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Abstract In modern short‐pulse fiber lasers, there is significant pulse breathing over each round trip of the laser loop. Consequently, averaged models cannot be used for quantitative modeling and design. Instead, lumped models, which are obtained by concatenating models for the various components of the laser, are required. As the pulses in lumped models are periodic rather than stationary, their linear stability is evaluated with the aid of the monodromy operator obtained by linearizing the round‐trip operator about the periodic pulse. Conditions are given on the smoothness and decay of the periodic pulse that ensure that the monodromy operator exists on an appropriate Lebesgue function space. A formula for the essential spectrum of the monodromy operator is given, which can be used to quantify the growth rate of continuous wave perturbations. This formula is established by showing that the essential spectrum of the monodromy operator equals that of an associated asymptotic operator. Since the asymptotic monodromy operator acts as a multiplication operator in the Fourier domain, it is possible to derive a formula for its spectrum. Although the main results are stated for a particular experimental stretched pulse laser, the analysis shows that they can be readily adapted to a wide range of lumped laser models.