Coherence in the output spectrum of frequency shifted feedback lasers

We present a detailed theoretical analysis, using correlation functions, of the coherence properties of the output from a frequency shifted feedback (FSF) laser seeded simultaneously by an external seed laser and by spontaneous emission (SE). We show that the output of a FSF laser is a cyclostationary process, for which the second-order correlation function is not stationary, but periodic. However, a period-averaged correlation function can be used to analyze the optical spectrum. From the fourth-order correlation function of the output of a Michelson interferometer we obtain the essential characteristics of the radio-frequency (RF) spectrum, needed for describing the use of the FSF laser for optical-ranging metrology. We show that, even for a FSF laser seeded by SE, the RF spectrum comprises a sequence of doublets, whose separation gives directly a measure of the length difference between the interferometer arms. This doublet structure is a result of the correlation of interference terms of individual components of the cyclostationary stochastic process. It is not seen in the optical spectrum of the FSF laser but is observable in the RF spectrum. We analyze the competition between SE and continuous wave (CW) seeding to obtain an analytical expression for the ratio of power in the discrete CW signal to the background continuum spectrum from SE. We show that, unlike mode competition in conventional lasers, where there occurs exponential selectivity, here there is a balance between the two fields; the power in the fields is related linearly, rather than exponentially, to the control parameters.