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 0566043ration, 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