1. INTRODUCTION

Over the years, there has been a lot of interest in flat lenses, a category including two competing technologies: diffractive lenses [1–4] and the “newcomer”—metalenses [5–13]. These lenses provide a compact and cost-effective solution that may replace conventional multi-element refractive lens designs for some applications, such as microscope objectives and cellphone camera lenses, and are thus of great interest to the community at large. A major obstacle in the path toward realizing the potential of such lenses is the large chromatic aberration associated with conventional diffractive lenses (CDLs), characteristic of diffraction gratings on which they are based. Metalenses are a sub-category of diffractive lenses, so they exhibit the same basic dispersion. To meet this challenge, achromatic metalenses (AMLs) [14–21] and achromatic diffractive lenses (ADLs) [22–25] have been developed. While these designs have shown achromatic behavior, they have not so far shown much utility for practical applications. This raises a question regarding their potential to provide the performance needed to meet the requirements of modern optical systems.

In a recent paper, the upper limit of achromatic metalens performance was explored [26]. It was shown that there is an inverse relation between the bandwidth over which the AML operates and the time delay it can provide. The time delay represents the “work” the lens is doing. We will show that the time delay is related to the Fresnel number of the lens, which will be defined in Section 2. In contrast to this, ADLs seem to suffer no such limitation. Recent papers have reported ADLs that operate over an extremely large wavelength bandwidth [27,28]. If so, what are the limitations of ADLs? This question has been raised by several authors [26–28], but to our knowledge has not been properly answered yet. In this mini-review we provide an answer to this question. Specifically, we show that ADLs suffer from a severe limitation, preventing them from providing near-diffraction-limited performance at high numerical aperture (NA) and broadband illumination. On the other hand, at low NA they can indeed cover a very large spectral band. We also discuss the limits of the spectral band that can be covered. Note that in this paper we focus on continuous spectrum applications, and not on discrete wavelength applications; see further discussion in Section 6.

Based on our analysis we compare the competing technologies of AMLs and ADLs. We show that ADLs have an advantage for low-NA applications with wide spectral range. AMLs on the other hand have an advantage for high-NA applications with modest spectral band. Both technologies still cannot provide near-diffraction-limited performance at high NA and wide spectral band, which is required, for example, for such applications as cellphone cameras and microscope objectives.

If what we say is true, how do we explain the many reports of moderate or even high-NA ADLs with near-diffraction-limited performance [27–32]? The answer is that the real limitation is not on NA but rather on the Fresnel number (FN), as will be explained. For most practical applications, the lens dimensions (aperture/focal length) should be orders of magnitude larger than the wavelength (e.g., 1 mm focal length and 1 µm wavelength). In these cases, the limitation of low FN translates to low NA. The reports mentioned above either have very small dimensions or provide resolution which is much lower than the diffraction limit, but mask this fact by presenting the resolution in terms of the full width at half-maximum (FWHM) instead of more relevant criteria for evaluating imaging lenses such as modulation transfer function (MTF) or Strehl ratio.

Based on our analysis one can gain significant insight into the physical mechanisms driving the different flat lens solutions, providing tools that can help find the best solution for a given application, and hopefully generate new knowledge based on novel solutions.

It is important to note that for imaging applications aberration correction over a field of view (FOV) is needed. In this paper we limit our discussion to single-element flat lens systems and their chromatic aberration limitation. It has been shown that flat lenses of moderate NA ($\sim 0.2$) and fairly wide FOV ($\lt \pm 40^{\circ}$) can be used over narrow spectral bands, by using a removed aperture stop [1,33]. If chromatic correction were achieved, the same method could be used for broadband applications.

The limitations discussed in this paper do not apply to a diffractive lens or metalens integrated into a refractive lens system (i.e., a hybrid lens system). Here the large chromatic aberration of the flat lens is being used to advantage, to correct the chromatic aberration of the refractive optics, so there is no need to achromatize the flat lens. Hybrid systems allow for a reduced number of elements in the optical system, i.e., improved compactness and cost. However, the “holy grail” of the field of flat lenses is a single-element system, that can be mass-produced at low cost compared to compound refractive lens systems. This is the focus of our paper.

2. ACHROMATIC METALENS PERFORMANCE LIMIT

We begin our discussion by briefly introducing the published analysis regarding metalens performance limits and expressing their results in terms of Fresnel number. There are three main methods used to create metalenses: truncated waveguide nanopillars, geometrical phase nanofins, and Huygens nanodisks [13]. We will focus on the truncated waveguide type, constructed of nanopillars, since this is the most commonly used method for achromatic metalens design.

Compensating for phase variations between the different wavelengths, so that they all have the same phase, will produce good efficiency for all wavelengths, but will not correct chromatic aberration. This is because for a diffractive optical element the local period is determined by the spatial phase gradient. The local period in turn determines the location of the diffraction orders via the grating equation, while the internal period profile determines only the amount of energy going to each order [34]. However, since the grating equation includes a wavelength dependence, different wavelengths going to the same diffraction order will be deflected to different angles, hence the chromatic aberration. To correct chromatic aberration, the location of the first order for the different wavelengths must be manipulated so that they coincide, meaning the effective local phase period (defined as the spatial range for which the phase is ramped from 0 to ${2}\pi$) must be different for different wavelengths. It turns out that to obtain achromatic focusing it is necessary to obtain the same group delay $\Delta T$, or optical path difference (OPD), for the different wavelengths [19,26]. This can be achieved by optimizing the dispersion of the nanostructure, such that despite of the physical structure being identical for all wavelengths, each wavelength will see a different phase profile.

The phase is related to the OPD and the time delay according to $\varphi = \frac{{2\pi}}{\lambda} \cdot {\rm OPD} = \frac{\omega}{c} \cdot {\rm OPD} = \omega \Delta T$, where $\lambda$ is the wavelength of light, $\omega$ is its angular frequency, and $c$ is the speed of light in vacuum. Therefore, achieving constant time delay necessitates obtaining phase which varies linearly with frequency over a range of frequencies. This is of course in addition to the requirement of inducing spherical phase at the nominal wavelength to achieve focusing, as is illustrated in Fig. 1(a). The larger the OPD or time delay that the lens must provide over its aperture, the larger the slope of the linear variation must be. This is illustrated in Fig. 1(b). To achieve this, one must engineer the nanopillar shape, so that it introduces such a frequency dependence. This is known as dispersion engineering. The phase induced by the truncated waveguide nanopillar is given by $\varphi (\omega) = \frac{\omega}{c}{n_{{\rm eff}}}(\omega)h$, where ${n_{{\rm eff}}}$ is the waveguide effective refractive index, and $h$ is the nanopillar height, as illustrated in Fig. 2. Therefore, the degree of freedom used to achieve the necessary large phase dispersion is a variation of the effective refractive index with frequency.

 

Fig. 1. (a) Illustration of achromatic metalens operation, which introduces a radially varying phase $\varphi$ at the nominal wavelength, and in addition must provide a wavelength-independent group delay, which is equal to $d\varphi /d\omega$. Reproduced from [26]. (b) Left panel shows the radial dependence of the phase function, for three wavelengths, designated by the RGB colored curves. The phase dispersion needed for three radial aperture points is shown in the right panel. It can be seen that the maximum phase dispersion slope increases as the nominal wavelength OPD increases. Reproduced from [19].

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Fig. 2. Dispersion-engineered waveguide. The dispersion is obtained by engineering the shape of the waveguide (which can be non-cylindrical and may include voids) so that the effective refractive index of the propagating mode varies widely depending on the wavelength/frequency of the light. ${\varphi _1}$ is the phase induced at wavelength ${\lambda _1}$, where the waveguide effective index is ${n_{\rm eff1}}$, and equivalently for ${\lambda _2}$.

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Shrestha et al. [19] set an upper limit for the bandwidth of a dispersion-engineered AML that depends on the maximum dispersion range that can be obtained with a specific nanoantenna library. Presutti and Monticone [26] generalized this limit by stating that the maximum refractive index dispersion range obtainable with any library will be between zero (i.e., effective refractive index independent of frequency) and ${\Delta}n = {n_a} - {n_b}$ (i.e., the effective refractive index is equal to that of the bulk nanoantenna material, ${n_a}$, for the highest frequency, and to the background index, ${n_b}$, for the lowest frequency). This results in Tucker’s upper limit for the time-delay-bandwidth product [35]:

As can be seen from Eq. (3), the FN of a lens is related to the maximum OPD provided by the lens ($d$), modulo ${\lambda _0}/2$ (or maximum phase provided by the lens, modulo $\pi$), which is the number of Fresnel zones in an equivalent Fresnel zone plate; see Fig. 3. The FN is, therefore, a good measure of how much “work” the lens is doing. We can express the maximum bandwidth of the lens in terms of the FN by substituting Eq. (2) into Eq. (1). Following this we can reverse the equation to express the maximum achievable FN for a given bandwidth. This results in Eq. (4) where $p = {h / {{\lambda _0}}}$ is the normalized height of the nanostructure, in units of the central wavelength ${\lambda _0}$. The expression on the right is obtained under the assumption that $\Delta \lambda \ll \lambda$. However, even if this is not the case, the correction factor is usually small (unless we have a very broad spectral range, such as those discussed later for ADLs, but as we will see, AMLs are not suitable for such spectral ranges). The larger the relative spectral range, the lower the maximum FN that can be achieved. In contrast, the larger the nanostructure height and its refractive index contrast with respect to its surrounding, the larger the FN that can be achieved:

 

Fig. 3. Fresnel number (FN) is the number of Fresnel zones in a lens. In the paraxial approximation this is equal to the maximum OPD, $d$, that the lens provides in units of half-wavelength.

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3. ACHROMATIC DIFFRACTIVE LENS LIMIT

Does the above delay-bandwidth limit apply also to ADLs? Not necessarily so. This is despite the fact that the propagation of light in ADLs is longitudinal (within the paraxial approximation) and not lateral, so the conditions mentioned in [26] seem to be unviolated. The difference originates from the different physical mechanisms that are used for the purpose of chromatic correction. In the case of a dispersion-engineered AML, the achromatic behavior is based on the dispersive nature of the truncated waveguides. Therefore, the analogy to the world of optical communications and delay lines is relevant. In the dispersion-engineered AML, all wavelengths are focused to the same diffraction order, and the dispersion allows the local period to vary, such that the dispersion in diffraction angle is compensated by increasing the local period with the increase in wavelength (see Fig. S6 in [16] and Fig. S7 in [37]). However, in the case of a multi-order ADL which is discussed next in our paper, the achromatic behavior is based on the variation in energy distribution among the various diffraction orders for different wavelengths, caused by the overall microscopic surface structure, rather than by a local effect related to a single point on the aperture. So, in practice, the chromaticity is compensated by the fact that different wavelengths are routed to different diffraction orders. This effect is NOT accounted for in the delay line analogy. These differences are illustrated in Fig. 4.

 

Fig. 4. (a) Conventional diffractive lens, operating at the first order of diffraction, showing associated longitudinal chromatic aberration $\Delta f$. (b) Multi-order diffractive (MOD) lens, operating around the $m$th order of diffraction, showing the ability to focus several harmonic wavelengths at the same point. Both types of lenses have a radially varying grating period $d$, which is independent of wavelength. (c) Achromatic metalens, which implements achromatic behavior at the first order of diffraction, by using high dispersion nanopillars, that effectively implement a different local period $d$ for different wavelengths, thus allowing them to come to focus at the same point.

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We now consider the case of a multi-order diffractive (MOD) lens [22,23], which is the forerunner of modern inverse-designed ADLs [38]. An MOD lens is a diffractive lens that is designed to operate at a high diffraction order $m$ for the nominal design wavelength. In such a case, the harmonic wavelengths $\lambda$, given by Eq. (5), will focus with high efficiency at the same point as the design wavelength ${\lambda _0}$, but at different diffraction orders $k$. This is illustrated in Fig. 4(b) for a case of three harmonic wavelengths, which focus at the nominal design order $m$ (green wavelength in the illustration), and the two neighboring orders $m - 1$ and $m + 1$ (red and blue wavelengths respectively), all located at the same plane. For discussion of diffraction efficiency and how to simulate and measure it, see Section 2 of Supplement 1.

An MOD lens cannot be perfectly achromatic over a spectral range, like an AML, but rather only at discrete wavelengths. However, it can still be called an achromatic lens, in the same way that a classical achromatic doublet is called so, despite it being corrected only at two discrete wavelengths, with secondary chromatic aberration in-between. From an optical designer’s point of view, it makes no difference if an aberration is perfectly corrected, or if it is corrected only at discrete points, but the residual aberration is smaller than the diffraction limit.

An MOD, like an AML, creates the same time delay for the different harmonic frequencies. This is achieved by applying a phase function that varies linearly with frequency, as does the AML [19]. However, as mentioned before, the change in the phase as a function of frequency is not provided by structural dispersion of the truncated waveguides, but rather by structural dispersion of the diffractive surface. This dispersion causes different harmonic frequencies to focus with optimal efficiency at different diffraction orders, according to Eq. (5) [22]. The phase function varies linearly with the order of diffraction $k$, and will therefore vary linearly with frequency, per the right-hand side of Eq. (5), where $\omega$ is the angular frequency associated with the wavelength $\lambda$. Thus, the same time delay at discrete resonant frequencies is achieved. This is described by Eq. (6), where ${\varphi _0}$ is the nominal phase function (for design wavelength ${\lambda _0}$ at design order $m$), and $\varphi$ is the phase function “seen” by a wavelength $\lambda$ (corresponding to frequency $\omega$) operating at order $k$:

What happens in between harmonic wavelengths? The energy of these non-harmonic wavelengths will be mostly divided between the two closest diffraction orders, so there is minimal loss of efficiency. However, the focus of these wavelengths will be shifted compared to that of the harmonic wavelengths. The maximum relative focal shift $\Delta f/f$ is given by Eq. (7) [22], where $\Delta {\lambda _{{\rm har}}} = {\lambda _0} - {{m{\lambda _0}} / {({m + 1})}} = {{{\lambda _0}} / {({m + 1})}}$ is the wavelength difference between neighboring harmonic wavelengths ${\lambda _0}$ and $\lambda$ of neighboring order $k = m + 1$, based on Eq. (5):

What limits the performance of an MOD lens is this residual chromatic focal shift. For the lens to be diffraction limited the maximum focal shift must be smaller than the diffraction-limited depth of focus. For the maximum focal shift, we will use half of $\Delta f$ given by Eq. (7), since half-way to the nearest resonant wavelength, more than 50% of the power will already shift to the nearby resonance. So, while there will be some power that is defocused by the full $\Delta f$, most of the power is only defocused by a maximum of $\Delta f/2$. The diffraction-limit condition is therefore given by Eq. (8), where the right-hand side is the diffraction-limited depth of focus [39]. This criterion is not intended to be exact, but rather to give an estimate of the maximum FN up to which we can expect to obtain near-diffraction-limited performance:

Substituting Eq. (3) into Eq. (8), we obtain the upper limit on the Fresnel number of a diffraction-limited MOD lens:

The design order $m$ is related to the zone depth $h$ (this is the feature height, analogous to the truncated waveguide height in an AML) by $m = {{h\Delta n} / {{\lambda _0}}} = p\Delta n$. Substituting this into Eq. (9) we obtain the following condition, which is analogous to the condition of Eq. (4) for metalenses:

The remarkable thing about this condition is that it is independent of spectral range. The only limit on the achievable FN is the zone depth and refractive index contrast. This is because for a high-order MOD lens (e.g.,  $m\sim{10}$), the chromatic blur is limited by the distance between neighboring harmonic wavelengths $\Delta {\lambda _{{\rm har}}}$, rather than by the overall spectral range. Note that this limit is only relevant to an MOD lens, and not to a CDL operating at a single order of diffraction. In the case of a CDL we typically operate over a spectral range smaller than $\Delta {\lambda _{{\rm har}}}$, thus achieving near-diffraction-limited performance at FN higher than the limit given by Eq. (10), but this will be a quasi-monochromatic mode of operation [40]. The case of a CDL is analyzed in Section 4.

The limit given by Eq. (10) is based on the chromatic aberration near the nominal wavelength ${\lambda _0}$ at the diffraction order $m$. For larger wavelengths, operating at lower diffraction order ($k \ll m$), the aberration will be larger [in Eq. (7), $m$ is replaced by $k$]. However, the diffraction-limited spot size will also be larger, which more than compensates for the increased aberration. For shorter wavelengths ($k \gg m$), the situation is reversed. The limit for wavelengths far from the nominal (both longer and shorter) comes out to be ${\rm FN }\lt m + {\lambda / {{\lambda _0}}}$ (where the FN is still given at the nominal wavelength ${\lambda _0}$). For large wavelengths satisfying ${\lambda / {{\lambda _0} \gt 1}}$, the maximal FN is now higher than that obtained from Eq. (10), so this is not a limiting factor. For shorter wavelengths satisfying ${\lambda / {{\lambda _0} \lt 1}}$, the limit can go as low as ${\rm FN} \lt m$. Assuming sufficiently large $m$, this is a negligible change which can be ignored.

Note that in our analysis we neglected material dispersion effects, therefore considering only diffractive chromatic aberration as described by Eq. (7). This is justified by the fact that in this paper we are dealing with “flat lenses,” i.e., small zone depth (of the order of a few micrometers), or low $m$. See Section 4 of Supplement 1 for additional discussion of high-order MOD lenses.

Equation (9) is based on the assumption that the lens is diffractive, hence its chromatic behavior is governed by Eq. (7). Under this assumption there will always be some diffractive chromatic aberration. However, in fact, if ${\rm FN }\lt 2m$ the lens has only one zone, i.e., it stops being a diffractive lens and becomes a refractive lens. This is because per Eq. (3) the FN is equal to the maximum OPD of the lens modulo ${\lambda _0}/2$, while $m$ is the zone depth in units of OPD modulo ${\lambda _0}$. This effect is demonstrated in the phase profiles shown in Fig. S1. In this case there is no chromatic aberration (other than that arising from material dispersion which we neglect). We must therefore update our condition on the maximum Fresnel number that can give diffraction-limited performance. While Eq. (10) is still valid for $m\; \lt \;{1}$ (which is the case if the nominal zone phase depth is less than ${2}\pi$), for larger values of $m$ the limit is now given by

It is worthwhile to note that in principle one can design a metalens to work as an MOD lens (i.e., operate around a high-diffraction order $m$). However, using current manufacturing technology, pushing for a high-order metalens is extremely challenging, mostly because of the large nanopillar aspect ratio required. In any case, the distinction we are making is not between metalenses and diffractive lenses per se, but between lenses where the chromatic correction is based on dispersion engineering of the nanostructures (necessarily metalenses), and lenses where the chromatic correction is based on multi-order operation (typically diffractive lenses but could also be a metalens). We will call the first type AML and the second type MOD/ADL.

If the maximum FN is not affected by the spectral range, what limits the spectral range an MOD can operate over? Let us look first at the upper limit for the wavelength. The maximum wavelength for which we can obtain 100% efficiency is that in which the MOD operates at the first order of diffraction, ${\lambda _1} = m{\lambda _0}$. At longer wavelengths the efficiency drops according to [34]

If we allow up to 50% drop in efficiency as the criterion for maximum wavelength, we obtain

Naively, one would expect that the maximal wavelength could be enormous. For example, for an MOD with $m = {100}$ and ${\lambda _0} = {1.5}\;{\unicode{x00B5}{\rm m}}$, the maximal wavelength is in the terahertz range. However, this is impractical for a couple of reasons. First, it is challenging to find materials with high transparency over such a broad spectral band. Furthermore, for such a high value of $m$, the scalar expression for diffraction efficiency [Eq. (12)] breaks down, and lower efficiencies are obtained [41]. In addition, a very high $m$ ($m\; \gt \;V$, where $V$ is the material Abbe number) leads to a semi-refractive lens, where material dispersion effects become dominant [42] (see Section 4 of Supplement 1). Lastly, a very high $m$ lens may no longer merit the description of being “flat.” In practice, moderate values of $m$ should be chosen for reasons of efficiency and manufacturability, and thus the maximal wavelength is expected to be limited to 1 order of magnitude larger than the design wavelength.

Now we move to the lower wavelength limit. Here we are limited by transverse sampling resolution. To obtain reasonable efficiency we must have at least two phase levels for each ${2}\pi$ phase induced in the shortest wavelength (this will result in 40.5% efficiency going to the desired order based on the scalar approximation [41]). If we can manufacture $N$ phase levels, this means [22]

If $\Lambda$ is the minimum transverse feature which can be manufactured by our machine, and assuming equally spaced phase levels, we obtain $N = a/\Lambda$, where $a$ is the minimal period. For an MOD lens with a certain NA for an infinitely distant object (at the nominal wavelength), $a$ is given by the grating equation [34]: $a = m{\lambda _0}/{\rm NA}$. Substituting the last two expressions into Eq. (14) results in

Incidentally, this is the same minimum wavelength that would be obtained for first-order operation at a given NA and transverse resolution of $\Lambda$, according to the Nyquist criterion [3,33].

The above limits were determined based on an MOD lens. Do they apply to an inverse-designed ADL? An inverse-designed ADL is designed numerically rather than analytically, and its aperture is divided into rings, whose heights are optimization variables. These heights are constrained to discrete values in the range of zero to the designated maximum depth $h$. The merit function reflects the efficiency and resolution in some manner, which are then maximized by the optimization algorithm [29]. This type of design method is described by Mait [43] as “direct design,” since the merit function represents the requested functionality (small spot size and large energetic efficiency). This is different from the design of an MOD, where the aim is to implement an ideal phase function, so it is an example of “indirect design.” In more recent papers these design methods are called inverse design and forward design, respectively. A conceptual illustration of the surface structure of an MOD lens and that of an inverse-designed ADL is shown in Fig. 5.

 

Fig. 5. MOD lens surface structure and conceptual illustration of ADL surface structure, for a maximum structure height matching ${ m} = {3}$. (a) 3D MOD lens profile. (b) Radial cross section of (a). (c) ADL lens surface, with random profile for illustration purposes, showing rings of constant width, with discrete phase values. (d) Radial cross section of (c).

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The possible advantage of “direct-design” methods is their ability to allow additional degrees of freedom. For example, in the case of a diffractive optical element used for a beam shaping application, the phase at the image plane, the amplitude outside a certain area of interest, and the efficiency (the last two are related in some way), can be used as free parameters [44]. The question is, for the case of an ADL, are there any additional degrees of freedom that can be used to improve upon the performance of an equivalent MOD? In an imaging lens, unlike beam shaping applications, the phase at the output cannot be considered a degree of freedom, since at the focal plane the beam is at a waist, so the phase is constant, aside from $\pi$ phase jumps between focal spot sidelobes. Amplitude and efficiency cannot be considered degrees of freedom, since the typical ADL should provide not only small spot size, but also the highest possible efficiency.

A degree of freedom utilized in AMLs is different phase offset for different frequencies [19]. Is this relevant for direct-designed ADLs? We think not, since as opposed to AMLs where a high-dispersion nanopillar at the on-axis point of the lens can provide the required phase piston, in an ADL there is no local dispersion mechanism that can control the phase piston.

The degree of freedom that direct optimization can allow is finding an optimal trade-off between different performance parameters, e.g.,  resolution and efficiency. While an MOD lens has near 100% efficiency, an ADL can possibly find a more optimal working point by trading-off resolution versus efficiency. However, since an improvement in resolution beyond that of a classical MOD will come at the expense of efficiency, we believe that at least as a rule of thumb, our resolution-based upper limit is relevant also for ADLs.

To validate the applicability of our proposed limit to both MODs and ADLs, we performed an empirical study based on published results. The results are presented in Section 5. This is the same methodology used by Presutti and Monticone [26]. They too derived a limit for forward-designed AMLs, based on an ideal hyperbolic phase function. They then proposed that this limit may be applicable to inverse-designed AMLs as well, and empirically showed compliance with their upper limit in a single case of an inverse-designed AML. While we acknowledge that we have not proven the applicability of our MOD-based limit to ADLs in general, we believe that it is a reasonable conjecture, based on the above discussion. We pose a further challenge to the community to fabricate and properly evaluate an ADL that exceeds this limit and will be happy to admit defeat in such a case, for the benefit of scientific and technological advance.

4. CHROMATIC FLAT LENS LIMIT

Having established upper limits for AMLs and ADLs, we can now assess their potential for improvement over a CDL. The chromatic focal shift of a CDL is given by [34]

Unlike the case of an ADL, here $\Delta \lambda$ is the full desired bandwidth, rather than the difference between two neighboring harmonic wavelengths. The actual longitudinal chromatic aberration [39] will be half of this shift since the CDL is optimized for the center wavelength ${\lambda _0}$. To obtain diffraction-limited performance we require that the longitudinal aberration be smaller than the depth of focus, as in Eq. (8). This results in

Comparing this to Eq. (4), we can see that they are similar, but the AML has an additional factor, related to the dispersion of the truncated waveguide. Therefore, the AML FN upper limit can be greater than that of the CDL, if nanoantenna height and index contrast are sufficiently large. Comparing this limit to that of an ADL, we see a major difference in that the ADL limit is independent of spectral range and is limited only by the profile height and index contrast. This ADL behavior is only relevant for spectral ranges broader than that of two neighboring harmonic wavelengths. For narrower spectral ranges it will function as a CDL, which can allow near-diffraction-limited performance at high Fresnel numbers for very narrow spectral ranges.

5. VALIDATION

To validate our ADL upper performance limit, we first compare it to a simulation of MOD lens performance, and then to a survey of published ADL designs. In Fig. 6 we show simulation results for MOD lenses with nominal design wavelength of 550 nm, spectral range 400–700 nm (with uniform spectral weighting), focal length 1 mm, and three numerical apertures: 0.05, 0.1, and 0.2. This simulation is repeated for two cases of zone depth: $m = {3}$ [Figs. 6(a)–6(c)] and $m = {6}$ [Figs. 6(d)–6(f)]. The first case corresponds to a maximum structure depth of 2.6 µm when manufactured in material with refractive index $n = {1.65}$. These parameters are used in several inverse-designed ADLs [25,27,30]. The second case corresponds to a double depth, which can provide three (or even four) point wavelength correction (RGB) in the visible band [Fig. 5(d)].

 

Fig. 6. MOD simulation results for $f = {1}\;{\rm mm}$ and 400–700 nm spectral range. (a) Focal shift and efficiency as a function of wavelength, for the various diffraction orders $k$, for design order $m = {3}$. The focal shift is only drawn for wavelengths with a diffraction efficiency greater than 0.5. The horizontal dashed–dotted lines represent the depth of focus for the three NAs simulated, for comparison with the focal shift. (b) PSFs for the three NAs, compared to the diffraction limit. Notice the increased sidelobes compared to the diffraction limit, but the unchanged FWHM. The 1D Strehl ratios are also noted. (c) MTFs corresponding to PSFs, with FNs compared to the maximum FN for which diffraction-limited performance can be obtained. (d)–(f) The same as panels (a)–(c), for design order $m = {6}$.

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In Figs. 6(a) and 6(d), the focal shift is shown as a function of wavelength, together with the efficiency of the relevant diffraction orders as a function of wavelength (the focal shift is shown only for wavelength points whose efficiency is greater than 0.5). Dashed–dotted lines corresponding to the depth of focus for the three different numerical apertures are also plotted. Based on this we can already expect near-diffraction-limited performance for NA = 0.05 and zone depth matching $m = {3}$, since the pink defocus line extends only slightly outside of the blue depth-of-focus lines, corresponding to this NA, in Fig. 6(a). However, for the higher NAs the behavior is far from diffraction limited. Based on the same consideration, for $m = {6}$ [Fig. 6(d)] we can expect diffraction-limited performance for NA = 0.05, near-diffraction-limited performance for NA = 0.1, but far from diffraction limit for NA = 0.2.

These expectations are confirmed by the corresponding simulated MTFs shown in Figs. 6(c) and 6(f) (for method of simulation, see Section 3 of Supplement 1). In these panels we also show the comparison between the maximum FN at which diffraction-limited performance can be obtained, calculated according to Eq. (11), and the actual FNs of the lenses at the three NAs. Diffraction-limited performance is obtained when ${\rm FN}\; \lt \;{{\rm FN}_{{\max}}}$ [NA 0.05 MTF in Figs. 6(c) and 6(f)]. In an intermediate case, where the FN is larger than ${{\rm FN}_{{\max}}}$ by a significant factor, but less than an order of magnitude, there is a significant drop in MTF but not devastating [NA 0.1 MTF curves in Figs. 6(c) and 6(f)]. Finally, when the FN is about an order of magnitude larger than ${{\rm FN}_{{\max}}}$, the resulting MTF is very low compared to the diffraction limit [NA 0.2 MTF curves in Figs. 6(c) and 6(f)]. For all cases we chose a nominal design wavelength of 550 nm. We checked and found that a different choice of design wavelength (e.g.,  600 nm) would affect the MTFs only slightly. This shows that choice of design wavelength has only a mild effect even for the case of low design order $m$ and is not expected to change our rule-of-thumb upper limit.

In Figs. 6(b) and 6(e) we show the point spread functions (PSFs) corresponding to the MTFs of Figs. 6(c) and 6(f). The PSFs are normalized so that their peak value is 1. For our analysis we use the one-dimensional (1D) version of the Strehl ratio, which is equal to the ratio of the line-spread-function (LSF) peak to that of the diffraction-limited LSF peak. This is as opposed to the standard two-dimensional (2D) Strehl ratio, which uses the PSF peaks [45,46]. While the 1D Strehl ratios, corresponding to the ratio of the area under the actual MTF to the area under the diffraction-limited MTF, drop significantly for the MTFs that are far from the diffraction limit (the 2D Strehl ratios are even lower than the 1D), the FWHM for all cases is nearly identical for the actual and the diffraction-limited spot. This shows why it is incorrect to assess performance based on FWHM, since the drop in MTF is a result of the increased sidelobes, and not of broadening of the central spot.

It is interesting to note that while doubling the feature height does give improved performance [panel (f) versus panel (c) in Fig. 6], the more dominant effect is that of the numerical aperture (differences between red, green, and blue MTF curves in each of the panels). The reason for this is that the FN is proportional to the square of the NA [Eq. (3)], while the FN upper limit varies only linearly with feature height [Eq. (11)].

We now move on to a survey of published ADL designs. The survey is not meant to be exhaustive, so it does not include all ADLs demonstrated so far. However, we believe it is more than sufficient to provide an understanding of the underlying physics and design considerations. The results of our survey are presented in Table 1. The table compares the following: FN (actual lens Fresnel number) to ${{\rm FN}_{{\max}}}$ (maximum diffraction-limited Fresnel number, based on Eqs. (10) or (11)—the larger of the two), ${\lambda _{{\min}}}$ (actual design minimum wavelength) to “${\lambda _{{\min}}}$ lim.” [minimum wavelength calculated according to Eq. (15)], and ${\lambda _{{\max}}}$ (actual design maximum wavelength) to “${\lambda _{{\max}}}$ lim.” [maximum wavelength calculated according to Eq. (13)]. These column headings are marked with olive green background and bold letters. The other columns give the rest of the design data.

Table 1. Published Design Parameters Compared to Parameter Limits Presented in This Paper

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The first two designs are MOD lenses. They were manufactured by diamond turning, so the effective feature size and number of phase levels are unknown. These parameters affect only the short wavelength limit, so we simply chose reasonable values (marked in orange). The rest of the designs are inverse-designed ADLs. The shown efficiency results are the reported simulated results, except for design no. 2, where the efficiency is a measured value, and is mostly due to silicon absorption in the operating spectrum. The values given represent average values over the relevant spectral range, based on the published results. Of course, the exact definition of the efficiency may vary slightly between publications (see Section 2 of Supplement 1). At any rate, the efficiency is not the main parameter of interest to us since we are looking at the trade-off between FN and resolution. The main point regarding efficiency is that it is quite high for all the examples.

Let us first look at the operating wavelength ranges. Almost all the designs are within the theoretical limits, or very nearly so. Only designs 12 and 13, which attempt to push the wavelength limits to obtain a very broad spectral range, significantly exceed the limits (cells marked in pink in rows 12 and 13). However, it is important to remember that these limits are not hard limits. They express the fact that beyond these wavelengths the efficiency is expected to be low. For the short wavelength limit of design 12, a drop in efficiency is indeed reported. For the long wavelength limits, the reported simulated efficiency is still high. If we take the example of design 13, the maximum “zone depth” in terms of phase for the longest wavelength (150 µm) is $2\pi /\lambda {\cdot}h{\cdot}\Delta n = 1.2\;\rm rad$. Even an ideal diffractive lens designed for first-order operation at this wavelength but limited to this zone depth would give an efficiency of less than 5%, as a result of the large detuning factor ($\alpha = 1.2/(2\pi)=0.19$). How can this be reconciled with the very high ($\gt\! 90\%$) simulated efficiency [27]? This remains to be seen.

We now look at the Fresnel numbers of the reported designs. Most of the designs exceed the theoretical limit, some by a small amount, but others by a large amount; these are marked in pink (in the “FN” and “${{\rm FN}_{{\max}}}$” columns). The theoretical limit reflects the maximum FN at which the lens can be diffraction limited, similar to the delay-bandwidth limit given in [26] for AMLs. Therefore, it should not come as a surprise that one can work at higher FNs. What we expect to see, however, is that at these higher FNs, the resolution will degrade accordingly (assuming relatively high efficiency is maintained, which is the case for all the designs in our survey). Unfortunately, most published resolution data relates only to FWHM, which is not an effective measure of resolution, as explained previously in this section. Fortunately, there are a few publications that do provide MTF data. To make this data meaningful for our purposes, we compared the simulated/measured MTF data (the data was manually adapted from published data, so it is not very accurate, but sufficient to demonstrate the principle) to the diffraction-limited MTF, which we simulated based on the provided lens parameters. From the examples shown in Fig. 7, corresponding to designs 1, 2, 8, and 11 of Table 1, it is clear that while it is possible to design ADLs whose FN exceeds the upper limit, the farther it is from the upper limit, the lower the resolution with respect to the diffraction limit. The Strehl ratio shown in the figure is again the one-dimensional Strehl ratio, which is the ratio of the pink-shaded area under the actual MTF to the area under the diffraction-limited MTF.

 

Fig. 7. (a)–(d) MTFs of ADL designs 1, 2, 8, and 11, respectively, compared to the diffraction-limited MTF. For each graph, the actual design FN is shown, compared to the ${{\rm FN}_{{\max}}}$ upper limit presented in this paper. The larger the FN relative to the upper limit, the lower the resolution relative to the diffraction limit.

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Based on Eq. (11) it would seem that there is no such thing as a diffraction-limited MOD lens. If the FN is smaller than ${{\rm FN}_{{\max}}}$ defined by Eq. (11) the lens is no longer diffractive, but rather refractive (an exception to this is if we work with small zone depth, so that $m\; \lt \;{1}$; however, this is not an MOD lens, but rather a CDL, and the efficiency will be compromised). Despite this, we can see that there are several designs in Table 1 (2, 3, and 5) where the FN is smaller than 2m, and nevertheless they are still presented as diffractive lenses. In the case of design 2, it is still an MOD lens, since it uses a landscape lens design, i.e., a removed aperture stop. Therefore, while the on-axis beam “sees” only a single zone, the off-axis beams use additional zones, as shown in Fig. 2 of [24]. Designs 3 and 5 are not MODs but rather inverse-designed ADLs. Since they do not have removed stops, and their FN is smaller than ${{\rm FN}_{{\max}}}$, they could have been implemented simply as a refractive lens, without need for inverse design. This is demonstrated in Section 5 of Supplement 1 for the case of design 3.

6. DISCUSSION

The upper limit on the Fresnel number for the different types of flat lenses may seem like a binary-type threshold, but in fact it provides much more than that. We have seen that a design can exceed this limit, but this will come at the expense of reduced Strehl ratio. Furthermore, the more the design exceeds the limit, the more severe the degradation in resolution, i.e., the lower the Strehl ratio. In short, there is a trade-off between flat lens focusing power (i.e., Fresnel number) and Strehl ratio.

For the case of an MOD, we can quantify this trade-off by simulating the Strehl ratio for different Fresnel numbers and maximum feature height (proportional to $m$). The result for the case of a lens operating over the visible spectrum (400–700 nm) is shown in Fig. 8(a). The zone phase depth divided by ${2}\pi$ ($x$-axis label) is equal to the design order $m$. As can be seen, the Strehl ratio increases for lower FNs and larger feature height. Not surprisingly, these trends reflect lower number of diffractive zones in the lens, and thus the MOD lens is becoming more refractive-like.

 

Fig. 8. (a), (b) 1D Strehl ratio of an MOD lens, $f = {1}\;{\rm mm}$, over the spectral range 400–700 nm, as a function of zone depth and Fresnel number (panel a), and zone depth and numerical aperture (panel b). (c), (d) Same as (a) and (b), for $f = {2}\;{\rm mm}$. Strehl ratio values are according to the color bar.

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The magnitude of the chromatic aberration relative to the diffraction limit determines the Strehl ratio. This relationship, in turn, is determined solely by the FN and $m$, per Eq. (9). Therefore, the result of Fig. 8(a) is independent of scaling of the lens, and can therefore be used as a general guide to what Strehl ratio can be achieved for given feature height and FN. This contrasts with the case of Strehl ratio as a function of NA [Fig. 8(b)], which is not scale invariant. To demonstrate this last statement, we have repeated the simulations of Figs. 8(a) and 8(b), for a focal length of 2 mm (instead of 1 mm). The results are shown in Figs. 8(c) and 8(d). As can be seen, panels (a) and (c) of Fig. 8 are identical (except for the bottom limit of the FN, which is different, since an NA of 0.025 corresponds to a higher FN for longer focal length), whereas panels (b) and (d) are not.

To understand what type of design (CDL, AML, or ADL) would be better for a certain application, we can look at the FN limit, and see which type gives a higher limit. The higher the limit, the better the performance is expected to be, even if the limit is exceeded. Of course, other factors, such as manufacturing complexity, cost, and other system requirements, must be considered, but the FN limit is a good start. To better understand the implications of this, let us look at a test case example of a cell phone camera lens design.

We would like to design an achromatic flat lens for the visible spectral range, which we will define here as ${\lambda _0} = {550}\;{\rm nm}$, $\Delta \lambda = {200}\;{\rm nm}$. The FN upper limit for a CDL, based on Eq. (17), is 550/200 = 2.75. For an AML, the largest $\Delta n$ is about 1 (${{\rm TiO}_2}$ truncated waveguides in background of glass or polymer), and the maximum antenna height is of the order of the wavelength, i.e., $p \approx 1$. Substituting these values into Eq. (4) results in ${{\rm FN}_{{\max}}} = {5.5}$ (using the approximate form; the accurate form gives 5.32, which is a negligible difference), i.e., double that of the CDL. Now let us look at an ADL. There we typically have $\Delta n \approx 0.65$ (when manufactured in photoresist) and $p \approx 6$. According to Eq. (10) we obtain ${{\rm FN}_{{\max}}} = {7.8}$. So, it seems an ADL is a slightly better choice for this application from a performance standpoint. Of course, the AML/ADL can be somewhat improved by increasing $p$ or ${\Delta}n$, but this comes at the expense of more complex manufacturing.

Is this useful for common flat lens applications? If we calculate the Fresnel number of a typical cellphone camera lens ($f = {4}\;{\rm mm}$, ${\rm NA} = {0.2}$, ${\lambda _0} = {550}\;{\rm nm}$) based on Eq. (3), we obtain ${\rm FN} = {291}$, which is more than an order of magnitude larger than the Fresnel numbers mentioned above. Thus, we can expect low resolution for an ADL-based camera lens, such as those shown in Figs. 7(a) and 7(c). Therefore, while AMLs and ADLs can provide an improvement over a CDL, they are still not the ultimate solution for many applications.

An important advantage of ADLs over AMLs is that they can have very large bandwidth. For example, if our central wavelength is 550 nm, using the same ${\Delta}n$ and $p$ as before, the maximum wavelength for an ADL, based on Eq. (13), is 7 times the central wavelength, so we can have an ADL that covers a very large spectral range of about 400–3850 nm (VIS-MWIR). This type of broad bandwidth ADL is demonstrated in recent work [27,28]. In essence, this stems from the ability of an ADL to operate at moderately high $m$ values, whereas the implementation of such $m$ values in AMLs is extremely challenging due to the deep sub-wavelength features that are needed.

Based on Eqs. (4) and (11), for AML and ADL, respectively, we can conclude that AMLs are better suited (i.e., give a higher ${{\rm FN}_{{\max}}}$) for narrow bandwidth applications (where they can provide high FN with good resolution), while ADLs are better suited for large bandwidth applications (where they are limited to low FN, or alternatively to poor resolution at high FN, but nonetheless will perform better than an equivalent AML). By comparing Eqs. (4) and (11) (assuming $m\; \gt \;{1}$) we find that an ADL is better for spectral ranges satisfying $\Delta \lambda /{\lambda _0} \gt 1$. In a real comparison, it is advisable to use $\Delta n$ based a realistic nanopillar library, rather than the optimistic assumption used in [26].

Interestingly, in a recent paper that compares the diffraction efficiency of a conventional (non-chromatically corrected) truncated-waveguide-type metalens to that of a CDL, it was found that a metalens has an advantage in angular coverage, including both high NA and large FOV, while a diffractive lens has an advantage in wavelength range coverage [48]. Based on this it may be possible to generalize the conclusion of the previous paragraph from achromatic metalenses and diffractive lenses to metalenses and diffractive lenses in general.

A well-known application for diffractive lenses is a refractive–diffractive doublet, for correction of chromatic aberration of refractive lenses. As mentioned in the introduction, for this application the upper limit on Fresnel number derived here is irrelevant, since the chromatic aberration is being used to advantage. In this context, an application for a dispersion-engineered metalens has been demonstrated [49] that allows correction of the secondary chromatic aberration. For this type of application, it seems there is an advantage to metalenses since the truncated waveguide dispersion can be tailored to need. However, such an application is beyond the scope of this paper, since the doublet is not a purely diffractive/flat lens, but rather most of the power is coming from the refractive component.

In our analysis, we focused on the case of continuous spectrum, which is typical for imaging systems. For discrete wavelength applications, the wavelength limits found in this paper apply, but not the FN limit. This is the case also for the AML upper limit presented in [26]. In the case of an AML, we no longer need such high dispersion. It is sufficient to have a large enough library of dispersive truncated waveguide designs so the desired phase functions can be implemented at a few specific wavelengths [50,51]. For an MOD, if the discrete wavelengths are harmonics, we no longer have chromatic blur. For both cases, it seems that the fewer and more widely spaced the wavelengths, the easier the design (feature height $h$ can be smaller), allowing to achieve higher FN alongside with good resolution. As the number of wavelengths and their density increase, we approach the limits presented in this paper. Calculation of relevant limits for the discrete wavelength case, and comparison of AML to ADL in this case, are left for future research.

7. CONCLUSION

The field of flat lenses in general, and achromatic flat lenses in particular, is being actively researched. A lot of excellent work has been done on methods of correcting the chromatic aberration of flat lenses. While the limitations of the current AML technology have been discussed, it was unclear what the limitations of ADL technology are. The purpose of this paper is to bridge this gap. We have shown that current ADL technology has a strong limitation on the maximum Fresnel number that can be achieved while preserving near-diffraction-limited resolution and high efficiency. This imposes a limit on the practical applicability of the technology for applications involving broadband illumination. It is our hope that this paper will help clarify some of the underlying physics, promote good engineering practice in the design, simulation, and testing of flat lenses, and provide a basis for innovative design ideas and applications.