Design and modelling of all-optical NAND gate using metal–insulator–metal (MIM) waveguides-based Mach–Zehnder interferometers for high-speed information processing

All the basic logic gates play a major role in carrying out the mathematical computation. The drawbacks of conventional electronics are alleviated by all-optical integrated circuits with a great application of high-speed computing and information processing. In this paper, plasmonic metal–insulator–metal (MIM) waveguides have an excellent property of propagating the surface plasmons beyond the diffraction limit up to deep sub-wavelength scale. All-optical NAND gate design is optimized by using MIM plasmonic waveguide-based Mach–Zehnder interferometers (MZIs) in the footprint of 36 µm × 8 µm that works at 1.55 µm operating wavelength. The better performance of the proposed device is achieved, such as the extinction ratio is 10.55 dB, insertion loss is obtained as 0.506 dB, and response time is 262 ps. The proposed design is verified by using the finite-difference time-domain (FDTD) technique and further analysis are carried out by mathematical computation and MATLAB simulation results.


Introduction
In the earlier days, the enrichment of changing information is attained by semiconductor devices, though predictably used this semiconductor technology gets menial due to some restraints like heat dissipation and interconnect delays (Kumar et al. 2015a). The telecommunication and network started research on photonic devices due to the increase of high deprivation of capacity and small size, in which the alteration of information is done through photons rather than electrons (Wu 2004). Due to miniaturization and with huge density devices, the diffraction of light becomes more susceptible. To overcome this problem, the restraining and regulating of light yonder diffraction of light can be contrived viable by using plasmonics, it attributes the combination of electrons and photons (Kumar et al. 2015b;Hu et al. 2008). Due to some unique features, the propagation has been achieved by surface plasmon polaritons (SPPs) in plasmonics. The synergy of electromagnetic waves and free electrons on the metals; produce the surface plasmon polaritons which is a restrained wave that interface between metal and dielectric (Kumar et al. 2017a). In the last few years, the researchers lead to scrutiny of various plasmonic waveguides based on SPPs such as metal-insulator-metal (MIM), insulator-metal-insulator (IMI), dielectric-loaded surface plasmon polaritons (DLSPP) (Zia et al. 2006;Dionne et al. 2006;Kumar et al. 2017b;Chen et al. 2009;Charbonneau et al. 2005). Among all these geometric waveguides, MIM waveguide has been preferred due to its exceptional properties like sub-wavelength confinement, very high grouping velocity and routing of light at nanoscale. By using semiconductor optical amplifier Mach-Zehnder interferometer (SOA-MZI) (Datta et al. 2015;Kim et al. 2006;Samanta and Mukhopadhyay 2012) and lithiumniobate (LiNbO 3 ); Zhang et al. 2016;Takeda et al. 2012) the researchers have been already implemented all the basic and universal logic gates. In this plasmonic NAND gate design, the optimization is achieved in terms of footprint. The proposed design is achieved by using only two MZIs but the earlier design of plasmonic and electro-optic NAND gate is implemented using four MZIs (Smith et al. 1976;Kumar et al. 2018;Yu and Zhang 2016). Using MIM waveguides, some designs of directional coupler (DC) and nonlinear Mach-Zehnder interferometers (MZIs) have been proposed and serving as a house for optical circuit designs. The significant phenomenon of DC is the coupling of optical signals; is beating for even-odd modes. By combining two DCs, an MZI is proposed with two linear waveguides in which one linear waveguide is filled with non-linear Kerr material to accomplish the exchange of light and named as nonlinear plasmonic MZI or, MIM waveguides based MZI. These MZIs are also used to design optical add drop multiplexers (OADM) in enabling greater connectivity in dense wavelength division multiplexing (DWDM). But these devices are prone to crosstalk thereby limits the performance of the device through component imperfections (Sanjeev and Kaler 2013a). One of the essential network elements in WDM networks is optical cross connect (OXC) nodes. The number of contributions leaked when an optical signal is passed through OXC will depend on the cross-connecting state of the OXC (Sanjeev and Kaler 2013b). All-optical components have a remarkable and vital role in modern photonic integrated circuits; like in same manner, logic gates play a major role in the electronic devices.
The outline of this paper presents a design of all-optical universal plasmonic NAND gate using MEH-PPV [poly(2-methoxy-5-(28-ethylhexyloxy)-PPV)] as a nonlinear Kerr material in its one arm of MZI (Swarnakar et al. 2018). The analysis of the proposed structure is carried out by using finite-difference time-domain (FDTD) method and results are verified in MATLAB (Scheuer and Orenstein 2005). The proposed work is simulated and verified by using commercial Opti-FDTD software supplied by Opti-Wave, Canada (https:// optiw ave. com/). Section 2 presents a description of nonlinear single MZI and its mathematical description and portraying the propagating of light through it. Section 3 comprises of design and simulation results of the NAND gate. Finally, Sect. 4 comprises the conclusion of work.

Design of metal-insulator-metal waveguides based Mach-Zehnder Interferometer
In this paper, metal-insulator-metal waveguides based MZI is designed by cascading of two DCs and two linear metal-insulator-metal (MIM) waveguides in which one linear (linear 2) waveguide is filled with nonlinear Kerr material to achieve switching of light as shown in Fig. 1. The structure of MZI has a size within the footprints of (32 × 3.2) µm. This is the basic optical device used toa great extent for the development of logic gates. MEH-PPV [poly(2-methoxy-5-(28-ethylhexyloxy)-PPV)] is used as the nonlinear Kerr material which gives enormous Kerr nonlinearity with permittivity of 2e −18 m 2 /v 2 (Pereda et al. 2003). The relative linear permittivity (εL) of Kerr material is 2.7225 with refractive index (RI) and response time are n = 1.65 and τ = 2.0 e − 15 s, respectively. The schematic of MIM waveguide contains top and bottom layers made up of metals sandwiched with a dielectric material in middle. The top metal layer is made of Silicon Oxynitride whose refractive index is 2.01 and the bottom layer is made of Silicon dioxide with RI = 1.22. The middle layer is filled with air as the dielectric material with RI = 1.0. The dispersion equation for three-layered MIM waveguide is given as Bader et al. (2002): where as є d and є m are the permittivity of dielectric and metal, respectively. k d and k m are the transverse wave number for dielectric and metal, respectively; where 'd' is the width of the linear insulator, which is taken as air. By using a Drude Lorentz model, the metal supporting SPPs are considered as silver (Pile et al. 2005): Fig. 1 Schematic of nonlinear Mach-Zehnder interferometer ω p = 140 × 1014 rad/s and γ = 0.13 × 1014 rad/s are bulk plasma frequency of silver and damping constant, respectively. By changing the RI of Kerr material, the intensities of light vary, which is called the Kerr effect, and the phase of the signal varies, which leads to switching of the light signal from one port to another port by a change in power of input signal. When continuous wave (CW) of wavelength 1550 nm excites transverse electric (TE) mode with an input power of E in = 3e 9 W/m (considered as logic '1') is given at input port 1, by obeying the principle of self-phase modulation (SPM), output signal obtained at the first output port as shown in Fig. 2b. When input power of E in = 0.7e 9 W/m (considered as logic '0') is given at input port 1, by obeying the principle of cross-phase modulation (XPM), output signal obtained at second output port as shown in Fig. 2c.
If an input, E in applied at input port, the output at A and B can be written (Pannipitiya et al. 2010;Kumar and Singh 2016) as: where α 1 is the attenuation constant of first DC, can be written as 1 = Further propagation of signal is from first DC off from points A and B through linear arms reaches at point C and D at output ports respectively, the input signal can be written as where as φ 2 has a phase difference in second linear arm due to Kerr material and φ 1 has a value equal to zero since in the second linear arm there is no phase difference occurs as detailed in Kumar et al. (2015c) as: where λ is the wavelength of signal and L is the length of the second linear arm of MZI and Polarization of light through single MZI, at a high-intensity signal, b low-intensity signal where ∆x and ∆y are the linear different RIs in the second linear arms of the MZI, because of modal birefringence with low and high intensities of a signal, respectively. Δn X and Δn Y are the RIs of nonlinear parts because of the induced birefringence. If the signal is launched with low intensity, then where E in s the intensity ofinput signal. When high intensity signal is launched, then due to XPM, the RI becomes where n 2 is the RI of the linear arm of the MZI containing Kerr material, (3) XXYY and (3) XXXX re the third-order susceptibility of the nonlinear Kerr material, b = 1/3 where the signal is purely electronic. Using Eqs. (4) and (7), phase shift becomes; where Δn L = n X − n Y and the Kerr coefficient is n 2B = 2n 2 (1-b). For high signal intensity, (∆φ ≠ 0) and the maximum, the transmission of signal through a first linear arm of MZI, with transmitivity written as When low intensityof light signal is fed into the input, it arrives at the output port can be written as Equations (9) and (10) gives the output of single MZI, one can calculate the power when high and low-intensity signal is fed at its first input port, respectively. This exclusive switching property of MZI has been used to design the proposed device.

Design, mathematical formulation and simulation results of all-optical NAND gate
The miniature of all-optical NAND gate is designed using two nonlinear MZI within the footprint of 36 µm × 8 µm as shown in Fig. 3. By using MIM waveguides, the proposed structure is designed which supports the plasmonic modes. To get desired operation of universal gate, the optical signals are given at first and second input port of MZI1 and MZI2, respectively. The second and first output ports of MZI1 and MZI2 provides phase shift leads to switching of light. To obtain the optical power at the output port, the two signals combined to get the output of the NAND gate.
The normalized power for output of NAND gate can be calculated for possible combinations as follows Kumar et al. (2015d): Hence the overall output of NAND gate is OUTPUT NAND = m 1 + m 2 + m 3, because there is no output signal for the remaining combinations of input signals.
The design of the all-optical NAND gate is verified using MATLAB and FDTD method. By cascading two MZIs, the layout of all-optical NAND gate as shown in Fig. 3. A continuous-wave (CW) in transverse electric (TE) mode, having a source of wavelength 1550 nm is fed with both input power of 0.7e 9 W/m and 3e 9 W/m are considered as low and high-intensity optical signals, respectively. The proposed design is within the mesh size of ∆x = 0.07711 µm and ∆z = 0.0772 µm. The perfect matched layer (PML) is used as the boundary conditions, with the reflection coefficient of e −12 . The propagation of the optical signal through the proposed design for all possible combinations of input signals is shown in Fig. 4. The timing diagram of the NAND gate is obtained through MATLAB simulation as shown in Fig. 5 is verified with the truth table as shown in Table 1. The different possible input combinations to verify the logic of NAND gate have been applied and discussed as per the following case study. Some operating parameters of plasmonic NAND gate are shown in Table 2. Relative output waveforms for all input combination of NAND gate: a "00"; b "01"; c "10"; d "11"

Case (a): A = 0, B = 0
In this case, A = 0, B = 0, means both signals are low-intensity signals fed from the first and second ports of MZI1 and MZI2. The outputs of these MZIs have arrived at the opposite port of the same MZI. The outputs are combined due to SPM and there after the output signal arrives at the output port (X) is considered as the output of this combination. Thus, the output of the NAND gate is "1" (as shown in Fig. 4a).

Case (b): A = 0, B = 1
At this point, A = 0 means low-intensity signal is fed from first input port of MZI1, due to SPM, the output signal arrives at the cross port of the same MZI1. For B = 1 means high-intensity signal is fed from second input port of MZI2, due to XPM, the output signal arrives at the first output port of the same MZI2. At the output port (X), a signal from the MZI1 is considered as the output of this combination. Thus, the output of NAND gate is "1" (as shown in Fig. 4b).

Case (c): A = 1, B = 0
Under this case, A = 1 means high-intensity signal is fed from the first input port of MZI1, due to XPM, the output signal arrives at the first output port of the same MZI1. For B = 0 means low-intensity signal is fed from the second input port of the MZI2, due to SPM, the output signal arrives at the first output port the same MZI2. At the output port (X) signal from the MZI2 is considered as the output of this combination. Thus, the output of NAND gate is "1" (as shown in Fig. 4c).

Case (d): A = 1, B = 1
For this case, A = 1 means high-intensity signal is fed from first input port of MZI1, due to XPM, the output signal arrives at the first output port of the same MZI1. For B = 1 means high-intensity signal is fed from the second input port of the MZI2, due to XPM, the output signal arrives at the second output port of the same MZI2. At the output port (X) no signal has arrived from both MZIs. Thus, the output of NAND gate is "0" (as shown in Fig. 4d). The output normalized powers for each combination of input signal for NAND gate is given in the Table 3. The threshold value of 0.3 is considered, below which any output power is considered as the logic '0' signal. Similarly, a threshold of 0.5 and above is considered as the logic '1' signal.
The relative waveforms of each input combination are given in below Fig. 5 to evaluate the output optical amplitude.
The blue line indicates the threshold limit, above which it is considered as logic '1'. The green line gives the threshold limit below which, it is taken as logic '0'.
From the above-discussed cases, we observed that the output of NAND gate is obtained only when both or either one of the inputs is fed with low intensity, which is exactly matched with the MATLAB timing diagrams shown in Fig. 6.
The response time is calculated for all the four input combinations of the proposed all-optical plasmonic NAND gate. The output power level requires some time to become stable. In the NAND gate for all the combinations, the response time almost equal. The response time of the proposed design all-optical NAND gate is in the range of 261.52 ps to 262.87 ps. The response time for all four combinations is shown in Fig. 7.
The extinction ratio is defined as the ratio of peak output power in ON state to the peak output power in OFF state. Hence the extinction ratio of this work is given as P ON = peak output power in ON state. P OFF = peak output power in OFF state. Insertion loss (IL) is defined as the ratio of peak input power to the peak output power. It is given as ER = 10 log 10 P ON P OFF = 10 log 10 1.486 0.131 = 10.55

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493 Page 10 of 13 P in peak input power. P out peak output power. The performance analysis of the all-optical plasmonic NAND logic gate is studied by calculating some parameters like extinction ratio (ER) and insertion loss (IL) are provided in Table 4. The ER is calculated as 10.55 dB and IR is observed as 0.506 dB. The performance and some key design parameters of the proposed design is compared with the earlier designs are shown in Table 5. The proposed all-optical plasmonic NAND gate is compared with existing plasmonic NAND gate, NAND gate designed by using semiconductor optical amplifier (SOA), and photonic crystal ring resonator. From Table 5, we can easily determine that the proposed structure has less size as compared to previous design and the extinction ratio, insertion loss, and response time are also optimized. IL = 10 log 10 P in P out = 10 log 10 3.37e9 3e9 = 0.506

Conclusion
In this paper, metal-insulator-metal (MIM) plasmonic waveguides are used to design the Mach-Zehnder interferometer (MZI). An MZI switching property is mainly used to design an all-optical NAND gate. The footprint of the designed gate is 36 µm × 8 µm at an operating wavelength of 1.55 µm. Some parameters are calculated like extinction ratio (ER) and insertion loss (IL) are 10.55 dB and 0.506 dB, respectively. The response time of the proposed all-optical NAND gate is in the range of 261.52 ps to 262.87 ps. As per Table 5, performance of the proposed gate is better than the earlier NAND gate. The proposed design is ultra-compact in nature and also useful for optical computing technologies. The ultrafast switching capability of the nonlinear Kerr material can be used for switching components in the applications requiring high speed communication. This device has the potential in the development of many PICs including other logic gates, combinational and sequential logic circuits. The study of the proposed design verification is done using mathematical computation, MATLAB simulation and carried out by FDTD results.