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

1. Introduction

Optical vortex beams, carrying orbital angular momentum (OAM), have been used in a variety of applications including optical tweezers and microscopy [1,2], trapping and guiding of cold atoms [3–5], controlling the chirality of twisted metal nanostructures [6,7], quantum communication [8,9], and spiral interferometry [10]. Even though optical vortex beams can be obtained by directly generating the Laguerre-Gaussian (LG) modes with circular symmetry, it is generally difficult to make a laser oscillation in a pure LG mode. Alternatively, the optical vortex beams are generated by converting the high-order Hermite-Gaussian (HG) modes into LG modes via the use of an astigmatic mode converter (AMC) [11–16]. Abramochkin and Volostnikov originally used an AMC formed by a matched pair cylindrical lenses to generate the so-called Hermite-Laguerre-Gaussian (HLG) beams [12]. The HLG beams, a continuous transformation lying between HG and LG modes, can successively be realized by rotating the cylindrical lens about the optical axis by an angle. The HLG modes display a plentiful evolution of point dislocations and edge dislocations. It has been theoretically verified that the HLG modes can be expressed as the superposition of the degenerate HG modes by using the SU(2) algebra [17]. Nevertheless, the superposition based on the HG modes makes the representation of the HLG mode extremely complicated and leads to the hindrance of calculation, even only for computing the intermediate order mode.

On the other hand, since the paraxial wave equation for the spherical laser cavity is identical to the Schrödinger equation for the 2D harmonic oscillator [18], the laser transverse modes have been employed to explore the wave functions with quantum-classical correspondence [19,20]. The selective excitation was used to generate the elliptical orbital modes, which are laser transverse modes to be well localized on the elliptical orbits. It has been universally found that wave functions localized on periodic orbits are associated with striking quantum phenomena such as conductance fluctuations in mesoscopic semiconductor billiards [21,22], oscillations in photo-detachment cross sections [23,24], and shell effects in metallic clusters [25,26]. So far, the relationship between the HLG modes and elliptical orbital modes has not been explored and connected.

In this work, we first use the SU(2) representation of the HLG mode to manifest the successive transformation between HG and LG modes by varying the angle parameters. We then use the mathematical formula of Schrödinger coherent state to construct the time-dependent HLG-based coherent state and to verify it to be given by a closed form of Gaussian wave packet. By using the explicit closed form, the elliptical orbital mode related to the HLG modes is analytically derived as an integral of the Gaussian wave-packet state over the corresponding orbit. On the other hand, we also confirm that the elliptical orbital mode can be expressed as the superposition of the degenerate HLG modes. We further use the superposition formula and the quantum Fourier transform to verify that the HLG mode can be inversely expressed as the superposition of the elliptical orbital modes with different ellipticities and rotations. The novelty is that the whole expression for the HLG mode is only related to the integration of Gaussian wave-packet states without involving Hermite and Laguerre polynomials. It is believed that the derived formula can provide an important insight into quantum physics and laser transverse modes with OAM [27–33].

2. SU(2) representation for Hermite-Laguerre Gaussian modes

The transverse HG modes in the spherical cavity are identical to the eigenfunctions in the 2D isotropic harmonic oscillator. In terms of the creation operator, the HG modes are given by [34]

ψn1,n2(HG)(x˜,y˜)=(a1)n1n1!(a2)n2n2!ψ0,0(x˜,y˜)
where
ψ0,0(x˜,y˜)=1πe(x˜+y˜)2/2,
a1=12(x˜x˜),
a2=12(y˜y˜),
x˜=2x/w(z), y˜=2y/w(z), w(z)=wo1+(z/zR)2, wo=λzR/π, zR is the Rayleigh range, λ is the photon wavelength, and n1 and n2 are the transverse orders in the x and y directions, respectively. Using the SU(2) algebra, the HLG modes, continuous transformation from HG to LG modes, can be generalized as [35]
Ψn1,n2(α,β)(x˜,y˜)=(b1)n1n1!(b2)n2n2!ψ0,0(x˜,y˜),
where
[b1b2]=[eiα/2cos(β/2)eiα/2sin(β/2)eiα/2sin(β/2)eiα/2cos(β/2)][a1a2],
α and β can be imaged as the azimuthal and polar angles for a point on the Poincaré sphere. It can be verified that the HLG modes with α=0 and β=0, i.e., Ψn1,n2(0,0)(x˜,y˜), correspond to the HG modes ψn1,n2(HG)(x˜,y˜). On the other hand, the HLG modes with α=π/2 and β=π/2, i.e., Ψn1,n2(π/2,π/2)(x˜,y˜), correspond to the LG modes ψn1,n2(LG)(x˜,y˜). Note that the azimuthal quantum number of LG mode is determined by l=n2n1 and the radial quantum number is given by the smaller of n1 and n2. Figure 1 shows the calculated patterns of HLG modes Ψn1,n2(α,β)(x˜,y˜) with n1=4 and n2=3 for several values of α and β. The transition from the HG mode to LG mode can be clearly seen by varying the values from α=0 and β=0to α=π/2 and β=π/2. The OAM per photon for the HLG mode Ψn1,n2(α,β)(x˜,y˜) can be analytically found to be given by <Lz>=(n1n2)sinαsinβ.

 

Fig. 1 Calculated patterns for Ψn1,n2(α,β)(x˜,y˜) with n1=4 and n2=3 for several values of α and β.

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3. Elliptical orbital modes

Schrödinger used the generating function of Hermite polynomials [36] to derive the Gaussian wave packet [37] as

g(x,u)=n=0unn!e|u|2/2[12nn!Hn(x˜)ex˜2/2]=e(x˜222ux˜+u2+|u|2)/2.
In terms of u=Nei(ωt+ϕ), the central peak of the Gaussian wave packet in Eq. (7) can be found to mimic the classical motion x˜=2Re(u)=2Ncos(ωt+ϕ), where the phase factor ϕ is related to the initial position. Using the representation of Schrödinger coherent state, the HLG-based coherent state can be expressed as
g(α,β)(x˜,y˜,u1,u2)=n1=0n2=0u1n1n1!u2n2n2!e|u1|2+|u2|22Ψn1,n2(α,β)(x˜,y˜),
where the parameters u1 and u2 are given by
[u1u2]=[N1ei(θ+ϕ/2)N2ei(θϕ/2)],
the variable θ represents ωt to be in the range of 0 to 2π, and the parameters N1, N2, and ϕ are related to the elliptical orbit. By substituting Eq. (9) into Eq. (8), the wave packet state g(α,β)(x˜,y˜,u1,u2) can be summed to give an expression in closed form,
g(α,β)(x˜,y˜,u1,u2)=n1=0n2=0N1n1/2n1!N2n2/2n2!eN1+N22Ψn1,n2(α,β)(x˜,y˜)ei(n1+n2)θei(n1n2)ϕ/2.
On the other hand, g(α,β)(x˜,y˜,u1,u2) can be summed to give an expression in closed form,
g(α,β)(x˜,y˜,u1,u2)=1πe(x˜222v1x˜+v12+|v1|2)2e(y˜222v2y˜+v22+|v2|2)2
with
[v1v2]=[eiα/2cos(β/2)eiα/2sin(β/2)eiα/2sin(β/2)eiα/2cos(β/2)][u1u2],
The explicit closed form in Eq. (11) is derived from substituting Eq. (5) into Eq. (8) to lead to
g(α,β)(x˜,y˜,u1,u2)=n1=0n2=0u1n1n1!u2n2n2!e(|u1|2+|u2|2)/2(b1)n1(b2)n2ψ0,0(x˜,y˜).
Then, substituting Eq. (6) into Eq. (13) and after cumbersome algebra, g(α,β)(x˜,y˜,u1,u2) can be organized as a separable form of a1 and a2,
g(α,β)(x˜,y˜,u1,u2)=m1=0m2=0v1m1m1!v2m2m2!e(|v1|2+|v2|2)/2(a1)m1(a2)m2ψ0,0(x˜,y˜),
which can be combined with Eq. (7) to obtain Eq. (11).

With the substitution of θ=ωt, the explicit closed form in Eq. (11) indicates that g(α,β)(x˜,y˜,u1,u2) is a Gaussian wave packet with the central peak moving in the elliptical orbit of x˜=2Re(v1) and y˜=2Re(v2). The stationary elliptical mode [38] derived from the wave packet g(α,β)(x˜,y˜,u1,u2) can be given by

ΦN1,N2(α,β)(x˜,y˜,ϕ)=12π02πg(α,β)(x˜,y˜,u1,u2)ei(N1+N2)θdθ.
On the other hand, Eq. (10) reveals that the Gaussian wave packet g(α,β)(x˜,y˜,u1,u2) is the superposition of all HLG modes Ψn1,n2(α,β)(x˜,y˜). Substituting both expressions of Eqs. (10) and (11) into Eq. (15) and using the orthogonality relation,
12π02πei(nn)θdθ=δn,n,
the stationary coherent state ΦN1,N2(α,β)(x˜,y˜,ϕ) can be derived as
ΦN1,N2(α,β)(x˜,y˜,ϕ)=12π02π1πe(x˜222v1x˜+v12+|v1|2)2e(y˜222v2y˜+v22+|v2|2)2ei(N1+N2)θdθ=K=N2N1N1(N1K)/2(N1K)!N2(N2+K)/2(N2+K)!eN1+N22ei(N1N2)ϕ/2ΨN1K,N2+K(α,β)(x˜,y˜)eiKϕ,
where the HLG modes ΨN1K,N2+K(α,β)(x˜,y˜) with n1+n2=N1+N2 have been regrouped by exploiting the new index K to express the superposed modes with n1=N1K and n2=N2+K. For a given (N1,N2,ϕ) and (α,β), the spatial intensity of the mode ΦN1,N2(α,β)(x˜,y˜,ϕ) is exactly concentrated on the elliptical orbit. The OAM per photon for the elliptical orbital mode ΦN1,N2(α,β)(x˜,y˜,ϕ) can be analytically proven to be

<Lz>=[(n1n2)sinαsinβ+2n1n2(cosαsinϕ+cosβsinαcosϕ)].

4. Discrete Fourier transform

The stationary coherent state ΦN1,N2(α,β)(x˜,y˜,ϕ) can be seen to be a superposition of N+1 degenerate HLG modes ΨN1K,N2+K(α,β)(x˜,y˜) with the phase factor eiKϕ in the weighting coefficient, where N1+N2=N. The concept of the discrete Fourier transform can be used to divide the phase parameter ϕ into N+1 different values as ϕn=2πn/(N+1) with n=0,1,,N. The conventional discrete Fourier transform {F0,F1,,FN} of a discrete function {f0,f1,,fN} and its inverse are given by

Fk=1N+1n=0Nfnei2πnk/(N+1),
fn=k=0NFkei2πnk/(N+1).
Using the discrete Fourier transform upon the quantum state [39], the set {ΦN1,N2(α,β)(x˜,y˜,ϕ)} can constitute a basis to represent the HG mode ΨN1K,N2+K(α,β)(x˜,y˜) in an inverse way. Using the orthogonality identity of complex exponentials:
1N+1n=0Nexp[i2π(KS)n/(N+1)]=δK,S,
the Fourier transform for Eq. (17) can be reduced to
1N+1n=0NΦN1,N2(α,β)(x˜,y˜,ϕn)ei(N1N2)ϕn/2eiSϕn.=N1(N1S)/2(N1S)!N2(N2+S)/2(N2+S)!eN1+N22ΨN1S,N2+S(α,β)(x˜,y˜)
Without loss of generality, we can use Eq. (22) for S=0 to obtain
ΨN1,N2(α,β)(x˜,y˜)=[N1N1/2N1!N2N2/2N2!eN2]11N+1n=0NΦN1,N2(α,β)(x˜,y˜,ϕn)ei(N1N2)ϕn/2.
Equation (23) indicates that the HLG mode ΨN1,N2(α,β)(x˜,y˜) can be interpreted as a summation of the generalized elliptical modes ΦN1,N2(α,β)(x˜,y˜,ϕn) which are given by an integral of the Gaussian wave-packet state over the classical orbit, as shown in Eq. (17).

The first case for the numerical demonstration is the modes ΦN1,N2(α,β)(x˜,y˜,ϕn) with α=0 and β=0. Note that the mode ΨN1,N2(α,β)(x˜,y˜) with α=0 and β=0 is the HG mode ψN1,N2(HG)(x˜,y˜). Figure 2 shows the calculated results for ΦN1,N2(α,β)(x˜,y˜,ϕn) by using the integral formula in Eqs. (17) with (N1,N2)=(8,7), (α,β)=(0,0), and ϕn=πn/8 with n=0,1,,15. The calculated elliptical modes ΦN1,N2(α,β)(x˜,y˜,ϕn) with (α,β)=(0,0) can be seen to have different ellipticities and different directions for the major axes. Applying the calculated ΦN1,N2(α,β)(x˜,y˜,ϕn) to Eq. (22), the resulting pattern for ΨN1,N2(α,β)(x˜,y˜), as shown in the central part of Fig. 2, can be seen to be the HG mode ψN1,N2(HG)(x˜,y˜).

 

Fig. 2 Calculated results for ΦN1,N2(α,β)(x˜,y˜,ϕn) by using the integral formula in Eq. (17) with (N1,N2)=(8,7), (α,β)=(0,0), and ϕn=πn/8 with n=0,1,,15. The central pattern for ψ8,7(HG)(x˜,y˜) obtained by substituting the calculated ΦN1,N2(α,β)(x˜,y˜,ϕn) into Eq. (22).

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The second numerical demonstration is the modes ΦN1,N2(α,β)(x˜,y˜,ϕn)with α=π/2 and β=π/2. For this case, the mode ΨN1,N2(α,β)(x˜,y˜) is the LG mode ψN1,N2(LG)(x˜,y˜). Figure 3 shows the calculated results for ΦN1,N2(α,β)(x˜,y˜,ϕn) by using the integral formula in Eq. (17) with (N1,N2)=(2,13), (α,β)=(π/2,π/2) and ϕn=πn/8 with n=0,1,,15. The calculated elliptical modes ΦN1,N2(α,β)(x˜,y˜,ϕn) with (α,β)=(π/2,π/2) can be seen to have different directions for the major axes but their ellipticities are the same. Applying the calculated ΦN1,N2(α,β)(x˜,y˜,ϕn) to Eq. (22), the resulting pattern for ΨN1,N2(α,β)(x˜,y˜), as shown in the central part of Fig. 3, can be seen to be the LG mode ψN1,N2(LG)(x˜,y˜). Figure 4 shows the calculated results for the case of ΨN1,N2(α,β)(x˜,y˜) with (N1,N2)=(4,11), (α,β)=(2π/5,2π/5) and ϕn=πn/8 with n=0,1,,15. The calculated elliptical modes ΦN1,N2(α,β)(x˜,y˜,ϕn) with (α,β) =(2π/5,2π/5)can be seen to have different ellipticities and different directions for the major axes. From Figs. 2–4, it is found that the decomposition of the HLG mode ΨN1,N2(α,β)(x˜,y˜) into the elliptical orbital modes ΦN1,N2(α,β)(x˜,y˜,ϕn) can reveal the symmetrical structure clearly.

 

Fig. 3 Calculated results for ΦN1,N2(α,β)(x˜,y˜,ϕn) by using the integral formula in Eq. (17) with (N1,N2)=(2,13), (α,β)=(π/2,π/2), and ϕn=πn/8 with n=0,1,,15. The central pattern for ψ2,13(LG)(x˜,y˜) obtained by substituting the calculated ΦN1,N2(α,β)(x˜,y˜,ϕn) into Eq. (22).

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Fig. 4 Calculated results with (N1,N2)=(4,11), (α,β)=(2π/5,2π/5), and ϕn=πn/8 with n=0,1,,15.The central pattern for Ψ4,11(α,β)(x˜,y˜) obtained by substituting the calculated ΦN1,N2(α,β)(x˜,y˜,ϕn) into Eq. (22).

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5. Conclusions

In summary, the SU(2) representation of the HLG mode has been used to manifest the successive transformation between HG and LG modes by varying the angle parameters. We have derived a closed form of Gaussian wave packet for the time-dependent HLG-based coherent state. We have further exploited the explicit closed form to derive the elliptical orbital mode as an integral of the Gaussian wave-packet state over the orbit. On the other hand, we have also dervied the elliptical orbital mode as the superposition of the degenerate HLG modes. We finally employ the derived formulae and the quantum Fourier transform to verify that the HLG mode is inversely expressed as the superposition of the elliptical orbital modes. The complete form for the HLG mode can clearly manifest the quantum-classical correspondence and provide an important insight into the laser transverse modes with OAM.

Funding

Ministry of Science and Technology of Taiwan (MOST107-2119-M-009-015).

Acknowledgments

This work was financially supported by the Research Team of Photonic Technologies and Intelligent Systems at NCTU within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan. This work is also supported by the Ministry of Science and Technology of Taiwan (Contract No. MOST107-2119-M-009-015).

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References

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  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]
  2. 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]
  3. Y. Song, D. Milam, and W. T. Hill, “Long, narrow all-light atom guide,” Opt. Lett. 24(24), 1805–1807 (1999).
    [Crossref] [PubMed]
  4. 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]
  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. 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]
  7. 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]
  8. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
    [Crossref]
  9. G. Gibson, J. Courtial, M. 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]
  10. 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]
  11. E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83(1–2), 123–135 (1991).
    [Crossref]
  12. 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]
  13. H. Sridhar, M. G. Cohen, and J. W. Noé, “Creating optical vortex modes with a single cylinder lens,” Proc. SPIE 7613, 76130X (2010).
    [Crossref]
  14. 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]
  15. 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]
  16. 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]
  17. 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]
  18. A. E. Siegman, Lasers (University Science Books, 1986).
  19. 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]
  20. Y. F. Chen and Y. P. Lan, “Observation of transverse patterns in an isotropic microchip laser,” Phys. Rev. A 67(4), 043814 (2003).
    [Crossref]
  21. 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]
  22. 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]
  23. 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]
  24. 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]
  25. M. Brack, “The physics of simple metal clusters: self-consistent jellium model and semiclassical approaches,” Rev. Mod. Phys. 65(3), 677–732 (1993).
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  26. W. A. de Heer, “The physics of simple metal clusters: experimental aspects and simple models,” Rev. Mod. Phys. 65(3), 611–676 (1993).
    [Crossref]
  27. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Vortex Hermite-Gaussian laser beams,” Opt. Lett. 40(5), 701–704 (2015).
    [Crossref] [PubMed]
  28. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(5), S157–S161 (2004).
    [Crossref]
  29. 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]
  30. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Elliptical Gaussian optical vortices,” Phys. Rev. A (Coll. Park) 95(5), 053805 (2017).
    [Crossref]
  31. A. Y. Okulov, “Vortex-antivortex wavefunction of a degenerate quantum gas,” Laser Phys. 19(8), 1796–1803 (2009).
    [Crossref]
  32. A. Y. Okulov, “Superfluid rotation sensor with helical laser trap,” J. Low Temp. Phys. 171(3–4), 397–407 (2013).
    [Crossref]
  33. A. Y. Okulov, “Laser singular Theta-pinch,” Phys. Lett. A 374(44), 4523–4527 (2010).
    [Crossref]
  34. S. Flügge, Practical Quantum Mechanics (Springer-Verlag, 1971), p. 107.
  35. 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]
  36. N. N. Lebedev, Special Functions & Their Applications (Dover, 1972).
  37. E. Schrödinger, “Der stetige Übergang von der Mikro-zur Makromechanik,” Naturwissenschaften 14(28), 664–666 (1926).
    [Crossref]
  38. J. Pollet, O. Méplan, and C. Gignoux, “Elliptic eigenstates for the quantum harmonic oscillator,” J. Phys. A 28(24), 7287–7297 (1995).
    [Crossref]
  39. R. Blüumel, Foundations of Quantum Mechanics: From Photons to Quantum Computers (Jones and Bartlett Publishers, 2010).

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]

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]

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

2015 (1)

2013 (4)

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

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]

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]

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

2009 (1)

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

2007 (2)

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]

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

2006 (2)

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]

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]

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)

Equations on this page are rendered with MathJax. Learn more.

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 .

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