1. Introduction

Bessel beams, a fascinating and distinctive type of light field, were firstly proposed as a unique solution to the Helmholtz equation by Durnin et al. [1,2]. In recent years, the Bessel beams with their abundant physical properties and extensive applications have attracted much attention [1–26]. As the propagation distance increases, Bessel beams maintain a constant transverse intensity distribution, meaning they exhibit non-diffracting characteristics throughout the propagation path, which is different from the typical Gaussian beams scattered during propagation. This unique behavior leads to a series of prospective applications including optical manipulation [3,4], micro drilling [5] and optical trapping [6,7]. In essence, Bessel beams are generated through the interference of plane waves in a specific manner, allowing precise control over the phase of light. Various methods have been developed for generating Bessel beams, including holograms [8–10], axicons [11–14], localized modes [15,16], guided modes [17], metallic subwavelength aperture [18], annular-type photonic crystals [19], metasurfaces [20–22] and cylindrical antenna [23]. Among these, metasurfaces have revolutionized the field of optics by providing unprecedented control over the behavior of light [27,28]. Additionally, Graded index materials (GIMs), characterized by their spatially varying phase distribution at the exit surface, have emerged as a potent method for generating Bessel beams [29–32]. One of the primary advantages of using GIMs is their ability to precisely control the phase distribution, a property emphasized by the generation of orbital angular momentum (OAM) beams [33]. However, the flow of light controlled by individual metamaterial is normally single-channel, which is hard to realize multichannel Bessel beams steering.

Zero-index materials (ZIMs), characterized by their unique electromagnetic properties [34–47], present an intriguing concept in optics. In traditional materials, light undergoes a change in direction and velocity when it passes from one medium to another due to different refractive index. ZIMs, exhibit no such change. Consequently, ZIMs can be utilized to realize energy tunneling [41,42], directional emission [43,44], wavefront shaping [45], cloaking [46,47], etc. More interestingly, a novel hybrid invisibility cloaks strategy of integrating metamaterials and metasurfaces together for more sophisticated and advanced functionalities has been proposed recently [48–50]. That is, the hybrid metamaterials can be more superior and possess more advanced functionalities than each of them alone. Inspired by above wonderful works, it is natural to think of using hybrid ideas to realize multi-channel high-efficiency Bessel beams that can travel in different directions at the same time.

In this work, we combine the superiors of the quasi-isotropic all-dielectric ZIM and highly efficient full-wave control of the all-dielectric sub-wavelength modulated GIM to realize multi-channel Bessel beams. The GIM can be used to steer the direction of the incident wavefront arbitrarily through suitable designed propagation phase distribution of the exit surface, which is a highly efficient full-wave (coverage of 360 degrees) control of the wavefront. Afterwards, all-dielectric ZIM based on triple degeneracy Dirac-like point in 2D photonic crystal (PC) is introduced to obtain uniformly distributed field accompanied by cloaking effect. At last, the hybrid metamaterials composed of ZIM and GIM are designed and multi-channel high-efficiency Bessel beams were demonstrated numerically at infrared region. Utilizing all-dielectric hybrid metamaterials, our results provided a novel strategy to realize multi-channel Bessel beams. Furthermore, the integration of ZIM and GIM components endows the system with impurity immunity properties and optical crosstalk immunity. This integration is of paramount significance in complex optical systems, where such features are crucial to ensuring successful application in real-world scenarios.

2. Design principle of hybrid metamaterials

In the following, we show the design principle of the hybrid metamaterials that can realize the multi-directional Bessel beams. The generation of Bessel beams through the hybrid metamaterial is shown in Fig. 1(a).

 

Fig. 1. (a) Schematic diagram of creating Bessel beams by using hybrid metamaterials. (b) Illustration of the Bessel beam formation principle in momentum space, showcasing the wavevector components ${k_x}$ and ${k_y}$ and the associated momentum change ($\varDelta k$) that contribute to directional emission.

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The photonic crystal function as an initial phase modulator, reshaping the wavefront emitted from a point source into a planar wave upon reaching the exit surface. This transformation is realized through the unique properties of ZIM, which permit negligible phase changes within the material [44]. According to the Snell’s law ${n_1}sin\theta_1 = {n_2}sin\theta_2 $, the direction of the emergent light aligns perpendicular to the exit surface. The metasurfaces provide specialized phase control, resulting in momentum change (Δk). The special spatial arrangement within the metasurfaces ensure that the outgoing wavefront assumes the characteristic profile of Bessel beams. Figure 1(b) provides a depiction of the underlying principles in momentum space. The point on the left signifies the isofrequency contour (IFC) for the ZIM, while the circular contour on the right depicts the isofrequency curve for air. The point source, situated within the ZIM, emits light that exits normally from the surface due to the unique electromagnetic properties of ZIM. Upon encountering the metasurface, there is an additional momentum change as represented by the arrows. Due to the conservation of tangential wavevector components, this momentum shift results in directional reflection. This directional emission is critical for creating Bessel beams, as it confines the propagation to a precise trajectory in free space, which is a hallmark of the non-diffracting nature of Bessel beams.

Following the design principle, we start by realizing a ZIM. A common approach is to create a lattice of dielectric rods or holes in another dielectric material. By controlling the size, shape and spacing of these structures, we can achieve precise control of the refractive index. When designed correctly, photonic crystals can form a Dirac cone at the Γ point in the band structure, indicating zero refractive index. Based on the parameter settings in Ref. [39], we adopted specific lattice constants and dielectric block dimensions to ensure zero refractive index in the designed 2D photonic crystal.

The band structure of a two-dimensional (2D) photonic crystal (PC) with square lattice structure are shown in Fig. 2(a). The unit cell, with lattice constants of a = 600 nm, consists of a square dielectric block (dielectric block length of b = 260 nm) with a relative permittivity (${\varepsilon _r}$) of 13.7. The dielectric structures are embedded within the dielectric background with ${\varepsilon _s}$= 2.25 [39]. The band structures are calculated for the transverse electric (TE) polarization with the electric field along the rod axis, which is implemented by adopting the software COMSOL Multiphysics. The Dirac cone-like dispersion (DLP) pattern prominently emerges at the center of the Brillouin zone at the normalized frequency range ƒ = 0.424${c_0}$/$a$, where ${c_0}$ denotes the speed of light in vacuum. This distinctive dispersion behavior is characterized by the intersection of two transverse bands, both exhibiting linear dispersion, with a flat quasi-longitudinal band. The corresponding eigenmodes at the Dirac-like point are shown in Fig. 2(b). This unique configuration leads to an intriguing result: effective ZIM. As shown in Fig. 2(c) and Fig. 2(d), we also show the uniform field distribution and defect immunity characteristics of ZIM. The electric field distribution was simulated by utilizing CST Microwave Studio. After the normal incident plane wave passing through the ZIM, the phase of the transmissive wave remains in perfect agreement with the incident wave, and the internal field within the ZIM exhibits remarkable uniformity, which persists even when there are defects inside the ZIM. Therefore, we construct the effective anisotropic ENZ media based on the two-dimension all-dielectric photonic crystal.

 

Fig. 2. (a) Band structure of the photonic crystal with the inset showing the unit cell, while the Dirac-like point is denoted by DLP. (b) The degenerate eigenmodes associated with the DLP. The numerical simulations with the uniform phase distribution(c) and defect immunity characteristic(d). (e) The phase distribution within a gradient refractive index medium. (f) and (g) depict the manipulation of light through the GIM and the subsequent formation of Bessel beams. (h) and (i) illustrate the advanced light control capabilities of the hybrid metamaterials and the generation of Bessel beams, respectively.

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In order to form Bessel beams, we also need suitable phase distribution like axicon lens. The GIM represent an effective and innovative approach for the generation of Bessel beams. The gradient refractive index metamaterial of our structure is composed of square dielectric units with different hole sizes. The total thickness of the GIM is 600 nm, consistent with one period of the ZIM. Each layer within the GIM is engineered to be 100 nm thick, much less than 1/14 of the operating wavelength. As shown in Fig. 2(e), the unit cell is composed of the square dielectric (${\varepsilon _r}$=13.7) with a side length of c and the central hole within the medium with a side length of d. Within the effective medium theory [51], the unit cell effective refractive index ${n_x} = \sqrt {n_1^2({1 - \kappa } )- \kappa } $, where the ratio $\kappa = {d^2}/{c^2}$ represents the area percentage of air in a unit cell. The effective refractive index, represented by the green dotted line, and the corresponding phase distribution, denoted by the solid black line, show a symmetric, triangular profile, suggesting a gradation in the hole sizes from smaller in the center to larger on the sides. It is worth noting that here a is 6 times of c. For each GIM unit cell (which is deep sub-wavelength), the gradient refractive index material achieves 10 degrees of phase control. In our design, 36 periods correspond to full phase control in 360 degrees. As shown in Fig. 2(f), the half of the linear gradient index metasurface structure, providing an additional momentum change ($\varDelta k$), results in the deflection of a plane wave as it passes through the metasurface, as indicated by ${k_0}$ to ${k_2}$. The beam deflection angle through the GIM is determined by the generalized Snell’s Law as follows [52]: ${n_1}\sin {\theta _1} - {n_2}\sin {\theta _2} = \frac{{d\varphi }}{{dx}}\frac{\lambda }{{2\pi }}$. Here, the GIM's unit cell width is 100 nm ($dx$) with a phase gradient of 10 degrees ($d\varphi $) and operates at a wavelength of 1415 nm. The resulting calculated deflection angle is approximately 23.14 degrees. This deflection is consistent with the phenomenon illustrated in Fig. 1(b), which is instrumental in the wavefront manipulation required for Bessel beams formation. Figure 2(g) shows the whole GIM structure, designed based on the phase profile from Fig. 2(e), modulates the phase of the incident wave to generate Bessel beams. Furthermore, Figs. 2(h)-2(i) demonstrate the successful application of the hybrid metamaterials attached to the right side of a zero-index photonic crystal. The interplay between the two materials is evident from the modulation of light and the resultant beams profile. The formation of the Bessel beams is visually confirmed through the characteristic concentric ring patterns and the maintenance of beams shape over the propagation distance, which are quintessential features of Bessel beams.

3. Properties of multi-directional Bessel beams based on the hybrid metamaterials

Building upon the construction of the hybrid metamaterials previously detailed, we now turn our focus to the investigation of its intrinsic properties. Specifically, figure 3(a) displays a schematic representation of total internal reflection occurring at the interface of the hybrid metamaterials. A beam of parallel light, incident from an external source, safeguarding the internal light sources from external photonic interference. This property is critical for maintaining the fidelity of on-chip optical communication, ensuring that signal integrity is preserved without contamination from stray light.

 

Fig. 3. (a) Schematic diagram of hybrid metamaterials with incident light being totally reflected, as indicated by the arrows. (b) The momentum space analysis of the reflected light, with tangential wave vectors not intersecting the zero-index IFC line, leading to the depicted reflection. (c) and (d) The numerically simulated field distribution for external light reflection by the metamaterials, evidencing isolation from external interference and Bessel beam profile with central defect, displaying self-reconstruction capability.

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Figure 3(b) displays the mechanism of total internal reflection at the hybrid metamaterials interface by momentum space analysis. After the interaction with the GIM, the plane wave acquires an additional momentum change ($\varDelta k$). The conservation of tangential wavevector components is graphically represented by dashed lines, which do not intersect with the IFC of the ZIM. This absence of intersection points represents that the wavevector of incident light does not support transmission modes within the zero-index medium, resulting in the observed reflection rather than transmission through the interface. The light is refracted at specific angles corresponding to the reflected wavevectors ${k_1}$ and ${k_2}$, which is related to the phase gradient design of the hybrid metamaterials. Subsequently, Fig. 3(c) presents a numerical simulation affirming the resilience of the hybrid metamaterial to external wave interference. This simulation demonstrates the effectiveness of the design in isolating the internal generation of the Bessel beams from external light sources, highlighting the crucial role of crosstalk prevention in maintaining the integrity of complex optical paths. Figure 3(d) showcases the inherent property of the Bessel beams produced by the internal light source: its immunity to path obstructions.

Based on the structure of the hybrid metamaterials, we present the generation of multi-channel Bessel beams. As illustrated in Figs. 4(a)–4(b), our numerical simulations present the creation and inherent defect immunity of dual-channel Bessel beams, a unique attribute afforded by the zero-refractive-index characteristic of the material, which plays a critical role in wavefront shaping while maintaining resilience to internal imperfections. We also show the four-channel Bessel beams generation. The self-healing properties of Bessel beams mean that even though the field is distorted by opaque defect, the original properties of the field can be restored after a short transmission distance. In the preceding discussion, we elucidated the theoretical foundation of our approach, wherein the arbitrary design of zero refractive index boundaries enables the generation of Bessel beams in any direction. Then we extend our investigations to the generation of four-beams Bessel beams. The field distributions of numerical simulation are presented in Fig. 4, showing the fascinating interference patterns that emerge from this configuration. Furthermore, our study highlights the robustness and impurity immunity exhibited by Bessel beams when subjected to perturbations and defects within the optical system in Figs. 4(d) and 4(e). This property is of paramount importance for practical applications, especially in real-world scenarios where optical systems are often exposed to imperfections and external disturbances.

 

Fig. 4. (a) Generation of dual-channel Bessel beams within the hybrid metamaterials. (b) The internal defect immunity characteristic of the dual-channel Bessel beam. (c)-(e) Extension to the four-channel Bessel beam generation, showcasing the advanced structuring capability of the system. (d) and (e) showing the robustness of the Bessel beams against defects, highlighting the self-healing nature and stability of the multi-channel configuration.

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In order to quantitatively analyze the impact of defects on the generation of Bessel beams, we analyzed numerical field simulations under different defect conditions. As shown in Fig. 5, we comprehensively analyzed the field intensity distribution featuring diverse shapes and radii within ZIM. Figures 5(a) and 5(b) display the electric field intensity when the ZIM is embedded in circular defects and square defects of different size. We observed that the profiles of the Bessel beam are largely preserved despite the introduction of defects. Notably, this property is observed even when the defect dimensions are considerable (corresponding to the dimensions of two-unit cells of the structure). For a more precise quantitative evaluation, we have analyzed the field strength along the white dashed line located 6.45 µm from the exit surface as presented in Figs. 5(a) and 5(b). From Fig. 5(c), it can be seen that with the increase in the radius of the circular defects, the magnitude of the normalized field progressively diminishes, ranging from unit down to 0.73. In Fig. 5(d), the square defects exhibit a slightly more pronounced influence compared to their circular counterparts. This effect is attributed to the larger scattering cross section of the square defects, which consequently reduces the magnitude of the normalized field from unit to 0.63. The robustness of Bessel beams to structural defects is numerically analyzed, highlighting their potential for applications that still require precise beam control in the presence of defects.

 

Fig. 5. The numerically simulated field distributions when the circle (a) and square (b) defects with different size within ZIM. (c) and (d) The normalized electric field |Ez| distributions along the x direction at the reference position that is marked by the while dashed lines in (a) and (b), respectively.

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4. Conclusion

In summary, based on the concept of hybrid metamaterials, we firstly realized multi-channel Bessel beams by using an all-dielectric ZIM with GIM. The unit cell of the GIM is far smaller than the wavelength, achieving deep sub-wavelength light field control. Our design not only ensures robustness against defects during beam propagation and generation but also significantly reduces optical crosstalk, enhancing the performance in complex optical paths. Importantly, this methodology allows for the creation of arbitrary channel Bessel beams by adjusting the design of the exit surface. Our results provide a new perspective to design novel optical devices with multi-channel and open novel routes to steering the electromagnetic waves in nano-scale structures.