2324 J. Opt. Soc. Am. A/Vol. 23, No 9/September 2006 Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field Tomasz M. Grzegorczyk, Brandon A Kemp, and Jin Au Kong Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received January 9, 2006; revised March 1, 2006; accepted March 16, 2006: posted April 6, 2006(Doc. ID 66668) The Mie theory and the Foldy-Lax multiple-scattering equations are applied to compute the scattered field of ane incidences. The Max. well stress tensor is then used to compute the force on each cylinder. Trapping and binding forces are studied as a fund of particle size, number, ittivity, and separation. Finally, the formulation is applied to a system of 20 particles, and the results show clear similarities with known experimental reports. The formu lation presented here extends the capabilities of modeling particle interaction and optical matter beyond the imple cases of the Rayleigh regime and two-particle systems. o 2006 Optical Society of America oCIS codes:290.4210,2602 20.7010 1 INTRODUCTION cal considerations have been limited to considering only The manipulation of small dielectric particles, typicall special cases where approximations could be used. Typi dielectric spheres, by electromagnetic waves wa first cally, a widely studied situation involves a single particle demonstrated in Ref. 1 and was immediately followed by in a Rayleigh regime: it has been shown that the force can important experimental observations such as optica be split into the gradient force and scattering force, 3-16 levitation and single-beam trapping. In such experi- respectively referring to the force due to the gradient of ments, the exerting force on the particles is due to the the field intensity and to the force due to the momentum wave to the particles, as was already predicted by James model approximates the scattered field of the particle by a C Maxwell. Levitation and trapping are associated with dipole radiation and therefore has to be used with caution two different physical origins: Levitation is due to the for larger particles or when multiple particle are consI scattering force of the impinging electromagnetic wave, ered in a close proximity. The scattering from large nd trapping is due to the gradient of the electric field in- spherical particles can be directly obtained from the Mie tensity acting on small particles. These two new effect theory; some researchers have used different approaches needed to be displaced without damage. 4 Optical binding study particles in the presence of a ground planel or th found applications in systems where minute particles such as the discrete dipole approximation(DDA) yas later reported,b as a third manifestation of the opti- multiple-multipole method to study cylindrical circular cal forces generated within a system of particles submit- and elliptical objects. However, the computation of the ted to an electromagnetic excitation. Optical binding, force on a system of multiple particles has not received which can become a dominant effect in the near field of much attention yet, despite some work on a particle in a the particles and can dictate their motion, can be under- complex environment such as a rough surface- and on stood as a secondary trapping effect, due not to the sole a system of two particles still in the Rayleigh regime22or incident field but to the scattered field from all the par- using the dda. The advantage of the approach used ticles in the system. Both trapping and binding have been Ref. 23 is the inherent advantage of DDA, namely, the recently experimentally verified on a collection of spheri- flexibility of studying particles of arbitrary shapes and cal polystyrene beads submitted to two different types of permittivities, but the disadvantages of the DDa are that incidence,, and a new trapping regime based on binding large particles require important computer resources forces has been reported in Ref 9 This is particularly true when the fields need to be com- The theoretical understanding of such puted in the close proximity of the particle boundary quires the computation of the force via either the Maxwell here a large number of dipoles have to be used in order stress tensor, or the lorentz force on bound currents to reduce the effects of surface discretization and charges(Ref. 12 and references herein). In both cases In the present paper, we present an exact theoretical e methods require the knowledge of the total electric model based on Mie theory to compute the force on an ar- and magnetic field around or within the particles due to bitrary number of dielectric particles under plane-wave the incident wave and the scattered waves from all the incidences. For the purpose of simplicity, we limit our- particles in the system. With an increasing number of selves to infinite dielectric cylinders and in-plane inci randomly positioned particles, the computation of the to- dences (i.e, with all the wave vectors perpendicular to the tal field quickly becomes nontrivial so that first theoreti- axes of the cylinders)with the electric field parallel to the 7/$15.00 o 2006 Optical Society of AmericaTrapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field Tomasz M. Grzegorczyk, Brandon A. Kemp, and Jin Au Kong Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received January 9, 2006; revised March 1, 2006; accepted March 16, 2006; posted April 6, 2006 (Doc. ID 66668) The Mie theory and the Foldy–Lax multiple-scattering equations are applied to compute the scattered field of an arbitrary number of infinite dielectric cylinders of arbitrary size, subject to in-plane incidences. The Max￾well stress tensor is then used to compute the force on each cylinder. Trapping and binding forces are studied as a function of particle size, number, permittivity, and separation. Finally, the formulation is applied to a system of 20 particles, and the results show clear similarities with known experimental reports. The formu￾lation presented here extends the capabilities of modeling particle interaction and optical matter beyond the simple cases of the Rayleigh regime and two-particle systems. © 2006 Optical Society of America OCIS codes: 290.4210, 260.2110, 020.7010. 1. INTRODUCTION The manipulation of small dielectric particles, typically dielectric spheres, by electromagnetic waves was first demonstrated in Ref. 1 and was immediately followed by important experimental observations such as optical levitation2 and single-beam trapping.3 In such experi￾ments, the exerting force on the particles is due to the transfer of momentum from the incident electromagnetic wave to the particles, as was already predicted by James C. Maxwell. Levitation and trapping are associated with two different physical origins: Levitation is due to the scattering force of the impinging electromagnetic wave, and trapping is due to the gradient of the electric field in￾tensity acting on small particles. These two new effects found applications in systems where minute particles needed to be displaced without damage.4 Optical binding was later reported5,6 as a third manifestation of the opti￾cal forces generated within a system of particles submit￾ted to an electromagnetic excitation. Optical binding, which can become a dominant effect in the near field of the particles and can dictate their motion, can be under￾stood as a secondary trapping effect, due not to the sole incident field but to the scattered field from all the par￾ticles in the system. Both trapping and binding have been recently experimentally verified on a collection of spheri￾cal polystyrene beads submitted to two different types of incidence,7,8 and a new trapping regime based on binding forces has been reported in Ref. 9. The theoretical understanding of such experiments re￾quires the computation of the force via either the Maxwell stress tensor10,11 or the Lorentz force on bound currents and charges (Ref. 12 and references herein). In both cases, the methods require the knowledge of the total electric and magnetic field around or within the particles due to the incident wave and the scattered waves from all the particles in the system. With an increasing number of randomly positioned particles, the computation of the to￾tal field quickly becomes nontrivial so that first theoreti￾cal considerations have been limited to considering only special cases where approximations could be used. Typi￾cally, a widely studied situation involves a single particle in a Rayleigh regime: it has been shown that the force can be split into the gradient force and scattering force,13–16 respectively referring to the force due to the gradient of the field intensity and to the force due to the momentum of the electromagnetic wave. However, the Rayleigh model approximates the scattered field of the particle by a dipole radiation and therefore has to be used with caution for larger particles or when multiple particles are consid￾ered in a close proximity. The scattering from large spherical particles can be directly obtained from the Mie theory; some researchers have used different approaches such as the discrete dipole approximation17 (DDA) to study particles in the presence of a ground plane18 or the multiple-multipole method to study cylindrical circular and elliptical objects.19 However, the computation of the force on a system of multiple particles has not received much attention yet, despite some work on a particle in a complex environment such as a rough surface20,21 and on a system of two particles still in the Rayleigh regime22 or using the DDA.23 The advantage of the approach used in Ref. 23 is the inherent advantage of DDA, namely, the flexibility of studying particles of arbitrary shapes and permittivities, but the disadvantages of the DDA are that large particles require important computer resources. This is particularly true when the fields need to be com￾puted in the close proximity of the particle boundary, where a large number of dipoles have to be used in order to reduce the effects of surface discretization. In the present paper, we present an exact theoretical model based on Mie theory to compute the force on an ar￾bitrary number of dielectric particles under plane-wave incidences. For the purpose of simplicity, we limit our￾selves to infinite dielectric cylinders and in-plane inci￾dences (i.e., with all the wave vectors perpendicular to the axes of the cylinders) with the electric field parallel to the 2324 J. Opt. Soc. Am. A/Vol. 23, No. 9/September 2006 Grzegorczyk et al. 1084-7529/06/092324-7/$15.00 © 2006 Optical Society of America