Berkeley ELENG 232 - Einstein's AB Coefficients Spontaneous Emission

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EE 232 Lightwave DevicesLecture 7: Einstein’s AB Coefficients, St EiiSpontaneous Emission Instructor: Ming C. WuUniversity of California, BerkeleyElectrical Engineering and Computer Sciences DeptElectrical Engineering and Computer Sciences Dept.EE232 Lecture 7-1©2008. University of CaliforniaEinstein’s AB CoefficientsstimRsponRR()()21 21 2 11()sponstimRAffff=−E2hνhνhν21R21Rhν12R()()21 21 2 1 2112 12 1 2 211()1()stimRBffPERBf fPE=−=−E1hνhνhνhν21 21For non-monochromatic light:() () : phPE n NE=1/ number of photons per unit volume per energy interval1 : Number of1kBpphhkTnω= photons per state (Bose-Einstein distribution)/1kBphkTeω−()322121338 :Number of states with photon energy E per unit volume, rbanENEhcπ=EE232 Lecture 7-2©2008. University of California per energy intervalEinstein’s AB Coefficients12 21 21At thermal equilibrium:spon stimRR R=+() () ()12 1 2 21 21 2 1 21 2 1 21 1() 1 1()EFBf f PE Af f Bf fPE−=−+−()() ()12121 2 1212112 1 2 21 2 112 211()11BBBEFkTEF EFkT kTAf fAePEBf f Bf fBeBe−−−−==−− −−2112 21321 2218( ) BrphEEkTAnENE nBe Bπ−⋅= ⇒−212121/3311kBBEEhkTkTAhc eBe Bω−⎛⎞=⎜⎟−⎝⎠−12 21Be B12 2112 213221 21 8()rBe BBBAnENEπ=EE232 Lecture 7-3©2008. University of California21 21213321()rNEBhc==Spontaneous Emission Spectra12 213221 2121338()rBBBAnENEπ====CBE2()1sponCVRff∝−Spontaneous Emission[]213321 21 21 21 2 121 2 1 21()() (1 )() ()spon sponabs absNEBhcRrEdEAffRrEdEBffPE==−==−hν[][]21 2 1 2121() ()Absorption coefficient:()() ()net netabsnetrRrEdEBffPErEdEnEdE Bf f gEdEα==−=−VBE1[]21 1 2 212121 21() ()()(/)()()rsponEdE Bf f gEdEPE c n crEgEα==−=−=21 2 1(1 )Af fnff−CBE2()stimnet C VRff∝−Stimulated Emission21()gE21222121 21 2132 381111() () []rsponrEFnffBcnErEgEhπΔ−=CBhνEE232 Lecture 7-4©2008. University of California2121 21 2132 3() () [ ]1BEFkTghcsmeVe−Δ−VBE1Spontaneous Emission and Gain Spectra for Various Temperatures21044×31044×41044×51044×JointDOS21044×31044×41044×51044×)JointDOS1 1.2 1.4 1.6 1.8011044×hvieV1.5V)1 1.2 1.4 1.6 1.8011044×hvieV1.5V)15−0V)eV)5eV)EmissionProbability150V)eV)5eV)FermiInversionFactor1 1.2 1.4 1.6 1.81.5−hvieV1.5 106×T=1KSpontaneous1 1.2 1.4 1.6 1.81.5−hvieV1.5 106×Gain0T 1 KT= 77KT = 300KSpontaneousEmissionSpectra0GainSpectraEE232 Lecture 7-5©2008. University of California1 1.2 1.4 1.6 1.81.5−106×hvieV1 1.2 1.4 1.6 1.81.5 106×hvieVSpontaneous Emission and Gain Spectra for ΔF (T = 300 K)21044×31044×41044×51044×i)JointDOS21044×31044×41044×51044×i)JointDOS1 1.2 1.4 1.6 1.8011044×hvieV1.5)1 1.2 1.4 1.6 1.8011044×hvieV1.55eV)05eV)55eV)6eV)ΔF= 1 5eVEmissionProbability15−05eV)55eV)6eV)FermiInversionFactorΔF= 1 5eV1 1.2 1.4 1.6 1.81.5−hvieV1.5 106×ΔF= 1.5eVΔF= 1.55eVΔF= 1.6eVSpontaneous1 1.2 1.4 1.6 1.81.5hvieV1.5 106×GainΔF= 1.5eVΔF= 1.55eVΔF= 1.6eV0SpontaneousEmissionSpectra0GainSpectraEE232 Lecture 7-6©2008. University of California1 1.2 1.4 1.6 1.81.5− 106×hvieV1 1.2 1.4 1.6 1.81.5−106×hvieVSpontaneous Emission Lifetime()211() ( )()() ()1 ()sponrgerrEfffEfEωρω ωτ=−====()2121222121 21 2132() ()1 ()81() ()eC VsponrEFkTffEfEnErE gEhcωπ−Δ=−==222210321()8()()()BkTercvrgggefnECeP E fhcfωπρωωω−⎛⎞=⋅−⎜⎟⎝⎠JG= ===32222210()18grrcvfhcnECePτπ⇒= ⋅⋅JG Flux_per_eV8 π⋅ nr2⋅ Eg2⋅2hb()3 2:= Flux_per_eV 6.078 1047×s4k=Typically ~rτ1 nsec2π⋅h_bar⋅()c⋅mkg⋅τ_r1Flux_per_eV1C0m06⋅ Ep⋅⋅:=EE232 Lecture 7-7©2008. University of California6τ_r 5.443 1010−× s=Alternative Form for Gain23220022 2212188cvrrrrhhcCePnE nhλπτπτλ⋅= ⋅=JG 02() ( )()8rggrrhgEfnλωρω ωπτ=−===EE232 Lecture 7-8©2008. University of


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Berkeley ELENG 232 - Einstein's AB Coefficients Spontaneous Emission

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