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