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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 26 — Dec. 21, 2009
  • pp: 23729–23735
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Utilizing critical angles in sensing partially ordered liquid crystal profile

Alaeddin S. Abu-Abed and Robert G. Lindquist  »View Author Affiliations


Optics Express, Vol. 17, Issue 26, pp. 23729-23735 (2009)
http://dx.doi.org/10.1364/OE.17.023729


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Abstract

This paper investigates a new approach for tracking nematic uniaxial liquid crystal (LC) profile in partially ordered LC based sensors. This approach utilizes measuring critical angles for total internal reflection (TIR) at the interface of optically isotropic and partially ordered LC film. The proposed optical transduction requires measuring of the ordinary critical angle and two extraordinary critical angles in orthogonal directions to report the LC degree of ordering and the director axis orientation.

© 2009 Optical Society of America

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

1. B. H. Clare and N. L. Abbott, “Orientations of nematic liquid crystals on surfaces presenting controlled densities of peptides: amplification of protein-peptide binding events,” Langmuir 21, 6451–6461 (2005). [CrossRef] [PubMed]

]. 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

2. H. Zhang, P. Guo, P. Chen, S. Chang, and J. Yuan, “Liquid-crystal-filled photonic crystal for terahertz switch and filter,” J. Opt. Soc. Am. B 26, 101–106 (2009). [CrossRef]

] and [3

3. T. Woliński, A. Czapla, S. Ertman, M. Tefelska, A. Domański, J. Wójcik, E. Kruszelnicki, and R. Dabrowski, “Photonic liquid crystal fibers for sensing applications,” IEEE Trans. Instrum. Meas. 57, 1796–1802 (2008). [CrossRef]

]. 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

4. S. Sridharamurthy, K. Cadwell, N. Abbott, and H. Jiang, “A Liquid crystal based gas sensor using microfabricated pillar arrays as a support structure,” Proc. IEEE Sensors Conference, 1044–1047 (2007).

]. 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

5. A. S. Abu-Abed and R. G. Lindquist, “Capacitive transduction for liquid crystal based sensors, part II: partially disordered systems,” IEEE Sens. J. 8, 1557–15642008). [CrossRef]

]. Tracking the LC director is also accomplishable using optical transduction methods. In [6

6. F. Yang and J. R. Sambles“Critical angles for reflectivity at an isotropic-anisotropic boundary,” J. Mod. Opt. 40, 1131–1142 (1993). [CrossRef]

], an optical technique utilizes the critical angle was developed to monitor the director axis in well ordered LC. Although the effort in [6

6. F. Yang and J. R. Sambles“Critical angles for reflectivity at an isotropic-anisotropic boundary,” J. Mod. Opt. 40, 1131–1142 (1993). [CrossRef]

] 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, ne, and the ordinary, no, 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

n=sinθcosϕx̂+sinθsinϕŷ+cosθẑ
(1)

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

7. A. S. Abu-Abed, “Optical waves in partially ordered anisotropic media,” Opt. Express 17, 1646–1651 (2009). [CrossRef] [PubMed]

]

nos=[2no2+ne23S2n̄Δn3]12,nes=[2no2+ne23+S4n̄Δn3]12
(2)

Where n̄=(no+ne)/2, Δn=ne-no. For a light wave propagating in a medium with refractive index n, the wavenormal k⃗, as in Fig. 1, can be given as

k=k(sinθkcosϕkx̂+sinθksinϕkŷ+cosθkẑ)
(3)

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

7. A. S. Abu-Abed, “Optical waves in partially ordered anisotropic media,” Opt. Express 17, 1646–1651 (2009). [CrossRef] [PubMed]

]

nes,eff=[nos2nes2nos2+2Sn̄Δn(k̂n)2]12
(4)

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

5. A. S. Abu-Abed and R. G. Lindquist, “Capacitive transduction for liquid crystal based sensors, part II: partially disordered systems,” IEEE Sens. J. 8, 1557–15642008). [CrossRef]

]

n̄xx2=nis2+S2n̄Δn(sin2θcos2ϕ13)n̄xy2=n̄yx2=S2n̄Δnsin2θsinϕcosϕ
n̄yy2=nis2+S2n̄Δn(sin2θsin2ϕ13)n̄yz2=n̄zy2=S2n̄Δnsinθcosθsinϕ
n̄zz2=nis2+S2n̄Δn(cos2θ13)n̄xz2=n̄zx2=S2n̄Δnsinθcosθcosϕ
(5)

with n 2 is=(2n 2 o+n 2 e)/3, where nis 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

5. A. S. Abu-Abed and R. G. Lindquist, “Capacitive transduction for liquid crystal based sensors, part II: partially disordered systems,” IEEE Sens. J. 8, 1557–15642008). [CrossRef]

]. 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.

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 ϕk 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 θi, θo, and θe 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

nisinθi=nossinθo
(6)

In this mode, we can investigate the existence of the TIR in three cases. (i) When ni<no, no TIR can be achieved, (ii) when ni>nis, TIR is achievable at any degree of ordering, and (iii) when noninis, 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(no/ni)≤θco<π/2. The ordinary critical angle (when existed) is given by

θco=sin1[2no2+ne2S2n̄Δn3ni2]12
(7)

The minimum achievable ordinary critical angle is θ co,min=sin-1(no/ni) at which S=1, where the maximum is θ co,max=sin-1(nis/ni), at which S=0. As an example, for LC E7 at 633 nm, the refractive indices are (ne,no)=(1.7472,1.5217). When the isotropic medium is flint glass (29% lead) with ni=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°≤θco≤90°. If the isotropic medium is selected to be a flint glass (55% lead) with ni=1.669, critical angles exist at all ordering states and their values are bounded by 65.75°≤θco≤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

S=3nis23ni2sin2θco2n̄Δn
(8)

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

nisinθi=nes,effsinθe
(9)

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 θe is not as easy as the ordinary case, since n es,eff also depends on θe. Substituting Eq. (4) in Eq. (9), and solving for θe gives

θe=tan1[n̄zz2Γ+csc2θinos2nes2n̄zz2ni2n̄zz2(n̄xy2sin2ϕk+n̄yy2sin2ϕk+n̄xx2cos2ϕk)+Γ2]
(10)

where Γ=Sn̄Δnsin2θ cos(ϕ-ϕk) 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), nẑ, nẑ, or when the plane of incidence is orthogonal with the nẑ plane (i.e. |ϕ-ϕk|=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

θce=sin1[nos2nes2n̄zz2ni2[n̄zz2(n̄xy2sin2ϕk+n̄yy2sin2ϕk+n̄xx2cos2ϕk)mΓ2]]12
(11)

4. LC profile monitoring

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

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 ni, such that ni>ne>no. 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, ki1 and ki2 represent the two incident wavevectors in the xz plane (ϕk=π) and the yz plane (ϕk=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 θce>θco). 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

θel,max(ϕk=π)=tan1(n̄zz2n̄xz2)andθe2,max(ϕk=3π2)=tan1(n̄zz2n̄yz2)
(12)

and the corresponding extraordinary critical angles are given by

θce1=sin1nesn̄zzni(n̄zz2+n̄xx2nos2)12andθce2=sin1nesn̄zzni(n̄zz2+n̄yy2nos2)12
(13)

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 ni=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°≤θco≤57.38° compared to the extraordinary critical angle 53.21°≤θce≤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, ni cannot be less than ne. A major issue in this method is in the case of weak anisotropy, i.e. S is small. In this case, θce and θco will become more closer to each other, yet, the difference is still recognizable. For instance, when θ=0 and |ϕ-ϕk|=π, θce-θco=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.
Fig. 4. The extraordinary critical angles sensitivity versus ϕ when θ=π/2, at different ordering degrees, and Δθco and Δθce versus the prism refractive index, ni.

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.

References and links

1.

B. H. Clare and N. L. Abbott, “Orientations of nematic liquid crystals on surfaces presenting controlled densities of peptides: amplification of protein-peptide binding events,” Langmuir 21, 6451–6461 (2005). [CrossRef] [PubMed]

2.

H. Zhang, P. Guo, P. Chen, S. Chang, and J. Yuan, “Liquid-crystal-filled photonic crystal for terahertz switch and filter,” J. Opt. Soc. Am. B 26, 101–106 (2009). [CrossRef]

3.

T. Woliński, A. Czapla, S. Ertman, M. Tefelska, A. Domański, J. Wójcik, E. Kruszelnicki, and R. Dabrowski, “Photonic liquid crystal fibers for sensing applications,” IEEE Trans. Instrum. Meas. 57, 1796–1802 (2008). [CrossRef]

4.

S. Sridharamurthy, K. Cadwell, N. Abbott, and H. Jiang, “A Liquid crystal based gas sensor using microfabricated pillar arrays as a support structure,” Proc. IEEE Sensors Conference, 1044–1047 (2007).

5.

A. S. Abu-Abed and R. G. Lindquist, “Capacitive transduction for liquid crystal based sensors, part II: partially disordered systems,” IEEE Sens. J. 8, 1557–15642008). [CrossRef]

6.

F. Yang and J. R. Sambles“Critical angles for reflectivity at an isotropic-anisotropic boundary,” J. Mod. Opt. 40, 1131–1142 (1993). [CrossRef]

7.

A. S. Abu-Abed, “Optical waves in partially ordered anisotropic media,” Opt. Express 17, 1646–1651 (2009). [CrossRef] [PubMed]

OCIS Codes
(130.6010) Integrated optics : Sensors
(160.3710) Materials : Liquid crystals
(230.3720) Optical devices : Liquid-crystal devices
(260.1440) Physical optics : Birefringence

ToC Category:
Optical Devices

History
Original Manuscript: August 31, 2009
Revised Manuscript: November 3, 2009
Manuscript Accepted: November 5, 2009
Published: December 11, 2009

Citation
Alaeddin S. Abu-Abed and Robert G. Lindquist, "Utilizing critical angles in sensing partially ordered liquid crystal profile," Opt. Express 17, 23729-23735 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-23729


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References

  1. B. H. Clare and N. L. Abbott, "Orientations of nematic liquid crystals on surfaces presenting controlled densities of peptides: amplification of protein-peptide binding events," Langmuir 21, 6451-6461 (2005). [CrossRef] [PubMed]
  2. H. Zhang, P. Guo, P. Chen, S. Chang, and J. Yuan, "Liquid-crystal-filled photonic crystal for terahertz switch and filter," J. Opt. Soc. Am. B 26, 101-106 (2009). [CrossRef]
  3. T. Woliński, A. Czapla, S. Ertman, M. Tefelska, A. Domański, J. Wójcik, E. Kruszelnicki, and R. Dabrowski, "Photonic liquid crystal fibers for sensing applications," IEEE Trans. Instrum. Meas. 57, 1796-1802 (2008). [CrossRef]
  4. S. Sridharamurthy, K. Cadwell, N. Abbott, and H. Jiang, "A Liquid crystal based gas sensor using microfabricated pillar arrays as a support structure," Proc. IEEE Sensors Conference, 1044-1047 (2007).
  5. A. S. Abu-Abed and R. G. Lindquist, "Capacitive transduction for liquid crystal based sensors, part II: partially disordered systems," IEEE Sens. J. 8, 1557-15642008). [CrossRef]
  6. F. Yang and J. R. Sambles, "Critical angles for reflectivity at an isotropic-anisotropic boundary," J. Mod. Opt. 40, 1131-1142 (1993). [CrossRef]
  7. A. S. Abu-Abed, "Optical waves in partially ordered anisotropic media," Opt. Express 17, 1646-1651 (2009). [CrossRef] [PubMed]

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