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 [13]. 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) [722].

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):

$$\varphi (r,\omega) = - \frac{\omega}{c}\left({\sqrt {{F^2} + {r^2}} - F} \right),$$
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 ${\omega _c}$:
$$\begin{split}\varphi (r,\omega) &= \varphi (r,{\omega _c}) + (\omega - {\omega _c}){\left. {\frac{{\partial \varphi (r,\omega)}}{{\partial \omega}}} \right|_{\omega = {\omega _c}}} \\ &\quad +\frac{1}{2}{(\omega - {\omega _c})^2}{\left. {\frac{{{\partial ^2}\varphi (r,\omega)}}{{\partial {\omega ^2}}}} \right|_{\omega = {\omega _c}}} + \ldots ,\end{split}$$
where the latter two terms are the group delay and the group-delay dispersion. As discussed by Chen et al. [7], when the focal length $F$ is frequency independent, Eq. (2) contains no higher-order terms than the linear one. Thus, to realize a perfectly achromatic metalens, working for both broadband pulses and incoherent light, the design should implement (i) a suitable frequency-independent phase pattern $\varphi (r,{\omega _c})$, (ii) a spatial pattern of group delay, and (iii) zero group-delay dispersion and higher-order terms. We note that the inclusion of an additional phase term in Eq. (1), a generic function of frequency but independent of position, would not influence the rest of our discussion. Its effect would be equivalent to adding a generic dispersive phase shifter in front of the lens, which might distort incident pulses, but would not prevent focusing (only the spatially varying terms affect focusing and imaging). We also note that while here we focus on metalenses for normal incidence, similar considerations apply to different phase functions in other metasurface devices (e.g., broadband beam deflection and splitting [23,24]).

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 [13]. 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, [2628], 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

$$\Delta T \Delta \omega \le \kappa ,$$
where $\kappa$ is a dimensionless quantity. Bounds on the TBP have been studied extensively in the field of slow light [2628]. In particular, as discussed below, different bounds have been derived in the literature under different assumptions, but $\kappa$ always depends on some general properties of the device, for example, its length and refractive-index contrast of the materials involved.

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.

 

Fig. 1. Delay-line model of a thin broadband metalens. Radially arranged delay lines provide a broadband signal the necessary group delay $\Delta T(r) $ to compensate for the difference in arrival times at the focus, while the phase pattern $\varphi (r) $ creates a spherical wavefront according to Eqs. (1) and (2).

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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

$$\Delta T(r) = \frac{{\partial \varphi}}{{\partial \omega}}(r) - \frac{{\partial \varphi}}{{\partial \omega}}(R)$$
$$= \frac{1}{c}\left({\sqrt {{F^2} + {R^2}} - \sqrt {{F^2} + {r^2}}} \right).$$
(In most cases, the group delay is equal to the actual time delay experienced by the signal, except in the presence of anomalous dispersion near resonances, which is, however, not a case of interest in this context, due to the strong absorption that unavoidably accompanies anomalous dispersion in passive systems [31].) The greatest delay is required at the center ($r = 0$) to compensate for the additional time taken by a signal arriving from the edge ($r = R$), such that the metalens design must achieve
$$\Delta {T_{{\rm max}}} = \frac{F}{c}\left({\sqrt {1 + {{(R/F)}^2}} - 1} \right).$$
This defines our required time delay in the TBP in Eq. (3).

Then, using the numerical aperture (NA) definition

$${\rm NA} = {n_b}\sin \theta = {n_b}\sin \left[{\rm arctan \left({\frac{R}{F}} \right)} \right],$$
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:
$$\Delta \omega \le \frac{{\kappa c}}{{F\left({\sqrt {1 + {{(R/F)}^2}} - 1} \right)}} = \frac{{\kappa c\sqrt {1 - {{({\rm NA}/{n_b})}^2}}}}{{F\left({1 - \sqrt {1 - {{({\rm NA}/{n_b})}^2}}} \right)}}.$$

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]):
    $$\kappa = 2.$$

    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 [29,3537]. 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
    $$\kappa = 2\pi \frac{L}{{{\lambda _c}}}({n_{{\rm max}}} - {n_{{\rm min}}}),$$
    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:
    $$\kappa = \frac{\pi}{{\sqrt 3}}\frac{L}{{{\lambda _c}}}{\eta _{\max}},$$
    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:

  • (i) for ultra-thin metalenses based on resonant meta-atoms (single resonance) [from (9)],
    $$\Delta \omega \le \frac{{2c}}{F}\Theta \left({\frac{{{\rm NA}}}{{{n_b}}}} \right),$$
  • (ii) for waveguide-based dielectric transparent metalenses [from (10)],
    $$\Delta \omega \le {\omega _c}\frac{{L\Delta n}}{F}\Theta \left({\frac{{{\rm NA}}}{{{n_b}}}} \right),$$
  • (iii) and for generic metalenses (not necessarily dielectric and lossless) of thickness larger than the wavelength [from (11)],
    $$\Delta \omega \le \frac{{{\omega _c}}}{{2\sqrt 3}}\frac{{L{\eta _{{\rm max}}}}}{F}\Theta \left({\frac{{{\rm NA}}}{{{n_b}}}} \right),$$
where we replaced ${\omega _c} = 2\pi c/{\lambda _c}$, and
$$\Theta \left({\frac{{{\rm NA}}}{{{n_b}}}} \right) = \frac{{\sqrt {1 - {{({\rm NA}/{n_b})}^2}}}}{{1 - \sqrt {1 - {{({\rm NA}/{n_b})}^2}}}}.$$

We 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 ${\rm NA}/{n_b} = 1$).

 

Fig. 2. Comparisons of published achromatic metalens designs against the derived bandwidth limits. (a) Limit based on the single-resonator TBP bound, given by Eq. (12). Not surprisingly, most data points exceed this bound except for some thin devices. (b) Limit based on Tucker’s TBP bound, given by Eq. (13). Each data point in the plots represent a single design, with each label corresponding to a specific row in Table 1. The performance of each metalens is represented in terms of numerical aperture and bandwidth. In order to compare vastly different designs against the bandwidth bounds, the bandwidths are normalized by $c$, $F$, and the corresponding $\kappa$ [see Eq. (8)]. In both panels, the lowest blue curve represents the function $\Theta ({\rm NA}/{n_b}) = c/F\Delta T_{\text{max}}$, where $\Delta T_{\text{max}}$ is the required time delay for ideal operation, given by Eq. (6). Fig. 2(b) includes both the upper bound for ideal metalenses with no aberrations, $\Delta {T_{{\rm err}}} = 0$ ($\Theta$, lower solid blue curve), and the bound for highly aberrated metalenses with an error $\Delta {T_{{\rm err}}} = 0.8\Delta T_{\text{max}}$ ($\Theta /0.2$, dashed blue curve) and $\Delta {T_{{\rm err}}} = 0.9\Delta T_{\text{max}}$ ($\Theta /0.1$, upper solid blue curve), which correspond to low values of the Strehl ratio according to Eq. (17). All design parameters and bandwidth values are given in Table 1.

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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:

$$S \approx {{\rm e}^{- {{\left({{k_0}\sigma} \right)}^2}}},$$
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:
$$S \approx \exp \left({- {{\left({{\alpha ^{- 1}}(\omega - {\omega _c})\Delta {T_{{\rm err}}}} \right)}^2}} \right).$$
Based on these considerations, it is then possible to approximately account for aberrations in our bandwidth bounds by substituting $\Delta T_{\text{max}}$ with $\Delta T_{\text{max}} - \Delta {T_{{\rm err}}}$ in Eq. (3) and expressing $\Delta {T_{{\rm err}}}$ in terms of $S$ by inverting Eq. (17). This leads to a looser bandwidth bound if the Strehl ratio decreases, suggesting that a relaxation of the imaging performance of the metalens allows for a broader bandwidth, as expected. We also stress that an imaging system is considered practically diffraction limited if $S \gt 0.8$ [6,3840], which implies that the bandwidth bound may be relaxed, to some degree, with respect to the ideal case, with only minimal deterioration of the imaging performance.
Tables Icon

Table 1. Summary of Design Parameters and Performance Values of Broadband Achromatic Metalenses in the Literaturea

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.

 

Fig. 3. Different bandwidth limits compared to the performance of a specific metalens design, from Ref. [9] (central wavelength ${\lambda _c} = 518$ nm): single-resonator limit [Eq. (9), dotted-dashed green curve]; Tucker’s limit [Eq. (10), dashed black], and Miller’s limit [Eq. (11), solid red] using the refractive-index/permittivity contrast considered in Ref. [9]; design-independent Tucker’s limit (dashed purple) with the highest refractive index for lossless dielectrics naturally available at optical frequency, $n \approx 4$. The inset includes the same curves and an additional curve (solid orange) for a design-independent version of Miller’s limit with the highest permittivity contrast (in magnitude) available at optical frequency $\eta \approx 100$ [26] (which may include the case of metallic materials as well as materials with loss and gain).

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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]

$$\frac{{\Delta \omega}}{{{\omega _c}}} \le \frac{1}{{L/{\lambda _c}(\varepsilon - {\varepsilon _b})}}{\left[{\log \frac{1}{{|\Gamma |}}} \right]^{- 1}},$$
where $\varepsilon = n_{{\rm avg}}^2$, and $\Gamma$ is the in-band reflection coefficient. If the equal sign is used, Eq. (18) represents the optimal trade-off between bandwidth and reflection reduction.
 

Fig. 4. Comparison of the bandwidth limit for achromatic performance (blue), based on Miller’s TBP, Eq. (14), and the Bode–Fano bandwidth limit on reflection reduction, Eq. (18) (orange), as a function of the product of permittivity contrast and normalized thickness: $\eta L/{\lambda _c}$. The limits are compared for various values of NA and in-band reflection coefficient $|\Gamma |$. As an example, we considered a dielectric metalens with $F = 100{\lambda _c}$ and ${\varepsilon _b} = 1$. In order to apply the Bode–Fano limit to the considered problem and compare it with Miller’s limit, we treat the metasurface as a homogeneous slab with averaged refractive index, and a permittivity contrast $\eta = \varepsilon - {\varepsilon _b} = {\eta _{{\rm max}}}$ (further details in the text). For a given value of NA and $|\Gamma |$, there is an optimal value of $\eta L/{\lambda _c}$, where the two limits intersect, that maximizes $\Delta \omega$.

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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).

References

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  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).

2020 (1)

2019 (9)

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

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]

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2017 (5)

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).
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2016 (4)

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C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110, 197401 (2013).
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2007 (1)

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2005 (1)

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Tucker, R. S.

Tünnermann, A.

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]

Vasquez, F. G.

Wang, J.

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]

Wang, J.-H.

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]

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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]

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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).
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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).
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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]

Yu, D.

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

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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]

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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).

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

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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]

Zhang, S.

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]

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Zhu, A. Y.

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]

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]

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).
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[Crossref]

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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).
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Figures (4)

Fig. 1.
Fig. 1. Delay-line model of a thin broadband metalens. Radially arranged delay lines provide a broadband signal the necessary group delay $\Delta T(r) $ to compensate for the difference in arrival times at the focus, while the phase pattern $\varphi (r) $ creates a spherical wavefront according to Eqs. (1) and (2).
Fig. 2.
Fig. 2. Comparisons of published achromatic metalens designs against the derived bandwidth limits. (a) Limit based on the single-resonator TBP bound, given by Eq. (12). Not surprisingly, most data points exceed this bound except for some thin devices. (b) Limit based on Tucker’s TBP bound, given by Eq. (13). Each data point in the plots represent a single design, with each label corresponding to a specific row in Table 1. The performance of each metalens is represented in terms of numerical aperture and bandwidth. In order to compare vastly different designs against the bandwidth bounds, the bandwidths are normalized by $c$, $F$, and the corresponding $\kappa$ [see Eq. (8)]. In both panels, the lowest blue curve represents the function $\Theta ({\rm NA}/{n_b}) = c/F\Delta T_{\text{max}}$, where $\Delta T_{\text{max}}$ is the required time delay for ideal operation, given by Eq. (6). Fig. 2(b) includes both the upper bound for ideal metalenses with no aberrations, $\Delta {T_{{\rm err}}} = 0$ ($\Theta$, lower solid blue curve), and the bound for highly aberrated metalenses with an error $\Delta {T_{{\rm err}}} = 0.8\Delta T_{\text{max}}$ ($\Theta /0.2$, dashed blue curve) and $\Delta {T_{{\rm err}}} = 0.9\Delta T_{\text{max}}$ ($\Theta /0.1$, upper solid blue curve), which correspond to low values of the Strehl ratio according to Eq. (17). All design parameters and bandwidth values are given in Table 1.
Fig. 3.
Fig. 3. Different bandwidth limits compared to the performance of a specific metalens design, from Ref. [9] (central wavelength ${\lambda _c} = 518$ nm): single-resonator limit [Eq. (9), dotted-dashed green curve]; Tucker’s limit [Eq. (10), dashed black], and Miller’s limit [Eq. (11), solid red] using the refractive-index/permittivity contrast considered in Ref. [9]; design-independent Tucker’s limit (dashed purple) with the highest refractive index for lossless dielectrics naturally available at optical frequency, $n \approx 4$. The inset includes the same curves and an additional curve (solid orange) for a design-independent version of Miller’s limit with the highest permittivity contrast (in magnitude) available at optical frequency $\eta \approx 100$ [26] (which may include the case of metallic materials as well as materials with loss and gain).
Fig. 4.
Fig. 4. Comparison of the bandwidth limit for achromatic performance (blue), based on Miller’s TBP, Eq. (14), and the Bode–Fano bandwidth limit on reflection reduction, Eq. (18) (orange), as a function of the product of permittivity contrast and normalized thickness: $\eta L/{\lambda _c}$. The limits are compared for various values of NA and in-band reflection coefficient $|\Gamma |$. As an example, we considered a dielectric metalens with $F = 100{\lambda _c}$ and ${\varepsilon _b} = 1$. In order to apply the Bode–Fano limit to the considered problem and compare it with Miller’s limit, we treat the metasurface as a homogeneous slab with averaged refractive index, and a permittivity contrast $\eta = \varepsilon - {\varepsilon _b} = {\eta _{{\rm max}}}$ (further details in the text). For a given value of NA and $|\Gamma |$, there is an optimal value of $\eta L/{\lambda _c}$, where the two limits intersect, that maximizes $\Delta \omega$.

Tables (1)

Tables Icon

Table 1. Summary of Design Parameters and Performance Values of Broadband Achromatic Metalenses in the Literaturea

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

φ ( r , ω ) = ω c ( F 2 + r 2 F ) ,
φ ( r , ω ) = φ ( r , ω c ) + ( ω ω c ) φ ( r , ω ) ω | ω = ω c + 1 2 ( ω ω c ) 2 2 φ ( r , ω ) ω 2 | ω = ω c + ,
Δ T Δ ω κ ,
Δ T ( r ) = φ ω ( r ) φ ω ( R )
= 1 c ( F 2 + R 2 F 2 + r 2 ) .
Δ T m a x = F c ( 1 + ( R / F ) 2 1 ) .
N A = n b sin θ = n b sin [ a r c t a n ( R F ) ] ,
Δ ω κ c F ( 1 + ( R / F ) 2 1 ) = κ c 1 ( N A / n b ) 2 F ( 1 1 ( N A / n b ) 2 ) .
κ = 2.
κ = 2 π L λ c ( n m a x n m i n ) ,
κ = π 3 L λ c η max ,
Δ ω 2 c F Θ ( N A n b ) ,
Δ ω ω c L Δ n F Θ ( N A n b ) ,
Δ ω ω c 2 3 L η m a x F Θ ( N A n b ) ,
Θ ( N A n b ) = 1 ( N A / n b ) 2 1 1 ( N A / n b ) 2 .
S e ( k 0 σ ) 2 ,
S exp ( ( α 1 ( ω ω c ) Δ T e r r ) 2 ) .
Δ ω ω c 1 L / λ c ( ε ε b ) [ log 1 | Γ | ] 1 ,

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