Geometric-Phase Waveplates for Free-Form Dark Hollow beams
GGeometric-Phase Waveplates for Free-Form Dark Hollow beams
Bruno Piccirillo, ∗ Ester Piedipalumbo,
1, 2, † and Enrico Santamato Department of Physics “E. Pancini”, Universit`a di Napoli Federico II,Complesso Universitario MSA, via Cintia, 80126, Fuorigrotta-Napoli, Italy INFN-Sezione di Napoli, Complesso Universitario MSA, via Cintia, 80126, Fuorigrotta-Napoli, Italy (Dated: May 5, 2020)We demonstrate the possibility to create optical beams with phase singularities engraved intoexotic intensity landscapes imitating the shapes of a large variety of diverse plane curves. Toachieve this aim, we have developed a method for directly encoding the geometric properties ofsome selected curve into a single azimuthal phase factor without passing through indirect encryptionmethods based on lengthy numerical procedures. The outcome is utilized to mould the optic axisdistribution of a liquid-crystal-based inhomogeneous waveplate. The latter is finally used to sculptthe wavefront of an input optical gaussian beam via Pancharatnam-Berry phase.
INTRODUCTION
Light sculpting has gained increasing importance inboth fundamental and applied optics [1]. Engraving sin-gularities in optical beams, in particular, has paved theway for multiple applications in both classical and quan-tum optics, most of which related to the angular momen-tum of light. Singular Optics has gradually become an in-dependent research field and now aspires to become a fun-damental cornerstone of modern photonics. Optical sin-gular beams have proven to be invaluable for non-contactmanipulation over micro- and nanoscale [2, 3] – whichhas enormous implications on modern nanophysics, crys-tal growth, metamaterials, to give just a few examples.Furthermore, the infinite dimensionality of the orbital an-gular momentum (OAM) space has paved the way for in-creasing data capacity of both free-space and fiber-opticcommunications [4] and for developing novel efficient pro-tocols for classical [5] as well as quantum informationprocessing [6–8]. No less important, optical singularitieshave been successfully utilized for super-resolution imag-ing [9, 10], on-chip optical switching [11–13], advancedmicroscopy [14, 15] and material machining [16–18].Needless to say the great potential of singular op-tics – and, more generally, of sculpted light – hasbeen progressively unlocked in time, through the de-velopment of increasingly efficient and versatile toolsfor shaping the optical wavefronts. The most promi-nent technologies currently available for shaping spa-tial modes are computer generated holograms (CGHs)displayed on spatial light modulators (SLMs) – basedon dynamic phase control – and Pancharatnam-Berryphase Optical Element (PBOEs) or geometric phase. In-deed, several methods are nowadays available to fabricategeometric-phase optical elements for wavefront-shaping,ranging from subwavelength metal stripe space-variantgratings [19], to multilayer plasmonic metasurfaces [20]and Spatially Varying Axis Plates (SVAPs) based on liq-uid crystals [21–26].In the present paper, we introduce a method for de- signing SVAPs enabling to generate scalar optical beamswith nonlinear azimuthal phase structures giving birthto phase singularities engraved within non-cylindricallysymmetric intensity profiles. Indeed, the cylindrical sym-metry typical of the intensity profile of helical beamssprings from their linear azimuthal phase profile, e i (cid:96)φ .Helical beams have helical wavefronts – hence the name– and carry an OAM of ¯ h(cid:96) per photon, (cid:96) being an in-teger number and φ the azimuthal polar angle aroundthe beam propagation direction. There are multiplefamilies of helical beams differing for their radial de-pendences. Well-known examples are Laguerre-Gaussian(LG) beams [27, 28], Bessel and Bessel-Gaussian (BG)beams [29] and the wider class of Hypergeometric-Gaussian (HyG) beams [30], to name just a few. A he-lical beam with an azimuthal index (cid:96) has an (cid:96) -fold ro-tational symmetry and its OAM spectrum accordinglyincludes only the component (cid:96) . Denoted as k the lightbeam wavevector, the azimuthal component of the lin-ear momentum is ¯ hk φ per photon: it does not dependon φ , but only on the distance from the beam axis.The energy flux is therefore rotationally invariant aroundthe beam axis, yielding the well-known cylindrically-symmetric doughnut-shaped profile. An azimuthallynonuniform k φ , in contrast, will break such symmetryand will give birth to an optical wavefront with a nonuni-form helical phase structure, which will result, in its turn,into a non-cylindrically symmetric intensity profile. AnOAM spectrum will broaden as a consequence of suchsymmetry breaking.To impart a nonlinear azimuthal structure, we havedeveloped a phase design method aimed at encoding thegeometric properties of some plane curve, in order to cre-ate an intensity profile imitating the shape of the curve.We presently demonstrate that such an approach enablesto directly determine the phase profile required to re-shape the intensity profile of a light beam, as well as itsOAM spectrum according to one’s wishes. Here, in fact,we avoid passing through indirect methods for encodingamplitude and phase of the target field into a single phasefunction [31]. Though, the price to be payed is that only a r X i v : . [ phy s i c s . op ti c s ] M a y some features of the intensity profile and of the OAMspectrum will be precisely determined. Despite these ap-parent limitations, our method spontaneously leads us tointroduce the concept of dark hollow beams with tailoredintensity profiles or “Free-Form Dark-Hollow” (FFDH)Beams. A detailed study of the optical properties ofFFDH beams will be reported elsewhere. Here we fo-cus the attention on the generation of such beams byusing the aforementioned SVAPs, of which q-plates [32]are probably the most famous examples. Liquid-crystalsbased SVAPs combine high conversion efficiency withexceptional manageability for overall high performance.Our SVAPs were fabricated adopting a “direct-write ap-proach”, as defined in Ref. [21]. However, we would liketo emphasize that our focus is presently on the methodworked out to determine the transmittance phase func-tion. Specifically, an arbitrary superposition of azimuthalmodes amounts to a complex function of φ with both anamplitude and a phase, i.e. (cid:88) (cid:96) c (cid:96) e i (cid:96)φ = A ( φ )e iΨ( φ ) . (1)Several approaches mostly based on Gerchberg-Saxtonalgorithm are usually adopted to obtain a pure phasefunction providing an acceptable approximation forEq. (1) [33]. In what follows, we describe a method todirectly generate a dark hollow beam in which the shapeof the dark zone is basically inherited from the shapeof a selected plane curve. This is achieved without re-curring to inverse algorithms such as those mentionedabove. They can be proved to be promising devices ofpotential interest for multiple applications ranging fromsuper-resolution microscopy, to directional selective trap-ping [34], as well as material processing, optical coro-nagraphy, not to mention the applications to classicaland quantum communications [35, 36]. As an example,we consider the case of Stimulated Emission Depletion(STED) microscopy, in which super resolution is achievedby the selective deactivation of fluorophores through anexcitation beam filling the internal zone of doughnut-shaped de-excitation spot. Replacing the doughnut withan FFDH beam, the illumination area would acquire anon-circular shape, suitable for optimally send photonsto zones where they are really required and/or to preventthem from damaging the surrounding areas. FREE-FORM AZIMUTHAL PHASE SHAPING
The question arises as to what extent the transverse in-tensity profiles or the OAM spectrum of a light beam canbe moulded by manipulating a purely azimuthal phasefactor e i ψ ( φ ) , ψ ( φ ) being an arbitrary function of theazimuthal coordinate φ . Such a phase factor does notenable to explore all the possible field distributions, nei-ther approximately, since ψ is assumed independent of the distance r from the beam axis [31, 37]. As abovementioned, in this work, we aim at introducing a toy -method based on geometric intuition to determine themost appropriate azimuthal phase factor e i ψ ( φ ) requiredto generate dark hollow beams having arbitrary shapesor, as we have baptized them, FFDH beams. To this pur-pose, we need a “dough cutter” for partitioning the planearound the beam axis into a number of sectors – “slicingthe doughnut” – and then distribute the transverse inten-sity of light among the several sectors according to one’swishes and necessities. Moulding the intensity of lightwithin each sector is necessary for tailoring the bound-aries of the dark region around the axis – “shaping thehole of the doughnut”. The portions of light within differ-ent sectors can be disconnected from each other or not.Metaphors aside, our “dough cutter” is the azimuthalcomponent ¯ hk φ ( φ ) of the photon linear momentum as afunction of φ , i.e. k φ ( φ ) = 1 r dψ ( φ ) dφ . (2)Assuming ψ ( φ ) proportional to the orientation angleΘ( φ ) of the unit normal to some plane curve γ describedby φ -dependent parametric equations, then all the rel-evant features of k φ ( φ ) can be gathered from the rota-tional symmetry properties of γ and from the local radiusof curvature – the latter being related to both k φ and itsderivative. Such a geometric approach has the advantagethat Θ( φ ) – and therefore the plane curve it comes from –needs not to be determined, on a case-by-case basis, as asolution of an inverse problem. Rather, it can be helpfulusing a representation of the curve in polar coordinates,with some free parameters that can be tuned to matchas much as possible the target intensity profile. Curve selection
Multiple choices are available. Good options are Lam´ecurves or their generalizations. A Lam´e Curve, alsoknown as a superellipse [38], is a closed curve retainingthe geometric properties of semi-major axis and semi-minor axis, typical of an ellipse, but with a differentshape. In polar coordinates it is described by the equa-tion ( a cos φ ) nn − + ( b sin φ ) nn − = ρ ( φ ) nn − , (3)where a , b , and n are positive reals.In 2003, J. Gielis introduced a single parametric equa-tion – dubbed the “superformula” – describing multipleplane curves, of the most varied kinds, to study forms inplants and other living organisms [39]. The mathemati-cal expression of the superformula, in polar coordinates,is ρ ( φ ) = (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos mφ a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin mφ b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:33) − n , (4)where ρ is the distance of a point of the curve γ fromthe origin of the coordinate system as a function of theazimuthal angle φ , m is an integer number, n , n and n are three integers controlling its local radius of curvatureand, finally, the positive real numbers a and b parameter-ize the radii of the circumferences respectively inscribedand circumscribed to the curve γ . For even m = 2 k ,Eq. (4) describes a curve γ k closing over the interval[0 , π ). γ k is rotationally symmetric by an angle 2 π/k .For odd m = 2 k +1, γ k +1 closes over the interval [0 , π ).When a = b and n = n , γ m exhibits an m -fold rota-tional symmetry C m . As varying all the free parametersin Eq. (4), the generated curves can be deeply diverse.No doubt the curves could be grouped according to acriterium based on the order of the their rotational sym-metry. For m = 4, a = b and n = n >
2, for instance,the superformula simply returns the superellipses first in-troduced by G. Lam´e in 1818 [38]. For fixed values of m , a and b , however, the signs and the absolute values of n , n and n can dramatically change the topologicalproperties of the curves. Besides, a peculiar feature ofthe superformula is the fact that, independently of m ,when n = n = 2, it always degenerates into a circum-ference when a = b , or into an ellipse otherwise. Here,we are not interested in the mathematical peculiarities ofthe superformula, but rather in taking advantage of its“shape-shifter” capabilities. Encrypting the geometrical properties of the selectedcurves into the optical phase. Be γ ( a, b, m, n , n , n ) thecurve described by the superformula for some values ofthe free parameters. The normal unit vector n = ( n x , n y )of the curve is given by( n x + i n y ) = ρ ( φ ) − i ˙ ρ ( φ ) ρ ( φ ) + i ˙ ρ ( φ ) e φ , (5)where ˙ ρ is the derivative of ρ with respect to φ . Denotedas Θ( φ ) the angle that n forms with the x − axis, we setthe optical phase ψ ( φ ) to be ψ ( φ ) = 2 Θ( φ ; a, b, m, n , n , n ) . (6)Consequently, as varying the free parameters in Eq. (4),multiple phase profiles can be designed and FFDHs ac-cordingly generated. The realized phase profiles exhibit amodulation having the same symmetry properties as thecurve γ . In the following we show that the m -fold sym-metry characterizing the phase modulation affects alsothe intensity profile of the generated beam. Light inten-sity, indeed, is expected to be equally partitioned amongthe m equally spaced sectors of the phase profile.In Fig. 1, this geometry-to-phase transfer procedure issketched in the case a = b = 1, m = 5, n = 1 / n = n = 4 /
3. The rippled helical wavefront arisingfrom Eq. (6) is shown in Fig. 2 ( B )) for the same valuesof the parameters and is compared to the smooth helicalwavefront corresponding to a doughnut beam with (cid:96) = 2Fig. 2 ( A )). The latter can be easily shown to come froma circumference.This structure primarily affects the OAM spectrum,which includes only the components ( (cid:96) − m ) ± k m , with k integer number (Fig. 3), (cid:96) being the OAM index cor-responding to the background helical mode. Specifically,in Fig. 3, it has been reported the OAM power spectrum | c l | of the generated FFDH. In classical optics, the quan-tity | c l | is the fraction of the total power of the opticalfield component carrying an OAM proportional to l . Inquantum optics, it is the probability that a photon in thebeam carries an OAM of ¯ hl . The actual values of | c l | , asreported in Fig. 3, have been determined numerically, byFourier expanding the azimuthal phase factor reportedin Eq. (5). The skew rays follow the paths dictated by k φ . FREE-FORM AZIMUTHAL (FFA) SVAPS
Let’s now focus the attention on the experimentalmethods for generating optical beams having the phasestructure prescribed by Eq. (6). To reshape a TEM laser beam according to our wishes, we opted in favorof a properly tailored SVAP. The latter is a half-waveretardation plate in which the direction-angle of ¯Θ( r, φ )of the optic-axis is spatially variant [23, 24, 26]. When acircularly-polarized input beam passes through the plate,it acquires a geometric phase factor e ± i2 ¯Θ( r,φ ) . The signin the exponent depends on the handedness of the in-cident beam polarization C ± = ( x ± i y ) / √
2, which isreversed by the SVAP [25]. For a comprehensive view ofthe mechanism underlying wavefront reshaping via Ge-ometric or Pancharatnam-Berry Phase, we address thereader to Ref. [25]. In essence, moulding the phase ofa SVAP amounts to pattern the optic-axis so that itsdirection-angle is locally equal to half the prescribed op-tical phase. In order to fabricate a liquid-crystal SVAPfor generating FFDH beams, the optic-axis angular dis-tribution must be set to¯Θ( r, φ ) = ψ ( φ )2 = Θ( φ ; a, b, m, n , n , n ) . (7)In Fig. 4, we show the optic-axis pattern of a SVAP cor-responding to Θ( φ ; a = 1 , b = 1 , m = 5 , n = 1 / , n =4 / , n = 4 /
3) (inset A ) and, for comparison, the contri-bution to such pattern due to the modulation only (in-set B ). In Fig. 5 ( A ), it is shown a microscope imageof the SVAP between crossed polarizers, with a birefrin-gent λ -compensator inserted between the SVAP and theanalyzer. The λ -compensator has a path difference of550 nm and therefore introduces a π retardation at thatwavelength. The fast axis forms a 45 ◦ angle to the axis ofthe analyzer. When the compensator is put in, the sam-ple changes its color depending on its orientation. Thechanges in color are based on optical interference. Thismethod fully unveils the optic axis pattern underlying theSVAP (Fig. 4 ( A )), because, differently from the simplecrossed-polarizers method, it enables to distinguish be-tween orthogonal orientations of the optic axis.Though pure-phase holograms displayed on SLM couldbe used to create FFDH beams, fabricating optical de-vices based on Geometric Phase have proved to be notonly the most performing choice, but also the most nat-ural, since the unit normal distribution deduced from agenerating curve is directly translated into an optic axispattern. As an example, we have here chosen curves gen-erated via superformula, to take advantage of a large va-riety of shapes grouped under the same equation. A sim-ilar method, however, can be applied to any other curveor family of curves. INTENSITY PROFILES
As above mentioned, by adding a periodical azimuthalphase modulation to the phase of a helical beam, thecylindrical symmetry typical of the intensity profile ofa doughnut is broken. In fact, each photon at distance r from the beam axis suffers a change in its azimuthallinear momentum k φ that depends periodically on theorientation of the meridional plane it starts from. As k φ has the same period as ρ ( φ ) in Eq. (4), the resultingtransverse intensity profile turns to be periodic as well.What’s more, the details of the profile of k φ are inher-ited from the azimuthal rate of change of the unit vectornormal to the curve, therefore also the inflections of theintensity profiles will be inherited from the local curva-ture of the generating curve. This enables to set a one toone correspondence between the geometric properties ofthe generating curve and the transverse intensity profileof the beam, especially as far as is concerned the darkregion. In Fig. 6 A , we show the intensity profile of thebeam experimentally generated for the values of the pa-rameters a = b = 1, m = 5, n = 1 / n = n = 4 / z = 1 m from the SVAP, for a circularly po-larized input TEM gaussian mode with plane wavefrontand radius w = (1 . ± .
04) mm. For comparison, ininset B of Fig. 6, it is shown the theoretical intensityprofile predicted by calculating the Fresnel transform ofthe optical field E e − x y w +2 i Θ( φ ;1 , , , / , / , / , (8)for the same values of the parameters. The faintstriped structure surrounding the core profile originatesby diffraction from the abrupt azimuthal changes in thetransverse phase profile shown in Fig. 5 B . CONCLUDING REMARKS
We have shown the possibility to generate dark hol-low beams with a large variety of intensity landscapesby using a single azimuthal phase factor without passingthrough numerical methods for the optical field encryp-tion. The method is based on a geometric approach inwhich the intensity profile around the beam axis is sup-posed to imitate the shape of a selected closed curve.Also the OAM spectrum is affected by the shape of thegenerating curve. If the generating curve has an m -foldrotational symmetry, the OAM spectrum will includeonly components with multiple of m within a global shiftdetermined by the OAM index of the unperturbed helicalmode. Liquid-crystals SVAPs turn to be the most naturalchoice to implement such method, since the unit vectornormal to the generating curve come to be copied overthe axis pattern. Applications of FFA SVAPs can be eas-ily devised, in particular, for manipulating non-sphericalobjects trapped by optical tweezers – as unwanted rota-tions of micro-objects could be avoided – as well as forincreasing contrast in optical coronagraphy – as properlytailored dark-hollow beams with line singularities alongradial directions could be exploited to split the intensitydistribution around the optical axis. CONFLICT OF INTEREST STATEMENT
The authors declare that the research was conducted inthe absence of any commercial or financial relationshipsthat could be construed as a potential conflict of interest.
AUTHOR CONTRIBUTIONS
All the authors contributed to develop the method fordesigning the Free-Form SVAPs introduced in the paper.BP fabricated and tested the SVAPs. All the authorscontributed to writing the paper.
FUNDING
This work was supported by the University of NaplesResearch Funding Program (DR n. 3425-10062015) andby the European Research Council (ERC), under grantno. 694683 (PHOSPhOR).
ACKNOWLEDGMENTS
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FIG. 1. Schematic of the encryption procedure of the symmetry properties of a plane curve into the azimuthal phase of alight beam. In inset A , as an example, we show t and n , i.e. the tangent and the normal unit vectors to the curve inthe point P respectively. n forms the angle Θ( φ ) with the horizontal axis. In inset B , we show the transverse phase profile ψ ( φ ) = 2 Θ( φ ; 1 , , , / , / , /
3) (Eq. (6)). The latter can be regarded as the superposition of the phase modulation 2 ∆Θ m ( φ )(inset C ) and the helical phase profile 2 φ (inset D ).FIG. 2. Helical wavefronts corresponding to a circumference (inset A ) and to a the curve represented in Fig. 1 A (inset B ). FIG. 3. OAM power spectrum arising from the azimuthal phase profile corresponding to the values of the parameters a = b = 1, m = 5, n = 1 / n = n = 4 / FIG. 4. Optic axis patterns deduced from Eq. (7) for the values of the parameters a = b = 1, m = 5, n = 1 / n = n = 4 /
3. Inset A : optic axis pattern for a SVAP imparting to an input beam the geometric phase 2 φ + 2∆Θ m ( φ )(Fig. 1 B ). Inset B : optic axis pattern for a SVAP imparting the geometric phase 2∆Θ m ( φ ) (Fig. 1 C ).FIG. 5. Experimental observation of the optic axis distribution of the SVAP ( a = b = 1, m = 5, n = 1 / n = n = 4 / A : microscope image of the SVAP between crossed polarizers + birefringent compensator plate at 45 ◦ . This imageunveils the optic axis pattern underlying the SVAP (Fig. 4 A ), which is displayed in the image overlay. The image was recordedilluminating with white light the sample, sandwiched between crossed polarizers, and inserting, between the sample and theanalyzer, a birefringent λ -compensator ( λ = 550 nm), having the optic axis rotated by 45 ◦ . The arrows in the lower left cornersketch the axes orientations of the input linear polarizer (black arrow), the output analyzer (red arrow) and the λ -compensator(blue arrow). Inset B : optical transverse phase profile associated to the optic axis pattern in inset A – the same as in Fig. 1 B – here replicated for the sake of comparison. FIG. 6. Comparison between the experimental (inset A ) and theoretical (inset B ) intensity profiles of the beam generatedthrough the SVAP having the optic axis pattern shown in Fig. 4 A at distance z = 1 m, for the values a = b = 1, m = 5, n = 1 / n = n = 4 / w = (1 . ± ..