Key Points:
- Metasurfaces offer a versatile platform for wavefront engineering and imaging applications owing to their subwavelength spatial resolution and degrees of freedom for multiplexing across wavevector, polarization, and wavelength
- While metalenses can be made at very small apertures (on the order of microns) or out to apertures occupying a full 100 mm wafer or larger, the aberrations at these scales are quite distinct. Researchers, students, or companies evaluating metasurface technologies should be aware of some of these basic differences, given the nonlinear scaling behavior of Seidel aberrations that can drive large reductions in performance as aperture size increases
- These effects will influence the commercialization path for metasurface technologies, limiting utility of single-element components at larger aperture sizes, and likely driving efforts towards multi-element solutions and integration with other components depending on application requirements
Overview
Metasurface flat optics can be designed and fabricated at apertures sizes over a wide range of length scales, with potential applications for lenses of only a few microns in diameter (e.g., integrated at the pixel level for enhancing light collection efficiency, polarization or color filtering and routing, etc.), at millimeter to centimeter scales in consumer electronics or beamshaping use cases, or very large apertures for long-range imaging applications (e.g., remote sensing or beamsteering). While market potential in these different verticals is debatable, on the technical side there are basic differences between these cases that are worth understanding, as they have implications on the utility and design tradeoffs of meta-optics.
In interactions with clients, students, and researchers over the years, there tend to be many of the same questions and misconceptions about metalenses when considering their performance and constraints as you change aperture size. In this post, we’ll detail some of the key considerations and often misunderstood aspects of metalens design as aperture size changes.
Central to this is a frequent line of questioning that we hear, which proceeds somewhat along the following lines. A researcher designs a metalens and measures its performance. They then want a comparable lens but at a longer focal length, say going from 0.5 mm to 5 mm. They use the same design methodology, the same meta-atom design, the same f/#, and the same process flow to fabricate the device. The larger lens then performs far worse in terms of spot size and associated modulation transfer function (MTF) and they ask why that’s the case.
These effects are not at all unique to metasurfaces but are characteristic of optical lenses more generally; however, as many students, researchers, and practitioners within the metasurface community come from non-optics backgrounds (myself included), it’s worth walking through some of these considerations as they come up regularly.
Larger Apertures Are More Aberrated (Usually)
So why do these larger lenses tend to perform worse than their smaller counterparts? There are several reasons, but chiefly, with all else equal, larger aperture optics exhibit more significant aberrations. Put simply, if you take a lens of a given f/#, and then scale everything up by a factor of 10, what do you expect to happen? On the one hand, you may expect to get the same MTF, after all it would appear that nothing has really changed as the new system is proportional to the original one. Or perhaps you expect performance to be 10x worse, whatever that may mean.
The answer is usually not so simple and is quite case specific; however, most of the time, performance from an aberrations perspective will get worse unless the original and scaled up versions are both diffraction limited. With well-corrected systems at the diffraction limit, a lens that is 10 or even 100x larger in aperture size will have the same spot size and same MTF. If you are not diffraction limited, however, your spot size can increase dramatically, and oftentimes scaling exhibits far worse than linear behavior in terms of changes in performance. Some Seidel aberrations scale linearly with aperture size (e.g., transverse astigmatism and Petzval field curvature), while others scale quadratically (e.g., coma and longitudinal spherical aberration), and others cubically (e.g., transverse spherical aberration). The net of this is that one should not expect the same spot size or MTF with a naïve scaling of a metalens design.
f/# (Not Just Aperture Size) Really Matters
The exception to this is when f/# is high. In this case, this puts the lens design in a regime where it is far closer to diffraction-limited performance. As Lohmann points out in his paper on lens scaling laws, this is why for real lens designs that are commercially available and used in practice, f/# tends to increase as aperture size increases, as it helps to balance aberrations. The designs that both maintain f/# while increasing aperture size require more extensive optical design and entail more surfaces and higher costs, driven by the fact that these components by nature of their extended aperture are far more challenging to correct.
Why Increase Aperture Size?
Why increase aperture size at all then if you expect performance to degrade? While spot size and MTF tend to degrade, there are other reasons to have a larger aperture. Often this is because one needs a higher magnification image, which means a longer focal length. At a larger aperture size and a correspondingly higher focal length, the resultant image on a sensor will provided a more magnified view of a scene, providing higher pixels per degree, which can be critical in many use cases (e.g., resolving small text on a distant object for optical character recognition applications, or distinguishing individual lines in a barcode). Increasing focal length while keeping the same aperture size will reduce image brightness though, and so often the increase in aperture helps to maintain signal-to-noise ratio (SNR) at the sensor.
A Larger Aperture Collects More Light, but It does Not Make Your Image Brighter (Except for One Special Case)
One reason to not increase aperture size is for the goal of making a brighter image. This is a common misconception. The quantity that actually matters for image brightness is the f/# and not the aperture size on its own. With all else equal, a larger diameter lens will not make your image brighter at all.
This may seem counterintuitive at first, as a larger aperture does collect more light; however, that collected light is also distributed over a wider area by exactly the same proportion, leading to the same incident intensity.
There is an exception to this, which is irrelevant for most practical camera systems. If a lens is diffraction limited and you are imaging a point source, then the point source image (i.e., the diffraction-limited point spread function) will get brighter as the aperture size increases for a fixed f/# (assuming the scaled up lens is also diffraction limited). This arises from the fact that in this case, the image area is always the same (equal to that of the diffraction-limited spot), so more power is directed into the same size focal spot.
In most camera systems, the optics are not diffraction limited but are instead designed with a spot size catered to the sensor pixel size. Furthermore, when imaging an extended object, and not an infinitesimal point source, the imaged area will increase in tandem with the lens scale factor. As such, it is really the f/# that is the quantity that matters for image brightness, both for more conventional imaging lenses as well as metalenses.
What Does This Mean for Metalens Designers?
So for those of us working on metalenses, what does this all mean? Ultimately, we cannot break this scaling behavior. These effects are explainable via aberration theory that applies equally to metasurface flat optics as it does to other lens systems. Metasurfaces can provide advantages in terms of facilitating higher NA (equivalent to higher space-bandwidth product), but as we scale across aperture size, we will still encounter these same challenges.
Going forward, this will have implications for how metasurface systems are developed. Given the limitations of a single surface and restrictions on performance with extended aperture sizes, metasurface-based systems will likely need to push to multi-element solutions or other architectures. Folded optical elements may provide a means for accomplishing this, helping to mitigate multi-element alignment challenges while correcting aberrations more effectively. Conversely, the scaling effects may direct metalens adoption more towards use cases where miniaturization is paramount and where the nonlinear Seidel scaling is less problematic but where meta-optics provide a manufacturable platform with enhanced space-bandwidth product compared to alternatives.
There are other challenges with large-aperture meta-optics that we did not touch on here, including computational limits for large area meta-optics, unit costs in manufacturing as components occupy a large portion of a wafer, etc. These are also important considerations and we’ll dive further into some of these topics in future posts.
References
Lohmann, Adolf W. “Scaling laws for lens systems.” Applied optics 28.23 (1989): 4996-4998.
Smith, Warren J. Modern Optical Engineering: The Design of Optical Systems. 3rd ed., McGraw-Hill, 2000. Chapter 3, “Aberrations”
She, Alan, et al. “Large area metalenses: design, characterization, and mass manufacturing.” Optics express 26.2 (2018): 1573-1585.
Colburn, Shane, Alan Zhan, and Arka Majumdar. “Varifocal zoom imaging with large area focal length adjustable metalenses.” Optica 5.7 (2018): 825-831.
Park, Joon-Suh, et al. “All-glass, large metalens at visible wavelength using deep-ultraviolet projection lithography.” Nano letters 19.12 (2019): 8673-8682.
Zhang, Lidan, et al. “High-efficiency, 80 mm aperture metalens telescope.” Nano letters 23.1 (2022): 51-57.
Hollows, Gregory, and Nicholas James. “System Throughput, f/#, and Numerical Aperture.” Edmund Optics Knowledge Center, Edmund Optics Inc., n.d.
Liang, Haowen, et al. “High performance metalenses: numerical aperture, aberrations, chromaticity, and trade-offs.” Optica 6.12 (2019): 1461-1470.