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在数值模拟中,建立了掺Yb3+双包层光纤非线性脉冲放大器的数值模型,其中光纤放大器中的色散和非线性过程由非线性薛定谔方程描述,光纤放大器中的增益由速率方程和传输方程来描述。为了简化复杂的理论模型,首先忽略了较弱的高阶色散和高阶非线性效应带来的影响,主要考虑脉冲在增益光纤中的群速度色散和自相位调制作用,非线性薛定谔方程可简化为公式(1);其次,在不考虑激发态吸收和背景损耗的条件下,由于信号光的脉冲间隔远小于上能级寿命,等效于连续信号光,可以用稳态方程式(2)~(5)来表示。稳态方程组是关于传输距离z的耦合常微分方程,可用四阶龙格-库塔公式(Runge-Kutta)以迭代的方法求解。通过结合公式(2)~(5)可算出脉冲在光纤长度z上的每一小段dz中放大的结果,然后计算得出脉冲在放大后的振幅,最后将其代入公式(1)中利用分步傅里叶算法[18]求解,并在整个增益光纤长度上依次迭代计算即可较为准确地模拟出放大脉冲在群速度色散和自相位调制共同作用下的自相似演化情况。
$$i\frac{{\partial A({\textit{z}},t)}}{{\partial {\textit{z}}}} = \frac{{{\beta _2}}}{2}\frac{{{\partial ^2}A({\textit{z}},t)}}{{\partial {t^2}}} - \gamma {\left| {A({\textit{z}},t)} \right|^2}A({\textit{z}},t)$$ (1) $${N_2}({\textit{z}}) = \frac{{\dfrac{{{\lambda _p}}}{{{A_p}hc}}{\sigma _a}({\lambda _p}){P_p}({\textit{z}}) + \dfrac{1}{{{A_s}hc}} \displaystyle\sum\limits_k {{\lambda _k}{\sigma _a}({\lambda _k}){P_s}({\lambda _k},{\textit{z}})} }}{{\dfrac{{{\lambda _p}}}{{{A_p}hc}}\left[ {{\sigma _a}({\lambda _p}) + {\sigma _e}({\lambda _p})} \right]{P_p}({\textit{z}}) + \dfrac{1}{\tau } + \dfrac{1}{{{A_s}hc}} \displaystyle\sum\limits_k {{\lambda _k}\left[ {{\sigma _a}({\lambda _k}) + {\sigma _e}({\lambda _k})} \right]{P_s}({\lambda _k},{\textit{z}})} }}{N_{Yb}}$$ (2) $${N_1}({\textit{z}}) = {N_{Yb}} - {N_2}({\textit{z}})$$ (3) $$ - \frac{{{\rm d}{P_p}({\textit{z}})}}{{{\rm d}{\textit{z}}}} = \left[ {{\sigma _e}\left( {{\lambda _p}} \right){N_2}({\textit{z}}) - {\sigma _a}({\lambda _p}){N_1}({\textit{z}})} \right]{P_p}({\textit{z}}){\varGamma _p}$$ (4) $$\frac{{{\rm d}{P_s}({\textit{z}})}}{{{\rm d}{\textit{z}}}} = \sum\limits_k {\left[ {{\sigma _e}\left( {{\lambda _k}} \right){N_2}({\textit{z}}) - {\sigma _a}({\lambda _k}){N_1}({\textit{z}})} \right]{P_s}({\textit{z}},{\lambda _k}){\varGamma _s}} $$ (5) 在公式(1)中,A(z,t)表示延时系脉冲的慢变包络,β2
和γ分别表示光纤中的二阶色散和非线性系数。在公式(2)~(5)中,为了简化运算,假设光纤沿长度的方向掺杂均匀,掺杂密度为NYb,信号光和泵浦光在光纤的横截面上均匀分布,面积分别为As和Ap,Γs和Γp分别表示信号光和泵浦光的功率填充因子。τ表示激光上能级寿命,N2 (z)和N1(z)表示激光上、下能级粒子数密度分布,Pp(z)和Ps(z)表示泵浦光和信号光的平均功率,λp表示泵浦光的中心波长,λk表示脉冲光谱被分成K个波段后的每个波段的中心波长,σa(λk)和σe(λk)分别表示在λk处Yb3+离子的吸收截面和发射截面,Ps(z,λk)表示λk处信号光平均功率。 在自相似放大理论中,常引入M因子来描绘自相似脉冲的演化程度[19],如公式(6)所示:
$${M^2} = \frac{{\displaystyle\int {{{\left( {{{\left| A \right|}^2} - {{\left| {{A_p}} \right|}^2}} \right)}^2}{\rm d}t} }}{{{{\displaystyle\int {\left| A \right|} }^4}{\rm d}t}}$$ (6) 式中:A为自相似放大脉冲的振幅包络;Ap为抛物线脉冲的振幅包络。M 因子表示在相同的脉冲能量和峰值功率下,脉冲时域强度|A|2与抛物线脉冲时域强度|Ap|2的差距。M 因子的数值越小,脉冲越接近抛物线形;M 因子的数值越大,脉冲越偏离抛物线形,即偏离自相似 演化。通常情况下,当 M≤0.04 时,即可近似认为脉 冲与抛物线形相一致,实现了自相似演化。
评价去啁啾后的放大脉冲质量通常用Strehl ratio(SR)表示[20],如公式(7)所示:
$$SR{\rm{ = }}\frac{{1/{{\displaystyle\int {\left| {{A_c}} \right|} }^2}{\rm d}t}}{{1/\displaystyle\int {{{\left| {{A_{TL}}} \right|}^2}} {\rm d}t}}$$ (7) 式中:Ac为压缩后放大脉冲的时域包络;ATL为脉冲光谱对应的傅里叶变换极限脉冲的时域包络。由公式(7)可以看出,Strehl ratio取1时表示光谱对应的傅里叶变换极限脉冲。同时,Strehl ratio越接近1,说明还原后的去啁啾放大脉冲越接近变换极限脉冲,放大过程中脉冲主要积累线性啁啾,自相似演化程度越高。
在超短激光脉冲自相似放大的研究中,已经较为详细地探讨了脉冲宽度对自相似演化过程的影响[21],发现脉冲宽度在1 ps附近具有较好的演化效果。基于以上研究成果,笔者在模拟中采用1.2 ps的无啁啾高斯脉冲作为种子源,脉冲重复频率50 MHz,剩余模拟参数见表1,未设定的参数将作为讨论条件出现在模拟中。
Effects of gain distribution on self-similar amplification of picosecond pulses
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摘要: 采用数值模拟的方法,研究了增益分布对皮秒脉冲自相似放大的影响。构建了掺镱光纤内超短激光脉冲自相似放大的理论模型,以不同的泵浦方式、光纤长度和总增益系数实现不同的增益分布,探究了不同增益分布对增益光纤中皮秒脉冲的自相似放大过程和结果的影响。结果表明,皮秒脉冲在不同的增益分布下存在最佳的自相似放大结果,可以得到近变换极限的百飞秒高质量脉冲输出。发现同一信号光脉冲在增益光纤中演化至自相似放大过程时,正向泵浦方式下的演化速度比反向泵浦快。对于不同的增益光纤长度和总增益系数,正向泵浦方式下的信号光自相似区域主要集中在低入射脉冲能量和长波长区域,反向泵浦方式下的信号光自相似区域主要集中在高入射脉冲能量和短波长区域。Abstract: The effects of gain distribution on self-similar amplification of picosecond pulses in a Yb-doped fiber laser system were studied by numerical simulation. Ultrashort laser pulses amplified in self-similar amplification theoretical model was established to analyze the impact of pump configuration, fiber length and total gain coefficient on the self-similar amplification evolution and laser output performance. Detailed numerical simulation reveals that the best self-similar amplification result can be found for different cases, where high-quality self-similar pulses with ~100 fs transform-limited pulse duration are obtained. It is demonstrated that the self-similar evolution speed in a forward-pumping scheme is faster than that in a backward-pumping scheme for a fixed seed pulse. Furthermore, the results indicate that for the self-similar amplifier with different fiber lengths and gain coefficients, the forward-pumping scheme shows better evolution results in lower seed energy and longer wavelength range, while the backward-pumping scheme is more suitable for the higher seed energy and shorter wavelength range.
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Key words:
- picosecond laser /
- self-similar amplification /
- fiber amplifier /
- gain distribution
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图 2 不同泵浦方式下放大脉冲的M因子与脉冲中心波长和入射能量关系的伪彩图。(a)正向泵浦,(b)反向泵浦。A和B点分别表示M值在(a)和(b)的最低点,即自相似演化的最佳点
Figure 2. M-factors versus pulse central wavelength and input energy of amplified pulses in different pump schemes. (a) Forward pump, (b) backward pump. Point A and B represent the minimum value of M-factors in (a) and (b), which are the best points of self-similar evolution
图 3 信号光脉冲在不同泵浦方式下的放大结果。(a) A点处的信号光脉冲在正向泵浦下传输到光纤2 m处的去啁啾时域强度,(b) B点处的信号光脉冲在反向泵浦下传输到光纤2 m处的去啁啾时域强度。黑色实线表示信号光脉冲,红色实线表示对应的变换极限脉冲。内插图分别为对应的光谱(左)和压缩前脉冲时域(右),其中黑线和红线代表信号光脉冲和抛物线拟合
Figure 3. Amplification results of seed pulses in different pump schemes. (a) De-chirped pulse profile of the seed pulse of point A transmitting to the end of the 2 m-long fiber in forward-pumping scheme, (b) de-chirped pulse profile of the seed pulse of point B transmitting to the end of the 2 m-long fiber in backward-pumping scheme. Black lines and red lines are seed pulses and transform-limited pulses. Insets: corresponding spectra (left) and amplified pulses before compression (right). Black lines and red lines are seed pulses and parabolic fittings
图 4 A点处的信号光脉冲在正、反泵浦方式下的演化过程。(a) 峰值功率(实线)与M因子(点划线),(b) 10 dB光谱宽度(实线)与脉冲宽度(点划线)。黑色曲线表示正向泵浦,红色曲线表示反向泵浦。(c) 正向泵浦下的光谱演化,(d) 反向泵浦下的光谱演化
Figure 4. Evolution of the seed pulse of point A in forward-pumping and backward-pumping schemes. (a) Peak power (solid) and M-factor (dash-dot), (b) 10 dB spectral width (solid) and pulse width(dash-dot). Black lines and red lines represent forward pump and backward pump. (c) Spectral evolution in forward-pumping scheme, (d) Spectral evolution in backward-pumping scheme
图 5 B点处的信号光脉冲在正、反泵浦方式下的演化过程。(a) 峰值功率(实线)与M因子(点划线),(b)10 dB光谱宽度(实线)与脉冲宽度(点划线),黑色曲线表示正向泵浦,红色曲线表示反向泵浦。(c) 正向泵浦下的光谱演化,(d) 反向泵浦下的光谱演化
Figure 5. Evolution of the seed pulse of point B in forward-pumping and backward-pumping schemes. (a) Peak power (solid) and M-factor(dash-dot), (b) 10 dB spectral width (solid) and pulse width(dash-dot). Black lines and red lines represent forward pump and backward pump. (c) Spectral evolution in forward-pumping scheme, (d) Spectral evolution in backward-pumping scheme
图 6 不同光纤长度下,放大脉冲的M因子与脉冲的中心波长和入射能量的关系。正向泵浦方式下,在(a) 2 m,(b) 3 m,(c) 4 m的光纤长度下放大脉冲的M因子伪彩图;反向泵浦方式下,在(d) 2 m,(e) 3 m,(f) 4 m的光纤长度下放大脉冲的M因子伪彩图
Figure 6. M-factors versus central wavelength and input energy of amplified pulses in different pump schemes with different fiber lengths. In forward-pumping scheme, the M-factors of amplified pulses with the fiber length of (a) 2 m, (b) 3 m, (c) 4 m. In backward-pumping scheme, the M-factors of amplified pulses with the fiber length of (d) 2 m, (e) 3 m, (f) 4 m
图 7 不同泵浦方式、不同总增益下放大脉冲的M因子与脉冲的中心波长和入射能量的关系。正向泵浦方式下,在(a) 20 dB,(b) 25 dB,(c) 30 dB的总增益下放大脉冲的M因子伪彩图;反向泵浦方式下,在(d) 20 dB,(e) 25 dB,(f) 30 dB的总增益下放大脉冲的M因子伪彩图
Figure 7. M-factors versus central wavelength and input energy of amplified pulses in different pump schemes with different total gain coefficients. In forward-pumping scheme, the M-factors of amplified pulses with the total gain coefficient of (a) 20 dB, (b) 25 dB, (c) 30 dB. In backward-pumping scheme, the M-factors of amplified pulses with the total gain coefficient of (d) 20 dB, (e) 25 dB, (f) 30 dB
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