The resolution of a conventional lens is governed by its numerical aperture (NA) while its light-concentration ability is governed by its f-number (f/#), both of which are dependent on the diameter and focal length of the lens. Hence, these properties for a conventional lens are inherently coupled. Here, we show that by utilizing the principle of free-form broadband diffractive optical elements (BDOEs), we can create ultra-flat optical elements that are able to concentrate incoming light over a fixed aperture to different sized focal spots engineered in the shape of a square. Hence, we are able to decouple the NA and f/# of a lens by being able to independently control the size of the input aperture and the resolution of these devices. Specifically, we designed, fabricated, and characterized three BDOE lenses operating in the visible band (450nm–750nm), each having f/# of 11.25, but with NAs of 0.00075, 0.0067, and 0.054, respectively. We further use simulations to emphasize that such decoupling is possible even at much higher NAs. Experiments confirm achromatic focusing and broadband imaging. One of the lenses exhibited a depth-of-focus almost 2 orders of magnitude larger than the diffraction limit. Such BDOE lenses can eliminate the need for anti-aliasing filters and could also be very useful in focal-plane arrays with large pixel sizes, where light collection efficiency needs to be maintained. Furthermore, by abandoning rotational symmetry, one can achieve free-form geometries in the focal spot, such as a square that can more closely match the geometry of the sensor pixel.
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An image sensor, measures the image of a scene at discrete point, which is also known as point sampling. This point sampling rate depends on the pixel pitch of the sensor. If the spatial frequency component present in an image is much higher than the sampling rate of the sensor, aliasing will occur . Aliasing will create artificial moire like pattern in the final image. It is very challenging to remove these patterns in post-processing, because it is very difficult to determine if the pattern is due to aliasing or it is really present in the scene . The situation is much worse when a color filter array e.g Bayer filter, is present. One possible solution is to use an anti-aliasing filter (AAF), which is basically an optical low pass filter (OLPF) or blur filter. In digital cameras, AAF can be implemented in multiple ways: 1) by using two layers of birefringent material such as lithium niobate, in front of the sensor, which spreads each optical point into a cluster of points, or 2) by micro vibrating the sensor element . Optical low pass filtering can also be an achieved by careful tweaking of the f-number (f/#) and numerical aperture (NA) of the imaging lens well. Adjusting f/# and NA is the most preferable anti-aliasing method as it does not require any additional components (see section 7 of the supplementary material for details).
The f-number (f/#) of a lens is a measure of its light concentration ability, which directly controls the image brightness. The f/# is defined as the ratio of the focal length (f) of the lens to the entrance pupil diameter (D). The f/# can be adjusted by tuning the entrance pupil diameter, so decreasing the f/# will increase the area of the aperture, effectively increasing the light throughput or light-collection ability . However, varying the f/# alters the numerical aperture (NA) of the lens as well. The NA is defined mathematically as follows:
We note that by careful design of diffractive optics, it is possible to realize a lens with focal spot of arbitrary size and shape, independent of the f/# and NAeff [5–6]. Realizing a focal spot with relatively well-defined size, shape and intensity distribution has potential applications in microscopy , lithography [8–9] and additive manufacturing . Moreover, by treating the phase at the focal spot to be a free parameter, it is possible to achieve extreme depth of focus  and achromaticity across huge bandwidths [12,13]. Modification of the focal spot can be accomplished via a spatial light modulator (SLM) [7,14], phase plates , deformable mirrors  and diffractive elements [17,18]. Multi-level diffractive elements [11–13,19] and metasurfaces [20–22], coupled with inverse design are also used. Regardless, generating non-circular-symmetric focal spot usually requires precise illumination condition [14–17,23], and almost all previous approaches are constrained by Eq. (1). A brief literature survey is presented in supplementary material section 2.
In this paper, we demonstrate three broadband DOE-based “freeform” flat lenses operating in the visible spectrum (λ = 450nm to 750nm), which we call the BDOE lens. Here, the meaning of “freeform” is twofold. First, the BDOE lens is free to attain any size and shape within the design space, which in our example is 4mm x 4mm x 2.6μm. This is different from our previous multi-level diffractive lens (MDL) where the constituent elements were concentric rings [11–13,19,24]. Secondly, the focal spot can also be of any size and shape as well. Specifically, we designed 3 lenses with 4mm aperture and focal length of 45mm (Fig. 1(a)), which yields a geometric f/# of 11.25, and a theoretical NA of 0.044. However, by reshaping the foal spot to be a square with side 400µm, 45µm and 5.6µm, we were able to achieve effective NAs of 0.00075, 0.0067 and 0.054, respectively (Fig. 1(b)). Our approach is not limited to small NAs and we showcase two simulated BDOE lenses with f/# of 2.5, but effective NAs of 0.3 and 0.03 illustrated in Figs. 1(c-f).
Previously we have demonstrated the concept of BDOE as a lens, whose point spread function (PSF) is structured in the form of an image [5,6]. To avoid confusion from non-imaging devices (which have sometimes been misidentified as lenses), by a lens we mean that the lens-makers equation is satisfied by the object and image distances, and images can be formed. The BDOE is comprised of square pixels, whose minimum width is determined by the fabrication technology, and whose heights are determined via a nonlinear optimization procedure that has been described previously [11–13,19,24]. Briefly, the goal of the optimization is to maximize the diffraction efficiency based on the target image, averaged over all the wavelengths of interest. In this work, the target image is a focal spot, whose geometry and size are selected independently of the f# of the lens. Specifically, we designed, fabricated and characterized three BDOE lenses to focus light into a spot size of 400µm (L1), 45µm (L2) and 5.6µm (L3) respectively as illustrated in Figs. 1(a, b). For simplicity, we constrained the pixel size to be 5µm with 100 possible height levels between 0 and 2.6µm. Assuming an average wavelength of 600nm and designed spot sizes of 400µm, 45µm, 5.6µm, the effective NA of the BDOE lenses are 0.00075, 0.0067 and 0.054, respectively. The optimized height profiles along with the simulated point-spread functions of the BDOE lenses are shown in Fig. 1(b).
For ease of fabrication, we limited our experiments to low NAeff. However, this is not a fundamental limitation as illustrated by two flat lens designs showcased in Figs. 1(c-f) with f/#=2.5, but NAeff of 0.3 (L4) and 0.03 (L5). All design parameters were the same as in L1-L3, except pixel width was 1µm, the number of pixels was 100 X 100 (lens aperture size = 100µm), and focal length = 250µm. Clearly, engineering the spot-size and thereby, decoupling f/# and NA is possible at higher NAeff as well.
3. Experiments and discussion
The BDOEs were fabricated by patterning a positive-tone, transparent photopolymer film (S1813, Microchem), which was spun-cast to a thickness of 2.6µm on top of a soda-lime-glass substrate (thickness∼0.5mm, diameter = 50mm). Laser gray-scale lithography was used for patterning [11–13]. Optical micrographs of the fabricated devices are shown in Fig. 2.
The focusing performance of each flat lens was characterized by recording the point-spread function (PSF) under broadband and narrowband wavelengths. Each lens 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). The PSF was recorded directly on a monochrome CMOS image sensor (DMM 27UP031- ML, The Imaging Source). The captured PSFs under narrowband primary colors (red, green, blue) illumination and broadband white light illumination are shown in Fig. 2. Achromatic focusing with spot size close to that predicted by simulation is observed (see supplementary information), although some of the discrepancies can be attributed to the fabrication errors. For characterizing the depth of focus (DOF), we placed the image sensor on a stage and captured the PSFs at different distances under broadband illumination (455 - 755nm). The results are illustrated in Fig. 3. Our experiments confirm a DOF of ∼20mm for L3, which is almost 200 times larger than that expected from diffraction.
Finally, we assembled a camera by placing the BDOE lens in front of a color image sensor (DFM 72BUC02-ML, The Imaging Source) and recorded still and video images as summarized in Fig. 4 (Visualization 1, Visualization 2, and Visualization 3 for L1-L3, respectively). The resolution chart was back illuminated by broadband (455-755nm) white light from the super-continuum source and the color objects were illuminated by white LED light. The exposure time was adjusted to ensure that the frames were not saturated. In addition, dark frame was recorded and subtracted from each image. Note that, all the color images have similar magnification. The resolution chart reveals the smallest resolved spatial frequencies as 0.28 lp/mm, 11.30 lp/mm and 64 lp/mm, for L1, L2 and L3, respectively. Each lens produced images at the approximately the same exposure time.
As indicated in Fig. 3, L3 is expected to have an extended DOF. In order to confirm this, we placed L3 at a fixed distance from the sensor (70mm) and adjusted the object distance from ∼70mm to 400mm. Visualization 4 confirms that the image remains in focus over most of this range, indicating dramatic increase in depth of field across the visible band. The field of view of the lens L3 was characterized by measuring the off-axis PSFs under broadband illumination (445 – 755nm). The angle of incidence of the plane wave illumination was adjusted from 0o to 30o, and the corresponding PSFs were recorded. As indicated in Fig. 5, off-axis aberrations become prominent at angles > 15o. The modulation-transfer functions (MTFs) were also extracted from the measured PSF and it also confirms a full field of view of 30o.
It is generally thought that the f/# and NA of a lens is inversely related and cannot be changed independently. Here, we show that this limitation can be overcome by reshaping the geometry of the focal spot via free-form broadband diffractive flat lenses. In this paper, we experimentally demonstrated three such lenses, where we have successfully decoupled the f/# and NA relationship by controlling the focal spot size. All our experimental results confirm excellent focusing and imaging performance over a broad wavelength range. Such BDOE lenses could be very useful for anti-aliasing in image sensors with large pixel sizes, where light collection efficiency needs to be maintained. Furthermore, by abandoning rotational symmetry, one can achieve free-from geometries in the focal spot, such as a square that can more closely match the geometry of the sensor pixel.
Office of Naval Research (N66001-10-1-4065).
We thank Brian Baker, Steve Pritchett for fabrication advice.
R.M. is co-founder of Oblate Optics, which is commercializing the subject technology. The University of Utah has applied for a patent covering the subject technology.
See Supplement 1 for supporting content.
1. A. Davies and P. Fennessy, Digital imaging for photographers, 4ed. (Focal Press, 2001).
2. W. J. Leler, “Human Vision, Anti-aliasing, and the Cheap 4000 Line Display,” SIGGRAPH Comput. Graph. 14(3), 308–313 (1980). [CrossRef] .
3. Julie Bentley and Craig Olson, Field Guide to Lens Design (SPIE, 2012).
4. John E Greivenkamp, Field Guide to Geometrical Optics (SPIE, 2004).
5. N. Mohammad, M. Meem, X. Wan, and R. Menon, “Full-color, large area, transmissive holograms enabled by multi-level diffractive optics,” Sci. Rep. 7(1), 5789 (2017). [CrossRef] .
6. 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]
7. S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. 109(3), 675–679 (2012). [CrossRef]
8. H.-Y. Tsai, H. I. Smith, and R. Menon, “Reduction of focal-spot size using dichromats in absorbance modulation,” Opt. Lett. 33(24), 2916 (2008). [CrossRef]
9. R. Menon, D. Gil, and H. I. Smith, “Photon-sieve lithography,” J. Opt. Soc. Am. A 22(2), 342–345 (2005). [CrossRef]
10. Gebirie Y. Belay, Michael Vervaeke, Sijie Liao, Piet Van den Ecker, Tom Craeghs, Hugo Thienpont, and Jürgen Van Erps, “Design and prototyping of beam shapers to generate circular or square top-hat beams of different size for additive manufacturing applications,” Proc. SPIE 11349, 113490M (2020). [CrossRef]
11. S. Banerji, M. Meem, A. Majumder, B. Sensale-Rodriguez, and R. Menon, “Extreme-depth-of-focus imaging with a flat lens,” Optica 7(3), 214–217 (2020). [CrossRef]
12. M. Meem, S. Banerji, A. Majumder, J. C. Garcia, P. W. C. Hon, B. Sensale-Rodriguez, and R. Menon (2020). Imaging from the Visible to the Longwave Infrared wavelengths via an inverse-designed flat lens. ArXiv:2001.03684 [Physics]. http://arxiv.org/abs/2001.03684
13. M. Meem, S. Banerji, A. Majumder, F. G. Vasquez, B. Sensale-Rodriguez, and R. Menon, “Broadband lightweight flat lenses for long-wave infrared imaging,” Proc. Natl. Acad. Sci. USA 116, 21375–21378 (2019). [CrossRef]
14. A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Colored point spread function engineering for parallel confocal microscopy,” Opt. Express 24(24), 27395–27402 (2016). [CrossRef]
15. X. Wang, B. Zhu, Y. Dong, S. Wang, Z. Zhu, F. Bo, and X. Li, “Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays,” Opt. Express 25(22), 26844–26852 (2017). [CrossRef]
16. D. M. Wulstein and R. McGorty, “Point-spread function engineering enhances digital Fourier microscopy,” Opt. Lett. 42(22), 4603–4606 (2017). [CrossRef]
17. T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express 16(10), 7203–7213 (2008). [CrossRef]
18. V. Pal, C. Tradonsky, R. Chriki, N. Kaplan, A. Brodsky, M. Attia, N. Davidson, and A. A. Friesem, “Generating flat-top beams with extended depth of focus,” Appl. Opt. 57(16), 4583–4589 (2018). [CrossRef]
19. 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]
20. H. Chung and O. D. Miller, “High-NA achromatic metalenses by inverse design,” Opt. Express 28(5), 6945–6965 (2020). [CrossRef]
21. R. Pestourie, C. Pérez-Arancibia, Z. Lin, W. Shin, F. Capasso, and S. G. Johnson, “Inverse design of large-area metasurfaces,” Opt. Express 26(26), 33732–33747 (2018). [CrossRef]
22. A. Zhan, R. Gibson, J. Whitehead, E. Smith, J. Hendrickson, and A. Majumdar, “Controlling three-dimensional optical fields via inverse Mie scattering,” Sci. Adv. 5(10), eaax4769 (2019). [CrossRef] .
23. H. Chen, S. Tripathi, and K. C. Toussaint, “Demonstration of flat-top focusing under radial polarization illumination,” Opt. Lett. 39(4), 834–837 (2014). [CrossRef]
24. 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]