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Spin-orbit coupling controlled near-field propagation and focusing of Bloch surface wave

Fu Feng, Shi-Biao Wei, Ling Li, Chang-Jun Min, Xiao-Cong Yuan, and Michael Somekh

Open Access Open Access

Abstract

Bloch surface wave (BSW) can be considered as the dielectric analogue of surface plasmon polariton (SPP) with less loss since it is sustained at the surface of a truncated dielectric multilayer. As dielectric materials show nearly no ohmic loss, BSW can propagates much farther compared to SPP, and thus is beneficial for planar optical devices. In this paper, we study the spin-orbital interaction between incident beam and BSW. We demonstrate that due to the spin-orbital coupling, the near-field properties of generated BSW can be controlled with a meta-antenna structure. The meta-antenna is composed of two gold nano-antennas oriented at 45° and 135° as a near-field coupler. By careful design of the meta-antenna, the generated BSW can be guided and focused depending on the chirality of the incident beam. Three examples of meta-antennas are demonstrated for chiral sensitive focusing, directional switching and asymmetric focusing. The proposed method can be applied as a design method for low-loss on-chip photonic devices.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Manipulating a surface electromagnetic wave is a key requirement for on chip optical circuit and miniaturization of compact optical system. For the most commonly used electromagnetic surface wave, the Surface Plasmon Polariton (SPP), the ability to tailor it plays a key role in numerous applications such as sensing [1–3], nano-circuitry [1–4] optical data storage [5,6], super-resolution imaging [7,8], plasmonic tweezers [9,10], in-plane communications [11], and in-plane data processing [12]. Since the field of SPP is partially localized in the metal, the propagation distance of such surface wave is limited by dissipation. This constitutes a severe limit for many existing and potential application especially for large scale planar photonic devices. On the other hand, the Bloch surface wave (BSW) is a surface electromagnetic wave that exists at the interface between a truncated periodic dielectric multilayer and the surrounding medium [13,14]. Similar to SPP, BSW also enables optical near-field confinement and enhancement [15,16]. Therefore, BSW has been considered as an alternative of SPP with nearly no loss, as dielectrics show a much lower absorption compare with metals. Recent researches demonstrate that BSW can effectively be manipulated to generate diffraction free surface wave [17] and can be reflected, divided by using gratings on the surface [18]. Also it can be focused by spiral or concentric rings [19]. Further, BSW wave can be either TE or TM polarized, while SPP can only be TM polarized. A TE polarized surface wave has many advantages compare to a TM polarized surface wave, for example, it has a much higher coupling efficiency to excitonic state in lower dimension materials [20]. It is well known that due to the peculiar transverse spin angular momentum specific of SPP, the spin-orbit coupling effect can be used to shape the near field of the SPP wave [21]. As BSW has also shown magnetic spin-orbital coupling effect [22], one can also control the near field of BSW via spin-orbital effect. Compared with the slab-waveguide platform which is also composed of dielectric materials and widely used in integrated optical devices [23–28], the BSW devices could be smaller and achieve processing of optical signals in subwavelength scale [29].

In this paper, we propose to use the spin-orbit coupling effect to control the near-field propagation and focusing of BSW in various ways, including near-field focusing such as focal length tuning, directional focusing, and asymmetric focusing. The spin-orbit coupling is achieved by a chiral meta-antenna structure composed by arrays of sub-wavelength metal nano-antennas. Different to scattering the light that is perpendicular to the longer axis of the antenna as for SPP excitation [30], here the BSW launching mechanism is different. When the nano-antenna is placed on a BSW dielectric substrate, the nano-antenna launches strong BSW for the polarization that is parallel to the longer axis of the nano-antenna and the launched BSW propagates perpendicularly to the incident polarization direction. Further, according to our study, the nano-antenna can scatter incident circular polarized beams and give rise to BSW with different phase delay depending on the orientation of nano-antenna due to the geometric phase principle. This characteristic is beneficial to produce meta-antennas with chiral responses. Our studies in the following sections show that meta-antennas can focus the BSW on a desired focal point with various tunable features, such as polarization switching ability, directional launching and asymmetric focusing.

2. Directional coupling of BSW by chiral structure

In this paper, a dielectric multilayer sustaining TE polarized BSW is investigated. The structure is made of alternating layers of SiO2 and TiO2, as schematically illustrated in Fig. 1(a). Except the top SiO2 layer, the thickness of SiO2 and TiO2 layer is set to be 110nm and 72nm, respectively. The top SiO2 layer is set to be 550nm. The entire structure is on a glass substrate and the material in contact with the substrate is TiO2. The optical refractive index of the glass is set to 1.52 and the optical index of TiO2 and SiO2 is set to 2.38 and 1.46 respectively. For a TE polarized BSW that propagates along Y axis, the wave can be entirely described by Ex, Hy and Hz fields as shown in Fig. 1(a). The dispersion relation of the BSW supported by this structure is calculated by transfer matrix method and shown in Fig. 1(b). It can be observed that two modes can be excited at the 633nm incident wavelength, one at 62.8° and one at 66.1°. The mode at 62.8° corresponds to the BSW mode that we are interested in, and the mode at 66.1° refers to a sub-surface mode or guided mode [31]. In this paper, the discussion is limited only on the BSW mode.

 figure: Fig. 1.

Fig. 1. (a) Schematic presentation of BSW substrate with a gold nano-antenna laid on top along X axis under light excitation, the generated BSW propagates along Y axis and thus can be described by Hy, Hz and Ex components; (b) Dispersion relation of BSW structure described by reflectivity as a function of incident angle of a plane wave illuminating from the bottom, whose two modes located at 62.8° and 66.1° are corresponding to BSW mode and guided mode, respectively. The dashed line indicates the 633nm wavelength.

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For a nano-antenna on a gold surface, it selectively scatters incident light that is polarized perpendicular to it, and gives rise to the SPP. We now investigate the BSW generation mechanism by a metal nano-antenna on top of a BSW substrate as shown in Fig. 2(a). In this study, a gold nano-antenna is placed on top of a BSW dielectric substrate as a scattering coupler. Compared to dielectric materials, the metal scatter works as a dipole emitter with higher coupling efficiency in this case, as metal contains large number of free electrons. The length and width of the nano-antenna is 250nm and 50nm, respectively, and the nano-antenna is made by a 50nm thick gold. For the following calculations, the antenna always follows the Y axis. The calculations in the following sections are performed by the finite-difference time-domain (Lumerical FDTD Solutions) simulations, the refractive index of gold is set to be 0.18 + 3.4i. A global mesh of λ/10 is applied to the entire structure and an additional mesh of λ/20 applied on the parts of gold, a perfectly matched layer is placed around the simulation area. An incident linear-polarized plane-wave beam of 633nm wavelength is placed 1µm away on top of the structure, and illuminates normally to the substrate.

 figure: Fig. 2.

Fig. 2. (a) Up: Schematic presentation of a single 250nm×50nm×50nm nano-antenna following Y axis laying on the BSW substrate under excitation; Down: |Hz| field distribution of BSW excited when the linear polarization of incident beam is set to X(left) and Y(right), respectively; (b) Angular distribution of BSW as (a), blue and red curve represent BSW generated by incident beam with X and Y polarization respectively. The BSW intensity generated by X polarized light is amplified for 20 times for clarity.

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Figure 2(a) presents the generated BSW |Hz| field distribution on the substrate. Note that as the BSW is polarized in TE mode, |Hz|, which exists in BSW but not in incident beam, is used here to characterize the properties of the launched BSW. It can be observed that, when X-polarized light (Fig. 2(a)-left) is used as incident beam, the generated BSW propagates mainly to the + Y and –Y directions. In contrast, for Y-polarized incident light (Fig. 2(a)-right), the generated BSW propagates mainly to the + X and –X directions. Also, qualitatively, it can be observed that the Y-polarized incident beam generates significantly stronger BSW compared to the X-polarized incident beam.

Figure 2(b) is the angular distributions of BSW on surface generated by linearly X (blue line) or Y polarized (red line) incident beam. For a nano-antenna along the Y axis, the nano-antenna launches strong BSW for the polarization that is parallel to the longer axis of the nano-antenna, and launched BSW is significantly diminished when the excitation light is polarized perpendicularly to the nano-antenna. The launched BSW propagates perpendicularly to the polarization direction of the incident beam. Note that, these characteristics are different to the situation of SPP launching, where the incident polarization is perpendicular to the axis of antenna and the generated SPP propagates along the polarization direction. And because there is no loss during the propagation of BSW, its theoretical propagation distance is large while the propagation distance of SPP is only about 5–10µm. To obtain a high BSW excitation efficiency, it is important to keep a high aspect-ratio for each nano-antenna, so that the antenna can be treated as a dipole emitter under laser excitation. However, the excitation efficiency decreases when aspect-ratio increases, there is thus a compromise to be taken into consideration. In our study, we always chose 250nm×50nm×50nm as length, width and depth of each nano-antenna to keep it a dipolar like emitter while maintaining the excitation efficiency.

In order to generate a chiral response, we then design a chiral meta-antenna structure including double nano-antennas, and study its response to circularly polarized light. In such structure, the length, width and thickness of the nano-antenna remain the same as in Fig. 2, but the longer axis of the two nano-antenna is set to 45° and 135° with respect to the X axis, respectively, as shown in Fig. 3(b). In theory, a left circularly polarized (LCP) light can be expressed as superposition of two orthogonal linear polarization components:

$${{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_{lcp}} = {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_x} + i{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_y} ,$$
where i represents a π/2 phase shift between the two polarization components ${{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_x} $ and ${{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_y} $. Here a single nano-antenna scatter works like a linear electric dipole, whose dipole momentum can be written as the projection of incident electric field on the longer axis of the nano-antenna. In the case of a nano-antenna oriented at an angle α (angle between the longer axis of the nano-antenna and X axis), the dipole momentum can be written as:
$$p = \left|{{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_x}} \right|\cos (\alpha )+ i\left|{{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_y}} \right|\sin (\alpha )= {E_0}\textrm{exp}({i\alpha } ),$$
where the amplitude E0 = |${{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_x} $ | = |${{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_y} $| for a circularly polarized light. From Eq. (2), it can be seen that, under left circularly polarized excitation, the momentum of excited dipole has a phase delay of α, which is actually the well-known geometric phase [21].

 figure: Fig. 3.

Fig. 3. (a) Distribution of real part of Hz of an individual nano-antenna oriented at 45° and 135° under LCP excitation; (b) Schematic presentation of a double-line meta-antenna composed of a 135° nano-antenna on the left (line A) and a 45° nano-antenna on the right (line B). Horizontally the distance between the two nano-antennas is D, and vertically the nano-antenna pairs are repeated by a period of d, the number of meta-antenna pairs can be chosen according to different applications; (c)–(d) Intensity field distribution of 9 double line meta-antenna under LCP and RCP excitation, respectively. The beam is incident normally from the top. D and d are set to be 1/4 λbsw and 300nm, respectively. The distance between adjacent meta-antenna pairs is set to be 2λbsw. The positions and orientations of the nano-antennas are schematically presented by golden sticks on white background in the center of the figures. The background is white because the electric field saturate in the center.

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As mentioned, each nano-antenna can be treated as a dipole source that lies on the substrate along longer axis of the nano-antenna. Such an approximation is often considered for analyzing radiated far field or surface field that are far away from the nanoslit [32,33]. By superposing light fields generated from each dipole, the electric-field component of the scattered light can be expressed as:

$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} \left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} } \right) = i\omega {\mu _0}\sum\nolimits_{m = 1}^N {\left( {{{\textbf {G}}_0}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ,{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_m}} \right) + {{\textbf {G}}_\sigma }\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ,{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_m}} \right)} \right){\textbf {P}}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_m}} \right){e^{i{\varphi _m}}}},$$
where ω is the angular frequency of excitation light, µ0 is the free-space permeability, N is the number of nano-antennas, and ${\textbf {P}}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_m}} \right) = ({{p_m}\cos {\alpha_m},{p_m}\sin {\alpha_m},0} )\delta \left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_m}} \right)$ denotes a dipole located at position rm. pm and αm indicate the amplitude and orientation of the m-th dipole source, respectively, as expressed in Eq. (2). The function ${{\textbf {G}}_0}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ,{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_m}} \right)$ and ${{\textbf {G}}_\alpha }\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ,{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_m}} \right)$ represent free-space and surface Green dyadic tensors for the three-dimensional Helmoholtz equations, respectively. The phase term φm represents the relative phase difference between dipole sources introduced by orientation of nano-antennas. It can be expressed as ${\varphi _m} = \pm {\alpha _m}$ for circularly polarized lights (+/− sign for left/right circularly polarized light, respectively) as explained in Eqs. (1) and (2).

We here consider the BSW field generated by a serious of nano-antennas with same orientation patterned in a single line along Y direction. When the width of the nano-antenna line is negligible compare to the propagation distance of the BSW, from Eqs. (3) and (2), the complex electric field of generated BSW can be expressed as a function of tilted angle α as:

$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} ({x,y} )= A(\alpha ){e^{i{\varphi _\alpha }}}{e^{ik|x |}},$$
where A(α) and φα are the initial amplitude and phase of generated BSW, k is the wave vector of generated BSW, and |x| is the distance to the dipole source in X direction. There is also ${\varphi _\alpha } = \pm {\alpha }$ for left/right circularly polarized lights according to Eq. (2). Similar to the case of SPP [30], it can be seen from Eq. (4) that the generated BSW from a single nano-antenna line can be considered as a quasi-plane wave whose initial amplitude and phase depend on the orientation of the nano-antennas.

Figure 3(a) shows the BSW generation under illumination of 633nm (LCP) incident beam when a nano-antenna is oriented at 45° and 135°, respectively. It can be observed that, as predicted the nano-antennas act as a dipole oriented along longer axis of the antenna, and the two dipole sources have a phase difference of π/2 (measured by putting a field monitor on the surface) determined by their orientation angles according to Eq. (2). These behaviors of BSW are similar to the case of SPP that has been proved to be able to generate chiral-sensitive direction-controlled surface wave [34,35]. It is also remarkable that the response of the above antennas to right circularly polarized (RCP) incident beam is almost same except with the inverted phase (phase difference of -π/2). Then we combine the two nano-antennas oriented at 45° and 135° to form a chiral meta-antenna structure as shown in Fig. 3(b) which has directional surface wave switching characteristics. Under LCP excitation, by using the quasi-plane wave approximation demonstrated in Eq. (4), the electric field of generated BSW at the dashed line (named target plane) on the structure can simply be expressed as follows:

$${{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_{Target}} = {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_A} \textrm{exp}({ikx} )+ {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_B} \textrm{exp}({ik({x + D} )} ),$$
where ETarget is the electric field at the target plane, EA and EB are the electric field generated by the nano-antennas at line A and line B, respectively, k is the wave-vector of BSW, and D is the distance between line A and B. As above mentioned, the BSW generated by 45° and 135° antennas has a phase delay of π/2, from Eq. (4), we can get ${{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_A} = A(\alpha ){e^{i\alpha }}$ and ${{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_B} = \; {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_A} \textrm{exp}({ - i\pi /2} )$, hence Eq. (5) can be re-written as:
$${{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_{Target}} = {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_A} \exp ({ikx} )+ {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_A} \textrm{exp}\left[ {ik({x + D} )- \frac{\pi }{2}} \right] = A(\alpha ){e^{i\alpha }}{e^{ikx}}\left[ {1 + {e^{i\left( {kD - \frac{\pi }{2}} \right)}}} \right].$$
From Eq. (6), the constructive interference condition is fulfilled when D = (n + 1/4)λbsw (n = 0,1,2…). As consequence, most BSW generated by LCP will propagate to the left side (target plane), and as the structure has an inverted response to the RCP incident beam, the BSW generated by RCP will propagate to the right side. To go a bit further, a destructive interference condition is fulfilled when D = (n + 3/4)λbsw, and in such case the BSW generated by LCP and RCP incident beam will propagate towards right and left, respectively.

Figures 3(c) and 3(d) show the response of meta-antenna presented to LCP and RCP incident beam, respectively. The simulated structure is like the structure shown in Fig. 3(b), where the distance D is set to 1/4 λbsw and the distance d is set to 300nm. Each line (A and B) is composed of 40 nano-antennas. Totally 9 double-line meta-antenna structures are used in this simulation to increase the efficiency of the structure, and the distance between adjacent double-line meta-antenna is set to be 2λbsw, where the effective wavelength of BSW λbsw is calculated to be 470nm from Fig. 1(b). It can be observed that, as predicted, under LCP/RCP light excitation, most of the generated BSW propagates to the left/right side of the structure, respectively. The interference fringes in Figs. 3(c) and 3(d) come from the interference of BSW generated by different double-line meta-antennas. To be more precise, each meta-antenna can be considered as an individual scatter, and the generated BSW from these different meta-antennas interfere during propagation and create the interference pattern as shown in Figs. 3(c) and 3(d). The interference fringes would disappear if only single double-line structure is used, however, the BSW generation efficiency decreases in this case. The extinction ratio, defined as the ratio between power of side with more generated BSW and the side with less generated BSW, is calculated to be 30. This proves that the designed chiral meta-antenna structure can indeed control the propagation direction of the generated BSW, this can be beneficial for on-chip manipulation of BSW.

3. Spin controlled near-field focusing of BSW

In order to further improve the degree of freedom of manipulating the BSW, we here introduce a simple way of controlling the near-field focusing of the BSW by combining the Fresnel zone plate (FZP) with the designed meta-antenna. A FZP introduce a discrete amplitude modulation along the Y direction and can thus control the focus of the BSW [36]. The FZP radius profile can be expressed as follow [37]:

$${R_m} = \sqrt {mf{\lambda _{BSW}} + \frac{{\lambda _{BSW}^2{m^2}}}{4}},$$
where m is a natural number describing each radius of the FZP, and f is the designed focal length. R1 and R2 shown in Fig. 4(a) correspond to Rm calculated from Eq. (7). FZP and meta-antenna introduced above can be integrated together by distributing the nano-antennas with a profile that follows the radius profile of a FZP.

 figure: Fig. 4.

Fig. 4. (a) Schematic presentation of meta-antenna integrated with FZP under circular polarized light incident beam excitation, the integration of FZP is simply by removing the nano-antennas fall into R1 < R < R2 (named “dark zone”, same operations should be applied for higher orders); (b) - (c) Intensity profile of meta-antenna integrated with FZP under LCP/RCP incident excitation respectively.

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Figure 4(a) is a schematic presentation of recombination of the meta-antenna with FZP distribution. By simply removing the nano-antennas fall in the “dark zone” of the FZP, the FZP can be integrated with the meta-antenna. Figures 4(b) and 4(c) show the intensity distribution of such structure under LCP and RCP excitation with designed different focal lengths. From Fig. 4(b), it can be observed that, for a LCP polarization excitation and a FZP with designed focal length of 25µm, most of generated BSW propagates towards the left side of the structure, and BSW is efficiently focused at 25µm from the center of the structure. This agrees well with our previous theoretical predictions. Weak BSW fields following the typical caustics patterns of the FZP lens can still be observed at the right side of the structure, due to the finite aspect-ratio of the nano-antennas, this can be improved by using nano-antennas with higher aspect ratio or by increasing the number of meta-antenna pairs. Although the structure cannot achieve 100% directional launching of BSW, the extinction ratio is calculated to be more than 20 in this case, which is still sufficient for many applications. Figure 4(c) shows the intensity distribution of generated BSW when the structure is under RCP excitation and the designed focal length of the FZP is set to be 15µm. The result shows that the structure works as predicted. And indeed, the focal length can be controlled by changing the design parameters of the FZP. Note that although it is not demonstrated here due to limited calculation capacity, but as the BSW substrate has no absorption at all in theory, the focal length of the FZP can be designed to have a much higher value which is important for planar optical devices.

4. Asymmetric near-field double focusing of BSW

It has been demonstrated in the previous section that together with the FZP the meta-antenna can focus the BSW at either the left or the right side of the structure depending on the handedness of the incident beam. It would be a further step forward for manipulation of the BSW if the BSW can be focused at the left and right side simultaneously with controllable asymmetric focal lengths for both sides. We here demonstrated that it can be achieved by using a hybrid FZP design. The main idea is to design three different nano-antenna pairs that can do left-side, right-side and both-side generation of BSW. As we have seen that a pair of nano-antennas with 45° and 135° orientations can launch BSW to the left under LCP excitation. Because of the symmetry, a reversed nano-antenna pair with 135° and 45° orientations will launch BSW to the right side under same excitation.

We thus propose three nano-antenna pairs as shown in Fig. 5(a). For a LCP incident beam, the BSW will propagate to the left, right and both sides through type I, II, II (brown, blue, green) meta-antenna, respectively. The design protocol of hybrid FZP is shown in Fig. 5(b), where two FZPs with different focal lengths are designed from Eq. (7) and combined to form the hybrid FZP. In the profile of hybrid FZP, for the part where there is only FZP I or FZP II, the orientations of the nano-antennas stay the same or reversed to the design in Fig. 3, respectively. For the part where FZP I and FZP II overlap, the orientation of the nano-antennas is set to 90° with respect to X axis as the type III shown in Fig. 5(a). For the part where there is no FZP I nor FZP II, the nano-antennas are removed. Figures 5(c) and 5(d) show the simulated intensity field distribution of such hybrid structure under LCP/RCP illumination respectively, and the designed focal length for FZP I and FZP II is set to 25µm and 15µm respectively. It is clearly observed that an asymmetric near-field focusing of BSW is achieved by this structure. Also, the position of asymmetric foci of the structure can be switched by changing the optical handedness due to the switchable launching characteristics of type I and II meta-antenna. This result demonstrates that by changing the design parameters, we can achieve on-chip asymmetric focusing of BSW by a hybrid FZP. This would show great potential on manipulation of BSW and can be used for planar optical devices, optical sensing and other applications.

 figure: Fig. 5.

Fig. 5. (a) Schematic presentation of Type I, II, III (brown, blue, green) meta-antenna; (b) Schematic presentation of design protocol of asymmetric focusing hybrid FZP lens; (c) - (d) Intensity field distribution of designed hybrid FZP under LCP and RCP excitation, respectively.

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5. Conclusion

In this paper, we theoretically demonstrate for the first time that a chiral metal nano-antenna pair can be used to manipulate the BSW due to the spin–orbit coupling of light and surface electromagnetic field. Compare with the most used surface plasmon polariton, BSW shows huge advantage because of low loss, and thus possible to make a large scale optical chip with such structure. By carefully design the meta-antenna, we can focus the BSW on a desired focal point with various tunable features such as polarization switch ability, directional launching and asymmetric focusing. We expect that the method demonstrated can be later applied to surface wave switch on dielectric based substrate, tunable on-chip focusing of electromagnetic wave, and further arbitrary shaping of optical near fields on a surface in a region much larger than the wavelength of surface wave. These characteristics are crucial for integrated optical devices.

Funding

National Natural Science Foundation of China (91750205, U1701661, 61427819, 61805165, 61905147, 11604219); Leading Talents Program of Guangdong Province (00201505); Natural Science Foundation of Guangdong Province (2016A030312010, 2017A030313351); (Shenzhen Science and Technology Innovation Commission (JCYJ20180507182035270, KQTD2017033011044403, KQJSCX20170727100838364, ZDSYS201703031605029); Hong Kong GRF (152478); Shenzhen University (2019073).

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Figures (5)
Fig. 1.
Fig. 1. (a) Schematic presentation of BSW substrate with a gold nano-antenna laid on top along X axis under light excitation, the generated BSW propagates along Y axis and thus can be described by Hy, Hz and Ex components; (b) Dispersion relation of BSW structure described by reflectivity as a function of incident angle of a plane wave illuminating from the bottom, whose two modes located at 62.8° and 66.1° are corresponding to BSW mode and guided mode, respectively. The dashed line indicates the 633nm wavelength.

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Fig. 2.
Fig. 2. (a) Up: Schematic presentation of a single 250nm×50nm×50nm nano-antenna following Y axis laying on the BSW substrate under excitation; Down: |Hz| field distribution of BSW excited when the linear polarization of incident beam is set to X(left) and Y(right), respectively; (b) Angular distribution of BSW as (a), blue and red curve represent BSW generated by incident beam with X and Y polarization respectively. The BSW intensity generated by X polarized light is amplified for 20 times for clarity.

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Fig. 3.
Fig. 3. (a) Distribution of real part of Hz of an individual nano-antenna oriented at 45° and 135° under LCP excitation; (b) Schematic presentation of a double-line meta-antenna composed of a 135° nano-antenna on the left (line A) and a 45° nano-antenna on the right (line B). Horizontally the distance between the two nano-antennas is D, and vertically the nano-antenna pairs are repeated by a period of d, the number of meta-antenna pairs can be chosen according to different applications; (c)–(d) Intensity field distribution of 9 double line meta-antenna under LCP and RCP excitation, respectively. The beam is incident normally from the top. D and d are set to be 1/4 λbsw and 300nm, respectively. The distance between adjacent meta-antenna pairs is set to be 2λbsw. The positions and orientations of the nano-antennas are schematically presented by golden sticks on white background in the center of the figures. The background is white because the electric field saturate in the center.

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Fig. 4.
Fig. 4. (a) Schematic presentation of meta-antenna integrated with FZP under circular polarized light incident beam excitation, the integration of FZP is simply by removing the nano-antennas fall into R1 < R < R2 (named “dark zone”, same operations should be applied for higher orders); (b) - (c) Intensity profile of meta-antenna integrated with FZP under LCP/RCP incident excitation respectively.

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Fig. 5.
Fig. 5. (a) Schematic presentation of Type I, II, III (brown, blue, green) meta-antenna; (b) Schematic presentation of design protocol of asymmetric focusing hybrid FZP lens; (c) - (d) Intensity field distribution of designed hybrid FZP under LCP and RCP excitation, respectively.

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Equations (7)

Equations on this page are rendered with MathJax. Learn more.

E ⇀ l c p = E ⇀ x + i E ⇀ y ,
p = | E ⇀ x | cos ⁡ ( α ) + i | E ⇀ y | sin ⁡ ( α ) = E 0 exp ( i α ) ,
E ⇀ ( r ⇀ ) = i ω μ 0 ∑ m = 1 N ( G 0 ( r ⇀ , r ⇀ m ) + G σ ( r ⇀ , r ⇀ m ) ) P ( r ⇀ − r ⇀ m ) e i φ m ,
E ⇀ ( x , y ) = A ( α ) e i φ α e i k | x | ,
E ⇀ T a r g e t = E ⇀ A exp ( i k x ) + E ⇀ B exp ( i k ( x + D ) ) ,
E ⇀ T a r g e t = E ⇀ A exp ⁡ ( i k x ) + E ⇀ A exp [ i k ( x + D ) − π 2 ] = A ( α ) e i α e i k x [ 1 + e i ( k D − π 2 ) ] .
R m = m f λ B S W + λ B S W 2 m 2 4 ,
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