《电动力学》课程参考文献：Radiation pressure and the distribution of electromagnetic force in dielectric media.pdf_大学文库

Radiation pressure and the distribution of electromagnetic force in dielectric media Armis R. Zakharian, Masud Mansuripur, and Jerome v Moloney Optical Sciences Center, The University of Arizona, Tucson, Arizona 85721 Abstract: Using the Finite-Difference-Time-Domain(FDTD)method, we compute the electromagnetic field distribution in and around dielectric media of various shapes and optical properties. With the aid of the constitutive relations, we proceed to compute harge current densities, then employ the Lorentz law of force to determine the distribution of force density within the regions of interest. For a few simple cases where analytical solutions exist, these solutions are found to be in complete agreement with our numerical results. We also analyze the distribution of fields and forces in more complex systems, and discuss the relevance of our findings to experimental observations. In particular, we demonstrate the single-beam trapping of a dielectric micro-sphere immersed in a liquid under conditions that are typical of optical tweezers C 2005 Optical Society of America OCIS codes:(2602110)Electromagnetic theory; (140.7010) Trapping. References L. M. Mansuripur, Radiation pressure and the linear momentum of the electromagnetic field, " Opt. Express 12 5375-5401(2004),http://www.opticsexpressorg/abstract.cfm?uri=opex-12-22-5375 2. D. A. White, "Numerical modeling of optical gradient traps using the vector finite element method, "J Compt. Phys.159,13-37(2000) 3. A. Mazolli, P. A M. Neto, and H M. Nussenzveig, " Theory of trapping forces in optical tweezers, Proc.Roy.Soc.Lond.A459,3021-3041(2003) 4. C. Rockstuhl and H P Herzig, "Rigorous diffraction theory applied to the analysis of the optical force on elliptical nano- and micro-cylinders, " J Op Appl.Op.6,921-31(2004) Ashkin, "Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime, Biophys.J.61,569-582(1992) 6. A. Rohrbach and E. Stelzer, "Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations, "AppL. Opt. 41, 2494(2002) 7. J D. Jackson, Classical Electrodynamics, edition (Wiley, New York, 1975) 8. M. Mansuripur, A. R. Zakharian, and J. V. Moloney, "Radiation pressure on a dielectric wedge, Opt. Express 13,2064-2074(2005),http://www.opticsexpress.orgabstract.cfm?uri=opex-13-6-2064 9. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles, "Opt. Lett. 11, 288-290(1986 10. A. Ashkin and J. M. Dziedzic, "Optical trapping and manipulation of viruses and bacteria, Science 235, 1517- 1 Introduction In a previous paper [1] we showed that the direct application of the Lorentz law of force in conjunction with Maxwells equations can yield a complete picture of the electromagnetic force in metallic as well as dielectric media. In the case of the dielectrics, bound charges and bound currents were found to be responsible, respectively, for the electric and magnetic components of the Lorentz force. When a dielectric medium is homogeneous and isotropic, or can be divided into two or more such regions- each with its own uniform dielectric constant E- the bound charges appear only at the surface(s)and/or the interface(s)between adjacent dielectrics. The bound currents, however, induced by the local E-field in proportion to the time-rate of change of the polarization P=&(E-D)E, are distributed throughout the medium The E-field of the light exerts a force on the induced charge density, while the local H-field #6863·$1500US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005 (C)2005OSA 4 April 2005/VoL 13, No. 7/OPTICS EXPRESS 2321

Radiation pressure and the distribution of electromagnetic force in dielectric media Armis R. Zakharian, Masud Mansuripur, and Jerome V. Moloney Optical Sciences Center, The University of Arizona, Tucson, Arizona 85721 armis@email.arizona.edu Abstract: Using the Finite-Difference-Time-Domain (FDTD) method, we compute the electromagnetic field distribution in and around dielectric media of various shapes and optical properties. With the aid of the constitutive relations, we proceed to compute the bound charge and bound current densities, then employ the Lorentz law of force to determine the distribution of force density within the regions of interest. For a few simple cases where analytical solutions exist, these solutions are found to be in complete agreement with our numerical results. We also analyze the distribution of fields and forces in more complex systems, and discuss the relevance of our findings to experimental observations. In particular, we demonstrate the single-beam trapping of a dielectric micro-sphere immersed in a liquid under conditions that are typical of optical tweezers. © 2005 Optical Society of America OCIS codes: (260.2110) Electromagnetic theory; (140.7010) Trapping. References 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?URI=OPEX-12-22-5375 2. D. A. White, “Numerical modeling of optical gradient traps using the vector finite element method,” J. Compt. Phys. 159, 13-37 (2000). 3. A. Mazolli, P. A. M. Neto, and H. M. Nussenzveig, “Theory of trapping forces in optical tweezers,” Proc. Roy. Soc. Lond. A 459, 3021-3041 (2003). 4. C. Rockstuhl and H. P. Herzig, “Rigorous diffraction theory applied to the analysis of the optical force on elliptical nano- and micro-cylinders,” J. Opt. A: Pure Appl. Opt. 6, 921-31 (2004). 5. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569-582 (1992). 6. A. Rohrbach and E. Stelzer, “Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,” Appl. Opt. 41, 2494 (2002). 7. J. D. Jackson, Classical Electrodynamics, 2nd edition (Wiley, New York, 1975). 8. 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?URI=OPEX-13-6-2064 9. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288-290 (1986). 10. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517- 1520 (1987). 1. Introduction In a previous paper [1] we showed that the direct application of the Lorentz law of force in conjunction with Maxwell’s equations can yield a complete picture of the electromagnetic force in metallic as well as dielectric media. In the case of the dielectrics, bound charges and bound currents were found to be responsible, respectively, for the electric and magnetic components of the Lorentz force. When a dielectric medium is homogeneous and isotropic, or can be divided into two or more such regions – each with its own uniform dielectric constant ε – the bound charges appear only at the surface(s) and/or the interface(s) between adjacent dielectrics. The bound currents, however, induced by the local E-field in proportion to the time-rate of change of the polarization P = εo(ε − 1)E, are distributed throughout the medium. The E-field of the light exerts a force on the induced charge density, while the local H-field (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2321 #6863 - $15.00 US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005

exerts a force on the induced current density. The net force can then be obtained by integrating these forces over the entire volume of the dielectric The above method, although involving no approximations, differs from the so-called rigorous"methods commonly used in computing the force of radiation [2-4]. The primary difference is that we sidestep the use of Maxwells stress tensor by reaching directly to bound electrons and their associated currents, treating them as the localized sources of electro nagnetic force under the influence of the lights E-and B-fields. We also avoid the use of the two-component approach, where scattering and gradient forces are treated separately, either in the geometric-optical regime of ray-tracing [5], or in the electromagnetic regime where the light's momentum and intensity gradient are tracked separately [6]. Unlike some of the other approaches, our method computes the force density distribution and not just the total force In this paper we use Finite-Difference-Time-Domain(FDTD) computer simulations to obtain the electromagnetic field distribution in and around several dielectric media. The theoretical underpinnings of our computational method are described in Section 2. Section 3 is devoted to comparisons with exact solutions in the case of a dielectric slab illuminated at normal incidence, and also in the case of a semi-infinite medium under a p-polarized plane- wave at oblique incidence. In Section 4 we show that a one-dimensional gaussian beam propagating in an isotropic, homogeneous medium, exerts either an expansive or a compressive lateral force on its host medium, depending on the beams polarization state. The case of a top-hat-shaped beam entering a semi-infinite dielectric medium at oblique incidence is also covered in Section 4. In Section 5 we study the behavior of a cylindrical glass ro illuminated by a(one-dimensional) Gaussian beam, paying particular attention to the effects of polarization on the force-density distribution. In Section 6 we analyze the single-beam trapping of a small spherical bead, immersed in a liquid, by a sharply focused laser beam both linear and circular polarization states of the beam are shown to result in strong trapping forces. a dielectric half-slab under a one-dimensional gaussian beam centered on one edge of the slab is studied in Section 7. General remarks and conclusions are the subject of Section 8 2. Theoretical considerations To compute the force of the electromagnetic radiation on a given medium, we solve Maxwells equations numerically to determine the distributions of the E-and H-fields(both inside and outside the medium). We then apply the Lorentz law F=pbE+Jb×B, where F is the force density, and Pb and Jb are the bound charge and current densities, respectively [7]. The magnetic induction B is related to the H-field via H. where uo=4I x 10 henrys/meter is the permeability of free space. In the absence of free charges V.D=0, where D=EE+P is the displacement vector, E =88542 x 10 farads/meter is the free-space permittivity, and P is the local polarization density. In linear media, D=EE, where a is the mediums relative permittivity; hence, P=E(E-DE When V D=0, the bound-charge density P,=-V P may be expressed as P,=EVE Inside a homogeneous, isotropic medium, E being proportional to D and V d=0 imply that P=0: no bound charges, therefore, exist inside such media. However, at the interface between two adjacent media, the component of d perpendicular to the interface, Di, must be continuous. The implication is that El is discontinuous and, therefore, bound charges exist at such interfaces; the interfacial bound charges will thus have areal density ob=E(E21-E11 Under the influence of the local E-field, these charges give rise to a Lorentz force densi F=2 Real(o E), where F is the force per unit area of the interface. Since the tangential E- field, Ell, is continuous across the interface, there is no ambiguity as to the value of eythat should be used in computing the force. As for the perpendicular component, the average El across the boundary, 2(E11+ E21), must be used in calculating the interfacial force [1, 8] #6863·$1500US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005 (C)2005OSA 4 April 2005/VoL 13, No. 7/OPTICS EXPRESS 2322

exerts a force on the induced current density. The net force can then be obtained by integrating these forces over the entire volume of the dielectric. The above method, although involving no approximations, differs from the so-called “rigorous” methods commonly used in computing the force of radiation [2-4]. The primary difference is that we sidestep the use of Maxwell’s stress tensor by reaching directly to bound electrons and their associated currents, treating them as the localized sources of electromagnetic force under the influence of the light’s E- and B-fields. We also avoid the use of the two-component approach, where scattering and gradient forces are treated separately, either in the geometric-optical regime of ray-tracing [5], or in the electromagnetic regime where the light’s momentum and intensity gradient are tracked separately [6]. Unlike some of the other approaches, our method computes the force density distribution and not just the total force. In this paper we use Finite-Difference-Time-Domain (FDTD) computer simulations to obtain the electromagnetic field distribution in and around several dielectric media. The theoretical underpinnings of our computational method are described in Section 2. Section 3 is devoted to comparisons with exact solutions in the case of a dielectric slab illuminated at normal incidence, and also in the case of a semi-infinite medium under a p-polarized planewave at oblique incidence. In Section 4 we show that a one-dimensional Gaussian beam propagating in an isotropic, homogeneous medium, exerts either an expansive or a compressive lateral force on its host medium, depending on the beam’s polarization state. The case of a top-hat-shaped beam entering a semi-infinite dielectric medium at oblique incidence is also covered in Section 4. In Section 5 we study the behavior of a cylindrical glass rod illuminated by a (one-dimensional) Gaussian beam, paying particular attention to the effects of polarization on the force-density distribution. In Section 6 we analyze the single-beam trapping of a small spherical bead, immersed in a liquid, by a sharply focused laser beam; both linear and circular polarization states of the beam are shown to result in strong trapping forces. A dielectric half-slab under a one-dimensional Gaussian beam centered on one edge of the slab is studied in Section 7. General remarks and conclusions are the subject of Section 8. 2. Theoretical considerations To compute the force of the electromagnetic radiation on a given medium, we solve Maxwell’s equations numerically to determine the distributions of the E- and H-fields (both inside and outside the medium). We then apply the Lorentz law F = ρb E + Jb × B, (1) where F is the force density, and ρb and Jb are the bound charge and current densities, respectively [7]. The magnetic induction B is related to the H-field via B = µ oH, where µ o = 4π × 10−7 henrys/meter is the permeability of free space. In the absence of free charges ∇ · D = 0, where D = εoE + P is the displacement vector, εo = 8.8542 × 10−12 farads/meter is the free-space permittivity, and P is the local polarization density. In linear media, D =εoε E, where ε is the medium’s relative permittivity; hence, P =εo(ε – 1)E. When ∇ · D = 0, the bound-charge density ρb = −∇ · P may be expressed as ρb = εo∇ · E. Inside a homogeneous, isotropic medium, E being proportional to D and ∇ · D = 0 imply that ρb = 0; no bound charges, therefore, exist inside such media. However, at the interface between two adjacent media, the component of D perpendicular to the interface, D⊥, must be continuous. The implication is that E⊥ is discontinuous and, therefore, bound charges exist at such interfaces; the interfacial bound charges will thus have areal density σ b =εo(E2⊥ − E1 ⊥). Under the influence of the local E-field, these charges give rise to a Lorentz force density F = ½ Real(σb E*), where F is the force per unit area of the interface. Since the tangential Efield, E| |, is continuous across the interface, there is no ambiguity as to the value of E| | that should be used in computing the force. As for the perpendicular component, the average E⊥ across the boundary, ½(E1 ⊥ + E2 ⊥), must be used in calculating the interfacial force [1,8]. (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2322 #6863 - $15.00 US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005

The only source of electrical currents within dielectric media are the oscillating dipoles, their(bound )current density J in a dispersionless medium being given by J s=dp/dt= Eo(E- l)dE/dr n this and the following equations, the underline denotes time dependence, symbols without the underline are used to represent the complex amplitudes of time-harmonic fields. Assuming time-harmonic fields with the time-dependence factor exp(i@n), one can rewrite q (2)as Jb=-iaE(E-1). The B-field of the electromagnetic wave exerts a force on the bound current according to the Lorentz law, namely, F=/Real(Jbx B ) where F is the (time-averaged) force per unit volume In FDTD simulations involving dispersive media, specific models (i.e, Debye, Drude, or Lorentz models) are used to represent the frequency dependence of the dielectric function @). Maxwell's equations are integrated in time over a discrete mesh, with the(model- dependent) dispersive behavior of the medium cast onto a time-dependent polarization vector We assume E(@)=Eo+AE(o), where the real-valued aoo denotes the relative permittivity of the medium in the absence of dispersion, the effects of dispersion enter through the complex valued function AE(@). The fraction of the polarization current density that embodies the contribution of Ae(@)is denoted by p. As the FDTD simulation progresses, determining the E-and H-fields as functions of time, Ip is computed concurrently by solving the relevant differential equations of the chosen model To derive an expression for the total (i.e, free carrier bound) current density i in a generally absorbing and dispersive medium, we begin with the following Maxwell equation Vx丑=aE+oDot Here the real-valued o denotes the material's conductivity for the free carriers. In our simulations, o is assumed to be a constant, independent of the frequency @; that is, the free carriers' conductivity is assumed to be dispersionless. Note that setting o=0 does not necessarily guarantee a transparent medium, as absorption could enter through the imaginary component of the complex permittivity, AE(o) Substituting for D in Eq ( 3)from the constitutive relation D=EE+P, we arrive at Eor=V×丑-(σE+Po)=V×丑-J, where, by definition, the total current density _ is given by D computations, Eq ( 3)is often rearranged as follows EnEo=V×旦-σE-Lp, where, for dispersive media(e.g, Debye, Drude, and Lorentz models), Eoo represents the frequency-independent part of the relative permittivity, while !p models dispersion (p=0 for non-dispersive dielectrics). When computing it is convenient to eliminate the time- derivative of E from Eq (5), as only one time-slice of E is normally stored during FDTD computations. From Eq (6), dE/Ot=(V XH-OE-Lp)Eoo, which, when substituted in Eq (5), yields J=(OE+Lp)lE+(1-llEoo)VXH Computing the total current density 2(i.e, the sum of the conduction and polarization current densities) at a given instant of time thus requires only the contemporary values of E, H, and Lp, which are readily available during the normal progression of an FDTD simulation #6863·$1500US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005 (C)2005OSA 4 April 2005/VoL 13, No. 7/OPTICS EXPRESS 2323

The only source of electrical currents within dielectric media are the oscillating dipoles, their (bound) current density Jb in a dispersionless medium being given by Jb = ∂P/∂t = εo(ε – 1)∂E/∂t. (2) (In this and the following equations, the underline denotes time dependence; symbols without the underline are used to represent the complex amplitudes of time-harmonic fields.) Assuming time-harmonic fields with the time-dependence factor exp(−iω t), one can rewrite Eq. (2) as Jb = −iω εo(ε – 1)E. The B-field of the electromagnetic wave exerts a force on the bound current according to the Lorentz law, namely, F = ½Real (Jb × B*), where F is the (time-averaged) force per unit volume. In FDTD simulations involving dispersive media, specific models (i.e., Debye, Drude, or Lorentz models) are used to represent the frequency dependence of the dielectric function ε(ω). Maxwell’s equations are integrated in time over a discrete mesh, with the (modeldependent) dispersive behavior of the medium cast onto a time-dependent polarization vector. We assume ε (ω) = ε ∞ +∆ε (ω), where the real-valued ε ∞ denotes the relative permittivity of the medium in the absence of dispersion; the effects of dispersion enter through the complexvalued function ∆ε (ω). The fraction of the polarization current density that embodies the contribution of ∆ε (ω) is denoted by J p. As the FDTD simulation progresses, determining the E- and H-fields as functions of time, J p is computed concurrently by solving the relevant differential equations of the chosen model. To derive an expression for the total (i.e., free carrier + bound) current density Ĵ in a generally absorbing and dispersive medium, we begin with the following Maxwell equation: ∇ × H = σ E + ∂D/∂t. (3) Here the real-valued σ denotes the material’s conductivity for the free carriers. In our simulations, σ is assumed to be a constant, independent of the frequency ω ; that is, the free carriers’ conductivity is assumed to be dispersionless. Note that setting σ = 0 does not necessarily guarantee a transparent medium, as absorption could enter through the imaginary component of the complex permittivity, ∆ε (ω). Substituting for D in Eq.(3) from the constitutive relation D = εoE + P, we arrive at εo ∂E/∂t = ∇ × H − (σ E + ∂P/∂t ) = ∇ × H − Ĵ , (4) where, by definition, the total current density Ĵ is given by Ĵ = ∇ × H − εo ∂E/∂ t. (5) In FDTD computations, Eq.(3) is often rearranged as follows: εoε ∞ ∂E/∂t = ∇ × H − σ E − J p , (6) where, for dispersive media (e.g., Debye, Drude, and Lorentz models), ε ∞ represents the frequency-independent part of the relative permittivity, while J p models dispersion (J p = 0 for non-dispersive dielectrics). When computing Ĵ , it is convenient to eliminate the timederivative of E from Eq.(5), as only one time-slice of E is normally stored during FDTD computations. From Eq.(6), εo ∂E/∂t = (∇ × H − σ E − J p) /ε ∞ , which, when substituted in Eq.(5), yields Ĵ = (σ E + J p)/ε ∞ + (1 − 1/ε ∞ )∇ ×H. (7) Computing the total current density Ĵ (i.e., the sum of the conduction and polarization current densities) at a given instant of time thus requires only the contemporary values of E, H, and J p , which are readily available during the normal progression of an FDTD simulation. (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2323 #6863 - $15.00 US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005

Given the electromagnetic fields E and H as functions of spatial coordinates and time, we compute the force density distribution by time-averaging the Lorentz equation, namel =(/TV(EV EE+JxuH)dr density P=VEE), and by the magnetic-field, IxB, can be readily identified in Ea, (op a% The time integral in the above equation is taken over one period of the(time-harmonic)li field. Contributions to the Lorentz force by the electric-field, pE (where the total charge 3. Comparison to exact solutions The case of a dielectric slab of index n and thickness d illuminated by a plane-wave at normal incidence was analyzed in our previous paper [1]. To fix the frame of reference, we assume that the slab is parallel to the xy-plane and the beam propagates in the -z direction. At normal incidence, there are no induced charges, and the electromagnetic pressure in its entirety may be attributed to the magnetic component of the Lorentz force. Inside the slab, a pair of counter-propagating plane-waves(along +=)interfere with each other and set up a system of fringes. The force is distributed within the fringe pattern, being positive (ie, in the +2 direction) over one-half of each fringe and negative over the other half. The net force is obtained by integrating the local force density through the thickness of the slab Figure 1 shows the computed force density F: inside a slab illuminated at normal incidence by a plane-wave of wavelength no=640nm and E-field amplitude Eo=1.0 V/m (Computed values of F and Fy were zero, as expected. The slab is suspended in free-space, and has refractive index n=2.0 and thickness d= 110nm. The total force along the =-axis(per unit cross-sectional area)is found to be F()d==-2481 pN/m2(4-=5.0 nm in our FDTD simulations)versus the exact value of.479 pN/m, obtained from theoretical considerations [1]. For a quarter-wave-thick slab, d= 80nm, the simulation yields j F()d==-3192 pN where the exact solution is. 188 pN/m2 1.0 n=2.0 n=1.0 Incident beam z lum] Fig. 1. Computed force density F:(per unit cross-sectional area) versus inside a dielectric slab illuminated with a normally incident plane-wave(o=0.64 um). The slab, suspended free-space, has n=2.0, d=110nm. The incident beam propagates along the negative =-axis As another example, consider a p-polarized plane-wave(1=650 nm)incident at 0=500 on a semi-infinite dielectric of refractive index n=3. 4. located in the region =<0. Figure 2 hows computed time-snapshots of the E: component of the field In Fig. 2(a)the transition of the refractive index from no=1.0 to n, =3. 4 is abrupt, and the force per unit area due to the #6863·$1500US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005 (C)2005OSA 4 April 2005/VoL 13, No. 7/OPTICS EXPRESS 2324

Given the electromagnetic fields E and H as functions of spatial coordinates and time, we compute the force density distribution by time-averaging the Lorentz equation, namely, = (1/T )∫ (E ∇ ·εoE + Ĵ ×µoH ) dt. (8) The time integral in the above equation is taken over one period of the (time-harmonic) light field. Contributions to the Lorentz force by the electric-field, ρE (where the total charge density ρ = ∇ ·εoE ), and by the magnetic-field, J ×B, can be readily identified in Eq.(8). 3. Comparison to exact solutions The case of a dielectric slab of index n and thickness d illuminated by a plane-wave at normal incidence was analyzed in our previous paper [1]. To fix the frame of reference, we assume that the slab is parallel to the xy-plane and the beam propagates in the –z direction. At normal incidence, there are no induced charges, and the electromagnetic pressure in its entirety may be attributed to the magnetic component of the Lorentz force. Inside the slab, a pair of counter-propagating plane-waves (along ± z) interfere with each other and set up a system of fringes. The force is distributed within the fringe pattern, being positive (i.e., in the +z direction) over one-half of each fringe and negative over the other half. The net force is obtained by integrating the local force density through the thickness of the slab. Figure 1 shows the computed force density Fz inside a slab illuminated at normal incidence by a plane-wave of wavelength λo = 640nm and E-field amplitude Eo = 1.0 V/m. (Computed values of Fx and Fy were zero, as expected.) The slab is suspended in free-space, and has refractive index n = 2.0 and thickness d = 110nm. The total force along the z-axis (per unit cross-sectional area) is found to be ∫ Fz(z) dz = −2.481 pN/m2 (∆z = 5.0 nm in our FDTD simulations) versus the exact value of −2.479 pN/m2 , obtained from theoretical considerations [1]. For a quarter-wave-thick slab, d = 80nm, the simulation yields ∫ Fz(z) dz = −3.192 pN/m2 , where the exact solution is −3.188 pN/m2 . Fig. 1. Computed force density Fz (per unit cross-sectional area) versus z inside a dielectric slab illuminated with a normally incident plane-wave (λo = 0.64 µm). The slab, suspended in free-space, has n = 2.0, d = 110nm. The incident beam propagates along the negative z-axis. As another example, consider a p-polarized plane-wave (λo = 650 nm) incident at θ = 50° on a semi-infinite dielectric of refractive index n = 3.4, located in the region z < 0. Figure 2 shows computed time-snapshots of the Ez component of the field. In Fig. 2(a) the transition of the refractive index from no = 1.0 to n1 = 3.4 is abrupt, and the force per unit area due to the 0 T n = 1.0 n = 2.0 n = 1.0 Incident beam (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2324 #6863 - $15.00 US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005

induced surface charges, computed in FDTD with Ay= A =5.0 nm, is(Fy, F-)=(1.696, 2. 468)N/m*, versus the exact value of(1.699, 2.467)pN/m. For n=2.0, FDTD simulations yield(Fr F)=(1.5903, 1.6475)pN/m2 versus the exact value of (1.5911, 1.6493)pN/m2 Figure 2(b)corresponds to a semi-infinite dielectric with a rapid (linear) transition of the refractive index from no= 1.0 to n=3.4 over a 40 nm-thick region. In this case, the force per unit sur face area computed in FDTD is(Fys F: )=(1.744, 2.638)pN/m We mention in passing that the case of a finite-diameter beam at Brewster's incidence on a dielectric wedge is also amenable to analytical as well as numerical solution, and that the numerically computed forces are in excellent agreement with the theoretical values [8] s(a) .s(b) 2^ ^^ n=3.4 n=1.0 3.4 n=10 Fig. 2. Time-snapshots of the E, component of a p-polarized plane-wave, Ao=0.65 um -<0. (a) Abrupt transition of the refractive index at ==0. (b)Linear transition of the refractive index from n,=1.0 to n,=3.4 over a 40 nm-thick region. The above results establish the accuracy of our numerical calculations, especially when transitions at the boundaries between regions of differing refractive indices, as was done in the case of Fig. 2(b), is a tool that is available in numerical simulations. Such smooth transitions at the boundaries may represent physical reality, or they may be used as an artificial tool to eliminate sharp discontinuities and singularities of the equations. To the extent that such smoothing operations do not modify the actual physics of the problem under consideration, they may be used with varying degrees of effectiveness 4. Force exerted by beams edge on the host medium We have shown in [1] that, among other things, the magnetic Lorentz force is responsible for a lateral pressure exerted on the host medium at the edges of a finite-diameter beam; the force per unit area at each edge (i.e, side-wall) of the beam is given by Here lEol is the magnitude of the E-field of a(finite-diameter) plane-wave in a medium of dielectric constant E If the E-field is parallel(perpendicular)to the beams edge, the force is ompressive(expansive); in other words, the opposite side-walls of the beam tend to push the medium toward(away from) the beam center. The"edge force" does not appear to be sensitive to the detailed structure of the beams edge, in particular, a one-dimensional Gaussian beam exhibits the edge force described by Eq(9)when its(magnetic)Lorentz force on the host medium is integrated laterally on either side of the beams center[1] Consider a one-dimensional Gaussian beam (uniform along x, Gaussian along y, and propagating in the negative z-direction) in a homogeneous host medium of refractive index n=2.0 the free-space wavelength of the cw beam is no=0.65um. Figure 3 shows time snapshots of the field profile(first row) and time-averaged force density distributions( second #6863·$1500US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005 (C)2005OSA 4 April 2005/VoL 13, No. 7/OPTICS EXPRESS 2325

induced surface charges, computed in FDTD with ∆y = ∆z = 5.0 nm, is (Fy, Fz) = (1.696, 2.468) pN/m2 , versus the exact value of (1.699, 2.467) pN/m2 . For n = 2.0, FDTD simulations yield (Fy, Fz) = (1.5903, 1.6475) pN/m2 versus the exact value of (1.5911, 1.6493) pN/m2 . Figure 2(b) corresponds to a semi-infinite dielectric with a rapid (linear) transition of the refractive index from no = 1.0 to n1 = 3.4 over a 40 nm-thick region. In this case, the force per unit surface area computed in FDTD is (Fy, Fz) = (1.744, 2.638) pN/m2 . We mention in passing that the case of a finite-diameter beam at Brewster’s incidence on a dielectric wedge is also amenable to analytical as well as numerical solution, and that the numerically computed forces are in excellent agreement with the theoretical values [8]. Fig. 2. Time-snapshots of the Ez component of a p-polarized plane-wave, λo = 0.65 µm, incident at θ = 50° on a semi-infinite dielectric of refractive index n = 3.4, located in the region z < 0. (a) Abrupt transition of the refractive index at z = 0. (b) Linear transition of the refractive index from no = 1.0 to n1 = 3.4 over a 40 nm-thick region. The above results establish the accuracy of our numerical calculations, especially when the surface charge is limited to a single pixel, as was the case in Fig. 2(a). Making smooth transitions at the boundaries between regions of differing refractive indices, as was done in the case of Fig. 2(b), is a tool that is available in numerical simulations. Such smooth transitions at the boundaries may represent physical reality, or they may be used as an artificial tool to eliminate sharp discontinuities and singularities of the equations. To the extent that such smoothing operations do not modify the actual physics of the problem under consideration, they may be used with varying degrees of effectiveness. 4. Force exerted by beam’s edge on the host medium We have shown in [1] that, among other things, the magnetic Lorentz force is responsible for a lateral pressure exerted on the host medium at the edges of a finite-diameter beam; the force per unit area at each edge (i.e., side-wall) of the beam is given by F (edge) = ¼εo(ε − 1)|Eo| 2 . (9) Here |Eo| is the magnitude of the E-field of a (finite-diameter) plane-wave in a medium of dielectric constant ε. If the E-field is parallel (perpendicular) to the beam’s edge, the force is compressive (expansive); in other words, the opposite side-walls of the beam tend to push the medium toward (away from) the beam center. The “edge force” does not appear to be sensitive to the detailed structure of the beam’s edge; in particular, a one-dimensional Gaussian beam exhibits the edge force described by Eq. (9) when its (magnetic) Lorentz force on the host medium is integrated laterally on either side of the beam’s center [1]. Consider a one-dimensional Gaussian beam (uniform along x, Gaussian along y, and propagating in the negative z-direction) in a homogeneous host medium of refractive index n = 2.0; the free-space wavelength of the cw beam is λo = 0.65µm. Figure 3 shows time snapshots of the field profile (first row) and time-averaged force density distributions (second (a) n = 3.4 n = 1.0 n = 3.4 n = 1.0 (b) (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2325 #6863 - $15.00 US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005

row) in the y z cross-sectional plane. The beam’s full-width at half-maximum-amplitude measured at the waist (located at z = +0.3 µm) is FWHM = 1.5 µm. Fig. 3. One-dimensional Gaussian beam in a homogeneous medium of refractive index n = 2.0. The beam waist is at z = 0.3µm, and the beam’s propagation direction is along the negative zaxis. (a) Time snapshot of Hx distribution for p-light, with superposed arrows depicting the (Ey, Ez) vector field. (b) Time snapshot of Ex distribution for s-light, with the (Hy, Hz) vector field superposed. (c) Force density distribution of the Fy component in the case of ppolarization, with the (Fy, Fz) vector field superposed. (d) Distribution of the force density component Fy for the s-polarized beam, with the (Fy, Fz) vector field superposed. In Fig. 3 the field is p-polarized on the left- and s-polarized on the right-hand-side. The upper-left frame shows a color-coded plot of the Hx distribution, on which the E-field vector (Ey, Ez) is superposed. The frame below it shows the force density plot of Fy, together with the (Fy, Fz) vector field in the case of p-polarization. The upper-right frame is a time snapshot of the Ex distribution, with the H-field vector (Hy, Hz) superposed. The frame below it shows Fy for the s-polarized beam, with the (Fy, Fz) vector field superposed. These force-density fields are computed using time-averages (over one period of the optical wave, T = λo/c) of the timeand space-dependent force density. Note that Fy is expansive for p-light and compressive for s-light. The integrated lateral force per unit area on each half of the beam, ∫ F(x, y, z) dy, is 1.6577 pN/m2 for p-light and 1.6581 pN/m2 for s-light. The theoretical value of the force per unit area on each edge of the beam given by Eq.(9) with Eo= 0.5V/m is F (edge) = ±1.66 pN/m2 . Another example of the same phenomenon (lateral force at the beam’s edge) is exhibited by a finite-diameter beam that enters a semi-infinite dielectric at oblique incidence. Figure 4 (a) (b) (c) (d) (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2326 #6863 - $15.00 US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005

shows time-snapshots of the field distributions for a linearly-polarized wave(o=0.65um) having a top-hat cross-sectional profile with smooth edges. The beam is incident from the free-space at inc=50 onto a semi-infinite dielectric of index ny=2.0; the dielectric fills the half-space :<0. The case of p-polarization is shown on the left-, that of s-polarization on the ght- hand side For the p-polarized beam, its angle of incidence being close to Brewster's angle BB=63.43, there is little reflectivity at the interface and most of the light is transmitted through to the dielectric medium, whereas for the s-polarized beam a goo fraction of the incident light is being reflected Fig. 4. Time snapshots of the field components for a linearly-polarized wave (o=0.65um) having a top-hat cross-sectional profile with smooth edges, incident at Binc=50% from free- dielectric of refractive index n,=2.0. located in the half-s field H, in the case of p-polarization, the superposed arrows represent the electric-field(En Er).(Right) Electric field Ex in the case of s-polarization, the superposed Computed force densities(Fy F: )inside the dielectric medium are displayed in Fig. 5; the interface region has been excluded to avoid, in the case of p-light, the high force region of the induced surface charges(For s-light the E-field is continuous at the boundary and, therefore no surface charges are induced. In both cases the force fields near the leading edge of the beam show oscillatory behavior, whereas the trailing edge is fairly smooth. Despite oscillations near the edge, the force is seen to be generally expansive for p-light and compressive for s-light, that is, the sign of the integrated lateral force on each side of the center is consistent with the theoretical arguments presented in [1]. In units of pN/m, the integrated force(JFy dy, JF: dy) for p-light is(-2.64, -1. 13)for the left-edge and(2.59, 1.096) for the right-edge. These edge forces are nearly identical in strength(to better than +1.5%) are orthogonal to the propagation direction within the dielectric, and are in fair agreement with the theoretical value of +(2.51, 1.04)PN/m2 obtained from Eq (9)with E=0.64 V/m The corresponding edge forces for the s-light depicted in Fig. 5, right-hand column,are (1.957, 0.821) for the left edge and(1.939,-0815)for the right edge. Again, this compressive force is orthogonal to the propagation direction, and is in reasonable agreement with the theoretical value of+(1.92, 0.8)pN/m2 obtained from Eq (9)with E.=0.56 V/m 5. Cylindrical rod illuminated by Gaussian beam Figure 6 shows time-snapshot plots of Hx, Ey, E: components of a p-polarized, one- dimensional Gaussian beam(no=0.65 um, amplitude FWHM=0.5 um), propagating in free- space along the negative :-direction the beams waist is at ==0.5 um. Given the amplitude profile as H( z=0.5um)=Ho exp[-wvlyo)"], the value of yo at the waist is 0.3um, which corresponds to a divergence half-angle 8=arctan(dIy)=35%. The small diameter of the beam at the waist thus results in its rapid divergence along the propagation direction. The cone angle of the beam, although large enough to exhibit lateral trapping of small dielectric objects, is not sufficient to produce vertical trapping [5], as will be seen below #6863·$1500US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005 (C)2005OSA 4 April 2005/VoL 13, No. 7/OPTICS EXPRESS 2327

shows time-snapshots of the field distributions for a linearly-polarized wave (λo = 0.65µm) having a top-hat cross-sectional profile with smooth edges. The beam is incident from the free-space at θ inc = 50° onto a semi-infinite dielectric of index ns = 2.0; the dielectric fills the half-space z < 0. The case of p-polarization is shown on the left-, that of s-polarization on the right-hand side. For the p-polarized beam, its angle of incidence being close to Brewster’s angle θ B = 63.43°, there is little reflectivity at the interface and most of the light is transmitted through to the dielectric medium, whereas for the s-polarized beam a good fraction of the incident light is being reflected. Fig. 4. Time snapshots of the field components for a linearly-polarized wave (λo = 0.65µm) having a top-hat cross-sectional profile with smooth edges, incident at θ inc = 50° from freespace onto a semi-infinite dielectric of refractive index ns = 2.0, located in the half-space z < 0. (Left) Magnetic field Hx in the case of p-polarization; the superposed arrows represent the electric-field (Ey, Ez). (Right) Electric field Ex in the case of s-polarization; the superposed arrows represent the magnetic-field (Hy, Hz). Computed force densities (Fy, Fz) inside the dielectric medium are displayed in Fig. 5; the interface region has been excluded to avoid, in the case of p-light, the high force region of the induced surface charges. (For s-light the E-field is continuous at the boundary and, therefore, no surface charges are induced.) In both cases the force fields near the leading edge of the beam show oscillatory behavior, whereas the trailing edge is fairly smooth. Despite oscillations near the edge, the force is seen to be generally expansive for p-light and compressive for s-light, that is, the sign of the integrated lateral force on each side of the center is consistent with the theoretical arguments presented in [1]. In units of pN/m2 , the integrated force ( ∫Fy dy, ∫Fz dy) for p-light is (−2.64, −1.13) for the left-edge and (2.59, 1.096) for the right-edge. These edge forces are nearly identical in strength (to better than ±1.5%), are orthogonal to the propagation direction within the dielectric, and are in fair agreement with the theoretical value of ±(2.51, 1.04) pN/m2 obtained from Eq. (9) with Eo = 0.64 V/m. The corresponding edge forces for the s-light depicted in Fig. 5, right-hand column, are (1.957, 0.821) for the left edge and (−1.939, −0.815) for the right edge. Again, this compressive force is orthogonal to the propagation direction, and is in reasonable agreement with the theoretical value of ±(1.92, 0.8) pN/m2 obtained from Eq. (9) with Eo = 0.56 V/m. 5. Cylindrical rod illuminated by Gaussian beam Figure 6 shows time-snapshot plots of Hx, Ey, Ez components of a p-polarized, onedimensional Gaussian beam (λo = 0.65 µm, amplitude FWHM = 0.5 µm), propagating in freespace along the negative z-direction; the beam’s waist is at z = 0.5 µm. Given the amplitude profile as Hx( y,z = 0.5µm) = Ho exp[−(y/yo) 2 ], the value of yo at the waist is 0.3µm, which corresponds to a divergence half-angle θ = arctan (λo/πyo) ≈ 35°. The small diameter of the beam at the waist thus results in its rapid divergence along the propagation direction. The cone angle of the beam, although large enough to exhibit lateral trapping of small dielectric objects, is not sufficient to produce vertical trapping [5], as will be seen below. (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2327 #6863 - $15.00 US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005

In this Gaussian beam’s path we placed a cylindrical rod (radius rc = 0.2 µm, index nc = 2.0) parallel to the x-axis at (y, z) = (0.2 µm, 0). The cylinder’s index changes linearly at the surface, from n = 1.0 to 2.0, within a 20 nm-thick transition layer (grid cell size = 5nm). Figure 7, left column, shows computed plots of |Hx | as well as force-density components Fy and Fz for this p-polarized beam. The integrated force is ( ∫ Fy( y, z) dydz, ∫ Fz( y, z) dydz) = (0.17, 0.21) pN/m without including the surface-charge contribution, and (−0.4, −0.58) pN/m when the surface-charge contribution is included. The net force is thus pulling the cylinder to the left while, at the same time, pushing it downward. The right-hand column of Fig. 7 shows the case of a s-polarized Gaussian beam (same parameters as in Fig. 6) illuminating the same glass cylinder as above. From top to bottom, the figure shows computed plots of |Ex | and force-density components Fy and Fz. Here surface charges are absent, and the force is purely due to the magnetic component of the Lorentz force. The integrated force is (∫ Fy( y, z) dydz, ∫ Fz( y, z) dydz) = (−0.14, −0.68) pN/m, which, as in the case of p-light, tends to pull the rod to the left while pushing it downward. The pull to the left is consistent with what has been observed in optical trapping experiments, but the downward push is contrary to the known behavior of single-beam traps (i.e., optical tweezers) [9,10]. We will see in the next section that vertical trapping benefits from the submersion of the particle in a liquid, where the index mismatch between the dielectric and its environment is reduced and the scattering force (caused by surface reflections) is minimized. More importantly, the results of the next section demonstrate the necessity of a sharper focus (i.e., a smaller beam waist diameter) [5] than has been used in the present simulations. (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2329 #6863 - $15.00 US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005