## Abstract

Metalenses have shown great promise in their ability to function as ultracompact optical systems for focusing and imaging. Remarkably, several designs have been recently demonstrated that operate over a large range of frequencies with minimized chromatic aberrations, potentially paving the way for ultrathin achromatic optics. Here, we derive fundamental bandwidth limits that apply to broadband optical metalenses regardless of their implementation. Specifically, we discuss how the product between achievable time delay and bandwidth is limited in any time-invariant system, and we apply well-established bounds on this product to a general focusing system. We then show that all metalenses designed thus far obey the appropriate bandwidth limit. The derived physical bounds provide a useful metric to compare and assess the performance of different devices, and they offer fundamental insight into how to design better broadband metalenses.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

The field of metasurfaces holds the promise of a revolution in many areas of optics and photonics. In principle, any optical system may be made flat and compact by replacing the conventional optics with ultrathin devices, with great potential benefits in terms of size, cost, and ease of fabrication [1–3]. While metasurfaces can achieve arbitrary wavefront transformations and may even be used for optical wave-based computing [4,5], “metalenses” specifically designed for focusing and imaging represent one of the most important classes of metasurfaces for practical applications. One of the main challenges in this context is the realization of thin metalenses operating over a broad wavelength range, with minimized chromatic aberrations. In conventional optics, it is possible to stack various lenses to correct chromatic aberrations [6], but at the price of making the overall system more bulky and costly. It is therefore remarkable that, in recent years, different groups have demonstrated metalenses with a fixed focal length over a large range of wavelengths, some even surpassing the achromatic performance of conventional lens systems (at least for normal incidence) [7–22].

Metalenses achieve focusing by changing the phase of an incoming plane wave, with a phase profile that must vary radially according to the following equation [1] (for normally incident light):

where $\omega$, $F$, $r$, and $c$ are the angular frequency, focal length, radial coordinate, and speed of light, respectively. In the general case, a spatial- and frequency-dependent phase profile, as in Eq. (1), can be Taylor expanded around a central frequency ${\omega _c}$:Early examples of optical metalenses were based on deeply subwavelength
“meta-atoms” (e.g. plasmonic dipole nano-antennas) operating near a
scattering resonance to achieve the phase delay $\varphi (r,{\omega
_c})$ over the smallest possible thickness (a
fraction of a wavelength), without any considerations of the linear and
higher-order terms in Eq. (2). While this strategy allows realizing arbitrarily thin
metasurfaces, these devices were highly dispersive with large chromatic
aberrations, due to their inherent resonant nature, and could operate at
either only a single frequency or a discrete set of frequencies [1–3]. In contrast, the most recent designs
at the time of writing have employed relatively thicker meta-structures
(still on the order of a wavelength) that function essentially as
microscopic waveguide segments. The waveguiding approach does not depend
on the phase delay obtained through near-resonant light interaction with a
scatterer, but rather on the *true time delay*
obtained via guided-wave propagation, thus allowing significantly larger
bandwidths. Conceptually, this new approach is more similar to early
examples of flat lenses at microwave frequencies, e.g., [25], than to the first versions of modern optical
metasurfaces based on resonant meta-atoms.

While the results obtained in recent works on broadband metalenses are remarkable, here we argue that there exists a strict physical bound on the chromatic properties of a metalens, which stems from the fact that it is not possible to impart an arbitrary group delay to a signal independently of its bandwidth. Indeed, the delay-bandwidth product is limited in any linear, time-invariant system, as recognized in several works, [26–28], and is related to the thickness of the device. Based on this concept, in the following section, we derive fundamental limits on the bandwidth of achromatic metalenses, and assess the performance of various existing designs against these limits.

## 2. RESULTS

*Time-bandwidth products*—Some attempts at
identifying limits on the bandwidth of specific metalens designs have been
recently made. For example, Shrestha *et al.*
[10] have derived a bound on
metalens bandwidths based on the range of dispersion properties covered by
a meta-structure library, and Fathnan and Powell [29,30] have
derived bandwidth limits on low-frequency metasurfaces composed of
printed-circuit impedance sheets. Here, instead, we are interested in a
fundamental limit, applicable in general to any metalens, regardless of
its specific implementation. With this goal in mind, we turn to the
concept of delay-bandwidth, or time-bandwidth, product (TBP): the TBP of a
device (a cavity, waveguide, etc.) is the product of the time delay, or
interaction time, $\Delta T$, experienced by the signal, and the
signal bandwidth $\Delta \omega$. Wave physics imposes an upper bound on
this quantity, which can be generally written as

In order to apply the concept of TBP and the associated bounds to our problem, we treat a metasurface lens as composed of one-dimensional delay lines. More specifically, we consider a rotationally symmetric radial array, or continuum, of delay-line buffers, such that the incident wave is delayed as a function of radius, as illustrated in Fig. 1. According to Eq. (2), each delay-line buffer must impart a suitable group delay to compensate for the difference in the arrival times of wavepackets at the focus. The lens must also impart a frequency-independent phase pattern, $\varphi (r,{\omega _c})$, to create a spherical wavefront at the output. This phase pattern can be implemented independently of the group-delay requirement, using, for example, the concept of geometric phase, as shown in Ref. [7]. We would like to note that the 1D model in Fig. 1 is an approximation, since a metasurface with finite thickness is not strictly an array of 1D structures; however, we expect that light normally incident on metasurfaces of thicknesses on the order of one wavelength would acquire time delay predominantly through longitudinal propagation, with little propagation along the lateral (radial) direction. For this reason, in what follows, we consider only metasurfaces with thicknesses smaller than about five free-space wavelengths. We further discuss this approximation and its implications in Section 3.

Consider a metalens with radius $R$ and focal distance $F$. From Eq. (1) and Eq. (2), the required relative group delay imposed by the lens at a radial position $r \le R$ is

Then, using the numerical aperture (NA) definition

where ${n_b}$ is the background refractive index, and the identity $1 + \mathop {\tan}\nolimits^2 (\rm arcsin x) = 1/(1 - {x^2})$, we can use (3) and (6) to set a limit on the metalens’ bandwidth based on its NA and geometrical properties:We note that the form of this limit is consistent with the bandwidth bound derived in Ref. [29] based on different considerations (Foster’s reactance theorem) for the case of metasurfaces composed of impedance sheets. Our results, however, are more general and apply to any type of metalenses, as we discuss in the following.

As mentioned above, different values of the upper bound $\kappa$ have been derived for different general classes of devices. Thus, depending on the type of metasurface (whether it is based on resonant meta-atoms or waveguiding structures), we can apply the appropriate TBP bound and derive the relevant bandwidth limit. We now identify three relevant cases, covering most types of metalenses.

- (i) If the metasurface has deeply subwavelength thickness, the only way to impart the necessary phase/time delay to incoming light is by interaction with resonant scattering meta-atoms. In this case, coupled-mode theory provides a geometry- and material-independent TBP for a single resonator (e.g., see [32]):
In this case, Eq. (3) becomes an equality, and it may also be written in terms of the $Q$ factor of the resonant meta-atoms, $\Delta T = 2Q/{\omega _c}$, as recognized in Refs. [33,34]. While the following goes beyond the scope of the present work, we also note that this case could be generalized to account for more than one resonator, as in the case of overlapped electric and magnetic dipole resonances used in Huygens’ metasurfaces [29,35–37]. However, the bandwidth bounds are not expected to significantly improve. In addition, while electric-dipole resonators can be made infinitesimally thin (e.g., printed dipole antennas), a magnetic-dipole resonator always requires a non-zero thickness, since a non-zero volume is necessary to establish a circulation of (conduction or polarization) current supporting a non-zero magnetic-dipole moment [31].

- (ii) For thick metasurfaces based on inclusions acting as truncated waveguides, the previous limit clearly does not apply, since wave-guiding structures cannot be treated as individual resonators. Tucker
*et al*. [27] provide a generally applicable time-bandwidth limit, valid for any one-dimensional, lossless, dielectric device that may be treated as a waveguide (the limit is strictly valid only if the fractional bandwidth is not too large, i.e., smaller than unity). The value of the upper bound $\kappa$ is given by where ${n_{{\rm max}}}$ and ${n_{{\rm min}}}$ are the maximum and minimum*effective*refractive indices of the device. This effective index $n(\omega)$ is defined by the dispersion relation for the mode of interest in the structure, $\beta (\omega) = \omega n(\omega)/c$, as if the structure was homogeneous. $n(\omega)$, ${n_{{\rm max}}}$, and ${n_{{\rm min}}}$ are generally different from the material indices that make up the device. However, in the case of a one-dimensional dielectric waveguide segment (or coupled segments, as in Ref. [7]), which is the case of interest for most modern metalenses, we can take ${n_{{\rm max}}}$ and ${n_{{\rm min}}}$ as the refractive indices of the dielectric material composing the waveguide and of the surrounding medium, respectively, because the guided-mode dispersion converges to the light line of the low-index material at low frequency, and to the light line of the high-index material at high frequency. We thus replace the difference term in Eq. (10) with $\Delta n = {n_{{\rm max}}} - {n_b}.$ - (iii) Finally, Miller [26] provides a similar, but much more general, time-bandwidth limit that is valid for a very broad class of one-dimensional structures (not necessarily dielectric) acting as delay lines: where ${\eta _{\max}} = |({\varepsilon _{\max}} - {\varepsilon _b})/{\varepsilon _b}|$ is the device’s maximum contrast in relative permittivity, with respect to the surrounding medium’s permittivity ${\varepsilon _b}$, at any frequency within the band of interest and at any position within the structure. The limit is very general, as it is independent of the device design, and is not based on the simplifying assumptions used in Tucker’s limit (transparent materials and well-defined group velocity). The limit strictly applies if the device length $L \gg {\lambda _c}$, where ${\lambda _c}$ is the band-center wavelength in the background medium, and if the fractional bandwidth is not too large. However, in practice, we have verified that if $L$ is just a few times larger than the longest wavelength in the device, Miller’s limit seems to apply, i.e., it is consistent with Tucker’s limit (which has no assumptions on length). If they both apply, Miller’s limit is close to Tucker’s, and exceeds it to some degree if ${\varepsilon _{{\rm max}}}\, {{{{\gtrapprox}}}}\, 6$.

These bounds on the TBP, within their limits of applicability, may be combined with Eq. (8) to obtain an upper bound on the bandwidth of achromatic metalenses. Most types of metalenses, for any thickness (smaller than a few wavelengths) and material composition, are covered by the three TBP bounds outlined above, leading to the following bandwidth limits:

where we replaced ${\omega _c} = 2\pi c/{\lambda _c}$, andWe compared these bandwidth limits to various broadband metalens designs available in the literature. Comparisons are shown in Fig. 2, using suitably normalized quantities, and are tabulated in Table 1. What immediately stands out is that the limits correctly predict the expected performance trend: for a larger NA, the achievable bandwidth shrinks because the required maximum time delay, $\Delta {T_{{\rm max}}}$ in Eq. (6), rapidly increases (and diverges at ${\rm NA}/{n_b} = 1$).

Not surprisingly, as seen in Fig. 2(a), only a few thin, subwavelength, metalenses obey the bandwidth limit based on the single-resonator TBP given by Eq. (9). Instead, all the metalens designs obey our limits based on Tucker’s or Miller’s TBP [see Fig. 2(b) and Table 1], including recent ultrabroadband metalenses obtained using free-form all-area optimization and inverse design [13].

At this point, it is important to note that although all the considered designs claim achromatic performance, some have non-negligible focal length variations, or do not disclose the exact field profile at the focal plane and the associated level of monochromatic aberrations. The focal field profile is relevant because the bandwidth limit derivation above assumes a diffraction-limited lens with no aberrations, i.e., Strehl ratio $S = 1$. The Strehl ratio is a measure of the wavefront aberration, defined as the ratio of the peak focal spot intensity to the maximum attainable intensity of an ideal lens. A metalens that does not achieve diffraction-limited focusing with $S = 1$ across the nominal operational bandwidth is not implementing the phase and time-delay profiles assumed above exactly; hence, it may surpass the bounds since the requirements are somewhat relaxed.

To quantify the effect of aberrations on our bandwidth bounds, we assume that the aberrations are not too large, which is the scenario of interest for imaging applications. In this case, the Strehl ratio is approximately independent of the nature of the aberration, and according to the “extended Maréchal approximation,” it can be estimated from the variance of the wavefront deformation with respect to an ideal spherical wavefront:

where $\sigma$ is the standard deviation of the spatial wavefront deformation, and ${k_0}$ is the free-space wavenumber [6,38,39]. In addition, the maximum peak-to-peak deformation of the wavefront can be related to the standard deviation as $\Delta {W_{{\rm max}}} \approx \alpha \sigma$, where the factor $\alpha$ depends on the type of aberration, and for a mixture of low-order aberrations (defocusing, etc.), $\alpha$ is on the order of a few units (e.g., $\alpha \approx 4.5$ [38]). This spatial error corresponds to a maximum phase error $\Delta {\varphi _{{\rm max}}} = \Delta {W_{{\rm max}}}{k_0}$. Thus, if a less-than-unity Strehl ratio is tolerated within the operational bandwidth of the metalens, an error in the implemented phase profile would be acceptable, which in turn would relax the requirements on the time delay $\Delta T_{\text{max}}$ and the associated bandwidth bounds. In particular, assuming no phase errors at the central frequency, if the implemented time delay is incorrect by a maximum amount $\Delta {T_{{\rm err}}}$, Eq. (2) indicates that the phase profile would be incorrect by an amount $\Delta {\varphi _{{\rm max}}} = (\omega - {\omega _c})\Delta {T_{{\rm err}}}$ at a certain frequency $\omega$, corresponding to a Strehl ratio:Since we do not have access to the field profiles of all the considered metalenses, in Fig. 2(b) we include both the bound for ideal metalenses with no aberrations (lower solid blue curve), and a bound for highly aberrated metalenses with an error $\Delta {T_{{\rm err}}} = 0.9\Delta T_{\text{max}}$ (upper solid blue curve), corresponding to low values of the Strehl ratio according to Eq. (17). Most published metalens designs are below the bound for ideal metalenses, with only a handful of designs exceeding this limit. However, the latter are all bound by the limit for aberrated metalenses with $\Delta {T_{{\rm err}}} = 0.8\Delta T_{\text{max}}$ (dashed blue curve), corresponding to a typical Strehl ratio ${\lt}0.5$ away from the central wavelength, which is consistent with the published results (we note that since the nominal $\Delta T_{\text{max}}$ depends on $F$ and NA, according to Eq. (6), the resulting Strehl ratio also depends on these quantities). Thus, in principle, even broader bandwidths could be achievable, but only at the expense of even higher aberrations and lower focal spot intensity.

Finally, in Fig. 3, we show an example of how a specific metalens design (from Ref. [9]) compares with the bandwidth limits described above, considering the case of no aberrations for simplicity. This metasurface, which is based on dielectric waveguide segments, has a much wider bandwidth than what would be achievable using a single-resonator-based design, as expected. In addition, its bandwidth performance is not too far from the appropriate upper bound (either Eqs. (10) or (11)) based on the employed materials and thickness. In other words, the dielectric metalens is using its thickness and refractive-index contrast almost optimally. Figure 3 also shows a design-independent version of both Tucker’s and Miller’s limit using the highest refractive-index and permittivity contrast naturally available at optical frequencies, for lossless dielectrics and generic materials, respectively. Further details are discussed in Section 3.

*Bandwidth limits on reflection
suppression*—For the sake of completeness, we briefly discuss
another important trade-off, between the bandwidth of operation of a
metalens and its transmission efficiency. The ability to transmit energy
efficiently requires, at a minimum, that the reflections are minimized,
namely, that the metalens is impedance matched with respect to the medium
in which the incident wave propagates (usually air or a transparent
substrate). While it is always possible to design a lossless
anti-reflection coating to achieve identically zero reflection (ideal
impedance matching) at a single frequency, a fundamental trade-off exists
between the reflection reduction and the continuous bandwidth over which
this reduction can be achieved. This fundamental limit on broadband
impedance matching is known as the Bode–Fano limit [45], which has been used for decades in microwave
engineering, but it applies equally well at optical frequencies [46]. This bound depends uniquely on the
linearity, passivity, time invariance, and causality of the scattering
system, and, most importantly, is independent of the employed
anti-reflection coating, regardless of its complexity (the matching
structure is only assumed to be lossless).

In order to apply the Bode–Fano limit to the problem under consideration, we approximate the metalens as a thin homogeneous slab with a refractive index equal to the average refractive index ${n_{{\rm avg}}}$ of the materials composing the structure. This is clearly a coarse approximation, but it allows us to get some general insight on this relevant design trade-off. In addition, we assume that we operate in the most favorable condition for impedance matching, i.e., we assume that the central frequency corresponds to a Fabry–Perot resonance of the slab, at which the reflection coefficient automatically goes to zero. The slab thickness $L$ is assumed to be smaller than or comparable to the wavelength. Under these approximations, the limit is given by [46]

Depending on the application under consideration, the maximum bandwidth over which a metalens can operate depends on both the achromatic focusing limit derived above and the impedance-matching limit. Interestingly, it is immediately clear that Eq. (18) is inversely proportional to the thickness $L$ and permittivity contrast $\eta$ of the device, while Eqs. (13) and (14) are directly proportional to these quantities. This suggests the existence of a trade-off between the ability to reduce reflections with an anti-reflection coating (maximizing transmission efficiency) and the ability to minimize chromatic aberrations for a metasurface operating over a broad continuous bandwidth. This trade-off is represented in Fig. 4: thicker devices or larger refractive-index contrasts lead to wider bandwidths for achromatic operation (blue curves), but narrower bandwidths over which the reflection coefficient can be reduced to a certain level (orange curves). In other words, achieving achromatic performance over a wider band requires a larger $\eta L/{\lambda _c}$, which, however, increases the minimum reflectance achievable over that band, as expected. If both efficiency and achromatic performance are equally important, an optimal value of $\eta L/{\lambda _c}$ may be identified depending on the specific application under consideration.

## 3. DISCUSSION AND CONCLUSION

Considering the bandwidth limits on achromatic metalenses discussed above, one may wonder what type of metalens design, for a fixed refractive-index/permittivity contrast and thickness, can get closest to the limit and why.

Interestingly, for a certain refractive-index contrast, a metasurface
design based on suitable dielectric waveguide segments seems to directly
provide a way to realize performance close to the upper bound for the
given thickness. Indeed, the guided-mode dispersion of a dielectric
waveguide converges to the light line of the low-index material at low
frequency, and to the light line of the high-index material at high
frequency. This provides an intermediate frequency window with low group
velocity and locally linear dispersion that is automatically close to the
optimal linear dispersion considered by Tucker *et
al.* [27] for an ideal
delay line for that level of contrast.

It is therefore not surprising that, even when considering free-form
all-area optimization of dielectric metalenses as in Ref. [13], the optimization tends to create a spatial
distribution of material with “channels” that resemble waveguide segments.
It is also not surprising that many of the designs we considered are
relatively close to the limit, as shown in Fig. 2(b), since many make use of the available length and
refractive-index contrast almost optimally. Using the largest naturally
available refractive index for a transparent material at optical
frequency, which is around three to four in silicon and germanium, would
certainly provide a wider bandwidth, but not an order-of-magnitude
improvement with respect to metalenses fabricated with lower values of
refractive index. Fig. 3 (purple
dashed curve) shows the bandwidth limit for this maximum value of lossless
refractive index, $n = 4$, compared to the bandwidth of the
metalens in Ref. [9]. Such a
bandwidth limit provides a design-independent upper bound for transparent
dielectric metasurfaces, which depends only on the thickness and the
desired focal length and NA. We also note that certain recently studied
materials, such as phase-change chalcogenides [47], have been shown to exhibit very large
refractive indices over broad bandwidths, which could be promising in the
context of achromatic metasurfaces; however, their non-negligible
absorption losses will unavoidably deteriorate the performance of the
device. Since it is unlikely that a much larger refractive-index contrast
could be achieved with *lossless* materials at
optical frequencies, the only way to improve the bandwidth performance
using transparent materials is to consider longer devices, or overcome the
limit by breaking its main assumptions, for example, time invariance, a
possibility that will be the subject of future works.

Considering much longer metalenses may also break the assumption of
one-dimensionality on which the limits above are based (see Fig. 1 and related discussion). An example of
this is the broadband metalens in Ref. [22], designed through free-form all-area optimization, whose
thickness is more than five free-space wavelengths (and even longer
considering the wavelength within the metasurface structure). Thus, this
metalens cannot be considered an array of one-dimensional delay lines as
in Fig. 1, since lateral
propagation can no longer be neglected. Indeed, this thick metasurface
manages to surpass our bounds to some degree, with relatively small
aberrations. In general, we expect that thicker metasurfaces or a stack of
metasurfaces, with a thickness of several wavelengths, may be designed to
optimally take advantage of the two- or three-dimensionality of the
system, increasing the path a wavepacket travels laterally, not just
longitudinally, which would in turn lead to wider achievable bandwidths.
In this context, we believe that all-area optimization is critical to
fully take advantage of the whole available *volume*.

In addition, one may also wonder whether it would be possible to
artificially increase the maximum available refractive index by realizing
an engineered metamaterial with an effective index much larger than the
one of the constituent materials. However, if the thickness of the
metalens is limited to approximately a wavelength or few wavelengths, the
meta-atoms must be very small to actually form an effective homogeneous
metamaterial, and not act as a discrete arrangement of elements. If we
choose, for example, the size of the meta-atoms to be $d \approx \lambda
/10$, a dielectric meta-atom would be largely
off-resonance even considering the largest refractive index, $n \approx 4$ (the first resonance of a high-index
dielectric sphere is of magnetic dipolar type, and it occurs when $d \approx \lambda
/n$). As a result, the effective permittivity
would not be too different from the average between the permittivity of
the inclusions and of the background, following standard mixing formulas
for non-resonant meta-atoms (e.g., see [48]). Using plasmonic materials would allow realizing deeply
subwavelength resonant meta-atoms and, therefore, a metamaterial with much
larger effective permittivity. This would, however, be accompanied by
large Lorentzian dispersion around an unavoidable absorption peak, which
would greatly reduce the bandwidth and efficiency of the device. Still,
Miller’s limit in Eq. (14),
which is based on the *magnitude* of the
permittivity contrast, does not preclude the possibility of achieving
better bandwidth performance at optical frequencies by using metallic
materials, for which the contrast can be as high as ${\eta _{\max}} \approx
100$ at near-infrared frequencies. In theory,
this would allow an order-of-magnitude improvement in bandwidth, as seen
in the inset of Fig. 3 (orange
curve), where we show Miller’s limit for ${\eta _{\max}} \approx
100$. This provides an ultimate upper bound on
the bandwidth of optical metalenses that may include any possible
material. However, there is no guarantee that this limit is tight, namely,
that it could be achieved with a physical design [26].

We also note that the derived bounds apply to the class of phase profiles given by Eq. (1) (plus a frequency-dependent reference phase term). Interestingly, it was recently shown that, if the phase profile is allowed to be freely optimized, an inverse-designed diffractive lens may achieve very wide bandwidths, with a focusing performance that is adequate for conventional imaging applications [49]. The analysis of this different type of thin lenses will be the subject of future works. Finally, we expect that the use of post-processing and, more generally, computational imaging techniques could enable broadband imaging even if the metalens itself does not perform achromatic focusing. This is demonstrated, for example, in Refs. [50,51], using extended depth of focus metalenses and computational reconstruction. In this context, our bounds would be crucial to assess whether a dispersion engineered metasurface is sufficient to achieve the desired bandwidth for the considered application, or whether post-processing would be beneficial or necessary.

To conclude, we believe that the fundamental bandwidth limits presented in this paper will prove useful to the many research groups working on metasurfaces to assess and compare the performance of different devices, and may offer fundamental insight into how to design broadband achromatic metalenses for different applications.

## Funding

National Science Foundation (1741694); Air Force Office of Scientific Research (FA9550-19-1-0043).

## Disclosures

The authors declare no conflicts of interest.

## REFERENCES

**1. **M. Khorasaninejad and F. Capasso, “Metalenses: versatile
multifunctional photonic components,”
Science **358**, eaam8100
(2017). [CrossRef]

**2. **M. L. Tseng, H.-H. Hsiao, C. H. Chu, M. K. Chen, G. Sun, A.-Q. Liu, and D. P. Tsai, “Metalenses: Advances and
applications,” Adv. Opt. Mater. **6**, 1800554
(2018). [CrossRef]

**3. **P. Lalanne and P. Chavel, “Metalenses at visible
wavelengths: past, present, perspectives,”
Laser Photon. Rev. **11**,
1600295 (2017). [CrossRef]

**4. **A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical
operations with metamaterials,”
Science **343**,
160–163 (2014). [CrossRef]

**5. **A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using
reflective plasmonic metasurfaces,” Nano
Lett. **15**,
791–797 (2015). [CrossRef]

**6. **M. Born and E. Wolf, *Principles of Optics*,
6th ed. (Pergamon,
1980).

**7. **W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic
metalens for focusing and imaging in the visible,”
Nat. Nanotechnol. **13**,
220–226 (2018). [CrossRef]

**8. **W. T. Chen, A. Y. Zhu, J. Sisler, Z. Bharwani, and F. Capasso, “A broadband achromatic
polarization-insensitive metalens consisting of anisotropic
nanostructures,” Nat. Commun. **10**, 1–7
(2019). [CrossRef]

**9. **M. Khorasaninejad, Z. Shi, A. Y. Zhu, W. T. Chen, V. Sanjeev, A. Zaidi, and F. Capasso, “Achromatic metalens over
60 nm bandwidth in the visible and metalens with reverse chromatic
dispersion,” Nano Lett. **17**, 1819–1824
(2017). [CrossRef]

**10. **S. Shrestha, A. C. Overvig, M. Lu, A. Stein, and N. Yu, “Broadband achromatic
dielectric metalenses,” Light: Sci.
Appl. **7**, 85
(2018). [CrossRef]

**11. **M. Ye, V. Ray, and Y. S. Yi, “Achromatic flat subwavelength
grating lens over whole visible bandwidths,”
IEEE Photon. Technol. Lett. **30**,
955–958 (2018). [CrossRef]

**12. **R. J. Lin, V.-C. Su, S. Wang, M. K. Chen, T. L. Chung, Y. H. Chen, H. Y. Kuo, J.-W. Chen, J. Chen, Y.-T. Huang, J.-H. Wang, C. H. Chu, P. Chieh Wu, T. Li, Z. Wang, S. Zhu, and D. P. Tsai, “Achromatic metalens array for
full-colour light-field imaging,” Nat.
Nanotechnol. **14**,
227–231 (2019). [CrossRef]

**13. **H. Chung and O. D. Miller, “High-NA, achromatic metalenses
by inverse design,” Opt. Express **28**, 6945–6965
(2019). [CrossRef]

**14. **N. Mohammad, M. Meem, P. Wang, and R. Menon, “Broadband imaging with one
planar diffractive lens,” Sci. Rep. **8**, 2799 (2018). [CrossRef]

**15. **S. Banerji, M. Meem, A. Majumder, F. G. Vasquez, B. Sensale-Rodriguez, and R. Menon, “Imaging with flat optics:
metalenses or diffractive lenses?”
Optica **6**,
805–810 (2019). [CrossRef]

**16. **S. Zhang, A. Soibel, S. Keo, D. Wilson, S. Rafol, D. Z. Ting, A. She, S. D. Gunapala, and F. Capasso, “Solid-immersion metalenses for
infrared focal plane arrays,” Appl. Phys.
Lett. **113**, 111104
(2018). [CrossRef]

**17. **S. Wang, P. C. Wu, V.-C. Su, Y.-C. Lai, M.-K. Chen, H. Y. Kuo, B. H. Chen, Y. H. Chen, T.-T. Huang, J.-H. Wang, R.-M. Lin, C.-H. Kuan, T. Li, Z. Wang, S. Zhu, and D. P. Tsai, “A broadband achromatic
metalens in the visible,” Nat.
Nanotechnol. **13**,
227–232 (2018). [CrossRef]

**18. **F. Balli, M. A. Sultan, S. K. Lami, and J. T. Hastings, “A hybrid achromatic
metalens,” arXiv:1909.07941
(2019).

**19. **B. Yu, J. Wen, X. Chen, and D. Zhang, “An achromatic metalens in the
near-infrared region with an array based on a single nano-rod
unit,” Appl. Phys. Express **12**, 092003 (2019). [CrossRef]

**20. **Q. Cheng, M. Ma, D. Yu, Z. Shen, J. Xie, J. Wang, N. Xu, H. Guo, W. Hu, S. Wang, T. Li, and S. Zhuang, “Broadband achromatic metalens
in terahertz regime,” Sci. Bull. **64**, 1525–1531
(2019). [CrossRef]

**21. **S. Wang, P. C. Wu, V.-C. Su, Y.-C. Lai, C. Hung Chu, J.-W. Chen, S.-H. Lu, J. Chen, B. Xu, C.-H. Kuan, T. Li, S. Zhu, and D. P. Tsai, “Broadband achromatic optical
metasurface devices,” Nat. Commun. **8**, 187 (2017). [CrossRef]

**22. **Z. Lin and S. G. Johnson, “Overlapping domains for
topology optimization of large-area metasurfaces,”
Opt. Express **27**,
32445–32453 (2019). [CrossRef]

**23. **D. Werdehausen, S. Burger, I. Staude, T. Pertsch, and M. Decker, “General design formalism for
highly efficient flat optics for broadband
applications,” Opt. Express **28**, 6452–6468
(2020). [CrossRef]

**24. **A. Ozer, N. Yilmaz, H. Kocer, and H. Kurt, “Polarization-insensitive beam
splitters using all-dielectric phase gradient metasurfaces at visible
wavelengths,” Opt. Lett. **43**, 4350–4353
(2018). [CrossRef]

**25. **D. M. Pozar, “Flat lens antenna concept
using aperture coupled microstrip patches,”
Electron. Lett. **32**,
2109–2111 (1996). [CrossRef]

**26. **D. A. B. Miller, “Fundamental limit to linear
one-dimensional slow light structures,” Phys.
Rev. Lett. **99**, 203903
(2007). [CrossRef]

**27. **R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers:
capabilities and fundamental limitations,” J.
Lightwave Technol. **23**,
4046–4066 (2005). [CrossRef]

**28. **J. Khurgin, “Bandwidth limitation in slow
light schemes,” in *Slow Light: Science and
Applications*, J. Khurgin and R. S. Tucker, eds. (Taylor & Francis
Group, 2008), chap. 15,
pp. 293–320.

**29. **A. A. Fathnan and D. A. Powell, “Bandwidth and size limits of
achromatic printed-circuit metasurfaces,” Opt.
Express **26**,
29440–29450
(2018). [CrossRef]

**30. **A. A. Fathnan, A. E. Olk, and D. A. Powell, “Broadband anomalous reflection
with dispersion controlled metasurfaces,”
arXiv:1912.03936 (2019).

**31. **J. D. Jackson, *Classical Electrodynamics*,
3rd ed. (Wiley,
1999).

**32. **S. A. Mann, D. L. Sounas, and A. Alù, “Nonreciprocal cavities and the
time-bandwidth limit,” Optica **6**, 104–110
(2019). [CrossRef]

**33. **H. Liang, A. Martins, B.-H. V. Borges, J. Zhou, E. R. Martins, J. Li, and T. F. Krauss, “High performance metalenses:
numerical aperture, aberrations, chromaticity, and
trade-offs,” Optica **6**, 1461–1470
(2019). [CrossRef]

**34. **E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “Controlling the sign of
chromatic dispersion in diffractive optics with dielectric
metasurfaces,” Optica **4**, 625–632
(2017). [CrossRef]

**35. **C. Pfeiffer and A. Grbic, “Metamaterial Huygens’
surfaces: tailoring wave fronts with reflectionless
sheets,” Phys. Rev. Lett. **110**, 197401
(2013). [CrossRef]

**36. **F. Monticone, N. M. Estakhri, and A. Alù, “Full control of nanoscale
optical transmission with a composite metascreen,”
Phys. Rev. Lett. **110**,
203903 (2013). [CrossRef]

**37. **A. Epstein and G. V. Eleftheriades, “Huygens’ metasurfaces via the
equivalence principle: design and applications,”
J. Opt. Soc. Am. B **33**,
A31–A50 (2016). [CrossRef]

**38. **J. C. Wyant and K. Creath, “Basic wavefront aberration
theory for optical metrology,” in *Applied
Optics and Optical Engineering, Volume XI*, R. R. Shannon and J. C. Wyant, eds.
(Academic, 1992),
Vol. 11.

**39. **J. W. Hardy, “Adaptive optics for
astronomical telescopes,” in *Adaptive Optics
for Astronomical Telescopes* (Oxford
University, 1998), chap.
4,
p. 104–134.

**40. **F. Aieta, P. Genevet, M. Kats, and F. Capasso, “Aberrations of flat lenses and
aplanatic metasurfaces,” Opt. Express **21**, 31530–31539
(2013). [CrossRef]

**41. **T. Siefke, S. Kroker, K. Pfeiffer, O. Puffky, K. Dietrich, D. Franta, I. Ohlídal, A. Szeghalmi, E.-B. Kley, and A. Tünnermann, “Materials pushing the
application limits of wire grid polarizers further into the deep
ultraviolet spectral range,” Adv. Opt.
Mater. **4**,
1780–1786
(2016). [CrossRef]

**42. **R. C. Devlin, M. Khorasaninejad, W. T. Chen, J. Oh, and F. Capasso, “Broadband high-efficiency
dielectric metasurfaces for the visible spectrum,”
Proc. Natl. Acad. Sci. USA **113**,
10473–10478 (2016). [CrossRef]

**43. **E. D. Palik, *Handbook of Optical Constants of
Solids* (Academic,
1998), Vol. 3.

**44. **R. Ferrini, M. Patrini, and S. Franchi, “Optical functions from 0.02 to
6 eV of AlxGa1-xSb/GaSb epitaxial layers,” J.
Appl. Phys. **84**,
4517–4524 (1998). [CrossRef]

**45. **R. Fano, “Theoretical limitations on the
broadband matching of arbitrary impedances,”
J. Franklin Inst. **249**,
57–83 (1950). [CrossRef]

**46. **F. Monticone and A. Alù, “Invisibility exposed: physical
bounds on passive cloaking,” Optica **3**, 718–724
(2016). [CrossRef]

**47. **S. Abdollahramezani, O. Hemmatyar, H. Taghinejad, A. Krasnok, Y. Kiarashinejad, M. Zandehshahvar, A. Alu, and A. Adibi, “Tunable nanophotonics enabled
by chalcogenide phase-change materials,”
arXiv:2001.06335 (2020).

**48. **S. A. Tretyakov, *Analytical Modeling in Applied
Electromagnetics* (Artech
House, 2003).

**49. **J. M. Meem, S. Banerji, A. Majumder, J. C. Garcia, P. W. C. Hon, B. Sensale-Rodriquez, and R. Menon, “Imaging from the visible to
the longwave infrared wavelengths via an inverse-designed flat
lens,” https://arxiv.org/abs/2001.03684
(2020).

**50. **S. Colburn, A. Zhan, and A. Majumdar, “Metasurface optics for
full-color computational imaging,” Sci.
Adv. **4**, eaar2114
(2018). [CrossRef]

**51. **L. Huang, J. Whitehead, S. Colburn, and A. Majumdar, “Design and analysis of
extended depth of focus metalenses for achromatic computational
imaging,” arXiv:2003.09599
(2020).