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Abstract
Optical vortices (OVs) with orbital angular momenta show promise for various optical applications. Spiral zone plates (SZPs) can generate a focused OV and are thus used in applications like edge-enhanced imaging. However, conventional SZPs have low diffraction efficiency because they operate based on amplitude modulation. This study proposes a liquid crystal spiral zone plate (LCSZP) that operates based on phase modulation and demonstrates the generation of a focused OV with high efficiency and electric tunability. The LCSZP has no threshold voltage owing to its homeotropic/planar hybrid alignment regions, and it can find applications in imaging, laser processing, and optical manipulation.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical vortices (OVs) with phase singularities have orbital angular momentum [1] that differs from the spin angular momentum associated with the circular polarization of light [2]. The orbital angular momentum is quantized in the unit of Dirac’s constant, ℏ [3], and it can take a value of infinity [4]. Therefore, OVs are considered promising for various optical applications [5,6], including laser processing [7–9], optical manipulation [10], optical communication [11], and imaging [12–16]. A spiral zone plate (SZP) is a diffractive optical element consisting of axisymmetric spiral arms that can generate a focused OV despite a single element [17]. Focused OVs generated SZPs are useful for high-contrast edge-enhancement imaging with low diffraction noise [14,18,19]. Conventional SZPs are usually fabricated using direct writing methods with lasers [20], electron beams [18,21], or focused ion beams [22,23]. In such SZPs, the diffraction efficiency is limited to 20% theoretically because they operate based on light amplitude modulation. SZPs based on light phase modulation can improve the diffraction efficiency and be realized using liquid crystals (LCs).
The forming methods of the spatial phase distribution of an OV have been variously reported. LC spatial light modulators (LCSLM), which can dynamically modulate a spatial light phase, are versatile OV generators. The efficiency is relatively high, typically in the 60%−85% range [24]. However, the LCSZP has limited applications in terms of compactness and productivity. OV generators based on diffractive optical elements, such as spiral phase plate [25], forked hologram [26], and q-plate [27], have the advantage of being thin and flat [28] and commonly have efficiencies ranging from 85% to over 95%. However, additional lenses are required when generating a focused OV using these OV generators, and the alignment between the phase singularity of the OV and the lens’s optical axis is very difficult. Therefore, the realization of SZPs based on diffractive optical elements is desired from viewpoints of compactness and a facile means of a focused OVs generation without the need for the alignment between phase singularity and lens’s optics axis.
LCs exhibit superior electro-optical properties derived from their crystallinity, fluidity, and large refractive index anisotropy, and diffractive optical elements with an LC provide high diffraction efficiency with electric tunability [29]. LC diffractive optical elements, applying an electric field (i.e., voltage), reorient the LC molecules in the bulk and thereby modulate the diffraction efficiency. When an electric field is applied to an LC cell with planar alignment layers on the substrates, the orientation of the LC molecules in the bulk changes with a driving threshold associated with the Fréedericksz transition. In contrast, in a sandwich-type LC cell with a homeotropic alignment layer on one substrate and a planar alignment layer on the opposite substrate, a homeotropic/planar hybrid alignment is induced. In the hybrid alignment region, the orientation of LC molecules in the bulk is tilted without an applied voltage. Thus, the LC cell with a hybrid alignment exhibits a better response against an electric field in that it has no driving threshold and can achieve a fast response time [30,31]. These unique characteristics are attractive for applications such as active optical diffractive elements and optical devices [32–34].
In this light, the present study proposes a liquid crystal spiral zone plate (LCSZP) with homeotropic and hybrid alignment regions. To form an SZP pattern with alternate homeotropic and hybrid alignment regions corresponding to adjacent spiral arms, a method that divides the homeotropic and planar alignment regions on the same substrate surface is developed. This method involves two steps. First, a patterned homeotropic alignment layer, namely, a silane self-assembled monolayer (SAM) with a long alkyl chain, is formed on a substrate using photolithography. Second, a planar photoalignment layer is spin-coated on the substrate. The SAM is hydrophobic, thus allowing two different alignment layers to be formed separately on the same substrate. The diffractive behavior of the LCSZP is simulated by the Fresnel diffraction theory. An LCSZP is fabricated using the abovementioned technique.
Experiments reveal that the LCSZP can generate a focused OV with high efficiency of 32.5%; therefore, this is 1.5 times the theoretical efficiency of conventional SZPs. Further, the LCSZP has a feature of diffractive behavior with electric tunability. In practice, this means that when an applied voltage varies, the diffraction efficiency changes and the generation of a focused OV can be switched. The focusing point corresponding to the wavelength indicating the maximum diffraction efficiency also is shifted. This focusing property allows for micro-focal length control and may be useful in optical manipulation, micro-machining, and biomedical sensing [35]. Therefore, this study can find applications in optical imaging, laser processing, and optical manipulation applications.
2. Comparison of conventional SZP and LCSZP
Figure 1(a) shows a conventional SZP with multiple spiral arms colored black and white. Its transmission function ${t_{\textrm{SZP}}}({r,\theta } )$ is obtained by binarizing a function that multiplies the radial Hilbert transform function $R{H_p}({r,\theta } )= \textrm{exp} ({jp\theta } )$ and the Fresnel lens transmission function $FL({r,\theta } )= \textrm{exp} \left( { - j\frac{{\pi {r^2}}}{{\lambda f}}} \right)$ as follows:
Fig. 1. (a) Structure of conventional SZP. Cross-section of LCSZP (b) without and (c) with applied voltage.
The amplitude of diffracted light through an SZP is denoted as $U({\rho ,\; \varphi ,\; z} )$, and it is calculated as
Figure 1(b) shows the cross-section of an LCSZP. An LCSZP has homeotropic and hybrid alignment regions corresponding to the black and white spiral arms in Fig. 1(a).
When linearly polarized light with the same azimuthal direction as the direction of the easy axis of the planar alignment layer is incident on an LCSZP, the transmission function of the LCSZP ${t_{\textrm{LCSZP}}}$ is represented as
When a voltage is applied between the substrates, although the LC in the hybrid alignment region is tilted according to the applied electric fields, that in the homeotropic alignment region maintains its original orientation, and the phase retardation in the hybrid alignment region changes. Thus, the diffraction efficiency changes depending on the applied voltage. When the phase retardation is full-wavelength, the incident light travels straight ahead without being diffracted. In other words, the LCSZP can switch the generation of a focused OV beam depending on the applied voltage.
The focal length of an LCSZP is
where l is the number of phase periods in the radial direction. The focal length is inversely proportional to an incident wavelength. According to Eq. (3), the retardation changes with voltage, and the diffraction efficiency depending on the wavelength also varies. Therefore, the focusing point, which indicates the maximum diffraction efficiency, shifts depending on applied voltages.3. Numerical simulation of light propagation through LCSZP by Fresnel diffraction
To confirm that the LCSZP can generate a focused OV, a diffraction image was calculated by the Fresnel diffraction theory based on algorithms using the fast Fourier transform [36]. A diffraction image with a monochromatic plane wave was obtained by the convolution of the LCSZP transmission function and the incident light wave. The convolution can be calculated by the fast Fourier transform.
We designed an LCSZP with an aperture of 1.27 mm, order of radial Hilbert transform of 5, and focal length of 46.6 mm at an incident wavelength of 532 nm. When a plane-wave-shaped beam with 532-nm wavelength was incident, the axial-intensity distribution in the direction of the propagation axis was obtained from the calculated diffraction images. The calculation range of the propagation distance between the aperture plane (i.e., LCSZP) and the observation plane was from 25.0 to 100.0 mm in 0.5-mm increments.
Figures 2(a) and (b) show the diffraction image at the focal length and axial-intensity distribution. The intensity profile of the diffraction image has a donut shape, with the light intensity at the center being zero. The intensity of zero has been maintained along the propagation axis. At the focal length, the intensity of the donut-shaped beam was the strongest, and the donut hole diameter was the tightest. The diffraction efficiency of LCSZP was calculated and obtained as 41%. The diffraction efficiency of conventional SZP was calculated as 20%. Therefore, LCSZP can generate a focused OV beam with higher efficiency than the diffraction efficiency of conventional SZPs.
Fig. 2. (a) Diffraction image at focal length and (b) axial-intensity distribution of generated focused vortex beam obtained by numerical simulation based on Fresnel diffraction theory.
4. Preparing an LCSZP
We have developed a method to form two different alignment regions—homeotropic and planar—on the same substrate surface and to fabricate an LCSZP that is clearly divided into homeotropic and hybrid alignment regions. First, a novolac-based positive photoresist film (Tokyo Ohaka Kogyo co., OFPR-800 LB) formed on an indium tin oxide substrate (Geomatech, 1006) was exposed to UV light using maskless lithography, and a photoresist film with the SZP pattern was obtained. A silane coupling agent (TCI, octadecyltrimethoxysilane) was deposited on the surface of the patterned photoresist film at 100 °C for over 30 min in a homemade vacuum chamber, with the result that silane SAM was formed on the substrate. Then, the substrate was dipped into an organic stripper (TKO, Stripper 106) at 80 °C for 2 min to remove the resist film, and then it was rinsed with isopropanol and distilled water to leave the patterned silane SAM. Second, an azo-based photoalignment agent (DIC, LIA-03) was spin-coated on the substrate at 5000 rpm for 30 s, and the solvent was slowly evaporated at 25 °C. Linearly polarized UV light with a wavelength of 365 nm and intensity of 70 mW/cm2 was irradiated on the substrate for 30 s to uniformly orient the photoalignment film toward the perpendicular direction of the exposed linear polarization direction. Through this procedure, a substrate with two different alignment layers on the same surface was fabricated. Next, the surface of an opposing substrate was formed silane SAM. Finally, the two substrates were used to fabricate a sandwich-type cell with a gap of 5 µm, and a nematic LC with positive dielectric anisotropy (LCC, E7) was filled into the cell at 80 °C, which induces the isotropic phase of the LC material, and gradually cooled to 25 °C.
Figures 3(a) and 3(b) show the polarization microscopy images observed in the fabricated sample without an applied voltage. When the alignment direction of the photoalignment film was in a diagonal position to the direction of the transmission axes of the crossed polarizers, the adjacent spiral arms showed alternate bright and dark regions. When the alignment direction was in an extinction position, the spiral arms were dark. Therefore, the LC in the adjacent spiral arms was considered to be alternately oriented to a homeotropic alignment and a hybrid alignment.
Fig. 3. Polarized microscopy images of fabricated sample without applied voltage at (a) a diagonal or (b) extinction position in the hybrid orientation region. P, A, and PA denote the polarizer, analyzer, and photoalignment direction, respectively.
Some line defects were seen in the hybrid alignment region. These were considered to reduce the diffraction efficiency. The line defects in the hybrid alignment region were formed in the bulk by the collision of splay or bent orientations, and they often appeared on the outside of the spiral arms owing to the anchoring force from adjacent homeotropic alignment regions [37]. However, the number of line defects could be reduced by the cooling process, which slowly changes from the isotropic phase to the LC phase. In addition, spiral-arc-shaped line defects were found to be formed at the boundary between homeotropic and hybrid alignment regions. These defects were considered to be formed at the interface between the LC and the alignment layer owing to the rapid orientation change from homeotropic to planar alignments on the substrate. However, these defects were expected to be removed by inducing a pretilt in the planar alignment layer. The aperture diameter of the sample was measured to be approximately 1.27 mm; therefore, the LCSZP was fabricated as designed.
A V-T curve was obtained when light with wavelengths of 480, 532, and 632 nm was incident on the sample at the hybrid alignment region, as shown in Fig. 4. The transmittance changed immediately when a voltage was applied to the sample. Thus, the LC orientation was a hybrid alignment. When voltages of 0.69 and 1.45 V were applied, transmittance was changed from minimum to maximum at a wavelength of 532 nm. When the transmittance is maximum, the retardation is half-wave, and when the transmittance is minimum, the retardation is full-wave. Accordingly, when the applied voltage changes from 0.69 to 1.45 V, the sample can switch the generation of a focused OV with a wavelength of 532 nm. The applied voltage, at which the retardation indicates half-wave at 480 nm and 632 nm, was 1.69 V and 1.22 V, respectively.
Fig. 4. V-T curves obtained when light with wavelengths of 480, 532, and 632 nm was normally incident on the hybrid alignment region of the sample.
5. Experiment and discussion
To demonstrate the generation of a focused OV beam, the diffraction image from the sample was observed using a CMOS camera (IDS, UI-3580CP-C-HQ) when linearly polarized light was normally incident on the sample. The linearly polarized light beam had a diameter of 2.0 mm, central wavelength of 532 nm, and the same polarization direction as the orientation direction of the photoalignment layer. A rectangular AC voltage with a frequency of 1.0 kHz was applied to the sample.
In the experiment, a beam with a Gaussian profile was incident on the sample. Figures 5(a) and 5(b) show the diffraction image at the focal length. When a voltage of 1.45 V was applied to the sample, the diffraction image was obtained as shown in Fig. 5(a). The diffraction image had a donut-shaped intensity profile; a spiral-shaped intensity profile was also confirmed around the donut-shaped one. When a voltage of 0.69 V was applied, the shape of the obtained diffraction image was close to that of the incident light intensity profile as shown in Fig. 5(b). Consequently, it was demonstrated that the LCSZP can generate an OV beam and can electrically switch the generation of OV.
Fig. 5. Diffraction images at the focal plane of sample and intensity profiles across center of diffraction images: (a) output beam for half-wave retardation condition and (b) output beam for full-wave retardation condition. Comparison of experimental and simulation results: (c) half-wave retardation condition and (d) full-wave retardation condition.
A spiral-shaped light with low intensity also appears around the donut-intensity profile. The appearance of the light intensity may be attributed to the phase difference gap and the spiral-arc-shaped line defects. When the phase difference is shifted, the diffraction efficiency is reduced, and light leakage occurs. It is considered that light propagating through a line defect is diffracted or scattered. For example, light propagating through a line defect is known to focus or diverge like a gravitational lens [37]. Thus, it is considered that light propagating through a spiral-arc-shaped line defect is spiral-shaped light.
The diffraction image was simulated when a beam with the same Gaussian profile as the incident beam used for the experiment was incident, and this image was evaluated through a comparison with the simulation result. The intensity profiles across the center of the diffraction images were extracted from the simulation and experimental results and shown in Fig. 5(c) and 5(d). The diameter of the donut hole was in good agreement with the simulation result. The shapes of the intensity profile were close to the simulation results, though the intensity profile was influenced by the spiral-arc-shaped line defects.
Although a light component that differed from the focused OV beam component appeared owing to the influence of the line defects in the hybrid alignment region at the boundary between the homeotropic and the hybrid alignment regions, a higher diffraction efficiency of 32.5% was obtained compared with that of the conventional SZP.
To investigate the focusing property, the diffraction images were observed before and after the focal length. The observation distance between the sample and the observation plane was shifted by ±20% of the theoretical focal length; therefore, for an incident wavelength of 532 nm, the distance range was 37.3 to 55.9 mm. Figure 6(a) shows the intensity profiles of the obtained diffraction images at observation distances of 37.3, 46.6, and 55.9 mm. Compared to before and after the focal length, the donut hole was the tightest, and the intensity was the highest at the focal length.
Fig. 6. Focusing properties of LCSZP: (a) intensity profiles of observed diffraction images at focal lengths of 46.6 and observation distances of 37.3 and 55.9 mm, and (b) Variation in the focusing point, which indicates the maximum diffraction efficiency when applying voltage (left) and diffraction images at focal lengths corresponding to wavelengths of 480 and 632 nm (right).
In addition, the focusing point was investigated when a voltage was applied to the sample; the result is shown in Fig. 6(b). The solid line indicates the simulation result obtained from Eq. (4), and the dots indicate the experimental results. When the incident wavelengths are 480, 532, and 632 nm, 1.69, 1.45, and 1.22 V were applied. These applied voltages were determined by the V-T curve shown in Fig. 4. The diffraction images at the focal planes for incident wavelengths of 480 and 632 nm are shown in Fig. 6(b) on the right. The focal length of light at each wavelength was in good agreement between the simulation and experimental results. Therefore, it is demonstrated that the proposed device can change the focusing point, which indicates the maximum diffraction efficiency, by applied voltages.
6. Conclusion
This study proposes a new type of SZP called LCSZP that can generate a focused OV with high efficiency, and a fabrication method was developed to form homeotropic and planar alignment regions on the same substrate to realize an LCSZP. A focused OV beam generation was shown through numerical simulations based on the Fresnel diffraction theory and demonstrated experimentally.
Several OV generating methods using LCSLM, spiral phase plates, and q-plate have been reported. The diffraction efficiency is relatively high. However, LCSZP requires a large external computational device. When spiral phase plates and q-plate were used, an additional lens was required, and the alignment between the phase singularity and the lens’s optical axis was difficult. SZP can facile generate a focused OV without the alignment, but the diffraction efficiency has been low. The proposed device has a theoretical efficiency of 41%; this is two times that of the conventional SZPs. Although the diffraction efficiency experimentally obtained 32.5% due to multiple defects of LC in the device, the diffraction efficiency is higher than conventional SZPs. In addition, the proposed device has electric tunability, which is not found in conventional SZPs. Thus, the LCSZP can find applications in optical fields such as imaging, laser processing, and optical manipulation.
Funding
Japan Society for the Promotion of Science (JP19H02581, JP20H00391, JP20H04672, JP21K18722).
Acknowledgment
The authors thank DIC Corporation for providing materials. This work was partly supported by MEXT KAKENHI (JP19H02581, JP20H00391, JP20H04672, and JP21K18722) and the MEXT LEADER Program.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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