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A new staggered field design and formulation for the one-dimensional propagator of the total-field/scattered-field source implementation in finite-difference time domain (FDTD) scattering simulations are presented. The new equations are based on split-field Maxwell's equations and the resulting technique extends the functionality of the multipoint auxiliary propagator to sourcing FDTD lattices hosting extended-stencil high-order algorithms. This technique virtually eliminates numerical dispersion, field location and polarization mismatches between propagator and main grid. The resulting machine accuracy-level leakage error from implementing this technique is confirmed for the standard low and high-order FDTD schemes as well as the M24 high-order algorithm. Normalized field leakage for all three algorithm implementations outside the total-field region was measured at below - 295 dB.