# Echelle Grating Spectroscopic Technology for High-Resolution and Broadband Spectral Measurement
## 4 Calibration  
The two-dimensional monochromatic images captured by the detector need to be converted into a one-dimensional spectrum (Figure 8) calibrated with respect to the wavelength and intensity, which is called spectral restoration. The accuracy of the wavelength calibration depends on the spectral restoration algorithm. Improving the computational accuracy as well as the operating rate are always the goals of calibration algorithms.  
In the early period, the wavelength position on a detector was calibrated by ray tracing with massive calculation. Later, it was optimized by mathematical methods. [PERSON] described the spectrum as its component cyclic orders and subsequently used the spectrographic optical axis as the fiduciary line to identify an unknown spectral line in terms of the position within its order cycle, with an accuracy of \(1/10^{5}\)[57]. [PERSON] proposed a method by which any line in an echelle spectrum was characterized by the ratio of \(l/L\), where \(L\) is the distance between two arbitrary marks at opposite ends of the prism spectral slit, and \(l\) is the distance of the echelle line from one of them. Its accuracy is better than \(1/10^{6}\)[58]. However, the calibration accuracy and speed are still insufficient. Researchers developed a method of building a simplified optical model to establish the relationship between the wavelength and pixel position. For example, the software, called EGRAM (for echellogram), mapped the spectrograms from echelle spectrographic systems to the first order [59]. [PERSON] deduced a detailed mathematical model suitable for real-time wavelength calibration of the refractive index of the prism, with an accuracy of \(10^{-3}\) nm for the entire spectral range [60]. Sometimes, the model has errors. [PERSON] combined the ray-tracing technique with the method of EGRAM [59] to solve this problem. The accuracy of the algorithm is improved by calculating the offset distance of the principal ray from the center of the image plane in the two-dimensional vertical direction and compensating the spectral line bending from the reflecting prism, and the maximum deviation is less than one pixel [44]. Because many algorithms need to know the locations and wavelengths of hundreds of lines, [PERSON] proposed a method that required neither the pixel locations of certain spectral features nor the true order number of each diffraction order. It is a fully automatic wavelength calibration method that uses duplicate spectral features in adjacent orders to establish the scale-invariant component of the wavelength solution [61].  
However, the actual image location may not be completely consistent with the theoretical wavelength coordinates, which may be caused by environmental changes (such as humidity, temperature, and atmospheric pressure), the processing of optical components, and adjustment errors. So, the model calibration is of significance. [PERSON] proposed a wavelength calibration method that simulated a conceptual spectrometer and could achieve sub-pixel accuracy of spectral line prediction positions in the temperature range of 5-35 \({}^{\circ}\)C [62], but this method ignored the effects of pressure and humidity. [PERSON] established a spectral simplified model that could invert systematic parameters, and the deviation between the corrected actual x- and y-coordinates and the model calculation was less than one pixel [63]. [PERSON] also proposed a time-saving simple spectral reduction algorithm. First, the least squares method (LSM) was used to associate the actual spectrogram with the theoretical model; then, the polynomial fitting method was used to improve the accuracy of the algorithm, and the wavelength error was less than 0.02 nm [64].  
Figure 8: Calibration diagram.