SCALORA et al PHYSICAL REVIEW E 73. 056604(2006) wells equations(17), compare with the temporal derivatives of the Abraham and Minkowski momenta, and plot the re- Abraham sults in Fig. 13. The figure comments itself, as once again the 4x10° Lorentz force is closely tracked only by the Abraham force 2x10° IV NEGATIVE INDEX MATERIALS In negative index materials, the Lorentz force density is slightly more complex because all the terms in Eq.(16)con 0 tribute. The resulting generalized expression for the longitu- dinal and transverse components of the Lorentz force den 6x10 sity, assuming both e and u are complex, may be written FIG. 13. The forces at play in the situation described in Figs and 12. Only the abraham force once again tracks the Lorentz force r,)=2(-1)e2H2-'(p-1)e门 very well during the entire time In fact, the figure shows that the Abraham and Minkowski +(a'2-1)H+(a-1) momenta may differ by as much as a factor of 2, as long as the pulse is still located inside the substrate. The two mo- menta converge to roughly the same value if the pulse is h2+ allowed to exit back into fre ee space The plateau between regions I and II represents the pulse in transit toward the mirror. Therefore, any detection scheme designed to discern ficant differences should detect motion of the reflec- tive membrane before the pulse exits back into vacuum. Just as we did before, we now determine the Lorentz force, Eq. (E-1)E(g+(ap-1)( (20), from the fields that result from the integration of Max- (a-1) HyME+H-H,Hg+ (24) For simplicity, we have retained only the lowest order terms 公会 A simple comparison reveals that Eq(24) reduces to Eq (20) when there are no magnetic contributions, as it should Each part of Eq.(24) displays a magnetic component that contains the longitudinal spatial derivative of the magnetic permeability. If u is discontinuous, some care should be ex- ercised when the volume integrals of Eq (24)are evaluated. As before, the fields found in Eq.(24)are calculated using Maxwells equations(17). Because typical dispersion lengths n negative index materials may easily exceed several hun- dred wavelengths [20-23]. Eqs. (17)may once again be sim- plified by retaining terms up to and including the first-order emporal derivatives on both fields. The reference wave- 34 82 length is now taken to be the plasma frequency, so that X Longitudinal Coordinate(in units Ap In our case and in our units, for incident 30 wave cycle FIG. 14. A pulse approximately 30 wave cycles wide( thus 30 pulses, Lp =p/K(@)I=30/1.6=562Xp, which justifies our from a negative index material at an angle of 150. We note that in Fig 14 we depict the typical dynamics that ensues as a result the Drude model, causality's only demand is that y#0. These con- ditions cause the pulse to refract anomalously in the upper quadrant, strate is magnetically active; we use the Drude model to while the pulse distorts in both real and Fourier space. The result is describe both a and u to enforce a causal response, and use a wave packet whose Poynting vector points forward in the direc- e(o)=(a)=1-1/(a2+iy) ackwards to- The pulse is incident at a 15 angle, and its carrier fre ward the entry surface [21]. We stop the pulse while it is still lo- quency is tuned at o=0.577, where both e(@)=u(@)=-2, cated inside the substrate and dEo/d0=duo/0-4. We choose y=10-, which 056604-10In fact, the figure shows that the Abraham and Minkowski momenta may differ by as much as a factor of 2, as long as the pulse is still located inside the substrate. The two mo￾menta converge to roughly the same value if the pulse is allowed to exit back into free space. The plateau between regions I and II represents the pulse in transit toward the mirror. Therefore, any detection scheme designed to discern any significant differences should detect motion of the reflec￾tive membrane before the pulse exits back into vacuum. Just as we did before, we now determine the Lorentz force, Eq.
20, from the fields that result from the integration of Max￾well’s equations
17, compare with the temporal derivatives of the Abraham and Minkowski momenta, and plot the re￾sults in Fig. 13. The figure comments itself, as once again the Lorentz force is closely tracked only by the Abraham force. IV. NEGATIVE INDEX MATERIALS In negative index materials, the Lorentz force density is slightly more complex because all the terms in Eq.
16 con￾tribute. The resulting generalized expression for the longitu￾dinal and transverse components of the Lorentz force den￾sity, assuming both  and
are complex, may be written as follows: f
r,t = 1 4r zˆi


* * − 1Ex * Hy −
*

 − 1ExHy *  +

*
* − 1Hy
Ex *
+
*

 − 1Hy *
Ex
− 

2
  Hy 2 + ¯  − 1 4r yˆi


* * − 1Ex * H −
*

 − 1ExH *  +

*
* − 1H
Ex *
+
*

 − 1H *
Ex
− 


*
 Hy * H +
*


 HyH *  + ¯ .
24 For simplicity, we have retained only the lowest order terms. A simple comparison reveals that Eq.
24 reduces to Eq.
20 when there are no magnetic contributions, as it should. Each part of Eq.
24 displays a magnetic component that contains the longitudinal spatial derivative of the magnetic permeability. If
is discontinuous, some care should be ex￾ercised when the volume integrals of Eq.
24 are evaluated. As before, the fields found in Eq.
24 are calculated using Maxwell’s equations
17. Because typical dispersion lengths in negative index materials may easily exceed several hun￾dred wavelengths 20–23, Eqs.
17 may once again be sim￾plified by retaining terms up to and including the first-order temporal derivatives on both fields. The reference wave￾length is now taken to be the plasma frequency, so that r =p. In our case and in our units, for incident 30 wave cycle pulses, LD
2  p 2 / k
˜ =302 /1.6=562p, which justifies our neglect of second and higher order temporal derivatives. In Fig. 14 we depict the typical dynamics that ensues as a result of integrating the system of equations
17 when the sub￾strate is magnetically active; we use the Drude model to describe both  and
to enforce a causal response, and use: 
˜=

˜=1−1/
˜ 2+i˜ . The pulse is incident at a 15° angle, and its carrier fre￾quency is tuned at =0.577, where both 
˜=

˜−2, and

˜/
˜ =


˜/
˜ 4. We choose =10−5, which FIG. 13. The forces at play in the situation described in Figs. 11 and 12. Only the Abraham force once again tracks the Lorentz force very well during the entire time. FIG. 14. A pulse approximately 30 wave cycles wide
thus 30 wave cycles in duration crosses an interface that separates vacuum from a negative index material at an angle of 15°. We note that in the Drude model, causality’s only demand is that 0. These con￾ditions cause the pulse to refract anomalously in the upper quadrant, while the pulse distorts in both real and Fourier space. The result is a wave packet whose Poynting vector points forward in the direc￾tion of propagation, and a wave vector that points backwards, to￾ward the entry surface 21. We stop the pulse while it is still lo￾cated inside the substrate. SCALORA et al. PHYSICAL REVIEW E 73, 056604
2006 056604-10