PHYSICAL REVIEW E 85, 046606 (2012)

Electromagnetic forces and internal stresses in dielectric media

Winston Frias* and Andrei I. Smolyakov Department of Physics and Engineering Physics, University of Saskatchewan, 116 Science Place Saskatoon, SK S7N 5E2, Canada (Received 16 August 2011; revised manuscript received 26 January 2012; published 23 April 2012) The macroscopic electromagnetic force on dielectric bodies and the related problem of the momentum conservation are discussed. It is argued that different forms of the momentum conservation and, respectively, different forms of the force density, correspond to the different ordering in the macroscopic averaging procedure. Different averaging procedures and averaging length scale assumptions result in expressions for the force density with vastly different force density profiles which can potentially be detected experimentally by measuring the profiles of the internal stresses in the medium. The expressions for the Helmholtz force is generalized for the dissipative case. It is shown that the net (integrated) force on the body in vacuum is the same for Lorentz and Helmholtz expressions in all configuration. The case of a semi-infinite medium is analyzed and it is shown that explicit assumptions on the boundary conditions at infinity remove ambiguity in the force on the semi-infinite dielectric.

DOI: 10.1103/PhysRevE.85.046606 PACS number(s): 03.50.De

I. INTRODUCTION This article is organized as follows. In Sec. II, the problem of the momentum exchange and the force is explored from Electromagnetic forces on charged particles result in an the microscopic point of view. In Sec. III, the Abraham and overall force imparted on a body that is called radiation Minkowski expressions for the momentum and the stress (ponderomotive) pressure. Though these forces are small, tensor of the electromagnetic field in a medium are reviewed their applications are becoming increasingly important (e.g., and discussed from the perspective of the momentum exchange in manipulations of nanoparticles, optical trapping, and ma- between a different subsystem. In Sec. IV, an expression for nipulation of biological cells [1–6]). In general, the force is the macroscopic force density is derived by using different determined by the momentum conservation equation in the ordering of the averaging length scales. The force density is medium. Therefore the force exerted by the electromagnetic compared with the Lorentz force density obtained from the field on a body is an intrinsic part of the problem of the macroscopic Maxwell’s equations. In Sec. V, the radiation momentum of light and the form of the electromagnetic force on a finite length slab is calculated. In Secs. VI and stress tensor in a material medium. The two most famous VII, the force on a dielectric coating film is calculated using expressions suggested for the momentum of light in the two different formulations. In Sec. VIII, the force on a slab medium are those postulated by Abraham and Minkowski, separated from a semi-infinite medium by an air gap of finite respectively, given by gA = E × H/c2 and gM = D × B. width is calculated. In Sec. IX, the forces on a finite length Different expressions for the momentum result in different dielectric slab with dissipation and on a semi-infinite region expressions for the stress tensor and different expressions for are calculated. In Sec. X, the summary and conclusions are the radiation force on a material medium. There have been given. numerous expressions for the ponderomotive force published in the literature [7–11] and a number of attempts were made to resolve the problem experimentally [9,12–14]. There also exist II. MOMENTUM EXCHANGE AND FORCES a number of review papers that extensively cover the theoretical AT THE MICROSCOPIC LEVEL discussions of this problem as well as experimental results and interpretations [9,12]. In contrast to most of the previous Conceptually, the exchange of momentum and, respec- work, we consider one particular aspect that is notably missing tively, the forces on charged particles are easiest to consider at in previous discussions. We analyze the spatial distribution of the microscopic level. The Maxwell equations in microscopic the radiation forces and show how different expressions for form are macroscopic force correspond to the average of microscopic 1 ∂e forces. We argue here that the introduction of the macroscopic ∇ × b = μ i + , (1) 0 c2 ∂t force inevitably requires one or another form of averaging ∂b and that the various expressions for the radiation forces cor- ∇ × e =− , (2) respond to different sizes of the sampling volume (averaging ∂t length scale). Therefore, different expressions for the forces ∇ · e =ρ/ε0, (3) represent different levels of resolution at which the forces can be measured. Different force densities result in internal ∇ · b = 0, (4) stresses existing at different length scales of the averaging volume. where ρ and i are the microscopic charges and currents due to all charged particles and e and b are the microscopic electric and magnetic fields. From Eqs. (1)–(4), one can write the *[email protected] conservation of the momentum of the electromagnetic field in

1539-3755/2012/85(4)/046606(12)046606-1 ©2012 American Physical Society WINSTON FRIAS AND ANDREI I. SMOLYAKOV PHYSICAL REVIEW E 85, 046606 (2012) the form, also that the local instantaneous fields e and b do not involve correlated fast scale processes in which quadratic averages ∂ × + ∇ · =− + × ε0 ( e b) t (ρe i b), (5) may contribute to the slow scale evolution of the fields. ∂t     1 1 1 = − 2 + − 2 t ε0 ee e I bb b I . (6) III. ABRAHAM AND MINKOWSKI FORMS OF THE 2 μ0 2 MOMENTUM CONSERVATION AND THE LORENTZ FORCE FROM MACROSCOPIC EQUATIONS The term on the right-hand side of Eq. (5) represents the momentum exchange between the electromagnetic field and The expressions for the momentum and forces at the the charged particles. This term, which is simply the total microscopic level, Eqs. (5) and (6), illustrate the momentum Lorentz force on a charged particle, appears with the opposite conservation between two subsystems: The momentum lost by sign in the conservation law for the mechanical momentum of the electromagnetic field appears as the particle momentum. particles: This momentum exchange represents the force on all particles. d  In the microscopic equations both subsystems are well defined m v = f ≡ ρe + i × b, (7) and the momentum exchange and force can be clearly identi- dt i i i fied. The situation becomes less transparent for macroscopic Maxwell’s equations. where mvi is the mechanical momentum of the ith particle = − Let’s consider macroscopic Maxwell’s equations in the and the sum is taken over all particles, ρ i qi δ (r ri ), = − = form: i i qi vi δ (r ri ), vi dri /dt. The microscopic Maxwell stress tensor t in Eq. (6) describes the flux of the momentum ∂B ∇ × E =− , (10) in the electromagnetic field. ∂t Despite its simplicity and transparency, the conservation ∂D of the momentum in the form given by Eqs. (5) and (7) ∇ × H = , (11) is impractical in most cases, except when one is interested ∂t in stationary momentum exchange for a finite-size body in ∇ · D = 0, (12) vacuum immersed in a harmonic electromagnetic field. In the latter case, the total force on the body is obtained by integration ∇ · B = 0. (13) over the whole body (over a closed surface in vacuum just outside the body) and averaging over a time period longer than Assuming ideal dielectric and neglecting free charges and the wave period, T>2π/ω. Then the total stationary force on currents, by manipulating Eqs. (10)–(13) in the same way as the body is in Eqs. (1)–(4), one can construct a conservation equation of     the form, =   =− ∂ × + ∇ · F f dV dv ε0 (e b) t ∂gM  ∂t + ∇ · TM =−fH , (14) ∂t =− t · dS, (8) S where where “” means averaging over time. For stationary pro- gM = D × B (15) cesses, the average of the time-dependent term is zero and the force becomes equal to the total flux of the momentum is the Minkowski momentum [15], into the body (represented by the stress tensor t). The stress T M = 1 (E · D δ ) − E D + 1 (H · B δ ) − H B (16) tensor is calculated at the surface of the body (on its vacuum i,j 2 ij i j 2 ij i j side). Therefore it becomes equal to the stress tensor of the is the Minkowski stress tensor, and macroscopic field in vacuum which can be relatively easily calculated: fH =−1 E2 ∇ε − 1 H 2 ∇μ (17)     2 2 1 2 1 1 2 T = ε0 EE − E I + BB − B I . (9) is the exchange term. 2 μ0 2 Equation (14) has the structure of the conservation of the density of the Minkowski momentum, the term ∇ · TM The problem of averaging (length and time scales) is already describes the flux of the momentum, and the exchange term, apparent in the steps involved in Eq. (8). Averaging over the which is known as the Helmholtz force, is given by fH . whole body volume and taking the boundary of the integration Further identical manipulations with Maxwell’s equation volume just outside of the body allows one to replace the result in the momentum conservation in the Abraham form microscopic fields with the macroscopic ones, which can be [16]: easily justified in vacuum. In this case t → T and the surface values for the macroscopic field and stress tensor T can easily ∂gA + ∇ · TA =−fA, (18) be determined by solving the macroscopic scattering problem ∂t for a harmonic electromagnetic field incident on the body. The full time derivative term will average to zero only for periodic where processes when the time average is done over a time scale gA = E × H/c2 (19) longer than the wave period, T>2π/ω. One has to assume

046606-2 ELECTROMAGNETIC FORCES AND INTERNAL STRESSES ... PHYSICAL REVIEW E 85, 046606 (2012) is the Abraham momentum, and medium. Of course the fields themselves, E and B, are different A = 1 · − − in vacuum and medium. In the latter, the fields are created by Ti,j (E D δij Ei Dj Ej Di ) 2 the polarization charges and magnetization currents ρm and + 1 · − − J , while in vacuum, E and B are the source free vacuum 2 (H B δij Hi Bj Hj Bi ) (20) m field determined by the Maxwell equations and boundary is the Abraham stress tensor. In this formulation, the exchange conditions. term is given by The momentum conservation in the form given in Eq. (28) 1 1 εμ − 1 ∂(E × H) has an attractive feature of clearly showing the momentum fA =− E2 ∇ε − H 2 ∇μ + , (21) 2 2 c2 ∂t exchange between the macroscopic electromagnetic field in the medium and the medium itself (via the polarization charge which is different from the Helmholtz force in Eq. (17) by the and the polarization and magnetization currents). The time addition of the last term, the so-called Abraham force [16]. variation of the momentum density and flux in Eq. (28) refer It is important to note at this point that both formulations of to the electromagnetic field part only. The exchange term fL the momentum conservation (14) and (18) are mathematically has the simple interpretation of the total force acting on the equivalent and both must be correct, from a mathematical polarization and magnetization charges and currents due to point of view, as long as we accept as correct the macroscopic the macroscopic fields. Equation (28) represents a momentum Maxwell equations (10)–(13). balance in a clearly defined subsystem of the macroscopic Another and very attractive formulation of the momen- electromagnetic field. This momentum balance is in the proper tum conservation, with transparent physical meaning, can form of the conservation law (e.g, density of the momentum in be obtained from the macroscopic Maxwell equations with a given volume, flux of the momentum through the boundary, explicit source terms due to the polarization charges and and sink/source terms). The sink/source which represents the magnetization current in the absence of free charges and momentum lost by the fields is easily identifiable with the force currents: applied to the other subsystem, the material medium. This ρm =−∇ · P, (22) is the total electromagnetic force applied to the polarization ∂P charges and magnetization currents. J = ∇ × M + . (23) m ∂t We should note that contrary to the Lorentz force formu- lation (28), both the Abraham and Minkowski formulations The source terms in Eqs. (22) and (23) create macroscopic of the momentum conservation do not show clear separation electric and magnetic field according to Maxwell equations, of two different subsystems with a momentum exchange A M A ∂B between them. Obviously, both g and g ,aswellasT ∇ × E =− , (24) and TM , contain a mixture of electromagnetic field terms ∂t E and B, and material terms P and M (or D and H)[17]. ∂E ∇ × B = μ0Jm + μ0 ε0 , (25) Respectively, the momentum exchange terms for both the ∂t Abraham and Minkowski expressions (sink/source terms) are

∇ · E = ρm/ε0, (26) not easily identifiable with the momentum exchange between two different subsystems. Note that for Lorentz force the ∇ · B = 0. (27) momentum density and momentum flux terms in Eq. (28) contain only the electromagnetic field terms [and not a mixture Straightforward identical manipulations with Eqs. (22)– of fields and material terms as in Eqs. (14) and (18). (27) produce the following form of the conservation law: A notable feature of the Abraham and Helmholtz ∂g (Minkowski) forces, fA and fH , is that in the stationary case, + ∇ · T =−fL, (28) ∂t when the time-dependent Abraham force can be neglected, the forces occur at the interfaces of the inhomogeneous material where medium. The Abraham and Helmholtz (Minkowski) force

g = ε0 E × B (29) densities in the stationary case are zero inside the body even in those cases when the electromagnetic field amplitude is is the electromagnetic field momentum, inhomogeneous. For a finite slab of a homogeneous dielectric 1 subject to an incident plane electromagnetic field, the force T = ε E · E δ − ε E E i,j 2 0 ij 0 i j density will be a sum of two delta-function-like contributions 1 1 at the front and back ends of the slab. Only the field amplitudes + B · B δij − Bi Bj (30) at the interfaces enter the expressions for the force. Of course, 2μ μ 0 0 the field amplitudes at the boundaries (and hence the total is the flux of the field momentum, and force) implicitly depend on the internal distribution of the electromagnetic field. fL = ρ E + J × B (31) One can notice that formal transformations between the is the Lorentz force acting on the polarization charges and Abraham and Minkowski conservation forms involve the the magnetization and polarization currents. It is important to transformation between the volume and surface contributions. note that E and B are the macroscopic fields calculated inside Therefore, after volume averaging (volume integration) certain the medium and that both the electromagnetic tensor and the volume contribution will appears only in the form of surface momentum density have the same form in both vacuum and terms. This situation is fully equivalent to the usual pressure

046606-3 WINSTON FRIAS AND ANDREI I. SMOLYAKOV PHYSICAL REVIEW E 85, 046606 (2012) force used in fluid dynamics. The volume (bulk) gradient where ρ and J are the charge and current densities inside a, pressure force describes the internal pressure force. After which are in general, rapidly changing functions of the spatial averaging over a given volume, the total force is simply given coordinates. The electric and magnetic fields, E and B,inside by the surface contributions. the volume a are produced by the sources internal and external In summary of this section, we would like to emphasize to the volume. The part of the force produced by the fields of again that as long as one accepts the validity of the macroscopic the internal sources (which change very rapidly inside a) will Maxwell equations in the form (10)–(13) and (1)–(4), all three cancel out due to Newton’s third law, and the net force will forms of the conservation law are mathematically correct since be then produced only by the fields due to the external (to they were obtained by formal mathematical transformations the volume) sources, E and B.Wesetthesizea, such that of Maxwell’s equations. The important question is how the external fields can be expanded around the center of the the exchange terms in Eqs. (14), (18), and (28), given by volume as Eqs. (17), (21), and (31) should be interpreted, how these E(r) ≈ E(r ) + (r − r ) · ∇E(r ), (34) exchange terms are related to some measurable physical 0 0 0 forces, and where these forces are applied (what is the spatial B(r) ≈ B(r0) + (r − r0) · ∇B(r0). (35) distribution of the force density). The latter expansion requires the condition adimensions of the dielectric body. As a result, any  internal stresses are averaged out and reduced to the remaining Fl = {ρl r · ∇E(0) + Jl × (B(0) + r · ∇B(0))} dV surface contributions [18]. We show also that the force density a  given by the Lorentz force acting on the induced charge and = · ∇ + × current densities ρ =−∇ · P and J = ∂P/∂t + ∇ × M,isa (pl )E(0) Jl B(0) dV volume force and suggests an averaging sample of the scale  a at which the induced macroscopic charges and currents vary + Jl × r · ∇B(0) dV. (36) more slowly than the fields in the volume of interest. The a latter corresponds to neglecting the internal fields created by = the individual charges and currents inside certain macroscopic In Eq. (36), the definition of the dipole moment pl averaging volume. a ρl r dV was used. One can further convert the two integrals in the right-hand side of Eq. (36). Using the continuity equation ∂ρl/∂t =−∇ · IV. THE AVERAGING OF THE MICROSCOPIC FORCE Jl, the first integral can be expressed as   In this section, the macroscopic force is obtained by ∂ρl averaging the microscopic forces acting on the point charges F1i = εij k Bk(0) ∇ · (Jlr)j dV + Bk(0) rj dV ∂t comprising the medium. The total force is defined as the sum a a ∂pl of the forces on all particles inside a given volume. This = ε j B (0), (37) summation (averaging) eliminates the internal forces acting ij k ∂t k between the particles inside the averaging volume sample. ∂p F = l × B. Obviously, changing the size of the averaging sample changes 1 ∂t the expression for the force density. We review two standard derivations of the average force [17,19] and consider the For the second integral, using [19] physical conditions under which one or another approach is   1 applicable. x · r Ji dV =− x × r × J dV , Consider a volume of size a, where a is much smaller a 2 a than the physical size of the medium but much larger than we have  the distance between microscopic charges, so there are several 1 charges present inside the volume. For a dielectric, in the F2i = εij k (r × J) dV × ∇ Bk(0) absence of free charges, the volume a involves one neutral 2 a j atom (or a small group of atoms clustered in the molecule). = εij k (m × ∇)j Bk(0), (38) The total force on this volume element is = × ∇ ×  F2 (m ) B, = E + × B Fl qi ( (ri ) vi (ri )) , (32) where m is the magnetic moment of the dipole. Combining the a results from Eqs. (36)–(38), the total force on the lth dipole is where the sum is taken over all charged particles inside the given by volume a. This volume is labeled by the index l,soF is the l ∂pl total force acting on the volume a. One can also write Eq. (32) Fl = (pl · ∇)E + × B + (ml × ∇) × B. (39) ∂t in the form,  It is important to note that in this derivation p and m are F = ρE + J × B dV, (33) l l a constants and the electromagnetic field is assumed to be slowly

046606-4 ELECTROMAGNETIC FORCES AND INTERNAL STRESSES ... PHYSICAL REVIEW E 85, 046606 (2012) varying on the length scale a. Now, one can group individual assumed to be uniform over the scale length a. The weight dipole elements into a larger sampling size Lm, aelectric charge within the averaging volume (polarization dividing the net force by the volume one gets the force density, charge density),  d + = + · ∇ f = (1/V) Fl w(r d) w(r) (d )w(r). (45) l Then, the integrals inside the brackets in Eq. (44) yield the ∂P macroscopic Lorentz force density which was obtained in the = (P · ∇)E + × B + (M × ∇)B. (40) ∂t previous section, Eq. (31) [17],   ∂P In this last equation, the density of polarization and fL = (−∇ · P) E + + ∇ × M × B. (46) magnetization vectors P and M are defined as [21] ∂t   P = (1/V) pl, M = (1/V) ml. The two expressions in Eqs. (40) and ((46) ) are related as l l fd = fL + ∇ · (PE − BM + B · MI) . (47) Note that in the transformation from Eqs. (39) to (40) Furthermore, the dipole force is related to the Helmholtz the electromagnetic field is assumed to be slowly varying on force via the length scale Lm (strictly speaking one requires that the ∇ ∇ d = H + ∇ · 1 · + 1 · gradients of the field E and B are approximately uniform f f 2 P EI 2 M BI . (48) on the length scale L ). Therefore, the dipole force in Eq. (40) m From Eqs. (47) and (48), it can be seen that the force is a result of the averaging over length scale L which involves m densities differ by the full derivative of a tensor. This tensor many individual dipoles with approximately uniform gradients should be interpreted as an internal pressure (stress) tensor that of the electromagnetic field over L . m can, in principle, be measured. As can be seen from Eqs. (47) It is worth noting that Eq. (40) is valid for dispersive and (48), upon integration over the whole volume of the body, media, provided that the dispersion of P(E,ω) and M(B,ω)are the internal pressure terms will vanish when the surface of the defined. In particular, for a plasma, where P =−ω2 E/ω2, pe integration volume is extended into vacuum where polarization the force density in Eq. (40) can be reduced to the familiar and magnetization are zero and all the expressions in Eqs. (14), expression for the ponderomotive or Miller force, proportional (18), (28), and (41) are equivalent. In the case of nonmagnetic to the gradient of E2, media, when M = B = 0, and for transverse waves when ω2 |E|2 ∇ · (PE) = 0, the dipole and the Lorentz force are equal, as can fd =− p ∇ . ω2 4 be seen from Eq. (47). For nonmagnetic media, the Helmholtz force differs from both the dipole and the Lorentz forces by a A conservation equation can be constructed that involves factor 1 ∇ (P · E). the dipole force fd similar to Eq. (28): 2 ∂g + ∇ · Td =−fd , (41) V. PONDEROMOTIVE FORCE DENSITY IN A ∂t DIELECTRIC SLAB OF THE FINITE LENGTH where As it was shown in the previous sections, in general, g = ε0 E × B, (42) for a lossless medium, the different formulations for the ε ponderomotive force may have a surface contribution as in d = 0 · − − Ti,j E E δij ε0 Ei Ej Pi Ej Eqs. (17) and (21), or a volume contribution as in Eqs. (40) 2 and (46). In the ideal case (absence of the dissipation) the B · B Bi Bj + δij − M · B δij − + Mi Bj . (43) surface part is represented by delta functions localized at the 2μ0 μ0 interfaces, where the dielectric index is discontinuous. In this section, we will calculate and compare the force profiles for a There exists a different ordering of the length scales leading nondissipative slab of a finite and fixed length d. to a different expression for the macroscopic force. In this We are concerned only with the interaction of monochro- common approach [17,22], an opposite assumption is made matic high-frequency electromagnetic fields with homoge- with respect to the relative length scale of the charge and neous dielectric, nonmagnetic media. In this case, the time current distribution and the electric field. In this approach the average quantities can be replaced with their respective aver- averaging is done over the volume size a  L by using some ages over a wave cycle. This way, the full time derivatives of weight function w [17,19], such that   the momentum density terms ∂t g and similar term in the force (such as Abraham force) will average to zero. Furthermore, f = ρ(r) w(r − s) d3s E(r) the conservation equations will reduce to the form, L   ∇ · T =−f. (49) + J(r) w(r − s) d3s × B(r), (44) L The volume size a involves many individual charges which The total force on the body can be calculated by integrating are summed up, but the electric and magnetic fields are the force density over the whole volume of the body (or

046606-5 WINSTON FRIAS AND ANDREI I. SMOLYAKOV PHYSICAL REVIEW E 85, 046606 (2012) equivalently, integrating the stress tensor over the surface of 0.15 the body) as in Eq. (8),

  2 | 0.1  =−   =  · 0 F f dV dV T dS. (50) E |

V s 1

k 0.05

Consider a nonmagnetic dielectric slab of thickness d.The 1) slab is characterized by a dielectric constant ε and bounded − 0 ε ( by vacuum (air) on both sides. Taking the z axis as normal to 0 −0.05 the slab and linearly polarized monochromatic light normally /ε ) incident on the slab, the electromagnetic fields E and B are z ( −0.1 given by f E = [E ,0,0]e−iωt, (51)  x  0 1 2 3 4 i ∂E k z/π B = 0,− x ,0 e−iωt. (52) 1 ω ∂z FIG. 1. (Color online) Density of the Lorentz force in a dielectric For the transverse waves in Eqs. (51) and (52), the electric slab, as calculated from Eq. (55). contribution from the dipole force (P · ∇)E and the Lorentz force (−∇ · P)E are zero. As a result, the dipole and Lorentz where R and T are the reflection and transmission force densities are the same and given by the force density coefficients. acting on the polarization current [10], As can be seen from Fig. 1, the Lorentz force density   has a standing wave spatial profile inside the slab, being 1 ∂P ∗ fd =fL= Re × B , (53) positive or negative according to the sign of sin(2k (d − z)). 2 ∂t 1 The total force oscillates as a function of the slab length and where P = ε0(ε − 1) E. In what follows, we refer to both only dielectric constant according to oscillations of the reflection as Lorentz force density/force. coefficient as shown in Figs. 2(a) and 2(b). Because the total For a dielectric slab, the electric field is given by force is proportional to |R|2, it is always positive (pushes the

ikv z −ikv z slab), despite the fact that the force density oscillates between Ex = (e + Re )E0 z>0, − positive and negative values. The total force is zero when when = ik1z + ik1z Ex (Ae Be )E0 0

046606-6 ELECTROMAGNETIC FORCES AND INTERNAL STRESSES ... PHYSICAL REVIEW E 85, 046606 (2012)

0 0.14

−0.01 2 2 | | 0 0 −0.02 E | E | 0

0.07 0 −0.03 F/ε F/ε

−0.04

0 0 1 2 3 4 5 −0.05 0 2 4 6 k1d/π k1d/π (a) FIG. 3. (Color online) The total force on the coating film as calculated from Eq. (62); ε1 = 1.5andε2 = 2.25. 0.6 is given by

ikv z −ikv z Ex = (e + Re )E0 z>0, 2 | − 0 ik1z ik1z Ex = (Ae + Be )E0 0 permittivity ε1 and thickness d on a semi-infinite substrate with dielectric constant This expression shows that the Lorentz force on the λ/4 ε2. The electromagnetic wave is incident from the vacuum coating is always negative, so that it pulls the coating from region on the left. For such a configuration, the electric field the substrate.

046606-7 WINSTON FRIAS AND ANDREI I. SMOLYAKOV PHYSICAL REVIEW E 85, 046606 (2012)

0.15 0.5

0.1 0 2

| 0.05 2 0 | 0 E −0.5 | 0 E 0 | 0

/ε −0.05 F H ) −1 2 z L ( F/ε F

f −0.1 −1.5 −0.15

−0.2 −2 0 0.5 1 1.5 0 1 2 3 4 5 z/d ε2

FIG. 4. (Color online) Force density in a coating-substrate system FIG. 5. (Color online) Force on a quarter-wavelength coating as from Eq. (54). The force density is zero inside the substrate (z>d); a function of the dielectric constant from Eqs. (64) and (72). ε1 = 1.5andε2 = 2.25. term in Eq. (67), plays the role of a “surface tension,” similar The force density in the coating-substrate system is shown to the surface tension at the interface between liquid and in Fig. 4. One important feature to note from Fig. 4 is that the gaseous phases. The surface contribution occurs as a result Lorentz force density is zero (in absence of dissipation) inside of the internal stress at such a boundary. the semi-infinite substrate, for z>d.Alternatively, this can be After the integration, these forces become illustrated by the running integral of the force density defined   Sε |E |2 by the expression, H =− 0 0 −  F1 (ε1 1) z  2   L = F (z) zˆS f (z )dz . (65) ε1 + ε2 + (ε1 − ε2) cos(2k1d) 0 × , (68) |g|2 The total force is accumulated in the coating layer only.     Sε |E |2 4(ε − ε )ε For z>d, the total integrated force in Eq. (65) remains H = H − 0 0 2 1 1 F2 F1 (69) constant [only the coating makes a contribution to the integral 4 |g|2 in Eq. (65)]. Therefore, the force density is zero in the Sε |E |2 = 0 0 (1 +|R|2 − ε |T |2). (70) semi-infinite region, which superficially may lead to a certain 2 2 paradox. It is obvious, however, that a single monochromatic Clearly these two expressions for Helmholtz force are wave, as in the region z>d,would not create any force in different from each other and different from the Lorentz force the semi-infinite region without dissipation. The case of a in Eq. (63). For the case of the λ/4 coating, Eqs. (68) and (69) semi-infinite dielectric region requires a special treatment and become it is discussed in Sec. IX.   Sε |E |2 √ H =− 0 0 − F1 ( ε2 1), (71) VII. HELMHOLTZ FORCE ON A DIELECTRIC 4 COATING FILM   Sε |E |2 √ H =− 0 0 − F2 ( ε2 1). (72) The calculation of the Helmholtz force on the coating 2 film may become ambiguous because of the delta-function It is interesting to note that the Lorentz force expression for contributions at the interfaces. the λ/4 coating is always negative, while the Helmholtz force Since the Helmholtz force density is a surface force density in Eq. (72) is negative for ε2 > 1 and positive for ε2 < 1. The and it is localized at the discontinuities of ε, the result for the effect of the boundary at z = d is to double the amplitude of total Helmholtz force will depend on whether the boundary theHelmholtzforceontheλ/4 layer. The forces on the λ/4 between ε1 and ε2 at z = d is included in the integration antireflective coating, calculated from Eqs. (64) and (72),are volume. One can define the force in two different ways, the shown in Fig. 5. first where the boundary is not included, and the second with The different expressions (64) and (72), resulting in the boundary included: different forces (and also in forces of a different sign),  can potentially be tested experimentally. Another important   1 d− ∂ε FH =− S |E |2 dz, (66) difference is a finite and oscillating force density for the 1 4 x ∂z  0 Lorentz force, Eq. (62), while the Helmholtz force has zero   1 d+ ∂ε volume density. The oscillating force density of the Lorentz FH =− | |2 2 S Ex dz force would result in oscillating internal stresses, contrary to 4 0 ∂z  + the uniform stress due to the Helmholtz force. This difference   1 d ∂ε = H − | |2 F1 S Ex dz, (67) offers an interesting possibility of experimental verification. 4 d− ∂z The net Helmholtz force is also different depending on whether where is an infinitesimally small parameter. A finite the λ/4 layer is fused to the substrate or freely placed next to contribution from the interface at z = d, given by the second it (with an infinitely thin air gap). The experiment can also be

046606-8 ELECTROMAGNETIC FORCES AND INTERNAL STRESSES ... PHYSICAL REVIEW E 85, 046606 (2012) conducted by measuring the force on the layer separated from 0.4 F L the substrate by an air gap of finite width. We analyze this configuration in the next section. 0.3

2 0.2 VIII. THE FORCE ON A DIELECTRIC FILM SEPARATED | FROM SUBSTRATE BY AN AIR GAP OF FINITE 0 E | 0.1 WIDTH 0

As it was noted in the previous section, due to surface F/ε 0 contributions, the net Helmholtz force is different depending on whether the λ/4 layer is fused to the substrate or freely −0.1 placed next to it. On the contrary, the net Lorentz force is the same independently on whether an infinitely thin air gap is −0.2 0 1 2 3 4 5 6 introduced between the coating and the substrate. This can be k L/π viewed as a limit case a finite air gap. The latter configuration v is also of interest for experimental studies. FIG. 6. (Color online) Total force on the dielectric layer as a Consider a slab of width d with dielectric constant ε , 1 function of the separation between the slab and the substrate; ε = separated from a semi-infinite region of dielectric constant 1 1.5, ε2 = 2.25 and k1d = π/2. ε2 by an air gap of width L. For this configuration, the electric field is given by Let us calculate the force on a semi-infinite region by

ikv z −ikv z using the results of the previous section [e.g., the force on a Ex = (e + Re )E0 z>0, semi-infinite region can be obtained by generalizing the result ik1z −ik1z Ex = (Ae + Be )E0 0

v | with periodicity of π/2, starting from kvL = π/4. In the 0 E → | 0.05 limit L 0, the force reduces to the negative value given 0 by (64).InFig.7, the force on the dielectric layer is drawn as a function of the width of the layer for different values of the F/ε phase kvL. 0

IX. THE FORCE ON A SEMI-INFINITE REGION AND ON −0.05 A FINITE SLAB WITH DISSIPATION 0 2 4 6 k1d/π It has already been noted in Sec. VI when analyzing the force on a coating, that the force density in the semi-infinite FIG. 7. (Color online) Total force on the dielectric layer as a substrate was zero. This may lead to a number of superficial function of the width of the layer for different values of the distance paradoxes. between the slab and the substrate.

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At first sight, the results in Eq. (76) and in Eq. (77) are 10 L f ,εi =0.5 obviously inconsistent with each other. This inconsistency H f ,εi =0.5 L occurs due to the nonregular nature of the semi-infinite 8 f ,εi =1.0 f H ,ε =1.0 dielectric case without dissipation due to failure to properly 2 i | account for the wave momentum at infinity. This is also 0

E 6 | apparent from the expression in Eq. (57): This expression 0 remains oscillatory for any d →∞. A finite length slab /ε ) 4 z without dissipation cannot be extended to infinity without ( explicit specification of the boundary condition for the wave f amplitude (and the wave momentum) at the opposite end of 2 the slab. →∞ 0 The natural way used to deal with the z behavior in a 0 0.2 0.4 0.6 0.8 1 semi-infinite region is to assume an infinitely small dissipation z/d in the dielectric [7,8,11,23,24]. In a semi-infinite medium, even with an infinitesimally small dissipation, all the momentum FIG. 8. (Color online) Lorentz force density and volume part of transmitted to the dielectric will be absorbed, thus generating the Helmholtz force density for the case of a dissipative slab as given = a finite force on the dielectric, while the force density goes by Eqs. (78) and (81); εr 2.25. to zero. Such calculations can easily be done starting from the case of a finite length slab with dissipation, with vacuum bulk (volume) part that depends on the imaginary part of the regions on both sides. Assuming that the wave vector and dielectric constant. The cumulative integral of the Helmholtz force density in the dielectric constant are complex such that k = kr + iki and ε = ε + iε, the Lorentz force density is given by the dissipative case, given by r i  z ε L 0 2 −2ki z 2 2ki z  H = H f =zˆ {(ki − εr ki + εi kr )(|A| e −|B| e ) F (z) zˆS f (z )dz , (82) 2 0 ∗ −2ikr z − 2(kr − εi ki − εr kr )Im(A Be )}. (78) has jumps at the points z = 0 and z = d, which correspond to − the effects of the surface contributions in the Helmholtz force The force density is localized in the region of the order of k 1 i density as can be seen in Fig. 9. and decays away from the boundary. Note that in the absence The total force on the dissipative slab is calculated by of dissipation (k = ε = 0), Eq. (78) becomes equivalent to i i integrating the expression in Eq. (81) over the length of the Eq. (54). The Lorentz force on the slab with dissipation is then slab. By solving for the fields inside the slab and applying obtained by integrating Eq. (78) over the length of the slab. boundary conditions, the Helmholtz force on a slab with The result of the integration for a slab of finite length with dissipation immersed in vacuum can be shown to be equal dissipation can be written in the form, to | |2 2 L Sε0 E0 2 2 Sε |E | F =zˆ (1 +|R| −|T | ). (79) FH =zˆ 0 0 (1 +|R|2 −|T |2). (83) 2 2 This expression is similar to the one obtained for the case of An important result is that this expression is identical to the the nondissipative slab, Eq. (57), however, in the presence of total Lorentz force, Eq. (79). It is worth noting that while the dissipation, the equality |R|2 +|T |2 = 1 is no longer valid, total force applied to the slab are the same in both formulations, where T is the transmission coefficient defined for the wave the spatial profiles for the Lorentz force and Helmholtz transmitted into the vacuum region on the right of the slab. expressions are dramatically different. A comparison of the It is important to note that the expression in Eq. (17) for the Lorentz force density and the volume part of the Helmholtz Helmholtz force is valid only for the case of real ε and μ and force density for the dissipative slab is shown in Fig. 8.Note thus cannot be used to analyze the dissipative case. We need to generalize the expression for the Helmholtz force to account for finite dissipation. From (16) one finds 0.4

1 ∂ 1 ∂ 2 0.3 H = − | fi Dj Ej Ej Dj . (80) 0 ∂x ∂x 0.2 2 i 2 i E | 0 0.1

Assuming, as for the Lorentz force, a complex dielectric /ε = + ) constant in the form ε εr iεi and the constitutive relation z 0 = ( D ε0εE, the time average of Eq. (80) reduces to F -0.1     1 ∂ 1 ∂ε -0.2 f H =− ε Im E E∗ − |E |2 r . (81) z 2 i x ∂z x 4 x ∂z 0 0.5 1 1.5 z/d It follows from (81) that in the general case of a complex ε, in addition to the surface part that depends on the real part of FIG. 9. (Color online) Integral of Helmholtz force density in a the dielectric constant, the Helmholtz force density acquires a dissipative slab as given by Eq. (82); ε = 2.25 + 0.1i.

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given by Eq. (17)], in the absence of dissipation, is exclusively 0.5 a surface force density and is localized at the boundaries, while the Lorentz force density given by Eq. (31) is a oscillating 0.4 volume force density as can be seen in Fig. 1. It has been 2 |

0 shown here that different expressions for the force density

E 0.3 | correspond to different assumptions on the averaging length 0 scale inherently assumed in any macroscopic model. 0.2 F/ε The identical transformations between different forms of the momentum conservation (respectively, between different 0.1 forms of the force density) involve transformations between volume and surface terms. The volume terms in the force 0 0 0.5 1 1.5 2 density correspond to the internal stress (pressure) forces. k d i Volume averaging leads to cancellation of internal forces so that only the surface (with respect to the averaging volume size) FIG. 10. (Color online) Force per unit area on a dissipative slab remains. Obviously, the assumed ordering of the averaging asgivenbyEqs.(79) and (83); ε = 2.25 + 0.1i. As can be seen, for length scales and the length scales for the electric charge k d  1, the force tends to a constant value. i and electromagnetic field distribution affect the structure of that the Helmholtz force has additional surface contribution the internal forces and eventually the expressions for the not shown Fig. 8. force density. Such difference is demonstrated explicitly via An example of the total force on a slab with dissipation as the direct averaging of microscopic forces as in Sec. IV. calculated in Eqs. (79) and (83) is given in Fig. 10 as a function This derivation is also valid for a dispersive medium, as in of the slab length. At small k d, the force amplitude oscillates plasma. i Two different expressions for the force density, dipole force and saturates to a constant value for large ki d. In the limit of large length (or strong dissipation), when (40) and Lorentz force (31), are derived directly in Sec. IV 2 by making different assumptions on involved length scales ki d  1, we have that |T | → 0 and the problem of a finite slab with dissipation and a semi-infinite medium (with in- for the electromagnetic field and charge distribution. For the finitely small dissipation) are equivalent. In this case, Eqs. (79) transverse electromagnetic waves, these two expressions are and (83) give [7] identical and produce the force density that follows a standing wave pattern inside the body as it is illustrated in Fig. 1.Itis Sε |E |2 worth noting here that, on the contrary, the Helmholtz force FL=FH =zˆ 0 0 (1 +|R|2). (84) 2 density for a nondissipative media is given by delta-function surface force. One can argue therefore that the Helmholtz force This force corresponds to the momentum of the electro- density corresponds to the averaging sample of the size of the magnetic field transmitted into the dielectric and eventually whole body. While the total force acting on a body immersed absorbed over the large (infinite) length. When the dissipation in vacuum is the same for different forms of the momentum is absent, the wave momentum is carried to infinity. The conservation, the difference in force density profiles and in expression in Eq. (84) is consistent with simple corpuscular internal stresses as discussed in this paper, in principle, can momentum balance similar to (60), assuming that all the be measured experimentally. The experimental measurements momentum inside the medium is eventually absorbed at of the internal stress (together with the mechanical stress infinity. The conservation of the momentum gives properties of the material) may provide the information on the F = hk¯ [Ni + NR] actual force density. The measurements conducted at different 2 resolutions can be used to confirm the point of view advocated = hk¯ Ni (1 +|R| ) in this paper: Different expressions for the force density exist at = 1 ε |E |2(1 +|R|2), (85) 2 0 0 different length scales corresponding to the different averaging where the wave vector k is in vacuum, and Ni = w/ (hω¯ ), (sampling) volumes. 2 w = ε0|E0| /2 . The total force on the dielectric plane slab was calculated in this paper for several configurations. For a single dissipative slab, this force is given by X. DISCUSSION AND SUMMARY ε |E |2 Maxwell equations lead to several possible forms of the F = 0 0 (1 +|R|2 −|T |2). (86) conservation equation for a momentumlike quantity, as in 2 Eqs. (14), (18), (28), and (41). All these forms are mathe- Here T is the wave transmission coefficient for the outgoing matically equivalent within the Maxwell macroscopic electro- wave in vacuum region on the right outside of the slab. This dynamics. However, the separation between the material and expression applies both for dissipative and nondissipative field parts of the momentum is not always clearly defined. As cases. We have shown that the same result can be obtained a result, the exchange term in these equations is not clearly from the Helmholtz force generalized to the dissipative case. identifiable as the density of the electromagnetic force acting In the nondissipative case, the condition |R|2 +|T |2 = 1 on the medium. It is shown here that different expressions for can be used, so that the force reduces to the expression 2 the force density with different structure have substantially F = ε0|E0||R| . Equation (86) is valid for arbitrary properties different spatial profiles (e.g., the Helmholtz force density of the plane slab dielectric, including those of metamaterials

046606-11 WINSTON FRIAS AND ANDREI I. SMOLYAKOV PHYSICAL REVIEW E 85, 046606 (2012) and negative refraction media. The net force always remains boundary, arising from the break in the homogeneity of the positive (pushes the slab away). space caused by the interface (the discontinuity in ε). The fact Equation (86) can also be used to determine the force on that this inhomogeneity is accompanied by a surface force a semi-infinite region. In a semi-infinite case, one needs to is a clear indication that the Helmholtz force corresponds to specify the fate of the momentum flux at infinity. For any real- the conservation of a pseudomomentum [27]. The striking istic conditions, even an infinitesimal dissipation will lead to difference in the force density profiles of the Lorentz and an eventual damping of the outgoing wave in the semi-infinite Helmholtz forces would result in different distributions of the region, so that T = 0inEq.(86). The resulting net force internal stresses that can be potentially detected in experiment. 2 F = ε0|E0|(1 +|R| )/2 is consistent with the corpuscular mo- We have calculated the total force on the dielectric slab mentum balance as discussed in Sec. IX: The total force is due coating as well as in the case of the slab separated from the to the absorption of the net wave momentum in the medium. It substrate by a finite air gap. For this case, the force amplitude is worth noting that the force density is infinitesimally small for oscillates and can be both positive or negative depending on vanishing (but finite) dissipation, while the integral for the total the slab width and distance from the substrate. This effect force remains finite. We have generalized the expression for may possibly be used for aggregation and separation of the Helmholtz force density to include dissipation and have nanoparticles. shown that the resulting Helmholtz force is identical to the Lorentz force. In addition to the negative delta-function surface ACKNOWLEDGMENTS contribution at the boundary, the Helmholtz force density has also a bulk contribution, which is positive, so the net force This work was supported by Natural Sciences and Engi- becomes positive as shown in Fig. 10. Without dissipation, neering Research Council of Canada. Stimulating discussions the Helmholtz force has only a negative part localized at the with A. Hirose are acknowledged with gratitude.

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