# -*- coding: utf-8 -*-
""" Geometric beam calculator for SCS.
Copyright (2021) SCS Team.
"""
import numpy as np
import matplotlib.pyplot as plt
from sympy.physics.optics import GeometricRay, FreeSpace, ThinLens
from sympy.solvers import solve
from sympy import symbols, simplify
from sympy.utilities.lambdify import lambdify
class GeoBeams:
"""The Beams object to compute beam propagation from source point to sample.
Inputs
------
elems : dict
dictionnaries of numerical values for the different parameters.
"""
def __init__(self):
# all distances in meters
self.elems = {
'dIHF': -56, # Intermediate Horizontal Focus position
'dEX': -30, # Exit Slit position
'dHFM': -3.35, # Horizontal Focusing Mirror position
'fHFM': 5.74, # HFM focal length
'dVFM': -2, # Vertical Focusing Mirror position
'fVFM': 5.05, # VFM focal length
'dBOZ': -0.23, # Beam splitting Off axis Zone plate position
'fBOZ_x': 0.25, # BOZ horizontal focal length
'fBOZ_y': 0.25, # BOZ vertical focal distance
'theta_grating': 0, # grating deflection angle in rad
'dSAMZ': 0.0, # Sample position
'ddetZ': 2.0, # Detector position
'WH': 0.8e-3, # BOZ horizontal width
'WV': 0.8e-3, # BOZ vertical width
'offaxis': -0.55e-3, # BOZ center to optical axis
'EXw': 100e-6, # Exit Slit width
'IHFw': 200e-6, # IHF source width
'x1': 0, # horizontal beam height at source
'theta1_x': 0, # horizontal beam angle at source
'x2': 0, # horizontal beam height at BOZ
'y1': 0, # vertical beam height at source
'theta1_y': 0, # vertical beam angle at source
'y2': 0 # vertical beam height at BOZ
}
self.compute_eqs()
def compute_eqs(self):
"""Compute the beam propagation equation between source and zone plate.
eq_x, eq_y: tuple, the horizontal x and vertical y equations to
compute the initial angle from the source to pass by the zone plate
position x2, given the source position at x1.
"""
dEX, dIHF, dVFM, dHFM, dBOZ = symbols('dEX, dIHF, dVFM, dHFM, dBOZ')
fVFM, fHFM, fBOZ_x, fBOZ_y = symbols('fVFM, fHFM, fBOZ_x, fBOZ_y')
x1, theta1_x, x2 = symbols('x1, theta1_x, x2')
y1, theta1_y, y2 = symbols('y1, theta1_y, y2')
# horizontal and vertical beam propagation from the
# source to the zone plate
self.HFM = simplify(
FreeSpace(-dHFM+dBOZ)*ThinLens(fHFM) *
FreeSpace(-dIHF+dHFM)*GeometricRay(x1, theta1_x))
self.VFM = simplify(
FreeSpace(-dVFM+dBOZ)*ThinLens(fVFM) *
FreeSpace(-dEX+dVFM)*GeometricRay(y1, theta1_y))
self.HFM_f = lambdify(self.elems.keys(), self.HFM, ['numpy'])
self.VFM_f = lambdify(self.elems.keys(), self.VFM, ['numpy'])
# beams for the zone plate first order (0th order is unfocused)
self.HBOZ = simplify(ThinLens(fBOZ_x)*self.HFM)
self.VBOZ = simplify(ThinLens(fBOZ_y)*self.VFM)
self.HBOZ_f = lambdify(self.elems.keys(), self.HBOZ, ['numpy'])
self.VBOZ_f = lambdify(self.elems.keys(), self.VBOZ, ['numpy'])
# solve which beam angle at the source theta1 ends on the
# zone plate at x2, given x1
self.theta1_x = solve(self.HFM[0] - x2, theta1_x)[0]
self.theta1_y = solve(self.VFM[0] - y2, theta1_y)[0]
self.theta1_x_f = lambdify(self.elems.keys(), self.theta1_x, ['numpy'])
self.theta1_y_f = lambdify(self.elems.keys(), self.theta1_y, ['numpy'])
def path(self):
"""Compute the horizontal and vertical beam propagation.
"""
x = []
z_x = []
# source
x1 = self.elems['x1']
angle1 = self.elems['theta1_x']
x += [x1]
z_x += [self.elems['dIHF']]
ray = GeometricRay(x1, angle1)
# KBS
ray = (ThinLens(self.elems['fHFM']) *
FreeSpace(-self.elems['dIHF'] + self.elems['dHFM'])*ray)
z_x += [self.elems['dHFM']]
x += [ray[0].evalf()]
# BOZ
ray = (ThinLens(self.elems['fBOZ_x']) *
FreeSpace(-self.elems['dHFM'] + self.elems['dBOZ'])*ray)
z_x += [self.elems['dBOZ']]
x += [ray[0].evalf()]
# detector
ray = FreeSpace(-self.elems['dBOZ']+self.elems['ddetZ'])*ray
z_x += [self.elems['ddetZ']]
x += [ray[0].evalf()]
y = []
z_y = []
# source
y1 = self.elems['y1']
angle1 = self.elems['theta1_y']
y += [y1]
z_y += [self.elems['dEX']]
ray = GeometricRay(y1, angle1)
# KBS
ray = (ThinLens(self.elems['fVFM']) *
FreeSpace(-self.elems['dEX']+self.elems['dVFM'])*ray)
z_y += [self.elems['dVFM']]
y += [ray[0].evalf()]
# BOZ
ray = (ThinLens(self.elems['fBOZ_y']) *
FreeSpace(-self.elems['dVFM']+self.elems['dBOZ'])*ray)
z_y += [self.elems['dBOZ']]
y += [ray[0].evalf()]
# detector
ray = FreeSpace(-self.elems['dBOZ']+self.elems['ddetZ'])*ray
z_y += [self.elems['ddetZ']]
y += [ray[0].evalf()]
return z_x, x, z_y, y
def plot(self):
fig, ax = plt.subplots(2, 1, figsize=(6, 4), sharex=True)
xmin, xmax = [np.Inf, -np.Inf]
ymin, ymax = [np.Inf, -np.Inf]
xs = np.empty((4,4))
ys = np.empty((4,4))
for k, (x2, y2) in enumerate(zip(
[self.elems['WH']/2, -self.elems['WH']/2],
np.array([self.elems['WV']/2, -self.elems['WV']/2])
+ self.elems['offaxis'])):
for kk, (x1, y1) in enumerate(zip(
[self.elems['IHFw']/2, -self.elems['IHFw']/2],
[self.elems['EXw']/2, -self.elems['EXw']/2])):
c = f'C{3*k+kk}'
self.elems['x1'] = x1
self.elems['x2'] = x2
self.elems['y1'] = y1
self.elems['y2'] = y2
self.elems['theta1_x'] = self.theta1_x_f(
*list(self.elems.values()))
self.elems['theta1_y'] = self.theta1_y_f(
*list(self.elems.values()))
z_x, x, z_y, y = self.path()
xs[2*k + kk, :] = 1e3*np.array(x)
ys[2*k + kk, :] = 1e3*np.array(y)
if xmin > x[-1]:
xmin = x[-1]
if xmax < x[-1]:
xmax = x[-1]
if ymin > y[-1]:
ymin = y[-1]
if ymax < y[-1]:
ymax = y[-1]
ax0in = ax[0].inset_axes([0.1, 0.65, 0.2, 0.25])
ax1in = ax[1].inset_axes([0.1, 0.65, 0.2, 0.25])
# fill_between to show bea255s
for k in range(4):
for kk in range(k+1, 4):
ax[0].fill_between(z_x, xs[k,:], xs[kk, :], color='C0')
ax0in.fill_between(z_x, xs[k,:], xs[kk, :], color='C0')
ax[1].fill_between(z_y, ys[k,:], ys[kk, :], color='C0')
ax1in.fill_between(z_y, ys[k,:], ys[kk, :], color='C0')
ax0in.set_xlim([-0.02, 0.1])
ax0in.set_ylim([-0.10, 0.10])
ax1in.set_xlim([-0.02, 0.1])
ax1in.set_ylim([-0.05, 0.15])
ax[0].set_ylabel('x (mm)')
ax[1].set_ylabel('y (mm)')
ax[1].set_xlabel('z (m)')
ax[0].axvline(self.elems['dSAMZ'], ls='--', c='k', alpha=0.5)
ax0in.axvline(self.elems['dSAMZ'], ls='--', c='k', alpha=0.5)
ax[1].axvline(self.elems['dSAMZ'], ls='--', c='k', alpha=0.5)
ax1in.axvline(self.elems['dSAMZ'], ls='--', c='k', alpha=0.5)
return fig
def plane_image(self, p0, n, ZPorder, Gorder=0):
"""Compute the BOZ image on a plane.
Inputs
------
p0: [x, y, z] point on the imaging plane
n: [X, Y, Z] plane normal vector
ZPorder: int, order of the zone plate, i.e. 0 or 1
Gorder: int, grating order, in [-1, 0, 1]
Returns
-------
list of 4 corners [x,y]
"""
xmin, xmax = [np.Inf, -np.Inf]
ymin, ymax = [np.Inf, -np.Inf]
for k, (x2, y2) in enumerate(zip(
[self.elems['WH']/2, -self.elems['WH']/2],
np.array([self.elems['WV']/2, -self.elems['WV']/2])
+ self.elems['offaxis'])):
for kk, (x1, y1) in enumerate(zip(
[self.elems['IHFw']/2, -self.elems['IHFw']/2],
[self.elems['EXw']/2, -self.elems['EXw']/2])):
self.elems['x1'] = x1
self.elems['y1'] = y1
self.elems['x2'] = x2
self.elems['y2'] = y2
self.elems['theta1_x'] = self.theta1_x_f(
*list(self.elems.values()))
self.elems['theta1_y'] = self.theta1_y_f(
*list(self.elems.values()))
# evaluate HFM/VFM or HBOZ/VBOZ
if ZPorder == 1:
bx = self.HBOZ_f(*list(self.elems.values()))
by = self.VBOZ_f(*list(self.elems.values()))
elif ZPorder == 0:
bx = self.HFM_f(*list(self.elems.values()))
by = self.VFM_f(*list(self.elems.values()))
else:
raise ValueError(
'ZPorder other than 0 or 1 not implemented')
# intersection beam with plane
l1 = np.array([x2, y2, self.elems['dBOZ']])
l12 = np.array([bx[1][0] + Gorder*self.elems['theta_grating'],
by[1][0], 1])
p = self.LinePlaneIntersection(p0, n, l1, l12=l12)
if xmin > p[0]:
xmin = p[0]
if xmax < p[0]:
xmax = p[0]
if ymin > p[1]:
ymin = p[1]
if ymax < p[1]:
ymax = p[1]
return np.array([[xmax, ymax], [xmax, ymin],
[xmin, ymin], [xmin, ymax]])
def LinePlaneIntersection(self, p0, n, l1, *, l2=None, l12=None):
""" Calculate the intersection point in space between a line (beam)
passing through 2 points l1 and l2 and a (sample) plane passing by
p0 with normal n
l1: [x,y,z] point on line
l2: [x,y,z] point on line
p0: [x,y,z] point on plane
n: plane normal vector
"""
assert not((l2 is None) and (l12 is None)), (
"Either l2 or l12 must be defined")
# plane parametrized as (p - p0).n = 0
# line parametrized as p = l1 + l12*d with d Real
if l12 is None:
l12 = l2 - l1
if np.dot(l12, n) == 0:
return [0, 0, 0] # line is either in the plane or outside of it
else:
d = np.dot((p0 - l1), n)/np.dot(l12, n)
return l1 + l12*d