1. Introduction

Many photonic elements in use today are optically birefringent, especially, those for altering or probing the polarization [1]. Non-linear crystals possess birefringence that is used to phase match and increase the efficiency of non-linear processes, such as second-harmonic generation or spontaneous parametric down-conversion [1,2]. Birefringence is the prominent property of liquid crystals (LCs), widely used today in optical and display applications, as it can be manipulated by external fields on diverse length scales, from tens of nano- to tens of micrometers as well as diverse time scales, from seconds to nanoseconds and even picoseconds [3–5]. This broad spatial and temporal tuneability makes LCs important building blocks of (micro)photonic circuits. As a consequence, a diverse range of micro-photonic applications using LCs have already been reported, such as LC lasers [6, 7], tunable microresonators [8], tunable lenses [9–11], tunable fiber-couplers [12], tunable fibers [13], tunable waveplates [14], optical sensors [15], curved optical solitons [16], modulators and switches [17,18]. A liquid crystal waveguide was realized using a silicon channel filled with LC in which the number of possible guided modes can be controlled using an electric field [19].

Nematic LCs or nematics are liquids in which the constituent parts, molecules of elongated shape, have a long-range orientational order but no long-range positional order. Nematic order is typically characterized by the average local orientation vector n called the director and a scalar nematic degree of order S. Alternatively, the two can be combined to form a single tensorial order parameter Q_ij, that is further related to generally anisotropic quantities such as electric or magnetic susceptibility [4]. Birefringence of nematics stems from orientationally dependent response of single molecules to external electric field at optical frequencies. In bulk, uniaxial nematic materials are thus distinguished by two different indices of refraction. The direction of n with respect to the light wavevector k of light, where |k| = 2π/λ, and its polarization E determine the two distinctive modes of light determined by the two refractive indices [4].

In chiral nematics, nematic molecules are chiral in nature or a chiral dopant is added to a non-chiral nematic [4]. They can exhibit a cholesteric phase, which is characterized by the director describing a helix with a finite pitch along the z-direction [4, 20]. For materials or regimes of high chirality i.e. pitch in the order of a few hundred nanometers, blue phases (BPs), can occur in a distinct temperature range and are characterized by effective twisting of the director in multiple directions, especially along the so-called double-twist cylinders (DTCs) (see Fig. 1). Indeed, these double-twist cylinders stack into structures with face-centered cubic or simple cubic lattice symmetry forming blue phase I and blue phase II, respectively. [4,21]. A variety of other chiral vortex and solitonic structures can form as well [22, 23]. In principle arbitrary birefringent profiles can also be created by in-situ laser polymerization [24]. In this work, we are interested in the photonic properties of a single double-twist cylinder, inside which the director is parallel to the cylinder axis at its center and then rotate uniformly for angle ϕ in the radial direction away from the axis.

 

Fig. 1 Double-twist cylinder as a photonic element. Azimuthally polarized light (electric field E_opt direction is illustrated with red arrows) propagating along the of double-twist birefringent profile (optical axis is represented by colored ellipsoids) observes radially dependent refractive index which can result in lensing and waveguiding.

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Vectorial beams of light have a cross-sectionally varying local electric (and magnetic) field direction i.e. polarization. In case of cylindrical symmetry, they can be formally obtained as solutions of the paraxial vector wave equation where two opposite cases arise. The first is the radially polarized, with electric field oriented in the radial direction and the second is the azimuthally polarized beam, with electric field pointed in in the perpendicular, azimuthal, direction at every point of the beam cross section [25]. An inherent property of radially polarized beams on focussing are the radially symmetric longitudinal electric field components that allow for beam waists below the standard Gaussian limit [26]. This property is used in atom and particle trapping [26], laser cutting [27], optical microscopy [28] and even for charged particle acceleration [29]. Vectorial beams can be generated from linearly polarized beams using conic Brewster prisms [30], few-mode optical fibers [31] or by special LC phase modulators [32]. With relation to liquid crystals, recent simulations show that vectorial beams could be generated using nematic defect lines [33]. It was reported that due to the symmetry matching of the radially polarized beam and radially escaped nematic profile, such beams experience a spatially variant refractive index that can lead to lensing and waveguiding [34]. Specifically radially escaped and radial birefringent profiles, were shown to perform as photonic waveguides, considering these inhomogeneous profiles surrounded by isotropic claddings of lower index of refraction [35–38] or by enclosing them inside the fiber cladding [39].

In this work, we report that birefringent double-twist cylinder profiles, the common birefringent structures in chiral or blue phase nematics, can perform as waveguiding or lensing photonic elements for azimuthally polarized beams. Effective refractive index for an azimuthally polarized beam inside such profile is always extraordinary, but the angle between the optical axis and local direction of polarization, changes continuously and symmetrically in the azimuthal direction, causing the light to effectively experience ordinary refractive index at the double-twist cylinder axis and extraordinary index at the profile edge. If material birefringence is assumed negative (positive), refractive index is greater (lower) at cylinder axis and lower (greater) at outer parts of the cylinder causing lensing effects on the incident beam. Specifically, we describe the lensing effects of double-twist profile in case of negative birefringence and show the role of input Laguerre-Gaussian beam width and especially polarization. We then extend the lensing effects to waveguiding and report on the main mechanism for losses occuring on waveguiding in double twist cylinders: change of polarization at the start of the nematic profile and outward pointing in-plane components of the Poynting vector.

2. Numerical method and implementation

Custom-developed finite-difference time-domain (FDTD) method was used for modelling flow of light through the birefringent double-twist nematic structure. This explicit method computes time evolution of electric and magnetic fields E and H using the Maxwell’s equations [40]

The components of the dielectric tensor ε_ij at a certain coordinate is connected to the local order parameter tensor Q_ij through the following relation

FDTD method is highly parallelizable and was implemented for calculations on graphics processing units (GPUs), which significantly reduces the computational time when dealing with large systems. A typical run involving 50 million data points takes around 8 hours on a machine with four Nvidia Titan GPUs. The simulation box is taken to be three-dimensional with typical size equal to x × y × z = 3.6 μm × 3.6 μm × 30 μm for lensing and a few times longer for waveguiding with the beam propagating mainly along the z direction. Resolution is chosen to be 30 nm/voxel. Simulation box sizes and resolution were optimized and chosen in a way that waveguides could be elongated as much as is possible with GPU memory restrictions. Two additional layers around the simulation box, with thicknesses of couple voxels each, are used to introduce external light sources (the incoming beam) and at other regions that any light reaching the boundary can radiate out of the simulation box, as implemented by the perfectly-matched layer (PML) [47]. We should though emphasize, that essentially all mode intensity (except at the beam source) is far from the boundaries as it is sustained by the birefringent profile and not the simulation box boundary conditions. Light and material parameters used in our simulations were chosen to optimize lensing and waveguiding regimes. The wavelength λ = 390 nm of input light, was chosen as it provides suitably low divergence inside the waveguides and lenses that are of ∼ μm size. Input beam was azimuthally polarized with Laguerre-Gaussian shape but without phase dependence in the cross-section. Such a beam was chosen as it is the most basic realizable beam with azimuthal polarization. Beam half-width was typically ∼ 1 μm, while the half-width of the double-twist profile was ∼ 2 μm for lenses and ∼ 1 μm for waveguides. Values of refractive indices n_e and n_o were taken to be 1.45 and 1.50, respectively. Nematic birefringence parameter, the difference between molecular extraordinary and ordinary refractive indices, was Δn = n_e − n_o = −0.05, where we note, that although possible [48], it is not a common value of birefringence at optical frequencies for nematics, as it is usually positive. No light absorption mechanism was introduced into the simulations. Also no dynamic effect, such as thermal fluctuations of the director field were considered. In real systems, further light losses would originate from liquid crystal light absorption, which is usually small in the visible range, as well as from light scattering from the director fluctuations. Out of the visible range, absorption properties are frequently strongly material dependent.

3. Results

3.1. Lensing

Spatially varying optical axis in a cross-section of a double-twist cylinder (see Fig. 1) provides a refractive index profile that can lead to polarization dependent lensing and partial waveguiding. It is particularly interesting when we consider azimuthally polarized beams and negative nematic birefringence, which we show leads to convergent lensing and long-range waveguiding. Azimuthally polarized beam is sent along a birefringent double-twist profile segment of thickness d and lensing emerges.

For lensing on birefringent profiles the profile thickness d and the profile of the birefringence Δn(r) = n_e(r) − n_o determine the phase retardation variation with radial distance. In a double-twist birefringent profile azimuthally polarized light observes ordinary refractive index n_o at the axis of the profile and n_e near its edges. Therefore Δn < 0 gives a focussing lens with f < 0. In Figs. 2(A)–2(D) light intensity profiles are shown. The beam incoming from the left side and travelling toward the right, reaches a beam waist of w₀ as it enters the birefringent profile. Lensing starts after the azimuthally polarized beam enters the double-twist profile. The focal spot size and the longitudinal position of the focus is determined by the ratio between the entry beam width and the profile width w₀/R. In Figs. 2(A)–2(D) the profile width 2R was kept constant and the input beam half-width w₀ was varied: w₀/R values are shown in Fig. 2(E) where longitudinal half-width profiles for cases A–D are plotted.

 

Fig. 2 Lensing on short segments of a double-twist nematic profile for different input beam widths. (A–D) Plots of light intensity at the central plane of the beam. Lensing of azimuthally polarized input beams with different input beam waists on identical double-twist profiles results in lensing with different numerical apertures. (E) Beam width profiles along the beam path for different input beam widths.

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Double-twist cylinders perform as polarization dependent lenses. Orthogonal to the azimuthal is the radial polarization for which polarization is radially directed at every point of the cross-section. Through superposition of polarization states, any other axially symmetric polarization states can be devised [25]. Figure 3 shows propagation of three beams with distinct vectorial polarizations-radial, azimuthal and their combination-beams through double-twist profiles. Clearly, the simulated light profiles show that lensing is optimal when beam is azimuthally polarized and non-existent when the input polarization is radial, with obvious differences in the width of the beam profiles.

 

Fig. 3 Lensing in birefringent double-twist cylinders for differently polarized input beams. In case of azimuthally polarized beam (A) lensing occurs, since the light experiences radially dependent refractive index. In case of radially polarized beam (B) the refractive index is uniform and no lensing occurs. In the intermediate case of radially-azimuthally polarized beam (C) only only partial lensing occurs. Double-twist profiles of the optical axis variation are shown in red, with green dashed line indicating the end of the profile region.

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A known property of radially polarized beams is the onset of a strong longitudinal E component when focussing the beam close to the width determined by the resolution limit [25]. In a symmetric case of azimuthal polarization, it is the magnetic field H that is radially oriented; therefore the magnetic field obtains a stronger longitudinal component H_z. This is shown in Fig. 4(A), where the beam entered the profile at the coordinate z₁, reached focus at z₂ and then expanded towards z₃ as marked in the subfigure below. In three smaller subfigures above, the longitudinal magnetic field component |H_z|² is shown in the cross-section of the beam at the three respective coordinates. We see that the longitudinal magnetic field component |H_z|² is strongest at the focal point and at the centre of the beam cross-section (r = 0). A smaller rise for the longitudinal amplitude of the electric field |E_z|²(r = 0, z) can be observed upon lensing as shown in Fig. 4(B) which occurs as a result of a change in the beam polarization state and decreases lensing efficiency and causes waveguiding losses. We will return to this point in more detail below.

 

Fig. 4 Longitudinal component of the magnetic field of azimuthally polarized beam. Lensing of azimuthally polarized beams results in occurence of strong longitudinal magnetic field component in the center of the beam. (A) Color plots of magnetic field component H_z in the cross section of the beam in three different positions along the beam path, z₁, z₂, z₃ (marked with white dashed lines on the lower plot) are shown, indicating that H_z is greatest at the centre of the beam cross-section at the beam waist. (B) Square of absolute amplitudes of longitudinal components H_z and E_z in the beam centre, marked with yellow line on the beam yz cross-section plot.

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3.2. Waveguiding

Lensing can be extended to light guiding inside a double-twist cylinder for an incoming azimuthally polarized beam. To test, whether such system can support propagating light eigenmodes, we performed simulations of beam propagation inside a double-twist nematic profiles with lengths of several hundred wavelengths. Laguerre-Gaussian beams with azimuthal polarization is used as a sensible initial guess of a possible eigenmode in such a waveguide and also were chosen as they are well experimentally realizable [25].

Waveguding of double-twist cylinders is presented in Fig. 5. Figure 5(A) shows longitudinal light intensity profiles for input azimuthally polarized Laguerre-Gaussian beams with different input beam widths. Simillarly as before, the coupling efficiency (i.e. how much light remains inside the profile) depends centrally on the ratio between the input beam waist 2w₀ and width of the double-twist cylinder 2R. In this case, the profile width was 6λ as shown in Fig. 5(A) and the optimal coupling regime occurs when w₀/R is slightly below one-half (w₀/R ∼ 0.4). When r is lower (w₀/R ≲ 0.25) or larger (w₀/R ≳ 0.7) initial losses become greater and light that is coupled into the fiber experiences large intensity fluctuations, that result from longitudinal focussing and defocussing of the beam inside the profile – see especially the first and last example in Fig. 5(A). Graph of relative beam power as a function of the travelled length in Fig. 5(C) shows that the azimuthally polarized Laguerre-Gaussian beams first experience losses as a consequence of a mismatch between their widths and the profile width. But additionally losses are observed even as the beam is further propagated inside the profile.

 

Fig. 5 Waveguiding in a double-twist cylinder birefringent profile. (A) Azimuthally polarized beams propagating inside the double twist cylinders for different input beam widths 2w₀. (B) Transversal components of the electric field E of the guided beam at three different representative positions along the beam path, denoted by green arrows. They indicate partial transformation of the initial azimuthal polarization to a polarization that is partially radial. (C) Beam power in units of initial beam power P₀ along the double twist cylinder for different input beam widths 2w₀. (D, E) Components of the Poynting vector averaged along the beam. Note the emergence of the radially twisted component of S in the transversal plane.

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For in-depth investigation of these losses, polarization of the beam along its path was observed. What we found is that initial azimuthal polarization acquires some degree of the radial polarization immediately after the beam enters the double-twist profile. We assumed in simulations, that azimuthally polarized beam entered the double-twist profile from an isotropic medium of refractive index n_iso = 1.475 – value chosen to reduce reflections at entry. At this boundary between two dielectric materials, boundary conditions for electric displacement field D and magnetic field B (not that different to H) need to be considered. As the longitudinal component of electric field density D_z needs to be preserved at the boundary $D_z^{\mathit{iso}}=D_z^{\mathit{DTC}}$. We take that in the initial isotropic layer the light polarization field E = E(r)e_ϕ is purely azimuthal but, using D = ε₀ε̳(r)E, it follows from the boundary condition that the longitudinal component of electric field intensity is non-zero in the double-twist profile

Losses can be alternatively described by the direction of energy flow inside a double-twist cylinder. Directional energy flux is represented with a Poynting vector defined as $\mathbf{S}=\frac{1}{2}\mathbf{E}\times \mathbf{H}$. Figures 5(D) and 5(E) show the profile of the transversal and longitudinal components of the Poynting vector averaged along the waveguide at a given moment. Clearly, the Poynting vector points mainly along the z direction; however, it has also a non-zero, outward pointing and twisted component in the x y plane. This is a direct consequence of the beam polarization changes described above.

To generalise, we have shown that, effectively, double-twist cylinders can perform as waveguides based on their birefringent profile for azimuthally polarized beams over lengths of several tens of micrometres with leakage losses of roughly 0.2% = −0.0087dB per one wavelength of travelled distance. The calculated values of the Poynting vectors can be used to estimate the magnitude of the radiated energy from such a waveguide. The radial component of the Poynting vector averaged along the waveguide (Fig. (5E)) is of the order of magnitude 3 × 10⁻³W/μm², giving effective radiated power per unit length of the waveguide ∼ 5 × 10⁻³W/μm ∼ 1 × 10⁻³W/λ. Note that these values agree within the order of magnitude with the energy losses shown in Fig. (5C).

3.3. Waveguiding along the curved double-twist cylinders

We explore light propagation along curved double-twist profiles as a part of a circle with curvature radius R_c, where nematic director at the centre of the curved profile follows its curvature. To characterise the losses due to curvature of the double twist cylinder, we calculate the ratio between power at a fixed travelled length of l = 60λ inside a curved profile and the input power P₀ for different curvature radii R_c. The ratio between the input beam waist at the entry and the diameter of the double-twist profile is set to 0.6, being within the optimal coupling regime (optimal beam width to profile width ratio). Note that setting non-optimal w₀/R will lead to larger initial losses and possible intensity and E_z oscillations.

In Fig. 6(B) dependence of the ratio between the output and input power on the curvature radius R_c is shown. We find that, losses increase notably after the curvature radius is smaller than a certain value, of the critical bend radius, which for the considered double-twist cylinder waveguides is of the order of ∼ 300λ. In standard (isotropic) waveguides critical radius is usually dependent on the numerical aperture, and thus on the difference of the refractive indices, of the fiber and cladding and, in case of multimode fibers, also on the propagating mode [49]. The field distribution inside curved waveguides also usually changes its shape due to curvature – it is forced toward the outer wall and some light leaves the waveguide. This effect can be also seen in Fig. 6(A) , where representative light intensity plots of an initially azimuthally polarized Laguerre-Gaussian beams inside a curved double-twist profile are shown.

 

Fig. 6 Waveguiding in curved birefringent double-twist profiles. (A) Light intensity of the azimuthally polarized beam inside a double-twist cylinder nematic structure curved with a radius of curvature R_c. For smaller radii of curvature substantial bending losses occur. (B) Ratio between the input and output beam power after travelling a fixed length of 60λ inside a curved nematic profile as a function of the radius of curvature R_c.

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4. Conclusion

In conclusion, we have shown that the birefringent profile of double-twist cylinders can perform as polarization dependent microlenses and as single-mode waveguides. Double-twist profile is transversed with an azimuthally polarized beam, for which the effective refractive index profile causes radially variable phase retardation. We have shown that in case of negative (positive) birefringence the double-twist cylinder profile of constant thickness can act as a converging (diverging) lens. Influence of input beam width and polarization on lensing performance were also demonstrated, showing that width of the input beam relative to the profile width, determines the beam width in focus and that lensing is most efficient in case of purely azimuthal input polarization.

Lensing was generalized into waveguiding of light inside the double-twist cylinders. Coupling efficiency was shown to be strongly dependent of the ratio r between the light-beam width (2w₀) and double-twist profile width (2R), showing effectively optimal coupling when r ∼ 0.9. The efficiency of such waveguides was considered by investigating losses along the waveguide, and the waveguide was found to be lossy with intensity losses of ∼ −0.0087dB per travelled wavelength. Losses were atributed to the emergence of the longitudinal component of the beam electric field, which appears upon entry of the azimuthally polarized beam into the birefringent profile of the double twist-cylinder. Potential losses from curvature of the double-twist cylinder profile are also determined, showing that losses increase significantly as curvature radius R_c drops below the critical value of waveguide curvature R_c ∼ 300λ.

Finally, the photonic system of a double-twist cylinder described here is an example of micro-scale polarization selective optical element. Due to liquid nature and good tunability with external fields –including external or self-light fields– such birefringent elements are interesting candidates for new generations of photonic circuits.