Bound modes properties

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).

We now examine the properties of bound modes and their fields. These results are derived from Maxwell s equations in Chapters 30 and 31. 

To summarize this part of the chapter, we have laid out inside the back cover physical quantities and waveguide parameters defining light propagation on a waveguide, together with bound-mode parameters. Table 11-1 summarizes the properties of bound modes. 

The basic properties of bound modes on optical waveguides were given in the previous chapter. In this chapter we display these properties explicitly for those few profiles which have exact solutions of Maxwell s equations. Our primary objective is to derive analytical expressions for the modal vector fields, which contain all the polarization properties of the waveguide discussed in Section 11-16. We pay particular attention to fundamental modes, since these... 

Radiation modes, like bound modes, obey nearly all of the general properties presented in Chapter 11. As we show below, the essential difference between bound and radiation modes is that there is no eigenvalue equation for radiation modes because of the relaxation of the boundary condition that the fields are evanescent as r -> oo. Furthermore, at large distances from the waveguide, their fields are oscillatory, or wavelike, and do not have the evanescent behavior of the bound-mode fields [1, 2],... 

In Section 11-4, we showed that each bound mode is orthogonal to every other bound mode and to all radiation modes. The orthogonality properties of radiation modes are derived in Section 31-3. On a nonabsorbing waveguide, the th and kth forward-propagating radiation modes obey... 

In this chapter we lay down the fundamental equations for an electromagnetic analysis of optical waveguides. From these equations we deduce elementary properties of the fields. The material presented here is used throughout Parts II and III, although it is abstracted mainly in Chapter 11 when discussing properties of bound modes. 

The radiation field of the scalar wave equation can be represented by the continuum of scalar radiation modes discussed above, or by a discrete summation of scalar leaky modes and a space wave. This is clear by analogy with the discussion of vector radiation and leaky modes for weakly guiding waveguides in Chapters 25 and 26. Scalar leaky modes have solutions P of Eq. (33-1) below their cutoff values when P becomes complex. Many of the properties of bound modes derived in this chapter also apply to leaky modes. For example, the orthogonality condition of Eq. (33-5a) applies to leaky modes, provided only that the cross-sectional area A. is replaced by the complex area A of Section 24-15 to ensure that the line integral of Eq. (33-4) vanishes. 

We have studied the vibrational properties of Au adatoms on the low-index faces of copper. From the position of new phonon modes, which are due to the presence of the adatom, it comes out that the gold adatom is weakly coupled with the atoms of Cu(l 11) for the directions parallel to the surface and tightly bound with those of Cu(lOO). These modes are found in lower frequencies than those of the Cu adatom. The temperature dependence of MSD s and relaxed positions of the Au adatom along the normal to the surface direction, reveal that this atom is more tightly bound with the (111) face and less with the (110) face. 

Possible modes of regulation of filament assembly may be anticipated from the basic properties of actin. We have shown that the tightly bound divalent metal ion (Ca or Mg ) interacts with the P- and y-phosphates of ATP bound to actin, and that the Me-ATP bidentate chelate is bound to G-actin in the A configuration. The nature of the bound metal ion affects the conformation of actin, the binding kinetics of ATP and ADP, and the rate of ATP hydrolysis. 

The transition of the compressibility, and other properties of the polyelectrolyte brnshes, is most likely accounted for in terms of the transition in the binding mode of the connterion to the polyelectrolytes, from the loosely bonnd state to the tightly bound one, which rednces inter- and intrachain repulsive interactions. The following snpports this ac-connt (1) At the critical density, = 0.20 chain/nm, the separation distance between polyelectrolyte chains, d, is 2.4 nm. This distance is close to the snm, 2.6 nm, of the chain diameter, 1.3 nm, and the size of two hydrated connterions, 1.32 nm, indicating that the abrupt... 

There has been considerable discussion regarding the mode of action of the sea cucumber and starfish saponins. Both the triterpene and steroidal glycosides inhibit both Na/K ATPase and Ca/Mg ATPase 06) possibly as a result of their aglycone structures. However, their detergent properties cause membrane disruption which will influence the activity of membrane-bound enzymes such as the ATPases. In investigating the actions of saponins on multilamellar liposomes, it was found that cholesterol serves as the binding site for such saponins and that cholesterol-free lip-somes are not lysed by saponins 107). 

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

The utilization of IR spectroscopy is very important in the characterization of pseudopolymorphic systems, especially hydrates. It has been used to study the pseudopolymorphic systems SQ-33600 [36], mefloquine hydrochloride [37], ranitidine HC1 [38], carbovir [39], and paroxetine hydrochloride [40]. In the case of SQ-33600 [36], humidity-dependent changes in the crystal properties of the disodium salt of this new HMG-CoA reductase inhibitor were characterized by a combination of physical analytical techniques. Three crystalline solid hydrates were identified, each having a definite stability over a range of humidity. Diffuse reflectance IR spectra were acquired on SQ-33600 material exposed to different relative humidity (RH) conditions. A sharp absorption band at 3640 cm-1 was indicative of the OH stretching mode associated with either strongly bound or crystalline water (Fig. 5A). The sharpness of the band is evidence of a bound species even at the lowest levels of moisture content. The bound nature of this water contained in low-moisture samples was confirmed by variable-temperature (VT) diffuse reflectance studies. As shown in Fig. 5B, the 3640 cm-1 peak progressively decreased in intensity upon thermal... 

Guided mode calculations were also carried out to compare the sensor response of several waveguide systems. In these simulations a model molecular monolayer is represented by a 2-nm thick layer with a refractive index of n 1.5. The optical properties of this model layer are typical of a dense layer of organic molecules on a substrate1 41, and are a reasonable approximation for a streptavidin protein layer bound to a biotinylated surface, the experimental model system we use to characterize our sensors. The ambient upper cladding was assumed to be water with a refractive index of n 1.32. For all examples, the lower cladding was assumed to be Si02 with an index of n 1.44. In the simulations, the effective index of... 

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