text:
\section{Introduction}\label{sec1}

The control of emission by nanoscale-size objects such as single atoms, quantum dots, dimers and oriented semiconductor polymer nanostructures is fundamentally important for applications in nanoscale optical manipulation, optical sensing, information processing and quantum communication~\cite{bd09,nh11,ra12}. The recent advances in nanophotonics have stimulated a series of experimental and theoretical works demonstrating that composite nano-systems can serve as nanoantennas~\cite{kk10,cv10,lc11,jl13}. Examples include structures composed of nanodimers, high-permittivity dielectric particles, metal particles, and atomic chains~\cite{ch03,hf04,mf07,pc08,hz12,sf13}. It has been demonstrated that these nano-systems may squeeze light into nanoscale volumes~\cite{sb10}, enhance the excitation and emission rate of individual emitters~\cite{ab06,kh06,mg07}, and tune the luminescence spectrum~\cite{rs08}, and the polarization~\cite{mt08}. Particularly interesting is the ability to control and tune the radiation pattern of a nanoantenna.

A related interesting problem is the ability of a composite structure of nanoparticles to work as a frequency filter which could route different frequencies of an incident beam into different directions. This feature has been demonstrated experimentally by Shegai {\it et al.}~\cite{sc11}. In the experiment, a pair of metallic nanoparticles, gold and silver, was deposited on glass at a very small distance. When illuminated with white light, the system scattered the red and blue components of the incident light into opposite directions.

Recently, we have developed a theory of directional emission for the somewhat related problem of directional light scattering by a system composed of two two-level atoms~\cite{la13}. We have shown that the system can operate as a directional nanonantenna provided that the atoms are not identical with unequal transition dipole moments or different transition frequencies. We have found that a difference between the transition dipole moments or between the transition frequencies of the atoms creates a phase shift between the dipole moments of the atoms which then leads to the directional light scattering. Thus, the crucial factor for the directionality is to use emitters of unequal dipole moments or different transition frequencies. In the experiment of Shegai {\it et al.}~\cite{sc11} the required phase difference between the dipole moments was achieved by using two nonidentical metallic nanoparticles of different plasmonic frequencies. It has also been shown that a metal-dielectric structure composed of a pair of dielectric and metal nanoparticles can lead to a directional light scattering~\cite{nk12}. These studies show that a constant phase shift between dipole moments can be achieved with nonidentical nanoparticles.

However, a constant phase shift between two oscillating dipoles could be achieved with identical nanoparticles. It is the purpose of this paper to demonstrate that a controlled directional emission can also be achieved in a system composed of two identical emitters. As we shall see below, it requires a driving field which could create a constant phase difference between the atomic dipoles or a constant difference between the field amplitudes at the positions of the atoms. We consider a system composed of two identical emitters located at short distances and driven by a laser beam with a structured phase and amplitude, an optical vortex beam like a Laguerre-Gaussian (LG) or Bessel beam~\cite{an08}. The use of an LG or Bessel beam can ensure either equal amplitudes and a constant phase difference or equal phases and a constant amplitude difference at the positions of the atoms.
 We are particularly interested in the directionality of the emission that is produced by the LG beam which is applied in the following manner: (i) a constant phase difference is produced at the positions of the atoms, and (ii) a constant amplitude difference is produced at the position of the atoms. We demonstrate that a controlled directionality of the emission can be achieved under these specific driving configurations. Especially, the atoms placed at short distances may operate as a highly directional nanoantenna. Depending on the phase or amplitude differences at the locations of the emitters, the system may operate as a two- or one-sided nanoantenna. A simple physical interpretation of the sources of the two- and one-sided emissions is given in terms of the collective states of the system.

The paper is organized as follows. In Sec.~\ref{sec2}, we review the basic properties of the laser field of a structured complex amplitude, an LG beam, and provide a simple explanation of how the beam could be applied to create a constant phase and/or amplitude difference between two dipoles located at different points. In Sec.~\ref{sec3} we describe the model in detail and discuss the method we use to to determine the radiation pattern of the system. The general expression for the radiation intensity is presented along with a brief discussion of its directional properties. In Sec.~\ref{sec4} we give illustrative figures of the directionality of the emission for two configurations of the LG beam at the positions of the atoms, either the same amplitudes and a constant phase difference or the same phases and a difference between the amplitudes. We discuss different cases and present a detailed analysis of the directional properties of the radiation pattern. Finally, in Sec.~\ref{sec5}, we summarize our results and conclude.

\section{Amplitude and phase difference in a Laguerre-Gaussian beam}\label{sec2}

We proceed now to show how we can create a scheme where two atoms (emitters) although irradiated by the same laser beam, may experience different phases and/or amplitudes of the beam. As mentioned above this can be achieved if the driving field has a structured phase and amplitude, i.e. a phase and amplitude which have a complex spatial dependence as in the case of an optical vortex beam like an LG or a Bessel beam~\cite{an08}. We choose to work with an LG beam whose interaction with a two-level atom result in a Rabi frequency given by~\cite{ba12}
\begin{align}
\Omega_{lp} &=\frac{\Omega_{00}}{\sqrt{1+z^{2}/z_{R}^{2}}}\sqrt{\frac{p!}{\left(|l|+p\right)!}}\left(\frac{r\sqrt{2}}{w(z)}\right)^{|l|} \nonumber\\
&\times L_{p}^{|l|}\left(2r^{2}/w^{2}(z)\right)e^{il\phi}e^{ikz} ,\label{v1}
\end{align}
where we have assumed that the beam propagates in the $z$-direction, $w(z)= w_{0}\sqrt{1+ z^{2} /z_{R}^{2}}$, $w_{0}$ is the beam waist, and $r$ is the position of the atom in the $xy$ plane. The numbers $l$ and $p$ are the mode indices with $l$ associated with the quantized angular momentum $l\hbar$ carried by the beam photon along the propagation axis and $p$  is the radial index associated with the number of intensity rings in the transverse plane.
The factor $L^{|l|}_{p}$ is the associated Laguerre polynomial while $\Omega_{00}=\Gamma\sqrt{2P/(\pi w_{0}^{2}I_{s}})$, with $P$ standing for the power of the laser beam, $\Gamma$ is the atomic excited state decay rate and $I_{s}$ the saturation intensity for the atomic transition. Since the expression (\ref{v1}) is given in the cylindrical coordinates we also have $r= \sqrt{x^{2} +y^{2}}$ and $\phi=\arctan(y/x)$.

First, we show how we can achieve different phases being experienced by two atoms located at different positions. Consider the example shown in Fig.~\ref{fig1} in which a system of two atoms is irradiated by a LG beam propagating along the $z$-direction. The shaded ringlike region shows schematically the spatial intensity distribution of the beam in the case of $p = 0$.
\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\includegraphics[height=5.5cm]{vasfig1}
\end{tabular}
\end{center}
\caption{(Color online) A Laguerre-Gaussian laser beam with a doughnut-like intensity profile interacting with two atoms. The beam travels along the $z$-axis but is displaced by a distance $d$ along the $x$-axis. The atoms distant $r_{12}$ from each other are located at positions $(0,r_{12}/2,0)$ and $(0,-r_{12}/2,0)$, respectively.}
\label{fig1}
\end{figure}

Assume that the interatomic axis has been displaced at a distance $d$ along the $x$-axis with respect to the center of an LG beam and the atoms are located at positions $(0,r_{12}/2,0)$ and $(0,-r_{12}/2,0)$. The Rabi frequencies $\Omega_{j}\, (j=1,2)$ experienced by the atoms can be easily determined from the general expression~(\ref{v1}), and are given by
\begin{align}
\Omega_{j} &= \frac{\Omega_{00}}{\sqrt{|l|!}} \left[\frac{2\left(r_{12}^{2}/4 +d^{2}\right)}{w_{0}^{2}}\right]^{\frac{|l|}{2}}
\exp\!\left[-\frac{\left(r_{12}^{2}/4 +d^{2}\right)}{w_{0}^{2}}\right]\nonumber\\
&\times \exp\!\left[-(-1)^{j}\, il\arctan\!\left(\frac{r_{12}}{2d}\right)\right] ,\label{v2}
\end{align}
where $j=1,2$. Clearly, there is a phase difference between the Rabi frequencies which is due to the azimuthal phase factors, and is given by
\begin{align}
\Delta\phi = 2l\arctan\left(r_{12}/2d\right) .\label{3}
\end{align}
The phase difference depends on the distance $r_{12}$ between the atoms, the lateral displacement $d$, and the helicity~$l$. By changing one of these parameters we can change the phase difference at will. It is easily seen that if there is no displacement along the $x$-axis, $d = 0$, and then the phase difference is equal to $\Delta\phi =l\pi$. Of course in a beam with a structured amplitude any change in these parameters will also result in a change in the magnitude of the Rabi frequencies. If we keep the beam propagation axis symmetrical with respect to the two atoms the field amplitudes experienced by the atoms will be the same.

The doughnut-like transverse intensity profile of the LG beam has a maximum at radial distances $r=w_{0}\sqrt{|l|/2}$, where $w_{0}$ is the beam waist at $z = 0$. The size of the beam waist can range from a few hundreds of microns down to half the wavelength of the laser beam. To create a phase difference equal to $\pi$ the angle subtended by the two atoms has to be equal to $\pi/|l|$. In this case the distance between the atoms is given by $r_{12} = w_{0}\sqrt{2|l|}\sin(\pi/2|l|)$. For very large values of the helicity $l$, the distance between the atoms is approximated by $r_{12}=w_{0}\pi/\sqrt{2|l|}$.  As we have shown in Ref.~\cite{la13}, for very short distances between the atoms their mutual interaction is very crucial for mode switching and routing. These can occur when the distance between the atoms is of the order of the wavelength of the laser radiation since in this case the interatomic dipole-dipole interaction is significant. Thus if, for example, we wish the distance between the atoms to be equal to $\lambda$ then the beam waist has to be $w_{0}=\lambda\sqrt{2|l|}/\pi$. For a value of $l=100$, which is experimentally achievable~\cite{fl12}, we get a beam waist of around $4.5\lambda$. Higher values of $l$ can lead to larger beam waists. In Ref.~\cite{fl12}, LG beams were produced with $l=300$. The use of such a beam with a waist equal to $7.8\lambda$ will ensure that two atoms separated by a distance equal to $\lambda$ experience Rabi frequencies with a phase difference equal to~$\pi$.

We now demonstrate how one can achieve different Rabi frequencies at the positions of the atoms. Since the mutual distance between the atoms has to be of the order of the wavelength for the system to work as a strongly directional antenna we must choose proper light fields with amplitude gradients of this order of magnitude. The first such case would be the use of a standing wave, which could be created along the $y$-axis. In this case we can arrange the configuration in such a way that one atom is at the node of the standing wave, thus its Rabi frequency is $0$, while the other one is at the anti-node of the standing wave so its Rabi frequency is the maximum possible. This configuration may have several restrictions in its operation since the distance between the node and the neighboring anti-node is always fixed and equal to $\lambda/4$. This fixed distance imposes a restriction on the distance between the atoms.

The structured beams could give us new opportunities since they could have spatial gradients of intensity which, with the proper choice of parameters, can be very sharp resulting in the desired different Rabi frequencies. The first such case is provided by an LG beam with a radial number $p$ different from $0$.  As noted by Plick {\it et al.}~\cite{pl13}, this number has been often overlooked in the literature with the attention of the researchers concentrated on the index $l$ which is related to the quantized orbital angular momentum carried by the photon. As we know a beam with a radial number $p$ has $p+1$ maxima (rings) in the transverse intensity profile. The distance between these rings depends on indices $l$ and $p$ and the choice of beam waist. The distance can be analytically calculated for generic $l$ and $p=1,2$.  As our numerical analysis has shown the intensity rings come closer as the value of the radial number $p$ increases while simultaneously we keep $l$ at the lowest possible value i.e. $l=1$. We chose here the case, where $p=20$ and $l=1$. In this case the intensity pattern on the transverse plane has 21 rings. In Fig.~\ref{fig2} we show the intensity versus the radial distance from the beam axis. The axis is scaled in units of the beam waist $w_{0}$. It is easily seen that the intensity maxima are very close to each other. The first and the second maxima are at a distance equal to $0.27w_{0}$. If we choose a value for the beam waist equal to $3.7\lambda$ then the distance between these maxima becomes equal to the wavelength of light $\lambda$. By placing the two atoms at these points we ensure that the Rabi frequency of the first atom can be three times larger than that of the second atom. By applying an LG beam with a higher values of the radial index the intensity maxima can come even closer so we can ensure different Rabi frequencies for atoms at a distance of the order of the wavelength for larger values of the beam waist.
\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\includegraphics[height=5.5cm]{vasfig2}
\end{tabular}
\end{center}
\caption{(Color online) Intensity pattern (in arbitrary units), up to a radial distance equal to the beam waist, in the case of a Laguerre-Gaussian laser beam with $l = 1$ and $p = 20$. The beam is assumed to propagate along the $z$ axis. In set: How the two atoms have to be irradiated by the beam in order to experience different light intensities and thus to ensure different Rabi frequencies.}
\label{fig2}
\end{figure}

The second case is obtained when we superpose two similar LG laser beams, propagating along the $z$-direction with opposite helicities, $l$ and $-l$. In this case we get a beam, which has a petal-like transverse intensity pattern with a number of $2l$ petals. The intensity pattern of this configuration, known in the literature as an optical Ferris wheel~\cite{pl13,fl07} is illustrated in Fig.~\ref{fig3}. The inset in Fig.~\ref{fig3} shows the intensity pattern on the first quadrant of the $xy$ plane with such a configuration for $l = 15$. It is shown that the intensity goes through $0$ as we move azimuthally from one point of maximum intensity to the next point of maximum intensity. The angle, which separates these two points is given, in a generic Ferris scheme, as $\Delta\phi =180^{\circ}/|l|$. To get two Rabi frequencies with a ratio, for example, equal to $10$, i.e. $\Omega_{1}/\Omega_{2} =10$, we can place one atom at a point of maximum intensity and the other atom at a point at an azimuthal angle $\theta = 84.3^{\circ}/|l|$. The length of the arc subtended by this angle, which is also the interatomic distance, is $r_{12}= (w_{0}/2)\sqrt{|l|/2}\sin(84.3^{\circ}/2|l|)$. If we use as $l=15$, then this angle is about $\theta = 5.6^{\circ}$ and the length of the corresponding arc is $r_{12} = 0.067w_{0}$. This length determines the distance between the two atoms. If we wish the distance to be equal to half of the wavelength we have to choose a beam waist equal to $7.46\lambda$. By using LG beams of a higher helicity we can ensure larger values of the beam waist. For example, for $l=100$ the beam waist has to be equal to $19.23\lambda$.
\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\includegraphics[height=5.5cm]{vasfig3}
\end{tabular}
\end{center}
\caption{(Color online) The two atoms located on the $xy$ plane are separated from each other by $r_{12}$ and are irradiated with an optical Ferris wheel beam propagating along the $z$-direction with helicity equal to $l = 15$. Atoms are at points where they experience different intensities of the field. Inset: Intensity pattern at the first quadrant of the $xy$ plane.}
\label{fig3}
\end{figure}

The Ferris wheel configuration has yet another interesting property. As has been shown~\cite{pl13,fl07}, if we change the frequency of one of the two beams comprising the Ferris wheel, then the intensity pattern rotates in space at an angular frequency $\omega_{F} = (\omega_{1}-\omega_{2})/2|l|$. When the pattern rotates the atoms will be periodically in regions of different intensities, thus their Rabi frequencies will vary in time, and this will result in a case where the intensity pattern emitted by the two atoms will rotate in space.

We should point out that the regime of very small beam waists, which we used in our numerical analysis, is associated with tightly focused light beams carrying orbital angular momentum. The use of such beams may introduce effects associated with the non-paraxial regime~\cite{nh12}, which we do not take into account in this paper.

\section{General formalism}\label{sec3}

We consider a system composed of two identical emitters located at fixed positions, at a distance $r_{12}$ from each other, irradiated by a coherent laser beam and interacting with the electromagnetic field. Each emitter may be simply a single atom, for example, or a quantum dot, or a larger object such as a dimer. If atoms are involved, this system can be realized in practice by optical methods, i.e. by considering that the atoms are placed at neighboring optical lattice sites or in microscopic optical traps~\cite{we11}. Alternatively, we can consider two ions trapped by electromagnetic fields~\cite{eb93}. The latter method has played an important role in the demonstration of interference effects in the light emitted by two ions~\cite{ib98}. Small fluctuations of the atom positions around the trap minima may obscure the effects of mode switching and light routing but we do not consider them in this work. If we wish to avoid the effects of spatial fluctuations, we can consider our atoms as generic two-level emitters (for example specially tailored nanoparticles) embedded in a material~\cite{sc11}. We focus on a single dipole transition between two non-degenerate energy levels of each emitter, the excited $\ket{e_{i}}$ and ground $\ket{g_{i}}$ levels, and refer to it simply as a two-level system. Thus, the emitters, according to the above description, can be modelled as dipoles. Our main concern here is on how the two emitters located close to each other and collectively interacting with the electromagnetic field can operate as a highly directional nanoantenna. We focus on the directional properties of the radiation field emitted by the system.

\subsection{Intensity of the radiation field}\label{sec3a}

The quantity of central interest is the intensity of the radiation field emitted by two atoms and detected in the far field zone of the system. The intensity can be written in terms of the positive and negative frequency parts of the electric field~as
\begin{align}
I(\vec{R},t) = \frac{R^{2}}{2\pi k_{0}}\langle \vec{E}^{(-)}(\vec{R},t)\cdot \vec{E}^{(+)}(\vec{R},t)\rangle ,\label{v4}
\end{align}
where $k_{0}=\omega_{0}/c$, $\vec{R}$ is a vector pointing in the direction of the detection of the field and $R$ is the distance between the radiating system and the detector. Here, we have introduced the factor $(R^{2}/2\pi k_{0})$ so that $I(\vec{R},t)d\Omega dt$ is the probability of detecting a photon at time $t$ inside the solid angle element $d\Omega$ around the direction $\vec{R}$ in the time interval $dt$.

The electric field radiated by the atoms can be expressed in terms of the atomic dipole moments. The negative and positive frequency parts of the field in the far field zone of the radiating atoms can be written as the sum of dipole fields,
\begin{align}
\vec{E}^{(-)}(\vec{R},t) &= -k_{0}^{2}\sum_{i=1}^{2}\frac{[\vec{R}_{i}\times(\vec{R}_{i}\times \vec{\mu}_{i})]}{R_{i}^{3}}S^{+}_{i}e^{ik\hat{R}\cdot \vec{r}_{i}} ,\nonumber\\
\vec{E}^{(+)}(\vec{R},t) &= -k_{0}^{2}\sum_{i=1}^{2}\frac{[\vec{R}_{i}\times(\vec{R}_{i}\times \vec{\mu}_{i})]}{R_{i}^{3}}S^{-}_{i}e^{-ik\hat{R}\cdot \vec{r}_{i}} ,\label{v5}
\end{align}
where $\vec{\mu}_{i}$ is the transition dipole moment, $S_{i}^{+}$ and $S^{-}_{i}$ are the usual raising and lowering operators of atom $i$, $\vec{R}_{i}$ is the position vector of atom $i$, and $\hat{R}$ is the unit vector in the direction of observation $(\hat{R}=\vec{R}/R)$.

When Eq.~(\ref{v5}) is substituted into Eq.~(\ref{v4}), the intensity of the radiation field measured in the direction $\vec{R}$ at time $t$ becomes~\cite{leh,ag74}
\begin{align}
I(\vec{R},t) = u(\hat{R})\sum_{i,j=1}^{2}\langle S^{+}_{i}(t)S^{-}_{j}(t)\rangle e^{k\hat{R}\cdot\vec{r}_{ij}} ,\label{v6}
\end{align}
where $u(\hat{R})=(3\Gamma/8\pi)\sin^{2}\vartheta$, with $\vartheta$ the angle between the direction of observation $\vec{R}$ and the direction of the atomic dipole moments, and $\Gamma$$-$ the spontaneous emission (damping) rate of the atomic transition. It is seen that the intensity of the radiation field is determined by the correlation functions of the atomic dipole operators. The intensity can be written in terms of four contributions involving the correlation functions and geometrical factors
\begin{align}
&I(\vec{R},t) = u(\hat{R})\left\{\langle S^{+}_{1}(t)S^{-}_{1}(t)\rangle + \langle S_{2}^{+}(t)S^{-}_{2}(t)\rangle\right. \nonumber\\
&\left. +\left[\langle S_{1}^{+}(t)S^{-}_{2}(t)\rangle\!+\!\langle S_{2}^{+}(t)S^{-}_{1}(t)\rangle\right]\!\cos(kr_{12}\cos\theta)\right. \nonumber\\
&\left. +\, i\!\left[\langle S^{+}_{1}(t)S^{-}_{2}(t)\rangle\!-\!\langle S_{2}^{+}(t)S^{-}_{1}(t)\rangle\right]\!\sin(kr_{12}\cos\theta)\right\} ,\label{v7}
\end{align}
where $\theta$ is the angle between $\vec{r}_{12}$ and the direction of observation $\hat{R}$.
The variation of the intensity with the observation angle $\theta$ is called the radiation pattern. Certain general features of the radiation pattern follow from Eq.~(\ref{v7}).
The first term expresses the intensity of the emitted radiation created by spontaneous emission from atom $1$. The second term expresses the intensity of the radiation spontaneously emitted by atom $2$. These two terms are always positive and independent of $\theta$. The contribution of these two terms obviously leads to a spherical shape of the radiation pattern. The third and fourth terms result from the interference between the radiation fields emitted by different atoms. If nonzero, these terms can lead to a non-spherical shape of the radiation pattern and the emitted radiation can exhibit a strong enhancement or reduction in a direction $\theta$ at which $\cos(kr_{12}\cos\theta)=\pm 1$ and/or $\sin(kr_{12}\cos\theta)=\pm 1$.

Let us look at some features of the $\cos(kr_{12}\cos\theta)$ and $\sin(kr_{12}\cos\theta)$ factors in $I(\vec{R},t)$ which define directions of maximum and minimum emission.
First, since the cosine and sine functions are shifted in phase by $\pi/2$, we see that the directions in which these two terms can enhance or reduce the intensity do not overlap. In particular, for atoms separated by a distance $r_{12}=\lambda/2$, the factor $\cos(kr_{12}\cos\theta)= 1$ for $\theta$ equal to $\pi/2$ and $3\pi/2$, while $\sin(kr_{12}\cos\theta) = 1$ for $\theta$ equal to $\pi/3$ and $5\pi/3$. Moreover, the number of directions in which the intensity can be enhanced or reduced increases with an increasing $r_{12}$. For instance, when $r_{12}=\lambda$, the factor $\cos(kr_{12}\cos\theta)= 1$ for $\theta$ equal to $0,\pi/2, \pi$, and $3\pi/2$, whereas $\sin(kr_{12}\cos\theta)=1$ for $\theta$ equal to $0.42\pi$ and  $1.58\pi$. Thus, if the goal is for the system to work as a highly directional nano-antenna emitting light only in a few directions, then the emitters should be kept at distances shorter than the resonant wavelength,  $r_{12}\leq \lambda$.

There is an another important difference in the directional properties of the two factors. It is easy see that if there is a direction $\theta$ in which $\cos(kr_{12}\cos\theta)$ is maximal $(=1)$, it is also maximal in the opposite direction $\theta +180^{\circ}$. This means that the factor $\cos(kr_{12}\cos\theta)$ has the property of concentrating the radiation along two opposite axial modes. We refer to this feature as an axial concentration of the radiation or a {\it two-sided} emission.
The directional property of $\sin(kr_{12}\cos\theta)$ is different. If there is a direction $\theta$ in which $\sin(kr_{12}\cos\theta)$ is maximal, it is also maximal in the direction $\theta^{\prime} = 360^{\circ} -\theta$. Thus, the factor $\sin(kr_{12}\cos\theta)$ has the property of concentrating the emission on one side of the system in two axial modes propagating in directions differing by $2\theta$. We refer to this feature as a spatial concentration of the radiation or a {\it one-sided} emission. The differences in the ways in which the factors $\cos(kr_{12}\cos\theta)$ and $\sin(kr_{12}\cos\theta)$ affect the radiation pattern are illustrated graphically in~Sec.~\ref{sec4}.

It is convenient, in particular for physical interpretation, to write the intensity in terms of the density matrix elements of the density operator of the two-atom system represented in the basis of the superposition (collective) states~\cite{fb13,dic,ft02}
\begin{align}
&\ket g =\ket{g_{1}}\ket{g_{2}} ,\quad \ket e =\ket{e_{1}}\ket{e_{2}} ,\nonumber\\
&\ket s =\frac{1}{\sqrt{2}}\left(\ket{e_{1}}\ket{g_{2}} + \ket{g_{1}}\ket{e_{2}}\right) ,\nonumber\\
&\ket a =\frac{1}{\sqrt{2}}\left(\ket{e_{1}}\ket{g_{2}} - \ket{g_{1}}\ket{e_{2}}\right) ,\label{v9}
\end{align}
where $\ket s$ and $\ket a$ are, respectively, the symmetric and antisymmetric combinations of the product of bare atomic states. Here, $\ket{e_{i}}$ and $\ket{g_{i}}$ represent the excited and ground states of atom $i$. It is easily shown that in terms of the density matrix elements the correlation functions appearing in Eq.~(\ref{v7}) are
\begin{align}
&\langle S_{1}^{+}(t)S^{-}_{1}(t)\rangle\!+\!\langle S_{2}^{+}(t)S^{-}_{2}(t)\rangle = \rho_{ss}(t)\!+\!\rho_{aa}(t)\!+\!2\rho_{ee}(t) ,\nonumber\\
&\langle S_{1}^{+}(t)S^{-}_{2}(t)\rangle\!+\!\langle S^{+}_{2}(t)S^{-}_{1}(t)\rangle = \rho_{ss}(t) - \rho_{aa}(t) , \nonumber\\
&i\left[\langle S^{+}_{1}(t)S^{-}_{2}(t)\rangle - \langle S^{+}_{2}(t)S^{-}_{1}(t)\rangle\right] = 2{\rm Im}[\rho_{as}(t)] .\label{v9a}
\end{align}
We see from Eq.~(\ref{v9a}) that the interference term proportional to $\cos(kr_{12}\cos\theta)$ will contribute to the intensity only when $\rho_{ss}(t)\neq \rho_{aa}(t)$, i.e. when the symmetric and antisymmetric states are unequally populated. Consequently, a reduction in the population of one of the two states relative to the population of the other state will be accompanied by an axial two-sided emission. On the other hand, the interference term proportional to $\sin(kr_{12}\cos\theta)$ will contribute to the intensity only when ${\rm Im}[\rho_{as}(t)]\neq 0$. Thus, a nonzero coherence between the symmetric and the antisymmetric states will be accompanied by a one-sided emission.

\subsection{Master equation}

Suppose that the emitters are located at positions, $\vec{r}_{1} = (0,-r_{12}/2,0)$ and $\vec{r}_{2} = (0,r_{12}/2,0)$ along the $y$ axis and are illuminated with a monochromatic laser beam of angular frequency $\omega_{L}$, a propagation wave vector $\vec{k}_{L}$, and of a variable intensity profile with spatially varying amplitude $\vec{E}_{L}(x,y,z)$ and phase $\phi_{L}(x,y,z)$, as shown in Fig.~\ref{fig4}. In a real experiment this might be an LG laser beam. The laser excites transitions between the two energy states $\ket{e_{i}}$ and $\ket{g_{i}}\, (i = 1, 2)$, separated by a frequency $\omega_{0}$. The population of the excited states of the emitters decays to the ground states at a rate~$\Gamma$.
\begin{figure}[h]
\centering{}\includegraphics[width=1.2\columnwidth]{vasfig4}
\caption{(Color online) Schematic diagram of the system composed of two emitters represented by transition dipole moments $\vec{\mu}_{1}$ and $\vec{\mu}_{2}$ and irradiated with a laser beam propagating in the direction perpendicular to the interatomic axis,~$\vec{k}_{L}\perp \vec{r}_{12}$. The laser has a cylindrically symmetric (doughnut shaped) intensity profile. With this variable intensity profile, the two emitters can be excited with arbitrary amplitudes and phases.}
\label{fig4}
\end{figure}

The emitters radiate by the process of spontaneous emission and the intensity of the emitted field is detected at a point $A$ distance $\vec{R}$ in the $yz$ plane. To explore the role of the structured beam profile in the controlled mode switching and directional emission, we choose the direction of propagation of the laser beam perpendicular to the interatomic axis, $\vec{k}_{L}\perp \vec{r}_{12}$, i.e. the laser beam propagates along the $z$-direction and has a structured amplitude in the $xy$ plane. If the driving field is in the form of a plane wave of a constant amplitude then the atoms experience the same amplitude and phase of the field. However, when the laser field is taken to have a variable intensity profile, then the atoms can experience different amplitudes and phases of the field; as a result, the radiative properties of the atoms may change.

Our purpose is to calculate the angular distribution of the radiation intensity, the radiation pattern of the field emitted by the driven atoms. Thus, according to Eq.~(\ref{v7}), we need to obtain the atomic correlation functions or the matrix elements of the reduced (atomic) density operator, which satisfies the master equation~\cite{leh,ag74}
\begin{align}
\frac{\partial\rho}{\partial t} = -\frac{i}{\hbar}\left[H_{0}+H_{L},\rho\right] + \left(\frac{\partial}{\partial t}\rho\right)_{S} +  \left(\frac{\partial}{\partial t}\rho\right)_{A} ,\label{e11}
\end{align}
where
\begin{align}
H_{0} = \hbar\Delta_{L}\left(A_{ss}\!+\!A_{aa}\!+\!2A_{ee}\right) +\hbar\Omega_{12}\left(A_{ss}-A_{aa}\right) ,
\end{align}
and
\begin{align}
H_{L} &= \frac{\hbar}{2\sqrt{2}}\left\{\Omega_{1}\left(A_{es}+A_{sg}+A_{ag}-A_{ea}\right)\right. \nonumber\\
&\left. +\, \Omega_{2}\left(A_{es}+A_{sg}-A_{ag}+A_{ea}\right) + {\rm H.c.}\right\} \label{e11a}
\end{align}
is the interaction Hamiltonian of the atoms with the driving laser field.
Here, $A_{nm}=\ket n\bra m$ are projection operator between the collective states,  $\Delta_{L}=\omega_{0}-\omega_{L}$ is the detuning of the laser field frequency $\omega_{L}$ from the atomic transition frequency $\omega_{0}$, and the quantities $\Omega_{1}$ and $\Omega_{2}$ are the Rabi frequencies of the laser field at the position of the atom $1$ and $2$, respectively. The quantity $\Omega_{12}$ depends on the distance between the atoms and gives the effect of the atomic interaction, the dipole-dipole interaction, on the shift of the energy levels of the system
\begin{align}
\Omega_{12} &= \frac{3}{4}\Gamma\left\{-\left[1-\left(\bar{\mu}\cdot\bar{r}_{12}\right)^{2}\right]\frac{\cos(k_{0}r_{12})}{k_{0}r_{12}}\right. \nonumber\\
&\left. + \left[1-3\left(\bar{\mu}\cdot\bar{r}_{12}\right)^{2}\right]\left[\frac{\sin(k_{0}r_{12})}{\left(k_{0}r_{12}\right)^{2}}+\frac{\cos(k_{0}r_{12})}{\left(k_{0}r_{12}\right)^{3}}\right]\right\} ,
\end{align}
where $k_{0}=\omega_{0}/c$, and $\bar{\mu}$ and $\bar{r}_{12}$ are unit vectors in the direction of the atomic dipole moment $\vec{\mu}$ and the interatomic axis $\vec{r}_{12}$, respectively.

The dissipative part of the master equation (\ref{e11}) consists of two terms corresponding to the two decay (fluorescent emission) channels~\cite{leh,fb13}; the symmetric channel $\ket e\rightarrow\ket{s}\rightarrow \ket g$ with an enhanced decay rate $\Gamma_{s}=\Gamma +\Gamma_{12}$:
\begin{align}
\left(\frac{\partial}{\partial t}\rho\right)_{S} =& -\frac{1}{2}\Gamma_{s}\left\{\left(A_{ee}+A_{ss}\right)\rho\right. \nonumber\\
&\left. +\rho\left(A_{ee}+A_{ss}\right) + A_{se}\rho A_{sg} + A_{gs}\rho A_{es}\right. \nonumber\\
&\left.- 2\left(A_{se}\rho A_{es} + A_{gs}\rho A_{sg}\right)\right\} ,\label{e12}
\end{align}
and the anti-symmetric channel $\ket e\rightarrow\ket{a}\rightarrow \ket g$ with a reduced decay rate $\Gamma_{a} = \Gamma -\Gamma_{12}$:
\begin{align}
\left(\frac{\partial}{\partial t}\rho\right)_{A} =& -\frac{1}{2}\Gamma_{a}\left\{\left(A_{ee}+A_{aa}\right)\rho\right. \nonumber\\
&\left. +\rho\left(A_{ee}+A_{aa}\right) + A_{ae}\rho A_{ag} +  A_{ga}\rho A_{ea}\right. \nonumber\\
&\left. - 2\left(A_{ae}\rho A_{ea} + A_{ga}\rho A_{ag}\right)\right\} .\label{e13}
\end{align}
Here, $\Gamma_{12}$ gives the effect of the atomic interaction on the damping rate of the system
\begin{align}
\Gamma_{12} &= \frac{3}{2}\Gamma\left\{\left[1-\left(\bar{\mu}\cdot\bar{r}_{12}\right)^{2}\right]\frac{\sin(k_{0}r_{12})}{k_{0}r_{12}}\right. \nonumber\\
&\left. + \left[1-3\!\left(\bar{\mu}\cdot\bar{r}_{12}\right)^{2}\right]\!\left[\frac{\cos(k_{0}r_{12})}{\left(k_{0}r_{12}\right)^{2}}-\frac{\sin(k_{0}r_{12})}{\left(k_{0}r_{12}\right)^{3}}\right]\right\} .
\end{align}

Since the Rabi frequencies $\Omega_{1}=|\Omega_{1}|\exp(i\phi_{1})$ and $\Omega_{2}=|\Omega_{2}|\exp(i\phi_{2})$ can be different, i.e. can have different magnitudes and phases, we write the interaction Hamiltonian (\ref{e11a}) in the form
\begin{align}
H_{L} = i\hbar\left[\Omega_{\alpha}\!\left(A_{es}\!+\!A_{sg}\right) + \Omega_{\beta}\!\left(A_{ea}\!+\!A_{ag}\right) - {\rm H.c.}\right] ,\label{e11b}
\end{align}
where
\begin{align}
\Omega_{\alpha} &= \left(\Omega_{d}\sin\phi_{d} -i\Omega_{0}\cos\phi_{d}\right)/\sqrt{2} ,\nonumber\\
\Omega_{\beta} &= \left(\Omega_{0}\sin\phi_{d} - i\Omega_{d}\cos\phi_{d}\right)/\sqrt{2} ,
\end{align}
in which
\begin{align}
\Omega_{0} = \frac{1}{2}\left(|\Omega_{1}| + |\Omega_{2}|\right) ,\quad \Omega_{d} = \frac{1}{2}\left(|\Omega_{1}| - |\Omega_{2}|\right) ,
\end{align}
and $\phi_{d} =(\phi_{1} - \phi_{2})/2$ is the phase difference between the Rabi frequencies of the atoms. Clearly, the transitions of the symmetric channel are driven at the Rabi frequency $\Omega_{\alpha}$, while the transitions of the antisymmetric channel are driven at the Rabi frequency $\Omega_{\beta}$. In general, both channels can be simultaneously driven by the laser.

\subsection{Equations of motion for the density matrix elements}\label{sec3c}

In the basis of the collective states the master equation leads to a set of 15 coupled differential equations for the density matrix elements. Among them we can distinguish eight equations for the density matrix elements determining transitions between the symmetric states
\begin{align}
\dot{\rho}_{ss} =& -\Gamma_{s}(\rho_{ss}-\rho_{ee}) + \left[\Omega_{\alpha}(\rho_{gs}-\rho_{se}) +c.c.\right] , \nonumber\\
\dot{\rho}_{ee} =& -2\Gamma\rho_{ee} +[\Omega_{\alpha}\rho_{se}-\Omega_{\beta}\rho_{ae} +c.c. ] ,\nonumber\\
\dot{\rho}_{sg} =& -\left(\frac{1}{2}\Gamma_{s}+i\Omega_{12}\right)\rho_{sg} -\frac{1}{2}\Gamma_{s}\rho_{es} +\Omega_{\alpha}(\rho_{gg}-\rho_{ss}) \nonumber\\
&-\Omega_{\alpha}^{\ast}\rho_{eg} -\Omega_{\beta}\rho_{sa} ,\nonumber\\
\dot{\rho}_{se} =& -\left[\frac{1}{2}\left(2\Gamma_{s}+\Gamma_{a}\right)+i\Omega_{12}\right]\rho_{se} + \Omega^{\ast}_{\alpha}(\rho_{ss}-\rho_{ee}) \nonumber\\
&+\Omega_{\alpha}\rho_{ge} -\Omega_{\beta}^{\ast}\rho_{sa} , \nonumber\\
\dot{\rho}_{eg} =& -\Gamma\rho_{eg} +\Omega_{\alpha}(\rho_{sg}-\rho_{es}) -\Omega_{\beta}(\rho_{ag}+\rho_{ea}) ,\nonumber\\
\dot{\rho}_{gs} =& -\left(\frac{1}{2}\Gamma_{s}-i\Omega_{12}\right)\rho_{gs} -\frac{1}{2}\Gamma_{s}\rho_{se} +\Omega^{\ast}_{\alpha}(\rho_{gg}-\rho_{ss}) \nonumber\\
&-\Omega_{\alpha}\rho_{ge} -\Omega^{\ast}_{\beta}\rho_{as} ,\nonumber\\
\dot{\rho}_{es} =& -\left[\frac{1}{2}\left(2\Gamma_{s}+\Gamma_{a}\right)-i\Omega_{12}\right]\rho_{es} + \Omega_{\alpha}(\rho_{ss}-\rho_{ee}) \nonumber\\ &+\Omega^{\ast}_{\alpha}\rho_{eg} -\Omega_{\beta}\rho_{as} , \nonumber\\
\dot{\rho}_{ge} =& -\Gamma\rho_{ge} +\Omega^{\ast}_{\alpha}(\rho_{gs}-\rho_{se}) -\Omega^{\ast}_{\beta}(\rho_{ga}+\rho_{ae}) ,\label{e16}
\end{align}
and seven equations determining transitions between antisymmetric states
\begin{align}
\dot{\rho}_{aa} =& -\Gamma_{a}(\rho_{aa}-\rho_{ee}) +\left[\Omega_{\beta}(\rho_{ga}+\rho_{ae}) + c.c.\right] ,\nonumber\\
\dot{\rho}_{ae} =& -\left[\frac{1}{2}(\Gamma_{s}\!+\!2\Gamma_{a})-i\Omega_{12}\right]\rho_{ae} -\Omega_{\beta}^{\ast}(\rho_{aa}\!-\!\rho_{ee}) \nonumber\\ &+\Omega_{\beta}\rho_{ge} +\Omega_{\alpha}^{\ast}\rho_{as} ,\nonumber\\
\dot{\rho}_{ag} =& -\left(\frac{1}{2}\Gamma_{a}-i\Omega_{12}\right)\rho_{ag} -\frac{1}{2}\Gamma_{a}\rho_{ea}
+\Omega_{\beta}(\rho_{gg}\!-\!\rho_{aa}) \nonumber\\
&+\Omega_{\beta}^{\ast}\rho_{eg} -\Omega_{\alpha}\rho_{as} ,\nonumber\\
\dot{\rho}_{as} =& -(\Gamma -2i\Omega_{12})\rho_{as} -\Omega_{\alpha}\rho_{ae} +\Omega_{\alpha}^{\ast}\rho_{ag} \nonumber\\
&+\Omega_{\beta}\rho_{gs} +\Omega_{\beta}^{\ast}\rho_{es} ,\nonumber\\
\dot{\rho}_{ea} =& -\left[\frac{1}{2}(\Gamma_{s}\!+\!2\Gamma_{a})+i\Omega_{12}\right]\rho_{ea} -\Omega_{\beta}(\rho_{aa}\!-\!\rho_{ee}) \nonumber\\ &+\Omega_{\beta}^{\ast}\rho_{eg} +\Omega_{\alpha}\rho_{sa} ,\nonumber\\
\dot{\rho}_{ga} =& -\left(\frac{1}{2}\Gamma_{a}+i\Omega_{12}\right)\rho_{ga} -\frac{1}{2}\Gamma_{a}\rho_{ae} +\Omega_{\beta}^{\ast}(\rho_{gg}\!-\!\rho_{aa}) \nonumber\\
&+\Omega_{\beta}\rho_{ge} -\Omega_{\alpha}^{\ast}\rho_{sa} ,\nonumber\\
\dot{\rho}_{sa} =& -(\Gamma+2i\Omega_{12})\rho_{sa} -\Omega_{\alpha}^{\ast}\rho_{ea} +\Omega_{\alpha}\rho_{ga} \nonumber\\
&+\Omega_{\beta}^{\ast}\rho_{sg} +\Omega_{\beta}\rho_{se} .\label{e17}
\end{align}
The remaining equation of motion for $\rho_{gg}$ is found from the closure relation $\rho_{gg} = 1-\rho_{ss}-\rho_{ee}-\rho_{aa}$.
We see from the equations that the transitions between the symmetric states are driven at the Rabi frequency $\Omega_{\alpha}$ and are coupled to the antisymmetric states with a strength proportional to $\Omega_{\beta}$. On the other hand, transitions between the antisymmetric modes are driven at the Rabi frequency $\Omega_{\beta}$ and are also coupled by $\Omega_{\beta}$ to the transitions between the symmetric states.
Thus, in the case of $\Omega_{\beta}=0$, the dynamics of the symmetric and anti-symmetric states are independent of each other.

For numerical analysis, it is convenient to write the set of differential equations in matrix form. When $\rho_{gg}$ is eliminated from the equations, we arrive at an inhomogeneous equation
\begin{align}
\frac{d}{dt}\vec{Y}  = -M\vec{Y} + \vec{P} ,\label{v22}
\end{align}
where $\vec{Y}$ is a column vector composed of the density matrix elements
\begin{align}
\vec{Y} =& \, {\rm col}(\rho_{ss},\rho_{ee}, \rho_{sg},\rho_{se},\rho_{eg},\rho_{gs},\rho_{es},\rho_{ge}, \nonumber\\
&\rho_{aa},\rho_{ae},\rho_{ag},\rho_{as},\rho_{ea},\rho_{ga},\rho_{sa}) ,
\end{align}
$\vec{P}$ is a column vector with nonzero elements
\begin{align}
P_{3}=\Omega_{\alpha},\, P_{6}=\Omega_{\alpha}^{\ast},\, P_{11}=\Omega_{\beta},\, P_{14}=\Omega_{\beta}^{\ast} ,
\end{align}
and $M$ is the $15\times 15$ matrix of the complex coefficients. It is convenient to express the matrix $M$ in block form as
 \begin{align}
  M = \left(
    \begin{array}{cc}
      S&B\\
      D&A
    \end{array}\right) ,\label{e19}
\end{align}
in which the block $S$ is an $8\times 8$ matrix of the coefficients of the eight equations involving the density matrix elements (\ref{e16}), block $A$ is a $7\times 7$ matrix of the coefficients of the seven equations involving the density matrix elements (\ref{e17}).

The off-diagonal blocks $B$ and $D$ are $9\times 6$ and $6\times 9$ matrices whose nonzero elements are
\begin{align}
B_{21} &=B_{37} = B_{53} =B_{55}=B_{74}=\Omega_{\beta} ,\nonumber\\
B_{25}&= B_{47} =B_{64}=B_{82}=B_{86}=\Omega_{\beta}^{\ast} .
\end{align}
and
\begin{align}
D_{12} &=-\Gamma_{a} ,\nonumber\\
D_{28} &=D_{46}=D_{68}=D_{74}=-\Omega_{\beta} ,\nonumber\\
D_{31} &= D_{32} =D_{52}=\Omega_{\beta} ,\nonumber\\
D_{22} &= -D_{35} =D_{55}=-D_{61}=-D_{62}=D_{73}=-\Omega_{\beta}^{\ast} .
\end{align}
Clearly, the nonzero elements of the off-diagonal blocks are only those involving $\Omega_{\beta}$ and its complex conjugate.  Note that $\Omega_{\beta}$ is different from $0$ only when the atoms experience different amplitudes and/or phases of the driving laser field.

\section{Radiation pattern}\label{sec4}

We now proceed to give illustrative examples of the directionality of the emission by a system of two identical two-level emitters driven by an LG beam. For this purpose, we numerically solve the set of the optical Bloch equations, Eq.~(\ref{v22}) to obtain the steady-state $(t\rightarrow \infty)$ values of the density matrix elements. The solutions are then applied to determine the steady-state radiation intensity. Written in spherical coordinates the radiation pattern of a two-atom system, Eq.~(\ref{v7}), is independent of the azimuthal angle~$\phi$. Consequently, the radiation pattern is displayed graphically in a polar form for a variety of different sizes of the system, i.e. different distances between the atoms. It can be regarded as the radiation pattern viewed from the top of the two-atom antenna. The following interatomic distances are chosen for specific consideration, $r_{12}=\lambda/4, \lambda/2, 3\lambda/4$ and $\lambda$. The choice of short distances $(r_{12}\leq \lambda)$ has been dictated by the fact that the directional character of the emitted field results from the interference between the electric fields emitted by the different atoms which, on the other hand, is pronounced for small~$r_{12}$. Moreover, at distances $r_{12}\leq \lambda$ the system emits light only in a few discrete directions, which justifies regarding the systems as a highly directional antenna. Radiation patterns are illustrated for two particularly interesting configurations of the driving field at the position of the atoms, i.e. when the driving field creates a phase difference only $(\phi_{d}\neq 0, \Omega_{d}=0)$, and an amplitude difference only $(\Omega_{d}\neq 0, \phi_{d}=0)$.
   \begin{figure}[h]
   \centering{}
   \includegraphics[width=.9\columnwidth]{vdaf5}
   \caption[example]{(Color online) Polar diagram of the radiation intensity as a function of the observation direction $\theta$ for the atomic separation $r_{12}=\lambda/4$ with $\Delta_{L}=0$, $\Omega_{1}=\Omega_{2}=0.2\Gamma$ and different values of the phase difference $\phi_{d}$: $\phi_{d} =0$ (dashed-dotted black line), $\phi_{d} =\pi/4$ (solid red line), and $\phi_{d} =-\pi/4$ (dashed blue line). The positions of the atoms with the transition dipole moments polarized perpendicular to the plane of the paper $(\bar{\mu}\perp\bar{r}_{12})$ are shown by filled (brown) circles.}
\label{fig5}
\end{figure}

Considering the phase difference first, we plot in Fig.~\ref{fig5} the angular distribution of the radiation intensity for the atomic separation $r_{12}=\lambda/4$ and phase differences, $\phi_{d}=0$ and $\phi_{d}=\pm\pi/4$. The case of $\phi_{d}=0$ corresponds to the dipole moments of the atoms driven with the same phase. The angular distribution of the emitted radiation is asymmetric with an enhanced emission in the direction perpendicular to the interatomic axis. Thus, for $\phi_{d}=0$, the system tends to radiate in the direction perpendicular to the interatomic axis. In other words, the system behaves as a two-sided antenna radiating along the direction perpendicular to the interatomic axis. Note that the pattern is symmetric around both the horizontal $(0,\pi)$ and the vertical $(\pi/2, 3\pi/2)$ axis.
The pattern varies with the phase difference $\phi_{d}$ with which the atoms are driven. We see that for the phase differences $\phi_{d}=\pm \pi/4$ the system tends to radiate along the interatomic axis with a striking asymmetry about the vertical axis. Depending on whether $\phi_{d}=\pi/4$ or $\phi_{d}=-\pi/4$, the emitted radiation is spatially concentrated in the left half $(\cos\theta <0)$ or in the right half $(\cos\theta >0)$ of the pattern. Thus, the direction of the emission reverses when $\phi_{d}$ reverses sign from $\pi/4$ to $-\pi/4$. Clearly, a phase difference between the atomic dipole moments dictates the direction of the emission. For $\phi_{d}=\pm\pi/4$ the system behaves as an one-sided antenna.  Another interesting observation is that the radiation patterns for $\phi_{d}=\pm\pi/4$ resemble very much the radiation pattern of a Yagi-Uda antenna~\cite{yu}. Similar pattern is also produced by a patch antenna~\cite{pa}.

The switching of the behavior of the system from a two-sided towards a one-sided antenna can be understood by invoking the effect of the angular factors $\cos(kr_{12}\cos\theta)$ and $\sin(kr_{12}\cos\theta)$ that determine the radiation pattern, Eq.~(\ref{v7}). As discussed in Sec.~\ref{sec3a}, the factor $\cos(kr_{12}\cos\theta)$ has the property of concentrating the radiation along two opposite axial modes, whereas the factor $\sin(kr_{12}\cos\theta)$ has the property of concentrating the emission along single axial modes. According to Eqs.~(\ref{v7}) and (\ref{v9a}), the factor $\cos(kr_{12}\cos\theta)$ contributes to the radiation pattern only when the symmetric and antisymmetric states of the system are unequally populated, $\rho_{ss}-\rho_{aa}\neq 0$. The factor $\sin(kr_{12}\cos\theta)$ contributes to the radiation pattern only when there is a nonzero coherence between these states, ${\rm Im}[\rho_{as}(t)]\neq 0$. It is easy to find that for the parameter values in Fig.~\ref{fig5}, $\rho_{ss}-\rho_{aa}\neq 0$ for both $\phi_{d}=0$ and $\phi_{d}=\pm \pi/4$, whereas ${\rm Im}[\rho_{as}(t)]=0$ for $\phi_{d}=0$, but ${\rm Im}[\rho_{as}(t)]\neq 0$ for $\phi_{d}=\pm\pi/4$ such that ${\rm Im}[\rho_{as}(t)]|_{\phi_{d}=\pi/4}=-{\rm Im}[\rho_{as}(t)]|_{\phi_{d}=-\pi/4}$. Thus, the switching of the behavior of the system from a two-sided towards a one-sided antenna is achieved by the creation of nonzero coherence between the symmetric and antisymmetric states.

In Fig.~\ref{fig6} we compare the angular distribution for $\phi_{d}=0$ with that for $\phi_{d}=\pi/2$ for the same atomic separation as in Fig.~\ref{fig5}. The phase difference $\phi_{d}=\pi/2$ corresponds to the dipole moments of the atoms driven with opposite phases, $\phi_{1}=\pi, \phi_{2}=0$. As discussed in the previous section, the case of $\phi_{d}=0$ corresponds to the situation where the laser field drives only the symmetric modes of the system, while in the case of $\phi_{d}=\pi/2$ the laser effectively drives only the antisymmetric modes. We see that the phase difference $\phi_{d}=\pi/2$ has the effect of forcing the atoms to radiate along the interatomic axis. Unlike the radiation patterns for the phase differences $\phi_{d}=\pm\pi/4$, this radiation pattern exhibits  highly pronounced lobes along the interatomic axis with zeros (nodes) in the emission occurring in the perpendicular directions, $\theta =\pi/2$ and $\theta =3\pi/2$. The opening angle of the lobes is about $120^{\circ}$, which is much smaller than that for $\phi_{d}=\pm\pi/4$. Behavior of this kind can be interpreted as being a consequence of the trapping of photons along the interatomic axis: a photon emitted by atom $1$ is absorbed by atom $2$ and then when emitted by atom $2$ is re-absorbed by atom $1$.
   \begin{figure}[h]
   \centering{}
   \includegraphics[width=.9\columnwidth]{vdaf7}
   \caption[example]{(Color online) Polar diagram of the radiation intensity as a function of the observation direction $\theta$ for the atomic separation $r_{12}=\lambda/4$ with $\Delta_{L}=0$, $\Omega_{1}=\Omega_{2}=0.2\Gamma$ and different phase differences $\phi_{d}$: $\phi_{d} =0$ (dashed-dotted black line), $\phi_{d} =\pi/2$ (solid red line).}
\label{fig6}
\end{figure}

We would like to stress here that although the emission for~$\phi_{d}=\pi/2$ is more focused along the interatomic axis than that for $\phi_{d}=\pi/4$, it is symmetrically distributed around the vertical axis. For this reason, the directionality of the emission for $\phi_{d}=\pi/4$, which is strongly asymmetric around the vertical axis, can still be regarded as more pronounced than that for $\phi_{d}=\pi/2$.
   \begin{figure}[h]
   \begin{center}
   \includegraphics[width=.9\columnwidth]{vdaf11}
   \end{center}
   \caption[example]{(Color online) Polar diagram of the radiation intensity as a function of the observation direction $\theta$ for the atomic separation $r_{12}=\lambda/2$ with $\Delta_{L}=0$, $\Omega_{1}=\Omega_{2}=0.2\Gamma$ and for different phase differences $\phi_{d}$: $\phi_{d} =0$ (solid black line), $\phi_{d} =\pi/4$ (dashed-dotted red line), and $\phi_{d} =\pi/2$ (dashed  blue line).}
\label{fig7}
\end{figure}

The directionality of the emission by the two-atom system can be improved by increasing the separation between the atoms. This is illustrated in Fig.~\ref{fig7} which shows the radiation pattern for the atomic separation $r_{12}=\lambda/2$ and various values of $\phi_{d}$. An improvement of the directionality shows up clearly in the presence of zeros in the emission for several phase differences $\phi_{d}$. The radiation patterns are composed of two well distinguished lobes whose directions and symmetry depend on $\phi_{d}$. A change of $\phi_{d}$ from $0$ to $\pi/2$ rotates the direction of the lobes by $\pi/2$; from the perpendicular to the parallel to the interatomic axis. For phases $0$ and $\pi/2$ the lobes are the same size but for any phase between these two values they are different sizes. Moreover, the opening angles of the lobes and their magnitudes also vary with $\phi_{d}$.

At a larger distance between the atoms a variation of the phase difference $\phi_{d}$ not only may lead to a change in the direction of the lobes but also may reduce the number of lobes. This is illustrated in Fig.~\ref{fig8}, which shows the radiation pattern for $r_{12}=3\lambda/4$ and two values of $\phi_{d}$. It is shown that the number of lobes depends on $\phi_{d}$ and is reduced from four to three when $\phi_{d}$ is varied from $0$ to $\pi/4$. Moreover, we see that upon changing the phase  $\phi_{d}$ from $0$ to $\pi/4$, the behavior of the system changes from that of a two-sided antenna to that of a one-sided antenna.
   \begin{figure}[h]
   \centering{}
   \includegraphics[width=.9\columnwidth]{vdaf8}
   \caption[example]{(Color online) Polar diagram of the radiation intensity as a function of the observation direction $\theta$ for the atomic separation $r_{12}=3\lambda/4$ with $\Delta_{L}=0$, $\Omega_{1}=\Omega_{2}=0.2\Gamma$ and different phase differences $\phi_{d}$: $\phi_{d} =0$ (dashed black line), $\phi_{d} =\pi/4$ (solid red line).}
\label{fig8}
\end{figure}

However, the reduction in the number of lobes seen for $r_{12}=3\lambda/4$ might not be seen for other distances between the emitters. For example, at distance $r_{12}=\lambda$, the variation of the phase $\phi_{d}$ from $0$ to $\pi/2$ changes the direction of the lobes and their shape, but the number remains the same. This is illustrated in Fig.~\ref{fig9} for various values of $\phi_{d}$ from $0$ to $\pi/2$. The figure shows that for $\phi_{d}$ equal to $0$ and $\pi/2$ the radiation pattern exhibits pronounced lobes in directions symmetrically located about the horizontal and vertical axis. The pattern becomes asymmetric for values of $\phi_{d}$ different from $0$ and $\pi/2$. Consequently, for $\phi_{d}=0, \pi/2$, the system behaves as a strongly directional two-sided antenna with the emission into narrow lobes. A change in phase from those of $0$ or $\pi/2$ causes the system to behave as an one-sided antenna.
   \begin{figure}[h]
   \centering{}\includegraphics[width=.9\columnwidth]{vdaf10}\\
   \centering{}\includegraphics[width=.9\columnwidth]{vdaf12}\\
   \caption[example]{(Color online) Polar diagram of the radiation intensity as a function of the observation direction $\theta$ for the atomic separation $r_{12}=\lambda$ with $\Delta_{L}=0$, $\Omega_{1}=\Omega_{2}=0.2\Gamma$, and different phase differences $\phi_{d}$. Top: The dashed black line represents $\phi_{d} =0$, the solid (red) line $\phi_{d} =\pi/2$. Bottom: The dashed black line represents $\phi_{d} =\pi/8$, the solid (red) line $\phi_{d} =\pi/4$.}
\label{fig9}
\end{figure}

The results presented in Figs.~\ref{fig5}$-$\ref{fig9} show clearly that in all cases considered it turns out that the system behaves as a two-sided directional antenna when the phase differences $\phi_{d}$ equals an even multiple of $\pi/4$. When $\phi_{d}$ equals an odd multiple of $\pi/4$, the system behaves as an one-sided antenna. It is particularly simple to interpret this behavior by referring to the expression for the radiation intensity, Eq.~(\ref{v7}). The intensity contains two interference terms, one proportional to $\cos(kr_{12}\cos\theta)$ and the other proportional to $\sin(kr_{12}\cos\theta)$. Radiation at an angle $\theta$ from atom $2$ has a distance $r_{12}\cos\theta$ farther to travel to the observation point than that from atom $1$. Its electric field is therefore retarded by an extra amount $kr_{12}\cos\theta$. The fields of the atoms interfere constructively if the phase $\phi_{2}$ at which the atom $2$ is driven relative to phase $\phi_{1}$ at which atom $1$ is driven, $\phi_{1}-\phi_{2} = 2\phi_{d}$, can compensate for the additional retardation. Thus, constructive interference occurs at angles $\theta$ such that
\begin{align}
kr_{12}\cos\theta = 2\phi_{d} .
\end{align}
For the phase difference $\phi_{d}=0$ or $\pi/2$, the interference term $\cos(kr_{12}\cos\theta)= \pm 1$ whereas $\sin(kr_{12}\cos\theta)=0$. In this case, the variation of the radiation intensity with $\theta$ is solely determined by the cosine term. Thus,  a two-sided emission takes place simply because $\cos\theta>0$ gives the same $\cos(kr_{12}\cos\theta)$ as $\cos\theta<0$. However, for the phase difference $\phi_{d}=\pi/4$, the interference term $\cos(kr_{12}\cos\theta)=0$ whereas $\sin(kr_{12}\cos\theta)=1$. Hence, the symmetry of the emission direction is broken simply because $\cos\theta>0$ gives $\sin(kr_{12}\cos\theta)$ positive whereas $\cos\theta<0$ gives $\sin(kr_{12}\cos\theta)$ negative. The radiation intensity then has different magnitudes in the $\cos\theta>0$ and $\cos\theta<0$ parts of the pattern. In physical terms one-sided emission reflects constructive and destructive interference effects on the two sides of the pattern.
   \begin{figure}[h]
   \centering{}
   \includegraphics[width=.9\columnwidth]{vdaf6}
   \caption[example]{(Color online) Polar diagram of the radiation intensity as a function of the observation direction $\theta$ for the atomic separation $r_{12}=\lambda/4$ with $\Delta_{L}=0$ and $\phi_{1}=\phi_{2}=0$ and for different Rabi frequencies experienced by the atoms, $\Omega_{1}=0.4\Gamma, \Omega_{2}=0$ (dashed black line), $\Omega_{1}=0, \Omega_{2}=0.4\Gamma$ [solid (red) line].}
\label{fig10}
\end{figure}

The above analysis of the radiation pattern has been focused on examples where the emitters experience the same amplitudes but different phases of the driving field.
Interesting behavior can be uncovered in the radiation pattern by considering a situation where the atoms experience different amplitudes of the driving field,
$\Omega_{d}\neq 0$. We illustrate this with an example in which the driving of the system is configured such that only one of the two atoms is driven by the laser field. Referring to the collective modes of the system discussion in Sec.~\ref{sec3c}, this corresponds to the situation where the symmetric and antisymmetric modes are driven by the same Rabi frequency, $\Omega_{\alpha}=\Omega_{\beta}$.
In Fig.~\ref{fig10} we show the radiation pattern for $r_{12}=\lambda/4, \phi_{d}=0$.
The radiation pattern is composed of a single lobe showing that the radiation tends to be mainly on one side of the system. Thus, the un-driven atom steers the system to radiate towards one side of the pattern, i.e. it shows a tendency to behave as an one-sided antenna. The direction of the lobe reverses when $\Omega_{d}$ reverses its sign. Again, this one-sided emission behavior can be understood by referring to the effect of the angular factor $\sin(kr_{12}\cos\theta)$ appearing in the expression for the radiation pattern. The contribution of this factor is determined by ${\rm Im}[\rho_{as}(t)]$. It is easily shown that ${\rm Im}[\rho_{as}(t)]\neq 0$ for $\Omega_{d}\neq 0$ such that ${\rm Im}[\rho_{as}(t)]|_{\Omega_{d}>0}=-{\rm Im}[\rho_{as}(t)]|_{\Omega_{d}<0}$.

\section{Summary}\label{sec5}

We have studied the directional properties of the radiation field emitted by a system of two identical two-level emitters located at short distances and driven by a laser beam with structured phase and amplitude. We have shown that the system can operate as a nanoantenna for controlled directional emission. We have calculated the radiation intensity of the field emitted by the system and have shown that a constant phase or amplitude difference at the positions of the emitters plays an essential role in the directivity of the emission. Polar diagrams have been presented showing the radiation patterns under various conditions of excitation and for various separations between the emitters. We have demonstrated that depending on the phase or amplitude difference at the positions of the emitters, the system can operate as a two- or as a one-sided directional nanoantenna. A two-sided directional emission is achieved with a symmetric driving where the emitters experience the same amplitude and a constant phase difference of the driving field. We have found a general directional property namely that when the phase difference at the positions of the emitters equals an even multiple of $\pi/4$, the system behaves as a two-sided antenna. On the contrary, when the phase difference equals an odd multiple of $\pi/4$, the system behaves as an one-sided antenna. We have also considered the case where the emitters experience the same phase but different amplitudes of the driving field and have found that the effect of different amplitudes is to cause the system to behave as an one-sided antenna radiating in one direction centered along
the interatomic axis.\\

\section*{Acknowledgments}

This research was funded by the National Plan for Science, Technology and Innovation (MAARIFAH), King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia, Award No. 11-MAT-1898-02.

redpajama_set_name: RedPajamaArXiv


AusPhotography:: Australia's Premier Photography Forum:: > .:GENERAL:. > :AP State Based Information and Meetups > ::ACT > Canberran's are there any around?
View Full Version : Canberran's are there any around?
Wondering if there are still any Canberra located photographers around?
Would be good to start to see some of your photos. The more you interact on the site, the more people will get to know you and about you and then want to meet up.
Yes, I am here. I haven't posted for awhile though.
Hundreds of us. We're just a quiet mob.
...and I'm a bit slow!
Hmm! Interesting thread! I hate to be cheeky, but there seems to be one (rather) missing!!
Semi Long time shooter recently joined, anyone here still active?
CT. The green colour like that of your name indicates active users.
I went to Canberra once. Does that count?
Yep, count me in. I'm relatively new to the forum and found this when looking around a bit. From down the southside, do a bit of gig photography and try my hand at some landscape work. Branching out into portrait work if I can.


text:
wind (n.1)
"air in motion," Old English wind "wind," from Proto-Germanic *winda- (source also of Old Saxon, Old Frisian, Middle Dutch, Dutch wind, Old Norse vindr, Old High German wind, German Wind, Gothic winds), from PIE *wē-nt-o‑ "blowing," suffixed (participial) form of root *we- "to blow."
Normal pronunciation evolution made this word rhyme with kind and rind (Donne rhymes it with mind and Thomas Moore with behind), but it shifted to a short vowel 18c., probably from influence of windy, where the short vowel is natural. A sad loss for poets, who now must rhyme it only with sinned and a handful of weak words. Symbolic of emptiness and vanity since late 13c.
I have forgot much, Cynara! gone with the wind. [Ernest Dowson, 1896]
Meaning "breath" is attested from late Old English; especially "breath in speaking" (early 14c.), so long-winded, also "easy or regular breathing" (early 14c.), hence second wind in the figurative sense (by 1830), an image from the sport of hunting.
Winds "wind instruments of an orchestra" is from 1876. Figurative phrase which way the wind blows for "the current state of affairs" is suggested from c. 1400. To get wind of "receive information about" is by 1809, perhaps inspired by French avoir le vent de. To take the wind out of (one's) sails in the figurative sense (by 1883) is an image from sailing, where a ship without wind can make no progress. Wind-chill index is recorded from 1939. Wind energy from 1976. Wind vane from 1725.
rest (n.1)
[sleep, repose, slumber] Old English ræste, reste "rest; a bed or couch; intermission of labor; mental peace, state of quiet or repose," from Proto-Germanic *rasto- (source also of Old Saxon resta "resting place, burial-place," Dutch rust, Old High German rasta, German Rast "rest, peace, repose"), a word of uncertain origin.
The original prehistoric signification of the Germanic noun was perhaps a measure of distance; compare Old High German rasta, which in addition to "rest" meant "league of miles," Old Norse rost "league, distance after which one rests," Gothic rasta "mile, stage of a journey." If so, perhaps a word from the nomadic period. But if the original sense was "repose," the sense was extended secondarily to "distance between two resting places."
Sense of "absence or cessation of motion" is from late 15c. The meaning "that on which anything leans for support, thing upon which something rests" is attested from 1580s, with specific senses developing later. In music, "an interval of silence," also the mark or sign denoting this, 1570s.
At rest "dead" is from mid-14c., on the notion of "last rest, the big sleep, the grave." The roadside rest stop for drivers on busy highways is by 1970. The colloquial expression give (something) a rest "stop talking about it" is by 1927, American English.
[R]est and repose apply especially to the suspended activity of the body ; ease and quiet to freedom from occupation or demands for activity, especially of the body ; tranquility and peace to the freedom of the mind from harassing cares or demands. [Century Dictionary]
resurrection (n.)
c. 1300, resureccioun, "the rising again of Christ after his death and burial," also a picture or image of this and the name of a Church festival commemorating it, from Anglo-French resurrectiun, Old French resurrection "the Resurrection of Christ" (12c.) and directly from Church Latin resurrectionem (nominative resurrectio) "a rising again from the dead," noun of action from past-participle stem of Latin resurgere "rise again, appear again" (see resurgent). Replaced Old English æriste; in Middle English sometimes translated as againrising.
The general or figurative sense of "a revival, a rising again" is from late 15c. Also used in Middle English of the rising again of the dead on the Last Day (c. 1300) and the future state which follows. Chaucer uses it of the opening of a flower, "whan that yt shulde unclose Agayn the sonne."
Resurrectionist, euphemism for "grave-robber," is attested from 1776. Resurrection pie was mid-19c. English schoolboy slang for a pie made from leftovers of previous meals; first attested 1831 as a Sheffield dialect term.
There was a dreadful pie for dinner every Monday; a meat-pie with a stony crust that did not break; but split into scaly layers, with horrible lumps of gristle inside, and such strings of sinew (alternated by lumps of flabby fat) as a ghoule might use as a rosary. We called it kitten pie—resurrection pie—rag pie—dead man's pie. We cursed it by night we cursed it by day; we wouldn't stand it, we said; we would write to our friends; we would go to sea. ["How I Went to Sea," Harper's Magazine, December 1852]
concord (n.)
early 14c., "agreement between persons, union in opinions or sentiment, state of mutual friendship, amiability," from Old French concorde (12c.) "concord, harmony, agreement, treaty," from Latin concordia "agreement, union," from concors (genitive concordis) "of the same mind," literally "hearts together," from assimilated form of com "with, together" (see con-) + cor (genitive cordis) "heart," from PIE root *kerd- "heart." Related: Concordial.
Meaning "a compact or agreement" is from late 15c. The village in Massachusetts (site of one of the opening battles of the Revolutionary War, April 19, 1775) was named in 1635, perhaps in reference to the peaceful dealings between the settlers and the local native tribes. The capital of New Hampshire was renamed for the Massachusetts town in 1763 (formerly it had been Pennycook, from a mangling of a native Algonquian word meaning "descent").
By the rude bridge that arched the flood,
Their flag to April's breeze unfurled,
Here once the embattled farmers stood
And fired the shot heard round the world.
[Emerson, from "Concord Hymn"]
The Concord grape was so called by 1853, from the Massachusetts town, where it was bred for the local climate and promoted by farmer Ephraim Wales Bull. It is mentioned, but not named in the "New England Farmer" of Oct. 26, 1850, in its acknowledgements:
From E. W. Bull, Concord, a lot of fine seedling grapes, which he produced by a cross of the Catawba with a native grape. It is very good, and partakes of the nature of its parents, having some of the vinous flavor of the Catawba, and a little of the acid peculiar to our native fruit.
region in Luzerne County, Pennsylvania, from Munsee Delaware (Algonquian) chwewamink "at the big river flat," from /xw-/ "big" + /-e:wam-/ "river flat" + /-enk/ "place." Popularized by 1809 poem "Gertrude of Wyoming," set amid wars between Indians and American settlers, written by Scottish author Thomas Campbell (1777-1844), who seems to have had a vague or defective notion of Pennsylvania geography:
On Susquehanna's side, fair Wyoming!
Although the wild-flower on thy ruin'd wall,
And roofless homes, a sad remembrance bring,
Of what thy gentle people did befall;
Yet thou wert once the loveliest land of all
That see the Atlantic wave their morn restore.
Sweet land! may I thy lost delights recall,
And paint thy Gertrude in her bowers of yore,
Whose beauty was the love of Pennsylvania's shore!
et cetera. Subsequently applied 19c. to other locations (in Kansas, Ohio, and Wisconsin), and to a western territory organized July 25, 1868 (admitted as a state 1890).
On the same day there was debate in the Senate over the name for the new Territory. Territories often keep their names when they become States, so we may be glad that "Cheyenne," to be pronounced "Shy-en," was not adopted. "Lincoln" was rejected for an obvious and, no doubt, sound reason. Apparently, nobody had a better name to offer, though there must be plenty of Indian words that could properly be used, and, for the present, the insignificant "Wyoming" is retained. [The Nation, June 11, 1868]
nutmeg (n.)
"hard aromatic seed of the fruit of a tree found in the East Indies," used as a spice on cookery, c. 1300, note-mug, from Old North French or Anglo-French *noiz mugue, from Old French nois muguete, an unexplained alteration of nois muscade "nut smelling like musk," from nois "nut" (from Latin nux, from PIE *kneu- "nut;" see nucleus) + Latin muscada, fem. of muscat "musky" (see muscat). Probably influenced in English by Medieval Latin nux maga (compare unaltered Dutch muskaatnoot, German muscatnuß, Swedish muskotnöt).
American English colloquial wooden nutmeg "anything false or fraudulent" is from 1827; Connecticut is called the Nutmeg State "in allusion to the story that wooden nutmegs are there manufactured for exportation." [John Russell Bartlett, "Dictionary of Americanisms," 1859]
At a dinner party, the other day, during a little playful discussion of Yankee character, a bland and benevolent-looking old gentleman at my side informed me that he had come to the conclusion that the wooden-nutmeg story was neither more nor less than a mischievous satire. "For," said he, "there would be such an amount of minute carving required to make a successful imitation of the nutmeg, that the deception would hardly pay the workman. For myself, I do not believe the cheat was ever practised." I thanked him in the name of my country for the justice done her, and assured him that the story of the Yankee having whittled a large lot of unsaleable shoe-pegs into melon seeds, and sold them to the Canadians, was also a base fabrication of our enemies. [Grace Greenwood, "Haps and Mishaps of a Tour in Europe in 1853"]
belittle (v.)
1781, "to make small, reduce in proportion," from be- + little (v.); first recorded in writings of Thomas Jefferson (and probably coined by him), Jefferson used it in "Notes on the State of Virginia" to characterize the view promoted as scientific by French naturalist Georges-Louis Leclerc Buffon that American species (including humans) were naturally smaller than and inferior to European ones, which Jefferson was at pains to refute. ("So far the Count de Buffon has carried this new theory of the tendency of nature to belittle her productions on this side of the Atlantic.") The word was roundly execrated in England, as be- is properly to be used only with verbs:
Belittle! What an expression! It may be an elegant one in Virginia, and even perfectly intelligible; but for our part, all we can do is to guess at its meaning. For shame, Mr. Jefferson! [European Magazine and London Review, August 1787; to guess was considered another Yankee barbarism]
Jefferson also sent Buffon a stuffed moose. The figurative sense of "depreciate, scorn as worthless" (as the reviewers did to this word) is from 1797 and is now almost the only sense. Related: Belittled; belittling.
Jefferson, as if disposed to assail the sovereignty of the English tongue as well as the sovereignty of the English sword, never hesitated to coin a word when it suited his purposes so to do; and though many of his brood are questionable on the ground of analogy and as intermixing languages; yet they were expressive, and became familiar. [Hugh Blair Grigsby, "The Virginia Convention of 1776," 1855]
clock (n.1)
"machine to measure and indicate time mechanically" (since late 1940s also electronically), late 14c., clokke, originally "clock with bells," probably from Middle Dutch clocke (Dutch klok) "a clock," from Old North French cloque (Old French cloke, Modern French cloche "a bell"), from Medieval Latin clocca "bell," which probably is from Celtic (compare Old Irish clocc, Welsh cloch, Manx clagg "a bell") and spread by Irish missionaries (unless the Celtic words are from Latin). Ultimately of imitative origin.
Wherever it actually arose, it was prob. echoic, imitating the rattling made by the early handbells of sheet-iron and quadrilateral shape, rather than the ringing of the cast circular bells of later date. [OED]
Replaced Old English dægmæl, from dæg "day" + mæl "measure, mark" (see meal (n.1)). The Latin word was horologium (source of French horologe, Spanish reloj, Italian oriolo, orologio); the Greeks used a water-clock (klepsydra, literally "water thief;" see clepsydra).
The image of put (or set) the clock back "return to an earlier state or system" is from 1862. Round-the-clock (adj.) is from 1943, originally in reference to air raids. To have a face that would stop a clock "be very ugly" is from 1886. (Variations from c. 1890 include break a mirror, kill chickens.)
I remember I remember
That boarding house forlorn,
The little window where the smell
Of hash came in the morn.
I mind the broken looking-glass,
The mattress like a rock,
The servant-girl from County Clare,
Whose face would stop a clock.
[... etc.; The Insurance Journal, January 1886]
diversity (n.)
mid-14c., diversite, "variety, diverseness;" late 14c., "quality of being diverse, fact of difference between two or more things or kinds; variety; separateness; that in which two or more things differ," mostly in a neutral sense, from Old French diversete "difference, diversity, unique feature, oddness:" also "wickedness, perversity" (12c., Modern French diversité), from Latin diversitatem (nominative diversitas) "contrariety, contradiction, disagreement;" also, as a secondary sense, "difference, diversity," from diversus "turned different ways" (in Late Latin "various"), past participle of divertere (see divert).
A negative meaning, "perverseness, being contrary to what is agreeable or right; conflict, strife; perversity, evil" existed in English from late 14c. but was obsolete from 17c. Diversity as a virtue in a nation is an idea from the rise of modern democracies in the 1790s, where it kept one faction from arrogating all power (but this was not quite the modern sense, as ethnicity, gender, sexual identity, etc. were not the qualities in mind):
The dissimilarity in the ingredients which will compose the national government, and still more in the manner in which they will be brought into action in its various branches, must form a powerful obstacle to a concert of views in any partial scheme of elections. There is sufficient diversity in the state of property, in the genius, manners, and habits of the people of the different parts of the Union, to occasion a material diversity of disposition in their representatives towards the different ranks and conditions in society. ["The Federalist," No. 60, Feb. 26, 1788 (Hamilton)]
Specific focus (in a positive sense) on race, gender, etc., "inclusion and visibility of persons of previously under-represented minority identities" is by 1992.
city (n.)
c. 1200, from Old French cite "town, city" (10c., Modern French cité), from earlier citet, from Latin civitatem (nominative civitas; in Late Latin sometimes citatem) originally "citizenship, condition or rights of a citizen, membership in the community," later "community of citizens, state, commonwealth" (used, for instance of the Gaulish tribes), from civis "townsman," from PIE root *kei- (1) "to lie," also forming words for "bed, couch," and with a secondary sense of "beloved, dear."
Now "a large and important town," but originally in early Middle English a walled town, a capital or cathedral town. Distinction from town is early 14c. OED calls it "Not a native designation, but app[arently] at first a somewhat grandiose title, used instead of the OE. burh"(see borough).
Between Latin and English the sense was transferred from the inhabitants to the place. The Latin word for "city" was urbs, but a resident was civis. Civitas seems to have replaced urbs as Rome (the ultimate urbs) lost its prestige. Loss of Latin -v- is regular in French in some situations (compare alleger from alleviare; neige from nivea; jeune from juvenis. A different sound evolution from the Latin word yielded Italian citta, Catalan ciutat, Spanish ciudad, Portuguese cidade.
London is the city from 1550s. As an adjective, "pertaining to a city, urban," from c. 1300. City hall "chief municipal offices" is first recorded 1670s; to fight city hall is 1913, American English. City slicker "a smart and plausible rogue, of a kind usu. found in cities" [OED] is first recorded 1916 (see slick (adj.)). City limits is from 1825.
The newspaper city-editor, who superintends the collection and publication of local news, is from 1834, American English; hence city desk attested from 1878. Inner city first attested 1968.

redpajama_set_name: RedPajamaCommonCrawl


text:
Irina Borissova (26 de agosto de 1995), es una luchadora kazaja de lucha libre. Ganó una medalla de plata en Campeonato Asiático de 2016. Segunda en el Campeonato Mundial de Juniores del año 2015.

Referencias

Luchadores de Kazajistán
Deportistas femeninas de Kazajistán
Kazajas del siglo XXI

redpajama_set_name: RedPajamaWikipedia


Is it true what they say? Vancouver is only a city because all the cars have to turn around there? I'm not really sure about that but there's at least one difference between Montreal and Vancouver when it comes to bikes. Vancouver likes its bikes the way it likes its architecture: lots of flash. Hence all the towers of concrete, steel and glass.
You often see a stack of $7000+ bikes leaning against the glass windows at Starbucks, or a glimmering arrow of fancy-bikes swish by on its way to Stanley Park, the North Shore, etc. Yes there are all sorts of bikes in Canada's most post-modern city but they are, relative to the bikes in Montreal, much more clean. The word 'shiny' comes to mind.
Speaking of spawning, where did all these fixies come from? Fixies are bikes that don't coast because the sprocket is screwed directly onto the hub of the rear wheel. There is no freewheeling on a fixie. This means that when the wheel moves forward–or backward–the chain also moves, so that the pedals are always in motion when the bike is moving. If you suddenly stop pedaling, the bike stops with you. Think of a clown pedaling forward and then backward while balancing on a bike… now you're thinking of a fixie. But now that clown is staring back at you in a happy, yet oddly menacing way, as it dawns on you that if fixies are about anything they are about revolution. The full turn. Start turning into finish into start… forward and backward becoming one… brangelina… that kind of thing.
Fixed gear bikes are associated with the origins of the bicycle, and with racing on the track.
I think, however, that the popularization of "fixies" is associated with bike couriers. I remember that couriers in the 90s started paring their bikes down to fixies to reduce the number of moving parts. Lo-fi, functional and cool. You would see them with those tiny u-locks or massive chain locks leaning against the office towers of business districts everywhere. People started to notice.
In French, 'courrier' means to run. Bike couriers are like the original bike racers. They run messages through the city. In the age of social networking and instant messaging, however, bike couriers are decidedly old-school. That said, they often deliver messages from the future. Literature and hollywood teach us (!) that when the high-tech cybernetic world collapses, bike couriers continue to pedal through the debris of the city. In the movie "I Am Legend", for example, that Prince of Belair guy is a bike courier who wakes up from a coma only to find the city inhabited by zombies due to a virus gone wild. More famously, in 1984 William Gibson penned the seminal cyberpunk novel, Neuromancer, as the first in his "Bridge Trilogy." A central character in this trilogy from the bespeckled author that popularized the term "cyberspace" is Chevette Washington…a bike courier.
So couriers deliver messages from the old school to the future world. They're the ones who make it through the ruins. So what is their message?
In an effort to bring things full circle, I want you to know that William Gibson lives in–you guessed it–Vancouver. The airy streets of Vancouver are (g)littered with fixies. Each time you see some bony hipster with skinny jeans and $450 messenger bag he's riding a fixie with colourful high-flange rims, with cards in the spokes or plastic toys on the bars.
The hipster wants everyone to know that even though they are part of the mature, working world, they are still young at heart in an old-school kind of way. They are on the edge of young and old; immature and mature. They are in this way, "edgy." Oh, and also in the way they dress. The bike courier's working vehicle has been appropriated by the tattooed fashionista! The revolution has been replaced by the "individual" whose bike is a sign of their inner irony with a side-order of "I used to be funky when I was young".
And now there is also what the Bike Snob NYC mockingly describes as "Fixed-Gear Theme Vacations: Getting There is Half the Fun, Filming It is the Other Half".
Personally, I go back and forth on the subject of the fixie. I own one and I'm sometimes embarrassed by the way it looks, but I like riding it. It's a bit shiny but in my defence I lived in Vancouver for years and love the place. I'd be less embarrassed by my bike in Vancouver. In Montreal the fixies as a whole are not as glitzy, though there are some rather refined nods to the old way of things. Yeah or Yawn? I find people talk about fixies the way they talk about Lance Armstrong. Lots of back and forth, pros and cons. It's your call.
Anyway, since William Blackburn is going to Austin, Texas to train, I'll dedicate this closing video to him.


In preparation for the Portland Metro Reptile Expo, I ordered some t-shirts to have for sale at my table. There are 130 of them available in 18 different designs. This is the debut of them--and a shift from stickers and magnets into apparel. There's an introductory price of $12 for one or $20 for two. Show price will be $15 each in the future.
There are also whispers of opening up a shop for products here on this domain. This is still in the talking stages, but if you're a fan and you regularly check for updates, consider this a sneak peek on gossip just for you.
Opening up a shop, though is going to take a really long time and tons of work. Essentially a re-brand, too. Currently working on stock on hand, and numbering everything so that order fulfillment runs smoothly.
In other news, my MOMMA is going to be head of the shipping department. She's a lot better at that kind of thing than I am, so it'll be good. She's pushed me to put things online again, and so we're working together to make this happen. She is pretty incredible, and I honestly don't know what I would be doing right now if she wasn't right here beside me.
No other news. Hope to see you at the show in Portland!


That drip of water, the moisture spots on the ceiling, the telltale water lines on the walls; it all indicates that you have moisture making its way inside of the house somewhere, but figuring out where is not always as easy as most homeowners think.
Even though it is easy to assume that water seeming to come in from overhead must be from a bad roof, this is not always the case. In fact, the roof often gets a lot of the blame when there is a water leak, but is less often the actual guilty component. Here are a few common construction problems that could have you pointing the finger at your roof mistakenly when there is really something else going on.
Inadequate Architecture - If you live in an older home, don't be surprised if a leak that you discover has nothing to do with faulty roofing materials, but the shape of the roof itself. If the pitch of a home's roof is not just right, it can lead water into places it should not be, or even cause pooling in certain specific areas. Even though you may not need a roofer for repairs, a professional roofer can take a look at your roof and help you determine if poor architectural design is the problem.
Inefficient Attic Space - If your attic is little more than a shabby storage area that is not at all climate controlled, moisture can be a big deal. Condensation can form inside of the attic and make its way down to the living space below. This is often a result of poor insulation, which allows the temperatures outdoors to invade the space and interact with the heated and cooled air from the underlying levels of the house. You may find drips running down the walls during periods of extremely hot or cold weather.
Non-Functioning Gutters - Most homeowners take their gutters for granted. However, gutters are a big deal when it comes to water and moisture, and if they are not functioning or clogged, you can most definitely see water inside of the house. The gutters are responsible for routing rainfall and other moisture away from the house. Therefore, if the gutters are not functional, water can end up in some pretty unusual places.
You may be tempted to place the blame on the roofing materials of your home if you see water inside of the house, but the roof may not always be to blame. Talk to a roof repair professional like one from Starkweather & Sons Roofing & Siding to help you diagnose the moisture issues you have inside of your home.


If you've been dreaming about booking your next meeting in Europe, now might be the perfect time. Meetings and events on the continent and the British Isles are effectively on sale.
The strength of the U.S. dollar, coupled with Europe's weak currencies, make it the perfect time to plan European meetings. "Travel to the Eurozone is becoming more inexpensive due to exchange rates and low inflation," says Jennifer Iduh, research project manager for the European Travel Commission. "This will have an impact on prices. Business travelers and organizers will benefit because they'll get a lot of value." The Eurozone includes most but not all of the 28 countries that are members of the European Union and that have adopted the euro.
One of the most important values affecting the meetings industry is at hotels. On average, hotel prices have gone down 6 percent from last year, according to Orbitz. What's more, prices at hotels in Western and Eastern Europe—in countries such as Italy, France, Belgium, Poland and Finland—have dropped by as much as 20 percent over the past year. In Paris, one of the meeting industry's favorite destinations, hotel prices are 10 percent lower than last year.
Airline ticket prices to Europe, meanwhile, have dropped 3 percent since 2014, and the overall dip in consumer prices throughout Europe has now made sightseeing trips, local restaurants and souvenirs less expensive.
Europe has always been desirable for meetings and events. Germany, Spain, France, Italy and other European countries are consistently among the most popular international association meeting destinations, according to the International Congress and Convention Association. Its rankings of the world's cities with the most international association meetings have always been topped by European capitals; the latest rankings from 2013 include Paris, Madrid, Vienna, Barcelona and Berlin in the top consecutive spots.
Besides rich cultural and historic attractions, Europe's developed infrastructure and services, widespread Internet connectivity and generally easy transportation make it a meetings magnet. It's a relatively safe and trusted location for international travelers, especially Americans. Its current affordability adds to its appeal.
Not surprisingly, those traveling to Europe soon will see real benefits. "This summer, travelers can look forward to getting away to Europe for a great price. From London to Paris, Dublin, Rome or Barcelona, even the most in-demand destinations are offering discounts and the dollar is stretching so much further," says Jeanenne Tornatore, Orbitz senior editor.
But what about the future, since meeting and event professionals work a year or more in advance when planning a destination event?
Planners with long lead times before meetings will still get plenty of value over the next few years, but they should consider booking Europe soon. The jury is still out on whether European currencies will grow slowly over the coming years or hold steady. Goldman Sachs predicts the euro will reach parity with the U.S. dollar by 2017. This means that the value of meetings in Europe shouldn't change much for at least a couple years. Of course, there's always a risk that the exchange rate could fluctuate in favor of the euro, but that's why locking in a price now, while the exchange rate is at its best, might be a good idea.
A recent report by Global Business Travel Association (GBTA) predicts a small increase in hotel prices over the next year, but this shouldn't change much.
"Overall in the region, the report predicted slightly lower attendee costs as suppliers are working harder to secure business," says Catherine McGavock, GBTA regional director for Europe.
So even with the possibility of slightly higher hotel prices, lower costs and favorable exchange rates will keep meetings planned in Europe at a discount for the foreseeable future.
European meeting planners, unfortunately, will find it more expensive to book meetings locally, though they may seek out Eastern European countries where inflation is lower. Also, international planners looking to book in the United States will find that American destination costs have gone up in price just as European destinations are going down.


text:
/**
 * Calls closure only after callback is called x times
 */
function after(closure, times) {
    return function () {
        if (--times <= 0) closure();
    };
}

module.exports = after;

redpajama_set_name: RedPajamaGithub


Don't waste your time on learning that doesn't work. Use my hacks to get fluent in English faster.
students improved their English with me.
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I am always taking photographs. It's a way for me to exercise my eye everyday, wether I'm using one of my many film cameras or my iPhone. I love to travel and make films so those two subjects turn up in my photos frequently. If you would like one of my photos hanging on your wall, contact me to order a print!
Trying to capture a movement consisting of almost a half million people is not easy.
This trip allowed me to reconnect with the perfectly capable camera in my pocket.
When you sleep next to the ocean, things get wet.
I have to admit, the beauty of the eastern United States blew me away.
If you havent packed into a car with your closest friends and driven across the country, you really should.


The song will go to radio next month, while the group is working on its first album in nearly 20 years with producers including Timbaland. They also plan to tour next year with a couple other artists.
"It will be the biggest R&B tour in music!" Dalvin told fans in a Facebook chat.
Watch their big comeback performance below.


Hello and welcome to full day Kindergarten!
My name is Mrs. Diehl and I am so excited to be teaching your children! I am originally from Syracuse, NY and moved with my husband to the Phoenix area in the summer of 2013. In 2009 I graduated with my Bachelors degree in Early Childhood Education from SUNY Cortland. From August 2010 - May 2012 I worked on Master's Degree classes in Special Education at Marywood University and I am planning on completing my Masters Degree here in Phoenix.
When I am not teaching I enjoy reading, swimming, watching Syracuse Basketball and traveling. Moving across the country was a great adventure and I am looking forward to all of our Kindergarten adventures!
In Person: Please feel free to stop by room 268 before or after school!


text:
Уйчу́р () — игральные карты, рисованные от руки, для народной игры муушиг, являющиеся произведением монгольской народной живописи конца XIX — начала XX века.

В соответствии с игрой на картах изображались 130 зверей, начиная с птиц и кончая львами. К примеру, в колоде присутствовало 64 вида птиц, от обыкновенного воробья до павлина, попугая, грифа и божественной птицы Гаруды, 32 вида зайцев, 16 видов оленей и так далее. Каждый художник не придерживался каких-либо строгих правил, и в каждую картинку вкладывал своё собственное решение темы, поэтому каждый лист являлся авторским произведением. Также на картах нередко изображались владыки земли и небес, добрые и злые духи в образах людей разных сословий.

В настоящее время собрание игральных карт можно увидеть в Музее изобразительных искусств в Улан-Баторе.

Ссылки

 Культура Монголии конца XIX — начала XX века. Живопись конца XIX — начала XX века

Монгольские народные игры
Искусство Монголии


text:
GOP Senator Warns Why HHS Nominee Will Be Bernie Sanders's 'Wingman'
Cortney O'Brien
@obrienc2
Posted: Jan 27, 2021 10:15 AM
Source: AP Photo/Eric Risberg
President Biden has nominated Xavier Becerra to serve as the next director of the Department of Health and Human Services. Sen. Mike Braun (R-IN) is playing soothsayer and predicting that Becerra is the wrong choice because he will be nothing more than Sen. Bernie Sanders's (D-VT) wingman. Becerra has supported some of the same radical measures like Medicare for All.
"Besides the fact that he's spent a career in Congress and has very little healthcare experience, I think he's going to be the perfect wingman for the Bernie Sanders approach to healthcare," Braun predicted on Fox Business's "Varney & Co." "I come from Main Street. I took on the healthcare industry in my own business: stressing transparency, getting rid of barriers to entry, getting my employees as healthcare consumers engaged in their own well-being. My main reservation is why would you not want to reform the broken healthcare system before you get the government more involved with it?"
That, he adds, is what happened with Obamacare, which was rushed through at the beginning of President Obama's first term.
"It turned into the Unaffordable Care Act because you had big health insurance and big government together: that never sounded like a good remedy to me," explains.
"I'd really like to see Becerra embrace reform measures before you embrace the Bernie Sanders approach of getting even more government involved in a broken system," Braun suggested.
The pro-life community has its own concerns about Becerra, citing his "aggressively pro-abortion" actions as the attorney general of California.
"Becerra is aggressively pro-abortion and a foe of free speech," said SBA List President Marjorie Dannenfelser. "As attorney general of California, he continued what his predecessor Kamala Harris started by persecuting citizen journalists who exposed Planned Parenthood's role in baby parts trafficking. Not only that, he went all the way to the Supreme Court to try to force California's pro-life pregnancy centers to advertise and refer for abortion – a policy the Court rejected as unconstitutional. In Congress, he even voted in favor of partial-birth abortion."
Becerra will testify before the Senate on Wednesday for his nomination hearings. Expect Sen. Braun to press him on his above concerns.
Xavier Becerra


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Amber Alert issued for missing Markham girl
UPDATE – The girl has been located, safe and sound, and her father has been taken into custody. For more information, click HERE.
The YRPS has requested an Amber Alert after a five-year-old girl from Markham was allegedly abducted by her father on Tuesday (March 19) afternoon.
According to police, Jannah Jaffri was removed from her school by her father, Soloman Jaffri, who is not supposed to have access to the child.
The girl is described as 4′ tall, with light brown, shoulder-length hair, brown eyes, wearing a pink sweater, white tank top, grey leggings, a navy blue coat and carrying a sparkly white unicorn backpack.
Jaffri, 25, is described as 6'1″ with an average build, short brown hair, a tattoo that says "Jannah" on his right arm, wearing a navy blue dress coat and black cargo pants.
Anyone who spots the father or the girl is asked to call 911.
Township will support efforts to re-establish a farmers' market in Sunderland
DRPS officers will be 'In the Zone' for traffic safety blitz
Human remains discovered near Lake Ontario in Oshawa
Few new details released by SIU in death of one-year-old boy


text:
Home/Reviews
Album, Single and EP Reviews
A Sudden Transition by Udi Levy
Title: A Sudden Transition
Catalogue Number: No catalogue number
Review Format: Download
In these days of fixing all the mistakes later, it is refreshing to discover a musician that takes his metier seriously enough to make an album sound like he could have done it all in one take. Udi Levy clearly has the credentials to pull off such a task and his album "Sudden Transition" not only demonstrates that but also changes his style from smooth jazz to that of the wonderful world of instrumental rock.
It doesn't take long for all that flair to float to the surface with "One For My Daddy'o" highlighting Mr Levy's ability to generate fret board fury. Control is also a strongpoint of his playing with, for example, "Countryside" demonstrating both discipline and sensitivity. That said, it is the more up-tempo songs that stick in the memory and there is, fortunately, more than enough of them to encourage you to join in with your very own air guitar. "Amends", in particular, would cause any rational man to wish he was an old school prog-rocker.
Whilst there were times when, no doubt due to Mr Levy's previous excursions into smooth jazz, this album threatened to transform itself into background music, that was more the fault of the songs than Mr Levy's dexterity. The fact remains, however, that I'd want him in my band.
The album is available from CD Baby.
www.facebook.com/udilevymusic
Reviewer: Bluesbunny
Review Date: July 23 2014
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text:
import { NgModule } from '@angular/core';

import { RouterModule } from '@angular/router';
import { SplitOptionPickerComponent } from './split-option-picker.component';
import { LuSelectModule } from '@lucca-front/ng/select';
import { LuOptionModule } from '@lucca-front/ng/option';
import { LuInputModule } from '@lucca-front/ng/input';
import { FormsModule } from '@angular/forms';

@NgModule({
	declarations: [
		SplitOptionPickerComponent,
	],
	imports: [
		LuSelectModule,
		LuOptionModule,
		LuInputModule,
		FormsModule,
		RouterModule.forChild([
			{ path: '', component: SplitOptionPickerComponent },
		]),
	],
})
export class SplitOptionPickerModule {}