Grzegorczyk et a ol. 23, No 9/September 2006/J. Opt. Soc. Am. 2329 (12b) where a is arbitrarily chosen such that the particles are not moved too much or too little between the first and sec. ond iterations (we typically tolerate a motion within a fraction of the particle radius). At the new set of positions the forces acting on each particle are computed again, and the process is reiterated until the correction terms on the 鵝翻 positions are negligible(typically less than 1% of the cur- rent position). It should be emphasized that the motion of the particles resulting from this iterative process is not trictly derived from the equations of motions of mechan cs but is merely a visual tool to estimate the evolution of the particles as a function of the forces acting on them when strong damping is present as in the case of a water background < The experiment with 20 particles was repeated mul- ole times, each time with a different set of initial posi- tions. An example of initial positions is given in Table 1 and illustrated in Fig. 6(a). Because of the large number of particles in the relatively small constrained surface the particles are often closely packed over single trapping sites and may compete for a site until an equilibrium is reached. The equilibrium is attained either when the par- ticles rearrange themselves to occupy separate traps, as shown in Fig. 6(b)(with the final positions given in Table 1)or when the total field is such that the effect of neigh- Fig. 6. Positions of 20 dielectric cylinders and field distributions boring traps on a conglomerate of particles produces no s cases:(a) random initial position in a three-plane-wave interference pattern (incident force, as shown close to the origin in Fig. 6(d). In both fi- field shown);(b) organized final position due to trapping and nal positions, it should also be noted that the particles do binding forces (incident field shown); (e)same as case(b)but with not always settle in regions of high field intensity, where the total field shown; (d)organized final position corresponding particle [see Fig. 1(a)]. This is a direct manifestation of a=0.15, set of initial positions different from that in case (a) the traps are predicted to be the strongest for a single 6∈a,∈=169 and N=10. The background patterns show the absolute the binding forces: The scattered field by all the particles value of the electric field (either incident or total field )on a scale from 0(black) to 3 V/m(white) Table 1. Initial Positions(xi, yi) and Final Positions(af, ye) of 20 Particles including all their interactions. modifies the incident-field ThreePlane- Wave Interference pattern pattern and thus modifies the location of the traps. The background pattern shown in Fig. 6(b), representing the yi(nm) incident field only, therefore only serves as a reference to 14891 59.30 00.08 120.94 see where the traps should have been if the scattered field -377.76 -38895 -410.69 -533.34 between the particles had been ignored. Obviously, the fi- -380.12 nal positions do not correspond to the traps due to the in- -335.74 7.45 -486.75 cident field, and therefore the gradient force approxi 106.53 396.91 18.50 tion used in the modeling of small particles cannot be 286.33 used in this case. Figure 6(c)shows the same case as Fig 278.84 301.18 6(b)with the background pattern corresponding to the to- 17949 -427.16 tal field instead of the incident field only. Interestingly, it 276.50 36289 219.12 346.56 is seen that the particles come to rest in regions of high 27925 91.55 204.00 111.49 total field intensity, indicating that for such size the gr dient force due to the total field is dominant Finally, we have confirmed that if the dielectric con 70.09 231 trast between the background medium and the particles is very low(typically, e-e=0.1), the traps due to the in 151.11 349.56 198.72 cident field single particle are still located in the high-field regions as in Fig. 1(a), and their strength is sig- 0637 nificantly increased. The immediate consequence is that 22498 art 416.51 -199.41 168.19 2597 within an area of (5a X5a) the particles evolve to almost 397.59 249.79 633.45 exactly occupy the positions of high field intensities due to he incident waves only, similar to what has been experi The corresponding positions are shown in Figs. 6(a)and 6(b). mentally reported in Refs. 7 and 31. This indicates neyl i+1 = yl i + f yl i ,
12b where  is arbitrarily chosen such that the particles are not moved too much or too little between the first and sec￾ond iterations (we typically tolerate a motion within a fraction of the particle radius). At the new set of positions, the forces acting on each particle are computed again, and the process is reiterated until the correction terms on the positions are negligible (typically less than 1% of the cur￾rent position). It should be emphasized that the motion of the particles resulting from this iterative process is not strictly derived from the equations of motions of mechan￾ics but is merely a visual tool to estimate the evolution of the particles as a function of the forces acting on them when strong damping is present as in the case of a water background. The experiment with 20 particles was repeated mul￾tiple times, each time with a different set of initial posi￾tions. An example of initial positions is given in Table 1 and illustrated in Fig. 6(a). Because of the large number of particles in the relatively small constrained surface, the particles are often closely packed over single trapping sites and may compete for a site until an equilibrium is reached. The equilibrium is attained either when the par￾ticles rearrange themselves to occupy separate traps, as shown in Fig. 6(b) (with the final positions given in Table 1) or when the total field is such that the effect of neigh￾boring traps on a conglomerate of particles produces no force, as shown close to the origin in Fig. 6(d). In both fi- nal positions, it should also be noted that the particles do not always settle in regions of high field intensity, where the traps are predicted to be the strongest for a single particle [see Fig. 1(a)]. This is a direct manifestation of the binding forces: The scattered field by all the particles, including all their interactions, modifies the incident-field pattern and thus modifies the location of the traps. The background pattern shown in Fig. 6(b), representing the incident field only, therefore only serves as a reference to see where the traps should have been if the scattered field between the particles had been ignored. Obviously, the fi- nal positions do not correspond to the traps due to the in￾cident field, and therefore the gradient force approxima￾tion used in the modeling of small particles cannot be used in this case. Figure 6(c) shows the same case as Fig. 6(b) with the background pattern corresponding to the to￾tal field instead of the incident field only. Interestingly, it is seen that the particles come to rest in regions of high total field intensity, indicating that for such size the gra￾dient force due to the total field is dominant. Finally, we have confirmed that if the dielectric con￾trast between the background medium and the particles is very low (typically, 
c−
0.1), the traps due to the in￾cident field on a single particle are still located in the high-field regions as in Fig. 1(a), and their strength is sig￾nificantly increased. The immediate consequence is that when a set of 20 such particles are randomly positioned within an area of
5a5a the particles evolve to almost exactly occupy the positions of high field intensities due to the incident waves only, similar to what has been experi￾mentally reported in Refs. 7 and 31. This indicates not Table 1. Initial Positions „xi,yi… and Final Positions „xf,yf… of 20 Particles in a Three-Plane-Wave Interference Patterna xi (nm) yi (nm) xf (nm) yf (nm) −148.91 −59.30 −200.08 −120.94 −377.76 −388.95 −410.69 −533.34 296.79 −380.12 179.12 −356.27 30.74 −335.74 −7.45 −486.75 −106.53 396.91 18.50 454.02 286.33 −85.47 398.75 16.19 278.84 301.18 219.68 374.49 −272.76 179.49 −427.16 245.14 −276.50 362.89 −219.12 346.56 279.25 91.55 204.00 111.49 −287.76 −231.91 −388.75 −230.97 42.83 17.06 −5.79 −2.15 70.09 231.64 −7.01 235.09 −310.92 3.13 −409.60 19.53 −151.11 −349.56 −198.72 −402.18 10.59 −162.72 2.95 −247.57 −101.12 118.56 −206.37 118.08 388.51 −224.98 416.51 −199.41 168.19 −235.86 193.02 −125.97 141.00 397.59 249.79 633.45 a The corresponding positions are shown in Figs. 6
a and 6
b. Fig. 6. Positions of 20 dielectric cylinders and field distributions (shown in the background) for various cases: (a) random initial position in a three-plane-wave interference pattern (incident field shown); (b) organized final position due to trapping and binding forces (incident field shown); (c) same as case (b) but with the total field shown; (d) organized final position corresponding to another set of initial positions different from that in case (a). In all cases, the parameters are =546 nm,
c=256
0,
=169
0, a=0.15 , and N=10. The background patterns show the absolute value of the electric field (either incident or total field) on a scale from 0 (black) to 3 V/m (white). Grzegorczyk et al. Vol. 23, No. 9/September 2006/J. Opt. Soc. Am. A 2329