Farfield pattern: spherical to cylindrical coordinate projection

The (complex) farfield emission pattern of a near field is defined on a spherical surface with a radius, $r$ approaching $infty$.

To project this spherical farfield emission onto a plane described using the cylindrical coordinate system one would need to project both the amplitude and the phase of the field. By projection, I don’t mean simply plotting a spherical surface on a 2D plane but rather more akin to using a lens to collect the farfield emission. In this case, one could imagine how different points on the spherical wavefront would need to travel difference distances before reaching the lens and thus the need for account for the relative phase change due to the relative difference in the path length.

The amplitude can projected using standard coordinate transformation.
While the additional phase due to the this projected can be considered by an additional phase term: $exp$(ikr $sin theta$)

For example, to project $E_theta$ in the spherical coordinates to the radial $E_rho$ component in the cylindrical coordinates:

$E_rho^{cyl} = E_{theta}^{sph}$ $cos$ $theta$ $exp$ ($ikr$ $sin theta$)

My question is what value of $r$ do I take? (Maybe not quite the "right" question).

Should there even be an $r$ dependence in this projection given how the farfield pattern is independent of $r$?

Different values of $r$ will result in different additional phases.
Given how the farfield pattern is independent of $r$ (for large enough $r$), I find it counterintuitive that there should be an $r$ dependence in phase term when projecting to the cylindrical coordinate system. After spending some time thinking about it, I think $r$ needs to take a certain value in order to actually perform the above calculation.

My attempt at understanding this is that even though, the farfield pattern is independent of $r$, it still has to be defined on a sphere with a finite $r$. Regardless of whether the farfield pattern is defined to be at $r$ $rightarrow$ $infty$ or at any finite large value, one can essentially take this to be a unit length and set $r = 1$ i.e. defining the farfield emission on a sphere with 1 unit radius. While $r = 1$ is still arbitrary, it is probably the most convenient value to take.

Indeed, this is what is being done in the farfield emission output results in my finite difference time domain (FDTD) simulations that I do. In other words, to perform the projection of the farfield emission onto the plane of the lens – including considerations for both the amplitude and the phase – I can take $r = 1$ as this value has been predefined in the simulation results. Is this understanding correct/accurate?

Would really appreciate it if someone could point me in the right direction to get my head around this issue.

Thanks.