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The conventional effective-index method corresponds to γ = 0, whereas Marcatili’s method corresponds to γ = 1. It can be proved that a much better accuracy for the propagation constant can be obtained, if the following expression for γ is employed (40): γ =1− ( 3 )W2 + ( 4 / 2 )W3 2W2 W3 + W2 + W3 4/ (55) where i = (n21 − n2i )/2n21 (i = 2, 3, 4), and W2 and W3 are the normalized parameters for the slab waveguide with thickness 2t. The method with γ given by Eq. (55) is called the effective-index method with built-in perturbation correction (39, 40).
With the assumption that the wave propagates in the z direction with a propagation constantβ, the transverse electric ﬁeld Et in a two-dimensional waveguide with refractive-index distribution n(x, y) satisﬁes the vector wave equation (32), which can be derived from Maxwell’s source-free equations: (42) where b is the normalized propagation constant deﬁned by Eq. (25). The value of m at b = 0, denoted by mc , can be calculated by substituting Eqs. (41) and (42) into Eq. , = (n21 − n22 )/2n22 , and f (x/d) is a normalized function 0 ≤ f ≤ 1 characterizing the shape of the proﬁle.
Not every value of β/k that satisﬁes Eq. (3) represents a physical guided wave. Suppose we travel in the z direction along with the wave at its phase velocity. , the same ﬁeld distribution in the transverse direction. The guided wave must therefore have a ﬁeld distribution in the x direction that is invariant in the z direction. However, according to the zig-zag wave model shown in Fig. 3(a), what we see also are plane waves that bounce up and down within the ﬁlm. To obtain a z-invariant ﬁeld distribution in the x direction, such plane waves must form a standing wave in the x direction.