text:
\section{Introduction}

Casimir-Lifshitz interactions \cite{Casimir48,Lifshitz55, Lifshitz61} are macroscopic dispersion forces that arise from quantum mechanical and thermal fluctuations of the electromagnetic field. {The name `dispersion forces'  originates from the fact that their properties are governed by the dielectric and magnetic dispersion of the interacting materials \cite{vdWBook,CasimirBook, BuhmannBookI, Woods2016Review}.} Casimir-Lifshitz forces share the same physical origin as the van der Waals-London forces, which conventionally denote the interaction in cases where retardation effects are negligible.

{These forces are studied for both fundamental and practical reasons. From a fundamental point of view, the Casimir-Lifshitz force plays an important role in the search for deviations from Newtonian gravitation in short range gravitation experiments. Its study can facilitate the precise comparisons between theoretical predictions and experimental data \cite{Sedmik2020}.  More practically, Casimir-Lifshitz interactions affect the actuation dynamics of nano- and micro-mechanical systems, such as switches, cantilevers, and actuators at a sub-micrometer length scale \cite{Serry1995,Broer2013, Broer2015, Mehdi2015, Tajik2018PRE, Munday2021Review}. The Casimir-Lifshitz interactions also underpin the stability of colloidal and biophysical macromolecular systems. These forces represent one of the pillars of the fundamental Deryaguin-Landau-Verwey-Overbeek theory of colloid stability \cite{Israelachvili}. }

{Precise evaluation of the Casimir-Lifshitz force between real materials requires a detailed knowledge of the materials' electromagnetic susceptibilities as inputs \cite{Lifshitz55, Lifshitz61}. Because this interaction is mediated by virtual photons, the frequency of which cannot be controlled directly, the Casimir-Lifshitz interaction is a broadband phenomenon. Consequently, the electromagnetic susceptibilities of the materials involved must be known - either theoretically or experimentally - as a function of frequency in a sufficiently broad range.
Another requirement to evaluate the Casimir-Lifshitz interaction  is solving Maxwell's equations for the given geometric configuration of the interacting bodies. {This latter aspect of the Casimir-Lifshitz calculations is the focus of this paper.} }

{The presence of anisotropies, either in the susceptibilities or the morphological anisotropy induced by the shape of the interacting bodies \cite{Hopkins2015}, implies the existence of not only forces but also  Casimir-Lifshitz torques between the interacting bodies \cite{Parsegiantorque}. Specifically, the dielectric anisotropy in the plane of reflection, which will exclusively concern us in what follows,  creates a dielectric contrast in the azimuthal direction, which gives rise to the  Casimir-Lifshitz torque attempting to align the orientations of both materials\cite{Ginzburg75}.   An exact analytical description of the Casimir-Lifshitz torque between two planar uniaxially anisotropic half spaces was originally based on the direct solution of the Maxwell equations and the pertaining dispersion equation by Barash more than four decades ago \cite{Barash1978}. An alternative methodology applicable to the Casimir-Lifshitz torque can be based on the solutions of the Maxwell's equations by interpreting them as an eigenvalue problem. This method is known as the transfer matrix method.  It is especially suited for describing electromagnetic field propagation through anisotropic materials \cite{Berreman1972, Yeh1979, Lekner1991}, and can be used to derive the Casimir torque between semi-infinite birefringent plates \cite{Shao2005,Philbin2008}. The results of the transfer matrix methodology were later shown to be consistent with the direct method of Barash \cite{Broer2019}. Some recent examples of the application of this method in the context of Casimir physics include biaxial materials, \cite{Emelianova2020} Weyl semi-metals, \cite{ChenLiang2020, Farias2020} and magnetic ferrite slabs \cite{Zeng2020}.  }

{The first experimental observation of Casimir-Lifshitz torques has been accomplished only recently, by relying on the anisotropic dielectric response of nematic liquid crystals \cite{Somers2018}. It seems that further experiments on Casimir-Lifshitz torques with liquid crystals are within experimental reach presently or in the near future. Apart from the Casimir-Lifshitz torques, liquid crystals exhibit diverse other Casimir fluctuation phenomena  and so might prove to be an abundant source of a variety of distinct Casimir phenomena \cite{Karimi_Pour_Haddadan_2014, Haddadan1}. The simplest way to theoretically describe the anisotropy of liquid crystals is by approximating them as uniaxial half spaces with a unique orientation \cite{Somers2015}. This essentially approximates the liquid crystal as a birefringent half space. In what follows we will refer to this approximation as {\sl 'half space approximation'}. This is indeed a valid approach if the liquid crystal is aligned, such as in the case of nematic liquid crystals \cite{Kornilovitch2012, *Kornilovitch2018}.
However, a cholesteric liquid crystal actually consists of a sequence of many thin, planar, uniaxially oriented layers, whose orientation varies periodically in space with a periodicity given by the cholesteric pitch \cite{Ryabchun2018}. This type of layered system has been shown to be amenable to the transfer matrix method \cite{Berreman1972} as electromagnetic waves in layered anisotropic media can be described by an algebra of 4$\times$4 matrices \cite{Veble2009}.

However, a major complication arises from the fact that the cholesteric liquid crystal consists of a large number of layers ($\sim10^4$) which generally implies a product of an equally large number  of 4$\times$4 matrices. Even though the non-retarded limit reduces the size of the matrices to 2$\times$2, it does not reduce the number of matrix multiplications \cite{Rudi2004,Kornilovitch2012,Bing2016, Bing2018}}. {Since the computational complexity of matrix multiplication increases linearly with the number of multiplications, an explicit analytical expression for the transfer matrix can reduce this complexity by up to four orders of magnitude. Hence here we propose a  way to analytically simplify this problem and derive an explicit expression for the composite transfer matrix for a specific planar multilayer structure that models a cholesteric liquid crystal. We have taken advantage of the facts that each layer is very thin, and that the difference in orientation between each successive layer is small \cite{Ryabchun2018} .} { This allows us to write an analytic result for the transfer matrix based on the Baker-Campbell-Hausdorff (BCH) formula \cite{HighamBook2008} as a second order expansion in this parameter, $\delta$, which is inversely proportional to the pitch length of the cholesteric liquid crystal. The form of the transfer matrix is then obtained as a function of the ordinary and extraordinary eigenvalues of the Maxwell equations, averaged over the orientations in the multilayer system.  {Consequently, our solutions enable us to analyze the interaction between two semi-infinite homochiral or heterochiral layers }and thus quantify macroscopically the effects of chirality. Our results point to a realistic possibility of measuring the effects of chirality in the macroscopic Casimir-Lifshitz torque for interaction geometries involving oriented cholesteric liquid crystals.} Note that this is different from effects of chirality of an intervening medium \cite{Barcellona2017, Jiang2019, Safari2020}.

The paper is organized as follows. After the introduction we will discuss the solutions of the Maxwell equations for a general planar anisotropic multilayer in section \ref{sec:Matrix}.  Next we will introduce the simplification for the cholesteric liquid crystal configuration in section \ref{sec:CLC}, the numerical implementation of which is then covered in section \ref{sec:Results}. Conclusions can be found in section \ref{sec:Conclusions}.

\section{Planar anisotropic multilayer}\label{sec:Matrix}

\subsection{Transfer matrix}

{Here we will formulate the theory of Casimir-Lifshitz interactions in a system composed of a one dimensional layering of finite thickness  anisotropic dielectric slabs. {(See \cref{fig:geometry})} {Later on, we will use this geometry to model a cholesteric liquid crystal.}  Our approach is based on an application of the transfer matrix method for the electromagnetic field modes propagating through  one-dimensional stratifications.  This method was  introduced as an algebra of 4 $\times$ 4 matrices by Berreman \cite{Berreman1972} and later reformulated by Yeh \cite{Yeh1979}. {Here we will follow the latter formulation, which is based on the change of basis transform in linear algebra. However, both formulations are equivalent \cite{Passler2017}.  }
 We assume that the anisotropic stratification coincides with the direction of the cholesteric axis, oriented along the $z$-direction and perpendicular to the translational $x$-$y$-plane of symmetry of the {laboratory coordinate} system. {(See \cref{fig:geometry})}}

We start by formulating Maxwell's equations as an eigenvalue problem, as we have done before for the case of single interfaces \cite{Broer2019}. Assume that the electromagnetic (EM) wave propagates within one of the anisotropic slabs labeled as $j$
whose anisotropic plane is facing the surface, which is defined as the $x$-$y$-plane in the laboratory coordinate system. Furthermore, the magnetic and electric anisotropy axes are assumed to be identical, implying that the electric permittivity and the magnetic permeability are given by:

\begin{equation}\label{eq:eps_tensor}
 \underline{\pmb{\varepsilon}}_j(\omega)=\begin{psmallmatrix}
	\varepsilon_{jx}\cos^2\theta_j+\varepsilon_{jy}\sin^2\theta_j&(\varepsilon_{jx}-\varepsilon_{jy})\sin\theta_j\cos\theta_j&0\\
	(\varepsilon_{jx}-\varepsilon_{jy})\sin\theta_j\cos\theta_j&\varepsilon_{jx}\sin^2\theta_j+\varepsilon_{jy}\cos^2\theta_j&0\\
	0&0&\varepsilon_{jz}
\end{psmallmatrix}
\end{equation}

\begin{equation}\label{eq:mu_tensor}
 \underline{\pmb{\mu}}_j(\omega)=\begin{psmallmatrix}
	\mu_{jx}\cos^2\theta_j+\mu_{jy}\sin^2\theta_j&(\mu_{jx}-\mu_{jy})\sin\theta_j\cos\theta_j&0\\
	(\mu_{jx}-\mu_{jy})\sin\theta_j\cos\theta_j&\mu_{jx}\sin^2\theta_j+\mu_{jy}\cos^2\theta_j&0\\
	0&0&\mu_{jz},
\end{psmallmatrix}
\end{equation}
where $\theta_j$ denotes the angle between the  optic axis of layer $j$  and the $x$-axis of laboratory's coordinate system. It must be stressed that the entries of both $\underline{\pmb{\varepsilon}}_j$ and $\underline{\pmb{\mu}}_j$ depend on frequency, but that the argument will be suppressed from now on. {Because of this dependence it is convenient to represent the electromagnetic fields in the frequency domain as well.  After also transforming to Fourier space, the {vectorial Maxwell} equations are  given by}

\begin{equation}\label{eq:kcross}
\mathbf{k}\times\mathbf{E}=\frac{\omega}{c}\mathbf{B},\qquad \mathbf{k}\times\mathbf{H}=-\frac{\omega}{c}\mathbf{D}
\end{equation}
where $\mathbf{B}=\underline{\pmb{\mu}}\cdot\mathbf{H}$ and $\mathbf{D}=\underline{\pmb{\varepsilon}}\cdot\mathbf{E}$. \cref{eq:kcross} is a system of six linear equations with six unknowns, four of which are independent. {Elimination of the $z-$component of $\mathbf{E}$ and $\mathbf{H}$  leads to the following 4 dimensional eigenvalue equation:  \cite{Rosa2008}}

\begin{equation}\label{eq:3Dk}
  \underline{\pmb{Q}}_j \pmb{\Psi}=q ~\pmb{\Psi},
\end{equation}
for the 4 dimensional EM field vector $\pmb{\Psi} = (E_x, E_y, H_x, H_y)^{T} $.  $\underline{\pmb{Q}}_j$ denotes a 4$\times4$ anti-diagonal block matrix whose explicit entries are given in the supplemental material, section I.

The four eigenvector solutions of \cref{eq:3Dk} represent the orientation dependent modes, which both can propagate forward and backward. They are denoted by $\pmb{\Psi}_{jo}^{\pm}$ and $\pmb{\Psi}_{je}^\pm$, where the subscripts $e$ and $o$ denote extraordinary and ordinary, respectively (for the case of birefringent media), and the superscripts $+$ and $-$ indicate forward and backward propagating modes, respectively. The mode eigenvalues are given by

\begin{subequations}
\begin{gather}
\label{eq:qjo}q^\pm_{jo}= \pm \sqrt{\varepsilon_{jy}\mu_{jx}\omega^2/c^2-(\mu_{jx}/\mu_{jz})k^2_\rho\cos^2(\theta_j-\eta)-k^2_\rho\sin^2(\theta_j-\eta)}\\
\label{eq:qje}q^\pm_{je}= \pm \sqrt{\varepsilon_{jx}\mu_{jy}\omega^2/c^2-(\varepsilon_{jx}/\varepsilon_{jz})k_\rho^2\cos^2(\theta_j-\eta)-k_\rho^2\sin^2(\theta_j-\eta)}
\end{gather}
\label{eq:eigenvalues}
\end{subequations}
where $k_\rho$ denotes the radial component of the wavevector, and $\eta$ represents the azimuthal component of the EM wave propagation direction.

\begin{figure}[!htbp]
 \includegraphics[width=0.9\textwidth]{./Fig1.eps}

 \caption{One dimensional stratification of $N$ finite thickness  anisotropic dielectric slabs representing an arbitrary planar multilayer stack.
 The stratified media, $i = 1,N$, with generally anisotropic layers are embedded in an isotropic bathing medium.  Each layer has an arbitrary material composition, thickness and orientation.  \label{fig:geometry}}
\end{figure}

Here we formulate a 4$\times$4 transfer matrix formalism for a one-dimensional stratified  system composed planar slabs of finite thickness with in-plane anisotropy (See Fig. \ref{fig:geometry}),  based on the methodology of Ref.\cite{Yeh1979}.

The matrix in \cref{eq:3Dk} has distinct eigenvalues so it is diagonalizable. This brings us to the so called \emph{propagation matrix}

\begin{equation}
\underline{\pmb{P}}_j\equiv\mathrm{diag}(\exp(-i q_{je} d_j), \exp(i q_{je} d_j), \exp(-i q_{jo} d_j), \exp(i q_{jo} d_j)).
\label{eq:propmat}
\end{equation}
Multiplication from the left of the matrix $\underline{\pmb{P}}_j$ by the EM field vector represents propagation within an anisotropic slab in the $z$-direction over a distance $d_j$, the thickness of the $j$th layer:
\[ \pmb{\Psi}(z+d_j)=\underline{\pmb{P}}_j\cdot\pmb{\Psi}(z),					\]
where $\pmb{\Psi}$ again denotes a 4 dimensional EM wave vector. $\underline{\pmb{P}}_j$ is a valid representation only in the eigenvector basis where the matrix is diagonal. Hence it is transformed to the  $xy$ basis as follows

\begin{equation}\label{eq:ChangeOfBasis}
\exp\left({-i\underline{\pmb{Q}}_{j}d_j}\right)= \underline{\pmb{S}}_j\cdot\underline{\pmb{P}}_j\cdot\underline{\pmb{S}}_j^{-1},
\end{equation}
where the columns of $\underline{\pmb{S}}_j$ consist of the eigenvectors of Maxwell's equations:

\begin{equation}
                     \underline{\pmb{S}}_j=  \begin{pmatrix}
                              \rule[-1ex]{0.5pt}{2.5ex} & \rule[-1ex]{0.5pt}{2.5ex} & \rule[-1ex]{0.5pt}{2.5ex} & \rule[-1ex]{0.5pt}{2.5ex}\\
                              \pmb{\Psi}_{je}^+ & \pmb{\Psi}_{je}^- & \pmb{\Psi}_{jo}^+ & \pmb{\Psi}_{jo}^-\\
                              \rule[-1ex]{0.5pt}{2.5ex} & \rule[-1ex]{0.5pt}{2.5ex} & \rule[-1ex]{0.5pt}{2.5ex} & \rule[-1ex]{0.5pt}{2.5ex}
                             \end{pmatrix}
\label{eq:SDef}
\end{equation}
which is analogous to what is called `rotation matrix' in Ref. \cite{Veble2009}. In \cref{eq:SDef} the normalization constants of the eigenvectors can be omitted because $\underline{\pmb{P}}_j$ is multiplied by the inverse of $\underline{\pmb{S}}_j$ from the right. The total transfer matrix that represents the EM wave propagation through all $N$ layers is given by

\begin{equation}
\underline{\pmb{T}}_N\equiv\mathrm{...} \underline{\pmb{S}}_{j-1}\cdot\underline{\pmb{P}}_{j-1}\cdot\underline{\pmb{S}}_{j-1}^{-1}\cdot \underline{\pmb{S}}_j\cdot\underline{\pmb{P}}_j\cdot\underline{\pmb{S}}_j^{-1}\cdot \underline{\pmb{S}}_{j+1}\cdot\underline{\pmb{P}}_{j+1}\cdot\underline{\pmb{S}}_{j+1}^{-1}\mathrm{...}
\equiv \prod_{j=1}^{N}\underline{\pmb{S}}_j\cdot\underline{\pmb{P}}_j\cdot\underline{\pmb{S}}_j^{-1}
\label{eq:TN}
\end{equation}
where the product symbol denotes matrix multiplication from the right for each subsequent index.

Next we have to consider what happens outside the slab, i.e. the cases $j=0$ and $j=N+1$. The EM field propagating through the isotropic embedding medium can be written as a linear combination of the $s$- and $p$-polarized modes.  By the same token as before, since the $s$- and $p$-polarized modes are the eigenmodes for  isotropic media, a matrix can be constructed with the $s$- and $p$-polarized mode eigenvectors as columns for $j=0$:

\begin{equation}
\underline{\pmb{S}}_0=\begin{pmatrix}
                              \rule[-1ex]{0.5pt}{2.5ex} & \rule[-1ex]{0.5pt}{2.5ex} & \rule[-1ex]{0.5pt}{2.5ex} & \rule[-1ex]{0.5pt}{2.5ex}\\
                              \pmb{\Psi}_{js}^+ & \pmb{\Psi}_{js}^- & \pmb{\Psi}_{jp}^+ & \pmb{\Psi}_{jp}^-\\
                              \rule[-1ex]{0.5pt}{2.5ex} & \rule[-1ex]{0.5pt}{2.5ex} & \rule[-1ex]{0.5pt}{2.5ex} & \rule[-1ex]{0.5pt}{2.5ex}
                             \end{pmatrix}
\label{eq:S0}
\end{equation}
the explicit form of which can be found in the supplemental material, section I. However, the case $j=N+1$ is a little different. {Here only forward propagating modes exist, which does not affect} {the matrix itself but instead only affects the vector on which the matrix acts}. {After all, in the $s-$ and $p-$mode basis, this information can be represented by two nonzero components, which represent the amplitudes of the forward propagating $s-$ and $p-$polarized modes, (see Section \ref{sec:Fresnel} for details).} Note that the matrices of \cref{eq:SDef,eq:S0} are singular at zero frequency. However, physically 'zero frequency' corresponds to a static limit where the frequency is small on the scale of the main absorption frequency of the materials. Hence within this limit it is possible to obtain the inverses of the matrices of \cref{eq:SDef,eq:S0}. {See also supplemental material, section V for more details on how to take the non-retarded limit}.

Since \cref{eq:TN} is valid in the laboratory $xy$ basis, it must be transformed to the $sp-$mode basis, which is the proper eigenmode basis for isotropic media. The total transfer matrix is hence

\begin{equation}
\underline{\pmb{M}}=\underline{\pmb{S}}_0^{-1}\cdot\underline{\pmb{T}}_N\cdot\underline{\pmb{S}}_{0}
\label{eq:M}
\end{equation}
where $\underline{\pmb{T}}_N$ is given by \cref{eq:TN}. For the general case considered here it is not possible to give an explicit closed expression for the matrix $\underline{\pmb{M}}$. As a test case, let us consider the limit of an infinitely thick slab. This limit does not directly and algebraically follow from \cref{eq:M} (see supplemental material, section III for details). It can be shown that for a semi-infinite slab, the transfer matrix simplifies to

\begin{equation}
\underline{\pmb{M}}=\underline{\pmb{S}}_0^{-1}\cdot\underline{\pmb{S}}_1.
\label{eq:Minf}
\end{equation}

\subsection{Lifshitz formula}\label{sec:Fresnel}

{The Casimir-Lifshitz interaction free energy can be obtained from the secular determinant of the modes,  whose zeros give the eigenfrequencies of the bound states. \cite{Veble2009} However, in its turn the secular determinant itself can be rewritten in terms of the Fresnel reflection coefficients that can be obtained from the transfer matrix formalism. This route then connects the transfer matrix formalism with the Casimir-Lifshitz interactions which we elucidate in what follows.}
 {\sl Grosso modo} one can follow either the approach based on the Green function tensors, closest to the original papers by Lifshitz et al \cite{Lifshitz55,*Lifshitz61}. (See e.g. Refs. \cite{Schwinger78, Lambrecht2006, Scheel2008}) or follow the {\sl heuristic derivation} by van Kampen et al. \cite{vanKampen68, Schram1973}, which was later shown to be actually exact \cite{Barash_1984}, and based on the summation over the allowed EM modes  within a given geometry, more akin to the original paper by Casimir \cite{Casimir48}. It can be shown that both approaches lead to the same result, \cite{Intravaia2012} namely the Lifshitz formula for dispersion interactions. Here we will take advantage of this result.

We assume that the matrix $\underline{\pmb{M}}$ from \cref{eq:M} is known and that its entries are given by $\underline{\pmb{M}}= \Big( M_{ik}\Big)$ with $i,k\in\{1,2,3,4\}$.  In such a case, the Fresnel reflection matrix elements can be shown to be given by \cite{Yeh1979,Passler2017}:

\begin{subequations}\label{eq:Fresnel}
\begin{gather}
r_{ss}=\frac{M_{21} M_{33}-M_{23} M_{31}}{M_{11} M_{33}-M_{13} M_{31}}
\label{eq:rss}\\
r_{sp}=\frac{M_{33} M_{41}-M_{31} M_{43}}{M_{11} M_{33}-M_{13} M_{31}}
\label{eq:rsp}\\
r_{ps}=\frac{M_{11} M_{23}-M_{13} M_{21}}{M_{11} M_{33}-M_{13} M_{31}}
\label{eq:rps}\\
r_{pp}=\frac{M_{11} M_{43}-M_{13}M_{41}}{M_{11} M_{33}-M_{13} M_{31}}.
\label{eq:rpp}
\end{gather}
\end{subequations}
We have verified that combining \cref{eq:Minf}  with   \cref{eq:Fresnel} restores the known Fresnel reflection matrix \cite{Lekner1991, Broer2019} for a semi-infinite birefringent plate.

To determine the Casimir energy and torque, \cref{eq:rss,eq:rsp,eq:rps,eq:rpp} must be inserted into the Lifshitz formula. The energy per unit area is given by:

\begin{equation}
 \frac{E_{Cas}}{A}=\frac{k_b T }{4\pi^2}\sum_{n=0}^{\infty}(1-\tfrac{1}{2}\delta_{0n})\int\limits_0^\infty\int\limits_0^{2\pi}\ln[D(k_\rho,\eta,\{\theta_j\}, \varphi, i\zeta_n)]k_\rho\mathrm{d} k_\rho\mathrm{d}\eta
\label{eq:Lifshitz}
 \end{equation}
where \cite{Lambrecht2006}
\begin{equation}
 D=\det(\underline{\pmb{I}}-\underline{\pmb{r}}_1(\{\theta_j\}, i\zeta_n)\cdot\underline{\pmb{r}}_2(\theta_1+\varphi, i\zeta_n)e^{-2k_3a})
\end{equation}
and the  reflection matrices are given by

\begin{equation}
 \underline{\pmb{r}}_q=\begin{pmatrix}
                        r_{q,ss}&r_{q,sp}\\
                        r_{q,ps}&r_{q,pp}
                       \end{pmatrix} \qquad q\in \{1,2\}.
\end{equation}
The subscript $q$ is merely a label for each multilayer stack. The entries of the matrices are given by \cref{eq:rss,eq:rsp,eq:rps,eq:rpp}. All quantities are evaluated at the imaginary Matsubara frequencies $ \zeta_n = \frac{2\pi n k_b T}{\hbar},$ so that each contribution to the Casimir energy decreases monotonically with $n$. The Casimir torque is then given by

\begin{equation}\label{eq:torque}
 \tau(a,\varphi)=-\frac{\partial E_{Cas}}{\partial \varphi}
\end{equation}
where $\varphi$ denotes the  angle between the optic axes of the layers of each stack closest to the gap. This result is equivalent to that of Ref. \cite{Veble2009} which uses the formalism of Berreman \cite{Berreman1972}, while here the same formalism has been used with the basis suggested by Yeh \cite{Yeh1979}.

\section{Modeling a Cholesteric liquid crystal}\label{sec:CLC}

\begin{figure}[!htbp]
\includegraphics[width=0.8\textwidth]{./Fig2.eps}

 \caption{Illustration of the continuum approximation used here to model a liquid crystal. It consists of layers with identical thicknesses $d$ but each with a slightly different orientation, each with identical increments $\delta\ll1$. Consequently, the multilayer structure manifests as a continuous inhomogeneity in the direction perpendicular to the plane of reflection (defined as the $z$-direction), where the orientation effectively behaves as a helix with pitch length $L$. }\label{fig:NLC}
\end{figure}

A cholesteric liquid crystal is modeled as a nonmagnetic uniaxial planar multilayer stack.   All anisotropic layers of equal thickness $d$ are made of identical materials, and only differ by their orientation $\theta_j$. We start by writing  \cref{eq:ChangeOfBasis} as

\[\exp({-i\underline{\pmb{Q}}_j d})= \underline{\pmb{S}}_j\cdot\underline{\pmb{P}}_j\cdot\underline{\pmb{S}}_j^{-1}\]
for a single slab with label $j$.

{Now let us formulate the conditions of validity of this approximation. In accordance with an actual cholesteric liquid crystal \cite{Ryabchun2018}, each layer is assumed to be infinitesimally thin, i.e. $d\rightarrow0$. Moreover, let the  difference in orientation between each subsequent layer, $\delta$, also be infinitesimally small, in such a way that this quantity parametrically describes a helix in space. Consequently, an infinitesimal change in $z$ within the liquid crystal is proportional to an infinitesimal change in the orientation: }

\begin{equation}\label{eq:helix}
    \mathrm{d} z=d=\frac{L}{2\pi} \delta,
\end{equation}
{where $L$ denotes the pitch length, in which the orientations of the layers complete one period, and $\delta\ll1$. Hence there is some freedom of choice, we can use either $z$ or $\theta$ as a (continuous) variable. In what follows we will use two variables interchangeably. Another assumption is that the total thickness $Nd$, where $N$ denotes the total number of layers, is infinite. This is justifiable because a typical thickness of  liquid crystals is of the order of several microns, and we will limit ourselves to separation distances of less than one micron ($a\ll Nd$), where the Casimir-Lifshitz torque is likely to be large enough to be measured \cite{Somers2018}. Finally, it will be assumed that the pitch length  of the cholesteric liquid crystal $L$, in which the orientations of the layers complete one period, is smaller than the total thickness. Hence at least one period is assumed to fit inside the crystal. We can summarize the condition for our approximation as follows:}

\begin{equation}\label{eq:condition}
    d\ll L \ll Nd,
\end{equation}
which means that the difference between the first orientation and last orientation is at least $2\pi$ radians. This is applicable to a cholesteric liquid crystal \cite{Ryabchun2018}. We also require $d$ to be small on the scale of the matrix norm of $\underline{\pmb{Q}}_j$:

\begin{equation}
||\underline{\pmb{Q}}_j||d\ll1\quad\text{for all } 1\leq j\leq N.
\label{eq:BCHCond}
\end{equation}
Physically, \cref{eq:BCHCond} can be understood as the thickness $d$  being much smaller than the typical values of the relevant wavelengths.   After all, in a continuous medium approach, the relevant wavelengths are assumed to be larger than the typical distances between atoms. Since the physical thickness is actually of the same order of magnitude, it can be approximated as a numerical integration stepsize in the context of a continuous medium approach.  The condition \cref{eq:BCHCond} allows us to use the Baker-Campbell-Hausdorff (BCH) formula \cite{HighamBook2008}. The second condition of \cref{eq:condition}  allows us to use \cref{eq:Minf} instead of \cref{eq:M}.

In section IV of the supplemental material we derive the following expression valid up to the second order in the angular deviation in the orientation between the neighboring uniaxial layers
\begin{equation}
\prod_{j=1}^{N}\exp\left(-id\underline{\pmb{Q}}_{j}\right)=\exp\Bigg(-id\sum_{j=1}^{N}\underline{\pmb{Q}}_{j}+O(\delta^2)\Bigg)\longrightarrow\exp\Bigg(-i\int\limits_{\min(z)}^{\max(z)}\underline{\pmb{Q}}(\theta)\mathrm{d} z + O(\delta^2)\Bigg),
\label{eq:NLayerDiscrete}
\end{equation}
where eventually we assumed that $j$ changes as a continuous variable associated with $z$. The full derivation is somewhat lengthy so we will summarize it here. In short, we use the Baker-Campbell-Haussdorff (BCH) formula \cite{HighamBook2008} up to third order in $\delta$. Then we combine the BCH formula with finite difference coefficients \cite{Fornberg1988} to determine that, three nearest neighbors to the right are needed to obtain the same accuracy as the BCH formula.

In order to evaluate the argument of the exponential matrix in \cref{eq:NLayerDiscrete}, and we will try to determine its diagonalized form. Let us focus on the leading order term: $-id\sum_{j=1}^{N} \underline{\pmb{Q}}_{j}$. Even though \cref{eq:ChangeOfBasis} tells us that each individual matrix $\underline{\pmb{Q}}_{j}$ is diagonalizable, it does not follow in general that their sum is also diagonalizable. However, within this particular approximation it is assumed that the commutators between the matrices $\underline{\pmb{Q}}_{j}$ are negligible and the matrices effectively commute, with the error of this approximation being of second order. Commuting  diagonalizable matrices can be shown to have the same eigenvectors. Consequently, each diagonalizable matrix $\underline{\pmb{Q}}_{j}$ can be transformed to the same eigenvector basis via one matrix, say $\underline{\pmb{S}}$. In other words, the matrices $\underline{\pmb{Q}}_{j}$ can be simultaneously diagonalized:
\[\sum_{j=1}^{N}\underline{\pmb{Q}}_{j} = \underline{\pmb{S}}\cdot\underline{\pmb{D}}\cdot\underline{\pmb{S}}^{-1}\]
where $\underline{\pmb{D}}$ is a diagonal matrix. This furthermore implies that:

\begin{equation}
\exp\left(-id\sum_{j=1}^{N}\underline{\pmb{Q}}_{j}\right)=\underline{\pmb{S}}\exp\left(-id\underline{\pmb{D}}+O(d^2)\right)\underline{\pmb{S}}^{-1}.
\label{eq:Simultaneous}
\end{equation}
 We know that the matrices $\underline{\pmb{S}}$ and $\underline{\pmb{D}}$ exist but now we want to determine their explicit form.  The physical interpretation of \cref{eq:Simultaneous} is that, in this approximation, the orientations of the different layers are averaged over the entire crystal. Since $d\rightarrow0$ and $N\rightarrow\infty$, it can be proposed that $d\sim1/N$, and the left hand side of \cref{eq:Simultaneous} becomes an average over all eigenvalues. Hence it follows that

\begin{equation}
\underline{\pmb{D}}=\frac{1}{N}\sum_{j=1}^{N}\text{diag}(q_{e}((\theta_j-\eta)),-q_{e}((\theta_j-\eta)),q_o,-q_o)
\label{eq:DDiscAvg}
\end{equation}
which changes to an average with respect to a continuous distribution where the sum is replaced by an integral. The approximation of \cref{eq:DDiscAvg} is of course simple and crude, its error being of second order in the stepsize $d$. It assumes that the planar multilayered system can be approximated as a single slab with an averaged orientation. Essentially the effect of the presence of multiple layers is treated additively in this case.

The transfer matrix for such a configuration is given by  (see \cref{eq:Simultaneous}):

\begin{equation}
\exp\left(-id\sum_{j=1}^{N}\underline{\pmb{Q}}_{j}\right)=\left<\underline{\pmb{S}}\right>\underline{\pmb{P}}_{tot}(\left<q_e\right>)\left<\underline{\pmb{S}}\right>^{-1},
\label{eq:avgSlab}
\end{equation}
where
\begin{equation}
\underline{\pmb{P}}_{tot}(\left<q_e\right>)=\text{diag}(\exp(-id_{tot}\left<q_e\right>),\exp(id_{tot}\left<q_e\right>,\exp(-id_{tot}q_o), \exp(id_{tot}q_o)).
\label{eq:Ptot}
\end{equation}
Here $d_{tot}=Nd$ represents the total thickness, and in the limit $d_{tot} \rightarrow \infty$,  $\underline{\pmb{P}}_{tot}$ tends to {${\underline{\pmb{P}}_{tot}\rightarrow\text{diag}(1,0,1,0)}.$} {(This does not follow directly algebraically, but from physical considerations, see Supplemental Material, section III ).} In \cref{eq:Ptot} the eigenvectors are the ones corresponding to $\left<q_e\right>$:

\begin{equation}
\left<\underline{\pmb{S}}\right>=\underline{\pmb{S}}(\left<q_e\right>,\left<\theta\right>),
\label{eq:Savg}
\end{equation}
where $\underline{\pmb{S}}$ is given by \cref{eq:SDef} and  $\left<q_e\right>$ denotes the extraordinary eigenvalue, averaged over the orientations:
\begin{equation}
 \left<q_e\right>=\frac{1}{N}\sum_{j=1}^{N}q_{e}((\theta_j-\eta)) \longrightarrow \frac{1}{N\delta}\int\limits_0^{N\delta}q_{e}(\theta-\eta)\mathrm{d}\theta.
\label{eq:qeavg}
\end{equation}
The rightmost expression follows after the continuity approximation that allows us to replace the sums over the discrete index $j$ by integrals over the continuous variable $\theta$.  We assume here that the pitch length, the length of one period, is {smaller than the total thickness, see \cref{eq:condition}}.

{Due to the periodicity of $q_e$,  $N\delta$ can be replaced by $\pi$, so that $q_{e}$ is averaged over one period and it can take}\\ {all possible real values}. {This is because the thickness of the liquid crystal is assumed to be larger than the pitch length.  (See \cref{eq:condition}) Consequently, at least one period can be fit into the crystal. The integer part of the ratio between the thickness and the pitch length will dominate the remainder, because the remainder represents a part of the liquid crystal located at an infinite distance from the plane of reflection. Setting $N\delta=\pi$}  also allows us to set $\eta=0$, because the integral becomes translation invariant over one period. In this case, the integral of \cref{eq:qeavg} can be evaluated analytically:

\begin{equation}
  \left<q_e\right>=\frac{1}{\pi}\int\limits_0^{\pi}q_{e}(\theta)\mathrm{d}\theta=\pm\frac{2}{\pi}\sqrt{\varepsilon_{1x}\left(\tfrac{k_\rho^2}{\varepsilon_{1y}}+\tfrac{\zeta^2}{c^2}\right)}E\left(\frac{k_\rho^2(\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}-1)}{\varepsilon_{1x}\left(\tfrac{k_\rho^2}{\varepsilon_{1y}}+\tfrac{\zeta^2}{c^2}\right)}\right),\label{eq:qeavg3}
\end{equation}
where $E(z)$ denotes the complete elliptic integral of the second kind, given by \cite{AbramowitzStegun}

\[E(z)=\int_{0}^{\tfrac{\pi}{2}}\sqrt{1-z\sin^2t}\mathrm{d} t,\]
the value of which is real for arguments $-1\leq z\leq1$.

To determine the average orientation, we must solve the following equation:
\begin{equation}\label{eq:qeavgEq}
  \left<q_e\right>=\pm\sqrt{\varepsilon_{1x}\tfrac{\zeta^2}{c^2}+\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}k_\rho^2\cos^2\left<\theta\right>+k_\rho^2\sin^2\left<\theta\right>}
\end{equation}
which  has four possible solutions:

\begin{equation}\label{eq:thsols}
 \left<\theta\right>=\pm\arccos\left(\pm\frac{\sqrt{k_\rho^2+\varepsilon_{1x}\zeta^2/c^2-\left<q_e\right>^2}}{k_\rho\sqrt{1-\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}}}\right).
\end{equation}
Here the different signs of $\left<\theta\right>$ represent the different chiralities of the liquid crystal: a positive $\left<\theta\right>$ is associated with right handed chirality and the minus sign corresponds to left handed chirality, if the axial wave vector component is positive. The different signs in the argument of the arc-cosine in \cref{eq:thsols} come from the different square roots in \cref{eq:qeavgEq},  associated with the two different propagation directions of the extraordinary modes in the liquid crystal. {However, since the quantity $\left<q_e\right>$ is an average along the helix, its sign must be interpreted as the direction of the helical spiral. Alternatively, the different signs of the square root of  $\left<q_e\right>$ in \cref{eq:qeavgEq} can be thought of as the location of the crystal: since the thicknesses of the crystals are assumed to be infinite, the negative square root represents the crystal being on the left and the positive square root represent the crystal being on the right. This distinction depends on the arbitrary choice of coordinates, and it is hence not physical.  } \cref{eq:thsols} provides four solutions whereas there are only two physically distinguishable shapes of the helix, namely left- and right handed. Therefore we need to establish how they can be physically interpreted. {We require that the physical $\left<\theta\right>$ is real. } First we have to verify that such solutions exist. {To this end we distinguish two cases: $\varepsilon_{1y}\geq\varepsilon_{1x}$ and $\varepsilon_{1y}\leq\varepsilon_{1x}$.}  In order for $\left<\theta\right>$ to be real, we need to check that

\[0\leq k_\rho^2+\varepsilon_{1x}\tfrac{\zeta^2}{c^2}-\left<q_e\right>^2\leq k_\rho^2\left(1-\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}\right),\]
which can be seen to be the case since

\[\min\left(1,\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}\right)\leq\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}\cos^2\theta+\sin^2\theta\leq\max\left(1,\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}\right)\]
for all real $\theta$. {Here it is assumed that $|\left<\theta\right>|\leq\pi$. } Hence the real value of the average within the domain $-\pi\leq\left<\theta\right>\leq\pi$ orientation is
\begin{subequations} \label{eq:thavg_left}
    \begin{equation}
        \left<\theta\right>_{L}=
                       \arccos\left(-\frac{\sqrt{k_\rho^2+\varepsilon_{1x}\zeta^2/c^2-\left<q_e\right>^2}}{k_\rho\sqrt{1-\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}}}\right)\\\label{eq:thetaLL}
    \end{equation}

    \begin{equation}
        \left<\theta\right>_{R}=                       -\arccos\left(-\frac{\sqrt{k_\rho^2+\varepsilon_{1x}\zeta^2/c^2-\left<q_e\right>^2}}{k_\rho\sqrt{1-\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}}}\right)\\\label{eq:thetaLR}
    \end{equation}
\end{subequations}
if the crystal is on the left, and
\begin{subequations} \label{eq:thavg_right}
    \begin{equation}
        \left<\theta\right>_{L}=
                       -\arccos\left(\frac{\sqrt{k_\rho^2+\varepsilon_{1x}\zeta^2/c^2-\left<q_e\right>^2}}{k_\rho\sqrt{1-\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}}}\right)\\\label{eq:thetaRL}
    \end{equation}
    \begin{equation}
        \left<\theta\right>_{R}=                       \arccos\left(\frac{\sqrt{k_\rho^2+\varepsilon_{1x}\zeta^2/c^2-\left<q_e\right>^2}}{k_\rho\sqrt{1-\tfrac{\varepsilon_{1x}}{\varepsilon_{1y}}}}\right)\\\label{eq:thetaRR}
    \end{equation}
\end{subequations}
if the crystal is on the right. Here  $\left<q_e\right>$ is given by \cref{eq:qeavg3}. The subscripts $L$ and $R$ in \cref{eq:thavg_right,eq:thavg_left}  denote the left- and  right handedness of the liquid crystal, respectively. Note that \cref{eq:thavg_right,eq:thavg_left} depend on frequency, which in turn affects the frequency dependence of the reflection matrix. Finally, \cref{eq:qeavg3,eq:thavg_right,eq:thavg_left} have to be inserted into \cref{eq:SDef} in order to obtain $\left<\underline{\pmb{S}}\right>_1$.

Now we move on to the next order term in the BCH expansion. Here the contribution of nested commutators is neglected, i.e. commutators are assumed to commute with other matrices. In order to take advantage of \cref{eq:avgSlab} we apply the BCH formula again to separate the BCH terms as follows (see Supplemental material, section IV for a detailed derivation):

 \begin{equation}
\exp\left(-i\frac{L}{\pi}\int\limits_0^{N\delta}\underline{\pmb{Q}}(\theta)\mathrm{d}\theta-3\frac{L^2}{\pi^2}\int\limits_{0}^{(N-3)\delta}\left[\underline{\pmb{Q}}(\theta),\frac{\partial \underline{\pmb{Q}}(\theta)}{\partial \theta}\right]\mathrm{d}\theta+O(\delta^3)\right)
 \label{eq:separated}
 \end{equation}
\[=\left<\underline{\pmb{S}}\right>_1\underline{\pmb{P}}_{tot}(\left<q_e\right>)\left<\underline{\pmb{S}}\right>_1^{-1}\exp\Bigg(-3\frac{L^2}{\pi^2}\int\limits_{j=1}^{(N-3)\theta}\left[\underline{\pmb{Q}}(\theta),\frac{\partial \underline{\pmb{Q}}(\theta)}{\partial \theta}\right]\mathrm{d}\theta\Bigg),\]
where we have taken advantage of the transfer matrix for the single interface case, \cref{eq:Minf}.
The argument of rightmost exponential matrix on the right hand side of \cref{eq:separated} is not jointly diagonalizable with the first BCH term. \footnote{The matrices may still be approximately jointly diagonalizable. However at this point it is not clear how to estimate this approximation, or what the common eigenvector basis should be.} However, this correction is expected to be small compared to the averaged factor on its left. This allows us to write it as a Taylor expansion. Hence we obtain

\begin{equation}
\
 \prod_{j=1}^{N}\exp\left(-id\underline{\pmb{Q}}_{j}\right)=\left<\underline{\pmb{S}}\right>_1\underline{\pmb{P}}_{tot}(\left<q_e\right>)\left<\underline{\pmb{S}}\right>_1^{-1}\left(\underline{\pmb{I}}-3\frac{L^2}{\pi^2}\int\limits_0^{(N-3)\delta}\left[\underline{\pmb{Q}}(\theta),\frac{\partial \underline{\pmb{Q}}(\theta)}{\partial \theta}\right]\mathrm{d}\theta+O(\delta^3)\right).
\label{eq:Taylor}
\end{equation}
The leading order term in \cref{eq:Taylor} can be understood as a slab with infinite thickness and an averaged orientation. The second order correction describes the effect due to the finite thicknesses of the layers and their slightly different orientations. For a liquid crystal of infinite thickness \cref{eq:Taylor} becomes

\begin{equation}
 \prod_{j=1}^{N}\exp\left(-id\underline{\pmb{Q}}_{j}\right)=\underline{\pmb{S}}_0^{-1}\left<\underline{\pmb{S}}\right>_1\left(\underline{\pmb{I}}-3\frac{L^2}{\pi^2}\int\limits_0^{(N-3)\delta}\left[\underline{\pmb{Q}}(\theta),\frac{\partial \underline{\pmb{Q}}(\theta)}{\partial \theta}\right]\mathrm{d}\theta+O(\delta^3)\right),
\label{eq:TaylorInfThick}
\end{equation}

Next we are faced with the task of writing \cref{eq:TaylorInfThick} as an expansion in $\delta$. Note that the integral absorbs one factor of $\delta$ and that the integration from 0 to $N\delta=\pi$ will not contribute to the integral (See supplemental material, section I for the explicit expressions for the commutator), so the integral is expected to be small $\sim\delta$. Within the approximations used here, it suffices to calculate the integral up to first order in $\delta$. (A more precise calculation would be beyond the approximation and hence be inconsistent). We will denote this first order coefficient by $\underline{\pmb{A}}$. Hence the transfer matrix is

\begin{equation}
 \prod_{j=1}^{N}\exp\left(-id\underline{\pmb{Q}}_{j}\right)= \underline{\pmb{S}}_0^{-1}\left<\underline{\pmb{S}}\right>_1\left(\underline{\pmb{I}}-3\frac{L^2}{\pi^2}\underline{\pmb{A}}\delta+O(\delta^3)\right),
 \label{eq:final}
\end{equation}
where $\underline{\pmb{A}}$ is given by the integral in \cref{eq:TaylorInfThick}, and its explicit expressions can be found in the supplemental material, section I. Note that the error of the BCH formula is of third order in $\delta$, as the higher order term \cref{eq:final} is actually quadratic in $\delta$. So this term does not have to follow the sign convention of \cref{eq:thavg_left,eq:thavg_right}. Since the approximate transfer matrix is known now, \cref{eq:Fresnel} can be applied to it to obtain the corresponding Fresnel matrix. Note that the factor of $L^2$ cancels out of the higher order correction for the Fresnel matrix. However, it's worth pointing out that $\delta$ is inversely proportional to $L$, (see \cref{eq:helix}) so that the final result still depends on the pitch length. We again refer to the supplemental material, section I for the explicit expressions for the corrections to the Fresnel matrix. It's worth noting that the Fresnel matrices are invariant under $\pmb{k}\rightarrow-\pmb{k}$, hence they are identical whether they are placed on the left or on the right (if they have the same orientation). However in general this is not necessarily true \cite{Bing2021}.

\section{Results}\label{sec:Results}
\begin{figure}[!htbp]
\includegraphics[width=0.49\textwidth]{./Fig3.eps}

 \caption{Dielectric functions of pure, nematic and doped, cholesteric 5CB. The dielectric function of nematic 5CB as proposed in Ref. \cite{Kornilovitch2012, Kornilovitch2018} (indicated by the black squares and blue triangles) is modified to account for the presence of a chiral dopant. This affects the static values (see the magenta crosses and the red circles), which were obtained from Ref. \cite{Kocakuelah2021}  }\label{fig:5CBeps}
\end{figure}
{Here we will use the results derived  in section \ref{sec:CLC} to calculate the Casimir-Lifshitz torque given by \cref{eq:torque}. The first configuration we consider is composed of a birefringent half-space, a finite vacuum gap of thickness 10 nm $<a<$1 $\mu$m, and a cholesteric half-space. Next we will investigate the combination of two cholesteric liquid crystals. The former in general should be accessible to experiments \cite{Somers2018}. While realistically, surface roughness generally plays a significant role at this range \cite{PeterRoughness}, the roughness effects have been considered elsewhere in detail \cite{d0paper, BroerEPL2011, BroerPRB2012} and we will ignore them here. We have verified our calculations by reproducing earlier results obtained by implementations of the Barash formula \cite{Munday2005, Somers2015, Thiyam2018} (See supplemental material, section V).}

{In order to calculate Casimir-Lifshitz interactions, the frequency dependent dielectric function of the interacting materials must be known. This is why we introduce it here first. We will ignore the temperature dependence of the dielectric function and assume a constant temperature of 293 K. This is because we are interested in the cholesteric phase only.}

{The dielectric function of a nematic liquid crystal can be described by a semi-empirical three band model \cite{Wu1991}:}

\begin{equation}\label{eq:eps5CB}
    \varepsilon_{N,x,y}(i\zeta)=\left(1+\sum_{k=1}^{3} \frac{C_{k,x,y}}{1+\tfrac{\zeta^2}{\omega_k^2}}\right)^2.
\end{equation}
{However,a pure 5CB crystal is nematic and not  cholesteric. It needs to be doped to reach a cholesteric phase, which is typically described  by a  Debye model \cite{delaFuente2014}:}

\begin{equation}\label{eq:epsD}
    \varepsilon_{D,x,y}(i\zeta)=\varepsilon_{\infty,x,y}+\frac{\varepsilon_{x,y}(0)-\varepsilon_{\infty,x,y}}{1+\zeta\tau}
\end{equation}
{Therefore we propose to model a doped nematic liquid crystal as the sum of these dielectric functions}

\begin{equation}\label{eq:Debye}
    \varepsilon_{x,y}(i\zeta)=\varepsilon_{D,x,y}(i\zeta)+\varepsilon_{N,x,y}(i\zeta)
\end{equation}
{However, the Debye relaxation processes take place on a much smaller frequency scale than the Matsubara frequencies at room temperature. More precisely, the smallest Matsubara frequency ($n=1$) at 293 K is six orders of magnitude larger than the typical values of the Debye relaxation frequencies. \cite{delaFuente2014,Kocakuelah2021}
 Therefore we propose to approximate it as the static term from \cref{eq:Debye}:}

\begin{equation}\label{eq:eps_tot}
\varepsilon_{x,y}(i\zeta)\approx\varepsilon_{D,x,y}(0)+\varepsilon_{5CB,x,y}(i\zeta)
\end{equation}
{in other words, the Debye processes only affect the static permittivity of the liquid crystal.} The different components of the dielectric functions for both the nematic and cholesteric case in  \cref{fig:5CBeps}.

\begin{figure}[!htbp]
\begin{subfigure}[t]{0.45\textwidth}
\centering
\includegraphics[width=\linewidth]{./Fig4a.eps}

 \caption{ }
 \label{fig:Left5CBBaTiO3}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\linewidth]{./Fig4b.eps}
 \caption{}
 \label{fig:Right5CBBaTiO3}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\linewidth]{./Fig4c.eps}
 \caption{ }
 \label{fig:BaTiO35CBphi}
 \end{subfigure}
 \hfill
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\linewidth]{./Fig4d.eps}
 \caption{ }
 \label{fig:BaTiO35CBdistance}
 \end{subfigure}
 \caption{The Casimir torque between a birefringent (BaTiO$_3$) crystal and a cholesteric liquid crystal. (a): a left handed facing a birefringent one. (b): a right handed  crystal facing a birefringent one. (c): the Casimir torque as a function of misalignment angle at a distance of $a=10$ nm. The different curves are: the triangles pointing right show the usual result for a nematic liquid crystal. Red circles: left handed crystal, $\delta=0$. Blue triangles pointing left: left handed crystal, $\delta\neq0$. Magenta crosses: right handed crystal, $\delta=0$. Black squares: right handed crystal, $\delta\neq0$. (d): The amplitude of the Casimir torque as a function of separation. The triangles pointing right show the nematic case. The cholesteric case is shown by the other two curves, of which the red circles represents $\delta=0$ and the black squares show the perturbed case. The inset of panel (d) shows the Casimir torque for the cases (a) (red circles) and (b) (blue crosses) at distances between 380 and 800 nm. }\label{fig:BaTiO35CB}
\end{figure}

{Now we are in a position to specify the materials, i.e. assign values to the parameters in \cref{eq:eps_tot}.  Let the cholesteric liquid crystal consist of 96 mass \% nematic 5CB (4- cyano-4'-pentyl-biphenyl) doped with 4 mass \% chiral dopant S811. \cite{Kocakuelah2021} The value of the static Debye term for this mixture was taken from a recent experiment \cite{Kocakuelah2021}. The dielectric function of 5CB  was established in Ref. \cite{Kornilovitch2012} based on data from Ref. \cite{Wu1993}. The optical data of barium titanate (BTO $\rm  Ba Ti O_3$) \cite{BERGSTROM1997} will be used to describe the semi-infinite birefringent half-space. For more details about the parameter values and their effect on the Casimir-Lifshitz interaction we refer to section VI of the supplemental material.}  Throughout this section we choose the value of $\delta$ (if nonzero) to be 0.03, which corresponds to a pitch length of 200 nm. (See \cref{eq:helix}).

\begin{figure}[ht]
\begin{subfigure}[t]{0.45\textwidth}
\centering
\includegraphics[width=\linewidth]{./Fig5a.eps}

 \caption{ }
 \label{fig:HomoChiralRight}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\linewidth]{./Fig5b.eps}
 \caption{ }
 \label{fig:HomoChiralLeft}
\end{subfigure}
\hfill
 \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\linewidth]{./Fig5c.eps}
\caption{ }
\hfill
\label{fig:HomoChiralPhi}
 \end{subfigure}
  \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\linewidth]{./Fig5d.eps}
 \caption{ }
 \label{fig:HomoChiral380-800nm}
 \end{subfigure}
  \caption{The Casimir torque between two cholesteric liquid crystals of identical chirality. (a): homochiral configuration of two left handed crystals (b): homochiral configuration of two right handed crystals. (c):  Casimir torque as a function of misalignment angle at a distance of $a=10$ nm. The different curves are: triangles pointing right: two nematic liquid crystals. Black crosses: unperturbed, case (a). blue squares: perturbative contribution to case (a) . Red circles: unperturbed case (b). orange circles: perturbative  contribution to case (b). (d): The Casimir torque as a function of misalignment angle for distances between 380 an 800 nm. The blue triangles represent case (a) and the red circles represent case (b).}\label{fig:HomoChiral}
\end{figure}

\begin{figure}
    \centering
\begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\linewidth]{./Fig6a.eps}
 \caption{ }
 \label{fig:HeteroChiralLR}
 \end{subfigure}
  \hfill
  \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\linewidth]{./Fig6b.eps}
 \caption{ }
 \label{fig:HeteroChiralRL}
 \end{subfigure}
  \hfill
  \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\linewidth]{./Fig6c.eps}
\caption{ }
\hfill
\label{fig:DeltaTau}
 \end{subfigure}
 \begin{subfigure}[t]{0.45\textwidth}
\includegraphics[width=\linewidth]{./Fig6d.eps}
 \caption{ }
 \label{fig:HeteroChiral380-800nm}
 \end{subfigure}
    \caption{The Casimir torque for the heterochiral case. (a): the left handed crystal is placed on the left, and the right handed crystal is on the right. (b): the reverse of panel (a). (c): several different contributions at 10 nm distance. Black crosses: difference between case (a) and case (b), including only lowest order terms. Red asterisks: idem, including higher order terms. Orange circles: perturbative contribution to case (a), blue squares:  perturbative contribution to case (b). (d): Casimir torque at distances between 380 and 800 nm, both for case (a) (black asterisks), and case (b) (red circles).  }\label{fig:HeteroChiral}
\end{figure}

The results are shown in \cref{fig:BaTiO35CB}. The upper two panels \cref{fig:Left5CBBaTiO3,fig:Right5CBBaTiO3}  display the two cases we distinguish here: the BaTiO$_3$ crystal can be faced with either a right handed or a left handed crystal. \cref{fig:BaTiO35CBphi} shows the Casimir torque as a function of the misalignment angle $\varphi$. For reference, we have included the usual case for a nematic liquid crystal, indicated by the cyan curve with triangles pointing to the right. The leading order term in \cref{eq:final} creates a shift in the angular dependence of the torque of about $\pi/2$ radians, making the proportionality closer to $\cos2\varphi$ than the usual $\sin2\varphi$. The sign of the torque changes depending on the left- or right handedness of the cholesteric liquid crystal, which is a qualitative chirality effect. While the leading order term horizontally shifts the graph, the next order term enhances the amplitude in only one direction, creating a clear deviation from the usual sinusoidal shape of the curve (even though it is still periodic with period $\pi$). For this reason we have verified that

\begin{equation}\label{eq:energy0}
     \int\limits_0^{\pi}\tau\mathrm{d}\varphi=0
\end{equation}
so that no energy is gained or lost during a rotation over a period. Note that the shape of the curve cannot be attributed to retardation effects, which are negligible at a distance of 10 nm. The higher order correction actually is most significant at short distances $\sim$ 10 nm, whereas retardation effects become more pronounced at larger distances. Finally, \cref{fig:BaTiO35CBdistance} shows the distance dependence of the amplitude of the torque. Not only is the amplitude enhanced significantly compared to the nematic case, it also decreases more slowly as a function of distance. Since the configuration of a birefringent and a semi-infinite liquid crystalline sample is presently experimentally accessible \cite{Somers2018}, this could make experimental observation of the torque easier at larger distances. The inset of \cref{fig:BaTiO35CBdistance} shows that the angular dependence remains the same at large distances (between 380 and 800 nm): the positions of the extrema and the nodes do not change as a function of distance. The higher order correction of the BCH formula (\cref{eq:final} is negligible in this separation range,  contributing significantly only at distances of $\sim$ 10 nm.

Next, we proceed to the case of two liquid crystals facing each other. Here we distinguish between the so called homo and heterochiral cases, shown in \cref{fig:HomoChiral,fig:HeteroChiral} respectively. In the former case two right handed or two left handed crystals face each other, and  in the latter case a left  handed crystal is faced with a right handed one.

First we will discuss the homochiral case, as shown in \cref{fig:HomoChiral}. \cref{fig:HomoChiralRight,fig:HomoChiralLeft} illustrate the possible configurations: two right- and two left-handed crystals. \cref{fig:HomoChiralPhi} shows a plot of the Casimir torque as a function of misalignment angle at a distance of 10 nm. For reference, the case of two nematic liquid (light blue triangles) has also been included. As in the previous case, (see \cref{fig:BaTiO35CB}), the leading order term shifts the usual sine to a cosine. However, unlike the previous case, here the higher order terms decrease the amplitude of the torque in one direction, which does give rise to a deviation from the sinusoidal form. This short distance is the only range where the finite pitch length effects, represented by the higher order term in \cref{eq:final}, actually contribute significantly. At this range, it seems as though the torque for the left handed configuration is opposite of that of the right handed one. However, at larger distances it can be seen that this is not the case. In \cref{fig:HomoChiral380-800nm} the Casimir torque is plotted as a function of the misalignment angle for distances between 380 and 800 nm. At this distance range, the higher order term of \cref{eq:final} contribute negligibly. Still even the lower order terms give rise to a significant deviation from the sinusoidal behavior from the previous case. Moreover, the position of the extrema changes with the distance, and the nodes are not on the x-axis. In other words, the angular dependence changes with the separation distance, which is qualitatively different from the previous case or the case of two birefringent materials. The left- and right-handed configurations do not exhibit opposite torques, but the phase difference between the curves changes as a function of distance as well. It is worth pointing out that, if a particular angle is chosen, near one of the nodes, it becomes possible to see the torque change sign as a function of distance. This kind of sign change has also been reported for biaxial materials \cite{Thiyam2018} and Weyl semi-metals \cite{ChenLiang2020}, for example.

Next we will address the heterochiral case shown in \cref{fig:HeteroChiral}. \cref{fig:HeteroChiralLR,fig:HeteroChiralRL} show the configurations of a left handed crystal on the left and a right handed crystal on the right, and the reverse configuration, respectively. These configurations are expected to be physically indistinguishable. To test this, we plot the different contributions to the torque as a function of misalignment angle again at 10 nm distance, see \cref{fig:DeltaTau}. It can be seen that the difference between the torque of \cref{fig:HeteroChiralLR} and \cref{fig:HeteroChiralRL} is non-negligible if the higher order term in \cref{eq:final} is omitted. (See the black crosses in \cref{fig:DeltaTau}) However, the inclusion of the higher order terms renders this difference negligible. (The red asterisks in \cref{fig:DeltaTau}). So even though the higher order corrections themselves are different (as indicated by the orange circles and blue squares in \cref{fig:DeltaTau}), the total torque is identical for both configurations. Finally, the torque is plotted as a function of the misalignment angle for distances of 380 to 800 nm in \cref{fig:HeteroChiral380-800nm}. Here the torque behaves qualitatively similar to the previous case, with the phase somewhere in between that of the torque for the two homochiral configurations of \cref{fig:HomoChiral380-800nm}. As expected, the torques for the configurations of \cref{fig:HeteroChiralLR,fig:HeteroChiralRL} are identical.

\section{Conclusions and outlook}\label{sec:Conclusions}
In order to understand the effects of cholesteric chirality on the Casimir-Lifshitz torque, we have modeled a cholesteric liquid crystal as a uniaxial planar multilayer system. The system consists of identical layers of equal thicknesses, but the orientation of each layer differs slightly from that of the adjacent one. This model allows us to derive an analytical simplification of the Fresnel matrices as an expansion in this small orientation difference, by means of the Baker-Campbell-Hausdorff (BCH) formula.

Numerically, we have obtained results that are appreciably different from the case of two semi-infinite birefringent plates, which is a valid approximation for nematic liquid crystals. \cite{Somers2015,Somers2018} In particular, in the case of a birefringent half space facing a cholesteric liquid crystal, the leading order term in the BCH expansion shifts the angular dependence of the torque by about $\pm\pi/2$ radians, depending on the chirality of the liquid crystal. At very short distances of about 10 nm, the higher order BCH term gives rise to a significant deviation from the usual sinusoidal shape of the torque as a function of misalignment angle. Moreover, the amplitude of the torque decreases more slowly as a function of distance than in the nematic case. Since the configuration of a birefringent crystal faced with a liquid crystal is presently experimentally accessible, \cite{Somers2018} we believe that especially this latter result can be useful for detecting the torque at large separation distances.

The case of two cholesteric liquid crystals consists of three physically different configurations: two homochiral ones (two left-handed crystals or two right-handed crystals), and one heterochiral one. In each case, the higher order term of the BCH expansion, which is associated with finite pitch length, contributes significantly at about 10 nm distance.  We have shown that it must be included to obtain physically consistent results in the heterochiral case. At larger distances, the angular dependence of the torque turns out to change as a function of distance, while again significantly deviating from the known sinusoidal behavior. Moreover, for certain values of the misalignment angle, the torque can change sign as a function of distance. Present technology does not yet make it possible to reach a sufficient degree of parallelism to measure the Casimir torque between two liquid crystals with satisfactory accuracy. \cite{Munday2021Review} However, present efforts directed at a resolution of this problem \cite{Sedmik2020} give us hope that it may be possible in the future to measure the torque for such a case.

Possible future endeavors include calculations beyond the approximations used here. For example, one could allow discontinuity in the inhomogeneous dielectric function. This would make it possible to study the effect of a thin water layer between two layers of the cholesteric liquid crystal. Alternatively, one could explore a larger distance range in which case one needs to investigate the effects of a finite total thickness. Finally, the approximate solution of the Maxwell equations obtained here may be of interest for the study of  cholesterics in different contexts, or for the understanding of chiral media in general.

\begin{acknowledgments}
We acknowledge funding from the Key project \#12034019 of the National Natural Science Foundation of China (WB and RP) and the Singapore Ministry of Education Academic Research Fund Tier 1 Grant No. RG160/19(S) (B-SL).
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
Trimbakeshwar
Kojoli - Nashik
Kojoli Population - Nashik, Maharashtra
Kojoli is a medium size village located in Trimbakeshwar Taluka of Nashik district, Maharashtra with total 112 families residing. The Kojoli village has population of 615 of which 312 are males while 303 are females as per Population Census 2011.
In Kojoli village population of children with age 0-6 is 113 which makes up 18.37 % of total population of village. Average Sex Ratio of Kojoli village is 971 which is higher than Maharashtra state average of 929. Child Sex Ratio for the Kojoli as per census is 1018, higher than Maharashtra average of 894.
Kojoli village has higher literacy rate compared to Maharashtra. In 2011, literacy rate of Kojoli village was 82.47 % compared to 82.34 % of Maharashtra. In Kojoli Male literacy stands at 86.72 % while female literacy rate was 78.05 %.
As per constitution of India and Panchyati Raaj Act, Kojoli village is administrated by Sarpanch (Head of Village) who is elected representative of village. Our website, don't have information about schools and hospital in Kojoli village.
Kojoli Data
Child (0-6) 113 56 57
Total Workers 312 170 142
Marginal Worker 12 4 8
In Kojoli village, most of the village population is from Schedule Tribe (ST). Schedule Tribe (ST) constitutes 65.85 % while Schedule Caste (SC) were 12.52 % of total population in Kojoli village.
In Kojoli village out of total population, 312 were engaged in work activities. 96.15 % of workers describe their work as Main Work (Employment or Earning more than 6 Months) while 3.85 % were involved in Marginal activity providing livelihood for less than 6 months. Of 312 workers engaged in Main Work, 112 were cultivators (owner or co-owner) while 137 were Agricultural labourer.

redpajama_set_name: RedPajamaCommonCrawl


text:
World wakes up to 'Brexit': 5 things to know
What we know folllowing the United Kingdom's historic vote to leave the European Union.
World wakes up to 'Brexit': 5 things to know What we know folllowing the United Kingdom's historic vote to leave the European Union. Check out this story on azcentral.com: http://usat.ly/28QUCHT
Jane Onyanga-Omara, USA TODAY Published 2:24 a.m. MT June 24, 2016 | Updated 7:07 a.m. MT June 24, 2016
The historic Brexit referendum passed, meaning Great Britain will leave the European Union. Find out what this means for the country and the global economy. USA TODAY
The sun rises over the Houses of Parliament and Big Ben in London, Britain, June 24, 2016.(Photo: HANNAH MCKAY, EPA)
LONDON — The British public voted to leave the European Union in a historic referendum Thursday. The departure is dubbed "Brexit" — Britain's exit from the 28-member bloc.
Here's what we know:
Morning! U.K. votes for 'Brexit'
The margin of victory for "Brexit" in the referendum was 52% to 48%, according to a final tally announced early Friday. "Leave" supporters argued that severing ties with the EU is the only way to reduce high levels of immigration and ensure the U.K.'s sovereignty over EU regulations. The "remain" camp said jobs, the economy, national security and the U.K.'s standing on the world stage would be compromised outside the bloc.
Cameron announces his resignation
David Cameron announced he will step down following the country's vote to leave the Eurpean Union. (June 24) AP
Prime Minister David Cameron, who led the campaign to remain in the EU, announced he will step down. "We should aim to have a new prime minister in place by the start of the Conservative Party conference in October," he said in a speech outside Downing Street on Friday soon after the vote tally was announced.
Historic 'Brexit' is here: What happens next
Cameron said he informed Queen Elizabeth II of his decision, adding that the will of the British people "must be respected." Among his possible successors is Boris Johnson, the former mayor of London who led the "leave" campaign.
U.K's Cameron vows to 'steady ship' before resigning
Pound plunges, global markets in turmoil
European Union leader Donald Tusk warns against 'hysterical' reactions after Britain votes to leave the EU. Newslook
The British pound fell 11% to a 31-year low and Britain's main stock index, the FTSE 100, nosedived 8.7% to 5,790 points shortly after the open Friday. U.S. stocks also nosedived at the open.
Markets were also volatile in Asia. Japan's Nikkei 225 average dropped nearly 1,300 points, or 8%, Friday. The Bank of England — the U.K.'s central bank — said it was monitoring developments closely and "has undertaken extensive contingency planning."
'No immediate change' for Britons, EU citizens
Londoners react to the result of Britain's EU referendum which saw 52 percent vote in favour of quitting the bloc, a margin of more than one million votes. Newslook
Cameron said there will be "no immediate change in circumstances" for British citizens living in other EU countries and EU nationals living in Britain. About 3 million people from other EU countries live in the U.K., and 1.2 million Britons live elsewhere in the EU.
One of the fundamental principles of the bloc is allowing the free movement of people and workers. Negotiations between the U.K. and the EU over the exit terms will take at least two years.
Emergency meetings called to deal with fallout
Emergency meetings were called around Europe Friday to deal with the fallout from the vote. "This result clearly has very significant implications for Ireland, as well as for Britain and for the European Union," a statement from the Irish government said ahead of an emergency cabinet meeting. German Chancellor Angela Merkel was expected to make a statement following a meeting with lawmakers in Berlin.
The European Parliament called a special session Tuesday. However Janet Yellen, the chair of the U.S. Federal Reserve, said Wednesday, before the referendum, that she did not plan any special meetings to respond to the result.
Contributing: Kim Hjelmgaard
'Brexit' vote on United Kingdom's future in EU
A journalist holds up a copy of the London Evening Standard newspaper as he takes part in a television broadcast outside the Bank of England in the City of London, on June 24, 2016 Matt Dunham, AP
Britain's Prime Minister David Cameron puts his arm around his wife Samantha after speaking outside 10 Downing Street on June 24, 2016. Cameron says he will resign by the time of party conference in the fall after Britain voted to leave the European Union after a bitterly divisive referendum campaign, according to tallies of official results Friday. Daniel Leal-Olivas, AP
Supporters of the 'Stronger In' Campaign react as results of the EU referendum are announced at a results party at the Royal Festival Hall in London on June 23, 2016. Rob Stothard, AFP/Getty Images
Supporters of the 'Stronger In' Campaign react as results of the EU referendum are announced at a results party at the Royal Festival Hall in London early in the morning of June 24, 2016. Rob Stothard, AFP/Getty Images
Leader of the United Kingdom Independence Party, or UKIP, Nigel Farage reacts outside the Leave EU referendum party at Millbank Tower in central London on June 24, 2016. Geoff Caddick, AFP/Getty Images
A pedestian looks at an electronic stock indicator in the window of a securities company in Tokyo on June 24, 2016. Tokyo stocks went into a hair-raising dive on Brexit fears, plunging more than eight percent on during the afternoon of June 24, as major firms including Toyota and mobile carrier SoftBank went into free fall. Kazuhiro Nogi, AFP/Getty Images
A trader sits at his desk under the day's performance board at the Frankfurt Stock exchange, the day after a majority of the British public voted for leaving the European Union, on June 24, 2016. Thomas Lohnes, Getty Images
An "In or Out" sign outside a house in Hangleton, England, on June 23, 2016. The United Kingdom held a referendum on whether to stay or leave the European Union, and millions voted in the bitterly-fought, knife-edge referendum that could tear up the nation's EU membership and spark the greatest emergency of the bloc's 60-year history. Glyn Kirk, AFP/Getty Images
City workers carry in a ballot box for counting at Gibraltar University after the close of polling stations on the day of the EU Referendum. Sean Gallup, Getty Images
Polling signs are removed at Llanishen Leisure Centre as voting comes to an end in Cardiff, Wales. Matthew Horwood, Getty Images
Referendum workers verify the validity of ballots in the verification center before counting at Gibraltar University after the close of polling stations. Sean Gallup, Getty Images
The North East region European Union referendum count takes place in Sunderland, United Kingdom. Ian Forsyth, Getty Images
A man wheels ballot boxes into Llanishen Leisure Centre in Cardiff, Wales. Matthew Horwood, Getty Images
Barbara House, who supports the Brexit, waters her plants near a poster calling for Britain to leave the EU in Headington. Millions of Britons began voting in a bitterly fought referendum that could tear up the island nation's EU membership and spark the greatest emergency of the bloc's 60-year history. Adrian Dennis, AFP/Getty Images
Satirical artist Kaya Mar poses with Brexit-themed artwork depicting UK Independence Party leader Nigel Farage, left, and former London mayor Boris Johnson in London. Leon Neal, AFP/Getty Images
UK Independence Party leader Nigel Farage poses for photographers as he leaves a polling station south of London as Britain holds a referendum to vote on whether to remain in or to leave the European Union. Ben Stansall, AFP/Getty Images
An elderly woman is reflected in a puddle of rain water as she arrives at a polling station at the Royal Hospital in Chelsea, west London, as Britain holds a referendum to vote on whether to remain in or to leave the European Union. Leon Neal, AFP/Getty Images
Voter Anne Whitman swims at Arundel Lido, set up as a polling station, in Arundel, southern England, as Britain holds a referendum on whether to stay or leave the European Union. Glyn Kirk, AFP/Getty Images
Jacqui Vaughan, left, and Sophie Allison, on horses Splash and Sharna, ride out of the driveway of a private residence used as a polling station, near Reading, west of London, as Britain holds a referendum to vote on whether to remain in or to leave the European Union. The rural house where Allison runs her livery business has been a polling station for local residents for more than 40 years. Adrian Dennis, AFP/Getty Images
A volunteer hands an unmarked ballot paper to a voter inside a bus being used as a temporary polling station in Kingston-Upon-Hull, northern England, as Britain holds a referendum to vote on whether to remain in or to leave the European Union. Lindsey Parnaby, AFP/Getty Images
British Prime Minister David Cameron and his wife, Samantha, leave after voting in the EU Referendum at the Central Methodist Hall in London. Facundo Arrizabalaga, EPA
Nuns leave a polling station after voting in the EU Referendum in London. Britons voted on whether to remain in or leave the European Union in a referendum on 23 June. Hannah Mckay, EPA
Chelsea pensioners enter a polling station to vote in the EU Referendum in London. Britons voted on whether to remain in or leave the European Union in a referendum on 23 June. Hannah Mckay, EPA
Several Gibraltarian voters wait to cast their ballot at a polling station in Gibraltar. Britons are deciding whether to remain in or leave the European Union. A.Carrasco Ragel, EPA
People arrive to cast their ballots in the EU referendum in Gibraltar. Sean Gallup, Getty Images
Jade Murphy stands outside a polling station at Notre Dame Primary School after voting in the EU referendum in Glasgow, Scotland. Jeff J Mitchell, Getty Images
A woman in a wheelchair with British and European Union flags shows her support for the United Kingdom to remain in the European Union on the day of the EU Referendum in Gibraltar. The United Kingdom and its dependent territories went to the polls to decide whether or not the United Kingdom will remain in the European Union. Sean Gallup, Getty Images
A man accompanied by his dog exits a polling station after voting in the EU referendum in Belfast, Northern Ireland. T Charles McQuillan, Getty Images
British Prime Minister David Cameron and his wife, Samantha, leave after voting in the EU referendum at the Central Methodist Hall in London on June 23, 2016. FACUNDO ARRIZABALAGA, EPA
Britain's Labour Party leader Jeremy Corbyn smiles as he arrives to cast his vote in the EU referendum at a polling station in Islington, north London. Daniel Leal Olivas, AP
Members of the public hold flags at a stay-in, pro-EU Referendum event in Parliament Square in London on June 19, 2016. Hayoung Jeon, European Pressphoto Agency
A bartender pulls a pint of beer decorated with EU Referendum beer taps in a pub in London on June 22, 2016. HANNAH MCKAY, EPA
British Prime Minister David Cameron reacts as he travels on his campaign bus from Bristol, England, on June 22, 2016, as the prime minister campaigns to avoid a Brexit ahead of the EU referendum. Rival sides threw their efforts into the final day of campaigning Wednesday, on the eve of Britain's vote on EU membership that will shape the future of Europe. Geoff Caddick, AFP/Getty Images
Former London Mayor, and "Vote Leave" campaigner Boris Johnson is pictured with a pint of beer ahead of meeting with members of the public and supporters in Piercebridge, England, on June 22, 2016, as he continued to campaign for a Brexit. Scott Heppell, AFP/Getty Images
A woman reads a British newspaper backing the 'Vote Remain' campaign on the London Underground on June 22, 2016. Hannah Mckay, European Pressphoto Agency
A "Leave" supporter poses in Clacton-on-Sea as UK Independence Party (UKIP) leader Nigel Farage visits on June 21, 2016. Justin Tallis, AFP/Getty Images
Liberal Democrat supporters gather in a park for a "Remain" rally by Liberal Democrat Leader Tim Farron in Carshalton on June 21, 2016. ADRIAN DENNIS, AFP/Getty Images
A "Leave" campaigner is seen in Clacton-on-Sea as UK Independence Party (UKIP) leader Nigel Farage visits on June 21, 2016. Justin Tallis, AFP/Getty Images
People walk past a pop-up "Brexit Tattoos" tattoo shop offering satirical anti-EU pro-Brexit tattoos for free, in central London on June 22, 2016. The shop, open for one day only, is the idea of the pro-EU "Remain" campaign. Justin Tallis, AFP/Getty Images
A 'Vote Leave' campaigner displays campaign buttons on his suit on June 22, 2016, in London. Jack Taylor, Getty Images
London Mayor Sadiq Khan delivers a speech at a 'Vote Remain' event in London on June 22, 2016. Will Oliver, European Pressphoto Agency
UK Independence Party (UKIP) leader Nigel Farage delivers a keynote speech in London on June 22, 2016. Facundo Arrizabalaga, European Pressphoto Agency
A man backing the 'Vote Remain' campaign walks along a street in London on June 22, 2016. Hannah Mckay, European Pressphoto Agency
A taxi driver holds a sticker for the "Vote Leave" pro-Brexit campaign as he drives past news media in London on June 22, 2016. Justin Tallis, AFP/Getty Images
A pro-EU banner being towed behind a plane passes Big Ben in the Houses of Parliament in London on June 22, 2016, on the final day of the EU referendum campaign before Britain goes to the polls to vote on continuing its membership in the EU. Alastair Grant, AP
An exterior view shows the Tate Modern Gallery illuminated with 'Vote Remain' in London on June 21, 2016. Will Oliver, European Pressphoto Agency
David Cameron announces resignation after U.K. votes for 'Brexit'
Read or Share this story: http://usat.ly/28QUCHT
Salary database prompts officials to address concerns of 'woefully underpaid'
Missing Northern Arizona man found dead in Wyoming river
ADOT employee accused of tampering with motor-vehicle records
Family, friends mourn 17-year-old killed in Phoenix hit-and-run
Arpaio 2020 billboards pop up — but who's behind them?
No monsoon storms expected in Phoenix-area for days


The Voices of the ElePHPant podcast has posted their latest interview with a member of the PHP community. In this latest show host Cal Evans talks with Todd Ross, a developer at Answers.com.
Cal and Todd talk about what the technology stack looks like at Answers.com, the "second largest PHP-based site" on the web. Technologies mentioned include Java (for search indexing), Varnish for caching and various database technologies. They also cover the talk he presented at this year's php[tek] (2016) about deploying large scale PHP applications.
You can listen (and watch) this episode either through the in-page audio or video player. If you'd like to listen offline, you can download the mp3 directly. If you enjoy the show, be sure to subscribe to their feed and follow them on Twitter for updates when the latest shows are released.


text:
Heritage Guide to The Geelong College
Heritage Home Page
To find information in this Guide please select one of the green coloured options.
To Select a Page Group when displayed, right click and select 'Open'.
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Archives Research Guide
Archives Catalog.
Image Gallery.
Year-by-Year Timeline.
CHAPMAN, Colin Burton
CHAPMAN, Professor Colin Burton
Colin Chapman, noted pharmacist and past dean of the Faculty of Pharmacy and Pharmaceutical Sciences (formerly the Victorian College of Pharmacy), was appointed an Emeritus Professor of Monash University in 2010. This appointment recognised Colin's dedicated service to both the faculty and the University over a long period.
After graduating with a Bachelor of Pharmacy from what was then the Victorian College of Pharmacy in 1970, Emeritus Professor Chapman undertook 12 months pre-registration training at Bull and Owen's Pharmacy. During this time he was a tutor resident at Geelong College assisting with students of the then Warrinn House. In 1969 he was called up to serve as a national serviceman in the Australian Regular Army, initially at Puckapunyal, Victoria. He undertook further work and studies with the army at Scheyville, NSW and Ingleburn, NSW, until his discharge in July 1972. In 1972 he married and travelled around Australia before returning to Melbourne to study Veterinary Science at Melbourne University He received a Bachelor of Veterinary Science graduating dux in 1976.
After a year with the Department of Agriculture He commenced studying for a PhD at the Attwood Veterinary Research Laboratory completing this in 1980. He later joined the then Victorian College of Pharmacy, Parkville as head of the School of Pharaceutics in 1987 and was appointed Dean and Director of the College in 1991.
During his period as head of the College of Pharmacy he negotiated its amalgamation with Monash University. After amalgamation he established the Monash Institute of Pharmaceutical Sciences, managed the extension of the Bachelor of Pharmacy program from 3 to 4 years to enable inclusion of clinical placements, enabled introduction of new graduate and postgraduate degrees and oversaw a significant increase in student numbers.
Rural initiatives included the Rural Entry Scheme for secondary students, compulsory rural placements for all pharmacy students at Monash, and the Monash University Pharmacy Scholarship for high achievers from country areas.
At various times, Colin was a member of the Therapeutic Goods Committee and chairman of the Committee of the Heads of Pharmacy Schools in Australia and New Zealand. He assisted in the establishment of the New Zealand and Australian Pharmacy Schools Accreditation Committee and wrote their competency guidelines. He also assisted with the establishment of the Victorian Pharmacy College Foundation in 2001.
In 1998, he was awarded an Honorary Fellowship of the Pharmaceutical Society of Australia Australia.
Following retirement from Monash University, Professor Chapman BPharm, Vic.I.C., BVSc (Hons) PhD Melb continues to work in the field of pharmacy, both locally and internationally. He has participated in projects in Indonesia and Saudi Arabia.
Colin Chapman was enrolled at Geelong College in 1960, departing in December 1966. At College, he was a member of the Swimming Team for two years from 1965 to 1966, vice captain of the 4th XVIII FootballTeam of 1964 and in 1965 and 1966 a member of the 2nd XVIII Football Team. In 1963, he was a member of the Athletics Team. He was both a House Prefect and School Prefect in 1966.
Sources: Monash University ( www.monash.edu/pharm/about/news-listing/archive/2011/colin-champan (sic)
© The Geelong College. Unless otherwise attributed, The Geelong College asserts its creative and commercial rights over all images and text used in this publication. No images or text material may be copied, reproduced or published without the written authorisation of The College.


Eighty-four-foot ketch. Five years. Forty thousand miles. The result: A boatload of handcrafted works of art and cultural artifacts. Jim, Vicki and Dylan's epic trip inspired HAVAIKI OCEANIC AND TRIBAL ART, which features a collection of treasures from the countries of Oceania; particularly the Solomon Islands, Papua New Guinea and Hawai'i. Many items are museum quality. Others are made from centuries-old practices. All are one of a kind, including the contemporary jewelry— influenced by primitive designs—prints and T-shirts. What's more, every table, sculpture, calabash, necklace—everything— comes with an interesting story.
Kaua'i is truly a special place as we believe you will soon discover. THE BEST GUIDEBOOK will help you enjoy the splendor of our island, the wide array of outdoor adventures and activities, and the warmth of our people.
The BEST GUIDEBOOK is a wonderful resource for information that will help you make the most of your trip to Kaua'i.


text:
Remote working video surveying: A solution for Sir Robert McAlpine during global crisis
Sir Robert McAlpine shapes how towns and cities develop and design and build some of the most historic buildings in the country.
As a consequence of the travel restrictions of the COVID-19 pandemic, Sir Robert McAlpine's Geotechnical Surveying Engineer who normally comes over from Ibiza for the summer months was unable to travel to carry out the walked Geotechnical Surveys for the Autolink A19 DBFO Contract.
Andy Sinclair, Project Manager for Sir Robert McAlpine on the contract explains, "We were trying to come up with an alternative to a walked survey by an Inspector or Engineer, so the visual aspect was the main reason why we needed to get footage of the survey being undertaken, which we could then share with our Geotechnical Engineer over the cloud." John McMahon, Highways Engineer continues, "We needed a solution to capture that data, so we thought video was a good format to send that information over to Ibiza so that he could utilise the videos and interrogate them to complete his survey for the year."
With a short time scale to scope and deliver the solution which would allow the Surveying Engineer to undertake his assessment of the condition of the Geotechnical assets on their network, Sir Robert McAlpine approached MGISS. "They had experience from other contacts from other industries, they made bespoke software to do what we wanted it to do, and that's important to us. MGISS understands the need to get the scope of the solution correct before any development is done, so the solution that does get put down on the table at the end of the project is exactly what you need."
John adds that, "We came to MGISS because we've used them in the past for other surveys, they've developed survey solutions for us for our VRS inspections, road signs, drainage – so we've got a history of working collaboratively with them. Plus, the speed was important because it was something that we hadn't attempted before; we needed to get it done through the summer months, we contacted MGISS to look at options and we've been able to capture the data on site in all weathers.
All the data was captured, sent over to MGISS, and they provided us with the arcGIS solution for the surveyor in Ibiza to be able to interrogate the videos, zoom in, pause and generally get the most out of the videos. It's gone really well."
"MGISS has adapted to our needs as they've changed throughout this 30 year contract"
It was vital for the solution for Sir Robert McAlpine to fit with their existing network, John explains, "It's really important that data is captured and secured, it's information that we have to compile and use during the contract and we also have to make sure that that data is in a fit shape to be handed back to the client at the end of the 30 year contract."
With a history of working together, it was important to MGISS to assess and serve the needs of Sir Robert McAlpine on this particular project. John reflects, "The positives is that in the relationship we've got with MGISS, we know they're going to react quickly and come up with what they know our requirements are, not what they want to sell us. Whenever we work together, we'll tell MGISS what we want to achieve and they'll come up with options, and then we'll agree on the scope before we go anywhere. Whenever we contact them they're always at the end of the phone, they're always willing to come to site, we've got a good relationship with them.
Working with MGISS allows us to supply a need that we didn't have previously, MGISS has adapted to our needs as they've changed throughout this 30 year contract, and it allows us to have a one-stop place where we can develop and adapt as a business."
Andy adds that, "Working with MGISS has allowed us to solve our short-term solutions, it's helped us to develop our thinking towards our medium-term problems, and the legacy now is that we've moved it on from the first year of video surveys into this year with a tweaked version of the solution. There's certainly been a lot of interesting features that we could look to develop into other surveys that we do, so it's been interesting and quite rewarding working with MGISS." Check out Sir Robert McAlpine here, www.srm.com.
Want to make an enquiry?
An end to guesswork: A new generation of technology for Laing O'Rourke
Improving data processing: An informed business decision for Crown Plus
Up to date on assets: A one-system asset management solution for Sir Robert McAlpine


It appears that Seth McFarlane has been vindicated in the eyes of a court after the dismissal of a lawsuit against him alleging that his 2012 comedic vulgar taking teddy bear film, Ted was too similar to a series of video sketches focusing on a character called "Charlie The Abusive Teddy Bear." A Los Angeles federal judge was satisfied with proof that the film's titular teddy bear was inspired through a creative process independent of the "Charlie" videos. It's certainly a huge relief for McFarlane as Ted 2 readies an imminent summer release.
Plaintiff and its counsel are now satisfied that, based on discovery produced in the action, the character Ted was independently created by Seth MacFarlane using his own efforts and creativity and was not copied from Plaintiff's Charlie character.
While the legal specifics of exactly what "satisfied" both plaintiff Bengal Mountain Productions and the Court was not revealed, it is clear that proof of the non-infringing nature of the Ted character had to be clear and definitive. In fact, a statement released by McFarlane's co-defendants in Media Rights Captal praises McFarlane's "genius" while dealing the parting shot saying that the lawsuit was "without merit and has been dismissed."
Undoubtedly, it serves as the most satisfying outcome for Seth McFarlane, whose notable work on animated hit shows like Family Guy and American Dad have long been subject to accusations ranging from being too derivative to straight up plagiarism. Yet, with Ted, we had a film that grossed an impressive $549 million worldwide, centering on an intriguing, but not entirely out from left-field concept of an anthropomorphic teddy bear who says and does hilariously offensive things. (And coincidentally, sounds a lot like MacFarlane's Family Guy character, Peter Griffin.) It's not surprising that there was something already out there that shared enough attributes to attract claims of being copycatted.
So, while we may not have all the dry legal details of the decision, it seems pretty clear from the outset that the lawsuit wouldn't stick. At the very least, this bit of free publicity will do nicely to help usher in Ted 2, which, to the chagrin of "Charlie," is set to hit theaters on June 26.


Dmytro Syzonov. Psycholinguistic Bases of Media Literacy: Considering the Issue of Media Texts Interpretation.
(2017) Science and education, 7, 82-88. Odessa.
The paper deals with psycholinguistic peculiarities of media communication. Media texts are analyzed with regard to extra- and intertextual factors such as multiculturalism, linguistic imperalism, media formats rennovation, etc. The experiment was based on the headlines of posts taken from Facebook and Twitter and was aimed at determining the emotions which arise in respondents reading them. 20 university students aged from 17 till 20 were surveyed by means of a questionnaire which is focused on the determination of respondents' psychology of emotions. The questionnaire consists of two units: with yes-no questions and with free-answers questions. The emotional scale was assessed by three values: negative, positive and neutral. According to the research outcomes, most of the respondents pay attention to the headline before reading the text of the post (95%) and assess its content according to the headline (90%) which shows headlines' manipulation effect. 55% of the respondents manifested positive emotions together with their positive assessment of the media text. Similarly, they manifested negative emotions together with their negative assessment of the text which shows the extra linguistic effect on the content of a media text, particularly political, moral, ethical or social. 25% of the respondents were neutral to the headlines and did not want to read the texts which shows the high level of their impassive attitude to the information flow. It means that people's emotions are affected by the headlines with positive or negative assessment. It has been found that psycholinguistic bases of media literacy are manifested in the understanding of the essence of a media text and its main assignment, in the analysis of conscious/subconscious impact on a respondent, in search for main methods and tactics with the help of which a recipient would not be a passive media consumer but a participant of active socio-political processes by means of mass media.
media psychology, media linguistics, psycholinguistics, media literacy, mass media, media planning, media monitoring.
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7. Syzonov, D.Y. (2015). Metody linhvistychnoho doslidzhennia mediatekstu: do problemy naukovoi interpretatsii [Methods of linguistic research of the mediatext: the problem of scientific interpretation]. Aktualmi problem Ukrainskoi linhvistyky: teoriia ta praktyka – Actual issues of Ukrainian linguistics: theory and practice, 30, 53-62. Kyiv [in Ukrainian].
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Bettering your homes interior has many advantages. These advantages include a more snug residing surroundings, more usable space, and more dwelling value in the even that you just decide to promote your house. Many individuals suppose that dwelling improvement and reworking is a long tedious project that will price loads of money. With a few tips and concepts, you will discover that this doesn't must be the case.
Individuals select to rework their dwelling for a lot of reasons. New adorning concepts that they like, dwelling improvement and making a new dwelling more satisfying are just a few. No matter what the reason being that you just select to vary up your house, you need to make certain the job does not put on you out half way by way of it and go away you with a half way remodeled dwelling.
Working room by room will be less complicated than engaged on the house as a complete. This will provide you with the chance to make every room the way you want it. You'll be able to fully end every room before deciding to start out one other project. This will provide you with time to loosen up and collect your ideas before starting the following room.
Deciding what theme you want every room to be will help when shopping for accessories for the rooms. Some rooms will be more enjoyable while others might be bolder for a more dramatic impact. Shade schemes together with accessorizing a room will aid you obtain this impact.
Thanks for visiting my blog, article above (Glamorous Solar System Quilt Inspiration With Regard To Our Property) published by Honora Payne at September, 28 2018.


The mixSorb S is the new development of 3P INSTRUMENTS for the investigation of the adsorption of gas and vapor mixtures! With the mixSorb S the mixSorb instrument series is completed, S stands for "small", that is for the investigation of smallest sample quantities by dynamic mixture sorption of gases and vapors, L stands for "Large", that means the investigation of larger quantities of industrial adsorbents.
The volume-optimized measuring system with an adsorber of 1 cc is available for the mixSorb S (8 bar) and the mixSorb SHP (HP for high pressure 68 bar). With this analyzer of the mixSorb series, you can now get the practice-relevant measurements in your lab. Adsorber volume of 1 cc – this allows the use of small sample quantities of e.g. MOFs or other "future" adsorbents for special separation processes.
Industrial adsorbents such as activated carbons and zeolites are usually available in sufficient amounts and should be investigated in representative quantities. Practical studies of the mixture gas and vapor sorption with the mixSorb L with a standard adsorber column of 100 cc are already being carried out at a number of device users in Germany and abroad. Thus, the mixSorb series established itself as a standard sorption instrument for laboratory tests of technical adsorbents. It impresses with its flexibility in both the hardware and in the control and evaluation/simulation software.
Concentration: Since CH4 (yellow) breaks through earlier than CO2 (red) we have some time, where only CH4 and He can be detected. Assuming a 100 % CH4 breakthrough, this would yield in a 50 % volume fraction at the detector.
Replacement: CO2 replaces CH4 at the adsorption sites, which yield in partial CH4 desorption and an increasing CH4 concentration at the adsorber outlet.
The next experiment shows a breakthrough curve of 10 % CO2 in CH4 at 40 bar ad 40 °C with a flow rate of 50 cm³/min on the same sample, measured with the internal TCD.
In a third example, we show that application-related measurements are possible with very small amounts of sample called SIFSIX-2-Cu-I that was provided by Prof. Zaworotko from Limerick University. A mass of 184 mg of sample were used in the separation of Acetylene from Ethylene (Pore chemistry and size control in hybrid porous materials for acetylene capture from ethylene; Science, 2016, 353 (6295), 141-144). A gas mixture of 10 % Acetylene and 65 % Ethylene in a Helium carrier gas were used at 1 bar of pressure and a flow rate of 40 cm³ min-1. It can be seen that the mixSorb S in combination with a mass spectrometer can deliver reliable scientific data with small sample amounts and under challenging analytical conditions. The observed rollover effect is less pronounced than for the example with CO2 and CH4. When plotting the molar fluxes over time it can be seen that this rollover can be entirely explained with the concentration effect. Therefore very little or no Ethylene had been adsorbed which speaks for a very good selectivity towards Acetylene for this sample.
The last example is a breakthrough experiment with 5 % CO2 in N2 at 5 bar and 40 °C on an activated carbon. The breakthrough curve was recorded, the loading of CO2 was determined and compared to mixture isotherm data, predicted with the IAST model by the 3P sim software out of pure component high pressure isotherms. The measured loading, determined with the mixSorb S shows an excellent agreement with the predicted mixture data.