Face recognition is an open-set problem requiring high discriminative power to ensure that intra-class distances remain smaller than inter-class distances. Margin-based softmax losses, such as SphereFace, CosFace, and ArcFace, have been widely adopted to enhance intra-class compactness and inter-class separability, yet they overlook the impact of noisy samples. By examining the distribution of samples in the angular space, we observe that clean samples predominantly cluster in the center region, whereas noisy samples tend to shift toward the peripheral region. Motivated by this observation, we propose the Exponential Angular Margin Loss (ExpFace), which introduces an angular exponential term as the margin. This design applies a larger penalty in the center region and a smaller penalty in the peripheral region within the angular space, thereby emphasizing clean samples while suppressing noisy samples. We present a unified analysis of ExpFace and classical margin-based softmax losses in terms of margin embedding forms, similarity curves, and gradient curves, showing that ExpFace not only avoids the training instability of SphereFace and the non-monotonicity of ArcFace, but also exhibits a similarity curve that applies penalties in the same manner as the decision boundary in the angular space. Extensive experiments demonstrate that ExpFace achieves state-of-the-art performance. To facilitate future research, we have released the source code at: https://github.com/dfr-code/ExpFace.