerted on the sheroid is contributed by rays of all orders, namely, T =R F R + R F R +R F T + R F T, =1 A1 where R is the vector from the article center O P to the intersection oint of the ray and the ellitical boundary and =1 F R and F T are the reaction forces exerted on the sheroid in resonse to the th order reflected and transmitted rays, resectively. The first terms in two brackets, R F R and R F T, reresent the torques induced in resonse to the reflected and transmitted rays of zeroth order =. They occur when the incident ray hits the sheroid at an incident angle i, and is refracted into sheroid via angle r,. Higher orders 1 mean the rays exeriencing 1 internal reflections. The rate of momentum contained in a roagating light ray is related to the ray s ower P by M = M mp k, A c where k is the unit vector in the roagation direction of the ray, c is light seed in the vacuum, and M m is the refractive index of the medium in which the ray roagates. The am