def _efieldn_linearized(x, y, sig_x, sig_y):
'''
Returns linearized field
'''
a = pm.sqrt(2) * sig_x
b = pm.sqrt(2) * sig_y
amplitude = 1. / (2. * np.pi * epsilon_0 * (a + b))
return x / a * amplitude, y / b * amplitude
def _efieldn_wb(x, y, sigma_x, sigma_y):
a = sigma_x * pm.sqrt(3)
b = sigma_y * pm.sqrt(3)
zs = x - 1j * y
if x**2 / (a)**2 + y**2 / (b)**2 <= 1:
chi = x / a + 1j * y / b
omegs = b * x / a - 1j * a * y / b
efield = 8.0*chi/(a+b) * \
(1.0-(2.0*zs+omegs)*chi/(3.0*(a+b)))
else:
zs = pm.abs(x) + 1j * pm.abs(y)
sqrt_diff = pm.sqrt(zs**2 - a**2 + b**2)
first_term = 2.0 * zs / (zs + sqrt_diff)
efield = 2.0 / zs * first_term * (zs + 2.0 * sqrt_diff) / (3.0 * zs)
efield = efield.real * pm.sign(x) - 1.0j * efield.imag * pm.sign(y)
denom = 4. * np.pi * epsilon_0
return efield.real / denom, efield.imag / denom
def _efieldn_kv_b(x, y, sigma_x, sigma_y):
'''
Field of a KV distrubition calculated as in here: https://cds.cern.ch/record/258225/files/P00020427.pdf
'''
a = sigma_x * pm.sqrt(2)
b = sigma_y * pm.sqrt(2)
if x == 0 and y == 0:
return 0, 0
elif (x / a)**2 + (y / b)**2 <= 1:
efield_x = 4.0 / (a + b) * x / a
efield_y = 4.0 / (a + b) * y / b
else:
zbar = x - 1j * y
efield = 4.0 / (zbar + pm.sqrt(zbar * zbar - a * a + b * b))
efield_x = efield.real
efield_y = -efield.imag
denom = 4 * np.pi * epsilon_0
return efield_x / denom, efield_y / denom
def _efieldn_kv_a(x, y, sigma_x, sigma_y):
'''
Field of a KV distrubition calculated as in here: https://cds.cern.ch/record/258225/files/P00020427.pdf
'''
a = sigma_x * pm.sqrt(2)
b = sigma_y * pm.sqrt(2)
if (x / a)**2 + (y / b)**2 <= 1:
efield_x = 4.0 / (a + b) * x / a
efield_y = 4.0 / (a + b) * y / b
else:
uxy = (x)**2 - (y)**2 - (a)**2 + (b)**2
vxy = uxy**2 + (2.0 * x * y)**2
efield_x = 4.0 / (a**2 - b**2) * (
x - pm.sign(x) / pm.sqrt(2.0) * pm.sqrt(uxy + pm.sqrt(vxy)))
uxy = (y)**2 - (x)**2 - (b)**2 + (a)**2
efield_y = 4.0 / (b**2 - a**2) * (
y - pm.sign(y) / pm.sqrt(2.0) * pm.sqrt(uxy + pm.sqrt(vxy)))
denom = 4 * np.pi * epsilon_0
return efield_x / denom, efield_y / denom
def _sig_sqrt(sig_x, sig_y):
return pm.sqrt(2 * (sig_x**2 - sig_y**2))