Profile clad parabolic

Fig. 1-10 Plots of the power-law profiles of Eq. (1-59) for various values of q. The clad parabolic profile corresponds lo q = 2 and the step profile to q = 00. The hyperbolic secant profile of Eq. (1-51) is shown dashed, and the dotted extension to the q = 2 core profile denotes the (infinite) parabolic profile of Eq. (1-44).

We showed in Section 2-13 that the transit time for a step-profile fiber is independent of the cross-sectional geometry. Consequently Eqs. (3-2) and (3-3) give the ray dispersion for step-profile fibers of arbitrary cross-section. We also found in Section 2-13 that the ray transit time for the noncircular, clad power-law profiles of Eq. (2-55) is identical to the transit time for the symmetric, clad power-law profiles in Table 2-1, page 40, i.e. dependent on only. Thus Eqs. (3-8) and (3-9) also give the optimum profile and minimum pulse spread for those noncircular profiles [5], which includes the clad parabolic-profile fiber of elliptical cross-section. In other words, ray dispersion on step-profilefibers of arbitrary cross-section and clad power-law profilefibers of noncircular cross-section is also given by the corresponding solutions for planar waveguides. 

Thus both Pbr and are smaller than the corresponding expressions for the step profile by a factor of q (q + 2). In particular, the clad parabolic-profile fiber (q = 2) accepts only half as much bound-ray power as the step-profile fiber (q = oo). 

This profile is given by the q = 2 clad parabolic-profile in Table 2-1, page 40. The total bound-ray power and source efficiency are obtained in an analogous manner to the step-profile expressions above. Hence... 

This expression gives jS(z) for the clad parabolic profile when q = 2, and for the step profile when q = oo. In the latter case we substitute for jS(z) from Eq. (5-2) and rearrange to obtain... 

To determine the transmission coefficient of Eq. (6-22) for the clad parabolic profile of Table 2-1, page 40, we recall from Eq. (2-19) that the radii of the inner and turning-point caustics are roots of the integrand. Hence the integrand is expressible as... 

Fig. 6-3 The normalized impulse response for a fiber with an absorbing cladding and (a) a step profile or (b) a clad parabolic profile. Fiber parameters in both cases are K = 70, A = 0.01 and co = 1 -5. Curves are for differing values of the product iiz/p [6],...

Spatial transient for clad parabolic-profile fibers 163... 

Example Diffuse illumination of clad parabolic-profile fibers 168... 

Example Ray dispersion on clad parabolic-profile fibers 171... 

In Sections 4-21 and 4-22, we showed that the shape of the impulse response on step and clad parabolic profiles is virtually rectangular. This conclusion is valid only in the spatial steady state. In the spatial transient, the power in tunneling rays manifests itself by adding a tail to the pulse. The power in the tail is large close to the source but becomes negligible at the onset of the spatial steady state [5]. 

We showed in Section 4-19 that the distribution function F ( , /) for both bound- and tunneling-ray power is given analytically for the clad parabolic profile fiber by Eq. (4-52), where... 

Fig. 8—4 Fraction of initial bound- and tunneling-ray power remaining on a clad parabolic-profile fiber illuminated by a diffuse source.

One of the simplest examples of a generalized parameter for graded-profile fibers is provided by the clad parabolic profile. The attenuation coefficient is the ratio of the transmission coefficient T to the ray half-period Zp of Table 2-1, page 40. We use the approximation of Eq. (7-16) for T and express in terms of V through the relationship inside the front cover. Thus... 

Fig. 8-5 (a) Fraction of initial tunneling-ray power remaining on a fiber illuminated by a diffuse source as a function of the generalized parameter G. The solid curve corresponds to the step profile and the dashed curve to the clad parabolic profile, (b) Fraction of initial bound-and tunneling-ray power remaining on the step profile. The soUd curve is the exact numerical result and the dashed curve is the generalized parameter value. 

The effects of tunneling rays on pulse broadening and impulse response on clad power-law profile fibers can be found quantitatively following the approach of the previous section. A simple example is the clad parabolic profile, for which the effective pulse broadening follows from Table 2-1, page 40, as [5]... 

The impulse response for the clad parabolic profile can be found using the method of the previous section, and is plotted in Fig. 8-6(b) [5]. The fraction of pulse power in tunneling rays is much smaller than it is on the step profile. This is in keeping with the fact that the diffuse source excites only one-third as much tunneling-ray power on the parabolic profile fiber, as we showed in Section 8-4. 

When the asymmetry is slight, it is sometimes possible to simplify the above analysis by treating the noncircular fiber as a small perturbation of a circular fiber. Thus, for example, the ray invariant I of the circular fiber can be replaced by an approximate invariant l(z) which varies very slowly along the noncircular fiber. The spatial transient on the elliptical, clad parabolic-profile fiber can be determined within this approximation. Details are given elsewhere [13]. 

Fig. 8-11 The fraction of initial bound- and tunneling-ray power remaining on an absorbing fiber illuninated by a diffuse source, for (a) the step profile and (b) the clad parabolic profile. The bottom curves correspond to ignoring all tunneling-ray power and the top curve y = 0 in (a) corresponds to zero tunneling-ray attenuation, flashed continuations of the curves are for the nonabsorbing fiber.

Classification of rays on the bend 9-2 Attenuation of light power 9-3 Example Step profile 9-4 Example Clad parabolic profile... 

Example Step and clad parabolic profiles References... 

We again consider the situation in Fig. 9-1. The straight waveguide has the clad parabolic profile of Table 1-1, page 19, and on the bend the core profile is defined by... 

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