Thescatteringmatrixtheoryhasbeendevelopedtocalculatethethird-ordernonlineareffectinsphere-grapheneslabstructures.Bydesigningstructuralparameters,wehavedemonstratedthattheincidentelectromagneticwavecanbewellconfinedinthegrapheneinthesestructuresduetotheformationofaboundstateinthecontinuum(BIC)ofradiationmodes.Basedonsuchaboundstate,third-harmonic(TH)generationandfour-wavemixing(FWM)havebeenstudied.ItisfoundthattheefficiencyofTHgenerationinmonolayergraphenecanbeenhancedabout7ordersofmagnitude.Itisinterestingthatwecandesignstructureparameterstomakeallbeams(thepumpbeam,probebeam,andgeneratedFWMsignal)beBICsatthesametime.Insuchacase,theefficiencyofFWMinmonolayergraphenecanbeenhancedabout9ordersofmagnitude.BoththeTHandFWMsignalsaresensitivetothewavelength,andpossesshighQfactors,whichexhibitverygoodmonochromaticity.BytakingsuitableBICs,theselectivegenerationofTHandFWMsignalsforS-andP-polarizedwavescanalsoberealized,whichisbeneficialforthedesignofopticaldevices.
