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