Spatial domain

Quadrupole mass spectrometers (mass filters) allow ions at each m/z value to pass through sequentially for example, ions at m/z 100, 101, 102 will pass one after the other through the quadrupole assembly so that first m/z 100 is transmitted, then m/z 101, then m/z 102 (or vice versa), and so on. Therefore, the ion collector (or detector) at the end of the quadrupole unit needs to cover only one point or focus in space (Figure 29.1a), and a complete mass spectrum is recorded over a period of time. The ions arrive at the collector sequentially, and ions are detected in a time domain, not in a spatial domain. 

It turns out that, in the CML, the local temporal period-doubling yields spatial domain structures consisting of phase coherent sites. By domains, we mean physical regions of the lattice in which the sites are correlated both spatially and temporally. This correlation may consist either of an exact translation symmetry in which the values of all sites are equal or possibly some combined period-2 space and time symmetry. These coherent domains are separated by domain walls, or kinks, that are produced at sites whose initial amplitudes are close to unstable fixed points of = a, for some period-rr. Generally speaking, as the period of the local map... 

The analogue between the Young- interferometer (interference pattern in the spatial domain) and the serrodyne modulated MZI (interference pattern in the time domain) is striking. An optimized Young interferometer17 can also show resolutions down to <3n 10 8. 

The fluidity of lipid bilayers permits dynamic interactions among membrane proteins. For example, the interactions of a neurotransmitter or hormone with its receptor can dissociate a transducer protein, which in turn will diffuse to interact with other effector proteins (Ch. 19). A given effector protein, such as adenylyl cyclase, may respond differently to different receptors because of mediation by different transducers. These dynamic interactions require rapid protein diffusion within the plane of the membrane bilayer. Receptor occupation can initiate extensive redistribution of membrane proteins, as exemplified by the clustering of membrane antigens consequent to binding bivalent antibodies [8]. In contrast to these examples of lateral mobility, the surface distribution of integral membrane proteins can be fixed by interactions with other proteins. Membranes may also be partitioned into local spatial domains consisting of networks... 

The raw data of the thermocouples consist of the temperature as a function of time (Fig. 8.9, left). In the raw data, the passing of the conversion front can be observed by a rapid increase in temperature. Because the distance between the thermocouples is known, the velocity of the conversion front can be determined. The front velocity can be used to transform the time domain in Fig. 8.9 (left) to the spatial domain. The resulting spatial flame profiles can be compared with the spatial profiles resulting from the model. The solid mass flux can also be plotted as a function of gas mass flow rate. The trend of this curve is similar to the model results (Fig. 8.9, right). 

This dependence has a character close to linear in the considered range of initial powers. By this way it is demonstrated that transmittance of the strueture is power dependent, and moreover, the transmittance-versus-power relation is different depending on the spatial domain over which the light beam intensity is integrated. 

Figure 22. Normalized pulse duration calculated by intensity integration within the spatial domain (—X , Xn) waveguide cross-section, depending on the dimensions of the...

It is important to realize that dispersion compensation can eliminate the high-order phase distortions (in the spectral domain) introduced by the objective lens, as discnssed above, but it cannot eliminate the scattering (in the spatial domain) that occnrs in depth imaging. Here we explore the nse of laser pulses that are dispersion compensated only before the medinm. In principle, it is possible to compensate for dispersion at greater depths, bnt if the dispersion of tissues is similar to that of pure water, it should be insignificant. Finally, we could titrate the amount of laser power nsed, increasing the intensity as the focal plane moves deeper into the tissue. 

In this chapter we explore several aspects of interferometric nonlinear microscopy. Our discussion is limited to methods that employ narrowband laser excitation i.e., interferences in the spectral domain are beyond the scope of this chapter. Phase-controlled spectral interferometry has been used extensively in broadband CARS microspectroscopy (Cui et al. 2006 Dudovich et al. 2002 Kee et al. 2006 Lim et al. 2005 Marks and Boppart 2004 Oron et al. 2003 Vacano et al. 2006), in addition to several applications in SHG (Tang et al. 2006) and two-photon excited fluorescence microscopy (Ando et al. 2002 Chuntonov et al. 2008 Dudovich et al. 2001 Tang et al. 2006). Here, we focus on interferences in the temporal and spatial domains for the purpose of generating new contrast mechanisms in the nonlinear imaging microscope. Special emphasis is given to the CARS technique, because it is sensitive to the phase response of the sample caused by the presence of spectroscopic resonances. 

It performs information embedding in the spatial domain and hence is sensitivity to geometric deformation such as image miss-alignment and to the presence of stains or scribbles on the printed media. 

The coefficients of the sines and cosines will be real for real data. Restoring a high-frequency band of c (unique complex) discrete spectral components to a low-frequency band of b (unique complex) spectral components will be the same (when transformed) as forming the discrete Fourier series from the high-frequency band and adding this function to the series formed from the low-frequency band. When applying the constraints in the spatial domain, the Fourier series representation will be used. 

The transient heat equation (Eq. 3.285) often serves as the model for parabolic equations. Here the solution depends on initial conditions, meaning a complete description of T(0, x) for the entire spatial domain at t =0. Furthermore the solution T(t,x) at any spatial position x and time t depends on boundary conditions up to the time t. The shading in Fig. 3.14 indicates the domain of influence for the solution at a point (indicated by the dot). 

The spatial domain is divided into discrete volumes defined by a mesh. The values of the independent variable, f are given at the mesh points, or nodes, by fj. The value of the dependent variable w in the volume surrounding the node is presumed to be represented by the value at the node, wj. The volume surrounding each node extends midway to the neighboring node that is, the radial extent of the volume extends from fj-1/2 to 77+1/2, where fj+i/2 = fj + /7+1). 

In summary, one may stress that the two-time-scale description on which the Kramers approach is based (see previously) clearly appears here in the time and spatial domains. During the first stage, the system relaxes rapidly and nonexponentially on a time scale rqs t and behaves as if there is no external force. On the longer time scale t", the system is characterized by the well-defined spatial equilibrium distribution, developed equilibrium values for the dynamical variables, and relaxes exponentially. 

The foundation for the development of these techniques is built on investigations into photon migration processes [2, 9]. Subsequent, detailed examination by Everall et al. [10, 11] demonstrated that the inelastically scattered (Raman) component decays substantially more slowly than its elastically scattered counterpart (i.e. the laser light) due to the regeneration of the Raman signal from the laser component. Discrimination between diffusely scattered photons and the ballistic and snake components is achieved by gating the detector in the temporal or spatial domain. 

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