Mode dispersion curve

Fig. 18. TM mode dispersion curves of fundamental (broken line) and harmonic waves (solid line) in waveguide of a hetero Y-type LB film consisting of C12PPy and arachidic acid. 

Using the refractive index value of the pyrazine LB film, we calculated the mode dispersion curves of the TM fundamental and the TM second-harmonic waves in the waveguide device composed of a waveguiding pyrazine layer and a fused quartz substrate when Nd YAG laser is used as a fundamental light (Fig. 18). These curves show that the Cerenkov type phase matching is possible in the range of the thickness from 410 nm to 510 nm. 

Figure 3. The guided mode dispersion curves used to determine phasematching possibilities (intersection of the fundamental and harmonic dispersion curves) for isotropic media.

Figure 6. The guided mode dispersion curves for a birefringent film and an optically isotropic substrate. Both the fundamental and harmonic curves are shown. The TE mode utilizes the ordinary refractive index and TM primarily the extraordinary index. Note the change in horizontal axis needed to plot both the fundamental and harmonic dispersion curves. Phase-matching of the TEq(co) to the TMo(2o>) is obtained at the intersection of the appropriate fundamental and harmonic curves.

Figure 7 Mode dispersion curves of a three-layer slab waveguide.

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

Figure 8.11 (a) Dispersion curve for CuCl(s) along [110] of the cubic unit cell, (b) Density of vibrational modes [3], Here L, T, A and O denote longitudinal, transverse, acoustic and optic. Reproduced by permission of B. Hennion and The Institute of Physics. 

In three dimensions, transverse and longitudinal optic and acoustic modes result. The dispersion curve for CuCl along [100] of the cubic unit cell [3] is shown in Figure 8.11(a) as an example. The number of discrete modes with frequencies in a defined interval can be displayed as a function of the frequency. This gives what is termed the density of vibrational modes or the vibrational density of states (DoS). The vibrational DoS of CuCl is given in Figure 8.11(b). 

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. 

For a molecular crystal, the internal modes tend to be q independent and thus appear as horizontal lines in Fig. 2.1 n is then equal to the number of molecules M in the cell, leading to a considerable simplification. The resulting dynamical matrix has 6M x 6M elements, considering both translational and rotational motions, and atom-atom potential functions may be used for its evaluation. Dispersion curves obtained in this manner for anthracene and naphthalene, are illustrated in Fig. 2.2. 

Fig. 24. The dispersion curves for USb energy plotted against wave-vector transfer Q (in units of 2 3t/a). The dashed lines represent the phonon dispersion and are based on the measured open points as well as on knowledge of phonons in NaCl structures. The magnetic modes are represented by solid squares (the collective excitation) and the hatched area (excitonic level). (Lander and Stirling )...

Before and after resonance, there are magnetization components of opposite sign 180° out of phase and in phase (+ u and - u direction) with the rf field B1 i (Fig. 1.8). At resonance, there is no magnetization in the u direction. If the receiver coil obtains the inductance current in phase with Bli (the u direction), a dispersion curve (Fig. 1.9) results, called the u mode. When the absorption or out of phase spectrum ( ) reaches its maximum (/ind (oj) = max.), the dispersion or in phase spectrum ( ) goes through zero and changes its sign, as illustrated in Fig. 1.9. 

Therefore the dispersion of the LO plasmon-phonon states is formally equivalent to the dispersion of the TO photon-phonon states, with 4irne2/m replacing k2 c2. When the plasmon-phonon frequency to is plotted against fn instead of k, dispersion curves for the LO modes are obtained which are similar to the polariton dispersion curves, the TO phonons showing no dispersion with /n. 

These coefficients can be evaluated for any biopolymer by taking an arbitrary chain length and determining the allowed values of frequencies for any set of boundary conditions. In our calculations we have assumed that the ends of the chain are fixed and we determine the frequency of the modes that permit an odd number of half wavelengths to be present on the chain. The eigenvectors for these frequencies are determined from the dispersion curves for an infinite chain. To make these calculations more compatible with experiment we have determined the absorption cross section which can be related to the Einstein coefficient by the following expression... 

Both A - and Ei-modes are Raman and IR active. The two nonpolar E2-modes E and E are Raman active only. The Bi-modes are IR and Raman inactive (silent modes). Phonon dispersion curves of wurtzite-structure and rocksalt-structure ZnO throughout the Brillouin Zone were reported in [106-108]. For crystals with wurtzite crystal structure, pure longitudinal or... 

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