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Abstract
Diffractive optics elements have exhibited many novel characteristics through various methods of employing Pancharatnam-Berry, or geometric, phase. One geometric-phase hologram (GPH) subset, consisting of a π-difference binary sampling, shows polarization-independent properties that are not present in the continuous GPH and the dynamic-phase binary analog. Here, we investigate the binary geometric-phase holograms (bin-GPHs) realized with anisotropic liquid crystal (LC) polymers. First, the optical properties of the ideal binary polarization grating are derived and simulated showing 81% cumulative first-order efficiency, polarization-independent diffraction when applying a π-switching scheme, innate odd (m = 2k + 1) diffractive orders, and variable polarization output. After, experimental results of two key bin-GPH elements, the binary polarization grating (Λ = 30μm) and binary geometric-phase lens (f/100), with π-offset regions and a 0.5μm transition pixel are presented. We found that the fabricated non-ideal bin-GPHs exhibit near-maximum theoretical polarization-insensitive diffraction efficiency and tunable polarization outputs. The simple, and scalable, fabrication of the anisotropic bin-GPH provides the potential for implementation within the next-generation near-eye displays for polarization-invariant beam-steering and waveguides.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Polarization-dependent performance can often have a negative effect on optical systems, from the exoplanet detection systems within large optical telescopes to near-eye displays overall throughput. Diffractive optical elements (DOEs) are critical optical components within these systems but bulky dynamic- and thin geometric-phase DOEs are both largely polarization sensitive [1–4]. Therefore, either polarization-based loss is accepted or additional components are used for tuning the input polarization of the light whether its passing through the atmosphere or being wave-guided through your headset. Additional optics increase system complexity and reduce throughput. For near-eye displays, polarization-independent thin-film DOEs could aid in the development of lightweight and compact optical structures integral to achieving the wearability needed for widely-used consumer extended-reality (XR) systems [5,6].
Geometric-phase holograms (GPH) are DOEs that permit highly-efficient manners of wavefront shaping and are commonly implemented with spatially-varying anisotropic liquid crystals [7] (LC) and metasurfaces. The most common form of GPHs are made with a continuously-varying phase profile in order to achieve close to ideal ($\eta _{\pm 1}\cong 100\%$) polarization-sensitive diffraction efficiency in the $\pm 1$-orders [8–17]. In contrast, the discrete phase profile consisting of strictly alternating $\pi$-phase steps has been theorized to have unique polarization-independent qualities [18]. Polarization-insensitive metasurfaces have been demonstrated with this technique [19–21]. However, this phase pattern can be replicated in anisotropic GPHs by limiting the local orientations to the equivalent discrete phase steps. We call this the binary geometric-phase hologram (bin-GPH). The bin-GPH could have many applications in remote sensing, near-eye displays, and optical communication systems.
In this paper, the binary phase patterning is replicated with anisotropic thin-films that have two discrete regions of differing phases, $\delta _1$ and $\delta_{2}$, which are controlled by the local orientation. Similar to the metasurface variants, the binary period can be used as a building block to create any arbitrary GPH. Two bin-GPH variants of common continuous GPHs, the polarization grating and the geometric-phase lens, were investigated to illustrate these principles. These variants were explored assuming a $\pi$-phase offset between $\delta _1$ and $\delta _2$; however, the arbitrary offset is addressed in the appendix. The Jones calculus analytical solution of the bin-GPH is presented under the monochromatic paraxial approximation. Finally, experimental results of the two bin-GPHs are shown.
2. Background
Geometric-phase holograms are a DOE subset that induce a spatially-varying phase factor to light waves incident on the thin-film or surface. The two predominate categories of GPHs are metasurfaces and patterned anisotropic thin-films. Metasurfaces employ subwavelength metallic or dielectric structures for resonance tuning of the wavefront [22–26]. The transmissive thin-films utilize Pancharatnam-Berry, or geometric, phase profiles in which the local phase, $\delta (x,y)$, is solely dependant on the local orientation angle, $\Phi (x,y)$, described by the relationship $\delta (x,y) = \pm 2\Phi (x,y)$. Often, GPHs are designed with continuous phase profiles due to their near 100$\%$ efficiency, polarization-selective characteristics, and strict adherence to the geometric-phase hologram theory. A continuous phase profile is achieved by sampling, $N$, the $0-2\pi$ phase change with at a rate of $N\geq 4$ [15,27]. Minimized sampling of two discrete regions ($N=2$) with a $\pi$-phase offset has been theorized as a degenerate state in geometric-phase DOEs with the unique characteristic of polarization independence that is not found at other sampling rates [18]. This can be explained by the $\pi$-offset binary phase patterning producing a phase scheme with indistinguishable $\pm$phase ramps due to $0-2\pi$ phase synchronicity.
In prior art, this GPH patterning has been demonstrated with metasurfaces through beamsplitters [19,20] and lenses [21]. These studies have shown geometric-phase metasurfaces with discrete $\pi$-phase offsets between the regions that exhibit polarization-invariant performance. Polarization-invariant diffraction has also been demonstrated in binary dynamic-phase anisotropic DOEs with orthogonal alignments [28–34]. However, the bin-GPH solely utilizes geometric phase.
3. Theory
3.1 Binary polarization gratings (bin-PGs)
The polarization grating (PG) is a standard GPH synonymous to the dynamic-phase prism along one direction. Polarization gratings are constructed with a continuous linear phase ramp through a spatially-varying optical axis ${\Phi (x) = \pi x /\Lambda }$, where $\Lambda$ is the constant grating period, with phase cycles of 2$\pi$ [12]. The synonymous binary polarization grating (bin-PG) is defined as possessing two equally-sized regions of arbitrary orientation, $\Phi _1$ and $\Phi _2$, with the transition occurring at $\frac {\Lambda }{2}$ [35]. The geometric phase profile of both PGs are determined by the varying orientations of the planar anisotropic LC films and adhere to the on-axis diffractive grating equation with normal incidence, $\sin (\theta )=\frac {m\lambda }{\Lambda }$. The binary-PG phase profile is realized through local period alignment of $\Phi$ as described by the nematic director profile:
This premise was investigated in a simplified $0^{\circ }-90^{\circ }$ patterning with paraxial Jones matrix methodology on an infinite grating [36]. The general $\pi$-offset grating can be described by fixing the nematic director profile to $\Phi _2 = \Phi _1 + 90^{\circ }$, leading to the Jones transfer matrix, $T_m$, as:
The $\Sigma \eta _{\pm 1}$ of the $\pi$-offset bin-PG, and therefore the maximum theoretical diffraction efficiencies of all bin-GPHs, is $81\%$ when $\zeta = \pi /2$. The $90^{\circ }$-orientation offset, or $\pi$-phase offset, displays the greatest first-order diffraction while also matching the binary geometric-phase scheme theorized to produce the polarization-invariant phenomenon [18]. Additional odd orders are invariably present. An overview of the $\pi$-phase offset bin-PG is shown in Fig. (1).
Fig. 1. (a). The ideal orientation profile of a bin-PG where one period consists of two separate regions ($N=2$) of uniform alignment and equal width. (b). The geometric phase output along the varying x-axis. (c). Unpolarized incident light is diffracted into the odd orders at an angle determined by the grating equation and the efficiency equations. The blue lines correspond to the nematic director in each region of the liquid crystal polymer that is aligned into the bin-PG pattern on a transmissive substrate. The alignment shown is that of the maximum diffraction efficiency with a $\pi$-offset between $\delta _{1}$ and $\delta _{2}$.
The polarization of the output electric field, $E_{out} = T_mE_{in}$, and is shown through the polarization ellipse angles, orientation angle ($\psi$) and ellipticity angle ($\chi$). These angles are derived through the set of equations, $\tan {2\psi } =\tan {2\alpha }\cos {\delta }$ and $\sin {2\chi } =\sin {2\alpha }\sin {\delta }$, where $\alpha = \arctan ({|E_y|/|E_x|})$. For the $\pi$-phase offset bin-GPH, the angles can be expressed as
assuming $0^{\circ }\leq \Phi _1\leq 180^{\circ }$ due to the liquid crystal 180$^{\circ }$ orientation congruence, $-45^{\circ }\leq \chi \leq 45^{\circ }$, and $0^{\circ }\leq \psi \leq 180^{\circ }$. The $\pi$-phase offset bin-PG output polarization is solely dependent on $\Phi _1$ and $E_{in}$ with $m$ having no impact on the state of the output polarization. The zero-order polarization output is equivalent to the input polarization - consistent with the GPH theory [15]. These characteristics of the non-zero orders are notably different from the continuous polarization grating, which possess a circular polarization selectivity. In addition, the $m=\pm 1$ outputs of continuous PGs are always left- or right-circularly polarized and possesses orthogonal polarization outputs [12]. Control of $\Phi _1$ in bin-PG fabrication provides an opportunity for tailoring of the output polarization given a known input. The output polarization can also be varied regionally without impacting the diffraction efficiency. The general characterizing equations for a bin-PG with an arbitrary offset can be found in the appendix.3.2 Binary geometric phase lens (bin-GPL)
The continuous geometric phase lens (GPL) is another fundamental GPH that can be recreated using the bin-GPH alignment scheme. The binary geometric phase lens (bin-GPL) is a direct comparison to theorized polarization-independent lenses [15,18] and the general theory has been implemented using metasurfaces [21]. The phase profile used to achieve the continuous GPL with no spherical aberration is:
where $f_0$ is the center focal length and $\lambda _0$ is the center wavelength [37]. The continuous orientation profile $\Phi {(x,y)}$ is transferred into the bin-GPL orientation profile ($\Phi _1 = 0$, $\Phi _2 = 90$) by: where mod and round are Matlab functions [38]. Under the paraxial approximation, the offset within the local binary period is the determinate of the bin-GPH efficiency. Therefore, the bin-GPL diffraction efficiency is equivalent to that shown in Eq. (4) and Eq. (5). When $\zeta = \pi /2$, the ${\pm }1$ waves of the bin-GPL will each maintain a cumulative $81\%$ diffraction efficiency at $\lambda _0$ with no zero-order leakage, invariant of the input polarization. Similar to the Fresnel zone plate, the presence of higher orders leads to the bin-GPL having additional focusing/defocusing waves of ${\pm }f_m$ approximately following:Chromatic focal dispersion and minimum grating period are consistent for all anisotropic GPLs, binary and continuous [37]. The basic characteristics of a bin-GPL are shown in Fig. (2).
Fig. 2. (a). A bin-GPL (f/200 at 532nm) orientation profile where $\Phi _2 = \Phi _1 + 90^{\circ }$. (b). The geometric phase output for a cross-section of the lens center. (c). The diffraction characteristics of a bin-GPL is shown. Incident light of any polarization is split into the odd ${\pm }f$ focusing/defocusing waves. The image is limited to the leakage and highest power odd orders ($m = 0,\pm 1,\pm 3$) are shown. Higher orders ($m\geq \pm 3$) will converge/diverge according to ${\pm }f_m$.
The polarization output of the bin-GPL, and all bin-GPHs, remains consistent under the paraxial approximation. Therefore, polarization output follows the analysis of the bin-PG and is equally dependent on $\Phi _{1}$ and $E_{in}$.
4. Fabrication
The fabrication of binary polarization gratings is analogous to that of continuous GPHs [39]. The use of direct-write holography provides the theoretical means to realize any arbitrary bin-GPH. The process begins with the application of a photo-alignment layer (PAL) LIA-CO01 (DIC Corp) onto 0.55mm thick D263 substrates that is then annealed ($130^{\circ }$C, $2$ min). After cooling, the substrate is placed on a 2D translation table within the direct-write holography system where it is then exposed with a linearly-polarized UV laser (Coherent Inc). The exposure parameters are designed to limit pixel cross-talk and apply $1 J/cm^2$ to achieve appropriate anchoring strength of the PAL [39]. The exposed pixel alignment angles are within $0.1^{\circ }$ of the design [40].
In the ideal case of bin-GPHs, there is an instantaneous transition between the contiguous phase regions. However, this phase step boundary breaks the liquid crystal elastic continuum creating many point defects which negatively impact the output polarization and efficiency [35]. A transition region is implemented between the two phase regions by introducing a biasing of the rotation direction [35]. The transition region is set to be a single biasing pixel of half the orientation difference, $\Phi _b =(\Phi _1 + \Phi _2)/2$. This process detracts from the ideal bin-GPH proportional to the width of the transition region [36]. Two biasing pixels are included into the local period at the locations x = 0, $\Lambda /2$. Higher resolutions are used to decrease the proportion of local period dedicated to the transition regions. A 0.5$\mu$m resolution has been achieved for fabrication of all bin-GPHs.
The anisotropic liquid crystal polymer (LCP) layer is applied using a reactive mesogen solution of RMM-A (Merck KGaA, $\Delta {n} = 0.15$ at 633nm) and propylene glycol methyl ether acetate (PGMEA) by spin-coating after the direct-write holography. The LCP is processed (60s @ 1000 rpm, cure: 75 s @ 190 mW of UV illumination from a 365-nm LED in dry nitrogen environment) with a $5\%$ solids concentration mixture. Due to the small feature size of the transition region, the film was fabricated with 14 sublayers. An optical thickness of approximately half-wave retardation ($\lambda _0=2{\Delta }nd$) at 532nm was achieved for maximum diffraction efficiency at that wavelength.
5. Results
5.1 Binary polarization grating
A $\pi$-offset bin-PG with a period of 30$\mu$m and a 0.5$\mu$m biasing pixel was fabricated. Micrographs of the bin-PG, shown in Fig. 3(a), have no visible liquid crystal defects. The bin-PG with a $\pi$-offset phase pattern displays an interference pattern of two synonymous regions in polarized micrograph images. The transition region shows a noticeable boundary due to the $45^{\circ }$ orientation offset.
Fig. 3. (a). Micrograph image of a fabricated 30$\mu$m $\pi$-offset bin-PG with a 0.5$\mu$m transition region. The alignment of the LCP is illustrated. The (b). diffraction pattern of a fabricated 30$\mu$m bin-PG with 532nm laser with a varying input polarizations. The diffraction efficiencies ($m={0,\pm 1,+2,+3}$) with (c). varying $\psi$ and (d). $\chi$ input.
Characterization of the bin-GPH was completed with a 532nm collimated green laser beam at normal incidence to the sample as show in Fig. 3(a). The diffraction efficiency of the m= $0,{\pm }1,{\pm }2,{\pm }3$ orders were measured using a Thorlabs PM100D power meter and S142C integrating sphere. A cumulative first-order diffraction efficiency $\Sigma \eta _{{\pm }1}= 80.2\%$ was recorded.
Imaging polarimetry was conducted on the bin-PG with a collimated 532nm LED input and a LUCID TRI050S1-PC camera. The bin-PG was aligned with the diffraction pattern parallel to the table and camera pixels. Figure (4) shows that the $\psi$ angle of the output polarization closely matches Eq. (6) even though ellipticity was measured. The imaging polarimetry across the rotation of the input retarder is shown in Fig. 4(b) and generally follows the expected output.
Fig. 4. (a) The measured bin-PG polarization outputs, $\psi$ and $\chi$, of m= $0,{\pm }1,+3$ with a varying $\psi$ input by polarizer rotation. (b) The measured bin-PG ellipticity output of m= $0,{\pm }1,+3$ with a varying $\chi$ input by retarder rotation.
5.2 Binary geometric phase lens
The second fabricated bin-GPH, shown in Fig. (5), is a $\pi$-offset f/100 bin-GPL with a 0.5$\mu$m transition pixel. The fabricated bin-GPL exhibits ever-present converging and diverging waves because of the polarization-insensitive diffraction. This is shown in Fig. 5(c-e) by the presence of the visible diverging and converging laser input onto the edge of the lens. The succulent in Fig. 6 is imaged through polarizations set at the input of the bin-GPL. In Fig. 6(b-e), the $m= +1$ focal point of the plant sustains regardless of the polarization inputs. However, the other orders are also present. This is contrasted with an equivalent f/100 GPL in Fig. 6(f-i) where the output is switched relative to the input polarization.
Fig. 5. The cross-polarized (a). center and (b). edge micrograph of a fabricated f/100 bin-GPL ($\Phi _1 = 0^{\circ }, \Phi _2 = 90^{\circ }$) with a 0.5$\mu$m transition region. The edge of bin-GPL diffraction pattern with incident light that is (c). linear (d). RCP and (e). LCP polarized. The edge of GPL diffraction pattern with incident light that is (f). linear (g). RCP and (h). LCP polarized.
Fig. 6. (a). The imaging of a succulent plant through a lens and polarizer. The images (b-e) show the succulent through a bin-GPL with various input polarizations. Images (f-i) show the succulent imaged through a conventional continuous GPL with various input polarizations.
6. Discussion
The fabricated bin-GPH elements here effectively use a single-pixel transition region to avoid disclination lines. However, even with this deviation from the ideal phase pattern, the experimental results thoroughly matched the bin-GPH theory. The average measured $\Sigma \eta _{{\pm }1}$ of the $\pi$-phase offset bin-PG was $80.2\%$, which is $99\%$ of the theoretical max of $\Sigma \eta _{{\pm }1}=81\%$. The transition region leads to a minor increase in polarization selectivity due to minimally increased sampling, $N=3$. A minor circularly-polarized selectivity can be seen in Fig. 3(b) with a ${\pm }3\%$ shift within the first order between RHC and LHC inputs. This effect is proportional to the relative size of the biasing pixel compared to the period. Additional discrepancy can be owed to the non-ideal retardation and bulk alignment. The experimental polarization results in Fig. 4(a) showed up to a 20$^\circ$ ellipticity deviation from theory. It is assumed that the ellipticity deviations are also accounted for by the non-ideal phase pattern ($N = 3$) and fabrication.
Only two common phase patterns, the bin-PG and bin-GPL, were assessed. The separate bin-GPH phase profiles provided validation that bin-GPH theory can be utilized in any DOE design. While the $\pi$-phase pattern was selected for the focus of the study, the binary geometric-phase theory is more general. The bin-GPH phase pattern can be implemented with any spatially-varying binary local period of LC orientation and offset.
Currently, direct-write holography is the best method for high resolution bin-GPHs. This method provides a means for any bin-GPH to be fabricated with the inclusion of biasing pixels. Even though the inclusion of the biasing pixels detracts from the ideal bin-GPH profile, it was necessary to produce a high-performance bin-GPH. Large feature sizes ($\Lambda = 30{\mu }m$, f/$\#$ = $100$) were used in fabrication to reduce the impact of the $0.5{\mu }m$ biasing pixel. Minimization of the biasing pixel is crucial for achieving ideal bin-GPHs with smaller feature sizes. Improved fabrication techniques and photoalignment materials are needed. This is a limitation specific to the anisotropic LC bin-GPH that is not present for metasurfaces or dynamic-phase DOEs.
7. Conclusion
We introduce the concept of the binary geometric-phase hologram (bin-GPH) and present theoretical analysis and experimental results. These geometric-phase structures utilize a binary LC alignment that has also previously shown polarization-independent diffraction in dynamic-phase anisotropic optics. A defect-free LC polymer bin-GPH phase pattern was achieved by the inclusion of a single transition pixel - a technique not required in continuous GPH elements. We demonstrate the polarization-independent capabilities of two bin-GPH elements manifested through the utilization of strict $\pi$-phase steps. The bin-PG with $\Lambda = 30{\mu }m$ experimentally exhibited a total first-order diffraction efficiency of $\Sigma \eta _{{\pm }1}= 80.2\%$ at $\lambda _0 = 532nm$, with a theoretical maximum of $81\%$, regardless of the input polarization. In addition, the output polarizations of all even diffractive orders are equivalent but possess a dependency on the input polarization and LC alignment angles. All characterization show a stark deviation from the strictly circularly-polarized outputs characteristic of continuous geometric-phase optics. To illustrate the adaptability of the bin-GPH theory, a second DOE was also produced, the f/$100$ bin-GPL, using the same $\pi$-offset polarization-independent phase pattern. The 2D anisotropic thin-film bin-GPH provides the building block for a new DOE subset synonymous to recent polarization-insensitive metasurfaces. Specifically, the polarization-independent diffraction efficiency of the $\pi$-offset phase scheme has potential for use in astronomy and AR/VR systems.
8. Appendix
The comprehensive equation for a general bin-GPH where the orientation of $\Phi _2$ is independent of $\Phi _1$ for all orders is:
Equivalent to the $\pi$-offset case, Eq. (11) is zero for all even values of $m$ regardless of $\Phi _1$ and $\Phi _2$ values. Therefore, it can be represented in a simplified from as
Funding
ImagineOptix Corporation (NCSU grant 2014-2450).
Acknowledgments
N. Zane Warriner thanks Dr. Micheal Kudenov for access to his lab and tools for polarimetry. We thank Merck KGaA for customized reactive mesogen materials.
Disclosures
The authors declare no conflicts of interest.
Data Availability
The data that support the findings of this study are available from the authors on reasonable request, see author contributions for specific data sets.
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