GENERALIZED PARAXIAL RESONATOR THEORY

This is a continuation from the previous tutorial - Complex paraxial wave optics

 

we will use the generalized paraxial-wave concepts of the preceding tutorial to analyze the Hermite-gaussian modes of generalized paraxial optical resonators. The resulting analysis will show how all such resonators can be classified into four categories of resonators whose modes are either real or complex gaussian in character, and either stable or unstable in behavior. It will also show how complicated multielement paraxial resonators can be analyzed based on knowledge only of their round-trip \(\text{ABCD}\) matrices, whether real or complex.

The results of this analysis will provide nearly exact descriptions of the Hermite-gaussian resonator eigenmodes for real and complex stable resonators, and will provide at least a great deal of insight into the mode properties of real and complex unstable resonators.

We will also examine some of the special features of multielement real stable resonators, and give an analysis of the general orthogonality properties of optical resonator eigenmodes.

 

 

1.  COMPLEX PARAXIAL RESONATOR ANALYSIS

As a general analytical model for this tutorial, we consider either a standing-wave cavity or a ring-laser cavity in which all the optical elements can be described as generalized complex paraxial elements. Any hard-edged stops or apertures in the resonator are thus ignored, or at least their effects are deferred for later consideration.

A complete round trip around such a resonator, including any end mirrors or internal soft apertures, can then be completely described by a complex total \(\text{ABCD}\) matrix. 

 

Self-Consistent Lowest-Order Gaussian Solutions

The analysis of such a resonator then proceeds as follows. We must select some specific reference plane inside the resonator—perhaps just before the output mirror—and then evaluate the total \(\text{ABCD}\) matrix for one complete round trip inside the cavity, starting from and returning to this reference plane, as illustrated in Figure 1. We then ask: is there a complex Hermite-gaussian beam whose \(\tilde{q}\) 

 

 

and \(\tilde{v}\) values will be self-consistent, that is to say, self-reproducing after one trip around this resonator? 

Let us focus initially on the lowest-order Hermite-gaussian mode, so that the parameter \(\tilde{v}\) is not involved. Also, let us assume in all situations from now on that q means the reduced value of \(\tilde{q}\), if it necessary to make the distinction.

We then ask specifically if there is a self-consistent value of \(\tilde{q}=\tilde{q}_1=\tilde{q}_2\) such that after one complete round trip this value will return to its initial value, as given by

\[\tag{1}\tilde{q}_2=\frac{A\tilde{q}+B}{C\tilde{q}_1+D}=\tilde{q}_1.\]

To find this self-consistent \(\tilde{q}\) value, or better the corresponding \(1/\tilde{q}\) value, we rewrite this as

\[\tag{2}\left(\frac{1}{\tilde{q}}\right)^2+\frac{A-D}{B}\left(\frac{1}{\tilde{q}}\right)+\frac{1-AD}{B^2}=0,\]

using the relationship \(AD-BC=1\). The solution to this quadratic equation then gives the two self-consistent solutions 

\[\tag{3}\frac{1}{\tilde{q}},\frac{1}{\tilde{q}_b}=\frac{D-A}{2B}\mp\frac{1}{B}\sqrt{\left(\frac{A+D}{2}\right)^2-1.}\]

These are the self-consistent \(\tilde{q}\) values or "eigen-\(\tilde{q}\)-values" of the resonator, which we will label from now on by the subscripts \(a\) and \(b\).

 

Confined Gaussian Solutions

A second important question is then whether at least one of these solutions represents a confined gaussian beam solution.

That is, we recall that the exponent in the Hermite-gaussian functions is given by 

\[\tag{4}-j\frac{\pi x^2}{\tilde{q}\lambda}\equiv-j\frac{\pi x^2}{R\lambda}-\frac{x^2}{w^2}\quad\text{or}\quad\frac{1}{\tilde{q}}\equiv\frac{1}{R}-j\frac{\lambda}{\pi w^2}.\]

If the fields of the beam are to die off at large radius, one or the other of \(1/\tilde{q}_a\) or \(1/\tilde{q}_b\) in Equation 3 must have a negative imaginary part, corresponding to the \(-x^2/w^2\) term with \(w^2\) being a real and positive quantity.

If this is not the situation, then the wave fields will not fall off as exp \(-x^2/w^2\) at large distances from the axis, and this wave obviously cannot represent a physical solution carrying finite power. 

Deciding which if either of the solutions in Equation 3. corresponds to a confined solution can become somewhat difficult in the completely general situation where \(A\), \(B\) and \(D\) may all be arbitrary complex quantities.

 

Perturbation-Stable Gaussian Solutions

Self-consistency after each round trip, together with a confined mode pattern, might seem to be enough to characterize a Hermite-gaussian wave and to determine whether it provides a physically meaningful eigensolution for an optical resonator or a periodic complex paraxial system.

But, as Casperson showed, we must ask still another question, namely, are these confined and self-consistent eigensolutions stable against perturbations? That is, if we start a wave around the cavity with an initial \(\tilde{q}\) value close to one of these eigensolutions, \(\tilde{q}_a\) or \(\tilde{q}_b\), but differing by a small amount \(\Delta\tilde{q}\), will this deviation \(\Delta\tilde{q}\) increase or decrease after one round trip? 

To analyze this, we can put an input beam with an initial value of either \(\tilde{q}_1=\tilde{q}_a+\Delta\tilde{q}_1\) or \(\tilde{q}_1=\tilde{q}_b+\Delta{q}_1\) into the round-trip propagation formula in Equation 1, and evaluate the output \(\tilde{q}_2\) to first order in \(\Delta\tilde{q}_1\), in the form

\[\tag{5}\begin{align}\tilde{q}_2&=\frac{A(\tilde{q}_{a,b}+\Delta\tilde{q}_1)+B}{C(\tilde{q_{a,b}+\tilde{q}_1})+D}\approx\left[\frac{A\tilde{q}_{a,b}+B}{C\tilde{q}_{a,b}+D}\right]\left[\frac{1}{C\tilde{q}_{a,b}+D}\right]^2\times\Delta\tilde{q}_1\\&=\tilde{q}_{a,b}+\Delta\tilde{q}_2,\end{align}\]

where we consider only small perturbations about either of the self-consistent solutions \(\tilde{q}_a\) or \(\tilde{q}_b\), so that \(\Delta\tilde{q}_1\), \(\Delta\tilde{q}_2\ll\tilde{q}_a\) or \(\tilde{q}_b\). 

The unperturbed portions of \(\tilde{q}_1\) and \(\tilde{q}_2\)  cancel out for either of the self-consistent eigensolutions \(\tilde{q}_a\) or \(\tilde{q}_b\); and so the input and output perturbations are related by

\[\tag{6}\Delta\tilde{q}_2=\left[\frac{1}{C\tilde{q}_{a,b}+B}\right]\times\Delta\tilde{q}_1\equiv\left[\frac{1}{A+B/\tilde{q}_{a,b}}\right]\times\Delta\tilde{q}_1.\]

But if we plug in the self-consistent values for \(\tilde{q}_a\) or \(\tilde{q}_b\), this then leads to perturbation growth ratios that are given for the \(\tilde{q}_a\) eigensolution by

\[\tag{7}\frac{\Delta\tilde{q}_2}{\Delta\tilde{q}_1}\bigg|_{\tilde{q}_1=\tilde{q}_a}=\left[\frac{1}{A+B/\tilde{q}_a}\right]^2=\left[\frac{A+D}{2}+\sqrt{\left(\frac{A+D}{2}\right)^2-1}\right]^2\equiv\lambda^2_a,\]

and for the \(\tilde{q}_b\) eigensolution by

\[\tag{8}\frac{\Delta\tilde{q}_2}{\Delta\tilde{q}_1}\bigg|_{\tilde{q}_1=\tilde{q}_b}=\left[\frac{1}{A+B/\tilde{q}_b}\right]^2=\left[\frac{A+D}{2}-\sqrt{\left(\frac{A+D}{2}\right)^2-1}\right]^2\equiv\lambda^2_b,\]

where the quantities \(\lambda_a\) and \(\lambda_b\) are in fact just the eigenvalues of the \(\text{ABCD}\) matrix. We will often speak of \(\lambda_a\) and \(\lambda_b\) as the perturbation eigenvalues corresponding to the self-consistent solutions \(\tilde{q}_a\) and \(\tilde{q}_b\), respectively, although in fact the respective perturbations actually grow as \(\lambda^2\) rather than as \(\lambda\).

Note that the \(\pm\) signs in the two Equations 7 and 8 for \(\lambda_a\) and \(\lambda_b\) are exactly reversed from the \(\mp\) signs in the corresponding self-consistent Equation 3 for \(\tilde{q}_a\) and \(\tilde{q}_b\)

 

The Perturbation Eigenvalues

It will again be a convenient shorthand to define a "complex \(\tilde{m}\) value" for a general complex paraxial resonator as 

\[\tag{9}\tilde{m}\equiv\frac{A+D}{2}.\]

Note that this \(\tilde{m}\) value is half the trace of the ray matrix. As such it is invariant under many transformations—particularly the choice of reference plane within the cavity.

The perturbation eigenvalues then take on the simple forms 

\[\tag{10}\lambda_a,\lambda_b=\tilde{m}\pm\sqrt{\tilde{m}^2-1},\]

and we can also see that for all values of \(m\)

\[\tag{11}\lambda_a\lambda_b\equiv1.\]

These eigenvalues have the same form as the ray matrix eigenvalues for stable or unstable periodic focusing systems that we developed from a purely geometric ray analysis in an earlier tutorial (Equation 39), except that they now represent the perturbation-stability eigenvalues, or the eigenvalues for growth and/or decay of small perturbations about the self-consistent waves, in a general complex paraxial optical system.

 

 

 

 

2. REAL AND GEOMETRICALLY STABLE RESONATORS 

Let us consider first the situation where the \(\text{ABCD}\) elements within a resonator are all purely real, i.e., there are no transversely varying gains or losses, and the purely real half-trace or \(m\) value of this matrix falls within the range

\[\tag{12}-1\leq m\leq1\quad\text{(geometrically}\;\text{stable}\;\text{situation).}\]

This situation represents a straightforward generalization of the stable two-mirror gaussian optical resonators discussed in an earlier tutorial. 

 

Confined Mode in a Real and Stable Resonator 

The self-consistent gaussian \(\tilde{q}\) values derived in the previous section can be written for this situation as 

\[\tag{13}\frac{1}{\tilde{q}_a},\frac{1}{\tilde{q}_b}=\frac{D-A}{2B}\mp j\frac{\sqrt{1-m^2}}{B}=\frac{1}{R}\mp j\frac{\lambda}{\pi w^2},\]

where \((D-A)\) is real, but the second term is in all situations imaginary. Depending upon the sign of \(B\), which can be either a positive or a negative quantity, one or the other of these two solutions then corresponds to a confined gaussian beam solution with a positive real spot size \(w\).

In other words, for any multielement optical resonator with a purely real \(\text{ABCD}\) matrix and \(m^2<1\), there will be a confined and self-consistent gaussian beam which can be fitted within the multielement paraxial resonator, in exactly the same way that a gaussian beam was fitted between the mirrors of the simple two-mirror resonator.

An example of such a gaussian mode for a stable multielement resonator is illustrated in Figure 2. The other eigensolution then corresponds to a nonphysical gaussian beam with a transversely increasing intensity, and hence can be ignored. 

The radius of curvature of the confined solution (actually, of either the confined or unconfined solutions) at the chosen reference plane is then 

\[\tag{14}R=\frac{2B}{D-A},\]

whereas the spot size is 

\[\tag{15}w^2_2=\frac{|B|\lambda}{\pi}\times\sqrt{\frac{1}{1-m^2}.}\]

We see that the \(|B|\) parameter for the generalized \(\text{ABCD}\) situation plays something like the same role as the cavity length \(L\) in the simple two-mirror resonators of before tutorial, whereas \(m\) (or \(m^2)\) plays something like the role of \(g_1g_2\) in the two-mirror resonator.

 

Eigenvalues for the Real and Geometrically Stable Resonator

Since \(|m|<1\), the perturbation eigenvalues of Equation 10. can be written in the form 

\[\tag{16}\lambda_a,\lambda_b=m\pm j\sqrt{1-m^2}=\cos\theta\pm j\sin\theta=\text{exp}(\pm j\theta),\]

where we introduce an angle \(\theta\) defined by

\[\tag{17}m\equiv\cos\theta\quad\text{or}\quad\theta=\cos^{-1}m=\cos^{-1}\left(\frac{A+D}{2}\right).\]

The eigenvalue for the confined eigensolution is thus either \(e^{j\theta}\) or \(e^{-j\theta}\), with the choice of sign depending on the sign of \(B\). This notation is exactly the same as the ray eigenvalues in Equations 45 and 46.

Small perturbations to the mode thus do not decrease on successive round trips, but neither do they grow, so that the system is considered stable.

We thus have the general conclusion that any paraxial resonator with \(m\) real and \(|m|\<1\) will have a confined and stable gaussian mode solution, such as that illustrated for a typical situation in Figure 2. Such resonators are both geometrically stable (in the periodic focusing sense) and perturbation-stable (in the sense introduced in the previous section).

This class of resonators represents a straightforward generalization of the ordinary two-mirror stable resonators considered in our previous tutorial Beam perturbation and diffraction.

Note that in a folded but planar multielement resonator (e.g., that in Figure 2), there may be some astigmatism; that is, the round-trip \(\text{ABCD}\) matrices may be different for rays in the plane of the paper and for rays perpendicular to this plane.

We must then evaluate the \(\text{ABCD}\) matrices, eigensolutions and eigenvalues separately and independently in the \(x\) and \(y\) transverse directions. (As a practical matter, resonators of this particular type are often used in dye lasers and other mode-locked lasers to get tightly focused waists between the two curved mirrors; and the resulting astigmatism in the off-axis mirrors and in various other Brewster angle plates is traded off to obtain near-zero astigmatism in the overall round trip.)

 

Higher-Order Hermite-Gaussian Modes

These real and geometrically stable resonators will also support a complete family of higher-order real Hermite-gaussian modes of the type described in our following tutorial WAVE OPTICS AND GAUSSIAN BEAMS, PHYSICAL PROPERTIES OF GAUSSIAN BEAMS and BEAM PERTURBATION AND DIFFRACTION provided that the mirror diameters in the real system are sufficiently large so that the diffraction losses are small.

Using the \(\text{ABCD}\) procedure described here thus makes it easy to find the gaussian mode parameters for multiple-element stable resonators, with any kind of real elements such as lenses, graded-index ducts, and so forth within the laser cavity.

Note that the eigenvalues of the resonator are independent of the choice of reference plane within the cavity, but the eigen-\(\tilde{q}\)-values depend upon the reference plane selected.

 

Modal Oscillations in Real Stable Gaussian Resonators

The fact that the perturbation eigenvalues in this class of resonators have magnitude unity actually means that perturbations in the stable gaussian mode will oscillate on successive round trips, neither growing nor dying out (at least, not in the ideal gaussian situation).

This means physically that if we launch a gaussian beam into such a resonator with a \(\tilde{q}\) value close to, but not exactly equal to, the confined self-consistent value, then on successive round trips the beam will "scallop" or oscillate about the self-consistent value in the quasi-sinusoidal fashion already shown in Figure 12.

The angle \(\theta\) gives the fraction of this oscillation that will be completed in one round trip, so that one complete oscillation of the beam will require essentially \(2\pi/\theta\) round trips. 

This same behavior can also be described in another basically equivalent fashion. Suppose again that a perturbed input wave near but not equal to the confined perturbation-stable eigenmode is sent into the resonator.

Such a perturbed eigenwave can be expanded as a superposition of the correct lowest-order gaussian eigenmode plus a small amount of higher-order Hermite-gaussian or Laguerre-gaussian mode components mixed in.

These various higher-order mode components will then propagate around the resonator with slightly different total phase shifts per round trip, because of the Guoy phase shifts we have discussed earlier.

These individual mode phase shifts will in fact differ by integer multiples of the angle \(\theta\), and hence the total field, produced by the superposition of these modes, will appear to oscillate or change in shape with a period equal to \(\theta\).

As a practical matter, any real stable resonator will have small but finite diffraction losses due to an outer aperture or finite mirror size; and these diffraction losses will be in general larger the higher the mode number.

In a real resonator, therefore, the higher-order mode components with \(m\geq1\) will gradually be filtered out, eventually reducing the fields in the resonator to the \(m=0\) component only. The scalloping in a real resonator will thus appear to damp out due to this diffraction filtering after a sufficient number of round trips.

 

 

3.  REAL AND GEOMETRICALLY UNSTABLE RESONATORS

We consider next an equally important but very different class of resonators— that is, those resonators in which the \(\text{ABCD}\) elements are still all real, but the resonator is geometrically unstable in the periodic focusing sense, so that

\[\tag{18}|m|=\bigg|\frac{A+D}{2}\bigg|>1\quad\text{(geometrically}\;\text{unstable)}.\]

This means the half-trace of the matrix is either m greater than \(+1\) (positive branch resonator) or \(m\) negative and less than \(-1\) (negative branch resonator).

 

Unstable Resonator Eigenwaves

The self-consistent eigenwaves in this situation are both purely real, and may be written in the form 

\[\tag{19}\frac{1}{\tilde{q}_a},\frac{1}{\tilde{q}_b}=\frac{D-A}{2B}\mp\frac{\sqrt{m^2-1}}{B}=\frac{1}{R_a}\quad\text{or}\quad\frac{1}{Rb}.\]

The formal solutions in this situation correspond to two purely spherical waves with radii of curvature \(R_a\) and \(R_b\) and with infinite width, i.e., with no gaussian transverse amplitude variation. These waves obviously violate the confinement condition of our analysis.

They are still of considerable practical importance, however. As we will see in a later tutorial they have a meaningful interpretation as zero-order solutions for the general class of hard-edged geometrically unstable resonators.

The matching perturbation eigenvalues for these two spherical waves are now also purely real quantities, i.e.,

\[\tag{20}\lambda_a,\lambda_b=m\pm\sqrt{m^2-1}=M\;\text{or}\quad 1/M,\]

where the geometric magnification \(M\) is a real number with magnitude \(|M|\) greater than unity. The eigenvalues and eigenwaves for either of these waves can also be rewritten in the alternative forms

\[\tag{21}\frac{1}{R_a},\frac{1}{R_b}=\frac{D-\lambda_a}{B},\frac{D-\lambda_b}{B},\]

and these can sometimes be useful expressions. 

 

Positive Branch Unstable Resonators

The labeling of these geometrically unstable solutions then becomes somewhat complicated, because geometrically unstable resonators must first be divided into positive-branch and negative-branch unstable resonators, depending upon whether \(M\) is greater than \(+1\) or less than \(-1\); and the two geometrical eigensolutions for either of these classes must then be separated into a magnifying and a demagnifying eigensolution. Which of these waves corresponds to eigensolution \(\tilde{q}_b\) or eigensolution \(\tilde{q}_b\) then depends upon which branch the resonator corresponds to.

Consider first the situation of \(m\) positive and greater than \(+1\). The two eigenvalues are then

\[\tag{22}\left.\begin{align}&\lambda_a=m+\sqrt{m^2-1}=M\\&\lambda_b=m-\sqrt{m^2-1}=1/M\end{align}\right\}\quad\left(\begin{array}&\text{positive}\;\text{branch}\\m>+1\end{array}\right),\]

where the geometric magnification \(M\) itself is also positive and greater than \(+1\). The two corresponding eigenwaves then have radii of curvature \(R_a\) and \(R_b\) as given in Equation 19 or 21.

Consider next a ray \(r_a\) which is perpendicular to the surface of the eigenwave \(R_a\). The displacement and slope of this ray crossing the reference plane at the start of any one round trip will then be related by \(r'_{a,1}=r_{a,1}/R_a\). After one round

 

 

trip, therefore, this ray will still be normal to the same spherical wavefront, but its displacement will now be reduced to 

\[\tag{23}\begin{align}r_{a,2}&=Ar_{a,1}+Br'_{a,1}=(A+B/R_a)r_{a,1}\\&=r_{a,1}/\lambda_a=r_{a,1}/M\qquad\text{(demagnifying}\;\text{eigenwave}\;\text{R}_a).\end{align}\]

In other words the transverse position of the ray will be demagnified by just the magnification M (Figure 3).

For a ray normal to the \(R_b\) eigensolution, on the other hand, the ray position will be magnified on each round trip by

\[\tag{24}\begin{align}r_{b,2}=r_{b,1}/\lambda_b=M\times r_{b,1}\qquad\text{(magnifying}\;\text{eigenwave}\;\text{R}_b).\end{align}\]

These two results indicate that one of the eigensolutions is a magnifying eigen-wave \(R_b\) which grows in transverse size, but keeps the same radius of curvature, on each round trip; whereas the other is a demagnifying eigenwave \(R_a\) which decreases in size on each round trip, but also preserves its radius of curvature. 

Suppose we send into such a resonator a beam that has wavefront curvature equal to one or the other of the unstable eigenwaves \(R_a\) or \(R_b\), but that has either a finite width or else a large gaussian spot size w; and then follow this beam through one or more trips around the resonator (or equivalently through one or more iterations of the associated \(\text{ABCD}\) matrix for the periodic system).

We will then find that the transverse spread, or the gaussian spot size, of one of these waves \((R_b)\) will magnify transversely by essentially the magnification M on each repeated round trip, whereas the other wave \((R_a)\) will demagnify in size by the inverse ratio \(1/M\) on each round trip, as in Figure 3.

We will therefore call these the magnifying and demagnifying eigensolutions or eigenwaves, for obvious reasons.

 

 

 

Negative Branch Unstable Resonators 

Suppose on the other hand that we consider a negative-branch resonator (Figure 4) for which \(m\) is negative and less than \(-1\). In this situation we will find that the two eigenvalues are 

\[\tag{25}\left.\begin{align}&\lambda_a=-|m|+\sqrt{m^2-1}=1/M\\&\lambda_b=-|m|-\sqrt{m^2-1}=M\end{align}\right\}\quad({\text{negative}\;\text{branch}}\;m<-1),\]

where \(M\) is now also a negative number in the range \(M<-1\). If we again trace the eigenwaves or eigenrays through this system, we will find that in this situation it is the \(R_a\) wave that magnifies in size (but inverts in sign) on each round trip, whereas the \(R_b\) wave also inverts but demagnifies in sign, as in Figure 4. 

For both the positive and the negative branch, therefore, one of the two eigenwaves magnifies and the other demagnifies in transverse width on each round trip, as illustrated in Figure 4. This is what we mean when we say that the system is geometrically unstable.

The negative branch differs from the positive branch only in different ordering of the \(a\) or \(b\) solutions, and in the fact that the wave inverts about the system axis, as well as becoming magnified after each pass, corresponding to the fact that \(M<-1\). 

This inversion about the axis on each pass for a negative branch resonator implies that there must be at least one (or in general, an odd number) of focal points inside the cavity, whereas positive branch resonators must have an even number (including zero) of such internal foci.

Negative-branch resonators, even though attractive in other ways, must usually be avoided in high-power lasers because of problems with optically induced breakdown at these focal points.

Note also that the magnifying wave going in the forward direction through the equivalent periodic lensguide is precisely equivalent to the demagnifying wave going in the reverse direction, and vice versa.

This point will become important in understanding asymmetric ring unstable resonators in later sections.

 

Perturbations in Real Unstable Resonators

Further examination of the wave solutions in the real unstable resonator situation will then reveal that in every case it is the magnifying wave solution that is associated with the perturbation-stable eigenvalue, and the demagnifying wave solution that is associated with the perturbation-unstable eigenvalue, whether these are the waves labeled by \(R_a\) or \(R_b\). 

That is, if we send into the system a spherical wavefront with a curvature close to the magnifying wavefront radius, but with some slight perturbation in curvature, that wavefront will grow in size transversely by the magnification \(M\) on each successive pass; but the deviation in curvature from the exact eigenwave will decrease on each successive pass by the ratio \((1/M)^2\).

Conversely, if the wavefront is initially close to the demagnifying solution but with some slight initial error, this error in the wavefront curvature will grow as \(M^2\) on each successive bounce, until the converging wave essentially "runs away," and eventually converts over into the diverging or magnifying wavefront. (This conversion generally occurs after only a rather small number of round trips in typical situations.)

If we follow the labeling conventions used in the preceding, in the positive-branch situation it is the a eigenwave that is demagnifying but perturbation unstable and the b eigenwave that is magnifying but perturbation stable; whereas for the negative-branch situation these subscripts must be reversed.

This general behavior clearly illustrates how the eigenvalues \(\lambda_a\) and \(\lambda_b\) are associated as much with the perturbation stability of the eigenwaves, as with the transverse magnification or demagnification of the waves on successive round trips.

 

Practical Properties of Real Unstable Resonators

These unbounded spherical-wave solutions for the real but geometrically unstable situation are clearly nonphysical as they stand. They do represent, however, a useful "zeroth-order" approximation to the real nongaussian modes of the so-called hard-edged unstable optical resonators (Figure 5) that are used in many higher-power lasers.

We will discuss this class of resonators in more detail in a later tutorial, but for the present we can say the following. 

If the magnifying wave expands on each round trip in a geometrically unstable resonator, it must eventually run into the mirror edges or the laser tube walls. These edges will then obviously have major effects on the wave, both in cutting off further transverse growth, and also in producing strong diffraction effects in the wavefront on the next round trip.

The resonator edges thus obviously play a very important role in real unstable resonators, and the diffraction effects due to these effects cannot be ignored, even in an approximate theory.

A correct analysis of real unstable resonators can thus be carried out in full detail only by doing a full diffraction or Huygens-integral analysis of the unstable resonator. 

As we will see in a later tutorial, however, experience shows that the magnifying eigenwave predicted by the simple \(\text{ABCD}\) analysis does give a very good first approximation for the basic wavefront radius of curvature (either \(R_a\) or \(R_b)\) and for the round-trip magnification \(M\) in a real hard-edged unstable resonator, even

 

 

when the strong edge-diffraction effects are taken into account. The exact transverse mode shape in such a resonator will look like the geometrically magnifying phase front, with an amplitude profile that has large Fresnel diffraction ripples and that expands by an amount \(M\) on each round trip, before being cut off by the finite aperture edges.

The detailed amplitude ripples can only be predicted from a full diffraction calculation taking into account the mirror edges, and the exact form of these ripples will depend very strongly on the exact aperture size.

These hard-edged unstable resonators can be very useful, and we will describe them in much more detail in the following tutorial. 

 

 

4. COMPLEX STABLE AND UNSTABLE RESONATORS 

Putting gaussian apertures or similar transverse amplitude variations into an optical resonator will lead in general to complex \(\text{ABCD}\) systems. The general solutions that we developed in the first section of this tutorial for the self-consistent eigenwaves and perturbation eigenvalues in a general complex paraxial resonator will then be given, once again, by 

\[\tag{26}\frac{1}{\tilde{q}_a},\frac{1}{\tilde{q}_b}=\frac{D-A}{2B}\mp\frac{1}{B}\sqrt{\left(\frac{A+D}{2}\right)^2-1,}\]

and

\[\tag{27}\lambda_a,\lambda_b=\frac{A+D}{2}\pm\sqrt{\left(\frac{A+D}{2}\right)^2-1,}\]

where \(A\), \(B\), and \(D\) may all be complex quantities. The conditions for a wellbehaved and physically real resonator eigenmode are again that at least one of these eigenwaves should be confined, and that the perturbation eigenvalue associated with that particular wave should have a magnitude \(|\lambda|\leq1\). 

When the individual matrix elements in Equations 26 and 27 may all potentially be complex numbers, it is not usually at all obvious by inspection which of the two eigenwaves is the confined solution; nor is it then obvious by inspection whether the eigenvalue associated with that confined wave has a magnitude less than or greater than unity.

In fact, the only way to answer these questions for a completely general complex paraxial resonator seems to be to calculate the complex \(\text{ABCD}\) elements and then examine the \(1/\tilde{q}\) and \(\lambda\) values for a specific system in close detail, to see whether or not they meet the necessary criteria.

 

 

 

Complex Perturbation-Stable Gaussian Resonators 

We can, however, gain considerable insight into complex paraxial resonator systems in the following fashion. It is possible to show quite generally (see the Problems) that if an arbitrarily small amount of transversely increasing loss (or transversely decreasing gain) is added anywhere inside any purely real \(\text{ABCD}\) system, whether geometrically stable or unstable, then one of the eigenwaves for that system will always be modified so as to become both a confined and perturbation-stable gaussian wave. 

In other words, an arbitrarily weak gaussian aperture, as in Figure 6, will act to convert any purely real resonator—whether it is geometrically stable or unstable to start with—into a complex perturbation-stable resonator.

Note that geometric stability and perturbation stability are thus quite separate and distinct concepts in the complex resonator.

Consider first the results for a geometrically stable resonator to which a weak gaussian aperture is added. An initially stable resonator will have one confined and one unconfined eigensolution, which we have indicated by points labeled \(1/\tilde{q}_a\) and \(1/\tilde{q}_b\) in Figure 7(a), together with matching eigenvalues \(\lambda_a\) and \(\lambda_b\), both of which lie on the unit circle in the complex \(\lambda\) plane.

Adding a weak positive gaussian aperture will then cause the eigenwaves and eigenvalues for this situation to move in the directions indicated by the arrows in the plots. In particular, for an initially stable resonator the confined eigensolution will always move so as to remain confined and become perturbation stable. 

If, on the other hand, we consider a geometrically unstable system, we initially have two unconfined eigenwaves \(R_a\) and \(R_b\) and two purely real eigenvalues \(\lambda_a\) and \(\lambda_b\), with \(\lambda_b\) being the perturbation-stable (and thus magnifying) solution in this example.

Adding the weak gaussian aperture will then always cause the perturbation-stable solution to remain perturbation-stable and become confined, as illustrated by the arrows in Figure 7(b).

 

Physical Example

Figure 8 shows an example of the kind of complex gaussian eigenmode that results for a resonator where the mirror curvatures by themselves lead to strongly unstable behavior in the geometric sense, but where the complex gaussian modes are perturbation-stabilized by a weak gaussian aperture.

One way of 

 

 

viewing the behavior of gaussian beams in this kind of resonator is to consider an initially injected gaussian beam with a curvature near the magnifying eigenwave of the unapertured resonator, but with a spot size intially small compared to the gaussian aperture.

Such a gaussian wavefront will then be magnified on each round trip by a factor close to the geometric magnification \(M\) as a result of the unstable defocusing effects.

This magnifying wave will eventually grow to a large enough diameter, however, that it will begin to run into the gaussian aperture. The divergence or the geometric magnification will then be limited and the wave trimmed back in size on each round trip by the "soft aperture" effects of the gaussian aperture or variable reflectivity mirror.

In fact, a gaussian beam with a spot size much larger than the aperture would be rapidly reduced in spot size on the first pass through the aperture.

The magnifying eigenwave thus stabilizes to a spot size which is constant (and perturbation-stable) on successive round trips, representing a balance between geometric magnification and soft aperture narrowing. 

This particular type of resonator, with a combination of geometric instability for round-trip magnification, plus gaussian aperturing for spot size and eigenwave stabilization, appears to hold very substantial promise for future development of large-diameter but well-controlled laser modes, as we will discuss in more detail in the following tutorial.

The development of such complex perturbation-stable gaussian resonators has been limited to date primarily by the difficulty in

 

 

obtaining practical gaussian apertures, but this is a problem which now seems to be finding practical solutions. 

 

Complex Perturbation-Unstable Resonators 

There is also the opposite situation, namely, a resonator with a transversely increasing transmission caused either by transversely decreasing loss or transversely increasing gain.

The opposite of the previous generalization then applies, namely, adding even a small amount of transversely increasing transmission will convert any purely real resonator, geometrically stable or unstable, into a complex and perturbation-unstable resonator. 

Even in this situation there will be one eigenwave which is transversely confined, and which might therefore seem to be a physically useful solution. If transversely increasing gain is present, however, this confined eigenwave will always turn out to be perturbation unstable: if it is disturbed even slightly, it will begin to grow in diameter.

The other wave which is perturbation stable, by contrast, will always turn out to be unconfined in the transverse direction. Hence, neither of these waves appears to be physically useful as a real resonator mode, and so such complex perturbation-unstable resonator modes appear to be of little interest for practical laser devices—at least not in their ideal form.

A brief summary of the properties of each of the general types of complex paraxial resonators is presented in Table 1.

 

Practical Considerations for Systems With Radially Increasing Gain

There can be some practical situations in which transversely increasing gain will be present in a resonator, at least over some finite range of diameter. In large-bore \(CO_2\) lasers, for example, increased heating of the laser gas at the center of the laser tube can lead to decreased gain on axis and increasing gain near the tube walls.

Gain saturation of almost any gain medium by a gaussian laser beam will also appear to reduce the gain more at the center than at the edges, even in an initially uniform gain medium. 

According to the complex resonator theory, even small effects of this type should then, at least in principle, cause even strongly stable real resonators to

 

TABLE 1. 

Complex Paraxial Resonator Types

 

become perturbation unstable. As a practical matter, however, it appears that the focusing effects of the lenses and mirrors in a real stable resonator are of substantially larger magnitude than the defocusing or destabilizing effects of weak radial gain increases.

What is probably of even more importance is that even weak diffraction effects from mirror edges or mode control apertures appear to have a stabilizing effect, though this is obviously not directly covered by the ideal \(\text{ABCD}\) theory.

The practical implication, therefore, is that weak and transversely bounded radial increases in gain do not appear to cause serious perturbations in otherwise stable laser resonators.

This entire subject has, however, not as yet been much investigated. 

 

Multiaperture Complex Resonators

We might finally, in the most general situation, even encounter resonators containing transversely decreasing transmission elements at some locations in the resonator, and transversely increasing or destabilizing elements at other places, so that both positive and negative apertures are encountered in a complete round trip. Will such a resonator then be overall complex-stable or complex-unstable?

It appears to be difficult to give any general answers regarding the resulting resonator behavior, other than by computing the total complex \(\text{ABCD}\) matrices for the resonator of interest, and then finding out by direct inspection into which class the overall system falls. 

An additional complication for resonators having both positive and negative gaussian apertures at different points around the resonator is the following. It is possible for such a resonator to have a mode which appears to be self-reproducing, confined and perturbation-stable when calculations are made starting from one reference plane within the resonator—for example, a reference plane just after a transversely decreasing aperture—but for the same eigenwave to appear as perturbation-stable but unconfined when the same resonator is analyzed starting from a different reference plane—for example, a reference plane just after one

 

 

of the transversely increasing apertures. The trajectory in the \(1/\tilde{q}\) plane for one example of such a resonator is shown in Figure 9. 

One primary point here is that beam power is in general not conserved in going through a gaussian aperture of either sign. In particular, it is possible for a confined gaussian beam having finite power, after passing through a transversely increasing aperture, to be converted into an unconfined beam carrying infinite power. All that is required is that the transverse gain increase more rapidly than the bounded input wave amplitude decreases.

Thus, a confined and perturbation-stable gaussian beam, in passing through a transversely increasing aperture, can under some conditions be transformed into an unconfined wave; and then later transformed back into a confined wave by a transversely decreasing aperture. 

The conclusion is that in those rare situations where transversely decreasing and increasing apertures are both present in a single resonator, we cannot find out whether the wave is confined at every plane merely by examining the solutions for the eigenwaves and perturbation eigenvalues at one reference plane.

The roundtrip \(\text{ABCD}\) matrix from only one reference plane does not contain sufficient information to determine this. We must instead follow the selected eigenwave completely round the resonator, step by step, and verify its confinement at every plane. 

 

 

5.  OTHER GENERAL PROPERTIES OF PARAXIAL RESONATORS

We can next analyze some further properties of complex paraxial resonators, including round-trip amplitude and phase changes; general properties of standing-wave and traveling-wave (ring) resonators; and the stability properties of higher-order modes in such resonators. 

 

Round-Trip Amplitude Changes and Phase Shifts

We consider next what happens to the complex amplitude coefficients in front of the Hermite-gaussian eigenmodes of a generalized paraxial resonator on each round trip.

We showed in the previous tutorial that when a generalized Hermite-gaussian mode passes through an arbitrary complex \(\text{ABCD}\) system, the amplitude coefficient for the wave changes by the complex ratio 

\[\tag{28}\frac{\tilde{a}_{2,n}}{\tilde{a}_{1,n}}=\left(\frac{1}{A+B/\tilde{q}_1}\right)^{n+1/2}.\]

But if \(\tilde{q}_1\)corresponds to either of the self-consistent eigenvalues \(\tilde{q}_a or \(\tilde{q}_b\), as it does for a resonator mode, this result says that the round-trip change in wave amplitude and phase for the mode in the resonator is given by the same factor as the perturbation eigenvalue.

If we work in only one transverse dimension, this factor is

\[\tag{29}\frac{\tilde{a}_{2,n}}{\tilde{a}_{1,n}}\bigg|_{\tilde{q}_1=\tilde{q}_a,\tilde{q}_b}=\left(\frac{1}{A+B/\tilde{q}_{a,b}}\right)^{n+1/2}=\lambda^{n+1/2}_{a,b},\]

whereas if we consider a complete \(\text{TEM}_{nm}\) higher-order Hermite-gaussian mode in two transverse dimensions, then we have

\[\tag{30}\frac{\tilde{a}_{2,nm}}{\tilde{a}_{1,nm}}\bigg|_{\tilde{q}_1=\tilde{q}_a,\tilde{q}_b}=\left(\frac{1}{A+B/\tilde{q}_{a,b}}\right)^{n+m+1}=\lambda^{n+m+1}_{a,b}.\]

 

Guoy Phase Shifts and Transverse Mode Frequencies 

The magnitudes of the eigenvalues \(\lambda_a\) and \(\lambda_b\) thus determine the roundtrip losses for the resonator eigenmodes (to be made up by the laser gain in an oscillating laser), whereas the phase angles of the eigenvalues give the phase shifts for the \(mn\)-th order Hermite-gaussian modes.

For example, in a real stable gaussian resonator the round-trip eigenvalue becomes simply 

\[\tag{31}\frac{\tilde{a}_{2,nm}}{\tilde{a}_{1,nm}}\bigg|_{\tilde{q}_1=\tilde{q}_a,\tilde{q}_b}=\text{exp}[\mp j(n+m+1)\theta],\]

where \(\cos\theta=m\). The angle \(\theta\) thus represents the multielement or real-\(\text{ABCD}\) generalization of the Guoy phase shift \(\psi\) for gaussian beams and two-mirror resonators that we discussed in Section 3. Higher-order modes once again have this phase shift increased by the factor \(n+m+1\).

A real gaussian resonator with no soft apertures has no diffraction losses. Hence, the magnitude of the amplitude coefficient remains unchanged, or the net round-trip amplitude gain is unity, for modes of all orders \(m\) or \(n\).

There is thus no transverse mode discrimination—at least not until some finite aperture is inserted into the cavity. 

For a complex stable resonator, i.e., one which contains soft apertures, the round-trip amplitude and phase shift will be given by the formulas developed in this section, where the applicable eigenvalue will be the perturbation eigenvalue for the confined and perturbation-stable mode.

Since this eigenvalue by definition has magnitude \(|\lambda|\leq1\), each higher-order mode is attenuated relative to lower-order modes by this eigenvalue raised to the appropriate power.

So long as no hard apertures are present, the perturbation eigenvalues are all that are needed to characterize completely the losses, the phase shifts, and the perturbation stability of Hermite-gaussian modes of all orders.

 

Higher-Order Hermite-Gaussian Modes

We have considered up to this point only the transformation and the perturbation-stability properties of the gaussian \(\tilde{q}\) parameter and of the mode amplitude coefficient \(\tilde{a}_{nm}\).

For higher-order modes, however, we also have to examine what happens to the complex spot size paramter \(\tilde{v}\) on successive round trips. 

The general transformation rule for the \(\tilde{v}\) parameter on one round trip is, as derived in the previous tutorial Complex paraxial wave optics.

\[\tag{32}\tilde{v}^2_2=(A+B/\tilde{q}_1)^2\times\tilde{v}^2_1+j\frac{4B}{k_1}(A+B/\tilde{q}_1).\]

If \(\tilde{q}_1\) has the values \(\tilde{q}_a\) or \(\tilde{q}_b\), then the self-consistent values of \(\tilde{v}_2=\tilde{v}_1=\tilde{v}\) are given by

\[\tag{33}\tilde{v}^2_{a,b}=\mp j\frac{B\lambda}{\pi}\times\sqrt{\frac{1}{\tilde{m}^2-1}},\]

where \(m\equiv(A+D)/2\), and where the upper and lower signs are consistent with the upper and lower signs in Equations 3 and 10 for the eigen-\(\tilde{q}\)-values and eigenvalues. 

If we then look at the perturbation stability of \(\tilde{v}_a\) and \(\tilde{v}_b\), we can find that whereas the perturbation stability for the \(\tilde{q}_a\) and \(\tilde{q}_b\) values is expressed by

\[\tag{34}\frac{\Delta\tilde{q}_2}{\Delta\tilde{q}_1}\bigg|_{\tilde{q_1}=\tilde{q}_a,\tilde{q}_b}=\lambda^2_a\quad\text{or}\quad\lambda^2_b\]

the perturbation stability analysis for the \(\tilde{v}\) values leads to exactly inverse results, namely,

\[\tag{35}\frac{\Delta\tilde{v}_2}{\Delta\tilde{q}_1}\bigg|_{\tilde{q}_1=\tilde{q}_a,\tilde{q}_b}=\frac{1}{\lambda^2_a}\quad\text{or}\quad\frac{1}{\lambda^2_b}=\lambda^2_b\quad\text{or}\quad\lambda^2_a.\]

In other words, any mode which is perturbation-stable in \(\tilde{q}\) is perturbation-unstable in \(\tilde{v}\).

This would seem to imply that no higher-order modes above \(n=2\) can exist and be simultaneously perturbation-stable in both \(\tilde{q}\) and \(\tilde{v}\) in a general complex paraxial resonator.

 

Physical Interpretation

The physical meaning of this result is the following. For purely real and stable systems, the eigenvalues \(\lambda_a\) and \(\lambda_b\) all have unity magnitude in any event. Both lowest and higher-order modes can then propagate with (marginal) stability in both \(\tilde{q}\) and \(\tilde{v}\) on repeated round trips, although we can expect that as soon as any finite aperture, soft or hard, is added to the resonator, the higher-order modes will begin to be filtered out, since they always spread out further. 

In complex but perturbation-stable systems, however, soft apertures are already present; and higher-order Hermite-gaussian or Laguerre-gaussian modes, because they spread farther out, will have higher losses.

Consider then the propagation of some higher-order mode \(\tilde{u}_n(x)\) with \(n\geq2\), so that both \(\tilde{q}\) and \(\tilde{v}\) are relevant.

Any slight perturbation of this mode can then be described as a coupling of some of the power from this particular \(n\)-th order mode into other Hermite-gaussian modes of both higher and lower index \(n'\). 

But any lower-order modes that are excited by the perturbation will have lower net losses than the original \(n\)-th order mode. Hence, as the beam goes around on successive round trips, the original \(n\)-th order mode will gradually die out relative to the lower-order modes; and in the long run only the lowest symmetric \((n=0)\) and antisymmetric \((n=1)\) modes will remain.

The higher-order Hermite-gaussian or Laguerre-gaussian modes are thus a mathematical possibility; but unless we put in specially shaped apertures (wires, point scatters, etc.), they do not represent long-term perturbation-stable solutions in competition with the lowest-order modes.

This higher-order mode suppression tendency of the complex stable resonators is, in fact, one of the most attractive features of this class of resonators.

 

Standing-Wave Resonators Versus Traveling-Wave Resonators

We can point out finally some significant distinctions that can be drawn between the eigenwaves in standing-wave and traveling-wave optical resonators. These distinctions will supplement, but in no way replace, the various considerations concerning confinement and stability that we have introduced in the previous section. 

Consider, for example, standing-wave resonators that are either geometrically stable or unstable. The unit cell for the equivalent lensguide in either situation is then necessarily symmetric about the two reference planes corresponding to the two end mirror surfaces of the standing-wave resonator.

If either of these mirror surfaces is taken as the reference plane for the \(\text{ABCD}\) calculation, the round-trip \(\text{ABCD}\) matrix necessarily has the symmetry property that \(A=D\).

From the analytical results derived earlier, we can see that for a real stable standing-wave resonator the two eigen-\(\tilde{q}\)-values are both purely imaginary on the end mirror surfaces, since \(D-A\equiv0\). This means in turn that the phase front of the eigenwaves is always exactly matched in curvature to the end mirror surfaces, and this implies in turn that the forward and reverse traveling waves in the stable standing-wave cavity are matched in curvature everywhere.

The standing wave is, in fact, a true standing wave (leaving out the possibility of different amplitudes for the forward and reverse waves due to finite mirror amplitude reflectivity or laser gain).

For real but unstable standing-wave resonators, on the other hand, the condition that \(A=D\) on the end mirror reference planes implies that the magnifying and demagnifying waves \(R_a\) and \(R_b\) (or vice versa) (a) have equal and opposite curvatures at the end mirror surface, and (b) neither of these curvatures matches the end mirror surface.

This is in fact a general characteristic of geometrically unstable resonators. We can also see, from a little further examination, that the geometrically demagnifying wave in the standing-wave situation is just the magnifying wave traveling in the reverse direction around the resonator, or in the reverse direction down the equivalent lensguide.

The magnifying and demagnifying waves are connected in essence simply by time reversal.

 

Traveling-Wave Resonators and Eigenmodes

Consider now ring-type or traveling-wave real stable and unstable resonators and their equivalent lensguides. For a ring laser there is in general no necessary requirement that the equivalent lensguides have any symmetry between forward and reverse directions, and hence there need not be any reference plane within the rings or the lensguides where the \(\text{ABCD}\) matrix will have equal diagonal elements, i.e., in general \(A\neq D\) at any plane.

In fact, if we use the same reference plane going in either direction around the ring, the inversion rules for \(\text{ABCD}\) matrices say that the \(\text{ABCD}\) matrix elements in the two directions are related by \(A_R=D_F\), \(D_R=A_F\), \(B_F=B_R\) and \(C_F=C_R\), where the subscripts refer to forward and reverse directions, respectively.

We will see later that the general propagation operators in the two directions, including finite aperture effects, are in general the mathematical transposes of each other. This does not mean that the \(\text{ABCD}\) matrices in the two directions are the matrix transposes of each other; but rather that when these matrices are inserted into the generalized Huygens' integral operator, this

 

 

operator in the reverse direction becomes the transpose of the integral operator in the reverse direction. 

 

General Properties of Traveling-Wave (Ring) Resonators 

Several general properties of the paraxial eigenmodes going in the two opposite directions around a ring can then be deduced from this. First, since the traces of the \(\text{ABCD}\) matrices going in the two directions are always identical, we can deduce that for either stable or unstable, real or complex resonators, the eigenvalues in the two opposite directions are always identical.

The losses and phase shifts in either direction around a ring laser cavity are always the same, even though the shapes of the eigenwaves in the two directions may be different. 

Second, for real stable resonators the perturbation-stable eigenwaves in the two directions are also the same, i.e., the eigenwave in one direction is just the reversed eigenwave in the other direction.

On the other hand, for real unstable resonators the magnifying and perturbation-stable eigenwave going in one direction is just the time-reversed demagnifying wave going in the other direction.

In the asymmetric unstable lensguide shown in Figure 10, for example, the wave \(\tilde{E}_F\) corresponds to the magnifying and perturbation-stable wave going in the forward direction.

The wave \(\tilde{E}_R\) then corresponds both to the demagnifying and perturbation-unstable wave going in the forward direction and also to the magnifying and perturbation-stable wave going in the reverse direction.

Note that although the wavefronts and beam profiles of these two waves are significantly different, the geometric magnifications are in fact the same.

The resonator parameters in Figure 10 have been chosen so that the magnifying wave in the forward direction has a collimated wavefront in the magnified section. This is not in general true of all or even most unstable resonators.

It occurs here only because the particular wave shown in Figure 10 corresponds to a confocal unstable resonator going in the forward direction. The same resonator is then obviously not confocal going in the reverse direction.

Note that this kind of directional asymmetry can only be accomplished in a ring resonator. There is no way in which this same situation could be accomplished in a standing-wave resonator.

 

Resonator Properties With Nonparaxial Apertures

Essentially all of the properties introduced in this section, although derived here only for ideal paraxial resonators with infinite mirrors and at most only soft apertures, remain either partially or completely true even for the exact eigenmodes of much more general resonators with finite mirror diameters or hard-edged apertures. 

If, for example, the underlying real paraxial elements in a resonator correspond to a geometrically stable system, then the exact eigenmodes in any standing-wave resonator will still have phase fronts that nearly coincide with the end mirror surfaces, and nearly correspond to pure standing-wave fields, even if the resonator contains finite apertures which produce small but finite diffraction losses.

Increasing amounts of diffraction loss will cause progressively increasing departures from these properties.

The exact eigenmodes in geometrically unstable resonators with finite apertures will also, as we have mentioned earlier, still have wavefront curvatures close to the magnifying spherical eigenwave, even for rather large diffraction losses.

The forward and reverse eigenmodes in the unstable traveling-wave situation will also continue to correspond to the magnifying and demagnifying forward solutions as described in the preceding, even for large diffraction losses. 

The exact eigenmodes in the forward and reverse direction for any travelingwave resonator, whether geometrically stable or unstable, will also continue to have exactly the same eigenvalues (and hence the same diffraction losses) in both directions even under very general conditions of finite apertures and large diffraction losses.

Several of these concepts will be discussed further, for more general resonator models, when we discuss the orthogonality properties of resonator modes in a later section.

 

 

6.  MULTIELEMENT STABLE RESONATOR DESIGNS

Many practical laser devices (such as, for example, mode-locked ion lasers, pulsed solid-state lasers, and tunable dye lasers) make use of multielement stable laser resonators which may contain a sizable number of lenses, curved and off-axis mirrors, ducts, Brewster-angle plates, and the like.

As we have already noted several times, the stable gaussian modes in these multielement cavities can be analyzed in one pass by multiplying out their round-trip \(\text{ABCD}\) matrices and then solving for their confined and perturbation-stable gaussian eigenwaves.

A simple interactive computer program can be of great assistance both in entering the various elements into such an analysis and then in calculating and displaying the results—especially since this procedure may often have to be done separately but simultaneously in two transverse dimensions.

In complicated resonators of this type, changing the values of different elements can have effects on the stable mode parameters that interact in a complex and sometimes sensitive fashion; so that even with computer assistance the design procedure for synthesizing an optimized laser resonator can be time-consuming and unclear.

Some general guidance in approaching the desired design values may then be of help; and this section will briefly review a few general principles for stable multielement resonator design.

 

Mode Sizes in Real Stable Resonators

In many situations we may wish to obtain a large spot size w at a certain plane inside a laser resonator, in order, for example, to extract maximum energy from a laser rod or tube; whereas in other situations a very small spot size may be needed to obtain maximum energy density in a saturable absorber or a dye laser cell or jet.

How can we control in general terms the spot size at selected planes in a multielement optical resonator? 

We might first recall that the gaussian spot size \(w\) for the stable eigenmode at the selected reference plane in a stable cavity is given by 

\[\tag{36}w^2=\frac{|B|\lambda}{\pi}\times\sqrt{\frac{1}{1-m^2}}=w^2_0\times\sqrt{\frac{1}{1-m^2},}\] 

where we use the abbreviation \(w^2_0\equiv|B|\lambda/\pi\). We can then observe that the "effective length" \(|B|\) for a multielement resonator will depend on the reference plane that is employed, whereas the half-trace m will be the same for all choices of reference plane.

In many situations, moreover, the effective round-trip length \(|B|\) will be of the same order as the physical cavity length, or shorter, so that \(w_0\) will be comparable to the (small) confocal spot size in a simple two-mirror resonator of comparable length.

To achieve a larger spot size \(w\gg w_0\), for example in order to fill a largediameter laser tube, we must then adjust \(m^2\) fairly close to unity, or close to the stability boundary for the cavity.

But we can then show that the sensitivity of this adjustment—that is, the sensitivity \(\delta w\) of the spot size \(w\) to small errors \(\delta m\) in the adjustment of the \(m\) parameter—will be given by 

\[\tag{37}\frac{\delta w}{w}=\frac{m^2}{2}\left(\frac{w}{w_0}\right)^4\frac{\delta m}{m}\approx\frac{1}{2}\left(\frac{w}{w_0}\right)^4\times\frac{\delta m}{m}\quad\text{for}\quad m^2\rightarrow1.\]

If we want to achieve, for example, a spot size w that is 10 times the "confocal value" \(w_0\), then the adjustment tolerance for whatever resonator parameter is being adjusted to vary \(m\) becomes \(\approx\) 5,000 times more sensitive than the allowable tolerance in the spot size \(w\) itself.

It is thus in general a difficult and tricky process to achieve a large spot size with a a large \(B\) parameter. 

To meet the opposite design objective, that is, to obtain an unusually small spot size, it is evident that adjustments of the \(m^2\) parameter offer little room for improvement; and we must find instead an overall resonator design that yields a sufficiently small \(B\) parameter at the desired reference plane. We will show how to accomplish either of these objectives shortly.

 

Intracavity Telescopes

One technique that can be useful for obtaining both small and large spot sizes is the use of an intracavity telescope or magnifier, usually with the focus set to infinity or not too far from there.

Consider, for example, a galilean telescope focused at infinity plus two adjoining free-space sections \(L_1\) and \(L_2\) as in the top part of Figure 11. The \(\text{ABCD}\) matrix going in the magnifying direction through this telescope is then 

\[\tag{38}\left[\begin{array}&A&B\\C&D\end{array}\right]\left[\begin{array}&M&ML_1+L_2/M+f_2-|f_1|\\0&1/M\end{array}\right],\]

where \(M\equiv f_2/|f_1|\) is the transverse magnification of the telescope. Note that the effective length on the smaller side of this telescope is equal to the actual physical length \(L_1\) multiplied by the telescope magnification \(M\), whereas the effective length on the larger side is the physical length divided by \(M\).

If we use instead a newtonian telescope, as in the lower part of Figure 11, then the \(\text{ABCD}\) matrix going in the forward direction through the telescope becomes

\[\tag{39}\left[\begin{array}&A&B\\C&D\end{array}\right]\left[\begin{array}&-M&(f_1+f_2)-(ML_1+L_2/M)\\0&-1/M\end{array}\right],\]

where in this situation \(-M\equiv-f_2/f1\) is the (negative) magnification through the telescope. One significant point here is that if we pick \(L_1\) and \(L_2\) to correspond to conjugate image points we can make the net value of the effective length \(B\) through the telescope be zero, or arbitrarily close to zero.

This condition obviously corresponds to the image relaying system, with transverse magnification \(M\), that we mentioned in an earlier section. 

More generally, an inverting or newtonian telescope can be used to insert a negative value of effective length into a resonator, in order to cancel out positive \(B\) contributions from other parts of the resonator. For a resonator to be geometrically stable, the net round-trip magnification must be unity.

This can be accomplished more or less automatically in a standing-wave cavity since the telescope will be traversed twice, going in opposite directions, so that the telescope magnification \(M\) will be canceled out.

 

Mode Spot Size Stability Against Internal Perturbations

One common cause of spot size instability in solid-state lasers can be the weak thermal focusing that occurs in solid laser rods when they are heavily pumped.

 

 

The combination of temperature rise and thermal expansion at the center of a cylindrical rod will produce an index increase due both to the thermal variation of the index of refraction and to thermally induced stress-optic effects The rod will then act like a weak index duct, with a sufficiently long focal length that it can usually be approximated as a weak thin lens.

For typical Nd:YAG rods, for example, the focal length of this thin lens decreases inversely with pumping power, and has a typical focal length of \(\approx1m\) for a few kW of input electrical power to the pumping lamps. 

The effect of this pump-power-dependent thermal focusing on mode spot size can then be eliminated to first order, while achieving the desired spot size in the rod, by the use of an intracavity telescope as illustrated in Figure 12.

In this cavity design the rod is placed close to a curved mirror of radius of curvature \(R\), with a telescope of magnification \(M\) placed close to the rod on the other side, spaced by a larger distance \(L_0\) from a flat mirror at the opposite end of the cavity.

In a typical cavity of this type the rod and telescope will be sufficiently short compared to the remainder of the cavity that we can lump their contribution to the cavity length into the rest of the cavity, and treat them as zero-length elements in a preliminary analysis.

The net ray matrix starting from a reference plane at the curved mirror end, going out through the demagnifying telescope to the flat mirror, and back again (but leaving out the curved mirror), will then have an \(\text{ABCD}\) matrix given by

\[\tag{40}\left[\begin{array}&A&B\\C&D\end{array}\right]=\left[\begin{array}&1/M&0\\0&M\end{array}\right]\times\left[\begin{array}&1&2L_0\\0&1\end{array}\right]\times\left[\begin{array}&M&0\\0&1/M\end{array}\right]=\left[\begin{array}&1&2M^2L_0\\0&1\end{array}\right].\]

 

 

In other words, the demagnifying telescope plus straight section looks like an equivalent section that is \(M^2\) as long as the actual physical cavity length. 

The effect of weak thermal focusing in the rod can then also be approximated as a small variation in the effective radius of curvature \(R\) of the right-hand mirror, and we can thus replace the multielement cavity by a simple two-mirror cavity of effective length \(L=M^2L_0\) with one flat and one curved mirror, as shown in Figure 12(b).

The spot sizes at the left and right-hand ends of the cavity are then given in terms of the equivalent g parameters by 

\[\tag{41}w^2_1=\frac{L\lambda}{\pi}\times\sqrt{\frac{g_2}{g_1(1-g_1g_2)}}\quad w^2_2=\frac{L\lambda}{\pi}\times\sqrt{\frac{g_1}{g_2(1-g_1g_2)}}.\]

Now, the spot size w2 at the curved mirror end is the spot size that we wish to stabilize against small variations in the effective mirror radius R and hence against small variations in the parameter \(g_2\equiv1-L/R_2\). The sensitivity of \(w_2\) to the latter parameter is then given by

\[\tag{42}\frac{\delta w_2}{w_2}=\frac{2g_1g_2-1}{4(1-g_1g_2)}\times\frac{\delta g_2}{g_2}\approx0\quad\text{for}\quad g_1g_2=1/2.\] 

For optimum sensitivity against thermal focusing effects, the resonator should therefore be designed to fall on the contour \(g_1g_2=1/2\) in the stability plane. 

A combination of cavity length \(L_0\), mirror radius \(R\) and telescope magnification \(M\) can thus be found which will simultaneously give the desired spot size \(w_2\) in the laser rod, and make this spot size stable to first order against thermal focusing in the rod.

In a more general version of this design procedure, a telescope focused at other than infinity can be employed, and a portion of the focusing power of the telescope used to change the effective radius \(R\) of the output mirror.

In a real system we would also want to carry out a more accurate analysis including the physical lengths and focal powers of all the elements, and examining the sensitivity of the system to, for example, slight changes in focal adjustment of the telescope.

The general criterion given here can be used, however, as the starting point for a fundamentally sound solution to this particular design problem.

 

Other Fundamental Cavity Designs

Various other concepts that are sometimes used in describing or synthesizing multielement cavities (or cavity subsections) with real \(\text{ABCD}\) matrices include the following: 

(1) A real \(\text{ABCD}\) system with \(C=0\) is often called a telescopic system, since any rays coming into the system parallel to the axis \(x'_1=0\) will emerge parallel to the axis \(x'_2=0\), as in a telescope focused to infinity.

We pointed out in the previous tutorial that all real and geometrically unstable systems (with \(|m|>1)\) will be telescopic going from one properly chosen real (curved) reference plane to another.

Geometrically stable systems (\(|m|<1)\), by contrast, are telescopic only in a complex ray sense, i.e., only if we extract out a complex gaussian wave from the input and output beams, or go from one "complex-curved reference plane" to another.

(2) Real \(\text{ABCD}\) systems with \(B=0\) can be referred to as self imaging or image relaying systems, since they transfer or image all the rays leaving a point \(x_1\) at the input to a point \(x_2=Ax_1\) after one round trip. (Such systems might also be called "round-trip confocal systems," since they transfer a focal point of the system back to the same focal point after one round trip.)

All real self-imaging or round-trip confocal systems are geometrically unstable \((|m|>1)\) except for the marginally stable situation where \(A=D=m=\pm1\).

(3) Cavities having the special limiting values of \(A=D=1\) and \(B=C=0\) are classified by some authors as degenerate cavities. They have the general properties that any arbitrary ray returns to its initial value after one round trip; and hence that any input beam pattern or image reproduces itself exactly after one round trip.

Systems with \(A=D=-1\) and \(B=C=0\) are then half degenerate: They repeat after two round trips, and convert into a degenerate cavity if we shift to an arbitrary off-axis ray as the optical axis.

Because such degenerate and half-degenerate cavities can support more or less arbitrary multimode transverse beam patterns, they are useful for applications in active imaging; in "Scan Lasers" whose direction of oscillation can be scanned within the laser cavity; in the regenerative amplification of distorted beam patterns; and in scanning interferometers or filters which do not require transverse mode matching.

(4) Finally, any system that has \(A=D=0\) (and hence \(m=0)\) will convert Huygens' integral into a simple Fourier transformation from the input wave to the output plane, going in either direction through the system.

Such systems might thus be referred to as Fourier transform systems. Systems with only \(A=0\) or only \(D=0\) accomplish a modified kind of Fourier transform in one direction only, and might thus be called "one-way Fourier transform systems."

The trace of m value of a system is, of course, invariant to changes in either the curvature or location of the reference plane, but the \(\text{ABCD}\) values themselves can be modified in various ways by changing either the curvature of the reference plane, as discussed in the previous tutorial or by moving to another reference plane within the system.

 

 

7.  ORTHOGONALITY PROPERTIES OF OPTICAL RESONATOR MODES 

To finish our discussion of general paraxial resonators, in this section we will consider the orthogonality properties of optical resonator eigenmodes.

Our analysis will include not only the ideal complex paraxial resonator model introduced at the beginning of this tutorial, but also a more general class of resonators having hard-edged apertures and other quite general nonparaxial apertures and elements.

In the derivation as we will carry it out in this section, these nonparaxial elements are all lumped at a single plane within the resonator, and described by a complex aperture transmission function \(\tilde{\rho}(s)\).

The results we will obtain probably apply, however, to a considerably more general class of resonators than just this single-aperture situation. 

 

"Normal Modes" in Transmission Lines and Resonators

The propagating modes in ordinary waveguides or transmission lines are commonly referred to as "normal modes" of these systems. The usual meaning of this phrase is that these eigenmodes are power-orthogonal to each other across the waveguide cross section, in the sense that 

\[\tag{43}\int_A\tilde{u}^*_n(s)\tilde{u}_m(s)ds=|\delta_{nm},\]

where \(\tilde{u}_n\) and \(\tilde{u}_m\) are two different eigensolutions in the transverse direction. These modes also generally provide a complete set of basis functions for expanding any propagating fields in the waveguide or in a resonant cavity having the same cross-section.

Optical resonator modes, however, are generally not orthogonal in this fashion, nor do they necessarily comprise a complete set.

Optical modes lack these desirable properties because the fundamental operator of which the optical modes are eigensolutions is not in general a hermitian operator; and nonhermitian operators are not necessarily guaranteed to have a complete and orthonormal set of eigensolutions.

Transverse eigenmodes in open-sided optical resonators and lensguides instead usually obey a biorthogonality relationship between the eigenmodes and a related set of transposed or adjoint modes.

We will show in this section that these adjoint modes represent in physical terms the transverse eigenmodes traveling in the opposite direction inside the same resonator or lensguide.

 

TABLE 2

Linear Operators and Their Adjoints 

 

Linear Operator Notation: Adjoints and Transposes

By way of introduction, let us first review some general properties of linear but not necessarily hermitian operators. Let \(L\) indicate a general linear operator, whether this means a differential, integral, or matrix operator.

Such an operator then acts on a function \(\tilde{u}\) (or in the matrix situation on a vector \(u\)) to produce some new function \(\tilde{u}'\), in the fashion 

\[\tag{44}L\tilde{u}=\tilde{u}'.\] 

Associated with any such linear operator \(L\) will then also be an adjoint or transposed operator \(L_T\), and an hermitian conjugate or hermitian adjoint operator \(L_H\)

The procedures for converting an operator to its transpose or its hermitian conjugate are illustrated in Table 2.

For example, if \(L\) is a matrix operator, then its transposed operator is obtained simply by interchanging the order of subscripts. For an integral operator the transpose is obtained by interchanging coordinates in the operator kernel.

For a differential operator the transpose is obtained by modifying each \(n\)-th order derivative term in the manner illustrated in the table.

The hermitian conjugate or hermitian adjoint operator for each of these situations is then simply the complex conjugate of the transpose operator.

A hermitian operator is then by definition an operator which is equal to its hermitian conjugate, so that \(L_H\equiv L^*_T=L\).

 

Operator Eigenfunctions and Eigenvalues

Most of the linear operators with which we work, whether hermitian or not, possess some set of eigenfunctions \(\tilde{u}_n\) and eigenvalues \(\tilde{\gamma}_n\) satisfying the eigenequation 

\[\tag{45}L\tilde{u}_n=\tilde{\gamma}_n\tilde{u}_n.\]

If a linear operator is hermitian, then it can be rigorously proven that its eigenvalues \(\tilde{\gamma}_n\) will all be purely real, and its eigenfunctions \(\tilde{u}_n\) will form a set that is complete and also orthonormal in the sense given earlier, namely,

\[\tag{46}\int\tilde{u}_n^*(s)\tilde{u}_m(s)ds=\delta_{nm}.\] 

The integration here is over the full range of the complete set of coordinates \(s\) that characterise the eigenfunctions.

If linear operator is not hermitian, however, then these properties cannot be proven in general, and may or not be true in individual specific situations. In this situation, the transposed operator \(L_T\) and the hermitian adjoint operator \(L_H\) will have separate and different sets of eigensolutions, call them \(\phi_n\) and \(\tilde{w}_n\) respectively, which satisfy the separate eigenequations 

\[\tag{47}L_T\phi_n=\tilde{k}_n\phi_n\quad\text{and}\quad L_H\tilde{w}_n=\tilde{\xi}_n\tilde{w}_n.\]

Since the two operators \(L_T\) and \(L_H\) are simply complex conjugates of each other, however, their eigensolutions are essentially the same set, with the relations

\[\tag{48}\phi_n\equiv\tilde{w}^*_n\quad\text{and}\quad\tilde{k}_n\equiv\tilde{\xi}^*_n.\]

It can also be shown that even for a nonhermitian operator, the eigenvalues of all three operators are related by 

\[\tag{49}\tilde{\xi}^*_n=\tilde{k}_n=\tilde{\gamma}_n.\]

That is, \(L\) and its transpose \(L_T\) always have the same eigenvalues even for nonhermitian operators. Their eigenfunctions \(\tilde{u}_n\) and \(\phi_n\) will not in general be complex conjugates, however, nor will they have any other simple relationship for a nonhermitian operator.

 

Biorthogonality

Instead of being power-orthogonal in the sense of Equation 43, the eigenfunctions \(\tilde{u}_n\) of a nonhermitian operator \(L\) will be biorthogonal to the eigenfunctions \(\phi_n\) of the corresponding transpose operator, in the form 

\[\tag{50}\int\tilde{u}_m(s)\phi_n(s)ds\equiv\int\tilde{u}_m(s)\tilde{w}^*_n(s)ds=\delta_n|_m.\]

(It is a matter of convenience whether we choose to employ the hermitian adjoint eigenfunctions \(\tilde{w}_n\) or the transpose eigenfunctions \(\phi_n\) in this relationship.)

A further useful relation between these eigenfunctions is that

\[\tag{51}\sum_n\phi_n(s)\tilde{u}_n(s_0)\equiv\sum_n\tilde{w}^*_n(s)\tilde{u}_n(s_0)=\delta(x-x_0),\]

where the sum is over the full set of eigenfunctions. From this relation, we can show that for either a matrix operator or an integral operator, the sum of the eigenvalues will be

\[\tag{52}\sum_n\tilde{\gamma}_n=\int_A\tilde{K}(s,s)ds=T_r[L],\]

where \(T_r\) means the trace of the matrix operator. This can be very useful for checking numerical calculations of optical resonator eigenmodes.

 

 

 

Optical Resonator Model 

To apply these orthogonality or biorthogonality concepts to optical resonators, we will use the slightly idealized model for an optical resonator shown in Figure 13. 

In this model we consider either a standing-wave cavity, or a ring cavity, or an equivalent periodic lensguide, which may contain an arbitrary collection of complex paraxial elements; but in which all finite mirrors, or hard-edged apertures, or other nonparaxial apertures or output coupling elements are lumped at a single transverse plane.

The transmission of the circulating wave through this plane is then described by a transverse transmission or reflection function \(\tilde{\rho}(s)\). 

That is, for a standing-wave resonator with some kind of finite output mirror, this function describes the transverse reflectivity variation \(\tilde{\rho}(s)\) of the output mirror. For a ring resonator or a periodic lensguide with some kind of internal aperture or spatial filter, on the other hand, \(\tilde{\rho}(s)\) is the transmission factor of the aperture between successive sections.

For a finite mirror or hard-edged aperture this function then has the value \(|\tilde{\rho}|=0\) outside the finite mirror or aperture edges. The remainder of the resonator is assumed to contain only an arbitrary sequence of complex paraxial optical elements.

For simplicity, we write the Huygens' integrals in one transverse dimension only, but this is easily generalized.

 

Forward and Reverse Propagation Operators

Let us first note that the propagation through the paraxial portions of the resonator, starting just after the aperture, traveling in the forward direction around the resonator, and ending just before the aperture, can be expressed by the standard complex paraxial Huygens' kernel

\[\tag{53}\tilde{K}(x,x_0)=\sqrt{\frac{j}{B\lambda_0}}\text{exp}\left[-j\frac{\pi}{B\lambda_0}(Ax^2_0-2xx_0+Dx^2)\right],\]

where \(A\), \(B\) and \(C\) are the complex matrix elements going around the resonator or along the lensguide in the forward direction. Note that this is not in general a hermitian integral kernel.

Suppose we reverse the direction of travel, however, and propagate the field in the reverse direction between these same two planes. The \(\text{ABCD}\) matrix elements going in the reverse direction through the same system are then given by \(A_R=D\), \(B_R=B\), \(C_R=C\), and \(D_R=A\); and the Huygens' kernel in the reverse direction is thus given by

\[\tag{54}\tilde{K}_R(x,x_0)=\sqrt{\frac{j}{B\lambda_0}}\text{exp}\left[-j\frac{\pi}{B\lambda_0}(Dx^2_0-2xx_0+Ax^2)\right],\]

But this is just the transpose of the kernel in the forward direction; so we can write (in two transverse directions)

\[\tag{55}\tilde{K}_R(s,s_0)=\tilde{K}(s_0,s)=\tilde{K}_T(s,s_0).\]

We see that (a) the Huygens' integral kernel is not a hermitian operator; and (b) propagating in the reverse direction through a given paraxial system is the transpose of propagating through the same system in the forward direction.

 

Handling the Nonparaxial Aperture 

We can now include the nonparaxial aperture in this same analysis by the following trick, namely, as shown in Figure 14, we will use as our initial reference plane a hypothetical plane located "halfway through" the transmitting aperture or reflecting mirror that is characterized by the aperture function \(\tilde{\rho}(s)\).

That is, we assume that to propagate once around the resonator the initial field \(\varepsilon_0(s_0)\) at this reference plane must be multiplied by half the mirror reflection function, or \(\tilde{\rho}^{1/2}(s)\); propagated on around the remainder of the cavity using Huygens' integral; and finally again multiplied by half the mirror reflection, or \(\tilde{\rho}^{1/2}(s)\), before becoming the resulting field \(\varepsilon(s)\) one full round trip later. (For a real curved mirror this is in fact equivalent to taking the reference plane on the mirror surface.)

The propagation operator for one complete trip around the resonator in the forward direction, starting out from and coming back to this reference plane, is then

\[\tag{56}\varepsilon(s)=\tilde{\rho}^{1/2}(s)\int\tilde{K}(s,s_0)\tilde{\rho}^{1/2}(s_0)\varepsilon_0(s_0)ds_0,\]

where \(\tilde{K}(s,s_0)\) is the forward Huygens' integral kernel as given in Equation 54, and where the integral is evaluated over the aperture midplane.

This integral is

 

 

obviously over all regions of the aperture or mirror midplane where the optical field is not zero, that is to say, all regions where \(|\tilde{\rho}(s)|\neq0\).

The resonator eigenmode equation going in the forward direction is then

\[\tag{57}\int\tilde{K}_F(s,s_0)\varepsilon^F_n(s_0)ds_0=\tilde{\gamma}^F_n\varepsilon^F_n(s),\]

where the overall kernel \(\tilde{K}_F(s,s_0)\) in the forward direction is 

\[\tag{58}\tilde{K}_F(s,s_0)=\tilde{\rho}^{1/2}(s)\times\tilde{K}(s,s_0)\times\tilde{\rho}^{1/2}(s_0).\]

This is clearly not in general a hermitian integral operator, and the transverse modes of optical resonators are thus in general not guaranteed to be normal modes in the usual sense.

If we go around the ring resonator or travel along the lensguide in the reverse direction, however, then we have potentially a different, transposed round-trip operator, such that the eigensolutions for the resonator or lensguide going in the reverse direction, call them \(\varepsilon^R_n(s)\), will be given by the eigenmode equation

\[\tag{59}\int\tilde{K}_R(s,s_0)\varepsilon^R_n(s_0)ds_0=\tilde{\gamma}^R_n\varepsilon^R_n(s),\]

where the reverse-direction kernel \(\tilde{K}_R(s,s_0)\) is given by

\[\tag{60}\begin{align}\tilde{K}_R(s,s_0)&=\tilde{\rho}^{1/2}(s_0)\times\tilde{K}(s_0,s)\times\tilde{\rho}^{1/2}(s)\\&\equiv\tilde{K}_F(s_0,s).\end{align}\]

In other words, even with the aperture included, these propagation operators are in general not hermitian, and the integral operator \(\tilde{K}_R\) for propagation through the lensguide or resonator in the reverse direction is just the transpose of the operator \(\tilde{K}_F\) for propagation in the forward direction. 

 

Biorthogonality of Resonator Eigenmodes

We can conclude therefore that the forward eigenmodes \(\varepsilon^F_n(s)\) of an optical resonator will in general not be orthogonal among themselves in the usual complex conjugate fashion.

Rather, the forward-going transverse modes \(\varepsilon^F_n(s)\) will be biorthogonal to the transverse modes \(\varepsilon^R_n(s)\) of the wave going in the reverse direction in the same resonator, as expressed mathematically by 

\[\tag{61}\int\varepsilon^R_n(s)\varepsilon^F_m(s)ds=\delta_{nm}.\]

Note that no complex conjugation is involved in the integrand.

This biorthogonality integral is derived thus far only for the special "midaperture" reference plane described by \(\tilde{\rho}(s)\).

Since the two sets of waves are traveling in opposite directions through the system, however, it is evident that we can shift to a reference plane immediately outside the aperture, on either side, by multiplying one set of modes by the partial reflection factor \(\tilde{\rho}^{1/2}(s)\) and dividing the other set by the same factor.

The net effect of this leaves the biorthogonality integral unchanged. The integral in Equation 61 thus holds immediately outside the aperture also.

In addition, given that the two sets of functions represent propagating waves traveling in opposite directions through a paraxial system, it is then not difficult to use the propagation properties of a general paraxial system to prove that biorthogonality at one transverse plane implies biorthogonality at all other planes within the resonator.

The integral in Equation 61 thus holds generally, at any plane within the resonator.

 

Physical Interpretation 

This biorthogonality relation between forward and reverse directions has the following physical interpretations for standing-wave and traveling-wave optical resonators.

First of all, in a standing-wave optical resonator with a 100% reflecting mirror at one end and the finite output mirror at the other end, the aperture midplane is actually on the surface of the output mirror.

The forward and reverse propagation paths around the resonator, starting from this mirror surface, are then equivalent and identical; and the equivalent lensguide is inherently symmetrical in the forward and reverse directions. 

A little consideration then shows that in such a standing-wave resonator the biorthogonality property means simply that the fields of the right-traveling wave component of any standing-wave eigenmode \(\varepsilon_n(x,z)\) are biorthogonal to the fields of the left-traveling wave components of all the other eigenmodes at the same transverse plane \(z\). 

In a ring cavity, on the other hand, the forward and reverse (or clockwise and anticlockwise) directions are physically distinguishable, and can correspond to a directionally asymmetric lensguide.

The forward and reverse waves can then have quite separate and distinct sets of eigenfunctions traveling in the two opposite directions around the ring or along the lensguide. Characteristic profiles for the oppositely traveling modes in an asymmetric ring unstable resonator are illustrated in Figure 15.

The geometric profiles of these modes are significantly different, as shown, with the forward mode in this particular asymmetric situation being confocal and the reverse mode nonconfocal.

The actual unstable resonator eigenmodes, which will be Fresnel-distorted variations on these zero

 

 

order geometric solutions, will have similar variations in the two directions. The eigenvalues will, however, be the same, as illustrated by the fact that the net magnification is \(M=2\) in both directions in Figure 15. These two sets of eigenfunctions are then biorthogonal to each other in exactly the fashion given in the preceding.

The nonhermitian character of the Huygens' integral operator may seem surprising, especially since Huygens' integral is derived from the wave equation operator \([\nabla^2-\mu\epsilon(\partial^2/\partial t^2)]\tilde{u}\), which is clearly hermitian. Suppose we consider, however, an asymmetric unstable lensguide with its forward and reverse modes as shown in Figures 10 or 15. The significant point is that the forward and reverse eigenmodes deliver power to opposite sides of the aperture.

More generally, in any real open-sided resonator there will be radiation leakage or output coupling fields that represent power flow out to infinity.

In terms of the complete wave equation in space and time, therefore, the operator is hermitian but the boundary conditions are not. This difficulty does not arise in simple waveguide problems where all the boundaries are closed and perfectly conducting, or else the fields all die away rapidly enough at infinity. When the optical resonator problem is separated into exp\((\pm jwt)\) terms, however, the individual operator applied only to the exp\((jwt)\) part of the fields turns out to be nonhermitian.

The derivation presented here has also been limited to the paraxial form of the Huygens integral.

All that is really necessary, however, is that the appropriate kernel or Green's function for propagation around the resonator obey the reciprocity property \(\tilde{K}_R(s,s_0)=\tilde{K}_F(s_0,s)\); and this will be generally true under

 

 

a very wide range of conditions. Hence the result presented here will be valid for a comparably wide range of resonator designs. 

  

Excitation of Resonator Eigenmodes 

The biorthogonality properties of optical resonator modes can lead to some perhaps surprising conclusions, particularly in unstable resonators. Suppose we wish to excite the maximum amplitude of the dominant mode in an unstable resonator or lensguide, as in Figure 16, using an injected signal of fixed total power. 

It might then seem that we would use the injected power most effectively by matching it as closely as possible to the wavefront that we want to excite, as shown in Figure 16(a).

In fact, however, the dominant mode of the unstable system will be excited most effectively, or with the largest mode amplitude, by matching the injected signal to the converging or adjoint wavefront, as shown in Figure 16(b), a process sometimes referred to as "adjoint coupling."

We can understand the necessity for this by supposing that the wave amplitude that is to be excited in the unstable system will be measured only after a large number of round trips, or after the wave has propagated through many sections of the equivalent lensguide, so that only the dominant mode in the system will be left.

It then seems more physically evident that if we want to get as much power as possible as far down the system as possible—which means putting as much of the excitation as possible into the dominant or lowest-loss mode—then we indeed want to focus the energy initially into a converging wave which will first travel inward before being spread back out by diffraction effects.

 

Power Normalization of Biorthogonal Modes

Part of the confusion in this situation arises from our usual concepts that we can compute the excitation of each individual "normal mode" in a system and then add up-the powers given to each normal mode to get the total power carried by the system.

In a biorthogonal system, however, we cannot do this because the eigenmodes are not in general power-orthogonal to each other—in fact, they are in general not orthogonal to each other at all, either with or without complex conjugation involved.

Hence one cannot compute the total power traveling along a nonhermitian lensguide as simply the sum of the powers in the individual eigenmodes.

We can normalize the individual forward or reverse eigenmodes \(\varepsilon^F_n\) and \(\varepsilon^R_n\) in various ways. However, we can also show that if these eigenmodes are biorthogonal in the form given in Equation 61, then it is in general not possible to also power-normalize both \(\varepsilon^F_n\) and \(\varepsilon^R_n\) such that the integral of \(|\varepsilon|^2\) is unity for both of these sets of functions separately.

We should perhaps note again that the Hermite-gaussian modes, which provide very close approximations for the real eigenmodes in stable gaussian resonators, are a complete set of normal modes in the usual sense, and hence we can apply all the usual orthonormality properties to these functions.

The diffraction losses in stable resonators are usually very small, and as a consequence the real modes in stable gaussian resonators are very nearly "normal modes" in the usual sense.

\[\tag{15}N_{eq}=\frac{a^2}{2R_0\lambda}=\frac{M^2-1}{2M}\frac{a^2}{B\lambda}=\frac{M^2-1}{2M^2}\times N_c\]