Eigenmodes

Generally speaking, when light is propagating in a uniform anisotropic medium, the direction of the electric field, and therefore the polarization state, will vary in space. Only when the electric field is in some special directions, known as the eigenmode, will its direction remain invariant in space, which will be proved to be true in this section. In the eigenmode, the electric field has the form 

In order to have non-zero solution, the determinant must be zero  

This equation is also called the eigen equation. Define 

Now we consider the eigenmodes, also referred to as normal modes. In component form. Equation (2.57) can be rewritten as three equations  

Draw a straight line which is through the origin and parallel to s. Cut a plane through the origin, which is perpendicular to s. This is plane is described by 

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From a mathematical point of view, conformations are special subsets of phase space a) invariant sets of MD systems, which correspond to infinite durations of stay (or relaxation times) and contain all subsets associated with different conformations, b) almost invariant sets, which correspond to finite relaxation times and consist of conformational subsets. In order to characterize the dynamics of a system, these subsets are the interesting objects. As already mentioned above, invariant measures are fixed points of the Frobenius-Perron operator or, equivalently, eigenmodes of the Frobenius-Perron operator associated with eigenvalue exactly 1. In view of this property, almost invariant sets will be understood to be connected with eigenmodes associated with (real) eigenvalues close (but not equal) to 1 - an idea recently developed in [6]. 

In the unperturbed case, say 7 = 70, let the system have two disjoint invariant sets Bi and B2 associated with two eigenmodes and invariant inea-... 

For more than two almost invariant sets one has to consider all eigenmea-sures corresponding to eigenmodes for eigenvalues close to one. In this case, the following lemma will be helpful. 

Oscillations of black holes. Non-radial oscillations of black holes can be excited when a mass is captured by the black hole. The so called quasinormal modes have eigenfrequencies and damping times which are characteristic of black holes, and very different of eigenfrequencies and damping times of quasi normal modes of stars having the same mass. Also the eigenmodes being different for a star and a black hole, the associated gw will also exhibit characteristic features. 

For Gaussian chains the spatial structure of the eigenmodes is given by the Rouse form... 

The dynamics of highly diluted star polymers on the scale of segmental diffusion was first calculated by Zimm and Kilb [143] who presented the spectrum of eigenmodes as it is known for linear homopolymers in dilute solutions [see Eq. (77)]. This spectrum was used to calculate macroscopic transport properties, e.g. the intrinsic viscosity [145], However, explicit theoretical calculations of the dynamic structure factor [S(Q, t)] are still missing at present. Instead of this the method of first cumulant was applied to analyze the dynamic properties of such diluted star systems on microscopic scales. 

Landauer proposed in 1957 the first mesoscopic theoretical approach to charge transport [176]. Transport is treated as a scattering problem, ignoring initially all inelastic interactions. Phase coherence is assumed to be preserved within the entire conductor. Transport properties, such as the electrical conductance, are intimately related to the transmission probability for an electron to cross the system. Landauer considered the current as a consequence of the injection of electrons at one end of a sample, and the probability of the electrons reaching the other end. The total conductance is determined by the sum of all current-carrying eigenmodes and their transmission probability, which leads to the Landauer formula of a ID system ... 

To be specific regarding the formalism, let uqia(t) denote the a component of the displacement field associated with wave vector q and eigenmode i at time t. In the absence of external forces, which can simply be added to the equation, the equation of motion for the coordinates that are not thermo-statted explicitly uqia would read as follows ... 

In Fig. 20 we show a theoretical dispersion plot using these parameters and a tensile stress = 2.7 x 10 dyn/cm. Due to the symmetry of the modes at X the stress tensor tpy does not affect the surface eigenmodes at this symmetry point. In addition, we have softened the intralayer force constant 4>ii in the first layer by about 10%. With these parameters, we find good agreement between experimental data and theoretical dispersion curves. 

Huang, Y.-Z., Guo, W.-H., and Wang, Q.-M., 2001, Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle microresonator,/EEB J. Quantum Electron. 37(1) 100-107. 

Key words eigenmode expansion microresonators mode matching mode solvers optical waveguide theory, photonic crystals, waveguide bends. 

An efficient formalism for the calculation of eigenmodes of the multilayer is known as the transfer matrix method . We will briefly outline its fundamentals. 

The fact that any field in the multilayer structure can be expressed by a superposition of its eigenmodes will be heavily used later as a basic principle of the film mode matching method. 

Sinee PMLs are lossy, the eigenmodes of multilayers with PML are generally eomplex, so that their effective indexes have to be localized in the eomplex plane. The great advantage of PML s is that they can be very easily ineorporated into the modeling software based on mode matching, and their behavior can be easily understood Irom their interpretation in terms of complex coordinate stretching. 

Different longitudinal sections in the structure may generally have very different refractive index profiles, and thus their eigenmode functions can strongly differ, too. As they have to be well-represented by the superposition of modes of other sections, the number M of considered eigenmodes needs to be sufficiently large (typically from tens to hundreds), and it means that many of these modes are evanescent in the direction of propagation (for their effective indexes, Re lV < 0). 

But it is true only if the complete system of eigenmodes is considered. In reality, we can use only a finite number M of eigenmodes in each section, and in this case Eq.(24) is not accurately satisfied. As it has been discussed... 

The Bloeh mode ean be defined as a wave eorresponding to the eigenmode of the transfer matrix of one period of the strueture. Let A is the transfer matrix deseribing wave transition from the left to the right of one period, calculated by sueeessive applications of Eq. (13) to each section and Eq. (13) to each interface between sections within the period. The Bloch mode then has to satisfy the condition... 

To find the eigenmode, we ehoose a suitable lateral position r] where the mode field is expected to be nonzero and calculate the values U+ and U of the immittanee matrix at that position starting from both sides as indicated in Fig. 5. Since the tangential field eomponents at that position must be eontinuous, we easily arrive to the set of linear equations... 

Zeros of its determinant determine the effective refractive indexes of the eigenmodes, and from the corresponding nontrivial solutions p (77 ) we can calculate the eomplete vectorial field distribution everywhere in the cross-section using Eqs. (36) - (40) or their more stable equivalents. 

As the refractive index within the slice depends on the vertical coordinate only, TE and TM modes can propagate independently in the slices, and Eqs (35) and (36) are fully applicable to this geometry without any change, too. Since we are interested in the propagation of waves in the azimuthal direction, it is advantageous to decompose the Hertz vectors into the eigenmodes of the slice as follows ... 

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