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# 克尔效应

## 克尔电光效应

${\displaystyle \Delta n=\lambda KE^{2))$

## 理论

### 直流克尔效应

${\displaystyle \mathbf {P} =\varepsilon _{0}{\boldsymbol {\chi ))^{(1)}\mathbf {E} +\varepsilon _{0}{\boldsymbol {\chi ))^{(2)}\mathbf {EE} +\varepsilon _{0}{\boldsymbol {\chi ))^{(3)}\mathbf {EEE} +\cdots }$

${\displaystyle P_{i}=\varepsilon _{0}\sum _{j=1}^{3}\chi _{ij}^{(1)}E_{j}+\varepsilon _{0}\sum _{j=1}^{3}\sum _{k=1}^{3}\chi _{ijk}^{(2)}E_{j}E_{k}+\varepsilon _{0}\sum _{j=1}^{3}\sum _{k=1}^{3}\sum _{l=1}^{3}\chi _{ijkl}^{(3)}E_{j}E_{k}E_{l}+\cdots }$

${\displaystyle P_{i}(\omega )=3\varepsilon _{0}\sum _{j=1}^{3}\sum _{k=1}^{3}\sum _{l=1}^{3}\chi _{ijkl}^{(3)}(\omega ;0,0,\omega )E_{j}(0)E_{k}(0)E_{l}(\omega )}$

${\displaystyle \chi _{1}=\chi _{iiii))$
${\displaystyle \chi _{2}=\chi _{jjkk))$
${\displaystyle \chi _{3}=\chi _{jkjk))$
${\displaystyle \chi _{4}=\chi _{jkkj))$

${\displaystyle \mathbf {E} _{0}=E_{0}{\hat {y))}$

${\displaystyle \mathbf {E} _{L}=(E_{x}{\hat {x))+E_{y}{\hat {y)))\cos(kz-\omega t)}$

${\displaystyle \mathbf {E} =E_{0}+\mathbf {E} _{L}=\mathbf {E} _{0}{\hat {y))+(E_{x}{\hat {x))+E_{y}{\hat {y)))\cos(kz-\omega t)}$

${\displaystyle P_{x}=3\varepsilon _{0}\chi _{xyyx}E_{0}E_{0}E_{x}=3\varepsilon _{0}\chi _{2}E_{0}E_{0}E_{Lx))$
${\displaystyle P_{y}=3\varepsilon _{0}\chi _{yyyy}E_{0}E_{0}E_{y}=3\varepsilon _{0}\chi _{1}E_{0}E_{0}E_{Ly))$

${\displaystyle \Delta n=n_{\parallel }-n_{\perp }\approx {\frac {3\varepsilon _{0}(\chi _{2}-\chi _{1})E_{0}E_{0)){2n))}$

${\displaystyle K\ {\stackrel {def}{=))\ {\frac {n_{\parallel }-n_{\perp )){\lambda _{0}E_{0}^{2))))$

### 交流克尔效应

${\displaystyle \mathbf {E} =E_{y}\cos(\omega t){\hat {y))}$

${\displaystyle P_{y}\simeq \varepsilon _{0}\left(\chi ^{(1)}+{\frac {3}{4))\chi ^{(3)}|E_{y}|^{2}\right)E_{y}\cos(\omega t)}$

${\displaystyle \chi =\chi _{\mathrm {LIN} }+\chi _{\mathrm {NL} }=\chi ^{(1)}+{\frac {3\chi ^{(3))){4))|E_{y}|^{2))$

${\displaystyle n=(1+\chi )^{1/2}=\left(1+\chi _{\mathrm {LIN} }+\chi _{\mathrm {NL} }\right)^{1/2}\simeq n_{0}\left(1+{\frac {1}{2{n_{0))^{2))}\chi _{\mathrm {NL} }\right)}$

${\displaystyle n=n_{0}+{\frac {3\chi ^{(3))){8n_{0))}|E_{y}|^{2}=n_{0}+n_{2}I}$

## 注释

1. ^ 注意到两种不同的各向同性，一种是晶体的光学各向同性，另一种是气体、液体、非晶体固体的结构各向同性

## 参考文献

1. ^ Weinberger, P. John Kerr and his Effects Found in 1877 and 1878 (PDF). Philosophical Magazine Letters. 2008, 88 (12): 897–907 [2014-05-20]. Bibcode:2008PMagL..88..897W. doi:10.1080/09500830802526604. （原始内容存档 (PDF)于2020-09-20）.
2. ^ 克尔效应与光开关，肖胜利 朱锋 郑好望 ，《现代物理知识》 2006年01期
3. ^ Melnichuk, Mike; Wood, Lowell T. Direct Kerr electro-optic effect in noncentrosymmetric materials. Phys. Rev. A. 2010, 82: 013821. Bibcode:2010PhRvA..82a3821M. doi:10.1103/PhysRevA.82.013821.
4. Geoffrey New. Introduction to Nonlinear Optics. Cambridge University Press. ISBN 978-1-139-50076-0.