## Observation and analysis of single and multiple high-order Laguerre-Gaussian beams generated from a hemi-cylindrical cavity with general astigmatism |

Optics Express, Vol. 21, Issue 23, pp. 28496-28506 (2013)

http://dx.doi.org/10.1364/OE.21.028496

Acrobat PDF (1441 KB)

### Abstract

We experimentally verified that anisotropic Hermite-Gaussian modes can be generated from a hemi-cylindrical laser cavity and can be transformed into high-order Laguerre-Gaussian modes using an extra-cavity cylindrical lens. We further combined the Huygens integral and the ABCD law to clearly demonstrate the transformation along the propagation direction. By controlling the pump offset and the pump size in hemi-cylindrical cavities, we experimentally observed the unique laser patterns that displayed the optical waves related to the coherent superposition of Laguerre-Gaussian modes.

© 2013 Optical Society of America

## 1. Introduction

1. J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A **70**(4), 042313 (2004). [CrossRef]

4. D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Effect of high-dimensional entanglement of Laguerre-Gaussian modes in parametric downconversion,” J. Opt. Soc. Am. B **26**(4), 797–804 (2009). [CrossRef]

5. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. **78**(25), 4713–4716 (1997). [CrossRef]

6. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**(11), 8185–8189 (1992). [CrossRef] [PubMed]

7. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express **12**(22), 5448–5456 (2004). [CrossRef] [PubMed]

8. M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: Mmodulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. **98**(8), 083602 (2007). [CrossRef] [PubMed]

9. A. E. Kaplan, I. Marzoli, W. E. Lamb Jr, and W. P. Schleich, “Multimode interference: Highly regular pattern formation in quantum wave-packet evolution,” Phys. Rev. A **61**(3), 032101 (2000). [CrossRef]

10. G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A **48**(1), 656–665 (1993). [CrossRef] [PubMed]

12. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. **42**, 219–276 (2001). [CrossRef]

13. K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, “Observation of the wave function of a quantum billiard from the transverse patterns of vertical cavity surface emitting lasers,” Phys. Rev. Lett. **89**(22), 224102 (2002). [CrossRef] [PubMed]

14. T. Gensty, K. Becker, I. Fischer, W. Elsässer, C. Degen, P. Debernardi, and G. P. Bava, “Wave chaos in real-world vertical-cavity surface-emitting lasers,” Phys. Rev. Lett. **94**(23), 233901 (2005). [CrossRef] [PubMed]

15. E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. **90**(20), 203901 (2003). [CrossRef] [PubMed]

16. I. Vorobeichik, E. Narevicius, G. Rosenblum, M. Orenstein, and N. Moiseyev, “Electromagnetic realization of orders-of-magnitude tunneling enhancement in a double well system,” Phys. Rev. Lett. **90**(17), 176806 (2003). [CrossRef] [PubMed]

17. M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A **43**(9), 5090–5113 (1991). [CrossRef] [PubMed]

19. S. Danakas and P. K. Aravind, “Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom),” Phys. Rev. A **45**(3), 1973–1977 (1992). [CrossRef] [PubMed]

20. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. **96**(1-3), 123–132 (1993). [CrossRef]

21. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer-generated holograms,” J. Mod. Opt. **45**(6), 1231–1237 (1998). [CrossRef]

22. N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A **25**(7), 1642–1651 (2008). [CrossRef] [PubMed]

23. A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express **13**(13), 4952–4962 (2005). [CrossRef] [PubMed]

24. Y. Chen, Y. Lan, and S. Wang, “Generation of Laguerre-Gaussian modes in fiber-coupled laser diode end pumped lasers,” Appl. Phys. B **72**(2), 167–170 (2001). [CrossRef]

25. M. P. Thirugnanasambandam, Yu. Senatsky, and K. Ueda, “Generation of very-high order Laguerre-Gaussian modes in Yb:YAG ceramic laser,” Laser Phys. Lett. **7**(9), 637–643 (2010). [CrossRef]

## 2. Experimental apparatus and propagation of astigmatic Hermite-Gaussian modes

_{4}laser. The cavity composed of a cylindrical mirror and a gain medium with coating is referred to as a simple astigmatic resonator [26

26. J. A. Arnaud and H. Kogelnik, “Gaussian light beams with general astigmatism,” Appl. Opt. **8**(8), 1687–1693 (1969). [CrossRef] [PubMed]

_{4}crystal with a length of

_{4}crystal was coated for antireflection at 1064 nm; the other side was coated as an output coupler with a reflectivity of 99%. The pump source was an 809 nm fiber-coupled laser diode with a core diameter of

_{0,6}mode along the propagation direction. The tomogram clearly displays the structural variation in the shape of the astigmatic high-order HG mode, which differed from that of the traditional HG mode.

## 3. Theoretical analyses for the propagation of astigmatic Hermite-Gaussian modes

10. G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A **48**(1), 656–665 (1993). [CrossRef] [PubMed]

27. J. Visser and G. Nienhuis, “Orbital angular momentum of general astigmatic modes,” Phys. Rev. A **70**(1), 013809 (2004). [CrossRef]

29. G. Nienhuis and J. Visser, “Angular momentum and vortices in paraxial beams,” J. Opt. A, Pure Appl. Opt. **6**(5), S248–S250 (2004). [CrossRef]

*d*can be given bywhere

*k*is the wave number and

*-*axis. By contrast, the divergence angle in the y direction was approximately 4 times larger than that in the x direction. The radius of curvature resulting from the thermal lens effect along the x-axis was

_{0,6}mode from

## 4. Transformation from astigmatic Hermite-Gaussian modes to Laguerre-Gaussian modes

20. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. **96**(1-3), 123–132 (1993). [CrossRef]

_{0,0}mode from the near field to the far field after transformation. The cavity length was fixed at

_{0,0}mode is transformed into LG

_{0,0}mode. To explain the transformation behavior of the mode with the result of Fresnel integral, we have to find the new basis which corresponds to the same axes (

*x*

_{1}and

*y*

_{1}axes) of the added cylindrical lens. Consequently, we expanded the rotated HG mode into a set of HG bases without rotation and determined the weighting coefficient to achieve the effect caused by an extra-cavity cylindrical lens with an angle

*d*-coefficient [31]. It revealed the equivalence of the basis in Eq. (6) and a basis represented by SU(2) transform. By substituting Eq. (4) into (6), the field distribution of the astigmatic HG modes traveling through a cylindrical lens with an angle of rotation

*d*is the distance of 2 cm from the beam waist to the extra-cavity cylindrical lens. The transverse profile of the astigmatic HG

_{0,0}mode can be converted to a circular-symmetric distribution at the far field by using of a single extra-cavity cylindrical lens. We employed the same concept to address the astigmatic high-order HG mode. Figures 5(a)–5(e) show the experimental transverse profile of the astigmatic HG

_{0,6}mode from the near field to the far field after transformation. The cavity length and the pump offset were fixed at

_{0,6}mode gradually became an elliptically shaped distribution at the position

_{0,47}by counting the nodes of the transverse pattern. Figures 7(a’)–7(d’) show the numerical results. The corresponding parameters of the numerical results were set to

32. E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. **90**(20), 203901 (2003). [CrossRef] [PubMed]

34. T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Generation of multi-axis Laguerre-Gaussian beams from geometric modes of a hemiconfocal cavity,” Appl. Phys. B **103**(4), 991–999 (2011). [CrossRef]

28. S. J. M. Harbraken and G. Nienhuis, “Modes of a twisted optical cavity,” Phys. Rev. A **75**(3), 033819 (2007). [CrossRef]

35. J. L. Blows and G. W. Forbes, “Mode characteristics of twisted resonators composed of two cylindrical mirrors,” Opt. Express **2**(5), 184–190 (1998). [CrossRef] [PubMed]

36. H. Weber, “Rays and fields in general astigmatic resonators,” J. Mod. Opt. **59**(8), 740–770 (2012). [CrossRef]

37. Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, “Devil’s staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces,” Phys. Rev. Lett. **96**(21), 213902 (2006). [CrossRef] [PubMed]

39. T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Generation of multi-axis Laguerre–Gaussian beams from geometric modes of a hemiconfocal cavity,” Appl. Phys. B **103**(4), 991–999 (2011). [CrossRef]

_{1,2}mode and HG

_{0,11}mode, usedto reconstruct the experimental result. The numerical result qualitatively agrees with the experimental result. Figure 8(c) depicts the phase distribution of the numerical result. The phase singularities were arranged at the center and the dark points of the outer ring of the transverse profile. The superposition of the high-order LG modes was demonstrated by using the astigmatic cavity and a single cylindrical lens.

37. Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, “Devil’s staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces,” Phys. Rev. Lett. **96**(21), 213902 (2006). [CrossRef] [PubMed]

## 5. Conclusion

## Acknowledgments

## References and links

1. | J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A |

2. | N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. |

3. | K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H.-A. Bachor, “Entangling the spatial properties of laser beams,” Science |

4. | D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Effect of high-dimensional entanglement of Laguerre-Gaussian modes in parametric downconversion,” J. Opt. Soc. Am. B |

5. | T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. |

6. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

7. | G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express |

8. | M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: Mmodulation, detection, and spatial quantum correlations,” Phys. Rev. Lett. |

9. | A. E. Kaplan, I. Marzoli, W. E. Lamb Jr, and W. P. Schleich, “Multimode interference: Highly regular pattern formation in quantum wave-packet evolution,” Phys. Rev. A |

10. | G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A |

11. | B. L. Johnson and G. Kirczenow, “Enhanced dynamical symmetries and quantum degeneracies in mesoscopic quantum dots: Role of the symmetries of closed classical orbits,” Europhys. Lett. |

12. | M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. |

13. | K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, “Observation of the wave function of a quantum billiard from the transverse patterns of vertical cavity surface emitting lasers,” Phys. Rev. Lett. |

14. | T. Gensty, K. Becker, I. Fischer, W. Elsässer, C. Degen, P. Debernardi, and G. P. Bava, “Wave chaos in real-world vertical-cavity surface-emitting lasers,” Phys. Rev. Lett. |

15. | E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. |

16. | I. Vorobeichik, E. Narevicius, G. Rosenblum, M. Orenstein, and N. Moiseyev, “Electromagnetic realization of orders-of-magnitude tunneling enhancement in a double well system,” Phys. Rev. Lett. |

17. | M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A |

18. | D. Dangoisse, D. Hennequin, C. Lepers, E. Louvergneaux, and P. Glorieux, “Two-dimensional optical lattices in a CO |

19. | S. Danakas and P. K. Aravind, “Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom),” Phys. Rev. A |

20. | M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. |

21. | J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer-generated holograms,” J. Mod. Opt. |

22. | N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A |

23. | A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express |

24. | Y. Chen, Y. Lan, and S. Wang, “Generation of Laguerre-Gaussian modes in fiber-coupled laser diode end pumped lasers,” Appl. Phys. B |

25. | M. P. Thirugnanasambandam, Yu. Senatsky, and K. Ueda, “Generation of very-high order Laguerre-Gaussian modes in Yb:YAG ceramic laser,” Laser Phys. Lett. |

26. | J. A. Arnaud and H. Kogelnik, “Gaussian light beams with general astigmatism,” Appl. Opt. |

27. | J. Visser and G. Nienhuis, “Orbital angular momentum of general astigmatic modes,” Phys. Rev. A |

28. | S. J. M. Harbraken and G. Nienhuis, “Modes of a twisted optical cavity,” Phys. Rev. A |

29. | G. Nienhuis and J. Visser, “Angular momentum and vortices in paraxial beams,” J. Opt. A, Pure Appl. Opt. |

30. | A. E. Siegman, |

31. | J. J. Sakurai, |

32. | E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. |

33. | Y. C. Lin, T. H. Lu, K. F. Huang, and Y. F. Chen, “Generation of optical vortex array with transformation of standing-wave Laguerre-Gaussian mode,” Opt. Express |

34. | T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Generation of multi-axis Laguerre-Gaussian beams from geometric modes of a hemiconfocal cavity,” Appl. Phys. B |

35. | J. L. Blows and G. W. Forbes, “Mode characteristics of twisted resonators composed of two cylindrical mirrors,” Opt. Express |

36. | H. Weber, “Rays and fields in general astigmatic resonators,” J. Mod. Opt. |

37. | Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, “Devil’s staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces,” Phys. Rev. Lett. |

38. | T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Three-dimensional coherent optical waves localized on trochoidal parametric surfaces,” Phys. Rev. Lett. |

39. | T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Generation of multi-axis Laguerre–Gaussian beams from geometric modes of a hemiconfocal cavity,” Appl. Phys. B |

**OCIS Codes**

(030.4070) Coherence and statistical optics : Modes

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

(140.3480) Lasers and laser optics : Lasers, diode-pumped

(140.4780) Lasers and laser optics : Optical resonators

**ToC Category:**

Physical Optics

**History**

Original Manuscript: September 13, 2013

Revised Manuscript: October 25, 2013

Manuscript Accepted: November 5, 2013

Published: November 12, 2013

**Citation**

T. H. Lu and Y.C. Wu, "Observation and analysis of single and multiple high-order Laguerre-Gaussian beams generated from a hemi-cylindrical cavity with general astigmatism," Opt. Express **21**, 28496-28506 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-28496

Sort: Year | Journal | Reset

### References

- J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A70(4), 042313 (2004). [CrossRef]
- N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett.93(5), 053601 (2004). [CrossRef] [PubMed]
- K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H.-A. Bachor, “Entangling the spatial properties of laser beams,” Science321(5888), 541–543 (2008). [CrossRef] [PubMed]
- D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Effect of high-dimensional entanglement of Laguerre-Gaussian modes in parametric downconversion,” J. Opt. Soc. Am. B26(4), 797–804 (2009). [CrossRef]
- T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.78(25), 4713–4716 (1997). [CrossRef]
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A45(11), 8185–8189 (1992). [CrossRef] [PubMed]
- G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express12(22), 5448–5456 (2004). [CrossRef] [PubMed]
- M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: Mmodulation, detection, and spatial quantum correlations,” Phys. Rev. Lett.98(8), 083602 (2007). [CrossRef] [PubMed]
- A. E. Kaplan, I. Marzoli, W. E. Lamb, and W. P. Schleich, “Multimode interference: Highly regular pattern formation in quantum wave-packet evolution,” Phys. Rev. A61(3), 032101 (2000). [CrossRef]
- G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A48(1), 656–665 (1993). [CrossRef] [PubMed]
- B. L. Johnson and G. Kirczenow, “Enhanced dynamical symmetries and quantum degeneracies in mesoscopic quantum dots: Role of the symmetries of closed classical orbits,” Europhys. Lett.51(4), 367–373 (2000). [CrossRef]
- M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt.42, 219–276 (2001). [CrossRef]
- K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, “Observation of the wave function of a quantum billiard from the transverse patterns of vertical cavity surface emitting lasers,” Phys. Rev. Lett.89(22), 224102 (2002). [CrossRef] [PubMed]
- T. Gensty, K. Becker, I. Fischer, W. Elsässer, C. Degen, P. Debernardi, and G. P. Bava, “Wave chaos in real-world vertical-cavity surface-emitting lasers,” Phys. Rev. Lett.94(23), 233901 (2005). [CrossRef] [PubMed]
- E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett.90(20), 203901 (2003). [CrossRef] [PubMed]
- I. Vorobeichik, E. Narevicius, G. Rosenblum, M. Orenstein, and N. Moiseyev, “Electromagnetic realization of orders-of-magnitude tunneling enhancement in a double well system,” Phys. Rev. Lett.90(17), 176806 (2003). [CrossRef] [PubMed]
- M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A43(9), 5090–5113 (1991). [CrossRef] [PubMed]
- D. Dangoisse, D. Hennequin, C. Lepers, E. Louvergneaux, and P. Glorieux, “Two-dimensional optical lattices in a CO2 laser,” Phys. Rev. A46(9), 5955–5958 (1992). [CrossRef] [PubMed]
- S. Danakas and P. K. Aravind, “Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom),” Phys. Rev. A45(3), 1973–1977 (1992). [CrossRef] [PubMed]
- M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun.96(1-3), 123–132 (1993). [CrossRef]
- J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer-generated holograms,” J. Mod. Opt.45(6), 1231–1237 (1998). [CrossRef]
- N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A25(7), 1642–1651 (2008). [CrossRef] [PubMed]
- A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express13(13), 4952–4962 (2005). [CrossRef] [PubMed]
- Y. Chen, Y. Lan, and S. Wang, “Generation of Laguerre-Gaussian modes in fiber-coupled laser diode end pumped lasers,” Appl. Phys. B72(2), 167–170 (2001). [CrossRef]
- M. P. Thirugnanasambandam, Yu. Senatsky, and K. Ueda, “Generation of very-high order Laguerre-Gaussian modes in Yb:YAG ceramic laser,” Laser Phys. Lett.7(9), 637–643 (2010). [CrossRef]
- J. A. Arnaud and H. Kogelnik, “Gaussian light beams with general astigmatism,” Appl. Opt.8(8), 1687–1693 (1969). [CrossRef] [PubMed]
- J. Visser and G. Nienhuis, “Orbital angular momentum of general astigmatic modes,” Phys. Rev. A70(1), 013809 (2004). [CrossRef]
- S. J. M. Harbraken and G. Nienhuis, “Modes of a twisted optical cavity,” Phys. Rev. A75(3), 033819 (2007). [CrossRef]
- G. Nienhuis and J. Visser, “Angular momentum and vortices in paraxial beams,” J. Opt. A, Pure Appl. Opt.6(5), S248–S250 (2004). [CrossRef]
- A. E. Siegman, Lasers (University Science, 1986).
- J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994).
- E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett.90(20), 203901 (2003). [CrossRef] [PubMed]
- Y. C. Lin, T. H. Lu, K. F. Huang, and Y. F. Chen, “Generation of optical vortex array with transformation of standing-wave Laguerre-Gaussian mode,” Opt. Express19(11), 10293–10303 (2011). [CrossRef] [PubMed]
- T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Generation of multi-axis Laguerre-Gaussian beams from geometric modes of a hemiconfocal cavity,” Appl. Phys. B103(4), 991–999 (2011). [CrossRef]
- J. L. Blows and G. W. Forbes, “Mode characteristics of twisted resonators composed of two cylindrical mirrors,” Opt. Express2(5), 184–190 (1998). [CrossRef] [PubMed]
- H. Weber, “Rays and fields in general astigmatic resonators,” J. Mod. Opt.59(8), 740–770 (2012). [CrossRef]
- Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, “Devil’s staircase in three-dimensional coherent waves localized on Lissajous parametric surfaces,” Phys. Rev. Lett.96(21), 213902 (2006). [CrossRef] [PubMed]
- T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Three-dimensional coherent optical waves localized on trochoidal parametric surfaces,” Phys. Rev. Lett.101(23), 233901 (2008). [CrossRef] [PubMed]
- T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, “Generation of multi-axis Laguerre–Gaussian beams from geometric modes of a hemiconfocal cavity,” Appl. Phys. B103(4), 991–999 (2011). [CrossRef]

## 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.