Functional Equation: $b(z)$ that solves $b(z)b(z^{-1})=1+a(z)a(z^{-1})$

Question

Given an analytic function $a:\mathbb C \to \mathbb C$, I need to find an analytic function $b:\mathbb C \to \mathbb C$ that satisfies

$$b(z)b\left(z^{-1}\right)=1+a(z)a\left(z^{-1}\right)$$

for all $z \in \mathbb C.$

Looks simple enough, but I don't know of a good way to start. Hints or references are welcome.