Analytic random-walk model for the coherence of a frequency comb

We present an analytical study of the frequency comb coherence due to random noise in the pulses phases. We derive a simple expression for the comb lineshape, which depends on a single parameter $N_{\text{eff}}$ with the physical meaning of number of coherent comb pulses, inversely proportional to the variance of the phase jumps between subsequent comb pulses. A comparison to the case of a cw-monomode laser with white noise frequency fluctuations is also presented.

Figure 1

Standard deviation of phase difference between $k$ pulses. Continuous red trace is the numerical simulation (average over 30 runs) where the independent phase jumps are taken from a Gaussian distribution with ${\sigma }_1=2\mathrm{mrad}$, corresponding to the model of a free-running comb considered in this paper. Dashed blue trace shows the effect of a phase feedback loop (shown in the inset and described in the text) characterized by the response time $\tau ={10}^3{T}_{\text{rep}}$. The number of coherent pulses is $N_{\text{eff}}=1/{\sigma }_1^2=2.5\times {10}^5$ for the case of no feedback, whereas it is $\infty$ in the case of the feedback loop [as for Eq. (5)]. The free-running random-walk behavior ${\sigma }_k={\sigma }_1\sqrt{k}$, and the feedback limit ${\sigma }_{\infty }={\sigma }_1\sqrt{\tau /T_{\text{rep}}}$ are also shown by light gray lines.

Figure 2

Comb-field intensity spectrum $Z_N$ [Eq. (8)] as a function of the normalized frequency detuning $x=\mathrm{\Delta }\nu /{\nu }_{\text{rep}}$ (distance from the carrier frequency measured in units of ${\nu }_{\text{rep}}$). (a) Ideal case (no phase fluctuations) $Z_N\left(x\right)={sin}^2\left(\pi Nx\right)/{sin}^2\left(\pi x\right)$; the comb modes appear on integers of $x$. The dimensionless comb-field intensity spectrum normalized to $N^2$ for one comb mode is shown for the three regimes: (b) $N\ll N_{\text{eff}}$ (ideal case), (c) $N=N_{\text{eff}}$, and (d) $N\gg N_{\text{eff}}$. Using a Brownian motion model to introduce random-phase fluctuations between successive comb pulses, the spectrum [gray lines in panels (b)–(d)] presents an irregular shape for $N\ge N_{\text{eff}}$. Averaging over such spectra leads to an analytic expression, shown by red lines, whose shape depends on the number of coherent pulses $N_{\text{eff}}=1/{\sigma }^2$. Around the integers the spectrum is well approximated by a sinc shape in panel (b) and by the Lorentzian curve (with $\mathrm{\Gamma }=1/2\pi N_{\text{eff}}$ as FWHM) in panel (d). These two asymptotic curves are plotted as dashed lines. Note the different scales in panels (a)–(c) ($1/N$ vs $2\pi /N_{\text{eff}}$).

Figure 3

In panels (a) and (b) the linewidth $\mathrm{\Delta }\nu$ (the FWHM of the mode peak), and the contrast $\rho$ (peak value over the plateau level of the spectrum), normalized by the corresponding asymptotic values for $N\to \infty$ ($\mathrm{\Delta }{\nu }_{\infty }$ and ${\rho }_{\infty }$), are shown as a function of $N/N_{\text{eff}}$. The corresponding behaviors for the ideal case (no phase fluctuations) are shown as gray lines.