def calc_a_coef(problem, boundary, eval_bc0, M, m1):
k = problem.k
R = problem.R
a = problem.a
nu = problem.nu
phi0_data = np.zeros(n_nodes, dtype=complex)
W = np.zeros((n_nodes, m1), dtype=complex)
for i in range(n_nodes):
th = th_data[i]
r = boundary.eval_r(th)
phi0_data[i] = eval_bc0(th)
for m in range(1, m1+1):
W[i, m-1] = jv(m*nu, k*r) * np.sin(m*nu*(th-a))
for m in range(M+1, m1+1):
W[:, m-1] = W[:, m-1] / jv(m*nu, k*R)
a_coef, residuals, rank, s = np.linalg.lstsq(W, phi0_data)
if rank != m1:
warnings.warn('Rank deficient')
return a_coef[:M], s
def compute_k2_vals(l_max, num_vals):
"""
Compute hyper radial infinite well K^2 eigenvalues for a well of unit radial width. The eigenvalues for a well with
parameter L = G + 3 D / 2
Compute square of zeros of J_{l+1/2}(x) for l = 0, 1/2, 1, ..., floor(l_max), floor(l_max)+1/2
:param l_max: Max l to find zeros of
:param num_vals: Total number of zeros to find for each l
:return K2: A 2*l_max + 1 by num_vals ndarray containing the computed squared zeros. K2[2*K + D-3] are the
eigenvalues for dimension D and hyper angular momentum L
"""
from numpy import arange, pi, zeros, zeros_like
from scipy.optimize import brentq
from scipy.special import jv
zro = zeros((2 * l_max + 1, num_vals), dtype=float)
z_l_m_1 = pi * arange(1,num_vals + l_max + 1)
z_l = zeros_like(z_l_m_1)
zz_l = zeros_like(z_l_m_1)
zro[0] = z_l_m_1[:num_vals]
for l in range(1, l_max + 1):
for i in range(num_vals + l_max - l):
zz_l[i] = brentq(lambda x: jv(l, x), z_l_m_1[i], z_l_m_1[i + 1])
z_l[i] = brentq(lambda x: jv(l + .5, x), z_l_m_1[i], z_l_m_1[i + 1])
z_l_m_1[:] = z_l[:]
zro[2 * l] = z_l[:num_vals]
zro[2 * l - 1] = zz_l[:num_vals]
if num_vals == 1:
zro = zro[:,0]
return zro**2
def besselmode(m, u, w, x, y, phioff=0):
"""Calculate the field of a bessel mode LP mode.
Arguments:
- m azimuthal number of periods (m=0,1,2,3...)
- u, w radial phase constant and radial decay constant
- x, y transverse coordinates
- phioff: offset angle, allows to rotate the mode in
the x-y plane
Returns:
- mode: calculated bessel mode
"""
xx,yy = np.meshgrid(x,y)
rr = np.reshape( np.sqrt(xx**2 + yy**2), len(x)*len(y))
phi = np.reshape( np.arctan2(xx,yy), len(x)*len(y))
fak = jv(m,u)/kn(m, w)
res = np.zeros(len(rr))
indx1 = rr<=1
res[indx1] = jv(m,u*rr[indx1])*np.cos(m*phi[indx1]+phioff)
indx2 = rr>1
res[indx2] = fak * kn(m, w * rr[indx2]) * np.cos(m * phi[indx2] + phioff)
res = res / np.max(np.abs(res))
return np.reshape(res, [len(y), len(x)])
def pumpistor_inductance(self, f=None, z_ext=None):
"""
Return the pumpistor inductance.
In the case of the non-degenerate case, a parent class must provide
a external_impedance method returning the impedance of the electrical
environment seen by the SQUID.
Parameters
----------
f : float, np.ndarray, optional
Signal frequency in hertz.
Is required in the non-degenerate case but optional for the
degenerate one.
z_ext : float, np.ndarray, optional
External impedance seen from the JPA point of view at the idler
frequency.
"""
# If z_ext is None, we return the pumpistor inductance of the
# degenerate case.
if z_ext is None:
return -2.*np.exp(1j*self.delta_theta())/self.delta_f()\
*cst.hbar/2./cst.e/self.I_c/abs(np.sin(self.F()))\
*self.phi_s/(2.*jv(1., self.phi_s)\
- 2.*np.exp(2j*self.delta_theta())*jv(3., self.phi_s))
else:
return cst.h/2./cst.e/np.pi/self.I_c/np.sin(self.F())**2./self.delta_f()**2.\
*(- 2.*np.cos(self.F())\
+ 1j*cst.h/2./cst.e/np.pi/self.I_c\
*2.*np.pi*(self.f_p - f)\
*(1./z_ext).conjugate())
def corner_exact(k,m):
"""
created Monday 11 July 2016
Function which vanishes at eigenenergies of circular corner.
"""
psi=jv(m,r1*k)-jv(m,r2*k)*yn(m,r1*k)/yn(m,r2*k)
return psi
def get_left_int_error(n, order):
a = 2
b = 30
intervals = np.linspace(0, 1, n, endpoint=True) ** 2 * (b-a) + a
discr = CompositeLegendreDiscretization(intervals, order)
x = discr.nodes
assert abs(discr.integral(1+0*x) - (b-a)) < 1e-13
alpha = 4
from scipy.special import jv, jvp
f = jvp(alpha, x)
num_int_f = jv(alpha, a) + discr.left_indefinite_integral(f)
int_f = jv(alpha, x)
if 0:
pt.plot(x.ravel(), num_int_f.ravel())
pt.plot(x.ravel(), int_f.ravel())
pt.show()
L2_err = np.sqrt(discr.integral((num_int_f - int_f)**2))
return 1/n, L2_err
def multiscale(self, ms, vm_in):
"""
Here we take the input as an phasor in the electrical RF domain
and the output is returned as a phasor in the optical domain
vm_in = [a0, a1, a2 ... an]
where
v(t) = sum_{k=0}^n a_k exp(j k w0) + c.c.
vm_out = [a0, a1, a2 ... an, a-n ... a-1]
where
E(t) = sum_{k=-n}^n a_k exp(j k w0)
Note:
Currently only the dc and fundamental component of the signal are used.
All other harmonics are assumed filtered out.
"""
#Output signal will be complex
mso = ms.copy()
mso.real = False
#Assume only fundamental harmonic and DC at input
a0 = vm_in[0]
a1,theta1 = np.abs(vm_in[1]), np.angle(vm_in[1])
#Jacobi-Anger expansion
k = mso.ms[:,np.newaxis]
b1 = special.jv(k, self.v1*a1) * np.exp(1j*self.v1*a0 + 1j*k*(theta1+0.5*pi))
b2 = special.jv(k, self.v2*a1) * np.exp(1j*self.phi + 1j*self.v2*a0 + 1j*k*(theta1+0.5*pi))
vm_out = self.eloss*self.Em*(b1 + self.a*b2)
return mso, vm_out
def bessel_series(a, b, z, maxiters=500, tol=tol):
"""Compute hyp1f1 using a series of Bessel functions; see (3.20) in
_[pop].
"""
Do, Dm, Dn = 1, 0, b/2
r = sqrt(z*(2*b - 4*a))
w = z**2 / r**(b + 1)
# Ao is the first coefficient, and An is the second.
Ao = 0
# The first summand comes from the zeroth term.
So = Do*jv(b - 1, r) / r**(b - 1) + Ao
An = Dn*w*jv(b + 1, r)
Sn = So + An
i = 3
while i <= maxiters:
if np.abs(Ao/So) < tol and np.abs(An/Sn) < tol:
break
tmp = Dn
Dn = ((i - 2 + b)*Dm + (2*a - b)*Do) / i
Do = Dm
Dm = tmp
w *= z/r
Ao = An
An = Dn*w*jv(b - 1 + i, r)
So = Sn
Sn += An
i += 1
# if i > maxiters:
# warnings.warn("Number of evaluations exceeded maxiters on "
# "a = {}, b = {}, z = {}.".format(a, b, z))
return gamma(b)*np.exp(z/2)*2**(b - 1)*Sn
def Rayleigh_1d(lamda=600,r=0.1*1.E-3,d= 5*1.E-3,D = 2):
a= 10 * 1.E-2
k=(2.*pi)/(lamda*1.E-9) #wavelength of light in vaccuum
X_Mmax=a/2. ; X_Mmin = -a/2.; Y_Mmin = X_Mmin; Y_Mmax=X_Mmax
N =400
X=linspace(X_Mmin, X_Mmax,N); Y=X # coordinates of screen
rm1=((X-d)**2+Y**2)**0.5
A1=k*r*rm1/D
I1=(ss.jv(1,A1)/A1)**2
rm2=((X+d)**2+Y**2)**0.5
A2=k*r*rm2/D
I2=(ss.jv(1,A2)/A2)**2
I=I1 + I2
fig = plt.figure(figsize=(7,5))
fig.suptitle('Fraunhofer Diffraction by circular aperture',fontsize=14, fontweight='bold')
ax1 = fig.add_subplot(111)
ax1.grid(True)
ax1.plot(X,I1, linewidth=2, alpha=0.8, label="I1")
ax1.plot(X,I2, linewidth=2, alpha=0.8, label="I2")
ax1.plot(X,I, linewidth=2, alpha=0.8, label="I = I1 + I2")
ax1.set_xlim(X_Mmin, X_Mmax)
ax1.set_xlabel(r'$X \ (m)$',fontsize=14, fontweight='bold')
ax1.set_ylabel(r'$I(X,Y)/I_0$',fontsize=14, fontweight='bold')
plt.legend()
plt.show()
def NormPart(self, Xsub):
"""
The von Mises and von Mises - Fisher Logarithmic Normalization partition function:...
is calculated in this method. For the 2D data the function is simplified for faster
calculation.
*** This function has been build after great research on the subject. However, some
things are not very clear this is always in a revision state until theoretically proven
to be correctly used.
Arguments
---------
Xsub: The subset of the data point are included in the von Mises-Fisher
distribution.
Output
------
The logarithmic value of the partition normalization function.
"""
# Calculating the r.
# The r it suppose to be the norm of the data points of the current cluster, not the...
# ...whole mixture as it is in the global objective function. ## Might this need to be...
# ...revised.
r = np.linalg.norm(Xsub)
# Calculating the Von Misses Fisher's k concentration is approximated as seen..
# ....in Banerjee et al. 2003.
dim = Xsub.shape[1]
# Calculating the partition function depending on the vector dimensions.
k = (r * dim - np.power(r, 3.0)) / (1 - np.power(r, 2.0))
# k=0.001 only for the case where the r is too small. Usually at the 1-2 first iterations...
# ...of the EM/Kmeans.
if k < 0.0:
k = 0.001
if dim > 3:
# This is the proper way for calculating the von Misses Fisher normalization factor.
bessel = np.abs(special.jv((dim / 2.0) - 1.0, k))
# bessel = np.abs(self.Jd((dim/2.0)-1.0, k))
cdk = np.power(k, (dim / 2.0) - 1) / (np.power(2 * np.pi, dim / 2) * bessel)
else:
# This is the proper way for calculating the vMF normalization factor for 2D vectors.
bessel = np.abs(special.jv(0, k))
# bessel = np.abs(self.Jd(0, k))
cdk = 1.0 / (2 * np.pi * bessel)
# Returning the log of the normalization function plus the log of the k that is used in...
# ...the von Mises Fisher PDF, which is separated from the Cosine Distance due to the log.
# The normalizers are multiplied by the number of all the X data subset, because it is...
# the global normalizer after the whole summations has been completed.
# NOTE: Still this need to be revised.
return np.log(cdk) + np.log(k) # * Xsub.shape[0]
def _chassat_integral(x, zeta=None, Lam=None, w=None, K1=None, K2=None, n1=None, n2=None, m1=None, m2=None):
"""Inner integral to compute the correlations (Chassat)"""
#compute bessel functions
j1 = jv(n1+1,x)
j2 = jv(n2+1,w*x)
j3 = jv(m1+m2,zeta*x)
j4 = jv(np.abs(m1-m2),zeta*x)
return x**(-14./3.) * j1 * j2 * (1. + (Lam / x)**2.)**(-11./6.) * (K1/w * j3 + K2/w * j4)
def bessel():
for i in range(1,360):
x = i / 6.28
y1 = spl.jv(1,x)
y2 = spl.yv(1,x)
y3 = spl.jv(3,x)
print("%7.3f, %7.3f, %7.3f, %7.3f" % (x, y1, y2, y3))
def func_deriv(x): if x==0: return 0 num=np.power(x,1+jv(0,x)) dnum=num*(-1*jv(1,x)*np.log(x)+(1+jv(0,x))/x) den=np.sqrt(1-x+100*(x**2)-100*(x**3)) dden=(-1+200*x-300*(x**2))/(2*den) df=(den*dnum-dden*num)/((den**2)) return df
def funSundePrimordial(p1, h1, a):
"""Função inicial para a curva teórica usando as curvas de bessel
Entrada:
a - espaçamento entre os condutores ou profundidade da camada
Saída:
resistividade para a profundidade "a"
"""
f = lambda m: exp(-2 * m * h1) * (jv(0, m * a) - jv(0, 2 * m * a))
return 2 * pi * a * quad(f, 0, Inf)[0]
def system(vec,V,Delta):
ru,iu,rw,iw = vec
u = ru+1j*iu
w = rw+1j*iw
first = u**2+w**2 - V**2
second = jv(0,u)/u/jv(1,u) - (1-Delta)*kv(0,w)/w/kv(1,w)
return np.real(first), np.imag(first),np.real(second), np.imag(second)
def eval_phi0(self, th):
a = self.a
nu = self.nu
k = self.k
r = self.boundary.eval_r(th)
p = lambda th: (th-a) * (th - (2*np.pi-a)) / (a*(2*np.pi - a))
return jv(3*nu/2, k*r) + (jv(nu/2, k*r) - jv(3*nu/2, k*r)) * p(th)
def plot4(V=10, nguess=100, wavelength = 10**-6, a = 5*10**-6, l=0, eps=10**-4):
lguess = [0]*nguess
for i in range(nguess):
lguess[i]=i*V*1.1/nguess
lguess.append(V)
functy = lambda x: (V**2-x*x)**(1/2)
functj = lambda x: x*jv(l+1, x)/jv(l, x)
functk = lambda x: functy(x)*kv(l+1, functy(x))/kv(l, functy(x))
funct = lambda x: functj(x)-functk(x)
lroots = [0] * len(lguess)
for i in range(len(lroots)):
lroots[i] = fsolve(funct, lguess[i])
#print lroots
lx = np.linspace(0,V+2, 10000)
plt.scatter(lx, functj(lx), s=.5)
plt.plot(lx, functk(lx))
#plt.plot(lx, funct(lx))
plt.ylim([0, max(functk(lx))])
#sorry this bit isn't particularly clean. Had to change strategies. Lazy. Etc.
def floatequals(a,b, eps=10**-6):
if type(b)==float:
b=[b]
for number in b:
if abs(a-number)<eps:
return True
return False
roots=[]
for i in range(len(lroots)):
if lroots[i]<0:
continue
if lroots[i]==lguess[i]:
continue
if floatequals(lroots[i], roots, eps=eps):
continue
roots.extend(lroots[i])
print roots
plt.scatter(roots, functj(roots), c='red')
#plt.show()
plt.scatter(roots, map(functk, roots), c='red')
title='Characteristic equation for a fiber optic, recall that Y^2=V^2-X^2'
plt.title(title, size='small')
plt.xlabel('X')
plt.ylabel('X*J_(l+1)(X)/J_l(X) and Y*K_(l+1)(Y)/K_l(Y)')
plt.savefig('hw9-4.png')
print 'the roots are ' + str(roots)
print min(roots)
print functk(0)
plt.show()
plt.close()
def CoreShell_ab(mCore,mShell,xCore,xShell):
# http://pymiescatt.readthedocs.io/en/latest/forwardCS.html#CoreShell_ab
m = mShell/mCore
u = mCore*xCore
v = mShell*xCore
w = mShell*xShell
mx = max(np.abs(mCore*xShell),np.abs(mShell*xShell))
nmax = np.round(2+xShell+4*(xShell**(1/3)))
nmx = np.round(max(nmax,mx)+16)
n = np.arange(1,nmax+1)
nu = n+0.5
sv = np.sqrt(0.5*np.pi*v)
sw = np.sqrt(0.5*np.pi*w)
sy = np.sqrt(0.5*np.pi*xShell)
pv = sv*jv(nu,v)
pw = sw*jv(nu,w)
py = sy*jv(nu,xShell)
chv = -sv*yv(nu,v)
chw = -sw*yv(nu,w)
chy = -sy*yv(nu,xShell)
p1y = np.append([np.sin(xShell)], [py[0:int(nmax)-1]])
ch1y = np.append([np.cos(xShell)], [chy[0:int(nmax)-1]])
gsy = py-(0+1.0j)*chy
gs1y = p1y-(0+1.0j)*ch1y
# B&H Equation 4.89
Dnu = np.zeros((int(nmx)),dtype=complex)
Dnv = np.zeros((int(nmx)),dtype=complex)
Dnw = np.zeros((int(nmx)),dtype=complex)
for i in range(int(nmx)-1,1,-1):
Dnu[i-1] = i/u-1/(Dnu[i]+i/u)
Dnv[i-1] = i/v-1/(Dnv[i]+i/v)
Dnw[i-1] = i/w-1/(Dnw[i]+i/w)
Du = Dnu[1:int(nmax)+1]
Dv = Dnv[1:int(nmax)+1]
Dw = Dnw[1:int(nmax)+1]
uu = m*Du-Dv
vv = Du/m-Dv
fv = pv/chv
dns = ((uu*fv/pw)/(uu*(pw-chw*fv)+(pw/pv)/chv))+Dw
gns = ((vv*fv/pw)/(vv*(pw-chw*fv)+(pw/pv)/chv))+Dw
a1 = dns/mShell+n/xShell
b1 = mShell*gns+n/xShell
an = (py*a1-p1y)/(gsy*a1-gs1y)
bn = (py*b1-p1y)/(gsy*b1-gs1y)
return an, bn
def hello_world():
# compute maxium
f = lambda x: -special.jv(3,x)
sol = optimize.minimize(f,1.0)
# plot
x = numpy.linspace(0,10,5000)
pyplot.plot(x, special.jv(3,x), '-', sol.x, -sol.fun, 'o')
# produce output
pyplot.savefig( PLOT_PATH + '/hello-world.png', dpi=96 )
def exact_solution(A, B, N):
"""Use scipy to construct exact solution to the Airy equation."""
# Setup Dedalus domain to get same grid
domain = dedalus_domain(N)
x = domain.grid(0)
# Compute Bessel function on grid
Jv = special.jv(A, x)
# Solve for coefficient using boundary condition
c = B / special.jv(A, 30)
y_exact = c * Jv
return y_exact
def P1(self):
U = self.U
W = self.W
return W**2/U**2 * kv(1, W)**2/jv(1, U)**2 * (
(1-self.s)**2 * (jv(0, U)**2 + jv(1, U)**2) +
(1+self.s)**2 * (jv(2, U)**2 - jv(1, U)*jv(3, U)) +
2*U**2/self.rb**2 * (jv(1, U)**2 - jv(0, U)*jv(2, U))
)
def eval_v(self, r, th):
"""
Using this function is "cheating".
"""
a = self.a
nu = self.nu
k = self.k
return (
jv(nu/2, k*r) * np.sin(nu/2 * (th-a)) +
jv(3*nu/2, k*r) * np.sin(3*nu/2 * (th-2*np.pi))
)
def eigenvalue_equation(cls, fiber, W):
U = sqrt(fiber.V**2 - W**2)
j0 = jv(0, U)
j1 = jv(1, U)
k1 = kv(1, W)
k1p = kvp(1, W, 1)
return (
j0/(U*j1) + (fiber.n**2 + fiber.nc**2)/(2*fiber.n**2) * k1p/(W*k1) - 1/U**2 +
sqrt(
((fiber.n**2 - fiber.nc**2)/(2*fiber.n**2) * k1p/(W*k1))**2
+ (cls._rboverrk(fiber, U, fiber.V)**2/fiber.n**2) * (1/W**2 + 1/U**2)**2
)
)
def get_r(M, sma, ecc):
### very fast series solution of Kepler's equation to 5th order. OK for low ecc of HST's orbit.
E = M + ( jv(1, 1.0 * ecc) * sin(1.0 * M) * 2.0 / float(1.0) +
jv(2, 2.0 * ecc) * sin(2.0 * M) * 2.0 / float(2.0) +
jv(3, 3.0 * ecc) * sin(3.0 * M) * 2.0 / float(3.0) +
jv(4, 4.0 * ecc) * sin(4.0 * M) * 2.0 / float(4.0) +
jv(5, 5.0 * ecc) * sin(5.0 * M) * 2.0 / float(5.0) )
T = 2.0 * numpy.arctan2(numpy.sqrt(1.0 + ecc) * numpy.sin(0.5 * E), numpy.sqrt(1.0 - ecc) * numpy.cos(0.5 * E))
R = sma * (1.0 - ecc ** 2) / ( 1.0 + ecc * numpy.cos(T) )
return R, T
def __init__(self,bc,face,FC):
#Input parameters
self.problem=bc.problem
self.N_cycles=self.problem.N_cycles
self.N_timesteps=self.problem.N_timesteps
self.nu=self.problem.nu
self.R=self.problem.radius[face]
self.T0=self.problem.T
self.dt=self.problem.dt
self.options=self.problem.options
#Load the Fourier Coefficients
infile_FC=open("./data/"+FC)
self.an=[];self.bn=[]
for line in infile_FC:
self.an.append(float(line.split()[0]))
self.bn.append(float(line.split()[1]))
self.omega=((2.*DOLFIN_PI)/self.T0)*self.N_cycles #omega
self.Q_mean=self.problem.bc_in_Qmean[face] #Mean flow rate
self.womn=self.R*(self.omega/self.nu)**0.5 #womersley number
if master:
print "Womersley Number is: :",self.womn
#Poieuille term constant
self.poiseuille_term_constant=((2.*self.an[0]*self.Q_mean)/(numpy.pi*self.R**2.))
#Pulsatile term constant
self.pulsatile_term_numerator1=[]
self.pulsatile_term_denominator=[]
for n in range(1,len(self.an)):
alpha=(n)**0.5*self.womn
self.pulsatile_term_numerator1.append( alpha*1j**1.5*jv(0,alpha*1j**1.5) )
self.pulsatile_term_denominator.append( alpha*1j**1.5*jv(0,alpha*1j**1.5)- 2*jv(1,alpha*1j**1.5) )
self.r=[] #store r values
self.pulsatile_for_all_r=[] #store the varying component of r
#Velocity term constant
self.velocity_term_constant=1./(numpy.pi*self.R**2.)*(self.Q_mean)
#flow rate term
self.t=[] #the flow rate series will remain the same for all 'r' values at a timestep
self.Qn_at_current_ts=0 #store the series for current timestep for multiple r values
#Directory to store the velocity profile
self.wom_dir="WomProf_"+self.options["case_name"]+"_womn"+str(self.womn)#Qmean"+str(self.an[0]*self.Q_mean)+"_R"+str(self.R)+"_nu"+str(self.nu)+"_ts"+str(self.N_timesteps/self.N_cycles)+"_cycles"+str(self.N_cycles)
self.n=0
def Mie_ab(m,x):
# http://pymiescatt.readthedocs.io/en/latest/forward.html#Mie_ab
mx = m*x
nmax = np.round(2+x+4*(x**(1/3)))
nmx = np.round(max(nmax,np.abs(mx))+16)
n = np.arange(1,nmax+1)
nu = n + 0.5
sx = np.sqrt(0.5*np.pi*x)
px = sx*jv(nu,x)
p1x = np.append(np.sin(x), px[0:int(nmax)-1])
chx = -sx*yv(nu,x)
ch1x = np.append(np.cos(x), chx[0:int(nmax)-1])
gsx = px-(0+1j)*chx
gs1x = p1x-(0+1j)*ch1x
# B&H Equation 4.89
Dn = np.zeros(int(nmx),dtype=complex)
for i in range(int(nmx)-1,1,-1):
Dn[i-1] = (i/mx)-(1/(Dn[i]+i/mx))
D = Dn[1:int(nmax)+1] # Dn(mx), drop terms beyond nMax
da = D/m+n/x
db = m*D+n/x
an = (da*px-p1x)/(da*gsx-gs1x)
bn = (db*px-p1x)/(db*gsx-gs1x)
return an, bn
def psf(xarray, yarray, a, R, l, I0):
# xarray and yarray are arrays
xv, yv = np.meshgrid(xarray, yarray)
q_grid = np.sqrt(xv ** 2.0 + yv ** 2.0)
x_arg_grid = (2.0 * np.pi * a * q_grid) / (l * R)
image = I0 * ((2.0 * special.jv(1, x_arg_grid)) / x_arg_grid) ** 2.0
return image
def eval_phi0(self, th):
a = self.a
nu = self.nu
k = self.k
r = self.boundary.eval_r(th)
return jv((self.K+1/2)*nu, k*r) * (th - a) / (2*np.pi - a)
def getftzer(Jzer,ngrid=128,Rpix=100):
'''
; Compute the Fourier Transform of Zernike mode
; ngrid = 128 ; grid half-size, pixels
; Rpix = 100 ; pupil radius in pixels
:param Jzer:
:return:
'''
x = np.arange(-ngrid,ngrid)
y = np.arange(-ngrid,ngrid)
theta = np.arctan2(x,y)
n,m = zern_num(Jzer)
f = np.roll(np.roll(dist(2*ngrid),
ngrid,
axis=0),
ngrid,
axis=1)/(2*ngrid)*Rpix
f[ngrid][ngrid] = 1e-3
ftmod = np.sqrt(n+1.0)*jv(n+1,2*np.pi*f)/(np.pi*f)
if m == 0:
zz = ftmod*np.complex(0,1.)**(n/2.)
else:
if (Jzer%2 == 0):
fact=np.sqrt(2.)*np.cos(m*theta)
else:
fact=np.sqrt(2.)*np.sin(m*theta)
zz = ftmod*fact*(-1)**((n-m/2.))*np.complex(0,1.)**m
return zz
def weights(self,nu,zeroArr):
"""
return the weights for quadrature
w_{ mu k} = Y_mu( pi xi_{nu k}/J_nu+1(Pi xi_{nu k})
"""
piXi = M.pi*zeroArr
return SS.yv(nu,piXi)/SS.jv(nu+1,piXi)
def __call__(self, k):
return k * self.pk(k) * jv(self.n, self.r*k)
def bem2cyl(wave_cond, discrete_cyl, vpot_cyl, trunc_ord):
""" Computes cylindrical amplitude coefficients from the velocity
potential values on a cylinder.
:param wave_cond: (water_depth, cfreq, wnum)
:type wave_cond: tuple
:param water_depth: water depth (m)
:type water_depth: float
:param cfreq: cyclic wave frequency (rad/s)
:type cfreq: float
:param wnum: wave number (rad/m) associated to that wave frequency and water depth
:type wnum: float
:param discrete_cyl: (radius_cyl, azimuth_cyl, axial_cyl)
:type discrete_cyl: tuple
:param radius_cyl: radius (m) of the cylinder
:type radius_cyl: float
:param azimuth_cyl: azimuthal discretization (rad) of the cylinder. An equispaced
discretization is assumed. Each element of the array is
an azimuth. Same axial discretization is assumed for each
azimuth
:type azimuth_cyl: 1D numpy array
:param axial_cyl: axial discretization (m) of the cylinder. Each element of the array is
the z-axial coordinate of a discrete point of the cylinder. Same
azimuthal discretization is assumed for each z-coordinate
:type axial_cyl: 1D numpy array
:param vpot_cyl: complex amplitude of the velocity potential (m**2/s) for each discrete point
of the cylinder and for the inputted water depth and wave frequency. shape
(extra axis, number of axial discretization, number of azimuthal
discretization)
:type vpot_cyl: 3D numpy array
:param trunc_ord: truncation order for the number of wave modes included.
Total number of wave modes is 2*trunc_ord+1
:type trunc_ord: int
"""
(water_depth, cfreq, wnum) = wave_cond
(radius_cyl, azimuth_cyl, axial_cyl) = discrete_cyl
dz = axial_cyl[1:] - axial_cyl[:-1]
dth = azimuth_cyl[1] - azimuth_cyl[0] # equispaced is assumed
rightz = all(dz > 0)
rightth = dth > 0
rightth2 = azimuth_cyl[-1] == 2. * np.pi + azimuth_cyl[0] or azimuth_cyl[
-1] == azimuth_cyl[0]
# Build integration domain
(z_cyl, th_cyl) = np.meshgrid(axial_cyl, azimuth_cyl, indexing='ij')
# Initialize
a_s = np.zeros((vpot_cyl.shape[0], 2 * trunc_ord + 1), dtype=complex)
for n_mode, mode in enumerate(range(-trunc_ord, trunc_ord + 1)):
integrand = vpot_cyl * np.cosh(wnum * (z_cyl + water_depth)) * np.exp(
-1j * mode * th_cyl)
# Integrate along th
int_th = (integrand[:, :, 1:] +
integrand[:, :, :-1]).sum(axis=2) * .5 * dth
if not rightth2: # add last paralepipede
int_th += (integrand[:, :, 0] + integrand[:, :, -1]) * .5 * dth
if not rightth:
int_th *= -1
# Integrate I_th along z
int_th_z = ((int_th[:, 1:] + int_th[:, :-1]) * dz).sum(axis=1) * .5
if not rightz:
int_th_z *= -1
# Cm
cntm = -1j * cfreq / (2 * np.pi * 9.809)
cntm *= 2 * np.cosh(wnum * water_depth)
cntm /= water_depth * (1 + np.sinh(2 * wnum * water_depth) /
(2 * wnum * water_depth))
cntm /= jv(mode, wnum * radius_cyl) - 1j * yv(mode, wnum * radius_cyl)
# amplitude coefficients
a_s[:, n_mode] = cntm * int_th_z
return a_s
def alpha_function_new(x, n, beta, R):
output = beta * ss.jv(n, x) + (x / R) * ss.jvp(n, x)
return output
def exo8():
v = np.linspace(0, 12, 12)
x = np.linspace(0, 20, 100)
v, x = np.meshgrid(v, x)
besselj = jv(v, x)
plotSurf(x, v, besselj, "x", "n", "Jn(x)", "Exo 8")
def loglkl(self, cosmo, data):
# Omega_m contains all species!
self.Omega_m = cosmo.Omega_m()
self.small_h = cosmo.h()
# needed for IA modelling:
if ('A_IA' in data.mcmc_parameters) and ('exp_IA'
in data.mcmc_parameters):
amp_IA = data.mcmc_parameters['A_IA'][
'current'] * data.mcmc_parameters['A_IA']['scale']
exp_IA = data.mcmc_parameters['exp_IA'][
'current'] * data.mcmc_parameters['exp_IA']['scale']
intrinsic_alignment = True
elif ('A_IA'
in data.mcmc_parameters) and ('exp_IA'
not in data.mcmc_parameters):
amp_IA = data.mcmc_parameters['A_IA'][
'current'] * data.mcmc_parameters['A_IA']['scale']
# redshift-scaling is turned off:
exp_IA = 0.
intrinsic_alignment = True
else:
intrinsic_alignment = False
# One wants to obtain here the relation between z and r, this is done
# by asking the cosmological module with the function z_of_r
self.r, self.dzdr = cosmo.z_of_r(self.z_p)
# Compute now the selection function p(r) = p(z) dz/dr normalized
# to one. The np.newaxis helps to broadcast the one-dimensional array
# dzdr to the proper shape. Note that p_norm is also broadcasted as
# an array of the same shape as p_z
if self.bootstrap_photoz_errors:
# draw a random bootstrap n(z); borders are inclusive!
random_index_bootstrap = np.random.randint(
int(self.index_bootstrap_low),
int(self.index_bootstrap_high) + 1)
#print('Bootstrap index:', random_index_bootstrap)
pz = np.zeros((self.nzmax, self.nzbins), 'float64')
pz_norm = np.zeros(self.nzbins, 'float64')
for zbin in xrange(self.nzbins):
#ATTENTION: hard-coded subfolder!
#index can be recycled since bootstraps for tomographic bins are independent!
fname = os.path.join(
self.data_directory,
'Nz_{0:}/Nz_{0:}_Bootstrap/Nz_z{1:}_boot{2:}_{0:}.asc'.
format(self.nz_method, self.zbin_labels[zbin],
random_index_bootstrap))
ztemp, hist_pz = np.loadtxt(fname, usecols=(0, 1), unpack=True)
shift_to_midpoint = np.diff(ztemp)[0] / 2.
spline_pz = itp.splrep(ztemp + shift_to_midpoint, hist_pz)
z_mod = self.z_p #+ self.shift_by_dz[zbin]
mask_min = z_mod >= ztemp.min()
mask_max = z_mod <= ztemp.max()
mask = mask_min & mask_max
pz[mask, zbin] = itp.splev(z_mod[mask], spline_pz)
pz_norm[zbin] = np.sum(0.5 * (pz[1:, zbin] + pz[:-1, zbin]) *
(self.z_p[1:] - self.z_p[:-1]))
self.pr = pz * (self.dzdr[:, np.newaxis] / pz_norm)
else:
# use fiducial dn/dz loaded in the __init__:
self.pr = self.pz * (self.dzdr[:, np.newaxis] / self.pz_norm)
# get linear growth rate if IA are modelled:
if intrinsic_alignment:
self.rho_crit = self.get_critical_density()
# derive the linear growth factor D(z)
linear_growth_rate = np.zeros_like(self.z_p)
#print(self.redshifts)
for index_z, z in enumerate(self.z_p):
try:
# for CLASS ver >= 2.6:
linear_growth_rate[
index_z] = cosmo.scale_independent_growth_factor(z)
except:
# my own function from private CLASS modification:
linear_growth_rate[index_z] = cosmo.growth_factor_at_z(z)
# normalize to unity at z=0:
try:
# for CLASS ver >= 2.6:
linear_growth_rate /= cosmo.scale_independent_growth_factor(0.)
except:
# my own function from private CLASS modification:
linear_growth_rate /= cosmo.growth_factor_at_z(0.)
# Compute function g_i(r), that depends on r and the bin
# g_i(r) = 2r(1+z(r)) int_r^+\infty drs p_r(rs) (rs-r)/rs
for Bin in xrange(self.nzbins):
# shift from first entry only useful if first enrty is 0!
#for nr in xrange(1, self.nzmax-1):
for nr in xrange(self.nzmax - 1):
fun = self.pr[nr:,
Bin] * (self.r[nr:] - self.r[nr]) / self.r[nr:]
self.g[nr, Bin] = np.sum(0.5 * (fun[1:] + fun[:-1]) *
(self.r[nr + 1:] - self.r[nr:-1]))
self.g[nr, Bin] *= 2. * self.r[nr] * (1. + self.z_p[nr])
#print('g(r) \n', self.g)
# Get power spectrum P(k=l/r,z(r)) from cosmological module
#self.pk_dm = np.zeros_like(self.pk)
kmax_in_inv_Mpc = self.k_max_h_by_Mpc * self.small_h
for index_l in xrange(self.nlmax):
for index_z in xrange(self.nzmax):
k_in_inv_Mpc = (self.l[index_l] + 0.5) / self.r[index_z]
if (k_in_inv_Mpc > kmax_in_inv_Mpc):
pk_dm = 0.
else:
pk_dm = cosmo.pk(k_in_inv_Mpc, self.z_p[index_z])
if 'A_bary' in data.mcmc_parameters:
A_bary = data.mcmc_parameters['A_bary'][
'current'] * data.mcmc_parameters['A_bary']['scale']
self.pk[index_l,
index_z] = pk_dm * self.baryon_feedback_bias_sqr(
k_in_inv_Mpc / self.small_h,
self.z_p[index_z],
A_bary=A_bary)
else:
self.pk[index_l, index_z] = pk_dm
'''
# Recover the non_linear scale computed by halofit. If no scale was
# affected, set the scale to one, and make sure that the nuisance
# parameter epsilon is set to zero
if cosmo.nonlinear_method == 0:
self.k_sigma[:] = 1.e6
else:
self.k_sigma = cosmo.nonlinear_scale(self.z_p, self.nzmax)
# Define the alpha function, that will characterize the theoretical
# uncertainty. Chosen to be 0.001 at low k, raise between 0.1 and 0.2
# to self.theoretical_error
if self.theoretical_error != 0:
for index_l in xrange(self.nlmax):
k = (self.l[index_l] + 0.5) / self.r
self.alpha[index_l, :] = np.log(1. + k[:] / self.k_sigma[:]) / (1. + np.log(1. + k[:] / self.k_sigma[:])) * self.theoretical_error
# recover the e_th_nu part of the error function
e_th_nu = self.coefficient_f_nu * cosmo.Omega_nu / cosmo.Omega_m()
# Compute the Error E_th_nu function
if 'epsilon' in self.use_nuisance:
for index_l in xrange(self.nlmax):
self.E_th_nu[index_l, :] = np.log(1. + self.l[index_l] / self.k_sigma[:] * self.r[:]) / (1. + np.log(1. + self.l[index_l] / self.k_sigma[:] * self.r[:])) * e_th_nu
# Add the error function, with the nuisance parameter, to P_nl_th, if
# the nuisance parameter exists
for index_l in xrange(self.nlmax):
epsilon = data.mcmc_parameters['epsilon']['current'] * (data.mcmc_parameters['epsilon']['scale'])
self.pk[index_l, :] *= (1. + epsilon * self.E_th_nu[index_l, :])
'''
Cl_GG_integrand = np.zeros_like(self.Cl_integrand)
Cl_GG = np.zeros_like(self.Cl)
if intrinsic_alignment:
Cl_II_integrand = np.zeros_like(self.Cl_integrand)
Cl_II = np.zeros_like(self.Cl)
Cl_GI_integrand = np.zeros_like(self.Cl_integrand)
Cl_GI = np.zeros_like(self.Cl)
dr = self.r[1:] - self.r[:-1]
# Start loop over l for computation of C_l^shear
# Start loop over l for computation of E_l
for il in xrange(self.nlmax):
# find Cl_integrand = (g(r) / r)**2 * P(l/r,z(r))
for Bin1 in xrange(self.nzbins):
for Bin2 in xrange(Bin1, self.nzbins):
Cl_GG_integrand[:, self.one_dim_index(
Bin1, Bin2
)] = self.g[:, Bin1] * self.g[:,
Bin2] / self.r**2 * self.pk[
il, :]
#print(self.Cl_integrand)
if intrinsic_alignment:
factor_IA = self.get_IA_factor(self.z_p,
linear_growth_rate,
amp_IA, exp_IA)
#print(self.eta_r[1:, zbin1].shape)
Cl_II_integrand[:, self.one_dim_index(
Bin1, Bin2
)] = self.pr[:,
Bin1] * self.pr[:,
Bin2] * factor_IA**2 / self.r**2 * self.pk[
il, :]
Cl_GI_integrand[:, self.one_dim_index(Bin1, Bin2)] = (
self.g[:, Bin1] * self.pr[:, Bin2] +
self.g[:, Bin2] * self.pr[:, Bin1]
) * factor_IA / self.r**2 * self.pk[il, :]
'''
if self.theoretical_error != 0:
self.El_integrand[1:, self.one_dim_index(Bin1, Bin2)] = self.g[:, Bin1] * self.g[:, Bin2] / self.r**2 * self.pk[il, :] * self.alpha[il, :]
'''
# Integrate over r to get C_l^shear_ij = P_ij(l)
# C_l^shear_ij = 9/16 Omega0_m^2 H_0^4 \sum_0^rmax dr (g_i(r)
# g_j(r) /r**2) P(k=l/r,z(r)) dr
# It is then multiplied by 9/16*Omega_m**2
# and then by (h/2997.9)**4 to be dimensionless
# (since P(k)*dr is in units of Mpc**4)
for Bin in xrange(self.nzcorrs):
Cl_GG[il, Bin] = np.sum(
0.5 *
(Cl_GG_integrand[1:, Bin] + Cl_GG_integrand[:-1, Bin]) *
dr)
Cl_GG[il, Bin] *= 9. / 16. * self.Omega_m**2
Cl_GG[il, Bin] *= (self.small_h / 2997.9)**4
if intrinsic_alignment:
Cl_II[il,
Bin] = np.sum(0.5 * (Cl_II_integrand[1:, Bin] +
Cl_II_integrand[:-1, Bin]) * dr)
Cl_GI[il,
Bin] = np.sum(0.5 * (Cl_GI_integrand[1:, Bin] +
Cl_GI_integrand[:-1, Bin]) * dr)
# here we divide by 4, because we get a 2 from g(r)!
Cl_GI[il, Bin] *= 3. / 4. * self.Omega_m
Cl_GI[il, Bin] *= (self.small_h / 2997.9)**2
'''
if self.theoretical_error != 0:
self.El[il, Bin] = np.sum(0.5 * (self.El_integrand[1:, Bin] + self.El_integrand[:-1, Bin]) * dr)
self.El[il, Bin] *= 9. / 16. * self.Omega_m**2
self.El[il, Bin] *= (self.small_h / 2997.9)**4
'''
'''
for Bin1 in xrange(self.nzbins):
Cl_GG[il, self.one_dim_index(Bin1, Bin1)] += self.noise
'''
if intrinsic_alignment:
self.Cl = Cl_GG + Cl_GI + Cl_II
else:
self.Cl = Cl_GG
#print(Cl_GG)
#print(self.Cl)
# Spline Cl[il,Bin1,Bin2] along l
for Bin in xrange(self.nzcorrs):
self.spline_Cl[Bin] = list(itp.splrep(self.l, self.Cl[:, Bin]))
# Interpolate Cl at values lll and store results in Cll
for Bin in xrange(self.nzcorrs):
self.Cll[Bin, :] = itp.splev(self.lll[:], self.spline_Cl[Bin])
# Start loop over theta values
for it in xrange(self.nthetatot):
ilmax = self.il_max[it]
self.BBessel0[:ilmax] = special.j0(self.lll[:ilmax] *
self.theta[it] * self.a2r)
self.BBessel4[:ilmax] = special.jv(
4, self.lll[:ilmax] * self.theta[it] * self.a2r)
# Here is the actual trapezoidal integral giving the xi's:
# - in more explicit style:
# for Bin in xrange(self.nzcorrs):
# for il in xrange(ilmax):
# self.xi1[it, Bin] = np.sum(self.ldl[il]*self.Cll[Bin,il]*self.BBessel0[il])
# self.xi2[it, Bin] = np.sum(self.ldl[il]*self.Cll[Bin,il]*self.BBessel4[il])
# - in more compact and vectorizable style:
self.xi1[it, :] = np.sum(self.ldl[:ilmax] * self.Cll[:, :ilmax] *
self.BBessel0[:ilmax],
axis=1)
self.xi2[it, :] = np.sum(self.ldl[:ilmax] * self.Cll[:, :ilmax] *
self.BBessel4[:ilmax],
axis=1)
# normalisation of xi's
self.xi1 = self.xi1 / (2. * math.pi)
self.xi2 = self.xi2 / (2. * math.pi)
# Spline the xi's
for Bin in xrange(self.nzcorrs):
self.xi1_theta[Bin] = list(itp.splrep(self.theta, self.xi1[:,
Bin]))
self.xi2_theta[Bin] = list(itp.splrep(self.theta, self.xi2[:,
Bin]))
# Get xi's in same column vector format as the data
#iz = 0
#for Bin in xrange(self.nzcorrs):
# iz = iz + 1 # this counts the bin combinations
# for i in xrange(self.ntheta):
# j = (iz-1)*2*self.ntheta
# self.xi[j+i] = itp.splev(
# self.theta_bins[i], self.xi1_theta[Bin])
# self.xi[self.ntheta + j+i] = itp.splev(
# self.theta_bins[i], self.xi2_theta[Bin])
# or in more compact/vectorizable form:
iz = 0
for Bin in xrange(self.nzcorrs):
iz = iz + 1 # this counts the bin combinations
j = (iz - 1) * 2 * self.ntheta
self.xi[j:j + self.ntheta] = itp.splev(
self.theta_bins[:self.ntheta], self.xi1_theta[Bin])
self.xi[j + self.ntheta:j + 2 * self.ntheta] = itp.splev(
self.theta_bins[:self.ntheta], self.xi2_theta[Bin])
# final chi2
vec = self.xi[self.mask_indices] - self.xi_obs[self.mask_indices]
#print(self.xi_obs[self.mask_indices], len(self.xi_obs[self.mask_indices]))
#print(self.xi[self.mask_indices], len(self.xi[self.mask_indices]))
# this is for running smoothly with MultiNest
# (in initial checking of prior space, there might occur weird solutions)
if np.isinf(vec).any() or np.isnan(vec).any():
chi2 = 2e12
else:
# don't invert that matrix...
# use the Cholesky decomposition instead:
yt = solve_triangular(self.cholesky_transform, vec, lower=True)
chi2 = yt.dot(yt)
return -chi2 / 2.
def model(self, x, p):
'''
Base Bessel function modulated spectrum model for micromotion.
Using definition from Pruttivarasin thesis.
p = [center, scale, gamma, offset, modulation depth]
'''
p[2] = abs(p[2]) # fwhm is positive
return p[3] + p[1]*p[2]*0.5*((sp.jv(-6,p[4])**2/((x - p[0] - 6*p[5])**2 + (0.5*p[2])**2)) +
(sp.jv(-5,p[4])**2/((x - p[0] - 5*p[5])**2 + (0.5*p[2])**2)) +
(sp.jv(-4,p[4])**2/((x - p[0] - 4*p[5])**2 + (0.5*p[2])**2)) +
(sp.jv(-3,p[4])**2/((x - p[0] -3*p[5])**2 + (0.5*p[2])**2)) +
(sp.jv(-2,p[4])**2/((x - p[0] -2*p[5])**2 + (0.5*p[2])**2)) +
(sp.jv(-1,p[4])**2/((x - p[0] - p[5])**2 + (0.5*p[2])**2)) +
(sp.jv(0,p[4])**2/((x - p[0])**2 + (0.5*p[2])**2)) +
(sp.jv(1,p[4])**2/((x - p[0] + p[5])**2 + (0.5*p[2])**2)) +
(sp.jv(2,p[4])**2/((x - p[0] + 2*p[5])**2 + (0.5*p[2])**2)) +
(sp.jv(3,p[4])**2/((x - p[0] + 3*p[5])**2 + (0.5*p[2])**2)) +
(sp.jv(4,p[4])**2/((x - p[0] + 4*p[5])**2 + (0.5*p[2])**2)) +
(sp.jv(5,p[4])**2/((x - p[0] + 5*p[5])**2 + (0.5*p[2])**2)) +
(sp.jv(6,p[4])**2/((x - p[0] + 6*p[5])**2 + (0.5*p[2])**2)))
def Floquet_edge2(self, kx, N):
"""
This function defines the Floquet edge states for case II
Parameters
----------
N = dimension of matrix
Returns
-------
Tridiagonal matrix
"""
# Parameter
vJ = self.J * self.S
vJc = self.Jc * self.S
v0 = 3 * vJ + vJc
Ep = 0.5 * sqrt(3 * self.Ey**2 + self.Ex**2 +
2 * sqrt(3) * self.Ey * self.Ex * cos(self.phi))
Em = 0.5 * sqrt(3 * self.Ey**2 + self.Ex**2 -
2 * sqrt(3) * self.Ey * self.Ex * cos(self.phi))
P1 = np.arctan(self.Ex * sin(self.phi) /
(self.Ey * sqrt(3) + self.Ex * cos(self.phi)))
P2 = np.arctan(self.Ex * sin(self.phi) /
(self.Ey * sqrt(3) - self.Ex * cos(self.phi)))
# Define H0
r0 = vJ * (jv(0, self.Ex) + jv(0, Em) * exp(1j * kx))
r0s = r0.conj()
H0 = np.array([[v0, -r0s, vJc, 0], [-r0, v0, 0, vJc],
[-vJc, 0, -v0, r0], [0, -vJc, r0s, -v0]])
# Define H1 and H-1
r1 = vJ * (jv(1, self.Ex) * exp(1j * self.phi) -
jv(1, Em) * exp(1j * kx) * exp(-1j * P2))
r1s = vJ * (-jv(1, self.Ex) * exp(1j * self.phi) +
jv(1, Em) * exp(-1j * kx) * exp(-1j * P2))
Hp = np.array([[0, -r1s, 0, 0], [-r1, 0, 0, 0], [0, 0, 0, r1],
[0, 0, r1s, 0]])
Hm = Hp.conj().T
# Floquet Hamiltonian for the diagonal matrix
com1 = dot(H0, Hm) - dot(Hm, H0)
com2 = dot(H0, Hp) - dot(Hp, H0)
com3 = dot(Hm, Hp) - dot(Hp, Hm)
Floquet_H0 = H0 - (1 / self.omega) * (com1 - com2 + com3)
r10 = vJ * jv(0, Ep)
H10 = np.array([[0, -r10, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0],
[0, 0, r10, 0]])
H20 = np.array([[0, 0, 0, 0], [-r10, 0, 0, 0], [0, 0, 0, r10],
[0, 0, 0, 0]])
r1p = vJ * jv(1, Ep) * exp(1j * P1)
r1m = -vJ * jv(1, Ep) * exp(1j * P1)
H1p = np.array([[0, 0, 0, 0], [-r1p, 0, 0, 0], [0, 0, r1p, 0],
[0, 0, 0, 0]])
H1m = H1p.conj().T
H2p = np.array([[0, 0, 0, 0], [-r1m, 0, 0, 0], [0, 0, 0, r1m],
[0, 0, 0, 0]])
H2m = H2p.conj().T
# Floquet Hamiltonian for the off-diagonal matrix
com11 = dot(H10, H1m) - dot(H1m, H10)
com21 = dot(H10, H1p) - dot(H1p, H10)
com31 = dot(H1m, H1p) - dot(H1p, H1m)
Floquet_H10 = H10 - (1 / self.omega) * (com11 - com21 + com31)
# Floquet Hamiltonian for the off-diagonal matrix
com12 = dot(H20, H2m) - dot(H2m, H20)
com22 = dot(H20, H2p) - dot(H2p, H20)
com32 = dot(H2m, H2p) - dot(H2p, H2m)
Floquet_H20 = H20 - (1 / self.omega) * (com12 - com22 + com32)
# Componenets of the tridiagonal matrices
c0 = diags(np.ones(N), 0).toarray()
cp = diags(np.ones(N - 1), 1).toarray()
cm = diags(np.ones(N - 1), -1).toarray()
# Tridiagonal Hamiltonian
H_tot = np.kron(c0, Floquet_H0) + np.kron(cp, Floquet_H10) + np.kron(
cm, Floquet_H20)
return H_tot
def _wignerd(j, m, n=0, approx_lim=10):
'''
Wigner "small d" matrix. (Euler z-y-z convention)
example::
j = 2
m = 1
n = 0
beta = linspace(0,pi,100)
wd210 = _wignerd(j,m,n)(beta)
some conditions have to be met::
j >= 0
-j <= m <= j
-j <= n <= j
The approx_lim determines at what point
bessel functions are used. Default is when::
j > m+10
# and
j > n+10
for integer l and n=0, we can use the spherical harmonics. If in
addition m=0, we can use the ordinary legendre polynomials.
'''
if (j < 0) or (abs(m) > j) or (abs(n) > j):
raise ValueError("_wignerd(j = {0}, m = {1}, n = {2}) value error.".
format(j, m, n) +
" Valid range for parameters: j>=0, -j<=m,n<=j.")
if (j > (m + approx_lim)) and (j > (n + approx_lim)):
#print('bessel (approximation)')
return lambda beta: jv(m - n, j * beta)
if (floor(j) == j) and (n == 0):
if m == 0:
#print('legendre (exact)')
return lambda beta: legendre(j)(cos(beta))
elif False:
#print('spherical harmonics (exact)')
a = sqrt(4. * pi / (2. * j + 1.))
return lambda beta: a * conjugate(sph_harm(m, j, beta, 0.))
jmn_terms = {
j + n: (m - n, m - n),
j - n: (n - m, 0.),
j + m: (n - m, 0.),
j - m: (m - n, m - n),
}
k = min(jmn_terms)
a, lmb = jmn_terms[k]
b = 2. * j - 2. * k - a
if (a < 0) or (b < 0):
raise ValueError(
"_wignerd(j = {0}, m = {1}, n = {2}) value error.".format(j, m, n)
+ " Encountered negative values in (a,b) = ({0},{1})".format(a, b))
coeff = power(-1., lmb) * sqrt(comb(2. * j - k, k + a)) * \
(1. / sqrt(comb(k + b, b)))
#print('jacobi (exact)')
return lambda beta: coeff \
* power(sin(0.5 * beta), a) \
* power(cos(0.5 * beta), b) \
* jacobi(k, a, b)(cos(beta))
jac=lambda x: np.array((-2 * .5 * (1 - x[0]) - 4 * x[0] *
(x[1] - x[0]**2), 2 * (x[1] - x[0]**2))))
"""prefer BFGS or L-BFGS"""
o.minimize(lambda x: .5 * (1 - x[0])**2 + (x[1] - x[0]**2)**2, [2, -1],
method='BFGS')
"""least squares - predict coefficients"""
x = np.array(range(10))
y = np.power(x, 2) - 2 * x + 1
A = np.c_[np.power(x, 2), x, np.ones(len(x))]
np.linalg.lstsq(A, y) #Gives 1, -2, 1 as first arg
"""f root"""
o.fsolve(lambda x: x**2 - 4, 5)
o.fsolve(lambda x: (x - 2)**2 - 4, 5)
"""-----------------integrate----------------------------------"""
"""Integration. Returns integral, error"""
f = lambda x: special.jv(2.5, x)
integrate.quad(f, 0, 10)
"""Integration by samples"""
x = np.linspace(0, 10, 10)
y = [f(xi) for xi in x]
integrate.simps(y, x)
"""-------------------interpolate--------------------------------"""
"""same can be done with splines"""
"""interpolation. Function by dots. Cubic best"""
y = [xi**2 + 2 * xi + 15 for xi in x]
x2 = np.linspace(0, 10, 100)
f = interp1d(x, y)
f2 = interp1d(x, y, kind='cubic')
plt.plot(x, y, 'o', x2, f(x2), '-', x2, f2(x2), '--')
plt.legend(['data', 'linear', 'cubic'], loc='best')
plt.show()
def tau2D_cylinder(self,
energyRange,
nk,
Uo,
m,
vfrac,
valley,
dk_len,
ro,
n=2000):
"""
This is a fast algorithm that uses Fermi’s golden rule to compute the energy dependent electron scattering rate
due cylindrical nanoparticles or pores extended perpendicular to the electrical current
"""
meff = np.array(m) * thermoelectricProperties.me
ko = 2 * np.pi / self.latticeParameter * np.array(valley)
del_k = 2 * np.pi / self.latticeParameter * dk_len * np.array(
[1, 1, 1])
N = vfrac / np.pi / ro**2
kx = np.linspace(ko[0], ko[0] + del_k[0], nk[0],
endpoint=True) # kpoints mesh
ky = np.linspace(ko[1], ko[1] + del_k[1], nk[1],
endpoint=True) # kpoints mesh
kz = np.linspace(ko[2], ko[2] + del_k[2], nk[2],
endpoint=True) # kpoints mesh
[xk, yk, zk] = np.meshgrid(kx, ky, kz)
xk_ = np.reshape(xk, -1)
yk_ = np.reshape(yk, -1)
zk_ = np.reshape(zk, -1)
kpoint = np.array([xk_, yk_, zk_])
mag_kpoint = norm(kpoint, axis=0)
E = thermoelectricProperties.hBar**2 / 2 * \
((kpoint[0, :] - ko[0])**2 / meff[0] + (kpoint[1, :] - ko[1])**2 / meff[1] + \
(kpoint[2, :] - ko[2]) ** 2 / meff[2]) * thermoelectricProperties.e2C
t = np.linspace(0, 2 * np.pi, n)
a = np.expand_dims(np.sqrt(2 * meff[1] / thermoelectricProperties.hBar**2 * \
E / thermoelectricProperties.e2C), axis=0)
b = np.expand_dims(np.sqrt(2 * meff[2] / thermoelectricProperties.hBar**2 * \
E / thermoelectricProperties.e2C), axis=0)
ds = np.sqrt((a.T * np.sin(t))**2 + (b.T * np.cos(t))**2)
cos_theta = ((a * kpoint[0]).T * np.cos(t) + (b * kpoint[1]).T * np.sin(t) + \
np.expand_dims(kpoint[2]**2, axis=1)) / \
np.sqrt(a.T**2 * np.cos(t)**2 + b.T**2 * np.sin(t)**2 + \
np.expand_dims(kpoint[2]**2, axis=1)) / np.expand_dims(mag_kpoint, axis=1)
delE = thermoelectricProperties.hBar**2 * \
np.abs((a.T * np.cos(t) - ko[0]) / meff[0] + \
(b.T * np.sin(t) - ko[1]) / meff[1] + (np.expand_dims(kpoint[2]**2, axis=1) - ko[2] / meff[2]))
qx = np.expand_dims(kpoint[0], axis=1) - a.T * np.cos(t)
qy = np.expand_dims(kpoint[1], axis=1) - b.T * np.sin(t)
qr = np.sqrt(qx**2 + qy**2)
tau = np.empty((len(ro), len(E)))
for r_idx in np.arange(len(ro)):
J = jv(1, ro[r_idx] * qr)
SR = 2 * np.pi / thermoelectricProperties.hBar * Uo**2 * (
2 * np.pi)**3 * (ro[r_idx] * J / qr)**2
f = SR * (1 - cos_theta) / delE * ds
int_ = np.trapz(f, t, axis=1)
tau[r_idx] = 1 / (N[r_idx] / (2 * np.pi)**3 *
int_) * thermoelectricProperties.e2C
Ec, indices, return_indices = np.unique(E,
return_index=True,
return_inverse=True)
tau_c = np.empty((len(ro), len(indices)))
tauFunctionEnergy = np.empty((len(ro), len(energyRange[0])))
for r_idx in np.arange(len(ro)):
tau_c[r_idx] = accum(return_indices,
tau[r_idx],
func=np.mean,
dtype=float)
for tau_idx in np.arange(len(tau_c)):
ESpline = PchipInterpolator(Ec[30:], tau_c[tau_idx, 30:])
tauFunctionEnergy[tau_idx] = ESpline(energyRange)
return tauFunctionEnergy
def analyticalTFpupil(x):
return pupilRadius**2 * scispe.jv(1, x) / x * 2 * np.pi
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
from scipy import special
from scipy import integrate
result = integrate.quad(lambda x: special.jv(4,x), 0, 20)
print(result )
#the following line is calculating Gaussian Integral using quad function
print("Gaussian integral", np.sqrt(np.pi), integrate.quad(lambda x: np.exp(-x**2),-np.inf, np.inf))
phi = 0.0 * pi + np.arctan(y / x)
if ((x < 0.0) & (y > 0.0)):
phi = 1.0 * pi + np.arctan(y / x)
if ((x < 0.0) & (y < 0.0)):
phi = 1.0 * pi + np.arctan(y / x)
if ((x > 0.0) & (y < 0.0)):
phi = 2.0 * pi + np.arctan(y / x)
j_x = q * -1.0 * r_cy * omega * np.sin(omega * t * Dt + theta_0)
j_y = q * 1.0 * r_cy * omega * np.cos(omega * t * Dt + theta_0)
j_rho = np.cos(phi) * j_x + np.sin(phi) * j_y
j_phi = np.cos(phi) * j_y - np.sin(phi) * j_x
E_rho_A = omega * mu_0 / (k_cut**2 * rho) * 1.0 * np.cos(
phi) * sp.jv(1, k_cut * rho) * np.sin(omega * t * Dt)
E_rho_B = omega * mu_0 / (k_cut**2 * rho) * -1.0 * np.sin(
phi) * sp.jv(1, k_cut * rho) * np.sin(omega * t * Dt)
E_phi_A = -1.0 * omega * mu_0 / (2 * k_cut) * np.sin(phi) * (
sp.jv(0, k_cut * rho) - sp.jv(2, k_cut * rho)) * np.sin(
omega * t * Dt)
E_phi_B = -1.0 * omega * mu_0 / (2 * k_cut) * np.cos(phi) * (
sp.jv(0, k_cut * rho) - sp.jv(2, k_cut * rho)) * np.sin(
omega * t * Dt)
A = A + Dt * (E_rho_A * j_rho + E_phi_A * j_phi)
B = B + Dt * (E_rho_B * j_rho + E_phi_B * j_phi)
P_Nmw = (pi * omega * mu_0 * zeta /
(2 * k_cut**4)) * (1.841**2 - 1.0) * sp.jv(
#A = np.array([[D[0], -D[0], 0],
# [D[0], -D[0]-D[1], D[1]],
# [0, D[2], -D[2]]])
#
#def dX_dt(sm, t=0):
# return np.dot(A,sm)
#
#t = np.linspace(0, 10, 100)
#X0 = np.array([10, 5, 20])
#X, infodict = integrate.odeint(dX_dt, X0, t, full_output=True)
#
#plt.plot(t,X)
#plt.xlabel('Time')
#plt.ylabel('X')
#plt.legend(['X1','X2','X3'])
#plt.savefig('/home/tomer/my_books/python_in_hydrology/images/ode_system.png')
#
from scipy import optimize, special
x = np.arange(0, 10, 0.01)
for k in np.arange(0.5, 5.5):
y = special.jv(k, x)
f = lambda x: -special.jv(k, x)
x_max = optimize.fminbound(f, 0, 6)
plt.plot(x, y, lw=3)
plt.plot([x_max], [special.jv(k, x_max)], 'rs', ms=12)
plt.title('Different Bessel functions and their local maxima')
plt.savefig('/home/tomer/my_books/python_in_hydrology/images/inverse.png')
plt.close()
zetaH = zetaH[0, :]
zetaV = zetaV[0, :]
rtol = htarg['rtol']
atol = htarg['atol']
nquad = htarg['nquad']
maxint = htarg['maxint']
pts_per_dec = htarg['pts_per_dec']
diff_quad = htarg['diff_quad']
a = htarg['a']
b = htarg['b']
limit = htarg['limit']
g_x, g_w = special.roots_legendre(nquad)
b_zero = np.pi * np.arange(1.25, maxint + 1)
for i in range(10):
b_x0 = special.j1(b_zero)
b_x1 = special.jv(2, b_zero)
b_h = -b_x0 / (b_x0 / b_zero - b_x1)
b_zero += b_h
if all(np.abs(b_h) < 8 * np.finfo(float).eps * b_zero):
break
xint = np.concatenate((np.array([1e-20]), b_zero))
dx = np.repeat(np.diff(xint) / 2, nquad)
Bx = dx * (np.tile(g_x, maxint) + 1) + np.repeat(xint[:-1], nquad)
BJ0 = special.j0(Bx) * np.tile(g_w, maxint)
BJ1 = special.j1(Bx) * np.tile(g_w, maxint)
intervals = xint / off[:, None]
lambd = Bx / off[:, None]
ang_fact = kernel.angle_factor(angle, ab, msrc, mrec)
# 1 Spline version
start = np.log(lambd.min())
stop = np.log(lambd.max())
def handleTE(st,
modes=[0, 0],
type_of_waveguide="Rectangular",
A=10,
B=5,
R=5):
if type_of_waveguide == "Rectangular":
x = np.linspace(0, A, 101)
y = np.linspace(0, B, 101)
X, Y = np.meshgrid(x, y)
M = int(modes[0])
N = int(modes[1])
par = TE_TM_Functions(M, N, A, B)
if M == 0 and N == 0:
st.error("m and n cannot be 0 at the same time")
return
u = np.cos(M * PI / A * X) * np.sin(N * PI / B * Y)
v = -1 * np.sin(M * PI / A * X) * np.cos(N * PI / B * Y)
fig, ax = plt.subplots()
plt.streamplot(X, Y, u, v, color="xkcd:azure")
plt.axis("scaled")
st.subheader("E field")
plt.xlim(0, A)
plt.ylim(0, B)
st.pyplot(fig)
u = np.sin(M * PI / A * X) * np.cos(N * PI / B * Y)
v = np.cos(M * PI / A * X) * np.sin(N * PI / B * Y)
fig, ax = plt.subplots()
plt.streamplot(x, y, u, v, color="red")
plt.axis("scaled")
st.subheader("H field")
plt.xlim(0, A)
plt.ylim(0, B)
st.pyplot(fig)
st.subheader("Values")
st.write(
pd.DataFrame({
"Parameter":
["Kc", "Fc", "Beta-g", "Vg", "Zin", "Zg", "lambda-g"],
"Value": [
par.Kc(),
par.Fc(),
par.beta_g(),
par.v_G(),
par.Z_in(),
par.Z_G_TE(),
par.lambda_G(),
],
"Unit": ["1/m", "Hz", "1/m", "m/s", "Ohm", "Ohm", "m"],
}))
else:
r = np.linspace(0, R, 101)
t = np.linspace(0, 2 * PI, 101)
T, RAD = np.meshgrid(t, r)
N = int(modes[0])
P = int(modes[1])
if P == 0:
st.error("p cannot be 0!")
return
X = special.jnp_zeros(N, P)
par = Circular_TE_TM_Functions(N, P, 2.3e-2)
U = special.jv(N, X[-1].round(3) / R * RAD) * np.sin(N * T)
V = special.jvp(N, X[-1].round(3) / R * RAD) * np.cos(N * T)
plt.axis("scaled")
fig, ax = plt.subplots()
plt.polar(2 * PI, R)
plt.streamplot(T, RAD, V, U, color="xkcd:azure")
plt.axis("scaled")
st.subheader("E field")
st.pyplot(fig)
st.markdown("**Scale: 5units = 2.3 cm**")
U = -1 * special.jv(N, X[-1].round(3) / R * RAD) * np.cos(N * T)
V = special.jv(N, X[-1].round(3) / R * RAD) * np.sin(N * T)
fig, ax = plt.subplots()
plt.polar(2 * PI, R)
plt.streamplot(T, RAD, V, U, color="red")
plt.axis("scaled")
st.subheader("H field")
st.pyplot(fig)
st.markdown("**Scale: 5units = 2.3 cm**")
st.subheader("Values")
st.write(
pd.DataFrame({
"Parameter":
["Kc", "Fc", "Beta-g", "Vg", "Zin", "Zg", "lambda-g"],
"Value": [
par.Kc_TE(),
par.Fc_TE(),
par.beta_g_TE(),
par.v_G_TE(),
par.Z_in(),
par.Z_G_TE(),
par.lambda_G_TE(),
],
"Unit": ["1/m", "Hz", "1/m", "m/s", "Ohm", "Ohm", "m"],
}))
def bessel_func(k):
return ((4 * spin - 2 *
(k - 1)) * ss.jv(0.25,
k * np.sqrt(2) * dip_const * xaxis) *
ss.jv(-0.25,
k * np.sqrt(2) * dip_const * xaxis))
def cohModelFunc(user_interface, context, queue, W_main, N, parameters,
parallel, debug):
user_interface.update_outputText("Starting Besinc model function...")
alpha = float(parameters[0])
try:
# parameters
M = N / 2
CL_pcohGS = None
if parallel:
#*******************************************************************
# PyOpenCL kernel function
#*******************************************************************
# KERNEL: CODE EXECUTED ON THE GPU
CL_pcohGS = Program(
context, """
__kernel void increase(__global float *res,
__global float *data,
const unsigned int N,
const unsigned int M,
const unsigned int i1,
const unsigned int j1,
const double x1,
const double y1,
const double a)
{
int row = get_global_id(0);
int col = get_global_id(1);
int y2 = -(row-M);
int x2 = (col-M);
double y22 = (double) y2;
double x22 = (double) x2;
double b1 = (double) x1-1*x2;
double b2 = (double) y1-1*y2;
double b = (double) sqrt(b1*b1+b2*b2);
//double J1 = (double) 1 - pown(a*b,2)/8.0 + pown(a*b,4)/192.0 - pown(a*b,6)/9216.0 + pown(a*b,8)/737280.0 - pown(a*b,10)/88473600.0 + pown(a*b,12)/14863564800.0 ;
double c = (double) a*b;
double zero = (double) 0.0;
double sum = 0;
int i;
for (i = 0; i < 80; i++)
{
double ii = (double) i;
double j1 = (double) 1+2*i;
double j2 = (double) i+1;
double j3 = (double) 2+i;
double arg = (double) a*b/2;
double actual = pow(-1,ii)*pow(arg,j1)/(tgamma(j2)*tgamma(j3));
sum = sum + actual;
}
if (c>zero)
{
sum = sum/c ;
}
else
{
sum = 1;
}
double data_const = (double) data[col + N*row ];
double final = (double) data_const*sum;
res[col + N*row ]= (float) final;
}
""").build()
#___________________________________________________________________
user_interface.update_outputText(
"PyOpenCl will be used. Starting Cycle...")
user_interface.update_outputText("__")
for i1 in range(0, N):
user_interface.update_outputTextSameLine(
str(round(i1 * 100. / N, 1)) + "% concluded (" + str(i1) +
"/" + str(N - 1) + ").")
for j1 in range(0, N):
if not count_nonzero(W_main[i1, j1]) == 0:
# Defining
result = zeros((N, N)).astype(float32)
data = copy.copy(W_main[i1, j1].real)
# Radius of point P1
x1 = j1 - M
y1 = M - i1
# creating memory on gpu
mf = mem_flags
# Result memory
result_gpu_memory = Buffer(context,
mf.READ_WRITE
| mf.COPY_HOST_PTR,
hostbuf=result)
# Data Memory
data_gpu_memory = Buffer(context,
mf.READ_ONLY
| mf.COPY_HOST_PTR,
hostbuf=data)
# Running the program (kernel)
CL_pcohGS.increase(queue, result.shape, None,
result_gpu_memory, data_gpu_memory,
int32(N), int32(M), int32(i1),
int32(j1), double(x1), double(y1),
double(alpha))
# Copying results to PCmemory
enqueue_copy(queue, result, result_gpu_memory)
# Copying results to matrices
W_main.real[i1][j1] = result
user_interface.update_outputTextSameLine("\r" +
str(round(100.0, 1)) +
"% concluded")
# Without PyOpenCL
else:
user_interface.update_outputText(
"PyOpenCl will NOT be used. Starting Cycle...")
for i1 in range(0, N):
user_interface.update_outputTextSameLine(
str(round(i1 * 100. / N, 1)) + "% concluded (" + str(i1) +
"/" + str(N - 1) + ").")
for j1 in range(0, N):
x1 = (j1 - M)
y1 = M - i1
for i2 in range(0, N):
for j2 in range(0, N):
x2 = j2 - M
y2 = M - i2
b = sqrt((x1 - x2)**2 + (y1 - y2)**2)
arg1 = alpha * b
if arg1 == 0:
y = 1
else:
y = (2 * jv(1, arg1) / arg1)
W_main.real[i1, j1, i2,
j2] = W_main.real[i1, j1, i2, j2] * y
user_interface.update_outputTextSameLine("\r" +
str(round(100.0, 1)) +
"% concluded")
except Exception as error:
user_interface.update_outputTextSameLine(str(error))
return W_main
def mag(
lam, z, p
): # Magnification as a function of wavelength, lens distance and optical offset
a = (((2.0 * math.pi) / (lam)) *
(math.sqrt(2 * rg / z))) * p # Dimensionless Bessel variable
return (4.0 * math.pi**2 * (rg / lam) * sci.jv(0, a)**2)
def a(n, x, mA, mB): # 散乱係数 an
return (mB * jv(n, mB * x) * jvp(n, mA * x) - mA * jv(n, mA * x) * jvp(n, mB * x)) / \
(mB * jv(n, mB * x) * h1vp(n, mA * x) -
mA * h1v(n, mA * x) * jvp(n, mB * x))
def calculate_FB_bases(L1):
maxK = (2 * L1 + 1)**2 - 1
L = L1 + 1
R = L1 + 0.5
truncate_freq_factor = 1.5
if L1 < 2:
truncate_freq_factor = 2
xx, yy = np.meshgrid(range(-L, L + 1), range(-L, L + 1))
xx = xx / R
yy = yy / R
ugrid = np.concatenate([yy.reshape(-1, 1), xx.reshape(-1, 1)], 1)
tgrid, rgrid = cart2pol(ugrid[:, 0], ugrid[:, 1])
num_grid_points = ugrid.shape[0]
kmax = 15
bessel = np.load(path_to_bessel)
B = bessel[(bessel[:, 0] <= kmax)
& (bessel[:, 3] <= np.pi * R * truncate_freq_factor)]
idxB = np.argsort(B[:, 2])
mu_ns = B[idxB, 2]**2
ang_freqs = B[idxB, 0]
rad_freqs = B[idxB, 1]
R_ns = B[idxB, 2]
num_kq_all = len(ang_freqs)
max_ang_freqs = max(ang_freqs)
Phi_ns = np.zeros((num_grid_points, num_kq_all), np.float32)
Psi = []
kq_Psi = []
num_bases = 0
for i in range(B.shape[0]):
ki = ang_freqs[i]
qi = rad_freqs[i]
rkqi = R_ns[i]
r0grid = rgrid * R_ns[i]
F = special.jv(ki, r0grid)
Phi = 1. / np.abs(special.jv(ki + 1, R_ns[i])) * F
Phi[rgrid >= 1] = 0
Phi_ns[:, i] = Phi
if ki == 0:
Psi.append(Phi)
kq_Psi.append([ki, qi, rkqi])
num_bases = num_bases + 1
else:
Psi.append(Phi * np.cos(ki * tgrid) * np.sqrt(2))
Psi.append(Phi * np.sin(ki * tgrid) * np.sqrt(2))
kq_Psi.append([ki, qi, rkqi])
kq_Psi.append([ki, qi, rkqi])
num_bases = num_bases + 2
Psi = np.array(Psi)
kq_Psi = np.array(kq_Psi)
num_bases = Psi.shape[1]
if num_bases > maxK:
Psi = Psi[:maxK]
kq_Psi = kq_Psi[:maxK]
num_bases = Psi.shape[0]
p = Psi.reshape(num_bases, 2 * L + 1, 2 * L + 1).transpose(1, 2, 0)
psi = p[1:-1, 1:-1, :]
# print(psi.shape)
psi = psi.reshape((2 * L1 + 1)**2, num_bases)
c = np.sqrt(np.sum(psi**2, 0).mean())
psi = psi / c
return psi, c, kq_Psi
def alpha_function(x, n, beta):
output = beta * ss.jv(n, x) + ss.jvp(n, x)
return output
# Bill Karr's Code for Problem 3 in Problem set 1
from __future__ import division
from scipy.special import jv
import numpy as np
# <codecell>
# Problem 3, Part A
bessel = np.array([jv(i, 20) for i in range(0, 51)])
diff = np.array([
(bessel[i] - (2 * (i - 1) * bessel[i - 1] / 20 - bessel[i - 2]))
for i in range(2, 51)
])
relError = np.array([
(bessel[i] - (2 *
(i - 1) * bessel[i - 1] / 20 - bessel[i - 2])) / bessel[i]
for i in range(2, 51)
])
print "PART A"
print "n & scipy approx & LHS - RHS & % error"
for i in range(0, 49):
print i + 2, "&", bessel[i], "&", diff[i], "&", 100 * abs(relError[i])
# <codecell>
# Problem 3, Part B
def J1(x):
return special.jv(1, x)
def b(n, x, mA, mB): # 散乱係数 bn
return (mA * jv(n, mB * x) * jvp(n, mA * x) - mB * jv(n, mA * x) * jvp(n, mB * x)) / \
(mA * jv(n, mB * x) * h1vp(n, mA * x) -
mB * h1v(n, mA * x) * jvp(n, mB * x))
# define domain
# z = np.arange(-L, L, dz) # 2N + 1 domain points
z = np.linspace(-L, L,
N + 1) # N points (used for now because I'm not zero padding)
print(len(z))
# define magnetic constants
B = 1.5
b = 0.1
epsilon = 1 - B / 2
## define integral components (eq. 2.18)
# bessel functions
bess_first = sp.jv(1, z)
bess_sec = sp.yn(1, z)
def mainIntegrand(S, c, z, N, L, b, B, epsilon):
# define S derivatives (spectral)
S_z = funcs.fftDeriv(S, z, order=1)
S_zz = funcs.fftDeriv(S, z, order=2)
# bessel functions
def I(domain, order=1):
# modified bessel of first kind
return sp.iv(order, domain)
def K(domain, order=1):
def makePSF(
wavelength=0.525,
NA=1.4,
nx=257,
nz=257,
dx=0.02,
dz=0.02,
RI=1.33,
immRI=1.5,
csRI=1.515,
csthick=170,
workingdistance=150,
particledistance=0,
num_basis=200,
num_samples=1000,
oversampling=1,
):
# dx/dz are output pixel sizes in microns
# RI is refractive index of sample
# workingdistance in microns, working distance (immersion medium thickness) design value
# particledistance in microns, particle distance from coverslip
# Size of the PSF array, pixels
ny = nx # square ... not really necessary until this becomes 3D again
ni, ni0 = tuplecheck(
immRI) # immersion medium RI experimental value, design value
ng, ng0 = tuplecheck(csRI) # coverslip RI experimental value, design value
tg, tg0 = tuplecheck(
csthick) # coverslip thickness experimental value, design value
# Precision control
# num_basis = 100 # Number of rescaled Bessels that approximate the phase function
# num_samples = 1000 # Number of pupil samples along radial direction
# oversampling = 2 # Defines the upsampling ratio on the image space grid for computations
# Scaling factors for the Fourier-Bessel series expansion
min_wavelength = 0.436 # microns
scaling_factor = (NA * (3 * np.arange(1, num_basis + 1) - 2) *
min_wavelength / wavelength)
# Place the origin at the center of the final PSF array
x0 = (nx - 1) / 2
y0 = (ny - 1) / 2
# Find the maximum possible radius coordinate of the PSF array by finding the distance
# from the center of the array to a corner
max_radius = round(sqrt((nx - x0) * (nx - x0) + (ny - y0) * (ny - y0))) + 1
# Radial coordinates, image space
r = dx * np.arange(0, oversampling * max_radius) / oversampling
# Radial coordinates, pupil space
a = min([NA, RI, ni, ni0, ng, ng0]) / NA
rho = np.linspace(0, a, num_samples)
# Stage displacements away from best focus
z = dz * np.arange(-nz / 2, nz / 2) + dz / 2
# Define the wavefront aberration
NArho2 = NA**2 * rho**2
OPDs = particledistance * np.sqrt(RI**2 - NArho2) # OPD in the sample
OPDi = (z.reshape(-1, 1) + workingdistance
) * np.sqrt(ni**2 - NArho2) - workingdistance * np.sqrt(
ni0**2 - NArho2) # OPD in the immersion medium
OPDg = tg * np.sqrt(ng**2 - NArho2) - tg0 * np.sqrt(
ng0**2 - NArho2) # OPD in the coverslip
W = 2 * np.pi / wavelength * (OPDs + OPDi + OPDg)
# Sample the phase
# Shape is (number of z samples by number of rho samples)
phase = np.cos(W) + 1j * np.sin(W)
# Define the basis of Bessel functions
# Shape is (number of basis functions by number of rho samples)
J = special.jv(0, scaling_factor.reshape(-1, 1) * rho)
# Compute the approximation to the sampled pupil phase by finding the least squares
# solution to the complex coefficients of the Fourier-Bessel expansion.
# Shape of C is (number of basis functions by number of z samples).
# Note the matrix transposes to get the dimensions correct.
C, residuals, _, _ = np.linalg.lstsq(J.T, phase.T, rcond=None)
# Which z-plane to compute
# z0 = 24
# The Fourier-Bessel approximation
# est = J.T.dot(C[:, z0])
b = 2 * np.pi * r.reshape(-1, 1) * NA / wavelength
def J0(x):
return special.jv(0, x)
def J1(x):
return special.jv(1, x)
# See equation 5 in Li, Xue, and Blu
denom = scaling_factor * scaling_factor - b * b
R = (scaling_factor * J1(scaling_factor * a) * J0(b * a) * a -
b * J0(scaling_factor * a) * J1(b * a) * a)
R /= denom
# The transpose places the axial direction along the first dimension of the array, i.e. rows
# This is only for convenience.
PSF_rz = (np.abs(R.dot(C))**2).T
# Normalize to the maximum value
PSF_rz /= np.max(PSF_rz)
return PSF_rz
def test_sp_special(v: int, x: float):
return jv(v, x)
def I(r):
if (r == 0):
return 1 / 2 # Using the limit found in the textbook.
else:
return (sp.jv(0, k * r) / (k * r))**2
out_file = "fresnel_fourier_1D.spec"
f = open(out_file, 'w')
header="#F %s \n\n#S 1 fresnel diffraction \n#N 3 \n#L X[m] intensity phase\n"%out_file
f.write(header)
for i in range(len(position_x)):
out = numpy.array((position_x[i], fieldIntensity[i],fieldPhase[i]))
f.write(("%20.11e "*out.size+"\n") % tuple( out.tolist()))
f.close()
print ("File written to disk: %s"%out_file)
#plots
from matplotlib import pylab as plt
from scipy.special import jv
sin_theta = position_x / distance
x = (2*numpy.pi/wavelength) * (aperture_diameter/2) * sin_theta
U_vs_theta = 2*jv(1,x)/x
I_vs_theta = U_vs_theta**2 * fieldIntensity.max()
plt.figure(1)
plt.plot(position_x*1e6,fieldIntensity,'-',position_x*1e6,I_vs_theta,)
plt.title("Fresnel Diffraction")
plt.xlabel("X [um]")
plt.ylabel("Intensity [a.u.]")
plt.show()
