Abstract

Negative propagation is an unusual effect concerning the local sign change in the Poynting vector components of an optical beam under free propagation. We report this effect for finite-energy Airy beams in a subwavelength nonparaxial regime. This effect is due to a coupling process between propagating and evanescent plane waves forming the beam in the spectral domain and it is demonstrated for a single TE or TM mode. This is contrary to what happens for vector Bessel beams and vector X-waves, for which a complex superposition of TE and TM modes is mandatory. We also show that evanescent waves cannot contribute to the energy flux density by themselves such that a pure evanescent Airy beam is not physically realizable. The break of the shape-preserving and diffraction-free properties of Airy beams in a nonparaxial regime is exclusively caused by the propagating waves. The negative propagation effect in subwavelength nonparaxial Airy beams opens new capabilities in optical traps and tweezers, optical detection of invisibility cloacks and selective on-chip manipulation of nanoparticles.

© 2012 OSA

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    [CrossRef]
  29. P. Vaveliuk, “Quantifying the paraxiality for laser beams from the M2-factor,” Opt. Lett.34, 340–342 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2012 (3)

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett.108, 163901 (2012).
[CrossRef] [PubMed]

Z. Chen, “Viewpoint: light bends itself into an arcs,” Phys.5, 44 (2012).
[CrossRef]

P. Vaveliuk and O. Martinez-Matos, “Physical interpretation of the paraxial estimator,” Opt. Commun.285, 4816–4820 (2012).
[CrossRef]

2011 (8)

2010 (3)

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett.35, 2082–2084 (2010).
[CrossRef] [PubMed]

M. I. Carvalho and M. Facão, “Propagation of Airy-related beams,” Opt. Express18, 21938–21949 (2010).
[CrossRef] [PubMed]

T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett.104, 197203 (2010).
[CrossRef] [PubMed]

2009 (6)

2008 (2)

2007 (8)

2005 (1)

A. Lencina and P. Vaveliuk, “Squared-field amplitude modulus and radiation intensity nonequivalence within nonlinear slabs,” Phys. Rev. E71, 056614 (2005).
[CrossRef]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264 –267 (1979).
[CrossRef]

Bagci, H.

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264 –267 (1979).
[CrossRef]

Bandres, M. A.

Baumgartl, J.

Bekenstein, R.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett.108, 163901 (2012).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264 –267 (1979).
[CrossRef]

Besieris, I. M.

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett.33, 207–209 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99, 213901 (2007).
[CrossRef]

Carvalho, M. I.

Chen, H.

Chen, Z.

Christodoulides, D. N.

Chumak, A. V.

T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett.104, 197203 (2010).
[CrossRef] [PubMed]

Ciz~már, T.

Cottrell, D. M.

Davis, J. A.

Dholakia, K.

Ding, J.

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett.33, 207–209 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99, 213901 (2007).
[CrossRef]

Facão, M.

Girard, C.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature Phys.3, 477–480 (2007).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1968).

Gutierrez-Vega, J. C.

Hazard, T. M.

Hillebrands, B.

T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett.104, 197203 (2010).
[CrossRef] [PubMed]

Janunts, N.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Yu. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
[CrossRef] [PubMed]

Kaminer, I.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett.108, 163901 (2012).
[CrossRef] [PubMed]

Kivshar, Yu. S.

W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Yu. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett.36, 1164–1166 (2011).
[CrossRef] [PubMed]

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Yu. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
[CrossRef] [PubMed]

Klein, A. E.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Yu. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
[CrossRef] [PubMed]

Kostylev, M. P.

T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett.104, 197203 (2010).
[CrossRef] [PubMed]

Lencina, A.

P. Vaveliuk, B. Ruiz, and A. Lencina, “Limits of the paraxial aproximation in laser beams,” Opt. Lett.32, 927–929 (2007).
[CrossRef] [PubMed]

A. Lencina and P. Vaveliuk, “Squared-field amplitude modulus and radiation intensity nonequivalence within nonlinear slabs,” Phys. Rev. E71, 056614 (2005).
[CrossRef]

Li, L.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett.107, 126804 (2011).
[CrossRef] [PubMed]

Li, T.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett.107, 126804 (2011).
[CrossRef] [PubMed]

Liu, W.

Liu, Y.

Lu, C.

Martinez-Matos, O.

P. Vaveliuk and O. Martinez-Matos, “Physical interpretation of the paraxial estimator,” Opt. Commun.285, 4816–4820 (2012).
[CrossRef]

P. Vaveliuk and O. Martinez-Matos, “Effect of ABCD transformations on beam paraxiality,” Opt. Express19, 25944–25953 (2011).
[CrossRef]

Mazilu, M.

McAuley, K. P.

Minovich, A.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Yu. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
[CrossRef] [PubMed]

Mintry, M. J.

Miroshnichenko, A. E.

Mitry, M. J.

Moret, M. A.

Morris, J. E.

Nemirovsky, J.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett.108, 163901 (2012).
[CrossRef] [PubMed]

Neshev, D. N.

W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Yu. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett.36, 1164–1166 (2011).
[CrossRef] [PubMed]

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Yu. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
[CrossRef] [PubMed]

Novitsky, A. V.

Novitsky, D. V.

Pertsch, T.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Yu. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
[CrossRef] [PubMed]

Quidant, R.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature Phys.3, 477–480 (2007).
[CrossRef]

Righini, M.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature Phys.3, 477–480 (2007).
[CrossRef]

Ruiz, B.

Ruiz, I.

Salandrino, A.

A. Salandrino and D. N. Christodoulides, “Viewpoint: Airy plasmons defeat diffraction on the surface,” Phys.4, 69 (2011).
[CrossRef]

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett.35, 2082–2084 (2010).
[CrossRef] [PubMed]

Salem, M. A.

Sandweg, C. W.

T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett.104, 197203 (2010).
[CrossRef] [PubMed]

Schneider, T.

T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett.104, 197203 (2010).
[CrossRef] [PubMed]

Segev, M.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett.108, 163901 (2012).
[CrossRef] [PubMed]

Serga, A. A.

T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett.104, 197203 (2010).
[CrossRef] [PubMed]

Shaarawi, A. M.

Shadrivov, I. V.

Siviloglou, G. A.

Slavin, A. N.

T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett.104, 197203 (2010).
[CrossRef] [PubMed]

Tiberkevich, V. S.

T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett.104, 197203 (2010).
[CrossRef] [PubMed]

Trudel, S.

T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett.104, 197203 (2010).
[CrossRef] [PubMed]

Vaveliuk, P.

Wang, H.-T.

Wang, S.

Wang, S. M.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett.107, 126804 (2011).
[CrossRef] [PubMed]

Wolff, S.

T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett.104, 197203 (2010).
[CrossRef] [PubMed]

Wu, B.-I.

B. Zhang and B.-I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett.103, 243901 (2009).
[CrossRef]

Yin, X.

Zebende, G. F.

Zelenina, A. S.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature Phys.3, 477–480 (2007).
[CrossRef]

Zhang, B.

B. Zhang and B.-I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett.103, 243901 (2009).
[CrossRef]

Zhang, B.-F.

Zhang, C.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett.107, 126804 (2011).
[CrossRef] [PubMed]

Zhang, P.

Zhang, X.

Zheng, Z.

Zhu, S. N.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett.107, 126804 (2011).
[CrossRef] [PubMed]

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264 –267 (1979).
[CrossRef]

Appl. Opt. (2)

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

Nature Phys. (1)

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature Phys.3, 477–480 (2007).
[CrossRef]

Opt. Commun. (1)

P. Vaveliuk and O. Martinez-Matos, “Physical interpretation of the paraxial estimator,” Opt. Commun.285, 4816–4820 (2012).
[CrossRef]

Opt. Express (6)

Opt. Lett. (10)

P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, “Plasmonic Airy beams with dynamically controlled trajectories,” Opt. Lett.36, 3191–3193 (2011).
[CrossRef] [PubMed]

D. M. Cottrell, J. A. Davis, and T. M. Hazard, “Direct generation of accelerating Airy beams using a 3/2 phase-only pattern,” Opt. Lett.34, 2634–2636 (2009).
[CrossRef] [PubMed]

A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett.34, 3430–3432 (2009).
[CrossRef] [PubMed]

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett.35, 2082–2084 (2010).
[CrossRef] [PubMed]

P. Vaveliuk, “Quantifying the paraxiality for laser beams from the M2-factor,” Opt. Lett.34, 340–342 (2009).
[CrossRef] [PubMed]

W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Yu. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett.36, 1164–1166 (2011).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett.33, 207–209 (2008).
[CrossRef] [PubMed]

P. Vaveliuk, B. Ruiz, and A. Lencina, “Limits of the paraxial aproximation in laser beams,” Opt. Lett.32, 927–929 (2007).
[CrossRef] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32, 979–981 (2007).
[CrossRef] [PubMed]

I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett.32, 2447–2449 (2007).
[CrossRef] [PubMed]

Phys. (2)

A. Salandrino and D. N. Christodoulides, “Viewpoint: Airy plasmons defeat diffraction on the surface,” Phys.4, 69 (2011).
[CrossRef]

Z. Chen, “Viewpoint: light bends itself into an arcs,” Phys.5, 44 (2012).
[CrossRef]

Phys. Rev. E (1)

A. Lencina and P. Vaveliuk, “Squared-field amplitude modulus and radiation intensity nonequivalence within nonlinear slabs,” Phys. Rev. E71, 056614 (2005).
[CrossRef]

Phys. Rev. Lett. (6)

T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett.104, 197203 (2010).
[CrossRef] [PubMed]

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Yu. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
[CrossRef] [PubMed]

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett.107, 126804 (2011).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99, 213901 (2007).
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I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett.108, 163901 (2012).
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Other (1)

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1968).

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Figures (4)

Fig. 1
Fig. 1

Map of �� vs. (x̃0, a). The points set the different configurations of Airy beams in this paraxial-nonparaxial map with the following values of the paraxial estimator. For Ai1: �� = 0.999996; For Ai2: �� = 0.92; For Ai3: �� = 0.75; For Ai4: �� = −0.58. The dashing line corresponds to �� = 0.

Fig. 2
Fig. 2

Features of a nonparaxial Airy beam for (a)–(b) configuration Ai2 and (c)–(d) configuration Ai3. (a) and (c) Energy flow density as a function of normalized spatial coordinates. (b) and (d) Profiles of both the total energy flow density and the energy flow density due to the propagating plane waves. Clearly, the contribution of evanescent waves is null for both configurations in the overall spatial range.

Fig. 3
Fig. 3

z-component of the time-averaged Poynting vector for a Airy beam in the configuration Ai4. (a) Such a magnitude as a function of normalized spatial coordinates. Sis negative at the green regions. (b) Profiles of both the total energy flow density and the energy flow density due to the propagating plane waves at z̃ = 0 and z̃ = 0.25 (inset). Clearly, both magnitudes differs due to a non-null coupling term between propagating and evanescent waves. This last term is the responsible by the negative propagation. Out of the region of influence of evanescent waves, Sand Spr are equivalent.

Fig. 4
Fig. 4

Squared-field amplitude modulus of an Airy beam for the configuration Ai4. (a) Such a magnitude as a function of normalized spatial coordinates. (b) Profiles of both the total squared-field amplitude modulus and that due to the propagating plane waves at z̃ = 0 and z̃ = 0.25 (inset). If 2π|U|2 is taken as the energy flux density, the negative propagation effect does not take place.

Equations (11)

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[ 2 / x 2 + 2 / z 2 + ε ( 2 π / λ 2 ) ] U ( x , z ) = 0 ,
U ( x ˜ , z ˜ ) = + 𝒰 ( p ˜ ; 0 ) e i 2 π z ˜ 1 p ˜ 2 e i 2 π p ˜ x ˜ d p ˜ ,
𝒰 ( p ˜ ; 0 ) = x ˜ 0 e ( a i 2 π p ˜ x ˜ 0 ) 3 .
U = U p r + U e v ,
U p r ( x ˜ , z ˜ ) = 1 1 𝒰 ( p ˜ ; 0 ) e i 2 π z ˜ 1 p ˜ 2 e i 2 π p ˜ x ˜ d p ˜ ,
U e v ( x ˜ , z ˜ ) = 1 𝒰 ( p ˜ ; 0 ) e 2 π z ˜ p ˜ 2 1 e i 2 π p ˜ x ˜ d p ˜ + 1 𝒰 ( p ˜ ; 0 ) e 2 π z ˜ p ˜ 2 1 e i 2 π p ˜ x ˜ d p ˜ .
S ( x ˜ , z ˜ ) = Im { U * U / z ˜ } .
S = S p r + S e v + S c r ,
S j = Im { U j * U j z ˜ } ; S c r = Im { U p r * U e v z ˜ + U e v * U p r z ˜ } ,
𝒫 = 1 1 / ( 32 π 2 a x ˜ 0 2 ) .
S e v ( x ˜ , z ˜ ) = 0

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