## Non-Gaussian modes in a HeNe laser

Optics Express, Vol. 17, Issue 24, pp. 21427-21432 (2009)

http://dx.doi.org/10.1364/OE.17.021427

Acrobat PDF (225 KB)

### 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

1. H. Kogelnik and T. Li, “Laser Beams and Resonators,” Appl. Opt. **5**(10), 1550–1567 (
1966). [CrossRef] [PubMed]

_{2}lasers [3

3. R. Gerlach, D. Wei, and N. M. Amer, “Coupling Efficiency of Waveguide Laser Resonators Formed by Flat Mirrors: Analysis and Experiments,” IEEE J. Quantum Electron. **20**(8), 948–963 (
1984). [CrossRef]

4. J. Henningsen, M. Hammerich, and A. Olafsson, “Mode Structure of Hollow Dielectric Waveguide lasers,” Appl. Phys. B **51**(4), 272–284 (
1990). [CrossRef]

## 2. Experiment

*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.

*d*can be expressed aswhere the

*g*-parameters are defined asWith the given radii of curvature this implies that the stability regions of the resonator are

*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

*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

3. R. Gerlach, D. Wei, and N. M. Amer, “Coupling Efficiency of Waveguide Laser Resonators Formed by Flat Mirrors: Analysis and Experiments,” IEEE J. Quantum Electron. **20**(8), 948–963 (
1984). [CrossRef]

4. J. Henningsen, M. Hammerich, and A. Olafsson, “Mode Structure of Hollow Dielectric Waveguide lasers,” Appl. Phys. B **51**(4), 272–284 (
1990). [CrossRef]

*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.

_{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.

^{th}eigenvalue as

*A*

_{i}, the round trip loss will be

*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.

*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.

*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

## Acknowledgement

## References and links

1. | H. Kogelnik and T. Li, “Laser Beams and Resonators,” Appl. Opt. |

2. | E. A. J. Marcatili and R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. |

3. | R. Gerlach, D. Wei, and N. M. Amer, “Coupling Efficiency of Waveguide Laser Resonators Formed by Flat Mirrors: Analysis and Experiments,” IEEE J. Quantum Electron. |

4. | J. Henningsen, M. Hammerich, and A. Olafsson, “Mode Structure of Hollow Dielectric Waveguide lasers,” Appl. Phys. B |

5. | D. A. Eastham, |

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(140.0140) Lasers and laser optics : Lasers and laser optics

(140.1340) Lasers and laser optics : Atomic gas lasers

(140.3410) Lasers and laser optics : Laser resonators

(140.3570) Lasers and laser optics : Lasers, single-mode

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: September 14, 2009

Revised Manuscript: November 2, 2009

Manuscript Accepted: November 4, 2009

Published: November 10, 2009

**Citation**

Jes Henningsen, "Non-Gaussian modes in a HeNe laser," Opt. Express **17**, 21427-21432 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21427

Sort: Year | Journal | Reset

### References

- H. Kogelnik and T. Li, “Laser Beams and Resonators,” Appl. Opt. 5(10), 1550–1567 (1966). [CrossRef] [PubMed]
- E. A. J. Marcatili and R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
- R. Gerlach, D. Wei, and N. M. Amer, “Coupling Efficiency of Waveguide Laser Resonators Formed by Flat Mirrors: Analysis and Experiments,” IEEE J. Quantum Electron. 20(8), 948–963 (1984). [CrossRef]
- J. Henningsen, M. Hammerich, and A. Olafsson, “Mode Structure of Hollow Dielectric Waveguide lasers,” Appl. Phys. B 51(4), 272–284 (1990). [CrossRef]
- D. A. Eastham, Atomic Physics of Lasers (Taylor & Francis, 1986).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.