Nanoscale liquid crystal polymer Bragg polarization gratings

We experimentally demonstrate nearly ideal liquid crystal (LC) polymer Bragg polarization gratings (PGs) operating at a visible wavelength of 450 nm and with a sub-wavelength period of 335 nm. Bragg PGs employ the geometric (Pancharatnam-Berry) phase, and have many properties fundamentally different than their isotropic analog. However, until now Bragg PGs with nanoscale periods (e.g., < 800 nm) have not been realized. Using photo-alignment polymers and high-birefringence LC materials, we employ multiple thin sublayers to overcome the critical thickness threshold, and use chiral dopants to induce a helical twist that effectively generates a slanted grating. These LC polymer Bragg PGs manifest 85-99% first-order efficiency, 19-29◦ field-of-view, Q ≈ 17, 200 nm spectral bandwidth, 84◦ deflection angle in air (in one case), and efficient waveguide-coupling (in another case). Compared to surface-relief and volume-holographic gratings, they show high efficiency with larger angular/spectral bandwidths and potentially simpler fabrication. These nanoscale Bragg PGs manifest a 6π rad/μm phase gradient, the largest reported for a geometric-phase hologram while maintaining a firstorder efficiency near 100%. c © 2017 Optical Society of America OCIS codes: (050.1950) Diffraction gratings; (050.6624) Subwavelength structures; (160.3710) Liquid crystals; (310.6845) Thin film devices and applications. References and links 1. L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” Opt. Acta 31(5), 579–588 (1984). 2. F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24(9), 584–586 (1999). 3. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27(13), 1141–1143 (2002). 4. 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Introduction
Polarization Gratings (PGs) are diffractive optical elements [1][2][3] comprising an in-plane anisotropy orientation that varies linearly along a surface, with a fixed anisotropy magnitude.They can be formed by azobenzene-containing polymers [1,4], "form-birefringence" in isotropic materials via high frequency surface profiles [5,6], plasmonic metasurfaces [7], and photo-aligned bulk liquid crystals (LCs) [8][9][10][11], and operate via the geometric phase [12][13][14][15].It has been shown in theory and experiment that they manifest compelling optical properties at large periods, including high (100%) efficiency into a single order and strong polarization selectivity.However, no one has yet experimentally demonstrated efficiency approaching 100% when the period is near or below the wavelength, especially for visible lightwaves.If successfully realized, this would be a fundamentally new means for high efficiency at very large diffraction angles, with unique constraints and benefits for many applications, such as augmented-reality systems [16,17], spectroscopy, optical telecommunications, polarimetry, front/back-lighting, nonmechanical beam steering, and remote optical sensing.

Background
Classical grating analysis [18,19] distinguishes Raman-Nath (thin) from Bragg (thick) grating regimes, irrespective of materials and fabrication methods.A dimensionless parameter is often used to distinguish between them: Q = 2πλd/ nΛ 2 , where Λ, λ, d and n are the period, vacuum wavelength, thickness, and average index, respectively.Traditionally, (isotropic) gratings in the Raman-Nath regime (Q < 1) have wide angular and spectral bandwidths, maximum singleorder efficiency of less than 34%, and produce many diffraction orders.Conversely, those in the Bragg regime (Q > 10) have sharply selective angular and spectral response, and can produce up to 100% efficiency into a single order when the incident wave travels along the Bragg angle within the grating medium: The most common gratings have been surface-relief-gratings [17,20] (SRGs) and volumeholographic-gratings [21] (VHGs).
The earliest report of a Bragg PG was an experimental study [4] employing a azo-polymer with low birefringence Δn (∼ 0.01), achieving ≥ 90% diffraction efficiency, with Λ = 2 μm, λ = 633 nm, d = 100 μm, θ B = 9.3 • , and Q ≈ 64.However, smaller periods would require much thicker films, e.g., d = 1 mm for Λ = λ at the same Q, which is unfeasible due to the manifestation of haze and absorption [22,23].Subsequent prior art [24,25] used numerical simulation and theoretical analysis to predict more generally that PG diffraction efficiency as high as 100% is possible in the Bragg regime for circularly polarized input and oblique incidence.
Nearly all prior art PGs fall into the Raman-Nath regime (Q < 1) when Λ >> λ.These PGs manifest high (∼ 100%) first-order diffraction efficiencies [1,[8][9][10], either from a non-chiral uniaxially birefringent material with d = λ/2Δn, or alternatively, in certain combinations of multiple chiral LC layers [26,27].Some works [24,[28][29][30][31] examined PGs formed by a single bulk nematic LC layer as Λ ∼ λ when Q 1.These studies concluded that high diffraction efficiency at large diffraction angles would require an unfeasibly high birefringence [28] (i.e., Δn > 0.4), would be impossible due to degradation of the polarization interference during recording [30], and/or would face a critical thickness limitation [29,32] (Λ/2 ≤ d C < Λ) beyond which the nematic LC cannot be directed into the PG pattern by alignment surfaces (e.g., a PG with Q = 10 and Λ = λ and n = 1.65 requires d = 2.6Λ, which is >> d C ).Two recent experimental reports have found ways to partially overcome some of these limitations.In one, a switchable PG (Λ = 900 nm, λ = 405 nm, and Q ≈ 2) was realized by aligning the LC with a metasurface [33], producing 35% peak efficiency.In another, a photo-aligned LC polymer PG (Λ = 1 μm, λ = 633 nm, and Q ≈ 6) produced 90% peak efficiency [34].A few years ago, we outlined in a patent application [35] the concept and demonstration of building up the required LC thickness for a Bragg PG with high efficiency in the infrared via a plurality (N) bulk LC sublayers and a single alignment layer.We also introduced how to use chiral dopants to achieve a grating slant.We employed reactive mesogens [36,37], also called a low-molecular weight LC polymer (LCP) network, wherein the thickness d i (i = 1..N) of each sublayer was thinner than d C .More recently, and apparently independently, a purely simulationbased study [38] examined a similar type of slanted bulk LC Bragg PG for transmissive cases when Λ > λ, which they called a polarization volume grating.
In this paper, we detail experimental results of Bragg PGs formed in bulk LCs with near-100% first-order efficiency at a blue wavelength with a subwavelength period.We also show how to control the slant angle to achieve on-axis peak efficiency, and characterize for the first time the polarization and spectrum of the diffracted orders.

Definitions and fundamental properties
The periodic structure of both switchable and polymerized LC PGs is embodied in the orientation of its optic axis (i.e., nematic director), where d is constant.The in-plane orientation angle Φ may vary both along the surface x and the thickness z, when a chiral nematic material is employed [26,27,38,39]: where φ is the twist angle of the layer and Λ x is the period at the surface (Fig. 1(a)).In Fig. 1(b) we show the calculated director profile using Eq. ( 2) for the parameters corresponding to the 3% chiral sample discussed below.This nematic director profile may be seen as a planar cholesteric structure with a vertical helix axis, but wherein the start angle of the helix varies linearly along the in-plane direction.It is exactly this in-plane variation that generates the gradient phase, i.e., diffraction, and it is the chiral twist that generates the grating slant.In this present work, nearly all (d i , φ i ) are identical, except for the optically negligible first sublayer.However, the reader should note that each sublayer has the potential to be individually controlled; this enables a wide design-space that allows for the fine tuning of optical characteristics for various applications, a topic for future work.Nevertheless, our focus here is on the fundamental properties of the simplest grating, i.e., one with at most a single slant.The slant angle θ G is The peak efficiency angle of incidence θ P is related to the slant angle by the equation where n in is the refractive index of the incident medium.For unslanted gratings (θ G = 0), peak diffraction efficiency occurs when the angle of incidence θ in = θ P = θ B .The pattern Φ(x, 0) may be formed via holographic lithography [8][9][10]15], direct-write laser scanning [15,40], or digital projection lithography [41].Note that the recording process always sets Λ x in the photo-alignment [42,43] layer (PAL), and that the volumetric period depends on this and the slant angle (Λ = Λ x cos θ G ). Two consequences of this can be uncovered by considering the conservation of momentum [18] (k m = mG + k in ), illustrated in Fig. 2. First, unlike slanted VHGs [21], all slanted LC Bragg PGs with the same Λ x will diffract into the same output angles.Second, the diffraction angle θ m for order m (for PGs in any regime) follows: where n out is the exit media refractive index.Our angle convention is shown in Fig. 2. For ideal PGs, only the first and zero diffraction orders [15], m = {−1, 0, +1}, can have non-zero efficiency.

Fabrication
Here, we aim to fabricate a series of LC polymer Bragg PGs with Λ = 335 nm for λ = 450 nm, where we vary the chiral material concentration to adjust φ and θ G .Fabrication begins with coating the azo-based PAL [43] LIA-CO01 (DIC Corp) on clean glass (D263) substrates (spin: 30 s @ 1500 rpm, bake: 60 s @ 130 • C), about 30 nm thick.For exposure, we employ two-beam polarization holographic lithography [8,9], using a solid-state 355 nm laser (Coherent Inc) arranged to provide two coherent, orthogonal, circularly polarized beams superimposed onto the PAL at ±32 • .Exposure energy was 4 J/cm 2 .The first LCP sublayer we coat serves the purpose of extending and enhancing the anchoring strength of the PAL to the second LCP sublayer.We use a first reactive mesogen solution, comprising solids RMM-A (Merck KGaA, Δn = 0.17 @ 450 nm) in solvent propylene-glycol-methyl-ether-acetate (PG-MEA from Sigma-Aldrich), with a 5% solids concentration.This is processed (spin: 60 s @ Fig. 3. Scanning electron microscope image of a slanted Bragg PG (3% chiral) with peak efficiency angle θ P = −3 • .700 rpm, cure: 30 s @ 190 mW of UV illumination from a 365 nm LED in dry nitrogen environment) to create approximately d 1 = 75 nm and φ 1 = 0.The next set of sublayers cause nearly all of the diffraction.For the second and subsequent sublayers, we use a second reactive mesogen solution, comprising 3% RMM-B (Merck KGaA, Δn = 0.28 and n = 1.68 @ 450 nm) in solvent PGMEA and in most cases a chiral nematic reactive mesogen RMM-C (Merck KGaA, helical pitch ∼ 400 nm and HT P ∼ 2.5 μm −1 , otherwise identical to RMM-B).This is processed (spin and cure steps identical to the first LCP sublayer except for 1000 rpm) to create d 2 = 55 nm.This was repeated 16 times, for a total of N = 18 and approximately d = 1000 nm.Thicknesses were measured by profilometer and ellipsometer.
To facilitate the the diffraction efficiency measurement for a wide range of incidence angles accurately, we laminated a high index substrate (n out = 1.74) onto the grating ( n = 1.68 @ 450 nm) with optical glue (NOA 170, Norland Products, Inc).The first diffraction order is then out-coupled with high index oil (n oil ≈ 1.7) and a Dove prism made from SF11 glass (n prism = 1.82).The original glass substrate (D263) will then serve as an endcap (n endca p = 1.53).

Results
To examine the nanoscale nematic director profile of the LCP, we prepared one sample with 3% chiral concentration for study with a scanning electron microscope (SEM).We first submersed it in liquid nitrogen, then broke it, and finally evaporated a 5 nm layer of gold onto it.The resulting cross-section of the fractured edge is shown in Fig. 3, from the same perspective as Fig. 1.The overall structure corresponds well to the expected nematic director profile of Eq. ( 2), where both the periodicity and grating slant are easily observed.The period and film thickness measured by the SEM software corresponds well to our other measurements.The SEM-measured slant angle is around −30 • (including the negative sign to be consistent with the coordinates), which can be used along with Eq. ( 4) to predict a peak efficiency angle θ P ≈ −3 • .This also shows good agreement with the optically measured value for this slant (i.e., purple curves discussed next), and with the nematic director predicted in Fig. 1(b).This same grating texture was observed over the entire 10 mm long edge of our SEM sample.
A series of LC polymer Bragg PGs were fabricated using a range of chiral concentrations, from 0% to 6%, in order to achieve different peak diffraction angles.The nonchiral sample (blue) is shown in Fig. 4 with a peak first-order efficiency of about 99%, defined as η −1 = P −1 /(P −1 + P 0 ) where P m is the output power of order m.Its corresponding field-of-view FOV was 29 • , defined as the full-width-half-max (FWHM) of the angular response.As chiral material is incorporated, the slant angle becomes nonzero and the peak angle shifts toward negative angles, as anticipated by Eq. ( 4), eventually meeting and exceeding the normal direction.These angular responses are also shown in Fig. 4. One sample (purple, 3% chiral) was nearly on-axis, with θ P = −3 • , η −1 = 88%, and FOV = 19 • .A photograph of these two samples at peak diffraction is also shown in Fig. 5(a) and 5(b).The general trends are summarized in Fig. 6, where η −1 , FOV , and Λ decrease slightly, while θ G and φ decrease substantially, all in a linear fashion with respect to θ P .
We also measured the spectral response of the zero-order for two samples with a spectrophotometer, and estimated the first-order efficiency.The result is shown in Fig. 7, where we observe the remarkably wide spectral bandwdith (FWHM) of ≥ 200 nm.
Two polarization measurements were performed.First, we measured the zero-and first-order efficiencies as the incident polarization was varied by rotating a quartz quarterwave (QW) plate receiving linearly polarized light, to vary the input from circular to linear and back to the orthogonal circular polarization.The result is shown in Fig. 8(a) for the nonchiral and nearly on-axis samples (blue and purple results in Fig. 4).The response shows that for both nonchiral and chiral Bragg PGs the highest and lowest efficiencies occur for left-and right-handed circular (LHC, RHC) input polarizations, respectively, as anticipated [4,24,25,35].It is notable that when the input is RHC, the Bragg PG is nearly transparent.The extinction ratio of the first-order for LHC/RHC input is about 400:1 and 55:1 for the nonchiral (blue) and chiral (purple) samples,  respectively.Second, we measured the output polarization states of the zero-and first-orders when the input polarization was set to achieve maximum efficiency.The result for these same  two samples across their respective FOV is shown in Fig. 8(b).As expected, the first-order handedness (RH) is also opposite the input LHC.Most notably, the nonchiral sample (blue) output a first-order with nearly circular polarization (ellipticity angle χ 1 ∼ 40 • ).Somewhat most surprisingly, the chiral sample (purple) output a nearly linearly polarized first-order.In both cases, the zero-orders are elliptically polarized.
Finally, we measured what occurs when total-internal-reflection (TIR) causes the first-order to reflect back onto the same grating multiple times.We illustrate in Fig. 9 the light path and the measured efficiencies of each interaction, relative to the input.After the first interaction, 88% of the incident wave was directed into a waveguiding angle, with nearly linear polarization.After TIR at the waveguide-air surface and returning for the second set of interactions, most of the light undergoes TIR at the endcap, and is equally split into its zero-and first-orders, where 41% diffracts out (m = +1 order).This apparently occurs because the two TIR events preserved the linear polarization of the wave.The remaining interactions out-couple a lower and lower fraction of the light, presumably because each adjusts the polarization.

Discussion
We selected this set of Λ and λ because blue is the most extreme visible color of a light-emittingdiode or laser light source, and also because this challenging sub-wavelength period is especially relevant to exit-pupil-expanders [17,20] (EPE) based on diffractive waveguides.All larger periods/wavelengths should be easier to realize.The unslanted version of this grating should have θ B = 42 • in air, meaning that light incident at this angle will diffract into θ −1 = −42 both n in = n out = 1, a remarkable 84 • deflection angle (between the zero-and first-orders).At normal incidence, the (on-axis) slanted version of this grating will diffract into θ −1 = 50.5 • inside a high index (n out = 1.74) substrate, according to Eq. ( 5).In order to experimentally verify Eq. ( 5), we measured the output diffraction angle at the peak angle incidence (θ in = θ P = −3 • ) in the 3% chiral sample, and found that θ −1 ≈ 45 • inside the exit medium (waveguide), which is very close to the analytic prediction θ −1 = 45.2 • .These PGs are solidly in the Bragg regime (Q ≈ 17) and Λ x is nearly 3000 lines/mm for all samples.Both unslanted and slanted Bragg PGs reach the maximum efficiency around half-wave thickness [38], which is similar to conventional Raman-Nath PGs [1,2,28].Interestingly, these LC polymer Bragg PGs employ a relatively-high Δn and therefore d is lower than VHGs [21].This leads to a much larger FOV and spectral bandwidth while maintaining high efficiency.As compared to SRGs, the efficiency and FOV are comparable, or even moderately larger [20]; nevertheless, a primary benefit may be the absence of development and etching steps in fabrication.Note that because the twist and thickness of each sublayer in LC polymer Bragg PGs can be independently adjusted, it should be possible to implement structures more complex than the single slant described here; for example, two slants to increase the FOV.
The transmittance, defined as T m = P m /P in , is nearly the same efficiency in our case, because the primary loss is the air-glass interfaces (about 5% each), and lower losses between the various glass, glue, and oil media (see Fig. 2).There was no haze observable by eye, and the absorbance of the LCP and PAL materials at 450 nm is < 1%.This work uncovers several surprises.First, while the input polarization response [Fig.8(a)] is the same as Raman-Nath PGs, the output polarization [Fig.8(b)] is unexpected: (i) it is not guaranteed to be circular, as in Raman-Nath PGs; and (ii) when the grating has a substantial slant, it is in general complex (depending at least on θ in , φ), and is sometimes almost purely linear.Second, it may be surprising to some that photo-alignment layers might support grating periods so small -nevertheless, at least one prior report [44] showed limited diffraction down to 200 nm periods.Third, a nonchiral material can apparently lead to θ P (= 38 • ) slightly less than predicted θ B (= 42 • ), i.e., blue curve in Fig. 4. Note we have repeatedly checked our period and angular response measurements; we also do not observe this behavior in our other work at Λ x ≥ 400 nm.This effect may be related to the appearance of pretilt in this nonchiral case.

Conclusion
We experimentally demonstrated the first LC polymer Bragg PGs at a visible wavelength (λ = 450 nm) and sub-wavelength period (Λ x = 335 nm) -the smallest for a high-efficiency PG of any kind.We assessed the use of a chiral nematic twisted LC material to achieve a slanted grating, and thereby adjust the angle of peak efficiency on-axis.These LC Bragg PGs manifest nearly ideal properties, including 85-99% efficiency, 19-29 • FOV , and have a spec-tral bandwidth of about 200 nm.They are strongly selective to input polarization, but in some cases produce unexpected output polarization.We believe the path is now viable for PGs development into many new small-period/large-angle applications, including augmented-reality systems, spectroscopy, optical telecommunications, polarimetry, front/back-lighting, nonmechanical beam steering, and remote optical sensing.