Step-profile approximation

The transmittance of the nonlinear step-like discontinuity in cylindrical waveguide has been evaluated under the assumption that profiles of low-intensity nonlinear modes can be approximated by profiles of linear modes. According to the results, nonlinear transmittance is less or greater than the linear one depending on waveguide parameters of the first and the second waveguides, Vi = kafafg 2= ka2(nf respectively. 

The composition profile is approximated by a step profile, with a uniform composition xf in the surface layer (0bulk phase x, at z>L. It is assumed that the total amount of liquid can be divided into two parts with the first constituting the homogeneous bulk phase (mole numbers in it n° = til -I- 2) and the remainder standing under the influence of the forces emanating from the solid surface causing adsorption (mole numbers, referred to unit mass of adsorbent, = n, -i- 2 the superscript a referring to adsorption) [17]. Simple mass balance considerations lead to the following expressions [12] ... 

The next step is to choose approximate concentration profiles as targets. There are many ways to select these initial vectors, and any method used to provide initial CR estimates can be useful for this purpose. Historically, the vectors obtained after performing varimax rotation onto the scores were used [47] and also the needle targets (i.e., vectors with only one non-null element equal to 1), which are the simplest representation of a peak-shaped profile [48, 75],... 

In Fig. 2.11 we depict the concentration profile of an ideal polymer near a flat wall and its replacement by a step profile with width S = 2R js/n (dashed). The simple approximation (2.55) reproduces the exact result within an accuracy of 1%. 

Sections of the dry reagent films were removed around the perimeter of the IDA electrodes with the use of a micro-manipulator tip and a Cambridge stereo zoom microscope (see below). A Dektak IIA step profiler was used to determine the approximate film thickness of the IDA electrodes. Electrodes were tested for shorts between the electrode fingers using a Fluke 87 multimeter prior to electrochemical testing. 

The approximations in these equations are for weakly guiding fibers and paraxial rays. Hence the optimum profile is close to parabolic. The pulse width is a factor of A/8, or dl/16, times that for the step profile, in Eq. (3-3), and is therefore considerably reduced. Since l/tj is one measure of the informationcarrying capacity of a waveguide, we deduce that capacity is increased by a factor of 8/A, or 16/0. We plot tj of Eq. (3-7) as the normalized time ctjzn o against q, corresponding to the solid curve in Fig. 3-3, for A = 0.01 or 0c = 0.14. There is a cusp at q pt, which means that ray dispersion is very sensitive to small variations about qopt- For example, when q = opt the pulse width increases by a factor of nearly 10. The normalized pulse width for a step profile with the same value of A is included for comparison. 

The approximate ray invariant for a slowly varying step-profile taper is expressed by Eq. (5-60). This relationship is accurate provided that the change 6p z) in taper radius over the local ray half-period is small. If the taper and fiber are weakly guiding then 0 (z) 1, and by generalizing the expression for Zp in Table 2-1, page 40, to slowly varying fibers, we deduce that Zp(z) = 2p(z)/0j(z) for meridional rays. In terms of the local taper angle Q(z), the slow-variation... 

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]... 

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 attenuation of tunneling-ray power in Eq. (7-3) depends on the product y(P, l)z, where the attenuation coefficient is the ratio of the transmission coefficient r to the ray half-period Zp. For the step profile, the latter is given in Table 2-1, page 40, and we use the linear approximation of Eq. (7-21) for T, which is an excellent approximation for all but the most weakly tunneling rays. If we express k in terms of the fiber parameter using the definition inside the front cover, then... 

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]... 

In Chapter 11 we discussed the fundamental properties of modes on optical waveguides. The vector fields of these modes are solutions of Maxwell s source-free equations or, equivalently, the homogeneous vector wave equations. However, we found in Chapter 12 that there are few known refractive-index profiles for which Maxwell s equations lead to exact solutions for the modal fields. Of these the step-profile is probably the only one of practical interest. Even for this relatively simple profile the derivation of the vector modal fields on a fiber is cumbersome. The objective of this chapter is to lay the foundations of an approximation method [1,2], which capitalizes on the small... 

The plot of Kg in Fig. 14-8(b) is virtually flat for q > I, with difference at = 1 of 5.1 per cent compared to = oo. In other words. Kg is virtually independent of profile shape in this range, and is well approximated by the step-profile value of 2.405. On rearranging Eq. (14-44)... 

Gaussian approximation 15-2 Example Gaussian proile 15-3 Example Step profile 15-4 Example Smoothed-out profiles 15-5 Field far from the fiber axis... 

We also discuss generalizations of the Gaussian approximation to other low-order modes. Finally, we briefly describe the equivalent step-profile approximation [4], and compare it with the Gaussian approximation. 

Q have approximately the same value of 1. This leads to Eq. (14-47X based on the step profile of volume 12, which is included in Table 15-2. 

The fundamental-mode properties of the weakly guiding, step-profile fiber were given in analytical form in the previous chapter, but, nevertheless, numerical solution of a transcendental eigenvalue equation is required. Within the Gaussian approximation the propagation constant is given explicitly, and all other modal properties have much simpler analytical forms, at the expense of only a slight loss of accuracy [4, 5]. 

Fig. 15-2 (a) Plots of the smoothed-out profiles of Eq. (15-9), where m = 0 is the Gaussian profile and m = oo is the step profile, and (b) the solid curves are the approximation of Table 15-2 for the distortion parameter, while the dashed curves are numerical solutions of the scalar wave equation [7]. 

We show in Table 15-3 that there is only a very small error between the approximate and exact values of U for the fundamental mode, calculated at the exact cutoff of single-mode operation for each profile [7]. The distortion parameter of Table 15-2 is plotted as the solid curves in Fig. 15-2(b) for the m = 2 and step profiles, together with the dashed curves for the exact values [7]. If Fj denotes the values of V for which there is no waveguide dispersion, i.e. D = 0, then... 

If we assume that spot size is independent of profile shape when V= we can use the Gaussian approximation to check if the consequence of this assumption is consistent with the exact value of V. The normalized intensity is independent of profile shape only if the spot size has the same value tq for all profiles. Thus we set K = in the ratio S/ a N,as given by Table 15-2,and use the step profile with V = 2405 as a reference profile which determines the value of r . On rearranging... 

With reference to Table 14-3, page 313, we assume that the fundamental-mode fields of an arbitrary profile fiber can be approximated by the fundamentalmode fields of some step-profile fiber, whose radial dependence is expressed by... 

If we use the step-profile approximation to determine the values of U for the fundamental modes of the Gaussian-profile fiber, then there is no limitation on the range of V, unlike the Gaussian approximation of Table 15-2. Furthermore, there is so little difference between the approximate and exact values of U, that a plot is indistinguishable from the dashed curve in Fig. 15-1 (b). Plots of the intensity distribution calculated from the step-profile approximation are virtually coincident with the exact solution away from the fiber axis, particularly for small values of V, whereas the Gaussian approximation leads to a significant error, as is evident in the V = 1.592 plot of Fig. 15-l(c). The Gaussian approximation has a far field which decreases too rapidly, as explained in Section 15-5, whereas the choice of field in Eq. (15-22) ensures the correct far field behavior for R > 1. 

Although the step-profile approximation is more useful in these two situations, the solutions of Eqs. (15-24) and (15-25) must be obtained numerically. The corresponding equations of Table 15-1 for the Gaussian approximation have the simple, exphdt forms of Table 15-2, and the far field is readily corrected, as we showed in Section 15-5. 

It is readily verified, by repeating the derivation in Table lS-1, page 339, for planar waveguides, that the above expression is the spot-size equation for the Gaussian approximation to the fundamental modes of a step-profile, planar waveguide of core half-width Py. 

Fig. 18-2 (a) The elliptically deformed, step-profile fiber has the same core cross-sectional area as the circular fiber, (b) The normalized birefringence Bp of Eq. (18-25), together with the Gaussian approximation Bg of Eq. (17-24), are plotted as a function of the fiber parameter. 

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