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An overlay of higher refractive index than the cladding is deposited on a long-period fiber grating (LPFG). This causes large attenuation-band shifts in the transmission spectrum, which permits the improvement of the sensitivity of the device to ambient and overlay refractive-index changes. To obtain maximum sensitivity for specific overlay and ambient refractive indexes, an optimum overlay thickness (OOT) must be selected. For complex overlay refractive indexes, there is an additional phenomenon of vanishing of the attenuation bands. This occurs for specific overlay-thickness values. The problem is analyzed with a numerical method based on linearly polarized (LP)-mode approximation and coupled-mode theory. Experimental results are contrasted with theoretical ones.