## 2 Methodology for Frequency Correction and Design for EO FP Etalon  
### Principle of EO Frequency Modulation  
The refractive index of EO crystals can vary with the applied electric field intensity. This property of EO crystals is called EO effect. The EO effect can be divided into linear and quadratic EO effects. EO frequency modulation in this paper used the linear EO effect.  
The linear EO effect of an EO crystal can be expressed by the Formula (1), where \(m\) = 1,2, \(\ldots\),6, and \(k\) = 1,2,3. \(n_{m}^{0}\) are the crystal's refractive indexes without any electric field. \(\gamma_{mk}\) are the components of the linear EO coefficient tensor. \(E_{k}\) is the applied electric-field intensity.  
\[\left\{\begin{array}{c}\Delta\Big{(}\frac{1}{n^{2}}\Big{)}_{m}=\left(\frac{ 1}{n^{2}}\right)_{m}-\left(\frac{1}{n^{2}}\right)_{m}^{0}=\gamma_{mk}E_{k}\\ \left(1/n^{2}\right)_{m}^{0}=\left\{\begin{array}{cc}1/\left(n_{m}^{0}\right) ^{2}&m=1,2,3\\ 0&m=4,5,6\end{array}\right.\end{array}\right. \tag{1}\]  
We computed the refractive-index change \(\Delta\big{(}1/n^{2}\big{)}_{m}\) of an EO crystal in Formula (1) according to the crystal's refractive-index ellipsoid equation. Formulas (2) and (3) are an EO crystal's refractive-index ellipsoid equation before and after an electric field is applied to the crystal.  
\[\frac{x^{2}}{\left(n_{1}^{0}\right)^{2}}+\frac{y^{2}}{\left(n_{2}^{0}\right)^{2} }+\frac{z^{2}}{\left(n_{3}^{0}\right)^{2}}=1 \tag{2}\]  
\[\begin{array}{c}\left(\frac{1}{\left(n_{1}^{0}\right)^{2}}+\gamma_{1k}E_{k} \right)x^{2}+\left(\frac{1}{\left(n_{2}^{0}\right)^{2}}+\gamma_{2k}E_{k} \right)y^{2}+\left(\frac{1}{\left(n_{3}^{0}\right)^{2}}+\gamma_{3k}E_{k} \right)z^{2}\\ \hskip 142.26378 pt+2\gamma_{4k}E_{k}yz+2\gamma_{5k}E_{k}xz+2\gamma_{6k}E_{k} xy=1\end{array} \tag{3}\]  
Those two Formulas (2) and (3) are in the principal-axis coordinate system. According to Formula (2), the principal axes of the refractive-index ellipsoid with no electric field are in the directions of coordinate axes. \(n_{m=1,2,3}^{0}\) are the EO crystal's refractive indexes along the ellipsoid's principal axes with no electric field.  
In Formula (3), \(k\) takes on the value 1, 2, or 3. The value means that the electric field \(E_{k}\) follows one coordinate axis to calculate the refractive-index variation quickly. After the electric field is applied to the EO crystal, if \(\gamma_{4k,5k,6k}\) are all equal to zero, the ellipsoid's three principal axes still follow the coordinate axes. In this case, Formula (3) can be transformed into  
\[\left\{\begin{array}{c}\frac{x^{2}}{n_{1}^{2}}+\frac{y^{2}}{n_{2}^{2}}+\frac {z^{2}}{n_{3}^{2}}=1\\ \frac{1}{\left(n_{m}\right)^{2}}\bigg{|}_{\text{m}=1,2,3}=\frac{1}{\left(n_{m }^{0}\right)^{2}}+\gamma_{mk}E_{k}\end{array}\right., \tag{4}\]  
where \(n_{m=1,2,3}\) are the EO crystal's refractive indexes along the \(x\), \(y\), and \(z\) axes in an electric field of \(E_{k}\).  
With [PERSON] expansion on the second equation in Formula (4), \(n_{m=1,2,3}\) can be written as Formula (5).  
\[n_{m=1,2,3}=n_{m}^{0}-\frac{1}{2}\left(n_{m}^{0}\right)^{3}\gamma_{mk}E_{k} \tag{5}\]  
Based on the Formula (5), the EO crystal's refractive index change is  
\[\Delta n_{m=1,2,3}=n_{m}-n_{m}^{0}=-\frac{1}{2}\left(n_{m}^{0}\right)^{3} \gamma_{mk}E_{k}\;. \tag{6}\]  
Finally, when we give the EO crystal an external electric field \(E_{k}\), the frequency variation of the oscillating longitudinal mode is as the Formula (7) describes. In this formula, \(L\) is the laser's optical oscillator length, \(l\) is the physical length of the EO crystal, \(v_{0}\) is the original frequency of the oscillating longitudinal mode, and \(N\) is this mode's order number.  
\[\Delta v=N\bigg{(}\frac{c}{2(L+l\Delta n_{m=1,2,3})}-\frac{c}{2L}\bigg{)} \approx\frac{v_{0}l}{2L}\Big{(}n_{m}^{0}\Big{)}^{3}\gamma_{mk}E_{k},\;m=1,2,3 \tag{7}\]