def periodic_envelope_propagation(field,omega0,s=1,z=None,omega=None,t=None):
import numpy as np
import expresso.pycas as pc
from .. import units
if z is None:
z = pc.Symbol('z')
if t is None:
t = pc.Symbol('t')
if omega is None:
omega = pc.Symbol('omega')
zindex = field.get_axis_index(z)
omegaindex = field.get_axis_index(omega)
omegamin,omegamax = field.bounds[omegaindex]
zmin, zmax = field.bounds[zindex]
sz = zmax - zmin
ukmin = float(field.evaluate( (omegamin-omega0)/units.c*sz ))
ukmax = float(field.evaluate( (omegamax-omega0)/units.c*sz ))
uzmin = 0
uzmax = 1
nz,ik = np.meshgrid(np.linspace(uzmin,uzmax,field.shape[zindex]),np.linspace(1j*ukmin,1j*ukmax,field.shape[omegaindex]))
factor = np.exp(-ik*nz/float(s))
del nz,ik
transform = field * factor
del factor
tdfield = fourier_transform(transform, omega, t, inverse=True)
del transform
tdfield.bounds[omegaindex] = [(b - tdfield.bounds[omegaindex][0]).evaluate() for b in tdfield.bounds[omegaindex]]
return tdfield
def __init__(self, settings, thread_count=1):
import expresso.pycas as pc
super(Fresnel2D, self).__init__(settings)
self._thread_count = thread_count
self._set_initial_field(settings)
pe = settings.partial_differential_equation
x, y, z = settings.partial_differential_equation.coordinates
R, = self._get_evaluators([pc.exp(pe.F * z.step)],
settings,
return_type=pc.Types.Complex)
D = pc.numpyfy(
self._evaluate(pc.exp(-pe.A * z.step * (pc.Symbol('kx')**2)),
settings))
if self._F_is_constant_in_z:
self.__R_step = R(*self._get_indices())
else:
self.__R = R
import numpy as np
fx = 2 * np.pi / (self._nx * self._get_as(x.step, float, settings))
kx = fx * (self._nx / 2. - np.abs(np.arange(self._nx) - self._nx / 2.))
self.__D_step = D(kx=kx, **self._get_indices_dict())
def __init__(self, sb, name):
self.name = name
self.symbol = getattr(sb, name)
self.min = getattr(sb, name + 'min')
self.max = getattr(sb, name + 'max')
self.step = getattr(sb, 'd' + name)
self.steps = getattr(sb, 'N' + name)
self.size = getattr(sb, 's' + name)
self.index = pc.Symbol('%s_i' % name, type=pc.Types.Natural)
def add_partial_differential_equation_category(settings, coordinates=None):
import expresso.pycas as pc
sb = settings.simulation_box
pde = settings.create_category(
'partial_differential_equation',
short_name='PDE',
info="parameters of the partial differential equation")
arg_attrs = [sb.coordinate_dict[x] for x in coordinates
] if coordinates is not None else sb.coordinates
pde.add_attribute('coordinates', arg_attrs)
x, y, z = [a.symbol for a in arg_attrs]
dx, dy, dz = [s.step for s in arg_attrs]
args = (x, y, z)
pde.create_function('A', args)
pde.create_function('C', args, pde.A)
pde.create_function('F', args)
pde.create_function('ra',
args,
pde.A * dz / dx**2,
info="finite difference paramter")
pde.create_function('rc',
args,
pde.C * dz / dy**2,
info="finite difference paramter")
pde.create_function('rf',
args,
pde.F * dz / 2,
info="finite difference paramter")
pde.create_function('u', args, info='solution to the PDE')
pde.lock('u')
pde.add_attribute(
'equation',
pc.equal(
pc.derivative(pde.u, z),
pde.A * pc.derivative(pc.derivative(pde.u, x), x) +
pde.C * pc.derivative(pc.derivative(pde.u, y), y) + pde.F * pde.u))
pde.create_key('u0',
pc.Function('u_0_PDE')(*args),
info="field initial condition")
pde.create_function('u_boundary', args, info="field boundary condition")
pde.create_key(arg_attrs[1].name + '0',
pc.Symbol(y.name + '_0_PDE'),
0,
info='Value to which the %s is set for 1D solvers' % y.name)
pde.lock()
return pde
def plot(arg, *args, **kwargs):
"""
Simple plot function for 1D and 2D coordinate arrays. If the data is complex, the absolute square value of the data will be plottted.
Parameters
-----------
arg: coordinate array
the input data
**kwargs: additional parameters to be passed to the plot functions
Returns
--------
plot: output of ax.plot for 1D and ax.imshow for 2D arrays
"""
import expresso.pycas
import numpy as np
from coordinate_ndarray import CoordinateNDArray
if isinstance(arg, expresso.pycas.Expression):
from .settings import Settings
if len(args) > 0 and isinstance(args[0], Settings):
settings = args[0]
args = list(args)
del args[0]
else:
settings = kwargs.get('settings')
if not settings:
raise ValueError(
'cannot plot expression: no settings provided')
del kwargs['settings']
arg = expression_to_array(arg, settings)
if isinstance(arg, (float, complex)):
print arg
return
elif not isinstance(arg, CoordinateNDArray):
if isinstance(arg, np.ndarray):
pc = expresso.pycas
arg = CoordinateNDArray(
arg, [(pc.S(0), pc.S(n)) for n in arg.shape],
[pc.Symbol('x_%i' % i) for i in range(len(arg.shape))])
else:
raise ValueError(
'cannot plot non CoordinateNDArray object. For plotting regular arrays please use the matplotlib.pyplot module.'
)
if not np.can_cast(arg.data.dtype, np.float128):
if np.all(arg.data.imag == 0): arg = arg.real
else: arg = abs(arg)**2
if len(arg.axis) == 1: return line_plot(arg, *args, **kwargs)
elif len(arg.axis) == 2: return image_plot(arg, *args, **kwargs)
else: raise ValueError("input array must be one or two dimensional")
def create_unit(name):
u = pc.Symbol("SI base unit " + name,
type=pc.Types.Real,
positive=True,
latex=r'\mathrm{%s}' % name,
repr=name)
globals()[name] = u
base_units.add(u)
add_metric_prefixes(name)
def __init__(self, settings, thread_count=None):
super(Fresnel3D, self).__init__(settings)
if thread_count == None:
import multiprocessing
thread_count = multiprocessing.cpu_count()
self._thread_count = thread_count
pe = settings.partial_differential_equation
x, y, z = settings.partial_differential_equation.coordinates
import expresso.pycas as pc
R, = self._get_evaluators([pc.exp(pe.F * z.step)],
settings,
return_type=pc.Types.Complex,
parallel=not self._F_is_constant_in_z)
if self._F_is_constant_in_z:
self.__R_step = R(*self._get_indices())
else:
self.__R = R
D = pc.numpyfy(
self._evaluate(
pc.exp(
-z.step *
(pe.A * pc.Symbol('kx')**2 + pe.C * pc.Symbol('ky')**2)),
settings))
import numpy as np
fx = 2 * np.pi / (self._nx * self._get_as(x.step, float, settings))
fy = 2 * np.pi / (self._ny * self._get_as(y.step, float, settings))
ky, kx = np.meshgrid(
fy * (self._ny / 2. - np.abs(np.arange(self._ny) - self._ny / 2.)),
fx * (self._nx / 2. - np.abs(np.arange(self._nx) - self._nx / 2.)))
self.__C_is_zero = settings.get_numeric(pe.C) == pc.S(0)
self.__D_step = D(kx=kx, ky=ky, **self._get_indices_dict())
self._set_initial_field(settings)
def __init__(self, settings, thread_count=1):
import expresso.pycas as pc
import numpy as np
pe = settings.partial_differential_equation
x, y, z = settings.partial_differential_equation.coordinates
xi = getattr(settings.simulation_box, x.name + 'i')
self.__ximin = float(settings.get_unitless(xi.subs(x.symbol, 0)))
self.__ximax = float(settings.get_unitless(xi.subs(x.symbol, x.max)))
self.__sx = self.__ximax - self.__ximin
super(FresnelCS, self).__init__(settings)
self._thread_count = thread_count
self._set_initial_field(settings)
R, = self._get_evaluators([pc.exp(pe.F * z.step)],
settings,
return_type=pc.Types.Complex)
D = pc.numpyfy(
self._evaluate(pc.exp(-pe.A * z.step * (pc.Symbol('kx')**2)),
settings))
if self._F_is_constant_in_z:
self.__R_step = R(*self._get_indices())
else:
self.__R = R
kx = hankel_freq(self._nx) * ((self._nx - 1) * 1. / self.__sx)
self.__D_step = D(kx=kx, **self._get_indices_dict())
self.__hankel_resample_matrix = hankel_resample_matrix(
self._nx,
(np.arange(self._nx) - self.__ximin) * (self._nx * 1. / self.__sx),
cache_key=(self._nx, self.__ximin, self.__sx),
xmax=self._nx)
def add_coordinate(settings, category, name):
import expresso.pycas as pc
x = category.create_key(name,
pc.Symbol(name, type=pc.Types.Real),
info='coordinate')
xmin = category.create_key('%smin' % name,
pc.Symbol('%s_min' % name),
info='minimum value')
xmax = category.create_key('%smax' % name,
pc.Symbol('%s_max' % name),
info='maximum value')
xi = pc.Symbol('%s_i' % name, type=pc.Types.Natural)
xif = category.create_key('%si' % name,
pc.Function('%s_i' % name)(x),
info='numerical index')
Nx = category.create_key('N%s' % name,
pc.Symbol('N_%s' % name, type=pc.Types.Natural),
info='numerical steps')
sx = category.create_key('s%s' % name,
pc.Symbol('s_%s' % name),
info='total size')
dx = category.create_key('d%s' % name,
pc.Symbol('Delta %s' % name),
info='step size')
setattr(category, name, xmin + xi * dx)
setattr(category, '%si' % name, (x - xmin) / dx)
setattr(category, 's%s' % name, xmax - xmin)
setattr(category, 'd%s' % name, sx / (Nx - 1))
category.lock(name, 'defined by %si' % name)
category.lock('s%s' % name, 'defined by %smin and %smax' % (name, name))
category.lock('d%s' % name, 'defined by s%s and N%s' % (name, name))
settings.unitless.add_key('%s_coordinate' % name, x)
def create_2D_frequency_settings_from(settings):
import expresso.pycas as pc
from .. import units
from ..plot import expression_to_array
from ..coordinate_ndarray import CoordinateNDArray
import numpy as np
settings.initialize()
settings = settings.copy()
sb = settings.simulation_box
we = settings.wave_equation
omega0 = settings.get_numeric(we.omega)
we.unlock('omega')
we.omega = None
freq_settings = create_paraxial_settings()
pde = freq_settings.partial_differential_equation
freq_settings.simulation_box.unlock()
freq_settings.simulation_box.remove_name('y')
omega = freq_settings.simulation_box.create_key('y', pc.Symbol('omega'))
fsb = freq_settings.simulation_box
freq_settings.simulation_box.y = fsb.ymin + fsb.yi * fsb.dy
n = settings.get_numeric(we.n.subs(sb.y, 0)).subs(omega,
abs(omega - omega0))
p = freq_settings.create_category('paramters', short_name="p")
p.create_function('n', (sb.x, sb.y, sb.z, omega), n)
p.create_symbol('k', (omega - omega0) / units.c)
p.create_symbol('k_0', omega0 / units.c)
pde.A = 1j / (2 * p.k_0)
pde.C = 0
pde.F = 1j / (2 * p.k_0) * (p.n**2 * p.k**2 - p.k_0**2)
xmin = settings.get_numeric(sb.xmin)
xmax = settings.get_numeric(sb.xmax)
freq_settings.simulation_box.xmin = xmin
freq_settings.simulation_box.xmax = xmax
freq_settings.simulation_box.Nx = settings.get_as(sb.Nx, int)
omegamin = settings.get_numeric(-2 * pc.pi * sb.Nt / sb.st)
omegamax = settings.get_numeric(2 * pc.pi * sb.Nt / sb.st)
freq_settings.simulation_box.ymin = omegamin
freq_settings.simulation_box.ymax = omegamax
freq_settings.simulation_box.Ny = settings.get_as(sb.Nt, int)
freq_settings.simulation_box.zmin = settings.get_numeric(sb.zmin)
freq_settings.simulation_box.zmax = settings.get_numeric(sb.zmax)
freq_settings.simulation_box.Nz = settings.get_as(sb.Nz, int)
freq_settings.unitless.create_key(
's', units.s,
settings.get_as(2 / (omegamax - omegamin) / units.s, float))
u0 = expression_to_array(settings.wave_equation.u0.subs([(sb.y, 0),
(sb.z, sb.zmin)]),
settings,
axes=(sb.x, sb.t))
u0hat = CoordinateNDArray(
np.fft.fftshift(np.fft.fft(u0.data, axis=1), axes=(1, )),
[(xmin, xmax), (omegamin, omegamax)], [sb.x, omega])
set_initial(freq_settings, u0hat)
freq_settings.partial_differential_equation.u_boundary = 0
return freq_settings
def create_symbol(name, value=None, info=None, **kwargs):
prefix = '_'.join(cat_path)
return cat.create_key(
name, pc.Symbol("%s_%s" % (name, prefix), **kwargs), value,
info)