We analyze the diffraction of the picture obtained by the subtraction of two in-line holograms recorded in two planes at distances of z and $z+\Delta z$ apart from the object plane. Our theoretical analysis reveals that the reconstructed field at the object plane is approximately equal to the Laplacian second-order differentiation of the object wave in transverse direction when $\Delta z$ is much smaller than the distance z, that is, the reconstructed image presents a high quality of edge enhancement. We further investigate the dependence of the edge-enhancement quality on the longitudinal differential distance $\Delta z$ and find that the reconstructed images have the sharpest edge-enhancement and high signal-to-noise ratio at the same time only when the value of $\Delta z∕z$ lies between 0.7% and 0.9% under our experimental condition. We also construct a criteria function, named the entropy of the image, to automatically focus the edge-enhanced image and demonstrate its adaptability in experiments.