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To be published in Applied Optics: Title:   Substrate aberration and correction for meta-lens imaging: an analytical approachAuthors:   Benedikt Groever,Charles Roques-Carmes,Steven Byrnes,Federico CapassoAccepted:   12 March 18Posted   13 March 18Doc. ID:   318972

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Research Article Applied Optics 1

Substrate aberration and correction for meta-lensimaging: an analytical approachBENEDIKT GROEVER1, CHARLES ROQUES-CARMES1,2, STEVEN J. BYRNES3, AND FEDERICOCAPASSO1,*

1Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA2Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA3Charles Stark Draper Laboratory, Cambridge, Massachusetts 02139, USA*Corresponding author: capasso@seas.harvard.edu

Compiled March 11, 2018

Meta-lenses based on flat optics enabled a fundamental shift in lens production - providing an easiermanufacturing process with an increase in lens profile precision and a reduction in size and weight. Herewe present an analytical approach to correct spherical aberrations caused by light propagation through thesubstrate by adding a substrate-corrected phase profile which differs from the original hyperbolic one. Ameta-lens encoding the new phase profile would yield diffraction-limited focusing and an increase of itsnumerical aperture of up to 0.3 without changing the radius or the focal length. In tightly-focused laserspot applications such as direct laser lithography and laser printing a substrate-corrected meta-lens canreduce the spatial footprint of the meta-lens. © 2018 Optical Society of America

OCIS codes: (050.1965) Diffractive lenses; (050.5080) Phase shift; (050.6624) Subwavelength structures; Meta-lenses, Metasurface.

http://dx.doi.org/10.1364/ao.XX.XXXXXX

1. INTRODUCTION

In recent years, metasurfaces have emerged as a new way tocontrol light through the optical properties of sub-wavelengthelements patterned on the flat surface of a substrate. A sub-wavelength element can locally change the amplitude, polariza-tion and phase of the incident electromagnetic wave to realizevarious optical functions in a compact configuration [1–6]. Ametasurface can be produced through a single lithography stepof nanofabrication which can easily be scaled to high-outputmanufacturing processes.

One of the most promising applications of metasurfaces isimaging. Here, each sub-wavelength element corresponds toone data point of the discretized phase profile of the meta-lens.Due to sub-wavelength spacing, light sees a near-continuousphase profile. During the fabrication, dimensions of the sub-wavelength elements are well controlled which enables highlens profile precision. A diffraction-limited focal spot can beachieved [7]. With conventional refractive lenses, the lens profileprecision depends on the accuracy of lens curvature which ismore difficult to control [8]. Future challenges of meta-lenses layin the correction of chromatic aberrations which can be reducedby several approaches such as refractive/diffractive compoundlenses [9], dispersion engineering of nanostructure resonances[10, 11] or designing a nanostructure that has control over groupvelocity delay and has the capability of 2π phase modulation

[12–14]. This would allow circumventing the current limitationsencountered in diffractive optics [15–17].

To fabricate a working meta-lens, a substrate is essential forarranging the sub-wavelength elements. But any transparentglass substrate has a refractive front surface. As of now, therefractive front surface did not play a major role in the develop-ment of meta-lenses, because most meta-lenses were designedto focus a normal incident beam [7, 18, 19]. However, in thegeneral point-to-point imaging configuration with finite objectand imaging distances, incidence is no longer normal. Refrac-tion occurs at the front surface of the substrate, where the frontsurface of the substrate is the surface with no meta-lens, whichleads to spherical aberrations. In this paper, we introduce ananalytical phase profile, which corrects spherical aberrationscaused by the front surface of substrate. Our approach alsooffers physical insight on how Seidel aberrations respond tometa-lens design changes. Previously such substrate correctionswere only performed with ray-tracing techniques [20]. Withoutthe correction of those aberrations, it is in general not possibleto get diffraction-limited focusing at high numerical aperture inpoint-to-point imaging systems.

Besides overcoming spherical aberrations, there are also otheradvantages. The substrate corrected phase profile allows mount-ing the meta-lens on the substrate side opposing the focal spot,hence the meta-lens is better protected from outside world con-taminants like dust and humidity which is important for applica-

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tions in microscopy [21, 22] where the lens is close the specimen.We also find that this configuration has a higher numerical aper-ture and therefore a smaller diffraction-limited focal spot. Thiswould be useful for focusing applications which require a tightlyfocused laser spot as direct laser lithography and laser printing[23].

Aberration correction plays a major role in the developmentof high resolution imaging systems for machine vision, computervision and microscopy applications. A single, planar lens cannotproduce a diffraction-limited spot along the focal plane mostlydue to Petzval field curvature and coma aberration [24, 25]. Re-cently, a diffraction-limited meta-lens doublet along the focalplane in the visible has been demonstrated [26]. For the devel-opment of new kinds of meta-lens objectives, ray tracing opti-mization is needed [26, 27]. Our analytical substrate-correctedphase profile can be used to predict better initial optimizationparameters [8, 28, 29].

2. HYPERBOLIC META-LENS (HML) AND SUBSTRATE-CORRECTED META-LENS (SCML)

The meta-lenses that are designed for focusing a normal inci-dent beam which remains collimated inside the substrate arecharacterized by a hyperbolic phase profile [24]:

ϕ(ρ) = −2π

λ0nm(

√ρ2 + f 2 − f ) (1)

This formula gives the required shift at each radial coordi-nate to achieve diffraction-limited focusing at the focal lengthf , λ0 is the (free space) illumination wavelength and nm is therefractive index of the ambient medium (nm = 1). Figure 1(a)shows the ray diagram of a meta-lens with a hyperbolic phaseprofile mounted on a substrate, referred to as hyperbolic meta-lens (HML) hereafter. Collimated light is incident from the frontsurface of the substrate. The meta-lens is mounted on the otherside of the substrate. The incident beam remains collimated inthe substrate. The HML performs diffracted-limited focusing.We refer to diffraction-limited focusing when all rays for a givenangle of incidence intersect the optical axis at the same position.Amplitude variations of the incoming electric field caused byabsorption or reflection only need to be considered once the sys-tem is diffraction-limited as they influence the shape of the focalspot, e.g. removal of the Airy disk caused by diffraction whichis known as apodization. Ray tracing is sufficient to achievediffraction-limited focusing. Figure 1(b) shows the HML per-formance when collimated light is incident from the meta-lensside. Here refraction in the substrate leads to a spherically aber-rated focal spot. In this paper, we present an analytical way todesign a meta-lens to compensate for the refractive propertiesof the substrate so that when light is incident from the meta-lens side it enables diffraction-limited focusing as shown in Fig.1(d). The meta-lens with a substrate-corrected phase profile,referred to as substrate-corrected meta-lens (SCML) hereafter,has a larger phase gradient than a HML of the same focal length(Fig. 1(c)). The larger phase gradient at the edge of the SCMLcorrects for light propagation through the substrate. Conversely,since the SCML is designed for the light to be collimated onthe meta-lens side, and if it is illuminated with collimated lightfrom the substrate side it results in a spherically aberrated fo-cal spot as shown in Fig. 1(e). The fact that the HML and theSCML are diffraction-limited only for one direction of incidenceis not specific to meta-lenses. Refractory plano-convex aspher-ical lenses, which are used to focus collimated light, have the

(a)

(b)

(f )

(c)

(d)

(e)

substrateHML

substrateSCML

NA of HML NA of HML (see Fig. 3(b) for SCML NA)(g)

Fig. 1. Focusing of hyperbolic meta-lens (HML) and substrate-corrected meta-lens (SCML) with incident collimated light fromdifferent sides of the substrate. (a) Ray diagram for diffraction-limited focusing with the HML. Light is incident from the frontsurface side of the substrate and every ray intersects the opticalaxis at the designed focal length. (b) Ray diagram for aberratedfocusing with the HML. In this configuration where light isincident from the backside of the substrate, spherical aberrationexists due to refraction at the front surface of the substrate. (c)Phase profile of the SCML and HML, Eq. (6) and (7). (d) Raydiagram for diffraction-limited focusing with the SCML. Light isincident from the meta-lens side. The steeper phase profile of theSCML takes the refraction at the front surface of the substrateinto account, in such a way that every ray intersects with theoptical axis at the same position. (e) Ray diagram for aberratedfocusing with the SCML. (f-g) CoC as a function of HML-SCMLmeta-lens radius, respectively, for light focused through differentsubstrate thickness d. For all meta-lenses depicted: f = 1.0mm, nsub = 1.46, nm = 1, for meta-lenses in (a-e): substratethickness d = 0.50 mm. For meta-lenses in (c): illuminationwavelength λ0 = 532 nm, the results in the other subfigures areindependent of the illumination wavelength as the 1/k0 factorin Generalized Snell’s law cancels the illumination wavelength.

same kind of directional behavior. In the following, we will referto the ”wrong” configuration for each meta-lens when it is illu-minated in such a way to produce a spherically aberrated focalspot; and to ”right” configuration when illumination results in adiffraction-limited spot.

The minimum achievable focal spot size for a meta-lens usedin the wrong configuration is shown in Fig. 1(f) and Fig. 1(g)respectively for the HML and the SCML. The figure of merit isthe Circle of Confusion (CoC). The circle containing all incidentrays is an important figure of merit for the size of the focal spot.The CoC is defined as the minimum circle, when moving alongthe optical axis. Images are sharpest when the image detector is

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at this position. As each meta-lens is only diffraction-limited fornormal incidence in the right configuration, we consider onlynormal incidence in the wrong configuration. A rule of thumbto characterize a diffraction-limited system is to verify that allthe rays fall within a disk of diameter dAiry = 1.22 (λ0/NA), NAis the numerical aperture. dAiry is roughly the diameter of thediffraction-limited Airy disk, so when this criterion is satisfied,the point-spread-function should be only modestly differentfrom the ideal Airy pattern. The diffraction-limited Airy diskas a function of radius for the HML and SCML is shown as ashaded area in Fig. 1(f-g). Figure 1(f) shows that the HML witha focal length of 1 mm used in the wrong configuration can stillcreate a diffraction-limited spot if the NA is smaller than 0.3.Conversely, Fig. 1(g) shows that the SCML with a focal lengthof 1 mm used in the wrong configuration (i.e. collimated onthe substrate side) can be diffraction-limited up to a radius of0.25 mm, corresponding to a NA of 0.3 to 0.4 depending on thethickness of the substrate (Fig. 3(b)).

The size of the Airy disk only depends on the NA of animaging system - not on the actual focal length or radius. Whenscaling the focal length and radius of the lens, the CoC scaleswith the same factor but the size of the Airy disk does not change.Therefore, it is harder to achieve diffraction-limited focusingwith a lens when the radius of the lens increases.

3. PHASE PROFILE OF A DIFFRACTION-LIMITEDPOINT-TO-POINT FOCUSING META-LENS

Now, we discuss the point-to-point imaging configuration. Theabove discussion (∞-to- f or f -to-∞) is a particular case of inwhich one point is at infinity. In a point-to-point imaging con-figuration, we need to consider an object point source locatedin front of the meta-lens which gets focused to an image point

x

zy

df

nsub

db

nm

(a) (b)df

dv

d

nsub

db

nmnm

x

zy

α β αsub αm βρ1

ρ2 ρρ αsub

Fig. 2. Point-to-point metasurface focusing. (a) Ray diagramof a meta-lens (dark green rectangles) mounted on an infinitelyextended substrate. The image (focal spot) is inside the sub-strate and the object point is in the ambient medium, bothon the optical axis. The designed front distance and the de-signed back distance are denoted as df and db respectively.(b) Ray diagram of a meta-lens mounted on a substrate of fi-nite thickness d and object and image spots in the ambientmedium. Light from a point source along the optical axis ata distance df from the meta-lens is focused to a point at dis-tance db from the meta-lens. The distance from the intersec-tion of the virtual rays (red lines) with the optical axis to themeta-lens is defined as virtual front distance dv. The virtualfront distance dv(ρ) is a function of the radial coordinate ρ.

on the other side of the meta-lens. We call the distance from theobject point source to the meta-lens the ”designed front distance”

df and the distance from the meta-lens to the image point the”designed back distance” db, (Fig. 2). The lens will work best (i.e.at the diffraction limit) when focusing light from the designedfront point to the designed back point or vice versa. In the fol-lowing we shall assume that the lens is operating in this way,with the object point source at the designed front position andthe image point at the designed back position. If the object pointsource is somewhere else on the optical axis, it will still generallybe focused, but the image point will have spherical aberrations.These spherical aberrations are not related to the substrate butrather are inevitable due to fix phase profile of the optical system.Their correction would require a variable imaging system, e.g.with varifocal meta-lenses [30], which can be tuned to track theobject position. Such systems are widely used in cameras. Toderive the phase profile of a meta-lens for diffraction-limitedpoint-to-point focusing, we shall first consider the case of a meta-lens mounted on an infinite extended substrate −∞ < z < 0 asshown in Fig. 2(a). The object point source and designed frontdistance are inside the substrate with a refractive index nsub.The image point and designed back distance are in the ambientmedium. We assume that Generalized Snell’s law applies at theinterface [1]:

1k0

dϕ

dρ= nm sin β− nsub sin α (2)

Here, k0 = 2π/λ0, α and β are α = arctan(ρ/df) and β =

arctan(−ρ/db) , respectively and√

x2 + y2 with x and y as theposition coordinates on the meta-lens while the optical axis is atx = y = 0 along the z-direction. Generalized Snell’s law is validonce the height of the dielectric meta-lens is sufficiently sub-wavelength, the transmission is near unity and the separationis sub-wavelength. With the identity: sin(arctan(x)) = x/(1 +x2)1/2, Eq. (2) can be written as:

1k0

dϕ

dρ= −nm

ρ√d2

b + ρ2− nsub

ρ√d2

f + ρ2

Integration of the equation above leads to:

ϕ(ρ) = −k0nm

√d2

b + ρ2 − k0nsub

√d2

f + ρ2 (3)

This is the required phase profile for diffraction-limited point-to-point focusing of a meta-lens mounted on an infinite extendedsubstrate. In the next step, we assume a substrate of thicknessd and d < df. We extend virtually to the optical axis (red linein Fig. 2(b)) the rays refracted into the substrate. The distancefrom the intersection of the virtual ray with the optical axis tothe meta-lens is the virtual front distance dv. The angle in thesubstrate αsub of the ray is given by:

sin αsub =ρ√

d2v + ρ2

Using Snell’s law, sin αsub = sin αmnm/nsub, at the front sur-face of the substrate yields:

nm

nsubsin αm =

ρ√d2

v + ρ2

Rewriting, sin αm:

nm

nsub

ρ1√(df − d)2 + ρ2

1

=ρ√

d2v + ρ2

ρ1 is an axillary variable with no physical meaning, used tosimplify the math. Important is the radial coordinate ρ, which is

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given by: ρ = ρ1 + ρ2. ρ1 can be rewritten using similar trianglesas: ρ1 = ρ(dv − d)/dv. Using this identity allows rewriting theprevious equation as follows:

nm

nsub

ρ

dv

(dv − d)√(df − d)2 +

ρ2

d2v(dv − d)2

=ρ√

d2v + ρ2

It can be rewritten as a 4th order polynomial as follows:

d4v + a3 d3

v + a2 d2v + a1 dv + a0 = 0 (4)

where

a3 = (−2d)

a2 = (d2 + ρ2)−(nsub

nm

)2(ρ2 + (df − d)2

)a1 =

[1−

(nsubnm

)2](− 2dρ2

)

a0 =

[1−

(nsubnm

)2](

d2ρ2)

Eq. (4) can only be solved analytically with great complica-tion as it cannot be reduced to a lower order. The analyticalsolution is shown in Code File 1 (Ref. [31]). The phase gradientfor diffraction-limited point-to-point focusing with a finite sub-strate meta-lens can be obtained by replacing the designed frontdistance in Eq. (2) with the virtual front distance:

1k0

dϕ

dρ= −nm

ρ√d2

b + ρ2− nsub

ρ√dv(ρ, df, d, nm, nsub)2 + ρ2

Integration of the equation above gives:

ϕ(ρ) = −k0nm√

d2b + ρ2− k0nsub

∫ ρ

0

ρ′√dv(ρ′, df, d, nm, nsub)

2 + ρ′2dρ′

(5)The first term of this equation is the phase profile of a HML

with focal distance db and the second term is the phase profile ofSCML with focal distance df. A meta-lens with the phase profilein Eq. (5) yields diffraction-limited point-to-point focusing. Thisis analogous to combining two aspherical lenses to a biconvexlens in refractive optics.

4. NUMERICAL APERTURE OF HML AND SCML

Consider now the phase profiles of two meta-lenses, HML andSCML with focal length f , the distance from the focal point tothe meta-lens plane. Then phase profiles are given respectivelyby:

ϕHML(ρ) = −k0nm

√f 2 + ρ2 (6)

ϕSCML(ρ) = −k0nsub

∫ ρ

0

ρ′√fv(ρ′, f , d, nm, nsub)2 + ρ′2

dρ′ (7)

Here, fv(ρ′, f , d, nm, nsub) is the virtual focal length definedas the virtual front distance dv by Eq. (4), dv = fv. The definitionof the focal length f should not be mistaken with the front focallength FFL, back focal length BFL or effective focal length EFL

which are different definitions of a focal length. We choose f asthe focal length, because it corresponds to the best figure of meritfor the spatial footprint of the whole system. The numericalaperture of SCML and HML are given by:

NAHML = nm sin β = nm sin[

arctan(r/ f )]

(8)

NASCML = nm sin α = nsub sin αsub = nsub sin[

arctan(r/ fv(r))](9)

where NA is as usual equal to sin β, with β the half-angle ofthe cone of rays converging to the image point. Here: fv(r) =fv(r, f , d, nm, nsub), r is the radius of the meta-lens. The NAof the SCML does not only depend on the radius and on thefocal length but also on thickness and on the refractive indexof the substrate. In the limit that the thickness of the substrated approaches zero or that the refractive index ratio (nsub/nm)approaches unity, the NA of the SCML approaches that of theHML.

10 -2 10 -1 10 0 10 1

Radius of meta-lens (mm)

0.01

0.02

0.05

0.1

0.25

0.5

1

NA

NA SCML , f=1mm, d=0.80mm

NA SCML , f=1mm, d=0.50mm

NA SCML , f=1mm, d=0.17mm

NA HML , f=1mm

(b)(a)

(c)

Fig. 3. Numerical Aperture NA of substrate-corrected meta-lens (SCML) in comparison with that the of hyperbolic meta-lens (HML). (a) NA of HML for different radii and NA ofSCML for different radii and substrate thicknesses d. In thelimit of small substrate thickness, the NA of the SCML ap-proaches the NA of the HML. (b) Numerical Aperture differ-ence, NASCML - NAHML, ∆NA, for different radii and substratethicknesses d. The maximum of ∆NA occurs for all SCMLsat a radius of 932.6 µm. (c) The maximum of ∆NA for dif-ferent substrate thicknesses d. The points which are equiv-alent in subfigures (b) and (c) are marked with red circles.For all meta-lenses depicted: f = 1 mm, nsub = 1.46, nm = 1.

Figure 3(a) shows the NA of the SCML and the HML witha glass substrate (nsub = 1.46) for different radii. As neitherthe SCML nor HML are immersed in a high-index liquid theNA cannot exceed a value greater than unity. The NA of allmeta-lenses continuously increases to the maximum value as theradius increases. A SCML has a larger NA than a HML with thesame radius and focal length. Similarly, a SCML with a thickersubstrate has a NA closer to unity than a SCML with a thinnersubstrate. Figure 3(b) shows how much larger the NA of a SCMLwith different substrate thickness is compared to that of a HML,the only limit to increasing substrate thickness being the focallength (d < f ), to remain in a geometry corresponding to Fig.2(b). At large and small radii the differences in the value of theNA between the different lenses reaches zero. The maximumdifference between the SCML and the HML for a focal lengthof 1 mm is reached at a radius of 932.6 µm. We verified that thisis true for all substrate thicknesses up to d = f . The maximum

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difference is larger for larger substrate thicknesses as shown inFig. 3(c). The NA increase from a HML to SCML can be as largeas 0.3, a limit set by a focal length equal to the thickness of thesubstrate. This enables a smaller focal spot in a more compactconfiguration.

5. MAXIMUM ANGLE OF INCIDENCE OF HML ANDSCML

Next, we consider off-axis illumination. Here, total internalreflection (TIR) fundamentally limits the functionality of the lensas beyond a certain angle light gets reflected instead of beingfocused. The maximum angle of incidence without TIR, θmax(referred to as maximum angle of incidence hereafter), is animportant figure of merit as it limits the field of view and thetransmission efficiency of a meta-lens. At the critical angle ofTIR light travels along the meta-lens surface as shown in Fig.4(a) [32, 33]. To calculate the critical angle, we begin by applyingGeneralized Snell’s law at the meta-lens surface [1]:

nm sin θm =1k0

dϕ

dρ

]ρ=−r

+ nsub sin θsub (10)

We apply it at the lower edge of the meta-lens which is theradial coordinate ρ with the largest phase gradient. Furthermore,at the critical angle of TIR, the angle in the ambient mediumis 90◦, sin θm = 1, and the angle in the substrate θsub can bereplaced with the maximum angle of incidence by applyingSnell’s law at the front surface of the substrate: nsub sin θsub =nm sin θm. The phase gradient at the edge of the HML can becalculated as follows:

1k0

dϕ

dρ

]ρ=−r

= nmr√

f 2 + r2= nm sin

[arctan

( r

f)]

= NA

With equation above and Eq. (10) a criterion for the maximumangle of incidence in terms of the NA of the HML and nm isgiven:

sin θmax =nm − NA

nm(11)

TIR at the lower edge of the HML, ρ = −r, corresponds tosin θm = 1 and an angle of incidence of θmax. TIR at the up-per edge of the HML, ρ = r, corresponds to sin θm = −1 andangle of incidence of −θmax. In an actual meta-lens the theoreti-cal value of the maximum angle of incidence θmax may not beachieved due to a discretization of the phase profile [1, 34], anon-zero thickness or a non-unity transmission [1]. The HMLonly has one possible surface for TIR as light at the front sur-face of the substrate goes from a lower index medium (ambientmedium) into a higher index medium (substrate). For the SCMLwe need to consider both surfaces for TIR. As shown in Fig. 4(b),let’s first consider the surface with the meta-lens [1]:

nsub sin θsub =1k0

dϕ

dρ

]ρ=−r

+ nm sin θmax

The gradient term in this equation is equal to the NA of theSCML. If this surface has TIR: sin θsub = ±1, then the criterionfor the maximum angle of incidence at the meta-lens surface isgiven by:

sin θmax =nsub − NA(r, f , d, nsub, nm)

nm(12)

This criterion can only be fulfilled if: sin θmax < 1 and hence:NA ≥ |nsub − nm|, otherwise the phase gradient at the edge ofthe meta-lens is too small to exhibit TIR.

Now, let’s consider TIR at the refractive front surface of thesubstrate shown in Fig. 4(c). TIR occurs at this surface whensin θm = 1, hence sin θsub = nm/nsub. Substituting this relationinto Eq. (10), eliminates nsub. The following criterion for themaximum angle of incidence can be obtained:

sin θmax =nm − NA(r, f , d, nsub, nm)

nm(13)

This constraint for the SCML max. angle of incidence is thatof an HML with the same NA. As the refractive index of thesubstrate is larger than that of ambient medium, TIR occurs firstat the front surface of the substrate, then at the SCML surface,hence Eq. (13) is a stricter criterion for the maximum angle ofincidence than Eq. (12).

substrateHMLTIR

SCML

(a) (b) (c)

(d)

-r -r -r

Eq. (12) Eq. (11) Eq. (13)

θmax θmaxθmax

(e)

Fig. 4. Total internal reflection (TIR) limitation for the maximumangle of incidence of hyperbolic meta-lens (HML) and substrate-corrected meta-lens (SCML). When TIR occurs, light propagatesalong the meta-lens and consequently limits the maximum angleof incidence. (a) Critical angle of TIR for the marginal ray (or-ange) at the lower edge of a HML (maximum angle of incidenceθmax, θmax = 17.03◦). When increasing the angle of incidencerays closer to the center of the meta-lens would be totally inter-nally reflected further. (b) Critical angle of TIR at the meta-lenssurface for the marginal ray at the lower edge of a SCML (θmax= 40.15◦). (c) Critical angle of TIR at front surface of the sub-strate for the marginal ray at the lower edge of a SCML (θmax= 11.42◦). The critical angle of TIR at the front surface of thesubstrate is always smaller than at the meta-lens and hence isa stricter constraint for maximum angle of incidence for theSCML. (d) Maximum angle of incidence of the HML and theSCML for different radii. The red circles have a correspondingdata point in subfigure (e) - same radius and substrate thick-ness. (e) Maximum angle of incidence of SCML for differentsubstrate thicknesses d. For all meta-lenses depicted: f = 1 mm,nsub = 1.46, nm = 1, for meta-lenses in (a-c): r = 1 mm, sub-strate thickness d = 0.50 mm, for meta-lenses in (e): r = 1 mm.

Figure 4(d) shows the maximum angle of incidence for theHML and SCML for different meta-lens radii and substrate thick-nesses when the focal length is 1 mm. SCMLs with a thicker

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substrate and larger radius (and hence a higher NA) have asmaller maximum angle of incidence than an HML with equiv-alent radius and focal length. Figure 4(e) shows the maximumangle of incidence for different substrate thicknesses when theradius and the focal length are both 1 mm. As the thicknessof the substrate increases the maximum angle of incidence isreduced. This analysis is particularly interesting in the design ofmeta-lens stacking that can correct off-axis aberrations.

6. SEIDEL ABERRATIONS OF HML AND SCML

If the HML and SCML are used in the correct configuration,Fig. 1(a) and Fig. 1(c) respectively, spherical aberrations are notpresent. But besides spherical aberrations, there are 4 othersmonochromatic aberrations: distortion, coma aberration, Pet-zval field curvature and astigmatism. All 5 monochromaticaberrations are known as Seidel aberrations. Because of sym-metry, astigmatism is not present in a radial symmetric opticalsystem which is infinity-corrected. Therefore, astigmatism is nei-ther present in the HMLs nor in the SCMLs. Coma aberration,Petzval field curvature and distortion are all present in singletlenses both in conventional refractive lenses and in single-layermeta-lenses. For meta-lenses, this is due to the fact, that thephases profile to achieve diffraction-limited imaging dependson the angle of incidence. Angle-multiplexed phase profileswhich are fully independent can only be implemented throughmulti-layered structures [35] as they provide enough degrees offreedom.

Coma aberration is best known for its resulting comet-shapedimage, when observing an off-axis point source. In ray optics, itis defined in terms of the radial coordinate ρ at a given angle ofincidence. The magnitude of coma is quantified by the distancefrom the chief ray to the intersection of two marginal rays, in aplane perpendicular to the optical axis. The two marginal rayshave the same angle of incidence as the chief ray and hit themeta-lens plane at coordinates ±ρ [36](see Fig. 5(a-b) and Fig.5(c-d) for the HML and SCML respectively). As the angle ofincidence increases, so does the magnitude of coma, and TIR iseventually encountered. We computed the magnitude of comafor different substrates and angles of incidence with a custom-made ray tracing code. The magnitude of coma is smaller forsmaller angles of incidence and smaller radical coordinates. Asa direct consequence of this observation, meta-lenses with lowerNA exhibit less coma than meta-lenses with a larger NA.

Figure 5(e) shows coma for the HML. We noticed that theHML has the same magnitude of coma as a meta-lens withno substrate. This is true as smaller angles in the substrate ofthe HML are compensated by the larger refractive index of thesubstrate (Generalized Snell’s law). For the same reason, coma ofthe HML is independent of the substrate thickness. Here, a meta-lens with no substrate is considered as a theoretical limit of thefinite substrate thickness case in the limit where the thicknessgoes to zero. At the maximum angle of incidence when TIRoccurs, the magnitude of coma for marginal rays at the edge ofthe HML is equal to the radius of the HML (ray diagram: Fig.4(a)).

Figure 5(f) shows coma for the SCML. For the SCML, themaximum coma is smaller than the radius of the SCML dueto the TIR limitation at the front surface of the substrate (raydiagram: Fig. 4(c)). For larger substrates, this effect occurs atsmaller angles of incidence and therefore leads to smaller coma.Figure 5(g-h) show the difference in coma between the HML andthe SCML at two different angles of incidence. The SCML has

substrateHMLComa

(a) (b) (c) (d)

(e)

(g)

Fig. 5aFig. 5b Fig. 4a

SCML

ρ = r ρ = r ρ = r

θ = 7.61̊θ = 3.81̊

ρ = 0.75r θ = 11.35˚ θ = 11.35˚

(f )

(h)

Fig. 5. Coma aberration for hyperbolic meta-lens (HML) andsubstrate corrected meta-lens (SCML). Coma is a function ofthe radial coordinate ρ and angles of incidence θ. (a-b) Raydiagram for coma of HML at an angle of incidence θ = 11.35◦

for radial coordinates ρ = 0.75r and ρ = r, (a) and (b) respectively.(c-d) Ray diagram for coma of the marginal rays at the edgeof a SCML for angles of incidence θ = 3.81◦ and θ = 7.61◦, (c)and (d) respectively. The magnitude of coma is smaller forsmaller angles of incidence and smaller radial coordinates. (e)Coma of the HML as a function of radial coordinate for differentangles of incidence with and without substrate. The red dotsindicate the corresponding ray diagram. The magnitude ofcoma at the edge of the HML is equal to the radius, Fig. 4(a).(f) Coma as a function of the radial coordinate for differentangles of incidence for the SCML. (g-h) Coma difference betweenthe SCML and the HML for different substrate thicknesses d atan angle of incidence of θ = 3.21◦ and θ = 6.42◦ for (g) and(h) respectively. θ = 6.42◦ is the maximum angle of incidencefor the SCML with a 0.80 mm thick substrate. For all meta-lenses depicted: f = 1.0 mm, r = 1.0 mm, nsub = 1.46, nm =1, for meta-lenses in (a-e): substrate thickness d = 0.50 mm.

less coma than the HML when the radial coordinate is small. Asthe radial coordinate increases, this behavior is reversed. Thiseffect is even larger when the substrate thickness is increased.

Figure 6(a-d) show the ray diagram for Petzval field curvatureand distortion of the HML and the SCML with different substratethicknesses. Petzval field curvature and distortion are definedby the focusing position of an oblique incident chief ray. Thefocusing position is defined as the intersection of two chief rayscrossing the meta-lens plane at positions ±ε.

Petzval field curvature is the distance along the optical axisfrom the focusing position to the flat vertical plane as ε ap-proaches zero [36]. Imaging sensors like CMOS and CDD areflat; therefore, the focal point before the flat vertical plane causesan aberrated focal spot on the imaging sensor. Intuitively, dis-tortion is the change in magnification across an image [36]. It isdefined as the relative difference of the actual image height tothe predicted image height:

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Research Article Applied Optics 7

0 0.1 0.2 0.3 0.4 0.5 0.6Petzval �eld curvature (mm)

0

10

20

30

40

Ang

le o

f inc

iden

ce (d

egre

e)

HMLmeta-lens with no substrateSCML, d=0.17mmSCML, d=0.50 mmSCML, d=0.80mm

-60 -50 -40 -30 -20 -10 0Distortion (%)

0

10

20

30

40

Ang

le o

f inc

iden

ce (d

egre

e)

HMLmeta-lens with no substrateSCML, d=0.17mmSCML, d=0.50mmSCML, d=0.80mm

(a) (b) (c) (d)

(e)

substrateHMLFocal surface

SCMLPetzval �eld curvatureImage heightPredicted image height

0.17 mm 0.50 mm 0.80 mm0.50 mm

Fig. 6(b) Fig. 6(a)

єє

є

є

є

є

є

є

(f )

Fig. 6(a)

Fig. 6(b)Fig. 6(c)

Fig. 6(c)Fig. 6(d) Fig. 6(d)

Fig. 6. Petzval field curvature and distortion of hyperbolic meta-lens (HML) and substrate-corrected meta-lens (SCML). (a-d)Ray diagram of two chief rays for the HML and SCML withdifferent substrate thicknesses d. When ε is very small the inter-section of those two chief rays defines the curved focal ”plane”,which deviates from the flat vertical plane (blue dashed line).(e) Petzval field curvature of meta-lenses depicted in (a-d). Pet-zval field curvature for a certain angle of incidence is definedas the focal length shift parallel to the optical axis. Referenceis normal incidence. The red dots indicate the correspondingray diagram. (f) Distortion of the same meta-lenses. Distor-tion is defined as the relative difference of the actual imageheight to the predicted image height, Eq. (14). The predictedimage height is the linear extrapolation of the image height atthe center with γ0 as proportionality constant. For all meta-lenses depicted: f = 1 mm, r = 1 mm, nsub = 1.46, nm = 1,for meta-lenses in (a-d): angle of incidence θ = 0◦ and θ = 40◦.

Distortion(%) =( ha − hp

ha

)· 100 (14)

Here, ha is the actual image height, hp is the predicted imageheight. The actual image height ha is the height of the focusingposition of the two chief rays. For a given angle of incidenceθ, the predicted image height is the extrapolated image heightfrom the image center: hp = γ0 tan(θ) with:

γ0 = f − d(nsub − nm

nm

)(15)

For an imaging system with working distance df, γ0 is relatedto the image center magnification M0 as follows: M0 = γ0/df.For most applications distortion is not a limitation as it can becorrected with post-processing techniques. Figure 6(e-f) showthat Petzval field curvature and distortion are smaller for theSCML than they are for the HML. A larger substrate thicknessdecreases both Petzval field curvature and distortion. At anangle of 40◦ incidence Petzval field curvature of the SCML witha 0.80 mm substrate is almost 3 times smaller compared to thatof the HML. The HML and meta-lens with no substrate areequivalent in terms of Petzval field curvature and distortion assmaller angles in the substrate of the HML are compensatedwith a larger refractive index of the substrate-just as previouslywith coma.

7. CONCLUSION

To conclude, the SCML exhibits certain advantages over theHML. The higher NA enables a smaller diffraction-limit in amore compact configuration; monochromatic aberrations likePetzval field curvature and distortion are reduced. The higherNA of the SCML comes at the cost of a smaller maximum an-gle of incidence. In microscopy, the meta-lens nanostructuresare better protected in the SCML configuration from outsideworld contaminants like dust and humidity because of the smallproximity to the specimen. In immersion microscopy, the SCMLwould allow designing the meta-lens in air and not in a liquid[37]. With dielectric metasurfaces this is easier because of thelarger index contrast. More generally, our work provides anintuitive platform towards designing meta-lens stacking, similarto the approach that has been developed by lens makers forbulky optical components, to systematically reduce aberrationsand thus give a new impulse to applications of meta-lenses inthe industry [38].

8. FUNDING INFORMATION

Air Force Office of Scientific Research (MURI, FA9550-14-1-0389,FA9550-16-1-0156). The authors disclose no competing financialinterest. The authors acknowledge Dr. Wei-Ting Chen for joiningdiscussions.

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