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

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    [CrossRef] [PubMed]
  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. Express19(11), 10293–10303 (2011).
    [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. B103(4), 991–999 (2011).
    [CrossRef]
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    [CrossRef] [PubMed]
  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.101(23), 233901 (2008).
    [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. B103(4), 991–999 (2011).
    [CrossRef]

2012

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

2011

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]

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]

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]

2010

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]

2009

2008

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]

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]

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]

2007

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]

S. J. M. Harbraken and G. Nienhuis, “Modes of a twisted optical cavity,” Phys. Rev. A75(3), 033819 (2007).
[CrossRef]

2006

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]

2005

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]

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]

2004

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]

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]

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

J. Visser and G. Nienhuis, “Orbital angular momentum of general astigmatic modes,” Phys. Rev. A70(1), 013809 (2004).
[CrossRef]

2003

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]

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]

2002

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]

2001

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

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]

2000

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]

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]

1998

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]

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]

1997

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]

1993

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]

G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A48(1), 656–665 (1993).
[CrossRef] [PubMed]

1992

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]

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]

1991

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]

1969

Allen, L.

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]

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]

G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A48(1), 656–665 (1993).
[CrossRef] [PubMed]

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]

Ando, T.

Aravind, P. K.

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]

Arlt, J.

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]

Arnaud, J. A.

Bachor, H.-A.

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]

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]

Barnett, S.

Bartlett, S. D.

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]

Battipede, F.

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]

Bava, G. P.

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]

Becker, K.

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]

Beijersbergen, M. W.

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]

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]

Blows, J. L.

Brambilla, M.

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]

Buchhave, P.

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]

Chen, Y.

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]

Chen, Y. F.

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]

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]

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, “Three-dimensional coherent optical waves localized on trochoidal parametric surfaces,” Phys. Rev. Lett.101(23), 233901 (2008).
[CrossRef] [PubMed]

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]

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]

Courtial, J.

Crawford, P. R.

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]

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]

Dalton, R. B.

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]

Danakas, S.

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]

Dangoisse, D.

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]

Davidson, N.

Debernardi, P.

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]

Degen, C.

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]

Delaubert, V.

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]

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]

Deng, J.

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]

Dholakia, K.

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]

Elsässer, W.

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]

Fabre, C.

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]

Fischer, I.

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]

Forbes, G. W.

Franke-Arnold, S.

Friesem, A. A.

Fu, J.

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]

Fukuchi, N.

Galvez, E. J.

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]

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]

Gensty, T.

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]

Gibson, G.

Gilchrist, A.

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]

Glorieux, P.

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]

Haglin, P. J.

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]

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]

Hara, T.

Harb, C.

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]

Harb, C. C.

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]

Harbraken, S. J. M.

S. J. M. Harbraken and G. Nienhuis, “Modes of a twisted optical cavity,” Phys. Rev. A75(3), 033819 (2007).
[CrossRef]

Harvey, M. D.

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]

Hennequin, D.

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]

Hirano, T.

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]

Huang, K. F.

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]

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]

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, “Three-dimensional coherent optical waves localized on trochoidal parametric surfaces,” Phys. Rev. Lett.101(23), 233901 (2008).
[CrossRef] [PubMed]

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]

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]

Inoue, T.

Ishaaya, A. A.

Janousek, J.

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]

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]

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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]

Kaplan, A. E.

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]

Kawase, D.

Kirczenow, G.

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]

Kogelnik, H.

Kuga, T.

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]

Lai, H. C.

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]

Lam, P. K.

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]

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]

Lamb, W. E.

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]

Lan, Y.

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]

Lan, Y. P.

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]

Langford, N. K.

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]

Lassen, M.

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]

Lepers, C.

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]

Lin, Y. C.

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]

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]

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, “Three-dimensional coherent optical waves localized on trochoidal parametric surfaces,” Phys. Rev. Lett.101(23), 233901 (2008).
[CrossRef] [PubMed]

Louvergneaux, E.

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]

Lu, T. H.

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]

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]

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, “Three-dimensional coherent optical waves localized on trochoidal parametric surfaces,” Phys. Rev. Lett.101(23), 233901 (2008).
[CrossRef] [PubMed]

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]

Lugiato, L. A.

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]

Marzoli, I.

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]

Matsumoto, N.

Miyamoto, Y.

Moiseyev, N.

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]

Morizur, J. F.

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]

Narevicius, E.

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]

Nienhuis, G.

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]

J. Visser and G. Nienhuis, “Orbital angular momentum of general astigmatic modes,” Phys. Rev. A70(1), 013809 (2004).
[CrossRef]

G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A48(1), 656–665 (1993).
[CrossRef] [PubMed]

O’Brien, J. L.

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]

Ohtake, Y.

Orenstein, M.

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]

Padgett, M. J.

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]

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]

Pas’ko, V.

Penna, V.

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]

Prati, F.

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]

Pryde, G. J.

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]

Pysher, M. J.

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]

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]

Rosenblum, G.

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]

Sasada, H.

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]

Sasaki, K.

Schleich, W. P.

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]

Senatsky, Yu.

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]

Shimizu, Y.

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]

Shiokawa, N.

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]

Si, Z.

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]

Soskin, M. S.

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

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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]

Su, K. W.

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]

Sztul, H. I.

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]

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]

Takeda, M.

Takeuchi, S.

Tamm, C.

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]

Tang, S.

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]

Thirugnanasambandam, M. P.

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]

Torii, Y.

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]

Treps, N.

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]

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]

Ueda, K.

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]

van der Veen, H. E. L. O.

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]

Vasnetsov, M.

Vasnetsov, M. V.

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

Visser, J.

J. Visser and G. Nienhuis, “Orbital angular momentum of general astigmatic modes,” Phys. Rev. A70(1), 013809 (2004).
[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]

Vorobeichik, I.

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]

Wagner, K.

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]

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]

Wang, S.

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]

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Appl. Opt.

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Figures (11)

Fig. 1
Fig. 1

Experimental setup for the generation of astigmatic HG beams using a diode-pumped microchip laser with an off-axis pumping scheme in a hemi-cylindrical cavity.

Fig. 2
Fig. 2

(a)–(e): Experimental tomographic transverse patterns of the astigmatic HG0,0 mode from the near field to the far field. (a’)–(e’): Numerically reconstructed patterns for the experimental results.

Fig. 3
Fig. 3

(a)–(e): Experimental tomographic transverse patterns of the astigmatic HG0,6 mode at z = 0 , 1.5 L , 3.0 L , 8.0 L , and far field. (a’)–(e’): Numerically reconstructed patterns for the experimental results.

Fig. 4
Fig. 4

(a)–(e): Experimental results of the transformation from astigmatic HG0,0 mode to LG mode. (a’)–(e’): Numerically reconstructed patterns for the experimental results.

Fig. 5
Fig. 5

(a)–(e): Experimental results for the transformation from the astigmatic HG0,6 mode to the LG mode. (a’)–(e’): Numerically reconstructed patterns for the experimental results.

Fig. 6
Fig. 6

(a): Phase distribution of a pure LG mode with an azimuthal index of 6. (b): Phase distribution of the numerical result for the transformed LG mode shown in Fig. 5(e’).

Fig. 7
Fig. 7

(a)–(d): Experimental results for the transformation from the astigmatic HG0,47 mode to the LG mode. The positions were z = 0 , 1 cm , 10 cm , and 100 cm . (a’)–(d’): Numerically reconstructed patterns for the experimental results.

Fig. 8
Fig. 8

(a): Experimental result for the transformation from the superposed astigmatic HG modes, HG1,2 mode and HG0,11 mode. (b): Numerically reconstructed pattern of (a). (c): Phase distribution of (b).

Fig. 9
Fig. 9

(a)–(d): Experimental transverse patterns of the flower-type modes at the far field. The cavity lengths are (a): 7.9 mm, (b): 6.9 mm, (c): 8.5 mm, (d): 7.5 mm.

Fig. 10
Fig. 10

(a)–(d): The far-field patterns for the superposition of the anisotropic HG modes in degenerate cavities without the added cylindrical lens corresponding to Figs. 9(a)9(d).

Fig. 11
Fig. 11

Numerical results for the patterns transformed from the superposed astigmatic HG modes at far field. (a): HG5,5 + HG5,15 + HG5,25 + HG5,35, (b): HG3,3 + HG3,14 + HG3,25 + HG3,36, (c): HG6,6 + HG6,22, (d): HG2,2 + HG2,9 + HG2,16 + HG2,23 + HG2,30 .

Equations (9)

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u n ( x,z )= i Bλ e ikd × u n ( x 0 , z 0 )exp[ i π( A x 0 2 2 x 0 x+D x 2 ) Bλ ] d x 0
u n ( x 0 , z 0 )= 2 2 n n! ω 0 π H n ( 2 x 0 ω 0 )exp[ i π x 0 2 λ q 0 ]
u n ( x,z )= 2 2 n n! ω 0 π ( 1 A+B/ q 0 ) n+ 1 2 ( ω ω 0 ) n H n ( 2 x ω )exp[ i π x 2 λq ]
u n,m ( x,y,z )= u n ( x,z ) u m ( y,z ) = 1 2 n+m1 πn!m! ω 0x ω 0y ( 1 A x + B x / q 0x ) n+1/2 ( 1 A y + B y / q 0y ) m+1/2 ( ω x ω 0x ) n ( ω y ω 0y ) m × H n ( 2 x ω x ) H m ( 2 y ω y )exp[ i π x 2 λ q x ]exp[ i π y 2 λ q y ]
M x ( z )= M y ( z )=( 1 z 0 1 ).
u n , m ( x 1 , y 1 , z ) = s = 0 N d s N 2 , n N 2 N 2 ( θ ) u s , N s ( x , y , z )
d s N 2 , n N 2 N 2 ( θ ) = s ! ( N s ) ! n ! ( N n ) ! × v = max [ 0 , s n ] min [ N n , s ] ( 1 ) v [ cos ( θ 2 ) ] n + s 2 v [ sin ( θ 2 ) ] m s + 2 v v ! ( N n v ) ! ( s v ) ! ( n s + v ) ! .
M x ( z ) = ( 1 z 0 1 ) ( 1 d 0 1 )
M y ( z ) = ( 1 z 0 1 ) ( 1 0 1 / f 1 ) ( 1 d 0 1 )

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