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Timing Jitter in Mode-Locked Lasers

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Spatio-Temporal Modeling and Device Optimization of Passively Mode-Locked Semiconductor Lasers

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Abstract

The timing jitter of a mode-locked laser quantifies the temporal pulse train stability in the presence of continuous external forcing. In this case, temporal stability refers to the pulse train’s ability to remain close to an ideal reference pulse train.

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Notes

  1. 1.

    Parts of the chapter have been published in [1].

  2. 2.

    In more mathematical terms, a passively mode-locked laser is an autonomous system, which is subject to a time shift symmetry. Any periodic solution therefore exhibits a neutral mode, which corresponds to the time shift symmetry. Perturbations in the direction of the time shift mode, will therefore remain in the system and continuous perturbations, such as spontaneous emission noise, accumulate over time.

  3. 3.

    The truncated high frequencies correspond to a sampling time, which is not sufficient to resolve the individual pulses and thus prevents a time-domain estimation of the timing jitter.

  4. 4.

    The pulse positions are determined as the center of mass of their intensity distribution. This measure is more robust against pulse-shape fluctuations than the pulse maximum [20].

  5. 5.

    Within the framework of the Floquet theory, i.e., the linear stability analysis of periodic orbits, a periodic orbit can be expanded into Floquet modes. The corresponding eigenvalues (Floquet multipliers) determine the stability and if applicable the damping of each mode.

  6. 6.

    This construction of the pulse train implicitly assumes an actively more-locked laser with a fixed clock time dictated by an external modulation. The added independent timing noise to each ideal pulse position is neither correlated nor accumulative.

  7. 7.

    Without the factor two, the integrated rms timing jitter \(\sigma _\mathrm{{rms}}\) is nonetheless a suitable measure of the timing fluctuations. The factor two only becomes relevant when \(\sigma _\mathrm{{rms}}\) is converted into the long-term jitter and compared to other long-term jitter estimation methods.

  8. 8.

    The power spectral density \(\mathcal {S}_{|E|^2}\) is symmetric with respect to the origin, i.e., \(\mathcal {S}_{|E|^2}(-\nu ) = \mathcal {S}_{|E|^2}(\nu )\), since \(|E|^2\) itself is real. Thus, the power of the noise components at the absolute frequency \(|\nu |\) is equally shared between the frequencies \(\nu \) and \(-\nu \).

  9. 9.

    Approximating the power P(h) contained within a harmonic becomes worse at higher harmonics, since the characteristic linewidth grows quadratically. The constant factor of two in Eq. (3.8) becomes especially relevant if the long-term timing jitter is to be estimated and compared against other methods.

  10. 10.

    The respective power spectra are computed from 50 realizations for integration times from 1 to 20 \(\mu \)s and from 20 realizations for integration times from 50 to 100 \(\mu \)s.

  11. 11.

    The ensemble expectation value \(\langle \delta T_k^2 \rangle _m\) is stationary, i.e., does not depend on k. This condition is fulfilled as long as the mode-locking state is stable. The opposite would imply the breakup of the mode-locking state and thereby also render the mean pulse period \(T_\mathrm{{C}}\) ill-defined. The timing deviations \(\{\Delta t_n\}\) form a weak-sense stationary process, since they accumulate the fluctuations \(\{\delta T_k\}\). The respective variance therefore depends on the number of accumulated increments \(\delta T_k\) and thus on the time.

  12. 12.

    With this knowledge, an educated guess can also be performed based on the laser parameters if one assumes that all correlations have sufficiently decayed within five lifetimes of the slowest quantity (\(e^{-5} \approx 0.007 < 1\%\)). For the considered laser, the gain section carrier reservoir with a lifetime on \(1\,\mathrm {ns}\) represents the slowest quantity. This corresponds to correlation time of \(\approx 5\,\mathrm {ns} \approx 66T_\mathrm{{C}}\) and yields an approximate minimum pulse separation \(n = 660\), which is of the correct order of magnitude.

  13. 13.

    The direct time-domain estimate of the long-term timing jitter could in principle be improved by using multiple data points and fitting the final relaxation to its equilibrium long-term value with the exponential decay \(\sigma _{\Delta t}(n) = A \exp (-\gamma _n n) + \sigma _\mathrm{{lt}}\). In practice, however, the utilized curve-fitting routines (SciPy 1.10) produced uncertainties in the fit parameters, especially in the constant offset \(\sigma _\mathrm{{lt}}\), which outweighed the potentially gained advantages. Note that this issue does not arise in the pulse-period fluctuation autocorrelation method, since the autocorrelation functions are guaranteed to decay to zero.

  14. 14.

    Low timing jitter values produce small repetition-rate linewidths, which are computationally costly to properly resolve.

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  1. Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany

    Stefan Meinecke

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Meinecke, S. (2022). Timing Jitter in Mode-Locked Lasers. In: Spatio-Temporal Modeling and Device Optimization of Passively Mode-Locked Semiconductor Lasers. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-96248-7_3

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