## Abstract

We report what is to our best knowledge the first observation of Mathieu–Gauss modes directly generated in an axicon-based stable resonator. By slightly breaking the symmetry of the cavity we were able to generate single lowest and high-order Mathieu–Gauss modes of high quality. The observed transverse modes have an inherent elliptic structure and exhibit remarkable agreement with theoretical predictions.

©2008 Optical Society of America

## 1. Introduction

Material processing with laser encompasses several applications such as cutting, welding, drilling, scribing, and surface modification processes. In each application, optimum results are obtained from a combination of particular properties of the beam. Therefore, it is necessary to shape the beam to suit the particular application. In this direction, the transverse structure of the modes supported by axicon-based resonators has been investigated with analytical, numerical, and experimental techniques [1, 2, 3, 4, 5]. Based on the fact that a refractive axicon transforms an incident plane wave into a Bessel beam, to our best knowledge, all studies of the axicon-based resonators have centered on the generation of circularly symmetric Bessel and Bessel-Gauss (BG) beams.

On the other hand, Mathieu beams constitute other family of nondiffracting beams that are solutions of the wave equation in elliptic coordinates [6], and have been successfully applied for example to photonic lattices [7], transfer of angular momentum using optical tweezers [8], and localized X-waves [9]. The Mathieu-Gauss (MG) beams are Mathieu beams apodized by a Gaussian transmittance [10], which carry a finite power and have been generated experimentally to a very good approximation using holographic methods [11]. The MG beams are formed as a superposition of fundamental Gaussian beams whose mean propagation axes lie on the surface of a cone, and whose amplitudes are modulated angularly by the angular Mathieu functions. Since the reflective axicon transforms an incident plane wave into a converging conical wave, then the axicon-based resonator is a natural device to generate not only BG beams, but also MG beams and parabolic-Gauss beams [10, 12].

In this paper, we report, to our best knowledge, the first observation of MG beams directly generated by an axicon-based stable resonator. The experiment employs a 10 W cw CO _{2} laser resonator composed by a copper axicon and a plane output coupler operating at 10.6 µm. It is found that when the symmetry of the cavity is slightly broken by some micrometers, the BG modes become elliptical transverse modes that can be identified as even-parityMG modes with a high quality. By the use of intracavity wire elements, that mainly obstruct the central part of the axicon, it is possible to obtain odd-parity MG modes as well. An important advantage of the axicon-based MG resonator, is that it does not require external optical elements or a special shape for the active medium, and it employs conventional mirrors and commercial axicons. The experimental results exhibit very good agreement with the theoretical predictions.

## 2. Mathieu-Gauss modes

We briefly describe the MG beams in order to get a physical insight. In free space, the complex amplitude of the *m*th-order even and odd MG beams propagating along the positive *z* axis of an elliptic coordinate system **r**=(*ξ,η,z*) is given by [10]

where Je_{m}(·) and Jo_{m}(·) are the *m*th order even and odd radial Mathieu functions, ce _{m}(·) and se_{m}(·) are the *m*th order even and odd angular Mathieu functions, GB(**r**)=*µ*
^{-1}exp(-*r*
^{2}/*µw*
^{2}
_{0}) is the fundamental Gaussian beam, µ(*z*)=1+*iz/(kw ^{2}_{0}/2), w*

_{0}is the Gaussian width at the waist plane

*z*=0, and

*k*is the wave number. In a transverse

*z*plane, the complex elliptic variables (

*ξ,η*) are related to the Cartesian coordinates (

*x,y*) by the relation: (

*x*+

*iy*)=

*f*

_{0}cosh(ξ+

*iη*), where

*f*

_{0}is the semifocal separation at the waist plane. The MG beams are characterized by the ellipticity parameter

*q*=

*k*/4, which carries information about the transverse wave number

^{2}_{t}f^{2}_{0}*k*and the ellipticity of the coordinate system through

_{t}*f*

_{0}. When

*q*→0, the foci of the elliptic coordinates collapse at the origin; therefore, the elliptic MG beams reduce to the circular BG beams. At any plane z the transverse shape of the power spectrum for the MG beams is given by|MG

_{m}(

*w*|

^{2}_{0}u/2i, w^{2}_{0}ν/2i;0)^{2}and corresponds to an annular ring whose mean radius and width are determined by

*k*and

_{t}*w*

_{0}respectively [10]. The ideal Mathieu nondiffracting beams are recovered when the Gaussian envelope tends to a plane wave, i.e.

*w*

_{0}→0. In this case, the power spectrum becomes an ideal Dirac delta circular ring.

## 3. Description of the cavity.

For the experiment we use the cavity configuration illustrated in Fig. 1(a). A detailed theoretical study of the axicon-based resonator for generating BG modes was presented in Ref. 4; it is the notation of this reference for the cavity variables that we use. The resonator consists of a refractive conical mirror (i.e. a diamond machined copper axicon) with characteristic angle *θ*
_{0}=0.5° and radius a=1 in, and a 2” ZnSe plane output mirror with reflectivity of 85%, separated a distance *L=a*/(2tan*θ*
_{0})≃1.45 m. The reflective axicon and the output coupler were mounted outside of the active medium to have the possibility of adjusting the alignment and length of the cavity, and also to avoid the surface damage of the cooper axicon which normally appears when it is in contact with the active medium. The active medium is a gas mixture of CO_{2}, N_{2} and He with a relation 1:1:8, respectively, and peak emission wavelength 10.6 µm. The medium was ionized with 2–5 KV AC and a variable electric current whose range is 40–80 mA. Figure 1(b) shows the schematic of the slow gas-flowing (5.6 ft ^{3}/min) CO_{2} laser system designed for this experiment. For stable operating conditions of 9.5 torr, 4 kV, and 60 mA, an average output power of 10 W was obtained from the axicon-based resonator. An increase in the efficiency was also obtained by adding a coolant jacket surrounding the tube containing the active medium, where a water-antifreeze mixture flows at -5 °C. Electrical and vibration isolation was achieved by mounting the laser system on an isolated granite table.

Adjusted for azimuthal symmetry, the laser emits fundamental *J*
_{0} (*k _{t}r*) Bessel modes characterized by concentric rings with a maximum spot at the optical axis, as shown in Fig. 2. The transverse spatial frequency of the Bessel mode is entirely determined by the axicon angle through the relation

*k*=

_{t}*k*sinθ0≃5170 m

^{-1}giving a separation of about π/kt=0.6 mm between consecutive fringes. We performed measurements at different

*z*planes to corroborate that the output beam maintains its nondiffracting transverse profile along the expected distance of

*L*=

*a*/(2tan

*θ*

_{0})≃1.45 m measured from the output coupler, see Fig. 1(a).

Additionally, it is known that if a beam is nondiffracting then its power spectrum (i.e. its two-dimensional Fourier transform) is a circular ring whose angular modulation is given by the angular spectrum of the original nondiffracting beam [13, 14, 15]. To verify this fact, we introduced a positive lens with focal distance *f*=25 cm in the propagation of the beam and recorded the pattern just at the focal plane of the lens. The measured image rings for the first two Bessel modes are displayed in Fig. 2. Both image rings have uniform intensity but different phases. The annular structure could be particularly useful in hole drilling. For example, to make holes with diameter of at least 0.1 in, the traditional procedure consists on moving the focused laser Gaussian spot along the hole circumference. The annular patterns allows to make the hole with only one shot, substantially reducing the drilling time per hole.

## 4. Observation of Mathieu-Gauss modes and physical discussion.

Once the resonator was emitting BG modes, to generate MG beams we were slightly breaking the symmetry of the cavity by tilting the output coupler by several microradians. The influence of the astigmatism onto the axicons was discussed previously in Refs. 16–18. Figure 3 shows the measured intensity distributions and their theoretical predictions of several even MG modes for different orders and ellipticities. To enable the observation of higher-order modes, it is necessary that the diameter of the resonator be sufficiently large to reduce the diffraction losses that affect them. Unlike the circularly symmetric BG modes, the MG modes consist of well defined elliptic and hyperbolic nodal lines with a dark elliptic spot on axis for *m*≥1. In general the patterns exhibit a tendency to be aligned to the *x* or *y* axes when the output coupler is tilted in this way. All modes in Fig. 3 were obtained by tilting the output coupler up to 40 mrad.

Figure 3 also shows the power spectra of the output beams recorded at the focal plane of a converging lens of focal distance *f*=25 cm. As expected, the patterns correspond to circular thin rings whose angular intensity variation is described by the angular Mathieu function, i.e. |ce_{m} (*θ,q*)|^{2}. The radius of the image ring depends on the lens and the axicon through the relation *r*
_{0}=*f* tan*θ*
_{0}, therefore the adjustment of the ring size can be easily done just by selecting the appropriate lens. For the parameters used in our experiment, we got *r*
_{0}=*f*tan*θ*
_{0}≃2.2 mm. We want to remark that the observation of this ring constitutes an indirect proof that the emitted mode belongs to the family of the nondiffracting beams.

The experimental intensity patterns shown in Fig. 3 were taken about 10 cm away from the output aperture. As the wavelength lies on the infrared range, a thermal plate kit supplied with an ultraviolet lamp was used to observe the transverse distributions of the output beam at real time. In the experiments we used several plates with a range sensitivity from 7.5 to 200 W/cm^{2} and 200–300 lines/mm resolution. Due to the small dimensions of the patterns –less than 5 mm– they were recorded using a commercial digital camera WAT-902DM. Like the BG modes, we corroborated that the MG modes preserve its nondiffracting behavior along a distance *L* from the output mirror. As depicted in Fig. 1(a), outside this invariance region, in the far field zone, the beam diverges and acquires a conical wavefront forming a doughnut structure with azimuthal intensity variation given by the angular even or odd Mathieu functions. In our experiment, we used a flat output coupler, but it is important to note that the radius of curvature of the output coupler is an additional parameter that could be modified to minimize the losses due to diffraction and to adjust the Gaussian width of the output beam. For a concave spherical output mirror the losses of the modes decrease as the radius of curvature decreases.

The experimental results reveal that the astigmatism in the resonator generated by the tilt of the output coupler induces MG modes with even parity. To force the laser to emit odd-parity modes, one or two perpendicular metallic crosshairs were introduced in the cavity just behind the center of the axicon. Figure 4 shows the odd modes obtained for *m*=1,2, and 3. The theoretical patterns shown in Figs. 3 and 4 were plotted from Eq. (1) choosing the ellipticity *q* and the width *w* for the best fit with the experimental results. In all cases *m* corresponds to the number of hyperbolic nodal lines in the intensity patterns. We emphasize the excellent agreement between the measured and theoretical patterns shown in Figs. 3 and 4.

For currents lower than the threshold operation current of 40 mA, the cavity generates time-varying complicated patterns resulting from the incoherent superposition of fundamental modes. For a higher operation current –within the range of 40 to 80 mA–, the output beam becomes stable and its shape depends on the alignment of the cavity only. The behavior of the output power as a function of the operating current is depicted in Fig. 5 for several gas pressures. Since BG and MG modes form two complete and equally valid families of beams for expanding an arbitrary Helmholtz-Gauss field with the same Gaussian envelope [10], the generation of one family of modes in the resonator could be interpreted as a coherent sum of degenerate modes of the other family simultaneously excited in the cavity. This implies that the modes are still linear and their properties do not depend on the nonlinearity of the gain medium. As far as we know, the patterns shown in Figs. 3 and 4 constitutes the first experimental evidence that stable resonators can be forced to single MG operation.

## Acknowledgments

We acknowledge financial support from Consejo Nacional de Ciencia y Tecnología (grant 82407), and from the Tecnológico de Monterrey (grant CAT141).

## References and links

**1. **J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. **190**, 117–122 (2001). [CrossRef]

**2. **A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based Bessel resonator: analytical description and experiment,” J. Opt. Soc. Am. A **18**, 1986–1992 (2001). [CrossRef]

**3. **J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, “Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis,” J. Opt. Soc. Am. A **20**, 2113–2122 (2003). [CrossRef]

**4. **M. Alvarez, M. Guizar-Sicairos, R. Rodríguez-Masegosa, and J. C. Gutiérrez-Vega, “Construction and characterization of CO_{2} laser with an axicon based Bessel-Gauss resonator,” Proc. SPIE **5708**, 323–331 (2005). [CrossRef]

**5. **R. I. Hernández-Aranda, S. Chávez-Cerda, and J. C. Gutiérrez-Vega, “Theory of the unstable Bessel resonator,” J. Opt. Soc. Am. A **22**, 1909–1917 (2005). [CrossRef]

**6. **J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. **25**, 1493–1495 (2000). [CrossRef]

**7. **Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, “Shaping soliton properties in Mathieu lattices,” Opt. Lett. **31**, 238–240 (2006). [CrossRef] [PubMed]

**8. **C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express **14**, 4182–4187 (2006). [CrossRef] [PubMed]

**9. **C. A. Dartora and H. E. Hernández-Figueroa, “Properties of a localized Mathieu pulse,” J. Opt. Soc. Am. A **21**, 662–667 (2004). [CrossRef]

**10. **J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A **22**, 289–298 (2005). [CrossRef]

**11. **C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. **45**, 068001 (2006). [CrossRef]

**12. **M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. **29**, 44–46 (2004). [CrossRef] [PubMed]

**13. **P. A. Bélanger and M. Rioux, “Ring pattern of a lens-axicon doublet illuminated by a Gaussian beam,” Appl. Opt. **17**, 1080–1086 (1978). [CrossRef] [PubMed]

**14. **M. Rioux and P. A. Bélanger, “Linear, annular and radial focusing with axicons and application to laser machining,” Appl. Opt. **17**, 1532–1536 (1978). [CrossRef] [PubMed]

**15. **J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, “Focusing evolution of generalized propagation invariant optical fields,” J. Opt. A **5**, 276–282 (2003). [CrossRef]

**16. **R. Akimoto, C. Saloma, T. Tanaka, and S. Kawata, “Imaging properties of axicon in a scanning optical system,” Appl. Opt. **31**, 6653–6657 (1992). [CrossRef]

**17. **Z. Bin and Z. Zhu, “Diffraction property of an axicon in oblique illumination,” Appl. Opt. **37**, 1080–1086 (1978).

**18. **A. Thanning, Z. Jaroszewicz, and A. T. Friberg, “Diffractive axicons in oblique illumination: analysis and experiments with comparison to elliptical axicons,” Appl. Opt. **42**, 9–17 (2003). [CrossRef]