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Multidimensional and multifunctional metasurface design using hybrid spin decoupling

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Abstract

Metasurfaces are nanometer-thick patterned interfaces that exhibit unprecedented control over the quintessential properties of light and provide a footing ground for many innovative optical effects and groundbreaking phenomena like metalenses, complex wavefront shaping, polarimetric sensing, etc. Often multifunctional metasurfaces enact a multitude of simultaneous functionalities by employing the photonic spin Hall effect (PSHE) that allows independent control of photons through spin-orbital interactions. However, the exhibited optical responses are locked to be opposite to each other, resulting in significant design complexities, cross-talk, and noise while adding more functionalities into a single device. Herein, we demonstrate multifunctional all-dielectric transmissive metasurfaces exploiting PSHE-based unique phase multiplexing as a generic designing method to provide independent control of orthogonal helicities, squeezing spin-dependent quad information channels with minimal observed noise and cross-talk. To authenticate the proposed concept, multifocal metalenses enabling spin-depended splitting in longitudinal and transverse directions are demonstrated, which generate two high-intensity focused spots under opposite handedness of the circularly polarized incidence and all four focus spots under the linearly polarized incidence of ultraviolet wavelengths. The proposed functional domain enhancement of metasurfaces with high-resolution phase modulation brings advances in compact multifunctional device design to the fields of microscopy, communication, data storage, imaging, etc.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Metasurfaces are artificially engineered ultrathin electromagnetic interfaces that exhibit the unprecedented ability to manipulate the fundamental properties of light (viz. phase, amplitude, and polarization) by implementing abrupt phase change through precisely engineered nanoscale building blocks. Their exceptional capability to arbitrarily control the light-matter interaction promises miniaturized, low-profile optical components for diverse applications in imaging, optical communication systems, optical manipulation, and sensing. Metasurfaces offer unique control over device design in optics and optical integration regimes [1] for numerous functionalities such as focusing [2], chiral imaging [3,4], aberration corrections [5,6], structure beam generation [79], optical knots [10], holography [11,12], absorbers [13,14], super-resolution lens [15,16] and tunable devices [17,18], to name only a few. Moreover, these devices can potentially be used in equally varied fields such as microscopy [1921], spectroscopy [2224], imaging [2528], communications [2932], bio-sensing [3335], human-machine interface [3639], and defense [40]. Thus, design techniques for such metasurfaces are also a multidimensional active research and development topic, including material innovations [41], nature-inspired design [42], artificial intelligence-based designs [43], and multifunctional optimizations [44], amongst other endeavors.

Multifunctional metasurfaces offer fertile research grounds to realize innovative applications with compactness and enhanced design dimensions. This furthers the cause of advanced integrated devices by alleviating the bottlenecks of complex systems’ design. Conventionally, multiple optical functionalities are integrated into a single metasurface using the wavelength, polarization, and angle multiplexing of incidence light [45] by implementing segmentation, interleaving, and layering design techniques [46,47]. The research community also exploited the harmonic nature of constituent building blocks to realize multifunctional devices such as achromatic lenses [48]. The earliest multifunctional metasurface designs stem from the interleaving and segmentation of propagation phase (PP) elements and exploited spatial multiplexing to achieve multifunctionality. Though functional and simple, such designs suffered from low angular resolution, meager efficiency, and noise for segmented design and speckle noise and capacity reduction for interleaved design [49].

To overcome the aforementioned shortcomings, polarization-based multiple function integration is demonstrated using PSHE, which refers to splitting opposite spin photons into separate planes orthogonal to the light propagation direction [50]. PSHE manifests itself by spin-orbital interactions that convert spin angular momentum into orbital angular momentum (OAM), though the effect is very weak [51]. The OAM of light is conserved, which leads to the splitting of photons based on their angular momentum. This effect has been observed in Pancharatnam-Berry (PB) phase and Rytov-Vladimirskii-Berry phase. The PB phase modulation effect is due to differences in effective refractive indexes along the propagation axis, which are asymmetric with greater refractive index along longer one. Due to such a difference, PSHE is observed as the opposite polarity photons split along the plane perpendicular to the propagation direction owing to asymmetric force. The efficiency of the PB phase-based PSHE phenomenon is quite high [52], and recently, 100% efficiency has been claimed for external PSHE using metasurfaces [53]. These developments spurred the design efforts to implement multifunctional devices based on PSHE [54], which can split orthogonal circular helicities of incident light. Earlier the independent phase control of orthogonal components still needed to be implemented. These PB-based designs act simultaneously as convex and concave lenses for opposite polarities forming real and virtual foci for the intended functionality [55].

To circumvent this limitation, orthogonal phases for both polarizations are multiplexed on the same interface e.g., interleaving [56]. This novelty in design offers little advantage over traditional design besides the ability to control output using polarization. Maguid et al. [57] used shared-aperture methodology to interleave and segment nanoantenna for a polarization-sensitive response. Li et al. [58] used the phase of the incident light to multiplex multiple vortices. Jiang et al. [59] used staked design and orbital angular momentum to multiplex multiple channels using a single metasurface. Zhang et al. used multiplexing and interleaving to combine multiple functionalities in a single metasurface [60]. Zhang et al. designed a geometric metalens to independently control orthogonal phases at opposite incident helicities using pure geometric phase [61]. All of these designs, though successful in implementing multifunctional behavior, sacrifice efficiency and/ or are complex in design and fabrication [45]. To further the degree of multifunctionality, the merger of propagation and geometric phase was proposed in [62], which allowed independent control of orthogonal helicities. Later, Wang et al. demonstrated the combination of PB and geometric phases to design multifunctional and even multidimensional metasurfaces [63]. All of these designs have a singular focus at either helicity. Wang et al. independently modulated two foci for orthogonal helicities by simultaneously interleaving propagation phase (PP) and geometric phase (PB) elements. These papers have used spin multiplexing to integrate multiple functionalities; however, the number of information channels is limited, complex to design, and prone to noise and cross-talk.

Herein, we present a multifocal and multidimensional all-dielectric transmissive metasurface platform exploiting the PSHE-based novel phase multiplexing technique that provides independent control of orthogonal helicities, squeezing spin-dependent quad information channels. The proposed platform exploits a combination of PP and PB (${\mathrm{\Psi}_{PG}}$) phase and pure PB (${\mathrm{\Psi}_{PB}}$) phase to independently manipulate two information channels at left-hand circularly polarized (LCP) and right-hand circularly polarized (RCP) incidence each, thus implementing spin-dependent quad information channels using merged phase ($\textrm{}{\mathrm{\Psi}_{MP}})$. The unique phase mergence technique employs independent PSHE decoupling sub-cells in an interleaved super-cell-based metasurface. The schematic representation of the proposed methodology is given in Fig. 1.

 figure: Fig. 1.

Fig. 1. Schematic illustration of the proposed multifocal and multifunctional metasurface platform for quad and independent information encoding. (a) Two independent functions for LCP incidence. (b) Two independent functions for RCP incidence. The inset illustrates the perspective view of the super-cell consisting nanoantenna of silicon nitride sitting on a sapphire substrate. Schematic of design of (c)Lens A forming four independent foci on axis (d) Lens B forming mixed off axis foci & (e) Lens C forming mixed on- and off-axis focal points

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As a demonstration of the multifunctional and multidimensional capabilities of the proposed metasurface platform, we numerically studied multiple metalenses with transverse, longitudinal, and mixed focusing using a combination of $\textrm{}{\mathrm{\Psi}_{PG}}$ and $\textrm{}{\mathrm{\Psi}_{PB}}$ by implementing three metasurface based lenses A, B and C. Figure 1 represents the schematic design for these lenses in Figs. 1 (c), (d) and (e) forming independent focal points in different configurations.

To the best of our knowledge, the presented design methodology explores the effects of phase mergence of PHSE techniques on a single metasurface to implement a multifunctional and multidimensional device that has yet to be studied. The presented numerical study shows that such a unique approach can yield a superior multifunctional design, which may find potential applications in imaging, communication, and information storage.

2. Design methodology

The proposed phase mergence technique is pictorially represented in Fig. 2. Information, represented by a multifocal lens, is mapped to phases where each phase is then mapped to orthogonal helicity, which are, in turn, mapped to different decoupling schemes, which are, finally, merged onto the single metasurface. To design the metasurface with novel phase mergence, we begin with the independent phases ${\varphi _{ {\pm} 1}}$ and ${\varphi _{ {\pm} 2}}\; $ encoding the independent information channels. The subscript ± denotes LCP and RCP helicity, respectively, whereas 1 and 2 denote the information channels resulting in a total of four independent phases to be merged on a single metasurface. The merged phase $\textrm{}{\mathrm{\Psi}_{MP}}$ is then given as,

$$\textrm{}{\mathrm{\Psi}_{MP}}({x,y} )= {\mathrm{\Psi}_{PG}}{{\mathbf 1}_{\boldsymbol A}}({x,y} )+ {\mathrm{\Psi}_{PB}}{{\mathbf 1}_{\boldsymbol B}}({x,y} )$$
$$\textrm{}{\mathrm{\Psi}_{MP}}({x,y} )= {f_{{\mathrm{\Psi}_{PG}}}}({{\varphi_{ + 1}}({x,y} ),\; {\varphi_{ - 1}}({x,y} )} ){{\mathbf 1}_{\boldsymbol A}}({x,y} )+ {f_{{\mathrm{\Psi}_{PB}}}}({{\varphi_{ + 2}}({x,y} ),\; {\varphi_{ - 2}}({x,y} )} ){{\mathbf 1}_{\boldsymbol B}}({x,y} )$$
where ${\mathbf 1_{{\boldsymbol A},{\boldsymbol B}}}({x,y} )$ are indicator functions that are one whenever $({x,y} )$ belongs to the set A/B and zero otherwise and ${f_{{\mathrm{\Psi}_{PG}}}}({{\varphi_{ {\pm} 1}}({x,y} )} )$ and ${f_{{\mathrm{\Psi}_{PB}}}}({{\varphi_{ {\pm} 2}}({x,y} )} )$ are phase mapping functions that map the phases on independent orthogonal helicities. These functions are defined below.

 figure: Fig. 2.

Fig. 2. The overview of proposed phase mergence technique. The functionality, in this case a multifocal lens, is mapped to independent phases which are merged using orthogonal helicities, onto a single metasurface.

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The ${\mathrm{\Psi}_{PG}}$ phase is a function of the structural and geometrical transformation of the nano-element. Using Jone’s calculus, the output phase of an anisotropic element due to birefringence is given as $\varphi + 2\sigma \theta $, where $\varphi $ is the structural phase defined by the dimensions of the nano-element and $\theta $ is the phase due to rotation of the structural element with respect to the principle axis. Using this result, which is given in detail in section I of the Supplement 1, ${f_{{\mathrm{\Psi}_{PG}}}}$ can be given as,

$${f_{{\mathrm{\Psi}_{PG}}}}({{\varphi_{ + 1}}({x,y} ),\; {\varphi_{ - 1}}({x,y} )} )= \left\{ {\begin{array}{{c}} {\varphi ({x,y} )\mapsto \max T({\varphi \textrm{|}L,W} )\qquad s.t.\; \varphi = \; {f_{structure}}({{\varphi_{ {\pm} 1}}({x,y} )} )}\\ {\theta ({x,y} )\mapsto R(\theta )\qquad \qquad \quad s.t.\; \theta = \; {f_{geom}}({{\varphi_{ {\pm} 1}}({x,y} )} )} \end{array}} \right.$$
which mathematically states that the phase mapping function is a combination of geometric optimization functions that maps $\varphi ({x,y} )$ to the structure of the nanoantenna through ${f_{structure}}({\varphi _{ {\pm} 1}}({x,y} )$ and $\theta ({x,y} )$ to the rotation of the nanoantenna through ${f_{geom}}({{\varphi_{ {\pm} 1}}({x,y} )} )$. The structural and rotational mapping functions are provided in the Supplement 1.

For other phase mapping function ${f_{{\mathrm{\Psi}_{PB}}}}({{\varphi_{ {\pm} 2}}({x,y} )} )$, PB phase is used to achieve independent RCP/LCP phase control by considering the far field electric field distribution. As the phase imparted to the interacting wave only depends upon the angle of the nano element's rotation, there is no structural dependence. For independent LCP/RCP focusing, the resultant far-field is a combination of LCP and RCP components for which the phase at the metasurface hence can simply be described as

$$\angle E = ta{n^{ - 1}}\left( {\frac{{\sin ({{\varphi_ + }} )+ \textrm{sin}({{\varphi_ - }} )}}{{\cos ({{\varphi_ + }} )+ \textrm{cos}({{\varphi_ - }} )}}} \right).$$
Where ${\varphi _ + }$ and ${\varphi _ - }$ represents general phases for LCP and RCP incidence, thus rotation of $\angle E/2$ imparts the required phases independently on the orthogonal circular helicities which gives ${f_{{\mathrm{\Psi}_{PB}}}}({{\varphi_{ {\pm} 2}}({x,y} )} )$ as,
$${f_{{\mathrm{\Psi}_{PB}}}}({{\varphi_{ + 2}}({x,y} ),\; {\varphi_{ - 2}}({x,y} )} )= R(\theta )\; s.t.\; \; \theta = ta{n^{ - 1}}\left( {\frac{{\sin ({{\varphi_ + }} )+ \textrm{sin}({{\varphi_ - }} )}}{{\cos ({{\varphi_ + }} )+ \textrm{cos}({{\varphi_ - }} )}}} \right).$$

The decoupling can be proven by simply applying the argument operator and expanding the resultant phase that would give ${\varphi _ + }({x,y} )+ {\varphi _ - }({x,y} )/2$ and is provided in section I of the Supplement 1.

The phase mergence then becomes

$${f_{{\mathrm{\Psi}_{MP}}}}({{\varphi_{ + 1,2}}({x,y} ),\; {\varphi_{ - 1,2}}({x,y} )} )= \left\{ {\begin{array}{{c}} {{f_{{\mathrm{\Psi}_{PG}}}}({{\varphi_{ + 1}}({x,y} ),\; {\varphi_{ - 1}}({x,y} )} ) \qquad ({x,y} )\; \in \boldsymbol A}\\ {{f_{{\mathrm{\Psi}_{PB}}}}({{\varphi_{ + 2}}({x,y} ),\; {\varphi_{ - 2}}({x,y} )} )\qquad{x,y} )\in \boldsymbol B} \end{array}} \right.\qquad $$
where
$${\boldsymbol A} = \textrm{\{ }({x,y} )\; \textrm{|}\; ({x + y} )= 2n\ast U\} $$
$${\boldsymbol B} = \textrm{\{ }({x,y} )\; \textrm{|}\; ({x + y} )= ({2n + 1} )\ast U\} $$
where U is the period of the cell arrangement on the metasurface and $n = 1,\; 2\; , \ldots ..$ is sequence of positive integers. To implement the lens, we define independent phases as
$${\varphi _{ {\pm} 1,2}}({x,y} )= \frac{{2\pi }}{\lambda }\left( {\sqrt {{{({x - {x_i}} )}^2} + {{({y - {y_i}} )}^2} + {f_i}^2} - {f_i}} \right)$$
where $i = 1,\; 2,\; 3,\; 4,$ and $({{x_i},{y_j}} )$ are coordinated of the focal points. ${\varphi _{ {\pm} 1,2}}({x,y} )$ represents phase at point $({x,y} )$ for ${i^{th}}$ functionality mapped to even and odd positions forming an interleaved structure with alternate PSHE-based helicity decoupling technique. All even-numbered nanoantennas (assuming the numbering starts at top left with element zero) are mapped using ${f_{{\mathrm{\Psi}_{PG}}}}({{\varphi_{ {\pm} 1}}({x,y} )} )$ mapping and odd-numbered are mapped using the ${f_{{\mathrm{\Psi}_{PB}}}}({{\varphi_{ {\pm} 2}}({x,y} )} )$.

3. Results & discussion

In this section detailed discussion for device design, simulation and results is presented. Firstly, Unit cell design simulations and results are discussed which is used to implement Eq. (7). The results and discussion for the implemented metalens follows afterwards.

The proposed hybrid spin-decoupling design employs two PSHE elements, where each element can independently control the phase of LCP and RCP incident light. To decouple the multiplexed optical functionalities, every element should act as a half-wave plate exhibiting maximum possible cross-polarization transmission efficiency. The proposed design follows a two-step unit cell optimization procedure for half-wave plate design and phase mapping. The required phase mapping function is derived previously and presented in Eq. (6) whereas the complete procedure of unit cell optimization is depicted in Fig. 3

 figure: Fig. 3.

Fig. 3. Numerical optimization of the fundamental building blocks. (a) Radius vs. period sweep to determine the optimum periodicity of the unit cell. (b) Length vs. width sweep for cross-polarization transmission. (c) Length vs. width sweep for co-polarization transmission. (d) Phase profile of cross-polarized transmitted light. (e) Perspective view of the Si3N4 nanoantenna sitting on the sapphire substrate and the complete phase coverage of each PSHE element vs. it’s in-plane rotation.

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. A bandgap-engineered silicon nitride (Si3N4) material is used as a best-suited candidate due to its excellent properties in the ultraviolet-visible (UV-Vis) band. The complex refractive index of the Si3N4 are measured through ellipsometry and is detailed in Figure S1 of the Supplement 1. The sufficient large value of refractive index and meager extinction coefficient allows the complete phase coverage while maintaining the maximum possible transmission profile in the desired band, respectively. The strong confinement of light within the nanoantenna and support of multimodal resonances due to optical properties at the design wavelength is motivated by applications in specifically Bio-sciences and related domains. This band finds usage in tissue imaging [64], sub-cellular [65], and plant imaging [66], to name a few. The resonances are simulated and reported in section II of Supplement 1.

As the first step in optimization, we determined the optimum value of the periodicity of the unit cell which allow the maximum possible constructive interference of the diffracted light. A sweep is performed over the diameter of the nanoantenna and the periodicity of the unit cell. The recorded transmission and the phase coverage is given in Fig. 3 (a) and (e), respectively. The phase coverage of the metasurface is determined by it dimensions as well as the separation between the nanoantenna. This phenomenon is well understood using index waveguide theory that models the change of phase of the propagating electromagnetic energy through abrupt transitions. The nanoantenna acts as an abruptly introduced waveguide in an otherwise uniform propagation media. The solution of Maxwell’s equations for such a transition introduces a phase change which also is effected by the neighboring similar structures. As analytical solution for large surfaces and number of interacting elements is not feasible numerical techniques are used. The results of transmission coefficient, both magnitude and phase, are recorded for different set of judiciously chosen values for the aforementioned sweep. From the results in Fig. 3(a), the broadest coverage of geometrical variation is achieved around 290 nm as indicated by the dashed line. At 290 nm, complete (0-2π) phase is covered in the least variation of geometric parameter. To optimize the structure acting as a half-wave plate, the length and width of the nanoantenna is swept and transmission profile of the co- and cross-polarized light is recorded and presented in Fig. 3 (b) and (c), respectively. The difference in transmission is due to the different propagation characteristics along the short and long dimensions of the element. In Fig. 3 (b), the co-polarized transmission behavior is illustrated whereas Fig. 3 (c) shows the transmission profile for the cross-polarized light. After careful analysis of these graphs, the region with maximum cross-polarized and minimum co-polarized transmission is selected and marked as solid white edged rectangle. For other (Ψ_PG) phase elements, a library of different sized elements is constructed that also mimics half-wave plate with varying efficiency. Each element of the library imparts a different phase with a realistic gap of 45 degrees between adjacent geometries. The selection is shown in Fig. 3 (b) and (c) using solid white colored circles. In Fig. 3 (d) the corresponding phase is shown using same solid white circles. For ${\mathrm{\Psi}_{PB}}$ phase, we selected a single size with high cross-polarization efficiency. Figure 3 (b) reports the various sizes selected for the lens design. Figure 3 (e) reports the phase sweep of the selected elements. Blue plot is for ${\mathrm{\Psi}_{PG}}$ phase element depicted in upper left corner of the figure. The upper axis shows different sized bricks for reference of blue curve. The grey curve is for PB element and it shows phase vs rotation. The second element is depicted in lower right corner. The complete $({0 - 2\pi } )$ phase is achieved for both elements. The optimized dimensions of the selected elements are given in section II of the Supplement 1.

As per previous discussion, spin-decoupled metasurfaces can be design by multiplexing the propagation phase and PB phase, simultaneously. For proof of the concept, we firstly demonstrated a metasurface capable of longitudinal spin splitting the four focus points at different lengths along propagation axis. For this purpose, using Eq. (7), an interleaved metalens is designed and simulated where each component encodes two phase functions for two focal points along the propagation axis (z-axis). Each of the element faithfully reproduces the intended functionality with minimum cross talk, as presented in Fig. 4. The numerical results validate the multifunctional capability of the proposed PSHE-based method. The constructive interference at the designed points only, indicated by the concentration of energy at the focal spots in the figure, is another indication of minimum cross talk.

 figure: Fig. 4.

Fig. 4. Longitudinal multifocal metalens simulation for different incident helicities. (a) Under the LP incidence. (b) For LCP incidence & (c) under RCP incidence. For clear visibility of the focusing spots, the insets show the focal points in transverse planes at corresponding values along the propagation axis. For LCP incidence same focal points are zoomed in and given in (d) for focal spot at 11.3 µm (e) for focal spot at 19.8 µm(d) for focal spot at 27.9 µm and (d) for focal spot at 36.1µm

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Each metasurface of $25 \times 25{\;\ \mathrm{\mu}}{\textrm{m}^2}$ is designed and numerically simulated. The four focal planes are set as ${\textrm{f}_{\textrm{RC}{\textrm{P}_1}}} = 17.5{\;\ \mathrm{\mu} \mathrm{m}},\textrm{}{\textrm{f}_{\textrm{RC}{\textrm{P}_2}}} = 27.5{\;\ \mathrm{\mu} \mathrm{m}},\textrm{}{\textrm{f}_{\textrm{LC}{\textrm{P}_1}}} = 12.5{\;\ \mathrm{\mu} \mathrm{m}},\textrm{and}{\textrm{f}_{\textrm{LC}{\textrm{P}_2}}} = 22.5{\;\ \mathrm{\mu} \mathrm{m}}$, respectively. The electric field intensity of the designed on-axis multifunctional metalens is shown in Fig. 4 . Under the linearly polarized (LP) incidence, the behavior of the diffracted light is shown in Fig. 4 (a) forming the four focal spots along the propagation axis. These spots are formed at 11.3 µm and 27.9 µm for LCP and 19.8 µm and 36.1 µm for RCP. Under the LCP incidence, the simulated results are shown in Fig. 4 (b), where the foal spots are formed at 11.3 µm and 27.9 µm, as expected. The two focal points along longitudinal axis clearly indicate spin decoupling in each of the unit cell element. Similarly, for RCP incidence, the designed focal points are produced as shown in Fig. 4 (c). The results reaffirm spin decoupling in hybrid design. The off-axis simulation confirms that the design is capable of faithfully encoding independent phases on spin de-coupled channels. Further such design channels can be mixed in a single design and can produce desired results faithfully.

The design simulation produced four focus points under LP incidence, two each for RCP and LCP components which is confirmed using respective incidence polarization, as the design of Lens A and the PSHE decoupling intended.

Design of Lens B demonstrates the multi-dimensional capabilities of the proposed technique by focusing four independent beams in four different transverse quadrants along the propagation axis. Using equation Lens B is designed to focus at focal point of $\textrm{f} = 12.5{\;\ \mathrm{\mu} \mathrm{m}}.$ Each of the focal point is at $\varDelta \textrm{x} = \textrm{}\varDelta \textrm{y} = 5{\mathrm{\mu} \mathrm{m}}$ from the origin of the plane at $10{\mathrm{\mu} \mathrm{m}}$. Hence all the focal points are ${\textrm{f}_{\textrm{RC}{\textrm{P}_1}}} = \textrm{}{\textrm{f}_{\textrm{RC}{\textrm{P}_2}}} = \textrm{}{\textrm{f}_{\textrm{LC}{\textrm{P}_1}}} = \textrm{}{\textrm{f}_{\textrm{LC}{\textrm{P}_2}}} = 12.5{\;\ \mathrm{\mu} \mathrm{m}}.$ The focal points for the designed lens are $({{x_{RCP1}},{y_{RCP1}}} )= ({ - 5{\mathrm{\mu} \mathrm{m}},0} ),({{x_{RCP2}},{y_{RCP2}}} )= ({ - 5{\mathrm{\mu} \mathrm{m}},0} ),\; ({{x_{LCP1}},{y_{LCP1}}} )= ({5{\mathrm{\mu} \mathrm{m}},0} ),({{x_{LCP2}},{y_{LCP2}}} )= ({0,5{\mathrm{\mu} \mathrm{m}}} )$ for a 25 µm by 25 µm lens. The multi-dimensional nature of the output show-cases the spin-decoupling can effectively encode multi-dimensional phases independently on each channel.

The simulated results are shown in Fig. 5. Again the results confirm that hybrid-PSHE technique has multi-dimensional capabilities as well. The focal spots formed has strong responses from constructive interference and remaining plane has minimal energy distribution. This indicates complete control of independent channels on the phase encoded and minimal cross talk. Figure 5 (a) shows the output with LP incidence. All four focus points are formed at the focal position of 11.3 ${\mathrm{\mu} \mathrm{m}}$. Two points are formed for LCP component and other two corresponds to RCP component. The locations of focus points are $({{\textrm{x}_{\textrm{RCP}1}},{\textrm{y}_{\textrm{RCP}1}}} )= ({ - 5{\mathrm{\mu} \mathrm{m}},0} ),({{\textrm{x}_{\textrm{RCP}2}},{\textrm{y}_{\textrm{RCP}2}}} )= ({ - 5{\mathrm{\mu} \mathrm{m}},0} ),\textrm{}({{\textrm{x}_{\textrm{LCP}1}},{\textrm{y}_{\textrm{LCP}1}}} )= ({5{\mathrm{\mu} \mathrm{m}},0} ),({{\textrm{x}_{\textrm{LCP}2}},{\textrm{y}_{\textrm{LCP}2}}} )= ({0,5{\mathrm{\mu} \mathrm{m}}} )$. Figure 5(b) shows results for LCP incidence. Two focal pints are formed at $({{\textrm{x}_{\textrm{RCP}1}},{\textrm{y}_{\textrm{RCP}1}}} )= ({ - 5{\mathrm{\mu} \mathrm{m}},0} ),({{\textrm{x}_{\textrm{RCP}2}},{\textrm{y}_{\textrm{RCP}2}}} )= ({0, - 5{\mathrm{\mu} \mathrm{m}}} ).$

 figure: Fig. 5.

Fig. 5. Transverse multifocal lens simulation for different incident helicities. (a) LP incidence (b) LCP incidence & (c) RCP incidence, (a), (b), (c) results are in XZ-planes. (d) XY- plane for LP incidence at z = 11.3 µm (e) XY- plane for LCP incidence at z = 11.3 µm (f) XY- plane for RCP incidence at z = 11.3 µm. Insets of zoomed in results have been added for clear visibility.

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First point is from first element and second comes from the other one. This proves that each element is able to transmit single desired helicity with minimum cross talk to other one. This is indicated by empty positions of corresponding orthogonal helicities in the LCP and RP incidences’ results. Figure 5(c) shows the result for RCP incidence. The results again confirm the multi-dimensional nature of the proposed methodology. The focal points are formed at (x_LCP1,y_LCP1) = (5µm,0),(x_LCP2,y_LCP2) = (0,5µm). Due to minimum cross talk corresponding RCP focal points are completely absent.

Finally, Lens C demonstrates multi-dimensional and multi-functional nature by focusing incident light at multiple transverse positions at multiple focal planes. The focal points are formed in xy-plane at different z-planes. This would prove that proposed technique can independently control phases for simultaneous longitudinal and transverse focusing. We use equation to calculate the phase required for each functionality and map it to one of the available spatial-helical channel. The focal planes of the lens are ${\textrm{f}_1} = 12.5\mathrm{\;\ \mu m\;\ and\;\ }{\textrm{f}_2} = 17.5{\;\ \mathrm{\mu} \mathrm{m}}$. The offset in transverse plane is set as $\varDelta \textrm{x} = \textrm{}\varDelta \textrm{y} = 5{\mathrm{\mu} \mathrm{m}}$. The multi-dimensional, multi-functional lens implementation establishes versatility of the proposed PSHE mergence.

Simulated results for the designed lens are given in Fig. 6. For LP simulation four focal points are formed at $({{\textrm{x}_{\textrm{RCP}1}},{\textrm{y}_{\textrm{RCP}1}},{\textrm{f}_1}} )= ({ - 5{\mathrm{\mu} \mathrm{m}},0{\mathrm{\mu} \mathrm{m}},11.3{\mathrm{\mu} \mathrm{m}}} ),({{\textrm{x}_{\textrm{RCP}2}},{\textrm{y}_{\textrm{RCP}2}},{\textrm{f}_2}} )= ({5{\mathrm{\mu} \mathrm{m}},0{\mathrm{\mu} \mathrm{m}},19.6{\mathrm{\mu} \mathrm{m}}} )$, $({{\textrm{x}_{\textrm{LCP}1}},{\textrm{y}_{\textrm{LCP}1}},{\textrm{f}_1}} )= ({5{\mathrm{\mu} \mathrm{m}},0{\mathrm{\mu} \mathrm{m}},11.3{\mathrm{\mu} \mathrm{m}}} ),({{\textrm{x}_{\textrm{LCP}2}},{\textrm{y}_{\textrm{LCP}2}},{\textrm{f}_2}} )$ = $({5{\mathrm{\mu} \mathrm{m}},0{\mathrm{\mu} \mathrm{m}},19.6{\mathrm{\mu} \mathrm{m}},} )$ for dimensions of 25 µm by 25 µm as given in Fig. 6 (a). Independent phase control of orthogonal helicities is achieved in PSHE mergence. The electric field intensities exhibit strong constructive interference responses from each helicity. Figure 6(b) shows response for the RP incidence. The focuses are formed at $({{\textrm{x}_{\textrm{RCP}1}},{\textrm{y}_{\textrm{RCP}1}},{\textrm{f}_1}} )= ({ - 5{\mathrm{\mu} \mathrm{m}},0{\mathrm{\mu} \mathrm{m}},11.3{\mathrm{\mu} \mathrm{m}}} )\textrm{and}({{\textrm{x}_{\textrm{RCP}2}},{\textrm{y}_{\textrm{RCP}2}},{\textrm{f}_2}} )= ({5{\mathrm{\mu} \mathrm{m}},0{\mathrm{\mu} \mathrm{m}},19.6{\mathrm{\mu} \mathrm{m}}} )$. Corresponding points for the LP incidence are non-existent showing minimal cross talk. Finally, Fig. 5(c) shows response for LCP incidence. The focuses are formed at $\textrm{}({{\textrm{x}_{\textrm{LCP}1}},{\textrm{y}_{\textrm{LCP}1}},{\textrm{f}_1}} )= ({5{\mathrm{\mu} \mathrm{m}},0{\mathrm{\mu} \mathrm{m}},11.3{\mathrm{\mu} \mathrm{m}}} )\; \textrm{and\; }({{\textrm{x}_{\textrm{LCP}2}},{\textrm{y}_{\textrm{LCP}2}},{\textrm{f}_2}} )= ({5{\mathrm{\mu} \mathrm{m}},0{\mathrm{\mu} \mathrm{m}},19.6{\mathrm{\mu} \mathrm{m}}} )$. Agreement with the intended design is verified in this simulation as well. The multi-dimensional and multi-functional capabilities of mixed PSHE techniques are hence established using on- and off- axis lens design.

 figure: Fig. 6.

Fig. 6. Longitudinal and transversal multifocal lens simulation for different incident helicities. (a) LP incidence (b) LCP incidence & (c) RCP incidence. All results are in XZ-planes. The insets show the focal points in transverse planes at corresponding z values. The empty regions indicate cross-polarization suppression and minimum cross talk. (d) zoomed in XZ – plane focal points (e) zoomed in XY – plane focal points

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4. Conclusion

In this paper multidimensional and multifunctional PSHE-based phase merger technique is proposed. Numerical simulations using different type of multifocal lenses demonstrated the feasibility of the proposed approach. The phase mergence is done using PB and propagation phase on two different elements which constitute a super cell in the proposed interleaved structure. Each Nano-element multiplexes orthogonal helicities using different PSHE phases allowing splitting of light based upon helicity and subsequently controlling each of the orthogonal helicity independently. The proposed approach is verified using longitudinal, transverse and mixed focal lenses. Simulations show promising results and verifies the proposed philosophy. The proposed approach can be further developed by investigating different types of Spin-Hall and their mutual interaction.

Funding

King Abdullah University of Science and Technology (Innovative Technologies Laboratories).

Acknowledgement

The authors would like to acknowledge research funding to the Innovative Technologies Laboratories from King Abdullah University of Science and Technology (KAUST).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

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Supplement 1       Revised supplemental-document

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)

Fig. 1.
Fig. 1. Schematic illustration of the proposed multifocal and multifunctional metasurface platform for quad and independent information encoding. (a) Two independent functions for LCP incidence. (b) Two independent functions for RCP incidence. The inset illustrates the perspective view of the super-cell consisting nanoantenna of silicon nitride sitting on a sapphire substrate. Schematic of design of (c)Lens A forming four independent foci on axis (d) Lens B forming mixed off axis foci & (e) Lens C forming mixed on- and off-axis focal points
Fig. 2.
Fig. 2. The overview of proposed phase mergence technique. The functionality, in this case a multifocal lens, is mapped to independent phases which are merged using orthogonal helicities, onto a single metasurface.
Fig. 3.
Fig. 3. Numerical optimization of the fundamental building blocks. (a) Radius vs. period sweep to determine the optimum periodicity of the unit cell. (b) Length vs. width sweep for cross-polarization transmission. (c) Length vs. width sweep for co-polarization transmission. (d) Phase profile of cross-polarized transmitted light. (e) Perspective view of the Si3N4 nanoantenna sitting on the sapphire substrate and the complete phase coverage of each PSHE element vs. it’s in-plane rotation.
Fig. 4.
Fig. 4. Longitudinal multifocal metalens simulation for different incident helicities. (a) Under the LP incidence. (b) For LCP incidence & (c) under RCP incidence. For clear visibility of the focusing spots, the insets show the focal points in transverse planes at corresponding values along the propagation axis. For LCP incidence same focal points are zoomed in and given in (d) for focal spot at 11.3 µm (e) for focal spot at 19.8 µm(d) for focal spot at 27.9 µm and (d) for focal spot at 36.1µm
Fig. 5.
Fig. 5. Transverse multifocal lens simulation for different incident helicities. (a) LP incidence (b) LCP incidence & (c) RCP incidence, (a), (b), (c) results are in XZ-planes. (d) XY- plane for LP incidence at z = 11.3 µm (e) XY- plane for LCP incidence at z = 11.3 µm (f) XY- plane for RCP incidence at z = 11.3 µm. Insets of zoomed in results have been added for clear visibility.
Fig. 6.
Fig. 6. Longitudinal and transversal multifocal lens simulation for different incident helicities. (a) LP incidence (b) LCP incidence & (c) RCP incidence. All results are in XZ-planes. The insets show the focal points in transverse planes at corresponding z values. The empty regions indicate cross-polarization suppression and minimum cross talk. (d) zoomed in XZ – plane focal points (e) zoomed in XY – plane focal points

Equations (9)

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Ψ M P ( x , y ) = Ψ P G 1 A ( x , y ) + Ψ P B 1 B ( x , y )
Ψ M P ( x , y ) = f Ψ P G ( φ + 1 ( x , y ) , φ 1 ( x , y ) ) 1 A ( x , y ) + f Ψ P B ( φ + 2 ( x , y ) , φ 2 ( x , y ) ) 1 B ( x , y )
f Ψ P G ( φ + 1 ( x , y ) , φ 1 ( x , y ) ) = { φ ( x , y ) max T ( φ | L , W ) s . t . φ = f s t r u c t u r e ( φ ± 1 ( x , y ) ) θ ( x , y ) R ( θ ) s . t . θ = f g e o m ( φ ± 1 ( x , y ) )
E = t a n 1 ( sin ( φ + ) + sin ( φ ) cos ( φ + ) + cos ( φ ) ) .
f Ψ P B ( φ + 2 ( x , y ) , φ 2 ( x , y ) ) = R ( θ ) s . t . θ = t a n 1 ( sin ( φ + ) + sin ( φ ) cos ( φ + ) + cos ( φ ) ) .
f Ψ M P ( φ + 1 , 2 ( x , y ) , φ 1 , 2 ( x , y ) ) = { f Ψ P G ( φ + 1 ( x , y ) , φ 1 ( x , y ) ) ( x , y ) A f Ψ P B ( φ + 2 ( x , y ) , φ 2 ( x , y ) ) x , y ) B
A = ( x , y ) | ( x + y ) = 2 n U }
B = ( x , y ) | ( x + y ) = ( 2 n + 1 ) U }
φ ± 1 , 2 ( x , y ) = 2 π λ ( ( x x i ) 2 + ( y y i ) 2 + f i 2 f i )
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