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Optimal design of the computational flat diffractive optical system

Zhe Wang, Mingxu Piao, Na Xie, Yuanming Zhao, Chengran Zhang, Dechao Ma, and Dongyi Yang

Open Access Open Access

Abstract

A design method of the computational flat diffractive computational flat diffractive optical system is presented to simplify the optical system structure and achieve high image quality. The aberration expression of the flat diffractive optical element (FDOE) is derived, and then computational imaging methods are used to eliminate the influence of off-axis aberration on image quality, so the field of view is expanded. Based on theoretical analysis, the FDOE is designed, and the field of view has been expanded from 2° to 5°. The results show that the detail resolution of the edge field of view is enhanced after restoration, and the modulation transfer function (MTF) of different subareas calculated using the slanted-edge method improved by an average of 0.17. The diffraction efficiency of the FDOE is greater than 95.75%. This method realizes the miniaturization and lightweight of the optical system, and provides new ideas for the integration of optical systems.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

In recent decades, imaging technologies of optical systems have rapidly developed towards small size, light weight and miniaturization. For conventional imaging system, it usually consists of multiple lenses to realize high image quality, so the system structure is complicated. Therefore, various single-plane imaging methods have been proposed to solve this problem. Metalens have garnered substantial attention due to their compact form and multifunctional characteristics. Metasurface consists of subwavelength-spaced phase shifters that can simplify optical systems by altering the amplitude, phase, and polarization of incident light [1–6]. In visible or near-infrared wavebands, the metalens is designed to achieve high efficiency and near-diffraction-limited focusing at a single wavelength. However, a challenge is presented by achieving high diffraction efficiency over a wide wavelength range, therefore the high quality of imaging at wide wavelengths is not possible. In addition, metasurface is difficult to manufacture, so it cannot apply in practical engineering [7–11]. A multi-order diffractive lens (MDLs) has been proposed, which uses a nonlinear optimization method to determine the height of each ring. The wavelength-averaged focusing efficiency of the MDLs is effectively improved to 40% of the focusing efficiency of the entire measurement in the visible band. However, other aberrations cannot be effectively corrected for efficient focusing [12–14].

The diffractive optical elements (DOEs) have high diffraction efficiency due to their unique imaging and temperature characteristics [15,16]. The diffraction efficiency of the Single-layer DOE (SLDOE) is larger than 80% in visible waveband. Compared with the metalens and the MDLs, the SLDOE can be processed easily by single point diamond turning or precision molding. However, high image quality cannot be achieved using the SLDOE alone due to severe chromatic aberration [17,18]. A multiorder diffractive optical element (MODE) is proposed, which is composed of multiple-order diffractive lens and diffractive Fresnel lens. It can focus several discrete wavelengths in the visible band with high diffraction efficiency over a small bandwidth of discrete wavelengths. However, a large and dense number of discrete wavelengths need to be calculated in order to achieve high diffraction efficiency in the visible band due to the small bandwidth corresponding to the discrete wavelengths [19,20].

This paper proposes a design method for flat diffractive optical element (FDOE) composed of a Fresnel surface and a diffraction surface. The relationship between the monochromatic aberration of the FDOE and the field of view is deduced. The analysis of aberration is used as a basis for calculating the accuracy of imaging to eliminate the effects of aberration. The influence of off-axis aberration is eliminated through point spread function (PSF) construction and the field of view of the element is expanded. The feasibility and effectiveness of the design framework are verified through design examples.

2. Design method

The design framework of the FDOE is shown in Fig. 1. The framework consists of two main parts: optical system design and computational imaging. In the optical system design, the FDOE are constructed using Fresnel surfaces and diffractive surfaces. In order to eliminate the influence of image quality with the extension of the field of view, a PSF model of the FDOE is established within the computational imaging design. A regularization term is introduced, and an iterative algorithm is employed to obtain the optimal solution for image restoration.

 figure: Fig. 1.

Fig. 1. Design framework of the FDOE

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2.1 Aberration theory analysis of the FDOE

In order to simulate the imaging performance of the FDOE, a comprehensive aberration analysis of the entire system is required. For the Fresnel surface, the equivalent surface formula is assumed to be:

$$x = \frac{{{C_1}{y^2}}}{{1 + \sqrt {1 - (1 + k){C_1}{y^2}} }} + \sum\limits_{i = 1}^8 {{a_{2i}}{y^{2i}}} $$
where ${C_1}$ is the vertex curvature of the quadratic surface, y is half of the diameter, ${a_i}$ is the aspheric coefficient. The aperture stop is on the FDOE, and the object is located at infinity. The monochromatic aberration coefficient of the Fresnel surface can be calculated by the following equation [21]:
$$\left\{ {\begin{array}{c} {{S_{IF}} = \frac{{{y^4}{K^3}}}{4}[{(\omega - 1){S^2} + {{(1 - \omega )}^{ - 1}}} ]}\\ {{S_{IIF}} = \frac{1}{2}J{y^2}{K^2}(1 + \omega )S}\\ {{S_{IIIF}} = {J^2}K}\\ {{S_{IVF}} = {J^2}K\omega }\\ {{S_{VF}} = 0}\\ {{C_{IF}} ={-} 2{y^2}(n - 1){C_1}/\nu }\\ {{C_{IIF}} = 0} \end{array}} \right.$$
where the bending variable $S = {C_1}/K$, optical power $K = (n - 1){C_1}$, $\omega = {n^{ - 1}}$, ${C_1}$ is the base curvature of the Fresnel surface, the Lach invariant is $J = nuy$, $\nu $ is the Abbe number, $y$ is the height of parallel rays at the edge. In Eq.(2), ${S_{IF}}$, ${S_{IIF}}$, ${S_{IIIF}}$, ${S_{IVF}}$ and ${S_{VF}}$ is the aberration coefficients of spherical aberration, coma, astigmatism, field curvature, and distortion, respectively.${C_{IF}}$, ${C_{IIF}}$ is axial chromatic aberration and lateral chromatic aberration respectively. From the Eq. (2), it can be seen that ${S_{VF}}$ and ${C_{IIF}}$ are equal to 0. The reason is that the FDOE coincides with the aperture stop, so the chief ray passes through the center of the FDOE. In order to correct the chromatic aberration of the Fresnel surface, the diffractive surface is proposed. The Abbe numbers of diffractive elements and Fresnel elements are opposite, so the chromatic aberration can be corrected by the FDOE without adding additional elements.

The optical path of the FDOE is analyzed, as shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. Edge incident light distribution diagram of the FDOE

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From Fig. 2, the angle of the Fresnel surface can be calculated as follows:

$$\left\{ {\begin{array}{c} {{\alpha_\textrm{1}} = \arctan x^{\prime}}\\ {{I_1} = {\alpha_\textrm{1}}}\\ {{I_1}^{\prime} = \arcsin (\frac{1}{n}\sin {I_1})}\\ {{U_1}^{\prime} = {\alpha_\textrm{1}} - {I_1}^{\prime}} \end{array}} \right.$$
where ${\alpha _\textrm{1}}$ is the angle of the slope of the point when parallel light is incident on the edge annulus of the Fresnel surface, ${I_\textrm{1}}$ is the incident angle, ${I_\textrm{1}}^{\prime}$ is the exit angle, ${U_1}^{\prime}$ is the angle between the intersection point of parallel light and the optical axis after passing through the Fresnel surface. After passing through the Fresnel surface, the diffraction surface is incident by the light. The angle of the diffraction surface can be calculated as follows:
$$\left\{ {\begin{array}{c} {{I_2} = {U_1}^{\prime} = {U_2}}\\ {{U_2}^{\prime} = {I_2}^{\prime} = \arcsin (n\sin {I_2})} \end{array}} \right.$$
where ${I_2}$ is the incident angle, ${I_2}^{\prime}$ is the exit angle, ${U_2}^{\prime}$ is the angle between the intersection point of light and the optical axis after passing through the diffraction surface. Monochromatic aberration coefficient of the diffraction surface can be calculated by the following equation [22]:
$$\left\{ {\begin{array}{c} {{S_{ID}} = \frac{{{y^\textrm{4}}K_D^3}}{4}\left[ {1 + {{(\frac{{{C_2}}}{{{K_D}}})}^2} + 3{{(\frac{{\sin {U_2} + \sin {U_2}^{\prime}}}{{\sin {U_2} - \sin {U_2}^{\prime}}})}^2}} \right]}\\ {{S_{IID}} ={-} {y^\textrm{2}}JK_D^2{{(\frac{{\sin {U_2} + \sin {U_2}^{\prime}}}{{\sin {U_2} - \sin {U_2}^{\prime}}})}^2}}\\ {{S_{IIID}} = {J^2}{K_D}}\\ {{S_{IVD}} = 0}\\ {{S_{VD}} = 0} \end{array}} \right.$$
where ${C_2}$ is the base curvature of the diffraction surface, $m$ is the diffraction order, ${K_D}$ is the optical power of the diffractive surface. The formula of ${K_D}$ is:
$${K_D} ={-} m\lambda {A_1}/\pi $$
where ${A_1}$ is the quadratic term coefficient in the phase polynomial. According to analysis Eq. (2) and (5), the monochromatic aberration coefficient of the FDOE is as follows:
$$\left\{ {\begin{array}{c} {{S_{IFDOE}} = \frac{1}{4}{y^4}\left[ {{K^3}(1 + \frac{1}{n}) - K_D^3A} \right]}\\ {{S_{IIFDOE}} = {y^2}J({K^2}\frac{{n + 1}}{{2n(n - 1)}} - K_D^2{B^2})}\\ {{S_{IIIFDOE}} = {J^2}(K + {K_D})}\\ {{S_{IVFDOE}} ={-} {J^2}K\frac{1}{n}}\\ {{S_{VFDOE}} = 0} \end{array}} \right.$$
where $A = 1 + {({C_2}{K_D})^2} + 3{B^2}$, $B = \frac{{\sin {U_2} + \sin {U_2}^{\prime}}}{{\sin {U_2} - \sin {U_2}^{\prime}}}$.

Due to the ${S_{IVD}}$ of the diffraction surface is 0, it is found the ${S_{IVF}}$ of the Fresnel surface cannot be corrected as the field of view increases by comparing Eq. (2) and Eq. (5). From Eq. (7), the ${S_{IIFDOE}}$ is proportional to J, while the ${S_{IIIFDOE}}$ and ${S_{IVFDOE}}$ are proportional to the square of J. The Lach invariant J is proportional to the field of view. The ${S_{IIIFDOE}}$ and ${S_{IVFDOE}}$ are increased significantly as the field of view angle increases. Therefore, correction of the ${S_{IIIFDOE}}$ and ${S_{IVFDOE}}$ are the focus of the computational imaging part, and other aberration terms are used as compensation factors.

2.2 Computational imaging of the FDOE

In traditional imaging system design, the primary objective is to achieve high image quality performance while adhering to design specifications. The emergence of computational imaging allows the imaging performance of the system to not be strictly conformed to the standards of traditional design. It can be achieved by applying convolution between the PSF and the real scene. The convolution formula is as follows:

$$g(x,y) = {x_{PSF}}(x,y) \otimes f(x,y) + \eta (x,y)$$
where $g(x,y)$ is the blurred image received by the detector, ${x_{PSF}}(x,y)$ is the PSF of the optical system, $f(x,y)$ is real scene, $\eta (x,y)$ is noise. After obtaining the simulated image, a deconvolution algorithm is applied to restore the simulated image. The aim of image restoration is to obtain an image that closely resembles the real scene. In this paper, $g(x,y)$, ${x_{PSF}}(x,y)$ and $\eta (x,y)$ are known, $f(x,y)$ is obtained through an inverse solution.

In general, diffraction effects are not considered and the PSF approximation is assumed to be spatially invariant within some partition of the full field of view for computational convenience [23]. In order to improve the recovery accuracy and consider the influence of diffraction effects, the PSF is used in this paper to vary spatially in the field of view range. A wavefront aberration model is constructed using Zernike polynomials to obtain the PSF in the image plane. Equally spaced samples are taken in the Y field of view direction and the wavefront is fitted to each sample point $Q$. The fitting coefficients are denoted as ${A_{i,Q}}$. Quadratic Lagrangian interpolation is used to fill in each subarea of the full field of view. The Zernike term coefficient at the FDOE pupil can be calculated as:

$$\begin{array}{c}{A_i} = {A_{i,Q - 1}}\frac{{(h - {h_Q})(h - {h_{Q + 1}})}}{{({h_{Q - 1}} - {h_Q})({h_{Q - 1}} - {h_{Q + 1}})}} + \\{A_{i,Q}}\frac{{(h - {h_{Q - 1}})(h - {h_{Q + 1}})}}{{({h_Q} - {h_{Q - 1}})({h_Q} - {h_{Q + 1}})}} + \\{A_{i,Q + 1}}\frac{{(h - {h_{Q - 1}})(h - {h_Q})}}{{({h_{Q + 1}} - {h_{Q - 1}})({h_{Q + 1}} - {h_Q})}}\end{array}$$
where h is the image height of the estimated point, ${h_Q}$ is the image height of sampling point $Q$. Zernike polynomials are employed in this paper to represent the wavefront aberration of the optical system. The wavefront aberration in the Y direction can be expressed as:
$${W_Y}(x,y) = {A_1}Z_0^0 + {A_3}Z_1^{ - 1} + {A_4}Z_2^0 + {A_5}Z_2^2 + {A_8}Z_3^{ - 1} + {A_9}Z_4^0$$
where ${A_1}$, ${A_3}$, ${A_4}$, ${A_5}$, ${A_8}$ and ${A_9}$ is the coefficient of each Zernike term, $Z_0^0$ is the translation term, $Z_1^{ - 1}$ and $Z_2^0$ is the tilt and off-axis defocus caused by field curvature respectively, $Z_2^2$ is the 0° astigmatism term, $Z_3^{ - 1}$ is the coma in the Y direction, $Z_4^0$ is the spherical aberration term.

The generalized pupil function ${P_Y}(x,y)$ in the Y field of view direction can be constructed using wavefront aberration, it can be expressed as:

$${P_Y}(x,y) = p(x,y)\exp [{ik{W_Y}(x,y)} ]$$
where k is the wave number, $k = 2\pi /\lambda $, $\lambda $ is wave, $p(x,y)$ can be calculated as:
$$p(x,y) = \left\{ \begin{array}{cc}1 & {{x^2} + {y^2} \le {\rho ^2}}\\ 0 & {otherwise} \end{array}\right.$$
where x and y is the pupil coordinates, $\rho $ is the pupil radius. The PSF in the Y field of view direction can be obtained by Fourier transforming ${P_Y}(x,y)$ and squaring the real part of the transformation result. It can be expressed as:
$${F_{PSF,Y}}(x,y) = {|{F[{{p_Y}(x,y)} ]} |^2}$$
where F is the Fourier transform. Since the FDOE discussed in this article is rotationally symmetric, the PSF for the entire field of view can be derived by rotating the Y-direction PSF. Only the PSF in a single direction needs to be constructed, so the calculation is simplified and the construction efficiency is improved, as demonstrated in Fig. 3.

 figure: Fig. 3.

Fig. 3. The schematic plot of the image simulation process.

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A regularization term is incorporated into the optimization criterion to facilitate image restoration and mitigate ill-posed behavior. The optimal approximate solution is computed using the least squares restoration method with spatially varying image constraints and the alternating direction method of multipliers (ADMM). The optimal criterion for inverse solution of the optical system degradation model is expressed as:

$$u = \min \{{\parallel {x_{PSF}}(x,y) \ast \hat{f} - g\parallel_2^2 + \gamma \parallel C\hat{f}{\parallel_p}} \}$$
where ${\parallel} {x_{PSF}}(x,y) \ast \hat{f} - g\parallel _2^2$ is the fidelity item, ${||{C\hat{f}} ||_P}$ is the smooth constraint, $\hat{f}$ is the optimal solution of the image in the restoration process, $g$ is the degraded image affected by off-axis aberration received by the detector, $\gamma$ is the weight of the smoothing constraint, $C$ refers to the Laplacian operator, P refers to different norm types. In order to obtain $\hat{f}$, the multiplier alternating method is used to iterate and complete the inverse solution of image degradation.

3. Design example

3.1 Optical system design

The specifications of the FDOE are listed in Table 1.

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Table 1. Specifications of the FDOE

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Aberration coefficients are calculated by substituting the system parameters into Eq. (7). Table 2 shows the aberration coefficients of the FDOE in 2° field of view and 5° field of view. The ${S_{IFDOE}}$ remains constant as the field of view increases, consistently at 0.00178. As discussed in Section 2.1, the changes in the ${S_{IIIFDOE}}$ and ${S_{IVFDOE}}$ are more significant than the ${S_{IIFDOE}}$ with an increase in the field of view. The ${S_{IIFDOE}}$ increases from 0.02991 to 0.07477, the increment is 0.04486. The ${S_{IIIFDOE}}$ increases from 0.07937 to 0.49607, the increment is 0.4167. The ${S_{IVFDOE}}$ changes from 0.05127 to 0.32047, the increment is 0.2692. Therefore, the correction of the ${S_{IIFDOE}}$ of the FDOE is implemented in the design stage, and the effects of the ${S_{IIIFDOE}}$ and ${S_{IVFDOE}}$ on image quality are eliminated in the computational imaging stage.

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Table 2. Aberration coefficients

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Each ring zone of the Fresnel surface can be expressed as follows:

$${\textrm{z}_f} = \frac{{c{r^2}}}{{1 + \sqrt {1 - ({1 + k} ){c^2}{r^2}} }} + \sum\limits_i^8 {{\alpha _i}{r^{2i}}} - \frac{{n - 1}}{n}h$$
where ${z_f}$ is the sag height at each point of the Fresnel surface, $n$ is the number of concentric rings, $h$ is the sag height of the aspheric base, $\frac{h}{n}$ is the tooth height of each ring. The groove accuracy of the Fresnel surface can reach 20 grooves per millimeter. In this paper, an equal tooth width design is used with a tooth width of 0.05 mm and a total of 100 rings. The schematic FDOE is shown in Fig. 4, and Fig. 4(a) is the Fresnel surface, and Fig. 4(b) is the diffractive surface.

 figure: Fig. 4.

Fig. 4. Simulation diagram of the FDOE. (a) Fresnel surface. (b) Diffraction surface.

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The MTF of the FDOE at 2° field of view is obtained by optimizing the Fresnel surface shape and diffraction phase coefficients to correct chromatic, coma, and spherical aberrations. As shown in Fig. 5(a), the MTF for all fields of view is larger than 0.46. The spot diagram of the 2° field of view is shown in Fig. 5(b), the system exhibits an Airy disk radius of 4.034µm, a maximum field of view with a root mean square (RMS) radius of 2.751µm. The diffuse spots in each field of view are smaller than the Airy disk area. After the field of view is enlarged, only coma is corrected through parameter changes, and astigmatism and field curvature are not considered. As is shown in Fig. 6(a), the MTF in the tangential direction of the FDOE at the 5° field of view is only 0.02. The spot diagram of the 5° field of view is shown in Fig. 6(b), the system exhibits an Airy disk radius of 2.751µm, the maximum field of view with an RMS radius of 11.626µm, the size of the diffuse spot is 2.7 times that of the Airy disk.

 figure: Fig. 5.

Fig. 5. Imaging quality of the FDOE with 2° field of view. (a) MTF. (b) Spot diagram.

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 figure: Fig. 6.

Fig. 6. Image quality of the FDOE with 5° field of view. (a) MTF. (b) Spot diagram.

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 figure: Fig. 7.

Fig. 7. Diffraction efficiency curve

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 figure: Fig. 8.

Fig. 8. Change curves of different Zernike coefficients within each field of view

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The diffraction efficiency needs to be discussed in the design of the diffraction surface. The diffraction efficiency of FDOE is analyzed by the following equation:

$${\eta _m} = \sin {c^2}\left\{ {m - \frac{H}{\lambda }[{n(\lambda ) - {n_0}(\lambda )} ]} \right\}$$
where $\sin c$ is the sine cardinal function; $H$ is the height of microstructure; ${n_0}(\lambda )$ is the refractive index of the space where the incident light is located; $n(\lambda )$ is the refractive index of the space where the emitted light is located. The height of microstructure is:
$$H = \frac{{{\lambda _0}}}{{n(\lambda ) - {n_0}(\lambda )}}$$
where ${\lambda _0}$ is the center wavelength. For the PMMA used in this paper, different refractive indexes correspond to different wavelengths:
$$n(\lambda ) = \sqrt {{a_0} + {a_1} \cdot {\lambda ^2} + {a_2} \cdot {\lambda ^{ - 2}} + {a_3} \cdot {\lambda ^{ - 4}} + {a_4} \cdot {\lambda ^{ - 6}} + {a_5} \cdot {\lambda ^{ - 8}}}$$
where ${a_0}$, ${a_1}$, ${a_2}$, ${a_3}$, ${a_4}$, ${a_5}$ are material parameters of the PMMA. The diffraction efficiency of the three wavelengths can be calculated according to Eq. (16) as shown in Fig. 13 and summarized in Table 3.

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Table 3. Evaluation results

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As shown in Table 3, the diffraction efficiency at the maximum wavelength is 96.88%, and the diffraction efficiency at the minimum wavelength is 95.75%. The diffraction efficiency of the FDOE in the working band is greater than 95.75%. The diffraction efficiency curve is shown in Fig. 7

3.2 Computational imaging analysis

In order to perform image restoration using deconvolution, the wavefront aberration model is obtained by fitting the sample points with Zernike polynomials, as shown in Section 2.2. The wavefront aberration model is used to construct the PSF model. The Zernike coefficient can be extracted from each field point of the simulation model established in Section 3.1. As shown in Table 4, the Zernike coefficients for 10 sampling points within the 5° field of view are provided. The change curves of different Zernike coefficients within each field of view are shown in Fig. 8.

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Table 4. Zernike coefficients for each sampling point with 5°

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As shown in Table 4, the ${A_9}$ changes from -0.003 to -0.005, a change of only 0.002. The ${A_5}$ changes from 0 to -0.356, with a change of 0.356. The ${A_4}$ changes from -0.122 to 0.349, with a change of 0.471. The ${A_8}$ changes from 0 to -0.082, with a change of 0.082. The variations in the ${A_5}$ and ${A_4}$ are 4.34 times and 5.74 times greater than ${A_8}$, respectively. As a result, the ${A_5}$ and ${A_4}$ are primarily corrected in the computational imaging component.

In this paper, the wavefront aberration model for each field of view is derived by incorporating the Zernike coefficients into the wavefront aberration, as given in Eq. (10). The wavefront aberration model is inserted into Eq. (11) to obtain the generalized pupil function. The Fourier transform is applied to the generalized pupil function using Eq. (13) to obtain the PSF model. As shown in Fig. 9, the PSF model obtained through the simulation software is compared with the PSF model constructed in this paper at 0°, 3.5° and 5° field of view angles respectively. The PSF models for 0°, 3.5°, and 5° fields of view are displayed respectively in Fig. 9(a), (b), and (c) which are generated by the simulation software. The PSF models for 0°, 3.5°, and 5° fields of view are displayed respectively in Fig. 10(a), (b), and (c) which are calculated by the model.

 figure: Fig. 9.

Fig. 9. PSF comparison of the FDOE. The PSF calculated by simulated software: (a) 0°, (b) 3.5°, (c) 5°.

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 figure: Fig. 10.

Fig. 10. PSF comparison of the FDOE. The PSF calculated by the model: (a) 0°, (b) 3.5°, (c) 5°.

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As shown in Fig. 9 and Fig. 10, the constructed model is the same as the model computed by the simulation software, so the method of constructing the model is accurate and can be used for subsequent inverse convolution processing.

The results of image simulation using the ISO12233 resolution chart are shown in Fig. 11. The center is revealed to be sharp while the edges are seen to be blurred. It can be inferred that this phenomenon is attributed to uncorrectable astigmatism and field curvature based on the analysis of the Zernike coefficients for the FDOE in Section 3.1. A comparison of the edge field of view of the ISO12233 resolution chart before and after restoration are shown in Fig. 12. The number “9” in the red box and the line in the green box are blurred, and the detail cannot be recognized in Fig. 12(a). After correction through computational imaging, the number “9” and the line appear clear and sharp in Fig. 12(b).

 figure: Fig. 11.

Fig. 11. Image simulation diagram of the ISO12233 resolution chart

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 figure: Fig. 12.

Fig. 12. Comparison images before and after edge field restoration. (a) edge field of view captured from the pre-restored resolution image; (b) edge field of view captured from the post-restored resolution image

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In order to objectively evaluate the image quality of the FDOE, the slanted-edge method is used to measure MTF. As shown in Fig. 12, areas 1 to 5 have been selected. As shown in Fig. 13, (a) and (b) are the comparison of MTF at 111lp/mm before and after each subarea of restoration. The MTF of area 3 changes from 0.049 to 0.234 with a maximum increment of 0.185. The MTF of area 1 changes from 0.15 to 0.29 with a minimum increment of 0.14. The average MTF improvement in each region is 0.17. It shows that the image quality of optical system with large field of view can be effectively improved by the computational imaging method proposed in this paper.

 figure: Fig. 13.

Fig. 13. Comparison of MTF before and after processing. (a) MTF of each area before processing; (b) MTF of each area after processing.

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Peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) are employed to assess the restored image. The evaluation results are presented in Table 5. PSNR is a traditional loss function, and it can evaluate two images at the pixel level. SSIM can evaluate the similarity of two images by calculating the mean and variance of the pixels of the two images and the covariance between the pixels of the two images respectively. An ideal recovered image has an SSIM equal to 1.

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Table 5. Evaluation results

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According to Table 5, the SSIM and PSNR values of the system after image restoration are higher than that before restoration. PSNR increased from 23.3916 to 34.9270 with an increment of 11.5354. SSIM increased from 0.8274 to 0.9884 in increments of 0.161.

4. Conclusion

In this paper, a design method of the FDOE is presented to achieve miniaturization and lightweight of the optical system. The relationship between the monochromatic aberration coefficient of the FDOE and the field of view is analyzed, and a computational imaging method is proposed to correct off-axis aberration. A design example is given to show the effect of the proposed method. The example is employed to simulate the FDOE with a focal length of 50 mm, f-number of 5, and 5° field of view. By comparing the processed image with the blurred image, the restored image exhibits sharper contours and higher stripe contrast. Through the use of two evaluation methods, PSNR and SSIM, increments before and after recovery are 11.5354 and 0.161, respectively. The application of the slanted-edge method for measuring MTF after restoration, each subarea of the image is significantly improved at 111lp/mm. An average MTF improvement of 0.17 is observed in each region. The results show that the method proposed in this article significantly improves the image quality of the FDOE in the 5° field of view. The diffraction efficiency of the FDOE is greater than 95.75%. It is high enough and will not affect the quality of the image. This paper describes an easy-to-process, high-quality planar imaging system that utilizes computational imaging to eliminate the effects of off-axis aberrations on image quality and provide new insights into the integration and miniaturization of optical systems.

Funding

National Natural Science Foundation of China (62105041).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (13)
Fig. 1.
Fig. 1. Design framework of the FDOE

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Fig. 2.
Fig. 2. Edge incident light distribution diagram of the FDOE

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Fig. 3.
Fig. 3. The schematic plot of the image simulation process.

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Fig. 4.
Fig. 4. Simulation diagram of the FDOE. (a) Fresnel surface. (b) Diffraction surface.

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Fig. 5.
Fig. 5. Imaging quality of the FDOE with 2° field of view. (a) MTF. (b) Spot diagram.

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Fig. 6.
Fig. 6. Image quality of the FDOE with 5° field of view. (a) MTF. (b) Spot diagram.

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Fig. 7.
Fig. 7. Diffraction efficiency curve

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Fig. 8.
Fig. 8. Change curves of different Zernike coefficients within each field of view

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Fig. 9.
Fig. 9. PSF comparison of the FDOE. The PSF calculated by simulated software: (a) 0°, (b) 3.5°, (c) 5°.

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Fig. 10.
Fig. 10. PSF comparison of the FDOE. The PSF calculated by the model: (a) 0°, (b) 3.5°, (c) 5°.

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Fig. 11.
Fig. 11. Image simulation diagram of the ISO12233 resolution chart

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Fig. 12.
Fig. 12. Comparison images before and after edge field restoration. (a) edge field of view captured from the pre-restored resolution image; (b) edge field of view captured from the post-restored resolution image

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Fig. 13.
Fig. 13. Comparison of MTF before and after processing. (a) MTF of each area before processing; (b) MTF of each area after processing.

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Tables (5)

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Table 1. Specifications of the FDOE

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Table 2. Aberration coefficients

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Table 3. Evaluation results

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Table 4. Zernike coefficients for each sampling point with 5°

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Table 5. Evaluation results

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Equations (18)

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x = C 1 y 2 1 + 1 − ( 1 + k ) C 1 y 2 + ∑ i = 1 8 a 2 i y 2 i
{ S I F = y 4 K 3 4 [ ( ω − 1 ) S 2 + ( 1 − ω ) − 1 ] S I I F = 1 2 J y 2 K 2 ( 1 + ω ) S S I I I F = J 2 K S I V F = J 2 K ω S V F = 0 C I F = − 2 y 2 ( n − 1 ) C 1 / ν C I I F = 0
{ α 1 = arctan ⁡ x ′ I 1 = α 1 I 1 ′ = arcsin ⁡ ( 1 n sin ⁡ I 1 ) U 1 ′ = α 1 − I 1 ′
{ I 2 = U 1 ′ = U 2 U 2 ′ = I 2 ′ = arcsin ⁡ ( n sin ⁡ I 2 )
{ S I D = y 4 K D 3 4 [ 1 + ( C 2 K D ) 2 + 3 ( sin ⁡ U 2 + sin ⁡ U 2 ′ sin ⁡ U 2 − sin ⁡ U 2 ′ ) 2 ] S I I D = − y 2 J K D 2 ( sin ⁡ U 2 + sin ⁡ U 2 ′ sin ⁡ U 2 − sin ⁡ U 2 ′ ) 2 S I I I D = J 2 K D S I V D = 0 S V D = 0
K D = − m λ A 1 / π
{ S I F D O E = 1 4 y 4 [ K 3 ( 1 + 1 n ) − K D 3 A ] S I I F D O E = y 2 J ( K 2 n + 1 2 n ( n − 1 ) − K D 2 B 2 ) S I I I F D O E = J 2 ( K + K D ) S I V F D O E = − J 2 K 1 n S V F D O E = 0
g ( x , y ) = x P S F ( x , y ) ⊗ f ( x , y ) + η ( x , y )
A i = A i , Q − 1 ( h − h Q ) ( h − h Q + 1 ) ( h Q − 1 − h Q ) ( h Q − 1 − h Q + 1 ) + A i , Q ( h − h Q − 1 ) ( h − h Q + 1 ) ( h Q − h Q − 1 ) ( h Q − h Q + 1 ) + A i , Q + 1 ( h − h Q − 1 ) ( h − h Q ) ( h Q + 1 − h Q − 1 ) ( h Q + 1 − h Q )
W Y ( x , y ) = A 1 Z 0 0 + A 3 Z 1 − 1 + A 4 Z 2 0 + A 5 Z 2 2 + A 8 Z 3 − 1 + A 9 Z 4 0
P Y ( x , y ) = p ( x , y ) exp ⁡ [ i k W Y ( x , y ) ]
p ( x , y ) = { 1 x 2 + y 2 ≤ ρ 2 0 o t h e r w i s e
F P S F , Y ( x , y ) = | F [ p Y ( x , y ) ] | 2
u = min { ∥ x P S F ( x , y ) ∗ f ^ − g ∥ 2 2 + γ ∥ C f ^ ∥ p }
z f = c r 2 1 + 1 − ( 1 + k ) c 2 r 2 + ∑ i 8 α i r 2 i − n − 1 n h
η m = sin ⁡ c 2 { m − H λ [ n ( λ ) − n 0 ( λ ) ] }
H = λ 0 n ( λ ) − n 0 ( λ )
n ( λ ) = a 0 + a 1 ⋅ λ 2 + a 2 ⋅ λ − 2 + a 3 ⋅ λ − 4 + a 4 ⋅ λ − 6 + a 5 ⋅ λ − 8
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