Modulation-phase

To improve the S/N ratio, the modulation signal is processed by amplification with a tuned amplifier using phase-sensitive detection. This means that the detected signal must not only be at the modulation frequency, but must also be in phase with the modulation. Since the amplifier itself can introduce a bit of phase shift, there is a phase control which, in principle, should be adjusted to maximize the signal amplitude. In practice, this control needs to be adjusted only rarely and in most cases the best approach is to leave it alone. 

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The modulated phase disappears for head sizes smaller than 1.14c7. Modulations of this kind have not been observed in experiments so far. 

For a certain choice of g cj)) and f 4>) the above equation can be solved analytically for simple spatially modulated phases, such as lamellar or hexagonal [22]. This is possible for the piecewise parabolic model of /(0) ... 

Figure 8 HRTEM image of the ineommensurately modulated phase observed at room temperature in Ti5oPd43Cr7- The modulations can be described by a sinusoidal wave with wave-vector 0.31 [IlOJbcc (courtesy Schwartz et al. )...

A phase modulation can also be expressed as frequency modulation. The corresponding frequency deviation is the time derivative of the modulated phase angle (Pm t). According to the basic relationships Afrequency deviation Af(f) with respect to the carrier frequency fg, commonly known as the Doppler frequency shift... 

Microwave frequency Center field Modulation frequency Modulation phase... 

Jassen T, Chapuis G, de Boissieu M eds (2007) Aperiodic crystals from modulated phases to quasicrystals. Oxford University Press, New York... 

Modulations are normally described as waves. The modulation wave can fit exactly with the underlying unmodulated component, or more precisely with the unit cell of the underlying component, in which case the structure is described as a commensurately modulated structure. In cases where the dimensions of the modulation are incommensurate (i.e., do not fit) with the unit cell of the underlying structure, the phase is an incommensurately modulated phase. Modulation changes are normally continuous and reversible. 

Figure 4.38 Idealized structure of SrxTiS3 modulated phases (a) an idealized TiS6 octahedron (the shaded faces are shared to form columns) (b) idealized hexagonal unit cell formed by columns of face-shared TiS6 octahedra (c) idealized unit cell formed by columns of face-sharing octahedra and Sr atoms (shaded) and (d) idealized structure of Sr8(TiS3)7.

The real structures of these phases are more complex. The coordination of the Ti atoms is always six, but the coordination polyhedron of sulfur atoms around the metal atoms is in turn modulated by the modulations of the Sr chains. The result of this is that some of the TiS, polyhedra vary between octahedra and a form some way between an octahedron and a trigonal prism. The vast majority of compositions give incommensurately modulated structures with enormous unit cells. As in the case of the other modulated phases, and the many more not mentioned, composition variation is accommodated without recourse to defects. ... 

The universal interferometric response of a balanced two-port interferometer is shown in Fig. 11.2 as a function of the fixed phase offset between the two waves. The maximum slope of the intensity curve occurs when the fixed phase offset between the waves is an odd integer of = re/2. These conditions of maximum slope are called the conditions of phase quadrature. There are two quadrature conditions per cycle, with opposite slopes and hence opposite signed responses to modulated phase. These are the positions of maximum phase-to-intensity transduction and are the operating points for interferometric detection of protein or DNA on spinning discs. 

If the intensity of the exciting light is modulated, the fluorescence light also shows a modulation at the same frequency cj, with a smaller degree of modulation and a phase shift against the modulation phase of the exciting light This phase shift is connected with the mean lifetime r of the excited state (exponential decay anticipated) by tg = CO T. 

Fig. 6. Experimental arrangement for lifetime measurements by the phase-shift method, using laser excitation. The laser beam is amplitude-modulated by a Pockel cell with analysing Nicol prism and a small part of the beam is reflected by a beam splitter B into a water cell, causing Rayleigh scattering. This Rayleigh-scattered light and the fluorescence light from the absorption cell are both focused onto the multiplier cathode PMl, where the difference in their modulation phases is detected. (From Baumgartner, G., Demtroder, W., Stock, M., ref. 122)).

The audio signal resulting from rf amplification and detection is amplified and detected in a phase sensitive manner by using the original modulation phase as a reference. Both mixer vacuum tubes 6S) and mechanical choppers are used. The resulting DC voltage is fed into a recorder. For sufficiently small audio modulation amplitudes the first derivative of the resonance absorption or dispersion results from the narrow-banding technique. 

Using the Euler-Lagrange variational method, one can find an analytical solution for the optimal modulation phase (p t), given a Markovian bath at long times (see Suppl. Info.). This yields... 

Polycatenars form a broken-layer type 2D density modulated phases because of the mismatch between the cross-section areas required for the terminal chains and for the mesogenic core. For such compounds layers become locally bent to provide more space for tails and, since the bending of layers is finite, this leads to... 

In the following sections we shall focus on the structure and properties of the two-dimensional phases formed by the bent-core liquid crystals. In Sect. 2 we describe the structure studies by the X-ray diffraction (XRD) method, optical studies, and the response of different structures to the external electric field. In Sect. 3 we give theoretical models of the director and layer structure in 2D modulated phases and discuss how to reconstruct electron density maps from XRD data. 

So far we have discussed 2D density modulated phases that are formed by deformation or breaking of the layers. However, there are also 2D phases with more subtle electron density modulations. In some cases additional peaks observed in the XRD pattern (Fig. 10) are related to a double layer periodicity in the structure. As double layer periodicity was observed in the bent-core liquid crystals formed by the asymmetric as well as symmetric molecules [22-25] it should be assumed that the mechanism leading to bilayers must be different from that of the pairing of longitudinal dipole moments of molecules from the neighboring layers, which is valid for smectic antiphases made by asymmetric rod-like molecules. 

Bent-core liquid crystals are especially interesting materials for basic research as in these systems the polar and tilt order are decoupled and polarization splay seems to be an inherent property of the system. Both effects lead to a variety of structures with unusual properties, e.g., formation of the 2D density modulated phases built of the smectic layers fragments. We have presented the current knowledge... 

In some cases, chiral resolution can be recognized by helical structure formation. A variety of helical filaments and fibers can be observed in the B7 phase, as shown in Fig. 10. The B7 phase is identified to be a polarization-modulated phase and the internal helical polarization modulation is responsible for the helix [15]. In contrast,... 

These results represent solid evidence for a general mechanism governing the phase behavior at surfaces and in thin films of modulated phases The interplay... 

FIGURE 3.20 (a) NSOM topography, (b) polarization modulation amplitude, and (c) polarization modulation phase images of rhodamine 110 microcrystals. The arrows appended to image c depict the transition dipole orientation in each crystal. 

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