3.1: Rate equations

A laser consists of two elements: a gain medium that amplifies light, and a resonator that confines the light to the amplifier. A very simplified view of gain in an optical amplifier is shown in Figure \(\PageIndex{1}\). The takeaway is that the amount of amplification supplied by the gain medium is not necessarily constant in time or space, because every every photon of light generated reduces the gain at the time and location where that photon was produced. As a result, laser gain is highly dynamic, which can make it quite difficult to control.

Figure \(\PageIndex{1}\): A simplified view of optical gain. The pink area represents a gain medium consisting of ions. The yellow spheres are ions that are able to emit a photon of light, while the dark red spheres are ions that cannot emit light. The top picture is the state of the medium before a single photon encounters an ion. The bottom picture is the state of the medium after a single photon encounters an ion.

In Figure \(\PageIndex{1}\) the amplification of a single photon is shown. In the top picture, a single photon encounters an ion capable of emitting light. On emitting (bottom picture), the ion cannot emit another photon, so it has become dark red, while the number of photons has doubled. As this process proceeds, the number of ions available to emit light decreases, reducing the potential for further light amplification. The balance between light emission and gain is described by rate equations

\[\begin{align} \frac{\textrm{d}N_1}{\textrm{d}t} &= I(N_2-N_1) - PN_1 + kN_2\label{eq:N1}\\ \frac{\textrm{d}N_2}{\textrm{d}t} &= -I(N_2-N_1) + PN_1 - kN_2\label{eq:N2}\\\frac{\textrm{d}I}{\textrm{d}t} &= I(N_2-N_1) -\alpha I - TI\label{eq:I}\end{align}\]

Equations \(\eqref{eq:N1}\)-\(\eqref{eq:I}\) are a set of coupled differential equations that describe the balance between gain and light intensity in an amplifier. \(N_2\) is the fraction of emitters available to emit, and \(N_1\) is the fraction of emitters unavailable. The total number of emitters is a constant (\(N_1 + N_2 = 1\). \(N_2\) is increased by the amount of power supplied to the amplifier (\(P\)), and is reduced by loss processes (\(k\)) and producing light. The intensity of the light, \(I\), is increased by amplification, which is proportional to \(N_2-N_1\), hence gain is only possible when \(N_2>N_1\). The light intensity is also reduced by some internal losses (\(\alpha\)), and emission to the outside world, which is given by \(T\).

To understand the behavior of the rate equations, let's simplify by assuming an amplifier with no output (\(T=0\)) and no internal losses (\(\alpha=k=0\)). Let us start with \(N_2=1\) and \(N_1 = P = 0\). The initial light intensity is some tiny value, corresponding to a single photon of light. The intensity of the light initially grows exponentially, but, then the gain drops and the amplifier saturates (see Figure \(\PageIndex{2}\)). Note that everything stabilizes at \(N_2=N_1\). Because there is no emission and no optical losses, the light intensity also stabilizes at a non-zero value. If we add loss processes (\(k>0\), \(\alpha>0\)), then \(N_2\) and \(I\) will reduce to zero.

Figure \(\PageIndex{2}\): A gain medium with initial conditions \(N_2 = 1\), and no additional excitation.

A laser should have an output. When we do this, the intensity will grow slower, and, once the gain is exhausted, the intensity will drop back to zero, as shown in Figure \(\PageIndex{3}\).

Figure \(\PageIndex{3}\): A gain medium with initial conditions \(N_2 = 1\), and some output (\(T>0\)).

Now to complete the picture, we will set \(P> 0\) and \(k>0\) so that the gain is refreshed and there are some internal losses. Because we are exciting the gain, we will start with \(N_2 = 0\). As shown in Figure \(\PageIndex{4}\), the laser only switches on when \(N_2>N_1\) for long enough.

Figure \(\PageIndex{4}\): In this case, the gain medium is excited continuously. \(N_2=0\) at \(t=0\) and rapidly grows. Only once it is very high does the intensity start to grow, causing the \(N_2\) population to reach a steady state just greater than\(N_1\).

As seen in Figure \(\PageIndex{5}\), the gain output stabilizes at a constant output, known as steady state conditions. These are the operating conditions for most lasers used in communications systems. One might be tempted to modulate the laser output by modulating the input power \(P=f(t)\). In early optical communication systems, this was, indeed, the case. To understand why this is no longer the case, consider the change of intensity in Figures \(\PageIndex{3}\) and \(\PageIndex{4}\). Note that the slopes are not very steep, indicating that the rise and fall times are quite slow. The time axis in these figures is, of course, related to the settings used in the calculation and should not be taken literally. However, the atomic processes and influence of the optical cavity (see next section) place quite stringent limits on how fast a pulse can build up or decay away in a laser.

Exercise \(\PageIndex{1}\)

The script below was used to generate Figures \(\PageIndex{2}-\PageIndex{4}\). Adjust the input power, \(P\), the loss \(k\leq 1\), and the output, \(T\leq 1\), to see how fast you can reach steady state at an output intensity greater than 0.2. Adjust only one parameter at a time while holding the rest fixed to observe how they influence the steady state behavior of the laser

• How do changes in \(P\) change the turn on time and output intensity?
• How do changes in \(k\) change the turn on time and output intensity?
• How do changes in \(T\) chane the turn on time and output intensity?

import numpy as np
from matplotlib import pylab as plt
from scipy.constants import c
from scipy.integrate import solve_ivp

#conditions for the simulation
in_cond = [1, 0, 1e-6]  #initial conditions. iorder is N1, N2, I
sigma = 0.05            #sigma: material property, leave as is
T = 0.02                #T: output 0-1
P = 0.05                #P: input power 0-infinity
loss = 0.0001           #k: losses 0-1
t_span = [0,1000]       #time over which the simulation will run. Adjust as necessary
                        #but always start at 0


#These are the coupled differential equations
def coupledDE(t, out, sig, pump, trans, k):
    ret = [0,0,0]
    ret[0] = sig*(out[1]-out[0])*out[2]-pump*out[0] 
    ret[1] = pump*out[0] - sig*(out[1]-out[0])*out[2] - k*out[1]
    ret[2] = sig*(out[1]-out[0])*out[2] - trans*out[2]
    return ret
 
#this line solves the differential equation
sol1 = solve_ivp(coupledDE,t_span, in_cond, args = (sigma, P, T, loss), max_step=0.1)

#plotting
fig, ax = plt.subplots(layout='constrained')
ax.plot(sol1.t, sol1.y[0,:], label = "$N_1$", color = 'xkcd:fuchsia')
ax.plot(sol1.t, sol1.y[1,:], label = "$N_2$", color = 'xkcd:brick')
plt.ylabel("Population density (normalized)")

ax.plot(sol1.t, sol1.y[2,:], label = "$I$", color = 'xkcd:azure')
plt.xlabel("Time (s)")
plt.ylabel("Intensity (arb.)")
plt.legend(loc = "upper right")

<matplotlib.legend.Legend at 0x7f089a117a60>

Answer

Increasing the input power changes the output power but does not change the rise time. Increasing the output causes the startup to delay, but the rise time is steeper. Once the output is too high, the laser never switches on. Increasing the loss reduces the output power and delays start up. It does not shorten the rise time. When the loss is high enough, the laser never reaches steady state and will eventually turn off.