Approximations higher-order modes

The mass sensitivity of an ST-quartz APM device was determined by depositing metal onto the unelectroded quartz surface, i.e., the side opposite the transducers. The plate mode velocity shift is plotted vs the surface mass density of deposited silver in Figure 3.35. As expected from the discussion above, the device is approximately twice as sensitive when higher-order modes (n 1) are excited than with the lowest-order (it = 0) mode. The mass sensitivity measured... 

The sensor response to DMMP vapor exposure (without any added dopant) is shown in Fig. 25. A percentage-response of approximately 2.1% was obtained, and the response time was 2 s [20]. This response is attributed to the leaking of the higher-order modes through the modified cladding of the fiber, due to the increased conductivity in polypyrrole film due to DMMP absorption. 

We ignore the small polarization corrections to P and Py, given by Eq. (13-11), because P f Py for isotropic, noncircular waveguides. This is an accurate approximation, provided the material anisotropy is not so minute as to be comparable to the small contribution of order due to the waveguide structure. The higher-order modes of the noncircular waveguide have the same form as the fundamental modes, except when the fiber is nearly circular, for reasons given in Section 13-9. 

The fundamental modes of the infinite parabolic profile fiber have a Gaussian spatial variation it is the exact solution of the scalar wave equation. Thus, the essence of the Gaussian approximation is the approximation of the fundamental-mode fields of an arbitrary profile fiber by the fundamentalmode fields of some parabolic profile fiber, the particular profile being determined from the stationary expression for the propagation constant in Table 15-1. Clearly this approach can be generalized to apply to higher-order modes, by fitting the appropriate solution for the infinite parabolic profile [9]. 

The higher-order mode approximation of Section 15-6 is inadequate for determining the range of single-mode operation of arbitrary clad-profile fibers. At cutoff of the 1 = ffi = 1 modes, the clad fiber fields satisfy Eq. (11-54) when n(R) = n, and thus have an R radial dependence in cyhndrical polar coordinates. The corresponding dependence of Eq. (15-17) is always exponential and is thus a poor approximation. To overcome this deficiency, we use the estimate for in Table 15-2 based on equal profile volumes. However, a much... 

In contrast to the planar waveguide, the core fields of the circular fiber, which are given in terms of Bessel functions in Table 12-3, page 250, do not represent a single family of rays [1]. However, the fields of higher-order modes, which have C/ > 1, are asymptotically equivalent to a single family. To demonstrate this, we first use the recurrence relations of Eq. (37-72) to express i in Table 12-3 in terms of J, and its derivative. Then, provided 1/ > 1 and 1/ — v s> we can substitute the far Debye approximations of Eq. (37-89). The azimuthal dependence on sin(v ) or cos(v ) is... 

In Section 24-18, we derived the power attenuation coefficient for tunneling leaky modes on a. step-profile, weakly guiding fiber. Here we show that, for higher-order modes, Eq. (24-36) is equivalent to the power attenuation coefficient of the corresponding skew tunneling rays. The argument of the Hankel functions in Eq. (24-36) is smaller than the order. Furthermore, we assume that / is sufficiently large that the order of both Hankel functions may be taken to be approximately /. Under these conditions, we can use the approximate forms of Eq. (37-90), and for simplicity we approximate x by the middle expression in Eq. (37-90b). Hence... 

Annihilation can also occur with the emission of three (or more) gamma-rays, and Ore and Powell (1949) calculated that the ratio of the cross sections for the three- and two-gamma-ray cases is approximately 1/370. Higher order processes are expected to be further depressed by a similar factor. A case in point is the four-gamma-ray mode, for which the branching ratio with the two-gamma-ray mode was shown by Adachi et al. (1994) to be approximately 1.5 x 10-6, in accord with QED calculations. 

We have now formulated the approximation and presented one way of actually calculating it, namely by means of Eqs. (42), (43) and Fig. 21. Another way of including much of the higher order effects in Eq. (42) is to calculate the wave functions for the excited electron m in a potential which directly includes a certain selection of the interaction matrix elements in Fig. 21. This is going to be our normal mode of operation, and the zeroth-order basis set will be chosen in the following way ... 

Now, the averaged model to order p is defined in terms of cm and (c) by the global equation (19) and the local equation (26). The local equation can be extended to any desired order in p by using higher order approximations of d. This form of the reduced model, expressed in terms of two concentration variables, will be referred to as the two-mode model , and is convenient for physical interpretation of various limiting cases as well as to extend the range of... 

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