Open Access
Add to BibSonomy
Abstract
It is generally assumed that correcting chromatic aberrations in imaging requires multiple optical elements. Here, we show that by allowing the phase in the image plane to be a free parameter, it is possible to correct chromatic variation of focal length over an extremely large bandwidth, from the visible (Vis) to the longwave infrared (LWIR) wavelengths using a single diffractive surface, i.e., a flat lens. Specifically, we designed, fabricated and characterized a flat, multi-level diffractive lens (MDL) with a thickness of ≤ 10µm, diameter of ∼1mm, and focal length of 18mm, which was constant over the operating bandwidth of λ=0.45µm (blue) to 15µm (LWIR). We experimentally characterized the point-spread functions, aberrations and imaging performance of cameras comprised of this MDL and appropriate image sensors for λ=0.45μm to 11μm. We further show using simulations that such extreme achromatic MDLs can be achieved even at high numerical apertures (NA=0.81). By drastically increasing the operating bandwidth and eliminating several refractive lenses, our approach enables thinner, lighter and simpler imaging systems.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Imaging is a form of information transfer from the object to the image plane. This can be accomplished via a lens that performs a one-to-one mapping [1], via an unconventional lens (such as one with a structured point-spread function or PSF) that performs a one-to-many mapping [2], or via no lens, where the light propagation performs a one-to-all mapping. In the first case, the image is formed directly. In the second case, the image is formed after computation, which can be especially useful, when encoding spectral [3,4], depth [5], polarization [6] or other information into the geometry of the PSF. Note that the modification of the PSF may be at the same scale as the diffraction limit [3–6] or it can even be much larger [7–10]. The image can be recovered in many cases in the no-optics scenario as well [11,12]. Intriguingly, machine learning may be employed to make inferences based on the acquired information (even without performing image reconstruction for human visualization), which has potential implications for privacy, among others [13,14].
The lens-based one-to-one mapping approach is preferred in many cases due to the high signal-to-noise ratio at each image point. When such a lens is illuminated by a plane wave, it forms a focused spot at its focal plane. When imaging at optical frequencies, only the intensity is measured. As a result, the phase of the field in the image or focal plane is a free parameter, i.e., it can be an arbitrary function of space and wavelength as long as the intensity distribution is localized to a spot (subject to satisfying the Helmholtz wave equation). Via back-propagation of the complex field to the lens plane, [15] it can be readily seen that the lens-pupil function is, in fact, not unique, also elaborated in section 1 of the Supplement 1 [16,17]. As such, the quadratic phase pupil function is just one of a large number of waves that can converge to a focus. It is also illustrative that early theoretical work pointed out that the pupil function for an ideal lens (at a single wavelength) must, in fact, have the phase and amplitude distribution of a converging dipole [18,19]. In this paper, we exploit such concepts and show that it is possible to search through all the possible (degenerate) lens-pupil functions to find one that achieves achromatic focusing (and imaging) over a large operating bandwidth of λ=0.45µm (blue) to 15µm (see section 1 of Supplement 1). We illustrate this via a single diffractive surface that is patterned with rings of width=8µm and heights varying from 0 to 10µm, fabricated within a polymer (positive-tone photoresist, see Fig. 1(a)). We refer to this diffractive surface as a multi-level diffractive lens (MDL). The focal length and diameter of the MDL are 18mm and 0.992mm, respectively. Figure 1(a) illustrates the phase shift imposed on an incident plane wave by the MDL at λ = 0.45µm, which is expressed as ψ= (2π/λ*h*(n-1)), where h is the MDL design height distribution and n is the refractive index at λ. The MDL pupil function is then expressed as exp(−iψ). We note that the maximum ring height of 10µm corresponds to a phase shift much larger than 2π for many of the wavelengths in our design. The amplitude and phase of the field distribution in the focal plane is also plotted in Fig. 1(a) for λ=0.45µm, and in Fig. 1(b) for λ=5µm and 15µm. All image sensors measure the square of the amplitude of the field distribution, i.e., the intensity distribution, and thereby, discard the phase. Here we show that a single diffractive surface can perform focusing and imaging that is substantially independent of wavelength over a range from visible to LWIR, something that is considered impossible in conventional lenses [20]. Furthermore, contrary to popular belief, we demonstrate via simulations that this extreme bandwidth can be achieved even at high numerical apertures.
Fig. 1. Achromaticity with a single surface is achieved by allowing phase in the focal plane to be a free parameter. (a) Schematic of a single surface multi-level diffractive lens (MDL) focusing collimated light to a focal spot. The MDL imparts its pupil function on the incident light. An image sensor discards the phase of the field in the focal plane, and only records the intensity. The height distribution (h) and the phase of the pupil function, ψ (λ=0.45µm) of a visible-LWIR MDL is shown. Note that the maximum h of 10µm is determined by fabrication and not the phase shift. The pupil function varies with λ. The normalized intensity distribution of the field in the focal plane at λ=0.45µm (top: cross-section through the center) is also shown. (b) The corresponding plots for λ=5µm and 15µm are shown. (c) Survey of operating bandwidths of exemplary metalenses.
It is important to note that in conventional refractive imaging systems, multiple lenses (sometimes made from different materials with differing dispersion properties) are used to correct for chromatic aberrations [20]. Not only are the individual refractive lenses thick and heavy, but multiple lenses require precise alignment during assembly. Metalenses have been proposed since the 1990s to mitigate these disadvantages [21,22]. More recently, the parabolic phase profile with group-velocity compensation has been applied to correct for chromatic aberrations [23]. Chromatic aberration can be corrected using phase-coded aperture and computational imaging as well [3,24–26]. Via a careful literature study (see summary in Fig. 1(c)), we conclude that the largest bandwidth demonstrated via a metalens currently is 2µm, i.e., from λ=3µm to 5µm [23]. However, the diameter of this metalens was only 30µm and it required ∼200nm features in a high-refractive-index material (GaSb). Needless to say, no metalens to date has ever reported imaging over the visible to the LWIR bands. We have already demonstrated separate MDLs operating in the ultraviolet (0.25µm to 0.4µm) [27], visible (0.45µm to 0.75µm) [28–31], near infrared (0.845µm to 0.875µm) [32], short-wave infrared (0.875µm to 1.675µm) [33], long-wave infrared (8µm to 12µm) [34] and terahertz (1000µm to 3000µm) [35,36] bands. In this paper, we show that a single MDL is actually able to image across the visible and the long-wave infrared bands, which we believe is the largest operating bandwidth for any flat lens demonstrated so far, and over an order of magnitude larger than the largest bandwidth demonstrated before.
2. Results and discussions
Our design methodology is similar to what has been reported previously [28–36]. To summarize, we maximize the wavelength-averaged focusing power of the MDL by selecting the distribution of heights of the rings that form the MDL. This selection is based upon a gradient-assisted direct-binary-search technique. We included a constraint of at most 100 height levels with a maximum individual height level of 10µm and minimum feature width of 8µm, which were dictated by our fabrication process. The material dispersion of a positive-tone photoresist, AZ9260 (Microchem) was assumed [34]. The height distribution of the designed MDL is shown in Fig. 1(a) (top-right inset), while the phase transmittance function (lens pupil function) at λ=0.45µm is shown in the bottom-left inset. As mentioned earlier, when this field is propagated to the focal plane, 18mm away from the MDL, the resulting intensity distributions are also shown in Fig. 1(a) (center inset). The corresponding intensity distribution plots for λ=5µm and λ=15µm are shown in Fig. 1(b), the intensity distributions are almost identical and simply scale with wavelength as expected, resulting in a single-surface lens that is achromatic from 0.45µm to 15µm. The simulated PSFs for all the wavelengths are depicted in Fig. S1 in Supplement 1.
The MDL was fabricated using grayscale lithography as reported previously [28–34]. We first utilized CVD-diamond as the support substrate as it is relatively transparent from the visible to the LWIR (see transmission spectrum in Fig. S3 in Supplement 1). However, the polycrystalline grains of the CVD diamond scatter light too much. To minimize this effect, we fabricated a pair of lenses of the same design: one on a glass wafer for λ=0.45µm to ∼2µm and another on a silicon wafer for λ = ∼2µm to 15µm. We emphasize that the patterned polymer was identical for all the lenses, i.e., the same MDL surface profile, fabricated on 3 different support substrates. Note that, the support substrate plays no role in the lens function, other than changing the transmission efficiency. Optical micrographs of these MDLs are shown in Fig. 2(a). The MDL on a glass wafer was placed in front of a silicon image sensor (DMM 27UP031-ML, Imaging Source) for visible and near-IR imaging. The average transmission efficiency in the visible and NIR wavelengths of the MDL with glass substrate was 85% (note that no anti-reflection coatings were used). The point-spread function (PSF) of the MDL was measured by illuminating it with a collimated beam from a tunable supercontinuum source (NKT Photonics SuperK Extreme with SuperK VARIA filter for visible wavelengths, 0.45µm-0.75µm and SuperK SELECT filter for near infrared wavelengths, 0.8µm-0.95µm) (Fig. S6 in Supplement 1) [28–33]. The wavelength of the source was tuned from 0.45µm to 0.95µm in steps of 50nm and bandwidth of 10nm. The focused spot at each wavelength was relayed with magnification (22.22X) onto the image sensor. The simulated and captured raw images from λ=0.45µm to 0.9µm are shown in Figs. 2(b) and 2(c). The infrared experiments were performed by placing the silicon-substrate MDL in front of a focal-plane array (FPA), three different FPAs were utilized for this purpose: 1) Zafiro640Micro, DRS Technologies: for wavelengths 1µm to 4µm; 2) Tau2, FLIR: for wavelengths 7µm to 8µm; and 3) PV320L, Electrophysics Scientific Imaging: for wavelengths 9µm to 11µm. The MDL was illuminated by collimated beam from a supercontinuum fiber source (SuperK Extreme/Fianium FIU6, NKT Photonics) for λ=1µm and 2µm, Mid-IR Supercontinuum laser SC4500, Thorlabs, for λ=3µm and 4µm and by a tunable quantum cascade laser (MIRcat-QTTM, DRS Daylight Solutions) for λ=7µm to 11µm (each with a pulse width of 100ns). The simulated and measured point-spread functions for representative wavelengths in the SWIR, MWIR and LWIR bands are shown in Figs. 2(d)–2(f). We emphasize that the focal length is 18mm for all wavelengths from 0.45µm to 15µm.
Fig. 2. Achromatic focusing by the MDL. (a) Optical micrographs of the fabricated MDLs. The device is the same, but the support substrate is glass for visible and NIR, while it is silicon for the other wavelengths. Simulated and Measured point-spread functions at the same focal plane, 18mm away from the MDL in (b) the visible spectrum and (c-f) the infra-red spectrum. A silicon CMOS image sensor and glass-substrate MDL were used for the (b) visible and (c) NIR images. A focal plane array and Si-substrate MDL were used for all images at λ>1µm (d-f). The size of each panel is 0.3mm, 0.3mm, 1mm, 1mm, 5mm and 5mm in (b), (c), (d), (e) at λ=3µm, rest of (e) and (f), respectively. The sizes for the simulation and experiment results are the same. Note that, the output power of the SuperK Extreme/Fianium FIU6, NKT Photonics, at λ=2µm is very low, which is why the PSF is noisy at this wavelength.
We measured the errors in the heights of the fabricated pixels from our process (see Fig. S7 in Supplement 1). In the future, it is possible to incorporate tolerance to fabrication errors as one of the metrics during the optimization-based design step, analogous to what was done previously for binary multi-wavelength diffractive lenses [4]. The diffraction-limited, measured and simulated full-width at half-maxima (FWHM) as function of wavelength are plotted in Fig. 3(a). The average strehl ratio across all the wavelengths is 0.74 (see section 12 in Supplement 1).
Fig. 3. Analysis of the MDL. (a) Strehl ratio (section 12 in Supplement 1) along with measured and simulated FWHM of the PSF as a function of wavelength. The error bar on the measured FWHM is equal to the pixel pitch of the sensor used. The average value, across all the wavelengths is 0.74. (b) Measured reconstructed wavefronts under broadband (0.4µm to 0.8µm), and narrowband (0.6µm, 0.9µm) illuminations. (c) Measured aberrations coefficients for both broadband and narrowband illuminations. (d) Measured shift in focal length as a function of wavelength (Vis-NIR). The average shift is 3%.
The wavefront aberrations in the visible and NIR bands were measured using a Shack-Hartmann wavefront sensor (Thorlabs, WFS 150-7AR). The wavefront aberrations were measured under broadband (0.4µm-0.8µm for the visible spectrum and at 0.8µm-0.95µm for the NIR spectrum) and under narrowband illuminations at 0.45µm, 0.5µm 0.55µm, 0.6µm, 0.65µm, 0.7µm, 0.75µm and 0.8µm with 50nm bandwidth, and at 0.85µm, 0.9µm and 0.95µm with 15nm bandwidth. Most importantly, the reconstructed wavefronts are quite similar for both the broadband and narrowband illuminations (Fig. 3(b)), confirming excellent achromaticity. The corresponding Zernike polynomial coefficients for measurements under 0.4µm-0.8µm, 0.8µm-0.95µm, 0.6µm and 0.9µm are shown in Fig. 3(c). The measurements confirm that the MDL indeed has low values for all aberrations (Table S1 in Supplement 1). Following a well-known procedure that calculates the focal length using the radius of curvature (ROC) measured from the WFS and the recorded distance between the test lens and the WFS, we calculated the change in focal length from the nominal value of 18mm as a function of wavelength (Fig. 3(d)) (details in section S2 in Supplement 1), confirming good achromatic behavior.
Next, we assembled a camera by placing the MDL in front of the appropriate image sensor. We characterized the imaging behavior of the MDL by capturing still and video images of various objects. The results are summarized in Fig. 4. See Supplement 1 for videos captured using the MDL with glass substrate under sunlight with NIR-cut filter, sunlight with vis-cut filter, white-LED light and with 0.85µm LED flashlight, respectively, and Supplement 1 using the MDL with Si substrate of a heated resistor coil. Note that corresponding visible, NIR and LWIR videos using the MDL with diamond substrate are in Supplement 1 (Table S2 in Supplement 1). The field of view of the MDL is estimated as 26.8° (see section 5 in Supplement 1). For majority of images, the distance between the MDL and the sensor was 19mm (i.e., the object distance) for all wavelengths, and the distance between the MDL and the object was 450mm for the visible and NIR bands, and 425mm for the IR bands. In each case, the exposure time was adjusted to ensure that the frames were not saturated. In addition, a dark frame was recorded and subtracted from the images. For the images in the SWIR and longer wavelengths, a hot-plate at temperature of approximately 200°C was placed behind the objects in order to image their silhouettes. The exceptions to this were the images of the soldering iron (3rd column in Fig. 4 at 180°C) and the heated resistor coil (only LWIR, 5th column in Fig. 4 at 150°C). The resolution-chart images in visible and NIR (Fig. 4) shows that the resolved spatial frequency is 28.5 line-pairs/mm, which corresponds to a spatial period of 17.5µm or about 8 times the sensor pixel size (2.2µm). This resolution corresponds approximately to the average FWHM over the visible-NIR bands (Fig. 3(a)). We also note that the visible image of the Macbeth color chart shows excellent color reproduction.
Fig. 4. Imaging from the visible to the LWIR using a single MDL. The visible and NIR images were taken using the silicon CMOS image sensor (DMM 27UP031-ML and DFM 72BUC02-ML, Imaging Source). The LWIR images were taken using a FLIR Tau2 camera sensor. The remaining images used the BST microbolometer focal plane array. The MDL with glass substrate was used for visible and NIR wavelengths, while the MDL with Si substrate was used for all other wavelengths. Note that all images are raw and unprocessed, and the bandwidth is limited by the spectral response of the sensors used. Also, see Visualization 1, Visualization 2, Visualization 3, Visualization 4, Visualization 5, Visualization 6, Visualization 7, and Visualization 8 for videos captured using the MDL under visible, IR and various illumination conditions. Note that for the SWIR-MWIR-LWIR images, the lower wavelength cut-off arises from transmission of the silicon substrate, and no filters were used.
Although in this demonstration, we used relatively low numerical aperture (NA), our approach is not limited by NA [37–39]. In section 9 of the Supplement 1, we show simulations of a NA=0.81 MDL that is achromatic from 0.45µm to 15µm.
3. Comparison to a pinhole camera
At λ=15μm, the diameter of the MDL is roughly the same as that of the first zone of a zone plate with focal length of 18mm. However, conventional zone plates are highly chromatic. In contrast, the MDL has focusing power due to the rings within its diameter, which enable the achromatic focusing over a much larger bandwidth. To drive home this point, we fabricated a pinhole with the same diameter as the MDL (0.992mm) and experimentally recorded the PSFs and images. These results summarized in Fig. 5 clearly show that a pinhole is not able to produce the resolution or quality of images that can be obtained with our MDL (see section 4 in Supplement 1).
Fig. 5. Comparison of MDL to a pinhole of same diameter (0.992mm). The PSFs were taken 18mm away from the both the MDL and pinhole. The visible and NIR images/PSF were taken with a silicon CMOS image sensor (LaserCam-HR, Coherent,Inc.) and a BST microbolometer focal plane array (PV320L, Electrophysics Scientific Imaging) was used for wavelengths λ >1µm. The MDL fabricated on a diamond substrate was used for comparison. Note that the pinhole and MDL images are at the same scale.
We also performed a comparison to a quadratic-phase profile lens that approximately follows our optimized phase profile (see Fig. S21 and section 10 in Supplement 1) and show that such a lens is not achromatic at the shorter wavelengths (visible and NIR). In other words, careful selection of all the ring heights is necessary to obtain the large bandwidth. We also acknowledge that more work is required to gain clear physical explanation of the mechanisms leading to such a large achromatic behavior. Furthermore, we show using simulations (see section 11 in Supplement 1) that the depth of focus of our MDL follows the diffraction-limited equation of λ/NA2, in contrast to other phase engineering approaches (such as the cubic phase plate).
4. Conclusion
A lens need not have a parabolic phase profile as is commonly understood. By removing this restriction, we enable numerous solutions to the lens design problem. Then, the final choice can be made based upon other requirements such as achromaticity (as we described here), extreme depth of focus [40], manufacturability, or minimization of aberrations, of weight, of thickness, of cost, etc. Our approach can be readily generalized to sub-wavelength diffractive optics (by employing full-wave diffraction models), which could be advantageous to manipulate the polarization states of light, [41] and could even be implemented in integrated-photonics platforms [42–44]. Another important application for extreme bandwidth flat lenses would be in imaging and focusing of ultra-short pulses (which have large spectral bandwidths). The next obvious question is what limits the bandwidth of such an MDL. Recent theoretical work has shown that the lens aperture limits achievable bandwidth [45]. We first note that our experiments actually exceed the limits described therein, and we attribute this to the fact that Ref. [45] only considered lenses with the conventional parabolic phase profile. Our previous work in computer-generated holography has clearly shown that performance can decrease with bandwidth [46]. Even so, initial simulations indicate that the bandwidth can be far larger than what we demonstrated here [47]. The space-bandwidth product (SBP) of the MDL, which is defined as the number of independent degrees of design freedom (lens-radius/ring-width*number of height levels) is expected to be the key limiting factor [48]. As per the channel-capacity theorem, we would expect that larger the SBP, the larger the operating bandwidth for a given field of view. By allowing for multiple height levels, the MDL enables far larger SBP when compared to binary diffractive lenses (including metalenses), which enables larger operating bandwidths. Exploring the trade-space of not only bandwidth, but also field of view and numerical aperture is clearly one of the important unexplored areas in the realm of flat optics.
Funding
National Science Foundation (ECCS #1351389, ECCS #1828480, ECCS #1936729); Office of Naval Research (N66001-10-1-4065).
Acknowledgement
We thank Brian Baker, Steve Pritchett and Christian Bach for fabrication advice, and Tom Tiwald (Woollam) for measuring dispersion of materials. RM acknowledges useful discussion with Henry Smith and Fernando Vasquez-Guevara. The support and resources from the Center for High Performance Computing at the University of Utah are gratefully acknowledged.
Disclosures
R. M. is co-founder of Oblate Optics, Inc., which is commercializing technology discussed in this manuscript. The University of Utah has filed for patent protection for technology discussed in this manuscript.
Supplemental document
See Supplement 1 for supporting content.
References
1. M. Born and E. Wolf, “Principle of optics7th edn,” (World Sci. Cambridge, 1999) (1999). [CrossRef]
2. Y. Shechtman, L. E. Weiss, A. S. Backer, M. Y. Lee, and W. Moerner, “Multicolour localization microscopy by point-spread-function engineering,” Nat. Photonics 10(9), 590–594 (2016). [CrossRef]
3. P. Wang and R. Menon, “Computational multispectral video imaging,” J. Opt. Soc. Am. A 35(1), 189–199 (2018). [CrossRef]
4. R. Menon, P. Rogge, and H.-Y. Tsai, “Design of diffractive lenses that generate optical nulls without phase singularities,” J. Opt. Soc. Am. A 26(2), 297–304 (2009). [CrossRef]
5. S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix pointspread function,” Proc. Natl. Acad. Sci. 106(9), 2995–2999 (2009). [CrossRef]
6. M. Khorasaninejad, W. T. Chen, A. Y. Zhu, J. Oh, R. C. Devlin, C. Roques-Carmes, I. Mishra, and F. Capasso, “Visible wavelength planar metalenses based on titanium dioxide,” IEEE J. Sel. Top. Quantum Electron. 23(3), 43–58 (2017). [CrossRef]
7. 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]
8. 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), 5789–5796 (2017). [CrossRef]
9. G. Kim, N. Nagarajan, E. Pastuzyn, K. Jenks, M. Capecchi, J. Shepherd, and R. Menon, “Deep-brain imaging via epi-fluorescence computational cannula microscopy,” Sci. Rep. 7(1), 44791–44798 (2017). [CrossRef]
10. G. Kim, N. Nagarajan, M. R. Capecchi, and R. Menon, “Cannula-based computational fluorescence microscopy,” Appl. Phys. Lett. 106(26), 261111 (2015). [CrossRef]
11. G. Kim and R. Menon, “Computational imaging enables a “see-through” lens-less camera,” Opt. Express 26(18), 22826–22836 (2018). [CrossRef]
12. G. Kim, K. Isaacson, R. Palmer, and R. Menon, “Lensless photography with only an image sensor,” Appl. Opt. 56(23), 6450–6456 (2017). [CrossRef]
13. G. Kim, S. Kapetanovic, R. Palmer, and R. Menon, “Lensless-camera based machine learning for image classification,” arXiv preprint arXiv:1709.00408 (2017).
14. Z. Pan, B. Rodriguez, and R. Menon, “Machine-learning enables image reconstruction and classification in a “see-through” camera,” OSA Continuum 3(3), 401–409 (2020). [CrossRef]
15. G. C. Sherman, “Integral-transform formulation of diffraction theory,” J. Opt. Soc. Am. 57(12), 1490–1498 (1967). [CrossRef]
16. F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54(12), 1481–1571 (1991). [CrossRef]
17. J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A 12(10), 2145–2158 (1995). [CrossRef]
18. J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71(1), 15–31 (1981). [CrossRef]
19. J. J. Stamnes and V. Dhayalan, “Focusing of electric-dipole waves,” Pure Appl. Opt. 5(2), 195–225 (1996). [CrossRef]
20. P. Rogers, “Aberrations of optical systems. wt welford adam hilger, 1986, isbn 0-85274-564-8, pp viii+ 284,£ 25.00,” Opt. Laser Technol. 19(3), 161 (1987). [CrossRef]
21. D. W. Prather, J. N. Mait, M. S. Mirotznik, and J. P. Collins, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15(6), 1599–1607 (1998). [CrossRef]
22. V. Levashov and A. Vinogradov, “Analytical theory of zone plate efficiency,” Phys. Rev. E 49(6), 5797–5803 (1994). [CrossRef]
23. S. Zhang, A. Soibel, S. A. Keo, D. Wilson, S. B. Rafol, D. Z. Ting, A. She, S. D. Gunapala, and F. Capasso, “Solid-immersion metalenses for infrared focal plane arrays,” Appl. Phys. Lett. 113(11), 111104 (2018). [CrossRef]
24. X. Zhao, J. Hu, Y. Lin, F. Xu, X. Zhu, D. Pu, L. Chen, and C. Wang, “Ultra-broadband achromatic imaging withdiffractive photon sieves,” Sci. Rep. 6(1), 28319 (2016). [CrossRef]
25. F. Heide, Q. Fu, Y. Peng, and W. Heidrich, “Encoded diffractive optics for full-spectrum computational imaging,” Sci. Rep. 6(1), 33543 (2016). [CrossRef]
26. S. Colburn, A. Zhan, and A. Majumdar, “Metasurface optics for full-color computational imaging,” Sci. Adv. , 4(2), eaar2114 (2018). [CrossRef]
27. S. Banerji and B. Sensale-Rodriguez, “Inverse designed achromatic flat lens operating in the ultraviolet,” OSA Continuum 3(7), 1917–1929 (2020). [CrossRef]
28. 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]
29. P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Sci. Rep. 6(1), 21545 (2016). [CrossRef]
30. N. Mohammad, M. Meem, B. Shen, P. Wang, and R. Menon, “Broadband imaging with one planar diffractive lens,” Sci. Rep. 8(1), 2799 (2018). [CrossRef]
31. M. Meem, A. Majumder, and R. Menon, “Full-color video and still imaging using two flat lenses,” Opt. Express 26(21), 26866–26871 (2018). [CrossRef]
32. S. Banerji, M. Meem, A. Majumder, F. V. Guevara, 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]
33. S. Banerji, M. Meem, A. Majumder, C. Dvonch, B. Sensale-Rodriguez, and R. Menon, “Single flat lens enabling imaging in the short-wave infra-red (swir) band,” OSA Continuum 2(10), 2968–2974 (2019). [CrossRef]
34. M. Meem, S. Banerji, A. Majumder, F. G. Vasquez, B. Sensale-Rodriguez, and R. Menon, “Broadband lightweightflat lenses for long-wave infrared imaging,” Proc. Natl. Acad. Sci. 116(43), 21375–21378 (2019). [CrossRef]
35. S. Banerji and B. Sensale-Rodriguez, “A computational design framework for efficient, fabrication error-tolerant, planar thz diffractive optical elements,” Sci. Rep. 9(1), 5801–5809 (2019). [CrossRef]
36. S. Banerji and B. Sensale-Rodriguez, “3d-printed diffractive terahertz optical elements through computational design,” in Micro-and Nanotechnology Sensors, Systems, and Applications XI, vol. 10982 (International Society for Optics and Photonics, 2019), p. 109822X.
37. M. Meem, S. Banerji, A. Majumder, J. C. Garcia, P. W. Hon, B. Sensale-Rodriguez, and R. Menon, “Imaging from the visible to the longwave infrared wavelengths via an inverse-designed flat lens,” arXiv preprint arXiv:2001.03684 (2020).
38. M. Meem, S. Banerji, C. Pies, T. Oberbiermann, A. Majumder, B. Sensale-Rodriguez, and R. Menon, “Large-area, high-numerical-aperture multi-level diffractive lens via inverse design,” Optica 7(3), 252–253 (2020). [CrossRef]
39. M. Meem, S. Banerji, A. Majumder, C. Pies, T. Oberbiermann, B. Sensale-Rodriguez, and R. Menon, “Inverse-designed achromatic flat lens enabling imaging across the visible and near-infrared with diameter > 3 mm and NA = 0.3,” Appl. Phys. Lett. 117(4), 041101 (2020). [CrossRef]
40. 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]
41. B. Shen, P. Wang, R. Polson, and R. Menon, “Ultra-high-efficiency metamaterial polarizer,” Optica 1(5), 356–360 (2014). [CrossRef]
42. B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nano photonics polarization beam splitter with 2.4×2.4 µm2 footprint,” Nat. Photonics 9(6), 378–382 (2015). [CrossRef]
43. B. Shen, R. Polson, and R. Menon, “Increasing the density of passive photonic-integrated circuits via nanophotonic cloaking,” Nat. Commun. 7(1), 13126–13129 (2016). [CrossRef]
44. A. Majumder, B. Shen, R. Polson, and R. Menon, “Ultra-compact polarization rotation in integrated silicon photonics using digital metamaterials,” Opt. Express 25(17), 19721–19731 (2017). [CrossRef]
45. F. Presutti and F. Monticone, “Focusing on bandwidth: achromatic metalens limits,” Optica 7(6), 624–631 (2020). [CrossRef]
46. G. Kim, J.-A. Dominguez-Caballero, and R. Menon, “Design and analysis of multi-wavelength diffractive optics,” Opt. Express 20(3), 2814–2823 (2012). [CrossRef]
47. S. Banerji, M. Meem, A. Majumder, B. Sensale-Rodriguez, and R. Menon, “Imaging over an unlimited bandwidth with a single diffractive surface,” arXiv preprint arXiv:1907.06251 (2019).
48. R. Shaw, “The application of fourier techniques and information theory to the assessment of photographic image quality,” Photogr. Sci. Eng. 6, 281–286 (1962).
