def VisualizeEigenfield(n=3,z=30,scale=50,\
a=30,*args,**kwargs):
zs=numpy.linspace(0,a*scale,200)
rs=numpy.linspace(-a*scale/2.,a*scale/2.,200)
from common.numerics import broadcast_items
from scipy.special import j0,j1
d=get_tip_eigenbasis_expansion(z,a=a,*args,**kwargs)
field_contributions=AWA(P[:,n-1],axes=[tip.LRM.qxs],axis_names=['$q/a$']).squeeze()
zs_norm=zs/float(a)
rs_norm=rs/float(a)
rs_norm,zs_norm=broadcast_items(rs_norm,zs_norm)
total_zfield=0
total_rfield=0
potential=0
for field_contrib,q,wq in zip(field_contributions,tip.LRM.qxs,tip.LRM.wqxs):
zpos_dependence=numpy.exp(-q*z/numpy.float(a))*numpy.exp(-q*zs_norm)*j0(q*rs_norm)
rpos_dependence=numpy.exp(-q*z/numpy.float(a))*numpy.exp(-q*zs_norm)*j1(q*rs_norm)
total_zfield+=(-field_contrib)*zpos_dependence*wq
total_rfield+=(-field_contrib)*rpos_dependence*wq
potential+=field_contrib*zpos_dependence*wq/q
total_zfield=AWA(total_zfield,axes=[rs,zs],axis_names=['r','z'])
total_rfield=AWA(total_rfield,axes=[rs,zs],axis_names=['r','z'])
potential=AWA(potential,axes=[rs,zs],axis_names=['r','z'])
potential-=numpy.mean(potential.cslice[:,z])
return total_zfield,total_rfield,potential
def RibbonTest(_N_sample_eigenbasis=100, _N_tip_eigenbasis=1):
show_eigs = False
run_test = True
xs, ys = np.arange(101), np.arange(101)
L = xs.max()
xv, yv = np.meshgrid(xs, ys)
eigpairs = {}
graphene_ribbon = False
if graphene_ribbon:
q0 = np.pi / L #This is for particle in box (allowed wavelength is n*2*L)
for n in range(1, _N_sample_eigenbasis + 1):
qx = n * q0
pw = AWA(planewave(qx, 0, xv, yv, x0=0, phi0=np.pi / 2),
axes=[xs, ys]) #cosine waves, this is for
eigpairs[qx**2] = pw / np.sqrt(np.sum(pw**2))
else:
q0 = 2 * np.pi / L #This is for infinite sample
for n in range(1, _N_sample_eigenbasis + 1):
qx = n * q0
pw = AWA(planewave(qx, 0, xv, yv, x0=0, phi0=np.pi / 2),
axes=[xs, ys]) #cosine waves
eigpairs[qx**2] = pw / np.sqrt(np.sum(pw**2))
pw2 = AWA(planewave(qx, 0, xv, yv, x0=0, phi0=0),
axes=[xs, ys]) #sine waves, This is for infinite sample
eigpairs[qx**2 + 1e-9] = pw2 / np.sqrt(np.sum(
pw2**2)) #This is for infinite sample
if show_eigs:
for i, q in enumerate(list(eigpairs.keys())):
if i < 5:
plt.figure()
plt.imshow(eigpairs[q])
plt.title("q={}".format(q))
plt.show()
if run_test:
q = 2 * np.pi / L * 20
Sample = SampleResponse(eigpairs,
qw=q,
N_sample_eigenbasis=_N_sample_eigenbasis)
Tip = TipResponse(Sample.xs,
Sample.ys,
q=q,
N_tip_eigenbasis=_N_tip_eigenbasis)
d = Sample.RasterScan(Tip)
plt.figure()
plt.imshow(np.abs(d['P']))
plt.title('P')
plt.colorbar()
plt.figure()
plt.imshow(np.abs(d['R']))
plt.title('R')
plt.colorbar()
plt.show()
def __call__(self,excitations,U,tip_eigenbasis):
if np.array(excitations).ndim==2: excitations=[excitations]
Exc=np.array([exc.ravel() for exc in excitations])
tip_eb=np.matrix([eigfunc.ravel() for eigfunc in tip_eigenbasis])
projected_result=np.dot(tip_eb.T,
np.dot(U,\
np.dot(self.D,\
np.dot(self.Phis,Exc.T))))
result=np.dot(self.Phis.T,\
np.dot(self.D,\
np.dot(self.Phis,Exc.T)))
result=np.array(result).T.reshape((len(excitations),)+self.phishape)
projected_result=np.array(projected_result).T.reshape((len(excitations),)+self.phishape)
return AWA(result,axes=[None,self.xs,self.ys]).squeeze(), AWA(projected_result,axes=[None,self.xs,self.ys]).squeeze()
def get_Erad(self,beta,zs=None,Nterms=None,interpolation='linear'):
if hasattr(beta,'__len__'):
if isinstance(beta,numpy.ndarray):
if not beta.ndim: beta=beta.tolist()
else: beta=numpy.array(beta)
if isinstance(beta,numpy.ndarray): beta=beta.reshape((len(beta),1,1))
if zs is None: zs=self.zs
Rs=self.evaluate_residues(zs,Nterms)#,interpolation)
Ps=self.evaluate_poles(zs,Nterms)#,interpolation)
#Should broadcast over freqs if beta has an additional first axis
#The offset term is absolutely critical, offsets false z-dependence arising from first terms
approach=numpy.sum(Rs*(1/(beta-Ps)+1/Ps),axis=-1)
axes=[zs]; axis_names=['z/a']
if hasattr(beta,'__len__'):
approach=approach.transpose()
if isinstance(beta,AWA):
axes=axes+[beta.axes[0]]
axis_names=axis_names+[beta.axis_names[0]]
else:
axes=axes+[None]
axis_names=axis_names+[None]
signals=AWA(approach,axes=axes,axis_names=axis_names).squeeze()
if not signals.ndim: signals=signals.tolist()
return signals
def get_tip_eigenbasis_expansion(z=.1,freq=1000,a=30,\
smoothing=0,reload_signal=True,\
*args,**kwargs):
"""Appears to render noise past the first 15 eigenvalues.
Smoothing by ~4 may be justified, removing zeros probably not..."""
# Rely on Lightning Rod Model only to load tip data from file in its process of computing a signal #
if reload_signal:
tip.verbose=False
signal=tip.LRM(freq,rp=mat.Au.reflection_p,zmin=z,amplitude=0,\
normalize_to=None,normalize_at=1000,Nzs=1,demodulate=False,\
*args,**kwargs)
tip.verbose=True
global L,g,M,alphas,P,Ls,Es
#Get diagonal basis for the matrix
L=tip.LRM.LambdaMatrix(tip.LRM.qxs)
g=numpy.matrix(numpy.diag(-tip.LRM.qxs*numpy.exp(-2*tip.LRM.qxs*z/numpy.float(a))*tip.LRM.wqxs))
M=AWA(L*g,axes=[tip.LRM.qxs]*2,axis_names=['q']*2)
#Smooth along s-axis (first), this is where we truncated the integral xform
if smoothing: M=numrec.smooth(M,axis=0,window_len=smoothing)
alphas,P=linalg.eig(numpy.matrix(M))
P=numpy.matrix(P)
Ls=numpy.array(P.getI()*tip.LRM.Lambda0Vector(tip.LRM.qxs)).squeeze()
Es=numpy.array(numpy.matrix(tip.LRM.get_dipole_moments(tip.LRM.qxs))*g*P).squeeze()
Rs=-Es*Ls/alphas**2
Ps=1/alphas
return {'Rs':Rs,'Ps':Ps,'Es':Es,'alphas':alphas,'Ls':Ls}
def __call__(self, x0, y0):
shift_by_dx = x0 - self.midx
shift_by_dy = y0 - self.midy
shift_by_nx = int(self.shape[0] * shift_by_dx / self.dx)
shift_by_ny = int(self.shape[1] * shift_by_dy / self.dy)
newJ=np.roll(np.roll(self.bigF,shift_by_nx,axis=0),\
shift_by_ny,axis=1)
output = newJ[self.shape[0]//2:(3*self.shape[0])//2,\
self.shape[1]//2:(3*self.shape[1])//2]
return AWA(output, axes=[self.xs, self.ys])
def InvertImage(image,nodes,normalize_to=mat.Au,Nterms=8,freq=1000):
row_len=image.shape[1]
signals=uncoil_image(image)
beta_norm=normalize_to.reflection_p(freq,q=1/(30e-7))
norm_signal=EigenfieldModel.get_signal_from_nodes(beta=beta_norm,nodes=nodes,Nterms=Nterms)
betas=EigenfieldModel.invert_signal(signals*norm_signal,nodes=nodes,Nterms=Nterms,select_by='pole')
betas_image=coil_image(betas,row_len=row_len)
if isinstance(image,AWA): betas_image=AWA(betas_image); betas_image.adopt_axes(image)
return betas_image
def GetExpansionApproachCurve(zs=numpy.linspace(.1,150,50),a=30,beta=1,Nterms=None,\
*args,**kwargs):
if hasattr(beta,'__len__'):
if not isinstance(beta,numpy.ndarray): beta=numpy.array(beta)
beta=beta.reshape((len(beta),1,1))
Rs=[]
Ps=[]
for i,z in enumerate(zs):
if i==0: kwargs['reload_signal']=True
else: kwargs['reload_signal']=False
d=get_tip_eigenbasis_expansion(z,a=a,*args,**kwargs)
Rs.append(d['Rs'])
Ps.append(d['Ps'])
Rs=numpy.array(Rs); Ps=numpy.array(Ps)
if not Nterms: Nterms=Ps.shape[1]
Rs=AWA(Rs[:,:Nterms],axes=[zs,None],axis_names=['Z','Term'])
Ps=AWA(Ps[:,:Nterms],axes=[zs,None],axis_names=['Z','Term'])
approach=numpy.sum(Rs/(beta-Ps)+Rs/Ps,axis=-1)
#beta*Rs/(poles-beta)=sig
#0=sig+beta*Rs/(beta-poles)
#0=sig/beta+Rs/(beta-poles)
axes=[zs]; axis_names=['Z']
if hasattr(beta,'__len__'):
approach=approach.transpose()
if isinstance(beta,AWA):
axes=axes+[beta.axes[0]]
axis_names=axis_names+[beta.axis_names[0]]
else:
axes=axes+[None]
axis_names=axis_names+[None]
approach=AWA(approach,axes=axes,axis_names=axis_names)
return {'Rs':Rs,'Ps':Ps,'signals':approach}
def get_signal_from_nodes(self,beta,nodes=[(0,1)],Nterms=None,interpolation='linear'):
if not hasattr(beta,'__len__'): beta=[beta]
if not isinstance(beta,AWA): beta=AWA(beta)
#`Frequency` axis will be first
if isinstance(beta,numpy.ndarray): beta=beta.reshape((len(beta),1,1))
#Weights apply across z-values
zs,ws=list(zip(*nodes))
ws_grid=numpy.array(ws).reshape((1,len(ws),1))
zs=numpy.array(zs)
#Evaluate at all nodal points
Rs=self.evaluate_residues(zs,Nterms,interpolation)
Ps=self.evaluate_poles(zs,Nterms,interpolation)
#Should broadcast over freqs if beta has an additional first axis
#The offset term is absolutely critical, offsets false z-dependence arising from first terms
approach=numpy.sum(numpy.sum(Rs*(1/(beta-Ps)+1/Ps)*ws_grid,axis=-1),axis=-1)#+Rs/Ps,axis=-1)
axes=[zs]; axis_names=['z/a']
if hasattr(beta,'__len__'):
approach=approach.transpose()
if isinstance(beta,AWA):
axes=axes+[beta.axes[0]]
axis_names=axis_names+[beta.axis_names[0]]
else:
axes=axes+[None]
axis_names=axis_names+[None]
signals=AWA(approach); signals.adopt_axes(beta)
signals=signals.squeeze()
if not signals.ndim: signals=signals.tolist()
return signals
def load_eigpairs(basedir=os.path.dirname("./"),eigpair_fname="UnitSquareMesh_100x100_1000_eigenbasis.h5"):
"""Normalization by sum always ensures that integration will be like summing, which is
much simpler than keeping track of dx, dy..."""
global eigpairs
eigpairs = dict()
path=os.path.join(basedir,eigpair_fname)
with h5py.File(path,'r') as f:
for key in list(f.keys()):
eigfunc=np.array(f.get(key))
eigfunc/=np.sqrt(np.sum(np.abs(eigfunc)**2))
eigpairs[float(key)] = AWA(eigfunc,\
axes=[np.linspace(0,1,eigfunc.shape[0]),\
np.linspace(0,1,eigfunc.shape[1])])
return eigpairs
def PlotHarmonicQuadratures(beta,amplitude=2,quadratures=quadratures,\
harmonic=3,**kwargs):
global quad_harmonic,ref_harmonic,harmonic_value
signal=EigenfieldModel(beta=beta)
harmonics={}
for quadrature_name,pair in quadratures:
quadrature,Nts=pair
harmonics_this_quad=[]
for Nt in Nts:
harmonics_this_Nt=EigenfieldModel.demodulate(signal,quadrature=quadrature,\
amplitude=amplitude,harmonics=[harmonic],Nts=Nt,**kwargs)
harmonic_value=harmonics_this_Nt.cslice[harmonic].squeeze().tolist()
harmonics_this_quad.append(numpy.complex(harmonic_value))
harmonics_this_quad=numpy.array(harmonics_this_quad,dtype=numpy.complex)
harmonics[quadrature_name]=AWA(harmonics_this_quad,axes=[Nts],axis_names=['$N_{quad}$'])
import itertools
figure()
markers=[None,'o','s','^','+','^']
markers = itertools.cycle(markers)
ref_quad='Gauss-Legendre'
ref_harmonic=harmonics[ref_quad][-1]
for quadrature_name in zip(*quadratures)[0]:
quad_harmonic=harmonics[quadrature_name]
marker = next(markers)
numpy.abs((quad_harmonic-ref_harmonic)/ref_harmonic)\
.plot(plotter=semilogy,marker=marker,label=quadrature_name,markersize=5)
ylabel('$S_%i\,\mathrm{rel.\,error}$'%harmonic)
leg=legend(loc='best',fancybox=True,shadow=True)
leg.get_frame().set_linewidth(.1)
xlim(0,200)
tight_layout()
return harmonics
def __init__(self, eigpairs, E, N=100, debug=True):
# Setting the easy stuff
eigvals = list(eigpairs.keys())
eigfuncs = list(eigpairs.values())
self.debug = debug
self.xs, self.ys = eigfuncs[0].axes
self.eigfuncs = AWA(eigfuncs,\
axes=[eigvals,self.xs,self.ys]).sort_by_axes()
self.eigvals = self.eigfuncs.axes[0]
self.phishape = self.eigfuncs[0].shape
self.E = E
self.N = N
# Setting the various physical quantities
self._SetUseEigenvalues(E)
self._SetEnergy()
self._SetSigma(10, 10)
self._SetCoulombKernel()
self._SetScatteringMatrix()
self.Us = []
def VisualizeEigenfieldDistributions(ns=[1,2,3],zs=[.1,5,15,30],a=30,*args,**kwargs):
zs.sort()
eigs=dict([(n,[]) for n in ns])
for i,z in enumerate(zs):
if i==0: kwargs['reload_signal']=True
else: kwargs['reload_signal']=False
d=get_tip_eigenbasis_expansion(z,a=a,*args,**kwargs)
for n in ns:
eigs[n].append(AWA(P[:,n-1],axes=[tip.LRM.qxs],axis_names=['$q/a$']).squeeze())
colors=list(zip(numpy.linspace(1,0,len(zs)),\
[0]*len(zs),\
numpy.linspace(0,1,len(zs))))
#return eigs
figure()
for i,n in enumerate(ns):
subplot(1,len(ns),i+1)
for j in range(len(zs)):
numpy.abs(eigs[n][j]).plot(plotter=semilogx,\
color=colors[j],\
label='$z/a=%1.1f$'%(zs[j]/numpy.float(a)))
props = dict(boxstyle='round', facecolor='white', alpha=0.8)
gca().text(0.67, 0.85, '$n=%i$'%n, transform=gca().transAxes, fontsize=24,bbox=props)
if i==0: ylabel('$E_n(q)$')
else:
yticks(numpy.arange(5)*.05,['']*5)
xts=xticks()
xticks(xts[0][1:])
xlim(1e-4,20)
ylim(0,.2)
grid()
tight_layout()
subplots_adjust(wspace=0)
def __init__(self, eigpairs, qw=44, N_sample_eigenbasis=100):
# Setting the easy stuff
eigvals = list(eigpairs.keys())
eigfuncs = list(eigpairs.values())
self.xs, self.ys = eigfuncs[0].axes
self.eigfuncs = AWA(eigfuncs,\
axes=[eigvals,self.xs,self.ys]).sort_by_axes()
self.eigvals = self.eigfuncs.axes[0]
self.phishape = self.eigfuncs[0].shape
self.qw = qw
self.N = N_sample_eigenbasis
sigma_1, sigma_2 = np.real(np.exp(2 * np.pi * 1j * 0.05)), np.imag(
np.exp(2 * np.pi * 1j * 0.05))
#lambda_p, L_p = 10,10
# Setting the various physical quantities
self._SetAlpha(sigma_1, sigma_2)
self._SetUseEigenvalues()
self._SetCoulombKernel()
self._SetScatteringMatrix()
self.Us = []
def demodulate(signals,amplitude=2,harmonics=list(range(4)),Nts=None,\
quadrature=numrec.GL):
"""Takes z-axis as first axis, frequency as final axis."""
global ts,wts,weights,signals_vs_time
#max harmonic resolvable will be frequency = 1/dt = Nts
if not Nts: Nts=4*numpy.max(harmonics)
if isinstance(quadrature,str) or hasattr(quadrature,'calc_nodes'):
ts,wts=numrec.GetQuadrature(N=Nts,xmin=-.5,xmax=0,quadrature=quadrature)
else:
ts,wts=numpy.linspace(-.5,0,Nts),None
if quadrature is None: quadrature=simps
zs=amplitude*(1+numpy.cos(2*numpy.pi*ts))
harmonics=numpy.array(harmonics).reshape((len(harmonics),1))
weights=numpy.cos(2*numpy.pi*harmonics*ts)
if wts is not None: weights*=wts
weights_grid=weights.reshape(weights.shape+(1,)*(signals.ndim-1))
signals_vs_time=signals.interpolate_axis(zs,axis=0,bounds_error=False,extrapolate=True)
signals_vs_time.set_axes([ts],axis_names=['t'])
integrand=signals_vs_time*weights_grid
if wts is not None:
demodulated=2*2*numpy.sum(integrand,axis=1) #perform quadrature
else: demodulated=2*2*quadrature(integrand,x=ts,axis=1)
axes=[harmonics]; axis_names=['harmonic']
if isinstance(signals,AWA):
axes+=signals.axes[1:]
axis_names+=signals.axis_names[1:]
demodulated=AWA(demodulated,axes=axes,axis_names=axis_names)
return demodulated
def __call__(self, x0, y0):
tip_eb = [t(x0, y0) for t in self.eigenbasis_translators]
return AWA(tip_eb, axes=[None, tip_eb[0].axes[0], tip_eb[0].axes[1]])
def CompareCarbonylTips(Ls=numpy.linspace(30,20e3,100),a=30,\
wavelength=6e3,amplitude=60,lift=30,\
geometries=['cone','hyperboloid'],\
taper_angles=[20],\
load=True,save=True,demo_file='CarbonylTips.pickle',\
enhancement_index=2):
carbonyl_freqs = numpy.linspace(1730, 1750, 10)
normalize_at_freq = numpy.mean(carbonyl_freqs)
freqs = [a / numpy.float(wavelength)
] #load frequency should be quite close to `sio2_freq`
Ls = Ls / float(a)
freq_labels = ['ED']
skin_depth = .05
##################################################
## Load or compute Carbonyl measurement metrics ##
##################################################
global d
d_keys=['max_s3','max_s2',\
'max_rel_absorption','max_absorption','max_freq',\
'charges0','charges_q0','Lambda0','enhancement',\
'signals','norm_signals','DCSD_contact','DCSD_lift',\
'psf_contact','psf_lift']
demo_path = os.path.join(root_dir, demo_file)
if load and os.path.isfile(demo_path):
Logger.write('Loading lengths data...')
d = pickle.load(open(demo_path))
else:
tip.LRM.load_params['reload_model'] = True
tip.LRM.geometric_params['skin_depth'] = skin_depth
d = {}
for i, geometry in enumerate(geometries):
for n, taper_angle in enumerate(taper_angles):
#If geometry is not conical, do not iterate to additional taper angles, these are all the same.
if geometry not in ['PtSi', 'hyperboloid', 'cone'] and n > 0:
break
tip.LRM.geometric_params['geometry'] = geometry
tip.LRM.geometric_params['taper_angle'] = taper_angle
signals_for_geometry = {}
for j, freq in enumerate(freqs):
signals_for_freq = dict([(key, []) for key in d_keys])
for k, L in enumerate(Ls):
tip.LRM.geometric_params['L'] = int(L)
Logger.write(
'Currently working on geometry "%s", freq=%s, L/a=%s...'
% (geometry, freq, L))
#Make sure to calculate on zs all the way up to zmax=2*amplitude+lift
carbonyl_vals=tip.LRM(carbonyl_freqs,rp=Carbonyl.reflection_p,Nqs=72,zmin=.1,a=a,amplitude=amplitude+lift/2.,\
normalize_to=mat.Si.reflection_p,normalize_at=normalize_at_freq,load_freq=freq,Nzs=30)
##Get overall signal with approach curve##
global carbonyl_sigs
carbonyl_sigs = carbonyl_vals['signals'] * numpy.exp(
-1j * numpy.angle(
carbonyl_vals['norm_signals'].cslice[0]))
carbonyl_rel_sigs = carbonyl_vals[
'signals'] / carbonyl_vals['norm_signals'].cslice[0]
ind = numpy.argmax(carbonyl_rel_sigs.imag.cslice[0]
) #Find maximum phase in contact
signals_for_freq['max_s3'].append(
carbonyl_vals['signal_3']
[ind]) #Pick out the peak frequency
signals_for_freq['max_s2'].append(
carbonyl_vals['signal_2']
[ind]) #Pick out the peak frequency
signals_for_freq['max_rel_absorption'].append(
carbonyl_rel_sigs[:, ind].imag.cslice[0]
) #Relative to out-of-contact on reference
signals_for_freq['max_absorption'].append(
carbonyl_sigs[:, ind].imag.cslice[0]
) #Absolute signal
signals_for_freq['max_freq'].append(
carbonyl_freqs[ind])
signals_for_freq['charges0'].append(
tip.LRM.charges0 /
(2 * numpy.pi * tip.LRM.charge_radii))
signals_for_freq['charges_q0'].append(
tip.LRM.charges.cslice[0] /
(2 * numpy.pi * tip.LRM.charge_radii))
signals_for_freq['Lambda0'].append(tip.LRM.Lambda0)
signals_for_freq['enhancement'].append(
numpy.abs(2 * tip.LRM.charges0[enhancement_index]))
signals_for_freq['signals'].append(
carbonyl_sigs[:,
ind]) #Pick out the peak frequency
signals_for_freq['norm_signals'].append(
carbonyl_vals['norm_signals'])
signals_for_freq['DCSD_contact'].append(
get_DCSD(carbonyl_sigs[:, ind],
zmin=0.1,
amplitude=60,
lift=0))
signals_for_freq['DCSD_lift'].append(
get_DCSD(carbonyl_sigs[:, ind],
zmin=0.1,
amplitude=60,
lift=lift))
##Isolate some point-spread functions##
rs = numpy.linspace(0, 5, 200).reshape((200, 1))
integrand_contact=AWA(tip.LRM.qxs**2*\
special.j0(tip.LRM.qxs*rs)*tip.LRM.get_dipole_moments(tip.LRM.qxs)*\
tip.LRM.wqxs,axes=[rs.squeeze(),tip.LRM.qxs])
integrand_contact=integrand_contact.interpolate_axis(numpy.logspace(numpy.log(tip.LRM.qxs.min())/numpy.log(10),\
numpy.log(tip.LRM.qxs.max())/numpy.log(10),1000),\
axis=1)
psf_contact = numpy.sum(integrand_contact, axis=1)
signals_for_freq['psf_contact'].append(
AWA(psf_contact,
axes=[rs.squeeze()],
axis_names=['r/a']))
integrand_lift=AWA(tip.LRM.qxs**2*numpy.exp(-tip.LRM.qxs*lift/numpy.float(a))*\
special.j0(tip.LRM.qxs*rs)*tip.LRM.get_dipole_moments(tip.LRM.qxs)*\
tip.LRM.wqxs,axes=[rs.squeeze(),tip.LRM.qxs])
integrand_lift=integrand_lift.interpolate_axis(numpy.logspace(numpy.log(tip.LRM.qxs.min())/numpy.log(10),\
numpy.log(tip.LRM.qxs.max())/numpy.log(10),1000),\
axis=1)
psf_lift = numpy.sum(integrand_lift, axis=1)
signals_for_freq['psf_lift'].append(
AWA(psf_lift,
axes=[rs.squeeze()],
axis_names=['r/a']))
progress = (i * len(Ls) * len(freqs) + j * len(Ls) +
k + 1) / numpy.float(
len(Ls) * len(freqs) *
len(geometries)) * 100
Logger.write('\tProgress: %1.1f%%' % progress)
for key in list(signals_for_freq.keys()):
if numpy.array(
signals_for_freq[key]
).ndim == 2: #Prepend probe length axis and expand these into AWA's
axes = [
a * Ls / numpy.float(wavelength),
signals_for_freq[key][0].axes[0]
]
axis_names = [
'$L/\lambda_\mathrm{C=O}$',
signals_for_freq[key][0].axis_names[0]
]
signals_for_freq[key]=AWA(numpy.array(signals_for_freq[key],dtype=numpy.complex),\
axes=axes,axis_names=axis_names)
else:
signals_for_freq[key]=AWA(numpy.array(signals_for_freq[key],dtype=numpy.complex),\
axes=[a*Ls/numpy.float(wavelength)],\
axis_names=['$L/\lambda_\mathrm{C=O}$'])
signals_for_geometry[freq_labels[j]] = signals_for_freq
geometry_name = geometry
if geometry in ['PtSi', 'hyperboloid', 'cone']:
geometry_name += str(taper_angle)
d[geometry_name] = signals_for_geometry
if not load and save:
Logger.write('Saving tip comparison data...')
file = open(demo_path, 'wb')
pickle.dump(d, file)
file.close()
return d
from common.baseclasses import AWA
from common.numerical_recipes import QuickConvolver as QC
import matplotlib.pyplot as plt
import numpy as np
image=AWA(np.zeros((101,101)),axes=[np.linspace(-.5,.5,101)]*2)
image[50,50]=1
kernel_function=lambda x,y: 1/np.sqrt(x**2+y**2+1e-8**2)
qc1=QC(size=(1,1),shape=(101,101),pad_by=.5,kernel_function=kernel_function)
result1=qc1(image)
result1-=result1.min() #overall offset, while correct, should not be meaningful
result1[result1==result1.max()]=0 #point at center is controlled by 1e-8
kernel_function_fourier=lambda kx,ky: 2*np.pi/np.sqrt(kx**2+ky**2+1e-8**2)
qc2=QC(size=(1,1),shape=(101,101),pad_by=.5,kernel_function_fourier=kernel_function_fourier)
result2=qc2(image)
result2-=result2.min() #overall offset is controlled by 1e-8
plt.figure();result1.cslice[0].plot()
result2.cslice[0].plot()
plt.gca().set_xscale('symlog',linthreshx=1e-2)
plt.show()
def PlotApproachCurves(zmax=30,materials={'Gold':(mat.Au,1000),\
'Carbonyl':(mat.PMMA,1735),\
r'$\mathrm{SiC}(\omega_{SO})$':(mat.SiC_6H_Ellips,945),\
r'$\mathrm{SiC}(\omega_{-})$':(mat.SiC_6H_Ellips,920),\
r'$\mathrm{SiC}(\omega_{+})$':(mat.SiC_6H_Ellips,970)},\
Nterms=15):
ordered=['Gold','Carbonyl',
r'$\mathrm{SiC}(\omega_{-})$',
r'$\mathrm{SiC}(\omega_{SO})$',\
r'$\mathrm{SiC}(\omega_{+})$']
colors=['c','g',(1,0,0),(.7,0,.7),(0,0,1)]
global beta,LRM_signals,EFM_signals,material_name
zs=numpy.logspace(numpy.log(.1/30.)/numpy.log(10.),numpy.log(zmax)/numpy.log(10.),100)
zs2=numpy.logspace(numpy.log(.1/30.)/numpy.log(10.),numpy.log(15)/numpy.log(10.),100)
LRM_signals={}
EFM_signals={}
for i,(material_name,pair) in enumerate(materials.items()):
material,freq=pair
beta=material.reflection_p(freq,q=1/(30e-7))
LRM_signals[material_name]=tip.LRM(freq,rp=beta,a=30,zs=zs*30,\
Nqs=244,demodulate=False,normalize_to=None)['signals']
LRM_signals[material_name].set_axes([zs])
EFM_signals[material_name]=EigenfieldModel(beta,zs=zs2,Nterms=Nterms)
if i==0: tip.LRM.load_params['reload_model']=False
tip.LRM.load_params['reload_model']=True
figure()
ref_signal=LRM_signals['Gold'].cslice[zmax]
subplot(121)
for material_name,color in zip(ordered,colors):
numpy.abs(LRM_signals[material_name]/ref_signal).plot(label=material_name,plotter=loglog,color=color)
numpy.abs(EFM_signals[material_name]/ref_signal).plot(color=color,marker='o',ls='')
ylabel('$|E_\mathrm{rad}|\,[\mathrm{a.u.}]$')
xlabel('$d/a$')
ylim(3e-1,3e1)
grid()
leg=legend(loc='lower left',fancybox=True,shadow=True)
leg.get_frame().set_linewidth(.1)
for t in leg.texts: t.set_fontsize(18)
subplot(122)
for material_name,color in zip(ordered,colors):
p_LRM=AWA(numpy.unwrap(numpy.angle(LRM_signals[material_name]/ref_signal)))
p_LRM.adopt_axes(LRM_signals[material_name])
p_EFM=AWA(numpy.unwrap(numpy.angle(EFM_signals[material_name]/ref_signal)))
p_EFM.adopt_axes(EFM_signals[material_name])
(p_LRM/(2*numpy.pi)).plot(label=material_name,plotter=semilogx,color=color)
(p_EFM/(2*numpy.pi)).plot(color=color,marker='o',ls='')
ylabel(r'$\mathrm{arg}(E_\mathrm{rad})\,/\,2\pi$')
xlabel('$d/a$')
ylim(-.05,.5)
grid()
gcf().set_size_inches([12.5,7],forward=True)
tight_layout()
subplots_adjust(wspace=.3)
run_test = True
global eigpairs
xs, ys = np.arange(101), np.arange(101)
L = xs.max()
xv, yv = np.meshgrid(xs, ys)
#eigpairs = load_eigpairs(basedir="../sample_eigenbasis_data")
eigpairs = {}
Nqs = 100
graphene_ribbon = True
if graphene_ribbon:
q0 = np.pi / L #This is for particle in box (allowed wavelength is n*2*L)
for n in range(1, Nqs + 1):
qx = n * q0
pw = AWA(planewave(qx, 0, xv, yv, x0=0, phi0=pi / 2),
axes=[xs, ys]) #cosine waves, this is for
eigpairs[qx**2] = pw / np.sqrt(np.sum(pw**2))
else:
q0 = 2 * np.pi / L #This is for infinite sample
for n in range(1, Nqs + 1):
qx = n * q0
pw = AWA(planewave(qx, 0, xv, yv, x0=0, phi0=np.pi / 2),
axes=[xs, ys]) #cosine waves
eigpairs[qx**2] = pw / np.sqrt(np.sum(pw**2))
pw2 = AWA(planewave(qx, 0, xv, yv, x0=0, phi0=0),
axes=[xs, ys]) #sine waves, This is for infinite sample
eigpairs[qx**2 + 1e-9] = pw2 / np.sqrt(np.sum(
pw2**2)) #This is for infinite sample
def invert_signal(self,signals,nodes=[(1,0)],Nterms=10,\
interpolation='linear',\
select_by='continuity',\
closest_pole=0,\
scaling=10):
"""The inversion is not unique, consequently the selected solution
will probably be wrong if signal values correspond with
"beta" values that are too large (`|beta|~>min{|Poles|}`).
This can be expected to break at around `|beta|>2`."""
#Default is to invert signal in contact
#~10 terms seem required to converge on e.g. SiO2 spectrum,
#especially on the Re(beta)<0 (low signal) side of phonons
global roots,poly,root_scaling
#global betas,all_roots,pmin,rs,ps,As,Bs,roots,to_minimize
if self.verbose:
Logger.write('Inverting `signals` based on the provided `nodes` to obtain consistent beta values...')
if not hasattr(signals,'__len__'): signals=[signals]
if not isinstance(signals,AWA): signals=AWA(signals)
zs,ws=list(zip(*nodes))
ws_grid=numpy.array(ws).reshape((len(ws),1)) #last dimension is to broadcast over all `Nterms` equally
zs=numpy.array(zs)
Rs=self.Rs[:,:Nterms].interpolate_axis(zs,axis=0,kind=interpolation,
bounds_error=False,extrapolate=True)
Ps=self.Ps[:,:Nterms].interpolate_axis(zs,axis=0,kind=interpolation,
bounds_error=False,extrapolate=True)
#`rs` and `ps` can safely remain as arrays for `invres`
rs=(Rs*ws_grid).flatten()
ps=Ps.flatten()
k0=numpy.sum(rs/ps).tolist()
#Rescale units so their order of magnitude centers around 1
rscaling=numpy.exp(-(numpy.log(numpy.abs(rs).max())+\
numpy.log(numpy.abs(rs).min()))/2.)
pscaling=numpy.exp(-(numpy.log(numpy.abs(ps).max())+\
numpy.log(numpy.abs(ps).min()))/2.)
root_scaling=1/pscaling
#rscaling=1
#pscaling=1
if self.verbose:
Logger.write('\tScaling residues by a factor %1.2e to reduce floating point overflow...'%rscaling)
Logger.write('\tScaling poles by a factor %1.2e to reduce floating point overflow...'%pscaling)
rs*=rscaling; ps*=pscaling
k0*=rscaling/pscaling
signals=signals*rscaling/pscaling
#highest order first in `Ps` and `Qs`
#VERY SLOW - about 100ms on practical inversions (~60 terms)
As,Bs=invres(rs, ps, k=[k0], tol=1e-16, rtype='avg') #tol=1e-16 is the smallest allowable to `unique_roots`..
dtype=numpy.complex128 #Double precision offers noticeable protection against overflow
As=numpy.array(As,dtype=dtype)
Bs=numpy.array(Bs,dtype=dtype)
signals=signals.astype(dtype)
#import time
betas=[]
for i,signal in enumerate(signals):
#t1=time.time()
#Root finding `roots` seems to give noisy results when `Bs` has degree >84, with dynamic range ~1e+/-30 in coefficients...
#Pretty fast - 5-9 ms on practical inversions with rank ~60 companion matrices, <1 ms with ~36 terms
#@TODO: Root finding chokes on `Nterms=9` (number of eigenfields) and `Nts=12` (number of nodes),
# necessary for truly converged S3 on resonant phonons, probably due to
# floating point overflow - leading term increases exponentially with
# number of terms, leading to huge dynamic range.
# Perhaps limited by the double precision of DGEEV.
# So, replace with faster / more reliable root finder?
# We need 1) speed, 2) ALL roots (or at least the first ~10 smallest)
poly=As-signal*Bs
roots=find_roots(poly,scaling=scaling)
roots=roots[roots.imag>0]
roots*=root_scaling #since all beta units scaled by `pscaling`, undo that here
#print time.time()-t1
#How should we select the most likely beta among the multiple solutions?
#1. Avoids large changes in value of beta
if select_by=='difference' and i>=1:
if i==1 and self.verbose:
Logger.write('\tSelecting remaining roots by minimizing differences with prior...')
to_minimize=numpy.abs(roots-betas[i-1])
#2. Avoids large changes in slope of beta (best for spectroscopy)
#Nearly guarantees good beta spectrum, with exception of very loosely sampled SiC spectrum
#Loosely samples SiO2-magnitude phonons still perfectly fine
elif select_by=='continuity' and i>=2:
if i==2 and self.verbose:
Logger.write('\tSelecting remaining roots by ensuring continuity with prior...')
earlier_diff=betas[i-1]-betas[i-2]
current_diffs=roots-betas[i-1]
to_minimize=numpy.abs(current_diffs-earlier_diff)
#3. Select specifically which pole we want |beta| to be closest to
else:
if i==0 and self.verbose:
Logger.write('\tSeeding inversion closest to pole %i...'%closest_pole)
reordering=numpy.argsort(numpy.abs(roots)) #Order the roots towards increasing beta
roots=roots[reordering]
to_minimize=numpy.abs(closest_pole-numpy.arange(len(roots)))
beta=roots[to_minimize==to_minimize.min()].squeeze()
betas.append(beta)
if not i%5 and self.verbose:
Logger.write('\tProgress: %1.2f%% - Inverted %i signals of %i.'%\
(((i+1)/numpy.float(len(signals))*100),\
(i+1),len(signals)))
betas=AWA(betas); betas.adopt_axes(signals)
betas=betas.squeeze()
if not betas.ndim: betas=betas.tolist()
return betas
def PlotIdealProbePerformance(ideal_length=404,wavelength=6e3,lift=30,\
approach=False,compare_spectra=True):
freq = 30 / wavelength
tip.LRM.geometric_params['L'] = ideal_length
if approach:
gold=tip.LRM(1180,rp=mat.Au.reflection_p,a=30,amplitude=60,\
normalize_to=mat.Si.reflection_p,Nqs=72,normalize_at=1738,Nzs=30,\
load_freq=freq)
p = numpy.exp(-1j * numpy.angle(gold['norm_signals'].cslice[120]))
figure()
approach = gold['sample_signal_v_time']
approach *= p
peak = abs(approach).max()
approach /= peak
approach += numpy.cos(2 * numpy.pi * tip.LRM.ts) * 10 + 50
numpy.abs(approach).plot(label='Approach on $Au$')
plot(-1 - tip.LRM.ts, numpy.abs(approach), color='b')
plot(-1 + tip.LRM.ts, numpy.abs(approach), color='b')
plot(-tip.LRM.ts, numpy.abs(approach), color='b')
plot(1 - tip.LRM.ts, numpy.abs(approach), color='b')
plot(1 + tip.LRM.ts, numpy.abs(approach), color='b')
approximation = gold['sample_signal_0'] / 2. + gold[
'sample_signal_1'] * numpy.cos(2 * numpy.pi * tip.LRM.ts)
approximation *= p
approximation = AWA(approximation, axes=[tip.LRM.ts])
approximation /= peak
numpy.abs(approximation).plot(label='1st Harmonic')
plot(-1 - tip.LRM.ts, numpy.abs(approximation), color='g', ls='-')
plot(-1 + tip.LRM.ts, numpy.abs(approximation), color='g', ls='-')
plot(-tip.LRM.ts, numpy.abs(approximation), color='g', ls='-')
plot(1 - tip.LRM.ts, numpy.abs(approximation), color='g', ls='-')
plot(1 + tip.LRM.ts, numpy.abs(approximation), color='g', ls='-')
xlabel('t/T')
ylabel('Near-field Signal [a.u.]')
leg = legend(loc='upper right', fancybox=True, shadow=True)
for t in leg.texts:
t.set_fontsize(18)
leg.get_frame().set_linewidth(.1)
grid()
tight_layout()
ylim(.35, 1.18)
if compare_spectra:
freqs = numpy.linspace(1100, 1900, 200)
layer = mat.LayeredMedia((mat.PMMA, 100e-7), exit=mat.Si)
PMMA=tip.LRM(freqs,rp=layer.reflection_p,a=30,amplitude=60+lift/2.,\
normalize_to=mat.Si.reflection_p,Nqs=72,normalize_at=1000,Nzs=30,\
load_freq=freq)
figure()
PMMA['signal_2'].imag.plot(color='b')
ylabel(r'$\mathrm{Im}[\,S_2\,]\,[\mathrm{norm.}]$', color='b')
tick_params(color='b', axis='y')
yticks(color='b')
xlabel(r'$\omega\,[cm^{-1}]$')
gca().spines['left'].set_color('b')
gca().spines['right'].set_color('r')
p = numpy.angle(PMMA['norm_signals'].cslice[0])
DCSD1 = get_DCSD(PMMA['signals']) * numpy.exp(-1j * p)
DCSD2 = get_DCSD(PMMA['signals'], lift=lift) * numpy.exp(-1j * p)
twinx()
(DCSD1.imag / numpy.abs(DCSD2)).plot(color='r')
ylabel(r'$\mathrm{DCSD}\,[\mathrm{norm.}]$', color='r', rotation=270)
tick_params(color='r', axis='y')
yticks(color='r')
tight_layout()
subplots_adjust(right=.85)
def __init__(self,zs=numpy.logspace(-3,2,400),Nterms_max=20,sort=True,
*args,**kwargs):
"""For some reason using Nqs>=244, getting higher q-resolution,
only makes more terms relevant, requiring twice as many terms for
stability and smoothness in approach curves...
(although overall """
self.zs=zs
self.Nterms_max=Nterms_max
#Take a look at current probe geometry
#Adjust quadrature parameters that assist in yielding smooth residues/poles
#Pre-determined as good values for Nterms=10
geometry=tip.LRM.geometric_params['geometry']
if self.verbose:
Logger.write('Setting up eigenfield tip model for geometry "%s"...'\
%geometry)
if geometry is 'cone': tip.LRM.quadrature_params['b']=.5
elif geometry is 'hyperboloid': tip.LRM.quadrature_params['b']=.5
global Rs,Ps
Rs=[]
Ps=[]
for i,z in enumerate(zs):
if i==0: kwargs['reload_signal']=True
elif i==1: kwargs['reload_signal']=False
d=get_tip_eigenbasis_expansion(z,a=1,*args,**kwargs)
Rrow=d['Rs']
Prow=d['Ps']
Rrow[numpy.isnan(Rrow)]=Rrow[numpy.isfinite(Rrow)][-1]
Prow[numpy.isnan(Prow)]=Prow[numpy.isfinite(Prow)][-1]
Rrow=Rrow[Prow.real>0]
Prow=Prow[Prow.real>0]
where_unphys_Ps=(Prow.imag>0)
Prow[where_unphys_Ps]-=1j*Prow[where_unphys_Ps].imag
Prow=Prow[:50]
Rrow=Rrow[:50]
if i>1 and sort:
#Ensure continuity with previous poles (could have done residues instead, but this works)
sorting=numpy.array([numpy.argmin(
numpy.abs((Prow-previous_P)-\
(previous_P-preprevious_P)))
for previous_P,preprevious_P in zip(previous_Prow,preprevious_Prow)])
Prow=Prow[sorting]
Rrow=Rrow[sorting]
Rs.append(Rrow[:Nterms_max])
Ps.append(Prow[:Nterms_max])
#Make sure to keep a reference to previous set of poles and residues
if sort:
previous_Prow=Prow
previous_Rrow=Rrow
if i>=1:
preprevious_Prow=previous_Prow
preprevious_Rrow=previous_Rrow
terms=numpy.arange(Nterms_max)+1
self.Rs=AWA(Rs,axes=[zs,terms],axis_names=['z/a','Term'])
self.Ps=AWA(Ps,axes=[zs,terms],axis_names=['z/a','Term'])
##Remove `nan`s from poles and residues##
#Best way to remove `nan`s (=huge values) is to replace with largest finite value
#Largest such value in Rs will be found in ratio to that in Ps, implying that
#beta in denominator of that term is irrelevant, so term+offset goes to zero anyway..
for j in range(Nterms_max):
Rrow=self.Rs[:,j]; Prow=self.Ps[:,j]
Rrow[numpy.isnan(Rrow)]=Rrow[numpy.isfinite(Rrow)][-1] #Highest value will be for largest z (end of array)
Prow[numpy.isnan(Prow)]=Prow[numpy.isfinite(Prow)][-1]
##Remove any positive imaginary part from poles##
#These are unphysical and are perhaps a by-product of inaccurate eigenvalues
#when diagonalizing an almost-singular matrix (i.e. g*Lambda)
#Just put them on the real line, at least it doesn't seem to hurt anything, at most it's more physical.
where_unphys_Ps=(self.Ps.imag>0)
self.Ps[where_unphys_Ps]-=1j*self.Ps[where_unphys_Ps].imag
if self.verbose: Logger.write('\tDone.')