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Abstract
Folded lenses offer advantages in terms of lightness and thinness, but they have limitations when it comes to correcting aberrations. In this paper, we propose a novel approach to address this issue by incorporating metasurfaces in the design of folded optical systems. Specifically, a folded refractive-metasurface hybrid annular aperture folded lens (AFL) is introduced. The structural characteristics of the AFL imaging system are analyzed to investigate the blocking ratio, thickness, and light collection capability of the ring aperture system. Additionally, a hybrid optical integration design using Zemax software is proposed for the metasurfaces. A quadruple-folded AFL working in the mid-infrared waveband is then designed. The superstructure surface is analyzed, and its processability is discussed. The results demonstrate that the reflective-metasurface hybrid AFL significantly improves the imaging quality of this type of optical system while meeting the required design accuracy.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The foundation of optical design is rooted in geometric optics, which uses aberration theory to quantify the imaging quality of an optical system. Traditionally, optical systems have to rely on methods such as incorporating additional lenses or using aspherical or freeform surfaces to correct aberrations. While effective, these approaches come with drawbacks including increased complexity, larger size, heavier weight, and higher costs. As optical systems strive for higher standards such as improved resolution, wider field of view, extended wavebands, and reduced size, weight, and component count, as well as the incorporation of special imaging functions like zoom and super resolution, new optical elements such as metasurfaces are being explored to address these challenges [1]. One promising solution is the integration of metasurfaces and traditional lenses in a hybrid optical system. This approach offers advantages such as reduced complexity, decreased material dependency, and simplified optimization of optical designs [2–4].
Infrared imaging systems, compared to visible waveband imaging systems, provide numerous advantages including enhanced information richness, improved anti-interference capabilities, and the ability to filter out irrelevant information. They enable the visualization of temperature information and facilitate more effective target acquisition, making them crucial components in various fields including astronomy, medical diagnosis, and military applications [5,6]. Similar to visible waveband imaging systems, infrared optical systems rely on different combinations of lenses and infrared materials to correct aberrations and meet imaging requirements. However, they face challenges in terms of complex optical systems that struggle to achieve high transmittance. For this reason, aperture folded lens (AFL) presents a viable solution for design and implementation. It consists of multiple annular reflective surfaces on the same substrate material, where light enters through the outermost ring, undergoes multiple reflections inside, and ultimately reaches the imaging plane. The AFL can be seen as a coaxial reflective optical system with an optical glass medium between the reflective mirrors [7]. Its simple structure eliminates the need for additional lens groups, resulting in a reduction in system length, volume, and design complexity [8,9].
However, the AFL’s substrate, composed of a single material, is susceptible to chromatic aberration, particularly lateral chromatic aberration [10]. Several methods have been proposed to address this issue. Firstly, low-dispersion materials like calcium fluoride (CaF2) can be utilized, although chromatic aberrations cannot be entirely eliminated due to the singular nature of the substrate material. Secondly, the middle medium of the AFL can be replaced with air or other optical materials [11], creating a reflective annular aperture ultra-thin imaging system that avoids chromatic aberration but increases challenges in terms of processing and adjustment. Thirdly, chromatic aberrations can be corrected by adding a lens group behind the AFL [12], but this structure hampers the miniaturization and weight reduction of the system. Lastly, adding a diffractive optical element (DOE) behind the annular aperture ultra-thin imaging system provides a solution [13], although the diffraction efficiency of the DOE is greatly influenced by the working environment and incident angle of light. The introduction of metasurfaces offers a new possibility for the AFL by allowing the manipulation of the phase and amplitude characteristics of electromagnetic waves using sub-wavelength structural units. This approach enables flexible modulation of the wavefront of light waves [14]. Metasurfaces have found applications in various areas, including optical holographic metasurfaces, vortex light generators, perfect absorbers, and detectors [15–20].
In this paper, we propose to integrate the metasurface structure into the AFL, forming a refractive-metasurface hybrid AFL structure for the mid-infrared waveband [21]. Based on the theoretical design and concepts of metasurfaces, we incorporate them into the AFL design, considering the optical path characteristics and introducing the light output position into the metasurface. The resulting refractive-metasurface hybrid AFL effectively corrects chromatic aberration in the mid-infrared waveband. Through a comparison of modulation transfer function (MTF) and spot diagrams before and after incorporating the metasurface, we observed a significant improvement in imaging quality, providing a means for optimizing the optical system. This refractive-metasurface AFL holds great potential for future applications in mobile phone lenses, wearable telescopes, portable telescopes, and other systems.
2. Imaging characterization of refractive-metasurface AFL
2.1 Working principle and design of metasurface
A metasurface, also referred to as a metalens, is new type of optical elements that possesses a unique capability to generate a phase range of 2π through its planar subwavelength structure. This enables precise control over the phase, amplitude and polarization of electromagnetic waves. The distinguishing feature of metasurfaces lies in their ability to manipulate electromagnetic waves at subwavelength and deep subwavelength scales while maintaining a flat and compact design, thus revolutionizing the manipulation of light fields.
There are two fundamental appoaches that govern the design of metasurfaces. The first one is the geometric phase, known as the Pancharatnam-Berry (PB) phase, which arises from the interaction between photon spin and orbit and depends solely on the rotation angle of the anisotropic structures [22]. With a PB metasurface, the phase of the output wave changes sign when the chirality of the incident circularly polarized wave is altered. In other words, this metasurface acts as a convex lens, effectively focusing left-handed circularly polarized waves while causing right-handed circularly polarized waves to diverge like a concave lens. Consequently, it becomes challenging to simultaneously focus and diverge both left- and right-handed circularly polarized waves, limiting the potential applications of metasurfaces relying solely on the PB phase. The second approach is the transmission phase, which arises from the difference in path length traveled by electromagnetic waves during transmission. The transmission phase is primarily determined by the inherent properties of the nano-structure, including materials and geometric parameters. The control of the phase is achieved by manipulating parameters such as the length (L), width (W), height (H), period (P), substrate height (t), and rotation angle (θ) of the nano-columns, as depicted in Fig. 1(a), which illustrates a schematic diagram of the unit structure designed for phase and polarization control. Figure 1(b) demonstrates an achromatic metasurface featuring cylindrical structures at its center, where the differently colored arrows represent incident light of varying wavelengths. Instead of dispersing the focal point across different positions based on wavelength, this metasurface can focus light of different wavelengths onto a single point.
Fig. 1. Schematic diagram of the metasurface. (a) structure of the nano-column; (b) achromatic metasurface featuring cylindrical structures
Phase plot of a focusing metasurface is expressed as
where, $R = \sqrt {{x^2} + {y^2}}$ is the arbitrary position on the metasurface, (x,y) is the distance to the center, f is the predetermined focal length.However, practical metasurface design is often limited to a single wavelength. When light with different wavelength enters the metasurface, the intended modulation functionality may deviate or even fail to achieve. Similar to the dispersion phenomenon caused by the refractive index of materials changing with the frequency of incident light in traditional optical elements, dispersion can lead to focal separation in lens systems, which can impact the performance and imaging quality of optical systems. To address this limitation, the concept of achromatic metasurfaces has been introduced. In theory, an achromatic metasurface can converge light at the same focus for different wavelengths, as shown in Fig. 1(b). To achieve broadband achromatic performance, the unit structures must provide phase compensation that maintains a fixed focal length across a range of wavelengths. For achromatic lenses operating in a wide waveband (λmin, λmax), Eq. (1) can be modified as
And the phase delay of ${\varphi _{\textrm{meta}}}({R,\lambda } )$ can be expressed as
To achieve broad waveband chromatic aberration correction, it is crucial to simultaneously satisfy the reference phase ϕmeta(R, λ) and the compensation phase C(λ). The reference phase is determined by the working wavelength 1/λ and is directly proportional to the phase difference between different incident wavelengths. The transmission phase response of each unit structure can be designed to compensate for this phase difference. On the other hand, the compensation phase C(λ) is optimized specifically for each wavelength to achieve wide waveband phase delay compensation. The Particle Swarm Optimization (PSO) algorithm can be employed to obtain the compensation factor. The PSO algorithm, proposed by Kennedy and Eberhart in 1995, is inspired by the foraging behavior of birds. It involves placing multiple physical particles in the search space of a function, where each particle has an associated objective function value at its current position. The movement of each particle in the search space depends on evaluating its own best position (i.e., the best fitness function value), as well as considering the influence of the surrounding population and multiple particles. After the particles have made their respective moves, the next iteration occurs. Over time, the particles collectively converge, similar to a flock of birds searching for food, towards the optimal fitness function value.
2.2 AFL design
The structure of the AFL closely resembles that of a conventional reflective optical system, consisting of concentric annular reflective surfaces fabricated on both the front and back surfaces of a substrate material. Light enters the system through the entrance annulus and undergoes multiple reflections by the mirrors, ultimately forming an image on the detector. Figure 2(a) illustrates a six-folded annular aperture system, where the light enters the AFL through the input ring and undergoes multiple reflections before forming an image on the detector. The arrows represent the incident light, while the blue, green, yellow, and purple colors represent the first, second, third, and fourth reflective mirrors, respectively. The final surface, where the metasurface is located, is indicated by the red color. Figure 2(b) demonstrates the relationship between the system's outer diameter and the blocking ratio for fixed effective apertures of 40 mm, 50 mm, and 60 mm.
Fig. 2. Traditional AFL design. (a) a six-folded annular aperture system; (b) the relationship between its outer diameter and blocking ratio
In order to analyze the light collecting efficiency of the circular aperture imaging system, it is crucial to take into account the presence of a central obstruction. This obstruction restricts the entry of light, allowing only the outermost annular region to contribute to the optical system. To simplify the analysis, the effective aperture diameter of the circular aperture is provided and can be expressed as
where, deffl represents the diameter of the entrance pupil that limits the amount of light entering the optical system, D refers to the outer diameter of the system which represents the overall size of the optical system, and α represents the aperture ratio that determines the amount of light that can be collected by the system and influences the depth of field and image brightness.When the effective aperture deffl is fixed, the relationship between the outer diameter D of the ultra-thin imaging system and the obscuration ratio can be observed in Fig. 2(b). As shown in the figure, it can be observed that as the obscuration ratio increases, the outer dimensions of the system need to increase significantly in order to maintain the effective aperture of the system.
The relation between the total thickness T and the number of light reflections N can be calculated from the system's focal length, the refractive index of the system’s substrate, and the number of light reflections in the annular aperture system is
where, Feffl represents the effective focal length of the annular aperture system, ns is the refractive index of the system substrate. From Eq. (5), it can be known that when the focal length is a certain value, the thickness of the AFL is related to the number of reflections and the refractive index of the substrate material.When the effective focal length of the system is provided, it is possible to calculate the total thickness of the system using Eq. (5), which relies on the refractive index of the substrate material and the number of reflections. For a given effective aperture diameter, Eq. (4) can be used to determine the relationship between the system’s outer diameter and obscuration ratio.
Increasing the number of reflections has several effects. The axial length of the imaging system decreases, while the obscuration ratio and outer aperture diameter experience significant growth. However, it is important to consider the challenges associated with a large system aperture in terms of component production and processing. Moreover, a large aperture can negatively impact the strength and stability of optical elements, especially in complex environmental conditions. Therefore, it is essential to account for both the number of reflections and the size of the outer aperture during the design process to achieve a suitable balance and ensure optimal system performance. The effective focal length of the system allows for the calculation of the total system thickness using Eq. (5), based on the refractive index of the substrate material and the number of reflections. Similarly, Eq. (4) can be used to determine the relationship between the outer diameter of the system and the obscuration ratio for a given effective aperture diameter. Increasing the number of reflections results in several effects: the axial length of the imaging system decreases, while the obscuration ratio and outer aperture diameter experience significant growth. However, it should be noted that having a large system aperture can present challenges in terms of component production and processing. Additionally, it can have a negative impact on the strength and stability of optical elements, particularly in complex environmental conditions. Therefore, it is crucial to consider both the number of reflections and the size of the outer aperture during the design process to achieve the right balance and ensure optimal system performance.
3. Design and discussion
3.1 AFL optical system design
We have developed an AFL specifically designed for the mid-infrared waveband, which serves as a noteworthy demonstration of our design capabilities. In order to achieve optimal performance, we took into account various factors such as the transparency of the substrate material, dispersion properties, and the complexity of the fabrication process. After careful evaluation, we concluded that CaF2 would be the most suitable choice for the substrate material. This material exhibits exceptional transmittance spanning the 0.11∼11 µm waveband, with transmittance exceeding 90% within the desired waveband. Additionally, CaF2 possesses robust mechanical strength and is highly compatible with diamond-turning processes, which greatly simplifies its processing. The design requirements for our AFL are listed in Table 1.
Table 1. Design requirement of AFL
We initially utilized ZEMAX software to generate the optical system, which followed the structure of the AFL but only consisted of a single reflective surface. To incorporate folded optics, we extended the distance from the first reflective surface to the image surface multiple times, ensuring equal lengths each time, and subsequently added additional reflective surfaces. The initial design featured a coaxial four-reflector configuration, as shown in Fig. 3(a). CaF2 was chosen as the low-dispersion material to fill the gaps between each reflective surface.
In addition, we conducted a comprehensive system optimization process by distributing the focal lengths of the light across the various reflective surfaces and adjusting the distances between each surface. Throughout this optimization, we carefully maintained the optical path structure and overall system length. To improve the imaging quality, we replaced each reflective surface with an even aspheric surface, following the standard surface type. The expression for the aspheric surface is as
The finalized optical structure of the AFL is presented in Fig. 3, showcasing a co-axial four-reflector configuration. To provide a comprehensive understanding of the AFL, Fig. 3(b) offers a solid model representation. The left image displays the front surface, which serves as the incident surface for the light, while the right image illustrates the back surface, functioning as the exit surface for the light. The red circle indicates the image plane, enabling a visual assessment of the optical performance of the system.
The lens data for the hybrid AFL optical structure is provided in Table 2. With regards to the system’s imaging performance, Fig. 4 presents the spot diagram showcasing the MTF for traditional AFL.
Table 2. Structure parameters of traditional AFL
In order to modify the wavefront of the light beam in intricate optical system designs, binary optical surfaces are utilized to implement specific structures on the refractive surface. Similarly, metasurfaces use fabricated geometrical components on the substrate surface to control the wavefront of the light beam through discretized phase control of the incident light. Therefore, the correlation between binary optical elements and electromagnetic metasurfaces can effectively be utilized for integrated system design.
However, since the ZEMAX software does not have a dedicated surface for metasurfaces, a binary optical element is used as a substitute for the metasurface element during optimization. In other words, system optimization is performed using “AFL + binary surface” instead of “AFL + metasurface.” The phase distribution provided by the binary optical element is acquired to subsequently design the metasurface. Consequently, the pre-designed metasurface is treated as a binary surface, and the optimized parameter settings are applied to obtain the parameters for the diffracted surface, as displayed in Table 3. The spot diagram and MTF are depicted in Fig. 5. Table 4 and Table 5 show the structure paremeter and parameters of aspheric coeffcients of the hybrid AFL with metasurface, separately.
Table 3. Parameters of aspheric coefficients of the traditional AFL
Table 4. Structure parameters of hybrid AFL
Table 5. Parameters of aspheric coefficients of the hybrid AFL
By examining Fig. 4 and Fig. 5, we can analyze the MTF and spot diagram before and after the incorporation of the metasurface. Figure 4(a) illustrates the MTF without the metasurface, with the solid line representing the MTF in the sagittal plane, the dotted line representing the MTF in the tangential plane, and the black line representing the diffraction limit. Additionally, Fig. 4(b) displays the point spread function without the metasurface. In contrast, Fig. 5(a) presents the MTF after the integration of the metasurface, while Fig. 5(b) depicts the spot diagram following the integration. A comparison between Figs. 5(a) and 5(b) allows for an evaluation of the imaging effect of the integrated metasurface. This comparison clearly reveals a significant enhancement in the imaging effect after the inclusion of the metasurface, effectively correcting the broadband chromatic aberration of AFL. The values of RMS radius and GEO radius for both structures can be found in Table 6. Overall, the designed hybrid AFL demonstrates its ability to achieve high imaging quality.
Table 6. RMS and GEO radii at different fields of view for AFL types
It can be seen that the RMS and GEO of the folded refractive-metasurface hybrid structure are smaller than those of the conventional refractive structure in all fields of view.
3.2 Metasurface design
Based on the phase delay requirements for the design, the metasurface at the outgoing can be designed. Consequently, high-transmission materials commonly used in the infrared range, such as silicon, germanium glass, and BaF2, are selected for the substrate and microstructure materials. Considering process design considerations, the metasurface is specifically fabricated using an all-silicon design, employing silicon for both the substrate and microstructure. According to the system integration design scheme, the phase distribution of the binary surface needs to be known, which is illustrated in Fig. 6, the phase to be provided by the metasurface, indicating the desired phase to be achieved by the metasurface.
To analyze the transmittance and phase response of the nano-columns, we employ the Finite-Difference Time-Domain (FDTD) method. Elliptical nano-columns made of silicon are placed on a square silicon substrate, as illustrated in Fig. 7(a). The height is fixed at H = 4500 nm, and the substrate size remains constant at Px = Py = 1650 nm. By varying the Rx and Ry values of the ellipses, we acquire distinct phase and transmittance distributions. The Rx and Ry parameters are scanned incrementally from 250∼650 nm in 4.5 nm steps. Figure 7(b) depicts the phase and transmittance distribution of the unit structure at a wavelength of 3.5 µm. To mitigate polarization effects, circular columns or cylinders with Rx =Ry are ultimately employed in constructing the metasurface. For further analysis, Figs. 7(c) and 7(d) stand for the unit microstructure phase sweep parameter and unit microstructure transmittance sweep parameter.
Fig. 7. Design of the metasurface. (a) micro-structure diagram of metasurface unit; (b) phase and transmittance distribution; (c) transmission distribution of unit micro-structure sweep; (d) phase distribution of unit micro-structure sweep parameters
Using the provided phase data, we designed a metasurface suitable for operation in the 3 to 5 µm wavelength range. The specific design parameters chosen include a center wavelength of 3.25 µm, a radius of 64 µm, and a focal length of 20 µm. For the simulation setup, we consider the near-field region in close proximity to the metasurface. A plane wave propagating along the z-axis is used as the input field for the overall metasurface structure. Perfect Matched Layer (PML) boundaries are defined along the X, -X, Y, -Y, Z, and -Z axes. By employing FDTD simulation, we obtained the near-field distributions of the outgoing electromagnetic waves from the metasurface. The FDTD solution includes a built-in 3-D far-field solver function that allows us to calculate the actual distribution of electromagnetic waves in the far field. This calculation provides valuable insights into the focusing effect achieved by the designed metasurface.
To evaluate the performance of the designed metasurface, we conduct simulations using incident wavelengths ranging from 3 to 5 µm. Specifically, we select five wavelengths within this range, excluding the central wavelength of 3.25 µm. The resulting focusing effect is illustrated in Fig. 8. In Figs. 8(a) and 8(b), the normalized distribution of light intensity is presented for the x-z and x-y planes, respectively. The red dashed line represents the focal length of 20 µm within the plane. It is evident that the metasurface effectively achieves the desired focusing effect.
Fig. 8. Focusing effect of the metasurface.(a) normalized distribution of x-z plane light intensity; (b) normalized distribution of x-y plane light focal spots
In addition, the field strength distribution at wavelengths of 3 µm (red curve), 3.25 µm (green curve), and 5 µm (red curve) in the x-z plane at x = 0 was analyzed. The focal positions at the three wavelengths are 19.7 µm, 19.9 µm, and 19.8 µm respectively. Since the central wavelength was set at 3.25 µm, the intensity is maximum at this wavelength, shown in Fig. 9.
The optimization design involves utilizing random data from the data table to generate an actual wavefront while considering the elimination of parameter combinations with low polarization conversion transmittance. The process entails comparing the ideal wavefront with the practical wavefront and achieving phase matching using data from the phase database. The optimization iterates continuously to minimize wavefront errors until satisfying the Rayleigh criterion. The Rayleigh criterion states that a wavefront can be deemed flawless if the maximum wavefront error between the ideal and practical wavefronts does not exceed a certain threshold. Additionally, in cases where the proportion of defective areas to the total wavefront area is small, these local defects can be disregarded.
3.3 Analysis of machining feasibility
To fabricate the AFL system, we employ a precise molding or single-point diamond turning technique. This is followed by the utilization of nanoscale imprinting or enhanced electron beam lithography to produce the metasurface. The integration of this metasurface with the AFL forms a hybrid AFL consisting of a metasurface and AFL, as depicted in Fig. 10. Enhanced electron beam lithography is a high-resolution nano-fabrication method used for creating nanoscale structures, such as metasurfaces. This technique surpasses conventional electron beam lithography in terms of resolution and processing speed, enabling faster and more precise fabrication.
The radius and phase exhibit a close to linear relationship. In order to ensure the accuracy of the processing trend while considering processing errors and theoretical overfitting, we linearly interpolate the radius-phase relationship. This results in a difference fitting map along with its corresponding focusing effect map, as illustrated in Fig. 11. The horizontal and vertical axes in the image represent a detector pixel size of 1025 × 1025.
Fig. 11. Interpolated fit plot. (a) raw data phase maps; (b) eight interpolation fit phase maps; (c) four-time interpolation to fit phase maps; (d) phase focussed renderings of raw data; (e) eight interpolation fit phase focus effects; (f) quadruple interpolation fit phase focus effects
In Fig. 10, it is evident that the 4th order, 8th order, and continuous phase effects exhibit similar patterns. Based on the design simulation data, it is recommended to select a cylindrical radius interval of 420 to 600 nm, which covers the entire 2π phase range. Furthermore, the original simulation data indicates a close linear relationship between the radius and phase. Therefore, the 4th-order processing characteristics are chosen. This involves selecting a cylindrical radius interval of 420 to 600 nm (equivalent to a diameter of 840-1200 nm) and a column height/hole depth of 4.5 µm (equivalent to a depth-to-width ratio of 3.7-5.4). These selected parameters approach the processing limit.
4. Conclusion
We have demonstrated a novel design of a refractive-metasurface hybrid AFL for the mid-infrared waveband. Our aim is to combat the chromatic aberration typically observed in traditional AFL designs. To achieve this, we introduced a metasurface constructed using phase transfer theory. The metasurface is then optimized using the particle swarm algorithm to achieve an achromatic aberration over a broad frequency range. We validated the optical properties of the metasurface by employing FDTD software. A comparative analysis of the spot diagram and MTF before and after integrating the metasurface reveals a significant improvement in image quality. With the inclusion of the metasurface, the MTF exceeds 0.373 at 17lp/mm, indicating enhanced imaging performance. To summarize, the integration of a metasurface into the hybrid AFL system offers several advantages, including compactness, miniaturization, and improved imaging quality. This research contributes to the development of miniaturized and lightweight military and civilian systems. Additionally, the precise phase control capabilities afforded by the metasurface address the issue of a small back focal length commonly encountered in AFL systems, presenting innovative solutions in this field.
Funding
National Natural Science Foundation of China (62375222, 61927810, 61905195); Northwestern Polytechnical University Central University Basic Research Business (310202011QD001); Shaanxi Fundamental Science Research Project for Mathematics and Physics (22JSQ025); Open Research Fund of Shaanxi Key Laboratory of Optical Remote Sensing and Intelligent Information Processing (KF20230101).
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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