Local critical angle

In Section 1-8 we introduced the notion of the complement of the local critical angle for graded-profile planar waveguides. By analogy, the complement of the local critical angle, 0c( ), for fibers is defined in terms of the profile by... 

In Chapter 3 we analysed pulse spread in terms of the variation in transit times between rays propagating at different angles 6fr) to the axial direction. Our analysis was limited to bound rays, whose directions lie in the range 0 < (r) < (r), where 6 (r) is the local critical angle of Eq. (2-29). However,... 

Delta is of the order of 10 and is typically of the order of a few milliradians. As long as the beam is incident below this critical angle, it is totally reflected and only an evanescent wave penetrates the substrate. This has two very important consequences. First, the penetration depth is of the order of 20 A and thus one can signiflcantly discriminate in favor of a surface-contained material. Compton and elastic scattering are also minimized. In addition, the reflection enhances the local intensity by as much as a factor of 4 as well as the effective path length. All of these factors combined enhance the surface sensitivty of the technique and when combined with solid-state fluorescence detection, submonolayer amounts of material can be detected. ... 

Bending of optical fibre causing a local exceeding of critical angle... 

A SAW can be generated by a longitudinal ultrasonic transducer by positioning the transducer at a critical angle 9(. = sin with respect to the local surface normal... 

Figure 2 shows reflection of chemical waves in the BZ system at a boundary created by differential oxygen inhibition. This reflection is instantaneous and independent of the specific local kinetics, in contrast to the delayed reflection observed in one-dimensional systems. Reflection takes place only if the incident wave comes from the medium with lower wave speed and the angle of incidence is larger than a critical value. The experiments employ very different angles of incidence, yet in all cases the angle of reflection was close to the critical angle, sin (vi/v5). 

Fig. 10. Molecular and electronic structure of cation 74. (a) Perspective view on the cation 74. (b) Coordination sphere of the Pb atom, selected interatomic distances (pm) and angles (deg) Pb-(center CC, C7), 281.7 Pb-(center C6 CT), 280.7 Pbl-C2, 227(2) Pbl-C2 221.1(15) Pbl-C8, 231.0(19) C6-C7, 133(2) CC-CT 132(2) (center C6, C7)-Pb-(center C6, C7 163.6. (c) Contour plots of the Laplacian distribution [V p(r)] in the plane containing the atoms Pb, C6 and C7. Solid and dotted lines designate regions of local charge concentration and depletion, respectively. The bond paths are indicated by the solid back lines, bond critical points are marked with a black square. (Reprinted with permission from Ref. 53. Copyright 2003, Wiely-VCH.)...

In contrast to measures of convergence that reflect a single local observable, for example, a torsion angle, some methods focus on the global sampling quality. For a simulation of a macromolecule, the distinction would be between asking "how well do I know this particular quantity " and "how well have I explored the conformational space of the molecule " The latter question is critical, in that if the conformational space is well sampled, most physical quantities should be known well. 

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