text:
\section{Introduction}
The finite-difference time-domain (FDTD) method is a powerful and flexible technique for the numerical integration of Maxwell's equations. It enables the rigorous study of optical, nanophotonic, and nanoplasmonic phenomena~\cite{taflove2005computational,taflove2013advances}. It uses centered finite differences applied to all electric and magnetic components of the electromagnetic field to achieve second-order integration accuracy. The algorithm originally proposed in 1966 by Yee~\cite{yee1966} solves the resulting set of coupled equations on three-dimensional numerical meshes, staggered in both space and time. Yee's approach remains today the most prominent numerical method for time-domain solutions of Maxwell's equations in complex settings. It is effectively used in commercial and research FDTD softwares~\cite{taflove2005computational,taflove2013advances} and is a key component of the particle-in-cell~\cite{Verboncoeur2005} and microscopic particle-in-cell~\cite{Varin2012,Peltz2012,Varin2014,Peltz2014} plasma simulation techniques.

As surprising as it seems, the inclusion of optical nonlinearity in the Yee-FDTD framework is not straightforward. It is easily demonstrated with the Amp\`ere's circuital law
\begin{align}\label{eq:ampere}
\nabla\times\mathbf{H} - \epsilon_0\frac{\partial\mathbf{E}}{\partial t} = \frac{\partial\mathbf{P}}{\partial t},
\end{align}
where the source term depends on a nonlinear susceptibility of the form
\begin{align}
\mathbf{P} = \epsilon_0(\chi^{(1)}\mathbf{E} + \chi^{(2)}\mathbf{E}^2 + \chi^{(3)}\mathbf{E}^3+\ldots).
\end{align}
This leads in Eq.~\eqref{eq:ampere} to a set of coupled nonlinear equations that are implicit in the electric field vector and whose solution is nontrivial~\cite{taflove2005computational,Goorjian1992}.
A formal solving approach, proposed by Greene and Taflove~\cite{taflove2013advances,Greene2006}, uses a recursive Newton method to obtain an approximate solution for $\mathbf{E}$. To ensure numerical convergence, it has to be performed over the entire numerical space at least a few times per time step. The Greene-Taflove iterative method is a rigorous way to include optical nonlinearity into the FDTD framework, but efficient implementation for solving three-dimensional problems is inherently complex. Typically, FDTD developers rely instead on explicit tricks whose implementation is simpler and computationally more efficient (see, e.g,~\cite{Maksymov2011}). But beyond the respective merits of the different implicit and explicit schemes, there is actually a need for transparent, physics-centric, nonlinear FDTD models that can provide a direct link between continuum optics and the quantum theory of optical polarizability. In this paper, we develop such a model based on the quantum two-level description of nonlinear optics.

As an alternative to the Greene-Taflove iterative technique, Gordon \emph{et al.} proposed the use of nonlinear oscillators whose integration in the Yee-FDTD framework is fully explicit and intuitive~\cite{Gordon2013}. Its connection to the Lorentz dispersion model---widely used to fit spectroscopic data---makes it a very appealing approach. However, time-domain integration of Gordon \emph{et al.}'s equations is unstable and should be used with care in time domain simulations like FDTD~\cite{VarinOE2015}. Nevertheless, we demonstrated previously that the inherent numerical instability of Gordon \emph{et al.}'s method is circumvented by introducing saturation to the nonlinearity term to mimic the dynamics of an atomic transition in the under-resonant limit~\cite{VarinOE2015}. However, the nonlinear oscillator equation presented in \cite{VarinOE2015} was introduced \emph{ad hoc}, without a formal mathematical proof. Moreover, in both \cite{Gordon2013} and \cite{VarinOE2015} the authors considered instantaneous nonlinearities only, while a rigorous treatment of nonlinear optics in dielectrics must include delayed contributions, in particular that associated with stimulated Raman scattering.

In this paper, we develop an approach for including optical nonlinearity in FDTD based on a nonlinear generalization of the Lorentz oscillator equation, commonly used to model linear optical dispersion (see, e.g.,~\cite{jackson1998electromagnetic}). It improves upon \cite{Gordon2013,VarinOE2015} by providing a formal derivation of the underlying oscillator equation, starting from the quantum mechanical equations for nonlinear optics in the two-level approximation, and by including a thorough discussion on how to deal with both the instantaneous (Kerr) and delayed (Raman) third-order nonlinearities. In particular, the proposed methodology is applied to the examples of second harmonic generation in periodic and plasmonic structures and to the propagation of intense femtosecond optical pulses in dielectrics.

This paper is divided as follows. First in Sec.~\ref{sec:nonlinear}, we present the nonlinear Lorentz model and its explicit integration in the Yee-FDTD framework. A formal derivation from the quantum mechanical two-level model equations is given in~\ref{appendix:A}. Next, in Sec.~\ref{sec:example1} we apply the nonlinear Lorentz model to the modelling of second harmonic generation (SHG) in periodically poled Lithium Niobate (PPLN). In Sec.~\ref{sec:3D}, we look at the SHG enhancement in a dielectric by a split-ring resonator nano-antenna. In Sec.~\ref{sec:example2}, we present the nonlinear Lorentz model methodology to deal with short laser pulse propagation in  centrosymmetric dielectrics and apply it to the particular case of optical solitons. In Sec.~\ref{sec:strong}, we discuss a particular form of the nonlinear Lorentz model for strong field applications and use it to model self-focusing in a Kerr medium. We finally conclude in Sec.~\ref{sec:conclusions}.

\section{The nonlinear Lorentz model}
\label{sec:nonlinear}
Optical dispersion is typically modeled with the following ordinary differential equation~\cite{jackson1998electromagnetic,fowles1975introduction,boyd2008nonlinear}
\begin{align}\label{eq:lorentz_drude}
\frac{d^2 \mathbf{P}}{d \tau^2} + \frac{\gamma}{\omega_0} \frac{d \mathbf{P}}{d \tau} +  \mathbf{P} = \epsilon_0\bar{\chi}^{(1)} \mathbf{E},
\end{align}
also known as the Lorentz oscillator equation,
where $\gamma$ and $\omega_0$ are respectively the collision and resonance frequencies, $\tau = \omega_0 t$ is the oscillator proper time, $\epsilon_0$ is the vacuum permittivity, and $\bar{\chi}^{(1)}$ is the \emph{static} linear electric susceptibility. Eq.~\eqref{eq:lorentz_drude} is known to have the following solution in the Fourier domain
\begin{align}\label{eq:lorentz_drude_ft}
\tilde{\mathbf{P}}(\omega) = \epsilon_0\left(\frac{\omega_0^2}{\omega_0^2 - \omega^2 - i\gamma\omega}\right)\bar{\chi}^{(1)} \tilde{\mathbf{E}}(\omega).
\end{align}
Going from Eq.~\eqref{eq:lorentz_drude} to Eq.~\eqref{eq:lorentz_drude_ft}, harmonic oscillation of the electric field $\mathbf{E}$ and polarization density $\mathbf{P}$ at the angular frequency $\omega$ was assumed to be as $e^{-i\omega t}$. Eq.~\eqref{eq:lorentz_drude_ft} is used extensively to fit spectroscopic data, where multiple equations are combined to obtain satisfying dispersion curves over a certain spectral range, e.g.,
\begin{align}\label{eq:lorentz_drude_sum}
n^2(\omega) = 1 + \sum_{k}\left(\frac{\omega_k^2}{\omega_k^2 - \omega^2 - i\gamma_k\omega}\right)\bar{\chi}^{(1)}_k,
\end{align}
where the subscript $k$ refers to the parameters of the $k\mathrm{th}$ oscillator. In the particular case of dielectrics where conductivity can be neglected (i.e., all $\gamma_k \simeq 0$), Eq.~\eqref{eq:lorentz_drude_sum} takes a particular form known as the Sellmeier formula~\cite{fowles1975introduction,Sellmeier}.

The use of the oscillator proper time $\tau$ in Eq.~\eqref{eq:lorentz_drude} emphasizes the direct relationship between the static polarization density (i.e., $\mathbf{P}$ in the limit where $d \mathbf{P}/d \tau \sim 0$) and the linear driving term on the right-hand side ($\epsilon_0\bar{\chi}^{(1)} \mathbf{E}$). Intuitively, nonlinear contributions to the polarization density are introduced by adding nonlinear source terms, e.g.,
\begin{align}\label{eq:nonlinear_lorentz_drude}
\frac{d^2 \mathbf{P}}{d \tau^2} + \frac{\gamma}{\omega_0} \frac{d \mathbf{P}}{d \tau} +  \mathbf{P} = \epsilon_0\left(\bar{\chi}^{(1)} \mathbf{E}+\bar{\chi}^{(2)}\mathbf{E}^2 + \bar{\chi}^{(3)} \mathbf{E}^3+\ldots\right).
\end{align}
A benefit of this formulation compared with, e.g., the anharmonic oscillator model~\cite{Gordon2013,Scalora2015,boyd2008nonlinear}, is a direct correspondence between the different numerical susceptibility parameters $\bar{\chi}^{(n)}$ and those commonly used in perturbative nonlinear optics, generally obtained by measurements or \emph{ab initio} calculations~\cite{boyd2008nonlinear}. The formal derivation of Eq.~\eqref{eq:nonlinear_lorentz_drude} from the quantum mechanical two-level atom model in the under-resonant, adiabatic-following, weak-field limit is given in~\ref{appendix:A}.

In Eq.~\eqref{eq:nonlinear_lorentz_drude}, the right-hand side is associated with linear and nonlinear scattering, as well as harmonic generation, self-modulation, and wave-mixing processes, whereas the left-hand side defines how strongly the medium gets polarized by the different driving frequencies (dispersion). We stress that $\mathbf{E}$ is the total electric field vector, i.e., $\mathbf{E} = \mathbf{E}_0 + \mathbf{E}_1e^{-i\omega t} + \mathbf{E}_2e^{-i2\omega t} + \ldots$ It thus represents not only the incident signal oscillating at the dominant angular frequency $\omega_L$ of the laser spectrum, but also contributions coming from linear scattering ($\sim e^{-i\omega_L t}$) as well as nonlinear scattering and wave mixing ($\sim e^{-i0\omega_L t},e^{-i\omega_L t} e^{-i2\omega_L t},e^{-i3\omega_L t},\ldots$).

For simplicity, in this paper the optical susceptibilities and oscillator parameters in Eq.~\eqref{eq:nonlinear_lorentz_drude} are represented by scalars. However, a general formulation of the nonlinear Lorentz model must use tensors to account for the polarization-dependent nature of light-matter interactions and allow for proper modelling of the anisotropic optical response of a particular medium (see, e.g., Sec.~\ref{sec:3D}). Similarly to the Sellmeier's approach [see, e.g., Eq.~\eqref{eq:lorentz_drude_sum}], multiple nonlinear Lorentz oscillators can be used to model optical materials, to fit both the linear and nonlinear dispersion (see~\ref{appendix:dispersion}). The use of the nonlinear Lorentz dispersion model Eq.~\eqref{eq:nonlinear_lorentz_drude} in FDTD simulations is demonstrated with examples in Secs.~\ref{sec:example1}-\ref{sec:strong}.

\subsection{Explicit numerical integration in the FDTD framework}\label{sec:integration}
We now demonstrate that integration of the nonlinear Lorentz model in the FDTD framework can be made fully explicit. Following the standard Yee procedure to express Maxwell's equations in terms of finite differences~\cite{taflove2005computational}, one gets the following discretized equations for the electric and magnetic field vectors:
\begin{subequations}
\begin{align}\label{eq:maxwell_fd_1}
\mathbf{H}^{n+1/2} &=  \mathbf{H}^{n-1/2} - \frac{\Delta t}{\mu_0}
\left(\nabla\times\mathbf{E}\right)^{n},\\
\mathbf{E}^{n+1} &=  \mathbf{E}^{n} + \frac{\Delta t}{\epsilon_0}\left[
\left(\nabla\times\mathbf{H}\right)^{n+1/2} - \mathbf{J}^{n+1/2}\right],\label{eq:maxwell_fd_2}
\end{align}
\end{subequations}
where $n$ is the time index of the discretized time $t = n\Delta t$, with $\Delta t$ being the numerical time step. We omitted the spatial indices voluntarily for simplicity (for details see, e.g.,~\cite{taflove2005computational}). In this ``free space fields'' formulation, all material properties are contained in the current density $\mathbf{J}^{n+1/2}$.

Using $d \mathbf{P}/d t = \mathbf{J}$, Eq.~\eqref{eq:nonlinear_lorentz_drude} is written in terms of two first-order ordinary differential equations whose discretization is done to match Yee's staggering scheme. This results in the following leapfrog integration equations for the material polarization and current densities:
\begin{subequations}
\begin{eqnarray}\label{eq:J_fd}
\mathbf{J}^{n + 1/2} &=& \frac{\left(1-\Gamma\right)}{
\left(1 + \Gamma\right)}\,\mathbf{J}^{n - 1/2} + \frac{\omega_0^2\Delta t}{
\left(1 + \Gamma\right)}\left(\bar{\mathbf{P}}^n - \mathbf{P}^n\right)\\
\mathbf{P}^{n + 1} &=&  \mathbf{P}^{n} + \Delta t\,\mathbf{J}^{n + 1/2}
\label{eq:P_fd}
\end{eqnarray}
\end{subequations}
where $\Gamma = \gamma\Delta t/2$ and $\bar{\mathbf{P}}^n = \epsilon_0[\bar{\chi}^{(1)}\mathbf{E}^n + \bar{\chi}^{(2)}(\mathbf{E}^n)^2 + \bar{\chi}^{(3)}(\mathbf{E}^n)^3+\ldots]$. It is seen immediately that in this form solving for $\mathbf{H}^{n+1/2}$ and $\mathbf{E}^{n+1}$, i.e., finding the electromagnetic field in the future, depends only on values in the past. Numerical integration of~Eqs.~(\ref{eq:maxwell_fd_1})-(\ref{eq:P_fd}) is thus fully explicit and does not require the Greene-Taflove iterative method~\cite{Greene2006}.

In the next sections~\ref{sec:example1},~\ref{sec:3D},~\ref{sec:example2}, and~\ref{sec:strong}, we elaborate more on coupling the nonlinear Lorentz model to the FDTD technique by solving specific problems associated with light propagation in second and third order nonlinear media.

\section{Example 1: second-harmonic generation in a periodically poled material}
\label{sec:example1}

As a first example, we consider quasi-phase-matched second harmonic generation (QPM SHG) in a periodic structure, more specifically in periodically poled lithium niobate (PPLN)~\cite{Fejer1992,Byer1997,boyd2008nonlinear}. Technically, PPLN QPM SHG crystals are engineered by depositing metal electrodes directly on the crystal surface to which a strong voltage is applied to align the molecules and break the natural crystal symmetry~\cite{Byer1997}. By switching periodically the polarity of the strong voltage, it is possible to create a periodic array of domains with alternating positive and negative second order susceptibilities ($\pm\chi^{(2)}$). The reversal period is chosen to optimize the conversion efficiency of an input laser beam into its second harmonic. The modulation of the linear index is usually assumed to have a negligible impact~\cite{Fejer1992}. Below, we show how FDTD modelling of QPM SHG can be performed with the help of the nonlinear Lorentz equation presented above.

For the demonstration, we considered a plane wave moving along the axis of an infinite PPLN crystal (see Fig.~\ref{fig:SHG}). For simplicity, we solved the corresponding 1D problem. To include both the linear and nonlinear responses of the periodically poled domains, we used polarization equations like the following:
\begin{align}\label{eq:2nd_order}
\frac{1}{\omega_k^2}\frac{d^2 P_k}{d t^2} +  P_k = \epsilon_0\left(\bar{\chi}^{(1)}_k E\pm \bar{\chi}^{(2)}_k E^2\right),
\end{align}
where the sign of $\bar{\chi}^{(2)}_k$ of the $k$th oscillator is chosen to account for the direction of the local average molecular alignment and associated average permanent dipole moment (see again Fig.~\ref{fig:SHG}). The linear refractive index is assumed to be constant throughout the medium.

\begin{figure}[tbp]
\centering
\includegraphics[width=0.75\columnwidth]{fig1}
\caption{Schematic representation of an electromagnetic plane wave impinging on an infinite PPLN crystal. Parameters $L$ and $\Lambda$ represent the distance away from the input face and the molecular orientation reversal period, respectively. The large-head black arrows indicate regions where the average permanent dipole moment is pointing up and down. With respect to the electric field $E_x$, this is associated with positive ($+\bar{\chi}^{(2)}_k$ for $\uparrow$) and negative ($-\bar{\chi}^{(2)}_k$ for $\downarrow$) second-order susceptibilities.}
\label{fig:SHG}
\end{figure}

{\renewcommand{\arraystretch}{1.5}
\begin{table}[b]
\centering
\caption{Oscillator parameters used to model QPM SHG in periodically poled MgO:LN with a second-order nonlinear Lorentz equation [see Eq.~\eqref{eq:2nd_order}]. Typical effective second order nonlinearity $d_{\mathrm{eff}}$ of commercial QPM crystals is in the $14-17\,\mathrm{pm/V}$ range, corresponding to $\bar{\chi}^{(2)} \simeq 30\,\mathrm{pm/V}$.}\label{table:MGOLN}
\vspace{0.25cm}
\begin{tabular}{|c|c|c|c|}
\hline
$k$ & $\bar{\chi}^{(1)}_k$ & $\bar{\chi}^{(2)}_k$ (pm/V) & $\omega_k$ (rad/s)\\ \hline\hline
1 & 2.4272 & 30  & $1.5494\times 10^{16}$ \\ \hline
2 & 1.4617 & 0  & $7.9514\times 10^{15}$ \\ \hline
3 & 9.6536 & 0 & $9.7766\times 10^{13}$ \\
\hline
\end{tabular}
\end{table}

\begin{figure}[ht]
\centering
\includegraphics[width=0.6\columnwidth]{fig2}
\caption{Spectral response of the three-band model used for FDTD calculations of SHG in PPLN. Here we assumed a Gaussian pulse with central wavelength $\lambda_0=1.064\,\mu\mathrm{m}$, laser intensity $10^8\,\mathrm{W/cm}^2$, and pulse duration 20~fs. The dashed curve labeled ``model" refers to the explicit integration (Sec.~\ref{sec:integration}) of the three-band model described in the main text. The full ``theoretical" curve refers to a polarization given by $P(\omega) = \epsilon_0\mathrm{TF}\{\bar{\chi}^{(1)}E(t) + \bar{\chi}^{(2)}E^2(t)\}$, where $\mathrm{TF}\{\}$ is the Fourier transform and $\bar{\chi}^{(1)}\equiv n^2(2\pi c/\lambda_0) - 1$, as from Eq.~\eqref{eq:Sellmeier_LiNbO3}. The other parameters, including $\bar{\chi}^{(2)}$, are defined in Table~\ref{table:MGOLN}.}
\label{fig:LiNbO3}
\end{figure}

Effective modelling of optical dispersion in lithium niobate (LN) is achieved by using three oscillator equations like Eq.~\eqref{eq:2nd_order}. We chose the linear part to match the Sellmeier formula for MgO doped LN (MgO:LN)~\cite{ZelmonJOSAB97}:
\begin{align}\label{eq:Sellmeier_LiNbO3}
n^2(\omega) = 1 + \sum_{k=1}^3\left(\frac{\omega_k^2}{\omega_k^2 - \omega^2}\right)\bar{\chi}^{(1)}_k.
\end{align}
Values for the different parameters are given in Table \ref{table:MGOLN}. A typical value for the nonlinear susceptibility of commercial QPM crystals was added to one of the band equations according to Eq.~\eqref{eq:2nd_order}. The effective strength of the model second-order polarization depends on the relative position of the driving frequency with respect to the resonance of the band. The choice of a particular band was defined by numerical tests (see Fig.~\ref{fig:LiNbO3}). In principle, the nonlinearity can be added to any of the bands, and even split between them, as long as the static parameters are scaled accordingly to give a correct, total second-order polarization (see also~\ref{appendix:dispersion}).

In Fig.~\ref{fig:SHG_results}, we compared the nonlinear-Lorentz/FDTD analysis to a conventional theoretical model for QPM SHG (see \ref{appendix:shg}). For FDTD modelling, we performed simulations of a 40-$\mu$m-thick medium slab in a
80-$\mu$m domain (ended by absorbing boundaries) discretized with a 4-nm resolution. A Gaussian laser pulse was initially outside the numerical domain and propagated through it for 0.8~ps with $\Delta t \simeq 13\,\mathrm{as}$. We sampled the electric field at 500 positions regularly spaced in the medium and took the absolute value of the Fourier transform of each of the 500 time traces [$E(t)$] to find the spectral amplitude of the second harmonic ($|E(2\omega_L)|$). These SH amplitudes were ultimately normalized by the amplitude of the incident electric field ($E_0 = \sqrt{2\eta_0 I}$, with $I$ being the laser intensity and $\eta_0$ the characteristic impedance of vacuum) and mapped to the corresponding position in the medium. Results are shown in Fig.~\ref{fig:SHG_results}. Using higher spatial and temporal resolution did not change the results significantly.

\begin{figure}[ht]
\centering
\includegraphics[width=0.6\columnwidth]{fig3}
\caption{Nonlinear-Lorentz/FDTD analysis of SHG in a QPM PPLN crystal. The field amplitude of the second harmonic signal as a function of the distance $L$ is compared with theory (see \ref{appendix:shg}). Laser pulse parameters were: intensity $5\times 10^{8}\,\mathrm{W/cm}^2$, duration 10 fs, and central wavelength $\lambda_0 = 1.064\,\mu\mathrm{m}$. To account for optical reflection at the input crystal face and different definitions of the field amplitude in both models, the intensity used in the theoretical model was rescaled to match the SH growth rate predicted by FDTD. The difference between the two models is barely noticeable.}
\label{fig:SHG_results}
\end{figure}

For a medium without a periodic structure (monolithic), the amplitude of the second-harmonic signal ($2\omega_L$) oscillates due to the wave vector mismatch with respect to the input signal ($\omega_L$). The oscillation period $\Lambda$ depends effectively on the mismatch parameter $\Delta k = (2\omega_L/c) [n(\omega_L) - n(2\omega_L)]$, while the amplitude of the oscillation is proportional to $\bar{\chi}^{(2)}I$~\cite{boyd2008nonlinear}. The excellent agreement between theory and FDTD in Fig.~\ref{fig:SHG_results} demonstrates that the nonlinear-Lorentz/FDTD analysis succeeds in reproducing quantitatively the dispersion, scattering, and wave mixing processes in SHG, as well as their interplay.

Quasi phase matching (QPM) consists in bypassing the wave vector mismatch by inverting the sign of $\chi^{(2)}$ periodically to make the SH signal grow continuously. With the parameters presented in Table~\ref{table:MGOLN}, the optimal sign reversal period is $\Lambda_{\mathrm{QPM,\, opt}} = 2\pi/\Delta k\simeq 5.9~\mu\mathrm{m}$. Periodic poling was implemented in both the theoretical model and FDTD by setting the sign of $\bar{\chi}^{(2)}$ with the spatial function $\mathrm{sign}[\sin(2\pi z/\Lambda_{\mathrm{QPM,\, opt}})]$, with $z=0$ corresponding to the input face of the crystal. The agreement between FDTD and theory for the QPM curves in Fig.~\ref{fig:SHG_results} demonstrates in addition the ability of the nonlinear-Lorentz/FDTD analysis to model linear and nonlinear optical processes correctly in the presence of a spatial modulation of the refractive index.

Summing up, in this section we have shown that quantitative insight into subwavelength nonlinear scattering and wave-mixing processes is possible with the nonlinear-Lorentz/FDTD approach. A three-dimensional generalization of the one-dimensional analysis we have provided is straightforward. Finite transverse beam profile and medium response anisotropy (birefringence) are easy to include as well. The flexibility and reliability of the nonlinear-Lorentz/FDTD method offer an exceptional potential for modelling SHG and other nonlinear phenomena in complex settings, e.g., for the analysis of SHG in metametarial devices~\cite{Lepetit2015a,Segal2015}. In the next section, we develop further on this topic by modelling in three dimensions the enhancement of SHG in the vicinity of a nanometric plasmonic structure.

\section{Example 2: SHG enhancement by a split-ring resonator}
\label{sec:3D}
In the previous section, we demonstrated the potential of the nonlinear-Lorentz/FDTD approach to model second-order optical processes in monolithic and periodic structures. To solve a one-dimensional formulation of the quasi-phase-matching problem, we used an effective, scalar nonlinear parameter $d_{\mathrm{eff}}$ to characterize the interaction between the incoming and scattered light with the nonlinear medium. However, for rigorous modelling of three-dimensional (3D) optics in solids, it is often necessary to consider the tensorial nature of both the linear and nonlinear susceptibilities. Below, we use the nonlinear-Lorentz dispersion model to perform 3D vectorial FDTD simulations of the SHG enhancement by a split-ring resonator (SRR) nano-antenna.

The nonlinear conversion of radiation into its second harmonic (SH) from a periodic array of SRRs~(see, e.g.,~\cite{Linden2012,Segal2015,OBrien2015}) is a representative application of plasmonic metamaterials. SRR arrays typically consist in gold nano-SRRs with thickness in the few-tens-of-nm range deposited on an optically transparent substrate (see, e.g.,~\cite{Niesler2009,Niesler2011,Linden2012}). The origin of the enhanced SHG emission has been much debated, and effectively depends on the specific alignment of the host (substrate) crystal planes with the gap of the SRR nano-antenna. Electromagnetic FDTD modelling is widely used for the design and optimization of SRR nano-antennas and to analyze experimental measurements.

FDTD modelling of SRR arrays is typically done by simulating a single SRR nano-antenna in a numerical domain surrounded by periodic boundaries in the directions parallel to the substrate surface to mimic an SRR array extending to infinity~(see, in particular,~\cite{Linden2012}). The 3D geometry of the computer model used here closely follows that found in~\cite{Niesler2009} and is shown in Fig.~\ref{fig:3D_SRR}. To define the linear dielectric constants of the nano-antenna (gold, Au) and the substrate (gallium arsenide, GaAs), we fit experimental data with oscillator equations in order to obtain reasonable refractive indexes and extinction coefficients at both the wavelength of the impinging laser pulse ($\lambda_L=1.5\,\mu\mathrm{m}$) and its second harmonic ($750\,\mathrm{nm}$). Details are given in Fig.~\ref{fig:GaAs_Gold} and below.

\begin{figure}[t]
	\centering
	\includegraphics[width=0.8\columnwidth]{fig4}
	\caption{Schematic representation of the 3D FDTD SRR simulation. To model the 2D array of gold (Au) SRRs nano-antennas deposited on a gallium arsenide substrate (GaAs) shown in (a), a single SRR unit cell in (b) is modelled with periodic boundary conditions in the $x$ and $y$ directions, and absorbing boundaries in $z$ (the direction of light propagation). The 3D simulation layout is given in (c), with the axis system showing the direction of the impinging plane wave ($k_z$) whose electric field is linearly polarized along the $x$ direction. All measures are in nanometers (nm). For the simulations we performed, the thickness of the gold SRR and the GaAs substrate is 30~nm~\cite{Linden2012}. The particular SRR shown in (b) has a resonance around $1.5~\mu\mathrm{m}$~\cite{BuschPR2007}.}
	\label{fig:3D_SRR}.
\end{figure}

For simplicity, we model here the gold optical response with a linear Drude polarization model, expressed below as a second-order, damped-oscillator equation:
\begin{equation}\label{eq:Drude}
\frac{d^2\mathbf{P}_\mathrm{Au}}{dt^2} +\gamma_p\frac{d\mathbf{P}_\mathrm{Au}}{dt} = \epsilon_0\omega_p^2\mathbf{E},
\end{equation}
where $\omega_p$ and $\gamma_p$ are the plasma and collision frequencies, respectively. Fit values for gold are given in Fig.~\ref{fig:GaAs_Gold}(b) (see also~\cite{EtchgoinJCP2007erratum}). It is readily observed that Eq.~\eqref{eq:Drude} has the same structure as Eq.~\eqref{eq:nonlinear_lorentz_drude} and that it integrates also explicitly in FDTD using the method presented in Sec.~\ref{sec:integration}.

Gallium arsenide (GaAs) is an interesting medium for the current demonstration. Its zincblende crystal structure possesses a cubic lattice that does not display linear birefringence and has a fairly simple second-order nonlinear dielectric susceptibility tensor~(see, e.g., \cite{boyd2008nonlinear}). Assuming the same crystal-SRR orientation as in~\cite{Niesler2009}, the static second-order polarization components in the transverse $x-y$ plane read:
\begin{subequations}\label{eq:P2_GaAs_static}
\begin{align}
	P_x^{(2)} & = \epsilon_0\bar{\chi}^{(2)}\left(E_zE_z - E_yE_y\right),\\
	P_y^{(2)} & = -2\epsilon_0\bar{\chi}^{(2)}E_xE_y.
\end{align}
\end{subequations}
Combining these two with the oscillator equation obtained from the fit of the experimental data for the linear dispersion [see Fig.~\ref{fig:GaAs_Gold}(a)], we can write a second-order nonlinear-Lorentz equation as follows:
\begin{align}\label{eq:GaAs}
\frac{1}{\omega_0^2}\frac{d^2\mathbf{P}_{\rm GaAs}}{dt^2} +\frac{\gamma_0}{\omega_0^2}\frac{d\mathbf{P}_{\rm GaAs}}{dt} + \mathbf{P}_{\rm GaAs} = &\epsilon_0\bar{\chi}^{(1)}\mathbf{E}\nonumber\\ &+ \epsilon_0\bar{\chi}^{(2)}\left[\left(E_zE_z - E_yE_y\right)\mathbf{i} - 2E_xE_y\,\mathbf{j}\right],
\end{align}
where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors along the $x$ and $y$ coordinate axes, respectively. Values for the linear parameters are found in the caption of Fig.~\ref{fig:GaAs_Gold}. The second-order parameter was set to the accepted value of
$\bar{\chi}^{(2)} = 7.4\times 10^{-10}\,\mathrm{m/V}$~\cite{boyd2008nonlinear}.

For the sake of simplicity, we have modeled here linear and nonlinear dispersion in one equation. In a more realistic scenario, different oscillator equations should be used to model accurately the frequency dependence of both the linear and nonlinear responses. More information is provided in \ref{appendix:dispersion}.

\begin{figure}[t]
	\centering
	\includegraphics[width=1.0\columnwidth]{fig5}
	\caption{Linear optical dispersion in (a) gallium arsenide (GaAs) and (b) gold (Au). Both the refractive index ($n = \mathrm{Re}\{\sqrt{\epsilon}\}$) and extinction coefficient ($\kappa = \mathrm{Im}\{\sqrt{\epsilon}\}$) are shown as solid lines and compared with the simplified numerical model (red dashed lines). Experimental data in (a) come from~\cite{ApsnesJAP1986,JellisonOM1992} (for photon energy above the band gap energy $E_g\simeq 1.4\,\mathrm{eV}$) and \cite{SkauliJAP2003} (below $E_g$). In (b), experimental data come from~\cite{JohnsonPRB1972}. Numerical model fit in (a) corresponds to $\epsilon_{\rm GaAs} = 1 + \bar{\chi}^{(1)}\omega_0^2/(\omega_0^2 - \omega^2 - i\gamma_0\omega)$ with $\bar{\chi}^{(1)} = 9.85$, $\omega_0 = 5.18\times 10^{15}\,\mathrm{rad/s}$, and $\gamma_0 = 6\times 10^{14}\,\mathrm{rad/s}$. Model fit in (b) corresponds to the Drude contribution given in~\cite{EtchgoinJCP2007erratum}, more specifically to $\epsilon_{\rm Au} = 1 - \omega_p^2/(\omega^2 + i\gamma_p\omega)$, with $\omega_p = 1.317\times 10^{16}\,\mathrm{rad/s}$ and $\gamma_p = 1.3\times 10^{14}\,\mathrm{rad/s}$. For integration in FDTD, both $\epsilon_{\rm GaAs}$ and $\epsilon_{\rm Au}$ are converted into time-domain equations by taking the inverse Fourier transform of the corresponding spectral-domain polarization density $\mathbf{P}(\omega) = [\epsilon(\omega) - 1]\mathbf{E}(\omega)$ (for further detail, see main text). In both cases, discretization of the resulting second-order differential equation is done as in Sec.~\ref{sec:integration}.}
	\label{fig:GaAs_Gold}.
\end{figure}

Inspection of the material models given above indicate that second harmonic should not be
generated in either the gold SRR nano-antenna or the GaAs substrate alone by a plane wave with its electric field polarized along $x$. This is confirmed in FDTD simulations with each of the materials considered separately [see Fig.~\ref{fig:SHG_SRR}(a)]. However, when both media are present, SHG emerges [see, again, Fig.~\ref{fig:SHG_SRR}(a)]. This is in agreement with the results reported in~\cite{Niesler2009}. Inspection of Eq.~\eqref{eq:GaAs} suggests that the presence of the second harmonic in $x$ arises from the $y$ and $z$ components of the electric field around the nano-antenna.

The spectral analysis of the average polarization density of gold and GaAs shows that even and odd harmonics are present in both media [see, again, Fig.~\ref{fig:SHG_SRR}(b)], although the gold polarization model is linear (and cannot produce harmonics on its own). This reveals a wave mixing effect that we interpret as follows. 1) The laser field at $\omega$ induces a linear plasmonic field around the gold SRR that drives SHG in GaAs. 2) The SH radiation from GaAs goes into the gold SRR and creates a second-order nanoplasmonic field. 3)~That second-order nanoplasmonic field acts back onto the GaAs substrate, where it can induce a linear polarization at $2\omega$ and mix with other frequencies through $\chi^{(2)}$. 4)~The radiation field from that first interaction loop can propagate into the SRR to induce various orders of nanoplasmonic fields, and the loop continues. Such complex coupling effects are hard to account for in a purely analytical treatment, but is fully embedded in FDTD simulations.

With this second example, we have shown that the nonlinear Lorentz model allows intuitive and flexible 3D FDTD modelling of nanophotonics and nanoplasmonics devices composed of several materials, including dielectrics with an anisotropic susceptibility tensor. We emphasize that current approaches to model the nonlinearity from metal nano-antennas are based on the classical Maxwell-Vlasov theory~\cite{ZengPRB2009,LiuJCP2010}, that gives a more detailed account of the metal plasma response.

\begin{figure}[t]
	\centering
	\includegraphics[width=1.0\columnwidth]{fig6}
	\caption{Spectral analysis of the 3D FDTD model described in Fig.~\ref{fig:3D_SRR}. In (a), the spectrum of the $x$ component of the radiated electric field recorded after the device indicates that SHG arises only when both media are present. With the current model, SHG from the GaAs substrate is triggered by the inhomogeneous field around the nano-antenna. The graph in (b) reveals the interplay between the two media that results in a complex
    wave-mixing process that is naturally accounted for by the nonlinear-Lorentz/FDTD model.}
	\label{fig:SHG_SRR}.
\end{figure}

\section{Example 3: short pulse propagation in centrosymmetric dielectrics}
\label{sec:example2}

In centrosymmetric dielectrics, the third order nonlinearity is responsible for third harmonic generation and the intensity-dependent modification of the refractive index (the optical Kerr effect). For laser pulses with a finite duration, an accurate description of these processes must include both the nearly-instantaneous electronic response (instantaneous Kerr) and a delayed component associated with stimulated molecular Raman scattering (Raman)~\cite{boyd2008nonlinear}. This is suitably represented by the following equation for the third-order polarization density~\cite{Agrawal}:
\begin{align}\label{eq:3rd_order_response}
\mathbf{P}^{(3)}(t) = \epsilon_0\chi^{(3)}\mathbf{E}\int_{-\infty}^t g(t - t')\left|\mathbf{E}(t') \right|^2dt',
\end{align}
where $g(t) = \alpha\delta(t) + (1-\alpha) h_{R}(t)$. Good approximate forms for the instantaneous Kerr and Raman response functions are, respectively, a Dirac delta function $\delta(t)$ and
\begin{align}\label{eq:Raman_function}
h_R(t) = \left( \frac{\tau_1^2 + \tau_2^2}{\tau_1\tau_2^2} \right) e^{-t/\tau_2}\sin(t/\tau_1),
\end{align}
with parameters $\tau_1$ and $\tau_2$ chosen to fit the Raman-gain spectrum~\cite{Agrawal}.
The balance between the instantaneous and delayed contributions is parametrized by $\alpha$. Eq.~\eqref{eq:3rd_order_response} is often written in terms of a time-dependent rotation/vibration coordinate $Q$ as
\begin{align}\label{eq:3rd_order_current_Greene}
\mathbf{P}^{(3)}(t) = \underbrace{\epsilon_0\chi^{(3)}\alpha\mathbf{E}^3}_{\text{instantaneous Kerr}} + \underbrace{\epsilon_0\chi^{(3)}(1 - \alpha) Q\mathbf{E}}_{\text{Raman}},
\end{align}
where
\begin{align}\label{eq:Q_convolution}
Q(t) = \int_{-\infty}^t h_R(t - t')\left|\mathbf{E}(t') \right|^2dt'.
\end{align}

Effectively, Eq.~\eqref{eq:Raman_function} is the retarded Green's function of a damped harmonic oscillator with resonance angular frequency $\omega_R = \sqrt{(\tau_1^2 + \tau_2^2)/\tau_1^2\tau_2^2}$ and damping constant  $\gamma_R = 1/\tau_2$. Then, an alternative equation to Eq.~\eqref{eq:Q_convolution} for the generalized rotation/vibration coordinate $Q$ is~(see also~\cite{Greene2006,Couairon2011,Kolesik2014})
\begin{align}\label{eq:Q}
\frac{d^2 Q}{dt^2} + 2\gamma_R\frac{d Q}{dt} + \omega_R^2 Q = \omega_R^2 \left|\mathbf{E}\right|^2.
\end{align}
Below, we use this differential form to avoid to compute the convolution.

For the integration of Eqs.~(\ref{eq:3rd_order_current_Greene}) and~(\ref{eq:Q}) in FDTD, Greene and Taflove~\cite{Greene2006,taflove2013advances} proposed to define a current density from two consecutive polarization densities, e.g.,
\begin{align}\label{eq:3rd_order_response_no_conv}
\mathbf{J}^{n+1/2}_{Raman} \propto \frac{Q^{n+1}\mathbf{E}^{n+1} - Q^{n}\mathbf{E}^{n}}{\Delta t},
\end{align}
which leads to implicit FDTD equations that need to be solved iteratively to obtain an approximate solution for $\mathbf{E}^{n+1}$~\cite{taflove2013advances,Greene2006}. Instead, explicit FDTD integration with delayed third-order nonlinearity can be done by replacing $\epsilon_0\bar{\chi}^{(3)}\mathbf{E}^3$ in Eq.~\eqref{eq:nonlinear_lorentz_drude} by the right-hand side of equation Eq.~\eqref{eq:3rd_order_current_Greene} to give the following nonlinear Lorentz equation:
\begin{align}\label{eq:nonlinear_3rd_order}
\frac{d^2 \mathbf{P}}{d \tau^2} + \frac{\gamma}{\omega_0} \frac{d \mathbf{P}}{d \tau} +  \mathbf{P} = \epsilon_0\bar{\chi}^{(1)} \mathbf{E} + \epsilon_0\bar{\chi}^{(3)}\left[\alpha\mathbf{E}^3 + (1 - \alpha) Q\mathbf{E}\right],
\end{align}
where the evolution of $Q$ is given by Eq.~\eqref{eq:Q}.

With $G = dQ/dt$, expressing Eqs.~(\ref{eq:Q}) and~(\ref{eq:nonlinear_3rd_order}) in terms of finite differences leads to the fully-explicit 4-step FDTD update sequence that follows:
\begin{enumerate}
\item Evaluate $\bar{\mathbf{P}}^n = \epsilon_0\bar{\chi}^{(1)} \mathbf{E}^n + \epsilon_0\bar{\chi}^{(3)}\left[\alpha (\mathbf{E}^n)^3 + (1 - \alpha) Q^n\mathbf{E}^n\right]$.
\item Update the current and polarization densities with Eqs.~(\ref{eq:J_fd})~and~(\ref{eq:P_fd}).
\item Update $Q$ with
\begin{subequations}
\begin{eqnarray}\label{eq:G_fd}
G^{n + 1/2} &=& \frac{\left(1-\Gamma_R\right)}{
\left(1 + \Gamma_R\right)}\,G^{n - 1/2} + \frac{\omega_R^2\Delta t}{
\left(1 + \Gamma_R\right)}\left[(\mathbf{E}^n)^2 - Q^n\right]\\
Q^{n + 1} &=&  Q^{n} + \Delta t\,G^{n + 1/2},
\label{eq:Q_fd}
\end{eqnarray}
\end{subequations}
where $\Gamma_R = \Delta t \gamma_R$.
\item Update the electromagnetic field with Eqs.~(\ref{eq:maxwell_fd_1}) and~(\ref{eq:maxwell_fd_2}).
\end{enumerate}

As a practical example, we considered the 1D propagation of a short optical pulse in fused silica and used a moving numerical window to track the pulse (the window moves with the pulse at the group velocity)~\cite{Arber2015}. Medium parameters are summarized in Table~\ref{table:Si02} (see table caption for other definitions used below).

\begin{table*}[htbp]
\centering
\caption{Parameters used to model optical pulse propagation in fused silica~\cite{Agrawal,MALITSON1965}. The third order susceptibility parameter $\bar{\chi}^{(3)}_k$ was obtained from the third order nonlinear index $n_2 = 2.59\times10^{-20}~\mathrm{m^2/W}$ measured at a reference wavelength of 1.5~$\mu\mathrm{m}$~\cite{Agrawal}. The corresponding nonlinear length is $L_\mathrm{NL} = (n_2I\omega_0/c)^{-1}\simeq 9.22\times 10^{12}~\mathrm{m}/I$, where $I$ is the laser intensity expressed in $\mathrm{W/m^2}$. The group velocity dispersion (GVD) parameter $\beta_2 \simeq -22.2~\mathrm{fs}^2/\mathrm{mm}$ is readily obtained from the Sellmeier equation, which allows to define the dispersion length $L_D = T^2/|\beta_2|\simeq 45~\mu\mathrm{m}\times (T[\mathrm{fs}])^2$, where $T[\mathrm{fs}]$ is the pulse duration in femtosecond.
}\label{table:Si02}
\vspace{0.25cm}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$k$ & $\bar{\chi}^{(1)}_k$ & $\omega_k$ (rad/s) & $\bar{\chi}^{(3)}_k$ (m$^2$/V$^2$) & $\alpha$ & $\omega_R$ (rad/s) &  $\gamma_R$ (1/s)\\ \hline\hline
1 & 0.69617 & $2.7537\times 10^{16}$ & $1.94\times 10^{-22}$ & 0.7 &  $8.7722\times 10^{13}$  & $3.1250\times 10^{13}$\\ \hline
2 & 0.40794 & $1.6205\times 10^{16}$ & 0  & 0   & 0  & 0\\ \hline
3 & 0.89748 & $1.9034\times 10^{14}$ & 0  & 0   & 0  & 0\\
\hline
\end{tabular}
\end{table*}

We recall that the propagation of intense and short light pulses in dielectrics is directly influenced by both the linear and nonlinear responses of the medium. We have shown in the previous examples that these effects are correctly accounted for by the nonlinear Lorentz model. In 1D and in the presence of anomalous dispersion ($\beta_2 < 0$), the interplay between group velocity dispersion (GVD) and self-phase modulation (SPM) can manifest itself in a spectacular way by the formation of temporal optical solitons~\cite{Agrawal}. In the case of a fundamental soliton ($N^2 = L_D/L_\mathrm{NL} \sim 1$), GVD and SPM are perfectly balanced and the pulse shape and spectrum remain constant during propagation. This particular optical effect was used by several authors to test nonlinear models in FDTD~\cite{Greene2006,Fujii2004,Goorjian1992b}. For a 10-fs pulse at a 1.5-$\mu\mathrm{m}$ wavelength in fused silica, the fundamental soliton condition is fulfilled when the laser intensity is on the order of $10^{11}~\mathrm{W/cm}^2$. Results shown in Fig.~\ref{fig:propagation} are in good agreement with this prediction and similar tests found in~\cite{Greene2006,Fujii2004,Goorjian1992b}. This demonstrates, again, the capability of the nonlinear-Lorentz/FDTD approach to model quantitatively linear and nonlinear optical processes and their interplay.

\begin{figure*}[ht]
\centering
\includegraphics[width=\columnwidth]{fig7}
\caption{Nonlinear-Lorentz/FDTD analysis of soliton propagation in fused silica. Color-coded are the pulse envelopes obtained by taking the absolute value of the Hilbert transform of the pulse snapshots at different propagation distances $z_0$ (see legend). In (a) where $I = 10^{8}\,\mathrm{W/cm}^2$, group velocity dispersion dominates and the initial pulse [$\propto\mathrm{sech}(t/t_0)$, with $t_0 = 10\,\mathrm{fs}$] spreads as it propagates. In (b) where $I = 3.25\times 10^{11}\,\mathrm{W/cm}^2$, the pulse propagates with nearly-constant width and amplitude over a distance greater than the soliton period $\pi L_D/2\simeq 7\,\mathrm{mm}$, suggesting that it is effectively a fundamental soliton~\cite{Agrawal}. The shift of the peak toward the left [$(z-z_0) < 0$] indicates that this soliton propagates at a slower group velocity than the pulse in (a). In (c) where $I = 10^{12}\,\mathrm{W/cm}^2$, nonlinearity dominates and the pulse self-phase modulates temporally before breaking up due to modulation instability. We stress that the origin of the $z$ axis, i.e., $z = z_0$, corresponds to a reference point moving at the linear group velocity.
}
\label{fig:propagation}
\end{figure*}

We emphasize that in previous works~\cite{Greene2006,Fujii2004,Goorjian1992b}, simulations were done with scaled parameters to ensure that optical solitons were observable within short propagation distances ($< 200~\mu\mathrm{m}$) with reasonable computational resources. In particular in~\cite{Fujii2004}, authors used $\chi^{(3)} = 7\times 10^{-2}\,\mathrm{m^2}/\mathrm{V^2}$ with a 1 V/m pulse peak electric field amplitude. With physically relevant material parameters like in the current work, this is equivalent to a laser intensity of $\sim 4.8\times 10^{13}\,\mathrm{W/cm}^2$, which is typically the light intensity inside a laser filament~\cite{Berge2007,couaironPR2007}. In this extreme regime the nonlinear material response is non-perturbative and accurate modelling of light-matter interaction must take material ionization and breakdown into account. Proper coverage of this complex topic is far beyond the scope of this paper. Nevertheless, in the next section we propose a particular form of the nonlinear Lorentz model suitable for strong-field applications, and test it in the context of self-focusing of a femtosecond laser pulse in a gas.

\section{Example 4: self-focusing in a dielectric}
\label{sec:strong}

For this last example, we consider self-focusing of an intense optical pulse in a dielectric due to the intensity-dependent modification of the refractive index associated with the third order nonlinearity. This phenomenon, where an optical pulse collapses onto itself, is typically associated with strong electric fields, with peak amplitudes beyond material breakdown thresholds. For both solids and gases, it is accepted that Kerr self-focusing is usually counter-acted by a rapid plasma build-up associated with a self-defocusing effect. In turn, laser filaments emerge when Kerr self-focusing and plasma defocusing balance each other~\cite{Berge2007,couaironPR2007}.

A rigorous study of laser filamentation is beyond the scope of this paper. However, there are still open questions related to laser filament dynamics and how to efficiently control the self-organization process~\cite{Berge2007,couaironPR2007}. Some of these questions are directly related to general optical phenomena like dispersion, third and fifth harmonic generation, the vectorial nature of light, and light propagation beyond the paraxial and slowly-varying-envelope approximations. All of which can be only fully accounted for by solving self-consistently Maxwell's equations. With the example provided below, we introduce a strong field version of the nonlinear-Lorentz/FDTD approach that can be used to explore this extreme phenomenon, provided that it would be complemented with proper models for field ionization and free-carrier dynamics.

In Table~\ref{table:strongfield}, we compiled different oscillator equations to integrate explicitly the optical Kerr nonlinearity in FDTD via Eqs.~(\ref{eq:J_fd}) and~(\ref{eq:P_fd}). In the under-resonant ($\omega_L \ll \omega_0$) weak-field ($\bar{\chi}^{(3)}|\mathbf{E}|^2\ll \bar{\chi}^{(1)}$) limit, they all trivially converge to the Lorentz dispersion model of Eq.~\eqref{eq:lorentz_drude_ft}. But we will see that their respective strong-field behaviour is significantly different. Their behaviour with respect to the driving frequency is addressed in~\ref{appendix:validity}.

{\renewcommand{\arraystretch}{1.5}
	\begin{table}[h]
		\centering
		\caption{Comparison between different nonlinear harmonic oscillator models that follow an equation like $\frac{d^2 \mathbf{P}}{d \tau^2} + \frac{\gamma}{\omega_0} \frac{d \mathbf{P}}{d \tau} +  \mathbf{P} = \bar{\mathbf{P}},$ where $\bar{\mathbf{P}}$ is a static polarization density that contains the nonlinearity and $\tau = \omega_0t$ is the oscillator proper time. The parameter $b$ below is the anharmonic constant, proportional to an effective $\chi^{(3)}$ (see \cite{Scalora2015,boyd2008nonlinear,Gordon2013} and \ref{appendix:A}). All these oscillator models can be integrated explicitly in FDTD via Eqs.~(\ref{eq:J_fd}) and~(\ref{eq:P_fd}), with the appropriate form for $\bar{\mathbf{P}}$. A numerical comparison is given in Figs.~\ref{fig:strong_field} and~\ref{fig:strong_field_I_scan} (see also~\ref{appendix:validity} for further analysis).}\label{table:strongfield}
		\vspace{0.25cm}
		\begin{tabular}{|p{2.0cm}|c|c|c|}
			\hline
			Name & $\bar{\mathbf{P}}$ & Refs.\\ \hline\hline
			Lorentz & $\epsilon_0\bar{\chi}^{(1)}\mathbf{E}$ &  \cite{fowles1975introduction}  \\ \hline
			Nonlinear Lorentz & $\epsilon_0\left(\bar{\chi}^{(1)} + \bar{\chi}^{(3)}|\mathbf{E}|^2\right)\mathbf{E}$ & Eq.~\eqref{eq:nonlinear_lorentz_drude}    \\ \hline
			Anharmonic & $\epsilon_0\bar{\chi}^{(1)}\mathbf{E} + b\mathbf{P}^3$ & \cite{Gordon2013,boyd2008nonlinear,Scalora2015}, Eq.~\eqref{eq:anharmonic_oscillator}\\ \hline
			Strong-field & see Eqs.~(\ref{eq:nonlinear_lorentz_sat}) and~(\ref{eq:nonlinear_lorentz_sat2}) & see also~\cite{VarinOE2015} \\
			\hline
		\end{tabular}
	\end{table}

\begin{figure*}[ht]
	\centering
	\includegraphics[width=\columnwidth]{fig8}
	\caption{Comparison between the nonlinear oscillator models summarized in Table~\ref{table:strongfield}. While in (a) their weak-field ($I = 10^6\,\mathrm{W/cm}^2$) solutions match perfectly, in (b) their strong-field behaviors ($I = 5\times 10^{13}\,\mathrm{W/cm}^2$) differ significantly. In (b), the strong-field Lorentz model characterized by Eqs.~(\ref{eq:nonlinear_lorentz_sat}) and~(\ref{eq:nonlinear_lorentz_sat2}) can effectively reproduce the polarization of the quantum mechanical two-level model (see~\ref{appendix:schrodinger}), whereas the nonlinear Lorentz model [see Eq.~\eqref{eq:nonlinear_lorentz_drude}] follows the instantaneous Kerr polarization $P = \epsilon_0\left(\bar{\chi}^{(1)}E + \bar{\chi}^{(3)}E^3\right)$. For strong fields, the integration of the anharmonic oscillator is unstable and fails at reproducing any of the two-level or instantaneous Kerr polarizations. For these tests, we used fictitious values for the oscillator parameters ($\gamma = 0$ and $\omega_0 = 3\times 10^{16}\,\mathrm{rad/s}$) and susceptibilities ($\bar{\chi}^{(1)} = 5\times 10^{-4}$ and $\bar{\chi}^{(3)} = 1.6\times 10^{-25}\,\mathrm{m}^2/\mathrm{V}^2$) similar to those for air at ambient temperature and pressure. The two-level model and the anaharmonic $b$ parameters were set so that all results agree in the weak-field limit shown in (a). The driving electric field was that of a 10-fs Gaussian pulse with $\lambda_L = 800\,\mathrm{nm}$.}
	\label{fig:strong_field}
\end{figure*}

\begin{figure*}[ht]
	\centering
	\includegraphics[width=\columnwidth]{fig9}
	\caption{Intensity-scaling analysis of the third-harmonic polarization amplitude obtained with the nonlinear harmonic oscillator models given in Table~\ref{table:strongfield} (see also caption of Fig.~\ref{fig:strong_field} for details). In (a), plotting $|P(3\omega)|$ as a function of $I^{3/2}$ emphasizes the three following things : 1. the perfect agreement between the nonlinear Lorentz and the instantaneous Kerr models, 2. the appearance of optical saturation in both the strong-field and quantum two-level models, and 3. the numerical instability of the anharmonic model. In (b), we plotted the relative error to the Kerr model for each of the other models. The relative error is defined as $100\times|P_{3,model} - P_{3,Kerr}|/P_{3,Kerr}$, where $P_{3,Kerr} = \epsilon_0\bar{\chi}^{(3)}E_0^3 = \epsilon_0\bar{\chi}^{(3)} (2\eta_0 )^{3/2}I^{3/2}$, and $P_{3,model}$ is given by the amplitude of the third-harmonic polarization $P(3\omega_L)$ for each of the numerical models (nonlinear, anharmonic, strong-field, and quantum).
	}
	\label{fig:strong_field_I_scan}
\end{figure*}

A rigorous treatment of strong-field optical phenomena requires quantum mechanics~(see, e.g.,~\cite{KolesikOptica2014}). Nevertheless, in a previous work~\cite{VarinOE2015} we have shown that the driving force of an oscillator can be chosen in a way to reproduce the polarization of a quantum mechanical two-level atom in the under-resonant limit. In terms of the formalism developed in the current paper, the strong-field oscillator of~\cite{VarinOE2015} reads
\begin{align}\label{eq:nonlinear_lorentz_sat}
\frac{d^2 \mathbf{P}}{d \tau^2} + \frac{\gamma}{\omega_0} \frac{d \mathbf{P}}{d \tau} +  \mathbf{P} = \epsilon_0\bar{\chi}^{(1)}\left(\frac{1}{1+\bar{\chi}^{(3)}|\mathbf{E}|^2/\bar{\chi}^{(1)}}\right)\mathbf{E}.
\end{align}
In the weak-field limit, it expands to
\begin{align}\label{eq:nonlinear_lorentz_sat_taylor}
\frac{d^2 \mathbf{P}}{d \tau^2} + \frac{\gamma}{\omega_0} \frac{d \mathbf{P}}{d \tau} +  \mathbf{P} = \epsilon_0\left(\bar{\chi}^{(1)}-\bar{\chi}^{(3)}|\mathbf{E}|^2 + \ldots\right)\mathbf{E}.
\end{align}
Comparing Eq.~\eqref{eq:nonlinear_lorentz_sat_taylor} with Eq.~\eqref{eq:nonlinear_lorentz_drude}, it appears that Eq.~\eqref{eq:nonlinear_lorentz_sat} effectively models the polarizability of a centrosymmetric material ($\bar{\chi}^{(2)}=0$) with a \emph{negative} Kerr nonlinearity. It is straightforward to deduce from the expected weak-field expansion for a \emph{positive} Kerr effect, i.e.,
\begin{align}\label{eq:nonlinear_lorentz_sat2_taylor}
\frac{d^2 \mathbf{P}}{d \tau^2} + \frac{\gamma}{\omega_0} \frac{d \mathbf{P}}{d \tau} +  \mathbf{P} = \epsilon_0\left(\bar{\chi}^{(1)}+\bar{\chi}^{(3)}|\mathbf{E}|^2 - \ldots\right)\mathbf{E},
\end{align}
that the corresponding strong-field oscillator equation should be
\begin{align}\label{eq:nonlinear_lorentz_sat2}
\frac{d^2 \mathbf{P}}{d \tau^2} + \frac{\gamma}{\omega_0} \frac{d \mathbf{P}}{d \tau} +  \mathbf{P} = \epsilon_0\bar{\chi}^{(1)}\left(2- \frac{1}{1+\bar{\chi}^{(3)}|\mathbf{E}|^2/\bar{\chi}^{(1)}}\right)\mathbf{E}.
\end{align}
The specific form of Eq.~\eqref{eq:nonlinear_lorentz_sat2} is here obtained \emph{ad hoc}, however it is in agreement with the formal solution presented in~\ref{appendix:A} where it is assumed that the population of the two-level atom is entirely in the ground state in the absence of a driving electric field [see, e.g., Eq.~\eqref{eq:nonlinear_lorentz}].

Although Eqs.~(\ref{eq:nonlinear_lorentz_sat}) and~(\ref{eq:nonlinear_lorentz_sat2}) are characterized by identical linear and nonlinear polarization spectral amplitudes $|\tilde{\mathbf{P}}(\omega)|$, they differ by the relative phase between the linear and the third harmonic signals. This difference is such that a laser beam propagating in a medium modelled with Eq.~(\ref{eq:nonlinear_lorentz_sat}) will experience self-defocusing ($-\bar{\chi}^{(3)}$), whereas in a medium modelled with Eq.~(\ref{eq:nonlinear_lorentz_sat2}) it will experience self-focusing ($\bar{\chi}^{(3)}$).

We compared the numerical behaviour of the models given in Table~\ref{table:strongfield} against the quantum mechanical two-level model given in \ref{appendix:schrodinger}. In the strong-field limit, the results in Fig.~\ref{fig:strong_field} clearly show that only the strong-field Lorentz model characterized by Eqs.~(\ref{eq:nonlinear_lorentz_sat}) and~(\ref{eq:nonlinear_lorentz_sat2}) can effectively reproduce the response associated with the quantum mechanical two-level equation, whereas the nonlinear Lorentz model follows an instantaneous Kerr polarization of the form $P = \epsilon_0\left(\bar{\chi}^{(1)}E + \bar{\chi}^{(3)}E^3\right)$.

Deeper insight was gained by plotting the amplitude of the third-harmonic polarization $|P(3\omega_L)|$ as a function of the laser intensity $I=E_0^2/2\eta_0$, where $\omega_L = 2\pi c/\lambda_L$ is the laser central laser frequency. In particular, we compared the numerically integrated models with the I-scaling behaviour expected from an instantaneous Kerr polarization, i.e.,  $P(3\omega_L) = \epsilon_0\bar{\chi}^{(3)}E_0^3 = \epsilon_0\bar{\chi}^{(3)} (2\eta_0 )^{3/2}I^{3/2}$, which is linear if plotted as a function of $I^{3/2}$. The results given in Fig.~\ref{fig:strong_field_I_scan} further emphasize the two observations of Fig.~\ref{fig:strong_field} as follows. 1. The nonlinear Lorentz model reproduces the instantaneous Kerr behaviour over the entire range of intensity considered with a relative error in the $10^{-5}$ range. 2. The strong-field model offers a reasonable approximation to the quantum two-level model. We recall that there is here an exceptional agreement between the dynamic nonlinear Lorentz and instantaneous Kerr models because simulations are performed in the under-resonant limit. The range of validity of the nonlinear Lorentz model with respect to the driving (laser) frequency is addressed in \ref{appendix:validity}.

\begin{figure*}[]
\centering
\includegraphics[width=0.6\columnwidth]{fig10}
\caption{Beam diameter and peak intensity of an 25-fs 800-nm laser pulse propagating in a fictitious gas medium modelled with a single strong-field oscillator equation like Eq.~\eqref{eq:nonlinear_lorentz_sat2}. Parameters are $\omega_0=2.9\times 10^{16}\,\mathrm{rad/s}$, $\gamma = 0$, $\bar{\chi}^{(1)}=6\times 10^{-4}$, and $\bar{\chi}^{(3)}=1.602\times 10^{-25}\,\mathrm{m}^2/\mathrm{W}$, which gives an optical response comparable to that of air at standard ambient temperature and pressure. Initial laser intensity was set to a high value for self-focusing to occur within a short distance to reduce the computation time.}
\label{fig:self_focusing}
\end{figure*}

\begin{figure*}[]
\centering
\includegraphics[width=1.0\columnwidth]{fig11}
\caption{Normalized electric field intensity ($\propto |\mathbf{E}|^2$) snapshots of self-focusing of the optical pulse of Fig.~\ref{fig:self_focusing}. Top row are FDTD results at different propagation distances. Bottom are transverse cuts of the top images along the plane of the peak intensity compared with results obtained with the paraxial equation given in \ref{apppendix:paraxial} (referred to as ``par.'' in the legend). All intensities are normalized to be directly comparable (corresponding values are found in Fig.~\ref{fig:self_focusing}). We see that at the early stage of the self-focusing process [(a),(b),(f), and (g)], the pulse remains symmetric and that the FDTD and paraxial models are in perfect agreement. But as the pulse collapses, it starts splitting longitudinally [see (d) and (e)], which stops prematurely the increase of the peak intensity, as observed in Fig.~\ref{fig:self_focusing}. In (i) and (j), it is also observed that the longitudinal splitting is coupled with important changes of the transverse profile, as compared with the paraxial model.}
\label{fig:self_focusing_frames}
\end{figure*}

From both Figs.~\ref{fig:strong_field} and \ref{fig:strong_field_I_scan} it appears that the anharmonic oscillator model is numerically unstable and it is not recommended for strong-field simulations. We stress that this numerical instability was also pointed out by independent authors (see, in particular,~\cite{Gordon2013,Scalora2015}). We emphasize that in the strong-field regime, it effectively fails at reproducing any of the two-level or instantaneous Kerr polarizations and their respective intensity-scaling behaviour. It is shown in \ref{appendix:A} that the anharmonic oscillator can also be derived from the two-level atom model equations. However, it stands on three important approximations: 1. the electric field is weak enough so that the nonlinearity can be treated as a small perturbation, 2. there is no population dynamics ($w\simeq -1$ and $dw/dt\simeq 0$), and 3. the nonlinear terms proportional to $\mathbf{P}^n$ with $n > 3$ are neglected (quadratic restoring force approximation). We identify that in the strong field limit the first and second approximations lead to inaccurate predictions of the optical nonlinearity even for moderate field strengths (see, e.g.,~\cite{VarinOE2015}), while the third one leads to numerical instability, in agreement with observations reported in~\cite{Gordon2013}. Our analysis thus suggests that the anharmonic (Duffing) oscillator [Eq.~\eqref{eq:anharmonic_oscillator}] should not be used for modeling nonlinear optical effects in moderate to intense fields with time-domain techniques like FDTD (see also~\cite{VarinOE2015}).

Ultimately, we modelled self-focusing of an optical pulse in a fictitious gas with Eq.~\eqref{eq:nonlinear_lorentz_sat2}. Two-dimensional FDTD simulations were compared against numerical results obtained with the nonlinear paraxial wave equation given in \ref{apppendix:paraxial}. In Fig.~\ref{fig:self_focusing}, it is effectively observed that at the beginning of the self-focusing process both the strong-field-Lorentz/FDTD and paraxial models agree perfectly. But as the beam collapses onto itself, significant differences arise. Effectively, it is shown that the paraxial equation predicts self-focusing of the pulse, followed almost immediately by defocusing. This is led by diffraction that then overcomes the self-focusing effect (which will eventually take over again to refocus, and so on). However, the 2D-FDTD results suggest a different scenario, where both the peak intensity and beam diameter level off and stay constant, at least for the propagation distance we covered. The smallest spot size observed in Fig.~\ref{fig:self_focusing} is approximately 16~$\mu$m ($\sim 20\lambda_L$). It is thus safe to assume that the paraxial approximation is not at the origin of the difference between the two models. Effectively, the sample snapshots of the self-focusing optical pulse given in Fig.~\ref{fig:self_focusing_frames} show that the finite duration of the pulse, although not so short (initially $\approx 25\,\mathrm{fs}$), plays a key role. The disagreement is thus better explained by longitudinal effects that are not considered in the nonlinear paraxial model.

We emphasize that it is well known that time dependence has to be included when dealing with optical pulses, in particular in the context of self-focusing and laser filamentation~\cite{Berge2007,couaironPR2007}. In that respect, the time-independent paraxial model is not ideal. We stress that we used it here for validation purposes. This is fully justified, as in Figs.~\ref{fig:self_focusing} and~\ref{fig:self_focusing_frames} there is a near-perfect agreement between the paraxial and nonlinear-Lorentz/FDTD models up to the collapse ($\sim 2.75\,\mathrm{cm}$). This demonstrates unambiguously the validity of the strong-field oscillator model of Eq.~\eqref{eq:nonlinear_lorentz_sat2} and its numerical stability in situations where the numerical integration of the anharmonic (Duffing) oscillator does not converge.

With this last example, we have shown again that the nonlinear-Lorentz method is a rigorous and transparent approach to the modelling of linear and nonlinear optical phenomena in the FDTD framework. Moreover, it provides both qualitative and quantitative insight in situations where simpler, approximate models miss important physical contributions. Complexity could have certainly been added to the paraxial equation, e.g., finite pulse duration and spectral shape, dispersion, vectorial treatment, non-paraxiality, ... but all these contribution are intrinsic to the nonlinear-Lorentz/FDTD model, in particular here in its strong-field formulation.

\section{Conclusions}
\label{sec:conclusions}

In this paper, we have presented an extension of the Lorentz dispersion model to include nonlinear optics in the FDTD framework. Compared with the current iterative techniques, its numerical integration is simple, intuitive, fully explicit, and flexible. A complete methodology was elaborated, supported by full-scale FDTD tests that were compared with known and accepted theoretical models and calculations. We have shown that three-dimensional FDTD implementation is straightforward due to the inherent vectorial nature of the model, with the possibility to include damping, restoring, and light-matter coupling in a tensorial form to allow effective modeling of anisotropic nonlinear response. Finally, we proposed and analyzed a formulation of the nonlinear Lorentz model suitable for strong field applications. The current work suggests that it should be used in replacement of the anharmonic (Duffing) oscillator that tends to be unstable and inaccurate for moderate to intense fields. Ultimately, the use of the strong-field nonlinear-Lorentz model in FDTD-based plasma simulation approaches like PIC and MicPIC promises insight into the light-matter interaction processes involved in situations where both the dielectric polarization and plasma dynamics are important, e.g., during laser filamentation and femtosecond micromachining of dielectrics.

\vspace{\baselineskip}
\textbf{Acknowledgments.}
The authors thank the EPOCH development and support team for their precious help.
C.V. thanks Antonino Cal\`a Lesina for stimulating discussions. This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Canada Research Chair in Ultrafast Photonics. EPOCH development was funded by the UK EPSRC grants EP/G054950/1, EP/G056803/1, EP/G055165/1 and EP/ M022463/1. Computations were made on the supercomputer Briar\'ee from Universit\'e de Montr\'eal, managed by Calcul Qu\'ebec and Compute Canada. The operation of this supercomputer is funded by the Canada Foundation for Innovation (CFI), the Minist\`ere de l'\'Economie, de la Science et de l'Innovation du Qu\'ebec (MESI) and the Fonds de recherche du Qu\'ebec - Nature et technologies (FRQ-NT). T.F. acknowledges financial support by the German Science Foundation (DFG) via SFB652/3 and SPP1840.

redpajama_set_name: RedPajamaArXiv


Now this album is a bloody treasure. Ed is a cult folk icon, a gay hippy outsider from back in the day whose descent into obscurity never dimmed his penchant for penning raw, heartfelt & lilting chunks of dream-like melancholia. He really does rip at your heartstrings with his "Leonard Cohen meets Daniel Johnston at Jonathan Richman's house" vocal style and beautifully naive harpsichord playing! The tunes have that lullaby quality in parts. He also sounds like he's playing to two disinterested couples in an almost empty dive at the end of the universe. His songs are warm, humorous, love-lorn, poetic & charming as hell. Also, due to the fact these are dusty home-recorded songs from the 80s, committed casually to 2 track tape, their lo-fi hiss adds another fascinating dimension to proceedings. You feel like you've stumbled on a magical relic even though on 'House of Embers' he kind of resembles an eccentric uncle attempting to re-create a Dylan-esque musical dimension on rustic acoustic & harmonica . I'm just as moved & exhilarated by this album as I am the similarly mystical UFO by Jim Sullivan - an altogether different collection of immaculate outsider folk wonderment.


text:
Engineering and manufacturing campus supports fight against Coronavirus
A state-of-the-art engineering and manufacturing campus close to the new Nightingale Hospital in Stratford says it has the perfect facilities for organisations looking to tackle the coronavirus outbreak.
The Centre for Engineering and Mechanical Excellence (CEME) in Rainham has a 12,500 sq ft business continuity hub available, along with 15 engineering workshops, laboratories and offices ranging from 200 sq ft- 2,000 sq ft. It also has the only fully equipped shared engineering space in London.
The centre's CEO Bill Williams said: "We have been extremely impressed by the way in which organisations have mobilised to fight Covid-19, from academics searching for a vaccine to car makers and engineers assembling ventilators. Gin distilleries have been making hand sanitiser while fashion and beauty brands have switched production to make masks and hospital gowns.
"We want people to be aware that we have the facilities and the desire to support them and the nation at this critical time. We also have a 600 space car park which could be utilised as a Covid-19 test centre, for example.
"There are very few people currently on site and those that are still working are providing critical support for the coronavirus outbreak and are following all the government guidelines."
The campus is the base for the NHS North East London Foundation Trust (NELFT), one of the key organisations involved in managing the outbreak locally.
CEME, an industrial charity, opened its doors in 2003 to manufacturing, engineering and technology companies. It has 200,000 sq ft of space in three large, modern buildings on a 19-acre secure site and also has a conference centre, event space and technical classrooms.
It is on the main A13 trunk road into London, 10 minutes from Canary Wharf and 10 minutes to the Dartford crossing, with a public bus terminus that links directly with the London Underground and Overground train services.
For further information, contact CEME's Commercial Manager Seb Coombs on 020 8596 5192, email seb.coombs@ceme.co.uk.

redpajama_set_name: RedPajamaCommonCrawl


text:
package org.xqqunit.system;

import static org.junit.Assert.assertEquals;

import org.junit.Test;

public final class ParameterTest {

	private static final String INPUT_FILE_PARAMETER_STRING = "XQQUNIT_INPUT_FILE";

	@Test
	public void getCorrectParameterByStringName()  {
		Parameter p = Parameter.byName(INPUT_FILE_PARAMETER_STRING);
		assertEquals("Parameter.INPUT_FILE mismatch", Parameter.INPUT_FILE, p);
		assertEquals("Parameter.INPUT_FILE.getName() mismatch", Parameter.INPUT_FILE.getName(), p.getName());
		assertEquals("Parameter and search string mismatch", INPUT_FILE_PARAMETER_STRING, p.getName());
	}
}

redpajama_set_name: RedPajamaGithub


Nicki selected close to 30 students to help out.
Chance The Rapper the new Chance Arts And Literature Fund for Chicago public schools in need.
Seahawks Michael Bennett To Donate Endorsement Money: "I Was Inspired By Chance"
Chance inspires Michael Bennett to donate.
"You are an example of the power of arts education," Michelle told Chance on twitter.
Chance live-streamed his press conference at Westcott Elementary School.
Bandcamp puts its money where its mouth is.
A fundraiser started in support of the victim of the Facebook Live video torture in Chicago has raised almost $170,000 in a week.
Kaepernick has made a real impact over the past year.
KD continues to give back to the OKC community.
The Weeknd is a charitable man.
The Weeknd making philanthropy moves.
50 Cent reportedly donated $100,000 to Autism Speaks after publicly making fun of a teen that has Autism.
Did Mark Zuckerberg Give Kanye West $1 Billion?
Did "The Life of Pablo" inspire Mark Zuckerberg to donate a huge sum of money toward Kanye's future "ideas"?
Nicki Minaj has supported Amber Rose's feminist march.
Chief Keef has put a hold on his hologram benefit concert.
Kanye West has made the biggest donation in the short history of Donda's House.
Migos and Quality Control donate $1000 to an Atlanta high school, and face criticism for it on Twitter.
Jay Z has pledged to make a financial contribution to Marina Abramovic's new art institute.


text:
Cooper announces aid for community-college students
RALEIGH (May 26, 2021) – A new directive from Gov. Roy Cooper will make $31.5 million available to help graduating high-school seniors attend North Carolina community colleges, as well as $12.5 million to help students who stopped out of college return to finish their degrees.
In all, Cooper announced Monday that he will direct $51 million in federal funds to help North Carolina students whose educations were disrupted by the coronavirus pandemic access and complete higher education.
The funds from North Carolina's share of the Governor's Emergency Education Relief Fund (GEER) will include:
$31.5 million for the Longleaf Commitment, to provide grants of at least $2,800 a year to cover tuition and fees for recent high-school graduates from low- to moderate-income households to attend community college for the next two years. Students must be North Carolina residents, graduate from a North Carolina high school this year and be a first-time college student. The grants will be offered only from the Fall 2021 to the Spring 2023 semester.
$12.5 million for Longleaf Complete, an effort to help students who have attended a community college, a UNC System school or independent college or university but left without a degree. The governor's office pointed out that more than 1 million North Carolinians are considered such "part-way home" students.
$5 million to rapidly expand mental-health services at UNC System campuses. The system says that 8 in 10 students said the pandemic negatively affected their mental health.
The governor's office noted that – unlike in other recessions – enrollment fell at North Carolina community colleges during the pandemic (as it did nationwide).
Enrollment in curriculum programs was down 20% in Fall 2020,1 and the pandemic posed a threat to the state's ability to reach a goal of 2 million North Carolinians ages 25-44 with a high-quality credential or degree by 2030.2
"The GEER funds announced today will provide much-needed relief for the state's community colleges and universities, help us continue to build and grow a successful and diverse workforce, and provide students equitable access to postsecondary education," Cooper said.
The governor said the directive is a first step toward a more robust NC Guarantee grant program that still must be negotiated with the General Assembly.
Under that proposal, Cooper wants to use American Rescue Plan funds to provide students from households with incomes up to $60,000 at least $6,000 in state and federal grants toward attendance at any UNC System institution or NC community college.
"While we work with legislators to fund the NC Guarantee, today's graduates need help immediately," Cooper said.
NC Community College System President Thomas Stith says he expects community colleges to lead the state's recovery from the pandemic.
"Education translates into opportunity, and I thank Gov. Cooper for his decision to use federal funds to extend higher-education opportunities for students to attend community colleges," Stith said.3
At a webinar Wednesday hosted by myFutureNC, officials from the NC Community College System and the governor's office emphasized that as long as an eligible student applies to a North Carolina community college and completes the Free Application for Federal Student Aid (FAFSA), the State Education Assistance Authority will automatically award funds to the student.
"There is not a grant application," said Dr. James "JW" Kelley, Associate Vice President for Student Services at NCCCS.
Geoff Coltrane, the Senior Education Advisor to Cooper, said undocumented students or students who are subject to Deferred Action for Childhood Arrivals (DACA) are not eligible for the grants, but they are eligible for aid through pandemic relief funds previously approved by Congress.
Kelley said colleges can work with students who have immigration issues to make a variety of other scholarships available as well.
More information on Longleaf Commitment Grants is available here.
1 NC Pandemic Recovery Office.
2 https://files.nc.gov/governor/documents/files/210524-GEER-II-Fact-Sheet_final-1.pdf. 3https://governor.nc.gov/news/governor-cooper-announces-514-million-education-grants-new-financial-aid-program.
Filed Under: 2021, myFutureNC, NC Community Colleges


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text:
The town of Gowran, co. Kilkenny, is situated in the march of that county far from any aid from the English and is surrounded by Irish enemies, who have lately burned it, and destoyed the lieges in the same, and daily plot to do so again. GRANT, in aid of the paving and enclosing of that town with a stone wall, and of other necessary works, to the burgesses and commons that, for 40 years, they and their successors may take from all articles for sale coming to that town the same customs as are taken in the town of Kilkenny by royal grants; such that the money arising from those customs is expended upon the enclosing and paving of the said town of Gowran, and on other necessary works, and not otherwise, and that an account of the same is rendered before the earl of Ormond annually, and not at the Ex.
RCH; BL, Egerton MS 78, p. 37; NLI, GO MS 193, p. 81.


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Every day, Google changes in its ranking algorithm. Some are very small tweaks; others seriously change the SERPs. This post of SEO Services Kolkata will focus on such areas that, in my experience, have a positive impact and a big return on investment.
The Google index is now going mobile first and you must have a mobile-friendly page, else you will likely take a hit in the SERPs.
That's especially true for websites using unwanted ads. If you aren't certain as to whether your website meets the criteria for being mobile friendly, log in to your Search Console account and view the Mobile Usability Report. Google will report all mobile issues there, so you can take the appropriate action and fix it.
When people use keywords or phrase to find anything and research actual search terms – that people enter into search engines.
1: Make a list of important, related topics based on what you know about your business.
2: Fill in that topic with keywords.
3: Research related search keyword terms.
Our SEO Company Kolkata will help to get best keyword research for our business.
Before deciding the meta description ask yourself if the title exactly describes the content of the website or is the describe strong enough to attract clicks?
If you have already created one, then consider updating it by utilizing best tools and proper practices.
Google always suggest the webmasters focus on their primary things on providing a better user experience.
As the algorithm gets "smarter", websites that do so are positioned to benefit the most. A good user experience goes much deeper than a good user experience writing clean code.
On your landing page, use descriptive anchor texts and add it to related pages with the same topics on the previous pages. Do not create too many links from a page because the fewer pages you link to have a few authority as you.
Searchmetrics reported that over a third of Google search results incorporated Rich Snippets that supported by Schema.
Schema.org is a collection of different HTML tags that can be added to a Website page. These tags create an enhanced description that appears in search results which commonly known as rich snippets.
We use Schema to create Rich Snippets for Organizations, Services, Events, Music, People, Products, Recipes, Review Ratings, and Videos.
Always respond to all your customer inquiries and take care of their issue. Provide important, accurate and rich content that is valuable and provides your visitors with what they exactly search for.
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text:
Homeepub
Men Don't Cry Ebook
Mukoma Wa Ngugi
Men Don't Cry invites us into the home of Mourad Chennoun in Nice, where his father spends his days fixing things in the backyard, his mother bemoans the loss of her natal village in Algeria, and the name Dounia is taboo.
When Mourad's father has a stroke, he is forced to rise above his fear of becoming an overweight bachelor, tied down to home by his mother's cooking, and take steps to bridge the gulf between his family and estranged sister.
This quest takes him to the Paris suburbs where he starts his teaching career, and falls into the world of undocumented Algerian toyboys and discovers that Douania has become a staunch feminist, aspiring politician and fierce assimilationist.
Can Mourad adapt to his new, fast-paced Parisian life and uphold his family's values?
A poignant coming-of-age story from the widely-acclaimed author of Kiffe Kiffe Tomorrow.
Unbury Our Dead With Song Ebook
Unbury our Dead With Song is a novel about four talented Ethiopian musicians – The Diva, The Corporal, the Taliban Man and Miriam, who are competing to see who can sing the best Tizita (popularly referred to as Ethiopian blues). Taking place in an illegal boxing hall in Nairobi, Kenya, the competition is covered by a US educated Kenyan journalist, John Thandi Manfredi, who writes for a popular tabloid, The National Inquisitor. He follows the musicians back to Ethiopia in order to learn more about the Tizita and their lives. As he learns more about the Tizita and the multiple meanings of beauty, he uncovers that behind each of the musicians, there are layered lives and secrets. Ultimately, the novel is a love letter to African music, beauty and imagination.
This Crazy Paradise Ebook
It takes a divorce for psychologist Dr. Onele Dlamini to realise she wants to write steamy love stories. She's given herself one month on a Greek-style resort on the West Coast of South Africa in which to prove she can do it. Daniel Lindo is the red-hot workman who arrives to fix her roof and ask for her help with his insomnia – over another woman. Onele really, really shouldn't play therapist here, but she's curious… there is more to this man's past than meets the eye.
Is Daniel the cure for Onele's writer's block? Or is she just going Crazy in Paradise?
The Seeing Place Ebook
Hot-shot producer Thuli Poni is holding auditions for her latest play in Cape Town. Andile Sebe is an up-and-coming star, just waiting for his big break. Thuli casts Andile and challenges him to link his painful past with the role he is portraying, leading him to open up to her. The two fall for each other and a passionate romance ensues.
But when auditions open for Sins of the Fathers, the most-watched TV show in South Africa, Thuli turns cold. Will she play a part in Andile's rise to fame, or will she hinder it?
This is a story about how love can triumph against the odds if we stay humble, take risks and are willing to learn. The Seeing Place offers a very different kind of romance – between a powerful woman and a man who wants something only she can offer.
Love Next Door Ebook
Amina Thula
When business analyst Abby finally moves into a place of her own, she is delighted to discover that her new apartment block is also home to a hunk-next-door. Kopano – teacher, swim coach, artist and all-round nice guy – seems too good to be true. Until Abby discovers evidence of a mystery girlfriend.
As neighbours Abby and Kopano spend more and more time together as 'just good friends', the attraction deepens and their close connection develops into something more intense.
But just as her love life finally seems on track, a fabulous career opportunity opens up for Abby in New York – and she is torn between making her career aspirations come true and leaving behind the man of her dreams…
Love Me Unconditionally Ebook
Ola Awonubi
Deola Banjoko has it all. A wonderful career, an accomplished fiancé and great plans for the future. But as the years pass and Deola fails to become pregnant, the foundations of the wonderful life they have built together begin to crumble, unable to withstand the pressures of life, family and cultural expectations.
Recovering from a broken heart, Deola takes up a chance to work in Nigeria. She crosses paths with businessman Femi Da Silva and ends up working as a consultant for his PR company. She soon learns that Femi has issues of his own, having been widowed ten years before, and tries to fight her attraction to him. Can Deola move beyond the pain of the past and the constrictions of societal pressures and find the unconditional love she is seeking?
A Man Who Is Not A Man
Thando Mgqolozana
A Man Who Is Not A Man recounts the personal trauma of a young Xhosa initiate after a rite-of-passage circumcision goes wrong. With frankness and courage, this powerful novel details the pain and lifelong shame this protagonist experiences as a result not only of the physical trauma, but the social ostracism from being labeled 'a failed man.' He decodes the mysteries of this long-standing cultural tradition and calls to account the elders for the disintegrating support systems that allow such tragic outcomes. But it is also through this life-changing experience that the protagonist is forced to find his strength and humanity and reassess what it really means to be a man.
Amina Thula (1)
Elnathan John (1)
Mukoma Wa Ngugi (2)
Ola Awonubi (1)
Thando Mgqolozana (1)


text:
Coronavirus update: For the latest up-to-date information please see our information page.
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News From Your Theatre
The Overtones to open 10th Anniversary Tour at the Victoria Theatre
The multi-platinum selling group tour the UK to celebrate 10 years as one of the country's most prolific live acts
A Night of Songs and Laughter with Rob Brydon
Award winning comedian Rob Brydon and his eight-piece band will take to the road in 2021 his new show, Rob Brydon – A Night Of Songs & Laughter
Jack and the Beanstalk Treasure Hunt Competition!
Category: Panto
Win a family ticket for 4 people to see Jack and the Beanstalk at the Victoria Theatre in December 2021 by taking part in our treasure hunt!
Lifeline grant for The Victoria Theatre Halifax
The Victoria Theatre Halifax has been awarded £301,330 from the government's £1.57 billion Culture Recovery Fund...
Q & A with Sir Ranulph Fiennes
Named by The Guinness Book of Records as 'the world's greatest living explorer', Sir Ranulph Fiennes is bringing his Living Dangerously tour to the Victoria Theatre in 2021
Stars of Britain's Got Talent The Noise Next Door return to Halifax
The U.K's premier improv comedy troupe The Noise Next Door to perform at the Victoria Theatre
Alan Carr, Not Again, Alan!
Join Alan on his new tour as he muses upon the things that make his life weird and wonderful.
Take A Chance on ABBA MANIA…
…..And you won't be disappointed! The world's longest running Abba Tribute Concert ABBA MANIA is coming to the Victoria Theatre Halifax.
An Interview with Rob Beckett ahead of his Wallop! Tour
Rob Beckett's comedy has long been packing a solid punch, so he's found the ideal title for his new touring show in Wallop coming soon to the Victoria Theatre Halifax!
The Australian Pink Floyd Show offer free tickets for NHS workers!
The Australian Pink Floyd Show offer a number of tickets for FREE for NHS workers for a 24 hour period for their concert at the Victoria Theatre Halifax.
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text:
Q: jdbc foreign key constraint I am creating a java application and I have connected to mysql database. I tried creating a table with foreign key and I got error Cannot add foreign key constraint below are my tables
execute("CREATE TABLE " + BBBBB + "("
                        + "         id int(11) NOT NULL AUTO_INCREMENT,\n"
                        + "         consultantID varchar(225) NOT NULL,\n"
                        + "         name varchar(225) NOT NULL,\n"
                        + "         gender varchar(225) NOT NULL,\n"
                        + "         age varchar(225) NOT NULL,\n"
                        + "         email varchar(225) NOT NULL,\n"
                        + "         mobile varchar(225) NOT NULL,\n"
                        + "         address varchar(225) NOT NULL,\n"
                        + "         PRIMARY KEY (id)"
                        + ")        ENGINE=InnoDB DEFAULT CHARSET=latin1"
                );

execute("CREATE TABLE " + AAAAA + "("
                        + "         id int(11) NOT NULL AUTO_INCREMENT,\n"
                        + "         patientID varchar(225) NOT NULL,\n"
                        + "         name varchar(225) NOT NULL,\n"
                        + "         gender varchar(225) NOT NULL,\n"
                        + "         age varchar(225) NOT NULL,\n"
                        + "         insurance varchar(225),\n"
                        + "         mobile varchar(225) NOT NULL,\n"
                        + "         address varchar(225) NOT NULL,\n"
                        + "         PRIMARY KEY (id)"
                        + ")        ENGINE=InnoDB DEFAULT CHARSET=latin1"
                );

execute("CREATE TABLE " + CCCCC + "("
                        + "         id int(11) NOT NULL AUTO_INCREMENT,\n"
                        + "         patientID varchar(225) NOT NULL,\n"
                        + "         consultantID varchar(225) NOT NULL,\n"
                        + "         time varchar(225) NOT NULL,\n"
                        + "         date varchar(225) NOT NULL,\n"
                        + "         renewCount INTEGER default 0,\n"
                        + "         FOREIGN KEY (patientID) REFERENCES AAAAA(pID),\n"
                        + "         FOREIGN KEY (consultantID) REFERENCES BBBBB(cID),\n"
                        + "         PRIMARY KEY (id)"
                        + ")        ENGINE=InnoDB DEFAULT CHARSET=latin1"

I got an error as stated above. additional codes would be added on request. Thanks

A: *

*in the third statement, pID and cID don't seem to exist, they should be patientID and consultantID

*to create the tables you are using the variables AAAAA and BBBBB, but in the third statement you are referring to them as the string constants "AAAAA" and "BBBBB", which might not be correct, you should always use the variables, for example:
" ... REFERENCES " + BBBBB + "(cID) ... "

redpajama_set_name: RedPajamaStackExchange


Guide Kim, a grumpy little girl raised by her grandmother on a journey to stop an evil corporation from destroying her village. Fluid gameplay allow players to choose their own pace, whether fast as a hack'n'slash or methodic as chess. Keen is a game for the tactics player, the puzzle fan, and anyone that loves a solid, old-school adventure.


Last week, a disastrous hurricane hit The Philippines, my motherland and my home. They say it is the strongest, most powerful storm ever recorded in history... and the most affected island just happens to be my mother's province, Tacloban, Leyte.
The last time there was a super storm (Ondoy in 2009), I was home. I was there when it happened, and I was there to help when people were trying to rebuild and get back on their feet.
Now, on a similar yet worse situation, I feel so helpless being so far away. All I can really do now is bring light into the situation, and use my blog as a tool to allow people to get informed and help with whatever they can.
The UN's hunger-fighting organization has allocated an immediate $2 million for Haiyan relief, with a greater appeal pending as needs become apparent. The UN organization is sending 40 metric tons of fortified biscuits in the immediate aftermath, as well as working with the government to restore emergency telecommunications in the area. Americans can text the word AID to 27722 to donate $10 or give online. Learn more here.
The Christian humanitarian organization that specifically supports families living in poverty is providing food, water and hygiene kits at the evacuation centers. World Vision was also still actively responding to last month's earthquake in Bohol, which fortunately was not struck by the eye of the storm.Learn more here.
The Christian hunger and poverty-fighting organization is allocating 100 percent of all disaster donations for relief efforts "to immediately meet the specific needs of disaster survivors." Text TYPHOON to 80888 to Donate $10 or give online.Learn more here.
The organization, which prioritizes kids' needs, has sent relief kits for children and families, including household cleaning items, temporary school tents and learning materials. Learn more here.
The international poverty-fighting nonprofit, which provides sustainable global food and water solutions, is distributing drinking water, buckets, soap and chlorine tablets. It's also providing sanitation equipment to help prevent waterborne diseases. Learn more here.
All I'm asking you to do is to give up one thing worth £3 this week and instead text UNICEF to 70 123 and donate the money to those who really need it.
When you think about it, it's a really small thing to do. For example, I usually buy a newspaper on a Sunday which costs around £2.50. This week instead of buying a paper I'll send a £3 donation to UNICEF - this really won't affect me but could make a huge difference to a person or a family affected by the typhoon.
Last thing (I promise!) - instead of liking this post please re-post it to your wall so that the idea can be spread to as many people as possible.
The Filipino Spirit is Waterproof. This is the understatement of the millennium. No matter how disastrous the calamity... we Filipinos always manage to get back up with a smile. Always.
I want to extend my personal thanks to everyone who's helped my country out... The Brits, the Australians, the Americans... EVERYONE who's offered a donation or a prayer... Thank you. Acts of kindness like these restores one's faith in humanity... and really reinforces one's belief that people are innately good.
If you want to help out, and don't know where to start... I hope this blog post points you towards the right direction... Please continue to think good thoughts about my country, and send your best positive energy towards the victims of this awful tragedy.


Six in 10 Ohio voters sat Tuesday's election out — the lowest turnout on record for a gubernatorial election.
Six in 10 Ohio voters sat Tuesday's election out, the lowest turnout on record for a gubernatorial election.
Preliminary results showed about 3.1 million Ohioans cast ballots in the November election, 40 percent of the nearly 7.75 million people registered. County elections boards also reported about 161,000 outstanding absentee and provisional ballots.
That's a record low for Ohio gubernatorial election turnout, which has ranged from 47.8 percent in 2002 to 62.3 percent in 1982, according to Ohio Secretary of State's Office records since 1978, the first year voter turnout was tracked. Only three elections in recent history had lower turnout and all were local, off-year elections.
Turnout could be adjusted as absentee and provisional ballots are counted, but its remains significantly lower than previous midterm elections. The average turnout for presidential elections has been 71.1 percent, compared with 54.8 percent in gubernatorial races and 44.7 percent in off-year, local elections.
Turnout in Ottawa County was 51.8 percent of registered voters — the lowest on record for a gubernatorial race, according to preliminary results. Voter turnout in 2010, when Kasich won his first term, was 53.9 percent.
Only three counties — Monroe, Shelby and Ottawa — had higher than 50 percent of registered voters cast ballots, according to preliminary results.
Experts predicted low voter turnout because of Democrat Ed FitzGerald's uncompetitive run for governor, which hurt the statewide ticket. Lower turnout tends to favor Republican candidates, according to John Green, director of the University of Akron's Bliss Institute of Applied Politics.
Republican gubernatorial candidates have won every race with less than 50 percent of Ohio voters casting ballots. On Tuesday, Republicans swept statewide races with Kasich beating FitzGerald by 31 percentage points, according to unofficial results.


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Home Greek news economy Sex Industry in Greece Grows Amidst the Financial Crisis
Sex Industry in Greece Grows Amidst the Financial Crisis
Since 2008, when the financial crisis first appeared in Greece, the number of women who work in the sex industry has dramatically increased. Women trade sex for money, illegally in most cases, in order to put food on the table for their children and for themselves.
Almost a third of Greek citizens are living below the poverty line and women, as well as children, are the most vulnerable groups who suffer from the austerity measures set by Greece's creditors. The large unemployment rate, which has surpassed 25%, has led more than 18,000 women –an increase compared to the 17,000 in 2012- to work as prostitutes. According to the Greek Centre for Social Sciences and Panteion University, the number of people selling sexual services in Greece has soared by 150% during the seven-year crisis, noted the International Business Times.
Human Rights Watch senior research assistant Eva Cossé noted: "Over the last few years, Athens has struggled with an increasing number of people who have lost their jobs, been evicted from their homes, and affected by poverty and social exclusion."
"As austerity measures have been carried out, HIV, suicide and depression have increased, and hundreds of thousands of people have been locked out of the health system altogether," Cossé added. "Some live on the streets and others go there to find drugs or to find clients for sex work."
Greece's legislation on the matter is not very helpful. Street prostitution is illegal, but sex work, for example in brothels is not. However, these brothels and the women working there, need to issue special licenses. Only 1,000 sex workers have been officially registered because the law provides that women must adhere to a long list of requirements in order to register as sex workers.
"The Greek government should implement a genuine public health approach to people who exchange sex for money, drugs or life necessities, and make sure that police officers respect the rights of women whose lives are hard enough already," said Cossé.
Panteion University
sex industry


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