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EOS response function
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Infrared electric field sampled frequency comb spectroscopy

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Although the temporal increment in the dual-comb EOS configuration is ≈5 fs (Δfr = 50 Hz), the finite pulse duration of the sampling pulse (10 fs) imposes an instrument response that must be removed by deconvolution to attain the true electric field waveform of the infrared radiation. As this measurement can be considered a convolution in time domain, the “true” electric field may be acquired by deconvolving the response function of EOS (59, 60). The electric field of the SFG is proportional to the envelope of the NIR sampling pulse, i.e., $ESFG∝deffexp[iΔkL]−1iΔk×ENIREMIR$, $Δk=1c(n(ωMIR)ωMIR−ng(ωMIR)ωNIR)$ is the phase mismatch in the SFG process, and ENIR is the sampling pulse envelope. By computing the phase mismatch and acquiring ENIR via FROG, a response function can be constructed (fig. S1). Deconvolving in the frequency domain, and inverse Fourier transforming the spectrum, provides the electric field measurement without the influence of the finite pulse duration in the sampling pulse. Because we operated far from phonon resonances in GaSe, deff was taken to be frequency independent. The phase-matching curves for the 30-μm-thick EOS crystal at two different angles are shown in fig. S3. The phase-matching bandwidth is factored into the EOS response function, and it is directly proportional to the sampling efficiency (fig. S1).

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