Fabrication of ideal geometric-phase hologramswith arbitrary wavefrontsJIHWAN KIM,1 YANMING LI,1,2 MATTHEW N. MISKIEWICZ,1 CHULWOO OH,1,3 MICHAEL W. KUDENOV,1 AND

MICHAEL J. ESCUTI1,*1Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, North Carolina 27606, USA2Currently at Apple Inc., Flat Panel Display, Cupertino, California 95014, USA3Currently at Intel Corporation, Technology and Manufacturing Group, Hillsboro, Oregon 97124, USA*Corresponding author: mjescuti@ncsu.edu

Received 12 August 2015; revised 9 October 2015; accepted 9 October 2015 (Doc. ID 247820); published 4 November 2015

Throughout optics and photonics, phase is normally controlled via an optical path difference. Although much lesscommon, an alternative means for phase control exists: a geometric phase (GP) shift occurring when a light wave istransformed through one parameter space, e.g., polarization, in such a way as to create a change in a second parameter,e.g., phase. In thin films and surfaces where only the GP varies spatially—which may be called GP holograms(GPHs)—the phase profile of nearly any (physical or virtual) object can in principle be embodied as an inhomo-geneous anisotropy manifesting exceptional diffraction and polarization behavior. Pure GP elements have had poorefficiency and utility up to now, except in isolated cases, due to the lack of fabrication techniques producing elementswith an arbitrary spatially varying GP shift at visible and near-infrared wavelengths. Here, we describe two methods tocreate high-fidelity GPHs, one interferometric and another direct-write, capable of recording the wavefront of nearlyany physical or virtual object. We employ photoaligned liquid crystals to record the patterns as an inhomogeneousoptical axis profile in thin films with a few μm thickness. We report on eight representative examples, including a GPlens with F/2.3 (at 633 nm) and 99% diffraction efficiency across visible wavelengths, and several GP vortex phaseplates with excellent modal purity and remarkably small central defect size (e.g., 0.7 and 7 μm for topological chargesof 1 and 8, respectively). We also report on a GP Fourier hologram, a fan-out grid with dozens of far-field spots, and anelaborate phase profile, which showed excellent fidelity and very low leakage wave transmittance and haze. Together,these techniques are the first practical bases for arbitrary GPHs with essentially no loss, high phase gradients(∼rad∕μm), novel polarization functionality, and broadband behavior. © 2015 Optical Society of America

OCIS codes: (050.1940) Diffraction; (090.2890) Holographic optical elements; (160.3710) Liquid crystals; (310.6860) Thin films, optical

properties; (350.5030) Phase.

http://dx.doi.org/10.1364/OPTICA.2.000958

1. INTRODUCTION

Phase shifts in light waves are ordinarily produced via an opticalpath difference (OPD) known as a dynamic phase effect. However,a change in other light wave parameters can also induce a corre-sponding phase shift, called a geometric phase (GP) effect [1–3].This can be caused by optical anisotropy [4,5], plasmonic metama-terials [6–8], propagation evolutions from reflections [9], or adia-batic waveguiding [10]. Elements incorporating these, includingpolarization holograms [11–13] and metasurface holograms [6–8,14,15], can produce light waves with complex amplitude, phase,and polarization by embodying inhomogeneous attenuation, dy-namic phase, and/or GP. In elements wherein only the GP variesspatially, exceptional diffraction and polarization behavior arises—most strikingly, they can theoretically produce nearly any wavefrontvariation and produce its conjugate just by changing input polari-zation [Fig. 1(a)]. This should be possible with 100% efficiency[13,16], e.g., when a birefringent layer has half-wave retardation,

since the desired phase profile is encoded in the optical axis orien-tation [Fig. 1(b)]. This allows for continuous optical phase shiftswithout phase resets [2] in stark contrast to conventional elements,wherein phase profiles are encoded as (usually) discrete OPDvariations in refractive index or thickness, limiting performance.We set forth that the class of pure GP elements is most properlydesignated GP holograms (GPHs), wherein attenuation and dy-namic phase are homogenous.

Although the promise of GPHs is great, the principal challengeremaining is their fabrication: achieving precise and pure inhomo-geneous GP control for arbitrary patterns at useful wavelengthswithout detrimental flaws. Most work to date has employed eithermaterial anisotropy (e.g., Refs. [12,17,18]) or subwavelengthstructures (e.g., Refs. [8,19,20]). Nearly all work has focused ona few GP wavefronts with spatial symmetry (i.e., in x, y, r, or ϕ),including GP prisms (i.e., polarization gratings) [17,20–24], GPvortex phase plates (i.e., q-plates or vortex retarders) [16,25,26],

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Corrected 4 November 2015

and GP lenses [2,27]. Other work has examined more generalphase profiles [18,28,29]. So far, experiment essentially matchestheory only in the simplest of these elements, the GP prism[23,24] and vortex phase plates [16,25,26], which each havelinear phase profiles. In all other cases, experimental results arelimited by one or more of the following: low efficiency, high scat-tering, small clear aperture, spurious leakage, or operation wave-lengths outside the visible or near-infrared.

Here we employ two fabrication methods that create trulygeneralized GPHs that enable highly efficient thin-film GP ele-ments with arbitrary phase profiles, for visible through infraredwavelengths. Two techniques are necessary because phase profilescommonly arise in two different ways: in some cases, a physicalobject generates the desired wavefront while, in other cases, equa-tions or numerical analyses determine a virtual object description.Both involve creating a linear polarization orientation map that isrecorded as an optical axis profile, using a linear photoalignmentpolymer [30] (LPP) and a polymerizable liquid crystal [17,23](LC), as illustrated in Fig. 1(b). In what follows, we begin withthe theoretical foundation of pure GPHs and then introduce ourfabrication methods and results.

2. GEOMETRIC PHASE HOLOGRAM THEORY

The GPHs we study here employ the Pancharatnam–Berry phase[4,5], and may be identified as follows: thin films with ananisotropy defined by an optical axis Φ�x; y� varying in the sub-strate plane, with a strictly constant net linear birefringence/dichroism magnitude (along z). Hasman et al. [19] were perhapsthe first to clearly explain that elements like these output at mostthree distinct waves (see also Ref. [31]) for each input wave withjχini polarization, δin�x; y� phase, and unity amplitude:

eiδin jχ ini !GPH

ffiffiffiffiffiffi

η�p

ei�δin�2Φ�jχ�i �ffiffiffiffiffi

η−p

ei�δin−2Φ�jχ−i� ffiffiffiffiffi

η0p

eiδin jχini: (1)

We identify the output terms as the primary (�), conjugate (−),and leakage (0) waves, where the polarizations χ� are always cir-cular and mutually orthogonal, and where coefficients η# are theefficiencies (i.e., fraction of input power coupled into their respec-tive waves). Most interestingly, the primary and conjugate wavesgain a phase shift �2Φ�x; y� which, in the case of a lens, causesone to converge and the other to diverge [Fig. 1(a)] with a positiveand negative focal length, respectively. While the anisotropy may

be birefringent and/or dichroic in general, here we study onlypure birefringence since it allows for η� � η− � 100%, i.e., com-plete coupling of the input into either the primary or conjugatewaves, or any combination selected by jχini. In this case, theefficiencies of the primary and conjugate waves are η� �jhχinjχ�ij2 sin2�Γ∕2� and the efficiency of the leakage wave isη0 � cos2�Γ∕2�, where Γ is the retardation. While the simplestlayer is uniaxially birefringent and homogeneous in z [Fig. 1(b)],other anisotropies [22,32] may be employed, resulting in differentη# but producing the same behavior otherwise [33].

3. TWO DIFFERENT FABRICATION METHODS

A. Interferometer Approach

The first method enables the recording of nearly any wavefrontas a GPH from a physical element, using a specialized interfero-metric principle. In conventional (intensity) interference, when aprimary (or “object”) wave with phase δ�x; y� is superimposedwith a similarly polarized and coherent reference wave, thenδ�x; y� is incorporated into the amplitude of the standing wave.In more general interference, however, where the primary andreference polarizations are not the same, the resulting standingwave has a complex distribution of both amplitude and polariza-tion [13,34–36]. More specifically, when the two waves areorthogonally circularly polarized, then a special situation results:the standing wave is approximately linearly polarized everywhere,with in-plane orientation ψ�x; y�. While this principle has longbeen used for polarization grating fabrication [17,21,23,24] withtwo plane waves leading to ψ�x; y� � �Δk · r�∕2, whereΔk is thewavevector difference, this also applies in general for nearly anynonplanar object wave. When the object wave has phase δ�x; y�,then the polarization map is substantially linearly polarized, withψ�x; y� � �δ�x; y� � Δk · r�∕2. While this principle was hypoth-esized by Marrucci et al. [16] and employed to some extent byRuiz et al. [37], neither they nor anyone else has yet proposedor demonstrated a general implementation.

The approach introduced here is the first to enable interfero-metric holographic recording of an arbitrary physical object’swavefront as a pure GP phase element. We employ a modifiedMach–Zehnder interferometer [Fig. 1(c)] to achieve thisδ�x; y� → ψ�x; y� recording principle. The first polarizing beamsplitter (PBS) separates the object/primary (O) and reference (R)arms into orthogonal linear polarizations, which are then recom-bined by a second PBS. A quarter-wave retarder transforms the

(a)

(b)

(c) (d)

Fig. 1. Geometric phase holograms. (a) Illustration of GP lens behavior. (b) Example of GPH cross-section with LC and LPP. (c) New modifiedMach–Zehnder interferometric fabrication. (d) New direct-write fabrication. Note that ψ�x; y� is the linear polarization orientation angle, Φ�x; y� is theoptical axis angle, and δ�x; y� is the phase angle.

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beams into orthogonal circular polarizations. The object to be re-corded must be placed before the second PBS so that its wavefrontinterferes with the reference, as illustrated for a lens in Fig. 1(c).For objects with wavefronts that diverge rapidly, lenses may beplaced immediately before and after the second PBS to relay theobject’s wavefront to the recording plane (not shown). Note thatboth O and R waves can be arranged parallel and collocated at thehologram plane. These constraints ensure that Δk � 0, allowingthe often preferable elimination of the diffraction grating term(Δk · r). Nearly any transparent and reflective dynamic phaseobject can be recorded in this way.

Photoaligned LC materials are especially well suited to GPHrecording because of several unique properties [30]. These in-clude: (i) any modest variation in the intensity of the standingwave will be eliminated and not recorded because the responseof LPP is the same beyond a threshold fluence; and (ii) LPP isable to record only the polarization orientation angle and notits ellipticity.

B. Direct-Write Approach

The second fabrication method involves directly writing a virtualwavefront (i.e., nearly any inhomogeneous phase shift) via con-tinuous scanning of a focused laser beam, partially introduced inRef. [38]. At least three degrees of freedom are needed to recordan arbitrary GPH, two for the spatial scanning and one for controlof the anisotropy orientation. Most of the previously studied scan-ning methods employ only two, typically one spatial and one ori-entational [26,39–41]. Of the prior scanning methods that usethree degrees of freedom [18,29,42], all employ a pixel-by-pixeldiscrete scan approach, a beam size not greater than those pixels,and one-shot exposure in each pixel, which altogether makesmoothly varying and continuous phase profiles unattainable.

Conversely, we postulate (and here demonstrate) that a continu-ous scan in all three dimensions achieves continuous optical axisprofiles and thus continuous phase profiles. Furthermore, anyresidue of the scan contours themselves can be eliminated if theyat least partially overlap and the recording material responds tothe average polarization—as occurs in the photoaligned LC.The LC to LPP interaction also makes it possible to create featuressmaller than the writing beam size, exceeding the resolution limitsfound in the prior scanning approaches. We implement thisprinciple with a direct-write system consisting of a polarizationrotator and a 2D translation system for spatial scanning as shownin Fig. 1(d).

4. RESULTS

These two techniques in principle enable the embodiment ofnearly any phase change as a GPH. To demonstrate this rangeof possibilities, we highlight eight representative experimental ex-amples. The resulting quality and complexity significantly exceedall prior GP elements.

A. Interferometer Results

In the first GPH example, we fabricated a GP lens [Fig. 2(a)],using the modified Mach–Zehnder interferometer and a refractivespherical lens. Its clear aperture diameter is 16 mm and it hasa focal length of 36 mm at 633 nm. The ideal phase shift[Fig. 2(a.i)] and its crossed polarizer texture [Fig. 2(a.ii)] are wellmatched by the corresponding experimental results [Figs. 2(a.i)and 2(a.iii)]. The unique GP lens characteristic behavior is easilyobserved in Fig. 2(a.iv), where a positive or negative focal lengthmay be selected by changing input polarization.

Fig. 2. Interference GPH. (a) Lens. (b) Axicon. (c) Large period prism (i.e., polarization grating). Columns: i, Simulated (curve) and measured (circles)primary (magenta) and conjugate (cyan) phases, where measurement error is less than�0.02 rad. ii, Theoretical optical axis profile, with simulated cross-polarizer texture. iii, Measured polarizing optical micrograph under crossed polarizers. iv, Far-field result. Scale bars in column iii indicate 500 μm.

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Additionally, we measured spatial power distribution at thefocal plane of the spot produced when the GP lens focuses aGaussian laser beam [Fig. 3(a)]. The resulting focused beamshows almost no deviation from I�r� ∝ exp�−2�x2 � y2�∕w2

0�,where w0 is the beam waist, indicating that the wavefront ofthe refractive lens was well captured by the GP lens (cf., Ref. [43]).

This result exceeds all prior diffractive lenses (e.g., etched-surface Fresnel [44] and previous GP [37,39] types) with regardsto efficiency, bandwidth, and scattering. In our results, 99% ofincident light is directed into the converging and/or divergingwaves, depending on input polarization. Because we used the two-layer broadband LC structure from our work with PGs [45] andmultitwist retarders [32], this high (real-world) transmittance ap-pears across the whole visible spectrum [Fig. 3(b)]. The lack ofspurious leakages can be directly attributed to the continuousnature of the geometric phase change embodied in the anisotropiclayer, and the lack of defects or other degradation [Fig. 2(a.iii)].Note that while the transmittance is achromatic, the focal lengthmanifests chromatic dispersion [shown in the inset of Fig. 3(b)] aswith all diffractive lenses.

Indeed, this feature enables fabrication of fast [i.e., low F-number (F∕#)] lenses. For example, a parabolic GP lens profilewill have f � �ΛminD∕�2λ� � �πD∕�λj∇δjmax�, where D isthe lens diameter and Λmin and j∇δjmax are the extreme valuesat the lens edge of the local grating period and the magnitudeof the phase shift gradient, respectively. Even though this GP lensreplays at 633 nm with F∕2.3, the refractive lens used for therecording at 325 nm was F∕4.5 (i.e., a lens with focal length100 mm illuminated with a beam of 22 mm). The F∕# uponreplay will be scaled by the ratio of the replay to recording wave-lengths, in this case �633 nm�∕�325 nm� ≈ 2. In Fig. 3(c), wecalculate the relationship in general: GP lenses that replay fastat visible and infrared wavelengths may be recorded (in the UV)using much slower physical lenses. Furthermore, as Ref. [46]highlights, GP lenses offer practical removal of lens aberration.

This GP lens also manifests excellent imaging properties. InFigs. 3(d) and 3(e), we arranged the GP lens and a circular polar-izer between a picture of our university logo on a computer screenand a digital camera, where one polarization leads to a reducedapparent size while the orthogonal polarization magnifies the logo(where pixels are clearly resolved).

We also fabricated a GP axicon [Fig. 2(b)] and GP prism[Fig. 2(c)], which both showed highly linear phase in the radialand horizontal directions, respectively. In the far field, the GPaxicon produces the expected ring [Fig. 2(b.iv)] for jχ�i with∼99% transmittance, and the GP prism manifests ∼99% trans-mittance in both primary and conjugate waves. Note that thesmall plateau in the apex of the axicon, as seen in the phase[Fig. 2(b.i)] and the crossed polarizer image [Fig. 2(b.iii)], arisesdirectly from the rounded region at the tip of the physical axiconthat was recorded. It is also worthwhile to note that the GP prism(i.e., a polarization grating) in Fig. 2(c.iii) has a very large gratingperiod (Λ � 0.56 mm), something difficult to accomplish other-wise. Its interferogram [Fig. 2(c.iv)], captured a few mm from theelement and analyzed by a linear polarizer, shows that the primarywave is propagating at a small angle from the leakage wave, andthat the wavefronts are highly uniform.

B. Direct-Write Results

In the first example of direct writing, we fabricated a set of GPvortex phase plates which produce optical vortices with topologi-cal charges of m � 1, 2, 4, and 8. For each, the target phaseprofile [Fig. 4(a.i)] was continuously scanned by a small Gaussianbeam (diameter 2w0 � 4 μm; uniform linear polarization) in aspiral pattern [Fig. 4(a.i)] with arms 2 μm apart, starting at a ra-dius of ∼2 μm. The angle Φ�x; y� was varied according to theideal optical axis profile [Fig. 4(a.ii)]. This led to substantial over-lap between neighboring scans, causing every location in the LPPto be exposed multiple times, with different polarization anglesand intensities, often from the peripheral parts of the Gaussian

Fig. 3. Detailed Results from GP Lens. (a) Measured (dots) focused beam profile of GP lens (blue) and recorded lens (red), compared with perfectGaussian result. (b) Measured transmittance of primary � conjugate (solid) and leakage (dashed) waves; inset: chromatic dispersion of focal spot.(c) Predicted replay versus recorded F∕#; dot indicates the GP lens reported here. (d) Imaging through the GP lens with jχ�i polarization, showingpositive focal length. (e) Same as (d), but with jχ−i polarization, showing negative focal length.

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beam. This results in two important consequences. First, the scanpattern is not observable in the LC orientation profile becausethe local nematic director aligns along the average polarizationdirection of the multiple exposures, equivalent to ψ�x; y�. TheLC alignment is smoothly varying [Fig. 4(a.iii)], without pixelsor discontinuities, and a continuous geometric phase profile isproduced. Second, the singularity is small because only the centerpoint is equally exposed to all linear polarizations, which results innull averaging. Remarkably, the size of the central defect in eachcase is significantly smaller (i.e., 0.7, 1.6, 5, and 7 μm, respec-tively) than the best reported [26,47,48], and smaller than or sim-ilar to the beam diameter used to record the pattern. As a result,the far-field vortex beam intensities [Fig. 4(a.iv)] show excellentpurity in each case.

In our second example of direct writing, we fabricated a lens-less GP Fourier hologram (with image focus at infinity). The tar-get far-field intensity was a fan-out grid with 63 spots of roughlyequal intensity (∼1.5% of the input) in the shape of an “E” offsetfrom the incident direction and a vertical line of 27 spots on axiswith half this intensity, shown in Fig. 4(b.iv). This corresponds tohalf of the similar grid in Ref. [49], and was chosen to emphasizethe unique features of the GP Fourier hologram. Using theGerchberg–Saxton algorithm [50,51], we calculated a 64 × 64pixel tile of the optimum phase shift [shown in Fig. 4(b.i)],and arranged this into an 8 × 8 array to form the overall target512 × 512 hologram. The GPH was recorded with a pixel spacingof 4.4 μm and a 6 μm beam diameter, using a continuous-motionraster scan [Fig. 4(b.i)].

The target [Fig. 4(b.ii)] and experimental [Fig. 4(b.iii)] crossedpolarizer textures correspond overall, but notably, the experimen-tal result is smooth and largely continuous despite the discretenature of the target. Nevertheless, when illuminated with a2 mm diameter 633 nm laser with χ�, the primary wave intensitymatches the target very well [Fig. 4(b.iv)], at least qualitatively.The leakage wave transmittance is low (∼1%) and scattering is

minimal due to the largely continuous nature of the recordedphase, both dramatically improved over prior Fourier GPH (cf.,Refs. [6,8,14,29,39,52]). As expected, the conjugate wave (illu-minated by χ−) is rotated 180°. Interestingly, when both theprimary and conjugate waves are illuminated using a linear inputpolarization, the output is the full fan-out grid (cf., Ref. [49]), asuperposition of both waves simultaneously [directly predictedby Eq. (1)].

5. DISCUSSION

A primary feature of GPHs is the lack of a physical differencebetween phase shifts of 0 and 2πp for any integer p due to thecyclical nature of the optical axis orientation. Conversely, phaseshifts in thin-film dynamic phase elements (e.g., SLMs, conven-tional diffractive optics) are frequently embodied with 2π resets inthe OPD, corresponding to discontinuities in physical quantitiessuch as thickness or refractive index. These discontinuities resultin unwanted diffracted light, reducing efficiency and introducingartifacts. The lack of such discontinuities [2] in the GPHs de-scribed here allows them to embody high-fidelity continuousphase shifts, resulting in high efficiency. Consequently, theirprimary limitation is not the phase shift δ range, but ratherthe phase shift gradient j∇δj since the latter is linked to physicalparameters. In our materials, j∇δj is limited principally by theelastic deformation of the LC molecules, and we currently con-sider the maximum extreme to be at least j∇δjmax ≤ π rad∕μm,since we reliably fabricate polarization gratings [53] as smallas Λmin ∼ 2 μm.

The reader may fairly ask, “Why introduce a new name for thisclass of elements?” Indeed, the elements qualifying as pure GPHshave been called patterned retarders, space-variant Pancharatnam–Berry phase optical elements [16,31], plasmonic metasurfaces[6–8], or any one of at least half a dozen different names alreadyin the literature (e.g., Refs. [11,13,15,18,25,26,29,33]). We offer

Fig. 4. Direct-Write GPH. (a) Vortex phase plate. (b) Fourier hologram fan-out grid. Columns: i, Simulated phase. ii, Theoretical optical axis profilewith simulated crossed polarizer texture. iii, Measured polarizing optical micrograph under crossed polarizers. iv, Far-field result. Scale bars in column iiiindicate 20 μm. The approximate writing beam size and scan pattern is shown in column i. The insets of (b.i), (b.ii), and (b.iii) show a magnified view ofthe 8 × 8 pixels of the top left corner.

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three reasons supporting “geometric phase” as the proper name be-yond simple semantics: First, the diversity of terminology for thesame photonic effect is evidence of at least some confusion on thephysics fundamental to them all, hindering effective comparisonand dissemination; second, this name all at once provides the clear-est distinction for a general audience from conventional holograms(both amplitude and dynamic phase types) and from polarizationholograms, of which GPH is a distinct subclass; and third, only thethree output waves in Eq. (1) are possible in all embodiments ofinhomogeneous pure geometric phase, including all those otherthan anisotropy (e.g., Refs. [2,7,10,15,33]).

Because the same LC and LPP materials used in the displayindustry may be employed, these elements are immediately viablefor real-world use. For the same reason, GPH of nearly any δ canbe created with η� � η− ≃ 100% efficiency across wide band-widths in the visible and infrared using multiple chiral LC layers[32]. Furthermore, GPH can be both individually made withswitchable LCs [17,53,54] and/or stacked with switchable polari-zation selectors so that exponentially many multistate wavefronts[55] can be nonmechanically generated based on the addition andsubtraction of each element’s geometric phase shift (e.g., a lensassembly with multiple focal lengths). Finally, note that while pre-dominantly continuous Φ�x; y� are reported here, discrete do-mains with boundaries on the order of a few μm are also feasible.

6. CONCLUSION

We have described and validated two novel methods that enablethe fabrication of pure GPHs with essentially ideal properties. Asa result, the phase profile of nearly any (physical or virtual) objectcan now be embodied as a smoothly varying inhomogeneous op-tical axis in a thin film, functioning as a highly efficient hologramwith unique polarization sensitivity. The first method, based on amodified Mach–Zehnder interferometer, is capable of recording aphysical object’s wavefront. As the primary example, we fabricateda high-quality GP lens with F∕2.3 (at 633 nm), 99% diffractionefficiency across the visible spectrum, and phase properties match-ing the refractive lens used during recording. We also report on aGP axicon and GP prism (i.e., polarization grating), which mani-fest highly linear phase in the radial and horizontal directions, alsowith 99% transmittance in both primary and conjugate waves.The second method, employing a direct-write laser scanner, iscapable of generating holograms with arbitrary wavefronts froma virtual object description (e.g., equations or tabular data). Usingthis method, we fabricated a set of GP vortex phase plates withexcellent modal purity and remarkably small central defect size(e.g., 0.7 and 7 μm for topological charges of 1 and 8, respec-tively). We also made a GP Fourier hologram, a fan-out grid with63 far-field spots, and an elaborate phase profile, which showedexcellent fidelity and very low (∼1%) leakage wave transmittanceand haze. These fabrication techniques transform the compellingbehavior of GPH from a curiosity to a robust capability in thephotonics toolbox, viable for a wide range of applications acrossmany disciplines.

Funding. ImagineOptix Corporation; National ScienceFoundation (NSF) (NSF PECASE ECCS-0955127).

Acknowledgment. The authors are grateful to Dr. RaviKomanduri for helpful discussions throughout this work.

See Supplement 1 for supporting content.

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