## Abstract

It is well known that beam distortions in a laser cavity can deteriorate the laser’s beam quality, but why and exactly how the impact of such distortions can strongly depend on details of the resonator setup has not been studied in detail. This article clarifies the issue with a simple resonant mode coupling model, explaining e.g. why strong beam quality deterioration is often observed near certain (but not all) resonator frequency degeneracies, while a resonator can be very tolerant to distortions in other cases. These findings lead to important conclusions, including practical guidelines for optimizing laser beam quality via cavity design.

©2006 Optical Society of America

## 1. Introduction

One of the most fundamental properties of laser beams is their potential to be focused to very small spots without excessive divergence. This property, called beam quality, is important for many laser applications and can be quantified, e.g., with the well known *M*
^{2} factor [1]. It is well known that the beam quality of lasers can be deteriorated by the influence of various kinds of distortions within the laser cavity, in particular by aberrations of the thermal lens in the laser crystal. For optimizing the beam quality, the emphasis is often placed on minimizing the intracavity distortions, e.g., by creating a smooth pump intensity distribution and using a laser crystal with a high power efficiency and good thermal conductivity. The complementary approach of making the laser less sensitive to distortions has been less explored. It is apparent, however, that such a “robust” laser design could in many cases be a better solution to the beam quality problems. The potential of this approach is impressively illustrated by an observation which has been known for many years: the beam quality of a laser can be strongly deteriorated when its cavity is operated close to a point where the frequencies of higher-order transverse modes are degenerate [2,3]. Such degeneracies can be created or avoided e.g. by changing an arm length of the laser cavity. The connection of beam quality effects to these degeneracies was recognized early on, and it has also been clear that the phenomenon is somehow related to increased scattering from the fundamental transverse mode into higher-order modes. Nevertheless, a convincing explanation of this phenomenon has to the best of my knowledge not been presented, and important questions remain open. For example, it has not been understood why certain degeneracies (normally of low and even orders) affect the beam quality much more strongly than others, and how exactly these effects are related to aberrations e.g. in the laser gain medium. In this paper I show that the observed effects can be convincingly explained and quantitatively described as a resonant mode coupling phenomenon. This understanding leads to important insight concerning the optimization of laser beam quality (particularly for diode-pumped solid state lasers) via cavity design. The previously known design guidelines, which basically deal with appropriate fundamental mode sizes in the gain medium and with operation near the center of a cavity stability region, are supplemented with well-founded additions.

## 2. Resonant transverse mode coupling

This section introduces a qualitative description of the mechanism which translates intracavity beam distortions into reduced beam quality of a laser; a more quantitative treatment and a discussion of the consequences are presented in the following sections.

For conceptual simplicity, consider a laser resonator where all distortions occur only in one transverse plane, while propagation in the rest of the resonator is free from distortions and can thus be described with an ordinary ABCD round-trip matrix. Without any distortions, for a given ABCD matrix we can calculate Hermite-Gaussian (or Laguerre-Gaussian) cavity modes, which are self-reproducing field configurations apart from mode-dependent phase changes (arising from the Gouy phase shift) and from possible amplitude losses. Any monochromatic light field in the laser resonator can be decomposed into such cavity modes. Distortions, which can be phase distortions (e.g., aberrations from thermal lensing in the gain medium) as well as radially varying gain or loss (see Section 5), now cause a coupling between the complex amplitude coefficients corresponding to the cavity modes. Assuming that most of the power is in the fundamental (Gaussian) cavity mode, the distortions will in each cavity round trip feed certain amplitude contributions from the fundamental mode into higher-order modes. The magnitude of those contributions can be quantified with overlap integrals, as shown in Section 3. A crucial point is that if the fundamental mode is resonant, the higher-order modes will in general be non-resonant due to the mode-dependent Gouy phase shift. For that reason, the amplitude increments from successive round trips add out of phase, so that accumulation of power in the higher-order modes is limited. We thus recognize that mode coupling is not very effective in this situation because it is non-resonant.

In degenerate cases, however, where the frequencies of certain higher-order modes coincide with those of axial (fundamental) modes, we have resonant coupling. The amplitude contributions from many round trips then add coherently, and the resonant higher-order modes can accumulate high powers. It is evident that this resonant excitation of higher-order modes will strongly deteriorate the beam quality of the laser, and that this effect occurs within a certain finite range around a degeneracy point. The width of these resonances depends on the coupling strengths and on the losses of higher-order modes, as discussed in Section 5.

Coupling from the fundamental mode to higher-order modes has already been discussed in the literature, e.g., in Ref. [4] (with incoherent coupling only) or in Ref. [5] (coherent coupling, but not in the context of laser beam quality). However, it appears that the beam quality deterioration at degeneracy [2,3] has not been described as a resonance effect, and accordingly the weak effective coupling in the usual non-degenerate situations has not been recognized as a result of an off-resonance condition.

Section 3 contains a simplified quantitative discussion, Section 4 treats the coupling coefficients in more detail, and Section 5 shows how the results of the paper have practical relevance for advanced laser design.

## 3. A simple model

In order to illustrate the ideas described above, we can consider a simplified case where only a single higher-order mode is interacting with the fundamental mode. Such a situation would occur when the corresponding overlap integral (see below) is much larger for a particular higher-order mode than for all others, or when only one mode is close to resonance. The mode coupling can then be described by the equation

where *a*
_{0} and *a*
_{1} denote the complex amplitudes corresponding to the fundamental and the higher-order mode, respectively, and the primed amplitudes are those after one cavity round trip. *C* is the complex coupling coefficient (related to distortions, calculated with some overlap integral, see section 4), *A* is a real factor describing possible net loss of the higher-order mode (for *A* <1), and Δ is the phase shift per round trip of the higher-order mode relative to that of the fundamental mode, which is determined by the difference of the Gouy phase shifts per round trip. The total coupling matrix is constructed from two parts: one matrix containing *C*, which is Hermitian and conserves the total energy in the modes, and another one which attenuates the higher-order mode (if *A*<1) and introduces a relative phase shift due to the mode-dependent Gouy phase shifts.

Equation (1) can be repeatedly applied to simulate the accumulation of power in the higher-order mode, starting from a situation where all power is in the fundamental mode. Alternatively, one can consider the eigenmodes of the coupled system, where the power in the higher-order mode can be used to quantify the beam quality degradation. We use the second method, i.e., calculate the eigenvectors and eigenvalues of the coupling matrix. We start with the assumption of zero net round-trip gain for both modes (*A*=1). Figure 1(a) shows the resulting mode frequencies as a function of the difference Δ of round-trip phase shifts; note that Δ equals 2π times the cavity round-trip time multiplied with the frequency deviation from resonance. The frequencies exhibit the familiar phenomenon of avoided level crossing [5]. The right graph shows that far from degeneracy, the eigenmode with least frequency shift correspond closely to the unperturbed fundamental. At degeneracy (Δ=0), on the other hand, the power is evenly distributed over the two modes.

For a more realistic case with significant losses of the higher-order mode (e.g., 3%, see Fig. 2), as can be caused e.g. by an aperture or by transversely varying gain in an end-pumped laser, the effect on the mode frequency is weaker, and so is the transfer of power to the higher-order mode (for the same coupling strength). This is basically the same effect as reported, e.g., in Ref. [5], where the finesse of a passive resonator was reduced by coupling to higher order modes, and attenuating the higher-order modes with a pin-hole restored the finesse for the fundamental mode. This suppression emphasizes again that we are dealing with a coherent interaction, where the energy transferred to the higher-order mode in any round trip depends strongly on the excitation of that mode and its phase.

Similar graphs (not shown here) for a weaker coupling and weaker loss of the higher-order mode exhibit narrower features, i.e., the beam quality reduction occurs in a narrower range around the frequency degeneracy. It is important to note that the width of the resonances depends both on the coupling strength and the attenuation of the higher-order mode.

It is straightforward to generalize this model to situations with many transverse modes. Due to various symmetries, resonances for multiple modes can coincide, giving rise to more complex degeneracies. In such situations, complicated mode patterns can arise, and they can sensitively depend on any additional asymmetries introduced e.g. by misalignment or thermal distortions.

If the intracavity beam distortions do not occur in a single plane, this can still be described with a coupling matrix; only the calculation of the matrix elements will be more complicated.

## 4. Overlap Integrals

The complex coupling coefficients are determined by overlap integrals of the mode functions and a complex function describing the distortion. We use a set of normalized and orthogonal “cold cavity” mode functions *E _{n}*(

*x, y*), where for notational simplicity we use a single index

*n*instead of a pair of indices, as normally used e.g. for Hermite-Gaussian modes. Any field distribution on

*E*(

*x, y*) in the reference plane can then be written as

where the complex coefficients *c _{n}* can be obtained from

The distortions in the reference plane are described by a multiplication of the mode function with the complex function *t*(*x,y*). We then have the function *E _{a}*(

*x,y*)=

*t*(

*x,y*)

*E*(

*x,y*) after the element, the coefficients of which are obtained from the coefficients before the element according to

with

We can thus simulate the propagation through the aberrative element by a multiplication of the vector *c _{n}* with the matrix

*a*. The relation of the coupling coefficients to overlap integrals has important consequences, as discussed in the following section.

_{nm}## 5. Discussion

We have seen that the deterioration of beam quality in situations with (nearly) degenerate modes can be understood as a resonance phenomenon, without referring to more subtle effects such as spatial hole burning [2]. The same mechanism can work both in single-frequency and in multimode lasers, as every frequency component can be considered separately. In the following we discuss a number of conclusions arising from this new insight.

Symmetries of mode functions and distortions will often make many of the coupling coefficients vanish. For example, consider situations with circular and well aligned pump and laser beams in an isotropic laser crystal. As the resulting thermal lens and its aberrations also have a circular symmetry, there will be no coupling between e.g. the Gaussian fundamental mode TEM_{00} and any Hermite-Gaussian mode with an odd index or a Laguerre-Gaussian mode with nonzero azimuthal index. As a consequence, a frequency degeneracy of such a mode with the fundamental mode cannot impair the beam quality. On the other hand, a TEM_{02} mode, for example, can easily have a non-vanishing coupling integral, so that its resonance must be avoided to maintain good beam quality. If the pump and laser modes are misaligned, the symmetry is broken and additional higher-order modes can couple to the fundamental and contribute to reduced beam quality in a way which has probably not been recognized before.

Also note that any distortion with a relatively smooth spatial distribution cannot lead to strong coupling of the fundamental mode to high-order modes. This is consistent with the experimental observation that low-order mode degeneracies have the strongest effect on the beam quality [3]. In conclusion, smooth and symmetric beam distortions will couple the fundamental mode only to a few low-order modes.

Furthermore, we recognize that not only phase distortions, but also spatially varying gain or loss can lead to non-vanishing overlap integrals. This means that even in the absence of any thermal lensing (e.g., for lasers with very small pump powers), coupling to higher-order modes can result from radially varying gain due to a radially varying excitation, which normally occurs due to radially varying pump and laser beam intensities. Such contributions will be stronger in lasers with high cavity losses and thus large transverse gain variations, and they are hard to eliminate altogether. They are in principle minimized for an end-pumped laser where the pump beam profile is such that the saturated gain stays constant in the transverse direction. However, this balance would depend on the output power.

It is widely believed that a maximum reduction of net gain for higher order modes, e.g. by an aperture or by using a pump beam radius well below the fundamental mode radius, should optimize beam quality. However, this belief is not well supported by experimental experience, and the new findings give two different reasons to question this belief from a theoretical viewpoint as well. First, a fast drop of gain in the radial direction will lead to large coupling to higher order modes, which may offset the advantage of a high net loss for those modes – particularly when the gain or loss changes abruptly, as with a hard aperture. Second, we have seen that a high net loss for higher order modes broadens the corresponding resonances, which may make it more difficult to avoid all resonances simultaneously.

It appears that the following recipe for a laser design should generally lead to a good trade-off between the mentioned aspects:

The pump intensity distribution should be relatively smooth and symmetric – even if thermal lensing effects could be ignored. A moderate drop of net gain in the radial direction is beneficial, because it somewhat attenuates higher order modes (see Fig. 2 in comparison to Fig. 1). However, this drop should be radially symmetric and not too strong, so that excessive mode coupling and a large width of resonances are avoided. (The optimum trade-off will depend on contributions from other aberrations, e.g. from the thermal lens.) The smooth gain variation will then lead to significant coupling only to a few modes with relatively low orders and with high symmetry. Significant excitation of such modes can be avoided by designing the laser cavity so that the corresponding mode degeneracies are avoided. For example, the mode degeneracy for a Hermite-Gaussian TEM_{nm} mode with the TEM_{00} mode occurs if (*n*+*m*)·*φ*
_{G} is an integer multiple of 2π, where *φ*
_{G} is the Gouy phase shift per cavity round trip; details can be obtained, e.g., from Ref. [3]. Avoiding only few such resonances should usually be feasible, and the estimation of overlap factors for given transverse gain and phase changes (or even only an analysis of symmetries) allows to identify those few resonances which must be avoided in a particular case.

Of course, these guidelines apply in addition to those which were previously known. In particular, the cavity should have a fundamental mode which approximately matches the pump intensity distribution in the laser crystal. Also, this mode size should not sensitively depend on the strength of the thermal lens; the cavity should be operated well within a stability zone [6]. If an initial cavity design appears suitable according to these criteria but turns out to be too close to “dangerous” mode degeneracies, these degeneracies have to be shifted by modifying the Gouy phase shift per round trip. This may in principle be achieved by inserting additional curved mirrors to create an additional focus within the cavity. Unfortunately, this procedure also shifts the stability zones. However, when the cavity design is optimized with a numerical procedure for maximizing a figure of merit, one can find a design which combines suitable stability zones with the avoidance of “dangerous” degeneracies by taking both into account in the merit function.

It is apparent that intracavity distortions with complicated shapes, as can arise, e.g., from inhomogeneities in a poor quality laser crystal or from defects in a laser mirror, introduce coupling to many higher-order modes, so that it will be difficult to avoid all the corresponding resonances at the same time. Advanced cavity designs should thus be most useful in situations where most of the intracavity beam distortions result from predictable and high-symmetry effects such as thermal lensing and the gain profile, so that the number of degeneracies to avoid is small and predictable.

## 6. Conclusions

The relation of laser beam quality to intracavity aberrations and other distortions such as e.g. radially varying gain has been explained as a resonance phenomenon in the context of coherent mode coupling. This picture not only explains the severe deterioration of beam quality near degenerate points of laser cavities, but also gives a better understanding of beam quality issues in more general situations. This provides additional guidance for designing laser cavities with a low sensitivity to beam distortions. In particular, one can predict which frequency degeneracies of higher order modes have to be avoided in a particular situation in order to optimize the beam quality of a laser.

## Acknowledgments

The author thanks G. Arisholm for a number of useful comments on the manuscript.

## References and links

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