Objective　M　id-　infrared　lasers　at　approxim　ately　3　μm　can　be　used　in　m　any　fields　such　as　m　edical　surgery,　trace　gas　detection,　rem　ote　sensing,　and　infrared　counterm　easures.　Such　laser　source　can　be　obtained　via　D　y3+　-　,　H　o3+　-　,　or　Er3+　-　doped　laser　active　m　edium.　T　he　developm　ent　of　D　y3+　-　and　H　o3+　-　doped　lasers　is　lim　ited　because　of　the　lack　of　a　suitable　pum　p　source.　In　contrast,　there　is　a　strong　absorption　band　of　Er3+　-　doped　crystals　at　approxim　ately　970　nm　and　they　can　be　pum　ped　by　w　ell-　developed　high-　pow　er　InG　aA　s　LD　s.　H　ow　ever,　the　transition　of　Er3+　ions　at　approxim　ately　3　μm　w　avelength　is　self-　term　inating　because　the　lifetim　e　of　the　initial　laser　level　is　considerably　shorter　than　that　of　the　term　inal　laser　level.　T　o　overcom　e　this“bottleneck”,　Er3+　ions　w　ith　a　high　atom　ic　fraction　of　30%　‒　50　%　are　typically　required　to　depopulate　low　er　laser　level　4I13/2.　T　he　therm　al　effects　increase　w　ith　increasing　doping　concentration.　Cascading　4I13/2→　4I15/2　near-　infrared　laser　em　ission　can　achieve　a　3-　μm　-　laser　output　at　an　extrem　ely　low　doping　concentration　but　at　the　expense　of　reducing　the　conversion　efficiency.　In　our　recent　study,　it　has　been　proven　that　the　cyclic　cascade　can　im　prove　the　efficiency　of　a　3-　μm　-　laser　in　Er∶Y　A　G　crystals,　w　ith　a　low　doping　concentration　at　room　tem　perature.　H　ow　ever,　the　m　axim　um　slope　efficiency　does　not　exceed　10%.　O　n　the　basis　of　the　cyclic　cascade,　im　proving　the　slope　efficiency　is　a　m　eaningful　attem　pt.　M　ethods　In　this　study,　w　e　theoretically　analyzed　the　tw　o　m　ain　factors　influencing　the　m　id　-　infrared　(M　IR)　laser　slope　efficiency　in　Er∶Y　A　G　crystals,　nam　ely,　quantum　efficiency　and　overlapping　efficiency.　W　e　analyzed　the　m　echanism　of　im　proving　the　quantum　efficiency　by　a　cyclic　cascade.　For　the　overlapping　efficiency,　w　e　sim　ulated　the　distributions　of　the　pum　p　light　and　laser　m　ode　field　and　analyzed　their　influence　on　the　overlapping　efficiency.　N　ext,　the　coating　param　eters　of　the　cavity　m　irror　w　ere　designed　according　to　the　relevant　characteristic　w　avelengths　in　the　cyclic　cascade　process.　Er∶Y　A　G　crystals　w　ith　doping　concentrations　(atom　ic　fractions)　of　7.5%,　10%,　15%,　and　25%　w　ere　used　in　the　experim　ent　and　cut　along　their　〈　111　〉　crystal　direction,　w　ith　cross-　sections　of　3　m　m　×　3　m　m.　T　he　lengths　of　the　Er∶Y　A　G　crystals　w　ith　atom　ic　fractions　of　7.5%,　10%,　and　15%　w　ere　10　m　m　and　2　m　m,　and　the　lengths　of　Er∶Y　A　G　crystals　w　ith　atom　ic　fraction　of　25%　w　ere　5　m　m　and　2　m　m.　T　he　pum　ping　source　w　as　a　stable　w　avelength　fibercoupled　laser　diode　w　ith　a　central　em　ission　w　avelength　of　976　nm.　A　spectrom　eter　w　as　used　to　m　easure　the　laser　output　spectrum　and　determ　ine　w　hether　a　cascade　oscillation　w　as　form　ed.　A　n　energy　m　eter　w　as　used　to　m　easure　the　output　energy　and　other　related　param　eters.　T　hen,　the　laser　output　slope　efficiency　w　as　calculated　to　analyze　the　influence　of　im　proving　the　overlapping　efficiency　by　optim　izing　the　crystal　length　on　the　slope　efficiency.　Results　and　Discussions　In　the　output　spectra　of　the　crystals　w　ith　four　doping　concentrations,　tw　o　em　ission　peaks　in　near-infrared　(1469　nm)　and　m　id-　infrared　(2937　nm)　bands　can　be　observed　(Fig.　5),　show　ing　that　the　near-　infrared　light　and　m　id-　infrared　light　have　form　ed　a　cascade　oscillation　in　the　cavity.　For　Er∶Y　A　G　crystals　w　ith　lengths　of　10　m　m　and　5　m　m,　the　threshold　is　greater　than　23　m　J,　w　hereas　for　Er∶Y　A　G　crystals　w　ith　a　length　of　2　m　m,　the　threshold　is　low　er　than　17　m　J.　T　he　threshold　of　Er∶Y　A　G　crystals　with　a　larger　length　is　generally　higher　than　2　mm　(Fig.　6　and　Table　1).　In　terms　of　the　2937-　nm　mid-　infrared　laser　output　slope　efficiency,　under　the　same　doping　concentration,　the　laser　slope　efficiency　of　the　2-　mm　long　crystal　is　significantly　improved　compared　with　that　of　the　longer　laser　crystal.　In　particular,　for　Er∶YAG　crystals　with　an　atomic　fraction　of　10%,　when　the　crystal　length　is　10　mm,　the　mid-　infrared　laser　threshold　is　23.4　mJ,　and　the　slope　efficiency　is　23.5%;　when　the　crystal　length　is　2　mm,　the　mid-　infrared　laser　threshold　is　11.4　mJ,　and　the　slope　efficiency　is　36.5%.　Compared　with　the　10-　mm　long　crystal,　the　threshold　is　reduced　by　51.3%,　and　the　slope　efficiency　is　increased　by　55.3%.　To　the　best　of　our　knowledge,　this　is　the　first　time　that　a　slope　efficiency　exceeding　the　Stokes　limit　has　been　obtained　for　low-　doped　Er∶YAG　crystals　at　room　temperature　(Fig.　6　and　Table　1).　Conclusions　A　high-　efficiency　cyclic　cascade　Er∶YAG　mid-　infrared　pulsed　laser　at　room　temperature　is　reported.　Based　on　the　cyclic　cascade,　using　the　optimized　cyclic　cascade　cavity,　the　influence　of　optimizing　the　crystal　length　to　improve　the　beam　overlap　on　the　slope　efficiency　of　the　mid-　infrared　laser　is　explored.　The　experimental　results　show　that　by　appropriately　shortening　the　crystal　length,　the　threshold　of　mid-　infrared　laser　oscillation　is　reduced　considerably,　and　the　slope　efficiency　is　significantly　improved.　Slope　efficiencies　of　2937　nm　mid-　infrared　lasers　up　to　36.5%　and　37.2%,　exceeding　the　Stokes　limit　of　33.2%,　are　achieved　in　Er∶YAG　crystals　with　a　length　of　2　mm　and　Er　atomic　fractions　of　25%　and　10%,　respectively.　To　the　best　of　our　knowledge,　this　is　the　first　time　that　an　Er∶YAG　crystal　with　a　low　doping　concentration　(atomic　fraction　of　10%)　is　used　for　a　high-　efficiency　3-　μm　mid-　infrared　laser　output　exceeding　the　Stokes　limit　at　room　temperature.　Moreover,　Er∶YAG　crystals　with　an　Er　atomic　fraction　of　10%　can　produce　less　thermal　effect　than　Er∶YAG　crystals　with　an　Er　atomic　fraction　of　25%　and　is　expected　to　obtain　a　higher　average　output　power.　This　study　can　help　to　promote　the　development　of　3-　μm　mid-　infrared　high　efficiency　high　average　power　lasers.　©　2023　Science　Press.　All　rights　reserved.