STABLE TWO-MIRROR RESONATORS

This is a continuation from the previous tutorial - Beam perturbation and diffraction

The simplest kind of optical resonator consists of just two curved mirrors set up facing each other. If the curvatures of these two mirrors correspond to a stable periodic focusing system, and if their transverse dimensions are large enough so that we can neglect edge-diffraction effects, then these mirrors can in essence trap a set of lowest-order and higher-order gaussian modes or beams that will bounce back and forth between the two mirrors. These trapped Hermite-gaussian modes form, to a first approximation, a set of resonant modes for the two-mirror cavity.

Simple two-mirror cavities such as this are widely used in practical lasers, and the properties of these stable gaussian resonator modes form part of the basic lore of laser physics. In this chapter, therefore, we give a fairly detailed account of these properties and of how they are derived from gaussian beam theory. In addition we give a brief survey of the (usually) small deviations from ideal gaussian beam behavior that occur because of finite mirror sizes, including in particular the finite diffraction losses in finite-diameter resonators.

In later chapters we will discuss the additional complexities that arise in analyzing multielement resonators which contain, for example, intracavity lenses or gaussian apertures, as well as the quite different and nongaussian modes associated with unstable optical resonators. Even in these situations, however, the stable two-mirror gaussian concepts introduced in this section will prove very useful in understanding and explaining the behavior of these more complex resonators.

 

 

1. STABLE GAUSSIAN RESONATOR MODES

Suppose we have a gaussian beam with a certain waist size and waist location, as in Figure 1, and suppose that we then fit a pair of curved mirrors to this beam at any two points along the beam, as also illustrated in Figure 1. If the radii of curvature of the mirrors are exactly matched to the wavefront radii of the gaussian beam at those two points, and if the transverse size of the mirrors is substantially larger than the gaussian spot size of the beam, each of these mirrors will in essence reflect the gaussian beam exactly back on itself, with exactly reversed wavefront curvature and direction.

 

 

 

 

 

 

These two mirrors can thus trap the gaussian beam as a standing wave between the two mirrors, with, if the mirrors are large enough in size, negligible diffraction or "spillover" losses past the edges of the mirrors. The two mirrors thus form an optical resonator which can support both the lowest-order gaussian mode, and also higher-order Hermite-gaussian or Laguerre-gaussian modes, as resonant modes of the cavity. We will see in this section that this simple description is, in essence, exactly what happens in elementary stable two-mirror gaussian resonators. 

 

Stable Two-Mirror Resonator Analysis 

In practice, instead of being given a gaussian beam and asked to fit mirrors to it, we are much more likely to be given two curved mirrors \(M_1\) and \(M_2\) with radii of curvature \(R_1\) and \(R_2\) and spacing \(L\), and asked to find the right gaussian beam that will just fit properly between these two mirrors. To analyze this situation we can use the model in Figure 2, assuming that the gaussian beam will have an (initially unknown) spot size \(\omega_0\) or Rayleigh range \(z_R\equiv\pi\omega^2_0/\lambda\), and that the mirrors will be located at distances \(z_1\) and \(z_2\) from the (initially unknown) location of the beam waist. 

The essential conditions are then that the wavefront curvature \(R(z)\) of the gaussian beam, as given by gaussian beam theory, must match the mirror curvature at each mirror, taking into account the specified mirror spacing \(L\). This

 

 

 

provides us with three equations, namely, 

\[\tag{1}\begin{align}R(z_1)=z_1+z^2_R/z_1=-R_1,\\R(z_2)=z_2+z^2_R/z_2=+R_2,\end{align}\]

and

\[\tag{2}L=z_2-z_1.\] 

The minus sign in the first of these equations arises because of a difference in the sign conventions that we use in in describing beam wavefronts or in describing resonator mirrors. The gaussian wavefront curvature \(R(z)\) is usually taken as positive for a diverging beam, or negative for a converging beam, traveling to the right; whereas the mirror curvatures \(R_1\) and \(R_2\) are usually taken as positive numbers for mirrors that are concave inward, i.e., as seen looking out from within the resonator, and as negative numbers for mirrors that are convex as seen from inside the resonator.

 

The g Parameters 

We must then invert these three equations in order to find the gaussian beam parameters \(z_R,z_1\), and \(z_2\) in terms of the specified mirror curvatures and spacing \(R_1,R_2\) and \(L\). Before doing this, however, it is customary to define a pair of "resonator \(g\) parameters," \(g_1\) and \(g_2\), which were introduced in the early days of laser theory to describe laser resonators, and have since become standard notation in the field. These parameters are given by 

\[\tag{3}g_1\equiv1-\frac{L}{R_1}\qquad\text{and}\qquad g_2\equiv 1-\frac{L}{R_2}.\]

We will see more of their physical significance later.

In terms of these parameters we can then find that the trapped gaussian beam in Figure 2 will have a unique Rayleigh range given by

\[\tag{4}z^2_R=\frac{g_1g_2(1-g_1g_2)}{(g_1+g_2-2g_1g_2)^2}L^2,\]

and that the locations of the two mirrors relative to the gaussian beam waist will be given by

\[\tag{5}z_1=\frac{g_2(1-g_1)}{g_1+g_2-2g_1g_2}L\qquad\text{and}\qquad z_2=\frac{g_1(1-g_2)}{g_1+g_2-2g_1g_2}L.\]

(Note that if mirror Mi is located to the left of the beam waist, so that the waist is inside the resonator as in Figure 2, then \(z_1\) as measured from the waist will be a negative number.)

It is also useful to write out the waist spot size \(\omega_0\), which is given by

\[\tag{6}\omega^2_0=\frac{L\lambda}{\pi}\sqrt{\frac{g_1g_2(1-g_1g_2)}{(g_1+g_2-2g_1g_2)^2}},\]

and the spot sizes \(\omega_1\) and \(\omega_2\) at the ends of the resonator, which are given by

\[\tag{7}\omega^2_1=\frac{L\lambda}{\pi}\sqrt{\frac{g_2}{g_1(1-g_1g_2)}}\qquad\text{and}\qquad\omega^2_2=\frac{L\lambda}{\pi}\sqrt{\frac{g_1}{g_2(1-g_1g_2)}}.\] 

These quantities depend only on the resonator g parameters defined in the preceding, and on the quantity \(\sqrt{L\lambda/\pi}\) which we will discuss in the following.

 

Resonator Stability Diagram

It is immediately obvious from Equations 4 to 7 that real and finite solutions for the gaussian beam parameters and spot sizes can exist only if the \(g_1,g_2\) parameters are confined to a stability range defined by 

\[\tag{8}0\leq g_1g_2\leq1.\]

We refer to this as a stability range because this is also exactly the condition required for two mirrors with radii \(R_1\) and \(R_2\) and spacing L to form a stable periodic focusing system for rays, as analyzed in Ray optics and ray matrices this tutorial.

In the early days of gaussian resonator theory this stability criterion was immediately translated into the resonator stability diagram shown in Figure 4. Every two-mirror optical resonator can then be characterized by the parameters \(g_1=1-L/R_1\) and \(g_2=1-L/R_2\), and hence represented by a point in the \(g_1,g_2\) plane. If this point falls in the shaded stable region, shown in Figure 4, the mirrors correspond to a stable periodic focusing system, and the resonator (if the mirrors are large enough transversely) will trap a family of lowest and higher-order gaussian modes with gaussian beam parameters given by Equations 4 through 7. Such a stable resonator will thus have a unique set of gaussian transverse resonator modes. 

If the point \(g_1,g_2\) instead falls in any of the unstable regions outside the shaded area, the mirrors will correspond to an unstable periodic focusing system, and no gaussian beam that will fit properly between the mirrors can be found. These mirror configurations correspond to the very different (but also very useful) unstable optical resonators that we will discuss in a later tutorial.

Optical ray theory and gaussian mode theory thus have a close connection, which we will study in more detail later on, even though the diffraction effects that are an integral part of gaussian beam theory are entirely neglected in the

 

 

 

optical ray theory. Note also that these distinctions between stability and instability depend only on the g parameters, and are (to first order) entirely independent of either the optical wavelength or the transverse size or dimensions of the resonator. In the following section we will examine in more detail the various types of resonators that occur in various regions of the stability diagram, and the various practical properties of these resonators. 

 

Resonator Circle Diagrams 

An alternative and less commonly used graphical method for interpreting the gaussian beam parameters in stable two-mirror resonators is the circle diagram of Deschamps. Suppose again that two mirrors of radii \(R_1\) and \(R_2\) are set up with spacing \(L\). If we then draw circles with diameters \(R_1\) and \(R_2\) tangent to the concave side of each of these mirrors, as shown in Figure 5, the intersection of these two circles is a necessary and sufficient condition for the existence of a stable gaussian mode in the resonator; and moreover the waist location and its relative size in the resonator is determined by the line joining the intersection of these two circles. 

 

 

 

 

2.  IMPORTANT STABLE RESONATOR TYPES 

To gain more insight into the general properties of stable gaussian resonators, let us now survey some of the characteristics associated with resonators at various different points of interest in the stability diagram introduced in the previous section. 

(1) Symmetric Resonators 

Perhaps the simplest resonator configurations to analyze are symmetric resonators, which have mirror curvatures \(R_1=R_2=R\), and hence g parameters \(g_1=g_2=g=1-L/R\). The waist of the gaussian resonant mode is then obviously in the center of the resonator, with waist and end mirror spot sizes given by

\[\tag{9}\omega^2_0=\frac{L\lambda}{\pi}\sqrt{\frac{1+g}{4(1-g)}}\qquad\text{and}\qquad\omega^2_1=\omega^2_2=\frac{L\lambda}{\pi}\sqrt{\frac{1}{1-g^2}}.\]

All these symmetric resonators obviously lie along the \(+45^\circ\) diagonal through the origin in the \(g\) plane, with an allowed range from \(g=1\) (planar mirror case), through \(g=0\) (symmetric confocal case), to \(g=—1\) (concentric or spherical case).

Figure 6 shows how the resonator spot sizes change as the g value is varied along this range, for example, by steadily increasing the mirror curvatures while keeping the mirror spacing fixed.

 

 

 

(2) Half-Symmetric Resonators

Another elementary system is the half-symmetric resonator of Figure 7, in which one mirror is planar, \(R_1=\infty\), and the other curved, so that \(g_1=1\) and \(g_2=g=1-L/R_2\). Such a resonator is obviously equivalent to half of a symmetric system that is twice as long. The waist in this situation will be located on mirror number 1, with spot sizes given by 

\[\tag{10}\omega^2_0=\omega^2_1=\frac{L\lambda}{\pi}\sqrt{\frac{g}{1-g}}\qquad\text{and}\qquad\omega^2_2=\frac{L\lambda}{\pi}\sqrt{\frac{1}{g(1-g)}}.\]

The allowed range for \(g_2=g\) is now from \(+1\) to \(0\), corresponding to a vertical line between the points \((1,1)\) and \((1,0)\) in the stability diagram. 

 

(3) Symmetric Confocal Resonator 

The central point in the stability diagram, and in some sense a central type of stable optical resonator, is the symmetric confocal stable resonator, which is characterized by the values \(R_1=R_2+L\) and \(g_1=g_2=0\) (Figure 8). This is referred to as a confocal resonator because the focal points of the two end mirrors (which are located at \(R/2\) out from the mirror) coincide with each other at the center of the resonator. We have already seen in Figure 10 that confocality 

 

 

corresponds to the condition for a gaussian beam in which the center of curvature of each mirror is located exactly on the opposite mirror. The two mirrors are thus spaced from each other by exactly two Rayleigh ranges or by exactly the waist length of the trapped gaussian beam.

The spot sizes at the center and at the end mirrors of a confocal resonator are then given by 

\[\tag{11}\omega^2_0=\frac{L\lambda}{2\pi}\qquad\text{and}\qquad\omega^2_1=\omega^2_2=\frac{L\lambda}{\pi}.\]

The spot sizes on the end mirrors thus correspond exactly to the scale factor \(\sqrt{L\lambda/\pi}\) that arises in all types of stable resonators, whereas the spot size at the central waist is smaller by \(1/\sqrt{2}\).

Table 1 gives some typical values of this spot size for resonators of different lengths at the typical wavelengths of 633 nm for the He-Ne laser and at \(10.6\mu m\) for the \(CO_2\) lasers. The Table also shows the laser tube diameter that might be associated with this length of resonator, using the rule of thumb that aperture diameter \(d=\pi\omega\).

 

 

 

TABLE 1. 

 

These mode diameters are significantly smaller than the diameters of the laser rods or tubes that we might want to use to obtain reasonable laser power outputs at these wavelengths.

Finding ways to increase the diameter of the stable gaussian modes (or finding new resonator designs which inherently have larger mode volumes) is one of the primary design objectives in most laser designs. 

The confocal resonator in fact has overall the smallest average spot diameter along its length of any stable resonator, although we will see that other resonators may have a smaller waist size at one spot within the resonator. The confocal resonator is also highly insensitive to misalignment of either mirror.

Tilting of either mirror still leaves the center of curvature located on the other mirror surface, and merely displaces the optic axis of the resonator by a small amount. The confocal resonator can thus be very useful, for example, as a trial resonator design when we are first attempting to obtain laser oscillation from a laser medium whose gain is small or uncertain.

The small mode size then means very small diffraction losses, and the alignment insensitivity means that critical mirror alignment should not be necessary to get the laser to oscillate. 

The confocal resonator is also useful for power or energy measurements, in which we simply want to know how much power or energy is available in some laser medium, without consideration of mode control requirements. A confocal 

 

 

resonator is then likely to oscillate in a combination of lowest and higher-order modes that will fill the entire volume of the laser medium and extract essentially all the stimulated emission available from the laser medium. 

The small average size of the confocal modes, on the other hand, means that the lowest-order or \(\text{TEM}_{00}\) confocal mode will not be very effective in extracting power from larger-diameter gain media. Multimode oscillation, as in the power measurement situation, will mean large far-field diffraction spreading of the laser output beam. 

 

(4) Long-Radius (Near-Planar) Resonators

Another elementary resonator configuration, and one that was used in many of the earliest laser devices, is the near-planar or long-radius stable resonator of Figure 9. A planar or flat-mirror resonator can be regarded as the limiting situation of a long-radius stable resonator as the radii of curvature of the two mirrors go to infinity. The resonator parameters then become \(R_1\approx R_2\approx\infty\) and \(g_1\approx g_2\approx 1\).  If we let \(R_1=R_2=R\), the spot sizes in this situation all become large and essentially equal, in the form 

\[\tag{12}\omega^2_0\approx\omega^2_1\approx\omega^2_2\approx\frac{L\lambda}{\pi}\times\sqrt{\frac{R}{2L}}\qquad\text{for}\;R\gg L.\]

In gaussian beam terms the long-radius resonator has a very long and large waist, of which the resonator encompasses only a very short central part. As the mirror radii become infinite the spot sizes become infinite also, though only very slowly, with the radius increasing as \((R/2L)^{1/4}\). The exactly planar resonator occurs right on the stability boundary, at \(g_1=g_2=1\) and so the gaussian theory fails at and beyond that point. 

Long-radius resonators, although they can have larger mode volumes, are generally avoided in practical laser designs because of their very great alignment difficulties. Since the centers of curvature of the mirrors are cantilevered far out beyond the ends of the resonator, at distances \(\pm R\), very delicate angular

 

 

 

alignment of the mirrors becomes necessary if the optical axis of the resonator (which passes through these two centers of curvature) is to be kept aligned within the center of the laser medium itself. Long-radius mirrors are also difficult to manufacture and to test. Note, for example, that for a 2.5 cm diameter mirror with a 50-m radius of curvature, the total sag at the center of the mirror relative to the edges is only \(\approx1.5\mu m\). At the same time the spot size enhancement factor for a laser resonator that is \(L=50\) cm long is only \((R/2L)^{1/4}\approx2.7\). 

 

(5) Near-Concentric Resonators

The near-concentric stable resonator is another design which is on the boundary of the stability region, and which can give large spot sizes at the end mirrors, but now with a vanishingly small spot size in the center of the resonator, as illustrated in Figure 10. 

For a near-concentric resonator, in which the cavity length \(L\) is less than the sum of the two radii \(R_1+R_2\) by the small amount \(\Delta L\), the resonator parameters are given by \(R_1\approx R_2\approx R=L/2+\Delta L\) and \(g_1\approx g_2=-1+\Delta L/R\). The spot size at the central waist is then given by 

\[\tag{13}\omega^2_0\approx\frac{L\lambda}{\pi}\times\sqrt{\frac{\Delta L}{4L}}\qquad\text{for}\;\Delta L\ll L,\] 

 

 

 

and the end-mirror spot sizes by 

\[\tag{14}\omega^2_1=\omega^2_2\approx\frac{L\lambda}{\pi}\times\sqrt{\frac{4L}{\Delta L}}\qquad\text{for}\;\Delta L\ll L.\] 

The mirror radii are now physically reasonable, and the spot sizes can be adjusted in operation by using translatable end-mirror mounts. The mirrors can then be pulled slowly apart in order to bring the resonator closer to or even across the stability boundary, by making the incremental length \(\Delta L\) small or even negative. 

The central portion of the resonator, where the spot size becomes very small, is then not very useful, at least for laser power extraction. More seriously, the mirror centers of curvature now become very close to each other at the center of the cavity, as illustrated in Figure 10. Hence this resonator again becomes very sensitive to large axis misalignments caused by very small mirror misalignments. 

 

(6) Hemispherical Resonators 

The resonator design that is by far the most commonly used in practical stable-resonator lasers, such as, for example, most medium and low-power gas lasers, is the near-hemispherical or half-concentric stable resonator, of Figure 11, for which the resonator parameters are \(R_1=\infty\) and \(R_2=L+\Delta L\), and hence \(g_1=1\) and \(g_2=\Delta L/L\approx 0\). This resonator is like half of a near-concentric 

 

 

resonator, with the very small spot size at the plane-mirror end given by 

\[\tag{15}\omega^2_0=\omega^2_1\approx\frac{L\lambda}{\pi}\times\sqrt{\frac{\Delta L}{L}}\qquad\text{for}\;\Delta L\ll L,\] 

and the large spot size at the curved mirror end given by 

\[\tag{16}\omega^2_2\approx\frac{L\lambda}{\pi}\times\sqrt{\frac{L}{\Delta L}}\qquad\text{for}\;\Delta L\ll L.\]

Again by making small adjustments in the resonator length the spot size \(\omega_2\) at the curved mirror end can be made as large as desired, whereas the spot size \(\omega_1=\omega_0\) at the flat mirror end becomes corresponding tiny.

The mode volume in a near-hemispherical resonator is then essentially in the shape of a cone, as in Figure 12. Lasers with near-hemispherical resonators are usually designed with the cavity somewhat longer than the active laser volume, and with the laser tube or rod placed near the large-diameter end of the cavity.

Readers may note in typical small internal-mirror He-Ne lasers, for example, that the discharge region is usually stopped well short of the flat-mirror end of the laser; and in some situations a tapered laser bore is even employed.

 

Hemispherical Laser Construction

The great advantage of the hemispherical design, however, is that the mode alignment difficulties in this design are largely if not completely eliminated. Consider, for example, the construction of an internal-mirror laser structure in which the mirrors are to be attached directly to the laser tube bore, as in Figure 13. The mechanical requirements are then first that the bore itself be fabricated with a sufficiently accurate length \(L\) compared to the mirror radius R to give the desired \(\Delta L\) and hence the desired spot size.

This is not, in general, a severe requirement. The second requirement is that the flat-mirror end of the bore be sufficiently perpendicular to the bore axis that the mirror normal will travel down the bore, which is again typically not a severe mechanical tolerance.

The curved mirror can thus be brought into alignment with the perpendicular axis through the bore either by angular adjustments, or alternatively by sideways translation of the curved mirror relative to the bore. Gas lasers can thus be aligned on production lines purely by translation of the curved mirror, if this proves simpler than providing an angular adjustment.

For all these reasons hemispherical resonator designs, or slightly more complicated variations, are used in many practical lasers. Lasers with external mirrors usually provide for angular adjustments in both mirror mounts, and possibly a small length adjustment in one of the mounts; whereas many small gas lasers have completely fixed or internally mounted mirrors.

 

 

 

 

 

(7) Concave-Convex Resonators 

Even with a hemispherical design, the spot size of the \(\text{TEM}_{00}\) mode in a stable gaussian resonator is often much smaller than we would like in order to extract energy efficiently from a larger-diameter laser medium. Any design in fact which operates close to the stability boundary can give larger mode sizes, but only at the expense of high sensitivity to small fluctuations in the mirror curvature or spacing. Such resonators are also likely to be highly sensitive to effects such as pump-power dependent thermal focusing effects in solid-state laser rods. 

By moving out into the regions of the stability diagram beyond \(g_1=1\) or \(g_2=1\), it is possible to have so-called concave-convex stable resonators such as are illustrated in Figure 15. In these resonators the waist lies outside the resonator, and the mode volume is comparatively large everywhere inside the resonator. Resonators such as these have found some practical use, but generally tend to require inconveniently long mirror radii and sensitive alignment procedures. 

 

(8) Unstable Confocal Resonators 

Finally, it is also possible to have resonators that are confocal but asymmetric, i.e., resonators in which the two mirrors have different radii of curvature \(R_1\) and \(R_2\) but their focal points still coincide, as in Figure 15. The spacing for a general asymmetric confocal resonator is 

\[\tag{17}R_1/2+R_2/2=L,\]

which can be translated into the condition

\[\tag{18}g_1+g_2=2g_1g_2.\]

Examination of the stability diagram will show that this condition corresponds to a contour or locus which is unstable everywhere in the \(g_1,g_2\) plane, except at the symmetric confocal point \(g_1=g_2=0\), and at the planar symmetric point \(g_1=g_2=1\) (The mirror focal points in the latter situation are coincident at infinity.) 

All asymmetric confocal resonators are thus unstable, as is also evident from the gaussian beam parameters in the previous section. In fact, we will see later that such confocal unstable resonators are of particular interest because one of the circulating beams in such a resonator is always a collimated beam, which can be particularly useful as the collimated output beam from the unstable resonator. The inset in Figure 15 shows how a typical ray diverges outward on successive bounces in such a resonator. 

The symmetric confocal resonator shown in Figure 8 is thus obviously located at a kind of singular point or saddle point in the stability diagram, since small deviations from this point in different directions can take we either into stable or unstable regions of the plane.

 

 

 

 

3.  GAUSSIAN TRANSVERSE MODE FREQUENCIES 

Because of the Guoy phase shift and its dependence on Hermite-gaussian mode number, the different transverse modes in a stable gaussian resonator have different resonance frequencies and transverse mode frequency shifts that are sometimes of practical interest. In this section, therefore, we derive and summarize the analytic formulas for these transverse modes. 

 

Transverse Mode Phase Shifts 

The total phase shift from one end of the cavity to the other, including the \(k(z_2-z_1)\equiv kL\) term and the Guoy phase shift terms, for an \(nm\)-th order Hermite-gaussian mode is given by 

\[\tag{19}\phi(z_2-z_1)=kL-(n+m+1)\times[\psi(z_2)-\psi(z_1)]\]

where the Guoy phase shifts \(\psi\) are related to the gaussian beam parameters by 

\[\tag{20}\psi(z_i)=\tan^{-1}(z_i/z_R).\]

If we use Equations 5 and 6 for \(z_1\) and \(z_2\) in terms of \(g_1\) and \(g_2\) from the first section of this chapter, it is possible to show (after a fair amount of algebra) that the total Guoy phase shift along the resonator length is given in terms only of the g parameters by the formula 

\[\tag{21}\psi(z_2)-\psi(z_1)=\cos^{-1}\pm\sqrt{g_1g_2}\]

where the \(+\) sign applies in the upper right quadrant \((g_1,g_2>0)\) and the \(—\) sign applies in the lower left quadrant. (Note that in the lower left quadrant, for example, near the concentric situation \(g_1=g_2=-1\), the resonator becomes substantially longer than the gaussian beam waist. The beam then picks up essentially the full \(180^\circ\) Guoy phase shift, which means that the cosine of this phase shift must approach \(-1.)\) 

 

Transverse Mode Frequencies

The resonance condition for a standing-wave cavity says that this one-way phase shift must be an integer number of half cycles, or the total round-trip phase shift must be an integer multiple of \(2\pi\), so that we must satisfy 

\[\tag{22}\frac{\omega L}{c}-(n+m+1)\cos^{-1}\pm\sqrt{g_1g_2}=q\pi,\qquad q=\text{integer}.\]

The resonance frequencies of the axial-plus-transverse modes in the cavity must thus be given by

\[\tag{23}\omega=\omega_{qnm}=\left[q+(n+m+1)\frac{\cos^{-1}\pm\sqrt{g_1g_2}}{\pi}\right]\times\frac{2\pi c}{p},\]

where \(p\equiv 2L\) is the round-trip distance or perimeter of the cavity. A little inspection will show that the Guoy phase shift factor appearing in this equation takes on the limiting values

\[\tag{24}\frac{\cos^{-1}\pm\sqrt{g_1g_2}}{\pi}\approx\left\{\begin{align}&0\quad\text{for}\;\text{the}\;\text{near-planar}\;\text{situation,}\;g_1,g_2\rightarrow1,\\&1/2\quad\text{for}\;\text{the}\;\text{near-confocal}\;\text{situation,}\;g_1,g_2\rightarrow0,\\&1\quad\text{for}\;\text{the}\;\text{near-concentric}\;\text{situation,}\;g_1,g_2\rightarrow-1.\end{align}\right.\]

Let us examine these results in a bit more detail.

 

Near-Planar (Long-Radius) situation 

For the near-planar situation the transverse mode frequencies \(\omega_{qnm}\) associated with a given axial mode \(q\) are all clustered on the high-frequency side of the associated axial mode frequency \(\omega_{q00}\), as shown in the top line of Figure 16, with equal spacings that are small compared to the axial mode spacing.

The mode spot size is large in this situation, and the resonator length is short compared to the Rayleigh range for the gaussian beam. The transverse modes therefore pick up very little additional Guoy phase shift. What phase shift they do pick up subtracts from the plane-wave \(\omega L/C\) term, and therefore a higher frequency is required to make up the \(q\pi\) total phase shift—in other words, the higher-order transverse mode frequencies are always on the high-frequency side of the associated axial mode. 

The axial-plus-transverse mode spectrum in the near-planar situation thus appears as the usual axial mode frequencies spaced by \(q\times2\pi c/p\), with higher-order transverse modes clustered as satellite modes on the high-frequency side of each axial mode.

If we think, for example, of fixing the resonator length L and gradually increasing the mirror curvatures \(R\) to bring the resonator gradually inward from the near-planar situation to the near-confocal situation, then the mode spot size gradually gets smaller; the transverse derivatives get larger; the Guoy phase shift contributions become larger; and the transverse mode spacings gradually broaden out, as illustrated in Figure 16.

The higher-order transverse modes \(\omega_{qnm}\) associated with a given axial mode q move out toward, and in fact pass above, the frequencies of the higher-frequency axial modes, e.g., the \(q+1,00\) and \(q+2,00\) and higher axial modes.

 

 

 

Confocal Resonators 

The confocal resonator represents the situation, in fact, where the 01 and 10 transverse modes associated with the \(q\)-th. axial mode move out to fall exactly halfway between the \(q\) and \(q+1\) axial modes; the \(q11,\;q02\) and \(q20\) modes of the \(q\)-th axial mode move out to coincide with the \(q+2,00\) mode; and so forth.

The confocal resonator thus represents a situation where all the even-symmetry transverse modes of the cavity are exactly degenerate at the axial mode frequencies of the laser; and all the odd-symmetry modes are exactly degenerate at the "half-axial" positions midway between the axial mode locations. 

 

Scanning Fabry-Perot Interferometers 

This degeneracy in confocal transverse mode frequencies is of considerable practical importance for scanning Fabry-Perot interferometers or optical frequency tunable filters. A scanning Fabry-Perot interferometer is a passive optical cavity whose length \(L\) can be scanned over a few optical half-wavelengths, usually by means of a piezoelectric crystal or piezoelectric stack mounted behind one of the end mirrors. The resonant frequencies of the cavity can thus be scanned over a few axial modes or free spectral ranges of the cavity. 

If the output signal from a laser is sent through such a cavity while it is being scanned, and the optical signal transmitted through the scanning interferometer is displayed on an oscilloscope, then a large detected signal will be seen every time the scanning cavity frequency equals one of the oscillation frequencies in the laser output. The scanning interferometer thus provides an electrically tunable filter for examining and displaying the frequency components in the laser signal.

One practical difficulty with such interferometers, however, is the existence of higher-order transverse modes in the scanning interferometer.

Suppose the laser input signal has only a single frequency component, but that the input laser beam

 

 

is misaligned or improperly mode matched to the scanning interferometer cavity. Then as the interferometer cavity is scanned, the laser signal will successively come into resonance with and excite different higher-order transverse modes of the interferometer (Figure 17); and these in turn will produce transmitted signals in the interferometer output.

The single laser input frequency will produce multiple spurious, or at least unwanted, apparent frequency components in the scanning interferometer display. 

One way to eliminate these spurious transmission signals is to carefully align and mode-match the laser signal into the interferometer cavity—generally a delicate and difficult task. An alternative and much more practical solution is to make the scanning Fabry-Perot interferometer be an exactly confocal cavity, so that all the higher-order transverse modes are exactly degenerate in frequency.

This then means that the input beam to such an interferometer need not be mode-matched into the interferometer in order to excite only a single transmission resonance. Rather the input beam can be misaligned and mismatched to a significant extent, and yet it will still excite resonance transmission at only a single frequency. 

Commercial scanning Fabry-Perot interferometers are thus very often designed as confocal resonators. A mismatched input beam to a confocal cavity still excites a mixture of lowest and higher-order transverse modes in the interferometer; but in the confocal situation their resonance frequencies are all exactly degenerate.

More precisely, the even-order transverse modes are all degenerate at the axial mode frequencies, whereas the odd-order modes are degenerate halfway in between.

One disadvantage of a confocal scanning interferometer is that the free spectral range of the interferometer is now only half the axial mode spacing, or c/4L in hertzian frequency.

The odd-order modes can be eliminated or at least greatly attenuated, however, by at least centering and aligning a spatially coherent input beam to the interferometer, even if proper mode matching is not obtained.

Observing how the scanning interferometer signal sharpens up for very small adjustments of the interferometer length about the confocal condition is also an interesting experimental demonstration. 

 

Concentric Resonator situation

If we continue to sharpen the mirror curvature so that the resonator approaches the concentric condition, the \(q01\) mode in Figure 16 as an example will move out until it approaches the \(q+1,00\) mode from the low-frequency side; and so forth. In the concentric limit, therefore, the axial-transverse mode spectrum will come again to look like a set of axial modes with closely spaced transverse mode satellites, but with these now clustered on the low-frequency side.

It is important to realize, however, that these clusters of modes, although they are closely grouped in frequency, in this limit actually represent different axial as well as transverse mode numbers.

 

Transverse Mode Beats

One of the best ways to observe experimentally the presence of transverse mode oscillations, in any resonator configuration, is to look for the transverse mode beats between whatever axial-transverse modes may be ocillating in a given laser, using a suitable photodetector and radio-frequency receiver or spectrum analyzer.

Suppose a laser is oscillating simultaneously in two such modes with indices \(q_1n_1m_1\) and \(q_2n_2m_2\). The output signal from this laser impinging on a suitable photodetector may then be written in the form

\[\tag{25}\varepsilon(x,y,t)=\tilde{u}_1(x,y)e^{jw_1t}+\tilde{u}_2(x,y)e^{jw_2t},\] 

where \(\tilde{u}_1(x,y)\) and \(\tilde{u}_2(x,y)\) are the transverse patterns of these two modes on the photodetector. The total photocurrent or photosignal that this optical field will produce from a typical square-law optical detector is then given by

\[\tag{26}\begin{align}i(t)&=\int\int|\varepsilon(x,y,t)|^2 dxdy\\&=\int\int|\tilde{u}_1(x,y)e^{jw_1t}+\tilde{u}_2(x,y)e^{jw_2t}|^2 dxdy\\&=I_{01}+I_{02}+I_{12}e^{j(w_2-w_1)t}+c.c.,\end{align}\]

where the \(dc\) currents \(I_{01}\) and \(I_{02}\) are given by

\[\tag{27}I_{01}\int\int|\tilde{u}_1(x,y)|^2\quad dA\quad\text{and}\quad I_{02}=\int\int|\tilde{u}_2(x,y)|^2\;dA,\]

and the complex phasor amplitude \(I_{12}\) is given by

\[\tag{28}I_{12}=\int\int\tilde{u}_1^*(x,y)\times\tilde{u}_2(x,y)dA.\] 

The integrals are taken over the active surface area of the photodevice; and it is assumed that the photodetector response averages over a few optical cycles, so that sum-frequency cross products at \(\omega_1+\omega_2\) can be ignored.

The total output signal from the photodetector thus consists of dc currents \(I_{01} and \(I_{02}\) due to each beam separately, plus a cross product or beat frequency term \(I_{12}\) between the two signals, in the form

\[\tag{28}i(t)=I_{01}+I_{02}+I_{12}\cos[(\omega_2-\omega_1)t+\phi_{12}],\]

where \(\omega_2-\omega_1\) is the difference frequency or beat frequency between the two oscillating modes.

If several such modes are oscillating, similar beat frequencies between any pair of the oscillating modes may be observed. For a stable gaussian resonator these difference frequencies will be given in general by

\[\tag{30}\begin{align}\omega_2-\omega_1&=\omega_{q_2n_2m_2}-\omega_{q_1n_1m_1}\\&=\left[(q_2-q_1)+(n_2-n_1+m_2-m_1)\frac{\cos^{-1}\pm\sqrt{g_1g_2}}{\pi}\right]\times\frac{2\pi c}{p}\\&=\Delta q\times\Delta\omega_{ax}+\Delta(n+m)\times\omega_{trans},\end{align}\] 

where \(\Delta q=q_2-q_1,\Delta_n=n_2-n_1\) and so on. The beat signal will thus contain components at various integral multiples of the axial mode spacing \(\Delta\omega_{ax}\) and the transverse mode spacing \(\Delta\omega_{trans}\).

 

Overlap Integrals and Orthogonality

Note that the magnitude of each beat signal will be given by the overlap integral between the transverse modes on the photodetector surface, or 

\[\tag{31}I_{12}=\bigg|\int\int\tilde{u}^*_1(x,y)\times\tilde{u}_2(x,y)dx\;dy\bigg|.\]

For ideal Hermite-gaussian modes (and to a lesser extent for real resonator modes), this overlap integral between different transverse modes will integrate out to zero because of the transverse orthogonality or near-orthogonality between different transverse modes.

This is only true, however, if the integral is taken over the full transverse cross section of the modes. This overlap integral will in general not vanish if the integration is taken only over part of the beam cross section.

The area of integration may be limited either because the photodetector itself has a limited area, or because of an aperture inserted in the beam in front of the detector.

Inserting such a partial aperture is, in fact, a standard technique for making visible transverse mode beats that are otherwise not seen.

Transverse mode beat frequencies (which are typically in the range from a few MHz to a few hundred MHz) can be measured with great accuracy, since any absolute frequency shifts in the laser oscillation due to mechanical vibrations or thermal expansion are essentially the same for both modes, and cancel out of the difference frequency.

Such mode beats thus provide both a convenient diagnostic for the presence of higher-order transverse-mode oscillations, and also a particularly good test for resonator mode theory.

Note that the transverse beat frequencies are directly tied to the Guoy phase shifts of the different modes, which are in turn directly tied to the transverse spatial derivatives and hence the transverse mode patterns of the modes.

Experiments that have been done on transverse mode beats in stable gaussian resonators have always yielded results in excellent agreement with theory, and thus have served as at least a strong indirect confirmation of the validity of the gaussian resonator mode theory.

 

 

4.  MISALIGNMENT EFFECTS IN STABLE RESONATORS

The effects of mirror misalignments on stable two-mirror resonators can be relatively complicated, since misalignment or misadjustment of either mirror both rotates and translates the optical axis of the resonator.

One way to handle such 

 

 

misalignments is to use the techniques for misaligned ray matrix systems discussed in this tutorial RAY OPTICS WITH MISALIGNED ELEMENTS. We can give in this section, however, a brief description of the axis displacement and misalignment produced in a simple two-mirror cavity by angular misalignment of either end mirror. 

 

Misalignment Analysis

The optical axis in a two-mirror resonator is by definition the line passing through the centers of curvature \(C_1\) and \(C_2\) of the two end mirrors. The quadratic phase curvatures of the two mirrors are centered on or normal to this axis. If the cavity also contains any kind of aperture (including the apertures defined by the mirrors themselves), rotation of an end mirror will translate the optical axis relative to this aperture or, alternatively, will cause the aperture to be effectively off center with respect to the resonator axis.

The presence of an off-center aperture will tend to produce resonator eigenmodes which are mixtures of the even and odd eigenmodes of the aligned resonator. Solving for the exact eigenmodes and their exact diffraction losses in this situation becomes a complicated calculation. 

Simple geometry can at least tell us how far the optical axis will be translated and rotated by a small angular rotation of either end mirror.

Let \(\theta_1\) and \(\theta_2\) be the small angular rotations of the two end mirrors and \(\Delta x_1\)and \(\Delta x_2\) be the small sideways translations of the new or misaligned optical axis at the point where it intercepts the end mirrors, as shown in Figure 18. (Alternatively, \(\Delta x_1\) and \(\Delta x_2\) can represent the off-center translations of the apertures at those two mirrors.) From Figure 18 and some simple geometry, we can then evaluate these displacements as

\[\tag{32}\begin{align}&\Delta x_1=\frac{g_2}{1-g_1g_2}\times L\theta_1+\frac{1}{1-g_1g_2}\times L\theta_2\\&\Delta x_2=\frac{1}{1-g_1g_2}L\Delta\theta_1+\frac{g_1}{1-g_1g_2}L\Delta\theta_2.\end{align}\]

One criterion for judging the seriousness of misalignment effects is then to compare these displacements \(\Delta x_1\) and \(\Delta x_2\) with the resonator spot sizes \(\omega_1\) and \(\omega_2\) at the same end mirrors.

The angular displacement of the resonator axis (which can be important in evaluating far-field pointing accuracy, for example) can also be evaluated from

\[\tag{33}\Delta\theta\equiv\frac{\Delta x_2-\Delta x_1}{L}=\frac{(1-g_2)\theta_1-(1-g_1)\theta_2}{1-g_1g_2}.\]

Note that the sensitivity of all these measures to angular misalignment blows up as \(g_1g_2\rightarrow1\), i.e., as the resonator design approaches the stability boundary on either the planar (long-radius) or the near-concentric sides of the stability region.

 

 

5.  GAUSSIAN RESONATOR MODE LOSSES

The gaussian beam results developed in this chapter thus far are based on the assumption that the resonator end mirrors are infinitely wide in the transverse direction, or at least extend out so far compared to the gaussian spot size of the gaussian modes that any aperture diffraction effects are entirely negligible. 

Introduction of a finite aperture into a stable gaussian resonator then modifies these results, though generally by a small amount if the aperture diameter is large compared to the gaussian spot size. In this section we will review briefly the mode distortions and diffraction losses that result from introduction of finite apertures or mirror sizes. 

 

Resonator Fresnel Number 

A very important parameter for discussing aperture effects in finite-diameter stable (or for that matter unstable) optical resonators is the resonator Fresnel number \(Nf\), which is commonly defined as follows.

Let \(2a\) represent the transverse width of the resonator end mirrors in the \(x\) or \(y\) directions in a one-dimensional strip mirror situation, or alternatively the diameter of the circular end mirrors in a circularly symmetric situation. The resonator Fresnel number \(N_f\) is then defined, just as in the previous chapter, by 

\[\tag{34}\text{resonator}\;\text{Fresnel}\;\text{number},N_f\equiv\frac{a^2}{L\lambda}.\]

This parameter is obviously the number of Fresnel zones across one end mirror, as seen from the center of the opposite mirror. There are also, however, a number of other significant interpretations of this parameter. 

We note first that the spot size on the end mirror of a symmetric confocal resonator of length L is given by \(\omega^2_1=L\lambda/\pi\), and that other stable resonators have spot sizes which differ from this only by some numerical factor which depends on the g values. We can therefore write the simple expression

\[\tag{35}\frac{\text{resonator}\;\text{mirror}\;\text{surface}\;\text{area}}{\text{confocal}\;\text{TEM}_{00}\;\text{mode}\;\text{area}}=\frac{\pi a^2}{\pi\omega^2_1}=\pi N_f.\]

In other words, the ratio of the resonator mirror area to the area of the lowestorder confocal mode, with the mode area defined for this purpose by \(\pi\omega^2_1\), is given by the resonator Fresnel number multiplied by \(\pi\).

We can express much the same point in a slightly different way by recalling from an earlier chapter that the outer radius of an n-th order Hermite-gaussian mode (for \(n>1\)) is given to a good approximation by

\[\tag{36}s_n\approx\sqrt{n}\omega_1\approx\sqrt{nL\lambda/\pi}.\]

We might ask therefore what is the largest-order Hermite-gaussian or Laguerre-gaussian mode (indicated by index \(N_{max}\)) that will still fit within the aperture of width or diameter \(2a\)? The answer is again

\[\tag{37}N_\text{max}=a^2/\omega^2_1=\pi N_f.\]

The resonator Fresnel number \(N_f\) is thus essentially an indicator of how large the resonator aperture is compared to the confocal mode size in that resonator, or alternatively a measure of the order of transverse modes we can go to before these higher-order modes begin to be significantly perturbed by the aperture edges.

 

Finite-Diameter Resonator Mode Losses

The exact transverse eigenmodes and eigenvalues of stable gaussian resonators with finite apertures must be calculated by finding the eigensolutions to the resonator integral equation using the appropriate Huygens' kernel with finite limits of integration, as described in earlier chapters.

Analytical solutions to these equations with finite mirror diameters are generally not available (with a few limited exceptions), and so the eigensolutions must usually be found numerically. This is most commonly done using some variation of the Fox and Li iterative technique described earlier.

The results of some of these calculations will be summarized briefly here, and in the References at the end of this section.

Figure 19 shows, for example, the power losses per bounce (that is, per one-way transit) in both confocal and planar resonators having circular finite-diameter mirrors, for the first few azimuthally symmetric and radially varying modes in each situation. (Similar curves could be calculated and plotted for modes of higher azimuthal index m; they would in general have similar shapes, and show significantly higher losses.)

In the confocal situation, for example, which we already know to be a very small mode diameter or low loss situation, we see that as soon as the resonator Fresnel number becomes greater than about unity, the diffraction losses for the \(\text{TEM}_{00}\) mode become very small, on the order of 1% per bounce or less. The higher-order \(\text{TEM}_{10}\)and \(\text{TEM}_{20}\) modes, which have higher-order radial variations, have larger losses at any given Fresnel number, but all of these losses

 

 

 

decrease very rapidly and become very small as the Fresnel number increases much above unity. 

For the planar (flat) mirror resonator, by contrast, the losses are significantly larger, although again these losses decrease with increasing Fresnel number. The right-hand plot shows, in fact, that when \(N\) becomes greater than about 10, the losses become less than \(\approx\)1% per bounce and continue decreasing rapidly with increasing Fresnel number. 

 

Finite-Diameter Mode Patterns 

The exact mode calculations for the confocal resonator situation will also show that when the Fresnel number becomes greater than about unity, the exact mode pattern over the central portion of the resonator becomes very similar to the gaussian beam pattern predicted by the infinite-mirror gaussian mode theory presented in the preceding sections of this chapter.

The exact mode losses in this limit will, however, actually be considerably smaller than the spillover losses that we might calculate by using the gaussian mode patterns and calculating the amount of energy going past the mirror edges on each bounce.

In other words, the exact mode patterns will distort near the mirror edges in such a way as to reduce the mode amplitude and the power losses at the mirror edges below even the (small) value predicted by the gaussian beam theory.

The planar resonator mode patterns can not approach such a gaussian limit, since the planar resonator is on the boundary of the stability region, where the gaussian spot size blows up to infinity.

The planar resonator mode will, however, have a relatively smooth radial mode pattern, something similar to a \(J_n(r)\) Bessel-function pattern with the first (or \(n\)-th) null of the Bessel function occurring at the mirror edges.

There will also be small Fresnel ripples on top of this pattern and a small but finite value at the mirror edges, with the amplitudes of both the Fresnel ripples and the mirror-edge value becoming increasingly small as the Fresnel number (i.e., the mirror diameter) of the resonator increases.

 

 

 

Mode Losses and Phase Shifts 

Figure 20. shows further results from a large number of additional exact mode calculations for finite-diameter circular-mirror resonators with different \(g\) values ranging from \(g=0\) (confocal resonators) to \(g=1\) (planar resonators). Plotted in this situation are both the loss per bounce (expressed in dB) and the phase angle of the one-way resonator eigenvalue, for both the \(\text{TEM}_{00}\) and \(\text{TEM}_{01}\) modes, i.e., the lowest-order radial modes for the azimuthal indices \(m=0\) and \(m=1\). 

Of particular interest here are the phase shifts of the eigenvalue \(\tilde{\gamma}\), where \(\tilde{\gamma}\) is the resonator eigenvalue for a one-way pass. These phase shifts are the exact versions of the Guoy phase shifts \(\psi(z_2)-\psi(z_1)\) given in the ideal gaussian limit by Equation 21.

They thus determine the exact transverse mode spacing in the finite-diameter mirrors. We see, for example, that the phase shift for the \(g=0\) confocal \(\text{TEM}_{00}\) mode is exactly \(90^\circ\), corresponding to the phase shift through the waist region (from \(-z_R\) to \(z_R\)) in the confocal situation; whereas the same phase shift for the \(\text{TEM}_{01}\) situation is \(180^\circ\), corresponding to \(\Delta m=1\).

More generally, we can see that in both the losses and the phase shifts, as \(N\) increases for a given \(g\) value, there is a shoulder or break point—admittedly, a rather soft shoulder—above which the mode losses decrease more rapidly, whereas the added phase shift becomes essentially constant at the value predited by the cos \(\sqrt{g_1g_2}\) formula for the ideal gaussian situation.

This break point represents the value of \(N\) at which the truncation of the gaussian mode fields by the finite-diameter mirrors becomes sufficiently weak that the gaussian approximation for the mode shape is exceedingly good. This diameter increases with increasing g value, and never occurs at all for the plane mirror situation where \(g=1\)

 

Approximate Formulas for Resonator Losses 

A number of approximate or emperical formulas for finite-diameter resonator losses have been developed by various researchers. In general these are formulas for the one-way power loss per pass \(\delta\equiv1-|\tilde{\gamma}|^2\) where \(\tilde{\gamma}\) is the single-pass eigenvalue. Examples of these formulas include: 

\[\begin{align}\tag{38}\text(1)\;\text{for}\;\text{confocal}\;\text{square}\;\text{mirrors},\\&\delta\approx8\pi\sqrt{2N}\text{exp}(-4\pi N)\qquad&\text{for}\; N\geq0.5,\\&\delta\approx1-16N^2\text{exp}[-2(2\pi N/3)]\qquad&\text{for}\;N\rightarrow 0;\\(2)\text{for}\;\text{confocal}\;\text{circular}\;\text{mirrors},\\&\delta\approx\pi^22^4N\text{exp}{-4\pi N}\qquad&\text{for}\;N\;\geq1,\\&\delta\approx1-16N^2\text{exp}[-2(2\pi N/3)^2]\qquad&\text{for}\;N\;\rightarrow0\\(3)\text{for}\;\text{planar}\;\text{strip}\;\text{mirrors},\\&\delta\approx0.12N^{-3/2}\qquad&\text{for} N\;\geq1;\\(4)\text{for}\;\text{planar}\;\text{circular}\;\text{mirrors},\\&\delta\approx0.33N^{-3/2}\qquad&\text{for}\;N\geq1.\end{align}\]

More complicated formulas are given in various publications by the Soviet researcher Vainshtein (or Weinstein in some English translations).

 

Experimental Verification

The numerical results shown in Figure 20. may be regarded in a sense as "computer experiments" which serve both to verify the ideal gaussian mode theory and to define its limits.

Moreover, numerous measurements that have been made of resonator spot sizes and of transverse mode beat frequencies serve to verify the gaussian mode theory in quite exact detail. (Note that the transverse mode beats in particular represent a subtle but very significant confirmation of the theory.)

Detailed experimental tests of the mode losses predicted by the exact theory have been quite limited, however. Such experiments are difficult because the losses are generally small, and thus easily obscured by other mirror and scattering losses, Brewster window losses, effects of cavity misalignment, and the like.

Figure 21. shows the very good comparison between experiment and theory for the \(\text{TEM}_{00}\) mode losses that was obtained in a near-hemispherical resonator with one flat mirror and curved mirror of radius 60 cm, as the mirror spacing was pulled apart toward the stability boundary at \(L=60\) cm. Note that the exact mirror spacing at which the losses reach some particular loss level as L is increased depends on the diameter or Fresnel number of the resonator, with

 

 

this value moving out toward the limiting value of \(L=600\) mm as the mirror diameter increases. 

We might say, in fact, that the stability diagram of Figure 4. has infinitely sharp edges between stable and unstable regions only if the mirror diameter is infinite. For finite-diameter mirrors the boundary line between stable and unstable regions is more like a fuzzy boundary region, where the resonator behavior changes over from ideal gaussian behavior, uninfluenced by the mirror edges, to a region where the mode pattern and mode losses are strongly influenced by the mirror or aperture edges. This boundary region has a width that decreases toward zero as the mirror diameter becomes very large. 

 

Hole-Coupled Resonators

A number of calculations and experiments have also been done on the modes of hole-coupled resonators, that is, stable gaussian resonators with coupling holes in the center of the end mirror.

Such mirrors can be comparatively simple to fabricate (for example by drilling a central hole in a polished copper or molybdenum mirror), and the use of hole coupling might seem attractive as one way of bypassing the difficulties of finding low-loss transparent mirror substrates and low-loss reflective coatings for high-power \(\text{IR}\) or \(\text{UV}\) lasers. 

The general conclusion from exact Fox and Li calculations of such resonators, however, is that this is not an effective method for obtaining significant output coupling while simultaneously maintaining good transverse mode performance. As soon as the diameter of a central coupling hole is made large enough to couple significant power out of the lowest-order gaussian mode in such a resonator, theory and experiment show that the resonator will convert to oscillation in a distorted lowest-order eigenmode—basically, a mixture of higher-order Hermite-gaussian modes—"which avoids the central hole and concentrates the modal energy in an annular ring around the hole.

Resonators quite generally, in fact, show a remarkable ability to distort their eigenmodes and, so to speak, "pull in their skirts" to minimize diffraction losses for whatever aperture or distortion may be placed in the resonator. The one form of hole coupling that may be effective is a large number of coupling holes, each individually small compared to the \(\sqrt{L\lambda/\pi}\) dimension for the resonator, distributed more or less uniformly over the entire surface of the end mirror.