Reconstruction of index profiles: the inverse WKB method

An elegant way of avoiding this inconvenience is to define a continuous effectiveindex

function, which is then used to construct a refractive index profile by numerically
solving the WKB equation [11]. This procedure involves two steps: (a) to construct
an effective index function N(q) from the discrete set of measured guided modes, and
(b) to determine the corresponding profile n(x) by inverting equation (3.73).
Having a set of ν experimentally measured modal indices N(m) (m = 0, 1, 2 . . . ,
ν − 1), a natural way to construct the artificial effective index function N(q) is to
fit the ν discrete modal positions to a polynomial of order (ν − 1). This can be
done for instance by Neville’s algorithm [12]. As an example of this implementation,
Figure 3.30 shows four measured mode indices (circles), and the associated effective
index function calculated by a polynomial fit of order 3 (continuous line).
In order to obtain the peak refractive index n0 at the surface we first need to estimate
the value q = q0, which is done by substituting xt = 0 into equation (3.73), where the
guided mode indices m are now denoted by the artificial set of modes of indices q.
In this case, by substituting n0 = n(0) = N(q) (= nf ) in equation (2.149), we obtain
φs = π. Substituting this value into equation (3.73), besides φt = π/2, and taking into
account that the WKB integral vanishes, we finally obtain that the value of q at the
surface is q0 = −0.75. Therefore, the refractive index at the surface is easily computed
by evaluating the index function, obtained by a polynomial fit, at q = −0.75:

Due to the inevitable existence of errors in measured effective indices, in practice it
is convenient to employ least-squares fitting to construct the effective index function for
the reconstruction of refractive index profiles. In this way, the influence of experimental
errors in the obtained index profile can be minimised.