# Stimulated Transitions: The Classical Oscillator Model

This is a continuation from the previous tutorial - ** optical fiber coatings**.

Our first major objective in this tutorial is to understand how optical signals act on atoms (or ions, or molecules) to excite resonance responses and to cause transitions between the atomic energy levels. In later tutorial we will examine how the excited atoms or molecules react back on the optical signals to produce gain and phase shift. Eventually we will combine these two parts of the problem into a complete, self-consistent description of laser action. For the minute, however, all we want to consider is what optical fields do to atoms.

The effect of a near-resonant applied signal on a collection of atoms can be divided into two parts. First, there is a resonance excitation of some individual transition in the atoms. This can be modeled by a resonant oscillator model, which leads to a resonant atomic susceptibility, among other things. In this chapter we will develop the classical electron oscillator model for an atomic resonance, and show how this model can lead to equations that describe all the essential features of a single atomic transition. In the next tutorial we will show in more detail how this purely classical model can in fact describe and explain even the most complex quantum-mechanical aspects of real atomic transitions.

The second aspect of the atomic response in real atoms is that, under the influence of an applied signal, atoms begin to make stimulated transitions between the upper and lower levels involved in the transition, so that the atomic level populations begin to change. These stimulated transition rates are described by the atomic rate equations that we introduced in the opening chapter of this book. We will discuss these rate equations in more detail in several later tutorials.

## 1. The Classical Electron Oscillator

Let us first review some of the important physical properties of real atoms. Note that throughout this tutorial, we will speak of "atoms" as a shorthand for simple free atoms, ions, or molecules in gases; or for individual laser atoms, ions, or molecules in solids or in liquids (such as the Cr^{3+} ions in ruby, or the Rhodamine 6G dye molecules in a laser dye); or even for the valence and conduction electrons responsible for optical transitions in semiconductors. Some of the important background facts about real atoms are as follows.

- Atoms consist in simplified terms of a massive fixed nucleus plus a surrounding electron-charge distribution, whether we think of this distribution as a fuzzy charge cloud, or as a set of electronic orbits, as shown in Figure 2.1(a), or as a quantum wavefunction.
- Atoms exhibit sharp resonances both in their spontaneous-radiation wavelengths and in their stimulated response to applied signals.
- These resonances are usually simple harmonic resonances—that is, there are usually no additional responses at exactly integer multiples of these sharp resonant frequencies.
- Most (though not all) atoms respond to the electric field of an applied signal rather than the magnetic field. In more technical terms, the strongest atomic transitions, and those most important for laser action, are usually of the type known as electric dipole transitions. (There do exist other types of atomic transitions, including some laser transitions, that are classified as magnetic dipole, electric quadrupole, or even higher order. Magnetic dipole transitions are described, using a different classical model, in a later tutorial.)

All these properties lead us to use the classical electron oscillator (CEO) model shown in Figure 2.1 as a classical model to represent a single electric-dipole transition in a single atom. With some simple extensions, which we will describe later, this CEO model will give a complete and accurate description of every significant feature of a real atomic quantum transition.

### Analysis of the Classical Electron Oscillator Model

The CEO model envisions that the electronic charge cloud in a real atom may be displaced from its equilibrium position with an instantaneous displacement \(x(t)\), as shown in Figure 2.1(b). Because of the positive charge on the nucleus, this displacement causes the electronic charge cloud to experience a linear restoring force \(-Kx(t)\). The electronic charge cloud is thus in many ways similar to a point electron with mass \(m\) and charge \(-e\) that is located in a quadratic potential well, with potential \(V=Kx^2(t)\), or that is attached to a spring with spring constant \(K\). An externally applied signal with an electric field \(\mathcal{E}_x(t)\) may also be applied to this charge cloud.

The classical equation of motion for an electron trapped in such a potential well, or suspended on such a spring, and subjected to an applied electric field \(\mathcal{E}_x(t)\), is then

\[\tag{1}m\frac{d^2x(t)}{dt^2}=-Kx(t)-e\mathcal{E}_x(t)\]

which we may write in more abstract form as

\[\tag{2}\frac{d^2x(t)}{dt^2}+\omega_a^2x(t)=-(e/m)\mathcal{E}_x(t)\]

The frequency \(\omega_a\) is then the classical oscillator's resonance frequency, given by \(\omega_a^2\equiv{K/m}\). We will equate this resonance frequency for the CEO model with the transition frequency \(\omega_{21}\equiv(E_2-E_1)/\hbar\) of a real atomic transition in a real atom.

More generally, we will identify any one single transition in an individual atom with a corresponding classical electron oscillator, so that from here on we will refer to real atoms or to individual classical oscillators almost interchangeably.

### Damping and Oscillation Energy Decay

The oscillatory motion of the electron in the CEO model, or of the charge cloud in a real atom, must be damped in some fashion, however, since it will surely lose energy with time. Hence we must add a damping term to the equation of motion in the form

\[\tag{3}\frac{d^2x(t)}{dt^2}+\gamma\frac{dx(t)}{dt}+\omega_a^2x(t)=-\frac{e}{m}\mathcal{E}_x(t)\]

where \(\gamma\) is a damping rate or damping coefficient for the oscillator. The electronic motion \(x(t)\) without any applied signal will then oscillate and decay in the fashion

\[\tag{4}x(t)=x(t_0)\exp[-(\gamma/2)(t-t_0)+j\omega_a'(t-t_0)]\]

where \(\omega_a'\) is the exact resonance frequency given by

\[\tag{5}\omega_a'\equiv\sqrt{\omega_a^2-(\gamma/2)^2}\]

The \(Q\) of an optical frequency transition in an atom will always be high enough to allow us to simplify life from now on by ignoring the difference between \(\omega_a\) and \(\omega_a'\). The energy associated with the internal oscillation in the CEO model, which we will write as \(U_a(t)\), thus decays as

\[\tag{6}U_a(t)=\frac{1}{2}Kx^2(t)+\frac{1}{2}mv_x^2(t)=U_a(t_0)e^{-\gamma(t-t_0)}\equiv{U_a(t_0)}e^{-(t-t_0)/\tau}\]

The decay rate \(\gamma\) is thus the energy decay rate, and the lifetime \(\tau\equiv\gamma^{-1}\) is the energy decay time for the oscillator model.

Both classical electron oscillators and real atomic transitions will always lose energy in part by radiating away electromagnetic radiation, in what we call spontaneous emission or fluorescence, at the transition frequency \(\omega_a\). This radiation of electromagnetic energy from the oscillating charge cloud, as shown in Figure 2.2, leads to a purely radiative part of the decay rate \(\gamma\), which we will call \(\gamma_{rad}\).

Real atomic transitions in many cases, however, also lose additional oscillation energy by other "nonradiative" mechanisms, such as collisions with other atoms, or the emission of heat vibrations into a surrounding crystal lattice. This additional energy loss leads to an additional nonradiative part of the total decay rate, which we will denote by \(\gamma_{nr}\). The total energy decay rate is then generally given by

\[\tag{7}\gamma\equiv\frac{1}{U_a}\frac{dU_a}{dt}=\gamma_{rad}+\gamma_{nr}\]

Note that the energy \(U_a\) we are talking about here is the energy associated with the internal charge cloud oscillation within the atom. This energy is quite distinct from other kinds of energy the atom may also possess, such as the kinetic energy of motion the same atom may possess if the atom as a whole is moving rapidly in a gas.

The energy decay rate for an atomic transition may thus include both radiative and nonradiative parts. Radiative decay, which is exactly the same thing as spontaneous electromagnetic emission or fluorescent emission from the atom, is always present, though sometimes very weak.

Nonradiative decay can also be present, sometimes much more strongly and sometimes much less strongly than the radiative part of the total decay, depending on individual circumstances.

The causes of nonradiative decay can include inelastic collisions of atoms with each other, or with the walls of a laser tube, so that the internal oscillating energy of the atoms gets converted into kinetic energy of the gas atoms, or goes into heating up the tube walls.

Nonradiative decay in solids or liquids can also involve the loss of energy from the electronic oscillation of the atoms into lattice vibrations and hence into heat in the surrounding crystal lattice in a solid.

The general property of all nonradiative atomic relaxation or decay mechanisms is that energy is lost from the internal oscillatory motion of the individual atomic charge clouds, and that this energy goes into simple heating up of surrounding gas atoms or tube walls or crystal lattices.

### Radiative Decay Rates

The purely radiative decay rate or spontaneous emission rate for a classical electron oscillator can be calculated from classical electromagnetic theory. The sinusoidally oscillating electron radiates energy outward exactly like an oscillating dipole antenna or an oscillating current source; and this energy is the spontaneous emission.

The resulting decay rate for a classical electron oscillator imbedded in an infinite medium of dielectric permittivity \(\epsilon\) is given by

\[\tag{8}\gamma_{rad,ceo}=\frac{e^2\omega_a^2}{6\pi\epsilon{m}c^3}\]

Note that according to the conventions used in this tutorial, \(\epsilon\) and \(c\) are the dielectric permeability and the velocity of light in any surrounding dielectric medium, and not necessarily the free-space values \(\epsilon_0\) and \(c_0\). This classical oscillator radiative decay rate has a value \(\gamma_{rad,ceo}\approx10^8\text{ sec}^{-1}\) for a visible frequency oscillator, compared to an oscillation frequency of \(\omega_a\approx4\times10^{15}\text{ sec}^{-1}\). Hence, the decay rate is very small compared to the oscillation frequency.

Real atomic transitions have radiative decay rates that are determined by quantum considerations. These rates for real atoms are different from the classical expression just given, and are, different for each different atomic transition. For many transitions, however, the real atomic decay rates for so-called strongly allowed transitions are of the same order of magnitude as the purely classical radiative decay rate for a CEO with the same resonance frequency.

### Microscopic Dipole Moments and Macroscopic Polarization

The next important step we must take is to go from microscopic individual atoms, represented by individual electron oscillators, to macroscopic electromagnetic effects in real laser materials. We do this by adding up the microscopic electric dipole moments from many individual atoms or classical oscillators to produce a macroscopic electromagnetic polarization in the laser material.

We first note that displacement of the electronic charge cloud of an atom away from its equilibrium position around the nucleus by an effective distance \(x(t)\) means that there is a displacement of the center of the negative electronic charge, with value \(-e\), away from the matching positive charge \(+e\) of the heavy and nearly immobile nucleus.

This displacement creates a microscopic electric dipole moment \(\mu_x(t)\) associated with that individual oscillator or atom, which is given by

\[\tag{9}\mu_x(t)=[\text{charge}]\times[\text{displacement}]=-ex(t)\]

as shown in Figure 2.3.

Let us then recall that in electromagnetic theory Maxwell's equations are written in the form

\[\tag{10}\begin{align}\boldsymbol{\nabla}\times\boldsymbol{\mathcal{E}}(\pmb{r},t)&=-\frac{\partial\pmb{b}(\pmb{r},t)}{\partial{t}}\\\boldsymbol{\nabla}\times\pmb{h}(\pmb{r},t)&=\pmb{j}(\pmb{r},t)+\frac{\partial\pmb{d}(\pmb{r},t)}{\partial{t}}\end{align}\]

together with the definitions

\[\tag{11}\begin{align}\pmb{d}(\pmb{r},t)&=\epsilon_0\boldsymbol{\mathcal{E}}(\pmb{r},t)+\pmb{p}(\pmb{r},t)\\\pmb{b}(\pmb{r},t)&=\mu_0\pmb{b}(\pmb{r},t)+\pmb{m}(\pmb{r},t)\end{align}\]

in which \(\pmb{p}(\pmb{r}, t)\) and \(\pmb{m}(\pmb{r},t)\) are the electric and magnetic polarizations, or dipole moments per unit volume, at point \(\pmb{r}\) and time \(t\).

The electric polarization \(\pmb{p}(\pmb{r}, t)\) at any point in an atomic medium is thus, by definition, the net electric dipole moment per unit volume in a small differential volume surrounding that point. In a laser medium in particular, this polarization \(\pmb{p}\) must be calculated by adding up the vector sum of the individual dipole moments \(\mu_x\) of all the atoms in that unit volume.

Consider, for example, a tiny volume of a laser medium containing a very large number of microscopic atoms or classical oscillators, as shown schematically in Figure 2.4. (Note that in a typical laser medium the density of atoms may be anywhere from \(10^{12}\) to \(10^{19}\) laser atoms/cm^{3}; so there may be anywhere from \(10^3\) to \(10^{10}\) atoms even in a tiny cube only 10 optical wavelengths on a side.) Let each atom in this volume be labeled by an index \(i\), and let each atom have an instantaneous electric dipole moment \(\mu_{xi}(t)=-ex_i(t)\).

This medium will then have a macroscopic electric polarization \(\pmb{p}\) around that point \(\pmb{r}\) in the medium whose \(x\) component is given by

\[\tag{12}p_x(\pmb{r},t)\equiv{V}^{-1}\sum_{i=1}^{NV}\mu_{xi}(t)\]

The volume \(V\) here can represent any small unit volume (but still containing many dipoles) surrounding the point \(\pmb{r}\), and \(N\) is the density of individual dipoles in that volume, so that \(NV\) is the total number of dipoles.

We could, to be more general, write both the microscopic dipole moments and the macroscopic polarization in this formula as vector quantities, in which case the macroscopic polarization \(\pmb{p}\) would be the vector sum over all individual dipoles \(\boldsymbol{\mu}_i\) within that volume. However, for now we are focusing only on the linearly polarized \(x\) components of \(\pmb{p}(\pmb{r}, t)\) and \(\boldsymbol{\mu}_i(t)\).

Also, in real materials both the applied field \(\boldsymbol{\mathcal{E}}(\pmb{r},t)\) and the polarization \(\pmb{p}(\pmb{r}, t)\) will in general be functions of position \(\pmb{r}\), though the changes in value will be very small compared to interatomic spacings. We will not be worrying about the spatial variation of this macroscopic polarization until later, however.

The step we have just taken, of going from individual microscopic atomic dipole moments \(\mu_{xi}\) to a macroscopic electric polarization \(p_x\), is a crucial step in the theoretical analysis of laser action.

To analyze the response of a laser material, we use quantum theory—or as a substitute we use the CEO model— to calculate the microscopic dipole moments of individual laser atoms. These responses are then summed over large numbers of such atoms per unit volume in a real laser medium to produce the macroscopic polarization.

This polarization then goes into Maxwell's equations to produce laser absorption, gain, and/or phase shift (as we will see later). We measure in the laboratory, or employ in laser devices, only the macroscopic effects of this atomic polarization. We seldom if ever observe the minute microscopic effects produced by one tiny single atom acting alone.

### Discussion

The primary concept introduced in this section is that we can use the classical electron oscillator model, with resonance frequency \(\omega_a\), as a substitute for a single atomic transition with transition frequency \(\omega_{21}\) in a single real quantum atom.

The very great utility of the CEO model for this purpose will become apparent in following sections. The essential accuracy of this simple classical model can, however, be further illustrated by the following point.

Suppose a classical oscillating electric dipole antenna is placed close to a reflecting metallic surface, or close to one or more dielectric layers or surfaces. The spatial radiation pattern, the radiative decay rate, and even the resonance frequency of the classical dipole will then all be changed by significant amounts.

This occurs, in classical terms, because the radiating dipole is influenced by its own radiated fields reflected back from the nearby surfaces. These effects are strongest, of course, when the oscillator is close to the surface, within one wavelength or less.

Experimental studies of exactly these same effects have also been carried out on real atomic transition dipoles, using real radiating atoms placed very close to dielectric or metal surfaces, with exactly the same results being obtained for the real atoms.

Such experiments have been carried out, for example, by using thin monomolecular layers of radiating dye molecules adsorbed onto dielectric films one wavelength or less thick attached to a reflecting silver surface or to another dielectric surface or layer.

The observed changes in the radiative behavior of these real atomic (or molecular) dipoles have been found to agree completely with theoretical calculations using purely classical models for both the radiating atoms and the electromagnetic fields.

## 2. Collisions and Dephasing Processes

The next important concept that we have to introduce—a particularly fundamental and important concept—is the effect of dephasing events, such as atomic collisions, on the oscillation behavior of classical oscillators or of real atoms.

### Coherent Dipole Oscillations

Any single microscopic electric dipole oscillator, when left by itself, obeys the equation of motion

\[\tag{13}\frac{d^2\mu_x(t)}{dt^2}+\gamma\frac{d\mu_x(t)}{dt}+\omega_a^2\mu_x(t)=(e^2/m)\mathcal{E}_x(t)\]

which is obtained by multiplying \(-e\) into both sides of Equation 2.3 and using Equation 2.9. Hence the oscillating moment of a single atom with no applied field \(\mathcal{E}_x\) present has the exponentially decaying sinusoidal form

\[\tag{14}\mu_x(t)=\mu_{x0}\exp[-(\gamma/2)(t-t_0)+j\omega_a(t-t_0)+j\phi_0]\]

where \(\mu_{x0}\) is the magnitude and \(\phi_0\) the phase (at time \(t_0\)) of the initial oscillation that has been set up in the dipole oscillator, perhaps by some pulsed applied signal.

We have already pointed out that even a small volume of laser material may contain a large number of laser atoms, or tiny oscillating dipoles. We might therefore label each individual atom or dipole by an index \(i\), and write the oscillating dipole moment of the \(i\)-th atom as

\[\tag{15}\mu_{x,i}(t)=|\mu_{x0,i}(t_0)|\exp[-(\gamma/2)(t-t_0)+j\omega_a(t-t_0)+j\phi_i]\]

where \(\phi_i\) is the phase angle of the \(i\)-th dipole oscillator at the starting time \(t_0\).

Now suppose first that these dipoles are all oscillating together, all at the same frequency, and more importantly all initially in phase—that is, all with the same value of \(\phi_i\) at the same reference time \(t_0\). Then the total dipole moment due to the vector sum of all these moments in some small volume will be

\[\tag{16}\mu_{x,\text{tot}}(t)=\sum_{i=1}^{NV}\mu_{x,i}(t)=NV\mu_x(t)\qquad\begin{cases}\text{all dipoles}\\\text{oscillating}\\\text{in phase,}\end{cases}\]

where \(\mu_x(t)\) is the moment of any one dipole; \(N\) is the density of dipoles (i.e., the number per unit volume); and \(NV\) is the total number of dipoles in a small volume \(V\). The macroscopic polarization, or the dipole moment per unit volume, will then be given by \(p_x(t)=\mu_{x,\text{tot}}(t)/V\), or

\[\tag{17}p_x(t)=N\mu_{x0}\exp\left[[-(\gamma/2)+j\omega_a](t-t_0)+j\phi_0\right]\qquad\begin{cases}\text{all dipoles}\\\text{oscillating}\\\text{in phase}.\end{cases}\]

The macroscopic polarization \(p_x(t)\) in the atomic medium will thus have the same natural oscillation frequency uja and the same energy decay rate \(\gamma/2\) as the individual dipoles. In this example its magnitude will also be \(N\) times as large as any one individual dipole—but if (and only if) the individual dipoles all keep oscillating unperturbed and with the same phases.

This macroscopic polarization when all the dipoles are oscillating in time-phase with each other may be rather large in real situations. The dipoles are then said to be oscillating coherently, or fully aligned with each other.

### Dephasing Effects: Random Collisions

This is not the usual situation with real atoms, however. There are almost always perturbation effects, or dephasing effects, which scramble or randomize the time-phases \(\phi_i\) of individual dipole oscillators, and which thereby cause the macroscopic polarization \(p_x(t)\) to become much smaller than the result given by the two preceding equations.

To understand this, let us look first at a very simple example of how a particular type of dephasing process, namely, randomly occurring and instantaneous dephasing events or "collisions," might operate to destroy the macroscopic polarization or coherent dipolar oscillation in a collection of atoms.

Figure 2.5 shows three assumed dipole moments, which we label \(\mu_{x,1}(t)\), \(\mu_{x,2}(t)\), and \(\mu_{x,3}(t)\), all oscillating initially in phase and at the same oscillation frequency. The total moment \(\mu_{x,\text{tot}}(t)\) as shown at the bottom of the figure, is then initially three times as large as the moment of any one dipole.

Suppose, however, that after random time intervals first one and then another of the dipoles suffers an instantaneous dephasing event or "collision," which does not reduce the amplitude of the oscillating moment, but does shift it to a new phase angle in time.

After each such collision the amplitude of the total moment is reduced, because the individual moments no longer add in phase. (The individual dipole oscillations will also slowly decay in amplitude themselves because of energy decay, as discussed in the previous section; but we have not illustrated this point here.) Random collisions thus gradually destroy the macroscopic polarization, even without any energy decay.

Figure 2.6 illustrates in another manner this difference between dipole oscillators that are "aligned" or oscillating coherently in phase, and randomly phased dipole oscillators, by showing the results of adding up three phasors that represent the amplitude and instantaneous time-phase of the three separate individual dipole oscillators.

In (a) the three phasors are fully aligned; in (b) and (c) they are gradually shifted in phase or "dephased" to produce a smaller and smaller resultant sum.

Note that these are phasor diagrams, in which the horizontal and vertical axes for each vector are the real and imaginary parts of the phasor amplitudes of the oscillating moments, or the cosine and sine parts of the sinusoidal oscillations.

These axes do not represent the vector coordinates of the dipoles in space, since we are talking here for the moment only about the \(x\) component of the dipole oscillations \(\mu_x(t)\).

### Large Numbers of Dipoles

Suppose that we add up the phasor amplitudes of a large number \(NV\) of dipoles, but with the phase angles \(\phi_i\) randomly distributed over all values between \(0\) and \(2\pi\).

This then becomes the standard statistical problem of adding up many randomly phased sine waves, and we can find that the resulting total dipole moment \(\mu_{x,\text{tot}}(t)\) in any small volume \(V\) will now be a random quantity; i.e., its amplitude and phase will vary randomly from one small volume to another.

Moreover, the total dipole moment in any small volume will have a mean value of zero, i.e.,

\[\tag{18}\langle\mu_{x,\text{tot}}(t)\rangle=0\qquad(\text{randomly phased dipoles})\]

but will have a root-mean-square value given by

\[\tag{19}\langle\mu_{x,\text{tot}}^2(t)\rangle^{1/2}=(NV)^{1/2}|\mu_x(t)|\qquad(\text{randomly phased dipoles})\]

where \(|\mu_x|\) refers to the value for any one single dipole by itself.

The quantity \(NV\) will be a very large number even for very small volumes \(V\). Hence the rms moment, or rms macroscopic polarization, for the randomly phased case, which is proportional to \((NV)^{1/2}|\mu_x|\), will be very much smaller than the possible coherent polarization of order \(NV|\mu_x|\) that could be produced by the same number of dipoles oscillating in phase. (The rms polarization in the randomly phased case is in fact essentially random noise; and this noise is essentially the same thing as the spontaneous emission from a collection of quantum atoms, although we will not discuss this topic here.)

### Dephasing Mechanisms

The crucial point, then, is that any effect which tends to randomize the oscillation phases \(\phi_i\) in a large collection of individual dipoles (such as are present in even a small volume of laser material) will act to destroy any coherent macroscopic polarization that may be present in this collection of dipole oscillators. Such dephasing effects do exist in real atomic systems, and understanding these additional dephasing effects is our primary task in this section.

These dephasing effects that cause the oscillation phases of individual atomic oscillators to become randomized, even though each dipole continues to oscillate with the same decaying amplitude and average frequency, are often referred to for simplicity as collisions. However, the actual physical processes that can cause dephasing of the individual internal atomic oscillations in a collection of atoms can include the following.

- Atoms (or ions or molecules) in gases, moving with their normal thermal or Brownian motion, can in fact make random physical collisions with each other, or with other gas atoms, or with the walls of the laser tube. Even if these collisions are elastic—that is, even if they do not take any energy away from the internal electronic oscillation energy of the atoms—in general such collisions between atoms will scramble and randomize the phases of the electronic oscillations inside the colliding atoms.
- For laser atoms in solids, the quantum energy-level spacings and hence the exact transition frequencies \(\omega_a\) of the laser atoms are affected by nearby host atoms, and hence depend on the exact distances to nearby atoms in the host crystal lattice. Thermal vibrations of the crystal lattice will modulate these distances slightly, and thus modulate the atomic transition frequencies \(\omega_a\) by small but random amounts with time. This is called phonon broadening, and it produces in turn a random phase modulation and hence a "phase smearing" of the dipole oscillations in the laser atoms.
- In materials where the laser atoms are sufficiently dense, the local time-varying electric (or magnetic) fields produced by any one oscillating dipole may spread out to, and be felt by, other neighboring laser atoms. The individual oscillating dipoles are then no longer totally independent, but become weakly coupled to each other through what is called dipolar coupling. This kind of weak coupling between individual resonant systems, even if they are all identical, always tends to randomize and to broaden the overall response of the collection. (This is true of weakly coupled resonant electric circuits, as well as weakly coupled resonant atoms.) This process of dipolar coupling is thus still another mechanism for producing random phase smearing in the atomic dipoles. Moreover, this dipolar coupling itself will be randomly modulated in atoms by the thermal motion of the atoms, whether by gas kinetics in gases or lattice vibrations in solids.

Whatever may be the physical cause, the net result of each of these physical processes is to randomize or "dephase" the phases of individual atomic oscillators with respect to each other. The coherent dipole oscillations get converted eventually into incoherent oscillations.

### Exponential Decay: The Dephasing Time \(T_2\)

A simple formula for the rate at which a macroscopic polarization \(p_x(t)\) will be destroyed by random dephasing events can be developed as follows. Suppose that not just a few dipoles, as in the preceding examples, but a very large number of individual (but identical) oscillators are involved.

Suppose also that the dephasing events for individual dipoles happen randomly both in their times of occurrence and in the phase changes they produce.

Let there then be some large number \(N_0\) of dipoles in a unit volume, all initially oscillating in phase, so that the magnitude of the initial polarization at a starting time \(t_0\) is

\[\tag{20}p_x(t_0)=N_0\mu_{x0}\]

At any later time \(t\gt{t_0}\), we can then divide these \(N_0\) dipoles into (a) a decreasing number of dipoles \(N(t)\) that have not yet suffered any collisions at all; and (b) an increasing number \(N_0-N(t)\) of dipoles that have suffered at least one collision, and perhaps more. The \(N(t)\) dipoles that have not yet undergone any collisions or dephasing events will then continue to oscillate in phase and to produce a macroscopic polarization

\[\tag{21}p_x(t)=N(t)\mu_x(t)=N(t)\mu_{x0}\cos\omega_{at}\]

Those dipoles that have suffered even one collision, however, will have phases that are entirely random (assuming, as is normally done, that the phase of a dipole oscillation is entirely randomized after each collision). Hence those dipoles will add up to produce no coherent polarization at all, on the average.

The coherent polarization after any time \(t\gt{t_0}\) thus comes entirely from the remaining uncollided dipoles. [A more precise statement is that the \(N\) dipoles oscillating coherently in phase will add up to produce a macroscopic polarization proportional to \(N\mu_{x0}\), whereas the \(N_0-N\) dipoles oscillating with random phases will add up to produce a macroscopic sum with a mean value of zero and an rms value proportional to \((N_0-N)^{1/2}\mu_{x0}\). Since the number of atoms involved in any atomic system is always very large, the latter quantity is negligible compared to the coherent part of the oscillation; and we can neglect the contribution from the randomly phased dipoles in the latter group.]

The number of uncollided dipoles \(N(t)\) will of course decrease steadily with time. How can we calculate the rate at which the number of uncollided atoms \(N(t)\) decreases? Let us suppose that collisions occur at a random rate of \(1/T_2\) collisions per atom per second. Then, the total number of collisions \(dN\) that members of this uncollided group will undergo in a little time interval \(dt\) about time \(t\), or the loss rate from the uncollided group \(N(t)\) in time \(dt\), will be given by

\[\tag{22}dN(t)=-\frac{N(t)}{T_2}dt\]

The size of the uncollided group will thus decay as

\[\tag{23}N(t)=N_0e^{-(t-t_0)/T_2},\qquad{t\gt{t_0}}\]

The coherent macroscopic polarization produced by these still uncollided oscillators will therefore also decay as

\[\tag{24}\begin{align}p_x(t)&=N(t)\mu_x(t)\\&=N_0e^{-(t-t_0)/T_2}\times\mu_{x0}\exp[-(\gamma/2)(t-t_0)+j\omega_a(t-t_0)+j\phi_0]\\&=p_{x0}\exp[-(\gamma/2+1/T_2)(t-t_0)+j\omega_a(t-t_0)+j\phi_0]\end{align}\]

In other words, the amplitude decay rate \(\gamma/2\) appropriate for the individual dipoles must be replaced by \(\frac{1}{2}\gamma+T_2^{-1}\) as the effective amplitude decay rate for the coherent polarization \(p_x(t)\), or

\[\tag{25}\left(\frac{\gamma}{2}\right)\left(\begin{split}\text{single-dipole}\\\text{decay rate}\end{split}\right)\quad\Rightarrow\quad\left(\frac{\gamma}{2}+\frac{1}{T_2}\right)\left(\begin{split}\text{macroscopic}\\\text{polarization}\\\text{decay rate}\end{split}\right)\]

It may seem slightly odd that in this substitution the dephasing rate \(1/T_2\) gets added to the quantity \(\gamma/2\equiv1/2\tau\), which is half the energy decay rate. The reason for this difference of a factor of two—which will continue to be with us— is essentially that \(1/T_2\) and \(1/2\tau\) are both decay rates for sinusoidal amplitudes like \(\mu_x(t)\) or \(p_x(t)\); whereas \(\gamma\) itself is an energy decay rate for the quantity \(|\mu_x(t)|^2\).

### Summary

The primary conclusion of this section, therefore, is that although the macroscopic polarization \(p_x(t)\), has the same resonance frequency as the individual microscopic dipole oscillations \(\mu_x(t)\), it may have a faster decay rate because of dephasing effects. Individual atomic dipole oscillations, in the intervals between dephasing events, can thus be described as obeying the equation of motion

\[\tag{26}\frac{d^2\mu_x(t)}{dt^2}+\gamma\frac{d\mu_x(t)}{dt}+\omega_a^2\mu_x(t)=(e^2/m)\mathcal{E}_x(t)\]

with an amplitude decay rate \(\gamma/2\). But the coherent polarization \(p_x(t)\) must be described as obeying the equation

\[\tag{27}\frac{d^2p_x(t)}{dt^2}+(\gamma+2/T_2)\frac{dp_x(t)}{dt}+\omega_a^2p_x(t)=(Ne^2/m)\mathcal{E}_x(t)\]

where there is an additional factor of \(1/T_2\) in the amplitude decay rate because the dephasing processes cause the oscillations of individual atoms to become randomized in phase at a rate \(1/T_2\). (Note again the difference of a factor of 2 between the \(\gamma\) and \(1/T_2\) terms.)

In the analysis we have presented here—which is in fact very similar to the approach in much more sophisticated quantum treatments—the time constant \(T_2\) thus has the physical significance of the mean time between dephasing events or collisions for any one individual atom, so that \(1/T_2\) is the collision frequency for any one individual atom. The time constant \(T_2\) is thus often called the collision time. This same time constant is often referred to more broadly as the dephasing time, or even the dipolar interaction time for \(p_x(t)\). In quantum analyses or in the Bloch equations for magnetic resonance, \(T_2\) is also called the off-diagonal or transverse relaxation time.

## 3. More on Atomic Dynamics and Dephasing

Since dephasing effects are a particularly important aspect of atomic dynamics, let us look a bit further at some of the additional consequences and varieties of dephasing effects in real atomic systems.

### Dephasing Effects Plus Applied Signals

One basic assumption in discussions of dephasing is that dephasing effects and applied signal fields such as \(\mathcal{E}_x(t)\) will act on individual dipoles simultaneously and independently—that is, we can simply add up their effects in computing the total internal motion of individual atoms. What, then, are the relative strengths of these two effects?

To explore this, let us consider, for example, the response of a single dipole oscillator subjected to an on-resonance sinusoidal applied field \(\mathcal{E}_x(t)=E_1\cos\omega_at\) during the period just after this atom has suffered a randomizing collision at \(t=t_0\). How rapidly can the applied signal \(\mathcal{E}_x(t)\) "pull" the individual dipole moment \(\mu_x(t)\) back into a coherent phase relationship with the applied signal, following a dephasing collision?

The problem here is clearly to solve the single-dipole equation of motion (Equation 2.26) with the specified applied signal and with an arbitrary initial condition on the phase (i.e., the position and velocity) of the oscillator at time \(t=0\). This can be done straightforwardly, although the exact solution is a bit messy. We know, however, that an equation of this type has a transient or homogeneous solution, independent of \(\mathcal{E}_x(t)\), of the form

\[\tag{28}\mu_x(t)=\mu_{x0}\exp[-(\gamma/2)(t-t_0)+j\omega_a(t-t_0)+j\phi_0]\]

(There is a minor approximation in this expression, namely, the replacing of \(\omega_a'\) by \(\omega_a\).) We will also show later that an on-resonance applied signal will produce a steady-state or forced sinusoidal solution of the form

\[\tag{29}\mu_x(t)=\text{Re}\left[-j\frac{e^2E_1}{m\omega_a\gamma}e^{j\omega_at}\right]=\text{Re}\left[\tilde{\mu}_{ss}e^{j\omega_at}\right]\]

where \(\tilde{\mu}_{ss}\equiv{j}(e/m\omega_a\gamma)E_1\) is the steady-state phasor amplitude of the motion produced by the field \(E_1\). Suppose we also define \(\tilde{\mu}_0\equiv\mu_{x0}\exp{j}\phi(0)\) as the complex phasor amplitude (magnitude and phase) of the sinusoidal motion of \(\mu_x(t)\) immediately after the collision. (The phase \(\phi(0)\) will take on random values for different dipoles after different collisions.)

The total solution for \(\mu_x(t)\) following any given collision will then be a linear combination of the forced plus transient solutions, with just enough transient solution included to meet the initial boundary condition at \(t_0\), or

\[\tag{30}\mu_x(t)=\text{Re}\left[\tilde{\mu}_{ss}+(\tilde{\mu}_0-\tilde{\mu}_{ss})e^{-(\gamma/2)(t-t_0)}\right]e^{j\omega_a(t-t_0)}\]

This says, in effect, that we can write \(\mu_x(t)\) in the form

\[\tag{31}\mu_x(t)=\text{Re}\left[\tilde{\mu}(t)e^{j\omega_a(t-t_0)}\right]\]

where \(\tilde{\mu}(t)\) is a slowly changing complex phasor amplitude given by

\[\tag{32}\tilde{\mu}(t)=\tilde{\mu}_{ss}+[\tilde{\mu}_0-\tilde{\mu}_{ss}]e^{-(t-t_0)/2\tau}\]

In other words, following a collision the phasor amplitude \(\tilde{\mu}(t)\) of the sinusoidal motion "pulls in" from the initial random post-collision value \(\tilde{\mu}_0\), toward the forced or steady-state value \(\tilde{\mu}_{ss}\), with an exponential time constant \(2\tau\). Note that this pull-in time constant does not depend at all on the strength of the applied field.

Now, we will see later that for most real laser transitions the dephasing time \(T_2\) is usually much shorter than the energy decay time \(\tau\); so the dephasing time constant \(T_2\) is also much shorter than the pull-in time constant \(\tau\) (or \(2\tau\)). Any individual atom is, therefore, very likely to be dephased again by another collision, after a short time \(\approx{T_2}\), well before it gets pulled completely into phase by the applied signal \(\mathcal{E}_x(t)\). In real laser systems, therefore, even with applied signals present, the motions of the individual dipoles are mostly dephased, or randomly phased, by the dephasing processes. A coherent applied signal \(\mathcal{E}_x(t)\) can usually only struggle to impose a small amount of phase ordering on this unruly bunch of oscillators.

Exceptions to this usual situation occur only for applied signals that are strong enough to produce the kind of Rabi flopping behavior that we will discuss in later tutorials. Most signals in common lasers are "weak signals" which do not produce this kind of behavior; and the dipole motion in these system will be mostly random, with a small fractional amount of signal-imposed coherent ordering.

### Dephasing by Random Frequency Modulation

Let us also look in a bit more detail at another type of dephasing that occurs in many solid-state laser materials.

The most graphic way of picturing dephasing effects in any collection of atoms is probably the kind of sudden, sharp, discrete, randomly occurring dephasing events or "collisions" that we have described above. An important alternative dephasing process for atoms, however, especially in crystals and other solids, is phonon broadening, or phonon frequency modulation of the atomic transitions, rather than genuine collisions between different atoms as in a gas.

In systems with phonon broadening (or with dipolar coupling as well) the dephasing process results not from sudden collisions, but from a more continuous but still random frequency modulation of each individual dipole's oscillation frequency. The net result, however, is essentially the same: dipoles that begin oscillating in phase gradually end up, after a time on the order of \(T_2\), with their phases completely randomized.

Consider, for example, the chromium Cr^{3+} ions in a ruby crystal or the neodymium Nd^{3+} ions in a Nd:YAG crystal or glass lattice, such as we showed in previous tutorials; and suppose the internal electronic charges of several such ions have been set oscillating in an internal dipole oscillation with the same initial phase.

Now, the surrounding lattice itself will also be vibrating slightly at any finite temperature, because of thermal agitation; so the spacing between each ion and its nearest neighbors in the lattice will be changing slightly in a random way that is different for each ion.

But for ions in solids, small changes in the lattice spacing will cause very small but finite shifts in the exact resonance frequency \(\omega_a\) of the transition in each ion. The sinusoidal dipole oscillations of the various ions, as a result, will proceed at slightly different and randomly changing frequencies; and the dipole oscillations will thus drift slowly and randomly out of phase with each other.

This same argument can hold for dipole oscillations in a crystal lattice, in a glassy solid, in a liquid, or in any condensed atomic medium. Dipoles initially oscillating in phase will eventually be converted to random phases.

It is not so evident here that this will lead to an exponential decrease in the coherent polarization component. The fact is, however, that the same assumption of exponential decay that we made for the macroscopic polarization is just as good an approximation for these situations also.

### Some Typical Numbers for Dephasing Effects

The magnitudes of the dephasing effects and the values of the dephasing time \(T_2\) exhibit very large variations in different kinds of atomic media. Recall first that visible transitions have oscillation frequencies on the order of \(6\times10^{14}\) Hz and thus oscillation periods of the order of \(10^{-15}\) sec.

The collision frequencies for atoms in real gases can vary over a wide range, depending on gas pressure; but values in the range of \(1/T_2\approx10^8\) to \(10^9\text{ sec}^{-1}\), or dephasing times of \(T_2\approx10^{-8}\) to \(10^{-9}\) sec at low pressures, are not uncommon. Energy decay times in the range of \(\tau\equiv\gamma^{-1}\approx10^{-5}\) to \(10^{-7}\) sec for transitions of interest are also reasonable. The general conclusion is thus that there are always an enormous number of optical cycles between each collision or dephasing event. The collision rate is usually an order of magnitude or more higher than the energy decay rate; so the \(1/T_2\) term in the polarization decay often dominates over the \(\gamma/2\) part of the decay.

For atoms in solids, the lattice vibrational frequencies that are excited by thermal agitation, and that cause the thermal frequency dephasing, range from zero up to \(\approx10^{13}\) Hz. The lattice modulation of the atomic transition frequencies is thus in general very fast compared to any measurements we might try to make on the atoms, but still slow compared to the actual transition frequencies \(\omega_a\). We must then ask not only how rapidly the lattice atoms vibrate, but also how strongly they modulate or shift the transition frequencies \(\omega_a\). The answer in typical lasers (e.g., ruby or Nd:YAG) is that these frequencies are shifted randomly by amounts on the order of \(10^{11}\) to \(10^{12}\) Hz.

Now, two dipoles having a random frequency difference of \(\omega_{a2}-\omega_{a1}\) will get \(2\pi\) out of time-phase with each other after an interval of \(2\pi/(\omega_{a2}-\omega_{a1})\) seconds. The effective \(T_2\) dephasing times for ionic transitions in solids are thus often of the order of \(10^{-11}\) to \(10^{-12}\) sec. The energy decay times in solids on good laser transitions are sometimes as slow as \(10^{-3}\) to \(10^{-4}\) sec. Again, the \(1/T_2\) dephasing component dominates, generally by a very large amount, over the \(\gamma/2\) energy decay rate.

### Coherent Versus Incoherent Decay

Suppose, as a final mental exercise, that a large number of atoms \(N_0\) are initially all oscillating and radiating together in phase, as we described earlier. (Preparing a group of atoms in this coherently phased initial condition is not always easy to accomplish, as we will see later. It generally requires very strong applied signals, applied in very short pulses.)

Given this initial preparation, all these atoms or oscillators will then radiate together as one giant coherent dipole. The initial value of this dipole will be \(N_0\mu_{x0}\), where \(\mu_{x0}\) is the dipole moment of one individual atom; and the rate at which this collection of coherently oscillating atoms radiates energy will be proportional to \(N_0^2|\mu_{x0}|^2\). The essential point is that all the dipoles are radiating coherently, that is, in time-phase with each other.

This coherence will, however, be destroyed by dephasing processes in a time of order \(T_2\), which for real atomic transitions is often very short (from nanoseconds down to less than picoseconds). Once the coherent oscillation is destroyed, after a few dephasing times \(T_2\), the individual dipoles will in general still be oscillating and radiating energy, since their energy decay time \(\tau\equiv\gamma^{-1}\) is generally longer (sometimes much longer) than the dephasing time \(T_2\). The individual dipoles will continue, in fact, to radiate energy through the \(\gamma_{rad}\) and \(\gamma_{nr}\) processes, but they now radiate individually and incoherently, with random phase relationships between the dipoles. The radiation that now comes out from the sample is essentially narrowband noise, or spontaneous emission, or fluorescence centered at the atomic transition frequency \(\omega_a\). It comes out in all directions, and with a narrowband but essentially noise-like spectrum. The power radiated is simply the sum of the individual powers radiated by the \(N_0\) individual dipoles, and hence is now proportional to \(N_0|\mu_{x0}|^2\) rather than to \(N_0^2|\mu_{x0}|^2\).

Suppose we perform an experiment in which we set a collection of atomic dipoles oscillating coherently, perhaps using some kind of pulsed applied signal, and then observe how the atoms radiate afterward. We can expect to see two transients: first the coherent transient radiation, which may be strong but very fast (time constant \(\approx{T_2}\)); and then the incoherent transient radiation, generally much weaker but longer-lived, corresponding to normal spontaneous emission or fluorescence (time constant \(\approx{T_1}\)). So-called coherent pulse or coherent free-induction decay experiments displaying the first type of behavior can be performed on atoms at optical frequencies. These experiments are generally rather difficult, however, requiring short but intense coherent laser pulses for excitation, together with high-speed detectors for the coherently radiated signals.

Much more common are ordinary fluorescent lifetime experiments, such as we will illustrate later. In these, a group of atoms are again set oscillating, but the excitation mechanism is some form of incoherent excitation, such as a pulse of broadband light from an incoherent flashlamp, or a short burst of current through a collection of gas atoms. There is no initial phase coherence to the excitation in these cases, and hence no coherent initial polarization to either radiate coherently or decay at the \(T_2\) rate. The radiation in this case comes entirely from incoherent spontaneous emission or fluorescence, and the measured decay rate will be simply the energy decay rate \(\gamma\equiv\gamma_{rad}+\gamma_{nr}\). Understanding the distinction between these coherent and incoherent types of processes is extremely important in understanding the atomic phenomena involved in lasers.

## 4. Steady-State Response: The Atomic Susceptibility

Our next task is to compute the steady-state response of a collection of oscillators or atoms to a sinusoidal applied signal, and to express this response as a linear resonant electric susceptibility.

### Phasor Analysis

Suppose that the electric field \(\mathcal{E}_x(t)\) applied to a collection of classical oscillators, or electric dipole atoms, is a sinusoidal signal with frequency \(\omega\);, which we write in the form

\[\tag{33}\mathcal{E}_x(t)=\text{Re}\left[\tilde{E}_xe^{j\omega{t}}\right]=\frac{1}{2}\left[\tilde{E}_xe^{j\omega{t}}+\tilde{E}_x^*e^{-j\omega{t}}\right]\]

In electrical engineering jargon the complex quantity \(\tilde{E}_x\) is a "phasor" whose magnitude and phase angle give the amplitude and phase of the real quantity \(\mathcal{E}_x(t)\). Suppose, for example, that the complex phasor \(\tilde{E}_x\) has the magnitude and phase angle \(\tilde{E}_x\equiv|\tilde{E}_x|e^{j\phi}\). Then the real field \(\mathcal{E}_x(t)\) will be given by \(\mathcal{E}_x(t)=\text{Re}[|\tilde{E}_x|e^{j(\omega{t}+\phi)}]=|\tilde{E}_x|\cos(\omega{t}+\phi)\), so that obviously \(|\tilde{E}_x|\) is the magnitude and \(\phi\) the phase angle (in time) of the cosinusoidal signal.

The steady-state response from a linear atomic system will then have the same sinusoidal form, i.e.,

\[\tag{34}p_x(t)=\text{Re}[\tilde{P}_xe^{j\omega{t}}]=\frac{1}{2}[\tilde{P}_xe^{j\omega{t}}+\tilde{P}_x^*e^{-j\omega{t}}]\]

so that a similar description will obviously prevail for the magnitude and phase angle of the real polarization \(p_x(t)\) and its complex phasor amplitude \(\tilde{P}_x\).

Both the \(e^{j\omega{t}}\) and the \(e^{-j\omega{t}}\) terms in these phasor expansions are needed to give the complete real fields; but in any linear system with a linear differential equation, such as we are considering here, the \(\tilde{E}_xe^{j\omega{t}}\) part of the applied field will be connected only to the \(\tilde{P}_xe^{j\omega{t}}\) part of the induced polarization, and similarly for the \(\tilde{E}_x^*e^{-j\omega{t}}\) and \(\tilde{P}_x^*e^{-j\omega{t}}\) parts of these quantities. Moreover, these separate responses in any real physical system will be simply the complex conjugates of each other, so that the complex-conjugate or \(e^{-j\omega{t}}\) terms really contain no additional information over and above the \(e^{j\omega{t}}\) terms.

Following the usual practice in phasor analyses, therefore, we will focus only on the \(e^{j\omega{t}}\) terms from now on. Moreover, for simplicity we will generally leave off the "Re" notation from now on and write the real fields in the form \(\mathcal{E}_x(t)=\tilde{E}_xe^{j\omega{t}}\), with the operation of taking the real part being understood.

If we put these sinusoidal phasor expansions into the equation of motion for \(p_x(t)\), Equation 2.27, and separate out the \(e^{j\omega{t}}\) terms, we obtain a relation between the complex phasor amplitudes:

\[\tag{35}[-\omega^2+j\omega(\gamma+2/T_2)+\omega_a^2]\tilde{P}_x=\frac{Ne^2}{m}\tilde{E}_x\]

which we will rearrange into the form

\[\tag{36}\frac{\tilde{P}_x}{\tilde{E}_x}=\frac{Ne^2}{m}\frac{1}{\omega_a^2-\omega^2+j\omega(\gamma+2/T_2)}\]

This is the linear steady-state relationship between the phasor polarization \(\tilde{P}_x\) induced in the collection of oscillators or atoms and the field \(\tilde{E}_x\) applied to them. In linear-system terms, it is the transfer function for the response of the atomic medium.

### Electric Polarization and Susceptibility: Standard Definitions

This transfer function is more commonly known as the electric susceptibility of the atomic medium, as produced by the polarization response of the atoms or oscillators. We can recall that the electric field \(\tilde{E}\), the electric polarization \(\tilde{P}\), and the electric displacement \(\tilde{D}\) in any arbitrary dielectric medium are related under all circumstances by the basic definition from electromagnetic theory

\[\tag{37}\tilde{D}=\epsilon_0\tilde{E}+\tilde{P}\]

In the more restrictive case of a linear and isotropic dielectric medium, the polarization \(\tilde{P}\) and the electric field \(\tilde{E}\) will also be related, by an expression which is conventionally written in the form

\[\tag{38}\tilde{P}(\omega)=\tilde{\chi}(\omega)\epsilon_0\tilde{E}(\omega)\]

so that the quantity \(\tilde{\chi}(\omega)\) defined by

\[\tag{39}\tilde{\chi}(\omega)\equiv\frac{\tilde{P}(\omega)}{\epsilon_0\tilde{E}(\omega)}\]

is the electric susceptibility of the medium, with \(\epsilon_0\) being the dielectric permeability of free space. We will adopt a slightly modified version of this definition a few paragraphs further on.

The relationship between the electric displacement \(\tilde{D}\) and the electric field \(\tilde{E}\) in a linear medium can then be written, using the standard definition of Equation 2.39, as

\[\tag{40}\tilde{D}=\epsilon_0(1+\tilde{\chi})\tilde{E}=\tilde{\epsilon}\tilde{E}\]

which means that the complex dielectric constant \(\tilde{\epsilon}(\omega)\) is given by

\[\tag{41}\tilde{\epsilon}(\omega)\equiv\epsilon_0(1+\tilde{\chi})\]

For a completely general description, the field quantities \(\tilde{D}\), \(\tilde{E}\), and \(\tilde{P}\) really should be treated as vector quantities in these relations; and in the more general linear but anisotropic case the susceptibility \(\tilde{\chi}\) then becomes a tensor quantity. For simplicity, however, let us stick with scalar notation at this point.

The electric susceptibility relating the applied signal \(\tilde{E}_x\) and the atomic polarization \(\tilde{P}_x\) in an atomic medium is very important in calculating laser gain, phase shift, and many other properties, as we will see shortly. Before going further with this discussion, however, we must introduce a slightly nonstandard definition of the electric susceptibility \(\tilde{\chi}\), which is peculiar to this tutorial, but which will turn out to be very useful in simplifying later formulas.

### Atomic Susceptibility: A Modified Definition

In a sizable fraction of the laser materials of interest to us, the resonant oscillators or the laser atoms that produce the resonant polarization \(\tilde{P}_x\) are not located in free space. Rather, these atoms are imbedded in a laser crystal, or perhaps in a glass or a liquid host material. In the laser material ruby, for example, the Cr^{3+} laser atoms that are responsible for the laser behavior are dispersed (at ≈ 1% density) in a host lattice of colorless AI_{2}O_{3}, or sapphire. In dye laser solutions the dye molecules, for example, Rhodamine 6G, are dissolved at perhaps \(10^{-3}\) molar concentration in a liquid solvent such as water or ethanol.

In all these devices, the host materials in the absence of the laser atoms are transparent dielectric materials that are nearly lossless at the laser wavelength, but have a relative dielectric constant \(\epsilon/\epsilon_0\) or an index of refraction \(n\) that is significantly greater than unity.

These materials will possess, therefore, a large nonresonant linear electric polarization \(P_\text{host}\) that is associated with the host material by itself, and that has no direct connection with the generally much weaker resonant polarization \(P_\text{at}\) that comes from the resonant response of the classical oscillators or from the resonant transitions in the laser atoms.

We can therefore write the total displacement vector in such a material in more detail as

\[\tag{42}\tilde{D}=\epsilon_0\tilde{E}+\tilde{P}_\text{host}+\tilde{P}_\text{at}\]

In this equation \(\tilde{P}_\text{host}\) refers to the large, broadband, linear nonresonant polarization associated with the host material by itself; whereas \(\tilde{P}_\text{at}\) refers to the weak, narrowband, linear resonant polarization produced by the classical oscillators or atoms imbedded in the host material. Following conventional electromagnetic notation, we can then define a nonresonant susceptibility \(\tilde{\chi}_\text{host}\) and a dielectric constant \(\epsilon_\text{host}\) for the host material according to the usual definitions, in the form

\[\tag{43}\tilde{P}_\text{host}=\tilde{\chi}_\text{host}\epsilon_0\tilde{E}\qquad\text{and}\qquad\epsilon_\text{host}=\epsilon_0(1+\tilde{\chi}_\text{host})\]

The total polarization can therefore be written as

\[\tag{44}\tilde{D}=\epsilon_0[1+\tilde{\chi}_\text{host}]\tilde{E}+\tilde{P}_\text{at}=\epsilon_\text{host}\tilde{E}+\tilde{P}_\text{at}\]

Note that in typical laser crystals or liquids the host dielectric constant (at optical frequencies) will have magnitude \(\epsilon_\text{host}/\epsilon_0\approx2\) to \(3\), so that the dimensionless host susceptibility will have magnitude \(\tilde{\chi}_\text{host}\approx1\) to \(2\). To put this in another way, the index of refraction of typical laser host materials, given by \(n_\text{host}\equiv\sqrt{\epsilon_\text{host}/\epsilon_0}\) will have values of \(n_\text{host}\approx1.5\) to \(2.0\) for typical liquids or crystals.

Suppose now that we were also to define a separate susceptibility \(\tilde{\chi}_\text{at}\) for the atomic or resonant oscillator part of the response in the laser medium, using the same conventional definition as given earlier, namely,

\[\tag{45}\tilde{\chi}_\text{at}=\tilde{P}_\text{at}/\epsilon_0\tilde{E}\qquad\left(\begin{split}\text{conventional}\\\text{definition}\end{split}\right)\]

Then we would end up with a result of the form

\[\tag{46}\tilde{D}=\epsilon_\text{host}\tilde{E}+\tilde{\chi}_\text{at}\epsilon_0\tilde{E}=\epsilon_\text{host}[1+(\epsilon_0/\epsilon_\text{host})\tilde{\chi}_\text{at}]\tilde{E}\qquad\left(\begin{split}\text{conventional}\\\text{definition}\end{split}\right)\]

Now, there will be many times in later tutorials when we will want to expand the bracketed factor involving \(\tilde{\chi}_\text{at}\) to various orders in \((\epsilon_0/\epsilon_\text{host})\tilde{\chi}_\text{at}\) since this quantity is always small compared to unity. If we follow this conventional definition for \(\tilde{\chi}_\text{at}\), using \(\epsilon_0\), we will end up carrying along perpetual factors of \(\epsilon_0/\epsilon_\text{host}\) to various powers in all these expressions.

To avoid this, we will instead consistently use in this book an alternative nonstandard definition for \(\tilde{\chi}_\text{at}\) which we obtain by writing

\[\tag{47}\tilde{P}_\text{at}\equiv\tilde{\chi}_\text{at}\epsilon_\text{host}\tilde{E}\qquad\left(\begin{split}\text{this tutorial's}\\\text{definition}\end{split}\right)\]

so that the atomic resonance part of the susceptibility is defined by the expression

\[\tag{48}\tilde{\chi}_\text{at}\equiv\frac{\tilde{P}_\text{at}(\omega)}{\epsilon_\text{host}\tilde{E}(\omega)}\left(\begin{split}\text{this tutorial's}\\\text{definition}\end{split}\right)\]

Note that we are not rewriting any laws of electromagnetic theory by doing this—we are merely introducing a slightly unconventional way of defining \(\tilde{\chi}_\text{at}\) for an atomic transition, in which \(\epsilon_\text{host}\) is used as a normalizing constant in the denominator, rather than \(\epsilon_0\) as in the standard definition. If we use this definition, as we will from here on, the total electric displacement in any laser material is then given by the simpler form

\[\tag{49}\tilde{D}=\epsilon_\text{host}\tilde{E}+\tilde{P}_\text{at}=\epsilon_\text{host}[1+\tilde{\chi}_\text{at}]E\qquad\left(\begin{split}\text{this tutorial's}\\\text{definition}\end{split}\right)\]

This alternative form avoids the factor of \(\epsilon_0/\epsilon_\text{host}\) in Equation 2.46. For simplicity, from here on we will also drop all the "host" subscripts and simply write \(\epsilon_\text{host}\) as \(\epsilon\).

To keep all this straight, just remember: from here on \(\epsilon\) is the dielectric constant of the host lattice or dielectric material without the laser atoms; whereas \(\tilde{\chi}_\text{at}\) defined according to the alternative definition of Equation 2.48, is the additional (weak) contribution due to the resonant atomic transition in the laser atoms. Of course, if the laser material is a dilute gas with \(\epsilon_\text{host}=\epsilon_0\), there is no difference anyway.

### Resonant Susceptibility: The Resonance Approximation

With this definition we can write the general susceptibility \(\tilde{\chi}_\text{at}\) for the resonant response in a collection of resonant oscillators or atoms by combining Equations 2.36 and 2.48 to obtain

\[\tag{50}\tilde{\chi}_\text{at}(\omega)\equiv\frac{\tilde{P}_x}{\epsilon\tilde{E}_x}=\frac{Ne^2}{m\epsilon}\frac{1}{\omega_a^2-\omega^2+j\omega\Delta\omega_a}\]

We have introduced here the important quantity

\[\tag{51}\Delta\omega_a\equiv\gamma+2/T_2\]

which we will shortly identify as the atomic linewidth (FWHM) of the atomic resonance. Since both \(\gamma\) and \(2/T_2\) are always small compared to optical frequencies, this linewidth \(\Delta\omega_a\) is very small compared to the center frequency \(\omega_a\) for essentially all transitions of interest in lasers—never more than 10% at absolute most, and usually much, much narrower. (In fact, fractional linewidths greater than a fraction of a percent occur in practice only in semiconductor injection lasers and organic dye lasers.)

We are most often interested only in the response of the atoms to signal frequencies \(\omega\) that lie within a few linewidths of either side of the resonant frequency \(\omega_a\). Within this region we can make what is called the resonance approximation by writing

\[\tag{52}\omega^2-\omega_a^2=(\omega+\omega_a)(\omega-\omega_a)\approx2\omega_a(\omega-\omega_a)\approx2\omega(\omega-\omega_a)\]

so that the frequency-dependent part of the susceptibility expression becomes

\[\tag{53}\frac{1}{\omega_a^2-\omega^2+j\omega\Delta\omega_a}\approx\frac{1}{2\omega(\omega_a-\omega)+j\omega\Delta\omega}\approx\frac{1}{2\omega_a(\omega_a-\omega)+j\omega\Delta\omega}\]

By using this we can then convert Equation 2.50 into the simpler resonant form

\[\tag{54}\tilde{\chi}_\text{at}(\omega)=\frac{-jNe^2}{m\omega_a\epsilon\Delta\omega_a}\frac{1}{1+2j(\omega-\omega_a)/\Delta\omega_a}\]

It is evident that this response will decrease rapidly compared to its midband value as soon as the frequency detuning \(\omega-\omega_a\) becomes more than a few times the linewidth \(\pm\Delta\omega_a\); and hence it really does not matter at all whether we use \(\omega\) or \(\omega_a\) in the denominator in the first part of this expression, so long as we do not tune away from \(\omega_a\) by more than, say, \(\pm10\%\).

### The Lorentzian Lineshape

The right-hand part of Equation 2.54 exhibits a very common frequency dependence known as a complex lorentzian lineshape. Since we will be seeing this frequency dependence over and over in the remainder of our tutorials, let us gain a little familiarity with its properties.

Suppose that, for simplicity, we define a normalized frequency shift away from line center by

\[\tag{55}\Delta{x}\equiv2\frac{\omega-\omega_a}{\Delta\omega_a}\]

so that \(\Delta{x}=0\) corresponds to midband and \(\Delta{x}=\pm1\) corresponds to a frequency shift of half a linewidth away from line center on either side. Then the complex lorentzian lineshape is given by

\[\tag{56}\tilde{\chi}_\text{at}(\omega)=-j\chi_0^"\frac{1}{1+2j(\omega-\omega_a)/\Delta\omega_a}=-j\chi_0^"\frac{1}{1+j\Delta{x}}\]

where

\[\tag{57}\chi_0^"\equiv\frac{Ne^2}{m\omega_a\epsilon\Delta\omega_a}\]

is the magnitude of the negative-imaginary value at midband.

Readers familiar with Fourier transforms will recognize that this complex lorentzian lineshape is simply the Fourier transform in frequency space of the exponential time decay of the polarization \(p_x(t)\). (Whether the \(-j\) factor in front of the \(1/(1+j\Delta{x})\) frequency dependence is to be considered part of the complex lorentzian lineshape or not is entirely a matter of style.) Note once again that in examining the frequency dependence of lorentzian transitions— for example, in solving some of the problems at the end of this section—the frequency dependence of the constant \(\chi_0^"\) can De entirely ignored; i.e., it makes no practical difference whether we use \(\omega\) or \(\omega_a\) in the denominator of Equation 2.57. This constant in front can be treated as entirely independent of frequency within the resonance approximation.

The real and imaginary parts of this complex lorentzian lineshape then have the forms

\[\tag{58}\tilde{\chi}_\text{at}(\omega)\equiv\chi'(\omega)+j\chi^"(\omega)=-\chi_0^"\left[\frac{\Delta{x}}{1+\Delta{x^2}}+j\frac{1}{1+\Delta{x^2}}\right]\]

where \(\tilde{\chi}'(\omega)\) and \(\tilde{\chi}^"(\omega)\) are the real and imaginary parts of this function, as plotted in Figure 2.8. The imaginary part of this response, or \(\chi^"(\omega)\), has a resonant response curve of the form

\[\tag{59}\chi^"(\omega)=-\chi_0^"\frac{1}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}=-\chi_0^"\frac{1}{1+\Delta{x^2}}\]

This lineshape is conventionally called the real lorentzian lineshape, with a response centered at \(\Delta{x}=0\) or \(\omega=\omega_a\), and with a full width between the half-power points \(\omega-\omega_a=\pm\Delta\omega_a\) given by

\[\tag{60}\Delta\omega_a=\gamma+2/T_2\]

The linewidth \(\Delta\omega_a\) is thus the full width at half maximum (FWHM) linewidth of the atomic transition. We will shortly identify \(\chi^"(\omega)\) as the absorbing (or amplifying) part of the atomic response.

The real part of the lorentzian susceptibility, or \(\chi'(\omega)\), has the frequency dependence

\[\tag{61}\chi'(\omega)=-\chi_0^"\frac{2(\omega-\omega_a)/\Delta\omega_a}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}=-\chi_0^"\frac{\Delta{x}}{1+\Delta{x^2}}\]

which has the antisymmetric or roughly first-derivative form shown in Figure 2.8. We will shortly identify this \(\chi'(\omega)\) part as the reactive, or phase-shift, or dispersive part of the atomic response.

Note that the literature on atomic transitions and lasers uses many different linewidth definitions for \(\Delta\omega\), \(\Delta{f}\), \(\Delta\lambda\), etc., which in different publications are sometimes defined as the half widths of resonance lines; sometimes as the full widths, as here; and sometimes even as rms linewidths, or \(1/e^2\) linewidths, or other exotic widths. We will be consistent in this text in always using a FWHM definition for any linewidth \(\Delta\omega\), \(\Delta{f}\), or \(\Delta\lambda\), unless we explicitly say otherwise.

### Magnitude of the Steady-State Atomic Response

Let us emphasize once more that the atomic response of a collection of atoms to an applied signal is coherent, in the sense that the steady-state induced polarization \(\tilde{P}(\omega)\) follows, in amplitude and time-phase, the driving signal field \(\tilde{E}(\omega)\) in the manner described by the complex susceptibility or transfer function \(\tilde{\chi}(\omega)\). The susceptibility \(\tilde{\chi}(\omega)\) given by Equations 2.50, 2.54 or 2.56 is a dimensionless quantity. We will see later that in essentially every case of interest to us, the numerical value of this quantity is very small compared to unity. Only for very large atomic densities, very strongly allowed transitions, and very narrow linewidths does the numerical magnitude of \(\tilde{\chi}\) approach unity; and these conditions are not normally all present at once in laser materials.

We might also note that the magnitude of the atomic susceptibility at midband is proportional to the inverse linewidth \(1/\Delta\omega_a\), where the linewidth \(\Delta\omega_a\) for classical oscillators is given by

\[\tag{62}\Delta\omega_a=\gamma_{rad}+\gamma_{nr}+2/T_2\]

As the dephasing time \(T_2\) becomes smaller, this linewidth becomes larger, and hence the relative strength of the induced atomic response decreases in direct proportion to the dephasing processes as measured by \(2/T_2\).

If there were no dephasing processes, so that \(T_2\rightarrow\infty\), then the applied signal field \(\tilde{E}\) could drive all the individual atomic dipoles to oscillate completely in phase, and would produce the largest possible induced response, limited only by the dipole energy decay rate. The dephasing processes associated with any finite \(T_2\) value, however, operate to "fight" the coherent phasing effects of the applied signal, and to reduce the coherent polarization that can be developed. The usual situation in most (though not all) laser materials is that the \(2/T_2\) dephasing term dominates over the energy decay rate \(\gamma\); as a result, the applied signal can produce only a small fractional coherent ordering of the dipole oscillation's steady state, working against the randomizing effects of the dephasing processes.

## 5. Conversion to Real Atomic Transitions

The classical electron oscillator results derived in the preceding sections can be converted into quantum-mechanically correct formulas for real atomic transitions in real atoms by making a few simple and almost obvious substitutions. These substitutions are briefly introduced in this section, and then discussed in more detail in the following series of tutorials.

### Substitution of Radiative Decay Rate

The first step in converting from the CEO model to real atomic transitions is to notice a similarity between the constant in front of the classical oscillator susceptibility expression of Equation 2.57 in the preceding section, which has the form

\[\tag{63}\chi_0^"\equiv\frac{Ne^2}{m\omega_a\epsilon\Delta\omega_a}\]

and the classical oscillator radiative decay rate that we introduced in Equation 2.8, which is given by

\[\tag{64}\gamma_\text{rad,ceo}=\frac{e^2\omega_a^2}{6\pi\epsilon{m}c^3}\]

In fact, if we substitute the second of these into the first, we can write the amplitude of the classical oscillator susceptibility at midband in the form

\[\tag{65}\chi_0^"=\frac{3}{4\pi^2}\frac{N\lambda^3\gamma_\text{rad,ceo}}{\Delta\omega_a}\]

In this form all the atomic and electromagnetic constants appearing in the classical oscillator model (charge \(e\), mass \(m\), and the dielectric constant \(\epsilon\)) drop out; and the resulting expression depends only on directly measurable properties of the classical oscillator, namely, the transition wavelength \(\lambda\), the density of oscillators \(N\), the radiative decay rate \(\gamma_\text{rad,ceo}\), and the linewidth \(\Delta\omega_a\) of the transition itself. This expression is a more fundamental and useful way of writing the susceptibility, since it is now equally valid for either classical oscillators or real atoms, provided only that we use the appropriate values of \(\lambda\), \(\gamma_\text{rad}\) and \(\Delta\omega_a\) in each case.

### Introduction of Population Difference

The second and more fundamental step in converting from classical oscillators to real atoms is to notice that the classical electron oscillator response we have derived here is proportional to the number density \(N\) of the classical oscillators. But we learned in the previous tutorials that the response on real quantum transitions is proportional to the population difference density \(\Delta{N}_{12}=N_1-N_2\) between the populations (atoms per unit volume) in the lower and upper levels of the atomic transition.

That is, a collection of classical oscillator "atoms" can only absorb energy, at least in steady state. Both quantum theory and experiments show, however, that when a signal is applied to a collection of real quantum atoms, the steady-state response is always such that the lower-level atoms absorb energy through upward transitions, but the upper-level atoms emit energy through downward transitions. The lower-level atoms thus act essentially like conventional classical oscillators, but the upper-level atoms act somehow like "inverted" classical oscillators.

The single most crucial step in converting our classical oscillator results to accurate quantum formulas for real atomic transitions is thus to replace the classical oscillator density \(N\) by a quantum population difference \(\Delta{N}_{12}\equiv{N_1}-N_2\), where \(N_1\) and \(N_2\) are the number of atoms per unit volume, or the "level populations," in the lower and upper energy levels. This substitution is the primary point where quantum theory enters the classical oscillator model.

### Quantum Susceptibility Result

If we make both of these substitutions, and also for simplicity leave off all the classical oscillator labels, then the resonant susceptibility expression for either a collection of classical oscillators or a real atomic transition is given by the same expression, namely,

\[\tag{66}\tilde{\chi}_\text{at}(\omega)=-j\frac{3}{4\pi^2}\frac{\Delta{N}\lambda^3\gamma_\text{rad}}{\Delta\omega_a}\frac{1}{1+2j(\omega-\omega_a)/\Delta\omega_a}\]

It will often be convenient to write this expression for the complex lorentzian susceptibility in the form

\[\tag{67}\begin{align}\tilde{\chi}_\text{at}(\omega)&=-j\chi_0^"\times\frac{1}{1+2j(\omega-\omega_a)/\Delta\omega_a}\\&=-\chi_0^"\left[\frac{2(\omega-\omega_a)/\Delta\omega_a}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}+j\frac{1}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}\right]\end{align}\]

where \(-j\chi_0^"\) is the given midband susceptibility, with magnitude given by

\[\tag{68}\chi_0^"=\frac{3}{4\pi^2}\frac{\Delta{N}\lambda^3\gamma_\text{rad}}{\Delta\omega_a}\]

This expression then becomes an essentially quantum-mechanically correct expressions for the resonant susceptibility of any real electric-dipole atomic transition, provided simply that we use in these formulas the real (i.e., measured) values of the parameters \(\lambda\), \(\gamma_\text{rad}\), \(\Delta\omega_a\), and \(\Delta{N}\) for that particular atomic transition.

**Discussion**

The preceding results thus say that the linear response to an applied signal, as expressed by \(\tilde{\chi}(\omega)\), for a collection of classical oscillators or for a real atomic transition, depends only on the following.

- For the classical case the number of oscillators \(N\lambda^3\), or for the quantum case the net population difference \(\Delta{N}\lambda^3\), contained in a volume of one wavelength cubed, where \(\lambda\equiv\lambda_0/n\) is the wavelength in the host crystal medium.
- The radiative decay rate \(\gamma_\text{rad}\) characteristic of that particular oscillator, or of that particular atomic transition. This is a very fundamental and important point: different transitions in real atoms will have very different strengths, as measured by their radiative decay rates. We see here that the induced or stimulated response on each such transition will be directly proportional to the spontaneous emission rate on that same transition. Oscillators that radiate strongly also respond strongly.
- The inverse linewidth \(1/\Delta\omega_a\) of that transition. This says in effect that there is a characteristic area under each such resonance (with a magnitude proportional to \(\Delta{N}\lambda^3\gamma_\text{rad}\)). Transitions that are broadened or smeared out by dephasing effects, or by other line-broadening mechanism, then have proportionately less response at line center.
- And finally, there is the complex lorentzian lineshape that gives the frequency variation of the atomic response as a system is tuned on either side of the resonance frequency.

Each of these points is fundamental, and will reoccur many times in discussions of real atomic responses later on.

### The Quantum Polarization Equation of Motion

We can also make the same substitutions in the differential equation of motion for \(p(t)\) in the time domain, and rewrite Equation 2.27 in the form

\[\tag{69}\frac{d^2p_x(t)}{dt^2}+\Delta\omega_a\frac{dp_x(t)}{dt}+\omega_a^2p_x(t)=\frac{3\omega_a\epsilon\lambda^3\gamma_\text{rad}}{4\pi^2}\Delta{N}(t)\mathcal{E}_x(t)\]

after which this also becomes an essentially quantum-mechanically correct equation for the induced polarization response \(p(t)\), or at least for its quantum expectation value, on a real atomic transition. We will make further use of this quantum equation in later tutorials.

Notice that after making this conversion to the quantum case, we now have a situation in which the population difference \(\Delta{N}(t)\) on the right-hand side of the equation may itself be an explicit function of time, as a result of stimulated transitions, pumping effects, and/or relaxation processes, rather than being a constant value \(N\) as in the classical case. This makes the quantum equation essentially nonlinear, as contrasted with the essentially linear character of the classical oscillator model.

We will see later that in most cases of interest in lasers, the rate of change of the population difference \(\Delta{N}(t)\) is slow compared to the inverse of the atomic linewidth \(\Delta\omega_a\). This represents the so-called rate-equation limit, in which we can validly solve the polarization differential equation of motion in a linear fashion, just as we did in this chapter, and thereby obtain the linear resonant sinusoidal susceptibility given above.

There are other situations, however, in which the applied signal becomes strong enough (or the transition linewidth is narrow enough) that we move into a large-signal regime where the time-variation of \(\Delta{N}(t)\) does become important.

In this large-signal regime it is no longer possible to solve Equation 2.69 as a simple linear differential equation; and hence the linear susceptibility \(\tilde{\chi}(\omega)\) is no longer an adequate description of the atomic response. We must instead solve the nonlinear polarization equation for \(p(t)\), Equation 2.69, together with a separate rate equation for the time variation of \(\Delta{N}(t)\) that we will derive in a later tutorial, in order to get the full large-signal atomic behavior. The result in the large-signal limit is a more complex form of behavior, commonly referred to as Rabi flopping behavior, which we will describe in more detail in a later chapter.

The polarization equation of motion is thus more general than the sinusoidal susceptibility results, which are valid only within the so-called "rate-equation limits." Most laser devices in fact operate in the rate-equation regime; but there are also more complex large-signal phenomena, often referred to as "coherent pulse phenomena," which occur only in the Rabi-frequency regime. Such coherent pulse effects can be demonstrated experimentally using appropriate high-power laser beams and narrow-line atomic transitions.

### Additional Substitutions

Let us finally give a brief but complete list of all the other steps that are necessary to convert the classical oscillator results derived in this tutorial into the completely correct quantum results for any real electric-dipole atomic transition. Converting the classical oscillator formulas to apply to a real atomic transition requires the following steps.

**1. Transition frequency.** Any single kind of atom will of course have numerous resonant transitions among its large number of quantum energy levels \(E_i\). The classical electron oscillator can model only one such transition between two selected levels, say \(E_i\) and \(E_j\gt{E}_i\), at a time. To treat several different signals applied to different transitions at different frequencies simultaneously, we must in essence employ multiple CEO models, one for each transition. The level populations \(N_i(t)\) in the different levels involved are then interconnected by rate equations, as we will discuss in a later section.

The classical resonance frequency \(\omega_a\) must be replaced by the actual transition frequency \(\omega_{ji}\) in the real atom, i.e.,

\[\tag{70}\omega_a\Rightarrow\omega_{ji}=\frac{E_j-E_i}{\hbar}\]

The actual transition wavelength \(\lambda\), measured in the laser host medium, must of course also be used.

**2. Atomic population difference.** The population difference must be replaced by the population difference on that particular transition, i.e.,

\[\tag{71}\Delta{N}\Rightarrow\Delta{N_{ij}}=N_i-N_j\]

where \(N_i\) is the lower-level and \(N_j\) the upper-level population density.

**3. Radiative decay rate.** The radiative decay rate \(\gamma_\text{rad}\) must be replaced by a quantum radiative decay rate appropriate to the specific \(i\rightarrow{j}\) transition under consideration, i.e.,

\[\tag{72}\gamma_\text{rad}\Rightarrow\gamma_{\text{rad},ji}\]

Every real atomic transition between two energy levels \(E_i\) and \(E_j\) will have such a characteristic spontaneous-emission rate, which is the same thing as the Einstein A coefficient on that transition, i.e., \(\gamma_{\text{rad},ji}\equiv{A}_{ji}\).

**4. Transition linewidth.** The linewidth \(\Delta\omega_a\) must be replaced by a linewidth \(\Delta\omega_{a,ij}\) characteristic of the real transition in the real atoms, i.e.,

\[\tag{73}\Delta\omega_a\Rightarrow\Delta\omega_{a,ij}\]

This involves using real-atom values for the linewidth contributions of both the energy decay rate, i.e., \(\gamma_{ij}\), and the dephasing time \(T_{2,ij}\) on that particular transition, as well as any other broadening mechanisms that may be present. We will say more later about what these real-atom values mean and how they are obtained. Note also that different \(i\rightarrow{j}\) transitions in a given atom may have quite different linewidths \(\Delta\omega_{a,ij}\).

**5. Transition lineshape.** More generally, the complex lineshape of \(\tilde{\chi}_\text{at}(\omega)\) for a real atomic transition may not be exactly lorentzian, although many real atomic transitions are. It may be necessary for some transitions to replace the lorentzian frequency dependence with some alternative lineshape or frequency dependence for \(\tilde{\chi}(\omega)\). Whatever this lineshape may be, however, the real and imaginary parts \(\chi'(\omega)\) and \(\chi^"(\omega)\) near resonance will almost always have lineshapes much like those in Figure 2.8.

**6. Tensor properties.** We assumed in previous sections a classical oscillator model that was linearly polarized along the \(x\) direction. We have thus derived essentially only one tensor component of the linear susceptibility, that is, the component defined by

\[\tag{74}\tilde{P}_x(\omega)=\tilde{\chi}_{xx}(\omega)\epsilon\tilde{E}_x(\omega)\]

The response of a real atomic transition may involve a more complicated and anisotropic (though still linear) response of all three vector components \(\pmb{P}(\omega)=[\tilde{P}_x,\tilde{P}_y,\tilde{P}_z]\) to the vector field components \(\pmb{E}(\omega)=[\tilde{E}_x,\tilde{E}_y,\tilde{E}_z]\). The susceptibility \(\tilde{\chi}(\omega)\) must then be replaced by a tensor susceptibility \(\boldsymbol{\chi}(\omega)\), i.e.,

\[\tag{75}\tilde{\chi}(\omega)\Rightarrow\boldsymbol{\chi}(\omega)\]

where \(\boldsymbol{\chi}(\omega)\) is a 3 x 3 susceptibility tensor defined by

\[\tag{76}\pmb{P}(\omega)=\boldsymbol{\chi}(\omega)\epsilon\pmb{E}(\omega)\]

We discuss the resulting tensor properties of real transitions in more detail later.

**7. Polarization properties.** The magnitude of the response of an atomic transition to an applied signal in the tensor case will also depend on how well the applied field polarization \(\pmb{E}\) lines up or overlaps with the tensor polarization needed for optimum response from the atoms. If the applied field is not properly polarized or oriented with respect to the atoms, the observed response will be reduced. We can account for this by replacing the numerical factor of 3 that appears in the susceptibility expression with a factor we call "three star," i.e.,

\[\tag{77}\frac{3}{4\pi^2}\Rightarrow\frac{3^*}{4\pi^2}\]

where the numerical value of this \(3^*\) factor (to be explained in more detail in the following tutorial) is \(0\le3^*\le3\).

**8. Degeneracy effects.** What appears to be a single quantum energy level \(E_i\) may be in many real atomic systems some number \(g_i\) of degenerate energy levels, i.e., separate and quantum-mechanically distinct energy states all with the same or very nearly the same energy eigenvalue \(E_i\). To express the net small-signal response summed over all the distinct but overlapping transitions between these degenerate sublevels, the population-difference term \(N_i-N_j\) for systems with degeneracy must be replaced by

\[\tag{78}\Delta{N}_{ij}=(N_i-N_j)\Rightarrow\Delta{N}_{ij}=(g_j/g_i)N_i-N_j\]

where \(E_i\) is the lower and \(E_j\) the upper group of levels; \(g_i\) and \(g_j\) are the statistical weights or degeneracy factors of these lower and upper groups of levels; and \(N_i\) and \(N_j\) are the total population densities in the degenerate groups of lower and upper levels.

**9. Inhomogeneous broadening.** Finally, additional line-broadening and line-shifting mechanisms, the so-called "inhomogeneous" broadening mechanisms, will often broaden and change the lineshapes of real atomic resonances, over and above the broadening due to energy decay and dephasing as expressed in the linewidth formula \(\Delta\omega_a=\gamma+2/T_2\). The homogeneous linewidth \(\Delta\omega_a\) then gets replaced (at least for certain purposes) by an inhomogeneous linewidth \(\Delta\omega_d\), i.e.,

\[\tag{79}\Delta\omega_a\Rightarrow\Delta\omega_d\]

When this happens, the lineshape often gets changed also, from lorentzian to something more like gaussian in shape; and the \(3^*/4\pi^2\) numerical factor in front of the susceptibility expression may be increased by ≈ 50%. What is meant by inhomogeneous broadening, and how these additional broadening mechanisms affect real atomic resonances, is described in the final section of the following tutorial.

With these conversion factors included, the basic polarization equation of motion and the resulting linear susceptibility formula for a real homogeneously broadened atomic transition become quantum-mechanically and quantitatively correct for real quantum atomic transitions.

The next tutorial introduces ** reflecting and catadioptric afocal lenses**.