Multifunctional volumetric meta-optics for color and polarization image sensors: supplementary material PHILIP CAMAYD-MUÑOZ, CONNER BALLEW, GREGORY ROBERTS, ANDREI FARAON*

Kavli Nanoscience Institute and Thomas J. Watson Sr. Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125 *Corresponding author: faraon@caltech.edu

Published 31 March 2020

I. Inverse design methodInthisworkwedesignthree-dimensionaldielectricstructures,

optimizedtoperformaspecifiedopticalscatteringfunction:inthiscase,focusingincidentplanewavestodifferentpositionsdependingonthe frequencyandpolarization.Thestructure isdefinedbyaspatially-dependent refractive index distribution within a

cubicdesignregion.Thisrepresentsanexpansivedesignspacewiththe capacity to express a broad rangeof complex opticalmulti-functionality.However,identifyingtheoptimalindexdistributionfor a given target function remainsa challenging inversedesignproblem,particularlyforstronglyscatteringdevices[1].Therefore,weadoptaniterativeapproachguidedbygradientdescent(Fig.S1)in order to efficiently generate complex 3D designs[2]: startingfrom an initial index distribution, we use full-wave simulations(FDTD) to calculate the sensitivity of the sorting efficiencywithrespecttoperturbationsoftherefractiveindex.Thesensitivitycanbe calculated from just two simulations, allowing efficientoptimizationofthree-dimensionaldeviceswithmodestresources.Basedonthesensitivity,wemodifytheinitialdesigninordertomaximize the performance while conforming to fabricationconstraints. Thisupdateprocess is repeateduntil theoptimizeddevicecanefficientlyperformthetargetfunction.

Forthedesignspresentedinthismanuscript,westartwithauniformrefractiveindexdistribution, .This

distribution is continually updated to maximize theelectromagneticintensityatthetargetlocationinthefocalplane,

. This figure ofmerit serves as a proxy for

sortingefficiencywhilesimplifyingthesensitivitycalculation.The

sensitivity, ,iscomputedfromtheelectromagneticfieldsin

twoFDTDsimulations:

(S1)

where aretheelectricfieldswithinthecubewhenilluminated

fromabovewith a planewave, and are the electric fieldswithinthecubewhenilluminatedfrombelowwithapointsourceatthetargetlocation.Thephaseandamplitudeofthepointsourcearegivenbytheelectricfieldatthetargetlocationintheforwardsimulation.Wesimultaneouslycalculatethesensitivityformultiple

n !x( )

n0!x( ) = 1

2 nmax + nmin( )

f n !x( )( ) =!E !x0( )

2

dfdn

!x( )

dfdn!x( ) = 2n !x( )Re

!E fwd ⋅

!Eadj{ }

!E fwd

!Eadj

Fig.S1.Scatteringelementsaretailoredtoperformatargetfunctionthroughgradientdescent.Startingfromaninitialdesign,wecomputethesensitivityofthesortingefficiencytochangesintherefractiveindexusing a pair of FDTD simulations. The design is then updated inproportion to the sensitivity. After many iterations, the resultingstructurefocusesincidentlightwithhighefficiency.

This document provides supplementary information to "Multifunctional volumetric meta-optics for color and polarization image sensors," https://doi.org/10.1364/OPTICA.384228.

incidentwavelengthsandpolarizationsacrossthevisiblespectrum,assigningeachspectralbandtoadifferentquadrant:red(600nm–700nm)green(500nm–600nm)andblue(400nm–500nm).Then we use the spectrally-averaged sensitivity to update therefractiveindexofthedevice:

(S2)

Thestepsize fixedatasmallfraction(typically =0.001)toensurethatthechangeinrefractiveindexcanbetreatedasaperturbation in the linear regime.The sensitivity is recalculatedaftereachupdate.

II. Fabrication constraints

A. Binary index Duringtheoptimizationprocess,wedirectlyenforceasetof

constraintsontheindexdistributionasrequiredbythefabricationprocess. In particular,we are designing high-contrast scatteringelementsconstructed fromtwomaterials.Althoughthegradientdescentalgorithmdetailedaboveproducesoptimizeddeviceswithgradientindex,wecanenforcethebinaryconditionbyintroducinganauxiliarydensity rangingfrom[0,1].Thedensityisrelated

to the refractive index distribution via a sigmoidal projectionfilter[3]:

(S3)

wheretheparameter controlsthefilterstrength.Forsmall ,theindexdistributionisequaltothedensityscaledtotherangeofavailable refractive index. For large , the sigmoid filterapproximatesaHeaviside function,and the indexdistribution ispushedtowardeitherextreme. Importantly, the filter function iscontinuouslydifferentiable,suchthatthesensitivitycanbewrittenin termsof thedensity: .Duringoptimizationwe can

parameterizethedesignusingthedensity and ,gradually

increasingthestrengthofthefilter.Atearlystagesandsmall ,thisis equivalent to the unfiltered case. Over time as the strengthincreases, the optimized index distribution is gradually pushedtowardabinarydesign,evenasthedensityremainscontinuous.

B. Minimum feature size In addition to material constraints, device designs must

conformtotheresolutionlimitsimposedbythefabricationprocess.Forexample,diffractionandproximitydosingeffectslimitelectronbeamlithographytoapproximately10nmfeatures.Weenforcethisminimumfeaturesizefordevicedesignsbyintroducinga“dilated”density , which represents the maximum density

withinaneighborhood ofeachpoint [4].

(S4)

For a sufficiently large exponent 𝑀, this operation

approximates morphological grayscale dilation. However it iscontinuously differentiable with respect to the arguments.Therefore,thesensitivitycanbewrittenintermsoftheun-dilated

density: !"!#(𝒙&&⃗ )

= . The neighborhood is

taken to be a circle,where the radius represents theminimumfeaturesize.

When combined with the projection filter, the dilationoperation eliminates small features in the final binary indexdistribution.Duringoptimization,thedeviceisparameterizedbythedensity ,whiletheindexisdefinedbythedilateddensity

:

(S5)

The neighborhood is taken to be a circle, where the radiusrepresentstheminimumfeaturesize.

ni+1!x( ) = ni

!x( )+α dfλ

dn!x( )λ

∑

α α

ρ!x( )

n !x( ) = P ρ!x( )( )

=12+tanh 2βρ

!x( )−β( )

2tanh β( )

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟nmax − nmin( )+ nmin

β β

β

dfdρ =

dfdn

dndρ

ρ!x( ) β

β

!ρ!x( ) ρ

!ʹx( )

Ω!x

!ρ!x( ) = D ρ

!x( )( ) = 1

Mρ!x '( )( )

M

Ω∑M

dfdρ !x( ) =

dfd !ρ !ʹx( )

d !ρ !ʹx( )dρ !x( )Ω

∑ Ω

ρ!x( )

!ρ!x( )

n !x( ) = P !ρ !x( )( ) = P D ρ!x( )( )( )

Ω

Fig.S2.Thedesignofoptimizedscatteringelementsconsistsofiterativeupdatestotherefractiveindexdistribution,guidedbythesensitivitytoperturbationsintheindex,andconstrainedbymaterialsandfabricationprocessesasoutlinedabove.

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C. Connected layered designs Someof the device designs are intended for fabrication by

multi-layer2D lithography, consistingof severalpatterned slabsthatareinvariantintheverticaldirection(Fig.2,3).Inthiscase,werestricttheoptimizationbyaveragingthecalculatedsensitivityintheverticaldirectionwithineachlayer.Ineffect,voxelswithineachlayeraregovernedbyashared2Dprofile.

For the microwave device (Fig. 3), the design is furtherconstrainedsothateachlayerisfullyconnectedwithnofloatingpieces. We directly impose connectivity by periodically addingbridges between disconnected islands within each layer. Thisinterventiondoesnot takesensitivity intoaccount, and typicallycausesasmalldecreaseindeviceperformance.Therefore,weonlyapplytheconnectivityconstraintonceper40iterations,allowingtheperformancetorecoverthereafter.

III. Sorting efficiency of 3D printed designThefreeformscatteringelementshowninFig.1isdesignedfor

high-resolution 3D printing. It consists of amonolithic polymercube(index=1.56),infiltratedbyaseriesofholes(Fig.S3a).Aswithall the designs presented in this manuscript, the device sortsincidentlighttodifferentlocationsdependingonthewavelengthandpolarization.Hereweshowthesortingefficiencyunderlinearlypolarizedillumination.Thevisiblespectrumisdividedintothreespectralbands(400nm–500nm,500nm–600nm,600nm–700nm),eachfocusedonadifferentquadrantinthefocalplane(Fig.S3b).Weimposemirrorsymmetryalongadiagonalplanebisectingtheredandbluequadrants.Thisguaranteesthatthegreenspectralbandwill focus orthogonal polarizations to opposite quadrants,while the red and blue spectral bands will be polarizationindependent.Fig.S3cshowsthesortingefficiencyforeachquadrantacross the visible spectrum, defined as the fraction of incidentoptical power that reaches the target quadrant. This designachievesgreaterefficiencythansimilar layereddevices(Fig.2d),whichimposestrongerrestrictionsonthedevicetopology.

IV. Dependence on device thicknessEfficientmultifunctional devices require a large number of

opticaldegreesof freedom,whichscaleswith theoveralldevicethickness.Inthecaseofspectral-andpolarizationsortingelements,thisrestrictstheperformanceofultrathindevices.Hereweshowhow complex sorting behavior emerges with increased devicethickness.Fig. S4 shows thesortingefficiencyof several layereddevices (similar to Fig. 2), optimized with overall thicknessesranging from 400 nm – 2000 nm. Thin devices are unable toeffectivelysortincidentlight.Asthicknessincreases,theefficiencyimproveswhile crosstalk isminimized. Webelieve that addingevenmorelayersthanshownherewillincreasetheperformanceoftheoptimizeddevicealthoughnot indefinitely. Theeffectof thenumberof layersusedaswellas the thicknessof the individuallayersisaninterestingtopicforfutureinvestigation.

V. Dependence on incident angleThedevicespresented in thismanuscriptweredesigned to

sort light at normal incidence. In realistic imaging systems, theillumination will span a range of angles that depends on thenumericalapertureof the imagingoptics.Forexample,a typicalsmartphone camera[5] has a numerical aperture of NA = 0.21,corresponding to an acceptance cone spanning ±7.8° in SiO2. In

Fig. S4. Sorting efficiency for increasing device thickness.EfficiencyspectraforthemultilayerdesignsshowninFig2,optimizedfor1–5devicelayers.Eachplotcorrespondstoadifferentfocalregion,includingthe cross-polarized sub-pixel shown in yellow.Darker lines indicateincreasingdevicethickness.

Fig.S3.Sortingefficiencyof3Dprintedscatteringelement.(a)Deviceconsistsofasinglepieceofpolymer,asshowninFig.1.Lightincidentfrom above is focused to different quadrants depending on thepolarizationandfrequency.(b)Intensitydistributionatthefocalplaneunderbroadband,linearlypolarizedillumination.Colorscorrespondtotheobservedhueforvisiblelight.Theupperrightquadrantisassignedtocross-polarizedillumination,andthereforeremainsdark.(c)Sortingefficiencyforeachofthefourfocalregions,definedasthefractionofincidentlightfocusedintothetargetquadrant.

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ordertoevaluatethedeviceperformanceunderrealisticconditions,wehavecharacterizedthesortingefficiencyofthelayereddesign(shown in Fig. 2) for different illumination angles. As the angledeviatesfromthesurfacenormaltheefficiencydecreases(Fig.S5).Thisisduetoanincreaseinscatteringintonon-targetedquadrantswithinthefocalplane,aswellasreflectionfromthesurface.Thedrop-off inperformancewith incidentangle isaconsequenceofdecorrelation in the angular scattering through disorderedmedia[6],whichscaleswiththethicknessofthescatteringmedium.Since the device maintains high performance near normalincidence, we define the steepest functional angle where theperformancedropstohalfofthemaximumefficiency,roughly±8°.Therefore,thisdesignmaybedeployedincolorandpolarizationfiltersinmobileimagingsystems.Therangeofincidenceanglescanbeextended to steeperanglesby reducing thedevice thickness.Furthermore,wemaydirectlyincludenon-normalincidenceintothedesignalgorithm[2],whichhassofaronlyconsiderednormalincidence.

VI. Microwave near field measurement WesamplethescatteredmicrowavenearfieldintheKaband

usingascanningantenna.AnincidentGaussianbeam(FWHM=25mm) isgeneratedbyavectornetworkanalyzercoupled to freespaceviaamicrowavehornantennaandfocusingmirror(Fig.S6).Theinputbeampassesthroughthecube,scatteringintothefarfield.Wesamplethelocalelectricfieldatameasurementplane62mmbeyondtheoutputapertureofthedeviceusingaWR-28waveguideflangeinordertorecoverthecomplexscatteringamplitudeS21.Byscanning thepositionof theprobeantenna in themeasurementplane,wecanmeasurethelocalelectricfieldprofile.Weapplyadeconvolutionfiltertoaccountfortheanisotropyoftheprobeandthe finite aperture of the device. The complex fields arecomputationallyback-propagatedtothefocalplaneofthedevice.This analysis is repeated for a range ofmicrowave frequencieswithin the Ka band (26–40 GHz), and for both orthogonalpolarizations of the input beam. To measure the scatteringparametersforanorthogonalpolarization,werotatethedevice90°.We also measure the cross-polarized fields by rotating thewaveguideflange90°.

VII. Power flow analysis Weexperimentallyvalidatethedesignofvolumetricscattering

elementsusingamicrowaveanalogbymeasuring the scatteredfields.Thisanalysisrevealscloseagreementbetweensimulatedandmeasured sorting efficiencies, roughly 40% across themeasurementspectrum.However,theperformanceislowerthanpreviouslydesigneddevices(Fig.1and2).Hereweaccountfortheremaining energy that is lost in the system. From full-wavesimulationswecanquantifyfoursourcesofloss:reflectionfromthedevice,obliquescatteringawayfromthefocalplane,andabsorptionwithinthematerial(Fig.S7a).Bothreflectionandobliquescatteringcontributeequallytotheoveralllossinthesystem,independentofwavelength (Fig. S7b). We believe that these losses can besignificantly mitigated by explicitly including them in theoptimization algorithm, or through the appropriate choice ofboundary conditions and anti-reflection coatings. Due to theagreementbetween simulatedandexperimental efficiencies,weconcludethatabsorptionplaysanegligiblerole,whichisconsistentwiththeextremelylowextinctioncoefficientofpolypropyleneintheKaband[7].

Fig. S5. Sorting performance at non-normal incidence. The layereddevice is optimized to sort light incident from above. The sortingperformancedegradesastheincidentangleincreases.Comparedtothedesignatnormalincidence,theefficiencyisreducedbyhalfatanglesbeyond±8°.

Fig. S6. Experimental characterization at microwave frequencies.Microwave device consists of 20 patterned polypropylene sheets,assembled into a cube. The cube is illuminated by a collimatedmicrowave source operating in theKa band (26.5 – 40GHz). Localelectricfieldsarecollectedbyascanningnear-fieldprobe,andback-propagatedtothefocalplane.

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Fig.S7.Power flowanalysisofmicrowavedevices. (a)The incidentmicrowave beam is deflected into different regions, contributing tolower overall sorting efficiency. (b)Distribution of optical power insimulations,includingtransmission,reflection,andobliquescattering.Thepowerisnormalizedtotheinputapertureofthedevice.

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