  We derive a determinant formula for the WKB exponential of singularly
perturbed Zakharov-Shabat system that corresponds to the semiclassical (zero
dispersion) limit of the focusing Nonlinear Schr\" odinger equation. The
derivation is based on the Riemann-Hilbert Problem (RHP) representation of the
WKB exponential. We also prove its independence of the branchpoints of the
corresponding hyperelliptic surface assuming that the modulation equations are
satisfied.
