# Characteristic Analysis of Compact Spectrometer Based on Off-Axis Meta-Lens
## 3 Results and Discussion
### Relationship between Structural Parameters and Evaluation Indexes of Spectrometer
In principle, this  
Figure 3: Influences of the orientation angle of the OP \(\beta\) on the dispersive properties of the meta-lens: **(a)** The effective spectral range \(\Delta\lambda\) and spectral resolution \(\delta\lambda\) as a function of \(\beta\). Arrows ascribe the curves to the left or right \(y\)-axis. The data for \(\beta\) close to \(-45^{\circ}\) (indicated by the gray box) has low accuracy because the OP is parallel to the optical axis at this specific case; **(b)** Ray tracing calculations for the meta-lens at wavelengths of 1525, 1550 and 1665 nm, respectively, indicating an optimal orientation angle \(\beta=-70^{\circ}\); **(c)** Displacement \(\Delta u\) and FWHM of the focal line as a function of wavelength at \(\beta=-70^{\circ}\); **(d)** Spectral resolution \(\delta\lambda\) as a function of wavelength at \(\beta=-70^{\circ}\). problem may be overcome by patterning the detector with another metasurface. In the following, we will restrict the discussions by using the condition \(\beta=\alpha\) (i.e., the OP is perpendicular to the optical axis). This will not lose the generality to illustrate the features of spectrometers that can be reproduced with different specifications. There are three factors that affect the performance of the meta-lens-based spectrometer besides \(\beta\): aperture diameter \(D\), focal length \(f\) and off-axis angle \(\alpha\). For the meta-lens, NA is expressed as  
\[\text{NA}=\sin(\frac{1}{2}\tan^{-1}\frac{4F\cos\alpha}{4F^{2}-1}), \tag{5}\]  
where the \(F\)-number is \(F=f/D\). The influences of \(D\),\(f\) and \(\alpha\) on \(\Delta\lambda\) and \(\delta\lambda\) are presented in Figure 4. Same as above, the lower and upper bounds of the blue band denote the minimum and maximum spectral resolution, respectively. According to the calculation results, the effective spectral range \(\Delta\lambda\) is linearly proportional to the focal length \(f\) but the spectral resolution \(\delta\lambda\) is nearly independent on \(f\). However, \(\Delta\lambda\) and \(\delta\lambda\) both show an inverse relation with the aperture diameter \(D\) and the off-axis angle \(\alpha\). The configuration with large \(f\) or small \(D\) and \(\alpha\) attains wider effective spectral range due to the smaller spherical aberration [43]. From the wave theory and the linear dispersion \(\text{d}u/\text{d}\lambda\) for \(\beta=\alpha\) (see Figure 2b), the spectral resolution can be estimated by [20]  
\[\delta\lambda=\frac{\text{d}\lambda}{f\{\sin^{-1}[(1+\frac{\text{d}\lambda}{ \lambda\text{d}})\sin\alpha]-\alpha\}}\times\frac{0.5\lambda}{\text{NA}}. \tag{6}\]For a large \(F\)-number (\(F>6\) in our case), Equation (5) can be simplified as  
\[\text{NA}\approx\frac{1}{2}\tan^{-1}\frac{4F\cos\alpha}{4F^{2}-1}\approx\frac{2F \cos\alpha}{4F^{2}-1}. \tag{7}\]  
Then the spectral resolution at the design wavelength \(\left.\delta\lambda\right|_{\lambda=\lambda_{\text{d}}}\) is  
\[\left.\delta\lambda\right|_{\lambda=\lambda_{\text{d}}}=\lim_{\text{d}\lambda \to 0}\delta\lambda=\frac{0.5\lambda_{\text{d}}^{2}}{f\tan\alpha\times\text{ NA}}=\frac{\lambda_{\text{d}}^{2}}{D\sin\alpha}[1-(\frac{1}{2F})^{2}]\approx\frac{ \lambda_{\text{d}}^{2}}{D\sin\alpha}. \tag{8}\]  
According to Figure 2c, it is reasonable to approximate the minimum spectral resolution to be equal to \(\left.\delta\lambda\right|_{\lambda=\lambda_{\text{d}}}\) for \(\beta=\alpha\). Then we conclude that the minimum spectral resolution is inversely proportional to \(D\) or the sine of \(\alpha\) and nearly independent of \(f\) for meta-lens with a large \(F\)-number. The analytical result according to Equation (8) provides almost the same variation tendency with the numerical results.