Waveguide dispersion

A basic waveguide structure, which is sketched in Fig. 1, is composed of a guiding layer surrounded by two semi-infinite media of lower refractive indices. The optical properties of the stmcture are described by the waveguiding layer refractive index Hsf, and thickness t, and by the refractive indices of the two surrounding semi-infinite media, here called (for cover) and (for substrate). Application of Maxwell s equations and boundary conditions leads to the well-known waveguide dispersion equation [6] ... 

It is well known that the dispersion in the optical fibers is divided into three parts, modal dispersion, material dispersion, and waveguide dispersion. In the case of the SI POF, the modal dispersion is so large that the other two dispersions can be approximated to be almost zero. However, the quadratic refractive-index distribution in the GI POF can dramatically decrease the modal dispersion. We have succeeded in controlling the refractive-index profile of the GI POF to be almost a quadratic distribution by the interfacial-gel polymerization technique (2). Therefore, in order to analyze the ultimate bandwidth characteristics of the GI POF in this paper the optimum refractive index profile is investigated by taking into account not only the modal dispersion but also the material dispersion. 

Here D(X) is in ps/nm-km, and X is in nanometers. For the range of interest to telecommunications, 1260-1700 nm, this is claimed to be accurate to better than 1%. With material dispersion essentially fixed, the profile of the core and cladding refractive indices may be designed to take advantage of the properties of waveguide dispersion to partially cancel material dispersion to produce fiber with optimized total dispersion. 

Waveguide dispersion results from how waves traveling along a fiber interact with the profile in refractive index. Let us first discuss single mode fiber, since multimode fiber is treated somewhat differently. Various index profiies for singie mode fibers are shown in Figure 5. 

For the sake of simplicity, the discussion of waveguide dispersion for such structures often proceeds with the approximation that the index does not vary with wavelength. For small A, a normalized propagation constant, b, may be given as... 

Waveguide dispersion may thus be controlled by appropriate index profile design through its influence on V, a parameter dependent on the specific index profile chosen, as well as b, and subsequently... 

FIGURE 7.7 Effective index and waveguide dispersion in single-mode fibers where is the cutoff... 

In general, the contributions to 3tj due to material and waveguide dispersion are not easily separated [5, 6]. Accordingly, we can only examine each in isolation from the other in special situations. 

The contribution to pulse spreading due to waveguide dispersion in isolation from material dispersion is found from Eq. (11-37) by assuming n is independent of A. The form of 3t depends on the refractive-index profile. Later, in Section 11-20, we express waveguide dispersion in terms of more convenient modal parameters. 

We discussed pulse spreading due to waveguide dispersion in Section 11-12. It is convenient to express the pulse spread 5tj of Eq. (11-37) in terms of the waveguide parameter V and the modal parameter Uj. Using the definitions inside the back cover, we first rewrite Eq. (11-37) in terms of the group velocity of Eq. (11-31) as... 

Consequently, pulse spreading due to waveguide dispersion is characterized by the product DVA. In practice, the source frequency or wavelength is prescribed, whereas p and A are design parameters. 

Thus waveguide dispersion is zero for the parabolic profile to this order. The next order correction is finite, as we showed in the previous section. 

Fig. 14-10 (a) The distortion parameter for the fundamental modes of clad power-law profiles and (b) the values of the fiber parameter at which waveguide dispersion vanishes, the q= oo line corresponding to the value for the step profile. 

We showed in the previous section that the distortion parameter D can have a zero for values of K of practical interest only if q > 2. Consequently, the vanishing of waveguide dispersion depends primarily on profile shape. 

In Section 14-10, we introduced the concept of profile volume. We showed in the case of clad power-law profiles of equal volume that some properties, such as the range of single-mode operation and the fundamental-mode intensity distribution are insensitive to profile shape, whereas other properties, such as waveguide dispersion, depend critically on profile shape. Within the Gaussian approximation, we can demonstrate directly the insensitivity of the intensity distribution to profile shape. 

If 0 < V < 2.592 the fiber is single moded. The pulse transit time of Eq. (11-36) is inversely proportional to the group velocity, and pulse spreading due to waveguide dispersion is proportional to VAD = A/where D is the scalar distortion parameter. The expression for D in Table 15-2 is plotted as the solid curve in Fig. 15-1 (d). Compared with the dashed curve, calculated numerically, the maximum relative error is 9.6% at F = 2.9, while at K = this error is 9.4%. There is no zero of waveguide dispersion. 

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