Modes coupling

27-1 Coupled mode equations 27-2 Weakly guiding fibers 27-3 Example Weak power transfer 

A fiber with z-dependent nonuniformities is no longer cylindrically symmetric and therefore cannot support individual modes of the unperturbed fiber. 

Nevertheless, the fields of the perturbed fiber at position z can be described by a superposition of the fields of the complete set of bound and radiation modes of the unperturbed fiber. An individual mode of the set does not satisfy Maxwell s equations for the perturbed fiber, and hence the perturbed fields must generally be distributed between all modes of the set. This distribution varies with position along the fiber and is described by a set of coupled mode equations, which determine the amplitude of every mode [1-3], 

The unperturbed fiber has refractive-index profile h(x, y) and the perturbed fiber has z-dependent nonuniformities described by n(x, y, z). Both fibers are weakly guiding and we assume h = n s n j everywhere. For reasons given in the previous section, we ignore all polarization effects and work solely with solutions of the scalar wave equation. For convenience we assume the electric field is x-polarized and set 

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. 

In Chap. 3, we mentioned circulatory systems which are described (after linearization) by asymmetric stiffness and/or damping coefficient matrices. Stability of these class of systems has been studied by many authors (see, e.g., [48, 49, 72, 73]). In Sect. 4.2.1 below, we give a classic example where a follower force causes flutter instability. In multi-DOF systems, friction force may act as a follower force and destroy the symmetry of the stiffness and damping matrices resulting in flutter instability known as the mode coupling instability mechanism. This mechanism was first used to explain brake squeal [7]. Ono et al. [74] and Mottershead and Chan [75] studied hard disk drive instability using a similar concept. In Sect. 4.2.2, we study the mode coupling instability mechanism in a simple 2-DOF system with friction. 

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The GLE can be derived by invoking the linear response approximation for the response of the solvent modes coupled to the motion of the reaction coordinate. 

B2.5.351 after multiphoton excitation via the CF stretching vibration at 1070 cm. More than 17 photons are needed to break the C-I bond, a typical value in IR laser chemistry. Contributions from direct absorption (i) are insignificant, so that the process almost exclusively follows the quasi-resonant mechanism (iii), which can be treated by generalized first-order kinetics. As an example, figure B2.5.15 illustrates the fonnation of I atoms (upper trace) during excitation with the pulse sequence of a mode-coupled CO2 laser (lower trace). In addition to the mtensity, /, the fluence, F, of radiation is a very important parameter in IR laser chemistry (and more generally in nuiltiphoton excitation) ... 

Early studies showed tliat tire rates of ET are limited by solvation rates for certain barrierless electron transfer reactions. However, more recent studies showed tliat electron-transfer rates can far exceed tire rates of diffusional solvation, which indicate critical roles for intramolecular (high frequency) vibrational mode couplings and inertial solvation. The interiDlay between inter- and intramolecular degrees of freedom is particularly significant in tire Marcus inverted regime [45] (figure C3.2.12)). 

The eigenvalues of this mabix have the form of Eq. (68), but this time the matrix elements are given by Eqs. (84) and (85). The symmetry arguments used to determine which nuclear modes couple the states, Eq. (81), now play a cracial role in the model. Thus the linear expansion coefficients are only nonzero if the products of symmebies of the electronic states at Qq and the relevant nuclear mode contain the totally symmebic inep. As a result, on-diagonal matrix elements are only nonzero for totally symmebic nuclear coordinates and, if the elecbonic states have different symmeby, the off-diagonal elements will only... 

Survey spectra using the MPI method are used primarily for quantification of surface components in inorganic materials, with a detection limit of ppm to ppb. The same mode coupled with SPI can be used for molecular characterization of polymer films. 

Fig. 9. A correlation chart for the observed/predicted ripple characteristics for the reptation, Rouse and polymer mode coupling models. The restation model gives the best correlation ( 1) between theory and experiment.

The friction coefficient of a large B particle with radius ct in a fluid with viscosity r is well known and is given by the Stokes law, Q, = 67tT CT for stick boundary conditions or ( = 4jit ct for slip boundary conditions. For smaller particles, kinetic and mode coupling theories, as well as considerations based on microscopic boundary layers, show that the friction coefficient can be written approximately in terms of microscopic and hydrodynamic contributions as ( 1 = (,(H 1 + (,/( 1. The physical basis of this form can be understood as follows for a B particle with radius ct a hydrodynamic description of the solvent should... 

Rostov, K.S. Freed, K.F., Mode coupling theory for calculating the memory functions of flexible chain molecules influence on the long time dynamics of oligoglycines, J. Chem. Phys. 1997,106, 771-783... 

Kreuwel H.J.M., Lambeck P.V., Beltman J.M.M., Popma T.J.A., Mode coupling in multilayered structures applied to a chemical sensor and a wavelength selective directional coupler, ProcECIO 87, Glasgow, UK, 1987,217-220. 

Figure 1. Schematic illustration of waveguiding structures, a ray-picture b modal intensity distribution obtained with the aid of Maxwell s equation c mode coupling c whispering gallery mode.

When the SRI is equal to the cladding one, the cladding layer becomes an infinitely extended medium and thus supports no discrete cladding modes. In this case, a broadband radiation mode coupling occurs with no distinct attenuation bands8. 

Koyamada, Y., Numerical analysis of core mode to radiation mode coupling in long period fiber gratings, IEEE Photon. Technol. Lett. 2001, 13, 308 310... 

To use the OFRR as a biosensing device, the optical resonant mode is excited and the resonant frequency is measured continuously in real time. The conceptual measurement setup is illustrated in Fig. 14.3. Laser light from a distributed feedback (DFB) laser is delivered to the OFRR using fiber optic cable. One method that has been used to excite the resonant modes is to place a tapered fiber optic cable with a diameter less than 4 pm in contact with the OFRR. The evanescent field of the tapered fiber overlaps with the evanescent field outside of the capillary wall, which enables mode coupling between the two media24. 

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