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The most intensively star-forming galaxies are extremely luminous atfar-infrared (FIR) wavelengths, highly obscured at optical and ultravioletwavelengths, and lie at $z\ge 1-3$. We present a programme of ${\it Herschel}$FIR spectroscopic observations with the SPIRE FTS and photometric observationswith PACS, both on board ${\it Herschel}$, towards a sample of 45gravitationally lensed, dusty starbursts across $z\sim 1-3.6$. In total, wedetected 27 individual lines down to 3-$\sigma$, including nine $[\rm C{\smallII}]$ 158-$\mu$m lines with confirmed spectroscopic redshifts, five possible$[\rm C{\small II}]$ lines consistent with their far-infrared photometricredshifts, and in some individual sources a few $[\rm O{\small III}]$88-$\mu$m, $[\rm O{\small III}]$ 52-$\mu$m, $[\rm O{\small I}]$ 145-$\mu$m,$[\rm O{\small I}]$ 63-$\mu$m, $[\rm N{\small II}]$ 122-$\mu$m, and OH119-$\mu$m (in absorption) lines. To derive the typical physical properties ofthe gas in the sample, we stack all spectra weighted by their intrinsicluminosity and by their 500-$\mu$m flux densities, with the spectra scaled to acommon redshift. In the stacked spectra, we detect emission lines of $[\rmC{\small II}]$ 158-$\mu$m, $[\rm N{\small II}]$ 122-$\mu$m, $[\rm O{\smallIII}]$ 88-$\mu$m, $[\rm O{\small III}]$ 52-$\mu$m, $[\rm O{\small I}]$63-$\mu$m, and the absorption doublet of OH at 119-$\mu$m, at high fidelity. Wefind that the average electron densities traced by the $[\rm N{\small II}]$ and$[\rm O{\small III}]$ lines are higher than the average values in localstar-forming galaxies and ULIRGs, using the same tracers. From the $[\rmN{\small II}]/[\rm C{\small II}]$ and $[\rm O{\small I}]/[\rm C{\small II}]$ratios, we find that the $[\rm C{\small II}]$ emission is likely dominated bythe photo-dominated regions (PDR), instead of by ionised gas or large-scaleshocks.

Far-infrared Herschel SPIRE spectroscopy of lensed starbursts reveals physical conditions of ionised gas