1. Introduction

The challenge in imaging faint planets near bright stars is to greatly reduce starlight and associated photon noise while efficiently transmitting nearby planet light. High scattered starlight levels limit achievable contrasts, so exoplanet imaging very close to stars requires a high degree of wave front correction as well as a coronagraph that transmits well at very small angular separations. Modifying the phase of the stellar focal plane point spread function rather than its intensity allows very small angles to be reached, and the vortex phase mask in particular, which applies an azimuthal phase ramp to the stellar Airy pattern, operates close to the ideal coronagraphic limit [1].

A vortex phase mask can be realized either as a scalar vortex phase mask based on an azimuthal thickness ramp, or as a vector vortex waveplate (VVW), i.e., a spatially variant half-wave plate based on geometric (Pancharatnam) phase [2]. The latter has the advantage that phase discontinuities are in principle confined to a small central region. Liquid crystals (LCs) and LC polymers (LCPs) allow for the fabrication of VVWs with a continuous rotation of the optical axis and at the spatial frequencies required for obtaining VVWs of a high topological charge, the number of optical axis rotations around the singularity axis [3]. Note that, due to symmetry, the optical axis orientation states differing by 180 degrees are equivalent. LCs are transparent in the visible and near-IR, as well as at select wavelength regions in the mid-IR, far-IR, and microwave spectra, depending on their molecular structures. Due to their high optical anisotropy, the half-wave phase retardation condition is achieved in thin material layers (~1 μm).

In a vortex coronagraph, the bright stellar Airy pattern is centered on the vortex axis. Achieving high contrast thus requires a vortex phase pattern accurate to radii much smaller than Airy disk sizes for typical Cassegrain telescope focal ratios. VVW fabrication technologies run into difficulties near the center of the vortex. In a central region of diameter d determined by technological as well as fundamental factors to be analyzed in this paper, the vortex orientation pattern is replaced by defects and microdomains of LC orientation that scatter and/or distort the phase of the light beam propagating through the area. LCP-based VVWs had earlier been produced with central defects as small as 25 - 30 μm. As this is comparable to near-IR Airy disk sizes for typical Cassegrain telescope focal ratios, the suppression of the light in the center of the stellar Airy disk has thus required the inclusion of a small opaque disk at the axis of the vortex. Further reducing the central vortex defect area could potentially remove the need for a central light blocking disk. This paper presents techniques that have yielded LCP VVWs with reduced central defect sizes, down to < 3 μm, more compatible with visible-wavelength operation.

2. Evaluation of the acceptable defect size

To answer the question of how small the central vortex defect size would need to be to obviate the need to cover it, the optical propagation code PROPER [4] was used to calculate the rejection of a perfect Airy pattern by an imperfect vortex phase mask, when used in a vortex coronagraph (a focal plane vortex phase mask followed by a pupil/Lyot stop matching the input pupil). For the calculations, we assumed zero phase in a central region that was either clear or covered by an opaque absorber of the same size. The results of these calculations are shown in Fig. 1 where it can be seen that the contrast improves markedly as the defect is made smaller, and as it is covered. In fact, the calculations for the leakage due to an uncovered defect area are consistent with a very simple estimate of the light leakage based on the assumption that all of the light in the central defect area is diffracted by a small central aperture of the size of the defect. For apertures small compared to the Airy pattern full width half maximum, only about (d/Fλ)² of the light traverses the central defect (F and λ are the f-number of the focusing system and the wavelength, correspondingly). Subsequently, only (d/Fλ)² of that light makes it through the downstream Lyot stop, leading to a net leakage remaining inside the Lyot stop of (d/Fλ)⁴. Thus, to reach approximately 10⁻⁸ contrast at an angle of about 2λ/D (where D is the mirror diameter) in the uncovered case, the size of the vortex defect needs to be as small as ~0.01Fλ, or d/λ < 0.01F. For the usual case of F < 100, we arrive at a most natural result, d < λ: the central defect essentially needs to be sub-wavelength in size. This is in the range of micrometers for the near-IR ground based case and submicrometers for the visible space-based case.

 

Fig. 1 Rejection of a perfect Airy pattern by an imperfect vortex phase mask of topological charge 2, when used in a vortex coronagraph (a focal plane vortex phase mask followed by a pupil/Lyot stop matching the input pupil). For the calculation, a perfect vortex phase pattern of charge 2 was assumed to apply everywhere except in a small central region of diameter d, wherein the phase was assumed to be zero. In the “covered” case, the transmission of this central region was further assumed to be zero.

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3. LCP VVW fabrication concepts

LCs and LCPs allow continuous radial or azimuthal modulation of the optical axis orientation in the plane of the waveplate. In particular, the optical axis orientation α of a LC material, Fig. 2 , can be made to vary with the azimuthal coordinate ϕ as α = q_ϕ ϕ where q_ϕ, the azimuthal wavevector, can be equal to q_ϕ = n/2 (n = ± 1, 2…) due to the continuity condition Δα(ϕ = 2π) = nπ. The optical properties of VVWs are characterized by the wavelength λ_w at which the half-wave phase retardation condition is met. At fulfillment of the half-wave retardation condition, the linear polarization of light passing through the waveplate rotates twice as fast as the optical axis orientation. For a circularly polarized beam, rotation of the optical axis orientation is exhibited in modulation of the phase equal to 2α = 2q_ϕ ϕ = nϕ. Thus, the topological charge m of the output beam is twice the azimuthal wavevector q_ϕ describing the VVWs structurally.

 

Fig. 2 (a) Optical axis orientation in a VVW: ϕ – azimuthal coordinate; α – the orientation angle of the optical axis. The optical axis orientation patterns are shown for (a) q_φ = 2 and (b) q_φ = 4 outputting a beam of singularity order m = 4 and 8, correspondingly.

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The desired modulation of the optical axis orientation of LCs is obtained by creating appropriate boundary conditions. First, this was realized by mechanically rubbing a polymer coated substrate [5]. This circular rotation process allows the creation of an axially symmetric alignment pattern for a VVW of q_ϕ = 1. Obtaining subwavelength defect sizes with the aid of mechanical rubbing appears also rather challenging. We therefore used the so-called photoalignment technique in the current study wherein a thin film of a material creates anisotropic alignment conditions for LC molecules under the influence of a polarized, typically UV, beam [6].Earlier work [7] had relied on coupled mechanical rotations and a slit-defined beam to lay out the vortex pattern of the desired charge. We employ a two-step fabrication technology that reduces reliance on mechanical rotation and narrow slits. In the first step, the technique of mechanical rotations is essentially modified to minimize the defect size of a VVW of q_ϕ = 1. Those modifications include illumination of the (still rotating) substrate (with the photoalignment layer) by the line focus produced by a cylindrical lens for which the lateral resolution of a few μm is set by the focal length of the lens. The modifications included also redistribution of the light energy from the axis of the beam to its peripheries thus equalizing the exposure energy over the area of the vortex pattern being recorded on the photoalignment layer [8]. Direct rotation-free replication technique based on a linear-to-axial polarization converter is further used for fabrication of VVWs of higher topological charge [3]. A VVW with a half-wave phase retardation condition at the wavelength of the light used for the photoalignment is an example of such a polarization converter. As propagation of a linear polarized (charge 0) beam through a VVW results in an output beam with double the topological charge of the VVW, this technique allows fabricating VVWs with higher topological charges avoiding mechanical motions. Other big advantages of the replication technique are the greatly reduced exposure time, the lack of a need for precise optical alignment, and the opportunity of using versatile imaging/projection systems.

4. Materials and processes

Fabrication of LC polymer VVWs involves two sets of materials: photoalignment materials and polymerizable liquid crystals (PLCs), referred to also as reactive mesogens, and LC pre-polymers. PAAD (Photo-Aligning Azo-Dyes) series materials (www.beamco.com) were used in most tests reported here, specifically, PAAD-22 and PAAD-27. These materials are based on azobenzene, which aligns with its long axis perpendicular to the polarization direction of light of UV and/or blue-green wavelengths. When illuminated by polarized light, a layer of these materials that is a few nanometers thick on a substrate thus produces anisotropic surface anchoring conditions. The molecules of LC materials deposited on such a substrate are oriented along the resultant anisotropy axis. The film such obtained is photopolymerized by an unpolarized UV light, typically in nitrogen atmosphere.

Both the size and the nature of the defect formed in the axial region of VVWs depend on the properties of the photoalignment material. In the course of substrate rotation, the central most area is subject to light of varying polarization due to the width of the line focus. Cinnamate-based materials (earlier available from Rolic Ltd) require a long exposure (~1 h @ 10 mJ/cm²) to a UV light (325 nm wavelength) [9]. Thus the substrate is effectively subject to an unpolarized light source near the center. Together with the irreversibility of the photoalignment process, this results in the creation of random light scattering structures.

In contrast, the alignment processes in the PAAD series photoalignment materials are reversible, and also fast, due to their higher photosensitivity. The typical photoalignment time for PAAD series materials is ~10 min for UV light of 365 nm wavelength and 10 mJ/cm² power density. As a result, the central area subject to the beam acquires fairly homogeneous alignment, avoiding generation and accumulation of defects.

Ultimately, the defect size of a VVW is determined by the anchoring strength of the photoalignment material. The elastic energy density, F, of LC optical axis deformations critically increases near the axis of the VVW: F ~K (gradϕ)² ~K(q_ϕ/d)², where K is the elastic constant of the LC [10]. The defect size therefore fulfills the condition d ~q_ϕ (KL/U_a)^1/2 where U_a is the anchoring energy for the alignment at the boundary layers, and L is the thickness of the LC layer. Thus, the stronger the anchoring, the smaller the size of the defect. For typical parameter values, K ~10⁻¹³J/cm, L ~1 μm, and U_a ~10⁻⁵ J/cm², we may expect a defect size as small as d ~0.01 μm.

High quality VVWs were typically produced on fused silica substrates, 1” in diameter, 3 mm thick, λ/10 surface quality, and anti-reflection (AR) coated (on both sides) for the wavelength of the light used for photoalignment. The photoalignment material was deposited in a spin coating process to form a uniform film thickness of ~20 nm. PLCs, specifically RMS03-001C from Merck, were spin coated on top of the photoalignment layer from a PGMEA (propylene glycol methyl ether acetate) solution. The thickness of the layer that determines the half-wave phase retardation condition is controlled by the speed and duration of the spin coating. The fabrication process is finalized by polymerizing the PLC with an unpolarized UV light (365 nm, mercury xenon lamp coupled to a quartz fiber) under a nitrogen atmosphere for 5 min. For tuning the waveplate to longer wavelengths, the process is repeated, with different spin coating conditions. The waveplate characteristics are sensitive to material processing conditions, and the PLC needed to be pre-processed to minimize/avoid the generation of defects by filtering and degassing. We verified that the fabrication process, incorporating spin coating, is repeatable. No appreciable changes in the specifications of VVWs can be observed when the VVWs are fabricated with the same materials under the same conditions.

To identify the time and the speed of spin coating of the PLC that yield a half-wave retardation at the target wavelength, we fabricated a series of waveplates with uniform optical axis orientation, for which we measured the transmission spectrum of the waveplate between parallel polarizers with its optical axis oriented at 45 degrees with respect to the polarizer/analyzer axes. The minimum transmission in the spectrum occurs at the wavelength corresponding to half-wave phase retardation condition. The plot of this wavelength as a function of rotation speed allows determination of the speed required for tuning to the desired wavelength. Since the orientation far from the center of the VVW can be considered locally homogeneous, the spectrum of peripheral regions obtained between parallel polarizers can be used as an indicator of the obtained phase retardation.

5. Fabricating master VVWs (q_ϕ = 1)

VVWs with q_ϕ = 1 are the basic elements needed by the replication technique for fabricating higher topological charge VVWs, and their quality is thus the critical factor for all offspring VVWs. Perfecting the fabrication technique for the parent q_ϕ = 1 VVWs, which still relies on mechanical rotation, is therefore key to developing high quality VVWs. Above we discussed materials issues; we now address alignment and illumination issues in the optical system used to fabricate the VVWs that are also critical to minimizing the defect size.

The setup for the fabrication of q_ϕ = 1 VVWs is shown schematically in Fig. 3 . UV radiation is expanded to a 60 mm beam diameter with the aid of plano-convex lenses L1 and L2. A pinhole of 20 μm diameter at the focus of lens L1 provides spatial filtering, and a 1” diaphragm D at the output of lens L2 passes the relatively uniform central beam area of intensity ≈10 mW/cm². Finally, a linear aperture 4 mm wide selects a linear beam stripe, which is focused onto the glass substrate by a cylindrical lens of 75 mm focal length, yielding a focused stripe with a half power width of roughly 7 μm. The polarization of the laser beam was set along the y-axis by a linear polarizer, i.e., perpendicular to the transmitted beam stripe, in order to obtain radial alignment boundary conditions in the photoalignment layer.

 

Fig. 3 Experimental setup for fabricating a VVW with q_φ = 1: L1, L2- plano-convex lenses of 40 mm and 1000 mm focal lengths, respectively; PH- pinhole; VF-VVW- vector vortex waveplate for beam shaping; P – polarizer; LA – linear aperture with 4 mm opening; CL – cylindrical lens with 70 mm focal length; Substrate − glass substrate coated with photoalignment layer; RS- precision rotation stage. The beam propagates along the z-axis.

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Alignment of the rotation axis of the rotation stage with the center of the beam stripe is critical for minimizing the defect size. It is equally important to position the substrate with the photoalignment layer in the focus of the beam. The substrate was thus attached to a high precision rotational stage mounted on a high precision linear stage used for positioning the substrate in the transverse direction (y-axis) with respect to the laser beam stripe. The system of stages was mounted such that the rotation axis could be aligned with the central part of the beam stripe (with accuracy ~0.2 μm). A second, manually-controlled, translation stage allowed positioning the substrate in the focus of the cylindrical lens in the beam propagation direction (z-axis). The high sensitivity of the resultant defect size to the transverse alignment (along y-axis) can be seen in Fig. 4 [8,11].

 

Fig. 4 Sensitivity of vortex pattern to lateral offset of the rotation axis of the substrate with respect to the beam axis: (a) 19.6 μm, (b) 6.04 μm; (c) 0.4 μm; (d) 0.

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The optimum values of the rotation speed, 250°/s, and duration, ~30 min, for 10 mW/cm² beam intensity were determined in a series of tests by controlling the quality (uniformity, defect size, etc.) of the product VVWs. Note that the exposure time for producing orientation patterns modulated at high spatial frequencies need to be much longer than the time required for uniform alignment, since longer exposure ensures higher anchoring energy [12].

Light energy received by the central part of the substrate is larger than at the periphery by a factor ~πl/w >> 1, where w is the beam size and l is the size of the waveplate under fabrication. Thus, if the periphery receives enough exposure for good photoalignment, the central part is highly overexposed, potentially degrading the alignment in a large area around the rotation axis. Indeed, in initial tests, reducing the exposure energy density resulted in smaller defect sizes, but at the expense of degraded alignment in the underexposed periphery. Reducing the exposure energy by a factor of nearly 20 (by reducing the light intensity from 10 mW/cm² to 0.6 mW/cm²) resulted in a factor of 4 – 6 decrease in defect size, from 20 μm to ~5 μm for VVWs based on PAAD, and from 200 μm to ~30 μm for VVWs based on ROP-103 (Rolic Ltd.). However, by simply reducing the illumination, the overall diameter of the VVW with good alignment quality was reduced by a similar factor, i.e., from 20 mm to ~4 mm.

To obtain a VVW with both a small central defect and a large overall diameter, the entire substrate must receive comparable net exposure energy. Variable density optical filters such as bullseye filters to be used for such a large attenuation at the axial region of the beam would result in a substantial attenuation of the flux because of absorption or reflection losses. Such filters thus reduce the efficiency of the photoalignment process. Moreover, commercially available variable filters did not yield the desired intensity gradients.

We therefore chose to redistribute the illumination from the vicinity of the beam’s axis to its outskirts. Specifically, this was accomplished by making use of a q_ϕ = 1 VVW, as such a wave plate produces an annular beam. Note that this vortex does not itself need to be perfect, as its role is merely to redistribute a uniform intensity outward (i.e., unlike the ultimate coronagraphic case, the initial illumination is not sharply peaked in a tiny central region). Thus, a small central imperfection in the initial vortex filter translates only to a slight bit of extra light near the center. (Likewise, the post-vortex non-uniform polarization across the stripe’s width is not an issue, because the downstream polarizer selects a linear polarization state.)

To efficiently redistribute the light energy from the center of the beam to its peripheries, a q_ϕ = 1 VVW designed for 325 nm wavelength was used. The defect size of this VVW designated for beam shaping was 20 μm, and it allowed obtaining q_ϕ = 1 VVWs with nearly 10 μm defect size and 20 mm diameter. Moreover, the fabrication process was reduced to 5-7 min due to full utilization of all the available light. Precise setting of the half-wave retardation condition at 325 nm wavelength is essential for efficient reshaping of the beam. The obtained VVW, in which the defect size was limited to 10 μm, was further used as a beam shaping element to fabricate q_ϕ = 1 VVWs with further reduced defect size. Such consecutive improvements allow obtaining a high quality, large area (~1”) VVWs with defect sizes less than 10 μm. Combined with fine adjustment of the position of the beam shaping VVW with respect to the beam axis, this approach ultimately yielded q_ϕ = 1 VVWs with defect sizes of ~5 μm when using a cylindrical lens of 75 mm focal length, and below 3 μm for a lens of 25 mm focal length (Fig. 5 ).

 

Fig. 5 Polarizing microscopy photo of a VVW with 3 μm defect size.

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6. Higher charge “offspring” VVWs

Having established the technique for reducing the defect size in charge 1 vortices, we next aimed at fabricating VVWs with higher topological charge with similarly small defect sizes. We used the replication technique discussed above briefly for this purpose. The setup here (Fig. 6 ) is essentially quite simple consisting of the beam cleanup/expanding system, a polarizer, and a master VVW. Note that the master VVW automatically provides the needed illumination redistribution, thus allowing for reduced exposure times of 5-7 min.

 

Fig. 6 Schematic of the setup for producing VVWs of high topological charge: L1, L2 – lenses for beam expansion; PH – pinhole; P – polarizer; VVW-PC – VVW used as a polarization converter; S – substrate coated with PAAD.

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Figure 7 shows photos of VVWs involved in the process of fabricating a VVW tuned to λw = 800 nm. The VVW of q_ϕ = 2 and λw = 325 nm, Fig. 7(a), produced with the substrate rotation technique, was used to print the VVW of q_ϕ = 4 and λw = 400 nm, Fig. 7(b). Adding two more LCP layers transforms the latter into a VVW with λw = 800 nm. The half-wave phase retardation condition for a beam of 800 nm wavelength is obtained for a polymer film thickness ~3 μm due to relatively small optical anisotropy of available PLCs, n_par – n_perp = 0.13 at 800 nm. The spin coating technology, however, allows obtaining only nearly 1 μm thick films with the materials used in the study. Reducing the spin coating time or speed to produce thicker coatings compromises the quality of the film, because of material viscosity. Therefore, in order to tune the VVW to IR wavelengths, more than one PLC film needs to be deposited on a substrate. The technique was tested first by making uniform half-wave phase retardation plates and cycloidal diffractive waveplates tuned to the red/IR part of the spectrum. The defect size for all resultant VVWs was less than 10 μm. Optimization of the replication setup, particularly, by closer positioning of the substrate with the photoalignment layer to the VVW-polarization converter would allow improving the defect size of the VVWs of higher singularity order.

 

Fig. 7 Photos of VVWs of (a) m = 2, λ_w = 325 nm, (b) m = 4, λ_w = 400 nm, and (c) m = 4, λ_p = 800 nm between crossed polarizers. The VVW shown in (b) is printed using the VVW shown in (a).

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7. Conclusion/prospects

Liquid crystals (LCs), particularly, LC polymers (LCPs) are the only material systems that allow fabrication of VVWs with continuous rotation of the optical axis orientation and at a high azimuthal frequency corresponding to a high topological charge of VVWs. Study of LC/LCP VVWs has been extensive in recent years, but their fabrication yielded VVWs with sizable central defects that scatter light and reduce contrast. However, by using (a) highly sensitive materials with reversible photoalignment, (b) VVWs for redistributing the writing beam intensity from the central area to the peripheries, and (c) VVWs as linear to axial polarization converters, the fabrication of vortex masks with a large overall diameter, small defect size, and a high topological charge is readily achievable. In particular, we have demonstrated the feasibility of developing 25 mm diameter VVWs with less than 3 μm sized defects.

Further improvement of materials, fabrication techniques, and VVW characterization techniques should allow manufacture of VVWs with even smaller singularity sizes, potentially down to the sub-μm regime. RLC series polymerizable liquid crystals (Beam Engineering) allow, for example, obtaining half-wave phase retardation condition at near the red edge of the visible spectrum with a single coating simplifying the technological process and improving the quality of the waveplates. Those materials are optimized for use with PAAD series photoalignment materials. Complex compositions of monomers, surfactants, and solvents have been developed for eliminating defects and textures of LC polymers obtained as coatings as well as optimized for use with PAAD series photoalignment materials.

The implications of the technology extend beyond astronomy. It is particularly interesting to apply the techniques discussed above to improving the quality of liquid crystal VVWs and their arrays due to the prospects of using their tunability for communication and display applications [13,14].

Further improvements would also include fabrication of broadband VVWs with a small defect size. Note that Pancharatnam-phase does not explicitly depend on wavelength. Essentially, VVWs can be made broadband following techniques developed for half-wave plates or cycloidal diffractive waveplates [3,8].