  We propose a graphene-on-grating nanostructure to enable second-order spatial
differentiation computation in terahertz (THz) region. The differentiation
operation is based on the interference between the direct reflected field and
the leakage of two excited surface plasmon polaritons counter-propagating along
the graphene sheet. With the spatial coupled-mode theory, we derive out that
the requirement for the second-order spatial differentiation is the critical
coupling condition. We numerically demonstrate such an analog computation with
Gaussian beams. It shows that the spatial bandwidth of the proposed
differentiator is large enough such that even when the waist radius of the
Gaussian beam is as narrow as ${{w}_{0}}=0.68\lambda $ ($\lambda $ is the
free-space wavelength), the accuracy of the differentiator is higher than 95\%.
The proposed differentiator is ultra-compact, with a thickness less than
$0.1\lambda $, and useful for real-time imaging applications in THz security
detections.
