《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