The broadband, coherent nature of narrow-linewidth fiber frequency combs isexploited to measure the full complex spectrum of a molecular gas throughmulti-heterodyne spectroscopy. We measure the absorption and phase shiftexperienced by each of 155,000 individual frequency comb lines, spaced by 100MHz and spanning from 1495 nm to 1620 nm, after passing through a hydrogencyanide gas. The measured phase spectrum agrees with Kramers-Kronigtransformation of the absorption spectrum. This technique can provide a fullcomplex spectrum rapidly, over wide bandwidths, and with hertz-level accuracy.
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