Field derivation

In the following, we will always assume an azimuthally symmetric TM-polarized field (where TM stands for transverse magnetic with respect to the propagation z-axis according to the coordinate reference frame reported in Fig. 1, left) and derive the tangential electric and magnetic field components, assuming that we want to generate an ideal BB/BGB/GB distribution for the vertical z component. As a result of the previous hypotheses (viz., H _z = 0 and ∂/∂ϕ = 0, ϕ being the azimuthal coordinate of a cylindrical reference frame {ρ, ϕ, z}), $E_\rho ,\; E_z,\; H_\phi$ are the only non-zero field components. (A time-harmonic dependence e^jωt of the fields is tacitly assumed and suppressed throughout the paper.)

Fig. 1.

On the left, a ray interpretation for the generation of a BGB from an inward cylindrical aperture field is represented. In contrast with truncated BBs, BGBs exhibit a Gaussian amplitude modulation (the density of the rays decreases according to the Gaussian modulation). In the middle, a perspective view of the transverse profile of an ideal BGB is reported. On the right, it is shown that the BGB transverse profile can be obtained through the product between an ideal BB and an ideal GB.

From Maxwell's equations in a cylindrical reference frame, the tangential electromagnetic field components can be expressed in terms of E _z through:

(1)$$\eqalign{ E_\rho & = -{1\over \rho}{\partial\over \partial z}\int_0^\rho E_z\rho^\prime {d}\rho^\prime\, ,\; \cr H_\phi & = -j{k_0\over \eta\rho}\int_0^\rho E_z\rho^\prime {d}\rho^\prime\, ,\; }$$

where k ₀ is the free-space wavenumber, η is the characteristic impedance of the medium and ρ^′ is a dummy variable of integration along the radial coordinate.

It is worth to stress here that BBs arise as ideal solutions of Helmholtz equation in a cylindrical reference frame. As a result, the field derivation here proposed for BBs leads to a full-wave electromagnetic solution. On the other hand, BGBs and GBs arise as asymptotic solutions of Helmholtz equation in the paraxial limit, namely the paraxial wave equation. Therefore, a full-wave electromagnetic solution for such kinds of beams does not strictly exist over the whole space. In this light, the following section only aims at showing conditions for correctly synthesizing the aperture fields to obtain along the longitudinal component of the electric field either a BB, or a BGB, or a GB, in the near-field region. This approach is demonstrated to be effective within a limited region located in the near field of the radiating aperture (criteria to determine this region are provided in [Reference Fuscaldo, Benedetti, Comite, Baccarelli, Burghignoli and Galli10]).

Bessel beam

An ideal, axially symmetric scalar BB is represented through the expression $J_0( k_\rho \rho ) e^{-jk_z z}$ where J ₀( · ) is the zeroth order Bessel function of the first kind, $k_\rho$ and k _z are the radial and vertical wavenumbers, respectively, related to k ₀ through the separation relation $k_z^2 + k_\rho ^2 = k_0^2$. This expression shows that the intensity of an ideal BB is invariant along the z-axis, thus revealing its nondiffracting nature [Reference Durnin1]. In practice, a BB has to be created through a finite aperture; a ray interpretation (see Fig. 1, left) reveals that a truncated BB does not undergo diffraction up to an axial distance known as nondiffractive range $z_{\mathrm{ndr}} = \rho _\mathrm{ap}\cot \theta _0$, $\theta _0 = \arcsin {( k_\rho /k_0) }$ being the axicon angle.

As commented at the beginning of this section, we derive the tangential field components for having an ideal BB on E _z, i.e., $E_z\propto J_0( k_\rho \rho ) e^{-jk_z z}$. As is well known [Reference Chávez-Cerda16, Reference Albani, Pavone, Casaletti and Ettorre17], in order to have ${E_z\propto J_0( k_\rho \rho ) e^{-jk_z z}}$, the aperture field, i.e., the electromagnetic field at z = 0, is not required to have a stationary Bessel-like character. Indeed, a ray interpretation [Reference Felsen18] of radiation from a cylindrical aperture (see Fig. 1, left) reveals that if we assume an inward traveling-wave aperture distribution (represented in cylindrical coordinates by a first-kind zeroth-order Hankel function $H_0^{( 1) }({\cdot})$), constructive interference between the emitting rays is expected to occur in a diamond-shaped region whose vertex is determined by the aperture radius ρ _ap and yields a nondiffractive range at z _ndr. It is worth to recall here that the inward ray bundle turns into an outward ray bundle upon crossing its caustic, i.e., the z-axis (as expressed by the properties of Hankel functions, $H_0^{( 1) }( -x) = -H_0^{( 2) }( x) \, ,\; x\in {\mathbb{R}}$), and that $2J_0( x) = H_0^{( 1) }( x) + H_0^{( 2) }( x)$ [Reference Abramowitz and Stegun19].

Therefore, if we assume the vertical component of the electric aperture field to be ${E_{z, \mathrm{ap}}\propto H_0^{( 1) }( k_\rho \rho ) }$, a TM-polarized BB will be generated from a radiating aperture, whose tangential components take the form [Reference Harrington20]:

(2)$$\eqalign{ E_{\rho, \mathrm{ap}}\left(\rho\right)& \propto j {k_z\over k_\rho} H^{( 1) }_1\left(k_\rho \rho\right)\, ,\; \cr H_{\phi, \mathrm{ap}}\left(\rho\right)& \propto {1\over k_\rho} H^{( 1) }_1\left(k_\rho \rho\right)\, ,\; }$$

where $H_{1}^{( 1) }({\cdot})$ is the first-kind first-order Hankel function.

Bessel-Gauss beam

Under the ray-optics, paraxial approximation, the expression of an ideal BGB is given in [Reference Gori, Guattari and Padovani7, equation (2.7)], and is not reported here for brevity. The aperture field required for generating a BGB is easily obtained from [Reference Gori, Guattari and Padovani7, equation (2.7)] for z = 0 and it is basically an apodized version of a BB, characterized by a Gaussian apodization function. The conditions for obtaining a BGB then follow straightforwardly. Indeed, we wish to have an axially symmetric vector BGB having a Gaussian-modulated vertical component over the aperture which leads to ${E_{z, \mathrm{ap}}\propto J_0( k_\rho \rho ) e^{-\rho ^2/2w_0^2}}$, where w ₀ is the beam waist parameter, i.e. the radial distance at which the modulus squared of a Gaussian aperture distribution has decayed 1/e with respect to its maximum amplitude (reached at ρ = 0). (Note that the definition of w ₀ may differ by a $\sqrt {2}$ factor when compared with the relevant literature.)

In order to generate a BGB distribution for the longitudinal component of the electric field, an additional Gaussian taper has to be accounted for in the expression of the aperture fields (see Fig. 1, middle, right). Thus, the resulting equations for the tangential field components are [Reference Fuscaldo, Benedetti, Comite, Baccarelli, Burghignoli and Galli10]:

(3)$$\eqalign{ E_{\rho, \mathrm{ap}}\left(\rho\right)& \propto j {k_z\over k_\rho} H^{( 1) }_1\left(k_\rho \rho\right)e^{-{1}/{2}\left({\rho}/{w_0}\right)^2}\, ,\; \cr H_{\phi, \mathrm{ap}}\left(\rho\right)& \propto {1\over k_\rho} H^{( 1) }_1\left(k_\rho \rho\right)e^{-{1}/{2}\left({\rho}/{w_0}\right)^2}\, . }$$

Equation (3) represents the tangential aperture distribution that is required to obtain a TM-polarized BGB for E _z and within the nondiffractive range.

Gaussian beam

An ideal GB is commonly defined through the expression $( -jz_{R}/\zeta ) e^{( -jk_0z-jk_0\rho ^2/2\zeta ) }$, where ζ = z + jz _R is the complex beam parameter [Reference Deschamps21] and $z_{R} = k_0w_0^2$ is the Rayleigh range.

The expressions of the tangential electromagnetic field components when E _z takes the form of an ideal GB are obtained in closed form from (1) and read:

(4)$$\eqalign{ E_\rho & = {-jz_{R}e^{-jk_0z}\over \rho} \left[1-e^{-jk_0\rho^2/2\zeta}\left(1-{\rho^2\over 2\zeta^2}\right)\right]\, ,\; \cr H_\phi & = {-jz_{Re^{-jk_0z}\over \eta\rho} \left(1-e^{-jk_0\rho^2/2\zeta}\right)\, ,\; }}$$

thus, on the aperture plane z = 0 we simply have:

(5)$$\eqalign{ E_{\rho, {ap}}( \rho) & \propto{1\over \rho} \left[1-e^{-\rho^2/2w_0^2}\left(1 + {\rho^2\over 2k_0^2w_0^4}\right)\right]\, ,\; \cr H_{\phi, {ap}}( \rho) & \propto{1\over \eta\rho} \left(1-e^{-\rho^2/2w_0^2}\right)\, . }$$

As is known, an ideal GB is only defined under the paraxial limit (i.e., in the limit $k_\rho \to 0$ and k _z → k ₀). Conversely, the leaky-wave approach proposed here only allows for having a non-zero, although relatively small value of $k_\rho$ (the reasons for this limitation will appear clear in the next section “Leaky-wave approach”). Nevertheless, the synthesis of a GB beyond the paraxial limit will allow for a significant comparison with the synthesis of a BB and a BGB that would be otherwise less interesting if investigated in the paraxial limit. Indeed, paraxial BBs and BGBs, although propagating over large depth of fields, retain larger beam waists [Reference Fuscaldo, Benedetti, Comite, Baccarelli, Burghignoli and Galli10], thus being less attractive for focusing applications.

Leaky-wave approach

It has recently been shown that radially periodic leaky-wave antennas (LWAs) provide for a simple means to generate either BBs [Reference Comite, Fuscaldo, Podilchak, Hilario Re, Gómez-Guillamón Buendía, Burghignoli, Baccarelli and Galli22, Reference Fuscaldo, Comite, Boesso, Baccarelli, Burghignoli and Galli23] or BGBs [Reference Fuscaldo, Benedetti, Comite, Baccarelli, Burghignoli and Galli10], in particular for microwave and millimeter-wave ranges.

Radially periodic LWAs are two-dimensional (2-D) LWAs consisting of a grounded dielectric slab centrally fed by a coaxial feed and covered with an annular strip grating (see Fig. 2(a)) [Reference Baccarelli, Burghignoli, Lovat and Paulotto24]. The radially periodic character of the structure allows for exciting a backward cylindrical leaky wave that is responsible for the near-field Bessel-like character exhibited by the vertical component of the electric field in the diamond-shaped region above the radiating aperture (the interested reader can find more details in [Reference Comite, Fuscaldo, Podilchak, Hilario Re, Gómez-Guillamón Buendía, Burghignoli, Baccarelli and Galli22, Reference Fuscaldo, Comite, Boesso, Baccarelli, Burghignoli and Galli23]). As opposed to classical BBs that are generated from a uniform aperture distribution, those generated by radially periodic LWAs intrinsically have a tangential aperture distribution of the type $H^{( 2) }_1( k_\rho \rho ) \propto \exp ( -j\beta \rho ) \exp {( -\alpha \rho ) }/\sqrt {\rho }$ [Reference Fuscaldo, Comite, Boesso, Baccarelli, Burghignoli and Galli23], where α is the attenuation (or leakage) constant of the radial wavenumber $k_\rho = \beta -j\alpha$, β being the leaky phase constant, related to the axicon angle through sinθ ₀ = β/k ₀. However, this typical exponential decay of leaky waves can easily be converted to recover the aperture distribution in (3), which is required to obtain a BGB. Indeed, by locally modulating the strip width w and/or the period Λ of the grating (see Fig. 2(a)) it is possible to produce the local variation of the leakage rate required for obtaining a Gaussian modulation from an exponentially decaying one (the interested reader can find details in [Reference Fuscaldo, Benedetti, Comite, Baccarelli, Burghignoli and Galli10]). (It is worth noting here that the excitation of a backward cylindrical wave, i.e., with βα < 0, allows for recovering the required inward character of the Hankel function characterizing the aperture distribution.)

Fig. 2.

(a) A perspective view of a radially periodic leaky-wave antenna (LWA) where the annular strip grating is modulated in order to support a BB/BGB/GB. A metallic disk is placed on top of the central coaxial feed for impedance matching purposes. (b) Radial profiles of the leakage rate α/k ₀ versus ρ/λ₀ required to obtain a BB (green line), a BGB (red line), and a GB (blue line) (parameters in the text). (c) Color map plots of the leakage rate α/k ₀ (left) and leaky phase constant β/k ₀ (right) as a function of the period Λ and the filling factor FF.

The synthesis technique for obtaining a BGB has been described in [Reference Fuscaldo, Benedetti, Comite, Baccarelli, Burghignoli and Galli10] taking cues from the standard leaky-wave approach originally applied to 1-D periodic LWAs [Reference Oliner and Johnson25] to obtain a target far-field radiation pattern, and more recently to 2-D radially periodic LWAs [Reference Gómez-Tornero, Blanco, Rajo-Iglesias and Llombart26, Reference Blanco, Gómez-Tornero, Rajo-Iglesias and Llombart27] to obtain focused beams in the radiative near-field region.

However, to the authors’ best knowledge, the leaky-wave approach has not yet been applied to 2-D radially periodic LWAs for synthesizing GBs at microwaves.To this aim, the exponential taper that intrinsically characterizes a leaky wave should be converted to the aperture distribution given by (5), as needed for having an ideal GB. However, this would require β = 0, which in turn implies a periodic LWA with the open stopband (OSB) suppressed [Reference Jackson, Caloz and Itoh28, Reference Baccarelli, Burghignoli, Comite, Fuscaldo and Galli29]. While the possibility to work at β = 0, with the OSB suppressed, is an important aspect that is worth to be investigated in the future, in this work, we will synthesize GBs operating as close as possible to the OSB, but without employing any technique for its mitigation/suppression [Reference Baccarelli, Burghignoli, Comite, Fuscaldo and Galli29]. As a consequence, the synthesized GB will somehow differ from an ideal GB. Nevertheless, we will readily show here that it is still possible to synthesize a Gaussian-like beam, even when β ≠ 0. Indeed, the expression of an ideal GB (4) evaluated for z = z ₀ < 0 instead of z = 0 (as in (5)) yields an aperture field that can be described with rays that point toward the axis of symmetry with an axicon angle θ ₀ ≠ 0°. More details about the procedure employed to select z ₀ will be given in section “Numerical results”.

The modulation of the leakage constant required along the radial direction to obtain the BB (green line), the BGB (red line), and the GB (blue line) solution is obtained by means of [Reference Fuscaldo, Benedetti, Comite, Baccarelli, Burghignoli and Galli10, equation (12)] and reported in Fig. 2(b) assuming ρ _ap = 15λ ₀ = 15 cm, at f ₀ = 30 GHz (λ ₀ being the vacuum wavelength), an axicon angle θ ₀ = 10° (for the BB and the BGB solution), and a normalized beam waist parameter ${\tilde {w}_0 = w_0/\rho _{\rm ap} = 0.2}$ (for the BGB and the GB solution).

The flexibility of the leaky-wave approach to obtain BBs, BGBs, and GBs can be inferred from Fig. 2(c) where we have evaluated the variation of the normalized leaky attenuation and phase constant for different values of Λ and the filling factor defined as FF = w/Λ, by means of an accurate method-of-moments (MoM) analysis [Reference Baccarelli, Burghignoli, Frezza, Galli, Lampariello, Lovat and Paulotto30].

A closer look to Fig. 2(c) reveals that the iso-line β/k ₀ = 0 corresponds to high values of α/k ₀ with abrupt variations, thus confirming the presence of an OSB and also preventing the possibility to use these values for accurately generating a GB. Conversely, values of β/k ₀ as low as − 0.17 correspond to regions where α/k ₀ varies smoothly in the range required for the synthesis of BBs, BGBs, and GBs (see Fig. 2(b)); this motivates the choice of θ ₀ = 10° (note that sin 10° ≃ 0.17) for the other examples. In brief, the choice of θ ₀ = 10° proves to be a good compromise among (i) the feasibility of the design process, (ii) the deviation from the paraxial condition, (iii) the focused character that can be obtained with nonparaxial BBs and BGBs.

Finally, the radial profiles of the leakage constant are reconstructed with the values of α/k ₀ that provide for the requisite of keeping β/k ₀ as constant as possible. The latter condition identifies a path over the color maps in Fig. 2(c) that requires the simultaneous variation of Λ and FF. We should highlight that this is a considerable difference with respect to the approach proposed in [Reference Fuscaldo, Benedetti, Comite, Baccarelli, Burghignoli and Galli10], where the period was assumed to be fixed (at which corresponds a path that follows a straight horizontal line over the color maps in Fig. 2(c)), and there were no degrees of freedom for reducing the variation of the leaky phase constant.

In the next section “Numerical results”, we will numerically evaluate the near-field distribution of |E _z|² for all the beam solutions described so far, also highlighting their advantages and disadvantages in terms of diffractive and focusing properties when these solutions are generated beyond the paraxial limit.

Numerical results

In this section, we aim at showing and comparing the different beam propagating features of BBs, BGBs, and GBs, as well as the agreement between the expected theoretical distributions and accurate numerical simulations.

Near-field distributions for the vertical component of the electric field E _z will be illustrated over the domain |ρ| < ρ _ap and 0 < z ≤ z _ndr. Such distributions can be obtained from the aperture fields, through the application of the equivalence theorem, i.e., the Huygens–Fresnel principle [Reference Harrington20]. Indeed, the aperture field can be replaced by an equivalent magnetic surface current density $M_\phi$ equal to ${M_\phi = -2E_{\rho , \mathrm{ap}}}$. Therefore, the vectorial near-field distribution is given by [Reference Harrington20]:

(6)$$\eqalign{ & {\bf E}\left({\bf r}\right) = 2\oint_{{\rm S_A}} \left(E_{\rho, \mathrm{ap}}( {\bf r}') {\bf u}_{\rho}\times\nabla G( {\bf r},\; {\bf r}') \right){d}{\bf r}'\, ,\; \cr & {\bf H}\left({\bf r}\right) = {\,j\over k_0\eta}\nabla\times{\bf E}\left({\bf r}\right)\, ,\; }$$

where u_ρ is the radial unit vector, r (r′) represents the observation (source) points, G(r, r′) = e ^{−jk|r−r′|}/(4π|r − r′|) is the free-space Green's function and the integration domain S _A is the aperture plane. Such radiation integrals have been evaluated numerically, carefully handling the singular behavior of the free-space Green's function for observation points at or close to the plane z = 0.

In order to make a fair comparison among the three different solutions, numerical results are obtained under the same operating conditions used in the previous section “Leaky-wave approach”.

In Figs 3(a)–3(c), the near-field distributions of (a) a truncated BB, (b) an ideal BGB, and (c) an ideal GB are reported. Comparison among Figs 3(a)–3(c) highlights the following aspects:

• • For the BB solution (see Fig. 3(a)), the beam waist remains almost constant up to the nondiffractive range, that ray optics (shadow boundaries are reported in white dashed lines) would predict to occur at around 80λ ₀. However, the field intensity rapidly decays beyond that distance and the beam also starts to considerably widen along the transverse direction.

• • For the BGB solution (see Fig. 3(b)), the beam waist is maintained almost constant over a shorter distance with respect to the BB solution, due to the effect of the Gaussian amplitude modulation of the aperture distribution. From the numerical results, one can infer that the effective nondiffractive range is reduced to approximately 20λ ₀ (see the white dashed line) in agreement with the formula reported in [Reference Fuscaldo, Benedetti, Comite, Baccarelli, Burghignoli and Galli10, equation (8)]. However, the BGB solution features a more localized character close to the axis of symmetry, and a considerable reduction of the intensity of the annular rings that instead characterizes the BB solution.

• • For the GB solution (see Fig. 3(c)), the beam waist changes size along z according to (4). Specifically, it is manifest from Fig. 3(c) that the beam waist doubles at the Rayleigh range that for a normalized beam waist $\tilde {w}_0 = 0.2$ should occur at around 56λ ₀. Since the GB is shifted along the z-axis of 50λ ₀ the Rayleigh range is achieved at around z = 100λ ₀. We should stress here that the GB is shifted of 50λ₀ because we decided to synthesize on the aperture (i.e., at z = 0), the expression of an ideal GB at z = −50λ ₀, in agreement with the motivation outlined in the previous section “Leaky-wave approach”. The specific choice of z ₀ = −50λ ₀ will be commented next.

Fig. 3.

Color maps of |E _z|² (normalized to its maximum and in dB scale) generated through an aperture size of radius ρ _ap = 15λ₀ with λ ₀ = 1 cm (f ₀ = 30 GHz) for (a)–(c) ideal BBs, BGBs, and GBs, respectively; (d)–(f) ideal synthesis of BBs, BGBs, and GBs, i.e., radiated by the aperture distributions in (2), (3), and (5), respectively; (g)–(i) leaky-wave synthesis of BBs, BGBs, and GBs, i.e. reconstructing the aperture distributions in (2), (3), and (5), respectively, with the values of β/k ₀ and α/k ₀ provided by the synthesis procedure for the planar radially periodic leaky-wave antennas of Fig. 2(a).

In Figs 3(d)–3(f), near-field distribution of |E _z|² (normalized to its maximum and in dB scale) has been numerically evaluated through (6) (projected onto the vertical unit vector u_z), using (2) for a BB (see Fig. 3(d)), (3) for a BGB (see Fig. 3(e)), and (5) for a GB (see Fig. 3(f)). The same results, but when the aperture field is synthesized through the leaky-wave approach described in section “Leaky-wave approach” are reported in Figs 3(g)–3(i)) for the BB, the BGB, and the GB solution, respectively.

The appreciable differences between the ideal distributions (see Figs 3(a)–3(c)) and the near-field distributions obtained by radiating the ideal aperture fields (Figs 3(d)–3(f)) are due to the diffraction effects that are tacitly neglected by the ray-optics approximation over which the ideal distributions are based. This disagreement is even more reasonable for a GB which is not a solution of the wave equation, but of its paraxial approximation which leads to a parabolic equation [Reference Deschamps21]. As a result, the field radiated by a Gaussian distribution is not expected to recover the ideal GB solution, as is manifest from the significant differences between Figs 3(c) and 3(f). Nevertheless, it is still possible to see that, far from the aperture plane (beyond the white dashed line), the radiated field starts to recover the main beam shape GB. In agreement with the choices we made, the beam waist that one would expect to occur at z = 0 from the standard definition of a GB is here obtained for z = 50λ₀ (see the dashed white line).

In this regard, we should recall further comment the choice of z ₀ = −50λ ₀ for the aperture-field synthesis. Indeed, if one takes the expression of a GB at z = z ₀ and calculate the phase gradient, one would obtain an expression for β as a function of ρ that, for z ₀ = −50λ ₀ is approximately in the range $-0.17< \beta _\rho < 0$. Since β = −0.17 represents a good compromise between the requirement of being not too close to the OSB, and not too far from the paraxial condition, the value of z ₀ = −50λ ₀ has been selected. It is also worth noting here that the inward character of the aperture distribution is manifest from Fig. 3(f). This aspect confirms that the discrepancies between Figs 3(f) and 3(c) are mainly due to the deviation from the paraxial condition.

On the other hand, a very good agreement is appreciable between the fields radiated assuming the ideal aperture distributions and those synthesized with the leaky-wave procedure (compare Figs 3(d)–3(f) with Figs 3(g)–3(i)). This aspect further corroborates the accuracy and flexibility of the leaky-wave approach, and also confirms that the appropriate synthesis of the GB solution is mostly limited by the difficulties in achieving the paraxial limit with the proposed structure. This consideration suggests future interesting pathways for the synthesis of GBs at microwaves with periodic LWAs implementing OSB suppression techniques. Finally, we show in Figs 4(a)–4(b), the synthesis of a GB assuming the aperture plane being at z = −150λ ₀. Comparison of these results with those of Fig. 3(c) shows that, far from the aperture plane, it is easier to obtain a good agreement between an ideal GB (Fig. 3(c)), its ideal synthesis (Fig. 4(a)), and its leaky-wave synthesis (Fig. 4(b)). We should stress that, although the agreement is obtained far from the aperture-plane (approximately after 100λ ₀) the reported domain is fully within the radiative Fresnel region, the Fraunhofer distance starting at around z = 1800λ ₀ for the proposed LWA.

Fig. 4.

(a)–(b) As in Figs 3(f) and 3(i), respectively, but now assuming the aperture plane being at z = −150λ₀.