  Counter-propagating all-normal dispersion (CANDi) fiber laser is an emerging
high-energy single-cavity dual-comb laser source. Its relative timing jitter
(RTJ), a critical parameter for dual-comb timing precision and spectral
resolution, has not been comprehensively investigated. In this paper, we
enhance the state-of-the-art CANDi fiber laser pulse energy from 1 nJ to 8 nJ.
We then introduce a novel reference-free RTJ characterization technique that
provides shot-to-shot measurement capability at femtosecond precision for the
first time. The measurement noise floor reaches 1.6x10-7 fs2/Hz, and the
corresponding integrated measurement precision is only 1.8 fs (1 kHz, 20 MHz).
With this new characterization tool, we are able to study the physical origin
of CANDi laser's RTJ in detail. We first verify that the cavity length
fluctuation does not contribute to the RTJ. Then we measure the integrated RTJ
to be 39 fs (1 kHz, 20 MHz) and identify the pump relative intensity noise
(RIN) to be the dominant factor responsible for it. In particular, pump RIN is
coupled to the RTJ through the Gordon-Haus effect. Finally, solutions to reduce
the free-running CANDi laser's RTJ are discussed. This work provides a general
guideline to improve the performance of compact single-cavity dual-comb systems
like CANDi laser benefitting various dual-comb applications.
