Pattern formation, optical

Most reports over the past 4 years have dealt with the manipulation of display-related parameters such as electro-optic response and alignment, but increasingly also with thermal effects, pattern formation, nanoparticle-liquid crystal compatibility (i.e., enhancing the stability of dispersions), and to some degree with nanoparticle organization. 

Preliminary investigations of the liquid crystal phase behavior of these gold nanoparticles initially revealed an enantiotropic nematic phase (based on polarized light optical microscopy and thermal analysis) as well as some pattern formation of the gold nanoparticles in TEM experiments [540, 541],... 

Finally, our group reported on gold nanoparticles decorated with bent-core liquid crystals showing pattern formation on TEM grids after slow solvent evaporation (18 in Fig. 22). These particles showed interesting self-assembly effects in different bent-core liquid crystal hosts (SmCPA and Colr) and slightly improved electro-optic effects such as shorter response time, x, and unaltered spontaneous polarization in the SmCPA host, but no mesophase formation [547]. 

Laser Cooling of Sohds, Carl E. Mungan and Timothy R. Gosnell Optical Pattern Formation, L. A. Lugiato, M. Brambilla, and A. Gatti... 

Figure 5, Measured particle traces superimposed onto the optically detected concentration pattern. Arrows mark the direction of particle movement during the early phase of the pattern formation (A), and during the transition to honeycomb patterns (B). Field of view 1.7 x 1,6 nmi. (Adapted with permission from reference 13. Copyright 2002 Oldenbourg Wissenschaftsverlag.)...

In both the cases considered, an optical contrast of the patterns observed in isotropic liquids is very small. Certainly, the anisotropy of Uquid crystals brings new features in. For instance, the anisotropy of (helectric or diamagnetic susceptibility causes the Fredericks transition in nematics and wave like instabilities in cholesterics (see next Section), and the flexoelectric polarizaticm results in the field-controllable domain patterns. In turn, the anisotropy of electric conductivity is responsible for instability in the form of rolls to be discussed below. All these instabilities are not observed in the isotropic liquids and have an electric field threshold controlled by the corresponding parameters of anisotropy. In addition, due to the optical anisotropy, the contrast of the patterns that are driven by isotropic mechanisms , i.e. only indirectly dependent on anisotropy parameters, increases dramatically. Thanks to this, one can easily study specific features and mechanisms of different instability modes, both isotropic and anisotropic. The characteristic pattern formation is a special branch of physics dealing with a nonlinear response of dissipative media to external fields, and liquid crystals are suitable model objects for investigation of the relevant phenomena [39]. 

The order parameter was arguably first introduced by Landau at the equilibrium thermodynamic level to study phase behavior. Order parameter models can also be motivated through classical bifurcation theory, and several physical systems have been recently modeled in this way. They include Rayleigh-Benard convection, Faraday waves,or pattern formation in optical systems. Close to a bifurcation point, these models asymptotically describe the system under study, but they are now routinely used, in a phenomenological feshion, to describe highly nonlinear phenomena. 

T. Ackeman and W. Lange, Optical pattern formation in alkali metal vapors Mechanisms, phenomena and use, Appl. Phys. B, 72, 1, 2000. 

The examples shown in this chapter are only a small part of the rich variety of behavior encountered in far-from-equilibrium chemical systems. Here our objective is only to show a few examples an extensive description would form a book in itself At the end of the chapter there is a list of monographs and conference proceedings that give a detailed descriptions of oscillations, propagating waves, Turing structures, pattern formation on catalytic surfaces, multistability and chaos (both temporal and spatiotemporal). Dissipative structures have also been found in other fields such as hydrodynamics and optics. 

Muller, A.-D. Muller, R Hietschold, M., Electrochemical pattern formation in a scanning near-field optical microscope. Applied Physics A Materials Science Processing 1998, 66, S453-S456. 

Figure 14.5 Stripe pattern formation by dip coating, (a-d) Schematic illustration of the formation of an aligned nanoparticles-stripe pattern by vertical deposition. During raising the substrate slowly (a, b) the water evaporates and the wet contact line breaks up into aggregates of nanoparticles (b, c). (e) Optical microscopy image of the waterfront, leading to ID stripes across the entire substrate, formed both for silver (f) and gold (g) nanoparticles. (Images taken from Ref 46.) Abbreviation-. ID, onedimensional.

Order parameter equations for spatio-temporal patterns developing in chemical reactions are treated in a recent book by Kuramoto [5], In this paper I should rather like to present two other examples, namely fluid dynamics and flames. Just to illustrate how such equations in extended media may look, I present a describing pattern formation in the convection instability. The order parameter (x,t) can be essentially considered as the deviation of the temperature field from a constant gradient. The corresponding patterns can be directly measured optically. The order parameter equation reads [6]... 

Optical pattern formation in liquid crystals is of considerable interest here, typically far-field patterns appear as result of optical field induced modulation of the director in directions transverse to the direction of propagation. Patterns have been observed in resonant cavities [47], systems with feedback mirrors [48], and in systems without feedback [49]. The propagation of a laser beam in a capillary filled with a nematic has been studied, showing self-focusing, undulation and filamentation [50]. Photographs, showing the beam profile as a function of intensity in a 1.5 mm capillary filled with the nematic mixture E209 are shown in Fig. 2. 

Such beam propagation phenomena have also been studied in the context of (self-started) pattern formations. ". Due to the combined effect of feedback and spatial phase modulation a single incident beam could break up into an array of beams and exhibit maity dynamical patterns and bistabihties/instabilities as various LC and optical parameters are varied. In Section 12.2, we will also discuss a special case where a linearly polarized incident beam self-converts dynamically into an orthogonally polarized beam and exhibits various oscillatory and chaotic behaviors without feedback. 

In this section, we shall consider rheo-optics in a broad sense so that the methods are aimed at elucidating the following two main themes (1) the stmctural or molecular origin of rheological properties, and (2) the so-called dissipative stmc-tures, which are defined as the stmctures developed under external fields, as open nonequilibrium phenomena in statistical mechanics. Theme (2) is generally important for understanding the pattern formation in nature exploration of the dissipative stmctures in the mesoscopic scale especially seems to be quite challenging. 

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