micro-particles. For example, the restoring force acting on a small glass bead (d= 460nm, n=1.5, 20= 532nm)trapped in the air was found to be roughly 10-20% stronger when the olarization was aligned with, rather than perpendicular to, the particle offset direction. This and related predictions(described in Section 3)now await experimental verification to confirm the validity of the assumptions underlying our theoretical model The radiation force on a small glass bead (d=460nm, n= 1.5, 10=532nm) immerse n water was evaluated using three different models for the partitioning of the radiation force between the solid particle and its liquid environment Method 1, which lumps together the in- duced bound charges on the solid and liquid sides of the interface, was ruled out on account of its unsubstantiated prediction of a strong polarization-dependence of the trap behavior. While methods /I and lll both predict the trapping of the bead under x-and y-polarized beams, the ra- tio of the maximum lateral restoring force for polarization directions parallel and perpendicular to the particle offset direction was found to be 0.92 for method lI and 1.2 for method lll. The dependence of the trap stiffness anisotropy on the physical model underlying the computation is thus seen to be large enough to enable one to accept or reject specific models. In the case of trapping polystyrene beads(n=1.57)in water under a 1o=1064nm focused beam, comparison with available experimental data strongly suggests that Method ll is the correct method of com- puting the force of radiation on objects immersed in a liquid environment (Method lll yielded stiffness anisotropies of.07, -0.004, -0.1, -0.25, -0.34, -0.06, -0.007 for particle diameters d 532nm, 690nm, 850nm, 1030nm, 1280nm, 1500nm, 1660nm, respectively. For small particles (d less than or equal to io), we found excellent agreement between Method II and experiment see Fig. 12. The disagreement between theory and experiment for larger particles may indi- cate the need for more accurate measurements in this regime. Alternatively, the deviation of the observed stiffness anisotropy of larger particles from model calculations may be a hint that the radiation forces acting on the surrounding liquid and/or the attendant hydrodynamic effects should not be ignored in such calculations Appendix Equivalence of total force(and torque)for two formulations of the Lorentz law The Lorentz law of force, F=g(E+VxB), may be written in two different ways for a medium in which the macroscopic version of Maxwells equations are satisfied. The two formulations F1(r)=-(VP)E+(OP/1)×B (A1) F2(r)=(PV)E+(P/d)×B (A2) In an isotropic, homogeneous medium of dielectric constant E where the electric displacement field D(r)=EoE(r)+P(r)=EDEE(r), one may write P(r)=Eo(E-DE(r). In the absence of free charges and free currents in such a medium, Maxwell's first equation,V D(r)=0, implies that the volume density of bound charges within the medium is zero, that is, Pb=-V.P(r)=0 This leaves for the Lorentz force density in the bulk of the medium, according to Eq (Al), only the magnetic contribution, namely, Fi(r)=(aP/ar)x B. Equation(A2 ), however, leads to an entirely different result. Using Maxwell's equations in conjunction with the Lorentz law as expressed in Eq (A2) yield F2(r)=780(E-1)V(E2+1EF2+EP2) (A3) In other words, the lorentz force density in the bulk of a homogeneous medium according to the second formulation is proportional to the gradient of the E-field intensity, irrespective of the state of polarization of the optical field #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 3672micro-particles. For example, the restoring force acting on a small glass bead (d = 460nm, n = 1.5, λ0 = 532nm) trapped in the air was found to be roughly 10-20% stronger when the polarization was aligned with, rather than perpendicular to, the particle offset direction. This and related predictions (described in Section 3) now await experimental verification to confirm the validity of the assumptions underlying our theoretical model. The radiation force on a small glass bead (d = 460nm, n = 1.5, λ 0 = 532nm) immersed in water was evaluated using three different models for the partitioning of the radiation force between the solid particle and its liquid environment. Method I, which lumps together the in￾duced bound charges on the solid and liquid sides of the interface, was ruled out on account of its unsubstantiated prediction of a strong polarization-dependence of the trap behavior. While methods II and III both predict the trapping of the bead under x- and y-polarized beams, the ra￾tio of the maximum lateral restoring force for polarization directions parallel and perpendicular to the particle offset direction was found to be 0.92 for method II and 1.2 for method III. The dependence of the trap stiffness anisotropy on the physical model underlying the computation is thus seen to be large enough to enable one to accept or reject specific models. In the case of trapping polystyrene beads (n = 1.57) in water under a λ 0 = 1064nm focused beam, comparison with available experimental data strongly suggests that Method II is the correct method of com￾puting the force of radiation on objects immersed in a liquid environment. (Method III yielded stiffness anisotropies of 0.07, -0.004, -0.1, -0.25, -0.34, -0.06, -0.007 for particle diameters d = 532nm, 690nm, 850nm, 1030nm, 1280nm, 1500nm, 1660nm, respectively.) For small particles (d less than or equal to λ0), we found excellent agreement between Method II and experiment; see Fig. 12. The disagreement between theory and experiment for larger particles may indi￾cate the need for more accurate measurements in this regime. Alternatively, the deviation of the observed stiffness anisotropy of larger particles from model calculations may be a hint that the radiation forces acting on the surrounding liquid and/or the attendant hydrodynamic effects should not be ignored in such calculations. Appendix. Equivalence of total force (and torque) for two formulations of the Lorentz law The Lorentz law of force, F = q(E+V×B), may be written in two different ways for a medium in which the macroscopic version of Maxwell’s equations are satisfied. The two formulations are: F1(r) = −(∇·P)E+ (∂P/∂t)×B (A1) F2(r)=(P· ∇)E+ (∂P/∂t)×B (A2) In an isotropic, homogeneous medium of dielectric constant ε where the electric displacement field D(r) = ε0E(r) +P(r) = ε0εE(r), one may write P(r) = ε0(ε −1)E(r). In the absence of free charges and free currents in such a medium, Maxwell’s first equation, ∇·D(r) = 0, implies that the volume density of bound charges within the medium is zero, that is, ρ b = −∇·P(r) = 0. This leaves for the Lorentz force density in the bulk of the medium, according to Eq.(A1), only the magnetic contribution, namely, F1(r)=(∂P/∂t) × B. Equation (A2), however, leads to an entirely different result. Using Maxwell’s equations in conjunction with the Lorentz law as expressed in Eq.(A2) yields: F2(r) = 1 4 ε0(ε −1)∇(|Ex| 2 +|Ey| 2 +|Ez| 2). (A3) In other words, the Lorentz force density in the bulk of a homogeneous medium according to the second formulation is proportional to the gradient of the E-field intensity, irrespective of the state of polarization of the optical field. #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 3672