清华激光原理第七讲(4月7日)

清华激光原理

第7讲 四能级系统单模速率方程的建立 多模速率方程的建立 均匀加宽工作介质的增益系数从速率方程出发推导小信号增益系数表达式 小信号增益曲线 增益线宽

均匀加宽工作介质的增益饱和大信号增益系数、强光作用下的弱光增益系数

非均匀加宽工作介质增益系数作业：4-7、4-13、4-20

清华激光原理

Two, three, four level lasers2 2 laser pump 1 1 3 0

3 2laser 1 2 laser

1

0

清华激光原理

100ns

10ns

0.98nm

与管壁碰撞

铒离子能级图

He + e He* 21S0,23S1

He* + Ne Ne* + He + DE

清华激光原理

§4.3 Rate equations examples (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 fiberE3

He -Ne,E3

Nd: YAGS32

S3 E22

E2w03 A30 S30 S21 A21 W21 W12

w13 A3 S31 1

A2 S2 w2 w11 1 1 2

× ×

E1S10

E1

E0

S31 S32

A31 S32

S30 , A30 S32E1 E0 KT

S21 A21

S21 A21

S10 较大

清华激光原理

1. Four levels system1) single mode oscillation (l th mode, n)E3 E2S32 S21 A21 W21 W12 dn3 w03 A30 S30 n0W03 n3 S32 A30 ) (ignore S30) E1 dt S10 dn2 n3 S32 n2 S 21 A21 ) n1W12 n2W21 E0 dt f A ~ n3 S32 n2 S 21 A21 ) n2 2 n1 21 g n ,n 0 )N l f1 nn dn0 ~ W21 B21g n ,n 0 ) N l hn n1S10 n0W03 n3 A30 dt

dNl Nl n2W21 n1W12 dt Rln0 n1 n2 n3 n

21 n ,n 0 )N l v (4.4.13)没有包括A21引起的光子数？

A21 ~ g n ,n 0 )N l nn

清华激光原理

dNl Nl n2W21 n1W12 dt Rl f2 N n2 n1 21 n ,n 0 ) vN l l f1 Rl

I1 I 0 e

t L c

Consider loss only

I t ) I 0 e I 0e

Photon lifetime 光子寿命

t

Rl

Rate equations for practical laser are set up according to the various physical processes Different expressions of rate equations for a same laserdn2 n2 f2 R2 n2 n1 21 n ,n 0 )vN l dt 2 f1 dn1 n n f R1 1 2 n2 2 n1 21 n ,n 0 )vN l dt 1 21 f1 E2

L c (2.1.14)

Rl

2 1

R2E1 R1

21Laser transition

R1, R2 is atom density pumped per second into E1 and E2, respectively.

1, 2 is lifetime at E1, E2, respectively; 21 is spontaneous emission (荧光 fluorescence)lifetime for E2 E1.

清华激光原理

E3 E2w03 A30 S30

S32 S21 A21 W21 W12

E1S10

dn2 n3 S32 n2 S 21 A21 ) n1W12 n2W21 dt dn1 n2 S 21 A21 ) n1W12 n2W21 n1S10 dt

E0

E2 R2 E1 R1

2

21

Laser transition

dn2 n2 f2 R2 n2 n1 21 n ,n 0 )vN l dt 2 f1 dn1 n1 n2 f2 R1 n2 n1

21 n ,n 0 )vN l dt 1 21 f1

1

Pump transitions

清华激光原理

2) Rate equations for multimode oscillationSerial number of mode

l l1 ,l2 ln1 2 n

Frequency of mode

n n l ,n l n l

Photon number of serial mode

N l N l1 , N l2 N ln

Method : 对应每个模式分别建立一个速率方程, 序数相应变化A21v 2 ~ 21 n ,n 0 ) g n ,n 0 ) 2 8 hn 0

dn2 f n3 S 32 n2 A21 S 21 ) n2 2 n1 21 n l ,n 0 )vN l dt f1 l dNl Nl f2 n2 n1 21 n ,n 0 )vN l dt f1 Rl

dNl1 dt

,

dNln dt

清华激光原理

g(n n0)

g'(n n0)

Reasons of simplificationg'(n n0)

g(n n0)

DnF n

Do not need to consider differences of the modes for studying question Neglect the difference of diffraction n loss in the modes

矩形面积=原谱线面积

Simplifications Line shape function

rectangle (矩形)

Same loss, same lifetime of photon for each mode

21 n l ,n 0 )vNl 21v Nl 21vNl l

According to the above simplified model, multimode rate equations for four levels system as following

清华激光原理

Multimode rate equations for four levels laser systemdn3 n3S32 n0W03 dt 1n3 S32 A30 )泵浦效率 1

S32 S32 A30

dn2 f2 n2 A21 n2 n1 21vN n3S32 n2 A21 S21 ) dt f1 2 dn0 A21 n1S10 n0W03 2 荧光量子效率 dt A21 S21 dN f N n2 2 n1 21vN dt f1 R E3 E2w03 A30 S30 S21 A21 W21 W12 S32

n0 n1 n2 n3 n

N – total photon density for all modesE0

E1S10

清华激光原理

E3E2w03 A30 S30

S32 S21 A21 W21 W12

S32 1 S32 A30

E1S10

E0

Pumping efficiency: the fraction of the total pump power that is useful in creating the population inversion

2

A21 A21 S21

Fluorescence (quantum) efficiency from E2 E1, depends solely on characteristics of medium, it is the upper limit for performance of any laser.

Total quantum efficiency (总量子效率) F 1 2

发射荧光的光子数从光泵吸收的光子数

(Photon numbers of emitting fluorescence) (Photon numbers absorbed from pumping)

清华激光原理

Rate equations Expression of gain coefficient (effecting factors) behavior of gain saturation (增益饱和)(for homogeneous and inhomogeneous medium)

§4.5 Gain coefficient in homogeneous broadening medium不计损耗

1、Small signal gain coefficient for steady state (four levels)Dn>0I I+d I

dz

I= NhnvdI g Idz

f2 dN N n2 n1 21 n ,n 0 )vN dt f1 Rl dz=vdt I= NhnvdI Dn 21 n ,n 0 ) Idz(1.3.6)

dz=vdtI I 0eg0z

g Dn 21 n ,n 0 )

清华激光原理

Gain coefficient

v 2 A21 ~ g Dn 21 n ,n 0 ) Dn g n ,n 0 ) 3 8 n 0

Discussing key factors dependent on gain coefficient

*1) Gain coefficient is proportional to population inversion Dn(four levels system as example)dn3 S W03 n3 0 n0W03 n3 S 32 A30 ) 稳态 n3S32 n0W03 32 dt dn2 A21 ~ Dn n2 Dn g n ,n 0 )N l n2 S 21 A21 ) n3 S 32 dt nn稳态 dn0 n1S10 n0W03 n1 S10 n0W03 n3 A30 dtS10 W03

Dn n2

n1 0

d Dn Dn Dn 21 n l ,n 0 ) vN l n0W03 dt 2

1 2 A21 S21

清华激光原理

dDn Dn Dn 21 n l ,n 0 )vN n0W03 dt 2*Reason: optical intensity in medium is very small, the change of Dn introduced by STE can be neglected in small signal case小信号

dDn 0 Dn 0 n0W03 dt 2dDn 0 0 Dn 0 n0W03 2 dt

n3 0, n1 0,阈值附近n2很小

稳态

Dn 0 nW03 2

g Dn 21 n ,n 0 )

g 0 Dn 0 21 n ,n 0 )

Small signal (unsaturated) gain coefficient g0 is dependent on Dn0, independent on optical intensity In small signal case, Dn0 also independent on optical intensity, dependent on pumping rate W03 Dn0 review:如何理解小信号情况？

清华激光原理

2) Gain curve -- g0 varying with frequency g 0 n )g Dn 21 n ,n 0 )g n ) Dn 21 n ,n 0 ) Dn0 0 0

v22 8 n 0

~ A21g n ,n 0 )

Small signal gain curve depends on spectrum line shape ~ g v ,n 0 ) (小信号增益曲线的形状完全取决于谱线线型函数) Small signal gain coefficient

Dn H 2 )2 0 0 g H n ) g H n 0 ) n n 0 )2 Dn H 2 )2v 2 A21 g n 0 ) Dn 21 Dn 2 4 2n 0 Dn H0 H 0 00 gH ) n

Dn 0 n0W03 2

21 -STE cross section at n0

0 g H n ) depends on 21 n ) and W03

Dn F

增益线宽

荧光线宽DnF (自发辐射线宽DnH )

n0

n

清华激光原理

激光器类型 氦氖 Nd:YAG 钕玻璃 若丹明 6G GaAlAs (0.85mm) InGaAsP (1.55mm)

荧光线宽(s-1) 1.5×109 1.95×1011 7.5×1012 5×1012~3×1013 1013 1012~1013

3) g0(n0) is proportional to l02, inversely proportional to spectral width3S

3.39mm

3P

632.8nm

l0 2 A21 g n 0 ) Dn 21 Dn 4 2 Dn D0 0 0

He-Ne

2P