Abstract
Keywords
0 Introduction
With the rapid development of communication technology, optical communication has been widely used in the market because of its large communication capacity, low transmission loss and large frequency bandwidth. The chaotic laser signal, which is highly sensitive to the initial conditions of the system and external interference, has strong randomness and is not easy to be deciphered. So now which it has been widely used in the field of secure optical communication. Meanwhile, the logic operations realized by chaotic signals generated by semiconductor lasers have attracted great attention currently. Compared with edge-emitting laser, Vertical Cavity Surface Emitting Laser (VCSEL) has the advantages of low production cost, small divergence angle, small size and low threshold current [
Most of the above solutions implemented static logic operations. In the past, our research group studied logic operations with code width of 10ns (the operation rate is 0.1 GHz) [
1 Theory and model
The system composition and detailed light path are shown in Fig. 1. Here, the normalized injection current μ is the sum of two square waves of dμ1 and dμ2 that encode two logic inputs I1 and I2, i.e., μ=dμ1(I1)+dμ2(I2). The transverse electric field E0 is represented by a square wave that encode the control signal Lc, the high level and low level of the square wave are represented by E02 and E01 respectively. When E0= E01, Lc=0; and Lc=1 if E0=E02. Supposing that the average value of x-PL intensity from the VCSEL defined as A, its threshold is fixed at A*. The steps of realizing reconfigurable chaotic logic operations are as follows
Figure 1.Schematic diagram for reconfigurable chaotic logic operations of VCSEL with optical feedback
1) The appropriate values for normalized injection current μ(t)[dμ1(t), dμ2(t)] and transverse electric field E0 (t) are selected to encode the logic inputs [I1(t), I2(t)] and control signal Lc(t);
2) Add the normalized injection current and control signal, I(t)= I1(t)+I2(t) and Lc(t);
3) Use an electronic logic calculator to make Lc(t) and I1(t) and I2(t) satisfying different logic operation relationships such as NOT, AND, NAND, OR, NOR, XOR and XNOR in different time periods; for the logic NOT operation, I1(t)=I2(t).
4) Calculate the average value A under each group of logic inputs.
5) Threshold mechanism for obtaining the logic output, i.e., with the fixed threshold A*, the logic out Xo=0 if A-A*≤0; Xo=1 if A-A*>0.
Next, the working principle of the system is presented. The light emitted by VCSEL passes through the Fiber Isolator (FI) and then is separated by the Fiber Polarization Beam Splitter (FPBS) into x-PL and y-PL. The x-PL is split into two beams by a Fiber Splitter (FPS), one of which is converted into an electric signal by Photodetector (PD). The electric signal is used to demodulate the logic output through Subtracter (SR). Another x-PL beam is directly injected into the PPLN crystal as o-light input. The y-PL converts the polarized direction to the direction of the z-axis of the crystal by Faraday Rotator 1 (FR1) and Half-Wave Plate 1 (HWP1), and then injects into crystal as e-light input. After linear electro-optic modulation, the x-PL output from the PPLN crystal passes through the Neutral Density Filter 1 (NDF1), and the y-PL passes through FR2, HWP2, and NDF2. Then they are feedback into the VCSEL together. The feedback delay is τ, and NDF1 and NDF2 are used to control the feedback strength of x-PL and y-PL respectively.
In order to realize dynamic and reconfigurable chaotic logic operations, it is necessary for the control signal to change synchronously with the logic inputs. To solve this problem, we propose the following scheme: since the normalized injection current μ=dμ1+dμ2, and dμ1 and dμ2 are encoded into two logic inputs I1 and I2 respectively. Here we use time-varying current source S1 to generate two identical electrical signals dμ1 and dμ3. Similarly, the same two electrical signals dμ2 and dμ4 are generated by S2. And dμ3 and dμ4, in turn, are encoded into two electric logic inputs of Electronic Logic Calculator (ELC) such as i1and i3. Due to dμ1=dμ3 and dμ2= dμ4, the logic sets of the signals i1 and i2 are synchronized with those of the signals I1 and I2. The logic output of the ELC is defined as R, which can control the Transverse Electric Field (TEF) E0 by Transverse Electric FieldController (TEFC). Here, R=0 is encoded in the low level E01, R=1 is encoded in the high level E02. If R=0, we obtain E0=E01 and Lc=0; when R=1, we have E0=E02 and Lc=1. Using the ELC, R and i1, i2 can perform different logic operations, so that Lc can implement different logic operations with I1 and I2 indirectly. It is noted that the switching rate of the control signal is determined by ELC. At present, the reconfigurable operation rate of ELC can reach more than 10 GHz, which can satisfy the needs of reconfigurable chaotic logic operations realized in this paper.
Due to the VCSEL subject to the delay feedback of its own light, the dynamic equations of x-PL and y-PL are represented as follows
In the above formulas, subscripts x and y mean x-PL and y-PL respectively; E represents the complex amplitude of light; N is the total carrier concentration; n is the difference in concentration between carriers with spin-up and carriers with spin-down; ω0 is the center frequency of light; βsp represents the spontaneous emission factor, which is also defined as noise intensity; ζxand ζy are a pair of gaussian white noises that are independent of each other and obey the standard normal distribution. The meanings and values of other physical quantities are shown in Table 1 below.
Parameter and symbol | Value | Parameter and symbol | value |
---|---|---|---|
Line-width enhancement factor a | 3 | Duty ratio R | 0.5 |
field decay rate k | 300 | Polar angle θ/π | 1/2 |
Spin relaxation rate γs/ns-1 | 50 | Azimuth φ | 0 |
Nonradiative carrier relaxation γe/ns-1 | 1 | Crystal temperature F/ K | 293 |
Dichroism γa/ns-1 | -0.1 | Poled period of crystal Λ/m-1 | 5.8×105 |
Birefringence γp/ns-1 | 2 | Crystal length L/mm | 15 |
Delay time τ/ns | 2 | Refractive index of o-light n1 | 2.24 |
Effective refractive index of active layer ng | 3.6 | Refractive index of e-light n2 | 2.17 |
Effective area of light spot SA/μm2 | 38.485 | Differential material gain g/ m3·s-1 | 2.9×10-12 |
Length of the laser cavity Lv/μm | 10 | Field confinement factor to the active region Γ | 0.05 |
Optical feedback strength kf | 1.13 | Volume of the active layer V/μm3 | 384.85 |
Code width T/ps | 600 | The noise intensity βsp | 2.5×109 |
Table 1. The main parameters of the system
The x-PL and y-PL from VCSEL are injected into the PPLN crystal and converted into o-light and e-light respectively, and their amplitudes satisfy the following relationship
The linear EO modulation effect occurs in the PPLN crystal, and coupled wave equation of o-light and e-light can be expressed as follows
where
and
Ux and Uy in Eqs. (4)~(10) represent the amplitudes of o-light and e-light respectively; TL represents the time it takes for light to travel back and forth in the laser cavity once; ℏ is the Planck constant; Δk=kx-ky+K1, kx and ky denote the wave vectors of o-light and e-light at ω0, and kx=2πn1vc/ω0, ky=2πn2vc/ω0,K1=2π/Λ, Λ is the poled period, where the coefficients d1, d2, d3, and d4 are presented in Ref.[17]. The meanings and values of other physical quantities are presented in Table 1. The o-light and e-light are converted into x-PL and y-PL respectively after being output from the crystal, and the conversion relationship can be expressed as follows
2 Results and discussions
The Eqs. (1)~(3) can be solved according to the fourth-order Runge-Kutta method and the system parameters in Table 1. Because μ=dμ1+dμ2, dμ1 and dμ2 are used to modulate binary logic inputs I1 and I2, respectively. The logic input sequence (I1, I2) has four combinations as (0, 0), (0, 1), (1, 0) and (1, 1), and the specific modulation rules are: when dμ1=0.74, I1=0, and when dμ1=0.75, I1=1. Similarly, when dμ2=0.74, I2=0; when dμ2=0.75, I2=1. Therefore, when logic input (I1, I2)=(0, 0), μ=dμ1+dμ2=1.48; while (I1, I2)=(0, 1) or (1, 0), μ=dμ1+dμ2 =1.49; and μ=dμ1+dμ2=1.50 if logic input (I1, I2)=(1, 1). The control signal Lc is modulated by the transverse electric field E0, it means that, if E0=E02=0.75 kV/mm, Lc=1; else if E0=E01=0.589 4 kV/mm, Lc=0. Moreover, the logic output Xo can be demodulated by performing difference processing between A and the threshold A* under each logic input, namely, if A≤A*, Xo=0, else if A>A*, Xo=1.
The threshold value A* determines the reliability of the logic operations. In order to obtain a suitable threshold, we adopt the following technical solutions: since Lc and (I1, I2) can form the basic logic operation relationships such as NOT, AND, OR, XOR, etc., for each of the above-mentioned logic operation relations, we have calculated the maximum average Amax of x-PL intensity when Lc=0, and the minimum average Amin under Lc=1, as shown in Table 2. It can be found from the table that in all the logic relationships that Lc and (I1, I2) satisfy, the maximum value of A when Lc=0 is Amax=0.003, and that’s minimum value when Lc=1 is Amin=0.021. Therefore, the threshold A* must meet 0.003<A*<0.021. Here we take A*=0.013, that is, if A≤0.013, Xo=0; otherwise Xo=1.
Logic operations | (I1, I2) = (0, 0) | (I1, I2) = (0, 1) / (1, 0) | (I1, I2) = (1, 1) | |||
---|---|---|---|---|---|---|
Lc | A | Lc | A | Lc | A | |
0 | Amax=0 | 0 | Amax=0 | 1 | Amin=0.021 | |
1 | Amin=0.031 | 1 | Amin=0.027 | 0 | Amax=0.0014 | |
0 | Amax=0.0014 | 1 | Amin=0.03 | 1 | Amin=0.026 | |
1 | Amin=0.034 | 0 | Amax=0 | 0 | Amax=0 | |
0 | Amax=0.003 | 1 | Amin=0.03 | 0 | Amax=0.002 | |
1 | Amin=0.022 | 0 | Amax=0 | 1 | Amin=0.048 | |
1 | Amin=0.026 | * | * | 0 | Amax=0.002 |
Table 2. For the different logic operations relationship between
When Lc and logic input satisfy different logic operation relationships, the system can implement different logic operations. In order to obtain logic AND operation, Lc needs to satisfy the AND operation relationship with I1 and I2 as shown in Fig.2(a). The solid green line represents E0, the red dotted line represents the μ, and the blue solid line in Fig.2 (b) denotes x-PL intensity Ix. When (I1, I2) equals (0,0), (0,1), (1,0), respectively, x-PL intensity Ix is very small, and Amax=0< A* (see Table 2), then Xo=0; when (I1, I2)=(1, 1), x-PL intensity Ix gets large, and Amin =0.021>A* (see Table 2), thus Xo=1, so the logic output is produced by demodulation as shown in Fig.2(c). In summary, the system has performed logic AND operation, namely Xo=I1·I2. In the same way, is satisfied in Fig.2(a), and the system realizes logic NAND operation as shown in Fig.2(b),(c). Table 3 is the truth table of AND and NAND.
Figure 2.The implementation of chaotic logic AND (left column) and NAND (right column) operations
Logic input | AND | NAND | ||
---|---|---|---|---|
(I1, I2) | Lc | Xo | Lc | Xo |
(0,0) | 0 | 0 | 1 | 1 |
(0,1) | 0 | 0 | 1 | 1 |
(1,0) | 0 | 0 | 1 | 1 |
(1,1) | 1 | 1 | 0 | 0 |
Table 3. The truth table of logic AND and NAND
Similarly, we further obtain OR, NOR, XNOR, XOR and NOT operations, as shown in Fig.3, 4 and 5. Table 4, 5 and 6 are the truth tables for the above five logic operations.
Figure 3.The implementation of chaotic logic OR (left column) and NOR (right column) operations
Figure 4.The implementation of chaotic logic XNOR (left column) and XOR (right column) operations
Figure 5.The implementation of chaotic logic NOT (left column) and reconfigurable operations (right column)
Logic input | OR | NOR | ||
---|---|---|---|---|
(I1, I2) | Lc | Xo | Lc | Xo |
(0,0) | 0 | 0 | 1 | 1 |
(0,1) | 1 | 1 | 0 | 0 |
(1,0) | 1 | 1 | 0 | 0 |
(1,1) | 1 | 1 | 0 | 0 |
Table 4. The truth table of logic OR and NOR
Logic input | XNOR | XOR | ||
---|---|---|---|---|
(I1, I2) | Lc | Xo | Lc | Xo |
(0,0) | 1 | 1 | 0 | 0 |
(0,1) | 0 | 0 | 1 | 1 |
(1,0) | 0 | 0 | 1 | 1 |
(1,1) | 1 | 1 | 0 | 0 |
Table 5. The truth table of logic XNOR and XOR
Logic input | NOT | |
---|---|---|
(I1, I2) | Lc | Xo |
(0,0) | 1 | 1 |
(1,1) | 0 | 0 |
Table 6. The truth table of logic NOT
The above-mentioned logic operations are performed under the condition that the control signal Lc and the logic input (I1, I2) form the static logic operation relationship. In order to show the reconfigurable ability of the system, relying on ELC to convert the logic relationship between Lc and (I1, I2) as shown in Fig.5(a), during the time of 3 ns~4.8 ns, 4.8 ns~7.2 ns, 7.2 ns~9.6 ns, 9.6 ns~12 ns, 12 ns~14.4 ns, 14.4 ns~16.8 ns, 16.8 ns~17.4 ns, Lc and (I1, I2) in turn form logic OR, XOR, XNOR, AND, NOR, NAND and NOT relationship, the reconfigurable logic operations are obtained as shown in Fig.5(b), (c).
The noise in the system has a significant impact on the reliability of logic operations [
In order to explore the influence of code width T on the reliability of logic operations, we further calculate the dependence of the success probability P on the T as shown in Fig. 7. From the picture we can find that with the gradual increase of the T, the curve changes from violent oscillation to smaller fluctuation, and finally becomes a stable straight line with P always equals 1, which indicates that the larger the T, the better the reliability of the logic operations. We also discover the value of P equals 1 when T takes some values within 579 ps, such as T=104.5 ps, or T=107.9 ps, or T=124.8 ps, etc., but the value of P is unstable because a slight change in the value of T will cause the P to oscillate. And P always equals 1 when T≥579 ps. Therefore, reliable and stable logic operations can be obtained as long as the value of T is at least 579 ps.
Figure 7.The success probability of reconfigurable chaotic logic operation under different code width
3 Conclusions
Based on chaotic system of the VCSEL feedback by its own light and the linear electro-optic modulation effect, we have realized chaotic AND, OR, NOT, XOR, NAND, NOR and XNOR logic operations. The system can also perform reconfigurable logic operations by transforming the logic operation relationship between control signal and logic inputs. The further research shows that under code width T=600 ps, the success probability P always equals 1 when βsp<2.75×109. The value of P is also greater than 0.92 even though 2.75×109<βsp<4×109. These results indicate that the reconfigurable chaotic logic operations have good anti-noise performance. Moreover, under βsp=2.5×109, the value of P is unstable when T takes values within 579 ps. And the logic operations become reliable and stable as long as the value of T is at least 579 ps. The above research results have great application and reference value for the development of faster and safer combinatorial logic operation devices, such as optical full adders, optical data selectors, etc., as well as for the construction of reconfigurable optical networks.
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