How to Get Well-Behaved Expressions

Consider an infinite array of elements oriented in the direction located in slab m of a stratified dielectric medium as shown in Fig. 3.9. As throughout [61], the elements are assumed to be excited by an external plane wave. The plane wave spectrum generated by this array can be decomposed into five wave modes as described in references 66-71. Four of these modes involve the reflection coefficient from the dielectric interfaces bounding the m, slab, while the fifth mode does not and is referred to as the direct mode. It is this mode that requires special treatment and is the reason it is revisited in detail here. The direct mode field at some point / on a test element also located entirely in slab m and oriented in the p direction may be written [66, 68] 

Examining the first integral in (3.69) and employing integration by parts, one may write 

Both integrals in (3.74) and (3.75) are well-behaved and will converge rapidly inside the double infinite summation, as will the terms evaluated at +h and —/i. However, these same two terms evaluated at l wiU exhibit convergence problems and must therefore be treated separately. It is beneficial at this point to decompose these two terms into their respective x, y, and z components. Making use of the fact that e may be written [W] 

Examining (3.77) and (3.79) reveals that both expressions converge nicely in the double infinite summation over k and n. Combining the y components, on the other hand, results in an expression that differs in form from the combined X and z components and may be expressed instead as 

The second and third terms in (3.80) are identical in form to those found in (3.77) and (3.79), and likewise they converge rapidly in the double infinite summation. The convergent nature of the first term, however, is not immediately apparent. Employing the same mathematical identity used in references 66 and 68, the double infinite summation of the first term in (3.80) involving may be written 

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