crystals

Article Broadband Multichannel Optical Vortex Generators via Patterned Double-Layer Reverse-Twist Polymer

Hanqing Zhang 1, Wei Duan 1,2,*, Ting Wei 1, Chunting Xu 1 and Wei Hu 1,* 1 National Laboratory of Solid State Microstructures, College of Engineering and Applied Sciences, Nanjing University, Nanjing 210093, China; [email protected] (H.Z.); [email protected] (T.W.); [email protected] (C.X.) 2 School of Instrumentation and Optoelectronic Engineering, Beihang University, Beijing 100191, China * Correspondence: [email protected] (W.D.); [email protected] (W.H.); Tel.: +86-25-83597400 (W.D & W.H.)  Received: 31 August 2020; Accepted: 27 September 2020; Published: 29 September 2020 

Abstract: The capacity of an optical communication system can be greatly increased by using separate orbital angular momentum (OAM) modes as independent channels for signal transmission and encryption. At present, a transmissive OAM mode generator compatible with wavelength division multiplexing is being highly pursued. Here, we introduce a specific double-layer reverse-twist configuration into liquid crystal polymer (LCP) to overcome wavelength dependency. With this design, broadband-applicable OAM array generators are proposed and demonstrated. A Damman vortex grating and a Damman q-plate were encoded via photopatterning two subsequent LCP layers adopted with oppositely handed chiral dopants. Rectangular and hexagonal OAM arrays with mode conversion efficiencies exceeding 40.1% and 51.0% in the ranges of 530 to 930 nm, respectively, are presented. This provides a simple and broadband efficient strategy for beam shaping.

Keywords: beam shaping; liquid crystals; orbital angular momentum; geometric phase

1. Introduction The optical vortex (OV), an optical field featured by a helical phase front [1], has attracted extensive attention over the past few decades. The phase factor of an OV eimϕ is proportional to the azimuth ϕ, where m is the topological charge. The uncertainty of the phase front in the center of the OV makes a phase singularity, leading to a donut-like intensity distribution [2]. The Poynting vector of the beam twists m times around its propagation axis in a single wavelength, and the spinning makes the beam carry orbital angular momentum (OAM), which is quantitively depicted by m. Due to the infinity of m, the OAM forms a Hilbert space with infinite dimensions and adds a new degree of freedom for the manipulation of light [2]. An OV can be generated by a liquid crystal device called a q-plate [3] where q = m/2 and shows great potential in non-contact manipulation [4], [5,6], and information encoding and storage [7]. Especially, with different OAM modes as independent channels for signal transmission or encryption, the transmission capacity of optical communication can be dramatically extended [8,9] via mode-division multiplexing [10]. OV mode conversion efficiency is highly dependent on wavelength, thus hindering its compatibility with present wavelength-division multiplexing techniques. Methods for highly-efficient and parallel OV processing in a broadband remain highly pursued. Many efforts have been made to tackle this challenge. Damman vortex gratings (DVGs) [11] were introduced to generate OV arrays with equal energy distribution among multiple diffraction orders that carry different OAM states. Though the uniformity among various OAM states has been

Crystals 2020, 10, 882; doi:10.3390/cryst10100882 www.mdpi.com/journal/crystals Crystals 2020, 10, 882 2 of 7 well-addressed, these DVGs still suffer from significant wavelength-dependent efficiency. To solve this problem, the phase retardation/shift of OV array generators should be continuously and precisely tuned. Liquid crystals (LCs) are excellent optical anisotropic materials for phase modulations due to their large and external field tunability. Previously designed LC-mediated DVGs [12] allowed for tunable optimized bands for OV generation via the electrical alternation of the output phase retardations. More recently, the geometric phase was discovered for the light reflected off planar cholesteric LC chiral superstructures. For such a device, the reflective efficiency is constant in the Bragg band and the phase can be freely manipulated via initial alignment control [13–16]. A Damman-encoded vortex grating was presented with a wide photonic band gap from green to red. In addition, such grating can transform a to a series of reflective wave carrying different OAMs in different orders with very good energy uniformity in a broadband to 116 nm [17]. Until now, the transmissive elements suitable for generating and separating of multiple OVs remained formidable. In this paper, we introduce a double-layer reverse-twist configuration to compensate for the dispersion over a large wavelength range. By further encoding a DVG phase profile into a photopatterned liquid crystal polymer (LCP), multichannel OV arrays were generated equally-efficiently. Thanks to the broadband spin-orbit interaction, uniform OV arrays were achieved in equally-high-efficiency across visible and part of near-infrared range. This work may provide a solution for broadband and arbitrary beam shaping in a transmissive way.

2. Materials and Methods

2.1. Design and Principle Damman grating (DG) is a typical binary that provides equal energy distribution among all desired diffraction orders, providing a promising strategy for parallel beam processing [18,19]. Similar to DG, a DVG is a binary hologram calculated from the interference between a tilt plane wave and an OV. Figure1a exhibits the phase pattern of a 2 2 DVG, which is a two-dimension (2D) × DVG integrated by two one-dimension (1D) DVG with different m in x and y directions. The phase distribution ϕ of a 1D DVG (grating vector along x for instance) is denoted as ϕ = 2πx/Λ + mϕ in the x direction, where Λ is the period, m is the topological charge, and ϕ is the azimuthal angle. Here, the 2 2 DVG is directly integrated by two orthogonal DVGs (mx = 1 and my = 3, normalized × phase transition point: x1 = y1 = 0.5 [20]). As shown in Figure1a, the black regions represent 0, while the white regions represent π. Such a phase pattern can be modulated via a geometric phase by in-plane controlling the optical axis orientation α:

ϕDVG(x, y) α = + α , (1) 2 0 where ϕDVG(x, y) is the phase function of the DVG and α0 is the initial angle and usually assumed to be zero. The transmission formula of the 2 2 DVG can be expressed as: × + +   X∞ X∞  i2π T(x, y) = exp iϕ (x, y) = Cn n exp i(m nx + myny)ϕ + (n x + ny y) , (2) DVG x, y x x Λ nx= ny= −∞ −∞

th where Cn is the coefficient of the n diffraction order [21]. As mx = 1 and my = 3, the incident beam will be mainly divided into 4 orders that carry the topological charge m = nx + 3ny. Special algorithms are used to calculate the number of phase transition points [20]. Here, for x1 = y1 = 0.5, only simple 2D fork gratings that can generate a rectangular 2 2 optical vortex array are demonstrated. A more × complicated matrix can be rationally designed by separately changing the number and values of the phase transition points. Crystals 2020, 10, 882 3 of 7 Crystals 2020, 10, x FOR PEER REVIEW 3 of 7

Figure 1. (a) The binary phase pattern of a 2 2 Damman vortex grating (DVG) with mx = 1 and my = 3. Figure 1. (a) The binary phase pattern of a 2× × 2 Damman vortex grating (DVG) with mx = 1 and my = (b) A schematic of liquid crystal (LC) directors in the yellow rectangle region labeled in (a). 3. (b) A schematic of liquid crystal (LC) directors in the yellow rectangle region labeled in (a). The transmittance of the DVG can be analyzed by a Jones matrix and expressed as: The transmittance of the DVG can be analyzed by a Jones matrix and expressed as: "exp-iΓ/2 0 # Γ   Γ cos2 α" sin2α # T = R-α exp( iΓ/2) 0 Rα = cos I - Γisin Γ cos 2α , sin 2α T = R( α) − R(α) = cos I i sin ,(3) (3) − 0exp0 expiΓ/2(iΓ/2) 2 2 −2 sin22α sin-cos2 2αα cos 2α − where R is the rotation matrix, I is the identity matrix, Γ = 2πΔnd/λ is the phase retardation, Δn is the R I π nd n birefringencewhere is the coefficient, rotation matrix, d indicatesis thetheidentity thickness matrix, of theΓ birefringent= 2 ∆ /λ islayer, the phaseand λ retardation,is the free-space∆ is the birefringence coefficient, d indicates the thickness of the birefringent layer, and λ is the free-space wavelength of incident light. Considering a circular polarized incidence, it can be expressed as Ein = 1wavelength1 of incident light. Considering a circular polarized incidence, it can be expressed as . Thus, the! output can be depicted as: √2 ±i 1 1 Ein = . Thus, the output can be depicted as: √2 i 1 Γ 1 1 Γ ±i2α 1 ± Eout = T·Ein = cos - isin e , (4) √2 2 ±i √2 2 ∓i when the halfwave condition satisfied, the incident !left-circularly  polarized ! light is handedness- 1 Γ 1 1 Γ i2α 1 Eout = T Ein = cos i sin e± , (4) converted and modulated with· the designed√2 phase.2 i Meanwhile,− √2 for2 right-circularlyi polarized light, the opposite phase modulation is carried out. ± ∓ whenThe theabove halfwave design could condition be easily satisfied, realized in the an incidentLC system left-circularly where the optical polarized axis orientation light is couldhandedness-converted be directly encoded and with modulated azimuthal with angle the cont designedrol of LC phase. directors Meanwhile, [22]. As shown for right-circularly in Figure 1b, photoalignmentpolarized light, theagents opposite indicated phase in modulation yellow are isuti carriedlized to out. perform the alignment, and the adjacent LC directorsThe above marked design in orange could be follow easily the realized local gu inidance an LC and system further where spread the opticalthis order axis to orientation the bulky LC.could It needs be directly to be encodednoticed that with here azimuthal a double-lay angleer control reverse-twist of LC directors configur [22ation]. As was shown adopted. in Figure Along1b, thephotoalignment z axis, the LC agentsdirector indicated rotates by in a yellow certain are angle utilized and then to perform rotates theback alignment, in the second and layer. the adjacent Adjacent LC LCdirectors domains marked with orthogonal in orange follow planar thealignment local guidance are represented and further by blue spread and this cyan. order These to thecolors bulky mean LC. theIt needs LC helixes to be whose noticed initial that hereorientations a double-layer are 0° (blu reverse-twiste) and 90° (cyan), configuration respectively, was adopted. with respect Along to the the xz axis.axis, Here, the LC instead director of discrete rotates by wavelength a certain angle compen andsation then with rotates birefringent back in the phase second retarders layer. Adjacent[23], the sameLC domains starting withand ending orthogonal polarization planar alignment with wavele arength-dependent, represented by blue different and cyan. polarization These colors revolution mean processesthe LC helixes are introduced whose initial to ov orientationsercome the dispersion are 0◦ (blue) [24,25]. and 90◦ (cyan), respectively, with respect to the xAaxis. DVGHere, is suitable instead for of discretegenerating wavelength a series of compensation OVs with different with birefringent topological phase charges, retarders while [23 a], Dammanthe same startingq-plate and(DQP) ending [12] polarization can produce with an wavelength-dependent, OV array with a same di fftopologicalerent polarization charge. revolution Here, a hexagonalprocesses are2D Damman introduced grating to overcome that consists the dispersion of three 1D [24 Damman,25]. gratings with relative angles of 60° is encodedA DVG to is a suitable q-plate for with generating m = 1. Figure a series 2a of OVsillustrates with di thefferent optical topological axis distributions charges, while of this a Damman DQP, whereq-plate the (DQP) color [variation12] can produce from black an OVto white array indicate with a sames the directions topological of charge.the opticalHere, axis a varying hexagonal from2D 0°Damman to 180°. gratingFigure that2b shows consists the of 3D three scheme 1D Damman of the double-layer gratings with reverse-twist relative angles configuration of 60◦ is encoded in the centerto a q-plate of the withhexagonalm = 1. DQP. Figure When2aillustrates a circularly the po opticallarized axisincident distributions light passes of thisthrough DQP, this where DQP, the a hexagonalcolor variation lattice from array black of OVs to white with indicatesorthogonal, the circul directionsarly polarized of the optical states axis will varying be generated from 0 and◦ to 180will◦ . carryFigure a 2topologicalb shows the charge 3D scheme of 1. of the double-layer reverse-twist configuration in the center of the hexagonal DQP. When a circularly polarized incident light passes through this DQP, a hexagonal lattice array of OVs with orthogonal, circularly polarized states will be generated and will carry a topological charge of 1.

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FigureFigure 2. 2. ((aa)) The The optical optical axis axis distributions distributions of of a a hexagonal hexagonal lattice lattice Damman Damman q-plate q-plate (DQP) (DQP) with with mm == 1.1. ((bb)) A A schematic schematic of of LC LC directors directors in in the the yellow yellow square square region labeled in ( a).

2.2.2.2. Sample Sample Preparation Preparation TheThe indium-tin-oxide indium-tin-oxide glass glass substrates substrates were were ultrasonically ultrasonically bathed bathed and and spin-coated spin-coated with with photoalignmentphotoalignment agent agent SD1 SD1 (a (a polarization-sensitive polarization-sensitive sulfonic sulfonic azo azo dye dye that that was was dissolved dissolved in in dimethylformamidedimethylformamide at at a aconcentration concentration of of 0.3 0.3 wt.%) wt.%) [26,27]. [26,27 ].After After curing curing at 100 at 100°C for◦C 10 for min, 10 min,the substratethe substrate was wasplaced placed at the at theimage image plane plane of ofthe the Digital Digital Micro-mirror Micro-mirror Device Device (DMD)-based (DMD)-based microlithographymicrolithography system system [28,29] [28,29] to to accomplish accomplish a a uniform uniform alignment, alignment, and and then then the the DVG DVG pattern pattern was was recordedrecorded with with the the orthogonal orthogonal polarization polarization exposure exposure step. step. For For the the DQP, DQP, a a 36-step 36-step partly partly overlapping overlapping exposureexposure method method was was performed performed to to ca carryrry out out the the director director distributions. TheThe LCP LCP [30] [30] precursors precursors (UCL017, (UCL017, Dai-Nippon Dai-Nippon In Inkk and and Chemicals, Chemicals, Tokyo, Tokyo, Japan, Japan, dissolved dissolved in in methylbenzenemethylbenzene at at a aconcentration concentration of of23 23wt.%) wt.%) mixed mixed with with right right and andleft chiral left chiral dopants dopants (R811 (R811 and S811 and atS811 a concentration at a concentration of 1.27%, of 1.27%, Jiangsu Jiangsu Hecheng Hecheng Disp Displaylay Technology, Technology, Nanjing, Nanjing, China), China), respectively, respectively, werewere spin-coated spin-coated ontoonto the the SD1 SD1 membrane, membrane, layer layer by by la layer,yer, at at a a speed speed of of 600 600 r/min, r/min, followed followed by by a a two two minutesminutes of of UV UV exposure exposure for for each each layer. layer. The The precur precursorsor with with opposite opposite chirality chirality was was spin spin coated coated at at a a speedspeed of of 1600 1600 r/min r/min to to precisely precisely control control the the double, double, right right and and left left chiral chiral dopant dopant layers’ layers’ thicknesses thicknesses to to bebe 1.59 1.59 and and 1.76 1.76 μµm,m, respectively. respectively. At At this this condition, condition, the the initial initial orientation orientation of of the the LCP LCP layer layer was was kept kept thethe same same to to that that of of the the termination one. The total LCP satisfiedsatisfied thethe halfwavehalfwave condition,condition, thus thus forming forming a abroadband broadband OV OV array array generator. generator.

3. Results and Discussion 3. Results and Discussion Furthermore, in order to verify the broadband property, the planar DVG satisfied the 532 nm Furthermore, in order to verify the broadband property, the planar DVG satisfied the 532 nm halfwave condition was fabricated and analyzed, in comparison to that of the double-layer reverse-twist halfwave condition was fabricated and analyzed, in comparison to that of the double-layer reverse- configuration that could not. By adjusting the left-circularly polarized light incidents on the DVG, twist configuration that could not. By adjusting the left-circularly polarized light incidents on the Figure3a, g is able to show the micrographs of the double-layer reverse-twist and planar 2 2 DVGs DVG, Figure 3a, g is able to show the micrographs of the double-layer reverse-twist and planar× 2 × 2 under crossed polarizers. Due to the discontinuity of LC directors at the boundary between neighboring DVGs under crossed polarizers. Due to the discontinuity of LC directors at the boundary between orthogonal domains, dark lines, whose patterns are consisted with the designed phase patterns, can be neighboring orthogonal domains, dark lines, whose patterns are consisted with the designed phase observed. Figure3b–f,h–l shows the di ffraction patterns of the DVGs captured by a charge-coupled patterns, can be observed. Figure 3b–f,h–l shows the diffraction patterns of the DVGs captured by a device (VTSE3S-2000, Vihent, Shanghai, China), with incident wavelengths of 473, 532, 700, 800, and charge-coupled device (Vihent, VTSE3S-2000), with incident wavelengths of 473, 532, 700, 800, and 950 nm, respectively. Due to the geometric phase modulation, the beam was diffracted to different orders 950 nm, respectively. Due to the geometric phase modulation, the beam was diffracted to different carrying designed topological charges, e.g., as indicated by white circles in Figure3b. As expected, orders carrying designed topological charges, e.g., as indicated by white circles in Figure 3b. As m expected,donut-like donut-like OVs were obtainedOVs were in obtained different diinff ractiondifferent orders, diffraction and those orders, with and larger thosevalues with exhibitedlarger m valueslarger phaseexhibited singularities larger phase and ringsingularities radii [31]. and Figure ring3 b–fradii indicates [31]. Figure nearly 3b–f constant indicates di ff ractionnearly econstantfficiency diffractionacross the rangeefficiency of 530–930 across the nm. range Meanwhile, of 530–930 for nm. the planarMeanwhile, DVG, for the the diff planarraction DVG, efficiency the diffraction reached a efficiencymaximum reached at 532 nm, a maximum in which the at halfwave532 nm, in condition which the was halfwave perfectly condition satisfied. Thewas eperfectlyfficiency decreasedsatisfied. Thesignificantly efficiency when decreased the wavelength significantly deviated when fromthe wa 532velength nm (Figure deviated3h–l). from Additionally, 532 nm (Figure thanks to3h–l). the Additionally,dependency ofthanks diffraction to the angle dependency on incident of diffra wavelengths,ction angle the on pattern incident became wavelengths, larger along the withpattern the becameincreasing larger wavelength. along with A the comparison increasing between wavelength. above A two comparison samples verified between both above the faithfultwo samples phase verifiedencoding both and the the faithful equally phase high broadbandencoding and effi ciencythe equally of the high double-layer broadband reverse-twist efficiency of DVGs. the double- layer reverse-twist DVGs.

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Figure 3. 3. TheThe micrographs micrographs of ( ofa) double-layer (a) double-layer reverse-twist reverse-twist and ( andg) planar (g) planar 2 × 2 DVGs 2 2 under DVGs crossed under Figure 3. The micrographs of (a) double-layer reverse-twist and (g) planar 2 × 2 DVGs× under crossed polarizers;crossedpolarizers; polarizers; (b (–bf–) fand) and ((b h(–h–fl–))l ) andthe the corresponding (correspondingh–l) the corresponding diffraction patternspatterns diffraction at at 473, 473, patterns 532, 532, 700, at700, 473,800, 800, and 532, and 950 700,950 nm, nm, 800, respectively.andrespectively. 950 nm, respectively.

Double-layerDouble-layer reverse-twist reverse-twist reverse-twist and and and planar planar planar DQPsDQPs DQPs were were alsoalso also demonstrated. demonstrated. demonstrated. The The incident The incident incident was was also wasalso left- left- also circularlyleft-circularlycircularly polarized polarized polarized light. light. light. TheThe The samplessamples samples revealedrevealed revealed hexagonalhexagonal hexagonal latticelattice lattice patternspatterns patterns consistent consistent consistent with with with the the designeddesigned optical optical axis axis distributions distributions underunder crossed crossed po polarizerslarizerslarizers (Figure(Figure 4a,g). 44a,g).a,g). The TheThe continuous continuouscontinuous color colorcolor or oror graygray scale scale variations variations variations in in in the the the micrographs micrographs micrographs were were due due to to thethe the azimuthalazimuthal azimuthal angle angle angle change change changess ofs of ofthe the the LC LC LC directors. directors. directors. TheThe diffraction di ffdiffractionraction patterns patterns presented presented in in Figure Figure 44b–f,hb–f,h–l–l–l clearlyclearlyclearly showshow a aa regular regularregular hexagonal hexagonalhexagonal lattice latticelattice of of of diffractiondiffdiffractionraction orders orders with with the the same same m m of of 1. 1. By By presettingpresetting thethe the latticelattice lattice symmetry, symmetry, symmetry, the the pitch pitch of of the the Damman Damman grating,grating, and and and the the the topological topologicaltopological charge charge charge of of of the the the q-plate,q-plate, q-plate, the the OVOV OV arraysarrays arrays could could could be be rationally rationally be rationally manipulated. manipulated. manipulated. An An obviouslyAnobviously obviously different different different dependency dependency dependency on on onwavelength wavelength wavelength betw betweeneeneen thethe the two two two different different different designs designs designs was was was also also also verified. verified. verified.

FigureFigure 4. 4.The The micrographs micrographs ofof ((aa)) double-layerdouble-layer reverse-twist reverse-twist and and (g (g) )planar planar hexagonal hexagonal lattice lattice DQPs DQPs Figureunderunder crossed4. crossed The micrographs polarizers; polarizers; (b( bof––f f)()a andand) double-layer ((hh–l) the corresponding reverse-twist diffraction diandffraction (g) planar patterns patterns hexagonal at at473, 473, 532, 532,lattice 700, 700, 800,DQPs 800, underandand 950 950crossed nm, nm, respectively. respectively.polarizers; ( b–f) and (h–l) the corresponding diffraction patterns at 473, 532, 700, 800, and 950 nm, respectively. Moreover,Moreover, di diffractionffraction eefficienciesfficiencies atat severalseveral wavelengths were were quantitively quantitively detected detected and and are are depicteddepictedMoreover, in in Figure Figure diffraction5 .5. TheThe eefficiencyefficienciesfficiency was was at defined several defined as wavelengths asthe the intensities intensities were of total ofquantitively total first firstorders ordersdetected divided divided byand all are by depictedalltransmitted transmitted in Figure energy. energy. 5. TheTheoretically, Theoretically, efficiency it was was it wasdefined65.6% 65.6% for as 2 ×the for 2 DVG intensities 2 2 [20]. DVG The of [20 measuredtotal]. The first measured dataorders for divided the data double- for by theall × transmitteddouble-layerlayer reverse-twist energy. reverse-twist Theoretically, and planar and planar DVGs it was DVGs are 65.6% depicted are depictedfor 2by × 2red DVG by and red [20]. andblack The black dots measured dots in Figure in Figure data 5a, 5forrespectively.a, respectively.the double- layerTheThe red reverse-twist red dots dots varied varied inand the inplanar rangethe range DVGs of 40.1–49.7% of are 40.1–49 depicted for.7% the byfor bandwidth redthe andbandwidth black of 530–930 dots of 530–930 nm,in Figure which nm, 5a, was whichrespectively. comparable was Thetocomparable the red theoretical dots tovaried the value theoretical in inthe a broadband.range value of in 40.1–49 a broadban The.7% diff ractiond.for The the ediffraction ffibandwidthciency ofefficiency theof 530–930 planar of the DVGnm, planar exhibitedwhich DVG was a exhibited a maximum of 42.7% at 532 nm, and it decreased gradually along with the wavelength comparablemaximum of to 42.7% the theoretical at 532 nm, value and itin decreaseda broadban graduallyd. The diffraction along with efficiency the wavelength of the planar away DVG from away from 532 nm, i.e., the halfwave condition. Figure 5b reveals the diffraction efficiencies of exhibited532 nm, i.e., a maximum the halfwave of 42.7% condition. at 532 Figurenm, and5b revealsit decreased the di graduallyffraction along efficiencies with the of double-layerwavelength double-layer reverse-twist and planar DQPs. A quite similar phenomenon was observed. The red awayreverse-twist from 532 and nm, planar i.e., DQPs.the halfwave A quite similarcondition. phenomenon Figure 5b wasreveals observed. the diffraction The red curve efficiencies kept over of curve kept over 51.0% in the range of 530–930 nm, while the maximum efficiency of 57.2% was double-layer51.0% in the rangereverse-twist of 530–930 and nm, planar while DQPs. the maximum A quite similar efficiency phenomenon of 57.2% was was reached observed. at 532 The nm red for reached at 532 nm for the planar DQP (black), and the efficiency decreased rapidly along with the curvethe planar kept DQPover (black),51.0% in and the the range efficiency of 530–930 decreased nm, rapidlywhile the along maximum with the efficiency wavelength of away57.2% fromwas reached at 532 nm for the planar DQP (black), and the efficiency decreased rapidly along with the

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532wavelength nm. The curves away from matched 532 nm. well The with curves the matched intensity we variationll with the of intensity the OV variation arrays in of Figures the OV3 arrays and4 . Thein results Figures vividly 3 and verified4. The results the broadband vividly verified OV array the broadband generations. OV array generations.

FigureFigure 5. The 5. The dependencies dependencies of of di diffractionffraction e ffiefficienciesciencies on on wavelength. wavelength. ( a) DVGs DVGs and and ( (bb) )DQPs. DQPs.

4. Conclusions4. Conclusions A double-layerA double-layer reverse-twist reverse-twist configuration configuration waswas introducedintroduced into the planar geometric geometric phase phase opticsoptics elements elements to to realize realize broadbandbroadband multichanne multichannell OV OV generators. generators. By encoding By encoding DVG and DVG DQP and phase DQP phaseprofiles profiles to photopatterned to photopatterned LCP LCPlayers, layers, multicha multichannelnnel rectangular rectangular and hexa andgonal hexagonal OV arrays OV arrayswere weregenerated generated equally-efficiently equally-efficiently with with mode mode conver conversionsion efficiencies efficiencies exc exceedingeeding 40.1% 40.1% and and 51.0%, 51.0%, respectively,respectively,across across a broadband a broadbandwavelength wavelength range range over over 400 400 nm. nm. TheseThese devices showed showed significant significant superioritiessuperiorities in thein the working working band band compared compared to to the the planar planar onesones whosewhose efficiency efficiency maximized maximized at at the the halfwavehalfwave condition condition and and dropped dropped when when deviating deviating from from thethe designeddesigned wavelength.wavelength. Thanks Thanks to to the the fidelityfidelity and and flexibility flexibility of photoalignmentof photoalignment technology, technology, arbitrary arbitrary phase phase patterns patterns could could be be encoded encoded to to the double-layerthe double-layer reverse-twist reverse-twist LCP systems.LCP systems. This This supplies supplies aneasy an easy but but effi cientefficient fabrication fabrication method method for broadbandfor broadband beam shapingbeam shaping elements, elements, thus paving thus paving a bright a bright way forway advanced for advanced photonic photonic applications applications such such as wavelength-division multiplexing-compatible mode encoding, high-density information as wavelength-division multiplexing-compatible mode encoding, high-density information storage, storage, and encryption. and encryption. Author Contributions: W.D. and W.H. conceived the original idea; H.Z., W.D., T.W., and C.X. performed the Author Contributions: W.D. and W.H. conceived the original idea; H.Z., W.D., T.W., and C.X. performed the experiments,experiments, analyzed analyzed the datathe data and wroteand wrote the manuscript; the manuscrip W.H.t; W.H. supervised supervised and directed and directed the research. the research. All authors All haveauthors read and have agreed read and to the agreed published to the published version of version the manuscript. of the manuscript. Funding:Funding:This This work work was was supported supported by theby the National National Natural Natura Sciencel Science Foundation Foundation of Chinaof China (No. (No. 61922038) 61922038) and and the Fundamentalthe Fundamental Research Research Funds Funds for the for Central the Central Universities Universities (14380170) (14380170) and and the Tangthe Tang Scholar Scholar program. program.

ConflictsConflicts of Interest: of Interest:The The authors authors declare declare no no conflict conflict of of interest. interest.

ReferencesReferences

1. 1.Coullet, Coullet, P.; Gil,P.; Gil, L.; Rocca,L.; Rocca, F. OpticalF. Optical vortices. vortices.Opt. Opt. Commun. Commun.1989 1989,, 7373,, 403–408.403–408. 2. 2.Allen, Allen, L.; Beijersbergen,L.; Beijersbergen, M.W.; M.W.; Spreeuw, Spreeuw, R.J.; R.J.; Woerdman, Woerdman,J.P. J.P. Orbital Orbital angularangular momentum of of light light and and the the transformationtransformation of laguerre-Gaussianof laguerre-Gaussian laser modes. modes.Phys. Phys. Rev.Rev. AA 1992, 45,, 8185–8189. 8185–8189. [PubMed] 3. 3.Marrucci, Marrucci, L.; Manzo,L.; Manzo, C.; Paparo, C.; Paparo, D. Optical D. spin-to-orbital Optical spin angular-to-orbital momentum angular conversionmomentum in inhomogeneousconversion in anisotropicinhomogeneous media. Phys.anisotropic Rev.Lett. media.2006 Phys., 96 ,Rev. 163905. Lett. 2006 [PubMed, 96, 163905.]

4. 4.Gecevicius,Gecevicius, M.; M.; Drevinskas, Drevinskas, R.; Beresna,R.; Beresna, M.; Kazansky,M.; Kazansky, P.G. SingleP.G. Single beam be opticalam optical vortex vortex tweezers tweezers with tunable with tunable orbital angular momentum. Appl. Phys. Lett. 2014, 104, 288–299. orbital angular momentum. Appl. Phys. Lett. 2014, 104, 288–299. 5. D’Ambrosio, V.; Spagnolo, N.; Re, L.D.; Slussarenko, S.; Li, Y.; Kwek, L.C.; Sciarrino, F. Photonic 5. D’Ambrosio, V.; Spagnolo, N.; Re, L.D.; Slussarenko, S.; Li, Y.; Kwek, L.C.; Sciarrino, F. Photonic polarization polarization gears for ultra-sensitive angular measurements. Nat. Commun. 2013, 4, 2432. gears for ultra-sensitive angular measurements. Nat. Commun. 2013, 4, 2432. 6. Cardano, F.; Massa, F.; Qassim, H.; Karimi, E.; Slussarenko, S.; Paparo, D.; Marrucci, L. Quantum walks 6. Cardano, F.; Massa, F.; Qassim, H.; Karimi, E.; Slussarenko, S.; Paparo, D.; Marrucci, L. Quantum walks and and wavepacket dynamics on a lattice with twisted photons. Sci. Adv. 2015, 1, e1500087. wavepacket dynamics on a lattice with twisted photons. Sci. Adv. 2015, 1, e1500087. 7. Kumar, A.; Prabhakar, S.; Vaity, P.; Singh, R.P. Information content of optical vortex fields. Opt. Lett. 2011, 7. Kumar, A.; Prabhakar, S.; Vaity, P.; Singh, R.P. Information content of optical vortex fields. Opt. Lett. 36, 1161–1163. 2011, 36, 1161–1163.

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8. Wang, Z.; Zhang, N.; Yuan, X.C. High-volume optical vortex multiplexing and de-multiplexing for free-space optical communication. Opt. Express 2011, 19, 482–492. 9. Tamburini, F.; Mari, E.; Sponselli, A.; Thid, B.; Bianchini, A.; Romanato, F. Encoding many channels on the same frequency through radio vorticity: First experimental test. New J. Phys. 2012, 14, 033001. 10. Awaji, Y.; Wada, N.; Toda, Y. Demonstration of spatial mode division multiplexing using Laguerre-Gaussian mode beam in telecom-wavelength. In Proceedings of the 2010 IEEE Photonics Society’s 23rd Annual Meeting, Denver, CO, USA, 7–11 November 2010. 11. Gibson, G.; Courtial, J.; Padgett, M.J.; Vasnetsov, M.; Franke-Arnold, S. Free-space information transfer using light beams carrying orbital angular momentum. Opt. Express 2004, 12, 5448–5456. 12. Chen, P.; Ge, S.J.; Duan, W.; Wei, B.Y.; Cui, G.X.; Hu, W.; Lu, Y.Q. Digitalized Geometric Phases for Parallel Optical Spin and Orbital Angular Momentum Encoding. ACS Photonics 2017, 4, 1333–1338. 13. Kobashi, J.; Yoshida, H.; Ozaki, M. Planar optics with patterned chiral liquid crystals. Nat. Photonics 2016, 10, 389–392. 14. Rafayelyan, M.S.; Tkachenko, G.; Brasselet, E. Reflective Spin-Orbit Geometric Phase from Chiral Anisotropic Optical Media. Phys. Rev. Lett. 2016, 116, 253902. [PubMed] 15. Kobashi, J.; Yoshida, H.; Ozaki, M. Polychromatic Optical Vortex Generation from Patterned Cholesteric Liquid Crystals. Phys. Rev. Lett. 2016, 116, 253903. 16. Barboza, R.; Bortolozzo, U.; Clerc, M.G.; Residori, S. Berry Phase of Light under Bragg Reflection by Chiral Liquid-Crystal Media. Phys. Rev. Lett. 2016, 117, 053903. 17. Chen, P.; Ma, L.L.; Duan, W.; Chen, J.; Ge, S.J.; Zhu, Z.H.; Lu, Y.Q. Digitalizing Self-Assembled Chiral Superstructures for Optical Vortex Processing. Adv. Mater. 2018, 30, 1705865. 18. Dammann, H.; Klotz, E. Coherent Optical Generation and Inspection of Two-dimensional Periodic Structures. J. Mod. Opt. 1977, 24, 505–515. 19. Fan, F.; Yao, L.; Wang, X.; Shi, L.; Srivastava, A.K.; Chigrinov, V.G.; Kwok, H.-S.; Wen, S. Ferroelectric Liquid Crystal Dammann Grating by Patterned Photoalignment. Crystals 2017, 7, 79. 20. Zhou, C.; Liu, L. Numerical study of Dammann array illuminators. Appl. Opt. 1995, 34, 5961–5969. 21. Yu, J.; Zhou, C.; Jia, W.; Hu, A.; Cao, W.; Wu, J.; Wang, S. Three-dimensional Dammann vortex array with tunable topological charge. Appl. Opt. 2012, 51, 2485–2490. 22. Chen, P.; Wei, B.Y.; Hu, W.; Lu, Y.Q. Liquid-Crystal-Mediated Geometric Phase: From Transmissive to natBroadband Reflective Planar Optics. Adv. Mater. 2019, 1903665. [CrossRef] 23. Samoylov, A.V.; Samoylov, V.S.; Vidmachenko, A.P.; Perekhod, A.V. Achromatic and super-achromatic zero-order waveplates. J. Quant. Spectrosc. Radiat. Transf. 2004, 88, 319–325. 24. Komanduri, R.K.; Lawler, K.F.; Escuti, M.J. Multi-twist retarders: Broadband retardation control using self-aligning reactive liquid crystal layers. Opt. Express 2013, 21, 404–420. [PubMed] 25. Tang, S.T.; Kwok, H.S. Mueller calculus and perfect polarization conversion modes in liquid crystal displays. J. Appl. Phys. 2001, 89, 5288–5294. 26. Chigrinov, V.G.; Pikin, S.A.; Verevochnikov, A.M.; Kozenkov, V.M.; Khazimullin, M.V.; Ho, J.Y.; Kwok, H.S. Diffusion model of photoaligning in azo-dye layers. Phys. Rev. E 2004, 69, 061713. 27. Chigrinov, V.G.; Prudnikova, E.; Kozenkov, V.M.; Kwok, H.S.; Akiyama, H.; Kawara, T.; Takatsu, H. Synthesis and properties of azo dye aligning layers for liquid crystal cells. Liq. Cryst. 2002, 29, 1321–1327. 28. Wu, H.; Hu, W.; Hu, H.C.; Lin, X.W.; Zhu, G.; Choi, J.W. Arbitrary photo-patterning in liquid crystal alignments using DMD based lithography system. Opt. Express 2012, 20, 16684–16689. 29. Duan, W.; Chen, P.; Ge, S.J.; Liang, X.; Hu, W. A Fast-Response and Helicity-Dependent Lens Enabled by Micro-Patterned Dual-Frequency Liquid Crystals. Crystals 2019, 9, 111. 30. White, T.J.; Broer, D.J. Programmable and adaptive mechanics with liquid crystal polymer networks and elastomers. Nat. Mater. 2015, 14, 1087–1098. 31. Yao, A.M.; Padgett, M.J. Orbital angular momentum: Origins, behavior and applications. Adv. Opt. Photonics 2011, 3, 161–204.

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