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氟化镁晶体微腔在毫米量级,激发的回音壁谐振模式过多,不易分辨,V型结构的两条楔边可以显著抑制极性角方向的模式,且V型结构角度越小,抑制能力越强。不同于规则的球形或柱形微腔[16],直接求解V型结构微腔的解析解是非常困难的。图2为使用有限元软件COMSOL对V型结构微盘腔的模式场分布的结果,每个模式用(n,l,m) 3个量子数来表示,其中,n为径向模式数,描述了光场的径向分布在腔内的波节数;l为角向模式数,描述了光场在赤道面的波节数;m为方位角模式数,描述了光场在子午面的波节数。n=1、l=m的模式为基模,具有最小的模式体积。n越大其模式的电场分布越靠腔内,模式体积越大;l-m越大,则模式分布越趋向两极,模式体积越大且能量分布越分散。
利用微腔克尔效应产生光频梳的结构如图3所示。微腔光频梳产生依赖于非线性参量过程[5],当泵浦激光耦合进入克尔微腔,腔内先通过简并 FWM 产生较优邻近模式的频率成分,新频率成分进而与泵浦频率成分发生非简并FWM以拓宽光谱。由于高Q值微腔极强的场增强因子和天然的滤波结构,新的频率成分获得参量增益并形成参量振荡,产生一系列等间隔的光谱梳齿线,即克尔微腔光频梳。
外部光场通过波导耦合进微腔的动力学模型可以通过Lugiato-Lefever方程[17] (LLE)描述:
$$ \begin{split} {\tau _r}\dfrac{{\partial \psi }}{{\partial t}} =& ({\alpha _i}L/2 + {T_c}/2 + i{\delta _0})A(t,T)+ i\displaystyle\sum\limits_{k \geqslant 2} {\frac{{{i^k}L}}{{{\beta _k}}}} \frac{{\partial A(\tau ,T)}}{{\partial {T^k}}} +\\ &i\gamma L{\left| {A(\tau ,T)} \right|^2}A(\tau ,T)+\sqrt {{T_c}} {A_{in}}\\[-10pt] \end{split} $$ (1) 式中:A(τ, T)描述慢变光场下的复振幅;τr对应光场沿微腔传输一周的时间,与自由光谱范围FSR呈倒数关系既有τr=1/FSR;αi表示线性损耗系数;δ0表示了泵浦激光与谐振频率间失谐系数;L表示微腔的腔长;Tc描述了泵浦光与微腔的功率耦合系数;βk描述了微腔的k阶色散;γ为非线性系数;Ain表示泵浦光的振幅。公式(1)对于腔内光场的演化有一定局限性,其在数值仿真中需要实际的微腔参数作为依据,不具有普遍性,因此对公式(1)进行归一化处理,在仅仅考虑二阶色散的情况下,归一化LLE方程式为:
$$\dfrac{{\partial \psi }}{{\partial t}} = \left[ { - (1 + i\delta ) - i\beta \dfrac{\partial }{{\partial {t^2}}} + i{{\left| \varPsi \right|}^2}} \right]\varPsi + f{}_0$$ (2) 式中:δ表示归一化失谐系数δ=δ0/α;β为归一化色散系数β=β2(2π/τr)2/2α;f0表示归一化泵浦光功率
${f_0} = $ $ \sqrt {\gamma L{T_c}/{\alpha ^3}} {A_{in}}$ ;α表示损耗系数;β2为群色散速度。归一化LLE方程中光场围绕腔体一周的时间τr变为了2π。能够看出,微腔内光场的演变过程主要受制于色散系数β、失谐系数δ、以及耦合进入微腔的泵浦光功率f0。此外,损耗系数α与Q值息息相关,Q值愈大,激发光频梳所需的激光振幅的阈值愈小。
Research on Kerr optical frequency comb generation based on MgF2 crystalline microresonator with ultra-high-Q factor
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摘要: 克尔光频梳具有等距分布的梳状光谱结构,在精密测量、光钟、相干光通信、微波和光学任意波产生、光谱学及天文光谱仪校准等方面有着重要的应用。首先,微腔光频梳与其他光频梳系统相比,具有集成性高、体积小、功耗低的优势,大大扩展了光频梳的应用;其次,通过超精密加工方法制备了品质因子Q值达到了4.8×107的氟化镁微腔,并且得到了干净规律、排列规则的谐振谱,自由频率范围为9.73 GHz,为产生低重复频率光频梳提供了条件;最后,根据实验结果结合Lugiato-Lefever方程分析了氟化镁微腔光频梳的产生过程,研究了泵浦功率对光频梳的影响,通过调整失谐参数得到了孤子态光频梳。并且通过色散调控优化了微腔的光场模式,为产生具有超光滑光谱的孤子光频梳创造了先决条件,提升了光频梳性能。Abstract: Kerr optical frequency comb has an equidistantly distributed comb-like spectral structure and has important applications in precision measurement, optical clocks, coherent optical communications, microwave and optical arbitrary wave generation, spectroscopy, and calibration of astronomical spectrometers. Firstly, compared with other optical frequency comb systems, the microresonator optical frequency comb has the advantages of strong integration, small size and good flexibility, which greatly expands the application of optical frequency combs. Secondly, a MgF2 microresonator with a quality factor up to 4.8×107 was prepared by an ultra-precision machining method, and a clean, regular and regularly arranged spectrum was obtained. The free frequency range was 9.73 GHz, which provides conditions for generating low repetition rate optical frequency combs. Finally, according to the experimental results and the Lugiato-Lefever equation, the generation process of the MgF2 microresonator optical frequency comb was analyzed, and the influence of the pump power on the optical frequency comb was studied. The soliton state optical frequency comb was obtained by adjusting the detuning parameters. In addition, the optical field mode of the microresonator was optimized through dispersion control, which creates a condition for generating a soliton optical frequency comb with an ultra-smooth spectrum and improves the performance of the optical frequency comb.
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Key words:
- optical microresonator /
- MgF2 /
- quality factor /
- optical frequency comb /
- soliton
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