# Characteristic Analysis of Compact Spectrometer Based on Off-Axis Meta-Lens
## 2 Design Principle of Spectrometer Based on Off-Axis Meta-Lens  
A schematic diagram of the meta-lens-based spectrometer is shown in Figure 1. For practice, a collimation light circuit is basically required for the proposed device (not shown in Figure 1). For the off-axis meta-lens which will focus the incident normal plane wave at a deflected angle \(\alpha\), the phase distribution \(\varphi_{\mathrm{d}}\) imparted by it follows  
\[\varphi_{\mathrm{d}}(x,y)=2\pi-\frac{2\pi}{\lambda_{\mathrm{d}}}(\sqrt{\left( x-f\sin\alpha\right)^{2}+y^{2}+\left(f\cos\alpha\right)^{2}}-f), \tag{1}\]  
where \(\lambda_{\mathrm{d}}\) is the design wavelength (corresponding to \(\lambda_{2}\) in Figure 1) and \(f\) is the focal length.  
Figure 1: Schematic illustration showing the configuration of the spectrometer based on an off-axis meta-lens. The design wavelength is \(\lambda_{2}\) and the off-axis angle is \(\alpha\). The actual focal plane denoted by the dashed line has a skew orientation angle with the optical axis. The angle between the \(u\)-axis and the \(x\)-axis is defined as the orientation angle of the output plane (OP) \(\beta\). The inset shows the definition of its sign. In addition to the global coordinate system (\(x\), \(y\), \(z\)) based on the meta-lens, a local coordinate system (\(u\), \(v\), \(w\)) based on the OP is adopted to simplify the deduction.  
Two kinds of chromatics dispersion are introduced according to this configuration--the longitudinal displacement along the optical axis (the line from the center of the meta-lens to the focal point of the design wavelength) and lateral displacement perpendicular to the optical axis [21; 40]. It can be predicted that the actual focal plane for light with different wavelengths (\(\lambda_{1}\), \(\lambda_{2}\) and \(\lambda_{3}\) indicated in Figure 1 as an illustration) has a skew orientation angle with the optical axis. The output plane (OP) where the detector is placed does not always coincide with the actual focal plane in practice. A local coordinate system (\(u\), \(v\), \(w\)) centered at the focal point of the design wavelength (\(f\) sin\(\alpha\), \(0\),\(f\) cos\(\alpha\)) is introduced at a relation with (\(x\), \(y\), \(z\)) by  
\[\left\{\begin{array}{l}x=u\cos\beta+w\sin\beta+f\sin\alpha,\\ y=v,\\ z=-u\sin\beta+w\cos\beta+f\cos\alpha,\end{array}\right. \tag{2}\]  
where \(\beta\) denotes the angle between the \(u\)-axis and the \(x\)-axis. At the designing wavelength, the meta-lens will work as a blazed grating. At other frequencies, diffraction will be the dominant factor to decide the output field. In practice, the meta-lenses are usually constructed using subwavelength elements that have ideally a linear dispersion so that there is a dominant diffraction order for different frequencies in the band and their refractive angles are solely decided by the local period. Therefore, in ray optics, the meta-lens could be represented by a constant and frequency-independent phase profile. This assumption is also valid for wave optics when a meta-lens of broadband response is employed, as we assumed, for example the one consisting of Pancharatnam-Berry phase elements working for circularly polarized waves [20; 21]. Then, the input field \(U_{\mathrm{i}}(x,y,0)\) is written as  
\[U_{\mathrm{i}}(x,y,0)=P(x,y)\exp(i\varphi_{\mathrm{d}}(x,y)), \tag{3}\]  
where \(P(x,y)\) is the pupil function of the meta-lens. It takes nonzero and unity value only inside the meta-lens aperture. Based on [PERSON]'s diffraction formula, the output field \(U_{\mathrm{o}}(u,v,0)\) is expressed by [41]  
\[\left\{\begin{array}{l}U_{\mathrm{o}}(u,v,0)=\frac{1}{\mu\lambda}\iint U_{ \mathrm{i}}(x,y,0)\frac{\exp(2\pi r/\lambda)}{r^{2}}(-u\sin\beta+f\cos\alpha) \mathrm{d}x\mathrm{d}y,\\ r=\sqrt{\left(u\cos\beta+f\sin\alpha-x\right)^{2}+\left(v-y\right)^{2}+\left(- u\sin\beta+f\cos\alpha\right)^{2}},\end{array}\right. \tag{4}\]  
where \(r\) denotes the distance from the source (\(x,y,0\)) to the observation point (\(u\), \(v\), \(0\)) (local coordinate system).