《电动力学》课程参考文献：Radiation pressure of light pulses and conservation of linear momentum in dispersive media

PHYSICAL REVIEW E 73. 056604(2006) Radiation pressure of light pulses and conservation of linear momentum in dispersive media Michael Scalora, Giuseppe D'Aguanno, Nadia Mattiucci, Mark J. Bloemer, Marco Centim Concita Sibilia, and Joseph W. Haus Charles M. Bowden Research Center, AMSRD-AMR-WS-ST, Research, Development, and Engineering Center, Redstone arsenal. alabama 3 5898-5000. USA Time Domain Corporation, Cummings Research Park, 7057 Old Madison Pike, Huntsville, Alabama 35806, USA INFM at Dipartimento di Energetica, Universita di Roma"La Sapienza, Via A. Scarpa 16, 00161 Roma, Italy Electro-Optics Program, University of Dayton, Dayton, Ohio 45469-0245, USA (Received 12 December 2005: revised manuscript received 1 March 2006; published 16 May 2006) We derive an expression for the Minkowski momentum under conditions of dispersive susceptibility ermeability, and compare it to the abraham momentum in order to test the principle of conservation of linear momentum when matter is present. We investigate cases when an incident pulse interacts with a variety of structures, including thick substrates, resonant, free-standing, micron-sized multilayer stacks, and negative index materials. In general, we find that for media only a few wavelengths thick the Minkowski and Abraham momentum densities yield similar results. For more extended media, including substrates and Bragg mirrors embedded inside thick dielectric substrates, our calculations show dramatic differences between the minkowski and Abraham momenta. Without exception, in all cases investigated the instantaneous Lorentz force exerted on the medium is consistent only with the rate of change of the Abraham momentum. As a practical example, we use our model to predict that electromagnetic momentum and energy buildup inside a multilayer stack can lead ground for basic electromagnetic phenomena such as momentum transfer to macroscopic media W. r results to widely tunable accelerations that may easily reach and exceed 100 m/s2 for a mass of 10-s g. Our results suggest that the physics of the photonic band edge and other similar finite structures may be used as a testing DO:10.103/ PhysReve.73.056604 PACS number(s): 42.25 Bs, 42.25 Gy, 42.70.Qs, 78.20.Ci . INTRODUCTION light in fact exerts pressure, and was later experimentally For the better part of two decades photonic band gap verified by Nichols and Hull [6]. A good perspective of the (PBG) structures have been the subject of many theoretical early history of the subject is given by Mulser [7J, who also and experimental studies. Since the pioneering work of Yablonovitch [1] and John [2]. investigations have focused lated Brillouin and Raman scattering, are radiation-pressure on all kinds of geometrical arrangements, which vary from driven phenomena. More recently, Antonoyiannakis and Pen- one-dimensional, layered stacks, more amenable to analyti- dry [8] examined issues related to forces present in photonic topologies that require a full vector Maxwell approach [3]. In dielectric material, a light beam attracts the interface. The ent aspect of this particular problem, namely the interaction crystals, and the authors go on to predict an attractive force between neighboring dielectric spheres. Povinelli et al. [9] acting with pulses of finite bandwidth. Interesting questions studied the effect s of radiation pressure in omni-directional arise as incident pulses are tuned near the band edge, where reflector waveguides. They showed that as light propagates electromagnetic energy and momentum become temporarily down the guide(parallel to the dielectric mirrors),radiation stored inside the medium. When tuned near the band edge, in pressure causes the mirrors to attract, and, in the absence of the absence of meaningful absorption, a pulse of finite band. any losses, the attractive force appears to diverge near the cut width can lose forward momentum in at least two ways:() off frequency. Tucker et al. [10] have in rated effects of by tuning inside the gap, which results in mirrorlike reflec- radiation pressure and thermal jitter in a hybrid environment, tions and maximum transfer of momentum and (i) by tuning composed of a Fabry-Perot resonator as part of a microme. a minimum. and the field becomes localized inside the stack. that radiation pressur re can cause si mall changes in the sepa- It has been shown that relatively narrow-band band optical ration of movable mirrors even at room temperature, leading pulses may be transmitted without scattering losses or shape to nonlinear shifts of the Fabry-Perot resonance and hyster- changes [4] insuring that momentum and energy storage in- esis loops In MEMS lasers, the authors suggest that nonlin- porary. Therefore, a structure ear radiation pressure effects may induce changes in the not fixed to the laboratory frame naturally acquires linear characteristic low-frequency chirp of the device [10] momentum in an effort to conserve it. In what follows we The issue of how much electromagnetic momentum is attempt to answer the following question: how much and transferred to macroscopic bodies is still a matter of debate, what sort of motion results from the interaction? primarily". because what is considered electromagnetic The issue of radiation pressure on macroscopic bodies and what mechanical is to some extent arbitrary., as noted rches all the way back to Maxwell [5], who realized that by Jackson [11]. There are two well-known expressions that 1539-3755/200673(5)/056604(12) 056604-1 @2006 The American Physical Society

Radiation pressure of light pulses and conservation of linear momentum in dispersive media Michael Scalora,1 Giuseppe D’Aguanno,1 Nadia Mattiucci,2,1 Mark J. Bloemer,1 Marco Centini,3 Concita Sibilia,3 and Joseph W. Haus4 1 Charles M. Bowden Research Center, AMSRD-AMR-WS-ST, Research, Development, and Engineering Center, Redstone Arsenal, Alabama 35898-5000, USA 2 Time Domain Corporation, Cummings Research Park, 7057 Old Madison Pike, Huntsville, Alabama 35806, USA 3 INFM at Dipartimento di Energetica, Universita di Roma “La Sapienza”, Via A. Scarpa 16, 00161 Roma, Italy 4 Electro-Optics Program, University of Dayton, Dayton, Ohio 45469-0245, USA
Received 12 December 2005; revised manuscript received 1 March 2006; published 16 May 2006 We derive an expression for the Minkowski momentum under conditions of dispersive susceptibility and permeability, and compare it to the Abraham momentum in order to test the principle of conservation of linear momentum when matter is present. We investigate cases when an incident pulse interacts with a variety of structures, including thick substrates, resonant, free-standing, micron-sized multilayer stacks, and negative index materials. In general, we find that for media only a few wavelengths thick the Minkowski and Abraham momentum densities yield similar results. For more extended media, including substrates and Bragg mirrors embedded inside thick dielectric substrates, our calculations show dramatic differences between the Minkowski and Abraham momenta. Without exception, in all cases investigated the instantaneous Lorentz force exerted on the medium is consistent only with the rate of change of the Abraham momentum. As a practical example, we use our model to predict that electromagnetic momentum and energy buildup inside a multilayer stack can lead to widely tunable accelerations that may easily reach and exceed 1010 m/s2 for a mass of 10−5 g. Our results suggest that the physics of the photonic band edge and other similar finite structures may be used as a testing ground for basic electromagnetic phenomena such as momentum transfer to macroscopic media. DOI: 10.1103/PhysRevE.73.056604 PACS number
s: 42.25.Bs, 42.25.Gy, 42.70.Qs, 78.20.Ci I. INTRODUCTION For the better part of two decades photonic band gap
PBG structures have been the subject of many theoretical and experimental studies. Since the pioneering work of Yablonovitch 1 and John 2, investigations have focused on all kinds of geometrical arrangements, which vary from one-dimensional, layered stacks, more amenable to analyti￾cal treatment, to much more complicated three-dimensional topologies that require a full vector Maxwell approach 3. In our current effort, in part we focus our attention on a differ￾ent aspect of this particular problem, namely the interaction of short pulses with free-standing, resonant structures inter￾acting with pulses of finite bandwidth. Interesting questions arise as incident pulses are tuned near the band edge, where electromagnetic energy and momentum become temporarily stored inside the medium. When tuned near the band edge, in the absence of meaningful absorption, a pulse of finite band￾width can lose forward momentum in at least two ways:
i by tuning inside the gap, which results in mirrorlike reflec￾tions and maximum transfer of momentum and
ii by tuning at a band edge resonance, where the transfer of momentum is a minimum, and the field becomes localized inside the stack. It has been shown that relatively narrow-band band optical pulses may be transmitted without scattering losses or shape changes 4, insuring that momentum and energy storage in￾side the structure is only temporary. Therefore, a structure not fixed to the laboratory frame naturally acquires linear momentum in an effort to conserve it. In what follows we attempt to answer the following question: how much and what sort of motion results from the interaction? The issue of radiation pressure on macroscopic bodies arches all the way back to Maxwell 5, who realized that light in fact exerts pressure, and was later experimentally verified by Nichols and Hull 6. A good perspective of the early history of the subject is given by Mulser 7, who also showed that resonant multiwave interactions, such as stimu￾lated Brillouin and Raman scattering, are radiation-pressure￾driven phenomena. More recently, Antonoyiannakis and Pen￾dry 8 examined issues related to forces present in photonic crystals and found that when traversing from a low to a high dielectric material, a light beam attracts the interface. The implications then extend to 3D
three-dimensional photonic crystals, and the authors go on to predict an attractive force between neighboring dielectric spheres. Povinelli et al. 9 studied the effects of radiation pressure in omni-directional reflector waveguides. They showed that as light propagates down the guide
parallel to the dielectric mirrors, radiation pressure causes the mirrors to attract, and, in the absence of any losses, the attractive force appears to diverge near the cut off frequency. Tucker et al. 10 have investigated effects of radiation pressure and thermal jitter in a hybrid environment, composed of a Fabry-Perot resonator as part of a microme￾chanical switching mechanism
MEMS. The authors found that radiation pressure can cause small changes in the sepa￾ration of movable mirrors even at room temperature, leading to nonlinear shifts of the Fabry-Perot resonance and hyster￾esis loops. In MEMS lasers, the authors suggest that nonlin￾ear radiation pressure effects may induce changes in the characteristic low-frequency chirp of the device 10. The issue of how much electromagnetic momentum is transferred to macroscopic bodies is still a matter of debate, primarily “¼because what is considered electromagnetic and what mechanical is to some extent arbitrary¼,” as noted by Jackson 11. There are two well-known expressions that PHYSICAL REVIEW E 73, 056604
2006 1539-3755/2006/73
5/056604
12 056604-1 ©2006 The American Physical Society

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

RADIATION PRESSURE OF LIGHT PULSES AND PHYSICAL REVIEW E 73. 056604(2006) ation, the typical situation that we describe may be com- pared to a scattering event, during which most of the pulse is D2(r,1)=ε(r,o)2(r,o)e located outside the structure. The consequence of this is that the Minkowski and the abraham momentum densities di play only small differences that decrease as pulse width is [a(r,ω)+b(r,an)a+c(r,a)a2+… increased(the medium contribution in Eq (4)above is lim- ed by the small spatial extension of the structure compare to spatial pulse width). In the current situation we compare the two expressions of momentum density because, unlike where E, (r, c)is the Fourier transform of E(r, t). Assuming the simpler Abraham expression, unusual conditions could that a similar development follows for the magnetic fields, it intervene to significantly alter the appearance and substance is easy to show that of the Minkowski momentum density in a way that depends on the nature of the medium and its dispersive properties, (r,a)E2(r,) hus creating circumstances that may help discriminate be D(r, t)=E(r, wo)E(r, t)+i tween the two quantities even in the transient regime. With these considerations in mind, we set out to derive general ized forms of the momentum densities, and a generali B (rd=u(r. o)y (r. +i u(r, od dr, (r, " Lorentz force density under conditions of dispersive e and u, with an eye also toward applications to NIMs [19], which we briefly treat later in the manuscript B(r, t)=u(r, woH(r, t) dr, oo)dH (r, t) IL. THE MODEI We emphasize that the field decomposition that highlights an We use the Gaussian system of units, and for the moment envelope function and a carrier frequency is done as a matter we assume a TE-polarized incident field of the form of convenience and should be viewed as a simple mathemati cal transformation because the field retains its generality. E=&(E(, z, t)e( -ky y-o)+cc) Substituting Eqs.(11)into the definition of the Minkowski momentum density we find D×B1 H=y( (y, z, t)e ik -ky ) -uo)+cc) if [e(oo)u ( wo)E,,+cc]J iE(on)du( where x, y, i are the unit directional vectors; E and H are real electric and magnetic fields, respectively; E0,z, t) a8 H +c.C.+ H, 0, z, 1), and H_(, z, t)are general, complex envelope functions; and k2=kcos 0, and k,=-ksine; k=k0=@o/c This choice of carrier wave vector is consistent with a pulse yf[e()u(wo)E,1';+cc] initially located in vacuum. We make no other assumptions about the envelope functions. The model that we adopt takes material dispersion(including absorption) into account and (ie( du(wo)s.dH2 makes virtually no approximations. Following Eqs.( 8), the displacement field D may be similarly defined as follows D=x(D,(, z, t)e2 -ky y-uo+cc ) and may be related to the +cc.+ electric field by expanding the complex dielectric function as a Taylor series in the usual way We have simplified the notation by dropping the spatial de- pendence in both e and u, and it is implied in what follows In contrast, the Abraham momentum density is, more simply, d8(r E×H abraka 2 d0- ①-0)+… a(r, oo)+b(r, wo)o+c(r, wo)o+ L-i(cH'+8.H-Ii 4rcvlErH'.+E H) Then, for an isotropic medium, a simple constitutive relation (13) may be written as follows: For relatively slowly varying dielectric functions, the terms 0566043

ration, the typical situation that we describe may be com￾pared to a scattering event, during which most of the pulse is located outside the structure. The consequence of this is that the Minkowski and the Abraham momentum densities dis￾play only small differences that decrease as pulse width is increased
the medium contribution in Eq.
4 above is lim￾ited by the small spatial extension of the structure compared to spatial pulse width. In the current situation we compare the two expressions of momentum density because, unlike the simpler Abraham expression, unusual conditions could intervene to significantly alter the appearance and substance of the Minkowski momentum density in a way that depends on the nature of the medium and its dispersive properties, thus creating circumstances that may help discriminate be￾tween the two quantities even in the transient regime. With these considerations in mind, we set out to derive general￾ized forms of the momentum densities, and a generalized Lorentz force density under conditions of dispersive  and
, with an eye also toward applications to NIMs 19, which we briefly treat later in the manuscript. II. THE MODEL We use the Gaussian system of units, and for the moment we assume a TE-polarized incident field of the form E = xˆ„Ex
y,z,tei
kzz−kyy−0t + c.c.…, H = yˆ„Hy
y,z,tei
kzz−kyy−0t + c.c.… + zˆ„Hz
y,z,tei
kzz−kyy−0t + c.c.…,
8 where xˆ ,yˆ ,zˆ are the unit directional vectors; E and H are real electric and magnetic fields, respectively; Ex
y ,z,t, Hy
y ,z,t, and Hz
y ,z,t are general, complex envelope functions; and kz= k cosi and ky=− k sini , k =k0=0 /c. This choice of carrier wave vector is consistent with a pulse initially located in vacuum. We make no other assumptions about the envelope functions. The model that we adopt takes material dispersion
including absorption into account and makes virtually no approximations. Following Eqs.
8, the displacement field D may be similarly defined as follows: D=xˆ
Dx
y ,z,tei
kzz−kyy−0t +c.c., and may be related to the electric field by expanding the complex dielectric function as a Taylor series in the usual way: 
r, = 
r,0 + 

r,
  0
 − 0 + 1 2 
2 
r,
2  0
 − 0 2 + ¯ = a
r,0 + b
r,0 + c
r,02 + ¯ .
9 Then, for an isotropic medium, a simple constitutive relation may be written as follows: Dx
r,t = −  
r,E ˜ x
r,e−it d = −  a
r,0 + b
r,0 + c
r,02 + ¯  E ˜ x
r,e−it d,
10 where E ˜ x
r, is the Fourier transform of Ex
r,t. Assuming that a similar development follows for the magnetic fields, it is easy to show that Dx
r,t = 
r,0Ex
r,t + i

r,0

Ex
r,t
t + ¯ , By
r,t =

r,0Hy
r,t + i


r,0

Hy
r,t
t + ¯ , Bz
r,t =

r,0Hz
r,t + i


r,0

Hz
r,t
t + ¯ .
11 We emphasize that the field decomposition that highlights an envelope function and a carrier frequency is done as a matter of convenience and should be viewed as a simple mathemati￾cal transformation because the field retains its generality. Substituting Eqs.
11 into the definition of the Minkowski momentum density we find gMinkowski = D B 4c = 1 4c zˆ
0
*
0ExH* y + c.c. + i*
0


0
 E* x
Hy
t + c.c. + i
*
0

0
 H* y
Ex
t + c.c. + ¯ − 1 4c yˆ
0
*
0ExH* z + c.c. + i*
0


0
 E* x
Hz
t + c.c. + i
*
0

0
 H* z
Ex
t + c.c. + ¯ .
12 We have simplified the notation by dropping the spatial de￾pendence in both  and
, and it is implied in what follows. In contrast, the Abraham momentum density is, more simply, gAbraham = E H 4c = 1 4c zˆ
ExH* y + E* xHy − 1 4c yˆ
ExH* z + E* xHz.
13 For relatively slowly varying dielectric functions, the terms RADIATION PRESSURE OF LIGHT PULSES AND¼ PHYSICAL REVIEW E 73, 056604
2006 056604-3

SCALORA et al PHYSICAL REVIEW E 73. 056604(2006) shown in Eq.(12)are usually more than sufficient to accu- rately describe the dynamics, even for very short pulses(a F(O drf(r, t) few wave cycles in duration), because typical dispersion lengths may be on the order of meters, as we will see below 1/ dD(r, t) dE(r, t) The expression for the force density function, Eq (1), in the B absence of free charges and free currents, may be written as {(V)×H×Bdr (16) f(r, t)=phone +c(V×M)×B In deriving Eq.(16)from Eq. (15), we have assumed that the E(V·E)+ tCl( ar magnetic permeability is approximately real and constant to show the basic contributions, including a surface term when +cV×B-cV×H×B the magnetic permeability is discontinuous. We will general (14) ize this expression later when we deal with negative index materials We have made use of the usual constitutive relationsh Next, substituting Eqs. 8)and (11)into Maxwells equa between the fields, namely D=E+4P and B=H+4M. tions yields the following coupled differential equations Equation(14)includes a Coulomb contribution from bound [21-23] and magnetic current densities, in order to allow application ae. a' d8. a a8 to magnetically active materials. The Coulomb term shown 4丌ar24m2ar3 may be expressed in a variety of ways. For example, using the first of Eqs. (11), and by using the condition V.D=0,one =iBlE(s)E-H, sin 8-H, cos 0+-+-, can show that, in the absence of absorption(e=e), the Cou lomb term takes the form ame E(ve. E)+e/r de nl +i Ar as-24 7,+ E|+… =i所()H-Ecos]- The presence of higher order terms is implied. The form 4丌a24m iven in Eq. (14)thus suggests that there is a Coulomb con- ribution if (i) the incident field has a TM-polarized compo- (17) nent;(ii) scattering generally occurs from a three- =i以(9)H2-Exin+ dimensional structure with complex topology that generates other field polarizations; and (iii) if the field has curvature in all three dimensions. Under some circumstances, one may d@E(5)] F[oE(E) ignore the Coulomb contribution, for example by consider- ing TE modes using our Eqs.(8), which lead directly to V.E=dE (, z, t)/dx=0. This is a sufficient condition that may be easily satisfied in problems that exploit one-or two- 到,y=画 dimensional symmetries, as we do here. It should be appar ent, however, that more complicated topologies and/or the consideration of TM-polarized incident fields are in need and the prime symbol denotes differentiation with respect to the general approach afforded by Eq. (14). In light of the frequency. B; is the angle of incidence. The following scaling previous discussion, we will first examine the case of TE- has been adopted: 5=z/Ar y=y/Ar T=ct/Ar B=2o, and polarized incident pulses, and in the last section of the manu- @=o/or where A, =I um is conveniently chosen as the ref- script we will briefly discuss results that concern a TM- erence wavelength. We note that nonlinear effects may be polarized pulse that traverses a single, ordinary dielectric taken into account by adding a nonlinear polarization to the interface. Therefore,for TE-polarized waves, Eq. (14)re- right-hand sides of Eqs.(17), as shown in Ref. [23],for duces to example As we pointed out after the constitutive relation Eq.(9) r(r,) 4m( CVd dE(r,)×B the development that culminates with Eqs.(17)assumes that (15) the medium is isotropic, a restriction that can be removed should the need arise, without impacting the relative simplic Using Maxwell's equations, the total force can then be cal- ity of the approach or method of solution. Beyond this fact culated as Eqs.(17)do not contain any other approximations, but they 056604-4

shown in Eq.
12 are usually more than sufficient to accu￾rately describe the dynamics, even for very short pulses
a few wave cycles in duration, because typical dispersion lengths may be on the order of meters, as we will see below. The expression for the force density function, Eq.
1, in the absence of free charges and free currents, may be written as f
r,t = boundE + 1 c 
P
t + c

M B = 1 4 E

· E + 1 4c 
D
t −
E
t  + c
B − c
H B.
14 We have made use of the usual constitutive relationships between the fields, namely D=E+4P and B=H+4M. Equation
14 includes a Coulomb contribution from bound charges, and contributions from bound dielectric polarization and magnetic current densities, in order to allow application to magnetically active materials. The Coulomb term shown may be expressed in a variety of ways. For example, using the first of Eqs.
11, and by using the condition
·D=0, one can show that, in the absence of absorption
* , the Cou￾lomb term takes the form 1 4− E

 · E + 0E


  0 · E + ¯  1 + 1  

  0 0 + ¯ . The presence of higher order terms is implied. The form given in Eq.
14 thus suggests that there is a Coulomb con￾tribution if
i the incident field has a TM-polarized compo￾nent;
ii scattering generally occurs from a three￾dimensional structure with complex topology that generates other field polarizations; and
iii if the field has curvature in all three dimensions. Under some circumstances, one may ignore the Coulomb contribution, for example by consider￾ing TE modes using our Eqs.
8, which lead directly to
·E=
Ex
y ,z,t/
x0. This is a sufficient condition that may be easily satisfied in problems that exploit one- or two￾dimensional symmetries, as we do here. It should be appar￾ent, however, that more complicated topologies and/or the consideration of TM-polarized incident fields are in need of the general approach afforded by Eq.
14. In light of the previous discussion, we will first examine the case of TE￾polarized incident pulses, and in the last section of the manu￾script we will briefly discuss results that concern a TM￾polarized pulse that traverses a single, ordinary dielectric interface. Therefore, for TE-polarized waves, Eq.
14 re￾duces to f
r,t = 1 4c c
B −
E
r,t
t  B.
15 Using Maxwell’s equations, the total force can then be cal￾culated as F
t = volume dr3 f
r,t = volume  1 4

D
r,t
t −
E
r,t
t  Bdr3 + 1 4 volume 


H Bdr3.
16 In deriving Eq.
16 from Eq.
15, we have assumed that the magnetic permeability is approximately real and constant to show the basic contributions, including a surface term when the magnetic permeability is discontinuous. We will general￾ize this expression later when we deal with negative index materials. Next, substituting Eqs.
8 and
11 into Maxwell’s equa￾tions yields the following coupled differential equations 21–23: 
Ex
+ i 
4
2 Ex
2 −  242
3 Ex
3 + ¯ = i
Ex − Hz sin i − Hy cos i  +
Hz
˜y +
Hy
 ,
Hy
+ i
4
2 Hy
2 − 242
3 Hy
3 + ¯ = i

Hy − Ex cos i  −
Ex
 ,
Hz
+ i
4
2 Hz
2 − 242
3 Hz
3 + ¯ = i

Hz − Ex sin i  +
Ex
˜y .
17 Here  = 
˜ 

˜  0 , 
= 
2 ˜ 

˜ 2  0 , = 
˜


˜  0 ,
= 
2 ˜


˜ 2  0 , and the prime symbol denotes differentiation with respect to frequency. i is the angle of incidence. The following scaling has been adopted: =z/r, ˜y=y /r, =ct/r, =2˜, and ˜ =/r, where r=1
m is conveniently chosen as the ref￾erence wavelength. We note that nonlinear effects may be taken into account by adding a nonlinear polarization to the right-hand sides of Eqs.
17, as shown in Ref. 23, for example. As we pointed out after the constitutive relation Eq.
9, the development that culminates with Eqs.
17 assumes that the medium is isotropic, a restriction that can be removed should the need arise, without impacting the relative simplic￾ity of the approach or method of solution. Beyond this fact, Eqs.
17 do not contain any other approximations, but they SCALORA et al. PHYSICAL REVIEW E 73, 056604
2006 056604-4

RADIATION PRESSURE OF LIGHT PULSES AND PHYSICAL REVIEW E 73. 056604(2006) may be simplified depending on the circumstances. For ex ample, in ordinary dielectric materials we may neglect sec ond and higher order material dispersion terms, which elimi- nates second and higher order temporal derivatives. As example, in the spectral region of interest, which includes the near IR range(800-1200 nm), the dielectric function(ac- tual data)of Si3 N4 [24] may be written as Incident e(a)=37798+0178980+00408 Transmitted sing this approximately linear dielectric susceptibility model, indeed we have a=a[oE(51/a0=0.One may then estimate the second-order dispersion length, de Reflected fined as LB)+/"(l. where T, is incident pulse width, The result is Lp-2X10A(o -2 mm)for an incident, five wave-cycle pulse(-15 fs); 200-15010050 100I50 approximately 8 mm for a ten wave-cycle pulse; and 1 m for 100-wave cycle(-300 fs) pulses. In comparison, typical FIG. 1. A 100 fs pulse interacts with a 60 um thick Si3 N4 sub- multilayer stacks and substrates that we consider range from strate. Both E and H fields are shown as the pulse is partly trans- a few microns to a few tens of microns in thickness, and so mitted and partly reflected from both entry and exit interfaces. Out neglect of the second-order time derivative and beyond is side the structure the fields overlap, while inside pulse compression completely justified, even for pulses only a few wave cycles due to group velocity reduction and conservation of energy causes in duration the magnetic field to increase its amplitude with respect to the in In the frequency range and the materials that we are con- cident field sidering, assuming for the moment that u=y=l, in our aled coordinate system the simplified version of Eq (12)is D×B dg(,s,), minkowski i(eaHy+cc) (2 P(7)= 84, s, r)dy (21) +cC.+ These components may be used to calculate the angle of (e2:+c.) refraction [21]. To simplify matters further, for the moment we assume that the pulse is incident normal to the multilayer surface, i.e. Py(T)=0 at all times, and focus our attention on (19) the longitudinal component. Finally, assuming no frictional or other dissipative forces are present, conservation of mo- magnetic materials, Eq(15)also simpl f(r, t) X(r, t) Pstructure(a)=Pg-Ps(r) (22) P=P(7=0)=d厂=4,,7=0)d5 is the 4A2IiB(e'-1E' Hy-iB(e-1)E-H momentum initially carried by the pulse in free space, before t enters any medium. The force may then be calculated as the temporal derivative of the momentum in Eq (22). C4A, B(e-1)2H2-iB(e-1)EHz In this section we consider the interaction of I MW/cm Gaussian pulse of the type E,0,,T=0) =Eoe-(5-50)2+y yna, and similarly for the transverse mag (a-1)=2+(a-1)- (20) netic field, with a 60 um thick Si, N4 substrate, as depicted in Fig. 1. Choosing w- 20 corresponds to a l/e width of ap Having defined the relevant momentum densities in Eqs. proximately 100 fs in duration, but we note that the exact (12)and(13)above, the total momentum can then be easily temporal duration of the pulse is not crucial. The spatial calculated. In general one has two components, one longitu- extension(both longitudinal and transverse)of the pulse in dinal and one transverse, as follows [11] free space may be estimated from the figure at about 40 um 056604-5

may be simplified depending on the circumstances. For ex￾ample, in ordinary dielectric materials we may neglect sec￾ond and higher order material dispersion terms, which elimi￾nates second and higher order temporal derivatives. As an example, in the spectral region of interest, which includes the near IR range
800–1200 nm, the dielectric function
ac￾tual data of Si3N4 24 may be written as 
˜ = 3.7798 + 0.178 98˜ + 0.044 08 ˜ .
18 Using this approximately linear dielectric susceptibility model, indeed we have = 
3 ˜ 
/
˜ 3  0 0. One may then estimate the second-order dispersion length, de- fined as LD
2  p 2 / k
˜ , where p is incident pulse width, and k
˜=
2 k/
˜ 2 . The result is LD
2 2103 r
or 2 mm for an incident, five wave-cycle pulse
15 fs; approximately 8 mm for a ten wave-cycle pulse; and 1 m for 100-wave cycle
300 fs pulses. In comparison, typical multilayer stacks and substrates that we consider range from a few microns to a few tens of microns in thickness, and so neglect of the second-order time derivative and beyond is completely justified, even for pulses only a few wave cycles in duration. In the frequency range and the materials that we are con￾sidering, assuming for the moment that
= =1, in our scaled coordinate system the simplified version of Eq.
12 is gMinkowski = D B 4c = 1 4c zˆ
ExH* y + c.c. + i 1 2

˜ H* y
Ex
+ c.c. + ¯  − 1 4c yˆ
ExH* z + c.c. + i 1 2

˜ H* z
Ex
+ c.c. + ¯ .
19 For nonmagnetic materials, Eq.
15 also simplifies to f
r,t = 1 c
P
r,t
t B
r,t = 1 4r zˆi
* − 1E* xHy − i
 − 1ExH* y + 
 − 1
Ex
H* y +
* − 1
E* x
Hy + ¯ = 1 4r yˆi
* − 1E* xHz − i
 − 1ExH* z + 
 − 1
Ex
H* z +
* − 1
E* x
Hz + ¯ .
20 Having defined the relevant momentum densities in Eqs.
12 and
13 above, the total momentum can then be easily calculated. In general one has two components, one longitu￾dinal and one transverse, as follows 11: P
= =− = d ˜ y=− ˜ y= g
˜y,, dy˜, P˜ y
= =− = d ˜ y=− ˜ y= g˜ y
˜y,, dy˜.
21 These components may be used to calculate the angle of refraction 21. To simplify matters further, for the moment we assume that the pulse is incident normal to the multilayer surface, i.e., P˜ y
=0 at all times, and focus our attention on the longitudinal component. Finally, assuming no frictional or other dissipative forces are present, conservation of mo￾mentum requires that the linear momentum imparted to the structure be given by Pstructure
= P 0 − P
,
22 where P 0=P
=0==− = d˜ y=− ˜ y= g
˜y ,, =0dy˜ is the total momentum initially carried by the pulse in free space, before it enters any medium. The force may then be calculated as the temporal derivative of the momentum in Eq.
22. A thick, uniform substrate In this section we consider the interaction of a 1 MW/cm2 Gaussian pulse of the type Ex
˜y ,, =0 =E0e−
−0 2 +y−2/w2 , and similarly for the transverse mag￾netic field, with a 60
m thick Si3N4 substrate, as depicted in Fig. 1. Choosing w20 corresponds to a 1/e width of ap￾proximately 100 fs in duration, but we note that the exact temporal duration of the pulse is not crucial. The spatial extension
both longitudinal and transverse of the pulse in free space may be estimated from the figure at about 40
m FIG. 1. A 100 fs pulse interacts with a 60
m thick Si3N4 sub￾strate. Both E and H fields are shown as the pulse is partly trans￾mitted and partly reflected from both entry and exit interfaces. Out￾side the structure the fields overlap, while inside pulse compression due to group velocity reduction and conservation of energy causes the magnetic field to increase its amplitude with respect to the in￾cident field. RADIATION PRESSURE OF LIGHT PULSES AND¼ PHYSICAL REVIEW E 73, 056604
2006 056604-5

SCALORA et al PHYSICAL REVIEW E 73. 056604(2006) (1le width). Inside the medium, the longitudinal spatial width is compressed by roughly a factor proportional to the Abraham momentum oup index, and from the figure it is clear that at some point the pulse is completely embedded inside the medium, so that a steady-state dynamics is reached after the entire pulse 0.6 crosses the entry surface. Once the pulse reaches the exit 0.4 toward the entry interface, so that the energy leaks out rela- o24 scribe some basic facts intrinsic to the event The index of refraction of the substrate at the carrier wavelength(o=l um)is n-2. The transmittance through the surface may be computed as the fraction of energy trans-2-o04 mitted with respect to the incident energy. When dispersion is present, the electromagnetic energy density may be gener- lized as follows [22 C0门=P+pHP+(c:2- Time (in units of 2) FIG. 2. Linear momentum transferred to the substrate using the iB Minkowski (dashed) and Abraham(solid)momenta, normalized (23) with respect to the total incident momentum. The Minkowski mo- 4 mentum predicts that the momentum transferred to the structure will be negative during and after the first interface crossing. The where a, =Re(a),y=Re(), and the symbol ' once again slab gains a linear momentum of -2/3 of the initial momentum means differentiation with respect to the frequency. The total energy can be calculated by integrating Eq.(23)over all upon crossing the first interface. The final momentum of the slab after most of the energy has leaked out, is close to 38%o of the initial space, namely Wr(=f__adEU, s, 1). Our calcula- momentum tions yield a transmittance consistent with the usual transmit tance function: T=4n/(n+1)(T=0.888 for n=2). Evalua- menta(dashed). The figure clearly shows that the Abraham tion of the linear momentum yields a momentum transfer and Lorentz forces overlap during the entire process, while through the first interface that is identical to the result of Eq. the Minkowski momentum never represents the Lorentz (2)obtained in the quantum regime [15], with the proper force to any degree, except in the trivial case of zero force. In positive sign. For example, for n=2 Eq. (2)predicts that 2/3 of the initial momentum is transferred to the substrate and that is precisely what we find(see Fig. 2). However, when it comes to the exit interface. the results differ somewhat from 一-- From Minko those quoted in Ref [15], but are qualitatively similar. In any ase,our calculated, final slab momentum is Pinal"0.38Po To illustrate this, in Fig. 2 we plot the total linear momentum -fIxIo gained by the substrate as a function of time at the expense of the fields, normalized with respect to the total initial mo mentum carried by the pulse, as calculated using Eqs.(12),r (13), and(22). Here we see that the abraham momentum is negative during the first interface crossing. This implies that to conserve momentum the substrate should move toward the pulse or equivalently, be attracted by it. Therefore, the abra m and Minkowski momenta predict that the substrate will move in opposite directions. However, the figure also shows hat when the pulse exits to the right of the substrate, the 0.2x100.4x10 0.6x101 total Minkowski momentum reacquires a positive value not too dissimilar from the abraham momentum. as now most of (open triangles) the pulse is located in free space. This small discrepancy is force calculated using the Abraham(solid) and Minkowski(dashed) due to the fact that a small fraction of the pulse still lingers momenta. It is evident that only the Abraham momentum leads to inside the substrate. as it reflects back and forth from the the Lorentz force and tracks it almost exactly during the entire ti entry and exit interfaces. as in- In Fig. 3 we show the longitudinal Lorentz force calcu- dicated by the sign change of the force, upon entry on exit lated using our Eq(20)(triangles)and compare it to the time from the substrate, and the Minkowski force has alwa derivatives of the Abraham(solid line) and Minkowski mo- site sign 056604-6

1/e width. Inside the medium, the longitudinal spatial width is compressed by roughly a factor proportional to the group index, and from the figure it is clear that at some point the pulse is completely embedded inside the medium, so that a steady-state dynamics is reached after the entire pulse crosses the entry surface. Once the pulse reaches the exit interface, most of it is transmitted as part of it reflects back toward the entry interface, so that the energy leaks out rela￾tively slowly from both sides of the substrate. We now de￾scribe some basic facts intrinsic to the event. The index of refraction of the substrate at the carrier wavelength
0=1
m is n2. The transmittance through the surface may be computed as the fraction of energy trans￾mitted with respect to the incident energy. When dispersion is present, the electromagnetic energy density may be gener￾alized as follows 22: U
˜y,, = r Ex 2 + r Hy 2 + ir
4 Ex *
Ex
− Ex
Ex *
 + ir
4 Hy *
Hy
− Hy
Hy *
 + ¯ ,
23 where r=Re
, r=Re
, and the symbol
once again means differentiation with respect to the frequency. The total energy can be calculated by integrating Eq.
23 over all space, namely WT
=−  dy˜−  dU
˜y ,, . Our calcula￾tions yield a transmittance consistent with the usual transmit￾tance function: T=4n/
n+1 2
T=0.888 for n=2. Evalua￾tion of the linear momentum yields a momentum transfer through the first interface that is identical to the result of Eq.
2 obtained in the quantum regime 15, with the proper positive sign. For example, for n=2 Eq.
2 predicts that 2/3 of the initial momentum is transferred to the substrate, and that is precisely what we find
see Fig. 2. However, when it comes to the exit interface, the results differ somewhat from those quoted in Ref. 15, but are qualitatively similar. In any case, our calculated, final slab momentum is Pfinal0.38P0. To illustrate this, in Fig. 2 we plot the total linear momentum gained by the substrate as a function of time at the expense of the fields, normalized with respect to the total initial mo￾mentum carried by the pulse, as calculated using Eqs.
12,
13, and
22. Here we see that the Abraham momentum is always positive, while the Minkowski momentum becomes negative during the first interface crossing. This implies that to conserve momentum the substrate should move toward the pulse or equivalently, be attracted by it. Therefore, the Abra￾ham and Minkowski momenta predict that the substrate will move in opposite directions. However, the figure also shows that when the pulse exits to the right of the substrate, the total Minkowski momentum reacquires a positive value not too dissimilar from the Abraham momentum, as now most of the pulse is located in free space. This small discrepancy is due to the fact that a small fraction of the pulse still lingers inside the substrate, as it reflects back and forth from the entry and exit interfaces. In Fig. 3 we show the longitudinal Lorentz force calcu￾lated using our Eq.
20
triangles and compare it to the time derivatives of the Abraham
solid line and Minkowski mo￾menta
dashed. The figure clearly shows that the Abraham and Lorentz forces overlap during the entire process, while the Minkowski momentum never represents the Lorentz force to any degree, except in the trivial case of zero force. In FIG. 2. Linear momentum transferred to the substrate using the Minkowski
dashed and Abraham
solid momenta, normalized with respect to the total incident momentum. The Minkowski mo￾mentum predicts that the momentum transferred to the structure will be negative during and after the first interface crossing. The slab gains a linear momentum of 2/3 of the initial momentum upon crossing the first interface. The final momentum of the slab, after most of the energy has leaked out, is close to 38% of the initial momentum. FIG. 3. Lorentz force calculated using Eq.
20
open triangles, force calculated using the Abraham
solid and Minkowski
dashed momenta. It is evident that only the Abraham momentum leads to the Lorentz force and tracks it almost exactly during the entire time. The Abraham force is always directed toward the substrate, as in￾dicated by the sign change of the force, upon entry and upon exit from the substrate, and the Minkowski force has always the oppo￾site sign. SCALORA et al. PHYSICAL REVIEW E 73, 056604
2006 056604-6

RADIATION PRESSURE OF LIGHT PULSES AND PHYSICAL REVIEW E 73. 056604(2006) Index Profile 5 FIG. 4. Scattering of a 600 fs incident pulse from a Position(microns) 4 um multilayer structure composed of 15 periods of Si3N4(125 nm)/Si0,(150 nm), having a mass m-10-g, and vol- FIG. 5. Electric and magnetic field localization properties of a ume V=4x 10-12 m3. Most of the pulse is always located in free light pulse tuned at the photonic band edge. The electric and mag- pace during the entire interaction, a fact that eventually causes the netic fields are spatially delocalized, resulting in small group and Abraham and Minkowski momenta to be similar energy velocities. The y-axis scaling reflects the magnitude of fields inside the structure relative to the input intensity. This kind Ref [15] the total momentum transferred to the structure is field localization and enhancement. which carries momentum calculated by performing the time integral of the total calcu- energy, is not available for simple Fabry-Perot etalons lated force. It is evident even from Fig. 3 that this procedure may, under the right circumstances, yield similar areas for multilayer stack. The fields are delocalized with minimum both the Minkowski and Abraham momenta, especially if overlap, leading to small group and energy velocities [4] one waits for the pulse to leave the structure. However, cal- Because the transverse field profiles do not change, we resort culation of the forces via direct integration of Maxwell,s to plotting just the longitudinal, axial cross section of the equation, accompanied by a direct evaluation of the Abraham pulse and Minkowski momenta, reveals unmistakable agreement In Fig. 6 we compare the Minkowski and Abraham mo between the Abraham and the Lorentz forces. Based on this menta calculated using Eqs.(12)and(13)for a pulse ap- example, our conclusion is that it is generally not possible proximately 100 fs in duration for two different conditions for the Minkowski momentum, the averaged momentum Eq. tuning at the band edge resonance, and inside the gap. The (3), or any other plausible definition of momentum that uses calculations show that when the carrier wavelength is tuned inside the gap, so that the structure acts like a mirror(the the fields, to accomplish the same thing in substrates or other transmittance is less than 10-3), there is effectively no diff similar extended media ence between the two momenta, and so the curves overlap II. PHOTONIC BAND GAP STRUCTURES 6. 0xI We now consider a typical finite multilayer sample. We ssume the stack is composed of 15 periods of generic, rep- 8 resentative, dispersive materials with dielectric constants e E4.5x10 2(as in SiO2) and e2=4 [as in Siy N4, and we use the b dispersion function of Eq (18)above]over the entire near IR range. Assuming a cross section of approximately 1 mm and a thickness of -4 microns(Sio2 layers are taken to be 53.0x10 150 nm thick, and that Si3N4 layers are 125 nm thick) volume of the structure is Ve4x10-12 m3 Using the known material densities of SiO, and Si3N4, the mass of the struc- E 1.5x10 ture can be estimated at m-10 g. For the moment we neglect the presence of a substrate, and assume the beam waist is at least several tens of wave cycles wide, so that we nay also neglect diffraction effects 2x10 carrier frequency of a narrow-band pulse [4] is Time(see) near the band edge. The structure is located FIG. 6. Abraham and Minkowski momenta for a 100 fs pulse near the origin, and with a spatial extension of only tuned inside the gap(solid line overlapped by open circles)and for 4 microns, it is evident that most of the pulse is located the same pulse tuned at the band edge resonance(thin solid and outside the structure most of the time: this is the primary ong dashes). The average momentum is also calculated(thin reason why the Minkowski and Abraham momentum densi- dashes ). The Abraham and Minkowski momenta yield similar re ties generally differ little during the interaction. In Fig. 5 we sults because in both cases the pulse is located mostly in free space ow the electric and magnetic field profiles inside the during the entire time, as the structure is only a few microns thick 056604-7

Ref. 15 the total momentum transferred to the structure is calculated by performing the time integral of the total calcu￾lated force. It is evident even from Fig. 3 that this procedure may, under the right circumstances, yield similar areas for both the Minkowski and Abraham momenta, especially if one waits for the pulse to leave the structure. However, cal￾culation of the forces via direct integration of Maxwell’s equation, accompanied by a direct evaluation of the Abraham and Minkowski momenta, reveals unmistakable agreement between the Abraham and the Lorentz forces. Based on this example, our conclusion is that it is generally not possible for the Minkowski momentum, the averaged momentum Eq.
3, or any other plausible definition of momentum that uses the fields, to accomplish the same thing in substrates or other similar extended media. III. PHOTONIC BAND GAP STRUCTURES We now consider a typical finite multilayer sample. We assume the stack is composed of 15 periods of generic, rep￾resentative, dispersive materials with dielectric constants 1 2
as in SiO2 and 24 as in Si3N4, and we use the dispersion function of Eq.
18 above over the entire near IR range. Assuming a cross section of approximately 1 mm2 and a thickness of 4 microns
SiO2 layers are taken to be 150 nm thick, and that Si3N4 layers are 125 nm thick, the volume of the structure is V410−12 m3 . Using the known material densities of SiO2 and Si3N4, the mass of the struc￾ture can be estimated at m10−5 g. For the moment we neglect the presence of a substrate, and assume the beam waist is at least several tens of wave cycles wide, so that we may also neglect diffraction effects. In Fig. 4 we show a typical scattering event when the carrier frequency of a narrow-band pulse 4 is tuned at the first resonance near the band edge. The structure is located near the origin, and with a spatial extension of only 4 microns, it is evident that most of the pulse is located outside the structure most of the time: this is the primary reason why the Minkowski and Abraham momentum densi￾ties generally differ little during the interaction. In Fig. 5 we show the electric and magnetic field profiles inside the multilayer stack. The fields are delocalized with minimum overlap, leading to small group and energy velocities 4. Because the transverse field profiles do not change, we resort to plotting just the longitudinal, axial cross section of the pulse. In Fig. 6 we compare the Minkowski and Abraham mo￾menta calculated using Eqs.
12 and
13 for a pulse ap￾proximately 100 fs in duration for two different conditions: tuning at the band edge resonance, and inside the gap. The calculations show that when the carrier wavelength is tuned inside the gap, so that the structure acts like a mirror
the transmittance is less than 10−3, there is effectively no differ￾ence between the two momenta, and so the curves overlap FIG. 4. Scattering of a 600 fs incident pulse from a 4
m multilayer structure composed of 15 periods of Si3N4
125 nm/SiO2
150 nm, having a mass m10−5 g, and vol￾ume V410−12 m3. Most of the pulse is always located in free space during the entire interaction, a fact that eventually causes the Abraham and Minkowski momenta to be similar. FIG. 5. Electric and magnetic field localization properties of a light pulse tuned at the photonic band edge. The electric and mag￾netic fields are spatially delocalized, resulting in small group and energy velocities. The y-axis scaling reflects the magnitude of the fields inside the structure relative to the input intensity. This kind of field localization and enhancement, which carries momentum and energy, is not available for simple Fabry-Perot etalons. FIG. 6. Abraham and Minkowski momenta for a 100 fs pulse tuned inside the gap
solid line overlapped by open circles and for the same pulse tuned at the band edge resonance
thin solid and long dashes. The average momentum is also calculated
thin dashes. The Abraham and Minkowski momenta yield similar re￾sults because in both cases the pulse is located mostly in free space during the entire time, as the structure is only a few microns thick. RADIATION PRESSURE OF LIGHT PULSES AND¼ PHYSICAL REVIEW E 73, 056604
2006 056604-7

SCALORA et al PHYSICAL REVIEW E 73. 056604(2006) 75x105 Lorentz Force Force Abraham 50X10 A Minkowski 2.5X10 9 1x10 Time(sec) 0.8x101212x101216x10 FIG. 7. Lorentz force (open triangles), Abraham force(solid curve), and Minkowski (dashed) force obtained from the time de- FIG. 8. Force versus time experienced by the free-standing rivative of the respective momenta, depicted in Fig. 6. Small differ- multilayer stack, corresponding to an incident, 600 fs pulse tuned at ences notwithstanding, the Abraham force tracks the Lorentz force resonance, just as in Fig. 6. The bandwidth of the present pulse is to better than one part in a thousand. However, the total integrated approximately six times narrower compared to that of Fig. 6, which areas under the Abraham and Minkowski curves, which represent leads to better field localization and smaller overall reflection. Tun- the total momentum transferred. have the same value to at least one ing the carrier frequency of the pulse at resonance leads to negative art in one thousand forces, with correspondingly negative accelerations, just as in the (solid line with open circles). The pulse remains mostly out- oscilaioig. 3. Depending on pulse duration and bandwidth, this ide the structure, as the penetration depth(or skin depth) dent pulses. Only the Abraham force is seen to accurately reproduce amounts to only a small fraction of a wavelength. the lorentz force Tuning the pulse at resonance results in field localizations similar to those of Fig. 5 and leads to slightly different and amounts of energy and momentum are always reflected For distinguishable curves. However, it is also clear that any dif- example, referring to Figs. 8 and 9, we find that the structure ferences are transient, as only a tiny portion of the pulse begins to move forward as energy and momentum are stored occupies the structure at any given time. For all intents and inside it. In addition to field localization effects, tuning a purposes either representation may be used to obtain the or- relatively narrow-band pulse at resonance guarantees that it der of magnitude of the total momentum transferred. Never- will reacquire nearly all of its initial forward momentum, heless, taking the time derivatives of the momenta shown in within the bounds dictated by the bandwidth of the pulse total force experienced by the structure, which we show in the direction of the structure, consistent with pr shes in ig. 7. The figure clearly shows that even though differences dictions [15, 16], and as the peak of the pulse spills over to are small, it is only the Abraham momentum that once again the right of the barrier the structure is pushed backward. This coincides almost exactly with an independent calculation of can be discerned by the fact that the instantaneous accelera- the Lorentz force, Eq.(20), even though the integrated areas under the curves yield almost identical results, with differ- ences in the range of one part in a thousand. In Figs. 8 and 9 we show the predicted force and displace ment, respectively, associated with a mass of 10-5g acted upon by a Gaussian pulse approximately 600 fs in duration. and peak power of 1 MW/cm, once again tuned at reso- nance. Although Fig 8 suggests remarkably high accelera EE三 1x10 tions, with maxima of -+5X cm/s(force/mass), Fig 9 suggests that the magnitudes of the displacement and as sociated velocity [calculated using the simple classical ex pressions X(t)=xo+uf+Jat and V(t=at] are tempered by the extremely short interaction times. We note that both 0.5X10 10X10 Abraham and Minkowski momenta yield similar results, due to the finite extent of the structure FIG.9. Longitudinal displacement that corresponds to the Abra Tuning at a band edge resonance produces a more pecu- ham force (and acceleration)shown in Fig. 8. The structure is liar and ostensibly more intriguing dynamics, as Figs. 6-9 pushed forward, returns to the origin, but eventually acquires for- suggest. While the pulse generally exerts a force always di- ward terminal velocity. The cavity stores energy and momentum, rected toward the structure upon entering and upon exiting with a relatively long tail that may keep the structure moving back he medium, using finite bandwidth pulses means that the and forth, depending on pulse bandwidth and tuning with respect to structure will be left with some residual momentum as finite the band edg 056604-8

solid line with open circles. The pulse remains mostly out￾side the structure, as the penetration depth
or skin depth amounts to only a small fraction of a wavelength. Tuning the pulse at resonance results in field localizations similar to those of Fig. 5 and leads to slightly different and distinguishable curves. However, it is also clear that any dif￾ferences are transient, as only a tiny portion of the pulse occupies the structure at any given time. For all intents and purposes either representation may be used to obtain the or￾der of magnitude of the total momentum transferred. Never￾theless, taking the time derivatives of the momenta shown in Fig. 6, for the pulse tuned at the band edge, results in the total force experienced by the structure, which we show in Fig. 7. The figure clearly shows that even though differences are small, it is only the Abraham momentum that once again coincides almost exactly with an independent calculation of the Lorentz force, Eq.
20, even though the integrated areas under the curves yield almost identical results, with differ￾ences in the range of one part in a thousand. In Figs. 8 and 9 we show the predicted force and displace￾ment, respectively, associated with a mass of 10−5 g acted upon by a Gaussian pulse approximately 600 fs in duration, and peak power of 1 MW/cm2 , once again tuned at reso￾nance. Although Fig. 8 suggests remarkably high accelera￾tions, with maxima of ±51010 cm/s2
force/mass, Fig. 9 suggests that the magnitudes of the displacement and as￾sociated velocity calculated using the simple classical ex￾pressions X
tx0+vt+ 1 2 at2 and V
tat are tempered by the extremely short interaction times. We note that both Abraham and Minkowski momenta yield similar results, due to the finite extent of the structure. Tuning at a band edge resonance produces a more pecu￾liar and ostensibly more intriguing dynamics, as Figs. 6–9 suggest. While the pulse generally exerts a force always di￾rected toward the structure upon entering and upon exiting the medium, using finite bandwidth pulses means that the structure will be left with some residual momentum, as finite amounts of energy and momentum are always reflected. For example, referring to Figs. 8 and 9, we find that the structure begins to move forward as energy and momentum are stored inside it. In addition to field localization effects, tuning a relatively narrow-band pulse at resonance guarantees that it will reacquire nearly all of its initial forward momentum, within the bounds dictated by the bandwidth of the pulse. Therefore, we find that the pulse centroid always pushes in the direction of the structure, consistent with previous pre￾dictions 15,16, and as the peak of the pulse spills over to the right of the barrier the structure is pushed backward. This can be discerned by the fact that the instantaneous accelera￾FIG. 7. Lorentz force
open triangles, Abraham force
solid curve, and Minkowski
dashed force obtained from the time de￾rivative of the respective momenta, depicted in Fig. 6. Small differ￾ences notwithstanding, the Abraham force tracks the Lorentz force to better than one part in a thousand. However, the total integrated areas under the Abraham and Minkowski curves, which represent the total momentum transferred, have the same value to at least one part in one thousand. FIG. 8. Force versus time experienced by the free-standing multilayer stack, corresponding to an incident, 600 fs pulse tuned at resonance, just as in Fig. 6. The bandwidth of the present pulse is approximately six times narrower compared to that of Fig. 6, which leads to better field localization and smaller overall reflection. Tun￾ing the carrier frequency of the pulse at resonance leads to negative forces, with correspondingly negative accelerations, just as in the case of Fig. 3. Depending on pulse duration and bandwidth, this oscillatory motion may be sustained by a well-timed train of inci￾dent pulses. Only the Abraham force is seen to accurately reproduce the Lorentz force. FIG. 9. Longitudinal displacement that corresponds to the Abra￾ham force
and acceleration shown in Fig. 8. The structure is pushed forward, returns to the origin, but eventually acquires for￾ward terminal velocity. The cavity stores energy and momentum, with a relatively long tail that may keep the structure moving back and forth, depending on pulse bandwidth and tuning with respect to the band edge. SCALORA et al. PHYSICAL REVIEW E 73, 056604
2006 056604-8

RADIATION PRESSURE OF LIGHT PULSES AND PHYSICAL REVIEW E 73. 056604(2006) Incident s小N4sN4 075 050 0.5xl0 0.25 40-2002040 Normalized Frequency(1/microns) FIG. 11. The same structure described in the text and in the FIG.10. Plane-wave transmittance(left y axis) and total mo- medium, chosen here to be Sis N4. The figure shows both the inci- mentum transferred (right y axis)for the structure described in Fig. dent pulse and the pulse reflected from the embedded mirror while 4. A changing transmittance and field localization properties near it is still located inside the entry substrate. Transmittance through the band edge lead to widely tunable total momentum transfer. he Bragg mirror is less than 10-3. tion changes sign(Fig 8). Furthermore, forward motion is almost compensated by its backward movement, and the We now examine the case of a photonic band gap struc structure tends to return to its original position(Fig. 9). ture immersed inside a background dielectric material whose However, the device is literally immersed inside fields that index of refraction is other than unity. In the example we the cavity. In general, the structure may oscillate about the section in the middle of a dielectric substrate, akin to a re- origin, or move forward, begin to turn back, and then move flective membrane immersed in a liquid, and tune the carrier forward again. In this case the oscillation is ultimately fol- frequency of the incident pulse inside the photonic band gap lowed by a forward terminal velocity. These dynamics, and to utilize the structure as a mirror. The advantage of this the ultimate direction of motion of a free-standing mass, for situation with respect to an ordinary metallic mirror is that the most part depend on the tuning condition with respect to we have no material absorption to consider or interpret, thus the band edge, and the bandwidth of the incident pulse. This leaving no doubt as to how the energy and momentum are particular example clearly does not exhaust the possibilities Finally, in Fig. 10 we plot the plane-wave transmittance of utilized. The situation is depicted in Fig. ll, where we show he structure (left y axis), calculated using the matrix transfer he incident and reflected pulses, and the multilayer stack technique. On the right y axis we plot the total linear mo- immersed inside a Si3Na-like background medium. The entry um gained by the multilayer stack versus normalized substrate is thick enough to contain the entire pulse, so that a frequency, as calculated using the Abraham momentum [Eq steady-state dynamics is reached. In Fig. 12 we show the (13)]. In this instance we again use 600 fs incident pulses to predicted momenta. The time evolution of the momentum widely tunable because of the diverse field localization prop- We stop the pulse while it remains inside the substrate, be- erties that occur near the band edge, with minimum but non- cause discrepancies between the momenta are largest there zero momentum transfer at resonance and mirrorlike reflec- tions and maximum momentum transfer when the pulse is gap. The total momentum transferred, and hence displace ents, may be increased in at least three ways: (i) by increas UI ing pulse duration, (ii) by sending a train of pulses, or (iii) by increasing pulse peak power. For example, a group of 10 pulses pushes the overall displacement in the nanometer range. If we increase pulse duration to 100 ps, then the struc ture's displacement becomes of the same order of magnitude required to observe interference effects due to radiation pres- sure in MEMS environments [10]. In general, the degree of sensitivity appears to be remarkably high, but it may be fur x1034x10135x10 ther increased by either decreasing the number of incident Time(sec) pulses or by reducing peak power. One may envision appli cations to ultra-high sensitive torsional balances and pressure FIG 12. Abraham(solid curve), Minkowski(short dashes), and gauges, for example average(long dashes)momenta for the case depicted in Fig. 11 056604-9

tion changes sign
Fig. 8. Furthermore, forward motion is almost compensated by its backward movement, and the structure tends to return to its original position
Fig. 9. However, the device is literally immersed inside fields that will continue to push and pull as long as light lingers inside the cavity. In general, the structure may oscillate about the origin, or move forward, begin to turn back, and then move forward again. In this case the oscillation is ultimately fol￾lowed by a forward terminal velocity. These dynamics, and the ultimate direction of motion of a free-standing mass, for the most part depend on the tuning condition with respect to the band edge, and the bandwidth of the incident pulse. This particular example clearly does not exhaust the possibilities. Finally, in Fig. 10 we plot the plane-wave transmittance of the structure
left y axis, calculated using the matrix transfer technique. On the right y axis we plot the total linear mo￾mentum gained by the multilayer stack versus normalized frequency, as calculated using the Abraham momentum Eq.
13. In this instance we again use 600 fs incident pulses to better resolve the resonances. The figure clearly suggests that the amount of momentum transferred to the structure is widely tunable because of the diverse field localization prop￾erties that occur near the band edge, with minimum but non￾zero momentum transfer at resonance, and mirrorlike reflec￾tions and maximum momentum transfer when the pulse is tuned inside the gap. The total momentum transferred, and hence displace￾ments, may be increased in at least three ways:
i by increas￾ing pulse duration,
ii by sending a train of pulses, or
iii by increasing pulse peak power. For example, a group of 106 pulses pushes the overall displacement in the nanometer range. If we increase pulse duration to 100 ps, then the struc￾ture’s displacement becomes of the same order of magnitude required to observe interference effects due to radiation pres￾sure in MEMS environments 10. In general, the degree of sensitivity appears to be remarkably high, but it may be fur￾ther increased by either decreasing the number of incident pulses or by reducing peak power. One may envision appli￾cations to ultra-high sensitive torsional balances and pressure gauges, for example. We now examine the case of a photonic band gap struc￾ture immersed inside a background dielectric material whose index of refraction is other than unity. In the example we place the same multilayer stack that we used in the previous section in the middle of a dielectric substrate, akin to a re- flective membrane immersed in a liquid, and tune the carrier frequency of the incident pulse inside the photonic band gap to utilize the structure as a mirror. The advantage of this situation with respect to an ordinary metallic mirror is that we have no material absorption to consider or interpret, thus leaving no doubt as to how the energy and momentum are utilized. The situation is depicted in Fig. 11, where we show the incident and reflected pulses, and the multilayer stack immersed inside a Si3N4-like background medium. The entry substrate is thick enough to contain the entire pulse, so that a steady-state dynamics is reached. In Fig. 12 we show the predicted momenta. The time evolution of the momentum tracks the pulse as it crosses the entry interface
I, impacts the mirror
II, and turns back toward the entry surface
III. We stop the pulse while it remains inside the substrate, be￾cause discrepancies between the momenta are largest there. FIG. 10. Plane-wave transmittance
left y axis and total mo￾mentum transferred
right y axis for the structure described in Fig. 4. A changing transmittance and field localization properties near the band edge lead to widely tunable total momentum transfer. FIG. 11. The same structure described in the text and in the caption of Fig. 4 is now embedded inside a background dielectric medium, chosen here to be Si3N4. The figure shows both the inci￾dent pulse and the pulse reflected from the embedded mirror while it is still located inside the entry substrate. Transmittance through the Bragg mirror is less than 10−3. FIG. 12. Abraham
solid curve, Minkowski
short dashes, and average
long dashes momenta for the case depicted in Fig. 11. RADIATION PRESSURE OF LIGHT PULSES AND¼ PHYSICAL REVIEW E 73, 056604
2006 056604-9

SCALORA et al PHYSICAL REVIEW E 73. 056604(2006) wells equations(17), compare with the temporal derivatives of the Abraham and Minkowski momenta, and plot the re- Abraham sults in Fig. 13. The figure comments itself, as once again the 4x10° Lorentz force is closely tracked only by the Abraham force 2x10° IV NEGATIVE INDEX MATERIALS In negative index materials, the Lorentz force density is slightly more complex because all the terms in Eq.(16)con 0 tribute. The resulting generalized expression for the longitu- dinal and transverse components of the Lorentz force den 6x10 sity, assuming both e and u are complex, may be written FIG. 13. The forces at play in the situation described in Figs and 12. Only the abraham force once again tracks the Lorentz force r,)=2(-1)e2H2-'(p-1)e门 very well during the entire time In fact, the figure shows that the Abraham and Minkowski +(a'2-1)H+(a-1) momenta may differ by as much as a factor of 2, as long as the pulse is still located inside the substrate. The two mo- menta converge to roughly the same value if the pulse is h2+ allowed to exit back into fre ee space The plateau between regions I and II represents the pulse in transit toward the mirror. Therefore, any detection scheme designed to discern ficant differences should detect motion of the reflec- tive membrane before the pulse exits back into vacuum. Just as we did before, we now determine the Lorentz force, Eq. (E-1)E(g+(ap-1)( (20), from the fields that result from the integration of Max- (a-1) HyME+H-H,Hg+ (24) For simplicity, we have retained only the lowest order terms 公会 A simple comparison reveals that Eq(24) reduces to Eq (20) when there are no magnetic contributions, as it should Each part of Eq.(24) displays a magnetic component that contains the longitudinal spatial derivative of the magnetic permeability. If u is discontinuous, some care should be ex- ercised when the volume integrals of Eq (24)are evaluated. As before, the fields found in Eq.(24)are calculated using Maxwells equations(17). Because typical dispersion lengths n negative index materials may easily exceed several hun- dred wavelengths [20-23]. Eqs. (17)may once again be sim- plified by retaining terms up to and including the first-order emporal derivatives on both fields. The reference wave- 34 82 length is now taken to be the plasma frequency, so that X Longitudinal Coordinate(in units Ap In our case and in our units, for incident 30 wave cycle FIG. 14. A pulse approximately 30 wave cycles wide( thus 30 pulses, Lp =p/K(@)I=30/1.6=562Xp, which justifies our from a negative index material at an angle of 150. We note that in Fig 14 we depict the typical dynamics that ensues as a result the Drude model, causality's only demand is that y#0. These con- ditions cause the pulse to refract anomalously in the upper quadrant, strate is magnetically active; we use the Drude model to while the pulse distorts in both real and Fourier space. The result is describe both a and u to enforce a causal response, and use a wave packet whose Poynting vector points forward in the direc- e(o)=(a)=1-1/(a2+iy) ackwards to- The pulse is incident at a 15 angle, and its carrier fre ward the entry surface [21]. We stop the pulse while it is still lo- quency is tuned at o=0.577, where both e(@)=u(@)=-2, cated inside the substrate and dEo/d0=duo/0-4. We choose y=10-, which 056604-10

In fact, the figure shows that the Abraham and Minkowski momenta may differ by as much as a factor of 2, as long as the pulse is still located inside the substrate. The two mo￾menta converge to roughly the same value if the pulse is allowed to exit back into free space. The plateau between regions I and II represents the pulse in transit toward the mirror. Therefore, any detection scheme designed to discern any significant differences should detect motion of the reflec￾tive membrane before the pulse exits back into vacuum. Just as we did before, we now determine the Lorentz force, Eq.
20, from the fields that result from the integration of Max￾well’s equations
17, compare with the temporal derivatives of the Abraham and Minkowski momenta, and plot the re￾sults in Fig. 13. The figure comments itself, as once again the Lorentz force is closely tracked only by the Abraham force. IV. NEGATIVE INDEX MATERIALS In negative index materials, the Lorentz force density is slightly more complex because all the terms in Eq.
16 con￾tribute. The resulting generalized expression for the longitu￾dinal and transverse components of the Lorentz force den￾sity, assuming both  and
are complex, may be written as follows: f
r,t = 1 4r zˆi


* * − 1Ex * Hy −
*

 − 1ExHy *  +

*
* − 1Hy
Ex *
+
*

 − 1Hy *
Ex
− 

2
  Hy 2 + ¯  − 1 4r yˆi


* * − 1Ex * H −
*

 − 1ExH *  +

*
* − 1H
Ex *
+
*

 − 1H *
Ex
− 


*
 Hy * H +
*


 HyH *  + ¯ .
24 For simplicity, we have retained only the lowest order terms. A simple comparison reveals that Eq.
24 reduces to Eq.
20 when there are no magnetic contributions, as it should. Each part of Eq.
24 displays a magnetic component that contains the longitudinal spatial derivative of the magnetic permeability. If
is discontinuous, some care should be ex￾ercised when the volume integrals of Eq.
24 are evaluated. As before, the fields found in Eq.
24 are calculated using Maxwell’s equations
17. Because typical dispersion lengths in negative index materials may easily exceed several hun￾dred wavelengths 20–23, Eqs.
17 may once again be sim￾plified by retaining terms up to and including the first-order temporal derivatives on both fields. The reference wave￾length is now taken to be the plasma frequency, so that r =p. In our case and in our units, for incident 30 wave cycle pulses, LD
2  p 2 / k
˜ =302 /1.6=562p, which justifies our neglect of second and higher order temporal derivatives. In Fig. 14 we depict the typical dynamics that ensues as a result of integrating the system of equations
17 when the sub￾strate is magnetically active; we use the Drude model to describe both  and
to enforce a causal response, and use: 
˜=

˜=1−1/
˜ 2+i˜ . The pulse is incident at a 15° angle, and its carrier fre￾quency is tuned at =0.577, where both 
˜=

˜−2, and

˜/
˜ =


˜/
˜ 4. We choose =10−5, which FIG. 13. The forces at play in the situation described in Figs. 11 and 12. Only the Abraham force once again tracks the Lorentz force very well during the entire time. FIG. 14. A pulse approximately 30 wave cycles wide
thus 30 wave cycles in duration crosses an interface that separates vacuum from a negative index material at an angle of 15°. We note that in the Drude model, causality’s only demand is that 0. These con￾ditions cause the pulse to refract anomalously in the upper quadrant, while the pulse distorts in both real and Fourier space. The result is a wave packet whose Poynting vector points forward in the direc￾tion of propagation, and a wave vector that points backwards, to￾ward the entry surface 21. We stop the pulse while it is still lo￾cated inside the substrate. SCALORA et al. PHYSICAL REVIEW E 73, 056604
2006 056604-10