## Abstract

The performance of all-optical XOR gate based on quantum-dot (QD) SOA MZI has been simulated. The saturation power, optical gain and phase response of a QD SOA has been analyzed numerically using a rate equation model of quantum dots embedded in a wetting layer. The calculated response is used to model the XOR performance. For the parameters used here, XOR operation at ~250 Gb/s is feasible using QD based Mach-Zehnder interferometers. The speed is limited by the relaxation time from wetting layer to the quantum dots.

© 2005 Optical Society of America

## 1. Introduction

Photonic logic operations, such as all-optical XOR operation, is important for all-optical signal processing such as bit pattern matching [1], pseudo random number generation [2] and label swapping [3]. As is the case for electronic logic gates, all-optical logic gates fundamentally rely on nonlinearities [4]. So far, methods utilizing the nonlinearities of optical fiber [1,5] and semiconductor optical amplifier (SOA)[6,7,8] have been used to demonstrate all optical XOR functionality. The all-optical logic gate based on the nonlinearities of optical fiber has the potential of operating at terabits per second due to very short relaxation times (<100 fs) of its nonlinearity. The disadvantages of optical fiber are its nonlinearity is weak and long interaction lengths or high control energy is required to achieve reasonable switching efficiency. On the other hand, semiconductor optical amplifier has the advantages of high nonlinearity and easy of integration. All optical logic XOR at speeds of 20 to 40 Gb/s have been demonstrated with semiconductor optical amplifier (SOA) loop mirror (SLALOM) [6], ultrafast nonlinear interferometer (UNI) [7], and SOA based Mach-Zehnder interferometer (SOA-MZI) [8]. The all-optical logical gate base on SOA-MZI is believed to be stable, compact and simple.

However, the operating speed of XOR is limited to ~100 Gb/s due to the response time of gain saturation in a regular SOA [9–11]. SOA with quantum dot active region is a promising candidate for faster speed of operation. In this paper, we simulated the high speed performance of a quantum-dot based SOA-MZI XOR optical logic gate. Using a simplified dynamic model, the rate equations for carrier density in a QD semiconductor optical amplifier (SOA)were derived and solved numerically for gain recovery and phase recovery. The analysis is then extended to the XOR gate performance of a Mach-Zehnder interferometer utilizing QD based SOAs.

## 2. Principle of operation

Because of its compact and stable structure, SOA-MZI based XOR gate is desirable for integration needed for complex logic circuits. Fig. 1 presents a schematic diagram for SOA-MZI and its principle of operation.

The optical XOR gate with SOA-MZI consist of a symmetrical MZI with two SOAs placed in upper and lower arms of the interferometer as shown in Fig. 1. To perform the XOR function as shown in the truth table, two optical control beams A and B carried by optical signals at different wavelengths λ_{1} and λ_{2} are sent into port A and B of the MZI separately. The wavelengths of the two data signals can also be same. The signal, a clock stream of continuous series of “1”s̀ or a CW beam is split into two equal parts and injected into the two SOAs. Initially the MZI is unbalanced i.e. when A=0 and B=0, the signal at port C traveling through the two arms of the SOA acquires a phase difference of π when it recombines at the output port, thus the output is “0”. When A=1, B=0, the signal traveling through the arm with signal A acquires a phase change due to the cross phase modulation (XPM) between the pulse train A and signal and the signal traveling through the lower arm does not have this additional phase change. This results in an output “1”. The same phenomenon happens if A=0 and B=1. However, when A=1 and B=1 the phase change for the signal traveling both arms are equal, hence the output is “0”. When the phase shift is optimum, the best contrast ratio is achieved at the output port.

## 3. Quantum dot model

Theory of the signal amplification in a Quantum-dot SOA has been developed [12]. Current injection in a quantum dot is schematically shown in Fig. 2. The carriers are injected into the wetting layers from which it makes a fast transfer to the quantum dot.

We assume all the Quantum-dots in SOA are identical and uniform, also there is only one confined energy level in conduction and valance band of each dot. We label the carrier density by *N*_{w}
in wetting layer, *N*_{id}
in the ^{i}th quantum dot, and *N*
_{imax} as the maximum carrier density that *i*th quantum-dot can sustain. The rate equations for carrier density are given as follows:

where *J* is the injection current density, *d* is the total wetting layer thickness, ${\tau}_{w\mathit{\to}d}^{-1}$ is the transition rate between wetting layer and ground state in quantum dot, ${\tau}_{\mathit{\text{wr}}}^{-1}$ is the carrier recombination rate in wetting layer, ${\tau}_{d\mathit{\to}w}^{-1}$ is the excitation rate from ground state to wetting layer, ${\tau}_{\mathit{\text{dr}}}^{-1}$ is the recombination rate in semiconductor dot. We now introduce two additional gain dynamic equations induced by carrier heating and spectral hole burning effects [13,14]:

${\tau}_{\mathit{\text{SHB}}}^{-1}$ is the carrier-carrier scattering rate while ${\tau}_{\mathit{\text{CH}}}^{-1}$ is the temperature relaxation rate. *ε*_{SHB}
and *ε*_{CH}
are the nonlinear gain suppression factors due to carrier heating and spectral hole burning [14], and *g*
_{i} is the gain in the for the quantum dot transition. The total gain is given by:

The equation for the intensity S(t, z) of the CW signal is given by:

Solution of (6) is:

where *G*(*t, z*)=*Exp*[*h*(*t*)] and *h*(*t*)=∫z0*g*(*t,z*)*dz*

Integration of Eq. (6) over z leads to

Hence, we can transform (1)–(4) into temporal gain forms:

where hmax is the maximum value of integrated gain. *h*
_{max}=${\int}_{0}^{\mathrm{z}}$
*a*(*N*
_{max}-*N*
_{tr})*dz*′, *N*_{tr}
is the carrier density at transparency, *a* is the differential gain of SOA, ${h}_{\mathit{in}}={\int}_{0}^{z}\frac{\alpha J{\tau}_{wr}}{ed}d{z}^{\prime}$, *ε*_{SHB}
and *ε*_{CH}
are the gain suppression factors owing to spectral hole burning and carrier heating effect respectively. The phase change equation is given by:

where *α* is the usual linewidth enhancement factor associated with the interband transitions and *α*_{CH}
is the linewidth enhancement factor for carrier heating. The linwidth enhancement factor for spectral hole burning (SHB) process, *α*_{SHB}
~0 [14]. Typical α values are in the 2 to 7 range and *α*_{CH}
~1 [15]. . In the QD model used here, the majority of the carriers are injected into the wetting layer from which they transfer to the QD energy levels. The carrier relaxation time from the wetting layer to the QD is ~0.5 to10 ps [16]. The wetting layer is populated by the injected current and it serves as a reservoir of carriers for the QD. Thus with increasing current more carriers are present for transitions in the QD. This results in increasing saturation power and faster gain recovery for QD amplifiers. The calculated results for optical gain as a function of output power for different injected current densities is shown in Fig. 3. With increasing current density, the saturation power increases. In the calculation, the parameters are set as follows [16,17]: *τ*_{w→d}
=6ps, *τ*_{wr}
=0.2ns, *τ*_{dr}
=0.4ns, *τ*_{d→w}
=10ns, *τ*_{SHB}
=100fs, *τ*_{CH}
=300fs, *ε*_{SHB}*=ε*_{CH}
=0.08ps, and, Γ=0.15.

The calculated gain and phase recovery curves following a 1.5 ps wide pulse with 1.8 pJ pulse energy for two injected current densities is shown in Fig. 4. The gain recovers faster at high current density.

## 4. Simulation of Mach-Zehnder interferometer and XOR operation

Output optical intensity after MZI at wavelength λ_{3} (see Fig. 1) depends on the phase changes the CW signal undergoes at the two arms of the MZI due to the co-propagating data signals. At the output of the MZI, the light at wavelength λ_{3} propagating through the two arms interfere and the time dependent XOR output intensity is described by the following basic interferometer equations:

In which, G_{1,2}(t) is time dependent gain and ϕ_{1,2}(t) is phase shift in the two arms of SOA-MZI. The phase shift ϕ_{1,2}(t) is related to the gain G_{1,2}(t) by the linewidth enhancement factors α and *α*
_{CH} from Eq. (13):

Figure 5 shows the XOR performance of a MZI with quantum dot SOAs̀. The top two traces are the input pulses as a function of time. The bottom trace is the XOR output. The data rate in Fig. 5 is 160Gb/s and the pulse width is 1.5 ps. In our calculation, we set parameters as follows [16,17]: *τ*_{w→d}
=6ps, *τ*_{wr}
=0.2ns, *τ*_{dr}
=0.4ns, *τ*_{d→w}
=10ns, *τ*_{SHB}
=100fs, *τ*_{CH}
=300fs, *ε*_{SHB}
=*ε*_{CH}
=0.08ps, Γ=0.15, line width enhancement factor *α*=5, *α*_{CH}
=1 amplifier maximum gain=20dB, injected current density=4kA/cm^{2}.

A quantity known as *Q* is widely used to characterize the signal quality for pseudo-random signals[17]. Q -factor is defined as:

$\overline{{P}_{1}}$ is the average power of output signal *“*
_{1}”, σ1 is the standard deviation of all “1”. $\overline{{P}_{0}}$ and *σ*
_{0} are defined analogously for output signal “0”. Here we utilize “pseudo-eye-diagrams” (PED) [18] to plot the signal in the presence of noise. The calculated “1”s̀ superimposed on each other in the presence of noise for a 2^{7}-1 pseudo-random input bit pattern is shown in Fig. 6. The two Fig.s are for data rates of 80 Gb/s, 160 Gb/s and 250 Gb/s are generated at *τ*_{w→d}
=6ps and *τ*_{w→d}
=1ps respectively. The full width at half maximum of the Gaussian pulse used in the calculation is 1/5 th of the pulse period. The calculated Q-values are shown. The Q-value decreases as the data rate is increased.

As shown in Fig. 7, under repetition rate of 160Gb/s and 250Gb/s, system quality factor Q increases when the injection current is increased. Above 2kA/cm^{2}, however, Q saturates. The saturated Q factor for 250Gb/s is ~6 for the parameters [15,16] used here. A value of ~6 for Q can result in bit-error-rate of less than 10^{-9} [19]. Hence, we conclude that ~250Gb/s is the speed limit for the parameters used here. Higher speed of operation is feasible for shorter relaxation time from the wetting layer to the QD level.

Q value is sensitive to the input pulse width as shown in Fig. 8. The injection level for the calculation in Fig. 6 is 4 kA/cm^{2}. Q decreases when increasing the pulse width because two neighboring pulses tend to overlap for wider pulse widths.

## 5. Summary and discussion

The performance of all-optical XOR gate based on quantum-dot (QD) SOA MZI has been simulated. The saturation power, optical gain and phase response of a QD SOA has been analyzed numerically using a rate equation model of quantum dots embedded in a wetting layer. The saturation power increases and the gain recovery time decreases with increasing injection current. The calculated response is used to model the XOR performance. For the parameters used here, XOR operation at ~250 Gb/s is feasible using QD based Mach-Zehnder interferometers. The primary reason for the faster response of QD semiconductor optical amplifier (SOA) compared to that for SOA with regular active region is due to the presence of the wetting layer. The wetting layer serves as a carrier reservoir layer. Carriers depleted by the injected optical pulse in the QD ground state are replaced by fast carrier transfer from the wetting layer. At higher injected current density there are more carriers in the wetting layer and hence the performance is better as shown by higher value of Q (Fig. 7). Thus the speed is limited by the relaxation time from wetting layer to the quantum dot state. The calculation in this paper has been carried out for relaxation times of 1 ps and 6 ps. Measurements of this relaxation time have indicated values in the 0.5 to 10 ps range [16].

The calculation in this paper assumes a Gaussian pulse with a full width at half maximum of 1/5 th of the bit period for Fig. 7. We believe this assures that overlap between the pulses is small for a pulse sequence. For larger pulse widths, the overlap will contribute to the “0” state and hence the signal to noise ratio values (Q -values) obtained will be smaller than that calculated here. The effect of larger pulse width relative to the bit period is shown in Fig. 8. It shows Q - value decreases as the pulse width becomes larger. At larger pulse energies more carriers will be depleted and hence a larger phase shift is produced which will increase the signal to noise ratio. Thus we expect a better performance at larger pulse energies. However, since the depleted carriers needs to be replaced by carriers from the wetting layer for high speed, the device also needs to operate with high wetting layer carrier density and hence higher current density.

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