o FDTD: Me d ll n=l d [ nm zoom 532 9 690-197 A Our experimental data points 1030 1280-2 -0. 1660 -16 180 -139 phere diameter [um] Fig. 12. Trap stiffness anisotropy s)=l-(Kx/Ky)versus particle diameter d, computed with method 11 for polystyrene micro-beads (n=1.57)trapped in water(ninc 1.33)under a 20= 1064nm laser beam focused through an oil-immersion ae 1. 4NA objective lens. Numer ically, Kr and K were evaluated with a forward finite-difference approximation at x-offset 50-100nm, offset discretization Ax= 50nm, and =-offset= =0 where F(x=0, =0)=0 The computed value of =o for each particle is listed on the right-hand side. The solid triangles (green) represent experimental data obtained with a system similar to that depicted in Fig. 7, erating at 2o= 1064nm, and suffering from chromatic (and possibly spherical) aberrations. The solid circles(blue) represent experimental data from Table ll of Rohrbach Ref. [9]. Although Rohrbach uses a 1. 2N.A water immersion objective in his experiments, the comparison is w ranted here because our simulated 1. 4NA focused beam, upon transmission from oil to water, oses its marginal rays and acquires characteristics that are not too far from those of a 1. 2NA diffraction-limited focused spot. (Our experimental methodology is similar to that of rohrbach, e only difference being that we use the oil-immersion objective depicted in Fig. 7(a), whereas Rohrbach uses a water-immersion objective. The agreement between theory and experiment is remarkable in this case, except for the d=1660nm particle (We repeated the simulation for the d= 166Onm particle using the ex- act NA used in Rohrbach's experiments; the resulting value of s/, however, did not change ver much. )While it is possible that the larger beads used in the aforementioned experiments have been described inaccurately (i.e, n and d deviating from the specifications), and while it is true that the aberrations of our own measurement system need to be understood and properly mod eled, we cannot rule out the possibility that the radiation forces acting on the surrounding liquid (and ignored in the simulations )could have impacted the results of these measurements We mention in passing that, in the "Rayleigh particle"limit(d< 2o/ninc), the Lorentz force tensity(eV·E)E+Jb× B reduces, in the dipole approximation,to(p·V)E+dp/ot×B,as suming linear dependence of the polarization p on the local E-field, namely, p(r, =aDE(r, n) and neglecting the small terms due to the inhomogeneity of the magnetic field. For the smaller particles, the experimental data was found in Ref [9] to be in good agreement with the com- puted results based on the Rayleigh-Gans approximation We have computed the radiation force of no= 532nm and 20= 1064nm linearly polarized laser beams focused through air (NA=0.9)and through water(NA N 1. 4)on various spherical #67575-$15.00USD Received 15 February 2006, revised 7 April 2006, accepted 10 April 2006 (C)2006OSA 17 April 2006/Vol 14, No 8/OPTICS EXPRESS 36710.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 micro-sphere diameter [μm] -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1 - κx /κy FDTD: Method II, n =1.57, ninc = 1.33, λ0 = 1064nm Cubic spline interpolation Rohrbach’s experimental data points Our experimental data points 532 -91 690 -197 850 -99 1030 -96 1160 -98 1280 -21 1400 -120 1500 -166 1660 -16 1800 -257 1900 -139 d [nm] z0 [nm] Fig. 12. Trap stiffness anisotropy sl = 1 − (κx/κy) versus particle diameter d, computed with method II for polystyrene micro-beads (n = 1.57) trapped in water (ninc = 1.33) under a λ0 = 1064nm laser beam focused through an oil-immersion ≈ 1.4NA objective lens. Numer￾ically, κx and κy were evaluated with a forward finite-difference approximation at x-offset = 50 − 100nm, offset discretization Δx = 50nm, and z-offset = z0 where Fz(x = 0,z0) = 0. The computed value of z0 for each particle is listed on the right-hand side. The solid triangles (green) represent experimental data obtained with a system similar to that depicted in Fig. 7, op￾erating at λ0 = 1064nm, and suffering from chromatic (and possibly spherical) aberrations. The solid circles (blue) represent experimental data from Table II of Rohrbach Ref. [9]. Although Rohrbach uses a 1.2NA water immersion objective in his experiments, the comparison is war￾ranted here because our simulated 1.4NA focused beam, upon transmission from oil to water, loses its marginal rays and acquires characteristics that are not too far from those of a 1.2NA diffraction-limited focused spot. (Our experimental methodology is similar to that of Rohrbach, the only difference being that we use the oil-immersion objective depicted in Fig. 7(a), whereas Rohrbach uses a water-immersion objective.) The agreement between theory and experiment is remarkable in this case, except for the d = 1660nm particle. (We repeated the simulation for the d = 1660nm particle using the ex￾act NA used in Rohrbach’s experiments; the resulting value of sl, however, did not change very much.) While it is possible that the larger beads used in the aforementioned experiments have been described inaccurately (i.e., n and d deviating from the specifications), and while it is true that the aberrations of our own measurement system need to be understood and properly mod￾eled, we cannot rule out the possibility that the radiation forces acting on the surrounding liquid (and ignored in the simulations) could have impacted the results of these measurements. We mention in passing that, in the ”Rayleigh particle” limit (d λ0/ninc), the Lorentz force density (ε0∇·E)E+Jb ×B reduces, in the dipole approximation, to (p·∇)E+∂p/∂t ×B, as￾suming linear dependence of the polarization p on the local E-field, namely, p(r,t) = α 0E(r,t), and neglecting the small terms due to the inhomogeneity of the magnetic field. For the smaller particles, the experimental data was found in Ref. [9] to be in good agreement with the com￾puted results based on the Rayleigh-Gans approximation. 5. Concluding remarks We have computed the radiation force of λ0 = 532nm and λ0 = 1064nm linearly polarized laser beams focused through air (NA = 0.9) and through water (NA ≈ 1.4) on various spherical #67575 - $15.00 USD Received 15 February 2006; revised 7 April 2006; accepted 10 April 2006 (C) 2006 OSA 17 April 2006 / Vol. 14, No. 8 / OPTICS EXPRESS 3671