Mean-Field Theory in Java

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4.2.3. Mean-Field Theory
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Mechanisms based on electromagnetic forces are promising since they naturally include the intensity and polarization state of the incident light eld. In the mean eld model developed by Pedersen and coworkers (Pedersen et al., 1998; Pedersen and Johansen, 1997), each chromophore is subject to a potential resulting from all the other chromophore dipoles in the material. Irradiation orients chromophores, and this net orientation leads to a potential that naturally aligns other chromophores. Furthermore, there is an attractive force between side-by-side chromophores that are aligned similarly. This leads to a net force on chromophores in illuminated areas, causing them to order and aggregate. Obviously this model predicts an accumulation of chromophores in the illuminated areas. Thus surface relief peaks will be aligned with light intensity maxima. Although this result does not agree with experiments in amorphous samples, it is consistent with many experiments on LC systems. The mean- eld model inherently includes intermolecular cooperativity and orientational order, and it appears natural that it would be manifest in mobile LC systems. The polarization state of incident light is
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explicitly included in the model, as it serves to align dipoles and thus enhance the mean- eld force. Because of this, even pure polarization patterns lead to gratings in this model. This mechanism appeals to the unique properties of azobenzenes only to explain the photoorientation of dipoles. If this mechanism were general, one would expect it to operate on nonisomerizing dipoles that had been aligned by other means, which has not yet been observed.
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4.2.4. Permittivity Gradient Theory
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A mechanism involving spatial variation of the permittivity, e, has been suggested by Baldus and Zilker (2001). This model assumes that a spatial modulation of the refractive index, hence permittivity, is induced in the lm. This is certainly reasonable, given the well-known photoorientation and birefringence gratings in azo systems. A force is then exerted between the optical electric eld and the gradient in permittivity. Speci cally, the force is proportional to the intensity of the electric eld in the mass transport direction and to the gradient of the permittivity: e0 ~ ~ f E 2 re 2 4:6
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Mass is thus driven out of areas with a strong gradient in e, which generally moves material into the dark (consistent with the phase relationship in amorphous systems). Here again the mechanism appears general: any system with spatial variation of refractive index should be photopatternable, yet this is not observed. This model would appear to require that adequate photoorientation precede mass transport. Most experiments indicate, however, that both orientational and surface relief phenomena begin immediately and continue concurrently throughout inscription. This model was used to explain SRG formation in pulsed experiments (Baldus et al., 2001; Leopold et al., 2000), where thermal effects were suggested as giving rise to the spatial variation of permittivity, but the resulting force was essentially identical. However, conventional laser ablation appears to be a simpler explanation for those results (Yager and Barrett, 2004).
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4.2.5. Gradient Electric Force
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Kumar and coworkers proposed a mechanism on the basis of the observation that an electric eld component in the direction of mass ow was required (Yang et al., 2006; Bian et al., 2000; Viswanathan et al., 1999a; Kumar et al., 1998). This force is essentially an optical gradient force (Chaumet and Nieto-Vesperinas, 2000; Ashkin, 1997, 1970). Spatial variation of light (electric eld intensity and orientation) leads to a variation of the material susceptibility, w, at the sample surface. The electric eld then polarizes the material. The induced polarization is related to the light intensity and local susceptibility: ~ ~ Pi e0 wij Ej 4:7
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Forces then occur between the polarized material and the light eld, analogous to the net force on an electric dipole in an electric eld gradient. The time-averaged force was derived to be (Viswanathan et al., 1999a) as follows: D E ~ ~ ~ f P r E 4:8 The grating inscription is related to the spatially varying material susceptibility, the magnitude of the electric eld, and the gradient of the electric eld. This theory was extended to include near- eld optical gradient forces, which have been used for patterning in some experiments (Ikawa et al., 2001). The gradient force model naturally includes the polarization dependence of the incident light, and reproduces essentially all of the polarization features of single-beam and SRG experiments. It has been pointed out (Natansohn and Rochon, 2002), however, that another analysis (Gordon, 1973) of forces exerted on polarizable media suggested a dependence on the gradient of the electric eld, but not its polarization direction. The gradient force theory requires azobenzene photochemistry to modulate susceptibility via photoorientation and also implicitly assumes that photoplasticization is enabling mass transport. It would appear, however, that the force density predicted by this model is much too small to account for mass transport in real systems. A straightforward analysis presented by Saphiannikova et al. (2004a) is described here. In the case of two circularly polarized beams (for instance), the force acting in the x-direction, according to the gradient electric force model, would be
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