<b>Flat Field Curvature Using Flat Metalenses</b>

<p dir="ltr">A short presentation on eliminating the meniscus singlet lens using metasurfaces on a thick substrate.</p><p dir="ltr"><b>Updates (Sep-22-2025):</b></p><ol><li>A second revision (Rev. 2) slide deck has been uploaded to summarize the notes from below. This is a work in progress.</li></ol><p dir="ltr"><b>Updates (Sep-21-2025):</b></p><ol><li>In thinking further, the experimental data for the 0.8 NA metalens in the paper below by Martins et al show that the quadratic phase profile resulted in a point spread function (PSF) with ~ twice the FWHM (Full Width at Half Maximum) of the hyperboloidal phase (actually, it should be referred to as the spherical wave phase profile). Yet if one models a bulk flat lens with high index and high radius, and at 0.8 NA, the spherical aberration would be much larger, resulting in much larger PSF size. This implies that flat metalenses probably cannot be wholly modelled by high index low curvature bulk lenses. One must use proper software that models the actual nanostructures. Something else happens at the interface between the substrate and nanostructures of metasurfaces that cannot be macroscopically assessed by simulating a bulk high index high radius surface.</li><li>One also notes that a flat metalens surface is a surface imposing discontinuous phase change [e.g., N. Yu et al, "<a href="https://doi.org/10.1126/science.1210713" rel="noreferrer" target="_blank">Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction</a>," <i>Science</i> <b>334</b>, 333 (2011)] (they are like antenna). As such, the optical path difference (OPD) between an ideal ray in a reference sphere and an actual ray exiting a metalens depends only on the phase difference between a spherical wave and the imposed phase on the metasurface. For example, if only parallel incident axial rays are considered, then spherical aberration is eliminated completely at the flat metasurface because the "hyperboloidal phase" imposed creates a converging spherical wave. This is equivalent to taking an optical path length (OPL) of R (the radius of the spherical wave) and subtracting it with R, the radius of exiting wave, which is the same as the converging spherical wave's radius. Equivalently, it is taking the difference between the ideal OPD with the imposed OPD [see, e.g., slide 7 of "<a href="https://dx.doi.org/10.13140/RG.2.2.22683.27682" rel="noreferrer" target="_blank">Breaking Down the Metalens Phase Profile</a>"]. In the case of a quadratic phase profile, we can take the difference between the spherical wave phase and the quadratic phase, yielding only higher-order terms -- such as the 4th order term (which is spherical aberration).</li><li>Following from #2 above, the derivation of Seidel coefficients for a flat metalens probably involves imposing a general phase function to 4th order at the exit pupil (this means it involves only the quadratic and 4th order term), so that when converted to a generalized optical path length, the difference between this and a spherical converging wave's radius is the OPD. One then identifies the five Seidel terms (and therefore coefficients) from the math. Thus, the question then becomes what OPD results from only a quadratic phase imposed axially, but there is an oblique beam incident?</li><li>Furthermore, since the actual phase introduced by a thick plano-convex spherical lens (with convex side on the right for a beam entering from left to right) is (2pi/Lambda) x [do - z(r)] x (n - 1), where do is the center thickness of the lens, n is the refractive index, "x" is a scalar product, and z(r) is the sag of the convex side, and if this phase is imposed on the flat surface of a metalens, then we can ask, "What Seidel coefficients arise from this imposition?" Note that this is different from imposing a single axial quadratic phase profile. Here, n is not infinite (e.g., we can let n = 1.5).</li></ol><p dir="ltr"><b>Updates (Sep-20-2025):</b></p><ol><li>In searching further, I did also find this interesting article: [<a href="https://doi.org/10.1021/acsphotonics.0c00479" rel="noreferrer" target="_blank">Martins, Augusto, Li, Kezheng, Li, Juntao et al. (5 more authors) (2020) On Metalenses with Arbitrarily Wide Field of View. ACS Photonics. pp. 2073-2079. ISSN: 2330-4022</a>]. There, the authors discuss their point that a flat metalens surface behaves as if it were a surface/interface with infinite index and infinite radius (zero curvature). In reading this, it seems that there should be a resulting zero for the Petzval curvature. The paper appears to discuss the very problem I have been pondering, which is what happens when a quadratic phase is imposed on a flat metasurface (however, I have not studied the paper fully). <b><i>In thinking further, I feel that since one can impose or design any phase profile onto a metasurface, then the resulting aberration for either the on-axis or oblique rays should depend on that imposed phase profile rather than a model of the metasurface being a flat one with infinite index and zero curvature. </i></b>It also seems important to continue thinking about how to measure the actual Petzval radius or infer it. We should note that the intersection of the sagittal and tangential curves does not necessarily correspond to the Petzval surface location when higher-order astigmatism and field curvature is present.</li></ol><p dir="ltr"><b>Updates (Sep-18-2025):</b></p><ol><li>Slide 5 needs correction. The thickness was primarily to produce the 30 mm effective focal length, because the quadratic term for both surfaces are equal (hence, a thickness was needed to generate sufficient refractive power). However, if the quadratic terms were equal but opposite (so as to make them "biconvex"), then the lens can actually be rather thin.</li><li>One notes that the current model is that of surfaces with high index (n = 10,000,000) with low surface base curvature, hence, yielding essentially zero third-order Petzval curvature. Perhaps for this reason, when the astigmatism was made zero at full field, the sagittal and tangential curves intersected close to the paraxial image plane. Thus, the low Petzval may have been not due to the menisci phase profiles for the lens. Rather, they could perhaps have been due to the high index with low surface curvature (for the current model). However, in a subsequent exercise, even when phase (rather than refraction) was used for a "Binary 2" type of surface in Zemax to model the two surfaces, the total astigmatism could be made zero, and the sagittal and tangential positions could be made to intersect the paraxial image plane. In this case, the 3rd-order astigmatism was nonzero (it was about 11 waves), which implied that there was nonzero 3rd and 5th order Petzval (even though Zemax computed zero Petzval -- perhaps due to the computation method, which likely involves the use of a base radius of curvature for the surfaces). The current (personal) hypothesis is that if a flat surface introduces phase to focus rays, then a different method is needed to arrive at Seidel coefficients, such as the Petzval coefficient, which may not necessarily involve radius of curvature and refractive index. More work here is needed.</li><li>This is a preliminary exercise and did not consider the similarities between flat metalens surfaces and diffractive optical elements (DOEs). In the latter, it is known by way of Sweatt [W. C. Sweatt, "Mathematical equivalence between a holographic optical element and an ultra-high index lens," J. Opt. Soc. Am. <b>69</b>(3), 486-487 (1979)] that a DOE surface is equivalent to an interface with high index and low curvature.</li></ol><p dir="ltr"><br></p>