SCALORA et al PHYSICAL REVIEW E 73. 056604(2006) one may use, one due to Minkowski [12]. the other perhaps ciated with the(apparently)mechanical momentum [13, 17] more familiar form due to Abraham [13]. The latter is gen- of the bound charges moving within the dielectric material erally believed to be the correct expression, even though the In earlier work, Gordon [18 had shown that in a low- Minkowski form follows from momentum conservation ar- density gas the Lorentz force density may be recast as guments in the presence of matter, beginning with Maxwell,s equations and the Lorentz force [11]. Nevertheless, the sub- r)=aVE2)+1 E×H) ject has been controversial, and the Minkowski expression is believed to be flawed, in part because it is connected to a stress-energy tensor that forces both the susceptibility and where d eq.(5)to the case of radiation reflected fiom o is the mediums polarizability. The au permeability to be independent of density and temperature to apply 11], an unphysical situation that argues against it. perfect conductor. Integrating over all volume, with the re- Our approach does not include the formulation of a stress quirement that the field go to zero at the conductors surface energy tensor, as is often done [8, 9, 14], for example, because (this condition is also valid for well-localized wave packets, hat may tend to obscure the problem rather than clarify it, whose boundary conditions are zero at infinity ), the first term while providing no more definitive answers one way or the on the right-hand side vanishes, and the sole contribution to other. In order to remove some of the ambiguities inherent in the total force is the definition of a stress-tensor, which has some degree of built-in arbitrariness, one may address the problem by di F(=Na dU-(E×H) rectly integrating the vector Maxwell,s equations in space tolune at and time in the presence of matter, using pulses of finite where N is the particle density extent to include material dispersion and finite response In the present work we derive expressions for the times, and by treating more realistic extended structures of Minkowski momentum density and for the Lorentz force finite length. The resulting fields may then be used to form density in the general case of dispersive e and u, and study various quantities of interest, such as the Lorentz force the interaction of short optical pulses incident on(i) dielec [15, 16], for example, so that a direct assessment may be tric substrates of finite length, (i)micron-sized, multilayer made regarding momentum conservation. In Ref. [15]. for structures located in free space and also embedded within a example, using a quantum mechanical approach, Loudon dielectric medium, and (i)a negative index material(NIM). showed that beginning with a Lorentz force density in ordi- a medium that simultaneously displays negative e and u nary materials (u=1), in the absence of free charges and [19]. Integrating the vector Maxwell equations in two- dimensional space and time, in all cases that we investigate we find that conservation of linear momentum and the Lor- r,t)= aP (1) for change of the Abraham momentum, regardless of the medium the momentum a photon delivers to a surface when incident and its dispersive properties, in regions of negligible absorp- from free space when absorption is absent is [ tion, namely, E×H ALL VOLUME 4TC where n is the index of the material and po is the initial momentum. Recently, Mansuripur [16] suggested that base where F(t=m is the instantaneous Lorentz force on his calculation of momentum transfer to a transparent slab even though they may be related to the Abraham momentum, sible definition of momentum density is neither the Abraham Eq. (3)above, or any other plausible definition, are capable nor the Minkowski momentum, rather, an average of the two momentum densities combined into a simple, symmetrized Investigated. They come close in situations where the size of form [16 the structure is much smaller compared to the spatial exten- he incident packet, or if reflections occur from 1(①D×BNm++田)m).() a mirror located in free space. In these cases the analysis of average 4Tc Assuming the usual constitutive relation D=E+4P. the ab- Once we establish the theoretical basis of our approach sence of dispersion, and that A=l, it is easy to show that Eq. we go on to examine the response of relatively thick sub- (3)reduces to [16] strates and micron-sized resonant structures. and then the P×HE×H (4) response of extended, NIM substrates, illuminated by pulses cles in dura stances, the spatial extension of the pulse may be several tens One may easily identify the second term on the right-hand of microns, which is much longer than the length of any side as the usual Abraham electromagnetic momentum den- typical multilayer structure [4]. Although the theoretical ap sity. The first term on the right-hand side of Eq.(4)is asso- proach that we develop will apply to pulses of arbitrary du-one may use, one due to Minkowski 12, the other perhaps more familiar form due to Abraham 13. The latter is gen￾erally believed to be the correct expression, even though the Minkowski form follows from momentum conservation ar￾guments in the presence of matter, beginning with Maxwell’s equations and the Lorentz force 11. Nevertheless, the sub￾ject has been controversial, and the Minkowski expression is believed to be flawed, in part because it is connected to a stress-energy tensor that forces both the susceptibility and permeability to be independent of density and temperature 11, an unphysical situation that argues against it. Our approach does not include the formulation of a stress￾energy tensor, as is often done 8,9,14, for example, because that may tend to obscure the problem rather than clarify it, while providing no more definitive answers one way or the other. In order to remove some of the ambiguities inherent in the definition of a stress-tensor, which has some degree of built-in arbitrariness, one may address the problem by di￾rectly integrating the vector Maxwell’s equations in space and time in the presence of matter, using pulses of finite extent to include material dispersion and finite response times, and by treating more realistic extended structures of finite length. The resulting fields may then be used to form various quantities of interest, such as the Lorentz force 15,16, for example, so that a direct assessment may be made regarding momentum conservation. In Ref. 15, for example, using a quantum mechanical approach, Loudon showed that beginning with a Lorentz force density in ordi￾nary materials

=1, in the absence of free charges and currents, f
r,t = 1 c
P
t B,
1 the momentum a photon delivers to a surface when incident from free space when absorption is absent is 15 PT = 2P0 n − 1 n + 1 ,
2 where n is the index of the material and P0 is the initial momentum. Recently, Mansuripur 16 suggested that based on his calculation of momentum transfer to a transparent slab via the application of boundary conditions, the most plau￾sible definition of momentum density is neither the Abraham nor the Minkowski momentum, rather, an average of the two momentum densities combined into a simple, symmetrized form 16: gaverage = 1 4c 
D BMinkowski +
E + HAbraham 2 .
3 Assuming the usual constitutive relation D=E+4P, the ab￾sence of dispersion, and that
=1, it is easy to show that Eq.
3 reduces to 16 gaverage =  P H 2c + E H 4c .
4 One may easily identify the second term on the right-hand side as the usual Abraham electromagnetic momentum den￾sity. The first term on the right-hand side of Eq.
4 is asso￾ciated with the
apparently mechanical momentum 13,17 of the bound charges moving within the dielectric material. In earlier work, Gordon 18 had shown that in a low￾density gas the Lorentz force density may be recast as f
r,t =  1 2 E2  + 1 c

t
E H,
5 where  is the medium’s polarizability. The author went on to apply Eq.
5 to the case of radiation reflected from a perfect conductor. Integrating over all volume, with the re￾quirement that the field go to zero at the conductor’s surface
this condition is also valid for well-localized wave packets, whose boundary conditions are zero at infinity, the first term on the right-hand side vanishes, and the sole contribution to the total force is F
t = N c volume dv

t
E H,
6 where N is the particle density. In the present work we derive expressions for the Minkowski momentum density and for the Lorentz force density in the general case of dispersive  and
, and study the interaction of short optical pulses incident on
i dielec￾tric substrates of finite length,
ii micron-sized, multilayer structures located in free space and also embedded within a dielectric medium, and
iii a negative index material
NIM, a medium that simultaneously displays negative  and
19. Integrating the vector Maxwell equations in two￾dimensional space and time, in all cases that we investigate we find that conservation of linear momentum and the Lor￾entz force are consistent only with the temporal rate of change of the Abraham momentum, regardless of the medium and its dispersive properties, in regions of negligible absorp￾tion, namely,

t Pmech + ALL VOLUME E H 4c dv = 0,
7 where F
t=
Pmech
t is the instantaneous Lorentz force. Thus, even though they may be related to the Abraham momentum, neither the Minkowski nor the average momentum density in Eq.
3 above, or any other plausible definition, are capable of reproducing the Lorentz force in any of the circumstances investigated. They come close in situations where the size of the structure is much smaller compared to the spatial exten￾sion of the incident wave packet, or if reflections occur from a mirror located in free space. In these cases the analysis of the dynamics reveals only transient, relatively small differ￾ences. Once we establish the theoretical basis of our approach, we go on to examine the response of relatively thick sub￾strates and micron-sized resonant structures, and then the response of extended, NIM substrates, illuminated by pulses several tens of wave cycles in duration. Under some circum￾stances, the spatial extension of the pulse may be several tens of microns, which is much longer than the length of any typical multilayer structure 4. Although the theoretical ap￾proach that we develop will apply to pulses of arbitrary du￾SCALORA et al. PHYSICAL REVIEW E 73, 056604
2006 056604-2