LASER Q-SWITCHING
This is a continuation from the previous tutorial - Laser spiking and mode competition
\(\text Q\)-switching is a widely used laser technique in which we allow a laser pumping process to build up a much larger than usual population inversion inside a laser cavity, while keeping the cavity itself from oscillating by removing the cavity feedback or greatly increasing the cavity losses—in effect by blocking or removing one of the end mirrors.
Then, after a large inversion has been developed, we restore the cavity feedback, or "switch" the cavity \(\text Q\) back to its usual large value, using some suitably rapid modulation method.
The result in general is a very short, intense burst of laser output which dumps all the accumulated population inversion in a single short laser pulse, typically only a few tens of nanoseconds long.
There are many practical applications, including laser ranging, laser cutting and drilling, and nonlinear optical studies, where such a short but intense laser pulse is much more useful and effective than the same amount of laser energy distributed over a longer time.
The \(\text Q\)-switching approach is therefore a technique of great practical importance in many different laser systems. In the present tutorial we examine the general characteristics of laser \(\text Q\)-switching, and some of the lasers and modulation techniques that are useful for \(\text Q\)-switching; and then review some of the fundamental analytical concepts that apply to laser \(\text Q\)-switching in actively, passively, and repetitively \(\text Q\)-switched lasers.
1. LASER Q-SWITCHING: GENERAL DESCRIPTION
The fundamental dynamics of laser \(\text Q\)-switching, or giant pulsing, are shown schematically in Figure 1. As illustrated there, we assume that the cavity loss in the laser cavity is initially set at some artifically high value—that is, at an artificially low value of the laser cavity \(\text Q_c\)—while the inversion, and hence the gain and the stored energy, in the laser medium are pumped up to a value much larger than normally present in the oscillating laser. In essence, we block one of the laser mirrors to prevent the build-up of oscillation, while the laser pumping process builds up the population inversion over some period of time to a larger than normal value.
The cavity loss is then suddenly lowered to a more normal value—in other words the cavity \(\text Q_c\) is suddenly "switched" to a higher value—with the result

FIGURE 1. Laser \(\text Q\)-switching, step-by-step.
that the round-trip gain after switching is much larger than the cavity loss. The initial spontaneous emission or noise level in the laser cavity then immediately begins to build up at an unusually rapid rate, soon developing into a rapidly rising and intense burst, or "giant pulse," of laser oscillation.
This oscillation burst rapidly becomes sufficiently powerful that it begins to saturate or deplete the inverted atomic population—in essence to "burn up" the inverted atoms— in a very short time.
The oscillation signal in fact rapidly drives the inversion down well below the new cavity loss level, after which the oscillation signal in the cavity dies out nearly as rapidly as it rose. The entire process is somewhat similar to an unusually rapid and intense "spike" of the type we described in the preceding tutorial.
The oscillation build-up interval, and particularly the output pulse duration, are generally much shorter than the pumping time during which the population inversion was created. The inversion built up during a long pumping time is thus dumped during a very short pulse duration. The peak power in the \(\text Q\)-switched giant pulse can be three to four orders of magnitude more intense than the \(\text{cw}\) long-pulse oscillation level that would be created in the same laser using the same pumping rate.

Q-Switching Methods
Figure 2 illustrates some of the more common \(\text Q\)-switching methods that are employed in practical laser systems, including the following standard techniques:
\((1)\) Rotating Mirror \(\text Q\)-Switching: The most direct and one of the earliest Q-switching methods is simply to mount one end mirror of the laser on a rapidly spinning motor shaft, so that the laser can oscillate only during the brief interval when the mirror rotates through an aligned orientation with respect to the opposite mirror.
This method, although cheap and simple, has numerous practical disadvantages. Even with the highest-speed motors (Waring Blendor motors were reputed to be extremely good!) this approach suffers from uncertain timing, slow switching speed, lack of reliability, and vibration and mechanical noise which lead to alignment difficulties in the direction perpendicular to the plane of rotation. To avoid the latter, a rotating 90° prism rather than a rotating mirror was often employed.
This method is now used if at all only on very long laser cavities at very long laser wavelengths—e.g., long \(\text{CO}_2\) or far infrared molecular lasers—where mirror alignment is less critical and other modulation techniques may be more difficult.
\((2)\) Electrooptic \(\text Q\)-Switching: An electrooptic modulator, as we described in the previous tutorial consists in general of an electrooptic crystal which becomes birefringent under the influence of an applied electrical voltage, plus one or more prisms or other polarizing elements inside the laser cavity.
In one form of electrooptic \(\text Q\)-switch, an applied voltage sufficient to make the Pockels crystal into a quarter-wave plate is initially applied. Energy circulating once around the laser cavity then has its polarization rotated by \(90^\circ\) about the cavity axis, so that all the circulating energy is coupled out of the cavity by the polarizing element after just one round trip.
Switching the cavity to a low-loss condition is then accomplished by suddenly turning this voltage off (referred to as "crowbarring" the voltage across the modulator).
As an alternative, a fixed quarter-wave element can create the highloss condition with no voltage applied, and the Pockels cell can then be switched on to cancel this birefringence; but this approach requires an additional element inside the laser cavity.
Electrooptic \(\text Q\)-switching provides the fastest form of \(\text Q\)-switching (switching time \(\leq10\;\text{ns})\), with precise timing, good stability and repeatability, and a large hold-off ratio (i.e., large insertion loss in the low-\(\text Q\) state). This approach requires, however, both a fairly expensive electrooptic crystal and a very fastrising high-voltage pulse source (at least several \(\text{kV}\) in a few tens of nanoseconds).
Nanosecond rise-time pulses at this voltage level are difficult to obtain, and can produce severe electrical interference in nearby electronic equipment.
In addition, this approach needs several elements inside the laser cavity, and these elements (particularly the Pockels crystal) may be both optically lossy and subject to optical damage at the high intensities inside the \(\text Q\)-switched laser.
\((3)\) Acoustooptic \(\text Q\)-Switching: We also have described in the previous tutorial the use of an acoustooptic modulator, in which the index grating produced by an rf acoustic wave Bragg-diffracts light out of the laser cavity. Acoustooptic modulators have the advantages of very low optical insertion loss, relatively simple rf drive circuitry, and ease of use for repetitive \(\text Q\)-switching at kHz repetition rates.
They have only relatively slow opening times, however, as well as low hold-off ratios. Hence they are primarily employed for lower-gain cwpumped or repetitively \(\text Q\)-switched lasers (as we will describe in more detail in a later section of this tutorial).
\((4)\) Passive Saturable-Absorber \(\text Q\)-Switching: Passive \(\text Q\) switching (also described in more detail in a coming section) uses some form of easily saturable absorbing medium inside the laser cavity. Laser inversion is built up by the pumping process until the gain inside the cavity exceeds this absorption, and laser oscillation begins to develop inside the cavity.
This oscillation at some relatively low level then rapidly saturates the absorber and thus opens up the cavity, leading to the development of a rapid and intense oscillation pulse. Saturable absorption using an organic dye solution in an intracavity cell is the most common form of passive \(\text Q\)-switching, although there are other systems as well.
Passive \(\text Q\)-switching is generally simple, convenient, and requires a minimum of optical elements inside the laser and no external driving circuitry. It is subject to some shot-to-shot amplitude fluctuations and timing jitter, however, and external apparatus must be synchronized to the timing of the laser pulse rather than vice versa.
In addition, the absorbing dyes may need careful initial adjustment and may be subject to chemical or photochemical degradation in use. Passive \(\text Q\)-switching is nonetheless quite widely used in practical \(\text Q\)-switched lasers.
\((5)\) Thin-Film \(\text Q\)-Switching: A somewhat unusual form of saturable absorber Q-switching is the use of a thin absorbing or metallic layer on a glass or mylar substrate, with the laser energy focused to a small spot on this layer.
When laser oscillation first begins to build up in this cavity, the thin absorbing coating very rapidly burns away and is vaporized as the laser oscillation builds up from noise. This makes a particularly simple and fast-opening \(\text Q\)-switch which may have applications in low-cost "throwaway" lasers; or it may be practical

to obtain repeated shots by moving the absorbing film by a small amount between shots.
2. ACTIVE Q-SWITCHING: RATE-EQUATION ANALYSIS
Even though \(\text Q\)-switched lasers involve high intensities and moderately short pulses, a simple rate-equation analysis is still virtually always adequate for describing \(\text Q\)-switched laser behavior.
In this section we derive therefore some simple and yet quite useful results for each of the important stages of laser \(\text Q\)-switching, using a simple rate-equation formulation.
Basic Rate Equations
The elementary rate equations for the cavity photon number \(\text n(t)\) and the inverted population difference \(\text N(t)\) in a \(\text Q\)-switched laser are, once again,
\[\tag{1}\frac{dn}{dt}=KN_n-\gamma_cn\quad\text{and}\quad\frac{dN}{dt}=R_p-\gamma_2N-2^*KnN,\]
where \(\gamma_c\equiv 1/\tau_c\) is the total cavity decay rate; \(R_p\) is the pumping rate and \(\gamma_2\) is the decay rate or recovery rate for the inverted population difference; and \(\text K\) is the coupling coefficient between photons and atoms as given in earlier sections.
There are two limiting situations for the saturation behavior of the population difference \(\text N(t)\equiv N_2(t)-\text N_1(t)\) in a \(\text Q\)-switched laser. In many \(\text Q\)-switched lasers the relaxation out of the lower state will be sufficiently fast that the lower-level population \(N_1(t)\) will remain \(\approx 0\) at all times, even during the \(\text Q\)-switched pulse, in which case the parameter \(2^*=1\).
In a few situations the lower laser level may be "bottlenecked" so that it cannot empty out during the laser pulse (for example, in \(\text Q\)-switched ruby lasers, where the lower laser level is in fact the ground level). The "bottlenecking parameter" then becomes \(2^*=2\) in order to take into account the filling up of the lower level by stimulated transitions from the upper level.

Figure 3. shows in somewhat more detail the various stages in the usual pulse-pumped active \(\text Q\)-switching process. As shown in this figure, we can divide the \(\text Q\)-switching process into a pumping interval and a pulse output interval, with the latter being subdivided more or less into a pulse build-up interval, and a pulse emission interval.
We will now look in more detail at each of these intervals, and give a simplified solution and interpretation of the rate equations for each regime.
Pumping Interval, and Population Build-Up
To model the simplest form of pulsed pumping in a \(\text Q\)-switched laser, let us suppose that the cavity feedback is completely blocked during the pumping interval, so that no oscillation occurs and \(\text n(t)\equiv0\); and that a pump pulse with constant intensity \(R_p\) is turned on at \(t=0\). The population rate equation during this interval then reduces to the simplified form
\[\tag{2}\frac{dN(t)}{dt}\approx R_p-N(t)/\tau_2,\]
with the solution
\[\tag{3}N(t)=R_p\tau_2[1-\text{exp}(-t/\tau_2)].\]
The inversion thus builds up toward a maximum inversion value \(R_p\tau_2\), as illustrated in Figure 4. This inversion may or may not be above the threshold inversion with the cavity blocked; but it preferably will be much above the threshold inversion after the cavity \(\text Q\) is switched.
Figure 4 makes it clear that there is not much point in supplying pumping power to a \(\text Q\)-switched laser for longer than about one or two population decay times \(\tau_2\) before the \(\text Q\)-switching takes place, since the inverted population no longer continues to grow after this length of time.
In fact, at any instant of time, any atoms pumped up more than 1 or 2 lifetimes earlier will already have decayed and no longer be available.

Pumping Efficiency
We can make this same point in a more practical fashion by assuming a pumping pulse with constant pumping energy and variable pulse duration. Suppose the laser pumping energy (e.g., the energy input to a laser flashlamp) is supplied by a square pump pulse of fixed total energy, although the duration Tp and peak height \(\text R_p\) of this pump pulse may be changed.
We can then say that this pumping energy lifts up a total number of inverted atoms given by \(\text N_p=R_pT_p\) during the pumping pulse. The inversion \(\text N_i\) that still remains at the end of the pump pulse, and hence at the initiation of the \(\text Q\)-switching buildup, is given, however, by
\[\tag{4}\text N_i=\text N(T_p)=N_p\times\frac{1-\text{exp}(-T_p/\tau_2)}{T_p/\tau_2}.\]
The normalized inversion \(\text N_i/N_p\) produced by a pump pulse of fixed total energy or fixed \(N_p\equiv R_pT_p\) is plotted versus normalized pump pulse duration \(T_P/\tau_2\) in Figure 5. To get better than \(75\%\) efficiency from the pumping pulse, we need to restrict its duration to less than about half of the excited-state lifetime \(\tau_2\).
For typical solid-state lasers with \(\tau_2\approx 200\) \(\mu s\) to 1 \(\text{ms}\), this means we should use pumping pulses with durations not longer than a few hundred microseconds at most. Such pulses can be obtained from typical laser flashlamps, though careful circuit design is required to minimize the inductance in the flashlamp circuitry.
The tendency of flashlamps to explode also increases substantially as the pulse duration is decreased below a few hundred \(\mu s\).
Note also that for most visible gas lasers, as well as organic dye lasers, the population lifetimes \(\tau_2\) are typically a few nanoseconds to a few tens of nanoseconds. The time over which population inversion can be built up and stored is then in general less than the time required for a \(\text Q\)-switched pulse to develop. \(\text Q\)-switching of these lasers is thus not a useful or realistic concept, because the storage time for inverted atoms is just too short.

Pulse Build-Up Time
At some point near the end of the pumping interval the \(\text Q\)-switch will be opened, dropping the cavity losses to a low value; and the \(\text Q\)-switched oscillation pulse will then begin to build up. There will be, however, a finite delay time or pulse build-up time, typically of a few tens to hundreds of nanoseconds duration, between the time when the cavity \(\text Q\)-switch is opened and when the \(\text Q\)-switched pulse appears.
This pulse build-up time is not precisely defined, since the \(\text Q\)-switch itself may have a finite opening time, and the resulting pulse has a finite risetime and duration. To make an estimate of the build-up time for a simple situation, however, we can assume that after an initial population inversion has been pumped up, the losses in the laser cavity are switched essentially instantaneously at \(t=0\) (somewhere near the end of the pumping interval) to some much lower value.
The laser will at that point have an inversion \(\text N_i\) which is some ratio \(r\) times larger than the threshold inversion \(\text N_\text{th}\) just after switching. That is, this initial inversion ratio \(r\) is defined by
\[\tag{5}r\equiv\frac{N_i}{N_\text{th}}\equiv\frac{\text{population}\;\text{inversion}\;\text{just}\;\text{after}\;\text{switching}}{\text{threshold}\;\text{inversion}\;\text{just}\;\text{after}\;\text{switching}}.\]
as illustrated in Figure 6. This initial inversion ratio, plus the cavity lifetime \(\tau_c\), are the primary factors controlling the performance of a \(\text Q\)-switched laser.
At this point the photon density in the cavity is extremely small, although it will begin to grow rapidly; and so we can neglect the rate at which inverted atoms are burned up by the cavity photons.
If the pulse build-up time (a few hundred nanoseconds typically) is also short compared to the upper-level lifetime \(\tau_2\), as it usually will be, then we can neglect relaxation and pumping effects also, and assume that the population inversion \(\text N(t)\) stays approximately equal to its initial value \(\text N_i\) all through the pulse build-up time. The photon rate equation then becomes to a good approximation
\[\tag{6}\frac{dn(t)}{dt}\approx K[N_i-N_{th}]n(t)\approx\frac{(r-1)}{\tau_c}n(t),\]
with the solution
\[\tag{7}n(t)=n_i\;\text{exp}[(r-1)t/\tau_c].\]
Unless some artificially injected external signal is present in the laser cavity (as we will discuss in a later chapter on injection locking), the initial noise value in the cavity will be equivalent to \(n_i\approx 1\) or a few photons, representing the initial spontaneous-emission noise excitation of the cavity mode.
Let us arbitrarily define the end of the pulse build-up interval as that point on the leading edge of the \(\text Q\)-switched pulse where the photon number \(n(t)\) just becomes equal to the steady-state photon number \(n_{ss}\) that would be present in the laser cavity if the laser could be continuously pumped at a pumping rate r times above its threshold after \(\text Q\)-switching.
We pick this point because it marks the point where the stimulated emission term \(\text{KN}(t)n(t)\) first begins to burn up inverted population density at a significant rate. Note that this point is still quite early in the leading edge of the \(\text Q\)-switched pulse, since the photon number rip at the peak of the \(\text Q\)-switched pulse will be \(n_p\gg n_{ss}\).
The build-up time \(\text T_b\) from the initial photon density \(n_i\) to a photon density \(n_{ss}\) is then given, from Equation 7, by
\[\tag{8}\frac{n_{ss}}{n_i}=\text{exp}\left[\frac{(r-1)T_b}{\tau_c}\right],\]
or, if we invert this,
\[\tag{9}T_b=\frac{\tau_c}{r-1}\times 1\text{n}\left(\frac{n_{ss}}{n_i}\right).\]
The ratio of initial to final photon numbers in this expression may vary over the range from \(\text n_{ss}/\text n_i\approx 10^8\) to \(10^{12}\). Even a difference of four orders of magnitude in this ratio, however, only makes a difference of \(\pm20\%\) in the logarithm of this ratio. Hence, we may rewrite this result, using an earlier formula for the cavity decay time \(\tau_c\), in the form
\[\tag{10}T_b\approx\frac{(25\pm5)}{r-1}\times\frac{T}{\delta_c},\]
where \(T=2L/c\) is the round-trip time inside the laser cavity and \(\delta_c\) is the fractional power loss per round trip due to cavity losses and output coupling.
As one practical example, we might consider a flash-pumped \(\text{Nd}\):\(\text{YAG}\) laser with a cavity 60 \(\text{cm}\) long, so that \(T=4\) \(\text{ns}\), and an output mirror transmission \(1-\text R=60\%\), or \(\delta_c=\text{ln}(1/R)=1.05\).
If this laser is initially pumped to three times threshold \((r=3)\), the build-up time will \(T_b\approx 50\) \(\text{ns}\).
By contrast, a longer \(\text{cw}\)-pumped \(\text{Nd}\):\(\text{YAG}\) laser or a low-gain \(\text{CO}_2\) laser might have a 2 \(m\) long cavity with \(\text T=12\) \(\text{ns}\), and only \(20\%\) output coupling or \(\delta_c=\text{ln}(1/0.8)\approx0.22\), and be pumped to only \(50\%\) above threshold \((r=1.5)\). The build-up time in this situation will be \(T_b\approx3\) \(\mu s\).
Pulse Output Interval
We can now in fact develop a quite accurate analytical solution that will describe not only the pulse build-up time, but the entire interval during which the \(\text Q\)-switched pulse is generated and the inverted population is "dumped." This entire period, including pulse build-up and decay, is almost always much too short for either any significant pumping or relaxation of the inverted atoms to occur.
We can therefore leave out the pumping and relaxation terms in the population equation, and approximate the rate equations in this pulse-output interval by
\[\tag{11}\frac{dn(t)}{dt}=K[N(t)-N_{th}]n(t)\quad\text{and}\quad\frac{dN(t)}{dt}\approx-2^*K_n(t)N(t),\]
with the initial conditions that \(N=N_i\equiv rN_{th}\) and \(n=n_i\approx1\) at the switching time \(t=t_i\) Dividing these two equations into each other then gives the single relation
\[\tag{12}\frac{dn}{dN}=\frac{N_{th}-N}{2^*N}.\]
This relation can then be integrated, starting from the time when the cavity \(\text Q\) is switched and going to any arbitrary time \(t\), in the form
\[\tag{13}2^*\int^{n(t)}_{ni}dn=\int^{N(t)}_{Ni}\left(\frac{N_{th}}{N}-1\right)dN.\]
The initial photon number \(n_i\approx 1\) is negligible compared to the very much larger values of \(n(t)\) anytime during the laser output pulse. Hence we can set the lower limit on the left-hand integral to zero, and then solve both integrals to obtain the implicit relation
\[\tag{14}2^*n(t)\approx N_i-N(t)-\frac{N_i}{r}1\text n\left(\frac{N_i}{N(t)}\right)\quad\text{where}\quad r\equiv\frac{N_i}{N_{th}}.\]
The only parameters in this expression are the initial inversion \(N_i\), and the ratio \(r\equiv N_i/N_{th}\) by which this initial inversion exceeds threshold. This expression can then be manipulated to yield a large number of useful results concerning the \(\text Q\)-switched pulse.
Q-Switched Pulse Energy
For example, the total energy in the \(\text Q-switched pulse is a very important parameter for many applications, such as laser ranging or drilling or cutting. To calculate this, suppose we let the time \(t\) in Equation 13 run until some final time well after the \(\text Q\)-switched pulse is over, when the pulse energy is well down in the tail of the pulse, so that the photon number approaches a final value \(n_f\approx 0\). (This time would be at the far right-hand edge of Figure 3, or Figure 7.) The pulse inversion N(t) must then also approach a final value \(N_f\), which is given from Equation 14 by
\[\tag{15}N_i-N_f-\frac{N_i}{r}1\text n\left(\frac{N_i}{N_f}\right)\approx 0,\]
which can be rewritten as
\[\tag{16}1-\frac{N_f}{N_i}-\frac{1}{r}1\text n\left(\frac{N_i}{N_f}\right)=0.\]
This formula implicitly gives the ratio of final to initial inversions, \(N_f/N_i\), as a function only of the initial inversion ratio \(r=N_i/N_{th}\), and nothing else.

The initial energy stored in the atoms in the upper laser level just at the end of the pumping pulse, and potentially convertible into laser photons, can be written as \(U_\text{initial}=N_i\hbar\omega_a\) (we assume \(N_{2i}\approx N_i\) and \(N_{1i}\approx 0\) at the start); whereas the same energy remaining in the upper energy level at the end of the \(\text Q\)-switched burst will be given by \(\text U_\text{final}=(N_i+N_f)\hbar\omega_a/2^*\) (note that if \(2^*=2\), at most half of the initial energy can be dumped). Since atomic relaxation during the \(\text Q\)-switching interval is negligible, there is no other place that the difference between these initial and final energies can have gone except into \(\text Q\)-switched laser output.
The total energy delivered to the Q-switched pulse will thus be just the difference between the initial and final stored energies in the upper laser level, or
\[\tag{17}\text Q\text -\text{switched}\;\text{output}\;\text{pulse}\;\text{energy},\text U_\text{out}=\text U_\text{initial}-\text U_\text{final}=\frac{N_i-N_f}{2^*}\hbar\omega_a.\]
This energy is not necessarily all delivered to useful output from the Q-switched laser, since some of it may be dissipated in internal cavity losses. In practice, however, the useful output coupling will typically be substantially larger than the internal cavity losses in most \(\text Q\)-switched lasers, and so most of this energy does come out in the output beam.
Suppose we thus define an energy extraction efficiency \(\eta\) for the conversion of initial stored energy into \(\text Q\)-switched pulse energy, as given by
\[\tag{18}\eta\equiv\frac{\text Q\text -\text{switched}\;\text{output}\;\text{energy}}{\text{initial}\;\text{inversion}\;\text{energy}}=\frac{U_\text{out}}{U_\text{initial}}=\frac{N_i-N_f}{2^*N_i}.\]
Equations 16 and 18 then combine to give a single implicit relation between the initial inversion ratio \(r\) and the energy extraction efficiency \(\eta\), namely,
\[\tag{19}r=\frac{1}{2^*\eta(r)}\text{ln}\left(\frac{1}{1-2^*\eta(r)}\right)\quad\text{or}\quad 1-2^*\eta(r)=\text{exp}[-2^*r\eta(r)].\]
The solution to this implicit equation is plotted in Figure 8.
The efficiency with which a \(\text Q\)-switched laser extracts energy from the initial inverted population depends only on the initial inversion ratio \(r\), the bottlenecking parameter \(2^*\), and nothing else.
Note that \(2^*\eta(r)\) rapidly approaches \(100\%\) for \(r\geq 2\). So long as the condition \(r\gg1\) is satisfied, the output energy from a \(\text Q\)-switched laser is largely independent of the exact cavity output coupling or almost any other design parameters, providing the coupling is not so large as to reduce \(r\) too close to unity.
A \(\text Q\)-switched laser which is only a small amount above threshold \((r\rightarrow 1\)) will, on the other hand, leave a large fraction of the initial inversion still in the upper-level atomic population, i.e., it will waste a good deal of the initial inversion.

FIGURE 8. Energy-extraction efficiency in a Q-switched laser as a function of the initial inversion ratio.
Peak Pulse Power
The power output at the peak of a \(\text Q\)-switched laser pulse is also of practical interest. The cavity photon number \(n(t)\) obviously reaches its peak value \(n_p\) (and the slope \(dn/dt\) becomes 0) at the instant when the inversion \(N(t)\) passes through the threshold value \(N_{th}\) on its way downward in Figure 7. If we put these values into Equation 14, we can rewrite this equation in either of the forms
\[\tag{20}n_p=\frac{r-1-\text{ln}\;r}{2^*r}N_i\quad\text{or}\quad n_p=\frac{r-1-\text{ln}\; r}{2^*}N_\text{th}.\]
The first of these equations says that at the peak of the \(\text Q\)-switched pulse the photon number \(n_p\) in the cavity is equal to the initial inversion \(N_i\) multiplied by a factor which approaches unity for \(r\gg1\) (and \(2^*=1)\), as illustrated in Figure 9.
In other words, if the initial inversion is very much above threshold, so that there is a very large initial gain and initial signal growth rate, then the \(\text Q\)-switched laser in essence first converts virtually all of the initially inverted atoms into photons bouncing around inside the laser cavity.
These photons then "leak" out of the cavity, at something like the cavity decay rate \(\gamma_c\), to produce the \(\text Q\)-switched output pulse.
The second form in Equation 20 relates the peak power or peak photon number \(n_p\) in the cavity to the threshold inversion \(N_\text{th}\) after switching, which is usually a fixed quantity for a fixed cavity output coupling, and also to the initial inversion ratio \(r\), which usually depends linearly on the pumping power or energy applied to the laser. The peak power output from a \(\text Q\)-switched laser can be written, using this form, as
\[\tag{21}P_p=\frac{n_p\hbar\omega_a}{\tau_c}=\frac{r-1-\text{ln}\;r}{2^*}\times\frac{N_{th}\hbar\omega_a}{\tau_c}.\]


This form is thus convenient for comparison with experimental results of peak pulse output versus input pump energy \(W_p\) or pump rate \(R_p\), such as in Figure 10.
The peak pulse powers are actually plotted here versus the pumping lamp current in two similar \(\text{Nd}\):\(\text{YAG}\) lasers pumped by a rather long-pulse low-current flashlamp. The general conclusion is clearly that increasing the initial excitation r increases the peak intensity inside the laser—up to the point, that is, where

FIGURE 11. \(\text Q\)-switched pulsewidth versus initial inversion ratio.
some component inside the cavity suffers optical damage and terminates the experiment.
Q-Switched Pulsewidth
A good approximation to the pulsewidth \(\tau_p\) for a \(\text Q\)-switched laser pulse can be obtained by dividing the total pulse energy by the peak pulse power to obtain
\[\tag{22}\tau_p\approx\frac{U_\text{out}}{P_p}=\frac{r\eta(r)}{r-1-\text{ln}\;r}\times\tau_c,\]
where \(\tau_c\) is the cavity decay time. The result will not be exactly equal to the FWHM pulsewidth of the pulse, but it will be close enough for most purposes.
Figure 11 plots this pulsewidth (which is about \(20\%\) smaller than the true \(\text{FWHM}\) pulsewidth) versus the initial inversion ratio \(r\). The pulsewidth comes down toward a limiting value equal to the cavity lifetime \(\tau_c\) as the initial inversion \(r\) is raised sufficiently far above threshold.
Figure 12 shows experimental measurements of the \(\text Q\)-switched pulsewidth versus lamp current in the same two \(\text{Nd}\):\(\text{YAG}\) lasers shown in Figure 10. It is evident, as predicted, that the pulsewidth decreases rapidly with increasing pump power or initial inversion \(r\).
Moreover, since the two laser cavities use the same laser rod and the same mirror reflectivities, the shorter cavity should have a correspondingly shorter cavity lifetime \(\tau_c\) and hence a shorter pulsewidth \(\tau_p\), exactly as illustrated by the data.

FIGURE 12. Pulsewidth versus pumping rate in two typical \(\text Q\)-switched \(\text{Nd}\):\(\text{YAG}\) lasers.
Exact Solutions for the Q-Switched Pulseshape
If we really wish to calculate exact solutions for the \(\text Q\)-switched pulse envelope \(n(t)\) and the population variation \(N(t)\) (exact, at least, within the approximations made earlier in this section), we can substitute Equation 14 for \(n(t)\) versus \(N(t)\) into the rate equation \(dN/dt\approx-\text{KN}n\) to obtain a single differential expression for \(N(t)\) versus \(t\). This equation can then be integrated, starting with \(N=N_i\) at the switching time \(t=0\), in the form
\[\tag{23}\gamma_c\int^t_0dt=-N_{th}\int^{N(t)}_{Ni}\frac{dN}{N[N_i-N-N_\text{th}\;\text{ln}(N_i/N)]},\]
which can be converted if desired into the dimensionless integral form
\[\tag{24}\frac{t}{\tau_c}=-\frac{1}{r}\int^N_1\frac{dy}{y[(1-y)+\text{ln}\;y]},\]
where \(N\equiv N(t)/N_i\).
No analytical solution for this integral seems to be available. The integral can be evaluated numerically without much trouble, however, to give \(N(t)/N_i\) versus \(t/\tau_c\); and these values can then be substituted into Equation 14 to give an exact solution for \(n(t)\) versus \(t/\tau_c\). Figure 13 illustrates typical \(\text Q\)-switched pulseshapes for different degrees of initial inversion plotted versus \(t/\tau_c\) (and normalized to the same peak power in each case).
The pulseshapes clearly become somewhat asymmetric for larger initial inversions, with a very fast rise time and a slower decay time.
With a very large initial inversion \(r\gg1\), in fact, the leading edge builds up with a growth rate \((r-1)\gamma_c\) which is substantially faster than the cavity decay rate 7C during most of the leading edge.
Once the pulse reaches its peak, however, the most it can do is to saturate the laser gain down to zero. In the trailing edge, therefore, the pulse intensity dies out with a decay rate which is at most the empty cavity decay rate \(\gamma_c\). The pulses thus acquire a fast leading edge and a slower trailing edge.

FIGURE 13. Exact \(\text Q\)-switched pulse shapes.
Experimental Results
Let's now compare all this analysis with some experimental results. Figure 14 reproduces measured data on the pulse delay, or pulse build-up time after \(\text Q\)-switching, and also on the \(\text Q\)-switched pulse width, for a \(\text{Nd}\):\(\text{YAG}\) laser built at Stanford University by Professor Robert L. Byer.
These quantities are plotted versus the electrical pumping energy applied to the flashlamp in the laser, which will be more or less directly proportional to the initial inversion in the rod before the \(\text Q\)-switch is opened.
To compare this data with theory, we need to know the exact threshold pumping energy of the laser, in order to calculate the initial inversion ratio r accurately; and it is difficult to obtain this number with precision from the data in its initial form. Since we know from Equation 10, however, that the pulse build-up rate scales linearly with inversion ratio, a useful tactic is to invert the measured build-up times and replot them in the form of \(1/T_b\) versus pumping energy, as shown in the inset to Figure 26.14. Doing this makes it clear that the threshold pumping energy in this situation (corresponding to \(r=1)\) is almost exactly 10 \(\text J\) input to the flashlamp.
The solid lines in Figure 14 then represent a simultaneous fit to the theory for both the build-up time and pulsewidth, assuming that \(r\) is proportional to the lamp energy relative to 10 \(\text J\), and using the known cavity lifetime \(\tau_c\) for this laser. It appears that the simple theory developed in this section does give a very reasonable description for real \(\text Q\)-switched lasers. The slight discrepancies at the

FIGURE 14. Q-switching of a \(\text{Nd}\):\(\text{YAG}\) laser: experiment versus theory.
largest energies are expected, in view of the finite \(\text Q\)-switch opening time and the difficulty in measuring the shortest times accurately in the real laser.
Multiple Pulse Problems
A \(\text Q\)-switch which opens too slowly will lead to multiple output pulses from a Q-switched laser. Figure 15 (based on computer simulations) shows some typical examples illustrating the effects of a slowly opening \(\text Q\)-switch in a typical laser system.
The cavity loss in these examples starts from an initial value \(20\%\) higher than the initial gain, and falls sinusoidally to a value \(80\%\) below the initial gain in an adjustable switching time or half-period \(\text T_{sw}\). The loss then rises back up toward its initial value.
In \((a)\), the switching time \(\text T_\text{sw}=25\) \(\text{ns}\) is optimized so that the output pulse occurs just as the cavity loss reaches its minimum value. (Note that the output pulse will not begin to build up at all until the loss drops below the gain, and then will have a continuously increasing growth rate during the build-up interval.
The simple analysis of build-up time given earlier, based on constant gain and loss values, thus does not apply here.)
In \((b)\), with \(\text T_\text{sw}=50\) \(\text{ns}\), the loss drops more slowly, so that the output pulse, though somewhat delayed compared to the first situation, actually occurs before the loss has reached its minimum value. Nearly all the initial inversion is still dumped by the \(\text Q\)-switched pulse, however.
In \((c)\), with \(\text T_{sw}\) increased to the much longer value of 150 \(\text{ns}\) (note the change in the horizontal time scale), the gain drops so slowly that the effective inversion just before the first \(\text Q\)-switched pulse occurs is only about \(r\approx1.5\). Because of

FIGURE 15. Multiple pulsing with a slowly opening \(\text Q\)-switch.
this, the initial inversion is only slightly more than half dumped by the first \(\text Q\)-switched pulse, leaving a sizable amount of upper-level population and gain remaining after this pulse.
The loss then continues to drop, soon falling below the residual gain value remaining in the cavity. The result is a weak but observable second \(\text Q\)-switched pulse, which actually occurs after the loss has reached its minimum value and is beginning to rise again.
Finally, \((d)\) shows the same sort of behavior with an even slower \(\text Q\)-switch, \(\text T_{sw}=250\) \(\text{ns}\). This situation leads to two clearly defined sequential \(\text Q\)-switched pulses, each of which dumps half or less of the initial inversion.
The behavior shown in these examples, though calculated for one particular sinusoidal switching profile, is quite general—a slowly opening \(\text Q\)-switch, whether the loss falls linearly, sinusoidally, or in some other fashion with time, will eventually cause the generation of multiple \(\text Q\)-switched pulses.
The existence of these secondary pulses can, of course, cause timing errors in optical radars, measurement errors in scientific experiments, and a variety of other difficulties in practical applications.
A good \(\text Q\)-switching device, whatever its nature, should open in a time significantly shorter than the build-up time for the desired output pulse. The opening time required in practical lasers is thus typically a few tens of nanoseconds or faster.
3. PASSIVE (SATURABLE ABSORBER) Q-SWITCHING
Passive \(\text Q\)-switching, making use of a saturable absorber inside the laser cavity, is another basic approach for generating \(\text Q\)-switched pulses. This approach can be one of the simplest forms of \(\text Q\) switching in practice, since it requires a minimum of elements inside the laser cavity, and no external pulse sources or control circuitry.
The \(\text Q\)-switching behavior depends critically, however, on the saturation properties of the gain medium and the saturable absorber, and these can deteriorate with time or with repeated laser shots. In addition, there is no direct control over the timing of the \(\text Q\)-switched pulse—one must synchronize external apparatus to the laser output pulse, rather than the reverse.
The most common examples of saturable-absorber \(\text Q\)-switching are various solid-state lasers \(\text Q\)-switched by organic saturable-absorber dye solutions, such as \(\text{Nd}\):\(\text{YAG}\) or \(\text{Nd}\):glass lasers mode-locked with various commercially available dyes having names like Eastman Kodak 9860 or 9740.
The possibilities of thinfilm single-shot \(\text Q\)-switches for such lasers have also been mentioned. Certain infrared \(\text{CO}_2\) lasers are also passively \(\text Q\)-switched using the saturable absorption properties of \(\text{SF}_6\) vapor, or hot unpumped \(\text{CO}_2\) vapor, or thin \(p\)-type \(\text{Ge}\) slabs.
Rate Equations For Passive Q-Switching
The simplest model for a passively \(\text Q\)-switched laser consists of a laser cavity mode with cavity photon number \(n(t)\); a saturable gain medium with population difference and coupling coefficient which we might call \(\text N_g(t)\) and \(\text K_g\); and a saturable absorbing medium with population difference \(\text N_a(t)\) and coupling coefficient \(\text K_a\).
The elementary rate equations describing this system are then the cavity photon number equation, which becomes
\[\tag{25}\frac{dn(t)}{dt}=[K_gN_g(t)-K_aN_a(t)-\gamma_c]n(t),\]
plus the usual rate equation for the gain medium, which can be written as
\[\tag{26}\frac{dN_g(t)}{dt}=R_p-\gamma_{2g}N_g(t)-K_gN_g(t)n(t),\]
plus a similar rate equation for the saturable absorber, which we write as
\[\tag{27}\frac{dN_a(t)}{dt}=-\gamma_{2a}[N_a(t)-N_{a0}]-K_aN_a(t)n(t).\]
Obviously \(\gamma_{2g}\) and \(\gamma_{2a}\) now mean the population recovery rates for the gain and the saturable absorber, respectively.
The saturable absorber is assumed to relax back toward an unsaturated value \(\text N_{a0}\) with a time constant \(\tau_a=1/\gamma_{2a}\).
Approximate Solution
The solutions to these equations for the passively \(\text Q\)-switched laser are even more strongly nonlinear than for an actively \(\text Q\)-switched laser, since the \(\text Q\)-switching process itself is controlled by the signal buildup in the laser. Simple analytic results like those we derived for active \(\text Q\)-switching in an earlier section are thus difficult if not impossible to find. Passive \(\text Q\)-switching is thus more often approached by numerical or experimental methods than by analytical methods.
There is, however, one relatively simple analytical criterion for good passive \(\text Q\)-switching behavior that can be derived from these equations, as follows. Suppose the laser pump power is turned on and begins to pump up the laser gain medium, until the laser gain exceeds the cavity loss plus the unsaturated absorber losses.
The photon density \(n(t)\) in the cavity will then start to build up from noise, and after a certain time the photon density n(t) will become large enough that it begins to saturate the saturable absorber.
Let the point where the saturable absorber just begins to saturate, and the \(\text Q\)-switched pulse just starts to develop, be called \(t=0\) and let the laser inversion just at this point be represented by \(N_{g0}\).
Now, in most \(\text Q\)-switched lasers the pumping and relaxation times for the gain medium are long compared to the \(\text Q\)-switching buildup and decay time (as we have already noted in an earlier section), so that the gain medium equation during the \(\text Q\)-switching interval can be simplified to
\[\tag{28}\frac{dN_g(t)}{dt}\approx-K_gN_g(t)n(t),\]
which has as a formal solution
\[\tag{29}N_g(t)-N_{g0}\;\text{exp}\left[-K_g\int^t_0n(t')dt'\right].\]
The physical significance of this approximation is that the gain is depleted by the integrated or cumulative effect of the photon flux \(n(t)\) which passes through the gain medium, rather than by the instantaneous intensity in the cavity (at least in the initial stages).
The recovery times \(\tau_a\) for saturable absorbers are, on the other hand, usually short (in the range of nanoseconds to picoseconds) compared to the \(\text Q\)-switched pulsewidths \(\tau_p\) in practical lasers (which are typically tens to hundreds of nanoseconds).
The absorber's population difference will then be given to a good approximation by the steady-state solution of the absorber rate equation, or
\[\tag{30}N_a(t)\approx\frac{N_{a0}}{1+(K_a/\gamma_{2a})n(t)}.\]
meaning that the saturable absorber saturates in an essentially instantaneous fashion during the \(\text Q\)-switched pulse.
The initial growth rate for the cavity photon number, just before saturation of either absorber or amplifier occurs, is then given by
\[\tag{31}\frac{dn(t)}{dt}\approx[K_gN_{g0}-K_aN_{a0}-\gamma_c]n(t)=\gamma_{g0}n(t),\]
where \(\gamma_{g0}\equiv K_gN_{g0}-K_aN_{a0}-\gamma_c\) is the initial growth rate for the photon number, before any \(\text Q\)-switching has occurred. The photon number \(n(t)\) thus grows initially from a (very small) starting value \(n_i\) in the form
\[\tag{32}n(t)\approx n_i\;\text{exp}(\gamma_{g0}t).\]
The population inversion \(N_g(t)\) can then be written, using the approximation of Equation 29, in the form
\[\tag{33}N_g(t)\approx N_{g0}\;\text{exp}[-K_gn(t)/\gamma_{g0}],\]
where this approximation should be valid at least the very early stages of \(\text Q\)-switching.
If the above results for \(N_a(t)\) and \(N_g(t)\) are used in the rate equation for the cavity photon number \(n(t)\), the growth rate of the cavity signal for very early times in the \(\text Q\)-switching period then can be written approximately as
\[\tag{34}\frac{1}{n(t)}\frac{dn(t)}{dt}=K_gN_{g0}\;\text{exp}\left(\frac{-K_gn(t)}{\gamma_{g0}}\right)-\frac{K_aN_{a0}}{1+(K_a/\gamma_{2a})n(t)}-\gamma_c.\]
If we expand each of the terms on the right-hand side of this equation to first order in \(n(t)\), we can write this to a first approximation as
\[\tag{35}\frac{1}{n(t)}\frac{dn(t)}{dt}\approx\gamma_{g0}+\left(\frac{K^2_aN_{a0}}{\gamma_{2a}}-\frac{K^2_gN_{g0}}{\gamma_{g0}}\right)n(t)+\cdots.\]
The photon number at first grows as \(\gamma_{g0}\); but as \(n(t)\) begins to increase, the growth rate changes as determined by the second term on the right-hand side of the equation.
The "Second Threshold" Condition
The criterion that controls the \(\text Q\)-switching behavior is whether the second term on the right-hand side of Equation 35 has a positive or a negative sign, so that slope of the growth curve for the signal intensity will turn increasingly upward or downward as the photon number \(n(t)\) increases.
In physical terms, the question is whether the saturable absorber term will saturate first, thereby allowing the net growth rate to turn upward in an expanding or \(\text Q\)-switching fashion, as shown in Figure 16; or whether the gain will begin to saturate first, so that the intensity in the laser will never turn upward, but will only turn downward with increasing \(n(t)\), so that a true giant pulse never develops.
An approximate analytical criterion for good \(\text Q\)-switching is thus that the coefficient of the \(n(t)\) term on the right-hand side of the preceding equation must be positive.
This criterion will be satisfied if
\[\tag{36}\frac{K^2_aN_{a0}}{K^2_gN_{g0}}>\frac{\gamma_{2a}}{\gamma_{g0}}.\]
This is a fundamental condition, sometimes called the "second threshold condition," that must be satisfied for good passive \(\text Q\)-switching. (The "first threshold" is the earlier point in the pumping pulse where the gain first exceeds the unsaturated loss, so that the intensity in the cavity can begin to grow at all; the second threshold is the break point where the growth curve for \(n(t)\) turns upward.)

FIGURE 16. Good passive or saturable-absorber \(\text Q\)-switching is dependent on the existence of a "second threshold."
To put this criterion into more readily understandable terms, we can note that the ratio of the initial growth and decay rates \(K_gN_{g0}\) and \(K_aN_{a0}\) is just the ratio of the initial or unsaturated gain and loss factors \(2a_{g0}L_g\) and \(2a_{a0}L_a\) in the laser medium and in the saturable absorber cell.
These are both experimentally measurable quantities. Also, from the two rate equations we can see that the ratio of the saturation intensities in the two media will be given by
\[\tag{37}\frac{I_\text{sat,gain}}{I_\text{sat,abs}}\equiv\frac{\sigma_a\tau_a}{\sigma_g\tau_g}=\frac{\gamma_{2g}/K_g}{\gamma_{2a}/K_a}=\frac{K_a\tau_a}{K_g\tau_g}.\]
This tells us the rather obvious fact that the gain and loss coupling coefficients are related by \(K_a/K_g\equiv\sigma_a/\sigma_g\), since stimulated transition coefficients \(\text K\) are directly proportional to the stimulated emission or absorption cross sections \(\sigma\) in the two media.
With these interpretations, the criterion for good passive \(\text Q\)-switching reduces to
\[\tag{38}\frac{\gamma_{g0}}{\gamma_{2a}}\times\frac{\sigma_a}{\sigma_g}\times\frac{2a_{g0}L_g}{2a_{a0}L_a}>1.\]
To put this in still more practical terms, we can note that the saturable absorber's decay rate and lifetime are related by \(\gamma_{2a}\equiv1/\tau_a\). Suppose in addition that the net exponential gain coefficient (power gain coefficient) for one complete round trip around the laser cavity in the initial stage, before any saturation takes place, is denoted by \(\delta_{g0}\). That is, we write the initial photon growth rate as
\[\tag{39}\frac{1}{n}\;\frac{dn}{dt}\bigg|_{t=0}\equiv\gamma_{g0}=\frac{e^{\delta_{g0}}-1}{T}\approx\frac{\delta_{g0}}{T},\quad\delta_{g0}\ll1,\]
where \(\text T\) is the round-trip transit time. The criterion then becomes, finally,
\[\tag{40}\delta_{g0}\times\frac{\tau_a}{T}\times\frac{2a_{g0}L_g}{2a_{a0}L_a}\times\frac{\sigma_a}{\sigma_g}>1.\]
The ratio of cross sections \(\sigma_a/\sigma_g\) in this expression can be quite large (for example, \(\sigma_a\approx 10^{-16}\;\text{cm}^2\) for a dye absorber versus \(\sigma_g\approx 10^{-19}\;\text{cm}^2\) for a solid-state laser). The ratio of time constants, on the other hand, is apt to be small \(\tau_a<0.1\;\text{ns}\), perhaps, whereas \(\text T\approx5\text{ns}\)).
The ratio of laser gain to saturable absorber loss, or \(2a_{g0}L_g/2a_{a0}L_a\), must be greater than unity, but perhaps not by any large factor. Finally, the net initial growth coefficient \(\delta_{g0}\) in a passively \(\text Q\)-switched laser may be fairly small, as the pump slowly pushes the laser above threshold. Fast pumping, to push \(\delta_{g0}\) upward rapidly, is thus desirable.
Some readers may wonder why the net growth coefficient \(\delta_{g0}\) just at the start of the \(\text Q\)-switching process appears in this formula. The answer is that the saturable absorber saturates, in our approximation, on an instantaneous intensity basis, whereas the laser gain medium saturates on a cumulative or integrated intensity basis.
We want the laser intensity to grow very rapidly in the initial stages, so that the cavity signal reaches the absorber saturation level before very much of the laser inversion has been burned up.
4. REPETITIVE LASER Q-SWITCHING
Another useful way to operate some lasers is to pump the laser medium on a continuous rather than pulsed basis, and then to repeatedly \(\text Q\) switch the laser cavity at a repetition rate which may range from a few hundred to several thousand pulses per second.
Practical examples of repetitively \(\text Q\)-switched lasers include the \(\text{CO}_2\) laser, using either a rotating mirror or a \(\text{GaAs}\) or \(\text{Ge}\) acoustooptic cell, and the \(\text{Nd}\):\(\text{YAG}\) laser, using a quartz acoustooptic \(\text Q\)-switch.
The lower power but higher repetition rate pulses from such lasers can be useful, for example, in micromachining, surgical applications, or scientific experiments.
Characteristics of Repetitively Q-Switched Lasers
Because of practical limitations on the average pump power that can be applied to a laser under \(\text{cw}\) conditions, the round-trip gain in a \(\text{cw}\)-pumped and \(\text Q\)-switched laser just before the \(\text Q\)-switch opens is usually rather modest (perhaps 20-\(40\%\) round-trip power gain in a \(\text{cw}\) \(\text{Nd}\):\(\text{YAG}\) or \(\text{CO}_2\) laser); and the inversion ratio just after the \(\text Q\)-switch opens is similarly modest (perhaps \(r\approx1.2\) to 1.4).
As a result the pulse peak powers tend to be much smaller in repetitively \(\text Q\)-switched operation than in flash-pumped and \(\text Q\)-switched operation of the same laser, and the pulse build-up times and pulse durations substantially longer (from perhaps several hundred nanoseconds up to a few microseconds).
These lower-intensity pulses can still be very useful, for example, in materials processing applications, where the cutting or drilling effect of even a weak \(\text Q\)-switched pulse tends to be much more efficient than the same energy delivered in a \(\text{cw}\) beam.
Repetitively \(\text Q\)-switched \(\text{Nd}\):\(\text{YAG}\) lasers can thus be useful for resistor trimming, or for scribing or cutting thin films or integrated circuits.
Experiments on nonlinear effects, such as second harmonic generation or stimulated Raman scattering, which depend much more on peak laser power than on average power, are also much more effectively performed with the same total energy delivered as a series of pulses than as a \(\text{cw}\) beam.
The lower gain and the slower dynamics in the cw-pumped situation make it possible to use \(\text Q\)-switching modulators which have smaller hold-off and slower opening times than in the flashpumped situation.
The premier example of this type of modulator is the acoustooptic \(\text Q\)-switch illustrated in Figure 2 of this chapter. Figure 17 shows how the radio-frequency signal applied to an acoustooptic modulator can be turned off to "open" the \(\text Q\)-switch (upper trace), and the resulting \(\text Q\)-switched pulse from a low-power \(\text{Nd}\):\(\text{YAG}\) laser (lower trace).
The long time delay between the \(\text Q\)-switch opening and the output pulse is partly build-up time, but largely acoustic wave propagation time from the acoustic transducer to the laser beam position.
The requirements of repetitively \(\text Q\)-switched lasers thus match very well with the capabilities of acoustooptic modulators. In addition the \(\text{cw}\)-pumped and repetitively \(\text Q\)-switched laser tends to retain much the same power stability and good transverse and longitudinal mode stability as characterizes the underlying \(\text{cw}\)-pumped laser.
Elementary Analysis of Repetitive Q-Switching
To analyze this mode of operation in the simplest situation, we can first note that the time interval needed for the build-up and emission of a \(\text Q\)-switched pulse after the \(\text Q\)-switch opens will typically be a few microseconds or less, whereas the repumping interval between pulses will almost always be \(\geqslant\) 100 \(\mu s\) (sometimes considerably longer).
Therefore we can divide the periodic repetitive operation into a \(\text Q\)-switching interval which is essentially of zero length, plus a repumping

FIGURE 18. One period of a repetitive Q-switching cycle.
interval of duration \(r_r\equiv 1/f_r\) where \(f_r\) is the \(\text Q\)-switching repetition rate, as illustrated in Figure 18.
If we take the simplest possible laser model (single cavity mode, ideal upper laser level, fast-emptying lower level, and fixed \(\text{cw}\) pumping rate which is \(r_{cw}\) times above the \(\text{cw}\) laser threshold for the low-loss condition), then the laser inversions \(N_i\) and \(N_f\) just before and just after the \(k\)-\(\text{th}\) \(\text Q\)-switched pulse will be given by the \(\text Q\)-switching energy relation
\[\tag{41}N_i^{(k)}-N_f^{(k)}=N_\text{th}\text{ln}\left(\frac{N_i^{k}}{N_f^{(k)}}\right).\]
But the final inversion \(N_f^{(k)}\) just after the \(k\)-\(\text{th}\) pulse and the initial inversion \(N_i^{(k+1)}\) just before the \(k+1\)-\(\text{th}\) pulse are also connected by the repumping relation
\[\tag{42}N_i^{(k+1)}=r_{cw}N_\text{th}+(N^{(k)}_f-r_\text{cw}N_\text{th}e^{-\gamma_2r_r},\]
where \(r_{cw}\) is the normalized \(\text{cw}\) pumping rate and \(\gamma_2\) is the population decay rate for the upper laser level. These quantities are illustrated in Figure 18.
If we normalize both these inversions to the threshold inversion \(\text N_\text{th}\), we can write both of these equations in the dimensionless forms
\[\tag{43}r_i^{(k)}-r_f^{(k)}=\text{ln}\left(r_i^{(k)}/r_f^{(k)}\right)\quad\text{and}\quad r_i^{(k+1)}=r_\text{cw}-(r_\text{cw}-r_f^{(k)})e^{-\gamma_2/f_r},\]
where \(r_i=N_i/N_\text{th}\) is the actual inversion ratio just before each \(\text Q\)-switched pulse \((r_i\leqslant r_\text{cw})\) as defined in an earlier section; and \(r_f=N_f/N_\text{th}\leqslant 1\) is the residual inversion ratio \((r_f\leqslant 1)\) just after each pulse.
Steady-State Solutions
Under steady-state operating conditions, these initial and final inversions will each remain the same on successive pulses, so that we can drop the \(k\) and \(k+1\) superscripts.
Suppose we also define the normalized energy \(\omega_\text{out}\) dumped

FIGURE 19. Variation of pulse energy and average power output with repetition rate in a repetitively \(\text Q\)-switched laser.
in each pulse by
\[\tag{44}\omega_\text{out}\equiv\frac{N_i-N_f}{N_\text{th}}=r_i-r_f.\]
In physical terms, this normalized energy \(\omega_\text{out}\) is the ratio of the population dumped during a single \(\text Q\)-switched pulse to either the population stored in the rod under \(\text{cw}\) oscillation conditions, or the population pumped up in one lifetime \(\tau_2\) when the pump is exactly at threshold. (It is also the same thing as \(r_i\eta(r_i)\) where \(\eta\) is the energy extraction efficiency defined in an earlier section.)
\[\tag{45}\omega_\text{out}=r_i(1-e^{-u_\text{out}}),\]
after which Equations 41 through 45 can be combined to obtain the normalized pulse energy \(\omega_\text{out}\) for a given pumping rate \(r_\text{cw}\) and normalized repetition frequency \(f_r/\gamma_2\) The resulting values of \(r_i\), together with \(\gamma_c\), will then determine the pulsewidth and the build-up time of the \(\text Q\)-switched pulse, using the formulas from Section 2. Typical results from this calculation are shown in Figure 19.
The total energy \(\text U_\text{out}\) extracted from the laser medium on each \(\text Q\)-switched pulse is then given by
\[\tag{46}U_\text{out}=(N_i-N_f)\hbar\omega_a=\omega_\text{out}(r_\text{cw},f_r)\times N_\text{th}\hbar\omega_a.\]
The average laser power output is this energy per pulse \(U_\text{out}\) times the repetition rate \(f_r\), or \(P_\text{av}=f_r U\text{out}\). The average power output from the same laser under \(\text{cw}\) conditions, on the other hand, would be \(P_\text{cw}=(r_\text{cw}-1)\gamma_2N_{th}\hbar\omega\).
The ratio of average power under \(\text Q\)-switching conditions to the potential \(\text{cw}\) power output is then
\[\tag{47}\frac{P_\text{av}}{P_\text{cw}}=\frac{f_r}{\gamma_2}\times\frac{\omega_\text{out}(r_\text{cw},f_r)}{r_\text{cw}-1}.\]

FIGURE 20. Experimental results for peak pulse power, average power, and pulse build-up time versus repetition rate in a \(\text Q\)-switched \(\text{Nd}\):\(\text{YAG}\) laser.
(Both of these powers are actually the total power extracted from the laser medium into the oscillating mode; the actual useful output power in both situations will be somewhat less because of internal cavity losses.)
Experimental Results
Figure 20 gives some experimental results showing how the peak \(\text Q\)-switched pulse power and the average power output depend on repetition rate or repetition frequency \(f_r\) for a typical repetitively \(\text Q\)-switched \(\text{Nd}\):\(\text{YAG}\) laser. The experimental results compare well with the simple theory just developed.
The obvious conclusions are that at very low repetition rates, \(f_r\leqslant\gamma_2\), the laser acts as a source of fixed-energy \(\text Q\)-switched laser pulses with energy corre-sponding to an inversion ratio \(r_i\approx r_{cw}\), and the average power increases directly with repetition rate; whereas at high repetition rates, \(f_r\geqslant \gamma_2\), the laser supplies a fixed average power very nearly equal to the power \(P_\text{cw}\) the same laser would supply under \(\text{cw}\) operating conditions.
The breakpoint between these regimes occurs at or very near the repetition frequency given by \(f_r\approx\gamma_2\).
5. MODE SELECTION IN Q-SWITCHED LASERS
Transverse and axial mode selection processes are generally less effective in \(\text Q\)-switched lasers than in continuous-wave \(\text{(cw)}\) or long-pulse types of lasers; and \(\text Q\)-switched lasers are therefore more likely than \(\text{cw}\) lasers to oscillate in several axial and/or transverse modes. Control of these modes can be a significant practical problem in \(\text Q\)-switched lasers used for certain applications.
Axial Mode Discrimination and Axial Mode Beating
When a laser, whether \(\text Q\)-switched or \(\text{cw}\), oscillates in two modes simultaneously, the output exhibits "mode beats" at the difference frequency between the two modes. Mode beats of this type are illustrated for a \(\text Q\)-switched solidstate laser in the oscilloscope traces in Figure 21.
The upper trace shows a clean, smooth \(\text Q\)-switched oscillation pulse corresponding to oscillation in only a single axial mode. The lower trace shows the same output with two axial modes oscillating simultaneously. The two modes in this situation are two adjacent axial modes separated by the observed beat frequency of 150 \(\text{MHz}\).

FIGURE 22. Mode control methods for \(\text Q\)-switched lasers.
This observed output simply represents the fact that the sum of two sinusoids at closely adjacent frequencies \(f_1\) and \(f_2\) appears to be a single sinusoid at the average frequency \((f_1+f_2)/2\), whose amplitude is modulated at the difference frequency \(f_d=|f_1-f_2|\).
If three or more equally spaced modes are present, the output will still be periodic at the intermode frequency, but with a more complicated periodic waveform.
Note that in \(\text Q\)-switched lasers with shorter cavities and/or with wider atomic lines, the mode beats between axial modes may be at very high frequencies, which are outside the passband of the photodetector/amplifier/oscilloscope combination used to observe the laser output.
A smooth laser output pulse does not therefore guarantee single-mode operation, unless the photodetection system employed is fast enough to resolve the mode beats within the pulse.
Axial Mode Control
Axial mode beats may be undesirable in some applications, and various techniques are therefore employed to promote single-axial-mode oscillation in \(\text Q\)-switched (as also in \(\text{cw}\)) lasers.
As shown in Figure 22, one common technique is to employ one or more resonant etalons within the laser cavity, so as to modulate the cavity losses and provide additional loss for all but the centermost mode, as discussed in an earlier chapter. Another technique, often employed in \(\text{CO}_2\) lasers among others, is to place a low-pressure, narrowband gain cell within the same cavity as the high-pressure broadband \(\text{TEA}\) gain cell that furnishes the primary laser power.
The low-pressure cell is then operated either in a long-pulse or possibly even \(\text{cw}\) mode, so as to furnish either a preferential narrowband gain region containing only a single axial mode, or possibly even to produce weak "pre-lasing" before the main \(\text Q\)-switch pulse occurs.

FIGURE 23. Mode build-up and mode competition in a \(\text Q\)-switched laser.
More sophisticated single-mode techniques include injecting a weak signal from a separate single-frequency laser into the \(\text Q\)-switched laser cavity so that one single cavity mode is given a large preferential initial excitation; or employing various complex feedback systems on the cavity length to ensure that one single axial mode is located exactly at the central gain maximum of the laser medium.
Mode Discrimination in Q-Switched Lasers
A characteristic feature of actively \(\text Q\)-switched lasers is that there is little or no gain saturation during the major portion of the oscillation build-up.
During the build-up of the oscillation pulse the laser gain is usually large compared to the cavity losses for both the preferred or lowest-loss cavity mode and for some number of higher-loss axial and transverse modes; and so all of these modes tend to grow at comparably rapid rates for most of the pulse build-up time.
Only very near the peak of the \(\text Q\)-switched pulse is the gain saturated down to near or below the cavity losses.
Figure 23 thus illustrates schematically how the preferred or lowest-loss mode (sometimes called the dominant mode) and also the next higher-loss mode may grow to nearly the same amplitude on a log scale during the \(\text Q\)-switched burst.
As the inversion becomes saturated down by the dominant mode, near the peak of the \(\text Q\)-switching process, the gain of the next higher-loss mode also decreases; and this mode will typically pass through its peak value slightly earlier,

FIGURE 24. Ratio of intensities in two competing modes versus number of round trips, for different values of the discrimination parameter \(\delta\).
and begin to decay slightly faster than the dominant mode. If the inversion is large, however, and the losses of the two modes nearly the same, the next higher-loss mode may reach nearly the same peak amplitude as the dominant mode.
The axial and transverse mode selection in a \(\text Q\)-switched laser may, therefore, not be as good as in \(\text{cw}\) or long-pulse lasers, where the mode discrimination process has a longer time to operate.
Furthermore, there may be a noticeable improvement in mode discrimination even during the \(\text Q\)-switched pulse itself.
Mode Discrimination Analysis
The conditions necessary to achieve good mode discrimination by the time the peak of the giant pulse occurs can be developed from a simple argument which applies in essentially the same way to mode discrimination between either multiple axial or transverse modes in the laser.
To show this, let two different modes in a \(\text Q\)-switched laser cavity be denoted by indices \(a\) and \(\beta\). The ratio of the intensities of these two modes after \(\text N\) round trips during the build-up phase, starting from the same noise level, will then be given by an expression like
\[\tag{48}\frac{I_a}{I_\beta}=\bigg|\frac{[R_1R_2\;\text{exp}(2a_mp_m-2a_0p)]_a}{[R_1R_2\;\text{exp}(2a_mp_m-2a_0p)]_\beta}\bigg|^N=e^{N\delta}.\]
The numerator and denominator of the middle term represent the net roundtrip gain minus loss values for the two individual modes; and we then assume that these two net gain minus loss values stand in the ratio \(e^{\delta}\), where \(\delta=0.01\), for example, represents a \(1\%\) difference in the two net round-trip gains.
This difference may result in practice from either a difference in actual laser gain, as for example with two different axial modes, one of which is farther from line center; or from a difference in cavity losses, as for example with two different transverse modes.
The number of round in a typical \(\text Q\)-switched laser during the build-up phase will typically be on the order of \(N\approx 20\) to 30. Figure 24 plots the ratio of intensities \(I_a/I_{\beta}\) that will have developed after a number \(\text N\) of round trips, starting from the same initial noise intensities, for different values of the gain ratio parameter \(\delta\).
The general conclusion is that if two different modes have only a few percent difference in gain, the mode discrimination between them after even 20 or 30 round trips will be very poor. For good mode discrimination in a \(\text Q\)-switched laser, at least a \(10\%\) to \(15\%\) difference in mode losses or gains is required.
Transverse Mode Control and Mode Distortions
The suppression of unwanted higher-order transverse modes in \(\text Q\)-switched lasers is much like transverse mode control in \(\text{cw}\) lasers. That is, the general approach is to use suitable resonator designs and intracavity apertures so that the lowest-order transverse mode has significantly lower diffraction losses than the next higher-order transverse mode.
This condition must, however, be more stringently enforced in a \(\text Q\)-switched laser than in a \(\text{cw}\) laser, for the reasons discussed above.
Transverse mode control may in some situations be more effective in passively rather than actively \(\text Q\)-switched lasers, because the dominant transverse mode which builds up first will preferentially saturate the saturable absorber cell in the transverse region which its fields are most intense.
The saturable absorber then becomes a kind of dynamic spatial filter, which lets the lowest-order mode build up while continuing to suppress higher-order modes.
The transverse mode dynamics can also be complicated in a \(\text Q\)-switched laser, however, by dynamic changes in the resonator parameters occurring during the \(\text Q\)-switched pulse, caused by dynamic gain saturation produced by the pulse. The gain medium saturates first in those portions of the transverse cross section where the laser field is most intense.
This may cause the high-gain laser medium to act like a rapidly changing aperture, which may modify and distort the transverse profile of the circulating energy on successive round trips. We may then expect dynamic changes in the output beam profile during the pulse itself, produced by both \((a)\) continued filtering out of higher-order transverse modes, and \((b)\) dynamic distortion of all the modes.
6. Q-SWITCHED LASER APPLICATIONS
We can review briefly in this section some typical performance figures for \(\text Q\)-switched lasers, and some of the practical applications for these lasers.
Typical Performance Results
As a practical example of laser \(\text Q\)-switching, we might consider a typical small-to-medium size solid-state laser, such as a \(\text Q\)-switched ruby, \(\text{Nd}\):\(\text{YAG}\) or Nd:glass laser oscillator.
The pump power input from the capacitor bank to the flashlamp (or lamps) in such a laser might be 20 to 200 \(\text J\) delivered in a pulse 100 \(\mu s\) long, corresponding to an electrical power input of 1 to 2 \(\text{MW}\) to the flashlamps.
This electrical input might be converted to laser output in a typical laser with an efficiency between \(0.1\%\) (typical) and \(1\%\) (absolute best), so that the laser output energy will be between 20 \(\text{mJ}\) and perhaps 2 \(\text J\) in a single laser shot.
If the laser oscillates in "long pulse mode" (no \(\text Q\)-switching), this corresponds to an average power output between 200 \(\text W\) and 20 \(\text{kW}\) during the 100 \(\mu s\) pumping period (although spiking effects may produce peak powers 10 to \(20\times\) as large).
Suppose the same energy is extracted in a \(\text Q\)-switched pulse which is only 100 \(\text{ns}\) long. The peak optical output power during this pulse then has a value somewhere in the range from 2 to 200 \(\text{MW}\) peak, or more than \(10^4\) larger than the power level from the same laser in long pulse mode (although this may be reduced somewhat by the fact that in general a laser will oscillate more efficiently in long-pulse than in \(\text Q\)-switched operation).
These numbers are quite typical for \(\text Q\)-switched solid-state lasers. If more energy per pulse is needed, either larger diameter rods and flashlamps can be employed or, more commonly, the initial \(\text Q\)-switched laser is followed by one or more stages of laser amplification.
Extraction of too much energy from a single oscillator is usually avoided because of self-focusing and optical-damage problems that occur in the laser mirrors, laser rods, and \(\text Q\)-switching elements if the \(\text Q\)-switched energy or peak pulse power is increased much further.
Q-Switched Laser Applications
Important applications for the short pulses and high peak powers provided by \(\text Q\)-switched lasers include:
\((a)\) Laser radars, or lidars: The short intense pulses generated by \(\text Q\)-switched lasers are ideal for optical ranging, optical radar, or "lidar" applications, as for example in tank or artillery range-finders. \(\text Q\)-switched solid-state lasers along with the associated detection electronics can be packaged into remarkably small and rugged units, not much bigger than a large pair of binoculars, for military field applications.
The pulses from larger \(\text Q\)-switched lasers can be transmitted though large telescopes to permit ranging off corner-cube retroreflector targets in orbiting satellites, or on the surface of the moon.
By integrating over many \(\text Q\)-switched shots, the distance to the moon can in fact be measured with relative accuracy of 10 \(\text{cm}\) or less, permitting precise calculations of the lunar orbit and of "moon-quakes."
By conducting accurately timed ranging experiments on an orbiting satellite from observing stations in different continents, we can calculate the satellite orbit to very high accuracy; we can detect perturbations in the orbit caused by mass perturbations in the Earth itself; and distances between the observing stations can be established with sufficient precision to potentially measure continental drifts. \(\text{FLAG}\)
\((b)\) Tunable laser radars: Laser ranging systems using tunable lasers, whose wavelengths can be tuned to specific atomic or molecular transitions, can be used for for pollution detection, aerosol measurement, ranging on clouds and measurement of optical visibility, and similar applications.
Similar measurements from orbiting space stations may eventually become possible, leading to world-wide pollution detection and weather forecasting applications.
\((c)\) Cutting and drilling applications: The intense pulses from even rather modest sized \(\text Q\)-switched lasers are also extremely effective for optical cutting, welding, and scribing applications. \(\text Q\)-switched lasers can be used for drilling diamonds for wire dies and instrument jewels; scribing semiconductor wafers so that they can be separated into individual chips; welding shut nuclear fuel rods, and trimming resistors and integrated circuits while simultaneously monitoring the electrical properties of these circuits.
Unique applications include drilling holes in small rubber baby bottle nipples or in plastic pipes used for trickle irrigation systems.
\((d)\) Basic scientific experiments, especially nonlinear optics: Finally, \(\text Q\)-switched lasers have found great many applications in basic physics experiments, particularly where we need to have high-peak power to produce some nonlinear or optical breakdown effect, while still having only relatively small total energy so as to avoid damage effects.
Examples of such physical phenomena include optical harmonic generation, stimulated Raman and Brillouin scattering, gas breakdown, and the shock heating of gases and other samples.