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Abstract
This paper proposes an achromatic annular folded lens (AFL) with a reflective-diffractive optical element (RDOE). We derive novel mathematical models of the diffraction efficiency and polychromatic integral diffraction efficiency (PIDE) of the RDOE and an expression for its microstructure height. An AFL with an RDOE made of an optical plastic substrate material is designed in the visible waveband. To minimize the influence of incident angle on the diffraction efficiency and PIDE, the microstructure height is optimized. The design results indicate that the lateral color of the AFL is corrected, the modulation transfer function considering the diffraction efficiency is larger than 0.25 at 111 cycles/mm for all field of views. The hybrid AFL outperforms the conventional refractive imaging system in terms of the system size, volume, and image quality under the same specifications. It can be used in new-generation miniaturized imaging systems.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the continuing trend toward increased miniaturization and intelligentialization, compact imaging systems are in demand. A conventional optical imaging system comprises multiple optical lenses, making it bulky. A compact imaging system with a high image quality is desired for visible and infrared applications. In recent years, many potential optical elements or technologies have been developed to replace the conventional imaging system, such as metasurfaces [1], liquid lenses [2], GRIN lenses [3], negative index lenses [4] and so on. However, the above elements or technologies have rarely been applied to practical engineering in current stage, because of their poor performance and difficulty in manufacturing [5,6].
An annular folded imaging system comprises multiple concentric reflective zones, and each zone having a different aspherical surface [7]. Two concentric reflective surfaces are formed as two sides of a single solid lens, thus significantly reducing the total track and weight. The annular folded lens (AFL) can be fabricated using a common processing method, such as single point diamond turning (SPDT), and therefore can be applied to practical engineering. However, the image quality of the AFL is affected by color aberration, particularly the lateral color aberration [8]. This is mainly because the substrate of the AFL is made of a single optical material, and the chromatic aberration cannot be corrected. To minimize the influence of chromatic aberration, either calcium fluoride (CaF2) is selected as the substrate material because of its low optical dispersion, or the AFL is combined with other components [8–10], which combinations make the system more complicated. For a reflective annular folded system, as the middle medium of two multiple concentric reflective surfaces is air, so no chromatic aberrations exist in this structure [11,12]. However, the two reflective surfaces should be separately processed using the SPDT, and it is difficult to eliminate alignment errors between the two reflective surfaces are difficult to guarantee.
The injection molding technology can be used to produce a large number of complex and accurate optical lenses in a fast and cost-effective manner. As a manufacturing technology, injection molding is competing with other technologies capable of making components with similar function [13,14]. If the AFL is manufactured using the injection molding technology, the cost can be significantly reduced, and the processing efficiency can be improved. Compared with CaF2 as the substrate material, the optical dispersion of optical plastics is more serious. To use an optical plastic as the substrate material of the AFL for injection molding processing, an imaging diffractive optical element (DOE) should be introduced to achieve achromatism [15–17]. For the AFL, the aperture stop is usually placed in the incident annular zone, and the DOE is introduced on a surface far from the aperture stop to correct lateral color aberration. However, the DOE cannot be placed on the final transmissive surface of the AFL, because an index-matching gel is applied between this surface and the sensor [18,19]. Therefore, the DOE is formed on an annular reflective surface, thus realizing a reflective-diffractive optical element (RDOE).
In this paper, we derive a mathematical model of the diffraction efficiency and polychromatic integral diffraction efficiency (PIDE) of the RDOE and the expression for its microstructure height. With an optical plastic as the substrate material, the image quality of the AFL with and without the RDOE is analyzed. Based on the incident angle range of the designed RDOE, the microstructure height is optimized, and the diffraction efficiency and PIDE are discussed. Modulation transfer function (MTF) curves considering the diffraction efficiency are given, and a system model of the hybrid AFL is presented.
2. Diffraction efficiency of reflective-diffractive optical elements
The diffraction efficiency is also a key parameter of the RDOE as in the conventional transmissive DOE. The $m$ th order diffraction efficiency of the RDOE can be written as
where ${E_{Rm}}$ is the reflected energy of the $m$ th diffraction order and ${E_{R0}}$ is the total incidence energy of the RDOE. When the RDOE is applied to an imaging system, its designed period width is generally much greater than the wavelength. Therefore, the scalar theory can be used to analyze the diffraction efficiency of RDOEs. The diffraction efficiency of an RDOE with a continuous surface in each period can be expressed as [20]: where $\sin c(x) = {{\sin (\pi x)} \mathord{\left/ {\vphantom {{\sin (\pi x)} {(\pi x)}}} \right.} {(\pi x)}}$. m is the diffraction order, and the first diffraction order plays the key role, so $m$=1. $\phi $ is the phase retardation of the RDOE. To facilitate discussion, the continuous surface can be modeled by $N$-level phase profiles, as shown in Fig. 1. For a given number of phase levels, the diffraction efficiency of the $m$ th diffraction order can be written as [21]:By substituting Eq. (6) into Eq. (5), we can simplify the expression as follows.
According to Eq. (4), the phase retardation $\phi $ in a period is given by where H is the microstructure height of the RDOE, and $H = Nt$. When the design wavelength ${\lambda _0}$ and design incident angle ${\theta _0}$ are determined, the microstructure height of the $N$-level RDOE can be calculated by setting $\phi $ equal to one By simplifying Eq. (9), we can obtain where ${n_0}$ is the substrate material index of the RDOE at the design wavelength ${\lambda _0}$. Substituting Eq. (10) into Eq. (8) gives the expression for $\phi $: By replacing the phase retardation $\phi $ in Eq. (3) with Eq. (11), we can express the diffraction efficiency of the RDOE with a binary structure as follows:The diffraction efficiency of the RDOE with a continuous profile can be written as
The PIDE stands for the illumination over the entire waveband, and the actual optical transfer function (OTF) of a hybrid optical system can be approximated as the product of the PIDE and the theoretical OTF. The actual OTF can be written as [22] where ${f_x}$ and ${f_y}$ denote the spatial frequencies. ${\overline \eta _m}$ denotes the PIDE of the RDOE, and it can be expressed as3. Design example and discussions
3.1 Conventional optical system
In a conventional optical imaging system, multiple optical elements are commonly used to realize some certain specifications. As a comparison example, a typical Petzval-type lens [23] is shown in Fig. 3. Table 1 lists the design specifications. The detector is a SONY IMX421LLJ CMOS color sensor, and the diagonal size of the active area is 11 mm. This system is composed of six lenses, and the total track is 100.33 mm. The MTF values for all field of views are greater than 0.1 at the Nyquist Frequency [24] of 111 cycles/mm. The minimum and maximum values of the MTF are 0.1 and 0.14, respectively.
Table 1. Design Specifications
3.2 Achromatic AFL design
The AFL is inspired from a modified version of conventional telescopes with additional folding to minimize the track length. According to the design specifications given in Table 1, an AFL is designed. To be suitable for mass production using injection molding processing, a commonly used optical plastic, namely polymethyl methacrylate (PMMA) [25], is selected as the substrate material of the AFL.
Because the center obscuration is large, larger diameter AFLs have increased resolution and energy collection. Therefore, the diameter of an AFL is usually greater than that of a conventional optical system under the same specifications. As the number of folds increases, the diameter and volume also increase [8]. Considering the size and length of the AFL, we selected a four-reflection structure, and the physical thickness L can be calculated as follows
where ${f^{\prime}}$ is the focal length, n is the refractive index of the substrate material, and R is the number of reflections. If PMMA is selected as the substrate material, the thickness of the four-reflection AFL is 23 mm for a focal length of 61.8 mm. To obtain an effective entrance aperture, as listed in Table 1, the obscuration ratio and outer diameter of the AFL should be determined. The effective entrance aperture diameter ${D_{eff}}$ of the AFL can be expressed as where ${\alpha _{obs}}$ is the obscuration ratio, and D is the outer diameter of the AFL. When $\alpha $ is large, the midspatial frequency contrast is significantly less. Therefore, we determine D as 75 mm, and $\alpha $ is calculated to be 0.7 using ${D_{eff}}$ and D. Based on the above design parameters, the initial design of the AFL can be realized by using the optical design software ZEMAX OpticStudio [26]. The optimization variables are set for the radius of each zone, spacing thickness between two annular folded surfaces, and aspheric coefficients of the annular zones. The even asphere surface type is selected in the AFL design, and its corresponding polynomial expansion can be written as [26]In Fig. 4(a), the abscissa represents the lateral color, and the vertical coordinate represents the field of view. As the field of view increases, the lateral color also increases linearly. The maximum lateral color value is 32µm at the edge field of view, which is almost 7 times larger than the pixel pitch of CMOS (4.5µm). In Fig. 4(b), the different colors represent different wavelengths. The blue, green, and red colors represent the spot diagrams corresponding to wavelength of 0.486µm, 0.558µm and 0.656µm, respectively. The centers of the spot diagrams at the three wavelengths do not coincide with each other, except for the axial field. Although other aberrations are existed, the image quality is mainly affected by the lateral color aberration.
To correct the lateral color aberration and improve the image quality of the initial AFL, the RDOE is introduced. Because of the aperture stop is set at the entrance annular zone, the RDOE should be placed at far from the stop, on the last annular reflective surface to effectively correct the lateral color aberration effectively. The binary 2 surface type is selected, and the distribution of the diffractive phase is as follows [26]
where m is the diffraction order, which is usually set to 1, ${A_i}$ is the polynomial coefficient, $\rho $ is the normalized radial aperture, which can be expressed as where ${r_{norm}}$ is the normal radius. The detail lens data of the hybrid AFL with the RDOE is listed in Table 2. The polynomial coefficients ${\alpha _i}$ of the even asphere and phase coefficient ${A_i}$ of the diffractive surface are listed in Table 3.Table 2. Lens Data of Hybrid AFL with RDOE
Table 3. Detail Data for Even Asphere
Figure 5 shows the optical system layout and solid model of the hybrid AFL with the RDOE. In Fig. 5(a), the total length of the hybrid AFL is 22 mm, and the transparent zone is 11.25 mm. By comparing the different optical structures, we can find that the total length of the conventional Petzval-type lens (Fig. 3(a)) is 4.6 times that of the hybrid AFL. Moreover, the Petzval-type system comprises 6 optical lenses, whereas the hybrid AFL has only one lens. As shown in Fig. 5(b), the yellow area represents the entrance light zone, and the silver area indicates the reflective coating. The lateral color and spot diagram of the hybrid AFL are shown in Fig. 6.
Fig. 6. Image quality evaluation of the final designed AFL. (a) Lateral color curve. (b) Spot diagram.
Figure 6(a) shows that the maximum value of the lateral color is 1.3µm in the full field of view, which is less than the pixel pitch of CMOS (4.5µm). The lateral color aberration of the hybrid AFL is corrected by the RDOE. As shown in Fig. 6(b), the centers of the spot diagrams at different wavelengths are coincide, thus significantly reducing the spot radius of the off-axial field is the tangential direction. Accordingly, the MTF of the AFL with the RDOE is also improved. However, the MTF result calculated using the ZEMAX cannot be directly used for imaging quality analysis, because the effect of the diffraction efficiency of the RDOE is ignored. In practice, the energy scattered into additional parasitic diffraction orders can limit the imaging performance of the hybrid AFL [27,28]. To accurately predict the image quality of the hybrid AFL, the diffraction efficiency and the PIDE should be calculated, and a modified MTF can then be obtained.
3.3 Diffraction efficiency analysis of the RDOE
From the deduced relationship in Section 2, we can conclude that the microstructure height of the RDOE is not only related to the wavelength and substrate material, but also related to the incident angle. Therefore, the diffraction efficiency and the PIDE are also related to the incident angle. For the designed hybrid AFL, when the field of view is different, the incident angle on the RDOE is changed. To obtain a reasonable PIDE within the designed range of the incident angles, the microstructure height of the RDOE should be optimized. The flow diagram of the optimization process is shown in Fig. 7.
The details of the optimization process are given as the follows. Firstly, the initial design incident angle ${\theta _0}$ is given, such as ${\theta _0} = {\theta _{\min }}$. According to the design wavelength ${\lambda _0}$ and corresponding refractive index of the substrate material ${n_0}$, the microstructure height H can be calculated using Eq. (10). Secondly, the PIDEs at minimum and maximum incident angles, ${\overline \eta _{\theta \textrm{min}}}$ and ${\overline \eta _{\theta \textrm{max}}}$, can be calculated using Eq. (15) based on the above microstructure height H. Finally, ${\overline \eta _{\theta \textrm{min}}}$ and ${\overline \eta _{\theta \textrm{max}}}$ are compared. If ${\overline \eta _{\theta \textrm{min}}}$=${\overline \eta _{\theta \textrm{max}}}$, the designed RDOE has a reasonable PIDE over the range of incident angles, and the optimized microstructure height ${H_{opt}}$ can be obtained. If ${\overline \eta _{\theta \textrm{min}}}$ and ${\overline \eta _{\theta \textrm{max}}}$ have the different values, the design incident angle ${\theta _0}$ is incremented by $\Delta \theta $, and the above process is repeated until a specific design incident angle is found for which ${\overline \eta _{\theta \textrm{min}}}$=${\overline \eta _{\theta \textrm{max}}}$.
According to the final design result of the hybrid AFL given in Section 3.2, the minimum and maximum incident angles of the RDOE are 14.4° and 25.2°, respectively. For this incident angle range, the angle selection is 21.6° using the optimization method, as shown in Fig. 7. Therefore, the design incident angle ${\theta _0}$ is 21.6° and the design wavelength ${\lambda _0}$ is 0.558µm. According to Eq. (10) in Section 2, the optimized microstructure height ${H_{opt}}$ of the RDOE is −0.201µm. Substituting ${H_{opt}}$ into Eqs. (13) and (15), we can determine the diffraction efficiency and PIDE for different incident angles, as shown in Fig. 8. The minimum diffraction efficiency and PIDE for the RDOE are listed in Table 4.
Table 4. Minimum Diffraction Efficiency and PIDE versus Incident Angle for RDOE
If the incident angle decreases from 21.6° to 14.4°, the minimum diffraction efficiency reduces from 92.59% to 87.44% (a reduction of 5.15%), and it can be obtained at a wavelength 0.486µm. If the incident angle increases from 21.6° to 25.2°, the minimum diffraction efficiency reduces from 92.59% to 90.45% (a reduction of 2.14%), and it can be obtained at a wavelength of 0.656µm. The maximum reduction in the PIDE is 0.34%. This shows that the influence of incident angle on the diffraction efficiency and the PIDE can be ignored, when the microstructure height of the RDOE is optimized. The actual MTF considering the diffraction efficiency can be calculated using the ZEMAX programming language (ZPL) macro [25], as shown in Fig. 9. Table 5 lists the MTF values over all the fields in the tangential and sagittal directions. The MTF values for all the field of views are greater than 0.25 at 111 cycles/mm, and the minimum and maximum values of the MTF are 0.254 and 0.29, respectively. Compared with the conventional Petzval-type system shown Fig. 3(b), the MTF of the AFL with the RDOE is increased by 2 times under the same specifications.
Table 5. MTF values with considering the diffraction efficiency at different field of views
Figure 10 shows the phase plot and line frequency with respect to the aperture for the RDOE. In Fig. 10, the blue line indicates the phase over the semi-diameter of the RDOE surface, and the red line indicates the line frequency of the RDOE surface and its unit is periods per millimeter. As shown in Fig. 10, the line frequency at the maximum semi-diameter of the RDOE is 39 period/mm, and the corresponding minimum period width is 25.6µm. Therefore, the AFL can be fabricated using a common processing method such as SPDT.
4. Conclusion
In this paper, we proposed a hybrid AFL with an RDOE, exhibiting a compact size and a high image quality. The novel mathematical model of the diffraction efficiency and PIDE of the RDOE is deduced, and the expression for the microstructure height is obtained. An AFL with an RDOE made of a PMMA substrate material is designed in the visible waveband. For the AFL without the RDOE, the maximum lateral color value is 32µm, and this mainly affects the image quality. When the RDOE is introduced to realize a hybrid AFL, the maximum lateral color value is only 1.3µm. To minimize the influence of the incident angle on the diffraction efficiency, an optimization process for the microstructure height is presented. When the incident angle is increased from 14.4° to 25.2°, the maximum reduction in the PIDE is only 0.34%, so the influence of the incident angle is ignored. The MTF considering the diffraction efficiency is greater than 0.25 at 111 cycles/mm for all field of views. We found that the total length of the conventional Petzval-type system is 4.6 times that of the hybrid AFL, and the MTF of the hybrid AFL is 2 times that of the conventional Petzval-type system under the same specifications. The hybrid AFL has obvious advantages in terms of the system size, volume and image quality. The results of in this study can be useful for the development of next-generation of miniaturized imaging systems.
Funding
China Government (51-H34D01-8358-13/16); Excellent Youth Foundation of Jilin Province Scientific Committee (20190103134JH); Education Department Project of Jilin Province (JJKH20181110KJ).
References
1. S. Banerji, M. Meem, A. Majumder, F. G. Vasquez, B. Sensale-Rodriguez, and R. Menon, “Imaging with flat optics: metalenses or diffractive lenses?” Optica 6(6), 805–810 (2019). [CrossRef]
2. L. Li, C. Liu, H. Ren, H. Deng, and Q. Wang, “Annular folded electrowetting liquid lens,” Opt. Lett. 40(9), 1968–1971 (2015). [CrossRef]
3. J. Nagar, S. D. Campbell, and D. H. Werner, “Apochromatic singlets enabled by metasurface-augmented GRIN lenses,” Optica 5(2), 99–102 (2018). [CrossRef]
4. M. Piao, Q. Cui, and B. Zhang, “Achromatic negative index lens with diffractive optics,” J. Opt. 17(2), 025608 (2015). [CrossRef]
5. V. Su, C. Chu, G. Sun, and D. Tsai, “Advances in optical metasurfaces: fabrication and applications,” Opt. Express 26(10), 13148–13182 (2018). [CrossRef]
6. A. C. Urness, K. Anderson, C. Ye, W. L. Wilson, and R. R. Mcleod, “Arbitrary GRIN component fabrication in optically driven diffusive photopolymers,” Opt. Express 23(1), 264–273 (2015). [CrossRef]
7. J. Ford, E. Tremblay, and S. Fainman, “Multiple reflective lenses and lens systems,” US Patent, 7898749B2 (2011).
8. E. J. Tremblay, R. A. Stack, R. L. Morrison, and J. E. Ford, “Ultrathin cameras using annular folded optics,” Appl. Opt. 46(4), 463–471 (2007). [CrossRef]
9. L. Li, D. Wang, C. Liu, and Q. Wang, “Ultrathin zoom telescopic objective,” Opt. Express 24(16), 18674–18684 (2016). [CrossRef]
10. A. Arianpour, G. M. Schuster, E. J. Tremblay, I. Stamenov, A. Groisman, J. Legerton, W. Meyers, G. A. Amigo, and J. E. Ford, “Wearable telescopic contact lens,” Appl. Opt. 54(24), 7195–7204 (2015). [CrossRef]
11. D. Cheng, X. Chen, C. Xu, Y. Hu, and Y. Wang, “Optical description and design method with annularly stitched aspheric surface,” Appl. Opt. 54(34), 10154–10162 (2015). [CrossRef]
12. R. Morrison, R. Stack, G. Euliss, R. Athale, C. Reese, J. Vizgaitis, J. Stevens, E. Tremblay, and J. Ford, “An alternative approach to infrared optics,” Proc. SPIE 7660, 76601Y (2010). [CrossRef]
13. X. Lan, C. Li, C. Yang, and C. Xue, “Optimization of injection molding process parameters and axial surface compensation for producing and aspheric plastic lens with large diameter and center thickness,” Appl. Opt. 58(4), 927–934 (2019). [CrossRef]
14. S. Bäumer, Handbook of Plastic Optics, Second Edition, (Wiley-VCH, 2010).
15. H. Zhang, H. Liu, A. Lizana, W. Xu, C. Juan, and Z. Lu, “Methods for the performance enhancement and the error characterization of large diameter ground-based diffractive telescopes,” Opt. Express 25(22), 26662–26677 (2017). [CrossRef]
16. Y. G. Soskind, “Diffractive optics technologies in infrared systems,” Proc. SPIE 9451, 94511T (2015). [CrossRef]
17. C. Bigwood and A. Wood, “Two-element lenses for military applications,” Opt. Eng. 50(12), 121705 (2011). [CrossRef]
18. G. Agranov, V. Berezin, and R. H. Tsai, “Crosstalk and microlens study in a color CMOS image sensor,” IEEE Trans. Electron Devices 50(1), 4–11 (2003). [CrossRef]
19. E. J. Tremblay, R. A. Stack, R. L. Morrison, J. H. Karp, and J. E. Ford, “Ultrathin four-reflection imager,” Appl. Opt. 48(2), 343–354 (2009). [CrossRef]
20. D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics Design, Fabrication, and Test (SPIE, 2004).
21. G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” MIT Lincoln Laboratory Technical Report 914 (1991).
22. D. A. Buralli and G. M. Morris, “Effects of diffraction efficiency on the modulation transfer function of diffractive lenses,” Appl. Opt. 31(22), 4389–4396 (1992). [CrossRef]
23. M. Laikin, Lens Design, Fourth Edition (CRC, 2007).
24. R. E. Fischer, B. Tadic-Galeb, and P. R. Yoder, Optical System Design, Second Edition (SPIE, 2008).
25. M. Bass, G. Li, and E. V. Stryland, Handbook of Optics Volume IV Optical Properties of Materials, Nonlinear Optics, Quantum Optics, Third edition, (McGraw-Hill, 2010).
26. Zemax Development Corporation, “OpticStudio User Manual,” (2016).
27. T. Wang, H. Liu, H. Zhang, H. Zhang, Q. Sun, and Z. Lu, “Effect of incidence angles and manufacturing errors on the imaging performance of hybrid systems,” J. Opt. 13(3), 035711 (2011). [CrossRef]
28. C. Bigwood, “New infrared optical systems using diffractive optics,” Proc. SPIE 4767, 1–12 (2002). [CrossRef]
