From 372628e8602e724a5cd0a6e211179255f4b75adb Mon Sep 17 00:00:00 2001 From: Ian Jauslin Date: Fri, 1 Nov 2019 00:45:19 -0400 Subject: [PATCH] Initial commit --- Costin_Costin_Jauslin_Lebowitz_2019.tex | 468 + Makefile | 55 + README | 43 + bibliography/bibliography.tex | 46 + figs/avgcurrent.fig/Makefile | 39 + figs/avgcurrent.fig/avgcurrent.dat | 9606 +++++++++++++++++++ figs/avgcurrent.fig/avgcurrent.gnuplot | 24 + figs/avgcurrent.fig/avgcurrent_zoom.gnuplot | 25 + figs/avgcurrent.fig/decay.dat | 53 + figs/avgcurrent.fig/decay.gnuplot | 34 + figs/avgcurrent.fig/save.dat | 487 + figs/cn.fig/Makefile | 40 + figs/cn.fig/cn.dat | 9361 ++++++++++++++++++ figs/cn.fig/cn.gnuplot | 25 + figs/cn.fig/cn_short.dat | 26 + figs/cn.fig/cn_short.gnuplot | 25 + figs/cn.fig/mc.dat | 606 ++ figs/cn.fig/mc_short.dat | 206 + figs/current.fig/Makefile | 38 + figs/current.fig/current01.gnuplot | 29 + figs/current.fig/current05.gnuplot | 29 + figs/current.fig/current18.gnuplot | 29 + figs/current.fig/current30.gnuplot | 29 + figs/current.fig/mc01.dat | 606 ++ figs/current.fig/mc05.dat | 606 ++ figs/current.fig/mc18.dat | 1206 +++ figs/current.fig/mc30.dat | 1206 +++ figs/omega.fig/Makefile | 32 + figs/omega.fig/omega.dat | 57 + figs/omega.fig/omega.gnuplot | 22 + figs/wavefunction.fig/Makefile | 32 + figs/wavefunction.fig/current.gnuplot | 28 + figs/wavefunction.fig/density.gnuplot | 28 + figs/wavefunction.fig/imag.gnuplot | 28 + figs/wavefunction.fig/mc.dat | 606 ++ figs/wavefunction.fig/real.gnuplot | 28 + libs/ian.cls | 673 ++ libs/iantheo.sty | 162 + libs/point.sty | 106 + 39 files changed, 26749 insertions(+) create mode 100644 Costin_Costin_Jauslin_Lebowitz_2019.tex create mode 100644 Makefile create mode 100644 README create mode 100644 bibliography/bibliography.tex create mode 100644 figs/avgcurrent.fig/Makefile create mode 100644 figs/avgcurrent.fig/avgcurrent.dat create mode 100644 figs/avgcurrent.fig/avgcurrent.gnuplot create mode 100644 figs/avgcurrent.fig/avgcurrent_zoom.gnuplot create mode 100644 figs/avgcurrent.fig/decay.dat create mode 100644 figs/avgcurrent.fig/decay.gnuplot create mode 100644 figs/avgcurrent.fig/save.dat create mode 100644 figs/cn.fig/Makefile create mode 100644 figs/cn.fig/cn.dat create mode 100644 figs/cn.fig/cn.gnuplot create mode 100644 figs/cn.fig/cn_short.dat create mode 100644 figs/cn.fig/cn_short.gnuplot create mode 100644 figs/cn.fig/mc.dat create mode 100644 figs/cn.fig/mc_short.dat create mode 100644 figs/current.fig/Makefile create mode 100644 figs/current.fig/current01.gnuplot create mode 100644 figs/current.fig/current05.gnuplot create mode 100644 figs/current.fig/current18.gnuplot create mode 100644 figs/current.fig/current30.gnuplot create mode 100644 figs/current.fig/mc01.dat create mode 100644 figs/current.fig/mc05.dat create mode 100644 figs/current.fig/mc18.dat create mode 100644 figs/current.fig/mc30.dat create mode 100644 figs/omega.fig/Makefile create mode 100644 figs/omega.fig/omega.dat create mode 100644 figs/omega.fig/omega.gnuplot create mode 100644 figs/wavefunction.fig/Makefile create mode 100644 figs/wavefunction.fig/current.gnuplot create mode 100644 figs/wavefunction.fig/density.gnuplot create mode 100644 figs/wavefunction.fig/imag.gnuplot create mode 100644 figs/wavefunction.fig/mc.dat create mode 100644 figs/wavefunction.fig/real.gnuplot create mode 100644 libs/ian.cls create mode 100644 libs/iantheo.sty create mode 100644 libs/point.sty diff --git a/Costin_Costin_Jauslin_Lebowitz_2019.tex b/Costin_Costin_Jauslin_Lebowitz_2019.tex new file mode 100644 index 0000000..5f4c367 --- /dev/null +++ b/Costin_Costin_Jauslin_Lebowitz_2019.tex @@ -0,0 +1,468 @@ +\documentclass{ian} + +\usepackage{graphicx} +\usepackage{array} + +\begin{document} + +\hbox{} +\hfil{\bf\LARGE +Exact solution of the Schr\"odinger equation\par +\hfil for photoemission from a metal\par +} +\vfill + +\hfil{\bf\large Ovidiu Costin}\par +\hfil{\it Department of Mathematics, The Ohio State University}\par +\hfil{\tt\color{blue}\href{mailto:costin.9@osu.edu}{costin.9@osu.edu}}\par +\vskip20pt + +\hfil{\bf\large Rodica Costin}\par +\hfil{\it Department of Mathematics, The Ohio State University}\par +\hfil{\tt\color{blue}\href{mailto:costin.10@osu.edu}{costin.10@osu.edu}}\par +\vskip20pt + +\hfil{\bf\large Ian Jauslin}\par +\hfil{\it Department of Physics, Princeton University}\par +\hfil{\tt\color{blue}\href{mailto:ijauslin@princeton.edu}{ijauslin@princeton.edu}}\par +\vskip20pt + +\hfil{\bf\large Joel L. Lebowitz}\par +\hfil{\it Department of Mathematics and Physics, Rutgers University}\par +\hfil{\tt\color{blue}\href{mailto:lebowitz@math.rutgers.edu}{lebowitz@math.rutgers.edu}}\par +\vskip20pt + +\vfill + + +\hfil {\bf Abstract}\par +\medskip +{\small +We solve rigorously the time dependent Schr\"odinger equation describing electron emission from a metal surface by a laser field perpendicular to the surface. +We consider the system to be one-dimensional, with the half-line $x<0$ corresponding to the bulk of the metal and $x>0$ to the vacuum. +The laser field is modeled as a classical electric field oscillating with frequency $\omega$, acting only at $x>0$. +We consider an initial condition which is a stationary state of the system without a field, and, at time $t=0$, the field is switched on. +We prove the existence of a solution $\psi(x,t)$ of the Schr\"odinger equation for $t>0$, and compute the surface current. +The current exhibits a complex oscillatory behavior, which is not captured by the ``simple'' three step scenario. +As $t\to\infty$, $\psi(x,t)$ converges with a rate $t^{-\frac32}$ to a time periodic function with period $\frac{2\pi}{\omega}$ which coincides with that found by Faisal, Kami\'nski and Saczuk (Phys Rev A {\bf 72}, 023412, 2015). +However, for realistic values of the parameters, we have found that it can take quite a long time (over 50 laser periods) for the system to converge to its asymptote. +Of particular physical importance is the current averaged over a laser period $\frac{2\pi}\omega$, which exhibits a dramatic increase when $\hbar\omega$ becomes larger than the work function of the metal, which is consistent with the original photoelectric effect. +} + +\vfill + +\tableofcontents + +\vfill +\eject + +\setcounter{page}1 +\pagestyle{plain} + + +\section{Introduction} +\indent In this note, we continue our investigation of the exact solution of the time-dependent Schr\"odinger equation describing the emission of electrons from a metal surface by an electric field \cite{CCe18}. +We use the Sommerfeld model of quasi-free electrons with a Fermi distribution of energies, confined by a step potential $U=\mathcal E_F+W$, where $\mathcal E_F$ is the Fermi energy and $W$ is the work function of the metal. +This setup is commonly used as a simple model for the process of emission, both for a constant field and an oscillating field \cite{FN28,YGR11,Fo16,Je17}. +In both cases one focuses attention on electrons, part of the Fermi sea, with energy $\frac12k^2$ (in atomic units), moving from the left towards the metal surface at $x=0$. +These are described by a wave function $e^{ikx}$, $k>0$, $x<0$. +When the electrons reach the metal surface they are either reflected or transmitted through the barrier. +\bigskip + +\indent The time evolution of the wave function of an electron in such a beam is described by the one dimensional Schr\"odinger equation, +\begin{equation} + i\partial_t\psi(x,t)=H(x,t)\psi(x,t)=-\frac12\Delta\psi(x,t)+\Theta(x)(U-Ex\cos(\omega t))\psi(x,t) + \label{schrodinger} +\end{equation} +where $\Theta(x)=0$ for $x<0$ and $\Theta(x)=1$ for $x>0$, $E$ is the electric field perpendicular to the surface, $\frac\omega{2\pi}$ is the frequency and we are using atomic units $\hbar=m=1$. +Here, we ignore the carrier wave and Shottky effect. +We also only focus on the direction that is orthogonal to the metal surface, ignoring the other two directions \cite{KSH11}. + +\indent In the absence of an external field, $E=0$, the Schr\"odinger equation (\ref{schrodinger}) has a ``stationary" solution $e^{-i\frac12k^2t}\varphi_0(x)$ in which there is, for $k^2<2U$, a reflected beam of the same energy and intensity as the incoming beam $e^{ikx}$ and an evanescent, exponentially decaying tail on the right, +\begin{equation} \label{initial} + \varphi_0(x)= + \left\{\begin{array}{ll} + e^{ikx}+R_0e^{-ikx}&\mathrm{for\ }x<0\\ + T_0e^{- \alpha x}&\mathrm{for\ }x>0 + \end{array}\right. +\end{equation} +Using the continuity of $\psi$ and its derivative at $x=0$ yields +\begin{equation} + \alpha:=\sqrt{2U-k^2} + ,\quad + 1+R_0=T_0=\frac{2ik}{ik-\sqrt{2U-k^2}} + . +\end{equation} +The current, +\begin{equation} + j(x,t):=\mathcal Im(\psi^*(x,t)\partial_x\psi(x,t)) +\end{equation} +is zero and no electrons leave the metal. +Note that $\varphi_0$ is not square integrable, which will complicate matters when it comes to solving the Schr\"odinger equation. + +\indent If we turn on a constant field, $E>0, \ \omega=0$, at $t=0$, the Schr\"odinger equation has a ``stationary'' solution ${\rm e}^{-\frac 12 k^2 t}\varphi_{E}(x)$ with $\varphi_E$ satisfying the equation +\begin{equation} + \frac12k^2\varphi_E=-\frac12\frac{\partial^2}{\partial x^2}\varphi_E+\Theta(x)(U-Ex)\varphi_E +\end{equation} +with incident beam $\Theta(-x)e^{ikx}$. +This equation was solved by Fowler and Nordheim \cite{FN28} in 1928 to describe the quantum tunneling of electrons through a triangular barrier. +Their solution, with some corrections, is commonly used to describe the stationary current produced by a field $E$ acting on the surface of a metal \cite{Je17}. +\bigskip + +\indent In \cite{CCe18} we solved the time-dependent Schr\"odinger equation (\ref{schrodinger}) with initial condition $\varphi_0(x)$ for the constant field. +We showed that $\psi(x,t)$ converges, as $t\to\infty$, to the Fowler-Nordheim (FN) solution ${\rm e}^{-\frac 12 k^2 t}\varphi_{E}(x)$ +The exact rate of convergence is $t^{-\frac32}$. +The ``effective'' time of approach to the FN solution was found to be, for realistic values of the parameters, of the order of femtoseconds. + +\indent Here, we investigate solutions of (\ref{schrodinger}) with $E>0$, $\omega>0$, $\psi(x,0)=\varphi_0(x)$ as given in (\ref{initial}). +Since we are interested in describing situations in which the laser is turned on for only a short time, with pulses as short as a few periods, understanding the role of the initial state is important. +This problem turns out to be considerably more difficult than the constant field case, $\omega=0$. +\bigskip + +\indent There are several studies in the literature about the solution of the time-dependent Schr\"odinger equation, see \cite{KLe18} and references therein. +The method most often used is the Crank-Nicolson \cite{CN47} numerical scheme. +In the case treated here, however, as we will discuss in more detail in the following, because of the discontinuity in the potential at $x=0$, the Crank-Nicolson algorithm is actually inaccurate for short times. +We will introduce an alternative scheme that does not suffer from the discontinuity of the potential. + +\indent Of particular note is the work by Yalunin, Gulden and Roper \cite{YGR11}, where the initial condition and potential are the same as in (\ref{schrodinger}) and (\ref{initial}). +Our result is quite different from theirs for short time, but the values we obtain at longer times seem to agree qualitatively with those obtained in \cite{YGR11} for short times. +Furthermore, in \cite{YGR11}, there does not seem to be a distinction between long and short times, whereas we find that for some important physical quantities, like the current averaged over a laser period, one has to wait many periods to be close to the asymptotic value. +\bigskip + +\indent We look for an exact solution of (\ref{schrodinger}) for $t>0$ such that $\psi(x,t)$ and its derivative $\partial_x\psi(x,t)$ are continuous at $x=0$ and be bounded for $|x|\to\infty$. +To do this, we first obtain the solutions for $x<0$ and for $x>0$ with general initial conditions and unknown time dependent boundary conditions $\psi(0,t)=\psi_0(t)$ and $\partial_x\psi(x,t)|_{x=0}=\psi_{x,0}(t)$. +A solution of (\ref{schrodinger}) with appropriate $x\to\pm\infty$ conditions will exist if and only if we can match these boundary conditions coming from the right and from the left. +This approach to solving such an equation as (\ref{schrodinger}) goes under the general name of initial-boundary-value (IBV) problem, developed in particular by Fokas for these types of problems \cite{Fo97,Ry18}. + +\indent We prove the existence of a solution of (\ref{schrodinger}) by showing that a necessary and sufficient condition for this is that $\psi_0(t)$ satisfies an integral equation. +The existence of the solution $\psi(x,t)$ for all $x$ and $t$ then follows from the proof that this integral equation has a unique solution, which we provide. +Expressions for $\psi$ and for the current $j(x,t)$ are given by explicit integrals involving $\psi_0(t)$ \cite{CCe}. +These can be evaluated numerically. +To do so accurately for both short and moderate times, we developed a numerical scheme which goes beyond the Crank-Nicolson algorithm \cite{CN47}, see section\,\ref{Sec3}. +\bigskip + +\indent $\psi(x,t)$ has the long time form $e^{-i\frac12k^2t}\mathcal U(x,t)$, where $\mathcal U$ is (up to a phase) the periodic in time solution of (\ref{schrodinger}) derived by Faisal et al \cite{FKS05,ZL16}. +We note that Faisal et al. did not prove the existence of a solution to their infinite set of equations. +Instead, they solved numerically a truncated set of these equations. +Our results prove the existence of such a solution. + +\indent The approach to this asymptotic form $\mathcal U$ goes like $t^{-\frac32}$. +$\mathcal{U}(x,t)$ consists of resonance terms which can be associated, as was done in \cite{FKS05}, with the absorption (emission) of different number of photons, see Section\,\ref{longtime}. +These resonances show up dramatically when considering the dependence of the average current on $\omega$ in the vicinity of $\omega_c=W$, as seen in early experiments on the photoelectric effect where light with frequency $\omega>\omega_c$ was necessary to get electron emission. + + +\section{Existence of solutions of the Schr\"odinger equation} +\indent Here we give an outline of the proof derived in \cite{CCe}. To simplify the notations we rescale: let +\begin{equation}\label{rescale} +\bar t=\omega t,\ \ \bar x=\sqrt{2\omega}x,\ \ \bar U=U\omega^{-1},\ \ \bar E=E\omega^{-3/2}2^{-1/2} +\end{equation} +Equation (\ref{schrodinger}) becomes +\begin{equation}\label{eq12} + i\partial_{\bar t}\psi(\bar x,\bar t)=-\partial^2_{\bar x}\psi(\bar x,\bar t)+\Theta(\bar x)\left( \bar U-2\bar E \bar x\cos(\bar t)\right)\psi(\bar x,\bar t) +\end{equation} +As mentioned earlier, we will solve (\ref{eq12}) with the general initial condition +$\psi(\bar x,0):=f(\bar x)$ separately for $\bar x<0$, $\psi_-(\bar x,\bar t)$ and for $\bar x>0$, $\psi_+(\bar x,\bar t)$, and then match the values of $\psi_\pm(0,\bar t)$ and $\partial_{\bar x}\psi_\pm(0,\bar t)$. +\bigskip + +\indent To obtain these solutions we first go from $\bar x$ to Fourier space. +We write +\begin{equation} + \hat\psi(\xi,\bar t)=\frac1{\sqrt{2\pi}}\int_{-\infty}^\infty d\bar x\ e^{-i\bar x\xi}\psi(\bar x,\bar t) + =\hat\psi_-(\xi,\bar t)+\hat\psi_+(\xi,\bar t) +\end{equation} +where $\hat\psi_\pm$ is the half-line Fourier transform along $\pm \bar x>0$. +\bigskip + +\indent The equation for $\hat\psi_-(\xi,\bar t)$ has the form +\begin{equation}\label{eq5} +i\frac{\partial \hat{\psi}_-}{\partial\bar t}=\xi^2 \hat{\psi}_-- \frac1{\sqrt{2\pi}} \psi_{\bar x,0}(\bar t)-i\xi \frac1{\sqrt{2\pi}} \psi_0(\bar t) +\end{equation} +where +\begin{equation} + \psi_0(\bar t):=\psi(0,\bar t) + ,\quad + \psi_{\bar x,0}(\bar t):=\partial_{\bar x}\psi(0,\bar t) +\end{equation} +are treated as unknown functions to be determined later by the matching conditions. +The solution of (\ref{eq5}) is given by +\begin{equation} \label{psim} + \hat{\psi}_-(\xi,\bar t)= {\rm e}^{-i\xi^2\bar t}\left\{ C_-(\xi) + \frac1{\sqrt{2\pi}} \int_0^{\bar t} {\rm e}^{i\xi^2\bar s}\ \left[ i\psi_{\bar x,0}(\bar s)-\xi \psi_0(\bar s) \right]\, d\bar s\right\} +\end{equation} +where +$C_-(\xi)=\int_{-\infty}^0{\rm e}^{-i\bar x\xi}f(\bar x)\, d\bar x$. +Inverting the Fourier transform we obtain, for all $\bar x<0$, +\begin{equation}\label{psiminus} +{\psi}_-(\bar x,\bar t)=h_-(\bar x,\bar t)+ \frac i{{2\pi}} \int_0^{\bar t} d\bar s\, \psi_{\bar x,0}(\bar s) H_0(\bar t-\bar s,\bar x)+ \frac i{{2\pi}} \int_0^{\bar t} d\bar s\, \psi_0(\bar s) \, \partial_{\bar x}H_0(\bar t-\bar s,\bar x) +\end{equation} +where +$$ +h_-(\bar x,\bar t) =\frac 1{{2\pi}}\int_{-\infty}^0 f(\bar y) H_0(\bar t,\bar x-\bar y)\, d\bar y, +\quad +H_0(\bar t,\bar r)= \frac {\sqrt{\pi}}{\sqrt{i}{\sqrt{\bar t}}} {\rm e}^{\frac{i\bar r^2}{4\bar t}} +$$ +\bigskip + +\indent The equation for $\hat{\psi}_+(\xi,\bar t)$ is given by +$$i\frac{\partial \hat{\psi}_+}{\partial\bar t}=-i 2\bar E \cos(\bar t)\, \frac{\partial \hat{\psi}_+}{\partial \xi} +\left( \xi^2+\bar U\right) \hat{\psi}_+(\xi,\bar t) + \frac1{\sqrt{2\pi}} \psi_{\bar x,0}(\bar t)+i\xi \frac1{\sqrt{2\pi}} \psi_0(\bar t) $$ +Solving for $\hat\psi_+(\xi,\bar t)$ and inverting the Fourier transforms, we obtain +\begin{equation}\begin{array}{>\displaystyle c}\label{calcpsip} +\psi_+(\bar x,\bar t)=h_+(\bar x,\bar t)+\\ + \frac1{{2\pi}} {\rm e}^{2i\bar E \bar x\sin(\bar t)}\int_{-\infty}^{\infty} du\,{\rm e}^{i\bar xu-\Phi(u,\bar t)} \int_0^{\bar t} \, {\rm e}^{\Phi(u,\bar s)}\left[ -i \psi_{\bar x,0}(\bar s)+2\bar E\sin\bar s\, \psi_0(\bar s)\right]\, d\bar s\\ ++ \frac1{{2\pi}} {\rm e}^{2i\bar E \bar x\sin(\bar t)}\int_{-\infty}^{\infty} du\,{\rm e}^{i\bar xu-\Phi(u,\bar t)} \int_0^{\bar t} \, {\rm e}^{\Phi(u,\bar s)}\,u\, \psi_0(\bar s)d\bar s +\end{array}\end{equation} +where +\begin{equation}\begin{array}{>\displaystyle c} +\Phi(u,\bar t)=i\left[ (u^2 +\bar U+2\bar E^2)\bar t - 4\bar E u \cos(\bar t) -\bar E^2 \sin(2\bar t) \right], \\ +h_+(\bar x,\bar t)=\frac 1{2\sqrt{i \pi }}{\rm e }^{2i\bar x\bar E\sin(\bar t)-i(\bar U+2\bar E^2)\bar t+i\bar E^2\sin2\bar t}\int_0^{\infty }d\bar y\, f(\bar y)\, \frac 1{\sqrt{\bar t}}\,{\rm e}^{\frac{i(\bar x-\bar y-4\bar E+4\bar E\cos(\bar t))^2}{4\bar t}} +\end{array}\end{equation} +\bigskip + +\indent Taking now the limit $\bar x\to0$ from left and right in (\ref{psiminus}) and in (\ref{calcpsip}) and setting $\psi_-(0,\bar t)=\psi_+(0,\bar t)$, and similarly for $\partial_{\bar x}\psi_\pm $ we obtain (in fact $\psi_-(0,\bar t)=\psi_0(\bar t)=\psi_+(0,\bar t)$ implies that $\partial_{\bar x}\psi_-(0,\bar t)=\partial_{\bar x}\psi_+(0,\bar t)$) one equation with the only unknown function $\psi_0(\bar t)$, which has to satisfy the integral equation + \begin{equation}\begin{array}{>\displaystyle c}\label{eqcontr} + \psi_0(\bar t)= h_+(0,\bar t)+ h_-(0,\bar t) -\frac{1}{\pi}\int_0^{\bar t}\left( \int_0^{\bar s} \frac{h_-(0,\bar s')}{\sqrt{\bar s-\bar s'}}\, d\bar s'\right) \, G_1(\bar s,\bar t)\,d\bar s\\ + +\frac{1}{2\pi}\int_0^{\bar t} \left( \int_0^{\bar s} \frac{\psi_0(\bar s')}{\sqrt{\bar s-\bar s'}}\, d\bar s'\right) \, G_1(\bar s,\bar t)\, d\bar s \\ + +\frac{\bar E}{\sqrt{i\pi}}\int_0^{\bar t} d\bar s \, \,\psi_0(\bar s)\, \frac 1{\sqrt{\bar t-\bar s}}\, \left( \sin\bar s\, + \frac{ \cos(\bar t)- \cos(\bar s)}{\bar t-\bar s}\right) \,{\rm e}^{iF_0(\bar s,\bar t)} +\end{array}\end{equation} +where +$$G_1(\bar s,\bar t)=\frac d{d\bar s}\frac{e^{iF_0(\bar s,\bar t)}-1}{\sqrt{\bar t-\bar s}}$$ +with +\begin{equation}\label{Fzero} + F_0(s,\bar t)=\frac{4\bar E^2 (\cos(\bar t)- \cos(\bar s))^2}{\bar t-\bar s} -(\bar U_2+2\bar E^2)(\bar t-\bar s) +\bar E^2(\sin 2\bar t-\sin 2\bar s) +\end{equation} +We then prove that the integral equation (\ref{eqcontr}) has a unique solution, which implies that (\ref{eq12}) does as well. +\bigskip + +\indent Setting the initial condition $f(\bar x)=\varphi_1(\bar x)$ (obtained from (\ref{initial}) after the rescaling (\ref{rescale})) +we obtain explicit functions in equation (\ref{eqcontr}) for $\psi_0(\bar t)$, +\begin{equation}\label{hminspec} +h_-(0,\bar t)= {{\rm e}^{-i{\bar k}^{2}\bar t}}\,\frac12\, \left[ 1+{\rm erf} \left( -i^{3/2}\bar k\sqrt {\bar t}\right) \right] +-R_0\, {{\rm e}^{-i{k}^{2}\bar t}}\,\frac12\, \left[ -1+{\rm erf} \left( -i^{3/2} \bar k\sqrt {\bar t}\right) \right] +\end{equation} +and +\begin{equation}\begin{array}{>\displaystyle c}\label{hpluspec} +h_+(0,\bar t)= T_0\,\exp[{-i(\bar U+2\bar E^2)\bar t+i\bar E^2\sin 2\bar t-4\,\bar E \alpha\,\cos\bar t +i{\alpha}^{2}\bar t+4\,\bar E \alpha}] \times \\ +\frac 1{2} \left[1- {\rm erf} \left(2\,i^{3/2}\bar E\frac{\cos\bar t-1}{\sqrt {\bar t}}+ \alpha \sqrt{i\bar t} \right) \right] +\end{array}\end{equation} + where +$$\bar k=k/\sqrt{2\omega},\ \ \ \alpha=\sqrt{\bar U-\bar k^2},\ \ \ 1+R_0=T_0=\frac{2i\bar k}{i\bar k-\sqrt{\bar U-\bar k^2}}.$$ + +\section{Short time behavior: approach to the stationary state}\label{Sec3} +\indent We carried out numerical solution of (\ref{eqcontr}) using (\ref{hminspec}) and (\ref{hpluspec}) for a variety of values of $E$. +To compare with the work in \cite{YGR11}, we take $\frac {\hbar^2k^2}{2m}=\mathcal E_F=4.5$\,eV, $W=5.5$\,eV. +Unless otherwise specified, we also take $\omega=1.55\ \mathrm{eV}$. +The laser period $\tau=\frac{2\pi}\omega$ is then equal to $2.7\ \mathrm{fs}$. +\bigskip + +\indent The details of the numerical algorithms, which take into account the singular behavior of $\psi(x,t)$ at $x=0$ and $t=0$, will be given in a separate publication. +\bigskip + +\indent Figure \ref{fig:wavefunction} shows the real and imaginary parts of $e^{i\frac{k^2}2t}\psi_0(t)$, as well as $|\psi_0(t)|^2$ and the normalized current $\frac1kj(0,t)$. +While $|\psi_0|^2$ seems more in phase with the real part of$e^{i\frac{k^2}2t}\psi_0$, the current looks more in phase with the imaginary part. +Figure \ref{fig:current} shows the values of the current $j(0,t)$ passing through the origin as a function of time for different strengths of the field, all at $\omega=1.55$\, eV. +We see there a change of behavior as $E$ increases from $E=1$\,V/nm to $E=30$\,V/nm (the Keldysh parameter $\gamma=\frac{2\omega}E\sqrt W$ goes from 18.6 to 0.62). +For large values of $E$, fast oscillations appear, which become faster and larger as $E$ grows. +The oscillatory behavior within one period for strong fields is also observed in the approximate solution \cite{YGR11}, though the details vary. +We have no simple physical explanation for these fast oscillations. +They do not fit in with the ``simple three step'' scenario \cite{KSK92,Co93,KLe18}. +We note that the height of the first maximum is linear in $E$, while its location is almost independent of $E$. +\bigskip + +\begin{figure} + \includegraphics[width=8cm]{real.pdf} + \hfill + \includegraphics[width=8cm]{imag.pdf} + \par\penalty10000\bigskip\penalty10000 + \includegraphics[width=8cm]{density.pdf} + \hfill + \includegraphics[width=8cm]{current.pdf} + \caption{ + The real (a) and imaginary (b) parts of $e^{i\frac{k^2}2t}\psi_0(t)$ the density (c) and the current (d) as a function of $\frac{t\omega}{2\pi}$ for 3 periods, with $E=15\ \mathrm V\cdot\mathrm{nm}^{-1}$, $\omega=1.55\ \mathrm{eV}$. + The dotted line represents the phase of the field: $\cos(\omega t)$. + } + \label{fig:wavefunction} +\end{figure} + +\begin{figure} + \includegraphics[width=8cm]{current01.pdf} + \hfill + \includegraphics[width=8cm]{current05.pdf} + \par\penalty10000\bigskip\penalty10000 + \includegraphics[width=8cm]{current18.pdf} + \hfill + \includegraphics[width=8cm]{current30.pdf} + \caption{ + The normalized current $\frac jk$ as a function of $\frac{t\omega}{2\pi}$ for $\omega=1.55\ \mathrm{eV}$ and various values of the electric field: $E=1,\ 5,\ 18.6,\ 30\ \mathrm{V}\cdot\mathrm{nm}^{-1}$. + The Keldysh parameter $\gamma=\frac{2\omega}E\sqrt W$ for these fields is, respectively, $18.6$, $3.73$, $1.00$ and $0.621$. + The dotted line represents the phase of the electric field: $\cos(\omega t)$. + As the field increases, fast oscillations in the current appear. + These fast oscillations mostly occur while the field is increasing. + } + \label{fig:current} +\end{figure} + +\indent In Figure \ref{fig:cn}, we show a comparison of our solution with the results obtained from a direct solution of (\ref{schrodinger}) via the Crank-Nicolson algorithm. +The agreement, especially for the location of the maxima and minima after very early times is very good. +At short times, however, the agreement is not so good. +This is expected since the Crank-Nicolson scheme is based on approximating derivatives by finite differences. +However, at short times, $\psi(0,t)\sim t^{\frac32}$, which has a singular second derivative at $0$, so $\partial_t\psi$ is poorly approximated at short times by finite differences. +Note, also, that the Crank-Nicolson method requires a cut-off in $x$, restricting $\psi(x,t)$ to $x\in[-a,a]$. +This causes distortions in $\psi$ due to reflections from these artificial boundaries, and so can only be used for short times (this can be avoided by using non-local boundary conditions, such as ``transparent boundary conditions''). +\bigskip + +\begin{figure} + \includegraphics[width=8cm]{cn.pdf} + \hfill + \includegraphics[width=8cm]{cn_short.pdf} + \caption{ + The current computed with our method (blue, color online), compared with the Crank-Nicolson algorithm (red), for $\omega=1.55\ \mathrm{eV}$ and $E=15\ \mathrm{V}\cdot\mathrm{nm}^{-1}$. + The maxima and minima seem to occur at the same time, and the agreement is somewhat good for $t>\frac\pi\omega$. + The graph on the right focuses on short times, for which the Crank-Nicolson algorithm produces a different, and quite surprising result: the current initially shoots down to negative values before rising back up, but the field is initially positive, which should drive the current up, not down. + The discrepancy is due to the fact that the wavefunction is singular at $t=0$ (it has a divergent second derivative), and that the Crank-Nicolson algorithm approximates derivatives using finite differences, which is good only for regular functions. + } + \label{fig:cn} +\end{figure} + +\indent Also shown in the figures is the running average of the current +\begin{equation} + \left_t:=\frac1\tau\int_{t-\tau}^t j(0,t) + ,\quad + \tau:=\frac{2\pi}\omega + . +\end{equation} +It is seen there that while the current appears to approach its long time value rather rapidly it does not do so fully until much later. +As seen in Figure \ref{fig:avgcurrent}, where we plot $\left$ for $\omega=6\ \mathrm{eV}$, $E=10\ \mathrm{V}\cdot\mathrm{nm}^{-1}$, $\gamma=9.6$, $\left_t$ is still not all that close to its asymptotic value even when $t\approx48\tau$. +\bigskip + +\indent The rate of the decay of the average current to its asymptotic value is evaluated in Figure \ref{fig:decay}. +In order to compute this rate without having to guess the asymptotic value, we proceed as follows. +At the end of every laser period $t_n=\frac{2\pi}\omega n$, we compute the minimal value $\mu_n$ and maximal value $M_n$ of the average current in the period $(\frac{2\pi}\omega(n-1),\frac{2\pi}\omega n]$. +The plot shoes $M_n-\mu_n$ as a function of $t$, and shows that $\left_t\approx\left_\infty+g(t)t^{-\frac32}$, where $g(t)$ is bounded and asymptotically constant. +\bigskip + +\begin{figure} + \includegraphics[width=8cm]{avgcurrent.pdf} + \hfill + \includegraphics[width=8cm]{avgcurrent_zoom.pdf} + \caption{ + The normalized average current $\frac1k\left_t$ as a function if $\frac{t\omega}{2\pi}$ for $\omega=6\ \mathrm{eV}$ and $E=10\ \mathrm{V}\cdot\mathrm{nm}^{-1}$. + The two plots show the same data, but the first one focuses on times between the 12th and 48th period. + Here, $\omega$ is large enough that absorbing one photon suffices to overcome the work function. + Even after 48 periods, the average current has not converged to its asymptotic value. + } + \label{fig:avgcurrent} +\end{figure} + +\begin{figure} + \hfil\includegraphics[width=8cm]{decay.pdf} + \caption{ + The convergence rate to the asymptote of the average current as a function of $\frac{t\omega}{2\pi}$ on a log-log plot for $\omega=6$ and $E=10\ \mathrm{V}\cdot\mathrm{nm}^{-1}$. + The dots are computed at the end of each period $t_n=\frac{2\pi}\omega n$, and their value is the difference between the maximun $M_t$ and the minimum $\mu_t$ of the normalized average current $\frac1k\left_t$ in the period immediately preceding $t_n$. + The red line is a plot of $0.0030\times(\frac{t\omega}{2\pi})^{-\frac32}$, which fits the data rather well. + } + \label{fig:decay} +\end{figure} + +\indent In Figure \ref{fig:omega}, we show an estimate of the asymptotic average current as a function of $\omega$ in the vicinity of the one-photon threshold $\omega_c=W$ for $E=10\ \mathrm{V}\cdot\mathrm{nm}^{-1}$. +As we argued above, the average current converges slowly to its asymptotic value. +In order to estimate this asymptotic value without computing the current at very large times, we actually compute the average of the average current over a laser period: +\begin{equation} + \left<\left\right>:=\frac1\tau\int_{T-\tau}^T dt\ \left_t + . +\end{equation} +We see that there is a steep increase in $\left<\left\right>$ as $\omega$ increases past $\omega_c$. +This is precisely what is observed in experiments on the photoelectric effect, where the emission of electrons from the metal surface has such a threshold \cite{Mi16}. +This became a key element in Einstein's ansatz of localized photons. +Here, in the semi-classical treatment of the laser field, this phenomenon appears as a resonance. +The ponderomotive term $\frac{E^2}{4\omega^2}$, which appears in the photon energy, see below, is negligible compared to the work function $W$. +\bigskip + +\begin{figure} + \hfil\includegraphics[width=8cm]{omega.pdf} + \caption{ + The average of the average of the normalized current $\frac1k\left<\left_t\right>$ after 12 periods as a function of $\omega$, for $E=10\ \mathrm{V}\cdot\mathrm{nm}^{-1}$. + After 12 periods, the average current $\left_t$ is still fluctuating quite a lot, so, in order to eliminate these fluctuations and get closer to the asymptotic value $\left_\infty$, we average the average current over a laser period. + } + \label{fig:omega} +\end{figure} + +\section{Long time behavior of $\psi(x,t)$}\label{longtime} +\indent The long time behavior of the system is given by the poles of the Laplace transform +$$\hat{\psi}(x,p)=\int_0^{\infty}{\rm e}^{-pt} \psi(x,t)\, dt$$ +on the imaginary axis, as can be seen by taking the inverse Laplace transform +\begin{equation} + \psi(x,t)=\int_{-i\infty}^{i\infty}dp\ e^{pt}\hat\psi(x,p) + . +\end{equation} +Other poles, which necessarily have negative real parts, give rise to exponentially decaying terms while branch cuts generally contribute $t^{-\frac32}$ terms to the approach to the asymptotic state. +As mentioned earlier, the time asymptotic of $\psi$ is of the form +\begin{equation}\label{4p1} +\psi(x,t)\sim {\rm e}^{\frac 12 i k^2 t} \bar{\psi}(x,t) +\end{equation} +where $\bar{\psi}(x,t)$ is periodic in $t$ with period $2\pi/\omega$ and $\frac 12 k^2$ is the energy of the electrons in the incoming beam. +This corresponds to the poles of $\hat{\psi}(p,t)$ on the imaginary axis being at +\begin{equation}\label{4p2} +p=-\frac 12 i k^2+i\omega n,\ \ \ n\mathrm{\ \ integer} +\end{equation} +\bigskip + +\indent We further show that $\bar{\psi}(x,t)$ coincides, after the change of gauge, with the wave function $\mathcal{U}(x,t)$ computed by Faisal et al \cite{FKS05}. +In that work it was assumed (\ref{schrodinger}) (in the magnetic gauge) has a solution of the form (\ref{4p1}) with an incoming beam $e^{ikx}$ for $x<0$, without considering any initial conditions. +Computing the residues at the poles on the imaginary axis we show that +\begin{equation}\label{4p3} +\bar\psi(x,t)\sim \left\{\begin{array}{>\displaystyle ll} + e^{ikx} + +\sum_{m\in\mathbb Z}e^{-im\omega t}e^{-ix\sqrt{k^2+2m\omega}}\mathcal R_m + &\mathrm{for\ }x<0 + \\[0.5cm] + e^{i\frac E\omega x\sin\omega t}\sum_{n,m\in\mathbb Z}e^{-in\omega t}g_{n-m}^{(\kappa_m)}e^{-\kappa_mx}\mathcal T_m + &\mathrm{for\ }x>0 + \end{array}\right. +\end{equation} +where +\begin{equation} + \kappa_m=\sqrt{2U-k^2+\frac{E^2}{2\omega^2}-2m\omega} + \label{kappa} +\end{equation} +and +\begin{equation} + g_{n-m}^{(\kappa_m)}=\frac\omega{2\pi}\int_0^{\frac{2\pi}\omega}dt\ e^{-i(n-m)\omega t}e^{\frac{i\frac{E^2}{4\omega^2}}\omega\sin(2\omega t)+\kappa_m\frac{2E}{\omega^2}\cos(\omega t)} + . +\end{equation} +This is exactly of the form considered in \cite{FKS05}. +$\mathcal R_n$ and $\mathcal T_m$ are computed by matching boundary values of $\psi(0,t)$ and $\psi_{x,0}(t)$ at $x=0$. +\bigskip + +\indent A physical interpretation of (\ref{4p3}), see \cite{FKS05}, is that an electron in a beam coming from $-\infty$ and moving in the positive $x$-direction, ${\rm e}^{ikx},\ k>0$, absorbs or emits ``$m$ photons'' and is either reflected, transmitted or damped. +Transmission occurs when $m\omega>U+\frac {E^2}{4\omega^2}-\frac{k^2}2\equiv\omega_c$. +The $\frac{E^2}{4\omega^2}$ in (\ref{kappa}) corresponds to the ponderomotive energy of the electron in the laser field. +Damping occurs when the inequalities are in the opposite direction. +$\omega_c$ is the minimum frequency necessary to push the electron with kinetic energy $\frac12k^2$ (in the $x$-direction) over the potential barrier of height $U-\frac12k^2=W$. +For $x\gg0$, the current in the $m$-photon channel will have kinetic energy $m\omega-\omega_c$ and the current will be given by $\sqrt{m\omega-\omega_c}$. +This will also be equal to the average current at large $t$, which is independent of $x$. +This explains the picture in Figure 4 for $m=1$. +The larger $m$ values necessary for smaller $\omega$ are difficult to see. +\bigskip + +\indent The asymptotic form (\ref{4p3}) is true for all initial conditions of the form $\Theta(-x)e^{ikx}+f_0(s)$ as long as $f_0(x)$ only contains terms which are square integrable. +The additional terms in $\psi(x,t)$ which come from $f_0(x)$ go to zero as $t\to\infty$. +This follows from the fact, proven in \cite{CCe}, that the Floquet operator associated to (\ref{schrodinger}) has no point spectrum. +\bigskip + + + +\vfill + +\delimtitle{\bf Acknowledgements} +We thank David Huse, Kevin Jensen and Donald Schiffler for very valuable discussions. This work was supported by AFOSR Grant No. FA9500-16-1-0037. O.C. was partially supported by the NSF-DMS (Grant No. 1515755). I.J. was partially supported by the NSF-DMS (Grants No. 31128155 and 1802170). J.L.L. thanks the Institute for Advanced Study for its hospitality. +\enddelim + +\eject + +\begin{thebibliography}{WWW99} +\small +\IfFileExists{bibliography/bibliography.tex}{\input bibliography/bibliography.tex}{} +\end{thebibliography} + + +\end{document} + diff --git a/Makefile b/Makefile new file mode 100644 index 0000000..7d888bd --- /dev/null +++ b/Makefile @@ -0,0 +1,55 @@ +PROJECTNAME=$(basename $(wildcard *.tex)) +LIBS=$(notdir $(wildcard libs/*)) +FIGS=$(notdir $(wildcard figs/*.fig)) + +PDFS=$(addsuffix .pdf, $(PROJECTNAME)) +SYNCTEXS=$(addsuffix .synctex.gz, $(PROJECTNAME)) + +all: $(PROJECTNAME) + +$(PROJECTNAME): $(LIBS) $(FIGS) + pdflatex -file-line-error $@.tex + pdflatex -file-line-error $@.tex + pdflatex -synctex=1 $@.tex + +$(PROJECTNAME).aux: $(LIBS) $(FIGS) + pdflatex -file-line-error -draftmode $(PROJECTNAME).tex + + +$(SYNCTEXS): $(LIBS) $(FIGS) + pdflatex -synctex=1 $(patsubst %.synctex.gz, %.tex, $@) + + +libs: $(LIBS) + +$(LIBS): + ln -fs libs/$@ ./ + +figs: $(FIGS) + +$(FIGS): + make -C figs/$@ + for pdf in $$(find figs/$@/ -name '*.pdf'); do ln -fs "$$pdf" ./ ; done + + +clean-aux: clean-figs-aux + rm -f $(addsuffix .aux, $(PROJECTNAME)) + rm -f $(addsuffix .log, $(PROJECTNAME)) + rm -f $(addsuffix .out, $(PROJECTNAME)) + rm -f $(addsuffix .toc, $(PROJECTNAME)) + +clean-libs: + rm -f $(LIBS) + +clean-figs: + $(foreach fig,$(addprefix figs/, $(FIGS)), make -C $(fig) clean; ) + rm -f $(notdir $(wildcard figs/*.fig/*.pdf)) + +clean-figs-aux: + $(foreach fig,$(addprefix figs/, $(FIGS)), make -C $(fig) clean-aux; ) + + +clean-tex: + rm -f $(PDFS) $(SYNCTEXS) + +clean: clean-aux clean-tex clean-libs clean-figs diff --git a/README b/README new file mode 100644 index 0000000..f7fb902 --- /dev/null +++ b/README @@ -0,0 +1,43 @@ +This directory contains the source files to typeset the article, and generate +the figures. This can be accomplished by running + make + +This document uses a custom class file, located in the 'libs' directory, which +defines a number of commands. Most of these are drop-in replacements for those +defined in the 'article' class. + +Some extra functionality is provided in custom style files, located in the +'libs' directory. + +The figures are generated from data computed using the 'photocomp' tool, which +has not been released yet, so the data for the figures is provided here. + + + +* Dependencies: + + pdflatex + TeXlive packages: + amsfonts + array + color + graphics + hyperref + latex + marginnote + pgf + standalone + GNU make + gnuplot + +* Files: + + Costin_Costin_Jauslin_Lebowitz_2019.tex: + main LaTeX file + + libs: + custom LaTeX class file + + figs: + source code for the figures + diff --git a/bibliography/bibliography.tex b/bibliography/bibliography.tex new file mode 100644 index 0000000..da72033 --- /dev/null +++ b/bibliography/bibliography.tex @@ -0,0 +1,46 @@ +\bibitem[Co93]{Co93}P.B. Corkum - {\it Plasma perspective on strong field multiphoton ionization}, Physical Review Letters, volume\-~71, issue\-~13, pages\-~1994-1997, 1993,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRevLett.71.1994}{10.1103/PhysRevLett.71.1994}}.\par\medskip + +\bibitem[CCe]{CCe}O. Costin, R. Costin, I. Jauslin, J.L. Lebowitz, {\it in preparation}.\par\medskip + +\bibitem[CCe18]{CCe18}O. Costin, R. Costin, I. Jauslin, J.L. Lebowitz - {\it Solution of the time dependent Schr\"odinger equation leading to Fowler-Nordheim field emission}, Journal of Applied Physics, volume\-~124, number\-~213104, 2018,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1063/1.5066240}{10.1063/1.5066240}}, arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/1808.00936}{1808.00936}}.\par\medskip + +\bibitem[CN47]{CN47}J. Crank, P. Nicolson - {\it A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type}, Mathematical Proceedings of the Cambridge Philosophical Society, volume\-~43, issue\-~1, pages\-~50-67, 1947,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1017/S0305004100023197}{10.1017/S0305004100023197}}.\par\medskip + +\bibitem[FKS05]{FKS05}F.H.M. Faisal, J.Z. Kami\'nski, E. Saczuk - {\it Photoemission and high-order harmonic generation from solid surfaces in intense laser fields}, Physical Review A, volume\-~72, issue\-~2, number\-~023412, 2005,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRevA.72.023412}{10.1103/PhysRevA.72.023412}}.\par\medskip + +\bibitem[Fo97]{Fo97}A.S. Fokas - {\it A unified transform method for solving linear and certain nonlinear PDEs}, Proceedings of the Royal Society of London A, volume\-~453, issue\-~1962, pages\-~1411-1443, 1997,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1098/rspa.1997.0077}{10.1098/rspa.1997.0077}}.\par\medskip + +\bibitem[Fo16]{Fo16}R.G. Forbes - {\it Field Electron Emission Theory}, Proceedings of Young Researchers in Vacuum Micro/Nano Electronics, IEEE, 2016,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1109/VMNEYR.2016.7880403}{10.1109/VMNEYR.2016.7880403}}, arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/1801.08251}{1801.08251}}.\par\medskip + +\bibitem[FN28]{FN28}R.H. Fowler, L. Nordheim - {\it Electron emission in intense electric fields}, Proceedings of the Royal Society of London A, volume\-~119, issue\-~781, pages\-~173-181, 1928,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1098/rspa.1928.0091}{10.1098/rspa.1928.0091}}.\par\medskip + +\bibitem[Je17]{Je17}K.L. Jensen - {\it Introduction to the Physics of Electron Emission}, Wiley, 2017.\par\medskip + +\bibitem[KSK92]{KSK92}J.L. Krause, K.J. Schafer, K.C. Kulander - {\it High-order harmonic generation from atoms and ions in the high intensity regime}, Physical Review Letters, volume\-~68, issue\-~24, pages\-~3535-3538, 1992,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRevLett.68.3535}{10.1103/PhysRevLett.68.3535}}.\par\medskip + +\bibitem[KSH11]{KSH11}M. Kr\"uger, M. Schenk, P. Hommelhoff - {\it Attosecond control of electrons emitted from a nanoscale metal tip}, Nature, volume\-~475, issue\-~7354, pages\-~78-81, 2011,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1038/nature10196}{10.1038/nature10196}}, arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/1107.1591}{1107.1591}}.\par\medskip + +\bibitem[KLe18]{KLe18}M. Kr\"uger, C. Lemell, G. Wachter, J. Burgdörfer, P. Hommelhoff - {\it Attosecond physics phenomena at nanometric tips}, Journal of Physics B: Atomic, Molecular and Optical Physics, volume\-~51, issue\-~17, number\-~172001, 2018,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1088/1361-6455/aac6ac}{10.1088/1361-6455/aac6ac}}.\par\medskip + +\bibitem[Mi16]{Mi16}R.A. Millikan - {\it A Direct Photoelectric Determination of Planck's $h$}, Physical Review, volume\-~7, issue\-~3, pages\-~355-388, 1916,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRev.7.355}{10.1103/PhysRev.7.355}}.\par\medskip + +\bibitem[Ry18]{Ry18}Y. Rybalko - {\it Initial value problem for the time-dependent linear Schrodinger equation with a point singular potential by the unified transform method}, Opuscula Mathematica, volume\-~38, number\-~6, pages\-~883-898, 2018,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.7494/OpMath.2018.38.6.883}{10.7494/OpMath.2018.38.6.883}}, arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/1601.08008}{1601.08008}}.\par\medskip + +\bibitem[YGR11]{YGR11}S.V. Yalunin, M. Gulde, C. Ropers - {\it Strong-field photoemission from surfaces: Theoretical approaches}, Physical Review B, volume\-~84, issue\-~19, number\-~195426, 2011,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRevB.84.195426}{10.1103/PhysRevB.84.195426}}.\par\medskip + +\bibitem[ZL16]{ZL16}P. Zhang, Y.Y. Lau - {\it Ultrafast strong-field photoelectron emission from biased metal surfaces: exact solution to time-dependent Schr\"odinger Equation}, Scientific Reports, volume\-~6, number\-~19894, 2016,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1038/srep19894}{10.1038/srep19894}}.\par\medskip + diff --git a/figs/avgcurrent.fig/Makefile b/figs/avgcurrent.fig/Makefile new file mode 100644 index 0000000..9bb20af --- /dev/null +++ b/figs/avgcurrent.fig/Makefile @@ -0,0 +1,39 @@ +PROJECTNAME=avgcurrent avgcurrent_zoom decay + +PHOTOCOMP_PATH= + +DATS=avgcurrent.dat decay.dat + +PDFS=$(addsuffix .pdf, $(PROJECTNAME)) +TEXS=$(addsuffix .tikz.tex, $(PROJECTNAME)) + +all: $(PDFS) + +$(PDFS): $(DATS) + gnuplot $(patsubst %.pdf, %.gnuplot, $@) > $(patsubst %.pdf, %.tikz.tex, $@) + pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@) + +avgcurrent.dat: save.dat + $(PHOTOCOMP_PATH)photocomp -p "V=10;EF=4.5;omega=6;n_periods=48;J=480;E=10e9" mc -s save.dat > $@ + +decay.dat: save.dat + $(PHOTOCOMP_PATH)photocomp -p "V=10;EF=4.5;omega=6;n_periods=48;J=480;E=10e9" mc_decay -s save.dat > $@ + +save.dat: + $(PHOTOCOMP_PATH)photocomp -p "V=10;EF=4.5;omega=6;n_periods=48;J=480;E=10e9" save > $@ + +install: $(PDFS) + cp $^ $(INSTALLDIR)/ + +clean-aux: + rm -f $(addsuffix .tikz.tex, $(PROJECTNAME)) + rm -f $(addsuffix .aux, $(PROJECTNAME)) + rm -f $(addsuffix .log, $(PROJECTNAME)) + +clean-dat: + rm -f $(DATS) save.dat + +clean-tex: + rm -f $(PDFS) + +clean: clean-aux clean-tex diff --git a/figs/avgcurrent.fig/avgcurrent.dat b/figs/avgcurrent.fig/avgcurrent.dat new file mode 100644 index 0000000..08d5da0 --- /dev/null +++ b/figs/avgcurrent.fig/avgcurrent.dat @@ -0,0 +1,9606 @@ +#parameters: V_=1.000000000000000e+01 eV, EF=4.500000000000000e+00 eV, omega_=6.000000000000000e+00 eV, theta= 0.000000000000000e+00*pi W=5.500000000000000e+00 eV, n_photons=0.934268975891369, E=1.000000000000000e+10 V/m +#dimensionless: V=1.000000000000000e+00 k=9.486832980505138e-01 epsilon=7.266837529084202e-02 omega=6.000000000000000e-01 m=1.000000000000000e+00, keldysh=7.216404962767393e+00 +#numerical precision (matching): n_periods=48 NC=12 gl_order=16 J0=1 J=480 +#numerical precision (Crank-Nicolson): bmin=-2.000000000000000e+02 bmax= 2.000000000000000e+02 nx_cn=4000 mu= 1.000000000000000e-01 +# +#% t real(psi0) imag(psi0) abs(psi0)^2 current avgcurrent + + 4.999999999999999e-03 9.232513943769471e-01 -9.731954366777020e-01 1.799502495189280e+00 7.861953596498260e-03 0.000000000000000e+00 + 9.999999999999998e-03 9.460653600161577e-01 -9.506697267880593e-01 1.798812594853786e+00 1.501065517229972e-02 0.000000000000000e+00 + 1.500000000000000e-02 9.684052903773249e-01 -9.275238780940320e-01 1.798109350865384e+00 2.164996006301676e-02 0.000000000000000e+00 + 2.000000000000000e-02 9.902450989905524e-01 -9.038038396166203e-01 1.797446736580554e+00 2.784688372657630e-02 0.000000000000000e+00 + 2.500000000000000e-02 1.011562890331630e+00 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-1.063163957623890e+00 1.727479004492896e+00 2.659098426801894e-01 1.340021965799104e-03 + 4.797499999999960e+01 7.993700305732271e-01 -1.045072559525161e+00 1.731169100451113e+00 2.644404668560649e-01 1.339960860837563e-03 + 4.797999999999959e+01 8.255282292654210e-01 -1.026351929635869e+00 1.734895140781373e+00 2.627774359821951e-01 1.339903198757013e-03 + 4.798499999999959e+01 8.512215259205537e-01 -1.007013066963587e+00 1.738653403225925e+00 2.609242304260652e-01 1.339849010094743e-03 + 4.798999999999960e+01 8.764349534772510e-01 -9.870673804314336e-01 1.742440241188443e+00 2.588842832555354e-01 1.339798353285120e-03 + 4.799499999999959e+01 9.011538503483205e-01 -9.665266802899350e-01 1.746252085709886e+00 2.566609736439400e-01 1.339751147769178e-03 diff --git a/figs/avgcurrent.fig/avgcurrent.gnuplot b/figs/avgcurrent.fig/avgcurrent.gnuplot new file mode 100644 index 0000000..8f4059c --- /dev/null +++ b/figs/avgcurrent.fig/avgcurrent.gnuplot @@ -0,0 +1,24 @@ +set ylabel "$\\frac1k\\left_t$" norotate +set xlabel "$\\frac{t\\omega}{2\\pi}$" + +set xtics 0,12 +set ytics 0,0.001 + +set xrange [0:48] + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 8,6 standalone + +# no key +set key off + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 3 +set style line 2 linetype rgbcolor "#DC143C" linewidth 3 +set style line 3 linetype rgbcolor "#32CD32" linewidth 3 +set style line 4 linetype rgbcolor "#4B0082" linewidth 3 +set style line 5 linetype rgbcolor "#DAA520" linewidth 3 + +set pointsize 0.6 + +plot "avgcurrent.dat" using 1:($1>1 ? $6 : 1/0) with lines linestyle 1 diff --git a/figs/avgcurrent.fig/avgcurrent_zoom.gnuplot b/figs/avgcurrent.fig/avgcurrent_zoom.gnuplot new file mode 100644 index 0000000..d8511c4 --- /dev/null +++ b/figs/avgcurrent.fig/avgcurrent_zoom.gnuplot @@ -0,0 +1,25 @@ +set ylabel "$\\frac1k\\left_t$" norotate +set xlabel "$\\frac{t\\omega}{2\\pi}$" + +set xtics 0,12 +set ytics 0.00130,0.00004 + +set yrange [0.00128:0.00138] +set xrange [12:48] + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 8,6 standalone + +# no key +set key off + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 3 +set style line 2 linetype rgbcolor "#DC143C" linewidth 3 +set style line 3 linetype rgbcolor "#32CD32" linewidth 3 +set style line 4 linetype rgbcolor "#4B0082" linewidth 3 +set style line 5 linetype rgbcolor "#DAA520" linewidth 3 + +set pointsize 0.6 + +plot "avgcurrent.dat" using 1:6 with lines linestyle 1 diff --git a/figs/avgcurrent.fig/decay.dat b/figs/avgcurrent.fig/decay.dat new file mode 100644 index 0000000..0ee6eb7 --- /dev/null +++ b/figs/avgcurrent.fig/decay.dat @@ -0,0 +1,53 @@ +#parameters: V_=1.000000000000000e+01 eV, EF=4.500000000000000e+00 eV, omega_=6.000000000000000e+00 eV, theta= 0.000000000000000e+00*pi W=5.500000000000000e+00 eV, n_photons=0.934268975891369, E=1.000000000000000e+10 V/m +#dimensionless: V=1.000000000000000e+00 k=9.486832980505138e-01 epsilon=7.266837529084202e-02 omega=6.000000000000000e-01 m=1.000000000000000e+00, keldysh=7.216404962767393e+00 +#numerical precision (matching): n_periods=48 NC=12 gl_order=16 J0=1 J=480 +#numerical precision (Crank-Nicolson): bmin=-2.000000000000000e+02 bmax= 2.000000000000000e+02 nx_cn=4000 mu= 1.000000000000000e-01 +# +# t mu_t M_t + 2.005000000000000e+00 -4.465249574292407e-05 2.892008997946130e-03 + 3.004999999999999e+00 6.806473586532007e-04 1.611344559925361e-03 + 4.004999999999997e+00 9.531666802876900e-04 1.430074084610480e-03 + 5.004999999999996e+00 1.112095215394091e-03 1.415755296437017e-03 + 6.004999999999995e+00 1.178723364609898e-03 1.434548730775139e-03 + 7.004999999999999e+00 1.230653978557152e-03 1.451154221498075e-03 + 8.000000000000005e+00 1.284082546921191e-03 1.456183724818763e-03 + 9.000000000000011e+00 1.320101942418750e-03 1.449409650442600e-03 + 1.000000000000002e+01 1.332128084560356e-03 1.433544083730075e-03 + 1.100000000000002e+01 1.328615144193408e-03 1.411834798831924e-03 + 1.200000000000003e+01 1.317802662449091e-03 1.388121233446665e-03 + 1.300000000000003e+01 1.305974610489081e-03 1.375961738972150e-03 + 1.400000000000004e+01 1.297012109589079e-03 1.364747654234101e-03 + 1.500000000000004e+01 1.292775935877827e-03 1.351347207493206e-03 + 1.600000000000005e+01 1.293822677757008e-03 1.343607304368543e-03 + 1.700000000000005e+01 1.299820410514768e-03 1.342878973466090e-03 + 1.800000000000006e+01 1.309385031252287e-03 1.347024476018631e-03 + 1.900000000000006e+01 1.320173072770978e-03 1.353534908845761e-03 + 2.000000000000007e+01 1.324465132367436e-03 1.360198367149491e-03 + 2.100000000000008e+01 1.330100620972184e-03 1.365310202042966e-03 + 2.200000000000008e+01 1.336044821630408e-03 1.367671972762798e-03 + 2.300000000000009e+01 1.338710049596377e-03 1.366696969752792e-03 + 2.400000000000009e+01 1.337973002387375e-03 1.362663493743734e-03 + 2.500000000000009e+01 1.334759175167112e-03 1.356744645818275e-03 + 2.600000000000006e+01 1.330279210201444e-03 1.350842056746174e-03 + 2.700000000000004e+01 1.325840812522160e-03 1.348671983114142e-03 + 2.800000000000002e+01 1.322654598995199e-03 1.345258918518337e-03 + 2.900500000000000e+01 1.321582620668878e-03 1.341996494328002e-03 + 3.000499999999998e+01 1.322844314348249e-03 1.340940035684045e-03 + 3.100499999999995e+01 1.325948216456572e-03 1.342052212740544e-03 + 3.200499999999993e+01 1.329986989195847e-03 1.344786288894575e-03 + 3.300499999999991e+01 1.333421117850701e-03 1.348239586830857e-03 + 3.400499999999989e+01 1.334922273485275e-03 1.351365520831539e-03 + 3.500499999999987e+01 1.337371756144218e-03 1.353285967255712e-03 + 3.600499999999985e+01 1.339340919372187e-03 1.353558131763480e-03 + 3.700499999999982e+01 1.339607534213708e-03 1.352269215976699e-03 + 3.800499999999980e+01 1.338259311876398e-03 1.349860941552041e-03 + 3.900499999999978e+01 1.335843353599698e-03 1.346881687730867e-03 + 4.000499999999976e+01 1.333173924311831e-03 1.344961245451776e-03 + 4.100499999999974e+01 1.331033693671063e-03 1.343619013767633e-03 + 4.200499999999972e+01 1.329937987977380e-03 1.341614035795568e-03 + 4.300499999999970e+01 1.330038327956477e-03 1.340500676012152e-03 + 4.400499999999968e+01 1.331216650002774e-03 1.340788379664775e-03 + 4.500499999999965e+01 1.333210987171334e-03 1.342234189388972e-03 + 4.600499999999963e+01 1.335622628018670e-03 1.344273805075085e-03 + 4.700499999999961e+01 1.336632341415729e-03 1.346277075142191e-03 + 4.799999999999959e+01 1.337964973402016e-03 1.347746441811054e-03 diff --git a/figs/avgcurrent.fig/decay.gnuplot b/figs/avgcurrent.fig/decay.gnuplot new file mode 100644 index 0000000..f50499d --- /dev/null +++ b/figs/avgcurrent.fig/decay.gnuplot @@ -0,0 +1,34 @@ +set ylabel offset 5,0 "$M_t-\\mu_t$" +set xlabel "$\\frac{t\\omega}{2\\pi}$" + +set xtics add("10" 10, "20" 20, "30" 30, "40" 40) +set mxtics 81 +set ytics add ("$10^{-5}$" 0.00001, "$10^{-4}$" 0.0001) + +set xrange [10:50] +set yrange [0.000007:0.00015] + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 8,6 standalone + +# no key +set key off + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 3 +set style line 2 linetype rgbcolor "#DC143C" linewidth 3 +set style line 3 linetype rgbcolor "#32CD32" linewidth 3 +set style line 4 linetype rgbcolor "#4B0082" linewidth 3 +set style line 5 linetype rgbcolor "#DAA520" linewidth 3 + +set pointsize 0.6 + +# logscale +set logscale xy + +# fit +f(x)=-1.5*x+b + +fit [1:] f(x) "decay.dat" u (log10($1)):(log10($3-$2)) via b + +plot 10**f(log10(x)) linestyle 2, "decay.dat" using 1:($3-$2) with points linestyle 1 diff --git a/figs/avgcurrent.fig/save.dat b/figs/avgcurrent.fig/save.dat new file mode 100644 index 0000000..6756e24 --- /dev/null +++ b/figs/avgcurrent.fig/save.dat @@ -0,0 +1,487 @@ +#parameters: V_=1.000000000000000e+01 eV, EF=4.500000000000000e+00 eV, omega_=6.000000000000000e+00 eV, theta= 0.000000000000000e+00*pi W=5.500000000000000e+00 eV, n_photons=0.934268975891369, E=1.000000000000000e+10 V/m +#dimensionless: V=1.000000000000000e+00 k=9.486832980505138e-01 epsilon=7.266837529084202e-02 omega=6.000000000000000e-01 m=1.000000000000000e+00, keldysh=7.216404962767393e+00 +#numerical precision (matching): n_periods=48 NC=12 gl_order=16 J0=1 J=480 +#numerical precision (Crank-Nicolson): bmin=-2.000000000000000e+02 bmax= 2.000000000000000e+02 nx_cn=4000 mu= 1.000000000000000e-01 +# +480 +12 7.055350578843251e-01 -1.118122180108418e+00 -2.640685032764412e-01 -1.553800164380189e-01 -7.304626492933619e-02 -2.573420862470561e-02 -3.786290652531523e-03 7.460077530076078e-03 -2.817454264016051e-04 1.011443432259879e-03 7.164966002685155e-05 6.926345132725265e-05 1.103362165328192e-05 6.706450770592399e-06 1.357417921660424e-06 -2.541372315622481e-07 1.322515969230063e-07 -1.474041749784050e-07 -1.618720930368805e-09 -2.096450428357976e-08 -2.062004079557157e-09 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%.tikz.tex, $@) + +mc.dat: + $(PHOTOCOMP_PATH)photocomp -p "V=10;EF=4.5;omega=1.55;n_periods=3;J=30;E=15e9" mc > $@ +cn.dat: + $(PHOTOCOMP_PATH)photocomp -p "V=10;EF=4.5;omega=1.55;n_periods=3;E=15e9;x=0;bmin_cn=-1000;bmax_cn=1000;nx_cn=16000" cn > $@ + +mc_short.dat: + $(PHOTOCOMP_PATH)photocomp -p "V=10;EF=4.5;omega=1.55;n_periods=0.0005;J=10;E=15e9" mc > $@ +cn_short.dat: + $(PHOTOCOMP_PATH)photocomp -p "V=10;EF=4.5;omega=1.55;n_periods=0.0005;E=15e9;x=0;bmin_cn=-1000;bmax_cn=1000;nx_cn=16000" cn > $@ + +install: $(PDFS) + cp $^ $(INSTALLDIR)/ + +clean-aux: + rm -f $(addsuffix .tikz.tex, $(PROJECTNAME)) + rm -f $(addsuffix .aux, $(PROJECTNAME)) + rm -f $(addsuffix .log, $(PROJECTNAME)) + +clean-dat: + rm -f $(DATS) + +clean-tex: + rm -f $(PDFS) + +clean: clean-aux clean-tex diff --git a/figs/cn.fig/cn.dat b/figs/cn.fig/cn.dat new file mode 100644 index 0000000..b414f51 --- /dev/null +++ b/figs/cn.fig/cn.dat @@ -0,0 +1,9361 @@ +#parameters: V_=1.000000000000000e+01 eV, EF=4.500000000000000e+00 eV, omega_=1.550000000000000e+00 eV, theta= 0.000000000000000e+00*pi W=5.500000000000000e+00 eV, n_photons=5.845650641658291, E=1.500000000000000e+10 V/m +#dimensionless: V=1.000000000000000e+00 k=9.486832980505138e-01 epsilon=4.219454049145666e-01 omega=1.550000000000000e-01 m=1.000000000000000e+00, keldysh=1.242825299143273e+00 +#numerical precision (matching): n_periods=3 NC=12 gl_order=16 J0=1 J=120 +#numerical precision (Crank-Nicolson): bmin=-1.000000000000000e+03 bmax= 1.000000000000000e+03 nx_cn=16000 mu= 1.000000000000000e-01 +# +# t |psi|^2 current + 0.000000000000000e+00 1.799999999999999e+00 -2.340555645717801e-16 + 3.206999481946238e-04 1.791089758250680e+00 -1.855588686813462e-02 + 6.413998963892475e-04 1.776683425298279e+00 -2.129933323183431e-02 + 9.620998445838713e-04 1.772394942684849e+00 -1.627657251041743e-02 + 1.282799792778495e-03 1.772946054279545e+00 -1.774460107562316e-02 + 1.603499740973119e-03 1.767449664422158e+00 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6.866483146544486e-02 + 2.999506615464316e+00 1.648801052887924e+00 6.871374182505140e-02 + 2.999827315412511e+00 1.648924990199241e+00 6.875922660174547e-02 diff --git a/figs/cn.fig/cn.gnuplot b/figs/cn.fig/cn.gnuplot new file mode 100644 index 0000000..01f3abc --- /dev/null +++ b/figs/cn.fig/cn.gnuplot @@ -0,0 +1,25 @@ +set ylabel "$\\frac jk$" norotate +set xlabel "$\\frac{t\\omega}{2\\pi}$" + +set xtics 0,0.5 +set ytics -0.24,0.12 + +set yrange [-0.24:0.24] + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 8,6 standalone + +# no key +set key off + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 3 +set style line 2 linetype rgbcolor "#DC143C" linewidth 3 +set style line 3 linetype rgbcolor "#32CD32" linewidth 3 +set style line 4 linetype rgbcolor "#4B0082" linewidth 3 +set style line 5 linetype rgbcolor "#DAA520" linewidth 3 + +set pointsize 0.6 + +plot "mc.dat" using 1:5 with lines linestyle 1, \ +"cn.dat" using 1:3 with lines linestyle 2 diff --git a/figs/cn.fig/cn_short.dat b/figs/cn.fig/cn_short.dat new file mode 100644 index 0000000..837fa1f --- /dev/null +++ b/figs/cn.fig/cn_short.dat @@ -0,0 +1,26 @@ +#parameters: V_=1.000000000000000e+01 eV, EF=4.500000000000000e+00 eV, omega_=1.550000000000000e+00 eV, theta= 0.000000000000000e+00*pi W=5.500000000000000e+00 eV, n_photons=5.845650641658291, E=1.500000000000000e+10 V/m +#dimensionless: V=1.000000000000000e+00 k=9.486832980505138e-01 epsilon=4.219454049145666e-01 omega=1.550000000000000e-01 m=1.000000000000000e+00, keldysh=1.242825299143273e+00 +#numerical precision (matching): n_periods=0 NC=12 gl_order=16 J0=1 J=120 +#numerical precision (Crank-Nicolson): bmin=-1.000000000000000e+03 bmax= 1.000000000000000e+03 nx_cn=16000 mu= 1.000000000000000e-01 +# +# t |psi|^2 current + 0.000000000000000e+00 1.799999999999999e+00 -2.340555645717801e-16 + 2.631578947368421e-05 1.799930823214389e+00 -1.844039686300892e-03 + 5.263157894736842e-05 1.799724094980058e+00 -3.673701905409824e-03 + 7.894736842105265e-05 1.799382209921303e+00 -5.474798810937662e-03 + 1.052631578947368e-04 1.798909120120437e+00 -7.233518956906041e-03 + 1.315789473684211e-04 1.798310277327859e+00 -8.936608439940258e-03 + 1.578947368421053e-04 1.797592553339864e+00 -1.057154369733800e-02 + 1.842105263157895e-04 1.796764139802528e+00 -1.212669338626712e-02 + 2.105263157894737e-04 1.795834429023747e+00 -1.359146694241804e-02 + 2.368421052631579e-04 1.794813877673240e+00 -1.495644762705038e-02 + 2.631578947368422e-04 1.793713855517177e+00 -1.621350811654198e-02 + 2.894736842105263e-04 1.792546481565754e+00 -1.735590696406148e-02 + 3.157894736842106e-04 1.791324450204648e+00 -1.837836456445107e-02 + 3.421052631578947e-04 1.790060850031863e+00 -1.927711757587828e-02 + 3.684210526315790e-04 1.788768978227520e+00 -2.004995108983447e-02 + 3.947368421052632e-04 1.787462153344070e+00 -2.069620818973520e-02 + 4.210526315789474e-04 1.786153529417339e+00 -2.121677689112625e-02 + 4.473684210526316e-04 1.784855914264545e+00 -2.161405480851675e-02 + 4.736842105263158e-04 1.783581594754897e+00 -2.189189223885764e-02 + 5.000000000000000e-04 1.782342171712728e+00 -2.205551468437851e-02 diff --git a/figs/cn.fig/cn_short.gnuplot b/figs/cn.fig/cn_short.gnuplot new file mode 100644 index 0000000..fc98b65 --- /dev/null +++ b/figs/cn.fig/cn_short.gnuplot @@ -0,0 +1,25 @@ +set ylabel "$\\frac jk$" norotate +set xlabel "$\\frac{t\\omega}{2\\pi}$" + +set xtics 0,0.00025 +set ytics -0.025,0.005 + +set yrange [-0.025:0.005] + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 8,6 standalone + +# no key +set key off + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 3 +set style line 2 linetype rgbcolor "#DC143C" linewidth 3 +set style line 3 linetype rgbcolor "#32CD32" linewidth 3 +set style line 4 linetype rgbcolor "#4B0082" linewidth 3 +set style line 5 linetype rgbcolor "#DAA520" linewidth 3 + +set pointsize 0.6 + +plot "mc_short.dat" using 1:5 with lines linestyle 1, \ +"cn_short.dat" using 1:3 with lines linestyle 2 diff --git a/figs/cn.fig/mc.dat b/figs/cn.fig/mc.dat new file mode 100644 index 0000000..f04d9fc --- /dev/null +++ b/figs/cn.fig/mc.dat @@ -0,0 +1,606 @@ +#parameters: V_=1.000000000000000e+01 eV, EF=4.500000000000000e+00 eV, omega_=1.550000000000000e+00 eV, theta= 0.000000000000000e+00*pi W=5.500000000000000e+00 eV, n_photons=5.845650641658291, E=1.500000000000000e+10 V/m +#dimensionless: V=1.000000000000000e+00 k=9.486832980505138e-01 epsilon=4.219454049145666e-01 omega=1.550000000000000e-01 m=1.000000000000000e+00, keldysh=1.242825299143273e+00 +#numerical precision (matching): n_periods=3 NC=12 gl_order=16 J0=1 J=30 +#numerical precision (Crank-Nicolson): bmin=-2.000000000000000e+02 bmax= 2.000000000000000e+02 nx_cn=4000 mu= 1.000000000000000e-01 +# +#% t real(psi0) imag(psi0) abs(psi0)^2 current avgcurrent + + 5.000000000000000e-03 9.011237076219065e-01 -9.920990592322099e-01 1.796284479767687e+00 4.066156482056959e-02 0.000000000000000e+00 + 1.000000000000000e-02 9.036959786184608e-01 -9.882977598906416e-01 1.793398883976038e+00 7.201570001853567e-02 0.000000000000000e+00 + 1.500000000000000e-02 9.071741251355965e-01 -9.846371696331014e-01 1.792475249138620e+00 9.663201291162658e-02 0.000000000000000e+00 + 2.000000000000000e-02 9.112296705251185e-01 -9.813857962466512e-01 1.793457593519989e+00 1.156836312659321e-01 0.000000000000000e+00 + 2.500000000000000e-02 9.156315488882216e-01 -9.786363650543179e-01 1.796110268326171e+00 1.300140368375609e-01 0.000000000000000e+00 + 3.000000000000000e-02 9.202092406537398e-01 -9.764097761814351e-01 1.800161097607213e+00 1.403024957858335e-01 0.000000000000000e+00 + 3.500000000000000e-02 9.248342007156113e-01 -9.746926619406401e-01 1.805344084054215e+00 1.471214533034481e-01 0.000000000000000e+00 + 4.000000000000000e-02 9.294087151999519e-01 -9.734546205103959e-01 1.811414458082664e+00 1.509624712076890e-01 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1.882101942762323e+00 6.408840260766342e-02 3.342071335995561e-05 + 2.979999999999999e+00 9.521767337989366e-01 -9.879690158146953e-01 1.882723308597869e+00 6.833843556032662e-02 3.335532641448861e-05 + 2.984999999999999e+00 9.537276700345069e-01 -9.869772183405738e-01 1.883720498112746e+00 7.178899557659296e-02 3.338063211604125e-05 + 2.989999999999999e+00 9.553098287718852e-01 -9.861250470424733e-01 1.885059477352689e+00 7.444714985380274e-02 3.345251436762203e-05 + 2.994999999999999e+00 9.569022877989215e-01 -9.854143472756992e-01 1.886703424211602e+00 7.634448274519776e-02 3.362352834079004e-05 diff --git a/figs/cn.fig/mc_short.dat b/figs/cn.fig/mc_short.dat new file mode 100644 index 0000000..ca68146 --- /dev/null +++ b/figs/cn.fig/mc_short.dat @@ -0,0 +1,206 @@ +#parameters: V_=1.000000000000000e+01 eV, EF=4.500000000000000e+00 eV, omega_=1.550000000000000e+00 eV, theta= 0.000000000000000e+00*pi W=5.500000000000000e+00 eV, n_photons=5.845650641658291, E=1.500000000000000e+10 V/m +#dimensionless: V=1.000000000000000e+00 k=9.486832980505138e-01 epsilon=4.219454049145666e-01 omega=1.550000000000000e-01 m=1.000000000000000e+00, keldysh=1.242825299143273e+00 +#numerical precision (matching): n_periods=0 NC=12 gl_order=16 J0=1 J=10 +#numerical precision (Crank-Nicolson): bmin=-2.000000000000000e+02 bmax= 2.000000000000000e+02 nx_cn=4000 mu= 1.000000000000000e-01 +# +#% t real(psi0) imag(psi0) abs(psi0)^2 current avgcurrent + + 2.500000000000000e-06 9.000000026987423e-01 -9.949873899004303e-01 1.799999910918607e+00 2.505123638021616e-05 0.000000000000000e+00 + 5.000000000000000e-06 9.000000079987396e-01 -9.949873039976499e-01 1.799999749514243e+00 5.001970807803625e-05 0.000000000000000e+00 + 7.499999999999999e-06 9.000000152074226e-01 -9.949871931480786e-01 1.799999541902053e+00 7.493428466466743e-05 0.000000000000000e+00 + 1.000000000000000e-05 9.000000240763519e-01 -9.949870622582740e-01 1.799999297398785e+00 9.980526327470780e-05 0.000000000000000e+00 + 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4.629722951322276e-03 0.000000000000000e+00 + 4.900000000000001e-04 9.000176156326871e-01 -9.948703373920385e-01 1.799798696671696e+00 4.652744483891761e-03 0.000000000000000e+00 + 4.925000000000001e-04 9.000177762375979e-01 -9.948694742552578e-01 1.799797268348604e+00 4.675760408109110e-03 0.000000000000000e+00 + 4.950000000000001e-04 9.000179374965607e-01 -9.948686092706376e-01 1.799795837527656e+00 4.698770740000343e-03 0.000000000000000e+00 + 4.975000000000001e-04 9.000180994089861e-01 -9.948677424445761e-01 1.799794404220531e+00 4.721775500910389e-03 0.000000000000000e+00 diff --git a/figs/current.fig/Makefile b/figs/current.fig/Makefile new file mode 100644 index 0000000..b80c183 --- /dev/null +++ b/figs/current.fig/Makefile @@ -0,0 +1,38 @@ +PROJECTNAME=current01 current05 current18 current30 + +PHOTOCOMP_PATH= + +DATS=mc01.dat mc05.dat mc18.dat mc30.dat +PDFS=$(addsuffix .pdf, $(PROJECTNAME)) +TEXS=$(addsuffix .tikz.tex, $(PROJECTNAME)) + +all: $(PDFS) + +$(PDFS): $(DATS) + gnuplot $(patsubst %.pdf, %.gnuplot, $@) > $(patsubst %.pdf, %.tikz.tex, $@) + pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@) + +mc01.dat: + $(PHOTOCOMP_PATH)photocomp -p "V=10;EF=4.5;omega=1.55;n_periods=3;J=30;E=1e9" mc > $@ +mc05.dat: + $(PHOTOCOMP_PATH)photocomp -p "V=10;EF=4.5;omega=1.55;n_periods=3;J=30;E=5e9" mc > $@ +mc18.dat: + $(PHOTOCOMP_PATH)photocomp -p "V=10;EF=4.5;omega=1.55;n_periods=3;J=60;E=1.8642379487149097e10" mc > $@ +mc30.dat: + $(PHOTOCOMP_PATH)photocomp -p "V=10;EF=4.5;omega=1.55;n_periods=3;J=60;E=30e9" mc > $@ + +install: $(PDFS) + cp $^ $(INSTALLDIR)/ + +clean-aux: + rm -f $(addsuffix .tikz.tex, $(PROJECTNAME)) + rm -f $(addsuffix .aux, $(PROJECTNAME)) + rm -f $(addsuffix .log, $(PROJECTNAME)) + +clean-dat: + rm -f $(DATS) + +clean-tex: + rm -f $(PDFS) + +clean: clean-aux clean-tex diff --git a/figs/current.fig/current01.gnuplot b/figs/current.fig/current01.gnuplot new file mode 100644 index 0000000..f4b179e --- /dev/null +++ b/figs/current.fig/current01.gnuplot @@ -0,0 +1,29 @@ +set title "$E=1\\ \\mathrm{V}\\cdot\\mathrm{nm}^{-1}$" +set ylabel "$\\frac jk$" norotate +set xlabel "$\\frac{t\\omega}{2\\pi}$" +# +## start ticks at 0, then every x +set xtics 0,1 +set ytics -0.008,0.004 + +set yrange [-0.011:0.011] +set y2range [-1.4:1.4] + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 8,6 standalone + +# no key +set key off + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 3 +set style line 2 linetype rgbcolor "#DC143C" linewidth 3 +set style line 3 linetype rgbcolor "#000000" linewidth 1 +set style line 4 linetype rgbcolor "#4B0082" linewidth 3 +set style line 5 linetype rgbcolor "#DAA520" linewidth 3 + +set pointsize 0.6 + +plot 0 with lines linestyle 3 ,\ + cos(2*pi*x) axes x1y2 linetype 0 linewidth 2,\ + "mc01.dat" using 1:5 with lines linestyle 1 diff --git a/figs/current.fig/current05.gnuplot b/figs/current.fig/current05.gnuplot new file mode 100644 index 0000000..2396021 --- /dev/null +++ b/figs/current.fig/current05.gnuplot @@ -0,0 +1,29 @@ +set title "$E=5\\ \\mathrm{V}\\cdot\\mathrm{nm}^{-1}$" +set ylabel "$\\frac jk$" norotate +set xlabel "$\\frac{t\\omega}{2\\pi}$" +# +## start ticks at 0, then every x +set xtics 0,1 +set ytics -0.04,0.02 + +set yrange [-0.05:0.05] +set y2range [-1.4:1.4] + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 8,6 standalone + +# no key +set key off + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 3 +set style line 2 linetype rgbcolor "#DC143C" linewidth 3 +set style line 3 linetype rgbcolor "#000000" linewidth 1 +set style line 4 linetype rgbcolor "#4B0082" linewidth 3 +set style line 5 linetype rgbcolor "#DAA520" linewidth 3 + +set pointsize 0.6 + +plot 0 with lines linestyle 3 ,\ + cos(2*pi*x) axes x1y2 linetype 0 linewidth 2,\ + "mc05.dat" using 1:5 with lines linestyle 1 diff --git a/figs/current.fig/current18.gnuplot b/figs/current.fig/current18.gnuplot new file mode 100644 index 0000000..46e8805 --- /dev/null +++ b/figs/current.fig/current18.gnuplot @@ -0,0 +1,29 @@ +set title "$E=18.6\\ \\mathrm{V}\\cdot\\mathrm{nm}^{-1}$" +set ylabel "$\\frac jk$" norotate +set xlabel "$\\frac{t\\omega}{2\\pi}$" +# +## start ticks at 0, then every x +set xtics 0,1 +set ytics -0.4,0.2 + +set yrange [-0.4:0.4] +set y2range [-1.4:1.4] + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 8,6 standalone + +# no key +set key off + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 3 +set style line 2 linetype rgbcolor "#DC143C" linewidth 3 +set style line 3 linetype rgbcolor "#000000" linewidth 1 +set style line 4 linetype rgbcolor "#4B0082" linewidth 3 +set style line 5 linetype rgbcolor "#DAA520" linewidth 3 + +set pointsize 0.6 + +plot 0 with lines linestyle 3 ,\ + cos(2*pi*x) axes x1y2 linetype 0 linewidth 2,\ + "mc18.dat" using 1:5 with lines linestyle 1 diff --git a/figs/current.fig/current30.gnuplot b/figs/current.fig/current30.gnuplot new file mode 100644 index 0000000..7e50180 --- /dev/null +++ b/figs/current.fig/current30.gnuplot @@ -0,0 +1,29 @@ +set title "$E=30\\ \\mathrm{V}\\cdot\\mathrm{nm}^{-1}$" +set ylabel "$\\frac jk$" norotate +set xlabel "$\\frac{t\\omega}{2\\pi}$" +# +## start ticks at 0, then every x +set xtics 0,1 +set ytics -0.8,0.4 + +set yrange [-0.85:0.85] +set y2range [-1.4:1.4] + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 8,6 standalone + +# no key +set key off + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 3 +set style line 2 linetype rgbcolor "#DC143C" linewidth 3 +set style line 3 linetype rgbcolor "#000000" linewidth 1 +set style line 4 linetype rgbcolor "#4B0082" linewidth 3 +set style line 5 linetype rgbcolor "#DAA520" linewidth 3 + +set pointsize 0.6 + +plot 0 with lines linestyle 3 ,\ + cos(2*pi*x) axes x1y2 linetype 0 linewidth 2,\ + "mc30.dat" using 1:5 with lines linestyle 1 diff --git a/figs/current.fig/mc01.dat b/figs/current.fig/mc01.dat new file mode 100644 index 0000000..668c3fa --- /dev/null +++ b/figs/current.fig/mc01.dat @@ -0,0 +1,606 @@ +#parameters: V_=1.000000000000000e+01 eV, EF=4.500000000000000e+00 eV, omega_=1.550000000000000e+00 eV, theta= 0.000000000000000e+00*pi W=5.500000000000000e+00 eV, n_photons=3.558597156973679, E=1.000000000000000e+09 V/m +#dimensionless: V=1.000000000000000e+00 k=9.486832980505138e-01 epsilon=2.812969366097111e-02 omega=1.550000000000000e-01 m=1.000000000000000e+00 +#numerical precision (matching): n_periods=3 NC=12 gl_order=16 J0=1 J=30 +#numerical precision (Crank-Nicolson): bmin=-2.000000000000000e+02 bmax= 2.000000000000000e+02 nx_cn=4000 mu= 1.000000000000000e-01 +# +# t real(psi0) imag(psi0) abs(psi0)^2 current avgcurrent + + 5.000000000000000e-03 9.869413613109341e-01 -9.086799130898394e-01 1.799752435119238e+00 2.719081453552717e-03 0.000000000000000e+00 + 1.000000000000000e-02 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/dev/null +++ b/figs/current.fig/mc05.dat @@ -0,0 +1,606 @@ +#parameters: V_=1.000000000000000e+01 eV, EF=4.500000000000000e+00 eV, omega_=1.550000000000000e+00 eV, theta= 0.000000000000000e+00*pi W=5.500000000000000e+00 eV, n_photons=3.803638601761316, E=5.000000000000000e+09 V/m +#dimensionless: V=1.000000000000000e+00 k=9.486832980505138e-01 epsilon=1.406484683048555e-01 omega=1.550000000000000e-01 m=1.000000000000000e+00 +#numerical precision (matching): n_periods=3 NC=12 gl_order=16 J0=1 J=30 +#numerical precision (Crank-Nicolson): bmin=-2.000000000000000e+02 bmax= 2.000000000000000e+02 nx_cn=4000 mu= 1.000000000000000e-01 +# +#% t real(psi0) imag(psi0) abs(psi0)^2 current avgcurrent + + 5.000000000000000e-03 9.871712211657485e-01 -9.078849796854442e-01 1.798762156236315e+00 1.358367476325980e-02 0.000000000000000e+00 + 1.000000000000000e-02 1.066385854453090e+00 -8.127942114708884e-01 1.797813220778229e+00 2.410687891270248e-02 0.000000000000000e+00 + 1.500000000000000e-02 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V_=1.000000000000000e+01 eV, EF=4.500000000000000e+00 eV, omega_=1.550000000000000e+00 eV, theta= 0.000000000000000e+00*pi W=5.500000000000000e+00 eV, n_photons=7.096774193548386, E=1.864237948714910e+10 V/m +#dimensionless: V=1.000000000000000e+00 k=9.486832980505138e-01 epsilon=5.244044240850757e-01 omega=1.550000000000000e-01 m=1.000000000000000e+00 +#numerical precision (matching): n_periods=3 NC=12 gl_order=16 J0=1 J=60 +#numerical precision (Crank-Nicolson): bmin=-2.000000000000000e+02 bmax= 2.000000000000000e+02 nx_cn=4000 mu= 1.000000000000000e-01 +# +#% t real(psi0) imag(psi0) abs(psi0)^2 current avgcurrent + + 2.500000000000000e-03 9.447653915340156e-01 -9.514668439260040e-01 1.797870800130933e+00 2.714803577257402e-02 0.000000000000000e+00 + 5.000000000000000e-03 9.879445210033273e-01 -9.051779142036058e-01 1.795381432942484e+00 5.049523400674741e-02 0.000000000000000e+00 + 7.500000000000000e-03 1.029267970409780e+00 -8.566811785961701e-01 1.793295196672392e+00 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-4.113957823433949e-01 1.967437192611349e+00 1.401159559512090e-02 3.235656081099774e-05 + 2.990000000000003e+00 -1.320560088965236e+00 -4.721711912171004e-01 1.966824582383247e+00 1.569375383939665e-02 3.434951900178509e-05 + 2.992500000000003e+00 -1.297390866446691e+00 -5.319679026450072e-01 1.966212909783823e+00 1.739545059083254e-02 3.631030602596878e-05 + 2.995000000000003e+00 -1.271507448330014e+00 -5.906582726211257e-01 1.965608386174475e+00 1.910129885923719e-02 3.823543124992152e-05 + 2.997500000000003e+00 -1.242964448464145e+00 -6.481171788563063e-01 1.965016497674434e+00 2.079732858867505e-02 4.012172674362482e-05 diff --git a/figs/current.fig/mc30.dat b/figs/current.fig/mc30.dat new file mode 100644 index 0000000..59ccecf --- /dev/null +++ b/figs/current.fig/mc30.dat @@ -0,0 +1,1206 @@ +#parameters: V_=1.000000000000000e+01 eV, EF=4.500000000000000e+00 eV, omega_=1.550000000000000e+00 eV, theta= 0.000000000000000e+00*pi W=5.500000000000000e+00 eV, n_photons=12.737441276310584, E=3.000000000000000e+10 V/m +#dimensionless: V=1.000000000000000e+00 k=9.486832980505138e-01 epsilon=8.438908098291331e-01 omega=1.550000000000000e-01 m=1.000000000000000e+00 +#numerical precision (matching): n_periods=3 NC=12 gl_order=16 J0=1 J=60 +#numerical precision (Crank-Nicolson): bmin=-2.000000000000000e+02 bmax= 2.000000000000000e+02 nx_cn=4000 mu= 1.000000000000000e-01 +# +#% t real(psi0) imag(psi0) abs(psi0)^2 current avgcurrent + + 2.500000000000000e-03 9.449714442231751e-01 -9.505802323290621e-01 1.796573808492007e+00 4.364004005855032e-02 0.000000000000000e+00 + 5.000000000000000e-03 9.885755687202802e-01 -9.029290521816595e-01 1.792562528344296e+00 8.105573999350746e-02 0.000000000000000e+00 + 7.500000000000000e-03 1.030424794711546e+00 -8.528805958341297e-01 1.789180568306713e+00 1.139785066502029e-01 0.000000000000000e+00 + 1.000000000000000e-02 1.070268585640750e+00 -8.007524239893189e-01 1.786679289934222e+00 1.431049042983159e-01 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-6.313124991171323e-01 2.149660299711059e+00 2.796433320660927e-02 3.387819763318998e-04 + 2.995000000000003e+00 -1.293258266178136e+00 -6.896747517266253e-01 2.148168206207260e+00 2.867338341277413e-02 3.357101469579941e-04 + 2.997500000000003e+00 -1.260507633899907e+00 -7.466672941903716e-01 2.146391543333514e+00 3.002257036503367e-02 3.327684677065902e-04 diff --git a/figs/omega.fig/Makefile b/figs/omega.fig/Makefile new file mode 100644 index 0000000..58585ef --- /dev/null +++ b/figs/omega.fig/Makefile @@ -0,0 +1,32 @@ +PROJECTNAME=omega + +PHOTOCOMP_PATH= + +DATS=omega.dat +PDFS=$(addsuffix .pdf, $(PROJECTNAME)) +TEXS=$(addsuffix .tikz.tex, $(PROJECTNAME)) + +all: $(PDFS) + +$(PDFS): $(DATS) + gnuplot $(patsubst %.pdf, %.gnuplot, $@) > $(patsubst %.pdf, %.tikz.tex, $@) + pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@) + +omega.dat: + $(PHOTOCOMP_PATH)photocomp -p "V=10;EF=4.5;omega_min=5;omega_max=6;n_omega=50;n_periods=12;J=120;E=10e9" mc_avgavgcurrent_omega > $@ + +install: $(PDFS) + cp $^ $(INSTALLDIR)/ + +clean-aux: + rm -f $(addsuffix .tikz.tex, $(PROJECTNAME)) + rm -f $(addsuffix .aux, $(PROJECTNAME)) + rm -f $(addsuffix .log, $(PROJECTNAME)) + +clean-dat: + rm -f $(DATS) + +clean-tex: + rm -f $(PDFS) + +clean: clean-aux clean-tex diff --git a/figs/omega.fig/omega.dat b/figs/omega.fig/omega.dat new file mode 100644 index 0000000..f937d64 --- /dev/null +++ b/figs/omega.fig/omega.dat @@ -0,0 +1,57 @@ +#parameters: V_=1.000000000000000e+01 eV, EF=4.500000000000000e+00 eV, omega_=1.549800000000000e+00 eV, theta= 0.000000000000000e+00*pi W=5.500000000000000e+00 eV, n_photons=4.570246362361185, E=1.000000000000000e+10 V/m +#dimensionless: V=1.000000000000000e+00 k=9.486832980505138e-01 epsilon=2.813332376726366e-01 omega=1.549800000000000e-01 m=1.000000000000000e+00, keldysh=1.863997401882817e+00 +#numerical precision (matching): n_periods=12 NC=12 gl_order=16 J0=1 J=120 +#numerical precision (Crank-Nicolson): bmin=-2.000000000000000e+02 bmax= 2.000000000000000e+02 nx_cn=4000 mu= 1.000000000000000e-01 +# +# omega n_photons avgavgcurrent mu_t M_t + + 5.000000000000000e+00 1.115208395170143e+00 -1.767254374839443e-06 -5.872759586088563e-06 3.688215275613987e-06 + 5.020408163265307e+00 1.110552135135230e+00 3.481769195644922e-06 -9.520100843546093e-07 9.255081504239878e-06 + 5.040816326530612e+00 1.105935061178769e+00 8.694844819654358e-06 3.995239195851555e-06 1.275579751852944e-05 + 5.061224489795919e+00 1.101356675491701e+00 8.799345752084708e-06 3.933492541232696e-06 1.224640794826752e-05 + 5.081632653061225e+00 1.096816488736612e+00 3.536546271238413e-06 -1.397979335805772e-06 8.522746047746839e-06 + 5.102040816326531e+00 1.092314019866977e+00 -2.678873432257001e-06 -7.040626921087629e-06 2.405571404876937e-06 + 5.122448979591836e+00 1.087848795951036e+00 -4.141432277868735e-06 -8.179726614834721e-06 8.328635003936747e-07 + 5.142857142857143e+00 1.083420352000169e+00 1.065329962725955e-06 -3.219042681999279e-06 7.194092044434094e-06 + 5.163265306122449e+00 1.079028230801629e+00 8.827945387434261e-06 3.894608130034307e-06 1.361133448724558e-05 + 5.183673469387755e+00 1.074671982755515e+00 1.183484927813807e-05 6.720377263410872e-06 1.480764494759398e-05 + 5.204081632653061e+00 1.070351165715842e+00 7.157858272739758e-06 2.032275766979137e-06 1.184457809771067e-05 + 5.224489795918367e+00 1.066065344835611e+00 -1.780531480642952e-06 -6.314231515448376e-06 3.710341161754937e-06 + 5.244897959183675e+00 1.061814092415737e+00 -6.974014215351573e-06 -1.081260460372574e-05 -2.354147606942842e-06 + 5.265306122448980e+00 1.057596987757747e+00 -2.919310525570610e-06 -6.939018900602607e-06 3.261584855438926e-06 + 5.285714285714286e+00 1.053413617020115e+00 7.805288821933059e-06 2.919606326911266e-06 1.322079040879357e-05 + 5.306122448979592e+00 1.049263573078148e+00 1.604608685234645e-05 1.091547032847531e-05 1.965592004543567e-05 + 5.326530612244898e+00 1.045146455387307e+00 1.342844042504459e-05 8.070509203493191e-06 1.742760293516599e-05 + 5.346938775510203e+00 1.041061869849881e+00 6.319329527833020e-07 -4.136336829806877e-06 6.406676540427386e-06 + 5.367346938775510e+00 1.037009428684893e+00 -1.134767642351268e-05 -1.506771058349229e-05 -5.844979965672390e-06 + 5.387755102040817e+00 1.032988750301173e+00 -1.067393364549286e-05 -1.421962846406221e-05 -4.722030737635214e-06 + 5.408163265306122e+00 1.028999459173493e+00 5.215086717154612e-06 6.836825598078386e-07 1.134663037774954e-05 + 5.428571428571429e+00 1.025041185721677e+00 2.441354215520772e-05 1.886703947235648e-05 2.858626902354553e-05 + 5.448979591836735e+00 1.021113566192615e+00 2.822567366351067e-05 2.215698191711490e-05 3.098415058909033e-05 + 5.469387755102041e+00 1.017216242545081e+00 6.863535715779913e-06 1.809450344973610e-06 1.214526030149579e-05 + 5.489795918367346e+00 1.013348862337299e+00 -2.727525178415409e-05 -3.065256869035489e-05 -2.028636911037119e-05 + 5.510204081632653e+00 1.009511078617158e+00 -4.093113725581106e-05 -4.323799845384540e-05 -3.387376664678247e-05 + 5.530612244897959e+00 1.005702549815022e+00 3.817865286531565e-06 -5.142462430566329e-07 9.578586005080775e-06 + 5.551020408163265e+00 1.001922939639051e+00 1.261566938428398e-04 1.153817323932535e-04 1.255952555714206e-04 + 5.571428571428571e+00 9.981719169729742e-01 3.136578059083432e-04 2.930605844124897e-04 3.021841533407623e-04 + 5.591836734693878e+00 9.944491557762352e-01 5.259860824154864e-04 4.944117334814721e-04 5.042492984297921e-04 + 5.612244897959185e+00 9.907543349864588e-01 7.164148553233928e-04 6.751394677101236e-04 6.854066404094009e-04 + 5.632653061224490e+00 9.870871384241715e-01 8.537858911063791e-04 8.058221638153582e-04 8.150518511841755e-04 + 5.653061224489796e+00 9.834472546997073e-01 9.353257382948215e-04 8.829420577167401e-04 8.927529991862424e-04 + 5.673469387755102e+00 9.798343771222539e-01 9.817646019054812e-04 9.270353579757282e-04 9.371733789868266e-04 + 5.693877551020408e+00 9.762482036109686e-01 1.018946634135041e-03 9.621784532074163e-04 9.716817519431893e-04 + 5.714285714285714e+00 9.726884366081232e-01 1.062632861711127e-03 1.003323643755882e-03 1.013021944472856e-03 + 5.734693877551020e+00 9.691547829942048e-01 1.112490823467546e-03 1.050680221296718e-03 1.060895473227898e-03 + 5.755102040816327e+00 9.656469540049427e-01 1.158259765555210e-03 1.094279665056827e-03 1.104186194449871e-03 + 5.775510204081632e+00 9.621646651501934e-01 1.191371799970765e-03 1.126273990852587e-03 1.135056870744205e-03 + 5.795918367346939e+00 9.587076361346428e-01 1.211442176666135e-03 1.145140636162794e-03 1.154860889304104e-03 + 5.816326530612245e+00 9.552755907802787e-01 1.225889650553572e-03 1.158509079321945e-03 1.168174717175902e-03 + 5.836734693877551e+00 9.518682569505817e-01 1.242507899351386e-03 1.174196719688448e-03 1.183717652034037e-03 + 5.857142857142856e+00 9.484853664763989e-01 1.263551131600086e-03 1.194064607016431e-03 1.204602932412491e-03 + 5.877551020408164e+00 9.451266550834464e-01 1.285371822702834e-03 1.215127891243085e-03 1.225267717535952e-03 + 5.897959183673469e+00 9.417918623214092e-01 1.301870093180957e-03 1.230772703738859e-03 1.240099864845858e-03 + 5.918367346938775e+00 9.384807314945884e-01 1.311323970089323e-03 1.239871121117092e-03 1.249887264888310e-03 + 5.938775510204081e+00 9.351930095940618e-01 1.316437740598745e-03 1.244633452909735e-03 1.254030609123569e-03 + 5.959183673469388e+00 9.319284472313192e-01 1.321803385993161e-03 1.249587109594438e-03 1.258853111919532e-03 + 5.979591836734695e+00 9.286867985733309e-01 1.330167086142919e-03 1.257309860267525e-03 1.267423484293473e-03 + 6.000000000000000e+00 9.254678212790179e-01 1.340116024296602e-03 1.266994280781395e-03 1.277256236322942e-03 diff --git a/figs/omega.fig/omega.gnuplot b/figs/omega.fig/omega.gnuplot new file mode 100644 index 0000000..4bd952c --- /dev/null +++ b/figs/omega.fig/omega.gnuplot @@ -0,0 +1,22 @@ +set ylabel "$\\frac1k\\left<\\left\\right>$" norotate offset -1,0 +set xlabel "$\\omega (\\mathrm{eV})$" + +set xtics 5,0.25 +set ytics 0,0.0005 + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 8,6 standalone + +# no key +set key off + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 3 +set style line 2 linetype rgbcolor "#DC143C" linewidth 3 +set style line 3 linetype rgbcolor "#32CD32" linewidth 3 +set style line 4 linetype rgbcolor "#4B0082" linewidth 3 +set style line 5 linetype rgbcolor "#DAA520" linewidth 3 + +set pointsize 0.6 + +plot "omega.dat" using 1:3 with lines linestyle 1 diff --git a/figs/wavefunction.fig/Makefile b/figs/wavefunction.fig/Makefile new file mode 100644 index 0000000..fe3fbbd --- /dev/null +++ b/figs/wavefunction.fig/Makefile @@ -0,0 +1,32 @@ +PROJECTNAME=real imag density current + +PHOTOCOMP_PATH= + +DATS=mc.dat +PDFS=$(addsuffix .pdf, $(PROJECTNAME)) +TEXS=$(addsuffix .tikz.tex, $(PROJECTNAME)) + +all: $(PDFS) + +$(PDFS): $(DATS) + gnuplot $(patsubst %.pdf, %.gnuplot, $@) > $(patsubst %.pdf, %.tikz.tex, $@) + pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@) + +mc.dat: + $(PHOTOCOMP_PATH)photocomp -p "V=10;EF=4.5;omega=1.55;n_periods=3;J=30;E=15e9" mc > $@ + +install: $(PDFS) + cp $^ $(INSTALLDIR)/ + +clean-aux: + rm -f $(addsuffix .tikz.tex, $(PROJECTNAME)) + rm -f $(addsuffix .aux, $(PROJECTNAME)) + rm -f $(addsuffix .log, $(PROJECTNAME)) + +clean-dat: + rm -f $(DATS) + +clean-tex: + rm -f $(PDFS) + +clean: clean-aux clean-tex diff --git a/figs/wavefunction.fig/current.gnuplot b/figs/wavefunction.fig/current.gnuplot new file mode 100644 index 0000000..43777ca --- /dev/null +++ b/figs/wavefunction.fig/current.gnuplot @@ -0,0 +1,28 @@ +set ylabel "$\\frac jk$" norotate +set xlabel "$\\frac{t\\omega}{2\\pi}$" + +set xtics 0,1 +set ytics -0.3,0.15 + +set yrange [-0.3:0.3] +set y2range [-1.4:1.4] + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 8,6 standalone + +# no key +set key off + +set label "\\bf (d)" at 2.65,0.28 + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 3 +set style line 2 linetype rgbcolor "#DC143C" linewidth 3 +set style line 3 linetype rgbcolor "#32CD32" linewidth 3 +set style line 4 linetype rgbcolor "#4B0082" linewidth 3 +set style line 5 linetype rgbcolor "#DAA520" linewidth 3 + +set pointsize 0.6 + +plot cos(2*pi*x) axes x1y2 linetype 0 linewidth 2 ,\ + "mc.dat" using 1:5 with lines linestyle 1 diff --git a/figs/wavefunction.fig/density.gnuplot b/figs/wavefunction.fig/density.gnuplot new file mode 100644 index 0000000..4c6d5d5 --- /dev/null +++ b/figs/wavefunction.fig/density.gnuplot @@ -0,0 +1,28 @@ +set ylabel "$|\\psi_0|^2$" offset -1 +set xlabel "$\\frac{t\\omega}{2\\pi}$" + +set xtics 0,1 +set ytics 1.6,0.1 + +set yrange [1.6:2] +set y2range [-1.4:1.4] + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 8,6 standalone + +# no key +set key off + +set label "\\bf (c)" at 2.65,1.987 + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 3 +set style line 2 linetype rgbcolor "#DC143C" linewidth 3 +set style line 3 linetype rgbcolor "#32CD32" linewidth 3 +set style line 4 linetype rgbcolor "#4B0082" linewidth 3 +set style line 5 linetype rgbcolor "#DAA520" linewidth 3 + +set pointsize 0.6 + +plot cos(2*pi*x) axes x1y2 linetype 0 linewidth 2 ,\ +"mc.dat" using 1:4 with lines linestyle 2 diff --git a/figs/wavefunction.fig/imag.gnuplot b/figs/wavefunction.fig/imag.gnuplot new file mode 100644 index 0000000..97a16a7 --- /dev/null +++ b/figs/wavefunction.fig/imag.gnuplot @@ -0,0 +1,28 @@ +set ylabel "$\\mathcal Im(e^{\\frac{ik^2}2t}\\psi_0)$" offset -1 +set xlabel "$\\frac{t\\omega}{2\\pi}$" + +set xtics 0,1 +set ytics -1.02, 0.02 + +set yrange [-1.02:-0.94] +set y2range [-1.4:1.4] + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 8,6 standalone + +# no key +set key off + +set label "\\bf (b)" at 2.65,-0.943 + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 3 +set style line 2 linetype rgbcolor "#DC143C" linewidth 3 +set style line 3 linetype rgbcolor "#32CD32" linewidth 3 +set style line 4 linetype rgbcolor "#4B0082" linewidth 3 +set style line 5 linetype rgbcolor "#DAA520" linewidth 3 + +set pointsize 0.6 + +plot cos(2*pi*x) axes x1y2 linetype 0 linewidth 2 ,\ +"mc.dat" using 1:3 with lines linestyle 5 diff --git a/figs/wavefunction.fig/mc.dat b/figs/wavefunction.fig/mc.dat new file mode 100644 index 0000000..f04d9fc --- /dev/null +++ b/figs/wavefunction.fig/mc.dat @@ -0,0 +1,606 @@ +#parameters: V_=1.000000000000000e+01 eV, EF=4.500000000000000e+00 eV, omega_=1.550000000000000e+00 eV, theta= 0.000000000000000e+00*pi W=5.500000000000000e+00 eV, n_photons=5.845650641658291, E=1.500000000000000e+10 V/m +#dimensionless: V=1.000000000000000e+00 k=9.486832980505138e-01 epsilon=4.219454049145666e-01 omega=1.550000000000000e-01 m=1.000000000000000e+00, keldysh=1.242825299143273e+00 +#numerical precision (matching): n_periods=3 NC=12 gl_order=16 J0=1 J=30 +#numerical precision (Crank-Nicolson): bmin=-2.000000000000000e+02 bmax= 2.000000000000000e+02 nx_cn=4000 mu= 1.000000000000000e-01 +# +#% t real(psi0) imag(psi0) abs(psi0)^2 current avgcurrent + + 5.000000000000000e-03 9.011237076219065e-01 -9.920990592322099e-01 1.796284479767687e+00 4.066156482056959e-02 0.000000000000000e+00 + 1.000000000000000e-02 9.036959786184608e-01 -9.882977598906416e-01 1.793398883976038e+00 7.201570001853567e-02 0.000000000000000e+00 + 1.500000000000000e-02 9.071741251355965e-01 -9.846371696331014e-01 1.792475249138620e+00 9.663201291162658e-02 0.000000000000000e+00 + 2.000000000000000e-02 9.112296705251185e-01 -9.813857962466512e-01 1.793457593519989e+00 1.156836312659321e-01 0.000000000000000e+00 + 2.500000000000000e-02 9.156315488882216e-01 -9.786363650543179e-01 1.796110268326171e+00 1.300140368375609e-01 0.000000000000000e+00 + 3.000000000000000e-02 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a/figs/wavefunction.fig/real.gnuplot b/figs/wavefunction.fig/real.gnuplot new file mode 100644 index 0000000..3fa1f0b --- /dev/null +++ b/figs/wavefunction.fig/real.gnuplot @@ -0,0 +1,28 @@ +set ylabel "$\\mathcal Re(e^{\\frac{ik^2}2t}\\psi_0)$" offset -1 +set xlabel "$\\frac{t\\omega}{2\\pi}$" + +set xtics 0,1 +set ytics 0.82,0.04 + +set yrange [0.82:0.98] +set y2range [-1.4:1.4] + +# default output canvas size: 12.5cm x 8.75cm +set term lua tikz size 8,6 standalone + +# no key +set key off + +set label "\\bf (a)" at 2.65,0.974 + +# set linestyle +set style line 1 linetype rgbcolor "#4169E1" linewidth 3 +set style line 2 linetype rgbcolor "#DC143C" linewidth 3 +set style line 3 linetype rgbcolor "#32CD32" linewidth 3 +set style line 4 linetype rgbcolor "#4B0082" linewidth 3 +set style line 5 linetype rgbcolor "#DAA520" linewidth 3 + +set pointsize 0.6 + +plot cos(2*pi*x) axes x1y2 linetype 0 linewidth 2 ,\ +"mc.dat" using 1:2 with lines linestyle 3 diff --git a/libs/ian.cls b/libs/ian.cls new file mode 100644 index 0000000..aec99ad --- /dev/null +++ b/libs/ian.cls @@ -0,0 +1,673 @@ +%% +%% Ian's class file +%% + +%% TeX format +\NeedsTeXFormat{LaTeX2e}[1995/12/01] + +%% class name +\ProvidesClass{ian}[2017/09/29] + +%% boolean to signal that this class is being used +\newif\ifianclass + +%% options +% no section numbering in equations +\DeclareOption{section_in_eq}{\sectionsineqtrue} +\DeclareOption{section_in_fig}{\sectionsinfigtrue} +\DeclareOption{section_in_theo}{\PassOptionsToPackage{\CurrentOption}{iantheo}} +\DeclareOption{section_in_all}{\sectionsineqtrue\sectionsinfigtrue\PassOptionsToPackage{section_in_theo}{iantheo}} +\DeclareOption{subsection_in_eq}{\subsectionsineqtrue} +\DeclareOption{subsection_in_fig}{\subsectionsinfigtrue} +\DeclareOption{subsection_in_theo}{\PassOptionsToPackage{\CurrentOption}{iantheo}} +\DeclareOption{subsection_in_all}{\subsectionsineqtrue\subsectionsinfigtrue\PassOptionsToPackage{subsection_in_theo}{iantheo}} +\DeclareOption{no_section_in_eq}{\sectionsineqfalse} +\DeclareOption{no_section_in_fig}{\sectionsinfigfalse} +\DeclareOption{no_section_in_theo}{\PassOptionsToPackage{\CurrentOption}{iantheo}} +\DeclareOption{no_section_in_all}{\sectionsineqfalse\sectionsinfigfalse\PassOptionsToPackage{no_section_in_theo}{iantheo}} +\DeclareOption{no_subsection_in_eq}{\subsectionsineqfalse} +\DeclareOption{no_subsection_in_fig}{\subsectionsinfigfalse} +\DeclareOption{no_subsection_in_theo}{\PassOptionsToPackage{\CurrentOption}{iantheo}} +\DeclareOption{no_subsection_in_all}{\subsectionsineqfalse\subsectionsinfigfalse\PassOptionsToPackage{no_subsection_in_theo}{iantheo}} +% reset point +\DeclareOption{point_reset_at_section}{\PassOptionsToPackage{reset_at_section}{point}} +\DeclareOption{point_no_reset_at_section}{\PassOptionsToPackage{no_reset_at_section}{point}} +\DeclareOption{point_reset_at_theo}{\PassOptionsToPackage{reset_at_theo}{point}} +\DeclareOption{point_no_reset_at_theo}{\PassOptionsToPackage{no_reset_at_theo}{point}} + +\def\ian@defaultoptions{ + \ExecuteOptions{section_in_all, no_subsection_in_all} + \ProcessOptions + + %% required packages + \RequirePackage{iantheo} + \RequirePackage{point} + \RequirePackage{color} + \RequirePackage{marginnote} + \RequirePackage{amssymb} + \PassOptionsToPackage{hidelinks}{hyperref} + \RequirePackage{hyperref} +} + +%% paper dimensions +\setlength\paperheight{297mm} +\setlength\paperwidth{210mm} + +%% fonts +\input{size11.clo} +\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm} +\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf} +\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt} +\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf} +\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit} + +%% text dimensions +\hoffset=-50pt +\voffset=-72pt +\textwidth=460pt +\textheight=704pt + + +%% remove default indentation +\parindent=0pt +%% indent command +\def\indent{\hskip20pt} + +%% something is wrong with \thepage, redefine it +\gdef\thepage{\the\c@page} + +%% array lines (to use the array environment) +\setlength\arraycolsep{5\p@} +\setlength\arrayrulewidth{.4\p@} + + +%% correct vertical alignment at the end of a document +\AtEndDocument{ + \vfill + \eject +} + + +%% hyperlinks +% hyperlinkcounter +\newcounter{lncount} +% hyperref anchor +\def\hrefanchor{% +\stepcounter{lncount}% +\hypertarget{ln.\thelncount}{}% +} + +%% define a command and write it to aux file +\def\outdef#1#2{% + % define command% + \expandafter\xdef\csname #1\endcsname{#2}% + % hyperlink number% + \expandafter\xdef\csname #1@hl\endcsname{\thelncount}% + % write command to aux% + \immediate\write\@auxout{\noexpand\expandafter\noexpand\gdef\noexpand\csname #1\endcsname{\csname #1\endcsname}}% + \immediate\write\@auxout{\noexpand\expandafter\noexpand\gdef\noexpand\csname #1@hl\endcsname{\thelncount}}% +} + +%% can call commands even when they are not defined +\def\safe#1{% + \ifdefined#1% + #1% + \else% + {\color{red}\bf?}% + \fi% +} + +%% define a label for the latest tag +%% label defines a command containing the string stored in \tag +\def\deflabel{ + \def\label##1{\expandafter\outdef{label@##1}{\safe\tag}} + + \def\ref##1{% + % check whether the label is defined (hyperlink runs into errors if this check is omitted) + \ifcsname label@##1@hl\endcsname% + \hyperlink{ln.\csname label@##1@hl\endcsname}{{\color{blue}\safe\csname label@##1\endcsname}}% + \else% + \ifcsname label@##1\endcsname% + {\color{blue}\csname ##1\endcsname}% + \else% + {\bf ??}% + \fi% + \fi% + } +} + + +%% make a custom link at any given location in the document +\def\makelink#1#2{% + \hrefanchor% + \outdef{label@#1}{#2}% +} + + +%% section command +% counter +\newcounter{sectioncount} +% space before section +\newlength\secskip +\setlength\secskip{40pt} +% a prefix to put before the section number, e.g. A for appendices +\def\sectionprefix{} +% define some lengths +\newlength\secnumwidth +\newlength\sectitlewidth +\def\section#1{ + % reset counters + \stepcounter{sectioncount} + \setcounter{subsectioncount}{0} + \ifsectionsineq + \setcounter{seqcount}0 + \fi + \ifsectionsinfig + \setcounter{figcount}0 + \fi + + % space before section (if not first) + \ifnum\thesectioncount>1 + \vskip\secskip + \penalty-1000 + \fi + + % hyperref anchor + \hrefanchor + % define tag (for \label) + \xdef\tag{\sectionprefix\thesectioncount} + + % get widths + \def\@secnum{{\bf\Large\sectionprefix\thesectioncount.\hskip10pt}} + \settowidth\secnumwidth{\@secnum} + \setlength\sectitlewidth\textwidth + \addtolength\sectitlewidth{-\secnumwidth} + + % print name + \parbox{\textwidth}{ + \@secnum + \parbox[t]{\sectitlewidth}{\Large\bf #1}} + + % write to table of contents + \iftoc + % save lncount in aux variable which is written to toc + \immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@sec.\thesectioncount\endcsname{\thelncount}} + \write\tocoutput{\noexpand\tocsection{#1}{\thepage}} + \fi + + %space + \par\penalty10000 + \bigskip\penalty10000 +} + +%% subsection +% counter +\newcounter{subsectioncount} +% space before subsection +\newlength\subsecskip +\setlength\subsecskip{30pt} +\def\subsection#1{ + % counters + \stepcounter{subsectioncount} + \setcounter{subsubsectioncount}{0} + \ifsubsectionsineq + \setcounter{seqcount}0 + \fi + \ifsubsectionsinfig + \setcounter{figcount}0 + \fi + + % space before subsection (if not first) + \ifnum\thesubsectioncount>1 + \vskip\subsecskip + \penalty-500 + \fi + + % hyperref anchor + \hrefanchor + % define tag (for \label) + \xdef\tag{\sectionprefix\thesectioncount.\thesubsectioncount} + + % get widths + \def\@secnum{{\bf\large\hskip.5cm\sectionprefix\thesectioncount.\thesubsectioncount.\hskip5pt}} + \settowidth\secnumwidth{\@secnum} + \setlength\sectitlewidth\textwidth + \addtolength\sectitlewidth{-\secnumwidth} + % print name + \parbox{\textwidth}{ + \@secnum + \parbox[t]{\sectitlewidth}{\large\bf #1}} + + % write to table of contents + \iftoc + % save lncount in aux variable which is written to toc + \immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@subsec.\thesectioncount.\thesubsectioncount\endcsname{\thelncount}} + \write\tocoutput{\noexpand\tocsubsection{#1}{\thepage}} + \fi + + % space + \par\penalty10000 + \medskip\penalty10000 +} + +%% subsubsection +% counter +\newcounter{subsubsectioncount} +% space before subsubsection +\newlength\subsubsecskip +\setlength\subsubsecskip{20pt} +\def\subsubsection#1{ + % counters + \stepcounter{subsubsectioncount} + + % space before subsubsection (if not first) + \ifnum\thesubsubsectioncount>1 + \vskip\subsubsecskip + \penalty-500 + \fi + + % hyperref anchor + \hrefanchor + % define tag (for \label) + \xdef\tag{\sectionprefix\thesectioncount.\thesubsectioncount.\thesubsubsectioncount} + + % get widths + \def\@secnum{{\bf\hskip1.cm\sectionprefix\thesectioncount.\thesubsectioncount.\thesubsubsectioncount.\hskip5pt}} + \settowidth\secnumwidth{\@secnum} + \setlength\sectitlewidth\textwidth + \addtolength\sectitlewidth{-\secnumwidth} + % print name + \parbox{\textwidth}{ + \@secnum + \parbox[t]{\sectitlewidth}{\large\bf #1}} + + % write to table of contents + \iftoc + % save lncount in aux variable which is written to toc + \immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@subsubsec.\thesectioncount.\thesubsectioncount.\thesubsubsectioncount\endcsname{\thelncount}} + \write\tocoutput{\noexpand\tocsubsubsection{#1}{\thepage}} + \fi + + % space + \par\penalty10000 + \medskip\penalty10000 +} + +%% itemize +\newlength\itemizeskip +% left margin for items +\setlength\itemizeskip{20pt} +\newlength\itemizeseparator +% space between the item symbol and the text +\setlength\itemizeseparator{5pt} +% penalty preceding an itemize +\newcount\itemizepenalty +\itemizepenalty=0 +% counter counting the itemize level +\newcounter{itemizecount} + +% item symbol +\def\itemizept#1{ + \ifnum#1=1 + \textbullet + \else + $\scriptstyle\blacktriangleright$ + \fi +} + + +\newlength\current@itemizeskip +\setlength\current@itemizeskip{0pt} +\def\itemize{% + \par\expandafter\penalty\the\itemizepenalty\medskip\expandafter\penalty\the\itemizepenalty% + \addtocounter{itemizecount}{1}% + \addtolength\current@itemizeskip{\itemizeskip}% + \leftskip\current@itemizeskip% +} +\def\enditemize{% + \addtocounter{itemizecount}{-1}% + \addtolength\current@itemizeskip{-\itemizeskip}% + \par\expandafter\penalty\the\itemizepenalty\leftskip\current@itemizeskip% + \medskip\expandafter\penalty\the\itemizepenalty% +} + +% item, with optional argument to specify the item point +% @itemarg is set to true when there is an optional argument +\newif\if@itemarg +\def\item{% + % check whether there is an optional argument (if there is none, add on empty '[]') + \@ifnextchar [{\@itemargtrue\@itemx}{\@itemargfalse\@itemx[]}% +} +\newlength\itempt@total +\def\@itemx[#1]{ + \if@itemarg + \settowidth\itempt@total{#1} + \else + \settowidth\itempt@total{\itemizept\theitemizecount} + \fi + \addtolength\itempt@total{\itemizeseparator} + \par + \medskip + \if@itemarg + \hskip-\itempt@total#1\hskip\itemizeseparator + \else + \hskip-\itempt@total\itemizept\theitemizecount\hskip\itemizeseparator + \fi +} + +%% prevent page breaks after itemize +\newcount\previtemizepenalty +\def\nopagebreakafteritemize{ + \previtemizepenalty=\itemizepenalty + \itemizepenalty=10000 +} +%% back to previous value +\def\restorepagebreakafteritemize{ + \itemizepenalty=\previtemizepenalty +} + +%% enumerate +\newcounter{enumerate@count} +\def\enumerate{ + \setcounter{enumerate@count}0 + \let\olditem\item + \let\olditemizept\itemizept + \def\item{ + % counter + \stepcounter{enumerate@count} + % set header + \def\itemizept{\theenumerate@count.} + % hyperref anchor + \hrefanchor + % define tag (for \label) + \xdef\tag{\theenumerate@count} + \olditem + } + \itemize +} +\def\endenumerate{ + \enditemize + \let\item\olditem + \let\itemizept\olditemizept +} + + +%% equation numbering +% counter +\newcounter{seqcount} +% booleans (write section or subsection in equation number) +\newif\ifsectionsineq +\newif\ifsubsectionsineq +\def\seqcount{ + \stepcounter{seqcount} + % the output + \edef\seqformat{\theseqcount} + % add subsection number + \ifsubsectionsineq + \let\tmp\seqformat + \edef\seqformat{\thesubsectioncount.\tmp} + \fi + % add section number + \ifsectionsineq + \let\tmp\seqformat + \edef\seqformat{\sectionprefix\thesectioncount.\tmp} + \fi + % define tag (for \label) + \xdef\tag{\seqformat} + % write number + \marginnote{\hfill(\seqformat)} +} +%% equation environment compatibility +\def\equation{\hrefanchor$$\seqcount} +\def\endequation{$$\@ignoretrue} + + +%% figures +% counter +\newcounter{figcount} +% booleans (write section or subsection in equation number) +\newif\ifsectionsinfig +\newif\ifsubsectionsinfig +% width of figures +\newlength\figwidth +\setlength\figwidth\textwidth +\addtolength\figwidth{-2.5cm} +% caption +\def\defcaption{ + \long\def\caption##1{ + \stepcounter{figcount} + + % hyperref anchor + \hrefanchor + + % the number of the figure + \edef\figformat{\thefigcount} + % add subsection number + \ifsubsectionsinfig + \let\tmp\figformat + \edef\figformat{\thesubsectioncount.\tmp} + \fi + % add section number + \ifsectionsinfig + \let\tmp\figformat + \edef\figformat{\sectionprefix\thesectioncount.\tmp} + \fi + + % define tag (for \label) + \xdef\tag{\figformat} + + % write + \hfil fig \figformat: \parbox[t]{\figwidth}{\leavevmode\small##1} + + % space + \par\bigskip + } +} +%% short caption: centered +\def\captionshort#1{ + \stepcounter{figcount} + + % hyperref anchor + \hrefanchor + + % the number of the figure + \edef\figformat{\thefigcount} + % add section number + \ifsectionsinfig + \let\tmp\figformat + \edef\figformat{\sectionprefix\thesectioncount.\tmp} + \fi + + % define tag (for \label) + \xdef\tag{\figformat} + + % write + \hfil fig \figformat: {\small#1} + + %space + \par\bigskip +} + +%% environment +\def\figure{ + \par + \vfil\penalty100\vfilneg + \bigskip +} +\def\endfigure{ + \par + \bigskip +} + + +%% start appendices +\def\appendix{ + \vfill + \pagebreak + + % counter + \setcounter{sectioncount}0 + + % prefix + \def\sectionprefix{A} + + % write + {\bf \LARGE Appendices}\par\penalty10000\bigskip\penalty10000 + + % add a mention in the table of contents + \iftoc + \immediate\write\tocoutput{\noexpand\tocappendices}\penalty10000 + \fi + + %% uncomment for new page for each appendix + %\def\seqskip{\vfill\pagebreak} +} + + +%% bibliography +% size of header +\newlength\bibheader +\def\thebibliography#1{ + \hrefanchor + + % add a mention in the table of contents + \iftoc + % save lncount in aux variable which is written to toc + \immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@references\endcsname{\thelncount}} + \write\tocoutput{\noexpand\tocreferences{\thepage}}\penalty10000 + \fi + + % write + {\bf \LARGE References}\par\penalty10000\bigskip\penalty10000 + % width of header + \settowidth\bibheader{[#1]} + \leftskip\bibheader +} +% end environment +\def\endthebibliography{ + \par\leftskip0pt +} + +%% bibitem command +\def\bibitem[#1]#2{% + \hrefanchor% + \outdef{label@cite#2}{#1}% + \hskip-\bibheader% + \makebox[\bibheader]{\cite{#2}\hfill}% +} + +%% cite command (adapted from latex.ltx) +% @tempswa is set to true when there is an optional argument +\newif\@tempswa +\def\cite{% + % check whether there is an optional argument (if there is none, add on empty '[]') + \@ifnextchar [{\@tempswatrue\@citex}{\@tempswafalse\@citex[]}% +} +% command with optional argument +\def\@citex[#1]#2{\leavevmode% + % initialize loop + \let\@citea\@empty% + % format + \@cite{% + % loop over ',' separated list + \@for\@citeb:=#2\do{% + % text to add at each iteration of the loop (separator between citations) + \@citea\def\@citea{,\ }% + % add entry to citelist + \@writecitation{\@citeb}% + \ref{cite\@citeb}% + }% + }% + % add optional argument text (as an argument to '\@cite') + {#1}% +} +\def\@cite#1#2{% + [#1\if@tempswa , #2\fi]% +} +%% add entry to citelist after checking it has not already been added +\def\@writecitation#1{% + \ifcsname if#1cited\endcsname% + \else% + \expandafter\newif\csname if#1cited\endcsname% + \immediate\write\@auxout{\string\citation{#1}}% + \fi% +} + +%% table of contents +% boolean +\newif\iftoc +\def\tableofcontents{ + {\bf \large Table of contents:}\par\penalty10000\bigskip\penalty10000 + + % copy content from file + \IfFileExists{\jobname.toc}{\input{\jobname.toc}}{{\tt error: table of contents missing}} + + % open new toc + \newwrite\tocoutput + \immediate\openout\tocoutput=\jobname.toc + + \toctrue +} +%% close file +\AtEndDocument{ + % close toc + \iftoc + \immediate\closeout\tocoutput + \fi +} + + +%% fill line with dots +\def\leaderfill{\leaders\hbox to 1em {\hss. \hss}\hfill} + +%% same as sectionprefix +\def\tocsectionprefix{} + +%% toc formats +\newcounter{tocsectioncount} +\def\tocsection #1#2{ + \stepcounter{tocsectioncount} + \setcounter{tocsubsectioncount}{0} + \setcounter{tocsubsubsectioncount}{0} + % write + \smallskip\hyperlink{ln.\csname toc@sec.\thetocsectioncount\endcsname}{{\bf \tocsectionprefix\thetocsectioncount}.\hskip5pt {\color{blue}#1}\leaderfill#2}\par +} +\newcounter{tocsubsectioncount} +\def\tocsubsection #1#2{ + \stepcounter{tocsubsectioncount} + \setcounter{tocsubsubsectioncount}{0} + % write + {\hskip10pt\hyperlink{ln.\csname toc@subsec.\thetocsectioncount.\thetocsubsectioncount\endcsname}{{\bf \thetocsubsectioncount}.\hskip5pt {\color{blue}\small #1}\leaderfill#2}}\par +} +\newcounter{tocsubsubsectioncount} +\def\tocsubsubsection #1#2{ + \stepcounter{tocsubsubsectioncount} + % write + {\hskip20pt\hyperlink{ln.\csname toc@subsubsec.\thetocsectioncount.\thetocsubsectioncount.\thetocsubsubsectioncount\endcsname}{{\bf \thetocsubsubsectioncount}.\hskip5pt {\color{blue}\small #1}\leaderfill#2}}\par +} +\def\tocappendices{ + \medskip + \setcounter{tocsectioncount}0 + {\bf Appendices}\par + \smallskip + \def\tocsectionprefix{A} +} +\def\tocreferences#1{ + \medskip + {\hyperlink{ln.\csname toc@references\endcsname}{{\color{blue}\bf References}\leaderfill#1}}\par + \smallskip +} + + +%% definitions that must be loaded at begin document +\let\ian@olddocument\document +\def\document{ + \ian@olddocument + + \deflabel + \defcaption +} + +%% end +\ian@defaultoptions +\endinput diff --git a/libs/iantheo.sty b/libs/iantheo.sty new file mode 100644 index 0000000..d33a93d --- /dev/null +++ b/libs/iantheo.sty @@ -0,0 +1,162 @@ +%% +%% iantheorem package: +%% Ian's customized theorem command +%% + +%% boolean to signal that this package was loaded +\newif\ifiantheo + +%% TeX format +\NeedsTeXFormat{LaTeX2e}[1995/12/01] + +%% package name +\ProvidesPackage{iantheo}[2016/11/10] + +%% options +\newif\ifsectionintheo +\DeclareOption{section_in_theo}{\sectionintheotrue} +\DeclareOption{no_section_in_theo}{\sectionintheofalse} +\newif\ifsubsectionintheo +\DeclareOption{subsection_in_theo}{\subsectionintheotrue} +\DeclareOption{no_subsection_in_theo}{\subsectionintheofalse} + +\def\iantheo@defaultoptions{ + \ExecuteOptions{section_in_theo, no_subsection_in_theo} + \ProcessOptions + + %%% reset at every new section + \ifsectionintheo + \let\iantheo@oldsection\section + \gdef\section{\setcounter{theocount}{0}\iantheo@oldsection} + \fi + + %% reset at every new subsection + \ifsubsectionintheo + \let\iantheo@oldsubsection\subsection + \gdef\subsection{\setcounter{theocount}{0}\iantheo@oldsubsection} + \fi +} + + +%% delimiters +\def\delimtitle#1{ + \par% + \leavevmode% + \raise.3em\hbox to\hsize{% + \lower0.3em\hbox{\vrule height0.3em}% + \hrulefill% + \ \lower.3em\hbox{#1}\ % + \hrulefill% + \lower0.3em\hbox{\vrule height0.3em}% + }% + \par\penalty10000% +} + +%% callable by ref +\def\delimtitleref#1{ + \par% +% + \ifdefined\ianclass% + % hyperref anchor% + \hrefanchor% + % define tag (for \label)% + \xdef\tag{#1}% + \fi% +% + \leavevmode% + \raise.3em\hbox to\hsize{% + \lower0.3em\hbox{\vrule height0.3em}% + \hrulefill% + \ \lower.3em\hbox{\bf #1}\ % + \hrulefill% + \lower0.3em\hbox{\vrule height0.3em}% + }% + \par\penalty10000% +} + +%% no title +\def\delim{ + \par% + \leavevmode\raise.3em\hbox to\hsize{% + \lower0.3em\hbox{\vrule height0.3em}% + \hrulefill% + \lower0.3em\hbox{\vrule height0.3em}% + }% + \par\penalty10000% +} + +%% end delim +\def\enddelim{ + \par\penalty10000% + \leavevmode% + \raise.3em\hbox to\hsize{% + \vrule height0.3em\hrulefill\vrule height0.3em% + }% + \par% +} + + +%% theorem +% counter +\newcounter{theocount} +% booleans (write section or subsection in equation number) +\def\theo#1{ + \stepcounter{theocount} + \ifdefined\ianclass + % hyperref anchor + \hrefanchor + \fi + % the number + \def\formattheo{\thetheocount} + % add subsection number + \ifsubsectionintheo + \let\tmp\formattheo + \edef\formattheo{\thesubsectioncount.\tmp} + \fi + % add section number + \ifsectionintheo + \let\tmp\formattheo + \edef\formattheo{\sectionprefix\thesectioncount.\tmp} + \fi + % define tag (for \label) + \xdef\tag{\formattheo} + % write + \delimtitle{\bf #1 \formattheo} +} +\let\endtheo\enddelim +%% theorem headers with name +\def\theoname#1#2{ + \theo{#1}\hfil({\it #2})\par\penalty10000\medskip% +} + + +%% qed symbol +\def\qedsymbol{$\square$} +\def\qed{\penalty10000\hfill\penalty10000\qedsymbol} + + +%% compatibility with article class +\ifdefined\ianclasstrue + \relax +\else + \def\thesectioncount{\thesection} + \def\thesubsectioncount{\thesubsection} + \def\sectionprefix{} +\fi + + +%% prevent page breaks after displayed equations +\newcount\prevpostdisplaypenalty +\def\nopagebreakaftereq{ + \prevpostdisplaypenalty=\postdisplaypenalty + \postdisplaypenalty=10000 +} +%% back to previous value +\def\restorepagebreakaftereq{ + \postdisplaypenalty=\prevpostdisplaypenalty +} + + +%% end +\iantheo@defaultoptions +\endinput diff --git a/libs/point.sty b/libs/point.sty new file mode 100644 index 0000000..9f983dc --- /dev/null +++ b/libs/point.sty @@ -0,0 +1,106 @@ +%% +%% Points package: +%% \point commands +%% + +%% TeX format +\NeedsTeXFormat{LaTeX2e}[1995/12/01] + +%% package name +\ProvidesPackage{point}[2017/06/13] + +%% options +\newif\ifresetatsection +\DeclareOption{reset_at_section}{\resetatsectiontrue} +\DeclareOption{no_reset_at_section}{\resetatsectionfalse} +\newif\ifresetatsubsection +\DeclareOption{reset_at_subsection}{\resetatsubsectiontrue} +\DeclareOption{no_reset_at_subsection}{\resetatsubsectionfalse} +\newif\ifresetattheo +\DeclareOption{reset_at_theo}{\resetattheotrue} +\DeclareOption{no_reset_at_theo}{\resetattheofalse} + +\def\point@defaultoptions{ + \ExecuteOptions{reset_at_section, reset_at_subsection, no_reset_at_theo} + \ProcessOptions + + %% reset at every new section + \ifresetatsection + \let\point@oldsection\section + \gdef\section{\resetpointcounter\point@oldsection} + \fi + %% reset at every new subsection + \ifresetatsubsection + \let\point@oldsubsection\subsection + \gdef\subsection{\resetpointcounter\point@oldsubsection} + \fi + + %% reset at every new theorem + \ifresetattheo + \ifdefined\iantheotrue + \let\point@oldtheo\theo + \gdef\theo{\resetpointcounter\point@oldtheo} + \fi + \fi +} + + +%% point +% counter +\newcounter{pointcount} +\def\point{ + \stepcounter{pointcount} + \setcounter{subpointcount}{0} + % hyperref anchor (only if the class is 'ian') + \ifdefined\ifianclass + \hrefanchor + % define tag (for \label) + \xdef\tag{\thepointcount} + \fi + % header + \indent{\bf \thepointcount\ - } +} + +%% subpoint +% counter +\newcounter{subpointcount} +\def\subpoint{ + \stepcounter{subpointcount} + \setcounter{subsubpointcount}0 + % hyperref anchor (only if the class is 'ian') + \ifdefined\ifianclass + \hrefanchor + % define tag (for \label) + \xdef\tag{\thepointcount-\thesubpointcount} + \fi + % header + \indent\hskip.5cm{\bf \thepointcount-\thesubpointcount\ - } +} + +%% subsubpoint +% counter +\newcounter{subsubpointcount} +\def\subsubpoint{ + \stepcounter{subsubpointcount} + % hyperref anchor (only if the class is 'ian') + \ifdefined\ifianclass + \hrefanchor + % define tag (for \label) + \xdef\tag{\thepointcount-\thesubpointcount-\thesubsubpointcount} + \fi + \indent\hskip1cm{\bf \thepointcount-\thesubpointcount-\thesubsubpointcount\ - } +} + + +%% reset point counters +\def\resetpointcounter{ + \setcounter{pointcount}{0} + \setcounter{subpointcount}{0} + \setcounter{subsubpointcount}{0} +} + + + +%% end +\point@defaultoptions +\endinput