Abstract

We describe a novel polarization interferometer which permits the determination of the refractive indices for circularly-polarized light. It is based on a Jamin-Lebedeff interferometer, modified with waveplates, and permits us to experimentally determine the refractive indices nL and nR of the respectively left- and right-circularly polarized modes in a cholesteric liquid crystal. Whereas optical rotation measurements only determine the circular birefringence, i.e. the difference (nL – nR), the interferometer also permits the determination of their absolute values. We report refractive indices of a cholesteric liquid crystal in the region of selective (Bragg) reflection as a function of temperature.

© 2014 Optical Society of America

1. Introduction

In order to determine the refractive index n of a material one can utilize a polarization interferometer which has two orthogonal polarization states in its arms that respectively pass through a sample and a reference. The first interferometer of this kind was developed by Jamin in 1856 [1,2] and improved later by Lebedeff [2–5]. Conventionally, the Jamin-Lebedeff interferometer uses orthogonal linear polarization states and is able to determine refractive indices of crystals and biological materials [3,6], improve the contrast in microscopic imaging [7,8] and measure refractive indices of metallic nanostructures [9].

Common-path interferometers exist that take advantage of circularly polarization states and that are able to measure not only conventional refractive indices [10–12], but also the circular birefringence, i.e. the difference (nL – nR) [13]. However, an interferometer that can determine the circular-polarization (CP) dependent refractive indices nL and nR has, to the best of our knowledge, not been reported. Here, L and R refer to respectively the left and right polarization states.

The refractive indices of an optically active medium can be directly determined by measuring the angles of refraction for each circular polarization component, if the medium is isotropic [14,15]. However, this is not possible, if the medium is a thin-film or if it has uniaxial symmetry, where it becomes necessary to use an ellipsometer or an interferometer.

The goal of this work is to demonstrate a polarization interferometer which is able to provide information of circularly polarized modes appearing in uniaxial optically active media. It allows us to determine the circular polarization states in a cholesteric liquid crystal (CLC), which constitutes a periodic chiral medium. We also show that changing the periodicity of the cholesteric liquid crystal with temperature allows us to measure the dispersion of nL and nR in the Bragg regime, even though the interferometer only uses monochromatic light at one wavelength.

The paper is organized as follows: Section 2 gives a short introduction to CLCs. Section 3 describes in detail the polarization interferometer that we have developed. Experimental details of the sample are found in section 4. The main experimental results concerning the refractive indices and optical activity of a CLC are presented in section 5, and our conclusions appear in section 6.

2. Cholesteric liquid crystals

Liquid crystals (LCs) are partially ordered fluids with anisotropic physical properties. The simplest liquid crystalline phase is the nematic phase in which the long axes of rod-like molecules tend to align along a common axis (Fig. 1(a)), the director n. If either the constituting molecules are chiral or if chiral dopant molecules are dissolved in a nematic phase, the director becomes spatially twisted describing a helical array (Fig. 1(b)) the period of which, i.e. the pitch P, typically ranges from several hundred nanometers to several microns (Fig. 1(c)) [16–21].

 figure: Fig. 1

Fig. 1 (a) In the nematic phase, rod-like molecules are oriented preferentially along the director n (black arrow). (b) The cholesteric phase reassembles a spatially twisted nematic phase with helical distribution of the director along the helical axis (dashed arrow) and period P. (c) The periodic twisted array of the cholesteric phase can develop over several microns. Hence, a considerable number of full turns of the helix is achieved with typical LC cells.

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The twist in turn leads to a spatial periodic modulation of the dielectric tensor εij and hence the particular optical properties of the CLC [17]. The twisted nematic is optically active and its circular birefringence (difference in the refractive indices for left and right circularly polarized light, Fig. 2(a)) gives rise to a strong optical rotatory power (ORP) and selective Bragg reflection as shown in Fig. 2(b) [22–25].

 figure: Fig. 2

Fig. 2 (a) Schematic representation of a cholesteric liquid crystal cell with imposed planar alignment. For a left-handed CLC, RHCP light propagating along the helix is transmitted without any change. In contrast, LHCP light is strongly reflected. ki, kr and kt, respectively represent the wavevectors of the incident, reflected and transmitted waves. The dashed arrows indicate the direction of propagation as well as the intensity of the wave. (b) Around the Bragg regime, a peak of reflection (solid line) is observed corresponding to the excitation of the k1 mode. Here, the difference in velocities of the diffracted and non-diffracted wave leads to a strong optical rotatory power (ORP) (dashed line). The ordinary axis corresponds to the wavelengths interacting with the CLC. (c) Dispersion of ω vs. k for a CLC. Four polarized normal modes appear, k1 and k2 (solid lines) propagating to the right and the other two equivalent modes (dashed lines) propagating to the left. k1 and k2 are left- and right-handed elliptically polarized modes, respectively. For k1, a stop band appears where the Bragg diffraction occurs. This is the region of interest of this work.

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An analysis of propagating waves in CLCs provides a detailed description of their optical properties [16–21]. In particular, dispersion relations of CLCs are well known and schematically shown in Fig. 2(c). Here, the normal modes of propagation k1 and k2 are left- and right-handed elliptically polarized. In the Bragg regime, a CLC with a left-handed helical twist strongly reflects left-handed circularly polarized light (LHCP) while right-handed circularly polarized light (RHCP) is transmitted almost unchanged [21]. This is illustrated in Fig. 2(a) and observed experimentally in a CLC e.g. in [26] Kim et al. where for wavelengths between 650 – 720 nm, ≈100% of RHCP is transmitted while ≈95% of LHCP is reflected.

As shown in Fig. 2(b), at the frequency ωB, the reflectance is maximal, and an inversion point appears such that the ORP changes sign because k1 also changes sign (Fig. 2(c)). As a consequence the diffracted waves k1 move faster (respectively slower) with respect to the non-diffracted wave k2 close to this vicinity [19]. This leads to the conventional definition of optical rotatory power [17]:

ρ=12(k1k2),
where the rotation angle ρ is typically given in rad/mm.

For circularly polarized light propagating with wavevectors k1 and k2, and characteristic refractive indices nL and nR (Re{k1} = (2π/λ)nL and Re{k2} = (2π/λ)nR), so that Eq. (1) may be rewritten as [18,20,21]:

ρ=πλ(nLnR),
Here nLnR is the circular birefringence.

According to de Vries [22] ρ can be determined by measuring the pitch P and the principal refractive indices of the CLC, a task which is, however, not straightforward [27]. ρ can also be measured using optical polarization rotation measurements [23,28–30]. However, any information regarding the absolute values of the underlying refractive indices in the CLC is lost.

If both nL and nR are known independently then one can always calculate ρ with the help of Eq. (2). To the best of our knowledge the values of nL and nR have never been measured in the cholesteric phase. Interestingly, theoretical work carried out on this topic predicts the form of the dispersion for nL and nR around the Bragg regime, but this has not been previously confirmed experimentally [18].

3. Circular polarization interferometry

3.1 Principle of the polarization interferometer

The Jamin-Lebedeff polarization interferometer [3,5,9] consists of two identical birefringent crystals placed between two polarizers. The first crystal splits linearly polarized light and the second recombines the orthogonal components which interfere at the second polarizer placed at the output. In between the two crystals, the two orthogonally polarized beams are spatially separated and used to determine the unknown refractive index n of a material of interest. A material with known nRef is placed in one arm of the interferometer, while the other beam propagates through the sample (with unknown n). After recombining the two beams, the relative phase shift Δϕ of the two beams is simply expressed as [2,3]:

Δϕ=2πλ(nnRef)d,
where λ is the wavelength in vacuum and d is the thickness of the sample. For convenience, the reference and the sample material are assumed to have the same thickness.

The intensity at the detector as a function of the analyzer angle θ with respect to the axis of transmission of the first polarizer is calculated as:

I(θ)=Iysin2θ+Ixcos2θ+2IyIxsinθcosθcosΔϕ,
with Ix and Iy being measured after the analyzer at θ = 0 and θ = π/2, respectively. If Ix = Iy = I0, then the interference fringes show good contrast and Eq. (7) takes the simple form:

I(±π/4)=I0(1±cosΔϕ).

The Jamin-Lebedeff interferometer can be used to investigate isotropic and anisotropic structures using linearly polarized light, but it is not possible to sense any optical activity (chirality). Our setup is described in Fig. 3 and contains appropriate polarization optics to determine refractive indices for the circular polarization states. Each QWP introduces the same retardance in each beam and consequently there is no relative phase shift between the two arms (Δϕ = 0), and the interference pattern in Eq. (4) as well as Eq. (3) still hold.

 figure: Fig. 3

Fig. 3 Experimental setup. A Glan-Thompson polarizer P linearly polarizes the incident beam, the first λ/2 waveplate is used to balance the intensity of the o and e polarized waves (in the interferometer arms). A λ/4 waveplate transforms the linearly polarized e and o waves into circular polarization states of opposite handedness. One of the circular polarization states is sent through the sample, the other through a reference. Rotating the λ/4 waveplate switches the circular polarization states. The second λ/4 waveplate converts the circular components to linear polarization states before the two waves are recombined in the second calcite crystal. With the compensator C (variable linear birefringent element) an additional phase shift can be introduced between the vertical and the horizontal polarization states so that the intensity measured after the analyzer (A) can be determined as a function of this known retardation. A measurement protocol may thus consist of the following steps: First the phase difference between the circular polarized states is nulled. Then the substrate (sample or cuvette) is measured relative to air.

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After setting the initial state of the interferometer (without sample) to Δϕ = 0, one can observe changes in intensity when the sample and/or the reference are placed in the optical path. An optical compensator is used to determine Δϕ between the two arms of the interferometer [3–9]. The phase shift together with known values of λ, d, nRef, and Eq. (3) directly give the value of n.

3.2 Interferometry with liquid crystals

For a CLC, we expect that there is a strong birefringence and mode structure close to the Bragg regime, which we now examine with the circular polarization interferometer. Two identical CLC cells are placed between the two QWPs. One polarized beam goes only through one of the two cells which is kept at constant temperature TRef in the isotropic phase and which serves as the reference state. At TRef and in the absence of any optical resonance, RHCP and LHCP are practically indistinguishable for the CLC, which in the isotropic state is characterized by a single scalar refractive index nRef. In fact, our experiments confirmed that at TRef, nRef is identical for both LHCP and RHCP waves.

The circularly polarized beam of the other arm propagates through a second CLC cell. For this sample, the temperature is controlled with a custom-built Peltier system which has a temperature resolution of ± 0.05 °C. Changing the temperature of the sample will introduce a LC phase change between the isotropic and the cholesteric phase and thus introduce an optical phase shift Δϕ(T) with respect to the reference cell.

Bringing the two cells to the same temperature in the isotropic phase, permits us to balance the interferometer arms (zero retardance Δϕ = 0). Any phase contribution stemming from the glass substrate of the cells is thereby also nulled. In accordance with Eq. (3), the temperature-dependent refractive indices nR(T) and nL(T) of the CLC can be written as:

nL,R(T)=λ2πdΔϕL,R(T)+nRef,

where Δϕ(T) is measured using the compensator as described above, λ and d are known parameters, and nRef has to be determined independently, but since nRef is the index of refraction of an isotropic index (liquid) conventional refractometry can be used [31].

To determine the response of the CLC to linearly polarized (LP) light, the two QWPs were conveniently reoriented so that the linearly polarized state of the orthogonal waves is not altered and each beam accumulates exactly the same phase shift after the QWPs. Due to the symmetric positioning of the two QWPs, it is possible to determine the phase difference between the two arms with LP light when the QWPs are removed.

One caveat is that in the vicinity of the Bragg regime, selective reflection of one of the polarization states also leads to a decrease in the interference fringe contrast. This effect does not affect Eq. (6) [3], but makes it more difficult to determine Δϕ. In order to compensate for any reflection loss, we placed a λ/2 waveplate between the polarizer and the first calcite crystal (Fig. 3). The waveplate can balance the intensities, but also introduces a phase shift of π for each of the waves. The spatial direction of the two orthogonal components is fixed after the first calcite and does not depend on the orientation of the HWP.

Finally, we ensured that the temperature difference does not cause any phase differences between the cells due to local changes in the air density. To achieve this, the temperature of each CLC cell was controlled by independent stages. A ventilation system connected to each stage helps to extract the excess of heat and assures that the temperature inside the interferometer remains stable.

4. Experimental

Cholesteryl oleyl carbonate (COC) obtained from Sigma Aldrich shows the following approximate transition temperatures [25, 31–33]: Isotropic–cholesteric ~36 °C, cholesteric–smectic A ~20 °C. As is customary, we use the reduced temperature scale (TI-Ch – T) where TI-Ch correspond to the isotropic–cholesteric transition temperature and T to any temperature in the cholesteric and smectic phase. COC is a left-handed chiral material [34] exhibiting Bragg reflection and strong ORP close to the cholesteric–smectic A transition. A CLC sample with homogeneous alignment is prepared using commercial cells from E.H.C. Co. LTd. Here, the glass plates are coated with a semiconductor thin film of indium tin oxide (ITO) and with rubbed polyimide alignment layers. The gap of the cell is 25 μm.

According to Eq. (4), phase measurements are sensitive to changes in intensity of the orthogonally polarized beams for which reason a laser with good power stability is used. The experiments presented here are carried out with a single frequency He-Ne laser operating in the stabilized intensity mode at 632.99 nm. The beam diameter is 0.7 mm in the TEM00 mode and the output power is 1.2 mW. The laser provides a stable output power with fluctuations of ± 0.2%/hr. as well as a typical frequency stability of ± 2 MHz/hr. Within this time scale, the interferometer is very sensitive and any phase shift can be determined very precisely. The use of matched optical components will further improve the sensitivity of the interferometer. The interferometer can detect changes in phase of 1° ± 0.5° which means that we can resolve changes in our CLC cell of δn ≈(5 ± 2.5)*10−5.

5. Main results

5.1 Phase measurements and Bragg reflection

After placing the sample between parallel polarizers, the transmission of a 25 μm cell is measured for LHCP, RHCP, and LP light at different temperatures where selective reflection occurs. Here, the transmission is independently determined for each polarization state. In Fig. 4(a), the images show large areas (≈25mm2) with uniform Bragg reflection. As expected, only the LHCP light is strongly reflected, and 50% of the LP light (as half its intensity is LHCP). We attribute the ~12% loss to reflections at the glass surfaces and scattering of the CLC [35,36]. The absorption at 633 nm is negligible [34]. In the smectic A phase, the transmitted signal decreases as it gives rise to scattering [21].

 figure: Fig. 4

Fig. 4 Temperature-dependent transmission (a) and phase Δϕ (b) measurements of COC at 633 nm. Change in color of the sample observed at TI-Ch –T ≈i) 10 °C; ii) 14.2 °C; iii) 14.35 °C; iv) 14.45 °C. As expected, the LHCP laser spot is strongly reflected in iv). (b) Along the region of selective reflection, an almost constant Δϕ value appears for RHCP and LP light. In contrast, the LHCP component experiences a discontinuity in the vicinity of the reflection maximum, i.e. at the Bragg frequency ωB. (c) Shows a map of the polarization state of the beam. The profile exhibits the C4 symmetry that appears at TI-Ch –T ≈14.35 °C. Changing the temperature from TI-Ch –T ≈14.45 °C to ≈14.55 °C rotates the direction of the profile (rotation of the arrows) by ≈45° near the Bragg frequency ωB.

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Large changes are observed in the Bragg regime for LHCP light as shown in Fig. 4(a). In Fig. 4(b), Δϕ measured at each temperature is referenced to the signal measured at Δϕ = 0. A large change of Δϕ is observed for LHCP light near the maximum of the Bragg reflection, while no significant phase difference is observed for RHCP. The former can in the vicinity of ωB be related to a change in orientation of the major axis of the nearly circularly polarized modes [16, 17]. Indeed, a full 2π-scan of the retardation introduced by the QWP in our setup clearly shows this effect (Fig. 4(c)). In the corresponding map, a line connecting two opposite peaks indicates the direction of either the long or short axis of elliptically polarized light going through the QWP. At TI-Ch –T ≈14.35 °C a profile with C4 symmetry (Schönflies notation) is observed indicating a small ellipticity. However, at TI-Ch –T ≈14.45 °C and TI-Ch –T ≈14.55 °C the profiles become asymmetric and hence elliptical. Here, a change in temperature of 0.1 °C turns a maximum into a minimum. Considering TI-Ch –T ≈14.45 °C as the temperature at which the maximum of reflection (at ωB) is observed, a change in orientation of the maxima (about 45°) means that the long axis of the elliptically-polarized light has been rotated at TI-Ch –T ≈14.55 °C. We find that the long axis of the nearly circularly polarized mode of COC rotates by ≈90° after TI-Ch –T ≈14.45 °C (or ωB). This kind of rotation of the polarization ellipse is predicted theoretically [16] and manifests itself in a change of the phase velocity of the RHCP wave at ωB.

5.2 LHCP and RHCP refractive indices and optical activity

Using the data of Fig. 4(b) and Eq. (6), it is now straightforward to calculate the refractive indices nL and nR in the region of selective reflection as shown in Fig. 5. In the isotropic phase, nRef = 1.49804 is taken from [31] Evans et al.

 figure: Fig. 5

Fig. 5 (a) Temperature-dependent refractive indices nL (triangles) and nR (dots) of the LHCP and RHCP modes of the cholesteric liquid crystal COC in the region of selective reflection. While variations of nR with temperature are small, nL changes abruptly around the Bragg frequency. The dashed and dotted lines are a guide to the eye. (b) Theoretical curves predicted for nL ∝Re{KL} (dashed line) and nR∝Re{KR} (dotted line) at different wavelengths (adapted and reproduced from [18] Chandrasekhar). (c) ORP ρ of COC measured at different temperatures in the region of selective reflection. The open circles show the ORP obtained with optical polarization rotation measurements. Using the values of nL and nR in Fig. 5(a) and Eq. (2), the ORP is determined from the interferometrically-determined circular birefringence of the COC (shown by the crosses in the graph). The continuous and dashed lines are a guide to the eye.

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In Fig. 5(a), nR shows a small change, whereas nL changes dramatically at ωB. To our knowledge, a theoretical prediction of the dispersion of nL and nR for a CLC is only available in [18] Chandrasekhar. The dispersion curves of nL (∝Re{KL}) and nR (∝Re{KR}) for a semi-infinitely extended Bragg medium, as they appear in [18], are schematically shown in Fig. 5(b). Clear similarities between the phenomenological theoretical curve and our experimentally determined nL can be seen in Fig. 5(a). The dispersive line shape of nL in the vicinity of the Bragg regime is seen in both graphs.

We note that the theoretical estimate in Fig. 5(b) assumes polychromatic light and hence shows a broad Bragg peak. Here, possibly due to a resonance region, the background level changes linearly with the wavelength. In COC this background level in nR and nL is not observed due to the very narrow interval of selective reflection (≈0.7 °C) and the fixed wavelength of the light source, Fig. 5(a). Hence, the dispersion of nR is negligible. However, as a consequence of diffraction of the waves near ωB, nL changes strongly. Since we use a monochromatic source it follows that our Bragg peak is narrower and correspondingly the dispersion of nL is sharper. Interestingly, dispersion of nL occurs because the pitch approximately matches the wavelength of the laser during the diffraction of the waves near ωB. Here, the change of P with temperature permits the incident wavelength to map the periodic structure of the CLC.

Finally, we can, as a control, also deduce the optical rotation from the interferometrically determined refractive indices nL and nR with the help of Eq. (2). Figure 5(c) shows excellent agreement between the ORP deduced from the absolute values of nL and nR obtained with the interferometer and direct optical polarization rotation measurements.

We observe the sign change of the ORP in the vicinity of the Bragg frequency, as expected.

6. Conclusions

In this work, we have implemented a polarization interferometer sensitive to circularly polarized light. It permits the direct determination of refractive indices for circularly polarized modes in optically active (chiral) materials. Using this interferometer, we have determined for the first time the refractive indices nL and nR of the normal modes responsible of the optical activity in cholesteric liquid crystals (CLCs). The results show good agreement with theoretical predictions [18] for the two refractive indices of a CLC near the Bragg regime. We also report measurements of the associated optical activity of the CLC. The interferometer offers the possibility to directly determine refractive indices not only of cholesterics, but also of other periodic chiral media, as well as chiral thin-films.

References and links

1. Jamin, “Neuer interferential-refractor,” Ann. Phys. Chem. 174(6), 345–349 (1856). [CrossRef]  

2. M. Francon, Optical Interferometry (Academic New York and London, 1966).

3. D. G. Stavenga, H. L. Leertouwer, and B. D. Wilts, “Quantifying the refractive index dispersion of a pigmented biological tissue using Jamin–Lebedeff interference microscopy,” Light: Sci. Appl. 2(9), e100 (2013). [CrossRef]  

4. R. Fleischmann, “Interferenzverfahren zur Messung der absoluten Phasen bei der Untersuchung absorbierender Medien,” Z. Phys. 29, 275–284 (1950).

5. S. B. Mehta and C. J. R. Sheppard, “Partially coherent image formation in differential interference contrast (DIC) microscope,” Opt. Express 16(24), 19462–19479 (2008). [CrossRef]   [PubMed]  

6. F. J. Schaefer and W. Kleemann, “High-precision refractive index measurements revealing order parameter fluctuations in KMnF3 and NiO,” J. Appl. Phys. 57(7), 2606 (1985). [CrossRef]  

7. A. F. Brown and G. A. Dunn, “Microinterferometry of the movement of dry matter in fibroblasts,” J. Cell Sci. 92(Pt 3), 379–389 (1989). [PubMed]  

8. Z. Kam, “Microscopic imaging of cells,” Q. Rev. Biophys. 20(3-4), 201–259 (1987). [CrossRef]   [PubMed]  

9. V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative refractive-index material,” Laser Phys. Lett. 3(1), 49–55 (2006). [CrossRef]  

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13. C. Chou, W.-C. Kuo, and C. Y. Han, “Circularly polarized optical heterodyne interferometer for optical activity measurement of a quartz crystal,” Appl. Opt. 42(25), 5096–5100 (2003). [CrossRef]   [PubMed]  

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References

  • View by:

  1. Jamin, “Neuer interferential-refractor,” Ann. Phys. Chem. 174(6), 345–349 (1856).
    [Crossref]
  2. M. Francon, Optical Interferometry (Academic New York and London, 1966).
  3. D. G. Stavenga, H. L. Leertouwer, and B. D. Wilts, “Quantifying the refractive index dispersion of a pigmented biological tissue using Jamin–Lebedeff interference microscopy,” Light: Sci. Appl. 2(9), e100 (2013).
    [Crossref]
  4. R. Fleischmann, “Interferenzverfahren zur Messung der absoluten Phasen bei der Untersuchung absorbierender Medien,” Z. Phys. 29, 275–284 (1950).
  5. S. B. Mehta and C. J. R. Sheppard, “Partially coherent image formation in differential interference contrast (DIC) microscope,” Opt. Express 16(24), 19462–19479 (2008).
    [Crossref] [PubMed]
  6. F. J. Schaefer and W. Kleemann, “High-precision refractive index measurements revealing order parameter fluctuations in KMnF3 and NiO,” J. Appl. Phys. 57(7), 2606 (1985).
    [Crossref]
  7. A. F. Brown and G. A. Dunn, “Microinterferometry of the movement of dry matter in fibroblasts,” J. Cell Sci. 92(Pt 3), 379–389 (1989).
    [PubMed]
  8. Z. Kam, “Microscopic imaging of cells,” Q. Rev. Biophys. 20(3-4), 201–259 (1987).
    [Crossref] [PubMed]
  9. V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative refractive-index material,” Laser Phys. Lett. 3(1), 49–55 (2006).
    [Crossref]
  10. Z.-C. Jian, J.-Y. Lin, P.-J. Hsieh, and D.-C. Su, “Measurements of material refractive index with a circular heterodyne interferometer,” Proc. SPIE 5856, 882–892 (2005).
    [Crossref]
  11. D.-I. Serrano-García, N.-I. Toto-Arellano, A. Martínez-García, J. A. Rayas-Álvarez, and G. Rodríguez-Zurita, “Simultaneous phase shifting interferometry based in a Mach-Zehnder interferometer for measurement of transparent samples,” Proc. SPIE 8011, 80110M (2011).
    [Crossref]
  12. J. A. Davis, B. M. L. Pascoguin, I. Moreno, and A. Nava-Vega, “Circular-polarization-splitting common-path interferometer based on a zero-twist liquid-crystal display,” Opt. Lett. 34(9), 1486–1488 (2009).
    [Crossref] [PubMed]
  13. C. Chou, W.-C. Kuo, and C. Y. Han, “Circularly polarized optical heterodyne interferometer for optical activity measurement of a quartz crystal,” Appl. Opt. 42(25), 5096–5100 (2003).
    [Crossref] [PubMed]
  14. A. Ghosh and P. Fischer, “Chiral Molecules Split Light: Reflection and Refraction in a Chiral Liquid,” Phys. Rev. Lett. 97(17), 173002 (2006).
    [Crossref] [PubMed]
  15. M. Pfeifer and P. Fischer, “Weak value amplified optical activity measurements,” Opt. Express 19(17), 16508–16517 (2011).
    [Crossref] [PubMed]
  16. P. Oswald and P. Pieranski, Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments (CRC Taylor & Francis Group, 2005) Chap. B VII.
  17. P. Yeh and C. Gu, Optics of Liquid CrystalsDisplays, 2nd ed. (John Wiley & Soncs Inc, 2010) Chap 7.
  18. S. Chandrasekhar, Liquid crystals 2nd ed. (Cambridge University, 1977) Chap 4.
  19. V. A. Belyakov, V. E. Dmitrienko, and V. P. Orlov, “Optics of cholesteric liquid crystals,” Sov. Phys. Usp. 22(2), 64–88 (1979).
    [Crossref]
  20. S. Chandrasekhar and K. N. Rao, “Optical Rotatory Power of Liquid Crystals,” Acta Crystallogr. A 24(4), 445–451 (1968).
    [Crossref]
  21. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals - 6th ed. (Clarendon Oxford, 1993) Chap 6.
  22. H. L. de Vries, “Rotatory Power and Other Optical Properties of Certain Liquid Crystals,” Acta Crystallogr. 4(3), 219–226 (1951).
    [Crossref]
  23. H. Stegemeyer and K.-J. Mainusch, “Optical rotatory power of liquid crystal mixtures,” Chem. Phys. Lett. 6(1), 5–6 (1970).
    [Crossref]
  24. L. Melamed and D. Rubin, “Selected Optical Properties of Mixtures of Cholesteric Liquid Crystals,” Appl. Opt. 10(5), 1103–1107 (1971).
    [Crossref] [PubMed]
  25. R. D. Ennulat, “The selective light reflection by plane textures,” Mol Crys. Liq. Cryst. 13(4), 337–355 (1971).
    [Crossref]
  26. C. Kim, K. L. Marshall, J. U. Wallace, and S. H. Chen, “Photochromic glassy liquid crystals comprising mesogenic pendants to dithienylethene cores,” J. Mater. Chem. 18(46), 5592 (2008).
    [Crossref]
  27. N. Bitri, A. Gharbi, and J. P. Marcerou, “Scanning conoscopy measurement of the optical properties of chiral smectic liquid crystals,” Phys. B 403(21–22), 3921–3927 (2008).
    [Crossref]
  28. A. S. Sonin, A. V. Tolmachev, V. G. Tishchenko, and V. G. Rak, “Optical activity of the planar texture of a number of cholesterol esters,” Sov. Phys. JETP 41(5), 977 (1976).
  29. F. Beaubois, J. P. Marceroua, H. T. Nguyen, and J. C. Rouillon, “Optical rotatory power in tilted smectic phases,” Eur. Phys. J. E 3(3), 273–281 (2000).
    [Crossref]
  30. P. E. Sokol and J. T. Ho, “Optical rotatory power near a cholesteric–smectic A transition,” Appl. Phys. Lett. 31(8), 487 (1977).
    [Crossref]
  31. M. Evans, R. Moutran, and A. H. Price, “Dielectric properties, refractive index and far infrared spectrum of cholesteryl oleyl carbonate,” J. Chem. Soc., Faraday Trans. II 71, 1854–1862 (1975).
    [Crossref]
  32. H. S. Tai and J. Y. Lee, “Phase transition behaviors and selective optical properties of a binary cholesteric liquid crystals system: Mixtures of cholesteryl carbonate and cholesteryl nananoate,” J. Appl. Phys. 67(2), 1001–1006 (1990).
    [Crossref]
  33. R. Somashekar and D. Krishnamurti, “Optical anisotropy of cholesteryl oleyl carbonate,” Mol. Crys. Liq. Cryst. 84(1), 31–37 (1982).
    [Crossref]
  34. M. Goh, S. Matsushita, and K. Akagi, “From helical polyacetylene to helical graphite: synthesis in the chiral nematic liquid crystal field and morphology-retaining carbonisation,” Chem. Soc. Rev. 39(7), 2466–2476 (2010).
    [Crossref] [PubMed]
  35. D. K. Yang, X. Y. Huang, and Y. M. Zhu, “Bistable cholesteric reflective displays: Materials and drive schemes,” Annu. Rev. Mater. Sci. 27(1), 117–146 (1997).
    [Crossref]
  36. C. Bohley and T. Scharf, “Depolarization effects of light reflected by domain-structured cholesteric liquid crystal,” Opt. Commun. 214(1-6), 193–198 (2002).
    [Crossref]

2013 (1)

D. G. Stavenga, H. L. Leertouwer, and B. D. Wilts, “Quantifying the refractive index dispersion of a pigmented biological tissue using Jamin–Lebedeff interference microscopy,” Light: Sci. Appl. 2(9), e100 (2013).
[Crossref]

2011 (2)

D.-I. Serrano-García, N.-I. Toto-Arellano, A. Martínez-García, J. A. Rayas-Álvarez, and G. Rodríguez-Zurita, “Simultaneous phase shifting interferometry based in a Mach-Zehnder interferometer for measurement of transparent samples,” Proc. SPIE 8011, 80110M (2011).
[Crossref]

M. Pfeifer and P. Fischer, “Weak value amplified optical activity measurements,” Opt. Express 19(17), 16508–16517 (2011).
[Crossref] [PubMed]

2010 (1)

M. Goh, S. Matsushita, and K. Akagi, “From helical polyacetylene to helical graphite: synthesis in the chiral nematic liquid crystal field and morphology-retaining carbonisation,” Chem. Soc. Rev. 39(7), 2466–2476 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (3)

S. B. Mehta and C. J. R. Sheppard, “Partially coherent image formation in differential interference contrast (DIC) microscope,” Opt. Express 16(24), 19462–19479 (2008).
[Crossref] [PubMed]

C. Kim, K. L. Marshall, J. U. Wallace, and S. H. Chen, “Photochromic glassy liquid crystals comprising mesogenic pendants to dithienylethene cores,” J. Mater. Chem. 18(46), 5592 (2008).
[Crossref]

N. Bitri, A. Gharbi, and J. P. Marcerou, “Scanning conoscopy measurement of the optical properties of chiral smectic liquid crystals,” Phys. B 403(21–22), 3921–3927 (2008).
[Crossref]

2006 (2)

V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative refractive-index material,” Laser Phys. Lett. 3(1), 49–55 (2006).
[Crossref]

A. Ghosh and P. Fischer, “Chiral Molecules Split Light: Reflection and Refraction in a Chiral Liquid,” Phys. Rev. Lett. 97(17), 173002 (2006).
[Crossref] [PubMed]

2005 (1)

Z.-C. Jian, J.-Y. Lin, P.-J. Hsieh, and D.-C. Su, “Measurements of material refractive index with a circular heterodyne interferometer,” Proc. SPIE 5856, 882–892 (2005).
[Crossref]

2003 (1)

2002 (1)

C. Bohley and T. Scharf, “Depolarization effects of light reflected by domain-structured cholesteric liquid crystal,” Opt. Commun. 214(1-6), 193–198 (2002).
[Crossref]

2000 (1)

F. Beaubois, J. P. Marceroua, H. T. Nguyen, and J. C. Rouillon, “Optical rotatory power in tilted smectic phases,” Eur. Phys. J. E 3(3), 273–281 (2000).
[Crossref]

1997 (1)

D. K. Yang, X. Y. Huang, and Y. M. Zhu, “Bistable cholesteric reflective displays: Materials and drive schemes,” Annu. Rev. Mater. Sci. 27(1), 117–146 (1997).
[Crossref]

1990 (1)

H. S. Tai and J. Y. Lee, “Phase transition behaviors and selective optical properties of a binary cholesteric liquid crystals system: Mixtures of cholesteryl carbonate and cholesteryl nananoate,” J. Appl. Phys. 67(2), 1001–1006 (1990).
[Crossref]

1989 (1)

A. F. Brown and G. A. Dunn, “Microinterferometry of the movement of dry matter in fibroblasts,” J. Cell Sci. 92(Pt 3), 379–389 (1989).
[PubMed]

1987 (1)

Z. Kam, “Microscopic imaging of cells,” Q. Rev. Biophys. 20(3-4), 201–259 (1987).
[Crossref] [PubMed]

1985 (1)

F. J. Schaefer and W. Kleemann, “High-precision refractive index measurements revealing order parameter fluctuations in KMnF3 and NiO,” J. Appl. Phys. 57(7), 2606 (1985).
[Crossref]

1982 (1)

R. Somashekar and D. Krishnamurti, “Optical anisotropy of cholesteryl oleyl carbonate,” Mol. Crys. Liq. Cryst. 84(1), 31–37 (1982).
[Crossref]

1979 (1)

V. A. Belyakov, V. E. Dmitrienko, and V. P. Orlov, “Optics of cholesteric liquid crystals,” Sov. Phys. Usp. 22(2), 64–88 (1979).
[Crossref]

1977 (1)

P. E. Sokol and J. T. Ho, “Optical rotatory power near a cholesteric–smectic A transition,” Appl. Phys. Lett. 31(8), 487 (1977).
[Crossref]

1976 (1)

A. S. Sonin, A. V. Tolmachev, V. G. Tishchenko, and V. G. Rak, “Optical activity of the planar texture of a number of cholesterol esters,” Sov. Phys. JETP 41(5), 977 (1976).

1975 (1)

M. Evans, R. Moutran, and A. H. Price, “Dielectric properties, refractive index and far infrared spectrum of cholesteryl oleyl carbonate,” J. Chem. Soc., Faraday Trans. II 71, 1854–1862 (1975).
[Crossref]

1971 (2)

L. Melamed and D. Rubin, “Selected Optical Properties of Mixtures of Cholesteric Liquid Crystals,” Appl. Opt. 10(5), 1103–1107 (1971).
[Crossref] [PubMed]

R. D. Ennulat, “The selective light reflection by plane textures,” Mol Crys. Liq. Cryst. 13(4), 337–355 (1971).
[Crossref]

1970 (1)

H. Stegemeyer and K.-J. Mainusch, “Optical rotatory power of liquid crystal mixtures,” Chem. Phys. Lett. 6(1), 5–6 (1970).
[Crossref]

1968 (1)

S. Chandrasekhar and K. N. Rao, “Optical Rotatory Power of Liquid Crystals,” Acta Crystallogr. A 24(4), 445–451 (1968).
[Crossref]

1951 (1)

H. L. de Vries, “Rotatory Power and Other Optical Properties of Certain Liquid Crystals,” Acta Crystallogr. 4(3), 219–226 (1951).
[Crossref]

1950 (1)

R. Fleischmann, “Interferenzverfahren zur Messung der absoluten Phasen bei der Untersuchung absorbierender Medien,” Z. Phys. 29, 275–284 (1950).

1856 (1)

Jamin, “Neuer interferential-refractor,” Ann. Phys. Chem. 174(6), 345–349 (1856).
[Crossref]

Akagi, K.

M. Goh, S. Matsushita, and K. Akagi, “From helical polyacetylene to helical graphite: synthesis in the chiral nematic liquid crystal field and morphology-retaining carbonisation,” Chem. Soc. Rev. 39(7), 2466–2476 (2010).
[Crossref] [PubMed]

Beaubois, F.

F. Beaubois, J. P. Marceroua, H. T. Nguyen, and J. C. Rouillon, “Optical rotatory power in tilted smectic phases,” Eur. Phys. J. E 3(3), 273–281 (2000).
[Crossref]

Belyakov, V. A.

V. A. Belyakov, V. E. Dmitrienko, and V. P. Orlov, “Optics of cholesteric liquid crystals,” Sov. Phys. Usp. 22(2), 64–88 (1979).
[Crossref]

Bitri, N.

N. Bitri, A. Gharbi, and J. P. Marcerou, “Scanning conoscopy measurement of the optical properties of chiral smectic liquid crystals,” Phys. B 403(21–22), 3921–3927 (2008).
[Crossref]

Bohley, C.

C. Bohley and T. Scharf, “Depolarization effects of light reflected by domain-structured cholesteric liquid crystal,” Opt. Commun. 214(1-6), 193–198 (2002).
[Crossref]

Brown, A. F.

A. F. Brown and G. A. Dunn, “Microinterferometry of the movement of dry matter in fibroblasts,” J. Cell Sci. 92(Pt 3), 379–389 (1989).
[PubMed]

Cai, W.

V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative refractive-index material,” Laser Phys. Lett. 3(1), 49–55 (2006).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar and K. N. Rao, “Optical Rotatory Power of Liquid Crystals,” Acta Crystallogr. A 24(4), 445–451 (1968).
[Crossref]

Chen, S. H.

C. Kim, K. L. Marshall, J. U. Wallace, and S. H. Chen, “Photochromic glassy liquid crystals comprising mesogenic pendants to dithienylethene cores,” J. Mater. Chem. 18(46), 5592 (2008).
[Crossref]

Chettiar, U.

V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative refractive-index material,” Laser Phys. Lett. 3(1), 49–55 (2006).
[Crossref]

Chou, C.

Davis, J. A.

de Vries, H. L.

H. L. de Vries, “Rotatory Power and Other Optical Properties of Certain Liquid Crystals,” Acta Crystallogr. 4(3), 219–226 (1951).
[Crossref]

Dmitrienko, V. E.

V. A. Belyakov, V. E. Dmitrienko, and V. P. Orlov, “Optics of cholesteric liquid crystals,” Sov. Phys. Usp. 22(2), 64–88 (1979).
[Crossref]

Drachev, V. P.

V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative refractive-index material,” Laser Phys. Lett. 3(1), 49–55 (2006).
[Crossref]

Dunn, G. A.

A. F. Brown and G. A. Dunn, “Microinterferometry of the movement of dry matter in fibroblasts,” J. Cell Sci. 92(Pt 3), 379–389 (1989).
[PubMed]

Ennulat, R. D.

R. D. Ennulat, “The selective light reflection by plane textures,” Mol Crys. Liq. Cryst. 13(4), 337–355 (1971).
[Crossref]

Evans, M.

M. Evans, R. Moutran, and A. H. Price, “Dielectric properties, refractive index and far infrared spectrum of cholesteryl oleyl carbonate,” J. Chem. Soc., Faraday Trans. II 71, 1854–1862 (1975).
[Crossref]

Fischer, P.

M. Pfeifer and P. Fischer, “Weak value amplified optical activity measurements,” Opt. Express 19(17), 16508–16517 (2011).
[Crossref] [PubMed]

A. Ghosh and P. Fischer, “Chiral Molecules Split Light: Reflection and Refraction in a Chiral Liquid,” Phys. Rev. Lett. 97(17), 173002 (2006).
[Crossref] [PubMed]

Fleischmann, R.

R. Fleischmann, “Interferenzverfahren zur Messung der absoluten Phasen bei der Untersuchung absorbierender Medien,” Z. Phys. 29, 275–284 (1950).

Gharbi, A.

N. Bitri, A. Gharbi, and J. P. Marcerou, “Scanning conoscopy measurement of the optical properties of chiral smectic liquid crystals,” Phys. B 403(21–22), 3921–3927 (2008).
[Crossref]

Ghosh, A.

A. Ghosh and P. Fischer, “Chiral Molecules Split Light: Reflection and Refraction in a Chiral Liquid,” Phys. Rev. Lett. 97(17), 173002 (2006).
[Crossref] [PubMed]

Goh, M.

M. Goh, S. Matsushita, and K. Akagi, “From helical polyacetylene to helical graphite: synthesis in the chiral nematic liquid crystal field and morphology-retaining carbonisation,” Chem. Soc. Rev. 39(7), 2466–2476 (2010).
[Crossref] [PubMed]

Han, C. Y.

Ho, J. T.

P. E. Sokol and J. T. Ho, “Optical rotatory power near a cholesteric–smectic A transition,” Appl. Phys. Lett. 31(8), 487 (1977).
[Crossref]

Hsieh, P.-J.

Z.-C. Jian, J.-Y. Lin, P.-J. Hsieh, and D.-C. Su, “Measurements of material refractive index with a circular heterodyne interferometer,” Proc. SPIE 5856, 882–892 (2005).
[Crossref]

Huang, X. Y.

D. K. Yang, X. Y. Huang, and Y. M. Zhu, “Bistable cholesteric reflective displays: Materials and drive schemes,” Annu. Rev. Mater. Sci. 27(1), 117–146 (1997).
[Crossref]

Jamin,

Jamin, “Neuer interferential-refractor,” Ann. Phys. Chem. 174(6), 345–349 (1856).
[Crossref]

Jian, Z.-C.

Z.-C. Jian, J.-Y. Lin, P.-J. Hsieh, and D.-C. Su, “Measurements of material refractive index with a circular heterodyne interferometer,” Proc. SPIE 5856, 882–892 (2005).
[Crossref]

Kam, Z.

Z. Kam, “Microscopic imaging of cells,” Q. Rev. Biophys. 20(3-4), 201–259 (1987).
[Crossref] [PubMed]

Kildishev, A. V.

V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative refractive-index material,” Laser Phys. Lett. 3(1), 49–55 (2006).
[Crossref]

Kim, C.

C. Kim, K. L. Marshall, J. U. Wallace, and S. H. Chen, “Photochromic glassy liquid crystals comprising mesogenic pendants to dithienylethene cores,” J. Mater. Chem. 18(46), 5592 (2008).
[Crossref]

Kleemann, W.

F. J. Schaefer and W. Kleemann, “High-precision refractive index measurements revealing order parameter fluctuations in KMnF3 and NiO,” J. Appl. Phys. 57(7), 2606 (1985).
[Crossref]

Klimeck, G.

V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative refractive-index material,” Laser Phys. Lett. 3(1), 49–55 (2006).
[Crossref]

Krishnamurti, D.

R. Somashekar and D. Krishnamurti, “Optical anisotropy of cholesteryl oleyl carbonate,” Mol. Crys. Liq. Cryst. 84(1), 31–37 (1982).
[Crossref]

Kuo, W.-C.

Lee, J. Y.

H. S. Tai and J. Y. Lee, “Phase transition behaviors and selective optical properties of a binary cholesteric liquid crystals system: Mixtures of cholesteryl carbonate and cholesteryl nananoate,” J. Appl. Phys. 67(2), 1001–1006 (1990).
[Crossref]

Leertouwer, H. L.

D. G. Stavenga, H. L. Leertouwer, and B. D. Wilts, “Quantifying the refractive index dispersion of a pigmented biological tissue using Jamin–Lebedeff interference microscopy,” Light: Sci. Appl. 2(9), e100 (2013).
[Crossref]

Lin, J.-Y.

Z.-C. Jian, J.-Y. Lin, P.-J. Hsieh, and D.-C. Su, “Measurements of material refractive index with a circular heterodyne interferometer,” Proc. SPIE 5856, 882–892 (2005).
[Crossref]

Mainusch, K.-J.

H. Stegemeyer and K.-J. Mainusch, “Optical rotatory power of liquid crystal mixtures,” Chem. Phys. Lett. 6(1), 5–6 (1970).
[Crossref]

Marcerou, J. P.

N. Bitri, A. Gharbi, and J. P. Marcerou, “Scanning conoscopy measurement of the optical properties of chiral smectic liquid crystals,” Phys. B 403(21–22), 3921–3927 (2008).
[Crossref]

Marceroua, J. P.

F. Beaubois, J. P. Marceroua, H. T. Nguyen, and J. C. Rouillon, “Optical rotatory power in tilted smectic phases,” Eur. Phys. J. E 3(3), 273–281 (2000).
[Crossref]

Marshall, K. L.

C. Kim, K. L. Marshall, J. U. Wallace, and S. H. Chen, “Photochromic glassy liquid crystals comprising mesogenic pendants to dithienylethene cores,” J. Mater. Chem. 18(46), 5592 (2008).
[Crossref]

Martínez-García, A.

D.-I. Serrano-García, N.-I. Toto-Arellano, A. Martínez-García, J. A. Rayas-Álvarez, and G. Rodríguez-Zurita, “Simultaneous phase shifting interferometry based in a Mach-Zehnder interferometer for measurement of transparent samples,” Proc. SPIE 8011, 80110M (2011).
[Crossref]

Matsushita, S.

M. Goh, S. Matsushita, and K. Akagi, “From helical polyacetylene to helical graphite: synthesis in the chiral nematic liquid crystal field and morphology-retaining carbonisation,” Chem. Soc. Rev. 39(7), 2466–2476 (2010).
[Crossref] [PubMed]

Mehta, S. B.

Melamed, L.

Moreno, I.

Moutran, R.

M. Evans, R. Moutran, and A. H. Price, “Dielectric properties, refractive index and far infrared spectrum of cholesteryl oleyl carbonate,” J. Chem. Soc., Faraday Trans. II 71, 1854–1862 (1975).
[Crossref]

Nava-Vega, A.

Nguyen, H. T.

F. Beaubois, J. P. Marceroua, H. T. Nguyen, and J. C. Rouillon, “Optical rotatory power in tilted smectic phases,” Eur. Phys. J. E 3(3), 273–281 (2000).
[Crossref]

Orlov, V. P.

V. A. Belyakov, V. E. Dmitrienko, and V. P. Orlov, “Optics of cholesteric liquid crystals,” Sov. Phys. Usp. 22(2), 64–88 (1979).
[Crossref]

Pascoguin, B. M. L.

Pfeifer, M.

Price, A. H.

M. Evans, R. Moutran, and A. H. Price, “Dielectric properties, refractive index and far infrared spectrum of cholesteryl oleyl carbonate,” J. Chem. Soc., Faraday Trans. II 71, 1854–1862 (1975).
[Crossref]

Rak, V. G.

A. S. Sonin, A. V. Tolmachev, V. G. Tishchenko, and V. G. Rak, “Optical activity of the planar texture of a number of cholesterol esters,” Sov. Phys. JETP 41(5), 977 (1976).

Rao, K. N.

S. Chandrasekhar and K. N. Rao, “Optical Rotatory Power of Liquid Crystals,” Acta Crystallogr. A 24(4), 445–451 (1968).
[Crossref]

Rayas-Álvarez, J. A.

D.-I. Serrano-García, N.-I. Toto-Arellano, A. Martínez-García, J. A. Rayas-Álvarez, and G. Rodríguez-Zurita, “Simultaneous phase shifting interferometry based in a Mach-Zehnder interferometer for measurement of transparent samples,” Proc. SPIE 8011, 80110M (2011).
[Crossref]

Rodríguez-Zurita, G.

D.-I. Serrano-García, N.-I. Toto-Arellano, A. Martínez-García, J. A. Rayas-Álvarez, and G. Rodríguez-Zurita, “Simultaneous phase shifting interferometry based in a Mach-Zehnder interferometer for measurement of transparent samples,” Proc. SPIE 8011, 80110M (2011).
[Crossref]

Rouillon, J. C.

F. Beaubois, J. P. Marceroua, H. T. Nguyen, and J. C. Rouillon, “Optical rotatory power in tilted smectic phases,” Eur. Phys. J. E 3(3), 273–281 (2000).
[Crossref]

Rubin, D.

Sarychev, A. K.

V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative refractive-index material,” Laser Phys. Lett. 3(1), 49–55 (2006).
[Crossref]

Schaefer, F. J.

F. J. Schaefer and W. Kleemann, “High-precision refractive index measurements revealing order parameter fluctuations in KMnF3 and NiO,” J. Appl. Phys. 57(7), 2606 (1985).
[Crossref]

Scharf, T.

C. Bohley and T. Scharf, “Depolarization effects of light reflected by domain-structured cholesteric liquid crystal,” Opt. Commun. 214(1-6), 193–198 (2002).
[Crossref]

Serrano-García, D.-I.

D.-I. Serrano-García, N.-I. Toto-Arellano, A. Martínez-García, J. A. Rayas-Álvarez, and G. Rodríguez-Zurita, “Simultaneous phase shifting interferometry based in a Mach-Zehnder interferometer for measurement of transparent samples,” Proc. SPIE 8011, 80110M (2011).
[Crossref]

Shalaev, V. M.

V. P. Drachev, W. Cai, U. Chettiar, H. K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negative refractive-index material,” Laser Phys. Lett. 3(1), 49–55 (2006).
[Crossref]

Sheppard, C. J. R.

Sokol, P. E.

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Sonin, A. S.

A. S. Sonin, A. V. Tolmachev, V. G. Tishchenko, and V. G. Rak, “Optical activity of the planar texture of a number of cholesterol esters,” Sov. Phys. JETP 41(5), 977 (1976).

Stavenga, D. G.

D. G. Stavenga, H. L. Leertouwer, and B. D. Wilts, “Quantifying the refractive index dispersion of a pigmented biological tissue using Jamin–Lebedeff interference microscopy,” Light: Sci. Appl. 2(9), e100 (2013).
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Z.-C. Jian, J.-Y. Lin, P.-J. Hsieh, and D.-C. Su, “Measurements of material refractive index with a circular heterodyne interferometer,” Proc. SPIE 5856, 882–892 (2005).
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Tai, H. S.

H. S. Tai and J. Y. Lee, “Phase transition behaviors and selective optical properties of a binary cholesteric liquid crystals system: Mixtures of cholesteryl carbonate and cholesteryl nananoate,” J. Appl. Phys. 67(2), 1001–1006 (1990).
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Tishchenko, V. G.

A. S. Sonin, A. V. Tolmachev, V. G. Tishchenko, and V. G. Rak, “Optical activity of the planar texture of a number of cholesterol esters,” Sov. Phys. JETP 41(5), 977 (1976).

Tolmachev, A. V.

A. S. Sonin, A. V. Tolmachev, V. G. Tishchenko, and V. G. Rak, “Optical activity of the planar texture of a number of cholesterol esters,” Sov. Phys. JETP 41(5), 977 (1976).

Toto-Arellano, N.-I.

D.-I. Serrano-García, N.-I. Toto-Arellano, A. Martínez-García, J. A. Rayas-Álvarez, and G. Rodríguez-Zurita, “Simultaneous phase shifting interferometry based in a Mach-Zehnder interferometer for measurement of transparent samples,” Proc. SPIE 8011, 80110M (2011).
[Crossref]

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D. G. Stavenga, H. L. Leertouwer, and B. D. Wilts, “Quantifying the refractive index dispersion of a pigmented biological tissue using Jamin–Lebedeff interference microscopy,” Light: Sci. Appl. 2(9), e100 (2013).
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Acta Crystallogr. A (1)

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D. K. Yang, X. Y. Huang, and Y. M. Zhu, “Bistable cholesteric reflective displays: Materials and drive schemes,” Annu. Rev. Mater. Sci. 27(1), 117–146 (1997).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

P. E. Sokol and J. T. Ho, “Optical rotatory power near a cholesteric–smectic A transition,” Appl. Phys. Lett. 31(8), 487 (1977).
[Crossref]

Chem. Phys. Lett. (1)

H. Stegemeyer and K.-J. Mainusch, “Optical rotatory power of liquid crystal mixtures,” Chem. Phys. Lett. 6(1), 5–6 (1970).
[Crossref]

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M. Goh, S. Matsushita, and K. Akagi, “From helical polyacetylene to helical graphite: synthesis in the chiral nematic liquid crystal field and morphology-retaining carbonisation,” Chem. Soc. Rev. 39(7), 2466–2476 (2010).
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Laser Phys. Lett. (1)

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D. G. Stavenga, H. L. Leertouwer, and B. D. Wilts, “Quantifying the refractive index dispersion of a pigmented biological tissue using Jamin–Lebedeff interference microscopy,” Light: Sci. Appl. 2(9), e100 (2013).
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Z.-C. Jian, J.-Y. Lin, P.-J. Hsieh, and D.-C. Su, “Measurements of material refractive index with a circular heterodyne interferometer,” Proc. SPIE 5856, 882–892 (2005).
[Crossref]

D.-I. Serrano-García, N.-I. Toto-Arellano, A. Martínez-García, J. A. Rayas-Álvarez, and G. Rodríguez-Zurita, “Simultaneous phase shifting interferometry based in a Mach-Zehnder interferometer for measurement of transparent samples,” Proc. SPIE 8011, 80110M (2011).
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P. Oswald and P. Pieranski, Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments (CRC Taylor & Francis Group, 2005) Chap. B VII.

P. Yeh and C. Gu, Optics of Liquid CrystalsDisplays, 2nd ed. (John Wiley & Soncs Inc, 2010) Chap 7.

S. Chandrasekhar, Liquid crystals 2nd ed. (Cambridge University, 1977) Chap 4.

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals - 6th ed. (Clarendon Oxford, 1993) Chap 6.

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Figures (5)

Fig. 1
Fig. 1 (a) In the nematic phase, rod-like molecules are oriented preferentially along the director n (black arrow). (b) The cholesteric phase reassembles a spatially twisted nematic phase with helical distribution of the director along the helical axis (dashed arrow) and period P. (c) The periodic twisted array of the cholesteric phase can develop over several microns. Hence, a considerable number of full turns of the helix is achieved with typical LC cells.
Fig. 2
Fig. 2 (a) Schematic representation of a cholesteric liquid crystal cell with imposed planar alignment. For a left-handed CLC, RHCP light propagating along the helix is transmitted without any change. In contrast, LHCP light is strongly reflected. ki, kr and kt, respectively represent the wavevectors of the incident, reflected and transmitted waves. The dashed arrows indicate the direction of propagation as well as the intensity of the wave. (b) Around the Bragg regime, a peak of reflection (solid line) is observed corresponding to the excitation of the k1 mode. Here, the difference in velocities of the diffracted and non-diffracted wave leads to a strong optical rotatory power (ORP) (dashed line). The ordinary axis corresponds to the wavelengths interacting with the CLC. (c) Dispersion of ω vs. k for a CLC. Four polarized normal modes appear, k1 and k2 (solid lines) propagating to the right and the other two equivalent modes (dashed lines) propagating to the left. k1 and k2 are left- and right-handed elliptically polarized modes, respectively. For k1, a stop band appears where the Bragg diffraction occurs. This is the region of interest of this work.
Fig. 3
Fig. 3 Experimental setup. A Glan-Thompson polarizer P linearly polarizes the incident beam, the first λ/2 waveplate is used to balance the intensity of the o and e polarized waves (in the interferometer arms). A λ/4 waveplate transforms the linearly polarized e and o waves into circular polarization states of opposite handedness. One of the circular polarization states is sent through the sample, the other through a reference. Rotating the λ/4 waveplate switches the circular polarization states. The second λ/4 waveplate converts the circular components to linear polarization states before the two waves are recombined in the second calcite crystal. With the compensator C (variable linear birefringent element) an additional phase shift can be introduced between the vertical and the horizontal polarization states so that the intensity measured after the analyzer (A) can be determined as a function of this known retardation. A measurement protocol may thus consist of the following steps: First the phase difference between the circular polarized states is nulled. Then the substrate (sample or cuvette) is measured relative to air.
Fig. 4
Fig. 4 Temperature-dependent transmission (a) and phase Δϕ (b) measurements of COC at 633 nm. Change in color of the sample observed at TI-Ch –T ≈i) 10 °C; ii) 14.2 °C; iii) 14.35 °C; iv) 14.45 °C. As expected, the LHCP laser spot is strongly reflected in iv). (b) Along the region of selective reflection, an almost constant Δϕ value appears for RHCP and LP light. In contrast, the LHCP component experiences a discontinuity in the vicinity of the reflection maximum, i.e. at the Bragg frequency ωB. (c) Shows a map of the polarization state of the beam. The profile exhibits the C4 symmetry that appears at TI-Ch –T ≈14.35 °C. Changing the temperature from TI-Ch –T ≈14.45 °C to ≈14.55 °C rotates the direction of the profile (rotation of the arrows) by ≈45° near the Bragg frequency ωB.
Fig. 5
Fig. 5 (a) Temperature-dependent refractive indices nL (triangles) and nR (dots) of the LHCP and RHCP modes of the cholesteric liquid crystal COC in the region of selective reflection. While variations of nR with temperature are small, nL changes abruptly around the Bragg frequency. The dashed and dotted lines are a guide to the eye. (b) Theoretical curves predicted for nL ∝Re{KL} (dashed line) and nR∝Re{KR} (dotted line) at different wavelengths (adapted and reproduced from [18] Chandrasekhar). (c) ORP ρ of COC measured at different temperatures in the region of selective reflection. The open circles show the ORP obtained with optical polarization rotation measurements. Using the values of nL and nR in Fig. 5(a) and Eq. (2), the ORP is determined from the interferometrically-determined circular birefringence of the COC (shown by the crosses in the graph). The continuous and dashed lines are a guide to the eye.

Equations (6)

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ρ= 1 2 ( k 1 k 2 ),
ρ= π λ ( n L n R ),
Δϕ= 2π λ (n n Ref )d,
I( θ )= I y sin 2 θ+ I x cos 2 θ+2 I y I x sinθcosθcosΔϕ,
I( ±π/4 )= I 0 ( 1±cosΔϕ ).
n L,R ( T )= λ 2πd Δ ϕ L,R ( T )+ n Ref ,

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