1. Introduction

Efficient light manipulation is crucial in advancing scientific research and current technologies. Conventional optical components and systems achieving these functions exhibit significant large footprints with complex implementations. These bulky structures are unfit for integration with state-of-the-art on-chip devices, thus necessitating the hunt for miniaturized and versatile devices. Metasurfaces are ultracompact engineered structures constructed from plasmonic or dielectric nanoantennas, enabling arbitrary and precise light control at subwavelength scales [1–3]. These metasurfaces are vital tools for breaking the traditional three-dimensional constraints of conventional bulky optical elements. As a result, metasurfaces have emerged as a pivotal foundation for a wide range of optical applications, including flat lenses [4–6], holographic imaging [7–11], meta-absorbers [12–14], reconfigurable intelligent surfaces [15–17], multifunctional and broadband structures [18–22], structural color filtering [23–25], OAM generation and sensing [26–28], and vortex beam generators [29–35]. While metasurfaces excel in single-function performance with remarkable efficiency, current strategies for achieving multifunctionality often lead to a decline in overall performance. Two main approaches for achieving multifunctionality are the unit cell-based local approach and the interleaved metasurface approach. In the local-based approach, a library or array of unit cells is constructed to achieve localized phase control by manipulating physical parameters within each unit cell. However, this technique often requires complex and time-consuming numerical analysis, leading to significant efficiency degradation, mainly when dealing with large deflection angles. In contrast, the interleaving method generates a metasurface for each desired function, offering an alternative approach to achieve multifunctionality. The interleaved method manifests notable inefficacy since the overall efficiency of the metasurface is inversely proportional to the number of functions incorporated. As the number of multiplexed functions increase, the efficiency of each optical phenomena is decreased accordingly [36–38].

The endless demand for internet connectivity and the transmission of vast amounts of data has driven scientists to devise innovative techniques to facilitate the transfer of large data volumes through optical communication channels. In this context, metasurface-based optical vortex beams carrying orbital angular momentum (OAM) have received significant attention [39–43]. It is widely recognized that OAM offers a dominant advantage in mode-division multiplexing by providing abundant spatial channels. Optical vortex (OV) beams, characterized by their possession of OAM, manifest as having a pitch-black core (also known as singularity) within an annular intensity profile and a helical phase front expressed as ${e^{j\ell \varphi }}$, where $\ell$ represents the topological charge and $\varphi$ denotes the azimuthal angle. Nevertheless, the use of OV beams faces two primary challenges. Firstly, the topological charge remains a constant integer value, which limits the multiplexing capability of OAM beams. This constraint becomes particularly pronounced on meta-devices since a finite range of topological charges can be accommodated due to size limitations. Secondly, the OV beam suffers from alterations in intensity pattern as it is highly dependent upon its topological charge; as the topological charge increases, so does the size of the annular intensity pattern [44]. To overcome this issue, the concept of perfect vortex (PV) beam was coined as a solution, that exhibits topological charge-insensitive intensity distribution. The annular intensity profile remains immune to the increment in topological charge, making them ideally suited for trigonometric topological charge modulation [33,45]. Due to above-mentioned limitation of metasurface design, most of the PV beam generation structures exhibits single optical phenomena, thus roadblock the development of integration and miniaturization.

In this study, we proposed an all-dielectric metasurface platform to realize spin-decoupled broadband concentric PV beam generation for the near infrared (NIR) spectrum. Here, a single PV beam is generated by multiplexing the phase profiles of the focusing lens, axicon and spiral phase plate. Compared to existing PV beam generating literature, we chosen trigonometric function to tailor the topological charge and to achieve the phase profile of spiral phase plate. The trigonometric function helps achieve such a spiral plate with a dynamically varying phase, which can also be dubbed an infinite topological charge. This trigonometric function will have an infinite topological charge between -1 and +1. It will allow us to carry out multi-bit extensive information processing and storage, whereas integer-based meta-devices only carry single-bit information [46]. In prior research efforts, researchers have often resorted to generating grafted beams to expand the number of topological charges, but this approach has led to complex phase modifications.

In contrast, our proposed metasurface platform offers the advantage of attaining an infinite topological charge without introducing complexity into the phase generation process. The trigonometric functions are introduced in the system by transforming the azimuthal angle $\varphi $. In addition to enhancing information-carrying capacity, we introduce multifunctionality by implementing a spin-decoupling approach to our single meta-device. This innovation results in the generation of a superimposed PV beam characterized by concentric rings. This single metasurface achieves precise outcomes for right circularly polarized (RCP), left circularly polarized (LCP), and linearly polarized (LP) light. In spin-decoupling, we take advantage of the linearity of Fourier transformations by adding the complex transmission of each function, resulting in a single phase without compromising efficiency, and functions are embedded without any complexity. For this purpose, we optimized our nanoantenna to cater the demands of optical communication applications, enabling it to operate efficiently across the broadband spectrum enveloping the “E” and “S” band regions. This spectral range spans wavelengths from $1460\,\textrm{nm}$ to $1565\,\textrm{nm}$, which is essential [47]. The motivation behind targeting these specific wavelengths lies in their alignment with the established optical communication infrastructure, facilitating wavelength division multiplexing in a core technology employed in erbium-doped fiber amplifiers for seamless integration with few-mode fiber systems.

The attenuation is practically negligible at these wavelengths, in contrast to the considerably higher attenuation observed at visible and smaller wavelengths [48]. We used silicon as base material for the nanoantenna and glass as the substrate; silicon was chosen due to its compatibility with CMOS technology and can be fabricated with already established fabrication processes. For the proof of concept, we designed and numerically studied multiple metasurfaces implementing trigonometric functions of $\vartheta .sin(\vartheta )$ and s$in2(\vartheta )$ and the periods of the axicons were kept at 4 µm and 6 µm. Our results showcased the generation of broadband superimposed PV beams alongside multifunctionality, thereby proving that trigonometric functions can facilitate multifunctionality. We believe this work will pave the way for applications like optical trapping, quantum processing, optical communication, and high-capacity information processing.

2. Theory and design

The pathway to achieving the PV beam begins by generating an OV beam, where the radius is rendered independent of the equation [49]. While this concept appears promising in theory, it poses practical implementation challenges. An optimal approach was demonstrated by “Vaity and Rusch”, who experimentally validated PV beam generation through the Fourier transformation of a “Bessel-Gaussian” beam [50]. In 2013, Zhang's research team integrated three crucial components – a lens, an axicon, and a spiral plate – onto a single metasurface to successfully achieve the PV beam through Fourier transformation [51]. Their approach involved a spiral plate, converting a “Gaussian” beam into an LG beam with a specific integer topological charge. The axicon facilitated the conversion of the LG beam into a “Bessel-Gaussian” beam, and the lens played a pivotal role in performing the Fourier transform. The phase profiles are as follows:

In our study, while keeping the phase equation for all other elements unchanged, we modify Eq. (2) to attain the desired infinite topological charge with the assistance of trigonometric functions. First, we separate the azimuthal angle and remove the $\ell $:

Secondly, we write the azimuthal angle as the function of the trigonometric function to achieve the infinite topological charge as written in Eq. (6) and depicted in Fig. 1.

 

Fig. 1. The working principle and designing procedure of the proposed work. (a) A constant line shows an integer topological charge. (b) Due to the integrated trigonometric function, a continuously changing topological charge justifies the infinite topological charge. (c) The final resulted phase for the PV beam, with conventional integer topological charge. (d) The final resulted phase for the PV beam, with infinite topological charge. (e) The functionality of our meta-device is demonstrated, showcasing its multifunctional capabilities by generating distinct outcomes in response to various input conditions. Here, for better understanding and representation, we have plotted Liner, LCP, and RCP output at different points but in reality, all the outputs will lie at the same focal point.

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The final phase come out to be as follows:

By implementing the Eq. (8), we achieve the expression that help us to get the spin-decoupled multifunctionality.

Here, A₁ and A₂ represent the arbitrary amplitudes of the diffracted light. Equation (8) introduces two channels for encoding spin-dependent information on the metasurface interface. In contrast to previously described symmetric and asymmetric helicity multiplexing methods, this approach enables the multiplexing of distinct phases in the initial stage, with the accumulated phase (${\varphi _{final}}$) subsequently encoded onto the metasurface interface. The design principle is illustrated in Fig. 1, where Fig. 1(A) describes the traditional constant topological charge behavior and Fig. 1(B) presents the topological charge behavior defined by the trigonometric function. Figure 1(C) and 1(D) indicates the corresponding phase profile to constant and trigonometric functional-based topological charge. Figure 1(E) showcases the working mechanism of the proposed metasurface where broadband NIR light is incident from the substrate side and the desired intensity distribution is observed on the output side.

The design process of the proposed metasurface commences with the careful selection of materials, with a primary consideration being transparency within the wavelengths of interest. Subsequently, we optimize the parameters of a rectangular nanoantenna, encompassing variables such as length, width, height, and the period of the substrate. The choice of material is determined by examining its refractive index (η) and extinction coefficient (k) across the wavelengths intended for utilization. The selection of silicon is validated due to its notable characteristics, including a high refractive index $({\mathrm{\eta } = 3.48} )$ and extinction coefficient $({\textrm{k} = 0} )$ for our designed wavelength. Initially, we optimize the period of the fundamental building blocks (nanoantenna) under the Nyquist criterion, aiming to mitigate higher-order modes and prevent unwarranted interference between neighboring nanoantenna. This nanoantenna optimization is conducted at the specified design wavelength of 1550 nm. During this initial optimization phase, we employ a cylindrical nanoantenna whose diameter varies across a range of periods. This approach significantly reduces computational power requirements, expediting optimizing the desired period. The outcome, including transmission intensity and phase profiles, aids in identifying the optimal period value that guarantees the highest achievable transmission efficiency and comprehensive phase coverage. In the subsequent phase, we substitute the cylindrical nanoantenna with a rectangular-shaped nanoantenna, systematically varying their dimensions in terms of length and width while maintaining a fixed period value. This geometric shape plays a pivotal role in achieving broadband performance.

We deeply analyze the numerically simulated dataset and strategically choose the dimensional parameters of the nanoantenna to maximize polarization conversion efficiency and check phase distribution for the desired spectrum. We have optimized the rectangular bar because it can function as a half-wave plate, allowing us to observe RCP light when LCP light is incident. The data for the material was obtained from [52]. The illustrations in Fig. 2(a) depict the perspective and top views of a rectangular-shaped nanoantenna, which measures 1100 nm in height and is positioned atop a glass substrate. The illustrations in Fig. 2(d)-Fig. 2(g)) show the broadband optimization results for our selected wavelengths and justify the selection of Length $\textrm{L} = 400 \, \textrm{nm}$ and Width $\textrm{W} = 190 \, \textrm{nm}$, also indicated by the red area in the figures, which is the maximum intensity area. The comprehensive analysis of the meta-device involved numerical simulations carried out using Ansys Lumerical Inc. FDTD Solution 2022 R1.4 [53]. For nanoantenna optimization, a 3D simulation model of the unit cell was constructed with perfectly matched boundaries along the z-axis and periodic boundaries along the x-axis and y-axis. Using periodic boundary conditions effectively replicates the modeled structure at regular intervals along the x-axis and y-axis, extending infinitely. This approach facilitates precise measurement of the conversion efficiency of the unit cell. During the simulation of the entire metasurface, we employ the perfectly matched layer (open boundary condition) in all directions. A PML boundary allows electromagnetic waves to travel through the boundary with negligible reflections, ensuring the results are free from undesirable reflections.

 

Fig. 2. The design procedure of the nanoantenna. (a) Illustration of dimensional parameters and different cross-sectional views of unit cell. (b) The complete phase distribution vs. in-plane rotation of the unit cell. (c) Polarization conversion efficiencies of the optimized building bock with $\textrm{H} = 1100\,\textrm{nm}$ height, $\textrm{L} = 400\,\textrm{nm}$ length, $\textrm{W} = 190\,\textrm{nm}$ width, and $\textrm{P} = 694\,\textrm{nm}$ period. The blue and pink curves present the co-polarization and polarization conversion efficiency, respectively, whereas the dotted green line exhibits the average polarization conversion efficiency of the optimized building block. (d-g) Detailed numerical optimization procedure opted to achieved a broadband functional antenna with single unique physical dimensions. The intersection of dotted line pointed by an asterisk identifies the selected dimensional parameters.

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3. Results and discussion

The precise numerical optimization helps us to get a distinctive set of dimensions for the nanoantenna, specifically, a length (L) of $400\,\textrm{nm}$ and a width (W) of $190\,\textrm{nm}$ within a nanoantenna period (P) of 694 nm. An in-plane rotation is applied while exposing it to left-handed circularly polarized light to validate the nanoantenna's intended function as a half-wave plate for targeted wavelengths. The resultant values of cross-polarization are depicted in Fig. 2(C). Ensuring comprehensive phase coverage is a critical consideration in the meta-device design. The nanoantenna must exhibit phase coverage across the entire span as it rotates from 0 to π or 0 to 180°. The Pancharatnam-Berry (PB) phase concept informs us that ${\theta _{PB}} = \theta /2$, implying that the complete 2π phase can be achieved by a simple rotation spanning from 0 to π. Figure 2(B) confirms that the nanoantenna will ensure phase coverage for our desired vital wavelengths, as the starting and ending points of the phase remain the same. To start the numerical simulation, we chose the first trigonometric function as${\varphi _{spira{l_\infty }}}({x,y} )= {\varphi _{azimuthal}} \times \textrm{sin}({{\varphi_{azimuthal}}} )$, and to get multifunctionality (ring in a ring shape), we embedded two different periods of axicons, with ${\mathrm{\rho }_1} = 4000\,\textrm{nm}$ and ${\mathrm{\rho }_2} = 6000\,\textrm{nm}$. The focal point of the lens has been chosen to $\textrm{be f} = 60{\;\ \mathrm{\mu} \mathrm{m}}$ and ${\mathrm{\lambda }_\textrm{d}} = 1550\,\textrm{nm}$. The size of the meta-device came to be $60\mathrm{\;\ \mathrm{\mu} m\;\ } \times \textrm{}60{\;\ \mathrm{\mu} \mathrm{m}}$, and the average transmission efficiency of our broadband meta-device is ${\mathrm{{\rm E}}_{\textrm{avg} = \textrm{}}}94\textrm{\%}$. The results for the first trigonometric function have been showcased in Fig. 3.

 

Fig. 3. The numerical simulated result for $\mathrm{\varphi }.\textrm{Sin}(\mathrm{\varphi } )$ at $\mathrm{\lambda } = 1470\,\textrm{nm}$. (a) When linear light has been shined and a ring in a ring structure, the XY intensity plot is observed. (b) The XY intensity plot when right-handed circularly polarized has been shined, and a ring pattern of ${\mathrm{\rho }_1} = 4000\,\textrm{nm}$ can be observed. (c) The XY intensity plot when left-handed circularly polarized has been shined, and a ring pattern of ${\mathrm{\rho }_2} = 6000\,\textrm{nm}$ can be observed. (d-f) The XZ intensity plots of the same ring patterns concerning their input light. The dotted white light indicates the focal plane (g-i). The normalized FWHM results for different inputs.

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The versatility of our device’s multifunctionality, particularly its capacity to employ trigonometric functions for Perfect Vortex (PV) beam generation, has been substantiated in Fig. 3. The efficiency of cross-polarization achieved at a wavelength of $\mathrm{\lambda } = 1470\,\textrm{nm}$ was determined to be $92\,\textrm{\%}$. Examining the trigonometric phase revealed unconventional behavior, manifesting irregularities in the spiral plate phase. These irregularities contributed to anomalies in the annular intensity distribution, resulting in minor distortions and subtle dispersions within the circular pattern. Figure 3(a)–3(c) presents the multifunctional outcomes associated with the trigonometric topological charge, revealing distinct outputs for various inputs. Figure 3(a) notably demonstrates the superposition of perfect vortex beams. Figure 3(d)–3(i) shows the XZ monitor results and the FWHM of the annular intensity profile. For further investigation, we simulated the designed multifunctional structures for an input wavelength of $\mathrm{\lambda } = 1550\,\textrm{nm}$ and the achieved intensity distributions are illustrated in Fig. 4.

 

Fig. 4. The numerical simulated result for $\mathrm{\varphi }.\textrm{Sin}(\mathrm{\varphi } )$ at $\mathrm{\lambda } = 1550\,\textrm{nm}$. (a) When linear light has been shined and a ring in a ring structure, the XY intensity plot is observed. (b) The XY intensity plot when right-handed circularly polarized has been shined, and a ring pattern of ${\mathrm{\rho }_1} = 4000\,\textrm{nm}$ can be observed. (c) The XY intensity plot when left-handed circularly polarized has been shined, and a ring pattern of ${\mathrm{\rho }_2} = 6000\,\textrm{nm}$ can be observed. (d-f) The XZ intensity plots of the same ring patterns concerning their input light. The dotted white light indicates the focal plane (g-i). The normalized FWHM results for different inputs.

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Figure 4 illustrates the capability of our single meta-device to operate effectively across a broadband range while maintaining high cross-polarization efficiency. Precisely, at a wavelength of $\mathrm{\lambda } = 1550\,\textrm{nm}$, the transmission efficiency was measured as $95\%$. Furthermore, the far-field intensity profiles of the designed meta-device at this wavelength showcase the generation of highly efficient broadband PV beams tailored for the trigonometric topological charge. Lastly, to check whether the generated beam is a PV beam and can perform in the broadband region, we choose another trigonometric function ${\varphi _{spira{l_\infty }}}({x,y} )= \sin 2({{\varphi_{azimuthal}}} )$, and broadband ability has been checked at different critical telecom wavelengths. The results for comparisons are showcased in Fig. 5.

 

Fig. 5. The broadband generation of trigonometric PV beams. (a-d) The broadband generation of PV beam with trigonometric function of $\varphi .\textrm{sin}(\varphi )$. (e-h) The broadband generation of PV beam with the trigonometric function of $\textrm{sin}2(\varphi )$.

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Figure 5 provides evidence supporting our assertions regarding the broadband ability of the meta-device, which operates effectively in the “E” and “S” bands of optical communication. These selected wavelengths are pivotal within the telecom spectrum, presently integral in the fiber communication ecosystem. The multifunctionality has also confirmed that the super-positioned PV beam occurs for all the wavelengths and different trigonometric functions. Finally, in Fig. 5, we substantiate the generated beam as a Perfect Vortex beam. This concept is confirmed by the consistent diameter of the annular intensity, even when employing superposition techniques. The dotted guideline has been included to highlight the unchanging diameter during superposition for various axicon periods like ${\mathrm{\rho }_1} = 4000\,\textrm{nm}$ and ${\mathrm{\rho }_2} = 6000\,\textrm{nm}$. In this study, the choice of trigonometric function was made voluntarily. Any trigonometric function such as $Sin(\vartheta ),Cos(\vartheta ),Sin2(\vartheta ),Cos(\vartheta )$ can be employed, whereas we abstain from utilizing $Tan(\vartheta )$ due to its undefined points in its property. But optical gradient forces of these trigonometric functions are different (OAM has different spin direction like clockwise or anticlockwise) and for your application you can choose trigonometric function that suits you and further guidance can be taken from the [46]. We choose $\vartheta .Sin(\vartheta )\; \; and\; Sin2(\vartheta )$, to show that a simple Sin(ϑ) function and a hard function like $Sin2(\vartheta )= 2.Sin(\vartheta ).Cos(\vartheta )$, both can generate a ring a ring structure, while maintaining Perfect Vortex beam properties.

4. Conclusion

In summary, a highly efficient broadband metasurface platform to exhibit spin-decoupled concentric perfect vortex beams, where each beam incorporates trigonometric functions to achieve infinite topological charges. The realized metasurfaces are functional for the “E” and “S” bands of the optical communication spanning from $1460\,\textrm{nm}$ to $1565\,\textrm{nm}$, and an impressive $94\,\textrm{\%}$ average polarization conversion efficiency achieved for the intended wavelength range. Thanks to its multifunctionality, the metasurfaces generate superimposed PV beams when exposed to linear input. It yields distinct annular intensity profiles for RCP and LCP incident light. Silicon has been chosen for the design of our meta-device due to its incorporation with the already established CMOS fabrication process. This research will open avenues for applications such as optical communication, quantum processing, and high-capacity information processing.