We propose a novel tomographic method, nonlinear Gaussian process tomography (nonlinear GPT), that uses the Laplace approximation to impose constraints on non-negative physical quantities, such as the emissivity in plasma optical diagnostics. While positive-valued posteriors have previously been introduced through sampling-based approaches in the original GPT method, our alternative approach implements a logarithmic Gaussian process (log-GP) for faster computation and more natural enforcement of non-negativity. The effectiveness of the proposed log-GP tomography is demonstrated through a case study using the Ring Trap 1 (RT-1) device, where log-GPT outperforms existing methods, standard GPT, and the Minimum Fisher Information (MFI) methods in terms of reconstruction accuracy. The results highlight the effectiveness of nonlinear GPT for imposing physical constraints in applications to an inverse problem.