Optical Transitions for Laser Amplifiers
This is a continuation from the previous tutorial - guided-wave all-optical modulators and switches.
The word laser is an acronym for light amplification by stimulated emission of radiation. However, the term laser generally refers to a laser oscillator, which generates laser light without an input light wave.
A device that amplifies a laser beam by stimulated emission is called a laser amplifier.
Laser light is generally highly collimated with a very small divergence and highly coherent in time and space. It also has a relatively narrow spectral linewidth and a high intensity in comparison with light generated from ordinary sources.
Due to the process of stimulated emission, an optical wave amplified by a laser amplifier preserves most of the characteristics, including the frequency spectrum, the coherence, the polarization, the divergence, and the direction of propagation, of the input wave.
In this tutorial, we discuss the characteristics of laser amplifiers.
Optical Transitions
Optical absorption and emission occur though the interaction of optical radiation with electrons in a material system that defines the energy levels of the electrons.
Depending on the properties of a given material, electrons that interact with optical radiation can be either those bound to individual atoms or those residing in the energy-band structures of a material such as a semiconductor.
In any event, the absorption or emission of a photon by an electron is associated with a resonant transition of the electron between a lower energy level, \(|1\rangle\) of energy \(E_1\) and an upper energy level \(|2\rangle\) of energy level \(E_2\), as illustrated in Figure 10-1. The resonance frequency, \(\nu_{21}\), of the transition is determined by the separation between the energy levels:
\[\tag{10-1}\nu_{21}=\frac{E_2-E_1}{h}\]
In an atomic or molecular system, a given energy level usually consists of a number of degenerate quantum-mechanical states, which have the same energy. Therefore, the energy levels \(|1\rangle\) and \(|2\rangle\) are generally characterized by degeneracy factors \(g_1\) and \(g_2\), respectively.
There are basically three types of processes associated with resonant optical transitions between two energy levels in a system: absorption, stimulated emission, and spontaneous emission, which are illustrated in Figure 10-1(a), (b), and (c), respectively.
Absorption and stimulated emission of photons are both associated with induced transitions between the energy levels caused by interaction of an electron with the existing optical radiation.
If an electron is initially in the lower level \(|1\rangle\), it can absorb a photon to make a transition to the upper level \(|2\rangle\). If an electron is initially in the upper level \(|2\rangle\), the optical radiation can stimulate it to emit a photon by making a downward transition to the lower level \(|1\rangle\). Irrespective of the presence or absence of any existing optical radiation, an electron initially in the upper level \(|2\rangle\) can also spontaneously relax to the lower level \(|1\rangle\) by emitting a spontaneous photon.
A photon emitted by stimulated emission has the same frequency, phase, polarization, and propagation direction as the optical radiation that induces the process. In contrast, spontaneously emitted photons are random in phase and polarization and are emitted in all directions, though their frequencies are still dictated by the separation between the two energy levels, subject to a degree of uncertainty determined by the linewidth of the transition.
Therefore, stimulated emission results in the amplification of an optical signal, whereas spontaneous emission merely adds noise to an optical signal. Absorption simply leads to the attenuation of an optical signal.
Spectral Lineshape
A resonant transition is selective of the frequency of the interacting optical field because the process is associated with absorption or emission of a photon whose frequency is determined by the energy change of the transition indicated in (10-1).
The spectral characteristic of a resonant transition is never infinitely sharp, however. The finite spectral width of a resonant transition is dictated by the uncertainty principle of quantum mechanics, but it can be understood intuitively without the details of quantum mechanics by following the line of reasoning in the material dispersion tutorial.
One important conclusion learned from these discussions is that any response that has a finite relaxation time in the time domain must have a finite spectral width in the frequency domain.
As we shall see below, the rate of the induced transitions between two energy levels in a given system is directly proportional to the spontaneous emission rate from the upper to the lower level in that system.
Therefore, it is a basic law of physics that any allowed resonant transition between two energy levels has a finite relaxation time constant because at least the upper level has a finite lifetime due to spontaneous emission.
Consequently, for each particular resonant transition between two energy levels, there is a characteristic lineshape function, \(\hat{g}(\nu)\), of finite linewidth that characterizes the optical processes associated with the transition.
The lineshape function is generally normalized as
\[\tag{10-2}\displaystyle\int\limits_0^\infty\hat{g}(\nu)\text{d}\nu=\int\limits_0^\infty\hat{g}(\omega)\text{d}\omega=1\]
where \(\hat{g}(\nu)=2\pi\hat{g}(\omega)\).
Homogeneous Broadening
If all of the atoms in a material that participate in a resonant interaction associated with the energy levels \(|1\rangle\) and \(|2\rangle\) are indistinguishable, their responses to an electromagnetic field are characterized by the same resonance frequency \(\nu_{21}\) and the same relaxation constant \(\gamma_{21}\). In such a homogeneous system, the physical mechanisms that contribute to the linewidth of the transition affect all atoms equally. Spectral broadening due to such mechanisms is called homogenous broadening.
From the discussions in the material dispersion tutorial, the spectral characteristics of a damped response characterized by a single resonance frequency and a single relaxation constant, such as that of a resonant interaction in a homogeneously broadened system, are described by the functions given in (176) [refer to the material dispersion tutorial].
As we shall see in a later tutorial, in the interaction of a material with an optical field, the absorption and emission of optical energy are characterized by the imaginary part \(\chi''\) of the susceptibility of the material. Therefore, the spectral characteristics of optical absorption and emission due to a resonant transition in a homogeneously broadened medium are described by the Lorentzian lineshape function of \(\chi''(\omega)\) given in (176) [refer to the material dispersion tutorial].
Using the normalization condition in (10-2), we find that the resonant transition between \(|1\rangle\) and \(|2\rangle\) has the following normalized Lorentzian lineshape function:
\[\tag{10-3}\hat{g}(\omega)=\frac{1}{\pi}\frac{\gamma_{21}}{(\omega-\omega_{21})^2+\gamma^2_{21}}\]
which has a FWHM of \(\Delta\omega_\text{h}=2\gamma_{21}\), or
\[\tag{10-4}\hat{g}(\nu)=\frac{\Delta\nu_\text{h}}{2\pi[(\nu-\nu_{21})^2+(\Delta\nu_\text{h}/2)^2]}\]
where \(\Delta\nu_\text{h}=\gamma_{21}/\pi\) is the FWHM of \(\hat{g}(\nu)\). We see that the spectrum has a finite width that is determined by the relaxation constant \(\gamma_{21}\).
The fundamental mechanism for homogenous broadening is lifetime broadening due to the finite lifetimes, \(\tau_1\) and \(\tau_2\), of the energy levels, \(|1\rangle\), and \(|2\rangle\), respectively, that are involved in the resonant transition.
The population in an energy level can relax through both radiative transitions and nonradiative transitions to lower levels.
Radiative relaxation is associated with population relaxation through spontaneous emission of radiation. The radiative relaxation rate of the transition from level \(|2\rangle\) to level \(|1\rangle\) is characterized by a constant \(A_{21}\), known as the Einstein A coefficient, or a time constant \(\tau_\text{sp}=1/A_{21}\), known as the spontaneous radiative lifetime between \(|2\rangle\) and \(|1\rangle\). Both \(A_{21}\) and \(\tau_\text{sp}\) are discussed in further detail later.
The total radiative relaxation rate, \(\gamma_2^\text{rad}\), of level \(|2\rangle\) is the sum of all radiative spontaneous transition rates from \(|2\rangle\) to other levels: \(\gamma_2^\text{rad}=\sum_iA_{2i}\).
The nonradiative relaxation rate accounts for all other population relaxation mechanisms that do not result in the emission of photons.
Therefore, the total relaxation rate is the sum of the radiative and nonradiative relaxation rates, and the lifetime of an energy level has both radiative and nonradiative contributions:
\[\tag{10-5}\gamma_2=\gamma_2^\text{rad}+\gamma_2^\text{nonrad},\qquad\frac{1}{\tau_2}=\frac{1}{\tau_2^\text{rad}}+\frac{1}{\tau_2^\text{nonrad}}\]
where \(\tau_2=1/\gamma_2\), \(\tau_2^\text{rad}=1/\gamma_2^\text{rad}\), and \(\tau_2^\text{nonrad}=1/\gamma_2^\text{nonrad}\).
The same concept can be applied to level \(|1\rangle\) to obtain similar relations for \(\gamma_1\) and \(\tau_1\).
Even though \(\tau_2\) is contributed by both radiative and nonradiative decay from level \(|2\rangle\), fluorescent emission from level \(|2\rangle\) decays at the total relaxation rate \(\gamma_2\) of the population in level \(|2\rangle\). Therefore, the decay time constant of the fluorescent emission associated with population relaxation from \(|2\rangle\) is \(\tau_2\), not \(\tau_2^\text{rad}\).
For this reason, the total lifetimes \(\tau_1\) and \(\tau_2\) are known as the fluorescence lifetimes of energy levels \(|1\rangle\) and \(|2\rangle\), respectively.
The contributions of various relaxation rates to the radiative and nonradiative lifetimes, and to the fluorescence lifetimes, of the upper and lower laser levels are summarized in Figure 10-2.
The nonradiative relaxation rate of an energy level is a function of external perturbations such as collisions and thermal vibrations. IT can therefore be changed by varying the conditions of the surrounding environment.
The minimum broadening is called natural broadening and is caused only by radiative relaxation when the nonradiative processes are eliminated. The linewidth due to natural broadening alone is
\[\tag{10-6}\gamma_{21}^\text{natural}=\frac{1}{2}(\gamma_1^\text{rad}+\gamma_2^\text{rad})=\frac{1}{2}(\frac{1}{\tau_1^\text{rad}}+\frac{1}{\tau_2^\text{rad}})\]
The total contribution of lifetime broadening to the linewidth due to both radiative and nonradiative relaxation processes is
\[\tag{10-7}\gamma_{21}^\text{life}=\frac{1}{2}(\gamma_1+\gamma_2)=\frac{1}{2}(\frac{1}{\tau_1}+\frac{1}{\tau_2})\ge\gamma_{21}^\text{natural}\]
These contributions to \(\gamma_{21}^\text{natural}\) and \(\gamma_{21}^\text{life}\) are also summarized in Figure 10-2.
Note that the linewidth is determined by the lifetimes of both upper and lower laser levels.
In the case when the lower laser level \(|1\rangle\) is the ground state of an atomic system, such as in the situation of the ruby emission line at 694.3 nm, we have \(\gamma_1=0\) and \(\tau_1=\infty\). Then, the linewidth due to lifetime broadening is solely determined by the lifetime of the upper laser level, \(\tau_2\).
Other mechanisms that affect all atoms equally can further increase the homogeneous linewidth without changing the fluorescence lifetimes \(\tau_2\) and \(\tau_1\) of the upper and lower laser levels.
One important mechanism is collision-induced phase randomization of the emitted radiation. Collisions among atoms in a gas or liquid and collisions of atoms with phonons in a solid normally have two possible effects.
One is reduction of the fluorescence lifetimes of the upper and lower laser levels by increasing the nonradiative relaxation rates. Such a process increases lifetime broadening; its effect is included in \(\gamma_{21}^\text{life}\) through the dependence of \(\gamma_{21}^\text{life}\) on \(\gamma_1^\text{nonrad}\) and \(\gamma_2^\text{nonrad}\) contained in \(\gamma_1\) and \(\gamma_2\), respectively.
Collisions can also increase a homogeneous linewidth without reducing the fluorescence lifetimes by simply interrupting the phase of the radiation emitted through radiative relaxation. This dephasing process, qualified by a linewidth-broadening factor \(\gamma_{21}^\text{dephase}\), is often more important then the lifetime-reduction process, resulting in a homogeneous linewidth that is significantly broader than the linewidth contributed by lifetime broadening.
Therefore, the homogeneous linewidth can increase both with pressure and with temperature in a gas medium, and with active-ion concentration and temperature in a liquid or solid medium.
In general, the homogeneous linewidth, including the contributions of such external mechanisms, is a function of pressure, \(P\), active-ion concentration, \(N\), and temperature, \(T\):
\[\tag{10-8}\gamma_{21}(P,N,T)=\gamma_{21}^\text{life}+\gamma_{21}^\text{dephase}\ge\gamma_{21}^\text{life}\ge\gamma_{21}^\text{natural}\]
Example 10-1
The energy levels of laser transitions, along with radiative transition rates and emission wavelengths, of Nd: YAG are shown in Figure 10-3. The upper level \(^4\text{F}_{3/2}\) relaxes radiatively to four lower levels. The lowest level \(^4\text{I}_{9/2}\) is the ground level of the system. In this example, we consider the dominant transition that takes place between the upper level \(^4\text{F}_{3/2}\), labeled level \(|2\rangle\), and the lower level \(^4\text{I}_{11/2}\), labeled level \(|1\rangle\), for the well-known Nd: YAG emission wavelength of \(\lambda=1.064\text{ μm}\). The relaxation of the upper level \(^4\text{F}_{3/2}\) is predominantly radiative with a fluorescence lifetime of \(\tau_2=240\text{ μs}\). The relaxation of the lower level \(^4\text{I}_{11/2}\) is nonradiative with a fluorescence lifetime of \(\tau_1=200\text{ ps}\).
(a) Find the spontaneous radiative lifetime \(\tau_\text{sp}\) between \(|2\rangle\) and \(|1\rangle\).
(b) Find the radiative and nonradiative relaxation rates, \(\gamma_2^\text{rad}\) and \(\gamma_2^\text{nonrad}\), and the corresponding lifetimes, \(\tau_2^\text{rad}\) and \(\tau_2^\text{nonrad}\), for the upper level \(|2\rangle\).
(c) Find the natural linewidth, \(\Delta\nu_\text{natural}\), and the lifetime-broadened homogeneous linewidth, \(\Delta\nu_\text{life}\).
(d) If the measured linewidth at room temperature is \(\Delta\nu=150\text{ GHz}\) with a homogeneously broadened component of \(\Delta\nu_\text{h}=120\text{ GHz}\), what is the linewidth-broadening factor \(\gamma_{21}^\text{dephase}\) due to dephasing through phonon collisions?
(a) Using the radiative transition rate between \(^4\text{F}_{3/2}\) and \(^4\text{I}_{11/2}\), we find that
\[\tau_\text{sp}=\frac{1}{A_{21}}=\frac{1}{1940}\text{s}=515\text{ μs}\]
(b) The total radiative relaxation rate is the sum of the radiative transition rates from \(^4\text{F}_{3/2}\) to all four lower levels. Therefore, the radiative relaxation rate and the radiative lifetime are, respectively,
\[\gamma_2^\text{rad}=\sum_iA_{2i}=3868\text{ s}^{-1},\qquad\tau_2^\text{rad}=\frac{1}{\gamma_2^\text{rad}}=259\text{ μs}\]
Note that \(\tau_2^\text{rad}\gt\tau_2=240\text{ μs}\), as expected. Because the total relaxation rate of the upper level is \(\gamma_2=1/\tau_2=4167\text{ s}^{-1}\), the nonradiative relaxation rate and the nonradiative lifetime are, respectively,
\[\gamma_2^\text{nonrad}=\gamma_2-\gamma_2^\text{rad}=299\text{ s}^{-1},\qquad\tau_2^\text{nonrad}=\frac{1}{\gamma_2^\text{nonrad}}=3.34\text{ ms}\]
(c) Because level \(|1\rangle\) relaxes only nonradiatively, \(\gamma_1^\text{rad}=0\) and \(\tau_1^\text{rad}=\infty\). Therefore,
\[\gamma_{21}^\text{natural}=\frac{1}{2}(\gamma_1^\text{rad}+\gamma_2^\text{rad})=\frac{1}{2}(0+3868)\text{ s}^{-1}=1.93\times10^3\text{ s}^{-1}\]
Using (10-7), we find that
\[\gamma_{21}^\text{life}=\frac{1}{2}(\frac{1}{\tau_1}+\frac{1}{\tau_2})=\frac{1}{2}(\frac{1}{200\times10^{-12}}+\frac{1}{240\times10^{-6}})\text{ s}^{-1}=2.5\times10^9\text{ s}^{-1}\]
From (10-3) and (10-4), we know that \(\Delta\nu_\text{h}=\gamma_{21}/\pi\). Using a similar relation, we find that
\[\Delta\nu_\text{natural}=\frac{\gamma_{21}^\text{natural}}{\pi}=616\text{ Hz},\qquad\Delta\nu_\text{life}=\frac{\gamma_{21}^\text{life}}{\pi}=796\text{ Mhz}\]
(d) For \(\Delta\nu_\text{h}=120\text{ GHz}\), we have \(\gamma_{21}=\pi\Delta\nu_\text{h}=3.77\times10^{11}\text{ s}^{-1}\). Therefore,
\[\gamma_{21}^\text{dephase}=\gamma_{21}-\gamma_{21}^\text{life}=3.745\times10^{11}\text{ s}^{-1}\]
Clearly, \(\gamma_{21}\approx\gamma_{21}^\text{dephase}\gg\gamma_{21}^\text{life}\) in this example.
Inhomogeneous Broadening
A resonant transition can be further broadened by inhomogeneous broadening if certain physical mechanisms exist that do not affect all atoms equally, causing energy levels \(|1\rangle\) and/or \(|2\rangle\) to shift differently among different groups of atoms. The resulting inhomogeneous shifts of the resonance frequency contribute to inhomogeneous broadening of the transition spectrum on top of the original homogeneous broadening.
If we express the homogeneous lineshape function given in (10-4) as \(\hat{g}_\text{h}(\nu,\nu_{21})\) to indicate explicitly that its resonance frequency is at \(\nu_{21}\), the homogeneously broadened spectrum of a group of atoms whose resonance frequency is shifted from \(\nu_{21}\) to \(\nu_k\) is \(\hat{g}_\text{h}(\nu,\nu_k)\). The distribution of atoms in the system can be described by a probability density function \(p(\nu_k)\) with
\[\tag{10-9}\displaystyle\int\limits_0^\infty{p(\nu_k)}\text{d}\nu_k=1\]
The probability that the resonance frequency of a given atom falls in the range between \(\nu_k\) and \(\nu_k+\text{d}\nu_k\) is \(p(\nu_k)\text{d}\nu_k\). Then, the overall spectral lineshape of the inhomogeneously broadened transition is
\[\tag{10-10}\hat{g}(\nu)=\displaystyle\int\limits_0^\infty{p(\nu_k)}\hat{g}_\text{h}(\nu,\nu_k)\text{d}\nu_k\]
The overall lineshape function obtained from (10-10) depends on the degree of inhomogeneous broadening in comparison to the homogeneous broadening of the atoms. Mathematically, it depends on the spread of the distribution \(p(\nu_k)\) in comparison to the homogeneous linewidth.
One possibility for inhomogeneous broadening is the existence of different isotopes, which have slightly different resonance frequencies for a given resonant transition. In this situation, \(p(\nu_k)\text{d}\nu_k\) represents the percentage of each isotope group among all atoms and (10-10) becomes simply the weighted sum of the isotope groups.
Other mechanisms for inhomogeneous broadening include the Doppler effect in a gaseous medium at a low pressure and the random distribution of active impurity atoms doped in a solid host.
The inhomogeneous frequency shifts caused by these mechanisms are usually randomly distributed, resulting in a Gaussian functional distribution for \(p(\nu_k)\). In an extremely inhomogeneously broadened system, the spread of this distribution dominates the homogeneous linewidth. Then, the transition is characterized by a normalized Gaussian lineshape:
\[\tag{10-11}\hat{g}(\nu)=\frac{2(\ln2)^{1/2}}{\pi^{1/2}\Delta\nu_\text{inh}}\exp\left[-4\ln2\frac{(\nu-\nu_0)^2}{\Delta\nu^2_\text{inh}}\right]\]
where \(\nu_0\) is the center frequency and \(\Delta\nu_\text{inh}\) is the FWHM of the inhomogeneously broadened spectral distribution.
In terms of the angular frequency, the normalized Gaussian lineshape is
\[\tag{10-12}\hat{g}(\omega)=\frac{2(\ln2)^{1/2}}{\pi^{1/2}\Delta\omega_\text{inh}}\exp\left[-4\ln2\frac{(\omega-\omega_0)^2}{\Delta\omega^2_\text{inh}}\right]\]
where \(\omega_0=2\pi\nu_0\) and \(\Delta\omega_\text{inh}=2\pi\Delta\nu_\text{inh}\).
Figure 10-14 compares the normalized Lorentzian lineshape function and the normalized Gaussian lineshape function of the same FWHM.
In Figure 10-4(a), we show \(\hat{g}(\nu)\) as expressed in (10-4) for the Lorentzian lineshape and in (10-11) for the Gaussian lineshape, both with a normalized area as defined in (10-2).
In Figure 10-4(b), the lineshapes that are normalized to the same peak value are shown.
Whether a medium is homogeneously or inhomogeneously broadened is often a function of pressure and temperature. In a gas at low pressure, the velocity distribution of the gas molecules in thermal equilibrium is characterized by the Maxwellian velocity distribution, which is a Gaussian function. This velocity distribution leads to a Gaussian distribution of the Doppler frequency shifts with a linewidth \(\Delta\nu_\text{D}\) given by
\[\tag{10-13}\Delta\nu_\text{D}=2\nu\left(\frac{2(\ln2)k_\text{B}T}{Mc^2}\right)^{1/2}=\frac{2^{3/2}(\ln2)^{1/2}}{\lambda}\left(\frac{k_\text{B}T}{M}\right)^{1/2}\]
where \(\lambda\) is the emission wavelength, \(k_\text{B}\) is the Boltzmann constant, \(T\) is the temperature in kelvins, and \(M\) is the mass of the atom or molecule that emits the radiation.
When this Doppler-broadening effect dominates, the Gaussian lineshape has an inhomogeneous linewidth of \(\Delta\nu_\text{inh}=\Delta\nu_\text{D}\).
When the pressure is increased, frequent collisions among the gas molecules cause the homogeneous linewidth to increase. At a certain pressure, the homogeneous linewidth \(\Delta\nu_\text{h}\) finally dominates the Doppler linewidth \(\Delta\nu_\text{D}\). Then the medium becomes predominantly homogeneously broadened.
Another good example is the linewidth associated with the impurity ions doped in a solid host, such as Nd: YAG or Nd: glass.
At low temperatures, the homogeneous linewidth of the \(\text{Nd}^{3+}\) ions is narrow. The lineshape is dominated by inhomogeneous shifts of the resonance frequency due to variations in the local environment of individual \(\text{Nd}^{3+}\) ions. As a result, the lineshape function is inhomogeneously broadened.
As the temperature increases, the homogeneous linewidth increases because of increased collisions of phonons with the ions.
At room temperature, the spectral line of Nd: YAG at 1.064 μm has a total linewidth of \(\Delta\nu\approx120-180\text{ GHz}\) with an inhomogeneous component of only about \(6-30\text{ GHz}\). Therefore, Nd: YAG becomes pretty much homogeneously broadened at room temperature.
In comparison, Nd: glass has a much larger inhomogeneous linewidth than Nd: YAG because the glass host provides a larger range of local variations than the YAG crystal. At room temperature, the same spectral line of Nd: glass appears at 1.054 μm with a total linewidth of \(\Delta\nu\approx5-7\text{ THz}\), which is predominantly inhomogeneously broadened.
Example 10-2
The emission at 632.8 nm wavelength of the HeNe laser is caused by radiative transitions in the Ne atoms. The linewidth of this emission is inhomogeneously broadened due to Doppler broadening. The atomic mass number of Ne is 20, and the typical gas temperature of a HeNe laser is about 400 K. Find the emission linewidth.
The mass of a Ne atom of mass number 20 is \(M=20\times1.67\times10^{-27}\text{ kg}\). Using (10-13), we find that the inhomogeneously broadened linewidth due to Doppler broadening is
\[\Delta\nu_\text{D}=\frac{2^{3/2}(\ln2)^{1/2}}{632.8\times10^{-9}}\times\left(\frac{1.38\times10^{-23}\times400}{20\times1.67\times10^{-27}}\right)^{1/2}\text{ Hz}=1.5\text{ GHz}\]
Transition Rates
The probability per unit time for a resonant optical process to occur is measured by the transition rate of the process. Because of the resonant nature of the interaction, the transition rate of an induced process is a function of the spectral distribution of the optical radiation and the spectral characteristics of the resonant transition.
The spectral distribution of an optical field is characterized by its spectral energy density, \(u(\nu)\), which is the energy density of the optical radiation per unit frequency interval at the optical frequency \(\nu\). The total energy density of the radiation is
\[\tag{10-14}u=\displaystyle\int\limits_0^\infty{u(\nu)}\text{d}\nu\]
The spectral intensity distribution, \(I(\nu)\), of the radiation is related to \(u(\nu)\) by the relation
\[\tag{10-15}I(\nu)=\frac{c}{n}u(\nu)\]
where \(n\) is the refractive index of the medium, and the total intensity is simply
\[\tag{10-16}I=\displaystyle\int\limits_0^\infty{I(\nu)}\text{d}\nu\]
Because an induced transition is stimulated by optical radiation, its transition rate is proportional to the energy density of the optical radiation within the spectral response range of the transition.
The transition rate for the upward transition from \(|1\rangle\) to \(|2\rangle\) associated with absorption in the frequency range between \(\nu\) and \(\nu+\text{d}\nu\) is
\[\tag{10-17}W_{12}(\nu)\text{d}\nu=B_{12}u(\nu)\hat{g}(\nu)\text{d}\nu\qquad(\text{s}^{-1})\]
whereas that for the downward transition from \(|2\rangle\) to \(|1\rangle\) associated with stimulated emission in the frequency range between \(\nu\) and \(\nu+\text{d}\nu\) is
\[\tag{10-18}W_{21}(\nu)\text{d}\nu=B_{21}u(\nu)\hat{g}(\nu)\text{d}\nu\qquad(\text{s}^{-1})\]
The spontaneous emission rate is independent of the energy density of the radiation. The spontaneous emission spectrum associated with a particular resonant transition is determined solely by the lineshape function of the transition:
\[\tag{10-19}W_\text{sp}(\nu)\text{d}\nu=A_{21}\hat{g}(\nu)\text{d}\nu\qquad(\text{s}^{-1})\]
The \(A\) and \(B\) constants defined above are known as the Einstein \(A\) and \(B\) coefficients, respectively.
The induced and the spontaneous transition rates for a given system are not independent of each other but are directly proportional to one another. Such a relationship was first obtained by Einstein by considering the interaction of blackbody radiation with an ensemble of identical atomic systems in thermal equilibrium.
The spectral energy density of blackbody radiation at a temperature \(T\) is given by Planck's formula:
\[\tag{10-20}u(\nu)=\frac{8\pi{n^3}h\nu^3}{c^3}\frac{1}{\text{e}^{h\nu/k_\text{B}T}-1}\]
where \(k_\text{B}\) is the Boltzmann constant.
In thermal equilibrium with blackbody radiation, the total induced transition rates are
\[\tag{10-21}W_{12}=\displaystyle\int\limits_0^\infty{W_{12}}(\nu)\text{d}\nu=B_{12}\int\limits_0^\infty{u(\nu)}\hat{g}(\nu)\text{d}\nu\]
and
\[\tag{10-22}W_{21}=\displaystyle\int\limits_0^\infty{W_{21}}(\nu)\text{d}\nu=B_{21}\int\limits_0^\infty{u(\nu)}\hat{g}(\nu)\text{d}\nu\]
The total spontaneous emission rate is
\[\tag{10-23}W_\text{sp}=\displaystyle\int\limits_0^\infty{W_\text{sp}}(\nu)\text{d}\nu=A_{21}\]
The rates associated with resonant transitions between two atomic levels \(|1\rangle\) and \(|2\rangle\) in the interaction with a radiation field of energy density \(u(\nu)\) are summarized in Figure 10-5 below.
If \(N_2\) and \(N_1\) are the population densities per unit volume of the atoms in levels \(|2\rangle\) and \(|1\rangle\), respectively, the number of atoms per unit volume making the downward transition per unit time accompanied by the emission of radiation in a frequency range from \(\nu\) to \(\nu+\text{d}\nu\) is \(N_2(W_{21}(\nu)+W_\text{sp}(\nu))\text{d}\nu\), and the number of atoms per unit volume making the upward transition per unit time assisted by the absorption of radiation in the same frequency range is \(N_1W_{12}(\nu)\text{d}\nu\). In equilibrium, both the blackbody radiation spectral density and the atomic population density in each energy level should reach a steady state, meaning that
\[\tag{10-24}N_2[W_{21}(\nu)+W_\text{sp}(\nu)]=N_1W_{12}(\nu)\]
This relation spells out the principle of detailed balance in thermal equilibrium. Therefore, the steady-state population distribution in thermal equilibrium satisfies
\[\tag{10-25}\frac{N_2}{N_1}=\frac{W_{12}(\nu)}{W_{21}(\nu)+W_\text{sp}(\nu)}=\frac{B_{12}u(\nu)}{B_{21}u(\nu)+A_{21}}\]
In thermal equilibrium at temperature \(T\), however, the population ratio of the atoms in the upper and the lower levels follows the Boltzmann distribution. Taking into account the degeneracy factors, \(g_2\) and \(g_1\), of these energy levels, we have
\[\tag{10-26}\frac{N_2}{N_1}=\frac{g_2}{g_1}\exp(-h\nu/k_\text{B}T)\]
for the population densities associated with a transition energy of \(h\nu\). Combining (10-25) and (10-26), we have
\[\tag{10-27}u(\nu)=\frac{A_{21}/B_{21}}{(g_1B_{12}/g_2B_{21})\text{e}^{h\nu/k_\text{B}T}-1}\]
Identifying (10-27) with (10-20), we find that
\[\tag{10-28}\frac{A_{21}}{B_{21}}=\frac{8\pi{n^3}h\nu^3}{c^3}\]
and
\[\tag{10-29}g_1B_{12}=g_2B_{21}\]
The spontaneous radiative lifetime of the atoms in level \(|2\rangle\) associated with the radiative spontaneous transition from \(|2\rangle\) to \(|1\rangle\) is
\[\tag{10-30}\tau_\text{sp}=\frac{1}{W_\text{sp}}=\frac{1}{A_{21}}\]
Therefore, the spectral dependence of the spontaneous emission rate can be expressed as
\[\tag{10-31}W_\text{sp}(\nu)=\frac{1}{\tau_\text{sp}}\hat{g}(\nu)\]
According to the relations in (10-28) and (10-29), the transition rates of both of the induced processes of absorption and stimulated emission are directly proportional to the spontaneous emission rate.
In terms of \(\tau_\text{sp}\), the spectral dependence of the induced transition rates between energy levels \(|1\rangle\) and \(|2\rangle\) can be generally expressed as
\[\tag{10-32}W_{21}(\nu)=\frac{c^3}{8\pi{n^3}h\nu^3\tau_\text{sp}}u(\nu)\hat{g}(\nu)=\frac{c^2}{8\pi{n^2}h\nu^3\tau_\text{sp}}I(\nu)\hat{g}(\nu)\]
for the transition from \(|2\rangle\) to \(|1\rangle\) associated with stimulated emission and
\[\tag{10-33}W_{12}(\nu)=\frac{g_2}{g_1}W_{21}(\nu)\]
for the transition from \(|1\rangle\) to \(|2\rangle\) associated with absorption.
Because \(W(\nu)\) is the transition rate per unit frequency according to the definition in (10-17) to (10-19), we have \(W(\nu)\text{d}\nu=W(\omega)\text{d}\omega\). Therefore, \(W_\text{sp}(\nu)=2\pi{W}_\text{sp}(\omega)\), \(W_{21}(\nu)=2\pi{W}_{21}(\omega)\), and \(W_{12}(\nu)=2\pi{W}_{12}(\omega)\).
Transition Cross Section
It is often useful to express the transition probability of an atom in its interaction with optical radiation at a frequency \(\nu\) in terms of the transition cross section, \(\sigma(\nu)\).
For transitions between energy levels \(|1\rangle\) and \(|2\rangle\), the transition cross sections \(\sigma_{21}(\nu)\) and \(\sigma_{12}(\nu)\) are defined through the following relations to the transition rates:
\[\tag{10-34}W_{21}(\nu)=\frac{I(\nu)}{h\nu}\sigma_{21}(\nu)\]
and
\[\tag{10-35}W_{12}(\nu)=\frac{I(\nu)}{h\nu}\sigma_{12}(\nu)\]
The transition cross section \(\sigma_{21}(\nu)\), which is associated with stimulated emission, is also called the emission cross section, \(\sigma_\text{e}(\nu)\), whereas \(\sigma_{12}(\nu)\), which is associated with absorption, is also called the absorption cross section, \(\sigma_\text{a}(\nu)\).
From (10-32), we find that
\[\tag{10-36}\sigma_\text{e}(\nu)=\sigma_{21}(\nu)=\frac{c^2}{8\pi{n^2}\nu^2\tau_\text{sp}}\hat{g}(\nu)\]
According to (10-29) and (10-33), \(g_1\sigma_{12}=g_2\sigma_{21}\). Therefore,
\[\tag{10-37}\sigma_\text{a}(\nu)=\sigma_{12}(\nu)=\frac{g_2}{g_1}\sigma_{21}(\nu)=\frac{g_2}{g_1}\sigma_\text{e}(\nu)\]
The transition cross sections have the unit of area in square meters but are often quoted in square centimeters.
Note that \(\sigma(\nu)=\sigma(\omega)\) because \(\sigma(\nu)\) is simply defined as the value of the transition cross section at the frequency \(\nu\) rather than as that per unit frequency, but \(W(\nu)=2\pi{W}(\omega)\) and \(\hat{g}(\nu)=2\pi\hat{g}(\omega)\).
Therefore, in terms of \(\omega\),
\[\tag{10-38}\sigma_\text{e}(\omega)=\sigma_{21}(\omega)=\frac{\pi^2c^2}{n^2\omega^2\tau_\text{sp}}\hat{g}(\omega)\qquad\text{and}\qquad\sigma_\text{a}(\omega)=\frac{g_2}{g_1}\sigma_\text{e}(\omega)\]
For the ideal Lorentzian and Gaussian lineshapes expressed in (10-4) and (10-11), respectively, the peak value of \(\hat{g}(\nu)\) occurs at the center of the spectrum and is a function of linewidth \(\Delta\nu\) only. By applying this fact to (10-36), the peak value of the emission cross section at the center wavelength \(\lambda\) of the spectrum can be expressed as
\[\tag{10-39}\sigma_\text{e}^\text{h}=\frac{\lambda^2}{4\pi^2n^2\Delta\nu_\text{h}\tau_\text{sp}}\]
for a homogeneously broadened medium with an ideal Lorentzian lineshape, and as
\[\tag{10-40}\sigma_\text{e}^\text{inh}=\frac{(\ln2)^{1/2}\lambda^2}{4\pi^{3/2}n^2\Delta\nu_\text{inh}\tau_\text{sp}}\]
for an inhomogeneously broadened medium with an ideal Gaussian lineshape.
In practice, the experimentally measured peak emission cross section usually differs from that calculated using these formulas because the spectral lineshape of a realistic laser gain medium is generally determined by a combination of many different mechanisms and, consequently, is rarely ideal Lorentzian or ideal Gaussian.
Nevertheless, these formulas provide a very good estimate for the peak value of the emission cross section. They also clearly indicate that the emission cross section varies quadratically with the emission wavelength but is inversely proportional to both the emission linewidth and the spontaneous radiative lifetime of the laser transition.
The characteristics of some representative laser materials are listed in Table 10-1 below. As seen from Table 10-1, the parameters vary over a wide range among different types of laser gain media. For example, the peak value of the emission cross section varies from \(6\times10^{-25}\text{ m}^2\) for Er : fiber to \(2.5\times10^{-16}\text{ m}^2\) for the Ar-ion laser, whereas the spontaneous emission linewidth varies from 60 MHz for CO2 to 100 THz for Ti : sapphire. The fluorescence lifetime varies from the order of 1 ns for semiconductor gain media to the order of 10 ms for Er : fiber.
Example 10-3
The emission at 632.8 nm wavelength of a HeNe laser is inhomogeneously broadened due to Doppler broadening with a linewidth of \(\Delta\nu\approx\Delta\nu_\text{inh}=\Delta\nu_\text{D}=1.5\text{ GHz}\). The spontaneous radiative lifetime is \(\tau_\text{sp}=300\text{ ns}\). Being a gas laser, the refractive index of the medium is \(n\approx1\). Find the peak emission cross section of the HeNe laser at this wavelength.
Using (10-40), we find the following peak emission cross section at \(\lambda=632.8\text{ nm}\) for the HeNe laser:
\[\sigma_\text{e}=\sigma_\text{e}^\text{inh}=\frac{(\ln2)^{1/2}\times(632.8\times10^{-9})^2}{4\times\pi^{3/2}\times1^2\times1.5\times10^9\times300\times10^{-9}}\text{ m}^2=3.3\times10^{-17}\text{ m}^2\]
This calculated result agrees, by a small difference of only \(10\%\), with the value of \(3.7\times10^{-17}\text{ m}^2\) quoted in the literature, which is listed in Table 10-1.
The next tutorial covers optical absorption and amplification.