Transverse circular-polarized Bessel beam generation by inward cylindrical aperture distribution

S. C. Pavone; M. Ettorre; M. Casaletti; M. Albani

doi:10.1364/OE.24.011103

1. Introduction

In recent years growing interest has been gained by the so-called diffractionless beams, among which Bessel beams assume a relevant importance, since their early introduction by Durnin [1]. On the other hand, several attempts have been made to practically generate Bessel beams [2,3], both at optical frequencies [4–6] and at radio frequencies [7–9].

Whereas Bessel beam launchers usually adopt a standing wave (Bessel function) aperture distribution [10], in [11–13] it has been shown that Bessel beams can be alternatively generated synthesizing an inward cylindrical traveling wave (Hankel functions) aperture distribution. Namely, in [13] a tangential electric field polarized along the radial unit vector ρ̂ and shaped as an inward cylindrical traveling wave $H_{1}^{(1)} (k_{ρ a} ρ)$ produces a Bessel beam with the longitudinal component (i.e. normal to the radiating aperture) of the electric field shaped as J₀(k_ρaρ)e^−jk_zaz, in a well-defined conical region centered on the axis of symmetry of the radiating aperture. With reference to [14], the excited vector Bessel beam is the N₀ TM vector wave function.

In this paper, we extend the results obtained in [13], showing that a tangential electric field aperture distribution polarized along a fixed unit vector p̂ generates a Bessel beam with the same arbitrary transverse polarization p̂. Again, with reference to [14], the excited vector Bessel beam is a proper superposition of M_±1 and N_±1 vector wave functions (see example 4 in [14] for vector Bessel beam linearly polarized along p̂ = x̂).

As in [13], the electromagnetic field radiated by an infinite aperture is calculated in a closed form and split in its Geometrical Optics (GO) and Space Wave (SW) contributions, that is written in closed form in terms of the incomplete Hankel functions [15] and can be asymptotically interpreted as a space wave arising from the aperture center. In addition, the case of finite radiating aperture is analyzed numerically, showing that the finiteness of the aperture implies a maximum non-diffractive range, beyond which the Bessel beam starts diffracting.

Finally, to implement a circular-polarized (CP) Bessel beam launcher with a transverse inward cylindrical traveling wave equivalent aperture distribution, a radial parallel plate waveguide (PPW) loaded with several slot pairs arranged on a spiral is designed and optimized. The slot pairs are orthogonally located and excited in quadrature [16], thus allowing the generation of a circular-polarized field.

The presented low profile launcher can profitably find practical application in the field of medical imaging and of ground penetrating radars (GPR), both at millimeter waves and at Terahertz.

2. Analytical formulation for infinite apertures

The geometry of the problem is shown in Fig. 1. An inward cylindrical traveling wave is assumed for the tangential electric field distribution over the radiating aperture E_t(ρ, z = 0) = E_t(ρ, z = 0)p̂, where p̂ is a complex unit vector identifying a generic polarization state, namely linear, circular or elliptical. A time dependence e^jωt, ω = 2πf being the angular frequency, is assumed and suppressed. The electric field radiated by such a tangential electric field aperture distribution [17] at the observation point r = ρ + zẑ, with ρ = ρρ̂ = ρ(cosϕx̂ + sinϕŷ), is

E (ρ, z) = \frac{1}{4 π} \int_{\infty e^{- j π}}^{+ \infty} {\tilde{E}}_{t} (k_{ρ}) [H_{0}^{(2)} (k_{ρ} ρ) \hat{p} + j \frac{k_{ρ}}{k_{z}} H_{1}^{(2)} (k_{ρ} ρ) (\hat{ρ} \cdot \hat{p}) \hat{z}] e^{- j k_{z} z} k_{ρ} d k_{ρ},

{\tilde{E}}_{t} (k_{ρ}) = 2 π \int_{0}^{+ \infty} E_{t} (ρ, z = 0) J_{0} (k_{ρ} ρ) ρ d ρ,

where k is the free-space wavenumber, k_ρ and

k_{z} = \sqrt{k^{2} - k_{ρ}^{2}}

are the radial and the longitudinal spectral variables, whereas J_n() and

H_{n}^{(m)} ()

are the n-order Bessel and Hankel functions of the m-th kind, respectively. By assuming an inward cylindrical traveling wave aperture distribution

E_{t} (ρ, z = 0) = H_{0}^{(1)} (k_{ρ a} ρ)

, with k_ρa = k sinθ_a (Fig. 2(a)), its Hankel transform (2) is calculated in closed form as

{\tilde{E}}_{t} (k_{ρ}) = - \frac{4 j}{k_{ρ}^{2} - k_{ρ a}^{2}} .

By following the analytical procedure outlined in the Appendix, the radiated field is written as the sum of the GO and SW contributions, namely

E (r) = E^{GO} (r) + E^{SW} (r),

where

\begin{array}{l} E^{GO} (ρ, z) = [2 J_{0} (k_{ρ a} ρ) U (θ_{a} - θ) + H_{0}^{(1)} (k_{ρ a} ρ) U (θ - θ_{a})] e^{- j k_{z a} z} \hat{p} \\ + j \frac{k_{ρ a}}{k_{z a}} [2 J_{1} (k_{ρ a} ρ) U (θ_{a} - θ) + H_{1}^{(1)} (k_{ρ a} ρ) U (θ - θ_{a})] e^{- j k_{z a} z} (\hat{ρ} \cdot \hat{p}) \hat{z}, \end{array}

and

\begin{array}{l} E^{SW} (r) = [H_{0}^{(2)} (k_{ρ a} ρ, w_{0}^{+}) e^{j k_{z a} z} + sgn (w_{0}^{-}) H_{0}^{(2)} (k_{ρ a} ρ, | w_{0}^{-} |) e^{- j k_{z a} z}] \hat{p} \\ - j \frac{k_{ρ a}}{k_{z a}} [sgn (w_{0}^{-}) H_{1}^{(2)} (k_{ρ a} ρ, | w_{0}^{-} |) e^{- j k_{z a} z} - H_{1}^{(2)} (k_{ρ a} ρ, w_{0}^{+}) e^{- j k_{z a} z}] (\hat{ρ} \cdot \hat{p}) \hat{z} \\ ~ \frac{2}{π k} \frac{1}{{cos}^{2} θ_{a} - {cos}^{2} θ} \frac{e^{- j k r}}{r} \hat{r} \times (\hat{p} \times \hat{z}), \end{array}

in which r̂ = sinθ cosϕx̂ + sinθ sinϕŷ + cosθẑ,

k_{z a} = \sqrt{k^{2} - k_{ρ a}^{2}} = k cos θ_{a}

,

H_{m}^{(n)} (Ω, w_{0})

are the m-th order, n-th kind incomplete Hankel functions [15],

w_{0}^{\pm} = {tanh}^{- 1} cos θ \pm {tanh}^{- 1} cos θ_{a}

, and U(·) denotes the unit step function.

Fig. 1 Pictorial representation of a tangential field distribution directed along a fixed p̂ unit vector. The reference system for the subsequent formulation is highlighted.

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Fig. 2 (a) Schematic representation of the electric field transverse component, according to the analytical formulation for infinite apertures. In the yellow region, a Bessel profile J₀(k_ρaρ) is achieved in the transverse component of the electric field (θ < θ_a), whereas in the red regions an inward cylindrical traveling wave aperture distribution is present. (b) Map of the magnitude of the transverse component of the electric field in a vertical plane orthogonal to the radiating aperture.

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The GO field comprises a longitudinal (p̂) and a transverse (ẑ) part, showing that an infinite aperture is able to generate a non-diffractive Bessel beam with the transverse component of the electric field shaped as a zero-th order Bessel beam in the region θ < θ_a around the z-axis (Fig. 2(a)). The SW contribution is also written in a closed-form by means of the incomplete Hankel functions, as in [13, 15], and its asymptotic evaluation reveals that it behaves as a spherical wave with an elevation pattern. On the shadow boundary (θ = θ_a) the non-uniform asymptotic evaluation fails, thus predicting a diverging field, whereas the exact expression provides the correct transitional behaviour of the SW. It is worth noting that, as revealed by its asymptotic expression, the SW contribution to the field decays as 1/r moving far from the radiating aperture, thus the most significant term at a sufficiently large distance from the aperture is the GO contribution. Figure 2(b) shows the magnitude of the transverse p̂-component of the electric field calculated accordingly to (5)–(6) for f = 60GHz and θ_a = 10°.

It is worth noting that once the polarization state p̂ is defined, similar results for the radiated field can be achieved using the vector approach proposed in [14]. However, such approach requires a combination of cylindrical modes to generate a defined Bessel beam. The proposed technique only requires an inward cylindrical aperture field distribution to generate arbitrary polarized Bessel beams.

In the next Section, the extension to the case of finite radiating aperture is discussed numerically.

3. Numerical field evaluation for finite apertures

The radiation of a finite inward cylindrical aperture distribution has been treated numerically as in [13]. The Hankel transform of the tangential electric field aperture distribution becomes

{\tilde{E}}_{t} (k_{ρ}) = - \frac{2}{k_{ρ}^{2} - k_{ρ a}^{2}} [2 j + π k_{ρ a} a H_{1}^{(1)} (k_{ρ a} a) J_{0} (k_{ρ} a) - π k_{ρ} a H_{0}^{(1)} (k_{ρ} a) J_{1} (k_{ρ} a)],

in which a is the radius of the radiating aperture.

The electromagnetic field radiated by such a distribution can be calculated by the numerical evaluation of the integral in (1). In Fig. 3(a) the transverse component of the electric field is schematically depicted for the case of a finite radiating aperture with radius a = 15λ at the operating frequency f = 60GHz, by imposing θ_a = 10°, then k_ρa = k sinθ_a ∼ 0.17k. Figure 3(b) provides the numerical evaluation of the radiated field for the same aperture.

Fig. 3 (a) Schematic illustration of the transverse component of the electric field radiated by a finite aperture implementing an inward cylindrical traveling wave distribution. In the central yellow region, the Bessel beam is generated, whereas in the red (resp. blue) regions an inward (resp. outward) cylindrical traveling wave is present. (b) Map of the magnitude of the transverse component of the electric field in a vertical plane orthogonal to the radiating aperture, obtained by evaluating numerically the integral (1), in which the expression of the tangential electric field aperture distribution for finite apertures (7) is used. (c) Magnitude of the transverse component of the electric field on the longitudinal z-axis at ρ = 0.

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As in [13], the Bessel beam is established inside the highlighted region close to the aperture axis of symmetry up to a finite distance, called non-diffractive range, namely NDR = acotθ_a ∼ 85λ. Beyond that distance the beam starts spreading out, thus the assumption of diffractionless Bessel beam is no more valid.

It is worth noting that on the longitudinal z-axis (ρ = 0), the field amplitude oscillates, as clear from Fig. 3(c). Such a phenomenon is due to the finiteness of the radiating aperture and consequent on-axis interference of the diffracted rays from the aperture rim.

4. Implementation of a circular-polarized Bessel beam launcher

In this Section, a practical implementation of a circular-polarized Bessel beam launcher is presented. A radial parallel-plate waveguide (PPW) filled with a foam of relative dielectric permittivity ε_r = 1.04 and thickness h = 1.575mm, loaded with several radiating slot pairs has been designed in order to synthesize an inward cylindrical wave aperture distribution of the form $E_{t} (ρ, z = 0) = H_{0}^{(1)} (k_{ρ a} ρ) \hat{p}$ , with $\hat{p} = (\hat{x} - j \hat{y}) / \sqrt{2}$ denoting the right-handed circular polarization (RHCP) unit vector. The excited vector Bessel beam polarization is in turn a RHCP, and its expansion in terms of vector wave functions [14] comprises a superposition of the M₁ and N₁; namely the expansion coefficients of (3) of [14] are $α_{n}^{(+)} = α δ_{n 1}$ and $α_{n}^{(-)} = 0$ , with δ_nm denoting the Kronecker symbol and α an arbitrary amplitude. Each pair comprises two orthogonal slots (making an angle of ±45° with respect to the radial direction), which are $λ_{d} / 4 = λ_{0} / (4 \sqrt{ε_{r}})$ apart to be fed in quadrature by the outward feeding wave in the PPW, thus radiating a RHCP (Fig. 4(a)). In addition, the slots are arranged along a spiral to recreate the same phase profile of the target inward traveling wave aperture distribution (Fig. 4(b)). The design parameters are those used in Sec. 3 for the numerical analysis of finite radiating devices.

Fig. 4 (a) Pair of radiating slots, generating a circular-polarized field. (b) Pictorial illustration of the radial PPW loaded with several slot pairs generating a circular-polarized radiated field. The geometry of the each slot pair is highlighted and the total equivalent magnetic dipole moment of the slots M_tot(ρ, ϕ) = M_in(ρ, ϕ) + M_out(ρ, ϕ) is graphically shown.

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The optimization technique developed in [16] and profitably used in [13], together with an in-house Method of Moments for the analysis of the electromagnetic field distribution [18] have been used to synthesize the focusing system.

The co-polar component (RHCP) of the electric field is presented together with its cross-polar component (LHCP), both in the ρ − z plane (Fig. 5(a)–(b)) and in the transverse x − y plane at z = NDR/2 = 42.5λ (Fig. 5(c)–(d)), showing an excellent polarization purity of the launcher. By comparing Fig. 5(a) to Fig. 3(b), a clear consistency between the synthesized field and the theoretical model is revealed.

Fig. 5 (a,c) Co-polar (RHCP) and (b,d) cross-polar (LHCP) components of the electric field radiated by the designed focusing system, evaluated in the ρ − z (a,b) and x − y (c,d) planes at z = NDR/2. The cross-polar component is almost everywhere less than −20dB, demonstrating a good rejection of the undesired cross-polarization.

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5. Conclusion

In this paper, the capability of an inward cylindrical traveling wave aperture distribution directed along a fixed unit vector p̂ to generate a non-diffractive Bessel beam in the transverse component of the electric field has been demonstrated. Moving from the analytical formulation for an infinite radiating aperture, the electromagnetic field has been split into the GO and SW contributions, showing that the SW contribution can be written in a closed form by using the definition of the incomplete Hankel functions and asymptotically reveals its spherical wave nature. Moreover, the analytical model for infinite apertures has been extended numerically to the case of finite radiating aperture. In the second part, a CP Bessel beam launcher has been designed and optimized. The achieved results clearly show a nice agreement with the theoretical analysis and a remarkable polarization purity.

Appendix

By using the vector potential F(r), according to its definition [19], the electric field can be calculated by differentiation as

E (r) = - \frac{1}{ε} \nabla \times F (r),

where

F (r) = ε \iint_{slots} E (r^{'}) \times \hat{z} \frac{e^{- j k R}}{2 π R} d S^{'},

in which R = |r − r′| is the distance between the source and the observation point.

The integral in (9) is the convolution of the tangential electric field and the scalar Green’s function for an unbounded semi-infinite medium. In the spectral domain, (1) can be written as (z > 0)

F (r) = \frac{ε}{4 π j} \hat{p} \int_{\infty e^{- j π}}^{+ \infty} {\tilde{E}}_{t} (k_{ρ}) H_{0}^{(2)} (k_{ρ} ρ) \frac{e^{- j k_{z} z}}{k_{z}} k_{ρ} d k_{ρ},

that can be related to the scalar potential S(r) introduced in [13] by the relation F(r) = εk_ρaS(r)(p̂ × ẑ), therefore the GO and SW contributions of the potential can be easily calculated by following the Appendix of [13] as

F^{GO} (r) = \frac{ε}{j k_{z a}} e^{- j k_{z a} z} [2 J_{0} (k_{ρ a} ρ) U (θ_{a} - θ) + H_{0}^{(1)} (k_{ρ a} ρ) U (θ - θ_{a})] (\hat{p} \times \hat{z}),

and

F^{SW} (r) = \frac{ε}{j k_{z a}} [H_{0}^{(2)} (k_{ρ a} ρ, w_{0}^{+}) e^{j k_{z a} z} - sgn (w_{0}^{-}) H_{0}^{(2)} (k_{ρ a} ρ, | w_{0}^{-} |) e^{- j k_{z a} z}] (\hat{p} \times \hat{z}) .

According to (8), the GO contribution to the electric field can be determined by differentiation, thus

\begin{array}{l} E^{GO} (ρ, z) = [2 J_{0} (k_{ρ a} ρ) U (θ_{a} - θ) + H_{0}^{(1)} (k_{ρ a} ρ) U (θ - θ_{a})] e^{- j k_{z a} z} \hat{p} \\ + j \frac{k_{ρ a}}{k_{z a}} [2 J_{1} (k_{ρ a} ρ) U (θ_{a} - θ) + H_{1}^{(1)} (k_{ρ a} ρ) U (θ - θ_{a})] e^{- j k_{z a} z} (\hat{ρ} \cdot \hat{p}) \hat{z}, \end{array}

whereas the SW contribution is

\begin{array}{l} E^{SW} (r) = [H_{0}^{(2)} (k_{ρ a} ρ, w_{0}^{+}) e^{j k_{z a} z} + sgn (w_{0}^{-}) H_{0}^{(2)} (k_{ρ a} ρ, | w_{0}^{-} |) e^{- j k_{z a} z}] \hat{p} \\ - j \frac{k_{ρ a}}{k_{z a}} [sgn (w_{0}^{-}) H_{1}^{(2)} (k_{ρ a} ρ, | w_{0}^{-} |) e^{- j k_{z a} z} - H_{1}^{(2)} (k_{ρ a} ρ, w_{0}^{+}) e^{j k_{z a} z}] (\hat{ρ} \cdot \hat{p}) \hat{z} \end{array}

To highlight the wave nature of the SW contribution to the electric field, the non-uniform asymptotic expansion of the incomplete Hankel functions is provided, namely

H_{ν}^{(2)} (Ω, w_{0}) ~ \frac{e^{j ν π / 2}}{π} \frac{cosh ν w_{0}}{j Ω sinh w_{0}} e^{- j Ω cosh w_{0}},

from which the SW contribution to the vector potential (11) is asymptotically evaluated as

F^{SW} (r) ~ \frac{2}{π k} \frac{1}{{cos}^{2} θ_{a} - {cos}^{2} θ} \frac{e^{- j k r}}{r} \hat{r} \times (\hat{p} \times \hat{r}),

which reveals the spherical wave nature of the space wave contribution of the electric field.

Acknowledgments

Mauro Ettorre would like to thank the University of Rennes 1 (Action incitative, projets scientifiques émergents 2016) for its support.

References and links

1. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]

2. M. R. Lapointe, “Review of non-diffracting Bessel beam experiments,” Opt. Laser Tech. 24, 315 (1992). [CrossRef]

3. Z. Li, K. B. Alici, H. Caglayan, and E. Ozbay, “Generation of an axially asymmetric Bessel-like beam from a metallic subwavelength aperture,” Phys. Rev. Lett. 102, 143901 (2009). [CrossRef] [PubMed]

4. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000). [CrossRef]

5. R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932–942 (1991). [CrossRef]

6. S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).

7. M. Ettorre and A. Grbic, “Generation of propagating Bessel beams using leaky-wave modes,” IEEE Trans. Antennas Propag. 60, 3605–3613 (2012). [CrossRef]

8. H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, Localized Waves, (John Wiley & Sons, 2007),Vol. 194.

9. H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, Nondiffracting Waves (John Wiley & Sons, 2013). [CrossRef]

10. A. Mazzinghi, M. Balma, D. Devona, G. Guarnieri, G. Mauriello, M. Albani, and A. Freni, “Large Depth of Field Pseudo-Bessel Beam Generation with a RLSA Antenna,” IEEE Trans. Antennas Propag. 62, 3911–3919 (2014). [CrossRef]

11. M. Ettorre, S. C. Pavone, M. Casaletti, and M. Albani, “Experimental Validation of Bessel Beam generation by using an inward Hankel aperture distribution,” IEEE Trans. Antennas Propag. 63, 2539–2544 (2015). [CrossRef]

12. S. C. Pavone, M. Ettorre, and M. Albani, “Analysis and Design of Bessel Beam launchers: Longitudinal Polarization,” IEEE Trans. Antennas Propag. (to be published).

13. M. Albani, S. C. Pavone, M. Casaletti, and M. Ettorre, “Generation of non-diffractive Bessel beams by inward cylindrical traveling wave aperture distribution,” Opt. Express 22, 18354–18364 (2014). [CrossRef] [PubMed]

14. Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams”, J. Mod. Opt. 42(8), 1555–1566 (1995). [CrossRef]

15. R. Cicchetti and A. Faraone, “Incomplete Hankel and modified Bessel functions: a class of special functions for electromagnetics,” IEEE Trans. Antennas Propag. 52, 3373–3389 (2004). [CrossRef]

16. M. Albani, A. Mazzinghi, and A. Freni, “Automatic design of CP-RLSA antennas,” IEEE Trans. Antennas Propag. 60, 5538–5547 (2012). [CrossRef]

17. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, 1972), Chap. 5.

18. M. Casaletti, R. Sauleau, M. Ettorre, and S. Maci, “Efficient Analysis of Metallic and Dielectric Posts in Parallel-Plate Waveguide Structures,” IEEE Trans. Microw. Theory Tech. 60, 2979–2989 (2012). [CrossRef]

19. C. Balanis, Advanced Engineering Electromagnetics (John Wiley & Sons, 2012).

Transverse circular-polarized Bessel beam generation by inward cylindrical aperture distribution

Abstract

1. Introduction

2. Analytical formulation for infinite apertures

3. Numerical field evaluation for finite apertures

4. Implementation of a circular-polarized Bessel beam launcher

5. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (5)

Equations (16)

Optics Express