Abstract

Laser beams can be made to form bright and dark intensity helices of light. Such helices have a pitch length on the order of a wavelength and may have applications in lithography and the manipulation of particles through optical forces. The formation of bright helices is more strongly constrained by optical resolution limits than that of dark helices, corresponding scaling laws are derived and their relevance for photo-lithography pointed out. It is shown how to arrange dark helices on a grid in massively parallel fashion in order to create handed materials using photo-lithographic techniques.

© 2012 OSA

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    [CrossRef]
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    [CrossRef]

2011 (2)

2009 (4)

S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17, 9818–9827 (2009).
[CrossRef] [PubMed]

I. V. Semchenko, S. A. Khakhomov, and S. A. Tretyakov, “Chiral metamaterial with unit negative refraction index,” Eur. Phys. J. Appl. Phys. 46, 032607 (2009).
[CrossRef]

X.-L. Qi and S.-C. Zhang, “Field-induced gap and quantized charge pumping in a nanoscale helical wire,” Phys. Rev. B 79, 235442 (2009).
[CrossRef]

K. Volke-Sepúlveda and R. Jáuregui, “All-optical 3D atomic loops generated with Bessel light fields,” J. Phys. B: At. Mol. Phys. 42, 085303 (2009).
[CrossRef]

2007 (5)

M. Bhattacharya, “Lattice with a twist: Helical waveguides for ultracold matter,” Opt. Commun. 279, 219–222 (2007).
[CrossRef]

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
[CrossRef]

A. Sihvola, “Metamaterials in electromagnetics,” Metamaterials 1, 2–11 (2007).
[CrossRef]

P. Exner and M. Fraas, “A remark on helical waveguides,” Phys. Lett. A 369, 393–399 (2007).
[CrossRef]

S. W. Hell, “Far-Field Optical Nanoscopy,” Science 316, 1153–1158 (2007).
[CrossRef] [PubMed]

2006 (3)

2005 (2)

O. Steuernagel, “Equivalence between focused paraxial beams and the quantum harmonic oscillator,” Am. J. Phys. 73, 625–629 (2005).
[CrossRef]

Y. B. Gaididei, P. L. Christiansen, P. G. Kevrekidis, H. Büttner, and A. R. Bishop, “Localization of nonlinear excitations in curved waveguides,” New J. Phys. 7, 52–52 (2005).
[CrossRef]

2004 (3)

J. Lekner, “LETTER TO THE EDITOR: Helical light pulses,” J. Opt. A: Pure Appl. Opt. 6, L29–L32 (2004).
[CrossRef]

M. Padgett, J. Courtial, and L. Allen, “Light’s Orbital Angular Momentum,” Phys. Today 57, 35–40 (2004).
[CrossRef]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: Knotted threads of darkness,” Nature 432, 165–165 (2004).
[CrossRef] [PubMed]

2003 (1)

D. Meschede and H. Metcalf, “Atomic nanofabrication: atomic deposition and lithography by laser and magnetic forces,” J. Phys. D: Appl. Phys. 36, R17–R38 (2003).
[CrossRef]

2002 (3)

K. Dholakia, G. C. Spalding, and M. MacDonald, “Optical tweezers: The next generation,” Phys. World 15, 31–35 (2002).

M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21–28 (2002).
[CrossRef]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101–1103 (2002).
[CrossRef] [PubMed]

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] [PubMed]

S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278–281 (2001).
[CrossRef] [PubMed]

S. Kawata, H.-B. Sun, T. Tanaka, and K. Takada, “Finer features for functional microdevices,” Nature 412, 697–698 (2001).
[CrossRef] [PubMed]

2000 (1)

M. Padgett and L. Allen, “Light with a twist in its tail,” Cont. Phys. 41, 275–285 (2000).
[CrossRef]

1999 (1)

J. M. Vaughan, “Interferometry, atoms and light scattering: one hundred years of optics,” J. Opt. A: Pure Appl. Opt. 1, 750–768 (1999).
[CrossRef]

1998 (1)

S. Chu, “Nobel Lecture: The manipulation of neutral particles,” Rev. Mod. Phys. 70, 685–706 (1998).
[CrossRef]

1994 (1)

M. Harris, C. A. Hill, and J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

1974 (1)

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

Allen, L.

M. Padgett, J. Courtial, and L. Allen, “Light’s Orbital Angular Momentum,” Phys. Today 57, 35–40 (2004).
[CrossRef]

M. Padgett and L. Allen, “Light with a twist in its tail,” Cont. Phys. 41, 275–285 (2000).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Alt, W.

S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278–281 (2001).
[CrossRef] [PubMed]

Arlt, J.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101–1103 (2002).
[CrossRef] [PubMed]

M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21–28 (2002).
[CrossRef]

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] [PubMed]

Baumann, S. M.

Becker, J.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Berry, M. V.

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

Bhattacharya, M.

M. Bhattacharya, “Lattice with a twist: Helical waveguides for ultracold matter,” Opt. Commun. 279, 219–222 (2007).
[CrossRef]

Bishop, A. R.

Y. B. Gaididei, P. L. Christiansen, P. G. Kevrekidis, H. Büttner, and A. R. Bishop, “Localization of nonlinear excitations in curved waveguides,” New J. Phys. 7, 52–52 (2005).
[CrossRef]

Boguslawski, M.

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] [PubMed]

Büttner, H.

Y. B. Gaididei, P. L. Christiansen, P. G. Kevrekidis, H. Büttner, and A. R. Bishop, “Localization of nonlinear excitations in curved waveguides,” New J. Phys. 7, 52–52 (2005).
[CrossRef]

Christiansen, P. L.

Y. B. Gaididei, P. L. Christiansen, P. G. Kevrekidis, H. Büttner, and A. R. Bishop, “Localization of nonlinear excitations in curved waveguides,” New J. Phys. 7, 52–52 (2005).
[CrossRef]

Chu, S.

S. Chu, “Nobel Lecture: The manipulation of neutral particles,” Rev. Mod. Phys. 70, 685–706 (1998).
[CrossRef]

Courtial, J.

M. Padgett, J. Courtial, and L. Allen, “Light’s Orbital Angular Momentum,” Phys. Today 57, 35–40 (2004).
[CrossRef]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: Knotted threads of darkness,” Nature 432, 165–165 (2004).
[CrossRef] [PubMed]

Dennis, M. R.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: Knotted threads of darkness,” Nature 432, 165–165 (2004).
[CrossRef] [PubMed]

Denz, C.

Dholakia, K.

K. Dholakia, G. C. Spalding, and M. MacDonald, “Optical tweezers: The next generation,” Phys. World 15, 31–35 (2002).

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101–1103 (2002).
[CrossRef] [PubMed]

M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21–28 (2002).
[CrossRef]

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] [PubMed]

Exner, P.

P. Exner and M. Fraas, “A remark on helical waveguides,” Phys. Lett. A 369, 393–399 (2007).
[CrossRef]

Fraas, M.

P. Exner and M. Fraas, “A remark on helical waveguides,” Phys. Lett. A 369, 393–399 (2007).
[CrossRef]

Gaididei, Y. B.

Y. B. Gaididei, P. L. Christiansen, P. G. Kevrekidis, H. Büttner, and A. R. Bishop, “Localization of nonlinear excitations in curved waveguides,” New J. Phys. 7, 52–52 (2005).
[CrossRef]

Galvez, E. J.

Gomer, V.

S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278–281 (2001).
[CrossRef] [PubMed]

Hahn, J. W.

Hamazaki, J.

Harris, M.

M. Harris, C. A. Hill, and J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Haus, H. A.

H. A. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, Heidelberg, 2000).

Hell, S. W.

S. W. Hell, “Far-Field Optical Nanoscopy,” Science 316, 1153–1158 (2007).
[CrossRef] [PubMed]

Hill, C. A.

M. Harris, C. A. Hill, and J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Jang, J.

Jáuregui, R.

K. Volke-Sepúlveda and R. Jáuregui, “All-optical 3D atomic loops generated with Bessel light fields,” J. Phys. B: At. Mol. Phys. 42, 085303 (2009).
[CrossRef]

Jung, H.

Kalb, D. M.

Kawata, S.

S. Kawata, H.-B. Sun, T. Tanaka, and K. Takada, “Finer features for functional microdevices,” Nature 412, 697–698 (2001).
[CrossRef] [PubMed]

Keen, S.

Kevrekidis, P. G.

Y. B. Gaididei, P. L. Christiansen, P. G. Kevrekidis, H. Büttner, and A. R. Bishop, “Localization of nonlinear excitations in curved waveguides,” New J. Phys. 7, 52–52 (2005).
[CrossRef]

Khakhomov, S. A.

I. V. Semchenko, S. A. Khakhomov, and S. A. Tretyakov, “Chiral metamaterial with unit negative refraction index,” Eur. Phys. J. Appl. Phys. 46, 032607 (2009).
[CrossRef]

Kim, S.

Kim, Y.

Kuhr, S.

S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278–281 (2001).
[CrossRef] [PubMed]

Leach, J.

Lee, J. Y.

Lekner, J.

J. Lekner, “LETTER TO THE EDITOR: Helical light pulses,” J. Opt. A: Pure Appl. Opt. 6, L29–L32 (2004).
[CrossRef]

Love, G. D.

MacDonald, M.

K. Dholakia, G. C. Spalding, and M. MacDonald, “Optical tweezers: The next generation,” Phys. World 15, 31–35 (2002).

MacDonald, M. P.

M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21–28 (2002).
[CrossRef]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101–1103 (2002).
[CrossRef] [PubMed]

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] [PubMed]

MacMillan, L. H.

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

Marrucci, L.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

Meschede, D.

D. Meschede and H. Metcalf, “Atomic nanofabrication: atomic deposition and lithography by laser and magnetic forces,” J. Phys. D: Appl. Phys. 36, R17–R38 (2003).
[CrossRef]

S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278–281 (2001).
[CrossRef] [PubMed]

Metcalf, H.

D. Meschede and H. Metcalf, “Atomic nanofabrication: atomic deposition and lithography by laser and magnetic forces,” J. Phys. D: Appl. Phys. 36, R17–R38 (2003).
[CrossRef]

Mineta, Y.

Morita, R.

Müller, M.

S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278–281 (2001).
[CrossRef] [PubMed]

Nye, J. F.

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

Oka, K.

Padgett, M.

M. Padgett, J. Courtial, and L. Allen, “Light’s Orbital Angular Momentum,” Phys. Today 57, 35–40 (2004).
[CrossRef]

M. Padgett and L. Allen, “Light with a twist in its tail,” Cont. Phys. 41, 275–285 (2000).
[CrossRef]

Padgett, M. J.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

Paterson, L.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101–1103 (2002).
[CrossRef] [PubMed]

M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21–28 (2002).
[CrossRef]

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] [PubMed]

Qi, X.-L.

X.-L. Qi and S.-C. Zhang, “Field-induced gap and quantized charge pumping in a nanoscale helical wire,” Phys. Rev. B 79, 235442 (2009).
[CrossRef]

Rose, P.

Saunter, C.

Schrader, D.

S. Kuhr, W. Alt, D. Schrader, M. Müller, V. Gomer, and D. Meschede, “Deterministic Delivery of a Single Atom,” Science 293, 278–281 (2001).
[CrossRef] [PubMed]

Semchenko, I. V.

I. V. Semchenko, S. A. Khakhomov, and S. A. Tretyakov, “Chiral metamaterial with unit negative refraction index,” Eur. Phys. J. Appl. Phys. 46, 032607 (2009).
[CrossRef]

Sibbett, W.

M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. 201, 21–28 (2002).
[CrossRef]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101–1103 (2002).
[CrossRef] [PubMed]

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] [PubMed]

Sihvola, A.

A. Sihvola, “Metamaterials in electromagnetics,” Metamaterials 1, 2–11 (2007).
[CrossRef]

Spalding, G. C.

K. Dholakia, G. C. Spalding, and M. MacDonald, “Optical tweezers: The next generation,” Phys. World 15, 31–35 (2002).

Spreeuw, R. J. C.

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M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101–1103 (2002).
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M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
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S. Kawata, H.-B. Sun, T. Tanaka, and K. Takada, “Finer features for functional microdevices,” Nature 412, 697–698 (2001).
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Figures (5)

Fig. 1
Fig. 1

(a) illustrates the ‘pancake’ scenario explaining that helices form with pitch heights which are integer multiples of λ/2. (b) shows the superposition 1 2 u 0 , 0 ( x , y , z ) + u 0 , 1 ( x , y , z ) yielding a single bright helix (red line) enveloping a single dark helix (black line) with the minimal pitch length λ/2 [colored mesh was numerically determined as the locations with 90% of peak intensity]. (c) shows the magnitude |E| of its focal field distribution illustrating quadratic variation of |E| around the maximum and linear variation around the minimum. In (b) and (c) the x- and y-axes are given in units of focal beam radius w0, z-axis in units of λ, in (c) the peak intensity is normalized to unity.

Fig. 2
Fig. 2

Superposition u 0 , 2 ( x , y , z ) + 1 2 u 1 , 0 ( x , y , z ) yields a pair of dark and bright helices with pitch λ (a); its focal intensity distribution (b) [all units as in Fig. 1, note that compared to Fig. 1 the helices’ orientation is reversed].

Fig. 3
Fig. 3

The frequency anisotropy A = ωx/ωη = |xE|/|ηE| of the transversal trapping potential of dark helices of a single-helix beam ∑(C). The anisotropy A < 1, A ≈ 1 if one uses small helix radii (C small); qualitatively the same behaviour was seen for bright helices in reference [14] [zR measured in units of λ].

Fig. 4
Fig. 4

The focal intensity distribution of superposition u0,2(x, y, z) + u0,0(x, y, −z) (a) and its logarithm for superposition u 0 , 2 ( x , y , z ) + 1 10 u 0 , 0 ( x , y , z ) (b) demonstrate that narrowly wound bright helices self-overlap and become ill defined whereas dark helices remain distinguishable [coordinate axes scaled in units of beam width w0].

Fig. 5
Fig. 5

A set of parallel equal Gaussian beams u0,0(xXG, yYG, z) travelling along the z-axis and with their beam axes centered on a set of grid points {(XG, YG, 0)} of a hexagonal lattice is arranged into three sub-lattices with different polarization orientations (beam centers and respective beam polarizations indicated by arrows in panel (a); nearest neighbour distance 2.3w0). Despite its tight packing this sub-lattice arrangement avoids destructive interference between beams of equal polarization and yields a roughly uniformly bright background, see panels (b) and (c). The focal intensity distribution of a single dark helix created from the superposition ( x , y , z ; 1 8 ) of Eq. (5) is displayed in panel (d) together with an artist’s impression of the location of its dark helix as a black line. Such dark helices can be inserted into the bright background without losing their contrast, (b) and (c), for details see section 5. Massively parallel or sparse implementation of dark helices with good contrast is possible. All position coordinate axes are scaled in units of beam width w0.

Equations (15)

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U p , l ( z R , λ ; r , ϕ , z ) = 2 ( 1 + δ 0 , l ) π p ! ( p + | l | ) ! × ( 2 r w ( z ) ) | l | L p , | l | ( 2 r 2 w ( z ) 2 ) ( e i l ϕ w ( z ) ) × exp [ r 2 w ( z ) 2 i ( 2 p + | l | + 1 ) ζ ( z ) + i k r 2 2 ρ ( z ) ] .
E ( x , y , z ; t ) = E ( x , y , z ; t ) P = [ C u p , l ( x , y , z ) + C u p , l ( x , y , z ) ] e i ω t P ,
u p , l ( x , y , z ) = U p , l ( x , y , z ) e i k z .
z l l = ( l l ) 2 π / ( 2 k ) = λ ( l l ) / 2 .
( x , y , z ; C ) = C u 0 , 0 ( x , y , z ) + u 0 , 1 ( x , y , z )
( x ( z ) + i y ( z ) ) = ( x 0 + i y 0 ) ( 1 + z 2 / z R 2 ) exp ( i χ ( z ) )
with the phase χ ( z ) = ( 2 ( 1 + p + p ) + | l | + | l | ) ζ ( z ) + k r 2 ρ ( z ) + 2 k z .
x E = 2 π exp ( C 2 / 2 ) / ( z R λ ) ,
y E = i x E ,
and z E = C ( 3 λ + C 2 λ + 4 z R π ) / 2 π z R λ y E ;
α = arctan ( y E z E ) = arctan ( 2 π z R λ C ( 3 λ + C 2 λ + 4 z R π ) ) .
V ( x , η ) | x E | 2 x 2 + | η E | 2 η 2 , where η E = η ( E ) with
η E = i x E 2 z R π + 9 C 2 λ 6 C 4 λ 24 C 2 π z R + c c 6 λ + 8 C 4 π z R + 16 C 2 π 2 z R 2 / λ 2 z R π
| δ E M ( δ r ) | = | E ( r M ) | E ( r M + δ r ) | κ | 2 ( δ r ) 2 + O ( ( δ r ) 3 ) .
| δ E m ( δ r ) | | E ( r m + δ r ) | 0 | γ | | δ r | + O ( ( δ r ) 2 ) .

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