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Recently, approximate adjoint sensitivity techniques with structured-grid electromagnetic solvers have been proposed. In this letter, we propose a central adjoint sensitivity formula for response sensitivity with structured-grid solvers, whose accuracy is an order higher than that of the preceding techniques. The formula is derived from a backward problem, which is considered here for the first time, and the forward problem utilized in the previous techniques. Our technique maintains the computational efficiency typical for adjoint-based methods. We illustrate it through applications with a frequency-domain transmission line method solver.