Fundamentals of Laser Oscillation
This is a continuation from the previous tutorial - mid-IR and infrared fibers.
This tutorial brings us finally to the complete laser oscillator: atoms, plus pumping and population inversion, plus signals and amplification, plus mirrors to provide feedback and oscillation.
In this tutorial we will develop formulas for some of the simpler aspects of laser operation, including the population inversion required to reach oscillation threshold; the pumping power density required to produce this inversion; the laser power output, and its dependence on output coupling and pumping power in simple cases; the difference between homogeneously and inhomogeneously broadened lasers; and the atomic-frequency pulling effects in a laser oscillator.
In the next tutorial we will then develop a set of coupled rate equations which link cavity photons to laser atoms, and laser atoms to cavity photons. Using these equations we will explore laser oscillation buildup and the remarkable threshold properties characteristic of the laser oscillator.
1. Oscillation Threshold Conditions
The basic requirement either for just reaching laser oscillation threshold, or for maintaining steady-state laser oscillation, is that the round-trip gain inside the laser cavity, including mirror reflections, must be exactly unity, modulo an integer number of multiples of \(e^{-j2\pi}\).
Only if the round-trip gain is exactly unity can the system maintain steady-state oscillation, in which the circulating signal inside the cavity neither grows nor decays on successive round trips.
(This assertion does leave out some extremely minute effects due to spontaneous emission or noise in the laser cavity, which are totally negligible in any of the following discussions.)
In the notation developed in the laser mirrors and regenerative feedback tutorial, unity round-trip gain requires that
\[\tag{1}\begin{align}\tilde{g}_\text{rt}(\omega)&\equiv{r_1}r_2(r_3\ldots)\times\exp\left[\alpha_m(\omega)p_m-\alpha_0p-j\frac{\omega{p}}{c}-j\Delta\beta_m(\omega)p_m\right]\\&=\exp[-jq2\pi]\end{align}\]
where \(q\) is an integer.
This can in turn be separated into an amplitude or magnitude condition, which says that at steady-state the round-trip gain must have magnitude unity, or
\[\tag{2}r_1r_2(r_3\ldots)\times\exp[\alpha_m(\omega)p_m-\alpha_0p]=1\]
and a phase or frequency condition, which says that at steady state the roundtrip phase shift must be an integer multiple of \(2\pi\), or
\[\tag{3}\frac{\omega{p}}{c}+\Delta\beta_m(\omega)p_m=q\times2\pi\]
The first of these conditions determines the population inversion density, and hence the pumping rate, needed to reach oscillation threshold. The second condition determines primarily the frequency \(\omega\) at which the laser must oscillate.
Threshold Inversion Density
Suppose we rewrite the amplitude condition in terms of the round-trip power gains and losses, since we usually speak of power gains and mirror power reflectivities \(R_i\) rather than voltage reflectivities \(r_i\) in practical discussions. The gain coefficient required to just reach threshold in the laser cavity is then given by
\[\tag{4}2\alpha_m(\omega)p_m=2\alpha_0p+\ln\left[\frac{1}{R_1R_2(R_3\ldots)}\right]\]
or, in terms of the "delta notation" we introduced in the laser mirrors and regenerative feedback tutorial,
\[\tag{5}2\alpha_m(\omega)p_m[\equiv\delta_m(\omega)=\delta_0+\delta_1+\delta_2+\ldots]\equiv\delta_c\]
Now, we can recall from earlier chapters that the laser gain coefficient for a lorentzian atomic transition is given by
\[\tag{6}\alpha_m(\omega)=\frac{3^*}{4\pi}\frac{\gamma_\text{rad}\lambda^2}{\Delta\omega_a}\frac{\Delta{N}}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}\]
or by a very similar expression for a gaussian atomic transition. The inversion density required either to reach threshold, or to maintain steady-state oscillation, in a cavity mode located at midband (\(\omega=\omega_a\)) on a lorentzian atomic transition, is thus given by
\[\tag{7}\Delta{N}=\Delta{N}_\text{th}\equiv\frac{2\pi}{3^*}\times\frac{\Delta\omega_a}{\gamma_\text{rad}}\times\frac{1}{\lambda^2}\times\frac{\delta_c}{p_m}\]
In order to have achieve oscillation with the lowest possible inversion density we want to have a laser system with the following characteristics:
- A narrow atomic linewidth \(\Delta\omega_a\).
- A strong radiative decay rate \(\gamma_\text{rad}\).
- A long wavelength \(\lambda\).
- Low cavity losses and output coupling, \(\delta_c\).
- A long gain medium \(p_m\).
The dependence on wavelength in particular agrees with the general observation that infrared lasers are usually fairly easy to obtain, whereas visible and UV lasers become progressively more difficult.
Not all of these criteria are essential; some are not even always desirable. For example, many useful laser materials have wide linewidths and small radiative decay rates, and many lasers work best with very large output couplings. They do, however, at least indicate which properties will make achieving laser action more or less difficult.
This same threshold condition can also be expressed much more simply in terms of the transition cross section \(\sigma\) given by \(2\alpha_m=\Delta{N}\sigma\), which leads to the particularly simple result
\[\tag{8}\Delta{N}_\text{th}=\frac{\delta_c}{\sigma{p_m}}\]
A large transition cross section and a small threshold inversion obviously go together.
Threshold Pump Power Density
Although the threshold inversion density is important, of more practical importance is the pump power density required to achieve this threshold inversion in a practical laser. We can express this threshold pump power density in a fairly general form, more or less independent of the particular pumping mechanism that is employed, using the general laser pumping model shown in Figure 12.1.
First of all, obtaining the threshold inversion density given by Equation 12.7 will require achieving an upper-laser-level population density \(N_{2,\text{th}}\) that is greater than this threshold inversion \(\Delta{N}_\text{th}\) by some ratio that depends, as shown in Figure 12.1, on how much population accumulates in the lower laser level, which depends in turn on how rapidly the lower laser level empties out.
Suppose that in order to achieve inversion atoms are pumped upward by a pumping power density (power per unit volume) \(P_p/V\) into one or more upper pump levels \(E_3\), from which some fraction of these atoms fall down into the upper laser level \(E_2\), with a net pumping rate of \(R_p\) atoms per unit volume per second into this upper laser level.
Let the effective energy gap across which atoms must be lifted by the pumping mechanism be given by \(\hbar\omega_p\), where \(\omega_p\) is the "pumping frequency" (whether or not the pumping is actually done optically using photons of energy \(\hbar\omega_p\) or by some other mechanism); and let the fraction of excited atoms that actually end up falling into the upper laser level be given by a pumping efficiency \(\eta_p\). (The remainder of the pumping power is wasted, either in lifting up atoms which drop back down through other paths, or simply in added heat dissipation in the medium.)
The effective pumping rate \(R_p\) and the upper level population density \(N_2\) which is created by this pumping rate are then given by
\[\tag{9}N_2=\frac{R_p}{\gamma_2}=\frac{\eta_pP_p}{\gamma_2\hbar\omega_pV}\]
where \(P_p/V\) is the total pumping power density (power per unit volume) going into the laser medium, and \(\gamma_2\) is interpreted as the total downward decay rate out of level 2 due to all decay mechanisms.
By combining this expression with the threshold inversion expression 12.7, we can find that a quite general expression for threshold pump power density is
\[\tag{10}\frac{P_{p,\text{th}}}{V}=\frac{1}{\eta_p}\times\frac{N_{2,\text{th}}}{\Delta{N_\text{th}}}\times\frac{\omega_p}{\omega_a}\times\frac{\gamma_2}{\gamma_\text{rad}}\times\frac{4\pi^2}{3^*}\times\frac{\hbar\Delta\omega_a}{\lambda^3}\times\frac{c\delta_c}{p_m}\]
This is a very important expression for calculating—or at least estimating—the performance characteristics and pumping requirements of a given laser system.
Practical Laser Pumping Requirements
The first four factors in this expression are all dimensionless ratios with values which may vary greatly from laser system to laser system, but which are never smaller than unity. Each factor can thus provide a criterion for searching for good laser systems. Each ratio does in fact come close to unity in certain particularly favored laser systems, though each of them is also more commonly much worse than unity in other systems.
If we look at each of these factors in turn, we can see that the criteria for a good laser medium—or at least one with low pump-power requirements—include, first, a good pumping efficiency \(\eta_p\) in terms of atoms lifted up per unit pump power applied to the medium.
Systems which show up well on this criterion include optically pumped dye lasers, solid state lasers, semiconductor lasers, and some gas lasers such as chemical lasers and the CO\(_2\) laser. Systems not favored by this criterion include many common gas-discharge lasers, such as the He-Ne or ion lasers, where only a small part of the pumping energy goes into lifting atoms into the desired upper laser levels.
In a He-Ne or Argon-ion gas laser, for example, the number of atoms actually pumped into the upper laser level is very small compared to photon units of electrical energy dissipated in the gas discharge medium, so that \(\eta_p\ll1\).
In these lasers most of the electrical energy input into the laser gas goes either into exciting unwanted atomic levels or just into heating up the gas atoms and the electrons.
In a laser-pumped dye laser, by contrast, the ratio of dye molecules pumped up to the upper laser level to laser pumping photons absorbed can be almost unity. Of course, these laser-generated pumping photons are themselves rather expensive photons to obtain.
A good laser system should also have a lower laser level that empties out rapidly, so that \(N_1\approx0\) and \(\Delta{N}\approx{N_2}\) (which, of course, works against three-level lasers like ruby). A good laser system should also have its upper pumping level not far above the upper laser level, and its lower laser level close to (but not right at) the ground level, so that the pump photons can be as small as possible.
Again this favors dye lasers, most solid-state lasers, and some gas lasers, and works strongly against other gas lasers like He-Ne and argon lasers, where we must lift the atoms up to levels that are \(\approx\) 20 eV or more above ground level, in order to get out \(\approx\) 2 eV photons across the laser transition.
Finally, we want a laser transition which is as close to purely radiative on the laser transition as possible, with no other radiative or nonradiative decay rates to leak off atoms, so that \(\gamma_2\approx\gamma_\text{rad}\). This favors the ruby laser, organic dye lasers, and to a lesser extent other solid-state and gas lasers.
Note in particular that if the condition \(\gamma_2\approx\gamma_\text{rad}\) is satisfied, then the actual value of the transition strength \(\gamma_\text{rad}\) drops out of the pump power density expression. A strong transition with a large \(\gamma_\text{rad}\) needs a smaller population inversion, according to our above results; but at the same time the faster decay makes this inversion harder to maintain continuously. Hence ruby and Rhodamine 6G are both very good visible laser systems, even though their values of \(\gamma_\text{rad}\) differ by some 6 orders of magnitude.
After all these factors are taken into account, the remaining factors in obtaining laser inversion are the final two terms in Equation 12.10. Laser action is always harder to obtain the wider the atomic linewidth (though at the same time wide linewidth is essential if we want to have tunable laser action).
We also see once again that the difficulty in obtaining laser action goes up very rapidly as the laser wavelength gets shorter, with the pump power density rising, other factors being equal, at least proportional to \(1/\lambda^3\).
In fact, in doppler-broadened gas lasers the doppler linewidth itself tends to increase as \(1/\lambda\), and hence the wavelength dependence of \(P_{p,\text{th}}\) will be more like \(1/\lambda^4\). A genuine X-ray laser will be very difficult to obtain, both or this reason and because of other factors as well, not the least of these being the lack of good X-ray mirrors.
Finally, the last term in the threshold pumping condition contains the cavity factors, i.e., to reach oscillation in a weak laser system we want the lowest possible cavity losses, and the longest possible gain medium.
2. Oscillation Frequency and Frequency Pulling
Let us next look at the frequency characteristics of laser oscillators: whether a laser will oscillate only in a single mode and at a single frequency, or in many modes at once, and also what the exact frequencies of these oscillations will be if atomic pulling effects are included.
The axial cavity mode whose frequency is located closest to the center of an atomic transition will of course normally see the highest gain, and will thus normally reach oscillation threshold first, before other modes located further from the atomic line center.
Suppose, however, that we turn up the gain or the pump power still further, beyond the point where the first cavity mode reaches threshold. Will additional axial (or transverse) modes then also begin oscillating? The answer to this question is that there are in general two idealized or limiting types of laser oscillation behavior for the remaining cavity modes, depending on whether the laser transition is homogeneously or inhomogeneously broadened.
We will first consider these two general classes of laser oscillation, before discussing the atomic pulling effects that occur for either class.
Ideal Homogeneous Lasers: Single Frequency Oscillation
In an ideally homogeneous laser transition, the atomic lineshape is fixed and identical for all the atoms in the laser medium. The magnitude of the gain and phase shift measured at any given frequency will move up and down as the population inversion \(\Delta{N}\) varies; but the lineshapes of \(\chi^"(\omega)\) and \(\chi'(\omega)\) versus frequency will remain unchanged—in essence the whole lineshape moves up and down together.
Suppose the midband gain in such a homogeneous laser medium is increased until the axial mode closest to line center just reaches threshold (i.e., gain just equals losses), as illustrated in Figure 12.2(a). This mode \(q\) can then begin to oscillate, whereas all the other modes (\(q-1\), \(q+1\), and \(q+2\)) are still below threshold and cannot oscillate. (Note that the gain actually exceeds the losses in the center portion of the atomic line, but there is no cavity mode located there to build up to oscillation.)
Even if we pump this laser harder, we cannot push the gain profile further up so as to cause the \(q+1\) mode to oscillate, as illustrated by the dashed gain profile in Figure 12.2(a). Such oscillation is not possible, at any rate, on a cw or steady-state basis, because then gain would exceed loss for the \(q\)-th mode, and the amplitude of this mode would grow continuously on successive round trips.
It may of course be possible to push the gain for several modes above the steady-state or threshold value on a transient basis, during initial turn-on or pulsed operation of the laser. Note also that the centermost axial mode may not be the first or preferred mode to oscillate, if special mode-control methods are used to increase the losses of this mode relative to another axial mode further out on the atomic gain profile.
An ideally homogeneous laser, therefore, should oscillate under steady-state conditions in only one preferred mode, the first mode to reach threshold; and the gain in the laser medium will be clamped at the level that just causes that mode to reach threshold. Pumping harder will make that preferred mode oscillate more strongly, as we will see very shortly, but will not increase the gain or start new modes oscillating.
Several practical factors, such as spatial hole burning, tend to weaken this conclusion in real lasers. In an ideal system, however, and to a sizable extent in many real lasers, a homogeneously broadened laser will tend to oscillate at only a single frequency, on its centermost (or most preferred) axial and transverse mode.
The experimental spectra in Figure 12.3, taken on a semiconductor injection laser, give an excellent illustration of how a single oscillating mode can emerge just above threshold from a cluster of regeneratively amplified axial-mode noise peaks just below threshold. Note the sharp change in the character and intensity of the output spectrum as the diode injection current is increased from \(I_o\) = 155 mA, just below threshold, to \(I_o\) = 162.5 mA, just above threshold.
The special characteristics of semiconductor diode lasers, including their very small cavity volume, high gain, strong spontaneous emission, broad linewidth, and wide axial-mode spacing, make it comparatively easy to obtain this type of experimental result.
In most other types of lasers the below-threshold output is relatively much weaker, and the threshold transition very much sharper, so that similar experiments become very much more difficult.
In homogeneous Lasers: Multi-Axial-Mode Oscillation
Doppler-broadened gas lasers, and other lasers with strongly inhomogeneous transitions, by contrast, can easily oscillate simultaneously on multiple frequencies or multiple axial modes within the atomic linewidth.
As we will show in a later tutorial, when the atomic gain in an inhomogeneous transition exceeds the loss, each axial mode for which this occurs saturates only that subgroup of atoms, or that particular spectral packet, whose atomic frequencies are in resonance with that particular oscillation frequency.
As a result, the laser "burns a hole" in the gain curve, and saturates the gain down to equal the loss, at each oscillating axial mode separately, as illustrated in Figure 12.2(b). Inhomogeneous lasers can thus oscillate simultaneously in many axial modes, with each mode oscillating almost independent of all the other modes.
For many common laser systems, there can be a substantial number of axial cavity modes within the atomic gain profile of the laser. Helium-neon lasers, for example, with a doppler linewidth \(\Delta{f}_d\approx\) 1,500 MHz and axial-mode spacings \(\Delta{f}_\text{ax}\approx\) 150 to 500 MHz, will typically have three to ten axial modes within the atomic linewidth; and because the line is strongly inhomogeneous, the laser can oscillate in all of these modes at once. Figure 12.4 shows, for example, five simultaneous axial-mode oscillation frequencies from a typical cw He-Ne laser.
A low-pressure CO\(_2\) laser, on the other hand, with only 60 to 100 MHz of combined doppler and pressure broadening, may have only one axial mode within its atomic linewidth. Far-infrared and submillimeter molecular gas lasers also generally have very narrow atomic lines (because they operate at low gas pressures and because the doppler broadening decreases as the transition frequency decreases); and so they usually have one (or even fewer) axial modes within their atomic linewidth.
A neodymium-YAG laser with an atomic linewidth of \(\Delta{f}_a\approx\) 4 cm\(^{-1}\) \(\approx\) 120 GHz will typically have hundreds of axial modes within the atomic gain curve. As a result this type of laser will often exhibit highly multimode oscillation under the transient conditions associated with short-pulse operation.
At the same time this laser can oscillate in only one mode or more often a few simultaneous axial modes under continuous-wave or cw conditions, because of the strongly homogeneous character of the atomic transition.
Spatial Hole Burning
The most significant effect leading to multimode operation even in spectrally homogeneous lasers is spatial inhomogeneity, and especially spatial hole burning is illustrated in Figure 12.5.
Suppose a linear or standing-wave laser is initially oscillating in the \(q\)-th axial mode. This leads to a standing-wave pattern for the field amplitude or optical intensity along the \(z\) axis, with peaks and nulls spaced by one-half optical wavelength (between each null). The inverted population in this laser will then be saturated in a similar spatially periodic fashion, as illustrated in Figure 12.5.
One of the effects of this saturation will be to produce a spatial inverted-population grating or gain grating, which will introduce cross-coupling between the forward and backward-traveling wave components of the \(q\)-th axial mode.
Of more importance at this point, however, is the fact that, at least near the center of the cavity, the standing-wave pattern of the \((q+1)\)-th mode—which squeezes one more half optical wavelength into the cavity length—will have its maximum intensity located just at the points that are left unsaturated by the \(q\)-th mode. [The same point is, of course, equally true for the \((q-1)\)-th axial mode.]
As a result of this, the gain competition between the two adjacent axial modes is much reduced; and both axial modes may well be able to oscillate simultaneously, even with a strongly homogeneous laser medium, by using in essence different groups of atoms.
Oscillation with any two adjacent axial modes at equal amplitudes will then saturate the population uniformly, at least in the center of the cavity, possibly discouraging the oscillation of any further axial modes.
This behavior is sometimes seen, for example, in solid-state lasers, such as the Nd:YAG laser, which often seem to prefer to oscillate at steady state in just two axial modes.
Unidirectional oscillation in a ring-laser cavity is one way of eliminating this kind of hole burning, thus giving a better chance of obtaining single-frequency operation. Placing a comparatively short section of active laser medium close to one of the end mirrors is another way to reduce the effectiveness of the spatial hole-burning process.
Exact Oscillation Frequencies, and Frequency Pulling Effects
Laser oscillation normally occurs in only a few preferred longitudinal and transverse modes of a laser cavity. The exact oscillation frequency of a laser will, however, be shifted away by a small amount from the resonance frequency of the corresponding "cold cavity" mode—that is, the resonance frequency of the cavity mode without laser material—because of small frequency pulling effects associated with the \(\chi'\) Part of the atomic susceptibility. Let us next look at how these pulling effects can be calculated.
The round-trip phase shift \(\phi(\omega)\) in a laser cavity, with the gain medium present, must satisfy the phase shift condition
\[\tag{11}\phi(\omega)\equiv\frac{\omega{p}}{c}+\Delta\beta_m(\omega)p_m=q2\pi\]
where the atomic phase shift term is normally given by
\[\tag{12}\Delta\beta_m(\omega)=\frac{\beta\chi'(\omega)}{2}=\frac{\omega\chi'(\omega)}{2c}\]
We thus obtain a general condition on the laser frequency given by
\[\tag{13}\frac{\omega{p}}{c}\times\left[1+\frac{p_m}{2p}\chi'(\omega)\right]=q2\pi\]
Figure 12.6 shows, in greatly exaggerated form, how the \(\Delta\beta_m(\omega)\) or \(\chi'(\omega)\) contribution appropriate to an amplifying transition causes a shift in the exact frequency at which the \(\phi(\omega)\) curve intersects the \(q2\pi\) resonance values.
Frequency-Pulling Expression
The magnitude of the \(\chi'(\omega)\) term in Equation 12.13 is usually small compared to unity; and the size of the pulling effect will be still further reduced if the length of the atomic medium pm is small compared to the overall cavity perimeter \(p\).
Given this assumption, we can invert Equation 12.13 and solve for the pulled laser oscillation frequency—let's call it \(\omega_q'\)—in the form
\[\tag{14}\begin{align}\omega&=\omega_q'=\frac{q2\pi{c/p}}{1+(p_m/2p)\chi'(\omega_q')}\\&\approx\frac{q2\pi{c}}{p}\times\left[1-\frac{p_m}{p}\frac{\chi'(\omega_q')}{2}\right]\\&=\omega_q+\delta\omega_q\end{align}\]
where \(\omega_q\equiv{q}2\pi{c/p}\) is the unpulled or "cold cavity" resonance frequency, and \(\delta\omega_q\) is the pulling of the resonance frequency by the atomic phase-shift effects. We can then write this (usually) small pulling effect as
\[\tag{15}\delta\omega_q\approx-\frac{p_m}{2p}\omega_q\chi'(\omega_q')\approx-\frac{\Delta\beta_m(\omega_q')p_m}{p/c}\]
But since the axial-mode spacing in the cavity is given by \(\Delta\omega_\text{ax}=2\pi{c/p}\), we may rewrite this as
\[\tag{16}\frac{\delta\omega_q}{\Delta\omega_\text{ax}}\equiv\frac{\text{pulling amount}}{\text{axial mode spacing}}\approx-\frac{\Delta\beta_mp_m}{2\pi}\]
Since the atomic phase-shift term \(\Delta\beta_mp_m\) will usually be small compared to \(2\pi\), the pulling of each mode will usually be small compared to an axial-mode interval. If this pulling term is also small compared to the atomic linewidth, then we can evaluate the pulling contribution \(\chi'(\omega)\) at the unpulled frequency \(\omega=\omega_q\) (which is much simpler to do) rather than at the pulled frequency \(\omega=\omega_q'\).
The magnitude of the reactive susceptibility \(\chi'(\omega)\) in an oscillating laser will, of course, depend on the degree of saturation of the atomic transition, and thus on how strongly the laser is oscillating, as well as on where the oscillating mode (or modes) are located within the atomic linewidth. The phase-shift expressions 12.13 to 12.16 serve primarily, however, to determine the exact frequency at which the laser must oscillate, rather than its power level or other characteristics.
Linear Dispersion Region
Figure 12.6 shows how the \(\chi'(\omega)\) term for an amplifying transition tends to shift each axial mode in toward the atomic transition frequency ua by an amount \(\delta\omega_q\) that depends on distance from the line center. Amplifying transitions always tend to "pull" cavity frequencies toward line center; absorbing transitions have \(\chi'\) values of opposite sign, and thus tend to "push" the cavity resonances away from line center.
For axial modes within the central part of an atomic line, where the value of \(\chi'(\omega)\) increases essentially linearly with the mode offset, modes that are further from line center tend to be pulled proportionately more strongly, in such a way that the axial-mode spacing between successive modes remains nearly constant, though decreased by the mode-pulling effect.
This is sometimes referred to as the linear dispersion region of the atomic transition (Figure 12.7), in contrast to the nonlinear dispersion region further out on the atomic gain profile, where the value of \(\chi'(\omega)\) begins to bend over and no longer increases linearly with frequency.
Frequency Pulling for Lorentzian Atomic Transitions
The frequency-pulling term for a lorentzian atomic transition can be rewritten in a particularly simple form by noting that for a lorentzian transition the atomic gain coefficient \(\alpha_m(\omega)\) and the atomic phase shift \(\Delta\beta_m(\omega)\) are related by
\[\tag{17}\Delta\beta_m(\omega)=2\frac{\omega-\omega_a}{\Delta\omega_a}\times\alpha_m(\omega)\]
Hence the frequency-pulling correction can be written as
\[\tag{18}\frac{\delta\omega_q}{\Delta\omega_\text{ax}}\approx-\frac{2\alpha_m(\omega_q)p_m}{2\pi}\times\frac{\omega_q-\omega_a}{\Delta\omega_a}\]
In most lasers the round-trip power gain coefficient \(2\alpha_mp_m\) will be considerably smaller than \(2\pi\); and for those axial modes near line center the quantity \(\omega_q-\omega_a\) will be only a few axial-mode intervals, and hence usually small compared to the atomic linewidth \(\Delta\omega_a\). Hence the frequency pulling of each mode will be only a small fraction of the axial-mode spacing.
As a numerical example, we might consider a He-Ne laser with 10% power gain per round trip (\(2\alpha_mp_m=0.1\)) and ten axial modes within the atomic linewidth (\(\Delta\omega_\text{ax}/\Delta\omega_a=0.1\)). The fractional pulling for the first axial mode on either side of line center will then be
\[\tag{19}\frac{\delta\omega_q}{\Delta\omega_\text{ax}}\approx-\frac{0.1}{2\pi}\times\frac{1}{10}\approx-1.6\times10^{-3}\]
This corresponds to \(\approx\) 100 kHz pulling out of a 150 MHz axial-mode spacing.
It is also possible, however, for pulling effects to become much larger in lasers that have both very high gain and very narrow atomic linewidth. One example of this is the very high gain 3.39 \(\mu\)m transition in narrow-bore He-Ne laser tubes.
Still Another Frequency-Pulling Formulation
Still another version of the frequency-pulling formula is worth presenting briefly, because of the additional insight it gives into frequency-pulling effects. For the lorentzian case we have discussed, we can also rewrite the round-trip phase-shift condition into the form
\[\tag{20}\frac{\omega{p}}{c}+2\alpha_mp_m\frac{\omega-\omega_a}{\Delta\omega_a}=q2\pi\equiv\frac{\omega_qp}{c}\]
where \(\omega_q\equiv{q}2\pi{c/p}\) is the unpulled or cold cavity frequency. We can then argue that at steady-state oscillation in a homogeneous laser the total cavity gain given by \(2\alpha_mp_m\) should just equal the cavity losses, given by \(\delta_c\), and so the frequency-pulling expression can be rewritten in the form
\[\tag{21}\frac{\omega_ap}{c\delta_c}\times(\omega-\omega_q)+\frac{\omega_a}{\Delta\omega_a}\times(\omega-\omega_a)=0\]
But the quantity \(\omega_p/c\delta_c\) that multiplies the first frequency difference in this expression is just the "cold cavity" \(Q_c\) value that we have defined earlier; and we can for the sake of symmetry define the ratio multiplying the second term as a kind of "linewidth \(Q_a\)" given by \(Q_a\equiv\omega_a/\Delta\omega_a\).
The frequency condition then takes on the particularly simple form
\[\tag{22}Q_c(\omega-\omega_c)+Q_a(\omega-\omega_a)=0\]
where, to make the notation symmetric, we have relabeled the axial-mode frequency \(\omega_q\) as the "cold cavity" frequency \(\omega_c\). This result says that the pulled cavity or oscillation frequency is given by the symmetric expression
\[\tag{23}\omega_c'=\frac{Q_c\omega_c+Q_a\omega_a}{Q_c+Q_a}\]
This says that, as Figure 12.8 shows, if we couple a cavity resonance with \(Q=Q_c\) to an atomic resonance with \(Q=Q_a\), the resulting oscillation frequency will lie somewhere between the two resonance frequencies \(\omega_a\) and \(\omega_c\), closer to whichever one has the higher \(Q\). The usual situation in most laser oscillators is that the cavity resonance has much the higher \(Q\) value; and hence the oscillation occurs essentially at the cavity frequency \(\omega_c\), but pulled slightly toward the atomic frequency \(\omega_a\).
Exactly the opposite situation can also occur, for example, in certain microwave masers or atomic clocks that have an extraordinarily narrow atomic line and a much wider cavity linewidth. In these we want the oscillation frequency \(\omega\) to occur as accurately as possible at the atomic frequency \(\omega_a\), with as little perturbation as possible by the cavity frequency \(\omega_c\). Equation 12.23 then tells how much error may result if the cavity frequency u>c is unavoidably detuned by some amount from \(\omega_a\).
Frequency Beating Measurements
Laser frequency-pulling effects, although typically very small, can be observed in the laboratory in a number of ways. In practice it can be quite difficult to measure the absolute frequencies of laser oscillators to very high precision (although the frequencies of highly engineered laser frequency standards can at present be measured to an absolute accuracy exceeding 1 part in 10\(^{10}\), and can be stabilized with relative accuracies several orders of magnitude higher). Laser frequency pulling is therefore seldom if ever measured on an absolute basis.
It is much easier, however, to measure relative laser frequencies by observing the difference or beat frequency between two different laser signals, assuming these frequencies are close enough together and stable enough with respect to each other to give a clean beat note, as is true in particular of different axial-mode resonances in the same laser cavity.
Difference frequency measurements between two laser oscillations can be accomplished most easily by simply allowing the two laser beams, carefully aligned to be parallel to each other, to fall on any conventional optical detector, such as a photodiode or photomultiplier, and then looking at the photodetector output for signals at the heterodyne or "beat" frequencies between the optical frequencies, as illustrated in Figure 12.9.
Since the optical detector is a square-law device—that is, its signal current is proportional to optical intensity, or to optical \(E\) field squared—the detector responds to two optical signals at, say, \(f_q\) and \(f_{q+1}\) in proportion not only to the dc intensities \(I_q\) and \(I_{q+1}\) at these two frequencies, but also to a sinusoidal heterodyne beat note of intensity \(\sqrt{I_qI_{q+1}}\) at the difference frequency \(f_{q+1}-f_q\). (Do not allow the heterodyne jargon here to obscure the elementary fact that the instantaneous amplitude of any signal consisting of the sum of two sinusoidal carriers at \(f_q\) and \(f_{q+1}\) is automatically modulated at the difference frequency \(|f_{q+1}-f_q|\).)
If we make such a measurement on the output beam from, say, a typical 30-cm long He-Ne laser, using a reasonably fast photodetector and a radio receiver or spectrum analyzer, we can easily detect the 500 MHz beats between several simultaneous axial modes in the laser, as illustrated in Figure 12.9.
If the experiment is done using a suitable radio-frequency spectrum analyzer, we can usually see two or three closely spaced beat notes between different pairs of axial modes, with the frequencies of the different axial-mode beats spread out by a few hundred kHz about the expected \(\Delta\omega_\text{ax}\) value of the laser.
These multiple axial-mode beats result from the slightly different pulling effects that occur for different axial modes in the laser cavity, plus more complicated inhomogeneous cross-pulling effects we have not discussed yet.
The observed beat spectral components will in fact jump about in frequency by small amounts as the axial modes shift together across the atomic gain profile because of thermal drift of the laser cavity, and as different modes suddenly turn on or off at the outer edges of the oscillation range.
Mode-beating experiments are thus an effective way of observing mode-pulling effects and any other small frequency-shifting effects in laser oscillators.
Frequency Beating Between Two Independent Lasers
Heterodyne beats can be observed between beams from two separate lasers as well, but with somewhat more difficulty. The spatial overlap and especially the angular alignment of the two laser beams must first be adjusted to a very high degree of precision to observe any beats. (In a single laser the angular alignment of different frequency modes is automatic.)
The two lasers must then be tuned close enough together in frequency, and the frequency jitter of each laser must be kept small enough, so the beat frequency is within the range of the photodetector and receiver; and we must then scan either the receiver or the lasers until we find where this initially unknown beat frequency is located.
Optical heterodyne measurements using sufficiently stable lasers are nonetheless commonly carried out, often with the assistance of automatic frequency control (AFC) loops to stabilize the difference frequency between the two lasers.
3. Laser Output Power
We will next calculate the power output that can be obtained from an oscillating laser, as a function of the output coupling and the pumping power, using a simple laser model. In this section we will limit the derivation to a lightly coupled laser oscillator—that is, a laser in which the reflectivities of the laser end mirrors are not too much less than unity.
Steady-State Homogeneous Saturation Equations
We emphasized earlier that the round-trip power gain for the signal intensity inside a laser cavity must be exactly equal to unity under cw steady-state conditions. If we assume, for example, a standing-wave laser cavity with a simple homogeneously saturable gain medium, the growth of the two oppositely traveling waves \(I_+(z)\) and \(I_-(z)\) inside this cavity will be given by the two equations
\[\tag{24}\frac{dI_+(z)}{dz}=[2\alpha_m(z)-2\alpha_0]I_+(z)\]
for the forward or \(+z\) wave, and
\[\tag{25}\frac{dI_-(z)}{dz}=-[2\alpha_m(z)-2\alpha_0]I_-(z)\]
for the reverse wave as illustrated schematically in Figure 12.10. (We will carry out all the calculations in this section for a linear or standing-wave cavity, leaving it to the reader to carry out the essentially similar calculations of power output and optimum output coupling for a ring-laser cavity.)
For a homogeneously saturable gain medium, which saturates at each transverse plane \(z\) according to the sum of the intensities \(I_+(z)\) and \(I_-(z)\) at that plane, the saturated gain coefficient as a function of position along the axis will be
\[\tag{26}2\alpha_m(z)=\frac{2\alpha_{m0}}{1+[I_+(z)+I_-(z)]/I_\text{sat}}\]
where \(\alpha_{m0}\) is the unsaturated gain coefficient. This assumption neglects interference or standing-wave effects between the right- and left-traveling waves. Nonetheless, it serves as an excellent first approximation, and gives results that agree very well with experiment.
Small Output Coupling Approximation
Suppose now that the end-mirror reflectivities \(R_1\) and \(R_2\) are both close to unity, so that the intensities \(I_+\) and \(I_-\) remain nearly constant along the length of the cavity, as in Figure 12.10. (The unsaturated gain per pass through the laser medium need not be small; but the net saturated gain per pass must be not much greater than unity.)
We can then make the approximation that
\[\tag{27}I_+(z)\approx{I_-(z)}\approx{I_\text{circ}}\]
where \(I_\text{circ}\), the one-way circulating intensity inside the laser cavity, is to first order independent of position inside the cavity. The saturated gain coefficient is then similarly independent of position along the cavity, and can be written as
\[\tag{28}2\alpha_m\approx\frac{2\alpha_{m0}}{1+2I_\text{circ}/I_\text{sat}}\]
The factor of 2 in the denominator arises, of course, because the laser medium sees equal intensities \(I_\text{circ}\) traveling in both directions along the cavity.
Steady-State Oscillation Condition
The threshold and/or the steady-state gain condition for the laser oscillator is then given by
\[\tag{29}2\alpha_mp_m\approx\frac{2\alpha_{m0}p_m}{1+2I_\text{circ}/I_\text{sat}}=2\alpha_0p+\ln\left(\frac{1}{R_1R_2}\right)\equiv\delta_0+\delta_1+\delta_2\]
The circulating intensity inside the cavity that must build up in order to saturate the gain factor \(2\alpha_{m0}p_m\) down to where it just equals the total cavity losses \(\delta_0+\delta_1+\delta_2\) is thus given by
\[\tag{30}I_\text{circ}=\left[\frac{2\alpha_{m0}p_m}{\delta_0+\delta_1+\delta_2}-1\right]\times\frac{I_\text{sat}}{2}\]
It is often convenient to define a threshold ratio \(r\) given by
\[\tag{31}r\equiv\frac{2\alpha_{m0}p_m}{\delta_0+\delta_1+\delta_2}=\frac{\text{unstaturated round-trip laser gain}}{\text{total round-trip cavity losses}}\]
The condition \(r=1\) then corresponds to threshold; and the value of \(r\ge1\) tells by how much the laser gain is pumped above threshold. The circulating intensity inside a standing-wave cavity can then be written as
\[\tag{32}I_\text{circ}=(r-1)\times\frac{I_\text{sat}}{2}\]
This expression, of course, has meaning only so long as the laser is above threshold, so that \(r\gt1\) or \(2\alpha_{m0}p_m\gt\delta_0+\delta_1+\delta_2\).
Laser Power Output
Now, for a lightly coupled laser cavity, the end-mirror transmissions are given by \(T_1=1-R_1\approx\delta_1\) and \(T_2=1-R_2\approx\delta_2\), so long as both \(\delta_1\) and \(\delta_2\) are reasonably small compared to unity. Normally in a laser we take the power output from one end of the cavity only. The total potentially useful output intensity (power per unit area) is really the power from both ends of the laser cavity, however, or
\[\tag{33}I_\text{out}=(\delta_1+\delta_2)\times{I_\text{circ}}=\delta_e\times{I_\text{circ}}\]
where we have defined one additional delta factor \(\delta_e\) (with "e" standing for "external") by the definition
\[\tag{34}\delta_e\equiv\delta_1+\delta_2=\text{external cavity coupling}\]
The value of \(\delta_e\) thus represents the total external coupling, or output coupling, through both ends of the laser cavity. At least for small coupling, \(\delta_e\) represents the total fractional power coupled out per round trip through the external mirrors (or whatever other output coupling mechanism might be employed).
The total output intensity, as a function of the unsaturated gain \(2\alpha_{m0}\), the internal cavity losses \(\delta_0\), and this external coupling factor \(\delta_e\), then becomes
\[\tag{35}I_\text{out}=\delta_e\left[\frac{2\alpha_{m0}p_m}{\delta_0+\delta_e}-1\right]\frac{I_\text{sat}}{2}\]
Figure 12.11 shows a typical example of how both the circulating power and the output power vary with external coupling.
Experimental Verification
Some representative experimental results for power output versus output mirror transmission are shown in Figure 12.12, both for two very low-gain He- Ne lasers, and for a considerably higher-gain argon-laser-pumped Rhodamine 6G dye laser. Note that the results for the dye laser agree very well with the predicted form derived earlier, even though the maximum output coupling of \(\approx\) 20% is becoming significant compared to unity.
Mirrors or output couplers with continuously variable transmission or output coupling are not readily available, especially for small values of output coupling; and performing experiments like those shown in Figure 12.12 with a series of different transmission mirrors on a low-gain, low-coupling laser can be difficult, because the laser must be readjusted and realigned with each change of mirrors, and because you can never be certain that all the mirrors are equally lossless and defect-free.
Figure 12.13 shows a device that uses two contra-rotating dielectric plates tilted near Brewster's angle, where the output reflectivity from each surface of the plates passes through zero, in order to produce a variable output coupling or insertion loss in a laser cavity.
(The same device is also very useful as a separate external optical attenuator; the use of two plates means that the optical axis of the laser beam suffers no net transverse displacement as the plates are rotated in opposite directions.)
Figure 12.14 then shows a careful measurement, made using one of these devices, of the circulating power inside a cw Nd:YAG laser, with results in excellent agreement with theory. Note that though the output power from this laser is only a few hundred milliwatts, the circulating power inside the cavity is several tens of Watts.
Optimum Output Coupling Factor
For any of the lasers shown in Figures 12.11 or 12.12 there is obviously a maximum allowable output coupling, given by \(\delta_{e,\text{max}}\equiv2\alpha_{m0}p_m-2\alpha_0p=\delta_{m0}-\delta_0\), beyond which the cavity is overloaded, so that total cavity losses exceed the available gain, and no oscillation is possible.
As the cavity coupling or end-mirror transmission is reduced below this value, both the circulating intensity and the output intensity increase with decreasing coupling. Below a certain optimum coupling factor \(\delta_{e,\text{opt}}\), however, the mirror transmission decreases faster than \(I_\text{circ}\) increases, and the power output decreases, eventually becoming zero at zero transmission through the end mirrors.
The laser at this point is, of course, still oscillating—in fact, oscillating the strongest of all—but with all its available power being uselessly dissipated in the internal cavity losses.
Figure 12.15 illustrates in more detail how the laser output intensity for a typical laser depends on the cavity output coupling, assuming a fixed value of 20% power gain per round trip, and varying amounts of internal cavity loss.
It is evident that for each different value of internal cavity loss there is a different optimum output coupling which maximizes the output power. It is also apparent that the optimum output coupling is always considerably smaller than the available gain, and that even very small internal losses have a very serious effect on the maximum useful output power available from the laser.
It is a straightforward calculation to evaluate the optimum output coupling for given values of unsaturated gain and internal cavity losses. Differentiation of the expression for output intensity given in Equation 12.35 with respect to the output coupling \(\delta_e\) gives for this optimum coupling
\[\tag{36}\delta_{e,\text{opt}}=\sqrt{2\alpha_{m0}p_m\delta_0}-\delta_0=\left[\sqrt{\delta_{m0}/\delta_0}-1\right]\delta_0\]
where \(\delta_{m0}\equiv2\alpha_{m0}p_m\). The dashed line in Figure 12.15 indicates the locus of these optimum coupling values.
One slightly unusual aspect evident from Figure 12.15 is that the optimum coupling value apparently goes to zero as the internal cavity losses go to zero, i.e., \(\delta_{e,\text{opt}}\rightarrow0\) as \(\delta_0\rightarrow0\). In the limiting case of zero internal losses, we would apparently get maximum output by using end mirrors with 100% reflectivity and zero transmission!
The explanation of this minor paradox is, of course, that as both \(\delta_0\) and \(\delta_e\) go to zero, the internal circulating power \(I_\text{circ}\) goes to \(\infty\); and the product of zero coupling times infinite circulating power leads to a finite power output. Real laser cavities will, of course, always have some small but finite losses, and so will always require an equally small but finite output coupling.
Optimum Output Power and Power Extraction Efficiency
If one adjusts a standing-wave laser oscillator for optimum output coupling, the output intensity with this optimum coupling is then given by
\[\tag{37}\begin{align}I_\text{out,opt}&=\left[\sqrt{2\alpha_{m0}p_m}-\sqrt{\delta_0}\right]^2\frac{I_\text{sat}}{2}\\&=\left[1-\sqrt{\delta_0/\delta_{m0}}\right]^2\times[2\alpha_{m0}L_mI_\text{sat}]\end{align}\]
where we have used \(p_m\equiv2L_m\). But the second term in the second line of this expression can be recognized as the same maximum available intensity from the laser medium that we obtained in our earlier discussion of laser amplification, that is, \(I_\text{avail}\equiv2\alpha_{m0}L_mI_\text{sat}\).
We can then identify the remaining factors in the output intensity formula as defining the power extraction efficiency \(\eta\) with which the laser oscillator extracts energy from the laser medium and converts it into useful power output.
In particular, for a standing-wave laser cavity with arbitrary gain, loss, and output coupling, the extraction efficiency will be given in general by
\[\tag{38}\eta(\delta_0,\delta_e)\equiv\frac{I_\text{out}(\delta_0,\delta_e)}{I_\text{avail}}=\left[\frac{\delta_e}{\delta_0+\delta_e}-\frac{\delta_e}{2\alpha_{m0}p_m}\right]\]
The maximum value of this extraction efficiency with optimum output coupling, or \(\delta_e=\delta_{e,\text{opt}}\), then becomes
\[\tag{39}\eta_\text{opt}=\left[1-\sqrt{\frac{\delta_0}{2\alpha_{m0}p_m}}\right]^2\]
which depends only on the ratio of internal losses to unsaturated gain.
The most significant aspect of these results is the extremely serious effect that even very small internal losses (\(\delta_0\ll2\alpha_{m0}p_m\)) will have on the useful power output. Figure 12.16 shows how this optimum extraction efficiency rapidly decreases as the ratio of internal losses to unsaturated gain increases.
Note, for example, that internal losses only one-tenth as large as the unsaturated gain will reduce the optimized output intensity to less than 50% of its maximum value; and internal cavity losses equal to half the unsaturated gain will reduce the energy extraction efficiency to \(\eta_\text{opt}\approx9\%\).
To put this another way, in a laser with 5% power gain per pass, to extract even 50% of the potentially available power output we must cut the internal cavity losses to \(\le\) 0.3%—something which can be very difficult to do in a real laser cavity.
If the internal losses can be made sufficiently small, however, then the extraction efficiency of a laser oscillator—in contrast to our earlier results for single-pass laser amplifiers—can approach 100% with optimum coupling.
A properly coupled low-loss oscillator can extract nearly all the power that is available in a laser material, something that is much more difficult to do if the same medium is used as a single-pass laser amplifier.
Power Output Versus Pumping
We showed in earlier chapters that in many real laser systems the unsaturated gain coefficient \(2\alpha_{m0}\) increases linearly with the pumping power applied to the laser, whereas the magnitude of the saturation intensity \(I_\text{sat}\) is most often independent of the pumping power.
If this is so, then we can view the dimensionless gain factor \(r\) that we defined earlier as also representing a dimensionless pumping ratio, which gives the amount that the laser is pumped above its oscillation threshold, i.e.,
\[\tag{40}r\equiv\frac{2\alpha_{m0}p_m}{\delta_0+\delta_e}=\frac{R_p}{R_{p,\text{th}}}=\frac{\text{pumping power}}{\text{threshold pump power}}\]
Equation 12.35 laser output intensity then becomes
\[\tag{41}I_\text{out}=\frac{(r-1)\delta_eI_\text{sat}}{2}=\left[\frac{R_p}{R_{p,\text{th}}}-1\right]\times\frac{\delta_eI_\text{sat}}{2}\]
The power output versus pumping power or pumping rate, at fixed coupling, for a great many lasers will therefore be zero up to a certain threshold pumping level corresponding to \(r=1\), and will then rise more or less linearly with pumping rate above this threshold.
To illustrate this point, Figure 12.17 shows the oscillation power outputs versus pump power input for some very different lasers, including two cw lamp-pumped solid-state lasers; a dc-current-pumped semiconductor diode laser; and a laser-pumped cw dye laser, operating with three different end-mirror transmissions on the laser.
These results are typical of many different experimental results for many different types of laser devices, all showing more or less linear variation of laser output, with pump input for substantial distances above their pump thresholds.
A particularly pretty illustration of several fundamental aspects of laser physics is also shown by the experimental results for two similar narrow-strip buried heterostructure injection diode lasers shown in Figure 12.18.
These lasers are both PbSnTe diodes, fabricated using liquid-phase epitaxy, with active regions 1 to 1.5\(\mu\)m thick, 2 to 5 \(\mu\)m wide, and 250 to 450 \(\mu\)m long, oscillating in a single transverse mode at wavelengths near 9.5 \(\mu\)m.
The experimental spectra clearly show
- The regeneratively amplified spontaneous emission at the axial-mode peaks just at or below threshold;
- The sudden changeover to essentially a single oscillating axial mode produced by a very small change in current just at threshold (note the changes in vertical magnification between adjacent spectra);
- The extreme linearity of the power output versus pumping current above threshold. The laser transition is clearly very homogeneous in its spectral behavior.
4. Large Output Coupling Analysis
The power-output analysis of the previous section was based on a weak-coupling approximation. A more accurate analysis of the power output from a homogeneous laser with arbitrarily large round-trip gain and output coupling was originally developed by W. W. Rigrod at Bell Telephone Laboratories, and is often referred to as the "Rigrod analysis."
As we will demonstrate in this section, for a laser with large unsaturated atomic gain the power output remains fairly constant over a very wide range of output coupling, so that critical adjustment of the output coupling is not as essential for reasonably good energy extraction as it is for a low-gain laser.
Analytical Formulation: The Rigrod Analysis
The analysis we will repeat here assumes a homogeneously saturable gain medium as in Section 12.3, but no distributed losses, so that \(2\alpha_0p\equiv0\). Following Rigrod's original notation, as shown in Figure 12.19, we use \(I_+(z)\) and \(I_-(z)\) to indicate the intensities traveling toward \(+z\) and \(-z\), respectively, in the cavity.
These intensities then grow with distance according to the equations
\[\tag{42}\begin{align}\frac{dI_+(z)}{dz}&=+2\alpha_m(z)I_+(z)\\\frac{dI_-(z)}{dz}&=-2\alpha_m(z)I_-(z)\end{align}\]
(The sign changes in the second equation because the waves is traveling in the \(-z\) direction.) For simplicity, let us assume that \(I_+\) and \(I_-\) are normalized to the saturation intensity \(I_\text{sat}\) of the medium. The gain coefficient \(\alpha_m(z)\) at any plane \(z\) then saturates according to the total intensity at that plane in the form
\[\tag{43}\alpha_m(z)=\frac{\alpha_{m0}}{1+I_+(z)+I_-(z)}\]
Writing the gain in this form takes into account the spatial variation of both the forward- and the backward-traveling waves in a high-gain cavity, but neglects any spatial hole burning or induced-grating coupling effects caused by standing waves or by interference between the forward- and the backward-traveling waves inside the laser cavity.
By combining the two derivatives in Equation 12.42, we can see that the product of the intensities in the two directions at any plane is constant, i.e.,
\[\tag{44}\frac{d}{dz}[I_+(z)I_-(z)]=-2\alpha_mI_+I_-+2\alpha_mI_+I_-=0\]
so that we can write at any plane
\[\tag{45}I_+(z)I_-(z)=\text{constant}=C\]
The differential equation for, say, the \(I_+(z)\) wave can then be written as
\[\tag{46}\frac{dI_+(z)}{dz}=\frac{2\alpha_{m0}I_+(z)}{1+I_+(z)+C/I_+(z)}\]
and this can be integrated over the length of the laser medium in the form
\[\tag{47}\int_{I_1}^{I_2}\left(1+\frac{1}{I_+}+\frac{C}{I_+^2}\right)dI_+=2\alpha_{m0}\int_{0}^Ldz\]
Carrying out the same procedure for the \(I_-(z)\) wave, and using the boundary conditions shown in Figure 12.19, then leads to the pair of expressions
\[\tag{48}\begin{align}2\alpha_{m0}L&=\ln\left(\frac{I_2}{I_1}\right)+I_2-I_1-C\left(\frac{1}{I_2}-\frac{1}{I_1}\right)\\2\alpha_{m0}L&=\ln\left(\frac{I_4}{I_3}\right)+I_4-I_3-C\left(\frac{1}{I_4}-\frac{1}{I_3}\right)\end{align}\]
In addition we have the mirror power reflection coefficients \(I_1=R_1I_4\) and \(I_3=R_2I_2\), and the two product relations at the end surfaces, namely, \(I_1I_4=I_2I_3=C\).
By combining all these relations, together with some minor manipulation, we can eliminate the constant \(C\) and obtain the result that, for example, the normalized intensity striking the right-hand mirror is
\[\tag{49}I_2=\frac{1}{(1+r_2/r_1)(1-r_1r_2)}\left[2\alpha_{m0}L-\ln\left(\frac{1}{r_1r_2}\right)\right]\]
where \(r_1\equiv{R_1}^{1/2}\) and \(r_2\equiv{R_2}^{1/2}\) are the voltage reflection coefficients of the mirror.
Power Output and Power-Extraction Efficiency
Let us now assume that mirror \(M_2\) is the output mirror of the laser, with output coupling \(T_2\) and reflection coefficient \(R_2\), and that any finite reflectivity \(R_1\) of the other mirror \(M_1\) represents unwanted or unavoidable losses in that mirror. Then the useful output intensity from the laser (with all intensities measured now in real intensity units) will be
\[\tag{50}I_\text{out}=T_2I_2=\frac{T_2I_\text{sat}}{(1+r_2/r_1)(1-r_1r_2)}\left[\ln{G_0}-\ln\left(\frac{1}{r_1r_2}\right)\right]\]
We know from previous sections that the maximum intensity that can be extracted from such a laser medium is
\[\tag{51}I_\text{avail}=2\alpha_{m0}L_mI_\text{sat}\equiv(\ln{G_0})I_\text{sat}\]
and hence the power-extraction efficiency, or the normalized output intensity, of the laser can be written as
\[\tag{52}\eta\equiv\frac{I_\text{out}}{I_\text{avail}}=\frac{T_2}{(1+r_2/r_1)(1-r_1r_2)}\left[1+\frac{\ln{r_1r_2}}{\ln{G_0}}\right]\]
There are no small-amplitude restrictions on the unsaturated gain \(G_0\) or the output coupling level in this formula.
Typical Results
Examples of power output versus coupling as given by this formula for two comparatively large values of one-way unsaturated power gain \(G_0\) are shown in Figure 12.20. For large values of \(G_0\) and for reflectivity \(R_1\) on the left-hand mirror not too much less than unity, the power output is roughly constant over a very wide range of output couplings. The exact output coupling level applied to a high-gain laser is thus not nearly as critical a factor as it is for a low-gain laser.
At the same time it is evident that even fairly small losses caused by the finite reflectivity \(R_1\lt1\) at the left-hand mirror do have a significant effect on the useful power output from the other end. Obtaining the maximum available power output clearly requires minimum internal losses (\(R_1\rightarrow100\%\)), and power output is generally maximized by smaller rather than larger coupling (\(T_2\le50\%\)).
We could obviously differentiate Equation 12.50 or 12.52 to find the optimum output coupling \(T_{2,\text{opt}}\) and the associated optimum output power. This yields, however, a transcendental equation for \(T_{2,\text{opt}}\) does not seem useful to discuss in more detail here. Internal cavity losses an internal absorption coefficient \(2\alpha_0\) could also be added to the differential equations for the laser, but these equations then become much more difficult to integrate. The general effects of small distributed losses in the laser cavity are best assessed by assuming them to be incorporated as part of the left-hand mirror reflectivity \(R_1\).
Many experimental examples of measured power output versus coupling in confirmation of the Rigrod analysis can be found in the literature. Figure 12.21 shows, for example, the power output versus output mirror reflectivity for a large atomic iodine photodissociation laser, intended as an amplifier (with a one-way power gain of 200 to 300 times), but used in these tests as an oscillator.
The next tutorial introduces digital equalization in coherent optical transmission systems.