J. of Electromagn. Waves and Appl., Vol. 20, No 6, 827-839, 2006 LORENTZ FORCE ON DIELECTRIC AND MAGNETIC PARTICLES B. A. Kemp, T M. Grzegorczyk, and J. A. Kong Research Laboratory of electronics Massachusetts Institute of Technology 77 Massachusetts Ave.. 26-305. MA 02139. USA abstract-The well-known momentum conservation theorem is derived specifically for time-harmonic fields and is applied to calculate the radiation pressure on 2-D particles modeled as infinite dielectric and magnetic cylinders. The force calculation results from the divergence of the Maxwell stress tensor and is compared favorably via examples with the direct application of Lorentz force to bound currents and charges. The application of the momentum conservation heorem is shown to have the advantage of less computation, reducing the surface integration of the Lorentz force density to a line integral of the Maxwell stress tensor. The Lorentz force is applied to compute the force density throughout the particles, which demonstrates regions of compression and tension within the medium. Further comparison of the two force calculation methods is provided by the calculation of radiation pressure on a magnetic particle, which has not been previously published. The fields are found by application of the Mie theory along with the Foldy-Lax equations, which model interactions of multiple particles 1 INTRODUCTION The first observation of optical momentum transfer to small particles n 1970 [1 prompted further experimental demonstrations of radiation pressure such as optical levitation 2, radiation pressure on a liquid surface 3, and the single-beam optical trap 4, to name a few Subsequently, theoretical models have been developed to describe the experimental results and predict new phenomena, for example 5-9 However, the theory of radiation pressure is not new. In fact, the transfer of optical momentum to media was known by Poynting [101 from the application of the electromagnetic wave theory of light. Still
J. of Electromagn. Waves and Appl., Vol. 20, No. 6, 827–839, 2006 LORENTZ FORCE ON DIELECTRIC AND MAGNETIC PARTICLES B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong Research Laboratory of Electronics Massachusetts Institute of Technology 77 Massachusetts Ave., 26-305, MA 02139, USA Abstract—The well-known momentum conservation theorem is derived specifically for time-harmonic fields and is applied to calculate the radiation pressure on 2-D particles modeled as infinite dielectric and magnetic cylinders. The force calculation results from the divergence of the Maxwell stress tensor and is compared favorably via examples with the direct application of Lorentz force to bound currents and charges. The application of the momentum conservation theorem is shown to have the advantage of less computation, reducing the surface integration of the Lorentz force density to a line integral of the Maxwell stress tensor. The Lorentz force is applied to compute the force density throughout the particles, which demonstrates regions of compression and tension within the medium. Further comparison of the two force calculation methods is provided by the calculation of radiation pressure on a magnetic particle, which has not been previously published. The fields are found by application of the Mie theory along with the Foldy-Lax equations, which model interactions of multiple particles. 1. INTRODUCTION The first observation of optical momentum transfer to small particles in 1970 [1] prompted further experimental demonstrations of radiation pressure such as optical levitation [2], radiation pressure on a liquid surface [3], and the single-beam optical trap [4], to name a few. Subsequently, theoretical models have been developed to describe the experimental results and predict new phenomena, for example [5–9]. However, the theory of radiation pressure is not new. In fact, the transfer of optical momentum to media was known by Poynting [10] from the application of the electromagnetic wave theory of light. Still
Kemp, Grzegorczyk, and Kong ongoing work seeks to model the distribution of force on media by electromagnetic waves [11 The divergence of the Maxwell stress tensor [15 provide established method for calculating the radiation pressure on a dielectric surface via the application of the momentum conservation theorem[16 An alternate method for the calculation of radiation pressure on material media by the direct application of the Lorentz law has been recently reported [12 The method allows for the computation of force density at any point inside a dielectric [13 by the application of the Lorentz force to bound currents distributed throughout the medium and bound charges at the material surface and the method has been extended to include contributions from magnetic media [14 A comprehensive comparison of the two methods applied to particles has not been previously published, and, consequently, there exists some doubt in regard to the applicability of one method or the other. In the present paper, we compare the force exerted on 2-D dielectric cylinders as calculated from the divergence of the Maxwell stress tensor and the distributed lorentz force. First the total time average force as given by the divergence of the Maxwell stress tensor [16] is derived from the Lorentz law and the Maxwell equations for time harmonic fields. We demonstrate the numerical efficiency of he stress tensor method by computing the force on a 2-D dielectric particle represented by an infinite cylinder submitted to multiple plane waves. Second, we give the formulation for the distributed Lorentz force as applied to dielectric and magnetic media[12-14. The numerical integration of the distributed lorentz force over the 2-D particle cross-section area demonstrates equivalent results, although the convergence is shown to be much slower than the stress tensor line integration. Third, both the Maxwell stress tensor and the distributed Lorentz force methods are applied to two closely spaced particles in the three plane wave interference pattern, the former method exhibiting obustness with respect to choice of integration path and the latter method providing a 2-D map of the Lorentz force density distribution within the particles. Finally, the first theoretical demonstration of the Lorentz force applied to bound magnetic charges and currents in a 2-D particle is presented 2. MAXWELL STRESS TENSOR The momentum conservation theorem [16 relates the total force or a material object in terms of the momentum of the incident and scattered fields at all times it is derived from the lorentz force law and the Maxwell equations. In the case of time-harmonic fields, the
828 Kemp, Grzegorczyk, and Kong ongoing work seeks to model the distribution of force on media by electromagnetic waves [11–14]. The divergence of the Maxwell stress tensor [15] provides an established method for calculating the radiation pressure on a dielectric surface via the application of the momentum conservation theorem [16]. An alternate method for the calculation of radiation pressure on material media by the direct application of the Lorentz law has been recently reported [12]. The method allows for the computation of force density at any point inside a dielectric [13] by the application of the Lorentz force to bound currents distributed throughout the medium and bound charges at the material surface, and the method has been extended to include contributions from magnetic media [14]. A comprehensive comparison of the two methods applied to particles has not been previously published, and, consequently, there exists some doubt in regard to the applicability of one method or the other. In the present paper, we compare the force exerted on 2-D dielectric cylinders as calculated from the divergence of the Maxwell stress tensor and the distributed Lorentz force. First, the total time average force as given by the divergence of the Maxwell stress tensor [16] is derived from the Lorentz law and the Maxwell equations for time harmonic fields. We demonstrate the numerical efficiency of the stress tensor method by computing the force on a 2-D dielectric particle represented by an infinite cylinder submitted to multiple plane waves. Second, we give the formulation for the distributed Lorentz force as applied to dielectric and magnetic media [12–14]. The numerical integration of the distributed Lorentz force over the 2-D particle cross-section area demonstrates equivalent results, although the convergence is shown to be much slower than the stress tensor line integration. Third, both the Maxwell stress tensor and the distributed Lorentz force methods are applied to two closely spaced particles in the three plane wave interference pattern, the former method exhibiting robustness with respect to choice of integration path and the latter method providing a 2-D map of the Lorentz force density distribution within the particles. Finally, the first theoretical demonstration of the Lorentz force applied to bound magnetic charges and currents in a 2-D particle is presented. 2. MAXWELL STRESS TENSOR The momentum conservation theorem [16] relates the total force on a material object in terms of the momentum of the incident and scattered fields at all times. It is derived from the Lorentz force law and the Maxwell equations. In the case of time-harmonic fields, the
Lorentz force on dielectric and magnetic particles 829 time-average force on a material body can be calculated from a single divergence integral. The proof of this fact is shown by derivation of the momentum conservation theorem with the only assumption that all fields have a force provides the fundamental relationship between lence electromagnetic fields and the mechanical force on charges and currents[16]. The time average Lorentz force is given in terms of the electric field strength E and magnetic flux density b by 了={E+了×B where p and represent the electric charge and current, respectively, Ref represents the real part of a complex quantity, and()denotes the complex conjugate. The Maxwell Equations p=V·D =V×H+iD relate the sources p and to the electric flux density D and magnetic field strength H. Substitution yields F=Re(v DE+(VxH)XB-Dx( B).(3) After applying the remaining two Maxwell equations B B=V×E the force can by expressed as 了=2{(,DE+(×E)xD+(,B)B+(xB)×B() The momentum conservation theorem for time harmonic fields is 了=2{V)} where f is the time average force density in N/m3, and the Maxwell 6 ()=5(D.E*+B*·H)1-DE-B In(7), DE*and B* H are dyadic products and is the(3×3) identity matrix. By integration over a volume V enclosed by a surface S and
Lorentz force on dielectric and magnetic particles 829 time-average force on a material body can be calculated from a single divergence integral. The proof of this fact is shown by derivation of the momentum conservation theorem with the only assumption that all fields have e−iωt dependence. The Lorentz force provides the fundamental relationship between electromagnetic fields and the mechanical force on charges and currents [16]. The time average Lorentz force is given in terms of the electric field strength E¯ and magnetic flux density B¯ by ¯f = 1 2 Re ρE¯∗ + J¯× B¯∗ , (1) where ρ and J¯ represent the electric charge and current, respectively, Re{} represents the real part of a complex quantity, and (∗) denotes the complex conjugate. The Maxwell Equations ρ = ∇ · D¯ J¯ = ∇ × H¯ + iωD¯ (2) relate the sources ρ and J¯ to the electric flux density D¯ and magnetic field strength H¯ . Substitution yields ¯f = 1 2 Re (∇ · D¯)E¯∗ + (∇ × H¯ ) × B¯∗ − D¯ × (iωB¯) ∗ . (3) After applying the remaining two Maxwell equations 0 = ∇ · B¯ iωB¯ = ∇ × E, ¯ (4) the force can by expressed as ¯f = 1 2 Re (∇·D¯)E¯∗+(∇×E¯∗)×D¯ +(∇·B¯∗)H¯ +(∇×H¯ )×B¯∗ . (5) The momentum conservation theorem for time harmonic fields is reduced to ¯f = −1 2 Re ∇ · T ¯¯(¯r) , (6) where ¯f is the time average force density in N/m3, and the Maxwell stress tensor is [16] T ¯¯(¯r) = 1 2 D¯ · E¯∗ + B¯∗ · H¯ ¯¯I − D¯E¯∗ − B¯∗H. ¯ (7) In (7), D¯E¯∗ and B¯∗H¯ are dyadic products and ¯¯I is the (3×3) identity matrix. By integration over a volume V enclosed by a surface S and
Kemp, Grzegorczyk, and Kong application of the divergence theorem, the total force F on the material enclosed by S is given by F=-5B中ds(n() (8) where n is the outward normal to the surface S. When applying( 8) to calculate the force on a material object, the stress tensor in(7)is integrated over a surface chosen to completely enclose the object We consider the two-dimensional(2-D)problem of a circular cylinder incident by three tE plane waves. The incident electric field pattern El shown in Fig. 1 is due to three plane waves with free space wavelength Ao=532 nm] incident in the(ay) plane at angles o=iT/2, 7/6, 11T/6rad] with the electric field polarized in the i-direction. The polystyrene cylinder (ep=2.56co) has a radius of a=0.3Ao and is centered at (ao, yo)=(0, 100)nm] in a background of water(Eb= 1.69Eo). The total field is obtained as the superposition of incident and scattered fields, the latter is calculated from application of the Mie theory [8, 9 Figure 1. Incident electric field magnitude [V/m] due to three plane waves of free space wavelength Ao 532 nm propagating at angles r/2, 7/6,11/6 rad. The overlayed 2D particle is a cylinder 2.56Eo) of radius a 03Ao and infinite length in z with center position(ro, yo)=(0, 100)nm. The background medium is characterized by Eb= 1.69E0
830 Kemp, Grzegorczyk, and Kong application of the divergence theorem, the total force F¯ on the material enclosed by S is given by F¯ = −1 2 Re S dS nˆ · T ¯¯(¯r) , (8) where ˆn is the outward normal to the surface S. When applying (8) to calculate the force on a material object, the stress tensor in (7) is integrated over a surface chosen to completely enclose the object. We consider the two-dimensional (2-D) problem of a circular cylinder incident by three TE plane waves. The incident electric field pattern |E¯| shown in Fig. 1 is due to three plane waves with free space wavelength λ0 = 532 [nm] incident in the (xy) plane at angles φ = {π/2, 7π/6, 11π/6} [rad] with the electric field polarized in the zˆ-direction. The polystyrene cylinder ( p = 2.56 0) has a radius of a = 0.3λ0 and is centered at (x0, y0) = (0, 100) [nm] in a background of water ( b = 1.69 0). The total field is obtained as the superposition of incident and scattered fields, the latter is calculated from application of the Mie theory [8, 9]. Figure 1. Incident electric field magnitude [V/m] due to three plane waves of free space wavelength λ0 = 532 nm propagating at angles {π/2, 7π/6, 11π/6} rad. The overlayed 2D particle is a cylinder ( p = 2.56 0) of radius a = 0.3λ0 and infinite length in z with center position (x0, y0) = (0, 100) [nm]. The background medium is characterized by b = 1.69 0
Lorentz force on dielectric and magnetic particles To calculate the total force on the cylinder shown in Fig. 1, the Maxwell stress tensor is applied to the total field. For the 2-D problem the divergence of the stress tensor is computed by a line integral, which we evaluate by simple numerical integration. The path chosen is a circle of radius R concentric with the particle and the integration steps (RAo are assumed constant. The numerical integration is computed a{CnN(o}P△C2{n列小 where N represents the total number of integration points and the values of on] result from the discretization of E[0, 27]. Figure 2 shows the force versus the number of integration points for an integration radius of R= 1.0la. The results show that the integration converges rapidly. Because the force is calculated by a divergence integral, the result does not depend on the value of R, provided enough integration points are chosen. To confirm this, the force was calculated Number of Integration Poin Figure 2. y-directed force Fy versus the number of integration points used in the application of the Maxwell stress tensor in Eq.( 9). The integration path, shown by the inset diagram, is a circle of radius of R=1.0la concentric with the cylinder of radius a= 0.3X0. The configuration is the same as shown in Fig. 1
Lorentz force on dielectric and magnetic particles 831 To calculate the total force on the cylinder shown in Fig. 1, the Maxwell stress tensor is applied to the total field. For the 2-D problem, the divergence of the stress tensor is computed by a line integral, which we evaluate by simple numerical integration. The path chosen is a circle of radius R concentric with the particle and the integration steps (R∆φ) are assumed constant. The numerical integration is computed by F¯ = −1 2 Re 2π 0 nˆ · T ¯¯(R, φ)Rdφ ≈ −R∆φ N n=1 1 2 Re nˆ · T ¯¯(R, φ[n]) , (9) where N represents the total number of integration points and the values of φ[n] result from the discretization of φ ∈ [0, 2π]. Figure 2 shows the force versus the number of integration points for an integration radius of R = 1.01a. The results show that the integration converges rapidly. Because the force is calculated by a divergence integral, the result does not depend on the value of R, provided enough integration points are chosen. To confirm this, the force was calculated 0 20 40 60 80 100 -2 0 2 4 x 10-18 p b a R y x Number of Integration Points Fy [N] Figure 2. yˆ-directed force Fy versus the number of integration points used in the application of the Maxwell stress tensor in Eq. (9). The integration path, shown by the inset diagram, is a circle of radius of R = 1.01a concentric with the cylinder of radius a = 0.3λ0. The configuration is the same as shown in Fig. 1.
832 Kemp, Grzegorczyk, and Kong for various choices of integration radius yielding zero for all R a and F=y21190. 10-8[N/m]for all R> a, which is in agreement with the value reported by [14] 3. LORENTZ FORCE ON BOUND CURRENTS AND CHARGES The Lorentz force can be applied directly to bound currents and charges in a lossless medium [14]. The bulk force density in (N/m1 computed throughout the medium by 1 Re{-iP×B D”}, here the electric polarization Pe =(Ep -EbE and the magnetic polarization Pm =-(up -ubH are given in terms of the background constitutive parameters (ub, Eb) and the particle constitutive parameters (up, Ep). The surface force density in [N/m R where the bound electric surface charge density is Pe = n.(El Eoeb [12 the bound magnetic surface charge density is Pm =n 1-Ho)ub [14], and the unit vector n is an outward pointing normal to the surface. The fields(Eo, Ho) and(E1, H1) are the total fields just inside the particle and outside the particle, respectively, and the fields in(11) are given by Eaug =(E1+ Eo)/2 and Haug=(H1+Ho)/2 The distributed Lorentz force is applied to the problem of Fig. 1 Because TE polarized waves are incident upon a dielectric particle the bound charges at the surface ero and the total force F obtained by integrating the bulk Lorentz force density foulk over the cross section of the cylinder. The numerical integration is performed by summing the contribution from M discrete area elements. The area elements△A=△x△ y are taken to be identical so that the numerical integration is F=/1()△A∑n(圆xm,(2 here the dielectric polarization Pe[m] and magnetic field Hm] are evaluated at each point indexed by m in the cross section of the cylinder. The y-directed force is plotted in Fig 3 versus the number of integration points. The integral converges much slower than the
832 Kemp, Grzegorczyk, and Kong for various choices of integration radius yielding zero for all Ra, which is in agreement with the value reported by [14]. 3. LORENTZ FORCE ON BOUND CURRENTS AND CHARGES The Lorentz force can be applied directly to bound currents and charges in a lossless medium [14]. The bulk force density in [N/m3] is computed throughout the medium by ¯fbulk = 1 2 Re{−iωP¯e × B¯∗ − iωP¯m × D¯ ∗}, (10) where the electric polarization P¯e = ( p − b)E¯ and the magnetic polarization P¯m = −(µp − µb)H¯ are given in terms of the background constitutive parameters (µb, b) and the particle constitutive parameters (µp, p). The surface force density in [N/m2] is given by ¯fsurf = 1 2 Re{ρeE¯∗ avg + ρmH¯ ∗ avg}, (11) where the bound electric surface charge density is ρe = ˆn · (E¯1 − E¯0) b [12], the bound magnetic surface charge density is ρm = ˆn · (H¯1 − H¯0)µb [14], and the unit vector ˆn is an outward pointing normal to the surface. The fields (E¯0, H¯0) and (E¯1, H¯1) are the total fields just inside the particle and outside the particle, respectively, and the fields in (11) are given by E¯avg = (E¯1 + E¯0)/2 and H¯avg = (H¯1 + H¯0)/2. The distributed Lorentz force is applied to the problem of Fig. 1. Because TE polarized waves are incident upon a dielectric particle, the bound charges at the surface are zero and the total force F¯ is obtained by integrating the bulk Lorentz force density ¯fbulk over the cross section of the cylinder. The numerical integration is performed by summing the contribution from M discrete area elements. The area elements ∆A = ∆x∆y are taken to be identical so that the numerical integration is F¯ = S dA1 2 Re{ ¯fbulk} ≈ ∆A M m=1 1 2 Re{−iωP¯e[m] × µ0H¯ ∗[m]}, (12) where the dielectric polarization P¯e[m] and magnetic field H¯ [m] are evaluated at each point indexed by m in the cross section of the cylinder. The ˆy-directed force is plotted in Fig. 3versus the number of integration points. The integral converges much slower than the
Lorentz force on dielectric and magnetic particles 83 gxk2xx× 5001000150020002500300035004000 umber of Integration forgthredie ct applicationof the versest z fe ce ot er o Thegcanfin patios is the same as shown in Fig. 1. The dashed line is the force computed from the stress tensor of (9) with 100 numerical integration points on a concentric circle of radius r= 1.01a line integral applied to the stress tensor, however the resulting force is F=y21191.10-8(N/ml, thus matching the result from the Maxwell tress tensor 4. FORCE ON MULTIPLE DIELECTRIC PARTICLES The Mie theory and the Foldy-Lax multiple scattering equations are applied to calculate the force on multiple particles incident by a known electromagnetic field [8, 9. We consider the same incident field shown in Fig. 1 with two identical dielectric particles centered at (a, y)=(0, 100)nm]and (a, y)=(0, 300)[nm] as shown in Fig 4. As before, the 2-D polystyrene particles are modeled as infinite dielectric cylinders(Ep=2.56eo)in water(Eb= 1.69Eo)with radius a=0.3A The Maxwell stress tensor is applied to calculate the force on each particle by taking an integration path that just encloses each particle as shown in Fig 4. The force for the particle at(a, y)=(0, 100)[nm is F= y1.6517.108[N/m], and the force on the particle at
Lorentz force on dielectric and magnetic particles 833 500 1000 1500 2000 2500 3000 3500 4000 1.9 2 2.1 2.2 2.3 2.4 x 10-18 Number of Integration Points Fy [N] Figure 3. yˆ-directed force Fy versus the number of integration points for the direct application of the Lorentz force of (12). The configuration is the same as shown in Fig. 1. The dashed line is the force computed from the stress tensor of (9) with 100 numerical integration points on a concentric circle of radius R = 1.01a. line integral applied to the stress tensor, however the resulting force is F¯ = ˆy2.1191 · 10−18 [N/m], thus matching the result from the Maxwell stress tensor. 4. FORCE ON MULTIPLE DIELECTRIC PARTICLES The Mie theory and the Foldy-Lax multiple scattering equations are applied to calculate the force on multiple particles incident by a known electromagnetic field [8, 9]. We consider the same incident field shown in Fig. 1 with two identical dielectric particles centered at (x, y) = (0, 100) [nm] and (x, y) = (0, −300) [nm] as shown in Fig. 4. As before, the 2-D polystyrene particles are modeled as infinite dielectric cylinders ( p = 2.56 0) in water ( b = 1.69 0) with radius a = 0.3λ0. The Maxwell stress tensor is applied to calculate the force on each particle by taking an integration path that just encloses each particle as shown in Fig. 4. The force for the particle at (x, y) = (0, 100) [nm] is F¯ = ˆy1.6517 · 10−18 [N/m], and the force on the particle at
Kemp, Grzegorczyk, and Kong Figure 4. Two particles centered at (a, y)=(100, 0),(300, 0)[nm are subject to the incident field pattern of Fig. 1. The integration paths for the Maxwell stress tensor applied to the two particles are shown by the dotted lines. The sum of the forces on the two individual articles obtained by the smaller two integration circles is equal to the force obtained by integrating over the large circular integration path (0,-300)m]isp=-01.4901018N/m. By taking the integration path surrounding both particles, the total force on the system composed of both particles is F= y20269. 10-9[N/ml which agrees with the sum of the individual forces. This example demonstrates that the divergence of the stress tensor gives the total force on all currents and charges enclosed by the integration path and hat the integration path needs not be concentric with the material For comparison with the stress tensor method, the Lorentz force is applied to bound electric currents in both particles. The distribution of force densities are shown in Fig. 5. Although the a -directed force integrates to zero for both particles due to symmetry, it can be seen that the local force densities vary throughout the particle. These forces act in compression or tension in the various regions of the particle. The total force on each particle is found by integration of the local force densities throughout the particles. The force for he particle at(a, y)=(0, 100)(nm] is F= y16500. 10-18(N/ml using 17, 534 integration points, and the total force on the particle at(x,y)=(0,-30{mm]isF=-91.4523·10-1N/m] using17,530 integration points, which is agreement with the results of the Maxwell stress tensor divergence
834 Kemp, Grzegorczyk, and Kong p p b y x Figure 4. Two particles centered at (x, y) = (100, 0), (−300, 0) [nm] are subject to the incident field pattern of Fig. 1. The integration paths for the Maxwell stress tensor applied to the two particles are shown by the dotted lines. The sum of the forces on the two individual particles obtained by the smaller two integration circles is equal to the force obtained by integrating over the large circular integration path. (x, y) = (0, −300) [nm] is F¯ = −yˆ1.4490 · 10−18 [N/m]. By taking the integration path surrounding both particles, the total force on the system composed of both particles is F¯ = ˆy2.0269 · 10−19 [N/m], which agrees with the sum of the individual forces. This example demonstrates that the divergence of the stress tensor gives the total force on all currents and charges enclosed by the integration path and that the integration path needs not be concentric with the material bodies. For comparison with the stress tensor method, the Lorentz force is applied to bound electric currents in both particles. The distribution of force densities are shown in Fig. 5. Although the ˆx-directed force integrates to zero for both particles due to symmetry, it can be seen that the local force densities vary throughout the particle. These forces act in compression or tension in the various regions of the particle. The total force on each particle is found by integration of the local force densities throughout the particles. The force for the particle at (x, y) = (0, 100) [nm] is F¯ = ˆy1.6500 · 10−18 [N/m] using 17, 534 integration points, and the total force on the particle at (x, y) = (0, −300) [nm] is F¯ = −yˆ1.4523 · 10−18 [N/m] using 17, 530 integration points, which is agreement with the results of the Maxwell stress tensor divergence
Lorentz force on dielectric and magnetic particles 835 Figure 5. Lorentz force density [10-N/m l for two particles positioned at (a, y)=(100, 0), (300, 0)nm] illuminated by the three plane waves of free space wavelength Ao= 532 nm as shown in Fig. 1 The top plots are(a) y-directed force fy and(b)i-directed force fr for the particle at (a, y)=(0, 100)[nm]. The bottom plots are (c)y-directed force fy and(d) -directed force fr for the particle at(r, y)=(0, -300)nm]. The total Lorentz force on the particles is obtained by summation of the force densities on the particle at (a, y)=(0,100)[nm)(F=y1.650010-8[N/m)and the particle at(x,y)=(0,-300)lnm](F=-91.4523·10-18N/m]). For boti particles, the a-directed force is zero due to symmetry 5. RADIATION PRESSURE ON A MAGNETIC PARTICLE The Lorentz force has been previously applied to bound magnetic charges and currents to calculate the radiation pressure on a magnetic slab [ 14. However, the method has not been used to determine the force on magnetic particles. In this section, we calculate the radiation pressure on a 2-D magnetic particle represented by an infinite cylinder incident by a single te plane wave
Lorentz force on dielectric and magnetic particles 835 Figure 5. Lorentz force density [10−4 N/m3] for two particles positioned at (x, y) = (100, 0),(−300, 0) [nm] illuminated by the three plane waves of free space wavelength λ0 = 532 nm as shown in Fig. 1. The top plots are (a) ˆy-directed force fy and (b) ˆx-directed force fx for the particle at (x, y) = (0, 100) [nm]. The bottom plots are (c) ˆy-directed force fy and (d) ˆx-directed force fx for the particle at (x, y) = (0, −300) [nm]. The total Lorentz force on the particles is obtained by summation of the force densities on the particle at (x, y) = (0, 100) [nm] (F¯ = ˆy1.6500 · 10−18 [N/m]) and the particle at (x, y) = (0, −300) [nm] (F¯ = −yˆ1.4523 · 10−18 [N/m]). For both particles, the ˆx-directed force is zero due to symmetry. 5. RADIATION PRESSURE ON A MAGNETIC PARTICLE The Lorentz force has been previously applied to bound magnetic charges and currents to calculate the radiation pressure on a magnetic slab [14]. However, the method has not been used to determine the force on magnetic particles. In this section, we calculate the radiation pressure on a 2-D magnetic particle represented by an infinite cylinder incident by a single TE plane wave
836 Kemp, Grzegorczyk, and Kong The incident plane wave Ei= iE;e ikor propagates in free space (E0, Ho) with a wavelength Ao= 2T/ko=640]. The 2-D magnetic particles(Eo, 3Ho) are infinite in the i-direction, with radius a. The Maxwell stress tensor and the distributed lorentz force methods are applied to calculate the total force on the particles. The direct application of the lorentz force requires the model of bound magnetic currents Mb=-wwPn in(10) and bound magnetic surface charges pm in(11). Agreement between the two methods is shown in Fig. 6 as a function of particle radius. The oscillations in force are a result of internal resonances. which is also evident for the case of dielectric and magnetic slabs incident by plane waves[] ensor 日 04 Figure 6. Radiation pressure on a magnetic cylinder (up=3po, Ep Eo) versus the radius The tE plane wave propagates in the i- direction in free space(ub 4o, Eb= E0), and the wavelength is 640(nm. The force is calculated by the divergence of the stress tensor (line) and the Lorentz force on bound currents and charges(markers There are noticeable differences between the results of the two methods for larger values of a as shown in Fig. 6. This is due to increased spacial variations in force distribution for particles on the order of a wavelength or larger. The slow convergence of the total Lorentz force has been observed for small dielectric particles as shown Fig 3 and becomes a major obstacle for obtaining many digits of accuracy from the Lorentz force for large particles. To illustrate this point, we compare the force calculated for the magnetic particle with radius a= 1000 nm with various number of integration points M
836Kemp, Grzegorczyk, and Kong The incident plane wave E¯i = ˆzEieik0x propagates in free space ( 0, µ0) with a wavelength λ0 = 2π/k0 = 640 [nm]. The 2-D magnetic particles ( 0, 3µ0) are infinite in the ˆz-direction, with radius a. The Maxwell stress tensor and the distributed Lorentz force methods are applied to calculate the total force on the particles. The direct application of the Lorentz force requires the model of bound magnetic currents M¯b = −iωP¯m in (10) and bound magnetic surface charges ρm in (11). Agreement between the two methods is shown in Fig. 6 as a function of particle radius. The oscillations in force are a result of internal resonances, which is also evident for the case of dielectric and magnetic slabs incident by plane waves [14]. 0 200 400 600 800 1000 0 0.2 0.4 0.6 0.8 1 x 10-17 a [nm] Fx [N/m] Lorentz Tensor Figure 6. Radiation pressure on a magnetic cylinder (µp = 3µ0, p = 0) versus the radius a. The TE plane wave propagates in the ˆxdirection in free space (µb = µ0, b = 0), and the wavelength is 640 [nm]. The force is calculated by the divergence of the stress tensor (line) and the Lorentz force on bound currents and charges (markers). There are noticeable differences between the results of the two methods for larger values of a as shown in Fig. 6. This is due to increased spacial variations in force distribution for particles on the order of a wavelength or larger. The slow convergence of the total Lorentz force has been observed for small dielectric particles as shown in Fig. 3and becomes a major obstacle for obtaining many digits of accuracy from the Lorentz force for large particles. To illustrate this point, we compare the force calculated for the magnetic particle with radius a = 1000 nm with various number of integration points N in
