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tons (photons are bosons), which predicts that in the presence of an electro-
magnetic field having a frequency close to that of the frequency of the photon
emitted in the transition between the upper excited and lower excited state,
the atom emission rate is enhanced and this enhancement is larger, the more
photons that are present in its vicinity. On the other hand, the rate of change
of the number of photons is equal to the rate generated from the decay of the
atoms due to stimulated emission, minus the decay due to the finite lifetime
of the photon in the resonating cavity. Putting all this together, one is led, in
the simplest approximation, to write what are called the rate equations for the
number of atoms in the excited state and for the photon numbers in the cavity.
These coupled equations, in their simplest forms, are given by:

dN = P − N − BnN (4.58)
dt τ
decay

dn =− n + BnN (4.59)
dt τ
cavity

where N is the normalized number of atoms in the atom’s upper excited state,
n is the normalized number of photons present, P is the pumping rate, τ decay is
the atomic decay time from the upper excited state, due to all effects except
that of stimulated emission, τ cavity is the lifetime of the photon in the resonant
cavity, and B is the Einstein coefficient for stimulated emission.
These nonlinear differential equations describe the dynamics of laser oper-
ation. Now come back to relaxation oscillations in lasers, which is the prob-
lem at hand. Physically, this is an interplay between N and n. An increase in
the photon number causes an increase in stimulated emission, which causes
a decrease in the population of the higher excited level. This, in turn, causes
a reduction in the photon gain, which tends to decrease the number of pho-
tons present, and in turn, decreases stimulated emission. This leads to the
build-up of the higher excited state population, which increases the rate of
change of photons, with the cycle resuming but such that at each new cycle
the amplitude of the oscillations is dampened as compared with the cycle just
before it, until finally the system reaches a steady state.
To compute the dynamics of the problem, we proceed into two steps. First,
we generate the function M-file that contains the rate equations, and then pro-
ceed to solve these ODEs by calling the MATLAB ODE solver. We use typical
numbers for gas lasers.
Specifically the function M-file representing the laser rate equations is
given by:

function yp=laser1(t,y)
p=30; %pumping rate
gamma=10^(-2); %inverse natural lifetime

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