An optically injected semiconductor laser can produce excitable multipulses. Homoclinic bifurcation curves confine experimentally accessible regions in parameter space where the laser emits a certain number of pulses after being triggered from its steady state by a single perturbation. This phenomenon is organized by a generic codimension-two homoclinic bifurcation and should also be observable in other systems. DOI: 10.1103/PhysRevLett.88.063901 PACS numbers: 42.65.Sf, 05.45. -a, 42.55.Px A system is called excitable when it produces a large nonlinear response to a small but sufficiently large perturbation from its stable equilibrium. When perturbed above a certain threshold, an excitable system makes a large excursion in phase space leading to a large amplitude pulse. It then settles back to the stable equilibrium in what is called the refractory period. Here we use the example of an optically driven semiconductor laser to show that an excitable response to a single stimulus may have a form of not just a single pulse but of a certain fixed number of pulses. How many pulses are produced depends on the laser's operating conditions.The notion of excitability was first introduced in biology to describe the spiking behavior of nerve cells [1] and later found in reaction-diffusion systems [2]. More recently different types of excitability were found in optical systems, ranging from nonlinear cavities with temperaturedependent absorption [3] to lasers with saturable absorber [4], optical feedback [5], and optical injection [6]. In lasers essentially two types of excitability are found [4]: the classic case of excitability on an invariant circle, and excitability in the vicinity of a homoclinic bifurcation, found first in lasers with saturable absorber [4] and then in multisection distributed feedback lasers [7]. All these examples of excitability were associated with a single-pulse response to a single perturbation. A double-pulse excitable response observed in a laser with optical feedback was explained as the result of noise on a two-dimensional model [5].Our main result is that a multipulse response to a single perturbation is a natural and deterministic phenomenon. It is an example of excitability near a homoclinic bifurcation, but in this case near an n-homoclinic bifurcation, where the homoclinic orbit (to a saddle focus) closes up not after the first but after the nth global loop; see Figs. 2 and 4 below. The key element is a generic bifurcation structure of tongues formed by n-homoclinic bifurcation curves in parameter space. These tongues are organized by special points called Belyakov bifurcation points [8,9], and inside each such tongue a different number of pulses will be triggered. We show multipulse excitability here for the example of an optically injected semiconductor laser, but, in fact, this phenomenon may occur in any at least three-dimensional system with Belyakov bifurcation points. Other examples of systems with Belyakov points are an atmospheric circulation model [10], a tritrophic food ch...