Mach-Zehnder Interferometer Based All-Optical Fredkin Gate G.K.Maity1 , S.P.Maity2 , T. Chattopadhyay3 and J.N.Roy3 2 3

1 Calcutta Institute of Technology, Uluberia , Howrah, W.B. India Department of Information Technology, Bengal Engineering College and Science University, Shibpur, Howrah, India Department of Physics, College of Engineering and Management, Kolaghat KTPP Township. Midnapur (east). 721171, W.B., India. [email protected]

Abstract. Conservative and reversible logic gates are widely known to be compatible with revolutionary computing paradigms such as optical and quantum computing. A fundamental conservative reversible logic gate is the Fredkin gate. This paper presents an optical circuit for realization of Fredkin gate in all-optical domain. Semiconductor optical amplifier (SOA) based Mach-Zehnder interferometer (MZI) can play a significant role in this field of ultra fast all optical signal processing. Key Words: Reversible logic, Fredkin gate, Mach-Zehnder interferometer (MZI), optical switch.

1

Introduction

Reversible logic is of increasing importance to many future computer technologies. Reversible circuits are those circuits that do not lose information and reversible computation in a system can be performed only when the system comprises of reversible gates [1-5]. These circuits can generate unique output vector from each input vector, and vice versa, that is, there is a one-to-one mapping between input and output vectors. The classical set of gates such as AND, OR, and EXOR are not reversible as they are all multiple-input single output logic gates. A gate is reversible if the gate’s inputs and outputs have a one-to-one correspondence, i.e. there is a distinct output assignment for each distinct input. Therefore, a reversible gate’s inputs can be uniquely determined from its outputs. Reversible logic gates must have an equal number of inputs and outputs. A gate is conservative if the Hamming weight (number of logical ones) of its input equals the Hamming weight of its output. A conservative reversible gate is a gate that is both conservative and reversible simultaneously. A consequence of a gate’s reversibility and conservability is that conservative reversible gates are zero preserving and ones-preserving. A conservative reversible logic gate effectively permutes its inputs to form its outputs. Conservative gates are not necessarily reversible as several inputs with equal Hamming weights can be mapped to a single output of the same Hamming weight. For example, consider the two-input, two-output logic gate where one output is the logical AND of A. Ghosh and D. Choudhury (Eds.): IConTOP 2009 c Department of Applied Optics and Photonics, University of Calcutta 2009 

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the inputs, and the other output is the logical OR of the inputs. This logic gate is conservative and irreversible. Fredkin gate is a fundamental conservative reversible logic gate [1-2]. This paper presents an optical circuit for realization of Fredkin gate in all-optical domain. Semiconductor optical amplifier (SOA) based Mach-Zehnder interferometer (MZI) can play a significant role in this field of ultra fast all optical signal processing. Simulation has been done with Mathcad-7.

2

SOA based MZI switch:

Mach-Zehnder interferometer (MZI) switch, as shown in Fig.1 and Fig.2, is a very powerful optical device to realize ultra fast all-optical switching. In this switch a semiconductor amplifier (SOA) is inserted in each arm of a MZ interferometer [6-8]. The pulsed signal at the wavelength λ1 enters to the upper arm through coupler C2 such that most power passes through upper arm. At the same time, the incoming signal pulse at the wavelength λ2 enters port-1, is split equally by this coupler C1 and propagates simultaneously in the two arms. The intensity transmission characteristics at port-3 and port-4 can be expressed as [9]  1 G1 {k1 k2 + (1 − k1 )(1 − k2 )RG − 2 k1 k2 (1 − k1 )(1 − k2 )RG cos(Δφ)} 4 (1)  1 T4 (t) = G1 {k1 (1 − k1 ) + (1 − k2 )k2 RG − 2 k1 k2 (1 − k1 )(1 − k2 )RG cos(Δφ)} 4 (2) G2 α 2 Where, RG = G and are the time dependent gain, Δφ(t) = − ln( ), G1 2 G1 α is the line width enhancement factor (taken 7.5 here), and are the ratios of the couplers C1 and C2 respectively. For simplicity we take . The output signal power at port-3 and port-4 are,

T3 (t) =

Pj (t) = Pip (t) ∗ Tj (t), j = 3, 4

(3)

Where Pip (t) is the power of the incoming signal pulse. When both beams are present simultaneously, the control pulse saturates SOA-1 on change in carrier density inside SOA. The gain of the SOA during this period is [10-11], G(t) =



1− 1−

1 G0

1    (t) exp − UUinsat

(4)

Where Usat is the saturation energy of the SOA and Uin (t) = 2

t −∞



Pin (t ) dt



. Here we consider a Gaussian Pulse Pin (t) = σE√inπ exp(− σt 2 ) as control signal, where is the input pulse energy, σ is the full width at half maximum (taken 2.8 here). Now the gain recovery is happened in SOA-1 with time constant . The momentarily gain during this time is [11]  G(t) = G0

G(ts ) G0

s) exp[− (t−t ] τ e

, t > ts

(5)

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where, G(ts ) is the gain after saturation of SOA-1. We show the gain change for SOA-1 in the Fig.3 (All the simulation and calculation is done with Mathcad7). Physically the pulse is so short that the gain has no time to recover [10]. Here in we take G0 = unsaturated amplifier gain =29.6 dB, τe = 95 ps, UEsat = 0.1. From the graph in the fig.3 we find ts = 5.5 ps and G(ts ) = 7.969 dB. The beam in

Fig. 1. MZI-based optical switch

the lower arm experiences the unsaturated amplifier gain (as there is no strong optical pulse to saturate the SOA-2) i.e.G1 = G2 , recombine at the coupler C3. So that Δφ ≈ π . Hence all one bits are directed toward the bar port (upper port-3 in the figure).In the presence of control pulse, the output pulse at port-3 and port-4 is shown in the Fig.4 and the transmitted intensity for both port are also shown in Fig.5. However, in the absence of the λ1 beam, the both the incoming signal beam in two arms experiences the same unsaturated amplifier gain G0 in both SOA (i.e G1 =G2 . ), recombine at the coupler C3. So Δφ=0 . From equation (3), we can say P3 (t)=) and the pulse only exits at the cross port (lower port-4 in the figure). In the absences of control pulse, the output pulse at port-3 and port-4 is shown in the Fig.6. Optical filters are placed in

Fig. 2. Schematic diagram of SOA based MZI optical switch

front of the output ports for blocking theλ1 signal. The MZ scheme is preferable over cross-gain saturation as it does not reverse the bit pattern and results in a higher on-off contrast simply because nothing exits from bar port during 0 bits . Now, it is clear that in the absence of control signal (λ1 ), the incoming signal (λ2 ) exits through cross-port (lower channel) of MZI as shown in Fig.1.In this case no light is present in the bar-port (upper channel). But in the presence of control signal, the incoming signal exits through bar port of MZI as shown

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in Fig.1. In this case no light is present in the cross port. In the absence of incoming signal, bar-port and cross- port receives no light as the filter blocks the control signal. Schematic block diagram of MZI is shown in Fig.2. In our earlier proposal we utilized the above characteristics of MZI to design all-optical binary logic operations [12]. In this present proposal, the above characteristics of MZI are exploited for designing well-known reversible logic gates Fredkin gate (FG).

Fig. 3. Gain variation in SOA-1

Fig. 4. Output pulse at Bar and cross port when Control pulse=1

3

Fredkin Gate (FG)

Fredkin gate is a (3*3) conservative reversible gate [1, 2, 4, 5]. It has three inputs (A, B, C) and three outputs (X, Y, Z) satisfy the relation as follows: X=A  Y = A¯ ∧ B ∨ (A ∧ C)  Z = (A ∧ B) ∨ A¯ ∧ C The truth table is in the table1. Schematic diagram is given in Fig.7.

(6)

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Fig. 5. Intensity Transmission characteristics for port-3 and port-4 when Control pulse=1

Fig. 6. Output pulse at Bar and cross port when Control pulse=0

4

Fredkin Gate (FG) with MZI

The MZI based circuit for all optical reversible Fredkin gate is given in Fig.8. Here the output X is directly taken from the input pulse A through a beam splitter (BS). Light from A incidents on beam splitter and split into two part. A part of light is passed through wavelength converter that converts the wavelength of light from λ2 to λ1 .Finally they are connected with MZI switches MZI-1 and MZI-2 through erbium doped fibre amplifier (EDFA) so that they can act as control signal to the switches. Other two inputs B and C are the two incoming signals of the MZI-1 and MZI-2 respectively. The bar port of MZI-1 (B1 ) and cross port of MZI-2 (C2 ) is combined by a beam combiner BC-1 to get the output

Fig. 7. schematic diagram of Fredkin Gate (FG).

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input output ABCXY Z 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 1 0 1 1 0 1 0 1 1 1 1 1 1 1 Table 1. Truth table of Fredkin Gate.

Fig. 8. Circuit for all-optical Fredkin Gate (FG). BC : Beam Combiner EDFA : Erbium Doped Fiber Amplifier Wavelength Converter (WC)

Y and also the bar port of MZI-2 (B2 ) and cross port of MZI-1 (C1 ) is combined by BC-2 to get the final output Z. Let us discuss the operation in details. Here, the presence of light is taken as 1 state and absence of light is taken as 0 state. (1) When A=B=C=0 i.e. no light is present at input , then the final outputs receives no light i.e. X=Y=Z=0. (2) When A=B=0 and C=1, then according to the working principle of MZI described in the section-2, only the lower channel of MZI-2 receives the light. Hence, C2=1 and B2=0 (as the control signal of MZI-2 is 0). In this case upper and lower channel of MZI-1 receives no light (as the input signal of MZI-1 receives no light). Hence, C1=B1=0 .Therefore X=0, Y=1, Z=0. It satisfies the second row of the truth table-1. (3) When A=C=0 and B=1, then control signal is absent but incoming signal is present at MZI-1. So lower channel receives light and hence B1=0 and C1=1. In this case both incoming and control signal is absent in MZI-2. Therefore, B2=C2=0. Hence X=Y=0, Z=1. (4) When A=0 and B=C=1, then both the MZI switches receive light in the incoming signal but receives no light at control signals. So B1=B2=0 and C1=C2=1. Hence X=0, Y=Z=1.

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(5) When A=1 and B=C=0, then both the MZI receives no light as incoming signal. So B1=B2=0 and C1=C2=0. Hence X=1, Y=Z=0. (6) When A=1, B=0 and C=1, then MZI-1 receives no light as incoming signal and MZI-2 receives light pulse in both the incoming and control signal. So B1=C1=0 and B2=1, C2=0. Hence X=1, Y=1 and Z=0. (7) When A=1, B=1 and C=0, then MZI-2 receives no light as incoming input signal and MZI-1 receives light pulse as both the incoming and control signal. So B2=C2=0 and B1=1, C1=0. Hence X=1, Y=0 and Z=1. (8) When A=1, B=1 and C=1, then both the MZI-1 and MZI-2 receives light pulse as incoming and control signal. So B1=1, C1=0 and B2=1, C2=0. Hence X=1, Y=1 and Z=1. Simulation is done in Mathcad-7 taking the same values used in the section2. The power of the input pulse is taken A=2.26 dBm, B=C=1.13 dBm. The result is shown in the Fig.9.

5

Conclusion

In this paper, MZI based circuit has been proposed and described for realization of Fredkin gate. It is all-optical in nature. Simulations are done. As Fredkin gate is the fundamental conservative reversible logic gate so the different arithmetic and logic operations in reversible system can be performed.

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Fig. 9. Simulation result of Fig.8 x- axis: Time (ps) y-axis: Power(dBm)

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