Image super-resolution (SR) is a critical technique for enhancing image quality, playing a vital role in image enhancement. While recent advancements, notably transformer-based methods, have advanced the field, infrared image SR remains a formidable challenge. Due to the inherent characteristics of infrared sensors, such as limited resolution, temperature sensitivity, high noise levels, and environmental impacts, existing deep learning methods result in suboptimal enhancement outcomes when applied to infrared images. To address these challenges, we propose a specialized Contourlet residual framework tailored for infrared images to restore and enhance the critical details from the multi-scale and multi-directional infrared spectra decomposition. It precisely captures and amplifies the high-pass subbands of infrared images, such as edge details and texture nuances, which are vital for achieving superior reconstruction quality. Moreover, recognizing the limitations of traditional learning techniques in capturing the inherent characteristics of infrared images, we incorporate a prompt-based learning paradigm. This approach facilitates a more nuanced understanding and targeted optimization process for infrared images by leveraging the semantic comprehension offered by the visual language model. Our approach not only addresses the common pitfalls associated with infrared imaging but also sets a new paradigm for infrared image SR. Extensive experiments demonstrate that our approach obtains superior results against the existing visible image SR models, attaining state-of-the-art performance.