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B=3; %stimulated emission coefficient
c=30; %inverse lifetime of photon in cavity

yp(1,1)=p-gamma*y(1,1)-B*y(1,1)*y(2,1);
yp(2,1)=-c*y(2,1)+B*y(1,1)*y(2,1);
The script M-file to compute the laser dynamics and thus simulate the relax-
ation oscillations is:

tspan=[0 3];
yin=[1 1];
[t,y]=ode23('laser1',tspan,yin);

subplot(3,1,1)
plot(t,y(:,1))
xlabel('Normalized Time')
ylabel('N')

subplot(3,1,2);
plot(t,y(:,2))
xlabel('Normalized Time')
ylabel('n')

subplot(3,1,3);
plot(y(:,1),y(:,2))
xlabel('N')
ylabel('n')

As can be observed in Figure 4.7, the oscillations, as predicted, damp-out
after a while and the dynamical variables reach a steady state. The phase dia-
gram, shown in the bottom panel, is an alternate method to show how the
population of the atomic higher excited state and the photon number density
reach the steady state.

Question: Compute analytically from Eqs. (4.58) and (4.59), the steady-state
values for the higher excited state population and for the photon number,
and compare with the numerically obtained asymptotic values.

In-Class Exercise

Pb. 4.40 By changing the values of the appropriate parameters in the above
programs, find separately the effects of increasing or decreasing the value of

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