# Characteristic Analysis of Compact Spectrometer Based on Off-Axis Meta-Lens
## 1 Introduction  
Recent progresses on micro/nano-photonics have become a turning point for the transformation of traditional optical systems [1]. Metasurfaces, ultrathin planar structures patterned by monolithic subwavelength elements on surfaces, provide a new way to transform the incident wavefront to the desired forms by controlling the amplitude, phase and polarization of electromagnetic fields [2; 3; 4; 5; 6]. Compared with their traditional counterparts, metasurfaces have emerged as unparalleled devices to realize miniaturized, on-chip integrated and multifunctional optical systems [7; 8; 9; 10]. Tremendous breakthroughs in the field of metasurface have given rise to expansive applications ranging from planar imaging lenses [8; 9; 11; 12; 13; 14], holograms [15; 16; 17; 18; 19], spectrometers [20; 21], nonlinear devices [22; 23] and even to invisible cloaks [24; 25; 26], etc.  
Spectroscopy, a fundamental scientific instrument which has developed for hundreds of years since [PERSON] elaborated his discover that the sunlight is made up of a mixture of colors in 1672 [27], is of significant importance to physics [28; 29; 30], astronomy [31; 32; 33] and chemistry [34; 35; 36]. As pivotal components in spectroscopy, traditional spectrometers usually employ isolated dispersive elements (mostly gratings or prisms) and focusing lenses, which makes the whole system bulky, complicated and unsuitable for integration [21]. To produce miniature and compact spectrometers, researchers have explored the possibilities of taking the advantages of metasurfaces such as its flexibility to design arbitrary phase response or convenience to be integrated to other optical system to create multifunctional meta-lens-based spectrometers [20; 21; 37; 38; 39]. Spectrometers based on off-axis meta-lenses was first explored by [PERSON] et al. [20], which unambiguously shows the potential of the technique, i.e., meta-lenses, as an alternative way to build spectrometers. However, a more detailed parametric analysis is required to fully understand the features of the two-dimensional device,especially clarifying the performance dependencies (spectral range and spectral resolution) on the system's structural parameters (lens aperture, focal length, off-axis angle and orientation angle of the detector plane). These information is crucial for the device design and the dispersive characteristics of off-axis meta-lenses need to be further illustrated.  
In this paper, we carry out a comprehensive characteristic analysis of off-axis meta-lens based compact spectrometers using both rigorous wave theory and geometrical ray optics. The dispersive properties of the off-axis meta-lens provide the technique foundation for spectroscopy. We first show that by optimizing the orientation angle of the output (detector) plane, the effective spectral range can be enhanced about three times than that of the common configuration where the output plane is placed perpendicular to the optical axis. From the numerical diffraction calculations compared with the analytical geometrical optics analysis, we then discuss the full picture about the structural dependence of the intrinsic performance of the device. It is found that to design a high-resolution spectrometer, the aperture diameter and off-axis angle of the meta-lens should be enlarged as much as possible. On the contrary, a broadband spectrometer can be designed at certain sacrifice of spectral resolution by reducing the lens's aperture and off-axis angle and increasing the focal length. Based on these features, we give two concrete device designs: one could cover the 800-1800 nm band with the spectral resolution of 2-5 nm and the other covering the 780-920 nm band with a finer spectral resolution of 0.15-0.6 nm. The current research provides a new perspective to design meta-lens-based spectrometers.