Design of basic logic gates using optical threshold logic

The circuit complexity in VLSI Technology is increasing day by day and the size of the electronic devices has reduced drastically. But there is a serious tunnelling problem beyond 5 nano meter technology [1–3]. Hence, strong emphasis has been given to find out suitable alternatives. One of these potential alternatives is optical computing [4–6].

Threshold logic gate (TLG)is one of the most suitable alternative of Boolean logic function. The novelty of this logic gate is that the same hardware can be used for various logic purposes, by only varying the threshold value and the weight of the input signals.

Threshold logic gates have been implemented in many ways using VLSI circuits as described by V. Beiu et al [7]. Hampel and Varshavsky had also reported TLG using discrete MOSFETs [8, 9].Energy efficient TLG has been designed using spintronic and Floating gate transistors [10, 11].TLG has also been designed using memristors to achieve high speed,low power and low fabrication area [12–14]. But modern Electronic Design Automation (EDA) tools are not adequate to design accurate models for all memristive devices. The fabrication of memristors is still a challenge in semiconductor industry.

While the first generation optical TLG was proposed in 1980s [15] using Bragg Diffraction modulators and diode detectors, a true all optical TLG was described only in detectors,a true all optical TLG was described in last decade only by Shaik et. al and Sharifi et al [16, 17]. This optical TLG was made using photonic crystals doped with nonlinear optical material. A photonic crystal is nothing but a periodic optical crystalline structure in micrometer or nanometer level that affects movement of photons .The doping creates minute optical resonant cavity within the crystal, while the nonlinearity of the dopant was used to shift the resonance frequency. Although the process is effective, its fabrication is really cumbersome. Also its fan-in and fan-out designs are critical.

In order to overcome these complications, an all optical TLG (OTLG) design utilizing Fabry–Perot cavity which is an optical system made with mirrors and lenses [18–20] has been reported here. In this optical cavity two reflecting surfaces have been used to achieve the goal.Optical waves can travel through this optical cavity only when they are in resonance with it. The proposed design uses optical resonance cavity as the central functioning segment and for threshold operation simple interferometric method has been used rather than any non-linear optical material. Therefore, the fabrication of such system is quite simple.

In the proposed design the weights of the input signals have been varied by placing polarizer-analyser combination in the path of the incident beams which actually corresponds to the input digital signals. Additionally, one can vary the number of incident beams corresponding to the number of digital inputs according to the basic requirements. Since the collector is a continuous parabolic reflector, there is no constraint on the number of input beams.

Adjusting the strength of the threshold input beam injected into the cavity, one can vary the threshold value of the TLG.

In proposed OTLG, just by changing the threshold value (1.5 and 0.5)OR gate and AND gate have been implemented.NOT gate also has been implemented by changing the alignment of the cavity. Threshold power determines the logic function of the system.

The scheme regarding the implementation of TLG using Fabry–Perot cavity discussed in this paper is completely new to the best of our knowledge.The main advantage of this system is there is no constraint on the number of inputs and same set up can give us different logic outputs just by changing the threshold power.

Finesse has been used here to simulate the Fabry–Perot cavity operation. The simulated data has been processed, as well as analytically checked using MATLAB.

Inputs are fed into the system using a simple Newtonian telescope type assembly, that is basically made up of a reflector ${\bf{m}}{\bf{T}}$ and a lens ${\bf{L}}$ as shown in figure 1. The light focused by the reflector acts as a point source for the lens L, because the distance between mT and L is the sum of their focal lengths.

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The Fabry–Perot resonator [18–20] is the second part of the system which is composed of two highly reflecting mirrors, namely flat mirror ${\bf{m}}1$ and concave mirror ${\bf{m}}2.$ The input signal from the first part reaches the cavity through a beam-splitter ${\bf{B}}{\bf{S}}$ and an electro-optic-modulator EOM [21]. The threshold signal, kept 180° out of phase to the input signal, is applied through the other appropriate end of the $BS.$ The TEM₀₀ mode transmitted through ${\boldsymbol{m}}2$ mirror gives the required output.

The Electro Optic Modulator (EOM) modulates the laser beam before entering into a 0.12 m long Fabry–Perot cavity. It helps to provide the required information for the purpose of cavity-locking which is basically a standard Pound-Drever-Hall (PDH) locking system [22–24].

The cavity parameters are shown in table 1.

Table 1. Cavity parameters.

Parameters	Expression	Value
Wavelength of laser beam	$\lambda$	1064 nm
The length of the cavity	$L$	0.12 m
Radius of curvature of the second mirror	$R$	200.12 m
Cavity free spectral range (FSR)	${\nu }_{0}=\displaystyle \frac{c}{2L}$	$1.25\times {10}^{9}\,{\rm{Hz}}$
The cavity stability factor	$g=1-\displaystyle \frac{L}{R}$	$0.9994$
The amplitude reflectivity of both the mirrors	${r}_{m1}={r}_{m2}$	$0.999$
The amplitude transitivity of the mirrors	${t}_{m1}\approx \sqrt{1-{r}_{m1}^{2}},$ ${t}_{m2}\approx \sqrt{1-{r}_{m2}^{2}}$	$0.001$
The beam waist of resonant beam	${\omega }_{0}={\left(\displaystyle \frac{\lambda }{2\pi }\sqrt{d\left(2R-d\right)}\right)}^{0.5}$ Here, d = 2 L	$1.288\,{\rm{mm}}$
The rayleigh length of the beam	${z}_{R}=\displaystyle \frac{\pi {\omega }_{0}^{2}}{\lambda }$	$4.8982\,{\rm{m}}$
Cavity finesse	${\mathcal F} =\pi \displaystyle \frac{\sqrt{{r}_{m1}{r}_{m2}}}{1-{r}_{mc}{r}_{m2}}$	$3140.02$

Finesse has been used for the simulation work, strictly considering orthogonal Hermite-Gaussian (HG) for the decomposition of the light beams. Here, $\left({\boldsymbol{m}},\,{\boldsymbol{n}}\right)$ th order HG mode beam can be expressed by equation (1).

$$\begin{eqnarray}&&\begin{array}{l}{U}_{nm}\left(x,\,y,\,z\right)={\left({2}^{n+m-1}\,\cdot \,n!m!\pi \right)}^{-1/2}\,\cdot \,\displaystyle \frac{1}{w\left(z\right)}\,\cdot \,\exp \,\left(i\left(n+m+1\right)\psi \left(z\right)\right)\,\\ \,\times \,\cdot \,{H}_{n}\left(\displaystyle \frac{\sqrt{2}x}{w\left(z\right)}\right)\,\cdot \,{H}_{m}\left(\displaystyle \frac{\sqrt{2}y}{w\left(z\right)}\right)\,\cdot \,\exp \left(-i\displaystyle \frac{k\left({x}^{2}+{y}^{2}\right)}{2{R}_{c}\left(z\right)}-\displaystyle \frac{\left({x}^{2}+{y}^{2}\right)}{{w}^{2}\left(z\right)}\right)\end{array}\end{eqnarray} \tag{ 1 }$$

where ${H}_{n}\left(x\right)$ is the Hermite polynomial of order n, $w(z)$ is spot size and ${R}_{c}\left(z\right)$ is radius of curvature of the beam that defines the beam parameter $q(z)$ at distance $z\,\,$which is shown in equation (2).

$$\begin{eqnarray}&&\displaystyle \frac{1}{q(z)}=\displaystyle \frac{1}{{R}_{c}\left(z\right)}-i\displaystyle \frac{\lambda }{\pi {w}^{2}\left(z\right)}\end{eqnarray} \tag{ 2 }$$

The longitudinal phase lag of $(m,n)$ th HG mode, i.e. the Gouy phase is given by equation (3).

$$\begin{eqnarray}&&\psi \left(z\right)=ta{n}^{-1}\left(\displaystyle \frac{z}{{z}_{R}}\right)\end{eqnarray} \tag{ 3 }$$

The beam entering into the FP cavity is composed of various modes ensuring $m+n\leqslant 5.$ Since the signal part of the incoming beam is assumed to be a plane wave, only the modes supported by the cavity will resonate.

The threshold beam inserted into the cavity is represented by its beam parameter ${q}_{th}\left(z\right)$ defined by equation (4).

$$\begin{eqnarray}&&\displaystyle \frac{1}{{q}_{th}\left(z\right)}=\displaystyle \frac{1}{{R}_{c\_th}\left(z\right)}-i\displaystyle \frac{\lambda }{\pi {{w}_{th}}^{2}\left(z\right)},\end{eqnarray} \tag{ 4 }$$

with beam waist ${{\boldsymbol{\omega }}}_{0\_{\boldsymbol{th}}}=0.04{\bf{m}}$ at a distance 0.1 m from insertion node, rendering radius of curvature of the wave front at n3 node (m2 mirror), which is utterly different from m2 mirror's curvature and therefore it doesn't resonate inside the cavity.

Linearly separable logical functions can be implemented using threshold logic gates just by changing the input weights and threshold values. In this paper the Fabry-Perot cavity is simulated using Finesse which has been called through MATLAB and also the data was further processed and stored. Finesse has been chosen for this purpose as it has well known ability to simulate this type of optical cavity along with Pound Driver Hall Locking System. Matlab's compatibility with Finesse and its powerful graphical representation has been used for better image clarity. Also it helped to calculate the analytic expressions and match them with the data produced by Finesse. Figure 1 demonstrates the exact schematic diagram which is implemented in Finesse. Here basic gates AND, OR and NOT have been implemented by setting the weights and threshold values using OTLG.

In this optical configuration, as the electrical input is digital in nature, for such intermittent input, one can define any amount of power as the unit power depending upon the design of the digital logic system. For the current design, 4 W threshold power is considered as 1 unit. The output of the system is determined by presence or absence of light, any optical power producing a signal more than system noise is considered as high (1).

3.1. OR gate

For a two-input OR gate the threshold value and its corresponding optical power is shown in table 2.

Table 2. Truth table of OR gate along with threshold value and optical output power .

OR Gate				OR Gate Optical Power (W)
A	B	Threshold	Output	A	B	Threshold	Output
1	1	1.5	1	1	1	6	0.8
1	0	1.5	1	1	0	6	0.6
0	1	1.5	1	0	1	6	0.6
0	0	1.5	0	0	0	6	0

Here the weights for the two inputs are 1 and the threshold value is 1.5 which corresponds to 6 watt optical power. An output power of 0.8 watt for (11) combination, 0.6watt for (01) and (10) combinations and 0 watt for (00) combination have been achieved which is shown in table 1. The variation of output amplitude and output power with respect to input power are shown in figures 2(a) and (b) respectively for OR gate.

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3.2. AND Gate

For two-input AND gate the threshold value and its optical power to be set as shown in table 3.

Table 3. Truth table of AND gate along with threshold value and optical output power.

AND Gate				AND Gate Optical Power (W)
A	B	Threshold	Output	A	B	Threshold	Output
1	1	0.5	1	1	1	2	0.8
1	0	0.5	0	1	0	2	0
0	1	0.5	0	0	1	2	0
0	0	0.5	0	0	0	2	0

Here the weights for the two inputs are 1 and the threshold value is 0.5 which corresponds to 2 watts optical power. The output power 0.8 watt for (11) combination and 0 watt for (01), (10) and (11) combinations are obtained which is shown in table 2.The variation of output amplitude and output power with respect to input power are shown in figures 3(a) and (b) respectively for AND gate.

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3.3. NOT gate

To achieve the NOT operation the cavity is slightly misaligned to make the cavity unlocked by properly choosing operating point. The desired output is obtained within the misalignment range of $\pm 25\mu rad$ for mirror m2, as shown in figures 4(a) and (b).

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Table 4 shows the truth table of NOT gate where the threshold power is 2 W. 3watt input power is applied as binary (1) which is more than threshold power 2watt. 1 watt input power is applied as binary (0) which less than threshold power 2watt.

Table 4. Truth table of NOT gate along with threshold value and optical output power.

NOT gate		NOT Gate optical power (W)
A	Output	A	Threshold	Output
1	0	1 (3 W)	2	Almost 0
0	1	0 (1 W)	2	0.32

Figure 4(a) shows the output power for (1) input and figure 4(b) shows the output power for (0) input.

Basic gates have been designed using Optical Threshold Logic Gate. The threshold logic operation is implemented using an optical reflector system for signal entry and air filled Fabry–Perot resonator as comparator. The advantage of such a system is, no component is needed to be changed for various logic operations. Initially, it has been checked that the system is working perfect as OTLG for various threshold input power. Then the basic OR and AND gates are designed and studied by setting the threshold input power at 1.5 watt and .5 watt respectively. NOT gate is also studied by slightly misaligning the cavity. Any linearly separable function that includes universal gates and majority gates, can easily be designed using this method. Such Majority gate design can be further utilized for designing Full Adder and many more complex optical logic circuits.

All data that support the findings of this study are included within the article.