## 2 Analysis of Imaging Properties of HDOE in Wide-Bandwidth Spectrum
### Analysis of Spectral Characteristics of HDOE  
HDOE has the same optical power at multiple discrete harmonic wavelengths and can achieve 100% diffraction efficiency, which is more advantageous than ordinary DOE in imaging in the bandwidth spectrum. Then, the HDOE was designed using commercial optical design software to achieve diffractive imaging in the full spectral range of visible light. We set the design center wavelength to the median of the visible light spectrum (\(\lambda_{0}\) = 555 nm), to which the human eye is most sensitive, and set the diffraction order of the center wavelength to the 10 th order (\(m=10\)). The purpose of this is to generate enough harmonics to cover the visible spectrum. The phase function of the HDOE is optimized as a continuous and increasing higher-order term function in commercial optical design software, which can be expressed as:  
\[\varphi(r)=m\sum_{i=1}^{N}A_{i}r^{2i}, \tag{1}\]  
where \(m\) is a constant that represents the diffraction design order of the center wavelength, \(N\) is the order of the phase function, \(A_{i}\) is the corresponding coefficient of the order \(i\), and \(r\) is the radial coordinate of HDOE. A more detailed description is given in the Supplementary File. The diffraction efficiency of HDOE at wavelength \(\lambda\) is given by the following formula [24]:  
\[\eta_{\text{HDOE}}(\lambda)=\sin\text{c}^{2}\left(\frac{m\lambda_{0}}{\lambda }-k\right), \tag{2}\]  
where \(\lambda_{0}\) is the central spectral wavelength and \(k=1,2,3,4\cdots\) is the diffraction order that occurs. The diffraction efficiency curve of the diffractive lens in the visible spectrum is shown in Figure 2a. The image captured by the detector of SHDCIS has a color cast phenomenon, which is caused by two reasons; one is that the diffraction efficiency of HDOE varies in the broadband spectrum, which means that the ability of HDOE to focus different light waves is different; the other one is that the quantum efficiency of the sensor varies with the light wave. The quantum efficiency of the sensor is the detection efficiency of the sensor to light waves, which is expressed as \(\eta_{\text{QE}}\). Since the absorption efficiency of the optical signal by  
Figure 1: Schematic diagram of SHCIS: degraded low-resolution images are measured by single-lens HDOE and color RGB detectors.The DDZMR network performs image restoration and image super-resolution on degraded images to generate high-quality images. the semiconductor material is wavelength dependent, \(\eta_{QE}\) varies with the wavelength. The curve of \(\eta_{QE}\) is calibrated and measured by the detector manufacturer, as shown in Figure 2b, where the horizontal axis is the wavelength and the vertical axis is the percentage. The spectral response efficiency to light waves of different wavelengths can be calculated by multiplying the diffraction efficiency of the HDOE by the quantum efficiency of the detector:  
\[\eta_{SHDCIS}(\lambda)=\eta_{\text{Harmonic}}(\lambda)\cdot\eta_{QE}(\lambda). \tag{3}\]  
Next, the response efficiency for the light passing through the optical system on the three RGB channels of the detector can be calculated:  
\[\kappa_{SHDCIS}{=}\int\limits_{\lambda_{1}}^{\lambda_{2}}\eta_{SHDCIS}d\lambda, \tag{4}\]  
where (\(\lambda_{1}\), \(\lambda_{2}\)) is the wavelength response range of one of the sensor's RGB channels. The calculated response efficiency ratio of the three channels of RGB was 1:1.3324:1.1123, and the contrast shift ratio of the image measured by the experiment was 1:1.2983:1.1006. The calculated value was approximately consistent with the experimentally measured value. The slight deviation may have been due to the HDOE fabrication error or the quantum efficiency measurement error of the sensor.