As presented at VirginiaTech on 2019-05-24HEADv1.0master

22 files changed, 1433 insertions, 0 deletions

diff --git a/figs/Millikan-Lauritsen_current.png b/figs/Millikan-Lauritsen_current.png
new file mode 100644
index 0000000..5645be8
--- /dev/null
+++ b/figs/Millikan-Lauritsen_current.png
diff --git a/figs/anim_laser/Makefile b/figs/anim_laser/Makefile
new file mode 100644
index 0000000..9b66e37
--- /dev/null
+++ b/figs/anim_laser/Makefile
@@ -0,0 +1,12 @@
+PROJECTNAME=animate
+
+all: animate.dat
+
+run: animate.dat
+	python3 animate.py animate.dat &
+
+animate.dat:
+	julia laser.jl > animate.dat
+
+clean:
+	rm -f animate.dat
diff --git a/figs/anim_laser/animate.py b/figs/anim_laser/animate.py
new file mode 100644
index 0000000..1e6995a
--- /dev/null
+++ b/figs/anim_laser/animate.py
@@ -0,0 +1,103 @@
+from matplotlib import pyplot as pl
+from matplotlib import animation
+import sys
+
+# read data
+# time dependent data
+frames=[]
+## asymptotic data (located in the first block)
+#asym=[]
+infile=open(sys.argv[1],'r')
+row=[]
+for line in infile:
+    if line=='\n':
+        frames.append(row)
+        row=[]
+    else:
+        dat=[]
+        for n in line.split():
+            dat.append(float(n))
+        row.append(dat)
+infile.close()
+
+
+# set up plot
+fig = pl.figure()
+pl.subplot(211)
+axr=fig.gca()
+asym_rho, = axr.plot([],[],linewidth=3.5,color='#00FF00')
+cn_rho, = axr.plot([],[],color='#FF0000')
+
+pl.subplot(212)
+axJ=fig.gca()
+asym_J, = axJ.plot([],[],linewidth=3.5,color='#00FF00')
+cn_J, = axJ.plot([],[],color='#FF0000')
+
+# plot ranges
+xmax=0
+maxyr=0
+maxyJ=0
+for frame in frames:
+    for i in range(len(frame)):
+        if frame[i][1]>xmax:
+            xmax=frame[i][1]
+        if frame[i][2]>maxyr:
+            maxyr=frame[i][2]
+        if frame[i][3]>maxyr:
+            maxyr=frame[i][3]
+        if frame[i][4]>maxyJ:
+            maxyJ=frame[i][4]
+        if frame[i][5]>maxyJ:
+            maxyJ=frame[i][5]
+xmin=0
+minyr=0
+minyJ=0
+for frame in frames:
+    for i in range(len(frame)):
+        if frame[i][1]<xmin:
+            xmin=frame[i][1]
+        if frame[i][2]<minyr:
+            minyr=frame[i][2]
+        if frame[i][3]<minyr:
+            minyr=frame[i][3]
+        if frame[i][4]<minyJ:
+            minyJ=frame[i][4]
+        if frame[i][5]<minyJ:
+            minyJ=frame[i][5]
+
+# animate
+def init_plot():
+    axr.set_ylim(minyr,maxyr)
+    axr.set_xlim(xmin,xmax)
+    axJ.set_ylim(minyJ,maxyJ)
+    axJ.set_xlim(xmin,xmax)
+
+    axr.vlines(0,minyr,maxyr,linestyles="dotted")
+    axJ.vlines(0,minyJ,maxyJ,linestyles="dotted")
+    axJ.hlines(0,xmin,xmax,linestyles="dotted")
+def update(frame):
+    axr.set_title("t=% .3f fs" % (frame[0][0]))
+    xdata=[]
+    ydata=[]
+    asym_data=[]
+    cn_data=[]
+    for i in range(len(frame)):
+        xdata.append(frame[i][1])
+        asym_data.append(frame[i][2])
+        cn_data.append(frame[i][3])
+    asym_rho.set_data(xdata,asym_data)
+    cn_rho.set_data(xdata,cn_data)
+
+    xdata=[]
+    ydata=[]
+    asym_data=[]
+    cn_data=[]
+    for i in range(len(frame)):
+        xdata.append(frame[i][1])
+        asym_data.append(frame[i][4])
+        cn_data.append(frame[i][5])
+    asym_J.set_data(xdata,asym_data)
+    cn_J.set_data(xdata,cn_data)
+anim = animation.FuncAnimation(fig, update, frames=frames, blit=False, interval=100, repeat=True, init_func=init_plot)
+#anim.save('laser_schrodinger.m4v', fps=10, extra_args=['-vcodec', 'libx264'])
+pl.show()
diff --git a/figs/anim_laser/laser.jl b/figs/anim_laser/laser.jl
new file mode 100644
index 0000000..78cabff
--- /dev/null
+++ b/figs/anim_laser/laser.jl
@@ -0,0 +1,391 @@
+using FFTW
+using Printf
+using SpecialFunctions
+using LinearAlgebra
+
+# numerical values
+hbar=6.58e-16 # eV.s
+m=9.11e-31 # kg
+V_=6.6 # eV
+I=1e17 # W/m^2
+EF=3.3 # eV
+omega_=1.5498*4 # eV
+
+# dimensionless quantities
+V=1
+k=sqrt(EF/V_)
+epsilon=sqrt(hbar^3*I/(8*m*V_*omega_^2))
+omega=omega_/V_
+
+#epsilon=8*epsilon
+#omega=4*omega
+
+nu=60
+ell=20
+
+# for Crank-Nicolson
+xmin_cn=-100
+xmax_cn=100
+nx_cn=4000
+tmax=10
+# mu=dt/dx^2
+mu=0.1
+nt=floor(Int,tmax/(mu*((xmax_cn-xmin_cn)/nx_cn)^2))
+# only save up to memt time steps
+memt=10000
+
+
+# asymptotic value of \varphi_t(x)
+function phi_asym(x,T,gkz,k,epsilon,omega,V,nu,ell)
+  # compute residues
+  res=residue(x,T,gkz,k,epsilon,omega,V,nu)
+  times=compute_times(omega,nu,ell)
+  N=(2*nu+1)*(2*ell+1)
+  fun=zeros(Complex{Float64},N)
+  for m in 1:N
+    for n in -nu:nu
+      fun[m]+=-1im*res[n+nu+1]*exp(-1im*(k*k+n*omega)*times[m])
+    end
+  end
+  return fun
+end
+# asymptotic value of \partial\varphi_t(x)
+function dphi_asym(x,T,gkz,k,epsilon,omega,V,nu,ell)
+  # compute residues
+  res=dresidue(x,T,gkz,k,epsilon,omega,V,nu)
+  times=compute_times(omega,nu,ell)
+  N=(2*nu+1)*(2*ell+1)
+  fun=zeros(Complex{Float64},N)
+  for m in 1:N
+    for n in -nu:nu
+      fun[m]+=-1im*res[n+nu+1]*exp(-1im*(k*k+n*omega)*times[m])
+    end
+  end
+  return fun
+end
+
+# return times
+function compute_times(omega,nu,ell)
+  N=(2*nu+1)*(2*ell+1)
+  times=zeros(N)
+  for j in 1:N
+    times[j]=pi/(omega*(nu+0.5))*(j-1)
+  end
+  return times
+end
+
+# \kappa_m^{(\sigma)}
+function K(m,sigma,V,k,epsilon,omega)
+  return pow(V+2*epsilon*epsilon-k*k-sigma-m*omega,0.5,pi/2)
+end
+# mathfrak q_k
+function Q(k,V)
+  return sqrt(V-k*k)
+end
+
+# T_0
+function T0(k,V)
+  return 2im*k/(1im*k-sqrt(V-k*k))
+end
+
+# compute g
+function g(kappa,epsilon,omega,N)
+  B=zeros(Complex{Float64},N)
+  # prepare vector of B's
+  for i in 1:N
+    t=2*pi/(omega*N)*(i-1)
+    B[i]=exp(1im*epsilon*epsilon/omega*sin(2*omega*t)-kappa*4*epsilon/omega*cos(omega*t))
+  end
+  # take the fft
+  fft!(B)
+  # correct by a factor 1/N
+  for i in 1:N
+    B[i]=B[i]/N
+  end
+  return(B)
+end
+
+# index of m-n
+function mmn(n,nu)
+  return (n>=0 ? n+1 : n+4*nu+2)
+end
+
+# compute residues at -ik^2-in\omega
+function residue(x,T,gkz,k,epsilon,omega,V,nu)
+  # compute residue
+  out=zeros(Complex{Float64},2*nu+1)
+  if x>=0
+    for n in -nu:nu
+      for m in -nu:nu
+	out[n+nu+1]+=1im*gkz[m+nu+1][mmn(n-m,nu)]*exp(-K(m,0,V,k,epsilon,omega)*x)*T[m+nu+1]
+      end
+    end
+  else
+    for n in -nu:nu
+      out[n+nu+1]=(n==0 ? 1im*(exp(1im*k*x)-exp(-1im*k*x)) : 0)
+      for m in -nu:nu
+	out[n+nu+1]+=1im*T[m+nu+1]*gkz[m+nu+1][mmn(n-m,nu)]*exp(-1im*x*pow(k*k+n*omega,0.5,-pi/2))
+      end
+    end
+  end
+
+  return out
+end
+# compute \partial residues at -ik^2-in\omega
+function dresidue(x,T,gkz,k,epsilon,omega,V,nu)
+  # compute residue
+  out=zeros(Complex{Float64},2*nu+1)
+  if x>=0
+    for n in -nu:nu
+      for m in -nu:nu
+	out[n+nu+1]+=-K(m,0,V,k,epsilon,omega)*1im*gkz[m+nu+1][mmn(n-m,nu)]*exp(-K(m,0,V,k,epsilon,omega)*x)*T[m+nu+1]
+      end
+    end
+  else
+    for n in -nu:nu
+      out[n+nu+1]=(n==0 ? -k*(exp(1im*k*x)+exp(-1im*k*x)) : 0)
+      for m in -nu:nu
+	out[n+nu+1]+=pow(k*k+n*omega,0.5,-pi/2)*T[m+nu+1]*gkz[m+nu+1][mmn(n-m,nu)]*exp(-1im*x*pow(k*k+n*omega,0.5,-pi/2))
+      end
+    end
+  end
+
+  return out
+end
+
+
+# matching condition for residue
+function residue_matching(k,epsilon,omega,V,nu)
+  # preliminary: g^{(\kappa_m^{(0)})}
+  # g^{(\kappa_m^{(0)})}_n=gkz[m+nu+1][n+1] if n>=0 and gkz[m+nu+1][n+4*nu+2] if n<0
+  gkz=Array{Array{Complex{Float64}},1}(undef,2*nu+1)
+  for m in -nu:nu
+    gkz[m+nu+1]=g(K(m,0,V,k,epsilon,omega),epsilon,omega,4*nu+1)
+  end
+
+  # solve matching condition
+  G=zeros(Complex{Float64},2*nu+1,2*nu+1)
+  v=zeros(Complex{Float64},2*nu+1)
+  for n in -nu:nu
+    for m in -nu:nu
+      G[n+nu+1,m+nu+1]=((K(m,0,V,k,epsilon,omega)-1im*pow(k*k+n*omega,0.5,-pi/2))*gkz[m+nu+1][mmn(n-m,nu)]-epsilon*(gkz[m+nu+1][mmn(n-m+1,nu)]-gkz[m+nu+1][mmn(n-m-1,nu)]))
+    end
+  end
+  v[nu+1]=-2im*k
+  
+  return (G\v,gkz)
+end
+
+
+# fractional power with an arbitrary branch cut
+function pow(x,a,cut)
+  if(angle(x)/cut<=1)
+    return(abs(x)^a*exp(1im*angle(x)*a))
+  else
+    return(abs(x)^a*exp(1im*(angle(x)-sign(cut)*2*pi)*a))
+  end
+end
+
+# current
+function J(phi,dphi,x,epsilon,omega,t)
+  if x>=0
+    return(2*imag(conj(phi)*(dphi+2im*epsilon*sin(omega*t)*phi)))
+  else
+    return(2*imag(conj(phi)*dphi))
+  end
+end
+
+# compute \psi_t(x) using Crank-Nicolson
+function psi_cn(V,omega,epsilon,k,xs,ts,memt)
+  nx=length(xs)
+  nt=length(ts)
+
+  # length of output vector
+  len=min(nt,memt)
+  # how often to write to output
+  freq=ceil(Int,nt/memt)
+
+  # initialize
+  psis=zeros(Complex{Float64},nx,len)
+  times=zeros(Float64,len)
+
+  # previous time step
+  psi_prev=zeros(Complex{Float64},nx)
+
+  # init
+  for i in 1:nx
+    times[1]=0
+    psis[i,1]=(xs[i]<0 ? exp(1im*k*xs[i])+(T0(k,V)-1)*exp(-1im*k*xs[i]) : T0(k,V)*exp(-Q(k,V)*xs[i]))
+    psi_prev[i]=psis[i,1]
+  end
+  # the boundary condition is that psi at the boundary is constant (nothing needs to be done for this)
+
+  # matrix structure of the Crank-Nicolson algorithm
+  # diagonals of M
+  M0=zeros(Complex{Float64},nx-2)
+  Mp=zeros(Complex{Float64},nx-3)
+  Mm=zeros(Complex{Float64},nx-3)
+  v=zeros(Complex{Float64},nx-2)
+
+  for j in 1:nt-1
+    # print progress
+    progress(j,nt-1,1000)
+
+    # t_j
+    t=ts[j]
+    # t_{j+1}
+    tp=ts[j+1]
+    dt=tp-t
+    for mu in 2:nx-1
+      # x_mu
+      x=xs[mu]
+      dx2=(xs[mu+1]-xs[mu])*(xs[mu]-xs[mu-1])
+      # tridiagonal matrix
+      M0[mu-1]=1im/dt-1/dx2-(x<0 ? 0 : (V-2*epsilon*omega*cos(omega*tp)*x)/2)
+      if mu<nx-1
+	Mp[mu-1]=1/(2*dx2)
+      end
+      if mu>2
+	Mm[mu-2]=1/(2*dx2)
+      end
+      # right side
+      v[mu-1]=1im*psi_prev[mu]/dt-(psi_prev[mu+1]+psi_prev[mu-1]-2*psi_prev[mu])/(2*dx2)+(x<0 ? 0 : (V-2*epsilon*omega*cos(omega*t)*x)*psi_prev[mu]/2)
+      # correct for boundary conditions
+      # assumes the boundary condition is constant in time!
+      if mu==2
+	v[mu-1]-=psi_prev[1]/(2*dx2)
+      end
+      if mu==nx-1
+	v[mu-1]-=psi_prev[nx]/(2*dx2)
+      end
+    end
+
+    M=Tridiagonal(Mm,M0,Mp)
+
+    # copy to psi_prev
+    psi_prev[2:nx-1]=M\v
+
+    # write to output
+    if j%freq==0
+      psis[:,Int(j/freq)+1]=psi_prev
+      times[Int(j/freq)+1]=tp
+    end
+  end
+
+  return (times,psis)
+end
+
+# returns the times in b that are closest to those in a
+# assumes the vectors are ordered
+function nearest_times(a,b)
+  out=zeros(Int,length(a))
+  pointer=1
+  for i in 1:length(a)
+    if pointer==length(b)
+      out[i]=length(b)
+    end
+    for j in pointer:length(b)-1
+      if b[j+1]>a[i]
+	out[i]=(abs(b[j+1]-a[i])<abs(b[j]-a[i]) ? j+1 : j)
+	pointer=out[i]
+	break
+      end
+      # all remaining points are beyond the range
+      if j==length(b)-1
+	out[i]=length(b)
+	pointer=out[i]
+      end
+    end
+  end
+  return out
+end
+
+# space derivative using Crank Nicolson
+function deriv_cn(phi,i,xs)
+  if i==1
+    return (phi[i+1]-phi[i])/(xs[i+1]-xs[i])
+  end
+  if i==length(phi)
+    return (phi[i]-phi[i-1])/(xs[i]-xs[i-1])
+  end
+  return (phi[i+1]-phi[i])/(xs[i+1]-xs[i])/2 + (phi[i]-phi[i-1])/(xs[i]-xs[i-1])/2
+end
+
+################################################
+
+
+# print progress
+function progress(j,tot,freq)
+  if (j-1)%ceil(Int,tot/freq)==0
+    if j>1
+      @printf(stderr,"\r")
+    end
+    @printf(stderr,"%d/%d",j,tot)
+  end
+  if j==tot
+    @printf(stderr,"%d/%d\n",j,tot)
+  end
+end
+
+# print animation data using Crank Nicolson and compare to Laplace
+function print_anim_cn_cmp()
+  @printf(stderr,"epsilon=% .8e omega=% .8e k=% .8e V=% .8e z=% .8e T=% .8e gamma=% .8e\n",epsilon,omega,k,V,4*sqrt(nu)*epsilon/sqrt(omega),2*pi/(omega_/hbar)*1e15,sqrt((V-k^2)/(4*epsilon^2)))
+  xmin=-10
+  xmax=10
+  nx=200
+
+  # array of times for the asymptote
+  times=compute_times(omega,nu,ell)
+
+  # positions at which to compute phi
+  xs=Array(0:nx_cn)*(xmax_cn-xmin_cn)/nx_cn.+xmin_cn
+  # times at which to compute phi
+  ts=Array(0:nt-1)*tmax/nt
+
+  # compute matching condition
+  (T,gkz)=residue_matching(k,epsilon,omega,V,nu)
+
+  # compute wave function
+  (t_psi,psi)=psi_cn(V,omega,epsilon,k,xs,ts,memt)
+
+  # nearest times using Crank Nicolson
+  indices_t=nearest_times(times,t_psi)
+  indices_x=nearest_times(Array(0:nx-1)*(xmax-xmin)/nx.+xmin,xs)
+
+  ps_cn=Array{Array{Complex{Float64},1}}(undef,nx)
+  ps_asym=Array{Array{Complex{Float64},1}}(undef,nx)
+  dps_cn=Array{Array{Complex{Float64},1}}(undef,nx)
+  dps_asym=Array{Array{Complex{Float64},1}}(undef,nx)
+  for i in 1:nx
+    progress(i,nx,1000)
+    x=xmin+(xmax-xmin)*(i-1)/nx
+    ps_cn[i]=Array{Complex{Float64}}(undef,length(times))
+    dps_cn[i]=Array{Complex{Float64}}(undef,length(times))
+    for j in 1:length(times)
+      # phase change in magnetic gauge
+      phase=(x<0 ? 1 : exp(-2im*epsilon*sin(omega*times[j])*x))
+      ps_cn[i][j]=phase*psi[indices_x[i],indices_t[j]]
+      dps_cn[i][j]=phase*deriv_cn(psi[:,indices_t[j]],indices_x[i],xs)-(xs[indices_x[i]]<0 ? 0 : 2im*epsilon*sin(omega*times[j])*psi[indices_x[i],indices_t[j]])
+    end
+    ps_asym[i]=phi_asym(x,T,gkz,k,epsilon,omega,V,nu,ell)
+    dps_asym[i]=dphi_asym(x,T,gkz,k,epsilon,omega,V,nu,ell)
+  end
+
+  # print values at each time
+  for j in 1:length(times)
+    if times[j]>tmax
+      break
+    end
+    for i in 1:nx
+      x=(xmin+(xmax-xmin)*i/nx)
+      # remove point at 0
+      if i!=Int(nx/2)
+	@printf("% .8e % .8e % .8e % .8e % .8e % 8e\n",times[j]*hbar/V_*1e15,x*sqrt(hbar^2*1.6e-19/(2*m*V_))*1e9,abs(ps_asym[i][j])^2,abs(ps_cn[i][j])^2,J(ps_asym[i][j],dps_asym[i][j],x,epsilon,omega,times[j]),J(ps_cn[i][j],dps_cn[i][j],x,epsilon,omega,times[j]))
+      end
+    end
+    print('\n')
+  end
+end
+
+
+print_anim_cn_cmp()
diff --git a/figs/animation/FN_base.jl b/figs/animation/FN_base.jl
new file mode 100644
index 0000000..af2a1ee
--- /dev/null
+++ b/figs/animation/FN_base.jl
@@ -0,0 +1,170 @@
+# fractional power with an arbitrary branch cut
+function pow(x,a,cut)
+  if(angle(x)/cut<=1)
+    return(abs(x)^a*exp(1im*angle(x)*a))
+  else
+    return(abs(x)^a*exp(1im*(angle(x)-sign(cut)*2*pi)*a))
+  end
+end
+
+# asymptotic airy functions
+# specify a branch cut for the fractional power
+function airyai_asym(x,cut)
+  if(abs(real(pow(x,3/2,cut)))<airy_threshold)
+    return(exp(2/3*pow(x,3/2,cut))*airyai(x))
+  else
+    ret=0
+    for n in 0:airy_order
+      ret+=gamma(n+5/6)*gamma(n+1/6)*(-3/4)^n/(4*pi^(3/2)*factorial(n)*pow(x,3*n/2+1/4,cut))
+    end
+    return ret
+  end
+end
+function airyaiprime_asym(x,cut)
+  if(abs(real(pow(x,3/2,cut)))<airy_threshold)
+    return(exp(2/3*pow(x,3/2,cut))*airyaiprime(x))
+  else
+    ret=0
+    for n in 0:airy_order
+      ret+=gamma(n+5/6)*gamma(n+1/6)*(-3/4)^n/(4*pi^(3/2)*factorial(n))*(-1/pow(x,3*n/2-1/4,cut)-(3/2*n+1/4)/pow(x,3*n/2+5/4,cut))
+    end
+    return ret
+  end
+end
+
+# solutions of (-\Delta+V-ip)phi=0
+# assume that p has an infinitesimal real part (and adjust the branch cuts appropriately)
+function phi(p,x,E,V)
+  return(airyai_asym(exp(-1im*pi/3)*(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi))
+end
+function dphi(p,x,E,V)
+  return(exp(-1im*pi/3)*E^(1/3)*airyaiprime_asym(exp(-1im*pi/3)*(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi))
+end
+function eta(p,x,E,V)
+  return(exp(-1im*pi/3)*airyai_asym(-(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi/2))
+end
+function deta(p,x,E,V)
+  return(-exp(-1im*pi/3)*E^(1/3)*airyaiprime_asym(-(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi/2))
+end
+
+# Laplace transform of psi
+# assume that p has an infinitesimal real part (and adjust the branch cuts appropriately)
+# for example, (1im*p-V)^(3/2) becomes pow(1im*p-V,3/2,-pi/2) because when 1im*p is real negative, its square root should be imaginary positive
+function f(p,x,k0,E,V)
+  T=2im*k0/(1im*k0-sqrt(V-k0*k0))
+  R=T-1
+
+  if x>=0
+    C2=-1im*T/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*((sqrt(V-k0*k0)+pow(-1im*p,1/2,pi/2))/(-1im*p+k0*k0)-2im*E^(-1/3)*pi*quadgk(y -> (pow(-1im*p,1/2,pi/2)*eta(p,0,E,V)-deta(p,0,E,V))*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1])
+    FT=2*E^(-1/3)*pi*(quadgk(y -> phi(p,x,E,V)*eta(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2))),0,x)[1]+quadgk(y -> eta(p,x,E,V)*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2))),x,Inf)[1])
+    main=C2*phi(p,x,E,V)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2)))+T*FT
+
+    # subtract the contribution of the pole, which will be added back in after the integration
+    pole=psi_pole(x,k0,E,V)/(p+1im*k0*k0)
+    return(main-pole)
+  else
+    C1=-1im*T*((sqrt(V-k0*k0)*phi(p,0,E,V)+dphi(p,0,E,V))/(-1im*p+k0*k0)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))+E^(-1/3)*quadgk(y -> phi(p,y,E,V)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1])
+    FI=-1im*exp(1im*k0*x)/(-1im*p+k0*k0)
+    FR=-1im*exp(-1im*k0*x)/(-1im*p+k0*k0)
+    main=C1*exp(pow(-1im*p,1/2,pi/2)*x)+FI+R*FR
+
+    # subtract the contribution of the pole, which will be added back in after the integration
+    pole=psi_pole(x,k0,E,V)/(p+1im*k0*k0)
+    return(main-pole)
+  end
+end
+# its derivative
+function df(p,x,k0,E,V)
+  T=2im*k0/(1im*k0-sqrt(V-k0*k0))
+  R=T-1
+
+  if x>=0
+    C2=-1im*T/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*((sqrt(V-k0*k0)+pow(-1im*p,1/2,pi/2))/(-1im*p+k0*k0)-2im*E^(-1/3)*pi*quadgk(y -> (pow(-1im*p,1/2,pi/2)*eta(p,0,E,V)-deta(p,0,E,V))*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1])
+    dFT=2*E^(-1/3)*pi*(quadgk(y -> dphi(p,x,E,V)*eta(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2))),0,x)[1]+quadgk(y -> deta(p,x,E,V)*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2))),x,Inf)[1])
+    main=C2*dphi(p,x,E,V)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2)))+T*dFT
+
+    # subtract the contribution of the pole, which will be added back in after the integration
+    pole=dpsi_pole(x,k0,E,V)/(p+1im*k0*k0)
+    return(main-pole)
+  else
+    C1=-1im*T*((sqrt(V-k0*k0)*phi(p,0,E,V)+dphi(p,0,E,V))/(-1im*p+k0*k0)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))+E^(-1/3)*quadgk(y -> phi(p,y,E,V)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1])
+    dFI=k0*exp(1im*k0*x)/(-1im*p+k0*k0)
+    dFR=-k0*exp(-1im*k0*x)/(-1im*p+k0*k0)
+    main=C1*pow(-1im*p,1/2,pi/2)*exp(pow(-1im*p,1/2,pi/2)*x)+dFI+R*dFR
+
+    # subtract the contribution of the pole, which will be added back in after the integration
+    pole=dpsi_pole(x,k0,E,V)/(p+1im*k0*k0)
+    return(main-pole)
+  end
+end
+
+# psi (returns t,psi(x,t))
+function psi(x,k0,E,V,p_npoints,p_cutoff)
+  fft=fourier_fft(f,x,k0,E,V,p_npoints,p_cutoff)
+  # add the contribution of the pole
+  for i in 1:p_npoints
+      fft[2][i]=fft[2][i]+psi_pole(x,k0,E,V)*exp(-1im*k0*k0*fft[1][i])
+  end
+  return(fft)
+end
+# its derivative
+function dpsi(x,k0,E,V,p_npoints,p_cutoff)
+  fft=fourier_fft(df,x,k0,E,V,p_npoints,p_cutoff)
+  # add the contribution of the pole
+  for i in 1:p_npoints
+    fft[2][i]=fft[2][i]+dpsi_pole(x,k0,E,V)*exp(-1im*k0*k0*fft[1][i])
+  end
+  return(fft)
+end
+
+# compute Fourier transform by sampling and fft
+function fourier_fft(A,x,k0,E,V,p_npoints,p_cutoff)
+  fun=zeros(Complex{Float64},p_npoints)
+  times=zeros(p_npoints)
+
+  # prepare fft
+  for i in 1:p_npoints
+    fun[i]=p_cutoff/pi*A(1im*(-p_cutoff+2*p_cutoff*(i-1)/p_npoints),x,k0,E,V)
+    times[i]=(i-1)*pi/p_cutoff
+  end
+
+  ifft!(fun)
+  
+  # correct the phase
+  for i in 2:2:p_npoints
+    fun[i]=-fun[i]
+  end
+  return([times,fun])
+end
+
+# asymptotic value of psi
+function psi_pole(x,k0,E,V)
+  if x>=0
+    return(1im*phi(-1im*k0*k0,x,E,V)*2*k0/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(k0*k0-V),3/2,-pi/2)-E^(-1)*pow(k0*k0-V,3/2,-pi/2))))
+  else
+    return((1im*k0*phi(-1im*k0*k0,0,E,V)-dphi(-1im*k0*k0,0,E,V))/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(-1im*k0*x)+exp(1im*k0*x))
+  end
+end
+function dpsi_pole(x,k0,E,V)
+  if x>=0
+    return(1im*dphi(-1im*k0*k0,x,E,V)*2*k0/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(k0*k0-V),3/2,-pi/2)-E^(-1)*pow(k0*k0-V,3/2,-pi/2))))
+  else
+    return(-1im*k0*(1im*k0*phi(-1im*k0*k0,0,E,V)-dphi(-1im*k0*k0,0,E,V))/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(-1im*k0*x)+1im*k0*exp(1im*k0*x))
+  end
+end
+
+# current
+function J(ps,dps)
+  return(2*imag(conj(ps)*dps))
+end
+
+# complete computation of the current
+function current(x,k0,E,V,p_npoints,p_cutoff)
+  ps=psi(x,k0,E,V,p_npoints,p_cutoff)
+  dps=dpsi(x,k0,E,V,p_npoints,p_cutoff)
+  Js=zeros(Complex{Float64},p_npoints)
+  for i in 1:p_npoints
+    Js[i]=J(ps[2][i],dps[2][i])
+  end
+  return(Js)
+end
diff --git a/figs/animation/Makefile b/figs/animation/Makefile
new file mode 100644
index 0000000..d582a40
--- /dev/null
+++ b/figs/animation/Makefile
@@ -0,0 +1,12 @@
+PROJECTNAME=animate
+
+all: animate.dat
+
+run: animate.dat
+	python3 animate.py animate.dat &
+
+animate.dat:
+	julia animate_compute.jl > animate.dat
+
+clean:
+	rm -f animate.dat
diff --git a/figs/animation/animate.py b/figs/animation/animate.py
new file mode 100644
index 0000000..aaa3a80
--- /dev/null
+++ b/figs/animation/animate.py
@@ -0,0 +1,127 @@
+from matplotlib import pyplot as pl
+from matplotlib import animation
+import sys
+
+# read data
+# time dependent data
+frames=[]
+# asymptotic data (located in the first block)
+asym=[]
+infile=open(sys.argv[1],'r')
+row=[]
+for line in infile:
+    # read first block
+    if len(asym)==0:
+        if line=='\n':
+            asym=row
+            row=[]
+        else:
+            dat=[]
+            for n in line.split():
+                dat.append(float(n))
+            row.append(dat)
+    # read other blocks
+    else:
+        if line=='\n':
+            frames.append(row)
+            row=[]
+        else:
+            dat=[]
+            for n in line.split():
+                dat.append(float(n))
+            row.append(dat)
+infile.close()
+
+
+# set up plot
+fig = pl.figure()
+#pl.subplot(211)
+axr=fig.gca()
+asym_rho, = axr.plot([],[],linewidth=3.5,color='#00FF00')
+rho, = axr.plot([],[],color='red')
+
+#pl.subplot(212)
+#axJ=fig.gca()
+#asym_J, = axJ.plot([],[],linewidth=3.5,color='#00FF00')
+#J, = axJ.plot([],[],color='red')
+
+# plot ranges
+xmax=0
+maxyr=0
+maxyJ=0
+for frame in frames:
+    for i in range(len(frame)):
+        if frame[i][1]>xmax:
+            xmax=frame[i][1]
+        if frame[i][2]>maxyr:
+            maxyr=frame[i][2]
+        if frame[i][3]>maxyJ:
+            maxyJ=frame[i][3]
+for i in range(len(asym)):
+    if asym[i][0]>xmax:
+        xmax=asym[i][0]
+    if asym[i][1]>maxyr:
+        maxyr=asym[i][1]
+    if asym[i][2]>maxyJ:
+        maxyJ=asym[i][2]
+xmin=0
+minyr=0
+minyJ=0
+for frame in frames:
+    for i in range(len(frame)):
+        if frame[i][1]<xmin:
+            xmin=frame[i][1]
+        if frame[i][2]<minyr:
+            minyr=frame[i][2]
+        if frame[i][3]<minyJ:
+            minyJ=frame[i][3]
+for i in range(len(asym)):
+    if asym[i][0]<xmin:
+        xmin=asym[i][0]
+    if asym[i][1]<minyr:
+        minyr=asym[i][1]
+    if asym[i][2]<minyJ:
+        minyJ=asym[i][2]
+
+
+# plot asymptotes
+asym_rho_datax=[]
+asym_rho_datay=[]
+for i in range(len(asym)):
+    asym_rho_datax.append(asym[i][0])
+    asym_rho_datay.append(asym[i][1])
+asym_rho.set_data(asym_rho_datax,asym_rho_datay)
+#asym_J_datax=[]
+#asym_J_datay=[]
+#for i in range(len(asym)):
+#    asym_J_datax.append(asym[i][0])
+#    asym_J_datay.append(asym[i][2])
+#asym_J.set_data(asym_J_datax,asym_J_datay)
+
+# animate
+def init_plot():
+    axr.set_ylim(minyr,maxyr)
+    axr.set_xlim(xmin,xmax)
+    #axJ.set_ylim(minyJ,maxyJ)
+    #axJ.set_xlim(xmin,xmax)
+
+    axr.vlines(0,minyr,maxyr,linestyles="dotted")
+    #axJ.vlines(0,minyJ,maxyJ,linestyles="dotted")
+def update(frame):
+    axr.set_title("t=% .3f fs" % (frame[0][0]))
+    xdata=[]
+    ydata=[]
+    for i in range(len(frame)):
+        xdata.append(frame[i][1])
+        ydata.append(frame[i][2])
+    rho.set_data(xdata,ydata)
+
+    #xdata=[]
+    #ydata=[]
+    #for i in range(len(frame)):
+    #    xdata.append(frame[i][1])
+    #    ydata.append(frame[i][3])
+    #J.set_data(xdata,ydata)
+anim = animation.FuncAnimation(fig, update, frames=frames, blit=False, interval=100, repeat=True, init_func=init_plot)
+#anim.save('schrodinger_barrier.mp4', fps=15, extra_args=['-vcodec', 'libx264'])
+pl.show()
diff --git a/figs/animation/animate_compute.jl b/figs/animation/animate_compute.jl
new file mode 100644
index 0000000..452320e
--- /dev/null
+++ b/figs/animation/animate_compute.jl
@@ -0,0 +1,66 @@
+using QuadGK
+using SpecialFunctions
+using FFTW
+
+# numerical values
+hbar=6.58e-16 # eV.s
+m=9.11e-31 # kg
+Vn=9 # eV
+#En=14e9 # V/m
+En=10e9 # V/m
+Kn=4.5 # eV
+
+V=1
+E=En*hbar/(2*Vn^1.5*m^0.5)*sqrt(1.60e-19)
+k0=sqrt(Kn/Vn)
+
+# rescale x to nm
+nm_scale=hbar*sqrt(1.6e-19)/sqrt(2*m*Vn)*1e9
+
+# cutoffs
+p_cutoff=20*k0
+p_npoints=256
+
+# airy approximations
+airy_threshold=30
+airy_order=5
+
+# xbounds
+xmax=10
+xmin=-10
+x_npoints=200
+
+include("FN_base.jl")
+
+# compute wave function
+ps=Array{Array{Array{Complex{Float64},1},1}}(undef,x_npoints)
+dps=Array{Array{Array{Complex{Float64},1},1}}(undef,x_npoints)
+for i in 1:x_npoints
+  print(stderr,i,'/',x_npoints,'\n')
+  x=xmin+(xmax-xmin)*i/x_npoints
+  ps[i]=psi(x,k0,E,V,p_npoints,p_cutoff)
+  dps[i]=dpsi(x,k0,E,V,p_npoints,p_cutoff)
+end
+
+# compute asymptotic values
+ps_asym=Array{Complex{Float64}}(undef,x_npoints)
+dps_asym=Array{Complex{Float64}}(undef,x_npoints)
+for i in 1:x_npoints
+  x=xmin+(xmax-xmin)*i/x_npoints
+  ps_asym[i]=psi_pole(x,k0,E,V)
+  dps_asym[i]=dpsi_pole(x,k0,E,V)
+end
+
+# print asymptotic values
+for i in 1:x_npoints
+  print((xmin+(xmax-xmin)*i/x_npoints)*nm_scale,' ',abs(ps_asym[i])^2,' ',J(ps_asym[i],dps_asym[i])/(2*k0),'\n')
+end
+print('\n')
+
+# print values at each time
+for j in 1:p_npoints
+  for i in 1:x_npoints
+    print(real(ps[i][1][j])*hbar/Vn*1e15,' ',(xmin+(xmax-xmin)*i/x_npoints)*nm_scale,' ',abs(ps[i][2][j])^2,' ',J(ps[i][2][j],dps[i][2][j])/(2*k0),'\n')
+  end
+  print('\n')
+end
diff --git a/figs/contour.fig/Makefile b/figs/contour.fig/Makefile
new file mode 100644
index 0000000..bade7d0
--- /dev/null
+++ b/figs/contour.fig/Makefile
@@ -0,0 +1,30 @@
+PROJECTNAME=contour
+
+PDFS=$(addsuffix .pdf, $(PROJECTNAME))
+
+all: $(PDFS)
+
+$(PDFS): poles.tikz.tex
+	pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@)
+
+poles.tikz.tex:
+	python3 contour.py > poles.tikz.tex
+
+install: $(PDFS)
+	cp $^ $(INSTALLDIR)/
+
+$(LIBS):
+	ln -fs libs/$@ ./
+
+clean-libs:
+	rm -f $(LIBS)
+
+clean-aux:
+	rm -f $(addsuffix .aux, $(PROJECTNAME))
+	rm -f $(addsuffix .log, $(PROJECTNAME))
+	rm -f poles.tikz.tex
+
+clean-tex:
+	rm -f $(PDFS)
+
+clean: clean-libs clean-aux clean-tex
diff --git a/figs/contour.fig/contour.py b/figs/contour.fig/contour.py
new file mode 100644
index 0000000..102df86
--- /dev/null
+++ b/figs/contour.fig/contour.py
@@ -0,0 +1,82 @@
+#!/usr/bin/env python3
+
+import cmath
+import math
+import scipy.special as sp
+from scipy import optimize as op
+import random
+import sys
+
+# number of roots
+nr_roots=4
+
+# size of plot
+plotsize_x=3
+plotsize_y=3
+# rescale plot (so roots are not too close together)
+plotsize_scale_x=1
+plotsize_scale_y=6
+
+# numerical values
+hbar=6.58e-16 # eV.s
+m=9.11e-31 # kg
+Vn=9 # eV
+En=20e9 # V/m
+
+V=1
+E=En*hbar/(Vn**1.5*m**0.5)*math.sqrt(1.60e-19)
+
+# sqrt with branch cut along iR_+
+def sqrt_p(x):
+    r,p=cmath.polar(x)
+    if(p<cmath.pi/2):
+        return(cmath.rect(math.sqrt(r),p/2))
+    else:
+        return(cmath.rect(math.sqrt(r),(p-2*math.pi)/2))
+
+# solution of (-\Delta+V-ip)phi=0
+def phi(p,x,E,V):
+    return(sp.airy(cmath.exp(-1j*math.pi/3)*(E**(1/3)*x-E**(-2/3)*(V-1j*p)))[0])
+# its derivative
+def dphi(p,x,E,V):
+    return(cmath.exp(-1j*math.pi/3)*E**(1/3)*sp.airy(cmath.exp(-1j*math.pi/3)*(E**(1/3)*x-E**(-2/3)*(V-1j*p)))[1])
+
+
+# the function whose roots are to be computed and its derivative
+def f(p):
+    return(sqrt_p(-1j*p)*phi(p,0,E,V)-dphi(p,0,E,V))
+def df(p):
+    return(-1j/(2*sqrt_p(-1j*p))*phi(p,0,E,V)+sqrt_p(-1j*p)*dp_phi(p,0,E,V)-dp_dphi(p,0,E,V))
+
+# derivatives of phi with respect to p
+def dp_phi(p,x,E,V):
+    return(1j*cmath.exp(-1j*math.pi/3)*E**(-2/3)*sp.airy(cmath.exp(-1j*math.pi/3)*(E**(1/3)*x-E**(-2/3)*(V-1j*p)))[1])
+def dp_dphi(p,x,E,V):
+    return(-1j*(x-(V-1j*p)/E)*phi(p,x,E,V))
+
+# check whether the root was already found
+def check(root,roots):
+    # reject if the root doesn't fit on the plot
+    if(plotsize_scale_x*root.real<-plotsize_x or abs(plotsize_scale_y*root.imag)>plotsize_y):
+        return(False)
+    for x in roots:
+        if(abs(root-x)<1e-12):
+            return(False)
+    return(True)
+
+# find roots
+roots=[]
+while len(roots)<nr_roots:
+    try:
+        root=op.newton(f, -random.random()*plotsize_x/plotsize_scale_x+(2*random.random()-1)*plotsize_y/plotsize_scale_y*1j, fprime=df)
+        if(check(root,roots)):
+            roots.append(root)
+            print("found "+str(len(roots))+" roots of "+str(nr_roots), file=sys.stderr)
+    except RuntimeError:
+        root=0
+    except:
+        break
+
+# print result
+for root in roots:
+    print("\\pole{(% .3f,% .3f)}" % (plotsize_scale_x*root.real,plotsize_scale_y*root.imag))
diff --git a/figs/contour.fig/contour.tikz.tex b/figs/contour.fig/contour.tikz.tex
new file mode 100644
index 0000000..953d8f0
--- /dev/null
+++ b/figs/contour.fig/contour.tikz.tex
@@ -0,0 +1,34 @@
+\documentclass{standalone}
+
+\usepackage{tikz}
+
+\def\pole#1{
+  \draw#1++(0.1,0.1)--++(-0.2,-0.2);
+  \draw#1++(-0.1,0.1)--++(0.2,-0.2);
+  \draw[color=red, line width=1pt]#1circle(0.3);
+}
+
+\begin{document}
+\begin{tikzpicture}
+
+% branch cut
+\fill[color=gray](-3.3,-0.1)--++(3.3,0)--++(0,0.2)--++(-3.3,0)--cycle;
+
+% axes
+\draw(-3.3,0)--++(6.6,0);
+\draw(0,-3.3)--++(0,6.6);
+
+% -ik0^2
+\pole{(0,-2)}
+
+% other poles
+\input{poles.tikz}
+
+% contour
+\draw[color=red, line width=1pt](-3.3,-0.3)--++(3.3,0);
+\draw[color=red, line width=1pt](0,-0.3)..controls(0.3,-0.3)and(0.3,0.3)..(0,0.3);
+\draw[color=red, line width=1pt](-3.3,0.3)--++(3.3,0);
+
+
+\end{tikzpicture}
+\end{document}
diff --git a/figs/emitter.jpg b/figs/emitter.jpg
new file mode 100644
index 0000000..d84ad1f
--- /dev/null
+++ b/figs/emitter.jpg
diff --git a/figs/fowler-nordheim.fig/FN_base.jl b/figs/fowler-nordheim.fig/FN_base.jl
new file mode 100644
index 0000000..af2a1ee
--- /dev/null
+++ b/figs/fowler-nordheim.fig/FN_base.jl
@@ -0,0 +1,170 @@
+# fractional power with an arbitrary branch cut
+function pow(x,a,cut)
+  if(angle(x)/cut<=1)
+    return(abs(x)^a*exp(1im*angle(x)*a))
+  else
+    return(abs(x)^a*exp(1im*(angle(x)-sign(cut)*2*pi)*a))
+  end
+end
+
+# asymptotic airy functions
+# specify a branch cut for the fractional power
+function airyai_asym(x,cut)
+  if(abs(real(pow(x,3/2,cut)))<airy_threshold)
+    return(exp(2/3*pow(x,3/2,cut))*airyai(x))
+  else
+    ret=0
+    for n in 0:airy_order
+      ret+=gamma(n+5/6)*gamma(n+1/6)*(-3/4)^n/(4*pi^(3/2)*factorial(n)*pow(x,3*n/2+1/4,cut))
+    end
+    return ret
+  end
+end
+function airyaiprime_asym(x,cut)
+  if(abs(real(pow(x,3/2,cut)))<airy_threshold)
+    return(exp(2/3*pow(x,3/2,cut))*airyaiprime(x))
+  else
+    ret=0
+    for n in 0:airy_order
+      ret+=gamma(n+5/6)*gamma(n+1/6)*(-3/4)^n/(4*pi^(3/2)*factorial(n))*(-1/pow(x,3*n/2-1/4,cut)-(3/2*n+1/4)/pow(x,3*n/2+5/4,cut))
+    end
+    return ret
+  end
+end
+
+# solutions of (-\Delta+V-ip)phi=0
+# assume that p has an infinitesimal real part (and adjust the branch cuts appropriately)
+function phi(p,x,E,V)
+  return(airyai_asym(exp(-1im*pi/3)*(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi))
+end
+function dphi(p,x,E,V)
+  return(exp(-1im*pi/3)*E^(1/3)*airyaiprime_asym(exp(-1im*pi/3)*(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi))
+end
+function eta(p,x,E,V)
+  return(exp(-1im*pi/3)*airyai_asym(-(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi/2))
+end
+function deta(p,x,E,V)
+  return(-exp(-1im*pi/3)*E^(1/3)*airyaiprime_asym(-(E^(1/3)*x-E^(-2/3)*(V-1im*p)),pi/2))
+end
+
+# Laplace transform of psi
+# assume that p has an infinitesimal real part (and adjust the branch cuts appropriately)
+# for example, (1im*p-V)^(3/2) becomes pow(1im*p-V,3/2,-pi/2) because when 1im*p is real negative, its square root should be imaginary positive
+function f(p,x,k0,E,V)
+  T=2im*k0/(1im*k0-sqrt(V-k0*k0))
+  R=T-1
+
+  if x>=0
+    C2=-1im*T/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*((sqrt(V-k0*k0)+pow(-1im*p,1/2,pi/2))/(-1im*p+k0*k0)-2im*E^(-1/3)*pi*quadgk(y -> (pow(-1im*p,1/2,pi/2)*eta(p,0,E,V)-deta(p,0,E,V))*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1])
+    FT=2*E^(-1/3)*pi*(quadgk(y -> phi(p,x,E,V)*eta(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2))),0,x)[1]+quadgk(y -> eta(p,x,E,V)*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2))),x,Inf)[1])
+    main=C2*phi(p,x,E,V)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2)))+T*FT
+
+    # subtract the contribution of the pole, which will be added back in after the integration
+    pole=psi_pole(x,k0,E,V)/(p+1im*k0*k0)
+    return(main-pole)
+  else
+    C1=-1im*T*((sqrt(V-k0*k0)*phi(p,0,E,V)+dphi(p,0,E,V))/(-1im*p+k0*k0)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))+E^(-1/3)*quadgk(y -> phi(p,y,E,V)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1])
+    FI=-1im*exp(1im*k0*x)/(-1im*p+k0*k0)
+    FR=-1im*exp(-1im*k0*x)/(-1im*p+k0*k0)
+    main=C1*exp(pow(-1im*p,1/2,pi/2)*x)+FI+R*FR
+
+    # subtract the contribution of the pole, which will be added back in after the integration
+    pole=psi_pole(x,k0,E,V)/(p+1im*k0*k0)
+    return(main-pole)
+  end
+end
+# its derivative
+function df(p,x,k0,E,V)
+  T=2im*k0/(1im*k0-sqrt(V-k0*k0))
+  R=T-1
+
+  if x>=0
+    C2=-1im*T/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*((sqrt(V-k0*k0)+pow(-1im*p,1/2,pi/2))/(-1im*p+k0*k0)-2im*E^(-1/3)*pi*quadgk(y -> (pow(-1im*p,1/2,pi/2)*eta(p,0,E,V)-deta(p,0,E,V))*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1])
+    dFT=2*E^(-1/3)*pi*(quadgk(y -> dphi(p,x,E,V)*eta(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2))),0,x)[1]+quadgk(y -> deta(p,x,E,V)*phi(p,y,E,V)*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2))),x,Inf)[1])
+    main=C2*dphi(p,x,E,V)*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2)))+T*dFT
+
+    # subtract the contribution of the pole, which will be added back in after the integration
+    pole=dpsi_pole(x,k0,E,V)/(p+1im*k0*k0)
+    return(main-pole)
+  else
+    C1=-1im*T*((sqrt(V-k0*k0)*phi(p,0,E,V)+dphi(p,0,E,V))/(-1im*p+k0*k0)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))+E^(-1/3)*quadgk(y -> phi(p,y,E,V)/(pow(-1im*p,1/2,pi/2)*phi(p,0,E,V)-dphi(p,0,E,V))*exp(-sqrt(V-k0*k0)*y)*exp(2im/3*(pow(E^(1/3)*y+E^(-2/3)*(1im*p-V),3/2,-pi/2)-E^(-1)*pow(1im*p-V,3/2,-pi/2))),0,Inf)[1])
+    dFI=k0*exp(1im*k0*x)/(-1im*p+k0*k0)
+    dFR=-k0*exp(-1im*k0*x)/(-1im*p+k0*k0)
+    main=C1*pow(-1im*p,1/2,pi/2)*exp(pow(-1im*p,1/2,pi/2)*x)+dFI+R*dFR
+
+    # subtract the contribution of the pole, which will be added back in after the integration
+    pole=dpsi_pole(x,k0,E,V)/(p+1im*k0*k0)
+    return(main-pole)
+  end
+end
+
+# psi (returns t,psi(x,t))
+function psi(x,k0,E,V,p_npoints,p_cutoff)
+  fft=fourier_fft(f,x,k0,E,V,p_npoints,p_cutoff)
+  # add the contribution of the pole
+  for i in 1:p_npoints
+      fft[2][i]=fft[2][i]+psi_pole(x,k0,E,V)*exp(-1im*k0*k0*fft[1][i])
+  end
+  return(fft)
+end
+# its derivative
+function dpsi(x,k0,E,V,p_npoints,p_cutoff)
+  fft=fourier_fft(df,x,k0,E,V,p_npoints,p_cutoff)
+  # add the contribution of the pole
+  for i in 1:p_npoints
+    fft[2][i]=fft[2][i]+dpsi_pole(x,k0,E,V)*exp(-1im*k0*k0*fft[1][i])
+  end
+  return(fft)
+end
+
+# compute Fourier transform by sampling and fft
+function fourier_fft(A,x,k0,E,V,p_npoints,p_cutoff)
+  fun=zeros(Complex{Float64},p_npoints)
+  times=zeros(p_npoints)
+
+  # prepare fft
+  for i in 1:p_npoints
+    fun[i]=p_cutoff/pi*A(1im*(-p_cutoff+2*p_cutoff*(i-1)/p_npoints),x,k0,E,V)
+    times[i]=(i-1)*pi/p_cutoff
+  end
+
+  ifft!(fun)
+  
+  # correct the phase
+  for i in 2:2:p_npoints
+    fun[i]=-fun[i]
+  end
+  return([times,fun])
+end
+
+# asymptotic value of psi
+function psi_pole(x,k0,E,V)
+  if x>=0
+    return(1im*phi(-1im*k0*k0,x,E,V)*2*k0/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(k0*k0-V),3/2,-pi/2)-E^(-1)*pow(k0*k0-V,3/2,-pi/2))))
+  else
+    return((1im*k0*phi(-1im*k0*k0,0,E,V)-dphi(-1im*k0*k0,0,E,V))/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(-1im*k0*x)+exp(1im*k0*x))
+  end
+end
+function dpsi_pole(x,k0,E,V)
+  if x>=0
+    return(1im*dphi(-1im*k0*k0,x,E,V)*2*k0/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(2im/3*(pow(E^(1/3)*x+E^(-2/3)*(k0*k0-V),3/2,-pi/2)-E^(-1)*pow(k0*k0-V,3/2,-pi/2))))
+  else
+    return(-1im*k0*(1im*k0*phi(-1im*k0*k0,0,E,V)-dphi(-1im*k0*k0,0,E,V))/(1im*k0*phi(-1im*k0*k0,0,E,V)+dphi(-1im*k0*k0,0,E,V))*exp(-1im*k0*x)+1im*k0*exp(1im*k0*x))
+  end
+end
+
+# current
+function J(ps,dps)
+  return(2*imag(conj(ps)*dps))
+end
+
+# complete computation of the current
+function current(x,k0,E,V,p_npoints,p_cutoff)
+  ps=psi(x,k0,E,V,p_npoints,p_cutoff)
+  dps=dpsi(x,k0,E,V,p_npoints,p_cutoff)
+  Js=zeros(Complex{Float64},p_npoints)
+  for i in 1:p_npoints
+    Js[i]=J(ps[2][i],dps[2][i])
+  end
+  return(Js)
+end
diff --git a/figs/fowler-nordheim.fig/Makefile b/figs/fowler-nordheim.fig/Makefile
new file mode 100644
index 0000000..a6e5e53
--- /dev/null
+++ b/figs/fowler-nordheim.fig/Makefile
@@ -0,0 +1,29 @@
+PROJECTNAME=asymptotic
+
+PDFS=$(addsuffix .pdf, $(PROJECTNAME))
+TEXS=$(addsuffix .tikz.tex, $(PROJECTNAME))
+
+all: $(PDFS)
+
+$(PDFS): $(addsuffix .dat, $(PROJECTNAME))
+	gnuplot $(patsubst %.pdf, %.gnuplot, $@) > $(patsubst %.pdf, %.tikz.tex, $@)
+	pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@)
+
+asymptotic.dat:
+	julia asymptotic.jl > asymptotic.dat
+
+install: $(PDFS)
+	cp $^ $(INSTALLDIR)/
+
+clean-aux:
+	rm -f $(addsuffix .aux, $(PROJECTNAME))
+	rm -f $(addsuffix .log, $(PROJECTNAME))
+
+clean-dat:
+	rm -f $(addsuffix .tikz.tex, $(PROJECTNAME))
+	rm -f short-time.dat
+
+clean-tex:
+	rm -f $(PDFS)
+
+clean: clean-dat clean-aux clean-tex
diff --git a/figs/fowler-nordheim.fig/asymptotic.gnuplot b/figs/fowler-nordheim.fig/asymptotic.gnuplot
new file mode 100644
index 0000000..3296380
--- /dev/null
+++ b/figs/fowler-nordheim.fig/asymptotic.gnuplot
@@ -0,0 +1,59 @@
+datafile="asymptotic.dat"
+
+## can also set the following options
+#set title ""
+set ylabel "$|\\psi_{\\mathrm{FN}}|^2$" tc ls 1 #norotate
+set y2label "$J_{\\mathrm{FN}}$" tc ls 2 #norotate
+set xlabel "$x$"
+#
+#set xrange[:]
+#set yrange [:]
+set y2range [0:0.004]
+#
+## start ticks at 0, then every x
+#set xtics 0,x
+#set ytics 0,x
+## puts 4 minor tics between tics (5 intervals, i.e. every 0.01)
+set mxtics 5
+set mytics 5
+set my2tics 5
+
+# default output canvas size: 12.5cm x 8.75cm
+set term lua tikz size 12.5,8.75 standalone
+# run
+## gnuplot gnuplots && gnuplot_tikz out/latext/minimizer.tex
+
+set key off
+
+# 3=1+2 draw bottom and left sides of the box
+#set border 3
+# don't show tics on opposite sides
+set xtics nomirror
+set ytics nomirror tc ls 1
+set y2tics nomirror tc ls 2
+
+# Mathematica colors:
+## 3f3d99 (dark blue)
+## 9c4275 (dark pink)
+## 9a8d3f (dark yellow)
+## 3d9956 (dark green)
+# My colors
+## 4169E1 (pastel blue)
+## DC143C (bright red)
+## 32CD32 (bright green)
+## 4B0082 (deep purple)
+## DAA520 (ochre)
+
+# 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
+
+set arrow to 0, graph 1 nohead lt 0
+
+plot datafile using 1:2 with lines linestyle 1 ,\
+ datafile using 1:3 with lines linestyle 2 axes x1y2
diff --git a/figs/fowler-nordheim.fig/asymptotic.jl b/figs/fowler-nordheim.fig/asymptotic.jl
new file mode 100644
index 0000000..fd1d492
--- /dev/null
+++ b/figs/fowler-nordheim.fig/asymptotic.jl
@@ -0,0 +1,46 @@
+using QuadGK
+using SpecialFunctions
+using FFTW
+
+# numerical values
+hbar=6.58e-16 # eV.s
+m=9.11e-31 # kg
+Vn=9 # eV
+En=14e9 # V/m
+Kn=4.5 # eV
+
+V=1
+E=En*hbar/(2*Vn^1.5*m^0.5)*sqrt(1.60e-19)
+k0=sqrt(Kn/Vn)
+
+# rescale x to nm
+nm_scale=sqrt(2*m*Vn)/hbar*1e9*sqrt(1.60e-19)
+
+# cutoffs
+p_cutoff=20*k0
+p_npoints=256
+
+# airy approximations
+airy_threshold=30
+airy_order=5
+
+# xbounds
+xmax=10
+xmin=-10
+x_npoints=200
+
+include("FN_base.jl")
+
+# compute asymptotic values
+ps_asym=Array{Complex{Float64}}(undef,x_npoints)
+dps_asym=Array{Complex{Float64}}(undef,x_npoints)
+for i in 1:x_npoints
+  x=xmin+(xmax-xmin)*i/x_npoints
+  ps_asym[i]=psi_pole(x,k0,E,V)
+  dps_asym[i]=dpsi_pole(x,k0,E,V)
+end
+
+# print asymptotic values
+for i in 1:x_npoints
+  print((xmin+(xmax-xmin)*i/x_npoints)*nm_scale,' ',abs(ps_asym[i])^2,' ',J(ps_asym[i],dps_asym[i]),'\n')
+end
diff --git a/figs/libs/Makefile b/figs/libs/Makefile
new file mode 100644
index 0000000..33b81e2
--- /dev/null
+++ b/figs/libs/Makefile
@@ -0,0 +1,28 @@
+PROJECTNAME=$(basename $(basename $(wildcard *.tikz.tex)))
+LIBS=$(notdir $(wildcard libs/*))
+
+PDFS=$(addsuffix .pdf, $(PROJECTNAME))
+
+all: $(PDFS)
+
+$(PDFS): $(LIBS)
+	echo $(LIBS)
+	pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@)
+
+install: $(PDFS)
+	cp $^ $(INSTALLDIR)/
+
+$(LIBS):
+	ln -fs libs/$@ ./
+
+clean-libs:
+	rm -f $(LIBS)
+
+clean-aux:
+	rm -f $(addsuffix .aux, $(PROJECTNAME))
+	rm -f $(addsuffix .log, $(PROJECTNAME))
+
+clean-tex:
+	rm -f $(PDFS)
+
+clean: clean-libs clean-aux clean-tex
diff --git a/figs/potential.fig/Makefile b/figs/potential.fig/Makefile
new file mode 120000
index 0000000..704310e
--- /dev/null
+++ b/figs/potential.fig/Makefile
@@ -0,0 +1 @@
+../libs/Makefile
\ No newline at end of file
diff --git a/figs/potential.fig/potential.tikz.tex b/figs/potential.fig/potential.tikz.tex
new file mode 100644
index 0000000..ac1ba36
--- /dev/null
+++ b/figs/potential.fig/potential.tikz.tex
@@ -0,0 +1,17 @@
+\documentclass{standalone}
+
+\usepackage{tikz}
+
+\begin{document}
+\begin{tikzpicture}
+
+\draw(-3,0)--(3,0);
+\draw(0,-1.5)--(0,3);
+
+\draw[color=gray,line width=1pt, densely dotted](-2.5,1.25)--++(2.5,0);
+
+\draw[line width=1.5pt](-2.5,0)--(0,0)--(0,2.5)--(1.975,-1.25);
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/figs/potential.fig/potential_square.tikz.tex b/figs/potential.fig/potential_square.tikz.tex
new file mode 100644
index 0000000..bd13467
--- /dev/null
+++ b/figs/potential.fig/potential_square.tikz.tex
@@ -0,0 +1,17 @@
+\documentclass{standalone}
+
+\usepackage{tikz}
+
+\begin{document}
+\begin{tikzpicture}
+
+\draw(-3,0)--(3,0);
+\draw(0,-1.5)--(0,3);
+
+\draw[color=gray,line width=1pt, densely dotted](-2.5,1.25)--++(2.5,0);
+
+\draw[line width=1.5pt](-2.5,0)--(0,0)--(0,2.5)--(2.5,2.5);
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/figs/potential.fig/potential_square_photonic.tikz.tex b/figs/potential.fig/potential_square_photonic.tikz.tex
new file mode 100644
index 0000000..8a35fb4
--- /dev/null
+++ b/figs/potential.fig/potential_square_photonic.tikz.tex
@@ -0,0 +1,22 @@
+\documentclass{standalone}
+
+\usepackage{tikz}
+
+\begin{document}
+\begin{tikzpicture}
+
+\draw(-3,0)--(3,0);
+\draw(0,-1.5)--(0,3);
+
+\draw[color=gray,line width=1pt, densely dotted](-2.5,1.25)--++(2.5,0);
+\draw[color=gray,line width=1pt, densely dotted](-1,2.00)--++(1,0);
+\draw[color=gray,line width=1pt, densely dotted](-1,2.75)--++(3.5,0);
+
+\draw[->,line width=0.75pt,color=red](-0.5,1.25)--++(0,0.75);
+\draw[->,line width=0.75pt,color=red](-0.5,2.00)--++(0,0.75);
+
+\draw[line width=1.5pt](-2.5,0)--(0,0)--(0,2.5)--(2.5,2.5);
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/figs/potential.fig/potential_square_thermal.tikz.tex b/figs/potential.fig/potential_square_thermal.tikz.tex
new file mode 100644
index 0000000..3e8fd22
--- /dev/null
+++ b/figs/potential.fig/potential_square_thermal.tikz.tex
@@ -0,0 +1,17 @@
+\documentclass{standalone}
+
+\usepackage{tikz}
+
+\begin{document}
+\begin{tikzpicture}
+
+\draw(-3,0)--(3,0);
+\draw(0,-1.5)--(0,3);
+
+\draw[color=gray,line width=1pt, densely dotted](-2.5,2.75)--++(5,0);
+
+\draw[line width=1.5pt](-2.5,0)--(0,0)--(0,2.5)--(2.5,2.5);
+
+\end{tikzpicture}
+\end{document}
+