• Acta Photonica Sinica
  • Vol. 50, Issue 5, 174 (2021)
Geliang XU, Jian XU, Lingli KONG, Qifeng HUANG, Xianting QIU, Yangyang GUO, and Feng CHENG
Author Affiliations
  • School of Electronic Engineering, Chaohu University, Hefei238000, China
  • show less
    DOI: 10.3788/gzxb20215005.0506008 Cite this Article
    Geliang XU, Jian XU, Lingli KONG, Qifeng HUANG, Xianting QIU, Yangyang GUO, Feng CHENG. Reconfigurable Optical Chaotic Logic Operations with Fast Rate of Picoseconds Scale[J]. Acta Photonica Sinica, 2021, 50(5): 174 Copy Citation Text show less

    Abstract

    In order to realize dynamic and reconfigurable optical chaotic logic operations, a specific technical scheme based on Vertical Cavity Surface Emitting Laser (VCSEL) feedback by its own light and linear electro-optic modulation effect has been proposed. The normalized injection current is modulated as logic input, the transverse electric field is modulated as control signal, and the logic output is demodulated by the difference between the average value and the threshold value of the x-polarized light intensity from the output of VCSEL. By transforming the logic operation relationship between control signal and logic input, the system can switch freely among basic logic operations such as NOT, AND, NAND, OR, NOR, XOR and XNOR. When the code width is 600 ps and the noise intensity is as high as 2.75×109, the success probability of the logic operation still equals 1, indicating that the system has good anti-noise performance. And when the noise intensity equals 2.5×109, the success probability always equals 1 if the code width is at least 579 ps. The above results have great reference value for the development of fast and stable combinational logic operation devices.

    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 1-6, etc. In addition, it can excite mutually orthogonal chaotic x-Polarized Light (x-PL) and chaotic y-Polarized Light (y-PL) under current or feedback injection of external light. The polarization switching and bistability can also be induced under suitable parameter conditions 7-11. Based on VCSEL's nonlinear dynamics, previous work has exploited different experimental schemes to realize optical logic gates. Based on polarization bistability, noise and tunable light injection, Masoller et al explored experimental schemes for random logic gate and all-optical logic gate 12-14. For laser amplitude modulation based on coupled feedback and parallel synchronization of laser chaos, Yan realized photoelectric composite logic gate and all-optical logic gate 3-4. In 2015, our research group reported the experimental scheme of optoelectronic composite logic gate based on polarization switching in Optics Express 10. In 2016, based on the generalized chaos synchronization theory and the polarization switching, we also realized the all-optical random logic gate 15. In 2017, using frequency detuning to control polarization bistability, we further obtained dynamic all-optical chaotic logic operations 16.

    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) 16, the operation rate needs to be improved. Here in the paper, based on the theory of linear electro-optic modulation, reconfigurable chaotic logic operations with fast rate and good anti-noise performance are realized by the scheme of VCSEL feedback by its own light in this paper. The second part below describes the basic theory and model in detail, the third part shows the experimental results and discussion, and the fourth part is the conclusions.

    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=dμ1I1)+dμ2I2). 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= E01Lc=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

    Schematic diagram for reconfigurable chaotic logic operations of VCSEL with optical feedback

    Figure 1.Schematic diagram for reconfigurable chaotic logic operations of VCSEL with optical feedback

    1) The appropriate values for normalized injection current μ(t)[dμ1t), dμ2t)] and transverse electric field E0 t) are selected to encode the logic inputs [I1t), I2t)] and control signal Lct);

    2) Add the normalized injection current and control signal, It)= I1t)+I2t) and Lct);

    3) Use an electronic logic calculator to make Lct) and I1t) and I2t) satisfying different logic operation relationships such as NOT, AND, NAND, OR, NOR, XOR and XNOR in different time periods; for the logic NOT operation, I1t)=I2t).

    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 E01R=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 i1i2 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

    ddtEx(t)Ey(t)=k1+ia{[N(t)-1]}Ex(t)Ey(t)±k1+iain(t)Ey(t)Ex(t)(γa+iγp)Ex(t)Ey(t)+kfEx(t-τ)Ey(t-τ)×exp(-iω0τ)+βspγeNζxβspγeNζy

    dN(t)dt=-γe{N(t)-μ+N(t)(|Ex(t)|2+|Ey(t)|2)+in(t)[Ey(t)Ex*(t)-Ex(t)Ey*(t)]}

    dn(t)dt=-γsn(t)-γe{n(t)(|Ex(t)|2+|Ey(t)|2)+iN(t)[Ey(t)Ex*(t)-Ex(t)Ey*(t)]}

    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 symbolValueParameter and symbolvalue
    Line-width enhancement factor a3Duty ratio R0.5
    field decay rate k300Polar angle θ1/2
    Spin relaxation rate γs/ns-150Azimuth φ0
    Nonradiative carrier relaxation γe/ns-11Crystal temperature F/ K293
    Dichroism γa/ns-1-0.1Poled period of crystal Λ/m-15.8×105
    Birefringence γp/ns-12Crystal length L/mm15
    Delay time τ/ns2Refractive index of o-light n12.24
    Effective refractive index of active layer ng3.6Refractive index of e-light n22.17
    Effective area of light spot SA/μm238.485Differential material gain g/ ms-12.9×10-12
    Length of the laser cavity Lv/μm10Field confinement factor to the active region Γ0.05
    Optical feedback strength kf1.13Volume of the active layer V/μm3384.85
    Code width T/ps600The noise intensity βsp2.5×109

    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

    Ux, y(0,t-τ)=ω0VSATLvcn1,2Ex, y(t-τ)

    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

    Ux, y(L,t-τ)=ρx, y(L,t-τ)exp(iβ0L)exp[iϕx, y(L,t-τ)]

    where

    ρx,y(L,t-τ)=Ux,y2(0,t-τ)cos2(vL)+γUx,y(0,t-τ)d1,3Uy,x(0,t-τ)v2sin2(vL)1/2

    ϕx, y(L,t-τ)=arctan±γUx, y(0,t-τ)-d1,3Uy, x(0,t-τ)vUx, y(0,t-τ)tan(vL)

    β0=Δk-d2-d42

    and

    v=(Δk+d2-d4)2+4d1d32

    γ=d4-d2-Δk2

    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+K1kx and ky denote the wave vectors of o-light and e-light at ω0, and kx=2πn1vc/ω0ky=2πn2vc/ω0K1=2π/Λ, Λ is the poled period, where the coefficients d1d2d3, 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

    Ex,y(t-τ)=SATLvcn1,2ω0VUx,y(L,t-τ)

    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 (I1I2) 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 (I1I2)=(0, 0), μ=dμ1+dμ2=1.48; while (I1I2)=(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 AA*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 (I1I2) 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 (I1I2) 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)
    LcALcALcA
    Lc=I1I20Amax=00Amax=01Amin=0.021
    Lc=I1I2¯1Amin=0.0311Amin=0.0270Amax=0.0014
    Lc=I1+I20Amax=0.00141Amin=0.031Amin=0.026
    Lc=I1+I2¯1Amin=0.0340Amax=00Amax=0
    Lc=I1I20Amax=0.0031Amin=0.030Amax=0.002
    Lc=I1I21Amin=0.0220Amax=01Amin=0.048
    Lc=I1¯(I2¯)1Amin=0.026**0Amax=0.002

    Table 2. For the different logic operations relationship between Lc and(I1, I2), the maximum average (Amax) of the x-PL intensity under Lc=0 and its minimum average (Amin) under Lc=1.

    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 (I1I2) 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 (I1I2)=(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,Lc=I1I2¯ 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.

    The implementation of chaotic logic AND (left column) and NAND (right column) operations

    Figure 2.The implementation of chaotic logic AND (left column) and NAND (right column) operations

    Logic inputANDNAND
    (I1, I2)LcXoLcXo
    (0,0)0011
    (0,1)0011
    (1,0)0011
    (1,1)1100

    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.34 and 5. Table 45 and 6 are the truth tables for the above five logic operations.

    The implementation of chaotic logic OR (left column) and NOR (right column) operations

    Figure 3.The implementation of chaotic logic OR (left column) and NOR (right column) operations

    The implementation of chaotic logic XNOR (left column) and XOR (right column) operations

    Figure 4.The implementation of chaotic logic XNOR (left column) and XOR (right column) operations

    The implementation of chaotic logic NOT (left column) and reconfigurable operations (right column)

    Figure 5.The implementation of chaotic logic NOT (left column) and reconfigurable operations (right column)

    Logic inputORNOR
    (I1, I2)LcXoLcXo
    (0,0)0011
    (0,1)1100
    (1,0)1100
    (1,1)1100

    Table 4. The truth table of logic OR and NOR

    Logic inputXNORXOR
    (I1, I2)LcXoLcXo
    (0,0)1100
    (0,1)0011
    (1,0)0011
    (1,1)1100

    Table 5. The truth table of logic XNOR and XOR

    Logic inputNOT
    (I1, I2)LcXo
    (0,0)11
    (1,1)00

    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 (I1I2) 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 (I1I2) 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 (I1I2) 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 18. Here, we introduce the success probability P to describe the reliability of the reconfigurable chaotic logic operations in Fig. 5(b). P is defined as the ratio between the number of the correct bits and that of total bits of the chaotic logic output. We calculate the dependence of the success probability P on the noise intensity βsp as shown in Fig. 6. We can find that the value of P equals 1 when βsp<2.75×109. The value of P is greater than or equal to 0.92 even 2.75×109<βsp<4×109. These show that the reconfigurable optical chaotic logic operations have good anti-noise performance.

    Image Infomation Is Not Enable

    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.

    The success probability of reconfigurable chaotic logic operation under different code width

    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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    Geliang XU, Jian XU, Lingli KONG, Qifeng HUANG, Xianting QIU, Yangyang GUO, Feng CHENG. Reconfigurable Optical Chaotic Logic Operations with Fast Rate of Picoseconds Scale[J]. Acta Photonica Sinica, 2021, 50(5): 174
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