Vector beams (VBs) are light fields with a spatially inhomogeneous polarization distribution and are essentially the superposition of the eigenstates of the total angular momentum coupling the spin and orbital angular momentum [1,2]. Owing to their unique properties, VBs have attracted extensive research interest over the last few decades and have found broad applications in classical and quantum sciences. Classical applications include optical particle trapping [3], laser manufacturing [4], and high-dimensional optical communication [5]. The inseparability of VBs in space and polarization degrees of freedom is similar to the local entanglement of a bipartite system [6]. As novel resources for quantum information protocols, VBs have been used in quantum key distribution (QKD) [7], quantum walks [8], and the characterization of quantum channels [9]. Furthermore, discovering new exotic phenomena in VBs, such as the polarization Möbius strip [10], singularity knots and links [11], and quasiparticle-like topological states in the form of skyrmions [12] and hopfins [13], has become an important frontier in research fields, providing new degrees of freedom of topological states for light-field manipulations. To date, VBs of conventional bulky sizes have been primarily generated using devices such as spiral phase plates (SPPs) [14], spatial light modulators (SLMs) [15], and digital micro-mirror devices (DMDs) [16]. As miniaturized and integrated devices, metasurfaces have activated the manipulation of VBs at the subwavelength scale. In particular, the generation of high-order Poincaré (HOP) beams has become a popular topic in this area [17–21].

Metasurfaces are two-dimensional (2D) planar arrays of metallic or dielectric nanoresonators, referred to as meta-atoms [22,23]. In the last decade, they have been used in various applications, such as metalenses [24], generation of VBs [25–27], holographic encryption [28], and polarization-entangled photon sources [29]. Recently, dynamic reconfigurable technologies of metasurfaces using different tunabilities [30] along with the advanced design methods using intelligent algorithms [31] have fostered novel forefront fields in metasurface research [32] and accelerated the development of miniaturized and multifunctional optical systems for commercialized applications [33–37]. Notably, the modulation of light source properties is a popular and important method of optical tunability [22,38], which has been widely used to realize vector holograms [39,40] and generate different types of VBs [18,41]; and in HOP beam generation, the troublesome and bulky combination of waveplate and polarizer to adjust the incident polarization [42] has been exploited to reconfigure the arbitrary polarization states [18,40,41,43]. In the last few years, the tunability method of longitudinal propagation distance [44] was developed, and significant advancements have been achieved in generating vectorial structured light, including VBs, and Bessel and vortex beams with topological charges and linear polarization states continuously variable along the distance [43,45–49]. However, in generating longitudinally tunable VBs, all metasurfaces used half-waveplate (HWP) meta-atoms, and only the cross-polarized components could be manipulated. Moreover, with the axicon-phase profiles directly imparted to the cross-left circularly polarized (LCP) and right circularly polarized (RCP) vortices of equal amplitudes, the generated VBs were limited to linear polarization states and a long propagation distance was required to complete the VBs over the entire equator of the HOP sphere.

In contrast, metasurfaces of quarter-waveplate (QWP) meta-atoms produce light waves containing both co- and cross-polarized components and provide flexible means to manipulate the output fields; they have attracted immense interest and have facilitated a variety of novel functionalities [50–55]. With the capabilities of multidimensional operations on the wavefronts, such as vortex generation, focusing, and deflection, metasurfaces of QWP meta-atoms to be designed for longitudinal modulation of the unequal weights of two orthogonal vortices to realize HOP beams with arbitrary elliptical polarization states would be an interesting and challenging subject. To achieve this objective, several issues need to be resolved, including performing the simultaneous longitudinal tunabilities of the co- and cross-polarized components of the meta-atoms, and tunabilities of the amplitudes of two orthogonal and conjugate vortices, contracting the prolonged tunable distance to enhance the intensity of the generated VBs, and providing the sound theoretical approach to analyze the VB fields of the proposed metasurface.

In this study, we proposed a dielectric QWP metasurface interleaved by two sub-metasurfaces to generate HOP beams with arbitrary elliptical polarization states using longitudinal tunability. The phase profiles of the two co- and cross-polarized components of the sub-metasurfaces were suitably designed under RCP light illumination. For the two co-polarized components, the propagation phases of the two sub-metasurfaces were configured to form the same helical, primary axicon, and hyperbolic-phase profiles, but secondary axicon-phase profiles of opposite signs. The geometric phase profiles of the two cross-polarized components are designed to include the required helical and secondary axicon-phase profiles. By suitably setting the combined propagation and geometric phases in consistency with the phase profiles for the co-polarized components, the two cross-polarized vortices were acquired to be opposite but orthogonal to their co-polarized counterparts. The superposed amplitudes of the two co-polarized and two cross-polarized vortices were made tunable with z. This ensured the propagation distance z as a new degree of freedom for modulating amplitude weights; and the generation of an HOP beam at an arbitrary latitude on the HOP sphere was realized. With the converging effect of the hyperbolic phase, the tunable span of the propagation distance was significantly shortened, and the intensities of the generated HOP beams were considerably enhanced. In addition, in the theoretical analysis of longitudinally tunable HOP beams generated by metasurfaces, we used the stationary-phase method and obtained an analytical solution for vectorial light fields of metasurfaces by combining defocus and deflections. To verify this method, we conducted simulations using the finite-difference time-domain (FDTD) method to calculate the light fields of the tunable HOP beams. We fabricated metasurface samples of amorphous silicon (a-Si:H) and experimentally demonstrated the generation of HOP beams. We designed novel metasurfaces of QWP meta-atoms to longitudinally modulate the output field and realized the generation of tunable HOP beams at variable latitudes on the meridian of an HOP sphere. This study adds the propagation distance as a new degree of freedom to VB manipulation with metasurfaces, achieves manipulation of the amplitudes of two circularly polarized (CP) vortices, and is of great importance for applications of metasurfaces and VBs in broad areas, such as optical micromanipulation, high-precision detection, and conventional and quantum communications.

Figure 1(a) shows a schematic of the longitudinally tunable manipulation of HOP beams with arbitrary elliptical polarization states using a QWP metasurface under RCP illumination, which are represented by the latitudes on a meridian on the HOP sphere. A schematic of the x-component intensity patterns of the VBs on the planes at different z values is shown in Fig. 1(a). Metasurface S comprises two interleaved sub-metasurfaces, SA and SB, and is represented as S=SA+SB. SA and SB are composed of QWP meta-atoms, and the physical quantities of the meta-atoms and sub-metasurfaces are designated with subscript A or B to distinguish them. Meta-atoms A and B are rectangular dielectric nanopillars of a-Si:H, and their side and top views are shown in Fig. 1(b). The length and width of the meta-atoms are denoted as L and W, respectively. The height H is constant, and the orientation angle θ of the meta-atom is the angle of the long side with respect to the x-axis. In metasurface design, meta-atoms are arranged on concentric circular rings. The increment in the radii of the two adjacent rings is the lattice period P, and meta-atoms A and B are alternately arranged on the same ring at intervals of P, as shown in Fig. 1(c).

For the incident RCP light with the chirality factor of σ=−1, its unit vector |R⟩ can be expressed as uσ=−1=[1σi]T=[1−i]T. The transmitted light field of meta-atom A or B contains the co-polarized component uco,A/B and the cross-polarized component ucr,A/B, whose unit vectors uσ=−1 and u−σ=1=[1−σi]T=|L⟩ have chirality factors σ=−1 and −σ=1, identical and opposite to that of the incident light, respectively. It is well known that only the propagation phase φp,A/B can be imparted on uco,A/B. In contrast, both the propagation phase φp,A/B and the geometric phase φg,A/B, or equivalently, the combined phase φp,A/B+φg,A/B, are imparted to ucr,A/B. Furthermore, the propagation and geometric phase profiles φp,A/B(r,ϕ,0) and φg,A/B(r,ϕ,0) for the metasurface are decomposed into different phase profiles for different functionalities, where (r, ϕ, 0) represents the coordinates of an arbitrary point in the metasurface plane in the discussion. Figure 1(d) shows schematic diagrams of the decomposed phase profiles and the combined propagation and geometric profiles, where the insets in the upper-right corners show enlarged views of the phase profiles in the center. In the following, we describe in detail the design of φp,A/B(r,ϕ,0) and the co-polarized component Eco,A/B(r,ϕ,0) of the transmitted light field of each sub-metasurface at z=0. First, the helical-phase profile was constructed to form a vortex wavefront by arranging meta-atoms of different sizes, which is represented by φph,A/B(ϕ)=lp,A/Bϕ=lpϕ, where the subscript h denotes the helical-phase profile. Second, a hyperbolic propagation phase profile φlens(r)=k(f−r2+f2) was constructed for the sub-metasurfaces to focus the wavefront of the generated beams and provide more concentrated energy, where k=2π/λ is the wavevector. Third, the primary axicon-phase profile φa,A/B(r)=−2πr/d1 [43] was constructed, which deflects the focused vortex beam to extend its existence by lengthening the focal depth. If the hyperbolic-phase profile is absent, this phase profile produces nondiffractive Bessel beams [56,57]. Thereafter, the secondary axicon-phase profiles of opposite signs, that is, ±δφpa(r)=∓2πr/d2, are introduced to the two co-polarized components Eco,A(r,ϕ,0) and Eco,B(r,ϕ,0), where d2 satisfies d2≫d1, and different deflections are provided for the two components. Thus, a z-component difference, Δkz,co, of the wavevectors is produced between the Eco,A(ρ,α,z) and Eco,B(ρ,α,z) on the plane at a distance z, owing to the imparted phase factors eiΔkz,coz/2 and e−iΔkz,coz/2 arising from the secondary axicon-phase profiles, respectively. Strictly speaking, Δkz,co is also related to φlens(r) and φa,A/B(r), as discussed in the following section. As a result, with the variable phase difference resulting from Δkz,co at different distances z, the interference of Eco,A(ρ,α,z) and Eco,B(ρ,α,z) yields a superposed co-polarized vortex beam with tunable intensity, which indicates that the propagation distance z is a degree of freedom to manipulate the amplitude of the vortex. Finally, the propagation phase constants, φ0,A=π/2 and φ0,B=0, are introduced to Eco,A(r,ϕ,0) and Eco,B(r,ϕ,0), respectively, and the amplitude of the superposed light field resulting from the interference of Eco,A(ρ,α,z) and Eco,B(ρ,α,z) varies in the form of cos(π/4+Δkz,coz/2), as shown by the red gradient color bar along the optical axis in Fig. 1(a). Overall, the propagation phase profile of each sub-metasurface is expressed as $$\begin{gathered} φp,A/B(r,ϕ,0)=φph,A/B(ϕ)+φlens(r)+φa,A/B(r)±δφpa(r)+φ0,A/B \\ =lpϕ+k(f−r2+f2)−2πrd1∓2πrd2+φ0,A/B. \end{gathered}$$(1)

Similarly, when designing the phase profile of the cross-polarized component Ecr,A/B(r,ϕ,0) at z=0, the geometric phase profile φg,A/B(r,ϕ,0)=2σθA/B(r,ϕ,0) with σ=−1 for RCP incidence is added, which is realized by rotating the orientation angle θ of the meta-atom. The helical geometric phase profile φgh,A/B(ϕ)=lgϕ with topological charge lgA=lgB=lg and lg=−2lp was designed; therefore, Ecr,A/B(r,ϕ,0) carries topological charge lg+lp=−lp, which is equal but opposite to that of Eco,A/B(r,ϕ,0). In addition, secondary axicon-phase profiles of ±δφga(r)=∓2δφpa(r)=±4πr/d2 were introduced to φg,A/B(r,ϕ,0), respectively. Thus, θA/B(r,ϕ,0) can be written as θA/B(r,ϕ,0)=qϕ∓2πrd2+θ0,(2)where q and θ0 are the rotational order and initial orientation angle of the meta-atoms, respectively, with q=lg/2 for σ=−1. Considering the above propagation phase profiles, the z-component difference, Δkz,cr, between the wavevectors of Ecr,A(ρ,α,z) and Ecr,B(ρ, α, z) is −Δkz,co. Therefore, the amplitude of the superposed light field varies with the factor sin(π/4+Δkz,coz/2), as shown by the green gradient color bar in Fig. 1(a). Thus, the phase profile of the cross-polarized component Ecr,A/B(r,ϕ,0) can be expressed as $$\begin{gathered} φp,A/B(r,ϕ,0)+φg,A/B(r,ϕ,0)=φp,A/B(r,ϕ)+φgh,A/B(ϕ)±δφga(r) \\ =−lpϕ+k(f−r2+f2)−2πrd1±2πrd2+φ0,A/B. \end{gathered}$$(3)

From the above analysis, we know that by varying the longitudinal coordinate z, the amplitudes of the co- and cross-polarized vortex fields Eco(ρ,α,z)=Eco,A(ρ,α,z)+Eco,B(ρ,α,z) and Ecr(ρ,α,z)=Ecr,A(ρ,α,z)+Ecr,B(ρ,α,z) vary with z in cosine and sine functions, respectively. Consequently, the superposition of Eco(ρ,α,z) and Ecr(ρ,α,z) generated a VB corresponding to an arbitrary latitude on the meridian of the HOP sphere.

With a subwavelength-order length and width, the rectangular nanopillar dielectric meta-atom had two axes of mirror symmetry along both laterals. An incident light linearly polarized along either axis passing through the meta-atom experiences different phase delays, resulting in form birefringence of the two axes acting as the fast and slow axes. Thus, a meta-atom can be considered as an arbitrary waveplate, and its Jones matrix can be written as [58] J(x,y)=R(−θ)[|Txx|eiφxx(x,y)00|Tyy|eiφyy(x,y)]R(θ),(4)where x=rcosϕ, y=rsinϕ, R(θ) is the rotation matrix, |Txx|, |Tyy|, φxx, and φyy represent the transmitted amplitudes and phases of the transmitted light field along the fast and slow axes of the meta-atom, respectively, the corresponding phase retardation is φxx−φyy, and the propagation phase of a meta-atom is φp=φxx. For the selected QWP meta-atoms, the approximation |Txx|≈|Tyy|≈1 is considered to hold, and phase retardation takes φxx−φyy=π/2.

When the QWP meta-atom is illuminated with LCP and RCP lights, the transmitted light fields are expressed as J(x,y)|L⟩=eiφp(x,y)|L⟩+eiφLR(x,y)|R⟩,(5)J(x,y)|R⟩=eiφp(x,y)|R⟩+eiφRL(x,y)|L⟩.(6)

The first terms on the right-hand sides of the above two equations are co-polarized components with phase φp(x,y), while the second terms are cross-polarized components with phases φLR/RL(x,y)=φp(x,y)+φg(x,y), where the geometric phase φg(x,y) is equal to 2σθ(x,y) and σ=±1 corresponds to |L⟩ and |R⟩ for illuminating light, respectively.

Combining Eqs. (4), (5), and (6), we obtain the following equations for φxx(x,y), φyy(x,y), and θ(x,y): φxx(x,y)=12[φLR(x,y)+φRL(x,y)]+π,(7)φyy(x,y)=12[φLR(x,y)+φRL(x,y)]+π2,(8)θ(x,y)=14[φLR(x,y)−φRL(x,y)].(9)

Substituting them into Eqs. (5) and (6), we can obtain the transmitted light field Et,A/Bσ(r,ϕ,0) for meta-atom A or B under the illumination of arbitrarily circularly polarized (CP) light uσ [52]: Et,A/Bσ(r,ϕ,0)=J(x,y)·uσ=12eiφp,A/Buσ+12ei(φp,A/B+π/2)ei2σθA/Bu−σ.(10)

The first term on the right-hand side of the above equation is practically the co-polarized component uco,A/B with unit vector uσ and phase φp,A/B, whereas the second term is the cross-polarized component ucr,A/B with unit vector u−σ and phase φp,A/B+2σθA/B, where 2σθ(x,y)=φg(x,y) is the geometric phase. Additionally, Et,A/Bσ(r,ϕ,0) denotes the transmitted light field of the sub-metasurface, and is written as Et,A/Bσ=−1(r,ϕ,0) for RCP light illumination.

For the QWP metasurface design, the meta-atom can be selected from the eight selected meta-atoms with a propagation phase (φp) closest to the desired phase, φp,A/B(r,ϕ,0), for the sub-metasurface given in Eq. (1). The meta-atoms were then arranged on the rings according to the orientation angle, θA/B(r,ϕ,0), as given in Eq. (2).

We now consider the light field EA/Bσ(ρ,α,z) produced by sub-metasurface SA or SB on the observation plane at a distance z. Based on the Rayleigh–Sommerfeld formula, the light field is expressed as [59] EA/Bσ(ρ,α,z)=iλ∫0r0∫02πEt,A/Bσ(r,ϕ,0)eiks·ssrdϕdr,(11)where r0 is the radius of the circular metasurface sample and s=zz^0+r−ρ is the position vector from point M(r,ϕ,0) to N(ρ,α,z) on the metasurface and observation planes, respectively, as shown in Fig. 2(a). Here, z^0 denotes the z-axis unit vector, and r and ρ are the position vectors in the metasurface and observation planes, respectively; s=|s|=z2+|r−ρ|2 represents the distance between points M and N; and ks=ks/s is the wavevector in vector form. The complex exponential factor, eiks·s, in the above equation is essentially eiks.

Because the output optical field Eσ(ρ,α,z) on the observation plane is a superposition of EAσ(ρ,α,z) and EBσ(ρ,α,z) of SA and SB, by substituting Eqs. (1), (2), and (10) into Eq. (11) and replacing the denominator s in the integrand with z, we obtain $$\begin{gathered} Eσ=−1(ρ,α,z)=EAσ=−1(ρ,α,z)+EBσ=−1(ρ,α,z)≈iei[kz+(kρ2/2z)]2λz∫0r0∫02πrdϕdreik(1/z−1/f)r2/2−ikr[ρcos(α−ϕ)]/z−i2πr/d1 \\ ×[eilpϕ(e−i2πr/d2+iφ0,A+ei2πr/d2+iφ0,B)uσ=−1 \\ +e−i2θ0+iπ/2e−ilpϕ(ei2πr/d2+iφ0,A+e−i2πr/d2+iφ0,B)u−σ=1], \end{gathered}$$(12)where the approximations f−r2+f2≈−r2/2f and sO′=z2+r2≈z+r2/2z resulting from f≫r and z≫r are used, and sO′ is the position vector from M(r,ϕ,0) to the center O′ of the observation plane. By calculating the integral over ϕ at the right-hand side of the above equation, and using integral representation of the Bessel function, Eσ=−1(ρ,α,z) can be further expressed as $$\begin{gathered} Eσ=−1(ρ,α,z)≈iei[kz+(kρ2/2z)]2λz∫0r0rdreikr2(1/2z−1/2f)−i2πr/d1 \\ ×[eilpα(e−i2πr/d2+iπ/2+ei2πr/d2)uσ=−1 \\ +e−i2θ0+iπ/2e−ilpα(ei2πr/d2+iπ/2+e−i2πr/d2)u−σ=1]Jlp(krρz). \end{gathered}$$(13)

In the integral in the above equation, the integrand is a complex function of r, and its first term, F1(r)=eikr2(1/2z−1/2f)−i2πr/d1−i2πr/d2+iπ/2uσ=−1rJlp(krρz), corresponds to the co-polarized component of SA. This implies that the factor eilpα is constant with respect to r and is written outside the integral sign. Here, F1(r) is a complex function; its amplitude function is rJlp(krρz), and phase is k(12z−12f)r2−2πrd1−2πrd2+π2. This phase has a unique point rc at which its derivative with respect to r is equal to zero, which is referred to as the radial stationary point of the phase [60]. As an approximate but convenient method for performing integration with a complex-valued integrand, the integral in the above equation is approximated to be proportional to the value of the integrand at the stationary point. This is termed the stationary-phase method, as derived in Ref. [61]. Applying this method also to solve the integrals for the co-polarized components of SB, and for the cross-polarized components of SA and SB in Eq. (13), respectively, and with the constant coefficient neglected, we have $$\begin{gathered} Eσ=−1(ρ,α,z)≈i2πrc[λ3f/z(f−z)]1/2Jlp(krcρ/z)eikz+ikrc2(1/2z−1/2f)−iC0z/[(f−z)d1] \\ ×eiπ/4eikρ2/2ze−iC0z/[(f−z)d2]+iπ/2+eiC0z/[d2(f−z)]eilpαuσ=−1+e−i2θ0+iπ/2eiC0z/[(f−z)d2]+iπ/2+e−iC0z/[(f−z)d2]e−ilpαu−σ=1, \end{gathered}$$(14)rc=λf(d1+d2)d1d2zf−z=C02πzf−z,(15)where C0=2πλf(1/d1+1/d2). Noticeably, in Eq. (14), the first and second terms with uσ=−1 on the right-hand side are Eco,A/B(ρ,α,z), respectively, whereas the other two terms with u−σ=1 are Ecr,A/B(ρ,α,z), respectively. By adding a term for identical CP, Eq. (14) can be further simplified to $$\begin{gathered} Eσ=−1(ρ,α,z)≈i2πrc[λ3f/z(f−z)]1/2Jlp(krcρ/z) \\ ×eiπ/4eikρ2/2zeikz+ikrc2(1/2z−1/2f)−iC0z/[d1(f−z)] \\ ×[cos(π4−C0d2zf−z)eilpαuσ=−1+e−i2θ0+iπ/2sin(π4−C0d2zf−z)e−ilpαu−σ=1], \end{gathered}$$(16)where the terms with uσ=−1 and u−σ=1 in Eq. (16) correspond to Eco(ρ,α,z) and Ecr(ρ,α,z), respectively. Considering Eq. (15), Eq. (16) can be written in the simple form of a standard HOP beam: Eσ=−1(ρ,α,z)≈F(ρ,z)(aReiΦ/2eilpαuσ=−1+aLe−iΦ/2e−ilpαu−σ=1),(17)where F(ρ,z) is the envelope function of Eσ=−1(ρ,α,z); aR=cosΘ/2 and aL=sinΘ/2 denote the amplitude coefficients of the RCP and LCP components, respectively, and Θ and Φ are the latitude and longitude for Eσ=−1(ρ,α,z) on the HOP sphere, respectively. F(ρ,z), Θ, and Φ are expressed as $$\begin{gathered} F(ρ,z)=i2π(z/λ)1/2ei(π/2−θ0)eik(z+ρ2/2z)⁢eikC02z/[8π2(f−z)]e−iC0z/[d1(f−z)] \\ ×Jlp(krcρ/z)[f/(f−z)]3/2(d1+d2)/d1d2, \end{gathered}$$(18){Φ=2θ0−π2Θ=π2−2C0d2zf−z.(19)

From Eqs. (17)–(19), it is clear that with constants d1 and d2, the parameters Θ, aR, and aL can be adjusted by changing the longitudinal distance z; thus, HOP beams are generated with tunable ellipticity of the polarization states, which is also represented by the latitude Θ on the meridian. Moreover, by setting θ0 for the meta-atoms, different metasurface samples can be designed to generate the HOP beams on meridians with different longitudes Φ. In particular, when θ0 is set to π/4, π/2, 3π/4, and π, the corresponding metasurfaces can be used to generate HOP beams on meridians Φ=0, Φ=π/2, Φ=π, and Φ=3π/2, respectively.

As discussed in Subsection 2.A, the differences between the z-components of the wavevectors Δkz,co=kz,co−A−kz,co−B and Δkz,cr=kz,cr−A−kz,cr−B are crucial for realizing the longitudinal tunability of the amplitude weights. Therefore, we further analyzed the effects of the hyperbolic and axicon phases on the deflection and wavevector components of the beams, providing more physical descriptions for the tunable manipulation of HOP beams. For incident-plane light waves propagating along z direction, the wavevector k is equal to kz^0. When the beam is deflected in cylindrical symmetry, the deflection angle of propagation with respect to the z-axis is τ, and the radial and z-components of the wavevector of the beam are expressed as kr=k⁢sinτe^r and kz=k⁢cosτz^0=k2−kr2z^0, respectively, where e^r denotes the radial unit vector on the metasurface plane.

In the following, we consider Eco,A(ρ,α,z), which is the first right-hand term in Eq. (14), as an example to analyze the wavevector k=kr+kz. It is well understood that for a wavefront with phase function φ(x,y,z), the wavevector can be obtained from k=∇φ(x,y,z) [62]. Accordingly, we have kz=∂φ/∂z. Subsequently, from the phase function φ1(z)=kz+krc2(1/2z−1/2f)−C0z(1/d1+1/d2)/(f−z) implicated in the term for Eco,A(ρ,α,z) in Eq. (14), kz,co−A=∂φ1(z)/∂z can be obtained, and followingly, kr,co−A can be obtained based on k2=kz,co-A2+kr,co-A2. Using rc and C0 in Eq. (15), and simply calculating the derivatives of the phase functions of Eco,A(ρ,α,z) and Eco,B(ρ,α,z), we obtain kz,co-A/B=kcosτco-A/B≈1−λ2f22d12(f−z)2(1±2d1d2),(20)kr,co-A/B=ksinτco-A/B≈kλff−z(1d1±1d2),(21)where τco-A/B is the deflection angle of the corresponding co-polarized component, and the term of 1/d22 is neglected in Eq. (20). From Eq. (21), τco-A/B can be approximated as follows: τco-A/B≈sinτco-A/B≈λff−z(1d1±1d2).(22)

In the above equation, λ/d1=sinτ1≈τ1 and ±λ/d2=±sinτ2≈±τ2 happen to be the deflection angles caused by the primary and secondary axicon-phase profiles φa,A/B(r)=−2πr/d1 and ±δφpa(r)=∓2πr/d2, respectively. Moreover, under the condition r≪f, the hyperbolic-phase profile is approximated as the quadratic function, i.e., φlens(r)≈−kr2/2f. Obviously, this profile is different from the linear function of the axicon-phase profiles. Because the profile φlens(r) converges the transmitted beams at the focal point f, it is deduced that it provides a deflection angle τf≈r/f for the light line passing through a meta-atom at a point of radius r. As shown schematically in Fig. 2(b), the deflection angle in Eq. (22) is also obtained as τco-A/B≈r/z, which indicates that the light lines transmitting from points with radius coordinate r intersect the optical axis at point z after the deflection, and the light field at point z is mainly contributed by the light waves near points with radius r. The correspondence between r and z correlated with the deflection angle τco-A/B is noteworthy. Thus, with the relations of τf≈r/f and τco-A/B≈r/z, Eq. (22) can be written as τco-A/B≈τf+τ1±τ2=rf+λd1±λd2.(23)

The above equation indicates that, for a single sub-metasurface, the deflection angle τ=τco-A/B obtained from kz,co-A/B for the wavefront with uσ=−1 in Eq. (14) causes the light wave from M(r,ϕ,0) to propagate along sO′ and intersect with the optical axis at longitudinal distance z. According to Eq. (23), τco-A/B is essentially the sum of the deflection angles τ1, ±τ2, and τf≈r/f of the two axicon and hyperbolic phases. Figure 2(b) shows a schematic of the deflected light lines transmitted through meta-atoms A and B. As shown in Fig. 2(b), by introducing the primary axicon phase, the transmitted beams are no longer focused at the focal point but are dispersedly converged within an appropriate range. In addition, the dispersed convergence of the beam enhances the light intensity in this range, meeting the aim of the metasurface design. Importantly, although Eq. (14) seemingly reaches a result identical to that from the geometric optics picture, as an approximate theoretical solution, it provides a far more generalized depiction for the tunable HOP beams produced at arbitrarily variable distances z by QWP metasurfaces, with both the focusing and deflecting effects employed. To the best of our knowledge, this is the first theoretical solution for defocused VB generation using metasurfaces [20].

Similarly, for the cross-polarized components Ecr,A(ρ,α,z) and Ecr,B(ρ,α,z), the deflection angle τcr-A/B is obtained as τcr-A/B=τco-A/B±τg,A/B=r/f+λ/d1±λ/d2; compared with kr,co-A/B and kz,co-A/B given in Eqs. (20) and (21), the components kz,cr-A/B and kr,cr-A/B of the wavevector can be obtained by replacing the sign ± therein with ∓.

In addition, when f=500μm and z≤96μm, the condition f≫z is approximately satisfied and the approximation of f/(f−z) becomes 1+z/f. Thus, from Eq. (20), we obtain $$\begin{gathered} kz,co-A/B≈k−kλ22d12[1+(2z/f)±(2d1/d2)] \\ =k−(kfd1±Δkz,d2), \end{gathered}$$(24)where kfd1 and Δkz,d2 are equal to kλ22d12[1+(2z/f)] and kλ2d1d2, respectively. The difference Δkz,co for Eco,A(ρ,α,z) and Eco,B(ρ,α,z) can be written as Δkz,co=kz,co-A−kz,co-B=−2Δkz,d2. Similarly, the difference Δkz,cr for Ecr,A(ρ,α,z) and Ecr,B(ρ,α,z) is Δkz,cr=kz,cr-A−kz,cr-B=2Δkz,d2, with the introduction of geometric phase profiles. Notably, unlike Δkz,co, Δkz,cr is also dependent on the secondary axicon-phase ±δφga(r). Summarizing these results, we obtain Δkz,co/cr=kz,co/cr-A−kz,co/cr-B≈∓k2λ2d1d2.(25)

Considering the propagation phase constants with φ0,A=π/2 and φ0,B=0, as depicted previously, the phase factors of Eco(ρ,α,z) and Ecr(ρ,α,z) are rendered to contain ei(k−kfd1)zcos(π/4+Δkz,coz/2) and ei(k−kfd1)zsin(π/4+Δkz,coz/2), respectively. Therefore, the amplitude weights of the co- and cross-polarized vortices can be modulated by adjusting the propagation distance z, and the HOP beams on the meridian can be generated.

The results are shown in Fig. 3 to intuitively demonstrate the contributions of different parameters to the generation of VBs using metasurfaces. Figure 3(a) shows the vortex beam intensity pattern of the co-polarized component in the x−z plane with a hyperbolic-phase profile of f=90μm, but without an axicon-phase profile. Under such conditions, the focused position of the VBs is approximately close to that generated by practically designed metasurfaces with f=500μm, d1=2μm, and d2=70μm. It can be seen that the focal depth of the pattern in Fig. 3(a) is very small, and the full modulation of HOP beam amplitudes from the north to south pole cannot be realized within the distance range corresponding to the focal depth. In Fig. 3(b), the left and right panels show the VBs’ RCP and LCP component intensity patterns of Ia,R and Ia,L in the x−z plane with the two axicon-phase profiles of d1=2μm and d2=70μm, but without the hyperbolic-phase profile, i.e., f→∞. From the patterns, it is observed that a longitudinal distance range as large as approximately 45 μm is required to complete the full modulation of the amplitudes of the LCP and RCP components from zero to one (or one to zero) to generate all the HOP beams on a meridian, owing to the unfocused beam setting. It should be noted that the variation in Ia,R with z arises from the interference of the co-polarized components of SA and SB, whereas the variation in Ia,L with z results from the interference of the cross-polarized components of SA and SB. In Fig. 3(c), the left and right panels show the intensity patterns IR and IL of the RCP and LCP components in the x−z plane, where the hyperbolic- and two-axicon-phase profiles are set with f=500μm, d1=2μm, and d2=70μm, respectively, which are the parameters for designing the practical metasurface samples. It is observed that within the range of the propagation distance z from 82 to 96 μm, the amplitudes of the two vortices vary continuously from one to zero (or from zero to one) with z, completing the generation of the HOP beams on a full meridian. Here, for direct comparisons in Figs. 3(b) and 3(c), a unified color bar is used for the light intensity patterns. In Fig. 3(d), the curves of IR and IL versus x at z=82μm, 91 μm, and 93 μm are given, respectively. These results demonstrate that the proposed metasurface can generate HOP beams on a full meridian within a significantly diminished longitudinal distance range of approximately 14 μm. However, the intensity of the generated HOP beams can also be enhanced by the convergence of the hyperbolic phase.

To design the metasurfaces, we simulated the light fields of the meta-atoms and metasurface samples using the FDTD method. The meta-atom was a rectangular nanopillar of a-Si:H with a height H of 480 nm and lattice period P of 380 nm, and the simulation of the transmitted light field was performed using parametric sweeps over side lengths varying from 80 to 330 nm in steps of 1 nm. The illuminating light with a wavelength of 800 nm was linearly polarized in the x- and y-directions along the side length. The refractive index and extinction coefficient are n=3.744 and κ=0, respectively. The results for the transmitted amplitudes Txx and Tyy, and the phase shifts φxx and φyy versus the side lengths were obtained. Eight QWP meta-atoms were selected with the conditions of the phase retardation φxx−φyy=π/2 and their propagation phase φxx covering the range of 0 to 2π with equal increments. The results are shown in Fig. 2(c), where the blue and black dashed lines represent linear fits of φxx and φyy, respectively. The curve of amplitude Txx for the meta-atoms is also shown. Table 1 lists the dimensions of the eight meta-atoms.Table 1.

Dimensions of Eight Meta-atoms^a

For the designed metasurface samples, the light fields were simulated on the observation plane at different propagation distances, z. In the FDTD simulations, the light fields were first calculated in the near-field plane at a distance of 1 μm behind the metasurface and subsequently projected onto the observation plane in the far-field. For sample S1, the light fields generated were first-order HOP beams with polarization states corresponding to the points on the prime meridian in red for Φ=0 from the north to the south pole on the HOP sphere, as shown in Fig. 4(a). The simulations were performed, and the beam intensity patterns are shown in the first image row in Fig. 4(b), where the latitude-point number for the beam is given in the first title row, and the corresponding longitudinal distance z and latitude Θ are given in the second and third title rows, respectively. The first and second title columns in Fig. 4(b) indicate the metasurface samples and transmitted directions of the analyzing polarizer, respectively. From the x-polarized component patterns for sample S1, we can observe the variation characteristics of the VBs with the propagation distance z. Initially, the pattern exhibits a doughnut-shaped intensity distribution. Then, with an increase in z, the pattern changes into two horizontal lobes, with the in-between dividing dark line gradually becoming clearer, and the two lobes reach the maximum contrast in the pattern at a distance z corresponding to the equatorial point. Subsequently, the lobes gradually blurred again and the pattern finally restored the shape of the doughnut. During this process, the latitude representing the polarization states of the beam varied from the north to the south pole. The characteristics of the pattern change consistently demonstrated the realization of tunable HOP beams on a meridian along the propagation distance z.

The aforementioned evolution of the x-component patterns indicates the varying weights of the two CP vortices in the HOP beam. To explicitly show the weight variation versus the longitudinal distance, the first and second rows of Fig. 4(d) show the intensity patterns of the RCP and LCP components, that is, IRCP and ILCP, in the beams generated by S1 at different propagation distances. The patterns are shown as unified color bars. On the plane at the distance z=82.0μm, the component intensity IRCP is the maximum while ILCP is close to zero, corresponding to the VB at the north pole of the HOP sphere with Θ=0, and this distance is considered as the starting position for the HOP beam to be generated. With an increase in z, IRCP decreases, and ILCP increases gradually. At z=91.4μm, the IRCP and ILCP are approximately equal, such that the polarization state of the VB corresponds to the intersection point of the prime meridian and the equator, that is, Θ=0.50π, with the largest contrast for the two lobes in the x-component pattern. At z=95.5μm, the ILCP was at its maximum, whereas the IRCP was approximately zero, corresponding to the HOP beam at the south pole with Θ=π. Figure 4(c) shows the curves of IRCP and ILCP versus Θ, where the intensity value of a curve point was averaged over the circular area encircled by the first-order zero-point of the beam doughnut, and normalized with respect to the average value of ILCP set as 0.5 at Θ=0.50π. The solid curves in Fig. 4(c) correspond to the theoretical results and are consistent with the simulated curves.

In Fig. 4(e), the left and right panels show the simulation results of the IRCP and ILCP in the x−z plane, respectively, with z ranging from 80 to 100 μm. These simulation results further demonstrated the generation of HOP beams at different latitudes on the prime meridian.

By setting the initial orientation angle θ0 of the meta-atoms, the other three samples, S2, S3, and S4, were designed to generate HOP beams corresponding to other meridians with longitudes Φ=π/2, Φ=π, and Φ=3π/2, respectively. The radii of S2, S3, and S4 and the other parameters were the same as those of S1. The x-polarized component patterns of these samples are shown in the second to fourth rows of Fig. 4(b), and the points on the HOP sphere representing the polarization states for the corresponding samples are drawn on the meridians in green, pink, and purple, respectively, as shown in Fig. 4(a). The fifth row in Fig. 4(b) shows the total intensity images of S3. The VBs on the equator generated by the four samples are the familiar radial, 45º-slanted, azimuthal, and 135º-slanted polarizations, respectively. Because they are the fundamental VB set of linearly polarized states corresponding to the equatorial points on the four meridians, their x-component and total intensity images are shown in Fig. 4(f) specifically, where their polarization state distributions are overlaid on the IA patterns.

Figure 5(a) shows the experimental optical path for generating and observing the longitudinal modulation of HOP beams on a meridian using a metasurface sample. A linearly polarized light beam with a wavelength of 800 nm emitted from the laser was attenuated by the neutral attenuator A to an appropriate value to avoid saturation of the measured intensity patterns and damage to the detector, and the light beam was linearly polarized in the vertical direction after being transmitted through polarizer P1. The beam was converted into CP light using a QWP. Subsequently, CP light illuminated the metasurface sample from the substrate direction. The sample was mounted on a 3D nanometer stage (PI E516) to adjust the observation distance. A microscope objective (MO) was placed behind the sample to image the VB patterns on the observation plane at distance z from the metasurface. The magnified VB patterns were recorded using an s-CMOS detector on the MO image plane. By precisely moving the longitudinal position of the sample with the 3D nanometer stage, the VB patterns on the planes at different z values, corresponding to the polarization states for different points on the meridian of the HOP sphere, were recorded using s-CMOS. An analyzing polarizer, P2, was placed in front of the s-CMOS. The intensity images of the different components were recorded by rotating its transmitted direction, and the total intensity patterns were obtained by removing P2.

To fabricate the sample, a 480-nm-thick a-Si:H film was deposited on fused silica using plasma-enhanced chemical vapor deposition (Plasmalab 100, Oxford), and a layer of electron beam resist (ZEP520A, Zeon Chemicals) was spin-coated onto the film. A thin layer of e-spacer 300Z (Showa Denko) was coated to prevent charging during subsequent electron-beam exposure. The designed patterns of the metasurface samples were imprinted on the resist using electron-beam lithography (Raith150), followed by development in ZED-N50. Subsequently, an Al film layer was deposited onto the substrate via electron-beam evaporation (Temescal BJD-2000), and the sample was soaked in a solvent (ZDMAC from Zeon Co.) to remove the resist. The remaining Al pattern was used as the etch mask. Therefore, the designed patterns were transferred to an a-Si:H film using an inductively coupled plasma (Oxford Plasmalab System 100). Finally, wet etching was performed to remove the remaining Al-etching masks.

Four metasurface samples, S1, S2, S3, and S4, were fabricated. Figures 5(b) and 5(c) show the top and side views of the scanning electron microscopy (SEM) images of S1, respectively. Notably, to avoid an adverse influence on the subsequent optical measurement of the VBs produced by the sample, the usual thin gold layer used for capturing the SEM image was not deposited on its surface. Thus, the contrast of the SEM images appears imperfect owing to the effect of the charge accumulation.

Four metasurface samples (S1−S4) were used to generate HOP beams on the four corresponding meridians. Figure 6(a) shows the total and component intensity images of the VBs on the equator produced by S1−S4, which are VBs with radial, 45° slanted, azimuthal, and 135° slanted polarizations, respectively. The left title column indicates the transmitted direction for the analyzing polarizer, P2. The orientations of the two lobes in the component intensity image for each sample were rotated in the transmitted direction of P2. For clear observation, a lobe in each component image is marked with a hollow arrow. In Fig. 6(b), the first to fourth image rows show the x-component intensity images of VBs corresponding to different latitudes on the four meridians with Φ values of 0, π/2, π, and 3π/2 for the HOP sphere generated by S1 to S4, respectively. The fifth row shows the total intensity IA images of the HOP beams generated by S3. These experimental results are consistent with the FDTD simulation results shown in Fig. 4. Thus, we experimentally demonstrated the feasibility of generating HOP beams on the corresponding meridians through propagation distance tunability using QWP metasurfaces. Unlike previous methods that use a combination of HWP and QWP to adjust the ellipticity of the incident light, our method utilizes unchanged incident CP light, which provides a simpler method for generating and detecting VBs and miniaturizing the optical system for optimized integration.