Non-linear dispersion

The existence of even parity states lying above the have been identified by two-photon absorption (TPA), two-photon fluorescence excitation [31], and electroabsorption (EA) spectroscopy. We measured the TPA spectrum of DOO-PPV by Z-scan [32] showing both the real and imaginary components of Figure 7-13 compares the linear absorption spectrum of DOO-PPV to the spectra of the real and imaginary molecular second hyperpolarizability. The imaginary component, y", is proportional to the TPA coefficient a2, while the real component, y, corresponds to the non-linear dispersion, 2- The y" spectrum shows a clear peak at % 3.2,5 eV, 0.5 eV above the peak of the l.B absorption band, and a shoulder at se3.5 eV, which may correspond to a second TPA band. The y spectrum clearly indicates dispersion near the two y" peaks, as expected from Kra-mers-Kronig analysis, and permits us to identify two distinct TPA bands. 

For pure elemental semiconductors like silicon, the strong electronic absorption at energies above Eg produces a small non-linear dispersion of the refractive index below Eg in silicon, n = 3.57 near Eg at room temperature (RT) and it steadily decreases to 3.42 for wavelengths near 12 pm and stays close to this value down to radio frequency energies (see also [20]). For these elemental crystals, the dielectric constant at energies below Eg is real and equal to n2. The refractive index is isotropic for cubic crystals, but for crystals with one anisotropic axis, like those of the wurtzite type, the refractive index for the electric field component of the radiation parallel to this axis (n//) is slightly different from that for the component perpendicular to this axis (njJ. 

The rheological characterisation of non-Newtonian fluids is widely acknowledged to be far from straightforward. In some non-Newtonian systems, such as concentrated suspensions, rheological measurements may be complicated by non-linear, dispersive, dissipative and thixotropic mechanical properties and the rheometrical challenges posed by these features may be compoimded by an apparent yield stress. 

The present study demonstrates that the analytic calculation of hyperpolarizability dispersion coefficients provides an efficient alternative to the pointwise calculation of dispersion curves. The dispersion coefficients provide additional insight into non-linear optical properties and are transferable between the various optical processes, also to processes not investigated here as for example the ac-Kerr effect or coherent anti-Stokes Raman scattering (CARS), which depend on two independent laser frequencies and would be expensive to study with calculations ex-plictly frequency-dependent calculations. 

The p.c.s. measurements were carried out using a Malvern multibit correlator and spectrometer together with a mode stabilized Coherent Krypton-ion laser. The resulting time correlation functions were analysed using a non-linear least squares procedure on a PDP11 computer. The latex dispersions were first diluted to approximately 0.02% solids after which polymer solution of the required concentration was added. 

The non-linear response of plastic materials is more challenging in many respects than pseudoplastic materials. While some yield phenomena, such as that seen in clay dispersions of montmorillonite, can be catastrophic in nature and recover very rapidly, others such as polymer particle blends can yield slowly. Not all clay structures catastrophically thin. Clay platelets forming an elastic structure can be deformed by a finite strain such that they align with the deforming field. When the strain... 

A note of caution should be sounded here. Whilst the curves shown in Figure 6.5 are characteristic of many charged dispersions it should be recalled that once we apply a sinusoid to a non-linear system the response need not be a sinusoid. As the strain is increased into the nonlinear region, the waveform passing through the sample becomes progressively distorted. The instrumental analysis in this case involves... 

If the concentration profile can be determined the moduli can be evaluated. In principle there is no reason why this should be a non-linear measurement, it depends upon the magnitude of the gravitational Peclet number. Buscall35 suggested that a low speed centrifuge could be used to apply different acceleration gradients to the dispersion. If the angular velocity of the rotor is cor and if X is the distance from the centre of the rotor to the top of the sediment then the pressure balance equation becomes... 

Here Uq is the linear refractive index and An is the nonlinear part of the refractive index. Linear dispersion of the refractive index is accounted for in (2.2), (2.3). In general, nonlinear dispersion has to be taken into consideration, too. However, due to non-resonant character of the electronic nonlinearity in fused silica it is possible, as a first approximation, to neglect the nonlinear dispersion. 

E.A. Romanova, L.A. Melnikov, Spatiotemporal dynamics of femtosecond pulses in non-linear optical waveguides with material dispersion , Optics and Spectroscopy 96, 90-95 (2004). 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

Raybon, G., et al. 320 Gbit/s Single-channel Pseudo-linear Transmission over 200 km of Non-zero Dispersion Fiber, Optical Fiber Communication Conference, post-deadline paper 29, March 2000. 

P. Gueret and J. P. Vigier, Non-linear Klein-Gordon equation carrying a non-dispersive soliton-like singularity, Lett. Nuovo Cimento 35(8) (Ser. 2), 256—259 (1982). 

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