3.2: Optical cavities

In addition to gain, a laser also requires feedback, which is provided by mirrors that recirculate the light through the gain medium. This is referred to as an optical cavity. The simplest version, shown in Figure \(\PageIndex{1}\), consists of two mirrors facing each other.

Figure \(\PageIndex{1}\): simplified diagram of a laser.

In Figure \(\PageIndex{1}\), the mirrors are separated by a length, \(l\). If we follow a light wave starting at the left hand mirror, it will travel to the partially reflecting mirror and return to the left hand mirror again (a roundtrip). That will introduce a phase shift of \(\delta\phi = 2\pi\frac{2l}{\lambda}\), where \(\lambda\) is the wavelength of the light. If \(\delta\phi = 2m\pi\), where \(m\) is an integer, then there is no apparent phase change, and constructive interference occurs, allowing the light intensity to build up in the cavity. On the other hand, for any other value, each round trip introduces a phase change that reduces the build up of the intensity. Thus, there is a regularly spaced interval of frequencies (\(f = \frac{c}{\lambda}\)) that can resonate in the cavity given by

\[\begin{equation} f_m = \frac{mc}{2l}\label{eq:longModes}\end{equation}\]

where \(c\) is the speed of light. The resonant frequencies are called the longitudinal modes of the cavity and the spacing between longitudinal modes is called the free spectral range (FSR). Like optical fibers, there are also spatial models, called transverse modes, but, we will consider that the laser is always operating in the fundamental spatial mode. If the mirrors were perfectly reflecting and there were no other losses, only wavelengths that exactly satisfy equation \(\eqref{eq:longModes}\) would resonate. However, apart from losses, the partially reflecting mirror ensures that light constantly leaves the cavity. Thus, it is possible to define an average number of round trips that a photon will make, meaning that \(\delta\phi\) also has a maximum value.

The consequence is that each longitudinal mode has a finite width that plays a role in defining how sharply the laser emission is defined. Figure \(\PageIndex{2}\) shows an example of the longitudinal modes for a Helium-Neon (HeNe) laser operating at 632 nm (length 20 cm, refractive index 1.0), and a laser diode operating at 1550 nm (length 5 mm, refractive index 3.2).

Figure \(\PageIndex{2}\): Longitudinal modes around the central frequency of a Helium Neon laser (A) and a laser diode (B). Note that the central frequency has been subtracted to make the horizontal axis readable.

The HeNe laser has a much closer mode spacing (smaller FSR) than the laser diode. The longitudinal modes of the HeNe laser appear to be delta functions because the reflectivity of the output coupling mirror must be quite high (~98%), while the gain of a laser diode is so high that the natural reflection from the end facets of the semiconductor material is sufficient.

When we combine the fact that the gain material also has a finite bandwidth with the longitudinal mode spectrum of the optical cavity, we get the potential output spectrum of a laser (shown in Figure \(\PageIndex{3}\)). However, when we measure the spectrum of a continuous wave laser (one that emits with a constant power), it is highly unlikely that a spectrum like that in Figure \(\PageIndex{3}\) will be observed.

Figure \(\PageIndex{3}\): The gain spectrum of a laser. The gain medium has a finite bandwidth, which reduces the gain of longitudinal modes that do not coincide with the peak of the gain spectrum.

There are two reasons for the observed emission spectrum being different from the gain spectrum. First, the amount of gain at any one moment is finite, and is dependent on the intensity of light. The mode with the most gain will start first, and because it has a higher intensity, will also grow fastest. This will reduce the gain available to the other modes, slowing their growth even more. This suppresses most modes. This is illustrated in Figure \(\PageIndex{4}\) that shows the results of a rate equation calculation for three competing longitudinal models with different gains.

Figure \(\PageIndex{4}\): Rate equation calculation for three longitudinal modes. All three modes have the same starting conditions, but experience slightly different gains.

Gain competition can be useful, since we can, by cavity design, increase the gain of the longitudinal mode that we want, then gain competition will ensure that it is the mode most likely to be observed at the output. The second reason that we do not observe the full gain spectrum at once, but also don't observe a stable single longitudinal mode is due to the fact that the gain is distributed in space. Each longitudinal mode forms a standing wave between the mirrors. A standing wave has nodes and anti-nodes in fixed spatial locations, which means spatial locations where the intensity is always high and spatial locations where the intensity is always low. Thus, there is not full competition for the same gain, and more than a single mode can oscillate (see Figure \(\PageIndex{5}\)). However, the overlap between the different modes is sufficient to suppress most modes at any one time.

Figure \(\PageIndex{5}\): Two standing waves. Note that the brown trace has peaks at 1 and 3 wavelengths. At these locations, the gain for the purple mode is lower than that of the brown.

Furthermore, the competition between modes rarely leads to a stable emission spectrum. As a mode grows, it exhausts the gain where its standing wave intensity is highest, reducing the gain that it experiences. The gain elsewhere is available, but the mode currently dominant does not have the highest gain in that location because the standing wave intensity is low. Thus, the mode begins to die away, and a new mode begins to gain strength. This leads to a dynamic emission spectrum with longitudinal modes constantly growing and dying.

Exercise \(\PageIndex{1}\)

The code to calculate the output of a laser with gain competition between three modes is below. Is it possible to get more than a single mode to start? How different can the gains (\(\sigma\) in the code) be?

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, 1e-6, 1e-6]  #initial conditions. iorder is N1, N2, I
###### these are the gain factors for the three different modes
sigma1 = 0.05            #sigma1: gain of mode 1
sigma2 = 0.03            #sigma2: gain of mode 2
sigma3 = 0.02            #sigma3: gain of mode 3
######
T = 0.02                #T: output 0-1
P = 15                    #P: input power 0-infinity
loss = 0.00000           #k: losses 0-1
t_span = [0,1500]       #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, sig1, sig2, sig3, pump, trans, k):
    ret = [0,0,0,0,0]
    ret[0] = sig1*(out[1]-out[0])*out[2]-pump*out[0] +sig2*(out[1]-out[0])*out[3] +sig3*(out[1]-out[0])*out[4]
    ret[1] = -sig1*(out[1]-out[0])*out[2] - k*out[1] -sig2*(out[1]-out[0])*out[3] -sig3*(out[1]-out[0])*out[4] + pump*out[0]
    ret[2] = sig1*(out[1]-out[0])*out[2] - trans*out[2]
    
    ret[3] = sig2*(out[1]-out[0])*out[3] - trans*out[3]
    ret[4] = sig3*(out[1]-out[0])*out[4] - trans*out[4]
    return ret
 
#this line solves the differential equation
sol1 = solve_ivp(coupledDE,t_span, in_cond, args = (sigma1, sigma2, sigma3, P, T, loss), max_step=0.1)

#plotting
fig, ax = plt.subplots(layout='constrained')
ax.plot(sol1.t, sol1.y[2,:], label = "$I_1$", color = 'xkcd:fuchsia')
ax.plot(sol1.t, sol1.y[3,:], label = "$I_2$", color = 'xkcd:brick')
ax.plot(sol1.t, sol1.y[4,:], label = "$I_3$", color = 'xkcd:azure')
plt.xlabel("Time (s)")
plt.ylabel("Intensity (arb.)")
plt.legend(loc = "upper right")

<matplotlib.legend.Legend at 0x7f2f5f225460>

Answer

It depends quite a bit on the settings of the numerical solver. When the gain factors are identical, all three modes will run. When they are within about 0.1% of each other, the modes will start up, but those with less gain will slowly decay with time. What this shows is that the steady state solution has not been found yet, and after a very long time, the mode with lower gain will still stop emitting.