lorentz force particles

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J. of Electromagn. Waves and Appl., Vol. 20, No. 6, 827839, 2006

LORENTZ FORCE ON DIELECTRIC AND MAGNETICPARTICLES

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong

Research Laboratory of ElectronicsMassachusetts Institute of Technology

77 Massachusetts Ave., 26-305, MA 02139, USA

AbstractThe well-known momentum conservation theorem isderived specifically for time-harmonic fields and is applied to calculatethe radiation pressure on 2-D particles modeled as infinite dielectricand magnetic cylinders. The force calculation results from thedivergence of the Maxwell stress tensor and is compared favorablyvia examples with the direct application of Lorentz force to bound

currents and charges. The application of the momentum conservationtheorem is shown to have the advantage of less computation, reducingthe surface integration of the Lorentz force density to a line integralof the Maxwell stress tensor. The Lorentz force is applied to computethe force density throughout the particles, which demonstrates regionsof compression and tension within the medium. Further comparisonof the two force calculation methods is provided by the calculationof radiation pressure on a magnetic particle, which has not beenpreviously published. The fields are found by application of the Mietheory along with the Foldy-Lax equations, which model interactionsof multiple particles.

1. INTRODUCTION

The first observation of optical momentum transfer to small particlesin 1970 [1] prompted further experimental demonstrations of radiationpressure such as optical levitation [2], radiation pressure on a liquidsurface [3], and the single-beam optical trap [4], to name a few.Subsequently, theoretical models have been developed to describe theexperimental results and predict new phenomena, for example [59].However, the theory of radiation pressure is not new. In fact, thetransfer of optical momentum to media was known by Poynting [10]

from the application of the electromagnetic wave theory of light. Still,


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828 Kemp, Grzegorczyk, and Kong

ongoing work seeks to model the distribution of force on media byelectromagnetic waves [1114].

The divergence of the Maxwell stress tensor [15] provides anestablished 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 onmaterial media by the direct application of the Lorentz law has beenrecently reported [12]. The method allows for the computation offorce density at any point inside a dielectric [13] by the applicationof the Lorentz force to bound currents distributed throughout themedium and bound charges at the material surface, and the methodhas been extended to include contributions from magnetic media [14].A comprehensive comparison of the two methods applied to particleshas not been previously published, and, consequently, there exists somedoubt in regard to the applicability of one method or the other.

In the present paper, we compare the force exerted on 2-Ddielectric cylinders as calculated from the divergence of the Maxwellstress tensor and the distributed Lorentz force. First, the totaltime average force as given by the divergence of the Maxwell stress

tensor [16] is derived from the Lorentz law and the Maxwell equationsfor time harmonic fields. We demonstrate the numerical efficiency ofthe stress tensor method by computing the force on a 2-D dielectricparticle represented by an infinite cylinder submitted to multipleplane waves. Second, we give the formulation for the distributedLorentz force as applied to dielectric and magnetic media [1214]. Thenumerical integration of the distributed Lorentz force over the 2-Dparticle cross-section area demonstrates equivalent results, although

the convergence is shown to be much slower than the stress tensor lineintegration. Third, both the Maxwell stress tensor and the distributedLorentz force methods are applied to two closely spaced particles in thethree plane wave interference pattern, the former method exhibitingrobustness with respect to choice of integration path and the lattermethod providing a 2-D map of the Lorentz force density distributionwithin the particles. Finally, the first theoretical demonstration of theLorentz 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 ona material object in terms of the momentum of the incident andscattered fields at all times. It is derived from the Lorentz force lawand the Maxwell equations. In the case of time-harmonic fields, the


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Lorentz force on dielectric and magnetic particles 829

time-average force on a material body can be calculated from a singledivergence integral. The proof of this fact is shown by derivation ofthe momentum conservation theorem with the only assumption thatall fields have eit dependence.

The Lorentz force provides the fundamental relationship betweenelectromagnetic fields and the mechanical force on charges andcurrents [16]. The time average Lorentz force is given in terms ofthe electric field strength E and magnetic flux density B by

f =1

2Re

E + J B

, (1)

where and

J represent the electric charge and current, respectively,Re{} represents the real part of a complex quantity, and () denotesthe complex conjugate. The Maxwell Equations

= D

J = H + iD(2)

relate the sources and J to the electric flux density D and magnetic

field strength

H. Substitution yields

f =1

2Re

( D)E + ( H) B D (iB)

. (3)

After applying the remaining two Maxwell equations

0 = B

iB = E,(4)

the force can by expressed as

f =1

2Re

( D)E+ ( E) D + ( B)H+ ( H) B

. (5)

The momentum conservation theorem for time harmonic fields isreduced to

f = 1

2Re T(r), (6)

where f is the time average force density in N/m3, and the Maxwellstress tensor is [16]

T(r) =1

2

D E + B H

I DE BH. (7)In (7), DE and BH are dyadic products and I is the (3 3) identity

matrix. By integration over a volume V enclosed by a surface S and


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application of the divergence theorem, the total force F on the materialenclosed by S is given by

F =

1

2 Re

SdS

n

T(r)

, (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) isintegrated over a surface chosen to completely enclose the object.

We consider the two-dimensional (2-D) problem of a circularcylinder incident by three TE plane waves. The incident electric field

pattern |E| shown in Fig. 1 is due to three plane waves with freespace wavelength 0 = 532 [nm] incident in the (xy) plane at angles = {/2, 7/6, 11/6} [rad] with the electric field polarized in thez-direction. The polystyrene cylinder (p = 2.560) has a radius ofa = 0.30 and is centered at (x0, y0) = (0, 100) [nm] in a background ofwater (b = 1.690). The total field is obtained as the superposition ofincident and scattered fields, the latter is calculated from applicationof the Mie theory [8, 9].

Figure 1. Incident electric field magnitude [V/m] due to threeplane waves of free space wavelength 0 = 532 nm propagating atangles {/2, 7/6, 11/6} rad. The overlayed 2D particle is a cylinder(p = 2.560) of radius a = 0.30 and infinite length in z withcenter position (x0, y0) = (0, 100) [nm]. The background medium is

characterized by b = 1.690.


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To calculate the total force on the cylinder shown in Fig. 1, theMaxwell 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, whichwe evaluate by simple numerical integration. The path chosen is a

circle of radius R concentric with the particle and the integration steps(R) are assumed constant. The numerical integration is computedby

F = 1

2Re

2

0

n T(R, )Rd

R

Nn=1

1

2Re

n T(R, [n])

,

(9)

where N represents the total number of integration points and thevalues of [n] result from the discretization of [0, 2]. Figure 2shows the force versus the number of integration points for anintegration radius ofR = 1.01a. The results show that the integrationconverges rapidly. Because the force is calculated by a divergenceintegral, the result does not depend on the value ofR, provided enoughintegration points are chosen. To confirm this, the force was calculated

0 20 40 60 80 100-2

0

2

4x 10

-18

p

ba

R

y

x

Number of Integration Points

Fy

[N

]

Figure 2. y-directed force Fy versus the number of integration pointsused in the application of the Maxwell stress tensor in Eq. (9). Theintegration path, shown by the inset diagram, is a circle of radius ofR = 1.01a concentric with the cylinder of radius a = 0.30. The

configuration is the same as shown in Fig. 1.


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for various choices of integration radius yielding zero for all R < a andF = y2.1190 1018 [N/m] for all R > a, which is in agreement withthe value reported by [14].

3. LORENTZ FORCE ON BOUND CURRENTS ANDCHARGES

The Lorentz force can be applied directly to bound currents andcharges in a lossless medium [14]. The bulk force density in [N/m3] iscomputed throughout the medium by

fbulk =

1

2 Re{iPe B

iPm D

}, (10)

where the electric polarization Pe = (p b)E and themagnetic polarization Pm = (p b)H are given in terms ofthe background constitutive parameters (b, b) and the particleconstitutive parameters (p, p). The surface force density in [N/m

2]is given by

fsurf =1

2 Re{eE

avg + mH

avg}, (11)

where the bound electric surface charge density is e = n (E1 E0)b [12], the bound magnetic surface charge density is m = n (H1 H0)b [14], and the unit vector n is an outward pointing normalto the surface. The fields (E0, H0) and (E1, H1) are the total fields justinside the particle and outside the particle, respectively, and the fieldsin (11) are given by Eavg = (E1 + E0)/2 and Havg = (H1 + H0)/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 isobtained by integrating the bulk Lorentz force density fbulk over thecross section of the cylinder. The numerical integration is performedby summing the contribution from M discrete area elements. The areaelements A = xy are taken to be identical so that the numericalintegration is

F =

S

dA1

2Re{fbulk} A

Mm=1

1

2Re{iPe[m] 0H

[m]}, (12)

where the dielectric polarization Pe[m] and magnetic field H[m] areevaluated at each point indexed by m in the cross section of thecylinder. The y-directed force is plotted in Fig. 3 versus the number

of integration points. The integral converges much slower than the


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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. y-directed force Fy versus the number of integration pointsfor the direct application of the Lorentz force of (12). The configurationis the same as shown in Fig. 1. The dashed line is the force computedfrom the stress tensor of (9) with 100 numerical integration points ona concentric circle of radius R = 1.01a.

line integral applied to the stress tensor, however the resulting force is

F = y2.1191 1018

[N/m], thus matching the result from the Maxwellstress tensor.

4. FORCE ON MULTIPLE DIELECTRIC PARTICLES

The Mie theory and the Foldy-Lax multiple scattering equations areapplied to calculate the force on multiple particles incident by aknown electromagnetic field [8, 9]. We consider the same incidentfield 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. Asbefore, the 2-D polystyrene particles are modeled as infinite dielectriccylinders (p = 2.560) in water (b = 1.690) with radius a = 0.30.

The Maxwell stress tensor is applied to calculate the force on eachparticle by taking an integration path that just encloses each particleas shown in Fig. 4. The force for the particle at (x, y) = (0, 100) [nm]

is

F = y1.6517 1018

[N/m], and the force on the particle at


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p

p

b

yx

Figure 4. Two particles centered at (x, y) = (100, 0), (300, 0) [nm]are subject to the incident field pattern of Fig. 1. The integrationpaths for the Maxwell stress tensor applied to the two particles areshown by the dotted lines. The sum of the forces on the two individualparticles obtained by the smaller two integration circles is equal to theforce obtained by integrating over the large circular integration path.

(x, y) = ( 0, 300) [nm] is F = y1.4490 1018 [N/m]. By takingthe integration path surrounding both particles, the total force onthe system composed of both particles is F = y2.0269 1019 [N/m],which agrees with the sum of the individual forces. This exampledemonstrates that the divergence of the stress tensor gives the totalforce on all currents and charges enclosed by the integration path andthat the integration path needs not be concentric with the materialbodies.

For comparison with the stress tensor method, the Lorentz force isapplied to bound electric currents in both particles. The distributionof force densities are shown in Fig. 5. Although the x-directed forceintegrates to zero for both particles due to symmetry, it can be seen

that the local force densities vary throughout the particle. Theseforces act in compression or tension in the various regions of theparticle. The total force on each particle is found by integrationof the local force densities throughout the particles. The force forthe particle at (x, y) = ( 0, 100) [nm] is F = y1.6500 1018 [N/m]using 17, 534 integration points, and the total force on the particleat (x, y) = (0, 300) [nm] is F = y1.4523 1018 [N/m] using 17, 530integration points, which is agreement with the results of the Maxwell

stress tensor divergence.


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Figure 5. Lorentz force density [104 N/m3] for two particlespositioned at (x, y) = (100, 0), (300, 0) [nm] illuminated by the threeplane waves of free space wavelength 0 = 532 nm as shown in Fig. 1.The top plots are (a) y-directed force fy and (b) x-directed forcefx for the particle at (x, y) = ( 0, 100) [nm]. The bottom plots are

(c) y-directed force fy and (d) x-directed force fx for the particleat (x, y) = (0, 300) [nm]. The total Lorentz force on the particlesis obtained by summation of the force densities on the particle at(x, y) = ( 0, 100) [nm] (F = y1.6500 1018 [N/m]) and the particleat (x, y) = (0, 300)[nm] (F = y1.4523 1018 [N/m]). For bothparticles, the x-directed force is zero due to symmetry.

5. RADIATION PRESSURE ON A MAGNETICPARTICLE

The Lorentz force has been previously applied to bound magneticcharges and currents to calculate the radiation pressure on a magneticslab [14]. However, the method has not been used to determine theforce on magnetic particles. In this section, we calculate the radiationpressure on a 2-D magnetic particle represented by an infinite cylinder

incident by a single TE plane wave.


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The incident plane wave Ei = zEieik0x propagates in free space

(0, 0) with a wavelength 0 = 2/k0 = 640 [nm]. The 2-D magneticparticles (0, 30) are infinite in the z-direction, with radius a. TheMaxwell stress tensor and the distributed Lorentz force methods are

applied to calculate the total force on the particles. The directapplication of the Lorentz force requires the model of bound magneticcurrents Mb = iPm in (10) and bound magnetic surface charges min (11). Agreement between the two methods is shown in Fig. 6 asa function of particle radius. The oscillations in force are a result ofinternal resonances, which is also evident for the case of dielectric andmagnetic slabs incident by plane waves [14].

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1x 10

-17

a [nm]

Fx[

N/m]

LorentzTensor

Figure 6. Radiation pressure on a magnetic cylinder (p = 30, p =0) versus the radius a. The TE plane wave propagates in the x-direction in free space (b = 0, b = 0), and the wavelength is640 [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 twomethods for larger values of a as shown in Fig. 6. This is due toincreased spacial variations in force distribution for particles on theorder of a wavelength or larger. The slow convergence of the totalLorentz force has been observed for small dielectric particles as shownin Fig. 3 and becomes a major obstacle for obtaining many digits ofaccuracy from the Lorentz force for large particles. To illustrate thispoint, we compare the force calculated for the magnetic particle with

radius a = 1000 nm with various number of integration points N in


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Table 1. Radiation pressure on a magnetic particle (a = 1000 [nm]).

Number of Integration Points Total Lorentz Force Fx

25, 952 4.8177 1018

[N/m]103, 336 4.9175 1018 [N/m]

282, 868 4.9498 1018 [N/m]

Table 1. For each calculation, 200 points are used for the calculation offorce on bound surface charges, which was determined to be enough for

the number of significant digits reported. The force obtained from thestress tensor approach is 5.3524 1018 [N/m], which converges within100 points. It is clear that for this particular case, the total Lorentzforce converges so slowly that it could hinder studies involving manyvariables or multiple particles [8, 9]. For many applications, however,the Lorentz force is useful for getting a picture of force distributioninside the particle, while the divergence of the stress tensor is muchmore efficient for obtaining the total force on the particle.

6. CONCLUSIONS

We have calculated the radiation pressure on 2-D particles from boththe divergence of the Maxwell stress tensor and the direct applicationof the Lorentz force to bound currents and charges. The Maxwell stresstensor was derived for time harmonic fields from the Lorentz force and

the Maxwell equations. The Lorentz force is applied directly to boundcurrents and charges used to model dielectric and magnetic materials.The advantage of the stress tensor approach for force calculation on2-D particles, is that it reduces a combination of surface integralover the bulk force density fbulk in (10) and a line integral over thesurface force density fsurf in (11) to the line integral (8) over theMaxwell stress tensor in (7). This reduction in computation has beendemonstrated by example. The Lorentz force, however, can give the

force distribution throughout the particle. Such distributions can beimportant when optical forces are applied to sensitive objects such asin biological applications. For large particles, the total force may bedifficult to obtain accurately from the Lorentz force density, whichwe illustrated by calculating the radiation pressure on a magneticparticle. In applications of optical binding [8] such as building opticalmatter [17], the Lorentz force can be applied to map force distributionswithin particles, while the stress tensor approach can be used to find

the total force on the particle lattice.


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ACKNOWLEDGMENT

This work is sponsored by the Department of the Air Force underAir Force Contract FA8721-05-C-0002. Opinions, interpretations,

conclusions and recommendations are those of the author and are notnecessarily endorsed by the United States Government.

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