Circular wave representation

K thus defines a static polarization/rotation—whether linear, circular or elliptical—on the Poincare sphere. The 2, r representation of the vector K gives no indication of the future position of K that is, the representation does not address the indicated hatched trajectory of the vector K around the Poincare sphere. But it is precisely this trajectory which defines the particular polarization modulation for a specific wave. Stated differently a particular position of the vector K on the Poincare sphere gives no indication of its next position at a later time, because the vector can depart (be joined) in any direction from that position when only the static 2, r coordinates are given. 

Dirichlet function, which is an approximation of Delta function, S x). Various approximate representations of Dirac delta function are provided in Van der Pol Bremmer (1959) on pp 61-62. This clearly shows that we recover the applied boundary condition at y = 0. Therefore, the delta function is totally supported by the point at infinity in the wave number space (which is nothing but the circular arc of Fig. 2.20 i.e. the essential singularity of the kernel of the contour integral). 

The procedure for constructing higher-order modes of noncircular fibers was established in Section 13-8. For each mode the transverse fields are identical in direction and form to the fundamental-mode fields of Table 16-1, except that F now denotes the appropriate higher-order solution of the scalar wave equation of Eq. (16-3). Only when the fiber cross-section is sufficiently close to circular is this representation inappropriate, as explained in Section 13-9. We quantify this transition in the following section. 

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