清華激光原理第七講(4月7日)

1、第第 7 講講 四能級(jí)系統(tǒng)單模速率方程的建立 多模速率方程的建立 均勻加寬工作介質(zhì)的增益系數(shù) 從速率方程出發(fā)推導(dǎo)小信號(hào)增益系數(shù)表達(dá)式 小信號(hào)增益曲線 增益線寬 均勻加寬工作介質(zhì)的增益飽和 大信號(hào)增益系數(shù)、強(qiáng)光作用下的弱光增益系數(shù) 非均勻加寬工作介質(zhì)增益系數(shù) 作業(yè)：4-7、4-13、4-20pump21laserlaserlaserTwo, three, four level lasers2103213210He + e He* 21S0，23S1He* + Ne Ne* + He + DE100ns10ns與管壁碰撞鉺離子能級(jí)圖0.98nm4.3 Rate equations examples

2、 (three, four levels system ) Differential equations describing the dynamics of population and photon density of various levels Physical basic for rate equations: Einstein relationshipRed ruby, Er3+ doped fiberw13A31S31S32A21S21w21w12E1E2E332313231SASS2121AS He -Ne, Nd: YAG2121323030,ASSAS10S較大w03A3

3、0S30S32S21A21W21E3E2E1E0W12S10KTEE011. Four levels system1) single mode oscillation (l th mode, n)3030301010021112221212323212121212123232303230303,AnWnSndtdnNgnAnffnASnSnWnWnASnSndtdnASnWndtdnlnnnnnnnn3210RlllNWnWndtdN121212)vNNgnAhNgBWlll02102102121,nnnnnnnnw03A30S30S32S21A21W21E3E2E1E0W12S10沒有包括沒

4、有包括A21引起的光子數(shù)？引起的光子數(shù)？(ignore S30)（4.4.13）)RlllRlllNvNnffnNWnWndtdNnn0211122121212, )RltcLteIeItIeII0001 Rate equations for practical laser are set up according to the various physical processes Different expressions of rate equations for a same laser)llvNnffnnnRdtdnvNnffnnRdtdn02111222121111021112222

5、22,nnnnR1, R2 is atom density pumped per second into E1 and E2, respectively.1, 2 is lifetime at E1, E2, respectively; 21 is spontaneous emission (熒光 fluorescence）lifetime for E2 E1.E2E1R2R12121cLRlConsider loss onlyPhoton lifetime光子壽命（2.1.14）Laser transition)llvN,nffnnnRdtdnvN,nffnnRdtdn02111222121

6、11102111222222nnnn)101212121212121212121212123232SnWnWnASndtdnWnWnASnSndtdnE2E1R2R12121PumptransitionsLaser transitionw03A30S30S32S21A21W21E3E2E1E0W12S102) Rate equations for multimode oscillation Serial number of mode Frequency of mode Photon number of serial mode nll ,ll21nlll,nnnn21nllllNN,NN21Me

7、thod : 對應(yīng)每個(gè)模式分別建立一個(gè)速率方程, 序數(shù)相應(yīng)變化)lllvN,nffnSAnSndtdn0211122212123232nn)RllllNvNnffndtdNnn0211122,dtdNdtdNnll,1)0202210218nnnnn,ghvA,DnFng(nn0)g(nn0)g(nn0)g(nn0)nSimplifications Line shape function rectangle (矩形) Same loss, same lifetime of photon for each mode)vNNvvNlllll2121021,nn矩形面積原譜線面積矩形面積原譜線面積R

8、easons of simplificationDo not need to consider differences of the modes for studying questionNeglect the difference of diffraction loss in the modesAccording to the above simplified model, multimode rate equations for four levels system as followingN total photon density for all modes泵浦效率泵浦效率030101

9、03232212211122213230303WnSndtdnSnAnvNnffndtdnSnWndtdnRNvNnffndtdN211122nnnnn32102121212SAA3032321ASS)30323ASn)21212SAnMultimode rate equations for four levels laser system熒光量子效率熒光量子效率w03A30S30S32S21A21W21E3E2E1E0W12S10Pumping efficiency: the fraction of the total pump power that is useful in creatin

10、g the population inversionFluorescence (quantum) efficiency from E2E1, depends solely on characteristics of medium, it is the upper limit for performance of any laser.3032321ASSw03A30S30S32S21A21W21E3E2E1E0W12S102121212SAATotal quantum efficiency (總量子效率總量子效率)21F 發(fā)射熒光的光子數(shù)從光泵吸收的光子數(shù)（Photon numbers of e

11、mitting fluorescence）（Photon numbers absorbed from pumping） Rate equations Expression of gain coefficient (effecting factors) behavior of gain saturation (增益飽和) (for homogeneous and inhomogeneous medium)4.5 Gain coefficient in homogeneous broadening medium 1、Small signal gain coefficient for steady

12、state (four levels）)RlNvN,nffndtdNnn0211122I= Nhnvdz=vdt)210,dInIdzn n D)021,nnngD不計(jì)損耗IdzdIg II+d IdzDn0 I= Nhnv dz=vdtzgeII00(1.3.6)030212021,8,nnnnngAvnngDD *1) Gain coefficient is proportional to population inversion Dn (four levels system as example)2100032,lld nnnvNn Wdtn nDD D212121SA Gain coe

13、fficientDiscussing key factors dependent on gain coefficient穩(wěn)態(tài))3030301010323212120212303230303AnWnSndtdnSnASnN,gnAndtdnASnWndtdnlDnnn030323WnSn0332WS03n030101WnSn穩(wěn)態(tài)0310WS01n2nn D2nnD *Reason: optical intensity in medium is very small, the change of D Dn introduced by STE can be neglected in small si

14、gnal case030200WnndtndDD)0302021,WnnvNndtndlDDDnn20302030000nWnWnndtndDDD In small signal case, Dn0 also independent on optical intensity, dependent on pumping rate W03 Dn0 穩(wěn)態(tài), 0, 013nn閾值附近n2很小)02100,nnngD)210,gnn n DSmall signal (unsaturated) gain coefficient g0 is dependent on Dn0, independent on

15、optical intensityreview：如何理解小信號(hào)情況？小信號(hào)2) Gain curve - g0 varying with frequency )021202002100,8,nnnnnngAvnngDD)021,nnngD Small signal gain curve depends on spectrum line shape (小信號(hào)增益曲線的形狀完全取決于譜線線型函數(shù))Small signal gain coefficient )HHAvnngnnnDDD2022120210004 ) )220200022HHHHggnnnnnnDD)0,nvg )n0g=21STE

16、cross section at n021 增益線寬 熒光線寬DnF （自發(fā)輻射線寬DnDnH ） )n0Hg0nFnDn20300Wnn D ) )02103Hgdepends onand Wnn3) g0(n0) is proportional to l02, inversely proportional to spectral width)200002102124DAgnnlnn D DD3.39mm632.8nm3S3P2PHe-Ne激光器類型激光器類型熒光線寬熒光線寬（s s-1-1）氦氖1.5109Nd:YAG1.951011釹玻璃7.51012若丹明 6G5101231013Ga

17、AlAs (0.85mm)1013InGaAsP (1.55mm)101210132、Gain Saturation（增益飽和）大信號(hào)情況 Under the action of incidence light with In1 , at n1 (considering STE process)1) Saturation of inversion population What is gain saturation？Gain coefficient is decreased as STE intensity becomes strong. Reason of gain saturation：W

18、hen the intensity is strong enoughDgnnWI221)03020, 121wnnvNndtndDDDnn10INh vnn) )1100032003221102211020,1,11sn wn wnnIIvNIhnnn nn nnnDD 穩(wěn)態(tài)=1/Is(n)0d ndtD10InnnnD D Dsaturation of inversion population Saturation intensity at n0)0220122011122nIIvvnsHHDDDDnnnnn2210nhIs)221210222002,82HHA vhnn n nnnnD D

19、)10032211020,1n wnIhnn nnD 大信號(hào)情況（4.5.7）1sIIn,1sIIn0nD)10101snnIInnnDD 01nn小信號(hào)情況20032nnn WD Population n2 is barely affected by STE (即可忽略受激輻射的影響)0,gn n Incidence light at same frequency with different intensity, different effect on saturation of Dn Incident light with same intensity at different freq

20、uency, different effect on saturation of Dn210000nnIIIInnssDDDDnn)0220122011122nIIvvnsHHDDDDnnnnn01nn01nn21101HsIInnnnD043nnDDsII 1nFrequency range of saturation actionwhenif(Stimulating wave) At central frequency n0, STE probability is biggest, saturation action is strongest.The frequency depart fr

21、om n0, STE probability is reduced, saturation action becomes weak.2) Saturation gain coefficient in homogeneous medium (均勻加寬介質(zhì)的飽和（大信號(hào)）增益系數(shù)) Gain coefficient of quasi-monochromatic light at frequency n1, with the intensity of In1)021,nnngD)121211020,8HHvgInA gnnn nn D )HHAvngnnnDD2022120004) )DDsHHHH

22、IIgIg11122,22012001nnnnnnnnSaturation gain coefficient(4.5.5)0220122011122nIIvvnsHHDDDDnnnnn)1022102,2HHHgvnn nnnDDwhere(4.5.12) ) D DsHHHHIIgIg11122,22012001nnnnnnnn)110 swhere IInnn，) )2,000nnHsHgIgsII 1n Saturation gain coefficient is dropped to one half of un-saturated gain when01nn212101HsIInnn

23、nDgH(n1)nn1n0 )10ng Gain and saturation effect is very weak, (can be neglected), if the frequency is outside the range of思考題：大信號(hào)增益曲線寬度與小信號(hào)增益曲線寬度是否相等？思考題：大信號(hào)增益曲線寬度與小信號(hào)增益曲線寬度是否相等？)22110003202,sdnd nnnvNn WdtdthIn nnDD D Physical meaning of saturation intensity Is ：sII 1n 受激輻射造成n2的減小可以與其它自發(fā)輻射和無輻射躍遷造成的衰

24、減可以相比擬 由于受激輻射，促使E2能級(jí)壽命減小；增益降低 不僅是增益飽和與否的判據(jù)，又是決定腔內(nèi)光強(qiáng)即激光輸出功率大小的重要參數(shù) 其值由激光工作物質(zhì)本身性質(zhì)決定He-Ne: 632.8nm 0.3 w/mm2 CO2: 10.6mm 2w/mm2Ar ion: 514.5nm 7w/mm2 討論此命題的物理背景: 激光器中某一模式頻率首先起振, 成為強(qiáng)光；別的模式剛起振(弱光), 強(qiáng)光模式對剛起振的弱光模式的影響)D11,nnIgnIv3. 在強(qiáng)光強(qiáng)光 In1n1 作用下的弱光弱光 n n 增益系數(shù) 頻率為n1,光強(qiáng)為In1強(qiáng)光 同時(shí)有一頻率為n 的弱光入射 求強(qiáng)光對弱光增益系數(shù)的影響，即弱

25、光增益系數(shù)會(huì)如 何變化？1,1nnIn定性分析定量分析)021nn,ngD)0220122011122nIIvvnsHHDDDDnnnnn強(qiáng)光作用下的反轉(zhuǎn)粒子數(shù)強(qiáng)光作用下的反轉(zhuǎn)粒子數(shù)強(qiáng)光不僅使自身增益系數(shù)下降,而且使弱光增益系數(shù)也以同一比例下降, 其結(jié)果是整個(gè)增益曲線下降（為什么？增益線寬是否改變）為什么？增益線寬是否改變）復(fù)習(xí)復(fù)習(xí)：注意理解并區(qū)分強(qiáng)光增益及強(qiáng)光作用下弱光增益變化：注意理解并區(qū)分強(qiáng)光增益及強(qiáng)光作用下弱光增益變化) )DDsHHHHIIgIg11122,220122010nnnnnnnnnnsII 1n01nn) )2,01nnnHHgIg強(qiáng)光作用下的弱光增益系數(shù) )n0Hg)1

26、nnI ,gH )nHg1n0n弱光增益曲線弱光增益曲線)00,n n nD(4.5.16)（飽和增益）4.6 非均勻加寬工作介質(zhì)的增益系數(shù)及增益飽和 非均勻加寬小信號(hào)增益曲線為高斯線型 )021202002100,8,nnnnnnDgAvnngDD ) )D220100102ln4expDDDggnnnnn)2102023212212ln242lnDDnnnnDDDgAv1、非均勻加寬大信號(hào)增益系數(shù): 思路： Dn 按表觀中心頻率分類 )0000000,nnnnnDDdgndnD)1,1nnIgi000nnnd0nHnD 這部分粒子發(fā)射中心頻率為 線寬為 的均勻加寬譜線。這部分粒子對頻率n1

27、， 光的增益貢獻(xiàn)為dg按均勻加寬增益系數(shù)計(jì)算 (4.5.12)1nI)dgIg1,1nn) )DDsHHHHIIgIg11122,22012001nnnnnnnn)HHAvngnnnnDD20221200004表觀中心頻率小信號(hào)增益系數(shù)表觀中心頻率小信號(hào)增益系數(shù)大信號(hào)增益系數(shù)大信號(hào)增益系數(shù) (4-5-12)(4-5-14) )0000000,nnnnnDDdgndnDnn0)1ngHnDn0n(n0)nn0)dn0dn0n0n1) DDDDsHHHDIId,gnAvdg1124222012020000212nnnnnnnnnn)000nnDdn)021n1n) DD DDsHHHDIIdgnA

28、vdg11242,22012020000212nnnnnnnnnn000nnnd的粒子發(fā)射中心頻率為n0, 線寬為DnH的譜線) D DDD022010002202021211112,24,sHDHHiIIdgnAvIgnnnnnnnnnnnn) dgIg1,1nn00nnHDnnDD)0100,nnnnDDgg(4-6-1)DDDDsHDHHiIIdgnAvIg1112,24,2201001220202121nnnnnnnnnnnn )000nnDdn201HnnnD )00nHg)012002122201001220202121,1812,24,111nnnnnnnnnnnnnnnnDssHDHHigIInAvIIdgnAvIgD D DDD) )D2201001012ln4exp11,111DsisiiIIgIIgIgnnnnnnnnn )21202120002ln4DDnnnDDAvng)2012ln40010DDeggDDnnnnn(4-6-2)12221112HsuduarctgauaaIInnD )10ng 非均勻加寬介質(zhì)大信號(hào)增益系數(shù)) )D2201001012ln4exp11,111DsisiiIIgIIgIgnnnnnnnnnsII1n大信號(hào)情況)11,1nnnIgIi增益飽和大信號(hào)增益系數(shù))siiIIgIg