XUV Polarimetry

XUV Polarimetry#

This example extends the previous one on Dynamical Magnetic X-ray Scattering by showcasing the ability to simulate with variable incoming and outcoing polarization in the XUV and soft-X-ray spectral range.

Added in version 2.4.0: Enable variable/elliptical polarization in XrayDynMag

With that it is possible to extend the simulations beyond common geometries of XMCD and transversal MOKE (T-MOKE) to also probe in longitudinal and polar MOKE geometry, L-MOKE and P-MOKE, respectively.

Since polarizing optics are hardly available/feasible in the XUV and soft-X-ray spectral range, L- and P-MOKE rely on the sample reflection itself close to the Brewster angle as polarization analyzer. Or in other words, the exact Brewster angle relies on the magnetization and polarization-dependent refractive index.

Setup#

Do all necessary imports and settings.

import udkm1Dsim as ud
u = ud.u  # import the pint unit registry from udkm1Dsim
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
u.setup_matplotlib()  # use matplotlib with pint units

Structure#

Refer to the structure example for more details.

In the XUV spectral range, the atomic form factors from the Henke tables are more suitable and can be selected by the keyword atomic_form_factor_source in the Atom constructor.

Ta = ud.Atom('Ta', atomic_form_factor_source='henke')
Ru = ud.Atom('Ru', atomic_form_factor_source='henke')

Si = ud.Atom('Si', atomic_form_factor_source='henke')
O = ud.Atom('O',  atomic_form_factor_source='henke')

SiO2  = ud.AtomMixed('SiO2')
SiO2.add_atom(Si, 0.33)
SiO2.add_atom(O, 0.67)

For the magnetic Fe atom experimental data for the charge and magnetic form factors is used.

Initialize dynamical magnetic X-ray simulations#

Please refer to the Dynamical Magnetic X-ray Scattering example for more details.

dyn_mag = ud.XrayDynMag(S, True)
dyn_mag.disp_messages = True
dyn_mag.save_data = False

incoming polarizations set to: sigma
analyzer polarizations set to: unpolarized

dyn_mag.energy = 54*u.eV
dyn_mag.theta = np.r_[45]*u.deg

Polarization settings#

In all following simulations the analyzer polarization is kept as unpolarized, which is equivalent to

dyn_mag.set_outgoing_polarization(0)

To enable variable/elliptical incoming polarization, the pol_in_state must be set to 5 in the set_incoming_polarization method.

Added in version 2.4.0: Enable variable/elliptical polarization in XrayDynMag

In this case a single tuple or list of tuples of the azimuth angle \(\alpha\) and the ellipticity \(e\) of the polarization can be input.

Note

\[\begin{split} 0° \leq & \alpha \leq +180° \\ & \alpha = \begin{cases} 0° & s\text{-polarization}\\ 90° & p\text{-polarization} \end{cases}\\ & \\ -1 \leq & e \leq +1 \\ & e = \begin{cases} -1 &\text{circular left}\\ 0 &\text{linear}\\ +1 &\text{circular right} \end{cases}\end{split}\]

alphas = np.r_[0:180:0.1]
dyn_mag.set_incoming_polarization(5, [(alpha*u.deg, 0) for alpha in alphas])

incoming polarizations set to: elliptical

T-MOKE#

The current experimental configuration already corresponds to the well-known T-MOKE geometry.

Fe.mag_gamma = 180*u.deg  # s-pol
Fe.mag_phi = 90*u.deg  # in-plane

dyn_mag.theta = np.r_[45]*u.deg

R, R_phi, _, _ = dyn_mag.homogeneous_reflectivity()
asym_tmoke = ((R-R_phi)/(R+R_phi)).flatten()

plt.figure()
plt.plot(alphas, asym_tmoke)
plt.axvline(90, color='gray')
plt.axhline(0, color='gray')
plt.xlabel(r'$\alpha$ (deg)')
plt.ylabel('Asymmetry')
plt.xlim(0, 180)
plt.title('T-MOKE')
plt.show()

Calculating _homogeneous_reflectivity_ ...
Elapsed time for _homogeneous_reflectivity_: 0.036515 s

../_images/e20f831a51d92352655d2e98ad496c364f14f6ed68f04e12f76e1c8fa3dcc85a.png

L-MOKE#

Fe.mag_gamma = 90*u.deg  # p-pol
Fe.mag_phi = 90*u.deg  # in-plane

dyn_mag.theta = np.r_[45]*u.deg

R, R_phi, _, _ = dyn_mag.homogeneous_reflectivity()
asym_lmoke = ((R-R_phi)/(R+R_phi)).flatten()

plt.figure()
plt.plot(alphas, asym_lmoke)
plt.axvline(90, color='gray')
plt.axhline(0, color='gray')
plt.xlabel(r'$\alpha$ (deg)')
plt.ylabel('Asymmetry')
plt.xlim(0, 180)
plt.title('L-MOKE')
plt.show()

Calculating _homogeneous_reflectivity_ ...
Elapsed time for _homogeneous_reflectivity_: 0.048053 s

../_images/5f7ba24a776d72e076f0b2b0c553351a50dde0bf9f4bd7161f1c28a40260b529.png

P-MOKE#

Fe.mag_gamma = 0*u.deg  # any
Fe.mag_phi = 0*u.deg  # out-of-plane

dyn_mag.theta = np.r_[45]*u.deg

R, R_phi, _, _ = dyn_mag.homogeneous_reflectivity()
asym_pmoke = ((R-R_phi)/(R+R_phi)).flatten()

plt.figure()
plt.plot(alphas, asym_pmoke)
plt.axvline(90, color='gray')
plt.axhline(0, color='gray')
plt.xlabel(r'$\alpha$ (deg)')
plt.ylabel('Asymmetry')
plt.xlim(0, 180)
plt.title('P-MOKE')
plt.show()

Calculating _homogeneous_reflectivity_ ...
Elapsed time for _homogeneous_reflectivity_: 0.033113 s

../_images/a75b7f687d75717434272f18b4922fec11b51b12588576ff44e6342b1ffa3907.png

dyn_mag.theta = np.r_[15:75:0.1]*u.deg
R, R_phi, _, _ = dyn_mag.homogeneous_reflectivity()
asym_pmoke_map = ((R-R_phi)/((R+R_phi)))

# 3D plotting
from matplotlib.colors import LightSource

fig = plt.figure(figsize=[8,6])
ax = fig.add_subplot(111, projection="3d")

X, Y = np.meshgrid(alphas, dyn_mag.theta[0, :].magnitude)
Z = asym_pmoke_map[0, :, :]

ls = LightSource(315, 10)
rgb = ls.shade(Z, cmap=plt.colormaps['RdBu_r'], vert_exag=10, blend_mode='hsv')

ax.plot_surface(X, Y, Z,
                linewidth=.1, antialiased=True, shade=False,
                facecolors=rgb, rstride=15, cstride=15)

ax.contour(X, Y, Z, zdir='z', offset=-0.07, cmap='RdBu_r', levels=50, alpha=0.75)

ax.set_ylabel(r'$\theta$ (deg)')
ax.set_xlabel(r'$\alpha$ (deg)')
ax.set_zlabel('Asymmetry')
ax.set_title('P-MOKE')

ax.set_xlim(0, 180)
ax.set_ylim(15, 75)
ax.set_zlim(-0.07, 0.07)

ax.view_init(elev=20., azim=115, roll=0, )
ax.set_proj_type('persp', focal_length=0.5)  # FOV = 157.4 deg
ax.set_box_aspect((3.5, 4, 4))
for axis in (ax.xaxis, ax.yaxis, ax.zaxis):
    axis.pane.fill = False
    axis._axinfo["grid"]["color"] = (0.1, 0.1, 0.1, 0.1)  # white grid lines

Calculating _homogeneous_reflectivity_ ...
Elapsed time for _homogeneous_reflectivity_: 0.177133 s

../_images/7edaf83846d90bc502371d70fd04b27e4bb9ffb69751e0e4c139fd596cd015eb.png

Comparison#

plt.figure()
plt.plot(alphas, asym_lmoke, label='L-MOKE')
plt.plot(alphas, asym_tmoke, label='T-MOKE')
plt.plot(alphas, asym_pmoke, label='P-MOKE')
plt.legend()
plt.axvline(90, color='gray')
plt.axhline(0, color='gray')
plt.xlabel(r'$\alpha$ (deg)')
plt.ylabel('Asymmetry')
plt.xlim(0, 180)
plt.show()

../_images/58ab1a8b61cb23f3bf9866c6d14781d8d02021f1b5dc23f372893887ceebab60.png

	(a) L-MOKE	(b) T-MOKE	(c) P-MOKE
magnetization \(\gamma\)	90°	0°	-
magnetization \(\phi\)	90°	90°	0°
polarization \(\alpha\)	\(\neq\) 90°	\(=\) 90°	\(\neq\) 90°