We now extend this result to the more usual or realistic case of a step change in the refractive index for a field entering a homogeneous linear medium from the vacuum. The main differ ence is that a backward traveling wave is generated at the vacuum/dielectric interface and the resulting change in momentum must be incorporated into the conservation law. Using the same incident field as in Fig. 1, Fig 3 shows the field entering the linear medium. The leading part of the field has entered the medium and a portion of that field has been reflected. The oscillations that are seen in the figure represent the interference of the reflection with the trailing part of the incident field. In Fig. 4, the forward propagating refracted field is entirely within the medium and the reflection has separated from the interface. Figure 5 shows the fields after the refracted field has exited the medium through a gradient-index layer that suppresses the secondary reflec- tion. at the end of the calculation the momentum of the transmitted field is found to be smaller than that of the incident field by the momentum of the reflection. In order to satisfy conservation of momentum, a permanent forward momentum of twice the momentum of the refected field must by imputed to the material. In this regard, conservation of electromagnetic momentum is analogous to momentum conservation in the elastic collision of a small object with a wall DISTANCE (wavelengths) Fig 3. Propagation of the vector potential from vacuum into a linear medium through a step increase in the refractive index. The shaded region is the profile of the index of refraction The field travels to the right and the horizontal axis is scaled to the wavelength. At any point in the calculation, conservation of the momentum(20)can be demonstrated if a forward momentum of twice the momentum of the reflected field is contributed by the material As soon as the field starts to enter the medium the momentum of the forward traveling field begins to decrease and the momentum of the backward traveling wave, initially zero, begins to grow as does the momentum of the material. Once the refracted field is entirely within the medium. the momentum of the refracted field remains the same until the field exits the medium hrough the antireflective layer, and thereafter. Meanwhile, the process of reflection is complete and the reflected field travels through the vacuum and is unchanged. We can conclude that there is no additional transfer of momentum to the material once the field is no longer incident on its surface. Consequently, the permanent transfer of momentum from the field to the material occurs at the point of reflection, the surface of the medium, and only while the field is present at the boundary and undergoing Fresnel reflection. There is no temporary material momentum because the momentum is fully accounted for at any time, particularly as the field exits the medium. For the approximate square pulse of vacuum width w, the momentum conservation #77863·S1500USD Received 18 December 2006, accepted 7 January 2007 (C)2007OSA 22 January 2007/ Vol 15, No. 2/OPTICS EXPRESS 721We now extend this result to the more usual or realistic case of a step change in the refractive index for a field entering a homogeneous linear medium from the vacuum. The main differ￾ence is that a backward traveling wave is generated at the vacuum/dielectric interface and the resulting change in momentum must be incorporated into the conservation law. Using the same incident field as in Fig. 1, Fig. 3 shows the field entering the linear medium. The leading part of the field has entered the medium and a portion of that field has been reflected. The oscillations that are seen in the figure represent the interference of the reflection with the trailing part of the incident field. In Fig. 4, the forward propagating refracted field is entirely within the medium and the reflection has separated from the interface. Figure 5 shows the fields after the refracted field has exited the medium through a gradient-index layer that suppresses the secondary reflec￾tion. At the end of the calculation, the momentum of the transmitted field is found to be smaller than that of the incident field by the momentum of the reflection. In order to satisfy conservation of momentum, a permanent forward momentum of twice the momentum of the reflected field must by imputed to the material. In this regard, conservation of electromagnetic momentum is analogous to momentum conservation in the elastic collision of a small object with a wall. -30 -20 -10 0 10 20 30 DISTANCE (wavelengths) 0.0 0.3 0.6 0.9 1.2 AMPLITUDE (arb. units) 1.0 1.1 1.2 1.3 1.4 1.5 1.6 REFRACTIVE INDEX (no units) Fig. 3. Propagation of the vector potential from vacuum into a linear medium through a step increase in the refractive index. The shaded region is the profile of the index of refraction. The field travels to the right and the horizontal axis is scaled to the wavelength. At any point in the calculation, conservation of the momentum (20) can be demonstrated if a forward momentum of twice the momentum of the reflected field is contributed by the material. As soon as the field starts to enter the medium, the momentum of the forward traveling field begins to decrease and the momentum of the backward traveling wave, initially zero, begins to grow as does the momentum of the material. Once the refracted field is entirely within the medium, the momentum of the refracted field remains the same until the field exits the medium through the antireflective layer, and thereafter. Meanwhile, the process of reflection is complete and the reflected field travels through the vacuum and is unchanged. We can conclude that there is no additional transfer of momentum to the material once the field is no longer incident on its surface. Consequently, the permanent transfer of momentum from the field to the material occurs at the point of reflection, the surface of the medium, and only while the field is present at the boundary and undergoing Fresnel reflection. There is no temporary material momentum because the momentum is fully accounted for at any time, particularly as the field exits the medium. For the approximate square pulse of vacuum width w, the momentum conservation #77863 - $15.00 USD Received 18 December 2006; accepted 7 January 2007 (C) 2007 OSA 22 January 2007 / Vol. 15, No. 2 / OPTICS EXPRESS 721