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以近年来较为完善的理想二倍频转换效率模型为基础,深入研究空间走离效应过程中的能量转换与光束分离情况,建立聚焦光束和平行传输光束两种不同泵浦结构下的二倍频转换效率模型。根据理想二倍频效率方程[25],在单通泵浦二倍频晶体过程中,可以得到泵浦光强度I的表达式:
$$ I = \frac{P}{{\pi \cdot {{{w}}_{\text{0}}}^2}} $$ (1) $$ I = \frac{1}{2}{{c }}{n_1}{\varepsilon _0}{\left| E \right|^2} $$ (2) 式中:P为泵浦光峰值功率;w0为泵浦光束在二倍频晶体上的束腰半径;c为光速;ε0为真空介电常数;n1为入射光折射率;E为光振幅强度。
结合公式(1)、(2)可求出光振幅强度。再将二倍频特征长度代入理想二倍频转换效率方程[26],得到理想二倍频转换效率nSHG的表达式:
$$ {\omega _1} = 2\pi \cdot \frac{c}{\lambda } $$ (3) $$ {{{l}}_{{\rm{SHG}}}} = \frac{{c{n_1}}}{{{\omega _1}{{{d}}_{{{{\rm{eff}}}}}}E}} $$ (4) $$ {{{n}}_{{\text{SHG}}}} = {\tanh ^2}\left(\frac{z}{{{l_{{\rm{SHG}}}}}}\right) = {\tanh ^2}\left(\frac{{z \cdot {\omega _1}{{{d}}_{{\rm{eff}}}}E}}{{c{n_1}}}\right) $$ (5) 式中:λ为泵浦光波长;ω1为入射光角频率;deff为二倍频非线性吸收系数;lSHG为二倍频特征长度;z为LBO晶体长度。
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为得到实际二倍频转换效率,需要在理想二倍频转换效率的基础上精确消除走离效应产生的效率损失。对二倍频晶体内部发生的能量转换和光束分离现象进行建模,光束尺寸始终不变的平行光束的二倍频实验光路如图1(a)所示。
图 1 平行光束泵浦:(a)二倍频光路图;(b)二倍频晶体内部光束分布
Figure 1. Pumping of parallel beam: (a) SHG optical path; (b) Internal beam distribution of SHG crystal
如果入射光是直线传播,光束半径保持不变,随着光束在晶体内的传播,基频光能量逐渐减小,基频光能量E0公式为:
$$ {{w}}(z) = {w_0} $$ (6) $$ I(z) = {I_0}\left(1 - {n_{{\rm{SHG}}}} \cdot \frac{{{z}}}{L}\right) $$ (7) $$ {E_0} = \int_0^L \; \pi \cdot w{(z)^2}I(z){\rm{d}}z $$ (8) 式中:w0为束腰半径;nSHG为理想二倍频转换效率;I0为入射光强;w(z)为光束半径;I(z)为沿二倍频晶体纵向z轴每一位置的光强。
如图1(b)所示,沿z轴不同位置的走离距离l(z)公式为:
$$ l(z) = z \cdot \tan (\alpha ) $$ (9) 式中:L为晶体长度;α为走离角度;二倍频晶体的入射面z=0。
在光束尺寸始终不变的平行光束泵浦下,根据光场分布得到基频光和二倍频光在空间上相互作用部分的能量E1、走离效应产生的效率损失比例nwalk、消除走离效应后的实际二倍频转换效率n,公式如下:
$$ {E_1} = \int_0^L \; \pi \cdot w(z) \cdot \left(w(z) - \frac{{l(z)}}{2}\right) \cdot I(z){\rm{d}}z $$ (10) $$ {{{n}}_{{\text{walk}}}} = 1 - \frac{{{E_1}}}{{{E_0}}} $$ (11) $$ {{n}} = \left( {1 - {n_{{\rm{walk}}}}} \right) \cdot {n_{{\rm{SHG}}}} = \frac{{{E_1}}}{{{E_0}}} \cdot {{{n}}_{{\rm{SHG}}}} $$ (12) -
由于部分倍频晶体非线性系数较低,难以在低功率密度下实现高效二倍频转换,在实验中需要对泵浦光束进行聚焦,聚焦光束的二倍频过程晶体内光束分离情况如图2所示。
二倍频晶体内基频光的束腰半径和发散角度由聚焦透镜焦距、基频光束半径、入射光波长和基频光束质量决定,设定束腰位置在二倍频晶体中心,根据高斯光束分布公式,二倍频晶体纵向z轴上各个位置的光斑半径w(z)和走离距离l(z)为:
$$ {w}_{0}=\frac{f\lambda {M}^{2}}{\pi R} $$ (13) $$ 2\theta =\frac{2\lambda }{\pi {w}_{0}} $$ (14) $$ w(z)=w_{0} \sqrt{1+\left(\frac{\lambda z}{\pi w_{0}^{2}}\right)^{2}} $$ (15) $$ l(z) = \tan (\alpha ) \cdot \left(z + \frac{L}{2}\right) $$ (16) 式中:束腰处z=0;w0为束腰半径;f为聚焦透镜焦距;λ为入射光波长;M2为基频光的光束质量因子;R为入射到聚焦透镜的基频光束半径;2θ为高斯光束发散角;L为晶体长度;α为走离角度。光束在晶体内的传播过程中,非线性变换随之发生,基频光能量逐渐减小,对光强度进行整合,得到基频光能量E0为:
$$ I(z) = {I_0}\left(1 - \frac{{{{{n}}_{{\text{SHG}}}}}}{L}\left(z + \frac{L}{2}\right)\right) $$ (17) $$ {E_0} = \int_{ - L/2}^{L/2} {\begin{array}{*{20}{c}} {} \\ {} \end{array}} \pi \cdot w{(z)^2}I(z){\rm{d}}z $$ (18) 式中:nSHG为二倍频转换效率;I0为入射光强;I(z)为沿z轴每一位置光强。
在泵浦光束聚焦晶体的条件下,根据光场分布得到基频光和二倍频光在空间上相互作用部分的能量E1、走离效应产生的效率损失比例nwalk、消除走离效应后的实际二倍频转换效率n,公式如下:
$$\begin{split} {E}_{1}=&{\displaystyle {\int }_{-L/2}^{L/2}\begin{array}{c}\\ \end{array}}\Bigg(2\cdot \mathrm{arccos}\left(\frac{l(z)}{2w(z)}\right)\cdot\\ & w{(z)}^{2}-l(z)\cdot {\sqrt{w{(z)}^{2}-\frac{l{(z)}^{2}}{4}}}_{}\Bigg)\cdot I(z){\rm{d}}z \end{split} $$ (19) $$ {{{n}}_{{{{\rm{walk}}}}}} = 1 - \frac{{{E_1}}}{{{E_0}}} $$ (20) $$ {{n}} = \left( {1 - {n_{{\rm{walk}}}}} \right) \cdot {n_{{\rm{SHG}}}} = \frac{{{E_1}}}{{{E_0}}} \cdot {{{n}}_{{\rm{SHG}}}} $$ (21) -
为了分析走离效应的影响,根据所建立的二倍频效率模型,参数条件为:10 ns脉宽的激光器为泵浦源,50 μm束腰半径为泵浦光束,7 mm长LBO晶体为二倍频晶体,二倍频光路如图3(a)所示。有无走离效应的二倍频效率曲线如图3(b)所示,黑色线是无走离效应的理想二倍频效率,红色线是考虑走离效应的二倍频效率。可以看出,走离效应使得二倍频晶体上的二倍频光与基频光逐渐偏离,呈现二倍频光束过于发散的现象,导致最佳二倍频效率明显降低。聚焦束腰50 μm的泵浦光束和半径50 μm平行传输泵浦光束的二倍频效率曲线如图3(c)所示,聚焦光束和平行传输光束的二倍频效率有着一定的差异,因此有必要对其进行分类讨论。
图 3 (a)二倍频光路图;(b)有无走离效应时,二倍频效率分别随泵浦光脉冲能量的变化;(c)聚焦光束和平行传输光束,二倍频效率分别随泵浦光脉冲能量的变化
Figure 3. (a) SHG optical path; (b) The SHG efficiency varies with the energy of the pump laser, with or without the walk-off effect; (c) The SHG efficiency of focused and parallel beams varies with the energy of the pump laser
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利用这个充分考虑走离效应的二倍频效率模型,分别以基频光各项参数为变量,分析其对二倍频效率的影响。根据公式(13)和(14),基频光的光束质量、光束半径以及聚焦透镜的焦距都会对聚焦后的束腰和发散角度产生影响,进而影响二倍频效率,所以将先对其逐一进行分析。参数条件为波长1064 nm基频光通过凸透镜聚焦到10 mm长度的LBO晶体进行二倍频,束腰位于LBO晶体中心位置。
首先,探究基频光光束质量M2对二倍频转换效率的影响,针对1、1.5和2的M2,泵浦参数为半径500 μm基频光通过100 mm焦距凸透镜的条件,分析在不同的M2条件下,二倍频效率随泵浦光脉冲能量的变化,如图4(a)所示。随着光束质量的优化,基频光以更低的能量达到最佳二倍频转换效率,但是光束质量更好的基频光束能量更趋近于高斯分布,能量集中于光束中心位置,因此,受到空间走离效应影响导致的光束分离现象更为明显,使其最佳二倍频转换效率在理论上小于光束质量较差的条件。在实际实验中,由于光束质量较差的光束分布并不均匀,达不到该模型理论上的二倍频效率。
图 4 二倍频效率随泵浦光脉冲能量的变化。(a)基频光不同光束质量;(b)聚焦透镜不同焦距;(c)不同基频光尺寸
Figure 4. The variation of SHG efficiency with pump laser energy. (a) Different M2 of fundamental laser; (b) Different focal lengths of focusing lenses; (c) Different radius of fundamental laser
其次,探究聚焦透镜的焦距对二倍频效率的影响,针对50、100、200 mm焦距的凸透镜,调整泵浦参数为对M2=1且半径500 μm的基频光进行聚焦的条件,二倍频效率随泵浦光脉冲能量的变化如图4(b)所示。透镜焦距越小越能更快达到最佳二倍频效率,因为更小的焦距会使聚焦后的束腰显著变小(焦距f=50、100、200 mm的束腰半径分别为33.9、67.7、135.5 μm),使走离距离相对束腰半径的比例提高,产生更加显著的走离效应,二倍频光与基频光在空间上的失配导致最佳二倍频转换效率下降。
进一步地,探究基频光束半径对二倍频效率的影响,针对300、500、1000 μm基频光束半径,调整泵浦参数为100 mm焦距凸透镜对M2=1的基频光束进行聚焦的条件,二倍频效率随泵浦光脉冲能量的变化如图4(c)所示。越大的基频光束尺寸,聚焦后的束腰就越小,因此也就越快达到最佳二倍频效率且相应的最佳二倍频效率更小。
直接分析在不同束腰尺寸的泵浦光束下,二倍频效率随泵浦光脉冲能量的变化,束腰半径分别采用50、100、200 μm,结果如图5(a)所示。随着束腰尺寸的缩小,功率密度不断提高,在更低能量泵浦下达到最佳二倍频转换效率。需要特别提及的是,束腰尺寸的缩小会使走离距离相对光束尺寸的比例提高,产生更加显著的走离效应,使得最佳二倍频转换效率随之下降。
图 5 二倍频效率随泵浦光脉冲能量的变化。(a)不同束腰尺寸;(b)不同发散角
Figure 5. The variation of SHG efficiency with pump laser energy. (a) Different waist radius; (b) Different divergence angle
最后,直接分析束腰位置的基频光发散角对二倍频效率的影响,针对0.02、0.005、0.0025 rad的发散角,二倍频效率随泵浦光脉冲能量的变化如图5(b)所示。发散角越大,基频光产生的束腰越小,能量密度相对更高,所以可以在更低能量达到最佳二倍频效率。但是更大的发散角使得倍频光和基频光的匹配效果更差,走离效应更为严重,导致最佳二倍频效率降低。
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二倍频晶体种类、晶体切割角度和晶体折射率也是影响二倍频效率的重要因素。不同二倍频晶体的走离角度不同,使得最佳二倍频效率会有明显的不同;不同晶体有效非线性系数的差距会影响达到最佳二倍频效率所需的峰值功率密度。所以,选取10 mm长的LBO、KDP和KTP三种常用二倍频晶体,采用1064 nm基频光在晶体内聚焦束腰100 μm的条件,不同二倍频晶体参数见表1,倍频效率和峰值功率密度的关系如图6(a)所示,随着峰值功率密度的增加,二倍频效率都是先增加后达到饱和。不同晶体由于非线性系数、折射率、走离角度等参数等存在差异,二倍频效率的增长趋势和最佳值也有着明显的差异,10 mm的KDP、LBO和KTP理想二倍频效率分别为76%、85%和93%。
表 1 二倍频晶体参数对比
Table 1. Comparison of SHG crystal parameters
Frequency-doubling crystal deff/m·V−1 1 064 nm refractive index Walk-off angle/mrad Crystal cutting angle Negative biaxial crystal LBO 0.821 4×10−12 1.626 7.03 θ=90°,φ=11.6° Biaxial crystal KTP 3.58×10−12 1.74 3.16 θ=90°,φ=23.5° Negative uniaxial crystal KDP 0.469×10−12 1.513 11.6 - 为了满足最佳相位匹配条件,在不同波长泵浦时,需要采用不同切割角度的二倍频晶体;不同波长的激光在晶体内的折射率不同,也会对倍频效率产生影响。因此,将晶体切割角度和折射率对二倍频效率的影响转换成波长对二倍频效率的影响来更加直观地分析。针对表2中所述600、800、1064 nm三种波段的激光泵浦10 mm长的LBO晶体,LBO选用各自波长最佳相位匹配角度,泵浦参数条件为基频光在晶体内聚焦束腰100 μm,二倍频效率和峰值功率密度的关系如图6(b)所示。随着不同波长激光泵浦功率密度的增加,二倍频效率均呈现增长逐渐变缓直至饱和的趋势。600、800、1064 nm激光在二倍频效率饱和之后的效率分别为66%、67%、85%,不同波长最佳二倍频效率存在差异是由于不同泵浦波长对应的二倍频晶体最佳切割角度不同,不同的切割角度使激光在晶体内部的走离角度有较大差异,进而导致所能达到的最佳二倍频效率不同。
表 2 泵浦与LBO晶体参数对比
Table 2. Comparison of pump and LBO crystal parameters
Negative biaxial crystal LBO deff/m·V−1 Refractive index Walk-off angle/mrad Crystal cutting angle Pump of 1 064 nm laser 0.821 4×10−12 1.626 7.03 θ=90°, φ=11.60° Pump of 800 nm laser 0.748 3×10−12 1.632 16.55 θ=90°, φ=31.65° Pump of 600 nm laser 0.424 2×10−12 1.646 16 θ=90°, φ=62.60° 图 6 二倍频效率随峰值功率密度的变化。(a)不同二倍频晶体;(b) LBO晶体不同泵浦波长
Figure 6. Changes in SHG efficiency and peak power density. (a) Different SHG crystals; (b) Different pump wavelengths of LBO crystal
晶体长度是影响二倍频效率的核心因素之一,为了探究二倍频晶体LBO长度对二倍频转换效率的影响,针对5、7、10、20 mm长度的LBO晶体,调整泵浦参数为1064 nm激光在晶体内束腰200 μm的条件,分析二倍频效率随泵浦光脉冲能量的变化情况,如图7(a)所示。可以发现晶体长度对二倍频效率有非常明显的影响,更长的二倍频晶体虽然会导致最佳二倍频效率的下降,但是可以实现更低泵浦能量下的高效二倍频。
为了进一步探究不同非线性晶体长度对二倍频效率的影响,针对KDP、LBO和KTP三种二倍频晶体,调整1064 nm泵浦光参数为功率密度1 GW/cm2且在二倍频晶体内聚焦束腰100 μm的条件,分析二倍频效率随晶体长度的变化关系,结果如图7(b)所示。相对最大的非线性系数使KTP晶体可以在3 mm长度就达到最佳二倍频效率,之后随着晶体长度的增加,走离效应的影响也逐渐增加,导致KTP二倍频效率缓慢降低。较小的非线性系数使得LBO晶体的二倍频效率增长较为缓慢,在11 mm长度达到最佳二倍频效率,随后二倍频效率同样逐渐降低,由于LBO晶体7.03 mrad的走离角大于KTP晶体3.16 mrad的走离角,LBO的最佳二倍频效率较低且之后二倍频效率降低幅度更大。受限于相对最小的非线性系数,KDP晶体的二倍频效率增长最为缓慢,15 mm长度才能达到最佳二倍频效率,而11.6 mrad这一相对最大的走离角度导致KDP晶体的最佳二倍频效率最低,并且严重的走离效应使得KDP晶体在大于20 mm之后在空间上完全相位失配,无法再直接计算其二倍频效率。受到空间走离距离不同的影响,文中预设条件下的KDP、LBO和KTP晶体理论最佳二倍频效率分别是58%、80%和97%,差距显著。该模型通过对有效非线性系数和走离角度的分析,可以便捷选出所需二倍频晶体的最佳长度。
图 7 (a) LBO晶体不同长度对应的二倍频效率随能量变化;(b)二倍频效率随晶体长度的变化
Figure 7. (a) The variation of SHG efficiency with pump laser energy for different LBO lengths; (b) The variation of SHG efficiency with crystal length
该模型对影响二倍频效率的多种重要因素进行了综合考量与分析,可以从整体角度对最佳二倍频晶体的选择提供帮助。在选择合适的二倍频晶体过程中通常需要考虑追求的指标是转换效率、最大输出能量还是输出波长,并且会受到泵浦光参数的限制。如果只是追求更高的转换效率,则只需要将不同非线性晶体的非线性系数和走离角度带入本模型进行对比分析,即可选出最合适的二倍频晶体,如图6(a)所示。如果追求更高的能量,那么需要在不同非线性晶体的损伤阈值极限能量时,代入所提模型去分析其二倍频效率(为了能够达到损伤阈值极限,二倍频晶体需要进行布儒斯特角切割而不是镀膜),最后计算即可得出在不同条件下能够产生最大二倍频光能量的非线性晶体。如果有特定的输出波长要求,则需要先选择符合该波长变换范围的非线性晶体,再将这些非线性晶体代入所提模型,对比其他参数优劣来做出选择,如图6(b)所示。除此之外,该模型可以灵活地调整泵浦光各项参数,如图4和图5所示,从而实现对非线性晶体二倍频效率的精准预测。
SHG efficiency of nonlinear crystal walk-off effect
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摘要: 应用非线性频率变换技术的短波固体激光器在先进研究、生物医疗和工业生产等领域广泛应用,而走离效应是影响非线性变换效率的关键因素。为了实现更高效率的二倍频转换,各种减小走离效应的结构优化方案被提出,多种二倍频效率模型被建立,但是这些模型在普适性和影响因素全面性上还有所欠缺。文中在理想二倍频效率模型的基础上,提出一种非线性晶体走离效应下的二倍频效率模型,对于空间走离过程进行了更为充分的研究,将二倍频过程的多种因素都进行了细化分析。该模型有两个优势:一方面可以从多种参数的角度更精确地对平行传输光束与聚焦光束的二倍频效率分别进行预测;另一方面可以实现最佳二倍频晶体种类和最佳二倍频晶体长度的选择。Abstract:
Objective Visible and ultraviolet laser output can be realized by solid-state lasers using second-harmonic generation (SHG) technology, which have a wide range of applications in many fields, including atmospheric exploration, biomedicine and industrial processing. The spatial walk-off effect that occurs inside the crystal in the SHG process can cause the spatial phase mismatch between the fundamental laser and the frequency-doubling laser, which lead to obvious loss of SHG efficiency restricting the SHG conversion efficiency. In order to further optimize the frequency doubling efficiency, the SHG theoretical model combined with influencing factors including the spatial walk-off effect needs to be improved. Therefore, combined with the spatial walk-off effect and SHG laser analysis, a hybrid model is proposed to study the parameters in the conversion process of the internal walk-off effect of nonlinear crystals. In this model, the influence of various parameters such as SHG crystal length, laser power density, SHG crystal type, laser wavelength on the SHG efficiency are carefully analyzed, which is of great significance for the optimization of SHG structure and the improvement of conversion efficiency. Methods Based on the ideal SHG conversion efficiency model, this study discussed the energy conversion and beam separation in the process of space walk-off effect, and establishes the SHG conversion efficiency model of two different pump structures, including focused beam and parallel transmission beam (Fig.1-2). The specific effects of various parameter changes of SHG crystals and fundamental laser have been discussed. Results and Discussions According to the established frequency-doubling efficiency model, the influence of the departure effect on the frequency-doubling efficiency has been discussed (Fig.3). The frequency-doubling efficiency was analyzed by using the fundamental frequency optical parameters of fundamental frequency light, including beam quality, beam radius, focal length of focusing lens, beam waist size and divergence angle (Fig.4-5). The frequency-doubling efficiency was analyzed by taking the crystal parameters of frequency-doubling crystal type and crystal cutting angle as variables, and it was found that the theoretical optimal frequency-doubling efficiency of KDP, LBO and KTP crystals was 58%, 80% and 97% (Fig.6). In addition, the influence of frequency-doubling crystal length on frequency-doubling efficiency is studied, and the optimal length selection model of frequency-doubling crystal is obtained (Fig.7). Conclusions A SHG efficiency model of nonlinear crystals with the walk-off effect is established. Aiming at the two different situations of parallel transmission pump beam and focused pump beam, the nonlinear transformation efficiency model of the SHG process is improved through the simulation analysis of the spatial walk-off effect, and the influence of various pump parameters and crystal parameters changes on the SHG efficiency has been discussed. The theoretical optimal SHG efficiency of KDP, LBO and KTP crystals is 58%, 80% and 97% due to the influence of different spatial walk-off distances. Compared with the traditional SHG efficiency model, this model has a more complete study of the spatial walk-off process, and comprehensively refines the analysis of various factors in the SHG process, which can more accurately predict the change of SHG efficiency. It would be helpful to achieve SHG efficiency optimization by adjusting pump parameters and crystal parameters in the future. The model can also be used to realize the selection of the optimal length of SHG crystal and the best SHG crystal type in different conditions, which can be applied to the nonlinear crystal selection process of scientific research and commercial short-wave solid-state lasers to achieve conversion efficiency improvement and cost reduction. -
Key words:
- nonlinear frequency conversion /
- walk-off effect /
- SHG crystal /
- SHG efficiency
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图 3 (a)二倍频光路图;(b)有无走离效应时,二倍频效率分别随泵浦光脉冲能量的变化;(c)聚焦光束和平行传输光束,二倍频效率分别随泵浦光脉冲能量的变化
Figure 3. (a) SHG optical path; (b) The SHG efficiency varies with the energy of the pump laser, with or without the walk-off effect; (c) The SHG efficiency of focused and parallel beams varies with the energy of the pump laser
表 1 二倍频晶体参数对比
Table 1. Comparison of SHG crystal parameters
Frequency-doubling crystal deff/m·V−1 1 064 nm refractive index Walk-off angle/mrad Crystal cutting angle Negative biaxial crystal LBO 0.821 4×10−12 1.626 7.03 θ=90°,φ=11.6° Biaxial crystal KTP 3.58×10−12 1.74 3.16 θ=90°,φ=23.5° Negative uniaxial crystal KDP 0.469×10−12 1.513 11.6 - 表 2 泵浦与LBO晶体参数对比
Table 2. Comparison of pump and LBO crystal parameters
Negative biaxial crystal LBO deff/m·V−1 Refractive index Walk-off angle/mrad Crystal cutting angle Pump of 1 064 nm laser 0.821 4×10−12 1.626 7.03 θ=90°, φ=11.60° Pump of 800 nm laser 0.748 3×10−12 1.632 16.55 θ=90°, φ=31.65° Pump of 600 nm laser 0.424 2×10−12 1.646 16 θ=90°, φ=62.60° -
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