  The magnetic and superconducting properties of a series of underdoped
$Ba_{1-x}Na_{x}Fe_{2}As_{2}$ (BNFA) single crystals with $0.19 \leq x\leq 0.34$
has been investigated with the complementary muon-spin-rotation ($\mu$SR) and
infrared spectroscopy techniques. The focus has been on the different
antiferromagnetic states in the underdoped regime and their competition with
superconductivity, especially for the ones with a tetragonal crystal structure
and a so-called double-$Q$ magnetic order. Besides the collinear state with a
spatially inhomogeneous spin-charge-density wave (i-SCDW) order at $x=0.24$ and
$0.26$, that was previously identified in BNFA, we obtained evidence for an
orthomagnetic state with a "hedgehog"-type spin vortex crystal (SVC) structure
at $x=0.32$ and $0.34$. Whereas in the former i-SCDW state the infrared spectra
show no sign of a superconducting response down to the lowest measured
temperature of about 10K, in the SVC state there is a strong superconducting
response similar to the one at optimum doping. The magnetic order is strongly
suppressed here in the superconducting state and at $x=0.34$ there is even a
partial re-entrance into a paramagnetic state at $T<<T_c$.
