# Electric-Dipole Transitions in Real Atoms

This is a continuation from the previous tutorial - ** single-mode fibers for communications**.

In the previous tutorial we developed the classical electron oscillator model for an atomic transition, and showed how it could lead to quantum-mechanically correct expressions for the equation of motion and for the resonant susceptibility on a single atomic transition in a real quantum atom.

In this tutorial we continue this discussion to show how, with some simple extensions, this same purely classical model can explain even the most complex quantum-mechanical aspects of real atomic transitions. We also give some typical numerical values and experimental examples of these properties in real laser transitions.

## 1. Decay Rates and Transition Strengths in Real Atoms

This section discusses in more detail the energy decay rates and the transition strengths of real atomic transitions, and their relationship to the purely classical oscillator model.

### Energy Decay Processes in Real Atoms

Real atoms of course have a large number of quantum energy levels, with many transitions and decay rates among these levels. The atoms in an upper energy level \(E_j\) in a collection of real atoms will relax to many different lower levels \(E_i\) via both radiative and nonradiative decay mechanisms, as illustrated in Figure 3.1.

The total rate at which atoms will decay from an upper energy level \(E_j\) through all downward relaxation paths may be expressed by a "rate equation" of the form

\[\tag{1}\frac{dN_j}{dt}=-\sum_{E_i\lt{E_j}}\gamma_{ji}N_j=-\gamma_jN_j=-N_j/\tau_j\]

where \(\tau_j\) is the total lifetime of the excited state \(E_j\), and \(\gamma_j\) is its total decay rate. The total decay rate \(\gamma_j\) is given by the sum over all the downward decay paths, i.e.,

\[\tag{2}\gamma_j\equiv\frac{1}{\tau_j}=\sum_{E_i\lt{E_j}}\gamma_{ji}=\sum_{E_i\lt{E_J}}[\gamma_{\text{rad},ji}+\gamma_{\text{nr},ji}]\]

so that this sum includes both radiative and nonradiative rates to all lower levels \(E_i\lt{E_j}\).

In the absence of any applied signals, the population \(N_j(t)\) of the upper level will thus decay with time in the exponential form

\[\tag{3}N_j(t)=N_j(t_0)e^{-\gamma_j(t-t_0)}=N_j(t_0)e^{-(t-t_0)/\tau_j}\]

The decay rate \(\gamma_j\) given by these equations is the quantum analog for level \(E_j\) of the energy decay rate \(\gamma\) in the classical oscillator model.

### Fluorescent Lifetime Measurements

The lifetime \(\tau_j\) of an upper energy level can be measured by observing the fluorescent emission from the upper level \(E_j\) to any other lower level \(E_i\) immediately after a short pulse of pumping light applied to a solid-state laser material, or a short current pulse sent through a gaseous atomic system, has lifted an initial number of atoms up into the upper level.

Figure 3.2 illustrates this kind of fluorescent lifetime measurement on a ruby sample, using a stroboscopic light source that produces repeated pumping pulses a few tens of μs long, and an optical filter that blocks most of the excitation light, so that only the exponentially decaying ruby fluorescence (\(\tau\approx\) 4.3 ms) reaches the detector.

The measured intensity of the fluorescent emission on some specific \(j\rightarrow{i}\) transition will be proportional to the radiative decay rate \(\gamma_{\text{rad},ji}\) on that transition and to the upper-level population as a function of time, i.e.,

\[\tag{4}I_\text{fl}(t)=\text{const}\times\gamma_{\text{rad},ji}N_j(t)\]

Since the upper-level population \(N_j(t)\) decays with an exponential decay rate equal to the total decay rate \(\gamma_j\), the measured exponential behavior for the fluorescent emission will be like

\[\tag{5}I_\text{fl}(t)=\text{const}\times{N_j}(t)=\text{const}\times{e}^{-t/\tau_j}\]

A fluorescence decay measurement thus measures only the total lifetime \(\tau_j\) of the upper level \(E_j\), not the radiative decay rate \(\gamma_{\text{rad},ji}\) on any individual transition (even though it is this radiative decay rate that produces the observed fluorescent emission).

### Nonradiative Decay Rates

It is important to distinguish between the radiative and the nonradiative parts of the total energy decay rate \(\gamma_{ji}\) on each downward transition. The total decay rate, just as for a classical oscillator, is the sum of both mechanisms; so the total decay rate on the \(j\rightarrow{i}\) transition is

\[\tag{6}\gamma_{ji}=\gamma_{\text{rad},ji}+\gamma_{\text{nr},ji}\]

The radiative part of this decay represents spontaneous emission of electromagnetic radiation, which is physically the same thing as fluorescence. Radiative decay is always present (although sometimes very weak) on any real atomic transition.

The nonradiative part of the decay represents loss of energy from the atomic oscillations into heating up the immediate surroundings in all other possible ways, such as into inelastic collisions, collisions with the laser tube walls, lattice vibrations, and so forth.

Nonradiative decay may or may not be significant on any specific transition, and its magnitude may change greatly for different local surroundings of the atoms (e.g., the nonradiative decay rate can be quite different for the same solid-state laser ion in different host lattices).

### Purely Radiative Decay in Real Atoms

The radiative part of the total decay rate on a real atomic transition has a very close analogy to the radiative decay rate of a classical oscillator.

An oscillating atom, like an oscillating classical dipole, will radiate electromagnetic energy (photons) at its discrete oscillation frequencies; this radiation will decrease or decay with time; and many real atomic transitions will radiate with an electric-dipole-like radiation pattern.

The radiative decay rate for a real atomic transition is exactly the same thing as the Einstein A coefficient for that transition, i.e., \(\gamma_{\text{rad},ji}\equiv{A_{ji}}\) where \(E_j\) is the upper and \(E_i\) the lower energy level involved. The numerical value of \(A_{ji}\) or \(\gamma_{\text{rad},ji}\) on any real atomic transition is given by the quantum-mechanical integral

\[\tag{7}A_{ji}=\frac{8\pi^2}{\epsilon\hbar\lambda^3}\left|\displaystyle\iiint\psi_j^*(\pmb{r})e\pmb{r}\psi_i(\pmb{r})d\pmb{r}\right|^2\]

which involves the dipole moment operator \(e\pmb{r}\) and the product of the quantum wave functions of the two quantum states involved. Hence the quantum radiative decay rate for a real atomic transition can in principle always be calculated (though not always easily) if the quantum wave functions \(\psi_i\) and \(\psi_j\) of the two levels are known. Calculated values for simpler atoms and molecules can also be found in handbooks and in the literature.

We pointed out earlier that the classical electron oscillator has a radiative decay rate given by

\[\tag{8}\gamma_{\text{rad},ceo}=\frac{e^2\omega_a^2}{6\pi\epsilon{m}c^3}\approx2.47\times10^{-22}\times{nf_a^2}\]

where \(n\) is the index of refraction of the medium in which the oscillator is imbedded, and the oscillation frequency is measured in Hz (\(\equiv\)cycles/second). A useful rule of thumb is that the purely radiative lifetime for a classical oscillator is approximately given by

\[\tag{9}\tau_{\text{rad},ceo}(\text{ns})\approx\frac{45\times[\lambda_0(\text{microns})]^2}{n}\]

where \(n\) is again the index of refraction and \(\lambda_0\) is the free-space wavelength in μm. (These two equations are among the few formulas in this tutorial where an index of refraction term is explicitly needed.)

For example, at the visible wavelength \(\lambda_0=500\) nm or 0.5 μm, the classical oscillator lifetime is \(\tau_{\text{rad},ceo}\approx11\) ns, or \(\gamma_{\text{rad},ceo}\approx10^8\text{ sec}^{-1}\). Note also the wavelength-squared dependence of this lifetime: infrared oscillators will have substantially longer lifetimes than UV or especially X-ray oscillators.

### Oscillator Strength

It is then a general rule that the radiative decay rate for any real atomic or molecular transition will always be slower than, or at best comparable to, the radiative decay rate for a classical oscillator at the same frequency, so that \(\gamma_{\text{rad},ji}\le\gamma_{\text{rad},ceo}\) or \(\tau_{\text{rad},ji}\ge\tau_{\text{rad},ceo}\).

We can also recall that the induced response to an applied signal of either a real atomic transition or a classical electron oscillator will be directly proportional to the radiative decay rate \(\gamma_\text{rad}\), with essentially the same proportionality constant in each case.

Because of this, it has become conventional to define a dimensionless oscillator strength as a measure of the strength of the response on a real atomic transition relative to the response of a classical electron oscillator at the same frequency.

This oscillator strength is defined formally for a transition from level \(j\) down to level \(i\) by

\[\tag{10}\mathcal{F}_{ji}\equiv\frac{\gamma_{\text{rad},ji}}{3\gamma_{\text{rad},ceo}}=\frac{\tau_{\text{rad},ceo}}{3\tau_{\text{rad},ji}}\]

A factor of 3 appears in this definition because of the polarization properties of real atoms compared to classical oscillators, in a fashion which will emerge later.

Some typical oscillator strengths for real atomic transitions are given in Table 3.1.

Note that strongly allowed transitions starting from the ground level of a simple atom in a gas to the first excited level of opposite parity—for example, the \(3s\rightarrow3p\) transition in Na, or the \(2s\rightarrow2p\) transition in a Li atom—have oscillator strengths very close to unity, and hence radiative decay rates close to the classical oscillator values.

These transitions are sometimes called the resonance lines of the atoms, since they show up very strongly in both the spontaneous emission and the absorption spectra of these atoms.

Other allowed electric-dipole transitions in the same atoms may be from \(10^{-2}\) to \(10^{-5}\) times weaker, and magnetic-dipole and electric-quadrupole transitions may have oscillator strengths of \(\mathcal{F}\approx10^{-7}\) or smaller.

Laser transitions in solids or in gaseous molecules typically have similarly weak oscillator strengths, whereas the strong visible singlet-to-singlet transitions in organic dye molecules, such as the Rhodamine 6G dye laser molecule, may have oscillator strengths near unity, and hence radiative decay rates close to the classical oscillator value (e.g., radiative decay times of several nanoseconds).

A strongly allowed atomic transition with oscillator strength of the order of unity will thus have a stimulated response to an applied signal of the same magnitude as a classical electron oscillator at the same frequency.

Very weakly allowed atomic transitions, on the other hand, may have an oscillator strength or response ratio as small as \(\mathcal{F}\approx10^{-6}\) to \(10^{-7}\) times weaker. So-called "forbidden transitions," or atomic transitions on which virtually no response can be obtained, will have \(\gamma_{\text{rad},ji}\ll\gamma_{\text{rad},ceo}\) and hence \(\mathcal{F}\rightarrow0\) in principle, although in fact the decay rate is never absolutely zero.

### Sum Rules, and Oscillator Strengths for Degenerate Transitions

When the upper and lower energy levels are degenerate, with degeneracy factors \(g_j\) and \(g_i\) (to be explained in a later section), the upward and downward oscillator strengths for a given transition are usually defined more precisely by

\[\tag{11}\mathcal{F}_{ji}|_\text{down}=-\frac{\gamma_{\text{rad},ji}}{3\gamma_{\text{rad},ceo}}\qquad\text{and}\qquad\mathcal{F}_{ij}|_\text{up}=+\frac{g_j}{g_i}\frac{\gamma_{\text{rad},ji}}{3\gamma_{\text{rad},ceo}}\]

With these more precise definitions also go quantum-mechanical sum rules, which say that the numerical sum of the oscillator strengths \(\sum_{j\ne{i}}\mathcal{F}_{ji}\) (including sign) from a given level \(E_j\) to all other levels above and below it in the same atom has some simple value, which is usually close to unity.

### Example: The Nd:YAG Laser Transition

The 1.06 μm transition in the Nd:YAG laser is not only of great practical importance, but can provide a good illustration of many of the practical factors that determine the radiative decay rate and the oscillator strength for a real atomic transition.

The solid arrow in Figure 3.3 shows the strong laser transition at \(\lambda_0=1.0642\) μm on the \(^4F_{3/2}\) to \(^4I_{11/2}\) group of transitions in Nd:YAG. (The dashed lines on the left in this figure indicate other transitions near 1.35 μm and 880 nm on which useful laser oscillation can also be obtained; the transitions from the \(^4F_{3/2}\) to \(^4I_{15/2}\) levels, with wavelengths near 1.8 μm, are very weak and oscillate only with difficulty if at all.)

The measured fluorescent lifetime of the \(^4F_{3/2}\) upper energy level (call this level \(E_2\)) in this material is \(\tau_2\approx\) 230 μs; so the total decay rate for this compound level or group of levels is \(\gamma_2=\gamma_\text{rad}+\gamma_\text{nr}\approx4350\text{ s}^{-1}\). The measured quantum efficiency for this level, however, defined as the ratio of radiative decay (photons emitted) to total decay (i.e., total atoms relaxing down) turns out to have an experimental value

\[\tag{12}\frac{\text{radiative decay rate}}{\text{total decay rate}}\equiv\frac{\gamma_\text{rad}}{\gamma_\text{rad}+\gamma_\text{nr}}\approx0.56\]

so the purely radiative decay rate is \(\gamma_\text{rad}\approx0.56\times4350\approx2435\text{ s}^{-1}\). (The quantum efficiency is measured by shining a calibrated light source onto the crystal, and making a difficult measurement of the total number of input photons absorbed compared to total fluorescent photons emitted.)

The upper level \(E_2\) in Nd:YAG really consists, however, of two distinct but closely spaced and partially overlapping levels (call them \(E_{2a}\) and \(E_{2b}\)), which are sometimes called the \(R_1\) and \(R_2\) levels, and which have an energy spacing of \(\approx80\text{ cm}^{-1}\).

The upper level \(E_{2b}\) is the actual upper laser level. These two levels at room temperature will have Boltzmann population ratios \(N_{2b}/N_2\approx0.4\) and \(N_{2a}/N_2\approx0.6\), and will be held to these ratios by fast relaxation processes between the two levels.

Both of these levels will then radiate spontaneously with different strengths to six different lower levels; so there are actually 12 closely spaced fluorescent lines from the two upper levels to the six lower levels in the 1.06 μm group, with the relative strengths of these lines varying by more than an order of magnitude.

The branching ratio, or the amount of spontaneous radiation on the actual 1.0642 μm laser transition, relative to the total radiative emission from both \(^4F_{3/2}\) levels to all lower levels, has been measured to be

\[\tag{13}\frac{\gamma_\text{rad}(1.0642\text{ }\mu{\text{m laser line}})\times{N_{2b}}}{\gamma_\text{rad}(\text{all }1.06\text{ }\mu\text{m lines})\times{N_2}}\approx0.135\]

Hence we can finally deduce that the purely radiative decay rate for the isolated YAG laser transition by itself is

\[\tag{14}\gamma_\text{rad}(1.0642\text{ }\mu\text{m})\approx(0.135/0.40)\times2.435\times10^3\approx820\text{ sec}^{-1}\]

This corresponds to a purely radiative lifetime of 1/820 sec \(\approx\) 1.22 ms (to be compared to the measured fluorescent lifetime of 230 μs).

The numbers quoted here represent a current best estimate for the value of \(\gamma_{\text{rad},ji}\) that should be used in formulas for the response on this particular Nd:YAG laser transition.

However, even in a system as heavily studied as Nd:YAG, these numbers are uncertain, largely because of the experimental difficulties of measuring accurately such quantities as the branching ratio and the absolute fluorescent quantum efficiency. There is no observable physical quantity anywhere in this system that actually decays with this radiative lifetime of 1.22 ms.

## 2. Line-Broadening Mechanisms in Real Atoms

Let us now consider a few of the more important line-broadening mechanisms responsible for the atomic linewidths \(\Delta\omega_a\) in real atoms. All of these mechanisms are, as we will see, basically extensions of those derived for the classical electron oscillator.

In this section we give more information on homogeneous line-broadening mechanisms in real atoms, and on how these relate to the CEO model.

### Homogeneous Broadening

All the line-broadening mechanisms we have considered thus far produce what is called homogeneous broadening. This means simply that all the energy-decay and dephasing mechanisms we have discussed thus far act on all the dipoles in a collection in the same way, so that the response of each individual oscillator or atom in the collection is broadened in the same fashion.

The homogeneous lorentzian linewidth (FWHM) that we derived for the stimulated response of a collection of classical oscillators is then

\[\tag{15}\Delta\omega_a=\gamma+2/T_2\]

where \(\gamma\) is the energy decay rate and \(1/T_2\) the rate at which "dephasing events" occur, whatever may be the cause of these dephasing events.

There do exist additional and basically different types of broadening effects called inhomogeneous broadening effects, which we will introduce in the last section of this chapter. Doppler broadening is one primary example of such an inhomogeneous broadening mechanism.

### Lifetime Broadening in Real Atomic Transitions

That part of the homogeneous linewidth \(\Delta\omega_a\) caused by the total energy decay rate \(\gamma=\gamma_\text{rad}+\gamma_\text{nr}\) is called lifetime broadening.

Lifetime broadening is basically a Fourier-transform effect. An exponentially decaying signal of the form \(\mathcal{E}(t)=\exp[-(\gamma/2+j\omega_a)t]\) for \(t\gt0\) has a complex lorentzian Fourier transform of the form \(\tilde{E}(\omega)=1/[1+2j(\omega-\omega_a)/\gamma]\), which has a FWHM linewidth \(\Delta\omega_a=\gamma\).

If dephasing effects are absent, only this lifetime broadening will remain. If in addition all nonradiative mechanisms are turned off, then only radiative decay will be left, and the linewidth will take on its minimum possible value \(\omega_a=\gamma_\text{rad}\).

This is called purely radiative lifetime broadening. This purely radiatively broadened condition may sometimes occur for real atoms in very low-pressure gases, where the atoms are highly isolated, and where no collisions or nonradiative effects can occur (although doppler broadening, to be discussed later, will also be present and of great importance in such a gas).

In a collection of real atoms, the transition at frequency \(\omega_{ji}\) between two energy levels \(E_j\) and \(E_i\) with total decay rates \(\gamma_j\) and \(\gamma_i\), respectively, will generally have a lifetime-broadening contribution that is given in a more exact analysis by

\[\tag{16}\Delta\omega_a=\gamma_i+\gamma_j+2/T_{2,ij}\]

where \(2/T_{2,ij}\) is the dephasing rate appropriate to that particular transition. The main point here is that in most cases the \(\gamma\) term in the classical oscillator linewidth is replaced by the sum of the upper-state and lower-state energy decay rates \(\gamma_i+\gamma_j\), so far as lifetime-broadening effects are concerned.

We have noted previously that energy decay rates \(\gamma_j\) for real atomic transitions take on widely different values, depending on both radiative and nonradiative processes. For strong visible-wavelength atomic transitions in gases, \(\gamma_\text{rad}\) may become as large as \(\approx10^7\) to \(10^8\text{ s}^{-1}\), leading to a lifetime-broadening contribution \(\Delta\omega_a/2\pi\) ranging from a few MHz to a few tens of MHz. This can be a significant source of homogeneous line broadening for a transition in a low-pressure gas.

For the Nd:YAG laser on the other hand, the upper-level energy decay time is \(\tau_j\approx230\) μs. This gives a lifetime-broadening contribution of only 700 Hz, which is absolutely insignificant compared to the enormously larger phonon-broadening dephasing contribution of \(\Delta\omega_a/2\pi\approx120\) GHz.

### Dephasing Collisions and Pressure Broadening in Gases

The primary dephasing events for atoms or molecules in gases are real collisions between the radiating atoms or molecules and various collision partners.

In a typical gas mixture atoms may collide with other atoms of the same kind (called "self-broadening"); with atoms of different kinds (called "foreign-gas broadening"); or with the tube walls (generally not of importance at optical frequencies).

The total collision-broadening contribution to the homogeneous linewidth of a given atomic transition will then be directly proportional to the density, or to the partial pressure, of each species that is present.

The homogeneous linewidth will therefore increase linearly with total gas pressure (assuming a constant gas mixture) in the general form

\[\tag{17}\Delta\omega_a=A+BP\]

where A and B are constants that are different for different atomic transitions and gas mixtures.

This behavior is naturally referred to as pressure broadening, and Equation 3.17 is sometimes referred to as the Stern-Vollmer equation. (The coefficients A and B used here have nothing at all to do with the Einstein A and B coefficients).

Figure 3.4 illustrates some measured homogeneous pressure-broadening results for the 10.6 μm laser transition in CO_{2} caused by CO_{2} molecules colliding with other CO_{2} molecules and also with He atoms or N_{2} molecules in various gas mixtures.

Note that here (as in many other common gases) a few tens of torr of total pressure gives a few hundreds of MHz of pressure broadening. Note also that the lifetime-broadening contribution in these mixtures is apparently negligible, as indicated by the essentially zero intercept of the pressure-broadening curves at zero pressure.

### Typical Numerical Values

The amount of dephasing and line broadening that actually occurs in a real collision between two atoms (or molecules, or ions) depends on how close the two partners come to each other; how their quantum wave functions overlap and interact with each other during the collision; and (to a slight extent) how fast the atoms are traveling.

The atomic wave functions that are involved are, of course, different for different energy states \(E_i\) or \(E_j\) of the colliding partners. Therefore the amount of pressure broadening, or the constant factor B in the Stern-Vollmer formula, can often be different for different transitions even in the same atom.

Pressure-broadening coefficients are often expressed in practice in units of MHz/torr or, in some cases, GHz/atmosphere, as in Table 3.2.

Collision-broadening coefficients are also sometimes given in the literature as frequency broadening (in various units) versus gas density \(N\) rather than gas pressure \(P\). It is then convenient to remember that

\[\tag{18}N(\text{atoms/cm}^3)=9.64\times10^{18}\frac{P(\text{torr})}{T(\text{K})}\]

for the relation between partial pressure and density of each species in a gas mixture.

The results for the CO_{2} laser transition in Table 3.2 and in Figure 3.4 also illustrate how the pressure-broadening coefficient, or the effective cross section of a gas molecule for dephasing collisions, can be different for different collision partners.

In a typical He:N_{2}:CO_{2} laser gas mixture, the total pressure broadening of the 10.6 micron CO_{2} laser transition must be written as an expression like

\[\tag{19}\Delta\omega_a(\text{CO}_2)=A+B_\text{He}P_\text{He}+B_{\text{N}_2}P_{\text{N}_2}+B_{\text{CO}_2}P_{\text{CO}_2}\]

where each \(P_x\) is the partial pressure of a different gas, and the pressure-broadening coefficients \(B_x\) have different values for each different collision partner.

### Phonon Broadening (FM Broadening) of Real Atoms in Solids

Another kind of homogeneous line broadening that is important for many solid-state laser transitions is phonon broadening. Phonon broadening refers to a rapid and random frequency modulation of the instantaneous atomic-transition frequency for an atom in a solid (or liquid) caused by high-frequency lattice vibrations in the surrounding crystal lattice.

This process is physically quite different from a discrete collision-type process having a mean time \(T_2\) between collisions, but the net result in terms of randomizing the phases and broadening the response of a collection of oscillators is very much the same, and can in fact be described by an effective dephasing time \(T_2\).

Phonon broadening does not depend directly on atomic density \(N\) as does pressure broadening. It does, however, depend strongly on lattice temperature, since the lattice vibrations result from thermal excitation of the lattice modes.

Figure 3.5 shows, for example, the linewidths of two common solid-state laser transitions plotted versus temperature. The 694 nm laser transition in ruby shows a residual inhomogeneous strain broadening at lower temperature, changing over to thermal FM or phonon broadening at higher temperatures, whereas the linewidth of the 1.06 μm laser transition in Nd:YAG shows strongly temperature-dependent thermal phonon broadening over essentially the entire range plotted.

The phonon-broadening contribution will become very small for temperatures below a few tens of degrees Kelvin. There may then be a residual linewidtli contribution of inhomogeneous type, which arises from residual static strains and imperfections in the solid-state material. This residual strain broadening may be quite different from sample to sample, depending on the perfection of individual crystal samples.

Note also that besides phonon broadening in these solids, there may also be a significant thermal shift of the exact center frequencies of the transitions, which can sometimes be useful (and sometimes not so useful).

### Dipolar Broadening

A third important mechanism that produces homogeneous dephasing and line broadening in certain materials at higher densities is dipolar broadening. Dipolar broadening results from the random interaction and coupling between nearby atoms through their overlapping dipolar electric or magnetic fields (Figure 3.6).

The random perturbation of each dipole oscillator by the random fields from its neighbors can cause a time-varying frequency shift in the exact resonance frequency of each such dipole; and this in turn leads to an effective dephasing and line broadening in a fashion somewhat similar to phonon broadening.

Dipolar broadening is not commonly of great importance in laser materials, since they do not usually have the combination of high atomic density and strong atomic dipoles needed to make dipolar broadening predominate over the collision or thermal phonon-broadening mechanisms.

However, dipolar broadening can be observed, for example, in rare-earth pentaphosphates and certain other solid-state materials that can have a high intrinsic density of rare-earth atoms, and that are sometimes used for miniature optically pumped solid-state lasers.

If these materials are carefully prepared and cooled to liquid-helium temperatures, where thermal phonon broadening becomes negligible, dipolar-broadened linewidths of a few kHz can be observed by various sophisticated experiments.

### Transit-Time Broadening

There can be some experiments in which the atoms in a gas move across the full width of the optical beam with which they are interacting in a transit time \(T_\text{tr}\) which is small compared with either the energy decay lifetime \(\tau=1/\gamma\) or the dephasing time \(T_2\) of the atoms.

In such a situation, it is this transit time which limits the duration of the coherent interaction between the atoms and the applied signals, and which thus determines a kind of effective lifetime broadening. This is generally referred to as transit-time broadening, with an effective linewidth contribution on the order of \(\Delta\omega_a\approx1/T_\text{tr}\).

Since the thermal velocity of an atom or molecule in a gas is typically on the order of \(\approx10^5\) cm/s, transit-time broadening will produce only a few hundred kHz of broadening for a beam width or interaction length even as small as a few mm.

Transit-time broadening thus becomes significant only in special situations, for example, very high-resolution molecular-beam experiments involving tightly focused optical beams and high-speed molecules.

Transit-time broadening must also sometimes be considered with larger gas cells in experiments using extraordinarily high-resolution laser frequency standards, very low-pressure gases, and very long-lived molecular absorption lines.

### Coherent Pulse Experiments: Dephasing Versus Energy Decay

As we have noted in earlier discussions, it is important, and somewhat subtle, to distinguish clearly between those effects involved in energy decay and those involved in line broadening and dephasing of real atoms.

We described earlier, for example, an excited-state lifetime measurement in which atoms were excited into an upper energy level \(E_j\), and the spontaneous emission or fluorescence on a downward transition \(E_j\rightarrow{E_i}\) was then observed.

This fluorescent emission is purely spontaneous emission, that is, incoherent random noise with a narrow spectrum (of width \(\Delta\omega_a\)) centered at the transition frequency \(\omega_{ji}\).

The excitation mechanism (pumping light or electric current) excites the atoms into level \(E_j\) in an incoherent fashion. The atoms then oscillate spontaneously at frequencies like \(\omega_{ji}\), but with no phase coherence between individual atoms.

We add the radiated powers from each atom (not the voltages) to get the total spontaneous emission. This emission comes out randomly in all directions, and has the statistical and spectral characteristics of narrowband random noise.

It is also possible, though usually much more difficult, to perform a more complicated experiment to demonstrate coherent atomic emission and the effects of dephasing on this coherent emission.

Suppose some incoherent excitation mechanism, such as a flash of light or a current pulse in a gas, excites some of the atoms in an atomic medium up into some excited level \(E_j\) or \(E_i\), or maybe even into a mixture of both.

Spontaneous emission will then start. But before the populations \(N_j\) or \(N_i\) have decayed away, let us send a strong but short coherent signal pulse at the transition frequency \(\omega_{ji}\) through the atoms. This pulse will then excite a coherent response \(p(t)\) in the atoms on the \(j\rightarrow{i}\) transition.

This induced polarization \(p(t)\) will be given by the transient solution of the polarization equation of motion (Equation 2.69), taking into account the applied signal pulse.

The applied signal pulse may be too short for the steady-state solution \(\tilde{P}(\omega)\) given by the linear susceptibility to be reached. But nonetheless, after the signal pulse passes through the collection of atoms, they will be left with a coherently oscillating macroscopic polarization \(p(t)\) in the medium.

The atoms have all been driven in phase by the same applied signal; and after it passes they will continue to oscillate coherently and in phase at least for a brief while.

In the jargon of quantum electronics, we say that the atoms have been "coherently prepared" or "transversely aligned" by the strong signal pulse. They will then continue to radiate coherently and in the same direction as the applied signal pulse.

This radiation, like the applied signal, will be spatially and temporally coherent radiation, not noise. The atoms will have some memory of how they were coherently excited by the signal pulse; and we must add vectorially the radiated voltage, not power, from each oscillating atomic dipole.

The amplitude of this coherent oscillation and radiation will, however, decay away at a total rate \((\gamma/2+1/T_2)\) because of the dephasing plus lifetime processes.

This decay will be faster—often very much faster—than the energy decay rates \(\gamma_i\) or \(\gamma_j\) of the level populations. If the dephasing rate \(1/T_2\) is rapid compared to \(\gamma_i\) and \(\gamma_j\), the coherent radiation will rapidly disappear, leaving behind the much weaker but longer-lasting incoherent spontaneous emission.

This kind of more sophisticated experiment is referred to generally as a "coherent pulse" experiment. The presence of a coherent initial signal pulse to set up the transient coherent polarization \(p(t)\) is essential. The exponentially decaying coherent radiation after the coherent signal pulse is turned off is often called "free induction decay."

Note that a very narrow atomic transition in a gas might have a linewidth \(\Delta\omega_a/2\pi\approx1\) MHz, so that \(T_2\approx300\) ns. Optical signal pulses shorter than this can be generated, and lifetimes this short can be measured with fast photodetectors; hence coherent-pulse measurements on such a transition are feasible.

In Nd:YAG, the 1.06 μm laser transition has an upper-level energy-decay lifetime of \(\tau_2\approx230\) μs. The transverse dephasing time of this transition (its inverse phonon-broadened linewidth) is, however, more like \(T_2\approx1\) psec at room temperature. This is simply too fast to be either excited or observed with conveniently available optical tools.

## 3. Polarization Properties of Atomic Transitions

The transitions between quantum energy levels in real atoms exhibit anisotropic vector characteristics, or tensor characteristics, in both their spontaneous emission behavior and their stimulated response; and we need to understand the tensor nature of this behavior in order to fully understand real atomic transitions.

In the simplest case, the response of a real atomic transition may be either linearly polarized or circularly polarized on different transitions. In the most general case, any single transition in an atom or molecule may have an elliptically polarized response relative to some specific set of \((x, y, z)\) axes. The induced response in all these situations must then be described by a tensor susceptibility connecting the vector signal field and the vector atomic polarization.

We can gain a great deal of insight into these tensor properties by examining the transitions in a collection of single free atoms (not molecules) when these atoms are placed in a dc magnetic field.

The dc field then both provides a reference axis and also Zeeman-splits the energy levels to eliminate all degeneracy in the system. In this section we will examine the behavior of such Zeeman-split transitions; in the next section we will introduce the general tensor-analytical method.

### Zeeman-Split Atomic Transitions

The simplest example of a real atomic transition is probably the transition between a single lower energy level \(E_1\) that is an \(S\) state, having quantized angular momentum \(J=0\) and an upper level \(E_2\) that is a \(P\) state, having quantized total angular momentum \(J=1\). (Such states are characteristic of isolated single atoms in gases.)

An angular-momentum value greater than 0 means that the upper level really consists of \(2J+1=3\) distinct quantum levels, which are degenerate in energy in zero magnetic field. These levels will, however, be split apart by a dc magnetic field \(B_0\) into 3 distinct energy levels labeled by \(M_J=1\), \(0\), and \(-1\), as illustrated in Figure 3.7. (This splitting into separate energy levels is, of course, known as Zeeman splitting.)

There are then three separate and distinct transitions from the upper levels to the lower level, at three slightly different transition frequencies as illustrated in Figure 3.7.

Figure 3.8 shows some real spectral lines recorded on photographic plates in a high-resolution spectrometer for various spontaneous emission lines from excited zinc or sodium atoms, with and without dc magnetic fields, illustrating both the simplest and more complicated types of Zeeman splitting.

### Pi and Sigma Transitions

If we study the polarization behavior of the central transition (from \(M_J=0\) to \(M_J=0\)) in the example shown in Figure 3.7, we will find that this transition behaves exactly like a dipole oscillator that is linearly polarized along the direction of the dc magnetic field, both in its spontaneous radiation and in its stimulated response to an applied signal.

That is, on this particular transition the atoms act just like our linearly polarized CEO model, with their linear axis along the dc field. No spontaneous emission comes out in the direction directly along the dc field axis, for example, since a linearly oscillating dipole does not radiate along its polarization axis; and there will be no stimulated response to applied \(E\) fields perpendicular to that direction. Such a linearly polarized \(\Delta{M}_J=0\) transition is often called a \(\pi\) transition.

The outer two lines in Figure 3.7 (connected to the \(M_J=+1\) and \(-1\) levels) are then found to be circularly polarized with respect to the magnetic field axis, with opposite senses of circularity in both their spontaneous emission and their stimulated responses. These circularly polarized lines are called \(\sigma_+\) and \(\sigma_-\) transitions.

We will need to use tensors to describe the susceptibility properties of these transitions. Before we discuss this, however, a brief summary of some of the quantum properties of these atomic transitions may be very useful in understanding both their polarization properties and the relationship between the quantum theory and the classical models of these transitions.

### Quantum Description of Atomic Transitions

In quantum theory, the quantum state of any real atom at time t is completely specified by a quantum wave function \(\psi(\pmb{r},t)\), where \(\pmb{r}\) indicates a general position in space.

The evolution of this wave function in space and time is governed, according to quantum theory, by Schrodinger's equation of motion. We can, at least in principle, solve Schrodinger's equation to find \(\psi(\pmb{r},t)\) for a given atom with given initial conditions and a given applied signal; and we will then know everything there is to know physically about that atom.

Any isolated quantum system such as a single atom will also have a special set of quantum energy eigenstates or "stationary states" with associated quantum wave functions \(\psi_j(\pmb{r})\). These wave functions \(\psi_j(\pmb{r})\) are time-independent solutions of Schrodinger's equation with no applied signal present.

Each such eigenstate corresponds to one of the energy levels and energy eigenvalues \(E_j\) of the atom. These stationary eigenstates then provide a basis set, or a set of normal modes, for expanding any quantum state of the atom at any time.

A real atom at any instant of time will in general not be in a single energy eigenstate or energy level. Rather, it will be in a time-varying quantum state mixture of two or more such eigenstates. The wave function for a single atom at any instant of time may then be written in general as

\[\tag{20}\psi(\pmb{r},t)=\tilde{a}_1(t)e^{-iE_1t/\hbar}\psi_1(\pmb{r})+\tilde{a}_2(t)e^{-iE_2t/\hbar}\psi_2(\pmb{r})+\ldots\]

where \(E_1\), \(E_2\), etc., are the energy eigenvalues. In the absence of an applied signal or any other external perturbation, the complex-valued expansion coefficients \(\tilde{a}_1(t)\), \(\tilde{a}_2(t)\), ... in this expansion will be constant in time, and there will be only the \(\exp(-iE_jt/\hbar)\) frequency factor associated with each eigenstate.

One key idea here is that an atom is generally not in just one energy level. Rather, each atom is generally in a mixture of levels. An individual atom with a quantum state like that in Equation 3.20 then has a probability \(|\tilde{a}_1|^2\) of being found in level \(E_1\); a probability \(|\tilde{a}_2|^2\) of being found in level \(E_2\); and so forth. Averaging these probabilities over many atoms gives the same net effect as if \(N_1\) atoms were in level \(E)1\), \(N_2\) atoms in level \(E_2\), and so on.

A second key point is that these state mixtures are "stationary," in the sense that the \(\tilde{a}_j\)'s do not change with time unless there is an external signal or external perturbation applied to the atom. The time-varying phase rotation factor \(\exp(-iE_jt/\hbar)\) associated with each term in the expansion is necessary to make \(\psi(\pmb{r},t)\) satisfy the Schrodinger equation in the absence of an applied signal; but these phase factors do not, of course, change the magnitudes of the coefficients.

### Physical Interpretation of the Quantum State

One physical interpretation for the wave function \(\psi(\pmb{r},t)\) of an electron charge cloud surrounding a fixed nucleus is that \(|\psi(\pmb{r},t)|^2\) gives the probability density for finding an orbital electron at point \(\pmb{r}\) at time \(t\). More generally, we can say that \(\rho(\pmb{r},t)\equiv-e|\psi(\pmb{r},t)|^2\) gives the value (more precisely, the "quantum expectation value") of the local charge density in the electron charge cloud around the atom.

If the wave function \(\psi(\pmb{r},t)\) is a mixture of, say, two energy states, the charge density in the atom has the form

\[\tag{21}\begin{align}\rho(\pmb{r},t)&=\left|\tilde{a}_1(t)e^{-iE_1t/\hbar}\psi_1(\pmb{r})+\tilde{a}_2(t)e^{-iE_2t/\hbar}\psi_2(\pmb{r})\right|^2\\&=|\tilde{a}_1(t)|^2|\psi_1(\pmb{r})|^2+|\tilde{a}_2(t)|^2|\psi_2(\pmb{r})|^2\\&\quad+\tilde{a}_1(t)\tilde{a}_2^*(t)\psi_1(\pmb{r})\psi_2^*(\pmb{r})\exp[i(E_2-E_1)t/\hbar]\\&\quad+\tilde{a}_1^*(t)\tilde{a}_2(t)\psi_1^*(\pmb{r})\psi_2(\pmb{r})\exp[-i(E_2-E_1)t/\hbar]\\&=\rho_\text{dc}(\pmb{r})+\rho_\text{ac}(\pmb{r},t)\end{align}\]

The key observation here is that the atomic charge density contains both two static parts, proportional to the individual level occupancies \(|\tilde{a}_j(t)|^2|\psi_j(\pmb{r})|^2\), and a sinusoidally oscillating component given by the mixed term or cross term

\[\tag{22}\rho_\text{ac}(\pmb{r},t)=\text{Re}[\tilde{a}_1(t)\tilde{a}_2^*(t)\psi_1(\pmb{r})\psi_2^*(\pmb{r})e^{i\omega_{21}t}]\]

This oscillating component inherently oscillates at the transition frequency \(\omega_{21}=(E_2-E_1)/\hbar\) between the two levels involved. There is in effect a natural quantum oscillating dipole moment in the real quantum atom, which can be compared with the oscillating moment \(\mu_x(t)\) of the CEO model. This is a quantum-mechanically predicted oscillation in the atom, at the transition frequency between any two occupied levels.

The magnitude of this oscillating component is proportional to the cross term \(\tilde{a}_1(t)\tilde{a}_2^*(t)\) between the two level occupancies, and it obviously decays away as the level occupancy coefficients \(\tilde{a}_1(t)\) and \(\tilde{a}_2(t)\) decay away, just as \(\mu_x(t)\) decays at rate \(\gamma\) in the classical oscillator.

The phase \(\phi_i\) of the atomic oscillation in the \(i\)-th atom depends on the phase-angle difference of the complex coefficients \(\tilde{a}_1=|\tilde{a}_1|e^{-i\phi_1}\) and \(\tilde{a}_2=|\tilde{a}_2|e^{-i\phi_2}\) in the combination \(\tilde{a}_1\tilde{a}_2^*=|\tilde{a}_1\tilde{a}_2|e^{i(\phi_2-\phi_1)}\). This phase can be randomized by dephasing processes that randomize the individual phases of \(\tilde{a}_1\) and \(\tilde{a}_2\), without necessarily changing the occupancies \(|\tilde{a}_1|^2\) or \(|\tilde{a}_2|^2\) of either level.

### Zeeman Transitions: Linear and Circular Dipoles

As a specific example of such an oscillating charge pattern and oscillating dipole moment in a real atom, let us examine the simple but realistic Zeeman-split example described earlier.

We will look in the following paragraphs at simplified three-dimensional representations of the volume charge distributions \(\psi(\pmb{r},t)\) that correspond to various eigenstates and state mixtures, keeping in mind that \(\psi(\pmb{r},t)\) itself is a complex function with a sign or phase angle as well as a magnitude at each point in space.

Figure 3.9 shows in schematic form, for example, the wave functions \(|\psi(\pmb{r},t)|^2\) for \(J=O\) eigenstate or \(S\) state (a spherically symmetric charge cloud); for a \(J=1\), \(M_J=0\) or \(P_0\) eigenstate (dumbbell shape); and for a \(J=1\), \(M_J=\pm1\) or \(P_{\pm1}\) eigenstate (toroidal ring).

Note that in the dumbbell the wave function \(\psi(\pmb{r})\) has opposite sign in the upper and lower lobes, whereas in the \(M_J=\pm1\) states the wave function has an \(\exp(\pm{j}\theta)\) phase variation around the torus.

### Linearly Polarized (pi) Transition

Suppose, then, that the quantum state \(\psi(\pmb{r},t)\) of an atom is a mixture of, say, the \(1S\) and the \(2P_0\) states of the hydrogen atom (the ball and the dumbbell in Figure 3.9; the transition between these two levels in the hydrogen atom is, in fact, the Lyman \(\alpha\) line at 1216 Å).

When the phases of the complex coefficients \(\tilde{a}_1(t)\) and \(\tilde{a}_2(t)\) are included, the complex-valued wave functions \(\tilde{a}_1\psi_1(\pmb{r})\) and \(\tilde{a}_2\psi_2(\pmb{r})\) associated with these states may interfere constructively and/or destructively at different points to create the total wavefunction \(\psi(\pmb{r},t)\); and this interference will, moreover, rotate through all possible phases at the transition frequency \(\omega_{21}\) because of the \(\exp(-iE_jt/\hbar)\) terms.

The upper part of Figure 3.10 shows what the total wave function \(|\psi(\pmb{r},t)|^2\) produced by summing and squaring the \(1S\) and the \(2P_0\) states will look like at successive times during one oscillation cycle of the \(\exp(j\omega_{21}t)\) variation.

The center of charge of the total atomic charge cloud clearly oscillates back and forth linearly along what is here labeled the \(z\) axis. The quantum atom with this particular mixture of \(1S+2P_0\) states acts exactly like a linearly oscillating dipole.

### Circularly Polarized (sigma) Transitions

The lower part of Figure 3.10 shows the same type of result when the lower state \(E_1\) is again a \(1S\) state, but the upper level \(E_2\) is now a \(2P_1\) state with \(M_J=+1\). Because of the \(\exp(+j\theta)\) variation of the \(P_1\) state wave function around the equatorial plane, the wave functions \(\psi_1\) and \(\psi_2\) corresponding to the "ball" and the "torus" interfere constructively on one side and destructively on the other side of the rotational axis, producing a cancellation on one side and a "bulge" on the other side.

As the coefficients \(\tilde{a}_1e^{-iE_1t/\hbar}\) and \(\tilde{a}_2e^{-iE_2t/\hbar}\) rotate in time-phase, however, the resulting bulge in the quantity \(|\psi_1(\pmb{r})+\psi_2(\pmb{r})|^2\) rotates about the \(z\) axis at the transition frequency \(\omega_{21}\). The atom radiates like an oscillator that is circularly polarized in the \(x\), \(y\) plane. The polarization of this rotation will be of opposite sense (opposite circularity) depending on whether the magnetic quantum number \(M_j=+1\) or \(-1\) in the upper level.

Figure 3.11 summarizes the polarization properties and the radiation characteristics into various directions of these simple Zeeman-split oscillating-electric-dipole charge distributions. These results represent the quantum-mechanical polarization properties of real atomic transitions. They obviously can be very well represented, however, by the kinds of purely classical electron oscillator models we have been developing.

These polarization properties of the quantum oscillations in the atomic wave functions determine both the spontaneous and the stimulated properties of the real atoms. That is, an atom whose charge distribution can oscillate only in a certain direction on a given transition will obviously respond only to applied fields that have the same direction or sense of polarization. Hence Figure 3.11 illustrates equally well both the stimulated response and the spontaneous emission properties of these transitions.

### Elliptically Polarized Transitions

Many real atomic transitions, particularly the transitions of isolated single atoms in gases, as well as many molecular transitions, will have either pure linear or pure circular polarization properties exactly like those illustrated in Figures 3.10 and 3.11.

Atomic transitions in crystals or in complex molecules may, however, have more complex polarization properties.

It turns out (though we will not attempt to illustrate this in detail here) that the most general possible polarization for either an electric-dipole or a magnetic-dipole type of atomic transition is an elliptical polarization in an arbitrarily oriented plane of polarization, with arbitrary ellipticity and arbitrary orientation of the elliptical axes in that plane.

Linear and circular polarization are then elementary limiting cases of this general form.

## 4. Tensor Susceptibilities

Real atomic transitions thus have a tensor character that must be taken into account to give a complete and accurate description of the stimulated response on these transitions. In this section we summarize these tensor aspects of electric (or for that matter magnetic) dipole transitions in real atoms.

### Tensor Susceptibility: Linear Dipole Oscillators

Suppose that a sinusoidal signal with frequency \(\omega\) on or near a single atomic transition is applied to a collection of real electric-dipole atoms. Then the steady-state vector polarization \(\pmb{P}(\omega)\) induced in the collection of atoms must be related to the vector field \(\pmb{E}(\omega)\) by a tensor equation of the form

\[\tag{23}\pmb{P}(\omega)=\boldsymbol{\chi}(\omega)\epsilon\pmb{E}(\omega)\]

where \(\boldsymbol{\chi}(\omega)\) is a 3 x 3 tensor form of the susceptibility \(\tilde{\chi}(\omega)\), with components \(\tilde{\chi}_{xx}(\omega)\), \(\tilde{\chi}_{xy}(\omega)\), and so forth. Let us first examine the tensor character of this susceptibility for some simple examples, to get a feeling for the nature of these tensor responses.

The most elementary example is the linear classical electron oscillator. For the classical oscillator we calculated the \(x\) component of polarization \(\tilde{P}_x\) induced by an \(x\)-polarized field component \(\tilde{E}_x\). In tensor notation this gives us only the \(xx\) tensor component of \(\boldsymbol{\chi}\), or

\[\tag{24}\tilde{P}_x(\omega)=\tilde{\chi}_{xx}(\omega)\epsilon\tilde{E}_x(\omega)\]

It is physically evident that no \(\tilde{P}_y\) or \(\tilde{P}_z\) polarization components will occur in the linear oscillator model (since the electron is by definition not free to move along those coordinates in the linear model); and also that no response will be induced in the linear model by field components \(\tilde{E}_y\) or \(\tilde{E}_z\). Hence we can write this response in expanded tensor or matrix form as

\[\tag{25}\pmb{P}(\omega)=\begin{bmatrix}\tilde{P}_x(\omega)\\\tilde{P}_y(\omega)\\\tilde{P}_z(\omega)\\\end{bmatrix}=\tilde{\chi}(\omega)\epsilon\begin{bmatrix}3&0&0\\0&0&0\\0&0&0\end{bmatrix}\begin{bmatrix}\tilde{E}_x(\omega)\\\tilde{E}_y(\omega)\\\tilde{E}_z(\omega)\\\end{bmatrix}\]

Following a pattern that we will use repeatedly in this section, we have separated the right-hand side of this equation into a dimensionless tensor part with a trace of magnitude 3, plus a purely scalar (but still complex) susceptibility \(\tilde{\chi}(\omega)\).

The scalar susceptibility part of this expression for a homogeneously broadened lorentzian transition will then have the usual form

\[\tag{26}\tilde{\chi}(\omega)=-j\frac{1}{4\pi^2}\frac{\Delta{N}\lambda^3\gamma_\text{rad}}{\Delta\omega_a}\frac{1}{1+2j(\omega-\omega_a)/\Delta\omega_a}\]

in which the factor of 3 has been left with the dimensionless tensor for reasons that will become apparent later. Subscripts \(ij\) might also be attached to each factor in Equations 3.25 and 3.26 if necessary to identify the specific transition in a real atom that is involved.

Note that the choice of the \(x\) axis for the direction of the linear response here is entirely arbitrary. We might choose to label the linear response as being along the \(y\) or the \(z\) axes, or along some more arbitrary linear axis.

If we made this last choice, the tensor would become more complicated in form, corresponding to an arbitrary rotation of the coordinate axes with respect to the \(x\), \(y\), \(z\) axes. It would still be, however, a purely real tensor.

### Circularly Polarized (Gyrotropic) Responses

Let us next consider circularly polarized transitions, such as the \(\sigma_\pm\) transitions we saw in the previous section. For a transition that is circularly polarized in the \(x\), \(y\) plane (which is true of many simple transitions in free atoms), the tensor susceptibility becomes

\[\tag{27}\pmb{P}=\begin{bmatrix}\tilde{P}_x\\\tilde{P}_y\\\tilde{P}_z\end{bmatrix}=\tilde{\chi}(\omega)\epsilon\times\frac{3}{2}\begin{bmatrix}1&\mp{j}&0\\\pm{j}&1&0\\0&0&0\end{bmatrix}\begin{bmatrix}\tilde{E}_x\\\tilde{E}_y\\\tilde{E}_z\end{bmatrix}\]

where \(\tilde{\chi}(\omega)\) is exactly the same as in Equation 3.26, and the factor of 3/2 is attached to the tensor part of this circularly polarized expression, in order to make its trace (i.e., its diagonal sum) have the same value of 3 as for the linearly polarized expression.

Suppose that the applied signal in this case is linearly polarized along the \(x\) axis, i.e.,

\[\tag{28}\tilde{E}_x=\tilde{E}_0\quad\text{and}\quad\tilde{E}_y=\tilde{E}_z=0\]

Then the induced polarization components will be

\[\tag{29}\tilde{P}_x=(3\tilde{\chi}\epsilon/2)\tilde{E}_0\quad\text{and}\quad\tilde{P}_y=\mp{j}(3\tilde{\chi}\epsilon/2)\tilde{E}_0\]

where \((3/2)\tilde{\chi}\epsilon\) is in general a complex-valued quantity. Hence the real polarization terms will be of the form

\[\tag{30}\begin{align}p_x(t)&=\text{Re}\left[\tilde{P}_xe^{j\omega{t}}\right]=|(3\chi\epsilon/2)E_0|\cos(\omega{t}+\theta)\\p_y(t)&=\text{Re}\left[\tilde{P}_ye^{j\omega{t}}\right]=\pm|(3\chi\epsilon/2)E_0|\sin(\omega{t}+\theta)\end{align}\]

where \(\theta\) is the net phase angle of \((3\tilde{\chi}\epsilon/2)\tilde{E}_0\). Although the applied signal field is linearly polarized, the induced polarization \(p(t)\) is circularly polarized in the \(x\), \(y\) plane, rotating from \(x\) to \(y\) for the \(+\) sign or from \(x\) to \(-y\) for the \(-\) sign.

The circularly polarized tensor form given in Equation 3.27 inherently leads to circularly polarized behavior of the induced polarization. This form is characteristic of \(\sigma\)-type electric-dipole transitions and many simple magnetic-dipole transitions, and is often referred to as a gyrotropic tensor response. As before, rotation to a different coordinate orientation will make the tensor appear more complicated, but the essential character will remain the same.

### Elliptically Polarized Responses

Suppose a sinusoidal electric field \(\tilde{\pmb{E}}\) is applied to an arbitrary two-level nondegenerate electric-dipole transition in a real atom. Such a transition will have a quantum dipole matrix element \(\tilde{\boldsymbol{\mu}}_{21}\) given by the integral

\[\tag{31}\tilde{\boldsymbol{\mu}}_{21}=-e\displaystyle\iiint\psi_2^*(\pmb{r})\times\pmb{r}\times\psi_1(\pmb{r})d\pmb{r}\equiv\begin{bmatrix}\tilde{\mu}_x\\\tilde{\mu}_y\\\tilde{\mu}_z\end{bmatrix}\]

between the upper and lower levels of the transition.

That is, \(\tilde{\boldsymbol{\mu}}_{21}\) may be interpreted as a column vector with elements given by the \(x\), \(y\), \(z\) vector components of the integral. The hermitian conjugate \(\tilde{\boldsymbol{\mu}}_{21}^\dagger\) of this column vector is then a row vector whose elements \([\tilde{\mu}_x^*,\tilde{\mu}_y^*,\tilde{\mu}_z^*]\) are the complex conjugates of the elements in the column vector.

An exact quantum analysis then says that the expectation value for the phasor amplitude \(\tilde{\boldsymbol{\mu}}\) of the dipole moment induced in the atom by the applied field will be given by

\[\tag{32}\begin{align}\tilde{\boldsymbol{\mu}}&=\text{const}\times\left(\tilde{\boldsymbol{\mu}}_{21}^\dagger\cdot\tilde{\pmb{E}}\right)\times\tilde{\boldsymbol{\mu}}_{21}\\&=\text{const}\times\begin{bmatrix}\tilde{\mu}_x^*&\tilde{\mu}_y^*&\tilde{\mu}_z^*\end{bmatrix}\cdot\begin{bmatrix}\tilde{E}_x\\\tilde{E}_y\\\tilde{E}_z\end{bmatrix}\times\begin{bmatrix}\tilde{\mu}_x\\\tilde{\mu}_y\\\tilde{\mu}_z\end{bmatrix}\end{align}\]

where the dot product is taken in the usual matrix-multiplication fashion between the row vector \(\tilde{\boldsymbol{\mu}}_{21}^\dagger\) and the column vector \(\tilde{\pmb{E}}\) with elements [\(\tilde{E}_x\), \(\tilde{E}_y\), \(\tilde{E}_z\)],

The induced macroscopic polarization \(\tilde{\pmb{p}}(\omega)\) in a collection of atoms will then be just the microscopic dipole moment \(\tilde{\boldsymbol{\mu}}\) in each individual atom, as given by Equation 3.32, summed over all the atoms in any small unit volume.

Equation 3.32 contains a scalar constant, times a scalar dot product, times the column vector \(\tilde{\boldsymbol{\mu}}_{21}\), which is the net vector quantity on the right-hand side of the equation.

Equation 3.32 says, therefore, that the induced response \(\tilde{\boldsymbol{\mu}}\) or \(\tilde{\pmb{p}}\) of the atoms will always have exactly the same polarization properties as the transition's dipole matrix element \(\tilde{\boldsymbol{\mu}}_{21}\), regardless of the polarization properties of the applied signal \(\tilde{\pmb{E}}\). That is, you can drive the atoms with any polarization \(\tilde{\pmb{E}}\) you want; but they will always respond with their own fixed, characteristic form of polarization, as given by \(\tilde{\boldsymbol{\mu}}_{21}\).

The magnitude of this induced response, however, will depend on the dot product between the applied field \(\pmb{E}\) and the hermitian conjugate of the moment \(\boldsymbol{\mu}_{21}\); and this dot product is mathematically the same thing as matrix multiplication between these two quantities.

By invoking the associative properties of matrix and vector multiplication, therefore, we can reorder Equation 3.32 into the alternative form

\[\tag{33}\tilde{\boldsymbol{\mu}}=\text{const}\times\tilde{\boldsymbol{\mu}}_{21}\times\tilde{\boldsymbol{\mu}}_{21}^\dagger\times\tilde{\pmb{E}}=\text{const}\times\begin{bmatrix}\tilde{\mu}_x\\\tilde{\mu}_y\\\tilde{\mu}_z\end{bmatrix}\times\begin{bmatrix}\tilde{\mu}_x^*&\tilde{\mu}_y^*&\tilde{\mu}_z^*\end{bmatrix}\times\begin{bmatrix}\tilde{E}_x\\\tilde{E}_y\\\tilde{E}_z\end{bmatrix}\]

In this reorganized form, the middle product \(\tilde{\boldsymbol{\mu}}_{21}\times\tilde{\boldsymbol{\mu}}_{21}^\dagger\) can now be interpreted as the matrix product, computed according to the usual rules, of the two vector (or matrix) quantities \(\tilde{\boldsymbol{\mu}}_{21}\) and its hermitian conjugate.

But the result of this multiplication will be a 3 x 3 matrix or tensor \(\pmb{T}\), often called a dyadic product, which we will write as

\[\tag{34}\pmb{T}\equiv\text{const}\times\tilde{\boldsymbol{\mu}}_{21}\times\tilde{\boldsymbol{\mu}}_{21}^\dagger=\text{const}\times\begin{bmatrix}\tilde{\mu}_x^*\\\tilde{\mu}_y^*\\\tilde{\mu}_z^*\end{bmatrix}\times\begin{bmatrix}\tilde{\mu}_x&\tilde{\mu}_y&\tilde{\mu}_z\end{bmatrix}=\begin{bmatrix}\tilde{t}_{xx}&\tilde{t}_{xy}&\tilde{t}_{xz}\\\tilde{t}_{yx}&\tilde{t}_{yy}&\tilde{t}_{yz}\\\tilde{t}_{zx}&\tilde{t}_{zy}&\tilde{t}_{zz}\end{bmatrix}\]

where the constant is some suitable normalization constant. Note that the \(nm\)-th element of the \(\pmb{T}\) matrix is obtained in the usual matrix-multiplication way, by multiplying the \(n\)-th row of the \(\tilde{\boldsymbol{\mu}}_{21}^\dagger\) column vector (just one element) times the \(m\)-th column of the \(\tilde{\boldsymbol{\mu}}_{21}\) row vector (also just one element).

Hence we can write the macroscopic polarization in a general tensor form as

\[\tag{35}\tilde{\pmb{p}}(\omega)=\text{const}\times\tilde{\boldsymbol{\mu}}_{21}\tilde{\boldsymbol{\mu}}_{21}^\dagger\times\tilde{\pmb{E}}(\omega)=\tilde{\chi}(\omega)\epsilon\times\pmb{T}\times\tilde{\pmb{E}}(\omega)\]

where the most general form of the susceptibility tensor \(\pmb{T}\) for a dipole transition is given by the dyadic product

\[\tag{36}\pmb{T}=\text{const}\times\tilde{\boldsymbol{\mu}}_{21}\tilde{\boldsymbol{\mu}}_{21}^\dagger\]

Suppose the transition matrix element \(\tilde{\boldsymbol{\mu}}_{21}\) is a column vector with elements \([1,-j,0]\) corresponding to RHCP motion in the \(x\), \(y\) plane. The hermitian conjugate \(\tilde{\boldsymbol{\mu}}_{21}^\dagger\) is then a row vector with elements \([1,+j,0]\), and the tensor susceptibility has the form

\[\tag{37}\pmb{T}=\frac{3}{2}\times\begin{bmatrix}1&-j&0\end{bmatrix}\times\begin{bmatrix}1\\j\\0\end{bmatrix}=\frac{3}{2}\times\begin{bmatrix}1&j&0\\-j&1&0\\0&0&0\end{bmatrix}\]

This is, of course, just the RHCP gyrotropic result given in Equation 3.27.

### Most General Tensor Form

Simple linearly polarized and circularly polarized responses are the most common and elementary forms for the tensor responses of electric-dipole and magnetic-dipole atomic transitions.

To obtain the most general possible form for a dipole susceptibility tensor, we can note that the quantum transition moment \(\tilde{\boldsymbol{\mu}}_{21}\) can have at most three complex-valued vector components, namely, \(\tilde{\mu}_x\), \(\tilde{\mu}_y\), and \(\tilde{\mu}_z\), or six independent real numbers. Using these values, we can then carry out the matrix multiplication of the dyadic product as defined in Equation 3.34 to obtain the most general tensor form \(\pmb{T}\).

It can then be shown that for any such dipole transition the most general allowed form of this dyadic-product response will be an elliptically polarized tensor response, with the resulting induced polarization \(\tilde{P}(\omega)\) having some arbitrary (but fixed) degree of ellipticity and arbitrary orientation of the elliptical axes in some reference plane which is itself arbitrarily oriented with respect to the \(x\), \(y\), \(z\) axes. This behavior is inherent in the mathematical form itself, independent of physical properties of the transitions.

There seems to be little point in writing out this general elliptical tensor form in more detail here.

If you wish to know what the resulting tensor looks like, first add together the tensor responses for two independent linear responses along the \(x\) and \(y\) axes, but with an arbitrary amplitude ratio and arbitrary phase angle between them. This will produce the tensor form for an arbitrary elliptical response in the \(x\), \(y\) plane.

Performing a conventional coordinate rotation from the \(x\), \(y\), \(z\) axes to an arbitrarily oriented set of new \(x'\), \(y'\), \(z'\) axes will then generate the most general possible form for the susceptibility tensor.

Note that the degree of ellipticity of the original ellipse, plus the orientation of this ellipse in space, accounts for four real parameters.

The normalization condition that the trace of the resulting tensor should be normalized to three, i.e., \(\tilde{t}_{xx}+\tilde{t}_{yy}+\tilde{t}_{zz}=3\), then accounts for the remaining two of the six real numbers mentioned above. (Alternatively, we could require only that the magnitude of the trace be unity, leaving an arbitrary overall phase shift in all the tensor elements.) There are thus really only four adjustable real parameters among the nine complex elements of the normalized susceptibility tensor.

### Tensor Axes

But what determines the direction of the relevant axes of polarization and the degree of ellipticity for a real transition in a real atom? A simplified answer is as follows.

Single atoms floating freely in a gas always have degenerate electronic energy levels, for example, the Zeeman levels described earlier (except, of course, for \(J=0\) or \(S\) states, which are not degenerate). In this situation we must apply some static perturbation, such as a dc magnetic field (Zeeman splitting) or a dc electric field (Stark splitting), to "break" this degeneracy and to separate the individual transitions into distinct transition frequencies. Each of these separate Zeeman-split transitions will then have a distinct type and direction of tensor polarization.

The direction of the static perturbation in this situation will determine one of the reference axes for the tensor susceptibility; this direction is often chosen to be the \(z\) direction. The dc field direction will thus serve as the reference axis for the tensor responses on these transitions. For free atoms in such a static field, the response is then always either linear along this \(z\) axis (\(\pi\) transitions) or else circularly polarized about it (\(\sigma\) transitions), so that no unique choice for the \(x\) and \(y\) axes is either necessary or possible.

Atoms in a crystal will have a more complex environment, with more clearly determined reference axes, but often with a lower order of symmetry. In a crystal, each individual atom will be imbedded in some surrounding lattice structure with a distinctive orientation in space. The orientation of this lattice structure gives the reference axes against which the polarization-tensor properties of the atomic transitions can be uniquely evaluated. The most general possible result, as already noted, is an elliptically polarized tensor response with respect to these axes.

Finally, in molecules the structural axes of the molecular structure itself give reference axes for the electronic transitions of the electron charge cloud of the molecule. As the molecule rotates, these axes rotate with it. If a simple molecule has only a single axis of symmetry (e.g., a diatomic molecule like N_{2}), all its electronic transitions are either linear along this axis or circular about it.

### Isotropic Responses?

An important observation is that it is not physically possible for the tensor response of a single, nondegenerate atomic or molecular transition to be isotropic (that is, to be linear and equal in all directions). That is, a single nondegenerate transition cannot have a tensor response of the form

\[\tag{38}\begin{bmatrix}\tilde{P}_x\\\tilde{P}_y\\\tilde{P}_z\end{bmatrix}=\tilde{\chi}(\omega)\epsilon\times\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\times\begin{bmatrix}\tilde{E}_x\\\tilde{E}_y\\\tilde{E}_z\end{bmatrix}\]

A response that is effectively isotropic in this fashion can, however, be obtained by averaging over a collection of atoms. There are two different ways in which this can occur, as follows.

- The response of each individual atom in a collection may be anisotropic, with one of the nondegenerate tensor forms given earlier; but the atoms may have their reference axes randomly oriented in all directions. This would be expected in a randomly oriented collection of gas molecules, for example, or in a noncrystalline material, such as a liquid, a powder, or a glass, in which the local surroundings for different atoms may be randomly oriented. Averaging over all directions of the atomic axes leads to an isotropic overall response as in Equation 3.38.
- The observed response may be the summation over a complete set of degenerate atomic transitions that are not resolved in frequency, because no external perturbation has been applied to break the degeneracy. These degenerate transitions all coincide in frequency, and hence cannot be separately excited. Adding up the small-signal tensor responses of such a complete set of overlapping degenerate transitions then leads to an isotropic response here also. (Or we could say that there is no way to define any unique reference axes in the atoms; so the atoms are in effect randomly oriented.)

In either situation the tensor response will have the apparently isotropic form given in Equation 3.38, where the scalar \(\tilde{\chi}(\omega)\) is again the same as in Equation 3.26, that is, without the factor of 3 in front.

To look at this in another way, suppose we have a collection of randomly oriented atoms, so that \(N/3\) of them will be in effect oriented along each axis. The linear response due to these atoms will then be the value of \(\tilde{\chi}(\omega)\), including the initial factor of 3, but with a population (or population difference) of only \(N/3\) instead of \(N\).

Therefore, the response along each axis will be given by the scalar \(\tilde{\chi}(\omega)\) formula in the form we have used in this section without the initial factor of 3 that appeared in \(\tilde{\chi}(\omega)\) in earlier tutorials.

The isotropic tensor form of Equation 3.38 is thus both mathematically simple and characteristic of certain common physical situations. It also has a trace equal to three—at least if we give the diagonal elements the simplest value of unity.

It is largely for this reason that we have adopted the convention that \(\text{Tr}[\pmb{T}]\equiv3\) in writing all of the preceding normalized tensor susceptibilities. The significance of this factor of 3 is discussed in more detail in the following section.

## 5. The "Factor of Three"

One of the more confusing and often-argued aspects of atomic transitions is the "factor of three" that appeared in Equations 3.10 and 3.11, in the definition of oscillator strength, as well as in the trace of the tensor susceptibility in the preceding section. This section gives a brief but accurate explanation both of how this factor arises and of how it must be included in the appropriate theoretical formulas.

### Tensor Power Transfer Rates

We showed that the tensor susceptibility for a real atomic transition can be written in the form

\[\tag{39}\boldsymbol{\chi}(\omega)=\tilde{\chi}(\omega)\pmb{T}=-j\chi_0^"\pmb{T}\quad(\text{at midband})\]

where \(\tilde{\chi}(\omega)\), or its midband value \(-j\chi_0^"\), a scalar susceptibility formula (without the numerical factor of 3); and \(\pmb{T}\) is a dimensionless tensor that we will always normalize to make

\[\tag{40}\text{Tr}[\pmb{T}]=3\]

Let us now do some energy-storage and power-transfer calculations. For example, the time-averaged rate of energy transfer per unit volume from an applied field \(\boldsymbol{\mathcal{E}}(t)\) to a collection of atoms through the induced polarization \(\pmb{p}(t)\) may be written as

\[\tag{41}\frac{dU_a}{dt}=\left\langle\boldsymbol{\mathcal{E}}(t)\cdot\frac{d\pmb{p}(t)}{dt}\right\rangle\]

For a steady-state sinusoidal response given by \(\pmb{P}(\omega)=\boldsymbol{\chi}(\omega)\epsilon\pmb{E}(\omega)\), this leads at midband, \(\omega=\omega_a\), to the time-averaged result

\[\tag{42}\frac{dU_a}{dt}=-\frac{\omega_a\chi_0^"\epsilon[\pmb{E}^*\cdot\pmb{T}\pmb{E}+\pmb{E}\cdot\pmb{T}^*\pmb{E}^*]}{4}\]

The multiplications on the right-hand side of this equation must be carried out using the standard rules for matrix multiplication, with vectors to the right of the dots considered as column vectors and quantities to the left of the dots considered as row vectors.

The time-averaged stored energy per unit volume in the same signal fields is, however,

\[\tag{43}U_\text{sig}=\left\langle\frac{1}{2}\epsilon|\boldsymbol{\mathcal{E}}(t)|^2\right\rangle=\frac{\epsilon[\pmb{E}^*\cdot\pmb{E}]}{2}\]

Hence a ratio of energy transfer rate (to the atoms) over energy stored (in the signal fields) may be written as

\[\tag{44}\frac{1}{U_\text{sig}}\frac{dU_a}{dt}=\omega_a\chi_0^"\times\left[\frac{\pmb{E}^*\cdot\pmb{TE}+\pmb{E}\cdot\pmb{T}^*\pmb{E}^*}{2\pmb{E}^*\cdot\pmb{E}}\right]\]

When the dimensionless ratio in the brackets on the right-hand side of this equation is calculated for various forms of the susceptibility tensor \(\pmb{T}\), as given in the previous section, and for various signal field polarizations \(\pmb{E}\), its value always turns out to be somewhere between a maximum of 3 and a minimum of 0. (Some examples are shown in Table 3.3.)

### The Factor of "Three-Star"

The dimensionless factor that multiplies \(\omega_a\chi_0^"\) in Equation 3.44 thus always ranges between 0 and 3, depending on the nature of the signal polarization and the normalized tensor response. In fact, for different situations this dimensionless factor takes on values as follows.

- For aligned atoms—that is, for any collection of atoms that have a nondegenerate transition, and all of whose atomic axes are aligned in parallel to give an identical tensor response—there is always some optimum signal field polarization that will give this dimensionless factor its maximum value of 3, and thus make \((1/U_\text{sig})(dU_a/dt)=3\times\omega_a\chi_0^"\).
- For such aligned atoms there is always also an "anti-optimum" signal polarization, for which the corresponding value is identically zero. (Linear dipole transitions have, in fact, an entire plane in which the induced response is identically zero.)
- Combining aligned atoms with any other signal polarization between the optimum and anti-optimum forms gives a value for the dimensionless factor somewhere between \(3\times\omega_a\chi_0^"\) and \(0\times\omega_a\chi_0^"\).
- For nonaligned (which is to say, randomly aligned) atoms, and hence an isotropic tensor response, the dimensionless response always has the value of unity, so that \((1/U_\text{sig})(dU_a/dt)=1\times\omega\chi_0^"\).
- In a similar manner, for randomly polarized signal fields combined with any atomic alignment, the dimensionless response is also always unity.

In our discussions from here on, it would be nice if we did not have to keep track of the explicit vector nature of the signals or the tensor nature of the atomic responses.

In order to do this, while allowing for the tensor nature of the atomic response, we will give this dimensionless factor in Equation 3.44 a name, and include it in the atomic susceptibility expression from now on.

That is, we will from now on in this book often write the susceptibility expression for a homogeneous lorentzian atomic transition in the form

\[\tag{45}\tilde{\chi}(\omega)=-j\frac{3^*}{4\pi^2}\frac{\Delta{N}\lambda^3\gamma_\text{rad}}{\Delta\omega_a}\frac{1}{1+2j(\omega-\omega_a)/\Delta\omega_a}\]

where the parameter \(3^*\) ("three-star") indicates what we will from now on call the "factor of three." This parameter, depending on circumstances, may have the numerical values:

- \(3^*=3\) for fully aligned atoms plus optimally polarized fields; or
- \(3^*=1\) either for randomly aligned atoms with arbitrarily polarized fields, or for randomly polarized fields with any atomic alignment; or
- \(3^*=0\) for fully aligned atoms and "anti-optimum" fields; or
- \(0\le3^*\le3\) for any intermediate case.

This notation will prove very convenient, especially since, as we will see, this same factor of \(3^*\) carries over into many other stimulated-transition and gain formulas as well.

## 6. Degenerate Energy Levels and Degeneracy Factors

In many real atomic systems, what appears to be a single atomic resonance in a collection of atoms, with a single transition frequency \(\omega_{21}\) between upper and lower energy levels \(E_2\) and \(E_1\), may in fact be the summation of a number of overlapping transitions, with different strengths and polarization properties, between distinct but degenerate sublevels of the upper and lower levels.

It is still possible in discussing the small-signal response of such a system to treat such a set of degenerate transitions as a single transition with an isotropic susceptibility.

This section shows, however, that we must modify the definition of population inversion on such a degenerate transition by adding certain lower-level and upper-level degeneracy factors, in order to take into account the unresolved degeneracies of both the upper and lower levels.

### Degeneracy Factors

Suppose, to be specific, that two apparently discrete energy levels \(E_1\) and \(E_2\) really each consist of \(g_1\) and \(g_2\) quantum-mechanically distinct sublevels, respectively, as shown in Figure 3.12.

The integers \(g_1\) and \(g_2\) are then called the statistical weights or degeneracy factors of the levels. Let \(N_1\) and \(N_2\) be the total populations in levels \(E_1\) and \(E_2\).

At thermal equilibrium the atoms in each level will then be divided equally among the sublevels, with populations \(N_1/g_1\) or \(N_2/g_2\) in each of the respective sublevels. (There are, moreover, very rapid relaxation processes that usually act to rapidly equalize the populations of degenerate sublevels, even if they are somehow perturbated from equal populations, for example, by a strong applied signal.)

Boltzmann's Law, which relates the relative populations of an upper and lower energy level at thermal equilibrium, then applies rigorously to each distinct energy sublevel. In other words, it says that for any pair of such sublevels the population ratio at thermal equilibrium must be

\[\tag{46}\frac{N_2/g_2}{N_1/g_1}=\exp\left(-\frac{E_2-E_1}{kT}\right)\]

Hence for the total level populations the Boltzmann ratio really must be written in the form

\[\tag{47}\frac{N_2}{N_1}=\frac{g_2}{g_1}\exp\left(-\frac{E_2-E_1}{kT}\right)\]

This is a more precise generalization of the Boltzmann Law. Note that as a consequence of this, a highly degenerate upper level might possibly have, at thermal equilibrium, a larger total population than a lower level that is less degenerate (that is, if \(g_2/g_1\gt\exp[(E_2-E_1)/kt]\)). This is not a population inversion in any sense, however—for example, it does not lead to net stimulated emission or gain, as we will now show.

### Net Susceptibility of a Degenerate Transition

To evaluate the overall stimulated response on a degenerate transition, we must sum over all the individual sub-transitions, as shown in Figure 3.12. Let us label all the upper sublevels by an index \(m\) that runs from \(m=1\) to \(m=g_2\), and all the lower sublevels by a similar index \(n\). The total response on the transition is then the sum over \(n\) and \(m\) of all transitions between all the sublevels \(E_{1n}\) and \(E_{2m}\).

The tensor susceptibility on any one such transition between level \(E_{1n}\) and level \(E_{2m}\) may then be written in the form

\[\tag{48}\boldsymbol{\chi}_{1n,2m}(\omega)=\tilde{g}(\omega)\times\gamma_{\text{rad},2m\rightarrow1n}\times\left(\frac{N_1}{g_1}-\frac{N_2}{g_2}\right)\times\pmb{T}_{1n,2m}\]

where \(\pmb{T}_{1n,2m}\) is the tensor response on that particular transition; \(N_1/g_1-N_2/g_2\) gives the population difference on that particular transition; \(\gamma_{\text{rad},2m\rightarrow1n}\) gives the strength of that particular transition; and the lineshape function \(\tilde{g}(\omega)\) can usually be assumed to be the same for all transitions, namely,

\[\tag{49}\tilde{g}(\omega)=-j\frac{1}{4\pi^2}\frac{\lambda^3}{\Delta\omega_a}\frac{1}{1+2j(\omega-\omega_a)/\Delta\omega_a}\]

(If different sub-transitions have different linewidths, this will complicate the following analysis, but probably not change the results.)

The total response on all the \(1n\rightarrow2m\) transitions can then be written as

\[\tag{50}\begin{align}\boldsymbol{\chi}_\text{tot}(\omega)&=\sum_{n=1}^{g_1}\sum_{m=1}^{g_2}\boldsymbol{\chi}_{1n,2m}(\omega)\\&=\tilde{g}(\omega)\times\sum_{n=1}^{g_1}\sum_{m=1}^{g_2}\gamma_{\text{rad},2m\rightarrow1n}\left(\frac{N_1}{g_1}-\frac{N_2}{g_2}\right)\times\pmb{T}_\text{av}\end{align}\]

where \(\pmb{T}_\text{av}\) is some averaged tensor susceptibility over all the transitions involved. This tensor will simply be isotropic if the average is over a complete set of degenerate transitions.

At the same time, the total radiative decay rate downward out of the upper level \(E_2\) will be given by

\[\tag{51}\frac{dN_2}{dt}=-\sum_{n=1}^{g_1}\sum_{m=1}^{g_2}\gamma_{\text{rad},2m\rightarrow1n}\left(\frac{N_2}{g_2}\right)\]

That is, we must sum over all radiative decay rates from all upper sublevels to all lower sublevels. This total downward rate may then be equated to an averaged or measured radiative decay rate that we will call \(\gamma_{\text{rad},2\rightarrow1}\) defined by

\[\tag{52}\frac{dN_2}{dt}=-\gamma_{\text{rad},2\rightarrow1}N_2\]

This will be the measured radiative decay rate for level \(E_2\) viewed as a single effective level without degeneracy taken into account. Since the level populations can be taken outside the sums in all the preceding equations, this averaged decay rate is given by

\[\tag{53}\gamma_{\text{rad},2\rightarrow1}\equiv\frac{1}{g_2}\sum_n\sum_m\gamma_{\text{rad},2m\rightarrow1n}\]

Combining Equations 3.48 to 3.53 then gives

\[\tag{54}\boldsymbol{\chi}_\text{tot}(\omega)=\tilde{g}(\omega)\times\gamma_{\text{rad},2\rightarrow1}\times\left(\frac{g_2}{g_1}N_1-N_2\right)\times\pmb{T}_\text{av}\]

If we absorb the tensor properties into a factor \(3^*\) as in the previous section, this may be written as a scalar susceptibility

\[\tag{55}\tilde{\chi}_\text{tot}(\omega)=-j\frac{3^*}{4\pi^2}\frac{\lambda^3\gamma_{\text{rad},2\rightarrow1}}{\Delta\omega_a}\left(\frac{g_2}{g_1}N_1-N_2\right)\frac{1}{1+2j(\omega-\omega_a)/\Delta\omega_a}\]

where \(\pmb{T}_\text{av}\) will normally be isotropic and \(3^*\) will be equal to unity.

This final result now looks exactly like the nondegenerate susceptibility expression in earlier sections, except that the population difference \(\Delta{N}\) is replaced by

\[\tag{56}\Delta{N}\Rightarrow\left(\frac{g_2}{g_1}N_1-N_2\right)\]

A more precise condition for population inversion and gain on an atomic transition is thus

\[\tag{57}\frac{N_2}{g_2}\gt\frac{N_1}{g_1}\]

and not just \(N_2\gt{N_1}\). In physical terms, there must be true population inversion on the individual sublevels, and not simply \(N_2\gt{N_1}\).

For degenerate transitions in gases, with randomly aligned atoms, the averaged tensor susceptibility \(\pmb{T}_\text{av}\) will in fact always be isotropic, leading to \(3^*=1\) in Equation 3.55. For degenerate transitions in solids the situation may be somewhat more complex, and a degenerate transition may still have some anisotropic character to its tensor response \(\pmb{T}_\text{av}\).

### Discussion

The main result of this section, then, is that the small-signal steady-state response on a degenerate transition is exactly the same as for a nondegenerate transition, except that the effective value of \(\gamma_{\text{rad},2\rightarrow1}\) must be employed, and the effective population difference becomes \(\Delta{N}\equiv(g_2/g_1)N_1-N_2\) rather than just \(N_1-N_2\). We will use this result where appropriate in future sections.

This result does assume that the various sublevels of each main level remain equally populated, so that we can assign an equal fraction \(N_j/g_j\) of the level population to each one of them.

For very strong signals, and perhaps also for very short pulses (short compared to \(T_2\)), some of the transitions between sublevels \(E_{1n}\) and \(E_{2m}\) will respond more strongly to an applied signal than will others, because of substantially different values of \(\gamma_{\text{rad},2m\rightarrow1n}\) as well as different polarization properties.

A strong applied signal will then cause atoms to flow from certain sublevels \(E_{1n}\) to other sublevels \(E_{2m}\) at quite different rates; and this difference will tend to unbalance the otherwise equal sublevel populations, especially if for some reason the relaxation between the sublevels is slowed down.

This kind of selective pumping between lower and upper sublevels, especially when the degeneracy has been slightly broken, is in fact an essential element of a spectroscopic technique referred to as optical pumping.

Unless the degeneracy between sublevels is at least partially broken, however, there will usually also be relaxation processes between sublevels that will tend to rapidly return the sublevel populations to equality. These so-called "cross-relaxation" processes can be especially fast, because no energy change is required to relax an atom from one sublevel to another sublevel within the same degenerate main level. Strong applied signals can thus override these relaxation processes, but only temporarily.

The general warning to be taken is the following: In considering the effects of very strong (or very short-pulse) signals, for example, in so-called "coherent pulse" experiments, a degenerate transition can no longer be treated as a slightly modified single transition. It must instead be treated in detail as a set of multiple, independent, though still closely coupled transitions all at the same frequency.

## 7. Inhomogeneous Line Broadening

As the final step in describing the resonant response of real atomic transitions, we must introduce an additional and important type of line broadening known as inhomogeneous broadening, of which doppler broadening is the premier example.

### Homogeneous Broadening

The steady-state response of a homogeneously broadened transition in a collection of oscillators or atoms is given by the complex lorentzian formula

\[\tag{58}\tilde{\chi}_h(\omega;\omega_a)=-j\frac{3^*}{4\pi^2}\frac{\Delta{N}\lambda^3\gamma_\text{rad}}{\Delta\omega_a}\frac{1}{1+2j(\omega-\omega_a)/\Delta\omega_a}\]

We have attached a subscript \(h\) to indicate that this is the usual form for a homogeneous transition; and we have added the second argument to \(\tilde{\chi}_h(\omega;\omega_a)\) to indicate the explicit dependence on the resonance frequency \(\omega_a\) along with the applied frequency or signal frequency \(\omega\).

This kind of broadening is called homogeneous broadening because the response of each individual atom in the collection is equally and homogeneously broadened. Many real atomic transitions, under appropriate conditions, exhibit exactly this lineshape.

### Inhomogeneous Broadening

In many other real atomic situations, however, different atoms in a collection of nominally identical atoms may, for various reasons, have slightly different resonant frequencies \(\omega_a\), such that the \(\omega_a\) values for different atoms are randomly distributed about some central value \(\omega_{a0}\). We must then think of the resonance frequencies \(\omega_a\) for different atoms as being randomly shifted by small but different amounts for each atom in the collection.

An applied signal passing through such a collection of atoms will then see only a total response due to all the atoms—it will have no way to pick out only those atoms with certain specific frequency shifts.

If the random shifting of the individual center frequencies is sizable compared to the linewidth \(\Delta\omega_a\) of each individual response, any measurement of the overall response from all the atoms in the collection will then give a smeared-out or broadened summation of the randomly shifted responses of all the individual atoms (see Figure 3.13).

The overall response of the collection of atoms will be substantially broadened, and the response at line center will be substantially reduced in amplitude. This general type of behavior is referred to as inhomogeneous broadening.

### Spectral Packets

That subgroup of atoms whose resonant frequencies \(\omega_a\) all fall within a range of roughly one homogeneous linewidth \(\Delta\omega_a\) about a given value of \(\omega_a\) is often referred to as a single spectral packet (or spin packet in magnetic-resonance jargon).

All the atoms in a single packet have essentially the same (homogeneous) response to an applied signal. The total response of an inhomogeneously broadened line is then the sum of the individual responses of all the spectral packets, each at a different resonance frequency.

If the individual packets are spread out in frequency about \(\omega_{a0}\) by an amount large compared to their individual homogeneous widths \(\Delta\omega_a\), as in Figure 3.13, the line is said to be strongly inhomogeneous.

If the inhomogeneous shifting is small compared to the homogeneous packet widths, the line will remain essentially homogeneous, and the amount of inhomogeneous broadening that does exist will be of little importance.

### Causes of Inhomogeneous Broadening

There are several possible causes of random resonance-frequency shifting and thus of inhomogeneous broadening in typical atomic systems.

- In gases, different atoms will have different kinetic velocities through space. This kinetic motion produces a doppler shift in the frequency of an applied signal as seen by the atom, or alternatively a doppler shift in the apparent resonance frequency oja of the atom as seen by the applied signal. This so-called doppler broadening is an important and widespread source of inhomogeneous broadening for optical-frequency transitions in atomic and molecular gases.
- In solids, laser atoms at different sites in a crystal may see slightly different local surroundings, or different local crystal structures, because of defects, dislocations, or lattice impurities. This produces slightly different values for the exact energy levels of the atoms, and thus slight shifts in transition frequencies. To the extent that the local lattice surroundings are similar for every atom but vibrate rapidly and randomly in time, they produce a dynamic homogeneous phonon broadening. To the extent that the surroundings are different from site to site but static in time, they produce a static inhomogeneous lattice broadening or strain broadening.

Other types of inhomogeneous broadening also exist (for example, inhomogeneous dc magnetic fields in magnetic resonance experiments), but these are two of the most important for optical transitions.

### Doppler Broadening

One of the most common examples of inhomogeneous broadening is doppler broadening of the resonance transitions in gases. The atoms in an atomic or molecular gas will have, in addition to their internal oscillations, thermal or Brownian kinetic motion through space, with a maxwellian distribution of kinetic velocities.

When an atom moving with velocity \(v_z\) as in Figure 3.14 interacts with a wave of signal frequency \(\omega\) traveling at velocity \(c\) along the \(z\) direction (for example, a wave traveling down the axis of a laser tube), the frequency of the wave as seen by the atom will be doppler-shifted to a new value \(\omega'\) given by

\[\tag{59}\omega'=(1-v_z/c)\omega\]

Resonance of the applied signal with the atomic transition in that particular atom will then occur when the doppler-shifted signal frequency \(\omega'=\omega(1-v_z/c)\) seen by the moving atom equals the atom's internal resonance frequency \(\omega_{a0}\).

From an alternative viewpoint, resonance will occur when the signal frequency \(\omega\) measured in the laboratory frame equals the shifted resonance value \(\omega_{a0}(1+v_z/c)\). In other words, as seen from the lab the resonance frequency of the atom appears to be doppler-shifted to a new value,

\[\tag{60}\omega_a=(1+v_z/c)\omega_{a0}\]

For an atom or molecule of mass \(M\) in a gas at temperature \(T\), the kinetic velocity \(v_z\) has a mean-square value given by \(M\langle{v_z^2}\rangle\approx{kT}\). Hence the average doppler shift for a moving gas atom will be of order

\[\tag{61}\frac{\omega_a-\omega_{a0}}{\omega_{a0}}\approx\sqrt{\frac{kT}{Mc^2}}\approx10^{-6}\left(\begin{align}\text{for typical atomic}\\\text{masses and temperatures}\end{align}\right)\]

The amount of doppler broadening in a real gas thus depends (but only rather slowly) on the kinetic temperature \(T\) of the gas and on the molecular weight of the atom or molecule involved.

### Doppler Lineshape

To be more precise, the distribution of axial velocities \(v_z\) in a gas at thermal equilibrium will be a maxwellian, or gaussian, probability distribution given by

\[\tag{62}g(v_z)=\left(\frac{1}{2\pi\sigma_v^2}\right)^{1/2}\exp\left(-\frac{v_z^2}{2\sigma_v^2}\right)\]

with an rms spread given by \(\sigma_v^2=kT/M\). The inhomogeneous distribution of shifted resonant frequencies, call it \(g(\omega_a)\), for a doppler-broadened atomic transition will then similarly have a gaussian form that can be written as

\[\tag{63}g(\omega_a)=\left(\frac{4\ln2}{\pi\Delta\omega_d^2}\right)^{1/2}\exp\left[-(4\ln2)\left(\frac{\omega_a-\omega_{a0}}{\Delta\omega_d}\right)^2\right]\]

as illustrated in Figure 3.15.

This expression has been written so that \(\omega_{a0}\) is the center frequency; and following our standard convention, the linewidth \(\Delta\omega_d\) has been defined to be the FWHM linewidth of the gaussian distribution, which means it must take on the form

\[\tag{64}\Delta\omega_d=\sqrt{\frac{(8\ln2)kT}{Mc^2}}\omega_{a0}\]

It is useful to remember that in units of electron volts, \(kT\) at room temperature is 1/40 of an electron volt or 25 meV; and \(Mc^2\) is the rest-mass energy of the atom, which for a single proton is \(\approx10^9\) eV. For an atom or molecule with an atomic number of 20, the fractional doppler broadening is thus

\[\tag{65}\frac{\Delta\omega_d}{\omega_{a0}}=\sqrt{\frac{(8\ln2)kT}{Mc^2}}\approx\sqrt{\frac{5.5\times25\times10^{-3}}{20\times10^9}}\approx2.6\times10^{-6}\]

or typically a few parts per million.

A visible laser transition will have a center frequency on the order of \(\omega_a/2\pi\approx6\times10^{14}\) Hz, and a doppler broadening on the order of \(\Delta\omega_d/2\pi\approx2\times10^9\) Hz \(\approx2\) GHz. The room-temperature doppler broadening of the He-Ne laser transition at 633 nm, in fact, is just about \(\Delta\omega_a/2\pi\approx1500\) MHz.

### General Analysis of Inhomogeneous Broadening

As a more general approach to inhomogeneous broadening, suppose we consider a large collection of nominally identical atoms, with the fractional number of atoms whose exact resonant frequency is between some value \(\omega_a\) and \(\omega_a+d\omega_a\) being given by

\[\tag{66}dN(\omega_a)=Ng(\omega_a)d\omega_a\]

where \(N\) is the total number of atoms. (We really should use the population difference \(\Delta{N}\) here, but let's write \(N\) instead for simplicity.) The function \(g(\omega_a)\) is thus the probability density distribution over the resonant frequencies \(\omega_a\), with the normalization that

\[\tag{67}N^{-1}\int_0^\infty{dN}(\omega_a)=\int_{-\infty}^{\infty}g(\omega_a)d\omega_a=1\]

The function \(g(\omega_a)\) is always very narrowly clustered about the center frequency \(\omega_{a0}\), so any portion of the analytic function \(g(\omega_a)\) extending below \(\omega_a=0\) can be ignored. It therefore makes negligible difference whether the lower limit in this normalization integral is actually \(0\) or \(-\infty\).

To calculate the overall complex susceptibility of any such collection of atoms, we must then multiply the homogeneous response \(\tilde{\chi}_h(\omega;\omega_a)\) produced by any one atom whose resonance frequency is \(\omega_a\) by the fractional number of atoms \(g(\omega_a)d\omega_a\) that have the same resonance frequency \(\omega_a\), as illustrated in Figure 3.13, and then integrate that response over all values of \(\omega_a\) in the form

\[\tag{68}\tilde{\chi}(\omega)=\int_{-\infty}^{\infty}\tilde{\chi}_h(\omega;\omega_a)g(\omega_a)d\omega_a\]

Suppose the distribution \(g(\omega_a)\) is gaussian, as it often is. The full-blown equation for the complex small-signal susceptibility of an inhomogeneously broadened transition thus becomes a gaussian distribution of frequency-shifted lorentzian lines. If we write this out in full, it takes the general form

\[\tag{69}\begin{align}\tilde{\chi}(\omega)&=-j\frac{3^*}{4\pi^2}\sqrt{\frac{4\ln2}{\pi}}\frac{N\lambda^3\gamma_\text{rad}}{\Delta\omega_a\Delta\omega_d}\int_{-\infty}^{\infty}\frac{1}{1+2j(\omega-\omega_a)/\Delta\omega_a}\\&\qquad\times\exp\left[-(4\ln2)\left(\frac{\omega_a-\omega_{a0}}{\Delta\omega_d} \right)^2\right]d\omega_a\end{align}\]

This rather messy integral must be evaluated each time an accurate calculation is needed of the susceptibility of an atomic transition in which doppler broadening is important. (Even this integral still ignores certain large-signal saturation and "hole-burning" effects.)

Inhomogeneous broadening in general, whether due to doppler broadening or to other mechanisms, is usually caused by some kind of random distribution of velocities, or defects, or whatever; and random distributions, whatever their cause, are very often gaussian in form (as sometimes expressed in the Central Limit Theorem).

We will therefore interpret the gaussian expression for doppler broadening in Equation 3.63 somewhat more broadly, and use it as a general expression for \(g(\omega_a)\) in any kind of inhomogeneous broadening. Similarly, we will use \(\Delta\omega_d\) as a general notation for the inhomogeneous linewidth of an inhomogeneously broadened distribution, whether this is due to doppler broadening or to some other cause.

### Strongly Homogeneous Limit

The integral in Equation 3.69 cannot be evaluated analytically, at least not for arbitrary ratios of inhomogeneous broadening \(\Delta\omega_d\) to homogeneous broadening \(\Delta\omega_a\). The limiting cases of strongly homogeneous broadening and strongly inhomogeneous broadening can, however, be handled, at least approximately, as follows.

Let us suppose first that the inhomogeneous broadening effects are small, which means either that the resonance frequencies of individual packets are shifted by very little compared to the homogeneous linewidth \(\Delta\omega_a\), or alternatively that the individual packets have a wide homogeneous linewidth \(\Delta\omega_a\) compared to the inhomogeneous linewidth \(\Delta\omega_d\). The inhomogeneous distribution, whether gaussian or otherwise, is then essentially a delta function, i.e.,

\[\tag{70}g(\omega_a)\approx\delta(\omega_a-\omega_{a0})\qquad\text{if }\Delta\omega_d\ll\Delta\omega_a\]

The integral in Equation 3.69 is now trivial, and physically obvious: the overall response is simply the unperturbed homogeneous form \(\tilde{\chi}_h(\omega;\omega_a)\). In effect there is no inhomogeneous or doppler broadening. This is commonly known as the strongly homogeneous limit.

As one practical example of this, consider the 10.6 μm TEA CO_{2} laser operating at atmospheric pressure. The inhomogeneous doppler broadening for this long-wavelength transition is \(\Delta\omega_d/2\pi\approx60\) MHz, whereas the homogeneous pressure broadening at one atmosphere is \(\Delta\omega_a/2\pi\approx6\) GHz or 6,000 MHz. The individual packets are thus \(\approx\) 100 times wider than the doppler broadening, and the line is essentially homogeneous.

### Strongly Inhomogeneous Limit

Now suppose instead that the inhomogeneous linewidth \(\Delta\omega_d\) is large enough to shift the spectral packets widely in frequency compared to their homogeneous linewidth \(\Delta\omega_a\), so that there are many packets within the overall linewidth. It is then possible in this limit to obtain an analytic approximation to Equation 3.69 that is reasonably accurate for the imaginary part \(\chi^"(\omega)\) of the overall inhomogeneous susceptibility, though not for the \(\chi'(\omega)\) part.

The approximation for the absorptive part of the overall susceptibility is obtained by expanding the complex lorentzian \(\tilde{\chi}_h(\omega;\omega_a)\) inside the general integral into its real and imaginary parts. In the limit as \(\Delta\omega_a\) becomes small, the \(\chi_h^"\) Part of the homogeneous function becomes roughly like a delta function, i.e.,

\[\tag{71}\frac{2}{\pi\Delta\omega_a}\frac{1}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}\approx\delta(\omega-\omega_a)\qquad\text{if }\Delta\omega_a\ll\Delta\omega_d\]

This lorentzian curve is not a very good delta function, since its wings fall off only as \(1/(\omega-\omega_a)^2\) far from line center, but it is adequate here. Putting this into the general equation and integrating over the delta function then gives for the \(\chi^"(\omega)\) part of the susceptibility

\[\tag{72}\chi^"(\omega)\approx-\sqrt{\pi\ln2}\frac{3^*}{4\pi^2}\frac{\Delta{N}\lambda^3\gamma_\text{rad}}{\Delta\omega_d}\exp\left[-(4\ln2)\left(\frac{\omega-\omega_{a0}}{\Delta\omega_d}\right)^2\right]\]

in the strongly inhomogeneous limit where \(\Delta\omega_d\gg\Delta\omega_a\).

This expression for a strongly inhomogeneous absorption line has the following interesting features in comparison with the usual homogeneous lorentzian absorption line.

- It has a gaussian, not a lorentzian, lineshape for the absorption profile of \(\chi^"(\omega)\), with a FWHM linewidth of \(\Delta\omega_d\), not \(\Delta\omega_a\).
- It has essentially the same constant factors in front as does the homogeneous lorentzian response for \(\chi^"(\omega)\) except that the response now varies as \(1/\Delta\omega_d\), instead of \(1/\Delta\omega_a\).
- But it has an extra numerical factor of \(\sqrt{\pi\ln2}\approx1.48\) in front of the other factors that appear in the lorentzian expression.

In fact, these three simple modifications convert the \(\chi^"(\omega)\) susceptibility expression for a homogeneous lorentzian transition into the corresponding expression for a strongly inhomogeneous gaussian or doppler-broadened transition.

Figure 3.16 shows lorentzian and gaussian (i.e., strongly homogeneous and strongly inhomogeneous) susceptibilities \(\chi^"(\omega)\) normalized to the same FWHM linewidth and the same area. Note that the gaussian absorption curve \(\chi^"(\omega)\) has a peak value that is \(\approx\) 50% higher, but that it drops off much faster in the wings than does the lorentzian. The integrated area under each curve is the same, since the smaller area in the wings of the gaussian profile is balanced by the 50% larger peak intensity at the center.

It is also interesting to note that the homogeneous packet linewidth \(\Delta\omega_a\) actually does not appear at all in the strongly inhomogeneous expression given in Equation 3.72. Measuring the \(\chi^"(\omega)\) response of a strongly inhomogeneous line tells you \(\Delta\omega_d\), but it does not give any information about the homogeneous linewidth \(\Delta\omega_a\) of the packets buried within the line—at least not to first order.

### Complex Susceptibility in the Strongly Inhomogeneous Limit

It is not possible to develop a similar approximation for the reactive part of the susceptibility, \(\chi'(\omega)\), in the strongly inhomogeneous limit. The reason is essentially that although \(\chi^"(\omega)\) varies like \(1/(1+\omega^2)\) in frequency, which is a weak delta function, the real part \(\chi'(\omega)\) varies like \(\omega/(1+\omega^2)\), which is not a delta function at all. There is thus no analytic approximation to the exact integral of Equation 3.69 for \(\chi'(\omega)\) even in the strongly inhomogeneous limit.

Figure 3.17 does show numerically computed plots of both \(\chi'(\omega)\) and \(\chi^"(\omega)\) for the strongly inhomogeneous gaussian limit, \(\Delta\omega_d\gg\Delta\omega_a\). The inhomogeneous susceptibility \(\chi'(\omega)\), though it cannot be analytically approximated, looks in general very much like the lorentzian case; i.e., it is antisymmetric and looks generally (though not exactly) like the first derivative of the \(\chi^"(\omega)\) curve.

### Intermediate Region: Voight Profiles and Their Uses

In the intermediate region where \(\Delta\omega_a\approx\Delta\omega_d\) and neither of the limiting approximations is valid, the general expression for \(\tilde{\chi}(\omega)\) in Equation 3.69 can only be integrated numerically and then plotted for different ratios of \(\Delta\omega_a/\Delta\omega_d\).

The lineshapes for the \(\chi^"(\omega)\) curves that are obtained in this region are obviously intermediate between lorentzian and gaussian lineshapes, and are generally referred to as Voight profiles. The exact shape of the Voight profile depends on both the homogeneous and the inhomogeneous linewidths, or, more precisely, on the ratio of these two linewidths.

Figure 3.18 shows, for example, the measured absorption profile for a molecular transition in carbon monoxide (the \(v^"=0\), \(J^"=11\) to \(v'=1\), \(J'=10\) transition) at a wavelength of \(\lambda=4.76\) μm or \(1/\lambda=2099\text{ cm}^{-1}\), as measured with a tunable laser in a 10:2:88 mixture of CO_{2}:H_{2}:Ar at a temperature of 3,340 K and a pressure of 0.195 atm. (These rather unusual conditions were obtained in a special shock-tube measuring apparatus.) This absorption profile is clearly best matched by a Voight profile somewhere intermediate between a gaussian and a lorentzian.

If we have an experimental plot of \(\chi^"(\omega)\) for a transition in this intermediate region that has been measured with sufficient accuracy, we can in fact deconvolve the lorentzian and gaussian contributions by fitting the measured curve to numerically a computed Voight profile with the proper ratio of \(\Delta\omega_a/\Delta\omega_d\).

Since we can predict the doppler linewidth for a given transition in a gas fairly accurately from the theoretical expression in Equation 3.64, we can then use the ratio of \(\Delta\omega_a/\Delta\omega_d\) from this kind of Voight profile determinations to derive the homogeneous linewidth \(\Delta\omega_a\), provided it is not too small compared to \(\Delta\omega_d\).

Figure 3.19 shows, for example, absorption data for various pressures of pure CO_{2} taken with a tunable CO_{2} laser and fitted to Voight profiles. (The absorption measurement technique used here was actually a more effective way of measuring weak absorptions, called photoacoustic spectroscopy.)

The top trace shows the laser tuning curve, and the middle traces show raw data, with frequency markers every 30 MHz of frequency tuning. The lower plot shows this data normalized and fitted to a series of Voight profiles with increasing amounts of lorentzian pressure broadening.

### The Transition From Doppler to Pressure Broadening

If we gradually increase the gas pressure in an absorption cell, the measured absorption profile of a transition in the gas atoms will change over from being doppler-broadened at low pressures (\(\Delta\omega_a\ll\Delta\omega_d\)) to being pressure broadened at high pressures (\(\Delta\omega_a\gg\Delta\omega_d\)).

Figure 3.20(a) shows, as one example, an apparatus for making accurate measurements of the absorption profiles of various CO_{2} gas mixtures at different pressures using a tunable CO_{2} laser. In (b) we see direct midband absorption data versus total gas pressure measured on a typical He:N_{2}:CO_{2} gas mixture, and (c) shows the atomic linewidth deduced from this data. Both curves illustrate the changeover from inhomogeneous doppler broadening at low pressures to homogenous pressure (collision) broadening at high pressures.

Figure 3.21 which shows a very similar variation with pressure of the absorption coefficient on a certain chemical laser transition in the mid-IR using deuterium fluoride (DF) molecules.

### An Alternative Notation: \(T_2\) and \(T_2^*\)

The lorentzian and gaussian lineshapes that we have developed in this section are often expressed in an alternative notation, which we can briefly summarize as follows.

In the scientific literature on magnetic resonance, where inhomogeneous broadening was first studied, as well as in other areas of resonance physics, the complex homogeneous lorentzian lineshape is often written in the alternative notation

\[\tag{73}\tilde{\chi}_\text{lor}(\omega)=-j\chi_0^"\frac{1}{1+jT_2(\omega-\omega_a)}\]

and the real lorentzian lineshape for a homogeneous absorption line is then commonly written in normalized form as

\[\tag{74}g_\text{lor}(\omega)=\frac{2}{\pi\Delta\omega_a}\frac{1}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}\equiv\frac{T_2}{\pi}\frac{1}{1+T_2^2(\omega-\omega_a)^2}\]

with the same normalization that \(\int{g}_\text{lor}(\omega)d\omega=1\). In this notation the FWHM homogeneous linewidth is usually written in the simpler form

\[\tag{75}\Delta\omega_a\equiv2/T_2\]

rather than as \(\Delta\omega_a=\gamma+2/T_2\).

In essence, the \(\gamma\) contribution to the homogeneous linewidth has been absorbed into an expanded definition of \(2/T_2\) that includes both the dephasing and lifetime-broadening contributions.

Then, in order to make the gaussian lineshape function \(g_\text{gauss}(\omega)\) have the same algebraic constants in front for the same normalization, the inhomogeneous function is written in the analogous form

\[\tag{76}g_\text{gauss}(\omega)\equiv\left(\frac{T_2^*}{\pi}\right)\exp\left[-\frac{T_2^{*2}(\omega-\omega_a)^2}{\pi}\right]\]

which satisfies the same normalization that \(\int{g}_\text{gauss}(\omega)d\omega=1\). The parameter \(T_2^*\) that has been introduced here is the inhomogeneous analog to the dephasing time \(T_2\) in the homogeneous case. It is related to the gaussian inhomogeneous linewidth \(\Delta\omega_d\) by

\[\tag{77}\Delta\omega_d\equiv\frac{\sqrt{4\pi\ln2}}{T_2^*}\approx\frac{3}{T_2^*}\]

The quantity \(T_2^*\approx3/\Delta\omega_d\) is thus the inhomogeneous (or gaussian) analog to the quantity \(T_2=2/\Delta\omega_a\) for the homogeneous (or lorentzian) lineshape.

### Physical Significance of \(T_2\) and \(T_2^*\)

If we leave out the complications involving the additional \(\gamma\) contribution, the time constant \(T_2\) is what we identified earlier as the homogeneous dephasing time. It defines the average time duration within which the coherent oscillations of two different atomic dipoles are likely to be permanently and irreversibly randomized by collisions, or by other homogeneous dephasing events.

The time constant \(T_2^*\) can be given an analogous interpretation as the inhomogeneous dephasing time due to inhomogeneous broadening mechanisms for a group of oscillating atoms.

Consider, for example, two atoms located in different spectral packets within a gaussian inhomogeneous line. The natural oscillation frequencies \(\omega_{a1}\) and \(\omega_{a2}\) of these two atoms will differ by an amount \(\omega_{a1}-\omega_{a2}\) that will typically be of order \(\approx\Delta\omega_d\).

Even without any homogeneous dephasing events, therefore, these two oscillating dipoles will get out of phase by one half-cycle after a length of time \(\delta{t}\) given by \((\omega_{a1}-\omega_{a2})\delta{t}=\pi\), or \(\delta{t}=\pi/(\omega_{a1}-\omega_{a2})\approx\pi/\Delta\omega_d\approx{T}_2^*\).

The time constant \(T_2^*\) is thus the time duration after which different packets within an inhomogeneous line are likely to have become dephased because of their different oscillation frequencies, even without any collisions or similar dephasing events.

The condition for a strongly inhomogeneous atomic line can be written in either of the alternative forms

\[\tag{78}\text{strongly inhomogeneous line:}\qquad\Delta\omega_d\gg\Delta\omega_a\qquad\text{or}\qquad{T_2^*}\ll{T_2}\]

Thus in a strongly inhomogeneous line the \(T_2^*\) dephasing of different packets because of different oscillation frequencies will happen much more rapidly than the homogeneous dephasing within each packet that is caused by \(T_2\).

In practical experiments, therefore, if all the different atoms or packets within a strongly inhomogeneous line are initially set oscillating coherently and in phase by means of some suitable initial preparation pulse, the coherent macroscopic polarization \(p(t)\) in the collection will disappear after the shorter time \(T_2^*\), not the longer homogeneous dephasing time \(T_2\), because of the inhomogeneous frequency-difference effects. In an inhomogeneous line under small-signal conditions, \(T_2^*\) and not \(T_2\) is the significant dephasing time.

For example, in the He-Ne 633 nm laser transition the doppler linewidth is \(\Delta{f}_d\approx1500\) MHz, and the inhomogeneous dephasing time is thus \(T_2^*\approx3/(2\pi\Delta{f}_d)\approx320\) psec. This must be compared with a homogeneous linewidth for individual packets of more like \(\Delta{f}_a\approx100\) MHz, and hence a homogeneous dephasing time of \(T_2\approx1/\Delta\omega_a\approx3.2\) ns.

One vitally important difference between strongly homogeneous and strongly inhomogeneous systems, however, is that the inhomogeneous dephasing after the time \(T_2^*\) is fundamentally reversible: the different oscillation phases \(\omega_{a1}t\), \(\omega_{a2}t\), and so forth, that develop for different atoms after a time \(t\) can in principle be "unwound" by certain sophisticated large-signal or coherent-pulse techniques. We will discuss these later, in connection with coherent photon echo experiments.

### Inhomogeneous Strain Broadening: Glass Laser Materials

Inhomogeneous broadening is also of considerable importance in certain solid-state laser transitions. Random strains, defects, and other site-to-site variations in solid-state laser materials can significantly change the local crystal fields seen by laser ions that are imbedded in these materials, and this in turn can randomly shift the exact resonance frequencies of laser atoms in those materials, sometimes by quite large amounts.

This type of inhomogeneous broadening predominates in inhomogeneous materials such as laser glasses at room temperature, or in more organized crystalline laser materials at very low temperatures (approaching liquid-helium temperatures), where it is no longer masked by the much larger phonon-broadening effects. Since this kind of broadening, often called strain broadening, is caused basically by random defects in the laser material, its magnitude may depend strongly on material growth and perfection, impurities, and annealing. It is thus not possible to give any general formulas, since the amount of strain broadening may vary from sample to sample of the same material.

If these random strains and defects have a gaussian distribution, the resulting inhomogeneous broadening effects can look and act much like doppler broadening, even though the underlying physical mechanism is totally different. The ratio of homogeneous linewidth \(\Delta\omega_a\) to inhomogeneous linewidth \(\Delta\omega_d\) will still be the crucial parameter in determining whether the transition will be strongly homogeneous, strongly inhomogeneous, or somewhere in between.

For example, the widely used yttrium aluminum garnet (YAG) crystal can be grown with high crystal quality. The linewidth of the Nd^{3+} ion in Nd:YAG laser crystals therefore exhibits only a small amount of inhomogeneous strain broadening. The laser transition is primarily phonon broadened and thus homogeneous at room temperature. Reducing the temperature to below liquid-nitrogen temperature (77 K) greatly reduces the phonon broadening and makes the residual strain broadening observable.

On the other hand, the same Nd^{3+} ion placed in a Nd:glass laser material, with its much larger amount of structural randomness, has a much larger inhomogeneous strain-broadening component, which is significant even at room temperature. This broadening is due to variations in the local crystal fields seen by the laser ions at different sites within the glassy material. The ratio of inhomogeneous to homogeneous broadening in Nd:glass laser materials is not fully understood and varies considerably (by at least a factor of three) from one glass composition to another.

The inhomogeneous linewidths in different glasses at room temperature, for example, vary over a linewidth range of from at least 40 to 120 cm^{-1}. (Linewidths this wide are more often expressed in units of cm^{-1} or wavenumbers than in more conventional units; remember that 1 cm^{-1} = 30 GHz.)

The homogeneous linewidth in the same materials varies over a range from 20 to 75 cm^{-1}, and is strongly correlated (for reasons that are not well understood) with the velocity of sound in the glass. This homogeneous linewidth reduces to \(\ll\) 1 cm^{-1} at 4.2 K, where the lattice vibrations and hence the homogeneous phonon broadening are reduced to nearly zero.

As a general rule, therefore, Nd:glasses are found to fall somewhere in the intermediate or mixed category between homogeneous and inhomogeneous broadening, with ratios of \(\Delta\omega_a/\Delta\omega_d\) ranging from 0.16 to 1.9 in different glasses.

### Far Outside the Resonance Linewidth: All Lines Become Homogeneous

Suppose we go out into the far wings of any atomic resonance transition, homogeneous or inhomogeneous, and measure the atomic response at 5 or 10 linewidths out from the line center. (Note that in any usual atomic transition we can do this and still be well within the "resonance approximation" we introduced earlier, so that the lorentzian and gaussian lineshapes will still apply.)

The gaussian response characteristic of an inhomogeneous transition—for example, a doppler-broadened transition—will then fall off as \(\approx\exp[-(\omega-\omega_a)^2]\), whereas the lorentzian response characteristic of a homogeneous transition—or of a homogeneous packet within an inhomogeneous transition—will fall off only at the much slower rate of \(\approx1/(\omega-\omega_a)^2\) for the \(\chi^"\) part of the susceptibility, or the even slower rate of \(\approx1/(\omega-\omega_a)\) for the \(\chi'\) part of the susceptibility.

Figure 3.22 shows, for example, the \(\chi^"(\omega)\) parts of the susceptibility plotted on the same frequency scale for a gaussian transition with a given linewidth \(\Delta\omega_d\) and for a lorentzian line—or a lorentzian packet within the gaussian line—whose linewidth \(\Delta\omega_a\) is only 1/5 as large as the gaussian linewidth \(\Delta\omega_d\).

This example makes it clear that if we go far enough out from line center, the lorentzian response, though it may be 20 or 30 dB down from the midband value, will clearly dominate over the gaussian response.

In other words, far enough out in the wings, all transitions—even strongly inhomogeneous transitions— once again appear to be homogeneous in character.

If we tune away from an inhomogeneous transition by a sufficient number of inhomogeneous lineshapes, the atomic response will be very weak, though possibly still measurable; and the lineshape of that response will look like a homogeneous lineshape characterized by the \(\Delta\omega_a\) of the individual spectral packets, rather than the \(\Delta\omega_d\) of the inhomogeneous frequency spreading.

For sufficiently strong transitions, this homogeneous response far out in the wings of an inhomogeneous transition can still be of interest, as we will see later on.

### Summary

The differences between homogeneous and inhomogeneous broadening in the central part of the atomic line play a very significant role in the performance of a laser material, especially when saturation effects are taken into account.

Many practical laser materials, particularly gases, are strongly inhomogeneous, but others are strongly homogeneous.

We will return to the detailed "hole burning" properties of inhomogeneous laser systems in a later tutorial.

The next tutorial is an introduction to the ** fundamentals of coherent fiber optic transmission technology**.