Added esis.flights.f1.optics.gratings.materials.multilayer_witness_fit() function.

Author: byrdie

docs/conf.py

@@ -36,6 +36,7 @@
     'sphinx.ext.autosummary',
     'sphinx.ext.intersphinx',
     'sphinx.ext.inheritance_diagram',
+    'sphinxcontrib.bibtex',
     'jupyter_sphinx',
     'sphinx_favicon'
 ]
@@ -106,6 +107,10 @@
 # https://github.com/readthedocs/readthedocs.org/issues/2569
 master_doc = 'index'
 
+bibtex_bibfiles = ['refs.bib']
+bibtex_default_style = 'plain'
+bibtex_reference_style = 'author_year'
+
 intersphinx_mapping = {
     'python': ('https://docs.python.org/3', None),
     'numpy': ('https://numpy.org/doc/stable/', None),

docs/index.rst

@@ -11,6 +11,13 @@ launched from White Sands Missile Range on September 30th, 2019.
 
     esis
 
+References
+==========
+
+.. bibliography::
+
+|
+
 Indices and tables
 ==================
 

docs/refs.bib

@@ -0,0 +1,43 @@
+@article{Kortright1988,
+  title={Amorphous silicon carbide coatings for extreme ultraviolet optics.},
+  author={Jeffrey B Kortright and David L Windt},
+  journal={Applied optics},
+  year={1988},
+  volume={27 14},
+  pages={
+          2841-6
+        },
+  url={https://api.semanticscholar.org/CorpusID:39248487}
+}
+
+@article{VidalDasilva2010,
+    author = {Vidal-Dasilva, Manuela and Aquila, Andrew L. and Gullikson, Eric M. and Salmassi, Farhad and Larruquert, Juan I.},
+    title = "{Optical constants of magnetron-sputtered magnesium films in the 25–1300 eV energy range}",
+    journal = {Journal of Applied Physics},
+    volume = {108},
+    number = {6},
+    pages = {063517},
+    year = {2010},
+    month = {09},
+    abstract = "{The transmittance of dc magnetron-sputtered Mg thin films was measured in the 25–1300 eV spectral range. Freestanding Mg films protected with Al layers were characterized ex situ. Transmittance measurements were used to obtain the extinction coefficient k of Mg films. The obtained k values along with the data available in the literature, and with interpolations and extrapolations for the rest of the spectrum, were used to obtain the real part of the index of refraction n by the Kramers–Krönig analysis. Sum-rule tests indicated a good consistency of the data.}",
+    issn = {0021-8979},
+    doi = {10.1063/1.3481457},
+    url = {https://doi.org/10.1063/1.3481457},
+    eprint = {https://pubs.aip.org/aip/jap/article-pdf/doi/10.1063/1.3481457/13212872/063517\_1\_online.pdf},
+}
+
+@inproceedings{Rebellato2018,
+author = {Jennifer Rebellato and Evgueni Meltchakov and Regina Soufli and S{\'e}bastien De Rossi and Xueyan Zhang and Fr{\'e}d{\'e}ric Auch{\`e}re and Franck Delmotte},
+title = {{Analyses of tabulated optical constants for thin films in the EUV range and application to solar physics multilayer coatings}},
+volume = {10691},
+booktitle = {Advances in Optical Thin Films VI},
+editor = {Michel Lequime and H. Angus Macleod and Detlev Ristau},
+organization = {International Society for Optics and Photonics},
+publisher = {SPIE},
+pages = {106911U},
+keywords = {optical constants, soft x-rays, extreme ultraviolet, EUV, multilayer coatings, solar physics, thin films, materials},
+year = {2018},
+doi = {10.1117/12.2313346},
+URL = {https://doi.org/10.1117/12.2313346}
+}
+

esis/flights/f1/optics/gratings/materials/_materials.py

@@ -1,20 +1,26 @@
 import dataclasses
 import pathlib
 import numpy as np
+import scipy
 import astropy.units as u
 import named_arrays as na
 import optika
 
 __all__ = [
     "multilayer_design",
     "multilayer_witness_measured",
+    "multilayer_witness_fit",
 ]
 
 
 def multilayer_design() -> optika.materials.MultilayerMirror:
     """
     The as-designed multilayer coating for the ESIS diffraction gratings.
 
+    Based on the analysis of :cite:t:`Rebellato2018`, this model uses
+    :cite:t:`Kortright1988` for the silicon carbide optical constants, and
+    :cite:t:`VidalDasilva2010` for the magnesium optical constants.
+
     Examples
     --------
 
@@ -66,8 +72,22 @@ def multilayer_design() -> optika.materials.MultilayerMirror:
             ax.set_axis_off()
     """
 
+    layer_oxide = optika.materials.Layer(
+        chemical="SiO2",
+        thickness=1 * u.nm,
+        interface=optika.materials.profiles.ErfInterfaceProfile(1 * u.nm),
+        kwargs_plot=dict(
+            color="gray",
+        ),
+        x_label=1.1,
+    )
+
     layer_sic = optika.materials.Layer(
-        chemical="SiC",
+        chemical=optika.chemicals.Chemical(
+            formula="SiC",
+            is_amorphous=True,
+            table="kortright",
+        ),
         thickness=10 * u.nm,
         interface=optika.materials.profiles.ErfInterfaceProfile(1 * u.nm),
         kwargs_plot=dict(
@@ -86,15 +106,18 @@ def multilayer_design() -> optika.materials.MultilayerMirror:
     )
 
     layer_mg = optika.materials.Layer(
-        chemical="Mg",
+        chemical=optika.chemicals.Chemical(
+            formula="Mg",
+            table="fernandez_perea",
+        ),
         thickness=30 * u.nm,
         interface=optika.materials.profiles.ErfInterfaceProfile(1 * u.nm),
         kwargs_plot=dict(
             color="pink",
         ),
     )
 
-    layer_sio2 = optika.materials.Layer(
+    layer_substrate = optika.materials.Layer(
         chemical="SiO2",
         thickness=10 * u.mm,
         interface=optika.materials.profiles.ErfInterfaceProfile(1 * u.nm),
@@ -105,13 +128,13 @@ def multilayer_design() -> optika.materials.MultilayerMirror:
 
     return optika.materials.MultilayerMirror(
         layers=[
+            layer_oxide,
+            layer_sic,
             dataclasses.replace(
-                layer_sio2,
-                thickness=1 * u.nm,
-                x_label=1.1,
+                layer_al,
+                thickness=4 * u.nm,
+                interface=optika.materials.profiles.ErfInterfaceProfile(1 * u.nm),
             ),
-            layer_sic,
-            dataclasses.replace(layer_al, thickness=4 * u.nm),
             layer_mg,
             optika.materials.PeriodicLayerSequence(
                 layers=[
@@ -121,9 +144,13 @@ def multilayer_design() -> optika.materials.MultilayerMirror:
                 ],
                 num_periods=3,
             ),
-            dataclasses.replace(layer_al, thickness=10 * u.nm),
+            dataclasses.replace(
+                layer_al,
+                thickness=10 * u.nm,
+                interface=optika.materials.profiles.ErfInterfaceProfile(1 * u.nm),
+            ),
         ],
-        substrate=layer_sio2,
+        substrate=layer_substrate,
     )
 
 
@@ -219,3 +246,189 @@ def multilayer_witness_measured() -> optika.materials.MeasuredMirror:
     )
 
     return result
+
+
+def multilayer_witness_fit() -> optika.materials.MultilayerMirror:
+    r"""
+    A multilayer stack fitted to the witness sample measurements given by
+    :func:`multilayer_witness_measured`.
+
+    This fit has five free parameters: the ratio of the thicknesses of
+    :math:`\text{Mg}`, :math:`\text{Al}`, and the :math:`\text{SiC}` to their
+    as-designed thickness, the roughness of the substrate, and a single
+    roughness parameter for all the layers in the multilayer stack.
+
+    Examples
+    --------
+
+    Plot the fitted vs. measured reflectivity of the grating witness samples.
+
+    .. jupyter-execute::
+
+        import numpy as np
+        import matplotlib.pyplot as plt
+        import astropy.units as u
+        import astropy.visualization
+        import named_arrays as na
+        import optika
+        from esis.flights.f1.optics import gratings
+
+        # Load the measured reflectivity of the witness samples
+        multilayer_measured = gratings.materials.multilayer_witness_measured()
+        measurement = multilayer_measured.efficiency_measured
+
+        # Isolate the angle of incidence of the measurement
+        angle_incidence = measurement.inputs.direction
+
+        # Fit a multilayer stack to the measured reflectivity
+        multilayer = gratings.materials.multilayer_witness_fit()
+
+        # Define the rays incident on the multilayer stack that will be used to
+        # compute the reflectivity
+        rays = optika.rays.RayVectorArray(
+            wavelength=na.geomspace(250, 950, axis="wavelength", num=1001) * u.AA,
+            direction=na.Cartesian3dVectorArray(
+                x=np.sin(angle_incidence),
+                y=0,
+                z=np.cos(angle_incidence),
+            ),
+        )
+
+        # Compute the reflectivity of the fitted multilayer stack
+        reflectivity_fit = multilayer.efficiency(
+            rays=rays,
+            normal=na.Cartesian3dVectorArray(0, 0, -1),
+        )
+
+        # Plot the fitted vs. measured reflectivity
+        fig, ax = plt.subplots(constrained_layout=True)
+        na.plt.scatter(
+            measurement.inputs.wavelength,
+            measurement.outputs,
+            ax=ax,
+            label="measurement"
+        );
+        na.plt.plot(
+            rays.wavelength,
+            reflectivity_fit,
+            ax=ax,
+            axis="wavelength",
+            label="fit",
+            color="tab:orange",
+        );
+        ax.set_xlabel(f"wavelength ({rays.wavelength.unit:latex_inline})")
+        ax.set_ylabel("reflectivity")
+        ax.legend();
+
+        # Print the fitted multilayer stack
+        multilayer
+
+    Plot a diagram of the fitted multilayer stack
+
+    .. jupyter-execute::
+
+        with astropy.visualization.quantity_support():
+            fig, ax = plt.subplots()
+            multilayer.plot_layers(
+                ax=ax,
+                thickness_substrate=20 * u.nm,
+            )
+            ax.set_axis_off()
+
+    """
+
+    design = multilayer_design()
+
+    measurement = multilayer_witness_measured()
+    unit = u.nm
+
+    reflectivity = measurement.efficiency_measured.outputs
+    angle_incidence = measurement.efficiency_measured.inputs.direction
+
+    rays = optika.rays.RayVectorArray(
+        wavelength=measurement.efficiency_measured.inputs.wavelength,
+        direction=na.Cartesian3dVectorArray(
+            x=np.sin(angle_incidence),
+            y=0,
+            z=np.cos(angle_incidence),
+        ),
+    )
+
+    normal = na.Cartesian3dVectorArray(0, 0, -1)
+
+    def _multilayer(
+        ratio_SiC: float,
+        ratio_Al: float,
+        ratio_Mg: float,
+        width_layers: float,
+        width_substrate: float,
+    ):
+
+        width_layers = width_layers * unit
+        width_substrate = width_substrate * unit
+
+        result = multilayer_design()
+
+        result.layers[1].thickness = ratio_SiC * design.layers[1].thickness
+        result.layers[1].interface.width = width_layers
+
+        result.layers[2].thickness = ratio_Al * design.layers[2].thickness
+        result.layers[2].interface.width = width_layers
+
+        result.layers[3].thickness = ratio_Mg * design.layers[3].thickness
+        result.layers[3].interface.width = width_layers
+
+        thickness_Al = ratio_Al * design.layers[~1].layers[0].thickness
+        result.layers[~1].layers[0].thickness = thickness_Al
+        result.layers[~1].layers[0].interface.width = width_layers
+
+        thickness_SiC = ratio_SiC * design.layers[~1].layers[~1].thickness
+        result.layers[~1].layers[~1].thickness = thickness_SiC
+        result.layers[~1].layers[~1].interface.width = width_layers
+
+        thickness_Mg = ratio_Mg * design.layers[~1].layers[~0].thickness
+        result.layers[~1].layers[~0].thickness = thickness_Mg
+        result.layers[~1].layers[~0].interface.width = width_layers
+
+        result.layers[~0].thickness = ratio_Al * design.layers[~0].thickness
+        result.layers[~0].interface.width = width_layers
+
+        result.layers.pop(0)
+
+        result.substrate.interface.width = width_substrate
+        result.substrate.chemical = "Si"
+
+        return result
+
+    def _func(x: np.ndarray):
+
+        multilayer = _multilayer(*x)
+
+        reflectivity_fit = multilayer.efficiency(
+            rays=rays,
+            normal=normal,
+        )
+
+        result = np.sqrt(np.mean(np.square(reflectivity_fit - reflectivity)))
+
+        return result.ndarray.value
+
+    x0 = u.Quantity(
+        [
+            1,
+            1,
+            1,
+            1,
+            1,
+        ]
+    )
+
+    bounds = [(0, None)] * len(x0)
+
+    fit = scipy.optimize.minimize(
+        fun=_func,
+        x0=x0,
+        bounds=bounds,
+    )
+
+    return _multilayer(*fit.x)

esis/flights/f1/optics/gratings/materials/_materials_test.py

@@ -10,3 +10,8 @@ def test_multilayer_design():
 def test_multilayer_witness_measured():
     r = esis.flights.f1.optics.gratings.materials.multilayer_witness_measured()
     assert isinstance(r, optika.materials.MeasuredMirror)
+
+
+def test_multilayer_witness_fit():
+    r = esis.flights.f1.optics.gratings.materials.multilayer_witness_fit()
+    assert isinstance(r, optika.materials.MultilayerMirror)