Radially polarized plasmonic vector vortex generated by a metasurface spiral in gold film

Yuqin Zhang; Ruirui Zhang; Xing Li; Li Ma; Chunxiang Liu; Changwei He; Chuanfu Cheng

doi:10.1364/OE.25.032150

1. Introduction

Amplitude, phase, and polarization are key quantities that enable light waves to be manipulated and are important topics in wavefield modification and functional optical element engineering applications. The generation of optical vortices, including both phase and vector vortices, is a typical example of such manipulation. Phase vortices exhibit helical wavefronts, and their photons carry certain orbital angular momenta with respect to the vortex cores. Vector vortices exhibit polarization singularities with doughnut-shaped spatial profiles, and at their singular cores, their intensities are minimized. The generation of common vector vortices in free-space optics has motivated numerous studies within the past few years, and several methods have been developed to produce vector vortex beams by using liquid crystal q-plates [1, 2], Pancharatnam–Berry phase elements [3, 4], interferometers [5–7], and spatial light modulators [8, 9]. A radially polarized vector beam is a special vector vortex beam that has its wavefield polarized in the direction of the singular core on the transverse plane, and such beams have been applied successfully in high-resolution imaging [10, 11] and tight focusing [12, 13]. In the surface plasmon polariton (SPP) field, the manipulation of wavefields is a more interesting topic because of the nanoscale size of plasmonic systems and the diversity of novel optical phenomena [14–17]. Various plasmonic structures on metal film such as nanoslit rings [18–20], Archimedes spirals [21–23], and chiral slits [24, 25] have been demonstrated to generate plasmonic phase vortices. This work has provided insight into the novel properties of plasmonic phase vortices and is responsible for enabling several exciting applications [26–28]. In most work on plasmonic phase vortex generation, only the wavefield component perpendicular to the metal surface, which is often expressed as the z-component of the wavefield, has been considered, although the characteristics of the wavefield polarized parallel to the metal surface are more interesting. The wavefield is a two-dimensional vector, and with proper manipulation, vector vortices can be generated readily.

Plasmonic metasurfaces are periodic arrays of structures artificially engineered and fabricated on metal films for phase, polarization, and SPP mode distribution manipulation. A few years ago, Lin et al. proposed an interesting metasurface of orthogonal-nanoslit-pair arrays [29] to realize the unidirectional propagation of SPPs on the surface. The superimposed SPPs were independent of linearly polarized light but dependent on the handedness of circularly polarized light [30], thereby overcoming the inconvenience of SPP manipulation using polarization dependence as in the common slit structure. Since a spiral provides destructive interference at the center [31], it would be easy to manipulate SPPs and generate vector wavefields with doughnut-shaped intensities.

In this study, we investigated, for the first time, the radially polarized vector plasmonic vortex produced by a metasurface spiral of orthogonal nanoslit pairs. Using the SPP excited by a slit element polarized perpendicular to the slit, we theoretically derived the expressions for the radial and vanished azimuthal components of the SPP generated by the orthogonal nanoslit pairs. We used the vector-form Huygens–Fresnel principle to calculate the SPP wavefield in the central area of the spiral. In the derivation, the radial increase of the spiral eliminates the phase variation with the azimuth induced by the change in orientation of the slit pair under circularly polarized illumination. With the obtained vector wavefield, the wavefield polarization was quantitatively analyzed using the Jones matrix of the metasurface spiral, and the formation of a radially polarized plasmonic vector vortex was demonstrated by both the polarization ellipses and the azimuth of the dark line of the wavefield components. The results were verified by performing full finite-difference time domain (FDTD) simulations and experimental measurements. Experimentally, we used a Mach–Zehnder interferometer system and scattering imaging to extract the SPP wavefields with the phase and polarization distributions [32]. Overall, the doughnut-shaped radially polarized vortex and wavefield distributions were reasonably consistent with the theoretical and simulated results.

2. Theoretical analysis of metasurface spiral and FDTD solutions

2.1 Geometries of the nanoslit element and wavefield vector

The metasurface spiral of the orthogonal-nanoslit-pair arrays is depicted in Fig. 1, where Figs. 1(a) and 1(b) show a scanning electron microscopy (SEM) image and diagram of the metasurface spiral, respectively. The inner column can be described using

R = R_{0} + \frac{λ_{s p p}}{2 π} θ .

where R₀ is the starting radius of the inner column and λ_SPP is the wavelength of the SPPs. The distance between the two columns is D, and that between two adjacent slits in the same column is S.

Fig. 1 (a) SEM image of the metasurface spiral etched using a focused ion beam (FIB). (b) Diagram of the metasurface for generating a vector vortex. (c) Illustration of the orientations and basic geometrical parameter of the slits. (d) Approximation of (c).

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Figure 1(c) is an illustration of a nanoslit pair composing the spiral. In polar coordinates, the radial and azimuth unit vectors are ${\hat{e}}_{r}$ and ${\hat{e}}_{θ}$ , respectively. We define the unit vector ${\hat{e}}^{(i)}$ of the slit in the inner column pointing toward the outer side of the spiral and the unit vector ${\hat{e}}^{(e)}$ of the slit in the outer column pointing toward the inner side of the spiral. Then, the normal unit vectors ${\hat{e}}_{n}^{(i)}$ and ${\hat{e}}_{n}^{(e)}$ of the two slits are defined as being perpendicular to the slits with a π/2 counterclockwise rotation from the slit unit vectors. Referring to the transform of the unit vectors between polar and Cartesian coordinates and considering the two slits at angles of 45° and 135° with ${\hat{e}}_{r}$ , the slit vectors can be written as

[\begin{array}{l} {\hat{e}}^{(i)} \\ {\hat{e}}^{(e)} \end{array}] = [\begin{matrix} \cos [θ + (π / 4)] & \sin [θ + (π / 4)] \\ - \sin [θ + (π / 4)] & \cos [θ + (π / 4)] \end{matrix}] [\begin{array}{l} \hat{i} \\ \hat{j} \end{array}] .

The normal unit vectors of the slits can also be expressed as

[\begin{array}{l} {\hat{e}}_{n}^{(i)} \\ {\hat{e}}_{n}^{(e)} \end{array}] = [\begin{matrix} - \sin [θ + (π / 4)] & \cos [θ + (π / 4)] \\ - \cos [θ + (π / 4)] & - \sin [θ + (π / 4)] \end{matrix}] [\begin{array}{l} \hat{i} \\ \hat{j} \end{array}] .

When the incident plane wave is circularly polarized, it can be written as $E_{i n} = (E_{i n 0} / \sqrt{2}) (\hat{i} \pm i \hat{j})$ , where ± represents the left and right circular polarizations and E_in₀ represents the amplitude. As the component of the incident wave perpendicular to the slit excites the SPP wave, the efficiency component for the inner slit is $E_{n}^{(i)} = [{\hat{e}}_{n}^{(i)} \cdot E_{i n}] {\hat{e}}_{n}^{(i)}$ and that for the outer slit is $E_{n}^{(e)} = [{\hat{e}}_{n}^{(e)} \cdot E_{i n}] {\hat{e}}_{n}^{(e)}$ . Thus, we have

E_{n}^{(i)} = - (E_{i n 0} / \sqrt{2})exp{\mp i [(π / 4) - θ]} {\hat{e}}_{n}^{(i)}

and

E_{n}^{(e)} = - (E_{i n 0} / \sqrt{2})exp{\pm i [(π / 4) + θ]} {\hat{e}}_{n}^{(e)} .

For convenience in describing the propagation of the SPP wave toward the center, we define the unit vector pointing toward the center as ${\hat{e}}_{n c} = - {\hat{e}}_{r}$ and its normal as ${\hat{e}}_{t c} = {\hat{e}}_{θ}$ . The component $E_{n c}^{(i)}$ of $E_{n}^{(i)}$ in the radial direction is

E_{n c}^{(i)} = - (1 / 2) E_{i n 0} exp{\mp i [(π / 4) - θ]} {\hat{e}}_{n c},

and the component

E_{n c}^{(e)}

of

E_{n}^{(e)}

for the outer slits is

E_{n c}^{(e)} = - (1 / 2) E_{i n 0} exp{\pm i [(π / 4) + θ]} {\hat{e}}_{n c} .

Considering the relative phase difference δ = k_SPP(R_e – R_i) = π/2 of the waves generated by the path difference D = λ_SPP/4 between the outer and inner slits, the radial component of the SPP polarized in the ${\hat{\overset{⌢}{e}}}_{n c}$ direction, which is the sum of the wave components produced by the two slits, can be written as

E_{n c} = E_{n c}^{(i)} + {\hat{E}}_{n c}^{(e)} e^{i π / 2} .

Substituting Eqs. (6) and (7) into Eq. (8),

E_{n c} = E_{n c}^{(i)} + {\hat{E}}_{n c}^{(e)} e^{i π / 2} = {\begin{cases} - E_{i n 0} exp{i [(π / 4) - θ]} {\hat{e}}_{n c}, for RCP \\ 0, for LCP \end{cases} .

The azimuthal component of the SPP waves produced by the pair of inner and outer slits can also be calculated. For the inner slit, the azimuthal component $E_{t c}^{(i)} = [{\hat{e}}_{θ} \cdot E_{n}^{(i)}] {\hat{e}}_{θ}$ of its SPP wave is

E_{t c}^{(i)} = - (1 / 2) E_{i n 0} exp{\pm i [θ - (π / 4)]} {\hat{e}}_{θ},

and

E_{t c}^{(e)} = [{\hat{e}}_{θ} \cdot E_{n}^{(e)}] {\hat{e}}_{θ}

for the outer slit is

E_{t c}^{(e)} = (1 / 2) E_{i n 0} exp{\mp i [θ + (π / 4)]} {\hat{e}}_{θ} .

Considering the phase delay due to the spacing s/2 = λ_SPP/4 for the outer slit, and superimposing $E_{t c}^{(i)}$ and $E_{t c}^{(e)}$ in the above two expressions, the total azimuthal SPP wave component can be expressed as

E_{t c} = E_{t c}^{(i)} + E_{t c}^{(e)} exp(i k_{spp} s / 2)=0,

for both RCP and LCP illumination. The vanished E_tc above and E_nc in Eq. (9) demonstrate that the superimposed SPP excitation under LCP illumination is zero, which indicates the fundamental importance of the circular polarization handedness matching the orientation of the slit pair. In this case, the phase factor ϕ(θ) = (π/4) – θ in Eq. (9) is directly dependent on the azimuth θ.

2.2 Huygens–Fresnel principle of SPP and the wavefield near the spiral center

The SPP wavefield near the spiral center can be calculated using the vector-form Huygens–Fresnel principle, which describes the propagation of SPP waves [33]. The wavefield at a point p(x_p, y_p) can be written as

E_{n c} (x_{p}, y_{p}) = - (i / \sqrt{λ_{s p}}) \int d l E_{n c} (x_{q}, y_{q})exp(i k_{s p} ρ)exp(i π / 4)/ \sqrt{ρ},

where E_nc(x_q, y_q) is the source wave at point q given in Eq. (9), ρ is the distance between p and q, and dl is the differential of the path as a line integral, as demonstrated in Fig. 1(d). The polar coordinates of points p and q are (R, θ) and (r, α), respectively, and the distance from point q to p can be written as

ρ = {[R^{2} + r^{2} -2 R r \cos (α - θ)]}^{1 / 2} .

In the area near the spiral center, the condition r << R is approximately satisfied and

ρ = R - r \cos (α - θ)= R - (x_{p} x_{q} + y_{p} y_{q}) / R .

Then, the components E_nc,x(x_p, y_p) and E_nc,y(x_p, y_p) can be written using the Fourier transform in O-xy coordinates:

[\begin{array}{l} E_{n c, x}^{} (x_{p}, y_{p}) \\ E_{n c, y}^{} (x_{p}, y_{p}) \end{array}] = - (i / \sqrt{λ_{s p} R}) F T {[\begin{array}{l} E_{n c}^{} (x_{q}, y_{q}) \cos θ \\ E_{n c}^{} (x_{q}, y_{q}) \sin θ \end{array}] exp(i π / 4)},

where FT represents the Fourier transform with the spatial frequencies f_x = x_p/(λ_spR) and f_y = y_p/(λ_spR). The transform integral is conducted along the path of the spiral. E_nc,x and E_nc,y given in the above transform can also be obtained in polar form by substituting Eq. (14) into Eq. (13):

[\begin{array}{l} E_{n c, x}^{} (r, α) \\ E_{n c, y}^{} (r, α) \end{array}] = - i / \sqrt{λ_{s p} R} \int d l [\begin{array}{l} E_{n c}^{} (R, θ) \cos θ \\ E_{n c}^{} (R, θ) \sin θ \end{array}] exp[- i k_{s p} r \cos (α - θ)]exp[i k_{s p} (R + π / 4)] .

In the above equation, dl = R₀dθ and, considering the metasurface spiral, R is given by Eq. (1) with R₀ as the starting radius. Then, the calculations can be further simplified since the θ-dependent part of the phase exp(ik_spR) caused by the spiral simply eliminates the phase term exp(-iθ) given in Eq. (9). Using the integral representations of Bessel functions, E_nc,x(r, α) can be obtained:

E_{n c, x}^{} (r, α) = (E_{n c 0} R_{0} π / 2)[J_{1} (k_{s p} r)exp(i α)- J_{- 1} (k_{s p} r)exp(- i α)],

where

E_{n c 0}^{} = i \sqrt{2} E_{i n 0} exp[i (k_{s p} R_{0} + π / 2)] / \sqrt{λ_{s p} R} .

Finally, the x-component of the radial SPP wavefield generated by the metasurface structure can be expressed as

E_{n c, x}^{} (r, α) = E_{n c 0} R π J_{1} (k_{s p} r) \cos α .

By performing similar calculations, E_nc,y(r, α) can be found:

E_{n c, y}^{} (r, α) = E_{n c 0} R π J_{1} (k_{s p} r) \sin α .

From Eqs. (20) and (21), both the x- and y-components of SPP wavefield can be obtained. The total electric field can be written as a superposition of the two different components when δ = 0, as will be discussed later. Therefore, the SPP intensity distribution near the center is given by $I^{t o t} = E_{n c, x}^{2} (r, α)+ E_{n c, y}^{2} (r, α)$ with $I^{t o t} \propto J_{1}^{2} (k_{s p} r)$ . It is clear that the wavefield distribution is proportional to the first-order Bessel function J₁.

All former analyses involve the metasurface spiral with the radius increased in a counterclockwise manner. For the spiral wherein the radius is decreased in a counterclockwise manner, the components in the SPP wavefield are expressed using an additional phase term exp(-iθ), and this term ensures that the SPP wavefields of J₁(k_spr) and J₃(k_spr) are similar to the results in Eq. (18). Then, the intensities appear as doughnut-shaped distributions with more complicated elliptical polarizations.

Using the analytical expressions, the intensity and phase distributions were calculated and are presented in Fig. 2. Figure 2(a) shows the map of the total intensity |E_nc,x(r, α)|² + |E_nc,y(r, α)|², and the inset depicts a schematic of the spiral metasurface for clarity. Figures 2(b) and 2(c) present the magnified component intensity maps of |E_nc,x(r, α)|² and |E_nc,y(r, α)|² in the central area of 0.8 µm × 0.8 µm, respectively, and Fig. 2(d) depicts the corresponding magnified map of the total intensity. Figures 2(e) and 2(f) are phase maps of E_nc,x(r, α) and E_nc,y(r, α), respectively. The doughnut-shaped total intensity distribution in Fig. 2(d) has an intensity of 0 at r = 0, and Fig. 2(d) also includes a magnified map of the doughnut-shaped total intensity distribution with the polarization map superimposed.

Fig. 2 Analytical and numerical simulation results for the metasurface spiral. It is illuminated by RCP light, as indicated by the white solid arrow in (a). (a–c) Intensity profiles of E_x + E_y, E_x, and E_y, respectively. (d) Polarization map superimposed on the magnified view of the white dashed square in (a). (e, f) Corresponding phase distributions of x- and y-components. (g–l) Corresponding simulation results.

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2.3 Polarizations of the SPP wavefields

The polarization distribution can be described by the ellipses of the wavefield components (E_x, E_y) [34], which are given by (E_x/E_nc,x)² + (E_y/E_nc,y)² – 2E_xE_y/(E_nc,xE_nc,y)cosδ = sin²δ, where δ is the phase difference between ϕ_x and ϕ_y in Figs. 2(e) and 2(f), respectively. When δ = 0 or π, the ellipses reduce to line segments, representing the formation of a linearly polarized wavefield at (r, α). In Fig. 2(d), the polarization line segments of E_nc(r, α) drawn based on Eqs. (20) and (21) are shown and clearly all point to the doughnut center, implying that E_nc(r, α) has a polarization singularity at the center and is a radial vector vortex wavefield of polarization order l = 1. l determines the variation of the polarization orientation Φ(α) with azimuth α according to [35]

Φ (α) = l α + Φ_{0},

where Φ₀ is the initial polarization orientation. Generally, the polarization conversion of the metasurface spiral as an optical device is expressed by the Jones vector and matrix. The Jones vector of a vector wavefield is

J_{l} (α) = [\begin{array}{l} \cos (l α + Φ_{0}) \\ \sin (l α + Φ_{0}) \end{array}] .

We also deduced l = 1 and Φ₀ = 0 for E_nc(r, α) from Eqs. (20) and (21). The elements of the Jones matrix of the metasurface can be obtained using the unit output wavefields under both RCP and LCP illuminations. With the wavefield E_nc(r, α) for RCP illumination,

[\begin{array}{l} J_{x x} J_{x y} \\ J_{y x} J_{y y} \end{array}] [\begin{array}{l} 1 \\ - i \end{array}] = [\begin{array}{l} \cos α \\ \sin α \end{array}] .

Similarly, with the vanishing output field for LCP light,

[\begin{array}{l} J_{x x} J_{x y} \\ J_{y x} J_{y y} \end{array}] [\begin{array}{l} 1 \\ i \end{array}] = [\begin{array}{l} 0 \\ 0 \end{array}] .

The zero output amplitude represented in Eq. (25) actually reflects the unidirectional propagation induced by the orthogonal nanoslit metasurface [29]. By solving four linear equations and considering the factors on the right-hand sides of Eqs. (20) and (21), the Jones matrix of the metasurface spiral can be obtained:

M = E_{n c 0} R π J_{1} (k_{s p} r) / \sqrt{2} [\begin{array}{l} \cos α i \cos α \\ \sin α i \sin α \end{array}] .

The above Jones matrix indicates that a circularly polarized wave is transformed into a radially polarized wave distribution. It is interesting to note that, according to the matrix, the illumination of a linearly polarized wave with arbitrary orientation can also be changed into a radially polarized wave distribution. Moreover, when a simple spiral slit is used as the SPP excitation structure under illuminations of LCP and RCP lights, the focused central spot and vortex may be formed, respectively. This is different from the above obtained properties for the formation of vector vortex and zero intensity under both RCP and LCP illuminations, respectively. Then, the spiral slit and the metasurface spiral have different Jones matrices, and their polarization properties are also different.

3. FDTD simulations and experimental demonstration

3.1 FDTD simulations of the metasurface spiral

To verify the theoretical results, we performed simulations of the metasurface spiral with FDTD Solutions. The physical parameters were chosen to be the same as those used in the experiment. The incident wave was RCP and was provided by a 632.8 nm He–Ne laser, and the corresponding surface plasmon wavelength λ_SPP was 600 nm. The thickness of the gold film was set to 200 nm, and the length L and width W of each of the slits were 200 nm and 50 nm, respectively. The distance between the inner and outer spirals D was 150 nm, and the distance between two adjacent slits on a spiral S was 300 nm. The starting radius of the inner spiral r₀ was 6 µm. In the simulations, the region for calculation was 15 μm × 15 μm × 1 μm and the size of the Yee cell was a cube of side length 8.7 nm. The following results were obtained at the gold film surface.

In Fig. 2(g), a map of the calculated total intensity is presented, and Figs. 2(h) and 2(i) depict the magnified intensity maps of E_x(x, y) and E_y(x, y) in the central area, respectively. Figure 2(k) shows a magnified version of the intensity map in Fig. 2(g) with the polarization linear segments drawn using the E_x(x, y) and E_y(x, y) data. Overall, the former analytical results very closely match these simulation maps.

The intensity maps of the x- and y-components actually show the intensities of the linear polarizations in the corresponding orientations. Both maps have two lobes divided by the maximal dark lines, and the dark lines more quantitively express the radial polarization features of the wavefields. For general vector wave distributions, the azimuth of the dark line of its polarization component is given by α_D = (θ_D – Φ₀)/l + π (k + 0.5)/l with the integer k corresponding to different dark lines, θ_D representing the azimuthal orientation of the polarization, and Φ₀ is the initial polarization orientation [35]. For E_x(x, y), the maximal dark lines are perpendicular to the x-direction, and α_D = π/2 and 3π/2 with k = 0 and 1, respectively. Then, Φ₀ = 0 and l = 1 could be deduced, and the values are satisfied by the dark lines of E_y(x, y) and the polarized component of arbitrary orientations. These are the typical characteristics of radially polarized wavefields and demonstrate the formation of wavefields by the spiral metasurface. We also deduced that the optical vortex was formed because of the polarization singularity at the core of doughnut-shaped intensity distribution of [J₁(k_spr)]². The possibility of obtaining the polarization states of the vortices with Φ₀ in different values in plasmonic fields can be realized under appropriate conditions. In principle, the orientation of nanoslit pairs on the spiral determines the polarization direction of the SPP wavefields, and adjusting the orientation of the nanoslit pairs may provide a probable way to achieve the azimuthal vector vortices with Φ₀ = π/2 and other hybrid vector vortices [35].

3.2 Experimental demonstration

In the experiment, a metasurface spiral sample was fabricated using an FIB system, and the SEM image was presented in Fig. 1(a). The experimental setup is illustrated schematically in Fig. 3. A Mach–Zehnder interferometer was employed to record both the intensity maps and interference patterns of the scattered SPP wavefield and reference wave. The He–Ne laser beam was linearly polarized with a wavelength λ = 632.8 nm, and the quarter-wave plate converted the beam into RCP light. A beam-splitter (BS1) split it into the object and reference arms, and an adjustable attenuator (A1) adjusted the beam amplitudes of the two arms to be appropriate for interference. The object beam illuminated the sample at normal incidence from the substrate side. The sample was placed on a three-dimensional transitional stage (TS) for fine position adjustment. The light wave scattered from the sample surface was captured by a microscopic objective (MO), and the enlarged image with high magnification was received by an S-CMOS (Zyla-5.5, 16-bit, 2560 × 2160 pixels, 6.5 μm × 6.5 μm pixel size). To eliminate the spherical wave effect in the image wave due to the high imaging magnification of the MO, the reference beam was expanded and filtered by a spatial pinhole filter (SPF), and the longitudinal position of the SPF was carefully adjusted. By blocking and transmitting the reference beam, the S-CMOS recorded the intensity and interference patterns, respectively. From the interference patterns, the amplitude and phase distributions of the imaged SPP wavefields could be reconstructed. Because of the larger k_x and k_y wavevectors [32, 36], the reconstructed patterns were generally larger than but similar to those obtained with leakage radiation or near-field scanning microscopy.

Fig. 3 Experimental setup. The desired circularly polarized light was generated by passing a laser beam through a properly oriented quarter-wave plate (QWP), then splitting it into two beams by using a beam splitter (BS1). One beam was used as the reference wave, while the other illuminated the sample (S) at normal incidence to excite the SPPs on the sample surface. The S-CMOS recorded both the scattered SPP pattern and the interference pattern after passing through the polarized analyzer (P1).

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The experimental results are presented in Fig. 4, and the double-ended arrows in the top-right corners of the maps in Figs. 4(b)–4(e) indicate the direction of P1, giving the polarizations of the component intensity maps. The smaller lower-right patterns are the magnified maps of the squared central parts in each image. Figure 4(a) shows the doughnut-shaped intensity pattern obtained without P1, which is typical of the vortex field. Figures 4(b)–4(e) depict the intensity maps extracted from the interference patterns with polarizations in the horizontal, diagonal, vertical, and antidiagonal directions, respectively. The extinction direction of the intensity distribution rotates along with the P1 orientation. Overall, this behavior is in accordance with the previous theoretical and simulated results, signifying that the vortex was radially polarized. An interesting phenomenon indicated in Figs. 4(b)–4(e), though not obvious, is that the light wave in the spiral is inhomogeneous. For example, in Fig. 4(b), the intensity is weak at the horizontal and vertical positions of the spiral. In fact, such inhomogeneous distribution originates from the interference of SPP waves excited by the nanoslits in the inner spiral propagating to both sides with those excited by the nanoslits in the outer spiral propagating to the inner side of the spiral. It may be deduced that, at the horizontal and vertical positions, the SPP waves have the maximum component E_y, and when a polarizer in the x-direction is used in the measurement, intensities appear to be weak at these positions. Such reasons are true for other components of SPP wavefields, as shown in Figs. 4(c)–4(e). The influence of inhomogeneity is minimized by the propagation of the SPP waves from the slits to the central area of the spiral.

Fig. 4 Experimental results for the metasurface spiral illuminated by RCP light. Intensity distributions obtained (a) without and (b–e) with the analyzer. The polarization orientation of the analyzer is indicated by the white arrow in the top-right corner of each picture. A magnified view of the white square part indicated in each of the top panels is shown in its inset. (f) Polarization map describing the polarization state of the central vortex. (g–j) Corresponding phase patterns of different component wavefields.

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The patterns in Figs. 4(f)–4(j) have the same area as the insets of Figs. 4(b)–4(e), 0.89 μm × 0.89 μm. Figure 4(f) depicts the polarization ellipses plotted at the 13 × 13 grid positions. Figures 4(g)–4(j) present the phase maps corresponding to the intensity maps and were plotted using the data from the interference patterns. When plotting the polarization ellipses in Fig. 4(f), the phase difference δ between E_x and E_y was necessary and the data of the phases in Figs. 4(g) and 4(i) were used. Intuitively, at most of the grid points, the ellipse reduces to or approximates linear segments pointing toward the center, experimentally demonstrating the formation of radially polarized vector vortices. Because the polarizer was adjusted between recording the interference patterns of E_x and E_y, a constant phase difference between them may have been introduced. Then, when drawing the ellipses, a whole-field constant was added to the phase differences δ to make the ellipses be close to the line segments. In addition, although the four phase maps are not as ideal as those in Fig. 2, where the phases are almost constant in a semicircular area, the phase distribution in half of the area near the middle dashed line varies very slowly and is almost constant. Overall, these results are consistent with the theoretical and FDTD simulation results.

4. Conclusions

We investigated a method of generating a radially polarized vector vortex in the SPP field produced by a metasurface spiral of orthogonal nanoslit pairs. We introduced a theory that models the field generated by a single orthogonal nanoslit pair. Combining the nanoslit pairs with spiral geometry to construct a metasurface spiral and using the Huygens–Fresnel principle, we derived expressions for the x- and y-components in the central point area of the metasurface spiral and obtained a doughnut-shaped intensity distribution. Simultaneously, the polarization ellipses in (E_x, E_y) coordinates were drawn and qualitatively demonstrated the formation of a radially polarized SPP vector vortex. Our analytical method provides physical insight into the characteristics of plasmonic component fields. To understand the polarization properties, the Jones matrix of the metasurface spiral was derived. The theoretical and simulation data were agreed closely with each other. In addition, we constructed a Mach–Zehnder interferometer system to measure the SPP wavefields. Overall, the extracted data maps for the distributions of the amplitude and phase as well as the radial polarization state were reasonably consistent with the theoretical and simulation results. We expect this work to facilitate the design of plasmonic polarization devices and manipulation of on-chip SPP fields.

Funding

National Natural Science Foundation of China (11574185, 11604183, 11647015); Project of Shandong Province Higher Educational Science and Technology Program (J16LJ09).

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Radially polarized plasmonic vector vortex generated by a metasurface spiral in gold film

Abstract

1. Introduction

2. Theoretical analysis of metasurface spiral and FDTD solutions

2.1 Geometries of the nanoslit element and wavefield vector

2.2 Huygens–Fresnel principle of SPP and the wavefield near the spiral center

2.3 Polarizations of the SPP wavefields

3. FDTD simulations and experimental demonstration

3.1 FDTD simulations of the metasurface spiral

3.2 Experimental demonstration

4. Conclusions

Funding

References and links

Cited By

Figures (4)

Equations (26)

Optics Express