Population Inversion and Optical Gain
This is a continuation from the previous tutorial - optical absorption and amplification.
From the discussions in the optical absorption and amplification tutorial, it is clear that population inversion is the basic condition for the presence of an optical gain.
In the normal state of any system in thermal equilibrium, a low-energy state is always more populated than a high-energy state, hence no population inversion.
Population inversion in a system can only be accomplished through a process called pumping by actively exciting the atoms in a low-energy state to a high-energy state.
If left alone, the atoms in a system will relax to thermal equilibrium. Therefore, population inversion is a nonequilibrium state that cannot be sustained without active pumping.
To maintain a constant optical gain, continuous pumping is required to keep the population inversion at a constant level.
This condition is clearly consistent with the law of conservation of energy: amplification of an optical wave leads to an increase in optical energy, which is possible only if there is a source supplying the energy.
Pumping is the process that supplies the energy to the gain medium for the amplification of an optical wave. There are many different pumping techniques, including optical excitation, electric current injection, electric discharge, chemical reaction, and excitation with particle beams.
The use of a particular pumping technique depends on the properties of the gain medium being pumped. The lasers and optical amplifiers of particular interest in photonic systems are made of either dielectric solid-state media doped with active ions, such as Nd : YAG and Er : glass fiber, or direct-gap semiconductors, such as GaAs and InP.
For dielectric media, the most commonly used pumping technique is optical pumping either with incoherent light sources, such as flashlamps and light-emitting diodes, or with coherent light sources from other lasers.
Semiconductor gain media can also be optically pumped, but they are usually pumped with electric current injection.
In this tutorial, we consider the general conditions for pumping to achieve population inversion for an optical gain. Detailed pumping mechanisms and physical setups are not addressed here because they depend on the specific gain medium used in a given application.
Rate Equations
The net rate of increase of population density in a given energy level is described by a rate equation. As we shall see below, pumping for population inversion in any practical gain medium always requires the participation of more than two energy levels.
In general, a rate equation has to be written for each energy level that is involved in the process. For simplicity but without loss of validity, however, we shall explicitly write down only the rate equations for the two energy levels, \(|2\rangle\) and \(|1\rangle\), that are directly associated with the resonant transition of interest.
We are not interested in the population densities of other energy levels but only in how those levels affect \(N_2\) and \(N_1\).
In the presence of a monochromatic, coherent optical wave of intensity \(I\) at a frequency \(\nu\), the rate equations for \(N_2\) and \(N_1\) are
\[\tag{10-61}\frac{\text{d}N_2}{\text{d}t}=R_2-\frac{N_2}{\tau_2}-\frac{I}{h\nu}(N_2\sigma_\text{e}-N_1\sigma_\text{a})\]
\[\tag{10-62}\frac{\text{d}N_1}{\text{d}t}=R_1-\frac{N_1}{\tau_1}+\frac{N_2}{\tau_{21}}+\frac{I}{h\nu}(N_2\sigma_\text{e}-N_1\sigma_\text{a})\]
where \(R_2\) and \(R_1\) are the total rates of pumping into energy levels \(|2\rangle\) and \(|1\rangle\), respectively, and \(\tau_2\) and \(\tau_1\) are the fluorescence lifetimes of levels \(|2\rangle\) and \(|1\rangle\), respectively.
The rate of population decay, including radiative and nonradiative spontaneous relaxation, from level \(|2\rangle\) to level \(|1\rangle\) is \(\tau_{21}^{-1}\).
Because it is possible for the population in level \(|2\rangle\) to relax to other energy levels also, the total population decay rate of level \(|2\rangle\) is \(\tau_2^{-1}\ge\tau_{21}^{-1}\).
Therefore, in general, we have
\[\tag{10-63}\tau_2\le\tau_{21}\le\tau_\text{sp}\]
Note that \(\tau_{21}^{-1}\) is not the same as \(\gamma_{21}\) defined in (10-8) [refer to the optical transitions for laser amplifiers tutorial]: \(\tau_{21}^{-1}\) is purely the rate of population relaxation from level \(|2\rangle\) to level \(|1\rangle\), whereas \(\gamma_{21}\) is the rate of phase relaxation of the polarization associated with the transition between these two levels.
In an optical gain medium, level \(|2\rangle\) is known as the upper laser level and level \(|1\rangle\) is known as the lower laser level.
The fluorescence lifetime \(\tau_2\) of the upper laser level is an important parameter that determines the effectiveness of a gain medium. Generally speaking, the upper laser level has to be a metastable state with a relatively large \(\tau_2\) for a gain medium to be useful.
Population Inversion
Population inversion in a medium is generally defined as
\[\tag{10-64}N_2\gt\frac{g_2}{g_1}N_1\]
According to (10-50) [refer to the optical absorption and amplification tutorial], however, this condition does not guarantee an optical gain at a particular optical frequency \(\nu\) if \(\sigma_\text{a}(\nu)\ne(g_2/g_1)\sigma_\text{e}(\nu)\) when the population in each level, \(|1\rangle\) or \(|2\rangle\), is distributed unevenly among its sublevels.
For this reason, when the condition for population inversion given in (10-64) is achieved in a medium, we may find an optical gain at an optical frequency \(\nu\) where \(\sigma_\text{a}(\nu)\le(g_2/g_1)\sigma_\text{e}(\nu)\), but at the same time find an optical loss at another frequency \(\nu'\) where \(\sigma_\text{a}(\nu')\gt(g_2/g_1)\sigma_\text{e}(\nu')\).
What really matters for an optical wave at a given frequency is the optical gain at that particular frequency.
Therefore, in the following discussions, we shall consider, instead of the condition in (10-64), the following condition:
\[\tag{10-65}N_2\sigma_\text{e}(\nu)-N_1\sigma_\text{a}(\nu)\gt0\]
which guarantees an optical gain at frequency \(\nu\), as the effective condition of population inversion as far as an optical signal at frequency \(\nu\) is concerned.
The pumping requirement for the condition in (10-65) to be satisfied depends on the properties of a medium.
For atomic and molecular media, there are three different basic systems. Each has a different pumping requirement to reach effective population inversion for an optical gain. The pumping requirement can be found by solving the coupled rate equations in (10-61) and (10-62).
Two-level system
When the only energy levels involved in the pumping and the relaxation processes are the upper and the lower laser levels \(|2\rangle\) and \(|1\rangle\), the system can be considered as a two-level system.
In such a system, level \(|1\rangle\) is the ground state with \(\tau_1=\infty\), and level \(|2\rangle\) relaxes only to level \(|1\rangle\) so that \(\tau_{21}=\tau_2\). The total population density is \(N_\text{t}=N_1+N_2\).
While a pumping mechanism excites atoms from the lower laser level to the upper laser level, the same pump also stimulates atoms in the upper laser level to relax to the lower laser level.
Therefore, irrespective of the specific pumping technique used, \(R_2=-R_1=W_{12}^\text{p}N_1-W_{21}^\text{p}N_2\), where \(W_{12}^\text{p}\) and \(W_{21}^\text{p}\) are the pumping transition probability rates, or simply the pumping rates, from \(|1\rangle\) to \(|2\rangle\) and from \(|2\rangle\) to \(|1\rangle\), respectively.
Under these conditions, (10-61) and (10-62) are equivalent to each other. The upward and downward pumping transition rates are not independent of each other but are directly proportional to each other because both are associated with the interaction of the same pump source with a given set of energy levels.
We take the upward pumping rate to be \(W_{12}^\text{p}=W_\text{p}\) and the downward pumping rate to be \(W_{21}^\text{p}=pW_\text{p}\), where \(p\) is a constant that depends on the detailed characteristics of the two-level atomic system and the pump source.
In the steady state when \(\text{d}N_2/\text{d}t=\text{d}N_1/\text{d}t=0\), we then find that
\[\tag{10-66}N_2\sigma_\text{e}-N_1\sigma_\text{a}=\frac{W_\text{p}\tau_2(\sigma_\text{e}-p\sigma_\text{a})-\sigma_\text{a}}{1+(1+p)W_\text{p}\tau_2+(I\tau_2/h\nu)(\sigma_\text{e}+\sigma_\text{a})}N_\text{t}\]
Using the relation in (10-41) [refer to the optical absorption and amplification tutorial], we find that, for optical pumping,
\[\tag{10-67}p=\frac{\sigma_\text{e}^\text{p}}{\sigma_\text{a}^\text{p}}=\frac{\sigma_\text{e}(\lambda_\text{p})}{\sigma_\text{a}(\lambda_\text{p})}\]
where \(\sigma_\text{a}^\text{p}\) and \(\sigma_\text{e}^\text{p}\) are the absorption and emission cross sections, respectively, at the pump wavelength.
In a true two-level system, shown in Figure 10-9(a) above, the energy levels \(|2\rangle\) and \(|1\rangle\) can each be degenerate with degeneracies \(g_2\) and \(g_1\), respectively, but the population densities in both levels are evenly distributed among the respective degenerate states.
In this situation, \(p=\sigma_\text{e}^\text{p}/\sigma_\text{a}^\text{p}=g_1/g_2=\sigma_\text{e}/\sigma_\text{a}\). Then, we find from (10-66) that
\[\tag{10-68}N_2\sigma_\text{e}-N_1\sigma_\text{a}=\frac{-\sigma_\text{a}}{1+(\sigma_\text{e}+\sigma_\text{a})(I/h\nu+W_\text{p}/\sigma_\text{a})\tau_2}N_\text{t}\lt0\]
No matter how a true two-level system is pumped, it is clearly not possible to achieve population inversion for an optical gain in the steady state.
This situation can be understood by considering the fact that the pump for a two-level system has to be in resonance with the transition between the two levels, thus inducing downward transitions as well as upward transitions.
In the steady state, the two-level system would reach thermal equilibrium with the pump at a finite temperature \(T\), resulting in a Boltzmann population distribution of the form given in (10-26) [refer to the optical transitions for laser amplifiers tutorial] without population inversion.
As discussed in the optical absorption and amplification tutorial and illustrated in Figure 10-7 [refer to the optical absorption and amplification tutorial], however, in many cases an energy level is actually split into a band of closely spaced, but not exactly degenerate, sublevels with its population density unevenly distributed among these sublevels.
A system is not a true two-level system, but is known as a quasi-two-level system, if either or both of the two levels involved are split in such a manner.
By pumping such a quasi-two-level system properly, it is possible to reach the needed population inversion in the steady state for an optical gain at a particular laser frequency \(\nu\) because the ratio \(p=\sigma_\text{e}^\text{p}/\sigma_\text{a}^\text{p}\) at the pump frequency \(\nu_\text{p}\) can now be made different from the ratio \(\sigma_\text{e}/\sigma_\text{a}\) at the laser frequency \(\nu\) due to the uneven population distribution among the sublevels within an energy level.
From (10-66), we find that the pumping requirements for a steady-state optical gain from a quasi-two-level system are
\[\tag{10-69}p=\frac{\sigma_\text{e}^\text{p}}{\sigma_\text{a}^\text{p}}\lt\frac{\sigma_\text{e}}{\sigma_\text{a}}\qquad\text{and}\qquad{W_\text{p}}\gt\frac{1}{\tau_2}\frac{\sigma_\text{a}}{\sigma_\text{e}-p\sigma_\text{a}}\]
Because the absorption spectrum is generally shifted to the short-wavelength side of the emission spectrum, as discussed in the optical absorption and amplification tutorial and demonstrated in Figure 10-8 [refer to the optical absorption and amplification tutorial], these conditions can be satisfied by pumping sufficiently strongly at a higher transition energy than the photon energy corresponding to the peak of the emission spectrum.
In the case of optical pumping, this condition means that the pump wavelength has to be shorter than the emission wavelength. Figure 10-9(b) above illustrates such a pumping scheme of a quasi-two-level system.
Indeed, many laser gain media, including laser dyes, semiconductor gain media, and vibronic solid-state gain media, are often pumped as a quasi-two-level system.
Three-level system
Population inversion in steady state is possible for a system that has three energy levels involved in the process. Figure 10-10 shows the energy-level diagram of an idealized three-level system.
The lower laser level \(|1\rangle\) is the ground state, \(E_1=E_0\), or is very close to the ground state, within an energy separation of \(\Delta{E}_{10}\ll{k}_\text{B}T\) from the ground state, so that it is initially populated.
The atoms are pumped to an energy level \(|3\rangle\) above the upper laser level \(|2\rangle\).
An effective three-level system satisfies the following conditions:
(1) Population relaxation from level \(|3\rangle\) to level \(|2\rangle\) is very fast and efficient, ideally \(\tau_2\gg\tau_{32}\approx\tau_3\), so that the atoms excited by the pump quickly end up in level \(|2\rangle\).
(2) Level \(|3\rangle\) lies sufficiently high above level \(|2\rangle\) with \(\Delta{E}_{32}\gg{k}_\text{B}T\) so that the population in level \(|2\rangle\) cannot be thermally excited back to level \(|3\rangle\).
(3) The lower laser level \(|1\rangle\) is the ground state, or its population relaxes very slowly if it is not the ground state.
Under these conditions, \(R_2\approx{W}_\text{p}N_1\), \(R_1\approx-W_\text{p}N_1\), and \(N_1+N_2\approx{N}_\text{t}\). Furthermore, \(\tau_1\approx\infty\) and \(\tau_{21}\approx\tau_2\).
The parameter \(W_\text{p}\) is the effective pumping transition probability rate for exciting an atom in the ground state to eventually reach the upper laser level. It is proportional to the power of the pump.
In the steady state with a constant pump, \(W_\text{p}\) is a constant and \(\text{d}N_2/\text{d}t=\text{d}N_1/\text{d}t=0\).
With these conditions, we find that
\[\tag{10-70}N_2\sigma_\text{e}-N_1\sigma_\text{a}=\frac{W_\text{p}\tau_2\sigma_\text{e}-\sigma_\text{a}}{1+W_\text{p}\tau_2+(I\tau_2/h\nu)(\sigma_\text{e}+\sigma_\text{a})}N_\text{t}\]
Therefore, the pumping condition for a constant optical gain under steady-state population inversion is
\[\tag{10-71}W_\text{p}\gt\frac{\sigma_\text{a}}{\tau_2\sigma_\text{e}}\]
This condition sets the minimum pumping requirement for effective population inversion to reach an optical gain in a three-level system.
This requirement can be understood by considering the fact that almost all of the population initially resides in the lower laser level \(|1\rangle\). To achieve effective population inversion, the pump has to be strong enough to depopulate sufficient population density from level \(|1\rangle\), while the system has to be able to keep it in level \(|2\rangle\).
In the case when \(\sigma_\text{a}=\sigma_\text{e}\), no population inversion occurs before at least one-half of the total population is transferred from level \(|1\rangle\) to level \(|2\rangle\).
Four-level system
A four-level system, shown schematically in Figure 10-11, is more efficient than a three-level system.
A four-level system differs from a three-level system in that the lower laser level \(|1\rangle\) lies sufficiently high above the ground level \(|0\rangle\), with \(\Delta{E}_{10}\gg{k}_\text{B}T\).
Therefore, in thermal equilibrium, the population in \(|1\rangle\) is negligibly small compared with that in \(|0\rangle\). Pumping takes place from level \(|0\rangle\) to level \(|3\rangle\).
An effective four-level system also has to satisfy the conditions concerning levels \(|3\rangle\) and \(|2\rangle\) discussed above for an effective three-level system.
In addition, it has to satisfy the condition that the population in level \(|1\rangle\) relaxes very quickly back to the ground level, ideally \(\tau_1\approx\tau_{10}\ll\tau_2\), so that level \(|1\rangle\) remains relatively unpopulated in comparison with level \(|2\rangle\) when the system is pumped.
Under these conditions, \(N_1\approx0\) and \(R_2\approx{W}_\text{p}(N_\text{t}-N_2)\), where the effective pumping transition probability rate \(W_\text{p}\) is again proportional to the pump power.
Then, (10-62) can be ignored because \(N_1\approx0\). In the steady state when \(W_\text{p}\) is held constant, by taking \(\text{d}N_2/\text{d}t=0\) for (10-61), we find that
\[\tag{10-72}N_2\sigma_\text{e}-N_1\sigma_\text{a}\approx{N_2}\sigma_\text{e}=\frac{W_\text{p}\tau_2\sigma_\text{e}}{1+W_\text{p}\tau_2+(I\tau_2/h\nu)\sigma_\text{e}}N_\text{t}\]
This results indicates that there is no minimum pumping requirement for an ideal four-level system that satisfies the conditions discussed above.
Real systems are rarely ideal, but a practical four-level system is still much more efficient than a three-level system. There is no minimum pumping requirement for population inversion in a four-level system because level \(|1\rangle\) is initially empty in such a system.
Optical Gain
When the condition in (10-65) is satisfied for a given system, an optical gain coefficient at a given optical frequency \(\nu\) can be evaluated with \(g=N_2\sigma_\text{e}-N_1\sigma_\text{a}\) according to (10-50) [refer to the optical absorption and amplification tutorial].
The optical gain coefficient is a function of the optical signal intensity, \(I\), as a result of the dependence of \(N_2\) and \(N_1\) on \(I\) due to stimulated emission that changes the population densities by causing downward transition from level \(|2\rangle\) to level \(|1\rangle\).
This effect causes saturation of the optical gain coefficient by the optical signal. For all three basic systems discussed above, the optical gain coefficient can be expressed as a function of the optical signal intensity, \(I\):
\[\tag{10-73}g=\frac{g_0}{1+I/I_\text{sat}}\]
where \(g_0\) is the unsaturated gain coefficient, which is independent of the optical signal intensity, and \(I_\text{sat}\) is the saturation intensity of a medium, which can be generally expressed as
\[\tag{10-74}I_\text{sat}=\frac{h\nu}{\tau_\text{s}\sigma_\text{e}}\]
The time constant \(\tau_\text{s}\) is an effective saturation lifetime of the effective population inversion. It can be considered as an effective decay time constant for the optical gain coefficient through the relaxation of the effective population inversion.
Both \(g_0\) and \(\tau_\text{s}\) are functions of the intrinsic properties of a gain medium, as well as of the pumping rate. They can be found from (10-66), (10-70), and (10-72) for the quasi-two-level, three-level, and four-level systems, respectively. The results are summarized below.
Quasi-two-level system:
\[\tag{10-75}g_0=(W_\text{p}\tau_\text{s}\sigma_\text{e}-\sigma_\text{a})N_\text{t}\]
\[\tag{10-76}\tau_\text{s}=\tau_2\frac{1+\sigma_\text{a}/\sigma_\text{e}}{1+(1+p)W_\text{p}\tau_2}\]
Three-level system:
\[\tag{10-77}g_0=(W_\text{p}\tau_\text{s}\sigma_\text{e}-\sigma_\text{a})N_\text{t}\]
\[\tag{10-78}\tau_\text{s}=\tau_2\frac{1+\sigma_\text{a}/\sigma_\text{e}}{1+W_\text{p}\tau_2}\]
Four-level system:
\[\tag{10-79}g_0=W_\text{p}\tau_\text{s}\sigma_\text{e}N_\text{t}\]
\[\tag{10-80}\tau_\text{s}=\frac{\tau_2}{1+W_\text{p}\tau_2}\]
The minimum pumping requirement for a medium to have an optical gain is clearly \(g_0\gt0\). It can be shown that the minimum pumping requirements obtained by applying this condition to (10-75) and (10-77) are identical to those given in (10-69) and (10-71) for the quasi-two-level and the three-level systems, respectively. As for the four-level system, both (10-72) and (10-79) clearly indicate that it has no minimum pumping requirement.
For a desired unsaturated gain coefficient of \(g_0\), the required pumping rate can be found by solving (10-75) and (10-76) for a quasi-two-level system, (10-77) and (10-78) for a three-level system, and (10-79) and (10-80) for a four-level system. The results are summarized below.
Quasi-two-level system:
\[\tag{10-81}W_\text{p}=\frac{1}{\tau_2}\frac{\sigma_\text{a}N_\text{t}+g_0}{(\sigma_\text{e}-p\sigma_\text{a})N_\text{t}-(1+p)g_0}\]
Three-level system:
\[\tag{10-82}W_\text{p}=\frac{1}{\tau_2}\frac{\sigma_\text{a}N_\text{t}+g_0}{\sigma_\text{e}N_\text{t}-g_0}\]
Four-level system:
\[\tag{10-83}W_\text{p}=\frac{1}{\tau_2}\frac{g_0}{\sigma_\text{e}N_\text{t}-g_0}\]
In the limit when \(p\rightarrow0\), a quasi-two-level system is identical to a three-level system. In the limit when \(p\rightarrow0\) and \(\sigma_\text{a}\rightarrow0\), a quasi-two-level system behaves like a four-level system. In the limit when \(\sigma_\text{a}\rightarrow0\), a three-level system behaves like a four-level system. For a quasi-two-level system, it is clearly desirable to choose a pump wavelength for which the value of \(p\) is as small as possible.
Unsaturated gain coefficient
The unsaturated gain coefficient is also known as the small-signal gain coefficient because it is the gain coefficient of a weak optical field that does not saturate the gain medium.
In the case of optical pumping with a pump quantum efficiency \(\eta_\text{p}\), the pump intensity required for a desired pumping transition probability rate can be found by using (10-41) [refer to the optical absorption and amplification tutorial] as
\[\tag{10-84}I_\text{p}=\frac{1}{\eta_\text{p}}\frac{h\nu_\text{p}}{\sigma_\text{a}^\text{p}}W_\text{p}\]
where \(h\nu_\text{p}\) is the energy of the pump photon.
The pump quantum efficiency \(\eta_\text{p}\) is the net probability of exciting an atom to the upper laser level by each absorbed pump photon. In general, \(\eta_\text{p}\le1\).
It is convenient to define a saturation pump intensity, \(I_\text{p}^\text{sat}\), for a laser amplifier for which \(W_\text{p}\tau_2=1\) as
\[\tag{10-85}I_\text{p}^\text{sat}=\frac{h\nu_\text{p}}{\eta_\text{p}\tau_2\sigma_\text{a}^\text{p}}\]
This is the pump intensity that pumps one-half of the population in a three- or four-level system, and about one-half in a quasi-two-level system, to the upper laser level.
At this level and above, absorption of the pump power is significantly saturated due to depletion of the ground-state population by pumping.
For a pump intensity of \(I_\text{p}\), we have
\[W_\text{p}\tau_2=I_\text{p}/I_\text{p}^\text{sat}\]
For a four-level system, we have \(g\gt0\) as long as the medium is pumped because there is no minimum pumping requirement. For a quasi-two-level or three-level system, we find that \(g\gt0\) only when the pumping level exceeds its minimum pumping requirement; below that, the medium has absorption for \(g\lt0\).
When the unsaturated gain coefficient is zero, the medium becomes transparent, or bleached, to the optical signal, neither absorbing it nor amplifying it.
A quasi-two-level or three-level system reaches transparency, or the bleached condition, at the following transparency pumping rate:
\[\tag{10-86}W_\text{p}^\text{tr}=\frac{1}{\tau_2}\frac{\sigma_\text{a}}{\sigma_\text{e}-p\sigma_\text{a}}\]
The pump intensity corresponding to the transparency pumping rate is the transparency pump intensity, \(I_\text{p}^\text{tr}\), which can be expressed as
\[\tag{10-87}I_\text{p}^\text{tr}=\frac{1}{\eta_\text{p}}\frac{h\nu_\text{p}}{\tau_2\sigma_\text{a}^\text{p}}\frac{\sigma_\text{a}}{\sigma_\text{e}-p\sigma_\text{a}}=\frac{\sigma_\text{a}}{\sigma_\text{e}-p\sigma_\text{a}}I_\text{p}^\text{sat}\]
For a quasi-two-level system, \(p\ne0\) in general. For a three-level system, we take \(p=0\) for (10-86) and (10-87). For a four-level system, \(I_\text{p}^\text{tr}=0\) because a four-level system has no minimum pumping requirement and is thus transparent without pumping.
It can be seen from (10-75) to (10-80) that for any system, \(g_0\) increases with pump intensity less than linearly because \(\tau_\text{s}\) decreases with pump intensity though \(W_\text{p}\) is linearly proportional to the pump intensity.
This dependence of \(\tau_\text{s}\) on the pump intensity is caused by the fact that as the pump excites atoms from the ground state to any excited state to eventually reach the upper laser level, it depletes the population in the ground state.
Consequently, as the pump intensity increases, fewer atoms remain available for excitation in the ground state, thus reducing the differential increase of the effective population inversion with respect to the increase of the pump intensity.
It can be shown by using the relations in (10-75), (10-77), and (10-79) that the unsaturated gain coefficient of any system can be expressed as a function of pump intensity in the following general form:
\[\tag{10-88}g_0=\frac{(\sigma_\text{e}-p\sigma_\text{a})N_\text{t}}{1+(1+p)I_\text{p}/I_\text{p}^\text{sat}}\left(\frac{I_\text{p}}{I_\text{p}^\text{sat}}-\frac{I_\text{p}^\text{tr}}{I_\text{p}^\text{sat}}\right)=\frac{(\sigma_\text{e}+\sigma_\text{a})N_\text{t}}{1+(1+p)I_\text{p}/I_\text{p}^\text{sat}}\frac{I_\text{p}}{I_\text{p}^\text{sat}}-\sigma_\text{a}N_\text{t}\]
For a quasi-two-level system, \(p\ne0\) and \(I_\text{p}^\text{tr}\ne0\). For a three-level system, \(p=0\) but \(I_\text{p}^\text{tr}\ne0\). For a four-level system, \(p=0\) and \(I_\text{p}^\text{tr}=0\).
Note that for a quasi-two-level system or a three-level system, (10-88) is also valid when \(I_\text{p}\lt{I}_\text{p}^\text{tr}\) for \(g_0\lt0\). In this situation, the medium has an absorption coefficient of \(\alpha=-g_0\) at the laser transition frequency.
As can be seen in (10-88), \(g_0\) varies with \(I_\text{p}\) sublinearly at high pumping levels due to the dependence of \(\tau_\text{s}\) on \(I_\text{p}\) as discussed above.
For a four-level system, however, the unsaturated gain coefficient varies approximately linearly with \(I_\text{p}\) at a low pumping level such that \(I_\text{p}/I_\text{p}^\text{sat}\ll1\).
For a quasi-two-level system or a three-level system, significant pumping is needed just to reach transparency, but the unsaturated gain coefficient also varies approximately linearly with \(I_\text{p}\) for small variations of the pump intensity near the transparency point.
Gain saturation
The optical gain coefficient is a function of the intensity of the optical wave traveling in the gain medium; it decreases as the optical signal intensity increases.
According to (10-73), the optical gain coefficient is reduced to one-half that of the unsaturated gain coefficient \(g_0\) when the optical signal intensity reaches the saturation intensity \(I_\text{sat}\). The smaller the value of \(I_\text{sat}\), the easier it is for the gain to become saturated.
For a quasi-two-level system, \(\tau_\text{s}=\tau_2(1-p\sigma_\text{a}/\sigma_\text{e})\) at transparency. For three-level and four-level systems, \(\tau_\text{s}=\tau_2\) at transparency. For all three system, \(\tau_\text{s}\lt\tau_2\) as the gain medium is pumped above transparency for a positive gain coefficient. Therefore, \(I_\text{sat}\) increases as the gain medium is pumped harder for a larger unsaturated gain coefficient.
Example 10-7
The ruby laser is a three-level system. As shown in Figure 10-12, it has two primary pump bands at 404 and 554 nm wavelengths, from the \(^4\text{A}_2\) ground state to the \(^4\text{F}_1\) and \(^4\text{F}_2\) states, respectively, both of which relax quickly to the \(^2\text{E}\) state so that \(\tau_{32}\ll\tau_2=3\text{ ms}\). The absorption cross sections for \(\mathbf{E}\perp{c}\) polarization at 404 and 554 nm pump wavelengths are both \(\sigma_\text{a}^\text{p}=2\times10^{-23}\text{ m}^{-1}\). Assume a \(100\%\) pump quantum efficiency of \(\eta_\text{p}=1\).
(a) Find the transparency pumping rate of a ruby crystal for the 694.3 nm transitioin with \(\mathbf{E}\perp{c}\) polarization. Find the transparency pump intensity for each pump band. What is the saturation intensity at transparency?
(b) A ruby crystal rod doped with 0.05 wt. \(\%\) Cr2O3 has a Cr concentration of \(1.58\times10^{25}\text{ m}^{-3}\). It is pumped for an unsaturated gain coefficient of \(5\text{ m}^{-1}\). Find the required pumping rate, the saturation intensity at this pumping rate, and the required pump intensity for each pump band.
(a)
From Example 10-4 [refer to the optical absorption and amplification tutorial], we found that \(\sigma_\text{a}=1.25\times10^{-24}\text{ m}^2\), from Example 10-5 [refer to the optical absorption and amplification tutorial], we found that \(\sigma_\text{e}=1.34\times10^{-24}\text{ m}^2\), so that we can find the transparency pumping rate by using (10-86) for this three-level system:
\[W_\text{p}^\text{tr}=\frac{\sigma_\text{a}}{\tau_2\sigma_\text{e}}=\frac{1.25\times10^{-24}}{3\times10^{-3}\times1.34\times10^{-24}}\text{ s}^{-1}=311\text{ s}^{-1}\]
The pump photons at 404 and 554 nm wavelengths have photon energies of \(h\nu_\text{p1}=(1.2398/0.404)\text{ eV}=3.069\text{ eV}\) and \(h\nu_\text{p2}=(1.2398/0.554)\text{ eV}=2.238\text{ eV}\), respectively. The transparency pump intensity for the 404 nm pump band is (by using (10-86) and (10-87))
\[I_\text{p}^\text{tr}=\frac{h\nu_\text{p1}}{\sigma_\text{a}^\text{p}}W_\text{p}^\text{tr}=\frac{3.069\times1.6\times10^{-19}\times311}{2\times10^{-23}}\text{ W m}^{-2}=7.64\text{ MW m}^{-2}\]
and that for the 554 nm pump band is
\[I_\text{p}^\text{tr}=\frac{h\nu_\text{p2}}{\sigma_\text{a}^\text{p}}W_\text{p}^\text{tr}=\frac{2.238\times1.6\times10^{-19}\times311}{2\times10^{-23}}\text{ W m}^{-2}=5.57\text{ MW m}^{-2}\]
For the three-level ruby, \(\tau_\text{s}=\tau_2=3\text{ ms}\) at transparency. The photon energy for \(\lambda=694.3\text{ nm}\) is \(h\nu=(1.2398/0.6943)\text{ eV}=1.786\text{ eV}=1.786\times1.6\times10^{-19}\text{ J}\). Therefore, from (10-74), the saturation intensity at transparency is
\[I_\text{sat}=\frac{h\nu}{\tau_\text{s}\sigma_\text{e}}=\frac{1.786\times1.6\times10^{-19}}{3\times10^{-3}\times1.34\times10^{-24}}\text{ W m}^{-2}=71.1\text{ MW m}^{-2}\]
(b)
For \(N_\text{t}=1.58\times10^{25}\text{ m}^{-3}\), we find that \(\sigma_\text{e}N_\text{t}=1.34\times10^{-24}\times1.58\times10^{25}\text{ m}^{-1}=21.17\text{ m}^{-1}\) and \(\sigma_\text{a}N_\text{t}=1.25\times10^{-24}\times1.58\times10^{25}\text{ m}^{-1}=19.75\text{ m}^{-1}\). Thus, using (10-82), we find that, for \(g_0=5\text{ m}^{-1}\), the required pumping rate is
\[W_\text{p}=\frac{1}{3\times10^{-3}}\times\frac{19.75+5}{21.17-5}\text{ s}^{-1}=510\text{ s}^{-1}\]
which is 1.64 times the transparency pumping rate of \(W_\text{p}^\text{tr}=311\text{ s}^{-1}\). Therefore, the required pump power is 1.64 times the transparency pump power: \(I_\text{p}=1.64I_\text{p}^\text{tr}=12.53\text{ MW m}^{-2}\) for the 404 nm pump and \(I_\text{p}=1.64I_\text{p}^\text{tr}=9.13\text{ MW m}^{-2}\) for the 554 nm pump. At this pumping level, \(W_\text{p}\tau_2=1.53\). Therefore, from (10-78), \(\tau_\text{s}=\tau_2(1+1.25/1.34)/(1+1.53)=2.29\text{ ms}\). Then, the saturation intensity is
\[I_\text{sat}=\frac{h\nu}{\tau_\text{s}\sigma_\text{e}}=\frac{1.786\times1.6\times10^{-19}}{2.29\times10^{-3}\times1.34\times10^{-24}}\text{W m}^{-2}=93.1\text{ MW m}^{-2}\]
It is clear from this example that a very high pump power is required just to bring a ruby crystal to transparency because of the fact that it is a three-level system. For this reason, it is only feasible to pump a ruby laser with a pulsed pump.
As a consequence, CW operation is never realized for the ruby laser. Ruby lasers are always operated in the pulsed mode, most notably in the Q-switched mode for the generation of giant pulses.
The situation is very different for four-level systems, such as Nd : YAG, or quasi-two-level systems, such as Ti : sapphire and Cr : LiSAF.
Spontaneous emission power
When the upper laser level of a gain medium is populated, there is spontaneous emission. The upper laser level population can be found by solving \(N_1+N_2=N_\text{t}\) and \(N_2\sigma_\text{e}-N_1\sigma_\text{a}=g\) simultaneously to have
\[\tag{10-89}N_2=\frac{\sigma_\text{a}N_t+g}{\sigma_\text{e}+\sigma_\text{a}}\]
This relation is valid for all systems, including the four-level system. Though we have used \(N_1+N_2=N_\text{t}\), which is not valid for a four-level system, to obtain this relation, (10-89) reduces to \(N_2=g/\sigma_\text{e}\) in the case of a four-level system, for which \(\sigma_\text{a}=0\).
Note that, in the case of a quasi-two-level or a three-level system, \(g=-\alpha\) when the medium is not sufficiently pumped to reach transparency.
Because the maximum value of the absorption coefficient is \(\alpha_0=\sigma_\text{a}N_\text{t}\), we find that \(N_2\ge0\) for any positive or negative values of \(g\).
Note also that \(g\) appearing in (10-89) is the saturated gain coefficient if stimulated emission exists in the medium.
According to the discussions in the optical transitions for laser amplifiers tutorial, the spontaneous emission power is proportional to \(N_2\) only and is independent of \(N_1\). Therefore, regardless of whether the medium has a gain or a loss, the spontaneous emission power density, which is defined as the spontaneous emission power per unit volume of the medium in watts per cubic meter, is
\[\tag{10-90}\hat{P}_\text{sp}=\frac{h\nu}{\tau_\text{sp}}N_2=\frac{h\nu}{\tau_\text{sp}}\frac{\sigma_\text{a}N_\text{t}+g}{\sigma_\text{e}+\sigma_\text{a}}\]
where \(g\) can be positive, for a medium pumped above transparency, or negative, for a medium below transparency. For a gain volume of \(\mathcal{V}\), the spontaneous emission power is \(P_\text{sp}=\hat{P}_\text{sp}\mathcal{V}\).
In the case when the gain is not saturated so that \(g=g_0\), we find from (10-75), (10-77), and (10-79) that \(\sigma_\text{a}N_\text{t}+g=W_\text{p}\tau_\text{s}\sigma_\text{e}N_\text{t}\). For a medium that is optically pumped with a pump intensity \(I_\text{p}\), we then have
\[\tag{10-91}N_2=\frac{W_\text{p}\tau_2}{1+(1+p)W_\text{p}\tau_2}N_\text{t}=\frac{I_\text{p}/I_\text{p}^\text{sat}}{1+(1+p)I_\text{p}/I_\text{p}^\text{sat}}N_\text{t}\]
Then, the spontaneous emission power density in the absence of gain saturation can be expressed as
\[\tag{10-92}\hat{P}_\text{sp}=\frac{h\nu}{\tau_\text{sp}}\frac{I_\text{p}/I_\text{p}^\text{sat}}{1+(1+p)I_\text{p}/I_\text{p}^\text{sat}}N_\text{t}\]
At transparency, \(g=g_0=0\). The spontaneous emission power density at transparency, which is known as the critical fluorescence power density, is
\[\tag{10-93}\hat{P}_\text{sp}^\text{tr}=\frac{h\nu}{\tau_\text{sp}}\frac{\sigma_\text{a}}{\sigma_\text{e}+\sigma_\text{a}}N_\text{t}=\frac{h\nu}{\tau_\text{sp}}\frac{I_\text{p}^\text{tr}/I_\text{p}^\text{sat}}{1+(1+p)I_\text{p}^\text{tr}/I_\text{p}^\text{sat}}N_\text{t}\]
For a gain volume of \(\mathcal{V}\), the critical fluorescence power is \(P_\text{sp}^\text{tr}=\hat{P}_\text{sp}^\text{tr}\mathcal{V}\).
Example 10-8
A ruby crystal doped with 0.05 wt. \(\%\) Cr2O3 for a Cr concentration of \(1.58\times10^{25}\text{ m}^{-3}\) as discussed in Example 10-7 is considered. Almost all of the population in the upper laser level of a ruby laser crystal relaxes radiatively by to the ground state so that \(\tau_\text{sp}=\tau_{21}=\tau_2=3\text{ ms}\). Find the critical fluorescence power density corresponding to transparency for the 694.3 nm line at 300 K. What is the spontaneous emission power density if the ruby crystal is pumped above transparency for a gain coefficient of \(5\text{ m}^{-1}\) for the 694.3 nm line? What is it if the crystal is insufficiently pumped so that it has an absorption coefficient of \(5\text{ m}^{-1}\) for the 694.3 nm line? If a ruby laser rod of 6 cm length and 4 mm cross-sectional diameter is uniformly pumped, what are the spontaneous emission powers in the three cases considered here?
When we consider transparency for the 694.3 nm transition, we take \(\sigma_\text{a}=1.25\times10^{-24}\text{ m}^2\) and \(\sigma_\text{e}=1.34\times10^{-24}\text{ m}^2\) for this transition at 300 K, which are obtained in Examples 10-4 and 10-5, respectively [refer to the optical absorption and amplification tutorial]. However, the spontaneous emission is broadband covering both emission lines at 692.9 and 694.3 nm. Therefore, we take an average photon energy of the two for \(h\nu=1.787\text{ eV}\). Then, we find from (10-93) the following critical fluorescence power density at the transparency point for the 694.3 nm line:
\[\hat{P}_\text{sp}^\text{tr}=\frac{1.787\times1.6\times10^{-19}\times1.25\times10^{-24}\times1.58\times10^{25}}{3\times10^{-3}\times(1.34\times10^{-24}+1.25\times10^{-24})}\text{W m}^{-3}=727\text{ MW m}^{-3}\]
When pumped for a gain coefficient of \(5\text{ m}^{-1}\) for the 694.3 nm line, we find from (10-90) that
\[\hat{P}_\text{sp}=\frac{1.787\times1.6\times10^{-19}\times(1.25\times10^{-24}\times1.58\times10^{25}+5)}{3\times10^{-3}\times(1.34\times10^{-24}+1.25\times10^{-24})}\text{W m}^{-3}=911\text{ MW m}^{-3}\]
When the crystal is insufficiently pumped so that there is an absorption coefficient of \(5\text{ m}^{-1}\), \(g=-5\text{ m}^{-1}\). Then from (10-90)
\[\hat{P}_\text{sp}=\frac{1.787\times1.6\times10^{-19}\times(1.25\times10^{-24}\times1.58\times10^{25}-5)}{3\times10^{-3}\times(1.34\times10^{-24}+1.25\times10^{-24})}\text{W m}^{-3}=543\text{ MW m}^{-3}\]
For a rod of 6 cm length and 4 mm cross-sectional diameter, the volume is \(\mathcal{V}=\pi\times(4\times10^{-3}/2)^2\times6\times10^{-2}\text{ m}^3=7.54\times10^{-7}\text{ m}^3\). Therefore, the critical fluorescence power is \(P_\text{sp}^\text{tr}=\hat{P}_\text{sp}^\text{tr}\mathcal{V}=548\text{ W}\). The total spontaneous emission power is \(P_\text{sp}=687\text{ W}\) for \(g=5\text{ m}^{-1}\) and \(P_\text{sp}=407\text{ W}\) for \(g=-5\text{ m}^{-1}\).
From the consideration of energy conservation, it is clear that the power required to pump the crystal to a particular state has to be at least, and most often far exceed, that emitted spontaneously by the crystal. Therefore, these numbers again show the high power required to pump a ruby laser crystal just to its transparency point.
For example, if the pumping efficiency is \(10\%\), the pump power required to pump this crystal to transparency is as high as 5.48 kW.
The next tutorial covers laser amplifiers.