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当忽略高阶色散和非线性效应时,光脉冲在DDF中的传输可由薛定谔方程[9-10]表示:
$ \begin{gathered} \mathrm{i} \frac{\partial A(z, T)}{\partial z}=-\mathrm{i} \frac{\alpha}{2} A(z, T)+ \\ \frac{\beta_{2}}{2} D(z) \frac{\partial^{2} A(z, T)}{\partial T^{2}}-\gamma|A(z, T)|^{2} A(z, T) \end{gathered} $
(1) 式中,A(z, T)为脉冲的包络的慢变振幅,T是随脉冲以群速度移动的参考系中时间的量度,z为传输的距离,α为光纤的损耗系数,β2为z=0时的2阶群速度色散(group velocity dispersion, GVD)参量,D(z)为色散变化参量,γ为非线性系数。当忽略光纤的损耗时,该方程便演变为以下的非线性薛定谔方程:
$ \begin{gathered} \mathrm{i} \frac{\partial A(z, T)}{\partial z}=\frac{\beta_{2}}{2} D(z) \frac{\partial^{2} A(z, T)}{\partial T^{2}}- \\ \gamma|A(z, T)|^{2} A(z, T) \end{gathered} $
(2) 此时脉冲在DDF中的演化主要受2阶色散和低阶非线性影响。通过傅里叶变换,将(2)式改写为以下形式:
$ \frac{\partial A(z, T)}{\partial z}=(\hat{D}+\hat{N}) A(z, T) $
(3) 式中, $ \hat{D}=\frac{-\beta_{2}}{2} D(z) \frac{\partial^{2}}{\partial T^{2}}$ ,表示2阶色散作用, $ \hat{N}=$$\gamma|A(z, T)|^{2} A(z, T) $ ,表示非线性作用。由分步傅里叶方法可知,当光场通过一小段距离h时,色散和非线性效应可以分别作用。故每经过一小段距离h后的光脉冲的慢变振幅A(z+h, T)可由下式表示:
$ A(z+h, T)=\exp (h \hat{D}) \exp (h \hat{N}) A(z, T) $
(4) 本文中,DDF 2阶色散随着光纤长度z以双曲型变化,即 $D(z)=\frac{1}{1+H z} $ ,其中H为增益系数,归一化D(0)=1,采用无啁啾的高斯脉冲作为输入脉冲,T0为初始脉宽, 输入脉冲表达式为:
$A(0, T)=\exp \left(\frac{-T^{2}}{2 T_{0}^{2}}\right) $
(5) 初始输入高斯脉冲的半峰全宽(full width at half maximum, FWHM)TFWHM=1ps,中心波长为1550nm,当z=0时,2阶色散系数β2=1.36ps2/km,非线性系数γ=3.5(W·km)-1,增益系数H=22km-1。通过计算色散长度Ld=1.0608km,非线性长度LNL=0.0068km,实验中设置光纤长度为4倍的色散长度Ld,此时由于光纤长度远大于非线性长度,GVD效应会对光脉冲传输起主要作用。通过前面的分步傅里叶法,可得到脉冲在DDF中不同位置的演化的数值仿真值。图 1和图 2中分别给出脉冲的时域演化趋势和输出脉冲的啁啾曲线。
从图 1中可以看到,随着传输距离的增大,时域内脉冲不断进行展宽。其展宽的原因在传输前期主要来自于2阶GVD效应,但随着2阶GVD效应的减小,非线性效应(self-phase modulation, SPM)会逐渐变强,当GVD和SPM满足一定关系[11]时, 就会产生具有线性啁啾的自相似脉冲,因此需要合理选择DDF的长度,保证光脉冲完成自相似演化。当光脉冲经过4倍的Ld之后,可以看到,输出脉冲的啁啾具有较大的线性范围,此时SPM效应所产生的非线性啁啾在整个脉冲范围内几乎被正常GVD效应线性化,可以认为此时脉冲已经完成了自相似演化。
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由于自相似脉冲在传输的过程中其GVD效应远大于SPM效应,因此对演化完成后的自相似脉冲进行色散补偿后,便可以对自相似脉冲进行压缩。对色散进行补偿[11-13]的做法是在DDF后引入一段带有相反色散的光纤。对于引入相反色散光纤进行压缩脉冲可以这样理解,对于无初始啁啾的高斯脉冲在经过适当距离的DDF后,产生具有线性啁啾的自相似脉冲后在进入反常色散的光纤时,可以看作是带有啁啾的高斯脉冲,其入射场可表示为:
$ U(0, T)=\exp \left[\frac{-(1+\mathrm{i} C) T^{2}}{2 T_{0}^{2}}\right] $
(6) 式中, C为线性啁啾参量,由于高斯脉冲经过了带有正常色散DDF,所以自相似脉冲感应的是正啁啾,因此啁啾参量C>0。当不考虑非线性效应时,脉冲在具有线性色散介质光纤时满足以下线性微分方程:
$ \mathrm{i} \frac{\partial U(z, T)}{\partial z}=\frac{\beta_{2}}{2} \frac{\partial^{2} U(z, T)}{\partial T^{2}} $
(7) 当输入脉冲为带啁啾的高斯脉冲时,可求得上述方程的解为:
$ \begin{gathered} U(z, T)=\frac{T_{0}}{\left[T_{0}^{2}-\mathrm{i} \beta_{2} z(1+\mathrm{i} C)\right]^{1 / 2}} \times \\ \exp \left\{\frac{-(1+\mathrm{i} C) T^{2}}{2\left[T_{0}^{2}-\mathrm{i} \beta_{2} z(1+\mathrm{i} C)\right]}\right\} \end{gathered} $
(8) 通过计算得到在传输距离为z时的脉冲宽度T1与初始脉冲脉宽T0存在以下关系:
$ T_{1}=\left[\left(1+\frac{C \beta_{2} z}{T_{0}^{2}}\right)^{2}+\left(\frac{\beta_{2} z}{T_{0}^{2}}\right)^{2}\right]^{1 / 2} T_{0} $
(9) 从(9)式可以看出,若初始脉冲啁啾与光纤的2阶色散参量符号相反时,在传输适当的距离内啁啾高斯脉冲会被压缩,且存在一个最窄的压缩脉宽T1,与光纤的2阶色散参量和长度有关。同样对于带有线性正啁啾的自相似脉冲也是一样,通过引入一段带有反常色散的光纤,在经过适当的距离后,就能够对自相似脉冲进行压缩。
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理论上,只要引入一段带有反常色散的光纤都能实现脉冲压缩的目的,因此本文中分别引入带有反常色散的色散补偿光纤(2阶色散为常量)和带有反常色散的线性渐增光纤来对自相似脉冲进行压缩,以探究不同类型的反常色散光纤对自相似脉冲的压缩影响。取色散补偿光纤的2阶色散β2=-1.36(ps2/km),非线性系数γ=1(W·km)-1;色散线性渐增光纤2阶色散β2(z)=β2(0)(1+pz),其中初始2阶色散系数β2(0)=-1.36(ps2/km),色散渐增系数p取为1km-1,非线性系数与色散补偿光纤保持一致。通过数值仿真可以得到自相似脉冲经过各段反常色散光纤后的脉冲压缩情况。图 3和图 4分别为自相似脉冲经过普通色散补偿光纤和色散线性渐增光纤的压缩示意图,图 5为自相似脉冲经过色散补偿和色散线性渐增光纤脉冲后的压缩对比图。
Figure 5. Partial enlarged view of pulse compression of dispersion compensation fiber and linearly increasing dispersion fiber
通过数值仿真,当获得最佳超短脉冲输出时,色散补偿光纤长度为97.8m,超短输出脉冲半峰全宽TFWHM=52.6fs,输出脉冲峰值功率为684.5W;而色散线性渐增光纤长度为93.0m,超短输出脉冲半峰全宽TFWHM=53.8fs,输出脉冲峰值功率为688.2W。可以看到, 当色散渐增系数p=1km-1时,色散线性渐增光纤与色散补偿光纤一样能将自相似脉冲压缩至50fs量级,但色散线性渐增光纤与色散补偿光纤所产生的脉冲基座略有不同,色散线性渐增光纤的基座波动相对较小,且色散线性渐增光纤的光纤长度也小于色散补偿光纤。因此,相比较于普通色散补偿光纤,色散线性渐增光纤能获取较高质量的自相似脉冲压缩。
为了进一步探究色散渐增系数p对压缩质量的影响,在保持其它参量不变的情况下改变p值,观察脉冲在色散线性渐增光纤中的演化情况。图 6是p为5km-1和10km-1时的最佳超短脉冲输出示意图。
Figure 6. Schematic diagram of pulse compression of fibers with linearly increasing dispersion at p=5km-1, 10km-1
表 1为自相似脉冲经过不同p值的色散线性渐增光纤后输出最短脉冲的数值模拟值。通过数值仿真可知,当p=5km-1时,压缩至最短脉冲所需的光纤长度缩短为81.0m,最短脉冲半峰全宽TFWHM=58.0fs; 当p=10km-1时,所需的光纤长度缩短为74.2m,最短脉冲半峰全宽TFWHM=61.8fs。由以上结果可以得出:当保持光纤其它参量不变时,增大色散线性渐增系数p可以明显缩短超短脉冲输出时的补偿光纤长度,有利于减小脉冲在传输过程中产生的损耗,从而有效提高自相似脉冲的压缩质量,但随着色散渐增系数p迅速提高,其输出脉冲的峰值功率和脉宽也会略微有所降低。最终在p=10km-1时获得功率为630.6W、脉宽为61.8fs的超短脉冲输出。
Table 1. Numerical simulation results of fibers with linearly increasing dispersion
p=1km-1 p=5km-1 p=10km-1 optimal compensation fiber length/m 93.0 81.0 74.2 full width at half maximum of ultrashort pulse output/fs 53.8 58.0 61.8 peak power of output pulse/W 688.2 655.1 630.6 -
为了验证其它类型的色散渐增光纤是否也有类似的规律,又引入了色散指数渐增光纤来压缩自相似脉冲。色散指数渐增光纤的2阶色散可以表示为:β2(z)=β2(0)exp(pz),其中初始2阶色散系数β2(0)=-1.36(ps2/km),与色散线性渐增光纤一样分别取p为1km-1,5km-1,10km-1,非线性系数与上述光纤保持一致,通过数值仿真得出自相似脉冲在色散指数渐增光纤中的演化结果。图 7是p分别为1km-1,5km-1,10km-1时色散指数渐增光纤的脉冲压缩示意图。
Figure 7. Schematic diagram of pulse compression of fiber with increasing dispersion index at p=1km-1, 5km-1, 10km-1
表 2为自相似脉冲经过不同p值的色散指数渐增光纤后输出最短脉冲的数值模拟值。当色散渐增系数p=1km-1时,压缩至最短脉冲所需的色散指数渐增光纤长度为92.5m,最短脉冲半峰全宽TFWHM=54.2fs,输出脉冲峰值功率为674.2W;当色散渐增系数p=5km-1时,压缩至最短脉冲所需的光纤长度为82.5m,最短脉冲半峰全宽TFWHM=59.4fs,输出脉冲峰值功率为643.3W;当色散渐增系数p=10km-1时,压缩至最短脉冲所需的光纤长度缩短为70.3m,最短脉冲半峰全宽TFWHM=64.4fs,输出脉冲峰值功率为604.4W。
Table 2. Numerical simulation results of fibers with increasing dispersion index
p=1km-1 p=5km-1 p=10km-1 optimal compensation fiber length/m 92.5 82.5 70.3 full width at half maximum of ultrashort pulse output/fs 54.2 59.4 64.4 peak power of output pulse/W 674.2 643.3 604.4 上述结果与利用色散线性渐增光纤压缩脉冲得出的结论不谋而合,因此可以认为: 当保持其它参量不变时,增大色散渐增系数,能有效地缩短脉冲压缩所需的补偿光纤长度,在实际的光纤线路中,缩短补偿光纤长度就意味着能避免更多的损耗,从而提高脉冲的压缩质量,得到高质量的超短脉冲输出,但随着色散渐增系数的迅速增加,输出功率和脉宽也会略微减小,因此需要合理地选择色散渐增系数,在保证脉冲压缩质量的前提下缩短压缩光纤的长度。同时, 在比较指数型色散渐增光纤和线性型色散渐增光纤时可以发现,当色散渐增系数相等时,线性型色散渐增光纤的压缩效果要比指数型色散渐增光纤的压缩效果好。
基于啁啾补偿技术的自相似脉冲压缩光纤设计
Design of self-similar pulse compression fiber based on chirp compensation technology
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摘要: 为了获得高功率优质的超短脉冲光源,利用色散渐增光纤产生的强线性啁啾对自相似脉冲进行了啁啾补偿光纤设计。首先利用色散补偿光纤得到了半峰全宽为52.6fs、峰值功率为684.5W的超短脉冲输出。在此基础上研究了色散渐增的补偿光纤设计,讨论了色散线性渐增光纤和色散指数渐增光纤对自相似脉冲的压缩影响。当色散渐增系数取1km-1,5km-1,10km-1时,通过数值仿真得出了最短的输出脉宽、峰值功率以及所需的补偿光纤长度。结果表明,色散渐增光纤能缩短补偿光纤的长度,减小脉冲压缩时产生的损耗,得到半峰全宽为61.8fs和64.4fs的高功率超短脉冲输出。这一结果对自相似脉冲压缩光纤的设计是有帮助的。Abstract: In order to obtain high power and high quality ultrashort pulse light source, the chirp compensation fiber design of self-similar pulse was studied for the first time using strong linear chirp generated by the dispersion decreasing fiber (DDF). Firstly, the ultra-short pulse output with half height full width of 52.6fs and peak power of 684.5W was obtained by using dispersion compensation fiber. On this basis, the design of dispersion increasing compensation fiber was studied, and the effects of dispersion-linear-increased fiber and dispersion-exponential-increased fiber on the compression of self-similar pulses were emphatically discussed. When the dispersion increasing coefficient was respectively set to 1km-1, 5km-1, 10km-1, the shortest output pulse width, peak power and the required compensated fiber length were obtained by numerical simulation. The results show that the dispersion increasing fiber can shorten the length of the compensated fiber greatly, which is beneficial to reduce the loss caused by pulse compression, and finally obtain high power ultrashort pulse output with half height full width of 61.8fs and 64.4fs. This result is helpful to the design of self-similar pulse compression fiber.
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Key words:
- fiber optics /
- dispersion increasing fiber /
- chirp compensation /
- self-similar pulse
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Table 1. Numerical simulation results of fibers with linearly increasing dispersion
p=1km-1 p=5km-1 p=10km-1 optimal compensation fiber length/m 93.0 81.0 74.2 full width at half maximum of ultrashort pulse output/fs 53.8 58.0 61.8 peak power of output pulse/W 688.2 655.1 630.6 Table 2. Numerical simulation results of fibers with increasing dispersion index
p=1km-1 p=5km-1 p=10km-1 optimal compensation fiber length/m 92.5 82.5 70.3 full width at half maximum of ultrashort pulse output/fs 54.2 59.4 64.4 peak power of output pulse/W 674.2 643.3 604.4 -
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