Abstract

We exploit the SU(2) representation of the Hermite-Laguerre-Gaussian (HLG) mode to manifest the successive transformation between Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) modes. We theoretically confirmed that the time-dependent coherent state for the HLG modes can be simplified as a closed form of Gaussian wave packet. We further employ the explicit closed form to originate an integral of the Gaussian wave-packet state over the elliptical orbit to represent the elliptical orbital mode with fractional orbital angular momentum. On the other hand, we also derive the elliptical orbital mode as the superposition of the degenerate HLG modes. By using the derived formulae and the quantum Fourier transform, the HLG mode is inversely expressed as the superposition of the elliptical orbital modes. The derived representation unambiguously reveals the connection between HLG modes and bundles of elliptical orbits.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2018 (1)

Y. F. Chen, C. C. Chang, C. Y. Lee, J. C. Tung, H. C. Liang, and K. F. Huang, “Characterizing the propagation evolution of wave patterns and vortex structures in astigmatic transformations of Hermite–Gaussian beams,” Laser Phys. 28(1), 015002 (2018).
[Crossref]

2017 (3)

C. C. Chang, Y. H. Hsieh, C. Y. Lee, C. L. Sung, P. H. Tuan, J. C. Tung, H. C. Liang, and Y. F. Chen, “Generating high-peak-power structured lights in selectively pumped passively Q-switched lasers with astigmatic mode transformations,” Laser Phys. 27(12), 125805 (2017).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Elliptical Gaussian optical vortices,” Phys. Rev. A (Coll. Park) 95(5), 053805 (2017).
[Crossref]

Y. F. Chen, C. C. Chang, C. Y. Lee, C. L. Sung, J. C. Tung, K. W. Su, H. C. Liang, W. D. Chen, and G. Zhang, “High-peak-power large-angular-momentum beams generated from passively Q-switched geometric modes with astigmatic transformation,” Photon. Res. 5(6), 561–566 (2017).
[Crossref]

2015 (1)

2013 (4)

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
[Crossref]

A. Y. Okulov, “Superfluid rotation sensor with helical laser trap,” J. Low Temp. Phys. 171(3–4), 397–407 (2013).
[Crossref]

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[Crossref] [PubMed]

2012 (1)

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012).
[Crossref] [PubMed]

2011 (2)

H. A. Nam, M. G. Cohen, and J. W. Noé, “A simple method for creating a robust optical vortex beam with a single cylinder lens,” J. Opt. 13(6), 064026 (2011).
[Crossref]

Y. F. Chen, “Geometry of classical periodic orbits and quantum coherent states in coupled oscillators with SU(2) transformations,” Phys. Rev. A 83(3), 032124 (2011).
[Crossref]

2010 (3)

A. Y. Okulov, “Laser singular Theta-pinch,” Phys. Lett. A 374(44), 4523–4527 (2010).
[Crossref]

V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant Optical Manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010).
[Crossref] [PubMed]

H. Sridhar, M. G. Cohen, and J. W. Noé, “Creating optical vortex modes with a single cylinder lens,” Proc. SPIE 7613, 76130X (2010).
[Crossref]

2009 (1)

A. Y. Okulov, “Vortex-antivortex wavefunction of a degenerate quantum gas,” Laser Phys. 19(8), 1796–1803 (2009).
[Crossref]

2007 (2)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

R. Brunner, R. Meisels, F. Kuchar, R. Akis, D. K. Ferry, and J. P. Bird, “Draining of the sea of chaos: role of resonant transmission and reflection in an array of billiards,” Phys. Rev. Lett. 98(20), 204101 (2007).
[Crossref] [PubMed]

2006 (2)

C. Bracher and J. B. Delos, “Motion of an electron from a point source in parallel electric and magnetic fields,” Phys. Rev. Lett. 96(10), 100404 (2006).
[Crossref] [PubMed]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Spiral interferogram analysis,” J. Opt. Soc. Am. A 23(6), 1400–1409 (2006).
[Crossref] [PubMed]

2004 (2)

2003 (1)

Y. F. Chen and Y. P. Lan, “Observation of transverse patterns in an isotropic microchip laser,” Phys. Rev. A 67(4), 043814 (2003).
[Crossref]

2002 (1)

Y. F. Chen and Y. P. Lan, “Observation of laser transverse modes analogous to a SU(2) wave packet of a quantum harmonic oscillator,” Phys. Rev. A 66(5), 053812 (2002).
[Crossref]

2001 (1)

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phy. Rev. A 63(6), 063401 (2001).
[Crossref]

1999 (1)

1997 (2)

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]

I. V. Zozoulenko and K. F. Berggren, “Quantum scattering, resonant states and conductance fluctuations in an open square electron billiard,” Phys. Rev. B Condens. Matter 56(11), 6931–6941 (1997).
[Crossref]

1995 (1)

J. Pollet, O. Méplan, and C. Gignoux, “Elliptic eigenstates for the quantum harmonic oscillator,” J. Phys. A 28(24), 7287–7297 (1995).
[Crossref]

1994 (1)

A. D. Peters, C. Jaffé, and J. B. Delos, “Quantum manifestations of bifurcations of classical orbits: an exactly solvable model,” Phys. Rev. Lett. 73(21), 2825–2828 (1994).
[Crossref] [PubMed]

1993 (3)

M. Brack, “The physics of simple metal clusters: self-consistent jellium model and semiclassical approaches,” Rev. Mod. Phys. 65(3), 677–732 (1993).
[Crossref]

W. A. de Heer, “The physics of simple metal clusters: experimental aspects and simple models,” Rev. Mod. Phys. 65(3), 611–676 (1993).
[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]

1991 (1)

E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83(1–2), 123–135 (1991).
[Crossref]

1926 (1)

E. Schrödinger, “Der stetige Übergang von der Mikro-zur Makromechanik,” Naturwissenschaften 14(28), 664–666 (1926).
[Crossref]

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(5), S157–S161 (2004).
[Crossref]

E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83(1–2), 123–135 (1991).
[Crossref]

Akis, R.

R. Brunner, R. Meisels, F. Kuchar, R. Akis, D. K. Ferry, and J. P. Bird, “Draining of the sea of chaos: role of resonant transmission and reflection in an array of billiards,” Phys. Rev. Lett. 98(20), 204101 (2007).
[Crossref] [PubMed]

Allen, L.

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]

Alpmann, C.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
[Crossref]

Aoki, N.

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012).
[Crossref] [PubMed]

Barnett, S.

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]

Berggren, K. F.

I. V. Zozoulenko and K. F. Berggren, “Quantum scattering, resonant states and conductance fluctuations in an open square electron billiard,” Phys. Rev. B Condens. Matter 56(11), 6931–6941 (1997).
[Crossref]

Bernet, S.

Bird, J. P.

R. Brunner, R. Meisels, F. Kuchar, R. Akis, D. K. Ferry, and J. P. Bird, “Draining of the sea of chaos: role of resonant transmission and reflection in an array of billiards,” Phys. Rev. Lett. 98(20), 204101 (2007).
[Crossref] [PubMed]

Bracher, C.

C. Bracher and J. B. Delos, “Motion of an electron from a point source in parallel electric and magnetic fields,” Phys. Rev. Lett. 96(10), 100404 (2006).
[Crossref] [PubMed]

Brack, M.

M. Brack, “The physics of simple metal clusters: self-consistent jellium model and semiclassical approaches,” Rev. Mod. Phys. 65(3), 677–732 (1993).
[Crossref]

Brunner, R.

R. Brunner, R. Meisels, F. Kuchar, R. Akis, D. K. Ferry, and J. P. Bird, “Draining of the sea of chaos: role of resonant transmission and reflection in an array of billiards,” Phys. Rev. Lett. 98(20), 204101 (2007).
[Crossref] [PubMed]

Chang, C. C.

Y. F. Chen, C. C. Chang, C. Y. Lee, J. C. Tung, H. C. Liang, and K. F. Huang, “Characterizing the propagation evolution of wave patterns and vortex structures in astigmatic transformations of Hermite–Gaussian beams,” Laser Phys. 28(1), 015002 (2018).
[Crossref]

C. C. Chang, Y. H. Hsieh, C. Y. Lee, C. L. Sung, P. H. Tuan, J. C. Tung, H. C. Liang, and Y. F. Chen, “Generating high-peak-power structured lights in selectively pumped passively Q-switched lasers with astigmatic mode transformations,” Laser Phys. 27(12), 125805 (2017).
[Crossref]

Y. F. Chen, C. C. Chang, C. Y. Lee, C. L. Sung, J. C. Tung, K. W. Su, H. C. Liang, W. D. Chen, and G. Zhang, “High-peak-power large-angular-momentum beams generated from passively Q-switched geometric modes with astigmatic transformation,” Photon. Res. 5(6), 561–566 (2017).
[Crossref]

Chen, W. D.

Chen, Y. F.

Y. F. Chen, C. C. Chang, C. Y. Lee, J. C. Tung, H. C. Liang, and K. F. Huang, “Characterizing the propagation evolution of wave patterns and vortex structures in astigmatic transformations of Hermite–Gaussian beams,” Laser Phys. 28(1), 015002 (2018).
[Crossref]

C. C. Chang, Y. H. Hsieh, C. Y. Lee, C. L. Sung, P. H. Tuan, J. C. Tung, H. C. Liang, and Y. F. Chen, “Generating high-peak-power structured lights in selectively pumped passively Q-switched lasers with astigmatic mode transformations,” Laser Phys. 27(12), 125805 (2017).
[Crossref]

Y. F. Chen, C. C. Chang, C. Y. Lee, C. L. Sung, J. C. Tung, K. W. Su, H. C. Liang, W. D. Chen, and G. Zhang, “High-peak-power large-angular-momentum beams generated from passively Q-switched geometric modes with astigmatic transformation,” Photon. Res. 5(6), 561–566 (2017).
[Crossref]

Y. F. Chen, “Geometry of classical periodic orbits and quantum coherent states in coupled oscillators with SU(2) transformations,” Phys. Rev. A 83(3), 032124 (2011).
[Crossref]

Y. F. Chen and Y. P. Lan, “Observation of transverse patterns in an isotropic microchip laser,” Phys. Rev. A 67(4), 043814 (2003).
[Crossref]

Y. F. Chen and Y. P. Lan, “Observation of laser transverse modes analogous to a SU(2) wave packet of a quantum harmonic oscillator,” Phys. Rev. A 66(5), 053812 (2002).
[Crossref]

Cohen, M. G.

H. A. Nam, M. G. Cohen, and J. W. Noé, “A simple method for creating a robust optical vortex beam with a single cylinder lens,” J. Opt. 13(6), 064026 (2011).
[Crossref]

H. Sridhar, M. G. Cohen, and J. W. Noé, “Creating optical vortex modes with a single cylinder lens,” Proc. SPIE 7613, 76130X (2010).
[Crossref]

Courtial, J.

de Heer, W. A.

W. A. de Heer, “The physics of simple metal clusters: experimental aspects and simple models,” Rev. Mod. Phys. 65(3), 611–676 (1993).
[Crossref]

Delos, J. B.

C. Bracher and J. B. Delos, “Motion of an electron from a point source in parallel electric and magnetic fields,” Phys. Rev. Lett. 96(10), 100404 (2006).
[Crossref] [PubMed]

A. D. Peters, C. Jaffé, and J. B. Delos, “Quantum manifestations of bifurcations of classical orbits: an exactly solvable model,” Phys. Rev. Lett. 73(21), 2825–2828 (1994).
[Crossref] [PubMed]

Denz, C.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
[Crossref]

Desyatnikov, A. S.

V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant Optical Manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010).
[Crossref] [PubMed]

Esseling, M.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
[Crossref]

Ferry, D. K.

R. Brunner, R. Meisels, F. Kuchar, R. Akis, D. K. Ferry, and J. P. Bird, “Draining of the sea of chaos: role of resonant transmission and reflection in an array of billiards,” Phys. Rev. Lett. 98(20), 204101 (2007).
[Crossref] [PubMed]

Fickler, R.

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Franke-Arnold, S.

Fürhapter, S.

Gibson, G.

Gignoux, C.

J. Pollet, O. Méplan, and C. Gignoux, “Elliptic eigenstates for the quantum harmonic oscillator,” J. Phys. A 28(24), 7287–7297 (1995).
[Crossref]

Hill, W. T.

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]

Hsieh, Y. H.

C. C. Chang, Y. H. Hsieh, C. Y. Lee, C. L. Sung, P. H. Tuan, J. C. Tung, H. C. Liang, and Y. F. Chen, “Generating high-peak-power structured lights in selectively pumped passively Q-switched lasers with astigmatic mode transformations,” Laser Phys. 27(12), 125805 (2017).
[Crossref]

Huang, K. F.

Y. F. Chen, C. C. Chang, C. Y. Lee, J. C. Tung, H. C. Liang, and K. F. Huang, “Characterizing the propagation evolution of wave patterns and vortex structures in astigmatic transformations of Hermite–Gaussian beams,” Laser Phys. 28(1), 015002 (2018).
[Crossref]

Izdebskaya, Y. V.

V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant Optical Manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010).
[Crossref] [PubMed]

Jaffé, C.

A. D. Peters, C. Jaffé, and J. B. Delos, “Quantum manifestations of bifurcations of classical orbits: an exactly solvable model,” Phys. Rev. Lett. 73(21), 2825–2828 (1994).
[Crossref] [PubMed]

Jesacher, A.

Jhe, W.

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phy. Rev. A 63(6), 063401 (2001).
[Crossref]

Kim, K.

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phy. Rev. A 63(6), 063401 (2001).
[Crossref]

Kivshar, Y. S.

V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant Optical Manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010).
[Crossref] [PubMed]

Kotlyar, V. V.

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Elliptical Gaussian optical vortices,” Phys. Rev. A (Coll. Park) 95(5), 053805 (2017).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Vortex Hermite-Gaussian laser beams,” Opt. Lett. 40(5), 701–704 (2015).
[Crossref] [PubMed]

Kovalev, A. A.

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Elliptical Gaussian optical vortices,” Phys. Rev. A (Coll. Park) 95(5), 053805 (2017).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Vortex Hermite-Gaussian laser beams,” Opt. Lett. 40(5), 701–704 (2015).
[Crossref] [PubMed]

Krenn, M.

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Krolikowski, W.

V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant Optical Manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010).
[Crossref] [PubMed]

Kuchar, F.

R. Brunner, R. Meisels, F. Kuchar, R. Akis, D. K. Ferry, and J. P. Bird, “Draining of the sea of chaos: role of resonant transmission and reflection in an array of billiards,” Phys. Rev. Lett. 98(20), 204101 (2007).
[Crossref] [PubMed]

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]

Kwon, N.

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phy. Rev. A 63(6), 063401 (2001).
[Crossref]

Lan, Y. P.

Y. F. Chen and Y. P. Lan, “Observation of transverse patterns in an isotropic microchip laser,” Phys. Rev. A 67(4), 043814 (2003).
[Crossref]

Y. F. Chen and Y. P. Lan, “Observation of laser transverse modes analogous to a SU(2) wave packet of a quantum harmonic oscillator,” Phys. Rev. A 66(5), 053812 (2002).
[Crossref]

Lee, C. Y.

Y. F. Chen, C. C. Chang, C. Y. Lee, J. C. Tung, H. C. Liang, and K. F. Huang, “Characterizing the propagation evolution of wave patterns and vortex structures in astigmatic transformations of Hermite–Gaussian beams,” Laser Phys. 28(1), 015002 (2018).
[Crossref]

C. C. Chang, Y. H. Hsieh, C. Y. Lee, C. L. Sung, P. H. Tuan, J. C. Tung, H. C. Liang, and Y. F. Chen, “Generating high-peak-power structured lights in selectively pumped passively Q-switched lasers with astigmatic mode transformations,” Laser Phys. 27(12), 125805 (2017).
[Crossref]

Y. F. Chen, C. C. Chang, C. Y. Lee, C. L. Sung, J. C. Tung, K. W. Su, H. C. Liang, W. D. Chen, and G. Zhang, “High-peak-power large-angular-momentum beams generated from passively Q-switched geometric modes with astigmatic transformation,” Photon. Res. 5(6), 561–566 (2017).
[Crossref]

Liang, H. C.

Y. F. Chen, C. C. Chang, C. Y. Lee, J. C. Tung, H. C. Liang, and K. F. Huang, “Characterizing the propagation evolution of wave patterns and vortex structures in astigmatic transformations of Hermite–Gaussian beams,” Laser Phys. 28(1), 015002 (2018).
[Crossref]

C. C. Chang, Y. H. Hsieh, C. Y. Lee, C. L. Sung, P. H. Tuan, J. C. Tung, H. C. Liang, and Y. F. Chen, “Generating high-peak-power structured lights in selectively pumped passively Q-switched lasers with astigmatic mode transformations,” Laser Phys. 27(12), 125805 (2017).
[Crossref]

Y. F. Chen, C. C. Chang, C. Y. Lee, C. L. Sung, J. C. Tung, K. W. Su, H. C. Liang, W. D. Chen, and G. Zhang, “High-peak-power large-angular-momentum beams generated from passively Q-switched geometric modes with astigmatic transformation,” Photon. Res. 5(6), 561–566 (2017).
[Crossref]

Meisels, R.

R. Brunner, R. Meisels, F. Kuchar, R. Akis, D. K. Ferry, and J. P. Bird, “Draining of the sea of chaos: role of resonant transmission and reflection in an array of billiards,” Phys. Rev. Lett. 98(20), 204101 (2007).
[Crossref] [PubMed]

Méplan, O.

J. Pollet, O. Méplan, and C. Gignoux, “Elliptic eigenstates for the quantum harmonic oscillator,” J. Phys. A 28(24), 7287–7297 (1995).
[Crossref]

Milam, D.

Miyamoto, K.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[Crossref] [PubMed]

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012).
[Crossref] [PubMed]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Morita, R.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[Crossref] [PubMed]

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012).
[Crossref] [PubMed]

Nam, H. A.

H. A. Nam, M. G. Cohen, and J. W. Noé, “A simple method for creating a robust optical vortex beam with a single cylinder lens,” J. Opt. 13(6), 064026 (2011).
[Crossref]

Noé, J. W.

H. A. Nam, M. G. Cohen, and J. W. Noé, “A simple method for creating a robust optical vortex beam with a single cylinder lens,” J. Opt. 13(6), 064026 (2011).
[Crossref]

H. Sridhar, M. G. Cohen, and J. W. Noé, “Creating optical vortex modes with a single cylinder lens,” Proc. SPIE 7613, 76130X (2010).
[Crossref]

Okulov, A. Y.

A. Y. Okulov, “Superfluid rotation sensor with helical laser trap,” J. Low Temp. Phys. 171(3–4), 397–407 (2013).
[Crossref]

A. Y. Okulov, “Laser singular Theta-pinch,” Phys. Lett. A 374(44), 4523–4527 (2010).
[Crossref]

A. Y. Okulov, “Vortex-antivortex wavefunction of a degenerate quantum gas,” Laser Phys. 19(8), 1796–1803 (2009).
[Crossref]

Omatsu, T.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[Crossref] [PubMed]

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012).
[Crossref] [PubMed]

Padgett, M.

Pas’ko, V.

Peters, A. D.

A. D. Peters, C. Jaffé, and J. B. Delos, “Quantum manifestations of bifurcations of classical orbits: an exactly solvable model,” Phys. Rev. Lett. 73(21), 2825–2828 (1994).
[Crossref] [PubMed]

Plick, W. N.

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Pollet, J.

J. Pollet, O. Méplan, and C. Gignoux, “Elliptic eigenstates for the quantum harmonic oscillator,” J. Phys. A 28(24), 7287–7297 (1995).
[Crossref]

Porfirev, A. P.

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Elliptical Gaussian optical vortices,” Phys. Rev. A (Coll. Park) 95(5), 053805 (2017).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Vortex Hermite-Gaussian laser beams,” Opt. Lett. 40(5), 701–704 (2015).
[Crossref] [PubMed]

Ramelow, S.

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Ritsch-Marte, M.

Rode, A. V.

V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant Optical Manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010).
[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]

Schrödinger, E.

E. Schrödinger, “Der stetige Übergang von der Mikro-zur Makromechanik,” Naturwissenschaften 14(28), 664–666 (1926).
[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]

Shvedov, V. G.

V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant Optical Manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010).
[Crossref] [PubMed]

Song, Y.

Sridhar, H.

H. Sridhar, M. G. Cohen, and J. W. Noé, “Creating optical vortex modes with a single cylinder lens,” Proc. SPIE 7613, 76130X (2010).
[Crossref]

Su, K. W.

Sung, C. L.

Y. F. Chen, C. C. Chang, C. Y. Lee, C. L. Sung, J. C. Tung, K. W. Su, H. C. Liang, W. D. Chen, and G. Zhang, “High-peak-power large-angular-momentum beams generated from passively Q-switched geometric modes with astigmatic transformation,” Photon. Res. 5(6), 561–566 (2017).
[Crossref]

C. C. Chang, Y. H. Hsieh, C. Y. Lee, C. L. Sung, P. H. Tuan, J. C. Tung, H. C. Liang, and Y. F. Chen, “Generating high-peak-power structured lights in selectively pumped passively Q-switched lasers with astigmatic mode transformations,” Laser Phys. 27(12), 125805 (2017).
[Crossref]

Takahashi, F.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[Crossref] [PubMed]

Takizawa, S.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[Crossref] [PubMed]

Tokizane, Y.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[Crossref] [PubMed]

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]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Toyoda, K.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[Crossref] [PubMed]

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012).
[Crossref] [PubMed]

Tuan, P. H.

C. C. Chang, Y. H. Hsieh, C. Y. Lee, C. L. Sung, P. H. Tuan, J. C. Tung, H. C. Liang, and Y. F. Chen, “Generating high-peak-power structured lights in selectively pumped passively Q-switched lasers with astigmatic mode transformations,” Laser Phys. 27(12), 125805 (2017).
[Crossref]

Tung, J. C.

Y. F. Chen, C. C. Chang, C. Y. Lee, J. C. Tung, H. C. Liang, and K. F. Huang, “Characterizing the propagation evolution of wave patterns and vortex structures in astigmatic transformations of Hermite–Gaussian beams,” Laser Phys. 28(1), 015002 (2018).
[Crossref]

C. C. Chang, Y. H. Hsieh, C. Y. Lee, C. L. Sung, P. H. Tuan, J. C. Tung, H. C. Liang, and Y. F. Chen, “Generating high-peak-power structured lights in selectively pumped passively Q-switched lasers with astigmatic mode transformations,” Laser Phys. 27(12), 125805 (2017).
[Crossref]

Y. F. Chen, C. C. Chang, C. Y. Lee, C. L. Sung, J. C. Tung, K. W. Su, H. C. Liang, W. D. Chen, and G. Zhang, “High-peak-power large-angular-momentum beams generated from passively Q-switched geometric modes with astigmatic transformation,” Photon. Res. 5(6), 561–566 (2017).
[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.

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(5), S157–S161 (2004).
[Crossref]

E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83(1–2), 123–135 (1991).
[Crossref]

Woerdemann, M.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
[Crossref]

Woerdman, J. P.

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]

Xu, X.

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phy. Rev. A 63(6), 063401 (2001).
[Crossref]

Zeilinger, A.

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Zhang, G.

Zozoulenko, I. V.

I. V. Zozoulenko and K. F. Berggren, “Quantum scattering, resonant states and conductance fluctuations in an open square electron billiard,” Phys. Rev. B Condens. Matter 56(11), 6931–6941 (1997).
[Crossref]

J. Low Temp. Phys. (1)

A. Y. Okulov, “Superfluid rotation sensor with helical laser trap,” J. Low Temp. Phys. 171(3–4), 397–407 (2013).
[Crossref]

J. Opt. (1)

H. A. Nam, M. G. Cohen, and J. W. Noé, “A simple method for creating a robust optical vortex beam with a single cylinder lens,” J. Opt. 13(6), 064026 (2011).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(5), S157–S161 (2004).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Phys. A (1)

J. Pollet, O. Méplan, and C. Gignoux, “Elliptic eigenstates for the quantum harmonic oscillator,” J. Phys. A 28(24), 7287–7297 (1995).
[Crossref]

Laser Photonics Rev. (1)

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
[Crossref]

Laser Phys. (3)

C. C. Chang, Y. H. Hsieh, C. Y. Lee, C. L. Sung, P. H. Tuan, J. C. Tung, H. C. Liang, and Y. F. Chen, “Generating high-peak-power structured lights in selectively pumped passively Q-switched lasers with astigmatic mode transformations,” Laser Phys. 27(12), 125805 (2017).
[Crossref]

Y. F. Chen, C. C. Chang, C. Y. Lee, J. C. Tung, H. C. Liang, and K. F. Huang, “Characterizing the propagation evolution of wave patterns and vortex structures in astigmatic transformations of Hermite–Gaussian beams,” Laser Phys. 28(1), 015002 (2018).
[Crossref]

A. Y. Okulov, “Vortex-antivortex wavefunction of a degenerate quantum gas,” Laser Phys. 19(8), 1796–1803 (2009).
[Crossref]

Nano Lett. (1)

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012).
[Crossref] [PubMed]

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Naturwissenschaften (1)

E. Schrödinger, “Der stetige Übergang von der Mikro-zur Makromechanik,” Naturwissenschaften 14(28), 664–666 (1926).
[Crossref]

Opt. Commun. (2)

E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83(1–2), 123–135 (1991).
[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]

Opt. Express (1)

Opt. Lett. (2)

Photon. Res. (1)

Phy. Rev. A (1)

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phy. Rev. A 63(6), 063401 (2001).
[Crossref]

Phys. Lett. A (1)

A. Y. Okulov, “Laser singular Theta-pinch,” Phys. Lett. A 374(44), 4523–4527 (2010).
[Crossref]

Phys. Rev. A (4)

Y. F. Chen, “Geometry of classical periodic orbits and quantum coherent states in coupled oscillators with SU(2) transformations,” Phys. Rev. A 83(3), 032124 (2011).
[Crossref]

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Y. F. Chen and Y. P. Lan, “Observation of laser transverse modes analogous to a SU(2) wave packet of a quantum harmonic oscillator,” Phys. Rev. A 66(5), 053812 (2002).
[Crossref]

Y. F. Chen and Y. P. Lan, “Observation of transverse patterns in an isotropic microchip laser,” Phys. Rev. A 67(4), 043814 (2003).
[Crossref]

Phys. Rev. A (Coll. Park) (1)

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Elliptical Gaussian optical vortices,” Phys. Rev. A (Coll. Park) 95(5), 053805 (2017).
[Crossref]

Phys. Rev. B Condens. Matter (1)

I. V. Zozoulenko and K. F. Berggren, “Quantum scattering, resonant states and conductance fluctuations in an open square electron billiard,” Phys. Rev. B Condens. Matter 56(11), 6931–6941 (1997).
[Crossref]

Phys. Rev. Lett. (6)

R. Brunner, R. Meisels, F. Kuchar, R. Akis, D. K. Ferry, and J. P. Bird, “Draining of the sea of chaos: role of resonant transmission and reflection in an array of billiards,” Phys. Rev. Lett. 98(20), 204101 (2007).
[Crossref] [PubMed]

A. D. Peters, C. Jaffé, and J. B. Delos, “Quantum manifestations of bifurcations of classical orbits: an exactly solvable model,” Phys. Rev. Lett. 73(21), 2825–2828 (1994).
[Crossref] [PubMed]

C. Bracher and J. B. Delos, “Motion of an electron from a point source in parallel electric and magnetic fields,” Phys. Rev. Lett. 96(10), 100404 (2006).
[Crossref] [PubMed]

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]

V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant Optical Manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010).
[Crossref] [PubMed]

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[Crossref] [PubMed]

Proc. SPIE (1)

H. Sridhar, M. G. Cohen, and J. W. Noé, “Creating optical vortex modes with a single cylinder lens,” Proc. SPIE 7613, 76130X (2010).
[Crossref]

Rev. Mod. Phys. (2)

M. Brack, “The physics of simple metal clusters: self-consistent jellium model and semiclassical approaches,” Rev. Mod. Phys. 65(3), 677–732 (1993).
[Crossref]

W. A. de Heer, “The physics of simple metal clusters: experimental aspects and simple models,” Rev. Mod. Phys. 65(3), 611–676 (1993).
[Crossref]

Other (4)

N. N. Lebedev, Special Functions & Their Applications (Dover, 1972).

S. Flügge, Practical Quantum Mechanics (Springer-Verlag, 1971), p. 107.

A. E. Siegman, Lasers (University Science Books, 1986).

R. Blüumel, Foundations of Quantum Mechanics: From Photons to Quantum Computers (Jones and Bartlett Publishers, 2010).

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Figures (4)

Fig. 1
Fig. 1 Calculated patterns for Ψ n 1 , n 2 (α,β) ( x ˜ , y ˜ ) with n 1 =4 and n 2 =3 for several values of α and β.
Fig. 2
Fig. 2 Calculated results for Φ N 1 , N 2 (α,β) ( x ˜ , y ˜ , ϕ n ) by using the integral formula in Eq. (17) with ( N 1 , N 2 )=(8,7), (α,β)=(0,0), and ϕ n =πn/8 with n=0,1,,15. The central pattern for ψ 8,7 (HG) ( x ˜ , y ˜ ) obtained by substituting the calculated Φ N 1 , N 2 (α,β) ( x ˜ , y ˜ , ϕ n ) into Eq. (22).
Fig. 3
Fig. 3 Calculated results for Φ N 1 , N 2 (α,β) ( x ˜ , y ˜ , ϕ n ) by using the integral formula in Eq. (17) with ( N 1 , N 2 )=(2,13), (α,β)=(π/2 ,π/2 ), and ϕ n =πn/8 with n=0,1,,15. The central pattern for ψ 2,13 (LG) ( x ˜ , y ˜ ) obtained by substituting the calculated Φ N 1 , N 2 (α,β) ( x ˜ , y ˜ , ϕ n ) into Eq. (22).
Fig. 4
Fig. 4 Calculated results with ( N 1 , N 2 )=(4,11), (α,β) =( 2π/5 , 2π/5 ), and ϕ n =πn/8 with n=0,1,,15.The central pattern for Ψ 4,11 (α,β) ( x ˜ , y ˜ ) obtained by substituting the calculated Φ N 1 , N 2 (α,β) ( x ˜ , y ˜ , ϕ n ) into Eq. (22).

Equations (23)

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ψ n 1 , n 2 (HG) ( x ˜ , y ˜ )= ( a 1 ) n 1 n 1 ! ( a 2 ) n 2 n 2 ! ψ 0,0 ( x ˜ , y ˜ )
ψ 0,0 ( x ˜ , y ˜ )= 1 π e ( x ˜ + y ˜ ) 2 /2 ,
a 1 = 1 2 ( x ˜ x ˜ ),
a 2 = 1 2 ( y ˜ y ˜ ),
Ψ n 1 , n 2 (α,β) ( x ˜ , y ˜ )= ( b 1 ) n 1 n 1 ! ( b 2 ) n 2 n 2 ! ψ 0,0 ( x ˜ , y ˜ ),
[ b 1 b 2 ]=[ e iα/2 cos(β/2) e iα/2 sin(β/2) e iα/2 sin(β/2) e iα/2 cos(β/2) ][ a 1 a 2 ],
g(x,u)= n=0 u n n! e |u | 2 /2 [ 1 2 n n! H n ( x ˜ ) e x ˜ 2 /2 ] = e ( x ˜ 2 2 2 u x ˜ + u 2 +|u | 2 )/2 .
g (α,β) ( x ˜ , y ˜ , u 1 , u 2 )= n 1 =0 n 2 =0 u 1 n 1 n 1 ! u 2 n 2 n 2 ! e | u 1 | 2 +| u 2 | 2 2 Ψ n 1 , n 2 (α,β) ( x ˜ , y ˜ ) ,
[ u 1 u 2 ]=[ N 1 e i(θ+ϕ/2) N 2 e i(θϕ/2) ],
g (α,β) ( x ˜ , y ˜ , u 1 , u 2 )= n 1 =0 n 2 =0 N 1 n 1 /2 n 1 ! N 2 n 2 /2 n 2 ! e N 1 + N 2 2 Ψ n 1 , n 2 (α,β) ( x ˜ , y ˜ ) e i( n 1 + n 2 )θ e i( n 1 n 2 )ϕ/2 .
g (α,β) ( x ˜ , y ˜ , u 1 , u 2 )= 1 π e ( x ˜ 2 2 2 v 1 x ˜ + v 1 2 +| v 1 | 2 ) 2 e ( y ˜ 2 2 2 v 2 y ˜ + v 2 2 +| v 2 | 2 ) 2
[ v 1 v 2 ]=[ e iα/2 cos(β/2) e iα/2 sin(β/2) e iα/2 sin(β/2) e iα/2 cos(β/2) ][ u 1 u 2 ],
g (α,β) ( x ˜ , y ˜ , u 1 , u 2 )= n 1 =0 n 2 =0 u 1 n 1 n 1 ! u 2 n 2 n 2 ! e (| u 1 | 2 +| u 2 | 2 )/2 ( b 1 ) n 1 ( b 2 ) n 2 ψ 0,0 ( x ˜ , y ˜ ) .
g (α,β) ( x ˜ , y ˜ , u 1 , u 2 )= m 1 =0 m 2 =0 v 1 m 1 m 1 ! v 2 m 2 m 2 ! e (| v 1 | 2 +| v 2 | 2 )/2 ( a 1 ) m 1 ( a 2 ) m 2 ψ 0,0 ( x ˜ , y ˜ ) ,
Φ N 1 , N 2 (α,β) ( x ˜ , y ˜ ,ϕ)= 1 2π 0 2π g (α,β) ( x ˜ , y ˜ , u 1 , u 2 ) e i( N 1 + N 2 )θ dθ .
1 2π 0 2π e i(n n )θ dθ = δ n, n ,
Φ N 1 , N 2 (α,β) ( x ˜ , y ˜ ,ϕ)= 1 2π 0 2π 1 π e ( x ˜ 2 2 2 v 1 x ˜ + v 1 2 +| v 1 | 2 ) 2 e ( y ˜ 2 2 2 v 2 y ˜ + v 2 2 +| v 2 | 2 ) 2 e i( N 1 + N 2 )θ dθ = K= N 2 N 1 N 1 ( N 1 K)/2 ( N 1 K)! N 2 ( N 2 +K)/2 ( N 2 +K)! e N 1 + N 2 2 e i( N 1 N 2 )ϕ/2 Ψ N 1 K, N 2 +K (α,β) ( x ˜ , y ˜ ) e iKϕ ,
< L z >=[ ( n 1 n 2 )sinαsinβ+2 n 1 n 2 (cosαsinϕ+cosβsinαcosϕ) ].
F k = 1 N+1 n=0 N f n e i2πnk/(N+1) ,
f n = k=0 N F k e i2πnk/(N+1) .
1 N+1 n=0 N exp[ i2π(KS)n/(N+1) ] = δ K,S ,
1 N+1 n=0 N Φ N 1 , N 2 (α,β) ( x ˜ , y ˜ , ϕ n ) e i( N 1 N 2 ) ϕ n /2 e iS ϕ n . = N 1 ( N 1 S)/2 ( N 1 S)! N 2 ( N 2 +S)/2 ( N 2 +S)! e N 1 + N 2 2 Ψ N 1 S, N 2 +S (α,β) ( x ˜ , y ˜ )
Ψ N 1 , N 2 (α,β) ( x ˜ , y ˜ )= [ N 1 N 1 /2 N 1 ! N 2 N 2 /2 N 2 ! e N 2 ] 1 1 N+1 n=0 N Φ N 1 , N 2 (α,β) ( x ˜ , y ˜ , ϕ n ) e i( N 1 N 2 ) ϕ n /2 .