## Intracavity transverse modes controlled by a genetic algorithm based on Zernike mode coefficients

Optics Express, Vol. 15, Issue 25, pp. 17051-17062 (2007)

http://dx.doi.org/10.1364/OE.15.017051

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### Abstract

A new adaptive optics (AO) system for controlling the mode profile of a diode-laser-pumped Nd:YAG solid laser has been set up in our laboratory. A 19-element piezoelectric deformable mirror (DM), which is used as the rear mirror of the solid-state laser, is controlled by a genetic algorithm (GA). To improve the system convergence rate, the GA optimizes the first 10 orders of Zernike mode coefficients rather than optimize 19 voltages on the DM. The transform matrix between the 19 voltages and the first 10 orders of Zernike mode coefficients is deduced. Comparative numerical results show that the convergence speed and the correction performance of the AO system based on optimizing Zernike mode coefficients is far better than that of optimizing voltages. Moreover, experimental results showed that this AO system could change TEM_{10}, TEM_{11}, and TEM_{20} transverse modes into a TEM_{00} mode successfully.

© 2007 Optical Society of America

## 1. Introduction

_{00}) output are indispensable in more and more scientific applications such as laser communication, high-precision laser machining, and so on. Therefore, there is an increasing demand for solid-state lasers to generate a TEM

_{00}mode. Unfortunately, since both thermal lenses and thermally induced birefringence are the main thermal effects in solid-state laser resonators and can destroy the output laser beam quality greatly, the thermal effects must be first compensated for in order to obtain the fundamental mode output. Thermally induced birefringence may be eliminated successfully through the careful selection of a natural birefringent crystal. The spherical component of the thermal lens can be eliminated also by designing the resonator cavity well [1

1. W. Lubeigt, G. Valentine, J. Girkin, E. Bente, and D. Burns, “Active transverse mode control and optimization of an all-solid-state laser using an intracavity adaptive-optic mirror,” Opt. Express **10**, 550–555 (2002). [PubMed]

2. D. Burns, G. J. Valentine, W. Lubeigt, E. Bente, and A. I. Ferguson, “Development of high average power picosecond laser systems,” Proc. SPIE **4629**, 129–143 (2002). [CrossRef]

_{00}mode output is to place a pinhole in its resonators. However, this way will abate the output power greatly and may cause the resonator to be misaligned once the pumping conditions change. Since thermal effects in the resonators change with varying pumping conditions [3], a promising way for compensating the thermally induced phase aberrations and thermal lenses is to adopt adaptive methods. It is known that the adaptive optics (AO) technique is a powerful technique that allows dynamic correction of phase aberrations. Although it was initially developed for astronomy, it can also be used in the field of lasers [4]. A 37-element membrane deformable mirror (DM) has been used successfully intracavity to optimize the laser modes [1

1. W. Lubeigt, G. Valentine, J. Girkin, E. Bente, and D. Burns, “Active transverse mode control and optimization of an all-solid-state laser using an intracavity adaptive-optic mirror,” Opt. Express **10**, 550–555 (2002). [PubMed]

5. W. H. Jiang and H. G. Li, “Hartmann-Shack wave-front sensing and wave-front control algorithm,” Proc. SPIE. **1271**, 82–93 (1990). [CrossRef]

^{19}[6

6. P. Yang, S. J. Hu, S. Q. Chen, W. Yang, B. Xu, and W. H. Jiang, “Research on the phase aberration correction with a deformable mirror controlled by a genetic algorithm,” J. Phys: Conf Series **48**, 1017–1024 (2006). [CrossRef]

## 2. The transform matrix between Zernike mode coefficients and voltages

7. W. H. Jiang, N. Ling, X. B. Wu, C. H. Wang, H. Xian, S. F. Jiang, Z. J. Rong, C. L. Guan, L. T. Jiang, Z. B. Gong, Y. Wu, and Y. J. Wang, “37-element adaptive optics experimental system and turbulence compensation experiments,” Proc. SPIE **2828**, 312–321 (1996). [CrossRef]

_{j}is the voltage applied on the jth actuator, V

_{j}(x,y) is the influence function of the jth actuator on the wave-front, and n is the number of actuator.

_{j}, y

_{j}) is the space position of the ith actuator; p is set to 2 and d is the distance between every two neighboring actuators; and x and y represent the value in the x-coordinate and the y-coordinate, respectively, of the orthogonal coordinate plane.

*b*0 is the piston coefficient;

*Z*(

_{k}*x,y*) and

*b*(k=1, 2… m) are the kth Zernike polynomials and their corresponding coefficients, respectively; and m is the Zernike order.

_{k}*U*, we investigate the correction capability of the 19-element DM through simulation first.

^{th}to the 14

^{th}and the 18

^{th}to 20

^{th}orders of Zernike aberrations, to some extent. As a result, choosing the first 10 orders of Zernike polynomial coefficients as the basis to deduce the matrix

*U*is suitable.

*ψ*(

*x,y*) with a size of 10000×1, and

*u*represents the voltage vector. According to Eq. (3) and Eq. (5) we can obtain:

_{max}and δ

_{min}are the maximum and minimum singular values, respectively. In fact, the condition number is the measurement of the ill-condition of the matrix. Traditionally, this value should be smaller than 20. Since the condition number is 4.47 calculated by Eq. (9), the capability of V (x,y) is very good. Once

*ψ*(

*x,y*) and V (x,y) are ascertained, the transform vector between the Zernike mode coefficients and the actuator voltages can be calculated:

^{+}is the generalized inverse matrix of V (x,y). Finally, the transform matrix

*U*between the first 10 orders of Zernike mode coefficients and 19 actuator voltages on the DM can be obtained by calculating Eq. (10) ten times:

*u*(i=1, 2…10)is a 19×1 vector.

_{i}*U*has been set up, the GA will optimize the first 10 orders of Zernike mode coefficients rather than the 19 voltages on the actuators. It should be noticed that since each coefficient is set to 0.1, the voltages calculated by (6) can not be applied directly on actuators before they are multiplied by a factor of 10.

## 3. The principle of the GA based on coefficients

_{C}. Finally, some chromosome positions of individuals are mutated randomly with a mutation rate of P

_{m}for introducing a new individual and avoiding the convergence to a local maximum. By going through this process, the GA will gradually find the optimum mirror shape that can yield the best fitness parameter [9

9. O. Albert, L. Sherman, G. Mourou, and T. B. Norris, “Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy,” Opt. Lett. **25**, 52–54 (2000). [CrossRef]

11. P. N. Marsh, D. Burns, and J. M. Girikn, “Practical implementation of adaptive optics in multiphoton microscopy,” Opt. Express. **11**, 1123–1130 (2003). [CrossRef] [PubMed]

## 4. Comparative simulations

## 5. Mode control experiments

1. W. Lubeigt, G. Valentine, J. Girkin, E. Bente, and D. Burns, “Active transverse mode control and optimization of an all-solid-state laser using an intracavity adaptive-optic mirror,” Opt. Express **10**, 550–555 (2002). [PubMed]

^{3+}doping concentration of the crystal was 0.8%, and the rod was surrounded by an antireflection-coated cooling sleeve. The largest repetition rates, the pumping currents of the pump head, were 100 Hz and 70 A, respectively. A 70% reflective mirror was used as the output coupler (OC), and the output power could be adjusted from 0 to 50 W. After being attenuated by an attenuator, the output beam first was reflected by a beam splitter (BS1) and then passed through a 1064 nm narrow filter before it was focused by a lens onto an infrared CCD camera. The intensity information of the focus light spot was acquired with a frequency of 25 Hz by a frame grabber. Using one part of the intensity information as the object function to maximize (within a selected center area), the industrial computer calculated the 10-element Zernike mode coefficients, which finally could be transformed into 19 voltages by the transform matrix

*U*obtained in Section 2. Finally, these voltages were amplified by a high-voltage amplifier (HVA) before being applied to DM actuators. A monitor that was placed behind another beam splitter (BS2) was used to watch the laser mode profile. A power meter was also employed to detect the output laser power in real time.

_{10}mode. During the course of optimization, we found that the TEM

_{10}mode converged toward a fundamental TEM

_{00}mode, shown in Fig. 9(b), after about 1 minute. This phenomenon may be explained as follows: as the DM changed its surface shape in the course of optimization, its curvature radius also changed, which resulted in a change of the resonator configuration and thereby established conditions for generating the TEM

_{00}mode more efficiently and the suppression of the higher order modes, to a great extent. The output power was reduced to 5.3 W after mode optimization was accomplished. A simple video of the TEM

_{10}optimization procedure is displayed in Fig. 10.

_{00}mode successfully when the mode structure became too complex. In order to control the mode structure efficiently in a relatively high-pumping current status, a variable-size pinhole was placed near to the OC of the resonator to restrict the complex high-order modes roughly.

_{20}mode distribution was changed into a TEM

_{00}mode after about 85 seconds. The relative power in the selective region of the CCD camera was increased from 1 to 5.6. The selective region could be altered by the program built into the computer; in this case, the size of the selected region was as large as one diffraction limit. As a consequence of optimization, the output power was increased from 3.2 W to 4.7 W.

_{11}mode distribution was also changed to the TEM

_{00}mode after optimization was finished, and the relative power in the selective region of the CCD camera was increased from 1 to 5.2. In this case, the size of the selected region was also set as large as one diffraction limit. The output power was brought down from 7.2 W to 5.8 W during this course. It took about 140 seconds to finish the optimization. A video about the TEM

_{11}mode optimization performance before and after the AO system is on is given in Fig. 13.

## 6. Conclusions

_{10}, TEM

_{20}, and TEM

_{11}modes into TEM

_{00}mode within 1 to 3 minutes, based on an industrial computer with a 400 MB CPU and 256 MB RAM. We believe that the speed of our system greatly depends on the industrial computer, and if the speed of the industrial computer is increased, the time of mode optimization procedure can be reduced to only a few seconds.

## Acknowledgments

## References and links

1. | W. Lubeigt, G. Valentine, J. Girkin, E. Bente, and D. Burns, “Active transverse mode control and optimization of an all-solid-state laser using an intracavity adaptive-optic mirror,” Opt. Express |

2. | D. Burns, G. J. Valentine, W. Lubeigt, E. Bente, and A. I. Ferguson, “Development of high average power picosecond laser systems,” Proc. SPIE |

3. | P. Yang, S. J. Hu, X. D. Yang, S. Q. Chen, W. Yang, X. Zhang, and B. Xu, “Test and analysis of the time and space characteristics of phase aberration in a diode-side-pumped Nd:YAG laser,” SPIE Proc. |

4. | J. W. Hardy, |

5. | W. H. Jiang and H. G. Li, “Hartmann-Shack wave-front sensing and wave-front control algorithm,” Proc. SPIE. |

6. | P. Yang, S. J. Hu, S. Q. Chen, W. Yang, B. Xu, and W. H. Jiang, “Research on the phase aberration correction with a deformable mirror controlled by a genetic algorithm,” J. Phys: Conf Series |

7. | W. H. Jiang, N. Ling, X. B. Wu, C. H. Wang, H. Xian, S. F. Jiang, Z. J. Rong, C. L. Guan, L. T. Jiang, Z. B. Gong, Y. Wu, and Y. J. Wang, “37-element adaptive optics experimental system and turbulence compensation experiments,” Proc. SPIE |

8. | D. E. Goldberg, Genetic |

9. | O. Albert, L. Sherman, G. Mourou, and T. B. Norris, “Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy,” Opt. Lett. |

10. | A. C. F. Gonté, A. Courteville, and R. Dändliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. |

11. | P. N. Marsh, D. Burns, and J. M. Girikn, “Practical implementation of adaptive optics in multiphoton microscopy,” Opt. Express. |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

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

(140.6810) Lasers and laser optics : Thermal effects

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: September 4, 2007

Revised Manuscript: October 22, 2007

Manuscript Accepted: October 26, 2007

Published: December 5, 2007

**Virtual Issues**

Vol. 3, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Ping Yang, MingWu Ao, Yuan Liu, Bing Xu, and WenHan Jiang, "Intracavity transverse modes controlled by a genetic algorithm based on Zernike mode coefficients," Opt. Express **15**, 17051-17062 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-17051

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### References

- W. Lubeigt, G. Valentine, J. Girkin, E. Bente, and D. Burns, "Active transverse mode control and optimization of an all-solid-state laser using an intracavity adaptive-optic mirror," Opt. Express 10, 550-555 (2002). [PubMed]
- D. Burns, G. J. Valentine, W. Lubeigt, E. Bente, and A. I. Ferguson, "Development of high average power picosecond laser systems," Proc. SPIE 4629, 129-143 (2002). [CrossRef]
- P. Yang, S. J. Hu, X. D. Yang, S. Q. Chen, W. Yang, X. Zhang, and B. Xu, "Test and analysis of the time and space characteristics of phase aberration in a diode-side-pumped Nd:YAG laser," Proc. SPIE. 6108182-191 (2005).
- J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).
- W. H. Jiang and H. G. Li, "Hartmann-Shack wave-front sensing and wave-front control algorithm," Proc. SPIE. 1271, 82-93 (1990). [CrossRef]
- P. Yang, S. J. Hu, S. Q. Chen, W. Yang, B. Xu, and W. H. Jiang, "Research on the phase aberration correction with a deformable mirror controlled by a genetic algorithm," J. Phys: Conf. Series 48, 1017-1024 (2006). [CrossRef]
- W. H. Jiang, N. Ling, X. B. Wu, C. H. Wang, H. Xian, S. F. Jiang, Z. J. Rong, C. L. Guan, L. T. Jiang, Z. B. Gong, Y. Wu, and Y. J. Wang, "37-element adaptive optics experimental system and turbulence compensation experiments," Proc. SPIE 2828, 312-321 (1996). [CrossRef]
- D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, 1st ed. (Addison-Wesley Publishing Company, Inc., Boston, 1989).
- O. Albert, L. Sherman, G. Mourou, and T. B. Norris, "Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy," Opt. Lett. 25, 52-54 (2000). [CrossRef]
- A. C. F. Gonté, A. Courteville, and R. Dändliker, "Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror," Opt. Eng. 41, 1073-1076 (2002). [CrossRef]
- P. N. Marsh, D. Burns, and J. M. Girikn, "Practical implementation of adaptive optics in multiphoton microscopy," Opt. Express. 11, 1123-1130 (2003). [CrossRef] [PubMed]

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