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. 2019 Dec 5;4(25):21459–21468. doi: 10.1021/acsomega.9b03158

Theoretical Prediction of Umbilics Creation in Nematic Liquid Crystals with Positive Dielectric Anisotropy

Atefeh Habibpourmoghadam 1,*
PMCID: PMC6921641  PMID: 31867541

Abstract

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Optically assisted electrical generation of umbilic defects, arising in homeotropically aligned nematic liquid crystal cells and known as topological templates for the generation of optical vortices, are reported in nematic liquid crystals with positive dielectric anisotropy in detail. It is shown that nematic liquid crystals with positive dielectric anisotropy can serve as a stable and efficient medium for the optical vortex generation from both linearly and circularly polarized input Gaussian beams. Hybrid cells made from a thin layer of nematic liquid crystal confined between a photoresponsive slab of iron-doped lithium niobate and a glass plate coated with an active material, i.e., indium tin oxide, were studied. Exposure to a laser beam locally induces a photovoltaic field in the iron-doped lithium niobate substrate, which can penetrate into the liquid crystal film and induce realignment of molecules. The photovoltaic field drives charge carrier accumulation at the interface of indium tin oxide with the liquid crystal, which effectively modifies the shape and symmetry of the electric field. The photovoltaic field has a continuous radial distribution in the transverse xy-plane, weakening with increasing distance from the light irradiation center, where the electric field is normal to the cell plane. Umbilics are created as a result of the liquid crystal tendency to realign parallel to the electric field. Numerical studies of the transmitted intensity profiles in between linear polarizers reveal optical vortex pattern (of four and eight brushes) characteristics for the umbilical defects. The application of crossed circular polarizers results in annular-shaped intensity patterns as a result of spin-to-orbital angular momentum conversions, which give rise to the optical vortices.

1. Introduction

Dynamical rotation of the electromagnetic field of light with respect to the axis of propagation is described by a vector quantity called angular momentum associated as spin angular momentum (SAM) and orbital angular momentum (OAM). While SAM is attributed to the light beam polarization as elliptical or circular, OAM is corresponding with the twisting of the beam wave front around the propagation axis (coincident with a topological phase singularity); such a beam is called an optical vortex (OV).1−6 The intensity profiles of OVs were made from donut-shaped concentric rings with zero-intensity at the center1−6 associated with the beam phase singularity. At a given wavelength λ, the phase front of an OV beam is composed of |l| intertwined helices along the propagation axis, where the integer azimuthal index l gives the amount of OAM carried by the wave per photon as lℏ.6,7

OVs are structurally stable in a homogeneous and isotropic medium due to invariant OAM.8 In nonlinear media (characterized with third-order nonlinear optical susceptibility χ(3) > 0), the lensing property imposes vortex solitons generation as self-trapped spatially localized beams keeping structural shape during propagation.9,10 During the past decades, OVs have found tremendous applications in different fields such as coronagraphy in astronomy,11,12 cryptography based on OAM states of photons in telecommunication systems,13−15 micromanipulation techniques in biological systems,16−18 and as depletion beams in stimulated-emission-depletion (STED) microscopy.19

Different methods were developed for generating OVs, such as employing a pair of cylindrical lenses,20 spiral phase plates,21−23 segmented deformable mirrors,24 computer-generated holograms (CGHs) in the form of spatial light modulators (SLMs),25 forked diffraction gratings,25−27 or spiral Fresnel lenses.28 Nematic liquid crystals (LCs), as outstanding candidates, have found applications in the electro-optical systems aimed at spin–orbit manipulation of light29−31 because of their self-organization properties, ease of controlling their optical axis, exhibiting a long-range orientational order, and exhibiting high optical anisotropy and birefringence as well.

OVs can be generated in the nematic liquid crystals by means of different mechanisms such as q-plates,32 nematic droplets,33 and umbilical defects (in short “umbilics”).34,35 The topological structures of umbilics were described by Rapini in 1973,36 resembling string-like objects in three dimensions (3D).35,36 In the LC, umbilics act as topological matter templates, which are able to convert an incoming circularly polarized light beam into a helical (vortex) light beam, the so-called photonic spin-to-orbital momentum couplers.37 Umbilics have been extensively studied in the nematic LCs with negative dielectric anisotropy (Δε < 0), where the LCs tend to realign perpendicular to the applied electric field. Here, it is shown how successfully the nematic LCs with positive dielectric anisotropy (Δε > 0), as initially aligned homeotropically, can be employed for the purpose of umbilics generation as well confined and without any disclination line. In our recent experimental study,38 it was shown that the planar anchoring condition on the cell walls can lead to the formation of disclination lines in the LC cells with a similar structure to the one studied here.

Hybridized LC cells made from a thin film of LC confined between an indium tin oxide (ITO)-coated glass slab and a z-cut ferroelectric-substrate-iron-doped lithium niobate (Fe/LiNbO3 or in short Fe/LN) were studied (Figure 1). Treating the cell confining walls (both ITO-coated glass and the interface of Fe/LN with the LC) can provide a homeotropic anchoring condition for the LC, corresponding with an unperturbed condition. In the recent experimental work of Kravets and co-workers,39 it was shown that the photoassisted dc electric field can induce spin–orbit optical vortex generation with high purity and efficiency (>90%). Here, light induced photovoltaic (PV) field in the Fe/LN has a distribution in the LC media. Although it is strong enough to induce the LC reorientation,38,40−42 photoinduced charge carriers in the single active ITO layer reshape the electric field profile in a promising way for optical vortex generation. In the work reported by Barboza et al.,43 a liquid crystal light valve (LCLV) was made by employing Bi12SiO20 (BSO) slab as a photosensible wall, which acts as an optical tunable impedance, if illuminated properly, by means of tuning the effective voltage in the LC media.

Figure 1.

Figure 1

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Schematic of exposed hybridized LC test cell with a laser beam with sub milliwatt power.

The umbilics in the nematic LC with positive dielectric anisotropy (Δε > 0) are created as the result of the competition between the elastic torque and electric torque applied by the optically assisted PV field. Previously, they were studied both experimentally and theoretically in a situation where the external applied torque on the LC was of the purely optical nature.44,45

For comparison, umbilics generated in the LCs with negative dielectric anisotropy, referred to as model umbilics, were studied. The simulation results showed that the topological defects generated in the nematic liquid crystal with positive dielectric anisotropy possess all of the fundamental features of the model umbilics.

2. Discussion

2.1. Simulation of Photogenerated Electric Field in the Hybridized LC Cell

Studies were conducted for the nematic LC of MLC-208738,40 with the positive dielectric anisotropy of Δε = 13.31 at zero frequency associated with the dc electric field, a birefringence of Δn = 0.076, and an average elastic constant of K = 14.7 pN. To obtain a visualization of the photovoltaic field distribution in the LC, simulation was conducted in a test cell filled with an isotropic medium with a dielectric constant of εiso = 10.16 (equal to the average dielectric constant of MLC-2087), as shown in Figure 2. LC realignments were studied by taking into account the anisotropy of media as discussed in Section 4. The simulation geometry can be also seen in Figure 2. As previously described,40 light irradiation of the LC test cell with Fe/LN substrates gives rise to a photoinduced charge carriers separation that can be described by a two-dimensional Gaussian distribution (σ = σ0e–(x2+y2)/w2) of positive charge carriers at the top surface of the Fe/LN substrate and a similar distribution of negative charge carriers at the base. The full width at half maximum (FWHM) w of the Gaussian charge carriers distributions was set equal to the FWHM (14 μm) of the exposure beam focus. The thickness of the Fe/LN substrate was set to 100 μm and its dielectric constant was considered anisotropic (ε∥z = 29 and ε⊥z = 85). The simulation geometry had a footprint of x × y = 60 × 60 μm2 in the transverse xy-plane.

Figure 2.

Figure 2

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Schematic of photogenerated electric field distribution in a hybridized LC test cell with Fe/LN substrate and ITO-coated glass plate. The electric field generated within the Fe/LN substrate induced a charge carrier accumulation in the active ITO layer. As a result, modeling was done by placing Gaussian charge carrier distributions in the upper and lower plane (xy-plane) of the Fe/LN substrates, as well as in the LC/ITO interface. The E-field is shown in the rear half of the volume to give a clear view of the center area in magnification.

The test cell was set up from an ITO-coated cover glass with a relative permittivity of about εSITO = 946−48 in the static electric field. The thickness of the ITO layer deposited at the glass surface was much smaller than both the LC layer and the Fe/LN substrate; it was supposed in the range of 100–300 nm in the simulations: A Gaussian charge distribution, with negative sign and the same FWHM as the ones attributed to the Fe/LN crystal, was supposed at the boundary of the ITO with the LC, as a result of charge accumulation induced via photovoltaic field within the ITO layer (the reasons are explained in more detail in Section 4 as well as in the Supporting Information). In modeling, the photovoltaic field was considered as the only source of the static electric field in the cell. The electric field exhibits distribution inside the glass plate because of the limited permittivity and negligible thickness of the ITO film. The charge density amplitudes σ0 of the Gaussian charge carrier distributions, and thus the magnitude of the static electric fields inside the LC layer, were increased stepwise in the simulations.

The field distribution obtained in this model is seen in Figure 2. It shows that the local electric field is normal to the sample plane (parallel to the z-direction) at the exposure spot center. One can thus expect that the initial homeotropic alignment of the LC (in the LC with positive dielectric anisotropy) is maintained here locally, which—for transmitted polarized light—presents a virtual continuous phase singularity: this region appears as a dark spot in the intensity distribution of transmitted light. Moreover, fringe fields were seen around the exposure spot center (the center of the Gaussian charge carrier distribution) and the electric field decays in radial direction. As can be seen in Figure 2, the electric field distribution has a continuous radial distribution in the transverse xy-plane. Such a field distribution can be expected to induce a radial distribution of the LC director in the cell corresponding to the director realignment in a splay umbilic. If so, the polarization of the light incident in the area surrounding the local phase singularity can be modulated, since the optical axis is no longer oriented parallel to the propagation direction. In the umbilics with transverse distribution of the local optical axis, this leads to the creation of optical vortices.35,49

2.2. Modeling of Director Field in a Splay Umbilic Observed in the Nematic LC with Negative Anisotropy (Model Umbilic)

Umbilics appear in the LC cell at voltages higher than threshold U > Ut (Ut = π(K3/ε0|Δε|)1/2),35,36 where K3 is the bend elastic constant and Δε = ε∥Ω – ε⊥ is the dielectric anisotropy at a frequency Ω defined as the difference between permittivity along and across the director direction, i.e., ε∥Ω and ε⊥, respectively. Continuity in the structure is in fact the main difference between an umbilical and a frank defect.36 Structure of an umbilic is specified by a continuous core a(r) called reduced amplitude, which is numerically obtained from36

2.2. 1

where a∞ is the reduced amplitude at infinity approaching the value of one35 and the parameter χ is inversely proportional to the core radius rc, defined as36

2.2. 2

where d is LC film thickness, K is the effective elastic constant determined from the nature of the defect (for example, K = K1 for splay umbilics), and Ũ = U/Ut is called the reduced voltage. The local orientation of a nematic liquid crystal is represented by a dimensionless director field n, where n and −n are physically equivalent due to the lack of polarity of molecules. Director field n (described as n = (nx, ny, nz)) can be introduced based on the projections in the transverse xy-plane (i.e., n⊥) and along the z-axis (i.e., nz), respectively, as n = (n⊥, nz). The director field in transverse plane n⊥ reads n⊥ = |n⊥|C(φ), where |n⊥| is amplitude and C is a unit vector giving local direction in the transverse xy-plane as a function of angle φ (φ = sϕ + φ0, where s gives the topological strength of defect, ϕ is the azimuthal angle, and φ0 is a constant value equal to zero for the case of splay umbilic, as shown in Figure 3a). In the splay umbilic, C(φ) gives a radial distribution in the transverse xy-plane (Figure 3). According to Rapini,36n⊥ can be obtained from

2.2. 3

where a(r) (0 ≤ a(r) ≤ 1) has the minimum value of zero at the center of defect. By considering n as an unit vector field, nz can be obtained from the tilt angle θ with respect to the z-direction (k-vector supposed parallel to the z-direction), as nz = cos(θ). θ can be calculated from36

2.2. 4

where θ∞ is the value of θ at large r, defined as36

2.2. 5

where the parameters K1 and K3 are elastic constants corresponding with splay and bend configurations, respectively.

Figure 3.

Figure 3

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(a) Demonstration of the coordinate system elements. (b) Simulated reduced amplitude a at a relative voltage of U/Ut = 1.12. Projection of the model director field in the xy-plane (n⊥): (c) side-on view, (d) top view. The parameter d corresponds to the cell gap.

The director field of the model umbilic in the LC with negative dielectric anisotropy (based on the model described in the literature35) projected in the xy-plane and demonstrated in three dimensions (3D) is shown in Figures 3 and 4, respectively. The LC cell is addressed with a homogenous electric field. The simulations were done for the nematic LC of MLC-207935 with the negative dielectric anisotropy of Δε = −6.1 at Ω = 1 kHz, a birefringence of Δn = 0.15 at 589 nm wavelength, and the elastic constants of K1 = 15.9 pN and K3 = 18.3 pN. At the center of the model umbilic, n⊥ has the minimum value of zero (Figure 4): because of no preferential orientation in this spot, the LC keeps its homeotropic alignment along the film associated to a topological phase singularity. The magnitude of n⊥ increases in the radial direction and reaches its maximum value far from the center of the film35 under the dominance of the reduced amplitude a. The distribution of the local optical axis is comparable to an optical q-plate with radial geometry50 suitable for vortex generation. Such a director pattern can generate optical vortices in a thin sample or if the LC has a low birefringence, since diffraction effects (which are unwanted in a q-plate) are then negligible for light propagating perpendicular to the sample plane.50,51

Figure 4.

Figure 4

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Simulated director field of the model umbilic defect in an homeotropically aligned LC cell in (a) three dimensions (3D) and (b) two dimensions (2D). The umbilic shows rotational symmetry around the z-axis.

3. Results

3.1. Formation of a Splay Umbilic in the Nematic LC with Positive Anisotropy

In the second step, the field-induced LC realignment was simulated in the MLC-2087 filled sample with Fe/LN substrate by using a Q-tensor approach by fully considering the anisotropic dielectric constant of the LC (MLC-2087) with ε∥0 = 16.81 and ε⊥ = 3.50 at zero frequency and using the one-constant approximation (this approach was described earlier40 in detail). For the simulation of the electric field,38 the following modification was necessary: charge accumulation is considered in the ITO layer induced by the photovoltaic voltage (discussed in the Supporting Information). In addition, the anchoring conditions were defined based on the test cells condition: at both the ITO and Fe/LN surfaces (covering the LC), homeotropic anchoring was considered.

The LC director field as n = (nx, ny, nz) was obtained from numerically calculating Q-tensor40,52−57

3.1. 6

where S and δij are the scalar order parameter and Kronecker’s delta function, respectively. Energy of the LC volume in static can be described in terms of the Q-tensor elements and their spatial derivatives, i.e., Qij and ∇Qij. According to Landau–de Gennes expansion, in the absence of any external electrical or optical stimuli, the energy density of the LC can be considered as a summation of Landau–de Gennes potential and elastic energy, i.e., ELdGP(Qij) and ELdG(∇Qij), repectively40,54

3.1. 7

where

3.1. 8
3.1. 9

here A, B, and C are material constants (where A is temperature dependent) and L/2 is the effective elastic constant K. In the LC volume, the photovoltaic field induced dipole moments on the LC molecules: since the LC had a positive dielectric anisotropy, the director field was locally rotated toward the field polarization direction, which leads to elastic torques in the LC. Anisotropic dielectric properties of the LC were considered by the permittivity tensor40,54

3.1. 10

where ε̅ = 1/3(ε∥0 + 2ε⊥) and, as mentioned before, Δε = (ε∥0 – ε⊥), with ε∥0 and ε∥ being the permittivity along and across the director direction at dc PV field, respectively. In addition, the electric energy density stored in the system was40,54

3.1. 11

where the electric displacement field was obtained from Di = ε0εijEj (Ej = ∂jU), where ε0 is the vacuum permittivity. EE could be retrieved in terms of the electric potential U and Q-tensor elements as40,52−55

3.1. 12

The total free energy in the whole volume of the LC V was calculated from40,54

3.1. 13

By minimizing the total free energy, the Euler–Lagrange equation40 was obtained, which gave director field distributions in the static condition

3.1. 14

where the effective elastic constant K = L/2 was employed in the one elastic constant approximation. In this approach, the distribution of Q was numerically calculated using finite elements (COMSOL Multiphysics). The data obtained were postprocessed in Matlab in a mesh of x × y × z = 124 × 124 × 124 data points.

The boundary conditions for the LC director field alignment at the confining surfaces of ITO and Fe/LN were supposed homeotropic with infinite anchoring energy. Qb at these boundaries was locally set as

3.1. 15

where ex, ey, and ez are unit vectors of the Cartesian coordinate system (x,y,z).58

The distribution of the photovoltaic field (static electric field) is shown in Figure 5. As seen in this figure, the magnitude of the static electric field was higher in the upper region of the sample (near the ITO-coated glass plate, section I) than in the lower part of the sample (near Fe/LN interface, section II) and had a minimum value at the vicinity of the (virtual) surface S (Figure 5) at the center of the LC layer.

Figure 5.

Figure 5

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(a) Electric field distribution (photovoltaic field) in the xz-plane of the simulated test cell in the thin ITO layer. (b) The LC-filled area is shown selectively. This area is divided into two sections (section I near the ITO surface, section II near the field generating Fe/LN slab), which are separated by a virtual surface S. A possible tendency to locally realign the director from the initial homeotropic alignment A to an reoriented state B is indicated for each section.

At photovoltaic voltage (U > 0), ITO acts as an active n-type semiconductor in contact with the LC as an insulator media. The structure of the cell is comparable with a metal–insulator–semiconductor (MIS)59 structure. As a result, it can be expected that the charge accumulation happens at the boundary of ITO with the LC. In detail, the mechanism can be explained as near the ITO/LC interface, the ITO conduction band (Ec) bends downward toward the Fermi level (Ef) as it was kept flat. Since the carrier density in the semiconductor is proportional to exp(−(Ec – Ef)/kBT) (where kB is Boltzmann constant and T is temperature), this reduction drives electron accumulation59 near the ITO/LC interface. According to the literature,60 the induced charge density can reach an order of magnitude larger than the bulk-free carrier density of the ITO, if biased properly. Since this mechanism can strongly affect the real and imaginary parts of the refractive index of ITO (exploited with a thickness in the range of some tens to hundreds nanometer), it has found applications in the phase and absorption modulation of surface plasmon polariton (SPP) in the visible60 and infrared48 (specifically in telecommunication61,62) wavelength ranges, respectively. Here, charge accumulation in the ITO layer desirably affects the structure of the electric field in the LC cell, making it suitable for umbilics formation.

Charge carrier density supposed in accumulation at the ITO/LC interface results in the convergence of electric field in the LC media with rotational symmetry, which in fact provides the vital condition for the generation of the fringe director field (shown in Figure 4) corresponding with the umbilical realignments. Since ITO had a relatively small relative permittivity in the static electric field,46−48 the tangent element of the electric field at the boundary of the ITO with LC is seen (Figure 5a). However, as expected, the electric field strength decreased in the ITO thin film as it goes to zero, far away from the charge carrier accumulation62 in the glass (Figures 2 and 5).

In the laser on state, when the induced voltage exceeds the threshold, the director field is locally realigned from the initial homeotropic alignment (A) to a realigned state (B) parallel (or antiparallel) to the electric field vector in section I and II. Since the LC molecules have no polarity, they are tilted by the electric field in a way to minimize the energy, which means that they always prefer to rotate with the smaller possible angle to go from an initial to a final alignment. For example, in the left region (Figure 5, at x < 0) of the LC-filled area, the electric field tends to realign the LC molecules clockwise (section I, x < 0) and counterclockwise in section II. The static condition is maintained, because at every point, the electric torque exerted on the LC molecules is compensated by the net elastic torque exerted by the neighboring molecules. As a result, the director field was continuously realigned in the cell, giving rise to the umbilical configuration corresponding with minimum total free energy per unit volume of the LC.

The electric field distribution had rotational symmetry in the xy-plane and it was therefore able to induce a continuous radial (Figure 6c) realignment of the LC director in this plane. In the area surrounding the exposure spot center, where the photovoltaic field is vertical, the radial alignment was supported by the elastic interactions in the LC imposed by the neighboring molecules’ realignment in the radial direction. As seen in Figure 6, the simulated director field has rotational symmetry, as expected. The alignment in the xz-plane (Figure 6b) showed all of the characteristics of a splay umbilic35,36 configuration: as in the model umbilic (Figures 3 and 4), homeotropic alignment was preserved at the defect center (x = 0). Moreover, the director field had splay deformations, which is seen in Figure 6a,b (in the xz-plane). The radial distribution of the LC director is best seen in a top view (Figure 6c). In addition to giving the LC director distribution, the simulation was also capable of explaining the generation of optical vortices in the samples (as discussed in the following section).

Figure 6.

Figure 6

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Simulated splay umbilic director field, obtained in the one-constant approximation, for a LC test cell with Fe/LN substrate and ITO-coated cover glass, which was filled with MLC-2087. The amplitude of the Gaussian charge density distribution was set to σ0 = 40 μC m–2. (a) Simulated director field. For clarity, the half volume of the LC layer is shown to give a clear view of the center of the realigned director field. (b) Director pattern in the xz-plane. (c) Top view.

3.2. Simulation of Output Intensity Patterns

The intensity distributions of the optical transmissions in between linear and circular polarizers were calculated. The local effective refractive index for probe light passing (propagating in the z-direction) through the LC was obtained from37,49,63,64

3.2. 16

where θ(x,y,z) is the tilt angle of the director with respect to the z-direction, no and ne are ordinary and extraordinary refractive indices equal to 1.48 and 1.55 for MLC-2087 (simulations, positive dielectric anisotropy) at a wavelength of λ = 589 nm, respectively.

The total optical phase difference Γ(x,y) between ordinarily and extraordinarily polarized light was obtained for each point in the xy-plane by calculating the optical phase change accumulated by transmission through the LC layer35,37,49,64 as

3.2. 17

where d is the cell gap (30 μm). Optical phase changes Γ were numerically obtained and compared to the corresponding results for umbilic defects (referred to as model umbilics) generated in the LC (MLC-207935) with negative dielectric anisotropy reported by Brasselet (in ref (35)) in the conventional LC cells35,36 under static electric fields (at the reduced voltages Ũ = U/Ut equal to 1.1235 and 1.25,35 respectively). Numerical studies were done by the assumption of the same LC film thickness of 30 μm and irradiation by a laser beam with the same λ = 589 nm wavelength. Both models revealed continuous Gaussian-like phase change profiles37 for both types of LCs, where Γ always has the minimum value of zero at the center of defects (Figure 7), driven by the homeotropic alignment, as it provides an isotropic spot for the light propagation in the LC. Although the size of the defect core in the model umbilic is a function of applied voltage (as by increasing the voltage, it decreases significantly),35 the core radius of the umbilic generated in the LC with positive anisotropy depends only on the FWHM of the incident laser spot.

Figure 7.

Figure 7

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Simulated phase change profiles Γ (a) and (b) in the model umbilical defect at the reduced voltages Ũ equal to 1.1235 and 1.25. (c, d) in the umbilics observed in the nematic LC with positive anisotropy corresponding with the induced charge densities equal to σ0 = 40 μC m–2 and σ0 = 60 μC m–2 in the early stage of OVs formation, respectively.

The polarization modulation properties of the director patterns were studied by Jones calculus. The local Jones matrices M(x,y)50 were obtained by considering the Jones matrix for an oriented (azimuth angle φ(x,y)) wave plate inducing an optical phase change Γ(x,y), where in general Γ was free to take arbitrary and locally varying values

3.2. 18

The Jones electric field vector notation for x-polarized input light is Inline graphic, which describes a linearly polarized plane wave, polarized in the x-direction (Ein(x) = E0e–i(ωt – k0z)x), with angular frequency ω and wave vector k0. The Jones vector of the transmitted light can be obtained from Eout(x,y) = MEin(x)

3.2. 19

The algebraic relations cos(2φ) = 1/2 (ei2sϕ + e–i2sϕ) and Inline graphic can be applied to recast eq 19, where, in the case of splay umbilic, s = 135,36

3.2. 20

where Inline graphic is the Jones matrix for circular polarizations (− for the right handed circular polarized light, + for left handed circular polarized light). This calculation showed34,35,65 that x-polarized input light was partially converted into two contra-circularly polarized helical fields (optical vortices) with the phase factor of ei2sϕ and e–i2sϕ through an equal conversion factor of |1/2 sin(Γ/2)|2.3535 The phase fronts in each optical vortex had a topological charge of |l| = 2s = 2 corresponding with |l|ℏ orbital angular momentum (OAM) per photon,6 which was also suggested for the optical vortex generation in the nematic LC with negative dielectric anisotropy in the literautre.34,35,37,65 In the spin–orbit space S ⊗ L2 with quantum orbital number |l| = 2, the circular polarized vortex photons can be expressed as |−1⟩S ⊗ |+2⟩L2 and |+1⟩S ⊗ |−2⟩L2, respectively, where |−1⟩S and |+1⟩S stand for the right (spin down)- and left (spin up)-circular polarization basis and |+2⟩L2 and |−2⟩L2 for the orbital angular momentum basis with the quantum orbital number ±2, respectively. As a result, the output beam can be characterized as an entangled state of the two vortex photons as (|−1⟩S ⊗ |+2⟩L2 + |+ 1⟩S ⊗|−2⟩L2)/2,34 which appeared as annular-shaped beams if no analyzer was used and as optical brushes if a linear polarizer was employed as the analyzer.34,65 For the case of circularly polarized input light, a similar analytical approach can be employed:35,37,49−51 The input circularly polarized beam (|∓1⟩S) can be converted to a single vortex beam with opposite helicity (|±1⟩S ⊗ |∓2⟩L2).34

A quick check in the amount of momentum carried by the photons shows that if the input beam was linearly polarized, the output total angular momentum per photon is expectedly zero because of the axisymmetry of the system (“q = 1” case). For the case of circularly polarized input beam, the conversion of the spin angular momentum (SAM) to the orbital angular momentum (OAM) happened while the net exchanged momentum is zero.49,51 Therefore, when a photon passes through such a q-plate, regardless of the polarization, total angular momentum is conserved, which indicates zero torque on the matter defect and maintenance of the defect stabilization.

The output intensity patterns in between crossed and parallel polarizers (referred to as π⊥ and π∥, respectively) were obtained from eq 19 as

3.2. 21
3.2. 22

where I0 corresponds to the input intensity. Considering energy conservation, the relation Ioutπ⊥ + Iout = 1 was always maintained, as expected. The output intensity distributions numerically calculated for the umbilic defect (Figure 6) are shown in Figure 8. Patterns of four and eight brushes were seen in between linear polarizers at σ0 = 50 and 76 μC m–2, respectively. The intensity distributions calculated for crossed linear polarizers are shown in Figure 8a for σ0 = 50 μC m–2 and in Figure 8b for σ0 = 76 μC m–2, respectively. The corresponding patterns obtained for parallel linear polarizers are shown in Figure 8c,d. The patterns of OVs were well defined and clearly distinguishable, which are frequently seen35,65 and reported for umbilics:35 the first and second quadruple of OVs (numbered relative to the center of defect) are made from four identical elongated and flattened patterns, respectively, as seen in Figure 8a,c and Figure 8b,d, respectively.

Figure 8.

Figure 8

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Simulated output intensities (at λ = 589 nm) in between (a, b) cross- and (c, d) parallel linear polarizers for the umbilics generated in the LC (MLC-2087) at (a, c) σ0 = 50 μC m–2 and (b, d) σ0 = 76 μC m–2, respectively.

The annular-shaped intensity distributions for the umbilic director reorientations were obtained for completeness: in between crossed circular polarizers, the output intensity was calculated by35

3.2. 23

where σ⊥ indicates crossed circular polarizers. The calculated output patterns are shown (Figure 9). Here, the LC film acts as a spin–orbit coupler (q-plate), which creates a phase singularity along the propagation axis of incident nonsingular light beams, which appeared as a central dark spot. As seen in Figures 8 and 9, the output intensity patterns exhibited a residual hexagonal symmetry because of the numerical calculations in the Cartesian coordinate system, and this issue can be simply avoided by simulating the LC cell in a cylindrical coordinate system.

Figure 9.

Figure 9

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Transmitted intensity profiles (at λ = 589 nm) in between crossed circular polarizers simulated for the umbilics induced in the LC (MLC-2087) corresponding with (a) σ0 = 50 μC m–2 and (b) σ0 = 76 μC m–2, respectively.

In the recent work by Calisto and co-workers,66 the theoretical studies indicate the possibility of generation of optical vortices in a LC media with positive elastic anisotropy (δ > 0), defined as δ = (K1 – K2)/(K1 + K2). In another work,67 it was shown that in the liquid crystal light valves (LCLVs), competition between the forcing of the external electric field induced by the inhomogeneous light profile and the elastic anisotropy can result in the swirling of the vortex arms. Here, although the numerical modeling was made with the assumption of one elastic constant approximation, the model can well describe the structure of umbilics. Since the induced electric field is of the order of 10 kV/m, which prevails phase jump angle in the core of the vortex as the observed structure with no bending and quite stretched in the radial direction in the core region, and characterized straight lines in between linear polarizers are expected (Figure 8).

4. Conclusions

Optically assisted electrical formations of the umbilical defects in the LC with positive dielectric anisotropy are reported. The generation of umbilics was observed in the hybridized LC cells made from a plan-parallel z-cut Fe/LN substrate and an ITO-coated covering glass. The photovoltaic field, induced in the Fe/LN substrate upon light irradiation, has distribution in the LC-filled region as suitable for the formation of the umbilical defects. The intensity profiles in the nematic LC with positive dielectric anisotropy were obtained from simulations based on the Q-tensor method. The local exposure to a tightly focused Gaussian laser beam can lead to the generation of umbilics with four- and eight-brush textures (depending on the induced charge densities in the Fe/LN) in between linear polarizers. Investigating the output intensity profiles suggests that the shape of brushes well resembles the shape of brushes seen in a splay umbilic. Annular-shaped intensity profiles were obtained by investigating the light transmission in between crossed circular polarizers, typically characterizing the output intensity patterns of OVs.

Simulations revealed that the proper electric field (photovoltaic field) distribution in the LC film was responsible for the creation of umbilics. Comparisons of the director field profiles (in three dimensions (3D) and in the transverse plane), as well as the phase change profiles with the corresponding profiles associated with the model umbilics (observed in the nematic LC with negative anisotropy), assure the accuracy of the prediction.

The generation of OVs induced by the umbilics was first discovered from the discussed theoretical studies and simulations. The theory predictions were checked experimentally for a test cell (with the same geometry as discussed) filled with the nematic LC of MLC-2087, and the formation of umbilics inducing OVs were verified.

This finding can lead to new efficient techniques based on the OVs generation in the self-engineered and controllable LC cells with tremendous applications. Thanks to the locally confined topological defects, generation of an array of OVs from a single LC cell is promising. Since the size of the defect core is determined from the full width at half maximum (FWHM) of the laser beam, it enables the generation of OVs with desired structures.

Acknowledgments

The author would like to appreciate Prof. Pawel Pieranski (Laboratoire de Physique des Solides, Université Paris-Sud) for his friendly interest and deep discussions. The author also would like to thank Samuel L. Schafforz (Department of Chemistry, Paderborn University) for his friendly help in providing the first experimental observations of the OVs in between linear polarizers from the test cells filled with the nematic LC with positive dielectric anisotropy (of MLC-2087) and Dr. Alexander Lorenz (Department of Chemistry, Paderborn University) for managing the observation of OVs in between crossed circular polarizers.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.9b03158.

  • Discussion on the charge carrier accumulation in the ITO thin film; photovoltaic electric field responsible for the charge accumilation; numerically obtained electric potential distribution; charge carrier density (PDF)

The research work was partially funded by the US Air Force Office of Scientific Research (AFOSR) through the European Office of Aerospace Research & Development (EOARD) Grant FA9550-18-1-7002 and partially by the German Research Council (DFG) Grant LO 1922/4-1, which are gracefully acknowledged.

The author declares no competing financial interest.

Supplementary Material

ao9b03158_si_001.pdf (172KB, pdf)

References

  1. Boyd R. W.; Padgett M. J.. Quantum Mechanical Properties of Light Fields Carrying Orbital Angular Momentum. In Optics in Our Time; Springer: Cham, 2016; pp 435–454. [Google Scholar]
  2. Bekshaev A. Y.; Vasnetsov M. V.; Soskin M. S. Description of the morphology of optical vortices using the orbital angular momentum and its components. Opt. Spectrosc. 2006, 100, 910–915. 10.1134/S0030400X06060166. [DOI] [Google Scholar]
  3. Pismen L.Vortices in Nonlinear Fields, From Non-Equilibrium Patterns to Cosmic Strings; Oxford University Press, 1999. [Google Scholar]
  4. Bezryadina A.; Neshev D. N.; Desyatnikov A. S.; Young J.; Chen Z.; Kivshar Y. S. Observation of topological transformations of optical vortices in two-dimensional photonic lattices. Opt. Express 2006, 14, 8317–8327. 10.1364/OE.14.008317. [DOI] [PubMed] [Google Scholar]
  5. Maleev I. D.; Swartzlander G. A. Composite optical vortices. J. Opt. Soc. Am. B 2003, 20, 1169–1176. 10.1364/JOSAB.20.001169. [DOI] [Google Scholar]
  6. Yao A. M.; Padgett M. J. Orbital angular momentum: origins, behavior and applications. Adv. Opt. Photonics 2011, 3, 161–204. 10.1364/AOP.3.000161. [DOI] [Google Scholar]
  7. Allen L.; Beijersbergen M. W.; Spreeuw R. J. C.; Woerdman J. P. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Phys. Rev. A 1992, 45, 8185 10.1103/PhysRevA.45.8185. [DOI] [PubMed] [Google Scholar]
  8. Nye J. F.Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations; CRC Press, 1999. [Google Scholar]
  9. Kivshar Y. S.; Luther-Davies B. Dark optical solitons: physics and applications. Phys. Rep. 1998, 298, 81–197. 10.1016/S0370-1573(97)00073-2. [DOI] [Google Scholar]
  10. Desyatnikov A. S.; Torner L.; Kivshar Y. S.. Optical vortices and vortex solitons, arXiv:nlin/0501026; 2005.
  11. Foo G.; Palacios D. M.; Swartzlander G. A. Optical vortex coronagraph. Opt. Lett. 2005, 30, 3308–3310. 10.1364/OL.30.003308. [DOI] [PubMed] [Google Scholar]
  12. Swartzlander G. A. Peering into darkness with a vortex spatial filter. Opt. Lett. 2001, 26, 497–499. 10.1364/OL.26.000497. [DOI] [PubMed] [Google Scholar]
  13. Mair A.; Vaziri A.; Weihs G.; Zeilinger A. Entanglement of the orbital angular momentum states of photons. Nature 2001, 412, 313. 10.1038/35085529. [DOI] [PubMed] [Google Scholar]
  14. Wang J.; Yang J. Y.; Fazal I. M.; Ahmed N.; Yan Y.; Huang H.; Ren Y.; Yue Y.; Dolinar S.; Tur M.; Willner A. E. Terabit free-space data transmission employing orbital angular momentum multiplexing. Nat. Photonics 2012, 6, 488. 10.1038/nphoton.2012.138. [DOI] [Google Scholar]
  15. Gibson G.; Courtial J.; Padgett M. J.; Vasnetsov M.; Pas’ko V.; Barnett S. M.; Franke-Arnold S. Free-space information transfer using light beams carrying orbital angular momentum. Opt. Express 2004, 12, 5448–5456. 10.1364/OPEX.12.005448. [DOI] [PubMed] [Google Scholar]
  16. Grier D. G. A revolution in optical manipulation. Nature 2003, 424, 810. 10.1038/nature01935. [DOI] [PubMed] [Google Scholar]
  17. Fazal F. M.; Block S. M. Optical tweezers study life under tension. Nat. Photonics 2011, 5, 318. 10.1038/nphoton.2011.100. [DOI] [PMC free article] [PubMed] [Google Scholar]
  18. Jeffries G. D.; Edgar J. S.; Zhao Y.; Shelby J. P.; Fong C.; Chiu D. T. Using polarization-shaped optical vortex traps for single-cell nanosurgery. Nano Lett. 2007, 7, 415–420. 10.1021/nl0626784. [DOI] [PMC free article] [PubMed] [Google Scholar]
  19. Farahani J. N.; Schibler M. J.; Bentolila L. A. Stimulated emission depletion (STED) microscopy: from theory to practice. Microsc.: Sci. Technol. Appl. Educ. 2010, 2, 1539–1547. [Google Scholar]
  20. Wang S.; Zhang S. L.; Li P.; Hao M. H.; Yang H. M.; Xie J.; Feng G.-Y.; Zhou S. H. Generation of wavelength-and OAM-tunable vortex beam at low threshold. Opt. Express 2018, 26, 18164–18170. 10.1364/OE.26.018164. [DOI] [PubMed] [Google Scholar]
  21. Turnbull G. A.; Robertson D. A.; Smith G. M.; Allen L.; Padgett M. J. The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate. Opt. Commun. 1996, 127, 183–188. 10.1016/0030-4018(96)00070-3. [DOI] [Google Scholar]
  22. Oemrawsingh S. S. R.; Van Houwelingen J. A. W.; Eliel E. R.; Woerdman J. P.; Verstegen E. J. K.; Kloosterboer J. G.; ’t Hooft G. W. Production and characterization of spiral phase plates for optical wavelengths. Appl. Opt. 2004, 43, 688–694. 10.1364/AO.43.000688. [DOI] [PubMed] [Google Scholar]
  23. Sueda K.; Miyaji G.; Miyanaga N.; Nakatsuka M. Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses. Opt. Express 2004, 12, 3548–3553. 10.1364/OPEX.12.003548. [DOI] [PubMed] [Google Scholar]
  24. Tyson R. K.; Scipioni M.; Viegas J. Generation of an optical vortex with a segmented deformable mirror. Appl. Opt. 2008, 47, 6300–6306. 10.1364/AO.47.006300. [DOI] [PubMed] [Google Scholar]
  25. Bazhenov V. I.; Vasnetsov M. V.; Soskin M. S. Laser beams with wave front screw dislocations. Pisma Zh. Eksp. Teor. Fiz. 1990, 52, 1037–1039. [Google Scholar]
  26. Bazhenov V. Y.; Soskin M. S.; Vasnetsov M. V. Screw dislocations in light wavefronts. J. Mod. Opt. 1992, 39, 985–990. 10.1080/09500349214551011. [DOI] [Google Scholar]
  27. Heckenberg N. R.; McDuff R.; Smith C. P.; Rubinsztein-Dunlop H.; Wegener M. J. Laser beams with phase singularities. Opt. Quantum Electron. 1992, 24, S951–S962. 10.1007/BF01588597. [DOI] [Google Scholar]
  28. Heckenberg N. R.; McDuff R.; Smith C. P.; White A. G. Generation of optical phase singularities by computer-generated holograms. Opt. Lett. 1992, 17, 221–223. 10.1364/OL.17.000221. [DOI] [PubMed] [Google Scholar]
  29. Matsumoto N.; Ando T.; Inoue T.; Ohtake Y.; Fukuchi N.; Hara T. Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators. J. Opt. Soc. Am. A 2008, 25, 1642–1651. 10.1364/JOSAA.25.001642. [DOI] [PubMed] [Google Scholar]
  30. Aleksanyan A.; Brasselet E. Vortex coronagraphy from self-engineered liquid crystal spin-orbit masks. Opt. Lett. 2016, 41, 5234–5237. 10.1364/OL.41.005234. [DOI] [PubMed] [Google Scholar]
  31. Soskin M. S.; Vasnetsov M. V. Singular optics. Prog. Opt. 2001, 42, 219–276. 10.1016/s0079-6638(01)80018-4. [DOI] [Google Scholar]
  32. Marrucci L.; Manzo C.; Paparo D. Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media. Phys. Rev. Lett. 2006, 96, 163905 10.1103/PhysRevLett.96.163905. [DOI] [PubMed] [Google Scholar]
  33. Brasselet E.; Murazawa N.; Misawa H.; Juodkazis S. Optical vortices from liquid crystal droplets. Phys. Rev. Lett. 2009, 103, 103903 10.1103/PhysRevLett.103.103903. [DOI] [PubMed] [Google Scholar]
  34. Brasselet E.; Loussert C. Electrically controlled topological defects in liquid crystals as tunable spin-orbit encoders for photons. Opt. Lett. 2011, 36, 719–721. 10.1364/OL.36.000719. [DOI] [PubMed] [Google Scholar]
  35. Brasselet E. Tunable optical vortex arrays from a single nematic topological defect. Phys. Rev. Lett. 2012, 108, 087801 10.1103/PhysRevLett.108.087801. [DOI] [PubMed] [Google Scholar]
  36. Rapini A. Statistic properties and shear-induced displacements. J. Math. Phys. 1973, 34, 629–633. 10.1051/jphys:01973003407062900. [DOI] [Google Scholar]
  37. Barboza R.; Bortolozzo U.; Clerc M. G.; Residori S.; Vidal-Henriquez E. Optical vortex induction via light–matter interaction in liquid-crystal media. Adv. Opt. Photonics 2015, 7, 635–683. 10.1364/AOP.7.000635. [DOI] [Google Scholar]
  38. Habibpourmoghadam A.; Jiao L.; Omairat F.; Evans D. R.; Lucchetti L.; Reshetnyak V.; Lorenz A. In Confined Photovoltaic Fields in a Photo-Responsive Liquid Crystal Test Cell. Proceedings of SPIE 10361, Liquid Crystals XXI. 2017; pp 1036112.
  39. Kravets N.; Podoliak N.; Kaczmarek M.; Brasselet E. Self-induced liquid crystal q-plate by photoelectric interface activation. Appl. Phys. Lett. 2019, 114, 061101 10.1063/1.5082598. [DOI] [Google Scholar]
  40. Habibpourmoghadam A.; Jiao L.; Reshetnyak V.; Evans D. R.; Lorenz A. Optical manipulation and defect creation in a liquid crystal on a photoresponsive surface. Phys. Rev. E 2017, 96, 022701 10.1103/PhysRevE.96.022701. [DOI] [PubMed] [Google Scholar]
  41. Lucchetti L.; Kushnir K.; Zaltron A.; Simoni F. Light controlled phase shifter for optofluidics. Opt. Lett. 2016, 41, 333–335. 10.1364/OL.41.000333. [DOI] [PubMed] [Google Scholar]
  42. Habibpourmoghadam A.; Lucchetti L.; Evans D. R.; Reshetnyak V. Y.; Omairat F.; Schafforz S. L.; Lorenz A. Laser-induced erasable patterns in a N* liquid crystal on an iron doped lithium niobate surface. Opt. Express 2017, 25, 26148 10.1364/OE.25.026148. [DOI] [PubMed] [Google Scholar]
  43. Barboza R.; Bortolozzo U.; Assanto G.; Vidal-Henriquez E.; Clerc M. G.; Residori S. Vortex induction via anisotropy stabilized light-matter interaction. Phys. Rev. Lett. 2012, 109, 143901 10.1103/PhysRevLett.109.143901. [DOI] [PubMed] [Google Scholar]
  44. Brasselet E. Singular optical manipulation of birefringent elastic media using nonsingular beams. Opt. Lett. 2009, 34, 3229–3231. 10.1364/OL.34.003229. [DOI] [PubMed] [Google Scholar]
  45. Brasselet E. Singular optical reordering of liquid crystals using Gaussian beams. J. Opt. 2010, 12, 124005 10.1088/2040-8978/12/12/124005. [DOI] [Google Scholar]
  46. Hamberg I.; Granqvist C. G. Evaporated Sn-doped In2O3 films: Basic optical properties and applications to energy-efficient windows. J. Appl. Phys. 1986, 60, R123–R160. 10.1063/1.337534. [DOI] [Google Scholar]
  47. Gao Q.; Li E.; Wang A. X. Comparative analysis of transparent conductive oxide electro-absorption modulators. Opt. Mater. Express 2018, 8, 2850–2862. 10.1364/OME.8.002850. [DOI] [Google Scholar]
  48. Yi F.; Shim E.; Zhu A. Y.; Zhu H.; Reed J. C.; Cubukcu E. Voltage tuning of plasmonic absorbers by indium tin oxide. Appl. Phys. Lett. 2013, 102, 221102 10.1063/1.4809516. [DOI] [Google Scholar]
  49. Barboza R.; Bortolozzo U.; Assanto G.; Residori S. Optical vortex generation in nematic liquid crystal light valves. Mol. Cryst. Liq. Cryst. 2013, 572, 24–30. 10.1080/15421406.2012.763206. [DOI] [Google Scholar]
  50. Marrucci L. Generation of helical modes of light by spin-to-orbital angular momentum conversion in inhomogeneous liquid crystals. Mol. Cryst. Liq. Cryst. 2008, 488, 148–162. 10.1080/15421400802240524. [DOI] [Google Scholar]
  51. Barboza R.; Bortolozzo U.; Assanto G.; Vidal-Henriquez E.; Clerc M. G.; Residori S. Harnessing optical vortex lattices in nematic liquid crystals. Phys. Rev. Lett. 2013, 111, 093902 10.1103/PhysRevLett.111.093902. [DOI] [PubMed] [Google Scholar]
  52. Lee G.-D.; Bos P. J.; Ahn S. H.; Kim K. H. Fast Q-tensor method for modeling the dynamics of defects in a liquid crystal director field. Phys. Rev. E 2003, 67, 041715 10.1103/PhysRevE.67.041715. [DOI] [PubMed] [Google Scholar]
  53. James R.; Willman E.; Fernández F. A.; Day S. E. Finite-element modeling of liquid-crystal hydrodynamics with a variable degree of order. IEEE Trans. Electron Dev. 2006, 53, 1575–1582. 10.1109/TED.2006.876039. [DOI] [Google Scholar]
  54. Tovkach O. M.; Conklin C.; Calderer M. C.; Golovaty D.; Lavrentovich O. D.; Viñals J.; Walkington N. J. Q-tensor model for electrokinetics in nematic liquid crystals. Phys. Rev. Fluids 2017, 2, 053302 10.1103/PhysRevFluids.2.053302. [DOI] [Google Scholar]
  55. Anderson J. E.; Waston P.; Bos P. J. Comparisons of the vector method and tensor method for simulating liquid crystal devices. Liq. Cryst. 2001, 28, 109–115. 10.1080/02678290010001455. [DOI] [Google Scholar]
  56. Kralj S.; Virga E. G.; Zumer S. Biaxial torus around nematic point defects. Phys. Rev. E 1999, 60, 1858–1865. 10.1103/PhysRevE.60.1858. [DOI] [PubMed] [Google Scholar]
  57. Mottram N.; Newton C. J. P.. Introduction to Q-tensor theory, arXiv: 1409.3542 [Cond-Mat. Soft]; 2014.
  58. Kralj S.; Majumdar A. Order reconstruction patterns in nematic liquid crystal wells. Proc. R. Soc. A 2014, 470, 20140276 10.1098/rspa.2014.0276. [DOI] [Google Scholar]
  59. Sze S. M.; Ng K. K.. Physics of Semiconductor Devices; John Wiley & Sons, 2006; pp 197–200. [Google Scholar]
  60. Feigenbaum E.; Diest K.; Atwater H. A. Unity-order index change in transparent conducting oxides at visible frequencies. Nano Lett. 2010, 10, 2111–2116. 10.1021/nl1006307. [DOI] [PubMed] [Google Scholar]
  61. Sorger V. J.; Lanzillotti-Kimura N. D.; Ma R. M.; Zhang X. Ultra-compact silicon nanophotonic modulator with broadband response. Nanophotonics 2012, 1, 17–22. 10.1515/nanoph-2012-0009. [DOI] [Google Scholar]
  62. Melikyan A.; Lindenmann N.; Walheim S.; Leufke P. M.; Ulrich S.; Ye J.; Vincze P.; Hahn H.; Schimmel T.; Koos C.; Freude W.; Leuthold J. Surface plasmon polariton absorption modulator. Opt. Express 2011, 19, 8855–8869. 10.1364/OE.19.008855. [DOI] [PubMed] [Google Scholar]
  63. Pavlin J.; Vaupotic N.; Cepic M. Direction dependency of extraordinary refraction index in uniaxial nematic liquid crystals. Eur. J. Phys. 2013, 34, 331–344. 10.1088/0143-0807/34/2/331. [DOI] [Google Scholar]
  64. Pereira A. E. C.; Rosato A. Transmission of nematic liquid crystals in electric fields. Rev. Bras. Fis. 1975, 5, 237–241. [Google Scholar]
  65. Brasselet E. Tunable high-resolution macroscopic self-engineered geometric phase optical elements. Phys. Rev. Lett. 2018, 121, 033901 10.1103/PhysRevLett.121.033901. [DOI] [PubMed] [Google Scholar]
  66. Calisto E.; Clerc M. G.; Kowalczyk M.; Smyrnelis P. On the origin of the optical vortex lattices in a nematic liquid crystal light valve. Opt. Lett. 2019, 44, 2947–2950. 10.1364/OL.44.002947. [DOI] [PubMed] [Google Scholar]
  67. Barboza R.; Bortolozzo U.; Clerc M. G.; Residori S.; Vidal-Henriquez E. Light–matter interaction induces a single positive vortex with swirling arms. Philos. Trans. R. Soc., A 2014, 372, 20140019 10.1098/rsta.2014.0019. [DOI] [PubMed] [Google Scholar]

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