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## Stability Analysis of the Model

Now, we will analytically examine the laser model in order to gain insight as to what is really going on beneath the pictures. First we will determine the fixed points of the system. The fixed points of a system are the points where the system is not changing as time goes on, that is, and are equal to zero at the fixed point (e, n.) That is,

Solving these equations yields the fixed point (e = 0, ).

Our next step is to calculate the stability of this fixed point. (There may be other fixed points, but we will not consider them here.) In order to do this, we will simplify the system by means of linearization, a process which describes the behavior of the system close to the fixed point. To linearize, we make a change of variables in our system of real equations (that is, using and ):

Substituting these values into our original system, we get:

Since we are only looking at the system close to the fixed point, is a relatively small change. One expects terms in , , or of higher order than one will be even smaller. We will therefore neglect these higher-order terms, leaving us with a linear system. That system is:

Notice that the term on the right hand side of the last equation is always less than zero if is positive, and greater than zero if is negative. This means that tends toward zero, a trend we call stability. It remains to be seen if e is stable. To do this, we will find a threshold value for J, called , at which the stability of e changes. Consider Eqn. (19) with :

Notice that the right-hand side of this equation is of the form , where

When this form is encoutered in a differential equation, we can re-write the equation (Eqn. 22) as:

If then will grow, and if a < 0 it will decay to zero. From this we see that the threshold value, , for stability occurs when a = 0, i. e.,

Solving this equation for gives the value for J at which the stability of e changes, as desired.

For , a is negative, which means e = 0 is linearly stable. On the other hand, for , e = 0 is unstable. This change in the stability of a fixed point is called a bifurcation.

Thus, when J becomes greater than , the fixed point becomes unstable, and a periodic solution of the system is born. Recall from our discussion of the trajectories of the system that if the value of J is too low, the value of e will approach 0 instead of a periodic solution. This corresponds exactly to the periodic solution born at . That is, if J is below , the laser will not begin lasing, rather, |e| will go to zero, as discussed earlier in Figures [2(a)-(b)]. This will become clear when we draw the bifurcation diagram of the system. A bifurcation diagram is a picture showing how the fixed points of a system change as a parameter is varied. Figures [3-4] show two bifurcation diagrams of our laser system.

Figure 3: Bifurcation diagram, J vs. N(=n)

Figure 4: Bifurcation diagram, J vs. ER(=A)

These bifurcation diagrams, drawn with AUTO, show the stability of the fixed point as the parameter J varies from 0 to 0.3, and all other parameters are fixed at the realistic values given earlier. Figure [3] shows how the value of the n component of the fixed point changes as J is increased, while Figure [4] shows how the A component of the fixed point changes as J changes. The thick line denotes stability of the fixed point, while the thin line denotes instability. The point at which the fixed point becomes unstable is exactly the value of . When J is above this value, the fixed point is unstable, when it is below this value, it is stable, just as we calculated earlier. What we did not calculate earlier is the existence of periodic solutions, denoted by the dots in the bifurcation diagrams. These periodic solutions show the type of behavior we saw in the orbits of the system: n settles to a certain value, and e rotates at a fixed radius in the complex plane. However, our bifurcation diagrams tell us even more than that. They tell us that no matter how high we turn up J, we will never get an increase in n. We will, however, get an increase in e, although our rate of gain of e decreases as J increases. This is highly useful information for understanding how our laser works! We now know that the periodic solution's value of n cannot be increased by simply adding more energy. We also know how much energy needs to be put into the system in order for the laser to begin lasing.

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