\pard\tx520\tx1060\tx1600\tx2120\tx2660\tx3200\tx3720\tx4260\tx4800\tx5320\f0\b\i0\ul0\fs32\fc0 The Rate-Equation Model
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One of the simplest descriptions of laser actions is the rate equation model. In this model we look at the rates of change of the number of atoms in the various states as well as the total energy in the cavity, as multivariable functions that depend not only on time but also on the actual values of these variables at any given time. In other words we use a system of coupled differential equations.\
Although a typical laser is a multi-level system, most lasers can be broadly classified into three or four level systems. For our studies we have considered a somewhat idealized four level laser as shown in Fig. 1a. The common Nd3+-glass and CO2 lasers are examples of 4-level lasers. The dynamics of the model are as follows.\
Atoms in the ground state (an infinite ground state is assumed) are pumped up into level 3 at a rate R. In the laboratory this can be done in several ways - electric discharges, strong light sources or even a pulse from an external laser. Level 3 typically represents a state that has a very short lifetime, so that the atoms decay rapidly and non-radiatively to level 2. The latter, sometimes know as a meta-stable state has a measurable life-time that is significantly larger than that of level 3. Level one also represents a state with a short life time. \
The net effect is that for appropriate values of R, there is a gradual build up of atoms in level 2 whereas any atom in level one quickly decays to the ground state thereby causing a population inversion between levels 1 and 2. This gives the necessary condition for lasing. The emitted light from this transition is confined in the laser cavity and amplified A fraction of this light is allowed to escape out of the cavity through the partially transmitting mirror M2. This forms an intense laser beam.\
For details about the transformation of the variables to dimensionless form, please see,
\b Andrews & Tilley
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\b\fs32 Computer Experiments
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\b Testing the code
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As mentioned before the differential equations shown in Eqs. 4 through 6 are coupled and do not have solutions in closed form. The natural choice for an method to solve these equations in numerical form was an order 4 Runge-Kutta alogorithm. The first step in the experiments was to test the code. For this purpose, we used the values suggested used in the original experment as described by Andrews and Tilly in their paper.\
t1 = T1/T2 = 0.5\
t2 = 1\
t0 = T0/T2 = 10\
Using these values we ran our program through three passes, each time starting at t=0 and going upto t=200. The scaled pumping rate for each run was r=0.5, r.1.5 and r=4 respectively. The initial values of n1, n2 and w were set at 0. The results are shown in Plate 1.1a-c.\
When the pump rate is half of the threshold value, there is virtually no lasing, and the energy in the cavity is almost entirely due to spontaneous emission and hence, rises to a very low value. But when the rate is greater than the threshold value lasing takes place. The lasing state is reached in two steps. First, the non-lasing solution given by eq. 8a is reached. But this is unstable and eventually the system switches to the stable, lasing soultion. N1, being independent of W, remains more or less constant during this process, and N2, switches from its higher, unstable value to the lower value as W becomes positive.\
It should be noted that there is a big difference in the numerical time scales of the number densities and the energy. It is here that the smallness of the value of A21'/A21 comes into play. As mentioned before this value represents the spontaneous decay into the lasing mode. The physical representation of this value is the noise that is subsequently amplified to switch the system into lasing state. If this value is ignored, the spontaneous part of eq. 18 disappears, i.e. there is no initial noise, and the model never switches to the lasing state!\
\b Q Switching
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One of the common ways of obtaining short pulses is Q switching. In this method, laser oscillations are prevented in the cavity by reducing the Q of the cavity while pumping energy in. This results in a high degree of population inversion. Then the Q is switched to a high value so that the energy in the cavity builds up rapidly, converting the energy of the upper state atoms as they decay, into a short pulse of laser light. In the model used here, this can be achieved by setting the initial values of n1 and n2 so that (n2 - n1) is high. The following starting values may be used.\
n1[0]=0;\
n2[0]=5;\
w[0]=0;\
r=0;\
0<=t<=0.1 in steps of 0.001\
t0=0.1\
t1=0.5\
A'=10^-3\
\b Relaxation Oscillation
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Another interesting aspect of pulsed lasers is relaxation oscillation. Here the laser energy is allowed to build up so rapidly that the population inversion is depleted. However, because of the coupled nature of Eqs 1 though 3, with respect to (N2-N1), this diminishes the lasing so that the energy falls and the population inversion builds up again and the process is repeated. This produces short oscillating pulses of energy. This can be achieved by using a relatively high pump rate (i.e. well above threshold). The results from several runs are shown in Plates 1.3a...\
As can be seen from these plates, the pulses usually have decreasing amplitudes which finally produce the steady lasing state. However, for appropriate values of T0 and R, it is possible to force the system into a limit cycle where the pump rate is sufficiently high to bring the depleted population inversion back to the same value as before and then the process repeats. Typical experimental parameters are:\
n1[0]=0;\
n2[0]=0;\
w[0]=0;\
r=10;\
0<=t<=0.1 in steps of 0.001\
t0=0.0005\
t1=0.9\
A'=10^-5\
\b Mode Locking
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When a pumping rate with a period comparable to the period of oscillation of the cavity relaxation is applied, the relaxations are prevented because the upper state is replenished at the same time as the lasing starts. This gives very sharp pulses of enegy with widths that are comparable to the width of the pump profile (usually a Gaussian). This is called mode locking. This can be produced by the same values as above, but with a periodic Gaussian pulse rate for r.\
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