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在此仿真工作中,采用线性腔结构来分析锁模的稳定性。振荡器中包含一个光纤布喇格光栅(fiber Bragg grating, FBG)、一段掺镱光纤(ytterbium-doped fiber, YDF)、一片SESAM和两段单模光纤(single mode fiber, SMF)。该空腔中的所有光纤均为保偏光纤。为了表征光栅的影响,其被拆分为滤波器和光学耦合器(optical coupler, OC), OC的输出耦合比为20∶80;YDF、SMF 1和SMF 2的长度分别为0.35 m、0.30 m和0.30 m。脉冲于腔内的传播循环过程如图 1所示。
采用修正的非线性薛定谔方程来模拟光纤激光器中的脉冲演化[19-20]:
$ \frac{\partial A}{\partial z}=-\frac{\mathrm{i}}{2} \beta_2 \frac{\partial^2 A}{\partial t^2}+\mathrm{i} \gamma|A|^2 A+\frac{g}{2} A+\frac{g}{2 \mathit{\varOmega}_{\mathrm{g}}{ }^2} \frac{\partial^2 A}{\partial t^2} $
(1) 式中: A为脉冲包络的电场振幅; z为脉冲传播距离; β2为光纤的2阶色散系数,设为22 ps2/km; γ为光纤的非线性系数,设为5.1 km/W;最后一项表示增益滤波对脉冲的影响,其中Ωg为增益带宽,设为40 nm; g为腔内的增益,在无源纤中,g=0 m-1,而有源光纤中,g可表示为[21]:
$ g=g_0 \exp \left(-E_{\mathrm{p}} / E_{\mathrm{s}}\right) $
(2) 式中:g0为小信号增益系数,被设为6.9 m-1;Ep和Es分别表示脉冲能量和脉冲的增益饱和能量,增益饱和能量是和抽运功率正相关的。
为了更精确地描述脉冲的演化行为,使用SA的时间相关模型来表征其对脉冲光的吸收,时间依赖函数S(t)满足方程[22]:
$ \frac{\partial S(t)}{\partial t}=-\frac{S(t)-A_0}{\tau_{\mathrm{r}}}-\frac{|A(t)|^2}{E_{\mathrm{a}}} S(t) $
(3) 式中: t为时间; A0为调制深度; τr为恢复时间; Ea为吸收体的饱和能量。考虑到SA具有固有损耗αs, SESAM的反射函数为:
$ R(t)=1-S(t)-\alpha_{\mathrm{s}} $
(4) 模拟中,脉冲从一个小的噪声信号开始演化,而其演化过程由分步傅里叶算法计算。如果脉冲于相邻两个循环之间的相对能量变化小于10-9,则认为脉冲达到锁模状态; 如果脉冲在2000次腔循环后不能收敛,则认为激光器无法锁模[20]。
在锁模激光器中,脉冲的稳定形成与滤波器带宽有着很大关系。当正啁啾脉冲于正色散光纤中演化时,由于自相位调制(self-phase modulation, SPM)效应,其频谱和时域形状将继续展宽[23]。滤波器可以为其频谱提供周期性窄化,但如果滤波器带宽过宽,当脉冲能量达到一定值时,容易发生劈裂。同时,过量的光谱展宽意味着脉冲的两侧具有更大的波长差异,这将加剧脉冲的进一步变形,此时腔内只能形成一些特殊的孤子或孤子簇。为了使激光器产生稳定、高质量的脉冲序列,需要使用带宽合适的滤波器。
首先,以光栅带宽和增益饱和能量为遍历量来计算脉冲的收敛区域,模拟中设A0=0.14;αs=0.04。为了研究SESAM的恢复时间对锁模的影响,计算了不同τr值下脉冲收敛区域的变化,其结果如图 2a~图 2d所示。图中,不同的颜色表示脉冲达到收敛状态时所经历的不同周期数,白色区域表示脉冲无法收敛。显然无论τr值为多少,收敛域只存在于计算域中的一小部分空间中。同时根据计算结果,相较于带宽较大的光栅,当光栅带宽较小时,脉冲可以支持更高的增益饱和能量,而当光栅带宽增大到一定程度后,则腔体不再产生稳定的锁模脉冲。这表明此类直腔光纤激光器在滤波器带宽较小时具有更好的锁模效果。
图 2 由光栅带宽、脉冲饱和增益和恢复时间组成的参数空间中的稳定锁模区域
Figure 2. Stable mode-locking region in parameters space composed of grating bandwidth, pulse saturation gain and recovery time
同时可以注意到,脉冲收敛区域随着SA恢复时间的增加而逐渐减小。为了进一步探讨这个问题,计算了恢复时间为无穷时的收敛域情况,其结果如图 2e所示。此时式(3)右侧的第1项可忽略,于是得到:
$ S(t)=A_0 \exp \left[-\int_{-\infty}^t \frac{\left|A\left(t^{\prime}\right)\right|^2 \mathrm{~d} t^{\prime}}{E_{\mathrm{a}}}\right] $
(5) 式中: t′表示t时刻之前的时间。很明显,与图 2d相比,其锁模区域大幅减少,因此可以得出结论: 恢复时间较小的SESAM可以获得更大的锁模区域,使用小恢复时间的SESAM会为腔体带来更好的调节能力和更好的稳定性。
将恢复时间设为18 ps、滤波带宽设为0.4 nm后,计算了A0-Es空间中的脉冲收敛区域,以此分析了SESAM的调制深度对锁模的影响, 如图 3所示。具备高调制深度的SESAM相较于低调制深度的SESAM具有更大的锁模区域,当A0从0.05增加到0.25时,支持锁模的最大饱和功率几乎增加了3倍,这说明高调制深度的SESAM更有利于锁模脉冲的产生。
基于保偏光纤结构的直腔耗散孤子锁模激光器
Linear cavity dissipative soliton mode-locked laser based on polarization-maintaining fiber structure
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摘要: 为了获得稳定的超短脉冲序列,采用分步傅里叶算法数值分析了滤波器带宽、可饱和吸收体调制深度和可保和吸收体恢复时间对锁模的影响,并搭建了一个利用半导体可饱和吸收镜作为可饱和吸收体的全保偏结构的掺镱锁模光纤激光器。结果表明,该腔体可以自启动产生光谱带宽为0.104 nm、重复频率为102.32 MHz的耗散孤子脉冲序列; 且出射的脉冲在放大过程中表现出缓慢的展宽速率, 激光腔搭建成功。该激光器在光纤探针、频率梳和参量光学等领域具有良好的应用前景。Abstract: To obtain the stable mode-locked pulse train, the effects of filter bandwidth, saturable absorber (SA) modulation depth, and SA recovery time on mode-locking were analyzed numerically by the split-step Fourier method. According to the calculation results, the laser is established. The oscillator can generate self-starting dissipative soliton pulses train with a spectrum bandwidth of 0.104 nm and a repetition rate of 102.32 MHz. Meanwhile, the pulse train shows a slow broadening rate and good shape-preserving ability in the amplification process. This study indicates that this laser will have great application prospects in fiber probes, frequency comb, parameters optic and other fields.
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Key words:
- fiber optics /
- ultrafast optics /
- saturable absorber /
- fiber amplifier
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图 5 锁模特性示意图
a—基重频锁模序列 b—脉冲光谱 c—射频谱 d—劈裂的脉冲序列 e—输出功率-抽运功率曲线 f—输出功率的时间稳定性测量
Figure 5. Mode-locked characteristics
a—fundamental frequency mode-locked pulse train b—spectrum at pump power c—radio-frequency spectrum d—double pulse train e—output power and pump power curve f— time stability measurement of output power
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