1. Introduction

Zone plates, Fresnel Lenses and other diffractive optical elements (DOEs or diffractive lenses) can be very thin and lightweight and can be designed to focus radiation at any wavelength over a very broad range from microwaves to gamma-rays. Modern microfabrication techniques have opened the way to their application in a wide variety of contexts.

Two broad classes of such optics may be distinguished. Metalenses rely on subwavelength structures to arrange that radiation from different parts of the lens arrives at a focal point with the same phase despite different physical path lengths. Diffractive lenses seek to achieve the same objective but by using a bulk property of the lens material, its refractive index, $\mu$, which leads to a phase retard of $(\mu -1)h/\lambda$ when radiation of wavelength $\lambda$ passes through a thickness $h$ of the material (note that in this paper phase differences are measured in units of $2\pi$ radians, i.e., in cycles or wavelengths, unless otherwise indicated). The relative advantages and domains of applicability of the two approaches have been debated in the literature [1–4].

Historically a major limitation has been that a particular diffractive lens provides good imaging with high efficiency only over a narrow band around the wavelength for which it has been designed. Recently various ways of designing achromatic diffractive lenses that alleviate or obviate this limitation have been studied leading to what have been termed Achromatic Metalenses (AMLs) and Achromatic Diffractive Lenses (ADLs). The present work concerns how the design of such lenses can be optimized, the limits to what can be achieved in this direction and how some published designs compare with those limits. Emphasis is on ADLs, although some aspects of the approach adopted could also be applied to metalenses.

Interest here is in lenses that are large in the sense of having a large number of Fresnel zones and that are thin, that is to say the range of heights is similar to the minimum necessary to provide any phase shift in the range $0-2\pi$ radians. Ideally such lenses should provide efficient focusing over an arbitrary, and preferably continuous, band. A number of recent publications [5–19] have described ADLs in which it seems that this has been achieved. Other work has led to limits on what is achievable that appear inconsistent with the reported performance of some of these designs. We here attempt to clarify what is behind the disparate conclusions.

This paper is organized as follows. The issues involved in the design of ADLs are first discussed (Section 2) and then methods of finding lens profiles that maximize some measure of performance are reviewed (Section 3) and compared (Section 4). Published theoretical limits on the maximum possible performance of ADLs are summarized in Section 5 and some rules indicating the performance that can be anticipated in practical cases are presented in Section 6. Both limits and anticipated values are then compared with the reported performance of some designs described in the literature (Section 7). Probable explanations of the discrepancies are discussed and finally conclusions summarized.

2. Lens design

The question arises of how, for given lens parameters such as size, focal length and desired spectral range, can one arrive at a lens design that approaches as closely as possible to the desired performance or at least to the performance limits mentioned above?

The design of a refractive DOE involves finding a radial height profile such that the phases of radiation arriving at a point in the focal plane from different parts of the lens are such as to yield the required focusing performance. Designs for ADLs typically consist of a series of concentric rings. Often the rings are assumed to be of constant width and of heights that are integral multiples of a basic step up to some maximum, $h_{max}$. The implications of this discretization are discussed in Section 6 but this formulation of the problem has the advantage that it leads to a well-defined situation – lens design consists of choosing the number of height steps for each ring to maximize some performance measure or Figure of Merit (FoM) such as the Strehl ratio, or a measure of efficiency or of the shape of the Point Source Response Function (PSF). For achromatic lenses the FoM may reflect the performance at a series of $N_\lambda$ wavelengths $\lambda _1, \lambda _2 \dots$, which in the limit form a continuum.

In common with other work in the area, scalar diffraction theory will be used here even though the structures involved are not necessarily large compared with the wavelength. Angles will be taken to be small and amplitude factors important only close to the lens will not be included, so results are strictly only applicable to high f-number systems. Uniform illumination of the lens with radiation of wavelength $\lambda$ parallel to the axis (Fig. 1) is assumed. The complex amplitude at off-axis distance $\rho$ in the image plane due to a narrow annular region of the lens of radius $r_j$ and area $A_j$ is then given, apart from a multiplying factor, by a $J_0$ Bessel function and a phase term

Here $d$ is the distance of the point in the image plane from the center of the lens and the first three phase shift terms are,

The wavelength dependence of the refractive index $\mu$ of the lens material and of $t_{2\pi }$ , the thickness needed for a phase shift of $2\pi$ radians, have been made explicit here but will be dropped below. $\phi _G$ and $\phi _L$ are the phase shifts due to the geometric path to the focal point and to passage through the lens respectively and $\phi _I$ accounts for the extra path to an off-axis point. $\alpha (\lambda )$ is a constant that defines zero phase and may depend on wavelength.

 

Fig. 1. At a given wavelength, for maximum response at a point in the image plane distance $\rho$ from the axis the phase of radiation on arrival from all radii $r$ in the lens surface should be the same.

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The wave amplitude at a particular position in the image plane is given by the sum over the lens area of the terms given in Eq. (1):

The intensity is given by the square of the amplitude:

When considering the on-axis response both $\rho$ and $\phi _I$ are zero so the $J_0$ term in Eq. (1) is unity. The amplitude on-axis in the focal plane will then have a maximum value if $\phi _G + \phi _L$ is the same for all $r$. For a single wavelength and in the limit of a large number of rings and of height steps this leads to an ideal diffraction-limited lens and if the $A_j$ are normalized such that $\sum _{j}A_j = 1$ then $I(\lambda,0)=1$. In the general case, with this normalization $I(\lambda,0)$ is equal to the on-axis intensity relative to that for an ideal lens, the Strehl ratio, $S$.

Typically the performance at a number of ‘design’ wavelengths is considered. The maximum on-axis response at all these wavelengths will then be obtained if it can be arranged that

It is instructive to consider points corresponding to the two sides of Eq. (7) in a multidimensional $(\phi _1, \phi _2,\ldots \phi _{N_\lambda })$ space where $N_\lambda$ is the number of wavelengths at which the optimization is to be performed. We will use $\mathscr {L}$ to refer to the locus in that space of the combinations of phase shifts $\phi _L$ that result from different allowed lens heights. For a simple case in which $t_{2\pi } = h_{max}$ for all wavelengths, $\mathscr {L}$ is the diagonal of the unit hypercube, along with repeats offset by integers in any or all of the axes. This is illustrated for $N_\lambda = 2$ by the solid red lines in Fig 2. More generally, $\mathscr {L}$ may have a slope other than unity (dotted red lines) and may be either longer or shorter than the diagonal, depending on $h_{max}$. If $h$ is limited to discrete levels then $\mathscr {L}$ is reduced to a series of points (red circles in the figure).

 

Fig. 2. The locii $\mathscr {L}$ and $\mathscr {G}$ in an example case where optimization is to be at just two wavelengths ($N_\lambda =2$). The solid red lines show the combinations of phase shifts $\phi _L$ that can be obtained in the simple case where $p(\lambda _1) = p(\lambda _2) = 1$, where $p = h_{max} / t_{2\pi }$. The dotted lines indicate a more complex case with $p(\lambda _1) = 0.85, p(\lambda _2) = 1.25$. The red circles indicate $\mathscr {L}$ if $h$ is limited to $N_S=16$ steps. The blue lines show $\alpha (\lambda ) - \phi _G(r, \lambda )$, with the circles indicating the effect of considering only finite width rings. For illustrative purposes it supposed that there are 70 rings in a lens for which the Fresnel numbers at the two wavelengths are 24 and 13. Lens design involves finding for each blue circle a red circle that is ‘close’ in a sense that depends on the FoM used. For $N_\lambda > 2$ a corresponding cube or hyper-cube has to be imagined and the probability of finding a good close point decreases.

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The blue lines in Fig. 2 show $\mathscr {G}$, the corresponding locus of ideal phase shifts $\alpha -\phi _G$, at different radii within the lens. The blue circles illustrate how, if the profile is constrained to be a set of rings, then $\mathscr {G}$ becomes a series of points. For rings of constant width, the points corresponding to rings far from the axis are more sparsely distributed than the inner ones. Each segment of $\mathscr {G}$ crossing the unit cell corresponds to radii in the lens within an adjacent pair of Fresnel zones. In general the number of lines crossing each axis will be equal to $N_F/2$, where $N_F$ is the Fresnel number at the corresponding wavelength. However if there is an exact integer relationship between the wavelengths some of the lines may overlay one another. In the absence of such a special relationship, and considering lenses that are large in the sense that the number of Fresnel zones is high, $\mathscr {G}$ will then be distributed rather densely and uniformly.

Achieving perfect focusing at all wavelengths would require that every point on $\mathscr {G}$ also lies on $\mathscr {L}$. Clearly for most points on $\mathscr {G}$ this cannot be the case and it will not be possible to choose a lens height $h$ to produce the desired phase shift at all wavelengths. Except where the lines of $\mathscr {L}$ and $\mathscr {G}$ cross, a compromise choice of height must be adopted, for example picking for a given $r$ the point on $\mathscr {L}$ that is ‘closest’ according to some distance measure. The lines in Fig. 2 are cyclically repeated outside the unit square because in some cases that point may be on one of the repeats.

The requirement for ideal focusing is only that $\phi$ be constant for a particular wavelength, not that it be zero. A different non-zero ‘target phase’ $\alpha (\lambda )$ can be adopted for each wavelength if this leads to a better solution. In Fig. 2 varying $\alpha (\lambda _1)$ or $\alpha (\lambda _2)$ (or both) simply results in a cyclic shift of $\mathscr {G}$ along the corresponding axis (or axes). For $N>2$ the same applies in the corresponding higher dimensional space. The $\mathscr {G}$ points will still be spread rather uniformly over the space and at each lens radius the chance of finding a compromise $h$ that works well at both (all) wavelengths will remain low.

An exception occurs when the wavelengths are integer fractions of some base wavelength. As noted above, some lines of $\mathscr {G}$ may then be overlaid to form a single line. This property is used in Harmonic diffractive lenses [20,21].

3. Optimization procedures and Figures of Merit

How can one choose a lens profile that leads to a performance as close as possible to that desired? Various publications [1,5,7,12,22–26] have advocated the use of a ‘(gradient-assisted-) Direct(-ed) Binary Search’ (DBS) to find the best ring heights, stepping each in turn in a direction that leads to an increase in some Figure of Merit (FoM). Starting from a random profile, the procedure is repeated iteratively until some convergence criterion is met. One problem with DBS is that the solution found may be a local maximum, not a global one. This is particularly important if at one or more of the design wavelengths $h_{max}>t_{2\pi }$. Multiple maxima in the FoM versus height are then likely and when using DBS the best solution can clearly be missed. However even for thinner lenses local maxima can occur. Hu et al. [27] propose the use of a preselected starting profile instead of a random profile, which speeds up convergence and reduces the probability of missing the global maximum.

In principle DBS can be used with any FoM. Possible FoMs include a measure of efficiency [12] or the Strehl ratio. These can be combined with a measure of the similarity of the PSFs to a target shape [14] but in practice if the Fresnel number and the number of rings in the lens are high then a PSF with a central peak whose width and form are close to those of the Airy pattern from an ideal lens tends to arise automatically. This is illustrated in Appendix 1 (Supplement 1), where it is shown that even a lens profile with random ring heights has a high probability of achieving ‘diffraction limited’ imaging in the sense that there is a central core to the PSF whose width and shape are similar to those for an ideal lens. However in that case both the efficiency and the contrast ratio will be very poor.

Use of an FoM based on the Strehl ratios, $S$, at the different design wavelengths, for example their mean, $\bar {S}$, has the computational advantage that only the on-axis response needs to be evaluated, not the full PSF. The optimization criteria used by Doskolovich et al. [8,17] and by Liu et al. [28] , which can be shown (Appendix 2, Supplement 1) to be equivalent, share this advantage. Both effectively maximize a parameter we term $Q$ which is the sum over the design wavelengths of the real part of the complex wave amplitudes. It is shown in Appendix 2 (Supplement 1) that optimizing $Q$ is equivalent to using as FoM $\bar {S}$ but with an additional term that turns out to be important,

Here $\sigma$ is the standard deviation of the (real parts of) the net on-axis wave amplitudes at the different design wavelengths. For a flat response $\sigma = 0$ so $\bar {S}= (Q/N_\lambda )^2$.

A difficulty that arises when using DBS or any other algorithm to optimize either $\bar {S}$ or average efficiency is a tendency to lead to designs in which one of the design wavelengths is favored at the expense of all the others (this effect is apparent in Fig. 8(a), in Section 7). The reason that this comes about is discussed in Section 6. Interestingly the $\sigma ^2$ term in (8) penalizes such designs, leading to a more uniform response. Another way of favoring a uniform response is to use as a FoM the minimum over the design wavelengths of $S$ (or of efficiency) rather than the mean.

The adoption of FoMs such as $Q$ that, for a given set of $\alpha$, are a linear combination of the contributions from different rings within the lens offers a particular advantage. Assuming given $\alpha$, the optimum height of each ring can then be found independently by a simple search over the available height steps. Doskolovich et al. [8,17] employ this approach while implicitly assuming the $\alpha$ to be zero so the algorithm that they use is extremely fast. In Section 3.1 below we discuss the advantage to be gained by varying the $\alpha$. It is seen to be of major importance only if the Fresnel Number is low.

The Doskolovich et al. algorithm can equally be used with any given set of $\alpha$ and Liu et al. [28] find optimized $\alpha$ using Simulated Annealing, applying the algorithm at each trial. In the explorations leading to the present paper a Genetic Algorithm (Appendix 3, Supplement 1) and a simple search over a large number of random sets of $\alpha$ have also both been found to be effective. $Q'$ is used here to indicate $Q$ further optimized in any of these ways.

Xiao et al. [29] optimize a parameter that they call $J_\omega (F)$ that is a measure of coherence at the focal point given the assumed input spectrum. Although not presented in those terms, this turns out to be identical to the mean Strehl ratio. $J_\omega (F)$ is maximized using a combination of a Genetic Algorithm and a Hook-Jeeves Algorithm, the latter being being similar to a DBS. This is combined with a smoothing operation which eliminates solutions with large aspect ratios in the height profile to improve manufacturability.

The use of efficiency as an FoM is of particular interest as lens designs discussed in the literature are frequently characterized by it. High efficiency is crucial in some applications, for example in Astronomy where every photon matters. A typical definition of efficiency is “the ratio of the power inside a diameter, $d_3$, equal to 3 times that of the corresponding diffraction limited spot to the power incident on the lens” [14] and the size of the diffraction limited spot is frequently characterized by its full-width at half-maximum (FWHM) [11]. Efficiency measured in this way will be referred to here as $\epsilon _3$. Losses due to partial reflection at interfaces are usually disregarded. The efficiency may be averaged over a specific set of wavelengths for which the design is optimized, or over a specified band.

When $N_F$ is large, particularly if $N_\lambda$ is not too small, the deviations from the ideal of the phases of the contributions from different rings will be large and close to randomly distributed (as discussed in Section 2). In such circumstances the arguments used in Appendix 1 (Supplement 1) apply and the PSF will have core that is similar in shape to an Airy disk. Consequently the ratio $\epsilon _3/S$ will be close to that for an ideal monochromatic lens, 0.855. To the extent that there is a fixed ratio between the two, optimizing $S$ is obviously rather similar to optimizing $\epsilon$ and is computationally much less demanding.

3.1 Phase as a free parameter

Considerable emphasis has often been placed on the fact that with the DBS method the $\alpha$ do not need to be constrained and it has been suggested that "phase as a free parameter" is the key to some remarkable reported results [12,13,24–25,30]. However various lines of argument and experience with simulations all point to the fact that while important for small lenses, the $\alpha$ have very little influence on the obtainable performance if the lens contains a reasonably large number of Fresnel zones.

One way to see that this is likely to be the case is to consider the problem of finding an optimum profile as one of fitting some ‘data’ (the $\phi _G$ dictated by the design requirements) with a ‘model’ ($\phi _L -\alpha$) described by parameters that are the lens heights at different radii and the $\alpha$. The number of independent points ‘data’ points is of the order of $N_F N_\lambda$ whereas the number of adjustable parameters is of the order of $N_F$ if the $\alpha$ are fixed, or of $N_F + N_\lambda -1$ if they are free. If $N_F \gg N_\lambda -1$ then the improvement in any goodness of fit measure due to the inclusion of the extra parameters is likely to be small. An analogy between the sum of squares of the phase errors and $\chi ^2$ suggests that the fractional improvement in $Q$ will be similar to the fractional reduction in the number of degrees of freedom and hence it is reasonable to expect a relationship

$Q$ is of course related to $\bar {S}$ (Eq. (8)) and in practical examples $\epsilon$ follows $S$ so Eq. (9) also provides an indication of the likely fractional improvement in efficiency.

Equation (9) can only be expected to be approximate but simulations show that it gives a general guide to behavior over 3 orders of magnitude of parameter space as shown in Fig. 3. They confirm that the fractional improvement in performance associated with allowing "phase as a free parameter” is small if $(N_\lambda -1)/N_F$ is low.

 

Fig. 3. The fractional improvement in $Q$ resulting from allowing phase as a free parameter. The change is measured relative to the mean of the values obtained with a large number of random sets of $\alpha$. The line shows the relationship Eq. (9). Blue symbols are for a lens with parameters from [11] but with modified focal length or number of design wavelengths (Table 1 column 1); red symbols are for an idealized version (Table 1 column 2). Squares show the result of changing $N_\lambda$; circles that of changing $N_F$ by altering the focal length.

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The same conclusion is reached by considering the locii $\mathscr {G}$ and $\mathscr {L}$ introduced in Section 2. For lenses with large $N_F$ the $\mathscr {G}$ points are largely uniformly distributed over a unit cube in an $N_\lambda$ dimensional space. Changing one or more of the $\alpha$ results in a cyclic shift of the $\mathscr {G}$ along the corresponding axis(-es) without changing the distribution and will not, on average, bring them much closer to the locus $\mathscr {L}$ that represents the combinations of phase shifts that are available for the allowable heights $h$.

Finally the simulations discussed below show that in example practical cases Eq. (9) holds, at least approximately. It is seen that while the differences between design approaches that do and do not assume particular fixed $\alpha$ values are important for low $N_F$, they become become negligible for high $N_F$.

4. Comparison of design approaches

To compare the different approaches to obtaining an ’optimum’ profile, we consider an idealized example. The design parameters were based on those of the ‘Short Wavelength Infra-Red’ lens of [11] but with certain changes as indicated in Table 1. In particular the material characteristics were assumed to be such that the maximum lens height corresponds to $t_{2\pi }$ over the entire range. The ring width was also reduced in order to be less than one Fresnel zone width at the edge of the lens, even at the shortest wavelength and focal length. Different focal lengths, corresponding to a wide range of Fresnel numbers, were considered,.

Table 1. Design parameters of an ADL described in the literature [11] and of an idealized version of it used as an example in Section 4. The lens material in the latter is assumed to have refractive index, $n$, such that $n-1\propto \lambda ^{-1}$. Parameters in common are : diameter=8.93mm, $\lambda _{min}\text {--}\lambda _{max}= 0.875 \text {--}1.675$$\mu$m, $h_{max}$ = 2.4 $\mu$m, $N_S=100$.

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As shown in Fig. 4, in the limit of large $N_F$ all design approaches lead to similar lens performance. The figure also shows that for lower $N_F$ the spread in $\bar {S}$ is much greater than the error bars showing the $1\sigma$ rms spread over 1000 random choices of $\alpha$ values would suggest. This indicates that the distribution is highly non-Gaussian. The various optimization procedures yield similar values except that in a few cases DBS failed to find a solution as good as the other methods and, interestingly, $\alpha \equiv 0$ appears to be a poor choice.

 

Fig. 4. $\bar {S}$, the Strehl ratio averaged over the 9 design wavelengths, for an idealized lens with the design parameters in Table 1, optimized by different methods. A range of focal lengths is considered leading to different Fresnel Numbers. $\alpha \equiv 0$ corresponds to the use of optimum $Q$. ‘Average’ shows the mean and rms $\bar {S}$ found during 1000 runs using optimum $Q$ but with different random sets of $\alpha$ values while ‘Best’ shows the highest value among those runs, i.e., $Q'$. The results of using a DBS algorithm and of a Genetic algorithm to optimize the $\alpha$ are also shown. The horizontal line shows the value $1/N_\lambda = 0.090$ expected using Eqn: (10).

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5. Performance limits

Presutti and Monticone [31] have considered limits on the bandwidth over which good imaging is possible with AMLs and noted that some ADL designs that have been published appear to breach those limits. More recently Engelberg and Levy [3] (EL21) have argued that ADLs are not necessarily constrained by those limits and alternative limits are deduced, expressed in terms of the maximum Fresnel Number of ADLs beyond which the resolution will be degraded. Of the examples taken from the literature for data and tabulated in EL21, almost half have Fresnel Numbers higher than those upper limits by a factor of 10 or more! It is argued there that in those cases the resolution, while typically described as "near diffraction limited", has been overestimated because Full Width at Half Maximum (FWHM) has been used as a measure rather than a more representative measure, such as half power diameter (HPD), that recognizes the power lost into wide wings of the response.

In a key paper Teichman [32] (T21) has used time-domain analysis to derive expressions that allow upper limits to be placed on the Strehl ratio of a thin lens illuminated with radiation having a given spectrum, typically covering an extended wavelength band. The analysis should be applicable to both AMLs and ADLs. In either case the key parameter is the span of delays that can be imparted to radiation within the thickness of the lens. The limit arises because the group delay for off-axis and axial rays will only be matched if the phases are matched at all frequencies (wavelengths) present in the illuminating spectrum. Because of the generality of their approach, by assuming a spectrum consisting of a number of narrow lines it can be used to place a limit on the mean of the Strehl ratios at a specified set of wavelengths. This possibility is used in Section 7.

Xiao et al. [29] (X22) discuss upper limits that can be placed on a function that they term $J_\omega (F)$ which is the mean over a band of frequencies of the mutual coherence function at the focal point. With the normalization used $J_\omega (F)$ is identical to the mean Strehl ratio. The general expression found for an upper limit on $J_\omega (F)$ (Equations S-15 of the Supplementary Material linked to that paper) requires extensive computation and by making some simplifying assumptions an approximate analytic form (their Equations S-17) is also given.

T21 and X22 both use an approach in which potentially inconsistent combinations of ring heights are considered, resulting in upper bounds that, while guaranteed to be no lower than is possible with any physically realisable design, are likely in practice to be considerably higher, leading to an optimistic impression of what may be attainable.

Kigner et al. [26] (Supplement linked to that paper), and Majumder et al. in more detail in a preprint [33], have claimed that limits such as those discussed above do not apply to their ADL lens designs. They assert that the only limit to bandwidth is that of the detector read-out chain. As discussed in Appendix 4 (Supplement 1), there appears to be a misconception here and this does not mean that the bounds discussed above do not apply.

6. Expected average Strehl ratio and efficiency

It is shown in Appendix 5 (Supplement 1) that, subject to certain conditions, after optimization the average of the Strehl ratios at the $N_\lambda$ design wavelengths is expected to be

If the form of the focal spot is similar to an Airy disk (as it has been argued above is likely) this implies

In addition to the assumption, as throughout this work, that small angle approximations and scalar diffraction theory apply these equations apply on condition that :

For a cylindrical lens Eq. (10) is unchanged but the ideal PSF is a $\operatorname {sinc}^2$ function not an Airy disc formally implying that Eq. (11) becomes

In practical cases the above conditions are unlikely to be all fully met. In particular $h_{max}$ cannot be equal to $t_{2\pi }$ at all wavelengths unless the lens material has a dispersion law such that $\mu -1 \propto \lambda ^{-1}$. Nevertheless in many circumstances where these conditions are at least approximately obeyed the relationships prove to offer good guidelines. This is illustrated in Fig. 5 which shows the results of some simulations of a lens with parameters based on one described in the literature but optimized at various numbers of design wavelengths.

 

Fig. 5. Plots of $\bar {S}$ and $\bar {\epsilon _3}$ as a function of $N_\lambda$ for a lens based on the parameters described in [11] but with various $N_\lambda$. In each case the design wavelengths are uniformly distributed over the range 0.875$\mu m$ to 1.675$\mu m$. The profiles were obtained by optimizing $Q'$. The lines correspond to Eqns. (10) and (11).

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The lens for which results are shown in Fig. 5 has $N_F= 620$ at the mid-wavelength. Figure 6 shows the effect of changing the $N_F$ of a lens by changing its focal length. In this case the lens is the idealized lens described below on Section 4, ensuring that conditions 2-6 are reasonably well met. The relationships are seen to break down when $N_F$ is below a few hundred.

 

Fig. 6. As Fig. 5 but for an idealized variation of the lens (Table 1, column 3) and for a number of different focal lengths and hence of $N_F$. The assumed lens characteristics meet reasonably well conditions 2 to 6 in Section 6. The $N_F$ quoted are those at the center of the wavelength range.

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In general terms, if either condition (1) or (2) is not held then Eqns. (10) and (11) lead to underestimates, while if (4) or (5) are not valid they overestimate what is possible. Values of $h_{max}$ too small to meet condition (3) will lead to lower values than predicted by these relationships and vice versa.

There is an important implication of the fact that the numerical factor in Eq. (10) is less than unity. An ideal lens optimized for monochromatic performance at any one of $N_\lambda$ design wavelengths will have $S=1$ at that wavelength and $S\ge 0$ at the others, so $\bar {S} > 1/N_\lambda$. A similar argument applies to efficiencies. This explains the tendency of lens designs obtained by using DBS and optimizing either the mean Strehl ratio or the mean efficiency to have a very non-uniform response - the best lens using either of these measures as a FoM would actually be a monochromatic one! As mentioned above, the $\sigma ^2$ term in Eq. (8) penalizes such solutions when $Q$ is used.

If the design wavelengths are well separated then the both the Strehl ratio and the efficiency will have peaks at those wavelengths and will fall to a low values between them (an example will be shown below in Fig. 8). Thus the averages over the whole band will be much lower than $\bar {S}$ and $\bar {\epsilon _3}$. It is shown in Appendix 5 (Supplement 1) that under conditions similar to those listed above and for uniformly distributed wavelengths the average of the Strehl ratio over the band $\lambda _{min}$ to $\lambda _{max}$ will be about

Unless $N_\lambda$ is very small or $\lambda _{max}/\lambda _{min}$ is very large a good approximation is given by

$\overline {S}_{band}$ is seen to depend on the distribution of the design wavelengths rather than on their number. This will also be true for $\bar {\epsilon _3}$ averaged over the whole band. So for a continuum spectrum the focussed energy for an ADL will tend to be similar to that for a monochromatic diffractive lens of similar proportions.

7. Comparison with published designs

We now consider the reported performance figures for some published designs in the light of the work presented here. Details of some ADLs described in the literature having high Fresnel numbers are summarized in Table 2. All have $N_F \ge 200$ wheras according to the relations given by Engelberg and Levy [3] (EL21) in each case the upper limit on the Fresnel number of a diffraction limited ADL lens with the given characteristics is 5 or less. Consequently performance is expected to be severely limited compared with an ideal lens and the key question is what is actually achievable.

Table 2. The analysis presented here applied to some high $N_F$ ADLs described in the literature. Shading indicates values that are particularly important or anomalous.

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We take as a principal example the Short Wavelength Infrared (SWIR) lens designed and fabricated by Banerjee et al. [11] the parameters of which, given in Table 1, have been used as the starting point for many of the simulations presented above. Banerjee et al. used a profile obtained by a DBS algorithm and intended to be achromatic in the short-wave infrared regime, 875nm to 1675nm. Results from their paper are duplicated in Fig. 7 and were described as showing “simulated and measured (based on curve fitting) average efficiency for $\lambda$ = 875 nm to 1375 nm (of) 91% and 35%, respectively”. The definition of efficiency used was that of $\epsilon _3$.

 

Fig. 7. Fig. 2(a) from Ref. [11] reprinted with permission © Optical Society of America, showing simulated and measured performance of a lens with the parameters in Table 1.

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There are many reasons to think that these results should be treated with caution. As noted above the Fresnel number is far too high for uncompromized operation but the reported simulated efficiency actually exceeds that of an ideal lens, for which $\epsilon _3$ would be 0.855. Furthermore, assuming a flat spectrum between 875 nm and 1675 nm, the Teichman limit on the Strehl ratio for such a lens is 0.020. Using the method of Xiao et al. one finds an even lower limit (0.008). With an an Airy disk PSF the average efficiency over this band should be even lower than this.

Although a continuous line is shown in Fig. 7 and the averaging is said to be over the band, it seems that that the figures quoted are actually intended to be averages over the 9 design wavelengths; in the ’curve fitting’ intermediate wavelengths were ignored. A later publication [19] does indeed refer to this lens as "operating in discrete spectral bands". Using the method of Teichman and assuming an input spectrum with 9 narrow lines at the design wavelengths one finds an upper limit for the average over those wavelengths which is higher than that for a band, but still only $\bar {S} = 0.36$ (or 0.41 using X22). Again $\epsilon _3$ is expected to be less than $\bar {S}$.

Figure 8 shows attempts to reproduce the design of this lens using some of the techniques reviewed here. In 8(a) are the results of an optimization by a DBS method using as a FoM the value of ${\epsilon _3}$, averaged over the design wavelengths. The efficiency is moderately high only at those wavelengths, with very uneven values at different wavelengths and an average of only 0.083, more than an order of magnitude lower than that claimed. optimizing $Q'$ leads to similar pattern but much more uniform (b), while the result of using a DBS with a FoM which is the minimum $\epsilon _3$ over the 9 wavelengths (c) is still is more uniform. The improvements in uniformity are obtained at the expense of very small compromises in mean $\epsilon _3$. In all cases both the band- and line-averaged Strehl ratios found are compatible with the corresponding T21 and X22 limits.

 

Fig. 8. Attempts to reproduce Fig. 7 by different techniques. The green markers indicate the $\epsilon _3$ values at the 9 design wavelengths. The red circles show the corresponding Strehl ratios on the same scale. Top: DBS with $\bar {\epsilon _3}$ as FoM. Middle: Optimum $Q'$. Bottom: DBS maximizing a FoM that is the minimum over the 9 design wavelengths of $\epsilon _3$. The optimum $Q'$ results were obtained using the method of Doskolovich et al. with random sets of $\alpha$ and taking the best of 10000 trials. However similar results were obtained with simulated annealing (as in [28]) or with a genetic algorithm.

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Even assuming that the 91% (35% measured) efficiency in Fig. 7 is intended to refer to the average just over 9 spot values, how can the difference between these figures and the $\sim$8% found here be explained? The discrepancy appears to arise because, although it is defined as the ratio of the energy within the focal spot to the “total incident power", the efficiency was calculated from the PSFs. If this is done rather than computing or measuring the actual power incident on the lens, then it is essential to take into account all of the energy arriving in the focal plane. An ADL of the type being considered here acts as a scattering screen and much of the energy is diffracted to very large off-axis angles. Failure to take this into account will result in the efficiency being greatly overestimated. The extent of the image plane from which the total power was estimated is unclear.

The average Strehl ratio and efficiency found in the present work for an ADL with these parameters and shown in Fig. 8 are both close to those predicted by Eqs. (10) and (11) respectively, small difference being attributable to the fact that conditions 1, 4 and 6 listed in Section 6 are not fully met.

A similar situation is found in the case of two long wavelength infrared ADLs described in [12]. The published efficiencies and the Strehl ratios estimated from published figures greatly exceed those predicted to be possible but the simulations reported here lead to values consistent with those expectations. Again it would appear that the efficiency was greatly overestimated because the incident flux was deduced from measurements in a limited region of the image plane.

Investigations of results described in [7] and in [17] for cylindrical ADLs (Table 2) illustrate a different sort of issue. The simulations performed for the present work lead to values of $S$ averaged over the design wavelengths (0.165, 0.156) that are compatible with the corresponding T21 upper limits but lower than might be expected from a naive application of Eq. (10) (0.27). This may be partly because two of the conditions assumed in deriving that equation ($p\sim 1$ and $q\ll 1$|) are far from being met. The two violations have opposite effects; apparently the latter dominates.

More important is that the published $\bar {S}$ and $\bar {\epsilon _3}$ values for these lenses, particularly those in [17], are much higher than could be reproduced in the present simulations. The reason for this emphasizes an important consideration in simulations of ADLs. It was implicitly assumed in [7] that the diffraction due to a ring within the lens can be described by the $\phi _L$ and $\phi _G$ at its center. Because in [17] the objective was to make a comparison with the work in [7], the same assumption was made (L. Doskolovitch, private communication). As noted in Section 6, with $q$ that are too large $\phi _G$ can vary widely across a ring and the net diffracted amplitude from some of the outer rings where this effect is most important will be overestimated. If $q\sim 2$, as occurs at some wavelengths with the parameters of this lens, the range of phases across a ring can cover $2\pi$ radians and the contribution of that ring can even drop to zero! With the outer rings making a reduced contribution, the efficiency of the lens is lowered and its effective size reduced, leading to a poorer FWHM. Without adequate sampling during both design and simulation these losses are not apparent and the performance is exaggerated.

The example in the penultimate column of Table 2 is one of the lens designs described by Liu et al. [28]. The reported performance is better than predicted by Eqs. (10) and (11) but the differences are at a level that is not unexpected given that $p = 2.4$, significantly greater than unity, and $N_F$ is only 200, which is in the range below which Fig. 6 suggests that the ’large’ assumption starts to break down. The authors of this paper clearly recognize the importance of considering the flux arriving far off-axis in the image plane when assessing efficiency. They point out inadequacies in other published work in this respect and state that their definition of efficiency is based on “ total transmission power (integrated on $(>10^3 \times )$ FWHM ...)”. Taking into account such a large region indeed mitigates the problems of using image plane data to assess incident power. However the importance of this issue is emphasized by the fact that even assuming that the area considered is that within a radius of $10^3$ FWHM, not a diameter, simulations show that in this instance $\sim 20{\% }$ of the scattered flux is still missed.

The final column of the table shows results for one of the lens designs generated by Xiao et al. [29]. The lens they term S5 meets all of the criteria for lenses considered here, being large, thin and designed to cover a wide continuous band. The published Strehl ratio used in the table is taken from the Supplementary Materials associated with that paper. This appears to correct a 9 times higher figure given in the main text but is still difficult to reconcile with theoretical limits - even one derived using the methods presented in the same paper. Attempts to duplicate the design were not successful.

Not shown in the table is extreme example of a large ADL is the 100 mm diameter, 200 mm focal length, lens described in [19] which is "designed to be achromatic across the entire visible band". No details are given in the brief Conference Proceedings either of the number of wavelengths considered during the design nor of the efficiency obtained. Attempts to simulate the design of such a lens show that because of the very high Fresnel number ( $\sim$20000 at the mid-wavelength) the band around each of the design wavelengths in which the response is enhanced is very narrow. Furthermore the 5 micron width of the rings exceeds the extent of a Fresnel zone in the outer part of the lens by a large factor ( $q$ up to 6 ) so much of the lens surface will contribute little except background fog. The nett effect is that according to simulations using various large values of $N_\lambda$ the efficiency averaged over the 400-800 nm band is estimated to be only $\sim 10^{-4}$ and the average Strehl ratio $\overline {S}_{band} \sim 2.5\times 10^{-5}$ is much less than that expected on the basis of Eq. (14) ($1.2\times 10^{-4}$).

Many of the discrepancies seen here are in estimations of efficiency. In fact to determine the efficiency from simulations it is only necessary to consider that part of the PSF within the region specified in whatever definition is used (a radius of 3 times the HWHM for $\epsilon _3$). Provided that the PSF, $I(x)$, is correctly normalized so that its central value would be unity for an ideal lens (i.e., is equal to the Strehl ratio) then the efficiency can be found directly from its radius-weighted integral relative to that for an ideal Airy disk PSF, $\mathcal {A}(x)$,

Practical measurements of the efficiency of real lenses are difficult because ideally the same sensor should be used to measure the brightness at the centre of the PSF and at the plane of the lens, demanding a high dynamic range. However once the Strehl ratio and the shape of the core of the PSF are obtained the above approach obviates the need for measurements of the low level scattered background.

8. Conclusions

Both ADLs and AMLs are ’flat’ lenses that can enable very compact optical configurations compared with compound refractive lens or reflective optics, though the latter offer much higher efficiency and better- or true- achromaticity respectively.

The work here has concentrated on ADLs but many of the same considerations apply to AMLs. For example for a metalens there will again be a locus $\mathscr {L}$ of the available combinations of $\phi _L(\lambda _i)$ discussed in in Section 2 and illustrated in Fig. 2. It will dictated by the library of available structures and may not simply be a straight line. Arguments exist (e.g., [32]) that suggest that ADLs will be subject to constraints analogous to those affecting ADLs but further work would be needed to investigate them.

One class of potential applications of ADLs is image forming. For imaging with a single 3-color detector, for example an RGB CCD, designing a lens for operation at $N_{\lambda }=3$ wavelengths may sometimes be adequate but using higher $N_{\lambda }$ will improve the color rendition. For continuum illumination Section 6 shows that the total amount of flux that is accurately focused depends little on $N_\lambda$ and will be low. Irrespective of the choice of $N_\lambda$ there will be a background fog due to unfocused radiation.

A second class of potential applications calls for radiation to be concentrated into a small spot or aperture. Situations in optical communications may arise where distinct carriers are to be concentrated simultaneously and a lens could be designed for those wavelengths. It has been shown that, ignoring extensive wings, the form of core of the PSF of the lenses considered tends to be the same as for an ideal lens. Thus the measures $\bar {S}$ and $\bar {\epsilon _3}$ whose optimization has been discussed are again the relevant ones and low average efficiency is to be expected.

It has been demonstrated here that if one attempts to design thin ADLs with reasonably large Fresnel number and wide overall pass band by considering the operation at a limited number of well-separated design wavelengths then the performance between those wavelengths must be expected to be poor. Consequently it is important not to confuse averages of spot values at selected wavelengths with averages over a band, which will typically be far lower.

Strehl ratios from the present simulations are all consistent with limits calculated as described in T21 and X22. Those limits are mostly are mostly considerably higher than obtained in practice, perhaps giving an optimistic impression of what is achievable. Even when they are not strictly applicable, in many circumstances Eqs. (10), (11) and (14) provide useful guidelines as to what Strehl rations and efficiencies are actually attainable. Published results are often not consistent with one or more of these limits and should then be treated with caution.

It seems likely that widely reported high efficiencies for ADLs are mostly due to failure to estimate correctly the power incident on the lens. Using observed or simulated intensities in the focal plane to deduce the incident power is misleading unless low surface brightness flux scattered to large off-axis angles can be accurately determined.

Another pitfall that can lead to simulations overestimating the performance that is truly attainable is inadequate sampling within a ring. When the ring width is comparable with, or greater than, the width of the narrowest Fresnel zones (i.e., $q\gtrsim 1$) failure to evaluate the phases at a sufficient number of sample radii within a ring can lead to greatly over-optimistic expectations.

In optimization, a FoM based on the Strehl ratio, $S$, can be used as a surrogate for efficiency as it can only be high if both the efficiency and the compactness of the PSF are good. Its use has the computational advantage that only the on-axis response needs be calculated, not an extended PSF. The parameters termed here $Q$ and $Q'$ which are implicit in the design approach suggested by Doskolovich et al. and by Liu et al., respectively, are similar to a measure of $\bar {S}$ but their optimization leads to a more uniform distribution across the design wavelengths. Design using $Q$ is particularly fast because the height at each radius can be determined independently and only on-axis phases are considered.

$Q'$ refers to the repeated use of $Q$ but wrapped in a further optimization procedure in which the best values of the $\alpha$ are found. The difference between the performance of designs based on $Q$ and on $Q'$ corresponds to the much-vaunted advantage of allowing "phase as a free parameter". However that advantage has been shown to be small except when the number of Fresnel zones is comparatively low.

In conclusion, it is possible to design ADLs to work at simultaneously at a number of wavelengths but their limitations must be understood. The efficiency at each of those wavelengths will be less than that of a single wavelength diffractive optic and the total focused flux is likely to be no higher. Furthermore, unless the incoming radiation is restricted to those wavelengths for which a lens has been optimized, an image will contain a background due to scattered radiation at intermediate wavelengths. The circumstances in which such devices offer advantages may therefore be limited.