RADIATION PRESSURE OF LIGHT PULSES AND PHYSICAL REVIEW E 73. 056604(2006) other plausible definition of momentum density cannot pro- vide a viable alternative including the Livens and Minkowski momentum densities 口 Transverse 多 V TM POLARIZATION We now briefly comment on the case of TM-polarized 92x10 incident pulses, which may be described as follows H=X(H1(y,)ek3-y-+c,) E=y(E (y, z, D)ei(2 2-ky -oot)+cc) +2(E2(,z,t)ek2-6y-)+cc (26) 1x1013 For simplicity, we assume that a 30-wave cycle (lle Time(sec) width) pulse is obliquely incident on a SiaN4 substrate. We FIG. 15. Longitudinal(empty triangles)and transverse(open use this material because for relatively short propagation dis- squares)Lorentz forces, compared to the same quantities computed tances we can neglect absorption(e=e) and assume weak from the Abraham momentum(solid thin curve: longitudinal; solid dispersion(a= dGe]/dalang). In this limit, the Coulomb thick curve: transverse). The arrows point to both the longitudinal term [i.e, the expr and transverse Livens forces Once again, the agreement between (1/4TE(VE)=-(E EE. The Lorentz force density the prediction of the Abraham momentum and the Lorentz force is Eq. (20) may then be E follows obvious, and it remains unmatched by all competing definitions, including the Minkowski momentum (not shown in the figure) rt,)≈(e a8.a8 iB(,Hx2-eH)-%42+ yields no discernable absorption for relatively short propaga tion distances. Inside the substrate, the energy and group iB(E,H-E.H) velocities are both positive and equal, with a value of ap- proximately V&=VE=c/4 [21]. The front of the transmitted wave packet distorts and refracts in a direction consistent y)+ with Snells law, but with a negative index of refraction. The pulse initially contains both longitudinal and transverse mo- menta, as it is launched at an angle with respect to the sur- Just as was the case for negative index materials, the volume face. Because the conditions found in a nim environment re much more stringent and unique compared to conditions integral of Eq (27)contains surface terms that result from a ment to test the conservation laws and. as before. care should be exercised in the evaluation of In magnetically active materials, one may also consider the surface integrals. Nevertheless, we once again find that competing definition of momentum density, namely g, only the Abraham momentum leads to an accurate descrip- (E X B)/4TC, due to Livens [25], which in our case be- tion of the Lorentz force, as expected. comes E×B1 V CONCLUSIONS tEH+uEH,+ i/, dH 2丌\oor For almost a century the debate regarding the basic elec tromagnetic conservation laws in macroscopic media has uEH, +uEH continued unabated, and it will probably continue for some time to come. In this study we have used a numerical ap proach to solve the vector Maxwells equation when disper on is present, and established that, under a variety of cir- cumstances. conditions. and media. the conservation of In Fig. 15 we show the longitudinal and transverse Abraham linear momentum may be understood solely in terms of the and Livens forces and compare them with the volume inte- Poynting vector and Abraham momentum density. Gordon ral of the Lorentz force, Eq.(24). We also calculate the showed [18] that our Eq (7)is in fact the relevant conserva- Minkowski force, but for clarity we do not show it in the tion law even in the presence of matter. The presence of figure. Once again we find that it is only the Abraham mo- magnetic activity must be taken into account with some cau- mentum that accurately describes the conservation of linear tion, but given the guidance that our numerical evidence pro- momentum every step of the way, even under these extreme vides, it should be possible to generalize Gordon's expres conditions. Therefore, one can reasonably conclude that any sion for arbitrary a and A 056604-11yields no discernable absorption for relatively short propaga￾tion distances. Inside the substrate, the energy and group velocities are both positive and equal, with a value of ap￾proximately Vg=VEc/4 21. The front of the transmitted wave packet distorts and refracts in a direction consistent with Snell’s law, but with a negative index of refraction. The pulse initially contains both longitudinal and transverse mo￾menta, as it is launched at an angle with respect to the sur￾face. Because the conditions found in a NIM environment are much more stringent and unique compared to conditions existing in ordinary materials, it provides an ideal environ￾ment to test the conservation laws. In magnetically active materials, one may also consider a competing definition of momentum density, namely gL =
EB/4c, due to Livens 25, which in our case be￾comes gL = E B 4c = 1 4c zˆ
* EHy * +
E* Hy + i 2


 E*
Hy
−

*
E
Hy *
 + ¯  − 1 4c yˆ
* EHz * +
E* Hz + i 2


 E*
Hz
−

*
E
Hz *
 + ¯ .
25 In Fig. 15 we show the longitudinal and transverse Abraham and Livens forces and compare them with the volume inte￾gral of the Lorentz force, Eq.
24. We also calculate the Minkowski force, but for clarity we do not show it in the figure. Once again we find that it is only the Abraham mo￾mentum that accurately describes the conservation of linear momentum every step of the way, even under these extreme conditions. Therefore, one can reasonably conclude that any other plausible definition of momentum density cannot pro￾vide a viable alternative, including the Livens and Minkowski momentum densities. V. TM POLARIZATION We now briefly comment on the case of TM-polarized, incident pulses, which may be described as follows: H = xˆ„Hx
y,z,tei
kzz−kyy−0t + c.c.… E = yˆ„Ey
y,z,tei
kzz−kyy−0t + c.c.… + zˆ„Ez
y,z,tei
kzz−kyy−0t + c.c.….
26 For simplicity, we assume that a 30-wave cycle
1/e width pulse is obliquely incident on a Si3N4 substrate. We use this material because for relatively short propagation dis￾tances we can neglect absorption
* and assume weak dispersion
=
˜ /
˜ 0 . In this limit, the Coulomb term i.e., the expression below Eq.
14 reduces to
1/4E

·E=−
E·
/4E. The Lorentz force density Eq.
20 may then be written as follows: f
r,t 
 − 1 4r zˆi
EyHx * − Ey * Hx − Hx
Ey *
t + Hx *
Ey
t  − 1 4

z 2 Ez 2 + ¯  −
 − 1 4r yˆi
Ez * Hx − EzHx * + Hx
Ez *
t + Hx *
Ez
t  + 1 4

z
ExEy * + Ez * Ey + ¯ .
27 Just as was the case for negative index materials, the volume integral of Eq.
27 contains surface terms that result from a longitudinal, dielectric discontinuity that cannot be ignored and, as before, care should be exercised in the evaluation of the surface integrals. Nevertheless, we once again find that only the Abraham momentum leads to an accurate descrip￾tion of the Lorentz force, as expected. VI. CONCLUSIONS For almost a century the debate regarding the basic elec￾tromagnetic conservation laws in macroscopic media has continued unabated, and it will probably continue for some time to come. In this study we have used a numerical ap￾proach to solve the vector Maxwell’s equation when disper￾sion is present, and established that, under a variety of cir￾cumstances, conditions, and media, the conservation of linear momentum may be understood solely in terms of the Poynting vector and Abraham momentum density. Gordon showed 18 that our Eq.
7 is in fact the relevant conserva￾tion law even in the presence of matter. The presence of magnetic activity must be taken into account with some cau￾tion, but given the guidance that our numerical evidence pro￾vides, it should be possible to generalize Gordon’s expres￾sion for arbitrary  and
. FIG. 15. Longitudinal
empty triangles and transverse
open squares Lorentz forces, compared to the same quantities computed from the Abraham momentum
solid thin curve: longitudinal; solid thick curve: transverse. The arrows point to both the longitudinal and transverse Livens forces. Once again, the agreement between the prediction of the Abraham momentum and the Lorentz force is obvious, and it remains unmatched by all competing definitions, including the Minkowski momentum
not shown in the figure. RADIATION PRESSURE OF LIGHT PULSES AND¼ PHYSICAL REVIEW E 73, 056604
2006 056604-11