Abstract

Experimental measurements of the nonplanar phase of scalar optical beams is achieved using polarization singularities and Stokes parameters. The product of beam intensity and its phase gradient is calculated to get the Poynting vector distribution in the beam cross section. Using these we propose a scheme and experimentally obtain generic fundamental spiral, node, and saddle topological structures in the Poynting vector field.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Hesselink, Y. Levy, and Y. Lavin, IEEE Trans. Vis. Comp. Graphics 3, 1 (1997).
    [CrossRef]
  2. R. S. Gallagher, Computer Visualization: Graphics Techniques for Engineering and Scientific Analysis (CRC, 1994).
  3. V. V. Isaeva, N. V. Kasyanov, and E. V. Presnov, BioSystems 109, 280 (2012).
    [CrossRef]
  4. C. Lamberti, A. Sarti, and F. Bertucci, Comp. Cardiol. 24, 227 (1997).
  5. M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. Aditi Simha, “Soft active matter,” arxiv.org:1207.2929 (2012).
  6. M. V. Berry, in Physics of Defects, Les Houches Lecture Series Session XXXV, R. Balian, M. Kliman, and J.-P. Poirier, eds. (North-Holland, 1981), p. 453.
  7. J. F. Nye, in Physics of Defects, Les Houches Lecture Series Session XXXV, R. Balian, M. Kliman, and J.-P. Poirier, eds. (North-Holland, 1981), p. 545.
  8. C. Nash and S. Sen, Topology and Geometry for Physicists (Academic, 1983).
  9. M. Monastyrsky, Topology of Gauge Fields and Condensed Matter (Plenum, 1993).
  10. B. G. Chen, P. J. Ackerman, G. P. Alexander, R. D. Kamien, and I. I. Smalyukh, Phys. Rev. Lett. 110, 237801 (2013).
    [CrossRef]
  11. R. M. Kiehn, Proc. SPIE 5866, 192 (2005).
    [CrossRef]
  12. G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector–vortex beams: a classical analogy of entanglement,” to be presented at Frontiers in Optics, Orlando, Florida, October6–13, 2013.
  13. I. Mokhun, in Optical Correlation Techniques and Applications, O. V. Angelsky, ed. (SPIE, 2007), Chap. 1.
  14. M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
    [CrossRef]
  15. I. Freund, M. Soskin, and A. Mokhun, Opt. Commun. 208, 223 (2002).
    [CrossRef]
  16. M. Soskin, V. Denisenko, and R. Egorov, J. Opt. A 6, S281 (2004).
    [CrossRef]
  17. A. V. Novitsky and L. M. Barkovsky, Phys. Rev. A 79, 033821 (2009).
    [CrossRef]
  18. A. Ya. Bekshaev and M. S. Soskin, Opt. Commum. 271, 332 (2007).
    [CrossRef]
  19. A. Ya. Bekshaev, K. Y. Bliokh, and M. S. Soskin, J. Opt. 13, 053001 (2011).
    [CrossRef]
  20. J. Leach, S. Keen, and M. J. Padgett, Opt. Express 14, 11919 (2006).
    [CrossRef]
  21. M. Yeganeh, S. Rasouli, M. Dashti, S. Slussarenko, E. Santamato, and E. Karimi, Opt. Lett. 38, 887 (2013).
    [CrossRef]
  22. Y. Miyamoto, A. Wada, T. Yonemura, and M. Takeda, J. Opt. 15, 044002 (2013).
    [CrossRef]
  23. V. Kumar and N. Viswanathan, Second International Conference on Optical Angular Momentum, abstract book (2013), p. 38.
  24. V. G. Denisenko, A. Minovich, A. S. Desyatnikov, W. Krolikowski, M. S. Soskin, and Y. S. Kivshar, Opt. Lett. 33, 89 (2008).
    [CrossRef]
  25. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  26. J. Nye, Natural Focusing and Fine Structure of Light (IoP, 1999).
  27. J. F. Nye, Proc. R. Soc. London 389, 279 (1983).
    [CrossRef]
  28. M. V. Berry, J. Opt. A 11, 094001 (2009).
    [CrossRef]
  29. M. V. Berry, J. Phys. A 38, L745 (2005).
    [CrossRef]

2013 (3)

B. G. Chen, P. J. Ackerman, G. P. Alexander, R. D. Kamien, and I. I. Smalyukh, Phys. Rev. Lett. 110, 237801 (2013).
[CrossRef]

M. Yeganeh, S. Rasouli, M. Dashti, S. Slussarenko, E. Santamato, and E. Karimi, Opt. Lett. 38, 887 (2013).
[CrossRef]

Y. Miyamoto, A. Wada, T. Yonemura, and M. Takeda, J. Opt. 15, 044002 (2013).
[CrossRef]

2012 (1)

V. V. Isaeva, N. V. Kasyanov, and E. V. Presnov, BioSystems 109, 280 (2012).
[CrossRef]

2011 (1)

A. Ya. Bekshaev, K. Y. Bliokh, and M. S. Soskin, J. Opt. 13, 053001 (2011).
[CrossRef]

2009 (3)

A. V. Novitsky and L. M. Barkovsky, Phys. Rev. A 79, 033821 (2009).
[CrossRef]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
[CrossRef]

M. V. Berry, J. Opt. A 11, 094001 (2009).
[CrossRef]

2008 (1)

2007 (1)

A. Ya. Bekshaev and M. S. Soskin, Opt. Commum. 271, 332 (2007).
[CrossRef]

2006 (1)

2005 (2)

R. M. Kiehn, Proc. SPIE 5866, 192 (2005).
[CrossRef]

M. V. Berry, J. Phys. A 38, L745 (2005).
[CrossRef]

2004 (1)

M. Soskin, V. Denisenko, and R. Egorov, J. Opt. A 6, S281 (2004).
[CrossRef]

2002 (1)

I. Freund, M. Soskin, and A. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

1997 (2)

L. Hesselink, Y. Levy, and Y. Lavin, IEEE Trans. Vis. Comp. Graphics 3, 1 (1997).
[CrossRef]

C. Lamberti, A. Sarti, and F. Bertucci, Comp. Cardiol. 24, 227 (1997).

1983 (1)

J. F. Nye, Proc. R. Soc. London 389, 279 (1983).
[CrossRef]

Ackerman, P. J.

B. G. Chen, P. J. Ackerman, G. P. Alexander, R. D. Kamien, and I. I. Smalyukh, Phys. Rev. Lett. 110, 237801 (2013).
[CrossRef]

Aditi Simha, R.

M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. Aditi Simha, “Soft active matter,” arxiv.org:1207.2929 (2012).

Alexander, G. P.

B. G. Chen, P. J. Ackerman, G. P. Alexander, R. D. Kamien, and I. I. Smalyukh, Phys. Rev. Lett. 110, 237801 (2013).
[CrossRef]

Alfano, R. R.

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector–vortex beams: a classical analogy of entanglement,” to be presented at Frontiers in Optics, Orlando, Florida, October6–13, 2013.

Barkovsky, L. M.

A. V. Novitsky and L. M. Barkovsky, Phys. Rev. A 79, 033821 (2009).
[CrossRef]

Bekshaev, A. Ya.

A. Ya. Bekshaev, K. Y. Bliokh, and M. S. Soskin, J. Opt. 13, 053001 (2011).
[CrossRef]

A. Ya. Bekshaev and M. S. Soskin, Opt. Commum. 271, 332 (2007).
[CrossRef]

Berry, M. V.

M. V. Berry, J. Opt. A 11, 094001 (2009).
[CrossRef]

M. V. Berry, J. Phys. A 38, L745 (2005).
[CrossRef]

M. V. Berry, in Physics of Defects, Les Houches Lecture Series Session XXXV, R. Balian, M. Kliman, and J.-P. Poirier, eds. (North-Holland, 1981), p. 453.

Bertucci, F.

C. Lamberti, A. Sarti, and F. Bertucci, Comp. Cardiol. 24, 227 (1997).

Bliokh, K. Y.

A. Ya. Bekshaev, K. Y. Bliokh, and M. S. Soskin, J. Opt. 13, 053001 (2011).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Chen, B. G.

B. G. Chen, P. J. Ackerman, G. P. Alexander, R. D. Kamien, and I. I. Smalyukh, Phys. Rev. Lett. 110, 237801 (2013).
[CrossRef]

Dashti, M.

Denisenko, V.

M. Soskin, V. Denisenko, and R. Egorov, J. Opt. A 6, S281 (2004).
[CrossRef]

Denisenko, V. G.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
[CrossRef]

Desyatnikov, A. S.

Egorov, R.

M. Soskin, V. Denisenko, and R. Egorov, J. Opt. A 6, S281 (2004).
[CrossRef]

Freund, I.

I. Freund, M. Soskin, and A. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

Gallagher, R. S.

R. S. Gallagher, Computer Visualization: Graphics Techniques for Engineering and Scientific Analysis (CRC, 1994).

Hesselink, L.

L. Hesselink, Y. Levy, and Y. Lavin, IEEE Trans. Vis. Comp. Graphics 3, 1 (1997).
[CrossRef]

Isaeva, V. V.

V. V. Isaeva, N. V. Kasyanov, and E. V. Presnov, BioSystems 109, 280 (2012).
[CrossRef]

Joanny, J.-F.

M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. Aditi Simha, “Soft active matter,” arxiv.org:1207.2929 (2012).

Kamien, R. D.

B. G. Chen, P. J. Ackerman, G. P. Alexander, R. D. Kamien, and I. I. Smalyukh, Phys. Rev. Lett. 110, 237801 (2013).
[CrossRef]

Karimi, E.

M. Yeganeh, S. Rasouli, M. Dashti, S. Slussarenko, E. Santamato, and E. Karimi, Opt. Lett. 38, 887 (2013).
[CrossRef]

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector–vortex beams: a classical analogy of entanglement,” to be presented at Frontiers in Optics, Orlando, Florida, October6–13, 2013.

Kasyanov, N. V.

V. V. Isaeva, N. V. Kasyanov, and E. V. Presnov, BioSystems 109, 280 (2012).
[CrossRef]

Keen, S.

Kiehn, R. M.

R. M. Kiehn, Proc. SPIE 5866, 192 (2005).
[CrossRef]

Kivshar, Y. S.

Krolikowski, W.

Kumar, V.

V. Kumar and N. Viswanathan, Second International Conference on Optical Angular Momentum, abstract book (2013), p. 38.

Lamberti, C.

C. Lamberti, A. Sarti, and F. Bertucci, Comp. Cardiol. 24, 227 (1997).

Lavin, Y.

L. Hesselink, Y. Levy, and Y. Lavin, IEEE Trans. Vis. Comp. Graphics 3, 1 (1997).
[CrossRef]

Leach, J.

Levy, Y.

L. Hesselink, Y. Levy, and Y. Lavin, IEEE Trans. Vis. Comp. Graphics 3, 1 (1997).
[CrossRef]

Liverpool, T. B.

M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. Aditi Simha, “Soft active matter,” arxiv.org:1207.2929 (2012).

Marchetti, M. C.

M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. Aditi Simha, “Soft active matter,” arxiv.org:1207.2929 (2012).

Marrucci, L.

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector–vortex beams: a classical analogy of entanglement,” to be presented at Frontiers in Optics, Orlando, Florida, October6–13, 2013.

Milione, G.

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector–vortex beams: a classical analogy of entanglement,” to be presented at Frontiers in Optics, Orlando, Florida, October6–13, 2013.

Minovich, A.

Miyamoto, Y.

Y. Miyamoto, A. Wada, T. Yonemura, and M. Takeda, J. Opt. 15, 044002 (2013).
[CrossRef]

Mokhun, A.

I. Freund, M. Soskin, and A. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

Mokhun, I.

I. Mokhun, in Optical Correlation Techniques and Applications, O. V. Angelsky, ed. (SPIE, 2007), Chap. 1.

Monastyrsky, M.

M. Monastyrsky, Topology of Gauge Fields and Condensed Matter (Plenum, 1993).

Nash, C.

C. Nash and S. Sen, Topology and Geometry for Physicists (Academic, 1983).

Nguyen, T. A.

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector–vortex beams: a classical analogy of entanglement,” to be presented at Frontiers in Optics, Orlando, Florida, October6–13, 2013.

Nolan, D. A.

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector–vortex beams: a classical analogy of entanglement,” to be presented at Frontiers in Optics, Orlando, Florida, October6–13, 2013.

Novitsky, A. V.

A. V. Novitsky and L. M. Barkovsky, Phys. Rev. A 79, 033821 (2009).
[CrossRef]

Nye, J.

J. Nye, Natural Focusing and Fine Structure of Light (IoP, 1999).

Nye, J. F.

J. F. Nye, Proc. R. Soc. London 389, 279 (1983).
[CrossRef]

J. F. Nye, in Physics of Defects, Les Houches Lecture Series Session XXXV, R. Balian, M. Kliman, and J.-P. Poirier, eds. (North-Holland, 1981), p. 545.

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
[CrossRef]

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
[CrossRef]

J. Leach, S. Keen, and M. J. Padgett, Opt. Express 14, 11919 (2006).
[CrossRef]

Presnov, E. V.

V. V. Isaeva, N. V. Kasyanov, and E. V. Presnov, BioSystems 109, 280 (2012).
[CrossRef]

Prost, J.

M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. Aditi Simha, “Soft active matter,” arxiv.org:1207.2929 (2012).

Ramaswamy, S.

M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. Aditi Simha, “Soft active matter,” arxiv.org:1207.2929 (2012).

Rao, M.

M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. Aditi Simha, “Soft active matter,” arxiv.org:1207.2929 (2012).

Rasouli, S.

Santamato, E.

Sarti, A.

C. Lamberti, A. Sarti, and F. Bertucci, Comp. Cardiol. 24, 227 (1997).

Sen, S.

C. Nash and S. Sen, Topology and Geometry for Physicists (Academic, 1983).

Slussarenko, S.

M. Yeganeh, S. Rasouli, M. Dashti, S. Slussarenko, E. Santamato, and E. Karimi, Opt. Lett. 38, 887 (2013).
[CrossRef]

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector–vortex beams: a classical analogy of entanglement,” to be presented at Frontiers in Optics, Orlando, Florida, October6–13, 2013.

Smalyukh, I. I.

B. G. Chen, P. J. Ackerman, G. P. Alexander, R. D. Kamien, and I. I. Smalyukh, Phys. Rev. Lett. 110, 237801 (2013).
[CrossRef]

Soskin, M.

M. Soskin, V. Denisenko, and R. Egorov, J. Opt. A 6, S281 (2004).
[CrossRef]

I. Freund, M. Soskin, and A. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

Soskin, M. S.

A. Ya. Bekshaev, K. Y. Bliokh, and M. S. Soskin, J. Opt. 13, 053001 (2011).
[CrossRef]

V. G. Denisenko, A. Minovich, A. S. Desyatnikov, W. Krolikowski, M. S. Soskin, and Y. S. Kivshar, Opt. Lett. 33, 89 (2008).
[CrossRef]

A. Ya. Bekshaev and M. S. Soskin, Opt. Commum. 271, 332 (2007).
[CrossRef]

Takeda, M.

Y. Miyamoto, A. Wada, T. Yonemura, and M. Takeda, J. Opt. 15, 044002 (2013).
[CrossRef]

Viswanathan, N.

V. Kumar and N. Viswanathan, Second International Conference on Optical Angular Momentum, abstract book (2013), p. 38.

Wada, A.

Y. Miyamoto, A. Wada, T. Yonemura, and M. Takeda, J. Opt. 15, 044002 (2013).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Yeganeh, M.

Yonemura, T.

Y. Miyamoto, A. Wada, T. Yonemura, and M. Takeda, J. Opt. 15, 044002 (2013).
[CrossRef]

BioSystems (1)

V. V. Isaeva, N. V. Kasyanov, and E. V. Presnov, BioSystems 109, 280 (2012).
[CrossRef]

Comp. Cardiol. (1)

C. Lamberti, A. Sarti, and F. Bertucci, Comp. Cardiol. 24, 227 (1997).

IEEE Trans. Vis. Comp. Graphics (1)

L. Hesselink, Y. Levy, and Y. Lavin, IEEE Trans. Vis. Comp. Graphics 3, 1 (1997).
[CrossRef]

J. Opt. (2)

A. Ya. Bekshaev, K. Y. Bliokh, and M. S. Soskin, J. Opt. 13, 053001 (2011).
[CrossRef]

Y. Miyamoto, A. Wada, T. Yonemura, and M. Takeda, J. Opt. 15, 044002 (2013).
[CrossRef]

J. Opt. A (2)

M. V. Berry, J. Opt. A 11, 094001 (2009).
[CrossRef]

M. Soskin, V. Denisenko, and R. Egorov, J. Opt. A 6, S281 (2004).
[CrossRef]

J. Phys. A (1)

M. V. Berry, J. Phys. A 38, L745 (2005).
[CrossRef]

Opt. Commum. (1)

A. Ya. Bekshaev and M. S. Soskin, Opt. Commum. 271, 332 (2007).
[CrossRef]

Opt. Commun. (1)

I. Freund, M. Soskin, and A. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

A. V. Novitsky and L. M. Barkovsky, Phys. Rev. A 79, 033821 (2009).
[CrossRef]

Phys. Rev. Lett. (1)

B. G. Chen, P. J. Ackerman, G. P. Alexander, R. D. Kamien, and I. I. Smalyukh, Phys. Rev. Lett. 110, 237801 (2013).
[CrossRef]

Proc. R. Soc. London (1)

J. F. Nye, Proc. R. Soc. London 389, 279 (1983).
[CrossRef]

Proc. SPIE (1)

R. M. Kiehn, Proc. SPIE 5866, 192 (2005).
[CrossRef]

Prog. Opt. (1)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
[CrossRef]

Other (11)

R. S. Gallagher, Computer Visualization: Graphics Techniques for Engineering and Scientific Analysis (CRC, 1994).

G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector–vortex beams: a classical analogy of entanglement,” to be presented at Frontiers in Optics, Orlando, Florida, October6–13, 2013.

I. Mokhun, in Optical Correlation Techniques and Applications, O. V. Angelsky, ed. (SPIE, 2007), Chap. 1.

M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. Aditi Simha, “Soft active matter,” arxiv.org:1207.2929 (2012).

M. V. Berry, in Physics of Defects, Les Houches Lecture Series Session XXXV, R. Balian, M. Kliman, and J.-P. Poirier, eds. (North-Holland, 1981), p. 453.

J. F. Nye, in Physics of Defects, Les Houches Lecture Series Session XXXV, R. Balian, M. Kliman, and J.-P. Poirier, eds. (North-Holland, 1981), p. 545.

C. Nash and S. Sen, Topology and Geometry for Physicists (Academic, 1983).

M. Monastyrsky, Topology of Gauge Fields and Condensed Matter (Plenum, 1993).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

J. Nye, Natural Focusing and Fine Structure of Light (IoP, 1999).

V. Kumar and N. Viswanathan, Second International Conference on Optical Angular Momentum, abstract book (2013), p. 38.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

(a) Lemon and (b) star polarization singular patterns.

Fig. 2.
Fig. 2.

Schematic of the experimental setup to generate a polarization-engineered vector–vortex beam field. P, polarizer; H, half-wave plate; Q, quarter-wave plate; BS, beam-splitter; PBS, polarization beam splitter; M, mirror; ϕ(x,y), phase component(s); ISP, imaging Stokes polarimeter. (Q1 is optional for getting polarization singularities.)

Fig. 3.
Fig. 3.

Phase map and corresponding normalized radial Poynting vector flow obtained using (a) a SL and (b) a CL.

Fig. 4.
Fig. 4.

Phase map and corresponding normalized Poynting vector flow for (a) node, (b) saddle, (c) and spiral.

Fig. 5.
Fig. 5.

Beam intensity profile, phase map, and corresponding Poynting vector flow for vortex beams with (a)–(c) l=+1, (d)–(f) l=+2, and (g)–(i) l=+3.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

[vxvy]=[V11V12V21V22][xy].
Sx,y=1kIx,yφx,y.
S0=a12+a22=I(0°,0°)+I(90°,0°),S1=a12a22=I(0°,0°)I(90°,0°),S2=2a1a2cos(φ)=I(45°,0°)I(135°,0°),S3=2a1a2sin(φ)=I(45°,90°)I(135°,90°),
Ex=[(ε)+(x+iy)],
Ey=[i(ε)i(x+iy)].
φ(x,y)=a(x2+y2)+bx2+ctan1(y/x),

Metrics