Time delay signature and bandwidth of chaotic laser output from semiconductor laser

Zhang Yi-Ning; Feng Yu-Ling; Wang Xiao-Qian; Zhao Zhen-Ming; Gao Chao; Yao Zhi-Hai

doi:10.7498/aps.69.20191881

Abstract

Authors and contacts

FullText HTML

1. 引　言
2. 理论模型
3. TDS的数值研究和分析
3.1 延迟时间 ${\tau _1}$ 对TDS的影响
3.2 外光注入系数 ${k_{{\rm{in}}}}$ 和反馈强度 ${k_{{\rm{f1}}}}$ 对TDS的影响
3.3 结果对比和分析
4. 带宽的数值研究和分析
4.1 外光注入系数 ${k_{{\rm{in}}}}$ 对带宽的影响
4.2 滤波器带宽 ${\varLambda _1}$ 对系统输出混沌激光带宽的影响
4.3 反馈强度 ${k_{{\rm{f1}}}}$ 对系统输出混沌激光带宽的影响
4.4 抽运因子 ${P_{\rm{m}}}$ 对系统输出混沌激光带宽的影响
5. 结　论

References

Cited By

Abstract

Semiconductor laser (SL) can output chaotic lasers under external disturbances such as optical injection or optical feedback, and the bandwidth can reach up to GHz magnitude. External-cavity feedback semiconductor lasers can output high-dimensional chaotic lasers and are considered to be better sources of chaotic entropy. However, due to external cavity feedback and other effects, it will give rise to obvious external cavity time delay signature (TDS) in the output chaotic laser, which restricts the application of chaotic lasers. On the other hand, the bandwidth of chaotic laser determines the transmission rate of confidential communication, and therefore TDS and bandwidth are two important parameters that will affect chaotic laser’s applications. Therefore, it is significant to take appropriate measures to suppress the TDS and increase the bandwidth of chaotic laser output by semiconductor laser. In this paper the output laser from a semiconductor laser with single optical feedback is partially injected to another semiconductor laser with double filtered optical feedback. Thus they form a semiconductor laser system with external optical injection and double filtered optical feedback, i.e. a master-slave laser system which is used to suppress the TDS of chaotic laser and investigate its bandwidth. We numerically investigate the influences of external light injection coefficient, feedback intensity, pumping factor, and filter bandwidth on TDS. Then the suppression effects of this system on TDS are analyzed and compared with those of semiconductor laser system with external optical injection and single optical feedback, those of semiconductor laser system with external optical injection and double optical feedback, those of semiconductor laser system with external optical injection and single filtered optical feedback, and those of semiconductor laser system with double filtered optical feedback. The results show that the proposed scheme in this paper has the best suppression effect on TDS. Then the bandwidth of the chaotic laser output from the system is investigated under the condition of parameters of effectively suppressing TDS. The results show that the system proposed in this paper can increase the bandwidth of the system output chaotic laser by properly selecting the parametric values, and the maximum bandwidth value of the obtained chaotic laser is about 8.8 GHz. The above investigations indicate the effectiveness of the proposed scheme. The results of this investigation are significant for the application of chaotic lasers.

Keywords:

Authors and contacts

Department of Physics, Changchun University of Science and Technology, Changchun 130022, China

Corresponding author: Feng Yu-Ling, FYLCUST@163.com

FullText HTML

1. 引　言

半导体激光器(semiconductor laser, SL)是B类激光器, 所以在光注入或光反馈等外部扰动下可以输出混沌激光^[1-5], 并且带宽可以达到GHz量级. SL输出的混沌激光可广泛应用在高速真随机数生成与应用^[6,7]、光时域反射仪^[8]、混沌保密通信^[9]和混沌激光雷达^[10]等领域. 外腔反馈半导体激光器能输出高维度的混沌激光, 被认为是较好的混沌熵源^[11-16]. 但由于外腔反馈等作用, 会导致输出的混沌激光中产生明显的外腔延时特征(time delay signature, TDS), 这就制约了混沌激光的应用, 例如用混沌激光作为物理熵源生成高数随机数的性能将会变差; 另一方面混沌激光的带宽决定了保密通信的传输速率. 所以采取适当的措施来抑制SL输出混沌激光的延时特征并提高其带宽是很有意义的. Li和Chan^[17]实验和数值研究了具有光纤布拉格光栅反馈腔的单模激光器的延时特征, 数值结果给出了混沌状态对应的参数区间, 实验证实了当光纤布拉格光栅相对于激光器自由振荡频率正失谐时, 可以观察到TDS的最佳抑制效果. 孙巍阳等^[18]提出了一种展宽半导体激光器混沌载波发射机带宽的方案, 数值研究了主从半导体激光器之间失谐频率、注入强度、主从激光器偏置电流和从激光器的反馈强度对混沌载波发射机带宽的影响. Schires等^[19]实验验证了在短反馈和长反馈相结合的情况下, 混合分布反馈半导体激光器可以输出高带宽的混沌光. Xu等^[20]用一个具有光纤随机光栅反馈的半导体激光器作为主激光器, 其输出注入到一个从激光器, 则从激光器能输出具有平坦功率谱的混沌光, 其带宽达到8.5 GHz. Nguimdo等^[21]数值研究了半导体激光器TDS与其外部反馈系数之间的关系, 结果证实通过调整相位和反馈强度, 可以很好地抑制延时特征. Zhao等^[22]实验和数值研究了具有延迟干涉自相位调制光反馈半导体激光器产生混沌激光的有效带宽和TDS, 在相位调制引起的频谱扩展和延迟干涉的非线性滤波的共同作用下, 可以在宽动态工作范围内产生具有平坦频谱并具有优异TDS抑制特性的宽带混沌. Brunner等^[23]通过实验和理论研究了光反馈半导体激光器输出混沌光的时间序列, 并从中提取到了时延特征信号. Uchida等^[24]研究表明, 通过光注入方式半导体激光器可以实现混沌信号带宽增强. Wu等^[25]的研究结果已证实, 在合适的反馈参数条件下, 双光反馈半导体激光器输出混沌信号的TDS可得到更为有效的抑制. 本文提出一个具有外光注入的双路滤波光反馈半导体激光器(semiconductor laser with external optical injection and double filtered optical feedback, SL-EOI-DFOF)系统降低半导体激光器输出混沌激光的延时特征值, 然后在TDS被有效抑制的条件下, 研究了系统输出混沌激光的带宽.

2. 理论模型

将具有单路外腔光反馈的分布反馈半导体激光器(distributed feedback semiconductor lasers, DFB-SL)作为主激光器, 将具有双路滤波外腔光反馈的DFB-SL作为从激光器, 所提方案的系统示意图如图1所示.

$图 1 SL-EOI-DFOF系统示意图\r\nFig. 1. Schematic diagram of the SL-EOI-DFOF system.$

图 1 SL-EOI-DFOF系统示意图

Fig. 1. Schematic diagram of the SL-EOI-DFOF system.

下载: 全尺寸图片幻灯片

图1中M-DFB-SL代表主激光器, S-DFB-SL代表从激光器, FC (optical fiber coupler)是光纤耦合器, VA (variable attenuator)是可调衰减器, ISO (optical isolator)是光隔离器. 光纤反射镜FR (fiber reflector)将光反馈回M-DFB-SL中, 在图1中, M-DFB-SL产生的激光通过FC0, VA1, ISO1及FC1注入到S-DFB-SL中, S-DFB-SL输出的激光经过FC1和FC2后分成两束, 其中一束经过FC3又分成两束, 分别通过VA2和VA3及滤波器Filter1和滤波器Filter2再反馈回S-DFB-SL中; 从FC2输出的另一束光经过ISO2后, 利用光电探测器PD (photodetector)将光信号转化成电信号, 之后将电信号输入到示波器OSC (oscilloscope)中.

图1所示系统的动力学速率方程为^[26,27]:

$\begin{split}\frac{{{\rm{d}}{E_{\rm{m}}}(t)}}{{{\rm{d}}t}} \; &= \frac{1}{2}( {1 + {\rm{i}}{\alpha _{\rm{m}}}} )\bigg[ {\frac{{{g_{\rm{m}}}({{N_{\rm{m}}}(t) - {N_{0{\rm{m}}}}} )}}{{1 + {\varepsilon _{\rm{m}}} | {{E_{\rm{m}}}{{\left( t \right)}^2}} |}} - \tau _{\rm{p}}^{ - 1}} \bigg]\\ & \times{E_{\rm{m}}}(t) \!+\! \frac{{{k_{\rm{f}}}}}{{{\tau _{{\rm{in}}}}}}{E_{\rm{m}}}(t \!-\! \tau )\exp \left( { - {\rm{i}}{\omega _{\rm{m}}}\tau } \right),\\[-15pt]\end{split}$

(1)

$\begin{split}\frac{{{\rm{d}}{N_{\rm{m}}}\left( t \right)}}{{{\rm{d}}t}} =\; & {P_{\rm{m}}}{J_{{\rm{th}}}} - \frac{{{N_{\rm{m}}}\left( t \right)}}{{{\tau _{\rm{N}}}}} \\ &- \left[ {\frac{{{g_{\rm{m}}}\left( {{N_{\rm{s}}}\left( t \right) - {N_{0{\rm{m}}}}} \right)}}{{1 + {\varepsilon _{\rm{m}}}{{\left| {{E_{\rm{m}}}\left( t \right)} \right|}^2}}}} \right]{\left| {{E_{\rm{m}}}\left( t \right)} \right|^2},\end{split}$

(2)

$\begin{split} \frac{{{\rm{d}}{E_{\rm{s}}}\left( t \right)}}{{{\rm{d}}t}} =\; &\frac{1}{2}\left( {1 + {\rm{i}}{\alpha _{\rm{s}}}} \right)\left[ {\frac{{{g_{\rm{s}}}\left( {{N_{\rm{s}}}\left( t \right) - {N_{0{\rm{s}}}}} \right)}}{{1 + {\varepsilon _{\rm{s}}}{{\left| {{E_{\rm{s}}}\left( t \right)} \right|}^2}}} - \tau _{\rm{p}}^{ - 1}} \right]\\ & \times{E_{\rm{s}}}\left( t \right) + \frac{{{k_{{\rm{f1}}}}}}{{{\tau _{{\rm{in}}}}}}{F_1}\left( t \right) +\frac{{{k_{{\rm{f2}}}}}}{{{\tau _{{\rm{in}}}}}}{F_2}\left( t \right)\\ & + \frac{{{k_{{\rm{in}}}}}}{{{\tau _{{\rm{in}}}}}}{E_{\rm{m}}}\left( t \right)\exp \left( {{\rm{i}}2{\text{π}}\Delta ft} \right),\\[-10pt] \end{split}$

(3)

$\frac{{{\rm{d}}{N_{\rm{s}}}\left( t \right)}}{{{\rm{d}}t}} = {P_{\rm{s}}}{J_{{\rm{th}}}} - \frac{{{N_{\rm{s}}}\left( t \right)}}{{{\tau _{\rm{N}}}}} - \frac{{{g_{\rm{s}}}\left( {{N_{\rm{s}}}\left( t \right) - {N_{0{\rm{s}}}}} \right)}}{{1 + {\varepsilon _{\rm{s}}}{{\left| {{E_{\rm{s}}}\left( t \right)} \right|}^2}}}{\left| {{E_{\rm{s}}}\left( t \right)} \right|^2},$

(4)

$\begin{split}\frac{{{\rm{d}}{F_1}\left( t \right)}}{{{\rm{d}}t}} =\; & {\varLambda _1}{E_{\rm{s}}}\left( {t - {\tau _1}} \right)\exp \left( { - {\rm{i}}{\omega _{\rm{s}}}{\tau _1}} \right)\\ &+ \left( {{\rm{i2{\text{π}}}}\Delta {f_1} - {\varLambda _1}} \right){F_1}\left( t \right),\end{split}$

(5)

$\begin{split}\frac{{{\rm{d}}{F_2}\left( t \right)}}{{{\rm{d}}t}} =\; & {\varLambda _2}{E_{\rm{s}}}\left( {t - {\tau _2}} \right)\exp \left( { - {\rm{i}}{\omega _{\rm{s}}}{\tau _2}} \right) \\ &+ \left( {{\rm{i}}2{\text{π}}\Delta {f_2} - {\varLambda _2}} \right){F_2}\left( t \right),\end{split}$

(6)

其中 ${E_{\rm{m}}}\left( t \right)$ 和 ${E_{\rm{s}}}\left( t \right)$ 分别为主激光器和从激光器的慢变电场复振幅, 脚标m和s分别代表主激光器和从激光器, $N\left( t \right)$ 为载流子密度, ${P_{\rm{m}}}$ 和 ${P_{\rm{s}}}$ 分别为主激光器和从激光器的抽运因子, $\Delta f = \dfrac{\omega _{\rm m} - \omega _{\rm s}}{2{\text{π}}}$ 是主激光器和从激光器中心场频率之间的频率失谐, ${\omega _{\rm{m}}}$ 和 ${\omega _{\rm{s}}}$ 分别为主激光器和从激光器的中心场角频率, $\tau$ 是主激光器的外腔延迟时间, ${k_{\rm{f}}}$ 是主激光器反馈腔的反馈强度, ${g_{\rm{m}}}$ 和 ${g_{\rm{s}}}$ 分别为主激光器和从激光器的微分增益系数, ${\alpha _{\rm{m}}}$ 和 ${\alpha _{\rm{s}}}$ 分别是主激光器和从激光器的线宽增强因子, ${k_{{\rm{in}}}}$ 是主激光器对从激光器的外光注入系数, ${\varepsilon _{\rm{m}}}$ 和 ${\varepsilon _{\rm{s}}}$ 分别是主激光器和从激光器的饱和增益系数, ${N_{{\rm{0 m}}}}$ 和 ${N_{{\rm{0 s}}}}$ 分别是主激光器和从激光器的透明载流子密度, ${\tau _{\rm{p}}}$ 是光子寿命, ${\tau _{\rm{N}}}$ 是载流子寿命, ${J_{{\rm{th}}}}$ 是阈值电流密度, 且 ${J_{{\rm{th}}}} = {{N_{{\rm{th}}}}/ {{\tau _{\rm{N}}}}}$ , ${N_{{\rm{th}}}} = {N_0} + {1 / {g{\tau _{\rm{p}}}}}$ , ${\tau _{{\rm{in}}}}$ 是光子在激光腔内的往返时间. 脚标1和2分别表示从激光器的反馈腔1和反馈腔2, ${\tau _1}$ 和 ${\tau _2}$ 分别是两个反馈腔的延迟时间, ${k_{{\rm{f1}}}}$ 和 ${k_{{\rm{f2}}}}$ 分别表示从激光器两个反馈腔的反馈强度, $\Delta {f_1}$ 和 $\Delta {f_2}$ 分别表示两个反馈腔内滤波器中心频率和从激光器中心频率之间的频率失谐, ${\varLambda _1}$ 和 ${\varLambda _2}$ 表示两个滤波器的带宽, ${F_1}\left( t \right)$ 和 ${F_2}\left( t \right)$ 分别表示从激光器两个反馈腔中反馈光的电场振幅.

为了描述混沌激光的延时特征, 一般利用自相关函数和互信息函数, 自相关函数(autocorrelation function, ACF)定义为^[28]

$\begin{split} & {C_I}\left( {\Delta t} \right) =\\ & \frac{{\left\langle {[I\left( {t + \Delta t} \right) - \left\langle {I\left( {t + \Delta t} \right)} \right\rangle ][I\left( t \right) - \left\langle {I\left( t \right)} \right\rangle ]} \right\rangle }}{{\sqrt {\left\langle {{{[I\left( {t + \Delta t} \right) - \left\langle {I\left( {t + \Delta t} \right)} \right\rangle ]}^2}} \right\rangle \left\langle {{{[I\left( t \right) - \left\langle {I\left( t \right)} \right\rangle ]}^2}} \right\rangle } }},\end{split}$

其中 $I\left( t \right) = E{\left( t \right)^2}$ 表示混沌光强度时间序列, $\left\langle \cdot \right\rangle$ 表示对时间求平均, $\Delta t$ 表示移动时间. 利用从激光器输出光的自相关函数曲线中的局部最大值来定量描述混沌输出的延时特征值 $\beta$ , 即TDS. 一般认为 $\beta <0.2$ 时, 自相关峰值较难辨别, 即认为混沌激光的延时特征得到了较好的隐藏^[29]. 互信息函数(mutual information, MI)定义为^[30]

$\begin{split}{ {MI}}\left( {\Delta t} \right) =\; & \sum\limits_{I\left( t \right),I\left( {t + \Delta t} \right)} {p[I\left( t \right),I\left( {t + \Delta t} \right)]}\\ & \times\lg \frac{{p[I\left( t \right),I\left( {t + \Delta t} \right)]}}{{p[I\left( t \right)]p[I\left( {t + \Delta t} \right)]}},\end{split}$

其中 $p[I\left( t \right), I\left( {t + \Delta t} \right)]$ 表示联合概率密度, $p[I\left( t \right)]$ 和 $p[I\left( {t + \Delta t} \right)]$ 分别表示边缘概率密度.

3. TDS的数值研究和分析

首先对本文提出的SL-EOI-DFOF系统数值研究外腔延迟时间 ${\tau _1}$ 对TDS的影响, 然后进一步研究外光注入系数 ${k_{{\rm{in}}}}$ 和反馈强度 ${k_{{\rm{f1}}}}$ 对TDS的影响, 最后在相同的参数条件下将SL-EOI-DFOF系统对TDS的抑制效果和具有外光注入的单路光反馈半导体激光器(semiconductor laser with external optical injection and single optical feedback, SL-EOI-SOF)系统、具有外光注入的双路光反馈半导体激光器(semiconductor laser with external optical injection and double optical feedback, SL-EOI-DOF)系统、具有外光注入的单路滤波光反馈半导体激光器(semiconductor laser with external optical injection and single filtered optical feedback, SL-EOI-SFOF)系统以及无光注入双路滤波光反馈半导体激光器(semiconductor laser with double filtered optical feedback, SL-DFOF)系统进行对比和分析.

3.1 延迟时间 ${\tau _1}$ 对TDS的影响

数值研究中的参数取值如下^[17,27]: $\alpha = 5.0$ , g = 8.4 × 10^–13 m³·s^–1, N₀= 1.4 × 10²⁴ m^–3, τ_p = 1.927 × 10^–12 s, τ_N = 2.04 × 10^–9s, ε = 2.5 × 10^–23 m³, P_m = 1.4, P_s = 1.6, k_f = 0.1, ∆f₁= ∆f₂= 20 GHz, k_f1 = 0.1, k_f2 = 0.2, τ₂ = 3 ns, k_in = 0.1, Λ₁ = Λ₂ = 20 GHz, ∆f = 3.0 GHz. 用以上参数值通过4阶龙格-库塔法对方程(1)—(6)进行数值求解, 得到 ${\tau _1}$ 分别为2.7, 2.8和2.9 ns时从激光器输出混沌激光的时间序列图、自相关曲线以及互信息曲线, 如图2所示.

$图 2 SL-EOI-DFOF在不同的延迟时间${\tau _1}$下输出混沌激光的(a1)−(a3)时间序列、(b1)−(b3)自相关曲线以及(c1)−(c3)互信息曲线　(a1)−(c1)${\tau _1} = 2.7\;{\rm{ns}}$; (a2)−(c2)${\tau _1} = 2.8\;{\rm{ns}}$; (a3)−(c3)${\tau _1} = 2.9\;{\rm{ns}}$\r\nFig. 2. Time series (a1)−(a3), ACF curves (b1)−(b3) and MI curves (c1)−(c3) of chaotic laser from the SL-EOI-DFOF at different delay times ${\tau _1}$: (a1)−(c1)${\tau _1} = 2.7\;{\rm{ns}}$; (a2) −(c2)${\tau _1} = 2.8\;{\rm{ns}}$; (a3)−(c3)${\tau _1} = 2.9\;{\rm{ns}}$.$

图 2 SL-EOI-DFOF在不同的延迟时间

${\tau _1}$

下输出混沌激光的(a1)−(a3)时间序列、(b1)−(b3)自相关曲线以及(c1)−(c3)互信息曲线　(a1)−(c1)

${\tau _1} = 2.7\;{\rm{ns}}$

; (a2)−(c2)

${\tau _1} = 2.8\;{\rm{ns}}$

; (a3)−(c3)

${\tau _1} = 2.9\;{\rm{ns}}$

Fig. 2. Time series (a1)−(a3), ACF curves (b1)−(b3) and MI curves (c1)−(c3) of chaotic laser from the SL-EOI-DFOF at different delay times

${\tau _1}$

: (a1)−(c1)

${\tau _1} = 2.7\;{\rm{ns}}$

; (a2) −(c2)

${\tau _1} = 2.8\;{\rm{ns}}$

; (a3)−(c3)

${\tau _1} = 2.9\;{\rm{ns}}$

下载: 全尺寸图片幻灯片

在图2(a1)—(a3)中, 光强的幅值随时间的变化呈现无规则的起伏, 说明此时半导体激光器输出的是混沌激光; 由图2(b1)—(b3)可见左边第一个尖峰几乎和纵轴重合, 这是由激光器的弛豫振荡引起的^[31], 其余的尖峰即为延时特征峰. 通过对比图2(b1)—(b3), 可见延时特征峰的峰值都小于0.2且图2(b2)(对应的 ${\tau _1} = 2.8\;{\rm{ns}}$ )中峰值最小, 所以当 ${\tau _1} = 2.8\;{\rm{ns}}$ 时TDS的抑制效果最好, 这是由于此时两个反馈腔的延时差 ${\tau _2} - {\tau _1} = 0.2\;{\rm{ns}}$ 等于半导体激光器的弛豫振荡周期 ${\tau _{{\rm{RO}}}} \approx 2{\text{π}}({g{E^2}} /\!$ ${\tau _{\rm{p}}})^{- 1/2}\approx$ 0.2 ns^{[17,25,27,32]}. 通过观察对比图2(c1)—(c3)以及图2(b1)—(b3)发现当自相关曲线的峰值都小于0.2, 即TDS被有效抑制时, 互信息曲线的峰值也都被有效抑制, 并且其峰值都小于自相关曲线的峰值, 这和相关文献的研究结果一致^[21,33,34]. 所以下面的研究中只利用自相关函数来描述系统输出混沌激光的TDS^[35].

3.2 外光注入系数 ${k_{{\rm{in}}}}$ 和反馈强度 ${k_{{\rm{f1}}}}$ 对TDS的影响

下面取对TDS抑制效果较好的延迟时间 ${\tau _1} = 2.8\;{\rm{ns}}$ , 其他参数取值与图2(b2)相同, 数值求解方程(1)—(6), 得到图3所示的系统输出混沌激光的延时特征值 $\beta$ 随外光注入系数 ${k_{{\rm{in}}}}$ 和反馈强度 ${k_{{\rm{f1}}}}$ 变化的二维图.

$图 3 SL-EOI-DFOF输出混沌激光延时特征值$\beta $随参数${k_{{\rm{in}}}}$和${k_{{\rm{f1}}}}$变化的二维图\r\nFig. 3. Two-dimensional maps of the time-delay characteristic $\beta $ in the parameter space of ${k_{{\rm{in}}}}$ and ${k_{{\rm{f1}}}}$ of chaotic laser from the SL-EOI-DFOF.$

图 3 SL-EOI-DFOF输出混沌激光延时特征值

$\beta$

随参数

${k_{{\rm{in}}}}$

和

${k_{{\rm{f1}}}}$

变化的二维图

Fig. 3. Two-dimensional maps of the time-delay characteristic

$\beta$

in the parameter space of

${k_{{\rm{in}}}}$

and

${k_{{\rm{f1}}}}$

of chaotic laser from the SL-EOI-DFOF.

下载: 全尺寸图片幻灯片

由图3可见, 当 ${k_{{\rm{in}}}}$ 在区间(0, 0.5)内时, ${k_{{\rm{f1}}}}$ 的变化对 $\beta$ 值的影响不明显, 这是由于此时注入光对输出混沌激光的延时特征值 $\beta$ 起主要作用; 随着 ${k_{{\rm{in}}}}$ 在该区间内的增大, 整体上看 $\beta$ 值先减小, 之后再增大, 这是由于当 ${k_{{\rm{in}}}}$ 在区间(0, 0.25)内增大时, 外部注入光对从激光器的扰动使其输出光的无序性增强, $\beta$ 值减小; 但随着 ${k_{{\rm{in}}}}$ 在(0.25, 0.5)内的继续增大, 此时主激光器对从激光器相当于是一个外腔, 会使从激光器出现弱周期性, 导致 $\beta$ 值又增大. 当 ${k_{{\rm{in}}}}$ 在区间(0.05, 0.25)内时, $\beta$ 值都是较小的, 都小于0.2, 即TDS被很好地抑制了.

3.3 结果对比和分析

为了表明本文所提出的SL-EOI-DFOF系统能更好地抑制TDS. 这里首先将本文提出的方案和SL-EOI-SOF系统、SL-EOI-DOF系统、SL-EOI-SFOF系统及SL-DFOF系统进行对比和分析. 对于SL-EOI-SOF系统而言, 方程(5)和方程(6)不存在, 取方程(3)中的 ${F_1}\left( t \right) = {E_{\rm{s}}}\left( {t - {\tau _1}} \right)\times$ $\exp \left( { - {\rm{i}}{\omega _{\rm{s}}}{\tau _1}} \right)$ , ${k_{{\rm{in}}}} = 0.1$ , ${k_{{\rm{f1}}}} = 0.1$ , ${k_{{\rm{f2}}}} = 0$ ; 对于SL-EOI-DOF系统而言, 方程(5)和方程(6)不存在, 取方程(3)中的 ${F_1}\left( t \right) = {E_{\rm{s}}}\left( {t - {\tau _1}} \right)\exp \left( { - {\rm{i}}{\omega _{\rm{s}}}{\tau _1}} \right)$ , ${F_2}\left( t \right) = {E_{\rm{s}}}\left( {t - {\tau _2}} \right)\exp \left( { - {\rm{i}}{\omega _{\rm{s}}}{\tau _2}} \right)$ , ${k_{{\rm{in}}}} = 0.1$ , ${k_{{\rm{f1}}}} = 0.1$ , ${k_{{\rm{f2}}}} = 0.2$ ; 对于SL-EOI-SFOF系统而言, 取方程(3)中的 ${k_{{\rm{in}}}} = 0.1$ , ${k_{{\rm{f1}}}} = 0.1$ , ${k_{{\rm{f2}}}} = 0$ ; 对于SL-EOI-DFOF系统而言, 取方程(3)中的 ${k_{{\rm{in}}}} = 0.1$ , ${k_{{\rm{f1}}}} = 0.1$ , ${k_{{\rm{f2}}}} = 0.2$ ; 对于SL-DFOF系统而言, 取方程(3)中的 ${k_{{\rm{in}}}} = 0$ , ${k_{\operatorname{f} 1}} = 0.1$ , ${k_{{\rm{f2}}}} = 0.2$ . 以上这5个系统中的其他参数的取值与图2(b2)相同, 数值求解方程(1)—(6)得到5个系统输出混沌激光的延时特征值 $\beta$ 随 ${P_{\rm{m}}}$ 的变化曲线, 如图4所示.

$图 4 SL-EOI-SOF, SL-EOI-DOF, SL-EOI-SFOF, SL-EOI-DFOF和SL-DFOF输出混沌激光的延时特征值$\beta $随${P_{\rm{m}}}$的变化\r\nFig. 4. Variations of the time delay characteristic values $\beta $ with ${P_{\rm{m}}}$ of chaotic laser from the SL-EOI-SOF, SL-EOI-DOF, SL-EOI-SFOF, SL-EOI-DFOF and SL-DFOF, respectively.$

图 4 SL-EOI-SOF, SL-EOI-DOF, SL-EOI-SFOF, SL-EOI-DFOF和SL-DFOF输出混沌激光的延时特征值

$\beta$

随

${P_{\rm{m}}}$

的变化

Fig. 4. Variations of the time delay characteristic values

$\beta$

with

${P_{\rm{m}}}$

of chaotic laser from the SL-EOI-SOF, SL-EOI-DOF, SL-EOI-SFOF, SL-EOI-DFOF and SL-DFOF, respectively.

下载: 全尺寸图片幻灯片

由图4可见, 在所选的控制参数区间范围内, SL-DFOF系统输出混沌激光的延时特征值 $\beta$ 远大于其他4个系统, 其原因是外部光注入可以有效抑制混沌激光的TDS^[36]; SL-EOI-DFOF和SL-EOI-SFOF系统输出混沌激光的延时特征值 $\beta$ 都小于0.2, 并且小于SL-EOI-SOF和SL-EOI-DOF系统输出混沌激光的 $\beta$ 值, 这是因为滤波器的滤波特性对TDS具有抑制作用^[37]. 并且可以看出本文提出的SL-EOI-DFOF系统在 ${P_{\rm{m}}} = 1.4$ 时, $\beta$ 达到极小值, 则下面的研究中取 ${P_{\rm{m}}} = 1.4$ .

下面针对SL-EOI-DFOF和SL-EOI-SFOF两个系统进一步证明本文所提出的SL-EOI-DFOF系统可以更好地抑制TDS. 滤波器带宽也是影响反馈光TDS的因素之一. 这里以滤波器带宽 ${\varLambda _1}$ 作为控制参数进行对比和分析. 对于SL-EOI-DFOF系统取方程(6)中的 ${\varLambda _2} = 20\;{\rm{GHz}}$ , 其他的参数取值与图4相同; 对于SL-EOI-SFOF系统则取方程(6)中的 ${\varLambda _2} = 0\;{\rm{GHz}}$ , 其他参数取值也与图4相同. 数值求解方程(1)—(6), 得到两个系统输出混沌激光的延时特征值 $\beta$ 随 ${\varLambda _1}$ 的变化, 如图5所示.

$图 5 SL-EOI-DFOF和SL-EOI-SFOF输出混沌激光的延时特征值$\beta $随${\varLambda _1}$的变化\r\nFig. 5. Variations of the time delay characteristic values $\beta $ with ${\varLambda _1}$ of chaotic laser from the SL-EOI-DFOF and SL-EOI-SFOF, respectively.$

图 5 SL-EOI-DFOF和SL-EOI-SFOF输出混沌激光的延时特征值

$\beta$

随

${\varLambda _1}$

的变化

Fig. 5. Variations of the time delay characteristic values

$\beta$

with

${\varLambda _1}$

of chaotic laser from the SL-EOI-DFOF and SL-EOI-SFOF, respectively.

下载: 全尺寸图片幻灯片

由图5可见, 在其他参数相同的情况下, 在所选的控制参数区间内, SL-EOI-DFOF系统输出混沌光的 $\beta$ 值随 ${\varLambda _1}$ 的变化平稳, 都小于0.12, 在 ${\varLambda _1} = 20~{\rm{GHz}}$ 时 $\beta$ 达到最小值; 并且可以看出在 ${\varLambda _1}$ 的大部分区间内SL-EOI-DFOF系统的 $\beta$ 值都小于SL-EOI-SFOF系统. 其原因是: 与单路滤波光反馈系统相比, 由于滤波器的滤波特性和混沌激光的维度, 采用双路滤波光反馈系统更有利于获得高混沌程度的混沌激光^[32]. 综合图4和图5, 比较而言SL-EOI-DFOF系统对TDS的抑制效果最好, 证明了本文提出的SL-EOI-DFOF系统对TDS抑制的有效性.

4. 带宽的数值研究和分析

在本文提出的SL-EOI-DFOF系统中TDS被有效抑制的基础上, 研究了其输出混沌激光的带宽随外光注入系数 ${k_{{\rm{in}}}}$ 、反馈强度 ${k_{{\rm{f1}}}}$ 、抽运因子 ${P_{\rm{m}}}$ 和滤波器带宽 ${\varLambda _1}$ 的变化规律.

4.1 外光注入系数 ${k_{{\rm{in}}}}$ 对带宽的影响

这里取 ${P_{\rm{m}}} = 1.4$ , ${\varLambda _1} = 20{\rm{GHz}}$ , ${k_{{\rm{in}}}}$ 分别选择0, 0.1, 0.2, 其他参数值与图5相同. 数值求解方程(1)—(6), 得到系统输出混沌激光的时间序列和功率谱如图6所示. 由图6(a1)—(a3)可见时间序列呈现无规则的起伏, 说明激光器此时输出的是混沌激光.

$图 6 SL-EOI-DFOF在不同的外光注入系数${k_{{\rm{in}}}}$下输出混沌激光的(a1)−(a3)时间序列以及(b1)−(b3)对应的功率谱　(a1), (b1)${k_{{\rm{in}}}} = 0$; (a2), (b2)${k_{{\rm{in}}}} = 0.1$; (a3), (b3)${k_{{\rm{in}}}} = 0.2$, 其中(b1)—(b3)中的虚线标示了混沌激光3 dB带宽的值\r\nFig. 6. Time series (a1)−(a3) and the corresponding power spectra (b1)−(b3) of chaotic laser from SL-EOI-DFOF at different external light injection coefficient ${k_{{\rm{in}}}}$: (a1), (b1) ${k_{{\rm{in}}}} = 0$; (a2), (b2) ${k_{{\rm{in}}}} = 0.1$; (a3), (b3)${k_{{\rm{in}}}} = 0.2$, the dashed lines in (b1)−(b3) indicate the value of the 3 dB bandwidth of the chaotic laser.$

图 6 SL-EOI-DFOF在不同的外光注入系数

${k_{{\rm{in}}}}$

下输出混沌激光的(a1)−(a3)时间序列以及(b1)−(b3)对应的功率谱　(a1), (b1)

${k_{{\rm{in}}}} = 0$

; (a2), (b2)

${k_{{\rm{in}}}} = 0.1$

; (a3), (b3)

${k_{{\rm{in}}}} = 0.2$

, 其中(b1)—(b3)中的虚线标示了混沌激光3 dB带宽的值

Fig. 6. Time series (a1)−(a3) and the corresponding power spectra (b1)−(b3) of chaotic laser from SL-EOI-DFOF at different external light injection coefficient

${k_{{\rm{in}}}}$

: (a1), (b1)

${k_{{\rm{in}}}} = 0$

; (a2), (b2)

${k_{{\rm{in}}}} = 0.1$

; (a3), (b3)

${k_{{\rm{in}}}} = 0.2$

, the dashed lines in (b1)−(b3) indicate the value of the 3 dB bandwidth of the chaotic laser.

下载: 全尺寸图片幻灯片

对图6(b1)—(b3)中的功率谱进行拟合, 得到平滑后的功率谱曲线 (见功率谱中的白色曲线), 可以看出, 随着 ${k_{{\rm{in}}}}$ 的增大, 系统输出混沌激光的功率谱变得平坦, 即带宽有明显的展宽, 经过分析得到图6(b1)—(b3)对应的3 dB带宽分别为4.33, 5.21和7.64 GHz. 即在所选参数条件下, 改变外光注入系数 ${k_{{\rm{in}}}}$ , 则激光器输出混沌激光的带宽随之增大.

为了展示混沌激光的带宽随外光注入系数 ${k_{{\rm{in}}}}$ 的整体变化趋势, 下面以 ${k_{{\rm{in}}}}$ 作为控制参数, 其他参数的取值与图6相同, 数值求解方程(1)—(6), 得到系统输出混沌激光的3 dB带宽随外光注入系数 ${k_{{\rm{in}}}}$ 的变化如图7所示.

$图 7 SL-EOI-DFOF输出混沌激光的带宽随${k_{{\rm{in}}}}$的变化\r\nFig. 7. Bandwidth versus ${k_{{\rm{in}}}}$ of chaotic laser from the SL-EOI-DFOF.$

图 7 SL-EOI-DFOF输出混沌激光的带宽随

${k_{{\rm{in}}}}$

的变化

Fig. 7. Bandwidth versus

${k_{{\rm{in}}}}$

of chaotic laser from the SL-EOI-DFOF.

下载: 全尺寸图片幻灯片

由图7可见, 当外光注入系数 ${k_{{\rm{in}}}}$ 在所选参数范围内逐渐增大时, 激光器输出混沌激光的带宽整体呈递增的趋势, 这里获得的带宽最大值约为8.5 GHz. 这是由于随着 ${k_{{\rm{in}}}}$ 的增加, 注入光的光强也随之增加, 主激光器对从激光器的扰动效果随之增大, 导致激光器输出混沌光的混沌程度增强, 从而使带宽增加^[38].

4.2 滤波器带宽 ${\varLambda _1}$ 对系统输出混沌激光带宽的影响

根据图3, 取对TDS有较好抑制的 ${k_{{\rm{in}}}} = 0.2$ , 其他参数取值与图7相同, 数值求解方程(1)—(6), 得到系统输出混沌激光的3 dB带宽随 ${\varLambda _1}$ 的变化如图8所示.

$图 8 SL-EOI-DFOF输出混沌激光的带宽随${\varLambda _1}$的变化\r\nFig. 8. Bandwidth versus ${\varLambda _1}$ of chaotic laser from the SL-EOI-DFOF.$

图 8 SL-EOI-DFOF输出混沌激光的带宽随

${\varLambda _1}$

的变化

Fig. 8. Bandwidth versus

${\varLambda _1}$

of chaotic laser from the SL-EOI-DFOF.

下载: 全尺寸图片幻灯片

由图8可见, 曲线变化缓慢, 即滤波器的带宽 ${\varLambda _1}$ 对系统输出混沌激光的带宽有影响, 但不明显, 说明滤波器的滤波特性对带宽的影响不大.

4.3 反馈强度 ${k_{{\rm{f1}}}}$ 对系统输出混沌激光带宽的影响

下面以反馈强度 ${k_{{\rm{f1}}}}$ 作为控制参数, 其他参数取值与图8相同, 数值求解方程(1)—(6), 得到系统输出混沌激光的3 dB带宽随 ${k_{{\rm{f1}}}}$ 的变化如图9所示.

$图 9 SL-EOI-DFOF输出混沌激光的带宽随${k_{{\rm{f1}}}}$的变化\r\nFig. 9. Bandwidth versus ${k_{{\rm{f1}}}}$ of chaotic laser from the SL-EOI-DFOF.$

图 9 SL-EOI-DFOF输出混沌激光的带宽随

${k_{{\rm{f1}}}}$

的变化

Fig. 9. Bandwidth versus

${k_{{\rm{f1}}}}$

of chaotic laser from the SL-EOI-DFOF.

下载: 全尺寸图片幻灯片

从图9可以看出, 当反馈强度 ${k_{{\rm{f1}}}}$ 在所选参数范围内逐渐增大时, 激光器输出混沌激光的带宽先增大, 而后缓慢地减小. 这是由于随着 ${k_{{\rm{f1}}}}$ 在区间(0, 0.05)内的增大, 系统进入混沌态, 混沌程度增强, 拓宽了带宽; 但是随着反馈强度 ${k_{{\rm{f1}}}}$ 在区间(0.05, 0.2)内的持续增大, 导致激光器输出混沌光的混沌程度减弱, 带宽减小.

4.4 抽运因子 ${P_{\rm{m}}}$ 对系统输出混沌激光带宽的影响

下面以抽运因子 ${P_{\rm{m}}}$ 作为控制参数, 其他参数取值与图9相同, 数值求解方程(1)—(6), 得到系统输出混沌激光的3 dB带宽随 ${P_{\rm{m}}}$ 的变化如图10所示.

$图 10 SL-EOI-DFOF输出混沌激光的带宽随${P_{\rm{m}}}$的变化\r\nFig. 10. Bandwidth versus ${P_{\rm{m}}}$ of chaotic laser from the SL-EOI-DFOF.$

图 10 SL-EOI-DFOF输出混沌激光的带宽随

${P_{\rm{m}}}$

的变化

Fig. 10. Bandwidth versus

${P_{\rm{m}}}$

of chaotic laser from the SL-EOI-DFOF.

下载: 全尺寸图片幻灯片

由图10可见, 当抽运因子 ${P_{\rm{m}}}$ 在(1.1, 1.7)范围内逐渐增大时, 激光器输出混沌光的带宽整体呈线性增加趋势, 这里获得的带宽最大值约为8.8 GHz (经计算此时 $\beta$ 值小于0.2). 这是由于: 随着 ${P_{\rm{m}}}$ 在区间(1.1, 1.7)内的增大, 主激光器对从激光器的扰动作用随之增强, 使得激光器输出混沌光的混沌程度增强, 从而拓宽了混沌激光的带宽.

根据以上研究可见, 通过适当选择参数的取值, 本文提出的方案可以提高系统输出混沌激光的带宽.

5. 结　论

本文采用SL-EOI-DFOF系统来抑制TDS并研究其带宽. 首先对外光注入系数 ${k_{{\rm{in}}}}$ 、反馈强度 ${k_{{\rm{f1}}}}$ 、抽运因子 ${P_{\rm{m}}}$ 和滤波器带宽 ${\varLambda _1}$ 对系统输出混沌激光TDS的影响进行了数值研究和理论分析. 结果表明: 在所选的参数区间内, $\beta$ 值随反馈强度 ${k_{{\rm{f1}}}}$ 的变化较缓慢, 并且给出了本文提出的系统输出混沌光的延时特征值 $\beta$ 随着外光注入系数 ${k_{{\rm{in}}}}$ 变化的过程中TDS被有效抑制的 ${k_{{\rm{in}}}}$ 取值参数区间, 进而通过对比和分析SL-EOI-SOF系统、SL-EOI-DOF系统、SL-EOI-SFOF系统、SL-EOI-DFOF系统以及SL-DFOF系统输出混沌光的延时特征值 $\beta$ 随抽运因子 ${P_{\rm{m}}}$ 以及滤波器带宽 ${\varLambda _1}$ 的变化曲线, 表明了光注入和滤波光反馈对TDS的有效抑制, 并且通过进一步分析阐明了本文所提出的SL-EOI-DFOF系统对TDS的抑制效果是最佳的; 然后在对TDS具有最佳抑制效果的参数条件下, 研究了外光注入系数 ${k_{{\rm{in}}}}$ 、滤波器带宽 ${\varLambda _1}$ 、反馈强度 ${k_{{\rm{f1}}}}$ 和抽运因子 ${P_{\rm{m}}}$ 对系统输出混沌激光带宽的影响并进行了物理分析, 结果表明: 在所选的参数条件下, 随着 ${k_{\operatorname{in} }}$ 的增加, 系统输出混沌激光的带宽也随之增大, 这是由于 ${k_{\operatorname{in} }}$ 的增加使得注入光的光强也随之增加, 主激光器对从激光器的扰动效果随之增大, 导致激光器输出混沌光的混沌程度增强, 从而使带宽增加; 系统输出混沌激光的带宽随 ${\varLambda _1}$ 的变化较缓慢; 随着 ${k_{{\rm{f1}}}}$ 的增大, 系统输出混沌激光的带宽先增大之后逐渐减小; 混沌激光的带宽随 ${P_{\rm{m}}}$ 的增加而增大, 这是由于在所选的参数范围内, 随着 ${P_{\rm{m}}}$ 的增大, 主激光器对从激光器的扰动作用增强, 使得激光器输出混沌光的混沌程度增强, 拓宽了带宽; 这里获得混沌激光带宽的最大值约为8.8 GHz. 所以对于本文提出的方案, 通过优化参数的取值, 可以在较大的参数区间内抑制混沌激光的TDS并使其带宽有所提高, 从而证明了本文所提方案的有效性. 本文的研究结果对于混沌激光的应用是有意义的.

References

[1]	Simpson T B, Liu J M, Gavrielides A, Kovanis V, Alsing P M 1995 Phys. Rev. A 51 4181 Google Scholar
[2]	Lin F Y, Liu J M 2003 Opt. Commun. 221 173 Google Scholar
[3]	Senlin Y 2009 J. Opt. Commun. 30 20 Google Scholar
[4]	Deng T, Xia G Q, Cao L P, Chen J G, Lin X D, Wu Z M 2009 Opt. Commun. 282 2243 Google Scholar
[5]	张明江, 刘铁根, 郑建宇, 王安帮, 王云才 2011 中国激光 4 136 Google Scholar Zhang M J, Liu T G, Zheng J Y, Wang A B, Wang Y C 2011 Chin. J. Lasers 4 136 Google Scholar
[6]	Uchida A, Amano K, Inoue M, Hirano K, Naito S, Someya H, Yoshimura K 2008 Nat. Photonics 2 728 Google Scholar
[7]	Metropolis N, Ulam S 1949 J. Am. Stat. Assoc. 44 335 Google Scholar
[8]	Wang Y, Wang B, Wang A 2008 IEEE Photonics Technol. Lett. 20 1636 Google Scholar
[9]	Argyris A, Syvridis D, Larger L, Annovazzi V 2005 Nature 438 343 Google Scholar
[10]	Lin F Y, Liu J M 2004 IEEE J. Quantum Electron. 40 815 Google Scholar
[11]	张继兵, 张建忠, 杨毅彪, 梁君生, 王云才 2010 物理学报 59 7679 Google Scholar Zhang J B, Zhang J Z, Yang Y B, Liang J S, Wang Y C 2010 Acta Phys. Sin. 59 7679 Google Scholar
[12]	Wu J G, Xia G Q, Tang X, Lin X D, Wu Z M 2010 Opt. Express 18 6661 Google Scholar
[13]	Jafari A, Sedghi H, Mabhouti K, Behnia S 2011 Opt. Commun. 284 3018 Google Scholar
[14]	Udaltsov V S, Goedgebuer J P, Larger L, Vladimir S, Cuenot J, William T, Rhodes 2003 Phys. Lett. E 308 54 Google Scholar
[15]	Rontani D, Locquet A, Sciamanna M, Citrin D S 2007 Opt. Lett. 32 2960 Google Scholar
[16]	Vicente R, Daudén J, Colet P, Toral R 2005 IEEE J. Quantum Electron. 41 541 Google Scholar
[17]	Li S S, Chan S C 2015 IEEE J. Quantum Electron. 21 541 Google Scholar
[18]	孙巍阳, 张胜海, 吴天安, 张晓旭 2017 激光与光电子学进展 54 208 Google Scholar Sun W Y, Zhang S H, Wu T A, Zhang X X 2017 Las. Optoelect. Prog. 54 208 Google Scholar
[19]	Schires K, Gomez S, Gallet A, Duan G, Grillot F 2017 IEEE J. Quantum Electron. 99 1 Google Scholar
[20]	Xu Y P, Zhang L, Lu P, Mihailov S, Chen L, Bao X Y 2018 Opt. Laser Technol. 109 654 Google Scholar
[21]	Nguimdo R M, Soriano M C, Colet P 2011 Opt. Lett. 36 4322 Google Scholar
[22]	Zhao A, Jiang N, Liu S 2019 Opt. Express 27 12336 Google Scholar
[23]	Brunner D, Porte X, Soriano M C, Fischer I 2012 Sci. Rep. 2 732 Google Scholar
[24]	Uchida A, Heil T, Liu Y, Davis P, Aida T 2003 IEEE J. Quantum Electron. 39 0 Google Scholar
[25]	Wu J G, Xia G Q, Wu Z M 2009 Opt. Express 17 20124 Google Scholar
[26]	Lang R, Kobayashi K 1980 IEEE J. Quantum Electron. 16 347 Google Scholar
[27]	卢东, 钟祝强, 夏光琼, 吴正茂 2016 光子学报 45 1014003 Google Scholar Lu D, Zhong Z Q, Xia G Q, Wu Z M 2016 Acta Photon. Sin. 45 1014003 Google Scholar
[28]	Udaltsov V S, Larger L, Goedgebuer J P, Locquet A, Citrin D S 2005 J. Opt. Technol. 72 373 Google Scholar
[29]	高飞, 李念强, 张力月, 欧阳康 2016 量子光学学报 22 289 Gao F, Li N Q, Zhang L Y, Ouyang K 2016 J. Quantum Opt. 22 289
[30]	孙巍阳, 张胜海, 吴天安, 张晓旭 2016 激光与光电子学进展 53 121406 Google Scholar Sun W Y, Zhang S H, Wu T A, Zhang X X 2016 Las. Optoelect. Prog. 53 121406 Google Scholar
[31]	Wang A, Yang Y, Wang B, Zhang B, Li L, Wang Y C 2013 Opt. Express 21 8701 Google Scholar
[32]	蒋龙, 夏光琼, 吴加贵, 肖平, 吴正茂 2012 中国激光 39 1202003 Google Scholar Jiang L, Xia G Q, Wu J G, Xiao P, Wu Z M 2012 Chin. J. Lasers 39 1202003 Google Scholar
[33]	Wu Y, Wang B, Zhang J, Wang A, Wang Y 2013 Math. Probl. Eng. 2013 1 Google Scholar
[34]	张建忠, 王安帮, 张明江, 李晓春, 王云才 2011 物理学报 60 094207 Google Scholar Zhang J Z, Wang A B, Zhang M J, Li X C, Wang Y C 2011 Acta Phys. Sin. 60 094207 Google Scholar
[35]	王永胜, 赵彤, 王安帮, 张明江, 王云才 2017 激光与光电子进展 54 111404 Google Scholar Wang Y S, Zhao T, Wang A B, Zhang M J, Wang Y C 2017 Las. Optoelect. Prog. 54 111404 Google Scholar
[36]	Simpson T B, Liu J M, Huang K F, Tai K 1997 Quantum Semiclassical Opt. 9 765 Google Scholar
[37]	江宁, 刘丁, 薛琛鹏, 邱昆 2015 中国科技论文 10 1640 Google Scholar Jiang N, Liu D, Xue C P, Qiu K 2015 China Sciencepaper 10 1640 Google Scholar
[38]	王云才, 张耕玮, 王安帮, 王冰洁, 李艳丽, 郭萍 2007 物理学报 56 4372 Google Scholar Wang Y C, Zhang G W, Wang A B, Wang B J, Li Y L, Guo P 2007 Acta Phys. Sin. 56 4372 Google Scholar

Cited By

期刊类型引用(10)

1.	庞爽，冯玉玲，于萍. 双路混沌光注入的相位调制光反馈半导体激光器的混沌特性. 光学学报. 2023(03): 164-173 . 百度学术
2.	于萍，冯玉玲，范健，庞爽，姚治海. 半导体环形激光器输出混沌光的时延特征和带宽的研究. 长春理工大学学报(自然科学版). 2023(02): 1-9 . 百度学术
3.	于萍，范健，毛洪存，冯玉玲，庞爽，姚治海. 混沌外光注入半导体环形激光器的混沌特性. 吉林大学学报(理学版). 2023(03): 671-680 . 百度学术
4.	庞爽，冯玉玲，于萍，姚治海. 自混沌光相位调制光反馈半导体激光器输出光的混沌特性. 物理学报. 2022(15): 52-61 . 百度学术
5.	颜森林. 激光局域网络的混沌控制及并行队列同步. 物理学报. 2021(08): 171-182 . 百度学术
6.	匡尚奇，郭祥帅，冯玉玲，李博涵，张依宁，于萍，庞爽. 半导体激光器系统输出混沌激光研究进展. 中国光学. 2021(05): 1133-1145 . 百度学术
7.	郑亮，李秀玲，王瑛剑. 半导体激光器混沌通信研究进展. 光通信技术. 2020(01): 1-5 . 百度学术
8.	张依宁，徐艾诗，冯玉玲，赵振明，姚治海. 光电反馈半导体激光器输出光的混沌特性. 光学学报. 2020(12): 125-132 . 百度学术
9.	颜森林. 激光混沌交叉发射与交替并行接收在保密通信中应用的基本理论与技术. 中国激光. 2020(09): 231-237 . 百度学术
10.	颜森林. 激光混沌并行串联同步及其在中继器保密通信系统中的应用. 物理学报. 2019(17): 61-71 . 百度学术

其他类型引用(7)

图 1 SL-EOI-DFOF系统示意图

Figure 1. Schematic diagram of the SL-EOI-DFOF system.

DownLoad: Full-Size Img PowerPoint

图 2 SL-EOI-DFOF在不同的延迟时间 ${\tau _1}$ 下输出混沌激光的(a1)−(a3)时间序列、(b1)−(b3)自相关曲线以及(c1)−(c3)互信息曲线　(a1)−(c1) ${\tau _1} = 2.7\;{\rm{ns}}$ ; (a2)−(c2) ${\tau _1} = 2.8\;{\rm{ns}}$ ; (a3)−(c3) ${\tau _1} = 2.9\;{\rm{ns}}$

Figure 2. Time series (a1)−(a3), ACF curves (b1)−(b3) and MI curves (c1)−(c3) of chaotic laser from the SL-EOI-DFOF at different delay times ${\tau _1}$ : (a1)−(c1) ${\tau _1} = 2.7\;{\rm{ns}}$ ; (a2) −(c2) ${\tau _1} = 2.8\;{\rm{ns}}$ ; (a3)−(c3) ${\tau _1} = 2.9\;{\rm{ns}}$ .

DownLoad: Full-Size Img PowerPoint

图 3 SL-EOI-DFOF输出混沌激光延时特征值 $\beta$ 随参数 ${k_{{\rm{in}}}}$ 和 ${k_{{\rm{f1}}}}$ 变化的二维图

Figure 3. Two-dimensional maps of the time-delay characteristic $\beta$ in the parameter space of ${k_{{\rm{in}}}}$ and ${k_{{\rm{f1}}}}$ of chaotic laser from the SL-EOI-DFOF.

DownLoad: Full-Size Img PowerPoint

图 4 SL-EOI-SOF, SL-EOI-DOF, SL-EOI-SFOF, SL-EOI-DFOF和SL-DFOF输出混沌激光的延时特征值 $\beta$ 随 ${P_{\rm{m}}}$ 的变化

Figure 4. Variations of the time delay characteristic values $\beta$ with ${P_{\rm{m}}}$ of chaotic laser from the SL-EOI-SOF, SL-EOI-DOF, SL-EOI-SFOF, SL-EOI-DFOF and SL-DFOF, respectively.

DownLoad: Full-Size Img PowerPoint

图 5 SL-EOI-DFOF和SL-EOI-SFOF输出混沌激光的延时特征值 $\beta$ 随 ${\varLambda _1}$ 的变化

Figure 5. Variations of the time delay characteristic values $\beta$ with ${\varLambda _1}$ of chaotic laser from the SL-EOI-DFOF and SL-EOI-SFOF, respectively.

DownLoad: Full-Size Img PowerPoint

图 6 SL-EOI-DFOF在不同的外光注入系数 ${k_{{\rm{in}}}}$ 下输出混沌激光的(a1)−(a3)时间序列以及(b1)−(b3)对应的功率谱　(a1), (b1) ${k_{{\rm{in}}}} = 0$ ; (a2), (b2) ${k_{{\rm{in}}}} = 0.1$ ; (a3), (b3) ${k_{{\rm{in}}}} = 0.2$ , 其中(b1)—(b3)中的虚线标示了混沌激光3 dB带宽的值

Figure 6. Time series (a1)−(a3) and the corresponding power spectra (b1)−(b3) of chaotic laser from SL-EOI-DFOF at different external light injection coefficient ${k_{{\rm{in}}}}$ : (a1), (b1) ${k_{{\rm{in}}}} = 0$ ; (a2), (b2) ${k_{{\rm{in}}}} = 0.1$ ; (a3), (b3) ${k_{{\rm{in}}}} = 0.2$ , the dashed lines in (b1)−(b3) indicate the value of the 3 dB bandwidth of the chaotic laser.

DownLoad: Full-Size Img PowerPoint

图 7 SL-EOI-DFOF输出混沌激光的带宽随 ${k_{{\rm{in}}}}$ 的变化

Figure 7. Bandwidth versus ${k_{{\rm{in}}}}$ of chaotic laser from the SL-EOI-DFOF.

DownLoad: Full-Size Img PowerPoint

图 8 SL-EOI-DFOF输出混沌激光的带宽随 ${\varLambda _1}$ 的变化

Figure 8. Bandwidth versus ${\varLambda _1}$ of chaotic laser from the SL-EOI-DFOF.

DownLoad: Full-Size Img PowerPoint

图 9 SL-EOI-DFOF输出混沌激光的带宽随 ${k_{{\rm{f1}}}}$ 的变化

Figure 9. Bandwidth versus ${k_{{\rm{f1}}}}$ of chaotic laser from the SL-EOI-DFOF.

DownLoad: Full-Size Img PowerPoint

图 10 SL-EOI-DFOF输出混沌激光的带宽随 ${P_{\rm{m}}}$ 的变化

Figure 10. Bandwidth versus ${P_{\rm{m}}}$ of chaotic laser from the SL-EOI-DFOF.

DownLoad: Full-Size Img PowerPoint

[1]	Simpson T B, Liu J M, Gavrielides A, Kovanis V, Alsing P M 1995 Phys. Rev. A 51 4181 Google Scholar
[2]	Lin F Y, Liu J M 2003 Opt. Commun. 221 173 Google Scholar
[3]	Senlin Y 2009 J. Opt. Commun. 30 20 Google Scholar
[4]	Deng T, Xia G Q, Cao L P, Chen J G, Lin X D, Wu Z M 2009 Opt. Commun. 282 2243 Google Scholar
[5]	张明江, 刘铁根, 郑建宇, 王安帮, 王云才 2011 中国激光 4 136 Google Scholar Zhang M J, Liu T G, Zheng J Y, Wang A B, Wang Y C 2011 Chin. J. Lasers 4 136 Google Scholar
[6]	Uchida A, Amano K, Inoue M, Hirano K, Naito S, Someya H, Yoshimura K 2008 Nat. Photonics 2 728 Google Scholar
[7]	Metropolis N, Ulam S 1949 J. Am. Stat. Assoc. 44 335 Google Scholar
[8]	Wang Y, Wang B, Wang A 2008 IEEE Photonics Technol. Lett. 20 1636 Google Scholar
[9]	Argyris A, Syvridis D, Larger L, Annovazzi V 2005 Nature 438 343 Google Scholar
[10]	Lin F Y, Liu J M 2004 IEEE J. Quantum Electron. 40 815 Google Scholar
[11]	张继兵, 张建忠, 杨毅彪, 梁君生, 王云才 2010 物理学报 59 7679 Google Scholar Zhang J B, Zhang J Z, Yang Y B, Liang J S, Wang Y C 2010 Acta Phys. Sin. 59 7679 Google Scholar
[12]	Wu J G, Xia G Q, Tang X, Lin X D, Wu Z M 2010 Opt. Express 18 6661 Google Scholar
[13]	Jafari A, Sedghi H, Mabhouti K, Behnia S 2011 Opt. Commun. 284 3018 Google Scholar
[14]	Udaltsov V S, Goedgebuer J P, Larger L, Vladimir S, Cuenot J, William T, Rhodes 2003 Phys. Lett. E 308 54 Google Scholar
[15]	Rontani D, Locquet A, Sciamanna M, Citrin D S 2007 Opt. Lett. 32 2960 Google Scholar
[16]	Vicente R, Daudén J, Colet P, Toral R 2005 IEEE J. Quantum Electron. 41 541 Google Scholar
[17]	Li S S, Chan S C 2015 IEEE J. Quantum Electron. 21 541 Google Scholar
[18]	孙巍阳, 张胜海, 吴天安, 张晓旭 2017 激光与光电子学进展 54 208 Google Scholar Sun W Y, Zhang S H, Wu T A, Zhang X X 2017 Las. Optoelect. Prog. 54 208 Google Scholar
[19]	Schires K, Gomez S, Gallet A, Duan G, Grillot F 2017 IEEE J. Quantum Electron. 99 1 Google Scholar
[20]	Xu Y P, Zhang L, Lu P, Mihailov S, Chen L, Bao X Y 2018 Opt. Laser Technol. 109 654 Google Scholar
[21]	Nguimdo R M, Soriano M C, Colet P 2011 Opt. Lett. 36 4322 Google Scholar
[22]	Zhao A, Jiang N, Liu S 2019 Opt. Express 27 12336 Google Scholar
[23]	Brunner D, Porte X, Soriano M C, Fischer I 2012 Sci. Rep. 2 732 Google Scholar
[24]	Uchida A, Heil T, Liu Y, Davis P, Aida T 2003 IEEE J. Quantum Electron. 39 0 Google Scholar
[25]	Wu J G, Xia G Q, Wu Z M 2009 Opt. Express 17 20124 Google Scholar
[26]	Lang R, Kobayashi K 1980 IEEE J. Quantum Electron. 16 347 Google Scholar
[27]	卢东, 钟祝强, 夏光琼, 吴正茂 2016 光子学报 45 1014003 Google Scholar Lu D, Zhong Z Q, Xia G Q, Wu Z M 2016 Acta Photon. Sin. 45 1014003 Google Scholar
[28]	Udaltsov V S, Larger L, Goedgebuer J P, Locquet A, Citrin D S 2005 J. Opt. Technol. 72 373 Google Scholar
[29]	高飞, 李念强, 张力月, 欧阳康 2016 量子光学学报 22 289 Gao F, Li N Q, Zhang L Y, Ouyang K 2016 J. Quantum Opt. 22 289
[30]	孙巍阳, 张胜海, 吴天安, 张晓旭 2016 激光与光电子学进展 53 121406 Google Scholar Sun W Y, Zhang S H, Wu T A, Zhang X X 2016 Las. Optoelect. Prog. 53 121406 Google Scholar
[31]	Wang A, Yang Y, Wang B, Zhang B, Li L, Wang Y C 2013 Opt. Express 21 8701 Google Scholar
[32]	蒋龙, 夏光琼, 吴加贵, 肖平, 吴正茂 2012 中国激光 39 1202003 Google Scholar Jiang L, Xia G Q, Wu J G, Xiao P, Wu Z M 2012 Chin. J. Lasers 39 1202003 Google Scholar
[33]	Wu Y, Wang B, Zhang J, Wang A, Wang Y 2013 Math. Probl. Eng. 2013 1 Google Scholar
[34]	张建忠, 王安帮, 张明江, 李晓春, 王云才 2011 物理学报 60 094207 Google Scholar Zhang J Z, Wang A B, Zhang M J, Li X C, Wang Y C 2011 Acta Phys. Sin. 60 094207 Google Scholar
[35]	王永胜, 赵彤, 王安帮, 张明江, 王云才 2017 激光与光电子进展 54 111404 Google Scholar Wang Y S, Zhao T, Wang A B, Zhang M J, Wang Y C 2017 Las. Optoelect. Prog. 54 111404 Google Scholar
[36]	Simpson T B, Liu J M, Huang K F, Tai K 1997 Quantum Semiclassical Opt. 9 765 Google Scholar
[37]	江宁, 刘丁, 薛琛鹏, 邱昆 2015 中国科技论文 10 1640 Google Scholar Jiang N, Liu D, Xue C P, Qiu K 2015 China Sciencepaper 10 1640 Google Scholar
[38]	王云才, 张耕玮, 王安帮, 王冰洁, 李艳丽, 郭萍 2007 物理学报 56 4372 Google Scholar Wang Y C, Zhang G W, Wang A B, Wang B J, Li Y L, Guo P 2007 Acta Phys. Sin. 56 4372 Google Scholar

[1]	Mu Peng-Hua, Chen Hao, Liu Guo-Peng, Hu Guo-Si. Chaotic time delay feature cancellation and bandwidth enhancement in cascaded-coupled nanolasers. Acta Physica Sinica, 2024, 73(10): 104204. doi: 10.7498/aps.73.20231643
[2]	Zhou Pei, Zhang Ren-Heng, Zhu Jian, Li Nian-Qiang. High-performance linear frequency-modulated signal generation based on optically injected semiconductor laser with dual-loop optoelectronic feedback. Acta Physica Sinica, 2022, 71(21): 214204. doi: 10.7498/aps.71.20221308
[3]	Pang Shuang, Feng Yu-Ling, Yu Ping, Yao Zhi-Hai. Chaotic characteristics of output light from semiconductor laser with self-chaotic phase modulation and optical feedback. Acta Physica Sinica, 2022, 71(15): 150502. doi: 10.7498/aps.71.20220204
[4]	Li Zeng, Feng Yu-Ling, Wang Xiao-Qian, Yao Zhi-Hai. Time delay characteristics and bandwidth of chaotic laser from semiconductor laser. Acta Physica Sinica, 2018, 67(14): 140501. doi: 10.7498/aps.67.20180035
[5]	Qi Jun-Feng, Zhong Zhu-Qiang, Wang Guang-Na, Xia Guang-Qiong, Wu Zheng-Mao. Characteristics of chaotic output from a Gaussian apodized fiber Bragg grating external-cavity semiconductor laser. Acta Physica Sinica, 2017, 66(24): 244207. doi: 10.7498/aps.66.244207
[6]	Yang Xian-Jie, Chen Jian-Jun, Xia Guang-Qiong, Wu Jia-Gui, Wu Zheng-Mao. Analyses of the time-delay signature and bandwidth of the chaotic output from a master-slave vertical-cavity surface-emitting laser dynamical system. Acta Physica Sinica, 2015, 64(22): 224213. doi: 10.7498/aps.64.224213
[7]	Ding Ling, Wu Zheng-Mao, Wu Jia-Gui, Xia Guang-Qiong. Unidirectional open-loop chaotic synchronization communication based on a semiconductor laser with double optical feedback. Acta Physica Sinica, 2012, 61(1): 014212. doi: 10.7498/aps.61.014212
[8]	Liang Jun-Sheng, Wu Yuan, Wang An-Bang, Wang Yun-Cai. Extracting the external-cavity key of a chaotic semiconductor laser with double optical feedback by spectrum analyzer. Acta Physica Sinica, 2012, 61(3): 034211. doi: 10.7498/aps.61.034211
[9]	Zhang Jian-Zhong, Wang An-Bang, Zhang Ming-Jiang, Li Xiao-Chun, Wang Yun-Cai. Elimination of time-delay signature in an external cavity semiconductor laser by randomly modulating feedback phase. Acta Physica Sinica, 2011, 60(9): 094207. doi: 10.7498/aps.60.094207
[10]	Ding Ling, Wu Jia-Gui, Xia Guang-Qiong, Shen Jin-Ting, Li Neng-Yao, Wu Zheng-Mao. Suppression of time delay feedback signatures in a semiconductor laser with double optical feedback. Acta Physica Sinica, 2011, 60(1): 014210. doi: 10.7498/aps.60.014210
[11]	Yang Ling-Zhen, Qiao Zhan-Duo, Wu Yun-Qiao, Wang Yun-Cai. Study of chaotic bandwidth in erbium-doped ring fiber laser. Acta Physica Sinica, 2010, 59(6): 3965-3972. doi: 10.7498/aps.59.3965
[12]	Zhu Zhang-Ming, Hao Bao-Tian, Li Ru, Yang Yin-Tang. A novel nanometer CMOS interconnect optimal model with target delay and bandwidth constraint. Acta Physica Sinica, 2010, 59(3): 1997-2003. doi: 10.7498/aps.59.1997
[13]	Cao liang-Ping, Xia Guang-Qiong, Deng Tao, Lin Xiao-Dong, Wu Zheng-Mao. Bidirectional chaos communication based on semiconductor laser with incoherent optical feedback. Acta Physica Sinica, 2010, 59(8): 5541-5546. doi: 10.7498/aps.59.5541
[14]	Yan Sen-Lin. Bandwidth enhancement of a chaotic semiconductor laser transmitter by cross-phase modulation. Acta Physica Sinica, 2010, 59(6): 3810-3816. doi: 10.7498/aps.59.3810
[15]	Zhao Yan-Feng. Chaos characteristics of the semiconductor laser with double external cavity optical feedback. Acta Physica Sinica, 2009, 58(9): 6058-6062. doi: 10.7498/aps.58.6058
[16]	Kong Ling-Qin, Wang An-Bang, Wang Hai-Hong, Wang Yun-Cai. Dynamics of semiconductor laser with optical feedback: Evolution from low-frequency fluctuations to chaos. Acta Physica Sinica, 2008, 57(4): 2266-2272. doi: 10.7498/aps.57.2266
[17]	Fan Yan, Xia Guang-Qiong, Wu Zheng-Mao. The self-correlation performance of semiconductor lasers with optical feedback and optical injection. Acta Physica Sinica, 2008, 57(12): 7663-7667. doi: 10.7498/aps.57.7663
[18]	Wang Yun-Cai, Li Yan-Li, Wang An-Bang, Wang Bing-Jie, Zhang Geng-Wei, Guo Ping. High frequency message filtering characteristics of semiconductor laser as receiver in optical chaos communications. Acta Physica Sinica, 2007, 56(8): 4686-4693. doi: 10.7498/aps.56.4686
[19]	Wang Yun-Cai, Zhang Geng-Wei, Wang An-Bang, Wang Bing-Jie, Li Yan-Li, Guo Ping. Bandwidth enhancement of semiconductor laser as a chaotic transmitter by external light injection. Acta Physica Sinica, 2007, 56(8): 4372-4377. doi: 10.7498/aps.56.4372
[20]	Liu Chong, Ge Jian-Hong, Chen Jun. Influence of external cavity feedback on the oscillating characteristics of a semiconductor laser. Acta Physica Sinica, 2006, 55(10): 5211-5215. doi: 10.7498/aps.55.5211

期刊类型引用(10)

1.	庞爽，冯玉玲，于萍. 双路混沌光注入的相位调制光反馈半导体激光器的混沌特性. 光学学报. 2023(03): 164-173 . 百度学术
2.	于萍，冯玉玲，范健，庞爽，姚治海. 半导体环形激光器输出混沌光的时延特征和带宽的研究. 长春理工大学学报(自然科学版). 2023(02): 1-9 . 百度学术
3.	于萍，范健，毛洪存，冯玉玲，庞爽，姚治海. 混沌外光注入半导体环形激光器的混沌特性. 吉林大学学报(理学版). 2023(03): 671-680 . 百度学术
4.	庞爽，冯玉玲，于萍，姚治海. 自混沌光相位调制光反馈半导体激光器输出光的混沌特性. 物理学报. 2022(15): 52-61 . 百度学术
5.	颜森林. 激光局域网络的混沌控制及并行队列同步. 物理学报. 2021(08): 171-182 . 百度学术
6.	匡尚奇，郭祥帅，冯玉玲，李博涵，张依宁，于萍，庞爽. 半导体激光器系统输出混沌激光研究进展. 中国光学. 2021(05): 1133-1145 . 百度学术
7.	郑亮，李秀玲，王瑛剑. 半导体激光器混沌通信研究进展. 光通信技术. 2020(01): 1-5 . 百度学术
8.	张依宁，徐艾诗，冯玉玲，赵振明，姚治海. 光电反馈半导体激光器输出光的混沌特性. 光学学报. 2020(12): 125-132 . 百度学术
9.	颜森林. 激光混沌交叉发射与交替并行接收在保密通信中应用的基本理论与技术. 中国激光. 2020(09): 231-237 . 百度学术
10.	颜森林. 激光混沌并行串联同步及其在中继器保密通信系统中的应用. 物理学报. 2019(17): 61-71 . 百度学术

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板