# Echelle Grating Spectroscopic Technology for High-Resolution and Broadband Spectral Measurement
## 2 Fundamental Characteristics
### Blaze in Full Spectrum  
By normalizing the diffraction intensity function of the echelle, the diffraction efficiency function can be expressed as:  
\[\begin{array}{l}\dot{I}(i,\theta)=\frac{I}{N^{2}I_{0}}=\left(\frac{\sin \phi_{b}}{\phi_{b}}\right)^{2}\left(\frac{\sin N\frac{\delta\mu}{\lambda}}{N \sin\frac{\delta\mu}{\lambda}}\right)^{2}=BF\cdot\left(IF/N^{2}\right)\\ \delta_{b}=\frac{\pi}{\lambda}b(\sin(i-\theta_{B})+\sin(\theta-\theta_{B}))\\ \delta_{d}=\frac{\pi}{\lambda}d(\sin i+\sin\theta)\end{array} \tag{9}\]  
where \(BF\) is the blaze function, and \(IF\) is the interference function. When the rays of \(\lambda_{b}\) are just blazed, the grating equation is:  
\[m\lambda_{b}=d(\sin i+\sin\theta_{b}) \tag{10}\]  
\(\theta_{b}\) is the diffraction angle of the rays of \(\lambda_{b}\).  
Let us investigate the diffraction of the wavelengths adjacent to the blaze wavelength. We assumed that \(\lambda_{+}\) and \(\lambda_{-}\) are slightly larger and smaller than \(\lambda_{b}\), respectively, as shown in Figure 1.  
Their corresponding grating equations are:  
\[\begin{array}{ll}m\lambda_{\pm}&=d(\sin i+\sin(\theta_{b}\pm\theta_{e\pm}) \,)\\ &=m\lambda_{b}-d\sin\theta_{b}(1-\cos\theta_{e\pm})\pm d\cos\theta_{b}\sin\theta _{e\pm}\end{array} \tag{11}\]Obviously, the diffraction efficiencies corresponding to different \(\lambda_{\pm}\) vary. If we choose a pair of \(\lambda_{\pm}\) where the \(BF\) value is 0.405, \(\theta_{\varepsilon+}+\theta_{\varepsilon-}=\lambda/(d\text{cos}\theta_{B})\). When \(\theta_{\varepsilon\pm}\) is very small, \(\theta_{\varepsilon\pm}\) is approximately:  
\[\theta_{\varepsilon\pm}=\frac{\lambda_{\pm}}{2d\text{cos}\theta_{B}} \tag{12}\]  
Thus, cos\(\theta_{\varepsilon\pm}\) is nearly equal to 1. When \(\theta_{b}\) = \(\theta_{B}\), \(\lambda_{\pm}\) can be described as:  
\[\lambda_{\pm}=\frac{m\lambda_{b}}{m\mp\frac{1}{2}} \tag{13}\]  
When \(m\) is large enough, there is:  
\[\lambda_{+}-\lambda_{-}=\frac{m\lambda_{b}}{m^{2}-1/4}=\frac{\lambda_{b}}{m} \tag{14}\]  
\(\lambda_{b}/m\) is exactly the free spectral range of the \(m\)-th order. It shows that the diffraction efficiency within the free spectral range is not lower than 40% (Figure 2).  
Next, we discuss the relationship between the blazed wavelength of different orders and the free spectral range. It is assumed that the blazed wavelength of the first order at the Littrow mode is \(\lambda_{B}\), and the blazed wavelength of the \(m\)-th order is \(\lambda_{bm}\). Because the diffraction directions of these two wavelengths are the same, they can be depicted as:  
\[\lambda_{B}=2d\text{sin}\theta_{B} \tag{15}\]  
\[\lambda_{bm}=\frac{1}{m}2d\text{sin}\theta_{B}=\frac{\lambda_{B}}{m} \tag{16}\]  
Figure 1: The diffraction schematic of the echelle.  
Figure 2: The \(BF\) function schematic of the echelle at order \(m\).  
From Equation (13), we can find that the upper limit of the free spectral range of the \(m\)-th order is just the lower limit of the free spectral range of the \((m+1)\)-th order:  
\[\frac{\lambda_{B}}{m{+}1/2}=\frac{\lambda_{B}}{(m+1)-1/2} \tag{17}\]  
The free spectral range is very narrow, and the rays from different diffraction orders overlap with each other. The wavelength distribution at each order is shown in Figure 3.  
The free spectral ranges of adjacent orders can be connected to form an entire broadband spectrum. The diffraction efficiency of all the wavelengths is higher than 40% (Figure 4), which is the characteristic of the blaze in the full spectrum.