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RADIATION PRESSURE OF LIGHT PULSES AND PHYSICAL REVIEW E 73. 056604(2006) Incident s小N4sN4 075 050 0.5xl0 0.25 40-2002040 Normalized Frequency(1/microns) FIG. 11. The same structure described in the text and in the FIG.10. Plane-wave transmittance(left y axis) and total mo- medium, chosen here to be Sis N4. The figure shows both the inci- mentum transferred (right y axis)for the structure described in Fig. dent pulse and the pulse reflected from the embedded mirror while 4. A changing transmittance and field localization properties near it is still located inside the entry substrate. Transmittance through the band edge lead to widely tunable total momentum transfer. he Bragg mirror is less than 10-3. tion changes sign(Fig 8). Furthermore, forward motion is almost compensated by its backward movement, and the We now examine the case of a photonic band gap struc structure tends to return to its original position(Fig. 9). ture immersed inside a background dielectric material whose However, the device is literally immersed inside fields that index of refraction is other than unity. In the example we the cavity. In general, the structure may oscillate about the section in the middle of a dielectric substrate, akin to a re- origin, or move forward, begin to turn back, and then move flective membrane immersed in a liquid, and tune the carrier forward again. In this case the oscillation is ultimately fol- frequency of the incident pulse inside the photonic band gap lowed by a forward terminal velocity. These dynamics, and to utilize the structure as a mirror. The advantage of this the ultimate direction of motion of a free-standing mass, for situation with respect to an ordinary metallic mirror is that the most part depend on the tuning condition with respect to we have no material absorption to consider or interpret, thus the band edge, and the bandwidth of the incident pulse. This leaving no doubt as to how the energy and momentum are particular example clearly does not exhaust the possibilities Finally, in Fig. 10 we plot the plane-wave transmittance of utilized. The situation is depicted in Fig. ll, where we show he structure (left y axis), calculated using the matrix transfer he incident and reflected pulses, and the multilayer stack technique. On the right y axis we plot the total linear mo- immersed inside a Si3Na-like background medium. The entry um gained by the multilayer stack versus normalized substrate is thick enough to contain the entire pulse, so that a frequency, as calculated using the Abraham momentum [Eq steady-state dynamics is reached. In Fig. 12 we show the (13)]. In this instance we again use 600 fs incident pulses to predicted momenta. The time evolution of the momentum widely tunable because of the diverse field localization prop- We stop the pulse while it remains inside the substrate, be- erties that occur near the band edge, with minimum but non- cause discrepancies between the momenta are largest there zero momentum transfer at resonance and mirrorlike reflec- tions and maximum momentum transfer when the pulse is gap. The total momentum transferred, and hence displace ents, may be increased in at least three ways: (i) by increas UI ing pulse duration, (ii) by sending a train of pulses, or (iii) by increasing pulse peak power. For example, a group of 10 pulses pushes the overall displacement in the nanometer range. If we increase pulse duration to 100 ps, then the struc ture's displacement becomes of the same order of magnitude required to observe interference effects due to radiation pres- sure in MEMS environments [10]. In general, the degree of sensitivity appears to be remarkably high, but it may be fur x1034x10135x10 ther increased by either decreasing the number of incident Time(sec) pulses or by reducing peak power. One may envision appli cations to ultra-high sensitive torsional balances and pressure FIG 12. Abraham(solid curve), Minkowski(short dashes), and gauges, for example average(long dashes)momenta for the case depicted in Fig. 11 056604-9tion changes sign Fig. 8. Furthermore, forward motion is almost compensated by its backward movement, and the structure tends to return to its original position Fig. 9. However, the device is literally immersed inside fields that will continue to push and pull as long as light lingers inside the cavity. In general, the structure may oscillate about the origin, or move forward, begin to turn back, and then move forward again. In this case the oscillation is ultimately followed by a forward terminal velocity. These dynamics, and the ultimate direction of motion of a free-standing mass, for the most part depend on the tuning condition with respect to the band edge, and the bandwidth of the incident pulse. This particular example clearly does not exhaust the possibilities. Finally, in Fig. 10 we plot the plane-wave transmittance of the structure left y axis, calculated using the matrix transfer technique. On the right y axis we plot the total linear momentum gained by the multilayer stack versus normalized frequency, as calculated using the Abraham momentum Eq. 13. In this instance we again use 600 fs incident pulses to better resolve the resonances. The figure clearly suggests that the amount of momentum transferred to the structure is widely tunable because of the diverse field localization properties that occur near the band edge, with minimum but nonzero momentum transfer at resonance, and mirrorlike reflections and maximum momentum transfer when the pulse is tuned inside the gap. The total momentum transferred, and hence displacements, may be increased in at least three ways: i by increasing pulse duration, ii by sending a train of pulses, or iii by increasing pulse peak power. For example, a group of 106 pulses pushes the overall displacement in the nanometer range. If we increase pulse duration to 100 ps, then the structure’s displacement becomes of the same order of magnitude required to observe interference effects due to radiation pressure in MEMS environments 10. In general, the degree of sensitivity appears to be remarkably high, but it may be further increased by either decreasing the number of incident pulses or by reducing peak power. One may envision applications to ultra-high sensitive torsional balances and pressure gauges, for example. We now examine the case of a photonic band gap structure immersed inside a background dielectric material whose index of refraction is other than unity. In the example we place the same multilayer stack that we used in the previous section in the middle of a dielectric substrate, akin to a re- flective membrane immersed in a liquid, and tune the carrier frequency of the incident pulse inside the photonic band gap to utilize the structure as a mirror. The advantage of this situation with respect to an ordinary metallic mirror is that we have no material absorption to consider or interpret, thus leaving no doubt as to how the energy and momentum are utilized. The situation is depicted in Fig. 11, where we show the incident and reflected pulses, and the multilayer stack immersed inside a Si3N4-like background medium. The entry substrate is thick enough to contain the entire pulse, so that a steady-state dynamics is reached. In Fig. 12 we show the predicted momenta. The time evolution of the momentum tracks the pulse as it crosses the entry interface I, impacts the mirror II, and turns back toward the entry surface III. We stop the pulse while it remains inside the substrate, because discrepancies between the momenta are largest there. FIG. 10. Plane-wave transmittance left y axis and total momentum transferred right y axis for the structure described in Fig. 4. A changing transmittance and field localization properties near the band edge lead to widely tunable total momentum transfer. FIG. 11. The same structure described in the text and in the caption of Fig. 4 is now embedded inside a background dielectric medium, chosen here to be Si3N4. The figure shows both the incident pulse and the pulse reflected from the embedded mirror while it is still located inside the entry substrate. Transmittance through the Bragg mirror is less than 10−3. FIG. 12. Abraham solid curve, Minkowski short dashes, and average long dashes momenta for the case depicted in Fig. 11. RADIATION PRESSURE OF LIGHT PULSES AND¼ PHYSICAL REVIEW E 73, 056604 2006 056604-9