def kramers_kronig_analysis(self,
zlp=None,
iterations=1,
n=None,
t=None,
delta=0.5,
full_output=False):
"""Calculate the complex
dielectric function from a single scattering distribution (SSD) using
the Kramers-Kronig relations.
It uses the FFT method as in [Egerton2011]_. The SSD is an
EELSSpectrum instance containing SSD low-loss EELS with no zero-loss
peak. The internal loop is devised to approximately subtract the
surface plasmon contribution supposing an unoxidized planar surface and
neglecting coupling between the surfaces. This method does not account
for retardation effects, instrumental broading and surface plasmon
excitation in particles.
Note that either refractive index or thickness are required.
If both are None or if both are provided an exception is raised.
Parameters
----------
zlp: {None, number, Signal}
ZLP intensity. It is optional (can be None) if `t` is None and `n`
is not None and the thickness estimation is not required. If `t`
is not None, the ZLP is required to perform the normalization and
if `t` is not None, the ZLP is required to calculate the thickness.
If the ZLP is the same for all spectra, the integral of the ZLP
can be provided as a number. Otherwise, if the ZLP intensity is not
the same for all spectra, it can be provided as i) a Signal
of the same dimensions as the current signal containing the ZLP
spectra for each location ii) a Signal of signal dimension 0
and navigation_dimension equal to the current signal containing the
integrated ZLP intensity.
iterations: int
Number of the iterations for the internal loop to remove the
surface plasmon contribution. If 1 the surface plasmon contribution
is not estimated and subtracted (the default is 1).
n: {None, float}
The medium refractive index. Used for normalization of the
SSD to obtain the energy loss function. If given the thickness
is estimated and returned. It is only required when `t` is None.
t: {None, number, Signal}
The sample thickness in nm. Used for normalization of the
SSD to obtain the energy loss function. It is only required when
`n` is None. If the thickness is the same for all spectra it can be
given by a number. Otherwise, it can be provided as a Signal with
signal dimension 0 and navigation_dimension equal to the current
signal.
delta : float
A small number (0.1-0.5 eV) added to the energy axis in
specific steps of the calculation the surface loss correction to
improve stability.
full_output : bool
If True, return a dictionary that contains the estimated
thickness if `t` is None and the estimated surface plasmon
excitation and the spectrum corrected from surface plasmon
excitations if `iterations` > 1.
Returns
-------
eps: DielectricFunction instance
The complex dielectric function results,
$\epsilon = \epsilon_1 + i*\epsilon_2$,
contained in an DielectricFunction instance.
output: Dictionary (optional)
A dictionary of optional outputs with the following keys:
``thickness``
The estimated thickness in nm calculated by normalization of
the SSD (only when `t` is None)
``surface plasmon estimation``
The estimated surface plasmon excitation (only if
`iterations` > 1.)
Raises
------
ValuerError
If both `n` and `t` are undefined (None).
AttribureError
If the beam_energy or the collection angle are not defined in
metadata.
Notes
-----
This method is based in Egerton's Matlab code [Egerton2011]_ with some
minor differences:
* The integrals are performed using the simpsom rule instead of using
a summation.
* The wrap-around problem when computing the ffts is workarounded by
padding the signal instead of substracting the reflected tail.
.. [Egerton2011] Ray Egerton, "Electron Energy-Loss
Spectroscopy in the Electron Microscope", Springer-Verlag, 2011.
"""
output = {}
if iterations == 1:
# In this case s.data is not modified so there is no need to make
# a deep copy.
s = self.isig[0.:]
else:
s = self.isig[0.:].deepcopy()
sorig = self.isig[0.:]
# Avoid singularity at 0
if s.axes_manager.signal_axes[0].axis[0] == 0:
s = s.isig[1:]
sorig = self.isig[1:]
# Constants and units
me = constants.value(
'electron mass energy equivalent in MeV') * 1e3 # keV
# Mapped parameters
try:
e0 = s.metadata.Acquisition_instrument.TEM.beam_energy
except:
raise AttributeError("Please define the beam energy."
"You can do this e.g. by using the "
"set_microscope_parameters method")
try:
beta = s.metadata.Acquisition_instrument.TEM.Detector.EELS.collection_angle
except:
raise AttributeError("Please define the collection angle."
"You can do this e.g. by using the "
"set_microscope_parameters method")
axis = s.axes_manager.signal_axes[0]
eaxis = axis.axis.copy()
if isinstance(zlp, hyperspy.signal.Signal):
if (zlp.axes_manager.navigation_dimension ==
self.axes_manager.navigation_dimension):
if zlp.axes_manager.signal_dimension == 0:
i0 = zlp.data
else:
i0 = zlp.integrate1D(axis.index_in_axes_manager).data
else:
raise ValueError('The ZLP signal dimensions are not '
'compatible with the dimensions of the '
'low-loss signal')
i0 = i0.reshape(
np.insert(i0.shape, axis.index_in_array, 1))
elif isinstance(zlp, numbers.Number):
i0 = zlp
else:
raise ValueError('The zero-loss peak input is not valid.')
if isinstance(t, hyperspy.signal.Signal):
if (t.axes_manager.navigation_dimension ==
self.axes_manager.navigation_dimension) and (
t.axes_manager.signal_dimension == 0):
t = t.data
t = t.reshape(
np.insert(t.shape, axis.index_in_array, 1))
else:
raise ValueError('The thickness signal dimensions are not '
'compatible with the dimensions of the '
'low-loss signal')
elif isinstance(t, np.ndarray) and t.shape and t.shape != (1,):
raise ValueError("thickness must be a HyperSpy signal or a number,"
" not a numpy array.")
# Slicer to get the signal data from 0 to axis.size
slicer = s.axes_manager._get_data_slice(
[(axis.index_in_array, slice(None, axis.size)), ])
# Kinetic definitions
ke = e0 * (1 + e0 / 2. / me) / (1 + e0 / me) ** 2
tgt = e0 * (2 * me + e0) / (me + e0)
rk0 = 2590 * (1 + e0 / me) * np.sqrt(2 * ke / me)
for io in range(iterations):
# Calculation of the ELF by normalization of the SSD
# Norm(SSD) = Imag(-1/epsilon) (Energy Loss Funtion, ELF)
# We start by the "angular corrections"
Im = s.data / (np.log(1 + (beta * tgt / eaxis) ** 2)) / axis.scale
if n is None and t is None:
raise ValueError("The thickness and the refractive index are "
"not defined. Please provide one of them.")
elif n is not None and t is not None:
raise ValueError("Please provide the refractive index OR the "
"thickness information, not both")
elif n is not None:
# normalize using the refractive index.
K = (Im / eaxis).sum(axis=axis.index_in_array) * axis.scale
K = (K / (np.pi / 2) / (1 - 1. / n ** 2)).reshape(
np.insert(K.shape, axis.index_in_array, 1))
# Calculate the thickness only if possible and required
if zlp is not None and (full_output is True or
iterations > 1):
te = (332.5 * K * ke / i0)
if full_output is True:
output['thickness'] = te
elif t is not None:
if zlp is None:
raise ValueError("The ZLP must be provided when the "
"thickness is used for normalization.")
# normalize using the thickness
K = t * i0 / (332.5 * ke)
te = t
Im = Im / K
# Kramers Kronig Transform:
# We calculate KKT(Im(-1/epsilon))=1+Re(1/epsilon) with FFT
# Follows: D W Johnson 1975 J. Phys. A: Math. Gen. 8 490
# Use a size that is a power of two to speed up the fft and
# make it double the closest upper value to workaround the
# wrap-around problem.
esize = 2 * closest_power_of_two(axis.size)
q = -2 * np.fft.fft(Im, esize,
axis.index_in_array).imag / esize
q[slicer] *= -1
q = np.fft.fft(q, axis=axis.index_in_array)
# Final touch, we have Re(1/eps)
Re = q[slicer].real + 1
# Egerton does this to correct the wrap-around problem, but in our
# case this is not necessary because we compute the fft on an
# extended and padded spectrum to avoid this problem.
# Re=real(q)
# Tail correction
# vm=Re[axis.size-1]
# Re[:(axis.size-1)]=Re[:(axis.size-1)]+1-(0.5*vm*((axis.size-1) /
# (axis.size*2-arange(0,axis.size-1)))**2)
# Re[axis.size:]=1+(0.5*vm*((axis.size-1) /
# (axis.size+arange(0,axis.size)))**2)
# Epsilon appears:
# We calculate the real and imaginary parts of the CDF
e1 = Re / (Re ** 2 + Im ** 2)
e2 = Im / (Re ** 2 + Im ** 2)
if iterations > 1 and zlp is not None:
# Surface losses correction:
# Calculates the surface ELF from a vaccumm border effect
# A simulated surface plasmon is subtracted from the ELF
Srfelf = 4 * e2 / ((e1 + 1) ** 2 + e2 ** 2) - Im
adep = (tgt / (eaxis + delta) *
np.arctan(beta * tgt / axis.axis) -
beta / 1000. /
(beta ** 2 + axis.axis ** 2. / tgt ** 2))
Srfint = 2000 * K * adep * Srfelf / rk0 / te * axis.scale
s.data = sorig.data - Srfint
print 'Iteration number: ', io + 1, '/', iterations
if iterations == io + 1 and full_output is True:
sp = sorig._deepcopy_with_new_data(Srfint)
sp.metadata.General.title += (
" estimated surface plasmon excitation.")
output['surface plasmon estimation'] = sp
del sp
del Srfint
eps = s._deepcopy_with_new_data(e1 + e2 * 1j)
del s
eps.set_signal_type("DielectricFunction")
eps.metadata.General.title = (self.metadata.General.title +
'dielectric function '
'(from Kramers-Kronig analysis)')
if eps.tmp_parameters.has_item('filename'):
eps.tmp_parameters.filename = (
self.tmp_parameters.filename +
'_CDF_after_Kramers_Kronig_transform')
if 'thickness' in output:
thickness = eps._get_navigation_signal()
thickness.metadata.General.title = (
self.metadata.General.title + ' thickness '
'(calculated using Kramers-Kronig analysis)')
thickness.data = te[
self.axes_manager._get_data_slice([(
axis.index_in_array, 0)])]
output['thickness'] = thickness
if full_output is False:
return eps
else:
return eps, output
def fourier_log_deconvolution(self,
zlp,
add_zlp=False,
crop=False):
"""Performs fourier-log deconvolution.
Parameters
----------
zlp : EELSSpectrum
The corresponding zero-loss peak.
add_zlp : bool
If True, adds the ZLP to the deconvolved spectrum
crop : bool
If True crop the spectrum to leave out the channels that
have been modified to decay smoothly to zero at the sides
of the spectrum.
Returns
-------
An EELSSpectrum containing the current data deconvolved.
Notes
-----
For details see: Egerton, R. Electron Energy-Loss
Spectroscopy in the Electron Microscope. Springer-Verlag, 2011.
"""
self._check_signal_dimension_equals_one()
s = self.deepcopy()
zlp_size = zlp.axes_manager.signal_axes[0].size
self_size = self.axes_manager.signal_axes[0].size
tapped_channels = s.hanning_taper()
# Conservative new size to solve the wrap-around problem
size = zlp_size + self_size - 1
# Increase to the closest power of two to enhance the FFT
# performance
size = closest_power_of_two(size)
axis = self.axes_manager.signal_axes[0]
z = np.fft.rfft(zlp.data, n=size, axis=axis.index_in_array)
j = np.fft.rfft(s.data, n=size, axis=axis.index_in_array)
j1 = z * np.nan_to_num(np.log(j / z))
sdata = np.fft.irfft(j1, axis=axis.index_in_array)
s.data = sdata[s.axes_manager._get_data_slice(
[(axis.index_in_array, slice(None, self_size)), ])]
if add_zlp is True:
if self_size >= zlp_size:
s.data[s.axes_manager._get_data_slice(
[(axis.index_in_array, slice(None, zlp_size)), ])
] += zlp.data
else:
s.data += zlp.data[s.axes_manager._get_data_slice(
[(axis.index_in_array, slice(None, self_size)), ])]
s.metadata.General.title = (s.metadata.General.title +
' after Fourier-log deconvolution')
if s.tmp_parameters.has_item('filename'):
s.tmp_parameters.filename = (
self.tmp_parameters.filename +
'_after_fourier_log_deconvolution')
if crop is True:
s.crop(axis.index_in_axes_manager,
None, int(-tapped_channels))
return s
def kramers_kronig_analysis(self,
zlp=None,
iterations=1,
n=None,
t=None,
delta=0.5,
full_output=False):
"""Calculate the complex
dielectric function from a single scattering distribution (SSD) using
the Kramers-Kronig relations.
It uses the FFT method as in [Egerton2011]_. The SSD is an
EELSSpectrum instance containing SSD low-loss EELS with no zero-loss
peak. The internal loop is devised to approximately subtract the
surface plasmon contribution supposing an unoxidized planar surface and
neglecting coupling between the surfaces. This method does not account
for retardation effects, instrumental broading and surface plasmon
excitation in particles.
Note that either refractive index or thickness are required.
If both are None or if both are provided an exception is raised.
Parameters
----------
zlp: {None, number, Signal}
ZLP intensity. It is optional (can be None) if `t` is None and `n`
is not None and the thickness estimation is not required. If `t`
is not None, the ZLP is required to perform the normalization and
if `t` is not None, the ZLP is required to calculate the thickness.
If the ZLP is the same for all spectra, the integral of the ZLP
can be provided as a number. Otherwise, if the ZLP intensity is not
the same for all spectra, it can be provided as i) a Signal
of the same dimensions as the current signal containing the ZLP
spectra for each location ii) a Signal of signal dimension 0
and navigation_dimension equal to the current signal containing the
integrated ZLP intensity.
iterations: int
Number of the iterations for the internal loop to remove the
surface plasmon contribution. If 1 the surface plasmon contribution
is not estimated and subtracted (the default is 1).
n: {None, float}
The medium refractive index. Used for normalization of the
SSD to obtain the energy loss function. If given the thickness
is estimated and returned. It is only required when `t` is None.
t: {None, number, Signal}
The sample thickness in nm. Used for normalization of the
SSD to obtain the energy loss function. It is only required when
`n` is None. If the thickness is the same for all spectra it can be
given by a number. Otherwise, it can be provided as a Signal with
signal dimension 0 and navigation_dimension equal to the current
signal.
delta : float
A small number (0.1-0.5 eV) added to the energy axis in
specific steps of the calculation the surface loss correction to
improve stability.
full_output : bool
If True, return a dictionary that contains the estimated
thickness if `t` is None and the estimated surface plasmon
excitation and the spectrum corrected from surface plasmon
excitations if `iterations` > 1.
Returns
-------
eps: DielectricFunction instance
The complex dielectric function results,
$\epsilon = \epsilon_1 + i*\epsilon_2$,
contained in an DielectricFunction instance.
output: Dictionary (optional)
A dictionary of optional outputs with the following keys:
``thickness``
The estimated thickness in nm calculated by normalization of
the SSD (only when `t` is None)
``surface plasmon estimation``
The estimated surface plasmon excitation (only if
`iterations` > 1.)
Raises
------
ValuerError
If both `n` and `t` are undefined (None).
AttribureError
If the beam_energy or the collection angle are not defined in
metadata.
Notes
-----
This method is based in Egerton's Matlab code [Egerton2011]_ with some
minor differences:
* The integrals are performed using the simpsom rule instead of using
a summation.
* The wrap-around problem when computing the ffts is workarounded by
padding the signal instead of substracting the reflected tail.
.. [Egerton2011] Ray Egerton, "Electron Energy-Loss
Spectroscopy in the Electron Microscope", Springer-Verlag, 2011.
"""
output = {}
if iterations == 1:
# In this case s.data is not modified so there is no need to make
# a deep copy.
s = self.isig[0.:]
else:
s = self.isig[0.:].deepcopy()
sorig = self.isig[0.:]
# Avoid singularity at 0
if s.axes_manager.signal_axes[0].axis[0] == 0:
s = s.isig[1:]
sorig = self.isig[1:]
# Constants and units
me = constants.value(
'electron mass energy equivalent in MeV') * 1e3 # keV
# Mapped parameters
try:
e0 = s.metadata.Acquisition_instrument.TEM.beam_energy
except:
raise AttributeError("Please define the beam energy."
"You can do this e.g. by using the "
"set_microscope_parameters method")
try:
beta = s.metadata.Acquisition_instrument.TEM.Detector.EELS.collection_angle
except:
raise AttributeError("Please define the collection angle."
"You can do this e.g. by using the "
"set_microscope_parameters method")
axis = s.axes_manager.signal_axes[0]
eaxis = axis.axis.copy()
if isinstance(zlp, hyperspy.signal.Signal):
if (zlp.axes_manager.navigation_dimension ==
self.axes_manager.navigation_dimension):
if zlp.axes_manager.signal_dimension == 0:
i0 = zlp.data
else:
i0 = zlp.integrate1D(axis.index_in_axes_manager).data
else:
raise ValueError('The ZLP signal dimensions are not '
'compatible with the dimensions of the '
'low-loss signal')
i0 = i0.reshape(np.insert(i0.shape, axis.index_in_array, 1))
elif isinstance(zlp, numbers.Number):
i0 = zlp
else:
raise ValueError('The zero-loss peak input is not valid.')
if isinstance(t, hyperspy.signal.Signal):
if (t.axes_manager.navigation_dimension
== self.axes_manager.navigation_dimension) and (
t.axes_manager.signal_dimension == 0):
t = t.data
t = t.reshape(np.insert(t.shape, axis.index_in_array, 1))
else:
raise ValueError('The thickness signal dimensions are not '
'compatible with the dimensions of the '
'low-loss signal')
elif isinstance(t, np.ndarray) and t.shape and t.shape != (1, ):
raise ValueError("thickness must be a HyperSpy signal or a number,"
" not a numpy array.")
# Slicer to get the signal data from 0 to axis.size
slicer = s.axes_manager._get_data_slice([
(axis.index_in_array, slice(None, axis.size)),
])
# Kinetic definitions
ke = e0 * (1 + e0 / 2. / me) / (1 + e0 / me)**2
tgt = e0 * (2 * me + e0) / (me + e0)
rk0 = 2590 * (1 + e0 / me) * np.sqrt(2 * ke / me)
for io in range(iterations):
# Calculation of the ELF by normalization of the SSD
# Norm(SSD) = Imag(-1/epsilon) (Energy Loss Funtion, ELF)
# We start by the "angular corrections"
Im = s.data / (np.log(1 + (beta * tgt / eaxis)**2)) / axis.scale
if n is None and t is None:
raise ValueError("The thickness and the refractive index are "
"not defined. Please provide one of them.")
elif n is not None and t is not None:
raise ValueError("Please provide the refractive index OR the "
"thickness information, not both")
elif n is not None:
# normalize using the refractive index.
K = (Im / eaxis).sum(axis=axis.index_in_array) * axis.scale
K = (K / (np.pi / 2) / (1 - 1. / n**2)).reshape(
np.insert(K.shape, axis.index_in_array, 1))
# Calculate the thickness only if possible and required
if zlp is not None and (full_output is True or iterations > 1):
te = (332.5 * K * ke / i0)
if full_output is True:
output['thickness'] = te
elif t is not None:
if zlp is None:
raise ValueError("The ZLP must be provided when the "
"thickness is used for normalization.")
# normalize using the thickness
K = t * i0 / (332.5 * ke)
te = t
Im = Im / K
# Kramers Kronig Transform:
# We calculate KKT(Im(-1/epsilon))=1+Re(1/epsilon) with FFT
# Follows: D W Johnson 1975 J. Phys. A: Math. Gen. 8 490
# Use a size that is a power of two to speed up the fft and
# make it double the closest upper value to workaround the
# wrap-around problem.
esize = 2 * closest_power_of_two(axis.size)
q = -2 * np.fft.fft(Im, esize, axis.index_in_array).imag / esize
q[slicer] *= -1
q = np.fft.fft(q, axis=axis.index_in_array)
# Final touch, we have Re(1/eps)
Re = q[slicer].real + 1
# Egerton does this to correct the wrap-around problem, but in our
# case this is not necessary because we compute the fft on an
# extended and padded spectrum to avoid this problem.
# Re=real(q)
# Tail correction
# vm=Re[axis.size-1]
# Re[:(axis.size-1)]=Re[:(axis.size-1)]+1-(0.5*vm*((axis.size-1) /
# (axis.size*2-arange(0,axis.size-1)))**2)
# Re[axis.size:]=1+(0.5*vm*((axis.size-1) /
# (axis.size+arange(0,axis.size)))**2)
# Epsilon appears:
# We calculate the real and imaginary parts of the CDF
e1 = Re / (Re**2 + Im**2)
e2 = Im / (Re**2 + Im**2)
if iterations > 1 and zlp is not None:
# Surface losses correction:
# Calculates the surface ELF from a vaccumm border effect
# A simulated surface plasmon is subtracted from the ELF
Srfelf = 4 * e2 / ((e1 + 1)**2 + e2**2) - Im
adep = (tgt /
(eaxis + delta) * np.arctan(beta * tgt / axis.axis) -
beta / 1000. / (beta**2 + axis.axis**2. / tgt**2))
Srfint = 2000 * K * adep * Srfelf / rk0 / te * axis.scale
s.data = sorig.data - Srfint
print 'Iteration number: ', io + 1, '/', iterations
if iterations == io + 1 and full_output is True:
sp = sorig._deepcopy_with_new_data(Srfint)
sp.metadata.General.title += (
" estimated surface plasmon excitation.")
output['surface plasmon estimation'] = sp
del sp
del Srfint
eps = s._deepcopy_with_new_data(e1 + e2 * 1j)
del s
eps.set_signal_type("DielectricFunction")
eps.metadata.General.title = (self.metadata.General.title +
'dielectric function '
'(from Kramers-Kronig analysis)')
if eps.tmp_parameters.has_item('filename'):
eps.tmp_parameters.filename = (
self.tmp_parameters.filename +
'_CDF_after_Kramers_Kronig_transform')
if 'thickness' in output:
thickness = eps._get_navigation_signal()
thickness.metadata.General.title = (
self.metadata.General.title + ' thickness '
'(calculated using Kramers-Kronig analysis)')
thickness.data = te[self.axes_manager._get_data_slice([
(axis.index_in_array, 0)
])]
output['thickness'] = thickness
if full_output is False:
return eps
else:
return eps, output
def test_closest_power_of_two():
assert closest_power_of_two(5) == 8
assert closest_power_of_two(13) == 16
assert closest_power_of_two(120) == 128
assert closest_power_of_two(973) == 1024
def fourier_log_deconvolution(self, zlp, add_zlp=False, crop=False):
"""Performs fourier-log deconvolution.
Parameters
----------
zlp : EELSSpectrum
The corresponding zero-loss peak.
add_zlp : bool
If True, adds the ZLP to the deconvolved spectrum
crop : bool
If True crop the spectrum to leave out the channels that
have been modified to decay smoothly to zero at the sides
of the spectrum.
Returns
-------
An EELSSpectrum containing the current data deconvolved.
Notes
-----
For details see: Egerton, R. Electron Energy-Loss
Spectroscopy in the Electron Microscope. Springer-Verlag, 2011.
"""
self._check_signal_dimension_equals_one()
s = self.deepcopy()
zlp_size = zlp.axes_manager.signal_axes[0].size
self_size = self.axes_manager.signal_axes[0].size
tapped_channels = s.hanning_taper()
# Conservative new size to solve the wrap-around problem
size = zlp_size + self_size - 1
# Increase to the closest power of two to enhance the FFT
# performance
size = closest_power_of_two(size)
axis = self.axes_manager.signal_axes[0]
z = np.fft.rfft(zlp.data, n=size, axis=axis.index_in_array)
j = np.fft.rfft(s.data, n=size, axis=axis.index_in_array)
j1 = z * np.nan_to_num(np.log(j / z))
sdata = np.fft.irfft(j1, axis=axis.index_in_array)
s.data = sdata[s.axes_manager._get_data_slice([
(axis.index_in_array, slice(None, self_size)),
])]
if add_zlp is True:
if self_size >= zlp_size:
s.data[s.axes_manager._get_data_slice([
(axis.index_in_array, slice(None, zlp_size)),
])] += zlp.data
else:
s.data += zlp.data[s.axes_manager._get_data_slice([
(axis.index_in_array, slice(None, self_size)),
])]
s.metadata.General.title = (s.metadata.General.title +
' after Fourier-log deconvolution')
if s.tmp_parameters.has_item('filename'):
s.tmp_parameters.filename = (self.tmp_parameters.filename +
'_after_fourier_log_deconvolution')
if crop is True:
s.crop(axis.index_in_axes_manager, None, int(-tapped_channels))
return s