Revealing topological phase in Pancharatnam-Berry metasurfaces using mesoscopic electrodynamics
Zhanjie Gao, Sandeep Golla, Rajath Sawant, Vladimir Osipov, Gauthier Briere, Stephane Vezian, Benjamin Damilano, Patrice Genevet, Konstantin E. Dorfman
RRevealing topological phase in Pancharatnam-Berry metasurfaces using mesoscopicelectrodynamics
Zhanjie Gao, ∗ Sandeep Golla, ∗ Rajath Sawant, Vladimir Osipov,
1, 3
Gauthier Briere, Stephane Vezian, Benjamin Damilano, Patrice Genevet, † and Konstantin E. Dorfman ‡ State Key Laboratory of Precision Spectroscopy,East China Normal University, Shanghai 200062, China Universit´e Cˆote d Azur, CNRS, CRHEA, rue Bernard Gregory, Valbonne 06560, France Holon Institute of Technology, 52 Golomb Street, POB 305 Holon 5810201, Israel
Relying on the local orientation of nanostructures, Pancharatnam-Berry metasurfaces are currentlyenabling a new generation of polarization-sensitive optical devices. A systematical mesoscopicdescription of topological metasurfaces is developed, providing a deeper understanding of the physicalmechanisms leading to the polarization-dependent breaking of translational symmetry in contrastwith propagation phase effects. These theoretical results, along with interferometric experiments,contribute to the development of a solid theoretical framework for arbitrary polarization-dependentmetasurfaces.
I. INTRODUCTION
Pancharatnam-Berry (PB) metasurfaces, made of pe-riodic arrangements of subwavelength scatterers or an-tennas, have been extensively studied over the last fewyears and are currently considered as a forthcoming sub-stitute of bulky refractive optical components [1, 2]. Thereflection and refractive properties of light at interfacescan be efficiently controlled by appropriately designingthe phase profile of these surfaces [3]. Several applica-tions of PB metasurfaces, ranging from coloring to therealization of multifunctional tunable/active wavefrontshaping devices, have been proposed[4]. As a result of thefascinating degree of the wavefront manipulation offeredby metasurfaces, this technology is currently burstingthrough the doors of industry, particularly driven by theirpotential application in redefining optical designs, suchas lenses [5–8], holography [9–11], polarimetry [12–14]and a variety of broadband optical components, includingfree-form metaoptics [15–19].Despite these applications, significant efforts are cur-rently being made in deriving proper theoretical frame-works to guide the design of complex components. Most ofthe disruptive attempts in controlling light-matter interac-tions rely on a fully vectorial Maxwell’s equations, such aseffective medium theories [20–22], and the comprehensiveunderstanding of their polarization responses generallyobtained using extensive numerical simulations, such asfinite element method [23] or finite-difference time domaintechniques [3, 24, 25], which often shows the quantita-tive simulation results but lacking of qualitative physicalinterpretations [26–28]. Another approach, Green’s func-tion method and diffraction theory for gratings, providespartial interpretation of a few diffractive properties ofmetasurfaces. The generalized Snell’s law can be then ∗ Zhanjie Gao, and Sandeep Golla contributed equally to this work. † [email protected] ‡ [email protected] understood as a maximum grating efficiency in a givendiffraction order [29, 30]. However, a vectorial theoret-ical framework is still required to clearly explain whythe generalized Snell’s law occurs in the cross-polarizedtransmitted fields in PB metasurface system in the -1stor 1st diffraction orders only. To overcome these difficul-ties, the concept of geometric phase (PB phase), whichis responsible for the conversion of the polarization statein the linearly birefringent medium [32–36], is introduced.Several works have shown that the transmission matrixwhich describes the birefringent response can be sepa-rated into co-polarized and cross-polarized beams in thecircular basis by applying the PB phase induced by theorientation of nano-antennas [31, 37, 38]. However, thisapproach does not originate from first principle derivationand is not capable of explaining other diffractive proper-ties of PB metasurfaces, such as the connection betweengeneralized Snell’s law and polarization conversion. Obvi-ously, each of these approaches just capture a part of thewhole physical mechanism. To fill the gap between theseconcepts and incomplete demonstrations, a theoreticalframework is highly needed to interpret all the diffractiveproperties of PB metasurfaces in a precise and systematicway.In this letter, we propose a systematic mesoscopic elec-trodynamical theory to study the polarization-dependentmetasurface, showing that the transmission of a co-polarized beam only acquires global phase associatedwith the antenna response, called “the propagation phasedelay”, while the transmission of a cross-polarized beam issensitive to both PB and propagation phases. We extendthis phase effect to a more general situation by decompos-ing the arbitrary polarization of a normally incident lightin circular basis, showing that each eigenstate acquiresan opposite phase delay due to the topological phaseretardation associated with the PB phase (see Eq. 10).Furthermore, we derive a fully electrodynamical expres-sion and conduct optical measurements to analyze andvalidate this theoretical framework describing the diffrac-tive properties of topological phase gradient metasurfaces a r X i v : . [ phy s i c s . op ti c s ] F e b [39, 40], including the physical mechanisms of the coex-istence of the zero and nonzero phase gradient leadingto the ordinary and generalized Snell’s law, and the uni-versal principles of co-polarization and cross-polarizationtransmission. μ m 1 μ m A CB x y 𝛿Φ + θ ′ δ 𝑬 || 𝑬 || 𝑬 ⊥+ 𝑬 ⊥− 𝜹𝚽 − θ z x 𝜙 FIG. 1. (A) Schematic explanation the transmission propertiesof the PB metasurface where E || and E ±⊥ denote co-polarized andcross-polarized beams, θ and θ are the incident and refractionangles, respectively. (B) Schematic of the nanostructure arrayused to generate both classical and anomalous refraction. Thearray consists of a repetition of a unit cell containing five rotatednanopillars with dimensions l x × l y × l z separated by subwavelengthdistances a and a along x and y , respectively. The rotation angle φ = − π N +1 . (C) Scanning electron micrograph of a representativeGaN-based PB metasurface. II. THE MESOSCOPIC MODEL
The topological phase occurring on the converted stateof polarization is generated after transmission across a PBmetasurface, as shown in Fig. 1A. To study these interfacephenomena, we consider non-magnetic PB metasurfacesand express the transmitted light starting from Maxwell’sequations for monochromatic light in the media [41] (CGSunits) ∆ E − ∂ c ∂t E = − π ∇ ( ∇ · P ) + 4 π∂ c ∂t P , (1)where we assume that the electric field E with frequency ω i is far detuned from any electric resonance and P denotes the polarization of the metasurface and substrateson both side (see Section S1.A in Supplementary Materials(SM) for more details).The metasurface can be represented by a lattice witha primitive cell consisting of 2 N + 1 Gallium Nitride(GaN) nanopillars distributed along the x coordinate,which corresponds to a reciprocal lattice vector G mn = πm (2 N +1) a e x + πna e y , where m, n are integers, a and a are the nearest spacing between the individual nanopillarsalong x and y directions, respectively, see Fig. 1B. Thelinear response of the individual nanopillars, at position j in the unit cell, is described by a polarization vector (seeSection S1.B in SM for more details) P j ( z, ρ , ω ) = N X m,n Z Q d κ f mn,j ( φ j ) E ( z, κ , ω ) e i ψ mn,j . (2)Here ψ mn,j = G mn · ( ρ − j a ) + κ · ρ describesonly the propagation phase, and the form-factorof the j -th element in the mn -th lattice unitecell is f mn,j ( φ j ) = ˜Ω( G mn ) / [ π (2 N + 1) a a ]where ˜Ω( G mn ) is the Fourier transform of geomet-ric shape factor Ω( ρ ) = H ( | x | ≤ l x / H ( | y | ≤ l y /
2) with Heaviside function H (condition) = { , when condition is true; 0 , when condition is false } .The momentum integration over κ runs over Q - the firstBrillouin zone. The coefficient N = χ π includes thenanopillars material susceptibility χ .The translational symmetry of the metasurface dictatesthe form of the solution which is given by E ( z, ρ ) = X m,n e i G mn · ρ Z Q d κ (2 π ) E mn ( z, κ ) e i κ · ρ . (3)Considering the thickness l z to be much smaller thanthe xy dimension of the metasurface, we neglect the E z and P z components in the model. An incoming planewave can be written as E in ( z, ρ ) = E i e i k zi z +i κ i · ρ where k zi = q ω i n i c − κ i with propagation condition ω i n i > cκ i and refractive index n i .The geometric anisotropy of the nanopillars can betaken into account by replacing the scalar susceptibil-ity χ by the diagonal 2x2 susceptibility tensor. Thetensor components along the x and y axes are given by χ x and χ y , respectively. Therefore, for the rectangularnanopillar oriented along x and y , the transmission ma-trix in momentum space is given by ˜ E = ˆ˜ T E i , wherethe transmitted electric field in the momentum space is˜ E = { ˜ E x , ˜ E y } , and the incident field is ˜ E i = { ˜ E xi , ˜ E yi } .The transmission matrix then reads (see Section S2 inSM for details) ˆ˜ T = ˜ t xx ˜ t xy ˜ t yx ˜ t yy , (4)where ˜ t ij , i, j = x, y defined in Eq. (S17) explicitly de-pends on the form-factor f mn,j . For the element orientedalong x and y axes, i.e. φ j = 0, the corresponding form-factor f mn,j ( φ j = 0) ≡ sin( ml x π (2 N +1) a x ) sin( nl y πa y ) / ( mnπ ).Since the metasurface consists of nanopillars rotatedaround z axes with the constant incremental angle φ j = − πj N +1 in Fig. 1, the corresponding rotation matrix R ( φ j )is given by ˆ R ( φ j ) = cos( φ j ) − sin( φ j )sin( φ j ) cos( φ j ) . (5)According to superposition principle, the transmissionmatrix of the metasurface can be obtained by summing thecontributions of individual nanopillars, given by ˜ E ( K ) = P j ˆ˜ T ( φ j ) ˜ E i , where the rotation-dependent transmissionmatrix is given by ˆ˜ T ( φ j ) = ˆ R † ( φ j ) ˆ˜ T ˆ R ( φ j ).Using Pauli algebra for two-component polarization ba-sis without explicit factorization of the additional propa-gation phase, the rotation-dependent transmission matrixreads 2 ˆ˜ T ( φ j ) = (˜ t xx + ˜ t yy ) ˆ I + i(˜ t xy − ˜ t yx )ˆ σ z + (˜ t xx − ˜ t yy )( e φ j ˆ σ − + e − φ j ˆ σ + )+ i (˜ t xy + ˜ t yx )( e φ j ˆ σ − − e − φ j ˆ σ + ) . (6)Here, ˆ σ ± = (ˆ σ x ± iˆ σ y ) / e ± φ j can be understood as the PBphase term [31, 37, 38].The transmitted field in the coordinate space (the formof ˆ T is listed in Eq. (S18)) can be consecutively writtenas E ( z, ρ ) = P mnj F mn,j ( z, ρ ) ˆ T ( φ j ) E i ( z, ρ ), where thepropagation factor is F mn,j = e i ψ mn,j [ e i k z z H ( z >
0) + e − i k z z H ( z < | κ = κ i , (7)where k z = q ω i n t /c − K || , K || = ( G x,mn + κ x ) +( G y,mn + κ y ) with momentum vectors of incident light κ i = κ xi e x + κ yi e y . A. Discussion of analytical results
Analogous to the Bragg scattering in solid crystals,constructive interference of the propagating wave on thesub-wavelength periodic structure changes the complexamplitude of the refracted and reflected waves due tocumulative scattering from different crystal planes (seeEq. (7)). The evanescent waves emerge when n = 0,whose momentum vectors satisfy ω i n t /c − K x − K y < n = 0 contains both effects of propagation and topologicalphases. The first line in Eq. (6) corresponding to theco-polarization component transmitted field contains onlythe propagation phase e i ψ mn,j embedded in the propa-gation factor F mn,j . The second and the third lines inEq. (6) yield the cross-polarization components which de-pend on both propagation and PB phases via e i ψ mn,j ± φ j .Due to the translation invariance, the PB phase of theindividual nanopillars is distributed uniformly between 0and 2 π such that P j e ± iΞ j ’ m = 0 , ±
1, where Ξ j = − G mn · j a ± φ j is the total phase. For m = 0,only the PB phase-independent co-polarized componentcan be observed. By calculating the x -dependent part ofpropagation phase ψ mn,j , we can find this componentcorresponds to the conventional diffraction which fol-lows the ordinary Snell’s law, n t sin( θ ) − n i sin( θ ) = 0where θ and θ are the incident and transmitted an-gles, respectively. For m = ±
1, the PB phase can-cels the j -dependent part of the propagation phase andonly cross-polarized components are detectable since G x, ± a j ≡ ∓ φ j . The remaining x -dependent prop-agation phase given by ( ± π (2 N +1) a + κ xi ) x yields the gener-alized Snell’s law which governs the anomalous refraction.For the light beam propagating in the xz plane, the refrac-tion angle is defined by sin( θ ) = cω i n t (cid:16) ± π (2 N +1) a + κ xi (cid:17) .Thus for x = [0 , (2 N + 1) a ], one can obtainsin( θ ) n t − sin( θ ) n i = ± λ (2 N + 1) a . (8)We now calculate the Fresnel coefficient and analyze thechiral transmission properties in the circular polarization(CP) basis: σ ± = ( e x cos( θ ) ± i e y ) / √ θ is the re-fraction angle. Following the derivation shown in SectionS3 and assuming the amplitude of the incident light is E s = cos θ e x + s i e y ( s = ± E = P mnj ( E + E e − s i2 φ j + E e s i2 φ j ) F mn,j ,where amplitudes E , E , and E are given by E = t E s + t − E − s , E = t − + E s + t E − s , E = t − E s + t −− E − s . (9)Here, the coefficients t ± and t ±± are given in Eq. (S25).The additional phase factor e ± s i2 φ j , the PB phase, origi-nates not only from the geometric rotation of the nanopil-lar in the unit cell, but also from the polarization of light.In other words, the additional phase ± sφ j relies on thesymmetry relation between the polarization of light andgeometric nanostructures of metasurface rather than thespecific coordinate system, which is a characteristic oftopological phase. In order to make it more clear and easyto compare with existing research results, we discuss andsummarize the selective transmission of cross-polarizedbeam for all possible chiral combinations of the inputpolarization and metasurface in Tab. 1. For the meta-surface with clockwisely rotating nanopillars depicted inFig. 1B, right CP (RCP) incident light splits into RCPlight and left CP (LCP) light, while LCP incident lightsplits into LCP light and RCP light. Furthermore, aswe demonstrated here, the PB phase term e − s i2 φ j of theoutput amplitude contributes to the effect of non-zerocross-polarized beam. If the entire metasurface is ro-tated counter-clockwise by π along z axis which means φ j = π N +1 , the constant phase term is written as e mφ j .Then the phase gradient of cross-polarized beam changesits sign. It is clear that the sign of the phase gradientis determined by the handedness of incident light andmetasurface. TABLE I. Cross-polarized transmission for different combina-tions of the input polarization and metasurface
Antenna rotation Input Output(order) Phase gradientClockwise σ + σ − (+1) λ (2 N +1) a σ − σ + (-1) − λ (2 N +1) a LP σ − (+1) λ (2 N +1) a σ + (-1) − λ (2 N +1) a Counter-clockwise σ + σ − (-1) − λ (2 N +1) a σ − σ + (+1) λ (2 N +1) a LP σ − (-1) − λ (2 N +1) a σ + (+1) λ (2 N +1) a LP denotes linear polarization.
For an arbitrary input polarization, we can decomposethe normally incident light in the CP basis as E in ≡ E || = ασ + + βσ − with β = √ − α and considering a normalincidence θ = 0. The transmitted light can be then recastas E = X j,m = ± ( t || F ,j E || + t ⊥ F m ,j M ( φ j ) E ⊥ ) , (10)where t || = t xx + t yy , t ⊥ = t xx − t yy , E || · E ⊥ = 0, and M ( φ j ) = (cid:18) e i2 φ j − e − i2 φ j (cid:19) . The corresponding trans-mission of the co- and cross-polarized beams for an arbi-trary polarization incident light are illustrated schemat-ically in Fig. 1A. Depending on the combination ofthe incident polarization and geometric rotation of thenanopillar, a cross-polarized retardation with a positiveor negative phase occurs, leading to self-constructive orself-destructive interference effects. Figure 1A indicatesthe relative phase retardation δ Φ ± , is a function of theinterface lateral displacement δ ( x ) between co- and cross-polarized beams. III. INTERFEROMETRIC MEASUREMENT OFTHE TOPOLOGICAL PHASE
Therefore, the PB phase results in the opposite phasedelays on the orthogonal CP components. The relevantphenomena, such as generalized Snell’s law, arbitrary po-larization holography [6, 31], optical edge detection [42]and the photonic spin Hall effect[43, 44], can be thusdescribed using our theory.. In the following, we focus ontopological phase characterization using the polarization-dependent translational symmetry breaking measurementbased on the Mach-Zehnder interferometer (MZI). TheGaN-based PB metasurface is used as a 50 /
50 CP beamsplitter in the performance of self-phase referencing. Tobetter understand the design of the birefringent nanostruc-
Phase delay A C o n v e r s i o n e ff i c i e n c y L x (10 -7 m) Phase L y ( - m ) . . . B L x (10 -7 m) L y ( - m ) . . . C T -30 -20 -10 0 10 20 30 Detector angle (in degrees) N o r m a li z e d t r a n s m i tt e d p o w e r D 𝜎 + 𝜎 − Incident angle θ E I FIG. 2. (A) Calculated polarization conversion efficiency (blue),co-polarization transmission(red), of the subwavelength array of PBnanopillars as function of the delay between polarization eigenstates.(B) and (C) Full wave numerical simulations performed to extractthe phase retardation between E x and E y components (B) andtransmission maps (C) as functions of length and width of thenanopillars. (D) Experimental measurements of the normalizedtransmission across a PB metasurface designed according to theguideline in (B) and (C) as a function of the incidence angle changesfor LCP ( σ − ) incidence. (E) Comparison between experiments andtheory of the anomalous refraction efficiency as a function of theincident angle, where I is the transmitted power. Parameters of thesimulations are a = 500nm, a = 400nm, l x = 260nm, l y = 85nm, l z = 632 . λ = 632 . n i = 1 .
61 + 0 . n t = 1 . − . χ x,y (see Eq. (S8) in SM) with ω = 2 .
75 PHz, ω = 1 .
71 PHz,and n eff = 1 . − .
01i account for the Fresnel coefficient at the firstinterface (see Section S3.1 in SM for details). ture, we theoretically calculate the co- and cross-polarizedscattering amplitudes of an array of identical nanopillarsas a function of the phase delay between x and y polar-ization, i.e. tuning the phase difference of the diagonalelements of susceptibility tensor which represents the ge-ometric anisotropy of the metasurface. As shown in Fig.2A, the ratio of the co- and cross-polarized transmissionamplitude reach 50 /
50 when the the phase difference ofthe diagonal elements of susceptibility tensor is π/ π/
2. In order to identify GaN nanopillars with π/ π/ x and y polarizations, full wavenumerical simulations is performed to extract the phaseretardation between E x and E y components and also thetransmission efficiency as function of length and width ofthe nanopillars in Fig 2B and C. The white lines indicatethe regions for which the phase delay between x and y polarizations is equal to π/ π/
2, needed to adjustamplitudes for the interferometric characterization of thePB phase. According to these theoretical prediction, di-mensions of GaN nanopillars used were length l x = 260nm, l y = 85 nm and height 800 nm. These dimensionsgenerate phase retardation 3 π/ π/ µm X250 µm array. The nanofabrication of metasurface was realizedby patterning a 800 nm thick GaN thin film grown ona double side polished c-plan sapphire substrate via aMolecular Beam Epitaxy (MBE) RIBER system. TheGaN nanopillars were fabricated using a conventional elec-tron beam lithography system (Raith ElphyPlus, ZeissSupra 40) process with metallic Nickel (Ni) hard masksthrough a lift-off process. To this purpose, a double layerof around 200 nm Poly(methyl methacrylate) (PMMA)resists (495A4 then 950A2) was spin-coated on the GaNthin film, prior to baking the resist at a temperature of125 ℃. E-beam resist exposition was performed at 20keV. Resist development was realized with 3 : 1 IsopropylAlcohol (IPA):Methyl isobutyl ketone (MIBK) and a 50nm thick Ni mask was deposited using E-beam evapo-ration. After the lift-off process in the acetone solutionfor 4 hours, GaN nanopillar patterns were created usingreactive ion etching (RIE, Oxford system) with a plasmacomposed of Cl CH Ar gases. Finally, the Ni mask on thetop of GaN nanopillars was removed by using chemicaletching with 1 : 2 solution of HCl:HNO .Three gratings were designed and fabricated with dif-ferent periodic arrangements of rotated nanopillars withperiods 2, 2 . µ m, respectively. The refractionproperties of these designed metasurfaces are measuredas the experimental verification of theoretically predicted50/50 PB metasurface beam splitter. The measurementshave been realized using a conventional diffraction setup,comprizing a Si-detector plugged into a lock-in amplifierto improve the dection signal to noise ratio. Acquiring therefracted signal as a function of the transmission angle,the detector rotates in a circular motion from −
30 to 30degrees. Spectral refraction response was obtained bysweeping the wavelength of a supercontinuum source cou-pled to a tunable single line filter in the range of 480 − − Piezo stage controller P h a s e [ d e g . ] P h a s e [ d e g . ] P h a s e [ d e g . ] B δ(x)[μm] FIG. 3. (A) (Left) Schematic of the interferometric measurement forthe characterization of the topological phase shift introduced by PBmetasurface as a 50 /
50 CP beam splitter. (Right) The interferencefringes displacement according to the phase gradient direction δx ,resulting from the topological phase delay shift introduced on theanomalous beam. (B) The measured phase delays as a function ofthe displacements are reported for 3 different gratings, with periodsΓ = 4 , . µm from top to bottom, respectively. /
50 when the incident wavelength changes as shownin Fig. 2D. As shown in Fig. 2E, the experimentallymeasured transmission efficiency of cross-polarized beamhas two well-resolved peaks around 15° and 48° whichis in agreement with analytically predicted diffractionefficiency (red curve).We have experimentally characterized the topologicalphase using a self-interferometric measurement in a MZIconfiguration, replacing a beam splitter by the meta-surface as shown in Fig. 3A. Phase retardation of theanomalous refracted signal as a function of the lateraldisplacement of the metasurface, introduced by the shift-ing of metasurface along the phase gradient along the x axes, is recorded by monitoring the displacement ofthe interferogram fringes on a CCD camera after carefulrecombination and adjustment of the polarization hand-edness. The piezo stage controller is utilized to achieveminute translation of the metasurfaces as required forphase characterization in experiments discussed in Fig.3B. In the present configuration, one arm of the MZI origi-nates from the first order refraction from the metasurface.In addition to the anomalous refraction, the metasurfaceimposes a phase Φ ± ( x ) = G ± ,x δ ( x ) , (11)which is proportional to the metasurface displacement δ ( x ) along the phase gradient direction x . We pro-pose to experimentally measure this phase by recom-bining both arms on a beam splitter, and record-ing the resulting intensity profile as a function ofthe translation distance. The transmitted light ofRCP/LCP incidence is E = P j ( t || σ ± e i ω i n t z/c − t ⊥ e i Φ ± ( x ) σ ∓ e i √ ω i n t /c − G ± ,x z ) H ( z > I + = I || + I ⊥ r cos(Φ ± ( x ))) . (12)Here I || = | P j t || | , I ⊥ = | P j t ⊥ | are the intensity ofco- and cross-polarized transmission respectively, and r = √ I || I ⊥ I || + I ⊥ . Then the interference fringes displacement shownin right side of Fig. 3A provides indirect, yet unambiguousand conclusive measurement of the PB phase. As shownin Fig. 3B, we observe the linear phase variations with thewrapping periods equal to the PB phase of the metasurfacein agreement with the our theoretical result in Eq. (8). CONCLUSION
In summary, we provide an in-depth analysis of topolog-ical PB metasurfaces by comparing experimental resultsobtained with spatially oriented subwavelength birefrin-gent nanostructures, with a mesoscopic theory. This work,which demonstrates the origin of both controllable phaseretardation effects, namely the propagation phase andthe PB phase, is a first step in developing an intuitiveunderstanding of topological and functional beam split-ters for future applications in quantum optics and theirimplementations in relevant quantum information proto-cols based on metasurfaces, which is an important futureresearch direction in this field [45–51].
ACKNOWLEDGEMENT
Z.G thanks S. Jiang, P. Saurabh, G. Zhu for valuablediscussions. Z.G, V.O and K.E.D gratefully acknowledgethe support from National Science Foundation of China(No. 11934011), Zijiang Endowed Young Scholar Fund,East China normal University and Overseas Expertise In-troduction Project for Discipline Innovation (111 Project,B12024). KD is grateful for the support of “F´ed´erationDoeblin”. P.G, R.S, and G.B acknowledge funding fromthe European Research Council (ERC) under the Eu-ropean Union’s Horizon 2020 research and innovationprogramme (Grant agreement no. 639109). [1] Genevet P, Capasso F, Aieta F, Khorasaninejad M, DevlinR. Recent advances in planar optics: from plasmonic todielectric metasurfaces.
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Zhanjie Gao, ∗ Sandeep Golla, ∗ Rajath Sawant, Vladimir Osipov,
1, 3
Gauthier Briere, Stephane Vezian, Benjamin Damilano, Patrice Genevet, † and Konstantin E. Dorfman ‡ State Key Laboratory of Precision Spectroscopy,East China Normal University, Shanghai 200062, China Universit´e Cˆote d Azur, CNRS, CRHEA, rue Bernard Gregory, Valbonne 06560, France Holon Institute of Technology, 52 Golomb Street, POB 305 Holon 5810201, Israel
I. ELECTRODYNAMICS OF METASURFACEA. Lattice model for light-metasurface interaction
The classical formulation of the light scattering problem is based on the Maxwell’s equations without externalcurrents and charges [1] ∇ · D = 0 , (1) ∇ × E = − c ∂ B ∂t , (2) ∇ · B = 0 , (3) ∇ × H = 1 c ∂ D ∂t . (4)We restrict ourself by considering non-magnetic media, so that the magnetic and diamagnetic fields are equal (thevacuum electric and magnetic permittivity are set to unity), B = H . Following the standard Maxwell’s equationsnotations, we use the vector of polarization, D = E + 4 π P . Therefore the system of Maxwell’s equations are given byEq. (1) in the main text. Information about the matter is solely contained in the functional dependence of P on E .The geometry of the system is chosen such that z axis of the laboratory coordinate system can be directed along thenormal to the surface and ρ = ( x, y ) is a two-dimensional vector in the plane of metasurface. Particular dependence of P on E is considered in the following sections. When the metasurface is placed at the interface between two isotropicmedia with different refractive index, the corresponding z -dependent terms read P ( z, ρ , t ) = H ( | z | ≤ l z / P ( E ) + H ( z < − l z / χ i E ( z, ρ , t ) + H ( z > l z / χ t E ( z, ρ , t ) . (5)Here and below Heaviside function H (condition) = { , when condition is true; 0 , when condition is false } . Then Eq.(1) in the main text for the domain z < − l z / z > l z / c i = c/n i and c t = c/n t , respectively, where n i,t = p πχ i,t denotes the refractive index of substrates onboth side (see Fig. (1)).The polarization in Eq. (1) vanishes in vacuum, and has non-zero values inside the media, which represents the thinlayer of metasurface. We consider a classical linear response model for the metasurface P ( z, ρ , t ) = χ ( ρ ) E ( z, ρ , t ).The Fourier transformation of the fields over ρ are defined as follows: E ( z, κ , ω ) = Z d ρ dt E ( z, ρ , t ) e − i κ · ρ +i ωt , E ( r , t ) = 1(2 π ) Z d κ dω E ( z, κ , ω ) e i κ · ρ − i ωt . (6)Polarization is expressed in terms of the field, P ( z, κ , ω ) = 1(2 π ) Z d κ ˜ χ ( κ − κ ) E ( z, κ , ω ) , ˜ χ ( κ ) = Z d ρ χ ( ρ ) e − i κρ . (7)The linear susceptibility function χ ( ρ ) is a periodic function of coordinates. It can be represented as a sum of the ∗ Zhanjie Gao, and Sandeep Golla contributed equally to this work. † [email protected] ‡ [email protected] a r X i v : . [ phy s i c s . op ti c s ] F e b FIG. 1.
Schematics of the reflection and refraction from the metasurface. The blue arrows represent the incidence light at the angle θ ,ordinary refraction at the angle θ and the ordinary reflection. The red arrows are the same but for the anomalous refraction and reflection. susceptibilities of the individual primitives shifted and rotated in xy plane. The reflection and transition is thereforedefined by the vectors of the reciprocal lattice G , where the function χ (∆ κ ) reaches its maxima. FIG. 2.
Schematics of the metasurface. The length and width of the antenna are l x and l y , respectively. The distance between the twoantennas in x and y axis are a and a , respectively. Along the x axis, for M = 5 (shown here) the antennas are rotated by the angle φ = − π (2 N +1) . B. Metasurface with translational symmetry
Since the length of the antenna is in the sub-wavelength scale and the metasurface is translation-invariant withoutconsidering the rotation of the antenna, as shown in Fig. 2, the susceptibility function χ ( ρ ) can be represented by χ ( z, ρ ) = χ X mn N X j = − N Ω( ρ − % Mm + j,n ) , χ = χ x χ y . (8)where N = ( M − / M , and the susceptibilities of the nanopillar along x and y directions are χ x = [4 πω − πω (1 − sin ( θ ) n eff ) − / ] − , χ y = [4 πω − πω (1 − cos ( θ ) n eff ) − / ] − with differentresonance frequencies ω and ω [2], effective refractive index of the nanopillar n eff and incident angle θ . The indicatorfunction Ω( ρ ) describes the basic geometric primitive, which has a rectangular shape:Ω( ρ ) = H ( | x | ≤ l x / H ( | y | ≤ l y / . (9)The lattice translation vector % m,n = m a x + n a y with lattice primitive translation vectors a x = (2 N + 1) a e x and a y = a e y . 2D spatial Fourier transformation of Eq. (8) is˜ χ ( κ ) = χ X n e − i n κa X m N X j = − N ˜Ω mnj ( κ ) e − i((2 N +1) m + j ) κa , (10)where ˜Ω mnj ( κ ) is the Fourier transform of Ω( ρ − % Mm + j,n ). Since the rotation of antenna in one unite cell is treatedby the rotation of the transmission matrix later by the form ˆ˜ T ( φ j ) = ˆ R † ( φ j ) ˆ˜ T ˆ R ( φ j ) in the main text, all the ˜Ω mnj ( κ )with different indexes m, n, j are same and the subscripts mnj will be omitted below.. By using the Poisson summationformula ∞ P n = −∞ F ( k − πn a ) = πa ∞ P n = −∞ ˆ F ( n πa ) e − i kn/a where ˆ F is the Fourier transformation of F , Eq. (10) reduceto ˜ χ ( κ ) = X m ,n δ (cid:18) κ x − πm (2 N + 1) a (cid:19) δ (cid:18) κ y − πn a (cid:19) F m n , (11)where F m n = χ (2 π ) (2 N +1) a a N P j = − N ˜Ω( κ ) e − i j κa . Taking the inverse Fourier transform, we obtain˜ χ ( ρ ) = χ (2 N + 1) a a X m ,n N X j = − N ˜Ω( G m n ) e i G m n · ( ρ − j a ) , where the reciprocal lattice vector G m n = πm (2 N +1) a e x + πn a e y . Substitution of the result (11) into (7) produces theseries for the polarization vector P ( z, κ , ω ) = χ (2 N + 1) a a X m ,n N X j = − N Z d κ δ ( κ − κ − G m ,n ) e − i j ( κ − κ ) a ˜Ω( κ − κ ) E ( z, κ , ω ) . (12)By taking inverse Fourier transform of Eq. (12), polarization vector is given by Eq. (2) in the main text as a sum overthe Brillouin zones. To define the nonuniform part of the system of Maxwell’s equations, we assume that the incidentlight has the form E ( z, ρ , t ) = E i e i k zi z +i κ i ρ − i ω i t where k zi = q ω i n i c − κ i with the conditions ω i n i > cκ i . Since thereare no other temporal characteristics except the ω i , all the time-derivatives can be replaced by the multiplication by − i ω i . We seek for the solution of the Maxwell’s equations (1) in the form of Eq. (4) in the main text. C. Thin metasurface limit
In zeroth approximation, we assume that the surface polarization is caused by the incident light only, so that P || = χ E in e i k zi +i κ i ρ and ∂ ∂z E || + ∇ ρ E || + ω i n i,t c E || = − π ∇ ρ ( ∇ ρ · P || ) − π ∇ ρ ∂∂z P z − π ω i c P || . (13)The following equations are written for the e i( G mn + κ ) ρ components of above equations. All the G mn + κ spatialcomponents of the ˜ E, ˜ P have the same subscripts m and n which will be further omitted. The corresponding thinpolarization component H ( | z | ≤ l z / P ( E ) is˜ H (k z ) P || (k z , κ ) = sin( ξl z )4 πξ χ f mn E || ,i δ ( κ i − κ ) ’ l z π χ f mn E || ,i δ ( κ i − κ ) , (14)where f m,n = P Nj = − N f mn,j e − i j G mn a , 2 ξ = k zi − k z , E || ,i is the x, y amplitudes of incident light. When the incidentangle or the thickness of the antenna l z is small which corresponds to ξ ≈ l z ≈
0, then sin( ξl z ) ξ ≈ l z . Consideringthe thickness of the metasurface to be small, we neglect the E z and P z components in the model and obtain12 π ( ω i n i,t c − K || − k z ) ˜ E x (k z , κ ) = ( K x − ω i c ) ˜ P x + K x K y ˜ P y , (15)12 π ( ω i n i,t c − K || − k z ) ˜ E y (k z , κ ) = ( K y − ω i c ) ˜ P y + K y K x ˜ P x , (16)where K x = G mn,x + κ x , K y = G mn,y + κ y , K || = ( G mn,x + κ x ) + ( G mn,y + κ y ) , ˜ E y (k z , κ ) , ˜ E y (k z , κ ) are function ofk z , κ need to be calculated, ˜ P || = l z χ f mn E || ,i δ ( κ i − κ ). II. REFLECTION AND REFRACTION FROM METASURFACES WITH TRANSLATIONALSYMMETRYA. Transmission matrix
We can now investigate how the light is transmitted through the metasurface. In the present model, we assumethat, for a typical design, individual primitives are spaced sparsely which eliminates a possibility of the inter-elementinteractions. In this case, one can calculate the transmission through the individual primitive and then sum over allthe elements of the unit cell and the entire metasurface. Consider the rectangular primitive with perfectly alignedsides along x and y coordinates such that the long side l y is along the y -axis and the short side l x is along the x -axis.We can thus recast Eqs. (15) - (16) in terms of the transmission matrix ˆ˜ T given by Eq. (4) in the main text (Due to e − i j G mn a is included in the propagation phase ψ mn,j , we omitted it in here.): ˜ E x,m,n ˜ E y,m,n = ˜ t xx ˜ t xy ˜ t yx ˜ t yy E xi E yi , where the elements of transmission matrix are given by˜ t xx = ˜ C r ˜ C ( K x − ω i c ) χ x , ˜ t xy = ˜ C r ˜ C ( K x K y ) χ y , ˜ t yx = ˜ C r ˜ C ( K x K y ) χ x , ˜ t yy = ˜ C r ˜ C ( K y − ω i c ) χ y . (17)Here ˜ C = − k z + ω i n i,t c − K || , ˜ C r = 2 πl z f mn,j δ ( κ i − κ ), κ i < G , = π (2 N +1) a e x + πa e y . In the physical space, thetransmission matrix ˆ T can be written as t xx = C r C ( K x − ω i c ) χ x ,t xy = C r C ( K x K y ) χ y ,t yx = C r C ( K x K y ) χ x ,t yy = C r C ( K y − ω i c ) χ y . (18)where C = r ω i n i,t c − K || (cid:12)(cid:12) κ = κ i , C r = i l z f mn,j / π | κ = κ i . III. DERIVATION OF THE FRESNEL COEFFICIENT
So far we have investigated the transmission properties of the metasurfaces assuming that the incident and transmittedlight are polarized linearly. We now consider a circularly polarized (CP) light. We therefore have to transform thesolution to the CP basis σ ± = ( e x cos( θ ) ± i e y ) / √
2, where θ is the refraction angle (see Fig. 1). For an ordinaryoperator ˆ A in the linear polarization basis, we define an operator ˆ¯ A in the CP basis ˆ A ( x,y ) → ( σ ± ) ˆ¯ A . For instance, therotation operator ˆ R ( φ j ) defined in Eq. (6) can be recast in the circular polarization basis as an operator ˆ¯ R ( φ j ) definedas follows ˆ¯ R ( φ j ) = e i φj (2 − sec θ − cos θ )+ e − i φj (2+sec θ +cos θ )4 e i φj (sec θ − cos θ )+ e − i φj ( − sec θ +cos θ )4 e − i φj (sec θ − cos θ )+ e i φj ( − sec θ +cos θ )4 e − i φj (2 − sec θ − cos θ )+ e i φj (2+sec θ +cos θ )4 . (19)Similarly the transmission matrix ˆ T in Eq. (5) takes the formˆ¯ T ≡ t yy + t xx − i t xy cos θ +i t yx sec θ t yy − t xx − i t xy cos θ − i t yx sec θ t yy − t xx +i t xy cos θ +i t yx sec θ t yy + t xx +i t xy cos θ − i t yx sec θ . (20)We then consider the in plane transmission condition t xy = 0 , t yx = 0. In the CP basis the corresponding operatorreads ˆ¯ T ( φ j ) = t j t j t j t j , (21)where t j = 4( t xx + t yy ) + e i2 φ j ( t xx − t yy )(cos θ − sec θ ) + e − i2 φ j ( t xx − t yy )(sec θ − cos θ )8 ,t j = e − i2 φ j ( t xx − t yy )(2 − sec θ − cos θ ) + e i2 φ j ( t xx − t yy )(2 + sec θ + cos θ )8 . The CP light with the incident angle θ can be recast as { ( ± θ cos θ ) , ( ∓ θ cos θ ) , } . The E mn,j component of the refracted light then reads E mn,j = E +1 + E +2 e − i2 φ j + E +3 e i2 φ j E − + E − e − i2 φ j + E − e i2 φ j F mn,j , (22)where E +1 = ( t xx + t yy )(cos θ sec θ ± , E +2 = ( t xx − t yy )(cos θ ± θ − ,E +3 = ( t xx − t yy )(cos θ ∓ θ + 1)8 ,E − = ( t xx + t yy )(cos θ sec θ ∓ , E − = ( t xx − t yy )(cos θ ± θ + 1)8 ,E − = ( t xx − t yy )(cos θ ∓ θ − . (23)Consider the general refraction law, the refraction angle satisfies sin θ = mλ a n i,t + n i sin θn i,t where m = 0 , ±
1. Using thefield amplitudes in the circular polarization basis E ± s = ( e x cos( θ ) ± si e y ) / √
2, which yields the general result: E = X mnj ( E + E e − s i2 φ j + E e s i2 φ j ) F mn,j (24)with E , E , E can be written as Eq. (9) where t ± = ( t xx + t yy )(cos θ sec θ ± ,t ±± = ( t xx − t yy )(cos θ ± θ ± ,E ± s = ( e x cos( θ ) ± s i e y ) / √ . (25)Since l y << λ and l x < λ , then t yy ∝ χ y can be neglected. In order to get a more clear and simper analytic expressionwithout effect physical picture, we discuss a 1D model in the following. Consider t xx = i f mn,j l z χ x ( K x − ω ic )2 q ω i n i,tc − K x ≈ t xx cos θ where t xx = − i f mn,j l z χ x and the refractive indexes in two sides of the metafursace are close n i = n t ≈
1, then E , E , E can be written as E = t E s + t − E − s ,E = t − + E s + t E − s ,E = − t − E s − t −− E − s , (26)where t ± = 14 t xx (cos θ ± cos θ ) ,t ±± = 18 t xx (cos θ ± θ ± . (27)It is clear that the amplitude of different components satisfy | t | >> | t − + | , | t − | and | t | >> | t − | , | t −− | , whenthe incident and refracted angles are small. The transmission light is in the form of E = X mnj ( t E s + t E − s e − s i2 φ j ) F mn,j . Combining with the constant phase term e mφ j in F mn,j ( z, ρ ) (antennas are rotated in a clockwise direction inFig. 2), for RCP incident light ( s = 1), only m = 0 , s = − m = 0 , − P j e ± i2 φ j ’ P j e ± i4 φ j ’ A. Derivation of the transmission efficiency
In order to describe experimental observations, the details of theoretical model are presented as follows. As shownin the Fig. 3, the transmission of the light traverses three medium: the air, the substrate, and the metasurface.We first consider light transmission through the air/substrate interface. Refraction angle satisfies ordinary Snell’slaw: sin( θ )sin( θ ) = n n i where n and n i are the refractive indexes of air and substrate, respectively. The x and y componentsof the electric field in the substrate satisfy the Fresnel formula E x = 2 cos( θ ) √ θ ) + n cos( θ )] , E y = i θ ) √ θ ) + n cos( θ )] . After passing through the substrate light interacts with the metasurface where the refraction is described by thegeneralized Snell’s law and the amplitude obeys Eq. (10) in the main text. In order to model the structural birefringence Θ ′ δ 𝑬 || 𝑬 || 𝑬 ⊥ + Θ z x n n 𝑖 n 𝑒𝑓𝑓 n 𝑡 Θ′′
FIG. 3.
Schematic of the light transmission through the PB metasurface. of the rectangular nano pillars we define the x nd y components of the susceptibility tensor of GaN nanopillar accordingto already defined after Eq. (S8). This functional form of the susceptibility Cartesian components with two resonantfrequencies ω and ω [2], which agrees with angle resolved photoluminescence experiments of the birefringent material.In addition, by expanding the angle-dependent susceptibility components in a Taylor series, one can obtain an angulardispersion similar to that observed in THz metasurfaces [3], where the phenomenological amplitudes in the expansionrepresent inter-particle coupling strengths. The transmission through the metasurface is further described by ourmesoscopic model (see Eq. (10)), which allows us to obtain the final anomalous refraction efficiency. IV. DESIGN OF THE POLARIZATION DEPENDENT BEAM SPLITTER
The interferometric measurement between normal and anomalous refraction imposes that the scattering propertiesof the interface should contain both zero and first diffraction order with similar amplitude and polarization. Thelatter can be addressed by inserting a quarter wave plate into the path of the one of the diffracted beams. The formercondition requires tuning the antenna scattering parameter. The amplitude and phase responses of the nanopillarsforming the PB metasurface are related to the length and width of the nanopillar with a constant height of 800 nm.To maximize the PB metasurface efficiency, the antenna should maximally convert the polarization from the oppositeorthogonal circular polarization, thus introducing a phase shift between ordinary and extraordinary axis of a nanopillar[4]. However, in contrast to the previous PB metasurface works seeking for the high performance devices [5], we areherein interested in quantifying the PB phase from the self-referenced interferometric measurements. Proper designis achieved by considering birefrigent nanopillar introducing π/ π/ π/ O ) substrate. The refractive indexfor GaN has the following Sellmeier like relation [6]: n ( λ ) = s A λ λ − ( λ G ) + B λ λ − ( λ H ) ,n e ( λ ) = s A e λ λ − ( λ Ge ) + B e λ λ − ( λ He ) . Here, A = 0 . B = 3 . A e = 0 . B e = 4 . λ G = λ Ge = 350 nm , λ H = 153 nm , λ He = 173 . nm . ForSapphire (Al O ), refractive index relation is referred from [7]. For λ = 632 . n = 1 . . nm wavelength polarized along x and y axis, impinging at normal incidence satisfying the perfectly matched layer (PML) conditions in the directionof the light propagation subject to periodic boundary conditions along all the in-plane directions. The use of PMLboundary conditions in the propagation direction results in an open space simulation while in-plane periodic boundaryconditions mimic a subwavelength array of the identical nanostructures.In figure 4, the polarization-conversion efficiency of a single GaN nanopillar is obtained by FDTD simulation. Ameta-atom is impinged with two plane waves sources (x and y-polarized) of varying wavelength from 480 nm to 680nm with the interval of 20 nm. The other simulation conditions are kept same as in the design simulation. To performinterference measurements as described in the main text, the design of the element is chosen to diffract 50% of theincident light on cross polarization. E ff i c i e n c y o f f i r s t o r d e r Wavelength in nm
Deflection vs Wavelength
Designwavelength
FIG. 4.
Polarization conversion efficiency of a single GaN nanopillar.
V. INTERPRETATION OF THE INTERFEROMETRIC EXPERIMENTS
Note, that Eq. (12) shows that MZI detects the displacement phase. Technically the displacement phase is apropagation phase, also known as detour phase, since the measurement involves translational motion alone. Howeveras has been already pointed out in the discussion following Eq. (7), the propagation phase contains two parts, one ofwhich cancels out the PB phase yielding non vanishing first diffraction order. The detour phase is therefore a remainingpart of the propagation phase which enters the Snell’s law. Note, however, that due to the translational invarianceand uniform distribution of PB phases in the unit cell, the magnitude of the displacement phase is equivalent to thePB phase. This equivalence is not accidental and follows directly from the metasurface design itself, and thereforecan be controlled at will. The MZI measurements thus provides although indirect yet unambiguous and conclusivemeasurement of the PB phase. [1] Landau LD, Lifshitz EM.
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FIG. 5. left: Measured angular deflection efficiency as a function of the incident wavelength for σ ++