Open Access
Add to BibSonomy
Abstract
We utilized inverse design to engineer the point-spread function (PSF) of a low-f-number, freeform diffractive microlens in an array, so as to enable extended depth of focus (DOF). Each square microlens of side 69 µm and focal length 40 µm (in a polymer film, n∼1.47) generated a square PSF of side ∼10 µm that was achromatic over the visible band (450 to 750 nm), and also exhibited an extended DOF of ∼ ± 2 µm. The microlens has a geometric f/# (focal length divided by aperture size) of 0.58 in the polymer material (0.39 in air). Since each microlens is a square, the microlens array (MLA) can achieve 100% fill factor. By placing this microlens array (MLA) directly on a high-resolution print, we demonstrated integral imaging with applications in physical security. The extended DOF preserves the optical effects even with expected film-thickness variations, thereby increasing robustness in practical applications. Since these multi-level diffractive MLAs are fabricated using UV-nanoimprint lithography, they have the potential for low-cost large volume manufacturing.
© 2022 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In integral or light field imaging, [1–4] an appropriately designed printed image (or a display) is placed at the focal plane of a micro-lens array (MLA) as illustrated in Fig. 1(a) [5–8]. When viewed from the observer side, due to the creation of multiple parallaxes in the reconstructed image produced by the MLA, a 3D-image, otherwise known as integral image, can be viewed. The details of the integral image change with the point of view (POV) of the observer giving the feeling of 3D and depth in the 2D image plane. This entire system can be compressed to enable integral imaging in a compact form-factor for applications in optical document security, such as in currency notes. The direct applications lie in incorporating a window with a special print and an MLA on top, that would offer an integral image viewable by an observer, as in Fig. 1(a). Since both the prints and the MLA structures are extremely difficult to forge, such applications would offer strong anti-counterfeiting properties. One example of such a system in use is the 3D security ribbon present on the US 100-dollar bill [9]. For such applications, the total thickness of the substrate is restricted by the performance requirements; additionally, high volume substrate printing processes are already highly optimized, but are constrained with respect to the maximum printable resolutions. Security within documents is a function of the complexity of the security features produced; and since print resolution cannot be easily increased beyond its current state, the complexity of an MLA security feature can be enhanced by enlarging the MLA apertures as has been noted in the context of auto-stereoscopic integral imaging [10]. Furthermore, by designing an MLA with an extended DOF, which is larger than the expected variation in the commercially available film thickness (∼2µm to 3µm), we can ensure that the optical effects are preserved even with these variations. In other words, the MLA is designed such that the focal spot does not vary significantly over the expected thickness variations in the underlying film thickness. We note that alternative approaches to solve these problems include the use of multiple MLAs (which is much thicker and requires complex alignment) [11] or the use of holographic optical elements (which tend to have poor performance under broadband illumination) [12].
Fig. 1. A free form microlens array (MLA) with engineered PSFs enables 3D integral imaging. (a) A 3D integral image is observed when the MLA is placed one focal length, f, away from an optimized high-resolution print. (b) Each microlens on the front surface of the flexible polymer film focuses incident collimated light to a square shaped spot on the back surface (thickness = f = 40 µm). (c) Photograph of the fabricated device (inset shows that the device is flexible) (d) Optimized height profile and (e) optical scanning-confocal microscope image of one microlens. and (f) widefield optical micrograph of an array of 3 × 3 close-packed microlenses (period = 70 µm and size of one lens = 69 µm).
In these applications, the total thickness of such a system from the print to the MLA is equal to the focal length of each microlens. As mentioned earlier, larger MLA apertures are desirable to enable more complex and higher resolution 3D images. However, in conventional refractive MLAs, it is challenging to maintain short focal lengths, while increasing the aperture diameter, i.e., the regime of small f-number (f/#) (ratio of focal length to aperture size). The challenges arise from: the large sag leading to low fabrication yield; the reduced depth of focus (DOF), and enhanced aberrations [13–16]. We note that Fresnel-type [17] and metalens-based MLAs suffer from low efficiencies or chromatic aberrations [18–21]. There is also limited demonstration of low f/# metalenses in the literature. We have provided a comparative literature survey in Section 1 of the supplement. Specifically, we report on the first demonstration of a visible-band low-f# (<f/0.7) MLA using diffractive optics. Here, we show that multi-level diffractive optics-based freeform MLAs [22–30] can solve these problems by enabling structured point-spread functions.
We showed previously that freeform diffractive MLAs can be engineered to produce structured PSFs [23,24]. Such PSF engineering decouples the effective numerical aperture (defined as the ratio of the size of the focused spot to half the center wavelength) of the MLA from its f/# and can enable extended depth-of-focus (DOF). These previous demonstrations were performed at a relatively large f/# (11.25). Here, we created an MLA with an order of magnitude reduction in f/# = 0.58 in a polymer (0.39 in air) and fabricated it directly on top of a polymer film of thickness 40 µm (which matches the focal length in the polymer) using UV-nanoimprint lithography. Note that the focal length in air is scaled down by the refractive index of the polymer film leading to the lower f# in air. Finally, we combined the MLA with high-resolution prints to demonstrate integral imaging. Since the DOF is relatively large (±2 µm), the MLA can be robustly assembled directly onto the print substrate. These MLAs are also achromatic over the entire visible band.
In a recent review paper, freeform surfaces are simply defined as ones with no axis of rotational invariance [31]. Since our diffractive MLAs do not have an axis of rotational invariance, they are in fact an example of diffractive freeform optics. This distinction from rotationally symmetric surfaces is critical as such surfaces allow for correction of aberrations that are not rotationally symmetric. Such surfaces additionally require distinct methods of manufacturing and metrology.
2. Results and discussion
From Fourier optics, we know that the PSF is the square of the absolute value of the complex field in the focal plane, U, which itself can be expressed as the convolution of the pupil function (T) and the transfer function that describes free-space propagation (H). This is typically written in terms of the Fourier Transform (FT):
Figure 1(c) shows a photograph of the fabricated MLA on a flexible substrate. The optimized height-profile (design) for one microlens is shown in Fig. 1(d), while an optical micrograph of one microlens obtained using a laser confocal microscope (Olympus LEXT OLS5000) is shown in Fig. 1(e). Figure 1(f) shows an optical micrograph of a portion of the MLA showing an array of 3 × 3 close-packed microlenses, obtained using a widefield optical microscope (Keyence VHX-5000). The designs were simulated using periodic boundary conditions, but for fabrication robustness, the period was chosen to be 70 µm, leading to a 0.5 µm transparent unpatterend region around each microlens. This is visible as the white lines in Fig. 1(f). As indicated by the photograph in Fig. 1(c), the 40 µm-thick film is very flexible and can be readily applied to high-resolution prints directly, as illustrated in the inset in Fig. 1(c). Atomic-force micrographs from different parts of the MLA were used to estimate an approximate error in the fabricated pixel heights of ∼7% (see section 3 of the supplement). The MLA (total size = 10 mm × 10 mm) was fabricated using UV-nanoimprint lithography. Briefly, a reverse master pattern (obtained by subtracting the designed height profile from the maximum height) was first fabricated in a silicon wafer using scanning-electron-beam lithography (SEBL) and etching. The minimum feature size for our design is the individual pixel size, equal to 345 nm, which can be readily fabricated by SEBL. The silicon master was imprinted into the imprint resist layer, (OS, NIL Technologies ApS), which was dispensed onto the polymer film (cellulose diacetate) and UV-cured (prior to mold separation) to create the final imprint. Figure 2(a) shows the atomic force micrograph of a location of the MLA where a feature close to the maximum height of 1 µm is present. A line scan through this micrograph indicated by the red line is shown in Fig. 2(b). Finally, Figs. 2(c-d) show the specialized “offset” prints that were used to produce the integral images with the MLA. Figure 2(e) shows a 3D micrograph of one microlens using the confocal microscope (Olympus LEXT OLS5000). A line scan through the center of the microlens, indicated by the black line is used to compare the design and measured heights as shown in Fig. 2(f). The fabricated heights match well with design, except some of the smaller features are not resolved by the optical microscope.
Fig. 2. (a) Atomic force micrograph of a section of the microlens, with a pixel close to the maximum height of 1 µm. (b) Line scan through the micrograph in (a), indicated by the red line. (c-d) Optical micrographs of the specialized “offset” prints that were used to produce the integral images with the MLA. (e) 3D optical micrograph of one microlens. (f) Comparison of the design and measured heights of the line scan through the center of the microlens, indicated by the black line in (e).
The simulated through-wavelength and through-focus PSFs of one microlens in the array are shown in Fig. 3(a). We note that the simulations assumed monochromatic illumination (except for the broadband results, which averaged multiple monochromatic simulations). The diffraction-limited DOF (in air) is given by $\lambda \left( {{{\left( {\frac{{2f}}{D}} \right)}^2} + 1} \right)$, where f is the focal length in air (40 µm/1.47 ∼ 27µm), and D (69µm) is the aperture size. For the center wavelength of 0.6µm, this gives a diffraction-limited DOF = 0.97µm. By engineering the PSF to be a square of size ∼10µm, the shape and size of the PSF can be maintained within a total DOF of ∼ 4µm. To confirm optical performance, the fabricated MLA was illuminated by an expanded and collimated beam from a super-continuum source (SuperK EXTREME EXW-6, NKT Photonics) coupled to a tunable filter (SuperK VARIA, NKT photonics). An objective lens and a tube lens were used to magnify the PSF (formed on the back surface of the substrate) by 21.5X onto a monochrome CMOS image sensor (DMM 27UP031- ML, The Imaging Source). The magnification system and sensor were translated together along the optical axis to measure the through-focus PSFs (section 4 of the supplement). Note that the MLA was mounted on a glass slide for structural rigidity and the glass provides good index matching with the cellulose diacetate film (n∼1.47, Clarifoil, Celanese). The measured PSFs at the plane of best focus and the two defocus (± 2 µm) planes are shown in Fig. 3(b), while the measured longitudinal PSFs are shown in Fig. 3(c) confirming excellent achromaticity. We recorded the PSF at various Z-locations and stitched the frames to produce a through focus video that is presented in Visualization 1. We note that the size of the measured PSFs agree well with simulations, although the details of the interior speckle do not. We attribute these differences to the scalar diffraction approximation, use of monochromatic illumination in simulations and fabrication errors in experiments. The PSFs at the best focal plane indicated by the dashed white line in Fig. 3(c), and the two defocus planes are shown in Fig. 3(b). Further details including an illustration, photograph of the measurement system, a wider field of view containing additional focal spots and measurements at additional wavelengths are presented in Section 3 of the Supplement.
Fig. 3. Comparing simulations to experiments: Broadband focusing. (a) Simulated and (b) measured through-focus and through-λ (bandwidth = 15 nm) point-spread functions (PSFs). (See Visualization 1 for measured through focus video). (c) Measured longitudinal PSFs indicating achromaticity (focal length is approximately independent of λ). The dashed white line locates the plane of best focus (used to obtain the PSFs in b). Insets in (a) and (b) show magnified views of the PSFs.
The broadband focus is plotted in Fig. 4(a), along with a magnified view of the bottom corner in Fig. 4(b). The latter shows significant light diffraction arising from the edges of the microlens. Since the size of the microlens was 69µm, while the center to center spacing in the array was 70µm, a 0.5µm unpatterned frame remained (as can be seen by the white lines in Fig. 1(e)). This frame not being part of the carefully optimized design, leads to undesired edge diffraction. In Figs. 4(c) and 4(d), we plotted the averaged vertical and horizontal linescans through the focus, respectively. The linescans indicate that the size of the main lobe remains remarkably consistent across a defocus range of 4µm, thereby confirming an extended DOF. Edge diffraction causes relatively large side-lobes close to the boundaries of the microlens aperture. These effects can be mitigated in the future by matching the period of the MLA and the size of each microlens precisely. Section 3 of the supplement includes the data for the individual wavelengths. We also performed a careful measurement of the PSF at two orthogonal linear polarizations, and concluded that the PSFs are mostly polarization invariant (see Fig. S6).
Fig. 4. Comparing size of focused spot. (a) Measured broadband spot in the focal plane. (b) Magnified inset at the bottom corner shows edge diffraction due to the 0.5 µm-wide unpatterned frame, which leads to large sidelobes near the edges of the microlens aperture. This is readily seen in the averaged (c) vertical and (d) horizontal cross-sections. Nevertheless, the mainlobe (central 10 µm square) matches well for simulations (dashed lines) and experiments (solid lines) across a defocus range of 4 µm.
Next, we assembled the MLA directly in contact with several high-resolution color prints. The prints were generated using standard banknote plate making and offset printing equipment; original designs were generated at 10,160 dpi (pixel size 2.5 µm), resolution losses in plate making and printing limit the minimum feature size to 10–15 µm. The feature designs incorporated both line and icon Moiré effects, which were selected for their tolerance to the expected effective resolution reduction caused by the print process and also for the simplicity of the resulting optical effects. Images and videos of the integral images were recorded using an iPhone 12 Pro Max camera, while the assembled print was illuminated from behind using ambient room light. Figure 5(a) shows a schematic illustration of the setup using the print and the MLA that can be a direct application of this system to optical document security, where a window with a print and MLA on top can be incorporated in the document and viewed by the observer in ambient light to see the 3D integral image. Note that since the MLA is smaller than the print as shown in Fig. 1(d), we could only image sections of each print at one time. First, we present simulated videos of integral imaging for each of three different designs as seen in Visualization 2, Visualization 3, and Visualization 4. Next, after fabrication of the prints for each of the three designs, we first show that without the MLA, there is no integral imaging, as seen in Visualization 5, Visualization 6, and Visualization 7 for each of the prints. Finally, we combine the MLA with each of the three prints and demonstrate integral imaging for each of these evidenced by the animated and moving 3D patterns that match well with the simulations as seen in Visualization 8, Visualization 9, Visualization 10, Visualization 11, Visualization 12 and Visualization 13. It is to be noted that some sections of the patterns in the simulations had different colors compared to the final prints. Nevertheless, the final animated integral image effect remains same. In Figs. 5(b-d) we extracted some frames from the videos to show the simulations (left panels) and corresponding experimental images (right panels) of the integral images formed with the three different print designs. However, in order to see the 3D effect, the assembly must be viewed or recorded in videos. We recorded videos as presented in the visualizations noted above and the images shown in Fig. 5 are screen captures of different frames of the videos. Each panel in Fig. 5(b-d) shows two sequential frames (top and bottom) from the simulations (left columns) and corresponding frames from the recorded videos (right columns). The color mismatch is due to the use of false colors in simulations. Nevertheless, the image geometries agree very well and full-color images (without any dispersive/rainbow effects) can be observed in all cases. The 3D integral images is easily viewed with the naked eye. Figure 5(e) shows sequential frames from Visualization 11 illustrating the change in the image with viewing angle. See sections 5 and 6 of the supplement for further details.
Fig. 5. (a) Schematic illustration showing the principle of integral imaging using the print + the MLA. (b-d) Simulated (Visualization 2, Visualization 3, and Visualization 4) and experimentally recorded (Visualization 5, Visualization 6, Visualization 7, Visualization 8, Visualization 9, Visualization 10, Visualization 11, Visualization 12 and Visualization 13) changing 3D integral images for three different print designs. (e) Frames captured from Visualization 11 for print design 3 illustrating the 3D integral image at different viewing angles. The MLA size is 10mm X 10mm in all cases.
3. Conclusion
Engineering the PSF was previously achieved at low f/# and narrowband illumination [29] or at large f/# and broadband illumination [21,22]. In this work, we showed PSF engineering at both low f/# (0.58) and for broadband illumination via a freeform MLA enabled by multi-level diffractive optics. 3D spatial engineering of the PSF enables an increase in DOF by a factor of over 4, which we achieve at the expense of creating a focal intensity distribution that is larger than the diffraction limit, but sufficiently small to resolve the high-resolution printed features (10-15 µm) for our application of document security via integral imaging. Furthermore, the MLA was nanoimprinted directly onto a 40 µm-thick polymer film, which can be readily integrated into high-resolution prints or displays for 3D integral imaging. We also showed an example of application of this to optical document security. Our approach for ultra-thin low f# microlens arrays are clearly applicable in 3D displays, lightfield cameras, wavefront sensors and many other aplications.
Funding
National Science Foundation (1936729); Office of Naval Research (N000141512316, N000141912458, N66001-10-1-4065).
Acknowledgments
Discussions with N. Hansson and S. Banerji are gratefully acknowledged.
Disclosures
MM and RM: Oblate Optics (I,E,P).
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. G. Lippmann, “Épreuves réversibles donnant la sensation du relief,” J. Phys. Theor. Appl. 7(1), 821–825 (1908). [CrossRef]
2. A. Lumsdaine and T. Georgiev, “Full resolution lightfield rendering,” Adobe Technical Report, Adobe Systems, 1–12 (2008).
3. T. Georgiev and A. Lumsdaine, “Reducing plenoptic camera artifacts,” Computer Graphics Forum 29(6), 1955–1968 (2010). [CrossRef]
4. N. Zeller, F. Quint, and U. Stilla, “Depth estimation and camera calibration of a focused plenoptic camera for visual odometry,” ISPRS J. Photogramm. Remote Sens. 118, 83–100 (2016). [CrossRef]
5. J. Arai, H. Kawai, and F. Okano, “Microlens arrays for integral imaging system,” Appl. Opt. 45(36), 9066–9078 (2006). [CrossRef]
6. X. Zhou, Y. Peng, R. Peng, X. Zeng, Y. Zhang, and T. Guo, “Fabrication of large-scale microlens arrays based on screen printing for integral imaging 3D display,” ACS Appl. Mater. Interfaces 8(36), 24248–24255 (2016). [CrossRef]
7. Y. Bok, H. G. Jeon, and I. S. Kweon, “Geometric calibration of micro-lens-based light field cameras using line features,” IEEE Trans. Pattern Anal. Mach. Intell. 39(2), 287–300 (2017). [CrossRef]
8. Q. Wang, H. Deng, T. Jiao, D. Li, and F. Wang, “Imitating micro-lens array for integral imaging,” Chin. Opt. Lett. 8(5), 512–514 (2010). [CrossRef]
9. https://www.uscurrency.gov/denominations/100
10. M. Dejean, V. Nourrit, and J.-L. de Bougrenet de la Tocnaye, “Object kinetics perception in auto-stereoscopic vision,” J. Opt. Soc. Am. A 36(11), C104–C112 (2019). [CrossRef]
11. A. Karimzadeh, “Integral imaging system optical design with aberration consideration,” Appl. Opt. 54(7), 1765–1769 (2015). [CrossRef]
12. S. Lee, C. Jang, J. Cho, J. Yeom, and B. Lee, “Viewing angle enhancement of an integral imaging display using Bragg mismatched reconstruction of holographic optical elements,” Appl. Opt. 55(3), A95–A103 (2016). [CrossRef]
13. J. Brückner, R. Duparré, P. Leitel, A. Dannberg, A. Bräuer, and Tünnermann, “Thin wafer-level camera lenses inspired by insect compound eyes,” Opt. Express 18(24), 24379–24394 (2010). [CrossRef]
14. C. Gassner, J. Dunkel, A. Oberdörster, and A. Brückner, “Compact wide-angle array camera for presence detection,” Proc. SPIE 10545, 105450B (2018). [CrossRef]
15. W. Wang, G. Chen, Y. Weng, X. Weng, X. Zhou, C. Wu, T. Guo, Q. Yan, Z. Lin, and Y. Zhang, “Large-scale microlens arrays on flexible substrate with improved numerical aperture for curved integral imaging 3D display,” Sci. Rep. 10(1), 11741 (2020). [CrossRef]
16. R. Ahmed and H. Butt, “Strain-multiplex metalens array for tunable focusing and imaging,” Adv. Sci. 8(4), 2003394 (2021). [CrossRef]
17. M. J. Moghimi, J. Fernandes, A. Kanhere, and H. Jiang, “Micro-fresnel-zone-plate array on flexible substrate for large field-of-view and focus scanning,” Sci. Rep. 5(1), 15861 (2015). [CrossRef]
18. R. J. Lin, V. C. Su, S. Wang, M. K. Chen, T. L. Chung, Y. H. Chen, H. Y. Kuo, J.-W. Chen, J. Chen, Y.-T. Huang, J.-H. Wang, C. H. Chu, P. C. Wu, T. Li, Z. Wang, S. Zhu, and D. P. Tsai, “Achromatic metalens array for full-colour light-field imaging,” Nat. Nanotechnol. 14(3), 227–231 (2019). [CrossRef]
19. 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]
20. Z. Yang, Z. Wang, Y. Wang, X. Feng, M. Zhao, Z. Wan, L. Zhu, J. Liu, Y. Huang, J. Xia, and M. Wegener, “Generalized Hartmann-Shack array of dielectric metalens sub-arrays for polarimetric beam profiling,” Nat. Commun. 9(1), 4607 (2018). [CrossRef]
21. T. Hu, Q. Zhong, N. Li, Y. Dong, Z. Xu, D. Li, Y. H. Fu, Y. Zhou, K. H. Lai, V. Bliznetsov, H. Lee, W. L. Loh, S. Zhu, Q. Lin, and N. Singh, “A metalens array on a 12-inch glass wafer for optical dot projection,” Opt. Fiber Commun. Conf. (OFC) 2020, OSA Technical Digest (Optical Society of America, 2020), paper W4C.3. (2020).
22. S. Banerji, M. Meem, A. Majumder, B. Sensale-Rodriguez, and R. Menon, “Super-resolution imaging with an achromatic multi-level diffractive microlens array,” Opt. Lett. 45(22), 6158–6161 (2020). [CrossRef]
23. M. Meem, A. Majumder, and R. Menon, “Multi-plane, multi-band image projection via broadband diffractive optics,” Appl. Opt. 59(1), 38–44 (2020). [CrossRef]
24. M. Meem, A. Majumder, and R. Menon, “Freeform broadband flat lenses for visible imaging,” OSA Continuum 4(2), 491–497 (2021). [CrossRef]
25. S. Banerji, M. Meem, A. Majumder, F. Guevara Vasquez, B. Sensale-Rodriguez, and R. Menon, “Imaging with flat optics: metalenses or diffractive lenses?” Optica 6(6), 805–810 (2019). [CrossRef]
26. M. Meem, S. Banerji, A. Majumder, F. Guevara Vasquez, B. Sensale-Rodriguez, and R. Menon, “Broadband lightweight flat lenses for longwave-infrared imaging,” Proc. Natl. Acad. Sci. 116(43), 21375–21378 (2019). [CrossRef]
27. S. Banerji, M. Meem, A. Majumder, B. Sensale-Rodriguez, and R. Menon, “Diffractive flat lens enables extreme depth-of-focus imaging,” Optica 7(3), 214–2017 (2020). [CrossRef]
28. M. Meem, S. Banerji, A. Majumder, C. Pies, T. Oberbiermann, B. Sensale-Rodriguez, and R. Menon, “Large-area, high-NA multi-level diffractive lens via inverse design,” Optica 7(3), 252–253 (2020). [CrossRef]
29. M. Meem, S. Banerji, A. Majumder, C. Pies, T. Oberbiermann, B. Sensale-Rodriguez, and R. Menon, “Inverse-designed flat lens for imaging in the visible & near-infrared with diameter > 3 mm and NA=0.3,” Appl. Phys. Lett. 117(4), 041101 (2020). [CrossRef]
30. S. Banerji, M. Meem, A. Majumder, F. Guevara Vasquez, B. Sensale-Rodriguez, and R. Menon, “Ultra-thin near infrared camera enabled by a flat multi-level diffractive lens,” Opt. Lett. 44(22), 5450–5452 (2019). [CrossRef]
31. J. P. Rolland, M. A. Davies, T. J. Suleski, C. Evans, A. Bauer, J. C. Lambropoulos, and K. Falaggis, “Freeform optics for imaging,” Optica 8(2), 161–176 (2021). [CrossRef]
32. S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. 106(9), 2995–2999 (2009). [CrossRef]
33. S. D. Bañasa, E. Separa, J. Engay, and Glückstadab, “Point spread function shaping using geometric analysis,” Opt. Commun. 427(15), 522–527 (2018). [CrossRef]
34. J. E. Greivenkamp, “Field Guide to Geometrical Optics,” SPIE Press (2004).
35. R. Berlich and S. Stallinga, “High-order-helix point spread functions for monocular three-dimensional imaging with superior aberration robustness,” Opt. Express 26(4), 4873–4891 (2018). [CrossRef]
