# Multiple-Diffraction Subtractive Double Monochromator with High Resolution and Low Stray Light
## 2 Principle  
### Multiple Diffractions  
A multiple-diffraction scheme is an effective way to enhance spectral resolution. The schematic is shown in Figure 1. The incident light is diffracted on the grating for the first time and then reflected by the mirror to the grating for multiple diffractions. Disregarding the area constraints of the mirror and reflective grating, this multiple diffraction process can be extended iteratively up to the (\(2n-1\)) th and (\(2n\)) th diffractions. Assuming the wavelength \(\lambda_{0}\), \(\lambda_{0}\), as indicated by the blue ray in Figure 1, the angle of incidence for the first diffraction is \(\alpha_{1}\), and the diffraction angle is \(\beta_{1}\). The diffraction complies with Equation (1):  
\[d[\sin\alpha_{1}+\sin\beta_{1}]=m\lambda_{0} \tag{1}\]  
When the mirror is perpendicular to the diffraction ray of \(\lambda_{0}\), \(\lambda_{0}\) returns to the grating and diffracts again along the original path. For the wavelength \(\lambda_{0}+\Delta\lambda\), as indicated by the red light, after the first diffraction, the diffraction angle is \(\beta_{1}+\Delta\beta_{1}\), complying with Equation (2):  
\[d[\sin\alpha_{1}+\sin(\beta_{1}+\Delta\beta_{1})]=m(\lambda_{0}+\Delta\lambda) \tag{2}\]  
After being reflected by mirror M, the light with a wavelength of \(\lambda_{0}+\Delta\lambda\) is diffracted onto the grating again, satisfying Equation (3):  
\[d[\sin(\beta_{1}-\Delta\beta_{1})+\sin(\alpha_{1}+\Delta\beta_{2})]=m(\lambda _{0}+\Delta\lambda) \tag{3}\]  
As \(\Delta\beta_{1}\) and \(\Delta\beta_{2}\) are extremely small, we can expand the trigonometric functions and obtain the angular dispersion of the double diffraction as follows:  
\[\frac{\Delta\beta_{2}}{\Delta\lambda}=\frac{2m}{d\cos\alpha_{1}} \tag{4}\]  
Figure 1: Angular schematic of multiple diffractions. Mirror M and reflective grating G enable multiple diffractions. Polychromatic light is represented by black; light of wavelength \(\lambda_{0}\) is blue, and light of wavelength \(\lambda_{0}+\Delta\lambda\) is red. (**a**) Schematic diagram of first and second diffractions. (**b**) Schematic diagram of (\(2n-1\))th and (\(2n\))th diffractions. (**c**) Overall optical path sequence of multiple diffractions. Compared with the angular dispersion of single diffraction, we have:  
\[\frac{\Delta\beta_{2}}{\Delta\lambda}=2\frac{\cos\beta_{1}}{\cos\alpha_{1}}\frac {\Delta\beta_{1}}{\Delta\lambda} \tag{5}\]  
According to Equation (5), when \(\alpha_{1}>\beta_{1}\), the angular dispersion of double diffraction is at least twice as great as that of single diffraction. Furthermore, the angular dispersion for N-time diffraction can be derived via mathematical induction. For the \(2n\)-th diffraction order, we have:  
\[\Delta\beta_{2n}\cos\alpha_{1}-\Delta\beta_{2n-1}\cos\beta_{1}=\frac{m\Delta \lambda}{d} \tag{6}\]  
Similarly, for the \((2n-1)\)-th diffraction order:  
\[-\Delta\beta_{2n}\cos\alpha_{1}+\Delta\beta_{2n+1}\cos\beta_{1}=\frac{m\Delta \lambda}{d} \tag{7}\]  
which can also be expressed as:  
\[-\Delta\beta_{2n-2}\cos\alpha_{1}+\Delta\beta_{2n-1}\cos\beta_{1}=\frac{m \Delta\lambda}{d} \tag{8}\]  
By combining the above Equations (4) and (6)-(8), we derive:  
\[ \begin{array}{l}\frac{\Delta\beta_{2n}}{\Delta\lambda}=2n\frac{m}{d\cos \alpha_{1}}\\ \frac{\Delta\beta_{2n-1}}{\Delta\lambda}=(2n-1)\frac{m}{d\cos\beta_{1}}\end{array}  \tag{9}\]  
Equation (9) presents the general formula for calculating the angular dispersion of multiple diffractions. For an even number of diffractions, the angular dispersion is inversely proportional to the cosine of the incident angle \(\alpha_{1}\). However, for an odd number of diffractions, the angular dispersion varies inversely with the cosine of the diffraction angle \(\beta_{1}\) for the first time. So, the spectral resolution of multiple diffractions depends not only on the diffraction numbers but also on the angle parameters.