# Characteristic Analysis of Compact Spectrometer Based on Off-Axis Meta-Lens
## 3 Results and Discussion
### Relationship between Structural Parameters and Evaluation Indexes of Spectrometer  
As mentioned above, the disorientation of the actual focal plane with the output plane restricts the effective spectral range of the meta-lens-based spectrometer. We simulate the output field of system with different orientation angles of the OP \(\beta\) while other parameters are kept constant as those used in Figure 2 and analyze the effective spectral ranges \(\Delta\lambda\) and spectral resolutions \(\delta\lambda\) of them (shown in Figure 3a). The lower and upper bounds of the blue band in Figure 3a denote the minimum and maximum spectral resolution, respectively, which roughly have a constant ratio. The spectral range \(\Delta\lambda\) reaches the maximum at about 140 nm (almost three times larger than that at \(\beta=45^{\circ}\) in Figure 2) when \(\beta\) is about \(-70^{\circ}\). According to the above discussions, it is obvious that the effective spectral range will reach maximum when the OP overlaps the actual focal plane because of the decrease of the aberration. On the other hand, when the OP is nearly perpendicular to the actual focal plane, the effective spectral  
Figure 2: Dispersive characteristics of a meta-lens-based spectrometer: (**a**) The focal line along the \(u\)-axis as a function of wavelength for the configuration with \(f=20\) mm, aperture diameter \(D=2\) mm, \(\alpha=45^{\circ}\) and \(\beta=45^{\circ}\); (**b**) FWHM and spectral resolution \(\delta\lambda\) of the focal line as a function of wavelength. Inset shows the field profile along the dashed line in (**a**); (**c**) Electric field intensity normalized by the maximum of the field intensity at the design wavelength on the OP at different wavelengths (from 1526 to 1574 nm at a constant interval of 1.5 nm). range losses its physical meaning. The spectral resolution \(\delta\lambda\) is nearly independent on \(\beta\) and primarily decided by the structural parameters of the meta-lens (they are the aperture diameter, the focal length and the off-axis angle). Although the linear dispersion will increase when the orientation of OP is overlapped with the actual focal plane, the size of the focal spot will also increase proportionally. Based on the above analysis, hereafter we go a step further to study the optimal value of \(\beta\) according to the ray tracing method. Adoption of the method is reasonable because the geometrical optics approximation is available for our low NA design. A commercial software Zemax 2005 (Zemax LLC, Kirkland, WA, USA, 2005) was used to trace the light path at three wavelengths (1525 nm, 1550 nm, 1665 nm), as shown in Figure 3b. We adopt the Zernike standard phase surface deduced from Equation (1) [42]. We choose 1525 and 1665 nm as simulation wavelengths because they are the lower and upper bounds of the effective spectral range. The ray tracing results suggest that the optimal orientation angle is \(\beta=-70^{\circ}\), which agrees exactly with the prediction of wave analysis. The displacement \(\Delta u\) and FWHM of the focal line as a function of the operation wavelength at \(\beta=-70^{\circ}\) are shown in Figure 3c. The position of the minimum FWHM of the focal line red-shifts to the wavelength 1640 nm due to the extremely oblique orientation angle relative to the optical axis. The displacement of the focal line is not linearly dependent on wavelength anymore. Figure 3d gives the spectral resolution \(\delta\lambda\) as a function of wavelength at \(\beta=-70^{\circ}\). The spectral resolution below 1.6 nm is obtained over a wide spectral range from 1450 to 1600 nm. Although the effective spectral range will be greatly enhanced by adjusting the orientation angle of the OP, the actual working efficiency of the device may degrade due to the reduced photon absorption efficiency of detector at grazing incident angles.