Energy flow characteristics of vector X-Waves

Mohamed A. Salem; Hakan Bağcı

doi:10.1364/OE.19.008526

1. Introduction

X-Waves are a class of Localized Wave (LW) solutions to the scalar wave equation, which have been extensively studied in the literature (see [1] for a historical review), while studies of LW solutions to the vector wave equation have focused only on constructing solutions using a single polarization type, mostly the transverse electric (TE)-polarization (see, [2–6] and references therein). In this letter, we work with a full vector representation of X-Waves obtained by superimposing localized TE and transverse magnetic (TM) polarization solutions of the wave equation. To the best of our knowledge, this is the first time where such a full vector representation is used in the context of LWs.

We show that the amplitudes of the Poynting vector components of full vector X-Waves can change their signs locally. We study this peculiar phenomenon and further show that it does not stem from the nature of the scalar X-Wave solution, but it is the result of the weighted superposition of TE- and TM-polarization components. We further show that this phenomenon does not 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-Wave at the centroid plane (the plane with maximum localization in the transverse direction) and derive the necessary conditions to obtain negative or zero energy flux density in the longitudinal direction in the vicinity of the centroid. Objects interacting with the X-Wave in this region may experience negative or zero energy flux. This particular result might be of importance for various electromagnetic and optical applications, such as traps and tweezers, where the behavior of micro- and nanoparticles are manipulated by the changing Poynting vector [8, 9], and detection of transformation optics invisibility cloaks [10, 11]. Furthermore, to illustrate the possibility of 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 are truncated in the transverse plane with an apertured perfect absorbing screen.

While X-Waves could generally be expressed as weighted frequency superposition of Bessel beams [1], we assert that the Poynting vector of full vector X-Waves behaves different from that of Bessel beams [12] as it maintains a localized profile in the transverse plane. We also show that the peculiar energy flow characteristics could be achieved in the time domain, not only in the 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 weighted by an appropriate spectrum that preserves the necessary undistorted propagation condition, ω = Vk_z + α, over the spectral variables k_ρ,k_z and ω. Here, V is the centroid velocity, α is a positive real parameter that quantifies the periodicity of local deformations [1], 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_ρ and k_z in the transverse and longitudinal directions, respectively, and c is the speed of light in free-space. In what follows, we employ the ‘standard’ LW spectrum [1],

{\tilde{Ψ}}_{n} (k_{ρ}, k_{z}, ω; m, α) = 2^{n} {(k_{z} + \frac{α}{V})}^{m} e^{- a V k_{z}} e^{- a α} δ (k_{ρ}^{2} - [\frac{ω^{2}}{c^{2}} - k_{z}^{2}]) δ (ω - [V k_{z} + α]),

which yields an 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, 13], thus any observed negative flow of energy would be uniquely due to the mixing of TE- and TM-polarizations.

Hence, the generalized scalar X-Wave solution in cylindrical coordinates (ρ,φ,z) in its integral form reads

Ψ_{n} (ρ, ϕ, z, t) = \int_{0}^{\infty} d k_{z} \int_{- \infty}^{\infty} d ω \int_{0}^{\infty} d k_{ρ} {\tilde{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)} .

The integral (2) is carried out analytically using formula (6.621.1) in [14], yielding

Ψ_{n} (ρ, ϕ, ζ) = e^{i n ϕ} \frac{{(γ ρ)}^{n}}{τ^{1 + m + n}} \frac{(m + n)!}{n!}_{2} F_{1} (\frac{(1 + m + n)}{2}, \frac{(2 + m + n)}{2}; 1 + n; - η),

where

γ = \sqrt{{(V / c)}^{2} - 1}

, 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.

Next, we derive the X-Wave solution to the vector wave equation using single-component Hertz vector potentials as [16]

E = \nabla (\nabla \cdot Π_{e}) - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} Π_{e} - μ_{0} \nabla \times (\frac{\partial}{\partial t} Π_{h}),

H = ɛ_{0} \nabla \times (\frac{\partial}{\partial t} Π_{e}) + \nabla (\nabla \cdot Π_{h}) - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} Π_{h},

where E and H are the electric field and magnetic field vectors, Π_e = A_eΨ_n(ρ,φ,ζ)ẑ and Π_h = A_hΨ_n(ρ,φ,ζ)ẑ are the electric and magnetic Hertz vector potentials, A_e and A_h are arbitrary complex amplitudes, 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.

To better illustrate the Poynting vector behavior of vector X-Waves, we will only consider the special 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 the mathematical analysis without any loss of generality, while preserving the contribution of both Hertz potentials, Π_e and Π_h, to each transverse component of the fields, 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

Ψ_{1} (ρ, ϕ, ζ) = 2 e^{i ϕ} \frac{\sqrt{1 + η - 1}}{γ ρ \sqrt{1 + η}} .

We should note here that in previous studies of vector X-Waves, only the real part of the Hertz vector potential is kept [4, 6, 7] as it exhibits a symmetric localization in space and time around the centroid. In the following analysis, we keep the complex vector potentials, after multiplication with the complex amplitudes to manipulate the contribution of the real and the imaginary parts in the scalar solution, and finally retain only the real part of the electric and magnetic fields, viz.

E_{ρ} (ρ, ϕ, ζ) = 2 ℜ {\frac{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 {\frac{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 {\frac{γ^{3} ρ e^{i ϕ}}{X^{2} \sqrt{1 + η}} A_{e}},

H_{ρ} (ρ, ϕ, ζ) = 2 ℜ {\frac{e^{i ϕ} γ}{X^{(5 / 2)}} [i Ξ A_{h} + V ɛ_{0} X A_{e}]},

H_{ϕ} (ρ, ϕ, ζ) = 2 ℜ {\frac{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 {\frac{γ^{3} ρ e^{i ϕ}}{X^{2} \sqrt{1 + η}} A_{h}},

where X = (γρ)² + τ ², Ξ = τ ² – 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 × 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

\begin{array}{l} S_{ρ} (ρ, ϕ) = & 6 \frac{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}]}, \end{array}

\begin{array}{l} S_{ϕ} (ρ, ϕ) & = 6 \frac{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}}]}, \end{array}

\begin{array}{l} S_{z} (ρ, ϕ) & = 2 \frac{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}}]} . \end{array}

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) acquire negative values at certain values of 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

P (ρ_{0}) = \int_{0}^{ρ_{0}} d ρ \int_{- π}^{π} d ϕ ρ S (ρ, ϕ),

The total energy flux at the centroid is computed using Eq. (15) when ρ ₀ → ∞ as

P_{ρ} = 0,

P_{ϕ} = \frac{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} = \frac{3 π}{2 a^{4} V^{3}} [ɛ_{0} | A_{e} |^{2} + μ_{0} | A_{h} |^{2}]

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

4 V [ɛ_{0} | A_{e} |^{2} + μ_{0} | A_{h} |^{2}] + (A_{h}^{I} A_{e}^{R} - A_{e}^{I} A_{h}^{R})

according to Eq. (17), thus facilitating the manipulation of the rotational momentum at the centroid by the choice of A_e and A_h values. Moreover, it can be shown from Eq. (15), that 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] if we aperture the propagating X-Wave by a fully absorbing screen, such that the propagation axis of the X-Wave coincides with the center of the aperture with radius ρ ₀, we can obtain non-positive values for the energy flux.

Figure 1 depicts the z-component of the Poynting vector of the zero-order vector X-Wave with a = 2 × 10⁻¹⁶ s and V = 1.5c at its centroid computed by Eq. (14) for two different superpositions of the TE and TM amplitudes, with $A_{e} = 1 / \sqrt{ɛ_{0}}$ and $A_{h} = 1 / \sqrt{μ_{0}}$ in Fig. 1(a) (Configuration 1), and $A_{e} = 1 / \sqrt{ɛ_{0}}$ and $A_{h} = i / \sqrt{μ_{0}}$ in Fig. 1(b) (Configuration 2). The figure shows the local variations of S_z and that it changes its sign for both configurations. Figure 2 presents the energy flux in the 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 $B = (A_{e}^{I} A_{h}^{R} - A_{h}^{I} A_{e}^{R}) / (ɛ_{0} | A_{e} |^{2} + μ_{0} | A_{h} |^{2})$ . 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 the TE- and TM-polarizations. Additionally, as the zeros of P_z as a function of 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 _{0_max}, designates the upper exclusive limit of the truncation radius to obtain a net negative energy flux in the z-direction and is given by

ρ_{0_{max}} = \frac{a V}{\sqrt{3} γ} \sqrt{- 4 - \frac{4 \times 2^{1 / 3} (P - 2)}{Q} + 2^{2 / 3} Q},

where

Q = {(- 16 + 3 P + i \sqrt{P [P [183 - 32 P] - 288]})}^{1 / 3}

, 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 _{0_max} and that P_z is positive for ρ ₀ > ρ _{0_max}.

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 $A_{e} = 1 / \sqrt{ɛ_{0}}$ and $A_{h} = 1 / \sqrt{μ_{0}}$ ; and Configuration 2 (b) is for the amplitudes $A_{e} = 1 / \sqrt{ɛ_{0}}$ and $A_{h} = i / \sqrt{μ_{0}}$ .

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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 $A_{e} = 1 / \sqrt{ɛ_{0}}$ and $A_{h} = 1 / \sqrt{μ_{0}}$ ; and Configuration 2 (solid line) is for the amplitudes $A_{e} = 1 / \sqrt{ɛ_{0}}$ and $A_{h} = i / \sqrt{μ_{0}}$ .

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We should clarify here that while truncation is necessary for physical realization of (approximations 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 particle located in the vicinity of the centroid of an X-Wave with negative energy flux density. The local behavior of the field near the particle will be similar to that of a wave field with negative energy flux; and the particle will experience the effects of being illuminated by a field with negative energy flow. Truncating the X-Wave by a perfect absorbing screen with an aperture with a radius less than ρ _{0_max} 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], where the negative propagating portion of the beam is highly diffractive, an X-Wave could be designed, by tweaking its parameters to change ρ _{0_max}, where the reflected portion maintains its localization for a greater propagation distance [18, 19].

4. Conclusions

In this study we have revealed a novel peculiar characteristic of full vector X-Waves - negative or zero energy flow along the propagation direction. By studying the expressions of the Poynting vector at the centroid of the zero-order X-Wave with first-order azimuthal dependence, we showed that all of its components can attain negative values over bounded areas of the centroid plane. Expressions of the total energy flux assert that the field maintains its transverse localization 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 amplitudes. Whereas along longitudinal direction, the total energy flux is always positive, thus no negative propagation could be achieved in a homogeneous medium. Nevertheless, the analysis of the Poynting vector expressions suggests that by truncating the wave field by a circular aperture in a perfectly absorbing screen in the radial direction, there exists a condition that permits negative energy flow in the longitudinal direction. No similar condition exists for the energy flux 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 flux density (even without truncation). These peculiar energy flux characteristics might be of use in applications, where manipulation of micro- and nanoparticles using electromagnetic fields due to the localized nature of the wave field, its strong gradient, and the flexibility in manipulating the propagation and energy flow characteristics of X-Waves, are needed. Among such applications are electromagnetic and optical tweezers and traps, where the variation in the Poynting vector 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 of transformation optics invisibility cloaks.

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Energy flow characteristics of vector X-Waves

Abstract

1. Introduction

2. Full vector X-Waves

3. Energy flow characteristics

4. Conclusions

References and links

Cited By

Figures (2)

Equations (20)

Optics Express