Consider an optical fiber whose axis is the $z$-axis. Assume that the electric field $e_z$ along this direction can be written (in cylindrical coordinates) as
$$e_z(r,\phi) = A(\phi)B(r)$$
$k_c$ is the transverse (orthogonal to $z$) propagation constant. It is defined as
$$k_c = \pm \sqrt{k_2^2 - k_z^2} = \pm j \sqrt{k_z^2 - k_2^2} = \pm j \alpha$$
In the cladding $k_2 < k_z$, so $k_c$ must be pure imaginary. Function $B(r)$ is therefore determined by the modified Bessel equation:
$$\frac{\mathrm{d}^2 B(r)}{\mathrm{d}r^2} + \frac{1}{r} \frac{\mathrm{d} B(r)}{\mathrm{d}r} + \left( \alpha^2 + \frac{\nu^2}{r^2} \right) B(r) = 0$$
The field is expected to attenuate as $r$ increases, so the only acceptable solutions for $B(r)$ are the modified Bessel functions of the second kind:
$$B(r) = I_{\nu}(\pm \alpha r)$$
Both $k_c = j \alpha$ and $k_c = -j \alpha$ lead to the modified Bessel equation above, so both $I_{\nu}(\alpha r)$ and $I_{\nu}(- \alpha r)$ are mathematically acceptable solutions.
Question 1: Are they interchangeable, or is there a difference between them?
My considerations: $I_{\nu}$ is an even or odd function if (respectively) $\nu$ is even or odd; if $\nu$ has any other value, $I_{\nu}$ is nor even, neither odd. $k_c = -j \alpha$ would lead to an $I_{\nu}$ whose sign changes when the order $\nu$ changes. Instead, $k_c = \alpha$ would lead to a uniformity in the sign of $I_{\nu}$ for any integer value of $\nu$. But I do not have a decisive argument.
Question 2: Are there any textbooks or websites which deal with this subject? It't very hard to find them.
This document, at pdf page 243, states that $k_c = j \alpha$ «is always positive imaginary», but it doesn't explain why.
