Graded-profile fibers

Integration of Eq. (2-13c) leads to the ray invariant for graded-profile fibers... 

Fig. 2-4 Ray paths within the core of a graded-profile fiber showing (a) a meridional path and (b) a skew path, together with their projections onto the core cross-section. The angle 0 (r) between the projection and the azimuthal direction is shown in (c).

A simple method for classifying rays on graded-profile fibers uses the ray equation to determine the range of values of the radial coordinate r for which rays can propagate. This is accomplished by expressing the radial component of the ray-path equation in Eq. (2-13a) as a relationship between r and z. We use Eq. (2-16) to replace ds by dz, and substitute for d/ds from Eq. (2-17). This leads to... 

Fig. 2-6 Section of a tunneling ray path on a graded-profile fiber. In (a) the core path touches the turning-point caustic at P. Radiation originates at Q in the cladding and propagates along QR tangential to the radiation caustic. The projection onto the fiber cross-section is shown in (b).

Each ray of the graded-profile fiber is characterized by the invariants and /. For most purposes in subsequent chapters the ray trajectory is unimportant, and it is sufficient to know only the values of the ray-path parameters. The... 

Here we consider examples of graded-profile fibers which lead to analytical expressions for some or all of the ray-path parameters of interest. We can use the paraxial approximation of Section 1-10 to simplify determination of the path length. The results are included in Table 2-1. 

This result holds approximately for graded-profile fibers as well, since n(r) = co- In practice n o s 1.5, Mq s 1, and thus a minimum of about 96 % of source power is transmitted into the core. Throughout the rest of this chapter we neglect this slight loss, but it can be accounted for by replacing the intensity /(0o) ofEq. (4-1) by r/(0o). 

Fig. 5-4 Step- and graded-profile fibers whose core radius and profile vary along their length, showing (a), (b) a ray which remains bound, and (c), (d) a bound ray which becomes a leaky ray along part of its path.

Starting from the redistribution of ray power due to a ray incident on an isolated scatterer, we derive an integral equation which governs the distribution of bound-ray power when many scatterers are present in the liber. For convenience we assume a step profile, but the derivation is readily extended to graded-profile fibers [11],... 

For fibers with parameters of practical interest, the dominant effect of material absorption is the attenuation of total pulse power, together with a slight modification of pulse shape. The change in pulse shape is due to the fact that rays with larger inclinations to the fiber axis on step-profile fibers or with turning points closer to the interface of graded-profile fibers lose relatively more of their power to the cladding. 

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 first recall from Section 2-7 the discussion of tunneUng ray paths on monotonic graded-profile fibers with minimum index in the cladding and maximum core index on the axis. Part of the trajectory of a narrow tube of identical tunneling rays is illustrated in Fig. 7-3 (a). Each ray touches the turning-point caustic at radius r, where is the larger root of Eq. (2-19) in... 

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... 

Example Hybrid modes on a graded-profile fiber... 

Love, J. D. (1984) Exact, analytical solutions for modes on a graded-profile fiber. Opt. Quant. Elect, (submitted). 

The modal propagation constant /3(z) satisfies the local eigenvalue equation for each value of z. This equation has the plane-wave form of Eqs. (36-12) and (36-13) for high-order modes on step- and graded-profile fibers. Using the relationships between mode and ray parameters in Table 36-1, page 695, we find that the local eigenvalue equation has the form of the adiabatic invariant of Eq. (5-41). 

The same general interpretation of the correction factor for step-profile fibers also holds for clad, graded-profile fibers that are weakly guiding. Profiles leading to exact solutions of Eq. (21-35) are discussed in Chapter 14, while the WKB solution of the homogeneous scalar wave equation of Eq. (35 4) can often be used to determine the correction factor for other profiles [2, 8]. 

We can use the Gaussian approximation for clad, graded-profile fibers. Thus Fq is given by Eq. (15-2) and tV = -1/ ) by Table 15-2, page 340, in terms of the spot size. 

We can evaluate the area factor for graded-profile fibers within the Gaussian approximation. We replace Fq with Eq. (15-2) and W with (V — taken from... 

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