# Laser Oscillation Dynamics and Oscillation Threshold

This is a continuation from the previous tutorial - ** hermetic optical fibers - carbon-coated fibers**.

In this tutorial we discuss a number of additional topics related to the elementary properties of laser oscillation. We consider first the oscillation build-up time with which coherent oscillation develops from noise in a laser cavity.

From this we develop a set of simple coupled rate equations which link cavity photons to laser atoms, and laser atoms to cavity photons. Using these equations we explore further the laser oscillation buildup and the remarkable threshold properties characteristic of the laser oscillator.

We then examine briefly some of the more complex laser cavities that are useful in practice, including multimirror laser cavities, ring-cavity lasers, bistable laser systems, and "lasers" with no cavity at all.

## 1. Laser Oscillation Buildup

The primary question to be addressed in this section is: How fast does the coherent oscillation in a laser cavity build up from noise, when the laser is first turned on?

### Oscillation Buildup Analysis

To answer this question, consider a laser cavity in which the laser gain exceeds the cavity losses, at least temporarily; and follow any small packet of signal energy through one complete round trip within the cavity, with a roundtrip time \(T=p/c\), as shown in Figure 13.1.

The growth in circulating intensity in one round trip, starting with intensity \(I_0\) at time \(t=0\), is then

\[\tag{1}I(T)=I_0\times{R_1}R_2(R_3...)\exp[2\alpha_mp_m-2\alpha_0p]=I_0\exp[\delta_m-\delta_c]\]

where all the notation has been defined in the preceding tutorials. The net growth after \(N\) round trips will be given by

\[\tag{2}I(NT)=I_0\times[R_1R_2(R_3...)e^{2\alpha_mp_m-2\alpha_0p}]^N=I_0\exp[N(\delta_m-\delta_c)]\]

which we can rewrite more generally as

\[\tag{3}I(t)=I_0\exp\left[\frac{\delta_m-\delta_c}{T}t\right]\]

since the circulating intensity will make \(N\) round trips in a time \(t=NT\). It can also be convenient to write this as

\[\tag{4}I(t)=I_0\exp[(\gamma_m-\gamma_c)t]\]

where we define the cavity growth and decay rates by

\[\tag{5}\gamma_m\equiv\frac{\delta_m}{T}=\frac{2\alpha_mp_m}{T}\qquad\text{and}\qquad\gamma_c\equiv\frac{\delta_c}{T}=\frac{2\alpha_0p+\ln(1/R_{\text{tot}})}{T}\]

where \(T_\text{tot}\equiv{R_1}R_2(R_3...)\).

The cavity lifetime or exponential decay time \(\tau_c\) for optical signals in the cavity in the absence of laser gain is then given by

\[\tag{6}\tau_c\equiv\gamma_c^{-1}=\frac{T}{\delta_c}\]

Since the round-trip time \(T\) for a typical laser cavity is 1 to 10 ns, and the round-trip cavity losses may range from 1% (\(\delta_c\) = 0.01) to, say, 70% (\(\delta_c\approx\) 1), typical "cold cavity" decay times (with no laser gain) will range from the order of 1 ns to the order of 1 μs.

### Oscillation Buildup

The signal in a laser cavity following a sudden initial turn-on of the laser gain will thus build up exponentially with time much as shown in Figure 13.2, starting from an initial noise level \(I_0\) which is usually very small, typically corresponding to only a few spontaneous-emission noise photons in the cavity.

This buildup will continue until the circulating intensity reaches a steady-state level \(I_\text{ss}\) with a very large number of photons in the cavity. This steady-state level corresponds to the oscillation level at which the laser gain is saturated down enough to just equal the total cavity losses (internal losses plus output coupling).

Figure 13.2 does assume either that the laser gain is very suddenly turned on to its full value at the start of the buildup interval, or else that some added cavity losses are suddenly turned off at this point, with a switching time short compared to the growth rate of the laser intensity.

This may not be true in many real lasers. In an He-Ne laser, for example, the turn-on time for the plasma discharge and thus the laser gain will be much slower than the oscillation buildup time.

In many other lasers, however, including E-beam-pumped excimer lasers, certain optically pumped lasers, and \(Q\)-switched lasers of all types, the gain can be switched on (or added losses switched off) in times short compared to the oscillation buildup time.

### Typical Oscillation Buildup Times

It is very convenient in discussions such as these to define a normalized inversion ratio \(r\) as the ratio of the initial unsaturated gain coefficient \(\delta_{m0}\) to the cold-cavity loss coefficient \(\delta_c\), or

\[\tag{7}r\equiv\frac{\delta_{m0}}{\delta_c}=\frac{\gamma_{m0}}{\gamma_c}=\frac{2\alpha_{m0}p_m}{2\alpha_0p+\ln(1/R_\text{tot})}\]

The oscillation buildup rate of Equations 13.3 or 13.4 can then be written in the form

\[\tag{8}I(t)=I_0\exp\left[\frac{r-1}{\tau_c}t\right]\]

The total intensity buildup from the initial noise level \(I_0\) to the final steady-state oscillation level \(I_\text{ss}\) is then given, to a good approximation, by

\[\tag{9}I_\text{ss}\approx{I_0}\exp\left[\frac{r-1}{\tau_c}T_b\right]\]

or the buildup time \(T_b\) is given by

\[\tag{10}T_b\approx\frac{\tau_c}{r-1}\ln\left(\frac{I_\text{ss}}{I_0}\right)\]

The ratio of final oscillation level to initial noise level in real lasers may range from \(I_\text{ss}/I_0\approx10^8\) to \(I_\text{ss}/I_0\approx10^{12}\), depending on the type of laser. Since this ratio appears only logarithmically in the buildup-time expression of Equation 13.10, however, and since its logarithm varies only from \(\ln(I_\text{ss}/I_0)\approx18\) to \(\ln(I_\text{ss}/I_0)\approx28\), an exact knowledge of this ratio is not essential.

The general conclusion, in fact, is that the oscillation buildup time \(T_b\) may range from \(\approx\) 10 to \(\approx\) 30 cavity decay times \(\tau_c\), depending on how far the laser is pumped above threshold.

Thus, for a rather long, low-loss He-Ne laser operated not far above threshold, with \(L\) = 1 m, \(T\) = 6 ns, \(\delta_c\) = 3%, and \(r\) = 1.1, the buildup time will be \(T_b\approx\) 50 μs. For a short, high-gain Nd:YAG laser pumped well above threshold, on the other hand, with \(L\) = 30 cm, \(T\) = 2 ns, \(\delta_c\) = 0.5, and \(r\) = 3, the buildup time shortens to \(T_b\approx\) 50 ns.

### More Exact Buildup Analysis

In some but by no means all lasers the laser gain will saturate more or less immediately with increasing light intensity \(I(t)\). Let us assume this, and also assume that the unsaturated growth rate \(\gamma_{m0}\) is not much greater than the cold-cavity decay rate \(\gamma_c\), or that the initial inversion ratio \(r\) is not much greater than 1, so that the degree of saturation at steady state will remain small.

We can then write the instantaneous growth rate in a standing-wave laser oscillator cavity in the approximate form

\[\tag{11}\gamma_m(t)\approx\frac{\gamma_{m0}}{1+2I(t)/I_\text{sat}}\approx\gamma_{m0}[1-2I(t)/I_\text{sat}]\]

where \(\gamma_{m0}\equiv{r}\gamma_c\) is the unsaturated laser growth rate.

A more exact equation for the oscillation buildup, including gain saturation, can then be written in the form

\[\tag{12}\frac{dI}{dt}=\gamma{I}-\beta{I^2}\]

where we have followed the notation commonly used in the literature, with \(\gamma\equiv\gamma_{m0}-\gamma_c\) being the unsaturated growth rate and \(\beta\equiv2\gamma_{m0}/I_\text{sat}\) the saturation coefficient. The solution to this more exact equation is

\[\tag{13}I(t)=\frac{I_0I_\text{ss}e^{\gamma{t}}}{I_\text{ss}+I_0(e^{\gamma{t}}-1)}\]

where \(I_0\) is the initial intensity at turn-on, corresponding usually to a few noise photons inside the laser cavity; and \(I_\text{ss}\equiv\gamma/\beta=(1-\gamma_c/\gamma_{m0})\times{I_\text{sat}/2}\) is the steady-state oscillation level at the end of the buildup period.

The time delay following gain turn-on needed to reach, say, half the final intensity is then given from this expression

\[\tag{14}T_b=\frac{1}{\gamma}\ln\left(\frac{I_\text{ss}-I_0}{I_0}\right)\approx\frac{\tau_c}{r-1}\ln\left(\frac{I_\text{ss}}{I_0}\right)\]

which is essentially the same result we obtained earlier.

### Experimental Results

Figure 13.3 shows two examples of experimental results for oscillation buildup times in gas lasers.

In Figure 13.3(a) a helium-neon laser which is initially oscillating at steady state is suddenly quenched by illuminating the He-Ne laser tube with a short but intense pulse of ultraviolet radiation from a xenon flashlamp.

This UV radiation efficiently pumps neon atoms from a lower-lying \(1s^5\) metastable level up into the lower level of the laser transition, thus destroying the laser gain and suddenly quenching the laser action, without significantly disturbing either the laser cavity or the laser discharge.

The gain then recovers rapidly as these atoms relax back out of the lower laser level, and the laser oscillation builds back up again, with varying time delays for different steady-state intensities, as shown. Note that the numerical time delays agree generally with the numerical estimate of \(\approx\) 50 μs given earlier.

Figure 13.3(b) shows the buildup of oscillation in an optically pumped far-infrared laser that employs formic acid vapor as the laser medium for oscillation at a wavelength of 743 μm.

The pumping power coming from the 9 μm CO\(_2\) laser that excites this laser is turned on very rapidly in step-function fashion, using an acousto-optic modulator that has a 70-ns rise time, much faster than the buildup time for the far-infrared oscillation.

The experimental results shown can then be fitted very accurately into Equation 13.13 using only a single intensity-scaling parameter and a laser gain coefficient \(\alpha_m\) that is directly proportional to the optical pumping power.

### Spiking Behavior

In many solid-state lasers, as well as other types of lasers, the excess gain that is present during the oscillation buildup period does not saturate immediately with increasing laser intensity, but decreases only after a certain time delay required for the circulating laser intensity \(I(t)\) to "burn up" the excess population inversion. Analysis of this situation requires a more exact set of equations, to describe the dynamics of the atomic populations as well as the cavity fields.

The oscillation buildup in this situation may not converge smoothly to the final steady-state value, but may instead exhibit a strong transient overshoot, followed by quasi periodic "spiking" or relation oscillation behavior, as illustrated in Figure 13.2. We will carry out a more detailed examination of this interesting but rather useless spiking behavior in a later tutorial.

## 2. Derivation of the Cavity Rate Equation

In this section we will extend the cavity growth-rate calculation developed in the preceding section to derive a "photon rate equation" for the signal intensity or the number of photons in each laser cavity mode, including—for the first time— the effects of spontaneous emission.

In the following section we will then combine these cavity rate equations with the atomic rate equations we have developed earlier to obtain a set of coupled cavity plus atomic rate equations which are simple, and yet extremely useful in analyzing many fundamental aspects of laser theory.

### Derivation of the Cavity Rate Equation

The exponential growth rate for the signal intensity inside a laser cavity derived in the previous section had the general form

\[\tag{15}I(t)=I_0\exp[(\gamma_m-\gamma_c)t]\]

If either of the coefficients \(\gamma_m\) or \(\gamma_c\) is time-varying, however—as they well may be in real cases—then we must convert this equation to the more general differential form

\[\tag{16}\frac{dI(t)}{dt}=[\gamma_m(t)-\gamma_c(t)]\times{I(t)}\]

The parameters \(\gamma_m\) or \(\gamma_c\) might become time-varying, for example, because the gain coefficient saturates, or because we deliberately modulate the cavity losses or cavity output coupling with time. Equation 13.15 is then a correct solution to the more general Equation 13.16 only when the two gain and loss rates are constant.

In particular, the gain coefficient \(\gamma_m\) will be directly proportional to the inverted population difference \(\Delta{N}(t)\equiv{N_2}(t)-N_1(t)\) on the laser transition; and this population difference will very likely change with time in a real laser.

We can take this dependence of \(\gamma_m\) on \(\Delta{N}\) into account by writing

\[\tag{17}\gamma_m(t)\equiv\frac{2\alpha_mp_m}{T}\equiv{K}\Delta{N(t)}\]

where all the other geometrical and atomic parameters of the system are absorbed into the constant \(K\).

At the same time that we do this, we can also conveniently express the total signal energy inside the laser cavity in dimensionless units by defining a "number of photons" \(n(t)\) in the cavity by

\[\tag{18}\begin{align}n(t)&=\text{"number of photons in the cavity"}\\&\equiv\left[\frac{\text{total signal energy in the cavity}}{\text{quantum of energym, }\hbar\omega}\right]\\&=\text{const}\times{I_\text{circ}}(t)\end{align}\]

It should be emphasized that we are not focusing any special attention on the photon nature of light by writing, this equation—the emphasis in laser analyses should almost always be on the wave rather than the particle nature of light.

Rather, we are simply expressing the total signal energy in the laser cavity in the convenient units of \(\hbar\omega\). Also, we will not really need any explicit formula for the constant appearing in the last line of Equation 13.18, although if such a formula is wanted, the photon number \(n(t)\) in a low-gain standing-wave cavity of length \(L\), cross-sectional area \(A\) and circulating intensity \(I_\text{circ}\) can be calculated to a good first approximation from

\[\tag{19}n(t)\approx\frac{2AI_\text{circ}(t)L}{\hbar\omega{c}}=\frac{2V_c}{\hbar\omega_ac}I_\text{circ}(t)\]

where \(V_c=AL\) is the volume of the cavity mode.

Equations 13.16, 13.17, and 13.18 can then be combined to give the cavity rate equation

\[\tag{20}\frac{dn(t)}{dt}=[K\Delta{N(t)}-\gamma_c]\times{n(t)}\]

or

\[\tag{21}\frac{dn(t)}{dt}=K[N_2(t)-N_1(t)]n(t)-\gamma_cn(t)\]

where \(N_1(t)\) and \(N_2(t)\) are the total number of atoms in the lower and upper levels of the laser transition. The first two terms on the right-hand side of Equation 13.21 then represent stimulated emission and absorption between the cavity mode and the atoms, while the third term represents the cavity losses plus output coupling.

Note that in earlier tutorials we have consistently used \(N_1(t)\) and \(N_2(t)\) to indicate atomic densities, or numbers of atoms per unit volume. In writing the cavity and atomic rate equations in this and following tutorials, however, it will be more convenient to let the symbols \(N(t)\) and \(\Delta{N}(t)\) represent the total numbers of atoms inside the laser cavity.

### Value of the Coupling Constant K

By using the formulas derived in earlier tutorials for the gain coefficient \(\alpha_m\), we can rewrite the constant \(K\) appearing in Equations 13.17, 13.20, and 13.21 in the form (for a lorentzian transition)

\[\tag{22}K\equiv\frac{2\alpha_mp_m}{T\Delta{N}}=\frac{3^*}{4\pi^2}\frac{\omega_a\gamma_\text{rad}\lambda^3}{\Delta\omega_aV_c}\]

where \(V_c\) is the volume of the cavity or, more precisely, of the cavity mode with which the atoms are interacting.

Note again that we are now using \(N_1\) and \(N_2\) to indicate the total numbers of atoms in the laser levels, so that \(\Delta{N(t)}/V_c\) is the volume-averaged inversion density with which the cavity mode interacts.

Equation 13.22 can be reduced to the particularly simple and useful form

\[\tag{23}K=\frac{3^*\gamma_\text{rad}}{p}\]

if we define a parameter \(p\), called the cavity mode number (no relation to the laser cavity perimeter \(p\)), given by

\[\tag{24}p\equiv\frac{4\pi^2V_c}{\lambda^3}\frac{\Delta\omega_a}{\omega_a}\]

This parameter has a very important physical significance, as we will now show.

### Frequency Distribution of Resonant Cavity Modes

Suppose we consider some arbitrarily shaped enclosure or cavity having closed and completely reflecting walls. Let us then calculate all the theoretically possible lowest and higher-order electromagnetic modes in this cavity; and plot the resonant frequencies of these modes as tic marks on a frequency scale, as shown in Figure 13.4.

Then, at some low frequency, corresponding to a wavelength on the order of the cavity dimensions, we will see the lowest-order resonant mode of the cavity, followed by a succession of higher-order resonant modes with successively higher resonant frequencies, as shown in Figure 13.4.

As we go to much higher frequencies, where the cavity dimensions become large compared to the resonance wavelengths, these resonant modes will become more and more closely spaced along the frequency axis, so that the mode distribution in frequency space will become very dense.

In fact, it can be shown that in any such enclosure or cavity, regardless of its exact shape, the number of resonant modes falling within a unit (radian) frequency interval, or the resonant mode density \(\rho(\omega)\) along the frequency axis, will be given by

\[\tag{25}\rho(\omega)=\frac{dp(\omega)}{d\omega}\equiv\left[\frac{\text{number of cavity modes, }dp}{\text{frequency range, }d\omega}\right]=\frac{8\pi{V_c}}{\lambda^3}\frac{d\omega}{\omega}\]

This formula will hold for any cavity shape, whenever the frequencies are high enough that the cavity dimensions become large compared to the resonant wavelengths.

This distribution function can be given another interpretation that does not even require the concept of resonant modes. Suppose we wish to describe an arbitrary electromagnetic field distribution in some large rectangular volume.

We can always expand such a field distribution using a Fourier-series expansion in all three spatial coordinates or, to put this in another way, we can expand the fields in a set of waves traveling in all possible directions through the volume. (This process is sometimes referred to as "box normalization" in electromagnetic theory or in quantum mechanics.)

The number of independent Fourier components, or of traveling-wave terms, call this number \(dp(\omega)\), needed to give a complete description of an arbitrary electromagnetic field within a large rectangular region of volume \(V_c\), assuming this field distribution is made up of frequency components lying within a frequency range \(d\omega\), is then given by exactly the same formula \(dp(\omega)=\rho(\omega)d\omega\) given in Equation 13.25.

### The Cavity Mode Number p

The mode number \(p\) appearing in the stimulated transition constant formula (Equations 13.23 or 13.24) can then be understood as the effective number of laser cavity modes lying within the atomic transition linewidth \(\Delta\omega_a\), as shown in Figure 13.5.

That is, this number is given by the formula

\[\tag{26}p\equiv\rho(\omega)\times\frac{\pi\Delta\omega_a}{2}=\frac{4\pi^2V_c}{\lambda^3}\frac{\Delta\omega_a}{\omega_a}\]

The effective frequency bandwidth multiplying the mode-density function \(\rho(\omega)\) in this equation is \((\pi/2)\) times the atomic linewidth \(\Delta\omega_a\) rather than just \(\Delta\omega_a\), because this is the width of an equivalent rectangular distribution having the same peak height and the same area as a lorentzian lineshape, as shown in Figure 13.5.

The linewidth \(\Delta\omega_a\) of any laser transition is always small compared to the transition frequency \(\omega_a\), but the cavity volume \(V_c\) of a normal laser cavity is always very much larger than a single cubic wavelength. The mode number \(p\) is thus normally an extremely large number for ordinary laser cavities, with values typically on the order of \(p\approx10^7\) to \(p\approx10^{10}\). We will learn more about the significance of this parameter very shortly.

### Frequency Dependence of the Coupling Coefficient K

Before going on to introduce the concept of spontaneous emission into a cavity mode, we should note that the value of the rate-equation coupling constant \(K\) given in Equations 13.22 and 13.23 is obviously the midband value, appropriate to a cavity mode tuned to the center of the atomic transition.

If we consider instead a cavity mode whose resonant frequency \(\omega_i\) is tuned off the atomic line center, then the response of the atoms to the cavity fields will be reduced by the atomic lineshape, and hence the coupling coefficient \(K(\omega_i)\) for that off-resonance mode will be reduced with a frequency dependence \(K(\omega_i)=2\alpha_m(\omega_i)p_m/T\) that is given for a lorentzian transition by

\[\tag{27}K(\omega_i)=K_0\times\frac{1}{1+[2(\omega_i-\omega_a)/\Delta\omega_a]^2}\]

or for a gaussian transition by

\[\tag{28}K(\omega_i)=\sqrt{\pi\ln2}\times{K_0}\times\exp\left[-4\ln2\left(\frac{\omega_i-\omega_a}{\Delta\omega_a}\right)^2\right]\]

where \(K_0\equiv{K(\omega_a)}\equiv3^*\gamma_\text{rad}/p\).

Suppose we sum the coupling coefficients \(K(\omega_i)\) over all the cavity modes \(\omega_i\) underneath an atomic transition, weighted by their frequency dependences, while also averaging the polarization factor \(3^*\) over all field or atomic polarizations.

This sum over all modes is, in essence, an integration over the mode density shown in Figure 13.5, weighted by the atomic transition lineshape shown in that figure, as given in Equations 13.27 or 13.28. Using either of these lineshapes, we can obtain the very fundamental result that

\[\tag{29}\sum_{\text{all modes}}K(\omega_i)\rightarrow\int_0^{\infty}K(\omega)\times\rho(\omega)d\omega\equiv\gamma_\text{rad}\]

This fundamental result, which says that the sum of \(K(\omega_i)\) over all cavity modes under an atomic linewidth is just equal to the \(\gamma_\text{rad}\) value for that transition, is in fact a very general result, completely independent of the atomic lineshape, the details of the laser cavity, or any other factors except the radiative decay rate \(\gamma_\text{rad}\) of the atomic transition.

### Introduction of Spontaneous Emission

We have thus far written the cavity rate equation for a single cavity mode in the form

\[\tag{30}\frac{dn}{dt}=K[N_2-N_1]n-\gamma_cn\]

where this form includes stimulated transitions (that is, stimulated-emission and stimulated-absorption terms), and also cavity loss terms, but not yet spontaneous-emission terms.

We must now take into account the process of spontaneous emission from the upper-level atoms into this cavity mode (as well as into every other cavity mode within the laser cavity volume).

That is, the atoms are spontaneously emitting in a noise-like fashion, at a rate directly proportional to the number of upper-level (but not lower-level) atoms, and independent of the number of photons n already in each cavity mode; and a small fraction of this spontaneous emission will have the right direction, polarization and frequency to feed directly into the cavity mode we are considering.

The rate equation for each individual cavity mode must thus be extended to the more complete form

\[\tag{31}\frac{dn}{dt}=K[N_2-N_1]n+K_\text{sp}N_2-\gamma_cn\]

where \(K_\text{sp}\) is a spontaneous-emission constant governing the rate of emission from the upper-level atoms into that particular cavity mode.

But, there is a fundamental result of quantum theory—one of the most fundamental principles of quantum electronics, in fact—which says that the spontaneous-emission rate from any given set of atoms into any one individual cavity mode is exactly equal to the stimulated-emission rate that would be produced from those same atoms by one photon of coherent signal energy present in the same mode.

For each cavity mode with resonance frequency \(\omega_i\), therefore, the stimulated and spontaneous transition constants involved in the interaction with a given set of atoms must necessarily be related by

\[\tag{32}K_\text{sp}(\omega_i)\equiv{K}(\omega_i)\]

for each and every cavity mode. Therefore, the cavity rate equation for each separate cavity mode, including spontaneous emission, can also be rearranged into the form

\[\tag{33}\frac{dn}{dt}=KN_2[n+1]-KN_1n-\gamma_cn\]

Written in this form the equation seems to say that, whereas the net atomic absorption rate is proportional to the instantaneous number of cavity photons \(n(t)\), the net emission rate is proportional to \(n(t)+1\), i.e., the number of cavity photons plus one.

### Spontaneous Emission: The "Extra Photon"

This "plus one" factor caused by the spontaneous emission sometimes leads laser workers to speak of an "extra photon" in the cavity mode—a photon that somehow causes only downward transitions.

It is important to understand, however, that this additional spontaneous-emission term in the cavity rate equation is much more accurately viewed as an incoherent or noise-like driving term which excites the cavity mode in a random or noise-like fashion, completely uncorrelated with the coherent stimulated-emission terms or with any cavity signal that may already be present.

This spontaneous-emission term thus acts as a fundamental quantum noise source in the cavity equations. It is this quantum noise source which is responsible for the ultimate noise figure of laser amplifiers, for example, and also for the quantum noise fluctuations in phase and amplitude that are present in even the most ideally stabilized laser oscillators or frequency standards.

One of the practical conclusions stemming from this is that it is impossible to make a laser amplifier—or, in fact, any other kind of amplifier—with an equivalent input noise power less than one noise photon per Hz of bandwidth.

### Derivation of the Spontaneous-Emission Coefficient

Let us now verify that this spontaneous emission rate for noise photons into each cavity mode corresponds exactly to the spontaneous atomic emission process that we have already discussed in earlier tutorials. We can recall first that the total relaxation rate out of any upper atomic level will normally include two different relaxation processes.

First of all, there will normally always be a purely radiative relaxation rate, or a spontaneous emission rate, \(\gamma_\text{rad}N_2\) on the \(2\rightarrow1\) transition we are considering. In addition, there may be (and usually will be) both nonradiative relaxation rates from level 2 down to various lower levels and possibly other purely radiative decay rates from level 2 down to lower levels other than level 1.

The purely radiative or spontaneous emission part of this relaxation on the \(2\rightarrow1\) transition can then be described in two different but physically equivalent fashions.

From a "free-space" viewpoint, each upper-level atom in the cavity volume has a certain probability per unit time \(\gamma_\text{rad}\) of radiating spontaneously at some frequency within the atomic lineshape, and into some random emission direction.

If we look into the open sides of the cavity from any external point, we will see this spontaneous emission coming out from all sides of the cavity, with a lineshape corresponding to the atomic transition lineshape, and with a total emission rate (in photons/second) into all directions given by \(\gamma_\text{rad}N_2\).

Essentially all this spontaneous emission is emitted out through the open sides of a normal laser cavity, although a very minute portion of it is radiated into the low-loss direction exactly along the cavity axis.

From an alternative "cavity mode" viewpoint, however, the atoms can be thought of as spontaneously radiating this same energy, not out into free space, but rather directly into each of the very large number of resonant cavity modes (mostly very lossy modes) whose frequencies lie within the atomic linewidth.

The total spontaneous-emission fields coming out of the cavity in all directions can then be viewed as the result of the very rapid leakage or diffraction loss from all of these cavity modes out the sides of the cavity.

In considering the total number of modes within a resonant cavity, we must keep in mind that all of the reasonably low-loss cavity modes—that is, those lowest and slightly higher-order axial-transverse modes that we have described elsewhere in this tutorial series—really represent only a very minute fraction of the total number of cavity modes associated with the cavity volume.

There will typically be within an atomic linewidth only a few, or at most a few hundred, low-order axial-plus-transverse modes which describe the radiation traveling in the lowloss directions very close to the cavity axis.

There are, however, some \(p=10^7\) to \(10^{10}\) other potential cavity modes, most of them having enormously high losses out the cavity sides, which are needed in principle to describe all possible field configurations traveling in all directions within the cavity volume and within the atomic linewidth.

From the second viewpoint, therefore, each of these laser cavity modes within the atomic linewidth should receive spontaneous emission at a rate given by Equation 13.33, with a \(K\) value \(K(\omega_i)\) appropriate to that particular cavity mode. But, nearly all of these modes have extremely fast decay rates out the side of the cavity, and hence this energy radiated from the atoms into all these modes is immediately radiated on out of the cavity in all directions.

From this viewpoint we must equate the total spontaneous-emission power coming from the atoms to the total spontaneous-emission power emitted into all these cavity modes. That is, suppose we label each cavity mode by its resonance frequency \(\omega_i\). Then we can write for the total spontaneous rate on the \(2\rightarrow1\) transition

\[\tag{34}\sum_{\omega_i}K_\text{sp}(\omega_i)N_2=\sum_{\omega_i}K(\omega_i)N_2=\gamma_\text{rad}N_2\]

But, we have already shown in Equation 13.29 that the summation in the middle term of this equation just adds up to the radiative decay rate \(\gamma_\text{rad}\), so that this "conservation of total spontaneous emission" is indeed verified.

We can now understand better, as well, why the midband interaction constant \(K\), or \(K_\text{sp}\), must have the value \(3^*\gamma_\text{rad}/p\) given in Equation 13.23.

If we assume, as is reasonable, that the geometrical and atomic factors determining the spontaneous-emission rate into each mode within the atomic lineshape are likely to be essentially the same, except for polarization factors and for the atomic lineshape itself, then we can approximate the total spontaneous rate into all of the \(\approx{p}\) modes within the main part of the atomic linewidth by

\[\tag{35}\gamma_\text{rad}=\sum_{\omega_i}K_\text{sp}(\omega_i)\approx{p}\times{K_0}\]

where \(K_0\equiv{K}(\omega_a)=K_\text{sp}(\omega_a)\) refers to the on-resonance value for some preferred lowest-loss cavity mode located near the center of the atomic line. If we put in a polarization factor \(3^*\), the \(K_0\) value for this preferred mode located close to the line center becomes

\[\tag{36}K_0=K_\text{sp}(\omega_a)=K(\omega_a)=\frac{3^*\gamma_\text{rad}}{p}\]

where \(3^*\) has a value appropriate to that particular mode. But this is just what we started with in Equation 13.23. The \(p\) in the denominator simply represents the fact that \(1/p\) of the total spontaneous emission from the upper level atoms goes into that one particular cavity mode.

## 3. Coupled Cavity and Atomic Rate Equations

We must now proceed to join the cavity rate equations developed in the preceding section to the atomic rate equations developed in earlier tutorials. The result will be a set of coupled cavity plus atomic rate equations that are very useful in describing laser threshold behavior, laser amplitude modulation, laser spiking and \(Q\)-switching, and a wide range of other laser phenomena.

### Atomic Rate Equations

The signal and noise photons in the cavity mode discussed in Section 13.2 were assumed to be interacting with a two-level atomic system having total populations \(N_1(t)\) and \(N_2(t)\) in the lower and upper levels, respectively.

Drawing on our results from earlier tutorials, we can then write a pair of atomic rate equations for these level populations in the same form as in earlier tutorials, that is

\[\tag{37}\begin{align}\frac{dN_1}{dt}&=-W_{12}N_1+W_{21}N_2+\left[\begin{split}\text{pumping}\\\text{terms}\end{split}\right]+\left[\begin{split}\text{relaxation}\\\text{terms}\end{split}\right]\\\frac{dN_2}{dt}&=W_{12}N_1-W_{21}N_2+\left[\begin{split}\text{pumping}\\\text{terms}\end{split}\right]+\left[\begin{split}\text{relaxation}\\\text{terms}\end{split}\right]\end{align}\]

The \(W_{12}N_1(t)\) and \(W_{21}N_2(t)\) terms are the stimulated-transition terms caused by the cavity fields. The exact form of the pumping and relaxation terms in each equation will depend on the details of the particular atomic system and how it is being pumped or excited.

But, we know that the stimulated-transition probabilities \(W_{12}\) and \(W_{21}\) that appear in these atomic rate equations are themselves directly proportional to the signal energy, or to the cavity photon number \(n(t)\), in the resonant cavity mode.

We can thus write these stimulated-transition probabilities (leaving out degeneracy effects for simplicity) as being directly proportional to \(n(t)\) in the form

\[\tag{38}W_{12}=W_{21}=K'n(t)\]

where \(K'\) is again a proportionality constant which contains all the other geometrical and atomic parameters.

The stimulated transition terms in the atomic rate equations can thus be written as

\[\tag{39}W_{21}N_2-W_{12}N_1=K'[N_2(t)-N_1(t)]n(t)\]

But every time an atom makes a signal-stimulated transition downward in the atomic rate equations, giving up an energy of \(\hbar\omega\), this energy must be delivered into one of the cavity modes, so that the cavity photon number must simultaneously go up by one unit of \(\hbar\omega\) in the cavity rate equation for that mode. The reverse argument must of course apply equally well to stimulated absorption transitions going in the opposite direction.

The stimulated-transition rates \(K(N_1-N_2)n\) and \(K'(N_1-N_2)n\) in the cavity and in the atomic rate equations must therefore be numerically identical; and hence the constants \(K\) and \(K'\) in front of these terms must be the same, so that in fact \(K'\equiv{K}\).

The form of the pumping and relaxation terms in Equations 13.37 will depend on the exact atomic system being considered. Suppose we now consider, as a simple but specific example, two upper atomic levels \(E_1\) and \(E_2\) such as we have considered in earlier tutorials, with a pumping rate \(R_p\) into the upper level, and with the usual relaxation rates (in the optical-frequency approximation) from the upper and lower levels downward.

The complete atomic rate equations for this system will then take on the form

\[\tag{40}\begin{align}\frac{dN_2}{dt}&=R_p-Kn[N_2-N_1]-\gamma_2N_2\\\frac{dN_1}{dt}&=Kn[N_2-N_1]+\gamma_{12}N_2-\gamma_{10}N_1\end{align}\]

where the coupling coefficient \(K=K(\omega_i)\) is exactly the same as in the corresponding cavity rate equation.

### Complete Coupled Cavity and Atomic Equations

We have shown in Section 13.32 how to write the cavity rate equation for any one individual cavity mode, including spontaneous emission, and have also pointed out that a real laser system will have \(\approx{p}\) cavity modes, each one of which is, at least in principle, able to interact with the atoms contained within the cavity volume.

The final result of this discussion is then that to describe properly, even within the rate-equation approximation, a laser cavity having a large number of resonant modes, each labeled by index \(i\), plus a set of atoms with populations \(N_1\) and \(N_2\), we must, at least in principle, write down a separate rate equation for each cavity mode individually, in the form

\[\tag{41}\frac{dn_i(t)}{dt}=K_iN_2(t)[n_i(t)+1]-K_iN_1(t)n_i(t)-\gamma_{ci}n_i(t)\]

where \(n_i\) is the cavity photon number, \(K_i\) the coupling constant, and \(\gamma_{ci}\) the cavity decay rate for the \(i\)-th cavity mode.

We must then also write a pair of rate equations for the atomic populations in the general form developed in this section, namely,

\[\tag{42}\begin{align}\frac{dN_2(t)}{dt}&=\sum_iK_in_i(t)[N_1(t)-N_2(t)]+\left[\begin{split}\text{pumping}\\\text{terms}\end{split}\right]+\left[\begin{split}\text{relaxation}\\\text{terms}\end{split}\right]\\\frac{dN_1(t)}{dt}&=-\sum_iK_in_i(t)[N_1(t)-N_2(t)]+\left[\begin{split}\text{pumping}\\\text{terms}\end{split}\right]+\left[\begin{split}\text{relaxation}\\\text{terms}\end{split}\right]\end{align}\]

The two atomic levels are, in other words, potentially coupled to the total set of \(p\) near-resonant cavity modes, as illustrated in Figure 13.6, as well as to whatever pumping and relaxation processes may be present.

Note that in these equations the stimulated-transition terms for the atoms must be summed over the total stimulated-transition effects produced by the signal fields in all the cavity modes acting on the atoms (or at least all those cavity modes that contain any significant number of photons).

Note also that no additional spontaneous-emission terms need be added to the atomic rate equations, because the transition rate due to spontaneous emission into all the cavity modes is already included in the purely radiative part of the relaxation terms.

### Idealized Single-Mode, Single-Level Rate Equations

Writing out the complete set of cavity rate equations for \(p\approx10^8\) cavity modes would be a daunting task, with or without the assistance of a computer.

Fortunately, in most real lasers we only need to write out explicitly the cavity rate equations for one or a few of the most favored or lowest-loss cavity modes, and not for the whole set of \(p\) such modes.

In fact, one of the most remarkable features of laser action is that a typical laser cavity having perhaps \(p\approx10^8\) individual and distinct cavity resonance modes can still oscillate in just one or a few of these cavity modes. So long as only one or a few cavity modes are excited with any significant number of photons \(n_i\), we need write down the rate equations for only those few modes.

In fact, the simplest possible laser model—but one that still contains all the essential physics—is to assume that there is just one lowest-loss (or highest-gain) preferred cavity mode that builds up any significant photon number \(n(t)\), so that we need write only one cavity rate equation.

The atomic rate equations can also be put into their simplest form by assuming that the relaxation rate downward out of level 1 is sufficiently fast that \(N_1\approx0\) under all circumstances, and that the pumping into level \(N_2\) can be described by a simple pumping rate \(R_p\).

The coupled cavity and atomic rate equations 13.41 and 13.42 will then reduce to their simplest possible combined form, namely,

\[\tag{43}\begin{align}\frac{dn}{dt}&=KN_2(n+1)-\gamma_cn\\\frac{dN_2}{dt}&=R_p-KN_2n-\gamma_2N_2\end{align}\]

This simple pair of equations is still surprisingly general, and we will use these two equations extensively to analyze several fundamental aspects of laser behavior in succeeding tutorials.

We might also just mention some of the limitations of the coupled cavity plus atomic rate equations for analyzing laser behavior, even in the case where we might write a larger number of cavity mode equations.

In particular, this approach is necessarily limited to the small-signal or rate-equation atomic regime, as described in earlier tutorials. No coherent-pulse effects can be included.

More important, this rate-equation approach completely ignores, or hides, all the phase information associated with the signal fields in each resonant cavity mode.

It also completely leaves out any spatial interference effects between modes, and thus any spatially inhomogeneous saturation effects or "spatial hole burning" that this may produce in the atomic level populations \(N_1\) and \(N_2\). Nonetheless, this rate-equation approach can be very useful in laser theory, as we will see.

## 4. The Laser Threshold Region

The almost discontinuous change in power output that occurs at threshold, when a laser suddenly breaks into oscillation, is one of the most remarkable feature of laser behavior. Several of the most significant aspects of this laser threshold behavior can be explained using a remarkably simple rate-equation model, as we will demonstrate in this section.

### Idealized Rate-Equation Analysis

To analyze laser threshold behavior we can use the highly idealized, and yet very realistic, laser model developed in Section 13.3, consisting of a single preferred or lowest-loss cavity mode with photon number \(n(t)\), plus an ideal two-level laser transition with upper-level population \(N_2(t)\). This upper level is assumed to be pumped at a steady (but adjustable) pumping rate of Up atoms/second, and to have a population decay rate \(\gamma_2\). Downward relaxation out of the lower laser level is assumed to be arbitrarily fast, so that \(N_1\approx0\) under all circumstances.

The coupled rate equations for this system, as developed in Section 13.3, are then

\[\tag{44}\begin{align}\frac{dn}{dt}&=K(n+1)N_2-\gamma_cn\\\frac{dN_2}{dt}&=R_p-KnN_2-\gamma_2N_2\end{align}\]

where \(\gamma_c\) is the cavity decay rate, and the coupling constant \(K\) is given, as in the preceding sections, by \(K\equiv3^*\gamma_\text{rad}/p\). As usual, \(\gamma_\text{rad}\) is the radiative decay rate on the laser transition, and the important quantity \(p\) is the (very large) number of resonant cavity modes within the cavity volume and transition linewidth. To simplify the results slightly, we will set \(3^*=1\) from here on.

### Steady-State Solutions Below Threshold

The steady-state solutions to Equations 13.44, when \(d/dt=0\) in both equations, can then be manipulated in several different ways. For example, the form that is most useful for understanding below-threshold behavior is to write the steady-state solution to the first of these equations in the form

\[\tag{45}n_\text{ss}=\frac{N_\text{ss}}{\gamma_c/K-N_\text{ss}}=\frac{N_\text{ss}}{N_\text{th}-N_\text{ss}}\]

and the solution to the second in the form

\[\tag{46}N_\text{ss}=\frac{R_p}{\gamma_2+Kn_\text{ss}}=R_p\tau_2\times\frac{1}{1+(\gamma_\text{rad}/\gamma_2)\times(n_\text{ss}/p)}\]

Equation 13.45 then says that the number of steady-state photons \(n_\text{ss}\) in the cavity mode will remain small, somewhere between zero and perhaps a few hundred, until the upper-level population \(N_\text{ss}\) is raised to within a fraction of a percent of a threshold inversion value \(N_\text{th}\), where this threshold inversion value is given by

\[\tag{47}N_\text{th}\equiv\frac{\gamma_c}{K}=\frac{\gamma_c}{\gamma_\text{rad}}p\]

This value is the threshold inversion we calculated earlier, at which (or very near which) laser oscillation begins. Equation 13.46 then says that in this same region, so long as the photon number \(n_\text{ss}\) remains very much less than \(p\), the upper-level population increases essentially in direct proportion to the pumping rate; i.e., \(N_2\approx{R_p}\tau_2\). The threshold pumping rate, at which the population inversion \(N_\text{ss}\) will just reach the threshold inversion \(N_\text{th}\) if this continues, is given by

\[\tag{48}R_{p,\text{th}}=\gamma_2N_\text{th}=\frac{\gamma_2\gamma_c}{\gamma_\text{rad}}p\]

It is convenient to define a normalized pumping rate relative to this threshold value by

\[\tag{49}r\equiv\frac{R_p}{R_{p,\text{th}}}=\frac{\gamma_\text{rad}R_p}{\gamma_2\gamma_cp}\]

The below-threshold region (\(r\lt1\)) is then described by the approximate results

\[\tag{50}\left.\begin{align}n_\text{ss}&\approx\frac{r}{1-r}\\N_\text{ss}&\approx{r}\times{N_\text{th}}\end{align}\right\}\qquad\text{below threshold},\quad{r\lt1}\]

as plotted versus \(r\) in Figure 13.7.

It is evident that until the pumping rate \(r\) becomes very close to the threshold value \(r=1\), the photon number in the cavity will be of order unity or a few orders of magnitude larger. Because the photon number \(n_\text{ss}\) will remain \(\ll{p}\) for \(r\lt1\), the saturation term \(1/(1+n_\text{ss}/p)\) in the denominator of the pumping Equation 13.47 will be negligible, and so \(N_\text{ss}\) will increase linearly with pumping rate \(R_p\) below threshold, as shown in Figure 13.7.

### Steady-State Behavior Above Threshold

We can also, however, rearrange the steady-state solutions to the same two rate equations 13.44 in the reversed forms

\[\tag{51}N_\text{ss}=\frac{\gamma_c}{K}\times\frac{n_\text{ss}}{n_\text{ss}+1}=\frac{n_\text{ss}}{n_\text{ss}+1}\times{N_\text{th}}\]

and

\[\tag{52}n_\text{ss}=\frac{R_p-\gamma_2N_\text{ss}}{KN_\text{ss}}=\frac{\gamma_\text{rad}p}{\gamma_2}\left[\frac{N_\text{th}}{N_\text{ss}}r-1\right]\]

From Equation 13.51 we can see that above threshold, or as soon as the photon number becomes very much greater than unity, the population inversion \(N_\text{ss}\) "clamps" at the threshold value \(N_\text{ss}\approx{N_\text{th}}\) (or, more precisely, at just a miniscule amount below \(N_\text{th}\)). At the same time, if \(N_\text{ss}\) is clamped at \(N_\text{th}\) then Equation 13.52 says that for \(r\gt1\) the cavity photon number is given by

\[\tag{53}n_\text{ss}\approx(r-1)\times\frac{\gamma_\text{rad}}{\gamma_2}\times{p}\]

For any reasonable ratio of \(\gamma_\text{rad}/\gamma_2\) and any pumping rate \(r\) above threshold, this says that

- The photon number \(n_\text{ss}\) will increase linearly with pumping power above threshold
- The photon number will be of the same order of magnitude as the mode number \(p\), which we have noted is a very large number (order of \(10^8\) to \(10^{10}\)) in most laser cavities.

The approximate formulas for the laser behavior above threshold are thus

\[\tag{54}\left.\begin{align}N_\text{ss}&\approx{N_\text{th}}\\n_\text{ss}&\approx(r-1)\gamma_\text{rad}p/\gamma_2\end{align}\right\}\quad\text{above threshold},\quad{r\gt1}\]

as illustrated in Figure 13.8. Note that the cavity photon number \(n_\text{ss}\) in the below-threshold region in Figure 13.8 is orders of magnitude smaller than the value above threshold, and does not even show up on the scale of the above-threshold photon number.

### Energy Transfer Rates Below and Above Threshold

Below threshold, all of the pumping power used in lifting atoms into the upper laser level is reemitted by the atoms as incoherent energy, in the form of radiative relaxation processes (spontaneous emission, or fluorescence), plus nonradiative relaxation processes (lattice phonons, wall collisions, and the like), with a combined relaxation rate of \(\gamma_2N_2\equiv\gamma_\text{rad}N_2+\gamma_\text{nr}N_2\).

The radiative part of this relaxation in particular can be pictured as a process in which the atoms spontaneously emit into all of the \(\approx{p}\) resonant modes within the cavity linewidth, and then the energy spontaneously emitted into these cavity modes immediately leaks out of the cavity into all directions as incoherent spontaneous emission.

As soon as the laser goes above threshold, however, the upper-level population \(N_\text{ss}\) clamps at the threshold value, and hence the incoherent relaxation out of this level (radiative plus nonradiative) also clamps just at the value it had at threshold.

All of the additional pumping power fed into the upper laser level above threshold then goes into, or is stolen by, the coherently oscillating cavity mode. The laser thus provides a kind of optical illustration of the maxim that "the rich get richer" (or perhaps "the coherent get more coherent").

To illustrate this point, let us assume for simplicity that the upper-level relaxation is purely radiative, so that \(\gamma_2=\gamma_\text{rad}\) and let us define \(P_\text{th}\equiv{R_{p,\text{th}}}\hbar\omega_a\) to be the total pumping power that is fed into the upper laser level just at threshold.

The total incoherent or spontaneous emission power \(P_\text{fluor}\) coming out of the atoms as they fluorescence into all directions, and the total coherent oscillation power \(P_\text{osc}\) coming out of the cavity in the one coherently oscillating cavity mode, will then be given, both below and above threshold, by the simple expressions

\[\tag{55}P_\text{fluor}\equiv\gamma_2N_\text{ss}\hbar\omega\approx\begin{cases}rP_\text{th}\qquad{r\le1}\\P_\text{th}\;\;\qquad{r\ge1}\end{cases}\]

and

\[\tag{56}P_\text{osc}\equiv\gamma_cn_\text{ss}\hbar\omega\approx\begin{cases}0\qquad\quad\qquad\;\;{r\le1}\\(r-1)P_\text{th}\qquad{r\ge1}\end{cases}\]

As illustrated in Figure 13.9, below threshold all the input power goes into incoherent emission; above threshold all the additional pumping power goes into the coherent oscillation output.

Of course, in any real laser system most of the pump power input is not used directly for exciting atoms to the upper laser level, but rather is wasted in pumping atoms up into unwanted levels or in heating up the laser medium. Nonetheless, all of what does go into the upper laser level is then converted into laser oscillation above threshold.

### Exact Results for the Threshold Region

The approximate results derived above are very useful for insight into the behavior of laser oscillation below and above threshold. We can, however, also obtain an exact expression for the cavity photon number \(n_\text{ss}\) versus pumping rate \(r\) that is valid for all values of \(r\) (within the very mild approximations of the rate-equation approach) by eliminating \(N_2\) between the two basic rate equations 13.44 and solving for \(n_\text{ss}\) versus \(r\).

Suppose for simplicity we assume again that \(3^*=1\) and in addition that \(\gamma_2=\gamma_\text{rad}\), i.e., that the upper level relaxes entirely by radiative relaxation into level 1. Then the exact steady-state solution to the two rate equations 13.44 at the start of this section is the rather innocuous-looking expression

\[\tag{57}n_\text{ss}=\left[(r-1)+\sqrt{(r-1)^2+4r/p}\right]\times\frac{p}{2}\]

Figure 13.10 is a plot of this expression over a range of pumping power centered about \(r=1\), showing how the cavity photon number jumps almost discontinuously from its below-threshold value of order unity, or slightly larger, to numbers of order \(p\), as the pumping rate increases by a very small amount, with a change of order \(1/p^{1/2}\), at threshold. Note the widely different logarithmic scales on the two axes of this figure.

It is virtually impossible to control the pumping rate \(R_p\) in a real laser to a precision of order \(1/p^{1/2}\); and it is equally difficult, for that matter, to measure the cavity photon number accurately over a dynamic range covering 8 or 10 orders of magnitude. Hence it is not surprising that when we gradually turn up the pump-power knob in a real laser, the onset of oscillation at the oscillation threshold point, where \(r\) passes through one, usually appears as an essentially discontinuous event.

### Oscillation Mode Discrimination

The sharpness of the photon number curve versus r, combined with the sudden clamping of the population inversion \(N_2\) at the threshold value (really just below the threshold value), also helps to explain how a laser cavity having some \(p\approx10^8\) or \(10^{10}\) potentially oscillating modes can actually oscillate and extract all the additional pumping input in just one preferred oscillating mode.

Suppose, for example, a laser cavity has a resonant mode #1 which is the "most preferred" mode, because it has the lowest losses and/or the best coupling to the laser atoms; plus a second cavity mode #2 which is slightly less preferred because it has higher losses or weaker coupling to the atoms or both.

Then, as Figure 13.11 shows, when the population inversion clamps at the threshold inversion for mode #1, the less preferred mode #2 will still be slightly below threshold (at least, in an ideal picture).

Hence this second mode will never be able to develop a sizable number of photons. The extraordinary sharpness of the threshold behavior and the large value of \(p\) help to explain how the photon number in mode #2 can always remain \(\ll{p}\), no matter how hard the laser is pumped, unless the difference in losses between the two modes is of order \(1/p^{1/2}\) or smaller.

The preceding description is, of course, highly idealized. In particular it neglects the spatial and spectral inhomogeneity effects that we discuss elsewhere in this tutorial. Only the population inversion in those atoms that are fully "seen" by mode #1 will be clamped at threshold.

If there are other atoms not fully seen and saturated by mode #1, but seen by mode #2, the population inversion on these other atoms can increase with increased pumping, and can pull mode #2 above threshold.

Most practical lasers will in fact oscillate in several, or even many, cavity modes at pumping levels well above threshold; and controlling or eliminating multimode oscillation is a continuing design problem in lasers.

Nonetheless, there are also real lasers which are sufficiently well controlled that they can generate large laser output powers in exactly one single laser cavity mode, in full agreement with our idealized model.

### Experimental Threshold Measurements on Injection Diode Lasers

Experimental measurements on lasers just at or below threshold are very difficult, both because of the sharpness of the threshold, and hence the extraordinary stability required in such experiments, and also because of the very weak signals emitted from a cavity containing only a few noise photons below threshold.

Semiconductor diode lasers, however, because of their very small cavity volume, can have a smaller than average mode density (\(p\approx10^5-10^6\)), giving them a comparatively "soft" threshold. Their very efficient direct-current pumping mechanism can also make threshold experiments somewhat simpler.

We have already seen in earlier tutorials how a single preferred axial mode can spring into oscillation, rising out of a cluster of amplified axial-mode noise peaks in an injection laser.

Figure 13.12 shows a more detailed measurement of how the output power in the dominant axial mode from an injection laser suddenly rises by a large amount as the laser current is increased by a very small amount just at threshold.

The light output below threshold in this situation may not represent a fully accurate measure of the photon number in this single preferred cavity mode, since the measurement apparatus may detect some of the below-threshold noise emission from other axial or near-axial cavity modes. Nonetheless, the general trend is clear.

Figure 13.13 also shows for two other injection laser diodes the sharp clamping of the upper-level population at threshold, as observed by measuring the spontaneous emission out the side or top of the diode as the diode current passes through threshold.

Figure 13.13(b) shows that if the face of the diode is scratched or damaged to prevent laser oscillation, the sharp "knee" at the threshold point disappears and the sidelight fluorescence continues to increase with increasing current.

The energy-level system in a semiconductor injection laser is both very broad and much more complicated than just a simple two-level system. Hence, for example, the fluorescent emission at longer wavelengths than the laser wavelength does not clamp as sharply.

This emission presumably comes from electronic levels slightly below the laser levels, whose population is not depleted or controlled as sharply by the laser action.

### Other Laser Threshold Measurements

It is also possible, with care, to make threshold measurements in other lasers, for example, in highly stabilized He-Ne lasers.

One preferred technique is to stabilize the laser pumping rate at a value well above threshold at the middle of the atomic gain profile; and then tune the cavity-mode frequency out to the point on the side of the atomic-gain curve where gain just equals loss.

By tuning the cavity resonance through a very small frequency range centered on this point, using piezoelectric-length tuning, we can pass smoothly and repeatedly from just below to just above threshold.

Figure 13.14 shows the normalized power output from a highly stabilized He-Ne laser (measured by photon counting techniques) as the normalized unsaturated gain or effective pumping rate r is varied about the threshold by \(\pm2\) parts in \(10^3\). It is possible to deduce from this data that the mode number for this cavity is \(p\approx5\times10^7\).

Finally, as another demonstration of the "clamping" phenomenon, Figure 13.15 shows the fluorescent emission below threshold and the laser emission above threshold (with greatly reduced detector sensitivity) for a group of closely adjacent transitions with a common upper level in an HgCl excimer laser.

The molecules in this laser are created in a \(v'=0\) excited state by a high-voltage electron beam passing through a high-pressure cell containing rare-gas mixtures with small traces of Hg and CCI\(_4\).

The left-hand diagram shows a small part of the energy-level structure of the HgCl molecule, with several of the spontaneous emission lines identified (note that these are different rotational quantum transitions or lines, not just different axial modes).

The \(v'=0\) to \(v^"=22\) transition is the strongest of these lines in fluorescence (strongest value of \(\gamma_\text{rad}\)), and it also reaches laser threshold first.

Once this transition oscillates, the population of the upper \(v'=0\) level is then clamped. (The populations of the lower \(v^"\lt22\) levels may also be increased once oscillations begin by cascading from the \(v^"=22\) level.)

The significant point is that, even with very much higher pumping, none of the other lines can be brought to threshold, except possibly for very weak transient oscillation of the next adjoining line at the highest pump level.

### Threshold Characteristics

A summary of the changes occurring in the cavity fields and in the output beam as a laser passes through threshold for a single mode thus includes the following.

- A sudden very large rise in power output in the oscillating mode.
- Clamping, more or less completely, of the upper-level population and hence of the sidelight fluorescence.
- A sudden sharp spectral narrowing, in which the frequency width of the signal radiation suddenly changes from broadband spontaneous emission (with essentially the bandwidth of the atomic transition) to emission of all the additional energy in one or a few essentially monochromatic axial modes.
- A sudden sharp spatial or output beam narrowing, in which instead of spontaneous fluorescent emission coming out randomly in all directions, the additional energy emerges as a more or less collimated, spatially coherent beam which is describable by (depending on the cavity) only one or a few transverse cavity modes.
- A hidden but important change in the statistical character of the laser radiation, from essentially gaussian random noise to a coherent amplitude-stabilized oscillation.

None of these last three items emerges directly from the rate-equation model used in this section. However, the fact is that the signal energy in the laser cavity below threshold is essentially random noise, with random phase and with random amplitude that varies about its mean value from instant to instant.

We refer to this as gaussian noise, because the instantaneous amplitudes of the cosine and sine frequency components of this noise are random variables with gaussian probability-density distributions and with no correlation between cosine and sine components.

(This means that the phase of the instantaneous phasor amplitude has a uniform distribution between 0 and \(2\pi\), whereas the magnitude of the phasor amplitude has a Rayleigh distribution.)

Above threshold, on the other hand, the laser oscillates (ideally) in a single mode with a coherent purely sinusoidal oscillation of the instantaneous \(E\) field, just like any coherent electronic oscillator at any frequency. The amplitude of the oscillating \(E\) field is highly stabilized (by the gain saturation feedback mechanism that stabilizes any laser oscillator), with only very small residual amplitude fluctuations about the mean value.

The phase of the optical oscillation is random, in the sense that there is no absolute phase or absolute clock to which a free-running oscillator will be stabilized. However, the phase of a good laser oscillator will stay essentially fixed for long periods of time (an enormous number of optical cycles), changing only through a slow random walk in absolute phase caused by small residual noise effects in the laser.

## 5. Multiple-Mirror Cavities and Etalon Effects

In this and the following section we consider a number of more complicated multiple-mirror cavity designs which can be used in practical lasers to help obtain various desirable laser properties such as bandwidth narrowing, axial-mode selection, or single-frequency laser operation.

### Intracavity Etalons for Frequency Tuning and Mode Control

We have already noted that in many kinds of lasers, including doppler-broadened gas lasers, most solid-state lasers, and organic dye lasers, the atomic gain profile can be much wider than the axial-mode spacing of the laser cavity; and the laser can then oscillate simultaneously over a broad spectrum of multiple axial modes, especially if the gain medium is at all inhomogeneously broadened.

It is then common practice, provided the laser gain is not too small, to insert a short, tilted intracavity etalon, or even several such etalons, inside the laser cavity, as shown in Figure 13.16, so that the narrowband frequency transmission of these etalons near resonance can provide frequency tuning and axial mode selection in the laser.

We have analyzed the transmission properties of such simple passive etalons in an earlier section.

In this particular situation, the tilt of the intracavity etalon must be kept small enough that it does not seriously reduce the transmission finesse of the etalon through transverse walk-off, yet large enough that the reflected waves from each side of the etalon pass out of the cavity and do not set up additional resonances with the other mirrors of the laser cavity.

The center frequency of such an etalon for transmission along the axial direction of the laser can then be varied by angle tuning, temperature tuning, piezoelectric tuning, or sometimes gas-pressure tuning.

### Multiple-Mirror Laser Cavities and Interferometers

Sometimes additional mirrors added to laser cavities may be deliberately aligned in resonance with the existing cavity mirrors to obtain multimirror laser cavities. The resonance frequency properties of such cavities then become more complicated, in ways that may be useful for various laser purposes.

Figure 13.17 illustrates a number of different multimirror cavity and interferometer designs that have been used or studied in connection with laser oscillators.

### Basic Analysis of the Three-Mirror Cavity

The simplest form of multimirror cavity is obviously the three-mirror cavity shown in Figure 13.17(a). The properties of such a resonant cavity with three or more mirrors are in general complex, and it may be useful to introduce briefly some of these complexities using a simple analytical model.

Suppose we consider first a general three-mirror cavity as shown in Figure 13.18. (Such a cavity, if two of the mirrors are closely enough spaced, can also be viewed as a two-mirror resonator with an etalon mirror on one end or the other.) As shown in Figure 13.18, let us assume this resonator has mirror reflectivities \(R_1\), \(R_2\), and \(R\), and use \(\tilde{g}_1\) and \(\tilde{g}_2\) to describe the round-trip gains inside the two cavity segments, leaving out the mirror reflectivities, so that

\[\tag{58}\tilde{g}_1\equiv\exp(-\alpha_1p_1-j\omega{p_1}/c)\qquad\tilde{g}_2\equiv\exp(-\alpha_2p_2-j\omega{p_2}/c)\]

with \(\alpha_1p_1\) and \(\alpha_2p_2\) representing the round-trip laser gains or losses, if any, inside each segment of the cavity. (Note that this is different from our earlier notation, where \(g_\text{rt}\) represented the complete round-trip gain inside an interferometer, including the end-mirror reflectivities.)

One simple way to analyze such a cavity is to use our earlier results to write down the complex amplitude reflectivity, call it \(r_2'\), looking into the \(R\), \(R_2\) section of this interferometer (of length \(L_2\)) from the left. We can then use this result for \(r_2'\) as the effective end reflectivity to write down the total reflectivity, call it \(r_1'\), looking into the \(R_1\), \(R\) cavity segment (of length \(L_1\)) again from the left.

The end result of this is that the total reflectivity looking into the three-section cavity from outside the \(R_1\) mirror can be written as

\[\tag{59}r_1'=\frac{r_1(1-rr_2\tilde{g}_2)-\tilde{g}_1(r-r_2\tilde{g}_2)}{1-rr_1\tilde{g}_1-rr_2\tilde{g}_2+r_1r_2\tilde{g}_1\tilde{g}_2}\]

If we then consider the denominator \(D(\omega)\) of this expression as a function of frequency, the complex values of \(\omega\) that give the roots of this denominator, i.e., that make

\[\tag{60}D(\omega)\equiv{r_1r_2}\tilde{g}_1(\omega)\tilde{g}_2(\omega)-rr_1\tilde{g}_1(\omega)-rr_2\tilde{g}_2(\omega)+1=0\]

will define the resonance frequencies and the decay rates for the resonant modes of this cavity.

### Basic Properties of Three-Mirror Laser Cavities

Depending on the relative reflectivities and spacings of the cavity mirrors, one might view a three-mirror cavity of this type either as two semi-independent resonant cavities of lengths \(L_1\) and \(L_2\) coupled together by transmission through the central mirror \(R\); or alternatively one might consider this as a single long cavity of total length \(L_1+L_2\) with an internal perturbation produced by the mirror \(R\); or as a single cavity of length \(L_1\) with an etalon mirror of length \(L_2\) on one end. Various different analytical approximations can then be used to calculate the cavity resonant frequencies and losses from Equation 13.60.

As a general rule, however, the resonance properties of the three-mirror cavity are sufficiently complex that the use of numerical solutions and computer display techniques can be very helpful, if not essential, in finding and understanding the resulting cavity modes. We will show here only one or two examples of such solutions, to illustrate the type of behavior that can result.

Figure 13.19, for example, shows how the resonant frequencies of a three-mirror cavity shift and how the mode losses change in a typical situation if we vary the reflectivity \(R\) of the central mirror, with fixed reflectivities \(R_1\) and \(R_2\) for the end mirrors.

The cavity segments are assumed to be lossless in these particular plots, except for the finite mirror reflectivities, and the height of each spectral component is proportional to the energy decay rate for that resonance component.

The longer cavity segment \(L_1\) in this particular situation is assumed to be three times as long as the shorter cavity segment \(L_2\) on a macroscopic length scale, so that the overall axial mode spectrum will repeat periodically with a period corresponding to the axial mode spacing \(2\pi\times{c}/2L_2\) of the shorter cavity.

The spectral behavior will also depend strongly, however, on how the axial modes of the two individual cavities are "micro-tuned" with respect to each other. Parts (a) and (b) of Figure 13.19 illustrate the variation in cavity spectrum with central mirror reflectivity \(R\) when the two individual cavities are adjusted so that an axial mode characteristic of the short cavity by itself falls either exactly on top of, or halfway in between, the axial modes of the longer cavity. The dashed horizontal lines indicate the mode losses that would occur for the overall cavity \(L_1+L_2\) with no central mirror.

Some examination of Figure 13.19 is worthwhile. There are, of course, four axial modes per repetition period, since the overall cavity length is four times \(L_2\). It is then evident that the coupling between the two cavity segments produces both large variations in loss, and also strong frequency pulling effects on the modes, and both of these effects depend strongly on how the two cavity segments are tuned relative to each other.

Figure 13.20 similarly shows the variation in mode losses and resonant frequencies for a cavity with fixed mirror reflectivities and macroscopic lengths \(L_1\approx4L_2\) as the shorter cavity is tuned through one of its separate axial mode intervals. This example corresponds in essence to an etalon-mirror cavity in which the etalon mirror (length \(L_2\)) is continuously scanned in frequency through one of its axial mode intervals.

Note that the location of the low-loss region in this spectrum tunes more or less continuously across one full mode interval of the etalon mirror as the etalon is tuned. No single axial mode of the combined cavity tunes in this fashion, however, and there is clearly a discontinuous "mode jump" in the lowest-loss mode in the middle of the tuning range.

If we wish to select and tune a single axial mode across the full tuning range in an etalon cavity, it is obvious that simply tuning the etalon mirror is not enough. We must instead somehow tune the lengths of both cavity segments simultaneously, so that the lowest loss mode of the overall cavity tracks the high-reflectivity region of the etalon mirror.

### Applications of Three-Mirror Laser Cavities

Simple three-mirror cavities as discussed in the preceding have found some direct applications in lasers.

If, for example, we cleave a short semiconductor diode laser into two segments somewhere near the center, carefully maintaining the alignment of the two sections, and then attach separate current leads to the two sections, the result has been called the "cleaved coupled cavity" or \(C^3\) type of injection laser.

Both the gain and the optical length of each segment can be individually controlled in this situation; and this design has been found to have potential advantages in maintaining a single axial mode, without "mode hopping" effects, over a wide range of injection currents.

Three-mirror or etalon-mirror cavities of this type do not usually provide the optimum design for achieving single-axial-mode operation in low-gain inhomogeneously broadened gas lasers, however, although etalon mirrors are often used in high-gain pulsed solid-state lasers.

One reason for this is that, as Figure 13.22 shows, a high-reflectivity or low-loss etalon normally provides a narrow transmission peak and thus a broad reflection band, whereas what is wanted for axial mode control is a narrow reflection peak.

For this same reason, the Michelsonmirror cavity design shown in Figure 13.17(c), which has a sinusoidally varying reflectivity versus frequency, is also usually a less than optimum design.

### Fox-Smith Interferometers, and Other Multimirror Designs

More complex but preferred cavity designs for axial mode selection are then provided by one or another of the alternative forms of the Fox-Smith interferometer shown in Figures 13.17 (d) and (e).

Note that in both forms most of the signal energy circulating in the primary cavity will be reflected out of this cavity at most frequencies, except at those frequencies where the secondary cavity becomes resonant and builds up a large internal amplitude.

This design thus does provide the desired narrow reflection peak as shown in Figure 13.22(b).

Reasonably good mode selection can also be obtained in low-gain lasers using the "vernier Michelson" cavity design shown in Figure 13.17(g). Here, high selectivity is obtained by placing a laser tube in each arm of the Michelson interferometer; and the vernier action results from interference effects between the two long arms, which are made very nearly but not exactly the same length.

### Cavity Back-Reflection Effects

We might also note once again that laser cavities are often very sensitive (in power output, frequency tuning, and oscillation stability) to any back-reflection of the laser signal from external optical components directly back into the laser cavity.

These back-reflection effects can of course be understood as multimirror cavity effects of the type described in this section, with the external cavity usually being both long and mechanically unstable.

If the external back-reflectivity is small, this means that the external cavity segment is very lossy, or has low effective reflectivity; but the very high \(Q\) of the oscillating laser cavity segment can still mean that the effects of even weak backscattered signals can be very significant.

## 6. Unidirectional Ring-Laser Oscillators

Ring-laser cavities were understood and demonstrated very early on, and have since been extensively developed for application in ring-laser gyroscopes. Ringlaser cavities possess one unique capability as compared to standing-wave cavities, namely they have the ability to oscillate, simultaneously or independently, in either of two distinct counter-propagating directions.

Ring resonators also possess a number of other attributes which can be very useful in several different laser and passive interferometer applications. Full appreciation of the advantages of ring resonators in optical applications has only emerged more recently, and it seems useful therefore to give a brief summary in this section of the special properties of ring-laser resonators.

### Example of a Unidirectional Ring-laser Cavity

Figure 13.23 shows, by way of example, a typical folded ring resonator design as used in a commercially produced cw dye laser.

The ring cavity in this example contains not only the dye-jet gain medium, several frequency control etalons and filters, and an astigmatism compensating element, but also a unidirectional device ("optical diode") which allows oscillation to occur in only one direction around the ring.

This figure also illustrates the array of diagnostic elements which are used, in conjunction with electronic feedback loops, to control the etalon elements, the piezo mirror control, a double galvoplate, and the birefringent optical filter, all needed to give single-frequency laser operation tunable over a wide tuning range.

The primary advantage to unidirectional oscillation in a ring laser such as this is that the purely traveling-wave rather than standing-wave operation eliminates spatial hole-burning effects, making the laser medium in effect much more homogeneous.

This in turn substantially increases the mode competition between adjacent axial modes, making it possible to pump the laser considerably further above threshold while maintaining single-frequency operation.

In addition, because the traveling-wave mode saturates the gain medium uniformly, with no spatial nodes along the axial direction, this mode can extract more power than would otherwise be obtained.

The combined effect can be an increase in single-frequency power output by more than an order of magnitude compared to what can be obtained using a standing-wave cavity in a typical dye laser example. Similar advantages can be obtained in other lasers, for example, pulsed solid-state lasers, as well.

### Other Attributes of Ring Resonators

Other potentially useful attributes of ring optical resonators include:

- Increased cavity design flexibility and alignment insensitivity. We will point out later on, in the resonator tutorials of this series, that a ring optical cavity provides increased flexibility in resonator design as compared to a standing-wave cavity, especially for unstable resonator designs.

In particular, a ring cavity can easily employ a comparatively short beam expansion section, using readily available short-focal-length optical elements, and then a long collimated beam section at large beam diameter, for obtaining full power extraction from large diameter laser gain media.

(Figure 13.23 shows, by contrast, the way in which very small focal spots, for use with intracavity dye jets or modulation elements, can equally easily be obtained inside a ring-laser resonator.)

Ring resonators also offer the possibility of using prisms of various sorts in place of mirrors in forming the ring; and this aspect has been used in ring-laser designs such as Figure 13.24.

A planar ring resonator also has the interesting attribute of being first order insensitive to misalignments in the plane of the ring. That is, when any element is misaligned by a small amount in the plane of the ring, the resonator mode can always respond by making small changes in beam position and direction to find a new closed and aligned path in the plane of the ring.

- Elimination of input feedback, and reduced sensitivity to back reflection. We have noted earlier that when an external signal is injected into a ring resonator or ring interferometer, the reflected-plus-transmitted signal from the input mirror goes off in a different direction, with no optical feedback directly back into the external signal source.

This can be very useful for laser injection locking experiments, where feedback from a high-power locked oscillator back into the much lower-power injection source can be a major experimental problem.

A unidirectional ring oscillator can also be less sensitive to feedback from external reflections placed in its output beam, since these reflections go into a non-oscillating direction in the ring.

Similar considerations apply when a passive ring resonator is used as a scanning interferometer or frequency filter for laser diagnostics or frequency stabilization. Elimination of feedback from the passive cavity in this situation can minimize instability effects in the laser being studied.

- Single-pass operation of intracavity elements. Intracavity elements, such as modulators, harmonic generation crystals, and the like, are excited in only one direction in a unidirectional ring-laser oscillator.

While this may in many situations reduce the modulation efficiency or harmonic generation efficiency of the element, it can also simplify certain experiments and permit more accurate measurements on intracavity experimental cells or samples.

The order in which optical elements are encountered is also inherently different going in the two directions around a ring. Going in one direction, for example, a wave may see first the laser-gain medium, then a saturable absorber, and then the output coupler, whereas the order is obviously reversed in the opposite direction.

This can be used to control power levels and saturation intensities in different elements, and can be a source of directional nonreciprocity in a ring-laser oscillator.

### Ring-laser Disadvantages

The primary disadvantage of the ring resonator for laser applications, leaving aside the additional complexity and structural requirements, is probably that the gain medium is traversed only once.

A low-gain laser will thus operate considerably closer to threshold, and have more stringent requirements on reducing the output coupling, and especially the internal losses, if good efficiency is to be maintained.

(Of course, other lossy intracavity elements other than mirrors are also encountered only once rather than twice per round trip.)

The astigmatism produced by off-axis reflection from cavity mirrors must also be taken into account in the resonator design. This astigmatism can even be an advantage in some situations, however, and can usually be compensated for when it is not.

### Techniques for Obtaining Unidirectional Oscillation

The mode competition between two potentially oscillating modes in a ring laser (or in any other multimode laser situation) is in general a complex problem.

As we will show in later sections, depending on mode losses, mode cross-coupling, and mode self-saturation and cross-saturation properties, competition between two modes may lead to stable single-mode operation; to simultaneous dual-mode operation; or to random jumping back and forth between the two potential modes.

This applies especially to the oppositely traveling waves in a ring resonator, and several different techniques for obtaining or improving unidirectional operation in ring resonators have been demonstrated.

One of the simplest of these is to employ an auxiliary external mirror, having partial or complete reflectivity, to reflect part of, say the CCW circulating output back into the CW direction, as illustrated in Figure 13.25

If this cavity attempts to oscillate in the CW direction, the resulting back-reflected signal will serve as an injected signal for the CCW direction, leading to a much stronger oscillation in the CCW direction.

This technique does work as intended, at least crudely and in some situations. If the laser medium is otherwise homogeneous, the preferred CW oscillation does grow at the expense of the CCW oscillation.

The CCW oscillation is not fully extinguished, however, but remains as a low-amplitude injection signal to drive the CW oscillation. Intensity ratios between the two directions in the range of 10:1 to 50:1 have been reported in typical situations.

For inhomogeneously broadened materials the technique may not work at all, especially when the centermost axial mode of the ring cavity is not at line center. In this situation the ring may oscillate with equal intensity in both directions, and may also oscillate in several axial modes at once.

This scheme is also sensitive to internal backscattering inside the ring, which can interact interferometrically with the external mirror. These interactions make the basic technique fundamentally unsound when finite backscattering is taken into account.

### Nonreciprocal Optical Diodes

A much preferable solution is to place a nonreciprocal optical isolator, sometimes referred to as an "optical diode," inside the ring cavity to introduce nonreciprocal losses in the two directions. Figure 13.26 shows the basic elements of such an optical diode.

The primary component is a Faraday rotation device using a transparent material with a finite Verdet constant placed in a dc magnetic field.

When a linearly polarized optical wave passes through such an element, its plane of polarization is rotated about the optical axis, with a direction of rotation which depends on the dc magnetic field direction but not on the direction of travel of the wave.

To make an optical isolator, a second purely reciprocal rotation element is added to cancel out the Faraday rotation going in one direction through the system. This reciprocal element may be an optically active crystal or liquid (e.g., quartz, or a sugar solution), or a birefringent crystal used as a partial wave plate.

The reciprocity properties of this system then mean that the total polarization effects due to the two elements going in the forward direction through the system can cancel each other, giving no net polarization change, whereas the effects of the two elements going in the opposite direction will add to give a finite modification of the wave polarization.

A linearly polarized wave going in the forward direction through the system can then pass unattenuated through a Brewster angle plate (or some other type of polarization-sensitive element) on successive round trips, whereas a wave going in the opposite direction, because of the net polarization rotation, will experience added loss on each round trip.

(It should be noted that this additional loss is in general not given simply by calculating first the polarization rotation of a linearly polarized wave in the optically active elements and then the transmission of the resulting wave through the Brewster plate.

We must instead use some form of polarization calculus to calculate the total propagation of two orthogonal polarization components around the ring, as well as the cross coupling between these polarization components in each optical element; and the use these results to find the two polarization eigenmodes and associated eigenvalues for the cavity.

A cavity which contains birefringent or polarizing elements will in general have two such mixed polarization eigenmodes, neither of which will generally be as lossy as predicted by the simple approach given in the preceding.)

Useful Faraday rotators for optical wavelengths are difficult to obtain in practice, primarily because the physical basis of Faraday rotation is the anisotropic tensor response \(\chi'(\omega)\) on the side of some very strong, and Zeeman split, atomic transition. Materials with large Verdet constants (i.e., large polarization rotation per unit of dc magnetic field) are thus most often also highly absorbing, whereas highly transparent materials typically have very small Verdet constants.

In practice the optical diodes used in ring lasers typically have Faraday rotations of a few degrees, and differential losses between the two directions of a percent or so. This additional insertion loss in the reverse direction is, however, enough to strongly suppress oscillation in the reverse direction, and produce highly selective oscillation in the forward direction only.

### Nonplanar Ring Resonators

We will also point out later in this tutorial series that nonplanar ring resonators can provide unique image rotation and also polarization rotation properties.

These properties have been employed recently to develop a unique monolithic solid-state laser with inherent unidirectional properties, as illustrated in Figure 13.27.

The material used here is Nd:YAG, which has both laser gain and a finite Verdet constant. A small crystal cut as shown in this figure then provides a nonplanar ring resonator which employs total internal reflection at all but one of its surfaces.

The polarization rotation inherent in the nonplanar ring path is then compensated in one direction, but not in the other, by the Faraday rotation produced by a dc magnetic field.

The crystal thus oscillates inherently in only one direction around the ring, and as a consequence also achieves high-quality single-frequency operation.

## 7. Bistable Optical Systems

The equations of motion for a laser oscillator, or more generally for any system of coupled fields and atoms, are intrinsically nonlinear (although we often make linear approximations to these equations). It has been increasingly realized in recent years that one can often obtain in such nonlinear systems interesting and fundamental types of bistable, multistable, self-pulsing, and even chaotic behavior.

In this section, therefore, we briefly introduce some of the interesting bistability properties of lasers and also of passive optical cavities. These bistability properties may someday find practical applications in optical computers, "optical transistors," or other optical signal-processing devices, although the real practicality of any such all-optical computer systems remains at present still in considerable doubt.

### Bistable Laser Oscillation

As the simplest example of a bistable laser oscillator, we can consider a laser cavity containing both a homogeneously saturable gain medium and a homogeneously saturable atomic absorber.

Suppose that at small signal levels the combined saturable and nonsaturable losses in this cavity exceed the unsaturated gain, so that the cavity cannot begin oscillating spontaneously starting from noise. The laser thus has one stable operating point in the totally quiescent condition, with no signal present.

Suppose, however, that the cavity losses saturate much more easily with increasing signal intensity than does the laser gain—that is, the absorbing atoms have a lower saturation intensity than do the amplifying atoms.

At high- enough signal intensities the saturated round-trip losses can then drop below the saturated round-trip gain, as shown in Figure 13.28. If this laser can ever start oscillating, therefore—perhaps with assistance from some externally injected signal—it will build up to a large circulating intensity and remain oscillating until it is turned off.

This simple system thus exhibits bistable behavior, with two stable steady-state operating points, as shown in Figure 13.29. There is also a third potential steady-state operating point where gain also equals loss, at the first crossing of the saturable loss and saturable gain curves. However, it can readily be shown that this is not a stable operating point for the laser.

A laser of this type can also exhibit a strong hysteresis in the variation of output power with pumping power, as illustrated in Figure 13.29. The results shown there are for a cw CO\(_2\) laser with an intracavity cell containing 25 mm of gaseous SF\(_6\) as a saturable absorbing medium.

When the pumping current \(I_0\) is turned up from zero, laser action cannot start until the laser gain exceeds the laser cavity losses plus the unsaturated losses of the SF\(_6\) cell.

Once the laser starts oscillating, however, the SF\(_6\) absorber cell is saturated, and we can then reduce the pumping current to a considerably lower value before the laser will suddenly drop out of oscillation.

### Analysis of a Nonlinearly Absorbing Cavity

As an even simpler example of a bistable optical system, we can consider a passive Fabry-Perot interferometer, of either the standing-wave or the ring-cavity type, containing only a simple passive saturable absorber, and driven by an externally applied optical signal.

Suppose such a passive Fabry-Perot cavity has input and output mirrors with reflectivities \(R_1=r_1^2=\exp(-\delta_1)\) and \(R_2=r_2^2=\exp(-\delta_2)\) and a round-trip power attenuation coefficient due to a saturable atomic absorber of \(\delta_m\equiv2\alpha_mp_m\).

If this cavity is driven by an externally incident signal \(I_\text{inc}\) which is tuned to the cavity resonant frequency, then the internal circulating signal \(I_\text{circ}\) and the transmitted signal field \(I_\text{trans}\) from the cavity will be given by the elementary interferometer relations derived in earlier sections.

In particular, if we assume that all of the loss factors are small compared to unity, then the incident, circulating, and transmitted intensities from this cavity will be related by the expressions

\[\tag{61}\frac{I_\text{trans}}{I_\text{inc}}\approx\frac{4\delta_1\delta_2}{[\delta_1+\delta_2+\delta_m(I)]^2}\equiv{T(I)}\]

and

\[\tag{62}\frac{I_\text{circ}}{I_\text{inc}}\approx\frac{4\delta_1}{[\delta_1+\delta_2+\delta_m(I)]^2}\equiv\frac{T(I)}{\delta_1}\]

where \(T(I)\equiv{I}_\text{trans}/I_\text{inc}\) is the intensity-dependent power transmission through the cavity. Let us assume that the internal atomic absorption saturates in the homogeneous fashion

\[\tag{63}\delta_m=\delta_m(I)=\frac{\delta_{m0}}{1+2^*I_\text{circ}/I_\text{sat}}=\frac{\delta_{m0}}{1+I}\]

with \(2^*\equiv1\) for a ring cavity and \(2^*\equiv2\) for a standing-wave cavity, and where \(I\equiv2^*I_\text{circ}/I_\text{sat}\).

Then by picking successively increasing values of the circulating intensity \(I_\text{circ}\) we can calculate first the intensity-dependent transmission gain \(T(I)\), and then calculate and plot the transmitted intensity \(I_\text{trans}\) versus the incident intensity \(I_\text{inc}\) for the interferometer.

If we assume for simplicity that the input and output couplings are the same, \(\delta_1=\delta_2\), then the power transmission \(T(I)\) through the interferometer can be written as

\[\tag{64}T(I)=\left[\frac{1}{1+R/(1+I)}\right]^2=\left[\frac{1+I}{1+R+I}\right]^2\]

where \(R\) is the ratio of unsaturated gain to total coupling, as defined by

\[\tag{65}R\equiv\frac{\delta_{m0}}{\delta_1+\delta_2}\]

It is further convenient to define normalized input and output signal intensities for the cavity by

\[\tag{66}I_1\equiv{E_1^2}\equiv\frac{2^*I_\text{inc}}{\delta_1I_\text{sat}}\qquad\text{and}\qquad{I_2}\equiv{E_2^2}\equiv\frac{2^*I_\text{trans}}{\delta_1I_\text{sat}}\]

and then to eliminate the internal circulating intensity \(I\equiv2^*I_\text{circ}/I_\text{sat}\) between the preceding equations. The input-output field relationship then takes the simple form

\[\tag{67}E_1=E_2\left[1+\frac{R}{1+E_2^2}\right]\]

Figure 13.30 shows the nonlinear input-output relationship that is produced by this type of saturable interferometer transmission. A multivalued input-output relation occurs in this simple situation only if the ratio of unsaturated losses to total cavity coupling has a value \(R\ge8\).

### Absorptive Bistability

A saturable-absorber cavity of this type will, as a consequence, exhibit the general type of bistable input-output hysteresis behavior shown in Figure 13.31.

That is, if we turn up the input intensity to this cavity, starting from low values, the circulating intensity inside the cavity will at first not be greatly enhanced, because of the sizable absorption losses and hence low finesse of the interferometer.

When the incident signal level reaches a certain value marked by the first turning point in Figure 13.31, however, the circulating intensity will become large enough to begin to saturate the absorption.

The cavity finesse will then begin to increase, which means the circulating intensity inside the cavity will also begin to increase for the same incident power level, thus causing a further increase in cavity finesse and in circulating intensity.

The cavity operating point will then suddenly jump upward in a discontinuous fashion to the upper branch, where the cavity losses are essentially saturated and the cavity finesse, circulating intensity, and transmitted intensity are all much larger than on the lower branch.

If the incident intensity is then reduced, the much higher finesse of the cavity on the upper branch makes it possible for the internal circulating intensity to remain above the saturation level even with much smaller input intensity.

The cavity will thus move back down along the upper branch, until at a certain point it drops discontinuously back to the lower branch. The portion of the input-output curve between these two discontinuities, marked by a dashed line, is unstable and cannot be a steady-state solution.

### Dispersive Optical Bistability

An analogous but physically different (and generally more useful) type of bistability can also occur in a passive interferometer cavity containing a nonlinearly dispersive rather than absorptive medium.

Consider, for example, a simple ring or standing-wave interferometer cavity containing an optical Kerr type of material in which the optical refractive index n changes as the optical intensity is increased, in the form for example \(n(I)=n_0+n_2I\), where \(I\) is the circulating intensity inside the cavity.

As the circulating intensity changes, therefore, the resonant frequency of the cavity will change, and this will in turn change the relationship between the input, circulating, and output intensities in a manner which can lead to a variety of complex bistable and multistable behavior.

The incident signal in this situation need not be tuned to the small-signal resonant frequency of the cavity. The nonlinear behavior in this system can instead be examined using the following simple graphical analysis. If we again make use of results from earlier tutorials, the power transmission \(T(I)\) through the cavity can be written as

\[\tag{68}T(I)\equiv\frac{I_\text{trans}}{I_\text{inc}}=\frac{1}{1+F\sin^2\phi(I)/2}\]

where \(F\) is the finesse of the cavity and \(\phi(I)\) is the round-trip phase shift. For a cavity filled with an optical Kerr material and excited at a free-space wavelength \(\lambda_0\) this phase shift will be given by

\[\tag{69}\phi(I)=\frac{2\pi{p}}{\lambda_0}[n_0+n_2I_\text{circ}]\]

where \(p\) is the round-trip length in the cavity.

Figure 13.32 plots this transmission gain \(T(I)\) through an optical cavity containing a lossless optical Kerr material, assuming that the cavity resonance is tuned well away from the applied signal frequency at small signal levels, and that the intensity is varied over a wide enough range to shift several axial modes of the cavity through the applied signal frequency. This curve will be shifted transversely, depending upon how the incident signal is tuned.

But, in addition the input, circulating, and output intensities are also related by the expressions \(I_\text{trans}=\delta_2I_\text{inc}=TI_\text{inc}\), or \(T(I)=\delta_2I_\text{circ}/I_\text{inc}\), which corresponds to a set of straight lines in the \(T(I)\) versus \(I_\text{circ}\) plot, with slopes which decrease with increasing input intensity \(I_\text{inc}\).

Figure 13.32 then shows, for example, how for a low incident intensity \(I_\text{inc}=I_1\) there is only one operating point at which these two formulas intersect. The cavity frequency in this situation is essentially unshifted from its low-intensity value, and the cavity is operating at low transmission, well off resonance.

At a larger incident intensity \(I_\text{inc}=I_2\), however, there are three potential operating points, one at low transmission well off resonance, and two others within the high-transmission resonance peak. Only two of these points, however, indicated by the solid circles, represent stable operating points.

At the outermost operating point, for example, the cavity is operating close to one of its axial mode resonances, but with the intensity-shifted cavity resonance frequency slightly below the input signal frequency.

If the incident power level increases slightly, then the circulating intensity increases, and this in turn drives the cavity resonance frequency slightly lower (by increasing the value of the intracavity index \(n\)).

But, this moves the cavity resonance frequency slightly further away from the applied signal frequency, causing the circulating power to decrease and thus partially canceling the effect of the increased incident power.

Operating points on the upper sides of the cavity resonances are thus stable, whereas operating points on the lower sides, by a similar argument, are perturbation unstable.

This same plot also shows that if the incident intensity is still further increased to the value \(I_\text{inc}=I_3\), multistable behavior with three or more potential operating points also becomes possible.

### Fluctuations and Self-Pulsing Phenomena

These two examples illustrate the elementary properties of purely absorptive and purely dispersive optical bistability or optical multistability.

More generally, if one considers an optical cavity containing a general two-level resonant atomic system, then we will have an even more complex situation, with a mixture of absorptive and dispersive nonlinear properties, depending upon how the atoms and the cavity are tuned.

The atomic system may also exhibit either homogeneous or inhomogeneous saturation behavior, depending upon its type; and at larger signal levels we may have to take into account Rabi flopping effects in addition to the nonlinear saturation behavior of the atoms.

The net result of all this is a very rich and complex variety of nonlinear behavior in passively excited optical cavities containing saturable atoms. In addition to the simple bistability and hysteresis behavior illustrated in the preceding, at larger input intensities many of these nonlinear optical systems will exhibit spontaneous periodic fluctuations, in which the cavity intensity jumps back and forth at a regular rate between two branches of the input-output curve, thus converting a steady-state input beam into a pulsed output beam.

These jumps have a close conceptual relationship to phase transitions in atomic systems, and to limit cycle behavior in other nonlinear systems. In fact, it has been realized in recent years that many different nonlinear systems, ranging from optical cavities to mechanical systems and fluid-flow problems, can all exhibit broadly similar nonlinear properties.

As we turn up the excitation intensity, or some other kind of gain parameter in a nonlinear system, we often see at first some kind of bistable or hysteresis behavior, as illustrated in the preceding.

This may be followed by a periodic pulsing behavior, and this periodic behavior may at higher intensities show a discontinuous jump in the pulsation frequency, often to half the previous frequency (referred to as period doubling).

At each of these discontinuities if we suddenly switch the incident intensity to a value well above or well below the discontinuity point, then the transition from one form of behavior to another occurs very rapidly.

If the incident intensity is only moved a very small amount beyond the transition point, however, then the transition from one branch to the other occurs only very slowly, a phenomenon referred to as critical slowing down.

### The Transition to Chaos

Finally, such nonlinear systems may often, as the excitation parameter is varied, suddenly jump to a very distinctive new form of behavior referred to, with good reason, as chaos.

In the chaotic region, even though the equations of motion for the system are entirely deterministic and may contain only a few parameters, and the system input is constant, still the resulting system behavior (e.g., the cavity output intensity) fluctuates wildly with time, in what seems to be a totally random fashion.

The power spectrum for the cavity amplitude fluctuations, for example, will apparently have a continuous distribution in frequency, with no observable discrete frequency components.

A passive optical cavity containing an ideal optical Kerr material, for example, with an externally applied cw signal, can pass through discrete regions of bistable behavior, then periodic fluctuations, then various sorts of period doublings, and then various sharply defined regions of chaotic behavior, as the incident signal intensity is slowly increased.

The turbulence which invariably develops in a fluid flow above a sharply defined Reynolds number is another elementary illustration of chaos. These chaotic phenomena do not seem to depend on any fundamental noise sources in the system, and exhibit striking similarities across widely different physical systems.

### Experimental Results

All of the preceding-mentioned nonlinear phenomena, including bistability, multistability, periodic fluctuations, period doubling, and chaos, have in recent years been both predicted and experimentally observed, although often only with some difficulty, in optical cavities. In general, nonlinear dispersive and mixeddispersive effects are more easily obtained (as well as more interesting) than purely absorptive effects.

For example, by combining tunable dye lasers with the very strong but narrow resonance transitions in metal vapors, such as sodium or rubidium cells, one can demonstrate many atomic nonlinear phenomena using reasonable input powers and detection speeds. It is difficult to envision practical optical signal-processing devices using this approach, however.

Optical Kerr effects are also often observed using molecular liquids such as CS\(_2\), although only with comparatively high optical powers and fast pulses. Certain semiconductors, such as GaAs and InSb, and also semiconductor quantum well structures, can also exhibit strong optical nonlinearities at wavelengths near or just beyond their optical absorption edges; and there is much interest in these materials for possibly practical bistable optical systems.

## 8. Amplified Spontaneous Emission and Mirrorless Lasers

Some laser systems have such extremely high gain that they need no mirrors— they can emit very bright and more or less quasi coherent beams out each end of the laser medium simply as a result of very high-gain amplification of their own internal spontaneous emission traveling along the length of the laser-gain medium.

Interstellar masers and x-ray lasers must also of necessity operate without mirrors, since no mirrors are available.

Interesting concepts that have been developed in connection with this kind of behavior include such terms as super radiance, super fluorescence, coherence brightening, and amplified spontaneous emission (ASE).

In this section we will attempt to give a brief classification and explanation of each of these terms, together with a brief summary of the useful properties of mirrorless laser systems.

### Coherently Oscillating Dipoles and Free Induction Decay

In developing this topic it may be easiest to begin with the more exotic and strongly coherent forms of behavior, and work down toward the simplest and most common kinds of mirrorless or ASE lasers.

The first part of the following discussion will thus be closely related to the coherent-pulse and coherent-dipole types of behavior that we also discuss in other sections of this tutorial series.

We have pointed out elsewhere, for example, that if a collection of two-level atoms is prepared such that the individual atomic dipoles are oscillating or precessing at least partly in phase with each other, then the associated macroscopic polarization \(p(t)\) in the collection of atoms will emit electromagnetic radiation in a coherent fashion—that is, the emitted radiation will be coherent or sinusoidal in time, with a time-phase determined by the initial preparation of the atoms.

This radiation will also have directional or spatially coherent properties determined by the relative phases with which the radiating atoms at different points are initially set oscillating.

Such a coherently prepared atomic system may occupy a volume that is large in terms of optical wavelengths. If the initial atomic oscillations are then prepared using, for example, a traveling optical pulse of sufficiently large intensity, the resulting coherent emission will emerge in the same direction of travel as the preparing pulse.

If the degree of initial coherence imposed on the individual oscillators is comparatively small, and if the atomic populations are initially either not inverted, or at most have small gain, then this coherent radiation, although brighter and more directional than the usual spontaneous emission, will be relatively weak; and the coherently radiated signal will decay in time with the appropriate dephasing time \(T_2\) in homogeneous systems, or \(T_2^*\) in inhomogeneous systems, until it disappears into the incoherent spontaneous emission background from the same atoms.

This particular kind of coherent atomic radiation is often referred to as simple free-induction decay. Free-induction decay can be demonstrated experimentally both in low-frequency magnetic resonance systems and in optical-frequency atomic systems using pulsed laser excitation, as we describe elsewhere in this tutorial series.

### Dicke Superradiance

At a time well before the invention of the laser, R. H. Dicke also considered analytically the situation in which a sizable number of atoms contained in a small volume \(V\) may all be oscillating with a very high degree of coherence between the individual dipole oscillations.

If such a volume contains \(N\) coherently oscillating atomic depoles, the macroscopic dipole moment within the volume will have magnitude \(N\mu_1\), where \(\mu_1\) is the oscillating moment of a single atom.

The rate of coherent radiative power emission from this volume will then be proportional to \((N\mu_1)^2\), in contrast to the usual incoherent form of spontaneous emission, where the emission rate is proportional only to the number of atoms \(N\).

The coherent emission from this small but coherently excited volume will emerge as a short burst or pulse of radiation with a duration proportional to \(1/N\), rather than as an exponential decay with a lifetime r independent of the number of atoms.

This specific type of small-volume, coherently prepared emission, with strong initial coherence, has come to be known as Dicke superradiance.

The atoms here are locked together, not only by their initial preparation all in phase with each other, but also by the strong coupling of all the atoms to each other through their common radiation fields.

Dicke superradiance of this type has been observed in specially prepared low-frequency magnetic resonance systems, but not, at least in its simplest form, in optical-frequency systems.

### Incoherently Prepared Dicke Superradiance

Suppose next that a two-level system having \(N\) atoms is initially prepared with a completely inverted population, i.e., \(N_1=0\) and \(N_2=N\), so that each of the atoms is completely in its upper energy level to start with.

The atomic system will then initially possess no coherent macroscopic polarization \(p(t)\), since the quantum expectation value for the dipole oscillation of each individual atom is zero if the atom is entirely in its upper (or for that matter, its lower) quantum state.

(As an alternative, the atoms may be prepared in an only partially inverted state, but with an incoherent preparation method, such that all the atomic dipoles are randomly phased and no coherent macroscopic polarization is initially present.)

Each upper-level in this situation will then begin to radiate spontaneously and incoherently through the purely quantum spontaneous emission processes that are represented by the radiative decay rate \(\gamma_\text{rad}\).

Note that this spontaneous emission process, although it can be modeled by a gaussian quantum noise source, can only be derived from a completely quantum analysis, in which the atoms and the electromagnetic field are both quantized.

Dicke then pointed out in his original paper that if the \(N\) atoms prepared in this inverted but incoherent fashion were all contained in a volume small compared to the emission wavelength cubed, the atoms would all be coupled together through their overlapping radiation fields.

As a consequence the individual atoms will not in fact continue to radiate independently and incoherently. Rather the initial spontaneous emission from any one atom (or, if you like, a small initial fraction of the spontaneous emission from all of the atoms) will tend to "capture" or entrain the oscillations in all the other atoms, in such a way that the inverted system can develop a very large and almost totally coherent macroscopic polarization.

As a result, this system, although initially incoherent, can still evolve into a coherent superposition, and can emit, after a certain time delay, almost exactly the same sort of \((N\mu_1)^2\) superradiant burst described in the preceding.

Because the coherent emission builds up initially from spontaneous emission noise, the phase angle of the radiation will be entirely random from shot to shot, and there will also be small random fluctuations in the delay time between initial preparation of the atoms and emergence of the superradiant burst.

In terms of the Bloch vector picture developed in a later tutorial in this series, the "super Bloch vector" describing the sum of all the dipoles in the small volume is initially oriented essentially antiparallel to the effective dc magnetic field, in the highest-energy, but metastable, orientation.

The effect of spontaneous emission is then to give a small initial disturbance, or effectively a small initial tilt angle to this vector. If suitable conditions are met, this Bloch vector will then precess outward, maintaining constant length, so that its tip stays on the surface of a sphere, and will eventually radiate all of its energy into a superradiant burst as the precessing vector passes from the inverted orientation through the equaorial plane and on to the lowest-energy orientation parallel to the effective dc magnetic field.

### Optical Extensions of Dicke Superradiance

In the simplest situation, the emergence of this type of superradiant emission from an initially inverted but incoherently prepared atomic system depends on the atoms being contained within a volume \(V\le\lambda^3\), so that there will be very strong coupling between all the atoms through their common radiation field. (Note that this coupling occurs only through the radiation fields; the quantum wavefunctions of the individual atoms need not be overlapping.)

This particular type of small-volume incoherently prepared superradiant emission has not yet been demonstrated in any optical system, largely because there seems to be no available atomic medium in which a sufficient number of suitable atoms can be assembled within a volume of the order of \(\lambda^3\).

Considerable attention has been given, however, to the more general situation in which a large number of inverted atoms, with strong initial population inversion but with no initial coherent polarization, are prepared in an extended region of space, most often in the form of a long cylindrical region, or pencil, having a Fresnel number on the order of unity.

This larger volume may then, depending on the size of the inversion, emit either simple amplified spontaneous emission (ASE), as we will describe below, or a kind of extended Dicke superradiance which has come to be referred to as super fluorescence.

### Pure Superfluorescence Behavior

Ideal or pure superfluorescence behavior, as described by a number of theories and experiments (see References), will occur in such systems only under rather specialized conditions in which the atomic transition is strong and narrow enough, and the inversion density large enough, so that the radiative coupling between atoms becomes very strong, in the same sense as in the superradiant experiments described in the preceding, even though the atoms are spread over a volume large compared to the radiation wavelength.

The necessary conditions for pure superfluorescent behavior are quite complex, but a key condition seems to be that the atomic gain coefficient must be large and the sample length small compared to the distance that radiation can travel in one inverse atomic linewidth, or one atomic dephasing time \(T_2\).

If all the atoms are to emit cooperatively, they must be able to communicate with each other strongly in a time short compared to their dephasing time.

To accomplish this, radiation coming from any one atom must be strongly amplified and transmitted to another atom, and the reverse, before either of these atoms has either radiated spontaneously or been dephased.

The principal experimental features of pure superfluorescence will then be an intense simultaneous burst of quasi coherent radiation coming out in a narrow cone from each end of the inverted pencil of atoms.

This pulse will have an intensity proportional to the initial number of atoms squared, and an angular spread which is roughly the aspect ratio of the inverted pencil of atoms. The pulse duration will be inversely proportional to the number of atoms, and the pulse will have a definite time delay following the initial preparation of the inverted atoms.

Since this emission will be initiated by random spontaneous emission in the atoms, there will again be small random variations in the time delay of the pulse from one experimental shot to another.

The essential features of pure superfluorescence are thus a delayed emission pulse, with intensity proportional to \(N^2\), emerging from a large-volume atomic collection that is prepared with no initial coherent polarization or oscillating dipole moment.

### Superfluorescence Experiments

The conditions needed to demonstrate ideal superfluorescence are fairly hard to obtain, and only a few experiments have displayed this effect in a clear and definite fashion thus far.

Perhaps the clearest experiment demonstrating pure superfluorescence was carried out on an upper-level atomic transition having a wavelength of 2.9 \(\mu\)m in low-pressure cesium vapor.

The upper level of this transition could be selectively populated at a high density by optical pumping from the cesium ground state using a tunable pulsed dye laser at 455 nm.

By using a low-pressure cesium cell it was possible to obtain a vapor with minimal collision broadening and long lifetime; and by using Zeeman splitting in a dc magnetic field together with selective pumping it was possible to populate the upper level of only a single strong transition corresponding to a near-ideal two-level atomic system.

The results obtained in these experiments were then in excellent agreement with the theoretical concepts outlined preceding.

### Amplified Spontaneous Emission (ASE) Lasers

We come finally to the most common form of mirrorless laser behavior, namely, "ordinary" amplified spontaneous emission or ASE, as illustrated in Figure 13.33.

Amplified spontaneous emission as used here refers to any situation in which the spontaneous emission coming from a distribution of inverted laser atoms is linearly amplified by the same group of atoms, with a gain which is sizable in at least one direction through the atoms, but the more complex features of superfluorescence are not present.

If the amplification along a long thin cylinder of inverted atoms is sufficiently large, for example, this can produce an output beam from each end of the laser medium which can be quite bright, powerful, and moderately directional, with a fair amount of spatial (but usually not temporal) coherence.

This radiation may become strong enough to produce significant saturation along the gain medium, and to extract the major portion of the inversion energy into the directional beams.

The inverted medium thus acts as a "mirrorless laser," with output characteristics that are intermediate between a truly coherent laser oscillator and a completely incoherent thermal source.

Examples of such mirrorless lasers can include many pulsed excimer lasers and visible and ultraviolet molecular lasers, such as the N\(_2\) laser at 337 nm or the H\(_2\) laser at 120 nm, especially when pumped by fast transverse discharges or by electron beam pumping.

Mirrorless laser action also occurs in certain very highgain infrared gas laser lines, such as the 3.39 \(\mu\)m line in He-Ne or the 3.51 \(\mu\)m line in He-Xe; in very high-gain dye laser amplifiers; and in high-gain semiconductor diode lasers in which the mirror reflection at the end of the laser is deliberately spoiled.

The enormously large and powerful natural masers and lasers which occur in interstellar space are also primary examples of mirrorless or ASE laser systems.

### Questions of Terminology

There has in the past been considerable inconsistency in the laser literature in the use of the various terms superradiance, superfluorescence, and amplified spontaneous emission; and many articles still refer to the type of mirrorless laser we are discussing here as "superradiant emission" or as a "superradiant laser system."

In most of the mirrorless lasers of practical interest, however, the laser medium can still be assumed to remain entirely in the rate-equation regime (with population saturation taken into account); and the emerging radiation can be accurately described merely as narrowband amplified gaussian noise.

The kinds of pulse delays and large coherent polarizations that are characteristic of superradiance or superfluorescence, and the dependence of peak pulse intensity on N\(^2\) as described in the preceding, do not appear in simple ASE lasers.

It seems preferable, therefore, to refer to these simpler systems in general either as amplified spontaneous emission systems or as mirrorless lasers, and to reserve the terms superradiance and superfluorescence for the more specialized phenomena described in the preceding.

### Practical Characteristics of Mirrorless ASE Lasers

Consider a long slender rod of inverted laser medium with length L and diameter 2a, as illustrated in Figure 13.33, and assume for simplicity that the laser transition is completely inverted with inversion population density \(N\). The spontaneous emission power going out into all directions from any small unit volume of this medium will then be given by \(N\gamma_\text{rad}\hbar\omega\).

If we neglect for the minute any gain saturation effects and assume a long slender rod with \(L\gg{a}\), the contribution to the amplified spontaneous intensity dl arriving at the output end of the rod from any small length \(dz\) near the input end of the rod can then be written as

\[\tag{70}dI=\frac{\pi{a^2}N\gamma_\text{rad}\hbar\omega}{4\pi{L^2}}e^{2\alpha_m(L-z)}dz\]

where \(2\alpha_m\equiv{N}\sigma\) is the power amplification coefficient in the rod. If we integrate the total spontaneous emission contribution coming from the entire length of the rod, this yields for the total ASE intensity at the output end

\[\tag{71}\begin{align}I&\approx\frac{N\gamma_\text{rad}\hbar\omega{a^2}}{4L^2}e^{2\alpha_mz}\int_{0}^ze^{-2\alpha_mz}dz\\&\approx\frac{\gamma_\text{rad}\hbar\omega{a^2}}{4\sigma{L^2}}e^{2\alpha_mL}\end{align}\]

In writing this we have assumed that the total gain \(e^{2\alpha_mL}\) along the rod is large, so that we can replace the upper limit of the integral in Equation 13.71 by infinity.

Most of the ASE intensity at each end of the rod then comes from just the first gain length \((2\alpha_m)^{-1}\) at the other end of the rod, and the solid angle of this emitting volume as seen from the other end of the rod is essentially constant at \(\pi{a^2}/L^2\).

We have shown earlier that the stimulated transition rate \(W_{12}\) produced by a wave of intensity \(I\) is given by \(W_{12}=\sigma{I}/\hbar\omega\). Thus, the ratio of the stimulated transition rate caused by the ASE to the spontaneous emission rate in the same atoms at the output end of the rod (in other words, at either end) can be written in the simple form

\[\tag{72}\frac{W_{12}}{\gamma_\text{rad}}\approx\left(\frac{a}{2L}\right)^2e^{2\alpha_mL}\]

which depends only on the aspect ratio \(a/L\) of the rod, and the overall gain coefficient \(2\alpha_mL\).

Although the radius to length ratio \(a/L\) of a typical laser medium is normally small, the exponential gain factor in Equation 13.72 means that as soon as the gain coefficient \(2\alpha_mL\) becomes larger than a few times unity, the stimulated emission rate from the atoms at each end of the laser medium due to amplified spontaneous emission from the other end will begin to exceed the purely spontaneous emission rate by a large ratio.

In other words, as soon as \(2\alpha_mL\gg2\ln(2L/a)\), the presence of ASE will begin to speed up the net emission rate, and thus to shorten the effective inversion lifetime of the laser medium, by a significant amount.

Obviously, this lifetime shortening due to ASE will become even more serious if the amplification length is large in more than one direction, for example, across the width of a flat gain slab, or across all three dimensions of a spherical or rectangular gain volume.

Since the saturation intensity in a simple homogeneous gain medium can be written at \(I_\text{sat}=\hbar\omega/\sigma\tau_2\), where \(\tau_2\) is the effective lifetime or repumping time for the upper laser level, then we can also write the ratio of the amplified spontaneous emission intensity to the saturation intensity at either end of the rod in the form

\[\tag{73}\frac{I}{I_\text{sat}}\approx\left(\frac{a}{2L}\right)^2\left(\frac{\tau_2}{\tau_\text{rad}}\right)e^{2\alpha_mL}\]

Again, as soon as the net gain coefficient \(2\alpha_mL\) becomes more than a few times unity, the ASE will surely become large enough to produce significant gain saturation and significant power extraction from the inverted gain medium.

### Saturation and Power Extraction

Calculating the total power extraction and the exact end-fire emission pattern from such a mirrorless laser system becomes a more complicated problem when gain saturation is taken into account, since we must take into account both the effects of the ASE on the atomic inversion and the effects of gain saturation back on the growth rate for the ASE.

Several numerical calculations for problems of this type are listed in the References, and Figure 13.34 illustrates the general type of behavior that occurs in a long narrow high-gain ASE laser when saturation is taken into account.

The basic result illustrated here is that under high-gain conditions, ASE coming from each end of the gain medium tends to heavily saturate the inversion over a sizable region at the opposite end of the medium, leaving a relatively narrow region or band of unsaturated gain only in the central region of the rod.

As we increase the pumping level or the initial unsaturated gain value in a system of this type, this central unsaturated gain region becomes narrower and narrower. The growth of the intensities in opposite directions along the rod is also, of course, no longer a simple exponential with distance, although there is still large gain from one end of the system to the other.

Casperson has further calculated the manner in which the total spontaneous emission flux from the ends of an ASE laser increases as we turn up the pumping power, or alternatively the gain length in the laser medium.

Typical results for a homogeneously broadened laser medium are shown in Figure 13.35, and generally similar results are obtained for inhomogeneously broadened lasers.

The parameter labeling the different curves in this figure is a dimensionless measure of the spontaneous emission rate, related to the value of \(1/p\) from our earlier rate-equation analysis. This parameter thus has a value much less than unity for most typical laser systems.

The primary interpretation to be made here is that, whereas an ASE laser cannot have the kind of extraordinarily sharp threshold behavior produced by the feedback from the mirrors in an ordinary laser cavity, mirrorless lasers can exhibit a kind of "soft threshold behavior" which may not appear too greatly different from true laser action.

### Temporal and Spatial Output

The output spectrum from a mirrorless laser, at least at low intensity, will consist of the incoherent spontaneous emission from the laser medium, which has a spectral lineshape corresponding to the atomic lineshape, as amplified by the atomic gain process.

The amplification process is also characterized by the same atomic linewidth or bandwidth, however, and we have pointed out earlier that the finite linewidth of the gain medium means that the spectrum will be significantly narrowed, typically down to values 2 to 5 times narrower than the atomic linewidth in a homogeneous system at high gain values.

When saturation effects are taken into account in inhomogeneously broadened media, this narrowing can be significantly reduced, especially at higher gains, because the inhomogeneous gain profile saturates first in the center, and only more gradually in the wings of the line. Typical results have also been calculated by Casperson, as shown in Figure 13.36.

The temporal output from a mirrorless ASE laser, regardless of its spectral width, will always consist of narrowband, highly amplified but still essentially random gaussian noise, rather than any kind of coherent or amplitude-limited sinusoidal oscillation.

Mirrorless lasers thus generally lack most of the important temporal coherence features associated with the sinusoidal amplitude-stabilized oscillation in a true laser oscillator.

The spatial pattern from the ends of an ASE laser will be a narrow cone with a cone angle defined by the aspect ratio of the laser rod, that is with a half-angle \(\Delta\theta\approx{a/L}\). If the rod is very slender, so that it has a Fresnel number \(N_f\equiv{a^2}/L\lambda\approx1\), then all of the end-fire emission will emerge in essentially a single transverse mode. The output beam from a sufficiently slender mirrorless laser can thus have a large degree of spatial coherence (although perhaps not so much total power because of the small rod diameter).

A larger diameter rod, with \(N_f\gt1\), will emit its radiation into a random superposition of essentially \(\pi{N_f^2}\) transverse modes, as we will discuss in later tutorials, and will thus have considerably less ideal spatial coherence (although such a system may still compare not unfavorably with a cavity-type laser having poor mode selection and thus also a large number of transverse modes).

### Coherence Brightening and Swept-Gain Operation

This combination of significant spatial coherence with some spectral narrowing is sometimes referred to as coherence brightening in the mirrorless laser output.

Coherence brightening is not a very precisely defined term, however, and much the same kind of coherence brightening is equally well observed in long slender superradiant or superfluorescent systems.

We can also note that in some laser systems with large spontaneous emission, high gain, and short upper-state lifetimes, the pumping process in the laser medium is carried out using some form of traveling-wave excitation which travels down the laser medium in one direction, at a velocity close to the velocity of light.

If the total gain is large enough, the traveling excitation pulse soon becomes accompanied by a pulse of amplified spontaneous emission, which travels just behind the excitation pulse, and extracts or "dumps" the inversion energy as fast as it is created.

This type of swept-gain laser action, which produces a beam coming out of only one end of the laser, is likely to be characteristic of many X-ray lasers, as well as many fast pulse UV lasers, since the inversion lifetime for the laser medium in these situations may be comparable to or shorter than the transit time for a light pulse down the length of the laser medium.

### Parasitic Laser Oscillation and ASE

Amplified spontaneous emission can also play a very much unwanted role in many large high-gain laser systems.

In cascaded multisection laser amplifiers, for example, such as are often used in laser fusion systems, spontaneous emission from the input end of the amplifier chain may, after amplification through the chain, become large enough to deplete the laser inversion, damage the target pellet, or even cause optical damage to components, before the desired optical signal can be sent through the amplifier chain.

Saturable absorbers, which absorb the weak spontaneous emission but pass the larger signal pulses, must often be placed between the sections in such amplifier chains in order to avoid severe ASE problems.

Parasitic oscillations which arise from a combination of amplified spontaneous emission and weak unintentional reflections from various internal surfaces can also be a serious problem in any large high-gain laser system.

In large glass disk amplifiers, for example, parasitic oscillations and ASE in directions running across the face of the disk, or around the rim of the disk, can be a serious problem; and in general the total energy storage volume of any large high-power laser device is often limited by a combination of parasitic oscillations and ASE.

We might note finally that the emission improvement and lifetime shortening that occurs in an inverted atomic system is the exact opposite of the emission reduction and lifetime extension that has long been known due to radiation trapping in strongly absorbing atomic systems.

The spectral narrowing in an ASE system is thus the reverse of the line reversal that is well known in such radiation trapped systems.

The next tutorial introduces ** nonlinear compensation for digital coherent transmission**.