Extraordinary wave

Figure 9. Top An isotropic atom/molecule and crystal with isotropic polarizabilities will give rise to a spherical wave surface and index ellipsoid. Middle Another isotropic atom/molecule and crystal with larger isotropic polarizabilities will give rise to a smaller spherical wave surface and a larger spherical index ellipsoid. Bottom An anisotropic atom/molecule and crystal with anisotropic polarizabilities will give rise to an ordinary spherical wave surface and an ellipsoidal extraordinary wave surface. The index ellipsoid will have major and minor axes.

The wave fronts transmitted within the crystal are the envelopes of all the surfaces representing the secondary wavelets thus the +1 wave fronts in the crystal are given by the common tangents to extreme secondary wavelets. We see from the figure that there are two parallel wave fronts travelling in the crystal represented by tu and lm for the ordinary and extraordinary waves respectively, and that the wave normal direction is common both to them and the incident waves. It is also clear that the two parallel wave fronts travel with different speeds for they are at different positions within the crystal the extraordinary wave fronts advance faster than the ordinary wave fronts (rn > rt). In order to locate the images of the dot formed by the two waves, we must now consider the direction of advance of a given point on the front physically this is what is meant by the ray directions within the crystal. 

It is apparent from this that the vibration direction for the transmitted extraordinary waves is not generally perpendicular to the ray direction. 

For optically uniaxial crystals we know that the refractive index values for extraordinary waves are variable, with that for ordinary waves fixed. We can link this observation with that concerning the vibration directions for the two waves travelling along a general wave normal direction the ordinary vibration direction is always perpendicular to the optic axis, while the extraordinary vibration is always in the plane containing the optic axis and wave normal direction. This suggests that we may connect the variation of the refractive index in the crystal with the vibration direction of the light. This concept allows a convenient representation of anisotropic optical properties in the form of a spatial plot of the variation of refractive index as a function of vibration direction. Such a surface is known as the optical indicatrix. 

The second case refers to generating circularly polarized light by means of birefringent retarders. If the azimuth a of the electric vector is set at 45° and 6 at kKll (vdAn = kl4) respectively, the emerging radiation turns out to be circularly polarized. Such an optical element is often called quarter wave plate since 6 = kitl2 means a quarter wave displacement between the ordinary and extraordinary wave. The necessity of setting the azimuth at -i-45° arises from the requirement for circular polarization of an electromagnetic wave, its electric vector needs two equal components E and on exit of the retarder. 

Analogously, for extraordinary waves with vector E lying in the same plane as the wavevector k and the optical axis of the superlattice, the dispersion relation is... 

The reflection spectra of grooved Si sample with the lattice constant of 6 pm are shown in Fig. 3. These spectra differ significantly for two polarizations, E and Ej., manifesting that e < no. The effective refractive indices for ordinary and extraordinary waves were found from the neighboring extremes at V and V2... 

The electric displacement of the ordinary wave is perpendicular to vand to 63. The electric displacement of the extraordinary wave lies in the plane (r 63). The directions of and are represented by vectors of norm 1,... 

A direct confirmation of the existence of these two branches has been found by Liao, Clark and Pershan from their Brillouin scattering experiments on a monodomain sample of jff-methyl butyl /K(p-methoxy-benzylidene)amino) cinnamate. This compound shows the nematic, smectic A and smectic B phases. Choosing both the incident and the scattered light to be polarized either as ordinary or extraordinary waves, they observed two peaks corresponding to the two modes, the angular dependence of which is in excellent agreement with the theory (fig. 5.3.11). 

A more common way of building an intensity modulator uses two LiNbOj crystals in tandem. The crystals are arranged in a set-up similar to that in an amplitude modulator and are positioned so that the external field is applied parallel to the optical axis and the light beam travels perpendicular to it. However, the crystals are positioned so that the x- and y-axes are arranged at an angle of 45° to each other. The entry beam is accurately polarised vertically so that its electric vector bisects the x-andy-directions (Figure 9.12). This will split into an ordinary and extraordinary ray, each with polarisation normal to the other as it enters the first crystal. The second crystal is oriented so that the ordinary and extraordinary rays in the first crystal, defined by their relative polarisation, swap and become the extraordinary and ordinary rays in the second crystal. In this way, the ordinary and extraordinary beams travel identical optical paths. On emerging, the ordinary and extraordinary waves will have a phase difference due to the imposed values of the external electric field and will be linearly polarised perpendicular to one another. 

Here 6 is the angle to the z-axis (the optic axis) and Index-matching is possible in this configuration with the fundamental propagating as the extraordinary wave (plane-polarised in a... 

Since is not ir/2, Poynting s vector for the extraordinary wave is not collinear with k the wave-vector (D is not parallel to E) and so the fundamental beam gradually walks away from the UV it has generated. This leads to an astigmatic UV beam. The extent of the walk-off Is given by the angle p which can be evaluated from the equation... 

For a wave incident from the cover (region C) as shown in Fig. 1, the ordinary and extraordinary waves correspond to TE- and TM-polarized waves, respectively. Classical (anisotropic) thin-film analysis can be applied to obtain the transmission and reflection coefficients (i.e., diffraction efficiencies) of the propagating zero-order waves. 

When the beam passes through the birefringent plate with thickness d under the angle against the plate normal, a phase difference = (In/X) dnQ—no) s with As = d/cosP develops between the ordinary and the extraordinary waves. Only those wavelengths Xm can reach oscillation threshold... 

This condition can be fulfilled in unaxial birefringent crystals that have two different refractive indices no and n for the ordinary and the extraordinary waves. The ordinary wave is polarized in the x-y-plane perpendicular to the optical axis, while the extraordinary wave has its -vector in a plane defined by the optical axis and the incident beam. While the ordinary index no does not depend on the propagation direction, the extraordinary index n depends on the directions of both E and k. The refractive indices Uo, and their dependence on the propagation direction in uniaxial birefringent crystals can be illustrated by the index ellipsoid defined by the three principal axes of the dielectric tensor. If these axes are aligned with the jc-, y-, z-axes, we obtain with n = the index ellipsoid. 

One distinguishes between type-I and type-II phase-matching depending on which of the three waves with coi, C02, C03 = coi C02 propagates as an ordinary or as an extraordinary wave. Type 1 corresponds to (1 e, 2 e, 3- 0) in positive uniaxial crystals and to (1 o, 2 o, 3 e) in negative uniaxial crystals, whereas type II is characterized by (1 o, 2 e, 3 o) for positive and (1 e, 2 o, 3 e) for negative uniaxial crystals [5.220]. Let us now illustrate these general considerations with some specific examples. 

In favorable cases phase-matching is achieved for 6 = 90°. This has the advantage that both the fundamental and the SH beams travel collinearly through the crystal, whereas for 9 90° the power flow direction of the extraordinary wave differs from the propagation direction k. This results in a decrease of the overlap region between both beams. 

If the pump is an extraordinary wave, collinear phase matching can be achieved for some angle 0 against the optical axis, if rip(0), defined by (5.119), lies between no(cop) and n icop). 

The reorienution of the director of a nematic liquid crystal induced by the field of a light wave is considered. An oblique (with respect to the director) extraordinary wave of low intensity yields the predicted and previously observed giant optical nonlinearity in a nematic liquid crystal. For normal incidence of the light wave on the cuvette with a homeotropic orientation of the nematic liquid crystal, the reorientation appears only at light intensities above a certain threshold, and the process itself is similar to the Fredericks transition. The spatial distribution of the director direction is calculated for intensities above and below threshold. Hysteresis of the Fredericks transition in a light field, which has no analog in the case of static fields, is predicted. 

As already noted, an extraordinary wave propagating at an angle to the director induces strong nonlinear optical effects at very low powers P (Refs. 3-9). When a light wave is normally incident on a cuvette with horneotropic orientation of the NLC, however, reorientation of the director is possible only above some threshold value of the power of the wave. The distortions of... 

We recognize, however, that the phase difference between the ordinary and extraordinary waves at z =7. is... 

The nonthreshold Frederiks effect occurs when the extraordinary wave (e-wave) makes an oblique angle with the director. It was shown theoretically and experimentally in Ref. 8 that this effect can be detected even in extremely weak light fields. 

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