We propose a photonic spiking neural network (SNN) based on excitable vertical-cavity surface-emitting lasers with an embedded saturable absorber (VCSELs-SA) for emulating the sound azimuth detection function of the brain for the first time. Here, the spike encoding and response properties based on the excitability of VCSELs-SA are employed, and the difference between spike timings of two postsynaptic neurons serves as an indication of sound azimuth. Furthermore, the weight matrix contributing to the successful sound azimuth detection is carefully identified, and the effect of the time interval between two presynaptic spikes is considered. It is found that the weight range that can achieve sound azimuth detection decreases gradually with the increase of the time interval between the sound arriving at the left and right ears. Besides, the effective detection range of the time interval between two presynaptic spikes is also identified, which is similar to that of the biological auditory system, but with a much higher resolution which is at the nanosecond time scale. We further discuss the effect of device variations on the photonic sound azimuth detection. Hence, this photonic SNN is biologically plausible, which has comparable low energy consumption and higher resolution compared with the biological system. This work is valuable for brain-inspired information processing and a promising foundation for more complex spiking information processing implemented by photonic neuromorphic computing systems.