Frequency combs generated by trains of pulses emitted from mode-locked lasers are analyzed when the center time and phase of the pulses undergo noise-induced random walk, which broadens the comb lines. Asymptotic analysis and computation reveal that, when the standard deviation of the center-time jitter of the nth pulse scales as n(p/2) where p is a jitter exponent, the linewidth of the kth comb line scales as k(2/p). The linear-dispersionless (p=1) and pure-soliton (p=3) dynamics in lasers are derived as special cases of this time-frequency duality relation. In addition, the linewidth induced by phase jitter decreases with power P(out), as (P(out))(-1/p).
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