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Abstract
In this paper, four rectangular slits with the same size and regular rotation angle are regarded as the meta-atom, arranged on circular contours, to create plasmonic vortex lenses (PVLs) solely based on the geometric phase. These PVLs can achieve the same purpose of exciting surface plasmon polariton (SPP) vortices with arbitrary combinations of topological charge (TC) when illuminated by circularly polarized (CP) light with different handedness as the traditional PVLs. Furthermore, they can generate SPP vortices with different TCs and specific constant or varying electric-field intensities when excited by linearly polarized (LP) light, which marks the first instance of this phenomenon solely through geometric phase manipulation. The TC can be dynamically altered by controlling the polarization order of the incident vector beam. These PVLs not only possess advantages in terms of device miniaturization and the creation of a more uniform vortex field, as compared to PVLs based on the transmission phase, but also offer a more straightforward design process in comparison to traditional structures that rely solely on the geometric phase.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Surface plasmon polaritons (SPPs) are electromagnetic waves confined to a metal-dielectric interface and are caused by the collective oscillation of free electrons. Because of their unique properties, such as short wavelength, strong enhancement of the local electric field, and evanescence, they are widely used in laser physics [1–5], lithography [6–8], biosensors [9–16], integrated photonic circuits [17–20], and so on. The field of a plasmonic vortex with a certain orbital angular momentum is an electromagnetic field localized on the metal surface and is widely used for particle manipulation [21,22], biochemical analysis [23], optical field detection [24], and high-security encryption [25].
Numerous plasmonic vortex lenses (PVLs) are designed to excite specific plasmonic vortices. The classic PVL is a simple circular slit excited by the circularly polarized (CP) light of different handedness to form an SPP vortex with topological charge (TC) $l ={\pm} 1$ [26]. To generate an SPP vortex with a greater TC, a structure with two rows of paired rectangular slits arranged in a circular pattern was proposed [27–29]. By varying the rotation of the slits as a function of position, excitation of the rectangular slits by CP light creates SPPs with different initial phases (i.e., geometric phases), thus producing an SPP vortex with a greater TC. Although an SPP vortex with a greater TC can be generated by CP light, the TC of SPP vortices can be reversed only by changing the handedness of the incident light. Note that reversing the TC does not change the spatial distribution of the SPP’s electric field intensity. The transmission phase (i.e., the spiral structure) overcomes this restraint [30–33]. To flexibly and freely combine TCs under CP light of different handedness, two rows of slit pairs can be arranged on the spiral profile [34–36]. Therefore, the transmission phase is most introduced in previous works to freely combine the TC of SPP vortices. However, an issue arises as the size of the PVL increases with the spiral radius, presenting a challenge for miniaturizing these devices [37], particularly those designed for longer wavelengths. Furthermore, SPP waves, especially those with shorter wavelengths, generated by the slits at the end of the spiral experience more attenuation due to the increased optical path they must traverse compared to those generated by the front slits. This ultimately leads to an uneven field distribution of the SPP vortex [38].
To address these challenges, recent studies have proposed some structures solely based on the geometric phase to interfere between different plasmonic vortices [39] or to combine rectangular slits of varying sizes [40] for generating SPP vortices with a flexible combination of TC under the CP light with different handedness. Although these methods do not introduce the transmission phase, the costly optimization process of the slit rotation and size as a function of position complicates the calculations [41]. Additionally, SPP vortices generated by these structures require strict CP light for excitation. If linearly polarized (LP) light with an arbitrary polarization direction is used to create an SPP vortex with a specific TC, the introduction of the transmission phase becomes necessary in PVLs [42,43]. This introduction is not conducive to achieving uniform SPP vortex fields and can impede device miniaturization, as mentioned earlier. A comprehensive comparison of the principles, implementation mechanisms, advantages, and disadvantages of PVLs in previous studies is shown in Table 1.
Table 1. Comparison of PVLs in previous works
This paper proposes PVLs, solely based on the geometric phase, made up of meta-atoms comprising four rectangular slits with varying rotation angles, large aspect ratios, and the same size. The separation between these four rectangular slits is set to ${\lambda _{\textrm{SPP}}}/4$ (${\lambda _{\textrm{SPP}}}$ is the SPP wavelength). The meta-atoms are arranged on a circular contour, i.e., the slits are placed on four concentric circles with the radial increments of ${\lambda _{\textrm{SPP}}}/4$ (see Fig. 1). The systematic variation of the orientation of rectangular slits within each meta-atom allows for the flexible combination of SPP-vortex TCs under the excitation of CP light with different handedness. Furthermore, the PVL, based solely on the geometric phase, can generate SPP vortices with different TCs and specific constant or changing electric-field intensities for the first time when illuminated by the LP light. The TC of the SPP vortex varies dynamically as the incident light shifts from a scalar beam to a vector beam with varying polarization order, offering a unique approach to dynamically altering the TC of an SPP vortex field. The design principle of such PVLs relies solely on the geometric phase, granting them advantages in terms of device miniaturization and achieving a more uniform vortex field. Additionally, the uniformity in size and regular rotation angles of the slits simplifies the optimization process for designing the structure. The principal derivation and simulations not only validate the structural design but also showcase the broader functions of our PVLs compared to previous work.
Fig. 1. (a) Schematic diagram of the PVL made up of meta-atoms to manipulate plasmonic vortex by only using geometric phase. (b) Top view of the PVL. Inset shows the details of a meta-atom and defines the structure parameters.
2. Theoretical framework
2.1 Excitation with CP light
A schematic diagram of manipulating the plasmonic vortex by a PVL and a detailed view of the PVL formed by meta-atoms with four rectangular slits of the same size and regular rotation angles separated by the distance $D = {\lambda _{\textrm{SPP}}}/4$ are shown in Fig. 1(a) and 1(b), respectively. Rectangular slits with large aspect ratios are one of the most commonly used cell structures for designing PVLs to generate and modulate the SPP field. When CP light is normally incident on a single rectangular slit, it excites SPPs with in-plane dipole characteristics, so such slits can be regarded as SPP point sources [44,45]. The SPP field excited by $m\textrm{th}$ meta-atom at point M in space can be expressed as
where ${E_{Mms}}$ is the SPP field generated by the $s\textrm{th}$ rectangular slit in the $m\textrm{th}$ meta-atom at point M. According to the Huygens–Fresnel principle in two-dimensional space, ${E_{Mms}}$ can be expressed asWhen $\sigma ={-} 1$,
When ${\theta _{m1}} = a{\alpha _m} + b$ and ${\theta _{m3}} = c{\alpha _m} + d$ ($a$ and c are arbitrary integer values, b and d ${\in} [{ - \mathrm{\pi },\mathrm{\pi }} ]$), the electric field at any point $q\; (\rho ,\; \phi ,\; z$) in space is
2.2 Excitation with LP light
When the incident is LP light, the incident electric field can be expressed as ${{\mathbf E}_{{\mathbf{in}}}}$=${E_{\textrm{in}}}({\textrm{cos}\varphi ,\textrm{sin}\varphi } )$, where $\varphi $ is the ane between the optical electric field and the $\textrm{x}$-axis. When ${\theta _{m2}} = {\theta _{m1}} + \mathrm{\pi }/4$, ${\theta _{m4}} = {\theta _{m3}} + \mathrm{\pi }/4$ and ${r_2} = r$, the SPP electric field formed at point M by the $m$th meta-atom is
When ${\theta _{m1}} = a{\alpha _m} + b$ and ${\theta _{m3}} = a{\alpha _m} + d$, it then follows from Eq. (7) that
When $b$ and d are any fixed values, each meta-atom of the PVL generates the same SPP electric-field amplitude at point M. Finally, an SPP vortex field with a relatively uniform spatial distribution of electric field intensity is formed, and the electric field at any point $q(\rho ,\phi ,z$) in space can be expressed as
Therefore, irradiation by LP light of arbitrary polarization direction forms an SPP vortex with TC of $l = ({1 - 2a} )$ and the polarization information of the incident beam is transformed into the phase information of the formed SPP wave, which leads to changing the polarization direction $\varphi $ does not change the TC and electric-field strength of the SPP vortex.
When changing ${\theta _{m4}}$ from ${\theta _{m3}} + \mathrm{\pi }/4$ to ${\theta _{m3}} - \mathrm{\pi }/4$, the electric field of the SPP at point M by the $m\textrm{th}$ meta-atom is
When $a + c = 1$, it then follows from Eq. (10) that
Thus, the SPP vortex field is formed with $l = ({1 - 2a} )$. And because the polarization information of the incident beam is transformed into the amplitude information of the formed SPP wave, the polarization orientation of the incident LP light affects the amplitude of the electric field of the SPP vortex.
3. Simulation results and discussion
To verify the feasibility of the proposed method, we implemented a finite-difference time-domain (FDTD) simulation. For all simulations in this work, the Yee cell was taken to be a cube with sides of 5 nm, and perfectly matched layer (PML) boundary conditions were used on all boundaries. The designed PVL consists of 120 optimized meta-atoms (the angular interval between the slits in the same ring is $\mathrm{\pi }/60$) etched into a thin gold film supported on a silicon dioxide substrate, as shown in Fig. 1, where only some of the rectangular slits are displayed. The gold film is 200 nm thick and prevents the direct transmission of incident light. This paper uses an excitation wavelength of 532 nm as an example to verify the simulation. The plane wave is normal to the PVL and impinges on the silicon dioxide side to produce an SPP with ${\lambda _{\textrm{SPP}}} = 470\; \textrm{nm}$ propagating over the surface of the gold film. The permittivity of the gold film is $- 4.28 - \textrm{i}1.62$. Because the longitudinal component plays a leading role in the electric field of the SPP, the intensity $E_z^2$. of the longitudinal electric field is used to approximate the results for the electric field of the SPP. Finally, the simulation results come from 5 nm above the surface of the gold film.
Rectangular slits of the same size but different spatial orientations were etched into four concentric rings with the radii of ${r_s}$ at equal circumferential intervals, and the interval between each circle is ${\lambda _{\textrm{SPP}}}/4$. The radius of the innermost ring is ${r_1} = 2\; {\mathrm{\mu} \mathrm{m}}$. To avoid the intersection of rectangular slits arranged on four concentric rings, the slit length is set to be $L = 80\textrm{nm}$, and the slit width is $W = 20\; \textrm{nm}$.
3.1 CP light incidence
When CP light is incident on the PVL, the parameters b and d describing the slit rotation angle do not affect the final SPP vortex field, so we set $b = d = \mathrm{\pi }/4$. Figure 2 shows the resulting intensity and phase of the electric field of the SPP vortices generated by LCP light and RCP light when the parameters a and c differ. Figures 2(a) –2(c) show that, when tuning a and holding c fixed (or tuning c and holding a fixed), the TC of the SPP vortex excited by LCP (RCP) light changes whereas the electric field of the SPP vortex excited by LCP (RCP) light remains unchanged. By tuning both a and c simultaneously, the TCs of different SPP vortices combine upon excitation by LCP and RCP light. As shown in Fig. 2(d), when $a = 1$ and $c = 2.5$, a combination of TCs of ${l_R} ={-} 1\; $ and ${l_L} = 4\; $ is formed. When $a = 1.5$ and $c = 2$, a combination of TCs of ${l_R} ={-} 2\; $ and ${l_L} = 3\; $ is formed, as shown in Fig. 2(e). Therefore, these PVLs can excite SPP vortices with arbitrary combinations of TC under the CP light with different handedness by tunning a and $c$.
Fig. 2. Spatial distribution of electric-field intensity and phase of SPP vortex generated by LCP and RCP light incident on different PVLs: (a) $a ={-} 0.5\textrm{}({{l_R} = 2} )$, $c ={-} 2\textrm{}({{l_L} ={-} 5} )$; (b) $a ={-} 0.5\textrm{}({{l_R} = 2} )$, $c = 0.5\textrm{}({{l_L} = 0} )$; (c) $a = 2\textrm{}({{l_R} ={-} 3} )$, $c = 0.5\textrm{}({l_L} = 0)$; (d) $a = 1\textrm{}({{l_R} ={-} 1} )$, $c = 2.5\textrm{}({{l_L} = 4} )$; (e) $a = 1.5\textrm{}({{l_R} ={-} 2} )$, $c = 2\textrm{}({{l_L} = 3} )$.
3.2 LP light incidence
Consider $b\; = \; \mathrm{\pi }/4$ and $d\; = \; 3\mathrm{\pi }/4$, and take $a = 1$ as an example first. Figures 3(a) and 3(b) show the distribution of the intensity of the longitudinal electric field excited by different PVLs of ${\theta _{m4}} = {\theta _{m3}} + \mathrm{\pi }/4$ (PVLI) and ${\theta _{m4}} = {\theta _{m3}} - \mathrm{\pi }/4$ (PVLII), when the orientation of the electric field of the incident LP light is 0, $\mathrm{\pi }/4$, $\mathrm{\pi }/2$, and $3\mathrm{\pi }/4$. These simulation results show that these two PVLs form electric fields with zero electric-field amplitude at the center when excited by LP light of these orientations. The phase distribution diagram at the upper corner of every intensity result shows that both two PVLs can form a vortex field with $|l |= 1$, and varying the polarization orientation of the LP light does not affect the TC. As the azimuthal angle increases counterclockwise, the phase undergoes a transition from $\mathrm{\pi }$ to $-{-}\mathrm{\pi }$, resulting in the TC of $l ={-} 1$, consistent with $l = 1 - 2a$. A comparison between Figs. 3(a) and 3(b) reveals that the distinction between the two structures lies in their sensitivity to the polarization direction of the incident light. For PVLI, the intensity of the SPP vortex field it generates is unaffected by changes in the polarization direction of the incident light, marking it polarization-insensitive. Conversely, the intensity of the SPP electric field generated by PVLII varies when the polarization direction of the incident light changes by $\mathrm{\pi }/4$, which means it is polarization-sensitive. This behavior is more clearly demonstrated in Figs. 3(c) and 3(d), where it can be seen that altering the polarization direction of the incident light has no impact on the amplitude of the electric field of the SPP vortex field formed by PVLI, whereas the amplitude of the electric field of the SPP vortex formed by PVLII varies approximately sinusoidally with changes in the polarization orientation of the incident light. The phase distribution results also indicate that when the polarization direction of the incident light changes by $\mathrm{\pi }/4$, the zero-phase position of the vortex field formed by PVLI shifts by about $\mathrm{\pi }/4$, donated as $- \varDelta \varphi /l$. This further validates that PVLI can convert the polarization direction of the incident beam into the phase information of the resulting SPP field. In contrast, the zero-phase position of the vortex field formed by PVLII remains constant as the polarization direction of the incident beam changes, which further proves that PVLII can convert the polarization direction of the incident beam into the amplitude information of the resulting SPP field. Since the SPP vortices excited by PVLI and PVLII differ solely in intensity under identical excitation conditions, we have chosen to analyze one case (PVLI) in greater detail in the subsequent studies.
Fig. 3. The distributions of electric field intensity of SPP vortex generated by LP light of varying polarization direction incident on (a) PVLI and (b) PVLII both with $a = 1\; ({l ={-} 1} )$, $b\; = \mathrm{\;\ \pi }/4$, and $d\; = \; 3\mathrm{\pi }/4$. Panels I-IV in (a) and (b) are the electric field intensity distributions when $\varphi $ changes from 0 to $3\mathrm{\pi }/4$ in steps of $\mathrm{\pi }/4$. The insets at the upper and lower left corners reflect the phase distribution of the SPP field and polarization state of the incident beam, respectively. The simulated amplitude of the SPP vortex field generated by the (c) PVLI and (d) PVLII as a function of the value of $\varphi $ (squares). The circles in (d) show a standard sine curve.
Figure 4 shows the spatial distribution of the intensity (phase) of the longitudinal electric field generated by the PVLIs, which are composed of slits with varying dimensions: 80 nm${\times} $10 nm, 80 nm${\times} $20 nm, and 85 nm${\times} $35 nm, when illuminated by the LP light with $\varphi = 0$. To avoid the intersection of rectangular slits, the number of meta-atoms in the third structure was reduced from 120 to 80. It can be seen from the phase results that all three structures can produce an SPP vortex with $l ={-} 1$. From the results of electric field intensity, it is found that with the increase of the slit size, the intensity of the SPP vortex also increases, that is, the conversion efficiency from incident light to SPP improves. Especially when the length of the slit remains at 80 nm, and the width is increased from 10 nm to 20 nm, the intensity experiences a nearly 75-fold increase. As the size is further increased, the rate of conversion efficiency growth slows down. The larger size is also helpful for processing and manufacturing. Moreover, a comparison of the intensity results reveals that the reduction in the aspect ratio of the slit due to an increase in size ultimately results in some non-uniformity in the SPP vortex. This is because the reduction of the aspect ratio of the slit transforms the slit from being equivalent to a single dipole into a superposition of two dipoles with orthogonal vibration directions. Therefore, the slit with dimensions of 80 nm${\times} $20 nm is chosen in our work.
Fig. 4. Spatial distribution of electric-field intensity of SPP vortex generated by LP light with $\varphi = 0$ incident on different PVLIs with $a = 1\textrm{}\; ({l = 0} ),\textrm{}b\textrm{} = \textrm{}\pi /4$, $d\textrm{} = \textrm{}3\pi /4$ composed of slits with different dimensions: (a) $L = 80\textrm{nm},\; W = 10\; \textrm{nm}$; (b) $L = 80\; \textrm{nm},\; W = 20\; \textrm{nm}$; (c) $L = 85\; \textrm{nm},\; W = 35\; \textrm{nm}$. (d)–(f) Spatial distribution of the phase of electric field corresponding to panels (a)–(c). The polarization state of the incident beam is given in the lower-left corner of panels (a)–(c).
Figures 5(a)–5(e) [5(f)–5(j)] show the spatial distribution of the intensity (phase) of the longitudinal electric field generated by the PVLI with $b = \mathrm{\;\ \pi }/4\textrm{}$ and $d = \textrm{}3\mathrm{\pi }/4$ for several different values of a. As can be seen from the results of phase, when the parameter a. varies from 0.5 to 2.5 in increments of 0.5, the SPP field transitions from a focus to a vortex, with l changing from 0 to −4. Additionally, the results for electric field intensity reveal that the radius of the uniformly distributed SPP vortex field increases with the increment of TC. It's worth noting that fractional-order SPP vortices, which can be seen as the superposition of integer vortices, can also be achieved by selecting non-integer values for a. As an example, we consider $a = 1.25$ ($l ={-} 1.5$). When LP light with polarization directions $\varphi = 0$, $\mathrm{\pi }/4$, $\mathrm{\pi }/2$, and $3\mathrm{\pi }/4$ are respectively incident on the PVLI, the spatial distributions of the genated SPP are shown in Fig. 6. These distributions exhibit two petal-shaped bright spots with high intensity and a circular dark spot, forming a bright, irregularly shaped ring with uneven brightness. Below the bright ring, an arc spot is also present. All these patterns align with the characteristics of the fractional-order vortex with $l ={-} 1.5$ as described in [46]. It is apparent that the spatial distribution and intensity of the excited SPP field remain almost constant with the change of the polarization directions of the incident light. This suggests that the PVLI, used to generate fractional-order SPP vortices, possesses the same polarization insensitivity as the PVLI employed for generating integer-order SPP vortices. Therefore, the control of integer or fractional TCs of the SPP vortices can be achieved by adjusting the parameter a.
Fig. 5. Spatial distribution of electric-field intensity of SPP vortex generated by LP light with $\varphi = 0$ incident on different PVLIs with $b\; = \mathrm{\;\ \;\ \pi }/4$, $d\; = \; 3\mathrm{\pi }/4$, and (a) $a = 0.5\; ({l = 0} )$, (b) $a = 1\; ({l ={-} 1} )$, (c) $a = 1.5\; ({l ={-} 2} )$, (d) $a = 2\; ({l ={-} 3} )$, (e) $a = 2.5\; ({l ={-} 4} )$. (f)–(j) Spatial distribution of the phase of electric field corresponding to panels (a)–(e). The polarization state of the incident beam is given in the lower-left corner of panels (a)–(e).
Fig. 6. Spatial distribution of electric-field intensity of SPP vortex generated by LP light of varying polarization direction incident on the PVLI with $a = 1.25\; ({l ={-} 1.5} )$, $b\; = \; \mathrm{\pi }/4$, and $d\; = \textrm{}3\mathrm{\pi }/4$: (a) $\varphi = 0$, (b) $\varphi = \mathrm{\pi }/4$, (c) $\varphi = \mathrm{\pi }/2$, (d) $\varphi = 3\mathrm{\pi }/4$. The polarization state of the incident beam is given in the lower-left corner of panels (a)–(d).
While the values of b and d will not affect the TC of the SPP vortex formed by the PVL, they do have the capability to alter the amplitude of the vortex field. Take PVLI with $a = 1$ and $b\textrm{} = \; \mathrm{\pi }/4$ as an example to illustrate. Figures 7(a)–7(d) [7(e)–7(h)] show the spatial distribution of the intensity (phase) of the SPP field when the LP light with $\varphi = 0$ illuminates PVLIs with $d$ = $5\mathrm{\pi }/12$, $\mathrm{\pi }/2$, $7\mathrm{\pi }/12$, and $3\mathrm{\pi }/4\textrm{}$ respectively. From the phase results, it is found that when d changes, the zero-phase position of the vortex electric field undergoes a rotation of $\varDelta d/l$. . This indicates that the values of the structural parameters $b\; $ and d of the PVL can be transferred to the phase of the SPP vortex field. In addition, it can be s.en from the intensity results that changing d also varies the intensity of the electric field of the SPP vortex. This shows that besides the phase information, b and d can also be transferred to the amplitude information of the SPP vortex field. To present the change in amplitude of the electric field of the SPP vortex more accurately and intuitively, Fig. 7(i) plots the amplitude of the electric field of the SPP vortex generated by PVLs with differences $b - d$ at intervals of $\mathrm{\pi }/12$. The result shows that the amplitude variation is almost a standard sine function. Therefore, fixing a and tuning the difference $b - d$ changes the electric-field amplitude wiout changing the TC of the SPP vortex field. It also can be observed that adjusting d has a synchronous impact on the intensity of the main lobe and sidelobes of the SPP field. The reason is that the spatial distribution of the SPP electric field generated by the PVL conforms to the Bessel function of the first kind. By applying Fourier transformation, the far-field Bessel beam can be transformed into a perfect vortex beam, allowing for the suppression of sidelobes [47–49]. This provides inspiration for our future work aimed at sidelobe suppression in SPP vortices.
Fig. 7. Spatial distribution of the electric-field amplitude of SPP vortex generated by LP light with $\varphi = 0$ incident on different PVLIs with $a\; = \; 1\; ({l ={-} 1} )$, $b\; = \; \mathrm{\pi }/4$, and (a) $d = 5\mathrm{\pi }/12$, (b) $d = \mathrm{\pi }/2$, (c) $d = 7\mathrm{\pi }/12$, (d) $d = 3\mathrm{\pi }/4$. (e)–(h) Spatial distribution of the phase of electric field corresponding to panels (a)–(d). (i) The amplitude of the SPP vortex field generated by PVLs as a function of the absolute value of the difference $b - d$ (black squares). Red circles show the standard sine curve. The polarization state of the incident beam is given in the lower-left corner of panels (a)–(d).
The polarization information of the incident light can be converted into phase information of the SPP vortex field by PVLI, so when the incident light is converted from a pure linearly biased scalar optical field to a vector beam with different polarization orientations at each point in space [the incident optical field can be expressed as ${{\mathbf E}_{{\mathbf{in}}}} = {E_{\textrm{in}}}({\textrm{cos}({p\alpha + {\alpha_0}} ),\; \textrm{sin}({p\alpha + {\alpha_0}} )} )$, where p is an arbitrary integer that indicates the polarization order of the incident vector optical field, $\alpha $ is the azimuth angle, and ${\alpha _0}$ is the initial polarization orientation when $\alpha = 0$], then the electric field of any point $q\; (\rho ,\phi ,z$) in space can be expressed as
Thus, the SPP vortex is formed with $l = ({1 - 2a + p} )$. Consider a PVLI with $a = 0.5$, $b = \mathrm{\;\ \pi }/4,$ and $d = \textrm{}3\mathrm{\pi }/4$ as an example. Figures 8(a)–8(d) [8(e)–8(h)] show the results of the intensity (phase) of the SPP field when the initial polarization direction ${\alpha _0} = 0$ and the order $p = 0$, 1, 2, and 3, respectively. When the polarization order of the incident vector beam changes, the TC of the electric field of the SPP vortex also changes accordingly. The result of phase-distribution shows that the phase changes from $- \mathrm{\pi }$ to $\mathrm{\pi }$ as the azimuthal angle increases counterclockwise, so the TC should be $l = 0$, 1, 2, and 3, which verifies the correctness of the theory. Therefore, the TC of the SPP vortex can be altered by controlling the polarization order of the incident vector optical field, which may be used to dynamically modulate the TC of the electric field of SPP vortices. Moreover, changing ${\alpha _0}$ does not affect the TC or the electric-field intensity of the SPP vortex but only shifts the zero-phase position of the SPP vortex (see Fig. S1, Supplement 1).
Fig. 8. Spatial distribution of electric-field intensity of SPP vortex generated by vector beams of varying polarization order and same ${\alpha _{0\textrm{}}}({{\alpha_0} = 0} )$ incident on the PVLI with $a = 1$, $b\; = \mathrm{\;\ \pi }/4$, and $d\; = \; 3\mathrm{\pi }/4$: (a) $p = 0\; ({l = 0} )$, (b) $p = 1\; ({l = 1} )$, (c) $p = 2\; ({l = 2} )$, (d) $p = 3\; ({l = 3} )$. (e)–(h) Spatial distribution of the phase of the electric field corresponding to panels (a)–(d). The polarization state of the incident beam is given in the lower-left corner of panels (a)–(d).
In summary, the TC of the SPP vortex can be controlled by tuning a and c under the excitation of either CP or LP light; the state of the intensity of the vortex field, whether it changes or remains constant with varying polarization directions of the LP light, depends on ${\theta _{m4}}$; the specific intensity of the SPP vortex excited by LP light is determined by the difference of b and d. Table 2 summarizes the performance of our proposed PVLs composed of meta-atoms with four rectangular slits of the same size and regular rotation angle and compares them with some classical devices for manipulating plasmonic vortices. In comparison with previous devices, our PVLs not only eliminate the need for introducing the transmission phase and complex optimization processes but also offer more comprehensive functions.
Table 2. Properties comparison of different PVLs for manipulating plasmonic vortex
4. Conclusion
In conclusion, we propose PVLs, solely based on the geometric phase, made up of meta-atoms comprising four rectangular slits with varying rotation angles, large aspect ratios, and the same size. These PVLs not only simplify the optimization process of designing the structure compared with the traditional PVL solely based on geometric phase but also allow the manipulation of the SPP vortex field excited by light with various polarizations. These PVLs not only have the capability to freely combine SPP-vortex TCs by irradiated with CP light with different handedness but also can form SPP vortices with different TCs and specific constant or changing electric-field intensity under the excitation of the LP light for the first time only using geometric phase. Moreover, when the incident optical field changes from an LP scalar optical field to a vector beam with varying polarization directions at different points in space, the TC of the resulting SPP vortex varies upon changing the polarization order of the vector beam, which can be used to dynamically modulate the TC of the SPP vortex field and detect the polarization order of the incident vector beam. The design principle based solely on geometric phase grants these PVLs distinct advantages in terms of device miniaturization and achieving a more uniform vortex field.
Funding
Ministry of Education collaborative project (B17023); Fundamental Research Funds for the Central Universities (30920010011); National Natural Science Foundation of China (61627802); Beijing Municipal Natural Science Foundation (Z190018); National Key Scientific Instrument and Equipment Development Projects of China (61827814).
Acknowledgments
This work was supported by the National Key Scientific Instrument and Equipment Development Projects of China (61827814), Beijing Municipal Natural Science Foundation (Z190018), National Natural Science Foundation of China (No. 61627802), Fundamental Research Funds for the Central Universities (No. 30920010011), Ministry of Education collaborative project (B17023), Nanjing University of Science and Technology.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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