Abstract: The vector form of X-Waves is obtained as a superposition oftransverse electric and transverse magnetic polarized field components. It isshown that the signs of all components of the Poynting vector can be locallychanged using carefully chosen complex amplitudes of the transverseelectric and transverse magnetic polarization components. Negative energyflux density in the longitudinal direction can be observed in a boundedregion around the centroid; in this region the local behavior of the wavefield is similar to that of wave field with negative energy flow. This peculiarenergy flux phenomenon is of essential importance for electromagneticand optical traps and tweezers, where the location and momenta of micro-and nanoparticles are manipulated by changing the Poynting vector, and indetection of invisibility cloaks.
References and links1. H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (J. Wiley & Sons, 2008).2. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2032 (1989).3. R. Donnelly and D. Power, “The behavior of electromagnetic localized waves at a planar interface,” IEEE Trans.
Antennas Propag. 45, 580–591 (1997).4. E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernandez-Figueroa, “On the localized
superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).5. A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).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).7. E. Recami, “On localized “X-shaped” superluminal solutions to maxwell equations,” Phys. Rev. A 252, 586–610
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using Bessel beams in the ray optics regime,” Opt. Express 18, 24287–24292 (2010).10. B. Zhang and B. I. Wu, “Electromagnetic detection of a perfect invisibility cloak”, Phys. Rev. Lett. 103, 243901
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(2011).12. A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector bessel beams,” J. Opt. Soc. Am. A 24,
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(1989).14. A. Jeffery, I. Gradshteın, D. Zwillinger, and I. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).
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15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathemat-ical Tables, 9th ed. (Dover, 1964), chap. 15.
16. J. D. Jackson, Classical Electrodynamics 3rd 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).18. M. A. Salem and H. Bagcı, “On the propagation of truncated localized waves in dispersive silica,” Opt. Express
18, 25482–25493 (2010).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).
1. Introduction
X-Waves are a class of Localized Wave (LW) solutions to the scalar wave equation, whichhave been extensively studied in the literature (see [1] for a historical review), while studiesof LW solutions to the vector wave equation have focused only on constructing solutions us-ing a single polarization type, mostly the transverse electric (TE)-polarization (see, [2–6] andreferences therein). In this letter, we work with a full vector representation of X-Waves ob-tained by superimposing localized TE and transverse magnetic (TM) polarization solutions ofthe wave equation. To the best of our knowledge, this is the first time where such a full vectorrepresentation is used in the context of LWs.
We show that the amplitudes of the Poynting vector components of full vector X-Waves canchange their signs locally. We study this peculiar phenomenon and further show that it does notstem from the nature of the scalar X-Wave solution, but it is the result of the weighted super-position of TE- and TM-polarization components. We further show that this phenomenon doesnot occur in the case of single polarization (TE or TM) LWs [2, 7] or axially-symmetric full-vector X-Waves. More specifically, we examine the Poynting vector of the full vector X-Waveat the centroid plane (the plane with maximum localization in the transverse direction) and de-rive the necessary conditions to obtain negative or zero energy flux density in the longitudinaldirection in the vicinity of the centroid. Objects interacting with the X-Wave in this region mayexperience negative or zero energy flux. This particular result might be of importance for vari-ous electromagnetic and optical applications, such as traps and tweezers, where the behavior ofmicro- and nanoparticles are manipulated by the changing Poynting vector [8,9], and detectionof transformation optics invisibility cloaks [10, 11]. Furthermore, to illustrate the possibilityof the negative energy flow (not only its density), by following the concept presented in [12],we show that one can observe wave fields with negative energy flow if the X-Wave fields aretruncated in the transverse plane with an apertured perfect absorbing screen.
While X-Waves could generally be expressed as weighted frequency superposition of Besselbeams [1], we assert that the Poynting vector of full vector X-Waves behaves different from thatof Bessel beams [12] as it maintains a localized profile in the transverse plane. We also showthat the peculiar energy flow characteristics could be achieved in the time domain, not only inthe frequency domain as in the case with Bessel beams.
2. Full vector X-Waves
The LW solutions to the scalar wave equation are derived by summing Bessel beams weightedby an appropriate spectrum that preserves the necessary undistorted propagation condition,ω = Vkz +α , over the spectral variables kρ ,kz and ω . Here, V is the centroid velocity, α is apositive real parameter that quantifies the periodicity of local deformations [1], ω is the angularfrequency with the dispersion relation k = ω/c, k is the magnitude of the wave vector k withthe components kρ and kz in the transverse and longitudinal directions, respectively, and c is
#142497 - $15.00 USD Received 10 Feb 2011; revised 26 Mar 2011; accepted 29 Mar 2011; published 18 Apr 2011(C) 2011 OSA 25 April 2011 / Vol. 19, No. 9 / OPTICS EXPRESS 8527
the speed of light in free-space. In what follows, we employ the ‘standard’ LW spectrum [1],
Ψn(kρ ,kz,ω;m,α) = 2n(
kz +αV
)me−aVkze−aα δ
(k2
ρ −[
ω2
c2 − k2z
])δ (ω − [Vkz +α]) , (1)
which yields an m-th order X-Wave when α = 0 or focus wave mode (FWM) otherwise. Here-after, we consider only X-Wave solutions as they are always causal and do not have any back-ward propagating components [2, 13], thus any observed negative flow of energy would beuniquely due to the mixing of TE- and TM-polarizations.
Hence, the generalized scalar X-Wave solution in cylindrical coordinates (ρ ,φ ,z) in its inte-gral form reads
Ψn(ρ ,φ ,z, t) =∫ ∞
0dkz
∫ ∞
−∞dω
∫ ∞
0dkρ Ψn(kρ ,kz,ω;m,α)Jn(kρ ρ)einφ ei(kzz−ωt). (2)
The integral (2) is carried out analytically using formula (6.621.1) in [14], yielding
Ψn(ρ ,φ ,ζ ) = einφ (γρ)n
τ1+m+n
(m+n)!n! 2F1
((1+m+n)
2,(2+m+n)
2;1+n;−η
), (3)
where γ =√
(V/c)2 −1, τ =(aV − iζ ), ζ = z−Vt, η =(γρ/τ)2, and 2F1(a,b;c;z) is the Gausshypergeometric function [15]. Expression (3) is the generalized m-th order scalar X-Wave withazimuthal dependence of order n.
Next, we derive the X-Wave solution to the vector wave equation using single-componentHertz vector potentials as [16]
E = ∇(∇ ·Πe)− 1c2
∂ 2
∂ t2 Πe −μ0∇×(
∂∂ t
Πh
), (4)
H = ε0∇×(
∂∂ t
Πe
)+∇(∇ ·Πh)− 1
c2
∂ 2
∂ t2 Πh, (5)
where E and H are the electric field and magnetic field vectors, Πe = AeΨn(ρ ,φ ,ζ )z and Πh =AhΨn(ρ ,φ ,ζ )z are the electric and magnetic Hertz vector potentials, Ae and Ah are arbitrarycomplex amplitudes, z is the unit vector in the z-direction, and μ0 and ε0 are the free-spacepermeability and permittivity. The expressions (4) and (5) are thus the generalized vector formof the electromagnetic X-Wave.
To better illustrate the Poynting vector behavior of vector X-Waves, we will only consider thespecial case of the spectrum Eq. (1) with m = 0; which corresponds to the zero-order X-Wave[17]. Furthermore, we focus on the solution with the azimuthal dependance n= 1 to simplify themathematical analysis without any loss of generality, while preserving the contribution of bothHertz potentials, Πe and Πh, to each transverse component of the fields, E and H. It shouldbe noted that such axially asymmetric solutions are essential to produce the negative energyflux density. Accordingly, the generalized scalar solution of the X-Wave Eq. (3) reduces to itszero-order expression as
Ψ1(ρ ,φ ,ζ ) = 2eiφ√
1+η −1γρ
√1+η
.
We should note here that in previous studies of vector X-Waves, only the real part of theHertz vector potential is kept [4, 6, 7] as it exhibits a symmetric localization in space and timearound the centroid. In the following analysis, we keep the complex vector potentials, aftermultiplication with the complex amplitudes to manipulate the contribution of the real and the
#142497 - $15.00 USD Received 10 Feb 2011; revised 26 Mar 2011; accepted 29 Mar 2011; published 18 Apr 2011(C) 2011 OSA 25 April 2011 / Vol. 19, No. 9 / OPTICS EXPRESS 8528
imaginary parts in the scalar solution, and finally retain only the real part of the electric andmagnetic fields, viz.
Eρ(ρ ,φ ,ζ ) = 2ℜ{
iγeiφ
X (5/2)[ΞAe + iμ0VXAh]
}, (6)
Eφ (ρ ,φ ,ζ ) = 2ℜ{
iγeiφ
X (5/2)[iXAe −V μ0ΞAh]
}, (7)
Ez(ρ ,φ ,ζ ) = 6ℜ{
γ3ρeiφ
X2√
1+ηAe
}, (8)
Hρ(ρ ,φ ,ζ ) = 2ℜ{
eiφ γX (5/2)
[iΞAh +Vε0XAe]
}, (9)
Hφ (ρ ,φ ,ζ ) = 2ℜ{
iγeiφ
X (5/2)[iXAh +Vε0ΞAe]
}, (10)
Hz(ρ ,φ ,ζ ) = 6ℜ{
γ3ρeiφ
X2√
1+ηAh
}, (11)
where X = (γρ)2 + τ2, Ξ = τ2 −2(γρ)2 and ℜ{F} is the real part of F .
3. Energy flow characteristics
In this section, we study the energy flow characteristics of the zero-order full vector X-Wave.This is achieved via the characterization of the Poynting vector, which is given by S = E×Hand represents the energy flux density. By considering only the Poynting vector at the centroidof the X-Wave (ζ = 0), we can write the expressions of its vector components in explicit formusing Eqs. (6)–(11) as
Sρ(ρ ,φ) =6γ4ρaV 2ξ
χ5
{sin(2φ)
[ε0ℜ{A2
e}+μ0ℜ{A2h}]
+2cos(2φ)[ε0AR
e AIe +μ0AR
h AIh
]}, (12)
Sφ (ρ ,φ) =6γ4ρaV
χ5
{2ξ
[AR
h AIe −AI
hARe
]
+V χ[ε0|Ae|2 +μ0|Ah|2
]
+V χ cos(2φ)[ε0ℜ{A2
e}+μ0ℜ{A2h}]
−V χ sin(2φ)[ε0ℑ{A2
e}+μ0ℑ{A2h}]}
, (13)
Sz(ρ ,φ) =2γ2
χ5
{2(γ2 +2)χξ
[AI
eARh −AR
e AIh
]
+V(χ2 +ξ 2)[ε0|Ae|2 +μ0|Ah|2
]
−V(ξ 2 − χ2)cos(2φ)
[ε0ℜ{A2
e}+μ0ℜ{A2h}]
+V(ξ 2 − χ2)sin(2φ)
[ε0ℑ{A2
e}+μ0ℑ{A2h}]}
. (14)
Here, χ = (aV )2 +(γρ)2, ξ = (aV )2 −2(γρ)2 and ℑ{F} is the imaginary part of F .We note here that even though the Poynting vector components given in Eqs. (12)–(14) ac-
quire negative values at certain values of ρ and φ , this does not necessary ensure negativeenergy flow. In order to determine the direction of the energy flow, we compute the energy fluxvector P by integrating the Poynting vector over the transverse plane as
P(ρ0) =∫ ρ0
0dρ
∫ π
−πdφρS(ρ ,φ), (15)
#142497 - $15.00 USD Received 10 Feb 2011; revised 26 Mar 2011; accepted 29 Mar 2011; published 18 Apr 2011(C) 2011 OSA 25 April 2011 / Vol. 19, No. 9 / OPTICS EXPRESS 8529
The total energy flux at the centroid is computed using Eq. (15) when ρ0 → ∞ as
Pρ = 0, (16)
Pφ =3π2γ
32(aV )4
{4V
[ε0|Ae|2 +μ0|Ah|2
]+(AI
hARe −AI
eARh
)}, (17)
Pz =3π
2a4V 3
[ε0|Ae|2 +μ0|Ah|2
](18)
From Eq. (16), we deduce that the there is no net energy flow in the radial direction and thecentroid propagates rigidly without changing its shape, irrespective of the amplitudes of thepolarizations, thus the fields maintain their transverse localization indefinitely. The total en-ergy flux in the φ -direction acquires the same sign of the quantity 4V
[ε0|Ae|2 +μ0|Ah|2
]+(
AIhAR
e −AIeA
Rh
)according to Eq. (17), thus facilitating the manipulation of the rotational mo-
mentum at the centroid by the choice of Ae and Ah values. Moreover, it can be shown fromEq. (15), that Pφ (ρ0) does not change its sign with ρ0. The total energy flow in z-direction isalways in the positive z-direction as inferred from Eq. (18). However, following the conceptpresented in [12] if we aperture the propagating X-Wave by a fully absorbing screen, such thatthe propagation axis of the X-Wave coincides with the center of the aperture with radius ρ0, wecan obtain non-positive values for the energy flux.
Fig. 1. A comparison between the z-component of the Poynting vector at the centroid planeof the zero-order vector X-Wave with n = 1, a = 2×10−16s and V = 1.5c for two differentconfigurations. Configuration 1 (a) is for the amplitudes Ae = 1/
√ε0 and Ah = 1/
√μ0; andConfiguration 2 (b) is for the amplitudes Ae = 1/
√ε0 and Ah = i/
√μ0.
Figure 1 depicts the z-component of the Poynting vector of the zero-order vector X-Wavewith a = 2× 10−16s and V = 1.5c at its centroid computed by Eq. (14) for two different su-perpositions of the TE and TM amplitudes, with Ae = 1/
√ε0 and Ah = 1/
√μ0 in Fig. 1(a)(Configuration 1), and Ae = 1/
√ε0 and Ah = i/
√μ0 in Fig. 1(b) (Configuration 2). The figureshows the local variations of Sz and that it changes its sign for both configurations. Figure 2presents the energy flux in the z-direction computed by Eq. (15) for the same X-Wave withthe same amplitude configurations as in Fig. 1. It is clear from the figure that backward prop-agation is only possible in the second configuration. The necessary condition to have a neg-ative energy flux in the z-direction can be derived from Eq. (14) as B < −V/(γ2 + 2), whereB = (AI
eARh −AI
hARe )/(ε0|Ae|2 +μ0|Ah|2). We should note here that this condition depends only
on the choice of the X-Wave peak velocity V in addition to the choice of the amplitudes of theTE- and TM-polarizations. Additionally, as the zeros of Pz as a function of ρ0 are the roots of a
#142497 - $15.00 USD Received 10 Feb 2011; revised 26 Mar 2011; accepted 29 Mar 2011; published 18 Apr 2011(C) 2011 OSA 25 April 2011 / Vol. 19, No. 9 / OPTICS EXPRESS 8530
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10−8
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
ρ0 (m)
Pz (
ρ 0) [a
.u.]
Configuration 1Configuration 2
ρ0 max
Fig. 2. A comparison between the net energy flux in the z-direction in a finite circular crosssection with radius ρ0 of the zero-order vector X-Wave with n = 1, a = 2× 10−16s andV = 1.5c. Configuration 1 (dashed line) is for the amplitudes Ae = 1/
√ε0 and Ah = 1/
√μ0;and Configuration 2 (solid line) is for the amplitudes Ae = 1/
√ε0 and Ah = i/
√μ0.
polynomial, the necessary condition to have a negative energy flux implies that there exists onlyone real root greater than zero on the ρ0-axis. This root, ρ0max, designates the upper exclusivelimit of the truncation radius to obtain a net negative energy flux in the z-direction and is givenby
ρ0max =aV√3γ
√−4− 4×21/3(P−2)
Q+22/3Q,
where Q = (−16+ 3P+ i√
P[P[183−32P]−288])1/3, P = 1+(γ2 + 1)B/V , P < 0 and thepositive values of the roots are to be chosen. Figure 2 also shows the location of ρ0max and thatPz is positive for ρ0 > ρ0max.
We should clarify here that while truncation is necessary for physical realization of (approx-imations to) the infinite-energy type LWs such as the X-Wave under consideration here [18],the effects of the negative energy flux can be observed without truncation. Consider a particlelocated in the vicinity of the centroid of an X-Wave with negative energy flux density. The localbehavior of the field near the particle will be similar to that of a wave field with negative energyflux; and the particle will experience the effects of being illuminated by a field with negativeenergy flow. Truncating the X-Wave by a perfect absorbing screen with an aperture with a ra-dius less than ρ0max results in the elimination of most of the ‘X-shaped arms’ while maintainingthe highly localized central region, which will carry energy that propagates in the reverse di-rection. In contrast to the truncated Bessel beam [12], where the negative propagating portionof the beam is highly diffractive, an X-Wave could be designed, by tweaking its parameters tochange ρ0max, where the reflected portion maintains its localization for a greater propagationdistance [18, 19].
4. Conclusions
In this study we have revealed a novel peculiar characteristic of full vector X-Waves - negativeor zero energy flow along the propagation direction. By studying the expressions of the Poynt-ing vector at the centroid of the zero-order X-Wave with first-order azimuthal dependence, weshowed that all of its components can attain negative values over bounded areas of the centroid
#142497 - $15.00 USD Received 10 Feb 2011; revised 26 Mar 2011; accepted 29 Mar 2011; published 18 Apr 2011(C) 2011 OSA 25 April 2011 / Vol. 19, No. 9 / OPTICS EXPRESS 8531
plane. Expressions of the total energy flux assert that the field maintains its transverse localiza-tion indefinitely as there is no net flux in the radial direction. Along the azimuthal direction,the energy flux could rotate in either direction depending on the choice of the polarizations am-plitudes. Whereas along longitudinal direction, the total energy flux is always positive, thus nonegative propagation could be achieved in a homogeneous medium. Nevertheless, the analysisof the Poynting vector expressions suggests that by truncating the wave field by a circular aper-ture in a perfectly absorbing screen in the radial direction, there exists a condition that permitsnegative energy flow in the longitudinal direction. No similar condition exists for the energyflux in the azimuthal direction.
It follows directly from the analysis, that objects placed in the vicinity of the peak of the X-Waves would interact with the local field and experience the effects of the negative energy fluxdensity (even without truncation). These peculiar energy flux characteristics might be of use inapplications, where manipulation of micro- and nanoparticles using electromagnetic fields dueto the localized nature of the wave field, its strong gradient, and the flexibility in manipulatingthe propagation and energy flow characteristics of X-Waves, are needed. Among such applica-tions are electromagnetic and optical tweezers and traps, where the variation in the Poyntingvector could be used to exert force on the particles and manipulate their locations and momenta.The negative energy flux characteristics of the X-Waves could also be used in the detection oftransformation optics invisibility cloaks.
#142497 - $15.00 USD Received 10 Feb 2011; revised 26 Mar 2011; accepted 29 Mar 2011; published 18 Apr 2011(C) 2011 OSA 25 April 2011 / Vol. 19, No. 9 / OPTICS EXPRESS 8532
