1. Introduction

Liquid crystal sensors technology has developed enthusiastically over the past decade. For instance, due to the increasing demands on sensors that meet the complexity of targeting biochemical agents, LC sensors have grown to become promising and expansive field of research and development in biochemical sensing applications. In these sensors, LCs have been demonstrated to be excellent, accurate, fast, and high sensitive candidates in detecting targeted chemical and biological analytes. The LC molecular alignment is altered by the presence of these analytes, where the surface driven LC orientational changes have proven to be highly effective in amplifying the presence of these analytes [1]. The long-range orientational order, high anisotropy, and the collective behavior of the nematic LC molecules allow for extremely low levels of targeted agents. Recently, LC sensors have also been involved in other applications such as optical filters, modulators and photonic crystal sensors, however, the research in these fields is promising and not yet elaborated in the literature, see for example [2] and [3]. Optical transduction mechanisms, with imaging capability and visual inspection, are common techniques used in LC sensors to detect the distortion of the LC film, e.g. [4]. An autonomous system utilizing capacitive transduction technique has been recently developed to track the LC profile deformations. Eliminating the visual inspection requirement in detecting the LC distortions lead to a simpler system with autonomous operation and reduced possible false alarms [5]. Tracking the LC director is also accomplishable using optical transduction methods. In [6], an optical technique utilizes the critical angle was developed to monitor the director axis in well ordered LC. Although the effort in [6] is limited to well ordered systems, LC films in many practical sensors are partially ordered. When a reduction in ordering occurs, this method can produce misleading results. Therefore, the contribution of the order parameter should be considered in the critical angles’ measurements. This paper aims to monitor a partially ordered nematic LC profile optically. Moreover, this paper engages the input of the order parameter along with the average molecular orientation in studying the LC profile via optical transduction procedure, that utilizes the TIR phenomenon, where a visual inspection is no longer required. In addition, this paper treats the average LC director axis and the incident light wave as arbitrary vectors in the space.

2. Liquid crystal profile

Liquid crystals have unique behavior as their several distinct optical properties exhibit interesting changes when subjected to external stimuli. In nematic LCs, optical field propagating in parallel or perpendicular with the molecule experiences the extraordinary, n_e, and the ordinary, n_o, refractive indices, respectively. The average orientation of the molecules is represented by the director axis n, as in Fig. 1, and can be expressed as

where θ and ϕ are the zenithal and azimuthal angle, respectively. A scalar quantity called “order parameter” and denoted by S is used to quantify the degree of the molecular ordering over the ensemble, where S has values in the range 0≤S≤1. Taking S into account, the average ordinary and extraordinary refractive indices in the principal axes, are given by [7]

Where n̄=(n_o+n_e)/2, Δn=n_e-n_o. For a light wave propagating in a medium with refractive index n, the wavenormal k⃗, as in Fig. 1, can be given as

where |k⃗|=nω/c, θ_k and ϕ_k are the zenithal and the azimuthal angles describing k⃗, respectively. In partially ordered LC, the effective refractive index can be expressed as [7]

where k̂ is a unit vector in the direction of k⃗, and k̂·n is the dot product between k̂ and n. To track the director axis with respect to the lab frame axes xyz, we need first to translate the refractive indices from the principal axes into the xyz axes, and take the average over the total ensemble. As a result, the individual refractive indices in the lab frame axes are given by [5]

with n ² _is=(2n ² _o+n ² _e)/3, where n_is is the LC refractive index in the isotropic phase. In fact, the LC, in general, is inhomogeneous. However, by statistically averaging the molecules’ orientations and evaluating the average ordering (order parameter), the LC can be treated as homogenous (in average), yet, it is anisotropic [5]. These refractive indices will be utilized to calculate the critical angles at an isotropic/LC interface, and then to track the LC profile parameters.

 

Fig. 1. The director axis, n, and the propagation wave vector, k⃗, in the lab frame axes, xyz.

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3. Critical angles calculation

When light wave, with s and p polarization states, strikes the boundary between an isotropic and anisotropic LC medium, the LC film will support double refraction. The refracted wave will have two polarization states and it is a mixture of the ordinary, O-wave, and extraordinary, E-wave. We will treat the plane of incidence as the plane twisted by ϕ_k out of the xz plane (counter clockwise), see Fig. 1, where the interface between the two media is the xy plane. Let ni be the refractive index of the isotropic medium and θ_i, θ_o, and θ_e are the angles of incidence and the refracted O- and E- wavenormals, respectively. These parameters will be used to evaluate the critical angles in both modes.

3.1. Ordinary mode critical angle

The refracted ordinary wavenormal is coincident with the associated light ray, and its direction is independent from the director axis orientation, yet, it depends on the molecular degree of ordering. Applying Snell’s law at the boundary between the incident wave and the refracted O-wave normal yields

In this mode, we can investigate the existence of the TIR in three cases. (i) When n_i<n_o, n_o TIR can be achieved, (ii) when n_i>n_is, TIR is achievable at any degree of ordering, and (iii) when n_o≤n_i≤n_is, the existence of the TIR depends on S. In the later case, TIR can be achieved only in the range 1≥S≥3(n ² _is-n ² _i)/(2n̄Δn), where the associated critical angle is sin^-1(n_o/n_i)≤θ_co<π/2. The ordinary critical angle (when existed) is given by

The minimum achievable ordinary critical angle is θ _co,min=sin^-1(n_o/n_i) at which S=1, where the maximum is θ _co,max=sin^-1(n_is/n_i), at which S=0. As an example, for LC E7 at 633 nm, the refractive indices are (n_e,n_o)=(1.7472,1.5217). When the isotropic medium is flint glass (29% lead) with n_i=1.569, critical angles exist when the ordering degree is in the range 1≥S≥0.405, where the ordinary critical angle is in the range 75.9°≤θ_co≤90°. If the isotropic medium is selected to be a flint glass (55% lead) with n_i=1.669, critical angles exist at all ordering states and their values are bounded by 65.75°≤θ_co≤73.52°. This discussion is also helpful in selecting the isotropic medium. The ordinary critical angle is crucial quantity in partially ordered systems as it helps in tracking the order parameter which can be expressed as

Although the ordinary critical angle provides information about the order parameter, it provides no information about the director axis, which impacts the extraordinary critical angle, as we will see next.

3.2. Extraordinary mode critical angle

The refracted extraordinary wavenormal satisfies Snell’s law as

where n _es,eff as in Eq. (4). It is now obvious that the direction of the E-wavevector depends on the director axis orientation as well as the ordering degree. Solving Eq. (9) for θ_e is not as easy as the ordinary case, since n _es,eff also depends on θ_e. Substituting Eq. (4) in Eq. (9), and solving for θ_e gives

where Γ=Sn̄Δnsin2θ cos(ϕ-ϕ_k) and -n̄Δn≤Γ≤n̄Δn. From Eq. (10), we notice that when Γ≥0, the denominator can be zero, therefore, θ _e,max=90°. In fact, Γ=0 occurs when either S=0 (full disorder), n‖ẑ, n⊥ẑ, or when the plane of incidence is orthogonal with the nẑ plane (i.e. |ϕ-ϕ_k|=π/2). On the other hand, when Γ<0, the maximum extraordinary angle of refraction, at which the critical angle occurs, is θ _e,max <90°. In this case, when the extraordinary mode is excited with the critical angle, the ray travels in the interface, however, the wavevector has a maximum angle of θ _e,max=tan^-1(n̄² _zz/|Γ|), which shows that, in general, tan^-1[(n ² _e+n ² _o)/(n ² _e -n ² _o)]≤θ _e,max≤90°. Under these conditions, solving Eq. (10) for the extraordinary critical angle gives

where m=0 when Γ≥0 and m=1 when Γ<0. Let us next check the impact of the isotropic medium on this critical angle. (i) When n_i < n_o, no TIR can be achieved, and (ii) when n_i > n_e, TIR can be obtained at any degree of ordering, yet, θ_ce depends on θ, ϕ, and S. (iii) When n_o≤n_i≤n_e, the existence of critical angles depends on the LC profile parameters. The minimum achievable extraordinary critical angle is θ _ce,min=sin^-1[n_o/n_i], which occurs when the LC film is well ordered, and the director is in the plane of incidence and coplaner with the interface, i.e. (θ=π/2). The maximum extraordinary critical angle is given by θ _ce,max=sin^-1[n_e/n_i] and occurs when the LC is well ordered and the director is orthogonal with the plane of incidence, |ϕ-ϕ_k|=π/2, or when the director is on the plane of incidence and perpendicular on the boundary (θ=0). In these cases, the critical angle occurs when θ _e,max=π/2. For the LC E7, when the substrate is flint glass (29% or 55% lead), the existence of θ_ce depends on S, θ, and ϕ. In both cases, |Γ|≤0.3686 and 82.2°≤θ _e,max ≤ 90°, where θ _e,max=82.2° occurs when S=1 and the director is in the plane of incidence and tilted by θ=48.93°.

4. LC profile monitoring

Based on the previous discussion, one ordinary critical angle measurement can track the order parameter, however, two extraordinary critical angles in two orthogonal directions can uniquely track the director axis. To accomplish this, we propose the sensor structure as shown in Fig. 2 to be utilized in tracking the LC profile parameters.

 

Fig. 2. Schematic of the LC sensor experimental arrangement, (a) Side view (b) Top view

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In this sensor, the LC is selected as a positive nematic, where the pyramid prism and the substrate, as in Fig. 2(a), are isotropic with equal refractive indices n_i, such that n_i>n_e>n_o. A circularly polarized laser beam is split into two beams using 50:50 non polarizing beam splitter. Circularly polarized light enables an equal excitation of the ordinary and extraordinary waves for each of the orthogonally directed beams. The two beams are directed to strike the LC film sandwiched between the prism and the substrate. In Fig. 2, k⃗_i1 and k⃗_i2 represent the two incident wavevectors in the xz plane (ϕ_k=π) and the yz plane (ϕ_k=3π/2)), respectively, and θ _i1 and θ _i2 are the corresponding angles of incidence. The ordinary angles of refraction are θ _o1=θ _o2, and the extraordinary angles of refraction are θ _e1 and θ _e2, see Fig. 3(a). One way to track the refracted waves is to monitor the transmission behavior. The transmission of beam 1 drops to approximately half power at the critical angle for the ordinary ray and then drops to zero at the critical angle for the extraordinary ray (since θ_ce>θ_co). Likewise, the transmission of beam 2 will respond in a similar manner, as shown in Fig. 3(b). In this case, the ordinary critical angles in both directions are given by Eq. (7), as for the extraordinary critical angles can be obtained when the angles of refraction, as in Fig. 3, are

and the corresponding extraordinary critical angles are given by

It is shown from Eq. (13) and Eq. (5) that the extraordinary critical angles depend on S and the director orientation. Solving Eq. (13) for θ and ϕ will give a unique solution in each octant.

5. Simulation and results

In this section, we will investigate the proposed method in tracking the LC profile parameters and study the sensitivity of this transduction mechanism. With LC E7, let us choose the pyramid prism from Germanate (Schott IRG 2) glass with n_i=1.9 at 633 nm. This selection of the prism will allow in tracking the LC profile at all degrees of ordering with the ordinary critical angle is bounded by 53.21°≤θ_co≤57.38° compared to the extraordinary critical angle 53.21°≤θ_ce≤66.86°. Figure 4 shows the sensitivity of the extraordinary critical angles versus ϕ at selected order parameters when the LC is homogeneously aligned, i.e. θ=π/2. The simulation results show that smaller prism index of refraction will result in greater difference between the minimum and the maximum allowable ordinary and extraordinary critical angles as in Fig. 4(b), however, n_i cannot be less than n_e. A major issue in this method is in the case of weak anisotropy, i.e. S is small. In this case, θ_ce and θ_co will become more closer to each other, yet, the difference is still recognizable. For instance, when θ=0 and |ϕ-ϕ_k|=π, θ_ce-θ_co=13.6° at full ordering compared to 2.6° at S=0.2, where this difference can still be measured accurately by precise angle measurement tool.

 

Fig. 3. (a) The incident and extraordinary refracted wave vectors in orthogonal planes and (b) the transmitted power diagram, assuming θ _ce1<θ _ce2.

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Fig. 4. The extraordinary critical angles sensitivity versus ϕ when θ=π/2, at different ordering degrees, and Δθ_co and Δθ_ce versus the prism refractive index, n_i.

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As an example of tracking the LC profile, let the measured ordinary critical angle be θ _co=54.73° which gives S=0.63, where the measured extraordinary critical angles θ _ce1=63.03° and θ _ce2=56.74° results in the director orientation of (θ,ϕ)=(60.14°,84.89°).

6. Conclusion

An optical transduction mechanism that utilizes TIR phenomenon in tracking the LC profile in partially ordered LC sensors has been presented. It is been proven that measuring the critical angle of the ordinary refracted wave can provide information about the LC ordering degree. On the other hand, measuring critical angles of two refracted extraordinary wave vectors in two orthogonal planes can uniquely track the director axis orientation. This method has potential applications in LC biological and chemical sensors.