2. The focusing factor was missing a factor in the $\phi$ direction.
   Previously, it assumed the focusing only takes place in the $\theta$ direction. But focusing also takes place in the $\phi$-direction, where the angular distance changes as $\sin \theta d\phi$. A straightforward example to show that the previous version was wrong is the ground-bounce experiment (vertical propagation) - previously, this would give a focusing factor of $n_{ice}/n_{surface}$ instead of $n_{ice}^2/n_{surface}^2$. However, for most neutrino trajectories the $\phi$-factor is very close to 1 as $n\approx n_{ice}$ for most of the ray path.
   Derivation in pdf attached
   
   $\phi$-factor only:
   

3. Other minor corrections:
   For some reason, a separate expression was derived for $dy/dz$ and used in the code that looks really complicated:
   
   I suggest to replace this with the much simpler expression derived already in C.12
   
   These were numerically verified to give the same result.

There is also a 'typo' in equation 10:

which is however correct in the appendix (C.19) and in the code

(i.e. the first $\sqrt{...}$ factor in (10) should be $1/\sqrt{...}$ instead)

I'm attaching a draft of the appendix of my thesis in which the focusing factor is derived.
focusing_factor_derivation.pdf

The above issues have been fixed in PR #649

As requested by @cg-laser , here are the checks on the impact of the new focusing factor (including the $\phi$-term) for a deep (100 m ) and a shallow (3 m) antenna in Greenland ice
focusing_factor_comparison.pdf

As expected, for the most part, this results in a slight decrease in the focusing factor (of up to 10%, though the 90th percentile is < 1 %).