Abstract

We investigate optical singularities in coaxial superpositions of two Laguerre–Gaussian (LG) modes with a common beam waist from the viewpoints of a general formulation of phase structure, experimental generation of various superposition beams, and evaluation of the generated beams’ fidelity. By applying a holographic phase-amplitude modulation scheme using a phase-modulation-type spatial light modulator, output fidelity beyond 0.960 was observed under several typical conditions. Additionally, an elliptic-type folded singularity, which provides a different class of phase structures from familiar helical singularities, was predicted and observed in a superposition involving two LG modes of both radially and azimuthally higher orders.

© 2010 Optical Society of America

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  1. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2009 (2)

2008 (1)

2007 (4)

Y. Ohtake, T. Ando, N. Fukuchi, N. Matsumoto, H. Ito, and T. Hara, “Universal generation of higher-order multiringed Laguerre–Gaussian beams by using a spatial light modulator,” Opt. Lett. 32, 1411–1413 (2007).
[CrossRef] [PubMed]

S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical Ferris wheel for ultracold atoms,” Opt. Express 15, 8619–8625 (2007).
[CrossRef] [PubMed]

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

J. Adachi and G. Ishikawa, “Classification of phase singularities for complex scalar waves and their bifurcations,” Nonlinearity 20, 1907–1925 (2007).
[CrossRef]

2006 (1)

2005 (2)

J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
[CrossRef]

2004 (2)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

2003 (1)

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

2001 (2)

M. V. Berry and M. R. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime,” J. Phys. A 34, 8877–8888 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. London, Ser. A 457, 2251–2263 (2001).
[CrossRef]

2000 (1)

1998 (1)

1997 (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

1993 (2)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

1982 (1)

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

1971 (1)

Adachi, J.

J. Adachi and G. Ishikawa, “Classification of phase singularities for complex scalar waves and their bifurcations,” Nonlinearity 20, 1907–1925 (2007).
[CrossRef]

Allen, L.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

Ando, T.

Arlt, J.

Arnold, A. S.

Barnett, S. M.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

Basistiy, I. V.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Baumann, S. M.

Bazhenov, V. Yu.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Berry, M. V.

M. V. Berry and M. R. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime,” J. Phys. A 34, 8877–8888 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. London, Ser. A 457, 2251–2263 (2001).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Coutial, J.

J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Dändliker, R.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

Dennis, M. R.

J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. London, Ser. A 457, 2251–2263 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime,” J. Phys. A 34, 8877–8888 (2001).
[CrossRef]

Ellinas, D.

Flossmann, F.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
[CrossRef]

Franke-Arnold, S.

Fukuchi, N.

Galvez, E. J.

Girkin, J. M.

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Hamazaki, J.

Hara, T.

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Herzig, H. P.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

Ina, H.

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Inoue, T.

Ishikawa, G.

J. Adachi and G. Ishikawa, “Classification of phase singularities for complex scalar waves and their bifurcations,” Nonlinearity 20, 1907–1925 (2007).
[CrossRef]

Ito, H.

Ivanovskyy, A. A.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

Jones, A. L.

Kalb, D. M.

Khonina, S. N.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

Kirk, J. P.

Kobayashi, S.

Kotlyar, V. V.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

Leach, J.

S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical Ferris wheel for ultracold atoms,” Opt. Express 15, 8619–8625 (2007).
[CrossRef] [PubMed]

J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

Lembessis, V. E.

MacMillan, L. H.

Maier, M.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
[CrossRef]

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Matsumoto, N.

Mineta, Y.

Morita, R.

Nye, J. F.

J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A 15, 1132–1138 (1998).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Öhberg, P.

Ohtake, Y.

Oka, K.

Padgett, M. J.

S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical Ferris wheel for ultracold atoms,” Opt. Express 15, 8619–8625 (2007).
[CrossRef] [PubMed]

J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000).
[CrossRef]

Rockstuhl, C.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

Salt, M. G.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

Schwarz, U. T.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
[CrossRef]

Skidanov, R. V.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

Soifer, V. A.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

Soskin, M. S.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Takeda, M.

Vasnetsov, M. V.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Wright, A. J.

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

J. Mod. Opt. (1)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

J. Phys. A (1)

M. V. Berry and M. R. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime,” J. Phys. A 34, 8877–8888 (2001).
[CrossRef]

New J. Phys. (2)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Nonlinearity (1)

J. Adachi and G. Ishikawa, “Classification of phase singularities for complex scalar waves and their bifurcations,” Nonlinearity 20, 1907–1925 (2007).
[CrossRef]

Opt. Commun. (4)

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. A (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Proc. R. Soc. London, Ser. A (2)

M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. London, Ser. A 457, 2251–2263 (2001).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Other (1)

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Polar coordinates around zero points on the cross section of light propagation: (a) for the zero point at the beam center, i.e., the origin O, and (b) for the zero point placed at off-centered position P apart from O by distance ρ.

Fig. 2
Fig. 2

Schematic contour plots of the phase profiles described by Eq. (18) for (a) ( l = 2 , l = 1 ) and (b) ( l = 2 , l = 1 ) .

Fig. 3
Fig. 3

(a) Schematic contour plot of the phase distribution for complex amplitude Eq. (21) around the neighborhood of a zero point indicated by Eq. (22) ( l = 2 and l = 0 ). (b) Bird’s-eye view of (a).

Fig. 4
Fig. 4

Schematic diagram of experimental setup: SF, spatial filter; CL, collimator lens; BS 1 3 , beam splitters; L 1 , 2 , convex lenses with focal lengths of f 1 and f 2 , respectively. Precise phase modulation is achieved by a liquid-crystal-on-silicon (LCOS)-type SLM. Optical path distance from the SLM surface to L 1 is adjusted as f 1 to construct the 4 f system.

Fig. 5
Fig. 5

Beam patterns and phase profiles of equal superposition of LG p = 0 l = 0 and LG p = 1 l = 0 modes. Fidelity evaluated from the inner product of experimental and theoretical amplitudes is 0.979.

Fig. 6
Fig. 6

Beam patterns and phase profiles of superposition of 3 LG p = 0 l = 1 + 2 LG p = 2 l = 1 modes. Fidelity evaluated from the inner product of experimental and theoretical amplitudes is 0.963.

Fig. 7
Fig. 7

Beam patterns and phase profiles of equal superposition of LG p = 0 l = 11 and LG p = 0 l = 3 modes. Fidelity evaluated from the inner product of experimental and fitted theoretical amplitudes is 0.967.

Fig. 8
Fig. 8

Beam patterns and phase profiles of equal superposition of LG p = 0 l = 5 and LG p = 0 l = 3 modes. Fidelity evaluated from the inner product of experimental and fitted theoretical amplitudes is 0.971.

Fig. 9
Fig. 9

Beam patterns and phase profiles of superposition defined by LG p = 1 l = 2 + 27 / 32 LG p = 0 l = 0 . Mode purity evaluated from the inner product of experimental and theoretical amplitudes is 0.968.

Fig. 10
Fig. 10

Beam patterns and phase profiles of superposition of Eq. (33). Fidelity evaluated from the inner product of experimental and theoretical amplitudes is 0.968.

Fig. 11
Fig. 11

Beam patterns and phase profiles of elliptic-type phase singularity. Mode purity evaluated from the inner product of experimental and theoretical amplitudes is 0.979.

Fig. 12
Fig. 12

Beam patterns and phase profiles of parabolic-type phase singularity. Mode purity evaluated from the inner product of experimental and theoretical amplitudes is 0.976.

Fig. 13
Fig. 13

Beam patterns and phase profiles of hyperbolic-type phase singularity. Mode purity evaluated from the inner product of experimental and theoretical amplitudes is 0.975.

Equations (44)

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u p l ( r , ϕ , z ) = ( 1 ) p w z [ 2 π p ! ( p + | l | ) ! ] 1 / 2 ( 2 r 2 w z 2 ) | l | / 2 exp ( r 2 w z 2 ) L p | l | ( 2 r 2 w z 2 ) e i l ϕ   exp [ i 2 z k w 0 2 r 2 w z 2 ] exp [ i ( 2 p + | l | + 1 ) arctan ( 2 z k w 0 2 ) ] ,
w z = w 0 [ 1 + ( 2 z k w 0 2 ) 2 ] 1 / 2 ,
u ( r , ϕ ) = u p = 1 l = 0 ( r , ϕ ) + u p = 0 l = 0 ( r , ϕ ) = π 1 / 2 w 0 e r 2 / 2 r 2 .
u ( r , ϕ ) = u p = 2 l ( r , ϕ ) + [ | l | + 2 2 ( | l | + 1 ) ] 1 / 2 u p = 0 l ( r , ϕ ) = [ π ( | l | + 2 ) ! ] 1 / 2 w 0 e r 2 / 2 r | l | e i l ϕ × ( r | l | + 2 ) 2 ( r + | l | + 2 ) 2 .
u ( r , ϕ ) = c p , l u p l ( r , ϕ ) + c p , l u p l ( r , ϕ ) = ( 2 π ) 1 / 2 e r 2 / 2 w 0 [ C p , l r | l | L p | l | ( r 2 ) e i l ϕ + C p , l r | l | L p | l | ( r 2 ) e i l ϕ ] .
u ( r , ϕ ) = ( 2 π ) 1 / 2 e r 2 / 2 w 0 r | l | e i l ϕ C p , l e i δ F ( r , ϕ ) ,
F ( r , ϕ ) = r | l | | l | L p | l | ( r 2 ) e i [ ( l l ) ϕ + δ ] + A L p | l | ( r 2 ) ,
Ψ ( ϕ ) = arctan ( Im   u ( ε , ϕ ) Re   u ( ε , ϕ ) ) = arctan ( ε | l | [ c 0 + O ( ε ) ] sin ( l ϕ ) ε | l | [ c 0 + O ( ε ) ] cos ( l ϕ ) ) l ϕ     ( ε 0 ) ,
( ρ , α + 2 π j l l )     ( j = 0 , 1 , , | l l | 1 ) ,
r 2 = ρ 2 + ε 2 + 2 ρ ε   cos   φ ,
cos   ϕ = 1 2 r ρ ( r 2 + ρ 2 ε 2 ) = 1 1 2 ε 2 ρ 2 sin 2 φ + O ( ε 4 / ρ 4 ) ,
sin   ϕ = ε ρ sin   φ + O ( ε 3 / ρ 3 ) ϕ = ε ρ sin   φ + O ( ε 3 / ρ 3 ) ,
u ( ε , φ ) = ( 2 π ) 1 / 2 e ρ 2 / 2 w 0 ρ | l | C p , l [ ρ F ( ρ , α ) cos   φ i ( l l ) ρ | l | | l | L p | l | ( ρ 2 ) sin   φ ] ε ρ + O ( ε 2 / ρ 2 ) ,
F ( ρ , α ) | r F ( r , ϕ ) | r = ρ , ϕ = α .
F ( ρ , α ) = 0 ,
e r 2 / 2 r | l | e i l ϕ = e ρ 2 / 2 ρ | l | + O ( ε / ρ ) .
Ψ ( φ ) = arctan ( ( l l ) ρ | l | | l | L p | l | ( ρ 2 ) ρ F ( ρ , α ) tan   φ ) + O ( ε / ρ ) ,
F ( r , ϕ ) = r | l | | l | e i ( l l ) ϕ + A = 0.
cos ( l l ) α = sgn ( A ) .
F ( ρ , α ) = ( | l | | l | ) ρ | l | | l |     ( ρ | l | | l | = | A | ) ,
Ψ ( φ ) = arctan ( l l | l | | l | tan   φ ) + O ( ε / ρ ) .
Ψ ( φ ) φ = 0 φ = 2 π = Ψ ( φ ) 0 π / 2 + Ψ ( φ ) π / 2 3 π / 2 + Ψ ( φ ) 3 π / 2 2 π sgn ( l l | l | | l | ) 2 π     ( ε 0 ) .
Ψ ( φ ) = sgn ( l ) φ + O ( ε / ρ ) ,
u ( r , ϕ ) = u 1 l ( r , ϕ ) + [ | l | ! ( | l | + 1 ) ! ] 1 / 2 ( | l | + 1 ) 2 4 u 0 l ( r , ϕ ) ,
F ( r , ϕ ) = r 2 ( r 2 | l | 1 ) e i ( l l ) ϕ + ( | l | + 1 ) 2 4 ,
( ρ , α ) = ( ( | l | + 1 ) / 2 , 2 π j / ( l l ) )     ( j = 0 , 1 , , | l l | 1 ) .
u ( x , y ) 2 ( | l | + 1 ) 2 x 2 ( | l | + 1 ) [ ( l l ) 2 + 1 ] y 2 + i ( l l ) ( | l | + 1 2 ) 3 / 2 y ,
Ψ ( φ ) = arctan ( ( l l ) 2 ε a   sin   φ 4 a 2 cos 2 φ + b 2 sin 2 φ ) ,
ρ ± = | l | + 1 ( 1 2 ± 1 2 ) 1 / 2 .
| u ( r , ϕ , z ) | 2 = π 2 1 w z 2 e 2 r z 2 { r z 4 + 2 ( 1 r z 2 ) [ 1 ( 1 + 16 z 2 k w 0 2 ) 1 / 2 ] } { π 2 1 w 0 2 e 2 r 0 2 r 0 4 ( z 0 ) π 2 1 w z 2 e 2 r z 2 ( r z 2 1 ) 2 ( z ) , }
u ( r , φ ) = ( 2 π ) 1 / 2 e r 2 / 2 w 0 r | l | e i l ϕ F ( r , ϕ ) ,
F ( r , ϕ ) = C p 1 , l e i δ 1 L p 1 | l | ( r 2 ) + C p 2 , l e i δ 2 L p 2 | l | ( r 2 ) + + C p N , l e i δ N L p N | l | ( r 2 )
F ( r , ϕ ) ( r ρ 1 ) n 1 ( r ρ 2 ) n 2 ( r ρ j ) n j ,
u ( r , ϕ ) = ( 2 π ) 1 / 2 e r 2 / 2 w 0 [ C 0 , l 1 r | l 1 | e i ( l 1 ϕ + δ 1 ) + + C 0 , l N r | l N | e i ( l N ϕ + δ N ) ] ,
F ̃ ( z ) = C ̃ 0 , l 1 z l 1 + C ̃ 0 , l 2 z l 2 + + C ̃ 0 , l N z l N ( z γ 1 ) n 1 ( z γ 2 ) n 2 ( z γ j ) n j ,
u ( r , ϕ ) = u p = 0 l = 4 ( r , ϕ ) 1 3 u p = 0 l = 2 ( r , ϕ ) + 1 2 6 u p = 0 l = 0 ( r , ϕ ) = 1 2 3 π e r 2 / 2 w 0 ( r e i ϕ + 1 ) 2 ( r e i ϕ 1 ) 2
r | l | | l | L p | l | ( r 2 ) sin   η = 0 ,
r | l | | l | L p | l | ( r 2 ) cos   η + A L p | l | ( r 2 ) = 0.
u ( x , y ) x 2 + y 2 + i y ,
u ( x , y ) x 2 + i y ,
u ( x , y ) x 2 y 2 + i y .
HG 20 + HG 02 + 2 HG 00 + i HG 01 ,
HG 20 + 1 2 HG 00 + i HG 01 ,
HG 20 HG 02 + i HG 01 .

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