## Generating a geometric mode for clarifying differences between an operator method and SU(2) wave representation

Optics Express, Vol. 15, Issue 20, pp. 12692-12698 (2007)

http://dx.doi.org/10.1364/OE.15.012692

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### Abstract

We study both numerically and experimentally a specific geometric mode, named VW mode, in an end-pumped Nd:YVO_{4} laser with a plano-concave cavity near the 1/3-degeneracy. Three theoretical methods are used to analyze the transverse profiles, the operator method [J. Opt. Soc. Am. A 22, 1559 (2005)], the SU(2) wave representation [Phys. Rev. A 69, 053807 (2004)], and the Fox-Li approach. Some differences among them are addressed and clarified. Moreover, the generating conditions for VW mode are found peculiar and its propagation character is demonstrated. By comparing the experimental mode patterns with the three numerical results, we conclude that the field of a geometric mode in the operator method should be extended to include those of the reverse directional trajectories and the SU(2) coherent state representation is found too specific to produce the fringes within some transverse patterns.

© 2007 Optical Society of America

## 1. Introduction

5. Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A **69**, 053807 (2004). [CrossRef]

7. J. Visser, N. J. Zelders, and G. Nienhuis, “Wave description of geometric modes of a resonator,” J. Opt. Soc. Am. A **22**, 1559–1566 (2005). [CrossRef]

7. J. Visser, N. J. Zelders, and G. Nienhuis, “Wave description of geometric modes of a resonator,” J. Opt. Soc. Am. A **22**, 1559–1566 (2005). [CrossRef]

7. J. Visser, N. J. Zelders, and G. Nienhuis, “Wave description of geometric modes of a resonator,” J. Opt. Soc. Am. A **22**, 1559–1566 (2005). [CrossRef]

_{4}laser operated near the 1/3-degeneracy with a plano-concave cavity. By observing the transverse patterns of the VW mode, we conclude the operator method in [7

**22**, 1559–1566 (2005). [CrossRef]

5. Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A **69**, 053807 (2004). [CrossRef]

## 2. The numerical results

_{1}g

_{2}= 1/4, six-bounce or three round-trip rays will construct a closed trajectory. We plot a specific one that is symmetric with the optical axis in Fig. 1(a), in which z = 0 is the position of the flat end mirror. In Fig. 1(a), the initial ray is indicated starting at (x, z) = (0, 0) with a slope of 2a/L, then it is reflected back to (x, z) = (a, 0) and subsequently retraces itself after two round trips. Here L is the cavity length and x is the transverse coordinate. Using the operator method [7

**22**, 1559–1566 (2005). [CrossRef]

*π*/180 and the position vector a = 0, we plot the mode trajectories in Fig. 1(b). This is equivalent to considering a fundamental Gaussian beam with its center at (x, z) = (0, 0) and with a slope q/k that retraces itself after three round trips. We can see that the mode trajectories in Fig. 1(b) are the same as the closed rays in Fig. 1(a), which are like overlapped V and W shapes, so that we call it the VW mode in this paper. The VW mode is similar to Fig. 3(a) of [7

**22**, 1559–1566 (2005). [CrossRef]

5. Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A **69**, 053807 (2004). [CrossRef]

8. J. Banerji and G. S. Agarwal, “Non-linear wave packet dynamics of coherent states of various symmetry groups,” Opt. Express **5**, 220–229 (1999). [CrossRef] [PubMed]

9. V. Buzek and T. Quang, “Generalized coherent state for bosonic realization of SU(2) Lie algebra,” J. Opt. Soc. Am. B **6**, 2447–2449 (1989). [CrossRef]

*τ*= exp(

*iϕ*), Φ

^{(HG)}

_{3p,0}(

*x,y,z*) is HG mode with the transverse mode index 3p in x direction and p = 0, 1, 2, ..., n. We depict the trajectory of the VW mode with the total wave function being Φ

_{n}(

*x,y,z*;

*ϕ*=

*π*/2)+Φ

_{n}(

*x,y,z*;

*ϕ*=-

*π*/2) and n = 20 in Fig. 1(c). We can see that Fig. 1(c) and 1(b) have the same geometric trajectories. However, in Fig. 1(b) only two regions where two beams propagate in the same direction have the interference fringes but the fringes on the end mirrors preserve for a long distance in Fig. 1(c) which is absent in Fig. 1(b).

12. C. H. Chen, P. T. Tai, M. D. Wei, and W. F. Hsieh, “Multibeam-waist modes in an end-pumped Nd-YVO_{4} laser,” J. Opt. Soc. Am. B **20**, 1220–1226 (2003). There is lack of *γ* in the last term of equation (3) in this reference. [CrossRef]

13. S. A. Collins, “Lens-system diffraction-integral written in terms of matrix optics,” J. Opt. Soc. Am. **60**, 1168–1174 (1970). [CrossRef]

^{+}

_{m}(x’) and E

^{-}

_{m+1}(x) are the electric fields of the m-th and the (m+1)-th round trips, respectively, at the planes immediately after and before the gain medium; x’ and x are the corresponding transverse coordinates,

*λ*is the wavelength of the laser, 2w is the aperture width on the reference plane that is set on the flat mirror end. To implement the integral by the Romberg method, we divided the 2 mm aperture width into 2048 segments. The used parameters of the gain medium and the cavity configuration are the same as in [12

12. C. H. Chen, P. T. Tai, M. D. Wei, and W. F. Hsieh, “Multibeam-waist modes in an end-pumped Nd-YVO_{4} laser,” J. Opt. Soc. Am. B **20**, 1220–1226 (2003). There is lack of *γ* in the last term of equation (3) in this reference. [CrossRef]

*θ*,2

*θ*,

*θ*and those at z = 6 cm are

*θ*/2 ,

*θ*,

*θ*/2. When two beams interfere by a small angle the fringe spacing is inversely proportional to this angle, which explains the data of fringe spacing.

## 3. Experimental setup and results

*a*cut 2.0 at% microchip crystal and an output coupler with radius of curvature of 20 cm having 10% transmission at the lasing wavelength of 1064 nm. One face of the crystal facing the pumping beam had a dichroic coating with greater than 99.8% reflection at 1064 nm and greater than 99.5% transmission at the pump wavelength of 808 nm; the other surface was made of antireflection layer at 1064 nm. The pump source was a 1-W fiber-coupled laser diode with a 200μm of core diameter and a numerical aperture of 0.22. A convergent lens with 50 mm focal length and a 20x objective lens were added after the fiber output a distance of 5 cm and 85 cm, respectively to focus the pump beam into the laser crystal. Between the two lenses, the pump beam was split into three beams by two 50/50 beam splitters and two reflective mirrors.

**22**, 1559–1566 (2005). [CrossRef]

**69**, 053807 (2004). [CrossRef]

9. V. Buzek and T. Quang, “Generalized coherent state for bosonic realization of SU(2) Lie algebra,” J. Opt. Soc. Am. B **6**, 2447–2449 (1989). [CrossRef]

## 4. Conclusion

_{4}laser with a plano-concave cavity near the degeneration point of g

_{1}g

_{2}= 1/4. The generating condition for the VW mode is somewhat peculiar that is the three pumping beam power must have a proper ratio. Our simulations match with the experiments very well. By observing the 3-spots pattern of the VW mode, we conclude the operator method in [7

**22**, 1559–1566 (2005). [CrossRef]

**69**, 053807 (2004). [CrossRef]

## Acknowledgments

## References and links

1. | I. A. Ramsay and J. J. Degnan, “A ray analysis of optical resonators formed by two spherical mirrors,” Appl. Opt. |

2. | B. Sterman, A. Gabay, S. Yatsiv, and E. Dagan, “Off-axis folded laser beam trajectories in a strip-line CO |

3. | D. Dick and F. Hanson, “M modes in a diode side-pumped Nd:glass slab laser,” Opt. Lett. |

4. | J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO |

5. | Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, “Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity,” Phys. Rev. A |

6. | R. Akis and D. K. Ferry, “Ballistic transport and scarring effects in coupled quantum dots,” Phys. Rev. B |

7. | J. Visser, N. J. Zelders, and G. Nienhuis, “Wave description of geometric modes of a resonator,” J. Opt. Soc. Am. A |

8. | J. Banerji and G. S. Agarwal, “Non-linear wave packet dynamics of coherent states of various symmetry groups,” Opt. Express |

9. | V. Buzek and T. Quang, “Generalized coherent state for bosonic realization of SU(2) Lie algebra,” J. Opt. Soc. Am. B |

10. | P. T. Tai, C. H. Chen, and W. F. Hsieh, “Direct generation of optical bottle beams from a tightly focused end-pumped solid-state laser,” Opt. Express |

11. | C. H. Chen, P. T. Tai, and W. F. Hsieh, “Bottle beam from a bare laser for single-beam trapping,” Appl. Opt. |

12. | C. H. Chen, P. T. Tai, M. D. Wei, and W. F. Hsieh, “Multibeam-waist modes in an end-pumped Nd-YVO |

13. | S. A. Collins, “Lens-system diffraction-integral written in terms of matrix optics,” J. Opt. Soc. Am. |

**OCIS Codes**

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

(140.3410) Lasers and laser optics : Laser resonators

(140.3580) Lasers and laser optics : Lasers, solid-state

(140.3295) Lasers and laser optics : Laser beam characterization

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: August 1, 2007

Revised Manuscript: September 10, 2007

Manuscript Accepted: September 13, 2007

Published: September 19, 2007

**Citation**

Ching-Hsu Chen and Chi-Feng Chiu, "Generating a geometric mode for clarifying differences between an operator method and SU(2) wave representation," Opt. Express **15**, 12692-12698 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-12692

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### References

- I. A. Ramsay and J. J. Degnan, "A ray analysis of optical resonators formed by two spherical mirrors," Appl. Opt. 9, 385-398 (1970). [CrossRef] [PubMed]
- B. Sterman, A. Gabay, S. Yatsiv, and E. Dagan, "Off-axis folded laser beam trajectories in a strip-line CO2 laser," Opt. Lett. 14,1309-1311 (1989). [CrossRef] [PubMed]
- D. Dick and F. Hanson, "M modes in a diode side-pumped Nd:glass slab laser," Opt. Lett. 16,476-477 (1991). [CrossRef] [PubMed]
- J. Dingjan, M. P. van Exter, and J. P. Woerdman, "Geometric modes in a single-frequency Nd:YVO4 laser,’’Opt. Commun. 188, 345-351 (2001). [CrossRef]
- Y. F. Chen, C. H. Jiang, Y. P. Lan, and K. F. Huang, "Wave representation of geometrical laser beam trajectories in a hemiconfocal cavity," Phys. Rev. A 69, 053807 (2004). [CrossRef]
- R. Akis and D. K. Ferry, "Ballistic transport and scarring effects in coupled quantum dots," Phys. Rev. B 59, 7529-7536 (1999). [CrossRef]
- J. Visser, N. J. Zelders, and G. Nienhuis, "Wave description of geometric modes of a resonator," J. Opt. Soc. Am. A 22, 1559-1566 (2005). [CrossRef]
- J. Banerji and G. S. Agarwal, "Non-linear wave packet dynamics of coherent states of various symmetry groups," Opt. Express 5,220-229 (1999). [CrossRef] [PubMed]
- V. Buzek and T. Quang, "Generalized coherent state for bosonic realization of SU(2) Lie algebra," J. Opt. Soc. Am. B 6, 2447-2449 (1989). [CrossRef]
- P. T. Tai, C. H. Chen, and W. F. Hsieh, "Direct generation of optical bottle beams from a tightly focused end-pumped solid-state laser," Opt. Express 12, 5827-5833 (2004). [CrossRef] [PubMed]
- C. H. Chen, P. T. Tai, and W. F. Hsieh, "Bottle beam from a bare laser for single-beam trapping," Appl. Opt. 43, 6001-6006 (2004). [CrossRef] [PubMed]
- C. H. Chen, P. T. Tai, M. D. Wei, and W. F. Hsieh, "Multibeam-waist modes in an end-pumped Nd-YVO4 laser," J. Opt. Soc. Am. B 20, 1220-1226 (2003). There is lack of in the last term of equation (3) in this reference. [CrossRef]
- S. A. Collins, "Lens-system diffraction-integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1174 (1970). [CrossRef]

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