LASER SPIKING AND MODE COMPETITION

This is a continuation from the previous tutorial - Laser dynamice the laser cavity equations

 

 

1.  LASER SPIKING AND RELAXATION OSCILLATIONS

As soon as the first ruby laser was operated, it was immediately evident that this laser did not wish to oscillate smoothly or continuously during the 1 millisecond or so duration of the pumping flash. Figure 1 shows in fact the pumping flash, and the extremely irregular and unstable laser oscillation that resulted, as reported in one of the publications by T. H. Maiman on the first successful operation of the ruby laser. (The initial transient in the laser output trace is electrical leakage from the f lashlamp trigger circuitry.) 

More careful examination of this laser output intensity on an expanded time scale, using a faster photodetector and oscilloscope, showed that the ruby laser output typically consisted of an irregular sequence of sharp narrow pulses or

 

 

 

 

 

 

 

"spikes," each a fraction of a microsecond wide and a few microseconds apart, as illustrated in Figure 2. Sometimes the spiking behavior jumped erratically and discontinuously as in the upper trace, due presumably to fluctuations or mode jumps in the laser itself; whereas under more stable conditions the spiking behavior appeared to gradually damp out as in the lower trace (which is taken in the trailing edge of the pumping flash). 

The ruby laser, for a variety of reasons—primarily because it is a three-level solid-state laser—almost always tends to spike in this very strong and irregular fashion, although under special conditions even ruby laser spiking can be controlled.

Figure 3 shows, for example, the pumping pulse (upper trace) and the output intensity (lower trace) from a very small ruby laser which is operated under very well-controlled quasi-\(\text{cw}\) conditions at liquid nitrogen temperature (77 \(\text{K})\) and end-pumped by a 514.5 nm beam from a \(\text{cw}\) argon laser.

The laser exhibits a strong spiking response when it first comes on, but this spiking behavior rapidly damps out (with only one visible "glitch") into an essentially \(\text{cw}\) oscillation.

It was soon found that other and better disciplined four-level solid-state lasers also exhibited a similar spiking behavior when they were first turned on. The initial spiking in these other lasers, however, almost always damps down fairly quickly into a decaying quasi-sinusoidal relaxation oscillation in the laser power output.

 

 

 

 

This behavior is well-illustrated in Figure 4, which shows typical transient turn-on behavior for three different \(\text{Nd}\):\(\text{YAG}\) lasers. 

 

Large-Amplitude Spiking Versus Relaxation Oscillations

Most lasers that exhibit such spiking, in fact, if they are operated with stable power supplies and in a quiet and stable environment, will eventually settle down to a fairly constant output intensity. Even in this limit, however, any small perturbation, such as a sudden change in pumping rate or cavity loss, will trigger a transient relaxation oscillation of the same general character, which will again die away exponentially in the same oscillatory fashion. 

The terminology used to describe this kind of transient laser behavior is not entirely uniform in the laser literature. We will generally use the term "spiking" to refer to the discrete, sharp, large-amplitude pulses that typically occur during the initial turn-on phase of many lasers.

We will then use "relaxation oscillations" to describe the small-amplitude, quasi-sinusoidal, exponentially damped oscillations about the steady-state amplitude which occur when a continuously operating laser is slightly disturbed, or into which the initial spiking behavior generally evolves.

Spiking and relaxation oscillations are phenomena that are characteristic of most solid-state lasers, semiconductor lasers, and certain other laser systems in which the recovery time of the excited state population inversion is substantially longer than the laser cavity decay time.

Most gas lasers do not satisfy this necessary condition, and as a result spiking and relaxation oscillations are not generally observed in most gas lasers.

 

Laser Rate Equations

Spiking is an example of laser behavior that can be accurately described using a very simple set of single-mode, single-atomic-level rate equations for the laser. Let us write these equations once again as an equation for the cavity photon number 

\[\tag{1}\frac{dn(t)}{dt}=KN(t)n(t)-\gamma_cn(t),\]

and an equation for the population inversion 

\[\tag{2}\frac{dN(t)}{dt}=R_p-\gamma_2N(t)-KN(t)n(t).\]

We are as usual describing the oscillation amplitude in the laser cavity by the photon number \(n(t)\) in the oscillating mode, and the instantaneous population inversion by the upper level population or population inversion \(N(t)\). The pumping rate for the laser inversion is then \(R_p\) and the atomic decay rate is \(\gamma_2\).

These two coupled equations are nonlinear because of the product term \(KN(t)n(t)\) in each equation; and it is therefore not surprising that they may under certain conditions exhibit relaxation oscillations in their evolution toward a steady-state. In fact, these two equations will describe both spiking and relaxation oscillations with more than adequate accuracy in nearly all four-level laser systems.

There do not seem to be, however, any simple analytic solutions to Equations 1 and 2 that apply during the period of strong spiking, when both the population \(N(t)\) and particularly the cavity photon number \(n(t)\) are changing in a rapid and quite nonlinear fashion.

 

Elementary Description of Laser Spiking

A reasonably accurate description of the spiking process when a solid-state laser is first turned on can be given by following the graphical argument illustrated in Figure 5, which shows the time evolution of a single laser spike. In Figure 5 we suppose that the pumping intensity has been turned on somewhat earlier, so that the population inversion \(N(t)\) is passing up through the threshold value \(N_{th}\) at the starting time \(t_1\) shown in this diagram, whereas the photon number in the laser cavity is still essentially zero at this time. (Note that the buildup time for the population inversion to first reach threshold in solid-state lasers is typically on the order of several hundred microseconds or longer.) 

So long as the population inversion \(N(t)\) is below the threshold value\(N_{th}\) (which represents also the steady-state oscillation value), the photon density in the laser cavity remains essentially at zero (or more accurately at roughly one noise photon per cavity mode).

As soon as the population inversion passes through the threshold value \(N_{th}\), however, at time ti in the figure, the laser gain then exceeds the loss and the photon number in the cavity begins to build up exponentially from noise.

The exponential growth rate for this build-up is \([N(t)/N_{th}-1]\gamma_c\), where \(\gamma_c\) is the cavity decay rate as defined earlier, and \(N(t)/N_{th}\) is the instantaneous ratio of laser gain to cavity loss. This ratio is continually increasing, , as the pump pushes the population inversion further above threshold.

The e-folding time for build-up of the photon number \(n(t)\) is thus on the order of the cavity decay time \(\tau_c\) or thereabouts, which might be several tens of nanoseconds in a typical laser cavity. The cumulative time for the cavity photon number to build up from the initial noise level to an observable output signal 

 

 

 

 

corresponding to perhaps \(10^8\) to \(10^{10}\) photons in the cavity is 20 to 30 times longer than this, in the range of hundreds of nanoseconds to perhaps a few microseconds in typical lasers. The buildup rate for \(n(t)\) due to stimulated emission may thus be hundreds of times faster than the buildup rate for \(N(t)\) produced by the pumping process.

As soon as the cavity photon number \(n(t)\) passes through the steady-state oscillation level \(n_{ss}\)—that is, the level which would correspond to continuous steady-state operation in that laser at that pumping rate—the signal intensity in the cavity is large enough to begin burning up excited state atoms at a faster rate than the pump supplies them. Beyond time \(t_2\) in Figure 5, therefore, the population inversion \(N(t)\) no longer continues to rise, but rather begins to be pulled rather rapidly downward.

The population inversion \(N(t)\) still exceeds the steady-state value \(N_{th}\), however; so the net gain in the cavity continues to be greater than unity. The cavity photon number therefore continues to rise rapidly.

In fact, it is only the last portion of this rise over many orders of magnitude that will be visible on a linear plot, or on a linear oscilloscope display.

The point at which the population inversion \(N(t)\) comes back just to the threshold or steady-state value \(N_{th}\), so that gain just equals loss, is also the point at which the photon density \(n(t)\) reaches its peak value (time \(t_3\) in the figure) and then begins to fall back downward. There is still a large signal intensity circulating around inside the laser cavity and burning up excited state atoms,

 

 

 

 

however, so that the population inversion \(N(t)\) continues to be driven downward below the steady-state level. The laser gain is now less than the cavity losses, however, so that there is a net loss in the laser cavity. The photon number \(n(t)\) therefore drops back down precipitously. 

The point where \(n(t)\) again reaches the steady-state oscillation level (point \(t_4\) in the figure) is also the point at which the population inversion reaches a minimum, after which the pump can again begin to build up the population inversion toward threshold.

The photon number \(n(t)\) continues to decrease down to negligible values, however, so the pumping back up of the population inversion by the pump source occurs essentially independently of the photon number in the cavity.

The laser spikes are thus steep and narrow, because of the rapid rates of rise and fall of the photon number \(n(t)\) in the cavity, with these rates being related to the cavity decay rate \(\gamma_c\). The spacing between pulses is somewhat longer, because it is determined by the more lethargic manner in which the pumping source is able to replenish the net population inversion \(N(t)\).

This kind of large-signal spiking behavior eventually damps down in most lasers toward a quasi-sinusoidal relaxation oscillation type of behavior. The spiking tends to damp down because neither the cavity photon number n(t) nor especially the population inversion \(N(t)\) drops all the way to zero following a spike. Hence each successive spike starts from initial conditions that come closer and closer to the steady-state behavior of the laser.

More accurate and detailed calculations of laser spiking behavior using the rate equations given in Equations 1 and 2 can only be made by solving these equations with the aid of an analog or digital computer.

Figure 6 shows calculated results using the rate equations for a hypothetical laser with atomic lifetime \(\tau_2=5\) \(\text{ms}\), cavity lifetime \(\tau_c=16\) \(\text{ns}\), and a pumping rate \(r\approx2600\), i.e., the laser is pumped very far above threshold.

Note that the interval between the early, large-amplitude spikes changes slightly from spike to spike, but is not greatly different from the period of the damped quasi-sinusoidal behavior toward which the laser system evolves.

Anyone who might undertake these kinds of numerical or computer solutions of the rate equations should note that they involve large and rapid changes of amplitude, particularly in the cavity photon number. Therefore solving Equations 1 and 2 to obtain accurate predictions on a numerical computer requires careful attention to

 

 

 

 

the numerical algorithm and the equation solving procedures that are followed.

 

Phase Plane Description

Another way to describe spiking behavior analytically is to plot values of the photon number \(n(t)\) and the population inversion \(N(t)\) as points in a phase plane which has \(N(t)\) as, say, the x axis and \(n(t)\) as the y axis, as in Figure 7. Dividing the two rate equations 1 and 2 into each other then gives the equation 

\[\tag{3}\frac{dn}{dN}=\frac{K(n+1)N-\gamma_cn}{R_p-KnN-\gamma_2N},\]

which gives the slope of the trajectory of \(n(t)\) versus \(N(t)\) passing through any point in the \(\text N\), \(n\) plane. 

Figure 7 shows the typical convergence toward a steady-state limit point for a moderately "spiky" laser. Starting from any initial point No, no in this plane, we can thus follow the spiking trajectory as it circles in toward the final convergence point at \(N_{th}\),\(n_{ss}\).

 

Relaxation Oscillations: Linearized Analysis

Once the spiking behavior in a laser oscillator has damped down to what are essentially small-amplitude fluctuations about the steady-state oscillation conditions in the laser, we can carry out a linearized small-signal analysis which gives simple analytic solutions for the relaxation-oscillation frequency and damping rate. 

To carry out this analysis we begin with the same rate equations as in Equations 1 and 2, and recall that the steady-state or dc solutions to these equations above threshold are given by

\[\tag{4}N_{th}=\gamma_c/K\quad\text{and}\quad n_{ss}=R_p/KN_{th}-\gamma_2/K=(r-1)\gamma_2/K.\]

Suppose that in the relaxation-oscillation regime the instantaneous photon number and population inversion in the laser will be not too distant from the steady-state values, so that we can write these quantities in the form

\[\tag{6}\begin{align}&\frac{dn_1(t)}{dt}=(r-1)\gamma_2N_1(t)\\&\frac{dN_1(t)}{dt}=-\gamma_cn_1(t)-r\gamma_2N_1(t).\end{align}\]

If we assume that the small quantities \(n_1(t)\) and \(N_1(t)\) vary as \(e^{st}\), then this leads to the secular determinant

\[\tag{7}\bigg|\begin{align}&s\quad-(r-1)\gamma_2\\&\gamma_c\quad s+r\gamma_2\end{align}\bigg|=0\]

and hence to the secular equation

\[\tag{8}s^2+r\gamma_2s+(r-1)\gamma_s\gamma_c=0.\]

The natural roots of the system, or the exponential decay rates and oscillation frequencies for the relaxation oscillation behavior, are therefore given by

\[\tag{9}s=s_1,s_2=-\frac{r\gamma_2}{2}\pm\sqrt{\left(\frac{r\gamma_2}{2}\right)^2-(r-1)\gamma_2\gamma_c.}\]

There are two different situations to discuss, depending on the sign of the quantity inside the square root.

\((a)\) Nonspiking Lasers : In most gas lasers, to take care of the nonspiking situation first, the atomic lifetime \(\gamma_2\) and the cavity decay rate \(\gamma_c\) may be of

 

 

the same order of magnitude. Let us suppose in fact that the cavity decay rate \(\gamma_c\) is somewhat slower, or that the laser is not too far above threshold, so that \((r-1)\gamma_2\gamma_c\) is smaller than \((r\gamma_2/2)^2\). The two natural roots of this equation are then 

\[\tag{10}s=s_1,s_2 \approx\left\{\begin{array}--r\gamma_2\\ [(r-1)/r]\gamma_c.
\end{array}\right.\]

The transient response of the oscillating laser to any kind of perturbation in this limit has two exponentially decaying roots, one of which corresponds essentially to a net population repumping rate \(r\gamma_2\), whereas the other corresponds to a net cavity build-up rate of \([(r-1)/r]\gamma_c\). The system is overdamped, so that any fluctuations die out in a double exponential form rather than an oscillatory fashion, with time constants corresponding roughly to the atomic and cavity lifetimes.

When a laser in this category is suddenly turned on, the laser oscillation will generally build up and converge toward the steady-state level with little or no overshoot, or at least without the kind of extreme relaxation oscillations associated with spiking types of lasers

\((b)\) Strongly Spiking Lasers : The alternative situation, which is characteristic of most solid-state and certain other lasers (for example, the iodine photodissociation laser), occurs whenever the atomic decay rate \(\gamma_2\) is very much slower (with a time constant of perhaps hundreds of microseconds) compared to the cavity decay rate \(\gamma_c\) (which is perhaps hundreds of nanoseconds).

In this situation the natural roots for the transient response of the system may be written in the form

\[\tag{11}\begin{align}s_1,s_2&\approx-\frac{r\gamma_2}{2}\pm j\sqrt{(r-1)\gamma_2\gamma_c-(\frac{r\gamma_2}{2})^2}\\&\equiv-\gamma_{sp}\pm j\omega'_{sp}.\end{align}\]

The system clearly has an exponentially damped sinusoidal response of the form

\[\tag{12}n(t)=n_{ss}+n_1e^{-\gamma sp^t}\cos\omega'_{sp}t,\]

in which \(\gamma_{sp}\equiv r\gamma_2/2\) gives the decay rate with which the relaxation-oscillation behavior dies out. The small-signal relaxation-oscillation frequency for the laser is then \(\omega'_{sp}\equiv\sqrt{\omega^2_{sp}-\gamma^2_{sp}}\approx\omega_{sp}\), where \(\omega_{sp}\equiv\sqrt{(r-1)\gamma_2\gamma_c}\). (We use the subscript \(sp\) for these quantities because \(\omega_{sp}\) is very often referred to as the "spiking frequency," although we are being more precise and calling it the "relaxation oscillation frequency.") The population inversion \(N(t)\) will have similar damped sinusoidal fluctuations about its steady-state value.

If we take typical values for, say, a \(\text{Nd}\):\(\text{YAG}\) laser pumped to \(50\%\) above threshold \((r=1.5)\), and assume \(\tau_2\approx230\) \(\mu\)sec for the atomic lifetime and \(\tau_c\approx30\) nsec for the cavity lifetime, we find a typical relaxation-oscillation frequency of

\[\tag{13}\omega_{sp}=\sqrt{(r-1)\gamma_2\gamma_c}\approx2\pi\times40\;\text{kHz}.\]

Figure 8 is an oscilloscope trace that shows a classic example of this type of ringing, or transient relaxation oscillation, in a \(\text{Nd}\):\(\text{YAG}\) laser. The dependence of the relaxation-oscillation frequency \(\omega_{sp}\) on \(\sqrt{(r-1)\gamma_2\gamma_c}\) has been checked experimentally in a number of lasers; indeed, measurements of \(\omega_{sp}\) have sometimes been used as a way to find values for the parameters \(r\), \(\gamma_2\) or \(\gamma_c\).

We can also define a \(Q\) factor for the relaxation oscillations, given by

\[\tag{14}Q_{sp}\equiv\frac{\omega_{sp}}{\gamma_{sp}}\approx\sqrt{\frac{4(r-1)\gamma_c}{r^2\gamma_2}}\approx10-100,\]

again in reasonable agreement with observations.

This analysis thus gives the resonant frequency \(\omega_{sp}\) and the damping rate \(\gamma_{sp}\) in the small-signal relaxation-oscillation regime for an ideal four-level laser system.

Note that if we identify the relaxation time or recovery time for the atomic population by \(\tau_2\equiv1/\gamma_2\) and an effective build-up time or recovery time for the cavity fields by \((r-1)/\tau_c\), then the relaxation-oscillation frequency is just the geometric mean of these, i.e.,

\[\tag{15}\omega_{sp}=\sqrt{\frac{r-1}{\tau_c}\times\frac{1}{\tau_2}}.\]

The repetition rate for the large-amplitude spikes in the strong spiking regime will generally be somewhat different (usually slower) than the small-amplitude relaxation-oscillation frequency \(\omega_{sp}\), but not usually by more than a factor of 2 or 3.

The damping rate for the large-amplitude spikes will also be different, but \(\omega_{sp}\) and \(\gamma_{sp}\) at least give a general indication of the resonant frequency and damping rate of the laser even for large-amplitude perturbations.

 

Ruby Laser Spiking

The ruby laser, since it is a three-level system, has a somewhat more complex set of rate equations than those used in this section. A linearized fluctuation analysis of the ruby laser must thus be carried out using the same approach as in this section but applied to the appropriate equations for the ruby system. 

Such an analysis shows that the oscillatory behavior in ruby has a quite similar relaxation-oscillation frequency \(\omega_{sp}\), but is much less damped than in four-level lasers, as illustrated experimentally by the weak damping observed in the typical ruby laser results shown at the beginning of this section.

The sudden random jumps that are typically observed in the ruby spiking behavior can be attributed to various sudden transient fluctuations in the laser, caused by such disturbances as pump fluctuations, acoustic vibrations, thermal expansion of the ruby rod, or the sudden turning on or off of one or more transverse or axial modes within the ruby laser.

Special ruby lasers built with very careful transverse mode control, and with unusually long laser cavities to increase the

 

 

 

 

cavity lifetime relative to the upper level lifetime, have been found to give much more regular and smoothly damped spiking behavior. 

 

Spiking in Semiconductor Injection Diode Lasers 

Semiconductor injection diode lasers also exhibit essentially the same kind of spiking behavior as ruby and \(\text{Nd}\):\(\text{YAG}\) lasers, but in a very different time or frequency domain, as illustrated by the curves in Figure 9.

In each of these curves the upper trace shows the driving current pulse through a \(\text{GaAs}\) injection diode laser, whereas the lower trace shows the laser oscillation output on the same time scale. Note, however, that this time scale is 2 nsec per division, or several orders of magnitude faster than in the previous illustrations. 

The physical processes in a typical semiconductor laser are somewhat more complex than, for example, in a \(\text{Nd}\):\(\text{YAG}\) laser, since a typical semiconductor laser will have a large distributed optical loss along the junction; a low mirror reflectivity at each end of the lasers; and a large lower-level absorption as well as a large current-pumped upper level amplification.

The spiking behavior in semiconductor lasers can nonetheless be at least roughly described by essentially the same rate equations as in Equations 1 and 2, but with very different values for the time constants and hence the eventual relaxation-oscillation frequency.

A typical GasAs injection laser may have a cavity which is \(L= 300\) \(\mu m\) long, with an index of refraction \(n=3.35\), and a distributed ohmic loss coefficient along the optical waveguide of \(2a_0\approx60\;cm^{-1}\). The end mirror reflectivities due to dielectric reflection at the cleaved ends will then be \(R_1=R_2\approx 0.3\); the roundtrip transit time in the cavity will be \(T=2nL/c_0\approx6.7\) psec; and the cavity lifetime will be \(\tau_c=T/[4a_0L+1n(1/R_1R_2)]\approx1.1\) psec.

The upper level lifetime for the inverted population in the p-n junction might be \(\tau_2=\gamma^{-1}_2\approx3\) nsec.

It is then clear that \(\tau_c\ll\tau_2\), which is the condition for relaxation oscillations and spiking. The relaxation-oscillation frequency for a \(\text{GaAs}\) pumped to 1.5 times threshold will then be very roughly given by

\[\frac{\omega_{sp}}{2\pi}\approx\frac{1}{2\pi}\sqrt{\frac{0.5}{3.3\times10^{-21}}}\approx2\;\text{GHz}.\]

This calculated result is in reasonable agreement with typical experimental results for such lasers.

 

Suppression of Spiking

Various attempts have been made to control or suppress spiking behavior, either by adding some kind of loss mechanism within the laser cavity whose loss increases with increasing photon number \(n(t)\)—an optical limiting element, so to speak—or by adding an external feedback loop with a photodetector and loss modulator within the cavity.

It can be shown analytically that adding even a small amount of fast-acting limiting effect or saturable gain in a spiking laser will strongly damp the spikes and relaxation oscillations. This approach is limited, however, by the inability to find any good, fast-acting, low-threshold optical limiter which will be effective at the power levels present in typical spiking lasers, and which will not add large amounts of unwanted loss at the normal operating point. (The laser gain medium itself provides a kind of slow-acting optical gain control or \(\text{AGC}\); but it is the time delay of this \(\text{AGC}\) that makes the relaxation oscillations possible.) 

The external feedback approach is quite feasible, but generally too complicated and expensive to be worth implementing, given the minor seriousness of the spiking phenomena. Stable mechanical design, acoustic isolation, and power supply stabilization are the keys to avoiding recurrent small-amplitude relaxation oscillations in a practical laser, such as the \(\text{cw}\) \(\text{Nd}\):\(\text{YAG}\) laser.

Spiking is thus in general more of a nuisance than a useful phenomenon. It does, however, provide a convenient illustration of the validity of the rateequation theory for laser oscillators. Measurements of the relaxation-oscillation frequency have also been used on occasion to calculate or verify the cavity lifetime \(\tau_c\) and the atomic lifetime \(\tau_2\) in laser oscillators.

The spiking behavior in semiconductor injection lasers has also been used to generate a single short pulse \((\approx 100\) psec or less) from an injection diode laser, by turning on the driving current very rapidly to create a strong initial spike and then turning the current off equally rapidly before the second and later spikes can form. This makes it possible to generate a single optical pulse significantly shorter than the width of the driving current pulse itself.

 

Gain Switching

Gain switching is another initial-buildup phenomenon in lasers which is generally very similar to spiking. In certain pulsed gas lasers, for example, pulsed or \(\text{TEA}\) \(\text{CO}_2\) lasers, it is possible to turn on the laser gain quite rapidly using a fast pulse of pumping current through the laser tube. (Fast gain turn-on is also accomplished in \(\text{TEA}\) or electron-beam pumped excimer lasers in the visible or near ultraviolet, and in other laser systems as well.) This fast pumping may cause the population inversion and gain to go considerably above threshold before the laser oscillation has time to build up from the initial noise level in the cavity.

 

 

The oscillation output of these lasers can then also exhibit a single, rather large initial output spike, possibly followed by a few additional weaker spikes, during the initial turn-on transient. This behavior, which is usually called "gain switching," is similar in character to spiking, and can be described by essentially the same rate equations. 

Figure 10 illustrates the kind of gain switching behavior that can be observed, for example, in pulsed \(\text{CO}_2\) lasers at \(\lambda=10.6\;\mu m\). The plots show the results of theoretical calculations for gain switching in a particular high-pressure \(\text{CO}_2\) laser. (These calculations happen to be a laser system in which the pumping power is supplied by a separate pulsed deuterium fluoride or \(\text{DF}\) laser; the \(\text{DF}\) laser radiation in the near infrared is absorbed by \(\text{DF}\) molecules contained in the \(\text{CO}_2\) laser gas mixture, and this pumping energy is then transferred to the \(\text{CO}_2\) molecules by collisional energy transfer.) The three traces show, from top to bottom, the assumed DF laser pump variation; the calculated \(\text{CO}_2\) laser output; and the time-varying average gain in the \(\text{CO}_2\) laser. Experimental results on this laser are in general agreement with these theoretical curves.

Figure 11 shows experimental results for two different discharge-pumped \(\text{CO}_2\) lasers. The left-hand plot shows the 500 nanosecond long current pulse in a typical transversely excited, atmospheric pressure \(\text{(TEA)}\) \(\text{CO}_2\) laser, together with the resulting laser output. In this laser the output more or less follows the fast pumping current pulse, except for a turn-on transient or gain spike on the leading edge.

 

 

 

 

The right-hand trace shows a somewhat unusual low-pressure \(\text{CO}_2\) laser result, in which the longitudinal discharge current is a very short but very intense current pulse only \(\approx50\) nsec long. This current pulse produces, however, a sizable population inversion which develops and persists in the low-pressure medium for a long time following the current pulse.

The laser thus begins oscillation with a sizable gain-switched pulse that occurs some 70 \(\mu\)sec after the end of the current pulse, and continues to oscillate for a sizable time after that. (The inversion in this particular laser system is maintained primarily by long-lived vibrationally excited \(N_2\) molecules, which are created in large numbers by the initial current pulse, and which gradually transfer their energy into pumping up the laser \(\text{CO}_2\) molecules.) 

Because the population lifetimes \(\gamma_2\) are shorter and closer to the cavity decay times \(\gamma_c\) in these situations than in solid-state lasers, these gain-switched relaxation oscillations are much more heavily damped than the spikes in typical solid-state lasers, and so the spiking behavior typically damps out after one or a very few spikes.

 

 

2.  LASER AMPLITUDE MODULATION

In this section we consider amplitude modulation of laser oscillators produced by modulating either the pumping rate or the cavity losses, for the most part at frequencies small compared to the axial mode spacing of the laser. Other more sophisticated amplitude modulation techniques, such as \(Q\)-switching and mode locking, will be considered in later tutorials. 

 

Small-Amplitude Pump Modulation

If a laser oscillator is inherently "spiky"—that is, if it has only a weakly damped relaxation oscillation behavior—we can expect that the response of the oscillator to weak modulation of any of the laser parameters will also exhibit a rather strongly resonant response at the natural relaxation-oscillation frequency of the laser.

To demonstrate this, suppose we assume the pumping rate applied to the laser is modulated sinusoidally at some low modulation frequency \(\omega_m\) in the form 

\[\tag{16}R_p(t)=R_{p0}+R_{p1}\cos\omega_mt.\]

 

 

 

 

We can then also expand the cavity photon number and population inversion in the forms 

\[\tag{17}\begin{align}&n(t)=n_{ss}+R_e\tilde{n}_1e^{j\omega_mt},\quad|\tilde{n}_1|\ll n_{ss}\\&N(t)=N_{th}+Re\tilde{N}_1e^{j\omega_mt},\quad|\tilde{N}_1|\ll N_{th}.\end{align}\]

By using the same rate equations as in the spiking analysis, and again linearizing by dropping all products of the small sinusoidal terms \(R_{p1}\), \(\tilde{n}_1\) and \(\tilde{N}_1\), we can obtain a linear transfer function between pump modulation and oscillation amplitude response, namely,

\[\tag{18}\frac{\tilde{n}_1}{R_{p1}}=\frac{\omega^2_{sp}/\gamma_c}{\omega^2_{sp}-\omega^2_{sp}+2j\gamma_{sp}\omega_m},\]

where \(\omega_{sp}=\sqrt{(r-1)\gamma_2\gamma_c}\) and \(\gamma_{sp}=r\gamma_2/2\) are the relaxation-oscillation resonant frequency and decay rate defined in the previous section.

Figure 12 shows the theoretical variation of the amplitude and phase of the pump modulation response for such a laser in which the relaxation oscillations are relatively undamped \((\omega_{sp}/\gamma_{sp}=10)\) The general form of this response is exactly like the response of any similar underdamped resonant linear system, i.e.,

 

 

 

 

\[\tag{19}\frac{\gamma_c\tilde{n}_1}{R_{p1}}\approx\left\{\begin{array}&1\quad &\omega_m\ll\omega_{sp},\\-j\omega_{sp}/2\gamma_{sp}\quad &\omega_m=\omega_{sp},\\-\omega^2_{sp}/\omega^2_m\quad &\omega_m\gg\omega_{sp}.
\end{array}\right.\]

That is, there is a quasi-dc response well below resonance; a resonance peak with a \(90^{\circ}\) shift at resonance; and a rapidly decreasing response with a \(180^{\circ}\) phase shift at frequencies above resonance.

Figure 13 shows experimental results for the pump modulation response in a small 1 \(\text{mW}\) \(\text{Nd}\):\(\text{YAG}\) laser pumped by an array of 64 semiconductor light emitting diodes. (Semiconductor \(\text{LEDs}\), excited with a few hundred \(\text{mA}\) of direct current and emitting efficiently at around 840 \(\text{nm}\), can provide an effective pump source for a small \(\text{Nd}\):\(\text{YAG}\) laser oscillating at 1064 \(\text{nm}\).

Although there are practical problems in obtaining sufficiently powerful and long-lived \(\text{LEDs}\) and coupling their emission into the \(\text{Nd}\):\(\text{YAG}\) rod, this kind of pumping can provide an efficient, small, portable and rugged \(\text{YAG}\) laser, which can be directly modulated by modulating the current through the \(\text{LEDs}\).) 

The results in Figure 13 indicate that a modulation index of only \(0.48\%\) in the diode pumping current produced a modulation index of \(\approx30\%\) in the laser output near the laser relaxation oscillation frequency of \(\approx12\;\text{kHz}\). Similar experiments on this laser at different dc drive levels also confirm (as has also been done on many other lasers) the \(\sqrt{(r-1)\gamma_2\gamma_c}\) formula for the resonance frequency of the relaxation oscillations.

We can expect that the laser oscillation level will be similarly affected by sinusoidal modulation of any other laser parameter, such as cavity loss, output coupling, or mirror alignment. Random noise fluctuations in the laser structure or surroundings or pumping system can thus produce large unwanted resonant fluctuations in the laser oscillation level at frequencies near the natural relaxation oscillation frequency.

In general, if we examine carefully the output spectrum of any naturally spiky laser, we can expect to find enhanced noise sidebands on the output signal at or near the resonance frequency \(\omega_{sp}\).

The increased modulation response near the spiking frequency can also become a source of signal distortion in semiconductor diode lasers used in amplitude modulation communication systems, as we will discuss a little later in this section.

 

 

 

 

Large-Amplitude Pump Modulation

The response of a "spiky" laser to cavity modulation or pump modulation will be linear at very small modulation depths, but will become nonlinear for larger modulation depths. For example, by increasing the pump modulation index in the diode-pumped \(\text{YAG}\) laser of Figure 13 to a larger level, and then tuning the modulation frequency anywhere in the range below or up to the resonance frequency \(\omega_{sp}\), we can obtain the alternative kind of controlled or entrained spiking behavior shown in Figure 14.

In this case the current through the pumping diodes is being modulated at a frequency around half the natural resonance frequency \(\omega_{sp}\), with a modulation amplitude of \(\pm 70\%\) around the threshold level for the laser (shown by the dashed line in the figure). The strong laser spike occurring near the end of each pumping cycle is evident. 

In general it is a common characteristic of resonant but nonlinear systems, like spiky lasers, to exhibit complex nonlinear relaxation oscillations, especially when driven with larger excitation amplitudes. Figure 15 shows additional forced spiking behavior caused by sinusoidal pump modulation, here in a small diodelaser- pumped \(\text{LiNdP}_4\text O_{12}\) laser. The average pumping rate in this situation is set at twice the laser threshold, with a peak sinusoidal modulation \(R_{p1}=0.55\times R_{p0}\) about that level.

The upper left figure shows a locked single-spike behavior at a modulation frequency slightly less than the natural spiking rate in this laser (the sweep rate in all three parts of Figure 15 is 20 \(\mu s\) per division).

In the lower left illustration the modulation frequency has been raised somewhat above the natural spiking frequency, and the laser responds by emitting a spike only every other modulation cycle.

In the right-hand trace, at an intermediate frequency, the laser spikes on every cycle, but with an amplitude and timing that shifts every alternate cycle. Other more complex harmonic and subharmonic entrained spiking patterns can be obtained at other modulation frequencies and modulation depths.

 

Injection Diode Lasers

These types of resonant response and locked spiking behavior become of very direct interest in semiconductor diode lasers, whose natural spiking frequency lies in the \(\text{GHz}\) rather than \(\text{kHz}\) range, as we showed in the preceding section. This is especially true when such lasers are operated as directly modulated light sources for optical communication links, with modulation frequencies 

 

 

 

in the same range. The resonance enhancement of the small-signal modulation response near \(\omega_m\approx\omega_{sp}\) can then distort sinusoidal communication signals; while the transient response to pulsed current injection becomes of great importance in pulsed binary modulation devices.

In a pulsed binary system we need to know, for example, how much the laser response to one modulation pulse will depend on closely preceding pulses, or whether the laser will respond to a string of several successive closely spaced pulses in a manner different from isolated individual pulses.

We can thus find numerous papers in the literature reporting both rate-equation simulations and experimental tests of the effects of spiking and relaxation-oscillation response to current modulation in injection diode lasers.

We illustrated in the previous section the kind of strong initial turn-on spikes that can be produced in an injection diode by a pumping current pulse with a fast-rising leading edge.

As an extension of this, if we drive a particularly spiky injection diode laser with a very intense and fast-rise current pulse, and then turn off this current pulse equally rapidly after the first spike, it becomes possible to generate a laser pulse substantially shorter than the driving current pulse (e.g., an optical pulse of duration 50 ps for a current pulse of duration \(\mbox{¿}500\) ps).

This provides a convenient and inexpensive low-power ultrashort optical pulse source for calibrating the response of optical-fiber systems, photodetectors, or fast oscilloscopes, or for measuring other ultrafast physical phenomena.

 

Cavity Loss and Coupling Modulation

Small-amplitude modulation of the cavity loss rate, i.e., making \(\gamma_c=\gamma_c(t)\), should lead to much the same kind of resonance response near the relaxation oscillation frequency as happens for pump power modulation; and this type of response has been experimentally demonstated as well. Large-amplitude modulation of the cavity loss or cavity coupling should also similarly lead to controlled nonlinear spiking behavior under appropriate conditions. 

 

 

 

Two of the most common forms of large-amplitude cavity modulation are cavity dumping, which we will discuss further in this section, and \(Q\)-switching, which we will discuss in more detail in the following tutorials. 

Cavity dumping is a technique in which the output coupling from the laser cavity is suddenly increased to a very large value—essentially as if one of the end laser mirrors had been removed—so that all the circulating energy inside the cavity is "dumped" into an output pulse which, for perfect dumping will be exactly one cavity round-trip time \(\text{T}\) in length, and contain all the signal energy in the cavity. Figure 16 shows one common technique by which this can be accomplished.

This laser cavity contains a polarizing beam splitter which makes the cavity oscillate normally with a vertically oriented linear polarization. In order to dump the cavity energy, the voltage across the electrooptic Pockels modulator is suddenly switched (in a time short compared to \(\text{T}\), or typically a few ns) to a value which makes this transparent crystal become birefringent, with a magnitude corresponding to a quarter-wave plate for single pass, or a half-wave plate for double pass.

The linearly polarized energy circulating in the cavity, as it passes through this plate going to the right and then coming back to the left, has its polarization converted into circular polarization striking the right-hand mirror, and then on into horizontal polarization coming back out of the Pockels crystal.

All the energy in this polarization coming back to the polarizing beamsplitter (which can either be an especially coated dielectric plate, as shown, or a polarizing prism, such as a Glan-Thompson prism) is then dumped out of the cavity as shown.

(As a practical matter, a fixed quarter-wave plate is often added to the Pockels modulator, and the Pockels modulator is then initially biased to the fixed voltage, typically several thousand volts, needed to cancel the fixed quarter-wave plate. Cavity dumping is then accomplished by suddenly switching off the voltage across the Pockels cell, leaving only the fixed quarter-wave plate, since it is often easier to short out or "crowbar" the Pockels cell voltage from a high initial value down to zero in a few ns than it is to switch the same voltage from zero up to the necessary high value in the same length of time.)

Figure 17 is an oscilloscope trace that shows the effect of cavity dumping on a laser cavity. This is a low-pressure \(\text{CO}_2\) laser pumped by a long current pulse starting at the left end of the trace. The laser oscillation then shows a particularly strong initial gain spike, followed after \(\approx 80\;\mu\)sec by a cavity dumping transient. The detector in this experiment was observing the energy circulating

 

 

 

inside the cavity (via leakage through one of the end mirrors), rather than the energy dumped into the output direction. Hence, the circulating energy is observed to drop nearly to zero, representing nearly complete cavity dumping. The recirculating energy then rapidly recovers with a second gain-switched spike. (The noise on the trace following this point probably represents a combination of electrical pickup noise from the spark gap needed to provide the \(\approx25\text{kV}\), 2 \(\text{ns}\) pulse applied to the \(\text{CdT}_e\) electrooptic modulator crystal, plus acoustic ringing in the \(\text{CdT}_e\) crytal caused by application of the electrical pulse.) 

This kind of cavity dumping has the disadvantages of requiring high voltages with fast rise times, applied to a crystal which is typically expensive, often optically lossy, and subject to optical damage at high powers.

It can, however, provide fairly complete dumping, with moderately accurate electronic timing, and the ability to handle higher optical powers than most other cavity dumping schemes.

 

Repetitive Cavity Dumping

If a cw laser oscillator is running in steady-state with a \(5\%\) output coupling mirror \((R=95\%)\), the circulating intensity inside the laser cavity is then 20 times as large as the cw output intensity from the laser. If this circulating intensity is suddenly cavity dumped, the peak power output during the dumped pulse can be 20 times as large as the average or \(\text{cw}\) power output from the laser. (In fact, by reducing the output coupling below its optimum value for maximum average output power, we can make both the circulating intensity and thus the "dumpable" peak power still larger.) 

If we further allow the intensity inside the cavity to build back up again, and then again dump the cavity, using repetitive cavity dumping, we can obtain most of the available power output from the laser medium in the form of repeated pulses which have substantially higher peak power than the average power from the laser.

With proper choice of repetition frequency, the average power in the dumped output can approach the full average power available with optimum coupling in \(\text{cw}\) operation; but the higher peak powers can make this energy much more effective in cutting, welding, and other nonlinear laser processes.

Cavity dumping in lower power \(\text{cw}\) lasers, including \(\text{cw}\) \(\text{Nd}\):\(\text{YAG}\) lasers and gas lasers such as argon-ion lasers, is often accomplished using the kind of acoustooptic modulation arrangement shown (somewhat foreshortened) in Figure 18. The acoustooptic modulator consists of a bar of quartz or some other optically low-loss material with a piezoelectric acoustic transducer attached to one end, inserted in the laser cavity at Brewster's angle to minimize optical reflection losses.

 

 

 

 

A radiofrequency pulse with a power level of perhaps 10 to 20 \(\text{watts}\), at a frequency typically in the range of 20 to 50 \(\text{MHz}\), sends a strong acoustic wave at the same frequency down the bar. 

The density and index of refraction variations associated with this acoustic wave then act as a Bragg diffraction grating, deflecting from 50 to \(95\%\) of the circulating optical intensity at a small angle, as shown by the dashed lines in the drawing. In the optical arrangement shown in Figure 18, the diffracted signals in both directions are recombined to produce the output beam. (By having the acoustic signal normally present and suddenly turning it off, the same arrangement can also function as an acoustooptic Q-switching system, as we will discuss in the following tutorial.)

In practice, the rise time for the cavity dumping process when the acoustic signal is turned on is limited by the transit time with which the leading edge of the acoustic signal can travel across the width of the laser beam, moving at the speed of sound. To reduce this rise time (which is about 15 \(\text{nsec}\) per 100 \(\mu m\) of laser beam width), the laser beam is often brought to a focus inside the cavity using the folded arrangement of curved mirrors shown in Figure 18. Acoustooptic cavity dumping is thus best suited to relatively long laser cavities.

The recovery time for the intracavity intensity to build back up following a single cavity dumping pulse depends on the laser, but may typically range from a fraction of a microsecond to several microseconds. Repetitive cavity dumping can thus provide a way to get trains of pulses with increased peak power compared to the \(\text{cw}\) laser output, at repetition rates ranging from \(\text{kHz}\) to potentially a few \(\text{MHz}\).

Cavity dumping is also often used in mode-locked lasers to select out a single ultrashort pulse. If a single mode-locked pulse is circulating inside the laser cavity, as is characteristic of mode-locked operation, turning on the cavity dumping modulator can dump that single pulse into the output port of the laser.

 

 

 

 

3.  LASER FREQUENCY MODULATION AND FREQUENCY SWITCHING 

Suppose that instead of an amplitude modulator, a phase modulator is placed inside a laser cavity. By a phase modulator we mean any device which can add a time-varying phase shift \(\phi_m(t)\) to the signal wave passing through it. Such modulators are generally referred to, more or less interchangeably, either as phase modulators or as frequency \(\text{(FM)}\) modulators, since the effects of such a modulator can be described as producing either frequency modulation sidebands on the laser oscillation signal or as causing a phase shift or frequency jump of the laser oscillation frequency. 

We will describe in this section the kinds of frequency modulation and frequency switching behavior that can be produced in a laser oscillator by a phase modulator which is driven either with sinusoidal modulation signals at frequencies that are generally low compared to the axial mode spacing frequency of the laser cavity, or with step-function modulation signals that may be considerably faster. Discontinuous frequency switching of the laser oscillation will then turn out to be one of the more useful forms of such modulation.

There are also more complex types of mode coupling or mode locking behavior that can occur when the phase modulation \(\phi_m(t)\) is driven with signals at or near the axial mode frequency interval. These mode-coupling effects will be considered separately in a later tutorial devoted to mode locking in general.

 

Optical Phase and Frequency Modulators

The most common form for an electrooptic phase modulator, as illustrated in Figure 19, is a so-called Pockels cell, consisting of a crystal of an electrooptic material such as potassium dihydrogen phosphate \(\text{(KDP)}\), ammonium dihydrogen phosphate \(\text{(ADP)}\), or lithium niobate \(\text{(LiNbO}_3)\), with electrodes located so as to apply a transverse voltage across the crystal. (If the material is described as \(\text{KD}^*\text P\) or \(\text{AD}^*\text P\), this means the crystal is deuterated, i.e., the dihydrogen is replaced by dideuterium.)

Electrooptic materials such as these have the property that applying a transverse or (sometimes) a longitudinal electric field (depending upon the particular crystal) causes a small but significant change in the optical index of refraction and hence in the optical phase shift through the crystal. 

Usually applying a voltage causes the index of refraction to increase for optical \(\text E\) fields polarized along one transverse axis, and to decrease for fields polarized along the orthogonal transverse axis, so that the crystal in general acquires an electrically induced birefringence.

 

 

 

 

If this electrically induced birefringence is combined with an optical beam that is polarized, for example, between these two axes, then the applied voltage can produce polarization rotation of the optical signal.

This rotation, combined with optical polarizers, can then produce amplitude modulation and switching, such as we discussed in the previous section.

Alternatively, with proper orientation of the electrooptic crystal, so that the optical signal is polarized along one or the other of the induced axes in the crystal, the net effect on the optical signal can be pure phase modulation, with no polarization rotation or amplitude modulation.

In Pockels devices this change in the index of refraction, and hence in the optical phase shift through the crystal, is linearly proportional to the voltage applied across or along the crystal. Other useful electrooptic crystals, all of which must lack inversion symmetry in their crystal structure, include gallium arsenide \(\text{(GaAs)}\) and cadmium telluride \(\text{(CdTe})\), both of which are useful for infrared applications.

There also exist optical Kerr cells in which an applied voltage produces an optical index change in a transparent but polarizable liquid placed between the electrodes. The index change in such liquid Kerr cells is generally much smaller, but rises as the applied modulation voltage squared.

 

Cavity Mirror Motions

The simplest way to change the phase of an optical signal at some given point is to change the optical path length the signal must travel to reach that point. One elementary way to accomplish this for a signal which bounces off an optical mirror is to move the mirror forward or backward in some mechanical fashion. A movable mirror is thus the simplest kind of phase modulator.

One way to move a mirror electrically is to use either a piezoelectric, magneto-strictive, or magneto-inductive mirror mount (i.e., a solenoid or loudspeaker cone). These electromechanical methods are simple and inexpensive, but are generally limited to low-modulation frequencies (in the audio range), and at least with the piezoelectric mount to very small motions (a few microns at the most). 

The simplest form of phase or frequency modulation inside a laser cavity is thus simply to move a cavity end mirror back and forth by an amount \(\delta L\), as in Figure 20, thereby producing a double-pass phase modulation \(\delta\phi_m\) given by

\[\tag{20}\delta\phi_m=\frac{2\omega_0\delta L}{c}=\frac{2\pi\delta p}{\lambda},\]

where \(\omega_0\) is the unmodulated laser oscillation frequency and \(\delta p=2\delta L\) is the change in the cavity perimeter \(p\). In the static limit this leads to a shift of the oscillation frequency given by

\[\tag{21}\omega_0+\delta\omega_0=q\frac{2\pi c}{p+\delta p}\approx\omega_0(1-\delta p/p),\]

which can be rewritten as

\[\tag{22}\frac{\delta\omega_0}{\Delta\omega_{a\text x}}=\frac{-\delta\phi_m}{2\pi}\approx-\frac{\delta p}{\lambda}\approx\frac{2\delta L}{\lambda}.\]

This emphasizes that a net mirror motion of one-half wavelength in the standingwave cavity, or an added phase shift of \(\delta\phi_m=2\pi\) in either type of cavity, will shift the laser's oscillation frequency by exactly one axial mode spacing \(\Delta\omega_\text{ax}\).

 

Phase-Modulated Laser Cavity Model

What effects are produced if a more general phase modulator of any of the preceding types is placed inside an oscillating laser cavity? To answer this question, we will use a simplified but still generally realistic model of the laser cavity as shown in Figure 21. To make the analysis apply equally well to standing-wave or ring resonators, let us define the quantity \(\phi_m(t)\) to be the double-pass phase modulation going through the modulator and back again in a standing-wave cavity, or the single-pass phase modulation going once through the modulator in a ring laser.

The net round-trip effects in the ring or standing-wave cavities will then be essentially the same, provided that the modulator for the standing-wave is placed immediately adjacent to an end mirror, as is usually done. 

Consider then a general time-varying phase modulation \(\phi_m(t)\), but suppose for simplicity that the modulation frequencies applied to the phase modulator are on the order of an axial mode spacing or smaller, whereas the laser gain curve is perhaps several axial modes wide.

The frequency-modulation sidebands produced by the intracavity modulator acting on the circulating light within the laser cavity will then all fall well within the atomic gain profile. The effect of the finite bandwidth of the laser medium on the modulation sidebands can then be ignored to first order, along with any other dispersion effects in the laser cavity.

Suppose we then consider the time-varying laser oscillation signal at some arbitrary plane inside the laser cavity, for example, the plane just at the output side of the phase modulator, as illustrated in the cavity diagrams.

The signal \(\mathcal{E}_2(t)\) leaving the phase modulator at time t is then just equal to the signal \(\mathcal{E}_1(t)\) incident on the modulator at the same time, multiplied by the modulator transfer function \(e^{j\phi_m(t)}\). (We lump all the phase modulation into a plane of zero thickness for simplicity.) But if we assume that the net round-trip gain for the radiation in the remainder of the laser cavity is exactly unity—which is equivalent to our assumption that atomic bandwidth and dispersion effects can be neglected—then the signal \(\mathcal{E}_1(t)\) arriving at the modulator at time \(t\) is equal to the signal \(\mathcal{E}_2(t-T)\)which left the modulator one round-trip transit time \(\text T\) earlier.

 

 

 

Hence we may write the fundamental relationship for phase modulation inside a laser cavity (at least for signals within the atomic linewidth) as 

\[\tag{23}\mathcal{E}_2(t)=\mathcal{E}_1(t)\times e^{j\phi_m(t)}\approx\mathcal{E}_2(t-T)\times e^{j\phi_m(t)}.\]

Let us assume that the optical signal inside the laser cavity has the general form

\[\tag{24}\mathcal{E}_2(t)\approx E_0\;\text{exp}\;j[\omega_0t+\phi_c(t)],\]

where \(E_0\) is the (approximately constant) signal amplitude; \(\omega_0\) is the unmodulated oscillation frequency or carrier frequency; and \(\phi_c(t)\) is the time-varying phase modulation of the laser cavity signal produced by the phase modulator. Equation 23 then translates to the self-consistency condition

\[\tag{25}E_0\;\text{exp}\;j[\omega_0t+\phi_c(t)]=E_0\;\text{exp}\;j[\omega_0(t-T)+\phi_c(t-T)+\phi_m(t)].\]

But upon using the fact that (by definition) the frequency \(\omega_0\) is an axial mode for which \(\omega_0T=q2\pi\), this reduces to the phase modulation condition

\[\tag{26}\phi_c(t)=\phi_c(t-T)+\phi_m(t).\]

This is the basic equation for analyzing low-frequency phase modulation effects in the laser cavity.

 

Sinusoidal Phase or Frequency Modulation

Let's consider first pure sinusoidal modulation of the intracavity phase modulator. Suppose the phase modulation is driven sinusoidally at a modulation frequency \(\omega_m\) with a peak phase deviation \(\Phi_m\) in the form

\[\tag{27}\phi_m(t)=\Phi_m\cos\omega_mt.\]

The oscillation signal in the laser cavity will then acquire a similar phase modulation of the form

\[\tag{28}\phi_c(t)=\left(\frac{\Phi_m}{2\sin\omega_mT/2}\right)\times\sin\omega_m(t+T/2)\]

This result may make more physical sense if we interpret it as a sinusoidal modulation of the instantaneous frequency \(\omega_i(t)\)of the laser, where we define instantaneous frequency in the manner introduced in an earlier tutorial, namely,

\[\tag{29}\begin{align}\omega_i(t)&\equiv\frac{d[\omega_0t+\phi_c(t)]}{dt}\\&=\omega_0+\frac{\Phi_m}{2\pi}\times\frac{\pi\omega_m}{\sin(\pi\omega_m\Delta\omega_\text{ax})}\cos\omega_m(t+T/2).\end{align}\]

It is clear that an applied phase modulation with peak amplitude \(\Phi_m\) inside the laser cavity produces a frequency modulation of the laser signal itself.

This frequency modulation reduces in the low-modulation-frequency limit to

\[\tag{30}\omega_i(t)\approx\omega_0+\frac{\Phi_m\Delta\omega_\text{ax}}{2\pi}\cos\omega_m(t+T/2)\quad\text{if}\;\omega_m\ll\Delta\omega_\text{ax}.\]

The peak frequency deviation of the laser signal is thus given by the quasi static frequency-tuning result \(\omega_i\approx\omega_0+(\Phi_m/2\pi)\omega_\text{ax}\), so long as the modulation frequency \(\omega_m\) is fairly small compared to the axial mode spacing \(\omega_\text{ax}\).

If, however, the modulation frequency ujm approaches the axial mode frequency \(\omega_\text{ax}\)—which means that the modulation sidebands produced on any one axial mode come close to the adjoining axial mode cavity resonances—then the modulation index and the peak frequency deviation diverge toward very large values, as shown in Figure 22.

At some point in this limit, the simplified analysis of this section will no longer apply, and we must use instead the kind of axial-mode-coupling analysis that we will develop to describe laser mode locking in a later tutorial.

 

Linear Phase Shift and Instantaneous Frequency Switching

Let us next consider the interesting effects that can be produced by a sudden or fast-rising step-function phase shift applied inside a laser cavity. Suddenly and rapidly moving one end mirror of a laser cavity to a new position is, at least in principle, the simplest way of producing such a fast-rising phase shift. Applying an additional round-trip phase shift of magnitude \(\Phi_m\) inside a laser cavity will shift the resonance frequency of the cavity, and hence eventually the laser oscillation frequency, over by an amount \((\Phi_m/2\pi)\omega_\text{ax}\).

There are interesting experiments in which we might wish to shift the oscillation frequency of a \(\text{cw}\) laser quite suddenly, by some small or large amount. How fast (and how far) can we actually jump the oscillation frequency inside an oscillating laser cavity? 

To understand how we can do this in an ideal fashion, suppose that the intracavity phase modulator \(\phi_m(t)\) is changed linearly during one cavity roundtrip transit time T from an initial value \(\phi_m(0)=0\) at time \(t=0\), to a final

 

 

 

 

(constant) value \(\phi_m(T)=\Phi_m\) exactly one round-trip time later, as illustrated in Figure 23. We thus have the phase modulation input 

\[\tag{31}\phi_m(t)=\left\{\begin{align}&0,\quad &t<0\\&\Phi_mt/T,\quad &0<\text t<\text T\\&\Phi_m,\quad&t>0.\end{align}\right.\]

A little examination will then show that the phase \(\phi_c(t)\) of the cavity oscillation signal itself will rise linearly with a time slope \(d\phi_c(t)/dt=\Phi_m/T\), beginning at \(t=0\) and continuing indefinitely. 

 

 

 

But this is the same as saying that the oscillation frequency \(\omega_i(t)\) of the laser changes essentially instantaneously at \(t=0\) from its original value \(\omega_0\) to a new quasi static value, as given by 

\[\tag{32}\omega_i(t)=\left\{\begin{align}&\omega_0,\quad &t<0\\&\omega_0+(\Phi_m/2\pi)\omega_\text{ax},\quad &t>0.\end{align}\right.\]

With this particular phase modulation input, therefore, we can instantaneously shift the oscillation frequency by any desired amount (even by several axial mode spacings if \(\Phi_m\gg 2\pi)\), provided only that \((a)\) the new phase shift is inserted linearly in time over exactly one round-trip time, and \((b)\) the resulting signal remains within the central part of the atomic gain profile of the laser medium.

 

Physical Interpretation: Linear Doppler Shift 

This conclusion seems to run contrary to the intuition of some laser students. Some people argue, for example, that changing the oscillation frequency of a laser cavity ought to take at least several laser cavity lifetimes \(\tau_c\), since "it should take time for the photons at the old frequency to die out, and the new oscillation photons at the new frequency to build up." Setting aside the introduction into this argument of particle-like photons (which is almost always a mistake in any laser analysis), we can understand this instantaneous frequency shift as follows. 

First, consider those portions of the recirculating cavity signal which pass through the phase modulator during the linear ramp interval. These waves receive a linear phase modulation as given by the middle line of Equation 31 for \(\phi_m(t)\).

Suppose we think of this phase modulation as being provided, for example, by a linearly moving end mirror in a standing-wave cavity (though it would be difficult in practice to move a real mirror this rapidly).

Then, the reader can easily determine that the frequency shift of Equation 32 is exactly equivalent to a doppler shift off this moving mirror. This linear phase modulation or doppler shift, in fact, frequency shifts the reflected wave by exactly the desired frequency jump between the new and old values.

Figure 24 then shows (in greatly exaggerated form) how the optical cycles in the cavity might appear shortly after the end mirror has started moving inward (i.e., shortly after the ramp of phase modulation has begun). That portion of the circulating signal reflected from the mirror has been doppler-shifted upward in frequency by the inward-moving mirror.

If the duration of the linear ramp lasts, as in our ideal situation, exactly long enough for all the circulating waves within the laser cavity to come around and be doppler-shifted by the requisite amount

 

 

 

 

exactly once, then all of the circulating radiation within the cavity will be shifted to the new frequency. The moving mirror then stops, having just exactly done its job.

The new oscillation frequency need not be an axial mode of the old cavity. It will, however, be automatically an axial mode of the new cavity, with its new cavity length changed by the effective amount \(\delta L=-\Phi_m\lambda/4\pi\).

Note that the results given here only apply to the instantaneous \(\mathcal{E}(t)\) field at one particular plane in the cavity, namely, just after the phase modulator. The same frequency shift only arrives at any other planes within the laser cavity after an appropriate transit time delay.

 

Exact Results: General Analysis

Modulation signals that do not exactly match the ideal ramp waveform of Figure 23 will produce more complex frequency shifting effects. Suppose the mirror moves the same total distance in a shorter time (i.e., shorter than the cavity round-trip time). The recirculating signal shortly after the mirror moves will then look something like the drawing in Figure 25 (again with the modulation effects greatly exaggerated).

We will then need to consider the Fourier transform of the signal circulating inside the cavity, as expanded in axial cavity modes of the new cavity length; and consider how these new axial mode amplitudes will grow or decay in the cavity following the switching time. 

An exact analysis of intracavity phase modulation with any waveform \(\phi_m(t)\)(still assuming a perfectly flat atomic gain profile) can be obtained as follows. Suppose we define the Fourier transforms of the modulator phase \(\phi_m(t)\) and of the resulting cavity signal phase \(\phi_c(t)\) in the form 

\[\tag{33}\Phi_x(p)=\int^{\infty}_{-\infty}\phi_x(t)e^{-j2\pi pt}dt,\] 

with the inverse transform

\[\tag{34}\phi_x(t)=\int^\infty_{-\infty}\Phi_x(p)e^{j2\pi pt}dp.\]

where \(x\) can be either m for the modulator or \(c\) for the cavity signal. From a standard theorem for Fourier transforms we can write the transform of the time-delayed cavity phase as

\[\tag{35}\int^\infty_{-\infty}\phi_c(t-T)e^{-j2\pi pt}dt=e^{-j\pi pT}\Phi_c(p).\]

 

 

 

 

Hence the basic relationship between \(\Phi_c(p)\) and \(\Phi_m(p)\) becomes, in the transform domain, 

\[\tag{36}\Phi_c(p)=\frac{\Phi_m(p)}{1-e^{-j2\pi pT}}=\frac{e^{j\pi Tp}\Phi_m(p)}{2j\sin\pi Tp}.\]

We can then invert this transform to find the phase modulation \(\phi_c(t)\) of the laser cavity signal produced by any arbitrary phase modulation \(\phi_m(t)\).

We may also define \(\Omega_i(p)\) to be the Fourier transform of the instantaneous frequency deviation \(\omega_i(t)-\omega_0\equiv d\phi_c(t)/dt\).

Another Fourier theorem then tells us that the instantaneous frequency modulation of the laser is given by the inverse transform of

\[\tag{37}W_i(p)=j2\pi\Phi_c(p)=\frac{\pi pe^{j\pi Tp}\Phi_m(p)}{\sin\pi Tp}.\]

Our earlier results are specific examples of what can be produced by specific phase modulations \(\phi_m(t)\).

 

Laser Frequency Switching Spectroscopy

Laser frequency switching, using exactly the electrooptic ramp technique described in Figure 23 and Figure 24, has become the basis of a very useful form of time-resolved spectroscopy, in which coherent optical transients are excited by fast frequency-switched laser signals. One elementary form for such an experiment is illustrated in Figure 26. 

The example illustrated here uses a cw dye laser, which has a very wide atomic line, so that it can potentially be frequency switched over a large range. The output from this laser is sent through an atomic absorption cell, with the laser initially tuned at or near the center of the moderately narrow atomic transition in the cell.

Hence this signal, during an initial preparation phase, excites a steady-state coherent polarization in the atoms on this transition. (This induced atomic polarization may involve all the atoms in a homogeneously broadened

 

 

 

 

transition, or just those atoms in a single spectral packet with which the signal is in resonance in an inhomogeneously broadened transition.) 

The laser frequency is then suddenly switched to a new value that is either entirely outside the atomic absorption profile, though still within the much wider linewidth of the dye laser itself, or at least that is tuned to a completely different spectral packet in the in homogeneously broadened absorption cell, as shown in Figure 27. 

So far as the initially excited atoms in the absorption cell are concerned, the effect of this frequency jump is the same as if the laser signal is suddenly turned off—the laser signal is now shifted so far off resonance as to be irrelevant. The oscillating dipole polarization of these atoms will then continue to oscillate and to radiate at the natural oscillation frequency of the atoms, but this polarization will die out with the decay time \(T_2\) associated with the atomic transition.

A very important aspect of this experiment is that these oscillating atoms were initially excited by the spatially coherent forward-traveling laser beam. Hence they oscillate and continue to radiate in the same forward direction and with the same collimated beam pattern as the exciting laser beam. This spatial as well as temporal coherence of the radiation persists—though its amplitude dies away with time constant \(T_2\)—even after the laser frequency is switched.

Following the switching time the photodetector in this experiment thus sees a linear combination of the frequency-shifted, constant-amplitude laser beam, plus the exponentially decaying radiation from the excited atoms at the original excitation frequency. These two signals mix or heterodyne in the square-law photodetector, with the frequency-shifted laser acting as a frequency offset local oscillator, and the atomic radiation as the signal.

The result is a beat frequency output from the photodetector at the difference frequency \(\omega_\text{old}\) - \(\omega_\text{new}\) as determined by the laser frequency jump, with an amplitude that decays away in proportion to the decaying radiation from the originally excited spectral packet.

 

 

 

 

Experimental Results 

Figure 28 illustrates exactly this kind of result. The top trace shows an essentially ideal free induction decay signal obtained from a very narrow line produced by \(P_r^{3+}\) ions in \(LaF_3\) crystals at liquid helium temperature. The beat note between the frequency-shifted laser signal and the initially excited and still radiating atoms is obvious. The laser frequency jump from inside to outside the atomic line is 3 \(\text{MHz}\) in this instance. 

In the second and third traces, a \(\text{cw}\) dye laser signal with a few \(\text{mW}\) of power at 589.75 \(\text{nm}\) is passed through a cell containing 30-\(\text{m}\) torr of \(I_2\) vapor, which has strong narrow atomic lines in the visible.

In the second trace the laser frequency has been suddenly frequency shifted by approximately 54 \(\text{MHz}\), starting and ending within the doppler-broadened linewidth of one of the discrete optical-frequency rotational-vibrational absorption lines of the \(I_2\) molecule. The decaying oscillation signal is the beat note at 54 \(\text{MHz}\) between the free-induction decay radiation from the originally excited molecules and the laser signal at its new frequency.

The third trace is a similar result but with a somewhat higher laser power and a longer time scale. The broader, heavily damped, negative-going oscillation after the free-induction signal has died out represents the "optical nutation" response of the new group of iodine molecules that are being driven into forced oscillation in the spectral packet at the new laser frequency after switching.

The combination of the spatial coherence of the free-induction-decay signal from the originally excited atoms, plus the sensitivity advantage of coherent optical heterodyne detection, plus the relative simplicity of laser frequency switching, all combine to make this a very attractive and important technique for observing coherent transients and measuring various relaxation times in atomic systems. Similar experiments with much faster switching, and switching over much larger frequency jumps, have also been demonstrated.

Other variations on this technique include switching the laser frequency into rather than out of the atomic line, in order to observe the turn-on transient for the atomic response; switching from one packet to another within an inhomogeneous atomic line; and switching away from and then back to a single spectral packet, to observe more complicated atomic dynamics.

Detailed observation of the resulting decay profiles can bring out information about hyperfine degeneracies within atomic lines; about the widths and shapes of the holes burnt in inhomogeneous lines; and about many other details of great interest to atomic and molecular spectroscopists.

 

 

4.  LASER MODE COMPETITION

We now turn to a rather different but also important kind of laser dynamic behavior, namely, the mode competition between simultaneously oscillating (or potentially oscillating) modes in a laser oscillator. 

Most laser cavities have the potential of oscillating in a large number of different modes, including different axial and transverse cavity modes; different directions in ring-laser cavities; and even different senses of polarization in cavities with internal mirrors and no Brewster windows.

Different modes will in general have different gains, losses and saturation parameters, and will compete for the available population inversion in the laser.

Oscillation in one mode will generally reduce the gain available for another mode, and in some situations may suppress the other mode entirely. The purpose of this section is to introduce some of the elementary concepts and analytical techniques that arise in discussing competition between laser modes.

 

Mode Competition Effects

Competition between modes in a laser cavity is in general a very complex problem. We may need to take into account among other things: 

• Self-saturation and cross-saturation effects between modes, both in the gain medium and in any saturable absorbing media that may be present.
• Possible injection locking and frequency pulling effects between modes caused by scattering effects or by intracavity modulators (as we will discuss in more detail in a later tutorial).
• The degree of spatial overlap between modes. Two different transverse or axial modes will in general be partially overlapping and partially separated in space, and thus will have some shared and some separate regions of population inversion.
• The degree of spectral overlap between modes, including whether the competing modes are at the same or different frequencies, and whether the atomic line is homogeneous or inhomogeneous. 

In addition we may sometimes need to consider the beating effects between modes at different frequencies, and the population pulsations that this can produce. If two modes of different frequencies are present, the total optical intensity at any point includes both a dc component due to each mode, plus an oscillating component at the difference frequency between modes.

The saturation effects due to this time-varying part will be different depending on whether the difference frequency is large or small compared to the atomic relaxation rates for the population inversion. 

 

Self-Saturation and Cross-Saturation Coefficients

To illustrate some of the elementary features of mode competition we will review in the remainder of this section the simplest form of pure intensity or gain competition between just two potentially oscillating modes.

We will consider in this discussion only the self-saturation and cross-saturation effects on the respective gains for these two modes, ignoring any back-scattering or cross-scattering effects that may couple one mode into the other.

As in so many other aspects of laser behavior, the fundamental concepts involved in the discussion are much older than the laser, but are particularly well-illustrated by the laser example.

Let us note first that the self-saturation and cross-saturation effects between any two simultaneously oscillating modes in a laser will depend very much both on the spatial overlap of the two modes, and also their spectral overlap and on whether the atomic transition involved is homogeneous or inhomogeneous.

For example, in the simplest situation, a completely homogeneous atomic transition being saturated by two incoherently related signals \(I_1\) and \(I_2\) at different frequencies \(\omega_1\) and \(\omega_2\), we can in general write the gain saturation for mode \(\#1\) in the form

\[\tag{38}a_m(\omega_1)=\frac{a_{m0}(\omega_1)}{1+k_{11}I_1+k_{12}I_2}\approx a_{m0}(\omega_1)\times[1-k_{11}I_1-k_{12}I_2],\]

and for mode \(\#2\) in the form 

\[\tag{39}a_m(\omega_2)=\frac{a_{m0}(\omega_2)}{1+k_{22}I_2+k_{21}I_1}\approx a_{m0}(\omega_2)\times[1-k_{22}I_2-k_{21}I_1].\]

The coefficients \(k_{11}\) and \(k_{22}\) then represent self-saturation of the gains of modes \(\#1\) and \(\#2\) by their own intensities in each situation, whereas the coefficients \(k_{12}\) and \(k_{21}\) represent the cross-saturation of the gain of mode \(\#1\) by the intensity of mode \(\#2\), and vice versa.

The values of these four factors will depend on the inverse of the saturation intensity in the laser medium, but they will also depend on spatial overlaps of the modes with the gain medium and with each other, and on lineshape factors that must be included to take account of how far each of the signals is off the resonance line center.

The simplifying approximation in the second term in each equation will then be valid provided the gain coefficient is not too strongly saturated by either signal.

Suppose the two signals are incoherently related (i.e., their intensities can simply be added to get the total intensity in the medium); that both of their frequencies are near the center of a strongly homogeneous atomic gain profile; and that their spatial patterns are essentially identical in the gain medium. We would then expect to find that the four coefficients all have approximately equal values, i.e., \(k_{11}\approx k_{22}\approx k_{12}\approx k_{21}\).

Suppose the atomic transition is strongly inhomogeneous, on the other hand, so that the two signals lie within quite different spectral packets or velocity groups. We would then expect to find only very weak cross-saturation effects, so that \(k_{12},k_{21}\ll k_{11},k_{22}\).

In the more general situation of mixed homogeneous and inhomogeneous broadening, the evaluation of the relative self-saturation and cross-saturation effects will require a complicated integration over the partial saturation effects of all the spectral packets, such as we will carry out in a later tutorial.

The self-saturation and cross-saturation effects between two modes can thus have quite different relative magnitudes in different situations. We might further recall that for two coherently related signals at the same frequency passing in opposite directions though a completely homogeneous transition, and thus producing significant hole burning and grating backscattering, we have earlier derived the results that the net gain (or absorption) for signal \(\#1\) depends on the two signal intensities (in the small-saturation approximation) in the form

\[\tag{40}a_m(\omega_1)\approx a_{m0}(\omega_1)\times\left[1-\frac{I_1+2I_2}{I_\text{sat}}\right].\]

That is, here the cross-saturation coefficients \(k_{12}\) and \(K_{21}\) are actually larger than the self-saturation coefficients \(k_{11}\) and \(k_{22}\) effects by a factor of 2.

 

Two-Mode Competition Analysis

To describe the competition between two potentially oscillating modes in a fairly general fashion, therefore, we might write the rate equations, or intensity growth equations, for the two assumed modes in the generalized form 

\[\tag{41}\begin{align}dI_1/dt\approx\gamma_{m1}I_1\times[1-k_{11}I_1-k_{12}I_2]-\gamma_{c1}I_1\\dI_2/dt\approx\gamma_{m2}I_2\times[1-k_{22}I_2-k_{21}I_2]-\gamma_{c2}I_2.\end{align}\]

This says that the two modes each have unsaturated growth rates \(\gamma_{m1}\) and \(\gamma_{m2}\) and cavity decay rates \(\gamma_{c1}\) and \(\gamma_{c2}\), respectively. The saturation effects on the growth rate for each mode caused by itself and by the opposite mode are then expressed in terms of the same self-saturation and cross-saturation parameters \(k_{ij}\), which are inversely proportional to the saturation intensity in the laser medium.

To be consistent with much of the published literature, however, we will instead rewrite Equations 41 in a notation used originally by Willis Lamb, and repeated later in the book by Sargent, Scully and Lamb, namely,

\[\tag{42}\begin{align}dI_1/dt=[a_1-\beta_1I_1-\theta_{12}I_2]\times I_1\\dI_2/dt=[a_2-\beta_2I_2-\theta_{21}I_1]\times I_2.\end{align}\]

The coefficients \(a_1\) and \(a_2\) are then obviously the small-signal or unsaturated gains minus losses for each mode, whereas the coefficients \(\beta_i\) and \(\theta_{ij}\) represent the self- and cross-saturation coefficients. We will now explore the transient and steady-state solutions to Equations 42 in some detail.

 

Steady-State Solutions

Steady-state solutions to Equations 42 obviously require either that the saturated gain factor \(a_i-\beta_i I_i-\theta_{ij}I_j=0\), or else that the corresponding intensity \(l_i =0\), in each of Equations 42. The condition for zero saturated gain for each mode is given by one of the linear relations 

\[\tag{43}I_1=(a_1/\beta_1)-(\theta_{12}/\beta_1)I_2\quad\text{and}\quad I_2=(a_2/\beta_2)-(\theta_{21}/\beta_2)I_1,\]

and each of these relations can in turn be represented by a straight line in the \(I_1,I_2\) plane, as illustrated in Figure 29. These two lines then may or may not intersect in the first quadrant of the \(I_1,I_2\) plane, as shown in Figures 30 and 31.

To develop this analysis further, let us indicate the origin of each of these lines on its own axis by the points \(\text O_1\equiv a_1/\beta_1\) and \(\text O_2\equiv a_2/\beta_2\); and the tip or termination of each vector, where it intercepts the opposite axis, by \(T_1=a_1/\theta_{12}\) and \(T_2=a_2/\theta_{21}\).

Then, whether or not the two lines intersect, the two points

 

 

 

 

\(\text O_1\) and \(\text O_2\) clearly represent at least potential single-mode steady-state operating points for the two-mode laser, corresponding to the operating conditions 

\[\tag{44}I_{1,SS}=\text O_1=a_1/\beta_1\quad\text{and}\quad I_{2,SS}=0\]

on one hand, and to 

\[\tag{45}I_{2,SS}=\text O_2=a_2/\beta_2\quad\text{and}\quad I_{2,SS}=0\] 

in the other. 

If the two vectors also have an intersection point in the first quadrant of the \(I_1\), \(I_2\) plane, as illustrated in Figures 30 and 31, then that intersection also represents a third potential two-mode steady-state operating point, which we can call \(0_3\). Both modes can potentially oscillate simultaneously at this point, with steady-state intensities given by

\[\tag{46}I_{1,SS}=\frac{a_1-(\theta_{12}/\beta_2)a_2}{(1-C)\beta_1}\quad\text{and}\quad I_{2,SS}=\frac{a_2-(\theta_{21}/\beta_1)a_1}{(1-C)\beta_2},\]

where \(\text{C}\) is a dimensionless coupling factor given by

\[\tag{47}C\equiv\frac{\theta_{12}\theta_{21}}{\beta_1\beta_2}.\]

The question now is, under what conditions will these three potential steady-state solutions be stable against small perturbations? 

 

Perturbation Stability Analysis

To evaluate this we can follow the usual approach of expanding the intensities \(I_1(t)\) and \(I_2(t)\) about the steady-state intensities in the form

\[\tag{48}I_1(t)=I_{1,SS}+\epsilon_1(t)\quad\text{and}\quad I_2(t)=I_{2,SS}+\epsilon_2(t),\]

and then look for the linearized growth or decay rates of the small perturbations \(\epsilon_1(t)\) and \(\epsilon_2(t)\) about each potential steady-state operating point.

Let us treat first the single-mode operating point at \(\text O_1\), where mode \(\#1\) is oscillating alone with intensity \(I_1=a_1/\beta_1\). The linearized differential equations for the small-signal perturbations about this point then become

\[\tag{49}\frac{d\epsilon_1(t)}{dt}\approx-a_1\epsilon_1(t)-\frac{\theta_{12}a_1}{\beta_2}\epsilon_2(t),\]

and

\[\tag{50}\frac{d\epsilon_2(t)}{dt}\approx\left[a_2-\frac{\theta_{21}a_1}{\beta_1}\right]\epsilon_2(t).\]

The requirement that mode \(\#2\) remain stable at zero amplitude at this operating point is clearly determined by the stability criterion that

\[\tag{51}\frac{\theta_{21}a_1}{\beta_1}>a_2\quad\text{or}\frac{\theta_{21}a_1}{\beta_1a_2}>1.\]

In geometric terms, this condition says that

\[\tag{52}\frac{O_1}{T_2}=\frac{a_1/\beta_1}{a_2/\theta_{21}}>1.\]

That is, for single-mode stability the origin \(\text O_1\)of the zero-gain line for mode \(\#1\) must be farther out on the \(I_1\) intensity axis than the tip \(T_2\) of the zero-gain line for mode \(\#2\).

The stability criterion for the single-mode solution at \(\text O_2\) is obviously the same as Equation 52, except that the indices are reversed, i.e.,

\[\tag{53}\frac{\text O_2}{\text T_1}\equiv\frac{\theta_{21}a_2}{\beta_2a_1}>1.\]

In geometric terms, it is clear that if the two zero-gain lines do not intersect, as in Figure 29, the origin of the outermost of the two vectors (vector \(\text O_1\rightarrow T_1\) in this illustration) is the stable solution. The laser oscillates only in mode \(\#1\), and this mode completely suppresses the less favored mode \(\#2\).

 

Dual-Mode Stability Analysis

When the two zero-gain vectors do intersect, as in Figures 30 and 31, two quite different situations can occur. First of all, if the origins \(\text O_i\) of both vectors lie inside the tips \(\text T_j\) of the opposite vectors on the \(\text I_i\) axes, as in Figure 30, then by the criteria just developed neither of the single-mode solutions \(\text O_1\) or \(\text O_2\) can be stable.

Hence, the dual-mode solution \(\text O_3\) must be the only stable solution. The mathematical criterion for this is that both \(\text O_1/\text T_2\) and \(\text O_2/\text T_1\) must be \(< 1\), which we can write in the combined form, 

\[\tag{54}\frac{\text O_1}{\text T_2}\times\frac{\text O_2}{\text T_1}=\frac{\theta_{12}\theta_{21}}{\beta_1\beta_2}\equiv C<1\quad(\text{weak}\;\text{coupling}).\]

This is generally referred to as weak coupling between the two oscillating modes. Both modes can oscillate simultaneously, sharing the same gain medium.

 

 

 

 

 

 

 For strong coupling, on the other hand, we have the opposite condition, namely,

 \[\tag{55}\text C\equiv\frac{\theta_{12}\theta_{21}}{\beta_1\beta_2}=\frac{\text O_1}{\text T_2}\times\frac{\text O_2}{\text T_1}>1\quad(\text{strong}\;\text{coupling}),\]

and in fact both \(\text O_1/\text T_2\) and \(\text O_2/\text T_1\) individually \(>1\), as illustrated in Figure 31. For strong coupling, both of the single-mode solutions \(\text O_1\) or \(\text O_2\) are stable, and hence presumably the dual-mode solution \(\text O_3\) is not.

To verify these conclusions mathematically, we must apply the perturbation expansion about the dual-mode solution at point \(\text O_3\). The linearized perturbation equations then become, after some algebra,

\[\tag{56}\begin{align}d\epsilon_1/dt=-\beta_1I_{1,SS}\epsilon_1-\theta_{12}I_{1,SS}\epsilon_2,\\d\epsilon_2/dt=-\theta_{21}I_{2,SS}\epsilon_1-\beta_2I_{2,SS}\epsilon_2.\end{align}\]

To see if these equations are stable, we assume potential variations in \(\epsilon_1(t)\) and \(\epsilon_2(t)\) of the form \(e^{st}\), and then evaluate the resulting secular determinant given by

\[\tag{57}\bigg|\begin{align}s+\beta_1I_{1.SS}\quad\theta_{12}I_{1,SS}\\\theta_{21}I_{2,SS}\quad s\beta_2I_{2,SS}\end{align}\bigg|=0.\]

This reduces, after some further algebra, to the secular equation

\[\tag{58}s^2+\left(\frac{a'_1+a'_2}{1-C}\right)s+\frac{a'_1a'_2}{1-C}=0,\]

where \(a'_1\) and \(a'_2\) are reduced gains given by \(a'_1=a_1-(\theta_{12}/\beta_2)a_2\) and \(a'_2=a_2-(\theta_{21}/\beta_1)a_1\). (These gains represent physically the net gains for mode \(\#1\) when mode \(\#2\) is oscillating by itself at its full single-mode strength, and vice versa.)

The roots of the secular equation are then given by

\[\tag{59}2_{S1},2_{S2}=-\left(\frac{a'_1+a'_2}{1-C}\right)+\sqrt{\left(\frac{a'_1+a'_2}{1-C}\right)^2-4\frac{a'_1a'_2}{C-1}}.\]

The weak-coupling situation corresponds to \(C<1\) and \(a'_1,a'_2>0\), which makes both of the roots in this equation negative. The intersection point \(\text O_3\) in the weak-coupling situation is therefore a stable dual-mode operating point, as we surmised in the preceding.

The strong-coupling situation, with \(C>1\) and \(a'_1,a'_2<0\), by contrast causes one of the roots to take on the positive value

\[\tag{60}2_{S1}=\bigg|\frac{a'_1+a'_2}{C-1}\bigg|+\sqrt{\bigg|\frac{a'_1+a'_2}{C-1}\bigg|^2+4\frac{a'_1a'_2}{C-1}}.\]

The dual-mode solution at point \(\text O_3\) is unstable in this situation. Even if the two modes somehow start oscillating with intensities appropriate to this point, the effect of any small perturbation will cause the laser to move to a condition where only one of the modes (either one) is oscillating by itself. A two-mode laser with strong coupling between modes is, in other words, a bistable system.

 

Phase-Plane Trajectory Plots

To illustrate the physical behavior of the mode competition in these various situations in a bit more detail, we can follow the lead of Lamb's original paper and plot trajectories for the coupled oscillators in the \(I_1,I_2\) plane. Suppose the two modes are initially set oscillating with initial intensities corresponding to various points in the \(I_1,I_2\) plane (ignoring the practical question of how this might be done). 

We can then integrate the basic differential equations of motion for \(dI_1/dt\) and \(dI_2/dt\) forward in time and follow the trajectory of \(I_1\) and \(I_2\) as these intensities converge to a final steady-state solution in the  \(I_1\), \(I_2\) plane. The three plots in Figure 32 illustrate the general nature of this behavior for single-mode oscillation, weakly coupled dual-mode oscillation, and strongly coupled or bistable single-mode oscillation. Note how the various trajectories converge toward the stable point or points indicated by the stability analysis given in the preceding.

 

 

 

 

 

 

A few practical examples may be useful to illustrate the application of these fundamental concepts of mode competition to real lasers: 

\((1)\) Homogeneously Broadened Ring Lasers: One elementary example of mode competition occurs in ring lasers, where there are two potential modes having the same resonance frequency but traveling in opposite directions around the ring (Figure 33). Consider in particular a ring laser in which the gain medium saturates homogeneously, and in which coupling between the two directions produced by backscattering effects inside the ring cavity is negligible. 

If the two oppositely traveling waves in such a cavity are at the same frequency, the cross-saturation effect between the two oppositely traveling waves will be larger by a factor of two than the self-saturation effect due to either wave alone, because of the self-induced grating effects we have described in an earlier tutorial. A homogeneously broadened ring laser, at least under ideal conditions, will thus correspond to a very strongly coupled situation with \(\text C=4\).

Such a ring should therefore oscillate, at least under ideal conditions, in only one direction at a time.

This can be observed experimentally in a number of lasers, although backscattering effects, deviations of the laser medium from ideal homogeneity, and other effects may either partially or completely overcome these strong coupling effects.

Note also that a ring laser with a homogeneous saturable absorber should presumably want to run in both directions simultaneously. This corresponds in fact to the very important situation of colliding-pulse saturable-absorber mode locking, which will be discussed in more detail in a later section.

\((2)\) Mode competition in inhomogeneously broadened atomic systems: In an inhomogeneous atomic transition, a strong signal saturates or "burns a hole" in only a single spectral packet, or a small group of atoms with which the signal is nearly or completely in resonance.

Another competing mode may interact primarily with atoms in a different, unsaturated spectral packet. Mode competition effects in simple inhomogeneous atomic transitions thus depend very much on whether the two mode frequencies do or do not fall within the same hole, i.e., within roughly one homogeneous linewidth of each other.

If this is not so, and the two modes burn totally separate holes, the coupling can be expected to be weak \((C\ll1)\), in agreement with the common observation of simultaneous multimode oscillation in inhomogeneous lasers. If the two modes come within less than a homogeneous linewidth of each other, the coupling can be expected to approach \(\text C\approx 1\) or neutral coupling.

If the two waves are at the same frequency but traveling in opposite directions (as in a ring laser) the coupling in the inhomogeneous situation can potentially become somewhat stronger, and can approach \(\text C\approx 2\) (rather than \(C=4\) as in the analogous homogeneous situation).

\((3)\) Doppler-broadened lasers, and ring-laser gyroscopes: Inhomogeneous mode competition effects can become considerably more complex with either standing-wave or ring lasers that employ doppler-broadened inhomogeneous atomic transitions. In the ring laser, for example, if there are two cavity modes with frequencies \(\omega_1\) and \(\omega_2\), each mode can burn a single hole at some velocity class in the doppler profile determined by both the frequency detuning \(\omega-\omega_0\) of the mode and the direction in which the mode travels around the ring. We must then average the self-saturation and cross-saturation effects over all the velocity classes in the doppler-broadened atomic transition.

Suppose that in a ring-laser gyroscope only a single axial mode falls within the doppler-broadened atomic gain profile, as in Figure 34. If the ring-laser gyro is rotated sufficiently rapidly, then this axial mode is split into two frequencies \(\omega_{cw}\) and \(\omega_{ccw}\), corresponding to clockwise and counter-clockwise traveling modes, so long as \(|\omega_{ccw}-\omega_{cw}|\) is larger than the locking range of the laser gyroscope caused by backscatter within the ring.

If this axial mode, which is split into two simultaneously oscillating modes in opposite directions, is tuned off the center of the doppler gain profile on either side, then the two counter-propagating waves will burn separate and independent holes on opposite sides of the maxwellian velocity distribution.

The coupling between the two modes is then weak \((\text C<1)\), and both modes can oscillate simultaneously, although with a slight intensity preference toward the mode nearest line center, as in the lower part of Figure 34.

If the axial mode is tuned near or through the atomic line center, however, the two holes merge. The coupling then moves toward neutral or even toward strong coupling \((\text C\geq 1)\). The result is that one or the other of the oppositely traveling modes goes out just at line center, as shown in Figure 34.

Note that this is purely an intensity competition effect, quite separate from the injection locking effects in laser gyroscopes, which are also troublesome and which we will discuss in a later tutorial.

In practical ring-laser gyroscopes we want to avoid this mode competition. To accomplish this, most practical helium-neon ring-laser gyroscopes use a laser tube filled with an isotopic mixture of \(\text{Ne}^{20}\) and \(\text{Ne}^{22}\). These two transitions, have gain curves that overlap, but with center frequencies that are 

 

 

 

 

slightly displaced in frequency by an isotopic shift. Thus, a single axial mode can never be at the center of both lines simultaneously. The combined effect of these two transitions is sufficient to avoid the cross-mode suppression since the two modes are never simultaneously at the center of both lines.

\((4)\) Zeeman lasers: Some interesting and complicated experiments have also been done to study mode competition in what are called "Zeeman lasers." These are in general gas lasers which use a inhomogeneous doppler-broadened line and a standing-wave cavity, but with no polarization selecting elements.

The experiments are normally done using internal mirror lasers, in which there are no Brewster windows, with the mirror surfaces and the mirrors inside the laser carefully selected to avoid even small polarization anisotropies in the mirror reflectivity. A constant dc magnetic field is also applied either across or along the laser tube. 

A laser of this type can then oscillate in two different modes representing standing-wave resonances with two different senses of polarization—usually two different senses of circular polarization \(I_+\) and \(I_-\)—at the same axial mode frequency. These experimental results also depend on the fact that gas laser transitions will normally have an angular momentum degeneracy, with an angular momentum quantum number Jf and hence \(2J'+1\) degenerate transitions for the upper laser level, and \(2J"+1\) transitions for the lower laser level.

The calculations of the self-saturation and cross-saturation effects in these lasers then become quite complicated since we must include such effects as the superposition of the multiple transitions between each of the degenerate upper and lower levels; the polarization-dependent selection rules for each of these subtransitions; the Zeeman splitting of the upper and lower levels produced by the constant magnetic field; and the doppler velocity distribution for each of these transitions.

The resulting calculations, while lengthy, are straightforward, and one can make experimental and theoretical comparisons for a variety of conditions. The results can provide a detailed test of spectroscopic theory, laser theory, and mode competition theory, all within a single experiment.