1. Introduction

Lenses based on diffractive optical elements (DOEs) and metalenses control phase or amplitude of light across an aperture to focus the light via engineered constructive interference of diffracted waves at the focal point. Both work by effectively delaying the wavefront in a spatially varying fashion. Thin lenses of sufficient diameter and focusing power, however, cannot fully compensate for variation in free-space time of flight to maintain and focus an intact incident wavefront. Instead, they combine successive wavefronts. Broadband light, due to its limited temporal coherence, reduces the effectiveness of such interference. We represent lenses using effective medium theory (for metalenses and DOEs with subwavelength features) and scalar diffraction and show that temporal coherence of the illumination imposes hard limits on the performance of lenses as measured by the Strehl ratio. The same bound will apply to DOEs and metalenses, including those using dispersion engineering.

Recent papers have described DOEs and metalenses with broadband focusing effects [1–7]. Others have used frequency-domain approaches to analyze diffractive lens performance [8–12]. A chief advantage of the time-domain approach reported here is that it separates the influences on performance of the properties of the illumination, the constraints of lens construction, and the optical requirements. Here, we explore the bounds of performance of thin lenses under broadband light using temporal coherence and show computations for band-limited white light. The model and bounds apply equally well to DOEs and metalenses, including dispersion-engineered metalenses such as described in [3]. Shrestha et al. considered the largest metalenses capable of avoiding chromatic aberration [5]. Presutti and Monticone analyzed time-bandwidth considerations in computing the maximum size of broadband metalenses near the diffraction limit [13] in the region well-approximated by the extended Marechal approximation [14]. Engelberg and Levy used frequency-domain analysis to develop theory for bandwidth limits to achieve near-diffraction-limited performance and demonstrated examples of performance degradation for real lenses violating those limits. They also discussed differences between classes of flat lenses including metalenses and multi-order diffractive lenses [15]. We consider lenses potentially distant from the diffraction limit. The general construct in [13] and [15] of treating a broadband lens as attempting to compensate for time-of-flight differences applies to our analysis as well, but we quantify and bound the consequences of failing to fully compensate, while [5], [13], and [15] analyze the conditions under which nearly full compensation is possible. Our analysis applies equally well to metalenses and other classes of diffractive optics.

We model a lens as a radially symmetric thin layer producing a time delay $\tau $ (denoted group delay in [13] and [15]) as a function of radial coordinate $\rho $. A given lens material/manufacturing system will provide a maximum span of admissible time delays T. Ignoring a piston mode (a uniform time delay across the aperture, which does not affect optical performance), we express the constraint as requiring $|\tau |\le T/2$. In the case of a diffractive element, under the approximations of effective medium theory and scalar diffraction, this would correspond to a dispersionless material with refractive index n invariant through the depth and thickness varying by up to $Tc/({n - 1} )$, where c is the speed of light in a vacuum. This treatment applies equally well to discrete DOEs composed of annular grooves on a substrate or continuously varying kinoform lenses. A metalens can be similarly described as having an achievable span of effective time delays (with or without dispersion engineering). In what follows, we use the Huygens-Fresnel formulation to compute a bound for the Strehl ratio at the focal point.

2. Strehl ratio at the focal point

For a plane-wave normally incident on a radially symmetric planar aperture with diameter D and effective optical path delay $\tau (\rho )$ (refer to Fig. 1), the Huygens-Fresnel formulation including obliquity for the resulting scalar field at an on-axis focal point a distance f from the aperture is [16]

 

Fig. 1. System schematic of a radially-symmetric thin lens of diameter D and focal length f characterized by its effective time delay ${\boldsymbol \tau }$. ${\boldsymbol \tau }$ can equivalently be given as a function of the radial coordinate ${\boldsymbol \rho }$ or the free-space time of flight ${\boldsymbol \xi }$ associated with a given point on the aperture.

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We also define the free-space optical path in units of the center wavelength ${\lambda _0}$ along the central ray and marginal ray, respectively, as

The time-domain approach is fundamentally equivalent to the more common frequency-domain description. In the limit of long-time averaging, different wavelengths contribute independently to intensity due to the orthogonality of Fourier components, so in Eq. (7), the numerator of $\gamma ^{\prime}$ as defined in Eq. (6) could be replaced with a wavelength-by-wavelength Fourier representation. The standard frequency-domain view arises from switching the order of integration to integrate first over space and then over frequency. The wavelength-specific Strehl ratio represents the normalized peak of the monochromatic point-spread function (PSF), which is equivalent to the integral over the wavelength-specific optical transfer function. Care must be taken combining wavelength-specific Strehl ratios, which have wavelength-specific normalizations, into a polychromatic Strehl ratio.

A key advantage of the time-domain approach is that by integrating over the spectrum first, the characteristics of the light are all precomputed, and the effects of the illumination and the lens can be treated separately. The frequency-domain approach computes the effect of the lens and illumination together for every possible wavelength and then assimilates the constitutive spectrum.

3. Limits of performance

The limited range of element time delays restricts the domain of the autocorrelation accessible for a given pair of rings on the aperture $({\xi ,\xi^{\prime}} )$. In particular, $|{\tau (\xi )- \tau ({\xi^{\prime}} )} |\le T$. The best achievable lens would be the maximum of Eq. (7) over all lens designs $\tau (\xi )$. Currently, inverse-design procedures, like that described in [1], attempt to iteratively optimize a given design. However, the resulting prescriptions are not easily characterized nor are they proven globally optimal. To our knowledge, there is no known and proven globally optimal design, which is why the bounds sought in this paper would be instructive.

To identify performance bounds to all designs, we look at potentially inconsistent designs by independently optimizing $\tau (\xi )$ and $\tau ({\xi^{\prime}} )$. In effect, for each aperture ring $\xi $, we credit maximal constructive interference at the focus from all other aperture rings $\xi ^{\prime}$, regardless of mutual compatibility with constructive interference between those other rings. In other words, $\tau ({\xi^{\prime}} )$ in the inner integral in Eq. (7) will in general be a different function for each value of $\xi $ in the outer integral.

Intensity is a linear sum of contributions, each from the conjugate product of the scalar field at a pair (including degenerate pairs) of aperture points. Consider three points on the aperture with scalar fields A, B, and C. The intensity computed from the Huygens-Fresnel integral adds the contributions AA*, AB*, AC*, BA*, etc. A real design only has three degrees of freedom in these contributions, namely the element properties at A, B, and C. If we ignore the fact that A must have the same properties when counting the contributions from AB* and CA*, we may be able to improve apparent performance even though such an element is unrealizable. The element with AB* and CA* independently optimized is guaranteed to perform at least as well as the element where A is constrained by reality to be identical in AB* and CA*. In Appendix A we consider a bound based on independently optimizing the contribution of every pair of points, treating AA*, AB*, BA*, etc. each as a fully independent variable. In what follows, we reimpose a measure of consistency and require A to be the same for AA*, AB*, and AC*, but allow it to differ for BA* and CA*.

Because these various unphysical optimization spaces include the real design space, any optimum over the unphysical spaces is guaranteed to perform at least as well as the optimum over the realizable design space. There is no guarantee that the bounds will not wildly overpredict performance—they could be very loose—but they cannot underpredict performance. In other words, the bounds are guaranteed to be rigorous but not tight. Later in the paper, we will show comparisons between the bounds and published elements and observe that the bounds are all within an order of magnitude of practical performance and often much closer.

Returning to the mathematical derivation, from Eq. (7), the Strehl ratio

As long as $G({\zeta ;T} )$ is strictly non-negative, which it should be so long as the lens has the ability to generate a $2\pi $ phase shift at the center frequency, the integrand in Eq. (9) is non-negative, and the integrity of the bound can be maintained while eliminating dependence of the inner integral on ${\tau _1}$ by expanding the limits of the inner integral to the maximum possible extent and using the minimum value of ${\tau _1}$ in the denominator. Thus,

For insight, let us look at the terms in Eq. (13). The first term in the {} in Eq. (13) represents the refractive contribution to focusing. The harmonic order $\mu $ places a limit on the region of the lens that can contribute to coherent superposition of light waves without resorting to phase folding. The refractive contribution consists of the incoherent superposition of all such regions. Hence, the refractive contribution comes from assuming $g({\psi ;\mu } )= 0$ outside of $[{ - \mu /2,\mu /2} ]$.

The second term in the {} in Eq. (13) represents the diffractive contribution, which comes from coherent superposition of successive wavefronts from different parts of the aperture and relies on phase folding. For a given temporal coherence of light, $g({\psi ;\mu } )$ quantifies the maximum possible diffractive contribution as a function of $\psi $, the number of waves of correction required, and $\mu $, the available correction without phase folding.

Next, we look at some limiting approximations. If the obliquity factor $\chi = f/\sqrt {{f^2} + {\rho ^2}} $ can be neglected, Eq. (13) reduces to

The scaling term $\mathrm{\Delta }\ell /\mu $, apparent in Eqs. (14) and (17), is the minimum number of coherent subapertures. The refractive contribution to the Strehl ratio will scale with the inverse of the number of incoherently combined refractive subapertures.

4. Band-limited white light

As a concrete example of the bounds derived in the previous section, we treat band-limited white light with center frequency ${\nu _0}$ and bandwidth $\mathrm{\Delta }\nu $. For band-limited white light with $B \equiv \mathrm{\Delta }\nu /{\nu _0}$,

 

Fig. 2. For band-limited white light, the figure shows ${\boldsymbol \gamma ^{\prime}}({{\boldsymbol \psi }/{{\boldsymbol \nu }_0}} )$, the CDC, where ${\boldsymbol \psi } = {{\boldsymbol \nu }_0}{\mathbf \Delta }{\boldsymbol t}$, along with ${\boldsymbol g}({{\boldsymbol \psi };{\boldsymbol \mu }} )$, the sliding maximum of ${\boldsymbol \gamma ^{\prime}}({{\boldsymbol \psi }/{{\boldsymbol \nu }_0}} )$ over the interval $\left[ {{\boldsymbol \psi } - \frac{1}{2}{\boldsymbol \mu },{\boldsymbol \psi } + \frac{1}{2}{\boldsymbol \mu }} \right]$, where ${\boldsymbol \mu }$ is the harmonic order of the lens. Note that ${\boldsymbol \gamma ^{\prime}}({{\boldsymbol \psi }/{{\boldsymbol \nu }_0}} )= {\boldsymbol g}({{\boldsymbol \psi };0} )$. From left to right, the panels show fractional bandwidth ${\boldsymbol B} = {\mathbf \Delta }{\boldsymbol \nu }/{{\boldsymbol \nu }_0}$ of 0.1, 0.5, and 1. Each panel shows results for ${\boldsymbol \mu } = $ 1, 5, and 10.

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Figure 3 shows the upper bound Strehl ratio for lenses of various sizes, harmonic orders, bandwidths, and f# under white-light illumination centered in the visible band around ${\lambda _0} = c/{\nu _0} = $ 510 nm. The full band from 400 to 700 nm corresponds to a fractional bandwidth of B = 0.55. The Strehl ratio upper bound decays a little more slowly than the reciprocal of the lens diameter, which would be the scaling predicted by Eq. (18).

 

Fig. 3. Upper bound for Strehl ratio as a function of lens diameter. (a) Various harmonic orders ${\boldsymbol \mu }$, for f/1 and band-limited white light in the visible spectrum from 400 to 700 nm corresponding to B = 0.55 and ${{\boldsymbol \lambda }_0} = $ 510 nm (b) Various fractional bandwidth B, for f/1, ${\boldsymbol \mu } = 2$, and ${{\boldsymbol \lambda }_0} = $ 510 nm (c) Various f#, for ${\boldsymbol \mu } = 2$, B = 0.55, and ${{\boldsymbol \lambda }_0} = $ 510 nm. In the far upper left corner of (c), the kinks in the curves indicate where the alternate bound from Appendix A transitions to the bound in the main body of this paper.

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We intended to compare these bounds with the performance of lenses in the recent literature. However, we found very few lenses making use of diffractive optics or metasurfaces with $\mathrm{\Delta }\ell /\mu $ significantly greater than one. In other words, there were few lenses at low enough f# and large enough diameter to require a significant number of subaperture zones (regions between phase foldings). References [3] and [13] compiled lists of recently published metalens designs. We computed the effective $\mathrm{\Delta }\ell /\mu $ for all the designs and found that they range from 0.3 to 8.5. This fits with their examination of the bandwidth limits of near-diffraction-limited lenses. Meem et al. have published results for a millimeter-scale fast broadband DOE [1].

Table 1 shows results for the lens from [1] as well as those cataloged by [3] and [13] for which $\mathrm{\Delta }\ell /\mu > 2$, $\mu \ge 1$, ${\ell _2} > \mu $, and $D/{\lambda _0} \ge 50$. Where available, we compare to reported Strehl ratios for the same lenses. For comparison of the bound to a physically realistic design, we also show the Strehl ratio for a dispersionless kinoform lens of order $m = \mu $ with $M = \mathrm{\Delta }\ell /\mu $ zones, which, from a modified version of Eq. (7) neglecting obliquity, is

Table 1. Survey of published lenses

View Table

5. Discussion

We use the Strehl ratio as our measure of performance but note that there are many other useful measures of optical performance. Although all definitions of the Strehl ratio with which we are familiar use the actual intensity at the focal spot relative to the intensity produced by a perfect lens, we have encountered a number of different normalization standards. In our usage, we normalize relative to the focal intensity of a perfect lens of the same aperture with 100% transmission. Some authors normalize with respect to a perfect lens of the same total transmission as the real lens—this is similar to the value attained by integrating the area under the modulation transfer function, which is normalized to have a value of one at zero spatial frequency (corresponding to total transmission), although there is also some nuance associated with 1D vs. 2D Strehl ratios [15]. Both these definitions preserve the connection between Strehl ratio and contrast. Some authors normalize the Strehl ratio with respect to a perfect lens of the same aperture with the same amount of total energy in the focal spot. This definition severs the connection with contrast because it normalizes out widely diffused light, but it preserves some connection to the spot size because, for the same total focused energy, a higher peak intensity corresponds to a tighter spot. The focusing efficiency (fraction of incident light confined to the focal spot) connects this measure of Strehl ratio to more classical definitions.

When dealing with refractive lenses, one would not expect much difference between the definitions because transmission is typically high, and loss of contrast is typically due to expansion or smearing of the focal spot. In diffractive optics and meta-optics, much loss of contrast can be due to zeroth-order diffraction and widely scattered light, so substantial loss of contrast can occur while retaining a near-diffraction-limited focal spot. With these lenses, the different Strehl ratio definitions can yield widely varying values with quite different interpretations and performance implications.

We neglected chromatic dispersion in the bounds as discussed in Appendix C. We also neglected off-axis effects and polarization in our analysis. For a system optimized for on-axis Strehl ratio, off-axis performance would likely be worse than on-axis. Shadowing effects, reflective losses, effective aperture size, and obliquity losses would all rise off-axis. Likewise, polarization sensitivity would not provide an opportunity for better optimization, but a design optimized for designated polarization conditions would likely perform more poorly under off-nominal polarization conditions. We have not considered developing bounds for performance off-axis or under various polarization assumption, but we believe that the bounds reported here would apply under such circumstances.

Off-axis performance and polarization-sensitive performance are important conditions differentiating metalenses from DOEs. Metalenses may have independently optimizable polarization and angle-dependent behavior, in which case they might be able to outperform DOEs under the right circumstances. Metalenses without independently optimizable angle and polarization behavior could significantly underperform relative to DOEs. Nothing in this analysis sheds light on these differences. Any bounding of performance under these off-nominal conditions would require detailed information about the limits of design freedom of the metalenses.

Application of the bounds to dispersion-engineered metalenses requires some explanation. Many metalenses utilize design libraries with more degrees of freedom than a typical diffractive element, allowing simultaneous tuning of phase delay and dispersion. Dispersion engineering refers to the combined use of these freedoms. In the framework of this analysis, the equivalent $\mu $ of a metalens is proportional to the largest effective time delay possible at the center wavelength while controlling dispersion. This is given by the maximum achievable group delay the meta-element library is capable of while retaining 2π phase delay capability. The metalens’s equivalent harmonic order is given by $\mu = {\nu _0}\frac{{\partial \phi }}{{\partial \omega }}$, where $\frac{{\partial \phi }}{{\partial \omega }}$ is the maximum group delay with 2π phase-delay capability. With this definition of $\mu $, the bounds derived here are equally applicable to metalenses.

A principal advantage of the approach presented in this paper is that it allows separation of the light properties and lens prescription. The CDC of broadband light $\gamma ^{\prime}$ can be precomputed and depends only on the properties of the illumination. Via the maximum time delay achievable by a DOE or metalens, $\gamma ^{\prime}$ can then easily be converted into the maximum achievable CDC for any required optical path difference. At that stage, the material system (design constraints) and light properties have been folded into the function $g({\psi ;\mu } )$. Then, the maximum Strehl ratio for any given focal length and aperture size can be computed as an integral over the function $g({\psi ;\mu } )$, where the optical requirements enter via ${\ell _1}$ and ${\ell _2}$. It is this separation that enables the bounding by allowing direct treatment of the broadband performance of the lens.

In frequency-domain approaches, the performance of the element is computed first as a function of the illumination wavelength and then integrated over the spectrum. This leaves no clear way to produce useful bounds—because a DOE can theoretically achieve perfect monochromatic performance, optimizing one wavelength at a time would produce a perfect lens. DOE designers must either optimize at one particular wavelength or use inverse-design procedures where they iteratively tweak the design and compute the broadband (or discrete multi-wavelength) performance by integrating over the spectrum. A time-domain approach like that described here might find practical application in speeding up inverse-design algorithms for broadband-diffractive elements using Eq. (7) with a precomputed $\gamma ^{\prime}$.

Intensity is a nonlinear function of the scalar field over the aperture, but it is a linear function of the Hermitian products of scalar fields at pairs of aperture positions (see Eq. (5)). Because the intensity can be computed as a weighted sum of these products (weights depend on the spreading of spherical wavelets and obliquity), the total intensity is bounded by the weighted sum over the maximum possible intensity contribution of each pair, which is proportional to $g({\psi ;\mu } )$. Pairs of aperture points cannot be designed independently—in reality, each aperture point is designed, and the pairs come from all combinations of points. In the bound described in Appendix A, no additional constraints are placed on the design independence of aperture point pairs. In the bound described in the main body of this paper, the independence of aperture point pairs is partly constrained to tighten the bound relative to realizable designs. In either case, the computed intensity based on independent optimization of aperture point pairs is guaranteed to match or exceed the performance of any actual lens. The bound is not guaranteed to be tight, but as shown in Table 1, the bound is within a factor of 3 for known, relevant achievable lenses (either a theoretical kinoform design or a specific design from the literature).

6. Conclusion

We used time-domain analysis to derive rigorous bounds for the performance of ultra-thin lenses as measured by the Strehl ratio. The bounds are broadly applicable because they are built on a small number of fundamental optical principles and basic assumptions. They apply equally well to classical, higher-order, and inverse-designed DOEs as well as metalenses. We computed the bounds for band-limited white light, but the bounding formulation allows computation of bounds for any given illumination. A survey of published lenses showed no violations of the bound and order-of-magnitude tightness. As shown in Fig. 3, even this conservative bound reveals considerable limitations in the expansion of DOEs and metalenses to large aperture and/or low f# applications. Diffraction-limited focal spot size may be achievable but with extremely low contrast ratio. While there may be specific applications in which such performance characteristics are acceptable, we feel that it is important to acknowledge these limitations when considering future aspirations for ultra-thin flat lens design. Aside from bounding, the time-domain method of computation may enable faster broadband inverse lens design.

Appendix A: Alternate bound

The bound in the main body of this paper takes one aperture ring at a time and bounds the net intensity contribution from that ring based on a lens optimized for contributions involving that ring. To calculate such a bound, however, a conservative approximation was made between Eqs. (9) and (10). For small lenses, that approximation can loosen the bound considerably and even generate Strehl ratio bounds greater than one.

Here we consider an alternate bound that relaxes some constraints by independently maximizing the intensity contribution from every pair of rings. Mathematically, this allows ${\tau _1}$ to vary in the inner integral, whereas the original bound forced ${\tau _1}$ to remain consistent throughout the inner integral. The alternate bound will underperform the main paper’s bound for large apertures but will outperform it for certain small apertures. The alternate bound is always less than one. Results reported in this paper always take the tighter of the two bounds.

Beginning with Eq. (8),

(22)$$\mathrm{\Phi } \le {\left( {\ln \frac{{{\ell_1}}}{{{\ell_2}}}} \right)^{ - 2}}\mathop \smallint \nolimits_{{\ell _2}{\lambda _0}/c}^{{\ell _1}{\lambda _0}/c} \; \frac{{d\xi }}{\xi }\mathop {\max }\limits_{{\tau _1}} \mathop \smallint \nolimits_{{\ell _2}{\lambda _0}/c}^{{\ell _1}{\lambda _0}/c} \; \frac{{d\xi ^{\prime}}}{{\xi ^{\prime}}}\mathop {\textrm{max}}\limits_{{\tau _2}} {\; }\gamma ^{\prime}({{\tau_1} - {\tau_2} + \xi - \xi^{\prime}} ), $$

and relaxing the consistency constraint on ${\tau _1}$ in the inner integral, we can move the maximization with respect to ${\tau _1}$ into the inner integral,

(23)$$\mathrm{\Phi } \le {\left( {\ln \frac{{{\ell_1}}}{{{\ell_2}}}} \right)^{ - 2}}\mathop \smallint \nolimits_{{\ell _2}{\lambda _0}/c}^{{\ell _1}{\lambda _0}/c} \; \frac{{d\xi }}{\xi }\mathop \smallint \nolimits_{{\ell _2}{\lambda _0}/c}^{{\ell _1}{\lambda _0}/c} \; \frac{{d\xi ^{\prime}}}{{\xi ^{\prime}}}\mathop {\textrm{max}}\limits_{{\tau _1},{\tau _2}} {\; }\gamma ^{\prime}({{\tau_1} - {\tau_2} + \xi - \xi^{\prime}} ).$$

Note that $|{{\tau_1} - {\tau_2}} |< T$, so

(24)$$\mathop {\textrm{max}}\limits_{{\tau _1},{\tau _2}} {\; }\gamma ^{\prime}({{\tau_1} - {\tau_2} + \xi - \xi^{\prime}} )= G({\xi - \xi^{\prime};2T} ). $$

Switching to nondimensional form including the dummy variable $\xi $ in the outer integral,

(25)$$\mathrm{\Phi } \le {\left( {\ln \frac{{{\ell_1}}}{{{\ell_2}}}} \right)^{ - 2}}\mathop \smallint \nolimits_{{\ell _2}}^{{\ell _1}} \; \frac{{d\xi }}{\xi }\mathop \smallint \nolimits_{{\ell _2} - \xi }^{{\ell _1} - \xi } \; \frac{{d\zeta }}{{\zeta + \xi }}{\; }g({\zeta ,2\mu } ). $$

The order of integration can be switched. Note that because we have not needed to inflate the limits of the inner integral, the picture looks like that in Fig. 4 with the center band in the right-hand panel collapsed. Thus, only two integration regions are required, leading to

(26)$$\mathrm{\Phi } \le {\left( {\ln \frac{{{\ell_1}}}{{{\ell_2}}}} \right)^{ - 2}}\left[ {\mathop \smallint \nolimits_0^{\mathrm{\Delta }\ell } d\zeta {\; }g({\zeta ,2\mu } )\mathop \smallint \nolimits_{{\ell_2}}^{{\ell_1} - \zeta } \; \frac{{d\xi }}{\xi }\frac{1}{{\zeta + \xi }}{\; } + \mathop \smallint \nolimits_{ - \mathrm{\Delta }\ell }^0 d\zeta {\; }g({\zeta ,2\mu } )\mathop \smallint \nolimits_{{\ell_2} - \zeta }^{{\ell_1}} \; \frac{{d\xi }}{\xi }\frac{1}{{\zeta + \xi }}{\; }} \right]$$

Using the approach described in the Appendix B (Eqs. (30) and (31)), this can be integrated to form the alternate bound,

(27)$$\mathrm{\Phi } \le 2{\left( {\ln \frac{{{\ell_1}}}{{{\ell_2}}}} \right)^{ - 2}}\mathop \smallint \nolimits_0^{\mathrm{\Delta }\ell } d\zeta {\; }g({\zeta ,2\mu } )\frac{1}{\zeta }\left( {\ln \frac{{({{\ell_1} - \zeta } )({{\ell_2} + \zeta } )}}{{{\ell_1}{\ell_2}}}} \right). $$

 

Fig. 4. Schematic of change of order of integration. The domain of integration is shown on the left. In Eq. (28), the domain is integrated along vertical strips. Changing the order of integration amounts to switching to integration along horizontal strips in Eq. (29), for which the domain must be broken into the three subdomains shown on the right.

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Appendix B: Derivation details

The following provides details of the derivation of Eq. (13) from Eq. (12) (reproduced here as Eq. (28)). Switching the order of integration requires observing that the two integrals represent an integration over a two-dimensional region depicted in Fig. 4.

(28)$$\mathrm{\Phi } \le {\left( {\ln \frac{{{\ell_1}}}{{{\ell_2}}}} \right)^{ - 2}}\mathop \smallint \nolimits_{{\ell _2}}^{{\ell _1}} \; \frac{{d\xi }}{\xi }\mathop \smallint \nolimits_{{\ell _2} - \xi - \mu /2}^{{\ell _1} - \xi + \mu /2} \; \frac{{d\psi }}{{\psi + \xi - \mu /2}}{\; }g({\psi ;\mu } ). $$

Breaking the domain into three subdomains and changing the order of integration in Eq. (28) yields

(29)$$\mathrm{\Phi } \le {\left( {\ln \frac{{{\ell_1}}}{{{\ell_2}}}} \right)^{ - 2}}\left[ \begin{array}{l} \int_{\mu /2}^{{\ell_1} - {\ell_2} + \mu /2} {d\psi {\; }g({\psi ;\mu } )} \int_{{\ell_2}}^{{\ell_1} - \psi + \mu /2} {\frac{{d\xi }}{\xi }\frac{1}{{\psi + \xi - \mu /2}} + } \\ \int_{ - \mu /2}^{\mu /2} {d\psi {\; }g({\psi ;\mu } )} \int_{{\ell_2}}^{{\ell_1}} {\frac{{d\xi }}{\xi }\frac{1}{{\psi + \xi - \mu /2}}} + \\ \int_{{\ell_2} - {\ell_1} - \mu /2}^{ - \mu /2} {d\psi {\; }g({\psi ;\mu } )} \int_{{\ell_2} - \psi - \mu /2}^{{\ell_1}} {\frac{{d\xi }}{\xi }\frac{1}{{\psi + \xi - \mu /2}}} \end{array} \right]. $$

Partial fraction expansion of the inner integrand results in

(30)$$\frac{1}{\xi }\; \frac{1}{{\psi + \xi - \mu /2}} = \frac{1}{{\psi - \mu /2}}\left( {\frac{1}{\xi } - \frac{1}{{\psi + \xi - \mu /2}}} \right), $$

which can be integrated exactly,

(31)$$\smallint d\xi \frac{1}{{\psi - \mu /2}}\left( {\frac{1}{\xi } - \frac{1}{{\psi + \xi - \mu /2}}} \right) = \frac{1}{{\psi - \mu /2}}\left[ {\ln \xi - \ln \left( {\psi + \xi - \frac{\mu }{2}} \right)} \right] = \frac{1}{{\psi - \mu /2}}\ln \frac{\xi }{{\psi + \xi - \mu /2}}$$

Exact computation of the inner integrals in Eq. (29) leads to

(32)$$\mathrm{\Phi } \le {\left( {\ln \frac{{{\ell_1}}}{{{\ell_2}}}} \right)^{ - 2}}\left[ \begin{array}{l} \int_{\mu /2}^{{\ell_1} - {\ell_2} + \mu /2} \begin{array}{l} \frac{{d\psi }}{{\psi - \mu /2}}{\; }g({\psi ;\mu } )\ln \frac{{({{\ell_1} - \psi + \mu /2} )({\psi - \mu /2 + {\ell_2}} )}}{{{\ell_1}{\ell_2}}} + \\ \int_{ - \mu /2}^{\mu /2} {\frac{{d\psi }}{{\psi - \mu /2}}{\; }g({\psi ;\mu } )\ln \frac{{{\ell_1}({\psi - \mu /2 + {\ell_2}} )}}{{{\ell_2}({\psi - \mu /2 + {\ell_1}} )}} + } \end{array} \\ \int_{{\ell_2} - {\ell_1} - \mu /2}^{ - \mu /2} {\frac{{d\psi }}{{\psi - \mu /2}}{\; }g({\psi ;\mu } )\ln \frac{{{\ell_1}({{\ell_2} - \mu } )}}{{({\psi - \mu /2 + {\ell_1}} )({{\ell_2} - \psi - \mu /2} )}}} \end{array} \right]$$

Taking the $\psi \to - \psi $ for the dummy variable in the last integral allows the first and last integrals to be combined. The second integral can be simplified given that $g({\psi ;\mu } )= 1$ for $|\psi |\le \mu /2$ because the CDC is always unity for zero time offset, which is achievable if the free-space time-of-flight difference is fully correctable by the element. Together, these lead to Eq. (13).

Appendix C: Material dispersion

Can dispersion increase the performance in a way that undermines the rigor of the bounds? If not, can a tighter set of bounds be derived by accounting for dispersion behavior? We believe that dispersion will reduce both the refractive and diffractive performance of the lens, such that the bounds derived above remain rigorous. In this context, we construe refractive performance as the contribution from subapertures over which the free-space path difference is fully compensated (without phase folding and neglecting dispersion) by the lens and diffractive performance as the contribution from constructive interference between such subapertures requiring phase folding.

Let us treat the refractive contribution first. With dispersion-free behavior, the lens is able to fully compensate for free-space optical-path differences within the refractive capacity of the material system ($g({\psi ;\mu } )= 1$ for $|\psi |< \mu /2$). Any dispersion would cause the CDC to degrade for any $\psi \ne 0$, although for thin lenses, the dispersive effect is likely to be small. Introducing the Abbe number $V \equiv ({n({{\nu_0}} )- 1} )/\mathrm{\Delta }n$, where $\mathrm{\Delta }n$ is the span of refractive index across the waveband, the maximum dispersive contribution to phase error at the edge of the waveband would be $\frac{{\pi \mu }}{{2V}}({2 + B} )$. For fractional bandwidth B << 2, this is approximately $\frac{{\pi \mu }}{V}$. When the harmonic order of the lens is much less than the Abbe number $\mu \ll V$, there is little contribution of dispersion to reducing the refractive contribution of the lens.

For the diffractive contribution, we consider what happens at phase jumps of 2π at the center wavelength. The dispersive effect can be thought of as analogous to expanding the bandwidth. A 2π phase jump at the center wavelength corresponds, at other wavelengths, to some deviation from 2π, increasing with increasing deviation from the center wavelength. Shorter wavelengths will experience a larger than 2π phase jump, and longer wavelengths will experience a smaller than 2π phase jump. This occurs because the same jump in time delay will correspond to a larger number of wavelengths of optical path at the shorter wavelengths. Dispersion, which increases the index at shorter wavelengths, will further increase the phase jump at short wavelengths and further decrease the phase jump at longer wavelengths. Thus, the spread of phase jumps experienced over the spectrum will increase just as it would for higher bandwidth, and the CDC will decrease for light at the focal point from aperture points on opposite sides of a phase jump.

The phase discrepancy between the center frequency and the edge of the band across jumps of maximum size can be expressed as $\mathrm{\Delta }\phi = \pi \mu \left[ {B + \frac{{B + 2}}{{2V}}} \right]$, where the second term in the brackets represents the marginal contribution to phase error from dispersion. Thus, for B << 2, the dispersive-phase-error contribution to diffractive focusing will be at most $\mathrm{\Delta }\phi \approx \pi \mu /V$. As long as B >> 1/V, however, decoherence due to broadband illumination will dominate diffractive losses before dispersion begins to contribute meaningfully. For higher order diffractive contributions accumulated over numerous phase jumps, the phase discrepancy for M such jumps will be $\mathrm{\Delta }\phi = M\pi \mu \left[ {B + \frac{{B + 2}}{{2V}}} \right]$. Accounting for the maximum phase discrepancy over the entire lens, $\mathrm{\Delta }\phi = \pi ({\mathrm{\Delta }\ell - \mu } )\left[ {B + \frac{{B + 2}}{{2V}}} \right]$. In either case, broadband rather than dispersive losses are likely to dominate diffraction losses for B >> 1/V for reasonable Abbe number and bandwidth.

Although dispersion-engineered metalenses may be able to approximate dispersion-free behavior, unless a DOE would operate in a regime where material dispersion effects significantly affect the refractive contribution, we would not expect significant differences between DOEs and metalenses for similar designs and similar harmonic orders. In sum, despite ignoring dispersion, since dispersion should detract from performance, we believe our bounds will remain rigorous, and since dispersive effects are expected to be small, we believe the tightness of the bounds will not suffer much.

Nevertheless, one could construct a version of $g({\psi ;\mu } )$ accounting for dispersion, representing the best achievable CDC given material/manufacturing constraints as a function of free-space optical-path difference. In relation to the stepped appearance of $g({\psi ;\mu } )$ in Fig. 2, we would expect the dispersive effect on refraction to turn the level steps into sloped ones and the dispersive effect on diffraction to create larger jumps between the steps as shown at exaggerated scale in Fig. 5.

 

Fig. 5. Qualitative effect of dispersion on best achievable CDC ${\boldsymbol g}({{\boldsymbol \psi };{\boldsymbol \mu }} )$.

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Funding

Defense Advanced Research Projects Agency (EXTREME Program).

Acknowledgments

Many discussions with Joe Mait and Predrag Milojkovic helped clarify this paper. The views, opinions, and findings should not be construed as representing the official position of either the Department of Defense or the sponsoring organization. Approved for public release, distribution unlimited.

Disclosures

The author declares that there are no conflicts of interest related to this article.

Data availability

No data other than those directly cited or appearing in-full were generated or analyzed in the presented research.

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