""" Basic usage of pyElli ===================== Basic usage of building a model and fitting it to measurement data of SiO2 on Si. """ # %% Import required libraries import elli from elli.fitting import ParamsHist, fit # sphinx_gallery_thumbnail_path = '_static/basic_usage.png' # %% # Reading data # ------------------------ # # We load the data from the generated # `NeXus file `_ # and select the angle we want to analyse. # You may set the ANGLE constant to 50 or 60 to select # other angles of incidence from the example file. # Additionally, we're cutting the wavelength axis to be in between 210 nm and 800 nm. # This is because we're using literature values for Si, # which are only defined in this wavelength range. ANGLE = 70 psi_delta = elli.read_nexus_psi_delta("SiO2onSi.ellips.nxs").loc[ANGLE].loc[210:800] # %% # Setting parameters # ------------------------ # # As an example we analyse an oxidation layer of SiO2 on Si. # Prior to defining our model, we have to set the parameters we want to use. # We're going to use a :ref:`Cauchy model ` for SiO2 and load the Si values from # `literature values `_. # The parameter names can be chosen freely, # but you have to use the exact same name in the later model definition. # The package uses lmfit as fitting tool and you may refer to their # `documentation `_ # for details on parameter definition. params = ParamsHist() params.add("SiO2_n0", value=1.452, min=-100, max=100, vary=False) params.add("SiO2_n1", value=36.0, min=-40000, max=40000, vary=False) params.add("SiO2_n2", value=0, min=-40000, max=40000, vary=False) params.add("SiO2_k0", value=0, min=-100, max=100, vary=False) params.add("SiO2_k1", value=0, min=-40000, max=40000, vary=False) params.add("SiO2_k2", value=0, min=-40000, max=40000, vary=False) params.add("SiO2_d", value=20, min=0, max=40000, vary=True) # %% # Load silicon dispersion from the refractiveindexinfo database # ------------------------------------------------------------- # You can load any material from the index # `refractiveindex.info `__, which is # embedded into the software (so you may use it offline, too). Here, we # are interested in the literature values for the silicon substrate. # First we need to load the database with ``rii_db = elli.db.RII()`` and # then we can query it with ``rii_db.get_mat("Si", "Aspnes")`` to load # this # `entry `__. rii_db = elli.db.RII() Si = rii_db.get_mat("Si", "Aspnes") # %% # Building the model # ------------------------ # # For simple parameter estimation, # the fit decorator (**@fit**) in conjunction with the model definition is used. # The fitting decorator takes a pandas dataframe containing # the psi/delta measurement data (**psi_delta**) and the model parameters (**params**) as an input. # It then passes the wavelength from measurement dataframe (**lbda**) # and the parameters to the actual model function. # # Inside the model function the optical model is built, # i.e. the Si literature values are loaded # and the fitting parameters are filled into the Cauchy dispersion. # For details on how to insert data into the Cauchy model or other optical dispersion models, # you may refer to the documentation of pyElli. # Please keep in mind that the parameters you use here # have to be defined in the parameter object **param**. # # From the dispersion model isotropic materials are generated # (could also be an anisotropic material, refer to the docs for an overview). # This is done by calling the :code:`elli.IsotropicMaterial(...)` function # with a dispersion model as a parameter # or simply calling :code:`.get_mat()` on a dispersion model. # These two approaches are equivalent. From these materials the layer is build, # which only consists of the SiO2 layer in this example. # The final structure consists of an incoming half-space, # the layers and an outgoing half space. Specifically, # typically the light is coming from air and finally gets absorbed by the bulk material, # in our example this is Si, i.e. we call :code:`elli.Structure(elli.AIR, Layer, Si)`. # # To provide simulated data, we have to evaluate the structure # by calling the :code:`evaluate(...)` function, # which takes the experimental wavelength array **lbda**, **ANGLE** under which the experiment # was performed and the solver to be used to solve the transfer-matrix problem. # Here, we use a simple 2x2 matrix approach, # which splits the interaction in s and p-parts and therefore cannot account for anisotropy. # There exist 4x4 matrix solvers as well. # You may refer to the :ref:`solver documentation ` for further details. # # Executing the cell below in a jupyter notebook displays a comparison of the simulated Ψ / Δ values # at the current parameter values with their measured counterparts. # Additionally, input fields for each model parameter are shown. # You may change the parameters and the calculated data will change accordingly. # For clarification the modeled data is shown with `_calc` postfix in the legend. @fit(psi_delta, params) def model(lbda, params): # Generate the cauchy model from the current lmfit parameters SiO2 = elli.Cauchy( params["SiO2_n0"], params["SiO2_n1"], params["SiO2_n2"], params["SiO2_k0"], params["SiO2_k1"], params["SiO2_k2"], ).get_mat() # get_mat() generates an IsotropicMaterial from the dispersion relation # Construct the layers you expect in your sample # Here, it only consists of one layer SiO2 in between air and Si. # We build the structure coming from air, through the layers, # represented as an array, and having Si as bulk material. structure = elli.Structure( elli.AIR, # Input medium [elli.Layer(SiO2, params["SiO2_d"])], # Overlayer structure Si, ) # Output medium / Substrate # The model should return the evaluation of the structure at the experimental wavelengths lbda, # the experimental angle ANGLE and it should define a solver to calculate the transfer matrix. return structure.evaluate(lbda, ANGLE, solver=elli.Solver2x2) # %% # Fit and plot fit results # ------------------------ # # The fit of the data can be executed by calling the fit() function on the model function, # which automatically gets attached by the @fit decorator in the cell above. # The following cell basically executes the fit and plots # a comparison between the measurement and fitted data. fit_stats = model.fit() model.plot() # %% # Extracting the optical properties from the fit # ---------------------------------------------- # # Since we want to extract the dispersion relation of a layer in our measured stack, # we can use our fitted parameters. # The fit parameters are contained in the fits output :code:`params` attribute, # i.e. :code:`fit_stats.params` for our example. # We can use it to call our dispersion relation we used in our model # (here it is a Cauchy dispersion relation) # and fill in our fitted value. # By calling :code:`get_dielectric_df()` we can get the dielectric function of the material, # which is plotted here as an example. :code:`get_dielectric_df` uses a default wavelength range # which can also be changed by inputting a wavelength array as a parameter. fitted_model = elli.Cauchy( fit_stats.params["SiO2_n0"], fit_stats.params["SiO2_n1"], fit_stats.params["SiO2_n2"], fit_stats.params["SiO2_k0"], fit_stats.params["SiO2_k1"], fit_stats.params["SiO2_k2"], ) fitted_model.get_dielectric_df().plot(backend="plotly") # %% # We can also call :code:`get_refractive_index_df()` # to get the refractive index of the material as dataframe. fitted_model.get_refractive_index_df().plot(backend="plotly") # %% # If you want to write your data to a file you simply call pandas :code:`to_csv(...)` # function to write a csv file, i.e. for the dielectric function this writes as fitted_model.get_dielectric_df().to_csv("SiO2_diel_func.csv") # %% # You may also access a single value of your optical model fit_stats.params["SiO2_n0"].value # %% # Our simply print the fitted values in a list together with their fitting errors fit_stats.params # %% # Show fit statistics # -------------------- # Now, we may also print out the fit statictics from the model fit in the cell above. # The fit statistics are simple `lmfit fit statistics `_, too. # Typically, one uses chi square values as a figure of merit for the fit quality. # It is stored in the chisqr attribute of the fit_stats variable we defined above. fit_stats.chisqr # %% # We can print the full fit statistics, too. fit_stats # %% # References # ---------- # `Here `_ # you can find the latest jupyter notebook and data files of this example.