Apochromatic singlets enabledby metasurface-augmented GRIN lensesJ. NAGAR,†,* S. D. CAMPBELL,† AND D. H. WERNER

Department of Electrical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, USA*Corresponding author: jun163@psu.edu

Received 6 September 2017; revised 20 November 2017; accepted 13 December 2017 (Doc. ID 306305); published 25 January 2018

Chromatic aberrations are a primary limiting factor in thin,high-quality imaging systems. Recent advances in nanoscalemanufacturing, however, have enabled the creation of metasur-faces: ultra-thin optical components with sub-wavelengthfeatures that can locally manipulate the wavefront phase.Meanwhile, there has been renewed interest in GRadient-INdex (GRIN) lenses due to the extended degrees of designfreedom they offer. When combined, these two technologiescan provide unparalleled imaging system performance whilerealizing drastic reductions in size, weight, and power. Throughparaxial theory and full ray tracing we produce a lens singletthat can achieve three-color (apochromatic) correction by em-ploying a metasurface-augmented GRIN. This apochromaticsinglet has the potential for application in high-quality opticalsystems. © 2018 Optical Society of America

OCIS codes: (080.2740) Geometric optical design; (110.2760) Gradient-

index lenses; (260.2030) Dispersion; (160.3918) Metamaterials;

(050.6624) Subwavelength structures.

https://doi.org/10.1364/OPTICA.5.000099

Correction of chromatic aberrations is crucial for high-quality im-aging applications and has been the study of intense research forcenturies [1]. Classical solutions for two- (achromatic) and three-wavelength (apochromatic) correction traditionally require refractivedoublets and triplets, respectively. While multiple optical elementsprovide additional degrees of design freedom and thus higher per-formance, one seeks to reduce the number of elements in a systemto minimize SWaP (size, weight, and power) which is of increasingimportance in a wide variety of applications, including IR imaging.Moreover, additional elements may lead to stricter manufacturingconstraints and alignment tolerances [1]. Therefore, a color-corrected singlet is highly desirable. To this end, aspherical meniscuselements with a blazed zone plate serving as the back surface wereshown to provide achromatic performance comparable to conven-tional doublets while offering considerable SWaP reduction [2]. Inaddition, achromatic singlets have been achieved by exploiting theincreased degrees of design freedom offered by inhomogeneousGRadient-INdex (GRIN) materials [3,4]. Along with polychro-matic aberration correction, GRIN lenses also offer advantages in

terms of high performance imaging over an extremely wide fieldof view [5]. While both conventional and GRIN lenses work byaltering the relative phase of an incident wavefront, these devicesrequire an appreciable size in order to deliver their behavior. In con-trast, metasurfaces comprised of arrays of subwavelength geometricfeatures can provide an abrupt change of phase in an ultrathin sur-face [6]. Furthermore, by intelligently arranging the geometric fea-tures throughout the metasurface any spatially varying phase changecan be achieved, thus enabling the wavefront to be transformed atwill. In fact, monochromatic correction over a wide field of view hasbeen achieved with metasurface components [7]. There have alsobeen attempts to correct for chromatic aberration with metasurfaceseither through dispersion engineering of the nanostructures [8] orcompensating between metallic and structural dispersions [9].While these techniques have reduced chromatic aberrations presentin their designs, their efficacy is heavily dependent on the choice ofnanostructure and require numerical optimization in the designprocess. Nevertheless, these studies showcase the potential for meta-surface-augmented lenses to increase optical performance whilemaintaining a minimal SWaP. However, in order to truly maximizethe performance of these hybrid lenses, one should exploit all pos-sible design degrees of freedom available, namely, the pairing ofmetasurfaces with GRINmaterials. This combination has been usedfor improving the depth of focus of an optical probe [10] and in thedesign of a high-quality beam splitter [11]. However, no work hasbeen done in combining the advantageous aspects of GRINlenses and metasurfaces for the purpose of color correction. Suchmetasurface-augmented GRIN lenses are feasible due to recentbreakthroughs in GRIN lens [12] and metasurface [6] manufactur-ing. This Letter presents a completely general analytical theory forchromatic correction with metasurface-augmented GRIN (MS-GRIN) lenses. Furthermore, it is shown that such hybrid lensescan achieve apochromatic color correction in a thin singlet, sug-gesting a potential for large realizable SWaP reduction. A theoreticalproof is presented for color-corrected MS-GRIN lenses, and its ac-curacy is validated with numerical ray tracing for several exampleconfigurations. To visualize the potential for three-color correctedMS-GRIN lenses, consider the example shown in Fig. 1. Thehomogeneous plano-convex singlet [Fig. 1(a)] possesses chromaticaberrations, while the plano-convex MS-GRIN lens [Fig. 1(b)]exhibits radically improved chromatic performance, a result ofchoosing the appropriate front curvature, GRIN distribution,and metasurface phase distribution. Note that these are cartoons

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to illustrate the concept and that the chromatic aberration inFig. 1(a) is exaggerated for clarity.

To understand the performance of such MS-GRIN lenses, onemust first review the properties of these three components. Ametasurface is a thin two-dimensional structure (of potentiallyarbitrary shape) comprised of subwavelength features that canbe engineered to provide local changes in the relative phase ofan incident wavefront [6]. Therefore, a metasurface can bethought of as an infinitesimally thin phase-gradient surface thatcan be analytically described by the generalized Snell’s law [13]

nt sin�θt� � ni sin�θi� �1

kdϕdρ

; (1)

where ϕ is the phase profile, ρ is the radial distance, and ni, nt andθi, θt are the refractive indices and angles relative to the opticalaxis in the incident and transmitted regions, respectively. For acircular lens of diameter D, to achieve a focal length fm at thenominal wavelength λm, the metasurface must impart the follow-ing phase profile [6]:

ϕ � −2π

λm

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρ2 � f 2

m

q− fm

�: (2)

Then, assuming the source is an on-axis object point at infinityand an f∕# greater than or equal to one, paraxial approximationsmay be employed to find the power of the metasurface

ϕm�λ� ≈λ

λmfm; (3)

where the subscript m denotes metasurface. The metasurfaceAbbe number νm and partial dispersion Pm are given by

νm � ϕ2

ϕ1 − ϕ3

� λ2λ1 − λ3

; Pm � ϕ1 − ϕ2

ϕ1 − ϕ3

� λ1 − λ2λ1 − λ3

; (4)

where the subscripts 1, 2, and 3 refer to the short, center, and longwavelengths in the band of interest, respectively. In the visibleregime, the three wavelengths typically chosen are the hydrogenF line (486.1 nm), helium d line (587.6 nm), and hydrogen Cline (656.3 nm). Using these wavelengths, it follows that νm �−3.45 and Pm � 0.5962, leaving the optical power as the onlydegree of freedom (DOF) associated with the metasurface.The negative Abbe number of the metasurface immediatelystands out here and is desirable not only for chromatic aberrationcorrection but for reducing monochromatic aberrations as well.

Note that a diffractive element such as a kinoform has similar opticalproperties to a metasurface only when using the empirical Sweattmodel [1]. However, metasurfaces, due to their subwavelength unitcell sizes, can truly be modeled as a continuous phase profile.Equations (3), (4) then follow directly using (1), (2) and paraxialapproximations. Furthermore, metasurfaces have a number ofadvantages over traditional diffractive elements including extremelyhigh efficiencies [14], multifunctional capabilities [15], and the abil-ity to explicitly engineer the dispersion [8]. Next, consider a GRINlens of diameter D and thickness T featuring a hybrid radial-axial(i.e., spherical) index distribution represented by

n�r; z� � n0 � Δnr�2

D

�2

r2 � Δnz�1

T

�z: (5)

Its power, Abbe number, and partial dispersion are given by [5]

ϕG � −Δnr2�2

D

�2

T ; νG � Δnr2Δnr1 − Δnr3

;

PG � Δnr1 − Δnr2Δnr1 − Δnr3

; (6)

where Δnri is the radial index difference at the i’th wavelength.The expressions for the GRIN Abbe number and partial disper-sions are calculated as ratios between powers similar to Eq. (4). Itshould be noted that a GRIN lens comprised of a binary materialcomposition whose spatially-dependent index is described bysimple volume-filling fraction mixing rules possesses an effectiveAbbe number that is dependent on the dispersion slopes of its twobase materials. Interestingly, a GRIN’s Abbe number can rangefrom positive to negative even if the Abbe numbers of its constitu-ent materials are both positive. Finally, the power, Abbe number,and partial dispersion of a plano-convex refractive lens with frontradius of curvature Rf are given by [1]

ϕf � nf 2− 1

Rf; νf � nf 2

− 1

nf 1− nf 3

; Pf � nf 1− nf 2

nf 1− nf 3

; (7)

where f is a subscript indicating respect to the front surface andnf is the effective index that must be used when there is an inho-mogeneous surface index distribution. It has been found empiri-cally that evaluating Eq. (5) at a radial position r � 0.48D∕2 toserve as the effective surface index nf in Eq. (7) leads to accurateresults. Interestingly, while comprised of the same base materialsas the GRIN, the surface actually possesses a different Abbe num-ber and partial dispersion than the GRIN volume. Combiningthis with the metasurface gives the designer three componentswith unique Abbe numbers and partial dispersions, which canthen be intelligently combined to achieve apochromatic color-correction, which ensures that the powers at the three designwavelengths are exactly matched. This condition can be satisfiedby the following set of equations [1] assuming a metasurface-backed GRIN with a front radius of curvature:

ϕf � ϕG � ϕm � ϕT ;

ϕfνf

� ϕG

νG� ϕm

νm� 0;

Pfϕfνf

� PGϕG

νG� Pm

ϕm

νm� 0: (8)

Next, substituting Eqs. (3), (4), (6), and (7) into (8) leads toexplicit expressions for the front radius of curvature, thickness,and metasurface power:

Fig. 1. Cartoon examples of (a) a homogeneous lens showing chro-matic aberrations and (b) an MS-GRIN lens exhibiting apochromaticperformance.

Letter Vol. 5, No. 2 / February 2018 / Optica 100

X ≡ νf �PG − Pm� � νG�Pm − Pf � � νm�Pf − PG�;

Rf ���

νf �PG − Pm�X

�ϕT

nf 2− 1

�−1

;

T � �D∕2�22jαr j

�νG�Pm − Pf �

nf 2− 1

�ϕT ;

fm ���

νm�Pf − PG�X

�ϕT

�−1

: (9)

Note, while the chosen plano-convex geometry has but a singlesolution, a continuum of solutions opens up when allowing backsurface curvature and/or introducing a metasurface on the frontsurface. This would enable the engineer to choose the best designfor a specific application by carefully considering trade-offs betweenSWaP, Δn, and curvature, similar to the analysis in [3]. The tradi-tional method of determining the suitability of three potentialmaterial components is to visualize a triangle plotted in P versusν space whose vertices are located at the partial dispersion and Abbenumbers of the three materials, respectively [1]. In order to satisfythe apochromatic conditions, these three points cannot reside on astraight line. A GRIN lens with both a front and back radius ofcurvature offers three degrees of freedom. However, because theGRIN material is a volume fraction mixture of the two materials,the three points will always be on a straight line, regardless of thechoice of materials, index distribution, and curvatures. Therefore,apochromatic color correction is theoretically impossible with abinary-blended GRIN singlet. However, the metasurface (Pm, νm)point only depends on the wavelengths chosen and is independentof material choice, allowing the apochromatic conditions to besatisfied. Furthermore, for minimization of monochromatic aberra-tions, it is desirable for two of the points to lie on a horizontal line,with a large difference in Abbe number between them, while theypossess approximately the same partial dispersions. Mathematically,the triangle area is a measure of the quality of the three chosencomponents for apochromatic correction [1]. A PMMA/PSmixturewith νG � 9.3 and PG � 0.71 has previously been used inGRIN applications [16]; however, examination of Eq. (8) showsthat the apochromatic conditions can only be satisfied if the opticalpowers of the GRIN and front surface are negative and positive,respectively. Unfortunately, these powers fight against each other,which results in larger monochromatic aberrations. Therefore,a better material choice is a H-BAF6/H-ZK5 mixture that possessesa negative GRIN Abbe number νG � −1.43, a partial dispersionPG � 0.7241, and similar coefficients of thermal expansion.

In the case of this material combination, both the GRIN and surfacepowers are positive and the resulting relative monochromatic aber-rations are extremely small. While the proposed material mixture ishypothetical, another goal of this study is to shed light on whatmaterial properties are desirable and inspire new fabrication tech-niques. To this end, recent advancements in GRIN manufacturing,including 3D printing [12,17], have made more material combina-tions accessible for the manufacture of GRIN lenses. For compari-son purposes, a refractive apochromatic triplet consisting ofPilkington-Optiwhite, S-FPL53, and L-BBH1 will be considered.These materials were chosen because they have the largest trianglearea in P versus ν space for conventional materials in the visibleregime using material specifications from [18]. A graphical exami-nation of the relative partial dispersions versus Abbe number for theMS-GRIN and refractive triplet is shown in Fig. 2. One can see thatthe MS-GRIN is a better choice for achieving apochromatic perfor-mance due to its larger triangle area.

The validity of the theory can be tested through the applica-tion of ray tracing. To this end, we employ reTORT (transfor-mation optics ray tracer), an arbitrary-GRIN-capable ray tracerdeveloped in-house [19] that has been rigorously validated againstCode V and OSLO for a wide variety of examples. In particular,reTORT is utilized here to evaluate the performance of aMS-GRIN comprised of H-BAF6/H-ZK5 whose index variationis given by Eq. (5) and restricted to a purely radial configuration.An infinite conjugate on-axis ray bundle is considered. When us-ing the GRIN and metasurface parameters given by the paraxialtheory directly, one finds that there is a slight discrepancy betweenthe theoretical prediction and the ray trace results. This is due tothe paraxial nature of the theory, which does not include aberra-tions or principle plane shifts. Thus, to ensure the apochromaticperformance of the MS-GRIN lens was maximized, a global op-timizer based on the covariance matrix adaptation evolutionarystrategy (CMA-ES) [20] was employed. Nevertheless, the lensparameters predicted by the paraxial theory agree quite well withthe optimized parameters. Figure 3 shows the ratio of the lensparameters predicted by Eq. (9) and those found by the optimizerfor various f∕# configurations. As can be seen, the error for all lensparameters is quite small above around f∕2, and all three ratiosapproach unity as the f∕# increases. Again, these results suggestthe theory provides an extremely accurate estimate of the lensparameters, which can then be used to drastically expedite opti-mizations. Next, the observed focal drift versus wavelength is

Fig. 2. P-ν map for the MS-GRIN and refractive triplet.Fig. 3. Ratio of MS-GRIN parameters predicted by analytical theorywith those optimized by CMA-ES versus f∕#: (a) Rf , (b) T , and (c) ϕm.

Letter Vol. 5, No. 2 / February 2018 / Optica 101

plotted in Fig. 4 for a standard GRIN, the MS-GRIN, the refrac-tive doublet, and the refractive triplet.

The refractive doublet and GRIN are both achromatic, withf∕δf values of −1e3 and 2.8e3, respectively, highlighting thesuperior performance of the GRIN. Meanwhile, the refractivetriplet and MS-GRIN are both apochromatic, with f∕δf valuesof 5.5e3 and 2e5, respectively. Furthermore, the triplet suffersfrom extreme spherochromatism while the MS-GRIN has a muchsmaller amount. To minimize these aberrations, an r4 GRIN termand two aspherical perturbations to the front surface were addedto the MS-GRIN. Moreover, air spacing between elements andtwo aspherical perturbations to both the front and back surfaceswere added to the refractive triplet. Both configurations featurea fully illuminated 12.7 mm lens diameter and are extremely fastoperating at f∕2. The performances of these lenses are comparedin Fig. 5. Specifically, Fig. 5(a) shows the modulation transferfunction (MTF) for both designs compared to the diffractionlimit at λd . As can be seen, the MS-GRIN is nearly diffractionlimited while the triplet suffers from poor performance. Finally,

Fig. 5(b) shows the Strehl ratio (which takes into account all aber-rations of the system) [16] for the two designs versus wavelength,highlighting the broadband diffraction-limited performance ofthe MS-GRIN. We envision that the extremely high-performingdevices enabled by the MS-GRIN will motivate research in newmanufacturing techniques.

This Letter presented, for the first time, a lens singlet that iscapable of apochromatic color correction. Such performance wasachieved through the addition of a metasurface coating on aplano-convex GRIN lens. Analytical theory was presented for thesemetasurface-augmented GRIN singlets, which was then testedagainst ray tracing to prove its efficacy. Future studies include off-axis monochromatic performance and lateral color correction.

Funding. Defense Advanced Research Projects Agency(DARPA) (HR00111720032).

†These authors contributed equally to this work.

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Fig. 4. Focal drift versus wavelength for the GRIN, MS-GRIN,refractive doublet and triplet. The shaded region corresponds to thediffraction-limited depth of focus.

Fig. 5. Comparison between optimized MS-GRIN and refractivetriplet. (a) MTF at λd . (b) Strehl ratio versus wavelength.

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