高阶色散对高斯脉冲在超常介质中传输的影响及色散的补偿

摘要: 文中对超常介质和一些常规介质中色散系数进行了对比研究，发现超常介质中的各阶色散系数大于常规介质的色散系数大约3个数量级，也即在信号的传输过程中不再能忽略高阶色散的影响.基于非线性薛定谔方程，研究了高斯脉冲在超常介质中传输及各阶色散对脉冲形状的影响.发现在常规超常介质中三阶色散所致脉冲分裂是一个非常严重的问题.通过调整超常介质的结构参数，找到了既可使二阶色散得以补偿、又可使得高斯脉冲传输120km而不出现分裂的真正可用于通信的情形.
- 超常介质 /
- 色散 /
- 非线性薛定谔方程 /
- /
- 高斯脉冲 /
- 色散补偿
Abstract: This paper compares the dispersion in metamaterial and in some conventional media. It is found that each order of the dispersion in metamaterial is larger in three orders of magnitude than that in conventional media, so that high-order dispersions have to be taken into consideration in the signal propagation. We analyze the impact of each order of the dispersion on the propagation of Gaussian light pulse based on the nonlinear Schrödinger equation and the beam propagation method (BPM). We find that third-order dispersion leads to a serious pulse splitting. A case is found in which Gaussian pulse can propagate in metamaterial to 120km without splits and second dispersion can be compensated by adjusting structure of metamaterial. This is significant to optical communications.
- metamaterial /
- dispersion /
- nonlinear Schrö /
- dinger equation /
- Gaussian pulse /
- dispersion compensation

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图 1 折射率n、三阶非线性系数和一阶、二阶、三阶、四阶色散系数分别随归一化频率 $\overline{\omega }$ 的变化曲线

注: 各图中子图为归一化频率为0.7附近的局部放大图

Fig. 1 Variations of refractive index, third-order nonlinear coefficient, first-order, second-order, third-order, and forth-order depression on $\overline {\omega }$

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图 2 (a)、(c)分别为几种常规介质的二阶色散和三阶色散随波长变化的曲线图; (b)、(d)为相应的二阶色散和三阶色散随归一化频率变化的曲线图

Fig. 2 (a), (c) Second-order and third-order dispersion in several conventional media; (b), (d) The corresponding relationship of (a), (c) with normalized frequency

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图 3 (a)当 $\beta _2 =0, \beta _3 =2.069\;8$ ps $^3\cdot$ km $^{-1}$ 时, 高斯脉冲沿超常介质 $z$ 方向的传输图; (b)高斯脉冲在0 km、21 km、22 km、23 km处的波形对比图

Fig. 3 (a) Gaussian pulse propagation when $\beta _2 =0, \beta _3 =2.069\;8$ ps $^3\cdot$ km $^{-1}$ ; (b)Comparison of pulse waveforms at 0 km、21 km、22 km、23 km

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图 4 (a)、(c)相应于表 1中两种情况的高斯脉冲传输图; (b)、(d)为相应的脉冲在不同传输距离的波形对比图

Fig. 4 (a), (c) Pulse propagation for two cases in Table 1; (b), (d) Corresponding waveform comparison at different propagation distances

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图 5 (a)、(c)相应于表 2中两种情况的高斯脉冲传输图; (b)、(d)为相应的脉冲在不同传输距离的波形对比图

Fig. 5 (a), (c) Pulse propagation for two cases in Table 2; (b), (d) Corresponding waveform comparison at different propagation distances

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图 6 (a)、(c)相应于表 3中两种情况的高斯脉冲传输图, 其中红线没有考虑四阶色散; (b)、(d)脉冲传输到21 km时相应的半高带宽处的局部放大图

Fig. 6 (a), (c) Pulse propagation for two cases in Table 3 and red line is for the case without $\beta _4 $ ; (b), (d) Partial waveform magnification at FWHM for $z=21$ km

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图 7 折射率n、三阶非线性系数和一阶、二阶、三阶、四阶色散系数分别随归一化频率 $\overline{\omega }$ 和 $\overline {\omega }_p $ 的变化曲线

Fig. 7 Variations of refractive index, third-order nonlinear coefficient, first-order, second-order, third-order, and forth-order depression on $\overline {\omega }$ and $\overline {\omega }_p $

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图 8 (a)高斯脉冲在复合超常材料M1+M2中色散补偿后传输120 km的脉冲波形变化图; (b)为(a)的俯视图, 即高斯脉冲能量扩散图

Fig. 8 (a) Waveform variation of Gaussian pulse' 120 km propagation in compound metamaterials M1+M2; (b) Top view of (a)

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表 1 $\beta _\textbf{2} ＜\textbf{0}$ 且 $ \vert \beta _\textbf{2} \vert $ 逐渐增大但 $ \beta _\textbf{3}$ 变化很小的色散数据

Tab. 1 Two sets of dispersion data with $\beta _2 <0$

$\overline {\omega }$	$\beta _2$ /(ps $^2\cdot$ km $^{-1}$ )	$\beta_3$ /(ps $^3\cdot$ km $^{-1}$ )	$L_D$ /km	${L}'_D $ /km
0.706 80	-1.258 9	2.067 6	19.86	60.46
0.706 75	-2.671 5	2.067 51	9.36	60.53

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表 2 $\beta _{\bf 2} >\bf 0$ 且 $ \beta _ {\bf 2} $ 逐渐增大但 $\beta _{\bf 3}$ 变化很小的色散数据

Tab. 2 Two sets of dispersion data with $\beta _2 >0$

$\overline {\omega }$	$\beta _2$ /(ps $^2\cdot$ km $^{-1}$ )	$\beta_3$ /(ps $^3\cdot$ km $^{-1}$ )	$L_D$ /km	${L}'_D$ /km
0.706 90	1.571 5	2.072 6	15.91	60.31
0.706 95	2.989 4	2.075 2	8.36	60.24

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表 3 $ \beta _{\bf 2} $ 的符号不同, 而 ${ \vert} { \beta}_{\bf 2 }{ \vert} $ 、 ${ \beta} _{\bf 3}$ 、 ${\beta} _{\bf 4} $ 的值接近相同的色散数据

Tab. 3 Two sets of dispersion data including $\beta_4 $

$\overline {\omega }$	$\beta _2$ /(ps $^2\cdot$ km $^{-1}$ )	$\beta_3$ /(ps $^3\cdot$ km $^{-1}$ )	$\beta _4$ /(ps $^4\cdot$ km $^{-1}$ )	$L_D$ /km	$L'_D$ /km
0.706 587	-7.264 8	2.056 9	0.003 7	3.44	60.77
0.707 10	7.253 2	2.082 8	0.003 7	3.45	60.02

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表 4 两组 $\beta _\textbf{2} $ 符号相反, 且 $ \vert \beta _\textbf{2} \vert $ 接近、 $ \beta_\textbf{3} $ 值较小色散数据

Tab. 4 Two sets of data with different $\beta _2 $

超常材料	$\overline {\omega }$	$\overline {\omega }_p $	$\beta_2$ /(ps $^2\cdot$ km $^{-1}$ )	$\beta _3$ /(ps $^3\cdot$ km $^{-1}$ )	$L_D$ /km	${L}'_D$ /km
M1	0.88	0.941 095	1.184 9	0.330 2	21.10	378.56
M2	0.88	0.941 240	-1.183 5	0.327 2	21.12	382.03

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