Broadband imaging with one planar diffractive lens

We demonstrate imaging over the visible band using a single planar diffractive lens. This is enabled via multi-level diffractive optics that is designed to focus over a broad wavelength range, which we refer to as an achromatic diffractive lens (ADL). We designed, fabricated and characterized two ADLs with numerical apertures of 0.05 and 0.18. Diffraction-limited focusing is demonstrated for the NA = 0.05 lens with measured focusing efficiency of over 40% across the entire visible spectrum (450 nm to 750 nm). We characterized the lenses with a monochromatic and a color CMOS sensor, and demonstrated video imaging under natural sunlight and other broadband illumination conditions. We use rigorous electromagnetic simulations to emphasize that ADLs can achieve high NA (0.9) and large operating bandwidth (300 nm in the visible spectrum), a combination of metrics that have so far eluded other flat-lens technologies such as metalenses. These planar diffractive lenses can be cost-effectively manufactured over large areas and thereby, can enable the wide adoption of flat, low-cost lenses for a variety of imaging applications.


Design methodology
Our broadband lens is comprised of concentric rings of varying heights as illustrated in Fig. 1(a). The widths of the rings may be varying as well. In this manuscript, the widths of the rings were kept the same. The diameter of the lens is determined by the numerical aperture of the design, focal length and the longest wavelength of operation. The field in the focal plane is computed using scalar diffraction theory at each wavelength and the corresponding focusing efficiency is also computed. The focusing efficiency is defined as the power focused within 3 X FWHM divided by the total incident power at a given wavelength. The goal of our optimization based design is to maximize the focusing efficiency averaged over all the wavelengths of interest by varying the heights of the rings comprising the lens. We refer to this as the figure of merit (FOM). We utilized the modified binary-search algorithm to perform this optimization (see flowchart below). At first, an initial guess of height distribution is generated (usually a random distribution). In one iteration, all ring-heights are perturbed in a pre-designed manner (a random sequence). A positive unit perturbation (+Δh) is tried. If the updated FOM is increased, then this perturbation is kept, otherwise a negative unit perturbation (-Δh) is applied to this groove. If the new FOM is calculated to increase, then this negative perturbation is kept, otherwise it proceeds to the next groove. The guessed height distribution is updated accordingly. One iteration stops when all grooves are traversed. Termination conditions guarantee convergence, such as a maximum number of total iterations or a minimum FOM improvement threshold between two iterations.
The key design parameters are described below.

Simulation of focusing performance
The simulated point-spread functions (focal spots) at the design wavelengths for the 2 lenses are summarized in Fig. S1. The measured focal spots (from fig. 1 of the main text) are also included for comparison. It can be seen that the measurements agree well with the simulations. These simulations were performed using scalar diffraction theory. The simulated focusing efficiency spectra of the 2 designed lenses are shown in Fig. S2. The simulated average efficiency is somewhat higher than the measured ones. The reasons for this is not clear at the moment and we strongly suspect the impact of fabrication errors as described below.

Impact of fabrication errors
We measured the heights of the fabricated design (NA=0.18) over 10 randomly selected rings and these are summarized in the top chart in Fig. S3 along with the corresponding ideal design heights. The estimated error has a mean of 968nm and standard deviation of 156nm. Using this information, we simulated the focusing efficiency for 3 random realizations, where the pixel height error was randomly drawn from an uniform distribution with the same mean and standard deviation as the measured values. The results are summarized in the bottom panel of Fig. S3 and provide more proof for our hypothesis that the fabrication errors are likely reason for the reduced focusing efficiency. Another possibility, which requires additional study is the surface roughness of the photoresist after development. Figure S3: Top: measured and designed zone heights for 10 randomly selected zones in the NA=0.18 lens. Bottom: Impact of errors in the zone heights on focusing efficiency.

Focal spot characterization setup
The flat lenses were illuminated with expanded and collimated beam from a SuperK EXTREME EXW-6 source (NKT Photonics) and the SuperK VARIA filter (NKT Photonics). The wavelength and bandwidth can be changed using the VARIA filter. The focal planes of the flat lenses were magnified using an objective (RMS20X-PF, Thorlabs) and tube lens (ITL200, Thorlabs) and imaged onto a monochrome sensor (DMM 27UP031-ML, Imaging Source). The setup is shown in Fig. S4. Here, f represents the focal length of the flat lens and w.d. (roughly 2mm) is the working distance of the objective. The gap between objective and tube lens was ~90 mm and that between the sensor and the backside of tube lens was about 148mm. The magnification of the objective-tube lens was 22.22X.  To experimentally determine the focusing efficiency, we used the same setup but now we replaced the monochrome sensor with a 400 µm core diameter fiber tip (P400-1-UV-VIS, Ocean Optics) which in turns was connected to a spectrometer (Jaz Spectrometer, Ocean Optics). The setup is shown in Fig. S5. The flat lens was illuminated with expanded and collimated beam from the SuperK (445nm to 755nm). The fiber tip was scanned in X and Y directions using motorized stages so that the fiber tip was aligned with the peak of the magnified focal spot of the flat lens. A slight adjustment in the Z direction was also made to ensure that the integrated signal on the spectroscope was maximum. A reference signal was recorded with light passing through the unpatterned photoresist. Focusing efficiency was then calculated using the following equation: Focusing efficiency = (spectrometer signal when fiber tip aligned to the peak of magnified psf) / [(reference signal) X (area of magnified lens aperture) / (area of fiber tip aperture)]

Imaging setup
For imaging experiment, the 1951 USAF resolution test chart (R3L3S1N, Thorlabs) was used as the object. The flat lenses were used for imaging the object on to the sensor. A diffuser was placed behind the USAF target. The experimental setup is shown in Fig. S6. The USAF target was illuminated with the design wavelengths with 10nm bandwidth and corresponding images were captured using a monochrome sensor (DMM 27UP031-ML, Imaging Source). The exposure time was adjusted to ensure that the images did not get saturated. In each case, a dark frame was recorded and subtracted from the USAF target images.   . S7 shows the photographs of the color camera and its components. The color camera consists of a color sensor (DFM 72BUC02-ML), the flat lens and an IR-cut filter. These components were put onto a cage mount which was placed inside a 3D printed enclosure. The IR-cut filter was used only for outdoor photography.