## Abstract

Single transverse mode oscillation is realized in a conventional HeNe laser outside the stability region of the optical resonator. Depending on the mirror separation different spatial modes can be generated. The mode volume of these modes is laterally limited by the diameter of the discharge capillary rather than by the beam waist of a stable Gaussian mode. Numerical solution of the Maxwell equations with appropriate boundary conditions shows good agreement with the observations. Such modes could potentially facilitate single transverse mode operation of waveguide lasers and fiber lasers.

© 2009 OSA

## 1. Introduction

Gas lasers with low or moderate gain usually employ stable optical resonators [1]. Within the stability range Gaussian modes can be found whose phase fronts exactly match the mirror surfaces, thus allowing a standing wave to exist between the mirrors. Depending on the symmetry, the modes may be Hermite-Gaussian (HG) or Laguerre-Gaussian (LG). The lateral extension of a mode depends on its order, being smallest for the fundamental mode. Thus, to achieve single transverse mode oscillation it is necessary to restrict the lateral dimensions, either by limiting the gain volume, or by inserting an aperture in the resonator. Such measures will restrict the power output, and a trade-off is therefore involved between output power and mode purity. The inconsistency between single transverse mode operation and a large mode volume is a general problem for laser resonators based on guiding structures, and in this paper we use results for a HeNe laser with variable resonator length for demonstrating a possible solution to this problem. In a region of mirror separations outside the stability range the laser is shown to oscillate in non-Gaussian modes, which display the character of dielectric waveguide modes propagating inside the discharge tube [2], with diffraction-like behavior outside. These hybrid modes exploit the entire available gain medium, but the loss discrimination between transverse modes is large enough to allow single mode operation in the lowest order mode. Similar modes have been studied previously in the infrared range where they facilitated single fundamental mode oscillation of hollow dielectric waveguide CO_{2} lasers [3,4]. However, they have not previously been studied in the visible range.

## 2. Experiment

The experiments are conducted with a HeNe gain tube of type Melles Griot model 05-LHB-670, total length 0.351 m, provided with one internal totally reflecting mirror of nominal radius of curvature *R*
_{1} = 0.6 m, and one Brewster window. The output mirror with nominal radius of curvature *R*
_{2} = 0.45 m has a transmission coefficient of 1.4%, and is mounted on a rail for convenient change of the mirror separation. Its exit surface is shaped such that the phase front of the beam is essentially flat when leaving the laser. The radius *a* of the discharge capillary is not divulged by the manufacturer, but from the geometrical outline of the beam close to the output, the radius of the exit aperture is found to be about 1.25 mm.

The stability criterion for the resonator with mirror separation *d* can be expressed as

*g*-parameters are defined as

*d*<

*R*

_{2}(region I) or

*R*

_{1}<

*d*<

*R*

_{1}+

*R*

_{2}(region II) where

*d*is the mirror separation. A measurement of the laser output as a function of resonator length yields the result shown in Fig. 1 . While the expected decrease in power is observed as the stability limits are approached, it is also obvious that oscillation continues into the unstable region. In this work we focus on the vicinity of region I, and Fig. 2 shows the mode patterns observed at a distance of 47 mm from the output coupler, imaged with a 25 mm focal length lens, as the resonator length gradually moves into the unstable region.

## 3. Theory

The unfolded resonator is shown in Fig. 3
. A mirror with radius of curvature *R* is represented by a converging lens with focal length *R*/2, *L* denotes the length of the capillary, and *x*
_{1,2} denotes the separation between the mirrors and the capillary. In order to solve the Maxwell

equations with appropriate boundary conditions we apply the procedure outlined in Refs [3,4]. The capillary is considered as a hollow dielectric waveguide as studied by Ref [2]. Such a waveguide may support a variety of low-loss propagating modes, and we express the field inside the capillary as a superposition of such modes. Choosing a basis of m waveguide modes, the field at the leftmost dotted line in Fig. 3 is thus represented by an m-component column vector.

At the end of the capillary, the field is launched into free space, where it is expressed as a superposition of LG modes. Each waveguide mode is expanded in a basis of n LG modes, and the transformation from the waveguide into free space is thus expressed as an n-by-m matrix. LG modes of any waist radius *w*
_{0} constitute a complete orthonormal set, and the choice of *w*
_{0} is therefore in principle arbitrary. However, the truncation errors associated with using a finite subset of modes depend on *w*
_{0}. If it is chosen too small, the truncation errors become large for the low-order waveguide modes, whereas the high-order modes suffer if *w*
_{0} is chosen too large. We have found the best compromise to be *w*
_{0} = 0.2*a*. In free space and through the lens the LG modes propagate according to the usual laws for Gaussian waves. Re-entry into the waveguide is described by an m-by-n matrix, and the resulting waveguide modes propagate through the waveguide with loss and phase shift as given in Ref [2]. Repeating this sequence completes the round trip.

Since the presence of the Brewster window inside the resonator will prevent oscillation in anything but linearly polarized modes, only such modes are included in the basis. Further we note that both waveguide and free space modes can be classified according to their azimuthal symmetry. The hybrid modes EH_{1m} are circularly symmetric with radial variation given by the zero order Bessel function. They have a maximum at the center and m-1 radial zeros, and will excite predominantly the free space TEM^{(0)}
_{m-1} modes. The composite modes TE_{0m} + EH_{2m} and TM_{0m} + EH_{2m} have a single azimuthal zero with angular variation given by cosθ or sinθ. They are degenerate and may lock together to produce a circularly symmetric intensity pattern often referred to as a donut mode. The radial variation is given by the first order Bessel function with zero at the center, and these modes will predominantly excite the free space TEM^{(1)}
_{m-1} modes. Modes of different azimuthal symmetry do not mix, and therefore the two cases can be treated separately.

Requiring the field to reproduce itself after one double pass of the resonator, apart from a complex phase factor, we get an eigenvalue equation whose solutions provide the eigenmodes of the resonator. In the following these modes will be referred to as hybrid modes. Denoting the i^{th} eigenvalue as *A*
_{i}, the round trip loss will be

Figure 4
shows the loss as a function of the separation *x*
_{2} between the external mirror and the capillary for the two lowest-loss hybrid modes with no azimuthal zeros (blue and green), and the two with one azimuthal zero (red and cyan), using in both cases bases of 10 waveguide modes and 10 free space modes. The capillary length *L* and the distance *x*
_{1} between the capillary and the fixed mirror are not accessible for measurement, but a best estimate is *L* ≈0.305 m and *x*
_{1} ≈0.010 m. Taking into account the slight enhancement of optical length associated with the refractive index of the Brewster window, we have *x*
_{2} ≈*d*−0.313 m. The best overall agreement is obtained by choosing *a* = 1.085 mm, and this suggests that the overall radius of the discharge tube is smaller than the 1.25 mm exit aperture radius.

According to Eq. (1) free space modes cannot exist for 0.456<*d*<0.600 m, corresponding to 0.143<*x*
_{2}<0.287 m, and this region of high loss is evident in Fig. 4. The five observed patterns shown in Fig. 2 and in the left column of Fig. 5
correspond to the crosses marked in the expanded graph of Fig. 4, with *x*
_{2} = 0.127, 0.137, 0.144, 0.149, and 0.154 m. The center column of Fig. 5 shows the beam profiles as measured by scanning a pinhole across the beam, and the rightmost column shows the calculated profiles.

Inside the stability regions the hybrid modes are essentially identical to the usual free space modes, and at *x*
_{2} = 0.127 m (*d* = 0.440 m) the laser oscillates multi-mode. As the stability limit is approached, the losses increase, and at *x*
_{2} = 0.137 m (*d* = 0.450 m) only the two hybrid modes with smallest loss survive. The calculated profile of Fig. 5 assumes 20% contribution from the lowest order mode which is similar to LG_{00} and 80% from the next higher order mode which is similar to the ring-shaped LG_{01}. Near the stability limit *x*
_{2} = 0.143 m (*d* = 0.456 m) the lowest order hybrid mode changes dramatically into a diffraction-like pattern. Close to the laser it appears as a system of concentric rings which gradually merge into a single, almost Gaussian profile far from the laser. As the resonator length is further increased, the ripples in the near field become less pronounced, while the far field remains essentially Gaussian with the same width until oscillation ceases at *x*
_{2} = 0.158 m (*d* = 0.471 m). In an intermediate range 0.146<*x*
_{2}<0.151 m (0.459<*d*<0.464 m) oscillation tends to jump to the next higher order ring-shaped hybrid mode despite its slightly higher loss. This, as well as the dominance of the ring mode at *x*
_{2} = 0.137 m, may be related to a radial gain variation, since in a HeNe laser collisions with the wall of the discharge capillary play a crucial role for the inversion [5]. While the qualitative progression of the beam profiles is well reproduced by the calculation, there are differences in the details. This is probably because 10 basis functions are insufficient to adequately reproduce the very finely structured details of the near field.

In Fig. 6
the circles show for *x*
_{2} = 0.144 m (*d* = 0.457 m) the waist radius of a Gaussian fitted to the measured far field profiles, while the solid blue line represents a fit to the calculated field. For comparison, the solid green and red lines represent the waist radii for resonator lengths in the two stable regions. The smaller beam divergence observed in the unstable region corresponds to a larger beam waist, and in fact the far field is very similar to that created by Fraunhofer diffraction of a plane wave in a circular aperture with radius *a* = 1.085 mm. In the near field region the more complicated output pattern shows qualitative features as expected for Fresnel diffraction. Figure 7
shows, for distances *x*
_{2} out to 6 m from the output coupler, a visualization of the output profiles of the fundamental and the first higher order hybrid mode, for the same five resonator lengths as in Fig. 2 and Fig. 5.

## 4. Conclusion

We have studied non-Gaussian modes in a HeNe laser close to and beyond the stability limit for free space modes. When moving into the unstable region there is a continuous transition from the free space modes into hybrid modes which propagate as dielectric waveguide modes inside the discharge capillary, while showing diffraction-like behavior outside. This transition is associated with a gradual increase in losses and a marked improvement in loss discrimination. Thus, over a range of resonator lengths outside the stable regions, single transverse mode oscillation can be achieved with a mode volume determined by the cross section of the discharge capillary, and hence not subject to the limitation imposed by the waist of a free space Gaussian mode. Such hybrid modes can be realized in lasers based on waveguide resonators where many transverse modes have low loss, and where single transverse mode operation is therefore difficult to achieve. Another possible application is in fiber lasers, where the required structure of waveguide and “free space” can be tailored directly in the glass matrix, and where one has the added advantage of being able to have gain in both sections.

## Acknowledgement

Helpful discussions with Dr. Jan Hald are acknowledged.

## References and links

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