Optical spin-dependent beam separation in cyclic group symmetric metasurface

Novelty and impact statement

Vortex beam generation by geometric phase is known. However, a systematic study of relating symmetry property of cyclic group Cnh and vortex beam profile is never reported. This work shows that non-constant azimuthal gradient of total phase is the key in controlling azimuthal interference pattern of vortex beam with asymmetric helical wavefront. Noting the importance of vortex beam communications, this work opens a novel way to manipulate interplay between spin- and orbital angular momentum for vortex beam profiling.

1 Introduction

Since the pioneering work of Hasman group on Pancharatnam-Berry (PB) phase to manipulate wavefront of optical beam, [1], [2], [3], [4], [5] application of geometric phase has been expanded to metasufrace to open a research field of flat optics. [6], [7], [8], [9], [10] PB phase is a geometric phase, associated not with optical path but with polarization, which can be utilized for beam deflection or vortex beam generation, when a cross-polarization scattering of circular polarized light (CPL) takes place in an array of sub-wavelength nano-rods. [1], [2], [3]

Regrading a helical wavefront formation, a circular array of sub-wavelength scatterers introduces a geometric phase linearly increasing along azimuthal direction, allowing spin-to-orbital angular momentum conversion to generate a vortex beam possessing topological charge. Examples include metal nano-rods patterned from computer-generated holograms, [3] liquid crystal q-plate, [11], [12] uniaxial crystal, [13] nano-slits, [14] spatial light modulator, [15] plasmonic metasurface, [16], [17] V-shaped antenna array, [18], bulls-eye plasmon antennas, [19] TiO2 nanostructures, [20] dielectric metasurface, [21], and ultra-thin metalenses [22].

Metallic nano-rod is of a rectangular shape having two principal axes of plasmonic excitations along the width and the length. When nano-rods of the same rectangular shape are arrayed head-to-tail in a circle, the geometric phase ΦPB(ϕ) introduced by two neighboring nano-rods increases linearly along azimuthal direction, leading to a helical wavefront formation to generate a vortex beam of topological charge ±2. By arranging the rotation angle of each nano-rod of circular array in a specific way, vortex beams of arbitrary topological charges can be generated [17].

Let’s consider a series of metallic nano-rods possessing varying width with a fixed length. Now, in contrast to the same plasmonic excitation along long-axis of the fixed length, plasmonic excitation along short-axis of varying width is not the same among nano-rods. When the series of metallic nano-rods are arrayed head-to-tail in a circle, the dynamical phase introduced by two neighboring nano-rods of different widths does not increase linearly along azimuthal direction. That is, the azimuthal gradient of dynamical phase ∇ϕΦD(ϕ)which is not a constant of ±2 but depends on the azimuthal angle ϕ. We decompose the total phase Φtot(ϕ) as a sum of constant and non-constant azimuthal gradient terms, Φtot=ΦPB+ΦD with ∇ϕΦPB=±2.

There are several ways to relate the feature of a spin-dependent beam separation in vortex beam generation with the presence of a non-constant azimuthal gradient of total phase ∇ϕΦtot, which is the main topic of this work. First, a partial-wave expansion of cross-polarization scattering amplitude in terms of vortex beams of different topological charge can be adopted to examine the interference between vortex beams. Second, the role of non-constant azimuthal gradient ∇ϕΦtot can be identified in providing an azimuthal shift of beam to result in a spin-dependent beam separation.

For a systematic study, we introduce nano-structure of tapered arc (TA) having varying width with a fixed length as a unit. Different sizes of TA are arrayed in a circle to obtain metasurface belonging to a cyclic group of Cnh, which is named as tapered arc cyclic group symmetric metasurface (TA-CGSM).

Note that in the cyclic group theory, Cnh denotes groups containing a horizontal (h) reflection plane, σh, in addition to the rotation axis, Cn. That is, Cnh refers to a cyclic group with n-fold rotational symmetry by an angle of 360°/n and horizontal (h) reflection symmetry in three dimensions. Experimental measurement of vortex beam profiles from cross-polarization scattering of CPL beam from Cnh TA-CGSM is carried out and a detailed analysis is presented to relate cyclic group symmetry property of metasurface and the generated vortex beam profile.

First of all, PB phase is examined in Poincaré sphere when cross-polarization scattering takes place from linear and circular arrays of nano-rods with uniform thickness in Section 2. We extend the study of PB phase to the example of circular array of nano-rods with non-uniform thickness in Section 3. Sample fabrication of Cnh TA-CGSM and spin-dependent beam profile measurement are presented in Section 4 including elucidation of symmetry properties of Cnh. In Section 5 we discuss spin-dependent beam separation in terms of partial-wave expansion, azimuthal interference, and a non-constant azimuthal gradient of total phase ∇ϕΦtot. Also wavelength dispersion of spin-dependent beam separation of vortex beam is discussed in terms of photonic spin Hall effect.

2 PB phase from linear and circular arrays of nano-rods with uniform thickness

We compare linear and circular arrays of eight sub-wavelength nano-rods in providing PB phase through cross-polarization scattering of CPL Gaussian beam. Eight nano-rods rotated in the xy-plane by an angle φ with respect to the neighboring one are schematically shown in Figure 1 (a) and (b) and inset. In Figure 1 (c) PB phase is illustrated in Poincaré sphere, where colored circles correspond to each nano-rod rotated by an angle φ providing a geometric PB phase of 2φ with phase factor of e+i2ϕ(e−i2ϕ) for LCP σ+(RCP σ−) incidence beam [2], [3].

Figure 1:

Array of nano-rods with a constant gradient PB phase and Poincaré sphere plot of Stokes parameters. Inset figure shows a schematic of nano-rod with length D and width d making an angle ϕ with x-axis. (a) Linear array of rotating nano-rods from one to eight. (b) Circular array of nano-rods from one to eight in the range of 0≤φ≤π and repeated in the range of π≤φ≤2π. (c) Poincaré sphere plot of Stokes parameters of optical beams scattered from nano-rods of (a) and (b) for LCP incident beam. Color code corresponds to the nano-rod number. (d,e) Spin-dependent beam deflection of cross-polarization scattered optical beam shown up as a spin-dependent beam separation. +x-direction deflection of RCP σ− beam for LCP σ+ incidence beam in (d) and wereas the opposite in (e). (f, g) Generation of vortex beam of topological charge l=±2. Topological charge l=+2 with polarization state of RCP σ− for LCP σ+ Gaussian incidence beam in (f), and topological charge l=−2 with polarization state of LCP σ+ for RCP σ− Gaussian incidence beam in (g).

In the linear array, a constant PB phase gradient leads to a beam deflection in cross-polarized scattering. The array of nano-rods with uniform thickness provide a constant linear gradient with opposite signs for the left and the right polarized light, deflecting the corresponding beams in opposite directions as Figure 1 (d) and (e).

In the circular array, on the other hand, a constant PB phase gradient takes place in azimuthal direction ϕ providing a spiral phase shift. [23] Importantly, as shown in Figure 1 (c), the polarization states in Poincaré sphere go through a circular trajectory twice providing the solid angle of ±8π, which corresponds to geometric phase of ±4π, generating vortex beam of topological charge l=±2, as shown in Figure 1 (f) and (g).

Optical wave scattered from the circular array acquires the geometric phase factor coming from a circular closed path C in Poincaré sphere Stokes parameter space, ∮dγ(C)=±4π with a constant azimuthal gradient ∇ϕΦPB=±2. The intensity profiles of I−+(r, ϕ)=|E−+|2 and I+−(r, ϕ)=|E+−|2 are degenerate even though helical handedness is opposite for l=±2, where {+−}={out,in} stands for left-circular polarization (LCP, σ+) scattering/right-circular polarization (RCP, σ−) incidence and {−+} vice-versa. In fact, E−+(r, ϕ)=J2(k0r)exp(−k02r2+i2ϕ) and E+−(r, ϕ)=J−2(k0r)exp(−k02r2−i2ϕ) with nth order Bessel function Jn(k0r).

Owing to the rotational invariance of circular array of nano-rods belonging to the cyclic group C∞h, the azimuthal gradient ∇ϕΦPB is constant, and PB phase is accumulated in sequence by an equal amount originating from two neighboring nano-rods during one full cycle, leading to a ϕ-independent scattered electric field J±2(r)exp(−k02r2).

3 Total phase from circular arrays of nano-rods with non-uniform thickness

Cross-polarization scattering amplitude is determined by optical response of plasmonic resonance of a nano-rod, which depends on oscillator strength and resonance frequency of the nano-rod. Let’s consider a nano-rod of length D and width d. See the inset of Figure 1 (a). Upon incidence of an optical beam, there occur plasmonic excitations along two principal axes, i. e., along both long- and short-axis.

When the long-axis is rotated by an angle of φ with respect to x-axis, the short-axis makes an angle of θ=φ+π2. This leads to phase factors of e+i2φ and e+i2θ=−e+i2φ for cross-polarization scatterings from the long- and short-axis excitations, respectively. Note that phases of 2φ are the same, while the scattering amplitudes are opposite in sign. Refer to (5) in Ref. [23]. In other words, there occurs a destructive interference between scatterings from the long- and short-axis excitations, introducing the same amount of geometric phase of 2φ.

Now we introduce a circular array of nano-rods with varying width d from # 1 to # 16 having a fixed length D, which belongs to the cyclic group C1h, as displayed in Figure 2 (a). Scattering amplitude from the long-axis is the same for nano-rods from # 1 to # 16. As the width d increases from # 1 to # 16, however, scattering amplitude from the short-axis excitation increases owing to larger oscillator strength and the plasmonic resonance closer to the optical wave frequency. The amount of a destructive interference increases between long- and short-axis scatterings, leading to asymmetric total cross-polarization scattering amplitude as the width d increases from # 1 to # 16 along azimuthal angle ϕ direction.

Figure 2:

Circular array of nano-rods with a non-constant azimuthal gradient total phase and Poincaré sphere plot of Stokes parameters. (a) Circular array of nano-rods with varying width d from # 1 to # 16 in the range of 0≤φ≤2π (b) Poincaré sphere plot of Stokes parameters of optical beams cross-polarization scattered from nano-rods of (a) for LCP σ+ incidence beam. Color code corresponds to the nano-rod number. (c,d) Generation of asymmetric vortex beam and spin-dependent beam separation. Bight and dark spots are separated along x-direction for both LCP σ+ in (c) and RCP (σ−) in (d) Gaussian incidence beams.

Polarization states of cross-polarization scattering are plotted on Poincaré sphere in Figure 2 (b). Solid angle subtended by two meridians of cross-polarization scatterings from two neighboring nano-rods with different widths d is not a constant but depends on azimuthal angular location ϕ of nano-rod, resulting in a spiral trajectory of polarization states from near the north-pole down to near the equator. This leads to Φtot possessing a non-constant azimuthal gradient. That is, ∇ϕΦtot(ϕ) depends on the azimuthal angle ϕ, and we can decompose the total phase Φtot(ϕ) as a sum of constant and non-constant azimuthal gradient terms,

with ΦPB(ϕ) possessing a constant azimuthal gradient ∇ϕΦPB(ϕ)=±2. Here we note that the presence of a non-constant azimuthal gradient term ΦD(ϕ) originates from ϕ-dependent width of TA plasmonic nanostructures, as discussed above.

Azimuthal angular feature of vortex beam profiles I−+(r, ϕ) and I+−(r, ϕ) is determined by ϕ-dependence of Φtot(ϕ). Circular array in Figure 2 (a) belongs to the cyclic group C1h. From the viewpoint of symmetry property, the lowering of rotational symmetry from C∞h to C1h leads to non-vanishing of ΦD. As a result, asymmetric helical wavefront is formed and spin-dependent separation takes place with asymmetric vortex beam profile. Bight and dark spots are separated along x-direction for both LCP σ+ and RCP (σ−) Gaussian incidence beams as displayed in Figure 2 (c) and (d). The reason why the separation takes place along x-direction will be discussed in terms of the magnitude of azimuthal gradient ∇ϕΦtot(ϕ) in Section 5.

4 Total phase from circular array of nano-rods with rotational symmetry of cyclic group Cnh

Now we introduce metasurface composed of circular array of TAs, which belong to a well-defined cyclic group Cnh to study the relationship between cyclic group symmetry property of metasurface and the generated vortex beam profile. Schematics of six different tapered arc cyclic group symmetric metasurfaces (TA-CGSM) are shown in Figure 3 (a).

Figure 3:

Cyclic group symmetric metasurfaces and a spin- dependent beam separation. (a) Designed TA-CGSM belonging to cyclic groups of C∞h, C1h, C2h, C3h, C4h, C5h, and C6h, respectively. Periodicity is denoted by p=600 nm, and widths of arcs are by w1=45 nm, w2=60 nm, w3=75 nm, w4=90 nm, w5=105 nm, w6=120 nm, w7=135 nm, and w8=150 nm. (b) Optical microscope images of fabricated TA-CGSMs belonging to belonging to cyclic groups of C∞h, C1h, C2h, C3h, C4h, C5h, and C6h, respectively. (c) SEM images of fabricated TA-CGSMs belonging to belonging to cyclic groups of C∞h and C1h with scale bars of 1 μm. (d) Experimental setup for a spin-dependent beam separation measurement. (e,f) Far-field intensity distributions scattered from TA-CGSM belonging to cyclic groups of C∞h, C1h, C2h, C3h, C4h, C5h, and C6h for I−+ and I+−, respectively, measured with λ = 1310 nm incidence Gaussian beam. (g) Plot of the intensity difference ΔI=I−+−I+−.

5 Discussion

5.1 Symmetry property of vortex beam profiles under σh and C2

Far-field electric fields of asymmetric vortex beams shown in Figure 3 (e) and (f) can be described as below.

E(r, ϕ)−+=A(r, ϕ)exp{iΦtot−+(ϕ)}

(2)E(r, ϕ)+−=B(r, ϕ)exp{iΦtot+−(ϕ)}

where the azimuthal gradient of Φtot(ϕ), ∇ϕΦtot(ϕ), is ϕ-dependent.

Regarding Φtot−+(ϕ) and Φtot+−(ϕ), we note that the spiral trajectory for {+−} is the mirror image of that for {−+} with the equator plane of Poincaré sphere as a mirror plane, i. e., the senses of rotation are opposite to each other. This leads to the relation of total phases Φtot−+(ϕ)=−Φtot+−(ϕ), the same in magnitude but opposite in sign. See Figure 4 (d), (e), (D), and (E). Furthermore, since Cnh TA-CGMS is invariant under σh operation, scattering amplitude of {−+} from Cnh TA-CGM is the same as that of {+−} from π-rotated TA-CGM, i. e., E(r, ϕ)−+=E(r, ϕ+π)+−. Accordingly, we obtain the relation of I−+(ϕ)=I+−(ϕ+π) between scattering intensities of {−+} and {+−}. See Figure 3 (e) and (f) and Figure 4 (a), (b), (A), and (B). This is true for Cnh (both even and odd n) TA-CGSMs.

Figure 4:

Analytical calculation and FDTD calculation of a spin-dependent beam separation for TA-CGSM. (a, b) Analytically calculated far-field intensity of I−+ and I+−, respectively. (c) Calculated ΔI=I−+−I+−. (d, e) Azimuthal distribution of phase for C∞h, C1h, C2h, C3h, C4h, C5h, and C6h TA-CGSM. (f-i) Partial-wave expansion of E−+ and E+− of C4h CGSM. (A, B) FDTD calculated far-field intensity of I−+ and I+−, respectively. (C) Calculated ΔI=I−+−I+−. (D, E) Azimuthal distribution of phase for C∞h, C1h, C2h, C3h, C4h, C5h, and C6h TA-CGSM. (j-m) Partial-wave expansion of E−+ and E+− of C5h CGSM.

On the other hand, Cnh (even n) TA-CGSMs is invariant under a simultaneous operations of σh and i, which is equivalent to the invariance of Cnh (even n) TA-CGSMs under C2 operation. That is, scattering amplitude from Cnh (even n) TA-CGSM is the same as that from π-rotated Cnh (even n) TA-CGSM. In other words, I+−(ϕ+π)=I+−(ϕ) and I−+(ϕ+π)=I−+(ϕ). Combining with the above relation I−+(ϕ)=I+−(ϕ+π), we obtain I−+(ϕ)=I+−(ϕ), which leads to that ΔI=I−+−I+− is null for Cnh (even n) TA-CGSMs, but not for Cnh (odd n) TA-CGSMs. See the plot of ΔI=I−+−I+− in Figure 3, similar to what is reported in Ref [24].

5.2 Partial-wave expansion and azimuthal interference pattern

5.2.1 Partial-wave expansion

TA-CGSM renders azimuthal gradient ∇ϕΦtot of cross-polarization scattering beam to be ϕ-dependent, leading to ϕ-dependent helical wavefront, equivalently, to ϕ-dependent scattering amplitude in far-field as shown in experimental measurements of Figure 3 (e) and (f). This indicates that the scattering amplitude is composed of vortex beams of topological charge not only l=2 but also l≠2.

Bessel-Gaussian (BG) beam satisfies a paraxial equation of propagation and it describes an azimuthal symmetric intensity distribution carrying orbital angular momentum. [25] We note two properties of BG beams relevant to the spatial beam profiles observed in our experiment for a series of TA-CGSMs. The first property is that BG beams can be built up by a superposition of decentered Gaussian beams, whose centers are positioned on a circle and the beam direction of which points to the apex of a cone. [26] This is related to C∞h TA-CGSM, where the azimuthal symmetric annular beam intensity results from a superposition of decentered Gaussian beams scattered from each plasmonic segment of C∞h TA-CGSM. The second property is that an asymmetric Bessel-Gaussian beam can be represented as a linear combination of symmetric Bessel-Gaussian beams. [25] This is related to C1h, C2h, C3h, C4h, C5h, and C6h TA-CGSMs, where the spatial beam profile has an annular shape possessing azimuthal angle depenedent intensity distribution.

Since C1h, C2h, C3h, C4h, C5h, and C6h TA-CGSMs are of a ring structure with cyclic group symmetry lowered from that of C∞h TA-CGSM, we adopt BG beams as basis in a partial-wave expansion to examine how the symmetry lowering of azimuthal distribution of nano-rods modifes BG beam profile of the scattering light from azimuthal symmetric C∞hTA-CGSM. Vortex beam of topological charge l is described by the lth order Bessel-Gaussian beam BGl(r)eilϕ, [27]

BGl(r)eilϕ=Jl(k0r)e−k02r2eilϕ

where nth order Bessel function Jn(x) satisfies a symmetry relation J−n(x)=(−1)nJn(x). Now we express ϕ-dependent scattering amplitude as a linear combination of Bessel-Gaussian beams with partial-wave expansion coefficient a˜l[25], [28].

(3)E(r, ϕ)−+=A(r, ϕ)eiΦtot−+(ϕ)=∑la˜lBGl(r)eilϕ

(4)a˜l≡aleiψl

where a˜l is the lth order expansion coefficient of Fourier expansion, A(r, ϕ)=|E(r, ϕ)| is the amplitude, and Φtot=ΦPB+ΦD=tan−1[Im(E)/Re(E)] is the total phase.

When the second term ΦD(ϕ) of (1) is absent, the azimuthal gradient ∇ϕΦtot reduces to +2, generating a vortex beam of topological charge +2. That is, a˜2=1 and a˜l=0 for l≠2 in (3). The presence of ΦD(ϕ) of (1) is responsible for additional generation of vortex beams of topological charge l≠2, and magnitude al and phase ψl of a˜l (l≠2) in (4) describes how additional vortex beams contribute to determining ϕ-dependence of scattering amplitude.

Vortex beam profile, i. e., far-field intensity, I−+(r, ϕ)=|E(r, ϕ)−+|2, is nothing but the interference pattern between vortex beam of l=2 and additional vortex beams of l≠2, originating from non-constant azimuthal gradient ∇ϕΦtot. While magnitude al is a measure of contribution of vortex beam of l≠2, phase ψl plays an important role in determining azimuthal distribution of intensity I−+(r, ϕ), that is, the azimuthal location of bright and dark spots observed in vortex beams. An expression similar to (3) holds for E(r, ϕ)+− with Φtot−+(ϕ)=−Φtot+−(ϕ) in (2) as discussed in Section 5.1.

The partial-wave expansion analysis was adopted to relate group order n of cyclic symmetry in the interference pattern of the partial waves and spin-dependent beam splitting. By employing (3) and (4), i. e., I(r, ϕ)−+=|E(r, ϕ)−+|2=|∑la˜lBGl(r)eilϕ|2, the partial-wave expansion coefficient a˜l≡aleiψl is obtained by a least-squared-fit of the experimentally measured intensity profiles I−+ as a function of azimuthal angle in Figure 3 (e) & (f), and the fitted intensity and phase of scattered fields are plotted in Figure 4 (a) & (b) and (d) & (e), respectively. The fit values of magnitude al and phase ψl of a˜l in (4) are listed in Figure 5 (g).

Figure 5:

Analytically calculated intensity, phase, and phase gradient as a function of the azimuthal angle ϕ. (a, b) Calculated far-field intensity Iij. (c, d) Calculated phase Φij. (e,f) constant phase gradient dΦij/dφ of C∞h (black) and nonuniform phase gradient of C1h (red). The plots are for {ij} configuration, where {ij} are detection and incidence circular polarization helicity, {−+} for (a, c, e) and {+−} for (b, d, f). Non-constant phase gradient takes place owing to ϕ-dependent ∇ϕΦD(ϕ). (g) Fit values of partial-wave expansion coefficient a˜l, the lth order expansion coefficient.

5.2.2 Azimuthal interference pattern and correlation between group order n of Cnh and topological charge l

Now we study the interference behavior between l=+2(l=−2) and partial-wave vortex beams having difference topological charge l≠+2(l≠−2) for {−+}({+−}) scattering originating from ϕ-dependent∇ϕΦD(ϕ).

In order to identify the correlation between additionally generated vortex beam of topological charge l and group order n of Cnh, we consider {−+} scattering where vortex beam of l=2 and additional vortex beam of l≠2 interfere each other. From (3) we have

E(r, ϕ)−+=A(r, ϕ)eiΦPB−+(ϕ)=BG2(r)ei2ϕ+aleiψl BGl(r)eilϕ

(5)=ei2ϕ[BG2(r)+alBGl(r)ei(l+2)ϕ+lψl],(l≠2)

In (5) the amplitude A(r, ϕ) is expressed as below.

(6)A(r, ϕ)=[2alBG2(r)BGl(r)cos{(l+2)ϕ+lψl}+BG2(r)2+al2BGl(r)2]12

Azimuthal angle dependence is described by the first term of (6), which is the interference term. As azimuthal angle ϕ increases from 0 to 2π, a constructive interference takes place at azimuthal angle ϕm=m⋅ϕ0 (m = integer) with ϕ0 satisfying the relation (l+2)ϕ0=2π. That is, there are l+2 numbers of bright spots at the intensity I−+(r, ϕ) of vortex beam with asymmetric helical wavefront. On the other hand, the interference should possess n-fold rotation symmetry dictated by group order n of Cnh. Therefore we have the relation between l and n as n=l+2 or

(7)l=2−n

By looking up the table listed in Figure 5 (g) we find that the relation of (7) holds for all Cnh TA-CGSMs. Note that the non-vanishing vortex beams of l=−1 for C1h and l=−2 for C2h have the helicity opposite to that of generated vortex beams of l=+1 for C1h and l=+2 for C2h, and the interference term of two vortex beams of opposite helicities +l and −l is not ϕ-dependent as can be seen easily in (5) when a˜−l is added.

Partial wave expansions with fit values listed in Figure 5 (g) are explicitly expressed as below for C4h TA-CGSM,

E(r, ϕ)−+=BG2(r)e+i2ϕ+0.24BG−2(r)e−i2ϕexp(+i3π2)

(8)=BG2(r)e+i2ϕ−i0.24BG2(r)e−i2ϕ,

E(r, ϕ)+−=BG−2(r)e−i2ϕ+0.24BG2(r)e+i2ϕexp(−i3π2)

(9)=BG2(r)e−i2ϕ+i0.24BG2(r)e+i2ϕ,

and for C5h TA-CGSM,

E(r, ϕ)−+=BG2(r)e+i2ϕ+BG−3(r)e−i3ϕexp(+iπ)

=BG2(r)e+i2ϕ−BG3(r)e−i3ϕexp(+iπ)

(10)=BG2(r)e+i2ϕ+BG3(r)e−i3ϕ,

E(r, ϕ)+−=BG−2(r)e−i2ϕ+BG3(r)e+i3ϕexp(−iπ)

(11)=BG2(r)e−i2ϕ−BG3(r)e+i3ϕ.

From Eqs. (9)–(10) the intensity and phase of the sum of l=±2 vortex and partial-wave vortex beams are plotted for {−+}({+−}) scattering in C4h TA-CGSM and C5h TA-CGSM as displayed in Figure 4 (f) & (g) (Figure 4 (h) & (i)) and Figure 4 (j) & (k) (Figure 4 (l) & (m)), respectively. We find that ΔI=I−+−I+−=0 for C4h and ΔI≠0 for C5h TA-CGSMs.

We also adopted finite difference time domain (Lumerical FDTD) method to calculate the far-field intensity (I−+, I+−) and phase (Φ−+, Φ+−) distributions of scattered field from Cnh metasurfaces. See Figure 4 (A-E) for FDTD calculation, which is in a perfect agreement with experimental results.

5.4 Wavelength dependence of spin-dependent beam separation

As noted in Section 3, there occurs a destructive interference between scatterings from the long- and short-axis excitations, which is responsible for vortex beam profiles when cross-polarization scattered from for Cnh TA-CGSMs. In a given Cnh TA-CGSM, TA has a varying width with the same fixed length, and scattering amplitude from plasmonic excitation along short-axis of the varying width depends on plasmonic resonance. This leads to that the spin-dependent beam separation in vortex beam generation depends on wavelength of incident Gaussian beam.

We performed experimental measurement of spin-dependent beam separation of C1h TA-CGSM by employing lasers with different wavelengths. Single-mode fiber pigtailed laser diodes with wavelengths λ=1310 nm, λ=730 nm, and λ=660 nm are adopted as normal-incidence Gaussian beam sources. Azimuthal interference pattern goes through clock-wise rotation as wavelength gets shorter as shown in beam profile measurement and FDTD simulation at the top panel of Figure 6.

Figure 6:

Wavelength-dependence of a spin-dependent beam separation. Wavelength-dependence of a spin-dependent beam separation in C1h TA-CGSM metasurface is shown at 1300 nm, 730 nm, and 660 nm. (a), (c), (e), Experimental measurements and (b), (d), (f), FDTD calculations for C1h TA-CGSM. g,h,i, FDTD calculations for C1h′ metasurface with incident Gaussian beam propagation along +z^.

It is known that geometric phase is non-dispersive. However, scattering amplitude from short-axis excitation depends on the relative spectral location of incident beam wavelength with respect to plasmonics resonance. Subsequently, the location of polarization state at Poincaré sphere depends on the incident beam wavelength. As a result, non-constant azimuthal gradient ∇ϕΦD is dispersive.

In order to confirm that dispersive ∇ϕΦD is responsible for wavelength-dependent azimuthal interference pattern from C1h TA-CGSM, we examined C1′ metasurface composed of two semicircles with different thicknesses as shown at bottom-left panel of Figure 6, which possesses σyz symmetry, i. e., reflection symmetry with respect to yz-plane. FDTD simulation of cross-polarization scattering shows that spin-dependent vortex beam separation takes place always along x-direction for all three different wavelengths as shown at the bottom panel of Figure 6. Scattering from one semi-circle results in one full circle of polarization states in Poincaré sphere, and thereby introduces 2π of PB phase for each semi-circle as discussed in Section 2. However, the difference in thickness of two semi-circles results in ∇ϕΦD≠±2 at ϕ=0 & π, regardless of incident beam wavelength. As a result, a vortex beam with asymmetric wavefront is generated, which is composed of vortex beams of l=±2 and l=±1 as shown in Tables 1, 2, and 3.

Table 1:

Table of lth order expansion coefficient for λ=1300 nm.

λ = 1300 nm		E−+					E+−
l		−2	−1	0	1	2	−2	−1	0	1	2
C1	al	0	0.02	0.01	0.1	1	1	0.1	0.01	0.02	0
	ψl	0	0	π/2	π	0	0	−π	−π/2	0	0
C1′	al	0	0.02	0	0.1	1	1	0.1	0	0.02	0
	ψl	0	0	0	π	0	0	−π	0	0	0

Table 2:

Table of lth order expansion coefficient for λ=730 nm.

λ = 730 nm		E−+					E+−
l		−2	−1	0	1	2	−2	−1	0	1	2
C1	al	0	0.02	0.01	0.1	1	1	0.1	0.01	0.02	0
	ψl	0	−π/5	3π/10	4π/5	0	0	−4π/5	−3π/10	π/5	0
C1′	al	0	0.02	0	0.1	1	1	0.1	0	0.02	0
	ψl	0	π/5	0	π	0	0	−π	0	−π/5	0

Table 3:

Table of lth order expansion coefficient for λ=660 nm.

λ = 660 nm		E−+					E+−
l		−2	−1	0	1	2	−2	−1	0	1	2
C1	al	0	0.02	0.01	0.1	1	1	0.1	0.01	0.02	0
	ψl	0	−2π/5	π/10	3π/5	0	0	−3π/5	−π/10	2π/5	0
C1′	al	0	0.02	0	0.1	1	1	0.1	0	0.02	0
	ψl	0	π/5	0	11π/10	0	0	−11π/10	0	−π/5	0

A careful examination of the bottom panel of Figure 6 shows that there occurs a beam deflection in vertical direction along with spin-dependent vortex beam separation in horizontal direction. This is an example of beam deflection according to generalized Snell’s law.[29], [30] That is, the existence of ∇ϕΦD along y-direction is equivalent to the presence of phase discontinuity along y-direction, giving rise to δk→ in linear momentum k→ of scattered vortex beam along y-direction, and a vertical deflection of vortex beam takes place, which is wavelength-dependent. In C1′ metasurface, the minimum |∇ϕΦtot−+|(|∇ϕΦtot+−|) takes place at ϕ=π(ϕ=0≡2π). Owing to σyz symmetry, the net gradient of ϕ^-dependent ∇ϕΦtot is located at ϕ=π or ϕ=0≡2π with direction along ϕ^ corresponding to −y direction for both {−+} and {+−} scatterings, which leads to δ<p→> along −y^.

Consequently, photonic spin Hall effect takes place as a spin-dependent separation of vortex beam in horizontal direction. Berry curvature is the topological magnetic monopole, B→(p→)=p^/p2, and Lorentz equation of motion in momentum space is δx→=σδp→ℒB→(p). In helical wavefront, mean momentum <p→> is along the vortex beam propagation direction and takes part in Lorentz equation of motion, i. e., δ<x→>=σδ<p→>ℒB→(< p→ >). Hence, the transverse shift owing to photonic spin Hall effect takes place in −x^ and +x^ directions for incident LCP (σ=+1) and RCP (σ=−1) Gaussian beams, respectively, as displayed in the bottom panel of Figure 6.[31], [32] The interpretation of spin-dependent beam separation for Cnh in terms of photonic spin Hall effect is not straightforward due the difficulty in identifying the net gradient of ϕ^-dependent ∇ϕΦtot.

Normalized horizontal (X) and vertical (Y) spin-dependent beam separation profiles, ΔI(X) and ΔI(Y) along the dashed coordinate axes, are displayed in two right panels, where ΔI=I−+(r, ϕ)−I+−(r, ϕ).

6 Conclusion

Tapered arc cyclic group symmetric metasurface is introduced to explore the details of optical spin-dependent beam separation in spin-to-orbital angular momentum conversion. Presence of non-constant azimuthal gradient of total phase is found to be responsible for azimuthal interference pattern in vortex beams with asymmetric helical wavefront. By identifying the role of non-constant azimuthal gradient of total phase in giving rise to spin-dependent beam separation, spatial separation and vortex beam profiling are achieved in a controllable manner.

Azimuthal interference pattern in vortex beams with asymmetric helical wavefront found to be related to symmetry property of cyclic group Cnh, and group order n determines topological charge l≠±2 of additional vortex beams. Our work enhances a fundamental understating of interplay between spin- and orbital angular momentum of light in metasurface. Also our finding provides a significant advantage in applications such as vortex multiplexing in optical communication and optical tweezers employing vortex beams.