Abstract

The vector form of X-Waves is obtained as a superposition of transverse electric and transverse magnetic polarized field components. It is shown that the signs of all components of the Poynting vector can be locally changed using carefully chosen complex amplitudes of the transverse electric and transverse magnetic polarization components. Negative energy flux density in the longitudinal direction can be observed in a bounded region around the centroid; in this region the local behavior of the wave field is similar to that of wave field with negative energy flow. This peculiar energy flux phenomenon is of essential importance for electromagnetic and optical traps and tweezers, where the location and momenta of micro-and nanoparticles are manipulated by changing the Poynting vector, and in detection of invisibility cloaks.

©2011 Optical Society of America

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References

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  1. H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (J. Wiley & Sons, 2008).
    [Crossref]
  2. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2032 (1989).
    [Crossref] [PubMed]
  3. R. Donnelly and D. Power, “The behavior of electromagnetic localized waves at a planar interface,” IEEE Trans. Antennas Propag. 45, 580–591 (1997).
    [Crossref]
  4. E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
    [Crossref]
  5. A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
    [Crossref]
  6. A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Scattering of X-waves from a circular disk using a time domain incremental theory of diffraction,” Prog. Electromagn. Res. 44, 103–129 (2004).
    [Crossref]
  7. E. Recami, “On localized “X-shaped” superluminal solutions to maxwell equations,” Phys. Rev. A 252, 586–610 (1998).
  8. Zh. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
    [Crossref] [PubMed]
  9. L. A. Ambrosio and H. E. Hernández-Figueroa, “Gradient forces on double-negative particles in optical tweezers using Bessel beams in the ray optics regime,” Opt. Express 18, 24287–24292 (2010).
    [Crossref] [PubMed]
  10. B. Zhang and B. I. Wu, “Electromagnetic detection of a perfect invisibility cloak”, Phys. Rev. Lett. 103, 243901 (2009).
    [Crossref]
  11. H. Chen and M. Chen, “Flipping photons backward: reversed Cherenkov radiation,” Materials Today 14, 24–41 (2011).
    [Crossref]
  12. A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector bessel beams,” J. Opt. Soc. Am. A 24, 2844–2849 (2007).
    [Crossref]
  13. E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. 37, 1604–1608 (1989).
    [Crossref]
  14. A. Jeffery, I. Gradshteǐn, D. Zwillinger, and I. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).
  15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1964), chap. 15.
  16. J. D. Jackson, Classical Electrodynamics3rd ed. (Wiley, 1999).
  17. J.-Y. Lu and J. F. Greenleaf, “Nondiffracting x waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
    [Crossref] [PubMed]
  18. M. A. Salem and H. Bağcı, “On the propagation of truncated localized waves in dispersive silica,” Opt. Express 18, 25482–25493 (2010).
    [Crossref] [PubMed]
  19. M. Zamboni-Rached, “Analytical expressions for the longitudinal evolution of nondiffracting pulses truncated by finite apertures,” J. Opt. Soc. Am. A 23, 2166–2176 (2006).
    [Crossref]

2011 (2)

H. Chen and M. Chen, “Flipping photons backward: reversed Cherenkov radiation,” Materials Today 14, 24–41 (2011).
[Crossref]

Zh. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
[Crossref] [PubMed]

2010 (2)

2009 (1)

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

2007 (1)

2006 (1)

2004 (2)

A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[Crossref]

A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Scattering of X-waves from a circular disk using a time domain incremental theory of diffraction,” Prog. Electromagn. Res. 44, 103–129 (2004).
[Crossref]

2003 (1)

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[Crossref]

1998 (1)

E. Recami, “On localized “X-shaped” superluminal solutions to maxwell equations,” Phys. Rev. A 252, 586–610 (1998).

1997 (1)

R. Donnelly and D. Power, “The behavior of electromagnetic localized waves at a planar interface,” IEEE Trans. Antennas Propag. 45, 580–591 (1997).
[Crossref]

1992 (1)

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting x waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[Crossref] [PubMed]

1989 (2)

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2032 (1989).
[Crossref] [PubMed]

E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. 37, 1604–1608 (1989).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1964), chap. 15.

Ambrosio, L. A.

Attiya, A. M.

A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Scattering of X-waves from a circular disk using a time domain incremental theory of diffraction,” Prog. Electromagn. Res. 44, 103–129 (2004).
[Crossref]

Bagci, H.

Besieris, I. M.

A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Scattering of X-waves from a circular disk using a time domain incremental theory of diffraction,” Prog. Electromagn. Res. 44, 103–129 (2004).
[Crossref]

Chen, H.

H. Chen and M. Chen, “Flipping photons backward: reversed Cherenkov radiation,” Materials Today 14, 24–41 (2011).
[Crossref]

Zh. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
[Crossref] [PubMed]

Chen, M.

H. Chen and M. Chen, “Flipping photons backward: reversed Cherenkov radiation,” Materials Today 14, 24–41 (2011).
[Crossref]

Ciattoni, A.

A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[Crossref]

Conti, C.

A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[Crossref]

Dartora, C. A.

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[Crossref]

Ding, J.

Donnelly, R.

R. Donnelly and D. Power, “The behavior of electromagnetic localized waves at a planar interface,” IEEE Trans. Antennas Propag. 45, 580–591 (1997).
[Crossref]

El-Diwany, E.

A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Scattering of X-waves from a circular disk using a time domain incremental theory of diffraction,” Prog. Electromagn. Res. 44, 103–129 (2004).
[Crossref]

Gradshtein, I.

A. Jeffery, I. Gradshteǐn, D. Zwillinger, and I. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

Greenleaf, J. F.

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting x waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[Crossref] [PubMed]

Hernández-Figueroa, H. E.

L. A. Ambrosio and H. E. Hernández-Figueroa, “Gradient forces on double-negative particles in optical tweezers using Bessel beams in the ray optics regime,” Opt. Express 18, 24287–24292 (2010).
[Crossref] [PubMed]

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[Crossref]

Heyman, E.

E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. 37, 1604–1608 (1989).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics3rd ed. (Wiley, 1999).

Jeffery, A.

A. Jeffery, I. Gradshteǐn, D. Zwillinger, and I. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

Lu, J.-Y.

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting x waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[Crossref] [PubMed]

Nóbrega, K. Z.

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[Crossref]

Novitsky, A. V.

Novitsky, D. V.

Porto, P. D.

A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[Crossref]

Power, D.

R. Donnelly and D. Power, “The behavior of electromagnetic localized waves at a planar interface,” IEEE Trans. Antennas Propag. 45, 580–591 (1997).
[Crossref]

Recami, E.

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[Crossref]

E. Recami, “On localized “X-shaped” superluminal solutions to maxwell equations,” Phys. Rev. A 252, 586–610 (1998).

Ryzhik, I.

A. Jeffery, I. Gradshteǐn, D. Zwillinger, and I. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

Salem, M. A.

Shaarawi, A. M.

A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Scattering of X-waves from a circular disk using a time domain incremental theory of diffraction,” Prog. Electromagn. Res. 44, 103–129 (2004).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1964), chap. 15.

Wang, H.-T.

Wu, B. I.

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

Zamboni-Rached, M.

M. Zamboni-Rached, “Analytical expressions for the longitudinal evolution of nondiffracting pulses truncated by finite apertures,” J. Opt. Soc. Am. A 23, 2166–2176 (2006).
[Crossref]

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[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.

Zheng, Zh.

Ziolkowski, R. W.

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2032 (1989).
[Crossref] [PubMed]

Zwillinger, D.

A. Jeffery, I. Gradshteǐn, D. Zwillinger, and I. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[Crossref]

IEEE Trans. Antennas Propag. (2)

E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. 37, 1604–1608 (1989).
[Crossref]

R. Donnelly and D. Power, “The behavior of electromagnetic localized waves at a planar interface,” IEEE Trans. Antennas Propag. 45, 580–591 (1997).
[Crossref]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting x waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[Crossref] [PubMed]

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

Materials Today (1)

H. Chen and M. Chen, “Flipping photons backward: reversed Cherenkov radiation,” Materials Today 14, 24–41 (2011).
[Crossref]

Opt. Express (2)

Phys. Rev. A (2)

E. Recami, “On localized “X-shaped” superluminal solutions to maxwell equations,” Phys. Rev. A 252, 586–610 (1998).

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2032 (1989).
[Crossref] [PubMed]

Phys. Rev. E (1)

A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[Crossref]

Phys. Rev. Lett. (1)

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

Prog. Electromagn. Res. (1)

A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Scattering of X-waves from a circular disk using a time domain incremental theory of diffraction,” Prog. Electromagn. Res. 44, 103–129 (2004).
[Crossref]

Other (4)

H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (J. Wiley & Sons, 2008).
[Crossref]

A. Jeffery, I. Gradshteǐn, D. Zwillinger, and I. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1964), chap. 15.

J. D. Jackson, Classical Electrodynamics3rd ed. (Wiley, 1999).

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Figures (2)

Fig. 1
Fig. 1 A comparison between the z-component of the Poynting vector at the centroid plane of the zero-order vector X-Wave with n = 1, a = 2×10−16 s and V = 1.5c for two different configurations. Configuration 1 (a) is for the amplitudes A e = 1 / ɛ 0 and A h = 1 / μ 0 ; and Configuration 2 (b) is for the amplitudes A e = 1 / ɛ 0 and A h = i / μ 0 .

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Fig. 2
Fig. 2 A comparison between the net energy flux in the z-direction in a finite circular cross section with radius ρ 0 of the zero-order vector X-Wave with n = 1, a = 2 × 10−16 s and V = 1.5c. Configuration 1 (dashed line) is for the amplitudes A e = 1 / ɛ 0 and A h = 1 / μ 0 ; and Configuration 2 (solid line) is for the amplitudes A e = 1 / ɛ 0 and A h = i / μ 0 .

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Equations (20)

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Ψ ˜ n ( k ρ ,   k z ,   ω ;   m ,   α ) = 2 n ( k z + α V ) m e − a V k z e − a α δ ( k ρ 2 − [ ω 2 c 2 − k z 2 ] ) δ ( ω − [ V k z + α ] ) ,
Ψ n ( ρ , ϕ , z , t ) = ∫ 0 ∞ d k z ∫ − ∞ ∞ d ω ∫ 0 ∞ d k ρ Ψ ˜ n ( k ρ , k z , ω ; m , α ) J n ( k ρ ρ ) e i n ϕ e i ( k z z − ω t ) .
Ψ n ( ρ , ϕ , ζ ) = e i n ϕ ( γ ρ ) n τ 1 + m + n ( m + n ) ! n ! 2 F 1 ( ( 1 + m + n ) 2 ,   ( 2 + m + n ) 2 ;   1 + n ;   − η ) ,
E = ∇ ( ∇ · Π e ) − 1 c 2 ∂ 2 ∂ t 2 Π e − μ 0 ∇ × ( ∂ ∂ t Π h ) ,
H = ɛ 0 ∇ × ( ∂ ∂ t Π e ) + ∇ ( ∇ · Π h ) − 1 c 2 ∂ 2 ∂ t 2 Π h ,
Ψ 1 ( ρ , ϕ , ζ ) = 2 e i ϕ 1 + η − 1 γ ρ 1 + η .
E ρ ( ρ , ϕ , ζ ) = 2 ℜ { i γ e i ϕ X ( 5 / 2 ) [ Ξ A e + i μ 0 V X A h ] } ,
E ϕ ( ρ , ϕ , ζ ) = 2 ℜ { i γ e i ϕ X ( 5 / 2 ) [ i X A e − V μ 0 Ξ A h ] } ,
E z ( ρ , ϕ , ζ ) = 6 ℜ { γ 3 ρ e i ϕ X 2 1 + η A e } ,
H ρ ( ρ , ϕ , ζ ) = 2 ℜ { e i ϕ γ X ( 5 / 2 ) [ i Ξ A h + V ɛ 0 X A e ] } ,
H ϕ ( ρ , ϕ , ζ ) = 2 ℜ { i γ e i ϕ X ( 5 / 2 ) [ i X A h + V ɛ 0 Ξ A e ] } ,
H z ( ρ , ϕ , ζ ) = 6 ℜ { γ 3 ρ e i ϕ X 2 1 + η A h } ,
S ρ ( ρ , ϕ ) = 6 γ 4 ρ a V 2 ξ χ 5 { sin ( 2 ϕ ) [ ɛ 0 ℜ { A e 2 } + μ 0 ℜ { A h 2 } ] + 2 cos ( 2 ϕ ) [ ɛ 0 A e R A e I + μ 0 A h R A h I ] } ,
S ϕ ( ρ , ϕ ) = 6 γ 4 ρ a V χ 2 { 2 ξ [ A h R A e I − A h I A e R ] + V χ [ ɛ 0 | A e | 2 + μ 0 | A h | 2 ] + V χ cos ( 2 ϕ ) [ ɛ 0 ℜ { A e 2 } + μ 0 ℜ { A h 2 } ] − V χ sin ( 2 ϕ ) [ ɛ 0 ℑ { A e 2 } + μ 0 ℑ { A h 2 } ] } ,
S z ( ρ , ϕ ) = 2 γ 2 χ 5 { 2 ( γ 2 + 2 ) χ ξ [ A e I A h R − A e R A h I ] + V ( χ 2 + ξ 2 ) [ ɛ 0 | A e | 2 + μ 0 | A h | 2 ] − V ( ξ 2 − χ 2 ) cos ( 2 ϕ ) [ ɛ 0 ℜ { A e 2 } + μ 0 ℜ { A h 2 } ] + V ( ξ 2 − χ 2 ) sin ( 2 ϕ ) [ ɛ 0 ℑ { A e 2 } + μ 0 ℑ { A h 2 } ] } .
P ( ρ 0 ) = ∫ 0 ρ 0 d ρ ∫ − π π d ϕ ρ S ( ρ , ϕ ) ,
P ρ = 0 ,
P ϕ = 3 π 2 γ 32 ( a V ) 4 { 4 V [ ɛ 0 | A e | 2 + μ 0 | A h | 2 ] + ( A h I A e R − A e I A h R ) } ,
P z = 3 π 2 a 4 V 3 [ ɛ 0 | A e | 2 + μ 0 | A h | 2 ]
ρ 0 max = a V 3 γ − 4 − 4 × 2 1 / 3 ( P − 2 ) Q + 2 2 / 3 Q ,

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