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MORE ON UNSTABLE RESONATORS

This is a continuation from the previous tutorial - Unstable optical resonators

Unstable resonators, as a consequence of both their practical utility and their complex analytical properties, have stimulated many clever extensions as well as detailed mathematical analyses. In this tutorial we review briefly some of these more advanced analytical techniques and inventions, and then introduce what may eventually become the most useful form of unstable resonator, namely, the "soft-edged" or gaussian variable reflectivity type of geometrically unstable resonator.

 

 

1.  ADVANCED ANALYSES OF UNSTABLE RESONATORS

We first introduce in this section several of the more advanced analytical techniques for treating unstable resonators, along with some more advanced concepts and design ideas related to unstable resonator modes. 

 

Advanced Analytical Techniques for Unstable Resonators 

The purely geometric analysis of the unstable resonator, leaving out diffraction or wave-optical aspects, is very simple and provides an excellent zero-order approximation to the wavefronts and mode patterns of the unstable resonator.

Numerical solution of the exact integral equation, by contrast, provides an effective but somewhat expensive method for predicting the exact eigenvalues and eigenmode properties of unstable resonators. 

We might then attempt to develop some sort of approximate analytical extension to the geometric analysis, putting in diffraction effects to a first-order level of approximation, so as to obtain more accurate answers than in the geometric limit, but in a simpler and more general fashion than purely numerical calculations. We might expect that such a first-order solution would become increasingly valid for larger values of equivalent Fresnel number \(N_{eq}\), or for vanishing values of the optical wavelength \(\lambda\). 

Success in finding any such approximation technique which is simultaneously simple and reasonably accurate has been limited, however. The wave-optical properties of the unstable resonator seem to be inherently complex, so that, for example, the complicated ripple structure of the eigenmodes and the crossing

 

 

FIGURE 1.  An optical resonator, with wave propagation in the \(z\) direction, can also be viewed as a waveguide, with very-high-order propagation in the \(x\) direction.

behavior of the circular-mirror eigenvalues persist even to very large Fresnel numbers \(N_{eq}\), in much the same way that the Gibbs phenomenon in a square-wave Fourier series approximation persists no matter how many terms are kept in the Fourier expansion. 

There are, however, a few analytical techniques developed for unstable resonators which we can review briefly, in part because they can be used to obtain useful results, and in part because of the insight they can yield into unstable resonator behavior.

These techniques include the so-called coupled-mode analysis and several versions of the asymptotic or virtual-source analysis.

 

The Coupled Mode or Transverse Waveguide Approach

To introduce this interesting and quite different form of unstable resonator analysis, we can consider the wave propagation in both directions along a finite open-ended length of a conventional microwave waveguide, as shown in Figure 1(a). For simplicity let us consider only a two-dimensional strip waveguide formed by two flat metallic planes parallel to the \(x,y\) plane, as in Figure 1(a). We view this waveguide as having finite length \(2a\) in the \(x\) or propagation direction, height \(L\) in the transverse or \(z\) direction, and (in the simplest situation) no variations in the \(y\) direction. 

From the conventional waveguide viewpoint the propagation direction in this waveguide is the x direction; and we can view the internal energy as being reflected back and forth between the open ends of the waveguide, which have a finite but not \(100\%\) reflectivity.

The waveguide section is thus a kind of leaky waveguide cavity, with energy loss out of both ends.

Suppose this waveguide is operated in an extremely high-order transverse mode, at a frequency very close to the cutoff frequency for that mode. It is well known in waveguide theory that the fields in the guide will then be made up essentially of plane wave components that are traveling at nearly \(90^\circ\) to the nominal propagation direction or \(x\) axis, with their individual \(k\) vectors nearly parallel to the \(z\) rather than the \(x\) axis. These nearly transverse plane waves will combine in such a fashion as to satisfy the transverse boundary conditions at the two metallic planes.

In this limit the electromagnetic energy is essentially bouncing back and forth between the two planes and only traveling very slowly in the \(z\) direction, in agreement with the well-known fact that the axial group velocity in a waveguide goes to zero at cutoff.

But from an optical-resonator viewpoint, this same truncated waveguide section then looks very much like a very short, very wide optical resonator with its width in the \(x\) direction and its length or axial direction in the \(z\) direction, as in Figure 1(b). This same structure as seen from the usual optical-resonator viewpoint has propagating waves which bounce back and forth between the two planes in the \(z\) direction, with some leakage out the open sides in the \(x\) direction.

In fact, the same identical electromagnetic fields can equally well be described either from the waveguide viewpoint as made up of very high-order modes traveling (with very low group velocity) in the \(\pm x\) directions, with leakage out the open ends of the structure; or from the optical-resonator viewpoint as optical waves traveling in the \(\pm z\) direction with leakage out the open sides of the structure.

 

Coupled Mode Analysis of Unstable Resonators

A planar optical resonator can thus be analyzed by treating it as a waveguide problem with propagation in the transverse direction; expanding the fields in very high-order \(x\)-directed waveguide modes; and then properly evaluating the mode reflectivity, mode conversion, and leakage output which takes place at the truncated open ends of the waveguide.

This approach is known as the coupled mode approach to optical-resonator theory. Evaluating the complex end reflectivity for an open-ended waveguide is one of the more complicated aspects of this approach. 

Conversion from a planar to an unstable (or stable) resonator is accomplished simply by curving the transverse planes as shown in Figure 1(c), and then analyzing this situation as a variable-height waveguide. If the waveguide height increases going toward the ends, as shown, this models an unstable resonator; if the height decreases it models a stable resonator.

If such a structure is viewed as a pair of curved strips with finite width \(2a\) in the \(x\) direction and finite average separation \(L\) in the z direction, then it obviously corresponds to a strip optical resonator. If the curved surfaces are viewed instead as circular disks with radius a in the \(y\), \(z\) plane and spacing \(L\) in the \(z\) direction, then this is a circular-mirror resonator, in which the waveguide modes must be radial waveguide modes, but such modes are also well-known in waveguide theory.

This coupled-mode waveguide approach to optical resonators has been successfully applied to planar, stable and unstable resonators by several groups in both the U.S. and USSR. It may seem bizarre to analyze the propagating modes in a "waveguide" where the waveguide structure is, say, \(5\;\text{mm}\) long, 1 meter wide, and operates at a frequency of \(5\times10^5\;\text{GHz}\); but there is no formal objection to doing so, and in fact the approach does work.

The one difficulty is perhaps that the analysis is fairly complicated; and in the end we must do nearly as much numerical work to obtain numerical answers as if we simply did a brute-force Fox and \(\text{Li}\) calculation.

 

Asymptotic Analysis of Unstable Resonators

A second analytical approach to unstable resonators, commonly called the asymptotic approach, is closer in spirit to the opening paragraphs of this section. It involves expanding the Huygens' kernel in the resonator integral equation in inverse powers of the Fresnel number \(N\), or the equivalent Fresnel number \(N_{eq}\), and then evaluating the Huygens' integral using techniques from the method of stationary phase.

The resonator eigenequation is then converted into a polynomial equation whose order depends on the number of terms retained in the expansion. Solving this polynomial leads, after some further algebra, to results for both the eigenvalues and eigenfunctions of the unstable resonator. 

The asymptotic approach was developed first in rectangular coordinates, for application to strip resonators. It has since been extended to circular resonators using Bessel function terms in the expansion. It provides a convenient if somewhat tricky way of obtaining solutions at larger Fresnel numbers, with a domain of validity that in fact seems to extend down to Fresnel numbers even as small as unity.

 

Virtual Source Theory for Unstable Resonators

The most recent, and perhaps most useful, analytical approximation for the unstable resonator is the virtual source theory, which is a modified version of the asymptotic approach. In this analysis, one envisions standing, say, just inside the output coupler of an unstable resonator (or just before the aperture plane in an equivalent unstable lensguide), looking back into the resonator or lensguide.

One then sees a series of repeated images of the output aperture, with different sizes and at different distances, corresponding to the real output aperture seen after multiple internal reflections in the resonator or multiple periods back down the lensguide. 

The field across the output aperture in this model is then approximated by the sum of the geometric field of a plane wave coming from a distant aperture located \(\mathcal{N}\) round trips earlier, where \(\mathcal{N}\) is some moderately large integer, plus the \(\mathcal{N}\) individual edge waves coming from that aperture and all the closer apertures, as seen looking backwards down the equivalent lensguide from the output aperture.

Since each of these apertures is seen looking back through a different number of round trips, each such aperture seems to be located at a different distance and with a different transverse magnification.

If the number of terms \(\mathcal{N}\) is made large enough, this combination of the plane wave plus aperture-edge terms becomes a very good analytical approximation for the real field in the resonator, and one can then manipulate this function to obtain a polynomial equation of order \(\mathcal{N}\), which then gives the actual eigenvalues and eigenmodes.  

 

FIGURE 2.  In rectangular (but not circular) unstable resonators, the lowest-loss eigenvalue will often separate from the remaining intertwined eigenvalues above a certain magnification-dependent value of \(N_{eq}\).

A suitable number of terms seems to be 

\[\tag{1}\mathcal{N}\approx\frac{1\text n(KN_{eq})}{1\text n\;M},\] 

where \(K\) is a constant on the order of a few hundred, and perhaps smaller.

The virtual source technique has been demonstrated first for strip resonators, where the Fresnel-integral edge-wavefunctions can be calculated with relative ease, and more recently for circular-mirror resonators as well.

Although the virtual source technique is similar in many ways to the asymptotic approach, it has the considerable advantage that we can attach direct physical significance to each individual term in the mathematical formulation.

 

Mode Separation Behavior

One striking property of rectangular (but not circular) unstable resonators, which was observed in early numerical calculations and has been confirmed in more recent analyses, is that above some fairly large magnification-dependent value of \(N_{eq}\) the eigenvalue for the lowest-loss eigenmode separates off from the other eigenvalues, and no longer continues to exhibit the mode crossing behavior at larger Fresnel numbers, as shown for a typical example in Figure 2. The diffraction loss for the lowest-loss mode does continue to exhibit a periodic dependence on \(N_{eq}\), however, leading to what is often referred to as "cusping behavior," as illustrated in Figure 2. 

Development of the asymptotic analysis for unstable resonators made it possible to explore this mode separation behavior at large Fresnel numbers in more detail. An approximate expression for the critical value of \(N_{eq}\) above which mode separation will occur in strip resonators has been found to be

\[\tag{2}N_{eq}(\text{crit})\approx\frac{11.5}{(1\text n\;\text M)^3}.\]

Even above this value of \(N_{eq}\), however, it is observed that the lowest eigenmode may separate off and remain separated for only 6 or 7 cusping points, after which this mode may drop down in eigenvalue magnitude, and a new lowest order eigenmode may emerge at a single isolated crossing point. For still larger \(N_{eq}\) values there will be a further series of regular cusping points; and so on.

No evidence of any similar mode separation behavior has ever been seen, however, for circular-mirror hard-edged unstable resonators at any values of \(M\) or \(N_{eq}\); and the asymptotic analysis for circular-mirror resonators indicates that the periodic crossing of the eigenvalue magnitudes should apparently continue forever in them.

It seems clear that this infinite oscillation is unique to circular mirror resonators, and is closely related to our earlier discussions in this tutorial APERTURE DIFFRACTION: CIRCULAR APERTURES regarding the unique diffraction properties of circular apertures.

Only for a perfectly circular mirror or aperture is the Fresnel number independent of azimuth, so that the edge wave contributions from the entire aperture edge will always add in phase on the system axis or resonator axis.

 

Eigenmode Behavior at Mode Crossing Points

Another odd aspect of unstable resonator mode behavior brought out by more detailed mode studies is that the eigenmode patterns of the two lowest loss symmetric eigenmodes in a strip resonator appear to become nearly if not exactly identical at the mode crossing points. At a Fresnel number slightly below a crossing point, for example, the lowest-loss \((n=0)\) mode pattern will generally have more of its intensity concentrated toward the center of the resonator and less in the outer wings, so that the diffraction loss is indeed smaller than the geometric value; whereas the next higher-order \((n=2)\) symmetric mode will have smaller intensity near the center and increasing intensity toward the edges. 

As the Fresnel number is adjusted toward a crossing point, however, both mode patterns will change so that at the crossing point they become virtually identical, even to fine details of the Fresnel diffraction structure. On the other side of the crossing point the two mode patterns diverge again, more or less exchanging shapes.

This same behavior is also observed at Fresnel numbers above the mode separation limit, where mode crossing no longer occurs. The mode patterns again become very similar at the "cusping points" where the two lowestmode loss values approach each other, even if they no longer intersect.

Several examples of this behavior are shown in Figure 3, which illustrates the \(n=0\) and \(n=2\) lowest-order mode patterns for a strip resonator as calculated using the asymptotic analysis introduced by Horwitz. The mode intensity patterns are plotted from the center to the mirror edges for a magnification \(M=2.9\) and various equivalent Fresnel numbers corresponding to a maximum separation point, a mode crossing point, and a cusping point.

If the two lowest-order mode patterns at a mode crossing point are generally similar (and if both have reasonbly good beam quality), then this result implies that it may not be all that essential in unstable resonator design to aim for a half integer equivalent Fresnel number so as to obtain maximum loss discrimination between modes. Even if the resonator operates at a crossing point, and oscillates

 

 

FIGURE 3.  Whenever the two lowest-order modes in a strip unstable resonator come near a "crossing" (or "cusping") point, the mode patterns of the two eigenmodes become nearly identical.

simultaneously in two modes, the far-field beam quality may not be significantly deteriorated by the multi-transverse-mode operation. 

 

Oscillation Build-Up in Pulsed Unstable Resonators

Unstable resonators can be particularly useful for very high-gain, large volume pulsed lasers, in some of which (such as the pulsed \(\text{Cu}\) vapor and other metal vapor lasers) the full duration of laser action may only include a few round trips \((\leq10)\) within the laser cavity. It is then important to understand how rapidly a good transverse mode pattern can be built up within an unstable optical resonator. 

The difference in round-trip diffraction losses between the lowest-order mode (or modes) and all the higher-order modes in an unstable resonator is, fortunately, usually quite large.

As a result, when laser oscillation begins to build up from noise in a high-gain unstable resonator, the lowest-order mode, or possible a few lowest-order modes, will rapidly outstrip all the higher-order and higher loss transverse modes; and the desired lowest-order mode pattern will be fully established typically within only a few round trips.

An approximate formula for the number of round trips \(\mathcal{N}_{rt}\) needed to establish a predominantly lowest-order mode distribution in a high-gain unstable resonator of mirror half-width a and length \(L\) is given by Anan'ev as

\[\tag{3}\mathcal{N}_{rt}\approx\frac{1\text n(a^2/L\lambda)}{1\text n\text(M)}.\] 

The reader may recognize this from our earlier discussions as essentially the number of round trips that would be required for spontaneous emission coming into the resonator with wavefront equal to the converging eigenwave to be focused down and then converted by diffraction back out into the diverging or magnifying eigenwave.

Experiments and numerical simulations by Jones and Perkins demonstrate the general validity of this approximate expression in Equation 3. It can thus be used as an estimate for the magnification M needed to ensure that reasonable mode discrimination will be established within the number of round trips n that will be possible in a very short pulse unstable resonator laser.

 

 

2.  OTHER NOVEL UNSTABLE RESONATOR DESIGNS

Unstable resonator designs have been modified in many clever ways in various attempts to expand their usefulness. In this section we discuss briefly some of the novel approaches that have been suggested and tried. 

 

Ring Unstable Resonators 

Ring unstable resonators, as contrasted to linear unstable resonators, provide much increased design flexibility and a number of design possibilities over and above the advantages possessed by ring resonators for laser applications generally. Several of these aspects are illustrated by Figure 4. 

A ring unstable resonator can be designed, for example, to have a short telescopic magnification section using conveniently available optical elements with short radii of curvature, and then to have much longer collimated regions which can fill large diameter laser tubes.

Negative-branch ring resonators can also be built with internal spatial filters, which can clean up the mode patterns and filter out some of the phase distortion effects caused by inhomogeneous elements in the resonator.

Ring resonators also offer the possibility of unidirectional oscillation, which can eliminate spatial hole burning effects and make injection locking easier. As we have previously pointed out, the bare cavity modes going in opposite directions around a ring resonator will always have exactly the same inherent diffraction losses.

The transverse spatial profiles for the modes going in opposite directions can be quite different, however, and this can make a substantial difference in how the mode fills a given gain medium or saturable absorber medium in the two directions. The effects that this will have on the mode competition between the two directions in an oscillating laser can be fairly subtle; but this does offer one approach toward achieving unidirectional oscillation in a ring laser.

Ring lasers are often designed using various sorts of folded or "\(Z\)" shapes in order to hit at least some of the mirrors at close to normal incidence, since this minimizes astigmatism from curved mirrors and permits standard mirror coatings to be used. Figure 4(d) shows one typical example of a folded ring laser cavity, here for a nitrogen-pumped pulsed dye laser. Folded rings of this sort also minimize the amount of real estate needed for a given cavity length.

 

 

FIGURE 4.  Examples of ring unstable resonator designs, (a) Unstable ring laser with short telescope section and long expanded regions, (b) Negative-branch unstable resonator with internal spatial filter, (c) Ring cavity with different interaction mode volumes in opposite directions, (d) Example of a ring dye laser design including an intracavity diffraction grating.

 

Self-Imaging Unstable Resonators 

It is possible to design a negative-branch ring unstable resonator (and only a ring resonator) such that each round trip corresponds to an image relay system, which images a magnified version of the coupling aperture back onto itself after each round trip. In the negative-branch confocal ring resonator shown in Figure 5, for example, the round-trip \(\text{ABCD}\) matrix can be written as 

\[\tag{4}\left[\begin{align}&A&B\\&C&D\end{align}\right]=\left[\begin{array}&-&|M|&f_1+f_2-|M|L_1-L_2/|M|\\&0&-1/|M|\end{array}\right],\]

 

 

FIGURE 5.  Self-imaging ring unstable resonator.

where the (negative) value of magnification is given by \(M=-f_2/f_1\). If the element spacings in this resonator satisfy the self-imaging condition \(f_1+f_2=|M|L_1+L_2/|M|\), then the canonical form of this resonator has \(B\equiv0\). Each round trip in this self-imaging unstable resonator then has an effective propagation length that is identically zero, and hence the resonator has infinite Fresnel numbers, i.e., \(N_c=N_{eq}=\infty\). 

The exact eigenmodes of this kind of "self-imaging resonator" are difficult to predict, since it is not clear at present how we should handle the limiting situation of \(N_{eq}\rightarrow\infty\) in a mathematical analysis. There is considerable doubt, for example, whether the limiting behavior of the Huygens' integral problem for \(N_{eq}\rightarrow\infty\) really connects smoothly to the geometric limit, particularly in the circular-mirror situation. It is generally believed, however, that the self imaging situation should give a particularly smooth and uniform lowest-order mode pattern in an unstable resonator.

 

Off-Axis Unstable Resonators

Another useful if somewhat inelegant variation in unstable resonator design is to use any one of a variety of off-axis unstable resonator configurations, as illustrated in Figure 6. The primary motivation for any of these off-axis unstable resonator designs is a more uniformly illuminated near-field pattern, so as to increase the intensity in the central lobe and reduce the intensity in the diffraction side lobes in the far field. This objective can in fact generally be achieved, at least to some extent, despite the irregular far-field patterns that we can expect from the kinds of near-field patterns shown. 

 

Off-Axis Mirror Positioning Equals Misalignment

The general properties of these off-axis unstable resonators can be understood from the following argument. Shifting the output mirror of an unstable resonator off axis in either transverse dimension is in fact exactly equivalent to 

 

 

FIGURE 6.  Unstable resonators with off-axis output mirrors.

 

 

FIGURE 7.  An off-axis unstable resonator is the same as a misaligned unstable resonator.

misaligning the resonator mirrors (or the resonator optical axis), as shown in Figure 7. A misaligned resonator of this type, so long as the optical axis still intercepts the output mirror, can then be viewed, at least approximately, as consisting of two half-resonators on opposite sides of the optical axis, with the two sides having the same magnification but different Fresnel numbers. 

Since the mode properties and the output coupling of an unstable resonator depend primarily on the magnification \(M\), and only secondarily on the Fresnel number \(N_{eq}\), we can expect that the near-field pattern for a misaligned unstable resonator will still consist of a roughly uniform (or perhaps tapered) near-field distribution which is magnified outward from the optical axis by the same magnification \(M\) on all sides of the output mirror, as illustrated in Figure 6.

The Fresnel ripple structures within these geometrically predicted patterns, and the exact eigenmodes and mode losses, will certainly depend on the amount of misalignment; but the zero-order near-field and far-field patterns will still be very much as illustrated.

These predictions turn out to be entirely correct. Misalignment of an unstable resonator modifies the exact eigenvalues, and changes the mode crossing behavior, in a complicated but essentially minor fashion.

As we might expect, moreover, the nature of these changes is closely related to the equivalent Fresnel numbers of the individual half resonators, e.g., whether the two halves of the resonator are both at (different) half-integer \(N_{eq}\) values, or whether one side is at a half-integer and the other at an integer value.

 

 

FIGURE 8.  Two examples of rectangular stable unstable resonator design.

"The basic conclusion is that rectangular aperture unstable resonators are quite insensitive to misalignment, in the sense that the lowest-loss mode continues to be essentially diffraction limited as long as the feedback mirror remains well within the output beam." 

The optimum form of this approach, for a low-gain laser system with rectangular symmetry, might be to use an astigmatic rectangular unstable resonator as shown in Figure 6(c), with the magnification in one transverse dimension, say, \(M_x\), made just enough greater than unity to give good mode discrimination and good filling of the mode volume, but very small output coupling.

The magnification \(M_y\) in the other transverse direction could then be made large enough to give good output coupling, with the resonator misaligned in this direction until the relative amount of power coming past one edge was negligible. The small fractional amount of power coming past three sides of the mirror could then be discarded, and the output power extracted from the fourth side as a completely unobstructed and essentially rectangular output beam.

 

 

FIGURE 9.  A large one-sided stable unstable glass slab laser.

 

Stable-Unstable Resonators 

In a further extension of this concept, there are certain laser systems in which the active medium occurs in the form of a thin slab or gain sheet, which may have a very small equivalent Fresnel number in one transverse direction, and a substantially larger width or equivalent Fresnel number in the other transverse direction.

Examples include slab glass or other solid-state lasers; certain flowing chemical lasers where the excited states relax very rapidly after leaving a flat array of supersonic nozzles; certain dye lasers where we might pump a thin flowing sheet of dye with a flashlamp; and various transverse discharge-pumped and e-beam pumped molecular and excimer lasers. 

A very effective technique—at the cost of some complexity in optical components—can then be to employ a laser cavity which is stable and supports a lowest-order \(\text{TEM}_0\) gaussian mode in one direction, but which is unstable (and probably misaligned in one-sided fashion) in the other dimension. Figure 8 shows some examples of how this concept can be applied to various lasers. The Soviet researchers who carried out early experiments using the large glass slab shown in Figure 23.9 reported, for example, diffraction-limited performance \((\Delta\theta\approx20\mu\text{rad})\) in the unstable transverse dimension, for a laser resonator with an equivalent Fresnel number of \(N_{eq}\approx7000!\)

 

Unstable Resonators in Semiconductor Diode Lasers

Another prime candidate for this type of stable-unstable resonator design would seem to be the semiconductor diode laser or injection laser, in which the gain region is inherently a very thin slab, with stable mode guiding perpendicular to the junction region, and no mode confinement in the plane of the junction (at least in the simplest form of injection diode structure).

Use of a curved back mirror to spread out the fields into an unstable mode in the plane of the junction should give simultaneously good transverse mode control and much more power output from a much larger junction area. 

Only one Soviet experiment on this type of unstable semiconductor laser action has been published to date, although other efforts may be in progress. There can be some practical difficulties in obtaining a cleaved (or polished) back mirror surface with the necessary unstable curvature on a typically very small injection diode laser.

 

 

FIGURE 10.  Design concepts for unstable resonators using diffraction gratings and corner reflectors.

 

 

FIGURE 11.  Shaped mirror edges intended to cancel the edge-wave interference and reduce the near field diffraction effects.

 

 

Grating Expanded Unstable Resonators 

Designers of tunable dye lasers realized some years ago that spatially dispersive elements, such as prisms and gratings, could also be used as spatial beam expanders, functioning in essence as one-dimensional psuedo telescopes. Figure 10 shows a few of the more exotic proposals for applying this concept to unstable resonators, using the beam-expanding and also output-coupling properties either of diffraction gratings or of dihedral prisms or corner retroreflectors. Few if any experiments with this type of resonator design seem to have been attempted to date. 

 

Minimizing Edge Wave Effects: Aperture Shaping

We have discussed in earlier sections how the near-field diffraction effects from hard-edged apertures can be minimized by changing the aperture shape so as to more or less cancel the edge waves from different parts of the aperture edges.

In our discussion of aperture diffraction, for example, we noted that changing the shape of a hard-edged aperture to anything other than perfectly circular greatly reduces the peak value of the on-axis ripple in the near-field diffraction pattern. Similarly, tapering the actual reflectivity of the edge itself, so that the amplitude or phase of the edge reflectivity varies significantly over a distance corresponding to one full Fresnel zone or more, will greatly reduce the net amplitude of the diffracted edge wave, and thus reduce the Fresnel ripples in the near-field diffraction pattern. 

Understanding of these principles has led to various attempts to improve the mode performance of hard-edged unstable resonators by shaping or smoothing

 

 

FIGURE 12.  Unstable resonators with stepped or rounded mirror edges.

the edges of the output mirror or output coupler. Figure 11 shows some of the modified circular aperture shapes that have been considered for unstable resonators. The difficulties in fabricating such special mirror shapes, however, as well as the analytical complexities they introduce into any exact mode calculations, make this concept seem relatively unattractive for practical unstable resonator designs. 

 

Unstable Mirrors With Tapered Phase Values

A somewhat more attractive approach to edge-wave cancellation might be to modify or taper the phase angle of the mirror reflection coefficient over a narrow region near the mirror edges, but still retain completely reflective optics. Techniques that have been suggested for this purpose include mirrors with rims or steps near the mirror edges and rolled-over mirror edges, as illustrated in Figure 12. 

The basic design principle here is to vary the reflection phase through one or more multiples of \(2\pi\) over a transverse or radial distance which includes at least one or more Fresnel zones as measured by the collimated Fresnel number in the resonator.

In general it has been found that this kind of edge modulation applied to one or more of the end mirrors does cause the eigenvalue for the lowest-loss eigenmode in an unstable resonator to separate from the higher-order modes at a lower value of equivalent Fresnel number \(N_{eq}\), as well as reducing the periodic rippling of the eigenvalue or the cavity coupling with increasing \(N_{eq}\), as illustrated for a typical situation in Figure 13.

 

Retroreflected Unstable Resonators

Still another in the list of more unsuccessful attempts to extend the unstable resonator concept is what might be called the "retroreflected unstable resonator," as illustrated in Figure 14. 

The general idea here is to use an additional plane mirror to reflect a portion of the emerging beam from an unstable resonator back into the resonator, where it will excite a converging wave that will demagnify down into the center of the resonator and then eventually emerge again as an addition to the magnifying wave.

Figure 14 shows a design where the inner portion of the annular output beam is retroreflected; an alternative design retroreflects the outer portion of the annulus and lets the inner portion escape.

 

 

FIGURE 13.  Rounding the mirror edges, as in Figure 12, can lead to a clear separation of the lowest-loss eigenvalue from all the higher order modes. (From Santana and Felsen.)

 

 

FIGURE 14.  Retroreflected unstable resonator concept.

 

Advantages that might be hoped for from this design concept include reduced output coupling for the same magnification, for use with low-gain lasers; and the development of an extended annular region with partially standing-wave fields, which could be useful in coupling to certain kinds of annularly shaped active gain media.

This general concept also faces a number of fundamental difficulties, however, including: 

• The detailed transverse mode behavior turns out to be (as we might expect) very sensitive to the exact phase with which the retroreflected energy is fed back into the converging paraxial eigenwave. Hence, the resonator performance is sensitive to changes in the relative mirror spacings \(L_1\) and \(L_2\) on the order of a small fraction of an optical wavelength. The cavity is also very sensitive to small changes in the relative angular alignments of the three mirrors.

• Under a wide range of conditions the flat annular mirror and the larger curved mirror at the opposite end of the cavity can form a stable two mirror resonator, which can support a number of high-order Laguerre gaussian modes with large radial and azimuthal indices in the annular region outside the normal output mirror.

• In contrast to the simplified geometric picture in Figure 14, the beams will inherently experience diffraction spreading in traveling down to the annular mirror and back to the normal output mirror. This, along with the increased number of edges and apertures in the resonator, greatly complicates the exact mode behavior (and mode calculations) in the resonator.

All these difficulties have generally made the retrorenection concept seem unattractive after further examination, although we will see even more complex continuations of this idea a few paragraphs further on.

 

Back-Reflection Effects in Unstable Resonators

The substantial influence on unstable resonator performance of any energy that is fed back into the converging or demagnifying eigenwave has also led other authors to note that unstable resonators can be (1) unusually sensitive to any internal apertures or partially reflecting surfaces (imperfect \(\text{AR}\) coatings, etc.) which may reflect energy directly back along the laser axis with the correct curvature and direction to go into the converging wave, and also (2) unusually sensitive to external surfaces or optical elements that retroreflect the collimated output beam directly back into an unstable oscillator cavity. 

 

Axicons, Split-Mode Resonators, HSURIAs, and Other Exotica

Many higher-power laser systems, including especially chemical and gasdy-namic lasers, tend naturally to generate a thin annularly shaped gain medium, produced perhaps by radial flow of combusting gases coming out from the outer surface of a cylindrical burner shaped rather like a large oil drum.

It is then a particularly difficult challenge to find a resonator structure, perhaps some variant of the unstable resonator, which can extract the energy from a large annular gain region, yet retain good transverse mode control and beam quality. This problem may, in fact, be essentially unsolvable. 

Attempts to meet this challenge have led first to the idea of splitting an unstable resonator in half at some point along its optical axis, leading to the concept of the "split-mode unstable resonator"; and then folding these halves back on themselves to create various varieties of coupled half-symmetric unstable resonators.

If we accomplish this splitting using an axicon structure with its

 

 

FIGURE 15 (a) and (b), Radial and toroidal split-mode resonators with internal axicons. (c) \(\text{HSURIA}\) structure using a waxicon. (d) \(\text{HSURIA}\) structure with reflaxicon.

point located on the optical axis, we can create an increasingly complicated set of exotic radial or annular resonator configurations, a few of which are illustrated in Figure 15. 

Even the nomenclature for these structures becomes exotic. As an example, the "half-symmetric unstable resonator with internal axicon" becomes known, obviously, as the \(\text{HSURIA}\). (The test for whether a laser engineer has received an adequate liberal education is whether this is referred to as "a \(\text{HSURIA}\)" or "an \(\text{HSURIA}\).") In the jargon that has developed, an axicon that reverses the beam direction (and thereby inverts inner and outer edges) is a "\(\text W\)-axicon" or "waxicon," whereas a reflective axicon that expands or contracts a beam while sending it on in the same direction is a "reflaxicon."

The region where the beam is brought back together again at reduced diameter, and where most of the transverse mode control presumably takes place, is the "compacted region."

Axicon structures of any type exhibit extremely high sensitivity not only to angular misalignment, but to transverse displacements of even a fraction of a wavelength with respect to the optical axis of the resonator system. If the axicon is to function simultaneously as a telescopic or focusing element, the axicon may also require a surface that is parabolic, toroidal, or some more complex combination.

The construction and alignment of any type of practical large high-power laser employing an axicon or other complex resonator design is thus extremely difficulty, and may be essentially impossible.

 

 

FIGURE 16.  Various forms of tapered reflectivity for laser resonator mirrors.

 

 

3.  VARIABLE-REFLECTIVITY UNSTABLE RESONATORS

We have saved until last what seems to be the best unstable resonator concept of all—the use of geometrically unstable resonator optics, in order to achieve increased mode volume and good transverse mode discrimination, combined with gaussian variable-reBectivity mirrors (or variable-transmission output couplers) to control both the mode performance of the unstable resonator and the transverse beam profile of the output beam as well. 

 

Unstable Resonators With Variable Reflectivity Mirrors

An obvious extension of the tapered-edge mirror concepts introduced in the previous section is to taper the magnitude of the mirror reflectivity from its maximum value down to zero over some annular region at the outer edge of the output coupling mirror, as shown in Figure 16, in order to smear out the edge diffraction effects.

Once again it is found both theoretically and experimentally, as shown in Figure 17, that this kind of amplitude tapering can lead to substantially improved mode discrimination as well as substantial reduction of the periodic cusping behavior with the equivalent Fresnel number \(N_{eq}\). 

Again, however, the primary problem with both phase and amplitude tapering is finding practical methods for tapering the mirror reflectivity which can be fabricated to adequate optical tolerances at reasonable cost, and which will also stand high output powers if necessary.

In addition, unwanted phase perturbations associated with an amplitude taper may distort the phase front of the lowest order mode, so that this mode, although it may have excellent mode separation and mode discrimination against higher-order modes, will have a nonplanar or nonspherical wavefront which cannot easily be converted into a well-collimated output beam.

 

 

FIGURE 17.  Typical difference in behavior between (a) a conventional hard-edged unstable resonator, and (b) the same resonator with tapered-reflectivity mirrors. (From Anan'ev and Shersto bitov.)

 

 

FIGURE 18.  Canonical formulation for a geometrically unstable resonator with gaussian aperture or gaussian variable-reflectivity mirror.

 

Gaussian Variable Reflectivity Mirrors (VRM) 

If we are going to taper the output-mirror reflectivity, however, then the simplest solution, at least analytically, would seem to be a gaussian reflectivity taper—that is, converting the output mirror into effectively a soft gaussian aperture, so that the resonator will become identically a complex gaussian or paraxial resonator of the type already discussed in this tutorial COMPLEX STABLE AND UNSTABLE RESONATORS.

Such a resonator will then have well-understood Hermite-gaussian or Laguerre-gaussian modes, which can be analyzed more or less exactly using the complex \(\text{ABCD}\) methods described in this tutorial Generalized Paraxial Resonator Theory. In the remainder of this tutorial we will summarize some of the useful design features of such complex geometrically unstable gaussian resonators, as well as some of the practical methods by which such resonators might be obtained.

 

Analysis of an Unstable Gaussian-Reflectivity Resonator

To analyze an unstable resonator with a gaussian variable-reflectivity mirror, we can simply replace the hard-edged aperture in the canonical model of Canonical Analysis for Unstable Resonators by an aperture with the appropriate amplitude transmission coefficient \(\tilde{\rho}(r)\). [The mirror reflection coefficient \(\tilde{\rho}(r)\) in the real laser becomes the aperture transmission coefficient in the equivalent lensguide model.] For the ideal gaussian situation we assume this reflection or transmission coefficient is given by 

\[\tag{5}\tilde{\rho}(r)=\text{exp}\left(\frac{a_2r^2}{2}\right)\equiv\left(-\frac{r^2}{w^2_a}\right),\]

so that

\[\tag{6}\frac{a_2}{2}\equiv\frac{1}{w^2_a}\quad\text{or}\quad w_a\equiv\sqrt{\frac{2}{a_2}}.\]

The radius \(w_a\) is then the \(1/e\) radius for amplitude transmission, or the \(1/e^2\) radius for intensity transmission, through the aperture. (For simplicity, we will consider cylindrically symmetric resonators through the rest of this section.) The round-trip complex ray matrix for this system is then given by

\[\tag{7}\left[\begin{align}&A&B\\&C&D\end{align}\right]=\left[\begin{array}&1&MB\\0&1\end{array}\right]\times\left[\begin{array}&&M&0\\&0&1/M\end{array}\right]\times\left[\begin{array}&1&0\\-j\lambda a_2/2\pi&1\end{array}\right]=\left[\begin{array}&M-j\lambda a_2B/2\pi&B\\-j\lambda a_2/2\pi M&1/M\end{array}\right]\]

Although the gaussian aperture does not have a discrete sharp edge, it is convenient to define an effective Fresnel number for the gaussian aperture, based on the radius \(w_a\) and the resonator's effective length \(B\), by the definition

\[\tag{8}N_{ga}\equiv\frac{w^2_a}{B\lambda}=\text{gaussian}\;\text{resonator}\;\text{Fresnel}\;\text{number}.\]

In practice we will usually be interested in resonators with comparatively large mode diameters, or large values of this Fresnel number \(N_{ga}\).

With this notation the \(\text{ABCD}\) matrix can be written as

\[\tag{9}\left[\begin{align}&A&B\\&C&D\end{align}\right]=\left[\begin{array}&M-j/\pi N_{ga}&B\\-j/\pi N_{ga}MB&1/M\end{array}\right],\]

and we can then write the complex m value associated with this resonator as

\[\tag{10}\tilde{m}\equiv\frac{A+D}{2}=\frac{1}{2}\left[M+\frac{1}{M}-\frac{j}{\pi N_{ga}}\right]\equiv m_r-j\frac{1}{2\pi N_{ga}}.\]

The quantity mr is then the real, geometrically unstable \(m_r\) value that would be associated with this resonator in the absence of the gaussian aperture. It is related to the geometric magnification \(M\) by the same formulas used in ordinary unstable resonators, namely,

\[\tag{11}M=m_r+\sqrt{m^2_r-1}\quad\text{or}\quad m_r=\frac{M^2+1}{2M},\]

assuming for simplicity that we consider only the positive branch from here on.

 

Eigenvalues and Eigenmodes

We will also assume for the remainder of this section that the gaussian aperture's Fresnel number \(N_{ga}\) will be large enough compared to unity so that the imaginary part \(j/2\pi N_{ga}\) of the complex \(\tilde{m}\) expression can be considered as a small first-order perturbation relative to the real \(m_r\) part.

In practical terms, this means we assume a comparatively weak gaussian aperture. But since in most practical situations we want a comparatively large-diameter mode, this assumption matches the desired design objective. 

The complex eigenvalue for the confined, magnifying, perturbation-stable solution in this gaussian resonator can then be approximated by

\[\tag{12}\begin{align}\tilde{\gamma}&=\tilde{m}-\sqrt{\tilde{m}^2-1}\\&=m_r-j/2\pi N_{ga}-\sqrt{(m_r-j/2\pi N_{ga})^2-1}\\&\approx\left(\frac{1}{M}\right)+j\left(\frac{1}{M^2-1}\right)\frac{1}{\pi N_{ga}}\quad\text{for}\;m_r\gg1/2\pi N_{ga},\end{align}\]

whereas the corresponding complex gaussian eigensolution is given by a gaussian beam with \(\tilde{q}\) parameter

\[\tag{13}\begin{align}\frac{1}{\tilde{q}}&=\frac{D-A}{2B}+\frac{\sqrt{\tilde{m}^2-1}}{B}=\frac{D-\tilde{\gamma}}{B}\\&=\approx-j\left(\frac{1}{M^2-1}\right)\frac{1}{\pi N_{ga}B}\equiv-j\frac{\lambda}{\pi w^2}\quad\text{for}\;m_r\gg1/2\pi N_{ga}.\end{align}\]

We see that to first order in \(1/\pi N_{ga}B\), adding the gaussian aperture to this unstable resonator leaves the round-trip eigenvalue at \(\tilde{\gamma}\approx1/M\), plus a small imaginary part; whereas it changes the magnifying paraxial eigenwave from an unbounded plane wave (as seen in this canonical formulation) to a gaussian beam which is still collimated, with infinite radius of curvature, but with a finite spot size.

This modal spot size \(w\) is in fact given by

\[\tag{14}w^2\approx(M^2-1)\times w^2_a\quad\text{for}\;m_r\gg1/2\pi N_{ga}.\]

The amplitude discrimination between the lowest and higher-order \(\text{TEM}_{mn}\) modes will thus be roughly \(|\tilde{\gamma}|\approx(1/M)^{m+n}\) on each round trip, and the lowest-order \(\text{TEM}_{00}\) mode will be a clean, collimated gaussian beam inside the laser cavity.

 

Output Coupling and Output Beam Profiles 

Although the transverse profile of the lowest eigenmode will be gaussian inside the laser cavity, the output beam will have this gaussian profile multiplied by the radially varying mirror transmission \(T(r)\equiv1-R(r)\) outside the cavity. (We will certainly want to use a lossless variable reflection technique, so that \(R+T=1\) everywhere.) The end-mirror reflectivity \(R(r)\) will have the general form 

\[\tag{15}R(r)\equiv|\tilde{\rho}(r)|^2=R_0\;\text{exp}[-2r^2/w^2_a],\]

 

 

FIGURE 19.  Output beam profiles from a \(\text{VRM}\) resonator with different output mirror transmission factors.

where \(R_0\) is the the central value of reflection coefficient and \(T_0\equiv1-R_0\) is the power transmission at the center of the mirror. 

Multiplying the radially decreasing gaussian beam profile inside the resonator by the radially increasing mirror transmission then gives for the output beam profile

\[\tag{16}I_\text{out}(r)=\left[1-R_0e^{-2r^2/w^2_a}\right]e^{-2r^2/w^2},\]

where \(w^2\approx(M^2-1)w^2_a\) is the beam spot size in the large-Fresnel-number limit. Figure 19 shows plots of the output mirror transmission profile \(1-R(r)\); the gaussian beam profile \(I(r)=|\tilde{u}(r)|^2\) just inside the output mirror; and the output intensity profiles \(I_\text{out}(r)=[1-R(r)]I(r)\) (normalized to the same peak values) for a magnification \(M\sqrt{2}\) and for several different choices of the central mirror reflectivity \(R_0\).

If we want to have an output beam profile without a significant dip in the center, we must use an output mirror or output coupler whose reflectivity at the center is less than unity, so that there is some finite transmission out of the cavity even at the center of the beam. 

 

 

FIGURE 20.  Maximally flat output beam profiles from VRM lasers with different geometric magnifications.

 

Radially Averaged Output Coupling

The power output of a laser oscillator depends strongly on the output coupling. The effective output coupling from a gaussian variable-reflectivity-mirror cavity depends upon the effective or average power reflectivity of the end mirror, integrated across the entire mode pattern, as given by 

\[\tag{17}\bar{R}=\frac{\int^\infty_02\pi rR_{0e}^{-2r^2/w^2_a}e^{-2r^2/w^2}dr}{\int^\infty_02\pi re^{-2r^2/w^2}dr}=\frac{R_0}{M^2},\]

(as we could already have deduced from the fact that the eigenvalue for the resonator is \(\tilde{\gamma}\approx1/M\;\text{and}\;1-|\tilde{\gamma}|^2\equiv\bar{R}.\) The essential point here is that the average reflectivity of the end mirror, or one minus the output coupling, is given by the central reflectivity value \(R_0\) of the output mirror, divided by the magnification \(\text M\) squared.

This output coupling value must be adjusted to fall in the optimum output coupling range for the particular type of laser under consideration.

 

Maximally Flat Output Beam Profiles

The primary tradeoff in designing a gaussian VRM laser is then that if the central reflectivity \(R_0\) is too small the laser oscillator will be overcoupled; whereas if \(R_0\) is greater than a certain value, the mirror transmission will increase with radius faster than the gaussian intensity of the eigenmode itself decreases, so that the output beam will acquire a dip or hole in the center. 

For the smoothest and most uniform output beam profile, we might wish to design a gaussian-reflectivity unstable resonator laser so that it operates just at the reflectivity \(R_0\) where the central dip is about to appear. It is then easy to show that the "maximally flat" condition at which the central dip just begins to occur is given by

\[\tag{18}R_0(\text{maximally}\;\text{flat})=\frac{1}{M^2},\]

whereas the effective or average reflectivity of the end mirror under maximally flat conditions is given by

 \[\tag{19}\bar{R}(\text{maximally}\;\text{flat})=\frac{1}{M^4}.\]

Figure 20. shows examples of these maximally flat output profiles for different values of \(\text{M}\). 

 

Gaussian Reflectivity Design Criteria

The design procedure for a gaussian-variable-reflectivity unstable resonator can then be outlined as follows: 

  • Select a value of geometric magnification \(M\) which will achieve an adequate mode intensity discrimination ratio \(1/M^2\) per round trip between the lowest-order and next higher-order mode on each round trip.
  • At the same time select a value of mirror-center reflectivity Ro which will, if possible, satisfy the maximally flat criterion \(R_0\leqslant 1/M^2\) yet still keep the average reflectivity \(\bar{R}=R_0/M^2\) high enough for good power extraction from the laser medium.
  • Finally, choose the width parameter \(w_a\) or the Fresnel number \(N_{ga}\equiv w^2_a/B\lambda\) of the gaussian aperture large enough so that the spot size \(w^2= (M^2-1)w^2_a\) inside the resonator adequately fills the gain medium.

In low gain lasers, the combination of \(\text M\) and \(R_0\) required to achieve both good mode discrimination and a maximally flat output beam profile may over-couple the laser. In this situation, it may be necessary to increase the central reflectivity \(R_0\) and accept some amount of intensity reduction in the center of the beam. It can then be shown that the intensity reduction at the center of the beam relative to that at the peak of the annular ring (at radius \(r_p\)) is given by 

\[\tag{20}\frac{I_\text{out}(r=0)}{I_{out}(r=r_p)}=\frac{M^2(1-R_0)(M^2R_0)^{1/(M^2-1)}}{M^2-1}\]

for values of \(R_0>1/M^2\).

 

Practical Variable-Reflectivity Mirrors and Couplers 

Any practical form of gaussian aperture or gaussian variable-reflectivity mirror \(\text{(VRM)}\) which can be employed in this kind of resonator will thus be a very useful optical element. Components of this type which have been proposed or demonstrated to date include: 

(1) Absorbing gaussian apertures. These can made using evaporated metal or absorptive coatings, exposed and developed photographic films, or absorptive filters with radially tapered density. Aside from the difficulty of fabricating such apertures, they have the practical problem that they absorb power and hence reduce laser efficiency, and they are likely to be damage-prone in high power lasers.

(2) Tapered-reflectivity dielectric mirrors. These can be made by preparing some kind of tapered-reflectivity and hence tapered-transmission, lossless dielectric coating on one end mirror of the laser. Such mirrors, if they became

 

 

FIGURE 21.  A diffraction grating with radially varying groove depth can function as a \(\text{VRM}\) coupler capable of handling large laser powers.

readily available, would be very useful and effective. The principal problem here is that such coatings appear at present to be difficult and expensive to make. There may also be unwanted phase distortions in the transmitted wave due to the tapered coatings. 

(3) Tapered-groove-depth diffraction gratings. A complex but effective type of variable-reflectivity mirror or coupler can be obtained by etching a diffraction grating into an otherwise \(100\%\) reflecting mirror. This grating should be fabricated with a constant grating period and hence diffraction angle, but with a groove depth and hence a diffraction efficiency which varies from center to edge.

We can then use this type of element in either of two ways, with the diffracted beam either providing the mirror feedback and the specular beam the output coupling, or vice versa, as illustrated in Figure 10.  An element of this kind is obviously difficult to fabriate initially, but once fabricated should be efficient and effective.

At least one successful grating-mirror of this type has been prepared by ion beam milling techniques and used with success in a high-power \(CO_2\) laser.

 

Radially Varying Birefringent Couplers

One of the most promising and effective recent solutions to the variable-reflectivity mirror or coupler problem is the radially varying birefringent coupling technique illustrated in Figure 22.

In this technique a birefringent element (or, in a ring resonator, an optically active element) with a radially increasing strength or thickness is placed inside a laser cavity. The wave passing through the element will then have its polarization rotated, or converted from linear to elliptical, by an amount which increases with radius from the center of the element.

A polarization-sensitive coupling element, such as a dielectric-coated beam splitter, a polarizing crystal, or a Brewster-angle plate, is then used to extract the rotated polarization component with a strength which increases radially at the desired rate.

In more advanced versions of this technique, two polarizing elements of opposite sign, shaped like positive and negative lenses, can be used in cascade to produce the radially varying birefringence yet cancel out any focusing effects due to either element alone.

If ordinary birefringent crystals are used, a variety of both radial profiles and central reflection values can be obtained by rotating one

 

 

FIGURE 22.  Radially varying birefringent coupling technique.

or both of these crystals about the resonator axis, as demonstrated by Byer and his co-workers.

This particular technique appears to have great potential, at least for low to medium power lasers in the visible and near \(\text{IR}\). The necessary elements can be relatively simple to fabricate and adjust, and can have low insertion losses and relatively high optical-damage thresholds at least in the visible region where suitable optical materials are available.

Finding suitable birefringent elements and polarizers does, however, become much more difficult for high average power lasers, especially in the infrared; and alternative solutions to the general variable-reflectivity coupler problem with similarly useful properties would be well worth inventing. 

 

 

 

 

 

 


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