Pulse Breathing Dynamics in a Mode-Locked Laser measured via SHG autocorrelation

Abstract

Pulse-to-pulse fluctuations in mode-locked lasers fundamentally limit applications from optical frequency combs to supercontinuum generation. While timing jitter has been extensively characterized, pulse amplitude and width fluctuations remain less accessible experimentally. We present a statistical autocorrelation method that demonstrates pulse breathing dynamics through Fano factor analysis of second-harmonic generation autocorrelation. This reveals a characteristic M-shaped Fano profile with maxima at the autocorrelation shoulders, a signature of pulse shape dynamics that is invisible to time-averaged diagnostics. Applying this method to a passively mode-locked oscillator, we measure a bound for the pulse width fluctuations of between 10.4 and 12.0\,fs ($\sim5.0-5.7\%$ of the 210 fs FWHM). This provides a theoretical projection of $\sim3\%$ spectral bandwidth fluctuations in photonic crystal fiber supercontinuum generation using this laser. This diagnostic capability opens the door to identifying and suppressing specific breathing mechanisms, paving the way for the design of ultra-stable oscillators required for precision frequency metrology.

Figures (6)

• Figure 1: Schematic representation of the noise contributions derived in Eq. \ref{['eq:variance_prop']}. (a) Amplitude noise causes intensity fluctuations that scale with the pulse envelope, maximizing at zero delay ($\tau=0$). (b) Pulse breathing (width fluctuations) changes the pulse duration. Crucially, this creates negligible intensity change at the pulse peak ($\tau=0$) and far wings, but maximum change at the pulse shoulders (steepest slopes).
• Figure 2: (a) Schematic of the non-collinear intensity autocorrelator. The input pulse is split into two replicas by a beam splitter (BS); one arm traverses a variable delay line ($\tau$) before both are focused into a BBO crystal. The background-free sum-frequency generation (SFG-green line) energy is measured as a function of delay to yield the autocorrelation trace. (b) Illustration of the interaction at zero delay ($\tau=0$): the pulses overlap temporally and spatially, generating the SFG signal (central beam). (c) At large delays (e.g., $\tau>300\,$fs), the pulses are temporally separated, and no SFG signal is produced. BS$=50:50\,$non-polarizing beamsplitter, BBO$=\beta-$barium borate.
• Figure 3: (a) Experimental second harmonic generation frequency resolved optical gating (SHG-FROG) measured directly from the laser output. (b) Reconstructed SHG-FROG trace calculated from the reconstructed pulse field. The low retrieval error (G-error$<0.5\%$) indicates excellent convergence between the measured and calculated spectrograms. (c) Temporal profile of the reconstructed pulse along with the retrieved phase. (d) Optical spectrum retrieved from the SHG-FROG trace by the COPRA algorithm Geib:19 (central wavelength $\sim$1030 nm, FWHM $\sim$5.4 nm).
• Figure 4: (a) Mean SFG autocorrelation versus delay. A sech$^2$ fit gives $R^2=0.995$ and a FWHM of $\sim318$ fs. (b) Variance, mean-square, and Fano factor versus delay. Pulse breathing contribution is manifested in the double-peak (M-shaped) structure, which is more pronounced in the Fano factor than in the variance, making the Fano profile a more sensitive breathing metric. (c) Three-component variance decomposition of the SHG autocorrelation signal. Grey points represent the measured variance at each delay. The solid red curve is the analytical model (Eq. \ref{['newvar']}) for $\rho=0$, giving $\sigma_w/w\approx5.0\%$ and a delay stage positioning noise of $\sigma_{\tau}\approx8.0$ fs. Dashed curves indicate the separate contributions from amplitude noise (blue), delay jitter (green), and pulse-width breathing (red). The fit is restricted to the central soliton core region ($|\tau|<180$ fs) where the sech$^2$ approximation is highly accurate ($R^2=0.995$); in the grey wing regions, the measured pulse decays faster than an ideal sech$^2$, so the model overestimates the variance. Assuming the strongly anti-correlated soliton limit, $\rho=-1$, gives an upper bound of $\sigma_w/w \approx 5.7\%$. (d) The allowed range for the intrinsic breathing ($\sigma_w/w$) as a function of the assumed correlation ($\rho$). The range is constrained by the measured variance peak position and Fano shoulder height. The intrinsic breathing is bounded between 5.0% ($\rho=0$, uncorrelated noise) and 5.7% ($\rho=-1$, soliton-area-theorem limit), corresponding to pulse FWHM fluctuations of $10.4-12.0$ fs.
• Figure 5: Frequency-resolved RIN power spectral density (PSD) measured at two autocorrelation delays. The blue curve ($\tau=0$ fs) represents the pure amplitude noise baseline, while the red curve ($\tau=120$ fs) contains additional contributions from pulse breathing and delay stage noise. The high-frequency plateaus (dotted lines), estimated from the 60–80 MHz region where deterministic laser dynamics are absent, are $2.5\times10^{-11}$ Hz$^{-1}$ and $7.6\times10^{-11}$ Hz$^{-1}$ respectively. The $\sim1.5\times$ increase expected from shot noise scaling alone ($\propto 1/\langle I\rangle$, given the two-thirds intensity ratio between the two delays) accounts for only half the observed $\sim3.0\times$ elevation, with the remaining broadband excess strongly indicating intrinsic pulse shape fluctuations. A distinct spectral peak near $\sim5$ MHz, visible in both traces but more prominent at $\tau=120$ fs, identifies pump diode RIN as a technical noise source that couples into pulse breathing dynamics.