week ending PRL96,113903(20 PHYSICAL REVIEW LETTERS 24 MAR Stable Optical Trapping Based on Optical Binding Forces Tomasz M. Grzegorczyk, Brandon A Kemp, and Jin Au Kong Research laboratory of electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 23 September 2005: published 24 March 2006) Various trapping configurations have been realized so far, either based on the scattering force or the radient force. In this Letter, we propose a new trapping regime based on the equilibrium between a scattering force and optical binding forces only. The trap is realized from the interaction between a single plane wave and a series of fixed small particles, and is efficient at trapping multiple free particles. The effects are demonstrated analytically upon computing the exact scattering from a collection of cylindrical particles and calculating the Lorentz force on each free particle via the Maxwell stress tensor. DOF: 10.1 103/PhysRevLett96. 113903 PACS numbers: 42.25.Fx. 41.20.Jb Since 1970, it has been known that the motion of small trary number of cylindrical particles based on the Mie dielectric particles can be controlled by laser beams [1]. theory and the Foldy-Lax multiple scattering equations The radiation pressure created by the lasers on the particles [16]. We use here this ability in order to optimize the induces a scattering force along the direction of the beam location of a series of fixed small particles and devise a propagation, and a gradient force along the gradient of the configuration where scattering and optical binding forces field intensity. The theoretical demonstration of the exis- can be manipulated to create a new regime of particle tence of a gradient force on small particles was offered in trapping Ref [2]using a dipole approximation in the formulation of The configuration we shall use to begin is a general the Lorentz force [3]. For strong intensity gradients, it was ization of the one studied in Ref [ 15], where two particles shown that the gradient force can become dominant, yield- are submitted to an incident plane wave propagating in the ing a negative force responsible for the trapping and stable y direction, as shown in the inset of Fig. 1(a). The forces in levitation reported in Refs. [4-6]. For weak intensity gra- the i and y directions( denoted Fr and Fy, respectively) dients, it was shown that a stable trap could still be realized computed either using the method of Ref [15] for circular by using two counter-propagating beams in order to bal- geometries or the one of Ref. [16] for cylindrical geome ance the two scattering forces [7]. Developing upon this tries exhibit a tapered oscillatory behavior, as already work, multiple trapping configurations have been pro- known from Ref [15] and illustrated in Fig. 1(a) for the posed, such as Talbot trapping [8], trapping with diffractive typical set of experimental parameters detailed in the optics [8], trapping with a spherical lens [9], dielectropho- caption of the figure. The oscillatory behavior of either resis trapping [10], and two- or three-beam trapping [11- Fr or Fy is a direct manifestation of the optical binding 13] which offers the control of the size of the traps typi- between the particles: the mean value of the force(dashed cally achieved by varying the incident angles of the three line in the figure) corresponds to the force on a single beams. In all these cases, the particles are trapped either particle, while the amplitude of the oscillation is directly because of the gradient force, or because of the balance related to the strength of the binding. The latter can be between the scattering forces and other external forces in explained upon considering N two-dimensional line scat the system. teers(equivalent to infinitely thin cylinders)and writing In this Letter, we propose an alternative trapping regime the corresponding Foldy-Lax multiple scattering equations based on optical binding forces, i.e., an alternative way to [17]. The total electric field E()=Eind (r)+ Escat(r)at induce a negative force. Unlike the known configurations location F=ix t yy is expressed as the sum of the inci- mentioned above, the present configuration can be realized dent and the scattered field from the N scatterers in the with a single plane wave. Since optical binding manifests system(located at positions Fe, tEll,., ND), where itself when multiple particles submitted to an incident Einc()=Melky is the incident field with unit amplitude and electromagnetic wave interact and scatter collectively, its calculation requires us to include all the multiple interac tions between all the particles. Optical binding on a system sa(=2∑ifH0(kF-E(行).(1) of two particles was first studied in Ref. [14] where the particle response was assimilated to a harmonic oscillator. The Hankel function of the first kind and zeroth order Later, Ref. [15] used a discrete dipole approximation to represents the two-dimensional Greens function and f is evaluate the binding forces from the Maxwell stress tensor the scattering amplitude of a cylinder, related to its polar from two particles. Recently, we have proposed an exact inability which can be evaluated numerically [15, 18 method to compute the optical binding between an arbi- E() is the total exciting field on particle j from all the 0031-9007/06/96(11)/113903(4)S23.00 113903-1 o 2006 The American Physical Society
Stable Optical Trapping Based on Optical Binding Forces Tomasz M. Grzegorczyk, Brandon A. Kemp, and Jin Au Kong Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 23 September 2005; published 24 March 2006) Various trapping configurations have been realized so far, either based on the scattering force or the gradient force. In this Letter, we propose a new trapping regime based on the equilibrium between a scattering force and optical binding forces only. The trap is realized from the interaction between a single plane wave and a series of fixed small particles, and is efficient at trapping multiple free particles. The effects are demonstrated analytically upon computing the exact scattering from a collection of cylindrical particles and calculating the Lorentz force on each free particle via the Maxwell stress tensor. DOI: 10.1103/PhysRevLett.96.113903 PACS numbers: 42.25.Fx, 41.20.Jb Since 1970, it has been known that the motion of small dielectric particles can be controlled by laser beams [1]. The radiation pressure created by the lasers on the particles induces a scattering force along the direction of the beam propagation, and a gradient force along the gradient of the field intensity. The theoretical demonstration of the existence of a gradient force on small particles was offered in Ref. [2] using a dipole approximation in the formulation of the Lorentz force [3]. For strong intensity gradients, it was shown that the gradient force can become dominant, yielding a negative force responsible for the trapping and stable levitation reported in Refs.[4 –6]. For weak intensity gradients, it was shown that a stable trap could still be realized by using two counter-propagating beams in order to balance the two scattering forces [7]. Developing upon this work, multiple trapping configurations have been proposed, such as Talbot trapping [8], trapping with diffractive optics [8], trapping with a spherical lens [9], dielectrophoresis trapping [10], and two- or three-beam trapping [11– 13] which offers the control of the size of the traps typically achieved by varying the incident angles of the three beams. In all these cases, the particles are trapped either because of the gradient force, or because of the balance between the scattering forces and other external forces in the system. In this Letter, we propose an alternative trapping regime based on optical binding forces, i.e., an alternative way to induce a negative force. Unlike the known configurations mentioned above, the present configuration can be realized with a single plane wave. Since optical binding manifests itself when multiple particles submitted to an incident electromagnetic wave interact and scatter collectively, its calculation requires us to include all the multiple interactions between all the particles. Optical binding on a system of two particles was first studied in Ref. [14] where the particle response was assimilated to a harmonic oscillator. Later, Ref. [15] used a discrete dipole approximation to evaluate the binding forces from the Maxwell stress tensor from two particles. Recently, we have proposed an exact method to compute the optical binding between an arbitrary number of cylindrical particles based on the Mie theory and the Foldy-Lax multiple scattering equations [16]. We use here this ability in order to optimize the location of a series of fixed small particles and devise a configuration where scattering and optical binding forces can be manipulated to create a new regime of particle trapping. The configuration we shall use to begin is a generalization of the one studied in Ref. [15], where two particles are submitted to an incident plane wave propagating in the y^ direction, as shown in the inset of Fig. 1(a). The forces in the x^ and y^ directions (denoted Fx and Fy, respectively) computed either using the method of Ref. [15] for circular geometries or the one of Ref. [16] for cylindrical geometries exhibit a tapered oscillatory behavior, as already known from Ref. [15] and illustrated in Fig. 1(a) for the typical set of experimental parameters detailed in the caption of the figure. The oscillatory behavior of either Fx or Fy is a direct manifestation of the optical binding between the particles: the mean value of the force (dashed line in the figure) corresponds to the force on a single particle, while the amplitude of the oscillation is directly related to the strength of the binding. The latter can be explained upon considering N two-dimensional line scatterers (equivalent to infinitely thin cylinders) and writing the corresponding Foldy-Lax multiple scattering equations [17]. The total electric field Er E incr E scatr at location r xx^ yy^ is expressed as the sum of the incident and the scattered field from the N scatterers in the system (located at positions r‘, ‘ 2 f1; ... ; Ng), where E incr ze^ iky is the incident field with unit amplitude and E scatr z^ X N j1 ifH1 0 kjr rjjEe j rj: (1) The Hankel function of the first kind and zeroth order represents the two-dimensional Green’s function and f is the scattering amplitude of a cylinder, related to its polarizability which can be evaluated numerically [15,18]. Ee j rj is the total exciting field on particle j from all the PRL 96, 113903 (2006) PHYSICAL REVIEW LETTERS week ending 24 MARCH 2006 0031-9007=06=96(11)=113903(4)$23.00 113903-1 © 2006 The American Physical Society
PRL96,113903(200 PHYSICAL REVIEW LETTERS week ending 24 MAR written a E()=2ey+ I-ifHO(FI-F2D (Ho(klF-F1D H0)(kF-2) The magnetic field is directly obtained from Faradays law and the force is obtained from the contour integration of 升↑ the Maxwell stress tensor [19), which has been shown to be equivalent to the computation of the Lorentz force from bound currents and charges [20, 21]. The leading term in I the expression of the two components of the force have a dependence of the form Ho(kr), which yields the tapere oscillatory behavior clearly visible in I The per of the oscillations is therefore directly related to the period of the Hankel function(e. g, with the parameters of Fig. 1 the Hankel function has a period of about 0.42 um, in 日 oe al agreement with the oscillations shown in Fig. 1(a)obtained without approximations), while their amplitude is modu lated by the scattering amplitude f and the number of lum The amplitude of the oscillations can therefore be in creased by either increasing the size of the particles(bear- ing in mind that the analogy with line scatterers would (d) become less accurate), or by increasing the number of particles in the system. It is the second approach that we exploit here: increasing the number of particles in the system increases the optical binding forces between parti cles, which can create a negative force Since the number of particles and their positions are arbitrary, the resulting forces can be potentially modulated at will. In particular we first show that using a single plane wave lik Fig. 1(a), the amplitude of the oscillation of Fy can be in the insets as function of its position along i due to a indicating that optical binding forces can be made strong r subplots correspond to different arrangement of fixed enough to cancel or invert the scattering force due to the particles on the left:(a)single particle, (b)three particles align in incident plane wave. In order to achieve this, the proper y,()nine particles aligned in A, (d)nine particles aligned in positions of the particles have to be determined by consid- All particles have a permittivity E,=2.25Eo, are embedded in a ering their scattering characteristics. As can be seen from background of Eb=1.69Eo, and have a radius of 10 nm. The the parameters indicated in the caption of Fig.1,the closely packed particles are separated by 21 nm. The dashed particles are very small compared to the wavelength in curve indicates the force on gle particle (F order to prevent the scattering force from dominating the 4.36X10-2N/m). The superposed triangles have been calcu- binding forces(although this property has not been used in lated by the method shown in Ref. [21] from the fields obtained the calculations of the forces [16), which indicates that from the commercial package CST Microwave StudIo their response can be approximated by a rayleigh radiation from a dipole in the z direction [22]. Consequently, a strong other particles including all interactions, solved from radiation is induced in the i direction, and binding phe- nomena are expected to be enhanced if multiple dipoles can be induced in 2. This can be achieved by placing EsG;)=Ein(r)+>iTfHO( of Figs. 1(b)and I(c), or by aligning particles along the i axis, as shown in Fig. I(d) and its inset. Such configura either iteratively or by matrix inversion. For two cylinders tions could potentially be achieved using forces from positioned at FI and F2 with y =0 in both cases like shown evanescent fields, as suggested in Refs. [15,23]. We prefer in the inset of Fig. 1(a), the total electric field can be the vertical configuration to the horizontal one for reasons l13903-2
other particles including all interactions, solved from E e j rj E incrj X N ‘1 ‘j ifH1 0 kjrj r‘jEe j r‘ (2) either iteratively or by matrix inversion. For two cylinders positioned at r1 and r2 with y 0 in both cases like shown in the inset of Fig. 1(a), the total electric field can be written as Er z^ eiky if 1 ifH1 0 kjr1 r2j H1 0 kjr r1j H1 0 kjr r2j : (3) The magnetic field is directly obtained from Faraday’s law and the force is obtained from the contour integration of the Maxwell stress tensor [19], which has been shown to be equivalent to the computation of the Lorentz force from bound currents and charges [20,21]. The leading term in the expression of the two components of the force have a dependence of the form H1 0 kr, which yields the tapered oscillatory behavior clearly visible in Fig. 1(a). The period of the oscillations is therefore directly related to the period of the Hankel function (e.g., with the parameters of Fig. 1, the Hankel function has a period of about 0:42 m, in agreement with the oscillations shown in Fig. 1(a) obtained without approximations), while their amplitude is modulated by the scattering amplitude f and the number of particles. The amplitude of the oscillations can therefore be increased by either increasing the size of the particles (bearing in mind that the analogy with line scatterers would become less accurate), or by increasing the number of particles in the system. It is the second approach that we exploit here: increasing the number of particles in the system increases the optical binding forces between particles, which can create a negative force. Since the number of particles and their positions are arbitrary, the resulting forces can be potentially modulated at will. In particular, we first show that using a single plane wave like in Fig. 1(a), the amplitude of the oscillation of Fy can be enhanced to actually reach zero or even negative values, indicating that optical binding forces can be made strong enough to cancel or invert the scattering force due to the incident plane wave. In order to achieve this, the proper positions of the particles have to be determined by considering their scattering characteristics. As can be seen from the parameters indicated in the caption of Fig. 1, the particles are very small compared to the wavelength in order to prevent the scattering force from dominating the binding forces (although this property has not been used in the calculations of the forces [16]), which indicates that their response can be approximated by a Rayleigh radiation from a dipole in the z^ direction [22]. Consequently, a strong radiation is induced in the x^ direction, and binding phenomena are expected to be enhanced if multiple dipoles can be induced in z^. This can be achieved by placing additional particles along the y^ axis, as shown in the insets of Figs. 1(b) and 1(c), or by aligning particles along the x^ axis, as shown in Fig. 1(d) and its inset. Such configurations could potentially be achieved using forces from evanescent fields, as suggested in Refs. [15,23]. We prefer the vertical configuration to the horizontal one for reasons FIG. 1. Force in the y^ direction (Fy) felt by the right particle shown in the insets as function of its position along x^ due to a single plane wave incidence with E ze^ iky at 0 632:8 nm. The four subplots correspond to different arrangement of fixed particles on the left: (a) single particle, (b) three particles align in y^, (c) nine particles aligned in y^, (d) nine particles aligned in x^. All particles have a permittivity p 2:250, are embedded in a background of b 1:690, and have a radius of 10 nm. The closely packed particles are separated by 21 nm. The dashed curve indicates the force on a single particle (Fy ’ 4:36 1023 N=m). The superposed triangles have been calculated by the method shown in Ref. [21] from the fields obtained from the commercial package CST Microwave Studio®. PRL 96, 113903 (2006) PHYSICAL REVIEW LETTERS week ending 24 MARCH 2006 113903-2
week ending PRL96,113903(2006) PHYSICAL REVIEW LETTERS 24 MAR we shall detail hereafter. The corresponding results show equilibrium around x =xo= 297.5 nm while Fy is small that as the number of particles increases, the amplitude of but not exactly zero. a quick inspection reveals that Fyis the oscillations of Fy increases and eventually can be such null at x and y= yo=0.4 nm(data-not shown- to yield a zero or a negative Fy(these results have been are very similar to those of Fig. 2 with Fy peaking at zero at confirmed numerically by the commercial package Cst the same location as null). This configuration there Microwave Studio(, as explained in the caption of Fig. 1). fore realizes a one-dimensional trap Consequently, a system like that shown in the inset of Finally we verify that the trap is also two-dimensional Fig. 1(c), where nine particles are fixed and aligned along located at(. yo), as is further confirmed in Fig 3(a)by A, can indeed cancel the scattering force due to the incident computing the force field on a free particle spanning the le use of optical binding forces (xy) plane. The force distribution clearly shows an attrac We pursue by noting that the forces shown in Fig. I are tion toward (xo, yo) induced by the scattering and optical symmetric in x and, in the particular case of Fig. 1(c), that binding forces in the system. The optical well thus created the first minimum has a magnitude comparable to the first can be evaluated by computing the inverse gradient of the maximum(Fe-1. 1 x 10-2N/m at x <155 nm and force, and is shown in Fig 4. The potential energy is seen to F,e+1.5x 10-22N/m at x= 430 nm). Hence, the lo- dip sharply at the location of the trap and is thus efficient cation of a second set of nine vertical particles can be trapping free particles. Figures 3 and 4 also show that as the ptimized such that the independent forces tend to cancel number of trapped particles increases(up to four in this each other by superposing their respective minima and case), the potential well is strengthened. This is clearly maxima seen in Fig. 4 with an increased number of trapped parti- In order to exactly compute the force in the new con- cles, and is also illustrated in Figs. 3(b)and 3(c)by show iguration and optimize the location of the particles, we define an inverse problem based on the forward equations used in this Letter and presented in Ref. [16]. Although the reasoning based on independent forces from the two ver- tical sets of particles is not exact, it still gives a good initial guess of the initial positions to be used in the optimization scheme. The optimization is therefore run with 19 identical particles as those of Fig. 1 [18 are fixed and one spans the (ry) plane] and requires the magnitude of F, to be less than 三孑i 10% of the force on a solitary particle(indicated by the dashed lines in Fig. I)over a wide x range at y =0. The x [nm] result of the optimization yields the force shown in Fig. 2 (6) for two vertical arrangement of nine particles separated by 595 nm. It is seen that the force F.(solid line in the figure) lies within the specified constraints over about 400 nm, which corresponds to about 67% of the range. In addition, the force F( dashed line in the figure)is seen to be positive forx∈[o0nm,297.5m] and negative for x∈ [297.5 nm, 500 nml, with an amplitude about 2 orders of magnitude larger than Fy. Fr therefore creates a stable 055x 0--500xnm FIG. 3. Field force on a free particle in the vertical walls of particles separated by 595 nm and (a) no trapped e on a free oving along y =0 in the particle, (b)a single trapped particle, (c) four trapped particles configuration shown in the inset Solid line: F X 1024 [N/m]: all clustered around (x, y)=(2975 nm, 0.4 nm). Physical pa- Dashed line: Fr X10-22[N/m]. Physical parameters are iden- rameters are identical to those of Fig. 1. The tail of the arrows tical to those of fig. l correspond to the location of the center of the particle l13903-3
we shall detail hereafter. The corresponding results show that as the number of particles increases, the amplitude of the oscillations of Fy increases and eventually can be such to yield a zero or a negative Fy (these results have been confirmed numerically by the commercial package CST Microwave Studio®, as explained in the caption of Fig. 1). Consequently, a system like that shown in the inset of Fig. 1(c), where nine particles are fixed and aligned along y^, can indeed cancel the scattering force due to the incident plane wave by the sole use of optical binding forces. We pursue by noting that the forces shown in Fig. 1 are symmetric in x and, in the particular case of Fig. 1(c), that the first minimum has a magnitude comparable to the first maximum (Fy ’ 1:1 1022 N=m at x ’ 155 nm and Fy ’ 1:5 1022 N=m at x ’ 430 nm). Hence, the location of a second set of nine vertical particles can be optimized such that the independent forces tend to cancel each other by superposing their respective minima and maxima. In order to exactly compute the force in the new con- figuration and optimize the location of the particles, we define an inverse problem based on the forward equations used in this Letter and presented in Ref. [16]. Although the reasoning based on independent forces from the two vertical sets of particles is not exact, it still gives a good initial guess of the initial positions to be used in the optimization scheme. The optimization is therefore run with 19 identical particles as those of Fig. 1 [18 are fixed and one spans the (xy) plane] and requires the magnitude of Fy to be less than 10% of the force on a solitary particle (indicated by the dashed lines in Fig. 1) over a wide x range at y 0. The result of the optimization yields the force shown in Fig. 2 for two vertical arrangement of nine particles separated by 595 nm. It is seen that the force Fy (solid line in the figure) lies within the specified constraints over about 400 nm, which corresponds to about 67% of the range. In addition, the force Fx (dashed line in the figure) is seen to be positive for x 2 100 nm; 297:5 nm and negative for x 2 297:5 nm; 500 nm, with an amplitude about 2 orders of magnitude larger than Fy. Fx therefore creates a stable equilibrium around x x0 297:5 nm while Fy is small but not exactly zero. A quick inspection reveals that Fy is null at x x0 and y y0 0:4 nm (data—not shown— are very similar to those of Fig. 2 with Fy peaking at zero at the same location as Fx is null). This configuration therefore realizes a one-dimensional trap. Finally we verify that the trap is also two-dimensional, located at (x0; y0), as is further confirmed in Fig. 3(a) by computing the force field on a free particle spanning the (xy) plane. The force distribution clearly shows an attraction toward (x0; y0) induced by the scattering and optical binding forces in the system. The optical well thus created can be evaluated by computing the inverse gradient of the force, and is shown in Fig. 4. The potential energy is seen to dip sharply at the location of the trap and is thus efficient at trapping free particles. Figures 3 and 4 also show that as the number of trapped particles increases (up to four in this case), the potential well is strengthened. This is clearly seen in Fig. 4 with an increased number of trapped particles, and is also illustrated in Figs. 3(b) and 3(c) by showFIG. 2. Force on a free particle moving along y 0 in the configuration shown in the inset. Solid line: Fy 1024 N=m; Dashed line: Fx 1022 N=m. Physical parameters are identical to those of Fig. 1. FIG. 3. Field force on a free particle in the presence of two vertical walls of particles separated by 595 nm and (a) no trapped particle, (b) a single trapped particle, (c) four trapped particles, all clustered around x; y297:5 nm; 0:4 nm. Physical parameters are identical to those of Fig. 1. The tail of the arrows correspond to the location of the center of the particle. PRL 96, 113903 (2006) PHYSICAL REVIEW LETTERS week ending 24 MARCH 2006 113903-3
week ending PRL96,1139 PHYSICAL REVIEW LETTERS 24 MAR tions are those of the author and are not necessarily en- horsed by the United States Government Phys.Rev.Le.24,156(1970) [2]J.P n, Phys. Rev. A 8, 14(1973). 型 [3]L. and E. Lifshitz, The Theory of Classical Fields (Pergamon Press, New York, 1975), VoL. 2, ISBN 0-08- 0l8176-7 [4]A. Ashkin and J M. Dziedzic, Appl. Phys. Lett. 19, 283 1971). [5] A. Ashkin, Phys. Rev. Lett. 40, 729(1978) 6 A. Ashkin, J M. Dziedzic, J.E. Bjorkholm, and s. Chu, Opt.Let11,288(1986) [7] A. Ashkin and J M. Dziedzic, Appl. Phys. Lett. 28, 333 FIG 4. Potential energy at x=2975 nm for y E[150, 150]nm (1976) and at y =0 for x E [100, 500]nm, for zero up to four trapped[81J.M.Fournier,M.M.Burns, and J.AGolovchenko, Proc particles around (xo, yo)=(2975 nm, 0.4 nm). The energy has SPIE Int. Soc. Opt. Eng 2406. 101(1995) en computed as the inverse gradient of the force field distri- [9]G. Roosen, B. Delaunay, and C. Imbert, J. Opt. 8, 181 bution shown in Fig 3. [10] R. Holzel, N. Calander, Z. Chiragwandi, M. wilander, ing the force hield in the (xy) plane for one and four trapped [111 J.M. Fournier, G. Boer, G. Delacretaz, P. Jacquot Rohner, and R. Salathe, Proc. SPIE-Int. Soc. Opt. particles as they present the same effect). In addition, ng.5514,309(2004) Fig.4 indicates that the well is quasiharmonic in both [12] A. Casaburi, G. Pesce, P. Zemanek, and A. Sasso, Opt directions so that the irradiance necessary to maintain a Commun.251,393(2005) given trapping despite the Brownian motion [13] S.A. Tatarkova, A E Carruthers, and K Dholakia, Phys can be estimated from Refs. [24, 25]. If one accepts a Rev.Le.89,283901(2002) standard deviation of 100 nm, which corresponds to five [14] M.M. Burns, J.M. Fournier, and J.A. Golovchenko, particle diameters but still maintains the trapping property Phys. Rev. Lett. 63, 1233(1989) of the configuration, the necessary irradiance is Io- [15] P C. Chaumet and M. Nieto-Vesperinas, Phys. Rev. B 64 0.4 u W/um This irradiance is here obtained for infi nitely long cylinders, i.e., two-dimensional particles, [16] T.M. Grzegorczyk, B A. Kemp, and J. A Kong( to be hich explains the difference with the irradiance typi published) [7 L. Tsang, J. Kong, and C. Ao, Scattering of cally achieved in experimental three-dimensional col Electromagnetic Numerical Simulations(Wiley, gurations New York, 2000) The design of the new trap, based on two sets of verti-[18] J. Venermo and A Sihvola, J Electrost. 63, 101(2005) ally arranged particles, is not restrictive to this geometry: [19] J. Stratton, Electromagnetic Theory (McGraw-Hill, a proper design of dielectric diffractive elements would New York,1941),ISBN0-07-062150-0. achieve similar properties by molding the electromagnetic [20] A.R. Zakharian, M. Mansuripur, and J.V. Moloney, Opt field within a controlled region of space. Multiple traps Express 13, 2321(2005) based on the optical binding force could therefore be [2l] B.A. Kemp, T M. Grzegorczyk, and J.A. Kong, Opt realized, as an extension of the present work. It is a pleasure to acknowledge many stimulating dis- [22] J.A. Ko Electromagnetic Wave Theory (EMW, cussions v ith Professor J -M. Fournier. This work is sp Cambridge, MA, 2005), ISBN 0-9668143-9-8. sored by NASA-USRA under Contracts No. NAS5-03110 [23] M. Nieto-Vesperinas, P C. Chaumet, and A. Rahmani, Phil. Trans. R. Soc. A 362, 719(2004). and No. 07605-003-055, and by the Department of the Air [24] S. Chandrasekhar, Rev. Mod. Phys. 15, 1(1943) Force under Air Force Contract No. FA8721-05-C-0002. [25] L Novotny, R.X. Bian, and X.S. Xie, Phys.Rev.Lett Opinions, interpretations, conclusions, and rece 645(1997) 113903-4
ing the force field in the (xy) plane for one and four trapped particles (data are not shown for two and three trapped particles as they present the same effect). In addition, Fig. 4 indicates that the well is quasiharmonic in both directions so that the irradiance necessary to maintain a given trapping accuracy despite the Brownian motion can be estimated from Refs. [24,25]. If one accepts a standard deviation of 100 nm, which corresponds to five particle diameters but still maintains the trapping property of the configuration, the necessary irradiance is I0 0:4 W=m2. This irradiance is here obtained for infi- nitely long cylinders, i.e., two-dimensional particles, which explains the difference with the irradiance typically achieved in experimental three-dimensional con- figurations. The design of the new trap, based on two sets of vertically arranged particles, is not restrictive to this geometry: a proper design of dielectric diffractive elements would achieve similar properties by molding the electromagnetic field within a controlled region of space. Multiple traps based on the optical binding force could therefore be realized, as an extension of the present work. It is a pleasure to acknowledge many stimulating discussions with Professor J.-M. Fournier. This work is sponsored by NASA-USRA under Contracts No. NAS5-03110 and No. 07605-003-055, and by the Department of the Air Force under Air Force Contract No. FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the author and are not necessarily endorsed by the United States Government. [1] A. Ashkin, Phys. Rev. Lett. 24, 156 (1970). [2] J. P. Gordon, Phys. Rev. A 8, 14 (1973). [3] L. Landau and E. Lifshitz, The Theory of Classical Fields (Pergamon Press, New York, 1975), Vol. 2, ISBN 0-08- 018176-7. [4] A. Ashkin and J. M. Dziedzic, Appl. Phys. Lett. 19, 283 (1971). [5] A. Ashkin, Phys. Rev. Lett. 40, 729 (1978). [6] A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, Opt. Lett. 11, 288 (1986). [7] A. Ashkin and J. M. Dziedzic, Appl. Phys. Lett. 28, 333 (1976). [8] J.-M. Fournier, M. M. Burns, and J. A. Golovchenko, Proc. SPIE Int. Soc. Opt. Eng. 2406, 101 (1995). [9] G. Roosen, B. Delaunay, and C. Imbert, J. Opt. 8, 181 (1977). [10] R. Ho¨lzel, N. Calander, Z. Chiragwandi, M. Willander, and F. F. Bier, Phys. Rev. Lett. 95, 128102 (2005). [11] J.-M. Fournier, G. Boer, G. Delacre´taz, P. Jacquot, J. Rohner, and R. Salathe´, Proc. SPIE-Int. Soc. Opt. Eng. 5514, 309 (2004). [12] A. Casaburi, G. Pesce, P. Zema´nek, and A. Sasso, Opt. Commun. 251, 393 (2005). [13] S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, Phys. Rev. Lett. 89, 283901 (2002). [14] M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, Phys. Rev. Lett. 63, 1233 (1989). [15] P. C. Chaumet and M. Nieto-Vesperinas, Phys. Rev. B 64, 035422 (2001). [16] T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong (to be published). [17] L. Tsang, J. Kong, K. Ding, and C. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, New York, 2000). [18] J. Venermo and A. Sihvola, J. Electrost. 63, 101 (2005). [19] J. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), ISBN 0-07-062150-0. [20] A. R. Zakharian, M. Mansuripur, and J. V. Moloney, Opt. Express 13, 2321 (2005). [21] B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, Opt. Express 13, 9280 (2005). [22] J. A. Kong, Electromagnetic Wave Theory (EMW, Cambridge, MA, 2005), ISBN 0-9668143-9-8. [23] M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, Phil. Trans. R. Soc. A 362, 719 (2004). [24] S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943). [25] L. Novotny, R. X. Bian, and X. S. Xie, Phys. Rev. Lett. 79, 645 (1997). −200 −100 0 100 200 300 400 500 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 Potential energy [× 10−29] y axis [nm] | x axis [nm] 0 1 2 3 4 FIG. 4. Potential energy at x’297:5 nm for y2 150;150 nm and at y ’ 0 for x 2 100; 500 nm, for zero up to four trapped particles around x0; y0297:5 nm; 0:4 nm. The energy has been computed as the inverse gradient of the force field distribution shown in Fig. 3. PRL 96, 113903 (2006) PHYSICAL REVIEW LETTERS week ending 24 MARCH 2006 113903-4