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Zernike多项式是定义在单位圆域上的正交多项式,因为它易与Seidel像差建立联系的特点,常被用作基底函数系对光束的波前像差进行拟合[18]。比如在激光系统中,利用Zernike多项式可以描述因腔镜失调、面形误差以及激光窗口热透镜效应等导致的波前畸变。Zernike多项式表示为[19]:
$ \left\{ \begin{array}{l} {Z_{{k_{\rm{e}}}}}(\rho ,\theta ) = \sqrt {2(n + 1)} {R_{n,m}}(\rho )\cos (m\theta ),(m > 0)\\ {Z_{{k_{\rm{o}}}}}(\rho ,\theta ) = - \sqrt {2(n + 1)} {R_{n,m}}(\rho )\sin (m\theta ),(m < 0)\\ {Z_k}(\rho ,\theta ) = \sqrt {n + 1} {R_{n,0}}(\rho ),(m = 0) \end{array} \right. $
(1) 式中,Z代表Zernike多项式,k为多项式阶数,下标ke代表k为偶数,下标ko代表k为奇数;n为径向自由度;m为角向频率,变化范围为-n, -n+2, …,n-2,n;ρ为单位圆上的半径,取值范围为[0, 1];极角θ的取值范围为[0,2π]。径向多项式Rn, m定义为:
$ \begin{array}{*{20}{c}} {{R_{n,m}}(\rho ) = }\\ {\sum\limits_{s = 0}^{\left( {n - \left| m \right|} \right)/2} {\frac{{{{( - 1)}^s}(n - s)!{\rho ^{n - 2s}}}}{{s!\left[ {\frac{{n + |m|}}{2} - s} \right]!\left[ {\frac{{n - |m|}}{2} - s} \right]!}}} } \end{array} $
(2) 基于Zernike多项式,则飞秒脉冲的波前像差φ(ω, r, θ)可以表示为[20]:
$ \varphi (\omega ,r,\theta ) = \sum\limits_{k = 1}^\infty {{c_k}} (\omega ){Z_k}(r/R,\theta ) $
(3) 式中,ω为激光角频率,r为径向坐标,ck为模式系数,R为激光光斑半径。由于飞秒脉冲的宽光谱特性,每个光谱元对应的模式系数可以表示为[20]:
$ {c_k}(\omega ) = {c_k}\left( {{\omega _0}} \right)\frac{\omega }{{{\omega _0}}} $
(4) 式中,ω0代表激光的中心频率。各阶Zernike多项式及对应的波前像差如表 1所示[19]。
Table 1. Zernike polynomial and the corresponding primary aberration
k n m Zk(ρ, θ) aberration 4 2 0 $\sqrt{3}$(2 ρ2-1) defocus 5 2 -2 $\sqrt{6}$ ρ2sin(2 θ) astigmatism 6 2 2 $\sqrt{6}$ ρ2cos(2 θ) astigmatism 7 3 -1 $\sqrt{8}$(3ρ2-2ρ)sin θ coma 8 3 1 $\sqrt{8}$(3ρ2-2ρ)cos θ coma 9 3 -3 $\sqrt{8}$ ρ3sin(3 θ) trefoil 10 3 3 $\sqrt{8}$ ρ3cos(3 θ) trefoil 11 4 0 $\sqrt{5}$ (6ρ4-6ρ2+1) spherical -
由于飞秒脉冲具有极高的峰值功率,因此对飞秒脉冲聚焦通常采用反射式的离轴抛物面镜进行。离轴抛物面镜无色散、色差,同时消除了光学非线性效应对超短脉冲激光的影响;而当入射光与光轴严格平行时,反射光可无球差的聚焦于焦点。离轴抛物面镜的面形误差以及调节精度对飞秒脉冲的聚焦特性具有重要影响[21],在本文中将离轴抛物面镜等效于理想的消色差透镜,从而可以利用瑞利-索末菲标量衍射理论重点讨论不同像差对飞秒脉冲聚焦特性的影响。
考虑两种情况:(1)入射飞秒脉冲具有均匀的强度分布;(2)入射脉冲具有高斯强度分布。这两种情况下的入射脉冲电场频域复振幅E可以分别表示为[21]:
$ \begin{array}{*{20}{c}} {E(\omega ,x,y,z = 0) = {E_0}\exp \left[ { - \frac{{{\tau ^2}{{\left( {\omega - {\omega _0}} \right)}^2}}}{4}} \right] \times }\\ {u(x,y)\exp [{\rm{i}}\varphi (\omega ,x,y,z = 0)]} \end{array} $
(5) $ \begin{array}{*{20}{c}} {E(\omega ,x,y,z = 0) = }\\ {{E_0}\exp \left[ { - \frac{{{\tau ^2}{{\left( {\omega - {\omega _0}} \right)}^2}}}{4}} \right]\exp \left( { - \frac{{{x^2} + {y^2}}}{{{R^2}}}} \right) \times }\\ {\exp [{\rm{i}}\varphi (\omega ,x,y,z = 0)]} \end{array} $
(6) 其中,
$ u(x,y) = \left\{ {\begin{array}{*{20}{l}} {1,\left( {\sqrt {{x^2} + {y^2}} \le R} \right)}\\ {0,\left( {\sqrt {{x^2} + {y^2}} > R} \right)} \end{array}} \right. $
(7) 式中, E0为常数,x, y, z为空间坐标,τ为脉冲持续时间(1/e2半宽),ω0为中心频率,R为光斑半径(高斯强度分布下为1/e2半径),φ为波前像差。对(5)式和(6)式进行傅里叶逆变换,可得入射脉冲的时域形状。模拟表明,波前像差φ(ω, x, y, z=0)项对飞秒脉冲时域分布的影响几乎可以忽略,故聚焦前脉冲的持续时间可直接对(5)式和(6)式中的exp[-τ2(ω-ω0)2/4]项进行傅里叶逆变换求解,其傅里叶逆变换为exp[-t2/τ2],式中t为时间,因此时域强度分布为exp[-2t2/τ2],可见τ即为脉冲持续时间(1/e2半宽)。根据瑞利-索末菲标量衍射理论,聚焦后飞秒脉冲场演化可以表示为[20]:
$ \begin{array}{*{20}{c}} {E\left( {\omega ,{x^\prime },{y^\prime },z} \right) = \frac{1}{{{\rm{i}}\lambda }}\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } E } (\omega ,x,y,z = 0) \times }\\ {\exp \left[ { - \frac{{{\rm{i}}k\left( {{x^2} + {y^2}} \right)}}{{2f}}} \right]\frac{{\exp \left( {{\rm{i}}k{r_{\rm{d}}}} \right)}}{{{r_{\rm{d}}}}}\cos {\theta _{\rm{d}}}{\rm{d}}x{\rm{d}}y} \end{array} $
(8) 式中,(x′, y′)为传输距离z处的空间坐标,f为抛物面镜焦距,λ为激光波长,k为激光波数(2π/λ),rd为衍射距离,θd为衍射角,它们分别表示为:
$ {r_{\rm{d}}} = \sqrt {{{\left( {{x^\prime } - x} \right)}^2} + {{\left( {{y^\prime } - y} \right)}^2} + {z^2}} $
(9) $ \cos {\theta _{\rm{d}}} = z/{r_{\rm{d}}} $
(10) 对(8)式进行数值求解,可求得衍射距离z处的频域复振幅分布E(ω, x′, y′, z),再对其进行傅里叶逆变换,可得时域电场:
$ \begin{array}{*{20}{c}} {E\left( {t,{x^\prime },{y^\prime },z} \right) = }\\ {\frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} }}\int_{ - \infty }^{ + \infty } E \left( {\omega ,{x^\prime },{y^\prime },z} \right)\exp ( - {\rm{i}}\omega t){\rm{d}}\omega } \end{array} $
(11) 因此, 衍射距离z处在一个光周期T的平均强度分布为:
$ I\left( {{x^\prime },{y^\prime },z} \right) = \frac{1}{T}\int_T {{{\left| {E\left( {t,{x^\prime },{y^\prime },z} \right)} \right|}^2}} {\rm{d}}t $
(12) 而I(x′, y′, z)正是所关心的聚焦强度分布。特别的,当z=f时,即为焦平面处的强度分布。
波前像差对超短飞秒激光脉冲聚焦特性的影响
Effect of wavefront aberration on focusing characterisitics of ultrashort femtosecond laser pulses
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摘要: 为了研究波前像差对超短飞秒激光脉冲聚焦特性的影响, 基于瑞利-索末菲标量衍射理论, 对比研究了在散焦、像散、慧差、三叶形像差以及球差等各类波前像差下, 均匀强度分布和高斯强度分布的超短飞秒激光脉冲的聚焦特性。结果表明, 波前像差对均匀强度分布的飞秒脉冲在焦平面处的光强分布具有明显的不利影响, 从而降低飞秒脉冲的聚焦峰值功率, 而对高斯强度分布的飞秒脉冲影响相对较小, 即在高斯强度分布下, 在焦平面处仍然有可能获得较好的、近衍射极限的聚焦光斑; 在非焦平面处, 即使初始脉冲具有高斯强度分布, 非焦平面处的光强分布受各类波前像差的影响也较为明显; 对于所研究的30fs(1/e2半宽度)超短脉冲, 波前像差对脉冲持续时间的影响几乎可以忽略。此研究结果对超短飞秒激光束的光束质量评估及聚焦特性分析具有实际的指导意义。Abstract: In order to study effect of wavefront aberration on focusing characteristics of ultrashort femtosecond laser pulses, based on Rayleigh-Sommerfeld scalar diffraction theory, focusing characteristics of ultrashort femtosecond laser pulses with uniform intensity distribution and Gaussian intensity distribution were compared and studied under different wavefront aberration, such as defocusing, astigmatism, coma aberration, trilobe aberration and spherical aberration. The results show that, wavefront aberration has an obvious adverse effect on intensity distribution of femtosecond pulses with uniform intensity distribution at the focal plane. The peak power of focusing femtosecond pulses is reduced. However, the influence of femtosecond pulses with Gaussian intensity distribution is relatively small. Under Gaussian intensity distribution, it is still possible to obtain better focusing spot near diffraction limit at focal plane. At the non-focal plane, even if the initial pulse has Gaussian intensity distribution, intensity distribution at the non-focal plane is also affected by various wavefront aberration. For 30fs (1/e2 half width) ultrashort pulse in this paper, the effect of wavefront aberration on pulse duration is almost negligible. The results are of practical significance for the evaluation of beam quality and the analysis of focusing characteristics of ultrashort femtosecond laser beams.
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Key words:
- ultrafast optics /
- focusing /
- wavefront aberration /
- pulse duration /
- diffraction theory
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Table 1. Zernike polynomial and the corresponding primary aberration
k n m Zk(ρ, θ) aberration 4 2 0 $\sqrt{3}$(2 ρ2-1) defocus 5 2 -2 $\sqrt{6}$ ρ2sin(2 θ) astigmatism 6 2 2 $\sqrt{6}$ ρ2cos(2 θ) astigmatism 7 3 -1 $\sqrt{8}$(3ρ2-2ρ)sin θ coma 8 3 1 $\sqrt{8}$(3ρ2-2ρ)cos θ coma 9 3 -3 $\sqrt{8}$ ρ3sin(3 θ) trefoil 10 3 3 $\sqrt{8}$ ρ3cos(3 θ) trefoil 11 4 0 $\sqrt{5}$ (6ρ4-6ρ2+1) spherical -
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