Laser Rate Equations
5.4 The Four-Level Laser System
Fast nonradiative transition
Pump
Laser
N1 N2
N3 N4
E4
E3
E2
E1 hνp
hν1
Fig. 5.3 A four-level system; the pump lifts atoms from level E1to level E4 from where they decay rapidly to level E3and laser emission takes place between levels E3and E2. Atoms drop down from level E2to level E1
system is shown in Fig5.3. Level 1 is the ground level and levels 2, 3, and 4 are excited levels of the system. Atoms from level 1 are pumped to level 4 from where they make a fast nonradiative relaxation to level 3. Level 3 which corresponds to the upper laser level is usually a metastable level having a long lifetime. The tran- sition from level 3 to level 2 forms the laser transition. In order that atoms do not accumulate in level 2 and hence destroy the population inversion between levels 3 and 2, level 2 must have a very small lifetime so that atoms from level 2 are quickly removed to level 1 ready for pumping to level 4. If the relaxation rate of atoms from level 2 to level 1 is faster than the rate of arrival of atoms to level 2 then one can obtain population inversion between levels 3 and 2 even for very small pump pow- ers. Level 4 can be a collection of a large number of levels or a broad level. In such a case an optical pump source emitting over a broad range of frequencies can be used to pump atoms from level 1 to level 4 effectively. In addition, level 2 is required to be sufficiently above the ground level so that, at ordinary temperatures, level 2 is almost unpopulated. The population of level 2 can also be reduced by lowering the temperature of the system.
We shall now write the rate equations corresponding to the populations of the four levels. Let N1, N2, N3, and N4be the population densities of levels 1, 2, 3, and 4, respectively. The rate of change of N4can be written as
dN4
dt =Wp(N1−N4)−T43N4 (5.44) where, as before, WpN1is the number of atoms being pumped per unit time per unit volume, WpN4is the stimulated emission rate per unit volume,
T43=A43+S43 (5.45)
5.4 The Four-Level Laser System 107 is the relaxation rate from level 4 to level 3 and is the sum of the radiative (A43) and nonradiative (S43) rates. In writing Eq. (5.44) we have neglected (T42) and (T41) in comparison to (T43), i.e., we have assumed that the atoms in level 4 relax to level 3 rather than to levels 2 and 1.
Similarly, the rate equation for level 3 may be written as dN3
dt =Wl(N2−N3)+T43N4−T32N3 (5.46) where
Wl= π2c2
ω3n20A32gl(ω)I1 (5.47) represents the stimulated transition rate per atom between levels 3 and 2 and the subscript 1 stands for laser transition; gl(ω) is the lineshape function describing the 3↔2 transition and I1is the intensity of the radiation at the frequencyω = (E3−E2)/. Also
T32=A32+S32 (5.48)
is the net spontaneous relaxation rate from level 3 to level 2 and consists of the radiative (A32) and the nonradiative (S32) contributions. Again we have neglected any spontaneous transition from level 3 to level 1. In a similar manner, we can write
dN2
dt = −Wl(N2−N3)+T32N3−T21N2 (5.49) dN1
dt = −Wp(N1−N4)+T21N2 (5.50) where
T21=A21+S21 (5.51)
is the spontaneous relaxation rate from 2→1.
Under steady-state conditions dN1
dt = dN2
dt =dN3
dt = dN4
dt =0 (5.52)
We will thus get four simultaneous equations in N1, N2, N3, and N4and in addition we have
N=N1+N2+N3+N4 (5.53)
for the total number of atoms per unit volume in the system.
From Eq. (5.44) we obtain, setting dN4/dt=0 N4
N1 = Wp
(Wp+T43) (5.54)
If the relaxation from level 4 to level 3 is very rapid then T43 Wpand hence N4N1. Using this approximation in the remaining three equations we can obtain for the population difference,
N3−N2
N ≈ Wp(T21−T32)
Wp(T21+T32)+T32T21+Wl(2Wp+T21) (5.55) Thus in order to be able to obtain population inversion between levels 3 and 2, we must have
T21 >T32 (5.56)
i.e., the spontaneous rate of deexcitation of level 2 to level 1 must be larger than the spontaneous rate of deexcitation of level 3 to level 2.
If we now assume T21T32, then from Eq. (5.55) we obtain N3−N2
N ≈ Wp
Wp+T32
1
1+Wl(T21+2Wp)/T21(Wp+T32) (5.57) From the above equation we see that even for very small pump rates one can obtain population inversion between levels 3 and 2. This is contrary to what we found in a three-level system, where there was a minimum pump rate, Wpt, required to achieve inversion. The first factor in Eq. (5.57) which is independent of Wl[i.e., independent of the intensity of radiation corresponding to the laser transition – see Eq. (5.47)] – gives the small signal gain coefficient whereas the second factor in Eq. (5.57) gives the saturation behavior.
Just below threshold for laser oscillation, Wl≈0, and hence from Eq. (5.57) we obtain
N
N ≈ Wp
(Wp+T32) (5.58)
whereN = N3−N2is the population inversion density. We shall now consider two examples of four-level systems.
Example 5.1 The Nd:YAG laser corresponds to a four-level laser system (seeChapter 11). For such a laser, typical values of various parameters are
λ0=1.06μm (v=2.83×1014Hz), v=1.95×1011Hz,
tsp=2.3×10−4s, N=6×1019cm−3, n0=1.82 (5.59) If we consider a resonator cavity of length 7 cm and R1=1.00, R2=0.90, neglecting other loss factors (i.e.,α1=0)
tc= − 2n0d
c ln R1R2 ≈8×10−9s
5.4 The Four-Level Laser System 109 We now use Eq. (4.32) to estimate the population inversion density to start laser oscillation corresponding to the center of the laser transition:
(N)t= 4v2n30 c3
1 g(ω)
tsp tc
= 4v2n30 c3 π2vtsp
tc
(5.60)
where for a homogenous transition (see Section 4.5)
g(ω0)=2/πω=1/π2v (5.61)
Thus substituting various values, we obtain
(N)t≈4×1015 cm−3 (5.62)
Since (N)t N, we may assume in Eq. (5.58) T32 Wpand hence we obtain for the threshold pumping rate required to start laser oscillation
Wpt≈(N)t
N T32≈(N)t N
1 tsp
=4×1015 6×1019× 1
2.3×10−4 ≈0.3 s−1
At this pumping rate the number of atoms being pumped from level 1 to level 4 is WptN1and since N2, N3and N4are all very small compared to N1, we have N1≈N. For every atom lifted from level 1 to level 4 an energy hvphas to be given to the atom where vpis the average pump frequency corresponding to the 1→4 transition. Assuming vp≈4×1014Hz we obtain for the threshold pump power required per unit volume of the laser medium
Pth=WptN1hvp≈WptNhνp
=0.3×6×1019×6.6×10−34×4×1014
≈4.8 W/cm3
which is about three orders of magnitude smaller than that obtained for ruby.
Example 5.2 As a second example of a four-level laser system, we consider the He–Ne laser (seeChapter 11). We use the following data:
λ0=0.6328×10−4cm(v=4.74×1014Hz), tsp=10−7s, v=109Hz, n0≈1
(5.63)
If we consider the resonator to be of length 10 cm and having mirrors of reflectivities R1=R2=0.98, then assuming the absence of other loss mechanisms (αl=0),
tc= −2n0d/c ln R1R2
≈1.6×10−8s (5.64)
For an inhomogeneously broadened transition (see Section 4.5)
g(ω0)= 2 ω
ln 2 π
1 2
≈1.5×10−10s
(5.65)
Thus the threshold population inversion required is
(N)t≈1.4×109cm-3 (5.66)
Hence the threshold pump power required to start laser oscillation is
Pth=WptN1(E4−E1)
≈(N)t tsp
hvp
(5.67)
where again we assume (N)tN and T32≈A32=1/tsp. Assuming vp≈5×1015Hz, we obtain
Pth=1.4×109×6.6×10−34×5×1015 10−7
≈50 mW/cm3
(5.68)
which again is very small compared to the threshold powers required for ruby laser.