Modal amplitudes

This is an exact system of equations that describes the evolution of modal amplitudes along the z-axis for the forward propagating field. A similar equation holds for the backward propagating component, of course. 

In this equation the matrices B and A represent the block matrices of (7), while the vector w contains all state-space variables at time t . In practice the spectral analysis is carried out using the mode shapes of the undamped equation of motion together with the non-dimensional modal amplitude vector w =[u,hvY. The time step h and the angular modal frequency co are combined into the non-dimensional frequency parameter = hco. When the first equation in (7) is multiplied by h the matrices B and A can be represented in non-dimensional form as... 

The spectral analysis of the extended state-space algorithm is based on the free vibration response of a single mode of the equations (26) and (27). In the non-dimensional generic format (9) this corresponds to the extended modal amplitude vector = u, hv, s, ht] and the matrices... 

As updating parameters are selected the decrease of Young s modulus in 9 distinct zones and the spring stiffness at the pendulum supports. The objective function comprises residuals of 4 natural frequencies and residuals of modal amplitudes and curvatures of the first and second bending modes. 

Figure 9 Comparison modal amplitudes and curvatures (bending mode I)...

Alternatively, modal amplitudes and velocities can be chosen as state variables, leading to a desirable decoupling of the state differential equations. 

For either numerical solution of the field equations by means of the finite element method or determination of a system of ordinary differential equations for modal amplitudes, the existence of a variational statement or weak form of the field equations is essential. For the complementary aspect of the problem concerned with the elastic field for a fixed boundary configuration, the powerful minimum potential energy theorem is available (Fung 1965). The purpose here is to introduce a variational principle as a basis for describing the rate of shape evolution for a fixed shape and a fixed elastic field. 

Table 11-1 Properties of bound modes. Summary of bound-mode properties, dropping the subscript j when only one mode is involved. A nonabsorbing waveguide is assumed so that e, h, can be taken to be real and e, are then imaginary. The modal amplitude coefficient, a, is defined by Eq. (11-2).

Table 12-5 Modal properties of the step-profile fiber. All quantities except are calculated from Table 11-1, page 230, Parameters are defined in Table 12-3 (a) and inside the back cover. The + and - refer to even and odd modes, R = rjp and a is the modal amplitude.

Table 13-2 Properties of bound inodes on weakly guiding waveguides. Parameters are defined inside the back cover. The modal amplitude a depends on the source of illumination, and A o is the core cross-section. We assume e, and F are real on nonabsorbing... 

The modal power flow along the fiber per unit cross-sectional area, or intensity, is given by the time-averaged Poynting vector S in Table 14-3, where a is the modal amplitude. To describe the change in this distribution with changes in V, we keep the total modal power P fixed, and define a normalized intensity S = S/P = S/ a N. Hence... 

Table 14-6 Modesofthe weakly guiding step-profile fiber. The vector modal fields are found by substitution into Table 14-1. Modalpoweris given by laPN, where a is the modal amplitude. Parameters are defined inside the back cover. 

We define S to be the intensity distribution when there is unit power in the fundamental mode, i.e. a iV = 1, where a is the modal amplitude and N the normalization. If we normalize S with the cross-sectional area within radius p, then Table 15-2 gives... 

Fields at the endface 20-3 Fields of the illuminating beam 20-4 Gaussian and uniform beams 20-5 Modal amplitudes and power... 

In order to determine the modal amplitudes of Eq. (20-2), we must know either E, or H( at the fiber endface. In general, it is not possible to determine analytical expressions for these fields. However, the problem is much simpler when the fiber is weakly guiding [2], as we show below. 

The current sources in Fig. 21-1 launch power into bound modes, and thus specify the modal amplitudes. Expressions for these amplitudes are derived from Maxwell s equations in Chapter 31. We find from Eqs. (31-35) and... 

We are primarily interested in radiation from the fundamental modes of bent, single-mode fibers. Within the weak-guidance approximation, the power radiated is insensitive to polarization, since p. Thus we can conveniently assume that the transverse electric field is parallel to the Z-axis in Fig. 23-2(a), i.e. orthogonal to the plane of the bend. Close to and within the core, the magnitude of the electric field on the bend is given by aj Fo (R) exp (ifiz), using the local-mode approxinution, where is the modal amplitude, Fq (R) is the... 

As we now have orthogonality relations and normalization expressions for leaky modes, results which were derived for bound modes in earlier chapters can simply be extended to apply to leaky modes. These include the perturbation expressions of Chapter 18, the modal amplitudes due to illumination in Chapter 20, and the excitation and scattering effects of current sources in Chapters 21 to 23. We give an example of leaky-mode excitation by a source in Section 24—23. 

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