Design method of nonsymmetric imaging systems consisting of multiple flat phase elements

Tong Yang; Dewen Cheng; Yongtian Wang

doi:10.1364/OE.26.025347

1. Introduction

Optical imaging systems play a very important role in our daily life, and it is of great significance in the areas of industry [1,2], education [3,4], defense [5], entertainment [6,7], etc. With the development of science and technology, people have higher requirements on modern imaging systems, such as larger field-of-view (FOV), larger aperture, smaller volume, fewer elements and lighter weight. In addition, special system configurations and functions are often required. Recently, non-traditional aspherical surfaces and freeform surfaces are increasingly used to correct aberrations and deal with the difficult design tasks [1,8–15]. Conventional imaging systems are based on geometric optical surfaces. However, these kinds of imaging systems has several intrinsic problems. First, the systems often use large bulks of optical materials or mirror substrate to form the surface shape, which increase the system mass and size. Secondly, for some compact system design tasks, the light inside the system has to be deflected at large angles, which lead to unphysical surface shapes and complex mathematical solution [16]. Thirdly, for system with nonsymmetric configurations, the alignment of the system is extremely difficult. The employment of freeform surfaces further increases the difficulty as they are nonrotationally symmetric and have no references.

Phase elements manipulate light in a way different from conventional imaging optical components based on true geometric surfaces. Among different kinds of phase elements, the flat or planar ones are of great significance. Compared with conventional geometric surface elements mentioned above, flat phase elements can effectively reduce the weight and volume of the total system. In addition, the alignment process of the system consisting of flat elements is much easier than the one with complex surface shapes. There are generally two kinds of phase elements. One is the diffractive element. It uses wavelength scale surface or phase structure to redirect an optical beam using diffraction rather than refraction or reflection [1,17]. Typical diffractive elements include diffractive optical element (DOE), holographic optical element (HOE), computer-generated hologram (CGH), etc. Therefore, we can realize flat phase element based on diffraction optics to control the wavefront. The other type of flat phase element is metasurface [18–21], which is a two-dimensional array of miniature, anisotropic, sub-wavelength scale scatters. This kind of flat and ultrathin optical element can arbitrarily control the wavefront by introducing spatial variations generated by effective dielectric and magnetic properties. Using both kinds of flat phase element, it is possible to extend the imaging system design field and realize the compact, light-weight, and easy-aligned optical imaging systems. In addition, it is easy to integrate multiple phase elements into a single flat element.

To realize an imaging system consisting of flat phase elements, the task of optical design is to determine the locations of the elements as well as their phase functions to get the best imaging performance. Many methods can be proposed for the design of phase functions (or phase profiles) of flat phase elements in an optical system. For the numeric-type diffractive elements, iterative optimization methods are often used [1,22,23], such as inverse fast Fourier transform (FFT) method, Gerchberg-Saxton (G-S) algorithm [24], simulated annealing, genetic algorithms, etc. However, these methods are often used for the design of beam shapers, spot array generators, etc, not for imaging optics. In addition, only one phase profile can be obtained using these methods for most cases. In general, closed-form or analytic-type phase functions are preferred in imaging optics, and optical design software are often used. Traditional design method for this kind of system often starts from a system with certain optical power, approximate system specifications and not too bad imaging performance. However, if we are designing novel imaging optics consisting of only flat phase elements, especially the systems with special nonsymmetric configuration and advanced system specifications, feasible starting points cannot be found in most cases. In this way we have to create such a system, or start the design from only a combination of simple planes with no phase functions. Therefore the design is very difficult and very likely to fail. To deal with the intrinsic problems of the traditional design methods, some researchers proposed direct or point-by-point optical design methods of phase elements, which can handle the design problems in another way. Hong et al proposed the design of near-eye visor using flat metasurface [16]. During the design the phase gradient on the flat phase element is calculated point-by-point directly to bend the light in different directions. However only one element is considered in their design. So this method is not suitable for the general multi-surface design problem. João Mendes-Lopes et al proposed the improved SMS method [25]. Kinoform surfaces (diffractive surfaces) are calculated together directly with the remaining surfaces of the optical system through a point-by-point design method. It realized the good control of two or three wavefronts using one or two surfaces. So there is a restriction on the number of field points considered in the design process. In addition, at present, the designs are restricted to conventional co-axis configurations. For an actual imaging system, it works for a certain object size and a certain width of light beam. So it is a basic requirement to consider the light rays of multiple fields and different pupil positions during the imaging system design. Furthermore, the design task generally requires multiple phase elements design in off-axis nonsymmetric configurations.

In this paper, we propose a novel design method and realize the design of off-axis nonsymmetric imaging systems consisting of flat phase elements. Here, without loss of generality, the word “element” in the design process refers to a single reflective phase element or a single phase surface (such as refractive surface). Compared with other traditional design methods of phase elements, the phase profiles or functions are generated point-by-point directly based on the given system specifications and configuration. The whole design process starts from an initial system using simple geometric planes. The dependence on existing starting points is significantly reduced and advanced design skills are not required. In comparison with other direct or point-by-point design methods, in our method, multiple phase elements can be designed and the rays of multiple fields and pupil positions are employed. The phase functions of the flat elements are designed quickly and effectively by connecting and integrating the real three-dimensional (3D) space and the phase function space. In addition, closed-form phase functions can be obtained which are easy to be integrated with optical design software and optimization. This method can be taken as a fast phase retrieval method to some degree. Using the proposed design framework we can realize the same functions and applications of advanced geometric optical design technologies (such as freeform optics technology), but can generate more compact, lighter-weight and easier aligned systems. To demonstrate the feasibility of the proposed design method, we present a compact example imaging system that exhibits high performance. This design framework can be applied in many areas, such as virtual reality (VR) and augmented reality (AR), miniature cameras, high-performance telescopy, microscopy, remote sensing, illumination design and beam shaping, etc.

2. Design method

2.1 Design setup

The whole design process consists of four steps. The first step is to perform the setup work for the design, including the establishment of initial system and defining the feature rays. Then, the phase functions of the phase elements are preliminarily constructed based on the given object-image relationship or ray-mapping relationship. Next, the imaging performance of the system is significantly improved by an iterative process of phase functions. Finally, the final system is obtained through an optimization process using optical design software.

An initial system consisting of true geometric planes is firstly established for the following design process. The locations and orientations of the planes are approximately determined based on the given requirements on system configuration, and the folding geometry of the light path should be also maintained. No phase function is applied onto these elements at this time. It means that the light will not converge or diverge by these flat geometric surfaces and the whole system is afocal. This simple initial system actually consisting of “flat geometric surfaces” is taken as the input for the subsequent design.

The phase functions of the flat phase elements are designed by a “point-by-point” strategy. The phase values at the points where the light rays intersect the flat phase element are calculated respectively and a closed form of phase function will be then generated for the final optimization. To fully characterize and design non-symmetric imaging systems and phase functions, the rays used in the design process should be sampled from the entire aperture and full FOV. This is also the basic principle for designing an actual imaging system works for a certain FOV and a certain width of light beam. Let us assume that TF fields among the full FOV are employed in the design. There are many ways to sample the rays in one field. Here the common polar ray grid for the generally used circular aperture is used, as shown in Fig. 1. The aperture of each field can be divided into TA angles in total. Along each radial direction from the aperture center, TP pupil positions are selected. So the total ray number is TR = TF × TA × TP. The goal of the system design task is to redirect the feature light rays R_i (i = 1…TR) to their corresponding ideal image points I_i_,ideal (i = 1…TR) on the image plane. The ideal image points can be easily determined based on the given image plane position and the system specifications (such as focal length, magnification, field angle, object height, etc).

Fig. 1 The polar ray grid for defining the feature rays of each field.

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2.2 Preliminary construction of phase functions

When the initial system has been established and the feature rays have been defined, the next step is to construct the unknown phase functions or profiles for the flat elements. The function graph or profile can be considered as a three-dimensional surface in R³. Here we may call it “phase function space”. In this step, the phase function are constructed by the discrete data points in the phase function space corresponding to the feature light rays and intersections in the real 3D space.

Considering the travelling of a general feature ray R_i (i = 1,2…TR) at a flat phase element. It comes from point S_i = [S_i,x, S_i,y, S_i,z]^T and is redirected at P_i = [x_i, y_i, z_i (x_i,y_i)]^T on the phase element to E_i = [E_i,x, E_i,y, E_i,z]^T, as shown in Fig. 2. The unit incident and outgoing direction vector can be written as r_i = (P_i−S_i)/|P_i−S_i| = [r_i,x, r_i,y, r_i,z]^T and r_i' = (E_i−P_i)/|E_i−P_i| = [r_i,x', r_i,y', r_i,z']^T respectively. All these points and vectors are expressed with respect to the global coordinate system Oxyz. The optical path length (OPL) from S_i to E_i can be written as follows [23,25]

O P L (S_{i}, E_{i}) = n | P_{i} - S_{i} | + n' | E_{i} - P_{i} | + \frac{m λ}{2 π} ϕ (P_{i}),

where m is diffraction order, λ is the wavelength, n and n' are the refractive indices preceding and after the phase surface. If we are dealing with reflective flat phase element located in air, then n = n' = 1. Here ϕ is the phase function or phase profile of this flat phase element (expressed in the local coordinate system of this element) [1,23,25]. It defines the phase shift or phase jump for the incident wavefront at each point, and this phase shift is for the m = + 1 diffraction order in general. For the light diffracted into other orders, the optical path length will change according to Eq. (1) and the rays will travel into different directions, which will be seen later from the general refractive/reflective equation. Note that ϕ is a function of x and y coordinate, not z. ϕ(P_i) represents the phase value (or phase jump, phase shift) on the phase element at point P_i. It has the unit of radian. When the phase shift is divided by 2π then multiplied by λ, it can be transformed into optical path length. The role of phase function or phase profile ϕ can be also defined with respect the eikonal functions of incident and diffracted wavefronts [25], which is in consistence of the concept of optical path length. In addition, Eq. (1) is given for the diffractive surface case. For metasurface, the concept of diffraction order generally may not exist and the true phase function or phase shift should be ϕ_meta = mϕ (m = 1). However, it does not affect the solving process of phase functions. So here without loss of generality we still adding order m during the following calculation.

Fig. 2 Propagation of one ray through the phase element. (a) Refractive flat surface. (b) Reflective flat surface.

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Assume the local coordinates of P_i on the flat phase element is (x_i^local, y_i^local, z_i^local). If the coordinate of local origin O respect to the origin of the global coordinates is (x_o, y_o, z_o), α is tilt angle of the local surface y-axis with respect to the global y-axis, and it is a left-handed rotation angle about the + x-axis. We have

[\begin{matrix} x_{i}^{l o c a l} \\ y_{i}^{l o c a l} \\ z_{i}^{l o c a l} \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & \cos α & - \sin α \\ 0 & \sin α & \cos α \end{matrix}] [\begin{matrix} x_{i} \\ y_{i} \\ z_{i} \end{matrix}] .

Based on Fermat’s principle, we have

\frac{\partial O P L}{\partial x} = 0, \frac{\partial O P L}{\partial y} = 0.

In addition, phase function ϕ is generally expressed in the local coordinates. So ϕ(P_i) = ϕ(x_i^local, y_i^local). Using the first differential operation of Eq. (3),

\begin{array}{l} n \frac{(x_{i} - S_{i, x})}{| P_{i} - S_{i} |} + n \frac{(z_{i} - S_{i, z})}{| P_{i} - S_{i} |} \frac{\partial z_{i} (x_{i}, y_{i})}{\partial x_{i}} + n' \frac{(x_{i} - E_{i, x})}{| E_{i} - P_{i} |} + n' \frac{(z_{i} - E_{i, z})}{| E_{i} - P_{i} |} \frac{\partial z_{i} (x_{i}, y_{i})}{\partial x_{i}} \\ - \frac{m λ}{2 π} [\frac{\partial ϕ (P_{i})}{\partial x_{i}^{l o c a l}} - \frac{\partial ϕ (P_{i})}{\partial y_{i}^{l o c a l}} \frac{\partial z_{i} (x_{i}, y_{i})}{\partial x_{i}} \sin α] = 0. \end{array}

It can be simplified into

[n' r_{i, z}' - n r_{i, z} + \frac{m λ}{2 π} \frac{\partial ϕ (P_{i})}{\partial y_{i}^{l o c a l}} \sin α] \frac{\partial z_{i} (x_{i}, y_{i})}{\partial x_{i}} + [n' r_{i, x}' - n r_{i, x} - \frac{m λ}{2 π} \frac{\partial ϕ (P_{i})}{\partial x_{i}^{l o c a l}}] = 0.

Similarly, using the second differential operation of Eq. (3), we have

[n' r_{i, z}' - n r_{i, z} + \frac{m λ}{2 π} \frac{\partial ϕ (P_{i})}{\partial y_{i}^{l o c a l}} \sin α] \frac{\partial z_{i} (x_{i}, y_{i})}{\partial y_{i}} + [n' r_{i, y}' - n r_{i, y} - \frac{m λ}{2 π} \frac{\partial ϕ (P_{i})}{\partial y_{i}^{l o c a l}} \cos α] = 0.

Combining the above two equations, we can get

[n' r_{i, y}' - n r_{i, y} - \frac{m λ}{2 π} \frac{\partial ϕ (P_{i})}{\partial y_{i}^{l o c a l}} \cos α] \frac{\partial z_{i} (x_{i}, y_{i})}{\partial x_{i}} - [n' r_{i, x}' - n r_{i, x} - \frac{m λ}{2 π} \frac{\partial ϕ (P_{i})}{\partial x_{i}^{l o c a l}}] \frac{\partial z (x_{i}, y_{i})}{\partial y_{i}} = 0.

The vector form of the general refractive/reflective equation can be obtained based on Eqs. (5)-(7)

n' (N_{i} \times r_{i}') = n (N_{i} \times r_{i}) + \frac{m λ}{2 π} [N_{i} \times T \nabla ϕ (P_{i})],

where

N_{i} = {[\frac{\partial z_{i} (x_{i}, y_{i})}{\partial x_{i}}, \frac{\partial z_{i} (x_{i}, y_{i})}{\partial y_{i}}, - 1]}^{T}

is the global unit surface normal at feature data point P_i in the real 3D space. Here we consider the conventional case whereby the optical system has plane symmetry about the meridional plane (yOz plane). As the surface is flat, the surface normal at any point on the surface can be expressed as N_i = [0, sinα, cosα]^T.

\nabla ϕ (P_{i}) = {[\frac{\partial ϕ (P_{i})}{\partial y_{i}^{l o c a l}}, \frac{\partial ϕ (P_{i})}{\partial x_{i}^{l o c a l}}, 0]}^{T} = {[\frac{\partial ϕ}{\partial x} | \begin{matrix} P_{i} \end{matrix}, \frac{\partial ϕ}{\partial y} | \begin{matrix} P_{i} \end{matrix}, 0]}^{T}

is the gradient vector of phase function at P_i. T is a transformation matrix for ∇ϕ(P_i) from local coordinate system to global coordinate system, which can be written as

T = [\begin{matrix} 1 & 0 & 0 \\ 0 & \cos α & \sin α \\ 0 & - \sin α & \cos α \end{matrix}] .

Equation (8) can be transformed into

n' r_{i}' - n r_{i} - \frac{m λ}{2 π} T \nabla ϕ (P_{i}) = k N_{i},

where k is a scale factor. From Eq. (8) and Eq. (10), we can find that the two vectors

n' r_{i}' - n r_{i} - \frac{m λ}{2 π} T \nabla ϕ (P_{i})

and N_i have the same direction but different norms. So k can be considered as the ratio of two vectors. Therefore, k is not pre-defined before the design process, but it varies with the different design cases and feature rays. Equation (10) can be expanded in scalar form to give

{\begin{array}{l} n' r_{i, x}' - n r_{i, x} - \frac{m λ}{2 π} \frac{\partial ϕ}{\partial x} | \begin{matrix} P_{i} \end{matrix} = 0 \\ n' r_{i, y}' - n r_{i, y} - \frac{m λ}{2 π} \frac{\partial ϕ}{\partial y} | \begin{matrix} P_{i} \end{matrix} \cos α = k \sin α \\ n' r_{i, z}' - n r_{i, z} + \frac{m λ}{2 π} \frac{\partial ϕ}{\partial y} | \begin{matrix} P_{i} \end{matrix} \sin α = k \cos α \end{array} .

It can be rearranged into matrix form

[\begin{matrix} \frac{m λ}{2 π} & 0 & 0 \\ 0 & \frac{m λ}{2 π} \cos α & \sin α \\ 0 & - \frac{m λ}{2 π} \sin α & \cos α \end{matrix}] [\begin{matrix} \frac{\partial ϕ}{\partial x} | \begin{matrix} P_{i} \end{matrix} \\ \frac{\partial ϕ}{\partial y} | \begin{matrix} P_{i} \end{matrix} \\ k \end{matrix}] = [\begin{matrix} - n' r_{i, x}' + n r_{i, x} \\ - n' r_{i, y}' + n r_{i, y} \\ - n' r_{i, z}' + n r_{i, z} \end{matrix}]

Using this equation, we can calculate

\frac{\partial ϕ}{\partial x} | \begin{matrix} P_{i} \end{matrix}

and

\frac{\partial ϕ}{\partial y} | \begin{matrix} P_{i} \end{matrix}

. The value of k can also be calculated. But it has no practical use in the design process. For all the above equations, if we are dealing with reflective flat phase element located in air, then n = n' = 1. The graph or profile of the phase function can be considered as a surface in R³ (or the phase function space). The point corresponding to P_i in the phase function space (on the phase profile) is denoted as P_i_,𝜙. According to the knowledge of differential geometry, the “surface normal” at P_i_,𝜙 in the phase function space can be written as N_i_,𝜙 =

{[\frac{\partial ϕ}{\partial x} | \begin{matrix} P_{i} \end{matrix}, \frac{\partial ϕ}{\partial y} | \begin{matrix} P_{i} \end{matrix}, - 1]}^{T}

. In this paper, we expect to get smooth and continuous closed-form of phase functions. Let us assume that P_𝜙 is a point on the graph of a smooth and continuous phase function; In addition, P*_𝜙 is also a point on the graph of this phase function and is in the small neighborhood of P_𝜙. According the knowledge of differential geometry, P*_𝜙 should be approximately on the “tangent plane” of P_𝜙. In this way, the spatial relationship between P_𝜙, P*_𝜙 and the surface normal N_𝜙 at P_𝜙 is approximately determined. During the preliminary construction process of the phase functions, the phase value of the discrete data points as well as the surface normal in the phase function space are calculated one-by-one. These phase values and surface normals should not only well control the light rays as desired, but should also satisfy the general geometric property of a smooth and continuous phase function depicted above in order to generate a valid solution. Based on this, we may use the following principle in calculating the points on the surface in the phase function space: If we have already obtain i data points in total, the (i + 1)th data point is chosen to be nearest point to the “point-cloud” consisting of the i data points. The (i + 1)th data point should be on the tangent plane of the nearest data point among the i data points that has been calculated already. In this way the geometric property of the phase function can be approximately maintained.

Based on the above design principle (used to find the next data point) as well as the Fermat’s principle and generalized refractive/reflective equation (used to control the light rays), we can proposed the detailed design steps for constructing the phase function. The phase functions of the flat phase elements are constructed one-by-one starting from the initial system. Detailed steps are as follows:

(1) Select the first phase element (j = 1).
(2) When constructing the phase function of the jth phase element, all the other phase functions as well as the locations of all the phase elements are kept the same with their current states. The intersection of the first feature ray R₁ with the flat phase element in the real 3D space can be defined as P₁. The projected point of P₁ into the phase function space is defined as P_1,𝜙, as shown in Fig. 3(a). This point has the same local x and y coordinates with P₁. We may set a phase value ϕ(P₁) for point P_1,𝜙 (or P₁). Note that the determining of this phase value alone has no impact on the propagation of light rays. In general, the phase values of all the points will be uniformly increased or decreased in the final step to make the phase value at the surface vertex (with zero local x and y coordinates) to be zero. It should be noted that R₁ means the first ray (i = 1) among the TR rays in total used in the construction process, but it is not necessarily the chief ray. However, without loss of generality, we may choose the chief ray of the central field among the full FOV to be the first ray R₁. In this case, P₁ and P_1,𝜙 are defined as the vertices of this phase element and phase function respectively.
(3) When we have obtained the ith feature ray R_i, the intersection P_i of R_i with the phase element, and the ith data point P_i_,𝜙 in the phase function space, it is expected that the corresponding feature ray R₁ should be redirected at P_i to its ideal image point I_i_,ideal on the image plane. The intersection of R₁ with the phase element preceding the current element in real 3D space is defined as point S_i, as shown in Fig. 4, which can be easily obtained through ray tracing. If the current jth phase element is the one neighboring to the image plane, I_i_,ideal can be taken as the end point E_i. A more general case is that there are other phase elements (assume the total number is q) located between the current phase element and the image plane. We may define the intersections of R_i with these flat surfaces as E*_i_,𝜉 (ξ = 1…q), whose local coordinates can be expressed as (x*_i_,𝜉, y*_i_,𝜉, 0). The phase functions of these elements are ϕ_ξ (ξ = 1…q) respectively. The refractive index between E*_i_,𝜉 and E*_i_{,𝜉 + 1} is n_𝜉. The OPL between P_i and I_i_,ideal can be written as $O P L (P_{i}, I_{i, i d e a l}) = n' | E_{i, 1}^{*} - P_{i} | + n_{q} | I_{i, i d e a l} - E_{i, q}^{*} | + \sum_{ξ = 1}^{q - 1} n_{i} | E_{i, ξ + 1}^{*} - E_{i, ξ}^{*} | + \frac{m λ}{2 π} \sum_{ξ = 1}^{q} ϕ_{ξ} (E_{i, ξ}^{*}) .$
Based on Fermat’s principle, the variation of OPL(P_i, I_i_,ideal) with respect to x*_i_,𝜉 and y*_i_,𝜉 (ξ = 1…q) should be zero. So we can establish differential equations and solve for x*_i_,𝜉 and y*_i_,𝜉. In this way the point E*_i_,1 on the flat surface neighboring to the current phase element can be obtained, and it is taken as the end point E_i. Using the method depicted above, we can calculate the “surface normal” N_i_,𝜙 at P_i_,𝜙, as shown in Fig. 3(b). The tangent plane of P_i_,𝜙 in the phase function space can be also obtained.

(4) In order to determine the next feature ray R_i₊₁ as well as its data point P_i₊₁ on the phase element in the real 3D space, we first calculate the intersections of the remaining rays (the rays which have not been used to calculate their corresponding projected data points in the phase function space) with the phase element in the real 3D space, as shown in the sketch plot Fig. 3(b). Calculate the distances from P_i to the intersections in the real 3D space. Find the shortest distance, and the corresponding feature ray and intersection are defined as the next feature ray R_i₊₁ and data point P_i₊₁, as denoted in Fig. 3(b).
(5) Calculate the distances from P_i₊₁ to P_η (η = 1…i), as shown in Fig. 3(c). Find the shortest distance, and the corresponding intersection and data point in phase function space are Q_i and Q_i_,𝜙 respectively. The surface normal and tangent plane of Q_i_,𝜙 in phase function space have already known. Then we can find the point on the tangent plane that has the same local x and y coordinates of P_i₊₁ and get the phase value. This point is taken as the next feature point P_i_+1,𝜙 in the phase function space. Then i increases by 1.
(6) Repeat steps (3)-(5) until all the data points as well as their surface normals in the phase function space have been calculated. Then we can fit the data points into a closed-form phase function. This is the output of the construction process for one phase element. There are many kinds of mathematical descriptions of closed-form phase functions. Among them, the polynomial expansion is a special and often-used one and it is also used in the design example in Section 3. Here a fitting method based on least-square algorithm considering both the coordinates and surface normals of the data points can be used to improve the fitting accuracy [26]. The sketch plot for the graph of phase function after the fitting process is given in Fig. 3(d).
(7) Until now, the phase function of the jth element is obtained. Then j increases by 1. Repeat steps (2)-(6) until the phase functions of all the phase elements have been constructed.

Fig. 3 Design steps of the preliminary construction stage of phase function for jth element. (a) Find the first data point P₁ in the real 3D space and its projected point P_1,𝜙 in the phase function space. (b) When P_i and P_i_,𝜙 have been obtained, the “surface normal” N_i_,𝜙 at P_i_,𝜙 can be calculated. Calculate the distances from P_i to the intersections of the remaining rays with the phase element in the real 3D space. Find the shortest distance and the corresponding feature ray and intersection are defined as R_i₊₁ and P_i₊₁. (c) Find the nearest point Q_i among all the used data points in real 3D space to P_i. The corresponding projected point in phase function space is Q_i_,𝜙. Find the projected point P_i_+1,𝜙 of P_i₊₁ on the tangent plane of Q_i_,𝜙 in the phase function space. (d) When all the data points as well as their surface normals in the phase function space have been calculated, fit the data points into a closed-form phase function considering both the coordinates and surface normals. The graph of the phase function is sketched here.

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2.3 Iterative design of phase functions

Using the above design steps, the phase functions can be constructed preliminarily. The phase value and surface normal in the phase function space at each discrete data point calculated by the proposed method can accurately redirect each ray to its ideal image point. However, the closed-form phase function is obtained through a fitting method considering both the coordinates and the surface normals of the data points. It should be noted that there is generally no solution that the 3D graph of phase function passes all the projected data points in the phase function space, and the “surface normals” of the actual phase function well match the normals obtained by calculation at the same time. The fitting process is an approximation. That is to say, the rays may not reach their corresponding ideal image points after preliminary construction, especially using only one phase element. However, this goal can be approximated achieved by constructing multiple phase elements in the system. In addition, an iteration process is used to regenerate the phase functions in order to improve the performance. The design result of the preliminary construction process is taken as the design input for the iteration stage. In each iteration step, the phase functions of the flat phase elements are also designed one-by-one. When constructing the phase function of one phase element, all the other phase functions as well as the locations of all the phase elements are also kept the same with their current states. The intersections of the feature rays with the phase elements can be obtained and their projected data points in the phase function space can be calculated. Keeping the coordinates of the data points unchanged, the “surface normals” of the data points in the phase function space can be recalculated using the method demonstrated in the Section 2.2. Then the phase function can be obtained using the fitting method considering both the coordinates and surface normal. This new phase function is used to replace the old one. This design process is used for other phase elements in the system and a single iteration step is completed. The iteration step can be repeated and the imaging performance will be significantly improved. This system can be taken as the starting point for further optimization.

Fig. 4 The schematic view of the intersections of one feature ray R_i with the phase plates in real 3D space.

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The flowchart of the whole preliminary construction and iteration process is given in Fig. 5. The green boxes show the general operations. Blue boxes represent the operations in the real 3D space and yellow boxes represent the operations in the phase function space. There is not a step to add or extract one of the phase elements during the point-by-point design process. The total number of phase elements in the system (TEle) is not changed (Of course these elements are evolved from initial geometric planes). #NumEle (from 1 to TEle) is the sequence number of the current phase element under design. The locations of the phase elements are not changed during the point-by-point design process. The whole design process can be also seen as a fast phase retrieval method to some extent.

Fig. 5 Flowchart of the design process for the construction and iteration stage.

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To increase the working efficiency, the whole point-by-point design process is implemented into a program. The directions of the feature rays and their intersections with the surfaces (including the image plane for evaluating the image quality) are obtained with real ray trace using optical design software CODE V. MATLAB is employed for data processing, feature points calculation, phase function graph fitting, system specifications inputting, and data display. As the CODE V API (application programming interface) uses the Microsoft Windows standard Component Object Model (COM) interface, users can execute CODE V commands using MATLAB, which supports Windows COM architecture. The final MATLAB program enables automatic design of the phase functions point-by-point.

3. Design example

In this section, an example is proposed to demonstrate feasibility of the design method depicted in Section 2. The example is an off-axis reflective imaging system consisting of flat phase elements. The system has a 53mm entrance pupil diameter and the FOV is 8° × 6°. The F-number of the system is 1.798 and the working wavelength is 1.06μm. In order to make the system compact, the light beams are fully folded and overlaid inside the system. The alignment process of the system consisting of flat elements is much easier than the one with complex geometric surface shapes. In addition, in traditional off-axis three-mirror systems, the system are consisted of discrete primary mirror, secondary mirror and tertiary mirror. In the design example demonstrated in this paper, the primary mirror and tertiary mirror are in fact integrated on a single flat phase element. The two different phase functions for the original primary and tertiary mirrors are fabricated onto the different areas of this single element. So the light beams are reflected twice (the first time and the third time) on this single element inside the system. Using this design concept the degrees of freedom for the system alignment process can be significantly reduced. In addition, compared with similar design strategy in the off-axis geometric optical design field [11,27,28], it is much easier to realize the integration of multiple elements into one in the design of flat phase element systems. Based on the predetermined system configuration and folding geometry, the initial system consisting of geometric planes with no phase functions is firstly established for the following design process, as shown in Fig. 6(a). The next step is to sample the feature light rays using polar ray grid. The aperture of each field is divided into TA = 16 angles in total. Along each radial direction from the aperture center, TP = 7 pupil positions are selected. We use 6 sample fields in half FOV during the design process: (0°, 0°), (0°, 3°), (0°, −3°), (4°, 0°), (4°, 3°), and (4°, −3°). Therefore, in total, TR = 6 × 16 × 7 = 672 rays are used in the preliminary construction stage and iterative design stage. The system works under diffraction order m = 1 if using diffractive surfaces. For the case using metasurface, the phase function ϕ_meta of each element equals to ϕ directly.

Fig. 6 Preliminary construction stage and iteration stage of the design example. (a) Initial system using simple geometric planes. (b) System layout after the preliminary construction stage. (c) The change of σ_RMS with iterations. (d) Distortion grid of the system after iterations. (e) System layout after iterations.

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Then, the phase functions of the three flat phase elements can be constructed using the method depicted in Section 2.2. The system after the three phase functions have been preliminary constructed is as shown in Fig. 6(b). We can use the root-mean-square (RMS) deviation σ_RMS of between the actual and ideal image points of the feature rays to evaluate the system performance. It is defined as:

σ_{R M S} = \sqrt{\frac{\sum_{ζ = 1}^{T R} σ_{ζ}^{2}}{T R}},

where σ_ς (ς = 1…TR) is the deviation of the image point for the ςth feature ray. This is a comprehensive evaluation index as it can indicate both the image quality and distortion. After preliminary construction, σ_RMS = 1.3778mm. This system is taken as the input for the further iterative design stage. The change of σ_RMS with iterations are given in Fig. 6(c). Here, the result of iteration step 0 denotes the result after preliminary construction. After iterations, σ_RMS equals to 0.1794mm, which decreased by 86.98% compared with the value before iterations. The distortion grid of the system is shown in Fig. 6(d). The layout of the system is as shown in Fig. 6(e) and it is taken as a good starting point for further optimization.

The final optimization of the system is conducted in optical design software CODE V. The type of phase function chooses to be XY polynomials. In general, using more and higher order surface coefficients means there are more design freedoms and it will lead to better imaging performance. However, the order should not be too high considering the fabrication and testing. Using a smaller number of surface coefficients to correct the system aberration is an optical design principle [1,29], especially for the design with complex surfaces. The system design with phase elements using polynomial phase function is similar with the design using XY polynomial freeform surface. As have been discussed in [29–31], there is a relationship between polynomial coefficients and aberrations. The polynomial surface up to 4th order corresponds to primary aberrations (or 4th order wave aberrations) approximately, which are the dominant aberrations in the optical system before aberration correction. As a result, during the point-by-point design stage of phase profiles, 4th order polynomial surface was used and it is adequate for a starting point design. During the optimization process, we still used 4th order XY polynomials in the beginning. However, during the design, we found the image quality was limited by higher-order aberrations (6th order wave aberrations). So we gradually upgrade the phase function and employed 6th order polynomials to correct the higher-order aberrations and achieve a satisfying result. As the optical system is symmetric about the YOZ plane, only the even terms of x are used. The form of phase function is

\begin{array}{l} ϕ (x, y) = A_{2} y + A_{3} x^{2} + A_{5} y^{2} + A_{7} x^{2} y + A_{9} y^{3} + A_{10} x^{4} + A_{12} x^{2} y^{2} + A_{14} y^{4} \\ + A_{16} x^{4} y + A_{18} x^{2} y^{3} + A_{20} y^{5} + A_{21} x^{6} + A_{23} x^{4} y^{2} + A_{25} x^{2} y^{4} + A_{27} y^{6}, \end{array}

where A_i is the coefficient of the xy terms. All the coefficients of the phase functions as well as the locations of the phase elements and image plane are set as variables during the optimization process. The error function type used in the optimization is the default transverse ray aberration type in CODE V. The integration of the “primary mirror” and “tertiary mirror” is maintained by using proper coplanar constraints. The effective focal length and distortion of the system are controlled by constraining the actual image points of the chief rays for different fields via real ray trace data. After quick optimization, we can get the final design result, as shown in Fig. 7(a). The modulation transfer function (MTF) curves of the system are all above 0.5 at 100 lps/mm spatial frequency for all the fields, as displayed in Fig. 7(b). The error function given by CODE V for the final design is 0.53 and the imaging performance is satisfactory. We also try the same design example using polynomials up to 4th and 5th order under the same design constraints. The error functions are 6.22 and 3.25 respectively. When polynomials higher than 6th order are used, the image quality improves a little. In addition, for the system consisting of phase elements, efficiency is important and some incident light may become stray or unwanted light after the phase element. For the diffractive surface case, the depth and shape of each diffractive zone should be optimized to improve the diffraction efficiency. In this way, highest amount of incident light can be diffracted into the desired order of diffraction. Using reasonable design or optimization method or strategy, the efficiency can be closed to 100% [1,23,25,32]. So the MTF curves show in Fig. 7(b) may drop by a very small amount. The optimization of efficiency is beyond the scope of this paper and it is a further research we hope to be done. The case is similar if the system uses metasurface as the phase elements. The distortion grid of the system is shown in Fig. 7(c).

Fig. 7 Final design result of the example. (a) System layout. (b) MTF plot. (c) Distortion grid.

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The conventional systems using non-flat elements, such as the example systems given in [11,27,28], use off-axis nonsymmetric configuration and nonsymmetric freeform surfaces, which have no alignment references. So the alignment process is very difficult. Many well-designed auxiliary surfaces or references have to be carefully fabricated and employed to assist the alignment [11,27], which in turn add the complexity of the system. However, if we use flat phase elements in the system, the elements themselves can be taken as the alignment references, which reduce the alignment difficulty. In addition, when using flat elements in the system, it is much easier to fabricate multiple elements into a single flat phase element (compared with the systems given in [11,27], as the complexity and difficulty for the data-handling and the fabrication are significant). In this way, the degree of freedom using the alignment will be significantly reduced. The above discussions demonstrate that the systems consisting of flat phase elements can markedly simplify the system alignment process.

For the easy-aligned design example given in [28], one element in the off-axis three-mirror system is a geometric-plane-mirror to reduce the alignment difficulty. But the imaging performance will be sacrificed. The system works at an F-number of 2 and the FOV of the system is 3° × 3°. The entrance pupil is 30mm. Compared with the example in [28], in the design example of this paper, as flat phase elements can also contribute the aberration correction, the specifications of the system in this paper (53mm entrance pupil diameter, 8° × 6° FOV, 1.798 F-number) are much superior and the imaging performance is better. These comparisons show the feasibility of our design consisting of flat elements.

4. Conclusions and discussions

We demonstrate a novel design method and realize the design of off-axis nonsymmetric imaging systems consisting of flat phase elements. The rays of multiple fields and pupil positions are employed and multiple phase elements can be designed within the proposed method. The design framework enables the whole design starting from an initial system using simple geometric planes. The phase functions of the flat elements are designed quickly and effectively via a point-by-point approach by connecting and integrating the real 3D space and the phase function space. In addition, closed-form phase functions can be obtained which are easy to be integrated with optical design software and optimization. Using the proposed design framework we can realize the same functions and applications of freeform optics, but generating more compact, lighter-weight and easier aligned systems. To demonstrate the feasibility of the proposed design method, we present a compact example imaging system that exhibits high performance. This design framework can be applied in many areas of imaging fields and even non-imaging field such as illumination design and beam shaping.

The goal of this paper is to propose a novel point-by-point design method of off-axis nonsymmetric imaging systems consisting of flat phase elements, but not to demonstrate methods that can generate systems lighter than the ones designed by other methods of flat phase elements. Other design methods, both traditional and novel point-by-point methods, may also generate lighter systems. In addition, flat phase elements such as diffractive optical elements and metasurface can realize the same functions and applications of systems consisting of geometric surfaces, but the elements can be made ultra-thin, which will lead to much lighter systems in general. However, the total mass of a system consisting of flat phase elements is highly dependent on the actual fabrication techniques and different specific conditions. So it is difficult to qualitatively calculate how much lighter the system using flat phase elements will be than traditional geometric surfaces system. Realizing and developing optical systems with satisfactory performance and the lightest weight is the goal of our future research.

The method demonstrated in this paper can be taken as a fast phase retrieval method to some degree. In addition, the proposed method in this paper can be easily extended to the design of imaging systems containing both geometric freeform surfaces and flat phase elements. This kind of system is also of important value and applications [33]. The elements in the system are also designed one-by-one. The phase element are designed with the method depicted in this paper and the freeform surface can be also generated by point-by-point design methods [28,34,35]. Currently, in the design process proposed in this paper, only one wavelength can be considered. To overcome the problem of strong chromatic aberrations, some other techniques such as harmonic DOE and multiwavelength metasurfaces [36–38] can be used and integrated in the design framework in the future.

Funding

National Natural Science Foundation of China (61805012, 61727808); National Key Research and Development Program of China (2017YFA0701201); Beijing Institute of Technology Research Fund Program for Young Scholars.

Acknowledgments

We thank Synopsys for the educational license of CODE V.

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Design method of nonsymmetric imaging systems consisting of multiple flat phase elements

Abstract

1. Introduction

2. Design method

2.1 Design setup

2.2 Preliminary construction of phase functions

2.3 Iterative design of phase functions

3. Design example

4. Conclusions and discussions

Funding

Acknowledgments

References

Cited By

Figures (7)

Equations (15)

Optics Express