Abstract

This paper proposes a new scheme for generating vortex laser beams from a laser. The proposed system consists of a Dove prism embedded in an unbalanced Mach-Zehnder interferometer configuration. This configuration allows controlled construction of p×p vortex array beams from Ince-Gaussian modes, IGe p, p modes. An incident IGe p, p laser beam of variety order p can easily be generated from an end-pumped solid-state laser system with an off-axis pumping mechanism. This study simulates this type of vortex array laser beam generation, analytically derives the vortex positions of the resulting vortex array laser beams, and discusses beam propagation effects. The resulting vortex array laser beam can be applied to optical tweezers and atom traps in the form of two-dimensional arrays, or used to study the transfer of angular momentum to micro particles or atoms (Bose-Einstein condensate).

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  31. K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, "Optical trapping of a metal particle and a water droplet by a scanning laser beam," Appl. Phys. Lett. 60, 807-809 (1992).
  32. P. H. Jones, E. Stride and N. Saffari, "Trapping and manipulation of microscopic bubbles with a scanning optical tweezer," Appl. Phys. Lett. 89, 081113 (2006).

2008 (2)

S.-C. Chu, "Generation of multiple vortex beams with specified vortex number from lasers with controlled Ince-Gaussian modes," Jpn. J. Appl. Phys. 47, 5297-5303 (2008)

S.-C. Chu, T. Ohtomo, and K. Otsuka, "Generation of donutlike vortex beam with tunable orbital angular momentum from lasers with controlled Hermite-Gaussian modes," Appl. Opt. 47, 2583-2591 (2008)
[CrossRef]

2007 (2)

2006 (3)

K. J. Moh, X. -. Yuan, W. C. Cheong, L. S. Zhang, J. Lin, B. P. S. Ahluwalia, and H. Wang, "High-power efficient multiple optical vortices in a single beam generated by a kinoform-type spiral phase plate," Appl. Opt. 45, 1153-1161 (2006)
[CrossRef]

T. Xu and S. Wang, "Propagation of Ince-Gaussian beams in a thermal lens medium," Opt. Commun. 265, 1-5 (2006).
[CrossRef]

P. H. Jones, E. Stride and N. Saffari, "Trapping and manipulation of microscopic bubbles with a scanning optical tweezer," Appl. Phys. Lett. 89, 081113 (2006).

2005 (2)

2004 (5)

2003 (1)

2002 (2)

2001 (3)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef]

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, 3401 (2001).
[CrossRef]

J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

1999 (3)

1998 (1)

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, D. N. Christodoulides, "Self-Trapping of Dark Incoherent Light Beams," Science 280, 889-892 (1998)
[CrossRef]

1996 (3)

E. L. Wooten, R. L. Stone, E. W. Miles, and E. M. Bradley, "Rapidly tunable narrow band wavelength filter using LiNbO3 unbalanced Mach-Zehnder interferometers," J. Lightwave Technol. 14, 2530-2536 (1996).
[CrossRef]

G. A. Turnball, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, "The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate," Opt. Commun. 127, 183-188 (1996).
[CrossRef]

K. T. Gahagan and G. A. Swartzlander, Jr, "Optical vortex trapping of particles," Opt. Lett. 21, 827-829 (1996).
[CrossRef] [PubMed]

1993 (1)

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, 123-132 (1993).
[CrossRef] [PubMed]

1992 (1)

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, "Optical trapping of a metal particle and a water droplet by a scanning laser beam," Appl. Phys. Lett. 60, 807-809 (1992).

1991 (1)

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
[CrossRef] [PubMed]

Allen, L.

G. A. Turnball, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, "The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate," Opt. Commun. 127, 183-188 (1996).
[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, 123-132 (1993).
[CrossRef] [PubMed]

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef]

Bandres, M. A.

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, 123-132 (1993).
[CrossRef] [PubMed]

Borwinska, M.

Bradley, E. M.

E. L. Wooten, R. L. Stone, E. W. Miles, and E. M. Bradley, "Rapidly tunable narrow band wavelength filter using LiNbO3 unbalanced Mach-Zehnder interferometers," J. Lightwave Technol. 14, 2530-2536 (1996).
[CrossRef]

Brambilla, M.

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
[CrossRef] [PubMed]

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef]

Chen, Z.

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, D. N. Christodoulides, "Self-Trapping of Dark Incoherent Light Beams," Science 280, 889-892 (1998)
[CrossRef]

Chervenkov, S.

Christodoulides, D. N.

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, D. N. Christodoulides, "Self-Trapping of Dark Incoherent Light Beams," Science 280, 889-892 (1998)
[CrossRef]

Chu, S.

Chu, S.-C.

S.-C. Chu, "Generation of multiple vortex beams with specified vortex number from lasers with controlled Ince-Gaussian modes," Jpn. J. Appl. Phys. 47, 5297-5303 (2008)

S.-C. Chu, T. Ohtomo, and K. Otsuka, "Generation of donutlike vortex beam with tunable orbital angular momentum from lasers with controlled Hermite-Gaussian modes," Appl. Opt. 47, 2583-2591 (2008)
[CrossRef]

Coskun, T. H.

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, D. N. Christodoulides, "Self-Trapping of Dark Incoherent Light Beams," Science 280, 889-892 (1998)
[CrossRef]

Dholakia, K.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef]

Dreischuh, A.

Dubik, B.

J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

Endo, M.

Fujioka, T.

Gahagan, K. T.

Gutiérrez-Vega, J. C.

Hill, W. T.

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, 3401 (2001).
[CrossRef]

Jones, P. H.

P. H. Jones, E. Stride and N. Saffari, "Trapping and manipulation of microscopic bubbles with a scanning optical tweezer," Appl. Phys. Lett. 89, 081113 (2006).

Julio, M. A.

Kamikariya, K.

Kawakami, M.

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, 3401 (2001).
[CrossRef]

Kitamura, N.

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, "Optical trapping of a metal particle and a water droplet by a scanning laser beam," Appl. Phys. Lett. 60, 807-809 (1992).

Koshioka, M.

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, "Optical trapping of a metal particle and a water droplet by a scanning laser beam," Appl. Phys. Lett. 60, 807-809 (1992).

Kurzynowski, P.

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, 3401 (2001).
[CrossRef]

Lugiato, L. A.

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
[CrossRef] [PubMed]

MacDonald, M. P.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef]

Masajada, J.

J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

Masuhara, H.

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, "Optical trapping of a metal particle and a water droplet by a scanning laser beam," Appl. Phys. Lett. 60, 807-809 (1992).

Milam, D.

Miles, E. W.

E. L. Wooten, R. L. Stone, E. W. Miles, and E. M. Bradley, "Rapidly tunable narrow band wavelength filter using LiNbO3 unbalanced Mach-Zehnder interferometers," J. Lightwave Technol. 14, 2530-2536 (1996).
[CrossRef]

Misawa, H.

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, "Optical trapping of a metal particle and a water droplet by a scanning laser beam," Appl. Phys. Lett. 60, 807-809 (1992).

Mitchell, M.

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, D. N. Christodoulides, "Self-Trapping of Dark Incoherent Light Beams," Science 280, 889-892 (1998)
[CrossRef]

Moh, K. J.

Nanri, K.

Neshev, D.

Ohtomo, T.

Orenstein, M.

J. Scheuer, M. Orenstein, "Optical vortices crystals: Spontaneous generation in nonlinear semiconductor microcavities," Science 285, 230-233 (1999).
[CrossRef] [PubMed]

Otsuka, K.

Padgett, M. J.

G. A. Turnball, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, "The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate," Opt. Commun. 127, 183-188 (1996).
[CrossRef]

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef]

Paulus, G. G.

Penna, V.

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
[CrossRef] [PubMed]

Piccirillo, B.

Prati, F.

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
[CrossRef] [PubMed]

Robertson, D. A.

G. A. Turnball, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, "The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate," Opt. Commun. 127, 183-188 (1996).
[CrossRef]

Saffari, N.

P. H. Jones, E. Stride and N. Saffari, "Trapping and manipulation of microscopic bubbles with a scanning optical tweezer," Appl. Phys. Lett. 89, 081113 (2006).

Santamato, E.

Sasaki, K.

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, "Optical trapping of a metal particle and a water droplet by a scanning laser beam," Appl. Phys. Lett. 60, 807-809 (1992).

Sasso, A.

Scheuer, J.

J. Scheuer, M. Orenstein, "Optical vortices crystals: Spontaneous generation in nonlinear semiconductor microcavities," Science 285, 230-233 (1999).
[CrossRef] [PubMed]

Schwarz, U. T.

Segev, M.

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, D. N. Christodoulides, "Self-Trapping of Dark Incoherent Light Beams," Science 280, 889-892 (1998)
[CrossRef]

Sibbett, W.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001).
[CrossRef]

Smith, G. M.

G. A. Turnball, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, "The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate," Opt. Commun. 127, 183-188 (1996).
[CrossRef]

Song, Y.

Stone, R. L.

E. L. Wooten, R. L. Stone, E. W. Miles, and E. M. Bradley, "Rapidly tunable narrow band wavelength filter using LiNbO3 unbalanced Mach-Zehnder interferometers," J. Lightwave Technol. 14, 2530-2536 (1996).
[CrossRef]

Stride, E.

P. H. Jones, E. Stride and N. Saffari, "Trapping and manipulation of microscopic bubbles with a scanning optical tweezer," Appl. Phys. Lett. 89, 081113 (2006).

Swartzlander, G. A.

Takeda, S.

Tamm, C.

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
[CrossRef] [PubMed]

Turnball, G. A.

G. A. Turnball, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, "The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate," Opt. Commun. 127, 183-188 (1996).
[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, 123-132 (1993).
[CrossRef] [PubMed]

Vella, A.

Walther, H.

Wang, S.

T. Xu and S. Wang, "Propagation of Ince-Gaussian beams in a thermal lens medium," Opt. Commun. 265, 1-5 (2006).
[CrossRef]

Weiss, C. O.

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991).
[CrossRef] [PubMed]

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, 123-132 (1993).
[CrossRef] [PubMed]

Wooten, E. L.

E. L. Wooten, R. L. Stone, E. W. Miles, and E. M. Bradley, "Rapidly tunable narrow band wavelength filter using LiNbO3 unbalanced Mach-Zehnder interferometers," J. Lightwave Technol. 14, 2530-2536 (1996).
[CrossRef]

Xu, T.

T. Xu and S. Wang, "Propagation of Ince-Gaussian beams in a thermal lens medium," Opt. Commun. 265, 1-5 (2006).
[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, 3401 (2001).
[CrossRef]

Yuan, X.

Zhang, D. W.

Appl. Opt. (4)

Appl. Phys. Lett. (2)

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, "Optical trapping of a metal particle and a water droplet by a scanning laser beam," Appl. Phys. Lett. 60, 807-809 (1992).

P. H. Jones, E. Stride and N. Saffari, "Trapping and manipulation of microscopic bubbles with a scanning optical tweezer," Appl. Phys. Lett. 89, 081113 (2006).

J. Lightwave Technol. (1)

E. L. Wooten, R. L. Stone, E. W. Miles, and E. M. Bradley, "Rapidly tunable narrow band wavelength filter using LiNbO3 unbalanced Mach-Zehnder interferometers," J. Lightwave Technol. 14, 2530-2536 (1996).
[CrossRef]

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

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

Jpn. J. Appl. Phys. (1)

S.-C. Chu, "Generation of multiple vortex beams with specified vortex number from lasers with controlled Ince-Gaussian modes," Jpn. J. Appl. Phys. 47, 5297-5303 (2008)

Opt. Commun. (4)

J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001).
[CrossRef]

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Figures (12)

Fig. 1.
Fig. 1.

Some analytical patterns of Ince-Gaussian modes.

Fig. 2.
Fig. 2.

(a) Schematic diagram of the interferometer configuration. (b) Dove prism setup in the proposed interferometer. The field polarization states at all stations are drawn in green.

Fig. 3.
Fig. 3.

Diagram of the simulation model of a half-symmetric cavity

Fig. 4.
Fig. 4.

(a) Amplitude distributions of selected IGe p, p modes from p=2 to 4. (b) (c) Amplitude and phase distributions of vortex array laser beams created by superposing the IGe p, p mode and its rotated replica with a π/2 phase delay. (d) The Interferogram (calculation of interference fringes of the vortex array laser beam with a tilted plane wave). The widow widths of all interferograms are half of Fig. 4(a), (b) and (c).

Fig. 5.
Fig. 5.

(a) Amplitude distributions of forced single IGe p, p mode oscillations in a simulated endpumped solid-state laser system (from p=5 to 10). (b) Amplitude distributions of the vortex array laser beams generated by superposing the IGe p, p mode and its rotated replica with a π/2 phase delay.

Fig. 6.
Fig. 6.

The (a) amplitude and (b) phase distribution of the resulting vortex array laser beam with the nodal lines of the IGe p, p mode and rotated IGe p, p mode plotted in blue. Red spots indicate marginal vortices in these field distributions.

Fig. 7.
Fig. 7.

(a) Simulated excited single IGe p, p mode oscillations from an end-pumped solid-state laser system with different azimuthal pumping beam shapes (plotted by red circles). (b) The vortex array laser beams generated by superposing the selected IGe p, p mode and its rotated replica, [IGe p, p ]T with a π/2 phase delay.

Fig. 8.
Fig. 8.

Three kinds of incident laser beams and its corresponding resultant vortex laser beams from Dove prism embedded Mach-Zehnder interferometer. Three kinds of incident laser beams are (a) HG10, 0 mode, (b) IGe 10, 10 mode and (c) tilted IGe 10, 10 mode.

Fig. 9.
Fig. 9.

Intensity distribution, phase distributions and the Interferogram (calculation of interference fringes of the vortex array laser beam with a tilted plane wave) of laser beams superposed by two sub-beams, i.e., the IGe 4, 4 modes and its rotated replica [IGe 4, 4 ]T, with a phase difference Δϕ of values ranging from π/8 to 2π in 16 steps.

Fig. 10.
Fig. 10.

Intensity distributions, phase distributions and the interferograms (calculations of interference fringes of the vortex array laser beam with a tilted plane wave) of laser beams superposed by two sub-beams, i.e., the IGe 4, 4 mode and its rotated replica [IGe 4, 4 ]T, with a power ratio τ for two sub-beams ranging from 0.5 to 1 in 6 steps.

Fig. 11.
Fig. 11.

Intensity distribution, phase distributions and the interferogram (calculation of interference fringes of the vortex array laser beam with a tilted plane wave). (a) After another 4 m of propagation behind CCD. (b) After passing through a focal lens and observing at the back focal plane of a lens with a focal length of 1.5 m.

Fig. 12.
Fig. 12.

(a) Phase errors resulting from inserting the λ/4 optical path length difference between two sub-beams in the interferometer. (b) Phase errors from function (a) ϕ2 (z), (b) ϕ3 (z), and (c) the sum of two phase function, ϕ2 (z) and ϕ3 (z). The passing IGe p, p laser beam waist sizes are indicated above each sub-figures.

Equations (15)

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I G p , m e ( r , ε ) = C [ w 0 w ( z ) ] C p m ( i ξ , ε ) C p m ( η , ε ) exp [ r 2 w 2 ( z ) ]
× exp i [ kz + { k r 2 2 R ( z ) } ( p + 1 ) ψ z ( z ) ] ,
I G p , m o ( r , ε ) = S [ w 0 w ( z ) ] S p m ( i ξ , ε ) S p m ( η , ε ) exp [ r 2 w 2 ( z ) ]
× exp i [ kz + { k r 2 2 R ( z ) } ( p + 1 ) ψ z ( z ) ] ,
U VL = I G p , p e + i × [ I G p , p e ] T ,
C p p ( i ξ , ε ) C p p ( η , ε ) = 0 ,
C p p ( η , ε ) = 0 .
C 2 n 2 n ( η , ε ) = r = 0 n A r cos 2 r η , p = 2 n and n int ,
{ ( p 2 + 1 ) ε A 1 = a A 0 , ( p 2 + 2 ) ε A 2 = p ε A 0 ( 4 a ) A 1 , ( p 2 + r + 2 ) ε A r + 2 = [ a 4 ( r + 1 ) 2 ] A r + 1 + ( r p 2 ) ε A r .
C 2 n + 1 2 n + 1 ( η , ε ) = r = 0 n A r cos ( 2 r + 1 ) η , p = 2 n + 1 and n int ,
{ ( p + 3 ) ε 2 A 1 = [ a ε 2 ( p + 1 ) 1 ] A 0 , ( p + 2 r + 3 ) ε 2 A r + 1 = [ a ( 2 r + 1 ) 2 ] A r + ( 2 r p 1 ) ε 2 A r 1 .
d 2 N d η 2 + ε sin 2 η d N d η + ( a p ε cos 2 η ) N = 0 ,
x 2 f ( z ) 2 cos 2 η y 2 f ( z ) 2 sin 2 η = 1 , ( η = ± η 1 , ± η 2 , or ± η p ) .
y 2 f ( z ) 2 cos 2 η x 2 f ( z ) 2 sin 2 η = 1 , ( η = ± η 1 , ± η 2 , or ± η p ) .
{ d ϕ 1 ( z ) dz = k d ϕ 2 ( z ) dz = k r 2 ( 1 z R 2 z 2 ) 2 ( z + z R 2 z ) 2 d ϕ 3 ( z ) dz = 1 ( 1 + z z R 2 ) z R .

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