Abstract

“The “focused doughnut”, a single-cycle electromagnetic perturbation of toroidal topology with inseparable time and spatial dependencies propagates at the speed of light in vacuum, as was shown by Hellwarth and Nouchi in 1996. While normal incidence reflection and refraction of conventional electromagnetic pulses in isotropic media do not lead to polarization changes, “focused doughnut” pulses undergo complex field transformations owing to the toroidal field structure and the presence of longitudinal components. We also demonstrate that “focused doughnuts” can interact strongly with structured media exciting dominant dynamic toroidal dipoles in spherical dielectric particles.”

© 2016 Optical Society of America

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References

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  1. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc, 1998).
  2. J. Brittingham, “Focus waves modes in homogeneous Maxwells equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983).
    [Crossref]
  3. T. T. Wu and R. W. P. King, “Comment on “Focus wave modes in homogenous Maxwell’s equations: Transverse electric mode,” J. Appl. Phys. 56(9), 2587–2588 (1984).
    [Crossref]
  4. P. Bélanger, “Packetlike solutions of the homogenous-wave equation,” J. Opt. Soc. Am. A 1(7), 723–724 (1984).
    [Crossref]
  5. A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57(3), 678–683 (1985).
    [Crossref]
  6. R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26(4), 861–863 (1985).
    [Crossref]
  7. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39(4), 2005–2033 (1989).
    [Crossref] [PubMed]
  8. J. Lekner, “Localized electromagnetic pulses with azimuthal dependence,” J. Opt. A: Pure Appl. Opt 6(7), 711–716 (2004).
    [Crossref]
  9. J. Lekner, “Helical light pulses,” J. Opt. A: Pure Appl. Opt 6(10), 29–32 (2004).
    [Crossref]
  10. S. Feng, H. G. Winful, and R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23(14), 385–387 (1998).
    [Crossref]
  11. S. Feng, H. G. Winful, and R. W. Hellwarth, “Spatiotemporal evolution of single-cycle electromagnetic pulses,” Phys. Rev. E 59(4), 4630–4649 (1999).
    [Crossref]
  12. S. Hunsche, S. Feng, H. G. Winful, A. Leitenstorfer, M. C. Nuss, and E. P. Ippen, “Spatiotemporal focusing of single-cycle light pulses,” J. Opt. Soc. Am. A 16(8), 2025–2028 (1999).
    [Crossref]
  13. R. W. Hellwarth and P. Nouchi, “Focused one-cycle electromagnetic pulses,” Phys. Rev. E 54(1), 889–895 (1996).
    [Crossref]
  14. X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. 242(4), 599–604 (2004).
    [Crossref]
  15. A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65(2), 805–813 (1989).
    [Crossref]
  16. R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,” IEEE Trans. Antennas Propag. 40(8), 888–905 (1992).
    [Crossref]
  17. C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E 71(2), 026603 (2005).
    [Crossref]
  18. I. J. Youngs, N. Bowler, K. P. Lymer, and N. Hussain, “Dielectric relaxation in metal-coated particles: the dramatic role of nano-scale coatings,” J. Phys. D: Appl. Phys. 38, 188 (2005).
    [Crossref]
  19. I. J. Youngs, N. Bowler, and O. Ugurlu, “Dielectric relaxation in composites containing electrically isolated particles with thin semi-continuous metal coatings,” J. Phys. D: Appl. Phys. 39, 1312 (2006).
    [Crossref]
  20. C. Vrejoiu, “Electromagnetic multipoles in cartesian coordinates,” J. Phys. A 35(46), 9911–9922 (2002).
    [Crossref]
  21. E. E. Radescu and G. Vaman, “Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles,” Phys. Rev. E. 65(4), 046609 (2002).
    [Crossref]
  22. T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Sci. Rep. 3(6010), 1510–1512 (2010).
  23. V. A. Fedotov, A. V. Rogacheva, V. Savinov, D. P. Tsai, and N. I. Zheludev, “Resonant transparency and non-trivial non-radiating excitations in toroidal metamaterials.l.,” Science 3, 2967 (2013).
  24. A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric Metamaterials with Toroidal Dipolar Response,” Phys. Rev. X 5(1), 011036 (2015).

2015 (1)

A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric Metamaterials with Toroidal Dipolar Response,” Phys. Rev. X 5(1), 011036 (2015).

2013 (1)

V. A. Fedotov, A. V. Rogacheva, V. Savinov, D. P. Tsai, and N. I. Zheludev, “Resonant transparency and non-trivial non-radiating excitations in toroidal metamaterials.l.,” Science 3, 2967 (2013).

2010 (1)

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Sci. Rep. 3(6010), 1510–1512 (2010).

2006 (1)

I. J. Youngs, N. Bowler, and O. Ugurlu, “Dielectric relaxation in composites containing electrically isolated particles with thin semi-continuous metal coatings,” J. Phys. D: Appl. Phys. 39, 1312 (2006).
[Crossref]

2005 (2)

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E 71(2), 026603 (2005).
[Crossref]

I. J. Youngs, N. Bowler, K. P. Lymer, and N. Hussain, “Dielectric relaxation in metal-coated particles: the dramatic role of nano-scale coatings,” J. Phys. D: Appl. Phys. 38, 188 (2005).
[Crossref]

2004 (3)

X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. 242(4), 599–604 (2004).
[Crossref]

J. Lekner, “Localized electromagnetic pulses with azimuthal dependence,” J. Opt. A: Pure Appl. Opt 6(7), 711–716 (2004).
[Crossref]

J. Lekner, “Helical light pulses,” J. Opt. A: Pure Appl. Opt 6(10), 29–32 (2004).
[Crossref]

2002 (2)

C. Vrejoiu, “Electromagnetic multipoles in cartesian coordinates,” J. Phys. A 35(46), 9911–9922 (2002).
[Crossref]

E. E. Radescu and G. Vaman, “Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles,” Phys. Rev. E. 65(4), 046609 (2002).
[Crossref]

1999 (2)

S. Feng, H. G. Winful, and R. W. Hellwarth, “Spatiotemporal evolution of single-cycle electromagnetic pulses,” Phys. Rev. E 59(4), 4630–4649 (1999).
[Crossref]

S. Hunsche, S. Feng, H. G. Winful, A. Leitenstorfer, M. C. Nuss, and E. P. Ippen, “Spatiotemporal focusing of single-cycle light pulses,” J. Opt. Soc. Am. A 16(8), 2025–2028 (1999).
[Crossref]

1998 (1)

1996 (1)

R. W. Hellwarth and P. Nouchi, “Focused one-cycle electromagnetic pulses,” Phys. Rev. E 54(1), 889–895 (1996).
[Crossref]

1992 (1)

R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,” IEEE Trans. Antennas Propag. 40(8), 888–905 (1992).
[Crossref]

1989 (2)

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65(2), 805–813 (1989).
[Crossref]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39(4), 2005–2033 (1989).
[Crossref] [PubMed]

1985 (2)

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57(3), 678–683 (1985).
[Crossref]

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26(4), 861–863 (1985).
[Crossref]

1984 (2)

T. T. Wu and R. W. P. King, “Comment on “Focus wave modes in homogenous Maxwell’s equations: Transverse electric mode,” J. Appl. Phys. 56(9), 2587–2588 (1984).
[Crossref]

P. Bélanger, “Packetlike solutions of the homogenous-wave equation,” J. Opt. Soc. Am. A 1(7), 723–724 (1984).
[Crossref]

1983 (1)

J. Brittingham, “Focus waves modes in homogeneous Maxwells equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983).
[Crossref]

Akturk, S.

X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. 242(4), 599–604 (2004).
[Crossref]

Basharin, A. A.

A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric Metamaterials with Toroidal Dipolar Response,” Phys. Rev. X 5(1), 011036 (2015).

Bélanger, P.

Besieris, I. M.

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65(2), 805–813 (1989).
[Crossref]

Bowler, N.

I. J. Youngs, N. Bowler, and O. Ugurlu, “Dielectric relaxation in composites containing electrically isolated particles with thin semi-continuous metal coatings,” J. Phys. D: Appl. Phys. 39, 1312 (2006).
[Crossref]

I. J. Youngs, N. Bowler, K. P. Lymer, and N. Hussain, “Dielectric relaxation in metal-coated particles: the dramatic role of nano-scale coatings,” J. Phys. D: Appl. Phys. 38, 188 (2005).
[Crossref]

Brittingham, J.

J. Brittingham, “Focus waves modes in homogeneous Maxwells equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983).
[Crossref]

Economou, E. N.

A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric Metamaterials with Toroidal Dipolar Response,” Phys. Rev. X 5(1), 011036 (2015).

Fedotov, V. A.

A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric Metamaterials with Toroidal Dipolar Response,” Phys. Rev. X 5(1), 011036 (2015).

V. A. Fedotov, A. V. Rogacheva, V. Savinov, D. P. Tsai, and N. I. Zheludev, “Resonant transparency and non-trivial non-radiating excitations in toroidal metamaterials.l.,” Science 3, 2967 (2013).

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Sci. Rep. 3(6010), 1510–1512 (2010).

Feng, S.

Gu, X.

X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. 242(4), 599–604 (2004).
[Crossref]

Hellwarth, R. W.

S. Feng, H. G. Winful, and R. W. Hellwarth, “Spatiotemporal evolution of single-cycle electromagnetic pulses,” Phys. Rev. E 59(4), 4630–4649 (1999).
[Crossref]

S. Feng, H. G. Winful, and R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23(14), 385–387 (1998).
[Crossref]

R. W. Hellwarth and P. Nouchi, “Focused one-cycle electromagnetic pulses,” Phys. Rev. E 54(1), 889–895 (1996).
[Crossref]

Hunsche, S.

Hussain, N.

I. J. Youngs, N. Bowler, K. P. Lymer, and N. Hussain, “Dielectric relaxation in metal-coated particles: the dramatic role of nano-scale coatings,” J. Phys. D: Appl. Phys. 38, 188 (2005).
[Crossref]

Ippen, E. P.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc, 1998).

Kaelberer, T.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Sci. Rep. 3(6010), 1510–1512 (2010).

Kafesaki, M.

A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric Metamaterials with Toroidal Dipolar Response,” Phys. Rev. X 5(1), 011036 (2015).

King, R. W. P.

T. T. Wu and R. W. P. King, “Comment on “Focus wave modes in homogenous Maxwell’s equations: Transverse electric mode,” J. Appl. Phys. 56(9), 2587–2588 (1984).
[Crossref]

Leitenstorfer, A.

Lekner, J.

J. Lekner, “Localized electromagnetic pulses with azimuthal dependence,” J. Opt. A: Pure Appl. Opt 6(7), 711–716 (2004).
[Crossref]

J. Lekner, “Helical light pulses,” J. Opt. A: Pure Appl. Opt 6(10), 29–32 (2004).
[Crossref]

Lymer, K. P.

I. J. Youngs, N. Bowler, K. P. Lymer, and N. Hussain, “Dielectric relaxation in metal-coated particles: the dramatic role of nano-scale coatings,” J. Phys. D: Appl. Phys. 38, 188 (2005).
[Crossref]

Nouchi, P.

R. W. Hellwarth and P. Nouchi, “Focused one-cycle electromagnetic pulses,” Phys. Rev. E 54(1), 889–895 (1996).
[Crossref]

Nuss, M. C.

Papasimakis, N.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Sci. Rep. 3(6010), 1510–1512 (2010).

Piché, M.

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E 71(2), 026603 (2005).
[Crossref]

Porras, M. A.

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E 71(2), 026603 (2005).
[Crossref]

Radescu, E. E.

E. E. Radescu and G. Vaman, “Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles,” Phys. Rev. E. 65(4), 046609 (2002).
[Crossref]

Rogacheva, A. V.

V. A. Fedotov, A. V. Rogacheva, V. Savinov, D. P. Tsai, and N. I. Zheludev, “Resonant transparency and non-trivial non-radiating excitations in toroidal metamaterials.l.,” Science 3, 2967 (2013).

Savinov, V.

A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric Metamaterials with Toroidal Dipolar Response,” Phys. Rev. X 5(1), 011036 (2015).

V. A. Fedotov, A. V. Rogacheva, V. Savinov, D. P. Tsai, and N. I. Zheludev, “Resonant transparency and non-trivial non-radiating excitations in toroidal metamaterials.l.,” Science 3, 2967 (2013).

Sezginer, A.

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57(3), 678–683 (1985).
[Crossref]

Shaarawi, A. M.

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65(2), 805–813 (1989).
[Crossref]

Soukoulis, C. M.

A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric Metamaterials with Toroidal Dipolar Response,” Phys. Rev. X 5(1), 011036 (2015).

Trebino, R.

X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. 242(4), 599–604 (2004).
[Crossref]

Tsai, D. P.

V. A. Fedotov, A. V. Rogacheva, V. Savinov, D. P. Tsai, and N. I. Zheludev, “Resonant transparency and non-trivial non-radiating excitations in toroidal metamaterials.l.,” Science 3, 2967 (2013).

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Sci. Rep. 3(6010), 1510–1512 (2010).

Ugurlu, O.

I. J. Youngs, N. Bowler, and O. Ugurlu, “Dielectric relaxation in composites containing electrically isolated particles with thin semi-continuous metal coatings,” J. Phys. D: Appl. Phys. 39, 1312 (2006).
[Crossref]

Vaman, G.

E. E. Radescu and G. Vaman, “Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles,” Phys. Rev. E. 65(4), 046609 (2002).
[Crossref]

Varin, C.

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E 71(2), 026603 (2005).
[Crossref]

Vrejoiu, C.

C. Vrejoiu, “Electromagnetic multipoles in cartesian coordinates,” J. Phys. A 35(46), 9911–9922 (2002).
[Crossref]

Winful, H. G.

Wu, T. T.

T. T. Wu and R. W. P. King, “Comment on “Focus wave modes in homogenous Maxwell’s equations: Transverse electric mode,” J. Appl. Phys. 56(9), 2587–2588 (1984).
[Crossref]

Youngs, I. J.

I. J. Youngs, N. Bowler, and O. Ugurlu, “Dielectric relaxation in composites containing electrically isolated particles with thin semi-continuous metal coatings,” J. Phys. D: Appl. Phys. 39, 1312 (2006).
[Crossref]

I. J. Youngs, N. Bowler, K. P. Lymer, and N. Hussain, “Dielectric relaxation in metal-coated particles: the dramatic role of nano-scale coatings,” J. Phys. D: Appl. Phys. 38, 188 (2005).
[Crossref]

Zheludev, N. I.

A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric Metamaterials with Toroidal Dipolar Response,” Phys. Rev. X 5(1), 011036 (2015).

V. A. Fedotov, A. V. Rogacheva, V. Savinov, D. P. Tsai, and N. I. Zheludev, “Resonant transparency and non-trivial non-radiating excitations in toroidal metamaterials.l.,” Science 3, 2967 (2013).

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Sci. Rep. 3(6010), 1510–1512 (2010).

Ziolkowski, R. W.

R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,” IEEE Trans. Antennas Propag. 40(8), 888–905 (1992).
[Crossref]

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65(2), 805–813 (1989).
[Crossref]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39(4), 2005–2033 (1989).
[Crossref] [PubMed]

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26(4), 861–863 (1985).
[Crossref]

IEEE Trans. Antennas Propag. (1)

R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,” IEEE Trans. Antennas Propag. 40(8), 888–905 (1992).
[Crossref]

J. Appl. Phys. (4)

J. Brittingham, “Focus waves modes in homogeneous Maxwells equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983).
[Crossref]

T. T. Wu and R. W. P. King, “Comment on “Focus wave modes in homogenous Maxwell’s equations: Transverse electric mode,” J. Appl. Phys. 56(9), 2587–2588 (1984).
[Crossref]

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57(3), 678–683 (1985).
[Crossref]

A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65(2), 805–813 (1989).
[Crossref]

J. Math. Phys. (1)

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26(4), 861–863 (1985).
[Crossref]

J. Opt. A: Pure Appl. Opt (2)

J. Lekner, “Localized electromagnetic pulses with azimuthal dependence,” J. Opt. A: Pure Appl. Opt 6(7), 711–716 (2004).
[Crossref]

J. Lekner, “Helical light pulses,” J. Opt. A: Pure Appl. Opt 6(10), 29–32 (2004).
[Crossref]

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

J. Phys. A (1)

C. Vrejoiu, “Electromagnetic multipoles in cartesian coordinates,” J. Phys. A 35(46), 9911–9922 (2002).
[Crossref]

J. Phys. D: Appl. Phys. (2)

I. J. Youngs, N. Bowler, K. P. Lymer, and N. Hussain, “Dielectric relaxation in metal-coated particles: the dramatic role of nano-scale coatings,” J. Phys. D: Appl. Phys. 38, 188 (2005).
[Crossref]

I. J. Youngs, N. Bowler, and O. Ugurlu, “Dielectric relaxation in composites containing electrically isolated particles with thin semi-continuous metal coatings,” J. Phys. D: Appl. Phys. 39, 1312 (2006).
[Crossref]

Opt. Commun. (1)

X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. 242(4), 599–604 (2004).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (1)

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39(4), 2005–2033 (1989).
[Crossref] [PubMed]

Phys. Rev. E (3)

S. Feng, H. G. Winful, and R. W. Hellwarth, “Spatiotemporal evolution of single-cycle electromagnetic pulses,” Phys. Rev. E 59(4), 4630–4649 (1999).
[Crossref]

R. W. Hellwarth and P. Nouchi, “Focused one-cycle electromagnetic pulses,” Phys. Rev. E 54(1), 889–895 (1996).
[Crossref]

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E 71(2), 026603 (2005).
[Crossref]

Phys. Rev. E. (1)

E. E. Radescu and G. Vaman, “Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles,” Phys. Rev. E. 65(4), 046609 (2002).
[Crossref]

Phys. Rev. X (1)

A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric Metamaterials with Toroidal Dipolar Response,” Phys. Rev. X 5(1), 011036 (2015).

Sci. Rep. (1)

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Sci. Rep. 3(6010), 1510–1512 (2010).

Science (1)

V. A. Fedotov, A. V. Rogacheva, V. Savinov, D. P. Tsai, and N. I. Zheludev, “Resonant transparency and non-trivial non-radiating excitations in toroidal metamaterials.l.,” Science 3, 2967 (2013).

Other (1)

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc, 1998).

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Figures (6)

Fig. 1
Fig. 1 Field topology and focusing properties of the transverse electric “focused doughnut” pulse. The electric E and magnetic H fields are represented by green and red arrows respectively. The effective wavelength parameter q1 and the pulse focal region depth q2 are indicated on both diagrams, along with the region of maximum energy concentration at z = 0, t = 0. The evolution of the pulse envelope is indicated by the black lines and arrows.
Fig. 2
Fig. 2 Spatial and temporal structure of the “focused doughnut” pulses. Panels (a) and (b) show respectively the normalized transverse electric and longitudinal magnetic fields along a yz cross-section at time t = 0. The characteristic parameters of the FD pulse in all cases are q2 = 100q1. All plots are generated from the analytical form of the FD pulse, Eqs. (13). The propagation properties of the TM pulse are identical to the TE case presented here, by replacing electric (magnetic) with magnetic (electric) fields. (c) and (d) show the evolution of the on-axis field component of both the real and imaginary pulses respectively as they propagate, demonstrating the transformations between the single and 1 1 2 cycle pulses.
Fig. 3
Fig. 3 Fourier spectrum of the real FD pulses. (a) shows the intensity of the Fourier spectrum for q2 = 100q1 at z = q2 and for four different radial positions ρ. (b) shows the evolution of the peak frequency as a function of ρ and z for z = q2 → 2q2.
Fig. 4
Fig. 4 Reflection of “focused doughnut” pulses from a perfect conductor. (a) Transverse electric (left) and longitudinal (right) magnetic field components of a transverse electric FD pulse before (t1) and after (t2) reflection. (b) Similar to (a) but for a transverse magnetic pulse. In both cases the parameters of the FD pulse are q2 = 100q1 and the boundary is located at a distance z = 20q1 from the focal point of the pulse (z = 0). All field components have been normalized to their maximum value.
Fig. 5
Fig. 5 Reflection and refraction of “focused doughnut” pulses at a vacuum-dielectric interface. (a) Transverse electric (left) and longitudinal magnetic field (right) components of a transverse electric FD pulse before (t1) and after (t2) incidence on the interface. (b) Similar to (a) but for a transverse magnetic pulse. In both cases the parameters of the FD pulse are q2 = 100q1 and the boundary is located at a distance z = 15q1 from the focal point of the pulse (z = 0). All field components have been normalized to their maximum value. The dielectric is considered to be semi-infinite with a refractive index n = 2.
Fig. 6
Fig. 6 Interactions of “focused doughnut” pulses with spherical dielectric nanoparticles. (a) and (b) show the xz cross-sections of the COMSOL simulation domain at a time t = 0. (a) shows the normalised transverse electric field of a TE FD pulse, and (b) shows the normalised longitudinal electric field of the TM FD pulse. The outline of the spherical nanoparticle is shown by the dotted line. (c) & (d) show the electric field intensity integrated over the volume of the nanoparticle as a function of frequency, when under excitation from a transverse electric (TE) and a transverse magnetic (TM) FD pulse respectively. (e) & (f) show the electric field distributions on an xz cross-section of the nanoparticle (see grid in the insets to (c) & (d)) at resonance positions (i)–(vi). (g) & (h) show the scattering intensity of the individual cartesian multipoles up to quadrupole order (electric dipole p, magnetic dipole m, toroidal dipole T, electric quadrupole Qe, and magnetic quadrupole Qm) for illumination with TE and TM FD pulses, respectively. In (c)–(d) & (g)–(h) dots correspond to simulation data points, while lines serve as eye guides.

Equations (10)

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E θ = 4 i f 0 μ 0 ε 0 ρ ( q 1 + q 2 2 i c t ) [ ρ 2 + ( q 1 + i τ ) ( q 2 i σ ) ] 3
H ρ = 4 i f 0 ρ ( q 2 q 1 2 i z ) [ ρ 2 + ( q 1 + i τ ) ( q 2 i σ ) ] 3
H z = 4 f 0 ρ 2 ( q 1 + i τ ) ( q 2 i σ ) [ ρ 2 + ( q 1 + i τ ) ( q 2 i σ ) ] 3
F ω = ( i π μ 0 f 0 ω | ω | sin φ 2 r c 2 ) e i ω r | ω | Q c
Electric dipole p = 1 i ω j d 3 r
Magnetic dipole m = 1 2 c ( r × j ) d 3 r
Toroidal dipole T = 1 10 c [ ( r j ) r 2 r 2 j ] d 3 r
Electric quadrupole Q α β e = 1 2 i ω [ r α j β + r β j α 2 3 δ α β ( r j ) ] d 3 r
Magnetic quadrupole Q α β m = 1 3 c [ ( r × j ) α r β + ( r × j ) β r α ] d 3 r
I total = 2 ω 4 3 c 3 | p | 2 + 2 ω 4 3 c 3 | m | 2 + 2 ω 6 3 c 5 | T | 2 + 4 ω 5 3 c 4 ( p T ) + ω 6 5 c 5 Q α β e Q α β e + ω 6 20 c 5 Q α β m Q α β m

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