Local plane waves

Atomic units will be used throughout. The explicit density functionals representing the different contributions to the energy from the different terms of the hamiltonian are found performing expectation values taking Slater determinants of local plane waves as in the standard Fermi gas model. Those representing the first relativistic corrections are calculated in the Appendix. 

Consider the isolated scatterer S in Fig. 5-7(b), and a ray incident on it. Scattering is usually described in terms of a differential scattering cross-section ffj, which determines the distribution of scattered power when a plane wave is incident on the scatterer [12], To relate to scattering in terms of rays, we recall our concept of a ray as a local plane wave, discussed in Section 35-3. We regard the local plane wave associated with the incident ray in Fig. 5-7(b) as part of an infinite plane wave propagating in the same direction. Accordingly we may set... 

The transmission coefficient T is found by using the local plane-wave description of a ray. We regard the local plane wave as part of an infinite plane-wave incident on a planar interface between unbounded media, whose refractive indices coincide with the core and cladding indices and of the waveguide, as shown in Fig. l-3(b). For the step interface, Tis identical to the Fresnel transmission coefficient for plane-wave reflection at a planar dielectric interface [6]. In the weak-guidance approximation, when s n, the transmission coefficient is independent of polarization, and is derived in Section 35-6. From Eq. (35-20) we have [7]... 

The refracting-ray transmission coefficient for skew rays on graded-profile fibers is given by Eq. (7-6) within the local plane-wave approximation, except that 0, = njl — a, where a is the angle between the incident, reflected or transmitted rays and the normal at the interface. We deduce from Eqs. (2-14), (2-16) and (2-17) that... 

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]... 

It should be noted that Eq. (7-19) cannot be deduced from the graded-profile result of Eq. (7-15) in the limit r,p - p. As explained in Section 6-9, the local plane-wave derivation requires that the profile vary slowly over a distance equal to the wavelength of light. If the profile steepens and approaches the step profile, this condition is violated and Eq. (7-15) is no longer accurate [15]. 

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]. 

Fig. 11-2 The electric field vector is orthogonal to the ray, or local plane-wave direction. On the step-profile fiber, the direction of e for (a) a meridional ray is parallel to a fixed direction, and for (b) a skew ray it changes direction at each reflection. On the parabolic-profile fiber the direction of e changes continuously along the skew-ray path (c).

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. 

We now consider illumination by light beams that are parallel to, or subtend only small angles 0i to the axis of the fiber. The fields of the beam at the endface of the fiber have the form of a local plane wave whose wave vector is parallel to the direction of the beam. If the electric field, Ej, is uniformly polarized parallel to the x-axis in Fig. 20-2 (a) and the beam is parallel to the x-z plane, with a symmetric amplitude distribution/(r), then... 

The restriction 1 is of little practical concern for weakly guiding fibers. In terms of the local plane-wave vector, each bound mode field makes an angle... 

For a given value of the fiber parameter V, there is an optimum inclination which maximizes the power entering a particular mode. In Fig. 20-4(a) these values of 0, correspond to the peak values of P/P. Now it is intuitive that maximum excitation should occur when 0j is similar to the characteristic angle 0, which the local plane-wave vector of the modal field makes with the fiber axis, as discussed in Section 36-2. Since 0 and 02 ttre small on a weakly guiding fiber, the relationship with the modal parameter U in Table 36-1, page 695, shows that 0j/02 = (0i/0c) (VIV). If we insert the values of 0J0 at the peaks in Fig. 20-4(a) and the corresponding values of V/U, from Fig. 14-4, we find that 0j/02 = 1. A more rigorous analysis shows the peaks in Fig. 20-4(a) for all but the fundamental modes are proportional to Jf+iiU) provided K>(/ [2]. For the HE, mode the maximum possible efficiency of 85 % occurs when 0, = 0 and KS3.8. 

Local plane-wave description of loss phenomena 671... 

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