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Curr. Opt. Photon. 2023; 7(1): 83-89

Published online February 25, 2023 https://doi.org/10.3807/COPP.2023.7.1.83

## Switchable Spatial Control of Linearly Polarized Light Based on a Liquid-crystal Optical Waveguide

Zhengtao Zha1,2 , Qianshu Zhang2

1School of Physics and Astronomy, China West Normal University, Sichuan, Nanchong 637009, China
2School of Electronic Information Engineering, China West Normal University, Sichuan, Nanchong 637009, China

Corresponding author: *zaktao@stu.cwnu.edu.cn, ORCID 0000-0001-9827-5480
**jackyzhang@cwnu.edu.cn, ORCID 0000-0001-8448-2366

Received: October 17, 2022; Revised: November 10, 2022; Accepted: November 26, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

This study reports a structure, based on a liquid-crystal (LC) core optical waveguide, for the switchable spatial control of linearly polarized light. The refractive indices of both left and right isotropic claddings in the waveguide are between the two principal-axis indices of the nematic liquid crystal (NLC). Numerical simulations demonstrate that the proposed structure can be operated in transmission mode or as an attenuator by controlling the on and off states of the applied voltage, whether the initial excitation is transverse electric (TE) linearly polarized light or transverse magnetic (TM) linearly polarized light. The design can also be used as an integrated optical polarizer, since only one type of linearly polarized light is always permitted to pass through the core zone of the NLC optical waveguide.

Keywords: Nematic liquid crystal, Optical switches, Optical waveguide, Spatial control

OCIS codes: (130.0130) Integrated optics; (160.3710) Liquid crystals; (230.7380) Waveguides, channeled

### I. INTRODUCTION

Liquid crystals (LCs) exhibit large electro-optic effects and high optical birefringence [1], which enable them to change the optical properties of waveguides more effectively than traditional electro-optic materials. Moreover, nematic liquid crystals (NLCs) possess characteristics such as fast response [2], low driving voltage [3], low cost [4], and high transmittance in the visible and near-infrared spectral regions [5]. These advantages render NLCs electro-optic materials with great application potential in photonics.

In recent years, as the design of new LC optical-waveguide structures has become a current research theme [6], the spatial polarization control of light beams by an NLC optical waveguide in the visible and near-infrared regions has drawn tremendous attention [615]. Asquini et al. [7] reported an LC-core waveguide with a SiO2/Si triangular-cross-section groove for controlling transverse magnetic (TM) polarized light, with a threshold voltage of 2 V. Wang et al. [9, 10] reported an LC-core channel waveguide with a semicircular groove, with a threshold voltage of about 4 V; this structure was not only expensive to manufacture [2], but also could only realize electrical tuning of the quasi-TM polarization mode. Kabanova et al. [11] reported a slab LC-core optical waveguide that could electrically control the propagation path of TM linearly polarized light; not only did the structure require photolithography to produce periodic comb electrodes, but also the slab waveguide was not conducive to integration [16]. Melnikova et al. [12] reported a slab LC optical waveguide with mutually orthogonal NLC directors in adjacent regions. The electrically switchable structure could split the unpolarized beam into two orthogonal polarization components to propagate in the waveguide independently, but the angle of incidence for the excitation light was severely limited by the condition of total internal reflection (TIR). Rutkowska et al. [13] reported an LC optical-waveguide structure to control the propagation direction of linearly polarized light, with an alternating multidomain LC orientation of 0° and 45°. However, the functional realization of this design not only depended on the nonlinear optical effects in LCs (e.g. optical-soliton propagation and self-focusing), but also required multistep ultraviolet (UV) exposure and periodic amplitude masks. Rushnova et al. [14] reported a slab LC optical waveguide with mutually alternating twist and planar orientations in adjacent areas, which could be set to operate in transmission mode or as an attenuator by changing the applied voltage. However, that waveguide structure could only tune transverse electric (TE) linearly polarized light, and required complicated orientation of multi-domains of the NLCs.

In this study we propose a structure, based on a channel NLC-core optical waveguide, for the switchable spatial control of linearly polarized light. The refractive indices of both left and right claddings in the waveguide are between the two principal-axis indices of the NLC, which is significantly different from previously reported designs [1719]. In the latter, the refractive indices of all claddings in the waveguide were less than the refractive index of the short axis of the NLC. When the initial excitation is TE (or TM) linearly polarized light, the operational mode of the proposed waveguide can always be switched by controlling the on and off states of the applied voltage. More specifically, this optical waveguide can operate in conventional transmission mode, or become an attenuator. The operating principle of the structure is discussed in the next section. The functionality of the waveguide is verified in the third section with a known full-vector finite-difference (FVFD) mode solver and beam-propagation method (BPM). The fourth section is a summary.

### II. PRINCIPLE

The simplified geometry of the proposed NLC-core optical waveguide is shown in Fig. 1(a), in which the width and thickness of the core region are w and h respectively. The material of the waveguide core is NLC E7 (Merck Co., NJ, USA). ne and no are the refractive indices of the two principal axes respectively, parallel and perpendicular to the LC director. The upper and lower boundaries of the waveguide’s core region both have a very thin photoalignment layer [not shown in Fig. 1(a)] to promote the homogeneous alignment of the E7. Additionally, both left and right cladding layers in the optical waveguide are composed of the isotropic negative photoresist AZ15nXT [2] with refractive index nAZ15nXT. Although the orientation layer at the sidewall cladding should also be considered, to ensure better orientation uniformity of the NLCs in the actual preparation, as previously reported [18] the ideal case is still considered here, for simplicity of theoretical analysis. The upper and lower isotropic cladding layers of the waveguide are both plexiglass (PMMA), with refractive index nPMMA. It is worth noting that the relation between the material refractive indices for the waveguide is shown in Eq. (1), which is key to realizing the functionality of the proposed NLC structure.

Figure 1.Schematic diagrams of the liquid crystal (LC) optical waveguide’s structure and LC orientation. (a) Schematic of the cross section of the nematic liquid crystal (NLC)-core optical waveguide, (b) deflection of the NLC director, (c) orientation of the NLC in different voltage states.

nPMMA<no<nAZ15nXT<ne

where the value of each refractive index at the wavelength of 1.55 μm is shown in Table 1.

Refractive indices for the nematic liquid crystal (NLC)-core optical waveguide

 Waveguide Materials E7 AZ15nXT PMMA Refractive Indices ne no nAZ15nXT nPMMA 1.697 [20] 1.502 [20] 1.58 [21] 1.481 [22]

### 2.1. Voltage off State (U = 0)

In the absence of an applied voltage between the indium tin oxide (ITO) electrodes, the anchoring effect of the photoalignment layer makes all of the NLC molecules align along the x-axis direction, as shown in Fig. 1(c). Therefore, TE linearly polarized light (where the electric field is parallel to the x-direction) will see the refractive index ne of the extraordinary wave (i.e. e wave), thereby exciting the quasi-TE mode in the waveguide. According to Eq. (1), the mode field of the quasi-TE mode can be confined within the core region of the NLC optical waveguide. However, TM linearly polarized light (where the electric field is parallel to the y-direction) sees the refractive index no of the ordinary wave (i.e. o wave), which will excite the quasi-TM mode in the NLC optical waveguide. Based on Eq. (1), the energy of the quasi-TM mode must be attenuated into the left and right claddings of the optical waveguide.

### 2.2. Voltage on State (U >> Uc)

When a voltage exceeding the threshold is applied between the ITO electrodes, the LC director n^ is deflected around the z-axis inside the xoy plane [23], as shown in Fig. 1(b). According to Frank-Oseen elastic continuum theory [24] and Euler-Lagrange variational theory [25], the deflection angle α(y) of the LC director can be obtained as follows:

yh=UcπU0αy cos2φ+ k 33 /k 11 sin2φ sin2 αm sin2φ dφ

where Uc=πk11/ε0Δε is the threshold voltage, U is the applied voltage, k11 and k33 are Frank elastic constants, ε0 is the dielectric permittivity of vacuum, and ∆ ε is the dimensionless dielectric anisotropy of the NLC. Also, αm is the maximum deflection angle of the LC director, the value of which can be determined by Eq. (3).

UUc=2π0αm cos 2φ+ k 33/k 11 sin 2φ sin 2 α m sin 2φ dφ

k11, k33, and ∆ ε of the E7 are 11.1 pN, 17.1 pN, and 13.8 respectively [26]. Thus the threshold Uc can be calculated to be about 0.95 V. Figure 2 shows the distribution of the deflection angle of the LC director under different applied voltages.

Figure 2.Variation of the deflection angle of the nematic liquid crystal (NLC) director with applied voltage. (a) Maximum deflection angle αm; (b) the deflection angle α.

It can be seen from Fig. 2 that the maximum deflection angle αm of the NLC increases with increasing applied voltage. Moreover, the deflection angle of the LC director has a gradient distribution in the thickness direction (y-direction) of the LC cell, due to the effect of the surface anchoring energy. However, one can clearly see that this gradient property becomes very weak when the applied voltage is much greater than the threshold (i.e. U >> Uc). Besides, in practice, photoalignment materials with weak anchoring energy [27] can be used as alignment layers to further weaken the gradient effect of the deflection angle.

Therefore, the maximum deflection angle αm of the director can be approximated as the deflection angles of all NLC molecules, when the applied voltage is much larger than the threshold value (specifically, when the voltage is greater than or equal to 5 times the threshold). Under these conditions the NLC of the waveguide’s core region will approximately have the uniform dielectric tensor shown in Eq. (4).

ε=ε0× ne2cos2αm+no2sin2αm cosαmsinαmne2no2 0 cosαmsinαmne2no2 ne2sin2αm+no2cos2αm 0 0 0 no2

In particular, we consider the case where αm is equal to 90°, that is, the NLC molecules have transitioned from homogeneous orientation to homeotropic alignment, as shown in Fig. 1(c). The TE linearly polarized light still excites the quasi-TE mode in the NLC optical waveguide, but it sees the refractive index no of the o wave. According to Eq. (1), the energy of the quasi-TE mode must be attenuated into the left and right claddings of the optical waveguide. However, the TM linearly polarized light sees the refractive index ne of the e wave, thereby exciting the quasi-TM mode in the waveguide. Consequently, the mode field of the quasi-TM mode can be confined inside the core region of the NLC optical waveguide.

It is noteworthy that mode hybridization exists in the waveguide if and only if the dielectric tensor [ε] of the NLC has nonzero off-diagonal terms (i.e. 0° < αm < 90°) [18, 23, 28]. Yet, only the two cases of αm = 0° and 90° are involved in this study. Therefore, the above analysis of the quasi-TE and quasi-TM modes based on the relation to the refractive index is reasonable.

In light of all analysis above, in the absence of an applied voltage (off state, with αm = 0°) the NLC optical waveguide excited by the TE linearly polarized light operates in the conventional transmission mode, but that excited by the TM linearly polarized light can be used as an attenuator. However, when the applied voltage is much greater than the threshold (on state, with αm = 90°) the NLC optical waveguide excited by the TE linearly polarized light becomes an attenuator, but that excited by the TM linearly polarized light operates in the conventional transmission mode.

### III. NUMERICAL RESULTS

The FVFD mode solver [29] can be employed to calculate the eigenmodes in an NLC optical waveguide with a transversely anisotropic dielectric tensor [see Eq. (4)]. The size of the calculation window is 8 μm × 8 μm, including that the cross section of the waveguide core is 5 μm × 5 μm, the widths of both left and right claddings are 1.5 μm, and the cross sections of both upper and lower cladding layers are 8 μm × 1.5 μm. The effect of the photoalignment layer on the optical properties of the waveguide can be neglected, since it is sufficiently thin [18]. Moreover, the Dirichlet boundary condition [30] is applied to the calculation window.

Numerical results indicate that the NLC optical waveguide is multimode, as expected. In particular, the optical waveguide supports more than 20 eigenmodes at the wavelength of 1.55 μm. Nevertheless, numerous reports [18, 19, 28] have demonstrated that such transversely anisotropic NLC-core optical waveguides mainly excite low-order modes. Figure 3 shows the electric field distributions of the low-order eigenmodes supported by the NLC optical waveguide under two applied voltage states, including the first two quasi-TE modes and the first two quasi-TM modes.

Figure 3.Electric field distributions of the first two quasi-TE modes and the first two quasi-TM modes, with their effective refractive indices: (a)–(d) U = 0 (off state, with αm = 0°), (e)–(h) U >> Uc (on state, with αm = 90°).

One can see from Figs. 3(a)3(d) that in the absence of an applied voltage, the region of the electric field distribution of the quasi-TE mode and that of the quasi-TM mode are significantly different. The former is mainly concentrated in the core zone of the waveguide, while the latter is mainly concentrated in the left and right claddings of the waveguide. The reasons for this phenomenon are that (1) the refractive indices seen by TE and TM linearly polarized light are ne and no respectively, and (2) the materials of the NLC optical waveguide also satisfy the special relationship between refractive indices shown in Eq. (1). On the contrary, when the applied voltage is much greater than the threshold, it can be seen from Figs. 3(e) and 3(f) that the electric field distributions of the quasi-TE modes are mainly concentrated in the left and right claddings of the waveguide. Simultaneously, the electric field distributions of the quasi-TM modes are well confined inside the core of the waveguide, as shown in Figs. 3(g) and 3(h). As mentioned in section 2, the refractive index seen by the TE linearly polarized light has changed from ne to no, while the refractive index seen by the TM linearly polarized light has varied from no to ne. Thus as the state of the applied voltage turns from off to on, the main distribution regions of the quasi-TE mode and the quasi-TM mode are interchanged.

The above results indicate that the main distribution region of the eigenmode field of the quasi-TE (or quasi-TM) mode can be changed by switching the on and off states of the applied voltage. Moreover, one can also find from Fig. 3 that whether the state of the applied voltage is on or off, the main distribution zones of the quasi-TE and quasi-TM modes are always opposite. This means that only one quasilinear polarization mode can be stably supported inside the core region of the NLC optical waveguide.

Furthermore, to see the operational characteristics of the NLC optical waveguide proposed in this study more intuitively, it is necessary to solve for the propagation field of the waveguide. Because this multimode waveguide mainly excites low-order modes (i.e. it can support the paraxial approximation), the FVFD-BPM [31] can be employed to simulate the propagation of light. The initial excitation incident from the center of the waveguide is TE or TM linearly polarized Gaussian light with a half-width of 2.15 μm. The length L of the waveguide is 20 μm and the propagation step is 0.1 μm. The maximum effective refractive index of the fundamental mode obtained above is used as the reference index, to ensure that the slowly-varying-envelope approximation (SVEA) of the BPM is satisfied. Moreover, the transparent boundary condition (TBC) [32] is adopted, to suppress the reflection of the radiation mode from the boundaries of the calculation-domain window. Furthermore, the partial propagation loss is defined in Eq. (5) to describe the operational mode of the NLC optical waveguide better (that is, it does not include the absorption loss of the NLC, nor the loss when the light is coupled into or out of the waveguide).

Loss=10log10Pout/Pin/L

where Pin and Pout represent the optical powers at the input and output ends of the waveguide respectively. Considering the features of the main distribution region of the eigenmode in Fig. 3, Fig. 4 shows for different voltage states the propagation field distributions of the two linearly polarized light waves in the y section (i.e. xoz plane) of the waveguide.

Figure 4.Propagation field distribution of linearly polarized light in the LC optical waveguide under different voltage states: (a), (b) U = 0 (off state, with αm = 0°); (c), (d) U >> Uc (on state, with αm = 90°).

In the voltage off state (U = 0), as can be seen from Fig. 4(a) the propagation field excited by the TE linearly polarized light achieves conventional transmission, with a partial propagation loss of about −2.167 dB/cm; however, from Fig. 4(b) the NLC optical waveguide excited by the TM linearly polarized light already exhibits energy leakage at the output end of the waveguide’s core region significantly (i.e. it becomes an attenuator), with a partial propagation loss estimated as up to −183.237 dB/cm. In the voltage on state (U >> Uc), one can see from Fig. 4(c) that the propagation field excited by the TE linearly polarized light turns into strong attenuation, with partial propagation loss of about −246.175 dB/cm, yet from Fig. 4(d) the NLC optical waveguide excited by the TM linearly polarized light operates in the conventional transmission mode, for which the partial propagation loss is estimated to be as low as −1.075 dB/cm.

These results indicate that whether the initial excitation is TE linearly polarized light [see Figs. 4(a) and 4(c)] or TM linearly polarized light [see Figs. 4(b) and 4(d)], the operational mode can be always switched by controlling the on and off states of the applied voltage. More specifically, the NLC optical waveguide can be operated in transmission mode or as an attenuator. Moreover, comparing Fig. 4(a) to Fig. 4(b), and Fig. 4(c) to Fig. 4(d), one can find that the operational modes of TE and TM linearly polarized light are always opposite, regardless of whether the voltage state is on or off. This shows that for both applied voltage states only one type of linearly polarized light is permitted to pass through the core region of the NLC optical waveguide. Therefore, the proposed channel NLC-core structure can also be used as an integrated optical polarizer.

It is worth noting that the partial propagation losses of TE and TM linearly polarized light are not equal, regardless of whether the NLC optical waveguide operates in transmission mode [Figs. 4(a) and 4(d)] or as an attenuator [Figs. 4(b) and 4(c)]. This is because there is a certain difference between the eigenmode field of the quasi-TE mode and that of the quasi-TM mode in the waveguide (see Fig. 3). Consequently, Gaussian light waves with the same half-width but different polarization directions excite the eigenmodes to different extents. One can also find from Figs. 4(a) and 4(d) that the amplitude distribution of the electric field strength exhibits slight periodic like undulations in the process of propagation. This phenomenon occurs because, except for the fundamental modes in such multimode NLC optical waveguides, the light wave may also excite other low-order modes (e.g. the first-order mode), to a small extent.

### IV. CONCLUSION

This research proposes a channel-type LC-core optical waveguide to realize the spatial control and switching of linearly polarized light, in which the refractive indices of both left and right claddings of the waveguide are between the two principal-axis indices of the NLC. The structure can not only switch operational mode (i.e. achieve conventional transmission or act as an attenuator) of TE/TM linearly polarized light by controlling the on and off states of the applied voltage, but also can be used as an integrated optical polarizer. This design mainly includes the following advantages: (1) The proposed structure is a rectangular waveguide, which is more beneficial to integration due to its cross section being smaller than that of a slab waveguide; (2) both TE and TM linearly polarized light waves can be tuned by the structure; and (3) the structure does not require periodic electrodes, nor a complicated multidomain orientation of the NLC. Furthermore, the proposed structure has great potential applications in optical wavelength division multiplexing (WDM) networks [33] and related research based on linearly polarized light [34].

The authors declare no conflicts of interest.

### DATA AVAILABILITY

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.

Applied Basic Research Project of Science and Technology Department of Sichuan Province (2014JY0024); Scientific Research Fund of Science and Technology Bureau of Nanchong City (19YFZJ0090); Scientific Research Foundation of China West Normal University (17YC056).

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### Article

#### Article

Curr. Opt. Photon. 2023; 7(1): 83-89

Published online February 25, 2023 https://doi.org/10.3807/COPP.2023.7.1.83

## Switchable Spatial Control of Linearly Polarized Light Based on a Liquid-crystal Optical Waveguide

Zhengtao Zha1,2 , Qianshu Zhang2

1School of Physics and Astronomy, China West Normal University, Sichuan, Nanchong 637009, China
2School of Electronic Information Engineering, China West Normal University, Sichuan, Nanchong 637009, China

Correspondence to:*zaktao@stu.cwnu.edu.cn, ORCID 0000-0001-9827-5480
**jackyzhang@cwnu.edu.cn, ORCID 0000-0001-8448-2366

Received: October 17, 2022; Revised: November 10, 2022; Accepted: November 26, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

This study reports a structure, based on a liquid-crystal (LC) core optical waveguide, for the switchable spatial control of linearly polarized light. The refractive indices of both left and right isotropic claddings in the waveguide are between the two principal-axis indices of the nematic liquid crystal (NLC). Numerical simulations demonstrate that the proposed structure can be operated in transmission mode or as an attenuator by controlling the on and off states of the applied voltage, whether the initial excitation is transverse electric (TE) linearly polarized light or transverse magnetic (TM) linearly polarized light. The design can also be used as an integrated optical polarizer, since only one type of linearly polarized light is always permitted to pass through the core zone of the NLC optical waveguide.

Keywords: Nematic liquid crystal, Optical switches, Optical waveguide, Spatial control

### I. INTRODUCTION

Liquid crystals (LCs) exhibit large electro-optic effects and high optical birefringence [1], which enable them to change the optical properties of waveguides more effectively than traditional electro-optic materials. Moreover, nematic liquid crystals (NLCs) possess characteristics such as fast response [2], low driving voltage [3], low cost [4], and high transmittance in the visible and near-infrared spectral regions [5]. These advantages render NLCs electro-optic materials with great application potential in photonics.

In recent years, as the design of new LC optical-waveguide structures has become a current research theme [6], the spatial polarization control of light beams by an NLC optical waveguide in the visible and near-infrared regions has drawn tremendous attention [615]. Asquini et al. [7] reported an LC-core waveguide with a SiO2/Si triangular-cross-section groove for controlling transverse magnetic (TM) polarized light, with a threshold voltage of 2 V. Wang et al. [9, 10] reported an LC-core channel waveguide with a semicircular groove, with a threshold voltage of about 4 V; this structure was not only expensive to manufacture [2], but also could only realize electrical tuning of the quasi-TM polarization mode. Kabanova et al. [11] reported a slab LC-core optical waveguide that could electrically control the propagation path of TM linearly polarized light; not only did the structure require photolithography to produce periodic comb electrodes, but also the slab waveguide was not conducive to integration [16]. Melnikova et al. [12] reported a slab LC optical waveguide with mutually orthogonal NLC directors in adjacent regions. The electrically switchable structure could split the unpolarized beam into two orthogonal polarization components to propagate in the waveguide independently, but the angle of incidence for the excitation light was severely limited by the condition of total internal reflection (TIR). Rutkowska et al. [13] reported an LC optical-waveguide structure to control the propagation direction of linearly polarized light, with an alternating multidomain LC orientation of 0° and 45°. However, the functional realization of this design not only depended on the nonlinear optical effects in LCs (e.g. optical-soliton propagation and self-focusing), but also required multistep ultraviolet (UV) exposure and periodic amplitude masks. Rushnova et al. [14] reported a slab LC optical waveguide with mutually alternating twist and planar orientations in adjacent areas, which could be set to operate in transmission mode or as an attenuator by changing the applied voltage. However, that waveguide structure could only tune transverse electric (TE) linearly polarized light, and required complicated orientation of multi-domains of the NLCs.

In this study we propose a structure, based on a channel NLC-core optical waveguide, for the switchable spatial control of linearly polarized light. The refractive indices of both left and right claddings in the waveguide are between the two principal-axis indices of the NLC, which is significantly different from previously reported designs [1719]. In the latter, the refractive indices of all claddings in the waveguide were less than the refractive index of the short axis of the NLC. When the initial excitation is TE (or TM) linearly polarized light, the operational mode of the proposed waveguide can always be switched by controlling the on and off states of the applied voltage. More specifically, this optical waveguide can operate in conventional transmission mode, or become an attenuator. The operating principle of the structure is discussed in the next section. The functionality of the waveguide is verified in the third section with a known full-vector finite-difference (FVFD) mode solver and beam-propagation method (BPM). The fourth section is a summary.

### II. PRINCIPLE

The simplified geometry of the proposed NLC-core optical waveguide is shown in Fig. 1(a), in which the width and thickness of the core region are w and h respectively. The material of the waveguide core is NLC E7 (Merck Co., NJ, USA). ne and no are the refractive indices of the two principal axes respectively, parallel and perpendicular to the LC director. The upper and lower boundaries of the waveguide’s core region both have a very thin photoalignment layer [not shown in Fig. 1(a)] to promote the homogeneous alignment of the E7. Additionally, both left and right cladding layers in the optical waveguide are composed of the isotropic negative photoresist AZ15nXT [2] with refractive index nAZ15nXT. Although the orientation layer at the sidewall cladding should also be considered, to ensure better orientation uniformity of the NLCs in the actual preparation, as previously reported [18] the ideal case is still considered here, for simplicity of theoretical analysis. The upper and lower isotropic cladding layers of the waveguide are both plexiglass (PMMA), with refractive index nPMMA. It is worth noting that the relation between the material refractive indices for the waveguide is shown in Eq. (1), which is key to realizing the functionality of the proposed NLC structure.

Figure 1. Schematic diagrams of the liquid crystal (LC) optical waveguide’s structure and LC orientation. (a) Schematic of the cross section of the nematic liquid crystal (NLC)-core optical waveguide, (b) deflection of the NLC director, (c) orientation of the NLC in different voltage states.

$nPMMA

where the value of each refractive index at the wavelength of 1.55 μm is shown in Table 1.

Refractive indices for the nematic liquid crystal (NLC)-core optical waveguide.

 Waveguide Materials E7 AZ15nXT PMMA Refractive Indices ne no nAZ15nXT nPMMA 1.697 [20] 1.502 [20] 1.58 [21] 1.481 [22]

### 2.1. Voltage off State (U = 0)

In the absence of an applied voltage between the indium tin oxide (ITO) electrodes, the anchoring effect of the photoalignment layer makes all of the NLC molecules align along the x-axis direction, as shown in Fig. 1(c). Therefore, TE linearly polarized light (where the electric field is parallel to the x-direction) will see the refractive index ne of the extraordinary wave (i.e. e wave), thereby exciting the quasi-TE mode in the waveguide. According to Eq. (1), the mode field of the quasi-TE mode can be confined within the core region of the NLC optical waveguide. However, TM linearly polarized light (where the electric field is parallel to the y-direction) sees the refractive index no of the ordinary wave (i.e. o wave), which will excite the quasi-TM mode in the NLC optical waveguide. Based on Eq. (1), the energy of the quasi-TM mode must be attenuated into the left and right claddings of the optical waveguide.

### 2.2. Voltage on State (U >> Uc)

When a voltage exceeding the threshold is applied between the ITO electrodes, the LC director $n^$ is deflected around the z-axis inside the xoy plane [23], as shown in Fig. 1(b). According to Frank-Oseen elastic continuum theory [24] and Euler-Lagrange variational theory [25], the deflection angle α(y) of the LC director can be obtained as follows:

$yh=UcπU∫0αy cos2φ+ k 33 /k 11 sin2φ sin2 αm − sin2φ dφ$

where $Uc=πk11/ε0Δε$ is the threshold voltage, U is the applied voltage, k11 and k33 are Frank elastic constants, ε0 is the dielectric permittivity of vacuum, and ∆ ε is the dimensionless dielectric anisotropy of the NLC. Also, αm is the maximum deflection angle of the LC director, the value of which can be determined by Eq. (3).

$UUc=2π∫0αm cos 2φ+ k 33/k 11 sin 2φ sin 2 α m − sin 2φ dφ$

k11, k33, and ∆ ε of the E7 are 11.1 pN, 17.1 pN, and 13.8 respectively [26]. Thus the threshold Uc can be calculated to be about 0.95 V. Figure 2 shows the distribution of the deflection angle of the LC director under different applied voltages.

Figure 2. Variation of the deflection angle of the nematic liquid crystal (NLC) director with applied voltage. (a) Maximum deflection angle αm; (b) the deflection angle α.

It can be seen from Fig. 2 that the maximum deflection angle αm of the NLC increases with increasing applied voltage. Moreover, the deflection angle of the LC director has a gradient distribution in the thickness direction (y-direction) of the LC cell, due to the effect of the surface anchoring energy. However, one can clearly see that this gradient property becomes very weak when the applied voltage is much greater than the threshold (i.e. U >> Uc). Besides, in practice, photoalignment materials with weak anchoring energy [27] can be used as alignment layers to further weaken the gradient effect of the deflection angle.

Therefore, the maximum deflection angle αm of the director can be approximated as the deflection angles of all NLC molecules, when the applied voltage is much larger than the threshold value (specifically, when the voltage is greater than or equal to 5 times the threshold). Under these conditions the NLC of the waveguide’s core region will approximately have the uniform dielectric tensor shown in Eq. (4).

$ε=ε0× ne2cos2αm+no2sin2αm cosαmsinαmne2−no2 0 cosαmsinαmne2−no2 ne2sin2αm+no2cos2αm 0 0 0 no2$

In particular, we consider the case where αm is equal to 90°, that is, the NLC molecules have transitioned from homogeneous orientation to homeotropic alignment, as shown in Fig. 1(c). The TE linearly polarized light still excites the quasi-TE mode in the NLC optical waveguide, but it sees the refractive index no of the o wave. According to Eq. (1), the energy of the quasi-TE mode must be attenuated into the left and right claddings of the optical waveguide. However, the TM linearly polarized light sees the refractive index ne of the e wave, thereby exciting the quasi-TM mode in the waveguide. Consequently, the mode field of the quasi-TM mode can be confined inside the core region of the NLC optical waveguide.

It is noteworthy that mode hybridization exists in the waveguide if and only if the dielectric tensor [ε] of the NLC has nonzero off-diagonal terms (i.e. 0° < αm < 90°) [18, 23, 28]. Yet, only the two cases of αm = 0° and 90° are involved in this study. Therefore, the above analysis of the quasi-TE and quasi-TM modes based on the relation to the refractive index is reasonable.

In light of all analysis above, in the absence of an applied voltage (off state, with αm = 0°) the NLC optical waveguide excited by the TE linearly polarized light operates in the conventional transmission mode, but that excited by the TM linearly polarized light can be used as an attenuator. However, when the applied voltage is much greater than the threshold (on state, with αm = 90°) the NLC optical waveguide excited by the TE linearly polarized light becomes an attenuator, but that excited by the TM linearly polarized light operates in the conventional transmission mode.

### III. NUMERICAL RESULTS

The FVFD mode solver [29] can be employed to calculate the eigenmodes in an NLC optical waveguide with a transversely anisotropic dielectric tensor [see Eq. (4)]. The size of the calculation window is 8 μm × 8 μm, including that the cross section of the waveguide core is 5 μm × 5 μm, the widths of both left and right claddings are 1.5 μm, and the cross sections of both upper and lower cladding layers are 8 μm × 1.5 μm. The effect of the photoalignment layer on the optical properties of the waveguide can be neglected, since it is sufficiently thin [18]. Moreover, the Dirichlet boundary condition [30] is applied to the calculation window.

Numerical results indicate that the NLC optical waveguide is multimode, as expected. In particular, the optical waveguide supports more than 20 eigenmodes at the wavelength of 1.55 μm. Nevertheless, numerous reports [18, 19, 28] have demonstrated that such transversely anisotropic NLC-core optical waveguides mainly excite low-order modes. Figure 3 shows the electric field distributions of the low-order eigenmodes supported by the NLC optical waveguide under two applied voltage states, including the first two quasi-TE modes and the first two quasi-TM modes.

Figure 3. Electric field distributions of the first two quasi-TE modes and the first two quasi-TM modes, with their effective refractive indices: (a)–(d) U = 0 (off state, with αm = 0°), (e)–(h) U >> Uc (on state, with αm = 90°).

One can see from Figs. 3(a)3(d) that in the absence of an applied voltage, the region of the electric field distribution of the quasi-TE mode and that of the quasi-TM mode are significantly different. The former is mainly concentrated in the core zone of the waveguide, while the latter is mainly concentrated in the left and right claddings of the waveguide. The reasons for this phenomenon are that (1) the refractive indices seen by TE and TM linearly polarized light are ne and no respectively, and (2) the materials of the NLC optical waveguide also satisfy the special relationship between refractive indices shown in Eq. (1). On the contrary, when the applied voltage is much greater than the threshold, it can be seen from Figs. 3(e) and 3(f) that the electric field distributions of the quasi-TE modes are mainly concentrated in the left and right claddings of the waveguide. Simultaneously, the electric field distributions of the quasi-TM modes are well confined inside the core of the waveguide, as shown in Figs. 3(g) and 3(h). As mentioned in section 2, the refractive index seen by the TE linearly polarized light has changed from ne to no, while the refractive index seen by the TM linearly polarized light has varied from no to ne. Thus as the state of the applied voltage turns from off to on, the main distribution regions of the quasi-TE mode and the quasi-TM mode are interchanged.

The above results indicate that the main distribution region of the eigenmode field of the quasi-TE (or quasi-TM) mode can be changed by switching the on and off states of the applied voltage. Moreover, one can also find from Fig. 3 that whether the state of the applied voltage is on or off, the main distribution zones of the quasi-TE and quasi-TM modes are always opposite. This means that only one quasilinear polarization mode can be stably supported inside the core region of the NLC optical waveguide.

Furthermore, to see the operational characteristics of the NLC optical waveguide proposed in this study more intuitively, it is necessary to solve for the propagation field of the waveguide. Because this multimode waveguide mainly excites low-order modes (i.e. it can support the paraxial approximation), the FVFD-BPM [31] can be employed to simulate the propagation of light. The initial excitation incident from the center of the waveguide is TE or TM linearly polarized Gaussian light with a half-width of 2.15 μm. The length L of the waveguide is 20 μm and the propagation step is 0.1 μm. The maximum effective refractive index of the fundamental mode obtained above is used as the reference index, to ensure that the slowly-varying-envelope approximation (SVEA) of the BPM is satisfied. Moreover, the transparent boundary condition (TBC) [32] is adopted, to suppress the reflection of the radiation mode from the boundaries of the calculation-domain window. Furthermore, the partial propagation loss is defined in Eq. (5) to describe the operational mode of the NLC optical waveguide better (that is, it does not include the absorption loss of the NLC, nor the loss when the light is coupled into or out of the waveguide).

$Loss=10log10Pout/Pin/L$

where Pin and Pout represent the optical powers at the input and output ends of the waveguide respectively. Considering the features of the main distribution region of the eigenmode in Fig. 3, Fig. 4 shows for different voltage states the propagation field distributions of the two linearly polarized light waves in the y section (i.e. xoz plane) of the waveguide.

Figure 4. Propagation field distribution of linearly polarized light in the LC optical waveguide under different voltage states: (a), (b) U = 0 (off state, with αm = 0°); (c), (d) U >> Uc (on state, with αm = 90°).

In the voltage off state (U = 0), as can be seen from Fig. 4(a) the propagation field excited by the TE linearly polarized light achieves conventional transmission, with a partial propagation loss of about −2.167 dB/cm; however, from Fig. 4(b) the NLC optical waveguide excited by the TM linearly polarized light already exhibits energy leakage at the output end of the waveguide’s core region significantly (i.e. it becomes an attenuator), with a partial propagation loss estimated as up to −183.237 dB/cm. In the voltage on state (U >> Uc), one can see from Fig. 4(c) that the propagation field excited by the TE linearly polarized light turns into strong attenuation, with partial propagation loss of about −246.175 dB/cm, yet from Fig. 4(d) the NLC optical waveguide excited by the TM linearly polarized light operates in the conventional transmission mode, for which the partial propagation loss is estimated to be as low as −1.075 dB/cm.

These results indicate that whether the initial excitation is TE linearly polarized light [see Figs. 4(a) and 4(c)] or TM linearly polarized light [see Figs. 4(b) and 4(d)], the operational mode can be always switched by controlling the on and off states of the applied voltage. More specifically, the NLC optical waveguide can be operated in transmission mode or as an attenuator. Moreover, comparing Fig. 4(a) to Fig. 4(b), and Fig. 4(c) to Fig. 4(d), one can find that the operational modes of TE and TM linearly polarized light are always opposite, regardless of whether the voltage state is on or off. This shows that for both applied voltage states only one type of linearly polarized light is permitted to pass through the core region of the NLC optical waveguide. Therefore, the proposed channel NLC-core structure can also be used as an integrated optical polarizer.

It is worth noting that the partial propagation losses of TE and TM linearly polarized light are not equal, regardless of whether the NLC optical waveguide operates in transmission mode [Figs. 4(a) and 4(d)] or as an attenuator [Figs. 4(b) and 4(c)]. This is because there is a certain difference between the eigenmode field of the quasi-TE mode and that of the quasi-TM mode in the waveguide (see Fig. 3). Consequently, Gaussian light waves with the same half-width but different polarization directions excite the eigenmodes to different extents. One can also find from Figs. 4(a) and 4(d) that the amplitude distribution of the electric field strength exhibits slight periodic like undulations in the process of propagation. This phenomenon occurs because, except for the fundamental modes in such multimode NLC optical waveguides, the light wave may also excite other low-order modes (e.g. the first-order mode), to a small extent.

### IV. CONCLUSION

This research proposes a channel-type LC-core optical waveguide to realize the spatial control and switching of linearly polarized light, in which the refractive indices of both left and right claddings of the waveguide are between the two principal-axis indices of the NLC. The structure can not only switch operational mode (i.e. achieve conventional transmission or act as an attenuator) of TE/TM linearly polarized light by controlling the on and off states of the applied voltage, but also can be used as an integrated optical polarizer. This design mainly includes the following advantages: (1) The proposed structure is a rectangular waveguide, which is more beneficial to integration due to its cross section being smaller than that of a slab waveguide; (2) both TE and TM linearly polarized light waves can be tuned by the structure; and (3) the structure does not require periodic electrodes, nor a complicated multidomain orientation of the NLC. Furthermore, the proposed structure has great potential applications in optical wavelength division multiplexing (WDM) networks [33] and related research based on linearly polarized light [34].

### DISCLOSURES

The authors declare no conflicts of interest.

### DATA AVAILABILITY

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.

### FUNDING

Applied Basic Research Project of Science and Technology Department of Sichuan Province (2014JY0024); Scientific Research Fund of Science and Technology Bureau of Nanchong City (19YFZJ0090); Scientific Research Foundation of China West Normal University (17YC056).

### Fig 1.

Figure 1.Schematic diagrams of the liquid crystal (LC) optical waveguide’s structure and LC orientation. (a) Schematic of the cross section of the nematic liquid crystal (NLC)-core optical waveguide, (b) deflection of the NLC director, (c) orientation of the NLC in different voltage states.
Current Optics and Photonics 2023; 7: 83-89https://doi.org/10.3807/COPP.2023.7.1.83

### Fig 2.

Figure 2.Variation of the deflection angle of the nematic liquid crystal (NLC) director with applied voltage. (a) Maximum deflection angle αm; (b) the deflection angle α.
Current Optics and Photonics 2023; 7: 83-89https://doi.org/10.3807/COPP.2023.7.1.83

### Fig 3.

Figure 3.Electric field distributions of the first two quasi-TE modes and the first two quasi-TM modes, with their effective refractive indices: (a)–(d) U = 0 (off state, with αm = 0°), (e)–(h) U >> Uc (on state, with αm = 90°).
Current Optics and Photonics 2023; 7: 83-89https://doi.org/10.3807/COPP.2023.7.1.83

### Fig 4.

Figure 4.Propagation field distribution of linearly polarized light in the LC optical waveguide under different voltage states: (a), (b) U = 0 (off state, with αm = 0°); (c), (d) U >> Uc (on state, with αm = 90°).
Current Optics and Photonics 2023; 7: 83-89https://doi.org/10.3807/COPP.2023.7.1.83

Table 1 Refractive indices for the nematic liquid crystal (NLC)-core optical waveguide

 Waveguide Materials E7 AZ15nXT PMMA Refractive Indices ne no nAZ15nXT nPMMA 1.697 [20] 1.502 [20] 1.58 [21] 1.481 [22]

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Wonshik Choi,
Editor-in-chief