2328 J. Opt. Soc. Am. A/Vol. 23, No 9/September 2006 of the particle. Such result could be confirmed by an nalysis in the Rayleigh regime but is outside the scope of this paper. The effect of the permittivity of the cylinders on the binding force is illustrated in Fig. 4. As expected the larger the permittivity contrast between the particle and the background, the stronger the binding. This effect is here confirmed far from the Rayleigh regime and can be directly understood from the Born approximation: the scattered field is directly proportional to (Ec-e) and is therefore reduced in a low-permittivity contrast system Finally, we analyze the amplitude of the force exerted n the right-hand particle as a function of the number of particles on the left. The results for one particle are iden- tical to the ones presented in Fig 4 with the correspond- ing parameters, while the cases of two and four particles x [nm are illustrated in the insets of Fig. 5. The computation is Fig. 5. Force in the i direction for varying number of cylinders performed here with N=20 to ensure good convergence of right). The case of one cylinder corresponds to Fig. 2, the cases of two and four cylinders correspond to the situations shown in the linder case. Two interesting conclusions can be drawn insets. The positions of the fixed cylinders are (x,y) from these results. First, the binding force is stronger the cases of two and four particles, respectively. The parameters two particles. This directly justifies the experimental veri- are A=546 nm, =2.5660.6-1.6960,@=0.3A, and N=20 fication that binding phenomena become measurable when multiple particles are scattering together. Second the binding force is almost identical in the three-particle system and the five-particle system. Hence, this result puts a limitation on the previous conclusion, which is re- lated to the incident field: in the configuration studied here, the scattered fields from two and four particles on the left along the x axis are almost identical. This can be directly understood from a generalization of the Mie theory, which indicates that a large particle has a strong forward scattering and low side scattering. The clusters of two and four particles behave in some sense like a single ∧AA八 large particle, having different forward-scattering fields and similar side-scattering fields. Hence, the similarity in the binding force would obviously not hold for ar x-directed incidence, in which case the particle on the right would be in the forward-scattering region of the clusters and would be affected differently by their scat tered fields Fig 3. Force x direction for the configuration of Fig. 2 as unction of the relative distance between parti eters are A=546 nm, e-2.56Eo, E=1. 69Eo, N=10, and radius a as 5 RANDOM ARRANGEMENT OF 20 ndicated in the labels PARTICLES within the limitation of in-plane incident waves, we re- produce here part of the experimental conclusions of Ref. 7. Toward thi return to the of three plane-wave incidences like those shown in Fig. 1, where all the parameters are as indicated in the caption of the figure. The size of the cylinders is taken to be a=0. 151 since we have seen in Fig. 1(a)that the incident interfer- ence pattern yields good traps. Within a region of space of ApUaNe (5a X 5a), we randomly position 20 particles and compute the force on each due to the incident field and the scat tered field from all the particles in the system. The posi tion of each particle is then adjusted in space by ar amount proportional to the force acting on it. Mathemati cally, if we denote by (x(, yo and vre,fy the position and x [nmI Fig 4. Force in the x direction for the configuration of Fig. 2 as force on particle t at iteration i, respectively, the updated function of the relative distance cles. The positions at iteration(i+1)are obtained from eters are A=546 nm, E=1.69Eo, a= N=10. The relative permittivity of the cylinders is as in the labels (12a)of the particle. Such result could be confirmed by an analysis in the Rayleigh regime but is outside the scope of this paper. The effect of the permittivity of the cylinders on the binding force is illustrated in Fig. 4. As expected, the larger the permittivity contrast between the particle and the background, the stronger the binding. This effect is here confirmed far from the Rayleigh regime and can be directly understood from the Born approximation: the scattered field is directly proportional to

c−
and is therefore reduced in a low-permittivity contrast system. Finally, we analyze the amplitude of the force exerted on the right-hand particle as a function of the number of particles on the left. The results for one particle are iden￾tical to the ones presented in Fig. 4 with the correspond￾ing parameters, while the cases of two and four particles are illustrated in the insets of Fig. 5. The computation is performed here with N=20 to ensure good convergence of the fields even for closely packed cylinders as in the four￾cylinder case. Two interesting conclusions can be drawn from these results. First, the binding force is stronger within the system of three particles than in the system of two particles. This directly justifies the experimental veri- fication that binding phenomena become measurable when multiple particles are scattering together.7 Second, the binding force is almost identical in the three-particle system and the five-particle system. Hence, this result puts a limitation on the previous conclusion, which is re￾lated to the incident field: in the configuration studied here, the scattered fields from two and four particles on the left along the xˆ axis are almost identical. This can be directly understood from a generalization of the Mie theory, which indicates that a large particle has a strong forward scattering and low side scattering. The clusters of two and four particles behave in some sense like a single large particle, having different forward-scattering fields and similar side-scattering fields. Hence, the similarity in the binding force would obviously not hold for an xˆ-directed incidence, in which case the particle on the right would be in the forward-scattering region of the clusters and would be affected differently by their scat￾tered fields. 5. RANDOM ARRANGEMENT OF 20 PARTICLES Within the limitation of in-plane incident waves, we re￾produce here part of the experimental conclusions of Ref. 7. Toward this purpose, we return to the case of three plane-wave incidences like those shown in Fig. 1, where all the parameters are as indicated in the caption of the figure. The size of the cylinders is taken to be a=0.15 since we have seen in Fig. 1(a) that the incident interfer￾ence pattern yields good traps. Within a region of space of
5a5a, we randomly position 20 particles and compute the force on each due to the incident field and the scat￾tered field from all the particles in the system. The posi￾tion of each particle is then adjusted in space by an amount proportional to the force acting on it. Mathemati￾cally, if we denote by
x
i ,y
i and
f x
i ,f y
i the position and force on particle
at iteration i, respectively, the updated positions at iteration
i+1 are obtained from xl i+1 = xl i + f xl i ,
12a Fig. 3. Force in the xˆ direction for the configuration of Fig. 2 as a function of the relative distance between particles. The param￾eters are =546 nm,
c=2.56
0,
=1.69
0, N=10, and radius a as indicated in the labels. Fig. 4. Force in the xˆ direction for the configuration of Fig. 2 as a function of the relative distance between particles. The param￾eters are =546 nm,
=1.69
0, a=0.3 , and N=10. The relative permittivity of the cylinders is as indicated in the labels. Fig. 5. Force in the xˆ direction for varying number of cylinders on the left (the force and the positions are for the particle on the right). The case of one cylinder corresponds to Fig. 2, the cases of two and four cylinders correspond to the situations shown in the insets. The positions of the fixed cylinders are
x,y =
0, ±170 nm and
x,y=
−170 nm, ±170 nm,
0, ±170 nm in the cases of two and four particles, respectively. The parameters are =546 nm,
c=2.56
0,
=1.69
0, a=0.3 , and N=20. 2328 J. Opt. Soc. Am. A/Vol. 23, No. 9/September 2006 Grzegorczyk et al.