# Laser Amplification Explained in Detail

This is a continuation from the previous tutorial - ** introduction to photosensitive fibers**.

In this tutorial we examine the other side of the laser problem—that is, what laser atoms do to applied signals, rather than what applied signals do to atoms. This tutorial is concerned primarily with continuous-wave or "cw" laser amplification: how inverted atomic transitions amplify optical signals; what determines the magnitude and bandwidth of this gain; how it saturates; and what phase shifts are associated with it.

In later tutorials we will consider pulse propagation and pulsed laser amplification and we will add the laser mirrors to these amplifying atoms, and be able to discuss laser oscillation and the generation of coherent laser radiation.

## 1. Practical Aspects of Laser Amplifiers

Let us begin with a few words about the practical interest in lasers as optical amplifiers, rather than as oscillators. Single-pass (and sometimes double-pass) laser amplifiers are used in many practical situations, primarily as power amplifiers, and seldom if ever as weak-signal preamplifiers. The reasons for this are generally the following.

### Laser Power Amplifiers

Large laser devices very often face severe stability problems associated with large electrical power inputs, optical damage problems, mechanical vibrations, cooling and heat-dissipation problems, acoustic noise, etc.

One common way to obtain high laser power output, simultaneously with good beam quality, short pulse length, excellent frequency stability, and good beam control, is to generate a stable input laser signal from a small but well-controlled laser oscillator. This signal can then be amplified through a chain of laser amplifiers, in what is commonly known as a master-oscillator-power-amplifier or MOPA system.

Figure 7.1 shows, as one rather extreme example, the sequence of parallel cascaded Nd:glass laser amplifiers used in a giant laser fusion system, in which a four-story-high "space frame" supports some twenty parallel Nd:glass laser amplifier chains.

Very large high-power CO_{2} laser amplifiers are also used in laser fusion experiments; and much smaller but very high-gain pulsed dye laser amplifiers and pulsed solid-state laser amplifiers are used to amplify tunable dye laser pulses or mode-locked solid-state laser pulses in many laboratory experiments.

The output signals in these devices will probably be much more stable than if the same laser amplifiers were converted into a single very high power laser oscillator. The primary defect in the MOPA approach, as we will see later, is that it is generally much less effective in extracting the available power in the large amplifier devices than if the large amplifiers were themselves converted into powerful but hard-to-control large oscillators.

### Lasers as Weak Signal Amplifiers

Laser amplifiers are in fact almost always used as power amplifiers, especially for pulsed input signals, almost never as preamplifiers to amplify weak signals in optical receivers, for reasons related primarily to noise figure, and secondarily to the narrow bandwidth of most laser amplifiers.

There exists, as a result of fundamental quantum considerations, an unavoidable quantum noise level in any type of coherent or linear amplifier, at any frequency, whether it be laser, maser, transistor, or vacuum tube. (By a coherent amplifier we mean one which preserves all the phase and amplitude information in the amplified signal; this includes any kind of linear coherent heterodyne detection.) This quantum noise level is roughly equivalent to one noise photon per second per unit bandwidth at the input to the amplifier.

The physical source of this quantum noise in a laser amplifier is the unavoidable spontaneous emission in the laser system, from the upper energy level of the laser transition, into the signal to be amplified.

There is, however, some equivalent source of spontaneous-emission-like noise in every other linear amplification mechanism, no matter what its physical nature. (If there were not, it would be possible to use such an amplifier to make physical measurements that would violate the quantum uncertainty principle.)

This "quantum noise" source is normally negligible at ordinary radio or microwave frequencies, but becomes much more significant at optical frequencies. As a result, any coherent optical amplifier, including a laser amplifier, is generally unsuitable for detecting very weak optical signals.

An incoherent detection mechanism, such as a photomultiplier tube, can detect much weaker optical signals, though of course at the cost of losing all phase information contained in the signal.

As we will see in this and later tutorials, laser amplifiers also generally have a quite narrow bandwidth, especially if any regenerative feedback is added to increase the laser gain.

As a consequence of both noise and bandwidth considerations, therefore, laser amplifiers are not used to any significant extent in optical communications receivers or other weak-signal-detection applications.

## 2. Wave Propagation in an Atomic Medium

Our first formal step toward understanding laser amplification will be to analyze the propagation of an ideal plane electromagnetic wave through an atomic medium which may contain laser gain or loss, as well as atomic phase-shift terms and possibly ohmic losses or scattering losses.

### The Wave Equation in a Laser Medium

We begin with Maxwell's equations for a sinusoidal electromagnetic field at frequency \(\omega\). The two basic Maxwell equations are

\[\tag{1}\nabla\times\pmb{E}=-j\omega\pmb{B},\qquad\nabla\times\pmb{H}=\pmb{J}+j\omega\pmb{D}\]

where the real vector field \(\boldsymbol{\mathcal{E}}(\pmb{r},t)\) as a function of space and time is given by

\[\tag{2}\boldsymbol{\mathcal{E}}(\pmb{r},t)=\frac{1}{2}\left[\pmb{E}(\pmb{r})e^{j\omega{t}}+c.c.\right]\]

and similarly for all the other quantities. If these fields are in a linear dielectric host medium which has dielectric constant \(\epsilon\), and which possibly contains laser atoms as well as ohmic losses, these quantities will also be connected by "constitutive relations" which may be written as

\[\tag{3}\pmb{B}=\mu\pmb{H},\quad\pmb{J}=\sigma\pmb{E},\quad\text{and}\quad\pmb{D}=\epsilon\pmb{E}+\pmb{P}_\text{at}=\epsilon[1+\boldsymbol{\chi}_\text{at}]\pmb{E}\]

where \(\pmb{P}_\text{at}\) and \(\boldsymbol{\chi}_\text{at}\) represent the contribution of the laser atoms imbedded in the host dielectric medium. The material parameters appearing in these equations include:

- The optical-frequency dielectric permeability \(\epsilon\) of the host medium, not counting any atomic transitions due to laser atoms that may be present.
- The magnetic permeability \(\mu\) of the host medium (which will be very close to the free-space value \(\mu_0\) for essentially all common laser materials at optical frequencies).
- The conductivity \(\sigma\), which is included to account for any ohmic losses in the host material.
- The resonant susceptibility \(\boldsymbol{\chi}_\text{at}(\omega)\) associated with the transitions in any laser atoms that may be present, where this \(\boldsymbol{\chi}_\text{at}(\omega)\) is defined in the slightly unconventional fashion we introduced in earlier tutorials.

We will assume here that the atomic transition in these atoms is an electric-dipole transition, and thus contributes an electric polarization \(\pmb{P}_\text{at}\) in the medium.

A magnetic-dipole atomic transition would contribute instead an atomic magnetic polarization \(\pmb{M}_\text{at}\) and thus a magnetic susceptibility \(\boldsymbol{\chi}_m\) in the \(\pmb{B}=\mu\pmb{H}\) expression. The net result in the following expressions would be essentially the same, however.

Substituting Equations 7.1-7.3 into a vector identity for \(\nabla\times\nabla\times\pmb{E}\), and then assuming that \(\nabla\cdot\pmb{E}=0\), gives

\[\tag{4}\begin{align}\nabla\times\nabla\times\pmb{E}&\equiv\nabla(\nabla\cdot\pmb{E})-\nabla^2\pmb{E}\\&=-j\omega\mu\nabla\times\pmb{H}\\&=-j\omega\mu[\sigma+j\omega\epsilon(1+\boldsymbol{\chi}_\text{at})]\pmb{E}\\&=\omega^2\mu\epsilon[1+\boldsymbol{\chi}_\text{at}-j\sigma/\omega\epsilon]\pmb{E}\end{align}\]

We can assume that \(\nabla\cdot\pmb{E}=0\) provided only that the properties of the medium are spatially uniform; and for simplicity we can drop the tensor or vector notation for \(\boldsymbol{\chi}\) and \(\pmb{E}\). This vector equation then reduces to the scalar wave equation

\[\tag{5}[\nabla^2+\omega^2\mu\epsilon(1+\tilde{\chi}_\text{at}-j\sigma/\omega\epsilon)]\tilde{E}(x,y,z)=0\]

where \(\tilde{E}(x,y,z)\) is the phasor amplitude of any one of the vector components of \(\pmb{E}\).

This equation will be the fundamental starting point for the analyses in this tutorial. We can immediately note as one important point that the atomic susceptibility term \(\tilde{\chi}_\text{at}\equiv\chi'+j\chi^"\) and the ohmic loss term \(-j\sigma/\omega\epsilon\) appear in exactly similar fashion in this expression.

### Plane-Wave Approximation

Let us consider now either an infinite plane wave propagating in the \(z\) direction, so that the transverse derivatives are identically zero, i.e., \(\partial/\partial{x}=\partial/\partial{y}=0\), or a finite-width laser beam traveling in the same direction (see Figure 7.2).

For a laser beam of any reasonable transverse width (more than a few tens of wavelengths) the transverse derivations will be small enough, and the transverse second derivatives even smaller, so that we can make the approximation

\[\tag{6}\left|\frac{\partial^2\tilde{E}}{\partial{x^2}}\right|,\left|\frac{\partial^2\tilde{E}}{\partial{y^2}}\right|\ll\left|\frac{\partial^2\tilde{E}}{\partial{z^2}}\right|\]

(we will justify this in more detail in the following section).

With this approximation, Equation 7.5 will reduce to the one-dimensional scalar wave equation

\[\tag{7}\left[\frac{d^2}{dz^2}+\omega^2\mu\epsilon(1+\tilde{\chi}_\text{at}-j\sigma/\omega\epsilon)\right]\tilde{E}(z)=0\]

We will also for simplicity drop the subscripts on the atomic susceptibility \(\tilde{\chi}_\text{at}\) from here on.

### Lossless Free-Space Propagation

Let us now consider the traveling-wave solutions to this equation, first of all without any ohmic losses or laser atoms. We will generally refer to this as "free space" propagation, although we are in fact including the electric and magnetic permeabilities \(\epsilon\) and \(\mu\) of the host dielectric medium if there is one.

For \(\tilde{\chi}_\text{at}=\sigma=0\) the one-dimensional wave equation reduces to

\[\tag{8}\left[\frac{d^2}{dz^2}+\omega^2\mu\epsilon\right]\tilde{E}(z)=0\]

If we assume traveling-wave solutions to this equation of the form

\[\tag{9}\tilde{E}(z)=\text{const}\times{e}^{-\Gamma{z}}\]

where \(\Gamma\) is a complex-valued propagation constant, then the wave equation reduces to

\[\tag{10}[\Gamma^2+\omega^2\mu\epsilon]\tilde{E}=0\]

The allowed values for the complex propagation factor \(\Gamma\) are thus given by

\[\tag{11}\Gamma^2=-\omega^2\mu\epsilon\qquad\text{or}\qquad\Gamma=\pm{j}\omega\sqrt{\mu\epsilon}=\pm{j}\beta\]

where the quantity \(\beta\equiv\omega\sqrt{\mu\epsilon}\) is the plane-wave propagation constant in the host medium.

The complete solution for the \(\mathcal{E}\) field in the medium may thus be written as

\[\tag{12}\begin{align}\mathcal{E}(z,t)&=\frac{1}{2}\left[\tilde{E}_+e^{j(\omega{t}-\beta{z})}+\tilde{E}_+^*e^{-j(\omega{t}-\beta{z})}\right]\\&\qquad+\frac{1}{2}\left[\tilde{E}_-e^{j(\omega{t}+\beta{z})}+\tilde{E}_-^*e^{-j(\omega{t}+\beta{z})}\right]\end{align}\]

In this expansion the first line on the right-hand side represents a wave traveling to the right (i.e., in the \(+z\) direction) with a complex phasor amplitude \(\tilde{E}_+\), and the second line represents a wave traveling to the left with phasor amplitude \(\tilde{E}_-\). The reader should be sure that the distinction between these two waves is clear and well-understood.

The "free-space" propagation constant \(\beta\) for these waves may then be written in any of the various alternative forms:

\[\tag{13}\begin{align}\beta&=\omega\sqrt{\mu\epsilon}=\frac{\omega}{c}=\frac{n\omega}{c_0}\\&=\frac{2\pi}{\lambda}=\frac{2\pi{n}}{\lambda_0}\end{align}\]

where the refractive index \(n\) of the host crystal is given by

\[\tag{14}n\equiv\sqrt{\mu\epsilon/\mu_0\epsilon_0}\approx\sqrt{\epsilon/\epsilon_0}\quad\text{if}\quad\mu\approx\mu_0\]

Note again that in the notation used in this tutorial, \(c_0\) and \(\lambda_0\) are the velocity of light and the wavelength of the radiation in vacuum, whereas \(c\equiv{c_0}/n\) and \(\lambda\equiv\lambda_0/n\) always indicate the corresponding values in the dielectric medium.

When we identify particular laser transitions we normally give the value of the wavelength in air, e.g., \(\lambda_0=1.064\) μm for the Nd:YAG laser.

(Note also that in very precise calculations there will even be a slight difference, typically on the order of ~ 0.03%, between the exact vacuum wavelength of a transition and the commonly measured value of the wavelength in air.)

### Propagation with Laser Action and Loss

Let us now include laser action (i.e., an atomic transition) and also ohmic losses in the wave propagation calculation. The one-dimensional wave equation then becomes

\[\tag{15}\left[\frac{d^2}{dz^2}+\beta^2(1+\tilde{\chi}_\text{at}-j\sigma/\omega\epsilon)\right]\tilde{E}(z)=0\]

If we assume a \(z\)-directed propagation factor \(\Gamma\) in the same form as before, this propagation factor now becomes

\[\tag{16}\Gamma^2=-\omega^2\mu\epsilon[1+\tilde{\chi}_\text{at}-j\sigma/\omega\epsilon]=-\beta^2[1+\tilde{\chi}_\text{at}-j\sigma/\omega\epsilon]\]

or

\[\tag{17}\Gamma=j\beta\sqrt{1+\tilde{\chi}_\text{at}-j\sigma/\omega\epsilon}=j\beta\sqrt{1+\chi'(\omega)+j\chi^"(\omega)-j\sigma/\omega\epsilon}\]

We include the specific dependence of \(\chi'(\omega)\) and \(\chi^"(\omega)\) on frequency to emphasize that, at least for atomic transitions, this quantity will normally be complex and will have a resonant lineshape, with frequency-dependent real and imaginary parts.

Under almost all practical conditions, both the susceptibility \(\tilde{\chi}_\text{at}(\omega)\) and the loss factor \(\sigma/\omega\epsilon\) will have magnitudes that are \(\ll1\). Hence the square root in Equation 7.17 can, with negligible error, be expanded in the form \(\sqrt{1+\delta}\approx1+\delta/2\) to give

\[\tag{18}\Gamma\approx{j}\beta\times\left[1+\frac{1}{2}\chi'(\omega)+j\frac{1}{2}\chi^"(\omega)-j\sigma/2\omega\epsilon\right]\]

From here on we will separate this into the four individual terms

\[\tag{19}\begin{align}\Gamma(\omega)&=j\beta+j\beta\chi'(\omega)/2-\beta\chi^"(\omega)/2+\sigma/2\epsilon{c}\\&=j\beta+j\Delta\beta_m(\omega)-\alpha_m(\omega)+\alpha_0\end{align}\]

where each of the factors on the first line matches up with the corresponding factor on the second line. The propagation of a \(+z\) traveling wave thus takes on the form

\[\tag{20}\mathcal{E}(z,t)=\text{Re}\tilde{E}_0\exp\{j\omega{t}-j[\beta+\Delta\beta_m(\omega)]z+[\alpha_m(\omega)-\alpha_0]z\}\]

when the effects of ohmic losses and an atomic transition are included.

### Propagation Factors

The significant factors in this complex wave propagation behavior are the following.

- The basic plane wave propagation constant. This is the basic wave propagation coefficient \(\beta\) in the host medium, which is given by \[\tag{21}\beta=\beta(\omega)=\omega\sqrt{\mu\epsilon}=\omega/c\] This propagation constant leads to a fundamental phase variation \(\phi(z,\omega)\equiv\beta{z}=\omega{z/c}=2\pi{z/\lambda}\). This phase shift with distance is large (many complete cycles) for any propagation length \(z\gg\lambda\), and increases linearly (and rapidly) with frequency as shown by the dashed straight line in Figure 7.3.
- The additional atomic phase shift. There is an added phase-shift factor \(\Delta\phi(z,\omega)\equiv\Delta\beta_m(\omega)z\) due to the atomic transition, where \(\Delta\beta_m(\omega)\) is given by \[\tag{22}\Delta\beta_m=\Delta\beta_m(\omega)=(\beta/2)\chi'(\omega)\] This phase shift is caused by and has essentially the same lineshape as the reactive part of the atomic susceptibility, \(\chi'(\omega)\), as illustrated by the additional asymmetric contribution to the total phase shift in Figure 7.3. Note that the sign of this term depends on the sign of the population difference \(\Delta{N}\), just as does the atomic gain or loss coefficient \(\alpha_m\). We have drawn the phase shift in Figure 7.3 assuming an inverted or amplifying population difference.
- The atomic gain or loss coefficient. There is an atomic gain (or loss) coefficient \(\alpha_m(\omega)\) due to the atomic transition, given by \[\tag{23}\alpha_m=\alpha_m(\omega)=(\beta/2)\chi^"(\omega)\] This gain (or loss) has the lineshape of \(\chi^"(\omega)\) as illustrated in the bottom curve of Figure 7.3. We noted in an earlier tutorial that if we followed the definition \(\tilde{\chi}_\text{at}\equiv\chi'+j\chi^"\), an absorbing transition produced a negative value of \(\chi^"\). We see here also that an amplifying transition will imply positive values for both \(\chi^"\) and \(\alpha_m\), but an absorbing atomic transition will imply negative values for both these quantities. Of course, we can always associate a suitable \(\pm\) sign with \(\alpha_m(\omega)\) to give it the proper sign for either absorbing or amplifying media.
- The ohmic or background loss coefficient. Finally, there is an ohmic or background loss coefficient \(\alpha_0\) due to the host medium itself. For pure ohmic conductivity in the host medium, this loss term is given by \[\tag{24}\alpha_0=\frac{\beta}{2}\frac{\sigma}{\omega\epsilon}=\frac{\sigma}{2\epsilon{c}}\] We will extend the interpretation of the coefficient \(\alpha_0\) in later equations, however, to represent any kind of broadband, background absorption or loss that may be present for the signal in the laser medium, whether this loss is due to ohmic conductivity in the host crystal, or to other loss mechanisms such as scattering or diffraction losses. This loss usually has no significant variation with frequency across the range of interest for a single laser transition.

The preceding four expressions summarize the laser amplification or atomic absorption properties, as well as the phase-shift properties, of any real atomic medium.

Recall that in cases of interest to us \(\tilde{\chi}_\text{at}(\omega)\) is virtually always caused by a very narrow resonant transition, with bandwidth \(\ll1\%\). Hence the linear frequency dependence of \(\beta(\omega)\) across the narrow linewidth of \(\tilde{\chi}_\text{at}(\omega)\) can be neglected in the \((\beta/2)\chi'(\omega)\) and \((\beta/2)\chi^"(\omega)\) products, and only the midband value of \(\beta\) need be used. Each of these terms will show up in more detail in later sections.

### Experimental Example

A set of measurements of absorption and phase shift made by Bean and Izatt on the 694 nm laser transition in ruby, without pumping or laser inversion, will give a particularly clean and striking experimental confirmation of the results we have just derived.

Let us recall that the laser transition in the ruby energy-level system terminates on the ground level, so that this transition will have a strongly absorptive population difference in the absence of any laser pumping. We have also noted earlier that the \(^4A_2\) ground state of the Cr\(^{3+}\) ion in ruby is actually two energy levels which are split, even in zero magnetic field, into two closely spaced sublevels separated by \(\Delta{E}=0.38\text{ cm}^{-1}=11.4\text{ GHz}\), as illustrated in Figure 7.4. (Each of these two sublevels is in fact also a doublet, which can be further split into two Zeeman levels using a dc magnetic field of a few hundred to a few thousand gauss.)

At liquid-nitrogen temperature the phonon broadening in a good sample of ruby becomes small enough (\(\Delta\omega_a\le2\pi\times6\) GHz) that the separate absorption lines from the two ground levels can be clearly resolved in the optical absorption spectrum of ruby. Bean and Izatt have in fact made careful measurements of the transitions from this split ground state to the first excited or \(R_1\) level in a ruby sample, measuring both the absorption coefficient \(\alpha_m(\omega)\) versus frequency, which is directly proportional to \(\chi^"(\omega)\), and the change in index of refraction \(\Delta{n}(\omega)\) relative to the background index \(n_0\), which is directly proportional to \(\chi'(\omega)\). Typical results of their experiments are shown in Figure 7.5.

These independent measurements of \(\chi^"(\omega)\) and \(\chi'(\omega)\) can then be fitted very closely by simply summing two partially overlapping complex lorentzian lineshapes, as illustrated both in Figure 7.5 and in the double-lorentzian curves in Figure 7.6. These latter curves represent the sum of two elementary lorentzian lines with a relative peak amplitude of 1.29 to 1, a resonance frequency spacing \(\omega_{a2}-\omega_{a1}=2\pi\times11.5\) GHz, and equal linewidths \(\Delta\omega_a=2\pi\times5.88\) Ghz.

The close agreement between theory and experiment that is obtained here demonstrates both the validity of the lorentzian lineshape analysis and the close relationship between the \(\chi'(\omega)\) and \(\chi^"(\omega)\) parts of the atomic response.

### Larger Atomic Gain or Absorption Effects

The analytical results in this section (and indeed in most of the rest of this tutorial series) are based on the approximation that \(|\tilde{\chi}_\text{at}-j\sigma/\omega\epsilon|\ll1\). There are in fact only a few optical situations where this approximation is not valid, and where the related Taylor approximation for the complex propagation constant \(\Gamma\) will no longer be valid.

These include:

- Absorption in metals and semiconductors. For propagation into a semiconductor or a metal (or reflection from their surfaces) at wavelengths shorter than the band edge, or frequencies \(\hbar\omega\) greater than the bandgap energy \(E_g\), the effective conductivity \(\sigma\) and the \(-j\sigma/\omega\epsilon\) term can become very large. Exact expressions for both the propagation factor \(\Gamma\) and the wave impedance must then be employed to calculate absorption coefficients and phase shifts (as well as surface reflectivities).
- Absorption on strong resonance lines in metal vapors. Another and more interesting situation where the atomic susceptibility term \(\tilde{\chi}\) can become quite large compared to unity is ground-state absorption on the very strong visible or near-UV resonance lines of alkali metal vapors, such as sodium or rubidium, or other metal vapors such as Hg or Cd, at vapor pressures of a few torr or even lower. One particularly common example of this is the pair of sodium D lines at 589.0 and 589.6 nm in the green portion of the visible spectrum. The special features in these situations are that the transitions are very strongly allowed (with oscillator strengths approaching unity); they are relatively narrow, being broadened by doppler broadening only; and they are all ground-state absorption lines, so that all the atoms are in the lower level of the absorbing transition.

As a result, the absorption per unit length at line center on one of these transitions can be extremely large. For example, at the inside surface of a window in a cell containing a moderate vapor pressure of Na or Rb or Hg, the vapor will be so highly absorbing that it will appear essentially metallic and very highly reflecting. This will hold true, however, only within the very narrow range of frequencies within the atomic linewidth (typically a few GHz).

Interesting experiments on optical propagation and atomic transition phenomena can often be done in such vapors, using tunable dye lasers to tune at or very close to these transitions. The practical applications of these phenomena are somewhat limited, however, by the narrow bandwidths, and also by the voracious appetite of the alkali metal vapors for consuming and destroying almost any conveniently available transparent window materials (not to mention seals and even the metal walls of the vapor cells).

## 3. The Paraxial Wave Equation

The next step in accuracy beyond the plane-wave approximation of Section 2 is the paraxial wave equation. This equation, which we will derive in this section, in fact leads to exactly the same results for axial propagation as in Section 2, but also makes it possible to handle transverse variations and diffraction effects of the optical beam profile.

The paraxial wave equation is, in fact, complete enough to describe essentially all laser amplification and laser propagation problems of practical interest in lasers; so it is used in a wide variety of laser and nonlinear optical calculations. It seems worthwhile therefore to derive the paraxial equation at this point, even though we will not need to use it until later in this tutorial series.

### Paraxial Wave Derivation

The full vector form of the wave equation from Section 2 is

\[\tag{25}[\nabla^2+\beta^2(1+\tilde{\chi}-j\sigma/\omega\epsilon)]\pmb{E}(x,y,z)=0\]

where \(\beta\) is the plane-wave propagation constant in the host medium, disregarding losses and/or atomic transitions. Suppose we now write any given vector component of this complex \(\pmb{E}\) vector in the form

\[\tag{26}\tilde{E}(x,y,z)\equiv\tilde{u}(x,y,z)e^{-j\beta{z}}\]

This says that the field \(\tilde{E}(x,y,z)\) is basically a traveling wave of the form \(\exp(-j\beta{z})\) in the \(+z\) direction. (We would, of course, write this as \(e^{+j\beta{z}}\) if the wave were traveling instead in the \(-z\) direction; so reversing the wave direction is the same thing as reversing the sign of \(\beta\) in all the following equations.)

This traveling wave may, however, have a transverse amplitude and phase variation, i.e., a dependence on \(x\) and \(y\) as contained in \(\tilde{u}(x, y, z)\); and this transverse profile \(\tilde{u}(x, y, z)\) will in general change slowly with propagation distance \(z\) as the wave grows, spreads, and/or changes in shape because of absorption and/or diffraction effects, as illustrated for a typical case in Figure 7.2. The very rapid phase variation \(\exp(-j\beta{z})=\exp(-j2\pi{z}/\lambda)\) due to the traveling-wave part of the propagation has, however, been factored out of \(\tilde{u}(x,y,z)\).

Putting the above form into the wave equation then yields

\[\tag{27}\nabla^2\tilde{E}=\left[\frac{\partial^2\tilde{u}}{\partial{x^2}}+\frac{\partial^2\tilde{u}}{\partial{y^2}}+\frac{\partial^2\tilde{u}}{\partial{z^2}}-2j\beta\frac{\partial\tilde{u}}{\partial{z}}-\beta^2\tilde{u}\right]e^{-j\beta{z}}\]

Now, we know in advance (or at least we can verify shortly) that the transverse beam profile \(\tilde{u}(x,y,z)\) for any reasonably well-collimated optical beam will change only rather slowly with distance \(z\) along the beam.

That is, the effects of both diffraction and atomic gain or loss on the beam profile \(\tilde{u}(x, y, z)\) will be fairly slow, at least compared with the variation of one complete cycle in phase that occurs in one optical wavelength \(\lambda\) because of the \(\exp(-j2\pi{z}/\lambda)\) term. Hence we will make the paraxial approximation that the \(z\) dependence of \(\tilde{u}(x, y, z)\) is particularly slow, especially in its second derivative, so that

\[\tag{28}\left|\frac{\partial^2\tilde{u}}{\partial{z^2}}\right|\ll\left|2\beta\frac{\partial\tilde{u}}{\partial{z}}\right|\equiv\frac{4\pi}{\lambda}\left|\frac{\partial\tilde{u}}{\partial{z}}\right|\]

and also that

\[\tag{29}\left|\frac{\partial^2\tilde{u}}{\partial{z^2}}\right|\ll\left|\frac{\partial^2\tilde{u}}{\partial{x^2}}\right|,\quad\left|\frac{\partial^2\tilde{u}}{\partial{y^2}}\right|\]

We will show shortly that these approximations can in fact be very well justified for beams of interest in lasers.

Making these approximations then allows us to drop the \(\partial^2\tilde{u}/\partial{z^2}\) term in the preceding equation, and thus reduce the wave equation to the so-called paraxial form

\[\tag{30}\nabla^2_t\tilde{u}-2j\beta\frac{\partial\tilde{u}}{\partial{z}}+\beta^2(\tilde{\chi}_\text{at}-j\sigma/\omega\epsilon)\tilde{u}=0\]

where the laplacian operator in the transverse plane is denoted by

\[\tag{31}\nabla^2_t\equiv\frac{\partial^2}{\partial{x^2}}+\frac{\partial^2}{\partial{y^2}}\]

This paraxial form is the desired and widely used ** paraxial wave equation**.

### Diffraction Effects Versus Propagation Effects

The paraxial wave equation may also be turned around into the equivalent form

\[\tag{32}\frac{\partial\tilde{u}(x,y,z)}{\partial{z}}=-\frac{j}{2\beta}\nabla^2_t\tilde{u}(x,y,z)-[\alpha_0-\alpha_m+j\Delta\beta_m]\tilde{u}(x,y,z)\]

where the loss term \(\alpha_0\) and the atomic susceptibility terms \(\alpha_m(\omega)\) and \(j\Delta\beta_m(\omega)\) are defined exactly as in Section 2.

Equation 7.32 neatly separates the axial rate of change of the complex wave amplitude \(\tilde{u}(x,y,z)\) into two terms: the \(\Delta^2_t\tilde{u}\) term, which represents ** diffraction effects**; and the \(\alpha_0\), \(\alpha_m\) and \(j\Delta\beta_m\) terms, which represent

**,**

*ohmic and atomic gain***,**

*loss***caused by \(\sigma\) and \(\tilde{\chi}_\text{at}\).**

*and phase shift effects*Since these diffraction and gain or phase-shift effects appear in the differential equation as separate and independent terms for the \(z\) variation of the beam profile, we can conclude that the atomic gain and phase-shift effects for a finite laser beam are to first order unaffected by diffraction effects, and are the same as for an infinite plane wave; and also that the diffraction effects on such a beam are to first order unaffected by ** spatially uniform** atomic gain or phase-shift effects.

Note also that the paraxial results for the gain and phase shift \(\alpha_m\) and \(\Delta\beta_m\) are exactly the same as the plane-wave results derived by expanding the square-root function for \(\Gamma\) to first order in \(\tilde{\chi}_\text{at}\) and in \(\sigma/\omega\epsilon\). The paraxial approximation invokes essentially the same physical approximation concerning \(z\)-axis propagation as does the Taylor expansion of the square root in Equation 7.18.

### Validity of the Paraxial Approximation

A simple analytical example may give somewhat more insight into the validity of the paraxial approximation. Many real laser beams have a gaussian transverse profile of the form

\[\tag{33}|\tilde{u}(x)|=\tilde{u}_0\exp\left(-\frac{x^2}{w^2(z)}\right)\]

where the gaussian spot size \(w=w(z)\) is a slowly varying function of axial distance \(z\). (The complex field will have some similar transverse phase variation or phase curvature as well, but for simplicity let us leave this out of the following discussion.) The transverse derivatives of this beam profile at any fixed plane \(z\) are then given by

\[\tag{34}\frac{1}{\tilde{u}}\frac{\partial\tilde{u}}{\partial{x}}=\frac{2x}{w^2}\quad\text{and}\quad\frac{1}{\tilde{u}}\frac{\partial^2\tilde{u}}{\partial{x^2}}=\left(\frac{2}{w^2}-\frac{4x^2}{w^4}\right)\approx\frac{2}{w^2}\]

where the final approximation is valid both on the optic axis and over most of the main part of the gaussian beam profile.

Suppose now that the gaussian spot size \(w\) equals 1 mm (which is a fairly slender beam), at a visible wavelength of \(\lambda\) = 500 nm. The term that represents diffraction effects in the paraxial wave equation will then have an approximate numerical magnitude

\[\tag{35}\left|\frac{1}{\tilde{u}}\frac{\partial\tilde{u}}{\partial{z}}\right|\approx-j\left|\frac{1}{\beta\tilde{u}}\frac{\partial^2\tilde{u}}{\partial{x^2}}\right|\approx\frac{\lambda}{\pi{w^2}}\approx10^{-1}\text{ m}^{-1}\]

In other words, this small but rather smooth beam will propagate about 10 meters or so before diffraction effects cause any major change in the beam profile \(\tilde{u}(x,y,z)\).

Suppose also that the amplitude gain or loss in the axial direction due to the \(\alpha_m\) or \(\alpha_0\) terms has an \(e\)-folding length somewhere between 10 cm and 1 m (which implies a rather large power gain or attenuation, of between 10 and 100 dB/meter). The gain term in the paraxial equation then has a magnitude

\[\tag{36}\left|\frac{1}{\tilde{u}}\frac{\partial\tilde{u}}{\partial{z}}\right|\approx\alpha_m\approx1\text{ to }10\text{ m}^{-1}\]

In this example at least, diffraction spreading occurs somewhat more slowly than amplification.

The normalized first axial derivative \((1/\tilde{u})(\partial\tilde{u}/\partial{z})\) that results from either gain or diffraction effects thus occurs at a rate somewhere between \(10^{-1}\) and \(10^1\text{ m}^{-1}\). The second derivative \((1/\tilde{u})(\partial^2\tilde{u}/\partial{z^2})\) in the axial direction will then have a magnitude corresponding to (at most) this rate squared, say,

\[\tag{37}\left|\frac{1}{\tilde{u}}\frac{\partial^2\tilde{u}}{\partial{z^2}}\right|\approx10^{-2}\text{ to }10^2\text{ m}^{-2}\]

Therefore the axial derivative contribution produced by this second derivative term, which we dropped in deriving the paraxial equation, if expressed in the same fashion as Equation 7.37, would be about

\[\tag{38}\left|\frac{1}{2\beta\tilde{u}}\frac{\partial^2\tilde{u}}{\partial{z^2}}\right|\approx\frac{\lambda}{4\pi}\left|\frac{1}{\tilde{u}}\frac{\partial^2\tilde{u}}{\partial{z^2}}\right|\approx5\times10^{-10}\text{ to }5\times10^{-6}\text{ m}^{-1}\]

The normalized second derivative given in Equation 7.38 is thus many orders of magnitude smaller than the other derivative terms given in Equations 7.35 through 7.37. The basic paraxial approximation is clearly very well-justified in this example, even for a wide range of different axial growth rates.

## 4. Single-Pass Laser Amplification

Let us look next at some of the practicalities of single-pass, small-signal amplification for a wave passing through an inverted laser medium

### Laser Gain Formulas

If a quasi-plane wave propagates through a length \(L\) of laser material, as in Figure 7.7, the complex amplitude gain or "voltage gain" in an inverted laser medium will be

\[\tag{39}\tilde{g}(\omega)\equiv\frac{\tilde{E}(L)}{\tilde{E}(0)}=\exp\{-j[\beta+\Delta\beta_m(\omega)]L\}\times\exp\{[\alpha_m(\omega)-\alpha_0]L\}\]

The first exponent on the right-hand side represents the total phase shift through the amplifier, and the second is the amplitude gain or loss.

Because signal power or intensity \(I(z)\) is proportional to \(|\tilde{E}(z)|^2\), the single-pass power or intensity gain going through the laser medium is

\[\tag{40}G(\omega)\equiv\frac{I(L)}{I(0)}=|\tilde{g}(\omega)|^2=\exp[2\alpha_m(\omega)L-2\alpha_0L]\]

(Note again that in this tutorial series symbols like \(\alpha_m\) and \(\alpha_0\) will always denote gain coefficients or loss coefficients for the field amplitude or "voltage" of a wave; and hence we will always write \(2\alpha\) for a power gain coefficient. In other books and papers in the literature, the symbol \(\alpha\) by itself often means the power gain or loss coefficient.)

In most useful laser materials the ohmic insertion loss coefficient \(\alpha_0\) will be small compared to the laser gain coefficient \(\alpha_m\); so for simplicity we will leave out the \(2\alpha_0L\) loss factor in most of the following equations. Also, for many transitions the laser lineshape will be lorentzian, so that the imaginary part of the susceptibility is given by

\[\tag{41}\chi^"(\omega)=\frac{\chi^"_0}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}\]

where \(\chi^"_0\) is the midband value. The power gain \(G(\omega)\) then has the frequency lineshape

\[\tag{42}G(\omega)=\exp\left[\frac{\omega{L}\chi^"_0}{c}\times\frac{1}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}\right]\]

where \(c\) is the velocity of light in the laser medium. Note that in this gain expression, the lorentzian atomic lineshape appears in the exponent. If the atomic lineshape were inhomogeneous and gaussian, then the gaussian lineshape would similarly appear in the exponent.

The quantity \(G(\omega)\) is power gain expressed as a number. To convert this to power gain in decibels, or dB, as often used in engineering discussions, we must use the definition that

\[\tag{43}G_\text{dB}(\omega)\equiv10\log_{10}G(\omega)=4.34\log_eG(\omega)=\frac{4.34\omega_aL}{c}\chi^"(\omega)\]

Therefore ** the power gain measured in dB has the same lineshape as the atomic susceptibility** \(\chi^"(\omega)\), whether this lineshape is lorentzian, gaussian, or whatever.

### Amplification Bandwidth and Gain Narrowing

Because the frequency dependence of \(\chi^"(\omega)\) appears in the exponent of the gain expression, the exponential gain falls off much more rapidly with detuning than the atomic lineshape itself.

The bandwidth of a single-pass laser amplifier is thus generally narrower than the atomic linewidth (see Figure 7.8); and this bandwidth narrowing increases (that is, the bandwidth decreases still further) with increasing amplifier gain.

The conventional definition for the bandwidth of any amplifier is the full distance between frequency points at which the amplifier power gain has fallen to half the peak value. This corresponds to "3 dB down" from the peak gain value in dB, if we recall that \(10\log_{10}0.5=-3.01\). For a lorentzian atomic line the amplifier 3 dB points are thus defined as those frequencies \(\omega\) for which

\[\tag{44}G_\text{dB}(\omega)=\frac{G_\text{dB}(\omega_a)}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}=G_\text{dB}(\omega_a)-3\]

or

\[\tag{45}(\omega-\omega_a)_\text{3dB}=\pm\frac{\Delta\omega_a}{2}\sqrt{\frac{3}{G_\text{dB}(\omega_a)-3}}\]

The full 3-dB amplifier bandwidth between these points is then twice this value or

\[\tag{46}\Delta\omega_\text{3dB}=\Delta\omega_a\sqrt{\frac{3}{G_\text{dB}(\omega_a)-3}}\]

Figure 7.9 plots this amplifier bandwidth, normalized to the atomic linewidth, as a function of the midband gain \(G_\text{dB}(\omega_a)\) for both lorentzian and gaussian atomic lineshapes.

The 3-dB amplification bandwidth is substantially smaller than the atomic linewidth, dropping to only 30% to 40% of the atomic linewidth at higher gains. This so-called gain narrowing at higher gains is significant in reducing the useful bandwidth of a high-gain laser amplifier.

### Amplifier Phase Shift

The total phase shift for a single pass through a laser amplifier can be written as \(\exp[-j(\beta+\Delta\beta_m)L]\equiv\exp[-j\phi_\text{tot}(\omega)]\), where the total phase shift \(\phi_\text{tot}(\omega)\) is given by

\[\tag{47}\phi_\text{tot}(\omega)\equiv\beta(\omega)L+\Delta\beta_m(\omega)L=\frac{\omega{L}}{c}+\frac{\beta{L}}{2}\chi'(\omega)\]

The first term gives the basic "free-space" phase shift \(\beta(\omega)L=\omega{L}/c=2\pi{L}/\lambda\) through the laser medium. This term is large and increases linearly with increasing frequency. The second term is then the small added shift \(\Delta\beta_m(\omega)L\) due to the atomic transition, as illustrated earlier in Figure 7.3.

Note that the magnitude of the added phase shift through a laser amplifier or absorber is directly proportional to the net gain or attenuation through the same atomic medium.

For a lorentzian atomic transition we can in fact relate the added phase shift in radians to the amplitude gain (or loss) factor \(\alpha_mL\) (the value of which is often said to be measured in units of ** nepers**) by the relation

\[\tag{48}\begin{align}\Delta\beta_m(\omega)L&=\left(2\frac{\omega-\omega_a}{\Delta\omega_a}\right)\times\alpha_m(\omega)L\\&=\frac{G_\text{dB}(\omega_a)}{20\log_{10}e}\times\frac{2(\omega-\omega_a)/\Delta\omega_a}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}\end{align}\]

In practical terms, the peak value of the added phase shift \(\Delta\beta_mL\) occurs at half a linewidth, or \(\pm\Delta\omega_a/2\), off line center on each side; and the added phase shift in radians at these peaks is related to the midband gain in dB by \((\Delta\beta_mL)_\text{max}=G_\text{dB}/40\log_{10}e\approx{G_\text{dB}}/17.4\).

### Absorbing Media

The results just discussed are for an amplifying laser medium. The same formulas and physical ideas apply equally well to an absorbing (uninverted) atomic transition, however, if we simply reverse the sign of \(\tilde{\chi}_\text{at}(\omega)\) and hence of both \(\alpha_m(\omega)\) and \(\Delta\beta_m(\omega)\).

Figure 7.10 plots, for example, the power transmission \(T(\omega)=\exp[-2\alpha_m(\omega)L]\) versus frequency through a material with a lorentzian absorbing atomic transition. [In this terminology the power transmission \(T(\omega)\) is the same as the power gain \(G(\omega)\), but with a magnitude less than unity, not greater than unity.]

Note that for very strong absorption the transmission curve "touches bottom" and then broadens with increased absorption strength. An absorbing transition thus has "absorber broadening" rather than the "gain narrowing" discussed earlier.

## 5. Stimulated-Transition Cross Sections

A very useful concept which we will next introduce for describing both stimulated transitions in absorbing atoms and laser amplification in laser media is the stimulated-transition cross section of a laser atom. If you have encountered the idea of a cross section previously in some other connection, you will find this concept straightforward; if not, you may have to pay close attention at first.

### Absorption and Emission Cross Sections

Suppose a small black (i.e., totally absorbing) particle with a capture area or "cross section" \(\sigma\) is illuminated by an optical wave having intensity or power per unit area \(I\equiv{P/A}\). The net power \(\Delta{P}_\text{abs}\) absorbed by this object from the wave will then be its capture area or cross section \(\sigma\) times the incident power per unit area in the wave, or

\[\tag{49}\Delta{P}_\text{abs}=\sigma\times(P/A)=\sigma{I}\]

(In this tutorial series we attempt to always use \(P\) to represent total power in watts, and \(I\) to represent intensity in watts per unit area. When given in conversation, values for laser beam intensities are almost universally expressed in dimensions of watts/cm\(^2\), although the correct mks unit for substitution into formulas is watts/m\(^2\).)

Consider next a thin slab of thickness \(\Delta{z}\) and transverse area \(A\), as in Figure 7.11, containing densities \(N_1\) and \(N_2\) of atoms in the lower and upper energy levels of some atomic transition.

Suppose we say that each lower-level atom has an effective area or cross section \(\sigma_{12}\) for power absorption from the wave, and similarly each upper-level atom has an effective cross section \(\sigma_{21}\) for "negative absorption" or emission back to the wave (since upper-level atoms must emit power rather than absorb it).

The total number of lower-level atoms in this slab will then be \(N_1A\Delta{z}\), and the total absorbing area that results from all the lower-level atoms will be the total number of atoms times the cross section per atom, or \(N_1\sigma_{12}A\Delta{z}\). (We assume the slab is thin enough and the atoms small enough that shadowing of any one atom by other atoms is negligible.) Similarly, the total effective "emitting area" that results from all the upper-level atoms will be \(N_2\sigma_{21}A\Delta{z}\).

The net power absorbed by the atoms in the slab from an incident wave carrying a total power \(P\) distributed over the area \(A\) will then be

\[\tag{50}\Delta{P}_\text{abs}=(N_1\sigma_{12}-N_2\sigma_{21})\times{P}\Delta{z}\]

Note that the area factors in the slab volume \(A\Delta{z}\) and in the power density \(P/A\) just cancel.

The quantities \(\sigma_{12}\) and \(\sigma_{21}\) that we have introduced here are the ** stimulated-transition cross sections** of the atoms on the \(1\rightarrow2\) transition, with \(\sigma_{12}\) being the

**and \(\sigma_{21}\) the**

*stimulated-absorption cross section***.**

*stimulated emission cross section*These cross sections, which have dimensions of area per atom, provide a very useful way of expressing the strength of an atomic transition, or the size of the atomic response to an applied signal.

### Cross Sections and Amplification Coefficients

The net growth or decay with distance caused by an atomic transition for a wave carrying power \(P\) or intensity \(I\) through an atomic medium can then be written as

\[\tag{51}\frac{dP}{dz}=-\lim_{\Delta{z}\rightarrow0}\left(\frac{\Delta{P}_\text{abs}}{\Delta{z}}\right)=-(N_1\sigma_{12}-N_2\sigma_{21})\times{P}\]

The relationship between upward and downward cross sections on a possibly degenerate transition is in fact given by \(g_1\sigma_{12}=g_2\sigma_{21}\), and so the preceding equation, if converted into units of intensity \(I(z)\) and population difference \(\Delta{N}\), can be written as

\[\tag{52}\frac{1}{I}\frac{dI}{dz}=-\Delta{N}_{12}\sigma_{21}=-[(g_2/g_1)N_1-N_2]\sigma_{21}\]

where \(\Delta{N}_{12}=(g_2/g_1)N_1-N_2\) as usual.

But the growth or decay rate for a wave passing through an absorbing or amplifying atomic medium may also be written as \(I(z)=I(z_0)\exp[-2\alpha_m(z-z_0)]\), which corresponds to the differential relation

\[\tag{53}\frac{1}{I}\frac{dI}{dz}=-2\alpha_m(\omega)\]

Hence we obtain the simple but very useful expression for the absorption (or amplification) coefficient \(\alpha_m\) on the \(1\rightarrow2\) transition in terms of only the population difference and cross section, namely,

\[\tag{54}2\alpha_m(\omega)=\Delta{N}_{12}\sigma_{21}(\omega)\]

To specify the loss or gain per unit length on an atomic transition, all we need to know is the atomic density and the transition cross section. Note that the absorption coefficient \(\alpha_m(\omega)\) and the cross section \(\sigma_{21}(\omega)\) must necessarily have the same dependence on the atomic lineshape; i.e., there is an atomic lineshape contained in \(\sigma_{21}(\omega)\), although usually only the numerical value at midband is stated as so many cm\(^2\).

In practice the gain coefficient \(2\alpha_m\) is commonly expressed in cm\(^{-1}\), the density \(N\) in atoms/cm\(^3\), and the cross section \(\sigma\) in cm\(^2\)/atom, which makes Equation 7.54 dimensionally consistent even without use of mks units. Both \(\Delta{N}\) and \(\alpha_m\) will, of course, change sign on an inverted laser transition.

### Formula for the Cross Section

One way to obtain a theoretical expression for the cross section \(\sigma\) of a transition is to combine Equation 7.54 with the gain formula 7.23 to obtain

\[\tag{55}\begin{align}2\alpha_m(\omega)&=\Delta{N}\sigma(\omega)=\frac{2\pi}{\lambda}\chi^"(\omega)\\&=\frac{3^*}{2\pi\lambda}\frac{\Delta{N}\lambda^3\gamma_\text{rad}}{\Delta\omega_a}\times\frac{1}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}\end{align}\]

The population difference \(\Delta{N}\) can then be canceled from both sides to give the midband result

\[\tag{56}\sigma(\omega_a)=\frac{3^*}{2\pi}\frac{\gamma_\text{rad}}{\Delta\omega_a}\lambda^2\]

The more general form of the cross-section expression for an arbitrary \(i\rightarrow{j}\) transition, including degeneracy, is

\[\tag{57}\sigma_{ji}(\omega_a)=\frac{g_i}{g_j}\sigma_{ij}(\omega_a)=\frac{3^*}{2\pi}\frac{\gamma_{\text{rad},ji}}{\Delta\omega_a}\lambda_{ij}^2\]

These expressions give the midband value of the stimulated-emission cross section for a lorentzian atomic transition. For a gaussian transition, we must replace \(\Delta\omega_a\) by \(\Delta\omega_d\) and put an additional numerical factor of \(\sqrt{\pi\ln2}\approx1.48\) in front.

Note again that the degeneracy factors appear in such a way that the expression \(N_i\sigma_{ij}-N_j\sigma_{ji}\) converts neatly into the form \(\Delta{N}_{ij}\sigma_{ji}\), where we use the degenerate form of the population difference \(\Delta{N}_{ij}=(g_i/g_j)N_i-N_j\) as defined earlier, and where \(\sigma_{ji}\) is the cross section in the downward direction.

The effective cross section \(\sigma_{21}(\omega)\) then decreases off line center with precisely the same lineshape as the absorption susceptibility \(\chi^"(\omega)\) or the stimulated-transition probability \(W_{21}(\omega)\). That is, we may have either the lorentzian expression

\[\tag{58}\sigma(\text{lorentzian})=\frac{3^*}{2\pi}\frac{\gamma_\text{rad}\lambda^2}{\Delta\omega_a}\frac{1}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}\]

or the gaussian expression

\[\tag{59}\sigma(\text{gaussian})=\sqrt{\pi\ln2}\frac{3^*}{2\pi}\frac{\gamma_\text{rad}\lambda^2}{\Delta\omega_d}\exp\left[-(4\ln2)\left(\frac{\omega-\omega_a}{\Delta\omega_d}\right)^2\right]\]

corresponding to the homogeneous or inhomogeneous limiting cases.

### Maximum Value of the Transition Cross Section

The stimulated-emission (or absorption) cross section provides a convenient and useful way to express the apparent "size" of an atom for interacting with an optical wave, as well as a convenient way to calculate the expected gain in laser systems.

Let us look first at the maximum possible value that any such cross section can have. The cross section will be maximal for a transition that has purely radiative lifetime broadening only, and no other line-broadening effects, so that \(\Delta\omega_a\equiv\gamma_\text{rad}\).

If the atoms all have their transition axes aligned and the incident fields are optimally polarized, so that \(3^*=3\), the cross section is then

\[\tag{60}\sigma_\text{max}=\frac{3\lambda^2}{2\pi}\approx\frac{\lambda^2}{2}\]

This says that the maximum cross section is roughly one wavelength square. For a visible transition this means

\[\tag{61}\sigma_\text{max}\approx0.5\times(5000\text{Å})^2\approx10^{-9}\text{ cm}^2\]

Now, the actual physical size of an atom, as measured, say, by the radius of its outermost Bohr orbit, is only a few Angstroms; yet its effective cross section for capturing radiation can be thousands of Angstroms in diameter. The physical explanation for this is essentially that the atom has an internal resonance which makes it act like a miniature dipole radio antenna, whose effective cross section for receiving radio or optical waves can be very much larger than the physical dimensions of the antenna, or the atom, by itself.

### Real Atomic Cross Sections

For more realistic atoms with realistic line-broadening effects and random orientations (\(3^*=1\)), the effective cross section at midband is given by values more like

\[\tag{62}\sigma(\text{lorentzian})\approx\frac{\lambda^2}{2\pi}\frac{\gamma_\text{rad}}{\Delta\omega_a}\]

or by

\[\tag{63}\sigma(\text{gaussian})\approx\frac{\lambda^2}{4}\frac{\gamma_\text{rad}}{\Delta\omega_d}\]

Note that the wavelength \(\lambda\) in these expressions is, as always, the wavelength in the laser medium. Table 7.1 gives some typical cross-section values for a few of the more useful laser transitions.

For example, on a strong but doppler-broadened visible laser transition in a gas with oscillator strength \(\mathcal{F}\approx1\), a radiative decay rate \(\gamma_\text{rad}\approx10^8\text{ s}^{-1}\), and a doppler linewidth \(\Delta_d/2\pi\approx2\times10^9\) Hz, the cross section will be \(\sigma\approx5\times10^{-12}\text{ cm}^2\). Experimentally this would be regarded as a very large cross section; the oscillator strengths and cross sections for transitions in real single atoms in gases are typically one to three orders of magnitude smaller.

Visible and near IR laser transitions in solid-state laser materials have much smaller cross sections, in the range \(\sigma=10^{-18}\) to \(10^{-20}\text{ cm}^2\). For a typical rare-earth laser transition in a solid, the wavelength might be \(\lambda=\lambda_0/n\approx0.6\) μm; the radiative decay rate might be 500 \(\text{sec}^{-1}\); and the linewidth might be 4 \(\text{cm}^{-1}\) or \(\Delta\omega_a\approx2\pi\times4\times30\) GHz. The resulting cross section will be \(\sigma\approx4\times10^{-19}\text{ cm}^2\). Visible transitions in the organic molecules used as laser dyes have very wide linewidths, but also very strong radiative decay rates, with oscillator strengths close to unity. This leads to considerably larger cross sections, in the range of \(\sigma\approx1\) to \(5\times10^{-16}\text{ cm}^2\).

### Transition Strength

The cross section \(\sigma\) is a measure of the strength of an atomic transition, as modified by the line-broadening effects of whichever linewidth mechanism \(\Delta\omega_a\) or \(\Delta\omega_d\) is dominant in determining the linewidth of the atomic response. If the cross section as a function of frequency is integrated across the entire linewidth, however, we obtain a so-called transition strength given by

\[\tag{64}S\equiv\int\sigma(\omega)d\omega=\frac{3^*\gamma_\text{rad}\lambda^2}{4}\]

which is the direct measure of the strength of the transition and is entirely independent of the lineshape of \(\sigma(\omega)\), whether it be lorentzian, gaussian, or otherwise. With degeneracy factors included this expression becomes

\[\tag{65}\int\sigma_{ji}(\omega)d\omega=\frac{g_i}{g_j}\int\sigma_{ij(\omega)}d\omega=\frac{3^*\gamma_{\text{rad},ji}\lambda^2}{4}\]

where \(j\) is the upper and \(i\) the lower level.

Measuring (carefully) the integrated absorption or cross section across the full linewidth of an atomic transition is thus one practical way of determining the radiative decay rate or Einstein \(A\) coefficient for that transition. Calculated or measured values of the integrated line strengths for different transitions are often given in handbooks and tables of atomic properties.

## 6. Saturation Intensities in Laser Materials

The amplification coefficient for a signal wave passing through a laser amplifier is proportional to the population difference on the amplifying transition. At the same time, however, for a strong enough input signal the stimulated transition rate may become large enough to saturate the population difference, and thus reduce the gain coefficient seen by the signal. This process is commonly referred to as saturation of the gain (or absorption) coefficient by the applied signal.

Saturation behavior in a laser amplifier (or for that matter an atomic absorber) can be expected therefore whenever the signal strength becomes strong enough for the signal itself to reduce the signal growth or attenuation rate. Understanding this kind of saturation behavior, which is very important in determining the performance of practical laser systems, is our objective here.

### Saturation of the Population Difference

A wave traveling through an atomic medium will grow or decay in intensity with distance through the medium according to the differential formula

\[\tag{66}\frac{dI}{dz}=\pm2\alpha_mI=\pm\Delta{N}\sigma{I}\]

where \(\sigma\) is the stimulated-transition cross section and the \(\pm\) signs apply to inverted or absorbing population differences. We have also shown that the population difference \(\Delta{N}\), whether emitting or absorbing, will often saturate with increasing signal strength in the homogeneous form

\[\tag{67}\Delta{N}=\Delta{N_0}\times\frac{1}{1+W\tau_\text{eff}}=\Delta{N_0}\times\frac{1}{1+I/I_\text{sat}}\]

where \(\Delta{N_0}\) is an unsaturated or small-signal inversion value; \(\tau_\text{eff}\) is an effective lifetime or recovery time for the transition; and \(I_\text{sat}\) is the ** saturation intensity**, or the value of signal intensity passing through the laser medium that will saturate the gain (or loss) coefficient down to half its small-signal or unsaturated value.

The saturation intensity is thus a parameter of great importance in practical laser materials; and our first task in this section is to derive a simple formula and some typical values for this quantity.

(Note also that in writing the \(W\tau_\text{eff}\) term we have omitted the factor of 2 that appears in the \(1+2WT_1\) denominator for the ideal two-level case, because the condition that \(N_1+N_2=\text{constant}\) does not apply on most laser transitions as it does for a simple ideal two-level system.)

### The Stimulated-Transition Probability

Obviously, the stimulated-transition probability \(W\) must be directly proportional to the signal intensity (power per unit area) \(I\) inside the laser medium, with a proportionality factor that can be obtained by the following argument.

The net power absorbed by the atoms in a thin slab of thickness \(\Delta{z}\) from an incident wave carrying total power \(P\) uniformly distributed over a transverse area \(A\) will be

\[\tag{68}\Delta{P}_\text{abs}=(N_1\sigma_{12}-N_2\sigma_{21})P\Delta{z}\]

(Note that \(N_1\) and \(N_2\) are, as usual, atoms per unit volume, and that the area factors in the slab volume \(A\Delta{z}\) and the power density \(P/A\) just cancel.) But from a rate-equation analysis the net power absorption by the atoms in the same slab can also be written as

\[\tag{69}\Delta{P}_\text{abs}=(W_{12}N_1-W_{21}N_2)A\Delta{z}\hbar\omega_a\]

where the energy per photon \(\hbar\omega\) must be included to convert the net stimulated transition rate in atoms/second into a net power-absorption rate.

Equating these two expressions, including possible degeneracy factors, then gives the relation

\[\tag{70}W_{21}=\frac{g_1}{g_2}W_{12}=\frac{\sigma_{21}}{\hbar\omega}\frac{P}{A}=\frac{\sigma_{21}}{\hbar\omega}\times{I}\]

or in simple terms

\[\tag{71}W\equiv\frac{\sigma{I}}{\hbar\omega}\]

This is a very useful and general relation which connects the cross section \(\sigma\), intensity \(I\), and stimulated transition probability \(W\). For degenerate transitions the upward and downward stimulated transition cross sections must obey the same relationship \(g_1\sigma_{12}=g_2\sigma_{21}\) as do the stimulated-transition probabilities \(g_1W_{12}=g_2W_{21}\).

### Saturation Intensity Derivation

The gain or absorption coefficient \(2\alpha_m\) for a homogeneous atomic transition will thus commonly saturate with increasing signal intensity in the form

\[\tag{72}2\alpha_m=\frac{2\alpha_{m0}}{1+I/I_\text{sat}}=\frac{\Delta{N_0}\sigma}{1+(\sigma\tau_\text{eff}/\hbar\omega)I}\]

The saturation intensity that reduces the small-signal absorption coefficient \(2\alpha_{m0}\equiv\Delta{N_0}\sigma\) down to half its small-signal value is thus given by

\[\tag{73}I=I_\text{sat}\equiv\frac{\hbar\omega}{\sigma\tau_\text{eff}}\]

From this formula, the saturation intensity is inversely proportional to the transition cross section \(\sigma\); that is, the larger the cross section, the easier the transition is to saturate. The saturation intensity is also inversely proportional to the recovery time \(\tau_\text{eff}\), because the longer the recovery time (the slower the recovery rate), the easier the transition is to saturate. In fact, an intensity \(I=I_\text{sat}\) basically means one photon incident on each atom, within its cross section \(\sigma\), per recovery time \(\tau_\text{eff}\).

Of course, if the signal being applied to either an amplifying or an absorbing atomic transition is tuned off line center, the stimulated transition rate and hence the degree of saturation produced by that signal will decrease in proportion to the atomic lineshape, since for a given intensity \(I\) the applied signal will be less effective in inducing transitions and thus causing saturation.

Suppose the transition has a homogeneous lorentzian lineshape, and suppose we use the normalized variable \(y=2(\omega-\omega_a)/\Delta\omega\) as a shorthand for the frequency detuning. The effective saturation of the atomic gain or loss coefficient \(\alpha_m\) by a signal of intensity \(I\) applied off line center will then be given by

\[\tag{74}2\alpha_m(\omega,I)=\frac{2\alpha_{m0}(\omega)}{1+(I/I_\text{sat})\times\frac{1}{1+y^2}}\]

where \(I_\text{sat}\) is the saturation intensity appropriate to a signal at midband. (Note that \(\alpha_{m0}\) here indicates the unsaturated or small-signal gain, not the midband gain.)

We must then take this frequency dependence into account either by retaining the explicit frequency dependence \(1/(1+y^2)\) in this formula in all further calculations, or by assuming that the saturation intensity itself becomes frequency dependent, with the effective saturation intensity for an off-resonance signal increasing by the amount

\[\tag{75}I_\text{sat}(\omega)=I_\text{sat}(\omega_a)\times\left[1+\left(2\frac{\omega-\omega_a}{\Delta\omega_a}\right)^2\right]\]

The effective saturation intensity goes up off line center, because the applied signal fields are less effective in inducing transitions; so a larger signal intensity is needed to produce a given amount of saturation. The most common procedure is to give a number for the midband-saturation intensity value, and then to include the frequency dependence explicitly in Equation 7.74.

### Saturation Broadening or Power Broadening

Suppose we tune a signal of fixed intensity \(I\) across an absorption line or a gain profile, and measure the saturated loss or gain coefficient \(\alpha_m(\omega,I)\) versus \(\omega\) using this fixed-intensity signal.

Then the complete frequency dependence for the gain coefficient (or the absorption coefficient) on a homogeneous atomic transition will include both the real lorentzian lineshape or frequency dependence of the atomic response itself, which will have the form \(1/(1+y^2)\), and the frequency dependence of the saturation behavior, which we have given in Equation 7.74.

Suppose we include both of these frequency dependences explicitly in the gain coefficient \(\alpha_m(\omega,I)\). We can then rewrite Equation 7.74 in terms of the midband gain coefficient and saturation intensity, with the explicit frequency dependences

\[\tag{76}\begin{align}2\alpha_m(\omega,I)&=\frac{2\alpha_{m0}(\omega_a)}{1+y^2}\times\frac{1}{1+(I/I_\text{sat})(1/1+y^2)}\\&=\frac{2\alpha_{m0}(\omega_a)}{1+I/I_\text{sat}+y^2}\end{align}\]

This can then be further rewritten in the equivalent form

\[\tag{77}2\alpha_m(\omega,I)=\frac{2\alpha_{m0}(\omega_a)}{1+I/I_\text{sat}}\times\frac{1}{1+[2(\omega-\omega_a)/\Delta\omega_b]^2}\]

where \(\Delta\omega_b\) is a power-broadened or saturation-broadened linewidth given by

\[\tag{78}\Delta\omega_b\equiv\sqrt{1+I/I_\text{sat}}\times\Delta\omega_a\]

That is, the measured lineshape for \(\alpha_m(\omega,I)\) will still be lorentzian, but it will now appear to have a broadened linewidth given by \(\Delta\omega_b\) rather than \(\Delta\omega_a\).

Note that the homogeneous linewidth of the transition has not really been broadened in any fundamentally new way; but the absorption lineshape measured by means of a tunable signal of fixed intensity \(I\) appears to be broadened because of stronger saturation and hence flattening down of the gain or loss coefficient at the middle of the line. This type of ** power broadening** of the atomic response appears in other laser situations as well.

### Numerical Values for Saturation Intensities

This saturation intensity, measured in watts per unit area, is very important in determining the large-signal saturation behavior of laser amplifiers and oscillators, as well as saturable absorbers.

A laser amplifier will become saturated and give little or no additional gain when the signal intensity passing through the laser material becomes of the order of the saturation intensity.

Similarly, the power level in a laser oscillator, at least under steady-state conditions, is going to build up to at most a few times the saturation intensity, at which point the gain in the laser medium will be saturated down to equal the losses in the laser cavity.

The saturation intensity is thus a very important measure of the amount of power per unit cross-sectional area that can be extracted from a practical laser device.

In practical terms a visible gas-laser transition might have, very approximately, \(\hbar\omega\approx10^{-19}\text{ J}\), \(\sigma\approx10^{-13}\text{ cm}^2\), \(\tau_\text{eff}\approx10^{-6}\text{ s}\), and hence \(I_\text{sat}\approx1\text{ W/cm}^2\). The oscillation power outputs from visible cw gas lasers do typically range from milliwatts to perhaps a few watts at most.

A solid-state laser, on the other hand, might have \(\sigma\approx10^{-19}\text{ cm}^2\), \(\tau_\text{eff}\approx10^{-3}\text{ sec}\), and hence \(I_\text{sat}\approx1\text{ kW/cm}^2\). A good cw Nd:YAG laser oscillator with an area \(A\approx0.3\text{ cm}^2\) can have a cw power output of a few hundred watts. Note that a typical liquid-dye laser might have \(\sigma\approx10^{-16}\text{ cm}^2\), and \(\tau_\text{eff}\approx10^{-9}\) sec, giving \(I_\text{sat}\approx1\text{ MW/cm}^2\).

Note also that the saturation-intensity value in general does not depend on the pumping intensity applied to the laser medium, since neither the cross section \(\sigma\) nor the effective recovery time (in most materials) depends directly on the pumping rate.

Pumping a laser medium harder generally creates more small-signal gain, which has to be saturated down further; but it does not change the saturation intensity involved in the saturation expression.

## 7. Homogeneous Saturation in Laser Amplifiers

As an optical signal passes through a laser amplifier, the signal intensity \(I(z)\) grows more or less exponentially with distance along the length of the amplifier. However, when the signal intensity begins to approach the saturation intensity for the laser medium, the population difference and hence the gain coefficient in the laser material begin to be saturated; the rate of signal growth with distance begins to decrease; and the signal intensity thus grows more slowly with distance.

In a single-pass laser amplifier such saturation effects begin first at the output end of the amplifier (see Figure 7.12), but only when the input signal is large enough that the amplified signal level at the output end has approached the saturation intensity of the laser medium. This saturation at the output end then causes the growth rate to decrease near the output end, and this in turn reduces the overall saturated gain from input to output as compared to the small-signal or unsaturated gain of the amplifier.

As we increase the input intensity to a laser amplifier, the intensity \(I(z)\) will reach the saturating range at an earlier point along the amplifier: the saturation region moves toward the input end as the input power is increased. The net result of this saturation behavior is that large-signal output is not a linear function of large-signal input. In this section we will analyze this behavior in a simple lossless single-pass amplifier, assuming cw signals and homogeneous saturation of the laser gain coefficient.

### Homogeneous Saturation Analysis

Suppose the laser gain coefficient in a single-pass laser amplifier saturates homogeneously, with unsaturated gain coefficient \(2\alpha_{m0}\), saturation intensity \(I_\text{sat}\) and, for simplicity, no linear losses, so that \(\alpha_0=0\). The basic differential equation governing the growth rate for the signal intensity along the amplifier thus becomes

\[\tag{79}\frac{1}{I(z)}\frac{dI(z)}{dz}=2\alpha_m(I)=\frac{2\alpha_{m0}}{1+I(z)/I_\text{sat}}\]

where \(\alpha_{m0}\) is the unsaturated gain coefficient and \(I_\text{sat}\) the saturation intensity of the laser medium. Obviously we can not simply integrate this equation to obtain an overall gain \(G=\exp(2\alpha_mL)\), since the gain coefficient \(\alpha_m\) varies with intensity \(I\) and hence with distance \(z\) along the amplifier length.

If, however, we assume an input intensity \(I_\text{in}\) at the input end \(z=0\) and an output intensity \(I_\text{out}\) at the output end \(z=L\), then this equation can be rearranged into the form

\[\tag{80}\int_{I=I_\text{in}}^{I=I_\text{out}}\left[\frac{1}{I}+\frac{1}{I_\text{sat}}\right]dI=2\alpha_{m0}\int_{z=0}^{z=L}dz\]

Both sides of this equation can then be integrated to obtain the expression

\[\tag{81}\ln\left(\frac{I_\text{out}}{I_\text{in}}\right)+\frac{I_\text{out}-I_\text{in}}{I_\text{sat}}=2\alpha_{m0}L=\ln{G_0}\]

where \(G_0\equiv\exp(2\alpha_{m0}L)\) is the small-signal or unsaturated power gain through the amplifier.

As it stands this result gives us an implicit relationship between the input and output intensities, the saturation intensity, and the unsaturated power gain \(G_0\) for the amplifier.

We can then obtain useful numbers from this equation in the following fashion. Suppose we define the actual power gain of the amplifier as the ratio of output over input, or \(G\equiv{I}_\text{out}/I_\text{int}\), under arbitrary saturation conditions. This gain cannot be written as \(\exp(2\alpha_mL)\), and its value in fact depends on the intensities \(I_\text{in}\) or \(I_\text{out}\). We can, however, use Equation 7.81 to write this overall power gain in the form

\[\tag{82}G\equiv\frac{I_\text{out}}{I_\text{in}}=G_0\times\exp\left[-\frac{I_\text{out}-I_\text{in}}{I_\text{sat}}\right]\]

which says that the value of the saturated gain \(G\) at a given value of \(I_\text{in}\) (or \(I_\text{out}\)) is reduced below the unsaturated value \(G_0\) by a factor that depends exponentially on the extracted intensity \(I_\text{out}-I_\text{in}\) relative to the saturation intensity \(I_\text{sat}\).

These results can then be manipulated into a variety of forms that can be used in different ways. For example, Equation 7.82 can be rewritten in the forms

\[\tag{83}G\equiv\frac{I_\text{out}}{I_\text{in}}=G_0\times\exp\left[-\frac{(G-1)I_\text{in}}{I_\text{sat}}\right]=G_0\times\exp\left[-\frac{(G-1)I_\text{out}}{GI_\text{sat}}\right]\]

The first of these forms can then be turned around to give the relationship

\[\tag{84}\frac{I_\text{in}}{I_\text{sat}}=\frac{1}{G-1}\ln\left(\frac{G_0}{G}\right)\]

which gives the input intensity in terms of the unsaturated gain \(G_0\) and saturated gain \(G\). But using the second form (or just multiplying both sides of Equation 7.84 by \(G\)) also gives the result

\[\tag{85}\frac{I_\text{out}}{I_\text{sat}}=\frac{G}{G-1}\ln\left(\frac{G_0}{G}\right)\]

which gives the output intensity as a function of the same quantities.

For any given value of unsaturated gain \(G_0\) we can then plug different values of saturated gain in the range \(1\lt{G}\lt{G_0}\) into Equations 7.84 and 7.85 to obtain paired values of normalized input intensity \(I_\text{in}/I_\text{sat}\) and output intensity \(I_\text{out}/I_\text{sat}\).

Figure 7.13 illustrates the resulting amplifier input-output intensity curves for two different small-signal gain values. Note that for each value, the actual gain \(G\) begins to be saturated below its small-signal value \(G_0\) even at output intensities well below the saturation intensity.

At high enough input intensities the gain always saturates down toward the limiting value \(G=1\), or 0 dB. For high intensities the amplifier transmission saturates down, not toward zero transmission, but toward unity transmission—that is, the amplifier (which is assumed to have zero ohmic losses) becomes essentially transparent at high enough input intensities.

### Power Extraction and Available Power

We might next ask how much intensity, or how much power per unit cross-section area, can be extracted from such an amplifier at different input-signal levels? By manipulating the preceding equations we can find that the power per unit area extracted from the amplifier—that is, the output power minus the input power, or the power really supplied to the wave by the amplifier—is given by

\[\tag{86}I_\text{extr}\equiv{I}_\text{out}-I_\text{in}=\ln\left(\frac{G_0}{G}\right)\times{I}_\text{sat}\]

The values of this quantity are illustrated by the dashed lines in Figures 7.13 and 7.14.

Note that for low input intensity and high gain (\(G\approx{G_0}\)), the output power and the extracted power are essentially the same—that is, we are putting in very little power at the input end compared to what we are getting out at the output end. As the amplifier begins to saturate, however, the extracted power approaches a limiting value, which is the maximum power available to be extracted from the amplifying medium. This maximum available power from the amplifier (per unit area) is given by the limiting value

\[\tag{87}I_\text{avail}=\lim_{G\rightarrow1}\ln\left(\frac{G_0}{G}\right)\times{I}_\text{sat}=(\ln{G_0})I_\text{sat}\]

This is the maximum power per unit area that is available in the laser medium to be given up to the amplified signal.

This expression for the available intensity in the laser medium has a simple physical interpretation. It can be rewritten, using earlier formulas, as

\[\tag{88}I_\text{avail}=2\alpha_{m0}L\times{I}_\text{sat}=(\Delta{N_0}\sigma{L})\times\left(\frac{\hbar\omega}{\sigma\tau_\text{eff}}\right)\]

Since intensity is already power per unit area, we can reduce this to available power per unit volume by writing it as

\[\tag{89}\frac{I_\text{avail}}{L}\equiv\frac{P_\text{avail}}{V}=\frac{\Delta{N_0}\hbar\omega}{\tau_\text{eff}}\]

This says that the maximum power output per unit volume that we can obtain from the laser medium is given by the initial or small-signal inversion energy stored in the medium, or \(\Delta{N_0}\hbar\omega\), times an effective recovery rate \(1/\tau_\text{eff}\). In other words, we can obtain the initial inversion energy \(\Delta{N_0}\hbar\omega\) once in every effective relaxation or gain recovery time \(\tau_\text{eff}\), which makes good physical sense.

### Power Extraction Efficiency

A major practical problem, however, is that this available power can be fully extracted only by heavily saturating the amplifier, in essence, by saturating the amplifier gain down until its saturated gain is reduced close to \(G=1\).

Suppose we calculate the input and output power, and the associated gain and extracted power, for a hypothetical single-pass amplifier having an unsaturated gain \(G_0=1,000=30\) dB and an available power \(P_\text{avail}=(\ln{G_0})AI_\text{sat}=1\text{ kW/cm}^2\).

With this amplifier we might hope to put in an input of 1 W and obtain an amplified output of 1,000 times larger, or close to 1 kW. The actual numbers for this case are, however, those shown in Table 7.2.

Note in particular that by the time the device is putting out 800 W, the actual gain has already been reduced from a small-signal gain of 1,000 or 30 dB down to 9 dB or approximately 8. Hence, to obtain this output of 800 W, we must drive the amplifier with an input not of 0.8 W but of 100 W. To get an actual output power equal to the nominally available 1,000 W, we must provide 220 W of inputs that is, we already need a fairly high-power preamplifier, just to extract the available power from this power amplifier.

An instructive way to demonstrate this same point is to define an energy-extraction efficiency as the ratio of actually extracted power to available power, or

\[\tag{90}\eta_\text{extr}\equiv\frac{I_\text{extr}}{I_\text{avail}}=\frac{\ln{G_0}-\ln{G}}{\ln{G_0}}=1-G_\text{dB}/G_{0,\text{dB}}\]

This energy-extraction efficiency, if plotted versus actual or saturated gain \(G\) in dB, is thus a straight line (see Figure 7.15). To extract even half the power potentially available in a cw amplifier, one must give up half the small-signal dB gain of the amplifier.

Driving a single-pass amplifier hard enough to extract most of the energy potentially available in the laser medium is thus a difficult problem, and this difficulty in obtaining full energy extraction is the principle defect in MOPA applications.

One way around this difficulty is to use multipass amplification, with the same beam sent through the amplifier medium several times, possibly from slightly different directions. Another solution (with its own difficulties) is to convert the amplifier into an oscillator, since oscillators are generally more efficient at extracting the available energy from a laser medium.

### Saturable Amplifier Phase Shift

If the intensity \(I(z)\) at any plane in an amplifier is sufficient to cause saturation of the population difference and thus the gain coefficient \(\alpha_m\), then it will also cause a similar saturation of the atomic phase-shift coefficient \(\Delta\beta_m\). The total phase shift through a linear amplifier may thus also change under saturation conditions.

To analyze this, we can note that the differential equation for the net phase shift \(\phi(z)\) for a signal of frequency \(\omega\) as a function of distance \(z\) along the amplifier can be written as

\[\tag{91}d\phi(\omega,z)=\frac{\omega{dz}}{c}+\Delta\beta_m(\omega,z)dz\]

The atomic phase-shift term \(\Delta\beta_m\) will depend on the degree of saturation or on the intensity \(I(z)\), and thus will vary with distance \(z\) along the amplifier.

For a homogeneous lorentzian atomic transition, by using the relationship between saturated gain and the saturated value of \(\Delta\beta_m\) we can integrate this equation and show that the overall phase shift is related to the input and output intensities in the form

\[\tag{92}\phi_\text{tot}(\omega,L)=\frac{\omega{L}}{c}+\frac{\omega-\omega_a}{\Delta\omega_a}\times\ln\left(\frac{I_\text{out}}{I_\text{in}}\right)\]

where \(I_\text{in}\) and \(I_\text{out}\) are the actual (saturated) values of input and output intensity at frequency \(\omega\). There is, of course, no atomic phase-shift contribution exactly at line center.

## 8.1 Transient Response of Laser Amplifiers

Let us look first at the very interesting topic of the linear transient response of a laser amplifier, for example, to a step-function or a delta-function type of signal input.

To understand what a laser amplifier does on a transient basis, we have to recall what the atoms in the laser amplifier do on a transient basis. The induced polarization on an atomic transition is linear in the applied signal field, at least within the rate-equation approximation. A laser amplifier is thus a linear system in its response to an applied signal, at least at low enough signal levels that no saturation effects occur.

The impulse response of a laser amplifier to a delta-function-like input pulse should therefore be the Fourier transform of the transfer function, or the complex voltage gain function \(\tilde{g}(\omega)\), of this linear system; and the response to a fast-rising step-function input should be the integral of this impulse response. In this section we will illustrate what this means for both passive absorbers and laser amplifiers.

### Step Response of an Atomic Absorption Cell

We can obtain a very instructive picture of the transient response both of an atomic cell and of the atoms themselves, by examining an ingenious optical-pulse generation experiment carried out by Eli Yablonovitch at Harvard University, using a CO_{2} laser and an atomic absorption cell filled with absorbing (that is, unpumped) hot CO_{2} vapor.

Imagine first that a step-function optical signal with carrier frequency tuned to the resonance frequency \(\omega_a\) is sent into an absorption cell having a large total attenuation \(2\alpha_mL\), where \(\alpha_m\) is the midband absorption coefficient for the atomic transition in the cell.

Suppose that the rise time for the leading edge of this optical signal is fast compared to the inverse linewidth \(T_2\equiv1/\Delta\omega_a\) of the atomic transition (but not compared to the optical carrier frequency \(\omega_a\) of the signal).

As the leading edge of the signal pulse sweeps past each point in the cell, there will then be a time delay of order \(\approx{T_2}\) before the absorbing atoms at that cross section can begin to respond to the applied signal, that is, before the induced sinusoidal polarization \(\tilde{P}(\omega)\) can build up to its steady-state value.

As a result, the leading edge of the pulse will sweep through the full length of the cell with essentially no attenuation. Only after the atoms have begun to respond can the cell begin to function as an absorber, and begin to attenuate the applied signal.

The output response of an absorbing cell to a step input with fast enough rise time will thus be a short pulse as shown in Figure 8.1, with essentially the same peak amplitude as the leading edge of the input signal, and with a duration that corresponds roughly to the time constant \(T_2\) before the atomic response and hence the attenuation of the cell can develop.

### Transient Response to Signal Turn-Off

An even more interesting observation is that an essentially similar pulse, again of duration \(\approx{T_2}\) but now of opposite sign, will also emerge from the cell just after the termination of the input signal, if we can suddenly turn off the input signal with a similarly fast fall time. The physical interpretation of this trailing-edge pulse is particularly instructive.

As one way of understanding it, consider the total signal that will be seen by an optical detector looking at the total transmitted output from the absorption cell. This detector in our example will see essentially no signal under steady-state conditions, during the main part of the input pulse, since we assume that the attenuation through the absorber cell is large.

One instructive (and correct) way of describing this situation is to say that the detector is actually seeing the superposition of the full original signal field that would be generated at the detector location by the input signal source without the absorber cell present, plus the additional fields that are produced at the detector by radiation from the induced polarization \(p(\pmb{r}, t)\) in the absorber cell itself. This induced polarization is coherently related to the applied signal; and because we assume an absorbing medium or an absorbing transition, the induced polarization reradiates in a way that will cancel (or nearly cancel) the original applied signal at the detector. (For a strongly absorbing cell this induced polarization exists, with steadily decreasing amplitude, only in the first few absorption lengths at the input end of the cell, after which the total field, applied plus reradiated, becomes small to negligible for the rest of the cell length.)

Suppose we suddenly turn off the applied signal, with a very fast fall time. After an appropriate time delay, corresponding to the travel time at the velocity of light from the applied signal source to the detector, the component of the applied sjgnal field due to the applied signal source itself will thus suddenly vanish. The atomic dipoles, however, will at that instant still be oscillating and radiating in coherent fashion; and, as we have described earlier, they will continue to oscillate and reradiate until the coherent polarization dies out with time constant \(T_2\) because of an appropriate combination of dephasing and energy decay.

Just before the applied signal is turned off, the net signal reaching the detector is essentially zero, since the applied signal field is almost totally canceled by the fields radiated by the absorber-induced polarization (for high insertion loss). Just after the signal turnoff the applied signal component is gone; but the atomic polarization \(p(t)\) and its dipole radiation contribution remains, at least for a time of order \(T_2\). The net signal at the detector, or at the absorber output, will thus suddenly jump up to an amplitude essentially equal to the unattenuated input signal, but with a phase 180° out of phase with the applied signal, as shown in Figure 8.1. In other words, suddenly turning off the input will also produce a short transient pulse at the output.

### Experimental Results

Turning an optical signal on or off with a rise time short compared to an atomic response time \(T_2\) requires an unusually fast electrooptic modulator and/or a very narrow atomic absorption line; so experiments of the type described here are not in general easy. Yablonovitch developed an ingeniously simple and also useful way to carry out such a demonstration, using the experimental system shown in Figure 8.2.

In this experiment the output from a pulsed TEA CO_{2} laser, which generates a 10.6 μm laser pulse with a pulsewidth of about 100 ns and a peak power of about 100 MW, was passed first through a lens pair which focused the incident beam down to a focal spot less than two wavelengths in diameter. The beam diverging from this focal spot was then recollimated by the second lens, and transmitted through an absorption cell several meters long and containing hot CO_{2} vapor, which automatically absorbs at the CO_{2} laser wavelength. The absorbing cell was heated in order to thermally populate the lower level of the CO_{2} absorption line, and the pressure could be changed in order to vary the pressure-broadened linewidth and hence the response time \(T_2\).

When this type of TEA laser is fired, the input intensity to the absorbing cell rises quite slowly, following the build-up time of the TEA laser itself, which has a rise time much too slow (\(\approx\) 100 ns) to produce any of the transient pulse effects we have discussed here. At a certain power level, however, the optical intensity in the focal spot can become sufficient to produce very rapid gas breakdown in air, leading to the almost instantaneous creation of a high-density plasma or "laser spark" (accompanied by a very loud sound wave). The electron density in this plasma almost immediately becomes so high that its index of refraction drops almost instantaneously to zero. This plasma spark then acts like a tiny but highly reflecting ball, which scatters essentially all the laser energy out of the beam. The gas breakdown thus provides in essence a self-actuated optical switch, which can completely shut off the transmitted laser beam with a fall time of less than 30 psec in practice.

Shutting off the incident signal using this technique produced the transient output pulse from the CO_{2} absorber cell shown in Figure 8.2(b). This pulse not only is interesting as a demonstration of laser dynamics, but can also be useful for subsequent experiments, since it has a peak power nearly equal to the incident TEA laser signal, and a pulsewidth which can be varied simply by changing the gas pressure in the absorber cell in order to change the time constant \(T_2\).

### Mathematical Analysis: The Step-Function Spectrum

To obtain a simple mathematical description of this pulse-generation process, consider a sinusoidal signal of the form \(\mathcal{E}_1(t)=E_s(t)\exp(j\omega_at)\) where \(E_s(t)\) is a unit step-function, so that the carrier signal is suddenly turned on (or off) with zero rise time at \(t=0\). Such a signal has an optical spectrum, or Fourier transform, of the form

\[\tag{8.1}\tilde{E}_s(\omega)=\int_0^{\infty}e^{-j(\omega-\omega_a)t}dt=\frac{1}{j(\omega-\omega_a)}\]

When this spectrum passes through an absorber cell with a lorentzian lineshape, the output spectrum is this input spectrum multiplied by the transfer function of the absorber cell or

\[\tag{8.2}\tilde{E}_2(\omega)=\tilde{E}_s(\omega)\times\exp\left[\frac{-\alpha_mL}{1+jT_2(\omega-\omega_a)}\right]\]

where we use the simplified formula that \(\Delta\omega\equiv2/T_2\) to define \(T_2\). The output signal is then given by the inverse Fourier transform

\[\tag{8.3}\begin{align}\mathcal{E}_2(t)&=\frac{1}{2\pi}\int_{-\infty}^{\infty}\tilde{E}_2(\omega)e^{j\omega{t}}d\omega\\&=e^{j\omega_at}\int_{-\infty}^{\infty}\frac{\exp[j2\pi{st}-\alpha_mL/(1+j2\pi{T_2}s)]}{j2\pi{s}}ds\end{align}\]

If we write this as \(\mathcal{E}_2(t)=E_s(t)\exp(j\omega_at)\), where \(E_2(t)\) is the output envelope for a step-function input, then Yablonovitch and Goldhar have noted that this integral can be evaluated in terms of Bessel functions \(J_m\) in the form

\[\tag{8.4}E_2(t)=e^{-\alpha_mL}-e^{-t/T_2}\sum_{m=0}^{\infty}\left(\frac{t}{T_2\alpha_mL}\right)^{m/2}J_m\left(2\sqrt{\frac{\alpha_mLt}{T_2} }\right),\quad{t\ge0}\]

for the boundary conditions corresponding to the fast turn-off case.

(In doing this analysis we have left out the \(e^{-j\omega{L}}\) phase shift through the amplifier length \(L\), since this would merely produce a propagation time delay \(t=L/c\) in the output pulse; i.e., the output signal given in Equation 8.21 should really occur starting at \(t\ge{L/c}\) and not \(t\ge0\).)

### Step-Function Spectrum

This Fourier analysis says in physical terms that turning a monochromatic signal on or off very rapidly will give the signal a spectrum with frequency components extending far into the wings on both the high-frequency and the low-frequency side of the carrier frequency \(\omega_a\).

This spectral broadening following the breakdown point was confirmed in the Yablonovitch experiments by using an infrared spectrometer to show that the sharply truncated signal following the breakdown point had a much wider power spectrum than the input TEA laser signal.

Figure 8.3 illustrates the resulting long and approximately \(1/\omega^2\) tails in the measured spectral density on both high-frequency and low-frequency sides of the CO_{2} laser wavelength.

An alternative physical approach to understanding the pulse-formation process on either the leading or trailing edges is thus the following. When the spectrally broadened signal shown in Figure 8.3 is transmitted through the absorber cell, the comparatively narrowband CO_{2} absorption cuts out only the narrow central portion of the spectrum, within about one linewidth \(\Delta\omega_a\) about the carrier frequency. In frequency terms, it is the unattenuated spectral wings that are transmitted through the absorber cell, with a hole cut out of the center, that produce the leading and/or trailing edge pulses.

It is also important to realize that this pulse behavior results entirely from the linear or small-signal transient behavior of the absorber cell—no nonlinear or large-signal or Rabi flopping phenomena are involved in these particular results. The trailing-edge pulse in particular is a classic example of what is generally referred to as free induction decay—that is, after the signal turnoff the oscillating dipoles are still freely oscillating, initially in phase but with decaying total macroscopic polarization, and in the process are inducing a matching pulse of decaying radiation out the end of the absorption cell. The Yablonovitch experiment thus gives a highly instructive illustration of the linear response behavior of an atomic transition, and of how this behavior can be variously interpreted using a transient time-domain viewpoint, a Fourier or frequency-domain viewpoint, or an interpretation based on the reradiation from an induced atomic polarization.

### Impulse and Step Responses of Laser Amplifiers

Essentially the same transient effects can also be produced in an inverted laser amplifier using an input signal that is either a short enough pulse (quasi delta function) or a step-function having fast enough rise or fall times. For the amplifier it is perhaps more instructive, in fact, to begin with the impulse response of the amplifier to a very short input pulse.

If the input signal to a laser amplifier has the form \(\mathcal{E}_1(t)=E_i(t)e^{j\omega_at}\), where the envelope \(E_i(t)\approx\delta(t)\) is a very short delta-function-like pulse, then the Fourier spectrum of this input signal will be essentially flat. The output signal, or the impulse response of the laser amplifier, will then be essentially the Fourier transform of the amplifier's complex voltage gain function, or

\[\tag{8.5}\tilde{E}_2(\omega)=\exp\left[\frac{+\alpha_mL}{1+jT_2(\omega-\omega_a)}\right]\]

(assuming that the amplifier has a lorentzian lineshape). An analysis by Bridges, Haus, and Hopf then shows that the inverse transform of this spectrum is given by \(\mathcal{E}_2(t)=E_2(t)e^{j\omega_at}\), where the output envelope \(E_2(t)\) is given by

\[\tag{8.6}E_2(t)=\delta(t)+\sqrt{\frac{\alpha_mL}{T_2t}}I_1\left[2\sqrt{\alpha_mLt/T_2}\right]e^{-t/T_2},\quad{t\ge0}\]

with \(I_1\) being the modified Bessel function of first order. (Changing the sign of the atomic absorption from \(-\alpha_m\) to \(+\alpha_m\) in the \(\sqrt{\alpha{L}t/T_2}\) argument changes the Bessel function from the oscillatory \(J_m\)'s given in Equation 8.4 to a growing \(I_1\)-type modified Bessel function.)

The physical interpretation of the first term in this analytical result, as shown in Figure 8.4, is that the impulse function itself will travel through the amplifier essentially unchanged—that is, neither attenuated nor amplified—since the atoms simply do not have time to respond or to build up oscillation and begin reradiating in steady-state fashion as the impulse rushes past.

The sinusoidal fields during the impulse do, however, give a finite impulse or "kick" to the atomic dipoles even during the brief passage of the pulse, so that the atoms are left with some induced oscillation or polarization \(p(t)\) following the passage of the pulse. The atomic dipoles then continue radiating, and thus produce the decaying free-induction tail which follows the impulse, as shown in Figure 8.4.

Because the atoms are inverted or amplifying, this induced tail is in phase with the impulse function, rather than 180° out of phase like the absorber turn-off tail. (In a certain sense, all the gain experienced by the input impulse occurs after the impulse itself has swept past.)

### Amplifier Step Response

The step response of a laser amplifier to a fast-rising input signal will be given by the integral of the impulse response. In other words, if \(E_2(t)\) as given in Equation 8.6 is the impulse response, then the output response produced by a step input will be given by

\[\tag{8.7}E_2(t)|_\text{step}=\int_0^tE_2(t')|_\text{impulse}dt'\]

Integrating over the delta-function part of the impulse response will cause the output step response to jump instantaneously from 0 to the input value at \(t=0\), as shown in Figure 8.5. This obviously represents the fast-rising leading edge sweeping through the amplifier with neither gain nor attenuation, just as in the absorber.

Integrating over the free-induction "tail" will then cause the output signal to continue to grow up to the steady-state amplified output value of \(e^{\alpha_mL}\), also as shown in Figure 8.5. Again, obviously the net gain of the amplifier comes in integrating over the area of the tail; and in fact the net voltage gain is just the total area in the impulse plus the tail, compared to the area in the impulse only.

### Experimental Results

Careful measurements of the step response of a laser amplifier have been made and compared with a similar but more detailed analysis in work done by Bridges, Haus, and Hopf.

These measurements were made using a low-pressure CO_{2} laser amplifier having a pressure-broadened homogeneous atomic linewidth of around 120 MHz or a \(T_2\) of \(\approx\) 2.6 ns. A fast-rising input pulse with less than 1 nanosecond rise time was obtained by passing a low-power cw laser beam through an electrooptic light modulator, and this signal was then reflected back and forth for five passes through the amplifier in order to obtain a net gain of greater than 20 dB. Since the gain-narrowed 3 dB bandwidth of the low-pressure CO_{2} amplifier was reduced to about 50 MHz, the 1 ns pulse rise time was short enough to approximate a step-function input, and the \(\approx\) 15 ns rise time of the amplifier's output signal could then be observed with a fast infrared detector and oscilloscope, as illustrated in Figure 8.6.

The results in this experiment are closely fit by a more exact version of the theory outlined above, as illustrated by the theoretical plot in Figure 8.6(b). In fact, several measurements of this type using different amplifier gas pressures yielded both the pressure-broadening coefficient for the CO_{2} laser transition, and the zero-pressure Doppler-broadened intercept of \(\approx\) 59 MHz.

## 8.2. Spatial Hole Burning, and Standing-Wave Grating Effects

When two waves traveling in different directions are simultaneously present in a laser medium, interference between these two waves will produce both frequency beating effects and standing-wave patterns in the optical intensity. These interference effects in turn may produce both temporal modulation and spatial variations in the amount of saturation in the laser medium.

Interference between two waves at the same frequency but traveling in different directions in particular can produce spatial hole burning effects, which can modify the saturation behavior of each wave independently, as well as induced grating effects, which can couple the two initially independent waves to each other. Nonlinear coupling between waves can in turn significantly modify the behavior of certain laser systems. In this section we will therefore introduce the fundamentals of spatial hole burning, and analyze the first-order coupling effects that spatial hole burning can produce in elementary two-wave situations.

### Wave Interference Effects

Consider a general situation in which two propagating waves with complex amplitudes \(\tilde{E}_1\) and \(\tilde{E}_2\), frequencies \(\omega_1\) and \(\omega_2\), and propagation vectors \(\beta_1\) and \(\beta_2\) are simultaneously present in an atomic medium. The total \(\mathcal{E}\) field intensity at any point in the atomic medium must then be written as

\[\tag{8.8}\begin{align}\mathcal{E}(z,t)&=\mathcal{E}_1(z,t)+\mathcal{E}_2(z,t)\\&=\text{Re}\left[\tilde{E}_1(z)\exp{j(\omega_1t-\beta_1z)}+\tilde{E}_2(z)\exp{j(\omega_2t-\beta_2z)}\right]\end{align}\]

and so the total optical intensity \(I(z,t)\), at any point \(z\) and any instant of time \(t\), must then in general be written in the form

\[\tag{8.9}\begin{align}I(z,t)=|\mathcal{E}(z,t)|^2=&|\tilde{E}_1(z)|^2+|\tilde{E}_2(z)|^2\\&\quad+\tilde{E}_1^*(z)\tilde{E}_2(z)e^{j[(\omega_2-\omega_1)t-(\beta_2-\beta_1)z]}+c.c.\end{align}\]

We see that the local intensity will contain, in addition to the average intensities \(|\tilde{E}_1|^2\) and \(|\tilde{E}_2|^2\) associated with the two waves separately, an interference term proportional to the dot product \(\tilde{E}_1^*\tilde{E}_2\). This interference term contains both a time-variation, at the "beat frequency" or difference frequency upbeat \(\omega_\text{beat}=\omega_2-\omega_1\) between the two signals, and a spatial variation, with a spatial periodicity given by \(\beta_2-\beta_1\).

### Temporal Interference Terms

The interference between two signals with different frequencies \(\omega_1\) and \(\omega_2\) will thus produce a time-varying intensity at each point in the atomic medium, with a sinusoidal frequency equal to the beat frequency \(\omega_\text{beat}\). What this time-varying intensity does to the atomic medium, and particularly to the local population difference \(\Delta{N}(t)\), depends on the difference frequency \(\omega_\text{beat}\) and especially on its value relative to the atomic time constants \(T_1\) and \(T_2\).

Often this sinusoidal modulation can be neglected, for several reasons. Suppose the difference frequency \(\omega_2-\omega_1\) between the two modes is large compared to any of the population recovery times \(\tau\) or \(T_1\), as it often is. (This difference frequency may, for example, represent an axial-mode beat frequency of several hundred megahertz or larger.) Then the time-varying part of this modulation will be so rapid that the atomic population difference will simply not respond to this frequency; and hence all the terms oscillating sinusoidally in time can be ignored.

In other situations the two waves \(\tilde{E}_1\) and \(\tilde{E}_2\) may have orthogonal polarizations, so that the vector dot product between them is zero. The interference terms that vary in time and space will then all be identically zero.

Finally, sometimes there may be not just two such ideal sine waves but in fact many such waves, with a significant spread in frequency. If this spread in frequency is sufficiently large—in other words, if the overall temporal coherence of the optical signal is not large—then the temporal interference effects between the multiple signals will tend to be washed out on the average, and only the total time-averaged intensity of the signals will be important.

### Cross-Modulation Effects

Note, however, that if any of the atomic properties, such as the gain or loss or phase shift in the atomic medium, do become significantly modulated at the difference frequency \(\omega_\text{beat}=\omega_2-\omega_1\), either by time-varying saturation effects or by other nonlinear mixing effects in the atomic medium, then the resulting modulation effects will produce frequency sidebands on both of the applied signals. In fact, the modulation of the \(\omega_2\) optical signal by the time-variations at \(\omega_\text{beat}\) will produce both an upper sideband at \(\omega_2+\omega_\text{beat}=2\omega_2-\omega_1\) and a lower sideband at \(\omega_2-\omega_\text{beat}=\omega_1\); while the \(\omega_1\) signal will similarly acquire an upper sideband at \(\omega_1+\omega_\text{beat}=\omega_2\) and a lower sideband at \(\omega_1-\omega_\text{beat}=2\omega_1-\omega_2\).

In other words, any type of nonlinear modulation or cross-saturation effects in the atomic response produced by the two signals will react back to couple or cross-modulate the two signals to each other (as well as to produce new nonlinear mixing frequencies in the system). These nonlinear mixing or cross-modulation effects can become extremely complex, and also quite important in coupling together different frequency signals either in a laser medium or in other kinds of nonlinear optical materials.

### Standing-Wave Interference Effects

Even if the two optical waves are at the same frequency, they will still produce spatial (though not temporal) cross-modulation and cross-coupling effects. That is, even if \(\omega_1=\omega_2\) the intensity \(I\) in Equation 8.26 will have a spatial variation of the form

\[\tag{8.10}I(z)=I_1(z)+I_2(z)+2\sqrt{I_1I_2}\cos[(\beta_2-\beta_1)z+\phi]\]

where \(I_1\) and \(I_2\) are the intensities of the two waves separately, and the sinusoidal standing-wave portion has a spatial phase angle \(\phi\) related to the relative phases of the two \(E\) fields.

If an intensity pattern of this form is present in a homogeneously saturable atomic medium, it will presumably produce a spatially varying saturation of the form

\[\tag{8.11}\frac{\Delta{N}(z)}{\Delta{N_0}}=\frac{1}{1+I(z)/I_\text{sat}}=\frac{1}{1+[I_1+I_2+2(I_1I_2)^{1/2}\cos(\Delta\beta{z})]/I_\text{sat}}\]

as illustrated in Figure 8.7. This spatial variation can then considerably complicate the analysis of gain saturation, as well as introduce complex wave-coupling effects in laser problems. The spatial variation of the gain (or loss) saturation in an atomic medium, as illustrated in Figure 8.7(b), is commonly referred to as ** spatial hole burning** in the medium.

There are many ways in which the spatial interference effects between two or more waves can be washed out, however, so that we can merely add intensities— that is, merely write \(I(z)=I_1+I_2\), without the cross term—in order to calculate the total atomic saturation, as we will do in a number of later analyses.

First of all, if the waves are in fact at different frequencies, then the interference fringes or standing-wave patterns produced by the two beams will move, or sweep through the material, because of the temporal part of the interference effect. If the beat frequency is large and the material response is slow—that is, if \(\omega_\text{beat}\gg1/T_1\), as it often is—then the spatial saturation effects tend to be washed out. Crossed polarizations will also eliminate interference effects—in isotropic though not in anisotropic materials.

Finally, if not just two ideal plane waves are present, but instead many components with different k vectors, then the standing-wave patterns between different waves can become sufficiently complex that many of the cross-coupling effects tend to be washed out on the average, and again only the average intensity in all the waves is significant.

### Two-Wave Coupling Effects

There are many situations, however, in which spatial coupling effects, or "induced grating effects," between signals can be quite important. Let us look somewhat further at an analysis of the most elementary form of this coupling.

The situation we will analyze here is the elementary case of two coherently related uniform plane waves at the same optical frequency passing in opposite directions through a homogeneously saturable atomic medium. The superposition of these two oppositely traveling waves in the medium produces a standing wave with intensity fringes whose period is equal to half the optical wavelength of either wave, as shown in Figure 8.7. If this intensity pattern occurs inside a saturable amplifying or absorbing medium, the net result is to create a greater degree of saturation at the peaks of the intensity profile and a lesser degree of saturation at the nulls.

The saturable medium thus develops a stratified character, and becomes in effect a volume interference grating or a volume hologram. (Such a grating produced by two waves traveling in more or less exactly opposite directions is sometimes referred to as a Lippman grating.) A simple analysis of this case will both demonstrate how to analyze nonlinear wave-interaction problems, and lead to some useful and not entirely obvious conclusions concerning the interaction between these two waves.

The general one-dimensional wave equation that applies in this situation can be written as

\[\tag{8.12}\frac{d^2\tilde{E}(z)}{dz^2}+\beta^2\tilde{E}(z)=-\omega^2\mu\tilde{P}(z)\]

where \(\tilde{E}(z)\) is the total electric field and \(\tilde{P}(z)\) is the polarization in the atomic medium. The two waves traveling in the \(+z\) and \(-z\) directions are then written in the form

\[\tag{8.13}\mathcal{E}(z,t)=\text{Re}\left[\tilde{E}_1(z)e^{j(\omega{t}-\beta{z})}+\tilde{E}_2(z)e^{j(\omega{t}+\beta{z})}\right]\]

where we allow the possibility that each complex wave amplitude \(\tilde{E}(z)\) may change with distance. Note that the plus sign in front of the propagation constant \(\beta\) in the second term means that this wave is traveling to the left or toward \(-z\).

We also assume that the atomic medium is a homogeneously saturable gain medium (or absorption medium) in which the signal is exactly on resonance. Hence the sinusoidal polarization \(\tilde{P}(z)\) at any position \(z\) can be written as

\[\tag{8.14}\tilde{P}(z)=\tilde{\chi}(\omega_a,z)\epsilon\tilde{E}(z)=j\chi^"(\omega_a,z)\epsilon\tilde{E}(z)\]

where the susceptibility \(\chi^"(\omega_a,z)\) at any point will be the saturated value given by the expression

\[\tag{8.15}\chi^"(\omega_a,z)=\frac{\chi_0^"}{1+I(z)/I_\text{sat}}\]

Note that \(I(z)\), the total intensity at position \(z\), will contain a sinusoidal standing-wave variation of the type we have written above.

### Small-Saturation Approximation

To proceed further at this point, we must make the approximation that the degree of saturation produced in the atomic medium is comparatively weak, so that we can use the mathematical approximation \(1/(1+I/I_\text{sat})\approx1-I/I_\text{sat}\) for \(I/I_\text{sat}\ll1\). We can then write the saturated susceptibility as

\[\tag{8.16}\chi^"(\omega_a,z)\approx\chi^"_0\times[1-I(z)/I_\text{sat}],\quad{I/I_\text{sat}}\le0.2\]

We make this approximation partly because it is often physically reasonable, but also because it would be much more difficult to proceed if we did not make it.

Putting the exact form for the intensity as given in Equation 8.9 into the wave equation 8.12 then expands this equation into the form

\[\tag{8.17}\begin{align}\left[\frac{d^2\tilde{E}_1}{dz^2}-2j\beta\frac{d\tilde{E}_1}{dz}\right]&e^{-j\beta{z}}+\left[\frac{d^2\tilde{E}_2}{dz^2}+2j\beta\frac{d\tilde{E}_2}{dz}\right]e^{+j\beta{z}}\approx-\beta^2\chi^"_0\\&\quad\times\left[1-\frac{|\tilde{E}_1|^2+|\tilde{E}_2|^2+\tilde{E}_1^*\tilde{E}_2e^{+2j\beta{z}}+\tilde{E}_1\tilde{E}_2^*e^{-2j\beta{z}}}{I_\text{sat}}\right]\\&\qquad\qquad\qquad\qquad\times\left[\tilde{E}_1e^{-j\beta{z}}+\tilde{E}_2e^{+2\beta{z}}\right]\end{align}\]

We can drop both of the second-derivative terms on the left-hand side of this equation, on the basis of the slowly varying envelope approximation, and then multiply out and match up the \(e^{-j\beta{z}}\) and \(e^{+j\beta{z}}\) traveling-wave terms on each side of this equation.

When we do this, we note that there is a product term between the \(\tilde{E}_1\tilde{E}_2^*e^{-2j\beta{z}}\) interference term in the saturation expression for \(\chi^"(\omega,z)\) and the left going wave term \(\tilde{E}_2e^{+j\beta{z}}\), and that this product term leads to an additional right going term \(\tilde{E}_1\tilde{E}_2\tilde{E}_2^*e^{-j\beta{z}}\) on the right-hand side of the equation. Similarly, there is a product of \(\tilde{E}_1^*\tilde{E}_2e^{+2j\beta{z}}\) times \(\tilde{E}_1e^{-j\beta{z}}\), which leads to an additional \(\tilde{E}_1^*\tilde{E}_1\tilde{E}_2e^{+j\beta{z}}\) term on the right-hand side. When all these cross-coupling and saturation terms are sorted out, the result is the pair of coupled equations

\[\tag{8.18}\frac{d\tilde{E}_1}{dz}\approx\pm\alpha_{m0}\left[1-\frac{|\tilde{E}_1|^2+2|\tilde{E}_2|^2}{I_\text{sat}}\right]\times\tilde{E}_1\]

and

\[\tag{8.19}\frac{d\tilde{E}_2}{dz}\approx\mp\alpha_{m0}\left[1-\frac{2|\tilde{E}_1|^2+|\tilde{E}_2|^2}{I_\text{sat}}\right]\times\tilde{E}_2\]

where we have used \((1/2)\beta\chi^"_0=\alpha_{m0}\), and where the upper or lower signs apply depending on whether the atomic medium is an amplifying or absorbing medium.

These equations are sometimes referred to as a third-order expansion for the atomic response, since the derivative terms for, say, the wave amplitude \(\tilde{E}_1\) contain not only a linear or first-order term proportional to \(\tilde{E}_1\), but also third-order nonlinear terms of the form \(\tilde{E}_1\tilde{E}_1^*\tilde{E}_1\) and \(\tilde{E}_2\tilde{E}_2^*\tilde{E}_1\). If we want to keep track of intensities only, we can also use these equations to obtain

\[\tag{8.20}\frac{dI_1}{dz}\approx\pm2\alpha_{m0}\left[1-\frac{I_1+2I_2}{I_\text{sat}}\right]\times{I_1}\]

and

\[\tag{8.21}\frac{dI_2}{dz}\approx\mp2\alpha_{m0}\left[1-\frac{2I_1+I_2}{I_\text{sat}}\right]\times{I_2}\]

The most interesting thing to note here is the extra factor of 2 that appears in each of the cross-saturation terms, as compared to the self-saturation terms, in the preceding equations.

These terms represent an important (and perhaps unexpected) result that emerges from this analysis. For some reason the cross-saturation between waves—that is, for example, the degree of gain saturation for wave #1 produced by the intensity of wave #2—is exactly twice the self-saturation of each wave by its own intensity.

### Grating Backscattering Effects

It is evident, in fact, both physically and from the way in which these terms arise in the equations, that the excess part of this cross-saturation effect is not additional "saturation" caused by the other wave, but results from a grating backscattering effect.

That is, in physical terms the oppositely traveling waves \(\tilde{E}_1\) and \(\tilde{E}_2\) combine to produce a standing wave, which in turn produces a set of stratified layers or a thick grating in the partially saturated medium. This grating has exactly the correct spacing so that a portion of wave #1 gets backscattered into wave #2 and vice versa (see Figure 8.8).

This physical picture of backscattering from a standing-wave grating also makes it physically reasonable that the excess cross-saturation effect should wash out if waves \(\tilde{E}_1\) and \(\tilde{E}_2\) are sufficiently incoherent. If waves \(\tilde{E}_1\) and \(\tilde{E}_2\) contain different frequency components, or have a partially incoherent spatial pattern, the clean standing-wave grating pattern will tend to wash out or average out, and the associated backscattering or wave-coupling effects will be reduced or eliminated.

Standing-wave grating effects can play a significant role in saturable absorber mode locking of lasers, and especially in the mode competition between simultaneously oscillating modes traveling in opposite directions in a ring laser cavity, as well as in other laser experiments.

## 8.3. More on Laser Amplifier Saturation

This section will present some additional details on laser amplifier saturation, including the effects of nonsaturable amplifier losses, saturation behavior in inhomogeneously broadened amplifiers, and transversely varying saturation effects.

### Amplifiers With Saturable Gain and Loss

Many practical laser amplifier systems will have both a homogeneously saturable gain coefficient am and a smaller but nonsaturating "ohmic" loss coefficient \(\alpha_0\) due to host-crystal absorption, impurities, scattering losses, excited-state absorption, and other effects. The differential equation for signal growth along the amplifier then becomes

\[\tag{8.22}\frac{dI}{dz}=\frac{2\alpha_{m0}I}{1+I/I_\text{sat}}-2\alpha_0I\]

If this equation is integrated through an amplifier of length \(L\), as we did for the lossless amplifier of Section 7, the more complicated relation connecting input and output signal intensities becomes

\[\tag{8.23}\ln\left[\frac{I_\text{out}}{I_\text{in}}\right]=2\alpha_{m0}L-2\alpha_0L+\left(\frac{\alpha_{m0}}{\alpha_0}\right)\ln\left[\frac{\alpha_{m0}-\alpha_0(1+I_\text{out}/I_\text{sat})}{\alpha_{m0}-\alpha_0(1+I_\text{in}/I_\text{sat})}\right]\]

or, in an alternative form,

\[\tag{8.24}\ln\left[\frac{G_0}{G}\right]=\left(\frac{\alpha_{m0}}{\alpha_0}\right)\ln\left[\frac{\alpha_{m0}-\alpha_0(1+I_\text{in}/I_\text{sat})}{\alpha_{m0}-\alpha_0(1+I_\text{out}/I_\text{sat})}\right]\]

where \(G\equiv{I_\text{out}}/I_\text{in}\) is the saturated gain at a given input intensity and \(G_0\equiv\exp(2\alpha_{m0}L-2\alpha_0L)\) is the net unsaturated gain minus loss. Either of these equations must be solved implicitly to find the input-output intensity relationship for an amplifier with given small-signal gain coefficient \(2\alpha_{m0}L\) and loss coefficient \(2\alpha_0L\).

### Maximum Effective Amplifier Length

The deleterious effects of even relatively small amounts of loss in a laser amplifier can be appreciated, however, without making detailed input-output plots, by noting that if we make an arbitrarily long amplifier using such a medium, the effective gain coefficient given by

\[\tag{8.25}\frac{dI}{dz}=\left(\frac{2\alpha_{m0}}{1+I/I_\text{sat}}-2\alpha_0\right)I=2\alpha_m'I\]

will eventually saturate down to give zero further growth, i.e., \(\alpha'_m\rightarrow0\), as the intensity flux in the amplifier approaches a maximum value \(I_\text{max}\) given by

\[\tag{8.26}I_\text{max}=(\alpha_{m0}/\alpha_0-1)I_\text{sat}\]

In other words, for a given small-signal gain \(\alpha_{m0}\) and loss \(\alpha_0\), there is a maximum possible signal flux which can build up in the amplifier. Alternatively, there is a maximum useful amplifier length given approximately by

\[\tag{8.27}I_\text{max}\approx\frac{1}{2(\alpha_{m0}-\alpha_0)}\ln\left[\frac{\alpha_{m0}-\alpha_0}{\alpha_0}\times\frac{I_\text{sat}}{I_\text{in}}\right]\]

Beyond this length the signal intensity never increases, because all the additional energy given to the signal by the inverted medium through stimulated emission is immediately absorbed and dissipated by the ohmic loss mechanisms.

Ohmic loss considerations can become important in solid state amplifiers when we wish to obtain particularly high gains and high pulsed flux densities. Losses due to scattering from materials imperfections, or to absorption by either weak ground-state or excited-state absorptions, can then limit the available gain and power output. Similar considerations apply also to dye laser amplifiers, and to high-power visible and ultraviolet gas laser devices, such as excimer lasers, in which both high power and high efficiency are very desirable.

High-power gas lasers in particular often employ moderately high-pressure mixtures of several different gases which are very highly excited by intense transverse arc discharges or electron-beam pumping. There may well be previously unknown or unexpected excited-state absorption lines in these mixtures that unfortunately overlap the desired laser transition; more than one originally promising high-power gas laser system has been eliminated by such unpleasant discoveries.

### Inhomogeneously Saturating Amplifiers

Although many high-power lasers do have the kind of homogeneous broadening that we have assumed in the discussion of saturation thus far, other useful laser materials (including most low-pressure doppler-broadened gas lasers) can be inhomogeneously broadened. We will learn in later tutorials that when a strongly inhomogeneous transition is subjected to a strong monochromatic signal, the gain (or loss) for that signal saturates in the inhomogeneous form

\[\tag{8.28}\frac{dI}{dz}=\frac{2\alpha_{m0}I}{(1+I/I_\text{sat})^{1/2}}\]

The square root in the denominator comes about because in an inhomogeneous line the applied signal saturates not only the spectral packet with which it is in exact resonance, but also other adjoining packets with which it has weaker interactions, thus "burning a hole" in the inhomogeneous line.

The analog to the input-output relationship for the homogeneous amplifier that we derived in the previous section then becomes

\[\tag{8.29}\int_{I=I_\text{in}}^{I=I_\text{out}}\left(\frac{1}{I^2}+\frac{1}{II_\text{sat}}\right)^{1/2}dI(z)=\int_{z=0}^{z=L}2\alpha_{m0}dz\]

Performing the complicated integration in this case leads to the result

\[\tag{8.30}\begin{align}\frac{\sqrt{1+I_\text{out}/I_\text{sat}}-1}{\sqrt{1+I_\text{out}/I_\text{sat}}+1}=&\frac{\sqrt{1+I_\text{in}/I_\text{sat}}-1}{\sqrt{1+I_\text{in}/I_\text{out}}+1}\\&\times\exp\left[2\alpha_{m0}L-2\sqrt{1+I_\text{out}/I_\text{sat}}+2\sqrt{1+I_\text{in}/I_\text{sat}}\right]\end{align}\]

This expression provides an implicit (and somewhat complex) way of computing \(I_\text{out}\) versus \(I_\text{in}\) for specified values of \(G_0\) and \(I_\text{sat}\) in the inhomogeneous case.

### Transversely Varying Saturation

The discussions of amplifier saturation thus far have also all been phrased in terms of the intensity (power per unit area) of the amplifying beam. If a beam has a flat transverse intensity profile, then we can simply multiply the intensity \(I(z)\) by the cross-sectional area \(A\) of the beam to get the total power \(P(z)\) at any plane.

Real laser beams, however, typically have nonuniform transverse intensity profiles, with the gaussian transverse intensity profile being a common example. When a nonuniform beam passes through a saturable amplifier, the more intense parts of the beam saturate more rapidly than the weaker portions. The beam profile can thus be distorted, with in general the higher-intensity peaks being flattened out relative to the weaker parts.

The transverse profile of the amplifying beam may also change with distance because of diffraction effects as the beam propagates through the amplifier. Usually, however, amplifiers will be short enough and/or the beam diameters large enough that the beam will not have significant diffraction spreading; so diffraction effects will be far less important than spatially nonuniform saturation effects in the amplifier. To gain some insight into the latter, let us consider a few simple points about such saturation effects, using an elementary gaussian beam profile.

### Gaussian Beam Saturation

The transverse intensity distribution in a cylindrically symmetric gaussian beam with spot size w may be written as

\[\tag{8.31}I(r)=\frac{2P}{\pi{w^2}}\exp\left(-\frac{2r^2}{w^2}\right)\]

(The factor of 2 appears in the exponent because \(w\) is conventionally defined as the \(1/e\) radius for the \(E\) field amplitude.)

The peak intensity of the gaussian is thus the same as if the total power \(P\) were uniformly distributed over an area \(A=\pi{w^2}/2\); and indeed we can show that if we consider a uniform-intensity beam having the same total power and the same intensity as the central peak intensity of the gaussian, then the effective area for this equivalent uniform beam appears to be \(A_\text{eff}=\pi{w^2}/2\). This may not, however, be the best choice of effective area for a gaussian beam when we want to calculate saturation and power extraction effects.

Let us assume, as is often reasonable, that the amplifying beam is collimated and that diffraction effects are small. Then in essence each elemental cross-sectional area of the beam amplifies and saturates according to its own local intensity \(I(r,z)\), independently of all other points in the cross section. The equation for local intensity in a homogeneously saturating amplifier will be

\[\tag{8.32}\frac{\partial{I(r,z)}}{\partial{z}}=\frac{2\alpha_{m0}I(r,z)}{1+I(r,z)/I_\text{sat}}\]

and the relation between input and output intensity profiles will be the same as in Equation 7.81, namely,

\[\tag{8.33}\ln\left[\frac{I_\text{out}(r)}{I_\text{in}(r)}\right]+\frac{I_\text{out}(r)-I_\text{in}(r)}{I_\text{sat}}=\ln{G_0}\]

Given an input beam profile \(I_\text{in}(r)\), we must in general solve this equation numerically for \(I_\text{out}(r)\) and then integrate to find the total input and output powers \(P_\text{in}\) and \(P_\text{out}\).

Suppose, however, that an amplifier with a gaussian input profile is either short enough or heavily saturated enough that its overall saturated gain is small. Then the beam profile will not change greatly with distance, and we may assume that the beam remains gaussian at every plane \(z\) through the amplifier. This differential gain equation may then be integrated over the amplifier cross section to give the result

\[\tag{8.34}\begin{align}\frac{dP(z)}{dz}&=\int_{0}^{\infty}\frac{\partial{I(r,z)}}{\partial{r}}2\pi{r}dr\\&=\frac{8\alpha_{m0}P(z)}{w^2}\int_0^{\infty}\frac{r\exp(-2r^2/w^2)dr}{1+[2P(z)/\pi{w^2}I_\text{sat}]\exp(-2r^2/w^2)}\\&=\pi{w^2}\alpha_{m0}I_\text{sat}\ln\left[1+\frac{2P(z)}{\pi{w^2}I_\text{sat}}\right]\end{align}\]

Consider first the short and weakly saturated case, where the gaussian beam power \(P\) is small compared to the quantity \(\pi{w^2}I_\text{sat}/2\). Then by expanding the logarithm to second order, we can calculate that the power extraction in a short length \(\Delta{z}\) will vary with incident power \(P\) in the form

\[\tag{8.35}\Delta{P}\approx2\alpha_{m0}P[1-P/\pi{w^2}I_\text{sat}]\Delta{z}\quad\begin{cases}\text{gaussian profile,}\\\text{weak saturation}\end{cases}\]

The analogous result for power extraction by a uniform beam having area \(A\) and total power \(P\), in the small-saturation limit, would be

\[\tag{8.36}\Delta{P}=\frac{2\alpha_{m0}P\Delta{z}}{1+P/AI_\text{sat}}\approx2\alpha_{m0}P[1-P/AI_\text{sat}]\Delta{z}\quad\begin{cases}\text{uniform profile,}\\\text{weak saturation.}\end{cases}\]

By comparing these two equations, we may conclude that in the weak-saturation limit the effective area of a gaussian beam for power extraction is not \(\pi{w^2}/2\) but \(A_\text{eff}\approx\pi{w^2}\). In physical terms, the outer wings of the gaussian beam (where much of the power is carried) are at low intensity, and thus do not saturate the laser medium. The gaussian beam therefore acts as if its area were larger than we might expect.

Consider next the heavily saturated (and hence still low-gain) gaussian amplifier when \(P(z)\gg(\pi{w^2}/2)I_\text{sat}\). The power extracted from the laser medium in an incremental length \(\Delta{z}\) may then be written as

\[\tag{8.37}\Delta{P}\approx\pi{w^2}\alpha_{m0}I_\text{sat}\ln\left[1+\frac{2P_1}{\pi{w^2}I_\text{sat}}\right]\quad\begin{cases}\text{gaussian profile,}\\\text{strong saturation.}\end{cases}\]

The corresponding expression for a uniform beam in the high-saturation limit will be

\[\tag{8.38}\Delta{P}\approx{A}\times{I_\text{avail}}\Delta{z}\approx2\alpha_{m0}I_\text{sat}A\Delta{z}\quad\begin{cases}\text{uniform profile,}\\\text{strong saturation.}\end{cases}\]

By comparing these we can get a rough idea of how the effective saturation area of the gaussian beam increases at high intensities, as more and more of the gaussian beam profile rises above the saturation-intensity level.

The next tutorial introduces ** spectrally efficient multiplexing of Nyquist-WDM**.