## Energy flow characteristics of vector X-Waves |

Optics Express, Vol. 19, Issue 9, pp. 8526-8532 (2011)

http://dx.doi.org/10.1364/OE.19.008526

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### 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 OSA

## 1. Introduction

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]

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]

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

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]

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

## 2. Full vector X-Waves

*ω*=

*Vk*+

_{z}*α*, over the spectral variables

*k*,

_{ρ}*k*and

_{z}*ω*. Here,

*V*is the centroid velocity,

*α*is a positive real parameter that quantifies the periodicity of local deformations [1

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

*ω*is the angular frequency with the dispersion relation

*k*=

*ω*/

*c*,

*k*is the magnitude of the wave vector

**k**with the components

*k*and

_{ρ}*k*in the transverse and longitudinal directions, respectively, and

_{z}*c*is the speed of light in free-space. In what follows, we employ the ‘standard’ LW spectrum [1

*Localized Waves* (J. Wiley & Sons, 2008). [CrossRef]

*m*-th order X-Wave when

*α*= 0 or focus wave mode (FWM) otherwise. Hereafter, we consider only X-Wave solutions as they are always causal and do not have any backward propagating components [2

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

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

*ρ,φ,z*) in its integral form reads The integral (2) is carried out analytically using formula (6.621.1) in [14], yielding where

*τ*= (

*aV –iζ*),

*ζ*=

*z*–

*Vt*,

*η*= (

*γρ*/

*τ*)

^{2}, and

_{2}

*F*

_{1}(

*a, b; c; z*) is the Gauss hypergeometric function [15]. Expression (3) is the generalized

*m*-th order scalar X-Wave with azimuthal dependence of order

*n*.

**E**and

**H**are the electric field and magnetic field vectors, Π

*=*

_{e}*A*Ψ

_{e}*(*

_{n}*ρ,φ,ζ*)

**ẑ**and Π

*=*

_{h}*A*Ψ

_{h}*(*

_{n}*ρ,φ,ζ*)

**ẑ**are the electric and magnetic Hertz vector potentials,

*A*and

_{e}*A*are arbitrary complex amplitudes,

_{h}**ẑ**is the unit vector in the

*z*-direction, and

*μ*

_{0}and

*ε*

_{0}are the free-space permeability and permittivity. The expressions (4) and (5) are thus the generalized vector form of the electromagnetic X-Wave.

*m*= 0; which corresponds to the zero-order X-Wave [17

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]

*n*= 1 to simplify the mathematical analysis without any loss of generality, while preserving the contribution of both Hertz potentials, Π

*and Π*

_{e}*, to each transverse component of the fields,*

_{h}**E**and

**H**. It should be noted that such axially asymmetric solutions are essential to produce the negative energy flux density. Accordingly, the generalized scalar solution of the X-Wave Eq. (3) reduces to its zero-order expression as

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]

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]

*X*= (

*γρ*)

^{2}+

*τ*

^{2}, Ξ =

*τ*

^{2}– 2(

*γρ*)

^{2}and ℜ{

*F*} is the real part of

*F*.

## 3. Energy flow characteristics

**S**=

**E**×

**H**and represents the energy flux density. By considering only the Poynting vector at the centroid of the X-Wave (

*ζ*= 0), we can write the expressions of its vector components in explicit form using Eqs. (6)–(11) as

*χ*= (

*aV*)

^{2}+ (

*γρ*)

^{2},

*ξ*= (

*aV*)

^{2}– 2(

*γρ*)

^{2}and ℑ{

*F*} is the imaginary part of

*F*.

*ρ*and

*φ*, this does not necessary ensure negative energy flow. In order to determine the direction of the energy flow, we compute the energy flux vector

**P**by integrating the Poynting vector over the transverse plane as The total energy flux at the centroid is computed using Eq. (15) when

*ρ*

_{0}→ ∞ as From Eq. (16), we deduce that the there is no net energy flow in the radial direction and the centroid propagates rigidly without changing its shape, irrespective of the amplitudes of the polarizations, thus the fields maintain their transverse localization indefinitely. The total energy flux in the φ-direction acquires the same sign of the quantity

*A*and

_{e}*A*values. Moreover, it can be shown from Eq. (15), that

_{h}*P*(

_{φ}*ρ*

_{0}) does not change its sign with

*ρ*

_{0}. The total energy flow in

*z*-direction is always in the positive

*z*-direction as inferred from Eq. (18). However, following the concept presented in [12

12. A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector bessel beams,” J. Opt. Soc. Am. A **24**, 2844–2849 (2007). [CrossRef]

*ρ*

_{0}, we can obtain non-positive values for the energy flux.

*z*-component of the Poynting vector of the zero-order vector X-Wave with

*a*= 2 × 10

^{−16}

*s*and

*V*= 1.5

*c*at its centroid computed by Eq. (14) for two different superpositions of the TE and TM amplitudes, with

*S*and that it changes its sign for both configurations. Figure 2 presents the energy flux in the

_{z}*z*-direction computed by Eq. (15) for the same X-Wave with the same amplitude configurations as in Fig. 1. It is clear from the figure that backward propagation is only possible in the second configuration. The necessary condition to have a negative energy flux in the

*z*-direction can be derived from Eq. (14) as

*B*< –

*V*/(

*γ*

^{2}+ 2), where

*V*in addition to the choice of the amplitudes of the TE- and TM-polarizations. Additionally, as the zeros of

*P*as a function of

_{z}*ρ*

_{0}are the roots of a polynomial, the necessary condition to have a negative energy flux implies that there exists only one real root greater than zero on the

*ρ*

_{0}-axis. This root,

*ρ*

_{0max}, designates the upper exclusive limit of the truncation radius to obtain a net negative energy flux in the

*z*-direction and is given by where

*P*= 1 + (

*γ*

^{2}+ 1)

*B/V*,

*P*< 0 and the positive values of the roots are to be chosen. Figure 2 also shows the location of

*ρ*

_{0max}and that

*P*is positive for

_{z}*ρ*

_{0}>

*ρ*

_{0max}.

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]

*ρ*

_{0max}results in the elimination of most of the ‘X-shaped arms’ while maintaining the highly localized central region, which will carry energy that propagates in the reverse direction. In contrast to the truncated Bessel beam [12

**24**, 2844–2849 (2007). [CrossRef]

*ρ*

_{0max}, where the reflected portion maintains its localization for a greater propagation distance [18

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]

## 4. Conclusions

## References and links

1. | H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., |

2. | R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A |

3. | R. Donnelly and D. Power, “The behavior of electromagnetic localized waves at a planar interface,” IEEE Trans. Antennas Propag. |

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. |

5. | A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E |

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. |

7. | E. Recami, “On localized “X-shaped” superluminal solutions to maxwell equations,” Phys. Rev. A |

8. | Zh. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. |

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 |

10. | B. Zhang and B. I. Wu, “Electromagnetic detection of a perfect invisibility cloak”, Phys. Rev. Lett. |

11. | H. Chen and M. Chen, “Flipping photons backward: reversed Cherenkov radiation,” Materials Today |

12. | A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector bessel beams,” J. Opt. Soc. Am. A |

13. | E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. |

14. | A. Jeffery, I. Gradshteǐn, D. Zwillinger, and I. Ryzhik, |

15. | M. Abramowitz and I. A. Stegun, |

16. | J. D. Jackson, |

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 |

18. | M. A. Salem and H. Bağcı, “On the propagation of truncated localized waves in dispersive silica,” Opt. Express |

19. | M. Zamboni-Rached, “Analytical expressions for the longitudinal evolution of nondiffracting pulses truncated by finite apertures,” J. Opt. Soc. Am. A |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(350.5500) Other areas of optics : Propagation

(350.7420) Other areas of optics : Waves

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: February 10, 2011

Revised Manuscript: March 26, 2011

Manuscript Accepted: March 29, 2011

Published: April 18, 2011

**Virtual Issues**

Vol. 6, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Mohamed A. Salem and Hakan Bağcı, "Energy flow characteristics of vector X-Waves," Opt. Express **19**, 8526-8532 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8526

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### References

- H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (J. Wiley & Sons, 2008). [CrossRef]
- R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2032 (1989). [CrossRef] [PubMed]
- R. Donnelly and D. Power, “The behavior of electromagnetic localized waves at a planar interface,” IEEE Trans. Antennas Propag. 45, 580–591 (1997). [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]
- 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]
- E. Recami, “On localized “X-shaped” superluminal solutions to maxwell equations,” Phys. Rev. A 252, 586–610 (1998).
- 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]
- 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]
- B. Zhang and B. I. Wu, “Electromagnetic detection of a perfect invisibility cloak”, Phys. Rev. Lett. 103, 243901 (2009). [CrossRef]
- H. Chen and M. Chen, “Flipping photons backward: reversed Cherenkov radiation,” Materials Today 14, 24–41 (2011). [CrossRef]
- A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector bessel beams,” J. Opt. Soc. Am. A 24, 2844–2849 (2007). [CrossRef]
- E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. 37, 1604–1608 (1989). [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 Electrodynamics 3rd ed. (Wiley, 1999).
- 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]
- M. A. Salem and H. Bağcı, “On the propagation of truncated localized waves in dispersive silica,” Opt. Express 18, 25482–25493 (2010). [CrossRef] [PubMed]
- 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]

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