Spin-Orbit Interactions of Light

Spin-Orbit Interactions of Light

Konstantin Y. Bliokh

Quantum Condensed Matter Group, Center for Emergent Matter Science (CEMS)

RIKEN, Wako-shi, Japan

796 NATURE PHOTONICS | VOL 9 | DECEMBER 2015 | www.nature.com/naturephotonics

Light consists of electromagnetic waves that oscillate in time and propagate in space. Scalar waves are described by their intensity and phase distributions. These are the spatial (orbital) degrees of

freedom common to all types of waves, both classical and quantum. In particular, a localized intensity distribution determines the posi-tion of a wave beam or packet, whereas the phase gradient describes the propagation of a wave (that is, its wavevector or momentum). Importantly, electromagnetic waves are described by vector fields1. Light therefore also possesses intrinsic polarization degrees of free-dom, which are associated with the directions of the electric and magnetic fields oscillating in time. In the quantum picture, the right- and left-hand circular polarizations, with the electric and magnetic fields rotating about the wavevector direction, correspond to two spin states of photons2.

Recently, there has been enormous interest in the spin–orbit interactions (SOI) of light3–6. These are striking optical phenomena in which the spin (circular polarization) affects and controls the spatial degrees of freedom of light; that is, its intensity distributions and propagation paths. The intrinsic SOI of light originate from the fundamental spin properties of Maxwell’s equations7,8 and, there-fore, are analogous to the SOI of relativistic quantum particles2,9,10 and electrons in solids11,12. As such, intrinsic SOI phenomena appear in all basic optical processes but, akin to the Planck-constant small-ness of the electron SOI, they have a spatial scale of the order of the wavelength of light, which is small compared with macroscopic length scales.

Traditional ‘macroscopic’ geometrical optics can safely neglect the wavelength-scale SOI phenomena by treating the spatial and polarization properties of light separately. In particular, these degrees of freedom can be independently manipulated: by lenses or prisms, on the one hand, and polarizers or anisotropic waveplates, on the other. SOI phenomena come into play at the subwavelength scales of nano-optics, photonics and plasmonics. These areas of modern optics essentially deal with nonparaxial, structured light fields characterized by wavelength-scale inhomogeneities. The usual intuition of geometrical optics (based on the properties of scalar waves) does not work in such fields and should be substituted by the full-vector analysis of Maxwell waves. The SOI of light represent a new paradigm that provides physical insight and describes the behaviour of polarized light at subwavelength scales.

In the new reality of nano-optics, SOI phenomena have a two-fold importance. First, the coupling between the spatial and polarization

Spin–orbit interactions of lightK. Y. Bliokh1,2*, F. J. Rodríguez-Fortuño3, F. Nori1,4 and A. V. Zayats3

Light carries both spin and orbital angular momentum. These dynamical properties are determined by the polarization and spatial degrees of freedom of light. Nano-optics, photonics and plasmonics tend to explore subwavelength scales and addi-tional degrees of freedom of structured — that is, spatially inhomogeneous — optical fields. In such fields, spin and orbital properties become strongly coupled with each other. In this Review we cover the fundamental origins and important applica-tions of the main spin–orbit interaction phenomena in optics. These include: spin-Hall effects in inhomogeneous media and at optical interfaces, spin-dependent effects in nonparaxial (focused or scattered) fields, spin-controlled shaping of light using anisotropic structured interfaces (metasurfaces) and robust spin-directional coupling via evanescent near fields. We show that spin–orbit interactions are inherent in all basic optical processes, and that they play a crucial role in modern optics.

properties must be taken into account in the analysis of any nano-optical system. This is absolutely essential in the conception and design of modern optical devices. Second, the SOI of light can bring novel functionalities to optical nano-devices based on interactions between spin and orbital degrees of freedom. Indeed, SOI provide a robust, scalable and high-bandwidth toolbox for spin-controlled manipulations of light. Akin to semiconductor spintronics, SOI-based photonics allows information to be encoded and retrieved using polarization degrees of freedom.

Below we overview the SOI of light in paraxial and nonparaxial fields, in both simple optical elements (planar interfaces, lenses, anisotropic plates, waveguides and small particles) and complex nano-structures (photonic crystals, metamaterials and plasmonics structures). We divide the numerous SOI phenomena into several classes based on the following most representative examples:

(1) A circularly polarized laser beam reflected or refracted at a planar interface (or medium inhomogeneity) experiences a transverse spin-dependent subwavelength shift. This is a manifestation of the spin-Hall effect of light13–20. This effect provides important evidence of the fundamental quantum and relativistic properties of photons16,18, and it causes specific polarization aberrations at any optical interface. Supplied with suitable polarimetric tools, it can be employed for precision metrology21,22.

(2) The focusing of circularly polarized light by a high-numeri-cal-aperture lens, or scattering by a small particle, generates a spin-dependent optical vortex (that is, a helical phase) in the output field. This is an example of spin-to-orbital angu-lar momentum conversion in nonparaxial fields23–31. Breaking the cylindrical symmetry of a nonparaxial field also produces spin-Hall effect shifts32–37. These features stem from funda-mental angular-momentum properties of photons8,38, and they play an important role in high-resolution microscopy35, opti-cal manipulations25,26,39, polarimetry of scattering media40,41 and spin-controlled interactions of light with nano-elements or nano-apertures29,34,37,42,43.

(3) A similar spin-to-vortex conversion occurs when a paraxial beam propagates in optical fibres44 or anisotropic crystals45–47. Most importantly, properly designing anisotropic and inho-mogeneous structures (for example, metasurfaces or liquid crystals) allows considerable enhancement of the SOI effects

1Center for Emergent Matter Science, RIKEN, Wako-shi, Saitama 351-0198, Japan. 2Nonlinear Physics Centre, RSPhysE, The Australian National University, Canberra, Australia. 3Department of Physics, King’s College London, Strand, London WC2R 2LS, UK. 4Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1040, USA. *e-mail: [email protected]

REVIEW ARTICLE | FOCUSPUBLISHED ONLINE: 27 NOVEMBER 2015!|!DOI: 10.1038/NPHOTON.2010.201

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Outline

!  Spin-orbit coupling in paraxial beams " Propagation in gradient-index media

" Reflection/refraction at an interface

!  Spin-orbit coupling in nonparaxial fields " Spin and orbital AM in free space

" Focusing and Scattering

"  Imaging and Microscopy

!  Spin-orbit coupling in anisotropic structures

Angular momentum of paraxial light

2. Extrinsic orbital AM (trajectory)

ext = ×L r kkc

rc

3. Intrinsic orbital AM (vortex)

Lint = r − r( )× k

= k / k

= 0 = 1

= −1 = 2

/ kσ=S k1. Intrinsic spin AM (polarization)

1σ = 1σ = −

Geometric phase

Angular momentum is intimately related to rotations. There is a natural coupling between the AM and rotations of the system. It can be described by geometric (Berry) phases. Remarkably, these can be introduced via both geometric and dynamical approaches seemingly unrelated to each other. However, geometric and dynamical aspects become unified on a deeper level of understanding revealing the geometrodynamical nature of the AM physics (cf. general relativity).

Geometric phase

,i ie E e Eσ σ σ σΦ − Φ→ →e e

2D:

ϕσΦ = −

x x

y y

( )x yiσ σ= +e e e

1σ = 1σ = −

dζ ζΦ = − ⋅∫ ΩJ

AM-rotation coupling: Coriolis / angular-Doppler effect

3D:

δω = − ⋅J Ω

Mashhoon, 1988; Garetz, 1979; Bialynicki-Birula, 1997; Hannay, 1998

Φ

zk

xk

E ⊥ κ

ykσ=S κ

φ

θ

C

( ) ( ) ii eσ σφθ φσ= +e κ e e

( )2 1 c

2

os

zd Sφ

θ

πσ

πσ

Φ

−

= −

=∫

dynamics, S:

Φ =σ A k( ) ⋅dk

C∫

A = 1− cosθ

k sinθeφ ,

F = ∂k×A = k

k 3 Berry connection, curvature (parallel transport)

geometry, E:

M. V. Berry, 1984

Geometric phase

Φ =σ A k( ) ⋅dk

C∫

The geometric phase appears in all situations with variations k-vectors (directions of propagation) of light. In this manner, the Berry connection plays the role of an effective gauge field (vector-potential) in the k-space.

A - “vector-potential”

F = ∂k×A = k

k 3- “magnetic field”

- “Aharonov-Bohm (Dirac) phase”

Geometric phase: Gauge fields

Rytov, 1938; Vladimirskiy, 1941; Ross, 1984; Tomita, Chiao, Wu, PRL 1986

X

YZ

e

w

v

te

w

v

tew

v

t 2ΩΩ

Spin-orbit Lagrangian and phase

  SOI Lagrangian from Maxwell equations

LSOI =σA ⋅ k = S ⋅ΩBliokh, 2008

LSOI = A ⋅ p ≈ e

4m2 σ E × p( ) A = S× p

2 p2 1− mE

⎛⎝⎜

⎞⎠⎟

,   for Dirac equation

Φ = LSOI∫ dτ

Berry phase:

Φ02Φ0

Liberman & Zeldovich, PRA 1992; Bliokh & Bliokh, PLA 2004, PRE 2004; Onoda et al., PRL 2004

k c = k∇ ln n , rc = κ c −σF × k c

  ray equations (equations of motion)

X

YZ +

Spin-Hall effect of light

σ k

k 3 × k = S× κ

c c c constσ= × + =J r k κ   AM integral

2ΔΔ

  Berry phase   Spin-Hall effect   Anisotropy

Bliokh et al., Nature Photon. 2008


Origin of the spin-Hall shift

The spin-Hall-effect equations of motion can be derived ab initio from Maxwell equations. Remarkably, the geometric Berry curvature acts as a real physical field and provides real “force”. This is the essence of geometrodynamics in the particle description. But what causes the Hall-effect shift in the wave description? It turns out that the shift is caused by the transverse gradient of the Berry phase for different plane waves forming a transversely confined beam. Indeed, in wave physics, any shift can be attributed to the relative phase between the interfering waves:

δΦ

Relative phase between two waves induces shift of the interference pattern (e.g., Aharonov-Bohm effect):

Origin of the Hall-effect shift

Relative phase gradient between many waves induces shift of the wave packet (e.g., Wigner time delay):

( )ωΦ

ωΔ = ∂ Φ


Phase gradient from spatial dispersion induces The real-space shift (e.g., Goos-Hänchen effect):

( ) ( )/Xk kθΦ =ΦXk

X = ∂ Φ


In a similar way, transverse Berry-phase gradient induces the spin-Hall-effect shift:

σ +

σ −

k

σ +σ −

( ) /yk kϕ σΦ ∝

ykY = −∂ Φ

Origin of the spin-Hall shift

Reflection/refraction at an interface

This can be illustrated by the simplest example of the spin Hall effect of light that arises upon the beam reflection/refraction at a plane interface:

Fedorov, 1955; Schilling, 1965; Imbert, 1972; …… Onoda et al., PRL 2004; Bliokh & Bliokh, PRL 2006; Hosten & Kwiat, Science 2008

Δr ∝σ

ImbertFedorov transverse shift Δr

σ +

σ −

( )ˆ / sin ,z yR κ θ⇒

cotyσκ θΦ = −

coszS σ θ=

⇒

Y = −∂ky

Φ = σ cotθ

c c :y yk→ +k k egeometry, phase:

ext := +J L S

dynamics, AM:

sin coszJ kYδ θ σ θ= − + ⇒ 0zJδ =

k

σ +σ −


Goos-Hänchen and Imbert-Fedorov shifts

spatial GH: angular GH:

spatial IF: angular IF:

Bliokh & Bliokh, PRE 2007; Hosten & Kwiat, Science 2008

Schilling, 1965; … Bliokh & Bliokh, PRL 2006

Goos, Hänchen, 1947; Artman, 1948

Ra, 1973; …; Merano et al., Nature Photon. 2009

rX

tX

tY

rY

rxK

txK

tyK

ryK


Remarkably, the beam shift and accuracy of measurements can be enormously increased using the method of quantum weak measurements:

Angstrom accuracy!


The incident beam is linearly polarized, i.e., in the superposition of and states. Reflection/refraction shifts these states in the opposite direction on a subwavelength distance:

σ + σ −

reflection


Placing an orthogonal linear polarizer at the output (post-selection), we will see double-hump profile with the beam-width (instead of wavelength!) splitting of maxima:

polarizer

0w

But the shift of the beam centroid is zero.

polarizer

0w


However slight non-orthogonality of the output polarizer (slightly elliptical or linear tilted) drastically deforms the output intensity profile:

This results in the beam-width centroid shift, which is proportional to the original Hall-effect shift !


Recently we showed that similar weak measurements of the beam shifts can be realized using surface plasmon polaritons (SPP):

An incident optical beam scattered by a single slit produces output SPP beams, akin to refraction at interface.


But the SPPs are linearly p-polarized. This provides the build-in polarization postselection for weak measurements. One has only to take almost orthogonal s-polarization (i.e., along the slit) in the incident light:

2 20expin y w⎡ ⎤Φ ∝ −⎣ ⎦ - input spatial profile

Ψ in Y − ε X =

−i − ε( ) R + i − ε( ) L

2 - input polarization ε 1

2out

L RX

+Ψ = = - output polarization

3ˆout inw

out in

iσσ

εΨ Ψ

= =Ψ Ψ

- weak value of helicity


As a result of the spin-orbit interaction, the output SPP beam profile becomes shifted:

Φout ∝ exp − y − Δ( )2

w02⎡

⎣⎢⎤⎦⎥ - output spatial profile

- complex beam shift Δ −σ w = − i

ε

The real and imaginary part of the complex shift yield the coordinate and momentum spin-Hall shifts:

Rey = Δ 202 Imyk w−= Δ


The results SPP beam fields in the real and momentum (Fourier) spaces at different input polarizations:

Small real and imaginary induces strong deformations and coordinate and momentum spin-Hall shifts in the SPP beams.

ε

Gorodetski et al., PRL 2012


The results are in perfect agreement with the FDTD and analytical calculations:

c22 2

c c

2 Re12 / 2

yk

χθ εθ χ ε θ

=+

cc 22 2

c

Im2 / 2

yk k χθ εθχ ε θ

= −+

Summary of the SOI in paraxial beams

" Geometrodynamics of ligth carrying intrinsic AM; Geometric force from Berry curvature

" Spin and orbital Hall effects; Total AM conservation

" Hall shifts arise from the transverse gradient of the Berry phase

" Beam refraction/reflection at a plane interface

" Weak measurements of the spin Hall effect; Plasmonic spin Hall effect

Orbit-orbit interaction and phase

Φ

zk

xk

phase ⊥ κyk

Lint = ℓκ

φ

θ

C

  OOI Lagrangian LOOI = A ⋅ k = Lint⋅Ω

X

YZ

w

v

te

w

v

tew

v

t

e

20 0

Bliokh, PRL 2006; Alexeyev, Yavorsky, JOA 2006; Segev et al., PRL 1992; Kataevskaya, Kundikova, 1995

Φ = LSOI∫ dτ   Berry phase

Orbital-Hall effect of light

σ →σ +   vortex-dependent shifts

X

YZ

0 1 2 –1 –2

Bliokh, PRL 2006

Fedoseyev, Opt. Commun. 2001; Dasgupta & Gupta, ibid. 2006

J = rc × k c + σ + ( )κ c

  AM conservation

GH and IF shifts for vortex beams

Bliokh et al. Opt. Lett. 2009 (theory); Merano et al., PRA 2010 (exper.)

spatial GH: angular GH:

spatial IF: angular IF:

X r +αK y

r

X t +αK y

t

Yt +αKx

t + YOOIt

Yr +αKx

r

1+ ( )Kx

r

1+ ( )Kx

t

1+ ( )K y

t

1+ ( )K y

r


Experimental measurements of the vortex beam shifts:

Fedoseyev, 2001; Dasgupta & Gupta, 2006 Bliokh et al. 2009; Merano et al., 2010

Spin and orbital AM in free space

We have shown that the spin Hall effect is related to the transverse Berry-phase gradient of the plane waves forming the beam. Furthermore, the main scale of the effect is the wavelength. Therefore, it is natural to expect that the SOI phenomena will become more significant in nonparaxial (e.g., tightly focused) fields. And, indeed, some strong spin-dependent AM effect occurs in tightly focused light:


Focusing of a circularly polarized light with SAM results in vortex component in the 3D nonparaxial field and nonzero OAM proportional to . This is spin-to-orbital AM conversion:

However, it turns out that the basic theory of photon AM has some fundamental difficulties for nonparaxial fields…

Y. Zhao et al. PRL 2007

σ


The total AM operator for photon is a sum of the OAM and SAM operators:

ˆ ˆˆ ˆˆ ˆ= × + ≡ +J r p S L Sˆ ,i= ∂kr ˆ ,=p k

( )ˆa aijijS iε= −

( )ˆˆ ,z z zijijL i S iφ ε= − ∂ = −

Eσ ∝ ex + iσ e y( )eiφ

  z-components and paraxial eigenmodes: - polarization, l - vortex

= 1

= −1 = 0

σ = 1

σ = −1

Nonparaxial problem 1

“the separation of the total AM into orbital and spin parts has restricted physical meaning. ... States with definite values of OAM and SAM do not satisfy the condition of transversality in the general case.” A. I. Akhiezer, V. B. Berestetskii, “Quantum Electrodynamics” (1965)

E ⋅k = 0 ⇒   transversality

constraint: 3D#2D E k( )⊥ κ

LE ⊥ κ, S

E ⊥ κ though JE ⊥ κ

E(k)

/ k=κ k

Nonparaxial problem 2

“in the general nonparaxial case there is no separation into -dependent orbital and -dependent spin part of AM” S. M. Barnett, L. Allen (1994)

Eσ ∝ eθ + iσ eφ( )eiσφeiφ ≡ eσ κ( )eiφ   nonparaxial

vortex beam

Lz = + γ σ , Sz = 1−γ( )σ

Calculation based on the Poynting energy flows led to

⇒

spin-to-orbital AM conversion?..

Y. Zhao et al. (2007)

Spin-dependent OAM and position signify the spin-orbit interaction (SOI) of light: (i) spin-to-orbital AM conversion, (ii) spin Hall effect.

( ) 2/ˆ ˆˆ k′ + ×= k Sr r spin-dependent position cf. M.H.L. Pryce, PRSLA (1948), etc.

⇒

Modified AM and position operators

ˆˆ ˆ ˆ′ ′= + = ×ΔL L r k   projected SAM and OAM

( )ˆ ˆ ˆˆ ˆ σ′ = − = ⋅ ≡S S κ κ S κΔ ( )ˆ ˆ= − × ×Δ κ κ S

,ˆˆ ˆ ′ ′= +J L S compatible with transversality:

cf. van Enk & Nienhuis, JMO (1994) Bliokh et al. PRA 2010

Helicity representation

( ) ( ) ( ) ( ) ( ) ( )ˆ ˆ , ,ˆ ˆ : , ,x zz y yzU R R Rφ θ φ + −= − − →e e e e eκ κ

⇒ †ˆ ˆˆ ˆU U→O O

⊥ ⊥⊥ ⊥

00

0 0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

3D # 2D reduction and diagonalization:

ˆ ′L = L − σA × k  all operators become diagonal !

′r = r − σA

ˆ σ′ =S κ

( )ˆ diag ,11, 0σ = −

cf. Bialynicki-Birula, PRD (1987), Skagerstam (1992), Berard & Mohrbach, PLA (2006)

Bliokh et al. PRA 2010


Let us apply these fundamental results to a simple example of nonparaxial vector field – vector Bessel beams. They have very simple spectrum – a circle in k-space – and we assume that all the waves have the same helicity:

E(k)

Eσ = Aσδ θ −θ0( )eiφ

real space field:


Calculating the SAM and OAM expectation values in the Bessel beam, , we arrive at:

O = Eσ ′O Eσ

Lz = +σ ΦB( )1 ,z BS σ= − Φ

E(k)

ΦB ≡ 2π ΦB = AB ⋅

C∫ dk = 2π 1− cosθ0( )

  Berry phase !

Thus, the spin-to-orbit AM conversion in nonparaxial fields originates from the Berry phase associated with the azimuthal distribution of partial waves.


All basic manifestations of the SOI of light can be immediately seen in the spin-dependent real-space intensity distribution of the field. The intensity of the circularly-polarized vector Bessel beam reflects the spin-to-orbital AM conversion:

Iσ ρ( )∝ Aσ 2

a2J2 ρ( ) + b2J+2σ

2 ρ( ) + 2abJ+σ2 ρ( )⎡⎣ ⎤⎦

1 / 2)/ 2( , ab = − Φ= Φ

,σ → a ,σ − b + 2σ ,−σ − 2ab +σ ,0⎡

⎣⎤⎦


The transverse intensity distributions show -dependent radii:

σ1σ = − 1σ =

= 4

= 1

k⊥R

σ = +σ Φ = Lz

The beam radius can be can be obtained from the quantization (with Berry phase) and fine SOI splitting of the caustic:

σ +σ −

Bliokh et al. PRA 2010

Y. Gorodetski et al., PRL (2008) [cf. QHE in graphene].

2B πΦ =0 / 2 ,θ π=

= 1

+σ ΦB = +σ1σ = − 1σ =


The spin-dependent radius can be demonstrated in a circular plasmonic cavity generating Bessel modes:

1σ = ±


+σ ΦB = +σ

1σ = − 1σ =

= 0

1σ = ±

Y. Gorodetski et al., PRL (2008)


Gorodetski et al. Nano Lett. 2009

Interesting application to spin-dependent resonant transmission [Ebbesen et al. + Zheludev et al.]:

the word “SPiN” written with right-handed spirals is lit upwhen illuminated by |σ-⟩ light. For |σ+⟩ illumination, thecontrast is reversed. Linearly polarized illumination resultsin the same transmission for all nanoapertures: as a result,the letters are indistinguishable from the background.

A spectral transmission through a similarly structured arrayof only right-handed spirals was measured by a spectrometer(Horriba Jobin Yvon, iHR320) and normalized by transmis-sion through a coaxial aperture array without corrugations.In Figure 1c a resonant peak corresponding to a Q-factor of9 is clearly observed for |σ-⟩ illumination around 570 nm.The light transmission at the peak was found to be enhancedby a factor of 16 relative to an uncorrugated aperture array.The spectral transmission of a |σ+⟩ illumination does notexhibit any substantial resonance. Observation of a consider-able spin-dependent transmission (with a ratio of about 3)is presented.

The excitation of electromagnetic eigenmodes inside ananoaperture by surface waves is constrained by the AMselection rule, which is given by

where lSM is the normalized OAM of the surface mode and

lGM is the normalized OAM of the guided mode inside thecoaxial structure. A conceptual scheme of the mechanism isdepicted in Figure 2.

The coaxial nanoapertures were designed to be a singlemode system, i.e., possessing a single allowed excitation forlGM ) (1 (the analysis of the guided mode is presented inthe Supporting Information). The spiral corrugation couplesthe incident light into a plasmonic wave and induces adynamic spiral phase according to the spiral grating pitch,lS. In this experiment (see Figure 1b) the corrugation structureadds a spiral phase with lS ) 2. We verified that only lS )2 provides spin-dependent transmission enhancement throughthe apertures. The exceptional transmission enhancementobserved in the experiment indicates that the eigenmode ofthe coaxial waveguide was properly excited by a surfacemode. The apparent difference between lGM ) (1 and thespiral charge lS ) 2 of the surface mode leads one to assumethat another mechanism induced an additional spin-dependentspiral phase that compensated for the excess of AM on theleft side of the selection rule in eq 1. This phase modifiesthe complete OAM of the surface mode to be lSM ) σin + lS

(σin is the incident spin), which now perfectly satisfies theAM selection rule of the system. Thus, the incident spin isconverted into the OAM, conserving the total AM of thesystem. This photon spin-orbit coupling through surfaceplasmon excitation originates from a geometric Berry phasearising in the system.18-21

The above effect can be regarded as a spatial angularDoppler effect (ADE).22 In analogy with a temporal ADE,where an observer at a reference frame rotating with a rateΩt registers a spin-dependent temporal frequency shift,23 herea rotation of the periodic surface corrugation in space withΩ" induces a spin-dependent spatial frequency shift. Ac-cordingly, the geometric phase arising from this spatialfrequency modulation can be easily calculated to be φg )-σin∫Ω" d", where the grating rotates along the coordinate" by Ω" ) dθ/d", where the angle θ indicates the localgroove’s orientation. In the structure presented in Figure 1bwhere " ) $, the geometric phase is simply given by thespin-dependent spiral phase φg ) -σin$ producing therequired additional OAM for the surface wave to satisfy the

Figure 1. Spin-dependent transmission through coaxial nanoap-ertures. (a) Light transmission measured from the element for|σ-⟩, |σ+⟩ + |σ-⟩ and |σ+⟩ illuminations. (b) Scanning electronmicroscopy (SEM) picture of the element used in the experiment.Spiral corrugations were either left-handed or right-handed,depicted with a counterclockwise or clockwise arrow, respec-tively. The magnified SEM pictures of a single element and acoaxial nanoaperture are also presented. (c) Spectral transmissionenhancement for |σ-⟩ (red line) and |σ+⟩ (blue line). The spectrumwas normalized by the transmission measured for coaxialapertures without corrugations.

lSM ) lGM (1)

Figure 2. The mechanism of the nanoaperture’s excitation con-trolled by the AM selection rules. The incident beam bears theintrinsic angular momentum of σin. The excited surface modeacquires the orbital angular momentum of lSM as a result of thespin-orbit interaction. The guided mode with lGM is excited onlyif the selection rule is satisfied.

Nano Lett., Vol. 9, No. 8, 2009 3017

AM selection rule (eq 1). Note that while polarization andchirality effects in anisotropic inhomogeneous nanoscalestructures were investigated previously,24-27 the mechanismof spin-orbit interaction was not distinguished.

In the second experiment the spin-orbit interactionmechanism and the AM selection rules were experimentallyverified by investigating the effect in a circularlysymmetricsachiralsnanostructure.

For this purpose we induced an external OAM carried bythe incident beam instead of adding the OAM through aspiral surface corrugation. We fabricated the element on thesame substrate, which consisted of an individual coaxialaperture with the same dimensions as before, surrounded byan annular coupling grating (see Figure 3a). This aperturewas illuminated by a green laser light (Verdi of Coherent; λ) 532 nm) whose phase was modulated by a spatial lightmodulator (SLM Hamamatsu - PPM X8267) to achieve aspiral phase with lext ) 0, (2. The circular symmetry of

the coupling corrugation does not induce a dynamic phase,which means that lS ) 0. However, due to the spin-orbitinteraction, the incident spin induces a spiral Berry phaseand, as before, is converted to the OAM component of thesurface mode. The external spiral phase modifies thecomplete OAM of the surface mode to be lSM ) σin + lext.Therefore, when the external OAM is zero, the resultingsurface mode AM is lSM ) (1, in which case the selectionrule is always satisfied and the transmission of the elementis undistinguished for distinct spin states. Moreover, provid-ing an external spiral phase of lext ) 2, the AM of the surfacemode will be either 1 or 3 for an incident spin |σ-⟩ or |σ+⟩,respectively. For lext ) -2, the surface mode AM will becorrespondingly, -3 for |σ-⟩ and -1 for |σ+⟩. As before,the best overlapping of the surface mode and the guidedmode is obtained for lSM ) (1; therefore, the transmittedintensity will be strongly dependent upon the incident spin.Figure 3b shows the measured transmissions for variouscombinations of lext and σin. The transmission ratio of thetwo spin states was shown to be approximately 3 which isclose to the ratio measured with the spiral corrugations. Thus,the suggested AM selection rule is verified for systems ofsubstantially different symmetry.

Additional understanding of the AM evolution in theenhanced resonant transmission can be obtained by analyzingthe AM of the light scattered from the nanoaperture. To dothis, we investigated the scattered light far behind theelement. The transverse electric field component of theguided mode28 with lGM ) 1, can be described by the Jonesvector

ErGM ) (Ex, Ey)

T ) E0(r)(cos #, sin #)Te-i#

where E0(r) is the radial field dependence. Note that theguided mode in a circular basis is given by

ErGM ) E0(r)[1/√2)|σ+⟩ + (1/√2)e-i2#|σ-⟩]

This field distribution corresponds to a vectorial vortex (seeFigure 4a) with a Pancharatnam topological charge of 1,resulting in a total AM of j ) 1. As was shown previously,29

such vectorial vortices are unstable and collapse uponpropagation. We have calculated the guided mode using finitedifference time domain algorithm (FDTD) and propagatedit to the far-field. The calculated far-field distribution wascompared to the experimental measurement (see Figure 4b-e).

As expected from the far-field of ErGM, we observed a

bright and a dark spot for the left and right circularpolarization components, respectively, with a good agreementbetween calculation and measurement. Thus, the total AMof light in our system is conserved also for the scattered light(j ) 1). The analysis of the AM of the scattered light enablescharacterizing the allowed excitations in nanoscale systems.

In summary, we presented the effect of spin symmetrybreaking via spin-orbit interaction, which occurs even inrotationally symmetric (achiral) structures. We explored asimple plasmonic nanostructure that can be resonantly excitedonly if an AM selection rule is fulfilled. This selection rulerepresents a matching of the angular momentum of theincident radiation with the excited plasmonic modes in the

Figure 3. The removal of the spin-degeneracy in circular corruga-tion by use of externally induced OAM. (a) Experimental setup. Alaser beam is modulated by a spatial light modulator (SLM) toobtain a spiral phase and then incident through a beam splitter (BS)onto a coaxial aperture with circular corrugation. The transmittedlight is captured in the image plane by the camera. The spiral phasewith lext ) 2, the measured intensity distribution across the incidentbeam, and the SEM picture of the element are presented in thefigure. (b) Intensity distribution cross sections captured by thecamera for different lext. The blue dashed lines correspond to |σ+⟩illumination and red solid lines correspond to |σ-⟩ illumination.The intensity was normalized by the transmission measured via acoaxial aperture without the surrounding corrugation (the horizontaldimension was scaled according to the optical magnification).

3018 Nano Lett., Vol. 9, No. 8, 2009


1σ = − 1σ =

Ohno & Miyanishi, OE 2006

Focusing, scattering, and imaging

Lz = +σ ΦB Sz =σ 1− ΦB( ) ,

Iσ ρ,z( )∝ a J ρ( ) 2

+ b J+2σρ( ) 2

+ 2 ab J+σ ρ( ) 2

The same spin-to-orbital AM conversion occurs in a high-NA focusing, with the same azimuthal Berry phase and spin-dependent intensity. The only difference is -averaging. θ


Spin-dependent intensity and radius of focal spot:

Bliokh et al., OE 2011; Bokor et al., OE 2005

1σ = −1σ = = 1

R

σ Lz

k sinθ=+σ 1− cosθ( )

k sinθ

Spin and orbital Hall effects

k⊥Y

σ = −γ +σ ΦB( )⇒   orbital and spin Hall effects of light

1σ = −

1σ =

= −4 = 0 = 4

( ),φ δ δ∈ −

  azimuthally truncated field (symmetry breaking along x)

B. Zel’dovich et al. (1994), K.Y. Bliokh et al. (2008)

K. Y. Bliokh et al., PRL (2008)

= 0Plasmonic half-lens produces spin Hall effect:

Y ~σ

1σ = −

1σ =

Plasmonic experiment

The SOI of light is a coupling between SAM and OAM. It is caused by the k-space distribution and geometric interference of plane waves.

1.  In cylindrical geometry polarization # vortex: spin-to-orbit AM conversion ~σ

+

2. In asymmetric fields polarization # position:

vortex # position: spin, orbital Hall effects

+

Δy ~σ


Summary of the SOI in nonparaxial fields

" Modified SAM, OAM, and coordinate operators compatible with the transversality

" Berry phase terms: Spin-dependent OAM and coordinates

" Spin-to-orbital AM conversion from the Berry phase between interfering plane waves

" Quantization of the beam radius (caustic)

" Spin and orbital Hall effects in asymmetric fields

These results can be applied to a number of systems involving nonparaxial fields: 1)  Tight focusing by high-NA lens

2)  Scattering by small particles

3)  High-NA microscopy and imaging



Similar redistribution of waves on the sphere in k-space occurs upon scattering by a small particle:

21 1 1

21 1 1

1 1 1 11

1

1

2

2

2

2 2

ˆ1

1 12

2

i i

i i

i i

b e a b e

b e a b e

a b e a b e

a

a

b

φ φ

φ φ

φ φ

− −

−

−⎛ ⎞+⎜ ⎟⎜ ⎟= +⎜ ⎟⎜ ⎟⎝

−

− ⎠

−

−

Π −

Bliokh et al., 2010, 2011

  dipole (Rayleigh) scattering E θ ,φ( )∝−r × r ×E0 0( )⎡⎣ ⎤⎦ = Π θ ,φ( )E0 0( )

†ˆˆ ˆˆzU UΠ = P= spherical projection:


The same spin-to-orbital AM conversion, but helicity is not conserved:

0,σ → 1

2(1+ a1) 0,σ − b1 2σ ,−σ − 2a1b1 σ ,0⎡⎣

⎤⎦

2

2

1 cos ,1 coszl

θσθ

−=+

2

2

2cos1 coszs

θσθ

=+

Integrating over all angles , we obtain: θ

- OAM and SAM -densities: θ

1 ,2zL σ= 1

2zS σ=

Bliokh et al., 2011; Moe & Happer, 1977


“Plasmonic Aharonov-Bohm effect” :

Gorodetski et al., PRB 2010


Microscopy involves a series of focusing and scatterings inside, but it can have paraxial input and output fields:

lens

scatterer

lens

Hence, the SOI phenomena in the output field can be analyzed via usual polarimetry.

Microscopy + Polarimetry = Far-field SOI imaging


Using 3D transformations of the field due to the focusing and scattering inside the system, we obtain 2D space-variant Jones matrix connecting incoming and outgoing rays:

( ) ( ) ( )01 2lens 0 lens 0 0

ˆ ˆ ˆ, , , ,iisJ e U d e Uθ φ θ φ θ φΦΦ −= ∫∫r r

Rodrigues-Herrera et al., PRL 2010


If the scatterer is placed precisely on axis, the output polarization describes the spin-to-orbital AM conversion:

2

2(0)ˆ

i

i

beba

Jae

ϕ

ϕ

−⎛ −∝

−⎞

⎜ ⎟⎝ ⎠

Stokes parameters:

0, 0, 2 ,a bσ σ σ σ→ − −


If the scatterer is placed a bit off-axis, the output polarization demonstrates giant spin Hall effect:

(0) s s

s s

ˆ ˆi i

i i

e ee e

J Jφ φ

φ φ

ρ ρρ

αρ

− ∗ −

∗

⎛ ⎞∝ + ⎜ ⎟

⎝ ⎠( )s s sk x iyρ = +

Macro-separation of spins induced by subwavelength shift of the particle !


High sensitivity of the SOI effects can be employed as a new method for subwavelength probing. We determined position of the scatterer using spin-Hall effect:

Rodrigues-Herrera et al., PRL 2010

λ /10

Summary of the SOI of light

" SOI is everywhere: propagation, refraction, reflection, focusing, scattering, diffraction, anisotropy, nonlinearity, etc.

" We can consider SOI as undesirable aberrations or employ it for fine manipulations with light using internal degrees of freedom.

"  In any case, SOI effects cannot be ignored anymore as we deal with nano-optics and subwavelength scales.

" AM theory, energy flows, and geometric phases provide efficient description of SOI.

SOI in anisotropic nanostructures

An efficient way of manipulation of SOI using artificial geometric phase was developed by E. Hasman et al.

Bomzon et al., OL 2002 [47 citations]

July 1, 2002 / Vol. 27, No. 13 / OPTICS LETTERS 1141

Space-variant Pancharatnam–Berry phase optical elementswith computer-generated subwavelength gratings

Ze’ev Bomzon, Gabriel Biener, Vladimir Kleiner, and Erez Hasman

Optical Engineering Laboratory, Faculty of Mechanical Engineering, Technion –Israel Institute of Technology, Haifa 32000, Israel

Received January 22, 2002

Space-variant Pancharatnam–Berry phase optical elements based on computer-generated subwavelength grat-ings are presented. By continuously controlling the local orientation and period of the grating we can achieveany desired phase element. We present a theoretical analysis and experimentally demonstrate a Pancharat-nam–Berry phase-based diffraction grating for laser radiation at a wavelength of 10.6 mm. © 2002 OpticalSociety of America

OCIS codes: 260.5430, 350.1370, 050.2770, 050.1970.

The Pancharatnam–Berry phase is a geometric phaseassociated with the polarization of light. When thepolarization of a beam traverses a closed loop on thePoincaré sphere, the f inal state differs from the initialstate by a phase factor equal to half of the V area, en-compassed by the loop on the sphere.1,2 In a typicalexperiment, the polarization of a uniformly polarizedbeam is altered by a series of space-invariant (trans-versely homogeneous) wave plates and polarizers, andthe phase that evolves in the time domain is measuredby means of interference.3,4

Recently, we considered a Pancharatnam–Berryphase in the space domain. Using space-variant(transversely inhomogeneous) metal stripe subwave-length gratings, we demonstrated conversion ofcircular polarization into radial polarization5 andshowed that the conversion was accomplished by aspace-variant phase modif ication of geometric originthat affected beam propagation.6 Previously, Bhan-dari suggested the use of a discontinuous spatiallyvarying wave plate as a lens based on similar geomet-ric phase effects.7 Recent studies have investigatedperiodic polarization gratings.8 – 10 These authorsshowed that the polarization of diffracted orders coulddiffer from polarization of the incident beam. Weintend to prove and to utilize a connection betweenthe properties of such polarization gratings and thespace-domain Pancharatnam–Berry phase.

In this Letter we consider optical phase elementsbased on the space-domain Pancharatnam–Berryphase. Unlike diffractive and refractive elements,the phase is not introduced through optical pathdifferences but results from the geometric phasethat accompanies space-variant polarization manipu-lation. The elements are polarization dependent,thereby enabling multipurpose optical elements thatare suitable for applications such as optical switching,optical interconnects, and beam splitting. We showthat such elements can be realized using continuouscomputer-generated space-variant subwavelengthdielectric gratings. The continuity of the gratings en-sures the continuity of the resulting field, thereby elim-inating diffraction associated with discontinuity andenabling the fabrication of elements with high diffrac-tion efficiency. We experimentally demonstrate

Pancharatnam–Berry phase diffraction gratings forCO2 laser radiation at a wavelength of 10.6 mm,showing an ability to form complex polarization-dependent continuous-phase elements.

Figure 1 illustrates the concept of Pancharatnam–Berry phase optical elements (PBOEs) on the Poincarésphere. Circularly polarized light is incident on awave plate with constant retardation and a continuousspace-varying fast axis whose orientation is denotedby u!x, y". We show that, since the wave plate isspace varying, the beam at different points traversesdifferent paths on the Poincaré sphere, resulting in aspace-variant phase-front modif ication that originatesfrom the Pancharatnam–Berry phase. Our goal isto utilize this space-variant geometric phase to formnovel optical elements.

It is convenient to describe PBOEs by use of Jonescalculus. In this formalism, a wave plate with aspace-varying fast axis is described by the operator

T !x, y" ! R#u!x, y"$J!f"R21#u!x, y"$ ,

where J!f" is the operator for a wave plate with retar-dation f, R is the operator for an optical rotator, andu is the local orientation of the axis at each point !x, y".

Fig. 1. Illustration of the principle of PBOEs by use of thePoincaré sphere.

0146-9592/02/131141-03$15.00/0 © 2002 Optical Society of America


Fig. 2. Geometry of the space-variant subwavelengthgrating as well as the DGPs for incident jR! and jL!polarizations.

Fig. 3. Measurements of the transmitted far f ield for thesubwavelength PBOE grating when the retardation is f !p"2 and f ! p, respectively, for incident (a) circular left,(b) circular right, (c) linear polarizations.

12 periods of d. We formed the grating with amaximum local subwavelength period of L ! 3.2 mm,because the Wood anomaly occurs at 3.24 mm forGaAs.11 We applied the grating to a 500-mm-thickGaAs wafer using contact printing and electron–cyclotron resonance etching with BCl3 to a nominaldepth of 2.5 mm to yield a retardation of f ! p"2. Bycombining two such gratings we obtained a gratingwith f ! p retardation. Figure 2 illustrates the ge-ometry of the grating, as well the DGP for incident jL!and jR! polarization states as calculated from Eq. (2).The DGPs resemble blazed gratings with oppositeblazed directions for incident jL! and jR! polarizationstates, as expected from our previous discussions.

After fabrication, we illuminated the PBOEs withcircular and linear polarization. Figure 3 shows theexperimental images of the diffracted f ields for the re-sulting beams, as well as their cross sections when theretardation was f ! p"2 and f ! p. When the in-cident polarization is circular and f ! p"2, close to50% of the light is diffracted according to a first-orderDGP (the direction of diffraction depends on the inci-dent polarization), whereas the other 50% remainedundiffracted to zero order as expected from Eq. (2).The polarization of the diffracted order has a switchedhelicity as expected. For f ! p, no energy appears inthe zero order, and the diffraction eff iciency is close to100%. When the incident polarization is linear Ei !1"

p2 #jR! 1 jL!$, the two helical components of the

beam are subject to different DGPs of opposite signand are diffracted to f irst order in different directions.When f ! p"2, the zero order maintains the originalpolarization, in agreement with Eq. (2), whereas for pretardation the diffraction is 100% efficient for bothcircular polarizations, and no energy is observable atzero order.

To conclude, we have demonstrated novel polar-ization-dependent optical elements based on thePancharatnam–Berry phase. Unlike conventionalelements, PBOEs are not based on optical path differ-ence but on geometric phase modification resultingfrom space-variant polarization manipulation.

E. Hasman’s e-mail address is [email protected].

References

1. S. Pancharatnam, Proc. Indian Acad. Sci. Sect. A 44,247 (1956).

2. M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).3. R. Simon, H. J. Kimble, and E. C. G. Sudharshan, Phys.

Rev. Lett. 61, 19 (1988).4. P. G. Kwiat and R. Y. Chiao, Phys. Rev. Lett. 66, 588

(1991).5. Z. Bomzon, V. Kleiner, and E. Hasman, Appl. Phys.

Lett. 79, 1587 (2001).6. Z. Bomzon, V. Kleiner, and E. Hasman, Opt. Lett. 26,

1424 (2001).7. R. Bhandari, Phys. Rep. 281, 1 (1997).8. F. Gori, Opt. Lett. 24, 584 (1999).9. C. R. Fernández-Pousa, I. Moreno, J. A. Davis, and

J. Adachi, Opt. Lett. 26, 1651 (2001).10. J. Tervo and J. Turunen, Opt. Lett. 25, 785 (2000).11. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, Opt.


33 (2001).13. Z. Bomzon, V. Kleiner, and E. Hasman, Opt. Commun.

192, 169 (2001).

Anisotropic structure with space-varying anisotropy axis:


This results in space-variant geometric phase:

J ∝ a −ibe2iϕ x ,y( )

−ibe−2iϕ x ,y( ) a

⎛

⎝⎜⎜

⎞

⎠⎟⎟

a = cos δ / 2( ) b = sin δ / 2( )










References










192, 169 (2001).










References










192, 169 (2001).


Polarization-dependent diffraction (spin-Hall effect):










References










192, 169 (2001).


Cf. work by F. Capasso et al. in Science 2011:

equations—how, indeed, one can achieve theequal-amplitude condition and the constant phasegradient along the interface through suitable de-sign of the resonators.

There is a fundamental difference between theanomalous refraction phenomena caused by phasediscontinuities and those found in bulk designermetamaterials, which are caused by either negativedielectric permittivity and negative magneticpermeability or anisotropic dielectric permittivitywith different signs of permittivity tensor com-ponents along and transverse to the surface (3, 4).

Phase response of optical antennas.The phaseshift between the emitted and the incident radia-tion of an optical resonator changes appreciably

across a resonance. By spatially tailoring the geom-etry of the resonators in an array and hence theirfrequency response, one can design the phasediscontinuity along the interface and mold thewavefront of the reflected and refracted beams innearly arbitrary ways. The choice of the reso-nators is potentially wide-ranging, from electro-magnetic cavities (9, 10) to nanoparticle clusters(11, 12) and plasmonic antennas (13, 14). Weconcentrated on the latter because of their widelytailorable optical properties (15–19) and the easeof fabricating planar antennas of nanoscale thick-ness. The resonant nature of a rod antenna madeof a perfect electric conductor is shown in Fig.2A (20).

Phase shifts covering the 0-to-2p range areneeded to provide full control of the wavefront.To achieve the required phase coverage whilemaintaining large scattering amplitudes, we usedthe double-resonance properties of V-shaped an-tennas, which consist of two arms of equal lengthh connected at one end at an angle D (Fig. 2B).We define two unit vectors to describe the ori-entation of aV-antenna: ŝ along the symmetry axisof the antenna and â perpendicular to ŝ (Fig. 2B).V-antennas support “symmetric” and “antisym-metric”modes (Fig. 2B, middle and right), whichare excited by electric-field components along ŝand â axes, respectively. In the symmetric mode,the current distribution in each arm approximates

Fig. 2. (A) Calculated phase and amplitude ofscattered light from a straight rod antenna made ofa perfect electric conductor (20). The vertical dashedline indicates the first-order dipolar resonance ofthe antenna. (B) A V-antenna supports symmetricand antisymmetric modes, which are excited, re-spectively, by components of the incident field alongŝ and â axes. The angle between the incident po-larization and the antenna symmetry axis is 45°.The schematic current distribution is representedby colors on the antenna (blue for symmetric andred for antisymmetric mode), with brighter colorrepresenting larger currents. The direction of cur-rent flow is indicated by arrows with color gradient.(C) V-antennas corresponding to mirror images ofthose in (B). The components of the scattered elec-tric field perpendicular to the incident field in (B)and (C) have a p phase difference. (D and E) An-alytically calculated amplitude and phase shift ofthe cross-polarized scattered light for V-antennasconsisting of gold rods with a circular cross sectionand with various length h and angle between therods D at lo = 8 mm (20). The four circles in (D) and(E) indicate the values of h and D used in exper-iments. The rod geometry enables analytical cal-culations of the phase and amplitude of the scatteredlight, without requiring the extensive numericalsimulations needed to compute the same quan-tities for “flat” antennas with a rectangular cross-section, as used in the experiments. The opticalproperties of a rod and “flat” antenna of the samelength are quantitatively very similar, when therod antenna diameter and the “flat” antennawidth and thickness are much smaller than thelength (20). (F) Schematic unit cell of the plasmonicinterface for demonstrating the generalized laws ofreflection and refraction. The sample shown in Fig. 3Ais created by periodically translating in the x-y planethe unit cell. The antennas are designed to haveequal scattering amplitudes and constant phasedifferenceDF = p/4 between neighbors. (G) Finite-difference time-domain (FDTD) simulations of thescattered electric field for the individual antennascomposing the array in (F). Plots show the scat-tered electric field polarized in the x direction fory-polarized plane wave excitation at normal in-cidence from the silicon substrate. The siliconsubstrate is located at z ≤ 0. The antennas are equally spaced at a sub-wavelength separation G/8, where G is the unit cell length. The tilted redstraight line in (G) is the envelope of the projections of the spherical wavesscattered by the antennas onto the x-z plane. On account of Huygens’s

principle, the anomalously refracted beam resulting from the superposi-tion of these spherical waves is then a plane wave that satisfies thegeneralized Snell’s law (Eq. 2) with a phase gradient |dF/dx| = 2p/G alongthe interface.

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that of an individual straight antenna of lengthh (Fig. 2B, middle), and therefore the first-orderantenna resonance occurs at h ≈ leff/2, whereleff is the effective wavelength (14). In the anti-symmetric mode, the current distribution in eacharm approximates that of one half of a straightantenna of length 2h (Fig. 2B, right), and thecondition for the first-order resonance of thismode is 2h ≈ leff/2.

The polarization of the scattered radiationis the same as that of the incident light whenthe latter is polarized along ŝ or â. For an ar-bitrary incident polarization, both antenna modesare excited but with substantially different am-plitude and phase because of their distinctive reso-nance conditions. As a result, the scattered lightcan have a polarization different from that of theincident light. These modal properties of theV-antennas allow one to design the amplitude,phase, and polarization state of the scattered light.We chose the incident polarization to be at 45°with respect to ŝ and â so that both the symmetricand antisymmetric modes can be excited andthe scattered light has a substantial componentpolarized orthogonal to that of the incident light.Experimentally, this allows us to use a polarizerto decouple the scattered light from the excitation.

As a result of the modal properties of theV-antennas and the degrees of freedom in choosingantenna geometry (h and D), the cross-polarizedscattered light can have a large range of phasesand amplitudes for a given wavelength lo; ana-

lytical calculations of the amplitude and phaseresponse of V-antennas assumed to be made ofgold rods are shown in Fig. 2, D and E. In Fig.2D, the blue and red dashed curves correspond tothe resonance peaks of the symmetric and anti-symmetric modes, respectively. We chose fourantennas detuned from the resonance peaks, asindicated by circles in Fig. 2, D and E, whichprovide an incremental phase of p/4 from left toright for the cross-polarized scattered light. Bysimply taking the mirror structure (Fig. 2C) of anexisting V-antenna (Fig. 2B), one creates a newantenna whose cross-polarized radiation has an ad-ditional p phase shift. This is evident by observingthat the currents leading to cross-polarized radia-tion are p out of phase in Fig. 2, B and C. A set ofeight antennas were thus created from the initialfour antennas, as shown in Fig. 2F. Full-wave sim-ulations confirm that the amplitudes of the cross-polarized radiation scatteredby the eight antennas arenearly equal,with phases inp/4 increments (Fig. 2G).

A large phase coverage (~300°) can also beachieved by using arrays of straight antennas (fig.S3). However, to obtain the same range of phaseshift their scattering amplitudes will be substan-tially smaller than those of V-antennas (fig. S3).As a consequence of its double resonances, theV-antenna instead allows one to design an arraywith phase coverage of 2p and equal, yet high,scattering amplitudes for all of the array elements,leading to anomalously reflected and refractedbeams of substantially higher intensities.

Experiments on anomalous reflection andrefraction. We demonstrated experimentally thegeneralized laws of reflection and refractionusing plasmonic interfaces constructed by peri-odically arranging the eight constituent antennasas explained in the caption of Fig. 2F. The spacingbetween the antennas should be subwavelengthso as to provide efficient scattering and to preventthe occurrence of grating diffraction. However, itshould not be too small; otherwise, the strong near-field coupling between neighboring antennaswould perturb the designed scattering amplitudesand phases. A representative sample with thedensest packing of antennas,G = 11 mm, is shownin Fig. 3A, where G is the lateral period of theantenna array. In the schematic of the experimen-tal setup (Fig. 3B), we assume that the cross-polarized scattered light from the antennas on theleft side is phase-delayed as compared with theones on the right. By substituting into Eq. 2 –2p/Gfor dF/dx and the refractive indices of siliconand air (nSi and 1) for ni and nt, we obtain theangle of refraction for the cross-polarized beam

qt,⊥ = arcsin[nSisin(qi) – lo/G]

Figure 3C summarizes the experimental resultsof the ordinary and the anomalous refraction forsix samples with different G at normal incidence.The incident polarization is along the y axis inFig. 3A. The sample with the smallest G corre-sponds to the largest phase gradient and the most

Fig. 3. (A) Scanning electron microscope (SEM)image of a representative antenna array fabricatedon a silicon wafer. The unit cell of the plasmonicinterface (yellow) comprises eight gold V-antennasof width ~220 nm and thickness ~50 nm, and itrepeats with a periodicity of G = 11 mm in the xdirection and 1.5 mm in the y direction. (B) Schematicexperimental setup for y-polarized excitation (electricfield normal to the plane of incidence). (C and D)Measured far-field intensity profiles of the refractedbeams for y- and x-polarized excitations, respective-ly. The refraction angle is counted from the normalto the surface. The red and black curves are mea-sured with and without a polarizer, respectively, forsix samples with differentG. The polarizer is used toselect the anomalously refracted beams that arecross-polarized with respect to the excitation. Theamplitude of the red curves is magnified by a factorof two for clarity. The gray arrows indicate thecalculated angles of anomalous refraction accordingto Eq. 6.

ð6Þ

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that of an individual straight antenna of lengthh (Fig. 2B, middle), and therefore the first-orderantenna resonance occurs at h ≈ leff/2, whereleff is the effective wavelength (14). In the anti-symmetric mode, the current distribution in eacharm approximates that of one half of a straightantenna of length 2h (Fig. 2B, right), and thecondition for the first-order resonance of thismode is 2h ≈ leff/2.

The polarization of the scattered radiationis the same as that of the incident light whenthe latter is polarized along ŝ or â. For an ar-bitrary incident polarization, both antenna modesare excited but with substantially different am-plitude and phase because of their distinctive reso-nance conditions. As a result, the scattered lightcan have a polarization different from that of theincident light. These modal properties of theV-antennas allow one to design the amplitude,phase, and polarization state of the scattered light.We chose the incident polarization to be at 45°with respect to ŝ and â so that both the symmetricand antisymmetric modes can be excited andthe scattered light has a substantial componentpolarized orthogonal to that of the incident light.Experimentally, this allows us to use a polarizerto decouple the scattered light from the excitation.

As a result of the modal properties of theV-antennas and the degrees of freedom in choosingantenna geometry (h and D), the cross-polarizedscattered light can have a large range of phasesand amplitudes for a given wavelength lo; ana-

lytical calculations of the amplitude and phaseresponse of V-antennas assumed to be made ofgold rods are shown in Fig. 2, D and E. In Fig.2D, the blue and red dashed curves correspond tothe resonance peaks of the symmetric and anti-symmetric modes, respectively. We chose fourantennas detuned from the resonance peaks, asindicated by circles in Fig. 2, D and E, whichprovide an incremental phase of p/4 from left toright for the cross-polarized scattered light. Bysimply taking the mirror structure (Fig. 2C) of anexisting V-antenna (Fig. 2B), one creates a newantenna whose cross-polarized radiation has an ad-ditional p phase shift. This is evident by observingthat the currents leading to cross-polarized radia-tion are p out of phase in Fig. 2, B and C. A set ofeight antennas were thus created from the initialfour antennas, as shown in Fig. 2F. Full-wave sim-ulations confirm that the amplitudes of the cross-polarized radiation scatteredby the eight antennas arenearly equal,with phases inp/4 increments (Fig. 2G).

A large phase coverage (~300°) can also beachieved by using arrays of straight antennas (fig.S3). However, to obtain the same range of phaseshift their scattering amplitudes will be substan-tially smaller than those of V-antennas (fig. S3).As a consequence of its double resonances, theV-antenna instead allows one to design an arraywith phase coverage of 2p and equal, yet high,scattering amplitudes for all of the array elements,leading to anomalously reflected and refractedbeams of substantially higher intensities.

Experiments on anomalous reflection andrefraction. We demonstrated experimentally thegeneralized laws of reflection and refractionusing plasmonic interfaces constructed by peri-odically arranging the eight constituent antennasas explained in the caption of Fig. 2F. The spacingbetween the antennas should be subwavelengthso as to provide efficient scattering and to preventthe occurrence of grating diffraction. However, itshould not be too small; otherwise, the strong near-field coupling between neighboring antennaswould perturb the designed scattering amplitudesand phases. A representative sample with thedensest packing of antennas,G = 11 mm, is shownin Fig. 3A, where G is the lateral period of theantenna array. In the schematic of the experimen-tal setup (Fig. 3B), we assume that the cross-polarized scattered light from the antennas on theleft side is phase-delayed as compared with theones on the right. By substituting into Eq. 2 –2p/Gfor dF/dx and the refractive indices of siliconand air (nSi and 1) for ni and nt, we obtain theangle of refraction for the cross-polarized beam

qt,⊥ = arcsin[nSisin(qi) – lo/G]

Figure 3C summarizes the experimental resultsof the ordinary and the anomalous refraction forsix samples with different G at normal incidence.The incident polarization is along the y axis inFig. 3A. The sample with the smallest G corre-sponds to the largest phase gradient and the most

Fig. 3. (A) Scanning electron microscope (SEM)image of a representative antenna array fabricatedon a silicon wafer. The unit cell of the plasmonicinterface (yellow) comprises eight gold V-antennasof width ~220 nm and thickness ~50 nm, and itrepeats with a periodicity of G = 11 mm in the xdirection and 1.5 mm in the y direction. (B) Schematicexperimental setup for y-polarized excitation (electricfield normal to the plane of incidence). (C and D)Measured far-field intensity profiles of the refractedbeams for y- and x-polarized excitations, respective-ly. The refraction angle is counted from the normalto the surface. The red and black curves are mea-sured with and without a polarizer, respectively, forsix samples with differentG. The polarizer is used toselect the anomalously refracted beams that arecross-polarized with respect to the excitation. Theamplitude of the red curves is magnified by a factorof two for clarity. The gray arrows indicate thecalculated angles of anomalous refraction accordingto Eq. 6.

ð6Þ

www.sciencemag.org SCIENCE VOL 334 21 OCTOBER 2011 335

RESEARCH ARTICLES

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Cf. work by X. Zhang et al. in Science 2013:

9. J. T. Ye et al., Nat. Mater. 9, 125 (2010).10. Y. Lee et al., Phys. Rev. Lett. 106, 136809 (2011).11. M. M. Qazilbash et al., Science 318, 1750 (2007).12. M. Liu et al., Nature 487, 345 (2012).13. F. J. Morin, Phys. Rev. Lett. 3, 34 (1959).14. M. Nakano et al., Nature 487, 459 (2012).15. M. Imada, A. Fujimori, Y. Tokura, Rev. Mod. Phys. 70,

1039 (1998).16. See supplementary materials on Science Online.17. J. Cao et al., Nat. Nanotechnol. 4, 732 (2009).18. N. F. Mott, Metal Insulator Transitions (Taylor & Francis,

New York, ed. 2, 1990).19. R. Restori, D. Schwarzenbach, J. R. Schneider,

Acta Crystallogr. B 43, 251 (1987).

20. W. H. Rosevear, W. Paul, Phys. Rev. B 7, 2109 (1973).21. G. Silversmit, D. Depla, H. Poelman, G. B. Marin,

R. De Gryse, J. Electron Spectrosc. Relat. Phenom.135, 167 (2004).

22. A. Janotti et al., Phys. Rev. B 81, 085212 (2010).23. R. Kötz, M. Carlen, Electrochim. Acta 45, 2483 (2000).

Acknowledgments: We thank V. Deline, A. Kellock,T. Topuria, and P. Rice for sample characterization,B. Hughes for help with device fabrication, and K. Martensand A. Pushp for useful discussions. Supported by theGraduate School “Material Science in Mainz” [DeutscheForschungsgemeinschaft (DFG) GSC 266] (T.G.), theMultidisciplinary University Research Initiative program of

the Army Research Office (grant W911-NF-09-1-0398)( J.J.), a Feodor Lynen Research Fellowship from the Alexandervon Humboldt Foundation (T.D.S.), and a stipend fromDFG (GR4000/1-1) (T.G.).

Supplementary Materialswww.sciencemag.org/cgi/content/full/339/6126/1402/DC1Materials and MethodsSupplementary TextFigs. S1 to S8Table S1References (24–31)

20 September 2012; accepted 24 January 201310.1126/science.1230512

PhotonicSpinHall Effect atMetasurfacesXiaobo Yin,1,2 Ziliang Ye,1 Junsuk Rho,1 Yuan Wang,1 Xiang Zhang1,2*

The spin Hall effect (SHE) of light is very weak because of the extremely small photon momentum andspin-orbit interaction. Here, we report a strong photonic SHE resulting in a measured large splittingof polarized light at metasurfaces. The rapidly varying phase discontinuities along a metasurface,breaking the axial symmetry of the system, enable the direct observation of large transverse motionof circularly polarized light, even at normal incidence. The strong spin-orbit interaction deviatesthe polarized light from the trajectory prescribed by the ordinary Fermat principle. Such a strong andbroadband photonic SHE may provide a route for exploiting the spin and orbit angular momentumof light for information processing and communication.

The relativistic spin-orbit coupling of elec-trons results in intrinsic spin precessionsand, therefore, spin polarization–dependent

transverse currents, leading to the observation ofthe spin Hall effect (SHE) and the emerging fieldof spintronics (1–3). The coupling between anelectron’s spin degree of freedom and its orbitalmotion is similar to the coupling of the transverseelectric andmagnetic components of a propagatingelectromagnetic field (4). To conserve total angularmomentum, an inhomogeneity of material’s in-dex of refraction can cause momentum transferbetween the orbital and the spin angular momen-tum of light along its propagation trajectory, re-sulting in transverse splitting in polarizations.Such a photonic spin Hall effect (PSHE) was re-cently proposed theoretically to describe the spin-orbit interaction, the geometric phase, and theprecession of polarization in weakly inhomoge-neous media, as well as the interfaces betweenhomogenous media (5, 6).

However, the experimental observation of theSHE of light is challenging, because the amountof momentum that a photon carries and the spin-orbit interaction between the photon and its me-dium are exceedingly small. The exploration ofsuch a weak process relies on the accumulationof the effect through many multiple reflections(7) or ultrasensitive quantumweakmeasurementswith pre- and postselections of spin states (8, 9).Moreover, the present theory of PSHE assumes

the conservation of total angular momentum overthe entire beam (5–9), which may not hold, es-peciallywhen tailoredwavelength-scale photonicstructures are introduced. In this work, we demon-strate experimentally the strong interactions be-tween the spin and the orbital angular momentumof light in a thin metasurface—a two-dimensional(2D) electromagnetic nanostructurewith designedin-plane phase retardation at thewavelength scale.In such an optically thin material, the resonance-induced anomalous “skew scattering” of light

destroys the axial symmetry of the system thatenables us to observe a giant PSHE, even atthe normal incidence. In contrast, for interfacesbetween two homogeneous media, the spin-orbitcoupling does not exist at normal incidence (5–9).

Metamaterial made of subwavelength com-posites has electromagnetic responses that largelyoriginate from the designed structures rather thanthe constituent materials, leading to extraordinaryproperties including negative index of refraction(10, 11), superlens (12), and optical invisibilities(13, 14). As 2Dmetamaterials, metasurfaces haveshown intriguing abilities in manifesting electro-magnetic waves (15, 16). Recently, anomalousreflection and refraction at a metasurface has beenreported (17, 18), and a variety of applications, suchas flat lenses, have been explored (19–22). How-ever, the general approach toward metasurfacesneglects the spin degree of freedom of light, whichcan be substantial in these materials. We show herethat the rapidly varying in-plane phase retardationthat is dependent on position along the metasurfaceintroduces strong spin-orbit interactions, depart-ing the light trajectory (S) from what is depictedby Fermat principles, S = SFermat + SSO (where SSO

1National Science Foundation Nanoscale Science and Engi-neering Center, 3112 Etcheverry Hall, University of Californiaat Berkeley, Berkeley, CA 94720, USA. 2Materials SciencesDivision, Lawrence Berkeley National Laboratory (LBNL), 1Cyclotron Road, Berkeley, CA 94720, USA.

*Corresponding author. E-mail: [email protected]

Fig. 1. (A) Schematic of the PSHE.The spin-orbit interaction originatesfrom the transverse nature of light.When light is propagating along acurved trajectory, the transversalityof electromagnetic waves requires arotation in polarization. (B) Trans-

verse polarization splitting induced by a metasurface with a strong gradient of phase retardation along the xdirection. The rapid phase retardation refracts light in a skewing direction and results in the PSHE. The strong spin-orbit interactionwithin the optically thinmaterial leads to the accumulationof circular components of the beam inthe transverse directions (y′ directions) of the beam, even when the incident angle is normal to the surface.

www.sciencemag.org SCIENCE VOL 339 22 MARCH 2013 1405

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9. J. T. Ye et al., Nat. Mater. 9, 125 (2010).10. Y. Lee et al., Phys. Rev. Lett. 106, 136809 (2011).11. M. M. Qazilbash et al., Science 318, 1750 (2007).12. M. Liu et al., Nature 487, 345 (2012).13. F. J. Morin, Phys. Rev. Lett. 3, 34 (1959).14. M. Nakano et al., Nature 487, 459 (2012).15. M. Imada, A. Fujimori, Y. Tokura, Rev. Mod. Phys. 70,

1039 (1998).16. See supplementary materials on Science Online.17. J. Cao et al., Nat. Nanotechnol. 4, 732 (2009).18. N. F. Mott, Metal Insulator Transitions (Taylor & Francis,

New York, ed. 2, 1990).19. R. Restori, D. Schwarzenbach, J. R. Schneider,

Acta Crystallogr. B 43, 251 (1987).

20. W. H. Rosevear, W. Paul, Phys. Rev. B 7, 2109 (1973).21. G. Silversmit, D. Depla, H. Poelman, G. B. Marin,

R. De Gryse, J. Electron Spectrosc. Relat. Phenom.135, 167 (2004).

22. A. Janotti et al., Phys. Rev. B 81, 085212 (2010).23. R. Kötz, M. Carlen, Electrochim. Acta 45, 2483 (2000).

Acknowledgments: We thank V. Deline, A. Kellock,T. Topuria, and P. Rice for sample characterization,B. Hughes for help with device fabrication, and K. Martensand A. Pushp for useful discussions. Supported by theGraduate School “Material Science in Mainz” [DeutscheForschungsgemeinschaft (DFG) GSC 266] (T.G.), theMultidisciplinary University Research Initiative program of

the Army Research Office (grant W911-NF-09-1-0398)( J.J.), a Feodor Lynen Research Fellowship from the Alexandervon Humboldt Foundation (T.D.S.), and a stipend fromDFG (GR4000/1-1) (T.G.).

Supplementary Materialswww.sciencemag.org/cgi/content/full/339/6126/1402/DC1Materials and MethodsSupplementary TextFigs. S1 to S8Table S1References (24–31)

20 September 2012; accepted 24 January 201310.1126/science.1230512

PhotonicSpinHall Effect atMetasurfacesXiaobo Yin,1,2 Ziliang Ye,1 Junsuk Rho,1 Yuan Wang,1 Xiang Zhang1,2*

The spin Hall effect (SHE) of light is very weak because of the extremely small photon momentum andspin-orbit interaction. Here, we report a strong photonic SHE resulting in a measured large splittingof polarized light at metasurfaces. The rapidly varying phase discontinuities along a metasurface,breaking the axial symmetry of the system, enable the direct observation of large transverse motionof circularly polarized light, even at normal incidence. The strong spin-orbit interaction deviatesthe polarized light from the trajectory prescribed by the ordinary Fermat principle. Such a strong andbroadband photonic SHE may provide a route for exploiting the spin and orbit angular momentumof light for information processing and communication.

The relativistic spin-orbit coupling of elec-trons results in intrinsic spin precessionsand, therefore, spin polarization–dependent

transverse currents, leading to the observation ofthe spin Hall effect (SHE) and the emerging fieldof spintronics (1–3). The coupling between anelectron’s spin degree of freedom and its orbitalmotion is similar to the coupling of the transverseelectric andmagnetic components of a propagatingelectromagnetic field (4). To conserve total angularmomentum, an inhomogeneity of material’s in-dex of refraction can cause momentum transferbetween the orbital and the spin angular momen-tum of light along its propagation trajectory, re-sulting in transverse splitting in polarizations.Such a photonic spin Hall effect (PSHE) was re-cently proposed theoretically to describe the spin-orbit interaction, the geometric phase, and theprecession of polarization in weakly inhomoge-neous media, as well as the interfaces betweenhomogenous media (5, 6).

However, the experimental observation of theSHE of light is challenging, because the amountof momentum that a photon carries and the spin-orbit interaction between the photon and its me-dium are exceedingly small. The exploration ofsuch a weak process relies on the accumulationof the effect through many multiple reflections(7) or ultrasensitive quantumweakmeasurementswith pre- and postselections of spin states (8, 9).Moreover, the present theory of PSHE assumes

the conservation of total angular momentum overthe entire beam (5–9), which may not hold, es-peciallywhen tailoredwavelength-scale photonicstructures are introduced. In this work, we demon-strate experimentally the strong interactions be-tween the spin and the orbital angular momentumof light in a thin metasurface—a two-dimensional(2D) electromagnetic nanostructurewith designedin-plane phase retardation at thewavelength scale.In such an optically thin material, the resonance-induced anomalous “skew scattering” of light

destroys the axial symmetry of the system thatenables us to observe a giant PSHE, even atthe normal incidence. In contrast, for interfacesbetween two homogeneous media, the spin-orbitcoupling does not exist at normal incidence (5–9).

Metamaterial made of subwavelength com-posites has electromagnetic responses that largelyoriginate from the designed structures rather thanthe constituent materials, leading to extraordinaryproperties including negative index of refraction(10, 11), superlens (12), and optical invisibilities(13, 14). As 2Dmetamaterials, metasurfaces haveshown intriguing abilities in manifesting electro-magnetic waves (15, 16). Recently, anomalousreflection and refraction at a metasurface has beenreported (17, 18), and a variety of applications, suchas flat lenses, have been explored (19–22). How-ever, the general approach toward metasurfacesneglects the spin degree of freedom of light, whichcan be substantial in these materials. We show herethat the rapidly varying in-plane phase retardationthat is dependent on position along the metasurfaceintroduces strong spin-orbit interactions, depart-ing the light trajectory (S) from what is depictedby Fermat principles, S = SFermat + SSO (where SSO

1National Science Foundation Nanoscale Science and Engi-neering Center, 3112 Etcheverry Hall, University of Californiaat Berkeley, Berkeley, CA 94720, USA. 2Materials SciencesDivision, Lawrence Berkeley National Laboratory (LBNL), 1Cyclotron Road, Berkeley, CA 94720, USA.


Fig. 1. (A) Schematic of the PSHE.The spin-orbit interaction originatesfrom the transverse nature of light.When light is propagating along acurved trajectory, the transversalityof electromagnetic waves requires arotation in polarization. (B) Trans-

verse polarization splitting induced by a metasurface with a strong gradient of phase retardation along the xdirection. The rapid phase retardation refracts light in a skewing direction and results in the PSHE. The strong spin-orbit interactionwithin the optically thinmaterial leads to the accumulationof circular components of the beam inthe transverse directions (y′ directions) of the beam, even when the incident angle is normal to the surface.

www.sciencemag.org SCIENCE VOL 339 22 MARCH 2013 1405

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isacorrectiontothelighttrajectoryraised

from

the

metasurface-in

ducedspin-orbitinteraction).

The

PSHEor

thespin-orbitinteractionarises

from

thenoncolinearm

omentumandvelocity(th

echange

oftrajectory)

oflight.W

henlightisprop-

agatingalongacurved

trajectory(Fig.1A

),the

time-varyingmom

entumalongthelightpathmust

introduce

ageom

etric

polarizationrotationtomain-

tain

thepolarizationtransverseto

itsnew

prop-

agationdirection(23),e%

¼−k

ð%e⋅k

Þ=k2.H

ere,e%

andkarethepolarizationvector

andthewave

vector,respectively,

kisthechange

oftheprop-

agationdirection,

andkistheam

plitude

ofthe

wavevector.T

herotationof

thepolarizationde-

pendso

nthehelicity

oflightandmay

beconsidered

ascircular

birefringence

with

apure

geom

etric

origin(23–25).Astheback-actionfrom

geom

et-

ricpolarizatio

nrotatio

ns,the

spin-orbitinterac-

tionalso

changesthepropagationpath

oflig

ht,

aswewill

show

inlatersections,resultin

gin

ahelicity-dependent

transverse

displacementfor

light;i.e.,ph

oton

icSH

E.

Foranordinary

interfacebetweentwohomo-

geneousmedia,w

henaGaussianbeam

impinges

onto

theinterfaceatnorm

alincidence,theaxial

symmetry

norm

alto

thesurfaceelim

inates

the

spin-orbitcoupling,andthetotalangularmom

en-

tumoftheentirebeam

isconserved.How

ever,by

designingametasurface

with

arapidgradient

ofphasediscontinuity∇F

alongtheinterfaceinthe

xdirection(Fig.1B),weintro

duce

astrong

spin-

orbitcouplingwhenthelight

isrefractedoffthe

interface.The

rapid,

wavelength-scalephasere-

tardationcanbe

incorporated

intheopticalpath

bysuitabledesign

oftheinterface(17,

18,26).

Such

aposition-dependentphasediscontinuity

removes

theaxialsym

metry

oftheinterfaceand,

therefore,allowsus

toobservethePS

HE,evenat

thenorm

alincident

angle.

Figure

1Bschemati-

cally

depictsthePS

HEforthelight

beam

thatis

refractedoffametasurface

with

rapidin-plane

phaseretardation.The

mom

entumconservationat

themetasurface

now

musttakeinto

accountthat

theposition-dependentphaseretardationandthe

inducedeffectivecircular

birefringencearedeter-

mined

bythegradientofthein-plane

phasechange

orthecurvatureof

theraytrajectory,ðk

$kÞ=k

2

(7),wherekdependson

therapidphasechange.

Foralin

earlypolarizedincidence,lig

htof

op-

posite

helicities

will

beaccumulated

attheop-

positeedgesof

ananom

alouslyrefractedbeam

inthetransverse

direction(Fig.1

B).The

faster

thein-plane

phasechanges,thestrongertheeffect

becomes.B

ecause

both

thelocalp

hase

response

andits

gradient

aretailo

redthroug

hmetam

ate-

rialdesign,the

opticalspin-orbitinteractionfrom

themetasurface

isstrong,broadband,and

wide-

lytunable.

Toexperim

entally

observethestrong

PSHE

atthemetasurfaces,weused

apolarization-resolved

detectionsetup(Fig.2),which

allowsprecise

measurementofS

tokesp

aram

eterso

fthe

refracted

beam

,providing

thespin-stateinform

ationof

the

light

inthefarfield.

Asupercontin

uum

light

source

isused

forb

roadband

measurement,and

thebeam

isfocusedonto

thesamplewith

aspot

size

of~5

0mm

.Our

metasurfacesconsistof

V-shapedgoldantennas.B

ychanging

thelength

andorientationof

thearmsof

theV-shapedstruc-

tures,thesubw

avelengthantennas

atresonance

intro

duce

tunablephaseretardations

betweenthe

incident

andtheforw

ard-propagatingfields.Fo

rlinearp

hase

retardationalongthexdirection,we

choseeightantennas

with

optim

ized

geom

etry

parameters.Wemeasuredsamples

with

different

phasegradients;ascanning

electro

nmicroscope

imageof

theantennaarraywith

aphasegradient

of4.4rad/mm

isshow

nin

Fig.

2Aas

oneex-

ample.Weused

alens

of50-m

mfocallength

tocollect

theanom

alouslyrefractedfar-field

trans-

mission

throughthemetasurface,and

weim

aged

this

transmission

onan

InGaA

scamera.

The

polarizationof

theincidenceislinearandcanbe

adjusted

ineitherthexorydirections

with

ahalf-

waveplate.The

polarizationstates

oftheanom

-alouslyandtheregularlyrefractedbeam

sare

resolved

atthefarfield

usingan

achrom

atic

quarter-waveplate,ahalf-waveplate,andapo-

larizer

with

anextinctionratio

greaterthan

105

forallw

avelengths

ofinterest.

Because

thepolarizationstateof

light

isun-

ambiguouslydeterm

ined

bytheStokes

vector

S

onaunitPo

incaré

sphere,theevolutionof

the

polarizationdueto

spin-orbitinteractioniswell

describ

edby

theprecession

oftheStokes

vector

onaPoincarésphere.The

helicity

orthehandedness

oflightisgivenby

thecircularStokes

S zparam-

eter

(circular

polarization)

S z¼

I sþ−I s−

I sþþ

I s−,where

I sþandI s

−aretheintensities

oftheanom

alous

refractionwith

circular

polarizationbasis,which

wereim

aged

successivelyusingthecamera.The

coordinatesof

theim

agerepresentthein-plane

wavevectorsof

therefractedbeam

s.The

right

circular

s+andleftcircular

s–polarizations

are

discrim

inated

bysetting

appropriately

theangle

ofthewaveplates

and

polarization

analyzer.

Using

thepolarization-resolved

detection,wecal-

culatedthecircularStokesparameteroftheanom

-alouslyrefractedbeam

from

measurementsfor

each

pixel;thisisshow

ninFig.3A

forx-polarized

incidence.The

in-plane

wavevector

dependence

Fig.

2.(A)S

canningelectro

nmicroscope

imageof

ametasurface

with

arapidphasegradient

inthe

horizontal(x)direction.Theperiodoftheconstituent

V-shaped

antennais180nm

.Eight

antennas

with

different

lengths,orientations,and

spanning

angles

werechosen

fora

linearp

hase

retardation,ranging

from0to2p

withp/4intervals.Scalebar,500nm

.(B)

Lightfromabroadbandsource

wasfocused

onto

thesamplewith

alens

(focallengthf=

50mm).Thepolarizationcanbe

adjuste

din

both

thexandy

directions

withahalf-wa

veplate.Theregularly

andanom

alously

refra

cted

far-fieldtra

nsmissionthrough

themetasurface

wascollected

usingalens

(f=

50mm)andimaged

onan

InGa

Ascamera.

The

polarizationsta

teofthetra

nsmissionisresolved

byusingan

achrom

aticquarter-w

aveplate(l/4),ahalf-

wave

plate(l/2),andapolarizer

with

ahigh

extinctionratio.P,polarize

r.

Fig.

3.(A)O

bservationofagiantSHE

:the

helicity

oftheanom

alously

refra

cted

beam

.The

incid

ence

isfro

mthesilico

nsid

eontothemetasurfaceandispolarized

alongthexd

irectionalongthephasegradient.

Theincid

ence

angleisatsurfa

cenorm

al.Red

andblue

representthe

rightandleftcircularp

olariza

tions,

respectively.(B)H

elicityoftherefra

cted

beam

with

incid

ence

polarized

alongtheydirection.

22MARC

H20

13VO

L33

9SC

IENCE

www.scien

cemag

.org

1406REPO

RTS

on March 21, 2013 www.sciencemag.org Downloaded from

is a correction to the light trajectory raised from themetasurface-induced spin-orbit interaction).

The PSHE or the spin-orbit interaction arisesfrom the noncolinear momentum and velocity (thechange of trajectory) of light. When light is prop-agating along a curved trajectory (Fig. 1A), thetime-varyingmomentum along the light pathmustintroduce a geometric polarization rotation tomain-tain the polarization transverse to its new prop-agation direction (23), e% ¼ −kð%e ⋅ kÞ=k2. Here, e%and k are the polarization vector and the wavevector, respectively, k is the change of the prop-agation direction, and k is the amplitude of thewave vector. The rotation of the polarization de-pends on the helicity of light andmay be consideredas circular birefringence with a pure geometricorigin (23–25). As the back-action from geomet-ric polarization rotations, the spin-orbit interac-tion also changes the propagation path of light,as we will show in later sections, resulting in ahelicity-dependent transverse displacement forlight; i.e., photonic SHE.

For an ordinary interface between two homo-geneous media, when a Gaussian beam impingesonto the interface at normal incidence, the axialsymmetry normal to the surface eliminates thespin-orbit coupling, and the total angular momen-tum of the entire beam is conserved. However, bydesigning a metasurface with a rapid gradient ofphase discontinuity ∇F along the interface in thex direction (Fig. 1B), we introduce a strong spin-orbit coupling when the light is refracted off theinterface. The rapid, wavelength-scale phase re-tardation can be incorporated in the optical pathby suitable design of the interface (17, 18, 26).Such a position-dependent phase discontinuityremoves the axial symmetry of the interface and,therefore, allows us to observe the PSHE, even atthe normal incident angle. Figure 1B schemati-cally depicts the PSHE for the light beam that isrefracted off a metasurface with rapid in-planephase retardation. The momentum conservation atthe metasurface now must take into account thatthe position-dependent phase retardation and theinduced effective circular birefringence are deter-mined by the gradient of the in-plane phase changeor the curvature of the ray trajectory, ðk$ kÞ=k2(7), where kdepends on the rapid phase change.For a linearly polarized incidence, light of op-posite helicities will be accumulated at the op-posite edges of an anomalously refracted beamin the transverse direction (Fig. 1B). The fasterthe in-plane phase changes, the stronger the effectbecomes. Because both the local phase responseand its gradient are tailored through metamate-rial design, the optical spin-orbit interaction fromthe metasurface is strong, broadband, and wide-ly tunable.

To experimentally observe the strong PSHEat themetasurfaces,we used a polarization-resolveddetection setup (Fig. 2), which allows precisemeasurement of Stokes parameters of the refractedbeam, providing the spin-state information of thelight in the far field. A supercontinuum lightsource is used for broadband measurement, and

the beam is focused onto the sample with a spotsize of ~50 mm. Our metasurfaces consist ofV-shaped gold antennas. By changing the lengthand orientation of the arms of the V-shaped struc-tures, the subwavelength antennas at resonanceintroduce tunable phase retardations between theincident and the forward-propagating fields. Forlinear phase retardation along the x direction, wechose eight antennas with optimized geometryparameters. We measured samples with differentphase gradients; a scanning electron microscopeimage of the antenna array with a phase gradientof 4.4 rad/mm is shown in Fig. 2A as one ex-ample. We used a lens of 50-mm focal length tocollect the anomalously refracted far-field trans-mission through the metasurface, and we imagedthis transmission on an InGaAs camera. Thepolarization of the incidence is linear and can beadjusted in either the x or y directions with a half-wave plate. The polarization states of the anom-alously and the regularly refracted beams areresolved at the far field using an achromaticquarter-wave plate, a half-wave plate, and a po-

larizer with an extinction ratio greater than 105

for all wavelengths of interest.Because the polarization state of light is un-

ambiguously determined by the Stokes vector S

on a unit Poincaré sphere, the evolution of thepolarization due to spin-orbit interaction is welldescribed by the precession of the Stokes vectoron a Poincaré sphere. The helicity or the handednessof light is given by the circular Stokes Sz param-eter (circular polarization) Sz ¼ Isþ − Is−

Isþ þ Is−, where

Isþ and Is − are the intensities of the anomalousrefraction with circular polarization basis, whichwere imaged successively using the camera. Thecoordinates of the image represent the in-planewave vectors of the refracted beams. The rightcircular s+ and left circular s– polarizations arediscriminated by setting appropriately the angleof the wave plates and polarization analyzer.Using the polarization-resolved detection, we cal-culated the circular Stokes parameter of the anom-alously refracted beam from measurements foreach pixel; this is shown in Fig. 3A for x-polarizedincidence. The in-plane wave vector dependence

Fig. 2. (A) Scanning electron microscope image of a metasurface with a rapid phase gradient in thehorizontal (x) direction. The period of the constituent V-shaped antenna is 180 nm. Eight antennas withdifferent lengths, orientations, and spanning angles were chosen for a linear phase retardation, rangingfrom 0 to 2p with p /4 intervals. Scale bar, 500 nm. (B) Light from a broadband source was focused ontothe sample with a lens (focal length f = 50 mm). The polarization can be adjusted in both the x and ydirections with a half-wave plate. The regularly and anomalously refracted far-field transmission throughthe metasurface was collected using a lens ( f = 50 mm) and imaged on an InGaAs camera. Thepolarization state of the transmission is resolved by using an achromatic quarter-wave plate (l /4), a half-wave plate (l /2), and a polarizer with a high extinction ratio. P, polarizer.

Fig. 3. (A) Observation of a giant SHE: the helicity of the anomalously refracted beam. The incidence isfrom the silicon side onto the metasurface and is polarized along the x direction along the phase gradient.The incidence angle is at surface normal. Red and blue represent the right and left circular polarizations,respectively. (B) Helicity of the refracted beam with incidence polarized along the y direction.

22 MARCH 2013 VOL 339 SCIENCE www.sciencemag.org1406

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In cylindrical geometry: vortex (spin-to-orbit conversion)

Bomzon et al., OL 2002 [143 citations]

November 1, 2002 / Vol. 27, No. 21 / OPTICS LETTERS 1875

Formation of helical beams by use of Pancharatnam–Berryphase optical elements

Gabriel Biener, Avi Niv, Vladimir Kleiner, and Erez Hasman

Optical Engineering Laboratory, Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel

Received June 17, 2002

Spiral phase elements with topological charges based on space-variant Pancharatnam–Berry phase optical el-ements are presented. Such elements can be achieved by use of continuous computer-generated space-variantsubwavelength dielectric gratings. We present a theoretical analysis and experimentally demonstrate spiralgeometrical phases for infrared radiation at a wavelength of 10.6 mm. © 2002 Optical Society of America

OCIS codes: 260.5430, 350.1370, 050.2770, 050.1970.

Recent years have witnessed a growing interestin helical beams that are exploited in a variety ofapplications. These include trapping of atoms andmacroscopic particles,1,2 transferring of orbital angu-lar momentum to macroscopic objects,1 – 3 rotationalfrequency shifting, the study of optical vortices,4 andspecialized alignment schemes. Beams with helical(or spiral) wave fronts are described by complexamplitudes u!r, v" ~ exp!2ilv", where r and vare the cylindrical coordinates, namely, the radialcoordinate and the azimuthal angle, respectively,and l is the topological charge of the beams. At thecenter, the phase has a screw dislocation, also calleda phase singularity or optical vortex. Typically,hellical beams are formed by manipulation of lightafter it emerges from a laser by superposition of twoorthogonal (nonhelical) beams or by transformationof Gaussian beams into helical beams by means ofcomputer-generated holograms,2,4 cylindrical lenses,and spiral phase elements (SPEs).5 Alternatively, ahelical beam can be generated inside a laser cavity byinsertion of SPEs into the laser cavity.6 The commonapproaches to forming SPEs are as refractive ordiffractive optical elements by use of a milling tool, byuse of a single-stage etching process with a gray-scalemask, or with a multistage etching process.6 Suchhelical beam formations generally are cumbersomeor diff icult to achieve or suffer from large numbersof aberrations, low efficiency, or large and unstableoptical system.

In this Letter we consider novel SPEs based on thespace-domain Pancharatnam–Berry phase.7,8 Unlikediffractive and refractive elements, the phase isnot introduced through optical path differences butresults from the geometrical phase that accompaniesspace-variant polarization manipulation. We showthat such elements can be achieved by use of continu-ous computer-generated space-variant subwavelengthdielectric gratings. Moreover, we experimentallydemonstrate SPEs with different topological chargesbased on a Pancharatnam–Berry phase manipulation,with axially symmetric local subwavelength grooveorientation, for CO2 laser radiation at a wavelength of10.6 mm.

Recently, space-variant polarization-state manipu-lations were demonstrated with computer-generatedsubwavelength structures.9 When the period of a

subwavelength periodic structure is smaller than theincident wavelength, only zeroth order is a propagatingorder, and all other orders are evanescent. The sub-wavelength periodic structure behaves as a uniaxialcrystal with the optical axes parallel and perpendicu-lar to the subwavelength grooves. Therefore, byfabricating quasi-periodic subwavelength structures,for which the period and the orientation of the sub-wavelength grooves were space varying, we achievedcontinuously rotating wave plates.10 Furthermore,we showed that such polarization manipulationsinevitably lead to phase modif ication of geometricalorigin results from local polarization manipulationand are in fact a manifestation of the geometricalPancharatnam–Berry phase.9,11 Optical elementsthat use this effect to form a desired phase frontare called Pancharatnam–Berry phase optical ele-ments (PBOEs).12 In this Letter we maintain thatthe formation of a PBOE with a spiral geometricalphase indicates an ability to form complex continuousPBOEs.

The PBOEs are considered wave plates with con-stant retardation and continuously space varying fastaxes, the orientation of which is denoted u!r, v". It isconvenient to describe PBOEs by using Jones calculus.We find the space-dependent transmission matrix forthe PBOE by applying the optical rotator matrix to theJones matrix of the subwavelength dielectric gratingto obtain, on a helical basis,12

T !r,v" !12

#tx 1 ty exp!if"$∑1 00 1

∏1

12

#tx 2 ty

3 exp!if"$∑

0 exp#i2u!r,v"$exp#2i2u!r,v"$ 0

∏, (1)

where tx and ty are the real amplitude transmission co-efficients for light polarized perpendicular and parallelto the optical axes, respectively, and f is the retarda-tion of the grating. Thus, for an incident plane wavewith arbitrary polarization jEin%, we find that the re-sulting field is

jEout% !p

hE jEin% 1p

hR exp#i2u!r,v"$ jR%

1p

hL exp#2i2u!r,v"$ jL% , (2)

0146-9592/02/211875-03$15.00/0 © 2002 Optical Society of America

1876 OPTICS LETTERS / Vol. 27, No. 21 / November 1, 2002

where hE ! j1/2 !tx 1 ty exp"if#$j2, hR ! j1/2 !tx 2ty exp"if#$ %EinjL&j2, and hL ! j1/2 !tx 2 ty exp"if#$ 3%EinjR&j2 are the polarization order coupling eff icien-cies, %ajb& denotes an inner product, and jR& ! "1 0#Tand jL& ! "0 1#T represent the right-hand and theleft-hand circular polarization components, respec-tively. From Eq. (2) one can see that the emergingbeam from a PBOE comprises three polarizationorders. The first maintains the original polarizationstate and phase of the incident beam, the second isright-hand circularly polarized and has a phase modi-fication of 2u"r, v#, and the third has a polarizationdirection and a phase modification opposite those ofthe former polarization order. Note that the polar-ization eff iciencies depend on the shape and materialof the groove as well as on the polarization state of theincident beam. For the substantial case tx ! ty ! 1and f ! p, an incident wave with jR& polarizationis subject to total polarization state conversion andresults in an emerging field:

jEout& ! exp!2i2u"r,v#$ jL& . (3)

An important property of Eq. (3) is that the phasefactor depends on the local orientation of the sub-wavelength grating. This dependence is geometricalin nature and originates solely from local changes inthe polarization state of the emerging beam. Thisdependence can be illustrated by use of a Poincarésphere with three Stokes parameters, S1, S2, andS3, representing a polarization state, as depicted inFig. 1(a). The incident right-hand polarized and thetransmitted left-hand polarized waves correspond tothe north and the south poles of the sphere, respec-tively. Inasmuch as the subwavelength grating isspace varying, the beam at different points traversesdifferent paths on the Poincaré sphere. For instance,the geodesic lines A and B represent different pathsfor two waves transmitted through element domainsof local orientations u"r, 0# and u"r, v#. Geomet-rical calculations show that the phase differencewp ! 22u"r, v# between states, corresponding tou"r, 0# and u"r, v# orientations, is equal to half of thearea V enclosed by geodesic lines A and B.9,11 Thisfact is in compliance with the well-known rule, pro-posed by Pancharatnam, for comparing the phases oftwo light beams with different polarizations7 and canbe considered an extension into the space domain ofthe rule that we mentioned.

To design a continuous subwavelength struc-ture with the desired phase modification, we de-fine a space-variant subwavelength grating vectorKg"r, v#, oriented perpendicular to the desiredsubwavelength grooves. Figure 1(b) illustratesthis geometrical definition of the grating vec-tor. To design a PBOE with a spiral geometricalphase we need to ensure that the direction ofthe grating grooves is given by u"r, v# ! lv'2,where l is the topological charge. Therefore, fromFig. 1(b), the grating vector is given by Kg"r, v# !K0"r, v# (cos!"l'2 2 1#v$r 1 sin!"l'2 2 1#v$v), where

K0 ! 2p'L"r, v# is the local spatial frequency of agrating with a local period L"r, v#.

To ensure the continuity of the subwavelengthgrooves we required that = 3 Kg ! 0, which resultedin a differential equation that could be solved toyield the local grating period. The solution to thisproblem yielded K0"r# ! "2p'L0# "r0'r#l'2, where L0is the local subwavelength period at r ! r0. Con-sequently the grating function fg (def ined suchthat Kg ! =fg) was then found by integration ofKg"r, v# over an arbitrary path to yield fg"r, v# !"2pr0'L0# "r0'r#l'221 cos!"l'2 2 1#v$'"l'2 2 1# forl fi 2 and fg"r, v# ! "2pr0'L0# ln"r'r0# for l ! 2.We then obtained a Lee-type binary grating to de-scribe the grating function,11 fg, for l ! 1, 2, 3, 4.The grating was fabricated for CO2 laser radiationwith a wavelength of 10.6 mm, with L0 ! 2 mm,r0 ! 4.7 mm, and a maximum radius of 6 mm, re-sulting in 2 mm # L"r# # 3.2 mm. We formed thegrating with a maximum local period of 3.2 mm inorder not to exceed the Wood anomaly of GaAs. Themagnified geometries of the gratings for four topologi-cal charges are presented in Fig. 2. The elementswere fabricated upon 500-mm-thick GaAs wafersby contact photolithography and electron-cyclotronresonance etching with BCl3 to a nominal depth of2.5 mm, resulting in measured values of retardationof f ! p'2 and tx ! ty ! 0.9. These values are closeto the theoretical predictions achieved by rigorouscoupled-wave analysis. The inset in Fig. 2 shows

Fig. 1. (a) Illustration of the principle of PBOEs by useof the Poincaré sphere; insets, local orientations of thesubwavelength grooves. (b) Geometrical definition of thegrating vector.

Fig. 2. Top, geometry of the subwavelength gratings forfour topological charges. Bottom, image of a typical grat-ing profile taken with a scanning-electron microscope.

SOI in anisotropic nanostructures November 1, 2002 / Vol. 27, No. 21 / OPTICS LETTERS 1877

Fig. 3. (a) Interferogram measurements of the spiralPBOEs. (b) The corresponding spiral phases for differenttopological charges.

Fig. 4. Experimental far-f ield images and their calculatedand measured cross sections for the helical beams with l !1 4.

a scanning-electron microscope image of one of thedielectric structures.

Following the fabrication, the spiral PBOEs wereilluminated with a right-hand circularly polarizedbeam, jR!, at 10.6-mm wavelength. To provide ex-perimental evidence of the resultant spiral phasemodif ication of our PBOEs we used a self-interfero-gram measurement, using PBOEs with retardationf ! p"2. For such elements the transmitted beamcomprises two polarization orders: jR! polarizationstate and jL! with a phase modification of 2ilv,according to Eq. (2). The near-f ield intensity distri-butions of the transmitted beams that were followed bya linear polarizer were then measured. Figure 3(a)shows the interferogram patterns for various spiralPBOEs. The intensity dependence on the azimuthalangle is of the form I ~ 1 1 cos#lv$, whereas thenumber of the fringes is equal to l, the topologicalcharge of the beam. Figure 3(b) illustrates the phasefronts that result from the interferometer analysis,indicating spiral phases with different topologicalcharges.

Figure 4 shows the far-field images of transmittedbeams that have various topological charges as wellas the measured and theoretically calculated crosssections. We achieved the experimental result byfocusing the beam through a 500-mm focal-length lens

followed by a circular polarizer. We used the circularpolarizer to transmit only the desired jL! state andto eliminate the jR! polarization order that appearedbecause of the insuff icient etched depth of the grating.Dark spots can be observed at the center of thefar-field images, providing evidence of phase singu-larity in the center of the helical beams. We foundexcellent agreement between theory and experiment,clearly indicating the spiral phases of the beams withdifferent topological charges.

To conclude, we have demonstrated the formationof helical beams by using space-variant Pancharat-nam–Berry phase optical elements based on com-puter-generated subwavelength dielectric gratings.The formation of the spiral phase by the PBOE issubject to control of the local orientation of the grating.This can be achieved with a high level of accuracyby use of an advanced photolithographic process. Incontrast, in the formation of a SPE based on refractiveoptics the phase is inf luenced by fabrication errorscaused by inaccuracy of the etched three-dimensionalprofile. We are currently investigating a photolitho-graphic process with which to achieve accurate controlof the retardation phase to yield only the desiredpolarization order.


References

1. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett,P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).

2. L. Allen, M. J. Padgett, and M. Babiker, in Progressin Optics, E. Wolf, ed. (Pergamon, London, 1999),Vol. XXXIX, p. 291.

3. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature412, 313 (2001).

4. Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr.,J. Opt. Soc. Am. B 15, 2226 (1998).

5. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kris-tensen, and J. P. Woerdman, Opt. Commun. 112, 321(1994).

6. R. Oron, N. Davidson, A. A. Friesem, and E. Hasman,in Progress in Optics, E. Wolf, ed. (Pergamon, London,2001), Vol. XDII, p. 325.

7. S. Pancharatnam, Proc. Ind. Acad. Sci. A 44, 247(1956).

8. M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).9. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, Opt.

Lett. 27, 285 (2002).10. Z. Bomzon, A. Niv, G. Biener, V. Kleiner, and E. Has-

man, Appl. Phys. Lett. 80, 3685 (2002).11. Z. Bomzon, V. Kleiner, and E. Hasman, Opt. Lett. 26,

1424 (2001).12. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, Opt.

Lett. 27, 1141 (2002).

δ = π


Cf. work by F. Capasso et al. in Science 2011:

efficient

light

scatterin

ginto

thecross-polarized

beam

s.Weobserved

that

theangles

ofanom

a-lous

refractionagreewellwith

theoretical

pre-

dictions

ofEq.

6(Fig.3C

).The

samepeak

positions

wereobserved

fornorm

alincidence,

with

polarizationalongthexaxisinFig.3A

(Fig.

3D).To

agood

approxim

ation,weexpectthatthe

V-antennaswereoperatingindependently

atthe

packingdensity

used

inexperim

ents

(20).The

purposeof

usingalargeantennaarray(~230mm

by230mm

)issolely

toaccommodatethesize

oftheplane-wave–lik

eexcitatio

n(beam

radius

~100mm

).The

periodic

antennaarrangem

ent

isused

hereforconvenience

butisnotnecessary

tosatisfy

thegeneralized

lawsof

reflectionand

refractio

n.It

isonly

necessarythat

thephase

gradientisconstantalongtheplasmonicinterface

andthatthescatterin

gam

plitudesof

theantennas

areallequal.

The

phaseincrem

ents

between

nearestneighborsdo

notneed

tobe

constant,if

onerelaxestheunnecessaryconstraintof

equal

spacingbetweennearestantennas.

The

angles

ofrefractionandreflectionare

show

nin

Fig.

4,A

andB,respectively,

asa

functionof

q iforboth

thesilicon-airinterface

(black

curves

andsymbols)andtheplasmonic

interface(red

curves

andsymbols)(20).In

the

rangeof

q i=0°

to9°,theplasmonic

interface

Fig.

4.(A)A

ngleof

refractio

nversus

angleof

incidenceforthe

ordinary(black

curveandtriangles)

andanom

alousrefractio

n(red

curveanddots)forthesamplewith

G=15

mm.Thecurves

are

theoretical

calculations

madeby

usingthegeneralized

Snell’s

lawforrefra

ction(Eq.

2),andthe

symbolsareexperim

entald

ataextra

cted

from

refra

ctionmeasurementsas

afunctionof

theangleof

incidence(20).T

heshaded

region

represents“negative”

refra

ctionforthecross-polarized

light,a

sillustra

tedintheinset.Theblue

arrows

indicatethemodified

criticalanglesfortotalinternalreflection.

(B)An

gleof

reflectionversus

angleof

incidencefortheordinary

(black

curve)

andanom

alous(re

dcurveanddots)reflectionforthe

samplewith

G=15

mm.The

topleftinsetisthezoom

-inview

.The

curves

aretheoretical

calculations

madeby

usingEq.4,

andthesymbols

areexperim

entaldata

extractedfrom

reflectionmeasurementsas

afunctio

nof

theangleof

incidence(20).The

shaded

region

represents“negative”

reflectionforthe

cross-polarized

light,asillustratedinthebotto

mrig

htinset.Theblue

arrowindicatesthe

criticalangleofincidenceabovewh

ichtheanom

alouslyreflected

beam

becomes

evanescent.Experimentswith

lasersem

ittingat

diffe

rent

wavelengthsshow

that

the

plasmonicinterfa

cesa

rebroadband,anom

alously

reflectingandrefra

ctinglight

froml≈5mm

tol≈

10mm

.

Fig.

5.(A)SEM

imageofaplasmonicinterfa

cethat

createsan

optical

vorte

x.Theplasmonic

pattern

consistsof

eightregions,

each

occupied

byone

constituent

antennaof

theeight-elementset

ofFig.

2F.The

antennas

arearranged

soas

togenerate

aphaseshift

thatvariesa

zimuthally

from0to2p,thus

producingahelicoidalscatteredwavefront.(B)

Zoom

-inviewof

thecenter

partof

(A).(C

andD)R

espectively,

measuredandcalcu

latedfar-fieldintensity

distributions

ofan

opticalvortexwith

topologicalchargeone.The

constant

background

in(C)isdueto

thethermalra-

diation.

(EandF)

Respectively,measuredandcalcu

-latedspiralpatternscreatedby

theinterferenceofthe

vortexbeam

andaco-propagatingGaussianbeam

.(G

andH)R

espectively,

measuredandcalcu

latedinterfer-

ence

patternswith

adislocatedfringecreatedby

the

interferenceofthevortexbeam

andaGaussianbeam

when

thetwoaretiltedwithrespecttoeach

other.The

circularb

ordero

fthe

interferencepattern

in(G)arises

fromthefinite

aperture

ofthebeam

splitteru

sedto

combine

thevortexa

ndtheGa

ussia

nbeam

s(20).The

sizeof(C)and

(D)is6

0mmby

60mm,and

thatof(E)

to(H)is30

mmby

30mm.

21OCTO

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www.scien

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.org

336

RESE

ARC

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on March 21, 2013 www.sciencemag.org Downloaded from

efficient light scattering into the cross-polarizedbeams. We observed that the angles of anoma-lous refraction agree well with theoretical pre-dictions of Eq. 6 (Fig. 3C). The same peakpositions were observed for normal incidence,with polarization along the x axis in Fig. 3A (Fig.3D). To a good approximation, we expect that theV-antennas were operating independently at thepacking density used in experiments (20). Thepurpose of using a large antenna array (~230 mmby 230 mm) is solely to accommodate the sizeof the plane-wave–like excitation (beam radius~ 100 mm). The periodic antenna arrangementis used here for convenience but is not necessaryto satisfy the generalized laws of reflection andrefraction. It is only necessary that the phasegradient is constant along the plasmonic interfaceand that the scattering amplitudes of the antennasare all equal. The phase increments betweennearest neighbors do not need to be constant, ifone relaxes the unnecessary constraint of equalspacing between nearest antennas.

The angles of refraction and reflection areshown in Fig. 4, A and B, respectively, as afunction of qi for both the silicon-air interface(black curves and symbols) and the plasmonicinterface (red curves and symbols) (20). In therange of qi = 0° to 9°, the plasmonic interface

Fig. 4. (A) Angle of refraction versus angle of incidence for the ordinary (black curve and triangles)and anomalous refraction (red curve and dots) for the sample with G = 15 mm. The curves aretheoretical calculations made by using the generalized Snell’s law for refraction (Eq. 2), and thesymbols are experimental data extracted from refraction measurements as a function of the angle ofincidence (20). The shaded region represents “negative” refraction for the cross-polarized light, asillustrated in the inset. The blue arrows indicate the modified critical angles for total internal reflection.(B) Angle of reflection versus angle of incidence for the ordinary (black curve) and anomalous (redcurve and dots) reflection for the sample with G = 15 mm. The top left inset is the zoom-in view. Thecurves are theoretical calculations made by using Eq. 4, and the symbols are experimental dataextracted from reflection measurements as a function of the angle of incidence (20). The shadedregion represents “negative” reflection for the cross-polarized light, as illustrated in the bottom rightinset. The blue arrow indicates the critical angle of incidence above which the anomalously reflectedbeam becomes evanescent. Experiments with lasers emitting at different wavelengths show that theplasmonic interfaces are broadband, anomalously reflecting and refracting light from l ≈ 5 mm to l ≈10 mm.

Fig. 5. (A) SEM image of a plasmonic interface thatcreates an optical vortex. The plasmonic patternconsists of eight regions, each occupied by oneconstituent antenna of the eight-element set of Fig.2F. The antennas are arranged so as to generate aphase shift that varies azimuthally from 0 to 2p, thusproducing a helicoidal scattered wavefront. (B) Zoom-inview of the center part of (A). (C and D) Respectively,measured and calculated far-field intensity distributionsof an optical vortex with topological charge one. Theconstant background in (C) is due to the thermal ra-diation. (E and F) Respectively, measured and calcu-lated spiral patterns created by the interference of thevortex beam and a co-propagating Gaussian beam. (Gand H) Respectively, measured and calculated interfer-ence patterns with a dislocated fringe created by theinterference of the vortex beam and a Gaussian beamwhen the two are tilted with respect to each other. Thecircular border of the interference pattern in (G) arisesfrom the finite aperture of the beam splitter used tocombine the vortex and the Gaussian beams (20). Thesize of (C) and (D) is 60mm by 60mm, and that of (E)to (H) is 30 mm by 30 mm.

21 OCTOBER 2011 VOL 334 SCIENCE www.sciencemag.org336

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Cf. work by L. Marrucci et al. in PRL 2011 (q-plates):

mentum from the spin form to the orbital form takes place,with matter playing only an intermediate role. We note thatthis is a rather counterintuitive process, in which the inputpolarization of light controls the shape of the output wavefront.

To illustrate these effects, let us consider the specificcase of a planar slab of a uniaxial birefringent medium,having an homogeneous birefringent phase retardation of! (half-wave) across the slab and an inhomogeneous ori-entation of the fast (or slow) optical axis lying parallel tothe slab planes. We assume that the light beam impinges onthe slab at normal incidence, so that the slab planes areparallel to the xy coordinate plane. Moreover, the opticalaxis orientation in the xy plane, as specified by the angle "it forms with the x axis, is assumed to be given by thefollowing equation:

"!r; ’" # q’$ "0; (1)

where q and "0 are constants. Equation (1) implies thepresence of a defect in the medium localized at the planeorigin, r # 0, similar to the typical defects spontaneouslyformed by nematic liquid crystals [19]. However, if q is aninteger or a semi-integer, there will be no discontinuity linein the slab. In the following, we will refer to inhomoge-neous birefringent elements having the above specifiedgeometry as q plates. A few examples of q-plate geome-tries for different values of q and "0 are shown in Fig. 2.

To analyze the effect of a q plate on the optical field, it isconvenient to adopt a Jones formalism. The Jones matrixM to be applied on the field at each point of the q-platetransverse plane xy is the following:

M # R!%"" & 1 00 %1

! "&R!""

# cos2" sin2"

sin2" % cos2"

!; (2)

where R!"" is the standard 2' 2 rotation matrix by angle

" (in the xy plane), and it is understood that " depends onthe point !r; ’" in the transverse plane according to Eq. (1).

A left-circular polarized plane wave, described by theJones electric-field vector Ein # E0 ' (1; i), after passingthrough the q plate, will be transformed into the followingoutgoing wave (up to an unimportant overall phase shift):

E out #M &Ein # E0ei2"1%i

# $# E0ei2q’ei2"0

1%i

# $:

(3)

The wave emerging from the q plate is therefore uniformlyright-circular polarized, as would occur for a normal half-wave plate, but it has also acquired a phase factorexp!im’", with m # 2q; i.e., it has been transformed intoa helical wave with orbital helicity 2q and orbital angularmomentum 2q@ per photon. It is easy to verify that, in thecase of a right-circular input wave, the orbital helicity andangular momentum of the outgoing wave are sign-inverted.In other words, the input polarization of the light controlsthe sign of the orbital helicity of the output wave front. Itsmagnitude jmj is instead fixed by the birefringence axisgeometry.

In passing through the plate, each photon being con-verted from left-circular to right-circular changes its spinz-component angular momentum from $@ to %@. In thecase of a q plate having q # 1, the orbital z-componentangular momentum of each photon changes instead fromzero to 2@. Therefore, the total variation of the angularmomentum of light is nil, and there is no net transfer ofangular momentum to the plate: The plate in this case actsonly as a ‘‘coupler’’ of the two forms of optical angularmomentum, allowing their conversion into each other. Thisexact compensation of the spin and orbital angular mo-mentum exchanges with matter is clearly related to thecircular symmetry (rotation invariance) of the q # 1 plate,as can be proved by general energy arguments or by avariational approach to the optical angular momentumfluxes [2]. If q ! 1, the plate is not symmetric and willexchange an angular momentum of *2@!q% 1" with eachphoton, with a sign depending on the input polarization.Therefore, in this general case, the angular momentum willnot be just converted from spin to orbital, but the spindegree of freedom will still control the ‘‘direction’’ of theangular momentum exchange with the plate, besides thesign of the output wave-front helicity.

To demonstrate these optical phenomena, we built a qplate with q # 1 using nematic liquid crystal (LC) as thebirefringent material. The LC was sandwiched betweentwo plane glasses, thus forming a planar cell. The LCcell thickness (about 1 #m) and material (E63 fromMerck, Darmstadt, Germany) were chosen so as to obtaina birefringence retardation of approximately a half-wave,at the working wavelength $ # 633 nm. Before assembly,the inner surfaces of the two glasses were coated with apolyimide for planar alignment, and one of them was

(a) (b) (c)

FIG. 2 (color online). Examples of q plates. The tangent to thelines shown indicates the local direction of the optical axis.(a) q # 1=2 and "0 # 0 (a nonzero "0 is here just equivalentto an overall rigid rotation), which generates helical modes withm # *1; (b) q # 1 with "0 # 0 and (c) with "0 # !=2, whichcan both be used to generate modes with m # *2. The last twocases correspond to rotationally symmetric plates, giving rise toperfect spin-to-orbital angular momentum conversion, with noangular momentum transfer to the plate.

PRL 96, 163905 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending28 APRIL 2006

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briefly pressed against a piece of fabric kept in continuousrotation. This procedure led to a surface easy axis (i.e., thepreferred orientation of LC molecules) having the desiredq ! 1 circular-symmetric geometry, as that shown inFig. 2(c).

In order to measure the wave-front shape of the lightemerging from the q plate, we set up a Mach-Zenderinterferometer. A He-Ne laser beam with a TEM00

Gaussian profile was split in two beams, namely, signaland reference. The signal beam was first circularly polar-ized with the desired handedness by means of a properlyoriented quarter-wave plate and then was sent through theLC q plate. The beam emerging from the q plate was thensent through another quarter-wave plate and a linear polar-izer, arranged for transmitting the polarization handednessopposite to the initial one, so as to eliminate the residualunchanged circular polarization (this step is not necessarywhen using a q plate having exactly half-wave retardation).Finally, the signal beam was superimposed with the refer-ence and, thus, generated an interference pattern directlyon the sensing area of a CCD camera. We used two differ-ent interference geometries. In the first, the reference beamwave front was kept approximately plane (more precisely,it had the same wave-front curvature as the signal beam),but the two beams were slightly tilted with respect to eachother. For nonhelical waves, this geometry gives rise to aregular pattern of parallel straight fringes. If the wave frontof the signal beam is helical, the pattern develops a dis-location (double, in this case, since q ! 1 yieldsm ! "2),with an orientation depending on the sign of m and therelative orientation of the two beams. In the second ge-ometry, the reference beam wave front was approximatelyspherical, as obtained by inserting a lens in the referencearm. For nonhelical waves, the resulting interference pat-tern is made of concentric circular fringes. If the wave frontof the signal beam is helical, the pattern takes instead theform of a spiral (a double spiral, for m ! "2), with ahandedness depending on the sign of m (counterclockwiseoutgoing spirals, seen against the propagation direction asin our case, correspond to a positivem). Figure 3 shows theCCD-acquired images of the interference patterns we ob-tained in the two geometries, respectively, for a left-circular [Figs. 3(a) and 3(c)] and right-circular [Figs. 3(b)and 3(d)] input polarizations. These results show unambig-uously that the wave front of the light emerging from the qplate is indeed helical with m ! "2 [i.e., as shown inFigs. 1(c) and 1(d)], as predicted, and that it carries anorbital angular momentum just opposite to the variation ofspin angular momentum associated with the polarizationoccurring in the plate.

It must be emphasized that all commonly used methodsfor generating helical modes of light (cylindrical lenses[3,15], spiral plates [16], and holographic elements [8]) areassociated exclusively with an exchange of orbital angularmomentum of light with matter (indeed, they all involve

inhomogeneous isotropic media) and do not involve thewave polarization at all. In all these methods, the chiralityof the medium structure is imprinted on the generated wavefront, whose orbital helicity is therefore fixed (althoughholographic spatial light modulators allow a slow dynami-cal control of the generated helicity, by modifying themedium spatial structure). In contrast, in the angular mo-mentum conversion process described here, the chirality ofthe generated wave front is determined by the input polar-ization handedness and can, therefore, be easily controlleddynamically.

The generation of electromagnetic helical waves basedon spatially nonuniform polarization transformations waspreviously reported by Biener et al., who used subwave-length diffraction gratings as birefringent elements andtherefore were limited to the midinfrared spectral domain[20,21]. However, Biener et al. did not discuss the conver-sion of optical angular momentum involved in the process.Moreover, the interference geometry adopted by Bieneret al. did not allow distinguishing between right- and left-handed helical wave fronts and, therefore, did not allowmeasuring the sign of the associated orbital angular mo-mentum. For this reason, the results reported here provideto our knowledge the first actual demonstration of the all-optical spin-to-orbital angular momentum conversion pro-cess, as well as the first demonstration of the sign inversion

FIG. 3. Interference patterns of the helical modes emergingfrom the LC cell (signal beam) after superposition with thereference beam. Upper [(a),(b)] panels refer to the plane-wavereference geometry, lower [(c),(d)] panels to the spherical-wavereference one. Panels on the left [(a),(c)] are for a left-circularinput polarization and those on the right [(b),(d)] for a right-circular one.


163905-3

Marrucci et al., PRL 2006 [221 citations]



"!r; ’" # q’$ "0; (1)



M # R!%"" & 1 00 %1

! "&R!""

# cos2" sin2"

sin2" % cos2"

!; (2)





# $# E0ei2q’ei2"0

1%i

# $:

(3)




(a) (b) (c)



163905-2



"!r; ’" # q’$ "0; (1)



M # R!%"" & 1 00 %1

! "&R!""

# cos2" sin2"

sin2" % cos2"

!; (2)





# $# E0ei2q’ei2"0

1%i

# $:

(3)




(a) (b) (c)



163905-2


Particular case: azimuthal or radial anisotropy (q=2)

Brasselet et al., OL 2009, PRL 2009

laser manipulation. The cout! -polarized images of the twindroplets were obtained under spectrally filtered (at 532 nm)cin!-polarized incident white light condenser illumination,where the units vectors c! ¼ ðx! iyÞ=

ffiffiffi2

prefer to the cir-

cular polarization basis. Therefore, the droplets pair, whichbehave as twin sources with the same polarization state,give rise to intensity fringes, as shown in Fig. 2. Straightfringes are obtained when the output polarization state isparallel to the input one, cin % cout ¼ 1, see Fig. 2(a),whereas curved fringes are evidenced when cin % cout ¼0, as shown in Figs. 2(b) and 2(c). This, respectively,manifests an azimuthally invariant wave front profile[Fig. 2(a)] and a spin-dependent azimuthally varying phase[Figs. 2(b) and 2(c)]. In the latter case, the intensity patternlocally resembles the tilted Young patterns observed indouble-slit interferences using optical vortices [23], asshown in the insets of Fig. 2 that display the correspondingtable-top double-slit experiments.

It is known that the coherent superposition of two or-thogonally polarized light fields produces a total field withspace varying polarization. Moreover, the polarization of abeam at any point is determined by the phase differencebetween the c! components. Assuming the incident polar-ization to be cinþ, the absolute azimuthal phase structure of

the cout' -polarized output field, !ð!Þ, is therefore quantita-tively retrieved from a spatially resolved polarimetricanalysis of the output beam [24]. For this purpose, weevaluated the four Stokes parameters S0 ¼ jExj2 þ jEyj2,S1 ¼ jExj2 ' jEyj2, S2 ¼ 2ReðE(

xEyÞ and S3 ¼2 ImðE(

xEyÞ of the output beam. Their reduced values si ¼Si=S0 (i ¼ 1, 2, 3), which all range between '1 and 1, areshown in Figs. 3(a)–3(c), respectively. By construction,!ð!Þ is equal to twice the azimuthal angle of the polariza-tion ellipse of the total output field, c ¼ ð1=2Þ)arctanðs2=s1Þ, hence ! ¼ 2c . The results are shown inFig. 4(a) (see markers), which exhibits a uniformly spira-ling phase that accumulates a 4" phase over a full turnirrespective of the distance from the center of the droplet.Hence, we conclude to the generation of a phase singular-ity with a topological charge ‘ ¼ 2. To understand theobservation of a charge two vortex, one notes that thefreely suspended optically trapped droplet is at rest, hencethe total light angular momentum (spinþ orbital) is con-served. Consequently, the spin angular momentum flippingis associated to the appearance of a 2@ orbital angularmomentum per photon.The same system can be used to generate polychromatic

vortices by using white light. Panel 1 in Fig. 4(b), showsthe characteristic polychromatic doughnut intensity pro-file. The reddish (blueish) outer (inner) part of the whitelight doughnut indicates a wavelength-dependent angularmomentum conversion process, which merely results frommaterial dispersion [25]. This is emphasized in panels 2, 3,and 4 of Fig. 4(b) where the red, green, and blue compo-nents of the vortex exhibit a broader angular spreading atlarger wavelengths [26].The trapping beam itself can be used for in situ and on-

demand vortex generation. A unique feature of the pro-posed technique is that the beam traps the droplets on-axis,and preserves their radial symmetry. Therefore this systemenables the generation of on-axis optical vortices withoutneed for the sensitive optical alignment associated withother techniques. For the purpose of demonstration, wetrapped droplets with various diameters using beams withvarious intensities. We then measured the power in each ofthe circular components of the output beam, and calculatedthe normalized powers P!=P0, where P! refers to theoutput circularly polarized components that are parallel

FIG. 2. Twin droplets interferences. Insets: table-top double-slit Young experiment using a Gaussian beam (a) and a chargetwo optical vortex [(b) and (c)]. Scale bar is 5 #m.

FIG. 3 (color online). Experimental spatially resolved Stokespolarimetry analysis of a single tweezed radial droplet withdiameter ’ 7 #m for quasimonochromatic condenser illumina-tion. (a) s1, (b) s2, (c) s3.

FIG. 1 (color online). Optical vortex generation from a radialnematic liquid crystal droplet. An incident light with circularpolarization c! having a smooth wave front profile (‘ ¼ 0) ispartly converted into the orthogonal polarization state c* thatbears a phase singularity (‘ ! 0). Insets: (a) crossed polarizersimage identifying a cylindrically symmetric director distributionand (b) radial structure of the director.

PRL 103, 103903 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending

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the opposite case, !a ,b"= !0,1", the solution is givenfor Eqs. (1) and (2) with E+↔E− and u↔v. The gen-eral solution is obtained by the linear superpositionwith amplitudes #a#2+ #b#2=1.

To elucidate the coupling between the SAM andOAM for Gaussian beams we note that uniaxial crys-tals have usually weak birefringence, no−ne$10−1–10−3, so that the anisotropy and its conse-quence, the spin-orbit coupling, can be considered asa perturbation. Introducing the average refractive in-dex n= !no+ne" /2 and the small parameter != !no−ne" /n"1 we obtain #o=#!1+! /2" and #e$#!1−3! /2", where #=kn /2, and we keep only the termsof the leading order in !. Applying such a procedureto Eqs. (1) and (2) we derive the following represen-tation of the general solution, !E+,E−"T$M!a ,b"TG,where G=−!i#w2 /Z"exp!i#r2 /Z" with Z=z− iz0, z0=#w2, and the transformation matrix

M = % C Se−2i$

Se2i$ C & = C!0 + ST, !3"

T = !x cos!2$" + !y sin!2$", !4"

where !0 is identity matrix and !x,y are Pauli spinormatrices, C=cos % and S=−i sin % with %=!#r2z /Z2.Solution in this form is valid everywhere in the crys-tal if the anisotropy is small, !"1.

The matrix representation given by Eqs. (3) and (4)allows one to explore the dynamics of polarizationconversion in clear details. Because matrix !0 doesnot change initial polarization state, the first termC!0 describes the loss of power of the input beam. Incontrast, the second term in Eq. (3), ST, shows thepower gain experienced by the circularly polarizedcomponent that is orthogonal to the initial one andthe appearance of OAM compensating the loss ofSAM. More precisely, the matrix T changes the hand-edness of circular polarization and describes the ap-pearance of a vortex with a double topological charge,#l#=2, with the sign opposite to the SAM.

Experimentally accessible quantities to retrievethe optical spin-to-orbital conversion are thereduced powers of two components, P±/P0= !2/&w2"''#E±#2dxdy, where P0 is the input power.These quantities are plotted in Fig. 1(a) in the case!a ,b"= !1,0" and two different beam waists. Theoret-ical curves are obtained [9,10] by using Eqs. (1) and(2), P±/P0= 1

2 (1± !1+z2 /L2"−1), with L=2#e#ow2 / !#o−#e"$z0 /!. The angular momenta normalized to thetotal angular momentum are shown in Fig. 1(b); theyare defined as follows: SAM±= ±P±/P0 and OAM±= l±P±/P0 with l+=0 and l−=2.

In our experiments we used uniaxial calcite crystalsamples that are cut perpendicularly to the opticalaxis into 10 mm'10 mm'z mm slabs for z=1. . .14 mm with steps of 1 mm. Linearly polarizedlight from a He–Ne laser operating at wavelength (=633 nm (no=1.656 and ne=1.458) is converted intocircular polarization using a quarter-wave plate. Thebeam is then focused by a lens !f=25 mm" onto the

sample whose optical axis coincides with the direc-tion of propagation. The output beam is collimated bya second lens !f=100 mm" and passes through a sec-ond quarter-wave plate and a polarizing beamsplit-ter, which allows us to separate its orthogonally po-larized double-charge optical vortex and fundamentalGaussian components.

The intensity distributions of the c+ !l=0" and c−

!l=2" circularly polarized components of a monochro-matic beam are shown in Fig. 2 for various propaga-tion distances z. Both circular components exhibit

0 4 8 12-0.5

0

0.5

1

z (mm)0 4 8 120

0.2

0.4

0.6

1

OAM−

SAM+

SAM−

(a)

OAM+

(b)

z (mm)

0.8

0/P

P ±

0/ PP+

0/ PP−

normalizedangularmomentum

Fig. 1. (Color online) Transfer of normalized (a) powersand (b) angular momenta between c+ (red, green, plus) andc− (black, blue, minus) circularly polarized components.Curves, theory; markers, experiment. The beam waist w=4.59 )m (squares, dashed curves) and w=11.02 )m(circles, solid curves).

0

1

-5 0 50

1

x (arb.units)-5 0 5 -5 0 5

l = 0

l = 2

z = 3 mm 6 mm 9 mm

l = 0

l = 2

x (arb.units) x (arb.units)Fig. 2. (Color online) Experimentally measured intensityprofiles for two components #E±#2 of the laser beam with (=633 nm and w=4.59 )m. The thin (red) curves in the bot-tom diagrams were calculated using Eqs. (1) and (2), andthe thick (black) curves are obtained by averaging the ex-perimental ring profiles over azimuth.

1022 OPTICS LETTERS / Vol. 34, No. 7 / April 1, 2009

the opposite case, !a ,b"= !0,1", the solution is givenfor Eqs. (1) and (2) with E+↔E− and u↔v. The gen-eral solution is obtained by the linear superpositionwith amplitudes #a#2+ #b#2=1.

To elucidate the coupling between the SAM andOAM for Gaussian beams we note that uniaxial crys-tals have usually weak birefringence, no−ne$10−1–10−3, so that the anisotropy and its conse-quence, the spin-orbit coupling, can be considered asa perturbation. Introducing the average refractive in-dex n= !no+ne" /2 and the small parameter != !no−ne" /n"1 we obtain #o=#!1+! /2" and #e$#!1−3! /2", where #=kn /2, and we keep only the termsof the leading order in !. Applying such a procedureto Eqs. (1) and (2) we derive the following represen-tation of the general solution, !E+,E−"T$M!a ,b"TG,where G=−!i#w2 /Z"exp!i#r2 /Z" with Z=z− iz0, z0=#w2, and the transformation matrix

M = % C Se−2i$

Se2i$ C & = C!0 + ST, !3"

T = !x cos!2$" + !y sin!2$", !4"

where !0 is identity matrix and !x,y are Pauli spinormatrices, C=cos % and S=−i sin % with %=!#r2z /Z2.Solution in this form is valid everywhere in the crys-tal if the anisotropy is small, !"1.

The matrix representation given by Eqs. (3) and (4)allows one to explore the dynamics of polarizationconversion in clear details. Because matrix !0 doesnot change initial polarization state, the first termC!0 describes the loss of power of the input beam. Incontrast, the second term in Eq. (3), ST, shows thepower gain experienced by the circularly polarizedcomponent that is orthogonal to the initial one andthe appearance of OAM compensating the loss ofSAM. More precisely, the matrix T changes the hand-edness of circular polarization and describes the ap-pearance of a vortex with a double topological charge,#l#=2, with the sign opposite to the SAM.

Experimentally accessible quantities to retrievethe optical spin-to-orbital conversion are thereduced powers of two components, P±/P0= !2/&w2"''#E±#2dxdy, where P0 is the input power.These quantities are plotted in Fig. 1(a) in the case!a ,b"= !1,0" and two different beam waists. Theoret-ical curves are obtained [9,10] by using Eqs. (1) and(2), P±/P0= 1

2 (1± !1+z2 /L2"−1), with L=2#e#ow2 / !#o−#e"$z0 /!. The angular momenta normalized to thetotal angular momentum are shown in Fig. 1(b); theyare defined as follows: SAM±= ±P±/P0 and OAM±= l±P±/P0 with l+=0 and l−=2.

In our experiments we used uniaxial calcite crystalsamples that are cut perpendicularly to the opticalaxis into 10 mm'10 mm'z mm slabs for z=1. . .14 mm with steps of 1 mm. Linearly polarizedlight from a He–Ne laser operating at wavelength (=633 nm (no=1.656 and ne=1.458) is converted intocircular polarization using a quarter-wave plate. Thebeam is then focused by a lens !f=25 mm" onto the

sample whose optical axis coincides with the direc-tion of propagation. The output beam is collimated bya second lens !f=100 mm" and passes through a sec-ond quarter-wave plate and a polarizing beamsplit-ter, which allows us to separate its orthogonally po-larized double-charge optical vortex and fundamentalGaussian components.

The intensity distributions of the c+ !l=0" and c−

!l=2" circularly polarized components of a monochro-matic beam are shown in Fig. 2 for various propaga-tion distances z. Both circular components exhibit

0 4 8 12-0.5

0

0.5

1

z (mm)0 4 8 120

0.2

0.4

0.6

1

OAM−

SAM+

SAM−

(a)

OAM+

(b)

z (mm)

0.8

0/P

P ±

0/ PP+

0/ PP−

normalizedangularmomentum

Fig. 1. (Color online) Transfer of normalized (a) powersand (b) angular momenta between c+ (red, green, plus) andc− (black, blue, minus) circularly polarized components.Curves, theory; markers, experiment. The beam waist w=4.59 )m (squares, dashed curves) and w=11.02 )m(circles, solid curves).

0

1

-5 0 50

1

x (arb.units)-5 0 5 -5 0 5

l = 0

l = 2

z = 3 mm 6 mm 9 mm

l = 0

l = 2

x (arb.units) x (arb.units)Fig. 2. (Color online) Experimentally measured intensityprofiles for two components #E±#2 of the laser beam with (=633 nm and w=4.59 )m. The thin (red) curves in the bot-tom diagrams were calculated using Eqs. (1) and (2), andthe thick (black) curves are obtained by averaging the ex-perimental ring profiles over azimuth.

1022 OPTICS LETTERS / Vol. 34, No. 7 / April 1, 2009

laser manipulation. The cout! -polarized images of the twindroplets were obtained under spectrally filtered (at 532 nm)cin!-polarized incident white light condenser illumination,where the units vectors c! ¼ ðx! iyÞ=

ffiffiffi2

prefer to the cir-

cular polarization basis. Therefore, the droplets pair, whichbehave as twin sources with the same polarization state,give rise to intensity fringes, as shown in Fig. 2. Straightfringes are obtained when the output polarization state isparallel to the input one, cin % cout ¼ 1, see Fig. 2(a),whereas curved fringes are evidenced when cin % cout ¼0, as shown in Figs. 2(b) and 2(c). This, respectively,manifests an azimuthally invariant wave front profile[Fig. 2(a)] and a spin-dependent azimuthally varying phase[Figs. 2(b) and 2(c)]. In the latter case, the intensity patternlocally resembles the tilted Young patterns observed indouble-slit interferences using optical vortices [23], asshown in the insets of Fig. 2 that display the correspondingtable-top double-slit experiments.

It is known that the coherent superposition of two or-thogonally polarized light fields produces a total field withspace varying polarization. Moreover, the polarization of abeam at any point is determined by the phase differencebetween the c! components. Assuming the incident polar-ization to be cinþ, the absolute azimuthal phase structure of

the cout' -polarized output field, !ð!Þ, is therefore quantita-tively retrieved from a spatially resolved polarimetricanalysis of the output beam [24]. For this purpose, weevaluated the four Stokes parameters S0 ¼ jExj2 þ jEyj2,S1 ¼ jExj2 ' jEyj2, S2 ¼ 2ReðE(

xEyÞ and S3 ¼2 ImðE(

xEyÞ of the output beam. Their reduced values si ¼Si=S0 (i ¼ 1, 2, 3), which all range between '1 and 1, areshown in Figs. 3(a)–3(c), respectively. By construction,!ð!Þ is equal to twice the azimuthal angle of the polariza-tion ellipse of the total output field, c ¼ ð1=2Þ)arctanðs2=s1Þ, hence ! ¼ 2c . The results are shown inFig. 4(a) (see markers), which exhibits a uniformly spira-ling phase that accumulates a 4" phase over a full turnirrespective of the distance from the center of the droplet.Hence, we conclude to the generation of a phase singular-ity with a topological charge ‘ ¼ 2. To understand theobservation of a charge two vortex, one notes that thefreely suspended optically trapped droplet is at rest, hencethe total light angular momentum (spinþ orbital) is con-served. Consequently, the spin angular momentum flippingis associated to the appearance of a 2@ orbital angularmomentum per photon.The same system can be used to generate polychromatic

vortices by using white light. Panel 1 in Fig. 4(b), showsthe characteristic polychromatic doughnut intensity pro-file. The reddish (blueish) outer (inner) part of the whitelight doughnut indicates a wavelength-dependent angularmomentum conversion process, which merely results frommaterial dispersion [25]. This is emphasized in panels 2, 3,and 4 of Fig. 4(b) where the red, green, and blue compo-nents of the vortex exhibit a broader angular spreading atlarger wavelengths [26].The trapping beam itself can be used for in situ and on-

demand vortex generation. A unique feature of the pro-posed technique is that the beam traps the droplets on-axis,and preserves their radial symmetry. Therefore this systemenables the generation of on-axis optical vortices withoutneed for the sensitive optical alignment associated withother techniques. For the purpose of demonstration, wetrapped droplets with various diameters using beams withvarious intensities. We then measured the power in each ofthe circular components of the output beam, and calculatedthe normalized powers P!=P0, where P! refers to theoutput circularly polarized components that are parallel

FIG. 2. Twin droplets interferences. Insets: table-top double-slit Young experiment using a Gaussian beam (a) and a chargetwo optical vortex [(b) and (c)]. Scale bar is 5 #m.

FIG. 3 (color online). Experimental spatially resolved Stokespolarimetry analysis of a single tweezed radial droplet withdiameter ’ 7 #m for quasimonochromatic condenser illumina-tion. (a) s1, (b) s2, (c) s3.

FIG. 1 (color online). Optical vortex generation from a radialnematic liquid crystal droplet. An incident light with circularpolarization c! having a smooth wave front profile (‘ ¼ 0) ispartly converted into the orthogonal polarization state c* thatbears a phase singularity (‘ ! 0). Insets: (a) crossed polarizersimage identifying a cylindrically symmetric director distributionand (b) radial structure of the director.

PRL 103, 103903 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending

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103903-2


Shitrit et al., Nano Lett. 2011

Published: April 22, 2011

r 2011 American Chemical Society 2038 dx.doi.org/10.1021/nl2004835 |Nano Lett. 2011, 11, 2038–2042

LETTER

pubs.acs.org/NanoLett

Optical Spin Hall Effects in Plasmonic ChainsNir Shitrit, Itay Bretner, Yuri Gorodetski, Vladimir Kleiner, and Erez Hasman*

Micro and Nanooptics Laboratory, Faculty of Mechanical Engineering, and Russell Berrie Nanotechnology Institute, Technion—IsraelInstitute of Technology, Haifa 32000, Israel

The spin Hall effect is a basic phenomenon arising from thespin!orbit coupling of electrons.1 In particular, the spatial

trajectory of the moving electrons is affected by their intrinsicangular momentum. More generally, the dispersion relation ofthe particles is modified in a spin-dependent manner.2 The spinHall effect is regarded as intrinsic when it arises due to Rashbacoupling (Berry curvature),3 while the extrinsic effect occurs dueto the spin!orbit-dependent scattering of electrons fromimpurities.4 The optical spin Hall effect (OSHE)—beam dis-placement and momentum shift due to the optical spin(polarization helicity)—was recently presented.5!9 The effectwas attributed to the optical spin!orbit interaction occurringwhen the light passes through an anisotropic and inhomoge-neous medium. Here, we present and experimentally observe theoptical spin Hall effects in coupled localized plasmonic chains.The locally isotropic optical spin Hall effect (OSHE-LI) isregarded as the interaction between the optical spin and thepath ξ of the plasmonic chain with an isotropic unit cell(Figure 1a). In contrast, the locally anisotropic optical spin Halleffect (OSHE-LA) occurs due to the interaction between theoptical spin and the local anisotropy of the unit cell, which isindependent of the chain path (Figure 1b). This latter mechan-ism expands the scope of the OSHE and provides an additionaldegree of freedom in spin-based optics.

Localized surface plasmons are nonpropagating excitations ofthe conduction electrons of metallic nanostructures coupled tothe electromagnetic field. The localized plasmonic modes werefound to play a key role in extraordinary light concentrationby metallic nanofeatures.10 A localized field can be formed bya nanoscale structure characterized by an omnidirectionalreemission. By introducing a coupled nanostructure chain, theindividual localized mode can be tailored to a collective mode!coupled localized mode of the whole chain, having a directionalreemission.11!13 Due to a low dimensionality of the chain, theexcitation of the plasmonic chain mode is polarization-depen-dent, which leads to anisotropic scattering.

A coupled plasmonic chain was fabricated by focused ionbeam (FIB; FEI Strata 400s dual beam system, Gaþ, 30 keV,

46 pA) etching upon a thin (200 nm) Au film, deposited on aglass substrate (Figure 2a, inset). The chain consisted of isotropicunit cells—coaxial nanoapertures with radii of 75 and 125 nm—arranged in a single line with a periodΛ of 470 nm. The fabricatedelementwas immersed in an index-matching oil to obtain symmetricconfiguration. The coaxial shape of the aperture was chosen due toits well-proven ability of plasmon localization.14 The transmissionspectrum of the coupled plasmonic chain was found to bear asignature of two modes centered at the wavelengths of 700 and780 nm corresponding to transversal and longitudinal polarizationexcitations,15 respectively (Figure 2a).

In order to study the OSHE-LI, we fabricated a chain whoselocal orientation with respect to the x coordinate θ = tan!1(dy/dx)varies linearly with x, explicitly, θ(x) = πx/a where a is therotation period of the structure (Figure 2b). This demand leadsto a chain with a route x(ξ),y(ξ) given by the function ξ =(a/2π) ln[(1 þ sin(πx/a))/(1 ! sin(πx/a))]. The localorientation of the curved chain induces local anisotropy variationin the scattered field. The experimental setup to study theOSHEs is introduced in Figure 2c. Figure 2d shows themeasuredOSHE-LI observed from the chain. The structure was sand-wiched between circular polarizers and illuminated with a con-tinuous wave Ti:sapphire tunable laser (Spectra-Physics 3900S)at a wavelength of 780 nm in order to excite the longitudinalmode. The intensity distribution was measured in the far-field,which corresponds to the momentum space. Polarization anal-ysis shows that the scattering from the curved chain comprisestwo components: ballistic and spin-flip. The ballistic componentdoes not experience any diffraction and maintains the polariza-tion state of the incident beam, while the spin-flip component,with an opposite spin state, undergoes diffraction. Spin-depen-dent beam deflection is observed in the experiment via orthogo-nal circular polarizers and corresponds to a momentum shift ofΔkx = !2σπ/a, where σ = (1 is the incident spin state.

Received: February 10, 2011Revised: April 11, 2011

ABSTRACT: Observation of optical spin Hall effects (OSHEs) manifested by aspin-dependent momentum redirection is presented. The effect occurring solely asa result of the curvature of the coupled localized plasmonic chain is regarded as thelocally isotropic OSHE, while the locally anisotropic OSHE arises from the interac-tion between the optical spin and the local anisotropy of the plasmonic moderotating along the chain. A wavefront phase dislocation was observed in a circularcurvature, in which the dislocation strength was enhanced by the locally anisotropiceffect.

KEYWORDS: Plasmonics, optical spin Hall effect, angular momentum, polarization

2039 dx.doi.org/10.1021/nl2004835 |Nano Lett. 2011, 11, 2038–2042

Nano Letters LETTER

The peculiarity of the observed effect lies in its geometricnature. Light scattering by a system with spatially nonuniformanisotropy has a close analogy with a scattering from a revolvingmedium,16 as was shown recently.17 Hence, the scattering by thebent chain ismost conveniently studied using a rotating referenceframe17,18 (u,v), which is attached to the axis of the localanisotropy of the chain and follows the chain’s route ξ(x,y)(Figure 1a). This is accompanied by spatial rotation of the framewith a rateΩξ = dθ(ξ)/dξ, where θ(ξ) is the orientation angle.As a result, a spin-dependent momentum deviation Δkξ =!2σΩξ which corresponds to an additional phase of φ =

RΔkξ

dξ = !2σθ, appears in the spin-flip scattered component. Theexperimentally observed spin Hall momentum deviation concurswith the expected correction ofΔkx =3xφ=!2σπ/a. This effectis regarded as the OSHE-LI.

When the chain unit cell is anisotropic itself, the localanisotropy is also allowed to be arbitrarily oriented along thepath. The reference frame attached to the unit cell anisotropy ispresented as the system (u0,v0) in Figure 1b. It was previouslyshown that plasmonic structures consisting of nanorods exhibit ahigh polarization anisotropy that follows the orientation of therods.19,20 An element consisting of randomly arranged butsimilarly oriented rectangular apertures with dimensions of80" 220 nm was fabricated (Figure 3a, top inset). The localizedmode resonance of this array was obtained at a wavelength of730 nm by measuring the transmission spectrum with linearpolarization excitation parallel to its minor axis. High anisotropyis clearly observed between the two orthogonal linear polariza-tion excitations, which results from the local anisotropy of thenanorod (Figure 3a). Next, we fabricated a straight chain ofsubwavelength nanorods with a period of 430 nm (Figure 3a,bottom inset) in correspondence with the momentum matchingcondition and obtained a narrow resonant line shape of thetransmitted light, when the structure was excited by a linearpolarization parallel to the nanorod’s minor axis (Figure 3a). Theorientation of the nanorods was then varied linearly along the xaxis to obtain a spatial rotation rate of Ω = dθ0/dx = π/a(Figure 3b). The far-field intensity scattered from the elementwas analyzed using the same setup as before and the beamdeflection of the spin-flip component corresponding to the spinHall momentum deviation of Δkx = !2σΩ was detected

(Figure 3c). This beam deflection is regarded as the OSHE-LAand it arises due to the rotation of the local unit cell’s anisotropyrather than the chain path curvature.

Our described mechanism paves the way for one to considerother path symmetries. According to Noether’s theorem,21 forevery symmetry there is a corresponding dynamical conservationlaw.22 When the structure symmetry, or more explicitly the chainpath ξ(x,y), is circular, the corresponding conservation rule is forthe angular momentum (AM). The AM of an optical beam can

Figure 1. Coupled localized plasmonic chains. (a) A plasmonic chainwith isotropic unit cell and rotating reference frame (u,v) which followsthe path ξ. (b) An anisotropy unit cell chain with a frame (u0,v0) attachedto the local anisotropy axis of the unit cell. The lab reference frame isindicated by (x,y).

Figure 2. (a) Transmission spectra of the plasmonic chain consisting ofcoaxial nanoapertures with inner/outer radii of 75/125 nm, with periodof Λ = 470 nm and a length of 84.6 μm. The blue and red lines/arrowscorrespond to transversal and longitudinal polarization excitations,respectively. The inset shows a scanning electron microscopy (SEM)image of the chain. (b) SEM image of a curved chain whose localorientation θ is varied linearly along the x axis with a rotation period of a= 9 μm, and a structure length of 135 μm. (c) Experimental setup forOSHE measurement. A laser light beam passes through a circularpolarizer (a linear polarizer followed by a quarterwave plate (QWP))and then is incident onto the element. The emerging light is collected byan imaging lens to a CCD camera after polarization analysis with acircular polarizer!analyzer. (d) The spin-dependent momentum devia-tion for the OSHE-LI, at a wavelength of 780 nm. The red and blue linesstand for incident right- and left-handed circularly polarized light,respectively (σin =(1). σout denotes the spin state of the scattered light.


Nano Letters LETTER

The peculiarity of the observed effect lies in its geometricnature. Light scattering by a system with spatially nonuniformanisotropy has a close analogy with a scattering from a revolvingmedium,16 as was shown recently.17 Hence, the scattering by thebent chain ismost conveniently studied using a rotating referenceframe17,18 (u,v), which is attached to the axis of the localanisotropy of the chain and follows the chain’s route ξ(x,y)(Figure 1a). This is accompanied by spatial rotation of the framewith a rateΩξ = dθ(ξ)/dξ, where θ(ξ) is the orientation angle.As a result, a spin-dependent momentum deviation Δkξ =!2σΩξ which corresponds to an additional phase of φ =

RΔkξ

dξ = !2σθ, appears in the spin-flip scattered component. Theexperimentally observed spin Hall momentum deviation concurswith the expected correction ofΔkx =3xφ=!2σπ/a. This effectis regarded as the OSHE-LI.

When the chain unit cell is anisotropic itself, the localanisotropy is also allowed to be arbitrarily oriented along thepath. The reference frame attached to the unit cell anisotropy ispresented as the system (u0,v0) in Figure 1b. It was previouslyshown that plasmonic structures consisting of nanorods exhibit ahigh polarization anisotropy that follows the orientation of therods.19,20 An element consisting of randomly arranged butsimilarly oriented rectangular apertures with dimensions of80" 220 nm was fabricated (Figure 3a, top inset). The localizedmode resonance of this array was obtained at a wavelength of730 nm by measuring the transmission spectrum with linearpolarization excitation parallel to its minor axis. High anisotropyis clearly observed between the two orthogonal linear polariza-tion excitations, which results from the local anisotropy of thenanorod (Figure 3a). Next, we fabricated a straight chain ofsubwavelength nanorods with a period of 430 nm (Figure 3a,bottom inset) in correspondence with the momentum matchingcondition and obtained a narrow resonant line shape of thetransmitted light, when the structure was excited by a linearpolarization parallel to the nanorod’s minor axis (Figure 3a). Theorientation of the nanorods was then varied linearly along the xaxis to obtain a spatial rotation rate of Ω = dθ0/dx = π/a(Figure 3b). The far-field intensity scattered from the elementwas analyzed using the same setup as before and the beamdeflection of the spin-flip component corresponding to the spinHall momentum deviation of Δkx = !2σΩ was detected

(Figure 3c). This beam deflection is regarded as the OSHE-LAand it arises due to the rotation of the local unit cell’s anisotropyrather than the chain path curvature.

Our described mechanism paves the way for one to considerother path symmetries. According to Noether’s theorem,21 forevery symmetry there is a corresponding dynamical conservationlaw.22 When the structure symmetry, or more explicitly the chainpath ξ(x,y), is circular, the corresponding conservation rule is forthe angular momentum (AM). The AM of an optical beam can

Figure 1. Coupled localized plasmonic chains. (a) A plasmonic chainwith isotropic unit cell and rotating reference frame (u,v) which followsthe path ξ. (b) An anisotropy unit cell chain with a frame (u0,v0) attachedto the local anisotropy axis of the unit cell. The lab reference frame isindicated by (x,y).

Figure 2. (a) Transmission spectra of the plasmonic chain consisting ofcoaxial nanoapertures with inner/outer radii of 75/125 nm, with periodof Λ = 470 nm and a length of 84.6 μm. The blue and red lines/arrowscorrespond to transversal and longitudinal polarization excitations,respectively. The inset shows a scanning electron microscopy (SEM)image of the chain. (b) SEM image of a curved chain whose localorientation θ is varied linearly along the x axis with a rotation period of a= 9 μm, and a structure length of 135 μm. (c) Experimental setup forOSHE measurement. A laser light beam passes through a circularpolarizer (a linear polarizer followed by a quarterwave plate (QWP))and then is incident onto the element. The emerging light is collected byan imaging lens to a CCD camera after polarization analysis with acircular polarizer!analyzer. (d) The spin-dependent momentum devia-tion for the OSHE-LI, at a wavelength of 780 nm. The red and blue linesstand for incident right- and left-handed circularly polarized light,respectively (σin =(1). σout denotes the spin state of the scattered light.


Nano Letters LETTER

have two components: an intrinsic component that is associatedwith the handedness of the optical spin, and an extrinsic (orbital)component that is associated with its spatial structure. In anoptical paraxial beam with a spiral phase distribution (φ = !lj,where j is the azimuthal angle in polar coordinates and theinteger number l is the topological charge), the total AM perphoton, in units of p (normalized AM), was shown to be j =σþ l.23We fabricated circular chains with a path parameter ξ= roj(ro is the chain radius) of coaxial apertures and rotating nanorodswith a rotation rateΩ =m/ro, so the local anisotropy orientationis θ0 = mj (m is an integer). The far-field intensity distributionsof the scattered spin-flip components were measured using theexperimental configuration described earlier (see panels a and bof Figure 4 for coaxial apertures and rotating nanorods with m =2, respectively). A characteristic dark spot in the center is clearlyseen in the images which is a signature of orbital AM. Moreover,it is evident that the radius of the dark spot measured with the

nanorod chain of m = 2 is larger than the one measured with thecoaxial apertures, corresponding to a higher orbital AM. Theobserved OSHEs and specifically the orbital AM obtained fromcircular chains are due to the collective interaction of thelocalized modes within the periodic plasmonic chains. To verifythe role of the interaction, we measured the spin-flip componentof the scattered light from a circular chain with random distribu-tion of coaxial apertures. As evident from Figure 4c, a bright spotwas obtained, indicating zero orbital AM in a disorderedplasmonic chain.

The orbital AM of the spin-flip component scattered from acircular chain is characterized by the strength of the dislocationand its helicity. Phase dislocations are singular field points suchthat the phase obtains a 2π-fold jump whenmaking a closed looparound them. The dislocation’s strength is the number of wavefronts that end at the phase dislocation point. Its absolute valueand sign (helicity) can be measured simultaneously by theinterference of two optical vortices.24,25 For this purpose, anelement consisting of two separated identical chains of nanorodsrotated along a circular path withm = 1 was fabricated (Figure 5a,b). The two chains, which behave as twin sources, give rise tointensity fringes, as shown in panels f and g of Figure 5. In themeasured interference patterns of the spin-flip components, twoadditional fringes emerge, indicating two phase dislocations (seethe guiding lines in panels f and g of Figure 5). When we alter theincident spin state, the antisymmetric forklike picture is reversed.The experimental patterns with an incident spin of σ = (1 aresimilar to calculated patterns resulting from the scalar interfer-ence of two identical optical vortices with topological charges ofl = (2, respectively (Figure 5d,e). The spin-dependent fringepatterns of a similar element with m = !1 (Figure 5c) were alsomeasured (not shown) to verify that the helicity of the phasedislocation corresponds to the rotation handedness of thenanorods. Moreover, the interference pattern of the ballisticcomponent did not comprise a phase dislocation, indicating zeroorbital AM. Each of the phase dislocations of the spin-flipcomponents predicted by the calculated patterns (triple fork,see panels d and e of Figure 5), breaks in the experiment into apair of fundamental phase dislocations (double fork, see panels fand g of Figure 5). The nongeneric vortex collapse to genericvortices is in accordance with the prediction in ref 26; therefore,the topological charge of the beam in this experiment is givenby the number of the fundamental phase dislocations, resulting in

Figure 3. (a) Transmission spectra of randomly arranged identicallyoriented rectangular apertures with dimensions of 80 # 220 nm in86 μm square array, and of a homogeneous chain with period Λ =430 nm, local orientation θ0 = 45! and a length of 86 μm. The red andblue lines/arrows correspond to linear polarization excitations paralleland perpendicular to the nanorod’s minor axis, respectively. The insetsshow SEM images of the structures described above. (b) SEM image of achain in which the nanorods’ orientation θ0 varies linearly along thex axis with a rotation period of a = 3.44 μm, and a structure length of86 μm. (c) The spin-dependent momentum deviation for the OSHE-LA, at a wavelength of 730 nm. The red and blue lines stand for incidentspin states σin = (1, respectively.

Figure 4. The OSHE-LI and OSHE-LA for circular chains. Themeasured far-field intensity distribution of the spin-flip componentscattered from a circular chain of ordered (a) and disordered (c) coaxialapertures at a wavelength of 780 nm and rotating nanorods withm = 2 ata wavelength of 730 nm (b). Bottom, SEM images of the chains withradii of ro = 5 μm. The spin Hall momentum deviation is accompaniedby a spiral phase-front with l = 2 and l = 4, for the OSHE-LI (a) andOSHE-LA (b), respectively. Note that no spin Hall momentum devia-tion is observed from the disordered chain (c).


Finally, E. Hasman also got in Science 2013 with new ideas:

Shitrit et al., Science 2013

of 3 mm (Fig. 4; see also movie S4). In the fig-ures, thin-film interference is used to show theevolution, by using the lamellae thickness h tosolve the Fresnel equations that determine theconstructive and destructive interference of re-flected light. After draining for 6.4 s, a singlebubble bursts, which causes a rapid collapse ofthe foam structure. Compared to the case in Fig. 3,in this example the typical bubble size is muchlarger, which makes a priori prediction of rup-ture events less predictable.

In this work, we have developed a multi-scale model of the interplay between gas, liquid,and interface forces for a dry foam, permittingthe study of the effects of fluid properties, to-pology, bubble shape, and distribution on drain-age, rupture, and rearrangement.We demonstratedthe model by analyzing cascading properties ofbubble rupture together with large-scale hydro-dynamics. Both the scale-separated model andthe underlying numerical algorithms are gen-eral enough to allow extension of the physics atindividual scales to include other phenomena,such as disjoining pressure, diffusive coarsening,and different types of surface rheology, includingliquid-gas interfaces with mobile/stress-free bound-ary conditions, surface viscosity, evaporation dy-namics, and heating. The multiscale modeling

and numerical methodologies presented here sug-gest a wide variety of related applications, suchas in plastic and metal foam formation.

References and Notes1. D. L. Weaire, S. Hutzler, The Physics of Foams (Oxford

Univ. Press, 2001).2. J. C. Bird, R. de Ruiter, L. Courbin, H. A. Stone, Nature

465, 759 (2010).3. J. Plateau, Statique expérimentale et théorique des

liquides soumis aux seules forces moléculaires(Gauthier-Villars, Trubner et cie., Paris, France, 1873).

4. D. L. Chopp, J. Comput. Phys. 106, 77 (1993).5. K. Polthier, Global Theory of Minimal Surfaces,

Proceedings of the Clay Mathematics Institute 2001Summer School, D. Hoffman, Ed. (AmericanMathematical Society, Providence, RI, 2005).

6. J. von Neumann, Metal Interfaces, C. Herring, Ed. (AmericanSociety for Metals, Cleveland, OH, 1952), pp. 108–110.

7. W. W. Mullins, J. Appl. Phys. 27, 900 (1956).8. R. D. MacPherson, D. J. Srolovitz, Nature 446, 1053 (2007).9. S. Hilgenfeldt, A. M. Kraynik, S. A. Koehler, H. A. Stone,

Phys. Rev. Lett. 86, 2685 (2001).10. A. Oron, S. H. Davis, S. G. Bankoff, Rev. Mod. Phys. 69,

931 (1997).11. T. G. Myers, SIAM Rev. 40, 441 (1998).12. S. A. Koehler, S. Hilgenfeldt, H. A. Stone, J. Colloid

Interface Sci. 276, 420 (2004).13. M. Durand, H. A. Stone, Phys. Rev. Lett. 97, 226101 (2006).14. R. I. Saye, J. A. Sethian, Proc. Natl. Acad. Sci. U.S.A. 108,

19498 (2011).15. R. I. Saye, J. A. Sethian, J. Comput. Phys. 231, 6051 (2012).16. Y. Kim, M.-C. Lai, C. S. Peskin, J. Comput. Phys. 229,

5194 (2010).

17. K. Brakke, Exp. Math. 1, 141 (1992).18. J. U. Brackbill, D. B. Kothe, C. Zemach, J. Comput. Phys.

100, 335 (1992).19. See supplementary materials on Science Online.20. C. J. W. Breward, P. D. Howell, J. Fluid Mech. 458, 379

(2002).21. P. D. Howell, H. A. Stone, Eur. J. Appl. Math. 16, 569 (2005).22. A. J. Chorin, Math. Comput. 22, 745 (1968).23. U. Kornek et al., New J. Phys. 12, 073031 (2010).

Acknowledgments: This research was supported in part bythe Applied Mathematical Sciences subprogram of the Office ofEnergy Research, U.S. Department of Energy, under contractDE-AC02-05CH11231, by the Division of MathematicalSciences of the NSF, and by National Cancer InstituteU54CA143833. Some computations used the resources ofthe National Energy Research Scientific Computing Center,which is supported by the Office of Science of the U.S.Department of Energy under contract DE-AC02-05CH11231.J.A.S. was also supported by the Miller Foundation at Universityof California, Berkeley, and as an Einstein Visiting Fellow ofthe Einstein Foundation, Berlin. R.I.S. was also supported by anAmerican Australian Association Sir Keith Murdoch Fellowship.

Supplementary Materialswww.sciencemag.org/cgi/content/full/340/6133/720/DC1Materials and MethodsSupplementary TextFigs. S1 to S5Table S1References (24–27)Movies S1 to S4

24 September 2012; accepted 8 April 201310.1126/science.1230623

Spin-Optical Metamaterial Routeto Spin-Controlled PhotonicsNir Shitrit, Igor Yulevich, Elhanan Maguid, Dror Ozeri, Dekel Veksler,Vladimir Kleiner, Erez Hasman*

Spin optics provides a route to control light, whereby the photon helicity (spin angular momentum)degeneracy is removed due to a geometric gradient onto a metasurface. The alliance of spin opticsand metamaterials offers the dispersion engineering of a structured matter in a polarizationhelicity–dependent manner. We show that polarization-controlled optical modes of metamaterials arisewhere the spatial inversion symmetry is violated. The emerged spin-split dispersion of spontaneousemission originates from the spin-orbit interaction of light, generating a selection rule based onsymmetry restrictions in a spin-optical metamaterial. The inversion asymmetric metasurface is obtainedvia anisotropic optical antenna patterns. This type of metamaterial provides a route for spin-controllednanophotonic applications based on the design of the metasurface symmetry properties.

Metamaterials are artificial matter struc-tured on a size scale generally smallerthan the wavelength of external stimu-

li that enables a custom-tailored electromagneticresponse of the medium and functionalities suchas negative refraction (1), imaging without an in-trinsic limit to resolution (2), invisibility cloaking(3), and giant chirality (4, 5). An additional twistin this field originates from dispersion-engineeredmetamaterials (6, 7). A peculiar route to modifythe dispersion relation of an anisotropic inhomoge-neous metamaterial is the spin-orbit interaction

(SOI) of light; that is, a coupling of the intrinsicangular momentum (photon spin) and the extrinsicmomentum (8–10). Consequently, the optical spinprovides an additional degree of freedom in nano-optics for spin degeneracy removal phenomenasuch as the spin Hall effect of light (9, 11–14).The chiral behavior originates from a geometricgradient associated with a closed loop traverseupon the Poincaré sphere generating the geo-metric Pancharatnam-Berry phase (15, 16), notfrom the intrinsic local chirality of a meta-atom(4, 5, 17). Specifically, spin optics enables thedesign of a metamaterial with spin-controlledmodes, as in the Rashba effect in solids (18–21).

The Rashba effect is a manifestation of theSOI under broken inversion symmetry [i.e., theinversion transformation r → –r does not pre-

serve the structure (r is a position vector)], wherethe electron spin-degenerate parabolic bands splitinto dispersions with oppositely spin-polarizedstates. This effect can be illustrated via a relativisticelectron in an asymmetric quantum well experienc-ing an effective magnetic field in its rest frame,induced by a perpendicular potential gradient ∇V, asrepresented by the spin-polarized momentum offsetDk º T∇V (18–21). In terms of symmetries, thespin degeneracy associated with the spatial inversionsymmetry is lifted due to a symmetry-breakingelectric field normal to the heterointerface. Similarto the role of a potential gradient in the electronicRashba effect, the space-variant orientation anglef(x,y) of optical nanoantennas induces a spin-splitdispersion of Dk = s∇f (22–24), where sT = T1 isthe photon spin corresponding to right and leftcircularly polarized light, respectively. We report onthe design and fabrication of spin-optical meta-material that gives rise to a spin-controlled disper-sion due to the optical Rashba effect. The inversionasymmetry is obtained in artificial kagome struc-tures with anisotropic achiral antenna configu-rations (Fig. 1, A and B) modeling the uniform(q = 0) and staggered (

ffiffiffi3

p!

ffiffiffi3

p) chirality spin-

folding modes in the kagome antiferromagnet(25–27). In the geometrically frustrated kagomelattice (KL), the reorder of the local magneticmoments transforms the lattice from an inversionsymmetric (IS) to an inversion asymmetric (IaS)structure. Hence, we selected the KL as a platformfor investigating the symmetry influence on spin-based manipulation of metamaterial dispersion.

It was previously shown that the localizedmode resonance of an anisotropic void antenna

Micro and Nanooptics Laboratory, Faculty of Mechanical Engi-neering and Russell Berrie Nanotechnology Institute, Technion–Israel Institute of Technology, Haifa 32000, Israel.


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is observed with a linear polarization excitationparallel to its minor axis (14, 23). We used thisanisotropy in artificial kagome structures, wherethe anisotropic antennas are geometrically ar-ranged in such a way that their principal axesare aligned with the original spin direction in themagnetic KL phases. These metamaterials withthe nearest-antenna distance of L = 6.5 mm wererealized using standard photolithographic tech-niques (24) on a SiC substrate supporting reso-nant collective lattice vibrations [surface phononpolaritons (SPPs)] in the infrared region. Wemeasured angle-resolved thermal emission spec-tra by a Fourier transform infrared spectrometerat varying polar and fixed azimuthal angles (q,ϕ),respectively, while heating the samples to 773 K(see Fig. 1C for the experimental setup). Thedispersion relation w(k) at ϕ = 60° of the q = 0structure (Fig. 1D) exhibits good agreement withthe standard momentum-matching calculation (28)[see (24) for the isotropic KL analysis]. How-ever, the measured dispersion of the

ffiffiffi3

p!

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p

configuration (Fig. 1E) reveals new modes as aresult of the inversion asymmetry of the struc-ture, which may give rise to an optical spin degen-eracy removal. By measuring the S3 componentof the Stokes vector representing the circular po-larization portion within the emitted light (24, 29),we observed the

ffiffiffi3

p!

ffiffiffi3

pspin-projected disper-

sion and found that the new modes yield a highly

polarized emission with opposite spin states and aRashba splitting of Dk = 2p/3L (Fig. 1, F and G).

The removal of the spin degeneracy requiresa spin-dependent correction to fulfill the momentum-matching equation. The spin-controlled dis-persion of an IaS metamaterial obeys the spin-orbit momentum-matching (SOMM) conditionkjje ðsÞ ¼ kSPP þ mG1 þ nG2 − sKi, asso-

ciated with two differentiated sets of recipro-cal vectors: structural and orientational. Here,

ðG1,G2Þ ¼ pL ðx þ y=

ffiffiffi3

p, −x þ y=

ffiffiffi3

pÞ are the

structural reciprocal vectors determined by the q =0 unit cell, whereas K1,2 ¼ p

3L ð−x ∓ffiffiffi3

pyÞ are

the orientational reciprocal vectors determined bya

ffiffiffi3

p!

ffiffiffi3

punit cell (Fig. 2A, inset), associated

with the local field distribution; kjje is the wave

vector of the emitted light in the surface plane; kSPPis the SPP wave vector; (m,n) are the indices of theradiative modes; and i ∈ 1,2 is the index of thespecific spin-dependent geometric Rashba term.Note that in the vector equation the sign of theorientational term is determined by the spin. More-over, the specific geometric Rashba correction is aresult of an arbitrary orientational vector choicefrom K1,2; based on this choice, the secondaryvector linearly depends on the primary Ki and boththe structural vectors G1,2. Hence, the SOMM con-dition arises from the combined contributions of

the structural and orientational lattices resem-bling the structural and magnetic unit cells in thekagome antiferromagnet; yet, this concept is gen-eral and can be tailored to metamaterials and meta-surfaces (30). Considering low modes of (m,n) ∈0,T1, we calculated the spin-dependent disper-sions at ϕ = 0° and 60° (Fig. 2, A and B, respec-tively), confirming the S3 measured dispersions(Fig. 1, F and G, respectively), with the opticalRashba spin split of 2p/3L. Such a spin-split dis-persion is due to a giant optical Rashba effect, asthe geometric Rashba correction is in the order ofmagnitude of the structural term; in particular, theanomalous geometric phase gradient arising dueto the space-variant antenna orientation in the in-vestigated IaS photonic system resembles the po-tential gradient in the electronic Rashba effect.

The above condition can be also derived fromsymmetry restrictions, where the representationtheory is applied to generate selection rules. If agiven structure is invariant under a translation fol-lowed by a rotation and both operators commute,then a spin-orbit coupling is expected. By ap-plying the representation theory formalism con-sidering these symmetry constraints, a momentumselection rule with a spin-dependent geometricRashba correction can be obtained. When thisprocedure is implemented for the

ffiffiffi3

p!

ffiffiffi3

pKL,

which is invariant under a translation of 2L tothe left followed by a rotation of 120° counter-clockwise, the SOMM condition is realized [see(24) for the detailed discussion].

In addition to the spin-projected dispersions,selection rules can also specify the direction ofthe surface wave excited at a given frequency(Fig. 2, C to F). Hence, this concept serves as aplatform for spin-controlled surface waves pos-sessing excellent potential for manipulation on thenanoscale based on a geometric gradient. Partic-ularly, the SOMM provides the basis for a newtype of IaS spin-optical metamaterials, supportingspin-dependent plasmonic launching for nano-circuits (31, 32) andmultiport in on-chip photonics.

The observed optical Rashba spin-split disper-sions reveal two obvious relations of (i) w(k,s+) ≠w(k,s–), which is a signature of inversion sym-metry violation, and (ii) w(k,s+) = w(–k,s–), whichis a manifestation of time reversal symmetry (Fig.1, F and G, and Fig. 2, A and B). Moreover, thespin-projected dispersions show a clear discrim-ination between the different IaS directions, whichis not observed in the degenerated intensity disper-sions. By measuring the angle-resolved emissionspectra at varying ϕ and fixed q, we obtainedthe strength of the optical Rashba effect pointingon the IS and IaS directions in the KL (24). TheS3 dispersions of the

ffiffiffi3

p!

ffiffiffi3

pstructure are

shown in Fig. 3, A and B, where the time reversalsymmetry is clearly seen; in addition, the IS di-rections of ϕ ∈ 30°,90°,150° are manifested bythe spin degeneracy, whereas in all other direc-tions the degeneracy is removed.

The symmetry-based approach offers an ex-tended condition because it also recognizesthe IS directions resulting in spin-degenerated

Fig. 1. Inversion symmetric andasymmetric metamaterials andoptical Rashba effect. (A and B)Optical microscope images of q = 0and

ffiffiffi3

p!

ffiffiffi3

pstructures, respectively.

Rhombuses represent the correspond-ing unit cells in real space. The inset in(A) is a scanning electron microscopeimage of the 1-by-6–mm2 antenna,

etched to a depth of 1 mm on a SiC substrate. (C) Schematic setup for the spin-projected dispersion basedon the spin-optical metamaterial symmetry. The thermal radiation polarization state is resolved with the useof a circular polarization analyzer [a quarter-wave plate (QWP) followed by a linear polarizer (P)]. ke, wavevector of the emitted light. (D and E) Measured intensity dispersions of thermal emission from q = 0 andffiffiffi3

p!

ffiffiffi3

pstructures, respectively. Yellow lines in (D) correspond to the standard momentum-matching

calculation; green lines in (E) highlight the new modes. c, speed of light. (F and G) Measured spin-polarizeddispersions of the

ffiffiffi3

p!

ffiffiffi3

pstructure along the IaS directions of ϕ = 0° and 60°, respectively.

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dispersions. As a reference, we measured the inten-sity dispersions of the q = 0 structure at ϕ = 30°,verifying the standard momentum-matching cal-culation without the geometric Rashba correction.Note that this direction is arbitrary because thisstructure is IS in all directions; however, it is aspecific IS direction in the

ffiffiffi3

p!

ffiffiffi3

pstructure. In

accordance with this condition, we did not observeany spin dependence in the measured

ffiffiffi3

p!

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p

spin-projected dispersion at this direction. This re-sult is supported by the SOMM condition (Fig.

3D), which is general because it distinguishesbetween the IS and IaS directions and tailors aspin-degenerated or spin-dependent output, respec-tively, and it reveals new modes observed in theffiffiffi3

p!

ffiffiffi3

pintensity dispersion (Fig. 3C) owing to

the spin-dependent geometric Rashba correction.The spin degeneracy removal was also shown

in the near-field associated with the orbital an-gular momentum variation along the IaS di-rections. We revealed a chain of vortices withalternating helicities in the artificial

ffiffiffi3

p!

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p

building blocks, carrying a spin-dependent space-variant orbital angular momentum arising fromthe spiral phase front of the SPPs [see (24) forthe detailed analysis]. The reported spin-basedphenomena in the near- and far-fields inspire thedevelopment of a unified theory to establish alink between the spin-controlled radiative modesand the metasurface symmetry properties to en-compass a broader class of metastructures fromperiodic to quasi-periodic or aperiodic. The de-sign of metamaterial symmetries via geometricgradients provides a route for integrated nanoscalespintronic spin-optical devices based on spin-controlled manipulation of spontaneous emission,absorption, scattering, and surface-wave excitation.

References and Notes1. R. A. Shelby, D. R. Smith, S. Schultz, Science 292, 77 (2001).2. J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).3. W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev,

Nat. Photonics 1, 224 (2007).4. A. V. Rogacheva, V. A. Fedotov, A. S. Schwanecke,

N. I. Zheludev, Phys. Rev. Lett. 97, 177401 (2006).5. S. Zhang et al., Phys. Rev. Lett. 102, 023901 (2009).6. Z. Jacob, L. V. Alekseyev, E. Narimanov, Opt. Express

14, 8247 (2006).7. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov,

I. Kretzschmar, V. M. Menon, Science 336, 205 (2012).8. V. S. Liberman, B. Y. Zel’dovich, Phys. Rev. A 46, 5199 (1992).9. K. Y. Bliokh, Y. Gorodetski, V. Kleiner, E. Hasman,

Phys. Rev. Lett. 101, 030404 (2008).10. N. M. Litchinitser, Science 337, 1054 (2012).11. O. Hosten, P. Kwiat, Science 319, 787 (2008).12. Y. Gorodetski, A. Niv, V. Kleiner, E. Hasman, Phys. Rev. Lett.

101, 043903 (2008).13. K. Y. Bliokh, A. Niv, V. Kleiner, E. Hasman, Nat. Photonics

2, 748 (2008).14. N. Shitrit, I. Bretner, Y. Gorodetski, V. Kleiner, E. Hasman,

Nano Lett. 11, 2038 (2011).15. M. V. Berry, J. Mod. Opt. 34, 1401 (1987).16. Z. Bomzon, V. Kleiner, E. Hasman, Opt. Lett. 26,

1424 (2001).17. J. K. Gansel et al., Science 325, 1513 (2009).18. G. Dresselhaus, Phys. Rev. 100, 580 (1955).19. E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960).20. K. Ishizaka et al., Nat. Mater. 10, 521 (2011).21. P. D. C. King et al., Phys. Rev. Lett. 107, 096802 (2011).22. N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner,

E. Hasman, Phys. Rev. Lett. 105, 136402 (2010).23. K. Frischwasser, I. Yulevich, V. Kleiner, E. Hasman,

Opt. Express 19, 23475 (2011).24. See supplementary materials on Science Online.25. D. Grohol et al., Nat. Mater. 4, 323 (2005).26. R. Moessner, A. P. Ramirez, Phys. Today 59, 24 (2006).27. W. Schweika, M. Valldor, P. Lemmens, Phys. Rev. Lett. 98,

067201 (2007).28. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff,

Nature 391, 667 (1998).29. E. Collett, Polarized Light: Fundamentals and Applications

(Marcel Dekker, New York, 1993).30. N. Yu et al., Science 334, 333 (2011).31. E. S. Barnard, R. A. Pala, M. L. Brongersma, Nat. Nanotechnol.

6, 588 (2011).32. A. Schiffrin et al., Nature 493, 70 (2013).

Acknowledgments: This research was supported by the IsraelScience Foundation.

Supplementary Materialswww.sciencemag.org/cgi/content/full/340/6133/724/DC1Materials and MethodsSupplementary TextFigs. S1 to S6References (33–37)

7 January 2013; accepted 13 March 201310.1126/science.1234892

Fig. 3. Symmetryanalysisbyspin-projectedmeasure-ments. (A and B) MeasuredS3 dispersions of the

ffiffiffi3

p!ffiffiffi

3p

structure with varying ϕat q = 12° and –12°, respec-tively. The inset highlightsthe IS directions. (C and D)Measured and calculatedintensity dispersions of theffiffiffi3

p!

ffiffiffi3

pstructure along

an IS direction of ϕ = 30°,respectively. The standardmomentum-matching andthe SOMM calculations in(D) are denoted by the yellowand green lines, respectively.

Fig. 2. Spin-orbitmomentum-matching and surface-wavecontrol. (A and B) Calcu-lated S3 dispersions of theffiffiffi3

p!

ffiffiffi3

pstructure at ϕ =

0° and 60°, respectively, viathe SOMM condition. Red andblue lines correspond to s+and s– spin states, respective-ly. (Inset) Reciprocal space ofq = 0 (yellow) and

ffiffiffi3

p!

ffiffiffi3

p

(blue) structures with the cor-responding reciprocal vectors.The dashed blue arrow in-dicates that only one of theffiffiffi3

p!

ffiffiffi3

preciprocal vectors

is required to set the disper-sion of a spin-optical meta-material, in addition to bothof the q = 0 reciprocal vec-tors. (C andD) Spin-controlledsurface-wave concept. Thescheme introduces the cou-pling of the two spin-dependentmodes depicted in (A) tosurface-wave modes via thedegree of freedoms of w,kjjin, and s; k

jjin is the compo-

nent of the incident wavevector kin parallel to the surface. (E and F) Vector summation representation of the SOMM conditionforecasting the direction of the surface wave for the aforementioned modes.

+

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The most recent extensions

Xiao et al., Nature Commun. 2015

The subwavelength confinement of surface plasmon polar-iton (SPP) has been revolutionizing the way we controllight at the nanoscale and holds promise for future optical

information technology and optoelectronics1–3. Couplingpropagating light to SPPs is therefore a primary interest forboth fundamental studies and practical on-chip applications4.Prisms, holographic gratings and tailor-made plasmonic particlesare commonly used for this purpose to compensate themomentum mismatch between SPPs and propagation waves5–9.However, these conventional methods usually provide a limiteddynamic tunability unless sophisticated electrical or optical tuningon the material and geometrical parameters is employed9–13. Onthe other hand, many attentions have been paid to the spin–orbitinteraction of light using geometric-phase-enabled opticaland plasmonic systems14–21. Together with the recent develop-ments of resonator-based22–27 and geometric-phase-enabledmetasurfaces28–33, it offers an alternative route to excite SPPsthrough spin–orbit interaction with opposite geometric phases forthe two spins. The associated spin-dependent phenomena can beregarded as the optical spin-Hall effect (OSHE)34–40 in a moregeneral context about spin splitting of orbitals18,32,33,41–43. Forexample, a flip of the incident spin (circular polarization) cancause a split of beam displacement18,42, or a reverse inpropagation direction32,33 of a propagating SPP. However, thetime-reversely related SPP profiles, from the opposite geometricphases, generated with the two normal incident spins in thesecases have so far only demonstrated simple and symmetricsplitting of the two spins, known as OSHE. Without a propergeometric phase design scheme, the generated SPP profiles fromOSHE are far from arbitrary and independent for the two spins. Itrefrains us to fully exploit the potential of OSHE and to allow thetwo spins to work cooperatively in a flexible manner.

Here we demonstrate coherent and independent control of SPPorbitals for the two opposite spins using multiple rings of nano-slots with properly designed orientations on a metasurface. Thesecontrols range from generating different focal spots to generatingindependent complicated profiles for the two spins. This is madepossible in this work by establishing a geometric phase matchingscheme, which gives the orientation profile of the nano-slots fromthe superimposition of the target SPP profiles for the two spins.This scheme provides us to achieve arbitrary OSHE. Resultingfrom this independence of spin splitting, we further demonstratethe two opposite spins can cooperate with each other. Forexample, we can dynamically tune the phases and amplitudes ofthe designated SPP orbitals. Such coherent control can furtherprovide us the capability to assemble a series of individuallydesigned ‘time’ frames as a motion picture being played back byrotating the linear polarization of the incident light. This is a formof spin-enabled coherent control44–46 and provides a unique wayin achieving tunable orbital motions in plasmonics. For example,it can be used as tip-free near-field scanning optical microscopy47,polarization-steering plasmonic tweezers48, and coherent inputsof SPP devices (coherent logic gate, transistor, etc.)49.

ResultsGeometric phase design scheme. Figure 1 shows our metasurfaceplatform. It consists of two silver films separated by a dielectricspacer. An array of nano-slots on the upper film is etched withspecific orientation profile a(x,y) (inset of Fig. 1a). A semi-conductor laser at 1,064 nm is normally shined on the metasur-face to generate a target SPP profile on the air-metal interface. Fora left/right-handed circular polarization (LCP/RCP) incidence,each nano-slot reradiates as an electric dipole carrying an addi-tional geometric phase ±2a (refs 27–33) in its cross-polarizationradiation. By requiring constructive interference in building up a

target SPP profile (see the derivation in Supplementary Note 1),we obtain the following function to design the a profile:

f! EzÞ ¼ !12

arg @xEz ! i@yEz! "

þ constant:#

ð1Þ

For a target SPP profile E!z , we can use a(x,y)¼ fþ E þz! "

for LCPincidence or equivalently a(x,y)¼ f& E&z

! "for RCP incidence.

This immediately imposes a usual restriction on the input targetSPP orbital of the two spins: Ez

þ ¼ E&z! "' for the thin layer of

plasmonic particles to generate the same set of orientation profile.This time-reversal relationship actually comes from the oppositesigns of the geometric phases for the two spins (the geometricorigin of OSHE). Therefore, the two orbital profiles of differentincident spins cannot be specified independently. We call it thedirect scheme (Fig. 1a).

On the other hand, if we are only interested in generating SPPprofiles within a slot-free region (the target region inside the ringof particles in Fig. 1), an arbitrary SPP profile without radiationinto this region can be added to the input argument ofequation (1). The orientation profile becomes different but stillgenerates the same SPP profile. For a particular spin, if wedecompose the SPP profile in the target region into inwardand outward radiating parts. For example, a point-like standingwave J0 kSP rj jð Þ ¼ ðH 1ð Þ

0 kSP rj jð ÞþH 2ð Þ0 kSP rj jð ÞÞ=2, we only need to

ensure the inward radiating part matches to the correspondinginward radiating part of the input argument Ez to equation (1). Inother words, the information capacity carried by the orientationprofile of the nano-slots is far from completely exhausted. Thisredundant information capacity by considering generating SPP inthe target region can then be used for generating another SPP

! = f + (E+z) = f _ (E –

z ) with E –z = (E +

z)* ! = f + (E+z + (E –

z )*) = f _ (E –z + (E +

z)*)

E –z

E –z

E +z

E +z

yv

y

u!

a

x

x

1,064 nm laser

Ag

AgLiiNbO

3 substrate

MgF2

100X/1.4Oil objective

"

w

Targetregion

a b

c d

Figure 1 | Arbitrary OSHE of SPP. (a,b) Identical nano-slots withorientation profile a(x,y) on the x& y plane with u, v as the local principalaxes of each particle. Light is incident normally on the surface. a can bedesigned by (a) direct substitution of the target SPP orbital Ez

þ/Ez& (for LCP

and RCP incident wave) into equation (1) with restriction Eþz ¼ E&z! "'

; or

(b) the modified matching rule equation (2) employed in this work to gainindependent control of local SPP orbitals generated within the ring of nano-slots. (c) Top-view scanning electron microscopy (SEM) image ofexperiment sample. The sample is with geometrical parameters defined asl¼ 500 nm, w¼ 50 nm and a¼ 706 nm. Scale bars, 4mm in (c).(d) Schematics of the experimental set-up.

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms9360

2 NATURE COMMUNICATIONS | 6:8360 | DOI: 10.1038/ncomms9360 | www.nature.com/naturecommunications

& 2015 Macmillan Publishers Limited. All rights reserved.

linear polarization angle f of the incident light. Here the orbitalsfor both spins are set to same one: a focal spot withE!z ¼ H 2ð Þ

0 kSP rj jð Þ. The generated SPP profile becomescos fð ÞJ0 kSP rj jð Þ with linear polarization x cosfþ y sinf. Again,we use the modified scheme to design and fabricate the sampleshown in Fig. 3a. Full wave simulations and experimentsare performed with varying f in Fig. 3c–h. The measuredintensity of the orbital (integrated around the focus spot)gradually varies from maximum to zero by tuning f from 0! to90!, with theoretical prediction cos2(f) shown in Fig. 3b. In fact,we can further tune the maximum intensity to occur at apolarization-selective angle x, by multiplying e!ix on the LCPand RCP target profiles. The generated SPP standing wavein the target region becomes cos f& xð ÞJ0 kSP rj jð Þ. This amplitudecontrol (with x) constitutes a single ‘pixel’ for furtherconstruction.

Such coherent control of orbitals including phase andamplitude can further allow us to design individual pictureframes, which will only light up at different and designatedpolarization angle f. By using geometric phases from the same setof atoms, we can construct motion pictures by tuning the incidentpolarization. Here we take an array of these pixels for illustration,and instruct the light to write an alphabet when the linearpolarization angle f is rotated. We set the target SPP profile as

E!z ¼XN

i¼1

e! ixi cos y& yið ÞH 2ð Þ1 kSP r& rij jð Þ; ð3Þ

where xi¼fmax(i& 1)/(N& 1) linearly increases along the path ofpixels with index i¼ 1 to N (Fig. 4a). It lights up the pixels insequence with an example of letter ‘b’ (Fig. 4b). Each pixel is adipole-like focal spot with its direction yi aligned with the strokejoining successive pixels. To illustrate the dynamic behaviour, weplot the ‘time’ frames of the simulated and measured SPP profileswith different values of f. As f increases, the profiles have theirmaximum intensity tracing the letter ‘b’ in the counter clock-wise

direction as designed (Fig. 4c–l). On the other hand, a ‘static’picture of the letter ‘b’ is revealed by a RCP incidence (Fig. 5a,b).The polarization-selectivity at each pixel is lost in such a case. Thetarget orbitals for both spins have the same amplitude (staticpicture) designed from equation (3) while the subtle difference in

–10 100 –10 0x / !SP x / !SP

c

e

0

y / !

SP

10

0

y / !

SP

d

f

–10

10

–10

LCP/RCP

10 –15 0 x / !SP

15 –15 150x / !SP

–15

15

0

15

15

0

b a

LCP RCP

g

i

k

j

l

h

LCP RCP

LCP/RCP

1.0

0.0 0.2 0.4 0.6 0.8

m = 0 m = 1

"i

|E |2

Figure 2 | Arbitrary spin-Hall effect. (a) Arbitrary spin splitting: LCP orbital to focus to two spots (red) while RCP orbital to focus a single spot (blue) (seetext for detailed specification). (b) Top-view SEM image of the fabricated sample: a ring of nano-slots (with radii from 6.6lSP to 10lSP, embedded with a

square array of particles separated by a¼ 2lSP/3). (c–f) are simulated Ezj j2 and experimental intensity profiles for LCP and RCP incidence. (g) Arbitraryspin-dependent SPP profiles with more complicated pattern: shining LCP generating a triangle pattern (red) and shining RCP generating a cross pattern

(blue). (h) Top-view SEM image of the fabricated sample (with radii from 10lSP to 13.3 lSP and a¼ 2lSP/3). (i–l) simulated Ezj j2 and experimental intensityprofiles for LCP and RCP incidence. Here we use same colour scales for different incident CP. Scale bars, 4mm in b,h respectively.

90 0

Theory Expt.

1

0

0 y

/ !S

P

10

0

y / !

SP

–10

10

–10 –10 100 –10 100

x / !SP x / !SP

–10 100x / !SP

LCP/RCP

c e d

f

a b

g h

1.0

0.0 0.2 0.4 0.6 0.8

#

#(°)

⎜Ez⎜2

(a.u

.)

|E |2

Figure 3 | Coherent control of amplitude of a single orbital by varying /.(a) Target SPPs for LCP and RCP orbitals both focus at (0,0). Scale bars,

4 mm in a. (b) Predicted (curve) Ezj j2 and measured intensity of SPP profiles

(symbols) at focus spot with varying f. (c–h) are the simulated Ezj j2andmeasured intensity profiles with varying f (white arrows). Here we usesame colour scale for SPP patterns with different incident polarizationangles.

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms9360

4 NATURE COMMUNICATIONS | 6:8360 | DOI: 10.1038/ncomms9360 | www.nature.com/naturecommunications


phases at the pixels, storing the time sequence information, is acomplicated version of coherent intensity control between thestatic pictures from the two spins. Apart from a letter ‘b’, a letter‘O’, a letter ‘N’ and a letter ‘U’ are also designed and fabricated forfurther illustrations, the simulated and measured static pictures ofwhich are presented in Fig. 5c–h for RCP incidence. Themeasured RMSD are all below 0.2 with correlation coefficientaround 0.5, which show a clear correlation with the designed SPPpattern. Their motion pictures are shown in SupplementaryFigs 9–11. Both static and motion pictures faithfully realize thedesigned sequence in writing the letters.

DiscussionIn the present work, we have established a generic geometric phasescheme for SPP generation with geometric phase elementson a metasurface. This scheme provides us the freedom toindependently control both the amplitude and phase of local SPPorbitals by two opposite spins. With this freedom, we havedemonstrated arbitrary plasmonic spin-Hall effect not only aboutsplitting of SPP orbitals but also about generating differentcontrollable shapes of SPP profiles. We note that such arbitrari-ness in SPP orbital control (independence for two incident spins,with arbitrary amplitude and phase control at the same time) isprovided by the spin nature of CP light together with our thegeometric phase matching scheme (equations (1–3)). The usage ofCP light allows us to avoid the so called ‘amplitude spatial-dispersion’ problem of nano-slot arrays that the transmittedamplitude and phase though such nano-slots cannot beindependently controlled. Although the V-shape and C-shapeantennas22,23,27, can resolve such locked amplitude and phaseproblem with linear polarized incident light, the generated profilesfor the two linear polarization in this case will then be unavoidablydependent on each other so that we lost the independence of thegenerated patterns between the two orthogonal polarizations.

a

1

LCP/RCP b

0

y / !

SP

15

0

y / !

SP

–15 15

–15 –15 150 –15 150

x / !SP x / !SP

–15 150 –15 150x / !SP x / !SP

–15 150x / !SP

c

h

d

i

e f g

j k l

1.0

0.0 0.2 0.4 0.6 0.8

" = 0°

" = "max

ii

N

#i = "max (i – 1) / (N – 1) "

|E |2

Figure 4 | Motion pictures played by polarization rotation angle. (a) The scheme of Motion pictures, which is consisted by N pixels. Polarization-selective angle xi increases along the path of pixels. (b) SEM image of fabricated sample of moving letter ‘b’, (with fmax¼ 120! and ring radii from 13.3lSP to

16.7lSP). Scale bars, 10mm in b. (c–l) Are the simulated Ezj j2 and measured intensity profiles with varying f from 0! to 120! (white arrows). The colourscales of SPP patterns with different incident polarization angle are the same.

0

y / !

SP

15

0

y / !

SP

–15 15

–15

–15 150 –15 150x / !SP x / !SP

d c

f e

0

15

–15

y / !

SP

b a

y / !

SP

0

15

–15

h g

1.0

0.0 0.2 0.4 0.6 0.8

|E |2

Figure 5 | The static pictures of motion pictures. (a,b) letter ‘b’,(c,d) letter ‘O’, (e,f) letter ‘N’, and (g,h), letter ‘U’ with RCP incidence are

the simulated Ezj j2 profile/the measured intensity profile. The measuredperformance merit RMSD (PPM) are 0.15 (0.42) for letter ‘b’, 0.17 (0.53)for letter ‘O’, 0.16 (0.56) for letter ‘N’ and 0.13 (0.53) for letter ‘U’,respectively.

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms9360 ARTICLE

NATURE COMMUNICATIONS | 6:8360 | DOI: 10.1038/ncomms9360 | www.nature.com/naturecommunications 5


The most recent extensions

Maguid et al., Science 2016 (Hasman)

APPLIED OPTICS

Photonic spin-controlledmultifunctional shared-apertureantenna arrayElhanan Maguid,1* Igor Yulevich,1* Dekel Veksler,1* Vladimir Kleiner,1

Mark L. Brongersma,2 Erez Hasman1†

The shared-aperture phased antenna array developed in the field of radar applications is apromising approach for increased functionality in photonics.The alliance between theshared-aperture concepts and the geometric phase phenomenon arising from spin-orbitinteraction provides a route to implement photonic spin-control multifunctionalmetasurfaces.We adopted a thinning technique within the shared-aperture synthesisand investigated interleaved sparse nanoantenna matrices and the spin-enabledasymmetric harmonic response to achieve helicity-controlled multiple structuredwavefronts such as vortex beams carrying orbital angular momentum.We used multiplexedgeometric phase profiles to simultaneously measure spectrum characteristics and thepolarization state of light, enabling integrated on-chip spectropolarimetric analysis.The shared-aperture metasurface platform opens a pathway to novel types of nanophotonicfunctionality.

Multifunctional planar systems that canperform a number of concurrent tasksin a shared aperture were recently intro-duced in the field of phased-array an-tennas for radar applications. In particular,

modern radio frequency sensing, communication,and imaging systems based on sparse phased-array antennas are able to operate at different

frequency bands, polarizations, scanning directions,etc. These capabilities are ascribed to subarraysof elementary radiators, sharing a common phys-ical area that constitutes the complete apertureof the phased array (1–4). The shared-apertureconcept can be adopted to realize multifunctionalphotonic antenna arrays. The recent implementa-tion of metasurfaces, or metamaterials of reduced

dimensionality, is of particular relevance, as itopens up new opportunities to acquire virtuallyflat optics. Metasurfaces consist of a dense ar-rangement of resonant optical antennas on ascale smaller than the wavelength of externalstimuli (5–10). The resonant nature of the light-matter interaction of an individual nanoantennaaffords substantial control over the local light scat-tering properties. Specifically, the local phase pick-up can be manipulated by tailoring the antennamaterial, size, and shape, as well as environment–antenna resonance shaping (11–15), or throughthe geometric phase concept (10, 16–20). The latterconcept is an efficient approach for achievingspin-controlled phase modulation, whereas thephoton spin is associated with the intrinsic angularmomentum of light (21). Such a geometric phasemetasurface (GPM) transforms an incident circularlypolarized light into a beam of opposite helicity,imprinted with a geometric phase fgðx; yÞ ¼2sTqðx; yÞ, where sT ¼ T1 denotes the polarizationhelicity [photon spin in ħ units (ħ is Planck’sconstant h divided by 2p)] of the incident light,and q(x, y) is the nanoantenna orientation (22).Here we incorporate the shared-aperture phasedantenna array concept and spin-controlled two-dimensional (2D) optics based on nanoantennas.By use of the geometric phase mechanism, the

1202 3 JUNE 2016 • VOL 352 ISSUE 6290 sciencemag.org SCIENCE

1Micro and Nanooptics Laboratory, Faculty of MechanicalEngineering, and Russell Berrie Nanotechnology Institute,Technion – Israel Institute of Technology, Haifa 32000,Israel. 2Geballe Laboratory for Advanced Materials, StanfordUniversity, 476 Lomita Mall, Stanford, CA 94305, USA.*These authors contributed equally to this work. †Correspondingauthor. Email: [email protected]

Fig. 1. Schematic of shared-aperture concepts. (A to C) Segmented (A), interleaved (B), and HR (C) 1D phased arrays. (D to F) Schematic far-fieldintensity distribution of wavefronts with positive (red) and negative (blue) helicities emerging from segmented (D), interleaved (E), and HR (F) GPMscomposed of gap-plasmon nanoantennas (inset). Here, l denotes the topological charge of the spin-dependent OAM wavefronts [for more details,see (22)].

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spin-orbit interaction phenomenon, a multifunc-tional GPM, is implemented (Fig. 1).Photonic GPMs with spatially interleaved phase

profiles are designed by the random mixing ofoptical nanoantenna subarrays, where each sub-array provides a different phase function in aspin-dependent manner. This approach makesuse of the peculiar ability of random patterns toyield extraordinary information capacity andpolarization helicity control via the geometricphase concept. We realized a high-efficiencymultifunctional GPM in which the phase andpolarization distributions of each wavefront are

independently controlled. These GPMs are basedon gap-plasmon resonator (GPR) nanoantennasthat consist of metal-insulator-metal layers thatenable high reflectivity by increasing the couplingbetween the free wave and the fundamental res-onator mode. Moreover, adjustment of the GPRnanoantenna’s dimensions enables the design ofa high-efficiency half-wave plate (22, 23). For thereflection mode of a single wavefront, we found adiffraction efficiency of ~79%, which is in goodagreement with the theory (10, 18, 19) and thefinite-difference time-domain (FDTD) simulation(fig. S1, D to F) (22). Then, to design a metasurface

generating multiple spin-dependent wavefrontscarrying orbital angular momentum (OAM), theGPR nanoantennas were randomly distributed intoequal interleaved subarrays. The nanoantennasof each jth subarray were oriented accordingto the relation 2qjðx; yÞ ¼ kjx þ ljϕ, to obtainmomentum redirection sTkj and topological charge(winding number) sTlj; ϕ is the azimuthal angle(Fig. 2A). We measured the far-field intensity dis-tribution by illuminating the metasurface withcircularly polarized light at a wavelength of 760 nm,and we observed three-by-two spin-dependentOAM wavefronts with the desired topological

SCIENCE sciencemag.org 3 JUNE 2016 • VOL 352 ISSUE 6290 1203

Fig. 2. Multiple wavefront shaping via interleaved GPMs. (A) Scanning elec-tronmicroscope image of a gap-plasmon GPM of apertureD = 50 mm. (B andC)Measured spin-flip momentum deviation of three wavefronts with differentOAMs. sT denotes the incident spin. k0 is the wave number of the incident beam.(D and E) Interference pattern of the spin-flipped components with a plane wave,observed fromdifferent GPMsgeneratingOAMwavefronts of l=±1 (D) and l=±2(E), for s− and sþ illumination. (F) Observed angular width of plane waves emer-

ging from shared-aperture GPMs of different types and numbers of generatedchannels, corresponding to different colors. a.u., arbitrary units; rad, radians. (G)Measured momentum deviations for the interleaved and segmented GPMs,wherein the intensity distributions of nine channels in (F) are presented along thedashed colored lines. (H and I) Efficiency per channel (H) and number of bits perchannel (I). Red dots and black triangles denote the calculation and experimentresults, respectively.The blue lines correspond to ~1/N2.

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reported spin-controlled multifunctional meta-surfaces based on shared-aperture approachesare studied in terms of information capacity bymeans of Wigner phase-space distribution toestablish a link between the Shannon entropyand the capacity of photonic system (fig. S8) (22).Spectropolarimetry is an application for the

simultaneous measurement of the spectrum andpolarization state of light and is widely used tocharacterize chemical compounds. Implementingspectropolarimetric devices with metasurfacesopens new avenues for on-chip detection. Re-cently, on-chip chiroptical spectroscopy (27) waspresented, measuring the differential absorp-tion between left- and right-circular polariza-tions. The device used in this experiment isunable to provide full polarization analysis. Fur-thermore, it has been shown that a metasurfaceallows for the simultaneous determination ofthe Stokes parameters (i.e., the polarization stateof light) (28, 29) without consideration of thespectral analysis. Here we propose a simple, fast,and compact technique, based on an interleavedGPM, that can simultaneously characterize thepolarization state and spectrum of a wave trans-mitted through a semitransparent object. Thespectropolarimeter metasurface (SPM) is com-posed of three interleaved linear phase profilesðfð1Þg ¼ sTkxx; f

ð2Þg ¼ sTkyy; f

ð3Þg ¼ sTkyyÞ asso-

ciated with different nanoantenna subarrays.When a probe beam with an arbitrary polari-

zation state impinges on the SPM, two beamsof intensities IsT , consisting of opposite helicitystates, and two additional beams emerge. Thelatter have identical polarizations, conjugatewith respect to the incident beam, and beingprojected onto linear polarizers at 0° and 45°determines the linearly polarized componentsIL0 and IL45, respectively (Fig. 4A). The Stokesparameters of the probed beam are then cal-culated by the expressions (28) S0 ¼ 2ðIsþþIs− Þ=h, S1 ¼ 2IL0=h − S0, S2 ¼ 2IL45=h − S0,andS3 ¼ 2ðIsþ− Is− Þ=h, where the coefficient (h) isdetermined by a calibration experiment (fig. S9).The SPM of 50-mm diameter was normally il-luminated by a supercontinuum light source,passing through the acousto-optic modulator,which enables the tenability of various wave-lengths. Figure 4B shows the measured andcalculated Stokes parameters on a Poincarésphere for an analyzed beam impinging theSPM at a wavelength of 760 nm with differentpolarizations, obtained by a polarization-stategenerator (linear polarizer followed by a rotatedquarter-wave plate). The spectral resolving poweris defined according to the Rayleigh criterionas l=Dl ¼ q=Mx;y

2, where Mx;y2 is the beam

quality parameter in each direction (x, y) and qis the number of phase-modulation periods (30).The resolving power of the SPM was measuredand found to be l=Dl≅13 (Fig. 4C). The obtainedvalue is in good agreement with the calculated

resolving power for SPM with q ≅ 17 periodsand a laser beam quality ofMx;y

2 ¼ 1:3. Wemea-sured the optical rotatory dispersion (ORD) forthe specific rotations of D-glucose (chiral mole-cule) and its enantiomer L-glucose that weredissolved in water with predetermined concen-trations (Fig. 4D) (22). The ORD of D-glucoseshows a good agreement with the values foundin the literature (31), whereas the L-glucose ORDis manifested by the opposite behavior as ex-pected. We achieved real-time on-chip spectro-polarimetry by exploiting the interleaved GPM.The reported alliance of the spin-enabled geo-metric phase and shared-aperture concepts shedslight on the multifunctional wavefront manip-ulation in a spin-dependent manner. The intro-duced asymmetric HR and interleaved GPMsconstitute suitable candidates to realize on-demand multifunctional on-chip photonics.

REFERENCES AND NOTES

1. D. M. Pozar, S. D. Targonski, IEEE Trans. Antenn. Propag. 49,150–157 (2001).

2. R. L. Haupt, IEEE Trans. Antenn. Propag. 53, 2858–2864(2005).

3. A. M. Sayeed, V. Raghavan, IEEE J. Sel. Top. Signal Process. 1,156–166 (2007).

4. I. E. Lager, C. Trampuz, M. Simeoni, L. P. Ligthart, IEEE Trans.Antenn. Propag. 57, 2486–2490 (2009).

5. A. Papakostas et al., Phys. Rev. Lett. 90, 107404 (2003).6. X. Yin, Z. Ye, J. Rho, Y. Wang, X. Zhang, Science 339,

1405–1407 (2013).

SCIENCE sciencemag.org 3 JUNE 2016 • VOL 352 ISSUE 6290 1205

Fig. 4. Spectropolarimeter metasurface. (A) Sche-matic setup of the spectropolarimeter. The SPM is illu-minated by a continuum light passing through a cuvettewith chemical solvent, then four beams of intensities Isþ ,Is− , IL45, and IL0 are reflected toward a charge-coupleddevice camera. (B) Predicted (red dashed curve) andmeasured (blue circles) polarization states, obtained by apolarization-state generator (a linear polarizer followedby a rotated quarter-wave plate), depicted on a Poincarésphere. (C) Measured far-field intensities for ellipticalpolarization at two spectral lines (with wavelengths of740 and 780 nm) and (inset) the corresponding resolvingpower (black line) and calculation (blue line) of the 50-μmdiameter SPM. (D) ORD for the specific rotations of D- andL-glucose. Black squares and red circles represent themeasured ORD of D- and L-glucose, respectively.Theblueline depicts the dispersion acquired from (31).

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