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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 Spring '14
 Physics, Light

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