Weak-guidance approximation

By using weakly guidance approximation, equation (16) can be reduced to the following form ... 

To illustrate the use of the formal results for graded profile waveguides, we consider examples of profiles which have analytical solutions for some or all of the ray-path quantities of interest, including solutions within the weak-guidance approximation. The more important parameters are also included in Table 1-1. 

Within the weak-guidance approximation, we eliminate A/(x) between Eqs. (1-40) and (1-43), whence correct to order A... 

The slight change in value of the optimum profile exponent from to q due to profile dispersion is given approximately by 5q = — 2p, assuming the weak-guidance expressions in Eqs. (3-8) and (3-20). Althou dq, is small, it nevertheless has a dramatic effect on pulse dispersion. We calculate the ray dispersion t = —... 

This is identical in form to the uniform fiber transit time of Table 2-1, page 40, with p — (0) and q replaced by q. Thus a slight variation in exponent on a slowly varying fiber in the weak-guidance approximation is equivalent to a uniform fiber with a different exponent. If we substitute Eq. (5-31) into Eq. (5-34), we find that correct to second order... 

In the weak-guidance approximation, the differential scattering cross-section and the total cross-section ff, , have the forms [16]... 

With the exception of the parabolic profile, the transmission coefficient of Eq. (6-22) cannot be expressed in closed form for the clad power-law profiles of Table 2-1, page 40. However, within the linear approximation, T is given by Eq. (6-24) with — pdn (r)/dr p replaced by where 6 = (2A) in the weak-guidance approxi-... 

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 transmission coefficient for reflection at the interface is derived in Section 35-9. Within the weak-guidance approximation, Eq. (35-36) gives [8]... 

Accordingly, we deduce from this result and Eqs. (4-16), (4-18) and (8-5b) that the diffuse source excites twice as much bound-ray power on a step-profilefiber as on a parabolic-profile fiber, while six times as much power goes into tunneling rays on a step-profile fiber as on a parabolic-profile fiber within the weak-guidance approximation. 

The disp>ersion due to bound rays on step-profile fibers is given by Eq. (3-3) in the weak-guidance approximation. If we include leaky rays, then only those tunneling rays with effectively zero attenuation are included. Since transit time is independent of skewness, i.e. independent of /, this is equivalent to reducing the lower limit on from to defined by Eq. (8-24b). Thus the difference in transit times between the fastest, on-axis bound ray (p = n ) and the slowest tunneling ray (jS = follows from Table 2-1, page 40, as [7]... 

As discussed at the beginning of this example, the modal fields of the weak-guidance approximation are a good approximation to the exact fields of the fiber only under... 

We discussed the accuracy of the weak-guidance approximation for the infinite parabolic profile in Section 14-4. If we impose the condition j8 = kn we deduce from Eq. (14-9) and Table 14-2 that V p- G q)A Y for the infinite power-law profiles to be weakly guiding. 

If V is below the cutoff value 2.405 of the second mode in iFig. 14-4, the fiber is single moded and only the even and odd fundamental modes can propagate. Both modes have the same propagation constant j8. Consequently the group velocity and the transit time of Eq. (11-36) are independent of polarization. In the weak-guidance approximation, the expression for in Table 14—3 follows from Eq. (13-17), and is plotted against V in Fig. 14—3(d) as the dimensionless quantity (n — c)/cA. 

The weak-guidance approximation, described in Chapter 13, greatly simplifies the determination of the modal fields of optical waveguides, because it depends on solutions of the scalar wave equation, rather than on vector solutions of Maxwell s equations. For circular fibers, with an arbitrary profile, the scalar wave equation must normally be solved by purely numerical methods. We discussed the few profiles that have analytical solutions in Chapter 14. These solutions, including those for profiles of practical interest such as the step and clad power-law profiles, are given in terms of special functions or by series expansions, which usually necessitate tables or numerical evaluation to reveal the physical attributes of the modes. 

We begin with a brief review of the weak-guidance approximation for fundamental modes on circular fibers. In Sections 13-2 and 13-4, we showed that the two fundamental modes are virtually TEM waves, with transverse fields that are polarized parallel to one of a pair of orthogonal directions. The transverse field components for the x- and y-polarized HEj j modes are given by Eq. (13-9) relative to the axes of Fig. 14-1. The spatial variation Fq (r) is the fundamental-mode solution (/ = 0) of the scalar wave equation in Table 13-1, page 288. Hence... 

The simplest example of a noncircular waveguide is the planar waveguide of Chapter 12, whose modes are either TE or TM, as explained in Section 11-16. For each TE mode the electric field lies in the cross-section and is uniformly polarized. Consequently the weak-guidance solution is identical to the exact solution for the field ey and the propagation constant. Both satisfy the scalar wave equation of Eq. (12-16), and examples with analytical solutions are given in Table 12-7, page 264. Within the weak-guidance approximation the... 

The profile defined by Eq. (16-1) is unphysical, but by following the reasoning of Section 14-4, we see that the weak-guidance approximation is an... 

The weak-guidance approximation requires A 1 and = kuco, as discussed in Section 14-4, and this ensures that modal power is confined close to the fiber axis. We deduce from Eq. (16-S) that the fiber parameter must be sufficiently large to satisfy... 

Consider a single-mode, elliptical fiber whose refractive-index profile rotates along its length, as shown in Fig. 19-2. We recall from Section 13-5 that in the weak-guidance approximation one fundamental mode of the cylindrically symmetric, elliptical fiber is plane polarized with its transverse electric field parallel to the x-axis in Fig. 19-2(a) and has propagation constant The other fundamental mode s field is parallel to the y-axis... 

We note that Eq. (20-11) can be derived directly from the scalar wave equation. In the weak-guidance approximation, the fields ey in Eq. (20-la) satisfy the scalar wave equation. Consequently, the above result follows directly from the orthogonality condition of Eq. (33-5b). 

We now examine the scattering of power into bound modes when an x-polarized HEj j mode is incident on an isolated nonuniformity within a single-mode, step-profile fiber. Within the weak-guidance approximation, we deduce from Eq. (22-26) and Table 14-3, page 313, that... 

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