The excitation of open homogeneous waveguides

Orthogonality Conditions for Leaky Modes

“…It follows from Eqs. (9) and (13) that the amplitudes C mκ may have poles at zeros of the functions 1/s m (κ) in the complex plane κ.…”

“…From their set, we will then separate the LM fields. The wave equation has particular solutions in the form of waves [9] with the field E x = Ψ(y) exp(iβ κ z), where the function Ψ(y) is determined by the relations Ψ(y) = u (1) κ exp[iκ 1 (y + d)] + v (1) κ exp[−iκ 1 (y + d)], y < −d; u (2) κ exp[−iκ 2 (y − d)] + v (2) κ exp[iκ 2 (y − d)], y > d (1) for |y| > d. Here, β κ = k 2 n 2 1 − κ 2 1 are the propagation constants of modes and κ 1 and κ 2 are the transverse wave numbers in the substrate and the superstrate, respectively (κ 2 = k 2 (n 2 2 − n 2 1 ) + κ 2 1 ). Since κ 2 is expressed via κ 1 , in what follows we will assume that all quantities are functions of κ = κ 1 .…”

“…Then the amplitudes v (1) κ and v (2) κ of reflected and transmitted waves are determined from the continuity conditions for the external and internal fields on the boundaries y = ±d. The relation between these functions can be written formally as the operator equation [9,10] S (u (1) κ , u (2) …”

“…Thus, the S-matrix divides the degenerate set of RMs into two branches. It follows from the properties of the S-matrix that the modes of these branches are orthogonal [9,10].…”

“…With allowance for the radiation condition [1,10], in the expression for β κ we choose the branch for which Im β κ > 0 if Im k > 0. The amplitude coefficients are written as [9][10][11] …”