Types of Nonlinear Processes

In nonlinear processes, the emission/luminescence intensity (I) does not increase proportionally with increase in excitation power density. Thus, it is nonlinear in intensity of applied light. 

The effect that the light itself induces as it propagates through the medium determines the different types of nonlinear processes and optical phenomena. These phenomena are usually only observed at very high light intensities and such nonlinearity requires the use of high-power pulsed lasers [20]. 

Up-conversion relies on sequential absorption and luminescence with intermediate steps to generate shorter wavelengths. Hence, the presence of more than one metastable excited state is required the intermediate metastable states act as excitation reservoirs. One typical example is ground-state absorption followed by inter-mediate-state excitation, excited-state absorption, and final-state excitation to give the up-conversion (the intermediate states and final states are real states) [1, 35], There are many types of up-conversion mechanisms such as excited-state absorption, energy transfer up-conversion and cooperative up-conversion. All these up-conversion processes can be differentiated by studying the energy dependence, lifetime decay curve, power dependence, and concentration dependence by experimental measurements [36-39]. 

In contrast, multiphoton absorption requires only one real excited state and the accumulation of the photons is via virtual states which need not correspond to any real electronic or vibrational energy eigenstate. The absorption of the photons is simultaneous, with the extra energy of the excited states corresponding to the sum of the energies of the incident photons. 

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The results of the studies.discussed in Section II,C permit calculations to be made of the time required for the flame to spread to the entire propellant surface. Once this phase of the motor-ignition process has been completed, the time required to fill the combustion chamber and establish the steady-state operating conditions must be computed. This can be done by the formal solution of Eq. (7). Because this equation is a Bernoulli type of nonlinear equation, the formal solution becomes... 

Hereafter, we will call this reaction Unear recycUng. One can also assume a nonlinear type of recycUng process [28] such that different chiral species by their encounter react back to achiral molecules ... 

Nonlinear Oscillations (Limit Cycles). We want to restrict ourselves to nonlinear oscillations of limit cycle type (LC), which means that we are only dealing with selfsustained oscillations. This type of nonlinear oscillations can only occur in nonconservative systems, it is a periodic process, which is produced at the expense of a nonperiodic source of energy within the system. 

We have presented two types of nonlinear IR spectroscopic techniques sensitive to the structure and dynamics of peptides and proteins. While the 2D-IR spectra described in this section have been interpreted in terms of the static structure of the peptide, the first approach (i.e., the stimulated photon echo experiments of test molecules bound to enzymes) is less direct in that it measures the influence of the fluctuating surroundings (i.e., the peptide) on the vibrational frequency of a test molecule, rather than the fluctuations of the peptide backbone itself. Ultimately, one would like to combine both concepts and measure spectral diffusion processes of the amide I band directly. Since it is the geometry of the peptide groups with respect to each other that is responsible for the formation of the amide I excitation band, its spectral diffusion is directly related to structural fluctuations of the peptide backbone itself. A first step to measuring the structural dynamics of the peptide backbone is to measure stimulated photon echoes experiments on the amide I band (51). 

Equations (27) and (28) or alternatively Eq. (31) provide the most general formal expression for any type of 4WM process. They show that the nonlinear response function R(t3,t2,t 1), or its Fourier transform (cum + a>n + (oq,com + tu ,aim), contains the complete microscopic information relevant to the calculation of any 4WM signal. As indicated earlier, the various 4WM techniques differ by the choice of ks and ojs and by the temporal characteristics of the incoming fields E, (t), E2(t), and 3(t). A detailed analysis of the response function and of the nonlinear signal will be made in the following sections for specific models. At this point we shall consider the two limiting cases of ideal time-domain and frequency-domain 4WM. In an ideal time-domain 4WM, the durations of the incoming fields are infinitely short, that is,... 

In this chapter, we developed a general theory of 4WM processes in terms of the nonlinear response function of the nonlinear medium R(t3,t2,t,). The response function is an intrinsic molecular property that contains all the microscopic information relevant to any type of 4WM process. The details of a particular 4WM experiment are contained in the external fields E t), E2(t), E3(t), and in the particular choice of the observable mode ks. The generated signal is calculated by convolving the response function with the external fields and choosing ks [Eqs. (27), (28), and (31)]. It is only at this stage that... 

Linear (or first-order) kinetics refers to the situation where the rate of some process is proportional to the amount or concentration of drug raised to the power of one (the first power, hence the name first-order kinetics). This is equivalent to stating that the rate is equal to the amount or concentration of drug multiplied by a constant (a linear function, hence linear kinetics). All the PK models described in this chapter have assumed linear elimination (metabolism and excretion) kinetics. All distribution processes have been taken to follow linear kinetics or to be instantaneous (completed quickly). Absorption processes have been taken to be instantaneous (completed quickly), follow linear first-order kinetics, or follow zero-order kinetics. Thus out of these processes, only zero-order absorption represents a nonlinear process that is not completed in too short of a time period to matter. This lone example of nonlinear kinetics in the standard PK models represents a special case since nonlinear absorption is relatively easy to handle mathematically. Inclusion of any other type of nonlinear kinetic process in a PK model makes it impossible to write the... 

The most common type of nonlinear kinetics arises when the rate of a process is determined hy Michaelis-Menten kinetics. The concentration relationship for Michaelis-Menten kinetics can he written in the general form. 

In the general case the above considerations predict deviations from the Tafel equation which have sometimes been observed They may arise from either the nonlinear E (different types of electrode processes to the general current equations (110,1V) and (111,IV) is possible by introducing of special models which allow, in particular, an estimation of the role of the dynamical factor ([Pg.297]

The theories of the electronic and ionic currents have some features in common. One may formulate models in which the current is limited by the injection into the film from the contacts of positively or negatively charged carriers, or one may consider an equilibrium state to exist across either or both interfaces. One may postulate space-charge limited currents, trapping, and recombination processes. One of the chief differences between the ionic and the electronic currents is that the average velocity of the ions is approximately exponentially dependent on the field for fields which produce experimentally observable ionic currents, whereas the average velocity of electrons is linearly dependent on the field at low fields with different types of nonlinearity at high fields. 

As demonstrated in this volume, the study of optical effects in liquid crystals is motivated by three main purposes. Firstly, it is used as a tool in the basic research of different liquid crystalline states. Secondly, in mesophases new types of optical nonlinearities occur or new aspects of nonlinear processes become apparent. The study of these effects contributes to the progress of nonlinear optics. Thirdly, liquid crystals are investigated from the point of view of applications in certain nonlinear optical devices. 

TWO DIMENSIONAL SIMULATION MODELS WITH A NON-LINEAR DIFFUSION TERM. The accumulation of the experimental data suggests that the investigated phenomenon may indeed be an authentic non-linear reaction/diffusion coupling process. Furthermore, the low sensitivity to the type of chemical reaction suggests that non-linearities in the transport processes are the dominant factors. Therefore, many of our simulation efforts have been directed towards this type of nonlinearity. We exemplify the approach with one model others will appear elsewhere., ... 

The steady-state photoexcitation density, Nss. is determined by a rate equation that contains generation (G) and recombination (R) rates. In our studies we have frequently dealt with two types of recombination processes. One is linear or monomolecular, and the other is nonlinear or bimolecular. The rate equation for the monomolecular recombination process is... 

Chaotic behavior requires a nonhnearity in the equations of motion. For conservative mechanical systems, of which computing classical trajectories is, for us, the prime example. Section 5.2.2.1, the nonlinearity is due to the anharmonicity of the potential. In chemical kinetics" there are two sources of nonlinearity. One is when the concentrations are not uniform throughout the system so that diffusion must be taken into account. The other is if there is a feedback so that, for example, formation of products influences the reaction rate, see Problem H. As we shall see, this type of nonlinearity occurs naturally in many surface reactions and this is why we chose catalytic processes as an example. In both mechanical and chemical kinetics systems there is one more way to add nonlinear terms and this is by an external perturbation. For surface reactions this additional control can be implemented, for example, by modulating the gas-phase pressures of reactants and/or products."... 

Transfer function models are linear in nature, but chemical processes are known to exhibit nonhnear behavior. One could use the same type of optimization objective as given in Eq. (8-26) to determine parameters in nonlinear first-principle models, such as Eq. (8-3) presented earlier. Also, nonhnear empirical models, such as neural network models, have recently been proposed for process applications. The key to the use of these nonlinear empirical models is naving high-quality process data, which allows the important nonhnearities to be identified. 

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