This paper presents the simulation and design of an alloptical subtractor using a quantumdot semiconductor optical amplifier MachZehnder interferometer (QDSOA MZI) structure consisting of two cascaded switches, the first of which produces the differential bit. Then the second switch produces the borrow bit by using the output of the first switch and the subtrahend data stream. Simulation results were obtained by solving the rate equations of the QDSOA. The effects of QDSOA length, peak power and current density have been investigated. The designed gate can operate at speeds of over 250 Gb/s. The simulation results demonstrate a high extinction ratio and a clear and wideopening eye diagram.
I. INTRODUCTION
Alloptical gates will be essential elements in future communications systems. Alloptical processing has many applications, such as wavelength conversation, adddrop multiplexing, regeneration clock recovery, simple bitpattern recognition, address recognition, signal processing, and packet synchronization [
1

3
]. Subtraction is one of the most important operations in Boolean functions and is used in binary subtraction, binary counting, arithmetic logic units, encryption, and decryption in security networks. Many designs for a single subtractor or subtractor with adder have been proposed by using a semiconductor optical amplifier (SOA) and periodicallypoled lithium niobate (PPLN) waveguide [
4
], microring resonator [
5
], exploiting fourwave mixing (FWM) in a SOA [
6
], a SOA basedMachZehnder interferometer (MZI) [
7
], a threestate system [
8
], a terahertz optical asymmetric demultiplexer (TOAD) [
9
], a singleslot waveguide [
10
], two SOAs [
11
], implementing nonlinear material [
12
], and a nonlinear directional coupler [
13
]. SOA with quantumdot active region is a promising candidate for ultrafast operations [
14

15
]. The high output power, low threshold current density, ultrafast response behavior, good temperature stability, and low noise level of the quantumdot SOA (QDSOA) have been demonstrated, and it has been emphasized that these elements can be utilized as building blocks of alloptical systems [
17

19
]. In this paper, using two QDSOA MZI, we design a subtractor that can operate and perform well at speeds of over 250 Gb/s.
II. REALIZATION OF ALLOPTICAL SUBTRACTOR
The subtractor is a combinational logic device that performs subtraction between two binary numbers. Its A and C inputs are for the minuend and subtrahend respectively. The differential and borrow outputs are D and B respectively. If A and C are equal then B=D=0, but if A=1 and C=0 then the outputs will be D=1, B=0. If A=0 and C=1 the outputs will be D=1 and B=1.
Figure 1
shows the structure of our proposed subtractor. In the first MZI a differential bit is generated for which XOR operation has been done between A and C. Next the borrow bit is generated in the second MZI in which the differential bit acts as a probe signal. The theoretical results were verified by numerical simulation.
The output powers of the gate are expressed as,
Configuration of an alloptical subtractor based on QDSOA MZI. C1, C2, C3, C4: (ideally 50:50) 3 dB couplers, BS: beam splitter, WSC: wavelengthselective coupler, QDSOA: quantumdot SOA.
where Pin (t) denotes probe power that is a continuous wave pulse,
G
_{1,2,3,4}
(
t
) are the instantaneous gains of
QD
−
SOA
_{1,2,3,4}
respectively, and
α
is the QDSOA linewidth enhancement factor. To calculate gain and power we must solve the threelevel rate equation for the electron transmission between the wetting layer (WL), exited state (ES), and ground state (GS) respectively, which are as follows [
20

22
].
and the photon rate equation is
where z is the longitudinal direction along the QDSOA length L, t is the local time, and
V
_{g}
≈ 8.3× 10
^{7}
m/s
,
S (z, t)= P (z, t)
/(
A
_{eff}
V
_{g}
hν
) is the photon density of the input data signal. g
_{max}
is maximum modal gain,
α
_{int}
is material absorption coefficient,
hν
is the photon energy, J is the injection current density, e is electron charge,
N
_{W}
is the electron density in WL, and h and f are the electron occupation probabilities in the ES and GS respectively.
N
_{Q}
is the surface density of QDs,
L
_{W}
is the effective thickness of the active layer, 𝜏
_{2w}
is the electron escape time from the ES to the WL, 𝜏
_{WR}
is the spontaneous radiative life time in WL, 𝜏
_{21}
is the electron relaxation time from the GS to the ES, and 𝜏
_{1R}
is the spontaneous relaxation life time in the QDs.
III. SIMULATION RESULTS AND DISCUSSION
In order to investigate the feasibility of the proposed design, we have applied the 4thorder RungeKutta method to solve equations (35) numerically and to calculate the gain coefficient. The input pulses are pseudorandom binary sequences at 250 Gb/s whose profile is Gaussian, that is
where
P
_{max}
is peak power and
T
_{FWHM}
is their full width at half maximum and less than 1.5 ps. By using Leibniz’s integral rule, we can interchange the integral and partial differential operators. By employing Leibniz’s rule for
S
(z, t), we have
and by integrating along the z variable in equation (7) we have
The next integration along the t variable in equation (8) gives
where
(
t
) is the average of
S
(z, t) along the t variable. By integrating and normalizing [0, z], and solving for the optical power at location z in terms of the input power for
S
(z, t) in equation (6), and using equation (9), we have:
where
g
=
g
_{max}
(2
f
−1) . In addition, the gain is calculated from
. The simulation results and their comparison to the results of other papers verify this method. It was found that our method can solve rate equations more quickly than previous methods [
18
,
19
]. The extinction ratio of the input signals was assumed to be 10 dB. The parameter values were taken from the literature on other QDSOAbased interferometric gates [
19

22
]. These are:
g
_{max}
= 14
cm
^{1}
,
N
_{Q}
= 5× 10
^{10}
cm
^{−2}
, 𝜏
_{W}
_{2}
= 3
ps
, 𝜏
_{W}
= 1
_{ns}
,
L
_{W}
= 0.25
μm
, 𝜏21 = 0.16
ps
, 𝜏
_{1R}
= 0.4
ns
, 𝜏
_{12}
= 1.2
ps
,
L
= 4
mm
,
P
_{max}
= 10
dBm
,
α
= 2 and
J
= 2.5
kA
/
cm
^{2}
. In order to assess the performance of an optical logic design, the most important parameters are extinction ratio, amplitude modulation, quality factor, pseudoeye diagram, and relative eye opening, which we will investigate. The extinction ratio (ER) is defined as
where
and
are the minimum and maximum values of the peak power of “1” and “0”, respectively [
23
]. In order to distinguish 1 unambiguously from 0 the ER must be over 10 dB [
23
].
Figure 2(a)
depicts the ER variation versus QDSOA length for three different peak power values of input data signals, where the other parameters are kept fixed. It is observed in the curves that for 4 <
L
<5
mm
,
ER
>14.5
dB
will result. The maximum values are shifted to the right as the QDSOA length becomes larger. The smaller QDSOA lengths need stronger control signals.
Figure 2(b)
shows the variation of ER with peak power for three different current densities. This shows that ER decreases with increasing peak power and increases with larger current density.
Figure 2(c)
shows the ER variation with current density for three different QDSOA lengths. For
J
>
2kA
/ cm
^{2}
we have
ER
>16
dB
because at higher injected current density there are more carriers in the wetting layer, and we have a better response.
ER variation versus (a) QDSOA length for three different peak powers, (b) peak power for three different current densities, and (c) current density for three different lengths.
The ratio between maximum and minimum peak power for signals corresponding to “1” is defined as
where
and
is the maximum peak power of “1” and “1” respectively [
24
]. The lower the AM, the more uniform is the level of the “1” outputs, and the smaller is the related pattern effect.
Figure 3(a)
illustrates the effect of QDSOA length on the AM for three different peak powers of input data signals. For
L
< 4
mm
a very small AM is obtained, which is very desirable, and AM increases with larger peak powers. In
Fig. 3(b)
variations of AM with peak power for three different current densities are shown. The peak power increase has an undesirable effect on AM, while the increase of current density has a desirable effect on it.
Figure 3(c)
shows the variation of AM with J for three different QDSOA lengths. It is observed that for
J
> 2
kA
/
cm
^{2}
we can achieve a small AM, and that AM increases with increasing of length.
AM variation versus (a) QDSOA length for three different peak powers, (b) peak power for three different current densities, and (c) current density for three different lengths.
The quality factor is defined as
where
m
_{1}
(
m
_{0}
) and σ
_{1}
(σ
_{0}
) are the average power and standard deviation of the circuit output at “1” (“0”) respectively [
25
].
Figure 4
shows that Q decreases for
L
> 4.5
mm
and a larger peak power results from the decrease in Q factor. The results show that with J > 2.5
kA
/
cm
^{2}
and
L
= 4
mm
we can achieve a larger Q.
Q variation versus (a) QDSOA length for three different peak powers, (b) peak power for three different current densities, and (c) current density for three different lengths.
A pseudoeye diagram is one of the best standards for assessing the output signals. A clear and open eye diagram is desired.
Figure 5
illustrates the effect of electron relaxation time from ES to GS on PED. A slow transition of the electron between ES and WL results in a slower cross gain modulation. Thus speed is limited by the relaxation time from the wetting layer to the quantumdot state.
Simulated pseudoeye diagrams of output waveforms for electron relaxation time from the ES to the GS (a) 0.16 ps, (b) 1 ps, and (c) 2 ps.
The relative eye opening is defined as
where
and
are the minimum and maximum powers for ones and zeros respectively [
10
]. In this design the related eye opening is over 90%.
Table 1
shows the relative eye opening for three QDSOA lengths and three electron relaxation times from ES to GS. The other parameters are kept fixed.
Variations of relative eye opening with L and 𝜏21
Variations of relative eye opening with L and 𝜏_{21}
According to
Figs. 2

5
and our discussions so far, QDSOAs with lengths smaller than 4.5 mm, show desirable responses but smaller lengths decrease unsaturated gain.
Table 2
shows gain values for five different QDSOA lengths. It was found that with current density of more than 2.5
kA
/
cm
^{2}
we can achieve a more suitable performance. Increasing the current density might not be practical for a QDSOA [
22
], however, and we see that with smaller peak power the output is better, so for good switching we choose a peak power of 10 dB.
Variation of QDSOA gain with length
Variation of QDSOA gain with length
According to the above mentioned explanations and using the combination of input parameters
L
= 4
mm
,
J
2.5
kA
/
cm
^{2}
,
P
_{max}
=10
dBm
and 𝜏
_{21}
= 0.16
ps
, we can achieve these values in the output: ER = 15.37 dB, AM = 0.052 dB, eyeopening = 97.09%, Q = 49.13 dB, and 20.82 dB of unsaturated gain for 250 Gb/s input signals.
Figure 6
shows input and output data streams for a 32bit pseudorandom binary sequence of input.
Input and output waveforms of the alloptical subtractor. (a) minuend input, (b) subtrahend input, (c) differential output, and (d) borrow output.
Table 3
illustrates the output parameters of the proposed structure, compared to those of similar recent structures. The last column shows the parameters of our designed structure, which demonstrates its better performance, due to our choice of optimal parameters for the structure, and use of the transmitting ports in both MZIs.
The comparison of parameters of the proposed optical gate with those of other papers
The comparison of parameters of the proposed optical gate with those of other papers
The main reason for the faster response of a QDSOA compared to a bulk SOA is the presence of the wetting layer. The wetting layer serves as a carrier reservoir: Carriers depleted by the injected optical pulse in the QD ground state are replaced from the wetting layer by fast carrier transfer. At a higher injected current density there are more carriers in the wetting layer, resulting in a better performance. Thus the speed of optical logic gates is limited by the carrier recovery time in the device. Careful adjustment of the peak power of the modulated signal and QDSOA bias current is necessary for errorfree performance at high data rates and for avoiding overlap in response.
IV. CONCLUSION
The effects of the most important parameters on the performance of the proposed gate are investigated using a model of three coupled equations for a QDSOA into which a Gaussian pulse with rate 250 Gb/s has been inserted. The results show that output signals have high ER and Q and low AM, and that the gate can operate well at speeds of over 250 Gb/s. In both MZIs we used the transmitting port. The reflecting port can also be used to extend the device in generating other logic gates.
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