Visual SLAM is particularly challenging in environments affected by noise, varying lighting conditions, and darkness. Learning-based optical flow algorithms can leverage multiple modalities to address these challenges, but traditional optical flow-based visual SLAM approaches often require significant computational resources.To overcome this limitation, we propose FMF-SLAM, an efficient multimodal fusion SLAM method that utilizes fast Fourier transform (FFT) to enhance the algorithm efficiency. Specifically, we introduce a novel Fourier-based self-attention and cross-attention mechanism to extract features from RGB and depth signals. We further enhance the interaction of multimodal features by incorporating multi-scale knowledge distillation across modalities. We also demonstrate the practical feasibility of FMF-SLAM in real-world scenarios with real time performance by integrating it with a security robot by fusing with a global positioning module GNSS-RTK and global Bundle Adjustment. Our approach is validated using video sequences from TUM, TartanAir, and our real-world datasets, showcasing state-of-the-art performance under noisy, varying lighting, and dark conditions.Our code and datasets are available at https://github.com/youjie-zhou/FMF-SLAM.git.