Abstract

A vectorial optical field generator (VOF-Gen) based on two reflective phase-only liquid crystal spatial light modulators enables the creation of an arbitrary optical complex field. In this work, the capabilities of the VOF-Gen in terms of manipulating the spatial distributions of phase, amplitude, and polarization are experimentally demonstrated by generating a radially polarized optical field consisted of five annular rings, the focusing properties of which are also numerically studied with vectorial diffraction theory. By carefully adjusting the relative amplitude and phase between the adjacent rings, an optical needle field with purely longitudinal polarization can be produced in the focal region of a high numerical aperture lens. The versatile method presented in this work can be easily extended to the generation of a vectorial optical field with any desired complex distributions.

© 2016 Chinese Laser Press

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References

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    [Crossref]
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    [Crossref]
  8. T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  22. Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express 23, 17701–17710 (2015).
    [Crossref]
  23. S. Tripathi and K. C. Toussaint, “Versatile generation of optical vector fields and vector beams using a noninterferometric approach,” Opt. Express 20, 10788–10795 (2012).
    [Crossref]
  24. I. Moreno, J. A. Davis, T. M. Hernandez, D. M. Cottrell, and D. Sand, “Complete polarization control of light from a liquid crystal spatial light modulator,” Opt. Express 20, 364–376 (2012).
    [Crossref]
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    [Crossref]

2016 (1)

S. Wang and Q. Zhan, “Reflection type metasurface designed for high efficiency vectorial field generation,” Sci. Rep. 6, 29626 (2016).
[Crossref]

2015 (1)

2013 (4)

W. Han, Y. Yang, W. Cheng, and Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21, 20692–20706 (2013).
[Crossref]

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[Crossref]

O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
[Crossref]

H. Guo, X. Weng, M. Jiang, Y. Zhao, G. Sui, Q. Hu, Y. Wang, and S. Zhuang, “Tight focusing of a higher-order radially polarized beam transmitting through multi-zone binary phase pupil filters,” Opt. Express 21, 5363–5372 (2013).
[Crossref]

2012 (3)

2011 (3)

W. Han, W. Cheng, and Q. Zhan, “Flattop focusing with full Poincaré beams under low numerical aperture illumination,” Opt. Lett. 36, 1605–1607 (2011).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref]

2010 (1)

2009 (3)

2008 (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

2004 (3)

2003 (1)

R. Dorn, S. Qubis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref]

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

1990 (1)

R. D. Romea and W. D. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D 42, 1807–1818 (1990).
[Crossref]

1983 (2)

J. R. Fontana and R. H. Pantell, “A high-energy, laser accelerator for electrons using the inverse Cherenkov effect,” J. Appl. Phys. 54, 4285–4288 (1983).
[Crossref]

R. Fontana and R. H. Pantell, “A high-energy laser accelerator for electrons using the inverse Cerenkov effect,” J. Appl. Phys. 54, 4285–4288 (1983).
[Crossref]

Abeysinghe, D. C.

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9, 4320–4325 (2009).
[Crossref]

Aieta, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

Antoniou, N.

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[Crossref]

April, A.

Bai, J.

Barnett, S.

Betzig, E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

Capasso, F.

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

Chen, W.

J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express 18, 21965–21972 (2010).
[Crossref]

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9, 4320–4325 (2009).
[Crossref]

Chen, Z.

Cheng, W.

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Choudhury, A.

Cottrell, D. M.

Courtial, J.

Davidson, M. W.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref]

Davis, J. A.

Dehez, H.

Ding, J.

Dorn, R.

R. Dorn, S. Qubis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref]

Ferrari, A. C.

O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
[Crossref]

Fontana, J. R.

J. R. Fontana and R. H. Pantell, “A high-energy, laser accelerator for electrons using the inverse Cherenkov effect,” J. Appl. Phys. 54, 4285–4288 (1983).
[Crossref]

Fontana, R.

R. Fontana and R. H. Pantell, “A high-energy laser accelerator for electrons using the inverse Cerenkov effect,” J. Appl. Phys. 54, 4285–4288 (1983).
[Crossref]

Franke-Arnold, S.

Gaburro, Z.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

Galbraith, C. G.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref]

Galbraith, J. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref]

Gao, L.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref]

Genevet, P.

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

Gibson, G.

Gucciardi, P. G.

O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
[Crossref]

Guo, H.

Han, W.

Hernandez, T. M.

Hu, Q.

Jiang, M.

Jones, P. H.

O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
[Crossref]

Kats, M. A.

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

Kimura, W. D.

R. D. Romea and W. D. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D 42, 1807–1818 (1990).
[Crossref]

Leuchs, G.

R. Dorn, S. Qubis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref]

Lin, J.

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[Crossref]

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Maragò, O. M.

O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
[Crossref]

Milkie, D. E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref]

Moreno, I.

Nelson, R. L.

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9, 4320–4325 (2009).
[Crossref]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

Padgett, M.

Pantell, R. H.

R. Fontana and R. H. Pantell, “A high-energy laser accelerator for electrons using the inverse Cerenkov effect,” J. Appl. Phys. 54, 4285–4288 (1983).
[Crossref]

J. R. Fontana and R. H. Pantell, “A high-energy, laser accelerator for electrons using the inverse Cherenkov effect,” J. Appl. Phys. 54, 4285–4288 (1983).
[Crossref]

Pas’ko, V.

Piché, M.

Planchon, T. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref]

Qian, B.

Qubis, S.

R. Dorn, S. Qubis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref]

Romea, R. D.

R. D. Romea and W. D. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D 42, 1807–1818 (1990).
[Crossref]

Sand, D.

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Sheppard, C. J. R.

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Sui, G.

Tetienne, J. P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

Toussaint, K. C.

Tripathi, S.

Vasnetsov, M.

Volpe, G.

O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
[Crossref]

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Wang, J.

Wang, S.

S. Wang and Q. Zhan, “Reflection type metasurface designed for high efficiency vectorial field generation,” Sci. Rep. 6, 29626 (2016).
[Crossref]

Wang, Y.

Weng, X.

Yang, Y.

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

Yu, N.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

Zeng, T.

Zhan, Q.

Zhang, Y.

Zhao, Y.

Zhuang, S.

Adv. Opt. Photon. (1)

Appl. Opt. (1)

J. Appl. Phys. (2)

J. R. Fontana and R. H. Pantell, “A high-energy, laser accelerator for electrons using the inverse Cherenkov effect,” J. Appl. Phys. 54, 4285–4288 (1983).
[Crossref]

R. Fontana and R. H. Pantell, “A high-energy laser accelerator for electrons using the inverse Cerenkov effect,” J. Appl. Phys. 54, 4285–4288 (1983).
[Crossref]

Nano Lett. (2)

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[Crossref]

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9, 4320–4325 (2009).
[Crossref]

Nat. Methods (1)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[Crossref]

Nat. Nanotechnol. (1)

O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
[Crossref]

Nat. Photonics (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Opt. Express (10)

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
[Crossref]

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, 5448–5456 (2004).
[Crossref]

Y. Zhang and J. Bai, “Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams,” Opt. Express 17, 3698–3706 (2009).
[Crossref]

J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express 18, 21965–21972 (2010).
[Crossref]

I. Moreno, J. A. Davis, T. M. Hernandez, D. M. Cottrell, and D. Sand, “Complete polarization control of light from a liquid crystal spatial light modulator,” Opt. Express 20, 364–376 (2012).
[Crossref]

S. Tripathi and K. C. Toussaint, “Versatile generation of optical vector fields and vector beams using a noninterferometric approach,” Opt. Express 20, 10788–10795 (2012).
[Crossref]

H. Dehez, A. April, and M. Piché, “Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent,” Opt. Express 20, 14891–14905 (2012).
[Crossref]

H. Guo, X. Weng, M. Jiang, Y. Zhao, G. Sui, Q. Hu, Y. Wang, and S. Zhuang, “Tight focusing of a higher-order radially polarized beam transmitting through multi-zone binary phase pupil filters,” Opt. Express 21, 5363–5372 (2013).
[Crossref]

W. Han, Y. Yang, W. Cheng, and Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21, 20692–20706 (2013).
[Crossref]

Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express 23, 17701–17710 (2015).
[Crossref]

Opt. Lett. (1)

Phys. Rev. D (1)

R. D. Romea and W. D. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D 42, 1807–1818 (1990).
[Crossref]

Phys. Rev. Lett. (2)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

R. Dorn, S. Qubis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref]

Sci. Rep. (1)

S. Wang and Q. Zhan, “Reflection type metasurface designed for high efficiency vectorial field generation,” Sci. Rep. 6, 29626 (2016).
[Crossref]

Science (1)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

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Figures (7)

Fig. 1.
Fig. 1. VOF-Gen. (a) Schematic diagram of experimental setup. P, polarizer; L, lens; M, mirror; QWP, quarter-wave plate; SF, spatial filter. (b) Illustration of the PR setup and the effective rotation of the QWP fast axis for the incident (upper) and the reflected (lower) beams due to the mirror imaging of the laboratory coordinates in dashed lines.
Fig. 2.
Fig. 2. Structure and parameters of the desired optical field.
Fig. 3.
Fig. 3. Overall phase patterns displayed on VOF-Gen.
Fig. 4.
Fig. 4. Intensity distribution for desired optical field in the pupil plane: (a) theoretical and (b) experimental results. (c) Averaged amplitude distribution for experimental and theoretical results.
Fig. 5.
Fig. 5. Intensity distribution for five ring structured radially polarized fields after a polarizer with transmission axis orientation indicated by the arrows at 0°, 45°, 90°, and 135°. The upper row shows the theoretical results, and the lower row is for the experimental results.
Fig. 6.
Fig. 6. (a) Interference setup to verify the π phase shift that consists of a beamsplitter (BS) and two mirrors. (b) The required phase pattern loaded on VOF-Gen for π shift verification. (c) Interference fringes patterns of π phase jump.
Fig. 7.
Fig. 7. Intensity distributions and the electric field distributions in the focal region of a high NA (NA=0.95) objective lens (the polarization direction for the ONF is represented by arrows). (a) ONF calculated with the theoretical amplitude values; (b) focal field calculated with the experimentally obtained amplitude values; (c) zoomed-in picture of the area enclosed within the red dashed box in (a); (d) zoomed-in picture of the area enclosed within the red dashed box in (b).

Equations (1)

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JSLM4(x,y)=A0(x,y)sin(δ2(x,y)2)ei(δ2(x,y)2+δ3(x,y)2+π)(cos(δ3(x,y)2+π2)eiδ4(x,y)sin(δ3(x,y)2+π2)),

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