Electromagnetic and Material Contributions to Stress, Energy, and Momentum in Metamaterials

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ADVANCEDELECTROMAGNETICS, VOL. 6, NO. 1, JANUARY 2017

Electromagnetic and material contributions to stress, energy, and momentum in metamaterials

Brandon A. Kemp1 and Cheyenne J. Sheppard2

1College of Engineering, Arkansas State University, Jonesboro, AR, USA 2College of Sciences and Mathematics, Arkansas State University, Jonesboro, AR, USA *corresponding author, E-mail: [email protected]

Abstract all use the common Minkowski fields. In this correspondence, we demonstrate modeling of the We demonstrate modeling of the field-kinetic and material field-kinetic and material response subsystem for dielectric response subsystem for various media and extend the mod- media and extend the models to negative index metamate- els to dispersive negative index metamaterials. It is shown rials, which necessarily include dispersion and loss. In this that neither the Minkowski or Abraham models are uni- model, we consider the total SEM tensor as a conservation versally correct, as demonstrated to describe metamaterials of energy and momentum such that under both the field-kinetic and wave SEM models for various applications such as negative refraction, perfect lensing, Ttotal = Tmech + Tcan = Tmech + Tmat + TFk , (1) and invisibility cloaking.

TFk + Tmat 6= TMin (2) 1. Introduction where Tmech represents an external mechanical input of Electromagnetic momentum has been debated since the work, Tmat is the material response subsystem, and TFk early 20th century. The so-called Abraham-Minkowski is the field-kinetic subsystem. We show that the field- debate is attributed to two independent stress-energy- kinetic subsystem as given by the Chu formulation rep- momentum (SEM) tensors postulated for mathematically resents the energy and momentum contained within the modeling electrodynamics of media [1]. Significant atten- fields [4], and the sum of the field-kinetic and material re- tion has been given to the momentum density expressions sponse subsystems represent the wave or canonical subsys- associated with the rival SEM tensors. The momentum den- tem [5], which reduces to the Minkowski subsystem only sity expressions were defined as being either D¯ × B¯ from under negligible dispersion [6]. Our conclusion is that nei- ¯ ¯ the Minkowski tensor or 0µ0E ×H from the Abraham ten- ther the Minkowski or Abraham models are universally cor- sor, where D¯ and B¯ represent the electric displacement and rect. We demonstrate our view applied to the physics of magnetic induction fields, E¯ and H¯ are the electric and mag- negative index materials under both the field-kinetic and netic fields, and 0 and µ0 are the permittivity and perme- canonical SEM models for applications such as negative re- ability of vacuum, respectively. Barnett presented a resolu- fraction, which is fundamental to perfect lensing and invis- tion to the photon momentum controversy in 2010, which ibility cloaking. First, we review the mathematical frame- related the Abraham density to the kinetic momentum of work of SEM tensors, relativistic invariance requirements, light and the Minkowski density to the canonical momen- and applicability to electrodynamics in Section 2. Second, tum of light [2]. However, the original debate was in re- the field-kinetic and canonical SEM tensors are determined gard to the relativistic invariance of the 4 × 4 SEM tensors; for a broad class of causal, isotropic media in Section 3. Barnett’s partial resolution only identified the kinetic and Then, we discuss the implications of negative refraction in canonical momentum densities. Section 4. Conclusions are presented in Section 5. Three other well-known SEM tensors have been pro- posed over the past century, and have found significant 2. SEM Tensors use in scientific and engineering modeling applications [3]. First, the Einstein-Laub tensor, like the Abraham and Energy and momentum continuity for a given system can Minkowski tensors, utilizes the Minkowski fields, and it be divided into subsystems and mathematically represented shares the Abraham momentum density. Second, the Am- as [1, 3] perian or “Lorentz” tensor derives from the Amperian for- ∂W (¯r, t) ϕ (¯r, t) = −∇ · S¯ (¯r, t) − j (3a) mulation or EB-representation and utilizes the Amperian j j ∂t fields. Third, the Chu formulation or EH-representation ¯ ¯ ¯ ∂Gj(¯r, t) was developed in the 1960’s, and the Chu SEM tensor uti- fj(¯r, t) = −∇ · T¯j(¯r, t) − (3b) ∂t lizes the Chu fields. It is important to note that the latter ¯ ¯ two formulations uniquely define the fields within matter, where fj is the force density, ϕj is the power density, T¯j while the Abraham, Minkowski, and Einstein-Lab tensors is the momentum flux or stress tensor, S¯j is the power flux, ¯ ¯ ¯ ¯ Gj is the momentum density, and Wj is the energy density. so that · Tj = Fj, where = [∇, ∂/∂(ict)], Fj = ¯ Each j may represent any subsystem. Closing the overall [−fj, −iϕj/c], and the SEM tensor is P ¯ P system, such that j fj = 0 and j ϕj = 0 indicates ¯ ¯ momentum and energy conservation, respectively. ¯ Tj −icGj Tj = i ¯ . (5) The choice of which momentum density, stress tensor, − c Sj Wj and force density are applied in any given problem are in- ¯ ¯ herently tied to the formulation of electrodynamics being Here, we have defined T¯j ≡ −T¯j. considered and the interpretation rendered. It is impossi- ble to sidestep the momentum controversy by employing a 2.1. Equivalence of total force Lorentz force density since, as the theoretical construction ¯ implies, each force density is tied to a momentum density The electromagnetic force density f and power density ϕ and stress tensor through a formulation of Maxwell’s equa- inside an object depend upon formulation applied. In gen- tions [3]. One may rearrange the continuity equations, but eral, the equations take the form of Eqs. (3), where the force densities are defined by the corresponding momentum den- such mathematical exercises should not be taken as reason ¯ ¯ for interpretation [4, 7]. For reference, Table 1 lists the lead- sity G and stress tensor T given in Table 1 and the power densities ϕ are defined by the corresponding energy density ing electromagnetic momentum densities and stress tensors ¯ from the literature. The corresponding force densities for W and power flux S given in Table 2. The total electromagnetic force F¯(t) on an object can be computed by in- the stationary [1] and fully relativistic formulations [3] can ¯ also be found in the literature. tegrating an electromagnetic force density f(¯r, t) over the volume V of the object. An exact, mathematically equivalent approach is to apply the divergence theorem to reduce Table 1: Leading electromagnetic momentum densities and the contribution of an electromagnetic stress tensor T¯ to an stress tensors. integral over the surface A with outward pointing area ele- ¯ G¯ T¯ ment dA enclosing the volume V so that the total force is ¯ ¯ 1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ calculated equivalently by Abr. 0µ0E × H 2 D· E + B· H I − DE − BH ¯ ¯ 1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Min. D × B 2 D· E + B· H I − DE − BH ∂ ¯ ¯ 1 ¯ ¯ −1 ¯ ¯ ¯ ¯ ¯ −1 ¯ ¯ ¯ ¯ ¯ ¯ Amp. 0E × B 2 0E· E + µ0 B· B I − 0EE − µ0 BB F (t) = − dV G(¯r, t) − dA · T (¯r, t). (6) ¯ ¯ 1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ˆV ∂t ˛A E-L 0µ0E × H 2 0E· E + µ0H· H I − DE − BH Chu µ E¯ × H¯ 1 E·¯ E¯ + µ H·¯ H¯ I¯ − E¯E¯ − µ H¯H¯ 0 0 2 0 0 0 0 In some cases, the total force may be equivalent between formulations although the force densities differ. The same As one considers the amount of work being done on argument may be applied to the transfer of energy since the the system, one also sees that the momentum and stress are energy continuity equation takes the same form. explicitly tied to the energy density and flux. This can be illustrated using the work-energy relation or the Relativistic 2.1.1. Equivalence of Total Force Principle of Virtual Power (RPVP) [4, 5]. In Section 2.2 it Fig. 1 (a) depicts an object surrounded by vacuum illus- will be demonstrated, instead, using a straightforward ap- trating which total force equations are equivalent at any plication of the Lorentz transformation. For now, we list point in time. Integrating the electromagnetic force den- for reference in Table 2 the leading electromagnetic energy sity f¯(¯r, t) over the volume enclosing the object as depicted densities and power flows corresponding to the momentum by the dashed line yields the total force and is given by quantities in Table 1. Eq. (6), where the tensor reduces to the unambiguous vacuum Maxwell stress tensor since the surface of integration Table 2: Leading electromagnetic energy densities and is outside the material. However, the momentum density G¯ power flows. still depends on the electromagnetic formulation applied. Therefore, any formulations which share a common mo- S¯ W mentum density will produce identical results for the total Abr. E¯ × H¯ 1 D·¯ E¯ + B·¯ H¯ 2 force on an object at all points in time. The implication ¯ ¯ 1 ¯ ¯ ¯ ¯ Min. E × H 2 D· E + B· H is that the Chu, Einstein-Laub, and Abraham force densi- Amp. µ−1E¯ × B¯ 1 E·¯ E¯ + µ−1B·¯ B¯ 0 2 0 0 ties are equivalent in terms of total force and the Amperian E-L E¯ × H¯ 1 E·¯ E¯ + µ H·¯ H¯ 2 0 0 formulation differs only in the modeling of magnetic me- Chu E¯ × H¯ 1 E·¯ E¯ + µ H·¯ H¯ 2 0 0 dia. The Minkowski SEM tensor will yield a different time varying force for dielectric and magnetic media. The continuity equations can be rewritten in four- A second example is an object submerged inside an- dimensional coordinates (r,¯ ict) so that other medium such as a dielectric fluid as shown in Fig. 1 (b). The total force remains the same for two formu- ¯ ¯ ¯ fj = · Tj, −icGj (4a) lations if their momentum densities are identical. However, ϕ S¯ we must clarify how the force is computed. The electro- −i j = · −i j ,W (4b) c c j magnetic force is determined by volume integration of the

12 sity averages to zero, both computations in Fig 1 (a) and (b) reduce to the divergence of the vacuum stress tensor and the force is unambiguously independent of the formulation applied [3, 5]. As a result, it is generally not possible to determine the time-domain force or energy equations from only the time-average quantities [13].

2.2. Relativistic invariance

The total force on an object is often measured in experi- Figure 1: Illustration of total force calculation within a vol- ment. However, this quantity is insufficient for determin- ume V within the dashed line. The total force on (a) an ing the correct interpretations of electrodynamics as was object in vacuum is computed by volume integration of a demonstrated in the previous section. Moreover, the force force density within a region which completely encloses the densities differ between the various formulations [14], but material object and (b) an object embedded is computed in it has been shown that many experimental observations can a similar way by considering a thin vacuum region between be modeled in the stationary limit (i.e. material velocity the two materials. goes to zero v → 0) by closing the system using a variety of subsystems [15]. In fact, the problem with the stationary force density f¯(¯r, t), and there will generally be both vol- approximation is the work-energy relation is always satis- ume and surface forces at the boundary, the latter due to ma- fied in a trivial manner [4]; F¯ · v¯ = Φ = 0 since the ve- terial discontinuities at the boundaries [8, 9]. Any ambigu- locity is zero in the limit. To determine which tensors are ity as to which of the materials to which the surface forces truly valid in a physical sense, we must look at relativistic should be assigned is removed by mathematically introduc- invariance. ing a thin layer of vacuum separating the two materials. The It is a fundamental tenet of modern physics that ten- volume integration is then applied to the submerged mate- sors which are not relativistic invariant cannot be energy rial. The tensor reduces to the vacuum stress tensor since momentum tensors. As an example, consider a region of the surface of the integration is in the vacuum region, and space occupied by media described mathematically by a lo- the forces applied to the two media are unambiguous. Tak- cal mass density and a velocity field. The local momentum ing the limit of the vacuum region to zero yields the un- vector may vary with position and time regardless of how perturbed field problem with the surface forces applied cor- the coordinate system is assigned. Our inability to measure rectly to the corresponding media [10]. In general, forces relativistic effects in any experiment may only be due to our will be distributed to both the submerged object and the limited measurement capabilities. This limitation does not submerging fluid. The conclusion is unchanged. The to- prevent the fundamental laws of physics from holding [4]. tal calculated time-domain force on an object is the same For example, consider two frames of reference S and with any electromagnetic subsystem that share a common 0 0 momentum density [3]. We will find that another conclu- S , whereas S moves with constant velocity v¯ =xv ˆ with sion from this thought exercise is important; once we iden- respect to S. The SEM tensor in S is tify the correct subsystem separation of field and matter it   is necessary to assign a force on the embedded object and Txx Txy Txz icGx the embedding material [10, 11, 12]. ¯  Tyx Tyy Tyz icGy  Tj =   . (8) When materials are excited by monochromatic waves,  Tzx Tzy Tzz icGz  the time-average force is generally the observable quantity iSx/c iSy/c iSz/c W of interest. The total average force on a material object due to time-harmonic fields is We want to know if the tensor is invariant under certain hF¯i = − dA¯ · hT¯i, (7) electrodynamic hypotheses. As a simple demonstration, we ˛A consider the Lorentz transform of the component where the momentum density does not contribute to the 0 2 2 time-average force of time-harmonic fields [3]. Txx = γ Txx + β (Sx/c + cGx) − β W , (9) The total average force is the integration of the average force density over a volume that includes the entire object. −1/2 All such computations will yield identical results regardless where β = v/c and γ = (1 − β) . To consider the five of the force density applied. This statement is a result of different hypotheses (i.e. Abraham, Minkowski, Amperian, the condition that all electromagnetic SEM tensors must be Einstein-Laub, and Chu), we must consider the Lorentz consistent with Maxwell’s equations [1]. To illustrate this transformation laws for the field variables within each field fact, one must only consider again the problem in Fig. 1 framework. under time-averaged conditions. Since the momentum den- The Lorentz transformation laws for the Minkowski

13 field variables E¯, H¯ , D¯, and B¯ are [16] Table 3: Transformed stress tensors. ¯0 ¯ ¯ ¯ E = Ek + γ E⊥ +v ¯ × B , (10a) 0 0 Txx = −Txx ¯0 ¯ ¯ ¯ 0 0 0 0 1 ¯0 ¯0 ¯0 ¯0 H = Hk + γ H⊥ − v¯ × D , (10b) Abr. DxEx + BxHx − 2 δxx(D · E + B · H ) 2 ¯0 ¯0 ¯0 ¯0 ¯ +γ β[0µ0(E × H )x − (D × B )x] ¯0 ¯ ¯ v¯ × H D = Dk + γ D⊥ + , (10c) Min. D0 E0 + B0 H0 − 1 δ (D¯0 · E¯0 + B¯0 · H¯0) c2 x x x x 2 xx Amp. E0 E0 + µ−1B0 B0 − 1 δ ( E¯0 · E¯0 + µ−1B¯0 · B¯0) v¯ × E¯ 0 x x 0 x x 2 xx 0 0 ¯0 ¯ ¯ E-L γ2[E0 D0 + H0 B0 B = Bk + γ B⊥ − 2 , (10d) x x x x c 2 −1 0 0 2 2 0 0 +2γ βc {(E¯ × H¯ )x + c β (D¯ × B¯ )x +cβ(H0 B0 + H0 B0 + E0 D0 + E0 D0 )} where the subscripts k and ⊥ denote parallel and perpen- y y z z y y z z − 1 (1 + β2){ E0 E0 + µ H0 H0 dicular components to the velocity v¯. These transforma- 2 0 x x 0 x x +γ2[( E0 E0 + E0 E0 + µ H0 H0 + µ H0 H0 ) tion laws are applied for the Abraham, Minkowski, and 0 y y 0 z z 0 y y 0 z z +c2β2( B0 B0 + B0 B0 + µ D0 D0 + D0 D0 ) Einstein-Laub formulations, which utilize the Minkowski 0 y y 0 z z 0 y y z z ¯0 ¯0 ¯0 ¯0 fields. The Lorentz transformation laws for the Chu field +2cβ{(0E × B )x + (D × µ0H )x}]}] Chu E0 E0 + µ H0 H0 − 1 δ ( E¯0 · E¯0 + µ H¯0 · H¯0) variables E¯, H¯, P¯, and M¯ are [16] 0 x x 0 x x 2 xx 0 0 ¯0 ¯ ¯ ¯ E = Ek + γ E⊥ +v ¯ × µ0H , (11a) ¯0 ¯ ¯ ¯ H = Hk + γ H⊥ − v¯ × 0E , (11b) tum density G¯. While each of the three SEM tensors are rel- v¯ × (P¯ × v¯) ativistically invariant, it is desirable to determine if one of P¯0 = P¯ + γ P¯ − , (11c) k ⊥ c2 the three is the kinetic momentum of light, which is the momentum representing only field contributions without con- v¯ × (M¯ × v¯) ¯ 0 ¯ ¯ tributions from the mass of the material. In 1953, Balazs M = Mk + γ M⊥ − 2 , (11d) c developed a thought experiment that allows for the deter- where it is important to note that the Chu field values E¯ mination of the kinetic momentum by studying the center- and H¯ differ from the Minkowski ones. We will make clear of-mass displacement of a material slab as an electromag- which field values are being applied within a given con- netic pulse passes [17], and number of researchers have pre- text. The Lorentz transformation laws for the Amperian sented versions of this thought experiment [2, 3, 11, 18, 19]. field variables E¯, B¯, P¯, and M¯ are [16] We consider a slab of dispersive, impedance-matched material surrounded by vacuum. The slab is characterized ¯0 ¯ ¯ ¯ E = Ek + γ E⊥ +v ¯ × B , (12a) by a group velocity vg = ∂ω/∂k and wave impedance is ¯ p p ¯0 ¯ ¯ v¯ × E η = µ/ = µ0/0. An electromagnetic wave pulse has B = Bk + γ B⊥ − , (12b) c2 an initial free space momentum Ei/c. The slab of thickness v¯ × (P¯ × v¯) d delays the pulse with respect to the free space path by P¯0 = P¯ + γ P¯ − , (12c) the distance L = (n − 1)d since the group velocity in k ⊥ c2 g the material is v = c/n , where n is the group velocity ¯ g g g ¯ 0 ¯ ¯ v¯ × (M × v¯) index of refraction. The required kinetic momentum of the M = Mk + γ M⊥ − , (12d) c2 material while the pulse overlaps spatially with the slab is Utilizing Eq. (9), we substitute the corresponding com- ¯ ponent values for S¯, W , T¯, and G¯ given in Tables 1 and Ei 1 pm = 1 − , (13) 2 along with the field transformations for reference frame c ng S to S0 using Eq. (10) for the Abraham, Minkowski, and Einstein-Laub tensors, Eq. (11) for the Chu tensor, and which is necessary in order to maintain uniform motion of Eq. (12) for the Amperian tensor. After some algebraic ma- the center-of-mass energy. The momentum of the slab is the nipulation, the results are summarized in Table 3. It can difference between the momentum of the incident pulse and be seen that only the Minkowski, Chu, and Amperian SEM the material momentum given by Eq. (13) as required by tensors retain their form after transformation. Such mathe- momentum conservation. Therefore, the electromagnetic matical exercises can be carried-out on the remaining tensor momentum of the pulse is the field-kinetic momentum components with similar results. The conclusion is that the Abraham and Einstein-Laub tensors are not valid SEM tensors [4]. 1 Ei pFk = . (14) ng c 3. Energy and momentum in media This momentum corresponds to the Chu momentum, and 3.1. Field-kinetic subsystem we, therefore, take the Chu SEM tensor as the field-kinetic The Minkowski, Amperian, and Chu SEM tensors differ in formulation [4]. Further discussion of this relationship is many regards, but one significant difference is the momen- given in Section 4 of this correspondence.

14 The Maxwell-Chu equations [3, 16, 20] charges in motion per-unit-volume. The momentum continuity equation for the material is [23] ∂ ∇ × H¯ − 0 E¯ = J¯e (15a) ∂t 1 ∂r¯ ∂r¯ 1 T¯ = N qr¯ · E¯ + m · − ω2 mr¯ · r¯ I¯ ∂ mat 2 ∂t ∂t 2 e0 ∇ × E¯ + µ0 H¯ = −J¯h (15b) ∂t ¯ 0∇ · E¯ = ρe (15c) − qr¯E µ ∇ · H¯ = ρ (15d) 0 h ∂r¯ G¯mat = N qr¯ × µ0H¯ − m separate the electric and magnetic fields from the material ∂t response. The Chu formulation models the material re- ¯ ¯ ∂r¯ ¯ fmat = −fFk + N − γem . (20a) sponses by effective electric current density Je, magnetic ∂t current density J¯h, electric charge density ρe, and magnetic charge density ρh. These quantities for moving media with Here, we recognize the terms proportional to qr¯E¯ as the local velocity field v¯ are defined as [3, 16, 20] 1 2 energy of the charge in the electric field, 2 m|∂r/∂t¯ | as 2 ωe0 2 ∂P¯ the kinetic energy of the particle, and 2 m|r¯| as the po- ¯ ¯ ¯ 2 Je ≡ + ∇ × [P × v¯] + J (16a) tential energy. This last term proportional to ω is due to ∂t e0 the spring-like restoring nature between the electron and an ∂M¯ J¯ ≡ µ + µ ∇ × [M¯ × v¯] (16b) oppositely charged nucleus, which is assumed to be much h 0 ∂t 0 more massive. ρ ≡ −∇ · P¯ + ρ (16c) e Equation (18) can be written in terms of the macro- ¯ ρh ≡ −µ0∇ · M, (16d) scopic polarization definition by P¯ = Nqr¯, giving the equation of motion as where M¯ is the magnetization, P¯ is the polarization, J¯ is the free current density, and ρ is the free charge density of ∂2 ∂ Nq2 + γ + ω2 P¯ = E¯ = ω2 E¯, (21) the given medium. The energy and momentum continuity ∂t2 e ∂t e0 m 0 ep equations for the field-kinetic subsystem are completed by the interaction terms [3, 16, 20] q Nq2 where ωep = m is the plasma frequency. The dielectric ¯ ¯ ¯ ¯ ¯ ¯ ¯ 0 fFk = ρeE + ρhH + Je × µoH − Jh × 0E(17a) material contributions to the energy are [1, 3, 6] ¯ ¯ ¯ ¯ ϕFk = Je · E + Jh · H (17b) S¯mat = 0 (22a) Next, we will derive the material response to the field- ¯ ¯ ¯ 1 ∂P ∂P 2 ¯ ¯ kinetic subsystem for dielectrics and isotropic metamate- Wmat = 2 · + ωe0P· P (22b) 20ωep ∂t ∂t rials with electric and magnetic response. γ ∂P¯ ∂P¯ ϕmat = −ϕFk + 2 · (22c) 3.2. Material response in dielectrics 0ωep ∂t ∂t

Consider a harmonic oscillator describing the motion of a The dielectric material contributions to the momentum ¯ bound charge under excitation of a time-harmonic field E. are [1, 3, 6] The motion can be used to determine the stationary dielectric response of the charge q given the damping factor γ and ¯ 1 ¯ T¯mat = P·¯ E¯ I¯− P¯E¯ (23a) the electric resonant frequency ωe0. The equation of motion 2 for r¯ =xx ˆ +yy ˆ +zz ˆ is [20] 1 1 ∂P¯ ∂P¯ + P·¯ E¯ + · − ω2 P·¯ P¯ I¯ 2 ω2 ∂t ∂t e0 ∂2 ∂ 0 ep m + γ + ω2 r¯ = qE¯. (18) ¯ ∂t2 e ∂t e0 ¯ ¯ ¯ 1 ¯ ∂P Gmat = P × µ0H − 2 ∇P· (23b) 0ωep ∂t If there are N such harmonic oscillators per volume of di- γ ∂P¯ electric, the material subsystem is given by [21, 22, 23] ¯ ¯ e ¯ fmat = −fFk − 2 ∇P· . (23c) 0ωep ∂t S¯mat = 0 (19a) The material subsystem and field-kinetic subsystem can be ¯ 1 ∂r¯ ∂r¯ 1 2 Wmat = N m · + ω mr¯ · r¯ (19b) combined to form the canonical subsystem as described 2 ∂t ∂t 2 e0 conceptually in Eq. (2). ∂r¯ ∂r¯ ϕmat = −ϕFk + N γe · . (19c) ∂t ∂t 3.3. Energy and momentum in NIMs 1 ∂r¯ ∂r¯ The energy terms consist of a kinetic energy term 2 m ∂t · ∂t By duality of the field-kinetic (i.e Chu) subsystem, the 1 2 and a potential energy term 2 ω0mr¯ · r¯ for each of N causal dielectric and magnetic response of a stationary

15 isotropic system can be formulated as [6, 24] In the limit of zero losses (γe → 0 and γm → 0) both I = 0 and µI = 0 and the energy density satisﬁes the ∂2 ∂ + γ + ω2 P¯ = ω2 E¯ (24a) well-known relation [20, 21, 25, 26] ∂t2 e ∂t e0 0 ep 2 1 ∂(ω) 2 1 ∂(µω) 2 ∂ ∂ 2 2 ¯ ¯ + γ + ω µ M¯ = µ ω H¯ (24b) hWcani = E + H . (29) ∂t2 m ∂t m0 0 0 mp 4 ∂ω 4 ∂ω The resulting material energy contributions are It is the rate of change in energy that appears in the energy continuity equation, which tends to zero upon cycle aver- ¯ Smat = 0 (25a) aging, and since h∂W/∂ti = 0, the resulting conservation ¯ ¯ ¯ 1 ∂P ∂P 2 ¯ ¯ equation Wmat = 2 · + ωe0P· P 20ωep ∂t ∂t ¯ ¯ ¯ 1 ¯ 2 ¯ 2 µ0 ∂M ∂M 2 ¯ ¯ −h∇ · Scani = ωI |E| + ωµI |H| (30) + 2 · + ωm0P· P (25b) 2 2ωmp ∂t ∂t

γe ∂P¯ ∂P¯ is generally regarded as the complex Poynting’s theorem, ϕ = −ϕ + · 1 ∗ mat Fk 2 where hS¯cani = <{E¯ ×H¯ } is the time average Poynting 0ωep ∂t ∂t 2 power. γ ∂M¯ ∂M¯ m The average momentum density, + 2 · , (25c) 0ωmp ∂t ∂t ωω2 and the momentum contributions are ¯ 1 ¯ ¯∗ ¯ 0 ep ¯ 2 hGcani = < D × B + k 2 2 2 2 2 |E| 2 (ω − ωe0) + γe ω ¯ 1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Tmat = P· E + µ0M· H I − PE − µ0M(26a)H µ ωω2 2 ¯ 0 mp ¯ 2 + k 2 2 2 2 2 |H| , (31) 1 (ω − ωm0) + γmω + P·¯ E¯ + µ M·¯ H¯ 2 0 ¯ ¯ satisfies [27] 1 ∂P ∂P 2 ¯ ¯ + 2 · − ωe0P· P 0ωep ∂t ∂t 1 k¯ ∂ ∂µ hG¯ i = < D¯ × B¯∗ + |E¯|2 + |H¯ |2 µ ∂M¯ ∂M¯ can 2 2 ∂ω ∂ω + 0 · − ω2 M·¯ M¯ I¯ ω2 ∂t ∂t m0 1 1 mp = E¯ × H¯ ∗ (32) G¯mat = P¯ × µ0H¯ + E¯ × µ0M¯ + P¯ × µ0M¯ 2 vgvp ¯ ¯ 1 ¯ ∂P µ0 ¯ ∂M − 2 ∇P· − 2 ∇M· (26b) when the losses approach zero, where vp = ω/k and 0ωep ∂t ωmp ∂t vg = ∂ω/∂k are the phase velocity and the group veloc- ¯ ¯ ¯ γe ¯ ∂P ity, respectively. The time average stress tensor is found to fmat = −fFk − 2 ∇P· 0ωep ∂t be ¯ µ0γm ¯ ∂M − ∇M· . (26c) ¯ 1 1 ¯ ¯∗ ¯ ¯ ∗ ¯ ¯ ¯∗ ¯ ¯ ∗ ω2 ∂t hTcani = < (D · E + B · H )I − DE − BH (33) mp 2 2 Under time-harmonic excitation, the stationary (i.e. v¯ → 0) Minkowski constitutive parameters can be approx- since the dispersive terms average to zero. Because the imated D¯ = (ω)E¯ and B¯ = µ(ω)H¯ . Each field phaser average rate of change in momentum density is zero (i.e. ¯ follows the relation A¯ = <{Ae¯ −iωt}. The constitutive pa- h∂G/∂ti = 0), the momentum conservation theorem for a rameters are monochromatic wave becomes 2 ! 1 ωep ¯ ¯ ¯∗ ¯ ¯ ∗ (ω) = 1 − −h∇ · Tcani = < ωI E × B − ωµI H × D (34) 0 2 2 (27a) 2 ω − ωe0 + iωγe ! ω2 where the tensor is the complex Minkowski tensor. The µ(ω) = µ 1 − mp . (27b) 0 2 2 force terms in Eq. (34) have been defined as the force den- ω − ωm0 + iωγm sity on free currents [28, 29]. The time-averaged energy density is " # 4. Discussion ω2 ω2 + ω2 0 ep e0 ¯ 2 hWcani = 1 + E 2 2 2 2 2 2 For illustration purposes, we consider a couple of examples. (ω − ωe0) + γe ω First, we consider the momentum density contribution to an " 2 2 2 # µ0 ωmp ω + ωm0 2 electromagnetic pulse. Then, we consider the momentum + 1 + H¯ (28). 2 2 2 2 2 2 (ω − ωm0) + γmω flux or stress tensor contribution to a continuous wave.

16 4.1. Electromagnetic Pulse A pulse is incident upon a medium with constitutive parameters µ(ω), (ω). For simplicity, we consider the medium to be dispersive, but lossless. The excitation energy of the pulse is [1, 3]

2 Eexc ≡ Ei − Er = Ei(1 − |R| ), (35) where Ei is the incident energy, Er is the reflected energy, and |R|2 is the reflectivity. The excitation energy is unambiguously defined in terms of the vacuum quantities. The spatial length of the transmitted pulse is decreased propor- −1 tional to the factor ng due to the change in velocity inside the medium. We can investigate the interpretation of the field-kinetic and canonical subsystems inside the medium. If we assume the field-kinetic momentum density de- Figure 2: Demonstration of a refracted Gaussian beam at a fined by the Chu SEM tensor, we see that the momentum matched negative refractive index boundary. The incident inside the material is medium z < 0 is vacuum (µ0, 0) and the transmitted is a matched left-handed medium (− , −µ ). The parameters 1 E 0 0 ¯ i 2 for incident TM field are amplitude H = 1 A/m, inci- p¯Fk = dzGFk (z) =z ˆ 1 − R 0 ˆ ng c z dent wavelength λ0 = 1064 nm, beam waist w = λ0, and 1 NA = 0.8. =z ˆ Eexc. (36) ngc

This is the momentum derived in Balazs’ thought experi- We may arrive at this result using another approach by ment. viewing each photon as a particle where the effective mass We may arrive at this result using another approach by increases when it enters the material. The force on the ma- viewing each photon as a particle with effective mass un- terial is due to the momentum exchange changed as it enters the material so that m = m = E/c2. 0 E E c In the vacuum region (z < 0), the velocity of the pulse is p¯i − p¯can = m0v¯0 − mv¯ =z ˆ 2 c − 2 c c /(ngnp) ng v¯0 =zc ˆ . At t = 0, the pulse enters the material and the E velocity becomes v¯ =zc/n ˆ g. The force on the material is =z ˆ (1 − np) . (39) due to the momentum exchange c

E E c In this view, the momentum of the pulse entering a dielec- p¯ − p¯ = m v¯ − mv¯ =z ˆ c − tric is increased due to the additional momentum of the ma- i kin 0 0 c2 c2 n g terial which is added to the electromagnetic pulse. These E 1 =z ˆ 1 − . contributions were described in detail in Section 3. c ng It is interesting what happens to the two quantities in negative index media (NIM). Note that for NIM, np < 0, Therefore, the field-kinetic subsystem gives the interpreta- but ng > 0. The field-kinetic momentum, being related to tion of constant effective mass. ng, remains positive in negative index media. The canoni- Alternately, if we assume the canonical momentum cal momentum on the other hand reverses direction due to density defined for a dispersive, lossless material from its proportionality to np. Note, however that in the presence Eq. (32) of losses, the direction of the canonical momentum can be W (z) parallel or anti-parallel to the energy flow [6]. G¯ (z) =zn ˆ n i (1 − |R|2), (37) can g p c 4.2. Continuous Wave we see that the momentum inside the material is Consider the problem of a continuous wave incident upon a negative index material (− , −µ ) from vacuum ( , µ ). ¯ Ei 2 0 0 0 0 p¯can = dzGcan(z) =zn ˆ p 1 − R The wave shown in Fig. 2 is a TM Gaussian beam incident ˆz c at π/6. Eexc =zn ˆ p . (38) The field-kinetic force on the surface is F¯ = c Fk xˆ1.6 pN/m. The interpretation of this force is that there ¯ This momentum, which is proportional to the phase veloc- is an equal and opposite force −FFk = −xˆ1.6 N/m that ity index of refraction, is generally measured inside ma- changes the direction of the incident photons from the +ˆz terials as a combination of field plus material contribu- and +ˆx direction to the +ˆz and −xˆ direction. The trans- tions [12, 28, 30, 31, 32]. mitted electromagnetic momentum has an identical zˆ com-

17 ponent as the incident. Therefore, it is the equal and op- [2] S. M. Barnett, “Resolution of the Abraham- posite force of the material on the electromagnetic waves Minkowski dilemma,” Phys. Rev. Lett., vol. 104, p. at the surface that causes the change in momentum in the 070401, 2010. xˆ direction. The form of the field-kinetic momentum remains the same for all types of medium, and this descrip- [3] B. A. Kemp, “Macroscopic theory of optical momen- tion was previously applied to describe the optical dynam- tum,” Prog. Optics, vol. 60, p. 437, 2015. ics of anisotropic metamaterial cloaks [33, 34]. Although [4] C. J. Sheppard and B. A. Kemp, “Kinetic-energy- the field-kinetic subsystem had yet to be uniquely identi- momentum tensor in electrodynamics,” Phys. Rev. A, fied, the Chu formulation was used to describe the optical vol. 93, p. 013855, 2016. dynamics of optical cloaks, which explains in retrospect how the forces on the optical paths were able to describe [5] C. J. Sheppard and B. A. Kemp, “Relativistic analy- the transformed paths. Such a description is also useful for sis of field-kinetic and canonical electromagnetic sys- perfect lenses where planar surfaces are used to negatively tems,” Phys. Rev. A, vol. 93, p. 053832, 2016. refract images [24]. The canonical SEM tensor provides an alternate view- [6] B. A. Kemp, J. A. Kong, and T. M. Grzegorczyk, point. In this case, the time-averaged force reduces to the “Reversal of wave momentum in isotropic left-handed divergence of the Minkowski stress tensor. The canonical media,” Phys. Rev. A, vol. 75, p. 053810, 2007. force on the surface is F¯can =z ˆ2.8 pN/m. The zero force in the xˆ direction means that the tangential component of [7] T. M. Grzegorczyk and B. A. Kemp, “Transfer of op- the wave momentum is conserved [6]. The change in the tical momentum: reconciliations of the abraham and normal component of the wave momentum is due to the ad- minkowski formulations,” Proc. SPIE, vol. 7038, p. ditional momentum from the material response as described 70381S, 2008. in Section 3. [8] M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Ex- 5. Conclusions press, vol. 12, p. 5375, 2004. We have shown that the field-kinetic subsystem as given by [9] B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, the Chu formulation represents the energy and momentum “Lorentz force on dielectric and magnetic particles,” contained within the fields, and the sum of the field-kinetic J. of Electromagn. Waves and Appl., vol. 20, p. 827, and material response subsystems represent the wave or 2006. canonical subsystem. The canonical subsystem reduces to the Minkowski subsystem only under negligible dispersion. [10] B. A. Kemp and T. M. Grzegorczyk, “The observ- We may conclude that neither the Minkowski or Abraham able pressure of light in dielectric fluids,” Opt. Lett., SEM tensors are universally correct. In fact, the Abraham vol. 36, no. 4, p. 493, 2011. SEM tensor violates the physical constraint of relativistic invariance. Our view was demonstrated the physics of neg- [11] C. J. Sheppard and B. A. Kemp, “Optical pressure ative index materials under both the field-kinetic and canon- deduced from energy relations within relativistic for- ical or wave SEM models for negative refraction, which is mulations of electrodynamics,” Phys. Rev. A, vol. 89, fundamental to perfect lensing and invisibility cloaking. no. 1, p. 013825, 2014.

[12] B. A. Kemp, “Subsystem approach to the electrody- Acknowledgement namics in dielectric ﬂuids,” Proc. SPIE, vol. 8458, p. 845803, 2012. This work was sponsored in part by the NSF Divi- [13] B. A. Kemp, “Comment on “revisiting the bal- sion of Electrical, Communications, and Cyber Systems azs thought experiment in the presence of loss: (Award No. ECCS-1150514) through the Faculty Early Ca- electromagnetic-pulse-induced displacement of a reer Development (CAREER) program, the Arkansas EP- positive-index slab having arbitrary complex permit- SCoR program provided by the National Science Foun- tivity and permeability”,” Appl. Phys. A, vol. 110, p. dations Research Infrastructure Improvement (Award No. 517, 2013. OIA-1457888), and the Arkansas Research Alliance Fel- lows program. [14] M. Mansuripur, A. R. Zakharian, and E. W. Wright, “Electromagnetic-force distribution inside matter,” References Phys. Rev. A, vol. 88, p. 023826, 2013.

[1] B. A. Kemp, “Resolution of the Abraham-Minkowski [15] N. G. C. Astrath, L. C. Malacarne, M. L. Baesso, debate: Implications for the electromagnetic wave G. V. B. Lukasievicz, and S. E. Bialkowski, “Unrav- theory of light in matter,” J. Appl. Phys., vol. 109, p. elling the effects of radiation forces in water,” Nat. 111101, 2011. Commun., vol. 5, p. 4363, 2014.

18 [16] P. Penﬁeld and H. A. Haus, Electrodynamics of Mov- [33] H. Chen, B. Zhang, Y. Luo, B. A. Kemp, J. Zhang, ing Media. Cambridge, MA: M.I.T. Press, 1967. L. Ran, and B.-I. Wu, “Lorentz force and radiation pressure on a spherical cloak,” Phys. Rev. A, vol. 80, [17] N. L. Balazs, Phys. Rev., vol. 91, p. 408, 195d. p. 011808, 2009. [18] R. Loudon, “Radiation pressure and momentum in di- [34] H. Chen, B. Zhang, B. A. Kemp, and B.-I. Wu, “Op- electrics,” Fortschr. Phys, vol. 52, p. 1134, 2004. tical force on a cylindrical cloak under arbitrary wave [19] M. G. Scullion and S. M. Barnett, “Optical momen- illumination,” Opt. Lett., vol. 35, p. 667, March 2010. tum in negative-index media,” J. Mod. Opt., vol. 55, p. 2301, 2008.

[20] J. A. Kong, Electromagnetic Wave Theory. Cam- bridge, MA: EMW Publishing, 2005.

[21] L. Brillouin, Wave Propagation and Group Velocity. New York: Academic Press, 1960.

[22] R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A, vol. 3, p. 233, 1970.

[23] R. Loudon, L. Allen, and D. F. Nelson, “Propagation of electromagnetic energy and momentum through an absorbing dielectric,” Phys. Rev. E, vol. 55, p. 1071, 1997.

[24] S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Mate- rials. CRC Press, 2008.

[25] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. New York: Pergamon, 1984.

[26] J. D. Jackson, Classical Electrodynamics. New York: John Wiley and Sons, 1999.

[27] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of and µ,” Sov. Phys. Usp., vol. 10, p. 509, 1968.

[28] R. Loudon, S. M. Barnett, and C. Baxter, “Radiation pressure and momentum transfer in dielectrics: The photon drag effect,” Phys. Rev. A, vol. 71, p. 063802, 2005.

[29] B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Op- tical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett., vol. 97, p. 133902, 2006.

[30] R. V. Jones and J. C. S. Richards, “The pressure of radiation in a refracting medium,” Proc. R. Soc. Lond. A., vol. 221, p. 480, 1954.

[31] R. V. Jones and B. Leslie, “The measurement of optical radiation pressure in dispersive media,” Proc. R. Soc. Lond. A., vol. 360, p. 347, 1978.

[32] G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, W. Ketterle, and D. E. Pritchard, “Pho- ton recoil momentum in dispersive media,” Phys. Rev. Lett., vol. 94, p. 170403, 2005.