UNSTABLE OPTICAL RESONATORS
This is a continuation from the previous tutorial - Generalized paraxial resonator theory
1. ELEMENTARY PROPERTIES
Before developing a more detailed analysis of unstable resonator eigenmodes, as we will do in following sections, let us look in this section at some of the elementary practical aspects of unstable resonators, including practical coupling methods, and the general structure of the near-field and far-field beam patterns of unstable resonator lasers, using a purely geometric approach.
Figure 1. illustrates the general characteristics of the simplest form of hard-edged, single-ended, positive-branch, standing-wave confocal unstable resonator. We have already pointed out that unstable resonators in general:
- Are derived from unstable periodic focusing systems.
-
Are characterized by a geometric magnification parameter \(M\), which may be either a positive or negative number with magnitude greater than unity.

- Can be classified into positive-branch and negative-branch varieties, depending upon the sign of \(\text{M}).
- Have characteristic magnifying (or diverging), and demagnifying (or converging) geometrical eigenwaves according to a purely geometric or paraxial analysis.
- Are perturbation-stable for the magnifying eigenwave and perturbation-unstable for the demagnifying eigenwave, according to the paraxial analysis.
- And, have a set of real transverse eigenmodes with mode properties that are basically similar to the magnifying geometrical eigenwave, but that are strongly influenced by the diffraction properties at the outer mirror or aperture edges in the resonator.
We have also noted that confocal unstable resonators are particularly useful because they produce a collimated output beam, as in Figure 1; and that the demagnifying geometrical eigenwave going in one direction through a geometrically unstable system corresponds to a reversed version of the magnifying paraxial eigenwave going in the opposite direction. We will build on these concepts in the following tutorial.
Output Coupling Methods For Unstable Resonators
One important feature of the hard-edged unstable resonator is that the magnified or diffracted energy coming past the outer edge of the output mirror becomes the useful output from the laser.
This means that totally reflecting optics can be employed, thus eliminating the sometimes troublesome problems of finding low-loss dielectric coatings and transparent and low-loss mirror substrates at infrared, mm-wave or ultraviolet wavelengths.
In fact, all-metal mirrors with internal water cooling channels are often employed to carry away the heat dissipation associated with mirror reflection losses in very high power lasers.
The output mirror must then be mounted so as to permit the output beam to pass around it. One way to do this is to mount the mirror on a small post extending from an output window behind the mirror, as in Figure 2(a).
Another technique is to use a "spider" or arrangement of transverse wires or struts, as in figure 2(b).
These struts should have as small an area as possible to minimize power losses and unwanted far-field diffraction effects.

A third technique, widely used in higher-power lasers, is to employ a diagonal "scraper mirror" to provide the output coupling, as shown in Figure 3. This has the advantages that no struts or posts are required; the output coupling element can be mounted entirely separately from the cavity end mirrors; there are no obstructions in the output aperture; and the output beam direction can even be steered to some extent, without upsetting the cavity alignment.
The major practical problems with this approach are the difficulties in cutting an aperture with clean and optically smooth edges in the scraper mirror, and some small difficulties in standing-wave cavities because the circulating beam in the cavity actually strikes the scraper mirror twice, coming from opposite sides.
Geometrical Output Coupling Value
Note that the output coupling from an unstable resonator depends, in the zero-order or geometric approximation, only on the magnification \(\text{M}\), and not at all on the transverse mirror diameters or the Fresnel numbers of the resonator. (We will see some more exact corrections to this statement in the following tutorials, but it is still basically a good approximation.)

In a one-dimensional or strip-mirror resonator, according to the purely geometric picture, the field expands transversely in one dimension by the factor \(\text{M}\), and thus decreases in field amplitude by the geometric eigenvalue \(\tilde\gamma_\text{geom}\equiv1/|M|^{1/2}\) on each round trip. The fractional power loss per round trip is thus given by
\[\tag{1}\text{power}\;\text{loss}\;\text{per}\;\text{round}\;\text{trip}=1-|\tilde\gamma_\text{geom}|^2=1-\frac{1}{M}\;\text{(strip}\;\text{resonator).}\]
For a two-transverse-dimensional mirror, the area expansion becomes \(\text{M}^2\), so that the geometric eigenvalue is \(\tilde\gamma_\text{geom}\equiv 1/M\) and the fractional power loss per round trip becomes
\[\tag{2}\text{power}\;\text{loss}\;\text{per}\;\text{round}\;\text{trip}=1-|\tilde\gamma_\text{geom}|^2=1-\frac{1}{M^2}\quad\text{(circular}\;\text{resonator)}.\]
The reader can verify from geometric arguments, in fact, that even in a double-ended unstable resonator (with output from both ends) the total geometric output coupling per round trip is entirely independent of the transverse shape or alignment of the end mirrors (provided the resonator axis passes through both mirrors), and also of how the total magnification and total coupling may be divided between the two end mirrors.
Since the output coupling, in the geometric approximation, depends only on the magnification and not on the mirror diameters, we can therefore always expand the diameter of the geometric mode in an unstable resonator so as to fill a laser rod or tube of almost any size, at fixed magnification or output coupling,

simply by expanding the size of the output mirror until the magnified beam diameter more or less fills the active volume of the laser medium.
The efficient power extraction which this permits, along with good transverse mode control, is the primary attractive feature of the unstable optical resonator.
Near-Field Output Beam Pattern
The near-field beam pattern just beyond the output mirror for a circular-mirror unstable resonator will then be an annular beam with an outer diameter roughly \(\text{M}\) times the inner diameter, as shown in Figure 4. In the idealized geometric analysis this beam will be uniform in amplitude and spherical in phase across the annular aperture.
In real hard-edged unstable resonators, as we will see in the following section, the radial intensity profile in the annular region will often be closer to a radially tapered intensity profile, with circular Fresnel diffraction rings of small to moderate strength imposed upon the average profile.
The phase variation across the output plane in a real resonator will nearly always be very close to the predicted wavefront for the magnifying geometrical eigenwave discussed in previous tutorial REAL AND GEOMETRICALLY UNSTABLE RESONATORS, with only small Fresnel-ripple deviations from this ideal wavefront.
The phase ripples in the real resonator, or the deviation from the ideal geometrical situation, will be sufficiently small compared to an optical wavelength that they will produce little deterioration in the far-field beam pattern compared to the uniform geometric situation.

Far-Field Output Beam Pattern
The far-field pattern from an unstable resonator laser, like the far-field pattern from a Casegrainian telescope, will no longer have a hole on axis as in the near field, but will have a central lobe on axis, surrounded by side lobes or diffraction rings of decreasing amplitude, as shown in Figures 4. and 5. The angular width of this central lobe will correspond more or less to the diffraction-limited far-field pattern for a slit or circular aperture corresponding to the outer width or diameter of the near-field pattern.

Because of the finite annular nature of the near-field pattern, however, the surrounding side lobes or rings will spread out over an angular width corresponding roughly to the width of the annulus itself.
A laser having low magnification and thus a narrow annulus will have a correspondingly broad distribution of side lobes or rings, with only a limited fraction of the total energy contained in the central lobe.
To obtain the narrowest overall far-field distribution for a given near-field aperture diameter, we thus want to use the largest possible magnification, so that the largest possible fraction of the near-field aperture is filled with radiation.
Some examples of the far-field beam patterns from an low-power unstableresonator \(\text{CO}_2\) laser, as observed using burn patterns on thermofax paper, are shown in Figure 5. Note that in the far-field patterns, the different patterns represent different exposure times, in order to bring out first the narrow central peak and then the surrounding circular ring patterns. Figure 6 illustrates burn patterns in lucite blocks from a somewhat higher-power \(\text{CO}_2\) laser, again illustrating the annular near field, including a central spot of Arago, and the (much overexposed) far-field pattern.
Advantages of Unstable Laser Resonators
The desire to operate at large magnification, in order to obtain a good far-field beam pattern, means that unstable resonators operate best at large output couplings. The simple hard-edged unstable resonator is thus best suited to laser oscillators that are characterized by large mode volume (in technical terms, large Fresnel number), and large round-trip gain, so that the oscillator can operate efficiently at large output coupling.
Unstable resonators are frequently employed on very high power or high energy lasers as well; but high gain is the primary criterion to make efficient use of the unstable resonator's useful characteristics.
For laser devices that may have large mode volume or large Fresnel number, but only low gain—for example, \(\text{cw}\) \(\text{CO}_2\) lasers—the situation is more difficult. The potential solutions in this situation seem to be to use a low-magnification unstable resonator and live with the output beam problem; or to use a folded multipass configuration through the gain medium in order to get the effective Fresnel number down and the gain up; or to use either a very long stable cavity or one adjusted perilously close to the stability boundary, in order to get increased stable gaussian spot size; or best of all, if possible, to use the variable-reflectivity-mirror techniques will describe in the future tutorial.
The advantages of the unstable resonator concept, when the necessary conditions are met, then include:
- Large, and controllable, mode volume.
- Controllable diffractive output coupling.
- Good transverse mode discrimination.
- Single-ended, all-reflective optics.
- Automatically collimated output beams
- Ease of alignment and adjustment.
- Efficient power extraction.
- Good far-field beam patterns.
Unstable resonators have to date found useful application in flash-pumped solid-state lasers, such as ruby and \(\text{Nd}\):\(\text{YAG}\) lasers; in flash-pumped and nitrogen-pumped dye lasers; in pulsed and \(\text{cw}\) \(\text{CO}_2\) lasers; in optically pumped sub-mm and far infrared lasers; in many very high power chemical and gasdynamic lasers; in many excimer lasers such as \(\text{KrF}\) and \(\text{XeF}\) lasers; in pulsed metal vapor lasers such as the \(\text{Cu}\) vapor laser; in optically pumped Raman lasers; and in semiconductor injection diode lasers.
Additional Discussion
The intuitive feeling of some laser researchers seems to be that an unstable resonator laser, perhaps because it is called "unstable," or because of its use of diffractive rather than transmissive coupling, is in some way special and fundamentally different from a more conventional laser using output coupling through a partially transmitting mirror.
It has sometimes been argued, for example, that an unstable resonator laser, especially one with large magnification, "cannot have axial modes." It may be worthwhile asserting, therefore, that an unstable resonator laser cavity having a certain percentage output coupling will not differ in any fundamental aspect of its behavior from any other kind of laser cavity having the same effective output coupling (except in the details of their respective mode shapes and volumes).
It is clear in principle—and has many times been verified experimentally— that applying either a stable or an unstable cavity resonator with the same percentage output coupling to the same laser medium will produce essentially the same total power output.
Unstable resonator lasers can be and have been internally modulated, Q-switched, mode locked, injection locked, and generally made to demonstrate the same properties as any other type of laser.
The unstable resonator, however, because of its much better transverse mode discrimination, will almost always produce a better far field beam profile and higher on-axis brightness in the far field than any other cavity design for those high-gain and large Fresnel number resonators to which it is best suited.
2. CANONICAL ANALYSIS FOR UNSTABLE RESONATORS
Our next important steps are to calculate the significant parameters and to calculate and examine some of the exact mode properties of unstable optical resonators, taking into account the strong diffraction effects that occur at the mirror edges.
Before examining any of these exact results, however, we will develop a simple but very general canonical formulation for analyzing almost all standard unstable resonator designs of interest.
We do this by transforming the round-trip Huygens' integral for a general unstable resonator into an equivalent collimated free-space form, and then examining the important physical parameters of this canonical model.
Huygens' Integral for Unstable Resonators
The objective here is to obtain a basic analytical formalism that will cover all but the most exotic forms of real unstable resonators, by considering a general unstable resonator with either a single output coupling mirror in the standing-wave situation, or a single output coupling plane or mirror in the ring situation.
Given the \(\text{ABCD}\) matrix for the full round trip around the cavity (see Figure 7), we can then define as usual the half-trace parameter \(m\equiv(A+D)/2\), with \(|m|>1\) for unstable resonators. This then leads to the round-trip geometric magnification \(\text M\) given by
\[\tag{3}M\equiv\begin{align}&\left\{m+\sqrt{m^2-1}\quad\text{positive}\;\text{branch},m>+1\\-m-\sqrt{m^2-1}\quad\text{negative}\;\text{branch},m<-1,\right.\end{align}\]
where \(|M|\) is also \(>1\) for unstable resonators.
For purposes of analysis, let us choose a reference plane \(z_0\) that is located just inside the output mirror or just before the coupling aperture, going in the outward direction; and then consider the propagation to a reference plane \(z_2\) at the same location one period or round-trip later, as shown in the bottom plot of Figure 7.
If we assume that the output mirror or coupling aperture has a finite width of 2a in one transverse dimension, we can then write the round-trip Huygens' integral for the resonator in that one transverse direction in the form
\[\tag{4}\tilde{u}_2(x_2)=\sqrt{\frac{j}{B\lambda_0}}\int^a_{-a}\tilde{\rho}(x_0)\tilde{u}_0(x_0)\text{exp}\left[-j\frac{\pi}{B\lambda_0}(Ax^2_0-2x_2x_0+Dx^2_2)\right]dx_0.\]
If the output mirror or coupler, rather than being a simple hard-edged aperture, has some form of variable reflection or transmission, we can take this into account by including the transmission function \(\tilde{\rho}(x_0)\) which multiplies the input function \(\tilde{u}_0(x_0)\) inside the integral, as in the orthogonality discussion in the previous tutorial.
In a real circular or other two-transverse-dimension resonator we will have to write the Huygens' integral in both transverse dimensions, and take proper

account of the actual transverse shape of the output mirror or coupler, whether it is square, rectangular, circular, or some more complex shape. To simplify the notation, however, let us write out the expressions here in only one transverse dimension, assuming a simple aperture of half-width (or, if it is circular, of radius) equal to \(a\).
Canonical Formulation
We can now convert the Huygens' integral of Equation 4. into a general canonical form by a simple transformation which begins by writing the input and output waves in the forms
\[\tag{5}\tilde{u}_0(x_0)\equiv\tilde{v}_0(x_0)\times\text{exp}\left[+j\frac{\pi(A-M)x^2_0}{B\lambda_0}\right],\]
and
\[\tag{6}\tilde{u}_2(x_2)\equiv\tilde{v}_2(x_2)\times\text{exp}\left[-j\frac{\pi(D-1/M)x^2_2}{B\lambda_0}\right].\]
These transformations are physically equivalent to extracting out the spherical curvature of the unstable resonator modes, thereby converting the magnifying wavefronts into colhmated wavefronts at both the input and output ends of the

cavity. The Huygens' integral given in Equation 4. then becomes
\[\tag{7}\tilde{v}_2(x_2)=\sqrt{\frac{j}{B\lambda_0}}\int^a_{-a}\tilde{\rho}(x_0)\tilde{v}_0(x_0)\text{exp}\left[-j\frac{\pi}{B\lambda_0}(Mx^2_0-2x_2x_0+x^2_2/M)\right]dx_0.\]
But this form for Huygens' integral corresponds to propagation through a simple collimated telescopic system with a ray matrix of the form
\[\tag{8}\left[\begin{align}A&B\\C&D\end{align}\right]\equiv\left[\begin{array}&M&B\\0&1/M\end{array}\right]\equiv\left[\begin{array}&1&MB\\0&1\end{array}\right]\times\left[\begin{array}&M&0\\0&1/M\end{array}\right].\]
The overall system, with the spherical curvatures extracted out, can thus be factored into the matrix product of a zero-length telescope of magnification M, plus a free-space section of length MB, as indicated by the matrix product in Equation 22.8 and by the drawing in Figure 8.
To express this same point in another way, we can move the input reference plane from the location \(z_0\) just before the telescope to a magnified input plane \(z_1\) just after the zero-length magnification step, by making the change of variables \(x_1\equiv Mx_0\) and \(dx_1\equiv M\;dx_0\).
The Huygens' integral of Equation 7. can then be converted to the form
\[\tag{9}\tilde{v}_2(x_2)=\sqrt{\frac{j}{MB\lambda_0}}\int^{M_a}_{-M_a}\frac{\tilde{\rho}(x_1/M)\tilde{v}_0(x_1/M)}{M^{1/2}}\text{exp}\left[-j\frac{\pi(x_1-x_2)^2}{MB\lambda_0}\right].dx_1\]
The general Huygens' integral for traveling completely around an arbitrary unstable resonator has thus been converted into the form of a purely free space Huygens' integral, which operates through a distance \(\text{MB}\) and across a full width of \(2\text M_a\), and which is applied to a transversely magnified version of the input wavefunction \(\tilde{\rho}(x_0)\tilde{v}_0(x_0)\).
We can use this integral as a canonical form for making calculations on any type of unstable resonator, with any number of intracavity paraxial elements, which falls within the above assumptions.

Collimated Fresnel Number
The effective Fresnel number, call it \(N_c\), which characterizes the effective free-space propagation distance in this canonical model is then obviously given by
\[\tag{10}N_c=\frac{(Ma)^2}{MB\lambda_0}=\frac{Ma^2}{B\lambda_0}\equiv\text{collimated}\;\text{Fresnel}\;\text{number}.\]
This so-called collimated Fresnel number \(N_c\) will determine the amount of numerical work it takes to propagate a wave once around the resonator. It will also determine the number of Fresnel diffraction ripples that we can expect to see in the output wave across the output aperture of the unstable resonator.
It is thus a second primary parameter which, along with the magnification \(M\), characterizes any real hard-edged unstable resonator.
Low-Fresnel-number resonators \((N_c\leq 5\;\text{or}\;10\), say) can generally be handled numerically with reasonable computer programs, whereas large-Fresnel-number resonators \((N_c\geq 100\), say) are very difficult to handle in any sort of exact numerical analysis.
Unfortunately, many unstable resonator lasers of practical interest have collimated Fresnel numbers \(N_c\) that are as large as this, or even much larger.
The Converging (Demagnifying) Geometrical Wave Solution
The magnifying geometrical eigenwave solution for an unstable resonator, viewed in the canonical formulation, is clearly just a collimated plane wave which passes through the aperture (or equivalently bounces off the finite output mirror); is expanded transversely by the magnification \(M\), remaining planar; and then

propagates through distance \(\text{MB}\) back to the same reference plane, as shown in the top sketch in Figure 9.
This same canonical system also has, however, a demagnifying geometrical eigenwave (or equivalently a magnifying eigenwave going in the opposite direction), which in general is not collimated, but instead has the general form shown in the lower part of Figure 9.
We can obtain the parameters of this demagnifying solution from a simple geometric analysis as follows.
Consider a spherical wave of radius \(R_0\) viewed at the input reference plane zo in the canonical model. When we magnify or stretch such a spherical wave in the transverse direction by a magnification factor M, its radius of curvature is multiplied by \(M^2\), since the phase lag \(\Delta\phi(x)\) at the outer edge of the beam lag should stay the same before and after magnification (Figure 10). If we use \(R_1\) for the radius of curvature of the wave at the reference plane \(z_1\) after the magnification step, then we can write this phase lag at the beam edge in each case as
\[\tag{11}\Delta\phi_0(x)|_{x=a}=\frac{\pi a^2}{R_0\lambda}=\Delta\phi_1(x)|_{x=Ma}=\frac{\pi(Ma)^2}{R_1\lambda},\]
so that obviously \(R_1=M^2R_0\).
But after this spherical wave propagates on through the free-space distance \(\text{MB}\) in Figure 8., it should then come back to the same reference plane \(z_2\) with radius \(R_3\equiv R_0\), as given by
\[\tag{12}R_2=R_1+MB=M^2R_0+MB=R_0\]
Hence the radius of the demagnifying eigenwave in Figure 9(b) must be given by
\[\tag{13}R_0=-\frac{MB}{M^2-1},\]
where this is evaluated at the reference plane just before striking the output mirror or aperture. (The radius is negative because the wave is a converging wave.)
Aperture Coupling Between the Geometrical Eigenwaves
The reader should keep in mind that the magnifying and demagnifying geometrical eigenwaves shown in Figure 9. are, strictly speaking, eigensolutions only for an unbounded or purely geometrical unstable system—not for a real, hard-edged resonator in which finite beam diameters and edge diffraction effects must be included.
The dominant mode pattern in a real unstable resonator is, however, generally quite similar in its basic properties to the magnifying geometrical eigenwave.
Now, when this kind of magnifying wave strikes the edges of the output aperture, we can expect that spherical (or cylindrical) edge waves will be scattered from the aperture edges into all directions, as we have discussed in this tutorial Beam perturbation and diffraction.
Some of this edge-wave energy, in particular, will be scattered into a direction which matches up with the demagnifying eigenwave traveling on beyond the aperture (or if you like, some rays are scattered from the aperture edge into a direction which feeds directly into the demagnifying eigensolution).
We might hypothesize then that the wave energy scattered by the aperture edges from the magnifying eigenwave into the demagnifying eigenwave direction will at first demagnify down toward the "core" of the unstable system, coming closer to the axis on each pass around the system.
After a relatively few round trips, however, this energy will have demagnified down into such a small diameter beam that diffraction spreading effects will become very important.
After a relatively few round trips, therefore, this "demagnified energy" will be turned back outward by diffraction effects, and in fact will be converted back into the magnifying eigenwave direction.
If this simple edge-wave description has any validity, we might then guess that the relative phase angle with which the demagnifying wave is excited by the aperture edges and fed back into the primary magnifying wave may be quite significant in determining the mode behavior of a real unstable resonator.
Equivalent Fresnel Number
It turns out, in fact, that this relative phase shift between the magnifying and demagnifying geometrical eigenwaves at the aperture edge is very significant in real unstable resonators.
Suppose we calculate this relative phase shift, as shown in Figure 11, and then express it in terms of what has become known as the equivalent Fresnel number \(N_{eq}\) for an unstable resonator, through the definition
\[\tag{14}\left[\phi_\text{mag}(x)-\phi_\text{demag}(x)\right]_{x=a}=\frac{\pi a ^2}{R_0\lambda}\equiv N_{eq}\times 2\pi,\]
or
\[\tag{15}N_{eq}=\frac{a^2}{2R_0\lambda}=\frac{M^2-1}{2M}\frac{a^2}{B\lambda}=\frac{M^2-1}{2M^2}\times N_c.\]
This equivalent Fresnel number \(N_{eq}\) turns out to be a very important alternative parameter for the behavior of hard-edged unstable resonators—perhaps more important than the collimated Fresnel number \(N_c\) to which it is related.
Equation 15, although derived using the canonical model, is quite general. Exactly the same expression for \(N_{eq}\) can also be derived, for example, in any

specific resonator by using
\[\tag{16}\begin{align}\left[\phi_\text{mag}(x)-\phi_\text{demag}(x)\right]_{x=a}&\equiv\left(\frac{1}{R_b}-\frac{1}{R_a}\right)\frac{\pi a^2}{\lambda}\\&=\left(\frac{D-1/M}{B}-frac{D-M}{B}\right)\frac{\pi a^2}{\lambda}=N_{eq}\times 2\pi.\end{align}\].
A physical interpretation of this formula at the outer edge of the output mirror for a standing-wave unstable resonator is also shown in Figure 11(b).
Confocal Unstable Resonators
Design formulas for various kinds of unstable resonators in terms of the three parameters \(N_{eq}\), \(N_c\) and \(M\) are presented in Table 1. In particular, for any confocal unstable resonator (of either positive or negative branch) we can show that the matrix B parameter is given by \(B=(M+1)L/M\).
Because of the collimated output beam in this situation, it is also convenient to define an "outer Fresnel number" \(N_0\) for a confocal resonator based on the actual length \(L\) and the actual outer diameter \(2Ma\) of the resonator, by
\[\tag{17}N_0\equiv\frac{(Ma)^2}{L\lambda}=\text{outer}\;\text{Fresnel}\;\text{number},\]
where \(2a\) is the width or diameter of the output mirror, or of the hole in the output coupler. This outer Fresnel number is the quantity most directly related to the diameter and length of the real laser device. The relations between this outer Fresnel number and \(N_{eq}\) and Nc are then given in Table 1.
3. CABLE ASSEMBLY FIBER OPTIC TFOCA II, 100M Reel, Military Style.
To find the exact eigenvalues and eigenmodes for a hard-edged unstable resonator, we must solve the exact resonator integral equation (preferably after converting it into canonical form).
This can be done either by using a Fox-and- Li type numerical procedure or by one of several complicated analytical methods (most of which end up requiring extensive numerical calculations in any case).
We will describe in this section a few results of such calculations for typical hard-edged unstable resonators.
Exact Mode Losses
There are only a limited number of published results for unstable optical resonator modes in the literature, partly because, as we will see, the results are complex and somewhat difficult to present, and partly because successful unstable resonator design can usually proceed using only a general understanding of unstable resonator properties, without the need for detailed design curves.
Unstable resonator calculations are most often carried out either for one-transverse-dimension "strip" resonators, or for two-transverse-dimension circular-mirror resonators. Strip resonator calculations are considerably more common, probably because the Huygens-integral or Fourier-transform calculations that are involved seem simpler (though it is arguable whether they are really any simpler than the circular situation done right).
Square or rectangular-mirror unstable resonators can then be treated as the product of two crossed strip resonators of appropriate widths.
TABLE 1.
Unstable Resonator Formulas
Each of these cases refers to a two-mirror standing-wave cavity of length \(L\), with mirror radii of curvature \(R_1\) and \(R_2\) taken positive for mirrors concave inward toward the resonator, and with \(2_{a_1}\) and \(2_{a_2}\) being the mirror widths for strip resonators, or the mirror diameters for circular mirrors. The reference plane is just before the output mirror \(M_2\), and the basic Fresnel number \(N\) is defined by \(N\equiv a^2/L\lambda\) where \(2a\equiv 2_{a_2}\) is the width or diameter of \(M_2\).
a. Symmetric double-ended resonator, one-way pass
Mirror radii \(R_1=R_2=R\); mirror half-widths \(a_1=a_2=a\)
Half-trace parameter \(m=g=1-L/R\)
Positive branch: \(R<0\), \(m>+1\), \(M=m+\sqrt{m^2-1}\)
Negative branch: \(0<R<L/2\), \(m<-1\), \(M=m-\sqrt{m^2-1}\)
One-way ray matrix:
\(\left[\begin{array}&A&B\\C&D\end{array}\right]\left[\begin{align}&1-2L/R&L\\&-2L/R&1\end{align}\right]=\left[\begin{array}&2m-1&L\\2(m-1)/L&1\end{array}\right]\)
Fresnel numbers \(N_c=MN\), \(N_{eq}=\sqrt{m^2-1}N=[(M^2-1)/2M]N\)
b. Single-ended resonator, full round trip, output mirror \(M_2\)
Mirror radii \(R_1\), \(R_2\) arbitrary; mirror half-widths \(a_1=\infty\), \(a_2=a\)
Parameters \(g_1=1-L/R1\), \(g_2=1-L/R_2\); \(m=2g_1g_2-1\)
Positive branch: \(g_1g_2>1\), \(m>+1\), \(M=m+\sqrt{m^2-1}\)
Negative branch: \(g_1g_2<0\), \(m<-1\), \(M=m-\sqrt{m^2-1}\)
Round-trip ray matrix:
\(\left[\begin{array}&A&B\\C&D\end{array}\right]=\left[\begin{array}&M&(M+1)L/M\\0&1/M\end{array}\right]\)
Outer Fresnel number \(N_0=(Ma)^2/L\lambda\)
Fresnel number \(N_c=[M^2/(M+1)]N=[1/(M+1)]N_0\)
Fresnel number \(N_{eq}=[(M-1)/2]N=[(M-1)/2M^2]N_0\)

The earliest unstable resonator calculations were published by Fox and \(\text{Li}\) in a somewhat obscure conference proceedings, at a time before the general properties of the unstable class of resonators were recognized. As shown in Figure 12(a), Fox and Li found that as soon as the parameter \(g=1-L/R\) for a simple symmetric strip resonator was moved outside the stable region \(-1\leq g\leq 1\), the behavior of the resonator diffraction losses with Fresnel number \(N=a^2/L\lambda\) changed dramatically.
The round-trip diffraction losses, rather than decreasing rapidly with increasing \(N\), approached a roughly constant value, with a periodic ripple about this value.
Repeating these calculations, but plotting them against the equivalent Fresnel number \(N_{eq}\) rather than the elementary Fresnel number \(N\), as in Figure 12(b), later made it clear that iVeq rather than \(N\) is indeed the significant parameter in accounting for this periodic loss behavior.
Adding lines corresponding to the geometric round-trip loss values \(1-|\tilde{\gamma}_\text{geom}|^2=1-1/M\) (for the strip situation)? as we have done in these figures, also makes clear that the real unstable resonator diffraction losses are similar but by no means exactly equal to the geometrically predicted values.
Mode-Crossing Behavior
In addition, it soon became clear that what appeared to be cusps in the early Fox and \(\text{Li}\) loss curves were in fact "mode crossings," or values of \(M\) and Neq at which two different modes had exactly the same diffraction losses.
The loss curves at these points, when plotted versus \(N_{eq}\), passed through each other, so that what was previously the lowest-order or lowest-loss mode gave up that priority to a separate and distinguishably different mode.
These crossing points are sometimes referred to as "mode degeneracies." Only the magnitudes and not the phase angles of the eigenvalues are equal at these crossing points, however.
More detailed examination shows that the two complex eigenvalues are always far apart in the complex plane, so that whereas the eigenvalue magnitudes may "cross," the complex eigenvalues do not "intersect" or become degenerate in any true sense.
This behavior creates some difficulties in devising a sensible labeling scheme for the different eigenvalues, since the lowest-order mode at one value of \(N_{eq}\) is generally not the lowest-loss or dominant mode at other values.
This is usually resolved by switching indices so that the lowest-loss mode at any given value of \(N_{eq}\) is by definition the \(n=0\) mode.
Eigenvalues For Circular-Mirror Resonators
Figures 13. and 14. show some of the results from an extensive series of eigenvalue calculations carried out by the author and H. Y. Miller on circular mirror hard-edged unstable resonators for a wide range of magnifications and Fresnel numbers. (Note that these calculations plot the eigenvalue magnitude \(|\tilde{\gamma}|\) versus \(N_{eq}\) whereas the previous figures plotted the loss per bounce \(1-|\tilde{\gamma}|^2\).)
These calculations were carried out for modes with both zero and first-order azimuthal variations, i.e., assuming fields of the form \(\tilde{u}_{pl}(r,\theta)=\tilde{u}_p(r)\times e^{jl\theta}\) with \(l=0\) and \(l=1\).
These results demonstrate that the periodic crossing of the eigenvalues near integer values of \(N_{eq}\) continues indefinitely, up to the largest values of \(M\) and \(N_{eq}\) that could be handled with the available computational resources.
At low Fresnel numbers, which correspond to smaller mirror diameters, only a very few radial modes fit within the unstable resonator, or at least the computations indicate that all higher-order modes have such high losses that their eigenvalues cannot be seen numerically.
As the Fresnel number increases, additional modes come up out of the high-loss region, with each such mode eventually becoming the lowest-loss mode and then oscillating and perhaps recurring as the dominant mode at irregular repetition periods.
The phase angles of the eigenvalues obviously rotate continuously through increasing negative multiples of \(2\pi\) as the Fresnel number increases.
This indicates that if we follow the trajectory of any one eigenvalue in the complex plane, each eigenvalue rotates continuously about the origin in a roughly circular orbit with increasing Fresnel number.
The dominant eigenvalues, however many of them there are, are always spaced by roughly equal angles in the complex plane.
The Half-Integer "Anti-Crossing Points"
Those values of \(N_{eq}\) halfway between the mode crossings, that is, near half-integer values of equivalent Fresnel number in Figures 13. and 14, might


seem to be the optimum operating points for an unstable resonator laser, since they combine the largest discrimination between lowest and higher-order modes with the lowest diffraction losses for a given magnification \(M\) (which permits operating at a larger magnification for a specified coupling value).
This observation is largely true, but has some qualifications, as will be pointed out later.
The general observation is that for circular-mirror unstable resonators the mode crossings and the intervening maximum separation points occur very near

to values of \(N_{eq}\approx k\) and \(k+1/2\), with \(k\) integer, whereas for strip resonators these points occur nearer to \(N_{eq}\approx k+7/8\) and \(k+3/8\). This difference is almost surely due to the fact that a strip resonator actually has different Fresnel numbers along different azimuthal directions, as pointed out in this tutorial BEAM PERTURBATION AND DIFFRACTION in connection with slit diffraction effects.
Output Coupling Approximations
The previous results and many similar calculations all show that the resonator diffraction losses or output coupling values at the eigenvalue peaks near the half-integer Fresnel numbers are substantially smaller than predicted by the simple geometric theory; and in fact the diffraction losses for the lowest-order mode at any value of \(N_{eq}\) are almost always less than or at most equal to the geometric value.
This represents, as we have noted earlier, the strong ability of optical resonator modes to "pull in their skirts" or to shape their eigenmode profiles so as to minimize diffraction losses out the sides of the resonator.
We have already noted that the geometric eigenvalue for a circular resonator is given by \(\tilde{\gamma}_\text{geom}=1/M\).
A purely emperical formula says that the eigenvalue magnitude near the larger half-integer peaks in the same situation is given approximately by
\[\tag{18}\tilde{\gamma}_\text{peak}\approx\sqrt{\frac{2M^2-1}{M^4}}\]
as shown in Figure 15. The peak near the lowest value \(N_{eq}\approx1/2\) generally has a value even slightly larger, or losses even slightly smaller, than this.

Exact Mode Patterns
Figures 16. and 17. illustrate some of the complicated forms taken on by the lowest-order transverse eigenmodes in a one-dimensional strip resonator for different values of the magnification \(M\) and equivalent Fresnel number \(N_{eq}\). The corresponding mode patterns for circular-mirror resonators would be equally if not more interesting, but few detailed calculations for circular-mirror patterns can be found in the literature.
The general features will always be very much the same for either the strip or the circular mirror.
The results shown in Figures 16. and 17. come from early calculations, and may contain some minor inaccuracies, but they correctly illustrate the general features of the unstable resonator lowest-order modes. For each case shown, the top figure illustrates the intensity profile across the strip resonator just inside the output mirror, with the vertical lines indicating the edges of the mirror. The next plot down then shows the phase angle of the same wavefront versus the transverse coordinate.
The complex field amplitude falling outside the vertical bars or mirror edges in these figures determines the near-field pattern that will be coupled out of the resonator past the mirror edges.
The third plot indicates the far-field beam pattern versus beam angle that will be produced (in one transverse dimension) by

the out-coupled beam pattern. This pattern is essentially the Fourier transform of the out-coupled near-field pattern. Finally, the bottom plot shows the cumulative or integrated power ("power in the bucket") within a given far-field angle.
These plots taken all together illustrate most of the primary features of unstable resonator modes:
(1) The overall shape of the mode inside the resonator is somewhere between roughly triangular and roughly gaussian for low Fresnel numbers \(N_{eq}\leq1)\), changing over to a generally squarish shape for large Fresnel numbers \(N_{eq}\gg1)\), with the intensity pattern extending out to roughly \(M\) times the mirror diameter in each case.
(2) Superimposed on this basic shape are complex patterns of Fresnel ripples, which become increasingly complex and contain increasingly high spatial frequencies with increasing values of the collimated Fresnel number \(N_c\). In many cases, especially for \(N_c\gg1\), one can see that there are just about \(N_c\) large-scale ripples across the full magnified width of the resonator, together with significant amounts of much higher-frequency but small-amplitude ripples superimposed on this.
(3) The near-field transverse phase variation shown in the second plot in each group is the transverse phase variation as calculated for a confocal resonator, or equivalently for the canonical formulation. This phase variation represents

therefore the phasefront difference between the exact unstable eigenmode and the spherical wavefront of the diverging geometrical eigenwave. It is evident that this deviation is small in all situations; i.e., the actual lowest-order wavefront is very close to the purely geometric eigenwave.
The finite average value of the phase angle in certain situations simply represents the finite phase angle of the resonator eigenvalue \(\tilde{\gamma}\), which has not been subtracted out of these plots.
(4) The far-held plots make clear the primary weakness of low-magnihcation unstable resonators: when the near-held output beam is a comparatively narrow annulus, or two comparatively narrow slits as it is here, the far-held pattern contains a large number of side lobes, with only a small fraction of the total beam energy in the centermost lobe.
Note that in all the far-field plots the beam angle is normalized to the diffraction angle \(\lambda/2Ma\) characteristic of the full magnified outer width of the near-field pattern; and the first zero of the centermost lobe always occurs at roughly \(\theta\approx\lambda/2Ma\).
The first break point or change in slope in the integrated power curve indicates the fractional amount of the total output power that is contained in the central lobe. Note that, for example, this central lobe only contains \(\approx\)15% of the total power for \(M=1.42\), rising to \(\geq\)70% for \(M=10\).
Also note how little the far-field patterns actually depend on the equivalent or collimated Fresnel numbers—for any given magnification \(M\) the far-field patterns, and especially the "power in the bucket" curves, are essentially independent of \(N_{eq}\) or \(N_c\).
Higher-Order Modes
Calculations of higher-order unstable resonator modes are also very sparse in the literature. Figure 18 shows one example of the three lowest-order symmetric and antisymmetric modes for a very low Fresnel number strip resonator, as calculated using in this case an expansion of the one-dimensional Huygens kernel in linear prolate functions.
We can readily see in Figure 18 how the lowest-order or \(n=0\) mode has pulled in its fields so that its diffraction losses are substantially less than would be predicted by the geometric theory. The higher-order modes, by contrast, have substantially larger diffraction spread, and thus greatly increased diffraction losses for the \(n=1\) and \(n=2\) modes.

Loaded Resonator Calculations
The question repeatedly arises as to how the modes of a laser cavity will be modified by the effects of a spatially varying gain within the resonator, as well as the effects of gain saturation in the laser medium. To explore this question, Figure 19 shows the oscillation mode patterns calculated for two of the same unstable resonators as in Figure 13 and 14, assuming that a thin saturable "gain sheet" is pasted on the surface of mirror #1 (the back mirror) in the unstable resonator.
In the Fox and \(\text{Li}\) procedure used for these calculations, the circulating field is multiplied by this gain sheet on each round trip; and then the gain itself is saturated in a homogeneous fashion by the local intensity at each point across the mirror before calculating the next round trip.
The resonator fields are then found to converge, after only a very few round trips in most unstable situations, to a self-consistent transverse field and transverse gain pattern which presumably represents the actual oscillating mode that would develop in such a laser.
By comparing these "loaded resonator" results with the corresponding "bare resonator" modes in Figures 13 and 14, we can see that the general character of the mode is very little changed, the primary difference being that some of the higher peaks are pushed down in amplitude by the local gain saturation that they produce in the loaded resonator.
This seems to be true in most loaded resonator calculations: transverse gain variations and gain saturation have only minor effects on the mode patterns, although any kind of local transverse phase variations will have large effect on the mode patterns and on the far-field beam spread in particular.
4. UNSTABLE RESONATORS: EXPERIMENTAL RESULTS
Detailed experimental studies and comparisons with theory are also surprisingly sparse for unstable resonators, probably due to several reasons. First, many of the development efforts on unstable resonators were focused on very large highpower chemical and gasdynamic lasers in the infrared, where careful diagnostics are technically difficult; and because of the nature of these development projects emphasis was focused on meeting project specifications rather than on detailed exploration of the unstable resonator itself.
In addition, there does not seem to be available any convenient, large-bore, high-gain, continuously operating visible laser which could serve as a test bed for careful unstable resonator studies.
The general observation nonetheless is that the experimental performance of unstable resonators agrees very well with theoretical calculations, with no significant disagreements between theory and experiment being found. We will

review briefly in this section a few of the more significant experimental studies that have been performed on unstable resonators.
Low-Power Unstable Resonator Experiments
Perhaps the most careful and detailed such experimental study was that performed by Freiberg, Chenausky and Buczek at the United Technologies Research Center on a low-power \(10.6\mu m\) \(\text{CO}_2\) laser, with results as shown in Figure 20.
As shown in part (a) of Figure 20, a thin beam splitter was placed inside the laser cavity so that it reflected out a few percent of the circulating intensity just before the output scraper mirror. Insertion of this beam splitter, together with appropriate imaging optics and a scanning \(\text{IR}\) camera, made it possible to study the circulating mode pattern inside the laser cavity, as well as the near-field and far-field beam patterns coupled out of the cavity.
Figure 20(b) shows two typical examples of the internal circulating mode pattern and the near-field beam pattern coming from the scraper mirror, for two different magnifications and (small) values of equivalent Fresnel number. The general similarity between these results and the low-Fresnel-number calculations of the previous section is evident.

A typical measurement of integrated far-field energy versus far-field beam angle is shown in Figure 20(c). The design value for this resonator was a radial magnification of \(M=1.29\). Both exact calculations and near-field experiments showed, however, that the near-field pattern was closer to a linearly tapered annular radial variation falling to zero at approximately \(1.36\) rather than \(1.29\) times the coupling hole diameter.
The far-field experimental results have been compared, therefore, to the theoretical far-field patterns for both a uniformly illuminated annular aperture corresponding to \(M=1.29\), and a linearly tapered (but uniphase) annular aperture with \(M=1.36\). It is evident that the farfield beam pattern is not greatly sensitive to the difference between these two distributions, with the actual experimental values falling neatly in between.
Finally, by measuring carefully with an optical power meter both the circulating power inside the cavity just before the coupling mirror and the actual out-coupled power, it was possible to measure experimentally the actual diffraction coupling from the unstable resonator laser under varying experimental conditions. Part (d) of this figure demonstrates the extent to which these measurements agree with the periodic loss variation expected for the unstable resonator near \(N_{eq}=1\).
Measurements on Higher-Power Lasers
Experimental measurements on higher-power unstable resonator lasers have usually been limited to near-field intensity profiles or burn patterns, which are generally rather uninformative; to measurements of total power or energy output, which generally yield the expected power output for the laser in question; and to observations of the far-field beam patterns, which almost always give something very close to the expected pattern of an intense near-diffraction-limited central spot with surrounding diffraction rings or lines.
Figure 21 shows, for example, the variation along one transverse direction of the far-field burn pattern of a very large \((\approx 500\;\text{J/shot})\) electron-beam-ionized \(\text{CO}_2\) \(\text{TEA}\) laser having a rectangular laser medium \(\approx 1\) m long and 15 by 20 cm in cross section, using an unstable resonator with a square output mirror of width \(2a=9.5\) cm and a magnification \(M=1.58\).
The agreement between theory and experiment for the width of the central lobe and the spacing of the diffraction side lobes is very good, considering the nature of this type of experiment. Many other generally similar experimental results for other unstable resonator lasers can be found in the literature.
Large-Scale Unstable Resonator Simulations
We might also describe briefly here some of the very large-scale numerical simulations or "computer experiments" that have been carried out for unstable resonator lasers. These calculations are in essence extended versions of the Fox and \(\text{Li}\) method, in which an optical signal is propagated through repeated round trips inside an unstable resonator, taking into account both the optical beam propagation, as modified by the resonator mirrors, the laser gain medium, internal phase perturbations, and other effects, and also the effects of the optical signals back on the laser medium and the resonator parameters, including gain saturation and repumping, heating and distortion of the laser medium, possible mirror distortions, and other nonlinear effects.
These simulations thus require both large-scale optical codes which propagate the circulating optical field from plane to plane within the unstable resonator, and also atomic, molecular and possibly chemical codes which calculate the nonlinear responses of the gain and phase media inside the resonator.
The usual procedure is to divide the laser resonator into several sections in the axial direction and to lump the net laser gain and phase distortion occurring in each section into a thin discrete "gain and phase sheet."
The optical wave is propagated through a free-space length equal to one section, and then multiplied by the lumped gain and phase shift associated with that section, through one complete round trip; after which the gain and phase for each section is recalculated based on the optical intensities in the previous round trip.
Figure 22 illustrates the kind of results that can be obtained from such a simulation. The plots in part (a) show the normalized intensity and phase profiles just inside the output coupler for the lowest-order eigenmode in an empty circular-mirror unstable resonator with \(N_{eq}=1.5\) and \(M=2.5\).
The circular beam pattern and flat phase front of the mode are evident. The successive plots then show the results of adding a transversely flowing saturable gain medium; flowing gain plus internal phase perturbations from two intra-cavity shock waves; and flowing gain plus a small mirror tilt.
Including the saturable gain clearly produces a large change in the mode intensity profile but only minor changes in the phase profile. Adding to this the phase perturbation due to the weak shock waves causes further severe amplitude distortion, but still only relatively minor phase distortion. Indeed, the far-field beam patterns for all three of these situations are essentially identical, despite the large differences in the near-field intensity patterns.
The primary effect of mirror misalignment is to tilt the output wavefront while leaving it still essentially planar. The far-field beam spot in this situation is steered to one side, but otherwise is essentially unchanged.
We have pointed out earlier that it may take a large number of sample points (perhaps on the order of \(8N_c\) points in each transverse direction) to handle adequately even the undistorted bare-resonator modes of an optical resonator. This number can be substantially increased by higher-spatial-frequency distortions within the resonator, and by the necessity to provide adequate "guard bands"

outside the resonator edges. The numerical work required in these simulations can thus become very substantial. Fast transform algorithms (fast Fourier or Hankel transforms) seem clearly the optimum way to handle the optical propagation steps.
On the fortunate side, the general experience is that large-scale simulations of this type appear to converge in only a small number (e.g., 5 or 10) of round trips for most unstable resonators, presumably because of the large transverse mode discrimination associated with unstable resonators.
