A ferroelectric liquid crystal (FLC) cell with continuously alignment structure is realized by a polarization hologram method for fabricating a Pancharatnam-Berry (PB) lens, which is employed as a concave/convex lens. The PB phase can be maintained by the optical axis in-plane switching; meanwhile, its diffraction efficiency can be tuned in a certain range by electrically controlling azimuthal angle and optical biaxiality of the smectic helical structure realized by deformed helix ferroelectric liquid crystals. The measured diffraction efficiency of the fabricated device is up to 87% and the response time can be 300μs with a low electric voltage. The FLC PB lens can have potential applications in existing optical devices and the realization of FLC with continuous alignment structure can be further used for other LC-based optical devices.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Liquid crystals (LCs), because of the easy tunability of the refractive index, have become very popular for photonic devices like beam multiplexing, optical interconnects, tunable gratings [1–4]. LC-Pancharatnam-Berry lens (PBL) has attractive properties of high optical throughput and large uniaxial optical birefringence [5–7]. It plays an important role in the optical science and engineering community where the circular polarization dependent devices are well utilized, including the capacity of optical communication system [8,9], virtual and augmented reality display . However, the existing LC-PBL suffers from slow switching speed limited to the level of millisecond and high-power consumption. Several ways to improve the response time are still in progress [11,12,29,30]. FLCs utilizing photo-alignment technology [3,25–28] represent a promising way for fast response PBL. However, current achievements are mainly focus on two-domain structure optical elements and it is very difficult to achieve multi-domain structure rather than two-domain structure FLC cells with the complexity of FLC alignment and fabrication process [13,15,16]. Moreover, many FLC modes are unswitchable for PB phase because of their optical axes sweep in the plane of the cell substrates [13–16]. Thus, using proper FLC mode for switchable PBL and exploring continuous domain structure FLC optical elements are necessary.
In this paper, we deal with deformed helix FLCs (DHFLCs)  that can offer fast response and continuously changing of light transmittance within certain range based on optical axes changing [18,19]. While computer controlled digital micromechanical device has been widely exploited to fabricate PBLs, which requires several exposure steps . DHFLC PBL, a space-variance alignment structure has been first realized employing a polarization hologram [20–23] using photoalignment, which only requires a single step exposure. The periodic distribution of the easy axis can be written, erased and rewritten for a different PBL vector based on the optically active alignment layer. This DHFLC PBL shows fast response time, tunable diffraction efficiency at lower driving voltages and rewritable focal distance. The so-fabricated FLC PBL provides good optical quality with diffraction efficiency around 87% and response time 300µs at 4V/µm. The proposed fabrication process can be quite simple for the employment in verity photonic devices.
2. Theoretical analysis
Surface stabilized FLC (SSFLC) and electrically suppressed helix FLC (ESHFLC) modes can be used to realize switchable two-domain structures such as gratings or Fresnel lens by in-plane switching. However, when it comes to PBL, the PB phase always exists because the optical axis just sweeps in the plane of the cell substrates. In this paper, DHFLC is used for FLC PBL. The PB phase is induced based on the geometrical trajectory sketched out in the polarization state space (Poincare sphere). The PB lens has a lens-like phase profile entirely established using PB phase. The transmission function of the general LC-PB optical elements is given by,Fig. 1(b) and f is the focal distance of the lens. The diffracted wave has a polarization orthogonal to the incident wave and its phase is spatially modulated by an angle 2α(y, z). Given that the input light is circularly polarized i.e. (+/− denotes LHC/RHC polarized), and the half wave condition is satisfied i.e. Γ = π, the expression of the output field in Eq. (1) is reduced to,Fig. 1(a).
Another variable that affects transmittance in Eq. (1) is the expression Γ = 2πΔnd/λ, where Δn is the effective index Δneff of FLC cell, d is the thickness of the LC layer, and λ is the wavelength of the incident light. In the schematic of a typical DHFLC cell shown in Fig. 1 (b) when there is no electric field, helix exists with helix pitch denoted as P0 and one important requirement for DHF effect existence is d >> P0. When there is an external electric field to DHFLC molecules, different from nematic LC, FLC molecules tend to align along its helical cone and azimuthal angle φ with respect to the electric field as well as maintain the geometric phase, thus change of Δneff shown in Fig. 1(c). In DHFLC, the electric induced φ has expression:
Thus Δneff modulated by external electrical field can be calculated numerically by averaging through the whole helix pitch, using the formula below:
In Eq. (5), n‖and n⊥can be regarded as ne and no without electric field depicted in left figure of Fig. 1(d). Moreover, when we consider the electric induced biaxiality of DHFLC molecule in right figure of Fig. 1(d), n‖ and n⊥ will be expressed as a function of (ne, no, n1, n2, n3). As the result, the induced biaxiality  will additionally rotate φ towords electric field thus decrease Δneff.
Finally, the transmittance of DHFLC PBL can be expressed by Eq. (1) in terms of Δneff and α(y, z).
3. Switchable FLC Panchiaratnam-Berry lens
In this work, FLC 587 (from P. N. Lebedev Physical Institute of Russian Academy of Sciences) is selected as the material for its high transmission in the visible spectrum and fast switching. The phase transitions sequence of this LC during heating up from the solid crystalline phase is Cr→12°C→SmC*→110°C→SmA*→127°C→Is, while cooling from smectic C* phase crystallization occurs around −10◦C–15◦C. The spontaneous polarization Ps and the tilt angle θ at room temperature are 150nC/cm2 and 36.5°C, respectively. It has been proved  that asymmetric boundary conditions avoid the competition between the aligning forces existing between the surfaces of cells and the FLC helix, which provides a better alignment quality. A polarization photosensitive alignment sulphonic azo-dye (SD1) is chosen as the alignment material for its high anchoring energy, lack of mechanical damages and minimized unwanted electronic charges. Under exposure from a polarized light source in the UV-to-blue spectrum, the SD1 molecule tends to orient perpendicularly to the polarization of the incident light.
Figures. 2(a) and (b) show the schematic for the fabrication procedure of the FLC PBL. A thin film quarter-wave plate (QWP)/ half-wave plate (HWP) LC polymer thin-film PBL is designed when diffraction efficiency is maximized at 450nm wavelength, which can be achieved when the quarter-wave condition or half-wave condition is satisfied at 450nm. In Fig. 2(a), the setup proposed in  have been first used to fabricate FLC cells that comprises a linear polarizer, an HWP followed by the LC polymer thin-film HWP PBL and a SD1 coated indiumtin-oxide (ITO) glass substrate. LC polymer HWP PBL is used as the LC photo-pattern mask. The recorded alignment profile for the fabricated FLC PBL resembles the optical axis distribution of HWP with half of the spatial period in a single step exposure. Therefore, it is a way in terms of increasing the diffraction angle of any existing LC-PBL while the features can be maintained. In Fig. 2(b), the setup comprises a linear polarizer, a QWP with its optical axis oriented at 45° with respect to the transmission axis of the linear polarizer, followed by the LC polymer thin-film QWP PBL and an ITO coated glass substrate with SD1 as alignment layer. Having a 45° offset, the recorded alignment profile is a replica of the inhomogeneous QWP. Thus, we can ‘copy’ the LC polymer thin-film PBL pattern using QWP in a simple, compact and time-efficient manner. Thereafter, the SD1 substrate is exposed by a 0.3W/cm2 450nm laser for 90 seconds for imprinting the PBL pattern. The cell has been assembled with the second ITO substrate without treatment for any alignment layer and the cell thickness is maintained at certain cell gap. In addition, cell gap was selected to satisfy the half-wave condition for FLC PBL behaving as a convex/concave lens. Based on the Δn of FLC 587 used in this work, the cell gap can be maintained at its first half-wave condition 1.5μm and its third half-wave condition ~5μm.
The optical axes of FLC molecules will have a continuous orientation illustrated in Fig. 2(c). Molecules will in plane aligned to satisfy PBL optical axis distributions shown in Fig. 3(d) without electric field.
4.1 Focal length and PBL pattern
The micro-structured alignment quality of the DHFLC PBL is confirmed from the micro-photograph captured under crossed polarizers, as shown in Figs. 3(a)-3(d). Good alignment quality is observed, which shows the theoretically expected profile of a gradual decrease in grating pitch away from the center with smooth variations in transmittance level and minimal defects. The molecules have been aligned with continuously orientation structure. The phase profile of the lens can be deduced from the intensity profile T(r) = sin2[2α(r)] and α(r) = kr2 = πr2/fλ (k = 12.34 × 106m−2). Thus, the focal distance of the PBL deduced from the phase profile of α(r) at 630nm is 40cm. The polarization dependence of the fabricated PBL is tested by illuminating a 633nm laser beam through a circular polarizer. By passing through a RHC polarizer in front of the PBL, and an observation screen at some distance away from the lens, the size of the laser beam is reduced as a result of the focusing effect from the lens. The focal length corresponds to the distance when the size of the laser beam is minimized. The experimentally measured focal length is 42cm, which is in good agreement with the result deduced from the phase profile of α(r). The diameter of the PBL D is measured 13mm. Thus, the F number, which is defined as F = f/D for the fabricated PBL is 30.1. Reducing the F number of PBL is attractive for many imaging and display applications. This can be achieved via the process shown in Fig. 2(a). When the reduced F-number PBL is designed when half-wave condition at 450nm is satisfied, the fabricated sample can in turn be used as the phase mask for a further reduction in the focal length. The “copied” pattern quality and diffraction efficiency may drop after several times of “copy” processes.
The converging properties of the PBL illuminated from RHC polarized light is demonstrated in Fig. 3(e). Conversely, when a LHC polarizer is in front of the lens, the PBL behaves as a diverging lens for which the beam size expands and the intensity of the laser beam decreases as shown in Fig. 3(f). The converging and diverging properties are observed when 10V electric field is given to a 1.5μm DHFLC cell.
4.2 Diffraction efficiency
The diffraction efficiency of PBL is a measurement of the amount of incident light that is focused onto the primary focal plane and can be experimentally determined using the setup similar to Fig. 1(a) by adding a photo-diode detector placed at the focal plane. Power P1 is recorded by photo-diode detector located at the focal point as the reference power when a 633nm laser beam propagates through a RHC polarizer and a PBL. Power of the 1st order diffracted light P2 is determined when a 2mm aperture stop and a LHC polarizer are placed in front of the detector to reject the zero order component. The measured diffraction efficiency for PBL can be then determined from the expression: η = (P2/P1) × 100%. The DHFLC cells exhibit the properties of hysteretic free and electro-optical response being non-sensitive to the driving voltage polarity where driving voltage pulses with rectangular alternative sign are applied, as shown in Fig. 4(a) existing in a rather broad range 0.2-4kHz . In Figs. 4(b)-4(c), the diffraction efficiency is measured under electric field frequency of 500Hz between −10V to 10V with cell gap maintained at 1.5μm and 5μm respectively. As discussed in Eqs. (1)-(5), when FLC cell gap is designed in first half-wave condition, diffraction efficiency tends to increase with the absolute value of electric field increasing. In Fig. 4(b), the diffraction efficiency for 1.5μm thick cell can be tuned from 0.87 to 0.45. Beyond the electrical operation range of DHFLC mode, the efficiency is stable because the cell operates as ESHFLC mode. DHF effect only exists when it is operated in the range of –Ec < E < Ec, where Ec is decided by V/dFLC with unit V/m discussed previously. Deducting from Eq. (3), the averaging ‹φ› through the whole helix pitch is ∝1/|E|, where E is constrained by the DHFLC operation range. For better tuning range of efficiency and larger voltage control flexibility, the cell gap dFLC = 5μm was experimentally used which is close to but not the exact value of its third order of half wave condition. The diffraction efficiency can be tuned in the range from 0.85 to 0.13 with cell gap of 5μm shown in Fig. 4(c). However, enlarger the cell gap may lead to worth alignment quality mainly because the alignment anchoring energy may be unable to well satisfy the good boundary condition of DHFLC.
The diffraction efficiency is unable to be 0% mainly because tilt angle θ cannot go π/2, resulting in that Δneff(λ, E) cannot become 0. This can be optimized by using DHFLC materials with larger tilt angle. In addition, the measured diffraction efficiencies for fabricated PBL samples are above 85% close to the theoretical limit of 100% diffraction efficiency. The deviation between the measured diffraction efficiency and the theoretical diffraction efficiency is primarily due to a small number of microscopic defects that appeared at the edge of the lens.
In the range of −4V to 4V, the cell operates at the deformed helix mode shown in Fig. 4(b) with measured response time and .
The display performance of the FLC PBL is demonstrated in Fig. 5 by observing the image of letter ‘NWPU’, where RHC and LHC polarizers are inserted in front of the PBL for showing the enlarged letter in Fig. 5(a) and minimized letter in Fig. 5(b), respectively.
Further research needs to be done to improve the alignment quality and realize smaller focal distance. The resolution is defined by resolution of SD1 and cell gap. Comparing these two constrained parameters, the resolution of SD1 can be sub-micrometer, while, the resolution of a cell with 1.5μm cell gap will be ~1.5μm. For DHFLC, this cell gap determined resolution is possible to be realized because its pitch is much smaller than the cell gap. Thus, the finest focal distance will be determined by the thinnest size of outside fringe of PBL to be larger than cell gap.
In conclusion, we demonstrate the DHFLC-PB lenses using photoalignment via. polarization hologram setup, which is cost effective and time efficient since the continuous alignment profile has been realized from a single step exposure based on simple optical elements. Theoretical analyses on focal distance and diffraction efficiency have been discussed and match well with experiment results. The fabricated PB lenses depict continuously changing of diffraction efficiency from up 87% to 45% with 1.5μm cell gap and 85% to 13% with 5μm cell gap. The focal distance measured with the incident light at 633nm is 42cm comparable to theoretical number 40cm. Moreover, high quality PBL with smaller F number can be achieved with little effort by half-wave polarization hologram method. The DHFLC PB lenses provide fast response time of 300µs at 4V/µm and therefore make them suitable for applications in imaging, display, beam manipulation and various optical setups that require fast response.
National Key R&D Program of China (2017YFA0303800); National Natural Science Foundation of China (NSFC) (6180031536); Fundamental Research Funds for the Central Universities (G2018KY0309).
The authors declare that “there are no conflicts of interest related to this article.”
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