## Abstract

We proposed the use of relative encircled power as a measure of
focusing efficiency [Optica **7**, 252
(2020) [CrossRef] ]. The
comment [Optica **8**, 1009
(2021) [CrossRef] ] has raised useful
questions, which we address briefly here and provide some
clarifications.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

In the pioneering work of Ref. [1], the authors demonstrated an impressive 80% diffraction efficiency. With no intention of minimizing this achievement, we first note that diffraction efficiency and focusing efficiency are not the same, and focusing efficiency was unfortunately not reported in Ref. [1]. When a diffractive element (including a lens) is illuminated by a plane wave, the transmitted light may be decomposed into basis functions. In the case of lenses, these basis functions are usually spherical waves, where the diffraction orders represent the center of these spheres (positive and negative signs refer to converging and diverging spherical waves, respectively) [2]. The diffraction efficiency is the ratio of power carried by one of these spherical waves to the total incident power. When a lens is illuminated normally by a plane wave, the intensity in the focal plane can be written as the sum of contributions from the different diffraction orders, $I(\rho) = {I_0} + {I_{+ 1}}(\rho) + {I_{- 1}}(\rho) + \ldots$, where the subscript denotes the order of diffraction, and $\rho$ is the radial coordinate in the focal plane. Let us take the example of a perfect lens, where the diffraction efficiency is 100% (into the $+1$ order), $I(\rho) = {I_{+ 1}}(\rho)$ and from Fourier optics, we know that this is the Airy function, ${I_{A0}}{\left({\frac{{2{J_1}(x)}}{x}}\right)^2},$ where, ${I_{A0}}$ is the incident intensity, ${J_1}(x)$ is the Bessel function of the first kind, and $x = \frac{{\pi \rho}}{{\lambda f\#}},$ where $\lambda$ is the wavelength and $f\#$ is the $f$-number of the lens. Focusing efficiency was first introduced as the fraction of incident power within a focal spot of radius equal to 3 times its full width at half-maximum (FWHM) [3]. We can generalize this by calculating the power within a spot of a general radius, $R$ as

The subscript $A$ refers to the ideal lens, whose point spread function (PSF) is the Airy function. From this expression, we can readily show that the focusing efficiency when $R = {\rm{FWHM}}$ of

the PSF is 50% (Fig. 1).**In other words, a lens with 100% diffraction efficiency has focusing efficiency of 50% (measured with**$\boldsymbol{R}\, \boldsymbol{=}\, {\textbf{FWHM}}$

**).**The focusing efficiency will be ${\gt}{\sim}{{90}}\%$, if $R = {{3}}\times{\rm FWHM}$. This example illustrates that reporting focusing efficiency without the value of $R$ is not sufficient to give the full picture.

The commentors take issue with our critique regarding Ref. [4]. The gist of our comment was that the focusing efficiencies reported in Ref. [4] are higher than what is theoretically predicted possible with the concept of unit-cell design, as clearly explained in Ref. [5], both for monochromatic and broadband cases. This is a good example of a confusion arising from inconsistency in the size of the focal spot. Since the size of the focused spot used to compute focusing efficiency was not explicitly reported in Ref. [4], it is as well possible that the focusing efficiency reported there used a different focal spot than what was used in Ref. [5] for calculation of the upper bounds of focusing efficiency (where radius corresponding to the first zero in the PSF was used). This example simply proves our point in Ref. [6] that relative encircled power is a better metric for a lens than just a single value of focusing efficiency. In fact, we would like to correct the comment that relative encircled power is only a measure of resolution, which is clearly not correct, as can be seen by the example of the Airy PSF above. In fact, the relative encircled power is a measure of both resolution and efficiency.

Next, we acknowledge the criticism from the commentors that the relative encircled power in Ref. [6] was overestimated because the total incident power was underestimated due to the limited frame size of our recorded image. We also note that this same mistake is likely present in the vast majority of the flat-lens publications, including the ones cited by the commentors in their review paper. For example, one can examine Fig. 4 of Ref. [3] or Fig. S1 of Ref. [7] to note that the total incident power is underestimated due to the limited field of view of the microscope objectives used in both cases, which in turn leads to an overestimation of the measured focusing efficiencies.

The commentors stated, referring to our work that “*1-cm,
0.9-NA metalens* (Ref. 2 in [6])
*is called small.”* There are three reasons why we
chose to ignore the metalens referred here. First, we note that this lens has ${\rm{NA}} = {0.78}$ and not 0.9. Second, this metalens is a
negative lens. Third, no experimental validation of the PSF or modulation
transfer function (MTF) of this lens was provided that could validate the
reported NA.

Finally, the commentors brought up an observation about a lack of scale in Fig. 2(d). We apologize for this oversight, as we naïvely assumed that the standard Air Force chart image is well known. Our results showed that the lines in Group 7, Element 6 (size ${\sim} 2 \;{\unicode{x00B5}}{\rm m}$) were resolved, which is consistent with the reported PSF and MTF data.

## Funding

National Science Foundation (ECCS 1936729); Office of Naval Research (N000141512316, N000141912458, N66001-10-1-4065).

## Acknowledgment

We would like to thank the authors of the comment for their thoughtful remarks. Discussion with O. Miller is also gratefully acknowledged.

## Disclosures

RM: Oblate Optics (I, E, P).

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