Nonlinear frequency response applications

The method presented in this chapter is based on nonlinear frequency response and the concept of higher-order frequency response functions, which have been proven as very convenient tools for analyzing weakly nonlinear systems. The basics for their application lay in the facts that ... 

Petkovska, M., Application of nonlinear frequency response to adsorption systems with complex kinetic mechanisms. Adsorption, 11, 497 502, 2000. 

Petkovska, M. and Seidel-Morgenstem, A., Nonlinear frequency response of a chromatographic column. Part I Application to estimation of adsorption isotherms with inflection points, Chem. Eng. Common., 192, 1300-1333, 2005. 

Cycled Feed. The qualitative interpretation of responses to steps and pulses is often possible, but the quantitative exploitation of the data requires the numerical integration of nonlinear differential equations incorporated into a program for the search for the best parameters. A sinusoidal variation of a feed component concentration around a steady state value can be analyzed by the well developed methods of linear analysis if the relative amplitudes of the responses are under about 0.1. The application of these ideas to a modulated molecular beam was developed by Jones et al. ( 7) in 1972. A number of simple sequences of linear steps produces frequency responses shown in Fig. 7 (7). Here e is the ratio of product to reactant amplitude, n is the sticking probability, w is the forcing frequency, and k is the desorption rate constant for the product. For the series process k- is the rate constant of the surface reaction, and for the branched process P is the fraction reacting through path 1 and desorbing with a rate constant k. This method has recently been applied to the decomposition of hydrazine on Ir(lll) by Merrill and Sawin (35). 

For the practical application to nonlinear optics, further, noncentrosymmetric LB films are required to possess not only large nonlinear optical response but excellent optical quality and thickness appropriate to optical devices. In this study, a family of pyrazine derivatives was found to be an LB film-forming material applicable to waveguide devices. The optical nonlinearity in the pyrazine LB films and the application of the pyrazine LB films to a frequency-doubling waveguide device is demonstrated in the latter part. 

One subject that attracted much attention is the nonlinear optical properties of these semiconductor nanoclusters [17], The primary objective is to find materials with exceptional nonlinear optical response for possible applications such as optical switching and frequency conversion elements. When semiconductors such as GaAs are confined in two dimensions as ultrathin films (commonly referred to as multiple quantum well structures), their optical nonlinearities are enhanced and novel prototype devices can be built [18], The enhancement is attributed mostly to the presence of a sharp exciton absorption band at room temperature due to the quantum confinement effect. Naturally, this raises the expectation on three-dimensionally confined semiconductor nanoclusters. The nonlinearity of interest here is the resonant nonlinearity, which means that light is absorbed by the sample and the magnitude of the nonlinearity is determined by the excited state... 

The nonlinear optical and dielectric properties of polymers find increasing use in devices, such as cladding and coatings for optical fibres, piezoelectric and optical fibre sensors, frequency doublers, and thin films for integrated optics applications. It is therefore important to understand the dielectric, optical and mechanical response of polymeric materials to optimize their usage. The parameters that are important to evaluate these properties of polymers are their dipole moment polarizability a, hyperpolarizabilities 0... 

From the form of the polarization it is clear that in order to observe any nonlinear optical effect, the input beams must not be copropagating. Furthermore, nonlinear optical effects through the tensor y eee requires two different input frequencies (otherwise, the tensor components would vanish because of permutation symmetry in the last two indices, i.e., ytfl eee = Xijy ) For example, sum-frequency generation in isotropic solutions of chiral molecules through the tensor y1 1 1 has been experimentally observed, and the technique has been proposed as a new tool to study chiral molecules in solution.59,61 From an NLO applications point of view, however, this effect is probably not very useful because recent results suggest that the response is actually very low.62... 

Positioner Application Positioners are widely used on pneumatic valve actuators. Often they provide improved process loop control because they reduce valve-related nonlinearity. Dynamically, positioners maintain their ability to improve control valve performance for sinusoidal input frequencies up to about one-hall of the positioner bandwidth. At input frequencies greater than this, the attenuation in the positioner amplifier network gets large, and valve nonlinearity begins to affect final control element performance more significantly. Because of this, the most successful use of the positioner occurs when the positioner response bandwidth is greater than twice that of the most dominant time lag in the process loop. 

Nonlinear optical activity reflects the nonlinear response of /r, (f- ) to electromagnetic radiation, which Eq. (7) shows to be governed by the first and second hyperpolarizabilities, p and y. A high level of such activity can have important applications in a variety of electro-optical devices,82,86,87 such as frequency converters, modulators, switches, etc. 

Second harmonic — Any nonlinear oscillating system produces higher harmonic oscillations. The second harmonic is the response having twice the frequency of the basic oscillation. The - current response of a faradaic electrode reaction (- faradaic reaction) to perturbations of the - electrode - potential is generally nonlinear, and thus higher harmonic oscillations of the - alternating current (AC) are produced in - AC voltammetry. Since the -> capacitive current is a much more linear function of the electrode potential, the capacitive contribution to higher harmonic currents are rather small which allows a desirable discrimination of theses currents in electro-analytical applications. 

Both TCSPC and frequency-domain fluorimetry are limited in time resolution by the response of available detectors, typically >25 ps. For cases in which higher time resolution is needed, fluorescence up-conversion can be used (22). This technique uses short laser pulses (usually sub-picosecond) both to excite the sample and to resolve the fluorescence decay. Fluorescence collected from the sample is directed through a material with nonlinear optical properties. A portion of the laser pulse is used to gate the fluorescence by sum frequency generation. The fluorescence is up-converted to the sum frequency only when the gate pulse is present in the nonlinear material. The up-converted signal is detected. The resolution of the experiment therefore depends only on the laser pulse widths and not on the response time of the detectors. As a result, fluorescence can be resolved on the 100-fs time scale. For a recent application of fluorescence up-conversion to proteins, see Reference 23. 

Of central importance for understanding the fundamental properties of ferroelec-trics is dynamics of the crystal lattice, which is closely related to the phenomenon of ferroelectricity [1]. The soft-mode theory of displacive ferroelectrics [65] has established the relationship between the polar optical vibrational modes and the spontaneous polarization. The lowest-frequency transverse optical phonon, called the soft mode, involves the same atomic displacements as those responsible for the appearance of spontaneous polarization, and the soft mode instability at Curie temperature causes the ferroelectric phase transition. The soft-mode behavior is also related to such properties of ferroelectric materials as high dielectric constant, large piezoelectric coefficients, and dielectric nonlinearity, which are extremely important for technological applications. The Lyddane-Sachs-Teller (LST) relation connects the macroscopic dielectric constants of a material with its microscopic properties - optical phonon frequencies ... 

Example 8.1 demonstrates that application of a large-amplitude potential perturbation to a nonlinear system results in harmonics that appear at frequencies corresponding to multiples of the fundamental or applied frequency. A second result of Example 8.1 is the observation that application of a leirge-amplitude potential perturbation to a nonlinear system changes both the steady-state current density and the fundamental current response. The implication of this result is that the impedance response will also be distorted by application of a laige-amplitude potential perturbation. 

In contrast, the nonlinearities in bulk materials are due to the response of electrons not associated with individual sites, as it occurs in metals or semiconductors. In these materials, the nonlinear response is caused by effects of band structure or other mechanisms that are determined by the electronic response of the bulk medium. The first nonlinear materials that were applied successfully in the fabrication of passive and active photonic devices were in fact ferroelectric inorganic crystals, such as the potassium dihydrogen phosphate (KDP) crystal or the lithium niobate (LiNbO,) [20-22]. In the present, potassium dihydrogen phosphate crystal is broadly used as a laser frequency doubler, while the lithium niobate is the main material for optical electrooptic modulators that operate in the near-infrared spectral range. Another ferroelectric inorganic crystal, barium titanate (BaTiOj), is currently used in phase-conjugation applications [23]. 

Experimental NMR data are typically measured in response to one or more excitation pulses as a function of the time following the last pulse. From a general point of view, spectroscopy can be treated as a particular application of nonlinear system analysis [Bogl, Deul, Marl, Schl]. One-, two-, and multi-dimensional impulse-response functions are defined within this framework. They characterize the linear and nonlinear properties of the sample (and the measurement apparatus), which is simply referred to as the system. The impulse-response functions determine how the excitation signal is transformed into the response signal. A nonlinear system executes a nonlinear transformation of the input function to produce the output function. Here the parameter of the function, for instance the time, is preserved. In comparison to this, the Fourier transformation is a linear transformation of a function, where the parameter itself is changed. For instance, time is converted to frequency. The Fourier transforms of the impulse-response functions are known to the spectroscopist as spectra, to the system analyst as transfer functions, and to the physicist as dynamic susceptibilities. 

Let us now consider more specifically the case of a medium possessing an excited state u) close in energy to that of the emitted harmonic, 2h( >. For practical application, this condition is generally more useful than resonance at the fundamental frequency, since the latter condition is likely to result in a substantial loss of pump power through conventional single-photon absorption. In view of its denominator structure, it is clearly the first term in Eq. (85) that will provide the major contribution to the nonlinear response tensor... 

Wave mixing of two electric fields can give rise to second-order effects of nonlinear optics [4]. One of these is the harmonic generation that converts the fundamental wavelength of a laser into its half (see Section 12.2.2). But, if electric fields at different frequencies are used, the response of a medium with sufficient second-order dielectric susceptibility can be frequency shifted to the sum and the difference of the two laser frequencies [4]. In particular, sum frequency generation (SFG) is often used to study surfaces and has found applications to examine catalytic combustion [9,36]... 

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