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Spin-to-Orbital Angular Momentum Mapping ofPolychromatic Light

Mushegh Rafayelyan, Etienne Brasselet

To cite this version:Mushegh Rafayelyan, Etienne Brasselet. Spin-to-Orbital Angular Momentum Mapping of Poly-chromatic Light. Physical Review Letters, American Physical Society, 2018, 120 (21), pp.213903.�10.1103/PhysRevLett.120.213903�. �hal-01804149�

https://hal.archives-ouvertes.fr/hal-01804149https://hal.archives-ouvertes.fr

Spin-to-Orbital Angular Momentum Mapping of Polychromatic Light

Mushegh Rafayelyan and Etienne Brasselet*

Univ. Bordeaux, CNRS, LOMA, UMR 5798, F-33400 Talence, France

(Received 4 July 2017; published 25 May 2018)

Reflective geometric phase flat optics made from chiral anisotropic media recently unveiled a promisingroute towards polychromatic beam shaping. However, these broadband benefits are strongly mitigated bythe fact that flipping the incident helicity does not ensure geometric phase reversal. Here we overcome thisfundamental limitation by a simple and robust add-on whose advantages are emphasized in the context ofspin-to-orbital angular momentum mapping.

DOI: 10.1103/PhysRevLett.120.213903

Since the invention of thin lenses by Fresnel in the 1820sin the framework of lighthouse optics [1], tremendousdevelopment of flat optics has taken place. Nowadays, thedesign of subwavelength-thin optical elements enablingalmost arbitrary intensity, phase, and polarization trans-formations is well established and nanofabrication toolsmade possible their realization [2–4]. In this context, thedevelopment of geometric phase optical elements hasaroused great interest. Indeed, two decades after theproposal by Bhandari [5], which was followed by firstexperimental realizations in the early 2000s [6,7], now suchoptical elements are products of the photonics industry andthe underlying physics relying on spin-orbit interactions oflight [8] is well understood.The principle of such devices is grasped by considering a

transparent slab of uniaxial medium with space-variant in-plane optical axis orientation angle ψðx; yÞ and uniformhalf-wave birefringent phase retardation. As shown byBhandari [5], an incident circularly polarized plane wavepropagating along the z axis emerges from the sample withthe orthogonal circular polarization state and additionalgeometric phase factor proportional to σψðx; yÞ whereσ ¼ $1 is the helicity of the incident field. The geometricalinformation ψðx; yÞ can thus be transferred to the wavefront of the output light field and reversed on demand byflipping the incident helicity (σ → −σ). Since the half-waveretardation condition is satisfied for a specific wavelength,these geometric phase optical elements are monochromaticby design.A few designs have been proposed to achieve achromatic

transmissive [9–11] or reflective [12] geometric phaseshaping though being restricted to 4π-helical phase bulkyshapers with poor on-axis spatial resolution. Still, high-resolution transmissive flat optics with enhanced poly-chromatic behavior can be achieved by introducing amultitwisted anisotropic structure [13]. Recently, a simpleroption has been proposed [14], which combines the twistednature of chiral liquid crystals (i.e., cholesterics) that maylead to circular Bragg reflection [15] with well-developed

liquid crystal surface patterning technology [16,17]. Thegeometric (Berry) phase nature of such (Bragg) reflectiveoptical elements made of chiral and anisotropic media hasbeen revealed independently in Refs. [18,19]. In particular,polychromatic optical vortex beam generation using“Bragg-Berry” mirrors [20–24] unveiled a promisingoption. However, as previously noticed in Ref. [20], flip-ping the incident helicity does not ensure geometric phasereversal, which is a major drawback. Here we report on asimple and robust way to solve this problem with alteringneither the polychromaticity nor the rotational invariance,in contrast to what happens at large incidence angle[20–22]. This is done by using a rear mirror [25], seeFig. 1(a), which displays a cholesteric slab sandwichedbetween a glass substrate (at z ¼ 0) and a standard mirror(at z ¼ L) that are associated with boundary layers provid-ing identical in-plane liquid crystal orientation patternsgiven by the spatially varying angle ψ surfðx; yÞ. Thedemonstration is made in the context of spin-to-orbitalangular momentum mapping, which covers various aspectsof classical and quantum optics [26], and for which thepolychromatic option is not available to date.Cholesterics are characterized by a helical structure with

pitch p that is the distance over which the local liquidcrystal optical axis rotates by 2π around the z axis and byhandedness χ ¼ $1; see Fig. 1(b). From the optics ofuniform cholesterics [i.e., ψ surfðx; yÞ ¼ ψ0 with ψ0 con-stant], light propagating along the helix axis experienceshelicity-preserved Bragg reflection for the incident helicityσ ¼ −χ. This is associated with a photonic band gapdefined by pn⊥ < λ < pnk, nk;⊥ being the refractive indexalong and perpendicular to the liquid crystal director[29,30]. On the other hand, light with σ ¼ þχ is trans-mitted through the cholesteric. In our case, such light isreflected by the rear mirror, which flips the helicity.Backward light is thus Bragg reflected without helicitychange by the back side of the cholesteric. Ensuinghelicity-flipped reflection from the rear mirror is eventuallytransmitted through the cholesteric at fixed helicity. The

PHYSICAL REVIEW LETTERS 120, 213903 (2018)

0031-9007=18=120(21)=213903(5) 213903-1 © 2018 American Physical Society

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process is sketched in Fig. 1(c) and discussed in Ref. [31].Arbitrarily polarized light is thus fully Bragg reflected foroptical frequencies inside the band gap; see Fig. 1(c).Outside the band gap, only a fraction of the incident light isBragg reflected, see Fig. 1(c), while the rest is reflected bythe structure with flipped helicity, as depicted in Fig. 1(d).The geometric Berry phase of Bragg reflected light,ΦBB,

strongly depends on the incident helicity, wavelength, andcholesteric orientational boundary conditions, see Fig. 2.However, the differential geometric phase ΔΦBB ¼ΦBBðψ0Þ −ΦBBð0Þ does not depend on the wavelength.Namely, ΔΦBB ¼ 2σψ0, and the generalization to space-variant boundary conditions gives ΔΦBB ¼ 2σψ surfðx; yÞ,which is the main result of this work.Experimentally, this is investigated by preparing a

10 μm-thick Bragg-Berry optical element made of thechiral liquid crystal material SLC79 (BEAM Co). Thechiral film is sandwiched between two glass substratescoated by an azobenzene-based surface-alignment layerproviding a q-plate design for the orientational boundaryconditions [32]. Namely, ψ surf ¼ qφ, with q ¼ 1 andφ ¼ arctanðy=xÞ. The cholesteric pitch is adjusted toprovide a Bragg reflection spectrum centered aroundλ≳ 530 nm and the photonic band gap width is measuredto be ≃75 nm. Without the back mirror add-on, this opticalelement behaves as a polychromatic Bragg-Berry vortexgenerator (BBVG) for σ ¼ −χ only, hence loosing all the

assets of a q plate [26,32], as shown in the left part of Fig. 3.In this figure, the direct observation of the reflectedintensity profile under incident helicity σ ¼ $χ and vari-ous spectral cases are reported. In contrast, the latter issue issolved in the presence of the back mirror, as shown in theright part of Fig. 3. This supports the creation of a Bragg-Berry q plate (BBQP) allowing broadband spin-to-orbitalangular momentum mapping.First, we recall the useful formalism of high-order

Poincaré spheres of integer order l [33], Pl; see Fig. 4.Each point on the sphere corresponds to a unique opticalstate Ψl having two degrees of freedom,

Ψlða; bÞ ¼ ae−ilφcþ þ beþilφc−; ð1Þ

(a)

(b)

FIG. 2. Bragg reflected geometric phase spectra for the sameparameters as in Fig. 1 for incident helicity σ ¼ ∓χ (a), (b).

(a)

(c) (d)

(b)

FIG. 1. (a) Generic mirror-backed Bragg-Berry optical element.(b) Helical cholesteric ordering for χ ¼ þ1. (c) Calculatedhelicity-independent reflectance spectra and principle of thehelicity-controlled geometric phase for a uniform surface align-ment ψ surf ¼ ψ0, for Bragg reflected light. (d) Same as in panel(c) for non-Bragg reflected light. Simulations are made using4 × 4 Berreman formalism [27,28] at normal incidence, withnglass ¼ 1.52, nk¼1.72, n⊥¼1.42, p¼340 nm, χ¼−1, L¼10p,d ¼ 0 and assuming a perfect mirror.

FIG. 3. Experimental comparison of the helicity-dependentvortex beam generation between BBVG and BBQP ( χ ¼ −1,q ¼ 1). Images refer to the reflected beam intensity at near-normal incidence (1° incidence angle) collected at ∼1 m from theelement illuminated by a circularly polarized collimated super-continuum laser beam with helicity σ, diameter ∼1 mm, withinterference filters centered on λ1 ¼ 500 and λ2 ¼ 550 nm with10 nm full width at half maximum transmission spectrum, or not(fλg). For each pair of panels, the images are recorded at fixedacquisition time for the camera. Insets refer to enhanced-signalimages to stress the absence of geometric phase shaping of theresidual Fresnel reflection from the BBVG glass substrate.

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where (a, b) are complex numbers, jaj2 þ jbj2 ¼ 1, andc$ ¼ ðx$ iyÞ=

ffiffiffi2

prefers to the circular polarization unit

vectors in the Cartesian unit basis (x, y, z). The mapping ofthe Poincaré sphere of polarization to high-order Poincarésphere, namely, P0 → Pl, is actually achieved inside thephotonic band gap of a BBQP with handedness χ of order qaccording to the unitary transformation

Ψ0ða; bÞ ⟶BBQP

χ;qΨ2qðbeiΦ1 ; aeiΦ2Þ; ð2Þ

where constants Φ1;2 depend on λ, material properties andconsidered geometry. Isomorphisms between ϕ and Φ1;2ensure the full mapping between P0 and P2q.Without lack of generality, experiments are made using a

collimated Gaussian laser beam with λ ¼ 532 nm anddiameter ∼1 mm following the setup of Fig. 5. Thereflected beam is analyzed by direct polarimetry (i.e.,without reference beam) and by interferential measure-ments (i.e., with coaxial reference beam). The predictedmapping given by Eq. (2) is illustrated by analyzing thereflected far field for a representative set of incident states,namely, ΨðiÞ0 ¼ ðS0; N0; H0; D0; V0; A0Þ. The case of inci-dent circularly polarized light is summarized in Fig. 6where the top or bottom rows refer to ΨðiÞ0 ¼ ðS0; N0Þ. Theobserved doughnut intensity profile is the result of on-axisoptical phase singularity. Maximum [Figs. 6(c) and 6(h)]and minimum [Figs. 6(d) and 6(g)] visibility of theinterference pattern, respectively, implies parallel andorthogonal circularly polarized reflected and referencefields. On the other hand, double-arm spirals reveal phasesingularities with topological charge $2, whose sign isinferred from the spiral handedness. This demonstrates thetransformations SðiÞ0 → N

ðrÞ2 and N

ðiÞ0 → S

ðrÞ2 . Similar analy-

sis is performed for an incident linear polarization,

ΨðiÞ0 ¼ H0, which is reported in the top row of Fig. 7. Inthat case, the reflected field is a superposition of contra-circularly polarized optical vortex beams with oppositetopological charges two and equal weights; see Figs. 7(c)and 7(d). This agrees with the expected transformationHðiÞ0 → Ψ

ðrÞ0 ðeiΦ1=

ffiffiffi2

p; eiΦ2=

ffiffiffi2

pÞ. More details are obtained

from spatially resolved polarimetric analysis of thereflected field (see Supplemental Material [25]) that unveilsan inhomogeneous linearly polarized beam carrying polari-zation singularity with topological charge 2, see Fig. 7(e),and average degree of linear polarization equals to 0.87.More generally, the transformations XðiÞ0 → X

ðrÞ2 with X ¼

ðH;D; V; AÞ are thus validated. Note that this implies atuning of the gap (see Fig. 5) to ensure Φ1 −Φ2 ¼ 0.The preceding remark suggests that the isotropic phase

delay Ξiso ¼ 8πd=λ associated with the optical path 4d ofthe reflected field component with helicity σ ¼ χ, seeFig. 1(c), enables control of the BBQP at fixed incidentpolarization. Indeed, a dephasing Ξiso between the twocontracircularly polarized components corresponds to anangular displacement by an angle Δϕ ¼ Ξiso along a line oflatitude on P2q. This is implemented by placing the rear

(a) (b)

FIG. 4. (a) High-order Poincaré sphere Pl where any stateΨlða; bÞ given by Eq. (1) corresponds to a point parametrized bythe angles (ξ, ϕ) with tanðξ=2Þ ¼ jb=aj and ϕ ¼ argða=bÞ. Sixparticular states are indicated and depicted in panel (b), where far-field intensity and polarization patterns are shown assuming anincident Gaussian beam. Nl and Sl, respectively, refer tocircularly polarized vortex fields associated with sz ¼ $1 andlz ¼∓ 1; Hl, Dl, Vl and Al, respectively, refer to space-variantlinearly polarized beams having azimuth angle of the form lφ,lφþ π=4, lφþ π=2 and lφþ 3π=4.

FIG. 5. Sketch of the experimental setup. Here the gapbetween rear mirror (RM) and BBVG corresponds to an actualexperimental situation. mirror (M); beam splitters (BS1;2;3); lenswith focal length f ¼ 0.5 m in f − f configuration with respectto the BBQP and the camera (L); polarizer (P); polarizationcontrollers (PC1;2;3).

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

(e) (f) (g) (h)

FIG. 6. Analysis of the reflected far field for ΨðiÞ0 ¼ S0(top row) and ΨðiÞ0 ¼ N0 (bottom row). Interference intensitypatterns are obtained by coaxial coherent superposition withcircularly polarized reference beam with helicity $1, namelyΨðrefÞ0 ¼ ðN0; S0Þ. Superscripts (i), (r), and (ref) refer to incident,reflected, and reference fields.

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mirror on a piezoelectric stage. The experiment is carriedout by using a linearly polarized incident beam, whichcorresponds to a revolving exploration of the equator ofP2q, see Fig. 8. The rotating four-lobe intensity pattern witha period that corresponds to ΔΞiso ¼ 8π agrees with theexpected 2π rotation of a 4jqj-lobe intensity pattern forΔΞiso ¼ 8jqjπ in the general case.Reconfiguration of the BBQP at fixed incident polari-

zation is also possible by imparting a birefringent phaseretardation Ξaniso to the back-reflected light. This allows usto turn on and off the geometric phase shaping of theincident light with σ ¼ χ by switching Ξaniso from 0 to π=2.Indeed, Ξaniso ¼ π=2 acts as a quarter-wave plate and itscombination with the rear mirror behaves as an effectivehelicity-preserving reflector, as sketched in Figs. 9(a) and9(b). Consequently, one can switch from high-orderPoincaré sphere Pl to hybrid-order Poincaré sphere [34]Pl;0 with Pl;l0 defining the optical states

Ψl;l0ða; bÞ ¼ ae−ilφcþ þ beþil0φc−; ð3Þ

and noting that Pl;l ≡ Pl. When l ≠ l0 one obtains so-called full Poincaré beams [35] that refers to inhomoge-neously polarized beam whose local polarization statesfully cover P0. Remarkably, our approach does not requireeither to prepare an incident field with non-zero orbital stateor detuning the geometric phase optical element, in contrastto the case of usual q plates. In our case, one can switch

between vectorial vortex beams (Ψl) and full Poincarébeams of various kinds (Ψl;0 or Ψ0;l depending on χ).This is illustrated in Figs. 9(c)–9(e) (Ξaniso ¼ 0) andFigs. 9(f)–9(h) (Ξaniso ¼ π=2).As emphasized by Eq. (2), the dispersion on the high-

order Poincaré sphere prevents to have spatially super-imposed space-variant polarization patterns (except forq ¼ 1=2) although axisymmetric intensity distributionand topological phase shaping is properly achieved foreach wavelength. Indeed, except for the north and southpole, any initial polychromatic state is mapped to a line oflatitude on the output high-order Poincare sphere, eachpoint corresponding to one wavelength. This could never-theless be corrected in a satisfying manner by adding adispersion compensation scheme, for instance, by insertinga slab with negative refractive index n̄¼−½ðn2⊥þn2kÞ=2'

1=2

and thickness L̄ ¼ L=2 placed in between the back-mirror

(a)

(e)

(b) (c) (d)

FIG. 7. Top row: Analysis of the reflected far field for ΨðiÞ0 ¼H0 with a presentation similar to that of Fig. 6. Bottom row:Reconstruction of the polarization azimuth angle (Φ) of thereflected far field from spatially resolved Stokes polarimetry forthe four distinct cases leading to HðrÞ2 , D

ðrÞ2 , V

ðrÞ2 , and A

ðrÞ2 .

(a) (b)

FIG. 8. Revolving exploration of the equator of P2q byisotropic phase control for q ¼ 1. Right part: Volumetric repre-sentation of the intensity patterns (without reference beam)obtained by setting PC1;3 as a linear polarizer, as Ξiso variesover ≃22π. Left part: transverse intensity profile snapshot.

(a)(c) (f)

(g)

(h)

(d)

(e)

(b)

FIG. 9. (a), (b) On-demand activation of the geometric phasebeam shaping of the incident light with σ ¼ χ by tuning thebirefringent phase delay Ξaniso. Far field intensity profiles of thereflected light for incident helicity σ ¼ $χ are shown in panels(c),(d) and (f),(g). Insets: interference patterns. The case ofincident linear polarization is shown in panels (e) and (h), whenPC3 is set as a linear polarizer (↕). Left or right panels refer toΞaniso ¼ ð0; π=2Þ, for q ¼ 1 and χ ¼ −1.

(a) (b)

(c)

FIG. 10. (a) Dispersion of the reduced latitude angle,ðϕ − hϕiλÞ=π on the high-order Poincaré sphere for any incidentpolarization state for L̄ ¼ L=2 with L ¼ 5p, 10p and 20p;λ0¼pðnkþn⊥Þ=2 refers to the central wavelength of the pho-tonic band gap and Δλ ¼ pðnk − n⊥Þ refers to the band gapwidth. Inset: standard deviation of ϕðλÞ in the photonic band gapversus the thickness L̄ of the compensating layer. (b), (c) Braggreflected geometric phase spectra for L ¼ 10p and incidenthelicities σ ¼ $χ. Material parameters are those of Fig. 1.

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and the Bragg-Berry slab of thickness L. The quality ofsuch a dispersion management is quantitatively addressedin Fig. 10(a) for three different values of L. The corre-sponding spectral dependence of the helicity-dependentreflected phase for the Bragg-reflected light is shown inFigs. 10(b) and 10(c), whose comparison with Fig. 2illustrates well the sought-after dispersion compensationinside the photonic band gap. Another option—bulkierbut easier to implement and that provides perfectcompensation—consists of process incident photons withopposite helicity by two independent Bragg-Berry vortexgenerators with identical q and opposite χ using a two-armconfiguration with equal optical path, as sketched in theSupplemental Material, Fig. S4 [25].In conclusion, we note that although the present work is

carried out in the specific framework of spin to orbitalangular momentum mapping using q-plate design, itapplies to all possible kinds of wave fronts that are shapedby the geometric phase. These results should therefore beuseful for the optics and photonics community at large,even when dealing at the single photon level.

This study has been carried out with financial supportfrom the French National Research Agency (ProjectNo. ANR-15-CE30-0018).

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