principles of lasers激光原理第七章 速率方程

2 2 p g P sp P P g P p
2
sp
Below and above the threshold
N2 NP
Rst
167
Rp RPth RP RPth RP
Carrier reservoir Rsp
Below the threshold
N 2 R p sp N P = R p p
gth 1 vg p Nth N 2th 1 vg p
2 1
Ignore the spon. emission
dN P N =vg gN P P dt p 1 N p (t ) N p (0) exp v g g t p
Three-level lasers: fiber lasers like Er (铒1550 nm), Yb (镱1030 nm) , Ho (钬2000 nm) , Tm (铥2000 nm) doped fiber lasers
Four-level laser
Assumptions: The carrier density (population) is uniform inside the gain medium The photon density (electric field) is uniform inside the cavity The light volume is the same as the gain medium volume Only one mode oscillates in the cavity Then, the rate equation is space independent. The lifetimes of level 3 and level 1 is very fast
The rate equations g d N g 2 1 g 1 R p 1 2 vg gN P N 2 Nt dt g1 g1 sp g1 dN P N 1 g2 vg gN P P N Nt dt p sp g1
g dN d N d g N 2 2 N1 1 2 2 dt dt g1 g1 dt
g2 N2 1 1 g2 g2 1 N N N N Nt t 1 2 g g g 1 sp sp 1 sp 1
1 1 vg m vg ln m 2 L R1R2 If R1 =R 2 1 vg 1 1 ln L R 1
The total output power from the two mirrors is
Pout
E
m
N P hV p vg m
Examples 7.1
fast pump fast
The rate equations dN 2 N R p W N 2 N1 2 dt sp dN P N N W N 2 N1 P 2 dt p sp N3 0 N1 0 N g const

159
Gain clamping:
Above the threshold, the gain does not increase with the pump rate, but remains the same as that at the threshold. (carrier population clamping) The photon density increases linearly with The rate equations dN =0 dN N the pump rate. R v gN dt dt dN N N N 2th The threshold pump rate is (Np>=0) =v gN N P = p R p dt N 2th 1 sp Rpth sp vg p sp
N3
N2
W N 2 N1 =W N = FN N P vg N N P vg N 2 vg gN P
N1 Ng
The rate equations dN 2 N2 R p vg gN P dt sp dN P NP N2 vg gN P dt p sp
165
The steady-state solutions are N 2 R p sp
N P = R p p
N2
NP
Both the carrier density and
the photon density increase
RP
The rate equations dN 2 N R p vg gN P 2 dt sp dN P N N =v g gN P P 2 dt p sp
gain coefficient g N
Photon lifetime
p
1 vgT
The output power
The energy inside the cavity
161
E N P hVp
The energy output (loss) rate of the two mirrors
gain coefficient
g g N = N 2 2 N1 g1
Chapter 7_L15
163
Lasing threshold and power
Four-level laser
Three-level laser
Four-level laser
At the threshold
v g gth N 2th R pth 1
Hale Waihona Puke Above the threshold
N 2 N 2th N P = p R p R pth
R2
R1 L
Round-trip time
Lo t 2 c
Loss coefficient
1 1 T i ln 2 L R1R2
Some relations
Photon flux (cm -2s -1 ) F=vg N P Energy density (J/cm3 ) N P h Light intensity(mW/cm 2 ) I vg N P h Fh vg
is small), and there is only spontaneous emission, then the rate equations become Under steady-state condition dN 2 N2 Rp dt sp dN 2 dN P 0; =0 dt dt dN P NP N2 = dt p sp
fast pump fast N1 Ng
158 N3
N2
The laser emission occurs at level 3 level 2
The population inversion N N 2 The gain g N N 2
N3 0，N1 0
Four-level laser
The rate equations dN 2 N R p vg gN P 2 dt sp dN P N N =v g gN P P 2 dt p sp
Below the threshold
Below the threshold, the stimulated emission is negligible (photon number
sp
N2
is small, and is negligible in most
discussions.
Rp is the pump rate is the spontaneous emission factor
Four-level laser
l
160
Optical length
Lo L l nrl
All the sublevels are strongly coupled and hence in thermal equilibrium.
N1
The rate equations dN 2 N R p v g gN P 2 dt sp dN P N N v g gN P P 2 dt p sp N1 N t N 2 N3 0
The carrier lifetime
1
164
To achieve lasing,
N p (t ) N p (0) vg g 1
sp nr Condition for possible continuous-wave lasing

1

1
p
Thus, the lasing threshold is
Four-level lasers: Ionic crystal lasers like Nd3+:YAG (Neodymium-doped Yttrium Aluminium Garnet 掺钕 钇铝石榴石 1064 nm); Gas lasers like CO2 (10.6 um), He-Ne (632.5 nm)
m
(Quasi-) three-level laser
fast pump
162
N3
N2
In a (quasi-) three-level laser, the lower laser level is the ground level, or a sub-level of the ground level.