Fundamental modes approximations

The theory developed for perfect gases could be extended to solids, if the partition functions of crystals could be expressed in terms of a set of vibrational frequencies that correspond to its various fundamental modes of vibration (O Neil 1986). By estimating thermodynamic properties from elastic, structural, and spectroscopic data, Kieffer (1982) and subsequently Clayton and Kieffer (1991) calculated oxygen isotope partition function ratios and from these calculations derived a set of fractionation factors for silicate minerals. The calculations have no inherent temperature limitations and can be applied to any phase for which adequate spectroscopic and mechanical data are available. They are, however, limited in accuracy as a consequence of the approximations needed to carry out the calculations and the limited accuracy of the spectroscopic data. 

Let us start with the short-time approximation in which we can use the symbolic manipulation computer program described in Appendix A to find the corrections coming from the quantum fluctuations of the fields. The operator formulas (94) and (95) are valid also for the degenerate downconversion because the two processes are governed by the same Hamiltonian, but now initially the second-harmonic mode is populated while the fundamental mode is initially in the vacuum state. Assuming that the pump mode at the frequency 2oo is in a coherent state fi0) (p0 = /Ni,exp(k )h)), we have... 

Quasistationary Values of the Quantum Fano Factors Fp and their Semiclassical Approximations Ff, Given by Eq. (54), for the Fundamental Mode in Nth-Harmonic Generation with N = 1 — 5 in No-Energy-Transfer Regime11... 

Numerical values of the quantum Fano factors in comparison to their semiclassical approximations for the fundamental mode, given by Eq. (56), are presented in their dependence on N in Table I and Fig. 8a. Analogously, those values for harmonics are presented in Fig. 8b and Table II as calculated by the numerical quantum method and from analytical semiclassical formula (57). It is seen that the approximate predictions of the Fano factors, according to (56)... 

The Neumann series or the successive-approximation methods provide formulations that, theoretically, should yield the particular solution of Eq. (252). Practical implementation of those methods calls for special care in order to avoid a contribution from the fundamental mode [the term of Eq. (254)]. A fundamental-mode contamination can result, for example, if the source condition of Eq. (253) is not satisfied exactly. Numerical round-... 

In the non-depleted wave approximation the efficiency t] for frequency-doubling by conversion of a guided fundamental mode into a guided second-harmonic mode using the nonlinear optical coefficient djj is given by... 

The large amplitudes in Figure 23 are in the direction perpendicular to the catenary plane. The peak coincides with a catenary resonance with a peak displacement of approximately 1.5 mm. The frequency of the fundamental mode of the head frame varies between 2.4 and 3.4 Hz depending on the rope tension. The head frame flexibility is likely to have some influence on the catenary amplitudes. 

Previous results were used for computer simulation of optical properties of the fabricated structures. By substitution to the RSoft software of measured polymer refractive index and the structure surface profile the inside and output optical field distribution were calculated using beam propagation method. The calculated output optical field is given in Fig 20C. In addition, calculation proved that the effective refractive index for fundamental mode is approximately 1.6276 at 632.8 ran. The prepared multimode structure supported 30 modes at wavelength 632.8 ran. 

It can be shown [5.1,5.24] that in nonfocal resonators with large Fresnel numbers N the field distribution of the fundamental mode can also be described by the Gaussian profile (5.32). The confocal resonator with d = R can be replaced by other mirror configurations without changing the field configurations if the radius Rf of each mirror at the position zo equals the radius R of the wavefront in (5.37) at this position. This means that any two surfaces of constant phase can be replaced by reflectors, which have the same radius of curvature as the wave front - in the approximation outlined above. 

It is also assumed that efficient nonlinear coupling occurs only between two guided modes propagating in the +z direction, the fundamental mode a and the SHG mode 2- In the slowly varying envelope approximation (SVEA), the governing equations become... 

In general the removal densities A (r) are complicated functions, but in the present calculation we represent these by rough approximations based on the fundamental mode from the one-velocity model. In the... 

That is, we assert that the fundamental modes of neutrons of all energies are in first approximation nearly the same. If we use this assumption in (8.311), we obtain... 

The fundamental modes of vibration of the methylene group are shown in the flgure of Vibration of the Methylene Group. The approximate spectral positions where these vibrations will absorb infrared radiation are also given. 

In the 2003 NEHRP Recommended Provisions (BSSC 2004) for buildings with passive dampers, and for the equivalent lateral force (ELF) analysis (linear static analysis) specifically, the response is defined by two modes the fundamental mode and the residual mode, which is used to approximate the combined effects of higher modes. For response spectrum analysis, higher modes are... 

At the mirror surfaces is = 1 b = b, which means that the phase front is identical with the mirror surface. [Due to diffraction this is not quite true at the mirror edges at larger distances from the axis, where the approximation (5.32) is not correct]. At the center of the resonator is Zq = 0 b becomes infinite. At the beam waist the constant phase surface becomes a plane z = 0. This is illustrated in Fig.5.8 which shows the phase fronts and intensity profiles of the fundamental mode at different locations inside a confocal resonator. It can be shown [5.15] that also in nonconfocal resonators with large Fresnel numbers N the field distribution of the fundamental mode can be described by the Gaussian profile (5.27). 

Approximations for U close to cutoff and far from cutoff can be derived, in particular, for the fundamental modes which are of importance for single-... 

We ignore the small polarization corrections to P and Py, given by Eq. (13-11), because P f Py for isotropic, noncircular waveguides. This is an accurate approximation, provided the material anisotropy is not so minute as to be comparable to the small contribution of order due to the waveguide structure. The higher-order modes of the noncircular waveguide have the same form as the fundamental modes, except when the fiber is nearly circular, for reasons given in Section 13-9. 

If V is below the cutoff value 2.405 of the second mode in iFig. 14-4, the fiber is single moded and only the even and odd fundamental modes can propagate. Both modes have the same propagation constant j8. Consequently the group velocity and the transit time of Eq. (11-36) are independent of polarization. In the weak-guidance approximation, the expression for in Table 14—3 follows from Eq. (13-17), and is plotted against V in Fig. 14—3(d) as the dimensionless quantity (n — c)/cA. 

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