9: Noise and Frequency Control

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Page ID: 44673

Franz X. Kaertner
Massachusetts Institute of Technology via MIT OpenCourseWare

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So far we only considered the deterministic steady state pulse formation in ultrashort pulse laser systems due to the most important pulse shaping mechanisms prevailing in todays femtosecond lasers. Due to the recent interest in using modelocked lasers for frequency metrology and high-resolution laser spectroscopy as well as phase sensitive nonlinear optics the noise and tuning properties of mode combs emitted by modelocked lasers is of much current interest. Soliton-perturbation theory is well suited to successfully predict the noise behavior of many solid-state and fiber laser systems [1] as well as changes in group- and phase velocity in modelocked lasers due to intracavity nonlinear effects [5]. We start off by reconsidering the derivation of the master equation for describing the pulse shaping effects in a mode-locked laser. We assume that in steady-state the laser generates at some position \(z\) (for example at the point of the output coupler) inside the laser a sequence of pulses with the envelope \(a(T = mT_r, t)\). These envelopes are the solutions of the corresponding master equation, where the dynamics per roundtrip is described on a slow time scale \(T = mT_R\). Then the pulse train emitted from the laser including the carrier is

\[A(T, t) = \sum_{m = -\infty}^{+\infty} a(T = m T_r, t) e^{j[\omega_c (t-mT_R + (\tfrac{1}{v_g} - \tfrac{1}{v_p}) 2mL)]}. \nonumber \]

with repetition rate \(f_R = 1/T_R\) and center frequency \(\omega_c\). Both are in general subject to slow drifts due to mirror vibrations, changes in intracavity pulse energy that might be further converted into phase and group velocity changes. Note, the center frequency and repetition rate are only defined for times long compared to the roundtrip time in the laser. Usually, they only change on a time scale three orders of magnitude longer than the expectation value of the repetition rate.