Open Access
Add to BibSonomy
Abstract
Imaging systems consisting of flat phase element such as diffractive optical element, holographic optical element, and metasurface have important applications in many fields. However, there is still a lack of generalized and efficient design methods of these systems, especially for systems with nonsymmetric configurations. We proposed a design method of imaging system consisting of flat phase elements based on confocal properties. The description of the generalized phase function for realizing point-to-point stigmatic imaging is derived. Given the focal length or magnification as well as the locations of the elements based on the design requirements, the phase functions can be calculated very fast and stigmatic imaging of the central field is realized. The systems can be taken as good starting points for further optimization, during which the rotationally symmetric or freeform phase terms can be added. Several design examples are demonstrated to validate the feasibility of the method. The proposed method increases design efficiency while decreasing the dependence on existing systems and skills significantly, and can be easily integrated into optical design software.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A growing trend in imaging optics is to use phase elements, such as diffractive optical elements (DOE), holographic optical elements (HOE), and metasurfaces. Compared with traditional geometric optical elements, phase elements can be made ultrathin, which lead to compact and light-weight systems. In addition, the use of phase elements can break the limit of traditional refractive and reflective equation. Light incident on the phase element can be redirected into “unconventional” directions compared with pure geometric elements, thus leading to compact and flexible configurations. DOE redirects an optical beam using diffraction. The fabrication methods of the surface relief wavelength-scale diffractive structures include diamond turning, direct laser writing and lithography techniques, etc. DOE (including refractive/diffractive and reflective/diffractive hybrid elements) have been successfully used in imaging optics such as infrared imaging systems [1,2], commercial cameras [3] and head-mounted display [4], where DOE plays an important role in correcting the chromatic aberration and thermal aberration. HOE is in fact a kind of DOE, but it is generally fabricated through laser interference recording using holographic materials such as silver halide and photopolymer. Due to the angular and wavelength selectivity, HOE has many advantages and have been used in the see-through augmented-reality (AR) systems [5–12] and other applications. Metasurface consists of arrays of subwavelength meta-atoms which can be considered as light resonators, and can be fabricated by electron beam lithography (EBL) and nanoimprint lithography (NIL), etc. Light wavefront can be modulated due to the effective dielectric and magnetic properties and light-matter interaction. In recent ten years, there is a fast development of imaging systems using metasurface [13–18].
For all the above kinds of the phase elements, the wavefront modulation effect can be considered to be induced by the local phase change or phase shift at the element. In this way, the effect of the phase element can be characterized by its phase function. The key to the design of phase elements in imaging optical systems is to obtain the corresponding phase functions, in order to realizing good imaging performance and required configuration, system specifications and applications. The imaging systems may be nonrotationally symmetric, and the design requirements may be high due to specific applications. Therefore, the design of such complex systems may be difficult. Traditional design method is based on firstly selecting proper starting points and then applying optimization. However, a key problem is that the existing systems are rare. In this way, much time and human effort will be spent on the system establishment, trial-and-error and optimization, which heavily relies on design experiences. Iterative optimization methods such as inverse fast Fourier transform (FFT) method, Gerchberg-Saxton (G-S) algorithm [19] can be used to retrieve the phase function, but they are often used for the beam shaping using one phase profile. There are also some other methods based on point-by-point imaging. For example, HOE can be designed simply by considering the interference of two beam (ideal spherical wave or plane wave). Metasurface can be designed by calculating the phase function using the equal optical path method. However, these methods only consider a single phase element or surface, and the system is generally co-axial. Yang et al. proposed a point-by-point design method of multiple phase elements in a nonrotationally symmetric system [20]. However, the computation time is relatively long, especially when the number of elements in the system is large, as the calculation of data points based on Fermat’s Principle will be difficult. This hinders the applications of the method in the solution space survey tasks.
For the starting point design, the most important goal is to generate a feasible system based on given paraxial design requirements and system structure, and the time cost should be minimal. Inspired by the properties of the geometric conic surfaces [21–24], in this paper, we proposed a design method of imaging system consisting of flat phase elements based on confocal properties. If specific phase element can realize point-to-point stigmatic imaging, a system realizing ideal imaging for one field point can be established by sequentially coinciding the focal points of adjacent elements. The derivation of the generalized phase function of an off-axis element for realizing point-to-point stigmatic imaging for all cases (refractive or reflective, real or virtual focal point) are demonstrated. Based on given paraxial design requirement (such as focal length and magnification) and locations of phase elements as well as object and image (structure requirements), the phase functions of all the phase elements can be calculated immediately. The method can be used for refractive, reflective and catadioptric nonsymmetric systems and the number of elements is not limited. In addition, geometric conic surface, which can also realize point-to-point stigmatic imaging, can be also integrated into this design method directly. The resulting system can be taken as a good starting point for further optimization. Rotationally symmetric phase terms or freeform phase terms can be added directly to the phase function to improve the imaging performance and correct the field-dependent aberrations. Multiple design examples are given in Section 3 to validate the feasibility and effect of the proposed method.
2. Method
2.1 Mathematical expression of phase function
The wavefront modulation effect or the phase shift effect of a flat phase element (such as DOE, HOE and metasurface) can be characterized by its phase function ϕ(x,y), which is similar with the surface shape expression of a traditional geometric optical element. For the confocal imaging optical system design, proper mathematical expression of phase function realizing point-to-point stigmatic imaging is needed, which is also needed for element fabrication.
Considering the stigmatic imaging from point S to point E by a flat phase element. The phase element can be refractive or reflective, and the points S and E can be real or virtual focal point. Therefore, there are eight different cases, which are plotted in Fig. 1. ns and ne are the refractive indices preceding and after the phase surface (plotted in blue). The solid lines refer to real focal point and dash lines refer to virtual focal point. The phase element has its local coordinate system in a three-dimensional space, and the origin O of the coordinate system coincide with the mathematical vertex of the phase function. The x and y coordinate in phase function ϕ=ϕ(x,y) are consistent with the x and y coordinate of the local coordinate system.
During the design, the chief ray of the central field intersects with the phase element at point O. The chief ray is the ray from a field angle bundle that passes through the center of the aperture stop. In fact, the chief ray of the central field is also the optical-axial-ray (OAR) [25], which is defined as the ray that connects the center of the object (central field) with the center of the aperture stop within the optical system. Here we consider the case that the system is symmetric about the YOZ plane. To obtain the phase function, we choose the chief ray and another random ray of the same field point, as shown in Fig. 2. The random ray intersects with the phase element at point P with the coordinates of (x, y). Assume that the optical path lengths for the chief ray and the random ray in object space are Ls,c and Ls,r respectively, and the optical path lengths for the chief ray and random ray in image space are Le,c and Le,r respectively. For stigmatic imaging, the optical path lengths for different paths from point S to E should be equal. Therefore, we have
Here we consider the light rays in “image space” as an example. The distance between two points can be calculated if the coordinates of the point are given. In addition, it should be noted that phase shift effect of the phase element should also be added into the total optical path length (here we add the phase shift effect into the optical path in image space). If point E is a real focal point, we have
Here, m is the diffraction order, λ is the wavelength. For DOE and HOE, m is generally chosen to be 1. For metasurface, the concept of diffraction order generally may not exist and the true phase function or phase shift should be ϕmeta = mϕ (m = 1). le is the “image distance” measured from the phase element to E along the z direction. The sign of the distance matches the traditional sign convention of geometric optics. The case, in which the distance measured from O to the focal point is towards the + z direction, corresponds to a positive value, and vice versa. θe is the angle between the local XOY plane and the plane perpendicular to the line OE. The value of the angle is positive if the angle is towards the −z direction from the XOY plane to the plane perpendicular to the line OE. If point E is a virtual focal point, we have
2.2 Confocal system design
Based on the mathematical expression given in previous section, confocal system design can be realized. For the ith element in the system, its end focal point Ei should be coincide with the start focal point Si + 1 of the (i + 1)th element. The first start point S1 is the object point of the central field and the final end focal point coincide with the image point on the image plane. In this way, stigmatic imaging is realized for this field, and there is no geometric aberration for this field (chromatic aberration is not considered in this paper). There will be always a node at the central field point in the aberration field. This is also true for non-rotationally symmetric confocal systems using decentered and tilted elements as well as non-rotationally symmetric phase functions. Therefore, field constant aberrations are corrected in confocal system. This conclusion can be also validated by using proof by contradiction. If field-constant aberrations still exist, aberrations will exist across the full FOV, including the central field point, which is inconsistent with the design result of the confocal system that there is no aberration for the central field point. Figure 3 shows several examples of confocal system consisting of stigmatic phase elements.
Fig. 3. Several examples of confocal system consisting of stigmatic phase elements. (a) Three-mirror imaging system. (b) Single lens. (c) Monolithic catadioptric afocal system.
The design goal of the confocal system can be summarized as follows: Given the system specifications (such as focal length and magnification), the global locations and orientations of most of the phase elements, and some other data (such as the locations of some focal points or the object and image distance of some elements) as design inputs, in order to fulfill the confocal conditions, the unknown parameters of the phase elements can be calculated. The key of the calculation is to get the relationship between the system specifications and the parameters of the phase elements. Using paraxial optical theory, the focal length or magnification of a coaxial confocal system can be approximately calculated based on ls, le, θs and θe of each element. Note that “coaxial” means that all the focal points are in a same line and this line is perpendicular with all the elements at the intersections. For example, for a coaxial imaging system with no intermediate image and the object is located at infinity, the paraxial focal length EFL of the system can be calculated as
Detailed explanations of the above two equations can be found in Supplement 1. However, for the common case that the system is off-axis in y direction and symmetric about the YOZ plane, only the paraxial EFL and Γ in x direction can be calculated by Eqs. (7) and (8). These two equations will be incorrect in y direction. To solve this problem, the paraxial system specifications can be calculated by tracing of a paraxial ray from the central field (or the on-axis field). For example, for a ray having a very small pupil coordinate Δh in y direction in object space relative to the chief ray, the relationship between the paraxial focal length EFLy in y direction, Δh and the corresponding pupil angle Δu at the image plane can be given by
This equation can be used in the calculation of the parameters of the phase elements. Details will be given in the design examples in Section 3. If related programs are written, the automatic parameter calculation process can be very fast and the corresponding system is obtained immediately, as it is a forward calculation process in general. The system can be taken as a good starting point for further optimization. Some examples are given in Section 3 to show the design process of confocal systems. In addition, as off-axis geometric conic surface can also realize point-to-point stigmatic imaging based on its mathematical properties, the confocal system design can be extended to the system containing both flat phase elements and geometric surfaces.
2.3 System optimization
Although stigmatic imaging is realized for one field point and the field constant aberrations are corrected using the above confocal system design approach, the field dependent aberrations can be large and the design constraints about system structure and system specifications may be not satisfied. In this way, further aberration balancing and optimization is needed to get good imaging performance. In addition, the phase functions should be upgraded. Similar with geometric imaging systems, in order to correct higher order aberrations and the aberrations induced by nonrotationally symmetric configuration as well as the phase functions depicted in Section 2.1, higher order rotationally symmetric phase terms and freeform phase terms should be added to the phase functions. The added phase function terms can be in any type, such as XY polynomials, Zernike polynomials, radial basis functions, etc. The initial phase function given by Eq. (6) can be also transformed into the same type with the added phase function terms in order to get a uniform phase function expression type. This process can be done using a fitting process considering both the value and gradient of the original phase function at multiple discrete data points [20].
During optimization, the focal length of the system can be calculated using ABCD matrix method and then controlled. The distortion of the system can be obtained using real ray trace data and then controlled. For off-axis systems, the system structures constraints and the light obscuration elimination constraints can be also established using real ray trace data, similar with the off-axis geometric system design. The constraints become stricter as the optimization proceeds. It should be noted that when the starting point has been generated and during optimization, the confocal conditions may no longer be satisfied. Achieving the required system specifications and structure as well as good imaging performance is the design goal. The phase functions obtained after final optimization can be used in the fabrication of phase elements. For DOE, corresponding diffractive structures can be designed and fabricated. For HOE, expected phase function can be obtained by using specifically designed recording system [8,9]. For metasurface, the size, height, and orientation of the meta-atoms array can be calculated or selected based on the theory of propagating phase or Pancharatnam-Berry phase (geometric phase).
3. Design examples
In this section, we present several design examples to show the effect and feasibility of the description of flat phase elements and proposed design method. Here we only consider the systems which is symmetric about the YOZ plane.
3.1 Off-axis two-mirror systems
Off-axis two-mirror system is a typical kind of reflective system. Here we present the design of unobscured off-axis two-mirror imaging system consisting of confocal flat phase elements without intermediate image point inside the system. We only consider the case that the primary mirror (M1) has a real end focal point E1 (coincide with the virtual start focal point S2 of the secondary mirror (M2)), as shown in Fig. 4. The calculation process of other cases is similar with this one. The end focal point E2 of M2 is the image point I. Only one field point is considered during the design. The design inputs include the location of the two elements (M1 and M2) and the ideal image point, the orientation (tilt angle) of M1 and the focal length of the system. It means that the system specifications and almost all the structure parameters have been given based on the design requirements. The orientation (tilt angle) of M2 is unknown and will be calculated in order to keep the focal length in y direction. The design outputs also contain the parameters of the phase functions.
The chief ray should pass the vertex of each element and the image point. As the location of phase elements and the image points are given, E1 (or S2) should be on the line connecting O1 and O2. Using Eq. (7) and the given focal length, the location of E1 (or S2) can be calculated. In this way, the focal length in x direction can be realized. In order to control the focal length in y direction, a paraxial ray is used. As shown in Fig. 4, the paraxial ray of the central field is deviated from the chief ray in y direction by a small value Δh (an exaggerated Δh is plotted for clarity). This ray intersects with M1 at A1 (can be easily calculated), and will be then redirected to the point E1 (or S2). In addition, the corresponding pupil angle Δu of this paraxial ray at the image plane can be calculated by Eq. (9). As the line O2E2 is known, the line corresponding to the paraxial ray in image space can be obtained. Then, the intersection A2 of this line with A1E1 can be calculated. A2 and O2 are both on M2. In this way, the orientation (tilt angle) of M2 can be then calculated. At this time, all the focal points as well as the locations and orientations of the phase elements are obtained. The parameters of the phase functions realizing stigmatic imaging can be easily calculated and the system is fully established.
Using the above method, three off-axis two-mirror systems with different folding geometries consisting of confocal flat phase elements are designed, as shown in Fig. 5. The focal lengths in x and y directions are both 300 mm and the entrance pupil diameter is 60 mm. Ideal imaging of the central field point is realized. These systems can be taken as good starting points for further optimization. A generalized automatic program is compiled in optical design software CODE V to realize the fast generation of the two-mirror system based on the given design inputs. In fact, there are many other combinations of the design inputs and outputs. For example, the orientation (tilt angle) of M2 is given while the orientation of M1 is unknown and will be calculated in order to keep the focal length in y direction. The calculation process is similar with the method depicted above.
Fig. 5. Off-axis two-mirror systems consisting of confocal flat phase elements. Three difference folding geometries are given. The focal lengths in x and y directions are both 300 mm and the entrance pupil diameter is 60 mm.
3.2 Off-axis three-mirror systems
Off-axis three-mirror system is also a typical kind of reflective system and has been intensively used in many applications. We firstly present the design of unobscured off-axis three-mirror imaging system consisting of confocal flat phase elements without intermediate image point inside the system. We only consider the case that M1 has a real end focal point E1 (coincide with the virtual start focal point S2 of M2) and M2 has a virtual end point E2 (coincide with the real start focal point S3 of the tertiary mirror (M3)), as shown in Fig. 6. The end focal point E3 of M3 is the image point I. For other cases the parameter calculation process will be similar. Only one field point is considered during the design. The design inputs include the locations of the three elements (M1, M2 and M3) and the ideal image point, the orientations (tilt angles) of M1 and M2, the focal length of the system, and le of M1. It means that the system specifications and almost all the structure parameters have been given based on the design requirements. The orientation (tilt angle) of M3 is unknown and will be calculated in order to keep the focal length in y direction. The design outputs also contain the parameters of the phase functions.
Fig. 6. Sketch of the off-axis three-mirror system without intermediate image consisting of confocal flat phase elements
The calculation process is similar with the process depicted in Section 3.1. As the location of phase elements and the image points are given, E1 (S2) should be on the line connecting O1 and O2, and E2 (S3) should be on the line connecting O2 and O3. Using Eq. (7) as well as the given focal length and le of M1, the location of all the focal points can be calculated. In this way, the focal length in x direction can be realized. In order to control the focal length in y direction, a paraxial ray is used. As shown in Fig. 6, the line A2E2 can be obtained. As the line O3E3 is also known, the line corresponding to the paraxial ray in image space can be obtained. Then, the intersection A3 of this line with A2E2 can be calculated. In this way, the orientation (tilt angle) of M3 can be calculated. The parameters of the phase functions can be easily calculated and the system is then fully established.
Using the above method, eight off-axis three-mirror systems (all the possible different folding geometries [26]) without intermediate image point consisting of confocal flat phase elements are designed, as shown in Fig. 7. The focal lengths in x and y directions are both 90 mm and the entrance pupil diameter is 30 mm. le of M1 is 80 mm. Ideal imaging of the central field point is realized. It should be noted that the size of the entrance pupil diameter doesn’t affect stigmatic imaging. These systems can be taken as good starting points for further optimization. In addition, conic surfaces can also realize point-to-point stigmatic imaging based on their mathematical properties. Therefore, conic surfaces and phase elements can be integrated into one system to realize confocal system and stigmatic imaging. Given the two focal point of a conic surface, the surface parameters can be calculated [22]. For example, for the eight systems given in Fig. 7, we also generated similar systems in which M2 is changed into off-axis conic surface (ellipsoid, the two focal points are both virtual). The locations of the mirrors are the same with the systems in Fig. 7 and the optical layouts are very similar. Stigmatic imaging and the required focal lengths in x and y directions are realized.
Fig. 7. Off-axis three-mirror systems (without intermediate image) consisting of confocal flat phase elements. Eight difference folding geometries are given. The focal lengths in x and y directions are both 90 mm and the entrance pupil diameter is 30 mm.
Using similar methods, system with an intermediate image point inside the system can also be generated. As shown in Fig. 8, off-axis three-mirror systems with all kinds of different folding geometries are generated. The intermediate image point is between M2 and M3. The confocal relationship of mirrors used here is: M1 has a real end focal point E1 (coincide with the virtual start focal point S2 of M2) and M2 has a real end point E2 between M2 and M3 (coincide with the real start focal point S3 of M3). The end focal point E3 of M3 is the image point I. The design inputs are similar with the case shown in Fig. 7. The difference is that the distance between the intermediate point and M2 is given and le of M1 is obtained through calculation. Stigmatic imaging with focal lengths in x and y directions are both 90 mm is realized, which also shows the feasibility and effect of the proposed confocal phase elements and the design method. It is also possible that the intermediate image point is between M1 and M2. One possible confocal case is: M1 has a real end focal point E1 between M1 and M2 and it coincides with the real start focal point S2 of M2. M2 may have a virtual end point E2 (coincide with the real start focal point S3 of M3). The end focal point E3 of M3 is the image point I. The distance between the intermediate point and M1 (or le of M1) can be given as a design input. For all the above design tasks given in this section, generalized automatic programs are compiled in optical design software CODE V to realize the immediate generation of these systems based on the given design inputs, which increases the design efficiency and reduce the human effort. In addition, for the three-mirror system design, there are also many other combinations of the design inputs and outputs. For example, the orientation (tilt angle) of M3 is given while the orientation of M2 is unknown and will be calculated in order to keep the focal length in y direction. The calculation process is similar with the method depicted above. As the system generation process is fast and versatile, the proposed method can be used in the solution space survey tasks.
Fig. 8. Off-axis three-mirror systems (with intermediate image) consisting of confocal flat phase elements. Eight difference folding geometries are given. The focal lengths in x and y directions are both 90 mm and the entrance pupil diameter is 30 mm.
We further designed an off-axis three-mirror system using flat phase elements with good imaging performance and reduced assembly difficulty. The field-of-view (FOV) of the system is 5°×4°. The focal length is 90 mm and the entrance pupil diameter is 30 mm. The working wavelength is 1064 nm. The system has no intermediate image and M2 is taken as the aperture stop. The folding geometry shown in Fig. 7(a) is used. M1 and M3 use the same flat substrate. In this way, the degree of freedom during system assembly is reduced by 6. In addition, M2 is parallel to the M1-M3 element. In this way, the assembly difficulty will be further reduced. If M2 is parallel to the M1-M3 element, and the global y coordinates of the image point and the vertex of M3 are equal, the focal length in x direction calculated by using Eq. (7) and the focal length in y direction calculated by using Eq. (9) will be equal. This can be easily proved using the properties of similar triangles shown in Fig. 6. We first designed the confocal starting point using the method depicted in Section 2 and Section 3.2 (all the tilt angles of elements are given as input). The design result realizing stigmatic imaging for the central field point is shown in Fig. 9(a). After adding the field points up to the full FOV, the system layout is shown in Fig. 9(b) and this system is taken as the good starting point for further optimization. XY polynomials phase terms up to the 8th order are added to phase function of each mirror. As the system is symmetric about the YOZ plane, odd terms of x are not used in optimization and their coefficients are zero. The final system layout is shown in Fig. 9(c). The MTF is above 0.6 at 80lps/mm across the full FOV and the maximum RMS wavefront error is 0.065λ (λ=1064 nm), as shown in Figs. 10(a) and 10(b). The distortion grid is shown in Fig. 10(c). Good imaging performance is achieved.
Fig. 9. Off-axis three-mirror system using flat phase elements with good imaging performance and reduced assembly difficulty. (a) The system consisting of confocal phase elements. Only one field point is plotted. (b) System layout of the starting point when the rays of the field points up to the full FOV are plotted. (c) Final system after optimization.
Fig. 10. Imaging performance analysis of the three-mirror system. (a) MTF plot. (b) RMS wavefront error. (c) Distortion grid.
For most of the imaging systems, the focal length in x and y directions should be the same (or approximately the same), and this should be one of the design goals. This is the case we discussed in the paper. If the focal length in x and y directions are required to be different, the design process is almost the same. For the confocal starting point design, the paraxial focal length in x direction can still be calculated using paraxial theory (such as Eq. (7)), and the paraxial focal length in y direction can still be calculated using the method given in Section 2.2 and Section 3. During optimization, the focal lengths in two directions can be individually calculated (for example, using ABCD matrix method) and then individually controlled. It should also be noted that the focal lengths in x and y direction will be the same for rotationally symmetric systems.
3.3 Refractive system
The proposed method can also be used in the design of refractive systems. Here we demonstrate a design example of coaxial rotationally symmetric single flat lens with the front and rear surface are both described by phase functions. The front surface is the aperture stop. The field-of-view (FOV) of the system is 60°. The focal length is 730µm and the entrance pupil diameter is 850µm. The working wavelength is 800 nm and the material of the is SiO2. The thickness of the flat phase element is 1 mm. Such system can be realized by a metalens [14].
We firstly generate a starting point using the proposed method to realize stigmatic imaging of the central field and the required focal length, as shown in Fig. 11(a). Given the thickness of the element, the image location and the focal length as inputs, the two phase functions can be calculated immediately. This system is taken as a good starting point. Then, rotationally symmetric phase terms Ai(x2 + y2)i (i = 2,3,4,5) are added to both phase functions during further optimization to achieve good image quality. The focal length and distortion of the system are controlled during design. After fast optimization, the final system with good image quality can be obtained, as shown in Fig. 11(b). The MTF of full FOV is above 0.65 at 200lps/mm, as shown in Fig. 12(a). The distortion of the marginal field is –13%, as shown in the distortion curve (Fig. 12(b)). The obtained phase functions can be used for the design of meta-atoms array under the theory of propagating phase or Pancharatnam-Berry phase.
Fig. 11. Layout of the single flat lens. (a) The starting point realizing stigmatic imaging of the central field. (b) The final system after optimization.
Fig. 12. Imaging performance analysis of the single flat lens system. (a) MTF plot. (b) Distortion curve.
Imaging systems consisting of phase element can be designed using nodal aberration theory (NAT). NAT has been successfully used to characterize the aberration property of nonsymmetric systems and guide the related design tasks [25–30], especially freeform optical design. As the system consisting of phase element with nonrotationally symmetric phase functions is similar with freeform optical system, NAT can be taken as a fundamental theory for such systems and can be used to characterize the aberration of these systems.
Current research of NAT is difficult to be used to generate systems consisting of phase elements (not geometric surfaces) directly based on given requirements on system structure and system specifications. In Ref. 26, a method to establish the off-axis starting point is proposed but traditional sphere is used. However, NAT and the method proposed in this paper can be well integrated for the design and optimization of using phase elements. In this paper, the proposed design method as well as the generalized phase function can be used to generate confocal systems immediately based on given focal length or magnification and system structure. Stigmatic imaging is realized and the field constant aberrations are well corrected. This system can be taken as a good starting point for further optimization. However, the field-dependent aberrations may be large. Freeform phase terms (similar with the freeform surface terms overlay on geometric base surface) can be further added to corrected these aberrations. Using NAT, designers can evaluate the aberration-correction feasibility of all the possible geometries of confocal starting points using the method proposed in Ref. 26, and good folding geometry can be selected. NAT can then guide the optimization process. The type and value of aberrations induced by freeform terms have been derived in Refs. 29 and 30. Using these knowledges, after analyzing the aberration fields after each design step, specific freeform terms can be added during optimization to improve the imaging performance [28] until satisfactory design result is achieved.
4. Conclusions and discussions
In this paper, we proposed a design method of imaging systems using confocal flat phase elements. The generalized phase function for realizing point-to-point stigmatic imaging for all cases (refractive or reflective, real or virtual focal point) is proposed. Based on given design requirements (including system specifications and the system structure), the phase functions of the elements can be calculated immediately and stigmatic imaging of the central field is realized. The resulting system can be taken as a good starting point for further optimization. Geometric conic surface can be also integrated into the confocal design process. Design examples are given to show the utility and effect of the proposed method. The proposed generalized method can be used in the design and development of novel, compact and nonsymmetric imaging systems using DOE, HOE and metasurface.
Phase elements and freeform optical surface can be integrated into imaging systems. The integration can be a combination of individual freeform optical elements and phase elements. Another integration method is that the freeform surface and the phase function are combined into a single element (such as FSP element and metaform optics) [31,32]. For example, metaform optics integrates the combined benefits of a freeform optic and a metasurface into a single optical component [32]. Such element can lead to advanced system specifications, compact structure, and good imaging performance. The confocal design method proposed in this paper may also be extended to the systems consisting of such elements. Point-to-point stigmatic imaging of for one element can be realized by defining and combining proper surface base expression and phase function. The total optical power and total ray deflection effect of an element can be distributed to the geometric surface base and phase function based on actual needs and manufacturing ability. Confocal systems can be generated very fast after direct calculation and they can be taken as good starting points for further optimization (the design and optimization method given in Refs. 31 and 32 can be used). Related methods will be explored in future research.
Chromatic aberration is crucial problem for the systems using phase elements such as DOE, HOE and metasurface. In this paper, only one wavelength can be considered during the confocal design. If multiple wavelengths or a spectral band is used in the system, chromatic aberration will be large and imaging performance of the system will be bad. However, this problem may be overcome if multiple configurations (each configuration corresponds to one wavelength) can be used in the design. For example, for the systems using HOE, multiple holographic layers for different wavelengths can be added on the substrate. For each wavelength, confocal system design and further optimization can be conducted, and the phase functions corresponding to different wavelengths are obtained. Each holographic layer does not affect other wavelengths due to wavelength selectivity. In this way, chromatic aberration is corrected and full color imaging can be realized. For DOE and metasurface, similar design process can be realized using harmonic DOE and multiwavelength metasurface.
Funding
National Key Research and Development Program of China (2021YFB2802100); Beijing Natural Science Foundation (1222026); National Natural Science Foundation of China (U21A20140, 62275019); Young Elite Scientist Sponsorship Program by CAST (2019QNRC001).
Acknowledgments
We thank Synopsys for the educational license of CODE V.
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.
Supplemental document
See Supplement 1 for supporting content.
References
1. A. P. Wood, “Design of infrared hybrid refractive–diffractive lenses,” Appl. Opt. 31(13), 2253–2258 (1992). [CrossRef]
2. B. Zhang, Q. Cui, M. Piao, and Y. Hu, “Design of dual-band infrared zoom lens with multilayer diffractive optical elements,” Appl. Opt. 58(8), 2058–2067 (2019). [CrossRef]
3. T. Nakai and H. Ogawa, “Research on multi-layer diffractive optical elements and their application to camera lenses,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., Vol. 75 of OSA Trends in Optics and Photonics Series (Optica Publishing Group, 2002), paper DMA2.
4. H. Li, X. Zhang, C. Wang, J. Zhang, L. Wang, and H. Qu, “Design of an off-axis helmet-mounted display with freeform surface described by radial basis functions,” Opt. Commun. 309, 121–126 (2013). [CrossRef]
5. C. Draper, C. Bigler, M. Mann, K. Sarma, and P. Blanche, “Holographic waveguide head-up display with 2-D pupil expansion and longitudinal image magnification,” Appl. Opt. 58(5), A251–A257 (2019). [CrossRef]
6. S. Kim and J. Park, “Optical see-through Maxwellian near-to-eye display with an enlarged eyebox,” Opt. Lett. 43(4), 767–770 (2018). [CrossRef]
7. C. Yoo, M. Chae, S. Moon, and B. Lee, “Retinal projection type lightguide-based near-eye display with switchable viewpoints,” Opt. Express 28(3), 3116–3135 (2020). [CrossRef]
8. C. Jang, O. Mercier, K. Bang, G. Li, Y. Zhao, and D. Lanman, “Design and fabrication of freeform holographic optical elements,” ACM Trans. Graph. 39(6), 1–15 (2020). [CrossRef]
9. T. Yang, Y. Wang, D. Ni, D. Cheng, and Y. Wang, “Design of off-axis reflective imaging systems based on freeform holographic elements,” Opt. Express 30(11), 20117–20134 (2022). [CrossRef]
10. T. Shu, G. Hu, R. Wu, H. Li, Z. Zhang, and X. Liu, “Compact full-color augmented reality near-eye display using freeform optics and a holographic optical combiner,” Opt. Express 30(18), 31714–31727 (2022). [CrossRef]
11. C. Chang, K. Bang, G. Wetzstein, B. Lee, and L. Gao, “Toward the next-generation VR/AR optics: a review of holographic near-eye displays from a human-centric perspective,” Optica 7(11), 1563–1578 (2020). [CrossRef]
12. J. H. Park and B. Lee, “Holographic techniques for augmented reality and virtual reality near-eye displays,” Light: Adv. Manuf. 3(1), 1–14 (2022). [CrossRef]
13. Z. B. Fan, H. Y. Qiu, H. L. Zhang, X. N. Pang, L. D. Zhou, L. Liu, H. Ren, Q. H. Wang, and J. W. Dong, “A broadband achromatic metalens array for integral imaging in the visible,” Light: Sci. Appl. 8(1), 67 (2019). [CrossRef]
14. A. Arbabi, E. Arbabi, S. M. Kamali, Y. Horie, S. Han, and A. Faraon, “Miniature optical planar camera based on a wide-angle metasurface doublet corrected for monochromatic aberrations,” Nat. Commun. 7(1), 13682 (2016). [CrossRef]
15. S. J. Byrnes, A. Lenef, F. Aieta, and F. Capasso, “Designing large, high-efficiency, high-numerical-aperture, transmissive meta-lenses for visible light,” Opt. Express 24(5), 5110–5124 (2016). [CrossRef]
16. P. Lalanne and P. Chavel, “Metalenses at visible wavelengths: past, present, perspectives,” Laser Photonics Rev. 11(3), 1600295 (2017). [CrossRef]
17. S. Colburn, A. Zhan, and A. Majumdar, “Varifocal zoom imaging with large area focal length adjustable metalenses,” Optica 5(7), 825–831 (2018). [CrossRef]
18. X. Zou, G. Zheng, Q. Yuan, W. Zang, R. Chen, T. Li, L. Li, S. Wang, Z. Wang, and S. Zhu, “Imaging based on metalenses,” PhotoniX 1(1), 2 (2020). [CrossRef]
19. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–242 (1972).
20. T. Yang, D. Cheng, and Y. Wang, “Design method of nonsymmetric imaging systems consisting of multiple flat phase elements,” Opt. Express 26(19), 25347–25363 (2018). [CrossRef]
21. C. Xu, “Study on Design Method and Application of Freeform Unobscured Off-axis Reflective Systems,” Ph.D. Thesis, Beijing Institute of Technology (2018).
22. E. M. Schiesser, N. Takaki, and B. Stone, “Representations of Off-Axis Conics for Lens Design,” Proc. SPIE 12078, 120781F (2021). [CrossRef]
23. D. Cheng, C. Xu, T. Yang, and Y. Wang, “Off-axis reflective imaging system design with a conicoid-based freeform surface,” Opt. Express 30(6), 9590–9603 (2022). [CrossRef]
24. J. Sasián, “Method of confocal mirror design,” Opt. Eng. 58(01), 1 (2019). [CrossRef]
25. K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26(6), 1503–1517 (2009). [CrossRef]
26. A. Bauer, E. M. Schiesser, K. P. Thompson, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9(1), 1756 (2018). [CrossRef]
27. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005). [CrossRef]
28. K. Fuerschbach, J. Rolland, and K. P. Thompson, “A new family of optical systems employing φ–polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011). [CrossRef]
29. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Theory of aberration fields for general optical systems with freeform surfaces,” Opt. Express 22(22), 26585–26606 (2014). [CrossRef]
30. T. Yang, D. Cheng, and Y. Wang, “Aberration analysis for freeform surface terms overlay on general decentered and tilted optical surfaces,” Opt. Express 26(6), 7751–7770 (2018). [CrossRef]
31. Y. Duan, T. Yang, D. Cheng, and Y. Wang, “Design method for nonsymmetric imaging optics consisting of freeform-surface-substrate phase elements,” Opt. Express 28(2), 1603–1620 (2020). [CrossRef]
32. D. K. Nikolov, A. Bauer, F. Cheng, H. Kato, A. N. Vamivakas, and J. P. Rolland, “Metaform optics: bridging nanophotonics and freeform optics,” Sci. Adv. 7(18), eabe5112 (2021). [CrossRef]
