## Abstract

A scheme to realize all-optical Boolean logic functions AND, XOR and NOT using semiconductor optical amplifiers with quantum-dot active layers is studied. nonlinear dynamics including carrier heating and spectral hole-burning are taken into account together with the rate equations scheme. Results show with QD excited state and wetting layer serving as dual-reservoir of carriers, as well as the ultra fast carrier relaxation of the QD device, this scheme is suitable for high speed Boolean logic operations. Logic operation can be carried out up to speed of 250 Gb/s.

© 2010 OSA

## 1. Introduction

In future high-speed optical communication systems, logic gates will play important roles, such as signal regeneration, addressing, header recognition, data encoding and encryption [1]. In recent years, people have demonstrated optical logic using different schemes, including using dual semiconductor optical amplifier (SOA) Mach-Zehnder interferometer(MZI) [2, 3], semiconductor laser amplifier (SLA) loop mirror [4], ultrafast nonlinear interferometer (UNI) [5], four-wave mixing (FWM) in SOA [6] and cross gain (XPM)/cross phase (XPM) modulation in nonlinear devices [7]. Among above schemes, the SOA based MZI has the advantage of being relatively stable, simple and compact. To the author’s knowledge, however, operation speed of these schemes are limited by no more than 40 Gb/s. In order to realize higher speed data processing, faster device and schemes are needed.

The emergence of quantum-dot (QD) SOAs in recent years provided a better device for signal processing at communication band. Up till now, such device has experimentally demonstrated high saturated output power and low noise figure [8, 9], ultrafast carrier relaxation between QD energy states [10, 11] and a much smaller carrier heating (CH) impact on gain and phase recovery [12]. In recent years, rate equations approach is widely used to simulate XOR logic operation based on QD-SOAs [13]. However, [13]’s simulation is only based on inter-band carrier transitions, while neglecting gain saturation and nonlinear effects, which dominate the device’s gain dynamics above certain input power level [11].

In this work, we present a model to simulate optical logic operations using QD-SOA-MZI. The model supposes two discrete QD energy levels acting in carrier dynamics and nonlinear effects affecting the gain and phase dynamics of the device. Results show with nonlinear effects (especially the ultrafast recovering spectral hole burning (SHB) effect) expediting the gain recovery dynamics, this logic gate can have an improved output quality at high speed operation. For example, the calculated quality Q factor is 7.6 using input pulse with 0.5pJ pulse energy and 1.5ps pulse width and pump current density 1.8kA/cm^{2} (compared to Q factor ~4.8 under the same operating condition as reported in [13]).

## 2. QD-SOA structures and rate equations

The device we study here is the commonly discussed InAs/GaAs QD-SOA, with InAs Stranski-Krastanov (SK) quantum dots embedded in GaAs layer [14, 15]. This type of device can provide ~15dB gain at wavelength 1550nm with noise figure as low as 7 dB [9]. The active layer of the device consists of alternately stacked InAs island layers and GaAs intermediate layers. This configuration can significantly increase areal dot density and the modal gain of the SOA [14] and the gain is nearly polarization independent [16]. In this work we used the 2-level QD model to simulate the carrier transitions in the device [17–19]. The transition between the wetting layer, the QD excited state (ES) and the QD ground state (GS) is schematically illustrated in Fig. 1 .

We used rate equations to describe gain and phase dynamics in QD device [20]. The carrier density dynamics can be described as:

*w*,

*h*and

*f*are the occupation probabilities of the wetting layer, the QD excited state and ground state, respectively;

*N*,

_{wm}*N*and

_{esm}*N*are the maximum possible carrier densities of each state;

_{gsm}*Г*is the active layer confinement factor,

_{d}*I*is the injected current,

*V*is the effective volume of the active layer,

*a*is differential gain,

*S*(t) is photon density in the active region.

The gain of QD-SOA is come from both carrier density dynamics and nonlinear processes like CH and SHB effects [12, 21], expressed in (4) as:

where*ε*and

_{CH}*ε*are the gain suppression factors of carrier heating and spectral hole burning effects, respectively.

_{SHB}The injected light and carrier heating effect both contribute to the cross-phase modulation between probe and data signal, thus a phase change to the probe is [20]:

where*G*is the gain factor of the device with

_{l}(t)=exp[g(t)l]*l*being the effective length of the active layer,

*α*and

*α*are the linewidth enhancement factors of the waveguide and carrier heating process, respectively.

_{CH}## 3. Operation Principles of all-optical logic gates using QD-SOA-MZI

A QD-SOA MZI is used to realize all-optical XOR, AND, NOT operations. As is shown in Fig. 2
, two identical QD-SOAs form the two arms of the interferometer, three optical data streams centered at different wavelengths are coupled into the two arms, where the control beam at *λ _{3}* is evenly split into two branches at port 3 and guided into the two QD-SOAs respectively. The two branches each interact with data stream A or B in QD-SOA and experience modulated gain and phase due to XGM and XPM processes.

The two beams recombine at port 4, the phase shifters give the two arms an additional π phase difference, after a band-pass filter which screens out wavelength component of *λ _{1}* and

*λ*, the interference result (at

_{2}*λ*) can be expressed as [3]:

_{3}*P*(

_{cb}*t*) is the time-dependent power of control beam.

*ϕ*and

_{1}*ϕ*are phases experienced by control beam in each arm expressed in (5).

_{1}Similar to the reported schemes of SOA-MZI based optical logic XOR operation [3], we put data streams A and B into port 1 and 2 respectively and a much weaker clock as control beam. When *A=B*, then *G _{1}=G_{2}*,

*ϕ*, output will be 0 according to (6); if

_{1}=ϕ_{2}*A≠B*,

*P*, and its temporal shape similar to the input control beam pulse as a result of fast gain response.

_{out}(t)≠0Similar to XOR, we used a similar MZI scheme [22] to realize logic AND operation. By putting data A to port 1 and using data B as control beam, we can get gain modulated pattern of data B out of port 4 resembling logic function A AND B. To make results better in quality, a low power CW light goes into port 2 to cancel out the background noise of data stream A.

If we use a clock as data B, and optical CW as control beam, we will get out of port 4 data pattern of A XOR “1”, which is the same in terms of truth value as “NOT A”.

The QD device’s amplified spontaneous emission (ASE) can degrade the signal-to-noise ratio (SNR) of transmitted signal by *SNR _{in}=F∙SNR_{out}*, where

*F*is the device’s noise figure. For QD-SOA with short active region (~1mm), the ASE produced are usually much smaller compared to the saturation output power [11], so it’s impact on SOA’s XGM and XPM processes are totally negligible. For this reason, we only added an additional gain factor of

*F*to the control beam’s input noise power through the device amplification.

## 4. Simulation results and output quality evaluation

We solved rate Eqs. (1-3) under different conditions. Parameters used are experimental results on QD-SOAs for central wavelength *1.55μm*: *τ _{wr}*=

*τ*,

_{esr}=200ps*τ*,

_{gsr}=50ps*τ*,

_{w-e}=3ps*τ*[20],

_{e-w}=300ps*τ*,

_{g-e}=10ps*Г*, linewidth enhancement factors

_{d}=10%*α=4*,

*α*[23, 24], QD energy levels’ densities of states

_{CH}=0.2*n*,

_{w}=5.4×10^{17}cm^{−3}*n*[18], QD areal density is

_{w}:n_{e}:n_{g}≈15:2:1*7.5×10*, saturated output power is

^{10}cm^{−2}*18dBm*at

*1.55μm*wavelength[8], device differential gain

*a=8.6×10*[15], effective length

^{−15}cm^{2}*l*=1.0mm, noise figure

*F=7dB*[8], transparency pump current density is

*~0.2kA/cm*[12], gain suppression factors are

^{2}*ε*,

_{CH}=0.5×10^{−23}m^{3}*ε*[25], which correspond to threshold input pulse energy for both nonlinear effects~0.47 pJ. Value of time constant

_{SHB}=7.5×10^{−23}m^{3}*τ*has been measured in many experiments, the smallest reported value is ~0.1ps [12] and the largest measured value goes up to several picoseconds [18].

_{e-g}2^{31}-1 PRBS signals are used as input signals, input data SNR is *20dB*. Results of XOR, AND, NOT logic are shown in Fig. 3
and 4
, the output eye-diagram is also plotted to show output quality. Quality of output can also be quantitatively evaluated by calculating quality factor, which is defined as *Q=(S _{1}-S_{0})/(σ_{1}+σ_{0})* [1]. Here

*S*and

_{0}*S*are the average peak powers of output “1”s and “0”s, respectively;

_{1}*σ*and

_{0}*σ*are their standard deviations. The output bit-error rate (BER) is related with quality factor by [1]:

_{1}*BER≈(2π)*.

^{-1/2}exp(-Q^{2}/2)/QAt fixed operation condition, the output qualities are the same according to simulation result. This is because the three gates use the same scheme and share the same noise origin.

The calculated quality factor shows significant dependence on injected current density, pulse width, *τ _{e-g}* and single pulse energy. Figure 5
shows the output quality’s dependence on injected current density and input pulse width. From the results we find that at low injected current density level (J<1.8 kA/cm

^{2}), the Q factor is lower and increases as current density increases. This can be explained as: with increased current density, more carriers are fed to the wetting layer, each QD energy level can recover faster to initial carrier density level after depletion following pulse injection and amplification. This reduces the pattern effect considerably. For higher current density (J>1.8 kA/cm

^{2}), the increase in J will have a smaller impact on the gain recovery process because of carrier saturation. Also, narrower the input pulse (less energy and hence less carrier depletion) also results in better performance (higher Q).

Figure 6
shows the calculated Q factor dependence on single pulse energy and carrier relaxation time between QD excited state and ground state. From the results we see a decrease in output quality when increasing single pulse energy of the input data and *τ _{e-g}*. As single pulse energy increases, the carrier density of the active region of the device is depleted more and takes longer time to recover to initial level, thus lead to bigger patterning effect and degrade the quality. The transition lifetime

*τ*determines the speed of gain and phase recovery in the active region, thus Q-factor is higher for shorter transition times at high operation speed.

_{e-g}## 5. Conclusion

In this paper we presented a model to simulate high speed all-optical logic gates using QD-SOA based Mach-Zehnder Interferometer. Results show that QD-SOA based MZI can perform logic operations such as AND, XOR and NOT at high bit-rate up to 250 Gb/s. The impact on the high speed output quality (Q-factor) by a number of parameters, including injected current density, transition lifetime *τ _{e-g}*, input pulse width and single pulse energy, are also studied and discussed. Results show that for operation speed as high as 250 Gb/s, the Q factor is typically above 7 and can reach 11 under best conditions. For best output quality, the logic system requires injected current to be sufficiently high (>1.8 kA/cm

^{2}) and single pulse energy not be too big (<1.0 pJ), narrower input pulse width (FWHM ~1.0ps) can also lead to better output quality.

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