Nonlinear frequency response

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. and Do, D.D., Nonlinear frequency response of adsorption systems general approach and special cases, in Fundamentals of Adsorption, Meunier, F., Ed., Elsevier, Paris, 1998, pp. 1189-1194. 

Petkovska, M., Nonlinear frequency response of nonisothermal adsorption systems. Nonlinear Dyn., 26, 351-370, 2001. 

Petkovska, M. and Petkovska, L.T., Use of nonlinear frequency response for discriminating adsorption kinetics mechanisms resulting with bimodal characteristic functions. Adsorption, 9, 133-142, 2003. 

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. 

Petkovska, M., Zivkovic, V., Kaufmann, J., and Seidel-Morgenstern, A., Estimation of adsorption isotherms using nonlinear frequency response of a chromatographic column — experimental study. The 2003 AlChE Annual Meeting, San Francisco, USA, November 16-21, 2003, Proceedings on CD. 

In order to illustrate some of the basic aspects of the nonlinear optical response of materials, we first discuss the anliannonic oscillator model. This treatment may be viewed as the extension of the classical Lorentz model of the response of an atom or molecule to include nonlinear effects. In such models, the medium is treated as a collection of electrons bound about ion cores. Under the influence of the electric field associated with an optical wave, the ion cores move in the direction of the applied field, while the electrons are displaced in the opposite direction. These motions induce an oscillating dipole moment, which then couples back to the radiation fields. Since the ions are significantly more massive than the electrons, their motion is of secondary importance for optical frequencies and is neglected. 

From such a treatment, we may derive explicit expressions for the nonlinear radiation in tenns of the linear and nonlinear response and the excitation conditions. For the case of nonlinear reflection, we obtain an irradiance for the radiation emitted at the nonlinear frequency of... 

Figure 6.1 Nonlinear optical responses, (a) Second-order SF generation, the transition probability is enhanced when the IR light is resonant to the transition from the ground state g to a vibrational excited state V. CO is the angular frequency of the vibration, (b) Third-order coherent Raman scheme, the vibrational coherence is generated via impulsive stimulated...

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. 

Forced oscillation is a well-known technique for the characterization of linear systems and is referred to as a frequency response method in the process control field. By contrast, the response of nonlinear systems to forcing is much more diverse and not yet fully understood. In nonlinear systems, the forced response can be periodic with a period that is some integer multiple of the forcing period (a subharmonic response), or quasi-periodic (characterized by more than one frequency) or even chaotic, when the time series of the response appears to be random. In addition, abrupt transitions or bifurcations can occur between any of these responses as one or more of the parameters is varied and there can be more than one possible response for a given set of parameters depending on the initial conditions or recent history of the system. 

Comments on NLO and Electrooptic Coefficients. Typically, the Pockels effect is observed at relatively low frequencies (up to gigahertz) so that slower nonlinear polarization mechanisms, such as vibrational polarizations, can effectively contribute to the "r" coefficients. The tensor used traditionally by theorists to characterize the second-order nonlinear optical response is xijk Experimentalists use the coefficient dijk to describe second-order NLO effects. Usually the two are simply related by equation 31 (16) ... 

The frequency dependence of SHG at simple metal surface has been the focus of a recent theoretical study of Liebsch [100]. Time-dependent density functional theory was used in these calculations. The results suggest that the perpendicular surface contribution to the second harmonic current is found to be significantly larger than had been assumed previously. He also concludes that for 2 a> close to the threshold for electron emission, the self-consistently screened nonlinear electronic response becomes resonantly enhanced, analogous to local field enhancement in the linear response near the bulk plasma frequency. 

In most cases, the propagation equations discussed in this chapter do not require a specific form of material response. However, for the sake of concreteness, as well as for discussion of numerical methods, we want to describe a generic model of nonlinear material response. We consider a nonmagnetic, dispersive medium with relative permittivity e that is a function of the transverse coordinates x, y and of the angular frequency u>... 

An important feature that affects the numerical solution strategy is that these equations are written in the spectral space, either in the three dimensional space of wave-vectors (/-propagated UPPE) or in a two-dimensional space of transverse wave-vectors plus a one dimensional angular-frequency space (z-propagated UPPE). At the same time, the nonlinear material response must be calculated in the real-space representation. Consequently, a good implementation of Fast Fourier Transform is essential for a UPPE solver. 

Another passive method is the transference function method (TFM) introduced by Muramatsu [6]. The method consists of an oscillator that drives a crystal through a known measuring impedance and a radiofrequency voltmeter which measures the transference modulus of the system. Muramatsu [6] neglected the effect of the parasitic capacitance and his expression for the quartz impedance resulted in a nonlinear relationship between the measured resistance R with the ac voltage divider and the value of R measured by an impedance analyser. Calvo and Etchenique [74] improved the method and introduced an analytical expression to fit the entire transfer function around resonance in order to obtain the same values of R, L and C as measured by a frequency response analyser. 

The three methods taken together allow time-resolved spectroscopic characterization of propagating modes with far greater ease and completeness than has been possible in the past. They facilitate a number of important possibilities including spatiotemporal control over propagating modes, excitation and characterization of nonlinear lattice responses, tunable terahertz-frequency spectroscopy, and terahertz frequency and bandwidth signal processing. 

The effect of the kiex2 term on the motion of Xe is written as Xeee, for example, and physically represents optical second harmonic generation (SHG), to which the polar phonons do not contribute because they are not able to drive or follow high-frequency responses. This nonlinearity... 

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