  We demonstrate mid-infrared (MIR) frequency combs at 10 GHz repetition rate
via intra-pulse difference-frequency generation (DFG) in quasi-phase-matched
nonlinear media. Few-cycle pump pulses ($\mathbf{\lesssim}$15 fs, 100 pJ) from
a near-infrared (NIR) electro-optic frequency comb are provided via nonlinear
soliton-like compression in photonic-chip silicon-nitride waveguides.
Subsequent intra-pulse DFG in periodically-poled lithium niobate waveguides
yields MIR frequency combs in the 3.1--4.1 $\mu$m region, while
orientation-patterned gallium phosphide provides coverage across 7--11 $\mu$m.
Cascaded second-order nonlinearities simultaneously provide access to the
carrier-envelope-offset frequency of the pump source via in-line f-2f nonlinear
interferometry. The high-repetition rate MIR frequency combs introduced here
can be used for condensed phase spectroscopy and applications such as laser
heterodyne radiometry.
