Ultra-broadband gradient-pitch Bragg-Berry mirrors

Mushegh Rafayelyan,1 Gonzague Agez,2 and Etienne Brasselet1, ∗

1Univ. Bordeaux, CNRS, LOMA, UMR 5798, F-33400 Talence, France2CEMES, CNRS, University Paul-Sabatier, F-31055 Toulouse, France

(Dated: November 13, 2018)

The realization of geometric phase optical device operating over a broad spectral range is usuallyconfronted with intrinsic limitations depending of the physical process at play. Here we proposeto use chiral nematic liquid crystal slabs with helical ordering that varies in three dimensions.Namely, gradient-pitch cholesterics endowed with in-plane space-variant angular positioning of thesupramolecular helix. By doing so, we show that the recently introduced Bragg-Berry mirrors [Opt.Lett. 41, 3972-3975 (2016)] can be endowed with ultra-broadband spectral range. Experimentaldemonstration is made in the case of ultra-broadband optical vortex generation in the visible do-main. These results offer practical solution to the polychromatic management of the orbital angularmomentum of light combining the circular Bragg reflection of chiral media with the Berry phase.

PACS numbers: 42.25.–p, 42.79.Dj, 42.70.Df,42.60.Jf

I. INTRODUCTION

A. Context

Beam shaping is a basic need for photonic technologies(e.g., optical imaging, optical information and commu-nication, optical manipulation, laser processing of ma-terials) and nowadays there are a lot of techniques tocontrol at will the spatial distribution of intensity, phaseand polarization. In general this involves the use of op-tical elements that are designed to operate efficiently fora given wavelength while user-friendly devices or someapplications preferably require large spectral bandwidth.

This may become challenging when dealing with op-tical phase structuring since the latter may be stronglydependent on wavelength. Here we deal with the elab-oration of broadband specific optical elements. Namely,anisotropic and inhomogeneous optical elements relyingon so-called geometric phase, see for instance the earlyproposal of a geometric phase lens by Bhandari [1]. Suchelements have great potential for polychromatic arbitraryphase shaping once combined with material structuringadd-on features, as discussed hereafter.

Let us consider an ideal lossless geometric phase op-tical element as a transparent dielectric half-wave platecharacterized by a transverse two-dimensional distribu-tion of the optical axis orientation angle, ψ2D(x, y). Sucha device imparts to an incident circularly polarized planewave propagating along the z axis a space-variant phase±2ψ2D(x, y) that is of a geometric nature, where the ±sign depends on the incident polarization handedness.The underlying physics is one of the numerous manifes-tations of the spin-orbit interactions of light [2]. No-ticeably, such a behavior is wavelength-dependent sincethe half-wave retardation condition is satisfied for a spe-cific wavelength. To solve such an issue, metamaterials

∗ etienne.brasselet@u-bordeaux.fr

offer an elegant option and allow considering the realiza-tion of broadband geometric phase flat optical elements.However, the realization of lossless scalable geometricphase metasurfaces operating in the visible domain re-mains technologically difficult. In practice, promisingdevelopments came from patterned liquid crystals tech-nologies where the possibility to achieve arbitrary two-dimensional orientational patterns ψ2D(x, y) [3, 4] canbe endowed with an additional degree of freedom alongthe third dimension, namely, ψ3D(x, y, z).

The principle of the latter three-dimensional approachfinds its roots in the early contributions by Destriau andProuteau [5] and Pancharatnam [6, 7] who proposed touse a discrete set of uniform waveplates (respectively, twoand three) oriented along specific direction in order toachieve achromatic waveplates. Nowadays, space-variantmulti-twists waveplates can be realized and various liquidcrystal geometric phase optical elements having enhancedachromatic performances [8] have been demonstrated ex-perimentally, for instance geometric phase prisms, lensesor vortex generators [9–11]. However, strictly speaking,the half-wave retardance condition is not realized but itsnear fulfillment is achieved over a broad spectral band-width.

Recently, an alternative approach has been exploredbased on the use of chiral nematic liquid crystals (alsocalled cholesterics), which are known to satisfy the half-wave retardance condition in reflection over large spectralbandwidth. The idea consists to combine helical struc-turing of the optical axis of cholesterics along the prop-agation direction with space-variant optical axis orienta-tion in the transverse plane. This can be described by

ψ3D(x, y, z) = 2πχz/p0 + ψsurf(x, y) , (1)

where χ = ±1 refers to the handedness of thesupramolecular helix and the pitch p0 corresponds to thedistance over which the director rotates by 2π around thez axis. In addition, ψsurf(x, y) refers the two-dimensionalorientational boundary condition at z = 0 and z = L, asshown in Fig. 1.

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To understand how such space-variant chiralanisotropic media associated with three-dimensionalpattern may impart to an incident circularly polarizedplane wave propagating along the z axis an opticalphase spatial distribution of the form ±2ψsurf(x, y), wefirst recall hereafter the basics of the optics of uniformcholesterics.

B. Background

Uniform cholesterics are characterized byψsurf(x, y) = ψ0 with ψ0 constant. They are well-known for their ability to reflect selectively one of thetwo circular polarization states for wavelengths belong-ing to the photonic bandgap defined by pn⊥ < λ < pn‖,where n‖,⊥ are respectively the refractive indices alongand perpendicular to the liquid crystal director [12, 13].This phenomenon is called the circular Bragg reflection[14]. More precisely, let us consider a circularly polarizedplane wave propagating towards +z that we describedby a complex electric field Eσ = E0 exp(−iωt + k0z)cσwhere E0 is a constant, ω is the angular frequency,t is the time, k0 is the free-space wave vector, andcσ = (x+ iσy)/

√2 is the circular polarization unit basis

in the unit Cartesian coordinate system (x,y, z) withσ = ±1 being the helicity of light. When a uniformcholesteric slab is thick enough with respect to thepitch and wavelengths inside the bandgap, a circularlypolarized light propagating along the cholesteric helixaxis is almost fully reflected if the helix formed by the tipof its electric field vector has the same handedness as thecholesteric helix [15], i.e., σ = −χ. On the other hand,light is almost fully transmitted when the optical andmaterial helices have opposite handednesses [15], i.e.,σ = +χ. We refer to Fig. 2 for an illustrated summaryof the relative helical structuring of the electric field andthe cholesteric molecular orientation in the Bragg [panel(a)] and non-Bragg [panel (b)] situations.

Importantly, Bragg reflection preserves the electricfield helix handedness, hence flips the spin angular mo-mentum projection along the z axis per photon fromsz = −χ~ (incident light) to sz = +χ~ (Bragg-reflectedlight). However, for wavelengths outside the bandgap,only a fraction of the incident light is Bragg-reflectedwhile the rest is transmitted without spin-flip. Finally,light is almost fully transmitted for σ = +χ without spin-flip [15]. The situation is quantitatively summarized inFig. 3 where blue and red arrows respectively refer tohelicity σ = ∓χ for the propagating light. In addition,for the considered cases, we also show numerical simula-tions of the reflectance (R) and transmittance (T ) spec-tra, which are demonstrated to be independent on ψ0.

The figures 3(b) and 3(c) also display the dependenceof the reflected (ΦR) and transmitted (ΦT ) phase spec-tra on ψ0, which are evaluated according to ΦR,T =arg(ER,T ), where ER,T are the complex reflected andtransmitted electric field amplitudes at z = 0 and z =L, respectively. These simulations unveil qualitatively[17] the possibility to shape the transverse distributionof the optical phase of the Bragg-reflected light. In-deed, arbitrary phase difference between two points in

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FIG. 3. (a) Illustration of the supramolecular helix with constant pitch p0 and handedness χ = +1. (b) Calculated reflectance(R) and phase (ΦR) spectra of the helicity-preserved Bragg-reflected light in the case of a uniform cholesteric slab having aconstant pitch, for all possible orientational boundary conditions 0 ≤ ψ0 ≤ 2π. (c) Same as in box (b) for the transmittance(T ) and phase (ΦT ) of the transmitted light, for which the two cases σ = ±χ must be considered separately. Simulations aremade using 4 × 4 Berreman formalism [16] with nglass = 1.52, L = 7 µm, n‖ = 1.71, n⊥ = 1.43, p = 350 nm, χ = +1 andconsidering normally incident plane wave.

the (x, y) plane can be provided by using space-variantorientational boundary conditions noting that ∆ΦR =ΦR(ψ0)−ΦR(0) = −2χψ0, see Fig. 3(b). That is to say, aspace-variant cholesteric slab characterized by ψsurf(x, y)according to Eq. (1) is expected to act as a reflectivespatial light modulator imparting a phase spatial distri-bution of the form ±2ψsurf(x, y) to an incident circularlypolarized light field with helicity σ = −χ. In contrast,∆ΦT = 0 for σ = ±χ and therefore no transverse phaseshaping is anticipated in transmission. Such a behaviorresults from the fact that the transmitted phase does notdepend on ψ0 although it is of course strongly dependenton wavelength, see Fig. 3(c).

The optical properties of above discussed space-variantcholesterics with constant pitch have been experimentallydemonstrated recently, with already various demonstra-tions in the framework of prisms and lenses [18], and op-tical vortex generators [19–21]. These recent works thusunveiled a novel class of geometric phase reflective opticalelements endowed with intrinsic polychromatic features.

C. ‘Bragg-Berry mirror’ terminology

Physically, the discussed geometric phase reflective op-tical elements rely on the combination of two distinct op-tical effects that respectively confer unique properties tothe fabricated optical devices. Namely, polychromatic re-sponse and helicity-controlled phase structuring. Indeed,

the latter characteristics respectively rely on (i) circu-lar Bragg reflection of chiral anisotropic media [14] and(ii) geometric (Berry) phase associated with the couplingbetween intrinsic optical spin angular momentum and ro-tations of coordinates [2]. This invited us to suggest the‘Bragg-Berry mirror’ terminology in a previous work [20]in order to emphasize these underlying key features.

However, it should be clear that such Bragg-Berry pla-nar mirrors are not standard flat reflectors since lightpreserves its helicity upon reflection and may acquire ar-bitrary space-variant phase transverse spatial distribu-tion. Also, we note that the geometric phase nature ofsuch a beam shaping has been experimentally identifiedin Ref. 19 with two independent experiments that un-derpin the universality of the geometric phase regardingtemporal and spatial continuous rotations, as discussedin Ref. 22. In addition, an experiment based on discretejump of orientation by an angle of π/2 leading to a πphase shift has been reported in Ref. 23.

D. Position of the problem

As said in section I B, uniform cholesterics withconstant pitch have only first-order circular photonicbandgap defined by the wavelength range n⊥p0 < λ <n‖p0, which is typically less than 100 nm in the visible do-main (liquid crystals have usually optical anisotropy upto n‖ − n⊥ ∼ 0.3). However, it is known that cholester-

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ics having non-uniform pitch may exhibit higher-orderphotonic bandgaps or broaden their photonic bandgap[24]. In particular, cholesteric helix characterized by apitch gradient allows to achieve circular Bragg reflectionover the full visible domain, which can be realized usingphoto-induced [25] or thermally induced [26] as experi-mentally demonstrated in earlier works. For recent in-depth overviews we refer to review papers [27, 28]. Theback-home message here is that the photonic bandgapcan be extended from n⊥p0 < λ < n‖p0 in the caseof constant pitch to n⊥pmin < λ < n‖pmax estimatedin the case of gradient-pitch cholesterics defined by a z-dependent pitch p(z) = pmax − (pmax − pmin)z/L [24].

In practice, the gradient-pitch bandwidth can be sev-eral times larger that the constant-pitch bandwidth. Ac-cordingly, we propose to combine the previously demon-strated topological beam shaping capabilities of constant-pitch Bragg-Berry mirrors with the ultra-broadband per-formances of gradient-pitch cholesterics. This would pro-vide with reflective geometric phase optical elements op-erating efficiently over the full visible spectrum. Ofcourse, such a principle could be extended to larger wave-length regions by appropriate setting of pitch values.

Our work is organized such as, firstly, we addressthe case of uniform gradient-pitch Bragg mirrors in sec-tion II, where the existence of geometric phase in theBragg-reflected light is explored numerically and demon-strated experimentally. Then, without loss of generality,we address in section III the experimental realization ofultra-broadband gradient-pitch Bragg-Berry mirrors inthe context of optical vortex generation. Finally, we sum-marize the study and discuss a few prospective issues.

II. UNIFORM GRADIENT-PITCH BRAGGMIRRORS

A. Geometric phase of reflected field: numerics

Following the numerical approach presented in sectionI B that allows to anticipate the existence of a geometricphase for the Bragg-reflected light in the case of uniformcholesterics with constant pitch [see Fig. 3(b)], we per-form a similar set of simulations in the case of uniformgradient-pitch cholesterics described by

ψ3D(z) = 2πχz/p(z) + ψ0 , (2)

with the space-variant pitch p(z) given by

p(z) = pmax − (pmax − pmin)z/L , (3)

where the values of pmin and pmax are chosen to haveidentical central wavelength for the photonic bandgap,which eases the comparison between the two situations.The results are presented in Fig. 4 that is organized in asimilar fashion as in Fig. 3.

As shown in Fig. 4(b) and 4(c), the expected sub-stantial increase of the photonic bandgap bandwidth forσ = −χ is observed. Noteworthy, regarding the reflectedphase spectrum for σ = −χ and the transmitted phasespectra for σ = ±χ, the conclusions are unchanged withrespect to the case of constant pitch. Namely, a non-uniform pitch neither prevents the existence of the ge-ometric phase nor its expression ∆ΦR = −2χψ0, whilesignificantly widens the photonic bandgap. Therefore,this underpins our proposition presented in section I Dand invites us to implement its experimental validation,which is performed in what follows.

5

B. Sample preparation and characterization

We used cholesteric oligomers from Wacker ChemieGmbH with n‖ = 1.72 and n⊥ = 1.42 effective extraor-dinary and ordinary refractive indices. The molecularstructure of the material consists of a siloxane cyclicchain to which is attached, via aliphatic spacers, twotypes of side chains. Namely, an achiral mesogen anda chiral cholesterol-bearing mesogen. The pitch of thehelical structure and therefore the reflection wavelengthdepends on the molar percentage of the chiral mesogenein the molecule. This percentage is 31% in the case ofSilicon Red compound (SR), 40% for Silicon Green com-pound (SG) and 50% for Silicon Blue compound (SB).The cholesteric phase appears between 40–50◦C, whichcorresponds to the glass-transition temperature range,and 180–210◦C, which corresponds to the clearing tem-perature range. By freezing the film in a glassy solidstate, the cholesteric structure and its optical propertiesare kept at room temperature in a perennial manner.

A three-layer sample was elaborated by stackingthree different 10 µm-thick cholesteric oligomers layers,namely, SB/SG/SR, as shown in Fig. 5(a), according tothe protocol described hereafter. The SR and SB filmswere prepared by confinement between a glass substratecovered by polyimide surface anchoring layers and a mi-croscope coverslip and the SG layer was confined betweentwo microscope coverslips. The samples were kept at80◦C (cholesteric phase) for 10 minutes to form textureswith the cholesteric helix being perpendicular to the sub-strates that follow either the orientational boundary con-ditions provided by uniform surface anchoring layer orthe planar degenerate orientation provided by the mi-croscope coverslips. Then the samples are placed in thefreezer for 15 minutes −18◦C in order to obtain glassysolid layers. At this temperature the coverslips can beeasily taken off and obtained free-standing SG layer was

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FIG. 5. Sketch of the initial three-layer stack of cholestericoligomer (SB, SG, SR) sandwiched between tow glass sub-strates coated with surface-alignment layers providing the ori-entational boundary conditions before (a) and after (b) ther-mal treatment that leads to promote the inter-diffusion of thethree layers. (c) Typical transmission spectrum of a fabri-cated uniform gradient-pitch cholesteric sample under unpo-larized illumination.

sandwiched between the SR and SB semi-free-standingfilms. Finally, the SB/SG/SR three-layer sample was an-nealed several times at 80◦C during 5 minutes to promotethe inter-diffusion of the three compounds and obtain aconstant gradient of the cholesteric pitch along z.

The resulting ultra-broadband circular Bragg reflec-tion is illustrated in Fig. 5(c) that shows the typi-cal transmission spectrum of a uniform gradient-pitchcholesteric sample. Indeed, the photonic bandgap cov-ers almost all the visible domain. Still, we note that thelinear gradient-pitch description given by Eq. (3) is anidealization. Indeed, the actual pitch profile along thez axis evolves during the diffusion process [29]. There-fore, in our case, it depends on the time at which thecholesteric has been quenched in the glassy state.

C. Dynamical geometric phase experiment

As discussed in a previous work [19] dealing withconstant pitch cholesteric structure, the experimentaldemonstration of the geometric phase shaping capabil-ities of Bragg-Berry mirrors can be retrieved by im-plementing a dynamical phase experiment on uniformconstant-pitch cholesterics characterized by ψsurf(x, y) =ψ0. Here we extend this approach to the case of the fab-ricated uniform gradient-pitch cholesteric sample.

The set-up is sketched in Fig. 6(a). The main ideais that the ψ0-dependent phase of the Bragg reflectedlight, see Fig. 3(b), can be probed by rotating the sam-ple around the helix axis of the cholesteric at frequencyf0. Indeed, according to the relationship ∆φR = −2χψ0

established in the static case, such a rotational motionimparts to the reflected beam a time-dependent reflectedphase with frequency 2f0. In contrast, the Fresnel reflec-tion at air/glass interfaces of the rotating sample (glasssubstrates are not anti-reflection coated) do not contain atime-dependent phase as is the case for standard mirrors.

In turn, following the protocol described in [19], theexperiment consists in the detection of the time-varyingsignal that results from the collinear interference betweenthe Bragg and Fresnel contributions to the total reflectedfield. The results are shown as solid lines in Fig. 6(b),6(c) and 6(d) for three wavelengths falling in the photonicbandgap, namely, λ = 488, 532 and 633 nm [Fig. 5(c)].We notice that the Bragg and Fresnel contributions haveorthogonal circular polarization states and different pow-ers. Therefore it is necessary to use a quarter-wave plateand a linear polarizer whose relative orientation is cho-sen to maximize the visibility of the interference signalrecorded by the photodetector, as shown in Fig. 6(a).

The power Fourier spectra of the latter signals reveala peak frequency at 2f0, as well as higher-order har-monics at 2nf0 with n > 1 integer as expected froma non-sinusoidal periodic signal. We also checked thatboth the Bragg or Fresnel contributions taken separatelydo not participate to the observed interferential beat-ing, as respectively shown by dashed and dotted lines in

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FIG. 6. (a) Dynamical geometric phase experimental set-up.Slightly oblique incidence (the angle α equals a few degrees)allows to readily analyze the reflected light. Interference spec-tral filters for λ = 488, 532 and 633 nm and having a 3 nmfull-width half-maximum transmission spectrum have beenused. Thin/thick arrows that respectively depict the Fres-nel/Bragg reflected light beam refers to incident light reflectedat air/glass interface and to circular Bragg reflection from thecholesteric. Sample is rotated at frequency f0 ' 0.05 Hz.(b,c,d) X = [s(t)−〈s(t)〉]/〈s(t)〉, s(t) being the data acquiredby the photodetector for the three selected wavelengths, and〈s(t)〉 the time average value of s(t). (e,f,g) Power Fourierspectrum of the signals shown in panels (b,c,d) respectively.Spectral signals of Bragg and Fresnel signals alone are muchweaker than the one resulting from interference between theFresnel and Bragg reflected beams.

Fig. 6(b)–(g). Still, residual dynamics appear dependingon the probed location of the sample, which we attributeto experimental imperfections of the cholesteric ordering,non-ideal assembling of the substrates and optical adjust-ments. However, none of these imperfections questionsthe revealed geometric phase imparted to the helicity-preserved reflected light from gradient-pitch cholesterics.

Although the temporal beatings of above experimentsdo not allow to discriminate in which direction the sam-ple rotates, we note that the Bragg reflected beam expe-riences a frequency shift ±2f0 where the ± sign dependson the sense of rotation of the sample and the cholesterichandedness. From a mechanical point of view, the latterblue (+ sign) or red (+ sign) frequency shift is associ-ated with a nonzero optical radiation torque that results

from optical spin angular momentum flipping. The lat-ter optical torque is respectively anti-parallel or parallelto the angular momentum vector of the rotating sample.In contrast, the Fresnel reflection at air/glass interfacesof the sample preserves the spin angular momentum asis the case for standard mirrors, hence it is neither as-sociated with a contribution to the total optical torqueexerted on the sample nor to a rotational frequency shift.

III. NON-UNIFORM GRADIENT-PITCHBRAGG MIRRORS

After the theoretical and experimental demonstrationsof the geometric phase shaping capabilities of gradient-pitch Bragg-Berry mirrors with uniform surface align-ment, here we address the case of ultra-broadband ge-ometric phase beam shaping using space-variant orien-tational boundary conditions, ψsurf(x, y). Without lackof generality we restrict our demonstration to the case ofoptical vortex generation in order to ease the comparisonwith previous results on patterned cholesterics with con-stant pitch [20, 21]. Namely, we consider a cholestericslab that combines gradient-pitch and so-called q-platefeatures [30], which is ideally described by

ψ3D(φ, z) = 2πχz/p(z) + qφ , (4)

where the space-variant pitch p(z) is given by Eq. (3)and φ is the polar angle in the (x, y) plane. We chooseq = 1, which produces Bragg-reflected optical vortexbeams with topological charge ±2 depending on the signof χ. Such a sample was prepared following the proto-col presented in section II B using photoalignment lay-ers that impose a two-dimensional radial distributionψsurf(φ) = φ. This situation thus corresponds to the gen-eralization of the design used in [20] by performing thelongitudinal structural upgrade p0 → p(z). This is illus-trated in Fig. 7 where corresponding three-dimensionalmolecular ordering are shown.

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7

A. White light optical vortex generation

Ultra-broadband optical vortex generation experimentis done using the set-up shown in Fig. 8(a). It corre-sponds to a Michelson interferometer scheme where thegradient-pitch Bragg-Berry mirror (GPBBM) with q = 1stands as one of the two mirrors of the interferometer. Alinearly polarized supercontinuum Gaussian beam withcontrolled polarization orientation is prepared using alinear polarizer (P1) and an achromatic half-wave plate(AHWP). This allows to control the power ratio of thetwo arms of the interferometer by using a polarizing beamsplitter (PBS). By doing so, the air/glass Fresnel contri-bution of the beam reflected off the sample and the ref-erence beam reflected off the standard mirror (M) haveorthogonal polarization states. Therefore, post-selectionof the Bragg-reflected beam is realized by using a linearpolarizer oriented along the x-axis (P2). Here, a non-trivial point is to benefit from the natural imperfection ofthe PBS. Indeed, for an ideal PBS the x-polarized lightreflected off M would return back to the laser source.

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However, in practice, PBS does not redirect a small frac-tion of that field, which enables the selective formation ofan interference pattern with the y-polarized contributionof the Bragg light reflected off the cholesteric sample.

The co-axial interference pattern is acquired by a cam-era (Cam) placed in the focal plane of a lens (L2). Thisallows imaging the far-field of the vortex beam generatedby the sample, which exhibits a doughnut shape inten-sity profile as expected, see Fig. 8(b). On the other hand,the curvature of the reference beam in the plane of thecamera, see Fig. 8(c), is adjusted by placing a lens (L1)nearby M, which allows to control the number of fringesin the field of view. The resulting interference patternis a two-arm spiral unveiling the generation of an opticalphase singularity with topological charge two Fig. 8(d).We note the unavoidable presence of a whitish blurringon the latter pattern, which is due to the fact that thevisibility of the spatial modulation of the intensity can-not be optimized for all the wavelengths at the same timesince Bragg reflection alters the spectrum. These obser-vations thus complete the experimental validation.

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Diffractiongrating

Bragglight

Lens

FIG. 9. (a) Sketch of experimental set-up for spectral anal-ysis. Slightly oblique incidence (α ∼ 1◦) allows to readily an-alyze the reflected light. The Bragg-reflected light is selectedby observing the sample between crossed linear polarizers,taking rid off the Fresnel reflection from the interface betweenair and the glass substrate. The analysis module refers to twodistinct options detailed in panels (b) and (c). (b) Dispersionmodule: Bragg light is reflected off a blazed wavelength re-flective diffraction grating (from Thorlabs, model GR13-0605)and projected on a black screen. The screen is then re-imagedon a camera. (c) Spectral filtering module: we use a set ofseven interference filters centered on λ = 400 to 700 nm bystep of 50 nm, with 10 nm full-width half-maximum trans-mission spectrum. The camera placed in the focal plane ofthe lens records the far-field of the generated vortex beam.

8

B. Optical vortex generation: spectral behavior

More precise analysis of the generated white light vor-tex beam by the gradient-pitch Bragg-Berry vortex mir-ror is performed using the experimental set-up depictedin Fig. 9. It consists to analyze spectrally the Bragg-reflected field off the sample. This is done by illuminatingthe vortex mirror at small oblique incidence (α ∼ 1◦) bya linearly polarized supercontinuum Gaussian beam andselecting the Bragg-reflected field only by placing an out-put linear polarizer whose direction is orthogonal to thatof the incident one, as shown in Fig. 9(a). The Bragg-reflected light beam is then analyzed in two different andcomplementary ways. On the one hand, its spectral con-tent is revealed by imaging a black screen on which isprojected the spectrally dispersed beam by using a re-flective diffraction grating, see Fig. 9(b). On the otherhand, the far field intensity profile for a discrete set ofwavelengths are recorded by a camera at the focal planeof a lens placed after an interference filter, see Fig. 9(c).

In order to appreciate the enhanced polychromatic ge-ometric phase shaping of the gradient-pitch Bragg-Berryvortex mirror, the experiments are compared with thecase of a constant-pitch Bragg-Berry vortex mirror withthe identical charge q = 1, which was prepared usingthe cholesteric liquid crystal SLC79 (from BEAM Co)possessing a ∼ 70 nm-width photonic bandgap centeredon 530 nm wavelength. Dispersion results are shown inFig. 10 where panel (a) refers to constant-pitch and panel(b) to gradient-pitch. As expected from previous obser-vation of circular Bragg reflection with several hundredsof nanometers bandwith (Fig. 5), these observations con-firm that transverse space-variant orientational structureof the cholesteric sample does not alter the bandwidthenhancement.

Similarly, the comparative study of the far field vortexbeam intensity profiles is displayed in Fig. 11. Note thatin the case of constant pitch, image recording requireslonger acquisition time outside the photonic bandgap,hence revealing signal-to-noise issues altering the ide-ally expected doughnut intensity pattern whatever thewavelength. The good quality of the vortex beam pro-file over the 400-700 nm range thus recalls the ultra-broadband features of the gradient-pitch structure. Also,

(a)

(b)

B = B5

B = B(D)

Bragg-BerryvortexmirrorsF = 1

FIG. 10. Experimental spectra of the Bragg-reflected beamoff a Bragg-Berry vortex mirror with q = 1 for constant-pitch(a) and gradient-pitch (b).

700nm650nm600nm550nm500nm450nm400nm

700nm650nm600nm550nm500nm450nm400nm

(a) B = B5

B = B(D)(b)

FIG. 11. Experimental set of spectral components for thefar-field intensity profiles of the generated vortex beam by aBragg-Berry vortex mirror with q = 1 for constant-pitch (a)and gradient-pitch (b).

wavelength-dependent diameter of the doughnut beam isassociated with the spectral dependence of the waist ofthe supercontinuum source.

C. Optical vortex generation: oblique incidence

The dependence of the Bragg-reflected field from agradient-pitch Bragg-Berry vortex mirror on the angle ofincidence is also explored experimentally and comparedto numerical expectations. We note that only the sim-plest situation of a uniform cholesteric with a constantpitch illuminated at normal incidence, which is presentedin Fig. 3, finds an analytical solution for its reflection andtransmission fields according to the Berreman method[31]. In order to deal with the case of oblique incidence,a not too cumbersome approximate analytical approachcalled coupled-mode theory has been developed [32, 33].However, when exact solution is sought, one has to handlethe problem within a numerical approach, for instance byusing the Berreman 4×4 matrix formalism [16], the Am-bartsumians layer addition modified method [34] or thefinite-difference time-domain method (FDTD) [35]. Inpresent study, the 4×4 Berreman approach is used.

Firstly, following the set-up of Figs. 9(a) and 9(b),we compare the reflectance spectra for almost normalincidence (α = 1◦) and oblique incidence (α = 30◦). Theresults are shown in Figs. 12(a) and 12(b). A substantialblue shift of the spectrum of several tens of nanometers isobserved, which recalls the usual behavior of cholestericsunder oblique incidence. This is qualitatively comparedwith simulations Fig. 12(c) performed in the case of auniform gradient-pitch cholesteric with ψ0 = 0.

Then, following the set-up of Figs. 9(a) and 9(c), wecompare the far field intensity profiles of the generatedvortex beams for α = 1◦ and α = 30◦, see Figs. 13(a)and 13(b). Qualitatively, the doughnut-shaped charac-teristic of the generated vortex beam is preserved as

9

G

300400500600700800l (nm)

(c) 1

0

6 = 1°

6 = 30°

6 = 1°

6 = 30°

(a)

(b)

488nm 532nm 633nm

Blueshift

Blueshift

FIG. 12. Experimental spectra of the Bragg-reflected beamoff a gradient-pitch Bragg-Berry vortex mirror with q = 1following the set-up of Figs. 9(a) and 9(b) for α = 1◦ (a) andα = 30◦ (b). The three ticks at λ = 488, 532 and 633 nm,which have been evaluated using interference filters, providewith a nonlinear ruler in order to appreciate quantitativelythe frequency shift of the spectrum. (c) Simulated reflectancespectra for external incidence angle α = 1◦ (green curve) andα = 30◦ (blue curve) for a uniform gradient-pitch cholestericwith ψ0 = 0. Numerical parameters are the same as in Fig. 4.

the incidence angle increases. In addition, a closer lookat the central part unveils a splitting of the opticalphase singularity with topological charge two into twosingularities with unit charge, see enlargement panel ofFig. 13(b). Such high-charge splitting is a well-knownpractical feature, especially in the presence of a smoothcoherent background [36], that affects even high-qualityhigh-charge vortex beams [37].

However, we notice that the vortex splitting is muchmore pronounced at oblique incidence than at normalincidence in which case we hardly evidence it with theused imaging device, as shown in the enlargement panelof Fig. 13(a). This is explained recalling that tilted il-lumination breaks the axisymmetry [20] and the symme-try breaking is all the more pronounced as the angle ofincidence is high. From the theoretical point of view,the latter features can be grasped from simulations ofthe reflectance and reflected phase of the Bragg-reflectedlight on a uniform gradient-pitch Bragg-Berry mirror asa function of ψ0. The calculated spectra are shown inFigs. 13(c) and 13(d), respectively. The data thus con-firm that the geometric phase shaping is unaltered bythe oblique incidence while the reflectance is substan-tially modulated as a function of ψ0. Still, there mightalso have other contributions to the observed high-chargesplitting that are not taken into account in the latter sim-ulations, for instance the actual three-dimensional mod-ulation of the chiral properties of the sample.

300400500600700800l (nm)

(d)

G

300400500600700800l (nm)

(c) 1

0

(a) (b)

2p

0

J5

ΦL

2p

0

J5

2p

0

6 = 1° 6 = 30°

FIG. 13. Experimental far field intensity profiles of theBragg-reflected beam off a gradient-pitch Bragg-Berry vortexmirror with q = 1 following the set-up of Figs. 9(a) and 9(c)for α = 1◦ (a) and α = 30◦ (b). An enlargement of the cen-tral part of panels (a) and (b) are shown, where the luminanceand contrast have been adapted to emphasize the presence ofthe splitting of the double-charge phase singularity into twounit charge phase singularities in the case α = 30◦. Simulatedreflectance (d) and reflected phase (e) spectra for external in-cidence angle α = 30◦ for a uniform gradient-pitch cholestericwith 0 < ψ0 < 2π. Numerical parameters are the same as inFig. 4.

IV. CONCLUSION

In this work we showed that chiral nematic liquid crys-tal with helical ordering that varies in three dimensionsenables the realization of a ultra-broadband geometricphase reflective flat optical elements, namely gradient-pitch Bragg-Berry mirrors. The selective spatial phasestructuring imparted to the Bragg-reflected beam off thedevice is determined by the surface orientational bound-ary conditions of the liquid crystal slab. On the otherhand, the polychromatic properties of the device is en-sured by the presence of a gradient of the pitch of the he-lical bulk molecular ordering. Experimental demonstra-tion has been performed by fabricating an optical vortexgenerator operating in the full visible domain. Topologi-cal, spectral and angular optical properties have been ex-perimentally determined and discussed in the frameworkof numerical simulations. These results extend the use ofpreviously introduced Bragg-Berry mirrors to very largespectral bandwidths that can be adjusted by appropriatechoice of the material properties.

Of course, although present results are restricted to thepolychromatic management of the orbital angular mo-

10

mentum of light, they could be extended to any phaseshaping by appropriate surface patterning of alignmentlayers using state-of-the-art techniques [3, 4]. Neverthe-less, it should be recalled that light scattering remains achallenging issue that need further work. Indeed, chiralgradients in the three spatial dimensions lead in practiceto substantial scattering losses. In the present case, thespecular Bragg reflectance for wavelengths falling in thephotonic bandgap is typically four times smaller than theideal value.

Also, note that other three-dimensional chiral struc-tures that are not nematic have been previously discussedin the context of Bragg-Berry mirrors, see for instanceRef. [38] where the case of space-variant blue phases sam-ples has been discussed. Finally, we would like to men-tion that we became aware of a similar study made using

photo-induced gradient-pitch cholesterics published veryrecently [39] after this work has been completed [40]. Inpractice, since present study has been made using glassygradient-pitch cholesterics, it offers an interesting com-plementary approach. Also, in contrast to Ref. [39], wenotice that present study provides with behavior of thepolychromatic beam shaping versus the incidence angleand in the far field, thus bringing a thorough overviewof the optical properties of gradient-pitch Bragg-Berryoptical elements.

ACKNOWLEDGMENT

This study has been carried out with financial sup-port from the French National Research Agency (projectANR-15-CE30-0018).

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