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Abstract
By incorporating a liquid crystal layer into waveguides created through direct laser writing, a tunable waveplate in a fused silica waveguide is successfully realized. In the experiment, a voltage is applied to the birefringent liquid crystal layer, inducing changes in the internal molecular arrangement of the nematic liquid crystal (NLC). Owing to the voltage-induced rotation of the NLC molecules, the polarization of the transmitted light becomes dependent on the applied external electric field. This novel approach to polarization manipulation paves the way for a new family of reconfigurable devices and complex photonic circuits based upon femtosecond-written waveguides.
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1. Introduction
The technology known as femtosecond laser direct writing (FLDW) enables the fabrication of waveguides within a glass chip in a full 3D manner [1–3], allowing for example the observation of a plethora of topological effects [4]. With this technology, light can be guided along specific 3D paths within the chip. Moreover, FLDW technology can also be used to fabricate several optical components, like e.g. directional couplers, within the same chip [2,5]. The incorporation of various optical components enables the integration of various optical functions into a single glass chip, resulting in integrated photonic circuits exploiting the whole volume of the chip [6–10]. As a result, traditional bulk optical assemblies can be significantly miniaturized.
However, a major drawback of FLDW circuits is reconfigurability, that is, once a structure is written into a chip, it cannot be modified without rewriting the entire optical circuit. The lack of modulators is particularly problematic when the integrated circuit is to be used to perform an experiment in which a parameter must be systematically changed, a signal needs to be processed, or even when small but unavoidable imperfections need to be compensated for. In recent years, some proposals have emerged for incorporating tunability into FLDW circuits. The most successful is to integrate a resistor acting as a heating element, in turn inducing a phase shift through the thermo-optical effect [11–13]. In other publications, the Kerr effect was utilized to adjust nonlinearly the coupling ratio in a directional coupler [14,15]. However, none of these works address the aspect of customizing the polarization according to one’s preferences. Therefore, the aim of this publication is to present a novel approach for incorporating adjustable waveplates into optical circuits inscribed using the FLDW technology. For this purpose, a layer of liquid crystal is embedded into the waveguide. In full analogy with optical modulators based upon NLC, the orientation of the liquid crystal is controlled by the applied voltage, in turn inducing a bias-dependent phase retardation between two orthogonal components of the optical field. In other words, the NLC layer acts as a waveplate encompassing a voltage-dependent retardation [16].
2. Fabrication of the sample
To demonstrate the effect of a liquid crystal layer on the polarisation state of light in the waveguide, a commercial liquid crystal cell (Thorlabs LCC1318-A) was used as a sample. The cell consists of two fused silica glass slabs with a thickness of 3 mm, which sandwich a $L=(8 \pm 0.8) \mathrm{\mu}\textrm{m}$ gap that is to be filled with liquid crystal. The facets of the fused silica glass in direct contact with the NLC are covered with Indium Tin Oxide (ITO) as electrodes, as well as anti-parallel polymide alignment layers imposing a (4.5±0.5)° pre-tilt angle on the adjacent NLC molecules. A sketch of the whole liquid crystal cell is shown in Fig. 1(a).
Fig. 1. a) Top view of the liquid crystal cell with waveguide (not in scale). b) Acquired intensity output of a monomodal waveguide at 808 nm.
The glass facets placed at the NLC-air interfaces are additionally coated with an anti-reflection coating. The top facet of the cell was polished so that waveguides could be inscribed transversely [1] into the bulk of the glass, perpendicular to the liquid crystal layer.
The waveguides were inscribed using an Amplitude Satsuma laser producing pulses of 300 fs length at 1030 nm wavelength. A typical measured mode field at 808 nm is shown in Fig. 1(b). The inscription setup is sketched in Fig. 2. Those pulses were externally frequency doubled to 515 nm wavelength, and then shaped using an anamorphic zoom system as described in Ref. [17]. The astigmatic beam shaping realized by the anamorphic zoom system serves to minimize the intrinsic birefringence of the waveguides. The pulse energy was set to 660 nJ using a half-wave plate and a polarizer. The laser pulses were injected into the glass slabs of the empty liquid crystal cell using a 20x long focal distance objective (Mitutoyo) with a numerical aperture of 0.42. Straight monomodal waveguides were inscribed 0.6 mm below the glass surface with a scanning velocity of 45 mm min−1 using a high-precision automatized translation stage (Aerotech ANT130-XY). During the inscription process, the waveguides were written with a gap of 0.2 mm around the central cavity hosting the liquid crystals, so that the electrodes and alignment layers of the cell were not damaged by the intense illumination. In terms of Gaussian full width half maximum waists, the mode width at $\lambda =808~$nm along $y$ and $z$ is $(11.0\pm 0.5)~\mu$m and $(9.5\pm 0.5)~\mu$m, respectively [see Fig. 1(b) for a picture of the fundamental mode]. From inverting the Helmholtz equation [18], we find that the induced refractive index profile can be modelled as a Gaussian profile of diameter ($2w$) $14~\mu$m, with a maximum change in the refractive index between $4\times 10^{-4}$ and $5\times 10^{-4}$. The guiding losses caused by the gap were estimated using the overlap integral to be in the range 0.03 dB to 0.04 dB for light between 633 nm and 808 nm wavelength.
Fig. 2. Sketch of the inscription setup ($\lambda /2$ - half-wave plate, Pol.- polarizer, SHG - non-linear crystal for second harmonic generation).
After inscribing the waveguides, electrodes were connected with a signal generator and the cell was filled with E7 mixture (Synthon Chemicals), the latter being in the nematic phase at room temperature. The schematic of the NLC layer is shown in Fig. 3. Optically speaking, the NLC are uniaxial materials, with the optic axis corresponding to the mean alignment direction of the molecules, the latter described by a vectorial field called director [19]. The refractive indices for field oscillating normal or parallel to the director are labelled as $n_\bot$ and $n_\|$, respectively. We define $\Delta \psi$ as the phase retardation between the $y$ (extraordinary) and $z$ (ordinary) component. From the optics of anisotropic media we find (see Fig. 3) [19]
As shown in Fig. 4(a), the angle $\theta$ between the wave vector and the optic axis does not move significantly from the initial value of $90^\circ$ minus the pre-tilt until the bias overcomes the threshold (around 0.75 V). The reorientation curve then becomes very steep. A saturation then occurs when electric field and molecules are almost aligned. Whereas the maximum rotation angle does not depend on the cell thickness $L$, the phase retardation $\Delta \psi$ depends strongly on $L$ in agreement with Eq. (1), see Fig. 4(b). The simulations show clearly how the uncertainty in the cell thickness is much more relevant than the uncertainty in the pre-tilt angles. Figure 4(c-d) finally show the behavior of the beam polarization at the cell output when a diagonal polarization is launched at the input. To describe the polarization of an optical beam with components $E_y$ and $E_z$ propagating along $x$, we use the normalized Stokes vector $\boldsymbol S=(S_1,S_2,S_3)$. After defining the intensity $S_0=|E_y|^2+|E_z|^2$, the Stokes parameters in our geometry are thus $S_1=\left (|E_y|^2-|E_z|^2\right )/S_0$, $S_2=2\ \text {Re}(E_y E_z^*)/S_0$, and $S_3=-2\ \text {Im}(E_y E_z^*)/S_0$. Essentially, $S_1$ measures the amount of horizontal or vertical polarization, $S_2$ the amount of diagonal or anti-diagonal polarization, whereas $S_3$ provides the helicity of the beam. The linear birefringence in the NLC layer is everywhere along the $yz$ coordinate axes, if the spatial walk-off is neglected. From Ref. [24] it is then $d\boldsymbol S/dx = b(x) \boldsymbol S \times \hat {S}_1$, where $b= k_0 \left [n_e(x) - n_\bot \right ]$ is the local birefringence. From Eq. (1), $\Delta \psi = \int _0^L{b(x)dx}$. Thus, $S_1$ is a constant of the motion: the polarization represented by $\boldsymbol S$ indeed behaves like a spin 1/2 particle undergoing a Larmor precession around a fictitious magnetic field proportional to the birefringence and parallel to $\hat {S_1}$ [25]. Given their periodicity from the retardation $\Delta \psi$, the Stokes parameters makes one whole oscillation between the extreme values -1 and 1 for each variation of $2\pi$ in $\Delta \psi$. The sharpness of the reorientation dynamics after the Freédericksz threshold results in a high sensitivity with respect to the applied bias, yielding much lower distances between the local minima and maxima of the Stokes parameters.
Fig. 3. Schematic of the NLC layer (a) without bias and (b) with an applied bias. The green ellipsoids represent the NLC molecules, whose long axis corresponding to the molecular director. In this simplified drawing the sizes along $x$ are not in scale. Inset: definition of the angle $\theta$ between the director and the wavevector, the latter being parallel to $x$.
Fig. 4. Electro-optical effect in a thin NLC cell. (a) Maximum rotation angle $\pi /2-\theta$, (b) retardation $\Delta \psi$, Stokes parameters (c) $S_2$ and (d) $S_3$ at the output versus the applied voltage $V$ (RMS value) when a diagonal polarization is entering the NLC cell. Wavelength is 808 nm. In (b-d) the NLC cell thickness $L$ is 7.2 $\mu$m (blue curves), 8 $\mu$m (black curves) and 8.8 $\mu$m (green curves). In (a) and (b) solid and dashed lines correspond to a pre-tilt angle of $4^\circ$ and $5^\circ$, respectively. In (c-d) the two values of the pre-tilt angle corresponds to the edge of the shaded areas. For the material parameters we took the values $\epsilon _{\parallel }^{LF}=19.6\epsilon _0$, $\epsilon _{\bot }^{LF}=5.1\epsilon _0$, $K_1=10.3~$pN, and $K_3=16.5~$pN [20,21].
The typical birefringence for E7 liquid crystal mixtures for visible light is around $\Delta n = 0.2$. More precisely, the refractive indices for the E7 mixture can be computed by using the model and data provided in Ref. [26]. At room temperature ($T=20^\circ$), for $\lambda =633~$nm we get $n_\bot =1.519$ and $n_\parallel = 1.738$, thus providing an optical birefringence $\Delta n = n_\parallel -n_\bot =0.218$; for $\lambda =808~$nm, $n_\bot =1.511$, $n_\parallel = 1.719$, in turn providing $\Delta n=0.208$.
3. Characterization set-up
The NLC cells embedding FLDW waveguides were characterized using the set-up sketched in Fig. 5. The waveguides were analyzed using the light of either a Helium-Neon laser emitting at 633 nm or a Titanium-Sapphire laser emitting at 808 nm. To maximize the electro-optical effect, the input polarization was set by a half-wave plate to be either diagonal or anti-diagonal, whereas the input power was limited to 0.5 mW to avoid the emergence of nonlinear effects in the NLC layer [19]. The input beam was coupled into the waveguides using a 4x microscope objective with a numerical aperture of 0.1 (Olympus). On the other side, the intensity profile transmitted at the waveguide output was captured by a polarimeter (PAX1000IR1, Thorlabs) using a 20x objective with a numerical aperture of 0.4 (Olympus). An iris diaphragm between the polarimeter and the 20x objective filters out any stray light emitted from the surrounding of the waveguide. To rotate the NLC molecules, a sinusoidal voltage with 1 kHz frequency was applied via the ITO electrodes. For NLC, alternate voltages are preferable to continuous ones to avoid migration of ions inside the NLC layer and, in turn, screening of the external field. The frequency is faster than the time response of the NLC: the alternate voltage thus behaves like a DC with an amplitude provided by the RMS (Root Mean Square) value. Dependence on the applied bias of the polarization of the output light is finally displayed by the polarimeter in terms of the three normalized Stokes parameters $S_1$, $S_2$ and $S_3$.
4. Experimental results
We first verified the well known rotation of the NLC molecules due to the applied bias. In essence, the external low-frequency electric field induces a dipole in the highly mobile NLC molecules [term proportional to $\epsilon ^\mathrm {LF}_\|- \epsilon ^{LF}_\bot$ in Eq. (2)], which in turn rotate to minimize their interaction energy with the external stimulus; such a rotation is contrasted by the mechanical torque associated with the changes in the NLC macroscopic alignment [terms depending on the elastic constants in Eq. (2)] and physically due to the microscopic forces between the NLC molecules [16,19]. Owing to the strong anisotropy of the NLC molecules, the rotation induces a strong phase modulation on the extraordinary component (parallel to the $y$ axis in our sample), in turn yielding a voltage-dependent phase retardation between the ordinary and the extraordinary polarizations provided by Eq. (1). Given that $\Delta n L\approx 1.6~\mu$m, the overall range spanned by the phase retardation will be around $5.5\pi$ for $\lambda =633~$nm, and around $4\pi$ for $\lambda =808~$nm.
To probe the electrically-induced phase retardation, we measured the Stokes parameters versus the applied voltage when a diagonally or anti-diagonally polarized laser beam at either 633 nm or at 808 nm is sent through the NLC film, without coupling to any waveguide, in different regions of the cell. In the experiments, all the reported voltages correspond to the RMS values. For this preliminary test, the objectives used to couple light into the waveguide and image the output of the waveguide were removed from the setup, that is, a collimated beam of size 0.8 mm (in case of the 633 nm light) and 1 mm (in case of the 808 nm light) was employed to probe the cell. The phase retardation retrieved from the Stokes parameters versus the applied voltage are plotted in Fig. 6 for the two wavelengths and diagonally- or anti-diagonally-polarized inputs. First of all, the output polarization clearly changes with the applied voltage. Stokes parameters and phase retardation are in good agreement with the theoretical predictions, corresponding to the shaded red areas [16]. The Freédericksz transition occurs at around 0.75 V, as can be seen by the abrupt change in the phase retardation $\Delta \psi$. Owing to the saturation of the reorientational torque when the molecules become parallel to the external field, the steepest change in retardation, and thus polarization, can be observed in the interval from 0.75 V to around 2.5 V. As expected, the beam polarization periodically oscillates between diagonal, right/left circular polarization, and anti-diagonal, the number of loops being determined by the ratio between the phase delay in the unbiased case and $2\pi$.
Fig. 6. Phase retardation at $\lambda =633~$nm (left panel) and $\lambda =808~$nm (right panel) measured from the Stokes parameters versus the RMS applied bias in a bare NLC cell illuminated with a collimated beam. Each curve with symbols corresponds to a different position along the transverse plane of the cell, always in the absence of waveguide. The red shaded area is the theoretical predictions in the tolerance range for the cell parameters.
In the following experiment, the objectives are placed back into the setup and diagonally polarized laser light of 633 nm is coupled into the waveguide. Four different waveguides written with the same inscription parameters are evaluated for statistical purposes. The output polarization versus the applied voltage for one waveguide is plotted in Fig. 7(a), whereas the shaded areas in Fig. 7(b) show how much the Stokes parameters vary between different realizations. Figure 7(a,b) clearly illustrate a significant polarization shift based on the applied voltage fully analogous with the case of bare NLC cell, proving the functionality of using this method for adaptive waveplates in FLDW integrated photonic circuits. The measured polarization when no voltage is applied differs from the polarization measured without the waveguide (cf. Figure 6) due to the birefringence of the waveguide. As shown from the low thickness of the shaded areas around 0 V in Fig. 7(b), the birefringence of the waveguides does not significantly change from realization to realization. With respect to the theoretical predictions, a non-vanishing $S_1$ appears for voltages between the Freédericksz threshold and slightly more than 2 V. This can be ascribed to transverse shift of the extraordinary component stemming from the spatial walk-off, given by approximately $\epsilon _a \sin (2\theta )\left /\left (2n_\bot ^2\right )\right.$. The experimental retardation $\Delta \psi$ is always larger than the calculated one. This suggests weak anchoring conditions at the interfaces NLC-glass, that is, the external electric field is capable of slightly rotating the molecules even in proximity of the alignment layer [19]. Noteworthy, by supposing small variations in the NLC parameters (namely the elastic constants), the agreement between experiments and simulations was not significantly improved. We also notice that the area spanned by the Stokes parameters widens as the voltage is increased. Figure 7(c) finally shows the phase retardation extracted from the Stokes parameters. The measured phase retardation is slightly shifted with respect to the theoretical prediction, but it retains the same shape. The agreement is slightly worse than the bare cell, probably due to spurious effects of the waveguide on the polarization, that is, the waveguide can be modelled like a uniaxial plate only within certain limits due to the inhomogeneous stress induced during the writing procedure. A 3D visualization of the rotation of the polarization state on the Stokes sphere is visualized in Fig. 8 [27]. As discussed above, the polarization moves along a single meridian defined by $S_1=0$, demonstrating how the system composed by the waveguide and the NLC layer behaves like a tunable wave plate.
Fig. 7. Voltage-tuning of the light polarization at the output of the waveguides for a probe beam at 633 nm. (a) Stoker parameters versus the applied bias experimentally measured (dashed lines with symbols) and the corresponding numerical prediction (solid lines). Blue, red, and green curves correspond to $S_1$, $S_2$ and $S_3$, respectively. (b) As in panel (a), but now the shaded areas show the variability when the output is measured in four different waveguides written with identical parameters. (c) Phase retardation $\Delta \psi$ versus the applied bias, the experimental results being retrieved from the Stokes parameters shown in panel (a) and (b). Black stars and purple shaded area are the experimental results shown in panel (a) and (b), respectively. Red shaded area is the range of variation for the theoretical prediction as discussed in Fig. 4, whereas the black solid line is the response if the nominal values for the cell are used.
Fig. 8. Position of the output polarization state on the Stokes sphere for selected voltages, when light of 633 nm is injected into the waveguide.
The same experiment was repeated with anti-diagonally polarized laser light at 808 nm, to demonstrate that this approach of adding adaptive polarization control to femtosecond laser direct written waveguides can be used for different wavelengths. The measured output polarization states in dependence of the applied voltage are plotted in Fig. 9(a). The most striking difference with respect to the red wavelength is that the polarization makes two instead of three loops around the Poincaré sphere, in agreement with Fig. 6. The output polarization when no voltage is applied to the cell is close to anti-diagonal polarized and changes to approximately left-circular polarization for 0.9 V applied voltage. When increasing the voltage to 1.15 V the output polarization changes to diagonally polarized, followed by right-circular polarization at 1.35 V. When further increasing the voltage to 1.6 V and 1.9 V, the polarization passes through the polarization states of anti-diagonal and left circular polarization, respectively. Diagonal polarization is achieved again at 2.5 V, followed by right circular polarization at 4 V. For voltages above 4 V, only very slow changes of the polarization state were measured due to the saturation in the molecular rotation.
Fig. 9. (a) Polarization modulation at 808 nm for light transmitted through a waveguide. Top: phase retardation; bottom: Stokes parameters versus the applied bias. In the top figure the red shaded area corresponds to the interval of the theoretical predictions, with the black solid line being the mean value; black symbols are the values derived from the experiments. In the bottom figure dashed lines with symbol and solid lines correspond to experiments and theory, respectively. Blue, red and green lines correspond to $S_1$, $S_2$ and $S_3$, respectively. (b) Polarization evolution drawn on the Poincaré sphere for applied voltages of up to 4 V.
5. Conclusions
We realized waveguides in the glass slides of commercial liquid crystal cells by permanently modifying the glass locally via the illumination with ultrashort intense laser pulses. In such a way, we created a hybrid photonic circuit where the waveguides localize the optical beam and the NLC acts like a tunable wave plate. We demonstrated the full modulation of the optical polarization at two different wavelengths in the visible, with results in good agreement with theoretical expectations. With respect to the field of femtosecond writing, the proposed technology can be a valid alternative with respect to thermo-optical modulator [11] due to a lower energy consumption and due to the potentially faster time responses, conjugated with a lower cross-talk between adjacent waveguides. With respect to NLC-based waveguides [28–31], the main advantage of our proposal is the minimization of optical losses due to the shorter propagation length in the NLC, the latter being intrinsically associated with a very strong Rayleigh scattering [19]. With respect to tunable integrated devices based upon NLC integrated on the top of silicon or glass waveguides [32,33], our structures can modulate the light in a more compact way, set aside the full 3D arrangement of waveguides typical of femtosecond-writing techniques.
Funding
Deutsche Forschungsgemeinschaft (259607349, GRK2101); H2020 Marie Skłodowska-Curie Actions (889525).
Acknowledgments
K.L. work was funded in the framework of the Max Planck School of Photonics supported by BMBF, Max Planck Society, and Fraunhofer Society. We acknowledge support by the German Research Foundation Projekt-Nr. 512648189 and the Open Access Publication Fund of the Thüringer Universitäts- und Landesbibliothek Jena.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. K. M. Davis, K. Miura, N. Sugimoto, et al., “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21(21), 1729–1731 (1996). [CrossRef]
2. A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B: At. Mol. Opt. Phys. 43(16), 163001 (2010). [CrossRef]
3. Y. Jia, S. Wang, and F. Chen, “Femtosecond laser direct writing of flexibly configured waveguide geometries in optical crystals: fabrication and application,” Opto-Electron. Adv. 3(10), 190042 (2020). [CrossRef]
4. M. Kremer, L. J. Maczewsky, M. Heinrich, et al., “Topological effects in integrated photonic waveguide structures,” Opt. Mater. Express 11(4), 1014–1036 (2021). [CrossRef]
5. G. Della Valle, R. Osellame, and P. Laporta, “Micromachining of photonic devices by femtosecond laser pulses,” J. Opt. A: Pure Appl. Opt. 11(1), 013001 (2009). [CrossRef]
6. S. Nolte, M. Will, J. Burghoff, et al., “Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics,” Appl. Phys. A: Mater. Sci. Process. 77(1), 109–111 (2003). [CrossRef]
7. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, et al., “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013). [CrossRef]
8. M. Tillmann, B. Dakić, R. Heilmann, et al., “Experimental boson sampling,” Nat. Photonics 7(7), 540–544 (2013). [CrossRef]
9. T. Meany, D. N. Biggerstaff, M. A. Broome, et al., “Engineering integrated photonics for heralded quantum gates,” Sci. Rep. 6(1), 25126 (2016). [CrossRef]
10. G. Corrielli, A. Crespi, and R. Osellame, “Femtosecond laser micromachining for integrated quantum photonics,” Nanophotonics 10(15), 3789–3812 (2021). [CrossRef]
11. F. Flamini, L. Magrini, A. S. Rab, et al., “Thermally reconfigurable quantum photonic circuits at telecom wavelength by femtosecond laser micromachining,” Light: Sci. Appl. 4(11), e354 (2015). [CrossRef]
12. Z. Chaboyer, A. Stokes, J. Downes, et al., “Design and fabrication of reconfigurable laser-written waveguide circuits,” Opt. Express 25(26), 33056–33065 (2017). [CrossRef]
13. F. Ceccarelli, S. Atzeni, C. Pentangelo, et al., “Low power reconfigurability and reduced crosstalk in integrated photonic circuits fabricated by femtosecond laser micromachining,” Laser Photon. Rev. 14(10), 2000024 (2020). [CrossRef]
14. L. J. Maczewsky, M. Heinrich, M. Kremer, et al., “Nonlinearity-induced photonic topological insulator,” Science 370(6517), 701–704 (2020). [CrossRef]
15. S. Mukherjee and M. C. Rechtsman, “Observation of Floquet solitons in a topological bandgap,” Science 368(6493), 856–859 (2020). [CrossRef]
16. D.-K. Yang and S.-T. Wu, Fundamentals of Liquid Crystal Devices (John Wiley Sons, 2014).
17. K. Lammers, M. Siems, M. Ehrhardt, et al., “Nanograting based birefringent retardation elements in integrated photonic circuits,” in Laser-based Micro-and Nanoprocessing XIV, Vol. 11268 (SPIE, 2020), p. 241–278.
18. L. McCaughan and E. Bergmann, “Index distribution of optical waveguides from their mode profile,” J. Lightwave Technol. 1(1), 241–244 (1983). [CrossRef]
19. F. Simoni and J. Love, Nonlinear Optical Properties of Liquid Crystals and Polymer Dispersed Liquid Crystals vol. 2 (World Scientific, 1997).
20. J. F. Strömer, E. P. Raynes, and C. V. Brown, “Study of elastic constant ratios in nematic liquid crystals,” Appl. Phys. Lett. 88(5), 051915 (2006). [CrossRef]
21. D. Bankova, N. Brouckaert, N. Podoliak, et al., “Characterization of optically thin cells and experimental liquid crystals,” Appl. Opt. 61(16), 4663–4669 (2022). [CrossRef]
22. H. J. Deuling, “Deformation of nematic liquid crystals in an electric field,” Mol. Cryst. Liq. Cryst. (1969-1991) 19(2), 123–131 (1972). [CrossRef]
23. J. D. Hunter, “Matplotlib: A 2d graphics environment,” Comput. Sci. Eng. 9(3), 90–95 (2007). [CrossRef]
24. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4 × 4 matrix calculus,” J. Opt. Soc. Am. 68(12), 1756–1767 (1978). [CrossRef]
25. H. Kuratsuji and S. Kakigi, “Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters,” Phys. Rev. Lett. 80(9), 1888–1891 (1998). [CrossRef]
26. J. Li, S.-T. Wu, S. Brugioni, et al., “Infrared refractive indices of liquid crystals,” J. Appl. Phys. 97(7), 073501 (2005). [CrossRef]
27. J. del Hoyo, L. M. Sanchez-Brea, and A. Soria-Garcia, “Open source library for polarimetric calculations "py-pol",” in Computational Optics 2021, Vol. 11875 (SPIE, 2021), p. 1187506.
28. M. Rüetschi, P. Grütter, J. Fünfschilling, et al., “Creation of liquid crystal waveguides with scanning force microscopy,” Science 265(5171), 512–514 (1994). [CrossRef]
29. A. d’Alessandro, B. Bellini, D. Donisi, et al., “Nematic liquid crystal optical channel waveguides on silicon,” IEEE J. Quantum Electron. 42(10), 1084–1090 (2006). [CrossRef]
30. A. Piccardi, A. Alberucci, U. Bortolozzo, et al., “Readdressable interconnects with spatial soliton waveguides in liquid crystal light valves,” IEEE Photon. Technol. Lett. 22(10), 694–696 (2010). [CrossRef]
31. U. S. Tripathi and V. Rastogi, “Liquid crystal based rib waveguide,” J. Lightwave Technol. 38, 1 (2020). [CrossRef]
32. W. D. Cort, J. Beeckman, T. Claes, et al., “Wide tuning of silicon-on-insulator ring resonators with a liquid crystal cladding,” Opt. Lett. 36(19), 3876–3878 (2011). [CrossRef]
33. S. R. Davis, G. Farca, S. D. Rommel, et al., “Liquid crystal waveguides: new devices enabled by> 1000 waves of optical phase control,” in Emerging liquid crystal technologies V, Vol. 11875 (SPIE, 2010), p. 78–91.
