Key Points:
- Metalenses are sensitive to the wavelength of incidence, leading to aberrations that produce a chromatic focal shift
- These aberrations arise from discrepancies in group length along different ray paths in the system, or they can alternatively be explained as arising from phase-wrapping discontinuities that are present when the incident wavelength deviates from the designed wavelength
- Extensive research into achromatic metalenses over the past decade or so has led to a number of mitigation options, including dispersion engineering, multi-element systems, multi-wavelength achromats, and computational imaging meta-optics. Each of these approaches present unique design tradeoffs and challenges that designers and researchers should be aware of when evaluating metasurface components for a target application
Overview
It’s well established that metasurfaces, as diffractive optics, exhibit significant chromatic aberrations. For metalenses, this manifests as a chromatic focal shift leading to color blur in images. When operated with a single wavelength, a metasurface does not need to contend with these challenges, but many real-world applications require much wider operating bandwidths. As such, there has been extensive research on achromatic metasurfaces, employing a number of different design strategies to correct or circumvent these effects.
In this article, we will not exhaustively explore all of these, but we will focus on several key approaches and emphasize the underlying mechanisms as to how chromatic aberrations arise in the first place for meta-optics. As there are a number of misconceptions about these effects and their causes even within the metasurface research community, our goal is to make these effects clear and understandable (e.g., the causes, the viability of different solutions, and other secondary effects arising from chromatic aberrations).
What are chromatic aberrations?
Chromatic aberration is a flaw or undesired behavior in a lens system arising from a change in operating wavelength. For example, if a metalens designed for green light is illuminated with blue light, the focal length increases, leading to a blurry image if the camera sensor is not correspondingly shifted to the appropriately displaced back focal plane. Both within and outside of metasurface optics, there are two main types of chromatic aberration: longitudinal and transverse. Taking a lens as an example again, longitudinal chromatic aberration refers to errors in its focusing behavior along the optical axis (i.e., longitudinal shifts of the focal plane as in our green metalens illuminated by blue light notional example).
Transverse chromatic aberration instead refers to effects where a change in wavelength induces a shift off axis (i.e., transverse to the optical axis). If we treat the z-axis as the central optical axis, this is seen in metalenses for off-axis fields, where there is not only a longitudinal chromatic aberration along z, but additionally a transverse chromatic aberration that shifts the focal point further from the desired position in the X-Y plane.
These effects are actually quite similar to those present in refractive elements and other diffractive components, though they tend to differ in magnitude and the means for correcting these aberrations look quite different in practice.
What causes chromatic aberrations?
There are a few different ways of describing chromatic aberrations, but a useful and general means of treating them is by analyzing differences in group length across a set of rays. The group length is essentially the total path length that a given ray traverses through a system, weighted by the refractive index of each medium. For example, for a slab of glass of thickness L and refractive index n, for a ray that passes normally through this slab, the group length contribution along the ray’s trajectory is the product nL. Now, say that we have two rays incident at the same position and incidence angle on the slab, but now the two rays differ in wavelength. As refractive index in general is a function of wavelength, this product n(λ)L will now differ for the two rays, even if just slightly, because the index will change but also the path length traversed can vary because the Snell’s law-governed deflected angle in the medium differs.
The reason this matters is because in an optical system, e.g., a lens design, in which we want all the rays to converge at some desired focal point, if we want the design to be achromatic (i.e., to not depend on wavelength), then the group length of all ray paths must be equal. This condition will ensure that the wavefronts associated with each ray path will be in phase with one another. In practice, owing to dispersion, these ray paths will differ, leading to chromatic aberration.
In an optical system, the group length divided by the speed of light is equal to the time it takes for light to propagate along that path. As such, discrepancies in group length amongst a set of incident rays is equivalent to there being discrepancies in the time duration for light to propagate through the system.
The same is true of meta-optics, even for a single surface. The difference is that a metasurface can itself induce group length changes through its meta-atom response, the amount of which depends on the design of its meta-atoms. Exploiting this capability to engineer specific chromatic responses is called dispersion engineering, which we’ll detail further later in this post.
Phase Wrapping Discontinuities Driving Chromatic Aberrations
Another perspective for interpreting metalens chromatic aberration is through phase wrapping discontinuities. For example, if we take a metasurface lens design for a specific wavelength λ and we then implement this optic with a set of nanoposts that span the 0 to 2π range, there will be a series of Fresnel zones, wherein the phase wraps every 2π. In general, the position of where this phase wrap occurs will change if the lens were instead designed for a different wavelength; however, for a static metasurface design, these phase wraps (i.e., typically corresponding to a wraparound in a meta-atom lookup table from maximum to minimum feature size), these phase wrap positions are fixed. This means that for off-design wavelengths, there are now discontinuous jumps in phase at each wrapping position in the layout. Within the first Fresnel zone, even a regular metasurface lens without any dispersion engineering is technically achromatic, but when extending to multiple zones (e.g., to increase aperture size and NA), only then do these phase errors become present.
This is an important effect to understand, as oftentimes the chromatic effects of a metasurface are attributed to resonant properties of meta-atoms alone. In a sense, this is true, but it is an incomplete and misleading representation of what is actually causing chromatic focal shifts. For many conventional meta-atom designs, changing wavelength will alter the local phase response, but to first order this change is just a linear change in phase proportional to the change in wavelength (i.e., assuming the nanoposts can be modelled as an effective medium if off resonance). As such, it conveys little intuition about chromatic aberration by simply stating that a meta-atom’s operation depends on wavelength. What matters more is the global effect of this on the spatial phase function. Locally, for a small change in wavelength, each meta-atom’s phase delay actually changes by the proportional amount required in order for the phase delay at that position to meet exactly that required for the new wavelength. The issue is that the designer typically will utilize a phase lookup table that only supports 0 to 2π phase shifts at a single wavelength, which means it is only possible to induce the correct RELATIVE phase delays for a single wavelength matched to that lookup table.
These types of errors are equivalent to those arising from discrepancies in group length, but they provide another perspective for understanding these effects. If, for example, a designer wanted to have a metalens that gave the same focal length for two distinct wavelengths, then they would need a phase lookup table that provides a phase shift up to the least common multiple of the wavelengths times 2π. For example, for a design with 400 nm and 800 nm wavelengths, there needs to be a nanopost design that induces up to a 4π phase shift at 400 nm and up to 2π at 800 nm. This condition ensures that phase wrapping conditions occur at the appropriate spatial locations for both wavelengths, and this approach was successfully reported in simulation previously. This principle can be extended to more wavelengths, but the requirements on the meta-atom design quickly become impractical from an aspect ratio perspective. In the below figure, FDTD-simulated results verify this dual-wavelength achromatic behavior based on eliminating phase wrapping errors, but these results are based on nanopost designs with aspect ratios exceeding 20:1.
Dispersion Engineering as a Solution
Revisiting the notion of needing all rays to have the same group length in order for a metalens to be achromatic, dispersion engineering is a technique that offers a means to do this. Typically, a metalens is designed for a single wavelength, and phase shifters that work for that frequency are mapped to positions in a layout. With dispersion engineering, not only is the phase shift at the target (typically central) wavelength in a band specified, but the required phase shift at other wavelengths in the band is also specified. In particular, the required phase delay as a function of frequency is expanded into a Taylor series. While a standard metalens only accounts for the zero-order term (i.e., you only need to consider single-frequency operation), the higher order terms specifically incorporate frequency dependence.
Physically, the first and second order terms correspond to the group delay and group delay dispersion, and there are several metasurface works in the literature exploiting these terms to realize achromatic metalens designs. The exact manner in which this is accomplished varies, but typically it entails utilizing a more complex meta-atom design with multiple resonant scatterers per unit cell. These more complex unit cell designs exploit both a propagation-based phase delay as well as a Pancharatnam-Berry (PB) phase delay that is largely wavelength independent. In relying on PB phase, this means that dispersion-engineered metalenses are almost always polarization sensitive, typically requiring circular polarization for their operation.
Limits of Dispersion Engineering
While there have been numerous experimental demonstrations of dispersion-engineered metalenses, confirming broadband focusing and imaging in both the visible, infrared, and thermal ranges, there are some stringent practical limitations to this approach. Firstly, as mentioned in the prior section, this approach largely depends on the incident light being circularly polarized, as unit cell designs often require PB phase elements. This doesn’t prevent dispersion engineering from being useful, but it does limit practicality as in imaging of ambient scenes, incident light is unpolarized. Secondly, while dispersion engineering can in theory support wide wavelength ranges (e.g., the full visible spectrum), it can only do so for a very low NA and very small aperture size. In practice, even with an NA less than 0.1, it is only feasible to realize an achromatic dispersion-engineered metalens with an aperture on the order of tens of microns while supporting the full visible spectrum. The reasons for this are subtle and come about from the underlying physical mechanism for the meta-atoms.
In order to impart the higher order terms in the Tayler expansion of the phase (i.e., the terms that will correct the zero-order phase shift as a function of frequency across the desired bandwidth), the meta-atoms rely on a resonance condition. As either the NA increases for a fixed aperture size, or the aperture size increases for a fixed NA (i.e., the focal length and diameter increase proportionally), the requirements for this resonance condition become more demanding. In particular, in order to realize the group delay and group delay dispersion terms accurately, this means that the quality factor of the resonance must increase. Eventually, with a high enough combination of NA, aperture size, and operating bandwidth, the quality factor becomes so high that it is impractical. For example, in order to realize achromatic millimeter scale metalenses, the quality factor requirements entail pushing to nanopost aspect ratios that are not possible to fabricate with existing tools reliably.
Furthermore, even for smaller scale metalenses, dispersion engineering can be quite limiting in practice as the unit cell designs are far more complicated than standard meta-atoms based on square or circular cross sections. As typically multiple scatterers are required per unit cell with dispersion engineering, the critical dimension requirements are substantially smaller than those of more conventional unit cell designs.
We’ll note that there are a number of works that have demonstrated achromatic metalenses that utilize modified approaches and implementations of what we described in this subsection, but in many cases, while not explicitly referred to as dispersion-engineered devices (e.g., they instead exploit the slope and intercept of the phase response in their design layouts), these are largely semantic differences, as they are functionally equivalent in terms of their underlying physical mechanism exploiting a resonant response to impart higher order terms in the Taylor expansion of the phase.
Multi-element Solutions
So far, we have primarily focused on meta-optical systems comprising a single component, examining the sources of chromatic aberration and how meta-atom designs may be adapted to realize achromatic metalenses. This, however, is not the only means of mitigating chromatic aberrations in meta-optics. It is possible to utilize even simple meta-atom designs (e.g., square or circular cross sections) that do not exploit high quality factor resonances or PB phase in order to make a broadband focusing element. The key insight that enables this is again an understanding that it is a uniformity in group length for all ray paths that makes a metalens achromatic. While dispersion engineering is one means to do this, one can alternatively exploit multiple metasurfaces that as a system will ensure that the trajectory that each ray travels along will be equal in group length to all other rays. This is a powerful insight in that it can significantly relax the requirements on the meta-atom design.
The challenge is instead shifted to a multi-element surface design, which can be nontrivial for more complex system requirements. Experimental demonstrations of achromatic beam deflectors using this technique have already been shown, utilizing a folded geometry of two meta-optics, and simulated versions of achromatic lenses have been shown as well. In practice, the realized metalens designs were limited to a narrow field of view, though single surface dispersion-engineered metalenses are also limited in this regard. Extending the field of view will likely require more than 2 metasurfaces, the design of which is nontrivial.
Multi-wavelength Solutions
It is worth mentioning that while truly broadband metalenses that are achromatic across a continuous band are often touted as a design goal, there are many applications where only a multi-wavelength solution (e.g., only needing two or three wavelengths) can be useful. This is more likely in display systems, or imaging systems where there is a paired source with a narrow bandwidth. Multi-wavelength metalenses have significantly relaxed design requirements compared to broadband achromatic counterparts. There is a wide range of design options for realizing these systems including stacked plasmonic metasurfaces where each individual metasurface only narrowly couples to a specific wavelength, spatial multiplexing schemes wherein subregions of a given metalens are dedicated to a particular wavelength; and dispersive phase compensation meta-atoms, which is similar in principle to dispersion engineering but only must support a few discrete wavelengths rather than a continuous band.
Computational Imaging Solutions
One last category we wanted to cover in this article entails meta-optics designs that operate in conjunction with computational imaging algorithms. In this case, a metasurface lens is not designed to be an achromatic lens but instead is designed to realize a point spread function (PSF) that is achromatic but does not necessarily focus to a single spot. This introduces some complexity at the system level, but it enables circumventing the requirement of an achromatic focus as long as the PSF has certain properties that facilitate high-quality imaging. These end-to-end designed meta-optics often yield complex phase profiles that are freeform and asymmetric in nature.
In particular, researchers have demonstrated experimentally that if a meta-optic exhibits a modulation transfer function (MTF) without any zeros out to a desired cutoff frequency, and that is largely invariant across the desired bandwidth, that the PSF can serve as an encoding kernel on an imaged scene, which can then be inverted using software. Several demonstrations have shown full-color imaging with ambient, white light with f/2 metalenses out to apertures as large as 1 centimeter, far higher than what is possible with dispersion engineering or alternatives. This approach has its own tradeoffs though. In particular, introducing computational imaging entails adding latency and power consumption to the camera pipeline, and under low SNR conditions, the deconvolution routines can lead to unacceptable noise amplification in images. That being said, recent experimental demonstrations with AI-enhanced postprocessing and extended aperture size metasurface designs have shown promising imaging results.
Opportunities and Challenges
Chromatic aberrations have been and continue to be a major challenge for metasurfaces, limiting their utility and broader commercial adoption beyond single-wavelength applications. A number of design methodologies exist to mitigate or circumvent these aberrations, including dispersion engineering, cascaded multi-element solutions that satisfy the uniform group length condition for all rays, metasurfaces co-designed with computational imaging software, and multi-wavelength solutions for discrete wavelengths rather than supporting a continuous achromatic band. As metasurface manufacturing methods advance, techniques such as dispersion engineering are likely to become more viable for moderately higher aperture sizes and NA, though significant improvements in aspect ratio and tolerances will need to be made for it to ever become feasible for millimeter-scale devices or beyond. Advances in denoising algorithms and end-to-end optimization routines are also likely to benefit computational imaging meta-optical systems. Multi-element solutions represent a promising direction going forward, but this approach will need to contend with many of the same efficiency and stray light challenges that other multi-element metasurface systems currently struggle with. Designers, researchers, and other practitioners should be aware of the challenges and range of design options for improving or circumventing these chromatic effects. We anticipate the options for correcting chromatic aberrations will improve with time given ongoing investments and research in the meta-optics space.
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