RADIATION PRESSURE OF LIGHT PULSES AND PHYSICAL REVIEW E 73. 056604(2006) Index Profile 5 FIG. 4. Scattering of a 600 fs incident pulse from a Position(microns) 4 um multilayer structure composed of 15 periods of Si3N4(125 nm)/Si0,(150 nm), having a mass m-10-g, and vol- FIG. 5. Electric and magnetic field localization properties of a ume V=4x 10-12 m3. Most of the pulse is always located in free light pulse tuned at the photonic band edge. The electric and mag- pace during the entire interaction, a fact that eventually causes the netic fields are spatially delocalized, resulting in small group and Abraham and Minkowski momenta to be similar energy velocities. The y-axis scaling reflects the magnitude of fields inside the structure relative to the input intensity. This kind Ref [15] the total momentum transferred to the structure is field localization and enhancement. which carries momentum calculated by performing the time integral of the total calcu- energy, is not available for simple Fabry-Perot etalons lated force. It is evident even from Fig. 3 that this procedure may, under the right circumstances, yield similar areas for multilayer stack. The fields are delocalized with minimum both the Minkowski and Abraham momenta, especially if overlap, leading to small group and energy velocities [4] one waits for the pulse to leave the structure. However, cal- Because the transverse field profiles do not change, we resort culation of the forces via direct integration of Maxwell,s to plotting just the longitudinal, axial cross section of the equation, accompanied by a direct evaluation of the Abraham pulse and Minkowski momenta, reveals unmistakable agreement In Fig. 6 we compare the Minkowski and Abraham mo between the Abraham and the Lorentz forces. Based on this menta calculated using Eqs.(12)and(13)for a pulse ap- example, our conclusion is that it is generally not possible proximately 100 fs in duration for two different conditions for the Minkowski momentum, the averaged momentum Eq. tuning at the band edge resonance, and inside the gap. The (3), or any other plausible definition of momentum that uses calculations show that when the carrier wavelength is tuned inside the gap, so that the structure acts like a mirror(the the fields, to accomplish the same thing in substrates or other transmittance is less than 10-3), there is effectively no diff similar extended media ence between the two momenta, and so the curves overlap II. PHOTONIC BAND GAP STRUCTURES 6. 0xI We now consider a typical finite multilayer sample. We ssume the stack is composed of 15 periods of generic, rep- 8 resentative, dispersive materials with dielectric constants e E4.5x10 2(as in SiO2) and e2=4 [as in Siy N4, and we use the b dispersion function of Eq (18)above]over the entire near IR range. Assuming a cross section of approximately 1 mm and a thickness of -4 microns(Sio2 layers are taken to be 53.0x10 150 nm thick, and that Si3N4 layers are 125 nm thick) volume of the structure is Ve4x10-12 m3 Using the known material densities of SiO, and Si3N4, the mass of the struc- E 1.5x10 ture can be estimated at m-10 g. For the moment we neglect the presence of a substrate, and assume the beam waist is at least several tens of wave cycles wide, so that we nay also neglect diffraction effects 2x10 carrier frequency of a narrow-band pulse [4] is Time(see) near the band edge. The structure is located FIG. 6. Abraham and Minkowski momenta for a 100 fs pulse near the origin, and with a spatial extension of only tuned inside the gap(solid line overlapped by open circles)and for 4 microns, it is evident that most of the pulse is located the same pulse tuned at the band edge resonance(thin solid and outside the structure most of the time: this is the primary ong dashes). The average momentum is also calculated(thin reason why the Minkowski and Abraham momentum densi- dashes ). The Abraham and Minkowski momenta yield similar re ties generally differ little during the interaction. In Fig. 5 we sults because in both cases the pulse is located mostly in free space ow the electric and magnetic field profiles inside the during the entire time, as the structure is only a few microns thick 056604-7Ref. 15 the total momentum transferred to the structure is calculated by performing the time integral of the total calcu￾lated force. It is evident even from Fig. 3 that this procedure may, under the right circumstances, yield similar areas for both the Minkowski and Abraham momenta, especially if one waits for the pulse to leave the structure. However, cal￾culation of the forces via direct integration of Maxwell’s equation, accompanied by a direct evaluation of the Abraham and Minkowski momenta, reveals unmistakable agreement between the Abraham and the Lorentz forces. Based on this example, our conclusion is that it is generally not possible for the Minkowski momentum, the averaged momentum Eq.
3, or any other plausible definition of momentum that uses the fields, to accomplish the same thing in substrates or other similar extended media. III. PHOTONIC BAND GAP STRUCTURES We now consider a typical finite multilayer sample. We assume the stack is composed of 15 periods of generic, rep￾resentative, dispersive materials with dielectric constants 1 2
as in SiO2 and 24 as in Si3N4, and we use the dispersion function of Eq.
18 above over the entire near IR range. Assuming a cross section of approximately 1 mm2 and a thickness of 4 microns
SiO2 layers are taken to be 150 nm thick, and that Si3N4 layers are 125 nm thick, the volume of the structure is V410−12 m3 . Using the known material densities of SiO2 and Si3N4, the mass of the struc￾ture can be estimated at m10−5 g. For the moment we neglect the presence of a substrate, and assume the beam waist is at least several tens of wave cycles wide, so that we may also neglect diffraction effects. In Fig. 4 we show a typical scattering event when the carrier frequency of a narrow-band pulse 4 is tuned at the first resonance near the band edge. The structure is located near the origin, and with a spatial extension of only 4 microns, it is evident that most of the pulse is located outside the structure most of the time: this is the primary reason why the Minkowski and Abraham momentum densi￾ties generally differ little during the interaction. In Fig. 5 we show the electric and magnetic field profiles inside the multilayer stack. The fields are delocalized with minimum overlap, leading to small group and energy velocities 4. Because the transverse field profiles do not change, we resort to plotting just the longitudinal, axial cross section of the pulse. In Fig. 6 we compare the Minkowski and Abraham mo￾menta calculated using Eqs.
12 and
13 for a pulse ap￾proximately 100 fs in duration for two different conditions: tuning at the band edge resonance, and inside the gap. The calculations show that when the carrier wavelength is tuned inside the gap, so that the structure acts like a mirror
the transmittance is less than 10−3, there is effectively no differ￾ence between the two momenta, and so the curves overlap FIG. 4. Scattering of a 600 fs incident pulse from a 4
m multilayer structure composed of 15 periods of Si3N4
125 nm/SiO2
150 nm, having a mass m10−5 g, and vol￾ume V410−12 m3. Most of the pulse is always located in free space during the entire interaction, a fact that eventually causes the Abraham and Minkowski momenta to be similar. FIG. 5. Electric and magnetic field localization properties of a light pulse tuned at the photonic band edge. The electric and mag￾netic fields are spatially delocalized, resulting in small group and energy velocities. The y-axis scaling reflects the magnitude of the fields inside the structure relative to the input intensity. This kind of field localization and enhancement, which carries momentum and energy, is not available for simple Fabry-Perot etalons. FIG. 6. Abraham and Minkowski momenta for a 100 fs pulse tuned inside the gap
solid line overlapped by open circles and for the same pulse tuned at the band edge resonance
thin solid and long dashes. The average momentum is also calculated
thin dashes. The Abraham and Minkowski momenta yield similar re￾sults because in both cases the pulse is located mostly in free space during the entire time, as the structure is only a few microns thick. RADIATION PRESSURE OF LIGHT PULSES AND¼ PHYSICAL REVIEW E 73, 056604
2006 056604-7