Local plane waves tunneling

We consider weakly-guiding fibers whose profiles decrease monotonically from a maximum index on the axis to the uniform cladding index at the interface. Within the local plane-wave approximation, the transmission coefficient for tunneling rays is given by Eq. (35-45) as [5,12,13]... 

The transmission coefficient for tunneling rays on a weakly guiding, step-profile fiber with core and cladding indices and is derived in Section 35-12 within the local plane-wave approximation. Thus Eq. (35-46a) gives [9,14]... 

For situations where the above assumption cannot be adopted, expressions for the transmission coefficient in the transition region between tunneling and refracting rays are available [4,8]. The values of Tare plotted as curve (i) in Fig. 7-2(b) for a skew leaky ray with I = 0.033 on a clad parabolic fiber. To the left of the vertical dashed line, the curve corresponds to tunneling rays and coincides with the local plane-wave expression of Eq. (7-18) as increases [8]. Similarly, to the r ght of the vertical dashed line, the curve corresponds to refracting rays Ind coincides with the local plane-wave expression of Eq. (7-6) as decreases. A similar transition occurs for skew leaky rays on a step-profile fiber [16]. 

In this section we generalize the analysis of plane-wave incidence at a planar interface, and consider the incidence of local plane waves at a caustic in a slowly varying graded medium. Our goal is the derivation of the power transmission coefficient for tunneling rays. 

Consider a medium whose continuous refractive-index profile n(x) varies slowly over a distance equal to a wavelength. A ray path in the region x < 0 of Fig. 35-4(a) touches the turning-point caustic at x = x,p. When there is no path in the region beyond the caustic, the ray is totally reflected from the caustic and no power is lost, i.e. T=0. However, if a transmitted ray originates at the radiation caustic x = x j, then optical tunneling occurs, as described in Chapter 7, and power is lost from the path at x,p to the path at x. If the local plane-wave fields have propagation constant P, then the values of x,p and x are determined by Eq. (35-31). 

The link between the ray tubes at the caustics is denoted by the hatched region of width 6z in Fig. 35-4(a). Power lost by the incident ray at x,p tunnels through this region and enters the transmitted ray at x j. At and close to the caustics, local plane-wave theory is inadequate, but, provided the caustics are not too close together, it is not necessary to know the fields in these regions in order to determine the transmission coefficient. We can link the solutions in Eq. (35-32) using the connection formulae of WKB theory, which are available in standard texts [7]. Hence... 

Thermal excitation of this mode leads at room temperature to the flipping of the buckled dimer such that temporarily averaging techniques such as scanning tunneling microscopy (STM) observe a symmetric dimer. For further discussion of the surface phonons, which are localized mainly within the first two atomic layers, it is illustrative to consider that in a simpHfied approach one expects 12 different modes at a given wave vector. This results from the three degrees of freedom, namely, motion in x, y, and z direction, per four atoms (in the first and second Si layer) within the (2x1) unit cell. The 12 phonon modes at the F point are sketched in Figure 8.2.25 [58]. Four modes, Aj — are polarized shear horizontally with respect to the mirror plane, which contains the Si-Si dimer bond and the surface... 

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