Laser \(=\) Light Amplification by Stimulated
Emission of Radiation. The whole subject is one idea pushed
hard: stimulated emission produces a photon identical to the one that caused
it, so if stimulated emission can be made to outrun spontaneous emission and
absorption, light amplifies itself.
Coherence — temporal and spatial — and the characteristics of
laser light.
The three radiative processes and the Einstein coefficients.
Components of a laser: active medium, pump, resonator; population
inversion.
Resonator modes, cavity lifetime, threshold.
Rate equations: why two levels can never work, and how three do.
The standard systems: ruby, Nd:YAG, He–Ne.
\(Q\)-switching, mode locking and applications.
11.1 Coherence
A wave that remains sinusoidal for an infinite time, or over infinitely
extended space, is said to be coherent:
\begin{equation}
E = E_{0}\sin(\omega t - kz).
\label{eq:coherent-wave}
\end{equation}
For such a wave there is a definite phase relation between the field at two
different instants, or at two different points in space.
For an incoherent wave the phase changes abruptly after a certain
interval of time, or after a certain distance. An ordinary lamp emits light in
pulses of duration \(\tau\): the wave stays sinusoidal for a time \(\tau\) and then
jumps in phase.
Figure 11.1. A coherent wave keeps its phase indefinitely. An incoherent wave
holds phase only for the coherence time \(\tau\), after which it jumps. Ordinary
sources are incoherent; laser light is not.
The two kinds of coherence
Temporal coherence measures the correlation of the phase at
different points along the direction of propagation. It tells
you how monochromatic the source is. Equation \(\eqref{eq:coherent-wave}\)
is the ideal case.
Spatial coherence measures the correlation of the phase at
different points transverse to the direction of propagation. It
tells you how uniform the phase is across a wavefront.
Coherence parameters
Coherence time \(\tau_{c}\): the time for which the wave stays
sinusoidal.
The manuscript prints the spectral-purity relation as
\(\lambda/\Delta\lambda = \Delta\nu/\nu\). That is upside down. Differentiating
\(\nu = c/\lambda\) gives \(|\Delta\nu| = c\,\Delta\lambda/\lambda^{2}\), hence
\(\Delta\nu/\nu = \Delta\lambda/\lambda\): the fractional widths in
frequency and in wavelength are equal. The reciprocal \(\lambda/\Delta\lambda\)
is the resolving power, which is the inverse of this quantity.
Characteristics of laser light
Very nearly perfectly monochromatic.
Coherent, both temporally and spatially.
Highly directional — very small divergence.
High intensity.
11.2 The Three Radiative Processes
Figure 11.2. The three radiative processes. In stimulated emission the emitted
photon matches the incident one in frequency, phase, polarization and
direction of travel — which is the whole basis of laser action.
Stimulated absorption
The atom sits in the ground state \(E_{1}\) and a photon of frequency
\begin{equation}
\nu = \frac{E_{2}-E_{1}}{h}
\label{eq:bohr-frequency}
\end{equation}
falls on it. The atom absorbs the photon and goes to \(E_{2}\). Because the
process is provoked by the incident photon it is called stimulated
absorption. The rate is
\begin{equation}
\frac{\dd N_{1}}{\dd t} \propto N_{1}\,u(\omega).
\label{eq:rate-abs}
\end{equation}
Absorption removes photons from the beam, so intensity is never built up this
way.
Spontaneous emission
An atom in \(E_{2}\) with an ordinary allowed-transition lifetime of about
\(10^{-8}\,\)s will drop to \(E_{1}\) of its own accord after that time, emitting a
photon of frequency \(\eqref{eq:bohr-frequency}\). Nothing external provokes it,
so this is not stimulated emission. The rate is
\begin{equation}
\frac{\dd N_{2}}{\dd t} \propto N_{2}.
\label{eq:rate-spont}
\end{equation}
Photons emitted spontaneously are random in direction and random in phase, so
again no coherent intensity is built up.
Stimulated emission
Now suppose the atom sits in an excited state \(E_{2}\) whose lifetime is much
longer — of order \(10^{-3}\,\)s, i.e.\ a metastable state. The
probability of spontaneous decay is then very small, and the atom waits. If a
photon of frequency \(\eqref{eq:bohr-frequency}\) arrives, it forces the atom down
to \(E_{1}\), and the emitted photon has the same frequency, phase, polarization
and direction as the incident one. The rate is
\begin{equation}
\frac{\dd N_{2}}{\dd t} \propto N_{2}\,u(\omega).
\label{eq:rate-stim}
\end{equation}
The two photons are emitted at the same time and from the same place, their
wavefunctions overlap, and they interfere constructively: intensity is built
up, and the beam is forced into a single direction. This is the origin of both
the high intensity and the low divergence of laser light.
For laser action we therefore need stimulated emission to dominate, which
requires
\begin{equation}
\boxed{\ N_{2} > N_{1}\ }
\qquad\text{(}\textbf{population inversion}\text{).}
\label{eq:inversion}
\end{equation}
11.3 Einstein Coefficients
Let \(N_{1}\) atoms occupy \(E_{1}\) and \(N_{2}\) atoms occupy \(E_{2}\), in a
radiation field of energy density \(u(\omega)\). Per unit time per unit volume:
Process
Number per unit time per unit volume
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