Propagation constant radiation modes

Fig. 13.5 Calculated propagation constants (ft) for the fundamental modes of glass MNFs with refractive indices of 1.46 (silica), 1.48 (fluoride), 1.54 (phosphate), 1.89 (germinate), and 2.02 (tellurite), respectively. A circle marked on each curve corresponds to the maximum radius of the single mode MF. Radiation wavelength is X 633 nm. Reprinted from Ref. 62 with permission. 2008 Optical Society of America...

This formally simple procedure is very difficult to perform, however. Because of radiation from the bend, the azimuthal propagation constant v to be found is complex. Since the bend radius of the waveguide is typically larger than the wavelength, the real part of v can be large, too. Moreover, a number of modes of each slice with very different values of their effective indexes are to be considered simultaneously. It causes very serious... 

Representation of the modal fields 25-2 Propagation constant 25-3 Radiation-field expansion 25-4 Orthogonality and normalization 25-5 Power of the radiation field 25-6 Excitation of radiation modes... 

The discrete set of bound-mode propagation constants means that the fields of the waveguide in the spatial steady state are given by the finite sum over all bound modes in Eq. (11-2). In contrast, each radiation and evanescent mode can take any of the continuum of propagation-constant values given in Table 25-1, and thus an integration over all values of is necessary. However, like bound modes, the total radiation field requires a summation over the subscript j of Eq. (25-1) to account for the transverse fields of different modes. However, rather than use / as the continuum variable, when P is real for radiation modes and imaginary for evanescent modes, we use instead the modal parameter Q, defined below, in order to simplify the notation. We take / to be the positive root of the inverse relation whence... 

The complete spatial variation of each radiation mode is given by Eq. (25-1), and the range of values of the propagation constant satisfies 0 p < kn. For a particular value of j8, the modal fields can be regarded as a Fourier superposition of the fields of a family of plane waves, all inclined at angle 6 to the waveguide axis. In the uniform cladding, 6 is related to j8 by... 

In Chapter 13 we used the polarization properties of the waveguide to determine the direction of e, and the correction S j to the scalar propagation constant Pj. However, the propagation constant p for radiation modes takes any value in the range 0 < jS < kn y and is therefore a continuous variable independent of waveguide polarization. Consequently, higher-order correc-... 

The scalar propagation constant Pj is determined from the eigenvalue equation. We have ignored the continuum of radiation modes in Eq. (27-1) since we shall be considering coupling between bound modes only. 

The first application of the conjugated form of the reciprocity theorem demonstrates orthogonality of bound, radiation and leaky modes of nonabsorbing waveguides. Consider two modes propagating in the forward direction with propagation constants Pj nd The subscripts jand k may refer to two different bound modes, but, in the case... 

Let T j and jSj be the unknown field and the propagation constant of the j th mode of the second waveguide, whose refractive-index profile is n(x, y). We express 4 as an eigenfunction expansion over the complete set of bound solutions 4 and radiation solutions of the first waveguide with profile n(x,y). Hence... 

The MO measurements provide information about the angular distribution of molecules in the x, y, and z film coordinates. To extract MO data from IR spectra, the general selection rule equation (1.27) is invoked, which states that the absorption of linearly polarized radiation depends upon the orientation of the TDM of the given mode relative to the local electric field vector. If the TDM vector is distributed anisotropically in the sample, the macroscopic result is selective absorption of linearly polarized radiation propagating in different directions, as described by an anisotropic permittivity tensor e. Thus, it is the anisotropic optical constants of the ultrathin film (or their ratios) that are measured and then correlated with the MO parameters. Unlike for thick samples, this problem is complicated by optical effects in the IR spectra of ultrathin films, so that optical theory (Sections 1.5-1.7) must be considered, in addition to the statistical formulas that establish the connection between the principal values of the permittivity tensor s and the MO parameters. In fact, a thorough study of the MO in ultrathin films requires judicious selection not only of the theoretical model for extracting MO data from the IR spectra (this section) but also of the optimum experimental technique and conditions [angle(s) of incidence] for these measurements (Section 3.11.5). 

In Eq. (12.25), 6 is the angle between the polarization of the incident radiation (A ) and the direction of propagation of the scattered wave k ), R is the position of the detector, is the dynamic polarizability of the segment, and p iv) is the density of modes of the incident radiation at frequency v (Eqs. 12.9,12.12, and B12.1.14). The factor sin(5)/IAI is the same factor that determines the amplitude of the field from an oscillating electric dipole (Figs. 3.1 and 3.2), and the fluorescence from an excited molecule whose transition dipole is oriented along a fixed axis (Sect. 5.9). The polarizability Uaa can be obtained from the difference between the dielectric constant of the solution and that of the pure solvent. 

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