def sphere_extinction_analytical(freqs, r):
"""Analytical expressions for a PEC sphere's extinction for plane wave
with E = 1V/m
Parameters
----------
freqs : ndarray
Frequencies at which to calculate
r : real
Radius of sphere
"""
from scipy.special import sph_jnyn
N = 40
k0r = freqs*2*np.pi/c*r
#scs = np.zeros(len(k0r))
#scs_modal = np.zeros((len(k0r), N))
ecs_kerker = np.zeros(len(k0r))
for count, x in enumerate(k0r):
jn, jnp, yn, ynp = sph_jnyn(N, x)
h2n = jn - 1j*yn
h2np = jnp - 1j*ynp
a_n = ((x*jnp + jn)/(x*h2np + h2n))[1:]
b_n = (jn/h2n)[1:]
#scs[count] = 2*np.pi*sum((2*np.arange(1, N+1)+1)*(abs(a_n)**2 + abs(b_n)**2))/x**2 #
#scs_modal[count] = 2*np.pi*(2*np.arange(1, N+1)+1)*(abs(a_n)**2 + abs(b_n)**2)/x**2 #
ecs_kerker[count] = 2*np.pi*np.real(np.sum((2*np.arange(1, N+1)+1)*(a_n + b_n)))/x**2
return ecs_kerker*r**2/eta_0
def bessy(self, n, z):
"""Given order (n) and argument (z), computes mad bessel related junk.
Argument can be a scalar or vector numeric.
"""
#http://docs.scipy.org/doc/scipy/reference/special.html
# Returns all orders of N for a single argument (ie if order is 5,
# returns n=0,1,2,3,4 for arg z. Pain in the ass actually...
# sph_jnyn computes jn, derivjn, yn, derivyn in one pass
jn, djn, yn, dyn = special.sph_jnyn(n,z) #Compute sph j/y in one pass
i = 1j
# Choose only the nth's bessel function
jn=complex(jn[n])
yn=complex(yn[n])
djn=complex(djn[n])
dyn=complex(dyn[n])
hn=jn + i*yn #VERIFIED!!!
dhn=djn + i*dyn #Derivative (d/dp (j + iy)) = (dj + idy)
Psi=z*jn
dPsi=z*djn + jn #Psi = p(jn) DPsi= jn + p j'n Chain rollin
Zi=z*hn
dZi=z*dhn + hn
Xi=-z*yn #Xi=-i*(Zi + Psi) Verified WORKING either way!
dXi=-(z*dyn + yn) #dXi=-i*(dZi + dPsi)
return(Psi, dPsi, Zi, dZi, Xi, dXi)
def set_all_layers(self, lab):
self.data_layers = [0]
for bnd in lab.boundaries():
r, rd, rdd = bnd.shape.R(self.knots)
rdr = rd / r
r2rd2 = r**2 + rd**2
Rad = [0, {}, {}]
Radd = [0, {}, {}]
kis = [0, bnd.k1, bnd.k2]
for i in [1, 2]:
krs = kis[i] * r
JY = array([special.sph_jnyn(self.n, kr)
for kr in krs])[:, :, :]
Rad[i] = {
'j': JY[:, 0, :],
'h': JY[:, 0, :] + 1j * JY[:, 2, :]
}
Radd[i] = {
'j': JY[:, 1, :],
'h': JY[:, 1, :] + 1j * JY[:, 3, :]
}
self.data_layers.append({'ki': kis,\
'r': r,\
'rd': rd,\
'rdd': rdd,\
'rdr': rdr,\
'r2rd2': r2rd2,\
'Rad': Rad,\
'Radd': Radd})
def bessy(self, n, z):
"""Given order (n) and argument (z), computes mad bessel related junk.
Argument can be a scalar or vector numeric.
"""
#http://docs.scipy.org/doc/scipy/reference/special.html
# Returns all orders of N for a single argument (ie if order is 5,
# returns n=0,1,2,3,4 for arg z. Pain in the ass actually...
# sph_jnyn computes jn, derivjn, yn, derivyn in one pass
jn, djn, yn, dyn = special.sph_jnyn(n, z) #Compute sph j/y in one pass
i = 1j
# Choose only the nth's bessel function
jn = complex(jn[n])
yn = complex(yn[n])
djn = complex(djn[n])
dyn = complex(dyn[n])
hn = jn + i * yn #VERIFIED!!!
dhn = djn + i * dyn #Derivative (d/dp (j + iy)) = (dj + idy)
Psi = z * jn
dPsi = z * djn + jn #Psi = p(jn) DPsi= jn + p j'n Chain rollin
Zi = z * hn
dZi = z * dhn + hn
Xi = -z * yn #Xi=-i*(Zi + Psi) Verified WORKING either way!
dXi = -(z * dyn + yn) #dXi=-i*(dZi + dPsi)
return (Psi, dPsi, Zi, dZi, Xi, dXi)
def get_phase_shift_from_logder(self,logder,E,r):
k = np.sqrt(2*self.mass*E)
kr = k*r
jl, jlp, yl, ylp = sph_jnyn(0,kr)
iyl = 1j*yl
iylp = 1j*ylp
sl = -(jl-iyl)/(jl+iyl) * (logder-kr*(jlp-iylp)/(jl-iyl))/(logder-kr*(jlp+iylp)/(jl+iyl))
return np.angle(sl)/2.
def get_phase_shift_from_logder(self, logder, E, r):
k = np.sqrt(2 * self.mass * E)
kr = k * r
jl, jlp, yl, ylp = sph_jnyn(0, kr)
iyl = 1j * yl
iylp = 1j * ylp
sl = -(jl - iyl) / (jl + iyl) * (logder - kr * (jlp - iylp) /
(jl - iyl)) / (logder - kr *
(jlp + iylp) /
(jl + iyl))
return np.angle(sl) / 2.
def jnyn( n, resolution ):
delta = numpy.pi / resolution
zTable = numpy.mgrid[delta:max( maxz_j( n ), maxz_y( n ) ):delta]
jTable = numpy.zeros( ( len( zTable ), n + 1 ) )
yTable = numpy.zeros( ( len( zTable ), n + 1 ) )
for i, z in enumerate( zTable ):
jTable[i], _, yTable[i], _ = special.sph_jnyn( n, z )
jTable = jTable.transpose()
yTable = yTable.transpose()
return zTable, jTable, yTable
def riccati_2(nmax, x):
"""Riccati bessel function of the 2nd kind
returns (r2, r2'), n=0,1,...,nmax"""
jn, jnp, yn, ynp = special.sph_jnyn(nmax, x)
hn = jn + 1j * yn
hnp = jnp + 1j * ynp
r0 = x * hn
r1 = hn + x * hnp
return np.array([r0, r1])
def calculate_mie_coefficients(n_max, x, m):
"""
Calculates the Mie coefficients.
:rtype : object
:param n_max:
:param x: size parameter
:param m:
"""
# calculate spherical bessels #
jn, djn, yn, dyn = sph_jnyn(n_max, x) # j(n, x), y(n, x)
jm, djm, ym, dym = sph_jnyn(n_max, m * x) # j(n, mx), y(n, mx)
# calculate spherical hankel #
hn, dhn = sph_hn(n_max, x) # h(n, x)
# calculate riccati bessel functions #
dpsi_n = [x * jn[n - 1] - n * jn[n] for n in range(0, len(jn))]
dpsi_m = [m * x * jm[n - 1] - n * jm[n] for n in range(0, len(jm))]
dzeta_n = [x * hn[n - 1] - n * hn[n] for n in range(0, len(hn))]
a_n = (m**2 * jm * dpsi_n - jn * dpsi_m) / (m**2 * jm * dzeta_n -
hn * dpsi_m)
b_n = (jm * dpsi_n - jn * dpsi_m) / (jm * dzeta_n - hn * dpsi_m)
return a_n, b_n
def _mie_abcd(m, x):
'''Computes a matrix of Mie coefficients, a_n, b_n, c_n, d_n,
of orders n=1 to nmax, complex refractive index m=m'+im",
and size parameter x=k0*a, where k0= wave number
in the ambient medium, a=sphere radius;
p. 100, 477 in Bohren and Huffman (1983) BEWI:TDD122
C. Matzler, June 2002'''
nmax = np.round(2+x+4*x**(1.0/3.0))
mx = m * x
# Get the spherical bessel functions of the first (j) and second (y) kind,
# and their derivatives evaluated at x at order up to nmax
j_x,jd_x,y_x,yd_x = ss.sph_jnyn(nmax, x)
# The above function includes the 0 order bessel functions, which aren't used
j_x = j_x[1:]
jd_x = jd_x[1:]
y_x = y_x[1:]
yd_x = yd_x[1:]
# Get the spherical Hankel function of the first type (and it's derivative)
# from the combination of the bessel functions
h1_x = j_x + 1.0j*y_x
h1d_x = jd_x + 1.0j*yd_x
# Get the spherical bessel function of the first kind and it's derivative
# evaluated at mx
j_mx,jd_mx = ss.sph_jn(nmax, mx)
j_mx = j_mx[1:]
jd_mx = jd_mx[1:]
# Get primes (d/dx [x*f(x)]) using derivative product rule
j_xp = j_x + x*jd_x
j_mxp = j_mx + mx*jd_mx
h1_xp = h1_x + x*h1d_x
m2 = m * m
an = (m2 * j_mx * j_xp - j_x * j_mxp)/(m2 * j_mx * h1_xp - h1_x * j_mxp)
bn = (j_mx * j_xp - j_x * j_mxp)/(j_mx * h1_xp - h1_x * j_mxp)
## cn = (j_x * h1_xp - h1_x * j_xp)/(j_mx * h1_xp - h1_x * j_mxp)
## dn = m * (j_x * h1_xp - h1_x * j_xp)/(m2 * j_mx * h1_xp -h1_x * j_mxp)
return an, bn
def jnyn(n, resolution):
delta = numpy.pi / resolution
z_table = numpy.mgrid[min(minz_j(n),minz_y(n)):max(maxz_j(n), maxz_y(n)):delta]
j_table = numpy.zeros((len(z_table), n+1))
jdot_table = numpy.zeros((len(z_table), n+1))
y_table = numpy.zeros((len(z_table), n+1))
ydot_table = numpy.zeros((len(z_table), n+1))
for i, z in enumerate(z_table):
j_table[i], jdot_table[i], y_table[i], ydot_table[i] \
= special.sph_jnyn(n, z)
j_table = j_table.transpose()
jdot_table = jdot_table.transpose()
y_table = y_table.transpose()
ydot_table = ydot_table.transpose()
return z_table, j_table, jdot_table, y_table, ydot_table
def set_all_layers(self, lab):
self.data_layers = [0]
for bnd in lab.boundaries():
r, rd, rdd = bnd.shape.R(self.knots)
rdr = rd / r
r2rd2 = r ** 2 + rd ** 2
Rad = [0, {}, {}]
Radd = [0, {}, {}]
kis = [0, bnd.k1, bnd.k2]
for i in [1, 2]:
krs = kis[i] * r
JY = array([special.sph_jnyn(self.n, kr) for kr in krs])[:, :, :]
Rad[i] = {'j': JY[:, 0, :], 'h': JY[:, 0, :] + 1j * JY[:, 2, :]}
Radd[i] = {'j': JY[:, 1, :], 'h': JY[:, 1, :] + 1j * JY[:, 3, :]}
self.data_layers.append({'ki': kis,\
'r': r,\
'rd': rd,\
'rdd': rdd,\
'rdr': rdr,\
'r2rd2': r2rd2,\
'Rad': Rad,\
'Radd': Radd})
def mie_abcd(m, x):
"""Computes a matrix of Mie coefficients, a_n, b_n, c_n, d_n,
of orders n=1 to nmax, complex refractive index m=m'+im",
and size parameter x=k0*a, where k0= wave number
in the ambient medium, a=sphere radius;
p. 100, 477 in Bohren and Huffman (1983) BEWI:TDD122
C. Matzler, June 2002
Inputs:
- m - scalar, refractive-index;
- x - 1D array, size parameter;
Outputs:
- an - 1D array, Mie coefficient;
- bn - 1D array, Mie coefficient.
Example:
>>> import scipy.special as ss
>>> m = 5.62 + 2.78j # refractive index of water at 20 [Celsius]
>>> d = 0.003 # drop diameter - 3 [mm] --> 0.003 [m]
>>> lam = 0.01 # wavelength - 10 [mm] --> 0.01 [m]
>>> x = np.pi * d / lam
>>> an, bn = mie_abcd(m, x)
>>> an
array([ 3.28284411e-01 -3.36266733e-01j,
3.55370117e-03 -2.36426920e-02j,
3.90046294e-05 -5.15173202e-04j,
3.89223349e-07 -6.78235070e-06j,
2.94035836e-09 -5.85429913e-08j,
1.66462864e-11 -3.54797147e-10j, 7.17772878e-14 -1.58903782e-12j])
>>> bn
array([ 7.69466132e-02 +1.28116779e-01j,
6.75408598e-03 +6.94984812e-03j,
1.99953419e-04 +8.05215919e-05j,
2.23969208e-06 +8.10161445e-08j,
1.39633882e-08 -2.58644268e-09j,
6.06975879e-11 -1.97461249e-11j, 2.01460986e-13 -8.42856046e-14j])
"""
nmax = np.round(2 + x + 4 * x ** (1.0 / 3.0))
mx = m * x
# Get the spherical bessel functions of the first (j) and second (y) kind,
# and their derivatives evaluated at x at order up to nmax
j_x, jd_x, y_x, yd_x = ss.sph_jnyn(nmax, x)
# The above function includes the 0 order bessel functions, which aren't used
j_x = j_x[1:]
jd_x = jd_x[1:]
y_x = y_x[1:]
yd_x = yd_x[1:]
# Get the spherical Hankel function of the first type (and it's derivative)
# from the combination of the bessel functions
h1_x = j_x + 1.0j * y_x
h1d_x = jd_x + 1.0j * yd_x
# Get the spherical bessel function of the first kind and it's derivative
# evaluated at mx
j_mx, jd_mx = ss.sph_jn(nmax, mx)
j_mx = j_mx[1:]
jd_mx = jd_mx[1:]
# Get primes (d/dx [x*f(x)]) using derivative product rule
j_xp = j_x + x * jd_x
j_mxp = j_mx + mx * jd_mx
h1_xp = h1_x + x * h1d_x
m2 = m * m
an = (m2 * j_mx * j_xp - j_x * j_mxp) / (m2 * j_mx * h1_xp - h1_x * j_mxp)
bn = (j_mx * j_xp - j_x * j_mxp) / (j_mx * h1_xp - h1_x * j_mxp)
return an, bn
def sph_hn(n, x):
# calculate spherical hankel, h(n,x) = j(n,x) + iy(n,x) #
jn, djn, yn, dyn = sph_jnyn(n, x)
hn = jn + 1j * yn
dhn = djn + 1j * dyn
return hn, dhn
def get_JnHn(n, x):
JnYn = array([special.sph_jnyn(n, xl) for xl in x])
return JnYn[:, :2, :], JnYn[:, :2, :] + 1j * JnYn[:, 2:, :]
def get_JnHn(n, x):
JnYn = array([special.sph_jnyn(n, xl) for xl in x])
return JnYn[:, :2, :], JnYn[:, :2, :] + 1j * JnYn[:, 2:, :]
