In this paper, a new scheme is proposed to realize reconfigurable and multifunction optical logic gates (XOR, XNOR, NAND, and OR) using a Mach–Zehnder interferometer with a tunable thermo-optic phase shifter (TOPS). The reconfigurable optical logic gates are realized by tuning the phase of an optical signal using TOPS without changing the physical device structure. The logical input “0” or “1” is considered corresponding to the phase of the optical signal at TOPS. The logical output of the proposed device depends on the light intensity at output ports. The device is designed on silicon on insulator (SOI) platform and the simulation result shows that the on–off extinction ratio is greater than 37 dB at 1550 nm and >25 dB for the C-band. Moreover, it has a low insertion loss of 0.09 dB at a wavelength of 1550 nm and <0.8 dB for the C-band window. The proposed optical logic gates can be a promising logical device for programmable photonic integrated circuits.

## I. INTRODUCTION

Programmable photonic integrated circuits (PICs) are a promising technology for high performance optical computing that requires reconfigurable devices, such as tunable couplers, phase shifters, and waveguide meshes.^{1} All-optical devices, such as optical splitters,^{2} optical switches,^{3} and optical logic gates,^{4} are fundamental components in optical computing and signal processing. These optical components are implemented in photonic integrated circuits to overcome the limitations of electronic components, such as the speed, bandwidth, and power consumption. Silicon photonics is a suitable platform for integrating all-optical components into a single chip.^{5} The advancement in the fabrication of silicon nano-photonic devices leads to high optical confinement, low losses, and complementary metal–oxide–semiconductor (CMOS) compatibility with electronics devices. Recently, the optical logic devices have been widely demonstrated for various applications, such as programmable logic arrays,^{6} nanowire networks,^{7} and optical computing.^{8} Optical computing is a significant research area that requires high performance optical signal processing and optical logical operations.^{9}^{,}^{10} The 2 × 2 optical gate is used to couple the modulated optical signal and control the power coupling in programmable photonic circuits.^{11} Optical logic gates can be implemented as multi-operand for optical computing^{12} and for N × N unitary transformation in quantum computing.^{13} Optical logic gates have been realized using various techniques having different device structures, such as semiconductor optical amplifiers,^{14} microring resonators on silicon nitride,^{15} ring resonators with graphene nanoribbons,^{16} hybrid plasmonic waveguides,^{17}^{,}^{18} cross-phase modulation effects on phase-shifted grating,^{19} 2D non-linear photonic crystals using nonlinear Kerr effects,^{20} the multimode interference (MMI) coupler with photonic crystals,^{21} and MMI using binary-phase shift-keyed signals.^{22}^{,}^{23} In the literature, optical logic gates have been reported using the electro-optic principle.^{24}^{,}^{25} While this effect yields fast switching and energy-efficient devices, the related free carrier absorption causes additional loss and crosstalk. However, we propose to use the thermo-optic effect to minimize the insertion loss and enhance the device’s extinction ratio, and also silicon has a high thermo-optic coefficient and ease of fabrication.^{26} The previously reported MMI-based optical logic gates require multiple optical signals and a binary phase-shift keyed signal at the input to realize logic “0” and “1.” However, instead of multiple optical signals (multilaser sources) and the phase-shifted signal at the input waveguide, we have employed a single optical signal input with a tunable phase shifter at the arms of the MMI-based Mach–Zehnder interferometer (MZI) to provide logical inputs. The phase of the optical signal is tuned using the thermo-optic phase shifter (TOPS) to realize other logic gates, and also two logic gates are realized simultaneously. The proposed logic gates work as a reconfigurable and multifunction logic device.

In this article, reconfigurable and multifunction MMI-based optical logic gates have been realized using the thermo-optic principle. The TOPS are implemented on a silicon waveguide to control the phase of an optical signal to obtain a specific logic gate without changing the device structure. The logical input “0” or “1” is assumed with respect to a phase shift of the optical signal at the MZI arms. The logical output of the proposed device depends on the light intensity (amplitude) at the output ports. Hence, the optical logic gates (XOR, XNOR, NAND, and OR) are realized by converting the input phase information to the output amplitude information. The proposed device has a low insertion loss and high extinction ratio that can be implemented as a logical device in programmable PICs for optical computing.

## II. DEVICE STRUCTURE AND PRINCIPLE

### A. Device structure of proposed optical logic gates

The schematic diagram of the proposed optical logic gates is shown in Fig. 1. The proposed device is composed of a cascaded 2 × 2 MMI coupler to form a Mach–Zehnder interferometer having an optical input at port 1 and two logical inputs “A” and “B” and logical outputs “X” and “Y” at port 2 and port 3, respectively. The general interference principle is utilized to design 2 × 2 MMI as a 3 dB coupler. It splits the optical light into two output ports equally, which is having π/2 phase difference between them. The thermo-optic phase shifter is implemented over the silicon nano-waveguide and is placed at both arms of MZI as PS-A and PS-B to provide logical inputs A and B, respectively. The second 2 × 2 MMI coupler is also used as a 3 dB coupler. The output of the MZI structure depends on the phase difference between the two arms of MZI. The phase of the optical signal is controlled using TOPS to realize the desired logic gates.

The optical logic gates have been designed using a cascaded MMI-based Mach–Zehnder interferometer couplers and tunable thermo-optic phase shifters (TOPS). The TOPS is implemented at the two arms of MZI, which works as inputs of optical logic gates. First, we have designed and optimized an MMI-based 3 dB coupler to have minimum insertion loss and higher bandwidth. Second, we have optimized the TOPS to achieve the desired phase shift at the MZI arms. The general interference principle of the MMI is employed to design a 3 dB coupler. In general interference, first N number of images is formed at a distance L = 3L_{π}/N, where the beat length L_{π} = 4n_{eff}W_{MMI}^{2}/3λ_{0}, and W_{MMI}, λ_{0}, and n_{eff} are the MMI width, operating wavelength, and effective index, respectively. The phase relation between the input and output optical signals in the N × N MMI waveguide is given by^{27}

where i = 1, 2, …, N are numbered as the bottom–top input port and j = 1, 2, …, N are the top–bottom output port.

### B. Design and optimization of MMI-based 3 dB coupler

The device structure of the MMI-based 3 dB coupler is shown in Fig. 2. The proposed structure has two input ports (1 and 4) and two output ports (2 and 3). The input and output access ports have linear tapered waveguides to minimize the reflection at the interface between the input waveguide and the multimode waveguide. We have optimized the dimension of the tapered waveguide as W_{s} = 0.5 *µ*m at one end, W_{t} = 3.2 *µ*m at the other end, and the tapered length L_{t} = 30 *µ*m. The width and length of MMI are optimized using the eigenmode expansion (EME) method.

The transmission characteristics of the MMI coupler are analyzed for TE polarization at different MMI widths and lengths, as shown in Figs. 3(a) and 3(b), respectively. The proposed MMI-based 3 dB coupler has an MMI width (W_{MMI}) of 12 *µ*m and a length (L_{MMI}) of 525 *µ*m. The maximum power is achieved at a length of 525 *µ*m. At this length, the MMI coupler has a maximum and nearly equal normalized power of 0.5 at two output ports 2 and 3. Thus, the proposed device works efficiently as a 3 dB coupler. The fabrication tolerance (having transmission normalized power >0.9) of the 3 dB coupler is W_{MMI} ± 100 nm and L_{MMI} ± 5000 nm.

The transmission spectrum of the proposed 3 dB coupler is investigated for a broader wavelength, as shown in Fig. 4. We have observed that the maximum power is obtained at a wavelength of 1550 nm. The insertion loss of the proposed device is 0.03 dB at a wavelength of 1550 nm and less than 0.45 dB at a wavelength of 1530–1565 nm (C-band).

### C. Optimization of thermo-optic phase shifter

The thermo-optic phase shifter (TOPS) is designed using the Titanium–Tungsten (TiW) alloy, and a cross-sectional view of TOPS is shown in Fig. 5. The TiW material is used in microheaters since it has a high bulk resistivity of 0.61 *μ*Ω m. The thickness of the TiW microheater (t_{h}) layer is chosen to be 200 nm, which is compatible and feasible with commercial fabrication foundries.^{28} It is deposited over SiO_{2} cladding layer of 2.2 *µ*m. The dimension of the microheater is a critical parameter to obtain the desired phase shift. The optical phase shift is highly sensitive to the width and length of the microheater. The TiW microheater has an optimized width (W_{h}) of 4 *µ*m and length (L_{h}) of 200 *µ*m. The heat and temperature analysis of TOPS is performed using a lumerical heat solver.^{29}

The temperature profile of TOPS is exported from the lumerical device module to mode solution (finite-difference eigenmode solver) to perform optical simulations. Using this mode solver, the effective index of the silicon waveguide is analyzed at different temperatures. Due to a change in the effective index of the waveguide, the mode propagation constant gets changed. Subsequently, the phase of the optical signal is varied corresponding to the change in temperature. Thus, the thermo-optic effect can be used to tune the effective index of the silicon waveguide. The silicon waveguide has a relatively higher temperature sensitivity having a thermo-optic coefficient dn/dT = 1.86 × 10^{−4} at room temperature at a wavelength of 1550 nm.^{30} In the proposed optical logic gates, the thermo-optic phase shifter is designed using a TiW microheater. The switching time of the tunable TOPS can be achieved below 12 *µ*s, as the TiW microheater is used to demonstrate a three-mode switch.^{31} Furthermore, the TiW heater with a doped silicon resistive heater can be used to reduce the switching time below 2.0 *µ*s.^{32} The phase shift (ϕ) of the TOPS can be calculated using the following equation:^{33}

where L_{h} is the length of the microheater, ∆T is the change in temperature, and λ_{0} is the operating wavelength.

The phase shifts and effective indices at different microheater powers are shown in Fig. 6. It is observed that the input power of 12.3 or 61.5 mW is required to obtain a phase shift of 90°; 24.6, 73.8, or 123 mW is required to obtain the phase shift of 180°; and 36.9 or 86.1 mW is required to obtain the phase shift of 270°.

The temperature profile of the microheater for a different input power is shown in Fig. 7. The variation of temperature at the waveguide region with the microheater power is observed from the temperature field profile. Due to the temperature change, the effective index of the waveguide gets changed. Thus, the phase of the optical signal can be tuned by varying the input microheater power.

## III. SIMULATION RESULTS AND DISCUSSION

The performance of the proposed optical logic gates is analyzed through the lumerical heat solver and the Eigenmode expansion (EME) method. The proposed device works as the multifunction and reconfigurable logic gates, i.e., we can realize two logic gates simultaneously. We can also reconfigure the phase of the optical signal using TOPS to realize other logic gates. In our proposed structure, the phase information is assumed as the logical input and the amplitude of the output signal is assumed as the logical output that depends on the phase difference at the MZI arms. Thus, the input phase information is converted into the output amplitude information.

### A. XOR and XNOR gates

The operational conditions and truth table of XOR and XNOR logic gates are shown in Table I. Here, we have realized two logic gates (XOR and XNOR) simultaneously. The power of TOPS is tuned to obtain the desired phase shift at the MZI arms. In the phase shifter PS-A, the power P_{A} = 12.3 mW is applied to achieve π/2 phase shift, and it corresponds to logic “0” and P_{A} = 36.9 mW is applied to achieve 3π/2 phase shift, and it corresponds to logic “1.” Similarly, in the phase shifter PS-B, the phase shift of π/2 corresponds to logic “0” and the phase shift of 3π/2 corresponds to logic “1.”

. | Logical input . | Logical output . | ||||
---|---|---|---|---|---|---|

Case . | P_{A} (mW) ΦA
. | A . | P_{B} (mW) ΦB
. | B . | X (XOR) . | Y (XNOR) . |

I | 12.3 π/2 | 0 | 12.3 π/2 | 0 | 0 | 1 |

II | 12.3 π/2 | 0 | 36.9 3π/2 | 1 | 1 | 0 |

III | 36.9 3π/2 | 1 | 12.3 π/2 | 0 | 1 | 0 |

IV | 36.9 3π/2 | 1 | 36.9 3π/2 | 1 | 0 | 1 |

. | Logical input . | Logical output . | ||||
---|---|---|---|---|---|---|

Case . | P_{A} (mW) ΦA
. | A . | P_{B} (mW) ΦB
. | B . | X (XOR) . | Y (XNOR) . |

I | 12.3 π/2 | 0 | 12.3 π/2 | 0 | 0 | 1 |

II | 12.3 π/2 | 0 | 36.9 3π/2 | 1 | 1 | 0 |

III | 36.9 3π/2 | 1 | 12.3 π/2 | 0 | 1 | 0 |

IV | 36.9 3π/2 | 1 | 36.9 3π/2 | 1 | 0 | 1 |

Case I: The phase of the optical signal at PS-A (the upper arm of MZI) is tuned to ϕ_{A} = π/2 by applying the power P_{A} = 12.3 mW that corresponds to the logical input “0,” the phase of the optical signal at PS-B (lower arm of MZI) is tuned to ϕ_{B} = π/2 by applying the power P_{B} = 12.3 mW that corresponds to the logical input “0,” and the logical output depends on the light intensity (amplitude) at the output ports, i.e., XOR (X) = 0 and XNOR (Y) = 1, as shown in Fig. 8(a).

Case II: The phase of the optical signal at PS-A is tuned to ϕ_{A} = π/2 by applying the power P_{A} = 12.3 mW that corresponds to the logical input “0,” the phase of the optical signal at PS-B is tuned to ϕ_{B}=3π/2 by applying the power P_{B} = 36.9 mW that corresponds to the logical input “1,” and the logical output is XOR (X) = 1 and XNOR (Y) = 0, as shown in Fig. 8(b).

Case III: The phase of the optical signal at PS-A is tuned to ϕ_{A} = 3π/2 by applying the power P_{A} = 36.9 mW that corresponds to the logical input “1,” the phase of the optical signal at PS-B is tuned to ϕ_{B} = π/2 by applying the power P_{B} = 12.3 mW that corresponds to the logical input “0,” and the logical output is XOR (X) = 1 and XNOR (Y) = 0, as shown in Fig. 8(c).

Case IV: The phase of the optical signal at PS-A is tuned to ϕ_{A} = 3π/2 by applying the power P_{A} = 36.9 mW that corresponds to the logical input “1,” the phase of the optical signal at PS-B is tuned to ϕ_{B=}3π/2 by applying the power P_{B} = 36.9 mW that corresponds to the logical input “1,” and the logical output is XOR (X) = 0 and XNOR (Y) = 1, as shown in Fig. 8(d).

### B. NAND and OR gates

The operational conditions and truth table of the NAND and OR logic gates are shown in Table II. Here, we have realized two logic gates (NAND and OR) simultaneously. In the phase shifter PS-A, the power P_{A} = 61.5 mW is applied to achieve a phase shift of π/2 and it corresponds to logic “0,” and P_{A} = 73.8 mW is applied to achieve a phase shift of π and it corresponds to logic “1.” Similarly, in the phase shifter PS-B, power P_{B} = 86.1 mW is applied to achieve 3π/2 phase shift and it corresponds to logic “0,” and P_{B} = 123 mW is applied to achieve π phase shift and it corresponds to logic “1.”

. | Logical input . | Logical output . | ||||
---|---|---|---|---|---|---|

Case . | P_{A} (mW) ΦA
. | A . | P_{B} (mW) ΦB
. | B . | X (NAND) . | Y (OR) . |

I | 61.5 π/2 | 0 | 86.1 3π/2 | 0 | 1 | 0 |

II | 61.5 π/2 | 0 | 123.0 π | 1 | 1 | 1 |

III | 73.8 π | 1 | 86.1 3π/2 | 0 | 1 | 1 |

IV | 73.8 π | 1 | 123.0 π | 1 | 0 | 1 |

. | Logical input . | Logical output . | ||||
---|---|---|---|---|---|---|

Case . | P_{A} (mW) ΦA
. | A . | P_{B} (mW) ΦB
. | B . | X (NAND) . | Y (OR) . |

I | 61.5 π/2 | 0 | 86.1 3π/2 | 0 | 1 | 0 |

II | 61.5 π/2 | 0 | 123.0 π | 1 | 1 | 1 |

III | 73.8 π | 1 | 86.1 3π/2 | 0 | 1 | 1 |

IV | 73.8 π | 1 | 123.0 π | 1 | 0 | 1 |

Case I: The microheater power is tuned to P_{A} = 61.5 mW for the phase shift ϕ_{A} = π/2 that corresponds to the logical input “0” and P_{B} = 86.1 mW for the phase shift ϕ_{B} = 3π/2 that corresponds to the logical input “0,” and the logical output is NAND (X) = “1” and OR (Y) = “0,” as shown in Fig. 9(a).

Case II: The microheater power is tuned to P_{A} = 61.5 mW for the phase shift ϕ_{A} = π/2 that corresponds to the logical input “0” and P_{B} = 123 mW for the phase shift ϕ_{B=}π that corresponds to the logical input “1,” and the logical output is NAND (X) = “1” and OR (Y) = “1,” as shown in Fig. 9(b).

Case III: The microheater power is tuned to P_{A} = 73.8 mW for the phase shift ϕ_{A} = π that corresponds to the logical input “1” and P_{B} = 86.1 mW for the phase shift ϕ_{B} = 3π/2 that corresponds to the logical input “0,” and the logical output is NAND (X) = “1” and OR (Y) = “1,” as shown in Fig. 9(c).

Case IV: The microheater power is tuned to P_{A} = 73.8 mW for the phase shift ϕ_{A} = π that corresponds to the logical input “1” and P_{B} = 123 mW for the phase shift ϕ_{B} = π that corresponds to the logical input “1,” and the logical output is NAND (X) = “0” and OR (Y) = “1,” as shown in Fig. 9(d).

The on–off extinction ratio (ER) is a significant parameter to analyze the performance of optical logic gates. The extinction ratio and insertion loss (IL) of XOR and XNOR logic gates have been investigated by varying the input wavelengths for different cases, as shown in Fig. 10. The proposed XOR and XNOR logic gates have an ER of >45 dB and IL of 0.08 dB at a 1550 nm wavelength, and ER of >25 dB and IL < 0.8 dB for the C-band window (1530–1565 nm).

Figure 11 shows the ER and IL of NAND and OR logic gates at different wavelengths. It is observed that the proposed NAND and OR logic gates have an ER of >37 dB and IL of 0.09 dB at a 1550 nm, and an ER of >25 dB and IL of <0.8 dB for the C-band. A comparative performance analysis of various optical logic gates is shown in Table III. Compared with other structures, the proposed device shows competitive results.

References . | Device . | Technology . | Logical function . | Insertion loss (dB) . | ER (dB) . |
---|---|---|---|---|---|

16 | Rectangular ring | Graphene | NOT, XOR, | ⋯ | 14.2, 19.5, |

resonator | nanoribbon | and XNOR | and 14.1 | ||

17 | Y-splitter | Hybrid plasmonic | OR, XOR, NOT | ⋯ | 26 |

waveguide | |||||

18 | Multi-functional | Angle manipulation | AND, NAND, NOR, | ⋯ | 12.9, 24.2, 4.5, |

plasmonic waveguide | and NOT | and 24.2 | |||

19 | Phase-shifted | Cross-phase | NOT, AND, NAND, | ⋯ | >5 |

grating | modulation | XOR, and XNOR | |||

20 | Photonic crystal | Nonlinear | XOR, XNOR, NAND | ⋯ | 25, 30, 37 |

Kerr effect | and NOT | and 20.7 | |||

21 | MMI-based optical | Photonic crystal | XOR, XNOR, NAND, | <0.3 for C band | 28.6, 28.6, 25 and |

logic gates | and OR | 26.6 for C band | |||

22 | Multimode interference | Semi-binary-phase | XOR, NAND, OR, | <0.14 for C band | 26, 24.7, 25.9, |

(multilogic) | shift-keyed | XNOR, and NOT | 26, and 25 dB | ||

24 | Reconfigurable optical logic | Electro-optic | AND, OR, XOR, | ∼0.15 dB at 1549.1 nm | >10 |

architecture (microring resonator) | principle | and XNOR | |||

This work | MMI-based MZI | Thermo-optic | XOR, XNOR, NAND, | 0.09 at 1550 nm | >37 at 1550 nm |

(Reconfigurable and multifunction) | phase shifter | and OR | <0.8 dB for C band | >25 for C band |

References . | Device . | Technology . | Logical function . | Insertion loss (dB) . | ER (dB) . |
---|---|---|---|---|---|

16 | Rectangular ring | Graphene | NOT, XOR, | ⋯ | 14.2, 19.5, |

resonator | nanoribbon | and XNOR | and 14.1 | ||

17 | Y-splitter | Hybrid plasmonic | OR, XOR, NOT | ⋯ | 26 |

waveguide | |||||

18 | Multi-functional | Angle manipulation | AND, NAND, NOR, | ⋯ | 12.9, 24.2, 4.5, |

plasmonic waveguide | and NOT | and 24.2 | |||

19 | Phase-shifted | Cross-phase | NOT, AND, NAND, | ⋯ | >5 |

grating | modulation | XOR, and XNOR | |||

20 | Photonic crystal | Nonlinear | XOR, XNOR, NAND | ⋯ | 25, 30, 37 |

Kerr effect | and NOT | and 20.7 | |||

21 | MMI-based optical | Photonic crystal | XOR, XNOR, NAND, | <0.3 for C band | 28.6, 28.6, 25 and |

logic gates | and OR | 26.6 for C band | |||

22 | Multimode interference | Semi-binary-phase | XOR, NAND, OR, | <0.14 for C band | 26, 24.7, 25.9, |

(multilogic) | shift-keyed | XNOR, and NOT | 26, and 25 dB | ||

24 | Reconfigurable optical logic | Electro-optic | AND, OR, XOR, | ∼0.15 dB at 1549.1 nm | >10 |

architecture (microring resonator) | principle | and XNOR | |||

This work | MMI-based MZI | Thermo-optic | XOR, XNOR, NAND, | 0.09 at 1550 nm | >37 at 1550 nm |

(Reconfigurable and multifunction) | phase shifter | and OR | <0.8 dB for C band | >25 for C band |

## IV. CONCLUSION

In this paper, optical logic gates (XOR, XNOR, NAND, and OR) are demonstrated by using a cascaded 2 × 2 MMI and thermo-optic phase shifter in a MZI configuration. The simulation results show that the proposed logic device has an insertion loss of <0.8 dB and an on–off extinction ratio of >25 dB for the C-band window. The proposed device works as a reconfigurable and multifunction device by properly adjusting the phase of the optical signal with TOPS. The proposed optical logic gates can be a useful logical device in programmable photonic integrated circuits for optical computing.

## ACKNOWLEDGMENTS

This research was supported by the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India, under Grant No. 2019/000373.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.