Normalization free-space’ radiation modes

We use Eq. (30-9) for the transverse-field components and determine the constants from continuity of e j, h j, e j and h j at the interface. With the aid of the Wronskian of Eq. (37-77) this leads to the expressions in Table 25-3 for the ITE and ITM modes. The orthogonality and normalization of each radiation mode is identical to the corresponding free-space normalization of Table 25-2 for reasons given above. Alternatively, we can parallel the derivation of Nj(Q) in Section 25-7 using the radiation-mode fields. 

In order to apply radiation modes, we must show how to construct their fields and normalization from Maxwell s equations for a specific waveguide. This is facilitated if we first construct the "free-space modes, i.e. the radiation modes of an unbounded medium of uniform refractive index [1]. The free-space modes are easier to construct than the radiation modes of a waveguide, and... 

Radiation modes with p kn, i e. 0 = n/2, are composed of plane waves at near normal incidence to the core-cladding boundary so that they are only weakly influenced by the core. Thus, radiation modes with P < kn are approximately free-space modes with ej ej and = h. The implicit conditions discussed below in Section 25-11 help facilitate this limiting condition. At the other extreme, when P = kn, i.e. 0, 0, the plane waves that form a radiation mode are at near grazing incidence and are significantly influenced by the fiber core. 

The combination of e j and eyj which satisfies Eq. (25-24a) is not unique. However, when Tj is constructed according to Eq. (25-21), the orthogonality of the free-space modes, like the normalization, automatically ensures orthogonality of the corresponding radiation modes [4]. This may be verified... 

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See also in sourсe #XX -- [ ]

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