## Utilizing critical angles in sensing partially ordered liquid crystal profile

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

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]

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]

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

## 2. Liquid crystal profile

*n*, and the ordinary,

_{e}*n*, refractive indices, respectively. The average orientation of the molecules is represented by the director axis

_{o}**n**, as in Fig. 1, and can be expressed as

*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]

*n*̄=(

*n*+

_{o}*n*)/2, Δ

_{e}*n*=

*n*-

_{e}*n*. For a light wave propagating in a medium with refractive index

_{o}*n*, the wavenormal

**k**⃗, as in Fig. 1, can be given as

**k**⃗|=

*nω*/

*c*,

*θ*and

_{k}*ϕ*are the zenithal and the azimuthal angles describing

_{k}**k**⃗, 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]

**k**̂ is a unit vector in the direction of

**k**⃗, and

**k**̂·

**n**is the dot product between

**k**̂ 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]

*n*

^{2}

*=(2*

_{is}*n*

^{2}

*+*

_{o}*n*

^{2}

*)/3, where*

_{e}*n*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

_{is}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]

## 3. Critical angles calculation

*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*out of the

_{k}*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*, and

_{o}*θ*are the angles of incidence and the refracted

_{e}*O*- and

*E*- wavenormals, respectively. These parameters will be used to evaluate the critical angles in both modes.

## 3.1. Ordinary mode critical angle

*O*-wave normal yields

*n*<

_{i}*n*,

_{o}*n*TIR can be achieved, (ii) when

_{o}*n*>

_{i}*n*, TIR is achievable at any degree of ordering, and (iii) when

_{is}*n*≤

_{o}*n*≤

_{i}*n*, the existence of the TIR depends on

_{is}*S*. In the later case, TIR can be achieved only in the range 1≥

*S*≥3(

*n*

^{2}

*-*

_{is}*n*

^{2}

*)/(2*

_{i}*n*̄Δ

*n*), 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

*θ*

_{co,min}=sin

^{-1}(

*n*/

_{o}*n*) at which

_{i}*S*=1, where the maximum is

*θ*

_{co,max}=sin

^{-1}(

*n*/

_{is}*n*), at which

_{i}*S*=0. As an example, for LC E7 at 633 nm, the refractive indices are (

*n*,

_{e}*n*)=(1.7472,1.5217). When the isotropic medium is flint glass (29% lead) with

_{o}*n*=1.569, critical angles exist when the ordering degree is in the range 1≥

_{i}*S*≥0.405, where the ordinary critical angle is in the range 75.9°≤

*θ*≤90°. If the isotropic medium is selected to be a flint glass (55% lead) with

_{co}*n*=1.669, critical angles exist at all ordering states and their values are bounded by 65.75°≤

_{i}*θ*≤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

_{co}## 3.2. Extraordinary mode critical angle

*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*is not as easy as the ordinary case, since

_{e}*n*

_{es,eff}also depends on

*θ*. Substituting Eq. (4) in Eq. (9), and solving for

_{e}*θ*gives

_{e}*Sn*̄Δ

*n*sin2

*θ*cos(

*ϕ*-

*ϕ*) and -

_{k}*n*̄Δ

*n*≤Γ≤

*n*̄Δ

*n*. 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**‖

**z**̂,

**n**⊥

**z**̂, or when the plane of incidence is orthogonal with the

**nz**̂ 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}(

*n*̄

^{2}

*/|Γ|), which shows that, in general, tan*

_{zz}^{-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

*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*, no TIR can be achieved, and (ii) when

_{o}*n*>

_{i}*n*, TIR can be obtained at any degree of ordering, yet,

_{e}*θ*depends on

_{ce}*θ*,

*ϕ*, and

*S*. (iii) When

*n*≤

_{o}*n*≤

_{i}*n*, the existence of critical angles depends on the LC profile parameters. The minimum achievable extraordinary critical angle is

_{e}*θ*

_{ce,min}=sin

^{-1}[

*n*/

_{o}*n*], 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. (

_{i}*θ*=

*π*/2). The maximum extraordinary critical angle is given by

*θ*

_{ce,max}=sin

^{-1}[

*n*/

_{e}*n*] and occurs when the LC is well ordered and the director is orthogonal with the plane of incidence, |

_{i}*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*depends on

_{ce}*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

*n*, such that

_{i}*n*>

_{i}*n*>

_{e}*n*. 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,

_{o}**k**⃗

_{i1}and

**k**⃗

_{i2}represent the two incident wavevectors in the

*xz*plane (

*ϕ*=

_{k}*π*) and the

*yz*plane (

*ϕ*=3

_{k}*π*/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*). 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

_{co}## 5. Simulation and results

*n*=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°≤

_{i}*θ*≤57.38° compared to the extraordinary critical angle 53.21°≤

_{co}*θ*≤66.86°. Figure 4 shows the sensitivity of the extraordinary critical angles versus

_{ce}*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*cannot be less than

_{i}*n*. A major issue in this method is in the case of weak anisotropy, i.e.

_{e}*S*is small. In this case,

*θ*and

_{ce}*θ*will become more closer to each other, yet, the difference is still recognizable. For instance, when

_{co}*θ*=0 and |

*ϕ*-

*ϕ*|=

_{k}*π*,

*θ*-

_{ce}*θ*=13.6° at full ordering compared to 2.6° at

_{co}*S*=0.2, where this difference can still be measured accurately by precise angle measurement tool.

*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

## 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 |

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 |

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. |

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. |

6. | F. Yang and J. R. Sambles“Critical angles for reflectivity at an isotropic-anisotropic boundary,” J. Mod. Opt. |

7. | A. S. Abu-Abed, “Optical waves in partially ordered anisotropic media,” Opt. Express |

**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

- 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]
- 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]
- 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]
- 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).
- 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]
- F. Yang and J. R. Sambles, "Critical angles for reflectivity at an isotropic-anisotropic boundary," J. Mod. Opt. 40, 1131-1142 (1993). [CrossRef]
- A. S. Abu-Abed, "Optical waves in partially ordered anisotropic media," Opt. Express 17, 1646-1651 (2009). [CrossRef] [PubMed]

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