Transverse electric field polarization

We consider the cylindrical nanowire geometry shown in Fig. 17.1, with an incident plane wave normal to the cylinder axis and with an amplitude Eg. This is the simplest case to solve analytically and the one most often treated in experimental spectroscopic investigations of single nanowires. Possible orientations of linearly polarized incident light with respect to the wire axis are bounded by two cases. The first is the transverse magnetic (TM) polarization where the electric field is polarized parallel to the wire axis, and the second is the transverse electric (TE) polarization where the electric field is polarized perpendicularly to the wire axis. In TM polarization, the condition of continuity of the tangential electric field is expected to maximize the internal field, while in TE polarization, the dielectric mismatch should suppress the internal field. The incident plane wave may be expanded in cylindrical functions as ... 

In ionic and partially ionic crystals optic vibrations are associated with strong electric moments and hence can interact directly with the transverse electric field of incident infrared electromagnetic radiation. In terms of the phenomenological theory of infrared dispersion, if , and D are the electric field, polarization and displacement vectors respectively, then... 

Fig. 2.4. A transverse electric field, indicated by the arrow pointed towards the right at the top of the figiu-e, tilts the apolar director as shown by double headed arrows in a specific direction due to the flexoelectric effect on a 90° twisted nematic cell. The tilting direction reverses if the field direction is reversed. The transmitted intensity mesisured with a polarized light beam traversing the cell vertically as indicated by the dashed line will be identical in the two cases. On the other hand, with an oblique beam, the transmitted intensities for the two tilted director structures will be different, and can be used to me siu-e the flexocoefficient (adapted from Kischfai et cU. ).

In an anisotropic medium, the polarization and induced current generally lie in a direction different from that of the electric field. In that case, it is possible, e.g., to induce a longitudinal ciurent with a purely transverse electric field. This situation can be treated by representing the dielectric function as a tenor [11]. 

The imaginary part of the complex refractive indices for left and right circularly polarized light relates to circular dichroism, that is differential absorption for light of different circular polarizations. It is treated in a similar manner to linear dichroism, except that the definition of principal components follows a different convention. For linear birefringence and dichroism the principal values of the complex refractive index relate to the electric field polarization direction which is transverse to the propagation... 

The higher-order modes of waveguides with noncircular cross-sections are constructed from each pair of solutions Pj (x, y) and Pg (x, y) of Eq. (13-8) and their corresponding scalar propagation constants and p. The transverse electric fields of these modes are polarized along the same optical axes as the fundamental modes of Section 13-5. There are two pairs of higher-order modes. Each pair has fields given by Eq. (13-10), with p and P(X) y) replaced by Pg and Pj(x,y) for one pair, and by p and Pj,(x,y) for the other pair. The polarization corrections Sp, Sp, SPy and SPy are obtained from Eq. (13-11) with the appropriate field substituted for e,. 

In general the spatial variation of the modal fields of the slightly asymmetric fiber is similar to that of the modal fields of the circular fiber, but the fields are polarized along axes which we can take to be parallel to the optical axes of the circular fiber. When the symmetry of the circular cross-section is broken, the transverse electric fields are no longer given exactly by the symmetrical and... 

The two fundamental, or HEj j, modes and all other pairs of HEi modes were discussed in Section 13-4. Each mode of a particular pair has a transverse electric field whose direction, or polarization, is parallel to one of an arbitrary pair of orthogonal directions in the fiber cross-section [1], Thus, these modes are uniformly polarized. For convenience we take one mode to be x-polarized and the other y-polarized in Fig. 14-1. There is only one solution of the scalar wave equation for these modes, corresponding to 1 = 0 in Eq. (14-4). The transverse fields, given by Eq. (13-9) and repeated in Table 14-1, ignore all polarization properties of the fiber. For future reference, we give the transformation of the components of these fields from cartesian to polar... 

The transverse fields of the fundamental modes in Table 14-1 contain no polarization effects due to the fiber. These effects are included in higher-order corrections through the expansions in Section 32-1. The second-order transverse electric field corrections satisfy the equation in Table 32-1, page 627. If we set f = R, substitute e or e,3 from Table 14-1 fore, and take = 51//A from Table 14-2, it is readily verified that the... 

The finite propagation constant corrections SPi discussed above are responsible for interference effects between pairs of modes with the same scalar propagation constant. For example, suppose the odd HE21 and TEqi modes are excited with equal power and all other modes have zero power. If we erroneously ignore all polarization effects, then bPj = 5p = 0, and the total transverse electric field of the fiber follows from Table 14-1 as... 

It follows from the above discussion that propagation of the two fundamental modes on the eUiptical fiber is similar to plane-wave propagation in an anisotropic medium discussed in Section (11 -23) [1 ]. In particular, the total transverse electric field E, of the two modes is elliptically polarized and its direction changes with distance z along the fiber. On the other hand there is no... 

If the unperturbed fiber is noncircular, the transverse electric field of each mode is polarized parallel to one of the optical axes, Xq or as discussed in Sections 13-5 and 13-8. The directions of the optical axes on the perturbed fiber, which determine the polarization of the perturbed mode fields, are found either by inspection or by the formal methods of Section 32-5. 

The polarization of the transverse electric field as the fiber changes from elliptical to circular cross-section is shown qualitatively in Fig. 18-3 (a) for the four / = 1 modes. Here we examine the transition region between the two extremes, as discussed in Section 13-9. The transition is described quantitatively by the parameter A of Table 13-1, page 288. Substituting from Eq. (18-28) and Table 14-6, page 319, the absolute value of A is given by [9]... 

The four fundamental modes of the two identical fibers are composed of pairs of symmetric and antisymmetric modes corresponding, respectively, to T+ and T of Eq. (18-33). By symmetry the transverse electric fields are polarized parallel to either the x- or y-axes, as shown in Fig. 18-5. If subscripts x,y denote polarization and +, — denote symmetry, then... 

We assume that only x-polarized modes are involved. Hence the magnitude E of the total transverse electric field is given everywhere by... 

The scalar propagation constants P+ and for the fundamental modes of the composite waveguide are given by Eq. (18-35) in terms of the fundamental mode propagation constant for either fiber in isolation and C of Eq. (18-42). We explained in Section 13-5 that polarization corrections are required to correctly distinguish between the propagation constants of each pair of fundamental modes associated with P+ or P-. To determine each correction, we substitute the approximate transverse electric field of Eq. (18-36) into Eq. (13-12), where I now denotes the interface of both fibers. Thus, in the notation of Section 18-12, and with the help of Eqs. (18-36) and (18-33), we obtain 5 by setting... 

Fig. 18-8 Orientation of the transverse electric field for (a) the x-polarized and (b) the y-polarized fundmental modes on an arbitrary step-profile fiber with an isolated, small nonuniformity.

Consider a single-mode, elliptical fiber whose refractive-index profile rotates along its length, as shown in Fig. 19-2. We recall from Section 13-5 that in the weak-guidance approximation one fundamental mode of the cylindrically symmetric, elliptical fiber is plane polarized with its transverse electric field parallel to the x-axis in Fig. 19-2(a) and has propagation constant The other fundamental mode s field is parallel to the y-axis... 

We are primarily interested in radiation from the fundamental modes of bent, single-mode fibers. Within the weak-guidance approximation, the power radiated is insensitive to polarization, since p. Thus we can conveniently assume that the transverse electric field is parallel to the Z-axis in Fig. 23-2(a), i.e. orthogonal to the plane of the bend. Close to and within the core, the magnitude of the electric field on the bend is given by aj Fo (R) exp (ifiz), using the local-mode approxinution, where is the modal amplitude, Fq (R) is the... 

In Section 13-5 we used physical arguments to show that the transverse electric field must be polarized along the optical axes of the weakly guiding waveguide. If and yo are unit vectors parallel to the optical axes, then the two polarizations are expressible as... 

The dielectric constant e(w) of a crystal is found by considering the equations of motion in a transverse electric field E = Eq exp[i(1 - wt)] applied from outside the crystal and for which t 0. For frequencies in the infrared and far-infrared regions, such fields can excite TO-modes at q 0 [1.35]. The equation of motion and the polarization are given by (4.82,87) but the effective field now includes the external field as well, that is. 

As witli tlie nematic phase, a chiral version of tlie smectic C phase has been observed and is denoted SniC. In tliis phase, tlie director rotates around tlie cone generated by tlie tilt angle [9,32]. This phase is helielectric, i.e. tlie spontaneous polarization induced by dipolar ordering (transverse to tlie molecular long axis) rotates around a helix. However, if tlie helix is unwound by external forces such as surface interactions, or electric fields or by compensating tlie pitch in a mixture, so tliat it becomes infinite, tlie phase becomes ferroelectric. This is tlie basis of ferroelectric liquid crystal displays (section C2.2.4.4). If tliere is an alternation in polarization direction between layers tlie phase can be ferrielectric or antiferroelectric. A smectic A phase foniied by chiral molecules is sometimes denoted SiiiA, altliough, due to the untilted symmetry of tlie phase, it is not itself chiral. This notation is strictly incorrect because tlie asterisk should be used to indicate the chirality of tlie phase and not tliat of tlie constituent molecules. 

The last attribute of tire electromagnetic field we need to discuss is wave polarization. The nature of tire transverse field is such tliat tire oscillating field disturbance (which is perjDendicular to tire propagation direction) has a particular orientation in space. The polarization of light is detennined by tire time evolution of tire direction of tire electric field... 

Vibration Diagram Method. In actuality the last cases above are not described accurately by this dipole array model because actual phases of the electric fields are significantly altered from those of linear waves. (A more realistic, but complex model is to consider amplitude and phase characteristics of the oscillating vertically polarized component of electric field resulting from rotation of a line of transverse dipoles of equal magnitude but rotated relative to each other along the line such that their vertical components at some reference time are depicted by Figure 2.) For this reason and to handle details of focused laser beams one must resort to a more mathematically based description. Fortunately, numerical... 

---