光子晶体光纤中超连续谱形成机制的理论研究
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摘要
本论文根据飞秒激光脉冲在光子晶体光纤中传输所满足的广义非线性薛定谔方程,采用解析分析和数值计算两种方法,从研究啁啾的角度,试图解释超连续谱在光子晶体光纤中的形成过程中,高阶非线性和高阶色散对其所产生的作用。以初始入射脉冲为双曲正割为例,推导出了自相位啁啾、脉冲内拉曼散射啁啾和自陡啁啾的表达式,用分步傅立叶方法数值模拟了自陡啁啾、群速度色散啁啾、三阶色散啁啾的演变图以及综合各效应所得的总啁啾(高阶色散精确到四阶)的演变图,详细分析了高阶非线性和高阶色散以及综合各效应对超连续谱的产生所起的作用,所得结论如下:
1)对非线性效应而言,自相位啁啾是对称传输,相应的频谱也是对称展宽;脉冲内拉曼散射啁啾发生了两个变化,一是啁啾中心向后沿漂移,二是前沿的负啁啾峰值的绝对值大于后沿正啁啾的峰值,可见,脉冲内拉曼散射效应使得频谱的中心向长波长处移动,并且红移的谱宽大于蓝移的谱宽;自陡啁啾随着传输的延长,啁啾中心快速平移且啁啾整体向后沿提升,尤其是后沿的正啁啾明显变得陡峭起来,啁啾主体渐渐集中在后沿,可见,自陡效应对频谱蓝移贡献很大。
2)对高阶色散效应而言,不管是在正常色散区还是在反常色散区,正的三阶色散所致的啁啾在后沿产生大幅振荡,负的三阶色散所致的啁啾在前沿产生大幅振荡,这种振荡打破了频谱的对称性。四阶色散所致啁啾与三阶色散啁啾相比,形状并没有明显的改变。
3)当综合各效应时,在反常色散区,传输初期,啁啾对称传输,自相位和群速度色散效应使得脉冲压缩产生高阶孤子,脉冲的峰值功率逐渐变高,故使得脉冲内拉曼散射效应和自陡效应开始起明显作用,脉冲内拉曼散射效应使得频谱中心红移,而自陡效应则引起了频谱的蓝移,随着传输的延长,三阶色散作用渐渐凸显,终于在高阶非线性和高阶色散的综合作用下,啁啾的前后沿发生严重的不对称变化,最后导致高阶脉冲分裂,正是高阶孤子的分裂,才使得能量得到重新分配,最后形成了超连续谱;在正常色散区,啁啾的变化显示,由于自相位效应使得脉冲渐渐展宽,脉冲内拉曼散射效应和自陡的作用较弱,而三阶色散啁啾在前沿或后沿的不对称振荡,使得频谱不对称展宽,当频谱进入反常色散区时, ISRS、FWM等效应会使得频谱的能量发生转移或产生新的频率,从而产生了超连续谱。
According to the ENLS equations which describe the transmission of femtosecond level laser pulse in photonic crystal fiber (PCF) , the effects of high-order nonlinearities and high-order dispersion on the supercontinuum generation in PCF are tried to explained from the angle of the study of chirp by means of mathematical and numerical computing. To take the hyperbolic-secant input pulse for example, the functions of SPM-caused chirp (self-phase modulation, SPM), ISRS-caused chirp (intra-pulse simulated Raman scattering, ISRS) and SS-caused chirp (self–steepening, SS) are resolved mathematically, and the evolutions of SS-caused chirp, TOD-caused chirp (third-order dispersion, TOD) are simulated. Also is the evolution of chirp caused by all effects above. Conclusions are as the followings.
As far as the nonlinearities are concerned, the symmetrical transmission of SPM chirp leads to the symmetrical broadness of frequency-spectrum; For the ISRS chirp, the central point of chirp shifts to the trailing edge, and the absolute peak values of the negative chirps at leading edge are greater than those of positive chirp at trailing edge. So ISRS makes the spectrum center shift to long wavelength side, and red-shifted spectrum bandwidth is broader than that of blue shift. The center of SS chirp shifts to trailing edge rapidly and the whole chirp is upgrading to trailing side with propagation. In particular, Self-steepening of the pulse causes a sharp drop at the trailing part of the pulse resulting in an asymmetric broadening of the spectrum of the pulse towards the blue.
For higher-order dispersions, not only in anomalous-dispersion region but also in normal-dispersion region, positive TOD-induced (Third-order dispersion, TOD) chirp develops oscillations with great amplitude in trailing edge, and for negative TOD chirp, oscillations with great amplitude are developed in leading edge, which break the spectral symmetry. FOD (Fourth-order dispersion, FOD) chirp change the shape slightly comparing with TOD chirp when FOD is considered.
When all effects are considered together, for one thing, in anomalous-dispersion regime, for initial propagation, the total chirp transmits symmetrically which shows that the combination of SPM and GVD compresses pulse and higher-order solitons are formed. Because of the increase of pulse-peak power, the effects ISRS and SS make function. With further propagation, TOD effect is obvious gradually, and the combination of higher-order nonlinearities and dispersions Lead to breakup of higher-order soliton into fundamental solitons, which results in the distribution of energy and SC spectrum is formed at last. For another thing, in normal-dispersion regime,the evolutions of chirp show that, because of SPM-induced broaden pulse, the effects of ISRS and SS are slight, and TOD effect make great contribution to the spectral dissymmetry broadness. After the SPM-induced broadened spectrum enters into the anomalous-dispersion regime, higher-order nonlinearities, such as ISRS and XPM, lead to the distribution of energy and new frequencies are developed, which form SC spectrum at last.
1)对非线性效应而言,自相位啁啾是对称传输,相应的频谱也是对称展宽;脉冲内拉曼散射啁啾发生了两个变化,一是啁啾中心向后沿漂移,二是前沿的负啁啾峰值的绝对值大于后沿正啁啾的峰值,可见,脉冲内拉曼散射效应使得频谱的中心向长波长处移动,并且红移的谱宽大于蓝移的谱宽;自陡啁啾随着传输的延长,啁啾中心快速平移且啁啾整体向后沿提升,尤其是后沿的正啁啾明显变得陡峭起来,啁啾主体渐渐集中在后沿,可见,自陡效应对频谱蓝移贡献很大。
2)对高阶色散效应而言,不管是在正常色散区还是在反常色散区,正的三阶色散所致的啁啾在后沿产生大幅振荡,负的三阶色散所致的啁啾在前沿产生大幅振荡,这种振荡打破了频谱的对称性。四阶色散所致啁啾与三阶色散啁啾相比,形状并没有明显的改变。
3)当综合各效应时,在反常色散区,传输初期,啁啾对称传输,自相位和群速度色散效应使得脉冲压缩产生高阶孤子,脉冲的峰值功率逐渐变高,故使得脉冲内拉曼散射效应和自陡效应开始起明显作用,脉冲内拉曼散射效应使得频谱中心红移,而自陡效应则引起了频谱的蓝移,随着传输的延长,三阶色散作用渐渐凸显,终于在高阶非线性和高阶色散的综合作用下,啁啾的前后沿发生严重的不对称变化,最后导致高阶脉冲分裂,正是高阶孤子的分裂,才使得能量得到重新分配,最后形成了超连续谱;在正常色散区,啁啾的变化显示,由于自相位效应使得脉冲渐渐展宽,脉冲内拉曼散射效应和自陡的作用较弱,而三阶色散啁啾在前沿或后沿的不对称振荡,使得频谱不对称展宽,当频谱进入反常色散区时, ISRS、FWM等效应会使得频谱的能量发生转移或产生新的频率,从而产生了超连续谱。
According to the ENLS equations which describe the transmission of femtosecond level laser pulse in photonic crystal fiber (PCF) , the effects of high-order nonlinearities and high-order dispersion on the supercontinuum generation in PCF are tried to explained from the angle of the study of chirp by means of mathematical and numerical computing. To take the hyperbolic-secant input pulse for example, the functions of SPM-caused chirp (self-phase modulation, SPM), ISRS-caused chirp (intra-pulse simulated Raman scattering, ISRS) and SS-caused chirp (self–steepening, SS) are resolved mathematically, and the evolutions of SS-caused chirp, TOD-caused chirp (third-order dispersion, TOD) are simulated. Also is the evolution of chirp caused by all effects above. Conclusions are as the followings.
As far as the nonlinearities are concerned, the symmetrical transmission of SPM chirp leads to the symmetrical broadness of frequency-spectrum; For the ISRS chirp, the central point of chirp shifts to the trailing edge, and the absolute peak values of the negative chirps at leading edge are greater than those of positive chirp at trailing edge. So ISRS makes the spectrum center shift to long wavelength side, and red-shifted spectrum bandwidth is broader than that of blue shift. The center of SS chirp shifts to trailing edge rapidly and the whole chirp is upgrading to trailing side with propagation. In particular, Self-steepening of the pulse causes a sharp drop at the trailing part of the pulse resulting in an asymmetric broadening of the spectrum of the pulse towards the blue.
For higher-order dispersions, not only in anomalous-dispersion region but also in normal-dispersion region, positive TOD-induced (Third-order dispersion, TOD) chirp develops oscillations with great amplitude in trailing edge, and for negative TOD chirp, oscillations with great amplitude are developed in leading edge, which break the spectral symmetry. FOD (Fourth-order dispersion, FOD) chirp change the shape slightly comparing with TOD chirp when FOD is considered.
When all effects are considered together, for one thing, in anomalous-dispersion regime, for initial propagation, the total chirp transmits symmetrically which shows that the combination of SPM and GVD compresses pulse and higher-order solitons are formed. Because of the increase of pulse-peak power, the effects ISRS and SS make function. With further propagation, TOD effect is obvious gradually, and the combination of higher-order nonlinearities and dispersions Lead to breakup of higher-order soliton into fundamental solitons, which results in the distribution of energy and SC spectrum is formed at last. For another thing, in normal-dispersion regime,the evolutions of chirp show that, because of SPM-induced broaden pulse, the effects of ISRS and SS are slight, and TOD effect make great contribution to the spectral dissymmetry broadness. After the SPM-induced broadened spectrum enters into the anomalous-dispersion regime, higher-order nonlinearities, such as ISRS and XPM, lead to the distribution of energy and new frequencies are developed, which form SC spectrum at last.
引文
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