Optical pulses that propagate in an optical fibre will acquire additional phase due to group velocity dispersion in the fibre. Suppose the centre frequency of the pulse is ωp; then, the component at frequency ω will acquire a relative phase after propagation of distance z51:

where E(0,w)=E0(2N)exp(iwt) is the electrical field of light at frequency ω and position z = 0, normalized to the photon number per unit time. Here, we have assumed a flat spectrum for the comb, and N is the total number of comb lines. Dλ is the group velocity dispersion parameter, and Dλ ≈ 18 ps/nm/km for the SMF-28 fibre at 1550 nm. For soliton frequency combs, (ω − ωp)/2π = n × fr for the n-th comb line from the spectral envelope centre, where fr is the comb repetition frequency. Therefore, the photocurrent generated in the photodiode is

where we have used k=mnark=a(rmrn+1)/(1r) to derive the term cos(2πfrt), and we have set 2N0 + 1 = N. Higher harmonics of the repetition frequency are neglected as they are beyond the detection limit of our photodiode. Considering IDC as the average photocurrent flowing through the load resistor RL, the detected mmWave power at frequency fr is as follows:

where we have defined d = Dλ × z as accumulated dispersion, and Γ is the PD power roll-off at the repetition frequency. This equation is the same as Eq. (2) in the main text. When dispersion is very small (d → 0), the detected mmWave power is approximated by

which is Eq. (1) in the main text.

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