Phase corrector

In order to compensate for the distortions in the wavefront due to the atmosphere we must introduce a phase correction device into the optical beam. These phase correction devices operate by producing an optical path difference in the beam by varying either the refractive index of the phase corrector (refractive devices) or by introducing a variable geometrical path difference (reflective devices, i.e. deformable mirrors). Almost all AO systems use deformable mirrors, although there has been considerable research about liquid crystal devices in which the refractive index is electrically controlled. 

The all-pass filter is also called a delay equalizer or phase corrector that passes aU frequencies of the signal without attenuation but introduces a phase shift. With Rp = Ri,... 

Figure 1 outlines the basic AO system. Wavefronts incoming from the telescope are shown to be corrugated implying that they have phase errors. Part of the light is extracted to a wavefront phase sensor (usually referred to as a wavefront sensor, WFS). The wavefront phase is estimated and a wavefront corrector is used to cancel the phase errors by introducing compensating optical paths. The most common wavefront compensator is a deformable mirror. The idea of adaptive optics was first published by Babcock (1953) and shortly after by Linnik (1957). 

This system of equations was solved by a predictor-corrector method for several values of a, / , and y using a digital computer. It was not possible to examine values of ft above 50 (a = 0.001, y = 0) as the method broke down because of accumulated errors. Up to these values, although a step is formed in the extent of reaction vs. time curve, the rate of acceleration in the third phase of the reaction was much slower than observed in the experiments. 

T. E. Simos, Predictor-corrector phase-fitted methods for y = i(x,y) and an applica-... 

As mentioned above, the goal of MD is to compute the phase-space trajectories of a set of molecules. We shall just say a few words about numerical technicalities in MD simulations. One of the standard forms to solve these ordinary differential equations i.s by means of a finite difference approach and one typically uses a predictor-corrector algorithm of fourth order. The time step for integration must be below the vibrational frequency of the atoms, and therefore it is typically of the order of femtoseconds (fs). Consequently the simulation times achieved with MD are of the order of nanoseconds (ns). Processes related to collisions in solids are only of the order of a few picoseconds, and therefore ideal to be studied using this technique. 

One should be careful with this procedure, as in principle it renders a Monte Carlo simulation a non-Markov process. The effect is likely to be benign, but the safest way to proceed is to take the corrector updates of the pressure only during the equilibration phase of the simulation (i.e., those cycles normally granted to allow the system to equilibrate to the new state conditions). In our experience the corrector iteration usually converges very quickly, well before the end of the equilibration period. As a check one can... 

No modifications to the predictor-corrector formulas are needed to conduct integrations of coexistence involving more than two phases. In such instances all dependent variables are updated simultaneously by applying the corrector formula to each of the governing differential equations the simulation then continues with the new values in the manner outlined above. 

A New Phase Fitted Method. - Consider the one free parameter symmetric three-step hybrid predictor-corrector explicit method ... 

Continuum correction often introduces overcorrection errors for particular combinations of matrix and analyte. Dozens of these errors have been summarized in the literature by Slavin and Carnrick (1988). Fig. 9 shows the problem that arises in tissue samples (fish in this case) when Se is determined. There is a large negative signal caused by phosphate absorption bands in the gaseous phase when a continuum corrector is used. The upper line is background. There is no problem when Zeeman correction is used. 

The well-established elastic-predictor/plastic-corrector return mapping algorithm can be utilized to obtain the inelastic responses of the microscale amorphous and crystalline phases. Here, we only outline the steps to be used. A detailed description of this solution algorithm can be foimd in References [103] to [105]. The return mapping technique is capable of handling both associative and nonassociative flow rules with variant tangent stiffnesses and results in a consistent solution approach [105]. It is noted that this algorithm is applicable to the material, intermediate, or spatial formulations. 

So-called predictor-corrector methods have been used in molecular dynamics from its inception. In this method, molecular forces at time t are used to predict a trial phase point f +i. This prediction requires information about the state of the system at times earlier than f , so that the method is not self-starting. The forces at the phase point f i are then calculated and are used to refine the prediction and generate the actual phase point The method can be formulated to high order (fourth- and fifth-order procedures have been used) and is therefore very accurate. Two evaluations of the forces are required per step. The method is relatively fast given its high accuracy. However, the storage requirements are quite large, so that the method can be inconvenient in some applications. 

FIGURE 19 (a) The corrector of Fig. 18 incorporated in a transmission electron microscope, (b) The phase contrast transfer function of the corrected microscope. Dashed line no correction. Full line corrector switched on, energy width (a measure of the temporal coherence) 0.7 eV. Dotted line energy width 0.2 eV. Chromatic aberration remains a problem, and the full benefit of the corrector is obtained only if the energy width is very narrow. [From Haider, M., et al. (1998). J. Electron Microsc. 47,395. Copyright Japanese Society of Electron Microscopy.]... 

T. E. Simos, Predictor Corrector Phase-Fitted Methods for Y" = F(X,Y) and an Application to the Schrodinger-Equation, International Journal of Quantum Chemistry, 1995, 53(5), 473 3. 

Figure 6 Two algorithms to conduct Gibbs-Duhem integrations. The pressure route (upper-left) corresponds to integration of Clapey-ron .s equation and entails iVPr-ensemble simulations of both phases (depicted here by the two boxes). The chemical potential route (upper-right) entails p.VT simulations. In both cases, a trapezoidal predictor-corrector integration. scheme is illustrated here the integration advances from point 0 to point 1...

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