Optical ellipsoid

In the helical structure, the optical ellipsoid of the smectic C phase rotates together with the tilt plane. Like in cholesterics, we can imagine that helical turns form a stuck of equidistant quasi-layers that results in optical Bragg reflections in the visible range. Therefore, like cholesterics, smectic C liquid crystals are onedimensional photonic crystals. However, in the case of SmC, the distance between the reflecting layers is equal to the full pitch Pq and not to the half-pitch as in cholesterics, because at each half-pitch the molecules in the SmC are tilted in opposite directions. Hence, we have a situation physically different from that in cholesterics. 

FIGURE 4.34. An initially homeotropic nematic layer with oblique light incidence at an angle ie. The electric vector of the TM light wave is in the plane of incidence. 9 z) characterizes the spatial distribution of the optical ellipsoid. I = R + T, where R is reflectivity and T is transmittance, provided that absorption is absent. 

Fig. 4.6 Conceptual image of optical compensation, (a) Optical ellipsoid of rugby ball model, (b) Optical ellipsoid of disc-like model, (c) Combination of (a) and (b) [30]...

To make evaluations more definite, we use optical and microwave experimental data, as well as calculations of molecular dynamics of certain simple liquids which usually fit the experiment. Rotation is everywhere considered as classical, and the objects are two-atomic and spherical molecules, as well as hard ellipsoids. 

Figure 4. Segment at the end of the semi-minor axis, b, of an ellipsoid (left), and use of a test plate and auxiliary convex mirror to shorten the optical path of the test (right).

If one of the stars in the binary is not a neutron star, then the tests become less precise. Suppose that one observes the optical light from the companion to a neutron star. In addition to the spectral information that allows measurement of P and i i, one also has photometric information (e.g., the total optical flux from the companion). The companion is distorted into a pear shape by the gravity of the neutron star, with the point towards the neutron star. Therefore, from the side there is more projected area and hence greater flux than from either end. If the orbit is edge-on (i = 90°) then the flux varies maximally if the orbit is face-on (i = 0°) then there is no variation. Therefore, by modeling the system one can estimate the inclination from the flux variations. This is called the method of ellipsoidal light curves (Avni Bahcall 1975). 

However, one must be careful because in an LMXB the optical emission from the accretion disk (whether in the outer, cool regions or as reprocessed X-ray emission) can outshine the companion by a large factor. This makes spectral lines difficult to measure and also complicates the ellipsoidal light curve technique. The ideal systems to study are therefore transient systems, which undergo periods of active mass transfer (often for a few weeks to a few months) before lapsing into quiescence, where there is little to no mass transfer. During quiescence, the companion is still distorted by the gravity of the neutron star, hence the flux variations still occur, but without any contamination by the accretion disk. There is a relatively new approach similar to this that... 

Theory. In the general case where rigid revolution ellipsoidal particles in solution possess both a permanent and an induced dipolar moment colinear with the particle optical axis, the theory derived by Tinoco predicts the following behaviour of the solution birefringence An(t) in the limit of weak electric field (6). 

Crystals, except those belonging to the cubic system, are anisotropic in this respect the force of repulsion varies with the orientation of the crystal with respect to the direction of the field. The graph representing vectorialiy the diamagnetic susceptibility in all directions in a crystal is an ellipsoid, whose orientation with respect to the unit cell is restricted by symmetry in exactly the same way as that of the optical indicatrix. Thus, for uniaxial crystals the magnetic ellipsoid an ellipsoid of revolution whose unique axis coincides with the threefold, fourfold, or sixfold axis of the crystal for orthorhombic crystals the ellipsoid has three unequal axes which necessarily coincide with the three axes of the crystal for monoclinie crystals the only restriction is that one of the principal axes of the magnetic ellipsoid must coincide with the b axis of the crystal while for triclinic crystals the orientation of the ellipsoid is not restricted in any way. 

The displacements of the framework atoms in the H structure of analcime from the symmetrized positions are listed in Table III. They are considerably larger than the displacements of 0.02-0.07 A found in synthetic zeolite NaA (3) which are the smallest displacements recorded so far in a pseudosymmetric structure. The displacement vectors in analcime can be related to the apparent temperature parameters of the A structure. Figure 3 shows the experimentally determined vibration ellipsoids of optically isotropic analcime (7). The strongly anisotropic ellipsoids are in reasonably good accord with the displacement vectors obtained independently by DLS. 

If we define resolution as the shortest distance corresponding to the formation of one interference fringe, Fig. 5 represents the resolution limit of any possible optical system using the corresponding geometric configuration. The diffraction limited resolution and the interference limited resolution are represented by the separation of the hyperboloids and the ellipsoids, respectively. 

If all three principal values are positive, the quadric surface is an ellipsoid with semiaxes a, = TfiVz, but if one or two of the principal values are negative the quadric surface is a hyperboloid. For example, the (relative) impermeability tensor 3 is defined by nfn, where k is the permittivity and n0 is the permittivity of free space. As for any symmetric 7(2) the components of 3 define the representation quadric I3ijxixj= 1, which here is called the indicatrix or optical index ellipsoid. Referred to principal axes the indicatrix is... 

Figure 12. Optical arrangement by Fuller and Griffiths (25) for diffuse reflectance spectroscopy P, paraboloidal mirror E, ellipsoidal mirror S, sample D, detector...

For 5 = 0 (limit point at infinity), one obtains the corresponding Coulomb modifications, see Refs. [36,42] for the complications at origin. Note also the strong dependence on the incident directions for "ellipsoidal" potentials yet the optical theorem holds, see Ref. [39]. 

The surfaces seen on these photographs are known as ray velocity surfaces. For all uniaxial anisotropic crystals, we have a double surface when the crystal has a positive optic sign (like quartz), the ellipsoid of revolution is enclosed by the sphere, but if the crystal is optically negative (like calcite), the ellipsoid encloses the sphere. The assumption that the form of the variation of velocity for the extraordinary disturbance in uniaxial crystals is ellipsoidal was... 

For uniaxial crystals, the optical indicatrix is a single-surfaced ellipsoid of revolution similar in shape to the extraordinary ray velocity surface. To construct the optical indicatrix for a particular example, say calcite, we construct the ellipsoid of revolution so that the radius... 

---