Optical Resonators
7.7 Pulsed Operation of Lasers
7.7.1 Q-Switching
7.7 Pulsed Operation of Lasers 165 imply high Q. Thus when the shutter is kept closed and suddenly opened, the Q of the cavity is suddenly increased from a very small value to a large value and hence the name Q-switching. For generating another pulse the medium would again need to be pumped while the shutter is kept closed and the process repeated again.
Figure7.12shows schematically the time variation of the cavity loss, cavity Q, population inversion, and the output power. As shown in the figure an intense pulse is generated with the peak intensity appearing when the population inversion in the cavity is equal to the threshold value. Figure7.13 shows a Q-switched pulse emitted from a neodymium–YAG laser. The energy per pulse is 850 mJ and the pulse width is about 6 ns. This corresponds to a peak power of about 140 MW. The pulse repetition frequency is 10 Hz, i.e., the laser emits 10 pulses per second. Using this phenomenon it is possible to generate extremely high power pulses for use in various applications such as cutting, drilling, or in nuclear fusion experiments.
We will now write down rate equations corresponding to Q-switching and obtain the most important parameters such as peak power, total energy, and duration of the pulse. We shall consider only one mode of the laser resonator and shall examine
t
t
t t Loss
Q
ΔNι ΔN ΔNτ
Pout
t = 0 Fig. 7.12 Schematic of the
variation of loss, Q value, population inversion, and the laser output power with time
Fig. 7.13 An acousto optically Q-switched output from an Nd:YAG laser. The average power of the pulse train is 15 W and the repetition frequency is 2 kHz with the pulse duration of 97 ns.
[Figure provided by Brahmanand Upadhyaya, RRCAT, Indore]
the specific case of a three-level laser system such as that of ruby. In Section 5.5 while writing the rate equations for the population N2and the photon number n, we assumed the lower laser level to be essentially unpopulated. If this is not the case then instead of Eq. (5.74) we will have
d(N2V)
dt = −KnN2+KnN1−T21N2V+RV (7.42) where the second term on the right-hand side is the contribution due to absorption by N1atoms per unit volume in the lower level. Since the Q-switched pulse is of a very short duration, we will neglect the effect of the pump and spontaneous emission during the generation of the Q-switched pulse. It must, at the same time, be noted that for the start of the laser oscillation, spontaneous emission is essential. Thus we get from Eq. (7.42)
dN2 dt = −
Kn
V N (7.43)
where
N=(N2−N1)V, N2 =N2V (7.44) and V is the volume of the amplifying medium. Similarly one can also obtain or the rate of change of population of the lower level
dN1 dt =
Kn
V N (7.45)
where N1 =N1V. Subtracting Eq. (7.45) from Eq. (7.43) we get
7.7 Pulsed Operation of Lasers 167 d(N)
dt = −2 Kn
V N (7.46)
We can also write the equation for the rate of change of the photon number n in the cavity mode in analogy to Eq. (5.78) as
dn
dt =Kn(N2−N1)− n tc +KN2
≈ Kn
V N− n tc
(7.47)
where we have again neglected the spontaneous emission term KN2. From Eq. (7.47) we see that the threshold population inversion is
(N)t= V Ktc
(7.48)
when the gain represented by the first term on the right-hand side becomes equal to the loss represented by the second term [see Eq. (7.47)]. Replacing V/K in Eqs.
(7.46) and (7.47) by (N)ttcand writing
τ = t tc
(7.49)
we obtain
d(N)
dτ = −2n N (N)t
(7.50)
and
dn
dτ =n N (N)t
−1
(7.51)
Equations (7.51) and (7.50) give us the variation of the photon number n and the population inversionN in the cavity as a function of time. As can be seen the equations are nonlinear and solutions to the above set of equations can be obtained
numerically by starting from an initial condition N
(τ =0)= N
i and n(τ =0)=ni (7.52) where the subscript i stands for initial values. Here ni represents the initial small number of photons excited in the cavity mode through spontaneous emission. This spontaneous emission is necessary to trigger laser oscillation.
From Eq. (7.51) we see that since the system is initially pumped to an inver- sionN >
N
t, dn / dτ is positive; thus the number of photons in the cavity increases with time. The maximum number of photons in the cavity appear when dn / dτ =0, i.e., whenN=
N
t. At such an instant n is very large and from Eq. (7.50) we see thatNwill further reduce below
N
tand thus will result in a decrease in n.
Although the time-dependent solution of Eqs. (7.50) and (7.51) requires numeri- cal computation, we can analytically obtain the variation of n withNand from this we can draw some general conclusions regarding the peak power, the total energy in the pulse, and the approximate pulse duration. Indeed, dividing Eq. (7.51) by (7.50) we obtain
dn d(N) = 1
2
N
t
(N) −1
Integrating we get n−ni=1
2
&
Ntln N
(N)i
+
N
i−N'
(7.53)
7.7.1.1 Peak Power
Assuming the only loss mechanism to be output coupling and recalling our discussion in Section7.5we have for the instantaneous power output
Pout=nhν tc
(7.54) Thus the peak power output will correspond to maximum n which occurs when N=
N
t. Thus Pmax= nmaxhν
tc
= hν 2tc
N
tln
N
t
(N)i
+
N
t− N
i
(7.55)
where we have neglected ni(the small number of initial spontaneously emitted pho- tons in the cavity). This shows that the peak power is inversely proportional to cavity lifetime.
7.7 Pulsed Operation of Lasers 169 7.7.1.2 Total Energy
In order to calculate the total energy in the Q-switched pulse we return to Eq. (7.51) and substitute for
N /
N
tfrom Eq (7.50) to get dn
dτ = −1 2
d N dτ −n Integrating the above equation from t=0 to∞we get
nf −ni= 1 2
N
i− N
f
− ∞ 0
ndτ
or ∞
0
ndτ = 1 2
N
i− N
f
−(nf−ni) (7.56)
where the subscript f denotes final values. Since ni and nf are very small in comparison to the total integrated number of photons we may neglect them and obtain
∞ 0
ndτ ≈ 1 2
N
i− N
f
Thus the total energy of the Q-switched pulse is
E= ∞ 0
Poutdt
=hν ∞ 0
ndτ
=1 2
N
i− N
f
hν
(7.57)
The above expression could also have been derived through physical argu- ments as follows: for every additional photon appearing in the cavity mode there is an atom making a transition from the upper level to the lower level and for every atom making this transition the population inversion reduces by 2. Thus if the population inversion changes from (N)i to (N)f, the num- ber of photons emitted must be 12
N
i− N
f
and Eq. (7.57) follows immediately.
7.7.1.3 Pulse Duration
An approximate estimate for the duration of the Q-switched pulse can be obtained by dividing the total energy by the peak power. Thus
td= E Pmax =
N
i− N f
(N)tln (N)t
(N)i
+((N)t−(N)i)tc (7.58)
In the above formulas we still have the unknown quantity N
fthe final inversion.
In order to obtain this, we may use Eq. (7.53) for t→ ∞. Since the final number of photons in the cavity is small, we have
N
i− N
f
= N
tln N
i
(N)f
(7.59) from which we can obtain
N
f for a given set of N
iand N
t
Example 7.11 We consider the Q-switching of a ruby laser with the following characteristics:
Length of ruby rod=10 cm Area of cross section=1 cm2 Resonator length=10 cm Mirror reflectivities=1 and 0.7
Cr3+population density=1.58×1019cm–3 λ0=694.3 nm
n0=1.76 tsp=3×10–3s g(ω0)=1.1×10–12s
The above parameters yield a cavity lifetime of 3.3×10–9 s and the required threshold population density of 1.25×1017cm–3. Thus
N
t=1.25×1018 Choosing
N
i=4 N
t=5×1018 we get
Pmax=8.7×107W Solving Eq. (7.59) we obtain
N
f ≈0.02 N
iThus E≈0.7J
7.7 Pulsed Operation of Lasers 171 and
td≈8 ns