## 2 Analysis of Imaging Properties of HDOE in Wide-Bandwidth Spectrum
### Analysis of Image Quality Degradation of HDOE Imaging  
The imaging degradation of HDOE is mainly caused by strong chromatic aberration and decreased diffraction efficiency. The effect of chromatic aberration can be analyzed in the impulse response function of the main order diffraction of the light wave in the HDOE. When the diffraction efficiency of the light wave decreases, a part of the energy will be scattered and dispersed on the image plane, forming stray light background radiation. Then, the effect on the reduction of diffraction efficiency can be analyzed by the impulse response of the light wave diffracted in the non-design order. According to the Fresnel diffraction theory, a light wave with an amplitude of 1 on the HDOE plane(\(x^{\prime},y^{\prime}\)) passes through the HDOE; the complex amplitude expression of the wave field on the image plane \((x,y)\) with distance \(z\) is [8]:  
\[U_{\lambda}(x,y,z)=\frac{e^{i\mathbf{z}z}}{i\lambda z}\cdot\iint t_{HDOE, \lambda}(x^{\prime},y^{\prime})e^{\frac{\mathbf{z}}{2}[(x-x^{\prime})^{2}+(y-y ^{\prime})^{2}]}dx^{\prime}dy^{\prime}, \tag{5}\]  
where \(\lambda\) is the wavelength of light, i is the wave number, and \(i=2\pi/\lambda\), \(t_{HDOE,\lambda}\) is the complex amplitude transmittance function of HDOE. The complex amplitude transmittance function is defined as the ratio of the complex amplitude of the outgoing light field to the complex amplitude of the incident light field on the plane of the element, which can express the influence of the element on the incident wavefront. \(t_{HDOE,\lambda}=e^{i[\phi_{HDOE,\lambda}(x^{\prime},y^{\prime})]}\), \(\phi_{HDOE,\lambda}\) is the phase delay function of HDOE, which can be derived from the phase function \(\varphi(r)\). For a detailed derivation see the supplementary File. Thus, the impulse response function of the main order diffraction of the light wave on the image plane is:  
\[h_{1}(x,y)=|\mathrm{U}(x,y)|^{2} \tag{6}\]  
When the diffraction efficiency is lower than 100%, the light waves are diffracted in other non-design orders, and a part of the energy is not concentrated in the focus, resulting in the background radiation of stray light. Consquently, the light field intensity function of the HDOE imaging system can be written as the superposition of the focusing impulse response \(h_{1}(x,y)\) and the background impulse response \(h_{\mathrm{BG}}\):  
\[I(x,y)=\left|h_{1}(x,y)\right|^{2}+\left|h_{\mathrm{BG}}(x,y)\right|^{2} \tag{7}\]  
The point spread function of HDOE is a normalized representation of the light field intensity function, and the normalization process can be expressed as:  
\[h(x,y)=\left|I(x,y)\right|_{1}=\left|\left|h_{1}(x,y)\right|^{2}+\left|h_{ \mathrm{BG}}(x,y)\right|^{2}\right|_{1} \tag{8}\]  
A further simplified approximation can be obtained:  
\[h(x,y)=\eta_{\mathrm{HDOE}}h_{1}(x,y)+(1-\eta_{\mathrm{HDOE}})h_{\mathrm{BG} }(x,y), \tag{9}\]  
where \(h_{\mathrm{BG}}(x^{\prime},y^{\prime})=\int_{-\infty}^{\infty}\int_{-\infty} ^{\infty}(1-\eta_{1})\delta(x-x^{\prime},y-y^{\prime})dx^{\prime}dy^{\prime}\), \(\delta(\cdot)\) is the unit impulse function. For a detailed derivation, please refer to the Supplementary File. Thus, the imaging degradation model of HDOE in the bandwidth spectrum can be expressed as:  
\[g=[\eta_{\mathrm{HDOE}}h_{1}(x,y)+(1-\eta_{\mathrm{HDOE}})h_{\mathrm{BG}}(x,y )]*f+n \tag{10}\]  
The image quality degradation model includes various influencing factors that affect diffraction imaging, where \((1-\eta_{\mathrm{HDOE}})h_{\mathrm{BG}}(x,y)\) represents the influence of stray light caused by the diffraction of light waves in non-design orders, \(\eta_{\mathrm{HDOE}}\) represents the influence of diffraction efficiency, and \(h_{1}(x,y)\) represents the influence of chromatic aberration. This model comprehensively and accurately describes the image degradation process of diffraction imaging and provides a useful reference for further image restoration research.