Fundamental properties of modes

11-1 Propagation constant and phase velocity 11-2 Symmetry properties of the field components 11-3 Field representation on nonabsorbing waveguides 11-4 Orthogonality relations and normalization 11-5 Orthonormal modes 

11-6 Stored electric and magnetic energies 11-7 Power flow 

11-8 Fraction of modal power in the core 11-9 Total guided power 11-10 Fraction of total power in the core 

11-14 Solution in terms of the longitudinal field components 11-15 Coupling of the field components 11-16 Hybrid nature of the modal fields 11-17 Propagation constant 

11-18 Modal cutoff 11-19 Number of bound modes 11-20 Distortion parameter 11-21 Relationship with ray invariants 

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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... 

Modes of weakly guiding waveguides obey the fundamental properties of modes delineated in Chapter 11, and mainly because of the approximate TEM nature of the modal fields, these properties have the simpler forms of TaHe 13—2. The expressions in the first column are in terms of the transverse electric field e, and apply to all weakly guiding waveguides. Those in the second column are for waveguides which are sufficiently noncircular that e, can be replaced by either of the two fields for noncircular waveguides in Table 13-1, while the third column is for circular fibers only, when e, is replaced by any one of the four linear combinations Ct, for circular cross-sections in Table 13-1. We emphasize that Table 13-2 applies to all modes. 

Fundamental, laminar, and turbulent burning velocities describe three modes of flame propagation (see the Glossary for definitions). The fundamental burning velocity, S, is as its name implies, a fundamental property of a flammable mixture, and is a measure of how fast reactants are consumed and transformed into products of combustion. Fundamental burning velocity data for selected gases and vapors are listed in Appendix C of NFPA68 (1998). 

Flat Panel Displays Advanced Organic Materials describes the display type device specifications and material development. With clear descriptions and diagrams, the reader is presented with the fundamental properties of liquid crystals and electroluminescent organic compounds, along with the mode of operation of the displays using them. 

Of central importance for understanding the fundamental properties of ferroelec-trics is dynamics of the crystal lattice, which is closely related to the phenomenon of ferroelectricity [1]. The soft-mode theory of displacive ferroelectrics [65] has established the relationship between the polar optical vibrational modes and the spontaneous polarization. The lowest-frequency transverse optical phonon, called the soft mode, involves the same atomic displacements as those responsible for the appearance of spontaneous polarization, and the soft mode instability at Curie temperature causes the ferroelectric phase transition. The soft-mode behavior is also related to such properties of ferroelectric materials as high dielectric constant, large piezoelectric coefficients, and dielectric nonlinearity, which are extremely important for technological applications. The Lyddane-Sachs-Teller (LST) relation connects the macroscopic dielectric constants of a material with its microscopic properties - optical phonon frequencies ... 

This book has two goals. One is to facihtate the understanding of the fundamental properties of crystallization and the impact of these properties on crystaUization process development. The second is to aid practitioners in problem-solving using actual industrial examples under real process constraints. This book begins with fundamental thermodynamic properties (Chapters 2 and 3), nucleation and crystal growth kinetics (Chapter 4), and process dynamics and scale-up considerations (Chapters 5 and 6). Subsequent chapters cover modes of crystallization operation cooling (Chapter 7), evaporation (Chapter 8), antisolvent (Chapter 9), reaction (Chapter 10), and special cases of crystallization (Chapter 11). As mentioned, real industrial examples are provided in each chapter. 

In this chapter, we shall first briefly survey the methods used for the synthesis or generation of paramagnetic hydride complexes, subsequently survey the known classes of stable open-shell hydride complexes, then review our current knowledge on the fundamental properties of the M-H bond in open-shell compounds, and finally examine the various established decomposition modes. No review articles specifically focusing on paramagnetic hydride compounds and their chemical reactivity appear to have been previously published. With few exceptions, this chapter will be limited to the analysis of monometallic species of the d elements. The f elements are treated in a separate chapter in this... 

The fundamental properties of excitation of CNS neurons were presented with a focus on what neural elements around the electrode are activated under different conditions. During CNS stimulation action potentials are initiated in the axons of local cells, even for electrodes positioned over the cell body. The threshold difference between cathodic and anodic stimuli arises due to differences in the mode of activation. Anodic stimuli cause depolarization of the axon and excitation via a virtual cathode, while cathodic stimuli cause hyperpolarization at the site of excitation and the action potential is initiated during repolarization. The threshold for activation of presynaptic terminals projecting into the region of stimulation is often less than or equal to the threshold for direct excitation of local cells, and indirect effects mediated by synaptic transmission may alter the direct effects of stimulation on the postsynaptic cell. The fundamental understanding provided by this analysis enables rational design and interpretation of studies and devices employing electrical stimulation of the brain or spinal cord. 

Poly(3-hexylthiophene), P3HT, is a good example to illustrate the impact of structural perfectness on fundamental properties of a polymer and performance of devices fabricated on its basis. In general, the asymmetry of 3-substituted thiophenes results in three possible coupling modes when two monomers are... 

A fundamental property of modal analysis is that the response can be accurately captured if a, relatively, small number of modes are... 

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