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1 ECE 240 for Cvity from ECE 240 (See Notes on ) Quntum Rdition in ECE 240 Lsers - Fll 2017 Lecture 11 1

2 Free Spce ECE 240 for Cvity from Quntum Rdition in The electromgnetic mode density in free spce is required for severl spects of lsers. The mode density dn/dν per unit frequency in volume V for both polriztions is given by modified form of (2.1.40) of M&E dn dν where ν 0 is the operting frequency.. = ρ ν = 8πν2 0 c 3 V, Using ν = ω/2π nd dν = dω/2π, the mode density in ngulr frequency is dn dω. = ρ ω = ω2 0 π 2 c 3 V, where c = c 0 /n is defined in the medium. Using λν = c, we cn lso write this in wvelength units s ( ) dn V 1 dλ = ρ λ = 8π λ 3 λ, where λ is defined in the medium. ECE 240 Lsers - Fll 2017 Lecture 11 2

3 for Cvity ECE 240 for Cvity from Quntum Rdition in This cvity is chrcterized by photon lifetime τ p. Also chrcterized by the finesse F. (See Lecture 10.) The cvity cn lso be chrcterized by qulity fctor Q defined in the sme wy s tht of filter so tht Q = ν 0 ν = ω 0 ω. The bndwidth ν of single-mode cvity is the pproximte frequency width of one mode. The reciprocl 1/ ν of the bndwidth is n estimte of the number of modes per unit frequency, which is the mode densityρ cv for the cvity ρ cv = 1 ω = Q ω 0, where we work with ngulr frequency units, which re more common. ECE 240 Lsers - Fll 2017 Lecture 11 3

4 Simple Estimte of Effect of the Cvity - ECE 240 for Cvity from Quntum Rdition in Most derivtions of the effect of cvity use single polriztion. Then the free-spce mode density is reduced by fctor of two. The rtio of the free spce mode density to the cvity mode density is then ρ cv Q/ω = 0 ρ ω 2V ω0 2/π2 c 3 = 1 ( ) V 4π Q λ 3. We see tht the this fctor depends on the volume of the cvity V s compred to the wvelength λ nd the photon lifetime in the cvity s expressed by Q. ECE 240 Lsers - Fll 2017 Lecture 11 4

5 from ECE 240 for Cvity from Quntum Rdition in Now plce bck-the-envelope results on firm foundtion by determining the effect of concentric resontor on the rdition emitted by dipole. Strt with clssicl model bsed on Mxwell s eqution for the rte of emission for dipole in free spce. Consider hrmonic dipole moment p(t) = e sin(ω 0 t) x where e is the electron chrge nd x is unit vector in the direction of the oscilltion of the chrge. The trnsverse prt of the electric field, clled the rdition field E rd (r, t), t position r from point dipole locted t position R cn be written s 1 E rd (r, t) = 1 ω 2 ( ) 0 e jk r R ( nr 4πɛ c 2 x) n r, r R where n r is unit vector in the direction of r R, c = c 0 /n is the speed of light in the medium, ɛ = ɛ 0 ɛ r is the permitivity in the medium, nd r R λ. Note tht this ssumption will limit the size of the cvity for which the nlysis is ccurte. 1 See Section 2.5 of M nd E first edition for complete derivtion ECE 240 Lsers - Fll 2017 Lecture 11 5

6 Power into Solid Angle - 1 ECE 240 for Cvity from Quntum Rdition in If the dipole is linerly polrized, then the rdited power per solid ngle is given by (See the second to lst line of (2.5.13) in M&E) ) 2 1 sin 2 θ, dp dω = 1 4πɛ 4π 1 c 3 where θ is the ngle between x nd r. If p(t) = Re[p 0 e jω0t ], then ( ) d Solve for 2 2 p(t) dt 2 ( ( d 2 p(t) dt 2 d 2 p(t) dt 2 = ω 2 0Re[p 0 e jω0t ] ) d 2 2 p(t) dt 2 = ω0 p cos 2 (ω 0 t + rg p 0 ). ECE 240 Lsers - Fll 2017 Lecture 11 6

7 Power into Solid Angle - 2 ECE 240 for Cvity from Quntum Rdition in Averging over time intervl tht is long s compred to the period T = 2π/ω 0, we hve cos 2 (ω 0 t + rg p 0 ) = 1/2. Therefore, for lossless time-hrmonic dipole we hve dp dω = 1 p 0 2 ω0 4 4πɛ 8π c 3 sin2 θ, where p 0 = ex is the mgnitude of the dipole moment. This is the power per solid ngle in one polriztion of dipole. ECE 240 Lsers - Fll 2017 Lecture 11 7

8 Totl Time-Averged Rdited Power ECE 240 for Cvity from Quntum Rdition in The time verge totl power rdited by the dipole in one polriztion is determined by integrting over solid ngle of 4π sterdins nd is (See (see (2.5.13) in M&E) P = 1 p 0 2 ω 4 2π π 0 4πɛ 8π c 3 dφ sin 3 θdθ. 0 0 = 1 ω0 4 4πɛ 3c 3 p 0 2 Substituting p 0 2 ω 4 0 /c3 = (4πɛ)3P into the previous eqution, we hve dp dω = 3P 8π sin2 θ. This expression reltes the totl time-verged rdited power P to the power per unit solid ngle. ECE 240 Lsers - Fll 2017 Lecture 11 8

9 ECE 240 for Cvity from Quntum Rdition in Strt with clssicl model of free (non-driven) hrmonic oscilltor given in Lecture 1, which cn be written in terms of the dipole moment p = ex In this form, we interpret τ sp = 1/σ. d 2 p dt 2 + σ dp dt + ω2 op = 0 The most generl solution for this is in the form of dmped oscilltion. The chrcteristic eqution is (cf. Lecture 1) The roots cn be written s x 2 + σx + ω 2 0 = 0 x i = σ/2 ± jω 0 1 (σ 2 /4ω 2 0 ). ECE 240 Lsers - Fll 2017 Lecture 11 9

10 ECE 240 Generl Solution for Cvity from Quntum Rdition in Choosing the positive root to gree with our convention for the time dependence, e jω0t, the generl solution is p(t) = Re [p 0 e σt/2 e jω0 1 (σ 2 /4ω 20 )t ] = p 0 cos (ω 0 Xt + rg p 0 ), where p 0 is the complex mplitude of the hrmonic dipole moment nd X = 1 (σ 2 /4ω 2 0 ). The time-verged rdited power is modified by including power loss term e σt P (t) = 1 ω0 4 4πɛ 3c 3 p 0 2 e σt Ifσ ω 0, then we cn define time-verge energy over single cycle of the oscilltion. This condition defines wek-coupling regime. In this regime, it is unlikely tht the emitted rdition is re-bsorbed by the dipole. Tht condition defines the strong-coupling regime. ECE 240 Lsers - Fll 2017 Lecture 11 10

11 ECE 240 Energy in Dmped Hrmonic Oscilltor for Cvity from Quntum Rdition in Using p = ex, we cn write the kinetic energy ( KE = 2 1 mv(t)2 = m dp(t) 2e 2 dt Similrly, we hve for the potentil energy PE = 1 2 kx(t)2 = m 2e 2 ω2 0p(t) 2 where k = mω0 2 hs been used. Therefore, the totl verge energy is ( ( ) ) E = m 2 2e 2 ω0p(t) 2 2 dp(t) +. dt Substituting p(t) into this eqution nd verging so tht cos 2 (ω 0 t) = 1/2, the KE nd PE ech hve n equl contribution, which cn be written s E(t) = mω2 0 2e 2 p 0 2 e σt. ) 2 ECE 240 Lsers - Fll 2017 Lecture 11 11

12 Equte the Energy with the Power ECE 240 for Cvity from Quntum Rdition in Now equte the time-verged energy loss to the integrl of the time-verged power loss so tht E(t) = P (t)dt mω 2 0 2e 2 p 0 2 e σt = 1 σ 1 ω0 4 4πɛ 3c 3 p 0 2 e σt Solving for 1/σ = t sp, we cn write the clssicl spontneous emission lifetime s 1 = σ = A = 1 2e 2 ω0 2 τ sp 4πɛ 3mc 3. Putting in vlues yields lifetime of bout 10 ns for n oscillting electron t wvelength of one hlf micron. The constnt 2e 2 /3mc 3 is chrcteristic time tht is on the order of the time it tkes light to trvel n electron rdius r 0 = e 2 /mc 2. ECE 240 Lsers - Fll 2017 Lecture 11 12

13 Quntum ECE 240 for Cvity from Quntum Rdition in The spontneous emission rte A nm between to energy levels n nd m is presented in E&M Section 7.6 nd is given by A nm = 1 4 4πɛ 3 D 2 nmω 3 0 hc 3 where D nm is the squred-mgnitude of the dipole moment so tht D nm = e 2 r nm 2 r nm is the expected vlue of the overlp between the two wvefunctions tht define the two energy sttes. (See Lecture 3.) The expression derived from quntum mechnics is equl to the clssicl expression if the clssicl potentil energy 1 2 kx2 = 1 2 mω2 0 x2 is set equl to one hlf the vcuum stte energy or 1 2 ( hω/2). Mking this substitution, we hve for the clssicl result 1 = σ = A = 1 4e 2 x 2 ω0 3 τ sp 4πɛ 3 hc 3. ECE 240 Lsers - Fll 2017 Lecture 11 13

14 Vlidity of Clssicl Tretment Quntum ECE 240 for Cvity from Quntum Rdition in Given the equivlence between the clssicl description nd the quntum description of spontnteous emission, the effect of the cvity on the spontneous emission rte bsed on clssicl nlysis. We lso dopt this pproch. However, we expect this nlysis to brek down for smll cvities much less thn the wvelength λ for two resons. The first is tht not ll terms of the dipole field re used. The second is tht the true quntum nture of the interction is not ccounted for. ECE 240 Lsers - Fll 2017 Lecture 11 14

15 Rdition in ECE 240 for Cvity We now consider how the clssicl dipole emission rte is ffected by plcing dipole in concentric resontor with equl rdii of curvture plced t ±. from r d Ω m 2b Quntum Rdition in Imge Ω side L=2 The origin is defined t the focl point of the mirror nd the center of the dipole is displced from the origin by r d = x d x + z d ẑ. This displcement results in n imge tht is offset from the origin by r d s is shown in the Figure. ECE 240 Lsers - Fll 2017 Lecture 11 15

16 Solid Angles ECE 240 for Cvity from Quntum Rdition in The solid ngle Ω side out the sides of the resontor nd twice the solid ngle Ω mir tht single mirror subtends is equl to 4π or Ω side + 2Ω mir = 4π. Suppose tht Ω mir is smll. Then using the geometry given in the Figure Ω mir 4πb 2 /L 2, where the prmeters re defined in Figure. If the resontor is stble, then the power incident on mirror tht subtends solid ngle Ω mir nd the power in the solid ngle Ω side is equl to the totl rdited power. Therefore, t n ngle θ = π/2 the power P mir emitted into the totl solid ngle 2Ω mir subtended by the mirrors cn be expressed in terms of the totl time-verged power emitted from the dipole ( ) dp P mir = 2Ω mir = 3P dω θ=π/2 4π Ω mir. ECE 240 Lsers - Fll 2017 Lecture 11 16

17 Totl Rdited Power Ignoring the Cvity ECE 240 for Cvity from Quntum Rdition in Ignoring the effect of the cvity, the power tht is emitted out the side of the cvity is then ( P side = P P mir = P 1 3 ) 4π Ω mir Note tht P mir is only the effect of the solid ngle subtended by the cvity mirrors nd does not include the effect of the cvity. ECE 240 Lsers - Fll 2017 Lecture 11 17

18 ECE 240 Effect of the Cvity-1 for Cvity from Quntum Rdition in The power P cv including the effect of the cvity hs two fetures. At resonnce nd for highly reflective cvity such tht R 1, the power inside the cvity is relted to the power outside the cvity P mir emitted through single mirror by P mir = (T /2)P cv where T = 1 R is the power trnsmission for one mirror. We then hve where nd the finesse of the cvity is P cv 2 P mir T = 2 1 R F F = 4R (1 R) 2 F = π 2 F π R 1 R (See Lecture 10.) ECE 240 Lsers - Fll 2017 Lecture 11 18

19 Effect of the Cvity-2 ECE 240 for Cvity from Quntum Rdition in The second feture is the frequency dependence of the cvity, which is given by the Airy function for tht cvity (see Lecture 10). Therefore, we cn write the rtio P cv/p mir s 2 P cv P mir F }{{} Power enhncement by cvity F sin 2 (kl) }{{} Frequency dependence (Airy function). (1) The first term in (1) is the enhncement of the power from the effect of the cvity nd is the energy storge effect of the resontor. The second term in (1) is the frequency dependence s expressed by the Airy function. 2 (See Ref. 3 for more detiled nlysis.) ECE 240 Lsers - Fll 2017 Lecture 11 19

20 Mximum Enhncement ECE 240 for Cvity from Quntum Rdition in The mximum enhncement of the power cused by the effect of the cvity occurs if sin 2 (kl) = 0 with ( Pcv P mir )mx 2 F = 1 R Q, where the reltionship between the finesse nd Q is discussed in Lecture 10 nd Verdeyen Section 6.3. ECE 240 Lsers - Fll 2017 Lecture 11 20

21 Cvity Q nd Finesse (repet from Lecture 10) ECE 240 for Cvity from Quntum Rdition in The Q of cvity is defined s Q = ν 0 ν = ω 0 ω This vlue differs from the definition of the finesse by fctor of Q F = ν 0/ ν ν fsr / ν = ν 0 = ν 0 ν fsr (c/2d) = 2d λ for resonble sized resontors. Using the definition of the finesse F π/(1 R) for R 1, we hve tht Q 1 1 R with the scling fctor being different thn the scling fctor for the finesse. ECE 240 Lsers - Fll 2017 Lecture 11 21

22 Minimum Enhncement (Enhibited) ECE 240 for Cvity from Quntum Rdition in The minimum vlue occurs if sin 2 (kl) = 1 with ( Pcv P mir )min 1 = 1 R F 2 Q 1. We see tht reltive to the rdition from dipole in free spce tht subtends solid ngle Ω mir, the rdition into cvity mode is enhnced or inhibited by fctor tht is proportionl to the Q of the cvity. ECE 240 Lsers - Fll 2017 Lecture 11 22

23 Totl Power Into the Cvity ECE 240 for Cvity from Quntum Rdition in Using nd we cn we cn write P mir = 3P 4π Ω mir P cv F P mir 1 + F sin 2 (kl), ( P cv 3P ) F 4π 1 + F sin 2 Ω mir (kl) We see tht the totl rdition rte into the cvity depends on both the solid ngle Ω mir nd the finesse (or the Q) of the cvity. ECE 240 Lsers - Fll 2017 Lecture 11 23

24 Mx/Min Totl Power ECE 240 for Cvity from Quntum Rdition in The mximum totl power in the cvity occurs if R 1. so tht ( ) 1 3 P mx = P 1 R 2π Ω mir. The minimum totl power into the cvity occurs P mx = 3P (1 R) Ω mir. 8π Note tht the minimum vlue is less thn the rte if there ws no cvity, so tht P cv = P mir so tht P min = P mir = 3P 4π Ω mir. ECE 240 Lsers - Fll 2017 Lecture 11 24

25 Vlidity of Anlysis ECE 240 for Cvity from Quntum These expressions, derived using clssicl nlysis, re good strting points for n nlysis. However, they re expected to brek down for cvities on the order of the wvelength λ becuse they do not include ll of the relevnt physics. Rdition in ECE 240 Lsers - Fll 2017 Lecture 11 25

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