Vortex Beam Encoded All-Optical Logic Gates Based on Nano-Ring Plasmonic Antennas

Vortex beam encoded all-optical logic gates are suggested to be very important in future information processing. However, within current logic devices, only a few are encoded by using vortex beams and, in these devices, some space optical elements with big footprints (mirror, dove prism and pentaprism) are indispensable components, which is not conducive to device integration. In this paper, an integrated vortex beam encoded all-optical logic gate based on a nano-ring plasmonic antenna is proposed. In our scheme, by defining the two circular polarization states of the input vortex beams as the input logic states and the normalized intensity of the plasmonic field at the center of the nano-ring as the output logic states, OR and AND (NOR and NAND) logic gates are realized when two 1st (1st) order vortex beams are chosen as the two input signals; and a NOT logic gate is obtained when one 1st order vortex beam is chosen as the input signal. In addition, by defining the two linear polarization states (x and y polarization) of the input vortex beams as the two input logic states, an XNOR logic gate is realized when two 1st order vortex beams are chosen as the two input signals.


Introduction
In the era of big data, the exchange of network data can only match the large capacity of data transmission if ultra-high speed is achieved. All-optical computing is an effective way to improve the data calculation and the rate of data exchange [1]. All-optical logic gates are key elements in all-optical computing and optical circuits [2]. Therefore, many schemes for optical logic gates have been proposed. These optical logic gates can be divided into two categories: one is based on nonlinear optics [3][4][5][6], and the other is based on linear coherence [7][8][9]. In comparison, the nonlinear optics-based optical logic gates depend on the small nonlinear susceptibility of conventional materials so that intense light power is needed, which is an obstacle for practical applications. Linear interference logic operation is achieved by the destructive or constructive interference of the two input signals in which the relative phase difference of the two input signals plays the key role, hence it can be realized under very low light power, even under single photon level [10]. In recent years, linear interference logic devices have aroused great interest among researchers.
In all-optical logic gates, the two input signals are two orthogonal light modes. In linear interference schemes, two path modes that propagate along different paths are usually used as the two input signal lights. Examples include the two counter-propagating incident beams in the coherent

Interactions between Light and Surface Plasmon Polaritons (SPPs) on Nano-Ring Plasmonic Antennas
To show our scheme in detail, we first give the theory of the interactions between light and surface plasmon polaritons (SPPs) on nano-ring plasmonic antennas. A nano-ring plasmonic antenna is a structure that allows us to derive an analytical expression for the plasmonic focal field, which is well studied in [40]. Our following description of the theory follows the derivation process of [40]. The 3D drawing and the top view of the nano-ring plasmonic antenna and the coordinates for our calculation are illustrated in Figure 1. A single ring slot is etched into a thin metal film deposited on silica substrate.
To fabricate this device, one should first plate the metal film on the silica substrate, then fabricate the ring slot by using focused ion beam lithography. Light illuminates the structure normally from the silica substrate side. Considering the incident light possessing both spin angular momentum (SAM) with spin quantum number σ and OAM with topological charge l, it can be expressed in cylindrical coordinates as where i = (−1) 1/2 is the imaginary unit, → e r and → e ϕ are the radial unit vector and angular unit vector respectively. According to the excitation conditions of SPPs, when the slot is sufficiently narrow, only the radial component of the incident light can couple to the SPPs. Thus, the SPPs are exited along an incremental length of the annular slot at point (r 0 , ϕ) which contributes to the plasmonic field at an observation point (R, θ) as a secondary source to generate a field increment given by where E 0 is a constant that is related to the coupling efficiency from the incident light to the SPPs, k r is the wave vector of the SPPs that propagate in the metal plane, and k z is the wave vector of the SPPs that propagate in the z-direction. In this paper, for simplicity, we ignore the propagation loss of the SPPs propagating in the metal plane, hence k r = 2π/λ SPP with λ SPP being the wave length of the plasma wave. If the observation point is near the origin point, we have R r 0 (for example R ≤ 2λ SPP when r 0 = 10λ SPP ). In this case it can be obtained that Hence, the total plasmonic field at the observation point is given by → E SPP (R, θ) = → e z E 0 e −k z z e ik r r 0 r 0 2π 0 e i(l+σ)ϕ e −ik r R cos (θ−ϕ) dϕ = → e z 2πi (l+σ) E 0 r 0 e −k z z e ik r r 0 J l+σ (−k r R)e i(l+σ)θ . (4) normally from the silica substrate side. Considering the incident light possessing both spin angular momentum (SAM) with spin quantum number  and OAM with topological charge l, it can be expressed in cylindrical coordinates as where 0 E is a constant that is related to the coupling efficiency from the incident light to the SPPs, r k is the wave vector of the SPPs that propagate in the metal plane, and z k is the wave vector of the SPPs that propagate in the z-direction. In this paper, for simplicity, we ignore the propagation loss of the SPPs propagating in the metal plane, hence 2 / ). In this case it can be obtained that Hence, the total plasmonic field at the observation point is given by Using Equation (4), we can calculate the intensity of the plasmonic field near the center of the nano-ring. It is determined by the total angular momentum. Here we should note that the above theoretical results are general. They will not be restricted by the particular material and thickness of the metal film, slot width, and light wavelength. Although these parameters could affect the coupling efficiency from the incident vortex beam to the SPPs and the propagation loss of the SPPs propagating in the metal plane, their effects can be condensed into the amplitude 0 E . Hence, they do not affect the intensity distributions normalized by

Design and Discussion
In this section, according to the above theory derivation, we will show how to construct alloptical logic gates by using vortex beams as the input signals and the different intensity of the SPPs at the center point of the nano-ring as the output logic states. Using Equation (4), we can calculate the intensity of the plasmonic field near the center of the nano-ring. It is determined by the total angular momentum. Here we should note that the above theoretical results are general. They will not be restricted by the particular material and thickness of the metal film, slot width, and light wavelength. Although these parameters could affect the coupling efficiency from the incident vortex beam to the SPPs and the propagation loss of the SPPs propagating in the metal plane, their effects can be condensed into the amplitude E 0 . Hence, they do not affect the intensity distributions normalized by |E 0 | 2 .

Design and Discussion
In this section, according to the above theory derivation, we will show how to construct all-optical logic gates by using vortex beams as the input signals and the different intensity of the SPPs at the center point of the nano-ring as the output logic states.

OR and AND Logic Gates
To realize OR and AND logic gates, two 1st order OAM beams (with l = −1) with the same amplitudes and initial phases are chosen as the two input signals. Similar to [21], the circular polarization states of the input OAM beams are utilized to denote the two input logic states. Right circular polarization (RCP) with σ = −1 and left circular polarization (LCP) with σ = 1 are defined as the input states "0" and "1" respectively. Under these definitions, the intensity of the plasmonic field near the center of the nano-ring for four input states "11" (two RCP incidences), "10/01" (one RCP and one LCP incidences) and "00" (two LCP incidences) can be calculated. The results are shown in Figure 2, where (a), (b) and (c) correspond to the input logic states "11", "10/01", and "00" respectively. It is noted that the results are normalized by the intensity of the center point of the nano-ring under the "11" input state. Extracting the normalized intensity of the center point under the four input logic states and putting them into Table 1, it can be seen that the OR logic gate is obtained by setting the relative intensity threshold within the range of 0-0.25 and using the intensity lower and higher than the threshold to denote the two output logic states "0" and "1" respectively, and the AND logic gate is obtained by setting the relative intensity threshold within the range of 0.25-1. Nanomaterials 2019, 9, x FOR PEER REVIEW 4 of 9

OR and AND Logic Gates
To realize OR and AND logic gates, two 1st order OAM beams (with l = −1) with the same amplitudes and initial phases are chosen as the two input signals. Similar to [21], the circular polarization states of the input OAM beams are utilized to denote the two input logic states. Right circular polarization (RCP) with 1    and left circular polarization (LCP) with 1   are defined as the input states "0" and "1" respectively. Under these definitions, the intensity of the plasmonic field near the center of the nano-ring for four input states "11" (two RCP incidences), "10/01" (one RCP and one LCP incidences) and "00" (two LCP incidences) can be calculated. The results are shown in Figure 2, where (a), (b) and (c) correspond to the input logic states "11", "10/01", and "00" respectively. It is noted that the results are normalized by the intensity of the center point of the nanoring under the "11" input state. Extracting the normalized intensity of the center point under the four input logic states and putting them into Table 1, it can be seen that the OR logic gate is obtained by setting the relative intensity threshold within the range of 0-0.25 and using the intensity lower and higher than the threshold to denote the two output logic states "0" and "1" respectively, and the AND logic gate is obtained by setting the relative intensity threshold within the range of 0.25-1.
The physical mechanism can be explained as follows: under the excitation, the plasmonic field near the center of the nano-ring is (  The normalized intensity of the plasmonic field near the center of the nano-ring for (a) input logic states "11", (b) input logic states "10/01", and (c) input logic states "00" in the realization scheme of OR and AND logic gates.

NOT Logic Gate
The NOT gate has only one input. To realize the NOT gate, the 1st order OAM beam (with l = 1) is chosen as the input signal. Similarly, the circular polarization states RCP (with 1    ) and LCP (with 1   ) are defined as the input logic states "0" and "1" respectively. The calculated intensity of the plasmonic field near the center of the nano-ring for the two input states is shown in Figure 3, Figure 2. The normalized intensity of the plasmonic field near the center of the nano-ring for (a) input logic states "11", (b) input logic states "10/01", and (c) input logic states "00" in the realization scheme of OR and AND logic gates. The physical mechanism can be explained as follows: under the excitation, the plasmonic field near the center of the nano-ring is (l + σ)th order vortex. Under l = −1, the plasmonic field resulting from the "0" input state of an input signal is − → e z 2πE 0 r 0 e −k z z e ik r r 0 J −2 (−k r R)e −i2θ . It is a 2nd order vortex whose intensity at the center point is zero. The plasmonic field resulting from "1" input state of an input signal is then → e z 2πE 0 r 0 e −k z z e ik r r 0 J 0 (−k r R). It contributes to the plasmonic field of the center point by → F = → e z 2πE 0 r 0 e −k z z e ik r r 0 J 0 (0). Since the excitation of SPPs is a linear process, the center point plasmonic field should be 2 → F when both the two input states are "1". Therefore, the intensity of the plasmonic field at the center point under input states "11", "10/01", and "00" are respective 4|F| 2 , |F| 2 and 0, corresponding to normalized intensity 1, 0.25 and 0, respectively.

NOT Logic Gate
The NOT gate has only one input. To realize the NOT gate, the 1st order OAM beam (with l = 1) is chosen as the input signal. Similarly, the circular polarization states RCP (with σ = −1) and LCP (with σ = 1) are defined as the input logic states "0" and "1" respectively. The calculated intensity of the plasmonic field near the center of the nano-ring for the two input states is shown in Figure 3, which is normalized by the intensity of the center point of the nano-ring under the "0" input state. Figure 3a,b are the results corresponding to the input logic states "0" and "1", respectively. The normalized intensity of the center point under the two input logic states are extracted out and put into Table 2. It can be seen that if denoting the two output logic states "0" and "1" with normalized intensity 0 and 1, NOT logic gate is obtained. The physical mechanism is that under l = 1, the plasmonic field resulting from "1" input state is a 2nd order vortex (l + σ = 1 + 1 = 2) whose intensity at the center point is zero, while plasmonic field resulting from "0" input state is → e z 2πE 0 r 0 e −k z z e ik r r 0 J 0 (−k r R), which contributes to the plasmonic field of the center point by → F = → e z 2πE 0 r 0 e −k z z e ik r r 0 J 0 (0). Thus, through normalized by |F| 2 , the normalized intensity of the center point corresponding to the input logic states "0" and "1" are 1 and 0, respectively. Nanomaterials 2019, 9, x FOR PEER REVIEW 5 of 9 which is normalized by the intensity of the center point of the nano-ring under the "0" input state. Figure 3a,b are the results corresponding to the input logic states "0" and "1", respectively. The normalized intensity of the center point under the two input logic states are extracted out and put into Table 2. It can be seen that if denoting the two output logic states "0" and "1" with normalized intensity 0 and 1, NOT logic gate is obtained. The physical mechanism is that under l = 1, the plasmonic field resulting from "1" input state is a 2nd order vortex (

NOR and NAND Logic Gates
To realize NOR and NAND logic gates, two 1st order OAM beams (with l = 1) with the same amplitudes and initial phases are chosen as the two input signals. The circular polarization states RCP (with 1    ) and LCP (with 1   ) are defined as the input logic states "0" and "1" respectively. The calculated intensity of the plasmonic field near the center of the nano-ring for four input logic states "11" "10/01", and "00" is given in Figure 4, which is normalized by the intensity of the center point of the nano-ring under the "00" input state. Figure 4a-c correspond to the input logic states "00", "10/01", and "11" respectively. The normalized intensity of the center point under the four input logic states is extracted out and put into Table 3. It can be seen that the NOR logic gate is obtained by setting the relative intensity threshold within the range of 0.25-1 and using the intensity lower and higher than the threshold to denote the two output logic states "0" and "1" respectively, and the NAND logic gate is obtained by setting the relative intensity threshold within the range of 0-0.25. Similarly, the physical mechanism here is that under l = 1, the plasmonic field resulting from the "1" input state is a 2nd order vortex ( . So, the plasmonic field at the center point under input states "00", "10/01", and "11" are respective 2 F r , F r , and 0, corresponding to normalized intensity 1, 0.25, and 0, respectively.

NOR and NAND Logic Gates
To realize NOR and NAND logic gates, two 1st order OAM beams (with l = 1) with the same amplitudes and initial phases are chosen as the two input signals. The circular polarization states RCP (with σ = −1) and LCP (with σ = 1) are defined as the input logic states "0" and "1" respectively. The calculated intensity of the plasmonic field near the center of the nano-ring for four input logic states "11" "10/01", and "00" is given in Figure 4, which is normalized by the intensity of the center point of the nano-ring under the "00" input state. Figure 4a-c correspond to the input logic states "00", "10/01", and "11" respectively. The normalized intensity of the center point under the four input logic states is extracted out and put into Table 3. It can be seen that the NOR logic gate is obtained by setting the relative intensity threshold within the range of 0.25-1 and using the intensity lower and higher than the threshold to denote the two output logic states "0" and "1" respectively, and the NAND logic gate is obtained by setting the relative intensity threshold within the range of 0-0.25.
Similarly, the physical mechanism here is that under l = 1, the plasmonic field resulting from the "1" input state is a 2nd order vortex (l + σ = 1 + 1 = 2) whose intensity at the center point is zero, and the plasmonic field resulting from the "0" input state of an input signal is → e z 2πE 0 r 0 e −k z z e ik r r 0 J 0 (−k r R), which contributes to the plasmonic field of the center point by → F = → e z 2πE 0 r 0 e −k z z e ik r r 0 J 0 (0). So, the plasmonic Nanomaterials 2019, 9, 1649 6 of 9 field at the center point under input states "00", "10/01", and "11" are respective 2 → F , → F , and 0, corresponding to normalized intensity 1, 0.25, and 0, respectively. Nanomaterials 2019, 9, x FOR PEER REVIEW 6 of 9 Figure 4. The normalized intensity of the plasmonic field near the center of the nano-ring for (a) input logic states "00", (b) input logic states "10/01", and (c) input logic states "11" in the realization scheme of NOR and NAND logic gates. Table 3. NOR and NAND logic gates.

XNOR Logic Gate
To realize the XNOR logic gate, two 1st order OAM beams (with l = 1) with the same amplitudes and initial phases are chosen as the two input signals. Two linear polarization states are used to represent the two input logic states. The x axis linear polarization denotes input logic state "0" and the y axis linear polarization denotes input logic state "1". These two linear polarization states can be x y e e e  and ê  are the unit vectors of x axis linear polarization, y axis linear polarization, LCP and RCP respectively. Thus, we can still calculate the intensity of the plasmonic field near the center of the nano-ring for the four input logic states "11" "10/01", and "00" according to Equation (4). The results are shown in Figure 5, which are normalized by the intensity of the center point of the nano-ring under the "00" input state. Figure 5a-c correspond to the input logic states "00", "10/01", and "11", respectively. The normalized intensity of the center point under the four input logic states are extracted out and put into Table 4. It can be seen that the XNOR logic gate is obtained if setting the relative intensity threshold within the range of 0.5-1 and using the intensity lower and higher than the threshold to denote the two output logic states "0" and "1" respectively. Figure 4. The normalized intensity of the plasmonic field near the center of the nano-ring for (a) input logic states "00", (b) input logic states "10/01", and (c) input logic states "11" in the realization scheme of NOR and NAND logic gates.

XNOR Logic Gate
To realize the XNOR logic gate, two 1st order OAM beams (with l = 1) with the same amplitudes and initial phases are chosen as the two input signals. Two linear polarization states are used to represent the two input logic states. The x axis linear polarization denotes input logic state "0" and the y axis linear polarization denotes input logic state "1". These two linear polarization states can be expanded as superpositions of the two circularly polarization states, i.e.,ê x = (ê + +ê − )/ √ 2 and e y = −i(ê + −ê − )/ √ 2, whereê x ,ê y ,ê + andê − are the unit vectors of x axis linear polarization, y axis linear polarization, LCP and RCP respectively. Thus, we can still calculate the intensity of the plasmonic field near the center of the nano-ring for the four input logic states "11" "10/01", and "00" according to Equation (4). The results are shown in Figure 5, which are normalized by the intensity of the center point of the nano-ring under the "00" input state. Figure 5a-c correspond to the input logic states "00", "10/01", and "11", respectively. The normalized intensity of the center point under the four input logic states are extracted out and put into Table 4. It can be seen that the XNOR logic gate is obtained if setting the relative intensity threshold within the range of 0.5-1 and using the intensity lower and higher than the threshold to denote the two output logic states "0" and "1" respectively. of the center point of the nano-ring under the "00" input state. Figure 5a-c correspond to the input logic states "00", "10/01", and "11", respectively. The normalized intensity of the center point under the four input logic states are extracted out and put into Table 4. It can be seen that the XNOR logic gate is obtained if setting the relative intensity threshold within the range of 0.5-1 and using the intensity lower and higher than the threshold to denote the two output logic states "0" and "1" respectively. Figure 5. The normalized intensity of the plasmonic field near the center of the nano-ring for (a) input logic states "00", (b) input logic states "10/01", and (c) input logic states "11" in the realization scheme of XNOR logic gate.

Conclusions
In conclusion, we have put forward a vortex beams encoded scheme to achieve all-optical logic gates based on nano-ring plasmonic antennas. OR, AND, NOT, NOR, NAND and XNOR logic gates are designed and discussed. Since our scheme is compatible with vortex beams, it may have potential applications in vortex beam-based all-optical computing. In addition, since the size of the nano-ring plasmonic antenna can be designed on the scale of several wavelengths of the plasma wave, our scheme is very suitable for device miniaturization and integration.