Design and Analysis of a Complete Full Adder Based on Metal-Insulator-Metal (MIM) Waveguide-Based Plasmonic Waves

In this paper, metal-insulator-metal (MIM) plasmonic waveguide structures and a rectangular cavity resonator at a central frequency of 1550 nm were used to propose a complete full adder. Under this circumstances, the system has a fast function with slight variations in real- time or near real-time manner, and this led to its minimum power consumption, while serving in various situations. In this full adder, we benefited from the property of combining resonant waves in the first and second modes, and we managed to obtain a high transmission coefficient in states where the output must be active. This complete full adder operates through designing 4-input AND, XOR, OR, and NOT logic gates, resulting in the design of a complete full adder with low manufacturing complexity and cost relative to ones designed through combining the conventional 2-input AND and OR gates. In comparison of three computational methods, finite ‐ difference time ‐ domain (FDTD) is a simple and versatile method. This method directly discretizes the time ‐ domain partial differential form of Maxwell's equations in various dimensions while using analytical solution in the remaining direction and solving the 3D scattering problem. Therefore, necessary simulations were conducted using FDTD software, and showed a good fit to the results predicted through approximations intended for theoretical relations.


Introduction
Control of light using light is the most fundamental topic in all-optical integrated circuits, which have many applications in all-optical telecommunication networksb [6,7]. Extensive research has been conducted on optical computing since the 1980s [8]. However, the intensity of the research activities decreased due to some limitations of materials used in the manufacture of optical chips that prevented the manufacture of small and inexpensive optical chips for laboratory researches. The main reason for this is the problem of optical diffraction in optical devices that could be overcomed using plasmonic waves [12,13]. With the advent of plasmonic structures as well as the approach of technology towards the integrity of optoelectronic circuits, manufacturing problems and phenomena that helped prevent further compression of the structure led to the study and use of plasmonic structures and plasmonic waves. In very small sizes lower than the wavelengths, surface plasmons provide a suitable basis for the realization and manufacture of all-optical devices due to the increased intensity of concentrated optical fields. MIM waveguides are very important structures in plasmonic devices due to guiding surface plasmons in the crosssection of metal-dielectric structure [16,18,29,30]. Because these waveguides not only support the propagation of modes with very small wavelengths and high group velocities, but also show their ability to guide the wave up to relatively long distances. Combination of these waveguides with nano-resonators in various shapes, which are side-coupled to them, creates a variety of new structures, which have many applications in all-optical devices [1,2]. The transmission medium in optical networks is the optical fiber; and the wavelength band, which can be used to transmit information, is 1550 nm with minimal losses in the third optical telecommunication window. Therefore, the full adder for a wavelength of 1550 nm (C-Band: 1530-1565nm) will be designed [6,31,32]. One of the major problems in the expansion of optical networks is the limitation of diffraction in the limited size of optical devices due to the wavelength of light transmitted by the network, which increases the sizes of optical devices relative to those of electronic devices. The diffraction limit can be broken using plasmonics [7]. Diffraction is a phenomenon, which occurs when a wave encounters an obstacle or a slit, and which is defined as the bending of (electromagnetic) waves around the corners of an obstacle or aperture and their propagation into the region of geometrical shadow of the obstacle. Using optical devices, like silicon photonic devices, it causes other problems such as cases where the Pockels constant equals zero. Although effects such as the thermo-optic effect and dispersion effect of free carriers have been dynamically employed to control the optical properties of silicon, these technologies have adverse effects on velocity, losses, and so on. It is very difficult to integrate high-speed optical devices using silicon technology [28]. [Palik, 1977] Using plasmonic waveguides offers a new opportunity to integrate electro-optic polymers in high-speed optical technology [8].
The term plasmonic has been explained based on the process of interactions between electromagnetic waves and conduction electrons in metals with nanoscale dimensions. A lot of plasmonic waveguides and optical equipment, based on the propagation of surface plasmons, have been presented in recent studies [3,4,21]. All-optical logic gates with plasmonic waveguide structures have been introduced and simulated in [3,21,23] using a waveguide Y-branch splitter, but the theory of the problem has not been addressed in it, and the optical power loss is very high in this structure as well. In an all-optical integrated circuit, it is necessary to have several logic gates working in series, and if each gate causes a lot of loss, the practical application will not be possible. In a recent work [4], a photonic crystal waveguide and the nonlinear properties of optical materials have been used to design all-optical logic gates. Although photonic crystal structures and non-linear light have important applications, in comparison to plasmonic structures, they have major disadvantages such as enlarged size of pieces and high power losses for logic gates. Kaboli and Akhlaghi [5] have benefited from full absorption in periodic plasmonic nanoparticles and the operation of pumping optical power to these nanoparticles to modify the output, and to create an AND logic gate. Although their gate is a new design, it has a very high manufacturing complexity for a simple logic gate, the size of the piece is large, and the power consumption is high due to the permanent pumping of optical power. In [6], the use of plasmonic waveguides and slot cavity resonators have been proposed for basic logic gates, and has used transmission coefficients for the amplitude of waves in waveguides.
Different sections of this paper are as follows: In Section 2, numerical methods and Disadvantage of MPB software package already used are expressed. Section 3 will extend the methodology. Section 4 discuses on the theoretical equations of plasmonic structures, which theoretical equations related to plasmonic structures will be described and the basic structure for designing the adder explained. Then, mathematical relations for the electric permittivity coefficient in a plasmonic structure, as well as the wave equation will be explained. Sections 5 and 6 present the results and discussion on OR logic gate and the conclusion, respectively.

Numerical Methods
Computational Electromagnetics provide an in-depth introduction of the three main fullwave numerical methods in computational electromagnetics (CEM); namely, the method of moment (MoM), the finite element method (FEM), and the finite-difference timedomain (FDTD) method. One of the most popular computational methods for solving too large to handle problems in finite element packages is FDTD (finite difference time domain), which gets around having to invert a large linear system by instead mimicking Maxwell's Equations in the time domain. By implementing a uniform mesh, the overhead associated with storing the location of each point is eliminated (though there are conformal FDTD methods which try to split the difference), allowing order-of-magnitude increases in the size of domains that can be solved, compared to finite-element approaches. The cost for this is that FDTD methods are far harder to use than finite-element approaches -the analysis of the data to deconvolve behavior as a function of frequency can be nontrivial, and numerical stability is not always assured. In general, it requires a couple of years of experience to get really reliable results out of 3D FDTD methods, even with modern packages (Taflove and Hagness, 2005;Sadiku, 2009;Sadiku, 2017). Therefore, a full analysis is supposed to develop a quick and deep understanding of the essentials of CEM and to solve real-life electromagnetic problems.

Frequency-Domain vs. Time-Domain
There are two common computational electromagnetic approaches to study the dielectric structures: frequency-domain and time-domain. We feel that each one has not only its own place in a researcher's toolbox but also its unique advantages and disadvantages. The MIT Photonic Bands (MPB) package is frequency-domain. That is, it does a direct computation of the eigenstates and eigenvalues of Maxwell's equations using a planewave basis. Each field computed has a definite frequency. In contrast, time-domain techniques iterate Maxwell's equations in time (

Disadvantage of Frequency-Domain vs. Time-Domain
A traditional disadvantage of frequency-domain methods was that you had to compute all of the lowest eigenstates, up to the desired one, even if you didn't care about the lower ones. This was especially problematic in defect calculations, in which a large supercell was used, because in that case the lower bands were "folded" many times in the Brillouin zone.

MPB Software Package
MPB is a software package to compute definite-frequency eigenstates of Maxwell's equations in periodic dielectric structures. It can compute optical dispersion relations and eigenstates for structures such as strip waveguides and optical fibers. MPB is well suited for the study of photonic crystals: periodic dielectric structures are exhibiting a band gap in their optical modes and prohibiting propagation of light in that frequency range. Highperformance 3D/2D (FDTD) Maxwell's solver for design, analysis, and optimization of nanophotonic devices and processes, which used in this work, is Version S2019A-R1 (8.21.1781) [26].

Disadvantage of MPB Software Package
The disadvantages of frequency-domain versus time-domain disappear to some degree in MPB. However, the targeted eigensolver method used in MPB still has poor convergence, so time-domain methods such as Meep still have an advantage here (Taflove and Hagness, 200; Kang Ning et al., 2017). FDTD can give almost arbitrarily accurate answers to the macroscopic Maxwell's equations for any geometry. This could be further improved by repeating many of the steps.

Methodology
In this paper, we will benefit from a plasmonic structure and a slot cavity resonator to design a full adder. To this end, first OR and AND logic gates needed for the design of the adder will be proposed. Then, necessary simulations will be conducted with the aid of Lumerical's FDTD software. This is based on the finite element method. Numerical simulation results will never give exactly the correct answer. Therefore, the steps should be taken to reduce the error, for example via reducing the error often involves increased simulation time and memory. In finite difference schemes (such as FDTD) is often assumed that the error in a simulation result always diminishes with decreasing grid size.

Theoretical Equations of Plasmonic Structures
For the implementation of logic gates in this study, we benefit from a metal-insulator-metal waveguide, where a nanoscale insulator is placed between two metal structures. When the distance between the two metals is very short, the surface plasmons can be stimulated, and the plasmonic waves, which propagate between the two metal surfaces, can be used as messengers. As compared to Lorentz model, the FDTD approach can also account for a large variety of materials such as Drude dispersion materials, perfect metal, second-order, and third-order materials. As a result, this variety of materials can well support multidimensional simulations. In this study, we assume the insulator is air, and the metal is Drude model silver in which the relative dielectric function for metals is expressed by Eq. (1) [9].
where, ∞ is the dielectric constant of the material at infinite frequency, shows the plasma frequency, and denotes the angular frequency of damping. The complex-valued dielectric function of evaporated and template stripped polycrystalline silver films (the Complex permittivity of silver in terms of frequency) are determined from 0.05 eV (λ = 25 μm) to 4.14 eV (λ = 300 nm) with a statistical uncertainty of less than 1% [26]. As for silver, these values are as follows: ∞ = 3.7, = 9.1 . , = .018 . .
Two waveguide modes (TM0 and TM1) for the electric field intensities are that propagate Electromagnetic waves between two points. Under this condition, the system has a fast function and minimizes power consumption.We assume the width of the waveguides to be 100 nm so that only the first and second modes (TM0, TM1) are stimulated and depending on the application of equipment, the first or second mode is used. By solving the Maxwell equations, we can reach the dispersion equation as shown in Equation (2).
where, is the dielectric constant of the insulator (which is equal to one for air), illustrates the dielectric constant of the metal (silver), and are numbers for the dielectric wave and the metal, respectively. If we assume the complex propagation constant in the plasmonic waveguide to be β, we can achieve Equations (3) and (4) by writing electrical and magnetic fields inside the insulator and the metal , using boundary conditions.
Through the simultaneous solution of Equations (2), (3) and (4), we can obtain the propagation constant in the waveguide (β). Take Figure 1, which has been used as a basis for this study, and which consists of an MIM waveguide and a rectangular slot with a length of L s . After the stimulation of the MIM waveguide, if the resonance conditions are met in the rectangular slot, the plasmonic waves will form inside the rectangular slot in a static form and in the form of sequential reciprocating motions. Resonance occurs when the phase shift caused by the sequential reciprocations in the slot (in addition to phase differences that occur when colliding with the end of each side of the slot) is equal to an integer multiple of ; that is, relation ∆ = + = is true. Note that is ignored due to its negligible value. If we assume the refractive index inside the plasmonic waveguides and the resonance slot to be , where 0 = 2 is the wave number in the vacuum, we will obtain the Equation (5) If the input port in Figure 1 is stimulated by an electromagnetic wave at optical frequencies, resonant static electric and magnetic fields in a rectangular slot can be obtained by solving the Maxwell equations. The intensity of the magnetic fields in this resonatory nano-slot for the m-th mode is obtained through Equation (6). . ) ] (6) where, ∆ is the distance from the input port to the middle of the rectangular nano-slot, and since waveguides usually reach the edge of the nano-slot, ∆ = 2 is usually true (of course, it can be less than this value too, and will not cause any problems). Given that we assumed the width of the waveguides to be very small, only the first and second modes are stimulated in the nano-slot. For instance, the magnetic field intensity equation for the first mode is as shown in Equation (7). From Equation (7), we derived the necessary idea to design the AND, OR, and XOR gates, which are necessary bases for the adder because if we assume two input waveguides, whose distances to the middle of the nano-resonator are asymmetric to each other; that is, for one of them, the distance is ∆ 1 , and for the other ∆ 2 = −∆ 1 , then the two resonated magnetic fields in the rectangular slot will neutralize each other, and the field inside the rectangular slot will become zero. In addition, by increasing the number of waveguides, and adjusting their distances to each other, the desired output parameters can be obtained.

5.Input OR Gate
There were two main reasons for choosing a resonance structure in this paper. The first and the most important reason is that there is a high rate of leakage from one input to other inputs in structures, where the input and output plasmonic waveguides are physically connected to each other. This causes a lot of errors especially when there are a large number of inputs, thus making it practically impossible to design a structure with a large number of inputs in a package. On the other hand, as suggested in this paper, as resonance enclosures are used, the inputs will be practically isolated from each other, and only a slight leakage due to resonance may occur between the inputs, which will not cause many errors in response to the output, and which will be negligible. The second reason is the possibility of using a large number of inputs in a resonance structure. In this section of our paper, the experiences of Mohsen Olyaee et al.'s (2019) paper on the structure of the OR gate are used. In study of these researches, with increasing number of inputs, it will not be geometrically and physically possible to design and connect the waveguides. Even if this can be done for a small number of inputs, the surface of the design piece will become very large, but in the resonance design, a larger number of inputs can be designed on a piece with a smaller surface.

Simulation structures and method
The structure and dimensions of a 4-input OR logic gate are illustrated in Figure 1. We assume the threshold of the optical electric field amplitude in the waveguides equal to 0.4. If the field amplitude is greater than 0.4, it will be assumed as a logical one, and if the field amplitude is less than 0.4, it will be assumed as a logical zero. We have four input ports. We consider the intensity of light (the intensity of the electromagnetic wave), i.e. the square of the amplitude, as the inputs, and the intensity of light in the output as the gate output. In OR gate , to set a measurement criterion for the field amplitude, we set the threshold value (E) to 0.4. Figure 1. The structure and dimensions of a 4-input OR logic gate

discussion and results
First, we obtain resonance wavelengths for the structure. The following graph is obtained for the transmission coefficient of the structure. If the transmission coefficient is greater than one, this is because when Lumerical software deals with several inputs, it will be somewhat problematic in terms of defining the transmission coefficient, but this curve is sufficient for our work, and determines the the optimal operating wavelength. Figure 2. The resonance wavelengths for the structure Considering the Figure 2, a wavelength of 1490 nm will be suitable for the gate's operation. Of course, we have resonance in a wavelength of 780 nm too, but since the telecommunication windows are within 1470-1550 nm, we consider the main wavelength at 1490 nm. In fact, the resonance wavelength is obtained from a Equation (8), which λ depends on the length of the rectangle (l) in the cavity.
where 780 nm is synonymous with the first resonance, and 1490 nm is synonymous with the second resonance. In State 1 (wavelength of 780 nm), we assume that all four inputs are active (we assume the electric field amplitude of the input electromagnetic wave at each of the four inputs as being equal to the normalized one). In this state, the amplitude of the optical electric field at the output is about 1.5 units as shown in Figure 3-b, and since it is greater than 1.3 units, it will be considered equal to a logical one as being expected too. The electric fields in the structure of the OR gate are illustrated in Figure 3-c, and we can see that the circuit has operated correctly. The structure of the OR gate is illustrated in Figure 3(a). In electromagnetic waves, the amplitude is the maximum field strength of the electric and magnetic fields. We assume the threshold of the optical electric field amplitude in the waveguides equal to 1.3 units. If the field amplitude is greater than 1.3, it will be assumed as a logical one, and if the field amplitude is less than 1.3, it will be assumed as a logical zero. In this state, the amplitude of the electric field intensity at the output port equals 1.3 units as shown in the Figure 3. Due to the symmetry of the shape in State 2, we assume that it is sufficient to arbitrarily activate one of the inputs with an amplitude of 1 while inactivating the other or setting it to zero. The field amplitude equals 0.65 in this state. Figure 3(a) shows the shape of the inputs and outputs, and Figure 3(b) shows the amplitude of the electric field at the output, which is about 1.1 units, and which represents a logical one because it is larger than the threshold. The electric fields in the structure of the OR gate are illustrated in Figure 3(c), and we can see that the circuit has operated correctly. In this state, due to the resonance of the field caused by the rectangular resonator, the waveguide side has inclined towards orange, which represents a field resonating into the waveguide. In the other states, where the two opposite inputs are stimulated, and the two other ones are off, the conditions are like those in the previous state. Hence, to avoid repetition, we avoid describing these states.
In State 3, we assume that two of the neighboring inputs are stimulated with a normalization power of 1, and the two others are inactive. As for this state, Figure 5(a) shows the shape of the inputs and outputs, and Figure 5(b) shows the amplitude of the electric field at the output, which is about 0.7 units, and which represents a logical one because it is larger than the threshold. The electric fields in the structure of the OR gate are illustrated in Figure 5(c), and we can see that the circuit has operated correctly. In this state, due to the resonance of the field caused by the rectangular resonator, the side of the Input-1 and Input-2 waveguides has inclined towards yellow, which represents a field resonating into these two waveguides. In State 4, we assume that only one of the inputs is active, and the rest are inactive. Since the structure is symmetric, there is no difference in which input you choose as the active input. In this state, we consider the structure of the logic gate as shown in Figure 6(a). The electric field at the output is shown in Figure 6(b), and Figure 6(c) shows the size of the fields in the gate structure using colored markers. In the final state, where all four input ports are zero, logically, none of the fields get into the output port. Therefore, the amplitude of the electric field at the output port is zero, which represents the correct function of the proposed 4-input OR logic gate. Table 1 shows the electric field amplitudes at the inputs and outputs of the proposed logic gate as well as the correct functioning states of the logic gate. The remaining combinations of the four inputs are not shown in Table 1 due to their structural symmetry.

Conclusion
With the aid of the logic gates designed, we managed to design a half adder circuit with a simple structure through propagation of plasmonic waves at an optical frequency and window of 1550 nm. In Section 2, we used a 4-input OR gate to design and simulate a full adder circuit, which used plasmonic waves to transmit signals. The 4-input gates, presented in this study for the first time, have a simple structure, and are manufactured at a low cost. This proposed design functions much better in comparison to similar designs in credible studies conducted on logic gates designed using plasmonic waves. For instance, the logic gates designed in [10] using plasmonic waves have thresholds variable relative to the phase of the input wave, and are very vague, which makes the output separation very difficult. Whereas, in the proposed designs in this study, the threshold limit is clearly defined, and the outputs are clearly defined as logical 0 and 1 relative to this threshold, which is a very important advantage in the proposed design. Using a rectangular resonator and plasmonic waves, a filter with a central wavelength of 1550 nm has been designed in [11], which has high electromagnetic wave losses due to a failure to adjust the dimensions and correctly optimize the width of the resonatory slot and the distance of the waveguides to the resonator's center. Under this circumstance, in the proposed design the system has a fast function with slight variations in real-time or near real-time manner, and this led to its minimum power consumption, while serving in various situations. Additionally, the use of optical devices, like silicon photonic devices, causes other problems, for example, cases where the Pockels constant equals zero. Although the thermo-optic effect and dispersion effect of free carriers have been dynamically employed to control the optical properties of silicon, these technologies have adverse effects on velocity, losses, costs, and so on. However, by optimizing the dimensions of the structure, we managed to reduce the losses, and achieve a transmission coefficient of about 0.62, thus reducing the losses down to 25% less than the design in [12].
Furthermore, in our idea, given that the number of input ports varies in different states, the transmission coefficient of the amplitude cannot, in practice, show the changes in the output. Under this condition, a wave radiates to one of the inputs and the transmission coefficient at the output equals to the ratio of the output wave amplitude to the input wave amplitude. In other words, when light is radiated to both input ports, the transmission coefficient at the output is equaled to the ratio of the output wave amplitude to the sum of the amplitudes of both waves at the two inputs. Although the previous researches' designed gates are new and high-efficiency in terms of some characteristics, they also have a very high manufacturing complexity for a simple logic gate, for example, the piece size is large and the power consumption is high due to the permanent pumping of optical power. Therefore, all the calculations for the design of the structure will face difficulties. For this, the intensity of the wave amplitude at the output port as well as a plasmonic structure accompanied by a slot cavity resonator to design a full adder should be considered. After proposing the OR and AND logic gates for the design of the full adder, necessary simulations are conducted with the aid of Lumerical's FDTD software.

Recommendations
FDTD software is a powerful tool for various physics and engineering applications. One of its disadvantage is that it may provide an accuracy with slightly more fault than other software in simulating large-scale electrical properties.