# Standing Wave Field Distribution in Graded-Index Antireflection Coatings
## 2 Theory and Model  
The graded-index AR coatings for linear index distribution, exponential index distribution, and polynomial index distribution are designed at the working wavelengths from 200 nm to 1200 nm of incident laser, which is the wavelength range for inertial-fusion research. For inertial-fusion research, to possess the maximum possible energy, the incident laser is usually perpendicular to the optical elements of laser system. Therefore, we only consider the case that the incident laser is perpendicular to the coatings in this study. Fused silica is taken as the substrate of the graded-index AR coatings, which is widely used in high power laser system [6]. In our design, the maximum index of graded-index AR coatings is close to the index of fused silica (\(n=1.45\)) and the minimum index is close to air (\(n=1\)) [18]. The transmittance of the graded-index AR coatings are calculated by Fresnel coefficients matrix method [28; 29]. To make the graded-index AR coatings working at wavelengths from 200 nm to 1200 nm, a numerical searching method is used to find the optimum matching thickness of coating. In this method, the transmittance of AR coating is the quantity to be optimized and the coating has maximum transmittance of designed wavelengths in optimum matching thickness of coating. The method is that we divide the index distribution of graded-index AR coating by many thin layers. The thickness of each thin layer is taken as \(d\). We make the numerical search of \(d\) from 0 to \(d0\), here \(d0\) is the maximum searching thickness of the thin layers. If the maximum transmittance of graded-index AR coating is happened as thickness of thin layers is \(d1\), the optimum matching thickness of AR coating will be \(N\times d1\), where \(N\) is the number of the thin layers. Since index distribution of idea graded-index coating is continuous, to approach this ideal index distribution, the thickness of thin layers should be small and the number of the thin layers \(N\) should be large. This is a rule for thickness \(d\) and number of layers \(N\). Because the thickness of AR coating is less than the wavelength of incident light (e.g., the thickness of single layer AR coating is less than a quarter of wavelength), the maximum overall searching thickness of coating \(N\times d0\) can be taken as the wavelength of incident light (it is a enough maximum searching thickness and the optimize thickness \(N\times d1\) of coating is in this searching range). It should be noted that thickness \(d\) and the number of the thin layers \(N\) are modeling parameters, but not are parameters for real fabrication. The aim of the modeling is to find the optimum matching thickness of graded-index AR coating. For real fabrication, one can redivide the obtained optimum matching thickness of AR coating. The more detailed discussion can be found in Section 3 (paragraph 3) and Section 4 (paragraph 1). The standing wave field distribution of graded-index AR coatings are obtained by the numerical calculation of electromagnetic wave equation. The electromagnetic wave equation for a vertical incident plane wave in coating is in the form [24]  
\[\frac{d^{2}\overset{\rightarrow}{E}}{dz^{2}}+\frac{\omega^{2}}{c^{2}}\varepsilon( z)\overset{\rightarrow}{E}=0, \tag{1}\]  
where, \(\varepsilon(z)\) is the relative permittivity of coating at position \(z\). Here \(z=0\) is the surface of coating, \(\overset{\rightarrow}{E}\), \(\omega\) and \(c\) are the electric field amplitude, angular frequency of laser and velocity of light in vacuum, respectively. The boundary conditions of Equation (1) are as follows:  
\[\left\{ \begin{array}{c}\overset{\rightarrow}{E}_{0}^{+}(1+r)=\overset{ \rightarrow}{E}(z=0^{+})\\ ik_{0}\overset{\rightarrow}{E}_{0}^{+}(1-r)=\frac{d\overset{\rightarrow}{E} }{dz}(z=0^{+})\end{array} \right., \tag{2}\]  
where, \(k_{0}=\frac{\omega}{c}\); \(r=\frac{\overset{\rightarrow}{E}_{0}^{-}}{\overset{\rightarrow}{E}_{0}^{-}}\) is reflection coefficient of coating which is obtained by Fresnel coefficients matrix method.