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

Abstract

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.

Time-frequency distribution of interferograms from a frequency comb in dispersive media

“…Also, the position of the coherence maximum or the brightest fringe will vary as a function of the delay distance. We have shown that [16], even if the medium is homogeneous, the position of the maximum coherence varies non-linearly for short delay distances and linearly for larger delay. In addition, our simulations also show that the correlation patterns are non-linearly broadened at short delay distances and that linear broadening replaces the non-linear broadening at larger distances.…”

“…A detailed description of our work can be found in [16]. The frequency spectrum emitted by a mode-locked laser consists of a comb of regularly spaced frequencies ω m = mω r + ω 0 , m = 0, 1, 2, ... where ω 0 is the common offset frequency and ω r is the repetition frequency f r expressed in angular notation…”

“…A detailed analysis of this function in case of an unbalanced interferometer in the presence of a homogeneous dispersive medium leads to the following equation [16] …”

“…A continuous model was therefore developed for a better understanding of the problem. We can rewrite the cross-correlation as [16] …”

“…, R and θ are radial and angular parameters which are functions of f X (t) [16]. These results have allowed us to define a particular function ζ (X) by…”