Liquid-crystal phase-only devices

MOLLIQ-08111; No of Pages 15 Journal of Molecular Liquids xxx (2017) xxx–xxx

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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq

Liquid-crystal phase-only devices José M. Otón a,⁎, Eva Otón b, Xabier Quintana a, Morten A. Geday a a b

CEMDATIC, ETSI Telecomunicación, Universidad Politécnica de Madrid, 28040 Madrid, Spain Nikon and Essilor International Joint Research Center Co., Ltd, KSP R&D Build, C10F-1032 3-2-1, Sakado, Takatsu-ku, Kawasaki-shi, Kanagawa 213-0012, Japan

a r t i c l e

i n f o

Article history: Received 28 September 2017 Accepted 31 October 2017 Available online xxxx

a b s t r a c t Phase-only devices based on liquid crystals are becoming popular solutions for a high number of applications in many diverse areas. The use of an electrooptic material such as liquid crystals makes the device tunable, opening up many different possibilities of light beam manipulations: beam steering, beam shaping, modulation, focusing, generation of vortices or vector beams, etc. The subject has received a significant boost in the last decade since many research labs working on liquid crystal displays – an area almost entirely transferred to electronic companies – have evolved to these devices, where new effects and applications are continuously being unveiled, and a number of issues affecting their performance are still unsolved. The work is a review of the most relevant kinds of photonic devices developed with this technology. A detailed study of every device and effect would have been too lengthy and possibly would have lacked coherence for dealing on too dissimilar topics. Therefore, the work is focused on passive devices, i.e., with no active matrix driving, thus circumventing the large field of spatial light modulators, which surely deserve a thorough review themselves, but have been already analyzed in many publications. © 2017 Elsevier B.V. All rights reserved.

IN MEMORIAM — Prof. Yuriy Reznikov We had the pleasure of meeting Prof. Reznikov for the first time N20 years ago. We have had occasional contacts with him and his group in many liquid crystal conferences, especially the biannual CLC Conferences organized by the WAT University of Warsaw. I was impressed since the first time for the deep knowledge of subjects and clarity of ideas shown by Prof. Reznikov, as well as his kind openness for sharing his expertise and findings and suggesting solutions to complex problems with no apparent effort. More than 10 years ago we intended to initiate a long-term collaboration between our labs; one of his collaborators spent several months in our premises supported by a research grant. Unfortunately the collaboration was curtailed by administrative difficulties. Yuriy will remain in our thoughts as an example to follow and a demonstration that heavy funding may help, but it is neither necessary nor sufficient to carry out high quality physics. Excellence in research, like that shown by Prof. Reznikov, is the one and only way to push forward the knowledge edges. 1. Introduction Phase-only devices (POD) are devices where materials or optical systems interact with light beams modifying just their phase, ⁎ Corresponding author. E-mail address: [email protected] (J.M. Otón).

i.e., without affecting other beam characteristics such as wavelength or intensity. Phase modifications in PODs can be roughly classified in two categories, those that modify the phase of the same point of the light beam with time – temporal delays – and those that modify different points of the wavefront – spatial delays. The outcome of the first class usually leads to modulators, tunable filters or interferometers, whereas the second class leads to beam steerers and beam shaping devices (Fig. 1). Please note that this classification is neither limited nor exclusive. Liquid crystals are materials particularly suited for phase manipulation. They feature an unusually high optical birefringence that can be easily modified by moderate external fields. The development of LCbased PODs has undergone a significant growth in the last decade, as the interest of many research laboratories has evolved from display to non-display applications of LCs [1]. At present, LC display applications are a topic mostly confined to R&D units of large consumer electronic companies. Most scientific or academic laboratories have drifted to a number of fields where LC optical anisotropy can be exploited for different applications. The economic impact of these applications, obviously being not that of LCDs, is certainly not negligible, and is steadily growing with time. Among others, it is worth mentioning applications in guided and wireless optical communications, light confinement, photonic integrated circuits, tunable filters and lenses, lens arrays, plenoptic and 3D cameras, optical switching, beam steering, spatial light modulators, computer-generated holography and many others. Strictly, all simple LC devices – i.e., devices made of one or several liquid crystal layers, without polymers dispersions, dye-doping

https://doi.org/10.1016/j.molliq.2017.10.148 0167-7322/© 2017 Elsevier B.V. All rights reserved.

Please cite this article as: J.M. Otón, et al., Liquid-crystal phase-only devices, J. Mol. Liq. (2017), https://doi.org/10.1016/j.molliq.2017.10.148

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J.M. Otón et al. / Journal of Molecular Liquids xxx (2017) xxx–xxx

only SLMs on active matrices is wavelength-selective switches (WSS, Fig. 2) presently used as reconfigurable optical add-drop multiplexers (ROADM), a keystone component of wavelength-division multiplexing (WDM) in current optical communication networks. Liquid-crystal ROADMs are based on liquid crystal on silicon (LCoS) AMLCDs [3], a well-known technology for ultra-high resolution projection displays that have ultimately become a crucial element in a rather unrelated application of LC PODs. The scope of this work shall be focused, perhaps a bit loosely, on passive – i.e., with no active matrices – POD elements, providing nonetheless a sufficiently wide overview of the topic, its application areas, and its recent developments. Within this scope, the chapter will be divided into several sections containing a theoretical approach and a number of phase configurations that eventually will result into the main kinds of PODs. 2. Analog vs. digital devices: the limits of phase delays As mentioned, LCs feature a very large birefringence – values like Δn = 0.2–0.3 are considered typical, materials showing birefringence above 0.5–0.6 can be found – being therefore ideal for phase delay devices. Phase shifts are a consequence of optical path (n·d) differences. Let us consider two beams propagating parallel to each other. If the physical distance d covered by both beams is identical, then the phase delay δ between the beams is solely due to the difference in refractive index along the path, δ¼

2π Δn d λ

ð1Þ

Fig. 1. The principle behind the graded index lens and prism.

nanoparticles and the like – are phase-only devices (even these can be PODs in some cases). Thin cells of undoped non-dispersed LCs neither absorb nor scatter impinging light, but merely modify its State of Polarization (SOP). The modifications in SOP arise from phase delays between orthogonal polarization components, which can be driven through external electric signals and eventually be transformed into amplitude modifications by external optical elements such as polarizers. In this work, we will restrict the scope of PODs to those devices that take advantage of the phase delay itself, rather than using the phase delay in the generation of other effects. Moreover, devices like active-matrix liquid crystal displays (AMLCDs) in which most LCDs are based, result functionally equivalent to phase-only spatial light modulators (SLMs) [2]. SLMs customarily reach phase delays of 2π or higher, whereas AMLCDs are usually limited to π. Furthermore, AMLCDs include polarizers while SLMs do not. Both kinds of devices are based on the same electrooptic principles and are driven in the same way; nevertheless, their application fields are dissimilar. SLMs deserve a thorough review by themselves, beyond the scope of this work. Just to mention a significant example of phase-

Choosing values for light wavelength and birefringence, it can be easily found that a 2π full wave retardation can be achieved with LC layers of few μm. Fig. 3 illustrates the delay induced by a 40 μm LC cell with varying applied voltage (two crossed polarizers have been added to the optical setup to show up transmission variations). Over 40π delay (i.e., 20 full waves) can be generated, as seen in Fig. 4. This retardation is more than enough for many applications, especially those related with interference, phase gratings, optical switching and optical modulation. It is rather modest, however, for other applications involving beam steering and beam shaping, such as graded-index prisms and lenses. Let us clarify this point with an example. A prism having a 4 × 4 mm square section will be used. A 4 mm glass prism (n = 1.5) being 40 μm thick on the base would induce a retardation about 80π, twice as much as the LC – simply because the refractive index difference between air and glass is 0.5, roughly 2 × the LC Δ n. Such a prism would have a wedge angle of tan−1

40μm ¼ 0:573∘ 4mm

Fig. 2. Sketch of a reconfigurable wavelength selectable switch.


ð2Þ


3

Fig. 3. Transmission vs. voltage for a 40 μm LC cell.

In the graded-index LC prism, the equivalent angle would be approximately one half, assuming Δ n = 0.25. Deviations of incident beams given by such a prism are tiny, b 0.5° within an incidence range 0°–60°. Increasing this deviation is a must in many applications. Increasing the internal prism angle of a glass prism is trivial; increasing the equivalent angle in a graded-index phase device is not feasible above a certain value that depends on a number of factors, but remains in the 100's μm anyhow. Therefore, the only effective way to increase significantly the angle – and the deviation – is to reduce the size of the device. If the size of the prism is reduced to, say, 100 μm in the example shown above, the equivalent angle becomes 10.9°, and the deviation angle oscillates between 5° and 10° in the above mentioned incidence range. Needless to say, this approach does not seem feasible for many applications employing light beams with diameters in the mm range. Fortunately, there is a straightforward solution that takes advantage of the cyclic behavior of phase shifts. A 2π shift is equivalent to a 0π shift; consequently the steadily increasing thickness or retardation of the devices can be substituted (Fig. 5) by a saw-tooth scheme in which thickness or phase shift goes down to zero every time it reaches 2π. The outcome – known as phase wrapping – is the well-known Fresnel lenses or prisms [4], where the thickness of the material is periodically modified, and the equivalent phaseshift version as shown in the figure. This approach has a drawback, however. The steps must be multiple of 2π, otherwise only a fraction of light is diffracted, while other fraction remains undiffracted (0th order). The full-length steadily increasing phase shift device (top) produces a quite limited deviation (right curves, not to scale). The second scheme on the top shows the Fresnel equivalent, where deviation is substantially increased but the device loses its analog tunability, since the teeth of the saw tooth must be multiple of 2π to avoid undiffracted light leakage on 0th order. The three schemes of the bottom are digital approaches made of different numbers of steps. Diffraction efficiency

approaches rapidly to 100% (Fig. 6) as the number of steps increases. The higher the number is, the closer to the analog device (i.e. only one 1st order gets light), but the lower the diffraction angle. The binary grating (Fig. 5, bottom) features the widest angle but light is halved between 1st diffraction orders at both sides. 3. Theoretical approach It is well known that solid crystals are, in general, anisotropic, i.e., its refractive index (RI) depends on the incident angle of the impinging light and the direction of oscillation of its electric field. In the general case, the RIs of the solid are presented by an index ellipsoid – also known as optical indicatrix: x2 y2 z2 þ þ ¼1 n2x n2y n2z

Depending on the crystal system and the symmetry elements, solid crystals may be biaxial, having three different RIs, nx ≠ ny ≠ nz, or uniaxial, having two different RIs, no = nx = ny ≠ nx = ne, where no and ne stand for ordinary and extraordinary RI, the name customarily employed. In the cubic system symmetry is so high that only one index exists; therefore cubic solids are isotropic. Liquid crystals, especially nematic LCs, are uniaxial or ‘almost’ uniaxial, characterized by an ordinary and an extraordinary RI. Their optical indicatrix has two identical axes, therefore becoming a spheroid, also called ellipsoid of revolution. These identical axes correspond to the ordinary RI, while the different axis is extraordinary. Most LCs, including all nematics, show positive birefringence Δ n, defined as Δ n = ne − no, i.e., the spheroid is prolate. Let us consider a polarized light beam impinging onto a uniaxial material (Fig. 7). Light inside the material propagates according to its RI, as ‘seen’ by the electric field of the impinging beam. It can be shown from Maxwell's equations that any polarization propagates as two linear components corresponding to the lowest and the highest RIs found by the light beam. These indices are evaluated by taking the indicatrix cross-section perpendicular to the beam. The section is an ellipse whose radii are the actual RIs. In uniaxial materials, the smallest radius is always no, the ordinary index, whereas the largest radius is an effective refractive index neff having an intermediate value between no and ne: 1 n2eff ðθÞ

Fig. 4. Phase delay vs. voltage for a 40 μm LC cell.

ð3Þ

¼

cos2 ðθÞ sin2 ðθÞ þ 2 no n2e

ð4Þ

where θ is the angle between the electric field and the spheroid optical axis. As the indicatrix has two ordinary RIs and one extraordinary RI, it follows that only one electric field component will be modified by changes in the impinging direction, precisely the one associated to neff, since the other component will always be no regardless the LC or beam reorientation. In other words, phase control and retardations in standard LC materials and configurations are restricted to one linear component of the incident light; the other component is not affected by LC reorientations.


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Fig. 5. Analog and digital phase gratings. Right: 0th and 1st order intensities. See text for details.

Before progressing further it is convenient to mention that some particular LC materials and mesomorphic phases in specific configurations are polarization independent, i.e., both linear components of the state of polarization (SOP) bear the same phase delays. Among these, the blue-phases of cholesteric LCs [5] have attracted much interest lately since they feature submillisecond switching times [6] that result interesting for many potential applications in photonic devices. These materials show a cubic – hence isotropic – arrangement; the external electric field is applied in such a way that produce deformations of the cubic lattice in the direction of light propagation, hence the material is maintained isotropic for the impinging light. The induced delay is therefore constant regardless the incoming SOP. Phase delays may be employed in different ways. Delays between ordinary and effective (variable) components of standard LC materials effectively modify the SOP of the outcoming light respect to the incoming one. This setup, associated to polarizers, is used in LC displays to modify light transmitted across the system. Blue phases and other polarization-independent materials/configurations demand a different strategy, often requiring to split light into two separate optical paths and to recombine it after one of the paths interacts with the switching

LC. It is customary to employ matrix formulation to express delays. The most common formulation is based in Jones matrices. Assuming two orthogonal linear light components, the Jones matrix is 3 2π exp − j 0 6 7 λ 7 ¼ LC ¼ expðjωt Þ6 4 5 2π 0 exp −j neff z λ # " 1 0 2π 2π exp j ωt− n0 z 0 exp j Δneff z λ λ 2

ð5Þ

The leading term exp[j(ωt − (2π/λ)n0z)] describing the temporal variation and the absolute phase retardation is common to the entire device. It is generally omitted and the above equation becomes: LC ¼

1 0

0 ejδ

ð6Þ

where δ is the phase retardation as defined in Eq. (1). Phase-only devices are usually set so that only the tunable eigenmode (the neff

Fig. 6. Ideal diffraction efficiency of a sawtooth blaze grating as a function of the number of steps.



5

with cell thickness h:

V c ¼ Ec h ¼ π

Fig. 7. Left: a uniaxial index spheroid or optical indicatrix. The optic axis lies parallel to ne. Right: any impinging beam splits into an ordinary and an effective component neff.

component in Eq. (5)) is excited by the incoming light beam, since the orthogonal eigenmode (no) is not modulated by any variation of LC orientation. This setup is opposite to display devices, where both eigenmodes must be equally excited in order to generate the highest SOP variations. In practice, this usually implies to work with polarized light parallel to the neff component in the phase-only case, and forming a 45° angle with both components in the display case. The advantage of representing the device in a matrix form is that the behavior of stacked devices or inhomogeneous devices can be easily described by matrix multiplication. This is interesting, for example, in describing the actual profile of an LC cell under external electric field. The switching profile of an LC between two electrodes is determined by two competing torques: the applied electrical field and the anchoring strength of the molecules at the alignment surfaces, transmitted to the LC bulk by elastic interactions. The 2D or even 3D switching pattern as a function of the applied field has been modeled many times elsewhere using finite elements methods (e.g., [7]), and is beyond the scope of this work. The light path is divided into tiny slabs in which the RI is assumed constant, and the average refractive index is calculated by multiplying all the slab matrices. Fig. 8 shows an example of reorientation profile for a given set of elastic constants. Pretilt at the walls is assumed to be null, i.e., the anchoring energy is large enough for the surface molecules not to be reoriented at any applied field. The critical field Ec is defined as

π Ec ¼ h

sffiffiffiffiffiffiffiffiffiffiffi K 11 ε0 Δε

sffiffiffiffiffiffiffiffiffiffiffi K 11 ε0 Δε

ð8Þ

No simple analytical expression exists for the θ(z) variation, hence numerical solutions like the finite elements already mentioned are chosen. The accuracy of the obtained profiles depends on a number of parameters, including the elastic constants, that are poorly estimated in many cases. In practical implementations, consequently, it is often adopted an empirical solution based on direct calibration of the device, stored in the driving electronics via a look-up table. Let us wrap up all these ideas with an example. The blue line of Fig. 9 is an experimental measurement of the average birefringence (BR) induced in a planar device upon reorientation with an external voltage. Once exceeding the Fréedericksz threshold, Δn is reduced in a clearly nonlinear manner, as the LC reorients with the applied field. In theory Δ n = 0 can be achieved if sufficient voltage is applied. However, this is not desirable in practical applications since the asymptotic approach to such a limit implies large voltage increments for little BR range gains. The maximum voltage that will be applied restricts the maximum BR range that can be achieved. If the whole BR range goes from Δ n = ne − no to zero, voltages about five times Vc are usually enough to cover nearly 90% of that BR range. In the example of the figure, the achieved BR range is about 0.2, enough for a 2π delay at visible and NIR wavelengths in LC cells with moderate (5–10 μm) thickness. If we intend to use that BR to prepare a digital phase grating like those shown in Fig. 5, we must employ equidistant phase retardations, hence equidistant variations of Δ n. Owing to the nonlinear response of the LC cell, these equidistant retardations (6 horizontal orange lines of Fig. 9) correspond to non-equidistant variations of applied voltage. The voltage that should be applied to every electrode is given by the intersections of horizontal lines with the experimental curve, as shown by the vertical orange arrows. The example of Fig. 9 shows a case of six steps. Dividing every interval into halves, a solution for 11 electrodes is obtained. Other possible electrode sets can be easily derived. Implementation of these voltages in the driving electronics would be carried out as a look-up table or a derivation of an empirical formula fitted to the experimental points. Complex devices with multiple electrodes that adopt many configurations are better controlled with the experimental fit, especially if temperature and wavelength variations are tolerated. Analog phase gratings (Fig. 5) can be designed as well. In this case, the voltage gradient of the resistive electrodes should have a profile as close as possible to the experimental curve. This is achieved by varying

ð7Þ

where K11 is the splay elastic constant. Ec is linked to the Fréedericksz voltage Vc, the threshold switching voltage, approximately constant

Fig. 8. Reorientation of a nematic planar LC with applied voltage. The cell walls are located at 0 and 1 in the horizontal axis. Ec is the critical electric field obtained when the Fréedericksz voltage is applied.

Fig. 9. Blue line: experimental BR of a planar nematic cell. Horizontal orange lines: BR steps selected for a phase grating. Vertical orange lines: voltages required for a 2π phase shift in a device with 6 electrodes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


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the frequency of the electric signals and the shape of the electrodes, as shown in following sections. Based on the same underlying principles exposed above, many families of phase-only devices for different purposes have been proposed [8]. A thorough description of every family would lead to an excessively lengthy work. Instead, specific examples of the main families will be given, along with more detailed descriptions of specific features where appropriate.

4. Adaptive optics and wavefront correction Although SLMs are explicitly not included in this review, it is worth mentioning some relevant fields that are usually covered with these devices. In fact, high definition SLMs showing enough retardation can mimic virtually any phase-only device, from lenses and prisms to phase gratings, holograms or filters. Yet these devices may be created from other structures that are being described here. Adaptive optics (AO) is a well-known technology to improve the performance of optical systems by correcting distortions of the incoming wavefront. The technique was described by the middle of the past century as a system to compensate atmospheric turbulences in astronomical observations. Nevertheless, practical devices began to deploy in the 90s, when the technology was available. AO primary relies on a wavefront sensor – usually a Hartmann-Shack unit – and a wavefront corrector. Ideally, corrections must be done in real time; actually there is always latency in the system since the wavefront sensor must send its measurements to the wavefront corrector via a control computer. Obviously the response time of the system must be small enough to consider the wavefront constant, what is related with the so-called coherence time. Deformable mirrors (DM) – reflective surfaces deformed by microelectromechanical or magnetic actuators – have been the preferred option for astronomical applications, since they feature high reflectivity, achromatic phase modulation and fast response. However, increasing the number of actuators in such a micromechanical system is challenging. The maximum number of actuators in current systems is in the 1000's range. LC-based AO devices in open-loop configurations are considered an alternative to DM systems for their high density of actuators. A number of hurdles have had to be overcome to achieve practical devices: polarization dependence, slow response, and wavelength-dependent phase retardation. At present, several LC-based AOs are being tested in astronomical observatories [9]. AO has also found applications in biological systems [10]. Retinal images of patients suffering presbyopia, myopia, or hyperopia must be corrected for the aberrations developed by the own eye. Some LCbased AO systems have been proposed [11] for this application; it has

been argued that LCs should be the preferred option in this case, whereas DMs are preferred in astronomical systems. LC-based AO devices are not restricted to biological and astronomical fields. Indeed, their use is continuously spreading to different subjects. It is worth mentioning their application to wireless optical communications in the atmosphere [12], and particularly their use in modern microscopy [13]. Wavefronts are often deformed by aberrations from the scrutinized specimen itself, especially if deep observations are required. Light travelling through the sample becomes deformed by spatial variations in the refractive index. Using AO one can generate a preaberrated wavefront that compensates for the aberrations of the specimen (Fig. 10). If the sample is shone with such a wavefront, the aberrations cancel out and a flat wavefront is obtained rendering a diffraction-limited image. At present, AO is employed in many fancy new kinds of microscopy, such as multiphoton, confocal, light sheet, or stimulated-emission depletion microscopes, to name a few. Beam shaping is a topic related to wavefront correction, in which specific spatial phase or amplitude patterns modify the incoming beam (e.g., a Gaussian beam) making it to deploy a different optical field. Static beam shapers can be prepared from lenses, patterned masks, plates or metasurfaces. However, using LCs brings the additional advantage of making the structure dynamic, i.e., switchable. Again an LC SLM could cope with the microstructured motifs required for beam shaping, but they are optically inefficient for this application, and their quality is limited by pixelization. An interesting alternative recently proposed [14] employs photopatterned plates to modify the azimuthal angle of the LC, resulting in a flexible inexpensive technology to create any arbitrary beam shaping devices. 5. Tunable filters Optical filters include an ample range of devices, including lowpass and high-pass filters (e.g., cold mirrors and hot mirrors), and band-pass filters. Tunable band-pass filters are one of the most challenging kinds, especially if the bandwidth needs to be narrow, as in wireless communications. Two main kinds of tunable filters for wireless communications are available: opto-acoustic filters and tunable interference filters, many of which are based on LCs. Two kinds of interference filters exist: • Fabry-Perot filters are based on positive interference of multiple reflections between two semi-reflecting parallel surfaces. • Birefringent filters are based on the interference between two light polarizations travelling with speeds determined by the ordinary and the extraordinary refractive index respectively.

Fig. 10. The principle of aberration correction in high resolution microscopes. Using a flat wavefront (a) the sample returns an aberrated one. If a preaberrated compensating waveform (b) is used, then the sample aberrations cancel out and a corrected wavefront (c) is obtained. From [13].



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Fig. 11. Structure of a 3-stage Lyot filter (a) and a Solc filter (b). From [17].

• Volume Bragg gratings are diffraction gratings in which there is a periodic variation of the refractive index, that selectively reflects one component of light, within a spectral band centered on a wavelength proportional to the pitch of the grating.

5.1. Fabry-Perot filters The wavelength (rather, the frequency comb) selected through an FP filter is directly related to the optical path n·d between the external parallel plates. Therefore, FP filters can be made tunable by modifying the optical path inside the cavity. This is achieved by one of two basic ways: either modifying the distance between the parallel plates or changing the refractive index of the internal medium. If distance is modified, required displacements are just fractions of a wavelength; typically, piezoelectric or silicon MEMS control is employed. The second category of tunable filters uses fixed parallel plates and tunes the refractive index of an electrooptic material (e.g., a liquid crystal) that fills the cavity between the plates; when voltage is applied, the RI varies. Tunable LC Fabry-Perot filters are commercially available (Scientific Solutions Inc.). These filters are made of stacked Fabry-Perot etalons, typically two [15]. The first one is a thick resolving etalon, defining the width of the peak, and the second is a thin suppression etalon, increasing the spacing between transmitted peaks, i.e., the free spectral range. The dynamic response time of these filters is not disclosed, but the documentation includes examples of LC devices as thick as 30 μm. Such thick devices should have response times in order of 10's or perhaps even 100's of milliseconds. Both approaches have advantages and drawbacks, the preferable option being dependent on the application. An interesting feature of RItuning devices, recently demonstrated, is the possibility of preparing extremely small devices arranged in arrays of microcavities [16] employing picoliter volumes of liquid crystal.

5.2. Birefringence filters Various implementations of birefringent filters [17] exist (Lyot, Solc, fanned Solc, Lyot-Ohmann), common to all is that they consist of rather complex stacks of birefringent media like LCs [18], occasionally (Lyot filters) interspaced with polarizers (Fig. 11). On the other hand, most interference filters require either the incoming light to be polarized or to split the impinging beam into two twin filter sets via a polarizer beam splitter. A liquid crystal tunable filter [19] works on the principle of polarization dispersion either circularly or linearly [20]. When light passes through a waveplate (e.g. a liquid crystal variable retarder) it will be retarded by a certain number of waves. When light of a different wavelength passes through the same waveplate, it will be retarded by a different number of waves. A tunable optical filter consists of multiple liquid crystal variable retarders, fixed retarders and polarizers all protected in a thermal isolated housing. Temperature control is important since the birefringence of the liquid crystal variable retarders is a function of temperature as well as voltage. For this reason, the entire assembly is temperature-controlled, so that ambient conditions do not affect calibration or switching speed. Each liquid crystal cell that goes into the tunable filter is made to be highly uniform in retardation in order to achieve the best uniformity of color across the clear aperture. Tunable filters based on both Solc and Lyot can simultaneously accommodate an extremely narrow bandwidth and a wide field of view. The NASA Langley Research Center has developed large aperture Solc prisms with LC filled cavity [21].

6. Optical vortices The orbital angular momentum (OAM) of a light beam is a component of angular momentum dependent on the field spatial distribution, but independent of the state of polarization. OAM (actually, internal OAM) is related to a helical or twisted wavefront. Helical modes of the

Fig. 12. Example of a spiral phase plate where a helical beam is generated. From [27].


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Fig. 13. SPP and SDL devices. Continuous SPPs with a) ℓ=1, and b) ℓ=2; c) 12 level digital SPP ℓ=1, and d) ℓ=2; e) diffractive lens; f) continuous SPP with ℓ=4; g) Continuous SDL; h) Binary SDL. Gray levels are phase values in the range (0, 2π), except in (h) where phases are 0 and π.

electromagnetic field are characterized by a wavefront that is shaped as a helix, with an optical vortex in the center, at the beam axis. Optical vortices have lately received much attention [22] because of their applications in particle trapping, image processing, special phase contrast microscopy, free-space communication, and astronomy. To obtain a field vortex, a spiral phase profile has to be obtained. The spiral phase profile can be expressed as exp(jℓθ), where θ denotes the azimuth angle and ℓ is the topological charge. The topological charge is related to the number of times the light twists in one wavelength. The number is always an integer, and can be positive or negative, depending on the direction of the twist. The higher the number of twists is, the faster the light spins around its axis. The easiest way to generate an optical vortex is by impinging a laser beam onto an external spiral phase element like a computer generated hologram [23], or a spiral phase plate (SPP) [24], which have an improved efficiency (Fig. 12). The ideal SPP has a continuous surface thickness topology that imposes the desired azimuthal phase. SPPs are usually manufactured with multilevel quantized phase values; high efficiencies can be obtained with these devices, however, the wavelength and the topological charge cannot be modified. As many other devices, SPPs can be implemented as liquid crystal spatial light modulators. These SPPs are tunable, but they suffer from a limited light efficiency due to the SLM pixelated structure. To avoid this issue, LC cells with a special patterned ITO pie structure can be manufactured. The conversion of a Gaussian laser beam into a doughnut beam with efficiencies near 100% can be achieved with these cells [25]. LC cells stacked together may bring larger topological charge numbers. Generation of optical vortices up to ℓ = 8 has been achieved but

Fig. 14. Experimental devices: (a) Liquid crystal SPP and (b) Liquid crystal SDL. From [27].

the efficiency is reduced due to the stacking process. There is a growing interest in developing efficient and programmable devices for the generation and manipulation of vortex beams [26]. New liquid crystal devices have been developed [27] and characterized for optical vortex generation: pie liquid crystal cells and spiral diffractive lenses. Such devices, manufactured with specific geometrical patterned electrodes, are able to generate optical vortices of different topological charge and may work using different illumination wavelengths. Spiral diffractive lenses (SDL) have been proposed for astronomical applications to allow viewing dim stellar objects in the region of a much brighter object [28]. They have been manufactured with LC pixelated displays; however, LC devices with specific SDL electrode structures render better performance. A close subject lately receiving attention is the generation of radially polarized beams [29]. LC devices able to generate radially polarized beams have been manufactured [30]. LC SPP devices can be used as pseudo-radial beam generators, i.e., the generated beam is not a pure radially polarized beam, but it is accompanied with a spiral phase distribution. 6.1. Design of liquid crystal SPPs and SDLs Fig. 13 shows several masks for vortex generation with SPPs and SDLs. The pie-shaped sectors generate SPPs with different topological charges; the SDL is obtained as a combination of an SPP and a Fresnel lens. Analog and digital solutions are shown. In the digital case, every segment is independently controlled to induce a different reorientation of the liquid crystal. Voltages are applied in a stepwise fashion, so that steadily increasing phase delays are obtained. Matching the last and the first segment to achieve a multiple of 2π is required for the SPP to generate. Examples of actual devices are shown in Fig. 14. The pictures have been obtained between crossed polarizers to show the different delays. Fig. 15 summarizes results for SPP and SDL devices. The SPP pie shape at the upper part of the figure is divided into twelve phase steps. For each wavelength, the twelve voltage levels are selected to produce twelve equidistant phase values between 0 and 2π (Fig. 15a– c) or 4π (Fig. 15d–f). The uppermost images correspond to the liquid crystal SPPs as seen between crossed polarizers. The images below correspond to the focalization of the laser beam. The lower half of Fig. 15 shows optical vortices generated with a liquid crystal SDL with different topological charges. Being SPPs associated to Fresnel lenses, SDLs can focus the laser beam, so that several positive



9

Fig. 15. Top: Generation of ℓ=1 (a, b, c) and ℓ=2 (d, e, f) vortex beams with an LC SPP device at three different wavelengths. Bottom: Same with an SDL device. Focal planes acquisitions for m = −3, −1, 0, +1 and +3 at the three selected wavelengths are shown. From [27].

Fig. 16. Experimental generation of pseudo-radial polarization beams.


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Fig. 17. Left, possible design of digital BS device with phase wrapping. Right, analog BS.

and negative vortices are generated within a single beam, their images being collected at different focal distances. All the images for the same wavelength can be collected in the same experimental set, by moving the camera forward and backward to the different planes where the vortex beams focus. It can be shown that binary SDLs bring maximum intensities to orders ±1 and ±3. Since the phase delay in binary SDLs is adjusted to be π radians; this implies that 0th order is absent. The pattern on the focal plane of the physical positive lens (m = 0) is mainly due to the interference of the two defocused ±1 orders.

6.2. Generation of pseudo-radially polarized beams The LC SPP devices can be used to generate pseudo-radially polarized beams as well. For that purpose, the LC waveplate with variable delay is sandwiched between two quarter-wave plates. This system acts as a polarization rotator, where the rotation angle is equal to half the phase retardation introduced by the LC waveplate. A radially polarized beam can be generated addressing the SPP device with ℓ= 2. The phase origin is taken at the vertical direction, parallel to the input linear polarization. Then the polarization is rotated an angle for every azimuth, generating a (pseudo) radial polarization. As mentioned above, the radial polarization is not perfect because there is an additional phase term as a consequence of the polarization system. This phase term, however, is not noticeable when analyzing the state of polarization. Fig. 16 shows a sequence resulting from imaging the SPP device with the above configuration onto a CCD camera, and progressively rotating the orientation of the analyzer. As the analyzer rotates, the images clearly show two opposite dark sectors, where a linear polarization perpendicular to the analyzer axis is being created. These dark sectors are rotating as the analyzer is rotated, what verifies the generation of the (pseudo) radial polarization beam.

7. Beam steering Beam steerers are devices capable of redirecting light beams, like a prism. The deviation can be made tunable if electrooptic materials such as liquid crystals are employed [31]. As shown in Section II when discussing phase gratings, tunable LC devices of regular size (say, some mm or cm) generate tiny deviation angles (say, b1°), even if thick cells having large phase shifts are used. These deviations may result adequate for many applications, but not for all. If more deviation is required, the simplest solution is to employ phase wrapping, creating saw-tooth profiles (blaze gratings) that may reach very large phase delays. The drawback is that phase wrapping must rely on 2π steps, thus precluding analog steering variations. In this section we will study both beam steering (BS) devices: a digital BS with variable phase wrapping, and an analog BS with voltage gradient extended to the whole cell. Two possible designs of these BS devices are shown in Fig. 17. The device at the left in Fig. 17 is made of parallel conductive tracks that are connected either to one or to two opposite edges of the plate (e.g., interleaved). The width of the tracks and the interpixel gaps depend on the working wavelength; typically, widths for visible and NIR are in the range of μm or 10's μm, whereas gaps are as narrow as possible to avoid fringing effects (Fig. 18). Therefore track densities would be about 50–500 lines per mm. Every line is independently driven. Saw-tooth profiles are obtained gathering groups of tracks in stepwise refractive indices [32]. The larger the number of tracks per group, the higher the diffraction efficiency (see Fig. 6), but the narrower the deviation angle, since the saw-tooth pitch increases accordingly. The device at the right of Fig. 17 is one of the possible approaches to construct analog phase gratings. In this case, the goal is to generate a voltage gradient along the surface of the external plates. The gradient

Fig. 18. Left, phase gratings with interleaved electrodes. The outer connections are shown at the right side. Right, steps made by gathering groups of 8 lines as seen between crossed polarizers.



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Fig. 19. Left, 1D digital BS. The active area, 7 × 7 mm at the center of the glass cell, features 1000 interleaved electrodes of 5.5 μm with 1.5 μm interpixel gaps. Two standard STN chip-on-flex IC drivers (black bars) control every electrode. Right, two orthogonal gradients in an analog BS for 2D steering. From [33].

is obtained by circulating a current along a resistive electrode. Alternatively, one or two parallel resistive external tracks, as in the Figure, can be used interconnected by conductive standard ITO electrodes and an additional layer to homogenize the resistive layer avoiding diffraction. The voltage gradient is obtained as in the first case, but optical properties are usually superior. Moreover, this solution is more versatile if one wants the voltage gradient to be nonlinear, e.g. to mimic the non-

linearity of the birefringence gradient. This is achieved by making variable either the width of the resistive tracks or the distance between tracks. Voltage gradients can extend along the whole cell, thus producing a single analog device, or segmented in several sections to generate quasianalog saw-tooth gratings. Additionally, it is possible to create two orthogonal voltage gradients on either external plate (Fig. 19, right), so

Fig. 20. A 40-μm analog BS with orthogonal voltage gradients on the external plates working as a 2D beam steerer. The pictures are taken between crossed polarizers. The arrows show voltage gradients. The black squares are scaled laser outputs corresponding to these gradients. From [33].


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8.1. Zonal and modal lenses

Fig. 21. Top, a continuous phase profile (red line) mimicked stepwise with a pixelated device. Bottom: Electric equivalent circuit of a modal lens having a resistive electrode (upper line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

that 2D beam steering can be achieved with a single cell. This solution [33] is simpler and more efficient than stacking cells for the same purpose. Fig. 20 shows an example of a 2D beam steering cell with orthogonal gradients in the outer plates. An x-y area can be reached by a laser beam steered by this device, as seen in the scaled black screens presented beside every gradient picture.

8. Lenses Liquid crystal lenses are a good choice when it comes to small, tunable devices. Their low weight and low-cost are interesting advantages over the traditional alternatives employing mechanical elements. LC technology appears to be dominant in the field of millimeter sized tunable lenses with no moving parts. In the last few years, a number of new applications for LC lenses have been created or suggested: tunable contact lenses [34], plenoptic cameras [35], cylindrical lenses for 3D imaging [36], devices for augmented reality [37], autostereoscopic devices [38] or tunable focal length eyeglasses [39]. LC lenses are based on the same principles as beam steerers. Like BSs, LC lenses can be mimicked with multipixel SLMs having 2π phase delays or more; moreover, digital and analog designs can be realized employing electrodes with high resistivity or specific shapes. Spherical lenses are usually achieved with circular electrodes having Fresnelgeometry blaze gratings [40], while parallel linear electrodes produce cylindrical lenses.

Fig. 22. LC modal lens work principle: applying a radial voltage gradient, a phase profile is obtained.

On the other hand, lenses can be obtained with a matrix of pixels, i.e., different electrodes drive different areas of the lens. Pixels may be arranged in a squared matrix, as in SLMs, or adopt specific shapes for developing the desired structure, e.g., a Fresnel lens. Fresnel lenses are manufactured as circular blaze gratings. To achieve a blaze structure and consequently better diffraction efficiency, the device uses groups of four electrodes. These lenses yield very good diffraction efficiencies although the wavefront becomes distorted due to pixelization. In either case, these lenses are called zonal [41], and their performance is limited by resolution and diffraction issues between adjacent zones. Zonal lenses reproduce the phase profile stepwise (Fig. 21). As shown in the beam steerer section, analog devices are possible as well, thereby producing continuous phase profiles that can yield, in principle, more accurate approaches to the actual profile. These modal lenses are based on voltage gradients usually created by resistive electrodes. The electric equivalent circuit (Fig. 21, bottom) of these devices is functionally equivalent to a transmission line. In the figure, R are slices of the resistive electrode while CLC and RLC account for the capacitive and resistive components of the LC layer. The voltage gradient can be made circular applying the same voltage to the low resistivity outer circumference. There are a number of alternatives in both, zonal and modal lenses. Thick dielectric liquid crystal lenses have a quite simple structure [42]. The cell consists of two substrates: the first substrate includes a pattern of two concentric circles (two electrodes); the second is the ground electrode. Applying a voltage between the two patterned electrodes, the electric field lines crosses the liquid crystal quasi-perpendicularly (because of the large distance between the electrodes and the liquid crystal) and the field density is higher in the circle borders than in the center, creating the desired lens profile. The idea of thick dielectrics has been used in designing wavefront correctors, using the dielectric as a shield to minimize the pixelization effect of the device [43]. Hole lenses are based on a hole-patterned LC structure [44], where one of the substrates is a thin (≈0.2 mm) glass coated with aluminum, with a hole-patterned electrode. The size of the hole (hence the size of the lens) is limited to 2 or 3 times the cell gap, so the device is only useful for microlenses. The hole-patterned electrode provides a nonlinear distribution of the electric field inside the liquid crystal layer, which causes a non-uniform reorientation of the liquid director and thus the lens effect. A dual-frequency nematic liquid crystal can be used: molecules are aligned at 45° approximately, with respect to the substrates, by obliquely deposited SiOx. Depending on the frequency of the applied field, the director realigns either towards the homeotropic state or towards the homogenous state. This allows controlling not only the absolute value of the focal length but also its sign. The disadvantage of these devices is that the spot diameter is very small (about 300 μm), what ultimately limits the actual applications. Polymer dispersed liquid crystal lenses are based on gradient polymer LC networks [45]. A standard LC cell is filled with a nematic liquid crystal, mixed with a photocurable precursor of a polymer. The cell undergoes an UV insolation process under a shadow mask. The polymer tends to form a lattice with smaller voids in the highly UV exposed areas – where LC forms smaller nanodroplets – and larger holes in the areas with weaker UV exposure, which are filled with larger nanoscale droplets. When a uniform voltage is applied, the liquid crystal confined in large droplets switches at lower voltages than the liquid crystal confined in the small droplets. Using a uniform applied voltage it is possible to obtain the desired profile of a tunable lens; however adjusting the lens profile becomes a tricky process, as it needs to be done during the UV insolation process. LC modal lenses are usually prepared in standard sandwich cells with ITO coated substrates. An electrode is obtained by photolithography in one or both substrates. The electrode consists of an ITO-free circle in the substrate center. The ITO-free area is covered with a conductive



polymer layer acting as high resistivity electrode (e.g., PEDOT:PSS) [46]. Applying a voltage to the device a phase profile is created. The profile can be adjusted with the voltage and frequency of the control signal. Aberrations are small for short focal distances. Calibration of these devices for a number of focal distances is usually cumbersome; however, once the desired parameters are obtained, the use of the lens is straightforward. One possible problem of these devices is the stability of the polymeric electrode and the repeatability of the conductivity values of the spin-coated layer. In practice, this is solved by calibrating every single device and associating it to a specific look-up table. High resistivity electrodes have been used for modal liquid crystal wavefront correctors as well [47], in order to minimize pixelization effects. LC lenses usually work for one polarization only. There are some exceptions like polymer-dispersed and blue-phase LC devices. However, polymer-dispersed devices require high driving voltages (often exceeding 100 V) and only reduced phase delays are achieved. Blue-phase are promising if phase stabilization and monodomain cells [6] are achieved. Stacking several cells may lead to polarizerfree LC lenses as well [48], however a high control of the thickness of each layer may be required. Other approaches include the use of cholesteric liquid crystals (CLCs). Among the proposed devices a recent solution employs tailored patterned substrates [49]. The patterns produce a spatial variation of the orientation of the CLC helices thus creating specific phase profiles. 8.2. Design of a modal liquid crystal lens As mentioned above, applying a voltage in a concentric gradient to the device, the LC molecules reorient progressively in a circular

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configuration. From the optical point of view, a ‘circular prism’ – rather, a cone – is created, i.e. a liquid crystal lens (Fig. 22). The phase profile intends to emulate a spherical lens. The focal length can be modified from infinity to a certain minimum focal length by changing the applied voltage gradient. The actual value depends on the liquid crystal birefringence and the lens diameter. A radial shaped voltage gradient is induced using a high resistivity layer in the active area. The actual LC switching shape is nonparabolic; however, tuning up the applied voltages and frequencies a quasi-parabolic phase profile can be achieved [50]. Among other advantages, this design avoids the need of complex electrode patterns in the active area. The focal length of the lens can be tuned by varying the applied voltage and frequency. Focal length also depends on the effective lens diameter, the beam wavelength and the delay between the edge and center of the lens. It can be expressed as: F¼

r2 2Δnd

ð9Þ

The focal length is proportional to the radius squared (determining the lens curvature) and inversely proportional to the birefringence and thickness of the lens. Substituting the LC birefringence, the diameter, and the cell thickness in Eq. (9), the minimum theoretical focal length can be obtained for any given LC lens device. For example, the commercial mixture ZLI-3449-100 (Merck), having Δn = 0.1325, mounted in 50 μm-thick cells, shows a minimum theoretical focal length of 472 mm for 5 mm Ø lenses and 169 mm for 3 mm Ø lenses. Fig. 23 is a sequence of pictures taken between crossed polarizers of a 5 mm diameter LC modal lens driven by increasing voltages. The voltage gradient induces interference rings to show up, developing

Fig. 23. Interference radial patterns with increasing voltage gradient in a single electrode lens.


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from the edge towards the center of the cell active area. The phase delay between consecutive rings is 2π. A large number of rings can be observed at high voltages. The higher the number of rings (and the closer they are), the higher the lens power is. A perfect aberration-free lens needs an almost uniform ring distribution from the edge to the center. With this homogeneous configuration, a positive dielectric anisotropy LC would be fully switched outside the lens and progressively less reoriented if getting closer to the lens center. This setup would give a convergent lens. Using the same structure with a negative dielectric anisotropy LC in homeotropic configuration would result in a divergent lens. An interesting alternative recently reported [51] is to employ LC materials whose dielectric anisotropy is strongly dependent on frequency. It has been demonstrated that analog tunable focal length can be achieved at a constant applied voltage merely varying the frequency. The advantage is that frequency switching is 10 times faster than amplitude modulation. The performance of the tunable lens can be tested with a wavefront analyzer, e.g., a Hartmann-Shack wavefront sensor. Fig. 24 shows the evolution of optical power (in diopters) with voltage for a cell of the same series as Fig. 23. Focal length goes from infinity (top left) to about 4 diopters (bottom right). One interesting side effect is that high voltages produce shrinkage of the lens diameter through saturation. This makes possible for the lens to achieve powers above the power given by the theoretical focus length. However, saturating voltages also flatten the central lens area, worsening its performance. This issue may be corrected, or at least alleviated, by reshaping the lens with varying frequency, generating acceptable spherical lens profiles.

9. Conclusions LC photonic devices for non-display applications, and specifically phase-only devices, are experiencing a tremendous development in this century. Many research laboratories are evolving from displays – still the largest application in this field, but mostly beyond basic or applied research – to non-display devices. Progress is still limited, but possibilities seem to be endless. In this work some of the more relevant phase-only photonic devices have been analyzed. The list was not intended to be complete, but descriptive of the many fields and subfields of the area. In the next five years, LC photonic devices will have to contend with other technologies for several multibillion applications. Possibly the most relevant would be a display application, i.e., to become the preferred technology for wearable and portable displays in the Internet of Things (IoT); at present, the required flexible displays are being designed with OLED technology; yet a reliable rollable or foldable technology for the new generation of mobile phones and wearable gadgets is not established yet. Besides this display technology, possibly the most important field for phase-only devices would be the integration in photonic integrated circuits (PICs). At present, the level of integration in PICs is dramatically low – seven orders of magnitude – compared to electronic integrated circuits. Besides the fundamental reasons for that difference, technologies that would increase the PIC flexibility or functionality with moderate increments of complexity would greatly help to the development of these devices. Liquid crystals can interact with transmissions of optical signals through the PIC waveguides, effectively producing filters, switches or modulators with relatively simple setups. Currently no other technology, except MEMS partially, seems to be so well positioned in this area as liquid crystals.

Fig. 24. LC lens wavefront profiles from a Hartmann-Shack sensor. Voltage increases left to right and top to bottom. Focal lengths are: ∞, 1100, 480, 420, 390, 350, 310, 290, 280 mm respectively. From [46].



References [1] Z. Zhang, Z. You, D. Chu, Fundamentals of phase-only liquid crystal on silicon (LCOS) devices, Light Sci. Applicat. 3 (2014), e213. [2] M. Choi, J. Choi, Universal phase-only spatial light modulators, Opt. Express 25 (19) (2017) 22253–22267. [3] M. Wang, L. Zong, L. Mao, A. Márquez, Y. Ye, H. Zhao, F.J. Vaquero-Caballero, LCoS SLM study and its application in wavelength selective switch, Photo-Dermatology 4 (2017) 22–37. [4] See for Example: E. Hecht Optics. 4th ed., Pearson Education Ltd., Harlow, UK, 2014. [5] S. Tanaka, H. Yoshida, Y. Kawata, R. Kuwahara, R. Nishi, M. Ozaki, Double-twist cylinders in liquid crystalline cholesteric blue phases observed by transmission electron microscopy, Sci. Rep. 5 (2015), 16180. [6] E. Otón, E. Netter, T. Nakano, Y.D. Katayama, F. Inoue, Monodomain blue phase liquid crystal layers for phase modulation, Sci. Rep. 7 (2017), 44575. [7] E.J. Willman, Three Dimensional Finite Element Modelling of Liquid Crystal ElectroHydrodynamics, Doctoral Thesis, University College, London, 2009. [8] M.J. Abuleil, I.Y. August, Y. Oiknine, A. Stern, I. Abdulhalim, Stokes polarimetry, narrowband filtering, and hyperspectral imaging using a small number of liquid crystal devices, Proc. SPIE Liquid Crystals XX 9940 (2016) 99400S. [9] Da-yu Li, Liquid crystal adaptive optics wavefront processor based on multi-GPU, Chin. J. Liq. Cryst. Displays 31 (5) (2016) 491–496. [10] T.G. Bifano, J.A. Kubby, S. Gigan, Adaptive Optics and wavefront control for biological systems, J. Biomed. Opt. 21 (2016), 121501. [11] L.B. Yang, L.F. Hu, D.Y. Li, Z.L. Cao, Q.Q. Mu, J. Ma, L. Xuan, Determining the imaging plane of a retinal capillary layer in adaptive optical imaging, Chin. Phys. B 25 (9) (2016) 094219-1–094219-7. [12] H. Kaushal, G. Kaddoum, Optical communication in space: challenges and mitigation techniques, IEEE Commun. Surv. Tutorials 19 (1) (2017) 57–96. [13] M.J. Booth, Adaptive optical microscopy: the ongoing quest for a perfect image, Light Sci. Applicat. 3 (2014), e165. [14] P. Chen, Y.Q. Lu, W. Hu, Beam shaping via photopatterned liquid crystals, Liq. Cryst. 43 (13–15) (2016) 2051–2061. [15] J. Lin, Q. Tong, Y. Lei, Z. Xin, D. Wei, X. Zhang, J. Liao, H. Wang, C. Xie, Electrically tunable infrared filter based on a cascaded liquid-crystal Fabry-Perot for spectral imaging detection, Appl. Opt. 56 (7) (2017) 1925–1929. [16] C. Stoltzfus, R. Barbour, D. Atherton, Z. Barber, Micro-sized tunable liquid crystal optical filters, Opt. Lett. 42 (11) (2017) 2090–2093. [17] D. Sánchez-Montero, I.P. Garcilópez, C. Vázquez, P. Contreras, A. Tapetado, P.J. Pinzón, in: M. Yasin, H. Arof, S.W. Harun (Eds.),Recent Advances in WavelengthDivision-Multiplexing Plastic Optical Fiber Technologies in Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications, 2015 www. intechopen.com/books/advances-in-optical-fiber-technology-fundamental-opticalphenomena-and-applications (InTech). [18] M. Abuleil, I. Abdulhalim, Narrowband multispectral liquid crystal tunable filter, Opt. Lett. 41 (9) (2016) 1957–1960. [19] O. Aharon, I. Abdulhalim, Design of Wide Band Tunable Birefringent Filters with Liquid Crystals, 5 (6), PIERS Online, 2009 555–560. [20] Y. Huang, S. Zhang, Optical filter with tunable wavelength and bandwidth based on cholesteric liquid crystals, Opt. Lett. 36 (23) (2011) 4563–4565. [21] N. Clark, Design and performance evaluation of sensors and actuators for advanced optical systems (keynote), NASA Langley Research Center, 2011http://ntrs.nasa.gov/ archive/nasa/casi.ntrs.nasa.gov/20110008060_2011008844.pdf. [22] J. Albero, P. García-Martínez, N. Bennis, E. Otón, B. Cerrolaza, I. Moreno, Liquid crystal devices for the reconfigurable generation of optical vortices, J. Lightwave Technol. 30 (18) (2012) 3055–3060. [23] N.A. Carvajal, C.H. Acevedo, Y. Torres-Moreno, Generation of perfect optical vortices by using a transmission liquid crystal spatial light modulator, Int. J. Theor. Phys. 2017 (2017) 6852019. [24] J.F. Algorri, V. Urruchi, B. García-Cámara, J.M. Sánchez-Pena, Generation of optical vortices by an ideal liquid crystal spiral phase plate, IEEE Electron Device Lett. 35 (8) (2014) 856–858. [25] J. Kobashi, H. Yoshida, M. Ozaki, Broadband optical vortex generation from patterned cholesteric liquid crystals, Mol. Cryst. Liq. Cryst. 646 (1) (2017) 116–124.

15

[26] L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Magali, F. Sciarrino, Spin-to-orbital momentum of light and its classical and quantum applications, J. Opt. 13 (2011), 064001. [27] Eva Otón, Adaptive Liquid-Crystal Phase-Only Passive Devices, Ph.D. Thesis, Technical University of Madrid, Spain, 2013. [28] A. Aleksanyan, E. Brasselet, Optical vortex coronagraphy using liquid crystal topological defects, Proc. SPIE 9940 (2016) 99400D. [29] J.A. Davis, D.M. Cottrell, B.C. Schoonover, J.B. Cushing, J. Albero, I. Moreno, Vortex sensing analysis of radially and pseudo-radially polarized beams, Opt. Eng. 52 (5) (2013) 050502-1–050502-3. [30] U. Ruiz, P. Pagliusi, C. Provenzano, G. Cipparrone, Vector beams generated by tunable liquid crystal polarization holograms, J. Appl. Phys. 121 (2017), 153104. [31] X. Wang, X. Huang, Z. Huang, L. Wu, J. Shang, Q. Qiu, S. Wu, Nonmechanical infrared beam steering using blue addressed quantum dot doped liquid crystal grating, Nanoscale Res. Lett. 12 (2017) 36–39. [32] E. Otón, J. Pérez-Fernández, D. López-Molina, X. Quintana, J.M. Otón, M.A. Geday, Reliability of liquid crystals in space photonics, IEEE Photonics J. 7 (4) (2015) 6900909. [33] J.M. Otón, X. Quintana, M.A. Geday, Aerospatial applications of liquid crystal photonics, 3rd Intl. Workshop on Liquid Crystals for Photonics LCP’10, Elche, Spain, September 2010. [34] J. De Smet, A. Avci, P. Joshi, D. Schaubroeck, D. Cuypers, H. De Smet, Progress toward a liquid crystal contact lens display, J. Soc. Inf. Disp. 21 (2013) 399–406. [35] Y. Lei, Q. Tong, X. Zhang, H. Sang, A. Ji, C. Xie, An electrically tunable plenoptic camera using a liquid crystal microlens array, Rev. Sci. Instr. 86 (5) (2015) 053101. [36] J.F. Algorri, V. Urruchi, N. Bennis, J.M. Sánchez-Pena, J.M. Otón, Cylindrical liquid crystal microlens array with rotary optical power and tunable focal length, IEEE Electron. Device. Lett. 36 (6) (2015) 582–584. [37] Y.J. Wang, P.J. Chen, X. Liang, Augmented reality with image registration, vision correction and sunlight readability via liquid crystal devices, Sci. Rep. 7 (2017) 43301–433-12. [38] J.F. Algorri, V. Urruchi, B. García-Cámara, J.M. Sánchez-Pena, Liquid crystal microlenses for autostereoscopic displays, Materials 9 (1) (2016) 36–52. [39] N. Hasan, A. Banerjee, H. Kim, C.H. Mastrangelo, Tunable-focus lens for adaptive eyeglasses, Opt. Express 25 (2017) 1221–1233. [40] X.Q. Wang, S.A. Kumar, M.W.A. Tam, Z.G. Zheng, D. Shen, V.G. Chigrinov, H.S. Kwok, Liquid crystal Fresnel lens display, Chin. Phys. B 25 (9) (2016) 094215. [41] G.D. Love, A.F. Naumov, Modal liquid crystal lenses, Liq. Cryst. Today 10 (1) (2000) 1–4. [42] S. Xu, Y. Li, Y. Liu, J. Sun, H. Ren, S.T. Wu, Fast-response liquid crystal microlens, Micromachines 5 (2014) 300–324. [43] M. Locktev, G. Vdovin, N. Klimov, S. Kotova, Liquid crystal wavefront corrector with modal response based on spreading of the electric field in a dielectric material, Opt. Express 15 (6) (2007) 2770–2778. [44] C.Y. Chien, C.H. Li, C.R. Sheu, An electrically tunable liquid crystal lens with coaxial Bi-focus and single focus switching modes, Crystals 7 (7) (2017) 209-01–209-12. [45] D. Li, J. Zheng, K. Gui, K. Wang, Y. Wang, Electrically controlled hole-patterned tunable-focus lens with polymer dispersed liquid crystal doped with Ag nanoparticles, Optik 127 (19) (2016) 7788–7793. [46] David Pérez-Medialdea, Liquid Crystal Photonic Devices Based on Conductive Polymers, Ph.D. Thesis, Technical University of Madrid, Spain, 2010. [47] Carlos Carrasco, Design and Manufacturing of Devices Based on Polymeric Liquid Crystals, Ph.D. Thesis, Technical University of Madrid, Spain, 2012. [48] H. Chen, Y. Wang, C. Chang, Y. Lin, A polarizer-free liquid crystal lens exploiting an embedded-multilayered structure, IEEE Photon. Technol. Lett. 27 (2015) 899–902. [49] J. Kobashi, H. Yoshida, M. Ozaki, Planar optics with patterned chiral liquid crystals, Nat. Photonics 10 (2016) 389–392. [50] A.K. Kirby, P.J.W. Hands, G.D. Love, Optical design of liquid crystal lenses: off-axis modelling, SPIE Proc. 5874 (2005) 1–10. [51] J.F. Algorri, N. Bennis, J. Herman, P. Kula, V. Urruchi, J.M. Sánchez Pena, Low aberration and fast switching microlenses based on a novel liquid crystal mixture, Opt. Express 25 (13) (2017) 14795–14808.
