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由于晶体分为七大晶系32个点群,描述其三阶非线性极化率张量χ(3)比较复杂[33]。因此,绝大多数研究集中在最简单的各向同性和立方对称性晶体材料。此节将介绍任意偏振光场通过厚/薄的三阶非线性光学介质的传播方程,给出各向同性/各向异性非线性折射率和双光子吸收系数。
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对于任意偏振的光矢量,包括标量光场和矢量光场,都可以分解为左右旋圆偏振光线性叠加的形式:
$$\vec E = \left( {\begin{array}{*{20}{c}} {{E_ + }{{\vec \sigma }_ + }} \\ {{E_ - }{{\vec \sigma }_ - }} \end{array}} \right)$$ (1) 式中:
${{\vec \sigma }_ + }$ 和${{\vec \sigma }_ - } $ 分别为左旋和右旋圆偏振基矢。对于任意偏振光在三阶非线性光学介质中传播时,利用缓变包络近似,可以得到介质中两个圆偏振光的耦合非线性薛定谔方程为[34]:
$$\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\dfrac{{\partial {E_ \pm }}}{{\partial r}}} \right) - 2i{\kappa _ \pm }\dfrac{{\partial {E_ \pm }}}{{\partial {\textit{z}}}} - \kappa _ \pm ^2{E_ \pm } + \varepsilon _ \pm ^{{\rm{eff}}}{\kappa ^2}{E_ \pm } = 0$$ (2) 式中:
$\varepsilon _ \pm ^{{\rm{eff}}} = 1 + 4\pi {\chi ^{(1)}} + 4\pi \chi _ \pm ^{{\rm{NL}}}$ 为有效极化率,${\chi ^{(1)}}$ 为线性极化率,$\chi _{ \pm }^{{\rm{NL}}}$ 为三阶非线性极化率;${\kappa _ \pm } = {n_{0, \;\pm }}\kappa = {n_0}\omega /c$ 为两圆偏振分量的波矢,${n_0}$ 为线性折射率,$\omega $ 为光的圆频率,$c$ 为真空中的光速。 -
假设非线性光学样品足够薄,使得样品内由于衍射或折射率改变对光束尺寸的影响可以忽略不计。在这种情况下,由公式(2)可得分别描述光强和相位的两个方程:
$$\dfrac{{\partial I}}{{\partial {\textit{z}}}} = - {\alpha _0}I - {\alpha _2}{I^2}$$ (3) $$\dfrac{{\partial \Delta {\phi _ \pm }}}{{\partial {\textit{z}}}} = k{n_{2, \pm }}I$$ (4) 式中:
${\alpha _0}$ 为线性吸收系数;${\alpha _2}$ 为双光子吸收系数;${n_{2, \pm }}$ 为与左右旋圆偏振相关的非线性折射率;$k = 2\pi /\lambda $ 为波矢,$\lambda $ 为光的波长。需要强调的是公式(2)~(4)适用于描述各向同性和立方对称性晶体材料,也就是说样品具有各向同性的线性光学系数(
${\alpha _0}$ 和${n_0}$ ),而三阶非线性光学系数可以是各向同性的或者是各向异性的。${n_{2, \pm }}$ 和${\alpha _2}$ 分别与三阶非线性极化率${\chi ^{(3)}}$ 的实部和虚部相关。通常,三阶非线性极化率系数
$\chi _{ijkl}^{(3)}$ 是一个四阶张量,可以用81个张量元来描述。对于空间高度对称的晶体来说,这种独立张量元的数量将大幅减少。以下将分别介绍各向同性和各向异性三阶非线性光学系数。 -
对于各向同性的三阶非线性光学介质,基于左右旋圆偏振正交基矢,可以获得与左右旋分量相关的各向同性非线性折射率
$n_{{\rm{2,}} \pm }^{{\rm{iso}}}$ [27]和各向同性双光子吸收系数$\alpha _{\rm{2}}^{{\rm{iso}}}$ [35-36]分别为:$$n_{2, \pm }^{{\rm{iso}}} = \frac{3}{{{\varepsilon _0}cn_0^2}}\left[ {2{\rm{Re}} (A) + {\rm{Re}} (B)\frac{{{{(1 \mp e)}^2}}}{{1 + {e^2}}}} \right]$$ (5) $$\alpha _2^{{\rm{iso}}} = \frac{{12\pi }}{{{\varepsilon _0}cn_0^2\lambda }}\left[ {2{\rm{Im}} (A) + {\rm{Im}} (B){{\left( {\frac{{1 - {e^2}}}{{1 + {e^2}}}} \right)}^2}} \right]$$ (6) 式中:
${\varepsilon _0}$ 为真空中的介电常数;$e = (|{E_ + }| - |{E_ - }|)/ (|{E_ + }| + |{E_ - }|)$ 为偏振椭圆的椭偏率;$A = \chi _{{\rm{1122}}}^{{\rm{(3)}}}$ 和$B = \chi _{{\rm{1221}}}^{{\rm{(3)}}}$ 为各向同性介质中三阶非线性极化率的两个独立张量元。对各向同性非线性,满足$\chi _{{\rm{1111}}}^{{\rm{(3)}}} = 2\chi _{{\rm{1122}}}^{{\rm{(3)}}} + \chi _{{\rm{1221}}}^{{\rm{(3)}}}$ 。根据产生光学非线性的物理机理不同,A与B的相对大小也不尽相同。具体来说,在分子取向非线性时,$B/A = 6$ ;在非共振的电子非线性情况下,$B/A = 1$ ;在电致伸缩或热致非线性时,$B/A = 0$ [33]。可以看出,各向同性三阶非线性光学系数与光场的椭偏率
$e$ 密切相关。公式(5)和(6)的特例如下:对圆偏振光($e = + 1$ 和−1分别对应左旋和右旋圆偏振光),非线性折射率和双光子吸收系数分别为$n_{2, \pm }^{{\rm{iso,cir}}} = 6{\rm{Re}} (A)/({\varepsilon _0}cn_0^2)$ 和$\alpha _2^{{\rm{iso,cir}}} = 24\pi {\rm{Im}} (A)/\left({\varepsilon _0}cn_0^2\lambda \right )$ ;对$e = 0$ 的线偏振光,可得$n_2^{\rm{0}} = 3{\rm{Re}}\left [\chi _{1111}^{(3)}\right]/({\varepsilon _0}cn_0^2)$ 和$\alpha _2^{\rm{0}} =12\pi {\rm{Im}}\left [\chi _{1111}^{(3)}\right] /({\varepsilon _0}cn_0^2\lambda )$ 。 -
为简化起见,以立方对称性晶体(例如BaF2和ZnSe)为例研究任意椭圆偏振光激发下的各向异性三阶非线性光学效应。此时三阶非线性极化率
${\chi ^{(3)}}$ 只有三个独立分量$\chi _{{\rm{1111}}}^{{\rm{(3)}}}$ 、$\chi _{{\rm{1122}}}^{{\rm{(3)}}}$ 和$\chi _{{\rm{1221}}}^{{\rm{(3)}}}$ 。接下来,假定任意光束正入射至薄晶体表面,并且沿[001]晶轴传播,晶体的取向角$\theta $ 是偏振椭圆的长半轴与[100]晶轴之间的夹角。在这种特定的情况下,可得到与左旋和右旋相关的各向异性三阶非线性折射率为[29]:$$n_{2, \pm }^{{\rm{ani}}} = n_2^0\left[ {1 \mp \frac{{2e\delta }}{{1 + {e^2}}} - \frac{\sigma }{2}{{\sin }^2}(2\theta )\frac{{{{(1 \mp e)}^2}}}{{1 + {e^2}}}} \right]$$ (7) 各向异性双光子吸收系数为[32]:
$$\alpha _2^{{\rm{ani}}} = \alpha _2^0\left[ {1 - \frac{{4{e^2}}}{{{{(1 + {e^2})}^2}}}\delta - \frac{1}{2}\sigma {{\sin }^2}(2\theta ){{\left( {\frac{{1 - {e^2}}}{{1 + {e^2}}}} \right)}^2}} \right]$$ (8) 其中,各向异性系数
$\sigma $ 和二向色性系数$\delta $ 分别为:$$\sigma = \frac{{\chi _{1111}^{(3)} - \chi _{1122}^{(3)} - 2\chi _{1221}^{(3)}}}{{\chi _{1111}^{(3)}}}$$ (9) $$\delta = \frac{{\chi _{1111}^{(3)} + \chi _{1122}^{(3)} - 2\chi _{1221}^{(3)}}}{{2\chi _{1111}^{(3)}}}$$ (10) 式中:
$n_2^{\rm{0}}$ 和$\alpha _2^{\rm{0}}$ 分别为线偏振光的电场矢量平行于[100]晶轴时的三阶非线性折射率和双光子吸收系数。有趣的是公式(7)和(8)有三种特殊情况:线偏振光(
$e = 0$ )时,可得$n_{\rm{2}}^{{\rm{lin}}} = n_2^{\rm{0}}[1 - \sigma {\sin ^2}(2\theta )/2]$ 和$\alpha _{\rm{2}}^{{\rm{lin}}} = \alpha _2^{\rm{0}}[1 - \sigma {\sin ^2}(2\theta )/2]$ [18];圆偏振光(即$e = \pm 1$ )时,可得$n_{\rm{2}}^{{\rm{cir}}} = n_2^{\rm{0}}(1 - \delta )$ 和$\alpha _{\rm{2}}^{{\rm{cir}}} = \alpha _2^{\rm{0}}(1 - \delta )$ [37];当材料具有各向同性的三阶非线性折射和双光子吸收(即$\sigma = 0$ )时,公式(7)和(8)分别简化为公式(5)和(6)[35, 37]。 -
为了表征材料的三阶非线性光学系数,人们广泛采用单光束Z-扫描技术,该技术具有实验简单、测量灵敏度高、可同时测量
$\chi _{1111}^{{\rm{(3)}}}$ 的大小和符号等优点[38]。在过去的近三十年里,传统Z-扫描技术得到了极大的改进,被用来表征三阶非线性极化率${\chi ^{(3)}}$ 的张量元[18, 29, 32, 37]。例如,通过对线偏振光Z-扫描信号与晶体取向之间的依赖关系,可以得到$\chi _{1111}^{{\rm{(3)}}}$ 和各向异性系数$\sigma $ [18];分别用线性偏振光和圆偏振光进行Z-扫描测量,以确定各向同性和立方对称性晶体中${\chi ^{(3)}}$ 的独立张量元的大小和符号[37];用任意椭圆偏振光Z-扫描表征各向异性非线性光学介质中的三阶非线性极化率张量[29, 32]。原理上,采用任意椭圆偏振光束在不同晶体取向角
$\theta $ 下分别进行闭孔和开孔Z-扫描测量,可以获得各向异性非线性光学系数$n_2^0$ 、$\alpha _2^0$ 、$\sigma $ 和$\delta $ 。也可以采用如下简单而有效的方案:首先,测量不同晶体取向角$\theta $ 下的线偏振光($e = 0$ )闭孔和/或开孔Z-扫描曲线;其次,计算出不同$\theta $ 值下的非线性折射率$n_{\rm{2}}^{{\rm{lin}}}$ 和/或双光子吸收系数$\alpha _{\rm{2}}^{{\rm{lin}}}$ ;第三,用$n_{\rm{2}}^{{\rm{lin}}} = n_2^0[1 - \sigma {\sin ^2}(2\theta )/2]$ 和$\alpha _{\rm{2}}^{{\rm{lin}}} = \alpha _2^0[1 - \sigma {\sin ^2}(2\theta )/2]$ 分别拟合$n_{\rm{2}}^{{\rm{lin}}} \sim \theta $ 和$\alpha _{\rm{2}}^{{\rm{lin}}} \sim \theta $ 曲线(如图1所示),获得参数$n_2^0$ 、$\alpha _2^0$ 和$\sigma $ ;第四,开展圆偏振光Z-扫描测量并提取非线性折射率$n_{\rm{2}}^{{\rm{cir}}}$ 和$\alpha _{\rm{2}}^{{\rm{cir}}}$ ;最后,借助于已知的$n_2^0$ 和$\alpha _2^0$ 值,利用$n_{\rm{2}}^{{\rm{cir}}} = n_2^0(1 - \delta )$ 和/或$\alpha _{\rm{2}}^{{\rm{cir}}} = \alpha _2^0(1 - \delta )$ 可得出二向色性系数$\delta $ 。 -
强激光束与物质相互作用导致了多种新颖的非线性光学效应。聚焦激光束提供了强光场。由于非线性光学效应与光场特性有关,因此,有必要首先对聚焦光束的焦场进行研究。此节介绍了径向偏振光、杂化偏振光和庞加莱光束的弱聚焦场,给出了任意矢量光场通过非线性克尔介质后的光场传播理论,讨论了三种矢量光场激发的各向同性/各向异性非线性折射效应。
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此节的目的是给出三种典型矢量光场(即,径向偏振光、杂化偏振光和庞加莱光束,其光强和偏振态分布如图2所示)通过薄透镜弱聚焦后的电场解析表达式,为理解光与物质的相互作用奠定基础。
图 2 (a)径向偏振光;(b)杂化偏振光;(c)柠檬形庞加莱光束及其偏振态分布
Figure 2. (a) Radially polarized beam; (b) Hybridly polarized beam;(c) Lemon Poincaré beam with polarization distributions
基于旁轴近似下的矢量瑞利-索墨菲公式,可得电场为[39]:
$$ \begin{split} \vec E(\rho ,\varphi ,{\textit{z}}) =& \left( {\begin{array}{*{20}{c}} {{E_ + }(\rho ,\varphi ,{\textit{z}}){{\vec \sigma }_ + }} \\ {{E_ - }(\rho ,\varphi ,{\textit{z}}){{\vec \sigma }_ - }} \end{array}} \right) = \dfrac{{ - {\rm{i}}k}}{{2\pi (f + {\textit{z}})}}{e^{ik(f + {\textit{z}})}}\cdot\\ & \displaystyle\int\limits_0^\infty {\int\limits_0^{2\pi } {\left( {\begin{array}{*{20}{c}} {{E_ + }(r,\phi ){{\vec \sigma }_ + }} \\ {{E_ - }(r,\phi ){{\vec \sigma }_ - }} \end{array}} \right)} } \times \exp \left( {\dfrac{{{\rm{i}}k{r^2}}}{{2(f + {\textit{z}})}}} \right)\cdot\\ & \exp \left( { - \dfrac{{{\rm{i}}k\rho r\cos (\phi - \varphi )}}{{f + {\textit{z}}}}} \right)r{\rm{d}}r{\rm{d}}\phi \\ \end{split} $$ (11) 式中:
${E_ \pm }(r,\phi )$ 为入射面电场的左右旋圆偏振分量。注意,坐标原点${\textit{z}} = 0$ 位于透镜的几何焦点处,而位置${\textit{z}} = - f$ 对应于透镜平面处。为了获得薄透镜对矢量光场聚焦的解析表达式,可以利用如下两个积分定理[4, 39]:
$$\int_0^{2\pi } {{e^{ \pm i(m\phi + {\varphi _0})}}\exp [{\rm{i}}x\cos (\phi - \theta )]} {\rm{d}}\phi = 2\pi {i^m}{J_m}(x){e^{ \pm i(m\theta + {\varphi _0})}}$$ (12) $$\int_0^\infty {{r^2}{e^{ \!- \!\beta {r^2}}}{J_m}(\gamma r){\rm{d}}r} \! =\! \dfrac{{\sqrt \pi }}{{4{\beta ^{3/2}}}}{e^{\! - \!t}}\left[ {2t{I_{(m \!-\! 2)/2}}(t)\!-\! (m \!- 1 + 2t){{\rm{I}}_{{\rm{m}}/2}}(t)} \right]$$ (13) 式中:
${J_m}( \cdot )$ 为第一类m阶贝塞尔函数;${\rm{Re}} [m] > - 1$ ;${\rm{Re}} [\beta ] > 0$ ;$\gamma > 0$ ;$t = {\gamma ^2}/(8\beta )$ ;${{\rm{I_m}}}( \cdot )$ 为修正的m阶贝塞尔函数。 -
对于径向偏振光,其入射平面的电场可以写成[40]:
$$\vec E(r,\phi ) = \left( {\begin{array}{*{20}{c}} {{E_ + }(r,\phi ){{\vec \sigma }_ + }} \\ {{E_ - }(r,\phi ){{\vec \sigma }_ - }} \end{array}} \right) = \frac{{A(r)}}{{\sqrt 2 }}\left( {\begin{array}{*{20}{c}} {{e^{ - i\phi }}{{\vec \sigma }_ + }} \\ {{e^{ + i\phi }}{{\vec \sigma }_ - }} \end{array}} \right)$$ (14) 式中:
$A(r)$ 为径向依赖的矢量光场横截面的振幅分布。为简化起见,笔者把径向偏振光和杂化偏振光被薄透镜聚焦时的振幅取为最低阶拉盖尔-高斯光束$A(r) \propto r/w \cdot \exp [ - {r^2}/{w^2} - ik{r^2}/(2f)]$ ,其中,$w$ 为输入本征高斯光束的半径,$f$ 为薄透镜的焦距。将公式(14)代入到公式(11),利用公式(12)和(13)的积分定理,可得径向偏振光的焦场为[28, 40]:
$$\!\vec E(\rho ,\varphi ,{\textit{z}}) \!= \!\!\left( {\begin{array}{*{20}{c}} {{\!\!\!E_ + }{{\vec \sigma }_ + }}\!\!\! \\ {{\!\!\!E_ - }{{\vec \sigma }_ - }} \!\! \end{array}} \right) = \sqrt {\dfrac{{8\eta }}{\pi }} \dfrac{{{E_0}{g_1}f}}{{{{\text{μ}} ^{3/2}}{w^3}(f + {\textit{z}})}}{e^{ik(f + {\textit{z}}) - 2\eta }}\left( {\begin{array}{*{20}{c}} {{\!\!\!e^{ - i\varphi }}{{\vec \sigma }_ + }}\! \!\!\!\\ {{\!\!\!e^{ + i\varphi }}{{\vec \sigma }_ - }}\!\! \!\! \end{array}} \right)$$ (15) 式中:
$\mu = 1/{w^2} + ik{\textit{z}}/[2f(f + {\textit{z}})]$ ;$\eta = {k^2}{r^2}/[8 \mu {(f + {\textit{z}})^2}]$ ;${E_0}$ 为矢量光场在焦点处的峰值电场振幅;${g_1} = 2.066\;37$ 为归一化常数。正如公式(15)所描述的,聚焦的径向偏振光具有局域线偏振,其光场分布在自由空间任意传播位置均保持初始偏振态分布。通过公式(15),聚焦的径向偏振光的光强可写为:
$$I(\rho ,{\textit{z}}) = \dfrac{{8{I_0}g_1^2{f^2}\sqrt {\eta {\eta ^*}} }}{{\pi {{\left| \mu \right|}^3}{w^6}{{(f + {\textit{z}})}^2}}}{e^{ - 2\eta - 2{\eta ^*}}}$$ (16) 式中:星号*为复数的复共轭;
${I_0} = \varepsilon /(\sqrt \pi g_1^2\omega _0^2\tau )$ 为焦点处的峰值光强,其中,$\varepsilon $ 为入射的能量,$\tau $ 为激光脉冲持续时间内最大值${e^{ - 1}}$ 处的半宽度,${\omega _0} = \lambda f/(\pi w)$ 为聚焦的矢量光场的腰斑半径。相应地,聚焦光场的瑞利长度为${{\textit{z}}_0} = k\omega _0^2/2$ 。 -
杂化偏振光在
${\textit{z}} = 0$ 平面上的电场分布可以表示为:$$\vec E(r,\phi ) = \left( {\begin{array}{*{20}{c}} {{E_ + }(r,\phi ){{\vec \sigma }_ + }} \\ {{E_ - }(r,\phi ){{\vec \sigma }_ - }} \end{array}} \right) = \frac{{A(r)}}{{\sqrt 2 }}\left( {\begin{array}{*{20}{c}} {({e^{i\phi }} - i{e^{ - i\phi }}){{\vec \sigma }_ + }} \\ {({e^{i\phi }} + i{e^{ - i\phi }}){{\vec \sigma }_ - }} \end{array}} \right)$$ (17) 相应地,可得到沿
$ + {\textit{z}}$ 方向传播的聚焦杂化偏振光的电场分布为[27]:$$\begin{split} \vec E(\rho ,\varphi ,{\textit{z}}) =& \left( {\begin{array}{*{20}{c}} {{E_ + }{{\vec \sigma }_ + }} \\ {{E_ - }{{\vec \sigma }_ - }} \end{array}} \right) = \sqrt {\frac{{2\eta }}{\pi }} \frac{{{E_0}{g_1}f}}{{{{\text{μ}} ^{3/2}}{w^3}(f + {\textit{z}})}}{e^{ik(f + {\textit{z}}) - 2\eta }}\\ &\left( {\begin{array}{*{20}{c}} {({e^{i\varphi }} - i{e^{ - i\varphi }}){{\vec \sigma }_ + }} \\ {({e^{i\varphi }} + i{e^{ - i\varphi }}){{\vec \sigma }_ - }} \end{array}} \right) \end{split}$$ (18) -
对于柠檬形庞加莱光束,考虑到它是由相互正交的左旋高斯光束与右旋最低阶拉盖尔-高斯光束叠加而成,其横向电场表达式可写为[41]:
$$\vec E(r,\phi ) = \left( {\begin{array}{*{20}{c}} {{E_ + }(r,\phi ){{\vec \sigma }_ + }} \\ {{E_ - }(r,\phi ){{\vec \sigma }_ - }} \end{array}} \right) = {A_0}\exp \left( { - \frac{{{r^2}}}{{{w^2}}}} \right)\left( {\begin{array}{*{20}{c}} {{{\vec \sigma }_ + }} \\ {\dfrac{{\sqrt 2 r}}{w}{e^{i\phi }}{{\vec \sigma }_ - }} \end{array}} \right)$$ (19) 式中:
${A_0}$ 为光束的振幅。类似地,在坐标原点为焦平面的情况下,沿着
$ + {\textit{z}}$ 方向传播的柠檬形庞加莱光束经过薄透镜聚焦后,其电场可表示为[42]:$$\vec E(\rho ,\varphi ,{\textit{z}}) \!=\! \left( {\begin{array}{*{20}{c}} {{\!\!\!\!E_ + }{{\vec \sigma }_ + }}\!\!\!\! \\ {{\!\!\!\!E_ - }{{\vec \sigma }_ - }}\!\!\!\! \end{array}} \right) \!=\! \dfrac{{{E_0}f}}{{{w^2}\mu (f + {\textit{z}})}}{e^{ik(f + {\textit{z}}) - 2\eta }}\left( {\begin{array}{*{20}{c}} \!\!{{{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\vec \sigma }_ + }} \\ {\dfrac{{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! k\rho {e^{i(\varphi - \pi /2)}}}}{{\sqrt 2 w\mu (f + {\textit{z}})}}{{\vec \sigma }_ - \!\!\!\!}} \end{array}} \right)\!\!\!$$ (20) 特别是,在焦平面(即
${\textit{z}} = 0$ )处,可得焦场的表达式为:$$\vec E(\rho ,\varphi ,0) = \left( {\begin{array}{*{20}{c}} \!\!{{E_ + }{{\vec \sigma }_ + }} \\ \!\!{{E_ - }{{\vec \sigma }_ - }} \end{array}} \right) \!=\! {E_0}{e^{ikf}}\exp \left( { - \frac{{{\rho ^2}}}{{\omega _0^2}}} \right)\left( {\begin{array}{*{20}{c}} {{{\!\!\!\!\!\vec \sigma }_ + }} \\ {\dfrac{{\!\!\!\!\!\!\sqrt 2 \rho {e^{i(\varphi - \pi /2)}}}}{{{\omega _0}}}{{\vec \sigma }_ - }}\!\!\!\! \end{array}} \right)$$ (21) 此外,经透镜聚焦的庞加莱光束在传输到远场时,光束的束腰半径
${\omega _{\textit{z}}}$ 随传输距离z的变化。因而,在任意平面处的束腰半径可表示为${\omega _{\textit{z}}} = {\omega _0}{(1 + {{\textit{z}}^2}/{\textit{z}}_0^2)^{1/2}}$ 。 -
假设具有任意偏振态分布的激光束在光学薄样品中沿
$ + {\textit{z}}$ 轴传播,该样品的各向同性线性吸收系数为${\alpha _0}$ ,三阶非线性折射率为${n_{2, \pm }}$ 。在薄样品(即$L < < {{\textit{z}}_0}$ ,其中$L$ 是样品的厚度)近似下,根据样品入射面的电场${E_ \pm }(\rho ,\varphi ,{\textit{z}})$ ,可得样品出射面的复电场为:$${\vec E_e}(\rho ,\varphi ,{\textit{z}}) = \left( {\begin{array}{*{20}{c}} {E_ + ^e(\rho ,\varphi ,{\textit{z}}){{\vec \sigma }_ + }} \\ {E_ - ^e(\rho ,\varphi ,{\textit{z}}){{\vec \sigma }_ - }} \end{array}} \right) = {e^{ - {\alpha _0}L/2}}\left( {\begin{array}{*{20}{c}} {{E_ + }{e^{i\Delta {\phi _ + }}}{{\vec \sigma }_ + }} \\ {{E_ - }{e^{i\Delta {\phi _ - }}}{{\vec \sigma }_ - }} \end{array}} \right)$$ (22) 式中:
$\Delta {\phi _ \pm }(\rho ,\varphi ,{\textit{z}}) = k{n_{2, \pm }}I(\rho ,\varphi ,{\textit{z}}){L_{{\rm{eff}}}}$ ,${L_{{\rm{eff}}}} = (1 - {e^{ - {\alpha _0}L}})/ {\alpha _0}$ 。此外,定义了焦点处的峰值非线性折射相移为${\varPhi _ \pm } = k{n_{2, \pm }}{I_0}{L_{{\rm{eff}}}}$ 和$\Delta {\phi _0} = kn_2^0{I_0}{L_{{\rm{eff}}}}$ 。注意,对于各向同性和各向异性非线性折射,三阶非线性折射率${n_{2, \pm }}$ 分别用公式(5)和(7)表示。基于矢量瑞利-索墨菲公式,可得到通过非线性光学介质后的任意矢量光场传播到远场平面的电场表达式为:
$$ \begin{split} {{\vec E}_a}({r_a},\vartheta ,{{d}}) =& \left( {\begin{array}{*{20}{c}} {E_ + ^a{{\vec \sigma }_ + }} \\ {E_ - ^a{{\vec \sigma }_ - }} \end{array}} \right) = - \frac{{ik{e^{ikD}}}}{{2\pi D}}\displaystyle\int\limits_0^\infty \int\limits_0^{2\pi } \left( {\begin{array}{*{20}{c}} {E_ + ^e{{\vec \sigma }_ + }} \\ {E_ - ^e{{\vec \sigma }_ - }} \end{array}} \right)\cdot \\ &\exp \left( {\frac{{{\rm{i}}k{\rho ^2}}}{{2D}}} \right) \exp \left( { - \frac{{{\rm{i}}k{r_a}\rho }}{D}\cos (\varphi - \vartheta )} \right)\rho {\rm{d}}\rho {\rm{d}}\varphi \\ \end{split} $$ (23) 式中:
$D = d - {\textit{z}}$ ,$d$ 为焦平面到远场平面之间的距离。通过电场表达式(23),可以进一步研究非线性光学效应对矢量光场的光强分布、偏振态分布、自旋角动量分布和偏振畸点演化等的影响[27-29, 42]。 -
径向偏振光经各向异性非线性克尔介质后,远场平面的光强和偏振态分布随非线性相移
$\Delta {\phi _0}$ 和各向异性系数$\sigma $ 的变化如图3所示[28]。在各向同性克尔非线性($\sigma = 0$ )的情况下,远场的光强图样呈现出多环结构,并且仍然维持着局域线偏振分布。存在各向异性非线性($\sigma \ne 0$ )时,远场的光强图样由原来圆对称的多环结构逐渐变化为具有四重旋转对称性的花瓣结构。这一现象可以解释如下。径向偏振光与具有各向异性的非线性克尔介质相互作用的过程中,产生了具有四重旋转对称性的额外相移,即$\Delta \phi \propto [1 - \sigma {\sin ^2}(2\theta )/ 2]$ 。这种额外的非线性相移影响着光场的传播行为,最终使得远场的衍射图样由圆对称的多环结构过渡到具有四重旋转对称性的方形结构。通过计算电场的斯托克斯参量,可图示出远场观测面的偏振态分布。可以看出,由各向异性非线性导致的自衍射光强图样,在光场横截面上存在丰富的偏振态(包括线偏振、左旋椭圆偏振以及右旋椭圆偏振),这与各向同性非线性光学效应导致的光强图样中仅仅存在局域线偏振分布完全不同[25]。此外,内环的偏振态分布几乎为线偏振,而外环却呈现杂化偏振分布。总之,利用各向异性折射非线性,可以改变径向偏振光的偏振态和自旋角动量分布[28]。 -
杂化偏振光经各向同性非线性克尔介质调制后,在不同取值下的
$B/A$ 和$\Delta {\phi _0}$ 下,远场的光强和偏振态分布如图4所示[27]。对于$B/A = 0$ 的情况,正如公式(5)所描述的,非线性折射率的左右旋部分与椭偏率无关,也就是说,杂化偏振光入射到各向同性非线性克尔介质后,非线性光学效应不影响入射光的偏振态分布,从而,在传输的过程中,透射光仍然保持着起初的偏振态分布。此外,远场的强度分布呈现为圆对称的多环结构。对于$B/A \ne 0$ 的情况,远场光强不再为圆对称的多环结构,而是呈现类似方形结构,并且存在四重旋转对称性,这种旋转对称是由杂化偏振光的偏振态分布的对称性引起的(见图2(b))。当$B/A = 1$ 时,随着相移$\Delta {\phi _0}$ 的增加,更多的光场能量衍射到外环。然而,当$B/A = 6$ 时,除了衍射光的边缘外,远场的光强分布几乎与$\Delta {\phi _0}$ 无关。在固定$\Delta {\phi _0}$ 为某一定值的情况下,随着$B/A$ 的比值从0逐渐增加到6,更多的光场能量衍射到内环,这与在整个非线性过程中,B的相对贡献量增加有关。远场的空间自相位调制效应可以理解为:具有偏振态结构分布的光场,通过各向同性非线性光学效应后,杂化偏振光会产生如公式(5)所描述的结构相位,这种结构相位对光束的传输行为起到调制作用,从而导致具有结构的远场强度分布。此外,空间局域偏振椭圆的取向角和椭偏率均随着$\Delta {\phi _0}$ 的变化而改变[27]。 -
图5给出了各向异性非线性克尔介质位于透镜焦平面处时,在不同晶体取向角
$\theta $ 情况下,远场观测平面上杂化偏振光的光强图样、偏振态分布和自旋角动量分布[29]。由于光学非线性引起的折射率变化,远场光强图样呈现一个中心暗斑,周围环绕着同心环结构。如图5的第一行所示,由于随方位角变化的椭偏率的杂化偏振光激发各向异性克尔非线性,使得光强具有四重旋转对称的正方形分布。有趣的是,当晶体相对于光场旋转时,即[100]晶轴从平行变为垂直于光场的x轴,更多的光能量被衍射到外环。与远场光强分布类似,远场观测平面上的偏振态分布和自旋角动量分布具有二重旋转对称性。对于$\theta = 0^\circ $ 和$\theta = 45^\circ $ 的特殊情况,在相对于x轴为$\varphi = m \cdot 45^\circ $ ($m = 0, 1, \cdot \cdot \cdot 7$ )的方位角方向上存在三个非线性本征偏振(即线偏振、左旋圆偏振和右旋圆偏振)。当各向异性非线性克尔介质的[100]晶轴平行或者垂直于光场的$x$ 轴时,这些杂化偏振光的局域本征偏振态不会发生变化,并且出射场的偏振态以及自旋角动量与入射光场存在很大的相似性。对于任意的晶体取向(例如$\theta = {30^\circ}$ ),远场的偏振态分布与未激发光学非线性时相比完全不同。这种具有偏振结构的光场与各向异性非线性介质相互作用,由于增加了晶体取向角这一新维度,对远场观测面的杂化偏振光的偏振态以及自旋角动量分布的调控将更加灵活。 -
聚焦的柠檬形庞加莱光束通过各向同性或各向异性非线性克尔介质后,传输到三个典型位置处的光强和偏振态分布如图6所示[42]。其中,非线性克尔介质放置在焦平面
${\textit{z}} = 0$ 处,三个典型的位置分别为$d = {{\textit{z}}_0}$ 、$2{{\textit{z}}_0}$ 和$20{{\textit{z}}_0}$ 。当介质无光学非线性时,庞加莱光束在传输过程中光强分布几乎保持不变,只发生整体的偏振旋转[41]。柠檬形庞加莱光束通过各向同性非线性克尔介质以后,如图6第一行所显示的,由于非线性衍射,在传输以及扩张的过程中,叠加有偏振分布的强度图样仍然保持着最初的圆对称性,其原因是这种结构的光束在各向同性自聚焦非线性介质中能稳定传输[23]。有趣的是,由于各向异性克尔非线性,柠檬形庞加莱光束传输到远场,光强图样由中心亮斑围绕的衍射环变成椭圆结构伴随着两段衍射环(如图6第二行所示)。这种由各向异性光学非线性引起的特殊的光强图样以及偏振态分布可以理解如下。柠檬形偏振结构光场通过各向异性折射非线性,导致了柠檬形庞加莱光束具有空间结构的非线性相移(见公式(7))。这种额外的相移来自于两个幅度不相等的左旋和右旋部分,分别为$\Delta {\phi _ + }$ 和$\Delta {\phi _ - }$ 。当光束传输到远场平面时,这种$\Delta {\phi _ + }$ 与$\Delta {\phi _ - }$ 之间的相位差值变得越来越大,导致了合成后的庞加莱光束横截面上的每个点都发生了非线性偏振旋转。此外,由于具有各向异性的光学非线性,引起了远场光强图样和偏振分布出现了对称性破缺。
Research progress of third-order nonlinear optical effects excited by vectorial light fields (Invited)
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摘要: 具有偏振结构分布的强激光与非线性光学材料相互作用导致了多种新颖的非线性光学效应,反映了材料的非线性光学特性,调制了光场的传播行为。笔者概述了矢量光场激发三阶非线性光学效应的研究进展。首先简要介绍了任意偏振光激发三阶非线性光学效应的理论,如非线性薛定谔方程、光束传播方程、各向同性和各向异性三阶非线性光学系数。也简要介绍了表征三阶非线性光学系数的 Z-扫描技术。在弱聚焦条件下,给出了诸如径向偏振光、杂化偏振光和柠檬型庞加莱光束这三种类型矢量光场的焦场表达式。其次,重点回顾了多种矢量光场激发的各向同性/各向异性三阶非线性光学效应,包括径向偏振光激发的各向异性非线性光学效应、杂化偏振光激发的各向同性和各向异性非线性光学效应、柠檬型庞加莱光束激发的各向同性/各向异性非线性光学效应。最后,简要讨论了矢量光场在非线性偏振旋转、光束整形、可控光场塌缩与成丝和光限幅方面的应用。Abstract: Polarization-structured intense laser interacting with nonlinear optical material results in many novel third-order nonlinear optical effects, reflects the nonlinear optical property of the material, and modulates the propagation behavior of the beam itself. Herein, the research progress of third-order nonlinear optical effects excited by vectorial light fields was reviewed. Firstly, the basic theory of third-order nonlinear optical effects excited by arbitrary polarized lights was briefly introduced, such as nonlinear Schrödinger equation, beam propagation equation, and isotropic and anisotropic third-order nonlinear optical coefficients. The Z-scan techniquefor characterizing third-order nonlinear optical coefficients was also introducd. Under the weak focusing condition, the expressions for the focal field of three types of vectorial light fields were provided, i.e., radially polarized beams, hybridly polarized beams, and lemon-type Poincaré beams. Secondly, the isotropic and/or anisotropic third-order nonlinear optical effects excited by a variety of vectorial light fields was revisited, includinganisotropic nonlinear optical effects induced by radially polarized beams, isotropic and anisotropic Kerr nonlinearities excited by hybridly polarized beams, isotropic and anisotropic nonlinear optical effects induced by lemon-type Poincaré beams. Lastly, the prospects of their applications of vectorial light fields in nonlinear polarizationrotation, beam shaping, controllable field collapsing filaments, and optical limiting were briefly discussed.
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Key words:
- vectorial light field /
- nonlinear refraction /
- two-photon absorption /
- isotropy /
- anisotropy
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图 8 具有高斯形光强分布的矢量光场通过各向异性双光子吸收器后的(a)光场图样和偏振态分布,(b)光强沿光场径向分布[32]
Figure 8. (a) Intensity patterns superimposed with polarization distributions and (b) intensity profiles along the diameter of the vectorial light fields with Gaussian intensity distribution after passing through the anisotropic two-photon absorber[32]
图 10 具有随机噪声的杂化偏振光(第一行)和径向偏振光(第二行)在各向同性克尔介质中的塌缩行为。这四列(从左到右)依次对应以衍射长度为单位的四个传播距离(
$\zeta = 0$ , 0.12, 0.24, 和0.36)。第一列还给出了偏振态分布[45]Figure 10. Collapsing behaviors of (the first row) hybrid polarized beam and (the second row) radially polarized beam with random noises propagating in isotropic Kerr media. The four columns (from left to right) correspond to four propagation distances in units of diffraction length (
$\zeta = 0$ , 0.12, 0.24, and 0.36). The distributions of state of polarization are also shown in the first column[45]图 11 径向偏振光(RPB)激发各向异性(σ=−1.5)和各向同性(σ=0)双光子吸收器、偏振方向沿[100]晶体轴的线偏振光(θ=0)激发双光子吸收器的光限幅效应[30]
Figure 11. Optical limiting effects of anisotropic (σ=−1.5) and isotropic (σ=0) two-photon absorbers using radially polarized beams, and of two-photon absorber using linearly polarized beams for its polarization direction along the [100] crystallographic axis (θ=0)[30]
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