Lateral shift transit time

Transit time for the lateral shift 10-6 Lateral shift in planar waveguides 10-7 Preferred ray directions... 

Thus the modal and ray transit times are equal only when tj - 1. This condition is satisfied only by those rays belonging to modes well above cutoff, i.e. when Vp U, or, equivalently, when 0 < 0c- Hence is inaccurate for arbitrary values of 9. This inaccuracy arises because the ray transit time ignores diffraction effects, which were discussed in Chapter 10. The step-profile planar waveguide is a special case, however, because all diffraction effects can be accounted for exactly by including the lateral shift at each reflection, together with recognizing the preferred ray directions. TWs was carried out in Section 10-6, and for rays, or local plane waves, whose electric field is polarized in the y-direction in Fig. 10-2, leads to the modified ray transit time of Eq. (10-13). If we use Table 36-1 to express 0, and 0(.in terms of U, Vand Wand substitute rj for TE modes from Table 12-2, we find that Eqs. (10-13) and (12-8) are identical since 0 = 0. It is readily verified that the same conclusion holds for TM modes and local plane waves whose magnetic field is polarized in the y-direction of Fig. 10-2. 

The nature of the rock record from the time of the first sedimentary rocks ( 3.8 billion years ago) to about 1 to 2 billion years ago suggests that the amount of oxygen in the Earth s atmosphere was significantly lower than today, and that there were continuous chemical trends in the sedimentary rocks formed and, more subtly, in hydrosphere composition. Figure 10.6 illustrates how the chemistry of rocks shifted dramatically during this transitional period. The source rocks of sediments during this time may have been more basaltic than later ones ... 

Figure 6 shows the master curves for the PS films with M of 4.9k and 140k drawn by horizontal and vertical shifts of each curve shown in Fig. 5 at the reference temperatures of 267 and 333 K, respectively [26]. The master curves obtained from the dependence of lateral force on the scanning rate were very similar to the lateral force-temperature curves, as shown in Fig. 3. Hence, it seems plausible as a general concept that the scanning rate dependence of the lateral force exhibits a peak in a glass-rubber transition. Also, it is clear that the time-temperature superposition principle, which is characteristic of bulk viscoelastic materials [35], can be applied to the surface relaxation process as well. Assuming that Uj has a functional form of Arrhenius type [36, 37], the apparent activation energy for the aa-relaxati(Mi process, A//, is given by ...

The field shape consists of two symmetric half-cycles with zero field amplitude at the pulse center. The upper panel of Fig. 5.13 shows the effective potential curves Vf R,t) obtained under the control field (to = 51 fs). The plotted curves at t = 44 fs represent the maximum shifts of the PECs in the first half-cycle of the control pulse, and those at t = 58 fs represent the maximum shifts in the later half-cycle of the control pulse. In the first half-cycle of the pulse, the control field, through the positive transition amplitude function /xn(7 ), acts to shift the Vn R) potential curve upwards, and thus the dynamical crossing position Rx t) is first shifted to the left. It reaches the leftmost position R ett at the height of the control field, and then is shifted right and restored to 7 cross by t = to- The lower panel of Fig. 5.13 shows the resulting time-dependence of Rx(t). The fact that the... 

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