E1 f h rspon n 2 induced emission rinduced n 2 u f

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Unformatted text preview: porFonality populaFon of ground state 5 Emissive transiFons Possible transi0ons Spontaneous emission E2 N2 radiaFon in the cavity E2 ! E1 f= h Rspon ! N 2 INDUCED EMISSION Rinduced ! N 2 u ( f , T ) E1 N1 Rdown = Rinduced + Rspon Rdown = N 2 [ B21u ( f , T ) + A21 ] Coefficients of proporFonality 6 Einstein coefficients Possible transi0ons Spontaneous emission A21 N 2 E2 N2 STIMULATED EMISSION B21 N 2 u ( f , T ) AbsorpFon B12 N1u ( f , T ) E1 N1 A21, B21 and B12 are called the Einstein coefficients and are very important for the operaFon of a laser. 7 Equilibrium •  At equilibrium rate of transiFons up = down Rup = Rdown = R Rup = B12 N1u ( f , T ) R N1 = B12 u ( f , T ) Rdown = N 2 [ B21u ( f , T ) + A21 ] R N2 = B21u ( f , T ) + A21 N1 u ( f , T ) B21 + A21 = N2 u ( f , T ) B12 N1 = e!!/kT N2 from slide 4 A21 = u ( f , T )[ B12 ehf /kT ! B21 ] 8 High temperature limit A21 = u ( f , T )[ B12 ehf /kT ! B21 ] 8! hf 3 1 u( f , T ) = Planck’s distribuFon: c 3 ehf /kT " 1 For high T: ehf / kT ! 1+ hf kT 8" f 2 u ( f , T ) ! 3 kT c At high temperatures: ehf / kT !1 Independent of T! 8! f 2 A21 = 3 kT [ B12 " B21 ] c Therefore: Einstein coeff.’s for sFmulated emission and absorpFon are equal: 12 B B12 ! B21 = 0 = B21 ! B 9 A and B Using B12 = B21 ! B we get: A = u ( f , T )[ ehf /kT ! 1] B B 1 = A u ( f , T )[ ehf /kT ! 1] 8! hf 3 1 u( f , T ) = c 3 ehf /kT " 1 B c3 SFmulated/Spontaneous: = A 8! hf 3 Note: sFmulated emission/spontaneous emission coefficients are ∝ f ­3, so shorter wavelength lasers are harder to make and operate than longer wavelengths… 10 Number of photons Rdown = N 2 [ Bu...
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