Mach-Zehnder Interferometer Based All OpticalReversible NOR Gates

Saurabh Kotiyal, Himanshu Thapliyal and Nagarajan RanganathanDepartment of Computer Science and Engineering

University of South FloridaTampa, FL

skotiyal@mail.usf.edu, hthapliy@cse.usf.edu, ranganat@cse.usf.edu

Abstract—Reversible logic has promising applications in dis-sipation less optical computing, low power computing, quantumcomputing etc. Reversible circuits do not lose information, andthere is a one to one mapping between the input and theoutput vectors. In recent years researchers have implementedreversible logic gates in optical domain as it provides highspeed and low energy computations. The reversible gates canbe easily fabricated at the chip level using optical computing.The all optical implementation of reversible logic gates are basedon semiconductor optical amplifier (SOA) based Mach-Zehnderinterferometer (MZI). The Mach-Zehnder interferometer hasadvantages such as high speed, low power, easy fabrication andfast switching time. In the existing literature, the NAND logicbased implementation is the only implementation available forreversible gates and functions. There is a lack of research in thedirection of NOR logic based implementation of reversible gatesand functions. In this work, we propose the NOR logic based alloptical reversible gates referred as all optical TNOR gate and alloptical PNOR gate. The proposed all optical reversible NOR logicgates can implement the reversible boolean logic functions withreduced optical cost and propagation delay compared to theirimplementation using existing all optical reversible NAND gates.The advantages in terms of optical cost and delay is illustrated byimplementing 13 standard boolean functions that can representall 256 possible combinations of three variable boolean function.

I. INTRODUCTION

Reversible logic is emerging as a promising computingparadigm among the emerging technologies. Reversible logichas applications in quantum computing, quantum dot cellularautomata, optical computing, etc [1], [2], [3], [4], [5], [6], [7].Reversible circuits does not lose information while perform-ing the computations. A reversible circuit must comprise ofreversible logic gates. Reversible logic also has applicationsin power-efficient nanocomputing [8], [9], [10]. In reversiblelogic there exists a unique one to one mapping betweenthe input and output vectors. The unused outputs are usedto maintain the reversibility of a reversible circuits and arereferred as the garbage outputs. The inputs that are regeneratedat the outputs are not considered as the garbage outputs [11].The constant inputs in the reversible circuits are called theancilla inputs.A photon can provide unmatched high speed and can store

the information in a signal of zero mass. These properties ofphoton have attracted the attention of researchers to implementthe reversible logic gates in all optical domain. The all optical

implementation of reversible logic gates could be useful toovercome the limits imposed by conventional computing, andis also considered as implementation platform for quantumcomputing [12], [13], [14], [15], [16]. In the recent years,researchers have implemented several reversible logic gatesin optical computing domain such as Feynman gate, Toffoligate, Peres gate and Modified Fredkin gate. The all opticalimplementation of reversible logic gates can be achieved usingsemiconductor optical amplifier (SOA) based Mach-Zehnderinterferometer (MZI) optical switches. The Mach-Zehnderinterferometer based implementation of reversible logic gatesprovides significant advantages such as high speed, low power,fast switching time, and ease in the fabrication [17], [3], [18].In the existing literature, the NAND logic based implemen-

tation of reversible logic gates and reversible boolean functionsare the only available implementations. This is due to the lackof research in the direction of NOR logic based reversiblelogic gates and functions. In this work, we propose two novelall optical reversible NOR logic gates referred as all opticalTNOR gate and all optical PNOR gate. The proposed TNORand PNOR gates are useful for NOR logic based implementa-tion of reversible boolean functions. The proposed all opticalreversible NOR logic gates can implement the reversibleboolean functions with reduced optical cost and propagationdelay compared to the implementation of reversible booleanfunction using all optical reversible NAND gates (NANDlogic based reversible gates are all optical Toffoli gate andall optical Peres gate). The optical cost of a reversible logicgate is defined as the number of MZI switches used in itsall optical implementation. We illustrated the advantages ofproposed all optical reversible NOR gates in terms of opticalcost and delay by implementing the 13 standard booleanfunctions[19]. The 13 standard boolean functions proposedin [19] can represent all possible 256 combinations of threevariable boolean functions.The paper is organized as follows: the basics of existing all

optical reversible logic gates is presented in Section II; SectionII illustrate the proposed all optical TNOR gate; Section IVillustrate the proposed all optical PNOR gate; The opticalcost and delay analysis of proposed reversible NOR gates arediscussed in Section V, while the conclusions are provided inSection VI.

2012 IEEE Computer Society Annual Symposium on VLSI

978-0-7695-4767-1/12 $26.00 © 2012 IEEE

DOI 10.1109/ISVLSI.2012.72

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II. BASICS OF ALL OPTICAL REVERSIBLE LOGIC

The Mach-Zehnder interferometer (MZI) based opticalswitch is widely used to implement reversible logic gates[3], [17], [18], [20]. The design of all optical MZI switchis shown in Fig. 1(a). The all optical MZI switch can bedesigned using 2 Semiconductor optical amplifier (SOA-1,SOA-2) and two couplers(C-1, C-2). The operating principleof MZI based all optical switch can be explained as follows:

In MZI switch, there are two inputs ports A and B and twooutput ports called as bar port and cross port, respectivelyas shown in Fig.1(a). At the input ports, the optical signalcoming at port B is considered as the control signal (λ2), andthe optical signal coming at port A is considered as incomingsignal( λ1). The working of a MZI can be explained as: (i)when there is an incoming signal at port A and the controlsignal at port B then there is a light present at the outputbar port and no light is present at the output cross port, (ii)in the absence of control signal at input port B and thereis a incoming signal at input port A, then the outputs ofMZI are switched and results in the presence of light at theoutput cross port and no light at the bar port. We considerno light or absence of light as the logic value 0. The abovebehavior of MZI based all optical switch can be written asboolean functions having inputs to outputs mapping as (A, B)to (P=AB, Q = AB), where A(incoming signal), B(controlsignal) are the inputs of MZI and P(Bar Port), Q(Cross Port)are the outputs of MZI, respectively. The block diagram ofMZI based all optical switch is shown in Fig. 1(b). the opticalcost and the delay (Δ) of MZI based all optical switch isconsidered as unity.

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Fig. 1. Mach-Zehnder interferometer (MZI) based all optical switch

A. All optical Feynman gate

The Feynman gate (FG) is a 2 inputs and 2 outputsreversible gate. It has the mapping (A, B) to (P=A, Q= A⊕ B) where A, B are the inputs and P, Q are the outputs,respectively. The Feynman is also referred as the Controlled-NOT gate (CNOT) as when the input A=1 then the output

generated at Q will be complement of input B that is Q=B.A Feynman gate can be implemented using 2 MZI based alloptical switch, 2 beam combiner (BC) and 2 beam splitter(BS) in all optical reversible computing [3]. As the workingof the beam combiner (BC) is to simply combines the opticalbeams while the beam splitter simply splits the beams intotwo optical beams, hence researchers do not consider them inthe optical cost and the delay calculations [21], [22]. Figure2(a) and 2(b) shows the block diagram and the all opticalimplementation of the Feynman gate. As the Feynman gatecan be implemented using 2 MZI based optical switchesthus the optical cost of Feynman gate is considered as 2.In the all optical implementation of the Feynman gate, twoMZIs switches works in parallel thus the delay of the opticalFeynman gate is considered as 1Δ.

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Fig. 2. Feynman gate and its all optical implementation (FG: Feynman gate,MZI: Mach-Zehnder Interferometer, BC: Beam Combiner, BS: Beam Splitter)

B. All Optical Toffoli Gate

The Toffoli gate is a 3 inputs and 3 outputs reversible gate.The inputs to outputs mapping of a Toffoli gate is (A,B,C) to(P = A, Q = B, R = A.B⊕C), where A, B, C are the inputsand P, Q, R are the outputs, respectively [3]. The Toffoli gate isone of the most popular reversible logic gate as it is known tobe universal for implementing the reversible boolean logic. AToffoli gate can works as a NAND gate when the value at inputC=1. When C=1, the outputs of the Toffoli gate transform asP = A, Q = B, R = A.B ⊕ 1 = A.B. Figure 3(b) showsthe Toffoli gate working as a NAND gate when the value ofinput signal C is set to one. An all optical Toffoli gate can beimplemented using 3 MZI based all optical switches, 1 beamcombiner (BC) and 4 beam splitters [3]. Figure 3(a) and Fig.3(c) shows the block diagram and all optical implementationof Toffoli gate, respectively. The optical cost of Toffoli gate isconsidered as 3 as the Toffoli gate can be implemented using3 MZI based all optical switches. The Toffoli gate has a delay

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of 2Δ as two MZI switches out of three MZI switches workin parallel.

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Fig. 3. Toffoli gate and its all optical implementation (TG: Toffoli gate, MZI:Mach-Zehnder Interferometer, BC: Beam Combiner, BS: Beam Splitter)

C. All Optical Peres Gate

The Peres gate is a 3x3 reversible logic gate with the inputsto outputs mapping as (A,B,C) to (P = A, Q = A ⊕ B,R = A.B ⊕ C), where A, B, C are the inputs and P, Q, Rare the outputs respectively [18]. An all optical Peres gate canbe implemented using 4 MZI based switches, 5 beam splitters(BS) and 3 beam combiners (BC). Figure 4(a) and Fig. 4(b)shows the block diagram and the all optical implementationof a Peres gate, respectively. The optical cost of Peres gateis considered as 4 as the all optical implementation of Peresgate requires 4 MZI based switches. The delay of Peres gateis 2Δ as two MZI switches works in parallel with the twoother MZI switches.

III. PROPOSED ALL OPTICAL TNOR GATE

The proposed all optical TNOR gate can work as a re-placement of existing NAND based all optical Toffoli gate.The TNOR gate can perform NOR based implementation ofreversible boolean functions in optical computing domain withreduced optical cost and delay. The all optical TNOR gate is a3x3 reversible logic gate having inputs to outputs mapping as(A,B,C) to (P = A,Q = B, R = (A+B)⊕c), where A, B, Care the inputs and P, Q, R are the outputs, respectively. Figure5(a) and Fig. 5(c) show the block diagram and the all opticalimplementation of all optical TNOR gate, respectively. Thetruth table of all optical TNOR gate is shown in Table I. The alloptical TNOR gate will work as a NOR gate when the value ofinput signal C is 1. When C=1, the outputs of the TNOR gatetransforms as P = A, Q = B and R = (A+B)⊕1 = A+B.The all optical TNOR gate working as a NOR gate is shown

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Fig. 4. Peres gate and its all optical implementation (PG: Peres gate, MZI:Mach-Zehnder Interferometer, BC: Beam Combiner, BS: Beam Splitter)

in Fig. 5(b). The all optical TNOR gate can be implementedusing 2 MZI based switches, 4 beam splitters (BS) and 2 beamcombiners (BC). The optical cost of TNOR gate is consideredas 2, since its optical implementation requires 2 MZI basedswitches. The all optical TNOR gate has delay of 1Δ as in itsoptical design two MZI switches works in parallel.

TABLE ITRUTH TABLE OF ALL OPTICAL TNOR GATE

A B C P = A Q = B R = (A+B)⊕ C

0 0 0 0 0 00 0 1 0 0 10 1 0 0 1 10 1 1 0 1 01 0 0 1 0 11 0 1 1 0 01 1 0 1 1 11 1 1 1 1 0

IV. PROPOSED ALL OPTICAL PNOR GATE

The proposed all optical PNOR gate is 3 inputs and 3 out-puts optical reversible gate with the inputs to outputs mappingas (A,B,C) to (P = A, Q = A ⊕ B, R = (A + B) ⊕ C),where A, B, C are the inputs and P, Q, R are the outputs,respectively. The proposed all optical PNOR gate can be usedas the replacement of existing NAND based all optical Peresgate. The truth table of all optical PNOR gate is shown in TableII. An all optical PNOR gate can be implemented using 4 MZIbased switches, 7 beam splitters (BS) and 3 beam combiners(BC). Figure 6(a) and Fig. 6(b) show the block diagram andthe all optical implementation of PNOR gate, respectively. Theoptical cost of PNOR gate is considered as 4 as its opticalimplementation requires 4 MZI based switches. The delay ofall optical PNOR gate is 1Δ, as in its optical design four MZIswitches works in parallel.

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TABLE IITRUTH TABLE OF ALL OPTICAL PNOR GATE

A B C P = A Q = A⊕B R = (A+B)⊕ C

0 0 0 0 0 00 0 1 0 0 10 1 0 0 1 10 1 1 0 1 01 0 0 1 1 11 0 1 1 1 01 1 0 1 0 11 1 1 1 0 0

V. OPTICAL COST AND DELAY ANALYSIS OF REVERSIBLE

NOR GATES

The proposed all optical TNOR gate and all optical PNORgate can provide NOR based implementation of reversibleboolean functions. In the existing literature, the all opticalToffoli gate and all optical Peres are used for NAND basedimplementation of reversible boolean functions. The proposedall optical reversible NOR gates have advantages comparedto the existing all optical reversible NAND gates in terms ofoptical cost and delay. The Table III shows the optical costand delay analysis of existing all optical reversible NANDgates and the proposed all optical reversible NOR gates. Theproposed all optical TNOR gate (Optical cost=2, Delay=1Δ)has reduced optical cost and delay compared to the existing alloptical Toffoli gate that has optical cost of 3 and delay of 2Δ.The proposed PNOR gate (Optical cost=4, Delay=1Δ) whichis the NOR logic based counterpart of the Peres gate (Opticalcost=4, Delay=3Δ) shows the significant reduction in delaywith the same optical cost.In the existing literature, researchers have proposed 13

standard boolean functions that can represent all 256 possi-ble combinations of three variable boolean function[19]. Toillustrate the advantages of proposed all optical reversible

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TABLE IIIOPTICAL COST AND DELAY OF ALL OPTICAL REVERSIBLE NAND LOGIC

GATES AND PROPOSED ALL OPTICAL REVERSIBLE NOR LOGIC GATES

Optical Cost DelayFeynman Gate [3] 2 1ΔToffoli Gate [3] 3 2ΔPeres Gate [18] 4 2ΔProposed TNOR Gate 2 1ΔProposed PNOR Gate 4 1Δ

NOR gates in terms of optical cost and delay, 13 standardboolean functions are implemented using the proposed gatesas well as the existing all optical reversible NAND gates. Thecomparison study is shown in Table IV. The table IV providesthe optical cost and delay analysis of all 13 standard booleanfunctions using (i) existing reversible NAND logic gates (alloptical Toffoli gate), (ii) proposed all optical reversible NORgates (all optical TNOR gate), (iii) using existing and proposedall optical reversible logic gates (all optical Toffoli gate andproposed all optical TNOR gate). The implementation of 13standard boolean functions using proposed all optical TNORgates has 20.57% improvement in terms of total optical cost,and 50.52% improvement in terms of total delay compared tothe implementation using existing all optical Toffoli gate. Theimprovement using the combination of all optical Toffoli gateand the proposed all optical TNOR gate shows the 37.32%improvement in terms of optical cost and 26.80% improvement

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in terms of delay compared to the implementation using onlyall optical Toffoli gate.For illustration purpose, implementation of a complex stan-

dard boolean function no. 13 F13 = ABC + ABC +ABC +ABC is shown in Fig. 7. Fig. 7(a) shows the NAND basedimplementation of standard boolean function no. 13 usingexisting all optical Toffoli gate, where the all optical Feynmangates are used to generate the copy of inputs and complement-ing the inputs. The all optical implementation shown in Fig.7(a) requires 3 all optical Feynman gates and 11 all opticalToffoli gates. The optical cost of NAND based implementationof standard function F13 = ABC + ABC + ABC + ABC

using existing all optical Toffoli gate and all optical Feynmangate can be computed as Optical Cost(TG) (F13)= Opticalcost of Feynman gate * No. of all optical Feynman gate +Optical cost of all optical Toffoli gate * Optical cost of Toffoligate = 3*2+11*3=39. The delay of all optical implementationof function no. 13 can be computed as 11Δ. The NORbased all optical implementation of function no. 13 using theproposed all optical TNOR gate is shown in Fig. 7(b). The alloptical implementation shown in Fig. 7(b) requires 3 all opticalFeynman gates and 11 proposed all optical TNOR gates.The optical cost of implementation using proposed all opticalTNOR gate can be computed as Optical Cost(TNORG) (F13)=3*2+11*2=28. The implementation using existing all opticalTNOR gate has delay of 6Δ. The all optical implementation ofstandard boolean function no. 13 using the combination of alloptical Toffoli gate and the proposed all optical TNOR gate isshown in Fig.7(c). Here the combination of all optical Toffoligate and proposed all optical TNOR gate represents that 1)At the appropriate places to realize the NAND functions theall optical Toffoli gate will be used and 2) At the appropriateplaces to realize the NOR functions the proposed all opticalTNOR gate will be used. Fig.7(c) requires 4 all optical Toffoligates and 6 proposed all optical TNOR gates. The optical costof standard boolean function implemented using combinationof all optical Toffoli gate and proposed all optical TNORgate can be computed as Optical Cost(TG+TNORG) (F13)=4*3+6*2=24. From Fig.7(c), the delay can be computed as11Δ.The all optical implementation of standard boolean function

no. 13 (F13 = ABC+ ABC+ABC+ ABC) using proposedall optical TNOR gate (Optical cost =28, Delay = 6Δ)has significant reduction in terms of optical cost and delaycompared to the all implementation using existing all opticalToffoli gate (Optical cost =39, Delay = 11Δ). Also the alloptical implementation using combination of all optical Toffoligate and proposed all optical TNOR gate (Optical cost =24,Delay = 11Δ) shows the significant improvement in terms ofoptical cost compared to the all optical implementation usingonly the existing all optical Toffoli gate (Optical cost =39,Delay = 11Δ). In all optical reversible domain the all opticalPeres gate does not have the minimal optical cost amongexisting 3X3 all optical reversible gates, thus the comparisonstudy of implementation using all optical Peres gate is notshown in this work.

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Fig. 7. All optical implementation of standard boolean function no. 13(F = ABC + ABC +ABC + ABC)

VI. CONCLUSION

In this work, we have proposed two new all optical re-versible NOR gates for NOR based implementation of re-versible boolean functions. The first all optical TNOR gatecan work as a replacement for existing all optical Toffoligate, while the second all optical PNOR gate can work asa replacement of all optical Peres gate. It is illustrated thatthe proposed all optical reversible NOR gates have significantadvantages over existing reversible NAND gates in termsof optical cost and delay. Thus, the proposed new opticalreversible NOR gates can be beneficial for minimizing theoptical cost and delay of the optical reversible circuits. Allthe optical reversible gates are functionally verified at the logiclevel using Verilog HDL. This is done by creating a Veriloglibrary of Mach-Zehnder interferometer, beam combiner andbeam splitter. In conclusion, this work advances the state ofthe art of reversible optical circuits by providing NOR logicbased reversible gates as an alternative to NAND logic basedreversible gates.

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TABLE IVOPTICAL COST AND DELAY ANALYSIS OF REVERSIBLE NOR LOGIC GATES BY IMPLEMENTING 13 STANDARD BOOLEAN FUNCTIONS

Implementation using Implementation using Implementation using allFunction No. Standard Function all optical proposed all optical optical Toffoli gate and

Toffoli gate TNOR gate proposed all optical TNOR gateOptical Cost Delay Optical Cost Delay Optical Cost Delay

1 F = ABC 6 4Δ 10 3Δ 6 4Δ2 F = AB 3 2Δ 6 2Δ 3 2Δ3 F = ABC +ABC 19 10Δ 16 5Δ 12 6Δ4 F = ABC + ABC 21 9Δ 16 5Δ 12 6Δ5 F = AB +BC 9 6Δ 6 2Δ 5 3Δ6 F = AB + ABC 16 9Δ 14 4Δ 10 6Δ7 F = ABC + ABC +ABC 30 15Δ 20 6Δ 20 12Δ8 F = A 3 2Δ 2 1Δ 2 1Δ9 F = AB +BC +AC 15 10Δ 14 4Δ 10 6Δ10 F = AB + BC 11 5Δ 12 3Δ 9 5Δ11 F = AB +BC + ABC 24 9Δ 12 4Δ 11 5Δ12 F = AB + AB 13 5Δ 10 3Δ 7 4Δ13 F = ABC + ABC +ABC + ABC 39 11Δ 28 6Δ 24 11Δ

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