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# -*- coding: utf-8 -*-
""" Geometric beam calculator for SCS.
Copyright (2021) SCS Team.
"""
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.colors import hsv_to_rgb
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
'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)
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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
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
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, conf1=None, conf2=None):
fig, ax = plt.subplots(2, 1, figsize=(6, 4), sharex=True)
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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])
axs = [ax, ax0in, ax1in]
if conf1 is not None:
self.elems['fBOZ_x'] = conf1['F_x']
self.elems['fBOZ_y'] = conf1['F_y']
self.elems['theta_grating'] = conf1['theta_grating']
# plot either a given conf or whatever the last calculation was
c_rr = hsv_to_rgb([10/360, 100/100, 100/100])
self._plot(axs, c_rr)
if conf2 is not None:
self.elems['fBOZ_x'] = conf2['F_x']
self.elems['fBOZ_y'] = conf2['F_y']
self.elems['theta_grating'] = conf2['theta_grating']
c_rb = hsv_to_rgb([220/360, 100/100, 100/100])
self._plot(axs, c_rb)
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)
ax0in.set_xlim([-0.1, 0.3])
ax0in.set_ylim([-0.3, 0.3])
ax1in.set_xlim([-0.1, 0.3])
ax1in.set_ylim([-0.3, 0.3])
ax[0].set_ylabel('x (mm)')
ax[1].set_ylabel('y (mm)')
ax[1].set_xlabel('z (m)')
return fig
def _plot(self, axs, color):
ax, ax0in, ax1in = axs
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])):
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]
for k in range(4):
for kk in range(k+1, 4):
ax[0].fill_between(z_x, xs[k, :], xs[kk, :],
color=color, alpha=0.2, linewidth=0)
ax0in.fill_between(z_x, xs[k, :], xs[kk, :],
color=color, alpha=0.2, linewidth=0)
ax[1].fill_between(z_y, ys[k, :], ys[kk, :],
color=color, alpha=0.2, linewidth=0)
ax1in.fill_between(z_y, ys[k, :], ys[kk, :],
color=color, alpha=0.2, linewidth=0)
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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