How to calculate the properties of spiral antena?
The shortest (and quite possibly only tractable) answer to your question can be found through simulation: For some antenna shapes, we can analytically calculate (i.e. we have formulas) what the fields look like, but in general, and especially in the case of things defined like yours, it's unlikely we find something analytically solvable.
That means that to figure out what the radiation pattern, impedance and phase pattern of this antenna looks like, you'd have to use an EM simulator. OpenEMS would be a free example. You'll probably get results faster if you know someone who has access to CST studio and experience using it.
I don't think anyone could look at this formula and tell you any advantages or disadvantages directly. Chances are it performs worse in most aspect than &quot;established&quot; wideband antenna types: over the last 50 years, many people have tried many antenna types, and if there's no evidence in literature your proposed antenna works well, then it probably doesn't.