From a discrete to a continuous model for interpulse interference with a frequency-comb laser

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We have investigated correlation patterns generated by a frequency-comb laser in a dispersive unbalanced Michelson interferometer and apply the developed formalism to the case of distance metrology. Due to group velocity dispersion, the position of the brightest fringe of the correlation pattern, which is used for distance determination, cannot be derived by simply using the definition of group refractive index of the dispersive medium. It is shown that the discrete spectrum of the optical frequency comb gives rise to correlation functions which can be represented by a series, namely a discrete correlation series. We have developed a general formalism, valid for any pulse train, extending the discrete model to a continuous model of cross-correlation functions using the Poisson summation. Our model is relevant for any offset and repetition frequency of the frequency comb. From the continuous cross-correlation model we show that, even for a homogeneous dispersive medium the position of the brightest fringe varies nonlinearly for small delay distances and stabilizes for longer ones. We have compared the theoretical results to measurements of pulse propagation in air for path-lengths up to 200 m.