1. Introduction

Imaging systems are utilized across a wide range of applications spanning high-power microscopy, casual photography, long range target detection and viewing distant stars. Countless lenses have been designed for each application space while innovations in materials and manufacturing drive continuous evolution of optical systems. Regardless of application, imaging optics are ubiquitously characterized by their effective focal length ($f$) and F-number ($F$) because these metrics characterize fundamental features of the lens. Ultimately, f describes the size of the image or its magnification along with the field-of-view (FOV) and spatial frequency of image sampling when coupled to a specific sensor. Likewise, F or the numerical aperture drive the diffraction-limited resolution and image irradiance. Another commonality across lens designs, they seek to minimize the size, weight and cost (SWaC) of the imager while retaining image quality sufficient for the application. Here, we introduce a new lens metric called the volumetric imaging efficiency (VIE) that is a fundamental lens characteristic, similar to f and F. ; the VIE concisely describes the imaging capability of a lens relative its size. It is a normalized quantity describing how close a lens comes to the fundamental limit of image resolution achievable in a given volume.

The fundamental metrics of f. and F. along with optical specifications like FOV, spectral band and optical transfer function do not provide any information about SWaC – yet these are often drivers in a design optimizatio These non-optical parameters are routinely included in merit functions or bounding values while relying on the designer’s intuition and experience in the art to arrive at a good lens design. Several metrics have been introduced to provide a framework for joint evaluation of optical performance along with non-optical parameters in lens design. Most have relied onhspace-bandwidth product familiarized by A. W. Lohmann, et al. [1], in 1989 as a measure of the number of resolvable spots (${N_{spots}}$.) that fit into an image. In 2008, O. Cakmakci, et al. [2], showed the design efficiency of >3000 lens designs by comparing ${N_{spots}}$. as a function of lens complexity. Similarly, ${N_{spots}}$ was used by P. Milojkovic and J. N. Mait [3] in 2012o coare conventional flat image sensors against curved sensors for wide angle imaging. S. C. Olson, et al. [4], introduced a whole-system metric in 2016 that included ${N_{spots}}$ along with mass, power consumption, frame rate, dynamic range and spectral coverage for comparison of long-range EOIR imaging sensors.

From a design perspective, it would be valuable to have a-priori knowledge of the minimum size, weight and cost that is possible for a given set of optical requirements; comparing designs against these limits would facilitate finding globally optimized solutions. The VIE is a new metric that supports this global optimization in the context of lens size. Weight and cost are not considered explicitly because they have a less direct connection to optical requirements but we generally expect these to reduce with size. That generalization may break down when different technologies are introduced; for example, introduction of aspheric surfaces can reduce the number of surfaces required along with volume and weight but may increase cost.

Desire to increase performance or reduce SWaC of imaging systems has motivated continuous efforts to develop new and improved optical technologies. In the last decade, developments in lens technology has spanned metasurface lenses [5–10], diffractive optical elements [11], dynamic gradient-index materials [12–14], freeform surfaces [15–17], transformation optics [18,19], printed lenses [20], highly-aspheric molded plastics [21,22] and technologies to utilize curved image surfaces [23–31]. Each of these are vying to improve optical systems and have demonstrated compelling advancements over conventional optics.

Here we introduce a simple metric, the $\textrm{VIE}$., that provides a convenient tool to evaluate lens performance in the context of compact imaging systems. We show that this high-level metric can illuminate the limitations of certain lens technologies and the opportunities to improve system performance or size enabled by other optical technologies. Specifically, we delineate the VIE limit of conventional bulk optics as a function of FOV and show that metasurface lenses or curved image surfaces can surpass that limit by orders of magnitude in wide angle lenses. This study y be extended to explore the opportunities enabled by additional optical technologies and along other parameters of the trade space, such as f and center wavelength ($\lambda $.). Furthermore, it may be used to explore the potential synergy of combining multiple technologies together as in the example by S. Colburn, et al., where they combined metasurfaces with freeform optics to make a varifocal zoom lens [32].

The VIE includes normalization by an idealized “thin-lens” reference matching the f, FOV and center wavelength ($\lambda $.) of the lens. The “thin-lens” reference has no volume and provides perfect, diffraction-limited imaging or its entire FOV. Of course, this is not a real lens and it is not physically realizable even using a metasurface because it requires a different phase profile for every part of the FOV; the required angle-dependent phase function is provided in Supplement 1. Such a reference represents a fundamental limit to the compactness and performance of a lens design; normalizing lens performance by this upper bound yields the concept of “efficicy” for imaging optics.

The VIE concisely characterizes the optical train alone – unlike whole-system imaging metrics such as the General Image-Quality Equation (GIQE) for National Imagery Interpretability Rating Scale (NIIRS) [33] and the Targeting Task Performance (TTP) metric from the Night Vision Integrated Performance Model (NVIPM) [34]. We show that the VIE facilitates direct comparison of lenses; even when they are designed for disparate applications ranging from long-range to wide-angle imaging in visible, infrared and thermal bands. We also show that it provides a framework to compare the impact that emerging optical technologies may have on future image systems.

In Sect. 2 we introduce the VIE metric starting from the volumetric channel density of a lens and introducing a matched reference lens. In Sect. 3 we present methods for computing VIE data from existing prescriptions and methods for designing metasurface prescriptions of potential lenses. Section 4 presents the VIE data for >2800 conventional lens designs (i.e. homogeneous, bulk optics imaging onto planar focal plane arrays) and identifies an empirical VIE limit for conventional optical trains. This empirical limit can be surpassed by introducing new optical technologies; this is demonstrated through the VIE for several metasurface lenses and lenses with highly-curved image surfaces. In Sect. 5.1 we introduce an improved reference lens needed for ultra-wide angle lenses. In Sect. 5.2 we discuss the scaling characteristics of VIE and show that VIE saturates to a maximum value when a given design is scaled down to become diffraction-limited rather than aberration-limited; this scale-independent VIE reflects the intrinsic compactness of a design. Section 5.3 addresses limitations of the VIE forumated here and suggests remedies to make it more accurate and versatile, for example to consider MTF, image sampling, and multispectral or multiaperture systems. Section 6 summarizes the conclusions from this work.

2. Theory

Evaluation of lens performance typically includes the RMS spot size, modulation transfer function (MTF), aberration analysis, ensquared energy, distortion mapping, etc. These provide a high-fidelity description of lens characteristics and are standard tools for design and optimization. Size, weight and cost of optical components and mounts are co-optimized according to application requirements. The completed design presents the best solution found given the system constraints and parameter space explored (i.e. materials, technology, etc.). It would be instructive to compare that result against what is possible, in principle, if those constraints were removed. The VIE metric provides this insight wherein VIE=1 (i.e. 100% efficiency) indicates a system is as good as it could ever be in diffraction-limited imaging. Such a metric can facilitate system design by ensuring globally optimized solutions and judicious selection of optical technologies.

The VIE utilizes an intermediate metric, the volumetric channel density (VCD), introduced here and defined as

The total number of resolvable spots (${N_{spots}}$) is determined as the image area ($A$) divided by the average spot area ($a$). Formally, it is ${N_{spots}} = \mathop \int \nolimits_{A} {a^{ - 1}}dA \approx A/<a>$ where A is the image area and the bounds of the surface integral while <a> is the average spotsize; in practice, <a> may be computed as a weighted average of the spotsize at a few field angles. The numerator, ${N_{spots}}$, represents the number of independent spatial channels available in an image [1–3,35]. Each spot, or channel, in the image would provide a depth of information according to the dynamic range and noise of the sampling detector as described by Shannon’s channel capacity [36,37]. Multiple spectral samples under each spot could be considered different channels, but this amounts to parsing the image irradiance in different ways without changing the net capacity of the system. Characterizing capacity of a system requires detailing the detector used to sample the image and this inherently couples lens performance to sampling. In contrast, the VIE aims to characterize the optical train on its own to evaluate the optics in a general way and facilitate side-by-side comparison of disparate lens designs and/or technologies. We utilize ${N_{spots}}$ to quantify optical performance – independent from constraints associated with sampling.

The denominator of the VCD is the optical volume ($V$) weighted by a 2/3 exponent. The “optical volume” is defined as the minimum volume that bounds all rays travelling through the system from the first surface to the image. We exclude the volume of any extra glass or mounting hardware to facilitate evaluation of the underlying optical limits. An example system with its optical volume (shaded) is shown in Fig. 1(a); notice the optical volume catches the outermost rays contributing to the image (i.e. excluding any vignetted rays) while the unshaded corners of glass are not counted. The volume exponent was chosen so that VCD is scale independent when it is diffraction limited. Whole system scaling of spacings, diameters and radii of curvature by a factor of M preserves F and diffraction-limited spot sizes. This scales f by M and the image area by ${M^2}$ so that the numerator of VCD (${N_{spots}}$) scales as ${M^2}$. Volume scales as ${M^3}$ and the 2/3 power on the denominator balances the scale.

 

Fig. 1. (a) An example lens system and its minimum optical volume (shaded) used to evaluate its volumetric imaging efficiency (VIE). (b) The idealized reference comprised of a thin-lens that provides diffraction-limited imaging across its field-of-view (FOV=${\pm} {\theta _{max}}$). (c) Generalization of the idealized reference imaging onto a curved image surface; retains finite volume as FOV approaches ${\pm} 90^\circ $. In all three configurations, the optical volume bounding all rays that contribute to the image is shaded.

Download Full Size | PDF

In rotationally symmetric lenses, we find that we can reliably compute the optical volume by tracing the upper- and lower-rim rays for the largest field angle, taking whichever one is farthest from the optical axis at each surface and using them to define a series of adjacent isosceles trapezoids that encompass the optical volume when rotated about the optical axis. Care must be taken to not double-count the associated volume in reflective or catadioptric lenses where the bounding rays make multiple paths through the same physical volume. We use ray-tracing while computing volume to facilitate precise comparisons; a few exceptions will be noted where the full prescription was not available. In practice, rough estimates of volume may be used to estimate VIE with little information about a system; for example, volume may be estimated based on track length and average diameter of the optical train.

We hypothesize that there is a fundamental upper limit to the value of VCD that depends on a system’s F, $\lambda $, and FOV. Furthermore, we consider the limit is given by an idealized imager consisting of a single thin-lens that provides diffraction limited imaging across its full (circular) FOV onto an image plane as shown in Fig. 1(b). This is the model adopted by P. Milojkovic and J. N. Mait in Ref. [3] during their comparison between flat and curved image sensors. In a similar fashion, we consider the variation in diffraction limited spot size with field angle ($\theta $) as it maps across a flat image plane. This spot area increases with $\theta $ as

The three cosine factors in R arises from: 1) reduction in effective aperture size, 2) increased distance to the focal plane and 3) oblique incidence of the diffraction limited cone onto the image surface. Considering the physical optics definition of F as the speed of focus that determines the diffraction limited spot size, the first two terms can be lumped into a quadratic reduction of F with $\cos \theta $. While this idealized system does not represent any real system, it is plausible that this represents the maximum VCD as it gives perfect (i.e. diffraction-limited) imaging using a thin-lens with zero volume. It also has an analytic solution.

We derive VCD of this idealized system. The number of resolvable spots (${N_{spots}}$) is found integrating the inverse of the spot area ($a$) across the image area ($A$) giving

Here D is the diameter of the thin-lens and ${\theta _{max}}$ is the maximum field angle or half of FOV corresponding to the edge of the image. The volume is a that of a truncated right circular cone given by:

We define the efficiency of a system as the ratio of its VCD to the maximum as

Where $\textrm{VC}{\textrm{D}_{\textrm{max}}}$ is calculated for a fast, $F = 1$, lens with the same $\lambda $ and FOV as the target system. The VIE compares VCD of a real system against that of a model system representing the upper-limit of what may be possible in diffraction-limited imaging. This normalization allows us to cast performance in terms of an intuitive “efficiency” and at the same time eliminates sensitivity to $f,\lambda $. and FOV – which can make it difficult to compare lenses with even small differences in these parameters.

The simplicity of the VIE and the diffraction-limited reference are key to its utility. The VIE is easy to compute (a full prescription is not needed) and it yields a single value that is easy to interpret without considering details related to an application like spectral band, focal length, etc. However, it loses fidelity by treati the blur-limited information capacity using the average spot size a without regard to MTF characteristics and sampling processes. For example, additional spatial information beyond ${N_{spots}}$ can be extracted using multiple sampling and super-resolution techniques. The present analysis also assumes a single spectral channel is used to sample the image; whereas multiple spectral samples within the same spot area would increase the available information, potentially, at the expense of reduced sensitivity, dynamic range or frame rate – considerations related to image sampling that is beyond the scope of this metric.

The VIE metric presented above is not suited well to characterize ultra-wide angle (UWA) lenses with full FOV larger than ${\sim} 120^\circ $. The problem originates from the thin-lens reference which provides zero angular magnification and distortion. In other words, the image height follows $h = f\tan {\theta _{max}}$ where ${\theta _{max}}$. is half the maximum FOV and f., the effective focal length, is equal to the back focal distance. The image height and optical volume become asymptotic as $\theta \to 90^\circ $ (i.e. a hemispherical field of view). In contrast, UWA lenses typically employ retrofocus designs and distortion to compress the wide-angle image into a smaller area.

To address UWA lenses with larger FOV, we introduce a better reference in Sect. 5.1 that uses an idealized thin-lens to provide diffraction-limited imaging onto a curved image surface rather than a flat one. Such a model system was considered in Ref. [3] to compare imagers using flat and curved sensors. This provides a better – a more fundamental – upper limit to the VCD of any lens with a given $f,\lambda $. and FOV up to full hemispherical ($2\pi $. steradians) imaging. However, the vast majority of lenses can be compared against the simpler reference imaging onto a flat surface as detailed above. Consequently, we present the flat reference for the standard VIE metric to avoid obfuscating the metric with a detail that only becomes critical in the UWA regime. Later, we detail the ultra-wide angle variation of the VIE metric (VIE’) in Sect. 5 using example lenses for context.

Last, the VIE is formed in the context of a single-aperture, single-band system. Departures from this may warrant modifications to the VIE metric, the VCD definition and choice of a meaningful reference system. While the spectral band width is considered in computation of the average spot size, the reference is based on a single wavelength at the center of the band. This is reasonable if the bandwidth is less than the center wavelength (i.e. $\mathrm{\Delta }\lambda /\lambda < 1$) but not for multispectral systems that include multiple bands including some combination of VIS, SWIR, MWIR, LWIR or THz, for example. In future work, we will investigate how the VIE concept can be applied to multi-aperture and multi-band systems.

3. Methods

Example lenses were collected from the ZEMAX ZEBASE 6 database, the Code V built-in library, various online lens databases, journal articles, patent publications and proprietary lens designs. Each system was evaluated “as is” without optimization or modification. The f, F, FOV and wavelength range were taken directly from the reference; occasionally a system was underspecified and was simply left out of the analysis. Any errors in the prescription (intentional or otherwise) would not be corrected, so some of the lenses do not represent state-of-the-art imaging optics. In most cases, VCD and VIE are computed using exact ray tracing through prescriptions recreated in ZEMAX; prescriptions were not available for a few example lenses and VIE was computed manually using the available information.

Matlab was used with the Zemax OptiStudio Application Programming Interface (ZOS-API) to compute VIE programmatically to save time and prevent manual errors. The algorithm used to batch process our library of Zemax lens designs is described in Supplement 1. Fifteen rays per wavelength were used to compute spot size (i.e. diameter) relative to its centroid at each field angle. The average spot size for every system was evaluated using three field angles: 1) on axis, 2) 70% of full FOV and 3) full FOV. The following calculations were performed for each field angle. The geometric spot size was computed as an RMS radius using equal weight at three wavelengths (short, center and long) spanning the designed spectral band of the lens. The resulting geometric spot size (determined by aberrations) was compared against the diffraction spot size (taken as the Airy disk radius) and the maximum value was taken to represent the combined effects of aberrations and diffraction. The Airy disk diameter was calculated using the field-dependent working F-number determined by the vergence of ray bundles to each image point. Specifically, the Airy disk radius is $d = 1.22\lambda F$. where $F = 1/({2\tan ({\theta /2} )} )$. and $\theta $ is the angle between the upper- and lower-rim rays in image space.

After the spot radii are determined for each field, the average spot radius across the image is calculated as ${R_{avg}} = \mathop \sum \nolimits_{i = 1}^3 {w_i}{R_i}$ where ${R_i}$ is spot radius at each field angle and ${w_i}$ is a field weight describing the relative fraction of the image associated with each ${R_i}$. We choose field weighting $({{w_1},{w_2},{w_3}} )= ({0.26,\; 0.33,0.41} )$ to reflect the areal balance of the 3 fields assuming a rectangular image sensor will be used to sample the image with a 4:3 aspect ratio and the full FOV spans the smaller dimension. This choice of weights is somewhat arbitrary but does not strongly affect the results. We tested an alternative choice of equal weights (1/3 each) without a noticeable change to observed trends, indicating the spot size typically does not vary significantly across the FOV as expected for optimized lenses. Finally, ${N_{spots}}$ was computed dividing the rotationally symmetric (i.e. circular) image area by the average spot size; the image radius was determined by the centroid location for rays from the full FOV.

The optical volume was computed by tracing upper- and lower-rim rays from the full FOV; these are reliably the bounding rays in rotationally symmetric systems. The y-z intercepts of both rays were calculated at each surface and the ray path between each surface were taken as the skewed edge of a right trapezoid whose right angles were located on the optical axis (i.e. rotational axis of symmetry); rotation of this right trapezoid around the optical axis accounts for one segment of the optical volume. The larger of the upper- or lower-rim ray was used to determine optical volume in each segment and these seamlessly changed roles at the stop. Adding up all of the segments, including the final segment to the image plane, yields the total optical volume, V. Curvature of the front surface was added as a spherical or aspheric cap that adds (or subtracts) the volume of glass in front (or behind) where the bounding ray hits the first surface. Care was taken not to double count volume in reflective systems that reuse portions of the optical volume. Together, ${N_{spots}}$ and V were used to compute VCD for each lens and compared against an $F = 1$ reference with the same $f$, FOV, and center $\mathrm{\lambda }$.

To further illustrate the utility of VIE as a comparative tool, we design monochrome multi-metasurface (MMS) lenses [38]. Each MS in a MMS lens is treated as a simple phase surface described using the first three radially symmetric Zernike polynomials as:

VIE/FOV contours were generated for lenses with various EFL and number of MSs. Contours were generated by programmatically stepping through FOV. All lenses were designed for use at ${\lambda _0} = 632\; nm$ but could be scaled trivially by ${\lambda _0}/\lambda $ to give the same geometric ray traces for any particular $\lambda $. For each configuration, initial optimization was performed manually starting at full FOV = 60$^\circ $ to create a good baseline design. FOV was stepped up or down and then re-optimized using the procedure outlined above.

4. Results

The utility of the VIE metric becomes apparent when we evaluate a large number of lens designs and plot them, for example, versus FOV as shown in Fig. 2. Figure 2 is a scatter plot of VIE vs FOV for 2,871 lenses; only 930 are visible in the region of interest shown. The red circle data points use conventional bulk optics with curved refractive or reflective surfaces and homogeneous materials to image onto a flat image plane. Approximately 1,900 lenses have low VIE < 0.1%; their existence (but not the value of their VIE) is indicated by a circle on the x-axis at the corresponding FOV. Note that FOV is plotted as a solid angle in steradians which grows quadratically with angular (i.e. horizontal or vertical) FOV and better represents the scaling of the visible field than the angular FOV. The scatter plot reveals a trend of decreasing VIE with FOV. The dashed black line identifies an empirical threshold for conventional lenses and clearly illustrates that VIE degrades exponentially with FOV. The empirical limit shown is $\textrm{VI}{\textrm{E}_{\textrm{empirical}}} = {10^{ - 0.594 \times FOV}}$. This was determined by forcing the limit to contain VIE = 1 at FOV = 0 and computing the decay rate to the system with VIE = 0.0136 at FOV = 3.14 sr; all systems in our study would then lie below this limit. Note that there is nothing fundamental about this limit and we expect it to evolve provided additional, extraordinary lens designs that push the boundary of performance for conventional bulk optics. For example, newer generations of mobile imagers with deep aspheric lenses enabled by molded plastics may already exceed this limit. However, it is surprising to observe such a clear exponential decay over a wide FOV and we expect this to exhibit more structure provided more data; specifically, Fig. 5 shows several UWA systems with VIE that depart from this trendline at very large FOV.

 

Fig. 2. A VIE verses FOV scatter plot of 2871 lens systems. Anonymous datapoints (red circles) show conventional lens designs collected from various databases, patents, papers and proprietary designs. The black dotted line illustrates an empirical upper limit for these systems. Curves show VIE of MMS designs from this work. Example lenses discussed in the text are highlighted with special markers. Crescents are curved image sensors from Refs. [23,24], the crossed circle is the gigapixel camera [25] and the blue diamond is a MS doublet [5]. Pentagons compare conventional (filled) and 5-MS (empty) telephoto lenses illustrated in Fig. 4.

Download Full Size | PDF

Figure 2 shows that conventional lenses can provide VIE close to 1 over a small FOV but have poor efficiency when deployed to cover a large field of view. In contrast, new technologies may surpass the limits of conventional optics and the VIE is a robust tool for comparing different technologies, in different systems, with different spectral bands and for different applications. For example, the filled (VIE=0.46, Ref. [23]) and open (VIE=3.45, Ref. [24]) crescent datapoints show the VIE for monocentric lenses designed to image onto highly curved image surfaces; these provide wide angle lenses with >100-fold improvement in VIE surpassing the bounds of conventional lenses. The midwave infrared (MWIR) design in Ref. [24] has VIE>1 and is shown at the top of the axis; both of these should be compared against the curved reference system suitable for ultra-wide angle lenses discussed in Sect. 5. Despite using bulk optics, this type of system is technologically different because it requires highly curved image sensors or some unusual relay to connect the curved image to flat sensors. This system eliminates the need to flatten the Petzval curvature – a major constraint in optical design. It also results in symmetric optical paths for all field angles making it easy to correct aberrations. The work by Milojkovic and Mait in Ref. [3] drew this comparison previously, showing the superior compactness and resolution of lenses imaging onto curved surfaces compared flat surfaces. The exceptional performance enabled using curved image surfaces has motivated pursuit of curved image sensors [26,29,30], curved relays based on coherent fiber bundles [31] and monolithic assemblies of flat sensors across a curved image [25]. These technologies aim to capitalize on the revolutionary performance improvement enabled by curved image surfaces.

The curves in Fig. 2 illustrate the VIE/FOV space accessible using MMS lenses based on our design results. This technology imbues optical power to flat surfaces. They are engineered surfaces comprised of meta-atoms that give unprecedented control over the phase imprinted onto an optical wavefront. Recent MS developments in flat lenses, single-surface achromats, ultra-thin doublets, and wide-angle lenses offer diffraction-limited focusing. Meanwhile, the volume of the MS is microscopic due to subwavelength thickness of meta-atoms so that lens volume is dominated by empty (i.e. optically inactive) spaces such as MS substrates and the space between them or the image. This offers the perfect combination of high-resolution imaging and minimum volume. In fact, the MS lens looks so much like our idealized thin-lens reference in Section 2 that we might expect ∼100% VIE. However, imperfect (i.e. sub-diffraction-limited) imaging across the FOV yields more modest VIEs.

We estimate VIE for some MS lens designs. In 2016, a MMS doublet (0.717mm, F/0.9, 850nm wavelength) was demonstrated by A. Arbabi, et al., [5] with a 60° full FOV (0.84 sr) that greatly exceeds a conventional doublet. It is also very compact as both MSs are fabricated on opposite sides of a single 1.25 mm thick substrate. Ray tracing and experimental characterization reveal performance that is nearly diffraction-limited. The doublet has a VIE = 0.2 and is shown as a blue diamond in Fig. 2. This is at the threshold of what can be achieved with conventional lenses. M. Y. Shalaginov, et al. [10]., demonstrated a pair of ultra-wide-angle MS lenses in 2020. One lens was designed for near-infrared (NIR) and the other was designed for MWIR imaging. Both comprise a single MS with a remote aperture and provide nearly diffraction-limiting performance over a FOV approaching a full $2\pi \; sr$. hemisphere. While full prescriptions are not available, we estimate VIE with a high degree of accuracy using the data provided and ${\theta _{max}} = 85^\circ $. . The NIR lens is F/2.5 with EFL = 2.5mm operating at ${\lambda _0} = 0.94\; \mu m$. and yields a VIE = 2.2. The MWIR is F/2 with EFL = 2mm operating at ${\lambda _0} = 5.2\; \mu m$ and yields a VIE 4.3. Notice theses VIE values are 2.2x and 4.3x larger than our reference thin-lens, respectively! Indeed these data points are off the chart (i.e. Fig. 2) toward high FOV and high VIE. This is a testament to the compact, wide-angle, high-resolution performance of these lenses; also indicates that the reference presented in Section 2 is not truly a “fundamental” upper limit. We discuss the limits of the VIE in Section 5 and introduce a more suitable reference for ultra-wide angle lenses such as these.

We created preliminary MMS lens designs for various EFL and number of MSs to illustrate the capability of this technology using the VIE metric. A single metasurface perfectly focuses a monochromatic axial plane wave using a phase profile given by ${\mathrm{\Phi}_{ideal}} ={-} 2\pi \left( {\sqrt {{f^2} + {\rho^2}} - f} \right)/{\lambda _0}$. However, this ideal phase only works for normal incident light and oblique rays suffer severe aberrations – primarily coma. This rapidly degrades the image away from the center of the FOV. A single MS lens (${\lambda _0} = 632$ nm, f = 10 mm, F/1) optimized over a 60$^\circ $ FOV provides poor imaging with VIE < 10⁻⁵. Inspired by multi-element lenses where each surface and space contribute degrees-of-freedom to mitigate aberrations, we designed MMS lenses and benchmarked the results against state-of-the-art designs, where possible. Figure 3(a), (b) and (c) show two-MS, three-MS and five-MS designs with VIE = 0.0068, 0.041 and 0.3, respectively. These are optimized for a 60$^\circ $ FOV (0.84 sr) and are indicated by a symbol on their respective contour lines in Fig. 2. Prescriptions for these lenses are provided in Supplement 1. We note that these are just phase surface designs and represent the first step of a complete metasurface prescription; there is no consideration of materials, meta-atom bases, meta-atom layout and physical optics propagation [9,39,40]. Also, we only consider monochromatic operation without concern over the dispersive properties of the surface, so these phase profiles could equivalently be prescriptions for diffractive optical elements. As such, these prescriptions are idealized and real MS implementations of these prescriptions will yield lower performance due to discretization and non-ideal scattering.

 

Fig. 3. 2D layouts of two (a), three (b) and five (c) MS designs with 10 mm focal length. VIE, ${N_{spots}}$ and V are shown below each lens.

Download Full Size | PDF

Next, we show behavior of MS lenses by generating VIE/FOV contours for various configurations. Each contour summarizes different lenses with the same focal length and number of MSs. Each lens is optimized at a design FOV. The examples in Fig. 3 represent single points on their respective contours. Contours corresponding to the configurations shown in Fig. 3 are plotted on Fig. 2. These represent a collection of two-MS, three-MS and five-MS lenses designed at ${\lambda _0} = 632$ nm with f = 10 mm and F/1. The VIE decreases as the FOV increases beyond 0.8 sr. The plateau at small FOV arises because the geometric spots become small and the system is limited by diffraction – this is the diffraction-limited region. We discuss VIE saturation and scaling behavior in Section 5.

Notice we require 5 surfaces to reach the VIE that Arbabi, et al. [5]., achieve for a f = 0.717 mm lens using two surfaces to cover a FOV = 0.84 sr. The discrepancy arises from the longer focal length in Fig. 3 examples and our f = 10 mm contours. Predictably, it is more difficult to correct aberrations in long focal length MS lenses; just like conventional lens design. Aberrations scale down with lens dimensions while diffraction spots remain fixed by the constant F-number making it easier to reach the diffraction limit in short EFL designs. We reduced design focal length to f = 1 mm and generated new contours for two-MS and three-MS lenses; results are plotted on Fig. 2 as dotted lines and the prescription optimized at 120$^\circ $ FOV is provided in Supplement 1. At 0.84 sr (60° FOV), the two-MS curve is comparable to but surpasses the result for Arbabi’s lens. Small differences between the two in f, F and ${\lambda _0}$ inhibit a direct comparison but, nonetheless, give confidence that our optimizations generally yield good MS lens designs – comparable to the very best MS lens demonstrations. Similarity of the results are welcome. Recall the goal of the VIE: to provide a framework for comparison of disparate lenses by using a self-referencing diffraction-limited system that absolves differences in f and ${\lambda _0}$. Both f = 1 mm contours surpass the empirical limit for conventional lenses at FOV > 0.8 sr. These contours show that MS lenses offer significant improvement over conventional lenses at wide FOV and enable access to a previously inaccessible regime of compact cameras with wide FOV using flat sensors. Of course, our optimization does not represent a global maximum and new MS lens designs may outperform our examples. New optimization methods such as inverse design [41,42] promise to better employ the extreme degrees-of-freedom afforded by MSs where phase can be controlled down to the meta-atom level.

Last, we designed a MMS telephoto f = 100 mm lens for the MWIR to perform a side-by-side comparison against a conventional MWIR lens. The comparison lens shown in the inset of Fig. 4(a) was chosen from our library after observing its high VIE in the MWIR (black filled pentagon in Fig. 2). This telephoto lens uses 8 refractive elements (16 surfaces) to provide f = 100 mm in a total track length of 91.75 mm. It is F/3.1 with FOV = 20$^\circ $ at ${\lambda _0} = 4\; \mu m$ and provides diffraction-limited performance shown by the MTF in Fig. 4(a). It has VIE = 0.868 with FOV = 0.1 sr. Our five-MS design (red open pentagon in Fig. 2) and its MTF are shown in Fig. 4(b). It has the same specs; ${\lambda _0} = 4\; \mu m$, f = 100 mm, F/3.1 and FOV = 20$^\circ $ or 0.1 sr. Prescriptions of each surface are provided in Supplement 1. The lens uses fewer surfaces to yield VIE = 0.947 – an increase by 8 percentage points. Design imposed slow variation of phase across these large diameter optics in order to produce fabricatable designs. The maximum spatial frequency on any surface is 0.2 $\mu {m^{ - 1}}$, corresponding to a minimum 2$\pi $ phase reset period of 5 $\mu m$. A meta-atom pitch of 500 nm allows spanning this range with a 10 meta-atom basis; this pitch is easy to achieve using deep-ultraviolet lithography, for example, ensuring a profile that could be fabricated.

 

Fig. 4. Comparison of MTF between a conventional MWIR lens(a) and 5-MS lenses(b). Ray trace layouts for each lens are inset. The MWIR lenses was selected from a Zemax database, Zebase 6. The VIE for each lens is plotted in Fig. 2 and discussed in the text.

Download Full Size | PDF

5. Discussion

The VIE metric provides a new framework to analyze and compare the compactness of imaging optics. It employs a basic comparison of resolution to volume captured by the VCD. Some form of this comparison is routine in optical design and the VCD simply provides a specific mold to cast that comparison. The real value of the VIE arises by introduction of a simple reference system that presents a “fundamental” upper limit to VCD based on diffraction. This reference bears the concept of an efficiency for imaging systems and presents a hard target to design toward. It also eliminates sensitivity to $\lambda $ and f facilitating comparison of disparate lenses. This paves the way to compare optical technologies, in general. Consider an application with a requirement on FOV, for example, the metric allows the designer to compare different lens technologies to meet the resolution and volume needs of the application. Likewise, if a designer seeks to leverage a particular technology, the metric provides a tool to identify suitable imaging architectures such as the f, the FOV or the use of multiple apertures. We quantified a VIE limit for conventional imaging systems and showed that new technologies (curved images and MS lenses) can beat that limit (c.f. Fig. 2). Furthermore, we showed where these technologies can beat conventional optics (i.e. wide FOV) and how much room there is for improvement. In this section we discuss noteworthy details, artifacts and limitations of the metric.

5.1 Reference lens

The reference lens introduced in this paper is not suitable for ultra wide-angle (UWA) lenses with FOV approaching or exceeding $2\pi $ sr. UWA optics like fisheye or panoramic (i.e. donut) lenses use non-linear mapping to compress a UWA image onto a sensor. In contrast, the reference system provides distortion-free, rectilinear mapping onto a flat image surface. It also does not provide angular magnification so that chief ray angles are preserved between object and image space. The resulting image height (h) per field angle ($\theta $) is $h = f\tan \theta $ which grows asymptotically as $\theta \to 90^\circ $; the image area grows unbounded with such large angles. When image diameter increases to be much larger than the thin lens, chief rays become more oblique increasing the working F-number and spot size. The reference cannot be applied against lenses imaging beyond a full hemisphere with FOV > $2\pi $ sr. Combination of these effects make the idealized thin-lens reference a bad choice for normalization in UWA applications.

The deficiency in the reference comes into focus when we look at three conventional UWA lenses from Refs. [43], [44], [45] with full FOV = 174$^\circ $, 178$^\circ $, 180$^\circ $ and VIE = 0.06, 0.29, $1.6 \times {10^{19}}$ respectively. These VIE are larger than expected from the empirical limit trend shown in Fig. 2; the last value is an artifact of dividing by zero when volume of the flat reference lens becomes infinite. UWA lenses are notoriously bulky, requiring a large volume of optics to compress the FOV into a small image area – we expect small efficiency. The reference is not well-behaved so that we get arbitrarily large VIE the depends sensitively on the FOV near 2$\pi $.

Recent UWA lens designs based on metasurfaces demonstrate VIE > 100%. M. Y. Shalaginov, et al., demonstrated MS lenses with > 170$^\circ $ FOV for the midwave and near infrared spectrum [10]. We estimate VIE = 4.3 and 2.2 for these lenses, respectively. It is not surprising that this technology beats the empirical limit for conventional lenses but it is surprising that it surpasses what we have loosely considered a “fundamental” upper limit given by our thin-lens reference. Indeed, MS lenses provide remarkable compact imaging that enable high efficiency and large FOV.

There is a better reference that provides a more robust upper limit to VCD: an idealized thin-lens that provides diffraction-limited imaging onto a curved surface with radius equal to f. See Fig. 1(c) for a schematic cross-section of this reference system. It still provides unity angular magnification but $f \cdot \theta $ mapping onto the curved image. Unlike the reference lens which images onto a flat surface, this “curved” reference is well-behaved in the UWA regime. It has a finite volume bounded by a hemisphere with $V < 2\pi {f^3}/3$. Chief rays remain perpendicular at large field angles minimizing spot size in the image. This represents a very good imaging system, perhaps the best that is possible within the constraints of diffraction – truly a “fundamental” upper limit to the VCD of an imager. The “curved” reference also has a simple analytical expression for $VCD_{max}^{\prime} = N_{spots}^{\prime}/{({V^{\prime}} )^{2/3}}$ where

Here the first two terms are similar to Eq. (4) while L’ is the total length from vertex of the cone to the apex of the image sensor and h is the height of the spherical cap as illustrated in Fig. 1(c), formulae are provided in Supplement 1. The last term is the volume of a spherical cap with radius f and height h. As before $\textrm{VCD}_{\textrm{max}}^{\prime}$ is independent of f for a constant $F = f/\textrm{D}$.

Figure 5 plots $\textrm{VCD} \le {\lambda ^2}$ for the flat reference used throughout this paper along with the “curved” reference, also $F = 1$. Multiplication by ${\lambda ^2}$ eliminates the wavelength dependence in both references via ${N_{spots}}$ and $N_{spots}^{\prime}$ (c.f. Equation (3) and (7). Similarly, we expect this to provide effective wavelength-normalization for the VCD of real lens designs so that we can compare lenses designed for different spectral bands. VCD for the flat reference degrades rapidly when FOV approaches 2$\pi $ sr due to asymptotic increase in volume without a commensurate growth in resolvable spots. In contrast, the curved reference is well-behaved and nearly constant at large FOV out to 2$\pi $ sr.

 

Fig. 5. Plot of the Volumetric Channel Density (VCD) expanded to cover ultra-wide angle (UWA) FOV. VCD is multiplied by the square of the center wavelength to facilitate direct comparison. VCD_max for the flat reference (blue, circles) is degraded by an unrealistic expansion of optical volume at large FOV; VCD’_max for the curved reference (red, squares) provides a better fundamental limit across all FOV up to 2$\pi $. VCD curves of the empirical limit for conventional lenses and MMS lenses are shown. Example lenses discussed in the text are labeled with the same makers used in Fig. 2. Additional UWA lenses include short focal length MS lenses for the NIR/MWIR [10] and conventional lenses from Refs. [43–47](01–05, respectively).

Download Full Size | PDF

The curved reference represents the real fundamental limit of VCD ($\textrm{VCD}_{\textrm{max}}^{\prime}$) and, in general, systems should be compared against this. With few exceptions, however, optical systems image onto flat surfaces and the flat reference is more intuitive than the curved reference. The flat reference ($\textrm{VC}{\textrm{D}_{\textrm{max}}}$) also provides a good estimate for $\textrm{VCD}_{\textrm{max}}^{\prime}$ for most lenses. Figure 5 shows, the flat and curved references are equivalent and $\textrm{VC}{\textrm{D}_{\textrm{max}}} \approx \textrm{VCD}_{\textrm{max}}^{\prime}$ for FOV < ∼0.5 sr. At 1.5 sr (${\pm} $40$^\circ $) FOV the $\textrm{VCD}_{\textrm{max}}^{\prime} \approx 2 \times \textrm{VC}{\textrm{D}_{\textrm{max}}}$ and at $\pi $ sr (${\pm} $60$^\circ $) FOV the $\textrm{VCD}_{\textrm{max}}^{\prime} \approx 3 \times \textrm{VC}{\textrm{D}_{\textrm{max}}}$; using $\textrm{VC}{\textrm{D}_{\textrm{max}}}$ instead of $\textrm{VCD}_{\textrm{max}}^{\prime}$ results in overestimating a system’s VIE by a factor of 2 and 3, respectively. This discrepancy grows to more than a factor of 10 by 5.2 sr and asymptotes approaching $2\pi $ sr. Clearly, in the UWA regime one must use the curved reference and $\textrm{VCD}_{\textrm{max}}^{\prime}$ but up to ∼$\pi $ sr the flat reference may provide sufficient accuracy for comparing lens designs or optical technologies. We elected to use the flat reference throughout this paper for its simplicity, to avoid obfuscating the metric with a detail that only becomes critical in the UWA regime. The data in Fig. 2 is replotted using $\textrm{VCD}_{\textrm{max}}^{\prime}$ in Fig. S2 of Supplement 1.

Figure 5 also shows some data carried over from Fig. 2 and re-plotted as $\textrm{VCD} \cdot {\lambda ^2}$ (instead of VIE) along with some new data for UWA lenses. It includes a curve synthesized from the empirical VIE limit through multiplying it by $\textrm{VC}{\textrm{D}_{\textrm{max}}}$. $\textrm{VCD}$ curves for the optimized MS lens designs are multiplied by ${\lambda ^2}$ and plotted for comparison. Symbols show the $\textrm{VCD}$ of example wide angle lenses. Open circles numbered 1-5 indicate VCD of conventional UWA lenses from Refs. [43–47]. Notice the MWIR monocentric lens (open crescent) and both UWA MS lenses from Ref. [10] (six-point stars) yield $\textrm{VCD}$ > $\textrm{VC}{\textrm{D}_{\textrm{max}}}$ corresponding to VIE > 1. Normalizing the MS lenses by $\textrm{VCD}_{\textrm{max}}^{\prime}$ yields VIE of 0.095 for the NIR lens and 0.186 for the MWIR lens; these are down from 2.2 and 4.3 when normalized using $\textrm{VC}{\textrm{D}_{\textrm{max}}}$, respectively.

5.2 Scaling analysis

We compute a family of curves showing the variation in VIE with scale factor for some existing lens designs. Here we aim to visualize the trade space between VIE and f. We chose several lenses that span a wide range of designed FOVs and exhibited high VIE near the empirical limit. System scaling is implemented in ZEMAX to scale all dimensions by a specified scale factor, M. System scaling preserves F but scales the ray trace allowing one to scale a system down below the diffraction limit or up above the diffraction limit; blur due to aberrations scale with the system dimensions whereas diffraction-limited spot sizes are fixed by F. We re-scaled each lens without any changes (i.e. we did not perform re-optimization at each scale factor). The results are shown in Fig. 6(a). All lenses were designed at a scale factor of $M = 1$ and exhibit the same trends; VIE increases scaling to small dimensions until it plateaus when the system becomes diffraction-limited.

 

Fig. 6. (a) Whole system scaling plots (i.e. constant F) for select conventional lenses that lie near the empirical limit. (b) FOV sweeps for the same systems shown in part (a); numeric label (identifies each system in our library) is positioned near the design FOV for the lens.

Download Full Size | PDF

The VCD of a diffraction limited system is independent of M. The numerator, ${N_{spots}}$, scales as ${M^2}$ because the image area scales by ${M^2}$ while spot size is constant. The denominator, ${V^{2/3}}$, also scales as ${M^2}$. This scale independence applies to both a real system (as long as it remains diffraction limited) and the idealized thin-lens reference. The ratio of these two quantities is the VIE and is also scale independent while the real system remains diffraction limited – this gives rise to the plateau at small M.

The VCD of an aberration limited system scales as ${M^{ - 2}}$ driven by volume scaling. Aberrated spot areas grow at the same rate as image area giving a constant ${N_{spots}}$ contribution to VCD. VIE also scales as ${M^{ - 2}}$ in this regime and this is clearly seen in Fig. 6(a) for large M. As expected, most lenses exhibit a transition from the aberration-limited regime to the diffraction-limited regime near their design dimension with $M = 1$.

Next we consider scaling the image size or, equivalently, varying the FOV of a system. We retain the prescription without any re-optimization and simply compute the VIE for various FOVs; aperture diameters were expanded to mitigate vignetting when possible. Example lenses were chosen near the empirical limit that cover a wide range of design FOVs. VIE verse FOV curves were generated for each system and are shown in Fig. 6(b). These curves replicate the general trend that VIE decreases with increasing FOV; often at a rate similar to the slope of our empirical limit. Scaling to smaller FOV without re-optimizing the design is generally robust but scaling to larger FOV is often limited due to issues with vignetting or mechanical tolerance. The mobile lens design (2870) using plastic molded optics with deeply aspheric surfaces is an extreme example where VIE falls off dramatically when extending the aspheric apertures to cover a larger FOV. Also note that this surpasses the empirical limit at smaller FOV indicating the exemplary compactness of this technology.

Last we scaled the speed of the lens by scaling the aperture diameter alone. Re-optimization was not performed and scaling the aperture up frequently caused excessive vignetting. We stopped increasing speed when more than 50% of rays were vignetted as this would result in excessively low image brightness. We observe similar results to the FOV variation shown in Fig. 6(b); increasing speed typically yields lower VIE. This likely occurs because the system was designed to balance good performance and efficient light collection using as large an aperture as possible. In most cases, we observed a minor improvement in VIE can be achieved by reducing the stop diameter. We also conducted speed scaling simultaneous with system scaling and generated heat maps for six different lenses showing VIE sensitivity with respect to these variations. These results are provided in the Supplement 1. In the next section, we discuss the limitations of the VIE because it does not consider vignetting, optical throughput and sampling.

5.3 Limitations

We introduced an upper bound to the VCD given by an F/1 thin-lens reference system that provides idealized diffraction-limited imaging onto a flat sensor at a specified wavelength and FOV. Then we recognized the volume scaling of that reference yields an unreasonable comparison for ultra-wide angle lenses. We identified the idealized thin-lens imaging onto a curved image surface provides a more robust upper bound to diffraction-limited imaging optics. This fictitious lens is useful for determining the “efficiency” of a real lens with FOV up to $2\pi $ sr. To compute an efficiency for lenses with larger FOV, we recommend using the $\textrm{VCD}_{\textrm{max}}^{\prime}$ for a $2\pi $ FOV; the VCD of the curved reference is nearly constant beyond 0.84 sr FOV. In fact, the maximum VCD can be roughly approximated as 200,000/${\lambda ^2}$ (spots/mm²)·µm⁻² across all FOVs (c.f. Fig. 5); above 0.1 sr, this is within a factor of 2 of the exact value. Note $\lambda $ has units of microns. While we introduce this as an upper bound for diffraction-limited imaging, we arbitrarily chose $F = 1$ for this reference as it represents a fast but readily achievable speed. A faster lens with $F < 1$ that provides diffraction-limited performance over its FOV may surpass this upper bound. However, it must also be extremely compact. We expect that folded systems or systems with a high telephoto ratio may be able to exceed unity VIE especially if they also incorporate advanced optical technologies.

The VIE framework laid out in Sect. 2 and its extension to the UWA regime in Sect. 5.1 employ a quasi-2D approximation that off-axis diffraction limited spots are circular. Projection of an Airy disk onto an oblique image surface would distort the spot shape in the flat reference system. A bigger error in our approximation that affects both reference lenses is the aperture shortening for off-axis rays. This shortening only occurs along the meridional plane and not the sagittal plane. The cosine factor we applied to the aperture should only apply to the meridional axis while the sagittal axis is not compressed. This generates astigmatism that would produce elliptical spots in the image. We approximate spot radius based on the slow (i.e. longer) axis of the ellipse and therefore underestimate $\textrm{VC}{\textrm{D}_{\textrm{max}}}$ and $\textrm{VCD}_{\textrm{max}}^{\prime}$. Raising the reference values would suppress the efficiency of every system by the same factor and will not affect global conclusions drawn using the approximations embedded in the present framework.

At present, we only analyze infinite conjugates and axially symmetric lenses. However, it is straightforward to apply this framework to non-axial lenses and, in particular, lenses employing freeform surfaces. It is also easy to consider finite conjugates in which the objects are not distant but then the FOV is better characterized by a linear dimension; this also complicates the comparison of dissimilar optical trains.

We also present the framework using RMS spot size, however, it can be adapted to use modulus of the optical transfer function (MTF) instead. One can adopt an MTF cutoff value, for example 10%, and identify the spatial frequency where the system and the diffraction limit fall to this value. A more rigorous MTF-based approach might integrate the MTFs up to a cutoff frequency that captures some fraction of area under the MTF curve, for example 90%. The inverse of the cutoff frequency may be used to estimate the spot size in VCD and $\textrm{VC}{\textrm{D}_{\textrm{max}}}$, respectively. We investigated MTF-based VCD and VIE analysis for a small subset of our library. We concluded that the precise choice of cutoff frequency or the way ${N_{spots}}$ are estimated will not significantly affect a comparison between different lens designs. The chosen method must provide a reasonable basis for quantifying resolution across the image and the same method should be applied to each lens in the comparison – along with the reference system. The methods outlined in Sect. 3 should be used for comparison against the VIE results presented in Fig. 2.

The VIE framework does not consider image sampling. This was an intentional choice and was made to provide a metric that only considers the optical train. Decoupling the optics from electro-optic sampling and processing of the image simplifies the analysis by ignoring numerous degrees of freedom (DOF) in image sensor technology. These DOF affect how much information may be generated or how valuable (e.g. uncorrelated) that information is. For static sensors, sampling DOF might include the pitch, fill-factor, quantum efficiency, integration time, well depth, read noise, the number of spectral channels, and their arrangement (i.e. Bayer pattern or tandem structures). Dynamic sensors such as hyperspectral, scanning or time-of-flight sensors add additional DOF related to time. The VIE may be adapted in a general way to include basic sampling characteristics related to uniform detector spacing and optical throughput. Uniform detector spacing is a practical necessity in focal plane arrays and the number of optical spots, ${N_{spots}}$, may be replaced by the number of detectors as long as the system provides optimal sampling across the image. Variation of the optical throughput across the image could also be considered, especially when vignetting is significant, to determine the capacity of each channel. Addition of a computational back end to perform post-processing and reconstruction for color correction, de-blurring, super-resolution, etc. may further improve capabilities of the imaging system – even circumventing the requirement to form a sharp image [48]. For example, S. Colburn, A. Zhan and A. Majumdar demonstrated color correction for metasurface lenses using computational imaging [49]. We also note that co-design of the optics, image sampling and reconstruction such as V. Sitzmann et. al., [50] and Z. Lin et. al., [51] provides end-to-end optimization of a single system; in contrast, the VIE metric provides a framework to evaluate optical technologies and lens designs globally, relative to each other and limits imposed by diffraction.

The metric does not prescribe intrinsic value to the bandwidth of a system. Most conventional lens designs in our library cover a large spectral bandwidth – typically corrected over the entire band of interest (e.g. VIS, NIR, MWIR, LWIR). VIE calculation uses the weighted average spot size across the band. We use uniform weighting at the shortest, central and longest wavelength specified by the system. This is compared against a diffraction-limited reference evaluated at the central wavelength to represent an average spot size; diffraction-limited spots will be larger (smaller) at the long (short) wavelength end of the band. A more fair comparison might use the longest wavelength with the largest diffraction-limited spot size. Identical lenses that only differ in bandwidth but provide the same ${N_{spots}}$ from the same volume will have the same VIE; yet, the larger bandwidth system is clearly “better”.

It is especially important to recognize the lack of bandwidth in the context of VIEs reported for our MMS designs. Correcting the strong dispersion intrinsic to MSs and realizing the prescribed phase profiles over a wide bandwidth is quite challenging. However, recent works have shown that chromatically corrected MSs can provide diffraction limited focusing [52–54]. In Ref. [55], F. Presutti and F. Monticone introduced a fundamental limit to the spectral bandwidth achievable in a single metasurface. Nonetheless, all MMS lenses presented in Figs. 2 and 5 are monochrome designs based on phase-surfaces approximations that represent the upper limit of these particular designs. They do not include physical optics effects such as scattering inefficiency, polarization sensitivity, field-dependent phase delays, etc. that would degrade VIE of these lenses. In general, we expect trade-offs between resolution, focal length, complexity (e.g. number of MSs, MS diameter, volume of accessible phase space, etc.) and bandwidth in MS lenses.

While our MMS lens designs only considered thin surfaces with spacing large compared to $\lambda $, the VIE metric can be applied to evaluate more complex volumetric metastructures. Recent work by T. D. Gerke, et al., [56] and M. Mansouree, et al., [57] have shown that meta-atoms distributed through a volume and across multiple closely-spaced layers can be used to exploit coupling between them and provide ultimate control of the propagating light field. Likewise, A. Zhan, et al., demonstrated that such volume elements can be produced by 3D printing novel meta-atom structures such as spheres [58]. This enables volume elements that modify the phase in a complex and dynamic way. The VIE can be applied here because it only depends on the light field at the image and the volume traversed by light, in this way it can be applied to imaging systems employing any optical technology.

Last, the VIE presented here only considers lenses with a single aperture. Yet multi-aperture systems provide a compelling mechanism to expand FOV. Doubling FOV by adding a replica of a single aperture system will double ${N_{spots}}$ and optical volume to yield the same VIE as the single aperture system. Expanding to Z copies of the single aperture increases FOV by a factor Z while retaining VIE – as long as each copy addresses unique portions of the field of regard. This allows extending limited FOV systems into large FOV applications while moving horizontally across Fig. 2. This approach is particularly interesting for MS lenses that readily provide exceptional short focal length lenses that are fast and compact; numerous apertures could scale to cover large FOVs at high resolution. The challenge will be assembling and mounting aperture arrays to span a large field of regard with minimal volume for mounting and packaging these arrays. The VIE does not consider the volume required to mechanically and electrically integrate a system; that is why it appears “free” to replicate apertures. In practice, adding apertures will increase the volume and complexity of the integration providing some practical penalty to this scaling. Nonetheless, this is an important degree of freedom to consider in order to realize compact, wide angle and high-resolution imagers.

6. Conclusions

In this paper, we introduce a new metric for optical systems that concisely describes their resolution per unit volume – the VCD. We also introduced two analytic models for the upper limit of VCD (i.e. $\textrm{VC}{\textrm{D}_{\textrm{max}}}$) in diffraction-limited systems using thin-lenses to image onto flat and curved image surfaces. Comparing a lens VCD against this limit introduces the volumetric imaging efficiency (VIE). VIE is a robust comparative tool that can be used to compare disparate imaging systems operating at different wavelengths and employing various technologies. We demonstrate this by plotting VIE versus FOV for 2871 lens designs employing bulk glass optics, molded plastics, curved images and MS lenses. We identified an empirical limit when bulk glass optics are used to produce flat images and showed that imaging onto curved surfaces can increase VIE by 100x in wide angle applications. Next we calculated VIE for existing MS lenses and programmatically designed MMS phase prescriptions to compare against the empirical limit for bulk lenses; short focal length MS lenses can surpass this limit by >10x but this is more difficult at longer focal lengths. However, we showed a 100mm MWIR telephoto lens with 5-MSs with VIE that surpasses a well-corrected conventional design with 16 bulk glass surfaces. We discuss the scaling properties of VIE and the transition from aberration-limited (scales as ${M^{ - 2}}$) and diffraction-limited (constant) regimes. Last, we discusses the limitations of the present VIE framework and analysis. In the future, we hope this framework can be expanded to consider the other two SWaC variables (weight and cost) in a generalizable manner, extended to consider other impactful technologies (freeform surfaces, GRIN materials, etc.) and analyzed along different degrees of freedom ($f$, λ, aperture number, etc.) to better quantify the complex trade space of optical design.