Imaging with flat optics: metalenses or diffractive lenses?

Recently, there has been an explosion of interest in metalenses for imaging. The interest is primarily based on their sub-wavelength thicknesses. Diffractive lenses have been used as thin lenses since the late 19th century. Here, we show that multi-level diffractive lenses (MDLs), when designed properly can exceed the performance of metalenses. Furthermore, MDLs can be designed and fabricated with larger constituent features, making them accessible to low-cost, large area volume manufacturing, which is generally challenging for metalenses. The support substrate will dominate overall thickness for all flat optics. Therefore the advantage of a slight decrease in thickness (from ~2{\lambda} to ~{\lambda}/2) afforded by metalenses may not be useful. We further elaborate on the differences between these approaches and clarify that metalenses have unique advantages when manipulating the polarization states of light.


INTRODUCTION
Lenses are fundamental to imaging systems. Conventional lenses exploit refraction to focus light [1]. As a result, there is a fundamental trade-off that increases the thickness and weight of optics with increasing numerical aperture (or resolution). As illustrated in Fig. 1a with the example of a simple plano-convex lens, larger bending angles require larger thicknesses. Recently, there has been significant interest in reducing the thickness and weight of lenses by exploiting diffraction.
In such "flat-lenses," focusing is achieved by spatially arranging "zones" that impart appropriate phase so as to achieve constructive interference of the transmitted waves at the focus [2,3]. As illustrated in Fig. 1b, larger bending angles may be achieved with no change in thickness, simply by decreasing the local period of the diffractive structure. In order to ensure constructive interference, each ray must be locally phase shifted to compensate for the variation in its total optical path length to the focus. In traditional diffractive lenses, this is achieved by engineering the path traversed by the ray within the diffractive lens itself, as illustrated in Fig. 1c. In comparison to travelling the same distance in air, the optical path delay for a thickness, t is Δ = (n − 1) t, which then corresponds to a phase shift of Δ/λ*2π, where n is the refractive index of the material and λ is the wavelength of light. In order to achieve a phase shift of 2π, t must be at least λ/(n-1) ~ 2λ for n=1. 5. It is noted that diffractive lenses with numerical aperture (NA) > 1 under water immersion were demonstrated more than a decade ago [4]. In order to increase the focusing efficiency, blazed or multi-level diffractive lenses (MDLs) were also developed to approximate the optimal continuous phase distribution (see Fig. 1d). In fact, it was widely recognized that close to 100% efficiency could be achieved with such blazed diffractive optics [5]. However, at high numerical apertures, there is a rapid drop in efficiency due to the resonance conditions [6,7]. It was also quite definitively shown that this drop could be avoided by parametric optimization of the geometry of the constituent structures of the diffractive lens using both simulations [7,8] and experiments [9,10]. Another Achilles heel for diffractive lenses has been their poor broadband performance, which was overcome for discrete wavelengths via harmonic phase shifts [11] and by using higher-orders of diffraction [12]. We extended this work to continuous broadband spectra using efficient numerical techniques [13][14][15] and advanced multi-level nanofabrication at visible [16][17][18][19][20] and Terahertz spectra [21]. Here, we combine this multi-level approach with parametric optimization to show that high efficiency at high numerical aperture is feasible for both narrowband and broadband operation, which we believe has not been clearly demonstrated before. Recently, metalenses were proposed as a means to reduce the overall thickness of the conventional diffractive lens to sub-wavelength regimes by exploiting magnified phase changes that can occur in resonators [22][23][24][25][26][27][28][29]. Rather than using traversed path to create a phase shift, appropriately designed subwavelength antenna elements could achieve the same effect (see Fig.   1e). In this report, we show that the advantages of metalenses might be vastly over-stated and that the decrease in thickness from about 2λ (achievable via MDLs) to less than λ (achievable by metalenses) may not be useful for the majority of applications. To emphasize this point, we show a photograph from the side-view (Fig. 1f) of a multi-level diffractive lens that is corrected for the visible spectrum. This device was fabricated in a polymer film atop a glass wafer (thickness~0.6mm) as shown in Fig. 1g [19,20]. We point out that the support substrate will dominate the overall thickness in all cases, and thereby obviate any advantage due to reduction in the device thickness.
We further make the case that MDLs can achieve the same or better optical performance when compared to metalenses. To illustrate this point, we first performed an exhaustive literature survey of metalenses that have been reported so far. A summary of this survey is included in the Supplementary Information. Then, we selected exemplary metalenses that operate in the narrowband and in the broadband spectral regimes at low, medium and high numerical apertures, and we designed MDLs having the same optical specifications (focal length, numerical aperture and operating wavelengths). Finally, we compared the focusing efficiencies of the MDLs to those of the corresponding metalenses. Table 1 summarizes the key results. The first 3 columns are the optical specifications. Comparing the focusing efficiencies in columns 6 and 9 confirm that MDLs indeed perform better than metalenses. For the MDLs, we used a commonly available polymer photoresist (Shipley S1813) as the constituent material, since it exhibits high transmission in most wavelength regimes of interest here (measured dispersion is included in the Supplementary Information), and we have previously fabricated several MDLs in this material [16][17][18][19][20]. In all cases, we assume unpolarized input light for the MDLs. Thirdly, we point out that the fabrication complexity of metalenses is far higher than those of MDLs. As can be seen in Table 1 (columns 4 and 7), the minimum feature widths required for metalenses are significantly smaller than those for MDLs. In addition, metalenses generally require high-index materials (see Tables S1 and S2 in the Supplementary Information), whereas MDLs can be fabricated in low-index polymers. It is important to appreciate that any  [29] transparent material can be used for the MDL. This allows MDLs to be mass manufactured at low cost via high-volume imprinting techniques [30].

RESULTS AND DISCUSSION
Our design methodology involves nonlinear optimization to select the heights of the constituent elements of the MDL in order to maximize focusing efficiency averaged over all wavelengths of interest as described previously [18][19][20]. In congruence with work in metalenses, we define focusing efficiency as the ratio of the power within a spot of diameter equal to 3 times the simulated full-width at half-maximum (FWHM) to the total incident power [23]. The point-  (

2) Broadband MDLs
One of the big advantages of MDLs as we have pointed out before is their good achromatic performance over broad spectral bands [18][19][20][21]. Here, we reiterate this claim by directly comparing MDLs with metalenses of the same optical specifications. Again, following the parameters from  (

3) Aberrations analysis
When illuminated by a normally incident uniform plane wave, an ideal lens will generate a perfectly spherical wavefront that converges to the ideal focus. Aberrations in an actual lens are defined as the difference between the actual wavefront from this ideal wavefront. Here, we use the simulated wavefront to analyze the aberrations that are present in MDLs. Using the Zernikepolynomials representation of aberrations, we can calculate the wavefront errors as illustrated in   mature mass manufacturing capabilities that exist in the holograms industry to create low-cost large-area flat optics, enabling a new era of ultra-lightweight, thin optical systems.

Methods
All MDL designs were obtained using nonlinear optimization using a modified gradient-descentbased search algorithm that maximized wavelength-averaged focusing efficiency. The PSF simulations were performed using commercially available FDTD software from Lumerical.

Acknowledgements
We thank Tom Tiwald from Woollam for assistance with dispersion measurements, and Apratim Majumder for assistance with FDTD modeling. RM and MM acknowledge funding from Office of Naval Research (#N66001-10-1-4065). BSR and SB acknowledge support from NSF CAREER award: ECCS #1351389.

Author Contributions
SB performed the design and simulations. MM performed fabrication and analyzed the results.
BSR provided design guidance and analyzed the results. RM provided design guidance and analyzed the results. All authors prepared and edited the manuscript.

Competing Interests Statement
RM 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.

Materials and Correspondence
Correspondence and materials requests should be addressed to RM at rmenon@eng.utah.edu.
Imaging with flat optics: metalenses or diffractive lenses?

Literature Survey
A brief historical review of metalenses is necessary to appreciate its importance with relevance to its counterparts. The concept of such nanostructured sub-wavelength structures is not new. It actually dates back to year 1995-1996 [1][2][3][4][5][6], when the initial demonstrations of graded effectiveindex artificial-dielectrics for visible frequencies yielded very disappointing results with low measured diffraction efficiencies. The reasons for failure can be attributed to the following reasons. Firstly, the design assumed "adiabatic" effective index gradient [2,4]. Secondly, improper understanding of the relation between local subwavelength gratings and artificial dielectrics [1][2][3]. Lastly, the modeling was challenging as it led to aspect ratios quite difficult to manufacture with the materials and patterning technologies, which operated during that time [5].
Fast-forward 15 years later, the field was again revived when a paper was published in Science that revisited Snell's law at the interface between two uniform media with the help of an ultrathin grating composed of metallic nano-antennas etched on the interface [7]. Shortly after this, another paper [8] was published which controlled the phase using the plasmonic dispersion inside a waveguiding slit in a metal which ultimately led to the focusing of the incident light beam. This was really an important result from the perspective that it showed that due to this resonance occurring at the interface, the phase delay is amplified in contrast to the propagation delay. Therefore, the constraint of having large aspect ratios can be now be considerably relaxed.
This possibility was already quite intuitive in the previous paper also; but was very subtle in the presentation to be noticed. The novelty, which underlined in both these two papers, was the fact that graded phases can now be implemented by carefully designed nano-structures specifically; nano-antennas.
To summarize, metalenses unprecedented success can be attributed to three main reasons. The first reason is that these metalenses can control the phase propagation delay through an effectiveindex modulation leading to a waveguiding effect of the transmitted wavefront. Secondly, these nanostructures can quite effectively also monitor the phase with graded sizes or orientations.
Both the reasons combined provide the advantage to have fine spatial sampling with subwavelength "unit-cells"; thereby providing a rapid and robust spatial variation of the wrapped phase at the outer zones of the lens. Lastly, the introduction of a resonance delay (occurrence of a plasmonic resonance at the interface leading to amplified phase delay in contrast to the propagation delay) [9][10][11][12] to implement resonant high-contrast metalenses by combining two resonances, each covering a standard phase range of π [13][14][15]. This also relaxed the constraint on having stringent aspect ratios in the designed metalenses. Later on, it was also shown that by using centrosymmetric or rotationally asymmetric structures, full wavefront control could be achieved with a Berry-phase vortex [16]. As an immediate result, many research groups across the world started demonstrating nanostructured metasurfaces having the ability to control the amplitude, phase, polarization, orbital momentum, absorption, reflectance, emissivity of light with high spatial resolution.
Following are some of the game changing publications in the area among many to have appeared in various reputed journals throughout the past decade and has been provided herein to give a perspective of how the field has evolved until today.  2. Measured Dispersion of Photopolymer S1813 Figure S1: Measured dispersion of S1813.

FDTD Simulations
The full wave FDTD simulations were carried out using Lumerical FDTD Solutions. The material properties (refractive index and absorption coefficient as a function of frequency) of the Photoresist (S1813) was imported into Lumerical directly as the structure's optical data. A ".lsf" script was written to replicate the lens geometry using the same dimensions, which was specified during the optimization process as depicted in Fig. S2 (a). An incident plane wave (type: diffracting [see link]) along the backward "y-axis" direction (with TM polarization) were used to illuminate the diffractive lens surface. Visualization of boundary conditions imposed during the simulation. A boundary condition of (c) "symmetric" set to x-min and (d) "anti-symmetric" set to z-min.
For the broadband excitation, the entire range or bandwidth of the pulse was defined for the appropriate design. The entire FDTD simulation region was considered from the back surface of the spherical lens right up to 1.5 times the distance from the focal plane. A Perfectly Matched Layer (PML) boundary condition set up in the x-max, y-max and both z-min and z-max directions. As seen from Fig. S2 (b-d) that due to the inherent symmetry of the designed structure, the x-min boundary was set to "symmetric" and the z-min boundary was set to "antisymmetric" which reduced the requirements by ¼ of the original simulation requirements in terms of both time and memory. We would like to emphasis here that we tried to impose radial symmetry but could not as it has not yet been made available in the software [see link].
The default mesh was used to simulate the structures instead of a very fine mesh to avoid the huge computation time. The mesh accuracy was kept at "3" which has a good tradeoff for precision and accuracy versus the time and memory requirement. Field monitors placed at different planes above the lens and along the vertical plane to observe the field profiles of the propagating electromagnetic radiation. From Fig. 6d of this paper, we estimate that the efficiency is defined as power within width of about 3 times FWHM divided by total incident power. The actual definition is not clearly stated.

Simulated Point Spread Functions (PSFs) [Broadband Lenses]
The

Aberrations Analysis
The Zernike polynomial coefficient was fitted over a circular shaped pupil. The calculation was done using the reference [57]. The fit was achieved with a least squares fit method. The indexing scheme used was Fringe. The following lists all the fitting coefficients for the designed lenses: