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.