Flat polarization-controlled cylindrical lens based on the Pancharatnam-Berrygeometric phase

Bruno Piccirillo,1, ∗ Michela Florinda Picardi,1, 2 Lorenzo Marrucci,1, 3 and Enrico Santamato1

1Dipartimento di Fisica “E. Pancini”, Universita di Napoli Federico IIComplesso Univ. Monte S. Angelo, via Cintia, 80126 Napoli, Italy

2Department of Physics, King’s College London, Strand, London, WC2R 2LS, United Kingdom3CNR-ISASI, Via Campi Flegrei 34, Pozzuoli (NA) 80078, Italy

The working principle of ordinary refractive lenses can be explained in terms of the space-variantoptical phase retardations they introduce, which reshape the optical wavefront curvature and henceaffect the subsequent light propagation. These phases, in turn, are due to the varying optical pathlength seen by light at different transverse positions relative to the lens centre. A similar lensingbehavior can however be obtained when the optical phases are introduced by an entirely differentmechanism. Here, we consider the “geometric phases” that arise from the polarization transforma-tions occurring in anisotropic optical media, named after Pancharatnam and Berry. The mediumanisotropy axis is taken to be space-variant in the transverse plane and the resulting varying geomet-ric phases give rise to the wavefront reshaping and lensing effect, which however depends also on theinput polarization. We describe the realization and characterization of a cylindrical geometric-phaselens that is converging for a given input circular polarization state and diverging for the orthogonalone, which provides one of the simplest possible examples of optical element based on geometricphases. The demonstrated lens is flat and only few microns thick (not including the supportingsubstrates); moreover, its working wavelength can be tuned and the lensing can be switched on andoff by the action of an external control electric field. Other kinds of lenses or more general phaseelements inducing different wavefront distortions can be obtained by a similar approach. Besidestheir potential for optoelectronic technology, these devices offer good opportunities for introducingcollege-level students to an advanced topic of modern physics, such as the Berry phase, with thehelp of interesting optical demonstrations.

This is an author-created, un-copyedited version of an article published in European Journal ofPhysics, vol. 38, (2017) 034007 (15pp). IOP Publishing Ltd is not responsible for any errors oromissions in this version of the manuscript or any version derived from it. The Version of Record isavailable online at DOI: 10.1088/1361-6404/aa5e11.

I. INTRODUCTION

In the last years, significant progress has been madein the field of “structured light”, that is light featur-ing a relatively complex spatial structure. This struc-ture may be in the intensity distribution (as in opticalimages) but also, and more interestingly, in the phaseand polarization distributions of light. The phase distri-bution in the plane transverse to the main propagationdirection defines in particular the wavefront structure,which in turn determines the subsequent propagation be-havior of the light. Various families of structured opti-cal beams with interesting properties have been demon-strated, including for example “non-diffracting” beams1

or helical-wavefront beams that carry well-defined valuesof the orbital angular momentum2. Structured light isalso finding many applications: for example, in Ref.3, itwas shown that a suitably wavefront-shaped light beamcan be focused inside even the most strongly scatter-ing objects, overcoming the limitations imposed by lightscattering to the depth at which optical imaging methodscan retain their resolution and sensitivity. Some conceptsand methods of the structured-light field have also beenextended beyond the electromagnetic wave realm, as inthe case of shape-invariant or helical electronic wavefunc-tions4,5 or acoustic beams6. The control of the wave-

front structure can be nowadays based on a quite var-ied toolbox of methods, ranging from the simplest re-fractive lenses to the complex “transformation optics”associated, for example, with metamaterial cloaking7,8.Associated optical devices include refractive optical el-ements, such as for instance, axicon lenses9; diffractiveoptical elements, such as computer-generated hologramsdisplayed on spatial light modulators10; plasmonic ordielectric metasurfaces11–14; and finally Pancharatnam-Berry phase optical elements (PBOEs), which have re-cently stimulated a growing interest15–17 and which arethe subject of the present paper.

The working principle of ordinary refractive elementsfor wavefront shaping, such as standard lenses, is to intro-duce a space-variant phase retardation via a transversely-varying optical path length. This can be either frommodulations of the thickness of the optical element, asin shaped glass lenses, or of the refractive index, as ingradient-index (GRIN) devices. As we will explain morein detail below, this phase can also be named “dynam-ical”, as it is determined by the duration (in time orspace) of the optical propagation process. In contrast,PBOEs exploit an additional phase retardation arisingfrom polarization manipulations, which is independentof the propagation length. More in detail, in a PBOE,at each point in the transverse plane, the light polariza-

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tion is made to change continuously along a given pathon the Poincare sphere by the action of a suitable lo-cal medium anisotropy. This sequence of polarizationtransformations in general gives rise to an extra opticalphase retardation, in addition to the standard dynam-ical phase, that depends only on the geometry of thepath of transformations on the Poincare sphere, hencethe name “geometric phase”. This phase is also calledPancharatnam-Berry (PB) phase, from the names of theresearchers who first investigated it15,18.

The first concept of a PBOE goes back to 1997, whenBhandari proposed the design of a geometrical-phase lens(GPL) consisting of a composite plate made of severalconcentric rings, each being a birefringent medium withhalf-wave retardation but having different orientations ofthe optic axis16, sandwiched between two standard (i.e.uniform) quarter-wave plates: a linearly-polarized opti-cal beam passing through this device would acquire ageometric phase that varies in steps with the transversedistance from the center, giving rise, for suitable designof the structure, to a lensing effect. The lens would befocusing or defocusing, with the same focal length, de-pending on the input polarization, i.e. it is “polarization-controlled”. In 2003, Hasman et al.19 experimentallydemonstrated the first polarization-dependent GPL, inthe form of a discretized PBOE manufactured with sub-wavelength gratings, operating in the mid-infrared. Thefirst visible-domain GPL was realized with liquid crys-tals patterned by micro-rubbing in 200520, although theauthors of this work do not make explicit reference tothe geometric PB phases to explain the device workingprinciple. In 2006, Roux proposed the design of a con-tinuous GPL for the visible domain that exploited theform birefringence of subwavelength grooves21. But onlyin 2015, Gao et al.22 have reported the fabrication ofliquid crystal GPLs exploiting the versatile polarization-holography alignment technique. At about the sametime, liquid crystal GPLs based on a similar technol-ogy were also manufactured and used by our group fora proof-of-principle experimental demonstration of theconcept of geometric-phase waveguides23.

In this paper, we illustrate the working principle, fabri-cation and characterization of a liquid-crystal cylindricalGPL fabricated via polarization-holography. The presentGPL is also electrically tunable, i.e. its birefringent re-tardation can be controlled continuously via an exter-nal alternate-current (AC) voltage of few volts (at a fre-quency of few kHz), so that it can be operated with highefficiency across the whole visible and near-infrared spec-trum. In addition, its optical action can be switched onand off by applying specific amplitudes of the same ACvoltage. In this paper we chose to adopt the cylindricallens geometry, as it involves only one-dimensional pat-terning and it is hence simpler to understand and char-acterize, but the same approach can be readily used torealize spherical lenses or more general optical phase el-ements.

The paper is organized as follows. In Sec. II, we intro-

duce the concept of PB optical phase. In Sec. III we ex-plain the working principle of PBOEs, which exploit thePB phase for controlling the optical wavefront, and thenprovide a mathematical description of the GPL. Next,in Sec.IV we describe the experimental work, includingthe fabrication method (Subsec. IV A), the proceduresadopted for the lens testing and characterization (polar-ization microscopy in Subsec. IV B, and a quantitativestudy of a Gaussian laser beam propagation and focus-ing after passing through the GPL (Subsec. IV C). In theconcluding section, some brief remarks on the possibleuse of these devices for teaching purposes are given.

II. THE PANCHARATNAM-BERRY PHASE

In quantum mechanics, the adiabatic theorem statesthat a gradual change of the Hamiltonian from an initialform Hi to a final form Hf makes the system preparedinitially in the (nondegenerate) nth eigenstate of Hi

slide continuously into the nth eigenstate of the Hamil-tonian Hf , provided that no energy-level crossing oc-curs24. This is the basis for example for the well-knownadiabatic approximation in molecular physics named asBorn-Oppenheimer approximation. In 1984, Berry ex-tended the adiabatic theorem by noting that, during thecontinuous transformation of the Hamiltonian, the eigen-states accumulate a global phase that can be given asthe sum of two terms25. The first, known as “dynamicalphase”, is the time integral of the (slowly varying) energyEn divided by the reduced Planck constant ~ = h/(2π),which is the obvious extension of the usual phase thatis accumulated in time for the case of a static Hamilto-nian and it clearly depends on the duration of the tem-poral evolution. The second term, in contrast, has nostatic analog and is determined only by the geometry ofthe path followed in the space of the parameters drivingthe slow evolution, while it is totally independent of thetime duration: it is therefore named “geometric phase”,or Berry phase, after its discoverer.

The Berry phase has been recognized as an impor-tant unifying concept across physics, not limited to quan-tum mechanics, with applications ranging from molecularspectroscopy to condensed matter, cold atoms, plasmaphysics, etc. For a comprehensive review about manifes-tations of Berry phase, see for instance Ref.26. In thefield of optics, the Berry phase has had a great impact,providing for example a unifying explanation for severalrecently understood “spin-orbit optical phenomena”27,28.In particular, the two most common manifestations of ge-ometric phases in optics incude the Rytov-Vladimirskii-Berry phase (also known as “spin redirection” phase),arising when varying the wavevector direction of a lightbeam, and the PB phase, that is induced – as mentionedin the introduction – by a sequence of polarization ma-nipulations occurring in anisotropic media16,29. In thispaper, we are concerned only with the latter.

The standard phase retardation accumulated by a

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FIG. 1. Elliptical polarization in birefringent mediumand average index. An arbitrary elliptically polarized wavetraveling in a birefringent medium can be always decomposedin two orthogonal linearly oscillating fields, one along the di-rection of the fast optic axis (here named ordinary wave) andthe other along the orthogonal slow optic axis (extraordinarywave). The ratio of the two projection amplitudes fixes theangle θ, from which the average index n controlling the dy-namical phase retardations can be defined.

light beam passing through a uniform medium is ∆Φ =2πnd/λ, where d is the medium thickness, λ the vac-uum wavelength and n the medium refractive index(the product nd is also named “optical path length”).If the medium is birefringent, for any given propaga-tion direction two distinct refractive indices can be de-fined, typically termed ordinary and extraordinary in-dices, here denoted no and ne. The dynamical phase foran optical wave passing through such a birefringent (uni-form) medium is then defined as the natural extensionof the standard optical phase, i.e. γd = 2πnd/λ, wheren = cos2 θno + sin2 θne is the average refractive index, θbeing the angle formed by the input linear polarizationdirection with the ordinary (fast) axis in the transverseplane. More generally, if the input polarization is ellip-tical, θ is similarly defined by the projection angle ofthe input electric field onto the ordinary axis (see Fig.1). In other words, the index average is weighted withthe relative intensities Io ∝ cos2 θ, Ie ∝ sin2 θ of the or-dinary and extraordinary components of the wave, i.e.n = (Iono + Iene)/(Io + Ie) . For example, when usingthe standard birefringent waveplates, such as half-waveor quarter-wave plates, θ is determined by the orienta-tion of the waveplate axis with respect to the input po-larization. In particular, in all cases in which the wave-plate axis is oriented at 45 with respect to a linear in-put polarization or if the input light is circularly polar-ized (in the latter case, for any orientation of the wave-plate), the average index reduces to the arithmetic meann = (no + ne)/2. Represented the polarization states aspoints on the Poincare sphere, all these cases correspondto rotating around an axis that is orthogonal to the di-rection defined by the initial point on the sphere, andhence they correspond to moving along a geodesic arc onthe sphere (i.e. an arc of a great circle)30.

Now, as mentioned in the introduction, besides thedynamical phase defined above, birefringent media in-troduce an extra phase shift that is independent of thepropagation length and is determined only by the geom-

etry of the polarization-evolution path followed on thePoincare sphere. More precisely, for any cyclic evolutionstarting and ending in the same polarization state, theextra phase over the cycle is given by the following simpleexpression: γg = −Ω/2, where Ω is the solid angle sub-tended by the closed path (positive if the cycle is counter-clockwise when seen from outside, negative if clockwise).This extra phase is just the PB geometric phase18,29. Forexample, a sequence of three waveplates may bring theinput polarization from a starting polarization A to B,then to C and finally back to A along three geodesic arcson the sphere. The PB phase γg is then proportionalto the solid angle Ω(A,B,C) of the “geodesic triangle”ABC on the Poincare sphere (see Fig. 2). If the pointsA,B,C are displaced, for example by a suitable rotationof the waveplates, but the connecting lines remain arcs ofgreat circles, the phase change will only be given by thevariations of γg, because the dynamical phase will stayconstant.

The same definition of PB phase can be also applied tothe phase difference between two distinct optical trans-formations sharing the same initial and final states, asthis can obviously be traced back to the phase variationof a single cyclic transformation by reversing one of thetwo transformations. This leads us directly to what isprobably the simplest example of a PB geometric phasethat can be introduced and easily controlled experimen-tally. This is associated to the transformation from agiven input circular polarization state (left or right) tothe opposite one, as obtained by the action of a singlehalf-wave plate whose birefringent optic axis orientationis specified by the angle α (measured with respect to afixed reference axis in the transverse plane). We are inter-ested in the phase obtained by this transformation for anarbitrary angle α, relative to that obtained for a referenceangle, say α0 = 0. This corresponds in turn to the phasedifference between two distinct transformations leadingfrom left to right circular polarization or vice versa. Fora circularly polarized input, the half-wave plate generatesa geodesic arc on the Poincare sphere, corresponding tothe “meridian” located at 45+ 2α (relative to the refer-ence direction), so that the two transformations subtenda solid angle ±4α and hence have a PB phase differencegiven by γg = ±2α, where the sign is determined by theinput polarization (in γg, it is + for left circular inputand − for right circular input, where we adopt the nam-ing convention corresponding to the point of view of thereceiver). Hence, the optical phase of the outgoing wavecan be controlled by simply rotating the half-wave plate.In contrast, the dynamical phase is independent of α, soit plays no role.

III. CONTROLLING THE OPTICALWAVEFRONT: GEOMETRIC-PHASE LENS

The simplest possible design of a PBOE is a birefrin-gent half-wave plate having a transversely space-variant

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FIG. 2. Poincare sphere and geodesic triangles. ThePoincare sphere can be used to represent all possible polar-ization states of light. The cartesian coordinates in this repre-sentation are the reduced Stokes parameters si = Si/S0 withi = 1, 2, 3, each ranging from −1 to +1. A closed curve onthe surface formed by a sequence of polarization transforma-tions determines a subtended solid angle Ω and the resultinggeometric PB phase accumulated in the process. Particularlyinteresting are the curves formed by “geodesic arcs”, becausefor such curves the dynamical phase is constant (for a fixedmedium thickness). A geodesic arc between two points on thesphere is the minimum-length path connecting the points onthe surface: it is a concept that generalizes the notion of a‘straight line’ for curved surfaces. In this figure, a geodesic‘triangle’ is depicted, connecting three points A, B and C withgeodesic arcs and subtending a solid angle Ω.

optic axis, so that the PB phase discussed in the last ex-ample of the previous section varies in the transverse di-rections and results into a reshaped optical wavefront. Inorder to work properly and obtain only phase effects, theinput light must be circularly polarized, and the outputwill be also circularly polarized with the opposite circu-lar polarization (if different polarizations are instead usedat input, one obtains a complex polarization transversestructure in the output). The resulting optical phaseelement has a uniform thickness, despite the arbitrar-ily large phase differences that can be induced acrossthe plate. Another peculiar feature of this approach isthat opposite handedness of the input circularly polar-ized light give rise to opposite phase retardations acrossthe plate, and hence to “conjugate” wavefront outputs.The working principle of the device is illustrated pictori-ally in Fig. 3.

The standard materials used for making birefringentoptics are crystals, whose optic axis cannot be patterned.In principle, one could obtain a “segmented” PBOE bysimply attaching together several pieces of a crystallinehalf-wave plate, properly shaped and oriented, althoughthis approach is clearly not very convenient. Modern ap-

FIG. 3. Working principle of a PBOE. A PBOE canbe made as a birefringent medium with a uniform half-wavebirefringent retardation but an optic axis that is space-variantin the transverse plane. An input circularly-polarized planewave passing through the medium will be transformed intothe opposite-handed circular polarization uniformly across theplate. However, the polarization transformations taking placein the medium are different at distinct points in the trans-verse plane and hence give rise to a space-variant transversePB phase and a correspondingly reshaped output wavefront.For example, given any two transverse positions P1 and P2,the polarization evolution across the medium (black lines) isdifferent and corresponds to two distinct meridians on thePoincare sphere, sharing the initial and final points (i.e. thepoles on the Poincare sphere, corresponding to opposite cir-cular polarizations). Hence, the two optical rays will acquirea relative PB phase difference given by half the solid anglesubended by these two meridians.

proaches exploit different technologies to obtain pattern-able birefringent media, such as manufacturing subwave-length gratings or more complex metasurfaces, or usingliquid crystalline materials. Liquid crystals are particu-larly well-suited for making PBOEs, as they are as bire-fringent as many ordinary crystals (actually, they typ-ically have a fairly large birefringence), but their opticaxis can be easily made non-uniform and controlled withpatterned surface treatments or external fields.

In order to describe mathematically the action ofthe PBOE, we can use the Jones-matrix formalism31,in which the polarization state is described by two-component complex vectors giving the amplitudes of twoprescribed orthogonal components of the total opticalfield. The choice of orthogonal polarization pair definesa basis in the Jones vector space. Anisotropic media arethen simply described by two-by-two matrices, namedJones matrices. In particular, a birefringent medium seenin the ordinary-extraordinary linear polarization basis (asin Fig. 1) is described by the following diagonal Jones ma-trix (excluding the contribution of the dynamical phase,

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which is the same for all polarization components):

L0 =

(e−iδ/2 0

0 eiδ/2

). (1)

where δ = 2π(ne−no)d/λ is the birefringent retardationand we assume ne > no without loss of generality. Fora half-wave retardation, δ = π. The Jones matrix for anarbitrary xy basis in the transverse plane can then beobtained from this by the action of the rotation matrix

R(θ) =

(cos θ − sin θsin θ cos θ

), (2)

where θ represents the angle of the medium fast (ordi-nary) axis with respect to the x axis. More precisely,this matrix will rotate the x axis into the fast axis andthe y axis into the slow axis. Hence, the Jones matrixfor a birefringent medium whose fast optic axis forms anangle θ with the x axis is Lθ = R(−θ) ·L0 ·R(θ).

In a PBOE, the optic axis depends on the trans-verse coordinates, i.e. θ = θ(x, y), so that this ma-trix L is also a function of x, y. We now make an-other basis transformation, from the linear-polarizationbasis to the circular-polarization (CP) basis. In par-ticular, we introduce the complex unit vectors of theleft-/right- (L/R) circularly polarized states as follows:

L = (x+ iy) /√

2, R = (x− iy) /√

2. The Jones ma-

trix of the PBOE in the L, R basis is then given by

T (x, y) = cosδ

2

(1 00 1

)−i sin

δ

2

(0 e−2iθ(x,y)

e2iθ(x,y) 0

).

(3)The PBOE is working properly – i.e. it is properly“tuned” – if δ = π. In such case, the first term inthe equation above vanishes and only the second termis present. A left-handed CP input wave with Jonesvector (1, 0) in the CP basis, after passing through thePBOE is simply converted into the opposite CP state(0, 1) and multiplied by the space-variant phase factorexp i2θ(x, y), which is the expected PB phase giving riseto the wavefront reshaping. If the input CP polariza-tion is right-handed, the output will be left-handed andthe phase factor becomes exp−i2θ(x, y), i.e. the phaseis sign-inverted and the conjugate wavefront is obtained.

When δ 6= π, i.e. the PBOE is “untuned”, only afraction sin2 δ/2 of the input intensity suffers the phasechange (which remains the same as for a tuned PBOE),while the remaining cos2 δ/2 is unaffected. The retar-dation δ depends on the medium thickness d, the wave-length λ and the refractive indices ne and no. The lasttwo, in particular, will depend on temperature and on theactual optic axis three-dimensional orientation, as for ex-ample its tilt with respect to the transverse plane xy. Inliquid crystals, this tilt can be controlled electrically, andthis yields a very convenient method for tuning δ and set-ting it to the desired value of δ = π for any given workingwavelength32.

FIG. 4. GPL optical structure. Optic axis distribution ofa spherical (a) [cylindrical (c)] GPL between crossed polariz-ers and the corresponding microscopic images taken betweencrossed polarizers (b) and (d)); dark fringes in the imagescorrespond to locations where the optic axis is aligned eitherparallel or orthogonal to one of the polarizers.

Let us now specialize the discussion to a PBOE specif-ically designed to act as a focusing or defocusing lens,that is a GPL. This corresponds to saying that thePB phase must be quadratic in the transverse coordi-nates x, y33,34. Hence, we may generally set θ(x, y) =(σ/4)[(x/r0x)2 + (y/r0y)2], where r0x and r0y are curva-ture radii along the two axes and the sign variable σ = ±1defines the rotation direction of the optic axis (i.e. if theoptic axis rotates clockwise or counter-clockwise whenmoving away from the origin). These parameters areconstructive properties of the GPL. However, σ can besign-inverted by simply flipping the device by 180, or byreversing the propagation direction of the light throughthe device. For r0x = r0y, the GPL is circularly symmet-rical and equivalent to a spherical lens, while for r0x = r0and r0y →∞, it becomes equivalent to a cylindrical lenswith the axis along y. These two geometries are shownin Fig. 4.

For an input CP wave, the phase retardation in-duced by this GPL device will be given by γg(x, y) =±(σ/2)[(x/r0x)2 + (y/r0y)2], where ± = + for an inputleft CP and ± = − for an input right CP. Therefore, thewavefront undergoes a change of curvature equivalent tothat provided by a lens of focal distances fi = ±σπr20i/λ,with i = x, y (see, for example, Refs.33 and34). Hence,the sign of the focal distance of a GPL, determining if thelens acts as converging or diverging, will depend on boththe side of the plate used as input and the handedness

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FIG. 5. Geometric-phase lens operation. The geometric-phase lens (GPL) with circular polarization input behaves asa converging (continuous-line rays) or diverging (dashed-linerays) lens depending on the input polarization handedness. Ifthe GPL is crossed in the opposite direction, the converging-diverging polarizations are swapped.

of the input CP wave. The polarization-controlled fo-cusing/defocusing operation of the GPL is schematicallyillustrated in Fig. 5.

IV. EXPERIMENT

A. Fabrication method

Our GPLs were fabricated by using polarization holog-raphy in combination with the photoalignment of ne-matic liquid crystal cells22,35. Our manufacturing proce-dure is aimed at realizing Ø1/2” lenses with prescribedFDs.

Liquid crystal cells were initially realized assemblingtwo parallel glass substrates separated by 6 µm with fixedspacers, which had been previously coated with a suit-able photoaligning film35. Before filling them with theliquid crystal, the empty cells were patterned by suitableexposure to an expanded laser beam (532 nm frequency-doubled Nd:YVO4) with a prescribed inhomogeneous dis-tribution of the linear polarization state corresponding tothe desired pattern of the GPL optic axis. This polar-ization pattern is obtained in turn by using the outputof a polarizing Mach-Zehnder interferometer (see Fig. 6)having a standard glass template lens inserted into oneof the arms. Let the phase difference between the Gaus-sian beam propagating in the reference arm and the lens-transformed Gaussian beam be denoted as ψ(x, y). Inpolarization holography, the interference between the twocircularly polarized beams having opposite helicities andphase difference ψ(x, y), instead of an intensity modula-tion, gives rise to a space-variant linear polarization with

FIG. 6. Schematic of the experimental apparatus forGPL photoalignment. The light beam at λ = 532 nmemitted by a frequency-doubled Nd:YVO4 laser is magnifiedto half-inch diameter and injected into a polarizing Mach-Zehnder interferometer. The light beams propagating alongthe two arms are linearly polarized along orthogonal direc-tions. A quarter waveplate is placed after the output polar-izing beam splitter in order to turn the two superimposedbeams into circularly polarized with opposite helicities . Thetemplate lens TL of (positive) focal lenth f is placed in one ofthe two arms, while the photosensitive GPL cell (before fill-ing it with liquid crystal) is located in the output beam afterthe quarterwave plate at approximately 2f distance from thetemplate lens.

a polarization direction oriented at angle ψ(x, y)/2 withrespect to the reference axis. This distribution of polar-ization axes was then directly transferred into the opticaxis of the GPL by exposing the photosensitive film ofthe coated glass slides to the interference field. After theexposure, the cell was filled with a nematic liquid crys-tal (commercial mixture E7, from Merck), by exploitingcapillarity. The resulting geometric phase provided bythe GPL is then 2θ(x, y) = ψ(x, y), that is the same asthe phase difference between the two beams inside theinterferometer.

Once the cell is assembled, the birefringent phase re-tardation δ of the GPLs can be controlled by applying a10 kHz square-wave AC electric voltage with adjustableamplitude32. The tuning condition δ = π (half-wave re-tardation) is obtained for ≈ 2.5 V peak-peak. The GPLcan also be optically “switched off” by applying the volt-age giving full-wave retardation (≈ 1.0 V).

Let us call f the focal distance of the template lens.The focal distance of the manufactured GPL needs notbe exactly the same as that of the template lens, but itcan be adjusted to a slightly different value f by placingthe cell at a distance f+ f from the template lens duringthe exposure stage. In fact, the radius of curvature of aGaussian beam transformed by a lens of focal distancef at a distance f + f from the lens is f up to terms of

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FIG. 7. Near field white-light image of the fabri-cated GPL between crossed polarizers. The equivalent-curvature parameter r0 = 160 µm of the GPL is reported.In inset (a), the effect of the insertion of the λ-object beforethe analyzer is displayed. This image reveals that two ad-jacent maxima correspond to distinct values of the refractiveindices and therefore correspond to orthogonal orientations ofthe optic axis. In inset (b) the deduced distribution of theoptic axis is also shown, noting the locations where splay andbend elastic reactions are induced in the liquid crystal.

order f2λ2/(n2π2w40), w0 being the beam waist of the

input beam on the lens. However, the dependence of thefocal distance of a GPL on 1/λ requires more attention,since usually the cell is photoaligned using a light beam ofwavelength λ that is different from the actual wavelengthλ′ of operation. To account for this mismatch, one cansimply place the cell at a distance f+λ′f(λ′)/λ from thetemplate lens. In particular, to fabricate a GPL havingfocal distance f(λ′) = f , the distance of photoalignmentis to be f(1 + λ′/λ).

In our experiment, we used a template cylindrical lensof focal distance f = 150 mm and fabricated a cylin-drical GPL of nominal focal distance f = 130 mm atλ = 632.8 nm.

B. Characterization of GPL via polarizationmicroscopy

To fully characterize the fabricated GPL and check thequality of the product with respect to the design spec-ifications, we measured the characteristic curvature pa-rameter r0. In Fig. 7, we show the near field white-lightimage of our GPL between crossed polarizers, recordedby a digital camera connected to a polarizing microscope(Axsioskop, Zeiss) via a C-mount adapter. The theoret-ical intensity profile along the x-direction perpendicularto the fringes can be obtained in the following way. Con-sidering a linearly polarized input field, say along the xaxis, the transmitted optical field across the GPL can be

computed with the help of the Jones-matrix formalism,using Eq. (3) with the parabolic expression of θ(x, y).The output field is then projected along the orthogonaldirection y. The resulting intensity profile along the x-direction perpendicular to the fringes is described by thefunction

I(x) = I0 sin2 δ/2 sin2 (x2

r20+ 2θ0), (4)

where θ0 is an angle representing a rigid rotation of theoptic axis distribution with respect to the axis of one ofthe crossed polarizers. The retardation δ includes thedependence on the wavelength, which turns out to befactorized with respect to the spatial modulation. Thisentails that the positions of intensity maxima xM (orminima xm) are fixed. When the central fringe of the pat-tern is dark (central minimum), as is the case in Fig. 7),θ0 = 0 and

x2M/r20 = (2h+ 1)π/2, (5)

h being an integer number. By fitting the position ofthe maxima xM (measured in pixel units, as obtainedfrom the CCD camera image) as a function of the or-der of interference h and accounting for the pixels-to-length conversion factor, we obtain the curvature radiusr0 = 160.9± 0.4 µm. The error is derived from the dataresidual deviations from the best-fit. From this, the pre-dicted focal distance of the GPL at λ = 632.8 nm isf = 128.45± 0.07 mm.

Measuring r0 through this procedure has not only themerit of returning an accurate estimation of the expectedfocal length f , but also of testing the correctness of thelens quadratic phase profile, since this is related to thespace-variation scale of the polarization fringes. Two ad-jacent maxima on the right (or on the left of the centralminimum) correspond to two orthogonal orientations ofthe optic axis of the GPL, as can be easily inferred bythe fringe pattern obtained inserting a λ-object in be-tween the sample and the output polarizer36 (see inset(a) in Fig. 7). The elastic reactions induced in the liquidcrystals around adjacent maxima are different in nature– splay or bend – (see inset (b) in Fig. 7). This observa-tion may qualitatively explain the slight odd-even effectseen in the decreasing of the fringe width.

C. Characterization of the GPL via beampropagation analysis

As the next step, the optical lensing effect of the GPLwas measured directly. A Gaussian beam (TEM00 inputmode) at λ = 632.8 nm generated by a Helium-Neon laserwas sent through the GPL and then imaged by a digitalcamera placed at various propagation distances after thelens. By analyzing the acquired images, we determinedthe corresponding beam radii wσi (z) with i = x (focus-ing/defocusing direction) and i = y (lens axis, with no

8

focusing effect) for both left and right input circular po-larizations. The input beam radius was also imaged andanalyzed, yielding a beam-waist radius w0 = 1.478 mm.All beam radii were obtained through Gaussian best-fitsof the acquired transverse profile images (see Figs. 8 (a)and (b)). As expected, for each of the two input CPstates, the beam radii vary as functions of the distancez from the lens: the radius along the y axis of the GPLremains unchanged for both polarization states (Figs. 8(a) and (b), dashed lines); conversely, the radius alongthe x axis perpendicular to the lens axis is increasingfor a given input circular polarization (defocusing mode)(Fig. 8 (a), continuous line), while it exhibits a minimumfor the orthogonal circular polarization (focusing mode)(Fig. 8 (b) continuous line). The beam radius wσx wasalso fitted as a function of the longitudinal coordinate zaccording to the law

wσx2 = wσ0x

2

[1 +

(z

zR

)2], (6)

zR being the Rayleigh range of the gaussian beam. Inthe divergent case (Fig. 8 (a)), the beam waist (originof z axis) is virtually located behind the lens and thequadratic profile of (w−x )2 as a function of z cannot beexperimentally observed around its minimum. Conse-quently, the best fit in this case is affected by larger un-certainties than in the case (w+

x )2, and this translatesinto larger best-fit-residuals assumed as errors on the ex-perimental data points. From the position of this mini-mum we can determine the actual focal distance of theGPL, which is found to be f = 130.3 ± 0.3 mm, veryclose to that predicted from the structural analysis. Thesmall discrepancy between the two values is likely due toour small underestimation of the input beam divergenceassociated with the assumption we made that the inputbeam is perfectly Gaussian.

V. CONCLUSIONS AND FINAL REMARKS

In summary, after introducing the fundamental con-cept of the Pancharatnam-Berry optical phase, in this pa-per we explained the working principle of optical phase el-ements that exploit this phase, i.e. the PBOEs. We thenrestricted our attention to the specific case of geometric-phase lenses and illustrated a manufacturing method thatmakes use of polarization holography for patterning theoptic axis of a liquid crystal cell using a standard glasslens as a template, leading to the preparation of a GPLthat has approximately the same optical features. Fol-lowing this approach, we prepared a cylindrical GPL andthen characterized it by means of a direct structural anal-ysis with polarized microscopy and by studying the in-duced focusing effects for different input circular polar-izations.

We propose that GPLs such as that demonstrated inthis paper might be used for teaching demonstrations or

FIG. 8. Beam radii beyond the GPL as functions of thedistance z from the lens. (a) Beam radii along x (4 fordata points; continuous line for theoretical model) and along y( for data points; dashed line for theoretical model based onGaussian beam optics) for one of the two circular-polarizationhandedness at input. (b) Beam radii along x (4 for datapoints; continuous line for the theoretical model) and along y( for data points; dashed line for theoretical model) for theopposite handedness at input. Error bars are estimated fromthe fit residuals. It is evident that the radius along y is alwaysunaffected by the lens, while the radius along x is decreasingto a minimum and the increasing again in the converging-lenscase obtained for one circular-polarization handedness andmonotonically increasing in the diverging lens case obtainedfor the opposite handedness.

labs organized within a class of modern optics, or in sup-port of an undergraduate-level introduction to the topicof Berry phase, within a course of modern physics orquantum mechanics. In the case of optics classes, we be-lieve that the observation that a flat and extremely thinelement can induce a very strong focusing, similar to thatof a thick lens, can be very surprising, and it can triggerin the students useful discussions and reasoning aboutthe role of optical phases in determining the subsequentpropagation. The polarization dependence of the lenswould also be surprising for most students, and it maygive rise to unusual double-imaging effects when appliedto natural (unpolarized) light, because a distinct imagemay be formed for each polarization component. Whenapplied to illustrating Berry phases, these lenses may be

9

used to vividly demonstrate the role that these phasesmay have in real-world effects.

Acknowledgments We thank Mr. Joseph Junca foruseful discussions concerning the fabrication and test-ing of the GPL. This work was supported by the Eu-ropean Research Council (ERC), under grant no. 694683(PHOSPhOR).

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