Calculating the impedance of a dielectric rod in free space to get a match to 377 ohms?

Neil Smith G4DBN

I'm doing some simulations of dielrod/dielspike horns using a simple taper from a round waveguide down to a round rod.  The plan is to set the diameter of the round rod extension such that it has the same impedance as free space, so there is no discontinuity at the end of the taper, and hopefully to minimise reflections.

Can anyone point me at a way to calculate the diameter of a dielectric rod guide to get the impedance of the terminal extension close to 377 ohms?  At the moment, I'm just fiddling with the diameter to see if I can reduce the sidelobes a little, but I'd prefer to set the rod to a known size, then adjust the length of the rod and the taper angle.

The dielectric plastic is Rexolite 1422, with a dielectric "constant"  of 2.53. 

Neil G4DBN

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