Guided wavefields in composite structures contain a wealth of information related to anomalies caused by interactions between waves and structural damage. Guided wave curvature has emerged as one of the most important damage indices due to its ability to extract spatial characteristics of the full wavefield. However, its practical application is limited by the classical modal curvature method, which requires central difference estimation. This approach is highly sensitive to measurement resolution and noise, resulting in poor stability and reduced reliability in damage identification. To address these limitations, a two-dimensional Fourier spectral method for guided wave curvature estimation is proposed. This study investigates the sensitivity of curvature derived from different guided wave modes, namely the symmetric and antisymmetric modes, in relation to damage detection. The two-dimensional Fourier transform is applied in the wavenumber domain to compute the modal curvature of the wavefields with improved numerical stability. The method is further assessed for its robustness to noise using experimental data obtained from composite structures. Compared with the classical central difference method, the proposed approach demonstrates significantly enhanced noise tolerance while preserving the accuracy of damage localization. The results validate that the two-dimensional Fourier spectral curvature method provides a stable and reliable means for detecting invisible damage in composite materials.