Free-field realizations of $\mathcal{N}=1$ supersymmetric $\mathcal{W}$-algebras associated with Lie superalgebras $B(n-1,n)$ and $B(n,n)$

We extend the recursive construction of free-field realizations proposed by Ide, Ito, and Miyazaki to the $\mathcal{N}=1$ supersymmetric $\mathcal{W}$-algebras $\mathcal{S}\mathcal{W}B(n-1,n)$ and $\mathcal{S}\mathcal{W}B(n,n)$, associated with the Lie superalgebras $B(n-1,n)$ and $B(n,n)$, respectively. We obtain explicit free-field realizations of $\mathcal{S}\mathcal{W}B(0,1)$, $\mathcal{S}\mathcal{W}B(1,1)$, $\mathcal{S}\mathcal{W}B(1,2)$, and $\mathcal{S}\mathcal{W}B(2,2)$. In particular, the realizations of $\mathcal{S}\mathcal{W}B(1,2)$ and $\mathcal{S}\mathcal{W}B(2,2)$ are new. Using these realizations, we compute central charges, conformal weights, and eigenvalues of the generating currents on free-field highest-weight states, and construct the local integrals of motion $I_1$, $I_3$, and $I_5$.