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Image Transmission Based on Spiking Dynamics of Electrically Controlled VCSEL-SA Neuron

Image Transmission Based on Spiking Dynamics of Electrically Controlled VCSEL-SA Neuron hv photonics Communication Image Transmission Based on Spiking Dynamics of Electrically Controlled VCSEL-SA Neuron Min Ni, Xiaodong Lin, Xi Tang, Ziye Gao, Luyao Xiao, Jun Wang, Fan Ma, Qiulan Zheng and Tao Deng * School of Physical Science and Technology, Southwest University, Chongqing 400715, China; min333@swu.edu.cn (M.N.); linxd@swu.edu.cn (X.L.); tx1982@swu.edu.cn (X.T.); zygao@swu.edu.cn (Z.G.); xly123456@swu.edu.cn (L.X.); wangjun369@swu.edu.cn (J.W.); mafan2020@swu.edu.cn (F.M.); zql456723@swu.edu.cn (Q.Z.) * Correspondence: dengt@swu.edu.cn Abstract: Based on the spiking dynamics of the electrically controlled vertical-cavity surface-emitting laser with an embedded saturable absorber (VCSEL-SA), we propose an image transmission system using two unidirectionally coupled VCSEL-SAs and numerically investigate the binary-to-spike (BTS) conversation characteristics and the image transmission performance. The simulation results show that, through electrically injecting the binary data to VCSEL-SA, the BTS conversation can be realized and the conversion rate of BTS highly depends on the injection strength and bias current. Thus, the image transmission can be realized in the proposed system. Moreover, the parameter mismatches between these two VCSEL-SAs have some effects on the image transmission performance, but the encoded images are still successfully decoded even under certain parameter mismatches. In addition, spiking patterns can be also stored and transmitted in the cascaded system with optoelectronic feedback loop. Keywords: image transmission; vertical-cavity surface-emitting lasers with an embedded saturable absorber (VCSEL-SA); parameter mismatches; binary to spike (BTS); neuromorphic computing system Citation: Ni, M.; Lin, X.; Tang, X.; Gao, Z.; Xiao, L.; Wang, J.; Ma, F.; Zheng, Q.; Deng, T. Image Transmission Based on Spiking Dynamics of Electrically Controlled 1. Introduction VCSEL-SA Neuron. Photonics 2021, 8, The human brain possesses powerful computing capability and low power consump- 238. https://doi.org/10.3390/ tion [1,2]. Due to the limitation of memory ability, data interaction bandwidth, and high photonics8070238 energy consumption, traditional computers with Von Neumann structure are unable to meet the growing computing needs [3]. Therefore, the in-depth study of brain–machine Received: 30 April 2021 interface technology and neural mimicry systems is conducive to solve these complex Accepted: 22 June 2021 computing problems, which have attracted wide attention [4–7]. Neural mimicry systems Published: 25 June 2021 without Von Neumann structure can simulate biological sensing and realize brain-like computing, which greatly improves the computing capability and reduces power consump- Publisher’s Note: MDPI stays neutral tion [2]. Therefore, this neuromorphic system can possess huge application potential in with regard to jurisdictional claims in processing some complex computing tasks including decision making, learning, sensory published maps and institutional affil- information processing, and pattern recognition [8]. Recently, photonic neuromorphic iations. devices have shown great application prospects in the field of high-speed neuromorphic computing because they can simulate the basic characteristics of biological neurons and provide ultrafine pulse dynamics up to eight orders of magnitude faster than biological neurons [9]. Copyright: © 2021 by the authors. In recent years, semiconductor lasers (SLs) have been viewed as promising candidates Licensee MDPI, Basel, Switzerland. for a neuromorphic photonic model because of its strong analogy with biological neurons This article is an open access article in terms of the underlying excitability mechanisms [10]. Among these SL-based photonic distributed under the terms and neuron models, the VCSEL-based neuron model has attracted extensive attentions because conditions of the Creative Commons VCSEL possesses some unique advantages such as low cost, low energy consumption, easy Attribution (CC BY) license (https:// integration into the 2d/3d array, high coupling efficiency of optical fiber, and compatibility creativecommons.org/licenses/by/ with the existing optical fiber system [11–16]. So far, controllable activation and inhibition 4.0/). Photonics 2021, 8, 238. https://doi.org/10.3390/photonics8070238 https://www.mdpi.com/journal/photonics Photonics 2021, 8, 238 2 of 11 of sub-ns spiking patterns based on commercial VCSELs have been realized [17,18]. More recently, communication of spiking patterns between two cascaded VCSEL neurons was theoretically and experimentally demonstrated [19]. In particular, an integrated two-stage excitable laser VCSEL-SA can be constructed by combining VCSEL with a saturable absorber (SA), which possesses similar advan- tages to that of VCSELs and can be viewed as a simple spike-based leaky integrate-and- fire (LIF) neuron model [20]. In VCSEL-SA, once the number of carriers in the active region of VCSEL-SA accumulates to exceed the exciting threshold, the spikes can be excited. Correspondingly, the number of carriers in the active region exhibits an evolu- tion tendency of abrupt decrease and then gradual recovery, and vice versa. Moreover, VCSEL-SA can generate shorter sub-ns pulses in comparison with the conventional neuron model [21–24]. Furthermore, the excitability threshold of photonic neurons can be adjusted within a certain range [20]. Up to now, the previous studies on VCSEL-SA-based photonic neuron models mainly focused on the optical stimulation method [21,22], but relevant researches on the electrically controlled stimulation method are relatively few. In particular, compared with optically controlled stimulation, electrically controlled stimulation is easy to control and insensitive to phase variation. In addition, due to the limitation of material and technology, the present all-optical neuron networks have some defects such as online training, nonlinear logic operation, and large-scale integration [5,25]. Therefore, some photonic neuron networks focus on the integration of functional units on a chip, while other off-chip units can be optically or electronically realized [26]. Moreover, some machine learning mechanisms such as STDP in the neuron network also need to adjust the weight through controlling the external electronic circuit [27]. Consequently, the combination of photonic integration technology with some mature electronic methods is still very impor- tant, which is helpful to prompt the realization of future all-optical neuron network. Based on the abovementioned considerations, the spiking dynamics and its application based on electrically controlled VCSEL-SA deserve investigation. In this paper, we propose an image transmission system based on two cascaded elec- trically controlled VCSEL-SAs for the first time and investigate the characteristics of image encoding, transmission, and decoding. Moreover, the influence of some typical internal parameters is considered. The results show that this system can successfully realize image transmission under certain parameter mismatches. After introducing an optoelectronic feedback loop, the spiking patterns can be successfully stored and transmitted in the cascade electronically controlled system. 2. Theoretical Model A schematic diagram of the image transmission system composed of two unidirec- tionally coupled VCSEL-SAs is shown in Figure 1. In this system, the input images are firstly encoded into binary codes and mixed with the bias current through a Biastee, which is electrically injected into VCSEL-SA1 to conduct the conversion of binary data to spike (BTS). Then, the output spike signals from VCSEL-SA1 are divided into two parts. One is electrically injected into VCSEL-SA2 after passing through an optical isolator (ISO), a variable attenuator (VA), and a photoelectric detector (PD), where ISO is used to guarantee the unidirectional coupling, VA is used to adjust the injection weight, and PD is used to convert optical signals to electrical signals. The other (the dashed line part) is only used to form an optoelectronic feedback loop, which is used to store the spiking patterns. Consequently, the output spiking signals from VCSEL-SA2 can be used to recover the transmitted images. Photonics 2021, 8, 238 3 of 11 Photonics 2021, 8, x FOR PEER REVIEW 3 of 12 96 Figure 1. Figure 1. S Schematic chematic diag diagram ram of im of image age transm transmission ission betw between een two cas two cascaded caded VC VCSEL-SAs. SEL-SAs. DC DC: : direct direct cu curr rrent, IS ent, ISO: O: opt optical ical 97 isolator, VA: variable attenuator, PD: photodetector. isolator, VA: variable attenuator, PD: photodetector. After considering the effect of the external electrical injection (stimulus) and neglecting 98 After considering the effect of the external electrical injection (stimulus) and neglect- the polarization effects, according to the typical coupled rate equations of a two-section 99 ing the polarization effects, according to the typical coupled rate equations of a two-sec- excitable laser with SA region and gain region, the modified rate equations of two cascaded 100 tion excitable laser with SA region and gain region, the modified rate equations of two electrically controlled VCSEL-SAs can be described as follows [20,24]: 101 cascaded electrically controlled VCSEL-SAs can be described as follows [20,24]: dN N mph dN N mph mph mph = G g (n n )N + G g (n n )N + V b B n , (1) ma ma ma ms ms ms ma m mr 0ma mph 0ms mph ma dt t =−ΓΓ g() nnN+ g(n−n )N− +V βB n  mph ma ma ma00 ma mph ms ms ms ms mph ma m mr ma (1) dt τ mph I + k y (t, Dt) + W P t t N 1a 1 1 f 1 f dn n 1ph 1a 1a Ik +  ψ t,Δt +WP t −τ N () () dn n = G g (n n ) 1 ph +11af 1 1 f , (2) 1a 1a 1a 01a 11 aa =−Γ gn −n − + dt () V t eV (2) 11 aa 1a 01a 1a 1a 1a dt V τ eV 11 aa 1a N I + W P t t dn n 2ph 2a 12 1 inj 2a 2a = G g (n n ) + , (3) 2a 2a 2a 02a IW +  Pt −τ N () dt dn V nt eV 21ai 21 nj 2a2ph 2a 2a 22 aa =−Γ gn −n − + () (3) 22 aa 2a 02a dt V τ eV 22 aaN 2a dn n I mph , ms ms ms = G g (n n ) + , (4) ms ms ms 0ms dt V t eV ms ms ms dn n I mph ms ms ms =−Γ gn −n − + () where the subscripts m (m = 1, 2) identify the VCSEL-SA1 and VCSEL-SA2, respectively, ms ms ms 0ms (4) dt V τ eV ms ms ms while the active and absorber regions are identified by subscripts a and s, respectively. N (t) denotes the total amount of photons in the cavity. Furthermore, the number mph 102 where the subscripts m (m = 1, 2) identify the VCSEL-SA1 and VCSEL-SA2, respectively, of carriers and the bias current are defined as n t and I. The term ky( t, Dt represents ( ) ) 103 while the active and absorber regions are identified by subscripts a and s, respectively. the electrically controlled input stimulus coupled into the gain region, where k and Dt 104 N (t) denotes the total amount of photons in the cavity. Furthermore, the number of mph separately denote the input strength and the temporal duration of perturbation. W is 105 carriers and the bias current are defined as n(t) and I. The term kψ(t,Δt) represents the the feedback weight, t is feedback delay, G is the confinement factor, g is the differential 106 electrically controlled input stimulus coupled into the gain region, where k and Δt sep- gain/loss, B is the bimolecular recombination term, and b is the spontaneous emission 107 arately denote the input strength and the temporal duration of perturbation. W is the coupling factor. W and t are the injection coupling weight and injection coupling delay 12 inj 108 feedback weight, τ is feedback delay, Γ is the confinement factor, g is the differential from VCSEL-SA1 to VCSEL-SA2, respectively. 109 gain/loss, B is the bimolecular recombination term, and β is the spontaneous emission The output power is proportional to the photon number N inside the cavity and can ph 110 coupling factor. W12 and τ are the injection coupling weight and injection coupling de- be described as inj h G hc mc ma 111 lay from VCSEL-SA1 to VCSEL-SA2, respectively. P (t)  N (t). (5) m mph t l 112 The output power is proportional to the photon number mph N inside the cavity and ph 113 can be described as In this work, we use the PSNR (peak signal-to-noise ratio) to evaluate the image transmission performances [28,29], which is calculated using the logarithm of mean square η Γ hc mc ma Pt ≈ N t () () error (MSE), representing the mean square error between the output images and the original mmph (5) τλ mph images. In MSE, grayscale images should feature M  N dimensions, whereas M  N  O dimensions should be considered in RGB color images, which can be described as 114 In this work, we use the PSNR (peak signal-to-noise ratio) to evaluate the image 115 transmission performances [28,29], which is calculated using the logarithm of mean M N O h i MSE = (I I ) , (6) 116 square error (MSE), representing the mean square error between the output images and å å å (x,y,z) (x,y,z) M N O x = 1 y = 1 z = 1 117 the original images. In MSE, grayscale images should feature M × N dimensions, whereas 118 M × N × O dimensions should be considered in RGB color images, which can be described where M and N denote image resolution, O denotes the number of image channels, and 119 as I represents the pixel value of the original image at the x, y coordinates and channel (x,y,z) Photonics 2021, 8, 238 4 of 11 z, in the same way that I represents the output image. Correspondingly, PSRN can be described as PSRN = 10log10(max /MSE), (7) where max is the highest scale value of the 8 bit grayscale 255. From Equations (6) and (7), it can be seen that the difference in pixel values at the same coordinates and channels induces an error. Moreover, a higher PSNR value leads to less distortion. Conversely, a smaller PSNR results in more differences in the pixel value between the two images. Generally, a PSNR over 30 dB indicates that the image quality is good, and that the distortion can be perceived but acceptable. 3. Results and Discussions Based on the fourth-order Runge–Kutta methods, the modified rate Equations (1)–(4) can be numerically solved. For simplicity, we adopted identical parameters for two VCSEL- SAs and the threshold current I = 2.4 mA for a solitary laser in this work. Moreover, to th operate the laser in excitable regime, the bias current was set lower than the threshold current, namely, I = 2 mA and I = 0 mA. The other used parameters were as follows [22,30,31]: a s l = 850 nm, G = 0.06, G = 0.05, t = 1.1 ns, t = 100 ps, t = 4.8 ps, h = 6.634  10 , a s a s ph 18 3 18 3 12 3 1 12 3 1 V = 2.4 10 m , V = 2.4 10 m , g = 2.9 10 m s , g = 14.5 10 m s , a s a s 24 3 24 3 15 3 1 4 n = 1.1  10 m , n = 0.89  10 m , B = 1  10 m s , b = 1  10 , h = 0.4, 0a 0s r c 8 1 c = 3  10 ms . 3.1. Spiking Coding In the biological nervous system, communication across different neurons can be realized through transmitting an action voltage or spikes. In particular, the spike-based information transmission system can conduct sparse and efficient information transfer via spikes [5]. Spikes are essentially binary events including 0 and 1. A VCSEL-SA neuron is only active when spike events come; otherwise, it remains idle. Therefore, the event-driven encoding method necessarily contributes to energy efficiency over a given period of time, as demonstrated in some spike-based information processing such as speech and image recognition [32–34]. Consequently, information encoding of neurons becomes a key issue in neuron science. In this work, the image is firstly encoded as binary data with the on–off keying (OOK) format, which is used as perturbation to inject into the first VCSEL-SA together with the bias current; then, the encoded spike sequence in response to the external stimulus can be generated. Firstly, the regular pulse stimulus is considered. Figure 2a–c show the spiking response of VCSEL-SA1 to the stimulus with the fixed k = 1.1  10 and different temporal durations Dt of 1.21 ns, 2.42 ns, and 3.63 ns. The red dashed lines and the blue solid lines denote the input stimulus and corresponding spiking response, respectively. The binary series of ones and zeros are presented at the top of each diagram in Figure 2. When a 0 bit message is input, no spikes can emerge, which is identical to a free-running laser. When a 1 bit message is input, spikes can be excited once the excitable threshold is exceeded and the SA is saturated, resulting in the rapid release of the accumulated photon energy; then, the gain is depleted. As a result, a conversion from binary data to spike (BTS) is successfully realized. Next, after taking into account that the BTS conversation rate limits the spike-based information transmission rate [22], Figure 3 gives the conversion rate variation with input strength under different bias currents. When the injection strength remains constant, a higher conversation rate can be obtained for higher bias current, which can be interpreted as the refractory period decreasing with increasing bias current; then, a smaller injection perturbation can meet the excitable threshold condition. Obviously, through controlling the injection strength and bias current, the BTS rate can be adjusted in a relatively large range even if the maximal conversion rate is limited by the slower of the two carrier lifetimes of the gain and SA [35], which can offer huge prospects for future neuromorphic computing. Photonics 2021, 8, x FOR PEER REVIEW 5 of 12 Photonics 2021, 8, 238 5 of 11 Photonics 2021, 8, x FOR PEER REVIEW 5 of 12 164 Figure 2. Time series for VCSEL-SA1 output corresponding to the input stimulus under different 165 temporal durations Δt of (a) 1.21 ns, (b) 2.42 ns, and (c) 3.63 ns, where red dashed lines and blue 166 solid lines denote the binary encoded stimulus and the spiking response trains, respectively. 167 Next, after taking into account that the BTS conversation rate limits the spike-based 168 information transmission rate [22], Figure 3 gives the conversion rate variation with input 169 strength under different bias currents. When the injection strength remains constant, a 170 higher conversation rate can be obtained for higher bias current, which can be interpreted 171 as the refractory period decreasing with increasing bias current; then, a smaller injection 172 perturbation can meet the excitable threshold condition. Obviously, through controlling 173 the injection strength and bias current, the BTS rate can be adjusted in a relatively large 164 Figure 2. Time series for VCSEL-SA1 output corresponding to the input stimulus under different Figure 2. Time series for VCSEL-SA1 output corresponding to the input stimulus under different 174 range even if the maximal conversion rate is limited by the slower of the two carrier life- 165 temporal durations Δt of (a) 1.21 ns, (b) 2.42 ns, and (c) 3.63 ns, where red dashed lines and blue temporal durations Dt of (a) 1.21 ns, (b) 2.42 ns, and (c) 3.63 ns, where red dashed lines and blue 175 times of the gain and SA [35], which can offer huge prospects for future neuromorphic 166 solid lines denote the binary encoded stimulus and the spiking response trains, respectively. solid lines denote the binary encoded stimulus and the spiking response trains, respectively. 176 computing. 167 Next, after taking into account that the BTS conversation rate limits the spike-based 168 information transmission rate [22], Figure 3 gives the conversion rate variation with input 2.5 169 strength under different bias currents. When the injection strength remains constant, a 170 higher conversation rate can be obtained for higher bias current, which can be interpreted 2.0 171 as the refractory period decreasing with increasing bias current; then, a smaller injection 172 perturbation can meet the excitable threshold condition. Obviously, through controlling 1.5 173 the injection strength and bias current, the BTS rate can be adjusted in a relatively large 174 range even if the maximal conversion rate is limited by the slower of the two carrier life- 1.0 Biasing Current:1.9mA 175 times of the gain and SA [35], which can offer huge prospects for future neuromorphic Biasing Current:2.0mA Biasing Current:2.1mA 176 computing. 0.5 0.000 0.001 0.002 0.003 0.004 0.005 2.5 k(a.u.) 178 Figure 3. Conversion rate variation with input strength k for various biasing currents. Figure 3. Conversion rate variation with input strength k for various biasing currents. 2.0 3.2. Image Transmission 179 3.2. Image Transmission 1.5 In natural biological neuron networks, different neurons can realize communication 180 In natural biological neuron networks, different neurons can realize communication based on the transmission of the excited or suppressed spiking signals amongst neighbor- 181 based on the transmission of the excited or suppressed spiking signals amongst neighbor- 1.0 Biasing Current:1.9mA ing neurons through their axons and dendrites. In particular, the precise timing of spikes 182 ing neurons through their axons and dendrites. In particular, the precise timing of spikes Biasing Current:2.0mA Biasing Current:2.1mA without destroying their temporal structure is necessary for the successful communica- 183 without destroying their temporal structure is necessary for the successful communica- 0.5 tion. Figure 4 shows the spiking responses and n variation of two cascaded VCSEL-SAs 184 tion. Figure 4 shows the spiking responses and variation of two cascaded VCSEL-SAs for different injection coupling weights, where the input strength k = 1.1  10 −3 . After 0.000 0.001 0.002 0.003 0.004 0.005 185 for different injection coupling weights, where the input strength k = 1.1 × 10 . After tak- k(a.u.) taking into account that the injection coupling delay has no effect on the output charac- 186 ing into account that the injection coupling delay has no effect on the output characteris- teristics except for the time shift of spiking response series in our work, t was set as inj 187 tics except for the time shift of spiking response series in our work, τ was set as 0 ns for 178 Figure 3. Conversion rate variation with input strength k for various biasing currents. inj 0 ns for simplicity. The red dashed lines and blue solid lines denote the binary encoded 188 simplicity. The red dashed lines and blue solid lines denote the binary encoded stimulus stimulus and the spiking response trains, respectively, while the green solid lines denote 179 3.2. Image Transmission the variation of the number of carriers. Under external stimulus, VCSEL-SA1 fires five 180 In natural biological neuron networks, different neurons can realize communication consecutive spike signals during the perturbation time, as shown in Figure 4a1 and b1. For 181 based on the transmission of the excited or suppressed spiking signals amongst neighbor- –1 a relatively low coupling weight of 0.007 mw , as shown in Figure 4a3, the fired spiking 182 ing neurons through their axons and dendrites. In particular, the precise timing of spikes pattern in VCSEL-SA1 can be propagated to VCSEL-SA2. However, only two or three 183 without destroying their temporal structure is necessary for the successful communica- spikes can be excited, and some spiking information is lost for this low coupling weight 184 tion. Figure 4 shows the spiking responses and variation of two cascaded VCSEL 1-SAs during the propagation. Then, upon increasing the coupling weight to 0.017 mw , as −3 185 for different injection coupling weights, where the input strength k = 1.1 × 10 . After tak- shown in Figure 4b1–b4, the spiking events can be entirely propagated from VCSEL-SA1 to 186 ing into account that the injection coupling delay has no effect on the output characteris- VCSEL-SA2. Hence, a proper coupling weight can guarantee the successful propagation of 187 tics except for the time shift of spiking response series in our work, τ was set as 0 ns for inj image-spiking patterns. This phenomenon can be interpreted as both the accumulated car- 188 simplicity. The red dashed lines and blue solid lines denote the binary encoded stimulus rier density in the active region and the excited threshold of laser, which is approximately Conversion Rate (Gbps) Conversion Rate (Gbps) Photonics 2021, 8, x FOR PEER REVIEW 6 of 12 189 and the spiking response trains, respectively, while the green solid lines denote the vari- 190 ation of the number of carriers. Under external stimulus, VCSEL-SA1 fires five consecu- 191 tive spike signals during the perturbation time, as shown in Figure 4a1 and b1. For a rel- –1 192 atively low coupling weight of 0.007 mw , as shown in Figure 4a3, the fired spiking pat- 193 tern in VCSEL-SA1 can be propagated to VCSEL-SA2. However, only two or three spikes 194 can be excited, and some spiking information is lost for this low coupling weight during -1 195 the propagation. Then, upon increasing the coupling weight to 0.017 mw , as shown in 196 Figure 4b1–b4, the spiking events can be entirely propagated from VCSEL-SA1 to VCSEL- Photonics 2021, 8, 238 6 of 11 197 SA2. Hence, a proper coupling weight can guarantee the successful propagation of image- 198 spiking patterns. This phenomenon can be interpreted as both the accumulated carrier 199 density in the active region and the excited threshold of laser, which is approximately expressed as n = (t h G g +1)/(t G g ) + n , thus determining the spiking ex- 200 expressed as nathresh = (τ η 0Γs gs +1 s )/(τ Γ g ) +a n a , thus determining the spiking exciting ph ph 0a athresh ph s ph a 0a 0s s a citing phenomenon. Once the n accumulates to exceed the exciting threshold, the spikes 201 phenomenon. Once the n accumulates to exceed the exciting threshold, the spikes can can be excited. Correspondingly, n exhibits an evolution tendency of abrupt decrease and 202 be excited. Correspondingly, n exhibits an evolution tendency of abrupt decrease and then gradual recovery, and vice versa. Figure 4 shows the corresponding evolution of n . 203 then gradual recovery, and vice versa. Figure 4 shows the corresponding evolution of n . From these diagrams, one can see that, for a small coupling weight, a relatively longer 204 From these diagrams, one can see that, for a small coupling weight, a relatively longer accumulated time of n to reach the exciting threshold is necessary due to the limitation of 205 accumulated time of n to reach the exciting threshold is necessary due to the limitation the refractory period in the VCSEL-SA neuron, and new stimulus events cannot excite the 206 of the refractory period in the VCSEL-SA neuron, and new stimulus events cannot excite spikes during the carrier recovery. Upon further increasing the accumulation time until 207 the spikes during the carrier recovery. Upon further increasing the accumulation time un- n exceeds the exciting threshold, a new spike is excited. Therefore, it is necessary for the 208 til n exceeds the exciting threshold, a new spike is excited. Therefore, it is necessary for successful spike pattern propagation to increase the coupling weight to a certain level. 209 the successful spike pattern propagation to increase the coupling weight to a certain level. (a) (b) 210 Figure 4. Spiking outputs (rows 1 and 3) and n evolution (rows 2 and 4) in a two cascaded VCSEL-SA system for different Figure 4. Spiking outputs (rows 1 and 3) and n evolution (rows 2 and 4) in a two cascaded VCSEL-SA system for different -3 -1 -1 211 coupling weights of 0.007 mw (a) and 0.017 mw (b), where Δt = 6.05 ns, k = 1.1 × 10 , τ = 0 ns. 1 1 3 inj coupling weights of 0.007 mw (a) and 0.017 mw (b), where Dt = 6.05 ns, k = 1.1  10 , t = 0 ns. inj 212 Next, we disc Next, we discuss uss the im the image age transmissio transmissionn performance performancbased e based on th on this pr is proposed oposed system. sys- 213 tem. Figure 5a,b show the transmission quality of image with 73 × 73 pixels under k = 1.1 Figure 5a,b show the transmission quality of image with 73 73 pixels under k = 1.1 10 . −3 214 × 10 From . From this diagram, this done iagra can m, one c see that an image see ttransmission hat image tran can smis be si successfully on can be s achieved uccessfube- lly 215 achieve tween two d bet cascaded ween twVCSEL-SA o cascaded neur VCSE ons L-SA n under euron suitable s under s conditions, uitable condit whichions further , which f verifies ur- the results in Figure 4b1–b4. To further investigate the feasibility of this proposed method 216 ther verifies the results in Figure 4b1–b4. To further investigate the feasibility of this pro- in image transmission, Figure 5c,d show the transmission results of high-resolution im- 217 posed method in image transmission, Figure 5c,d show the transmission results of high- 3 1 −3 1 218 resol age with ution im 512 age  512 wipixels, th 512 × 51 wher 2 e pikxe =ls 1.1 , where  10 k = 1 and .1 × W10 = and 0.017 W mw 12 = 0.01 . Obviously 7 mw− . Obvi , the- feasibility of the high-resolution image transmission based on spiking dynamics of elec- 219 ously, the feasibility of the high-resolution image transmission based on spiking dynamics tronically controlled VCSEL-SA can be demonstrated to a certain extent even though the 220 of electronically controlled VCSEL-SA can be demonstrated to a certain extent even image transmission with higher resolution is not considered due to the limited computing 221 though the image transmission with higher resolution is not considered due to the limited ability of our computer, which can open a new window for future high-speed information 222 computing ability of our computer, which can open a new window for future high-speed transmission of high-resolution images or full HD videos. In addition, we should note that, 223 information transmission of high-resolution images or full HD videos. In addition, we due to limitation of BTS conversation rate, high-speed image transmission can introduce 224 should note that, due to limitation of BTS conversation rate, high-speed image transmis- higher BERs. 225 sion can introduce higher BERs. Generally, there exists a certain difference between two used lasers. Therefore, it is necessary to investigate the effect of several typical parameter mismatches on the spik- ing dynamics, and Figure 6 demonstrates the images transmission performance under a fixed coupling weight of 0.017 mw , where several typical parameter mismatches in- cluding t , t , and I are considered. For simplicity, the parameter value of VCSEL-SA1 ph a a is fixed while some parameters of VCSEL-SA2 are adjusted. The relative mismatched parameters are defined as Dt = t t /t , Dt = (t t )/t , and ph 2ph 1ph 1ph a 2a 1a 1a DI = (I I )/I . From Figure 6, one can see that the image transmission is fea- a 2a 1a 1a sible when the two lasers are in a certain parameter mismatch range. However, when the parameter mismatch exceeds a certain level, the image PSNR decreases with the increase in mismatch degree, and the image is distorted accordingly. Moreover, com- pared with I and t , mismatched t has a relatively smaller influence on the image a a ph transmission performance. Photonics 2021, 8, x FOR PEER REVIEW 7 of 12 Photonics 2021, 8, 238 7 of 11 Photonics 2021, 8, x FOR PEER REVIEW 7 of 12 227 Figure 5. Transmission results between two cascaded VCSEL-SA neurons for different resolution 228 images of 73 × 73 pixels (a,b) and 512 × 512 pixels (c,d), where (a,c) correspond to the original im- 229 age and (b,d) correspond to the transmitted image. 230 Generally, there exists a certain difference between two used lasers. Therefore, it is 231 necessary to investigate the effect of several typical parameter mismatches on the spiking 232 dynamics, and Figure 6 demonstrates the images transmission performance under a fixed -1 233 coupling weight of 0.017 mw , where several typical parameter mismatches includ- 234 ing τ , τ , and I are considered. For simplicity, the parameter value of VCSEL-SA1 is ph a a 235 fixed while some parameters of VCSEL-SA2 are adjusted. The relative mismatched pa- 236 rameters are defined as ∆τ = (τ - τ /τ , ∆τ =(τ - τ /τ , and ∆I =(I - I )/I . ph 2ph 1ph 1ph a 2a 1a 1a a 2a 1a 1a 237 From Figure 6, one can see that the image transmission is feasible when the two lasers are 238 in a certain parameter mismatch range. However, when the parameter mismatch exceeds 227 Figure 5. Transmission results between two cascaded VCSEL-SA neurons for different resolution Figure 5. Transmission results between two cascaded VCSEL-SA neurons for different resolution 239 a certain level, the image PSNR decreases with the increase in mismatch degree, and the 228 images of 73 × 73 pixels (a,b) and 512 × 512 pixels (c,d), where (a,c) correspond to the original im- images of 73 73 pixels (a,b) and 512 512 pixels (c,d), where (a,c) correspond to the original image 229 240 age and (b,d) correspond to image is distorted accord the transmitted image. ingly. Moreover, compared with I and τ , mismatched τ has a a ph and (b,d) correspond to the transmitted image. 241 a relatively smaller influence on the image transmission performance. 230 Generally, there exists a certain difference between two used lasers. Therefore, it is 231 necessary to investigate the effect of several typical parameter mismatches on the spiking 232 dynamics, and Figure 6 demonstrates the images transmission performance under a fixed -1 233 coupling weight of 0.017 mw , where several typical parameter mismatches includ- 234 ing τ , τ , and I are considered. For simplicity, the parameter value of VCSEL-SA1 is ph a a 235 fixed while some parameters of VCSEL-SA2 are adjusted. The relative mismatched pa- 236 rameters are defined as ∆τ = (τ - τ /τ , ∆τ =(τ - τ /τ , and ∆I =(I - I )/I . ph 2ph 1ph 1ph a 2a 1a 1a a 2a 1a 1a 237 From Figure 6, one can see that the image transmission is feasible when the two lasers are 238 in a certain parameter mismatch range. However, when the parameter mismatch exceeds 239 a certain level, the image PSNR decreases with the increase in mismatch degree, and the 240 image is distorted accordingly. Moreover, compared with I and τ , mismatched τ has a a ph 241 a relatively smaller influence on the image transmission performance. 243 Figure 6. PSNRs of output images from VCSEL-SA2 for different parameter mismatches of (a) τ , (b) τ , and (c) I , where Figure 6. PSNRs of output images from VCSEL-SA2 for different parameter mismatches of (a) t , ph ph a 244 the first (b ,) setcond, , and third, (c) I and , wher foe ur the th colu first, msecond, ns respecti thir vely d, and corr fourth espond to columns −10%, respectively −5%, 0%, 5%corr , and 10% espond parameter mismatches. to 10%, a a 5%, 0%, 5%, and 10% parameter mismatches. Figure 7 shows the PSNR variations of the output images with different coupling weight between the two lasers and corresponding transmitted images. From this diagram, one can see that PNSR gradually increases upon increasing the coupling weight, and then stabilizes at a certain level. When the coupling weight is relatively small, the transmitted images are seriously detorted and become very blurred, as shown in Figure 7(b1,b2). With increasing coupling weight, the images can be successfully transmitted, as shown in Figure 7(b3,b4). Moreover, Figure 8 gives the transmitted images for different coupling weight under 7% mismatched parameters of t (a), t (b), and I (c). Upon increasing the ph a a coupling weight, the black spots in the image disappear and then the image becomes clear. Correspondingly, the PSNR of the output image increases. Obviously, typical parameter 243 Figure 6. PSNRs of output images from VCSEL-SA2 for different parameter mismatches of (a) τ , (b) τ , and (c) I , where ph a mismatches have some impact on the image transmission performance in our proposed 244 the first, second, third, and fourth columns respectively correspond to −10%, −5%, 0%, 5%, and 10% parameter mismatches. Photonics 2021, 8, x FOR PEER REVIEW 8 of 12 Photonics 2021, 8, x FOR PEER REVIEW 8 of 12 245 Figure 7 shows the PSNR variations of the output images with different coupling 246 weight between the two lasers and corresponding transmitted images. From this diagram, 245 Figure 7 shows the PSNR variations of the output images with different coupling 247 246 one can weight between the two la see that PNSR grad sers ually an inc d correspond reases upon ing t incre ran asing smit t ted im he coup ages. F ling wei rom t ght his , a dia nd t gr h a en m, 248 stabilizes at a certain level. When the coupling weight is relatively small, the transmitted 247 one can see that PNSR gradually increases upon increasing the coupling weight, and then 249 images are seriously detorted and become very blurred, as shown in Figure 7(b1,b2). With 248 stabilizes at a certain level. When the coupling weight is relatively small, the transmitted 250 increasing coupling weight, the images can be successfully transmitted, as shown in Fig- 249 images are seriously detorted and become very blurred, as shown in Figure 7(b1,b2). With 251 ure 7(b3,b4). Moreover, Figure 8 gives the transmitted images for different coupling 250 increasing coupling weight, the images can be successfully transmitted, as shown in Fig- 252 weight under 7% mismatched parameters of τ (a), τ (b), and I c . Upon increasing 251 ure 7(b3,b4). Moreover, Figure 8 gives the transmitted images for different coupling ph a a Photonics 2021, 8, 238 8 of 11 252 weight under 7% mismatched parameters of τ (a), τ (b), and I (c . Upon increasing 253 the coupling weight, the black spots in the image disappear and then the image becomes ph a a 254 clear. Correspondingly, the PSNR of the output image increases. Obviously, typical pa- 253 the coupling weight, the black spots in the image disappear and then the image becomes 255 rameter mismatches have some impact on the image transmission performance in our 254 clear. Correspondingly, the PSNR of the output image increases. Obviously, typical pa- 256 proposed cascaded system. By suitably increasing the coupling weight, the image propa- 255 rameter mismatches have some impact on the image transmission performance in our cascaded system. By suitably increasing the coupling weight, the image propagation 257 gation robustness to the parameter mismatches can be efficiently enhanced [33,36]. 256 proposed cascaded system. By suitably increasing the coupling weight, the image propa- robustness to the parameter mismatches can be efficiently enhanced [33,36]. 257 gation robustness to the parameter mismatches can be efficiently enhanced [33,36]. 259 Figure 7. (a) PSNRs of output images from VCSEL-SA2 under different coupling weights; (b) the transmitted images for Figure 7. (a) PSNRs of output images from VCSEL-SA2 under different coupling weights; (b) the transmitted images for −1. 260 different coupling weights of (b1) 0.005, (b2) 0.010, (b3) 0.0155, and (b4) 0.020 mw 259 Figure 7. (a) PSNRs of output images from VCSEL-SA2 under different coupling weights 1 ; (b) the transmitted images for different coupling weights of (b1) 0.005, (b2) 0.010, (b3) 0.0155, and (b4) 0.020 mw . −1. 260 different coupling weights of (b1) 0.005, (b2) 0.010, (b3) 0.0155, and (b4) 0.020 mw 262 Figure 8. PSNRs of output images from VCSEL-SA2 under 7% parameter mismatches of (a) τ , ph 263 262 (b Figure 8. ) τ , and PS (c)NRs of I , where th output images from VCSEL- e first, second, third, fou SA2 under 7% parameter mismatches rth, and fifth columns respectively correspond to of (a) τ , a ph Figure 8. PSNRs of output images from VCSEL-SA2 under 7% parameter mismatches of (a) t , ph −1 264 0.009, 0.011, 0.013, 0.015, and 0.017 mw coupling weight. 263 (b) τ , and (c) I , where the first, second, third, fourth, and fifth columns respectively correspond to (b) t , and (c) I , where the first, second, third, fourth, and fifth columns respectively correspond to a a −1 264 0.009, 0.011, 0.013, 0.015, and 0.017 mw coupling weight. 0.009, 0.011, 0.013, 0.015, and 0.017 mw coupling weight. In a practical information transmission system, the device errors can also affect the information transmission performance, and different error correction methods have been adopted to assure successful information transmission [37,38]. Here, we further adopted the 8B10B conversion method to optimize the system communication performance. The decoded images before and after adopting the 8B10B method are shown in Figure 9, where 3 1 k = 1.1  10 and W = 0.015 mw . From these diagrams, one can see that, under our simulation conditions, the decoded image quality is significantly improved after adopting this error correction method, which indicates that our proposed system can be applied in future image transmission after adopting a suitable error correction method. Moreover, the Photonics 2021, 8, x FOR PEER REVIEW 9 of 12 265 In a practical information transmission system, the device errors can also affect the 266 information transmission performance, and different error correction methods have been 267 adopted to assure successful information transmission [37,38]. Here, we further adopted 268 the 8B10B conversion method to optimize the system communication performance. The 269 decoded images before and after adopting the 8B10B method are shown in Figure 9, where −3 −1 Photonics 2021, 8, 238 9 of 11 270 k = 1.1 × 10 and W12 = 0.015 mw . From these diagrams, one can see that, under our 271 simulation conditions, the decoded image quality is significantly improved after adopting 272 this error correction method, which indicates that our proposed system can be applied in 273 future image transmission after adopting a suitable error correction method. Moreover, additional simulation results demonstrate that this proposed image transmission scheme 274 the additional simulation results demonstrate that this proposed image transmission has relatively good robustness to noise under our simulation conditions. 275 scheme has relatively good robustness to noise under our simulation conditions. Figure 9. The decoded images from VCSEL-SA2 without (a) and with (b) 8B10B conversion method, 277 Figure 9. The decoded images from VCSEL-SA2 without (a) and with (b) 8B10B conversion -1 where W = 0.015 mw . 278 method,12 where W12 = 0.015 mw . 3.3. Storage of Spiking Patterns 279 3.3. Storage of Spiking Patterns Lastly, after adding the optoelectrical feedback loop to the first laser, the storage 280 Lastly, after adding the optoelectrical feedback loop to the first laser, the storage properties of image spiking patterns are discussed. Figure 10 shows the output time series 281 properties of image spiking patterns are discussed. Figure 10 shows the output time series of two cascaded VCSEL-SAs as response to the input stimulus (red dashed line), where 282 of two cascaded VCSEL-SAs as response to the input stimulus (red dashed line), where 1 1 4 W = 0.010 mw and W = 0.017 mw . An injected rectangular pulse with k = 3  10 f 12 −1 −1 −4 283 Wf = 0.010 mw and W12 = 0.017 mw . An injected rectangular pulse with k = 3 × 10 and Δt and Dt = 8 ns was used to encode the injection perturbation. From these diagrams, one 284 = 8 ns was used to encode the injection perturbation. From these diagrams, one can see can see that, under an external stimulus, a three-spike burst is fired repetitively by VCSEL- 285 that, under an external stimulus, a three-spike burst is fired repetitively by VCSEL-SA1 SA1 with a fixed time interval corresponding to the feedback delay. This phenomenon 286 with a fixed time interval corresponding to the feedback delay. This phenomenon can be can be interpreted as the first external perturbation firing three spikes for VCSEL-SA1, 287 interpreted as the first external perturbation firing three spikes for VCSEL-SA1, which can which can repetitively stimulate the VCSEL-SA1 through the added feedback loop. Con- 288 repetitively stimulate the VCSEL-SA1 through the added feedback loop. Consequently, a sequently, a repeated spiking response can be observed, as shown in Figure 10b. More- 289 repeated spiking response can be observed, as shown in Figure 10b. Moreover, the spike over, the spike responses of VCSEL-SA1 can be transmitted to VCSEL-SA2, as shown in 290 responses of VCSEL-SA1 can be transmitted to VCSEL-SA2, as shown in Figure 10c. Ob- Figure 10c. Obviously, the encoded spike information can be successfully stored in the 291 viously, the encoded spike information can be successfully stored in the electrically con- electrically controlled cascaded system under this condition, which can be applied in future 292 trolled cascaded system under this condition, which can be applied in future complex complex spiking pattern processing systems. 293 spiking pattern processing systems. Figure 10. Storage of spiking patterns in two cascaded VCSEL-SAs with optoelectronic feedback Figure 10. Storage of spiking patterns in two cascaded VCSEL- 1 1 4 under W = 0.010 mw , W = 0.017 mw , k = 3  10 and t = 2 ns, where (a) injected pulse, f 12 inj −1 −1 −4 SAs with optoelectronic feedback under Wf = 0.010 mw , W12 = 0.017 mw , k = 3 × 10 and τ =2 𝑠𝑛 , inj (b) spike trains from VCSEL-SA1 and (c) spike trains form VCSEL-SA2. where (a) injected pulse, (b) spike trains from VCSEL-SA1 and (c) spike trains form VCSEL-SA2." 4. Conclusions In conclusion, we demonstrated an image encoding and transmission system based on two electrically controlled vertical-cavity surface-emitting lasers with an embedded Photonics 2021, 8, 238 10 of 11 saturable absorber (VCSEL-SA). The simulation results show that, the conversion rate from binary code to spiking signal is highly dependent on the input strength and bias current. Under suitable conditions, the encoding images can be successfully transmitted in the proposed photonic neuron system. Moreover, typical parameter mismatches have some impact on the image transmission performance, and suitably increasing the coupling weight can improve the system robustness to parameter mismatches to a certain extent. Additionally, spiking patterns can be efficiently stored and transmitted in the electroni- cally controlled cascaded system. This work is valuable for the future construction and application of large-scale neural networks based on photonic neurons. Author Contributions: M.N. was responsible for the numerical simulation, analyzing the results, and the writing of the paper; X.T., Z.G., L.X., J.W., F.M. and Q.Z. were responsible for writing and revising the manuscript; X.L. and T.D. were responsible for the discussion of the results and reviewing/editing of the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (Grant No. 61875167), the Natural Science Foundation of Chongqing City (CSTC 2019jcyj-msxm X0136), the Fundamental Research Funds for the Central Universities of China (XDJK2020B053), and the National Training Program of Innovation and Entrepreneurship for Undergraduates College Students’ Innovation Fund of Southwest University under Grant No. 202010635089. Institutional Review Board Statement: “Not applicable” for studies not involving humans or animals. Informed Consent Statement: “Not applicable” for studies not involving humans. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available as the data also forms part of an ongo- ing study. Conflicts of Interest: The authors declare no conflict of interest. References 1. Yang, J.Q.; Wang, R.; Ren, Y.; Mao, J.Y.; Wang, Z.P.; Zhou, Y.; Han, S.T. 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Image Transmission Based on Spiking Dynamics of Electrically Controlled VCSEL-SA Neuron

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hv photonics Communication Image Transmission Based on Spiking Dynamics of Electrically Controlled VCSEL-SA Neuron Min Ni, Xiaodong Lin, Xi Tang, Ziye Gao, Luyao Xiao, Jun Wang, Fan Ma, Qiulan Zheng and Tao Deng * School of Physical Science and Technology, Southwest University, Chongqing 400715, China; min333@swu.edu.cn (M.N.); linxd@swu.edu.cn (X.L.); tx1982@swu.edu.cn (X.T.); zygao@swu.edu.cn (Z.G.); xly123456@swu.edu.cn (L.X.); wangjun369@swu.edu.cn (J.W.); mafan2020@swu.edu.cn (F.M.); zql456723@swu.edu.cn (Q.Z.) * Correspondence: dengt@swu.edu.cn Abstract: Based on the spiking dynamics of the electrically controlled vertical-cavity surface-emitting laser with an embedded saturable absorber (VCSEL-SA), we propose an image transmission system using two unidirectionally coupled VCSEL-SAs and numerically investigate the binary-to-spike (BTS) conversation characteristics and the image transmission performance. The simulation results show that, through electrically injecting the binary data to VCSEL-SA, the BTS conversation can be realized and the conversion rate of BTS highly depends on the injection strength and bias current. Thus, the image transmission can be realized in the proposed system. Moreover, the parameter mismatches between these two VCSEL-SAs have some effects on the image transmission performance, but the encoded images are still successfully decoded even under certain parameter mismatches. In addition, spiking patterns can be also stored and transmitted in the cascaded system with optoelectronic feedback loop. Keywords: image transmission; vertical-cavity surface-emitting lasers with an embedded saturable absorber (VCSEL-SA); parameter mismatches; binary to spike (BTS); neuromorphic computing system Citation: Ni, M.; Lin, X.; Tang, X.; Gao, Z.; Xiao, L.; Wang, J.; Ma, F.; Zheng, Q.; Deng, T. Image Transmission Based on Spiking Dynamics of Electrically Controlled 1. Introduction VCSEL-SA Neuron. Photonics 2021, 8, The human brain possesses powerful computing capability and low power consump- 238. https://doi.org/10.3390/ tion [1,2]. Due to the limitation of memory ability, data interaction bandwidth, and high photonics8070238 energy consumption, traditional computers with Von Neumann structure are unable to meet the growing computing needs [3]. Therefore, the in-depth study of brain–machine Received: 30 April 2021 interface technology and neural mimicry systems is conducive to solve these complex Accepted: 22 June 2021 computing problems, which have attracted wide attention [4–7]. Neural mimicry systems Published: 25 June 2021 without Von Neumann structure can simulate biological sensing and realize brain-like computing, which greatly improves the computing capability and reduces power consump- Publisher’s Note: MDPI stays neutral tion [2]. Therefore, this neuromorphic system can possess huge application potential in with regard to jurisdictional claims in processing some complex computing tasks including decision making, learning, sensory published maps and institutional affil- information processing, and pattern recognition [8]. Recently, photonic neuromorphic iations. devices have shown great application prospects in the field of high-speed neuromorphic computing because they can simulate the basic characteristics of biological neurons and provide ultrafine pulse dynamics up to eight orders of magnitude faster than biological neurons [9]. Copyright: © 2021 by the authors. In recent years, semiconductor lasers (SLs) have been viewed as promising candidates Licensee MDPI, Basel, Switzerland. for a neuromorphic photonic model because of its strong analogy with biological neurons This article is an open access article in terms of the underlying excitability mechanisms [10]. Among these SL-based photonic distributed under the terms and neuron models, the VCSEL-based neuron model has attracted extensive attentions because conditions of the Creative Commons VCSEL possesses some unique advantages such as low cost, low energy consumption, easy Attribution (CC BY) license (https:// integration into the 2d/3d array, high coupling efficiency of optical fiber, and compatibility creativecommons.org/licenses/by/ with the existing optical fiber system [11–16]. So far, controllable activation and inhibition 4.0/). Photonics 2021, 8, 238. https://doi.org/10.3390/photonics8070238 https://www.mdpi.com/journal/photonics Photonics 2021, 8, 238 2 of 11 of sub-ns spiking patterns based on commercial VCSELs have been realized [17,18]. More recently, communication of spiking patterns between two cascaded VCSEL neurons was theoretically and experimentally demonstrated [19]. In particular, an integrated two-stage excitable laser VCSEL-SA can be constructed by combining VCSEL with a saturable absorber (SA), which possesses similar advan- tages to that of VCSELs and can be viewed as a simple spike-based leaky integrate-and- fire (LIF) neuron model [20]. In VCSEL-SA, once the number of carriers in the active region of VCSEL-SA accumulates to exceed the exciting threshold, the spikes can be excited. Correspondingly, the number of carriers in the active region exhibits an evolu- tion tendency of abrupt decrease and then gradual recovery, and vice versa. Moreover, VCSEL-SA can generate shorter sub-ns pulses in comparison with the conventional neuron model [21–24]. Furthermore, the excitability threshold of photonic neurons can be adjusted within a certain range [20]. Up to now, the previous studies on VCSEL-SA-based photonic neuron models mainly focused on the optical stimulation method [21,22], but relevant researches on the electrically controlled stimulation method are relatively few. In particular, compared with optically controlled stimulation, electrically controlled stimulation is easy to control and insensitive to phase variation. In addition, due to the limitation of material and technology, the present all-optical neuron networks have some defects such as online training, nonlinear logic operation, and large-scale integration [5,25]. Therefore, some photonic neuron networks focus on the integration of functional units on a chip, while other off-chip units can be optically or electronically realized [26]. Moreover, some machine learning mechanisms such as STDP in the neuron network also need to adjust the weight through controlling the external electronic circuit [27]. Consequently, the combination of photonic integration technology with some mature electronic methods is still very impor- tant, which is helpful to prompt the realization of future all-optical neuron network. Based on the abovementioned considerations, the spiking dynamics and its application based on electrically controlled VCSEL-SA deserve investigation. In this paper, we propose an image transmission system based on two cascaded elec- trically controlled VCSEL-SAs for the first time and investigate the characteristics of image encoding, transmission, and decoding. Moreover, the influence of some typical internal parameters is considered. The results show that this system can successfully realize image transmission under certain parameter mismatches. After introducing an optoelectronic feedback loop, the spiking patterns can be successfully stored and transmitted in the cascade electronically controlled system. 2. Theoretical Model A schematic diagram of the image transmission system composed of two unidirec- tionally coupled VCSEL-SAs is shown in Figure 1. In this system, the input images are firstly encoded into binary codes and mixed with the bias current through a Biastee, which is electrically injected into VCSEL-SA1 to conduct the conversion of binary data to spike (BTS). Then, the output spike signals from VCSEL-SA1 are divided into two parts. One is electrically injected into VCSEL-SA2 after passing through an optical isolator (ISO), a variable attenuator (VA), and a photoelectric detector (PD), where ISO is used to guarantee the unidirectional coupling, VA is used to adjust the injection weight, and PD is used to convert optical signals to electrical signals. The other (the dashed line part) is only used to form an optoelectronic feedback loop, which is used to store the spiking patterns. Consequently, the output spiking signals from VCSEL-SA2 can be used to recover the transmitted images. Photonics 2021, 8, 238 3 of 11 Photonics 2021, 8, x FOR PEER REVIEW 3 of 12 96 Figure 1. Figure 1. S Schematic chematic diag diagram ram of im of image age transm transmission ission betw between een two cas two cascaded caded VC VCSEL-SAs. SEL-SAs. DC DC: : direct direct cu curr rrent, IS ent, ISO: O: opt optical ical 97 isolator, VA: variable attenuator, PD: photodetector. isolator, VA: variable attenuator, PD: photodetector. After considering the effect of the external electrical injection (stimulus) and neglecting 98 After considering the effect of the external electrical injection (stimulus) and neglect- the polarization effects, according to the typical coupled rate equations of a two-section 99 ing the polarization effects, according to the typical coupled rate equations of a two-sec- excitable laser with SA region and gain region, the modified rate equations of two cascaded 100 tion excitable laser with SA region and gain region, the modified rate equations of two electrically controlled VCSEL-SAs can be described as follows [20,24]: 101 cascaded electrically controlled VCSEL-SAs can be described as follows [20,24]: dN N mph dN N mph mph mph = G g (n n )N + G g (n n )N + V b B n , (1) ma ma ma ms ms ms ma m mr 0ma mph 0ms mph ma dt t =−ΓΓ g() nnN+ g(n−n )N− +V βB n  mph ma ma ma00 ma mph ms ms ms ms mph ma m mr ma (1) dt τ mph I + k y (t, Dt) + W P t t N 1a 1 1 f 1 f dn n 1ph 1a 1a Ik +  ψ t,Δt +WP t −τ N () () dn n = G g (n n ) 1 ph +11af 1 1 f , (2) 1a 1a 1a 01a 11 aa =−Γ gn −n − + dt () V t eV (2) 11 aa 1a 01a 1a 1a 1a dt V τ eV 11 aa 1a N I + W P t t dn n 2ph 2a 12 1 inj 2a 2a = G g (n n ) + , (3) 2a 2a 2a 02a IW +  Pt −τ N () dt dn V nt eV 21ai 21 nj 2a2ph 2a 2a 22 aa =−Γ gn −n − + () (3) 22 aa 2a 02a dt V τ eV 22 aaN 2a dn n I mph , ms ms ms = G g (n n ) + , (4) ms ms ms 0ms dt V t eV ms ms ms dn n I mph ms ms ms =−Γ gn −n − + () where the subscripts m (m = 1, 2) identify the VCSEL-SA1 and VCSEL-SA2, respectively, ms ms ms 0ms (4) dt V τ eV ms ms ms while the active and absorber regions are identified by subscripts a and s, respectively. N (t) denotes the total amount of photons in the cavity. Furthermore, the number mph 102 where the subscripts m (m = 1, 2) identify the VCSEL-SA1 and VCSEL-SA2, respectively, of carriers and the bias current are defined as n t and I. The term ky( t, Dt represents ( ) ) 103 while the active and absorber regions are identified by subscripts a and s, respectively. the electrically controlled input stimulus coupled into the gain region, where k and Dt 104 N (t) denotes the total amount of photons in the cavity. Furthermore, the number of mph separately denote the input strength and the temporal duration of perturbation. W is 105 carriers and the bias current are defined as n(t) and I. The term kψ(t,Δt) represents the the feedback weight, t is feedback delay, G is the confinement factor, g is the differential 106 electrically controlled input stimulus coupled into the gain region, where k and Δt sep- gain/loss, B is the bimolecular recombination term, and b is the spontaneous emission 107 arately denote the input strength and the temporal duration of perturbation. W is the coupling factor. W and t are the injection coupling weight and injection coupling delay 12 inj 108 feedback weight, τ is feedback delay, Γ is the confinement factor, g is the differential from VCSEL-SA1 to VCSEL-SA2, respectively. 109 gain/loss, B is the bimolecular recombination term, and β is the spontaneous emission The output power is proportional to the photon number N inside the cavity and can ph 110 coupling factor. W12 and τ are the injection coupling weight and injection coupling de- be described as inj h G hc mc ma 111 lay from VCSEL-SA1 to VCSEL-SA2, respectively. P (t)  N (t). (5) m mph t l 112 The output power is proportional to the photon number mph N inside the cavity and ph 113 can be described as In this work, we use the PSNR (peak signal-to-noise ratio) to evaluate the image transmission performances [28,29], which is calculated using the logarithm of mean square η Γ hc mc ma Pt ≈ N t () () error (MSE), representing the mean square error between the output images and the original mmph (5) τλ mph images. In MSE, grayscale images should feature M  N dimensions, whereas M  N  O dimensions should be considered in RGB color images, which can be described as 114 In this work, we use the PSNR (peak signal-to-noise ratio) to evaluate the image 115 transmission performances [28,29], which is calculated using the logarithm of mean M N O h i MSE = (I I ) , (6) 116 square error (MSE), representing the mean square error between the output images and å å å (x,y,z) (x,y,z) M N O x = 1 y = 1 z = 1 117 the original images. In MSE, grayscale images should feature M × N dimensions, whereas 118 M × N × O dimensions should be considered in RGB color images, which can be described where M and N denote image resolution, O denotes the number of image channels, and 119 as I represents the pixel value of the original image at the x, y coordinates and channel (x,y,z) Photonics 2021, 8, 238 4 of 11 z, in the same way that I represents the output image. Correspondingly, PSRN can be described as PSRN = 10log10(max /MSE), (7) where max is the highest scale value of the 8 bit grayscale 255. From Equations (6) and (7), it can be seen that the difference in pixel values at the same coordinates and channels induces an error. Moreover, a higher PSNR value leads to less distortion. Conversely, a smaller PSNR results in more differences in the pixel value between the two images. Generally, a PSNR over 30 dB indicates that the image quality is good, and that the distortion can be perceived but acceptable. 3. Results and Discussions Based on the fourth-order Runge–Kutta methods, the modified rate Equations (1)–(4) can be numerically solved. For simplicity, we adopted identical parameters for two VCSEL- SAs and the threshold current I = 2.4 mA for a solitary laser in this work. Moreover, to th operate the laser in excitable regime, the bias current was set lower than the threshold current, namely, I = 2 mA and I = 0 mA. The other used parameters were as follows [22,30,31]: a s l = 850 nm, G = 0.06, G = 0.05, t = 1.1 ns, t = 100 ps, t = 4.8 ps, h = 6.634  10 , a s a s ph 18 3 18 3 12 3 1 12 3 1 V = 2.4 10 m , V = 2.4 10 m , g = 2.9 10 m s , g = 14.5 10 m s , a s a s 24 3 24 3 15 3 1 4 n = 1.1  10 m , n = 0.89  10 m , B = 1  10 m s , b = 1  10 , h = 0.4, 0a 0s r c 8 1 c = 3  10 ms . 3.1. Spiking Coding In the biological nervous system, communication across different neurons can be realized through transmitting an action voltage or spikes. In particular, the spike-based information transmission system can conduct sparse and efficient information transfer via spikes [5]. Spikes are essentially binary events including 0 and 1. A VCSEL-SA neuron is only active when spike events come; otherwise, it remains idle. Therefore, the event-driven encoding method necessarily contributes to energy efficiency over a given period of time, as demonstrated in some spike-based information processing such as speech and image recognition [32–34]. Consequently, information encoding of neurons becomes a key issue in neuron science. In this work, the image is firstly encoded as binary data with the on–off keying (OOK) format, which is used as perturbation to inject into the first VCSEL-SA together with the bias current; then, the encoded spike sequence in response to the external stimulus can be generated. Firstly, the regular pulse stimulus is considered. Figure 2a–c show the spiking response of VCSEL-SA1 to the stimulus with the fixed k = 1.1  10 and different temporal durations Dt of 1.21 ns, 2.42 ns, and 3.63 ns. The red dashed lines and the blue solid lines denote the input stimulus and corresponding spiking response, respectively. The binary series of ones and zeros are presented at the top of each diagram in Figure 2. When a 0 bit message is input, no spikes can emerge, which is identical to a free-running laser. When a 1 bit message is input, spikes can be excited once the excitable threshold is exceeded and the SA is saturated, resulting in the rapid release of the accumulated photon energy; then, the gain is depleted. As a result, a conversion from binary data to spike (BTS) is successfully realized. Next, after taking into account that the BTS conversation rate limits the spike-based information transmission rate [22], Figure 3 gives the conversion rate variation with input strength under different bias currents. When the injection strength remains constant, a higher conversation rate can be obtained for higher bias current, which can be interpreted as the refractory period decreasing with increasing bias current; then, a smaller injection perturbation can meet the excitable threshold condition. Obviously, through controlling the injection strength and bias current, the BTS rate can be adjusted in a relatively large range even if the maximal conversion rate is limited by the slower of the two carrier lifetimes of the gain and SA [35], which can offer huge prospects for future neuromorphic computing. Photonics 2021, 8, x FOR PEER REVIEW 5 of 12 Photonics 2021, 8, 238 5 of 11 Photonics 2021, 8, x FOR PEER REVIEW 5 of 12 164 Figure 2. Time series for VCSEL-SA1 output corresponding to the input stimulus under different 165 temporal durations Δt of (a) 1.21 ns, (b) 2.42 ns, and (c) 3.63 ns, where red dashed lines and blue 166 solid lines denote the binary encoded stimulus and the spiking response trains, respectively. 167 Next, after taking into account that the BTS conversation rate limits the spike-based 168 information transmission rate [22], Figure 3 gives the conversion rate variation with input 169 strength under different bias currents. When the injection strength remains constant, a 170 higher conversation rate can be obtained for higher bias current, which can be interpreted 171 as the refractory period decreasing with increasing bias current; then, a smaller injection 172 perturbation can meet the excitable threshold condition. Obviously, through controlling 173 the injection strength and bias current, the BTS rate can be adjusted in a relatively large 164 Figure 2. Time series for VCSEL-SA1 output corresponding to the input stimulus under different Figure 2. Time series for VCSEL-SA1 output corresponding to the input stimulus under different 174 range even if the maximal conversion rate is limited by the slower of the two carrier life- 165 temporal durations Δt of (a) 1.21 ns, (b) 2.42 ns, and (c) 3.63 ns, where red dashed lines and blue temporal durations Dt of (a) 1.21 ns, (b) 2.42 ns, and (c) 3.63 ns, where red dashed lines and blue 175 times of the gain and SA [35], which can offer huge prospects for future neuromorphic 166 solid lines denote the binary encoded stimulus and the spiking response trains, respectively. solid lines denote the binary encoded stimulus and the spiking response trains, respectively. 176 computing. 167 Next, after taking into account that the BTS conversation rate limits the spike-based 168 information transmission rate [22], Figure 3 gives the conversion rate variation with input 2.5 169 strength under different bias currents. When the injection strength remains constant, a 170 higher conversation rate can be obtained for higher bias current, which can be interpreted 2.0 171 as the refractory period decreasing with increasing bias current; then, a smaller injection 172 perturbation can meet the excitable threshold condition. Obviously, through controlling 1.5 173 the injection strength and bias current, the BTS rate can be adjusted in a relatively large 174 range even if the maximal conversion rate is limited by the slower of the two carrier life- 1.0 Biasing Current:1.9mA 175 times of the gain and SA [35], which can offer huge prospects for future neuromorphic Biasing Current:2.0mA Biasing Current:2.1mA 176 computing. 0.5 0.000 0.001 0.002 0.003 0.004 0.005 2.5 k(a.u.) 178 Figure 3. Conversion rate variation with input strength k for various biasing currents. Figure 3. Conversion rate variation with input strength k for various biasing currents. 2.0 3.2. Image Transmission 179 3.2. Image Transmission 1.5 In natural biological neuron networks, different neurons can realize communication 180 In natural biological neuron networks, different neurons can realize communication based on the transmission of the excited or suppressed spiking signals amongst neighbor- 181 based on the transmission of the excited or suppressed spiking signals amongst neighbor- 1.0 Biasing Current:1.9mA ing neurons through their axons and dendrites. In particular, the precise timing of spikes 182 ing neurons through their axons and dendrites. In particular, the precise timing of spikes Biasing Current:2.0mA Biasing Current:2.1mA without destroying their temporal structure is necessary for the successful communica- 183 without destroying their temporal structure is necessary for the successful communica- 0.5 tion. Figure 4 shows the spiking responses and n variation of two cascaded VCSEL-SAs 184 tion. Figure 4 shows the spiking responses and variation of two cascaded VCSEL-SAs for different injection coupling weights, where the input strength k = 1.1  10 −3 . After 0.000 0.001 0.002 0.003 0.004 0.005 185 for different injection coupling weights, where the input strength k = 1.1 × 10 . After tak- k(a.u.) taking into account that the injection coupling delay has no effect on the output charac- 186 ing into account that the injection coupling delay has no effect on the output characteris- teristics except for the time shift of spiking response series in our work, t was set as inj 187 tics except for the time shift of spiking response series in our work, τ was set as 0 ns for 178 Figure 3. Conversion rate variation with input strength k for various biasing currents. inj 0 ns for simplicity. The red dashed lines and blue solid lines denote the binary encoded 188 simplicity. The red dashed lines and blue solid lines denote the binary encoded stimulus stimulus and the spiking response trains, respectively, while the green solid lines denote 179 3.2. Image Transmission the variation of the number of carriers. Under external stimulus, VCSEL-SA1 fires five 180 In natural biological neuron networks, different neurons can realize communication consecutive spike signals during the perturbation time, as shown in Figure 4a1 and b1. For 181 based on the transmission of the excited or suppressed spiking signals amongst neighbor- –1 a relatively low coupling weight of 0.007 mw , as shown in Figure 4a3, the fired spiking 182 ing neurons through their axons and dendrites. In particular, the precise timing of spikes pattern in VCSEL-SA1 can be propagated to VCSEL-SA2. However, only two or three 183 without destroying their temporal structure is necessary for the successful communica- spikes can be excited, and some spiking information is lost for this low coupling weight 184 tion. Figure 4 shows the spiking responses and variation of two cascaded VCSEL 1-SAs during the propagation. Then, upon increasing the coupling weight to 0.017 mw , as −3 185 for different injection coupling weights, where the input strength k = 1.1 × 10 . After tak- shown in Figure 4b1–b4, the spiking events can be entirely propagated from VCSEL-SA1 to 186 ing into account that the injection coupling delay has no effect on the output characteris- VCSEL-SA2. Hence, a proper coupling weight can guarantee the successful propagation of 187 tics except for the time shift of spiking response series in our work, τ was set as 0 ns for inj image-spiking patterns. This phenomenon can be interpreted as both the accumulated car- 188 simplicity. The red dashed lines and blue solid lines denote the binary encoded stimulus rier density in the active region and the excited threshold of laser, which is approximately Conversion Rate (Gbps) Conversion Rate (Gbps) Photonics 2021, 8, x FOR PEER REVIEW 6 of 12 189 and the spiking response trains, respectively, while the green solid lines denote the vari- 190 ation of the number of carriers. Under external stimulus, VCSEL-SA1 fires five consecu- 191 tive spike signals during the perturbation time, as shown in Figure 4a1 and b1. For a rel- –1 192 atively low coupling weight of 0.007 mw , as shown in Figure 4a3, the fired spiking pat- 193 tern in VCSEL-SA1 can be propagated to VCSEL-SA2. However, only two or three spikes 194 can be excited, and some spiking information is lost for this low coupling weight during -1 195 the propagation. Then, upon increasing the coupling weight to 0.017 mw , as shown in 196 Figure 4b1–b4, the spiking events can be entirely propagated from VCSEL-SA1 to VCSEL- Photonics 2021, 8, 238 6 of 11 197 SA2. Hence, a proper coupling weight can guarantee the successful propagation of image- 198 spiking patterns. This phenomenon can be interpreted as both the accumulated carrier 199 density in the active region and the excited threshold of laser, which is approximately expressed as n = (t h G g +1)/(t G g ) + n , thus determining the spiking ex- 200 expressed as nathresh = (τ η 0Γs gs +1 s )/(τ Γ g ) +a n a , thus determining the spiking exciting ph ph 0a athresh ph s ph a 0a 0s s a citing phenomenon. Once the n accumulates to exceed the exciting threshold, the spikes 201 phenomenon. Once the n accumulates to exceed the exciting threshold, the spikes can can be excited. Correspondingly, n exhibits an evolution tendency of abrupt decrease and 202 be excited. Correspondingly, n exhibits an evolution tendency of abrupt decrease and then gradual recovery, and vice versa. Figure 4 shows the corresponding evolution of n . 203 then gradual recovery, and vice versa. Figure 4 shows the corresponding evolution of n . From these diagrams, one can see that, for a small coupling weight, a relatively longer 204 From these diagrams, one can see that, for a small coupling weight, a relatively longer accumulated time of n to reach the exciting threshold is necessary due to the limitation of 205 accumulated time of n to reach the exciting threshold is necessary due to the limitation the refractory period in the VCSEL-SA neuron, and new stimulus events cannot excite the 206 of the refractory period in the VCSEL-SA neuron, and new stimulus events cannot excite spikes during the carrier recovery. Upon further increasing the accumulation time until 207 the spikes during the carrier recovery. Upon further increasing the accumulation time un- n exceeds the exciting threshold, a new spike is excited. Therefore, it is necessary for the 208 til n exceeds the exciting threshold, a new spike is excited. Therefore, it is necessary for successful spike pattern propagation to increase the coupling weight to a certain level. 209 the successful spike pattern propagation to increase the coupling weight to a certain level. (a) (b) 210 Figure 4. Spiking outputs (rows 1 and 3) and n evolution (rows 2 and 4) in a two cascaded VCSEL-SA system for different Figure 4. Spiking outputs (rows 1 and 3) and n evolution (rows 2 and 4) in a two cascaded VCSEL-SA system for different -3 -1 -1 211 coupling weights of 0.007 mw (a) and 0.017 mw (b), where Δt = 6.05 ns, k = 1.1 × 10 , τ = 0 ns. 1 1 3 inj coupling weights of 0.007 mw (a) and 0.017 mw (b), where Dt = 6.05 ns, k = 1.1  10 , t = 0 ns. inj 212 Next, we disc Next, we discuss uss the im the image age transmissio transmissionn performance performancbased e based on th on this pr is proposed oposed system. sys- 213 tem. Figure 5a,b show the transmission quality of image with 73 × 73 pixels under k = 1.1 Figure 5a,b show the transmission quality of image with 73 73 pixels under k = 1.1 10 . −3 214 × 10 From . From this diagram, this done iagra can m, one c see that an image see ttransmission hat image tran can smis be si successfully on can be s achieved uccessfube- lly 215 achieve tween two d bet cascaded ween twVCSEL-SA o cascaded neur VCSE ons L-SA n under euron suitable s under s conditions, uitable condit whichions further , which f verifies ur- the results in Figure 4b1–b4. To further investigate the feasibility of this proposed method 216 ther verifies the results in Figure 4b1–b4. To further investigate the feasibility of this pro- in image transmission, Figure 5c,d show the transmission results of high-resolution im- 217 posed method in image transmission, Figure 5c,d show the transmission results of high- 3 1 −3 1 218 resol age with ution im 512 age  512 wipixels, th 512 × 51 wher 2 e pikxe =ls 1.1 , where  10 k = 1 and .1 × W10 = and 0.017 W mw 12 = 0.01 . Obviously 7 mw− . Obvi , the- feasibility of the high-resolution image transmission based on spiking dynamics of elec- 219 ously, the feasibility of the high-resolution image transmission based on spiking dynamics tronically controlled VCSEL-SA can be demonstrated to a certain extent even though the 220 of electronically controlled VCSEL-SA can be demonstrated to a certain extent even image transmission with higher resolution is not considered due to the limited computing 221 though the image transmission with higher resolution is not considered due to the limited ability of our computer, which can open a new window for future high-speed information 222 computing ability of our computer, which can open a new window for future high-speed transmission of high-resolution images or full HD videos. In addition, we should note that, 223 information transmission of high-resolution images or full HD videos. In addition, we due to limitation of BTS conversation rate, high-speed image transmission can introduce 224 should note that, due to limitation of BTS conversation rate, high-speed image transmis- higher BERs. 225 sion can introduce higher BERs. Generally, there exists a certain difference between two used lasers. Therefore, it is necessary to investigate the effect of several typical parameter mismatches on the spik- ing dynamics, and Figure 6 demonstrates the images transmission performance under a fixed coupling weight of 0.017 mw , where several typical parameter mismatches in- cluding t , t , and I are considered. For simplicity, the parameter value of VCSEL-SA1 ph a a is fixed while some parameters of VCSEL-SA2 are adjusted. The relative mismatched parameters are defined as Dt = t t /t , Dt = (t t )/t , and ph 2ph 1ph 1ph a 2a 1a 1a DI = (I I )/I . From Figure 6, one can see that the image transmission is fea- a 2a 1a 1a sible when the two lasers are in a certain parameter mismatch range. However, when the parameter mismatch exceeds a certain level, the image PSNR decreases with the increase in mismatch degree, and the image is distorted accordingly. Moreover, com- pared with I and t , mismatched t has a relatively smaller influence on the image a a ph transmission performance. Photonics 2021, 8, x FOR PEER REVIEW 7 of 12 Photonics 2021, 8, 238 7 of 11 Photonics 2021, 8, x FOR PEER REVIEW 7 of 12 227 Figure 5. Transmission results between two cascaded VCSEL-SA neurons for different resolution 228 images of 73 × 73 pixels (a,b) and 512 × 512 pixels (c,d), where (a,c) correspond to the original im- 229 age and (b,d) correspond to the transmitted image. 230 Generally, there exists a certain difference between two used lasers. Therefore, it is 231 necessary to investigate the effect of several typical parameter mismatches on the spiking 232 dynamics, and Figure 6 demonstrates the images transmission performance under a fixed -1 233 coupling weight of 0.017 mw , where several typical parameter mismatches includ- 234 ing τ , τ , and I are considered. For simplicity, the parameter value of VCSEL-SA1 is ph a a 235 fixed while some parameters of VCSEL-SA2 are adjusted. The relative mismatched pa- 236 rameters are defined as ∆τ = (τ - τ /τ , ∆τ =(τ - τ /τ , and ∆I =(I - I )/I . ph 2ph 1ph 1ph a 2a 1a 1a a 2a 1a 1a 237 From Figure 6, one can see that the image transmission is feasible when the two lasers are 238 in a certain parameter mismatch range. However, when the parameter mismatch exceeds 227 Figure 5. Transmission results between two cascaded VCSEL-SA neurons for different resolution Figure 5. Transmission results between two cascaded VCSEL-SA neurons for different resolution 239 a certain level, the image PSNR decreases with the increase in mismatch degree, and the 228 images of 73 × 73 pixels (a,b) and 512 × 512 pixels (c,d), where (a,c) correspond to the original im- images of 73 73 pixels (a,b) and 512 512 pixels (c,d), where (a,c) correspond to the original image 229 240 age and (b,d) correspond to image is distorted accord the transmitted image. ingly. Moreover, compared with I and τ , mismatched τ has a a ph and (b,d) correspond to the transmitted image. 241 a relatively smaller influence on the image transmission performance. 230 Generally, there exists a certain difference between two used lasers. Therefore, it is 231 necessary to investigate the effect of several typical parameter mismatches on the spiking 232 dynamics, and Figure 6 demonstrates the images transmission performance under a fixed -1 233 coupling weight of 0.017 mw , where several typical parameter mismatches includ- 234 ing τ , τ , and I are considered. For simplicity, the parameter value of VCSEL-SA1 is ph a a 235 fixed while some parameters of VCSEL-SA2 are adjusted. The relative mismatched pa- 236 rameters are defined as ∆τ = (τ - τ /τ , ∆τ =(τ - τ /τ , and ∆I =(I - I )/I . ph 2ph 1ph 1ph a 2a 1a 1a a 2a 1a 1a 237 From Figure 6, one can see that the image transmission is feasible when the two lasers are 238 in a certain parameter mismatch range. However, when the parameter mismatch exceeds 239 a certain level, the image PSNR decreases with the increase in mismatch degree, and the 240 image is distorted accordingly. Moreover, compared with I and τ , mismatched τ has a a ph 241 a relatively smaller influence on the image transmission performance. 243 Figure 6. PSNRs of output images from VCSEL-SA2 for different parameter mismatches of (a) τ , (b) τ , and (c) I , where Figure 6. PSNRs of output images from VCSEL-SA2 for different parameter mismatches of (a) t , ph ph a 244 the first (b ,) setcond, , and third, (c) I and , wher foe ur the th colu first, msecond, ns respecti thir vely d, and corr fourth espond to columns −10%, respectively −5%, 0%, 5%corr , and 10% espond parameter mismatches. to 10%, a a 5%, 0%, 5%, and 10% parameter mismatches. Figure 7 shows the PSNR variations of the output images with different coupling weight between the two lasers and corresponding transmitted images. From this diagram, one can see that PNSR gradually increases upon increasing the coupling weight, and then stabilizes at a certain level. When the coupling weight is relatively small, the transmitted images are seriously detorted and become very blurred, as shown in Figure 7(b1,b2). With increasing coupling weight, the images can be successfully transmitted, as shown in Figure 7(b3,b4). Moreover, Figure 8 gives the transmitted images for different coupling weight under 7% mismatched parameters of t (a), t (b), and I (c). Upon increasing the ph a a coupling weight, the black spots in the image disappear and then the image becomes clear. Correspondingly, the PSNR of the output image increases. Obviously, typical parameter 243 Figure 6. PSNRs of output images from VCSEL-SA2 for different parameter mismatches of (a) τ , (b) τ , and (c) I , where ph a mismatches have some impact on the image transmission performance in our proposed 244 the first, second, third, and fourth columns respectively correspond to −10%, −5%, 0%, 5%, and 10% parameter mismatches. Photonics 2021, 8, x FOR PEER REVIEW 8 of 12 Photonics 2021, 8, x FOR PEER REVIEW 8 of 12 245 Figure 7 shows the PSNR variations of the output images with different coupling 246 weight between the two lasers and corresponding transmitted images. From this diagram, 245 Figure 7 shows the PSNR variations of the output images with different coupling 247 246 one can weight between the two la see that PNSR grad sers ually an inc d correspond reases upon ing t incre ran asing smit t ted im he coup ages. F ling wei rom t ght his , a dia nd t gr h a en m, 248 stabilizes at a certain level. When the coupling weight is relatively small, the transmitted 247 one can see that PNSR gradually increases upon increasing the coupling weight, and then 249 images are seriously detorted and become very blurred, as shown in Figure 7(b1,b2). With 248 stabilizes at a certain level. When the coupling weight is relatively small, the transmitted 250 increasing coupling weight, the images can be successfully transmitted, as shown in Fig- 249 images are seriously detorted and become very blurred, as shown in Figure 7(b1,b2). With 251 ure 7(b3,b4). Moreover, Figure 8 gives the transmitted images for different coupling 250 increasing coupling weight, the images can be successfully transmitted, as shown in Fig- 252 weight under 7% mismatched parameters of τ (a), τ (b), and I c . Upon increasing 251 ure 7(b3,b4). Moreover, Figure 8 gives the transmitted images for different coupling ph a a Photonics 2021, 8, 238 8 of 11 252 weight under 7% mismatched parameters of τ (a), τ (b), and I (c . Upon increasing 253 the coupling weight, the black spots in the image disappear and then the image becomes ph a a 254 clear. Correspondingly, the PSNR of the output image increases. Obviously, typical pa- 253 the coupling weight, the black spots in the image disappear and then the image becomes 255 rameter mismatches have some impact on the image transmission performance in our 254 clear. Correspondingly, the PSNR of the output image increases. Obviously, typical pa- 256 proposed cascaded system. By suitably increasing the coupling weight, the image propa- 255 rameter mismatches have some impact on the image transmission performance in our cascaded system. By suitably increasing the coupling weight, the image propagation 257 gation robustness to the parameter mismatches can be efficiently enhanced [33,36]. 256 proposed cascaded system. By suitably increasing the coupling weight, the image propa- robustness to the parameter mismatches can be efficiently enhanced [33,36]. 257 gation robustness to the parameter mismatches can be efficiently enhanced [33,36]. 259 Figure 7. (a) PSNRs of output images from VCSEL-SA2 under different coupling weights; (b) the transmitted images for Figure 7. (a) PSNRs of output images from VCSEL-SA2 under different coupling weights; (b) the transmitted images for −1. 260 different coupling weights of (b1) 0.005, (b2) 0.010, (b3) 0.0155, and (b4) 0.020 mw 259 Figure 7. (a) PSNRs of output images from VCSEL-SA2 under different coupling weights 1 ; (b) the transmitted images for different coupling weights of (b1) 0.005, (b2) 0.010, (b3) 0.0155, and (b4) 0.020 mw . −1. 260 different coupling weights of (b1) 0.005, (b2) 0.010, (b3) 0.0155, and (b4) 0.020 mw 262 Figure 8. PSNRs of output images from VCSEL-SA2 under 7% parameter mismatches of (a) τ , ph 263 262 (b Figure 8. ) τ , and PS (c)NRs of I , where th output images from VCSEL- e first, second, third, fou SA2 under 7% parameter mismatches rth, and fifth columns respectively correspond to of (a) τ , a ph Figure 8. PSNRs of output images from VCSEL-SA2 under 7% parameter mismatches of (a) t , ph −1 264 0.009, 0.011, 0.013, 0.015, and 0.017 mw coupling weight. 263 (b) τ , and (c) I , where the first, second, third, fourth, and fifth columns respectively correspond to (b) t , and (c) I , where the first, second, third, fourth, and fifth columns respectively correspond to a a −1 264 0.009, 0.011, 0.013, 0.015, and 0.017 mw coupling weight. 0.009, 0.011, 0.013, 0.015, and 0.017 mw coupling weight. In a practical information transmission system, the device errors can also affect the information transmission performance, and different error correction methods have been adopted to assure successful information transmission [37,38]. Here, we further adopted the 8B10B conversion method to optimize the system communication performance. The decoded images before and after adopting the 8B10B method are shown in Figure 9, where 3 1 k = 1.1  10 and W = 0.015 mw . From these diagrams, one can see that, under our simulation conditions, the decoded image quality is significantly improved after adopting this error correction method, which indicates that our proposed system can be applied in future image transmission after adopting a suitable error correction method. Moreover, the Photonics 2021, 8, x FOR PEER REVIEW 9 of 12 265 In a practical information transmission system, the device errors can also affect the 266 information transmission performance, and different error correction methods have been 267 adopted to assure successful information transmission [37,38]. Here, we further adopted 268 the 8B10B conversion method to optimize the system communication performance. The 269 decoded images before and after adopting the 8B10B method are shown in Figure 9, where −3 −1 Photonics 2021, 8, 238 9 of 11 270 k = 1.1 × 10 and W12 = 0.015 mw . From these diagrams, one can see that, under our 271 simulation conditions, the decoded image quality is significantly improved after adopting 272 this error correction method, which indicates that our proposed system can be applied in 273 future image transmission after adopting a suitable error correction method. Moreover, additional simulation results demonstrate that this proposed image transmission scheme 274 the additional simulation results demonstrate that this proposed image transmission has relatively good robustness to noise under our simulation conditions. 275 scheme has relatively good robustness to noise under our simulation conditions. Figure 9. The decoded images from VCSEL-SA2 without (a) and with (b) 8B10B conversion method, 277 Figure 9. The decoded images from VCSEL-SA2 without (a) and with (b) 8B10B conversion -1 where W = 0.015 mw . 278 method,12 where W12 = 0.015 mw . 3.3. Storage of Spiking Patterns 279 3.3. Storage of Spiking Patterns Lastly, after adding the optoelectrical feedback loop to the first laser, the storage 280 Lastly, after adding the optoelectrical feedback loop to the first laser, the storage properties of image spiking patterns are discussed. Figure 10 shows the output time series 281 properties of image spiking patterns are discussed. Figure 10 shows the output time series of two cascaded VCSEL-SAs as response to the input stimulus (red dashed line), where 282 of two cascaded VCSEL-SAs as response to the input stimulus (red dashed line), where 1 1 4 W = 0.010 mw and W = 0.017 mw . An injected rectangular pulse with k = 3  10 f 12 −1 −1 −4 283 Wf = 0.010 mw and W12 = 0.017 mw . An injected rectangular pulse with k = 3 × 10 and Δt and Dt = 8 ns was used to encode the injection perturbation. From these diagrams, one 284 = 8 ns was used to encode the injection perturbation. From these diagrams, one can see can see that, under an external stimulus, a three-spike burst is fired repetitively by VCSEL- 285 that, under an external stimulus, a three-spike burst is fired repetitively by VCSEL-SA1 SA1 with a fixed time interval corresponding to the feedback delay. This phenomenon 286 with a fixed time interval corresponding to the feedback delay. This phenomenon can be can be interpreted as the first external perturbation firing three spikes for VCSEL-SA1, 287 interpreted as the first external perturbation firing three spikes for VCSEL-SA1, which can which can repetitively stimulate the VCSEL-SA1 through the added feedback loop. Con- 288 repetitively stimulate the VCSEL-SA1 through the added feedback loop. Consequently, a sequently, a repeated spiking response can be observed, as shown in Figure 10b. More- 289 repeated spiking response can be observed, as shown in Figure 10b. Moreover, the spike over, the spike responses of VCSEL-SA1 can be transmitted to VCSEL-SA2, as shown in 290 responses of VCSEL-SA1 can be transmitted to VCSEL-SA2, as shown in Figure 10c. Ob- Figure 10c. Obviously, the encoded spike information can be successfully stored in the 291 viously, the encoded spike information can be successfully stored in the electrically con- electrically controlled cascaded system under this condition, which can be applied in future 292 trolled cascaded system under this condition, which can be applied in future complex complex spiking pattern processing systems. 293 spiking pattern processing systems. Figure 10. Storage of spiking patterns in two cascaded VCSEL-SAs with optoelectronic feedback Figure 10. Storage of spiking patterns in two cascaded VCSEL- 1 1 4 under W = 0.010 mw , W = 0.017 mw , k = 3  10 and t = 2 ns, where (a) injected pulse, f 12 inj −1 −1 −4 SAs with optoelectronic feedback under Wf = 0.010 mw , W12 = 0.017 mw , k = 3 × 10 and τ =2 𝑠𝑛 , inj (b) spike trains from VCSEL-SA1 and (c) spike trains form VCSEL-SA2. where (a) injected pulse, (b) spike trains from VCSEL-SA1 and (c) spike trains form VCSEL-SA2." 4. Conclusions In conclusion, we demonstrated an image encoding and transmission system based on two electrically controlled vertical-cavity surface-emitting lasers with an embedded Photonics 2021, 8, 238 10 of 11 saturable absorber (VCSEL-SA). The simulation results show that, the conversion rate from binary code to spiking signal is highly dependent on the input strength and bias current. Under suitable conditions, the encoding images can be successfully transmitted in the proposed photonic neuron system. Moreover, typical parameter mismatches have some impact on the image transmission performance, and suitably increasing the coupling weight can improve the system robustness to parameter mismatches to a certain extent. Additionally, spiking patterns can be efficiently stored and transmitted in the electroni- cally controlled cascaded system. This work is valuable for the future construction and application of large-scale neural networks based on photonic neurons. Author Contributions: M.N. was responsible for the numerical simulation, analyzing the results, and the writing of the paper; X.T., Z.G., L.X., J.W., F.M. and Q.Z. were responsible for writing and revising the manuscript; X.L. and T.D. were responsible for the discussion of the results and reviewing/editing of the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (Grant No. 61875167), the Natural Science Foundation of Chongqing City (CSTC 2019jcyj-msxm X0136), the Fundamental Research Funds for the Central Universities of China (XDJK2020B053), and the National Training Program of Innovation and Entrepreneurship for Undergraduates College Students’ Innovation Fund of Southwest University under Grant No. 202010635089. Institutional Review Board Statement: “Not applicable” for studies not involving humans or animals. Informed Consent Statement: “Not applicable” for studies not involving humans. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available as the data also forms part of an ongo- ing study. Conflicts of Interest: The authors declare no conflict of interest. References 1. Yang, J.Q.; Wang, R.; Ren, Y.; Mao, J.Y.; Wang, Z.P.; Zhou, Y.; Han, S.T. 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Journal

PhotonicsMultidisciplinary Digital Publishing Institute

Published: Jun 25, 2021

Keywords: image transmission; vertical-cavity surface-emitting lasers with an embedded saturable absorber (VCSEL-SA); parameter mismatches; binary to spike (BTS); neuromorphic computing system

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