# 克爾效应

（重定向自克尔效应

## 克爾電光效應

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

## 理論

### 直流克爾效應

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

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

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

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

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

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

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

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

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

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

### 交流克爾效應

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

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

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

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

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

## 註釋

1. ^ 注意到兩種不同的各向同性，一種是晶體的光學各向同性，另一種是氣體、液體、非晶體固體的結構各向同性

## 參考文獻

1. ^ Weinberger, P. John Kerr and his Effects Found in 1877 and 1878 (PDF). Philosophical Magazine Letters. 2008, 88 (12): 897–907. Bibcode:2008PMagL..88..897W. doi:10.1080/09500830802526604.
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.