RESEARCH ARTICLE SUMMARY◥

METASURFACES

Matrix Fourier optics enablesa compact full-Stokespolarization cameraNoah A. Rubin, Gabriele D’Aversa, Paul Chevalier, Zhujun Shi,Wei Ting Chen, Federico Capasso*

INTRODUCTION: Polarization describes thepath along which light’s electric field vector os-cillates. An essential quality of electromagneticradiation, polarization is often omitted in itsmathematical treatment. Nevertheless, polari-zation and its measurement are of interest inalmost every area of science, as well as in imag-ing technology. Traditional cameras are sensitiveto intensity alone, but in a variety of contexts,knowledge of polarization can reveal featuresthat are otherwise invisible.Determination of thefull-Stokes vector—the most complete descrip-tion of light’s polarization—necessitates at leastfour individualmeasurements. This results inoptical systems that are often bulky, reliant onmoving parts, and limited in time resolution.

RATIONALE: We introduce a formalism—matrix Fourier optics—for treating polarization

in paraxial diffractive optics. This formalism isa powerful generalization of a large body ofpast work on optical elements in which polari-zationmay vary spatially.Moreover, it suggestsa path to realizing many polarization devicesin parallel using a single optical element.We canthen design diffraction gratings whose ordersbehave as polarizers for an arbitrarily selectedset of polarization states, a new class of opticalelement. The intensity of light on a set of diffrac-tion orders is then dictated by the polarizationof the illuminating light,making these gratingsimmediately applicable to full-Stokes polariza-tion imaging.

RESULTS:We theoretically investigate thesegratings and develop an optimization schemefor their design. Our diffraction gratings wererealizedwith dielectricmetasurfaces inwhich

subwavelength, anisotropic structures providefor tunable polarization control at visible fre-quencies. Characterization of the fabricatedgratings shows that they perform as designed.Notably, an arbitrary set of polarizations maybe analyzed by a single unit cell, in contrast topast approaches that relied on interlacing ofseveral individually designed diffraction grat-ings, increasing the flexibility of these devices.These gratings enable a snapshot, full-Stokes

polarization camera—acamera acquiring imagesin which the full polar-ization state is known ateach pixel—with no tradi-tional polarization opticsand nomoving parts (see

panel A of the figure). Polarized light from aphotographic scene is incident on the gratinginside of a camera. The polarization is “sorted”by the specially designed subwavelengthmeta-surface grating.When combinedwith imagingoptics (a lens) and a sensor, four copies of theimage corresponding to four diffraction ordersare formed on the imaging sensor. These copieshave each, effectively, passed through a differ-ent polarizer whose functions are embeddedin the metasurface. The four images can beanalyzed pixel-wise to reconstruct the four-element Stokes vector across the scene. Severalexamples are shown at 532 nm, both indoorsand outdoors. The figure depicts an examplephotograph of two injection-molded plasticpieces, a ruler and a spoon (illuminated by alinearly polarized backlight), that show in-built stresses (see panels C to E of the figure)that are not evident in a traditional photo-graph (panel B). The camera is compact, re-quiring only the grating (which is flat andmonolithically integrated, handling all thepolarization analysis in the system), a lens,and a conventional CMOS (complementarymetal–oxide–semiconductor) sensor.

CONCLUSION:Metasurfaces can thereforesimplify and compactify the footprint of opticalsystems relying on polarization optics. Our de-sign formalism suggests future research direc-tions in polarization optics.Moreover, it enablesa snapshot, full-Stokes polarization imaging sys-temwith nomoving parts, no bulk polarizationoptics, and no specially patterned camera pixelsthat is not altogether more complicated than aconventional imaging system. Our hardwaremay enable the adoption of polarization imag-ing inapplications (remote sensing, atmosphericscience, machine vision, and even onboard au-tonomous vehicles)where its complexitymightotherwise prove prohibitive.▪

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The list of author affiliations is available in the full article online.*Corresponding author. Email: capasso@seas.harvard.eduCite this article as N. A. Rubin et al., Science 365, eaax1839(2019). DOI: 10.1126/science.aax1839

A B C

D E

Metasurface-based polarization camera. (A) Photographic scenes contain polarized lightthat is invisible to traditional, intensity-based imaging, which may reveal hidden features. Ourcamera uses a metasurface (inset) that directs incident light depending on its polarization,forming four copies of an image that permit polarization reconstruction. (B to E) A plastic rulerand spoon are photographed with the camera. (B) A monochrome intensity image (given bythe S0 component of the Stokes vector) does not reveal the rich polarization informationstemming from stress-birefringence readily evident in (C) to (E), which show a raw exposure,azimuth of the polarization ellipse, and the S3 component of the Stokes vector that describescircular polarization content, respectively.

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RESEARCH ARTICLE◥

METASURFACES

Matrix Fourier optics enablesa compact full-Stokespolarization cameraNoah A. Rubin1, Gabriele D’Aversa1,2, Paul Chevalier1, Zhujun Shi3,Wei Ting Chen1, Federico Capasso1*

Recent developments have enabled the practical realization of optical elements inwhich the polarization of light may vary spatially. We present an extension of Fourieroptics—matrix Fourier optics—for understanding these devices and apply it to the designand realization of metasurface gratings implementing arbitrary, parallel polarizationanalysis.We show how these gratings enable a compact, full-Stokes polarization camerawithout standard polarization optics. Our single-shot polarization camera requires nomoving parts, specially patterned pixels, or conventional polarization optics and mayenable the widespread adoption of polarization imaging in machine vision, remotesensing, and other areas.

Polarization refers to the path traversed bylight’s electric field vector. As a fundamen-tal characteristic of light, polarization andits measurement are of great interest inalmost all areas of science and in imaging

technology as well. Traditionally, polarization isspoken of as a property of a beam of light. How-ever, advances in the last few decades in holo-graphic media, micro- and nanofabrication, andother areas have enabled the practical realiza-tion of optical elements with tailored, spatiallyvarying polarization properties, even on a sub-wavelength scale, at optical frequencies. In thesedevices, the polarization state of light can bevaried controllably, point-to-point across an op-tical element.Work of this nature now has an extensive lit-

erature across several disciplines of optics undervarious names, including diffractive optics (1, 2),polarization holography (3–6), and nanophoton-ics (metasurfaces) (7, 8), in addition to apprecia-ble attention from the liquid crystal community(9, 10). These devices exhibit different behaviordepending on the polarization of illuminatinglight, so a natural question that arises is how theycan be designed to implementmany polarization-dependent functions in parallel.We consider a generalization of work in this

area, which we call matrix Fourier optics, thatsuggests new design strategies for the design ofpolarization optics not previously achievable ina single element. In particular, we apply it to thedesign of metasurface diffraction gratings that

can analyze several arbitrarily specified polariza-tion states in parallel.

Matrix Fourier optics

We present a general way of viewing diffrac-tion from a polarization-dependent obstacle. Inthe plane-wave expansion (or angular spectrum)picture of optics, an electromagnetic disturbanceU ðx; yÞ in a plane can be considered as beingformed from the interference of many planewaves incident at different angles. An individualplane wave in this set is characterized by its in-plane wave-vector ðkx; kyÞ and a weight given bythe Fourier transform of the optical field as

Aðkx; kyÞ ¼∬þ∞

�∞U ðx; yÞeiðkxxþkyyÞdx dy ð1Þ

Each plane wave with its weightAðkx; kyÞ canbe individually propagated forward in space toform the field at a different position (Fig. 1A).This formalism can treat light’s interaction withsimple obstacles. If a planar obstacle havinga transmission function tðx; yÞ (e.g., Young’sdouble slit, or a diffraction grating) is illuminatedby an incident field of magnitude E0 (which mayalso be spatially-varying), the field immediatelyfollowing the obstacle, tðx; yÞE0, can be handledin this way.This intuitive understanding of optical wave

propagation is the basis of the field broadly knownas Fourier optics (11) and underpinsmuch ofmod-ern optical physics, including imaging and holog-raphy. Notably, however, this picture is a scalarone and does not include light’s polarization.Nonetheless, it is possible to conceive of opticalelements in which polarization-dependent prop-erties vary as a function of space. Then, instead ofthe scalar transmission function tðx; yÞ, we may

consider that an optical element is associatedwithamatrix-valued function ~Jðx; yÞ.Here, ~Jðx; yÞdenotes the local Jones matrix (12), the way inwhich the Jones polarization vector is modified,at each point. (In this work, matrix quantities aredenoted by a tilde.)We consider that such an optical element

with a Jones matrix profile ~Jðx; yÞ is illuminatedby a field described by the Jones vector ketjE0i (in acknowledgment of its polarized nature).The field immediately after the obstacle is thengiven by ~Jðx; yÞjE0i. This field, too, can be prop-agated by plane-wave expansion as given by theFourier transform

jAðkx; kyÞi¼∬

þ∞

�∞~Jðx; yÞjE0ieiðkxxþkyyÞdx dy ð2Þ

which is now vector-valued. If the incident waveis a normally incident, uniformplane-wave, it willcarry no space-dependence and can be removedfrom the integral. The only integral to be eval-uated, then, is given by

~A ðkx; kyÞ ¼∬þ∞

�∞~J ðx; yÞeiðkxxþkyyÞdx dy ð3Þ

which is a Fourier integral distributed acrosseach of the four elements of the 2 × 2 Jonesmatrix yielding a Fourier coefficient ~A ðkx; kyÞwhich is itself a Jones matrix (Fig. 1B).This matrix Fourier transform is physically

significant. Instead of an amplitude and phase,as in traditional Fourier optics, a given direc-tion ðkx; kyÞ is associated with a polarization-dependent behavior given by the Jones matrixoperator ~Aðkx; kyÞ. In a sense, this descriptiondecouples the optical element described by~Jðx; yÞ from the polarization of the illuminatinglight: The interaction of all possible incidentpolarizations with ~Jðx; yÞ is handled at once bythe matrix Fourier transform ~A ðkx; kyÞ.In this work, we specialize to polarization-

dependent diffraction gratings—that is, elementsin which ~Jðx; yÞ is a periodic function and theangular spectrum ~Aðkx; kyÞ is discrete, yieldingdiffraction orders. Diffraction gratings simplyreveal the consequences of this matrix Fourieroptics, in particular, that it can be inverted andused as a design strategy for multifunctionalpolarization optics. Suppose there is a set ofdesired polarization devices to be realized (e.g.,polarizers, waveplates, optically active elements,or any behavior that can be described by a Jonesmatrix) having the Jones matrices f~Jkg and a setof diffraction orders f‘g . The Fourier seriesexpansion

~J ðx; yÞ ¼Xk→∈f‘g

~JkeiðkxxþkyyÞ ð4Þ

represents a single optical element realizing allof these (potentially complicated) polarization-dependent functions in parallel with each orderimplementing its own polarization device (Fig. 1C).In the case of a grating, the Fourier transform

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1Harvard John A. Paulson School of Engineering and AppliedSciences, Harvard University, Cambridge, MA 02138, USA.2Section de Physique, École Polytechnique Fédérale deLausanne, CH-1015 Lausanne, Switzerland. 3Department ofPhysics, Harvard University, Cambridge, MA 02138, USA.*Corresponding author. Email: capasso@seas.harvard.edu

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Eq. 3 becomes a Fourier series in Eq. 4 with theoptical elements f~Jkg as coefficients. We refer tosuch a diffraction grating as a matrix grating.This way of viewing polarization-dependent

diffraction has not seen wide acknowledgmentor use in the literature of polarization [withsome limited exceptions in diffractive optics(13, 14) and polarization ray-tracing (15, 16)].In a common strategy, optical elements—especiallyin the field of metasurfaces—are often designedso that when a given polarization is incident, theoutput polarization state is uniform across theelement and a scalar phase profile is imparted.When the orthogonal polarization is incident,the output polarization state is again uniformand a second phase profile is experienced. In thisway, optical elementswith different functions forchosen linear, circular, and arbitrary elliptical po-larization bases (7, 8, 17) can be realized. Notably,what is widely referred to as the “geometric” or“Pancharatnam-Berry” phase, at least as it relatesto this problem, is a subcase of that approach,being used to create scalar phase profiles forcircularly polarized light (2, 9, 18–21). In otherpast work, polarization is understood to varywith space, but a particular incident polariza-tion state is assumed (22, 23). These past designstrategies are subcases of the matrix approachpresented here.

Parallel polarization analysis by unitarypolarization gratings

The formalism presented in the last section isgeneral. Equation 4 provides a prescription forthe realization of a diffractive element imple-menting arbitrary polarization behavior, butdoes not specify the nature of the desired opti-cal functions (contained in f~Jkg) or how the re-sulting ~Jðx; yÞ should be practically realized.We focus on cases in whichf~Jkg, the behaviors

implemented by the grating orders, are polariza-tion analyzers. This choice is not a fundamentalone, but it is highly relevant for practical appli-cations: Analyzers are perhaps the most fun-damental polarization element and, moreover,devices that project light onto different states ofpolarization are of practical use in polarimetry,the measurement of light’s polarization state(24). A diffraction orderk behaving as a polarizercan be described by a Jonesmatrix that is a dyadic(outer product) expressed as

~Jk ¼ akjpkihqkj ð5Þ

Equation 5 describes a Jones matrix analyzingfor the chosen Jones vector jqki in accordancewith Malus’ law: When jqki is incident, inten-sity transmission is maximum. If instead the or-thogonal polarizationjq⊥kiwithhq⊥kjqki ¼ 0arrives,the output light is quenched. The light emergingfrom ~Jk will carry the polarization state jpki. (Atraditional polarizer familiar from laboratory ex-perience has jpki ¼ jqki). Finally,ak is a complex(scalar-valued) weight.Once the desired analyzers f~Jkg are specified

(Eq. 5), the grating ~Jðx; yÞ can be derived by Eq. 4.What physical optical element should implement

the resulting ~Jðx; yÞ? A highly convenient real-ization medium takes the form of metasurfaces(25, 26)—subwavelength-spaced arrays of nano-photonic phase shifters—composed of dielectricpillars possessing formbirefringence (7,8). Locally,the Jones matrix of these metasurfaces may bewell-approximatedby a linearly birefringentwave-plate (8) given by

~Jðx; yÞ ¼Rðqðx; yÞÞ

�eifxðx;yÞ 00 eifyðx;yÞ

�Rð�qðx; yÞÞ ð6Þ

Stated differently, such a metasurface can re-alize a sampled matrix grating where ~Jðx; yÞ islocally of the form of Eq. 6. The use of dielectricmetasurfaces obeying Eq. 6 is, again, not a fun-damental choice, but an especially convenientone: fx, fy, and q are all easily and continuouslyadjusted by varying the dimensions and angularorientation of a simple dielectric pillar, easilyfabricated lithographically (R is a 2 × 2 rotationmatrix). We refer to metagratings structuredfrom these elements simply as “gratings,” but incontrast to more generic diffraction gratings,it should be understood that the gratings herepossess special polarization properties owing totheir subwavelength features.By inspection, Eq. 6 describes a Jones matrix

that is (i) unitary everywhere—that is, ~J† ~J ¼ 1

for all ðx; yÞ, with 1 the 2 × 2 identity matrix—and more specifically, (ii) linearly birefringent,having linear polarizations as its eigenvectorsfor all ðx; yÞ.From the completely general matrix picture

presented we have made two specific choices thatthe diffraction orders of the grating should behaveas analyzers and that the grating should locally re-

semble Eq. 6, implying the above two constraints.But are these choices mathematically consistentwith one another? That is, for a general set of dif-fraction orders f‘g implementing the polariza-tion analyzers ~Jk ¼ akjpkihqkj for k ∈ f‘g, willthe ~Jðx; yÞmandated by Eq. 4 obey the form ofEq. 6 for all ðx; yÞ?This question is a matrix analog of extensive

work conducted on phase-only gratings (27, 28).We show in the supplementary text (section S1)that the linear birefringence required by Eq. 6 im-plies that the ~Jk must be symmetric mandatingthat each take the form ~Jk ¼ akjq�kihqkj. That is,the output polarization of each analyzer must bethe complex conjugate of the Jones vector beinganalyzed for; this is equivalent to a mirroringabout the equatorial plane of the Poincaré sphere,a change in sign of the third (chiral) Stokes com-ponent. Physically, this means that the analyzerleaves the polarization ellipse’s shape unchangedwhile reversing the handedness of its rotation, ageneralization of a key conclusion of (7) and (8).Accepting this constraint guarantees linear

birefringence (that is, matrix symmetry) for allðx; yÞ, but not necessarily unitarity everywhere.This unitarity restriction in particular impliesthat, if we insist on a matrix grating with exactlyn orders that are polarization state analyzers, theform of Eq. 6 can only be matched everywhereif n = 2 and the polarizations analyzed for arestrictly orthogonal (as proven in supplementarytext section S1) (17).

Design strategy and optimization

If we insist on using a single-layer metasurfaceobeying Eq. 6 everywhere, then wemay not haveall light confined to an arbitrary set of diffractionorders acting as polarization state analyzers. But

Rubin et al., Science 365, eaax1839 (2019) 5 July 2019 2 of 8

Fig. 1. Matrix Fourier Optics.(A) A scalar optical field Uðx; yÞhas a plane-wave expansionAðkx; kyÞ obtained by Fouriertransform (F ) that permits itspropagation forward in space.(B) An obstacle implementing

a 2 × 2 Jonesmatrix ~Jðx; yÞ thatvaries with space (and thuscontains four scalar functions,which are schematically repre-sented as different planeshere) has a Fourier-domain

representation ~Aðkx; kyÞ that isalso a Jones matrix. ~Aðkx; kyÞencodes the behavior of direc-tion ðkx; kyÞ as a function ofincident polarization. (C) If~Jðx; yÞ represents a periodicgrating, its orders aredescribed by Jones matrix

coefficients ~Jðkx ;kyÞ, eachdescribing an independentpolarization device, forinstance, a configuration ofbirefringent plates.The grating implements many such devices in parallel.

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what if some light is allowed to leak into otherdiffraction orders? Then, the ~Jk of some orderscould behave as analyzers for arbitrary polar-ization states with a limited amount of leakageinto other diffraction orders (implementing theirown ~Jk) so that the overall grating formed in Eq. 4is still unitary everywhere. In other words, thegrating can function as a coupled system inwhichneighboring diffraction orders compensate fordesired polarization-dependent behavior on aselected few in a way that preserves overall uni-tarity for all ðx; yÞ.Here, we seek to design a grating ~Jðx; yÞ that

locally obeys Eq. 6 with a set of diffraction ordersf‘g each having ~Jk that are analyzers for anarbitrarily specified set of polarization statesfjqkig as defined in Eq. 5 with as little light aspossible leaking into diffraction orders out-side off‘g. This is a question of optimization. Wedefine the quantities Ikq ¼ hqkj~J †

k~J kjqki , the

power on diffraction order k when the pre-ferred polarization jqki is incident, and Ikq⊥ ¼hq⊥kj~J†k~Jkjq⊥ki, the power on diffraction order kwhen the orthogonal polarization jq⊥ki with

hq⊥kjqki ¼ 0 is incident. The sumXk∈f‘g

Ikq should

be maximized to keep as much light as possiblein the orders of interest. Simultaneously, thecontrast of each order given by h ¼ ðIkq � Ikq⊥ Þ=ðIkq þ Ikq⊥ Þ—the polarization sensitivity of the orderto the desired polarization jqki—should be asclose to unity as possible to constrain the ~Jk toact as the desired analyzers (in the sense of Eq. 5).This can be addressed with gradient-descent.We reserve detailed discussion of this optimiza-tion to the supplementary text [sections S1 andS2, where we also describe a path to an analyt-ical formof the solution using variationalmethods(17, 27, 28)].Many previous works have considered dif-

fraction gratings capable of splitting, and thusanalyzing, light on the basis of its polarizationstate. However, these works have generally takena scalar approach, seeking to impart oppositeblazed grating phase profiles on orthogonal po-larization states. Consequently, several individ-ually designed gratingsmust be interlaced (oftencalled “spatialmultiplexing” or “shared aperture”)(29–32) or, equivalently, cascaded in series (1, 33)to create a grating whose orders analyze for anymore than two polarization states. This is in-herently problematic: These configurations can-not implement polarimetrywith any less than sixmeasurements (whereas four is the minimumrequired), compromising sensor space.Moreover,the polarization states of the analyzers cannot bearbitrarily specified. Interlacing different gratingsalso introduces unwanted periodicity, mandatingloss of light to out-of-plane diffraction. Thematrixapproach of this work shows that interlacing isnot necessary—all functions can be integratedinto a single grating—and moreover, affordspossibilities not achievable by interlacing. Thetetrahedron grating that we will present, as a sim-ple example, is not possible by simple interlacingas none of its four analyzer states are orthogonalto any of the others.

ExperimentTetrahedron grating designGratings implementing parallel polarization anal-ysis are of practical interest for Stokes polarime-try (24, 34) which requires a minimum of fourprojective polarization state measurements. Formaximum fidelity of Stokes vector reconstruction,these four states should be as distinct from oneanother as possible, accomplished by choosinganalyzer states corresponding to a tetrahedroninscribed in the Poincaré sphere (polarizationstate space) (35–37). We use the matrix formal-

ism and optimization scheme described aboveto design a two-dimensional diffraction gratinganalyzing for these four polarization states onits innermost four orders. The leftmost panelof Fig. 2A gives a map of these orders and thepolarization ellipses ðfjqkigÞ they are designedto analyze for in k-space. These diffractionorders and desired polarization states can be fedinto the aforementioned optimization, yielding anumerical ~J ðx; yÞ that is locally of the form ofEq. 6 (unitary and linearly birefringent). It isthen straightforward to map the locally required

Rubin et al., Science 365, eaax1839 (2019) 5 July 2019 3 of 8

Fig. 2. Matrix gratings forarbitrary parallel polarizationanalysis. (A) A 2D gratingunit cell is designed to analyzefour polarization statescorresponding to a tetrahedroninscribed in the Poincarésphere. On the left is a map ofthe diffraction orders andthe polarization ellipses theyanalyze in k-space.The designed11 × 11 element grating unitcell containing TiO2 rectangularpillars implementing this atl ¼ 532 nm is shown in design(middle) and as-fabricated[scanning electron micrograph(SEM), right]. Scale bar, 1 mm.(B) The grating is illuminated bylight whose polarization isvaried while recording theoutput polarization on a singlediffraction order with a full-Stokes polarimeter permittingreconstruction of the order’s4 × 4 experimental Mueller

matrix ~Mðm;nÞ. (C) Thepolarization contrast of eachorder is shown. Each orderis labeled by the polarizationellipse it analyzes, and resultsfrom the analytical gratingdesign (as-optimized, red), afull-wave simulation of thegrating (blue), and experiment(green) are given. (D) Thepolarizations analyzed by eachorder [for which they havethe contrasts given in (C)] ofthe tetrahedron grating areshown on the Poincaré spherealongside a tetrahedronindicating the desired analyzerpolarizations as predicted by the optimization, a full-wave simulation, and as-measured. (E) A set of1024 uniformly sampled input polarization states on the Poincaré sphere can be operated on by

an experimentally determined Mueller matrix ~Mðm;nÞ to form a distorted set of states depicting output

polarization as a function of input. Each sphere on the right represents the behavior of one of thefour engineered grating orders, whose designed analyzer polarizations are shown. On each, a bluearrow denotes the measured analyzer polarization jqðm;nÞi from (C), a red arrow the equator-mirrored

polarization jq�ðm;nÞi, and a green arrow the average output polarization over all points which, according

to the formalism here, should overlap with the red one. (In some cases, the overlap of red and greenarrows is too close to permit visual discrimination.)

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Jones matrix (fx , fy , q) to geometries of actualstructures by referring to a library of such struc-tures in a material platform/wavelength regimeof interest (8).In this work, that material platform is TiO2

pillars fabricated with an e-beam lithographyand atomic layer deposition technique exten-sively documented elsewhere (38). This permitsoperation at technologically important visiblewavelengths, aiding the camera application dis-cussed below. We stress, however, that neitherTiO2 nor visible wavelengths are central to thiswork. The grating unit cells presented here have11 such elements to a side with an interelementseparation of 420 nm so that the diffraction angleatl ¼ 532 nm isqDe6:6°(paraxial). The subwave-length spacing of the pillar elements themselvesassures that radiative orders may only stem fromthe collective unit cell of 11 × 11 elements. Thegrating unit cell designed by optimization for thetetrahedron case is shown in Fig. 2A (top andbottom, respectively), both in design and as-fabricated (electron micrograph, right). Thisunit cell is tessellated hundreds of times to createa grating.

Mueller matrix polarimetry and resultsIn polarization optics, the Jones and Muellermatrices describe polarization-dependent behav-ior. Of the two, only theMuellermatrix is directlyobservable, being a description of optical inten-sities rather than electric fields. To that end, weperform Mueller matrix polarimetry, that is, theexperimental determination of the 4 × 4Muellermatrix ~Mðm;nÞ of each of the four grating ordersðm;nÞ of interest (as a 2D grating, the orders arelabeled by two integer indices). The grating isilluminated with laser light at l ¼ 532 nm inseveral (at least four) different input polarizationstates with known Stokes vectors fS→ing. As thepolarization switches between the different knowninputs, a full-Stokes polarimeter canbe placed onthe order of interest to record the correspondingoutput Stokes vectors, fS→outg. The matrix ~M ðm;nÞlinking fS→ing to fS→outg can then be numericallydetermined from the data in the least-squaressense. This is sketched in Fig. 2B and discussed inmore detail in supplementary text section S3 (17).In analyzing the results, it is difficult tomake a

direct comparison of matrix quantities. Instead,because each order of interest is designed to act

as a polarization analyzer, it makes sense to askthe following questions: Do the orders act asanalyzers? For which polarizations? And, withwhat efficiency?The first question is addressed by Fig. 2C,

which plots the polarization contrast of eachdiffraction order. Each group of bars in Fig. 2Ccorresponds to the diffraction order designed toanalyze for the polarization ellipse shown belowit. Each group contains three color-coded bars:one for the numerically optimized ~Jðx; yÞ, onefor a full-wave simulation, and one for measure-ment. On the left of Fig. 2D, the contrast (thenormalized difference between the intensity onthat order when its preferred polarization is inci-dent versus the orthogonal one) is plotted. Anideal polarizer would have 100% contrast. Thenumerically optimized result predicts near per-fection (100%). In a simulation of the gratingdesign, the contrast decreases somewhat, andthen again as measured in actuality. However,the four orders of the measured grating all showpolarization contrasts in excess of 90%—the de-signed orders of the grating do act as polarizers(analyzers).Second, it must be verified that the orders act

as analyzers for the polarization states specifiedin the design. In Fig. 2D, the polarization statesfor which each grating order has maximum out-put intensity (as-optimized, in simulation, andas-measured) are plotted on the Poincaré spherealongside the tetrahedron representing the goalof the design. It can be seen that these analyzerpolarizations are close to their desired counter-parts (there are no notable mixups in Fig. 2D,whereby one analyzer polarization lands very faraway appearing to be closer to another, falselyexaggerating correspondence).Figure 2, C and D, shows that the metasur-

face grating implements four arbitrarily specifiedanalyzers in parallel with no polarizers or wave-plates, lending validity to the matrix approachunderlying its design.We address the question ofefficiency in the supplementary text (section S3)(17). The grating’s diffraction efficiency cannot bequantifiedwith a single number—it is polarizationdependent. Averaged over all possible input polar-izations, efficiency in terms of power diffractedinto the four orders of interest over incident powerexceeds 50%, high enough to enable practical use.The data contained in Fig. 2, C and D, are de-

rived from the first row of the Mueller matrix,which dictates the intensity S0 of the outgoingbeam. The remainder of the Mueller matrix con-trols the output beam’s polarization state. Thiscan be visualized by individually operating on aset of input Stokes vectors that uniformly samplethe Poincaré sphere (1024 dots) with each order’smeasured Mueller matrix ~M ðm;nÞ and plottingthe set of Stokes vectors that result as a new,distorted set of dots. This is shown for each of thefour orders of interest in Fig. 2E on four spheres.If each behaved as a perfect analyzer (compare toEq. 5), all polarizations would be mapped into asingle point in the distorted diagram at the statecorresponding to jpki. The grating is not perfectin experimental reality, so the spheres in Fig. 2E

Rubin et al., Science 365, eaax1839 (2019) 5 July 2019 4 of 8

Fig. 3. Metagrating full-Stokes polarizationcamera. (A) A matrixmetagrating (as in Fig. 2) isintegrated with an asphericlens to image four diffractionorders onto four quadrantsof a CMOS imaging sensor.A ray trace of one diffractionorder is shown. The camera isdesigned to image far-awayobjects, so each colorcorresponds to differentparallel ray bundles incidenton the grating from differentangles over a T5° FOV. Notshown: A 10-nm bandpassfilter at 532 nm and an aper-ture in front of the grating tolimit the FOV to preventoverlap of the four subimages.(B) Each copy of the imageon each quadrant has beenanalyzed along a differentpolarization. Pixel-wisedifferences in intensity can beused to synthesize a singlepolarization image of thescene in which the full Stokesvector is known at each point.(C) A clearer side view ofthe ray trace in (A). Blue,orange, and green correspondto ray bundles incident at +5°, 0°, and −5°, respectively. (D) Optical microscope image of the gratingsample, which is 1.5 mm in diameter and shows Fresnel zones owing to the weak, polarization-independent lensing effect imposed on top of the metagrating. An SEM inset shows the subwavelength,form-birefringent TiO2 pillars comprising the metasurface. (E) The imaging system in (A) can bepackaged into a practical, portable prototype with adjustable focus. The essential part of the camera,as shown in (C), is about 2 cm long.The prototype here is larger for ease of optomechanical mounting.

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contain points distributed over the entire sphere;the extent to which these points are concentratedin one direction and extremely sparse elsewhereillustrates that each diffraction order does in-deed act as an analyzer. For each grating order inFig. 2E, a blue arrow corresponds to the polariza-tion being analyzed (copied fromFig. 2D) (jqðm;nÞiin the notation of Eq. 5). A green arrow showsthe polarization on the grating order averagedover the set of incident polarizations. Finally, ared arrow depicts jq�ðm;nÞi , which has flippedhandedness with respect to jqðm;nÞi and is thusmirrored about the equator of the sphere. Ac-cording to the picture presented above, the linearbirefringence of the grating elements impliesthat the output must necessarily take on theform of jq�ðm;nÞi. This aspect of the theory is sup-ported by the close overlap of the red and greenarrows in each plot of Fig. 2E.In the supplementary text, results are given

for a second grating, an octahedron grating, inwhich six diffraction orders act as analyzers (17).

Full-Stokes polarization imaging

We now apply these matrix gratings in an areaof great practical interest. If paired with four de-tectors, the tetrahedron grating presented above

could function as a single-component, compact,and integrated full-Stokes polarimeter (a sensorto measure the polarization state of a beam), anarea where metagratings (29–32) and integratedapproaches (39, 40) have recently attracted con-siderable interest (34). However, for a variety ofapplications (41–44), a polarization camera, orimaging polarimeter, is of even more utility. Apolarization camera captures the Stokes vectorat each point in an image. In the case of the four-element Stokes vector, this necessitates fourindependent image acquisitions along indepen-dent polarization directions, whichmay be takensequentially in time (“division-of-time,” limitingtemporal resolution and often requiringmovingparts), by patterning a focal plane array withmicropolarizers (“division-of-focal-plane,” requir-ing expensive fabrication, usually without of-fering full-Stokes vector determination, andmandating loss of photons to absorptive micro-polarizer elements), or by simultaneous captureof the image along four paths each with indepen-dent polarization optics (“division-of-amplitude,”substantially increasing system bulk and com-plexity) (41).Grating-based approaches are not new to po-

larimetry (24, 45) (and imaging polarimetry),

particularly in the liquid crystal (46) and, morerecently, metasurface literatures (29–32). Owingto the limitations of previous approaches, how-ever, multiple gratings—either cascaded in seriesor patterned adjacent to one another—are re-quired to implement the measurements neces-sary for full-Stokes vector determination. Thematrix gratings here are free from this complica-tion and full-Stokes imaging can be implementedwith a single polarization element, promisingwide-ranging applications inmachine vision andremote sensing.

Design of an imaging system

The task here is to integrate a metagrating into aphotographic imaging system. The tetrahedrongrating described above is chosen because itoffers full-Stokes determination with only fourmeasurements, the minimum necessary. Wedeveloped an imaging system composed of thisgrating (implemented with a TiO2 metasurfaceas above) followed by an aspheric lens ( f= 20mm,whose choice is discussed in supplementary textsection S4) and a standard monochrome com-plementarymetal–oxide–semiconductor (CMOS)imaging sensor. This is depicted in Fig. 3A.Relative to Fig. 2A, the grating is rotated by 45°

Rubin et al., Science 365, eaax1839 (2019) 5 July 2019 5 of 8

Fig. 4. Full-Stokes polarization imagery. Indoor (A to C) and outdoor(D and E) photography with the camera depicted in Fig. 3. In eachcase, the raw unprocessed exposure, S0 (the traditional monochromeintensity image), the azimuth of the polarization ellipse [in degrees,given by arctanðS2=S1Þ], and the degree of polarization (DOP, given byffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

S21 þ S2

2 þ S23

q=S0) are shown. See text for detail on each case. Indoor

images were acquired with exposures on the order of 100 ms, outdoorimages with exposures on the order of 10 ms. In both cases, polarizationimagery can be acquired at video framerate. The bright disk in the centerof each raw exposure is zero-order light that does not interact with thegrating and thus forms a defocused image of the scene. All images are at a

single color (green, l e 532 nm).

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so that each of its orders forms an image of thescene on one quadrant of the imaging sensor.Each quadrant then contains a version of thephotograph analyzed along its characteristicpolarization—these images can be simultaneouslyacquired and the Stokes vector S

→reconstructed

pixel-wise (Fig. 3B), forming a monochromaticpolarization image.Simple ray tracing is used to optimize all as-

pects of the system: the separation of the gratingand the asphere, the separation of the asphereand the imaging sensor, and finally, the gratingperiod (and thus, the diffraction angle qD). Thegoal of the optimization is to take parallel raybundles over aT5° field-of-view (FOV)—which areassumed to emerge from a very distant object—and focus them within the bounds of a quad-rant of the sensor. An azimuthally symmetric,polarization-independent (scalar) phase profileis added on top of the matrix grating during thedesign and is experienced by all diffraction orders.This produces a weak lensing effect that aids inimaging. The design focuses on just one gratingorder. However, the other three will be imaged totheir respective quadrants by default because thesystem is rotationally symmetric.A real ray trace of the imaging system is shown

in Fig. 3A with a clearer side view in Fig. 3C. Theray colors correspond to parallel bundles inci-dent at different angles on the matrix grating.The optimized grating hasN = 10 elements at a420-nm pitch, yieldingqD ¼ 7:3° at l ¼ 532 nm.

In Fig. 3D a microscope image of the 1.5-mmsample is shown. The sample is surrounded bymetal to block stray light and displays Fresnelzones stemming from the weak lensing impartedon top of the grating (whose periodicity is toosmall to see at this magnification).Finally, the entire system can be packaged

into a prototype for practical use in polarizationphotography as shown in Fig. 3E with variablefocus. The size of this prototype is much largerthan is strictly necessary to permit easy opto-mechanical mounting—as is shown in Fig. 3C,the functional part of the system is only ~2 cmlong. An aperture in front of the camera permitscontrol of the FOV so that the four copies of theimage do not overlap, and a 10-nm bandpassfilter at 532 nm behind the sample prevents thedispersive nature of the grating from interferingwith imaging. This is important to note— if abroad band of illuminating wavelengths wereintroduced to the imaging system, the gratingwould effectively smear the images together, withspatial and spectral information colocated on thesensor. The bandwidth of dye filters used in con-ventional color sensors (~100 nm for the Bayerfilter) is too wide to effectively address this. Forcolor or hyperspectral polarization imaging, adifferent approach could be employed, such as theuse of a tunable filter at the input or incorporationof the grating into a pushbroom scanning systemwhere only one space dimension is acquired at atime (as is common in remote sensing).

Polarization imagingThe metagrating camera can then be used forpractical full-Stokes photography. The raw sen-sor acquisition approximates Fig. 3B. To form apolarization image, the four image copiesmust bealigned (registered) to one another, forming thevector I

→ ¼ ½ I0 I1 I2 I3 �T of measured intensityfrom the four quadrants at each pixel. If the po-larizations analyzed for by each diffraction orderare precisely known (calibration), the Stokes vec-tor at each pixel can be computed as S

→ ¼ A�1 I→,

where A is a matrix whose rows are these ana-lyzer Stokes vectors. Image registration and polar-imetric calibration are addressed simultaneouslywith an angle-dependentmethod (supplementarytext section S4) (17).In a polarization image,S

→ ¼ ½ S0 S1 S2 S3 �T isknown at each pixel, which does not admit easyvisualization. Instead, images can be formed fromscalar quantities derived from the Stokes vector.In the literature of polarization imaging, two pa-rameters of the polarization ellipse are commonlyused. The first is the azimuth angle,arctanðS2=S1Þ,which yields the physical orientation of the po-larization ellipse. The second is the degree ofpolarization, given by p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

S21 þ S22 þ S23p

=S0,which quantifies the degree to which light is (oris not) fully polarized. Finally, an image can beformed of justS0 which, as the intensity of thelight, yields a traditionalmonochromephotograph.In Fig. 4, exemplar photographs captured by

the polarization imaging system are shown. Foreach, images of the raw acquisition on the sen-sor, S0, the azimuth angle, and the degree ofpolarization (DOP) are shown. In all of theseimages, the illuminating light is unpolarizedand diffuse, that is, incident from all directions.The indoor images in Fig. 4 are taken underdiffuse light-emitting diode illumination,whereasoutdoor images are acquired in broad daylight.In the raw exposures, a bright disk in the cen-ter represents zero-order light that does notinteract with the grating and thus forms a de-focused version of the image (some stray light isalso present).Next, we describe each example. Figure 4A

constitutes a simple test—a paper frame holdingeight sheets of polarizing film whose axes arearranged radially outward with image-forminglight allowed to transmit from behind. A tra-ditional photograph sees no difference betweenthe sheets (S0), but an image of the azimuth ac-curately reveals their angular orientations.More-over, the DOP image shows that light passingthrough the sheets is highly polarized relative tothe surrounding surface.Figure 4, B to E, examine the polarization de-

pendence of specular reflection (47). Unpolarizedlight becomes partially polarized upon specularreflection in a direction perpendicular to theplane of incidence. In Fig. 4B, a conventionalplastic soda bottle is imaged head-on. Froman intensity image (S0 ), the bottle’s curvedshape could not be ascertained a posteriori.The azimuth image, however, displays smooth,continuous change around the top of the bottleevidencing its conical shape and could be used

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Fig. 5. Polarization imaging of S3. Example imagery from the camera making explicit use of S3, partof its full-Stokes capability. All images are acquired with a backlight linearly polarized at 45°. In eachexample, raw sensor acquisition, S0 (traditional intensity image), and S3 are shown. (A) 3D glassesare seen to contain opposite circular polarizers, invisible to the traditional intensity image. (B and C) Alaser-cut acrylic piece is stressed by hand-squeezing and displays stress birefringence evident in theS3 image. (D) An injection-molded plastic part (a tape dispenser) has complex, in-built stressesthat are visible in the S3 channel of the polarization image. Edge artifacts in these images and thosein Fig. 4 are discussed in the supplementary text (section S4) (17).

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in three-dimensional (3D) reconstruction; in-deed, the polarized nature of specular reflectionhas been used as a means of depth imaging(47–50). Figure 4C illustrates the same conceptfor a more complicated object— the face of oneof the authors. Its 3D nature is not discerniblefrom the S0 image (as opposed to, say, a mereprinted photograph of a face), but the azimuthimage traces its contour and could be used in3D facial reconstruction.Figure 4, D and E, depict outdoor scenes. In

Fig. 4D, a bicycle is seen parked on a grassyfield after rain on the Harvard campus. In frontof it is a boundary between grass and asphalt, aswell as a puddle. In the azimuth image, a strongdelineation is seen between the wet pavement,where the polarization direction is well-definedparallel to the ground, and the grass and puddle,where the azimuth is somewhat random becausethe reflection is diffuse. Moreover, the azimuthreveals the presence of an asphalt walkway inthe rear of the image, which is difficult to see inthe intensity image. In Fig. 4E, a row of cars isphotographed. Cars illuminated by sunlight are afavorite target in polarization imaging literaturebecause the windshields tend to yield strongpolarization signatures. This is seen for all threecars in the azimuth image, where thewindshieldsand auto bodies have definite polarization azi-muths and in the DOP image where the wind-shields are highly polarized relative to the restof the car and background.We note, however, that the examples in Fig. 4

make only indirect use of S3, the chiral Stokescomponent, through the DOP. Many existing po-larization cameras, particularly using the division-of-focal-plane approach, are unable to measureS3 (42). Figure 5, however, directly demonstratesthe full-Stokes capability of the camera. In Fig. 5,objects are illuminated from behind by lightlinearly polarized at 45° from a liquid crystalcomputer display. In each case, the raw sensoracquisition, S0, and S3 are shown. Figure 5A de-picts a pair of 3D glasses whose frames containopposite circular polarizers. This is not visiblein the traditional intensity image S0, but isreadily seen in theS3 image where each lens hasa T1 value. In Fig. 5B, a piece of acrylic is held infront of the camera, displaying little chiral com-ponent. Upon squeezing (Fig. 5C), no change isperceptible in the traditional photograph, butconsiderable information is now visible in S3stemming from stress birefringence (the photo-elastic effect). This is also evident in Fig. 5Dwhere an injection molded plastic part—a tapedispenser—displays an in-built, complicateddistribution of stress in S3 not visible with tra-ditional photography. Visualization of stress fieldsis an important application of polarization imag-ing, one that is readily accomplished with thecamera presented here.These examples demonstrate powerful appli-

cations in machine vision, remote sensing, andother areas. The camera presented here requiresno conventional polarization optics, offers simul-taneous data acquisition with no moving parts,does not require an imaging sensor specially

patterned with micropolarizers, and is compactand potentially mass-producible. Moreover, thecamera’s simplicity—just a single grating with animaging lens—suggests that these gratings couldbe designed around existing, conventional imag-ing systems to create polarization-sensitive ones.

Conclusion

Wehave introducedageneralpictureofpolarization-dependent diffraction frompolarization-sensitiveobstacles. This matrix Fourier optics describesoptical elements that can enactmanypolarization-dependent functions in their diffraction patterns.In this work, we applied this picture to the caseof periodic gratings analyzing arbitrary polar-izations in parallel, characterized these gratings,and showed how they enable a polarization cam-erawithno additional polarization optics,movingparts, or specialized sensors. Practical polarizationphotography with this camera was demonstrated.Our work generalizes a large body of research

in polarization-sensitive diffractive optics andmetasurfaces. Its emphasis is on the collectivebehavior of many elements at once as expressedin the Fourier transform, rather than the point-by-point consideration of each element alone,and thus advances design strategies in theseareas beyond simple phase profiles. Moreover,this work suggests several interesting directionsof research in multifunctional polarization op-tics. For instance, although polarization analyzers(polarizers) have been studied here, the matrixapproach is quite general: Gratings implement-ing a wide variety of polarization operators ontheir orders [e.g., waveplates, optically activemedia, and possibly more exotic behavior (51)]could be envisioned. The constraint of linearlybirefringent elements is by no means fundamen-tal, andwith the freedomafforded by lithographicfabrication, elements with more complex polar-ization responses (52) or multilayer structurescould be used to realize these behaviors. Thiswork illustrates the ability of metasurfaces tomarkedly simplify the architecture of systemsusing polarization optics. Gratings enabled bythe approach here present a simpler means offull-Stokes polarization imaging that can be eas-ily extended to other imaging systems and wave-lengths. These compact, lightweight, and passivedevices could enable the widespread adoption ofpolarization imaging.

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ACKNOWLEDGMENTS

We acknowledge M. Tamagnone, Y.-W. Huang, V. Ginis, S. Kheifets,Z. Li, A. Zaidi, and A. Dorrah (all of Harvard University) for helpfulcomments. Funding: N.A.R. acknowledges support from the

National Science Foundation Graduate Research FellowshipProgram (GRFP) under grant no. DGE1144152. This work wassupported by the Air Force Office of Scientific Research undergrant nos. FA9550-18-P-0024, FA9550-16-1-0156, and FA9550-14-1-0389 (MURI). We also acknowledge support from aPhysical Sciences and Engineering Accelerator grant fromHarvard University’s Office of Technology Development and agift from the Google Accelerated Science team. F.C.acknowledges support from King Abdullah University of Scienceand Technology (KAUST) Office of Sponsored Research (OSR)under grant no. OSR-2016-CRG5-2995. This work wasperformed in part at the Center for Nanoscale Systems (CNS), amember of the National Nanotechnology CoordinatedInfrastructure (NNCI), which is supported by the NationalScience Foundation under NSF award no. 1541959. CNS ispart of Harvard University. Author contributions: N.A.R.conceived the experiment and fabricated the gratings. G.D.wrote grating design codes and performed experimentalcharacterization. N.A.R. and Z.S. clarified theoretical aspectsof the work. W.T.C. contributed to camera design. N.A.R.and P.C. assembled the camera prototype, wrote camerasoftware, and acquired the images in Figs. 4 and 5. N.A.R. andF.C. wrote the manuscript. F.C. supervised the research.Competing interests: A provisional patent application(US 62/834,343) has been filed on the subject of this work.Data and materials availability: All data needed to evaluatethe conclusions in the paper are present in the paper and/or thesupplementary materials.

SUPPLEMENTARY MATERIALS

science.sciencemag.org/content/365/6448/eaax1839/suppl/DC1Materials and MethodsSupplementary TextFigs. S1 to S22Tables S1 and S2Movie S1References (53–57)

28 February 2019; accepted 8 May 201910.1126/science.aax1839

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Matrix Fourier optics enables a compact full-Stokes polarization cameraNoah A. Rubin, Gabriele D'Aversa, Paul Chevalier, Zhujun Shi, Wei Ting Chen and Federico Capasso

DOI: 10.1126/science.aax1839 (6448), eaax1839.365Science 

, this issue p. eaax1839Scienceprovide a simplified route for polarization imaging.full-Stokes compact polarization camera without conventional polarization optics and without moving parts. The results

designed a metasurface-basedet al.a polarizer and analyzer setup rotating to reveal the degree of polarization). Rubin (withinformation from a scene. Conventional polarimeters can be bulky and usually consist of mechanically moving parts

Imaging the polarization of light scattered from an object provides an additional degree of freedom for gainingA metasurface polarization camera

ARTICLE TOOLS http://science.sciencemag.org/content/365/6448/eaax1839

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