《电动力学》课程参考文献：Single-beam trapping of micro-beads in polarized light_Numerical simulations.pdf_大学文库

Single-beam trapping of micro-beads in polarized light: Numerical simulations A R Zakharian, P. Polynkin, M. Mansuripur, and J.V. Moloney College of Optical Sciences, University of Arizona, Tucson, Arizona 85721 Abstract: Using numerical solutions of Maxwell's equations in conjunc- tion with the Lorentz law of force, we compute the electromagnetic force distribution in and around a dielectric micro-sphere trapped by a focused laser beam. Dependence of the optical trap's stiffness on the polarization state of the incident beam is analyzed for particles suspended in air or immersed in water, under conditions similar to those realized in practical optical tweezers. A comparison of the simulation results with available experimental data reveals the merit of one physical model relative to two competing models, the three models arise from different interpretations of the same physical picture O 2006 Optical Society of America OCIS codes: (2602110)Electromagnetic theory; (1407010)Optical trapping,(0004430)Nu- merical computation References and links L, M. Mansuripur, "Radiation Pressure and the linear momentum of the electromagnetic field, Opt. Express 12, 2064-2074(2005),http://www.opticsexpress.org/abstract.cfm?id=8301/adielectricwedge,"opt.Express13, 2. M. Mansuripur, A oney, "Radiation Pressure or 3. M. Mansuripur, "Radiation Pressure and the linear momentum of light in dispersive dielectric media, Opt Express13.2245-225002005),htp/w 4. M. Mansuripur, "Angular momentum of circu electric media, Opt. Express 13, 5315- 324(2005),http://www.opticsexpress.org/abstract.cfm?id=84895 5. S M. Barnett and R. Loudon. "On the electromagnetic force on a dielectric medium .submitted to J Phys. B At Mol. Phys. (January 2006) 6. A.R. Zakharian, M. Mansuripur and J.V. Moloney, "Radiation Pressure and the distribution of the electromagnetic force in dielectric media," Opt. Express 13, 2321-2336(2005), 7. R Gauthier, "Computation of the optical trapping force using an FDTD based technique, Opt. Express 13 J.App.Phys.91,5474-5488(2002) 9. A Rohrbach, "Stiffness of Optical Traps: Quantitative agreement between experiment and electromagnetic the Phys.Rev.Let.9%5,168102(2005) 10. W.H. Wright, G. Sonek, and M.w. Berns, "Radiation trapping forces on microspheres with optical tweezers Appl.Phys.Lett63,715-717(1993) I1. W.H. Wright, G.J. Sonek, and M.w. Berns, "Parametric study of the forces on microspheres held by optica Opt33,1735-1 12. A. Rohrbach and E. H. K. Stelzer, "Trapping forces, force constants, and potential depths for dielectric spheres rations,”Appl.Opt.41,2494-2507(2002) 13. D. Ganic, X Gan and M. Gu, "Exact radiation trapping force calculation based on vectorial diffraction theory Opt.Express12,2670-2675(2004),http://www.opticsexpress.org/abstract.cfm?id=80240 14. P.w. Barber and S.C. Hill, Light Scattering by Particles: Computational Methods(World Scientific Publishing o.1990) #67575-$15.00USD Received 15 February 2006, revised 7 April 2006, accepted 10 April (C)2006OSA 17 April 2006/Vol 14, No 8/OPTICS EXPRESS

Single-beam trapping of micro-beads in polarized light: Numerical simulations A.R. Zakharian, P. Polynkin, M. Mansuripur, and J.V. Moloney College of Optical Sciences, University of Arizona, Tucson, Arizona 85721 armis@u.arizona.edu Abstract: Using numerical solutions of Maxwell’s equations in conjunction with the Lorentz law of force, we compute the electromagnetic force distribution in and around a dielectric micro-sphere trapped by a focused laser beam. Dependence of the optical trap’s stiffness on the polarization state of the incident beam is analyzed for particles suspended in air or immersed in water, under conditions similar to those realized in practical optical tweezers. A comparison of the simulation results with available experimental data reveals the merit of one physical model relative to two competing models; the three models arise from different interpretations of the same physical picture. © 2006 Optical Society of America OCIS codes: (260.2110) Electromagnetic theory; (140.7010) Optical trapping; (000.4430) Numerical computation References and links 1. M. Mansuripur, “Radiation Pressure and the linear momentum of the electromagnetic field,” Opt. Express 12, 5375–5401 (2004), http://www.opticsexpress.org/abstract.cfm?id=81636. 2. M. Mansuripur, A.R. Zakharian and J.V. Moloney, “Radiation Pressure on a dielectric wedge,” Opt. Express 13, 2064–2074 (2005), http://www.opticsexpress.org/abstract.cfm?id=83011. 3. M. Mansuripur, “Radiation Pressure and the linear momentum of light in dispersive dielectric media,” Opt. Express 13, 2245–2250 (2005), http://www.opticsexpress.org/abstract.cfm?id=83032. 4. M. Mansuripur, “Angular momentum of circularly polarized light in dielectric media,” Opt. Express 13, 5315– 5324 (2005), http://www.opticsexpress.org/abstract.cfm?id=84895. 5. S. M. Barnett and R. Loudon, “On the electromagnetic force on a dielectric medium,” submitted to J. Phys. B: At. Mol. Phys. (January 2006). 6. A.R. Zakharian, M. Mansuripur and J.V. Moloney, “Radiation Pressure and the distribution of the electromagnetic force in dielectric media,” Opt. Express 13, 2321–2336 (2005), http://www.opticsexpress.org/abstract.cfm?id=83272. 7. R. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Opt. Express 13, 3707–3718 (2005), http://www.opticsexpress.org/abstract.cfm?id=83817. 8. A. Rohrbach and E.H.K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474–5488 (2002). 9. A. Rohrbach, “Stiffness of Optical Traps: Quantitative agreement between experiment and electromagnetic theory,” Phys. Rev. Lett. 95, 168102 (2005). 10. W.H. Wright, G.J. Sonek, and M.W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715–717 (1993). 11. W.H. Wright, G.J. Sonek, and M.W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994). 12. A. Rohrbach and E. H. K. Stelzer, “Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,” Appl. Opt. 41, 2494–2507 (2002). 13. D. Ganic, X. Gan and M. Gu, “Exact radiation trapping force calculation based on vectorial diffraction theory,” Opt. Express 12, 2670–2675 (2004), http://www.opticsexpress.org/abstract.cfm?id=80240. 14. P.W. Barber and S.C. Hill, Light Scattering by Particles: Computational Methods (World Scientific Publishing Co. 1990). #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 3660

1. Introduction Computation of the force of radiation on a given object, through evaluation of the electro- magnetic field distribution according to Maxwells equations, followed by a direct application of the Lorentz law of force, has been described in Ref [1]-[4]. In particular, for an isotropic piecewise-homogeneous dielectric medium, the total force was shown to result from the force of the magnetic field acting on the induced bound current density, JbxB=[(1-1/E)VXHXB and from the force exerted by the electric component of the light field on the induced bound charge density at the interfaces between media of differing relative permittivity E. The contri- bution of the E-field component of the Lorentz force is thus specified(Ref [Im) by the force density PbE=(EoV.EE.(Note: As far as the total force and torque of radiation on a solid object are concerned, Barnett and Loudon Ref [5] have recently shown that an altemative for- mulation of the lorentz law -one that is often used in the radiation pressure literature- leads to exactly the same results. We discuss the relation between these two formulations in the Appen dix, and extend the proof of their equivalence to the case of solid objects immersed in a liquid which is a prime concern of the present paper. In a previous publication Ref [6] we described the numerical implementation of the afore- mentioned approach to the computation of the Lorentz force, based on the Finite-Difference Time-Domain(FDTD) solution of Maxwell's equations (An alternative application of the FDTD method to problems of radiation pressure may be found in Ref [7].)Our application of the FDTD method to the computation of the force exerted by a focused laser beam on a spherical dielectric particle immersed in a liquid showed that the forces experienced by the liquid layer immediately at the particle's surface could impact the overall force experienced by the particle. a detailed discussion of the(conceptual) separation of the bound charges on the surface of the particle from the bound charges induced in the surrounding liquid at the solid-liquid interface is given in Ref. [2]. The analysis indicated that the contribution to the Pbe part of the Lorentz force by the component of the E-field perpendicular to the interface can be computed in different ways, depending on the assumptions made concerning the nature of the electromagnetic and hydrodynamic interactions between the solid particle and its liquid In the present study we apply the FDTD method to analyze the polarization dependence of the interaction between a focused laser beam and a small spherical particle trapped either in the air or in a liquid host medium(water). In the latter case, results from different methods of be compared, with the goal of quantifying the effects that might be possible to differentiate in experiments Ref [8]-[9]. The polarization dependence of optical tweezers has been studied in the past, and the dependence of trap stiffness on polarization direction is well documented Ref [10]-[13]. The goal of the present paper is not a re-evaluation of the existing models, but rather a demonstration of the applicability of our own new model Ref [1]-[4 to the problem of trap stiffness anisotropy. Also, upon comparing our numerical results with experiment data, we gain further insight into the nature of the Lorentz force acting on solid objects liquid environments, and identify a preferred interpretation of the proposed physical model In a nutshell, there exists an ambiguity as to the nature of the effective force at the surface of the sphere, with at least three different models in contention. Carefully examining the trapping orce on a dielectric sphere immersed in water provides enough information to establish one of the three competing models and rule out the other two The paper is organized as follows. Section 2 discusses example cases used to validate our ted stiffness of the trap two orthogonal linear polarizations of a beam illuminating a micro-sphere suspended in air and in water, respectively. Summary and conclusions are presented in section 5 #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 3661

1. Introduction Computation of the force of radiation on a given object, through evaluation of the electromagnetic field distribution according to Maxwell’s equations, followed by a direct application of the Lorentz law of force, has been described in Ref. [1]-[4]. In particular, for an isotropic, piecewise-homogeneous dielectric medium, the total force was shown to result from the force of the magnetic field acting on the induced bound current density, J b ×B = [(1−1/ε)∇×H]×B, and from the force exerted by the electric component of the light field on the induced bound charge density at the interfaces between media of differing relative permittivity ε. The contribution of the E-field component of the Lorentz force is thus specified (Ref. [1]) by the force density ρbE = (ε0∇ · E)E. (Note: As far as the total force and torque of radiation on a solid object are concerned, Barnett and Loudon Ref. [5] have recently shown that an alternative formulation of the Lorentz law - one that is often used in the radiation pressure literature - leads to exactly the same results. We discuss the relation between these two formulations in the Appendix, and extend the proof of their equivalence to the case of solid objects immersed in a liquid, which is a prime concern of the present paper.) In a previous publication Ref. [6] we described the numerical implementation of the aforementioned approach to the computation of the Lorentz force, based on the Finite-Difference Time-Domain (FDTD) solution of Maxwell’s equations. (An alternative application of the FDTD method to problems of radiation pressure may be found in Ref. [7].) Our application of the FDTD method to the computation of the force exerted by a focused laser beam on a spherical dielectric particle immersed in a liquid showed that the forces experienced by the liquid layer immediately at the particle’s surface could impact the overall force experienced by the particle. A detailed discussion of the (conceptual) separation of the bound charges on the surface of the particle from the bound charges induced in the surrounding liquid at the solid-liquid interface is given in Ref. [2]. The analysis indicated that the contribution to the ρbE part of the Lorentz force by the component of the E-field perpendicular to the interface can be computed in different ways, depending on the assumptions made concerning the nature of the electromagnetic and hydrodynamic interactions between the solid particle and its liquid environment. In the present study we apply the FDTD method to analyze the polarization dependence of the interaction between a focused laser beam and a small spherical particle trapped either in the air or in a liquid host medium (water). In the latter case, results from different methods of computing the contributions to the net force by bound charges at the solid-liquid interface will be compared, with the goal of quantifying the effects that might be possible to differentiate in experiments Ref. [8]-[9]. The polarization dependence of optical tweezers has been studied in the past, and the dependence of trap stiffness on polarization direction is well documented Ref. [10]-[13]. The goal of the present paper is not a re-evaluation of the existing models, but rather a demonstration of the applicability of our own new model Ref. [1]-[4] to the problem of trap stiffness anisotropy. Also, upon comparing our numerical results with experimental data, we gain further insight into the nature of the Lorentz force acting on solid objects in liquid environments, and identify a preferred interpretation of the proposed physical model. In a nutshell, there exists an ambiguity as to the nature of the effective force at the surface of the sphere, with at least three different models in contention. Carefully examining the trapping force on a dielectric sphere immersed in water provides enough information to establish one of the three competing models and rule out the other two. The paper is organized as follows. Section 2 discusses example cases used to validate our numerical computations. In sections 3 and 4 we compare the computed stiffness of the trap for two orthogonal linear polarizations of a beam illuminating a micro-sphere suspended in air and in water, respectively. Summary and conclusions are presented in section 5. #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 3661

Ey, EZ, HX n surf ,△FDTD,2D,A=5nm △,△FDID,2D,△=5mm FDTD.3D.△=1020nm Fig. 1. Surface force integrated over the left-half of the cylinder as function of the cylinder's refractive index ncy The incident plane-wave has vacuum wavelength 2o=0.65um, and the incidence mediums refractive index is ninc= 1.0 (solid lines)or ninc = 1. 3(dashed lines). The exact results of the Lorentz-Mie theory(circles)are compared with those of our FDTD-based method(triangles and crosses). In the case of cylinder immersed in water, the charges induced at the solid-liquid interface were lumped together, then subjected to the average E-field at the nterface; in other words, no attempts were made in these calculations to distinguish the force of the light's E-field on the solid surface from that on the adjacent liquid. For each FDtD mulation the grid resolution A is indicated 2. Comparison to exact solutions To validate our numerical procedures, we first consider the problem of computing the surface forces exerted by a plane-wave incident on a dielectric cylinder, with propagation direction be- ing perpendicular to the cylinder axis. We consider in the Y Z-plane of incidence a p-polarized plane-wave(Ey, Es, Hr), for which the electric-field component normal to the cylinder surface El, is discontinuous; see Fig. 1. The free-space wavelength of the incident light is 20=0.65um and the radius of the cylindrical rod is rev=1.Oum. The surface forces computed in our FDTD simulations can be compared with those obtained from the Lorentz-Mie theory of light scat- tering, Ref [14. In computing the Lorentz force of the E-field on the interfacial(bound) harges induced on the cylinder surface we used the average(Ref. [2]) of the E-field nor mal to the surface(the tangential component of the E-field is continuous across the interface) the surface-charge density is assumed to be proportional to the discontinuity of the ei field at the interface. While in the Lorentz-Mie theory El at the surface is readily available, on the FDTD grid the cylinder surface is approximated as a discrete staircase, and the surface force density components(Fyur/Eur) are evaluated along the coordinate axes, Ref (6]. Figure I shows, for two different cases, the surface force integrated over one-half of the cylinder sur- face(180°≤6≤360°), as function of the refractive index norl of the cylinder. The agreement between exact theory and numerical simulation is remarkable. Our numerical discretization er ror is of the order of 1-2% for nay ranging from 1.5 to 3. 4, consistent with the 30-60 points per wavelength discretization and first-order convergence due to staircase approximation of the geometry. For the high index-contrast case of n inc/ncw=1/3.4 the error(Fig. 1, A)was found to be dominated by the inadequate convergence to a time-harmonic state; the error was reduced (Fig. 1, v) when we increased the integration time In the second test problem we used the exact solutions for radiation pressure distribution in a solid dielectric prism illuminated by a Gaussian beam of light at Brewster's incidence, Ref[2] #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 3662

ninc ncyl F z surf F y surf Ey,Ez,Hx r cyl Z Y 1.5 2 2.5 3 ncyl -12 -10 -8 -6 -4 -2 [pN/m] Fy surf, theory, ninc=1.0 Fz surf, theory, ninc=1.0 FDTD, 2D, Δ=5nm Fy surf, theory, ninc=1.3 Fz surf, theory, ninc=1.3 FDTD, 2D, Δ=5nm FDTD, 3D, Δ=10-20nm , , Fig. 1. Surface force integrated over the left-half of the cylinder as function of the cylinder’s refractive index ncyl. The incident plane-wave has vacuum wavelength λ0 = 0.65μm, and the incidence medium’s refractive index is ninc = 1.0 (solid lines) or ninc = 1.3 (dashed lines). The exact results of the Lorentz-Mie theory (circles) are compared with those of our FDTD-based method (triangles and crosses). In the case of cylinder immersed in water, the charges induced at the solid-liquid interface were lumped together, then subjected to the average E-field at the interface; in other words, no attempts were made in these calculations to distinguish the force of the light’s E-field on the solid surface from that on the adjacent liquid. For each FDTD simulation the grid resolution Δ is indicated. 2. Comparison to exact solutions To validate our numerical procedures, we first consider the problem of computing the surface forces exerted by a plane-wave incident on a dielectric cylinder, with propagation direction being perpendicular to the cylinder axis. We consider in the Y Z-plane of incidence a p-polarized plane-wave (Ey,Ez,Hx), for which the electric-field component normal to the cylinder surface, E⊥, is discontinuous; see Fig. 1. The free-space wavelength of the incident light is λ 0 = 0.65μm and the radius of the cylindrical rod is rcyl = 1.0μm. The surface forces computed in our FDTD simulations can be compared with those obtained from the Lorentz-Mie theory of light scattering, Ref. [14]. In computing the Lorentz force of the E-field on the interfacial (bound) charges induced on the cylinder surface we used the average (Ref. [2]) of the E-field normal to the surface (the tangential component of the E-field is continuous across the interface); the surface-charge density is assumed to be proportional to the discontinuity of the E ⊥ field at the interface. While in the Lorentz-Mie theory E⊥ at the surface is readily available, on the FDTD grid the cylinder surface is approximated as a discrete staircase, and the surface force density components (Fsur f y ,Fsur f z ) are evaluated along the coordinate axes, Ref. [6]. Figure 1 shows, for two different cases, the surface force integrated over one-half of the cylinder surface (180◦ ≤ θ ≤ 360◦), as function of the refractive index ncyl of the cylinder. The agreement between exact theory and numerical simulation is remarkable. Our numerical discretization error is of the order of 1-2% for ncyl ranging from 1.5 to 3.4, consistent with the 30-60 points per wavelength discretization and first-order convergence due to staircase approximation of the geometry. For the high index-contrast case of ninc/ncyl = 1/3.4 the error (Fig. 1, ) was found to be dominated by the inadequate convergence to a time-harmonic state; the error was reduced (Fig. 1, ) when we increased the integration time. In the second test problem we used the exact solutions for radiation pressure distribution in a solid dielectric prism illuminated by a Gaussian beam of light at Brewster’s incidence, Ref. [2]. #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 3662

Table 1. Definitions of the surface charge and surface force densities for methods l, II and lll. Method 0(E1-E21)oE81-E21)(Ex⊥-E21) E E 2(E1+E21)/2o2(E1+E21)/22(Eg1+E2L) When the prism is immersed in water, the interfacial charges induced on the solid and liquid ides of the interface must be distinguished from each other. The interaction of these charged layers with the local E-field determines their contribution to the net force acting on the prism Following the notation introduced in Ref [2] we enumerate in Table I the three approaches to the computation of the surface force density acting on the solid side of the solid-liquid interface Here(Fur/ fsr are the surface force density components parallel and perpendicular to the solid's surface; O2 is the surface charge density belonging to the solid side, El, E11, E21 are, respectively, the E-field components parallel to the interface, normal to the interface on the iquid side, and normal to the interface on the solid side; and EoEgl=EoE Ell=EoE2E21 is the electric displacement field Di normal to the interface Method Ill isolates the force acting on the solid side of the interface by introducing a small (artificial) gap between the solid and the liquid, then using the gap field Egl(derived from the continuity of the normal component of the displacement field D)to evaluate both the charge density o2 and the average E-field that acts on the solid; this corresponds to Eq (21)in Ref. [2]. While the conceptual introduction of a small gap at the solid-liquid interface is essential for the calculation of o2, it gives rise to a mutual attractive force between the two charged layers thus pushed apart. This force, which is included in the total force experienced by the solid object as calculated by Method lll,is probably inactive in practice and should be excluded. In Method II the contribution of this attractive force between the two charged layers (induced on the solid and liquid sides of the interface)is removed by using(E11+E21)/2 as the effective perpendicular field acting on the surface charge density o2 on the solid side of the interface; see Eq (24)in Ref [2]. In Method n ninc=1.33 F(△=20m)1663483×10-22383095 64 224231.08 Fy( exact)‖16664.05×10-224531 F(△=20mm)-0.0473.13×10-002016 +b=90° Table 2. Radiation force on the prism of Fig. 2, computed with Fig. 2. Dielectric prism the exact method and with the FDTD simulations. A p-polarized (Hr, Ey, E-)Gaussian beam illuminates the prism at Brewster's angle 0B. The net radiation force exerted on the prism is denoted medium of refractive by(Fy, F=-). The value of the mesh parameter A used in each #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 3663

Table 1. Definitions of the surface charge and surface force densities for methods I,II and III. Method I II III σ2 ε0(E1⊥ −E2⊥) ε0(Eg⊥ −E2⊥) ε0(Eg⊥ −E2⊥) Fsur f σ2E σ2E σ2E Fsur f ⊥ σ2(E1⊥ +E2⊥)/2 σ2(E1⊥ +E2⊥)/2 σ2(Eg⊥ +E2⊥)/2 When the prism is immersed in water, the interfacial charges induced on the solid and liquid sides of the interface must be distinguished from each other. The interaction of these charged layers with the local E-field determines their contribution to the net force acting on the prism. Following the notation introduced in Ref. [2] we enumerate in Table 1 the three approaches to the computation of the surface force density acting on the solid side of the solid-liquid interface. Here (Fsur f ,Fsur f ⊥ ) are the surface force density components parallel and perpendicular to the solid’s surface; σ2 is the surface charge density belonging to the solid side; E,E1⊥,E2⊥ are, respectively, the E-field components parallel to the interface, normal to the interface on the liquid side, and normal to the interface on the solid side; and ε 0Eg⊥ = ε0ε1E1⊥ = ε0ε2E2⊥ is the electric displacement field D⊥ normal to the interface. Method III isolates the force acting on the solid side of the interface by introducing a small (artificial) gap between the solid and the liquid, then using the gap field Eg⊥ (derived from the continuity of the normal component of the displacement field D) to evaluate both the charge density σ 2 and the average E-field that acts on the solid; this corresponds to Eq.(21) in Ref. [2]. While the conceptual introduction of a small gap at the solid-liquid interface is essential for the calculation of σ2, it gives rise to a mutual attractive force between the two charged layers thus pushed apart. This force, which is included in the total force experienced by the solid object as calculated by Method III, is probably inactive in practice and should be excluded. In Method II the contribution of this attractive force between the two charged layers (induced on the solid and liquid sides of the interface) is removed by using (E1⊥ +E2⊥)/2 as the effective perpendicular field acting on the surface charge density σ2 on the solid side of the interface; see Eq.(24) in Ref. [2]. In Method ninc n y z Gaussian beam F y Fz B B B B Fig. 2. Dielectric prism of refractive index n immersed in a host medium of refractive index ninc. Net force ninc = 1 ninc = 1.33 [pN/m] I II III Fy (Δ = 20nm) 16.63 4.83×10−3 22.38 30.95 Fy (Δ = 10nm) 16.64 —— 22.42 31.08 Fy (exact) 16.66 4.05×10−3 22.45 31.09 Fz (Δ = 20nm) -0.047 3.13×10−2 -0.02 -0.16 Fz (Δ = 10nm) -0.026 —— -0.016 -0.001 Fz (exact) 0.0 0.0 0.0 0.0 Table 2. Radiation force on the prism of Fig. 2, computed with the exact method and with the FDTD simulations. A p-polarized (Hx,Ey,Ez) Gaussian beam illuminates the prism at Brewster’s angle θB. The net radiation force exerted on the prism is denoted by (Fy,Fz). The value of the mesh parameter Δ used in each simulation is indicated. #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 3663

I the normal components of the E-field on the liquid and solid sides of the interface are used to compute the normal component of the average E-field as well as the induced charge density at the interface. In other words, the charges on the solid and liquid sides of the interface are lumped together, then subjected to the forces of the tangential E-field, which is continuous at the interface, and the perpendicular E-field, which is discontinuous at the interface(and therefore, in need of averaging) Method 1, which corresponds to Eq (27)in Ref [2], is also the method used in the case of a cylindrical rod immersed in water discussed earlier in conjunction We use a Gaussian beam(2o=0.65um)having a full-width of 7 8um at the 1/e point of the field amplitude, incident on the prism at Brewster's angle 0B =arctan(n/ninc); see Fig. 2. In the case of incidence from the free-space, n inc 1.0, the prism has refractive index n= 1.5, while for nine=1.33(water)the prism index is n=1995. The incident field amplitude's peak value is Eo=10/ninc V/m. Table 2 compares the total radiation force(consisting of the surface force density integrated over both surfaces, and the volume force density integrated over the prism volume)exerted on the prism, obtained with the exact method and with the FDTD simulations As shown in Fig. 2, the total force has a component Fy directed along the bisector of the prisms apex, and a second component F: perpendicular to this bisector in the plane of incidence. (The exact value of Fi is zero in all cases. For a prism immersed in water(ninc=1.33)the results obtained with the aforementioned methods 1, 11, and Ill of force computation are tabulated The numerical results are generally in good agreement with the exact solutions, the largest disagreements occur in the case of Method I used in conjunction with the immersed prism, where(due to the specific set of parameters chosen for the simulation) the exact value of F happens to be so small (less than 10-pN/ m) that it falls below the numerical accuracy of our simulations. In the computations with method lll the separation of the charges on the liquid and solid sides of the interface was simulated using an actual air-gap(width=2A). The numerical and exact solutions in general are in good agreement with less than 1% error in the net force magnitude when△≤20nm 3. Single-beam optical trap in the air We present computed results of the electromagnetic force exerted on a dielectric micro-sphere at and near the focus of a laser beam in the free space(n inc= 1.0). The incident beam, obtained by focusing a 20=532nm(or 1064nm) plane-wave through a 0.9NA, 5mm focal-length objective lens, propagates in the negative =-direction, as shown in Fig. 3. The plane-wave entering the abjective's pupil is linearly polarized along either the x-axis or the y-axis. The total power of the incident beam is P=S_dxdy= 1.oW. Figure 4 shows the Poynting vector distribution (S-field) in the XZ-plane for a micro-sphere of refractive index n=1.5 and diameter d 460nm, offset by(x,y, =)-offset=(250, 0, 50)nm from the focal point of the y-polarized incident beam. Positive(negative)values of the z-offset represent the particle being displaced into the converging(diverging) half-space of the focused beam. Upon scattering from the micro-sphere the large positive momentum acquired by the incident light along the x-axis, seen in Fig. 4, results in a net Fr force directed towards the beam axis. Figure 5 shows the computed(Fr, F) force components experienced by the micro-sphere as funct ere cente from the focal point of the lens ( -offset=0). The top( bottom) row shows the case of an x- polarized (y-polarized)incident beam. Due to symmetry, y-offset is set to zero, and offsets along the positive half of the x-axis are the only ones considered. The y component of the force was found to be zero(within the numerical accuracy of the simulations) for these linearly-polarized beams. Figure 5 indicates that the particle is trapped laterally, but not vertically along the z axis, since F- <0 for the entire range of offsets shown. (Note: A 460nm-diameter glass bead weighs approximately 1.0fN. If the focused laser beam shines on the bead from below, and #67575-$15.00USD Received 15 February 2006, revised 7 April 2006, accepted 10 April (C)2006OSA 17 April 2006/Vol 14, No 8/OPTICS EXPRESS

I the normal components of the E-field on the liquid and solid sides of the interface are used to compute the normal component of the average E-field as well as the induced charge density at the interface. In other words, the charges on the solid and liquid sides of the interface are lumped together, then subjected to the forces of the tangential E-field, which is continuous at the interface, and the perpendicular E-field, which is discontinuous at the interface (and, therefore, in need of averaging). Method I, which corresponds to Eq.(27) in Ref. [2], is also the method used in the case of a cylindrical rod immersed in water discussed earlier in conjunction with Fig. 1. We use a Gaussian beam (λ0 = 0.65μm) having a full-width of 7.8μm at the 1/e point of the field amplitude, incident on the prism at Brewster’s angle θ B = arctan(n/ninc); see Fig. 2. In the case of incidence from the free-space, ninc = 1.0, the prism has refractive index n = 1.5, while for ninc = 1.33 (water) the prism index is n = 1.995. The incident field amplitude’s peak value is E0 = 103/ninc V/m. Table 2 compares the total radiation force (consisting of the surface force density integrated over both surfaces, and the volume force density integrated over the prism volume) exerted on the prism, obtained with the exact method and with the FDTD simulations. As shown in Fig. 2, the total force has a component Fy directed along the bisector of the prism’s apex, and a second component Fz perpendicular to this bisector in the plane of incidence. (The exact value of Fz is zero in all cases.) For a prism immersed in water (ninc = 1.33) the results obtained with the aforementioned methods I, II, and III of force computation are tabulated. The numerical results are generally in good agreement with the exact solutions; the largest disagreements occur in the case of Method I used in conjunction with the immersed prism, where (due to the specific set of parameters chosen for the simulation) the exact value of Fy happens to be so small (less than 10−2 pN/m) that it falls below the numerical accuracy of our simulations. In the computations with method III the separation of the charges on the liquid and solid sides of the interface was simulated using an actual air-gap (width = 2Δ). The numerical and exact solutions in general are in good agreement with less than 1% error in the net force magnitude when Δ ≤ 20nm. 3. Single-beam optical trap in the air We present computed results of the electromagnetic force exerted on a dielectric micro-sphere at and near the focus of a laser beam in the free space (ninc = 1.0). The incident beam, obtained by focusing a λ0 = 532nm (or 1064nm) plane-wave through a 0.9NA, 5mm focal-length objective lens, propagates in the negative z-direction, as shown in Fig. 3. The plane-wave entering the objective’s pupil is linearly polarized along either the x-axis or the y-axis. The total power of the incident beam is P = Szdxdy = 1.0W. Figure 4 shows the Poynting vector distribution (S-field) in the XZ-plane for a micro-sphere of refractive index n = 1.5 and diameter d = 460nm, offset by (x,y,z)-offset=(250,0,50)nm from the focal point of the y-polarized incident beam. Positive (negative) values of the z-offset represent the particle being displaced into the converging (diverging) half-space of the focused beam. Upon scattering from the micro-sphere, the large positive momentum acquired by the incident light along the x-axis, seen in Fig. 4, results in a net Fx force directed towards the beam axis. Figure 5 shows the computed (Fx,Fz) force components experienced by the micro-sphere as functions of the sphere center’s offset from the focal point of the lens (y-offset = 0). The top (bottom) row shows the case of an xpolarized (y-polarized) incident beam. Due to symmetry, y-offset is set to zero, and offsets along the positive half of the x-axis are the only ones considered. The y component of the force was found to be zero (within the numerical accuracy of the simulations) for these linearly-polarized beams. Figure 5 indicates that the particle is trapped laterally, but not vertically along the zaxis, since Fz < 0 for the entire range of offsets shown. (Note: A 460nm-diameter glass bead weighs approximately 1.0 f N. If the focused laser beam shines on the bead from below, and #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 3664

lateral trap anisotropy sl, evaluated at x-offset = 50nm (with z-offset adjusted to yield maximum lateral trapping force Fx), is seen in Fig. 6 to be positive for small particle diameters, negative when the particle size is comparable to the wavelength of the trap beam, and weakly positive for d > 1.4μm. (b) -1.0 x (μm) +1.0 -1.0 x (μm) +1.0 +1.0 y (μm) -1.0 (c) (d) (a) Glass Plate (n = 1.47) Water (n = 1.33) Oil (n = 1.47) x z Objective E Fig. 7. (a) A linearly-polarized Gaussian beam having wavelength λ0=532nm and 1/e (amplitude) radius r0= 4.0mm is focused through a 1.4NA oil immersion objective lens. The lens has focal length f =3.0mm and aperture radius (at the entrance pupil) Ra=2.85mm; the refractive index of the immersion oil is 1.47. A glass plate of the same index as the oil separates the oil from the water (nwater=1.33), where the focused spot is used to trap various dielectric beads. The marginal rays are lost by total internal reflection at the glass-water interface; there is also some degree of apodization due to Fresnel reflection at this interface, but the phase aberrations induced by the transition from oil/glass to water are ignored in our calculations. The neglect of the spherical aberrations thus induced is justifiable, so long as the trap is not too far from the oil/water interface, Ref. [12]. (b)-(d) Logarithmic plots of the intensity distribution for x-, y-, and z-components of the E-field at the focal plane. The relative peak intensities are |Ex| 2:|Ey| 2: |Ez| 2≈ 1000 : 9 : 200. The total intensity distribution at the focal plane, I(x,y) = |Ex| 2 +|Ey| 2 +|Ez| 2, (not shown) is elongated in the x-direction; the full-width at halfmaximum intensity (FWHM) of the focused spot is 300nm along x and 196nm along y. The transmission efficiency of the oil/glass-to-water interface is 92.6%; that is, the overall loss of optical power to total-internal and Fresnel reflections at this interface is 7.4%. #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 3667

4. Single-beam optical trap in water In computing the optical trap properties in a liquid host medium(refractive index n inc=1.33), a collimated laser beam having 2o=532nm(or 1064nm)was focused through an NA N 1.4 immersion objective designed for diffraction-limited focusing within an immersion liquid of refractive index noil=1. 47; see Figs. 7, 8. In our first set of simulations corresponding to no 532nm, the spherical particle has d=460nm, n=1.5, and the total power of the incident beam is P=1. ow. For Methods 1, Il, and Ill of computing the surface-force discussed in section 2 Figs, 9-1I show the net force components(Fx, Fs)exerted on the micro-sphere illuminated by x-or y-polarized light. For most of the offset range considered in Fig. 9, the Fx force component computed with method I for x-polarized light is opposite in direction to the Fr computed for y-polarized light, indicating the impossibility of lateral trapping with x-polarization. Such a marked difference in the behavior of Fr for different polarization states exhibited by method ontradicts the experimental observations which show trapping(with similar strengths)for both polarization states. We conclude, therefore, that Method I is unphysical and must be abandoned The root of the problem with Method I can be traced to the lumping together of the solid and liquid charges at the interface, which weakens the negative contribution of Fx, thus allowing the positive contribution by the magnetic part of the Lorentz force(acting on the micro-sphere volume)to push the particle away from the focal point. In contrast, for y-polarized light, the contribution of the magnetic Lorentz force to Fr is negative, thus enabling lateral trapping I x um] y [um Fig8 Computed distributions of the electric field intensity EP(units: [2/m)in the xs- and yz-planes near the focus of a 2 o=532nm beam in water. The focused spot is obtained by sending an x-polarized plane-wave through an oil-immersion N 1. 4NA objective. The oil water are separated by a thin glass slide, index-matched to the immersion oil, Fig. 7. The force distributions computed with Method /l(Fig. 10)and Method Ill (Fig. I1) show qualitatively similar behavior for the two polarization states, and indicate trapping along botha and z-directions. Fr is strongest at lateral offset values on the order of one micro-sphere radius While both methods /I and ll result in trapping of the micro-bead, the ratio of the maximum lateral restoring forces Fx(sampled inside the dashed rectangles in Figs. 10 and 11)for x-and polarized beams is found to be 0.92 for method ll and 1.2 for method Ill. Thus, although Figs 10 and Il exhibit qualitatively similar behavior, their quantitative estimates of the restoring forces are sufficiently different to enable one to distinguish Method l from Method Ill based #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 3668

4. Single-beam optical trap in water In computing the optical trap properties in a liquid host medium (refractive index n inc = 1.33), a collimated laser beam having λ0 = 532nm (or 1064nm) was focused through an NA ≈ 1.4 immersion objective designed for diffraction-limited focusing within an immersion liquid of refractive index noil = 1.47; see Figs. 7, 8. In our first set of simulations corresponding to λ 0 = 532nm, the spherical particle has d = 460nm, n = 1.5, and the total power of the incident beam is P = 1.0W. For Methods I, II, and III of computing the surface-force discussed in section 2, Figs. 9-11 show the net force components (Fx,Fz) exerted on the micro-sphere illuminated by x- or y-polarized light. For most of the offset range considered in Fig. 9, the Fx force component computed with method I for x-polarized light is opposite in direction to the Fx computed for y-polarized light, indicating the impossibility of lateral trapping with x-polarization. Such a marked difference in the behavior of Fx for different polarization states exhibited by method I, contradicts the experimental observations which show trapping (with similar strengths) for both polarization states. We conclude, therefore, that Method I is unphysical and must be abandoned. [The root of the problem with Method I can be traced to the lumping together of the solid and liquid charges at the interface, which weakens the negative contribution of F sur f x , thus allowing the positive contribution by the magnetic part of the Lorentz force (acting on the micro-sphere volume) to push the particle away from the focal point. In contrast, for y-polarized light, the contribution of the magnetic Lorentz force to Fx is negative, thus enabling lateral trapping.] Fig. 8. Computed distributions of the electric field intensity |E| 2 (units: [V2/m2]) in the xzand yz-planes near the focus of a λ0 = 532nm beam in water. The focused spot is obtained by sending an x-polarized plane-wave through an oil-immersion ≈ 1.4NA objective. The oil and water are separated by a thin glass slide, index-matched to the immersion oil, Fig. 7. The force distributions computed with Method II (Fig. 10) and Method III (Fig. 11) show qualitatively similar behavior for the two polarization states, and indicate trapping along both xand z-directions. Fx is strongest at lateral offset values on the order of one micro-sphere radius. While both methods II and III result in trapping of the micro-bead, the ratio of the maximum lateral restoring forces Fx (sampled inside the dashed rectangles in Figs. 10 and 11) for x- and ypolarized beams is found to be 0.92 for method II and 1.2 for method III. Thus, although Figs. 10 and 11 exhibit qualitatively similar behavior, their quantitative estimates of the restoring forces are sufficiently different to enable one to distinguish Method II from Method III based #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 3668

Fz IpNI (Fx, Fz) E 000050.100150200250300.0010015020025030 Fx IpN] FZ PN (Fx. Fz) 00000.100.150200250300.00.100.150200253005D.10015020025030 offset lrm Fig 9. Plots of the net force components (Fr, F), computed with Method 1, for a glass micro- bead (d= 460nm, n=1.5)versus the offset from the focal point. The host medium is water (ninc = 1.33), 20= 532nm, the objective lens NA is a 1.4, and the incident beams power P=1.0W. Top row: x-polarization, bottom row: y-polarization. The non-trapping behavior of Method used in these calculations on experimentally determined values of stiffness anisotropy Figure 12 shows the dependence of the trap stiffness anisotropy s/=l-(K/Ky)on bead diameter d, computed with Method ll for polystyrene micro-beads(n=1.57) trapped in water (nine 133)under a 1o=1064nm laser beam focused through an oil-immersion 1. 4NA obJective lens(with a glass slide used to separate oil from water, as shown in Fig. 7). For d 0. For larger particles (850 1400nmm uperimposed on Fig. 12 are two sets of experimental data. The green triangles correspond to measurements carried out with a system similar to that depicted in Fig. 7, operating at 20=1064nm. In practice, this system suffered from chromatic(and possibly spherical)aberra tions; defects that are not taken into account in our computer simulations. The agreement with theoretical calculations is good for the d= 1260nm particle, but not so good for d= 1510nm and d=1900nm particles. The second set of experimental data(solid blue circles in Fig. 12) from Table Il of Rohrbach Ref. 9]. These were obtained with a 1. 2NA water immersion objective, corrected for all aberrations and, therefore, presumably operating in the diffrac tion limit. All other experimental parameters were the same as those used in our simulations Considering the loss of marginal rays at the oil/water interface, apodization due to Fresnel reflection losses, and the relatively small beam diameter at the entrance pupil of the simu lated lens, our focused spot should be fairly close to that used in Rohrbach's experiments #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 3669

Fz Fx Fz Fx (Fx,Fz) (Fx,Fz) Fig. 9. Plots of the net force components (Fx,Fz), computed with Method I, for a glass microbead (d = 460nm, n = 1.5) versus the offset from the focal point. The host medium is water (ninc = 1.33), λ0 = 532nm, the objective lens NA is ≈ 1.4, and the incident beam’s power is P = 1.0W. Top row: x-polarization, bottom row: y-polarization. The non-trapping behavior of x-polarized beam, which is contrary to experimental observations, indicates the invalidity of Method I used in these calculations. on experimentally determined values of stiffness anisotropy. Figure 12 shows the dependence of the trap stiffness anisotropy sl = 1 − (κx/κy) on bead 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 (with a glass slide used to separate oil from water, as shown in Fig. 7). For d 0. For larger particles (850 1400nm. Superimposed on Fig. 12 are two sets of experimental data. The green triangles correspond to measurements carried out with a system similar to that depicted in Fig. 7, operating at λ0 = 1064nm. In practice, this system suffered from chromatic (and possibly spherical) aberrations; defects that are not taken into account in our computer simulations. The agreement with theoretical calculations is good for the d = 1260nm particle, but not so good for d = 1510nm and d = 1900nm particles. The second set of experimental data (solid blue circles in Fig. 12) is from Table II of Rohrbach Ref. [9]. These were obtained with a 1.2NA water immersion objective, corrected for all aberrations and, therefore, presumably operating in the diffraction limit. All other experimental parameters were the same as those used in our simulations. (Considering the loss of marginal rays at the oil/water interface, apodization due to Fresnel reflection losses, and the relatively small beam diameter at the entrance pupil of the simulated lens, our focused spot should be fairly close to that used in Rohrbach’s experiments.) #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 3669