def fullDet(b, nu):
a = rho * b
i = ivp(nu, u1*a) / (u1*a * iv(nu, u1*a))
if nu == 1:
k2 = -2
k = 0
else:
k2 = 2 * nu / (u2 * b)**2 - 1 / (nu - 1)
k = -1
X = (1 / (u1*a)**2 + 1 / (u2*a)**2)
P1 = jn(nu, u2*a) * yn(nu, u2*b) - yn(nu, u2*a) * jn(nu, u2*b)
P2 = (jvp(nu, u2*a) * yn(nu, u2*b) - yvp(nu, u2*a) * jn(nu, u2*b)) / (u2 * a)
P3 = (jn(nu, u2*a) * yvp(nu, u2*b) - yn(nu, u2*a) * jvp(nu, u2*b)) / (u2 * b)
P4 = (jvp(nu, u2*a) * yvp(nu, u2*b) - yvp(nu, u2*a) * jvp(nu, u2*b)) / (u2 * a * u2 * b)
A = (n12 * i**2 - n32 * nu**2 * X**2)
if nu == 1:
B = 0
else:
B = 2 * n22 * n32 * nu * X * (P1 * P4 - P2 * P3)
return (n32 * k2 * A * P1**2 +
(n12 + n22) * n32 * i * k2 * P1 * P2 -
(n22 + n32) * k * A * P1 * P3 -
(n12*n32 + n22*n22) * i * k * P1 * P4 +
n22 * n32 * k2 * P2**2 -
n22 * (n12 + n32) * i * k * P2 * P3 -
n22 * (n22 + n32) * k * P2 * P4 -
B)
def Psi(self, r, neff, wl, nu, C):
u = self.u(r, neff, wl)
if neff < self.maxIndex(wl):
psi = (C[0] * jn(nu, u) + C[1] * yn(nu, u) if C[1] else
C[0] * jn(nu, u))
psip = u * (C[0] * jvp(nu, u) + C[1] * yvp(nu, u) if C[1] else
C[0] * jvp(nu, u))
else:
psi = (C[0] * iv(nu, u) + C[1] * kn(nu, u) if C[1] else
C[0] * iv(nu, u))
psip = u * (C[0] * ivp(nu, u) + C[1] * kvp(nu, u) if C[1] else
C[0] * ivp(nu, u))
# if numpy.isnan(psi):
# print(neff, self.maxIndex(wl), C, r)
return psi, psip
def __delta(self, nu, u1r1, u2r1, s1, s2, s3, n1sq, n2sq, n3sq):
"""s3 is sign of Delta"""
if s1 < 0:
f = ivp(nu, u1r1) / (iv(nu, u1r1) * u1r1) # c
else:
jnnuu1r1 = jn(nu, u1r1)
if jnnuu1r1 == 0: # Avoid zero division error
return float("inf")
f = jvp(nu, u1r1) / (jnnuu1r1 * u1r1) # a b d
if s1 == s2:
# b d
kappa1 = -(n1sq + n2sq) * f / n2sq
kappa2 = (n1sq * f * f / n2sq -
nu**2 * n3sq / n2sq * (1 / u1r1**2 - 1 / u2r1**2)**2)
else:
# a c
kappa1 = (n1sq + n2sq) * f / n2sq
kappa2 = (n1sq * f * f / n2sq -
nu**2 * n3sq / n2sq * (1 / u1r1**2 + 1 / u2r1**2)**2)
d = kappa1**2 - 4 * kappa2
if d < 0:
return numpy.nan
return u2r1 * (nu / u2r1**2 + (kappa1 + s3 * sqrt(d)) * 0.5)
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 _hefield(self, wl, nu, neff, r):
self._heceq(neff, wl, nu)
for i, rho in enumerate(self.fiber._r):
if r < rho:
break
else:
i += 1
layer = self.fiber.layers[i]
n = layer.maxIndex(wl)
u = layer.u(rho, neff, wl)
urp = u * r / rho
c1 = rho / u
c2 = wl.k0 * c1
c3 = nu * c1 / r if r else 0 # To avoid div by 0
c6 = constants.Y0 * n * n
if neff < n:
B1 = jn(nu, u)
B2 = yn(nu, u)
F1 = jn(nu, urp) / B1
F2 = yn(nu, urp) / B2 if i > 0 else 0
F3 = jvp(nu, urp) / B1
F4 = yvp(nu, urp) / B2 if i > 0 else 0
else:
c2 = -c2
B1 = iv(nu, u)
B2 = kn(nu, u)
F1 = iv(nu, urp) / B1
F2 = kn(nu, urp) / B2 if i > 0 else 0
F3 = ivp(nu, urp) / B1
F4 = kvp(nu, urp) / B2 if i > 0 else 0
A, B, Ap, Bp = layer.C[:, 0] + layer.C[:, 1] * self.alpha
Ez = A * F1 + B * F2
Ezp = A * F3 + B * F4
Hz = Ap * F1 + Bp * F2
Hzp = Ap * F3 + Bp * F4
if r == 0 and nu == 1:
# Asymptotic expansion of Ez (or Hz):
# J1(ur/p)/r (r->0) = u/(2p)
if neff < n:
f = 1 / (2 * jn(nu, u))
else:
f = 1 / (2 * iv(nu, u))
c3ez = A * f
c3hz = Ap * f
else:
c3ez = c3 * Ez
c3hz = c3 * Hz
Er = c2 * (neff * Ezp - constants.eta0 * c3hz)
Ep = c2 * (neff * c3ez - constants.eta0 * Hzp)
Hr = c2 * (neff * Hzp - c6 * c3ez)
Hp = c2 * (-neff * c3hz + c6 * Ezp)
return numpy.array((Er, Ep, Ez)), numpy.array((Hr, Hp, Hz))
def circle_wg_tm_mn(mesh, m=31, n=11, phi=90.0, radi=25.0):
x, y = mesh[0], mesh[1]
r, t = np.sqrt(x**2 + y**2), np.arctan(y / x)
b = radi
x0_mn = jn_zeros(m, n)[-1]
x1_mn = jnp_zeros(m, n)[-1]
if m == 0:
e_m = 1
else:
e_m = 2
func = np.sqrt(e_m / np.pi)
func *= 1 / np.abs(jvp(m + 1, x0_mn))
func *= (-jvp(m, x0_mn * r / b, 0) * np.cos(m * t) + m /
x0_mn * jvp(m, x0_mn * r / b, 0) / r * np.sin(m * t))
return func
def get_coefficients(Nterms, wavenumber_b, wavenumber_d, radius, epsilon_d,
epsilon_b):
"""Summarize the method."""
n = np.arange(-Nterms, Nterms + 1)
kb, kd = wavenumber_b, wavenumber_d
a = radius
an = (-jv(n, kb * a) / hankel2(n, kb * a) *
((epsilon_d * jvp(n, kd * a) /
(epsilon_b * kd * a * jv(n, kd * a)) - jvp(n, kb * a) /
(kb * a * jv(n, kb * a))) /
(epsilon_d * jvp(n, kd * a) /
(epsilon_b * kd * a * jv(n, kd * a)) - h2vp(n, kb * a) /
(kb * a * hankel2(n, kb * a)))))
cn = 1 / jv(n, kd * a) * (jv(n, kb * a) + an * hankel2(n, kb * a))
return an, cn
def lpConstants(self, r, neff, wl, nu, A):
u = self.u(r, neff, wl)
if neff < self.maxIndex(wl):
W = constants.pi / 2
return (W * (u * yvp(nu, u) * A[0] - yn(nu, u) * A[1]),
W * (jn(nu, u) * A[1] - u * jvp(nu, u) * A[0]))
else:
return ((u * kvp(nu, u) * A[0] - kn(nu, u) * A[1]),
(iv(nu, u) * A[1] - u * ivp(nu, u) * A[0]))
def cutoffHE2(b):
nu = 2
a = rho * b
i = ivp(2, u1*a) / (u1*a * iv(2, u1*a))
X = (1 / (u1*a)**2 + 1 / (u2*a)**2)
P1 = jn(nu, u2*a) * yn(nu, u2*b) - yn(nu, u2*a) * jn(nu, u2*b)
P2 = (jvp(nu, u2*a) * yn(nu, u2*b) - yvp(nu, u2*a) * jn(nu, u2*b)) / (u2 * a)
P3 = (jn(nu, u2*a) * yvp(nu, u2*b) - yn(nu, u2*a) * jvp(nu, u2*b)) / (u2 * b)
P4 = (jvp(nu, u2*a) * yvp(nu, u2*b) - yvp(nu, u2*a) * jvp(nu, u2*b)) / (u2 * a * u2 * b)
A = 2 * nu / (u2 * b)**2 - 1 / (nu - 1)
return (n22 / n32 * (i*P1 + P2) * (n12*i*P3 + n22 * P4) +
(A * i * P1 + A * P2 + i*P3 + P4) * (n12*i*P1 + n22*P2) -
n32 * nu**2 * X**2 * P1 * (A * P1 + P3) -
n22 * nu * X * (nu * X * P1 * P3 + 2 * P1 * P4 - 2 * P2 * P3))
def te0G(self, y):
'''
find the zeros of this function to find the TE0 modes
works only for mu_h=mu_c=1
'''
aa = 4.0j * special.jvp(0, z) * np.log(y) / special.jv(0, y) / np.pi
bb = self.cyl_n * y**2 - 4.0j / y / self.host_n
return aa + bb
def solve_for_AB(k, R1, R2):
# solve the system for A,B
#
# / J'_m(k R1) Y'_m(k R1) \ /A\ -- /0\
# \ J'_m(k R2) Y'_m(k R2) / \B/ -- \0/
M = numpy.zeros((2, 2))
M[0][0] = special.jvp(1, k * R1, 1) # Jprime_1 at R1
M[0][1] = special.yvp(1, k * R1, 1) # Yprime_1 at R1
M[1][0] = special.jvp(1, k * R2, 1) # Jprime_1 at R2
M[1][1] = special.yvp(1, k * R2, 1) # Yprime_1 at R2
# close but no cigar to zero, otherwise it returns x=(0,0)
b = numpy.array([1.e-16, 1.e-16])
x = numpy.linalg.solve(M, b)
return x[0], x[1]
def cutoffHE1(b):
a = rho * b
i = ivp(1, u1*a) / (u1*a * i1(u1*a))
X = (1 / (u1*a)**2 + 1 / (u2*a)**2)
P = j1(u2*a) * y1(u2*b) - y1(u2*a) * j1(u2*b)
Ps = (jvp(1, u2*a) * y1(u2*b) - yvp(1, u2*a) * j1(u2*b)) / (u2 * a)
return (i * P + Ps) * (n12 * i * P + n22 * Ps) - n32 * X * X * P * P
def Er(self, r, z):
"""Radial field component dV/dr"""
r /= self.r0
z /= self.r0
s = 0
for (n,c) in enumerate(self.cns):
if n==0:
continue
s += self.cns[n]*exp(-bessel_j0n[n]*z)*bessel_j0n[n]*special.jvp(0,bessel_j0n[n]*r)
return s/self.r0
def test_bessel():
x = np.linspace(0, 20, 100)
for n in range(2):
plt.plot(x, jn(n, x), 'r-')
plt.plot(x, jvp(n, x), 'b-')
#plt.ylim(0,3.5)
plt.show()
def test_cylindrical_bessel_j_prime():
order = np.random.randint(0, 10)
value = np.random.rand(1).item()
assert (roundScaler(NumCpp.bessel_jn_prime_Scaler(order, value),
NUM_DECIMALS_ROUND) == roundScaler(
sp.jvp(order, value).item(), NUM_DECIMALS_ROUND))
shapeInput = np.random.randint(20, 100, [
2,
])
shape = NumCpp.Shape(shapeInput[0].item(), shapeInput[1].item())
cArray = NumCpp.NdArray(shape)
order = np.random.randint(0, 10)
data = np.random.rand(shape.rows, shape.cols)
cArray.setArray(data)
assert np.array_equal(
roundArray(NumCpp.bessel_jn_prime_Array(order, cArray),
NUM_DECIMALS_ROUND),
roundArray(sp.jvp(order, data), NUM_DECIMALS_ROUND))
def cutoffHE1(b):
a = rho * b
i = ivp(1, u1 * a) / (u1 * a * i1(u1 * a))
X = (1 / (u1 * a)**2 + 1 / (u2 * a)**2)
P = j1(u2 * a) * y1(u2 * b) - y1(u2 * a) * j1(u2 * b)
Ps = (jvp(1, u2 * a) * y1(u2 * b) - yvp(1, u2 * a) * j1(u2 * b)) / (u2 * a)
return (i * P + Ps) * (n12 * i * P + n22 * Ps) - n32 * X * X * P * P
def hybrid_mode_HE_graphical(r_0, h_0, log_n, f0_0):
f0 = f0_0 * 1e12
r = r_0 * 1e-6
h = h_0 * 1e-6
epsilon_0 = 1
epsilon_2 = epsilon_0
e1, e2 = drude_model_Si(log_n, f0)
# e1,e2 = drude_lorentz_model_Si(11.68,0,0,f0)
epsilon_1 = e1
c = 2.99792458e8
lam = c / f0
k0 = 2 * pi / lam
n = 1
m = 0
beta = pi / h
alpha = sqrt(k0**2 * epsilon_1 - beta**2)
gamma1 = alpha
gamma2 = sqrt(beta**2 - k0**2 * epsilon_0)
u = gamma1 * r
v = gamma2 * r
# eta1 = jvp(1,u)/(u*j1(u)) + kvp(1,v)/(v*k1(v))
# eta2 = (epsilon_1*k0**2)*jvp(1, u) / (u * j1(u)) + (k0**2)*kvp(1, v) / (v * k1(v))
eta1 = jvp(n - 1, u) / (u * jvp(n, u)) - n / u**2
eta2 = 1j * hankel1(n - 1, 1, 1j * v) / v / hankel1(n, 1,
1j * v) - n / v**2
f_res_1 = abs(
(eta1 + eta2) * (k0**2 * epsilon_1 * eta1 + k0**2 * epsilon_2 * eta2))
f_res_2 = (n**2 * beta**2 * (1 / u**2 + 1 / v**2)**2)
# eta1 = jvp(n-1,u)/(u*jvp(n,u)) + hankel1(n-1,v)/(v*hankel1(n,v))
# eta2 = (epsilon_1*k0**2)*jvp(n-1, u) / (u * jvp(n,u)) + (k0**2)*hankel1(n-1, v) / (v * hankel1(n,v))
# eta3 = (beta**2) * (1/u**2 + 1/v**2)**2
#
# f_res_1 = abs(eta1*eta2)
# f_res_2 = eta3
return f_res_1, f_res_2
def integrand_nopoles(a, p_bar, nu):
#minimum value of a to keep p_perp real; equation 27
a_min = (nu / nu_c) * (1. - np.cos(theta) * p_bar) / np.sqrt(1. - p_bar**2.)
#check if a > a_min
if(a <= a_min):
return 0.
#gamma in terms of a, p_bar; equation 23
gamma = (nu_c / nu) * a / (1. - np.cos(theta) * p_bar)
#p_parallel, equation 21
p_parallel = gamma * m * c * p_bar
#p_perp, equation 22
p_perp = m * c * np.sqrt(gamma**2. * (1. - p_bar**2.) - 1.)
#this is gamma as a function of p_perp and p_parallel, equation 28
gamma_check = np.sqrt((p_perp/(m * c))**2. + (p_parallel/(m * c))**2. + 1.)
#Omega, equation 8
Omega = omega_c / gamma
omega = 2. * np.pi * nu
#z, equation 5
z = omega * np.sin(theta) * p_perp / (gamma * m * c * Omega)
#M, equation 6
M = (np.cos(theta) - (p_parallel / (gamma * m * c))) / np.sin(theta)
#N, equation 7
N = p_perp / (gamma * m * c)
#bessel function term in integrand (eq. 16); note that the 1/sin(pi * a) is taken out
# so that we can use the CPV integration method
bessel_term = np.pi * special.jv(a, z) * special.jvp(-a, z)
#jv'(-a, z) blows up with a, and jv(a, z) goes to zero;
#they limit to -1/(pi*z) * sin(pi*a) as a -> inf
if(np.isnan(bessel_term) == True):
bessel_term = - 1./z * np.sin(np.pi * a)
jacobian = - m**3. * c**3. * (nu_c / nu) * gamma**2. / (p_perp * (1. - np.cos(theta) * p_bar))
#dp_perp * dp_parallel from equation
dp_perp_dp_para = jacobian #*dadp_bar, which is notation we don't use for a numerical integral
#d^3p from equation 26
d3p = 2. * np.pi * p_perp * dp_perp_dp_para
ans = M * N / Omega * Df(gamma, nu) * bessel_term * d3p
return ans
def far_fields(radius, frequency, r, theta, phi):
"""
Calculate the electric and magnetic fields in the far field of the aperture.
:param r: The range to the field point (m).
:param theta: The theta angle to the field point (rad).
:param phi: The phi angle to the field point (rad).
:param radius: The radius of the aperture (m).
:param frequency: The operating frequency (Hz).
:return: The electric and magnetic fields radiated by the aperture (V/m), (A/m).
"""
# Calculate the wavenumber
k = 2.0 * pi * frequency / c
# Calculate the wave impedance
eta = sqrt(mu_0 / epsilon_0)
# Calculate the argument for the Bessel function
z = k * radius * sin(theta)
# Calculate the Bessel function
bessel_term1 = 0.5 * ones_like(z)
index = z != 0.0
bessel_term1[index] = jv(1, z[index]) / z[index]
# Calculate the Bessel function
jnpz = 1.84118378134065
denominator = (1.0 - (z / jnpz)**2)
index = denominator != 0.0
bessel_term2 = 0.377 * ones_like(z)
bessel_term2[index] = jvp(1, z[index]) / denominator[index]
# Define the radial-component of the electric far field (V/m)
e_r = 0.0
# Define the theta-component of the electric far field (V/m)
e_theta = 1j * k * radius * jv(1, jnpz) * exp(
-1j * k * r) / r * sin(phi) * bessel_term1
# Define the phi-component of the electric far field (V/m)
e_phi = 1j * k * radius * jv(1, jnpz) * exp(
-1j * k * r) / r * cos(theta) * cos(phi) * bessel_term2
# Define the radial-component of the magnetic far field (A/m)
h_r = 0.0
# Define the theta-component of the magnetic far field (A/m)
h_theta = 1j * k * radius * jv(1, jnpz) * exp(
-1j * k * r) / r * -cos(theta) * cos(phi) / eta * bessel_term2
# Define the phi-component of the magnetic far field (A/m)
h_phi = 1j * k * radius * jv(1, jnpz) * exp(
-1j * k * r) / r * sin(phi) / eta * bessel_term1
# Return all six components of the far field
return e_r, e_theta, e_phi, h_r, h_theta, h_phi
def EH_fields(self, ri, ro, nu, neff, wl, EH, tm=True):
"""
modify EH in-place (for speed)
"""
n = self.maxIndex(wl)
u = self.u(ro, neff, wl)
if ri == 0:
if nu == 0:
if tm:
self.C = numpy.array([1., 0., 0., 0.])
else:
self.C = numpy.array([0., 0., 1., 0.])
else:
self.C = numpy.zeros((4, 2))
self.C[0, 0] = 1 # Ez = 1
self.C[2, 1] = 1 # Hz = alpha
elif nu == 0:
self.C = numpy.zeros(4)
if tm:
c = constants.Y0 * n * n
idx = (0, 3)
self.C[:2] = self.tetmConstants(ri, ro, neff, wl, EH, c, idx)
else:
c = -constants.eta0
idx = (1, 2)
self.C[2:] = self.tetmConstants(ri, ro, neff, wl, EH, c, idx)
else:
self.C = self.vConstants(ri, ro, neff, wl, nu, EH)
# Compute EH fields
if neff < n:
c1 = wl.k0 * ro / u
F3 = jvp(nu, u) / jn(nu, u)
F4 = yvp(nu, u) / yn(nu, u)
else:
c1 = -wl.k0 * ro / u
F3 = ivp(nu, u) / iv(nu, u)
F4 = kvp(nu, u) / kn(nu, u)
c2 = neff * nu / u * c1
c3 = constants.eta0 * c1
c4 = constants.Y0 * n * n * c1
EH[0] = self.C[0] + self.C[1]
EH[1] = self.C[2] + self.C[3]
EH[2] = (c2 * (self.C[0] + self.C[1]) - c3 *
(F3 * self.C[2] + F4 * self.C[3]))
EH[3] = (c4 * (F3 * self.C[0] + F4 * self.C[1]) - c2 *
(self.C[2] + self.C[3]))
return EH
def EH_fields(self, ri, ro, nu, neff, wl, EH, tm=True):
"""
modify EH in-place (for speed)
"""
n = self.maxIndex(wl)
u = self.u(ro, neff, wl)
if ri == 0:
if nu == 0:
if tm:
self.C = numpy.array([1., 0., 0., 0.])
else:
self.C = numpy.array([0., 0., 1., 0.])
else:
self.C = numpy.zeros((4, 2))
self.C[0, 0] = 1 # Ez = 1
self.C[2, 1] = 1 # Hz = alpha
elif nu == 0:
self.C = numpy.zeros(4)
if tm:
c = constants.Y0 * n * n
idx = (0, 3)
self.C[:2] = self.tetmConstants(ri, ro, neff, wl, EH, c, idx)
else:
c = -constants.eta0
idx = (1, 2)
self.C[2:] = self.tetmConstants(ri, ro, neff, wl, EH, c, idx)
else:
self.C = self.vConstants(ri, ro, neff, wl, nu, EH)
# Compute EH fields
if neff < n:
c1 = wl.k0 * ro / u
F3 = jvp(nu, u) / jn(nu, u)
F4 = yvp(nu, u) / yn(nu, u)
else:
c1 = -wl.k0 * ro / u
F3 = ivp(nu, u) / iv(nu, u)
F4 = kvp(nu, u) / kn(nu, u)
c2 = neff * nu / u * c1
c3 = constants.eta0 * c1
c4 = constants.Y0 * n * n * c1
EH[0] = self.C[0] + self.C[1]
EH[1] = self.C[2] + self.C[3]
EH[2] = (c2 * (self.C[0] + self.C[1]) -
c3 * (F3 * self.C[2] + F4 * self.C[3]))
EH[3] = (c4 * (F3 * self.C[0] + F4 * self.C[1]) -
c2 * (self.C[2] + self.C[3]))
return EH
def order1(self, ps, v0_in):
"""
Equation assuming O(1) periodic input
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
pso = ps.copy()
pso.real = False
#Assume only fundamental harmonic and DC at input
a0 = v0_in[0]
a1,theta1 = np.abs(v0_in[1]), np.angle(v0_in[1])
#Jacobi-Anger expansion
k = pso.ms
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))
bp1 = special.jvp(k, self.v1*a1) * np.exp(1j*self.v1*a0 + 1j*k*(theta1+0.5*pi))
bp2 = special.jvp(k, self.v2*a1) * np.exp(1j*self.phi + 1j*self.v2*a0 + 1j*k*(theta1+0.5*pi))
v0_out = self.eloss*self.Em*(b1 + self.a*b2)
#Flag and store the O1 solution
self.O1_solution = (ps,v0_in,b1,b2,bp1,bp2)
self.O1_complete = True
return pso, v0_out
def norm(self, w, h, alpha, a, b):
pol, n, m = alpha
en = 1 if n == 0 else 2
if self.clad(w).real < -1e6:
radius = self.r
if pol == 'E':
u = jnp_zeros(n, m)[-1]
jnu = jv(n, u)
jnpu = 0.0
else:
u = jn_zeros(n, m)[-1]
jnu = 0.0
jnpu = jvp(n, u)
return np.sqrt(a**2 * np.pi * radius**2 / en *
(1 - n**2 / u**2) * jnu**2 +
b**2 * np.pi * radius**2 / en * jnpu**2)
# ac = a.conjugate()
# bc = b.conjugate()
u = self.samples.u(h**2, w, self.fill(w))
jnu = jv(n, u)
jnpu = jvp(n, u)
v = self.samples.v(h**2, w, self.clad(w))
knv = kv(n, v)
knpv = kvp(n, v)
# uc = u.conjugate()
# jnuc = jnu.conjugate()
# jnpuc = jnpu.conjugate()
# vc = v.conjugate()
# knvc = knv.conjugate()
# knpvc = knpv.conjugate()
val_u = 2 * np.pi * self.r**2 / en
val_v = val_u * ((u * jnu) / (v * knv))**2
# val_v = val_u * (uc * u * jnuc * jnu) / (vc * v * knvc * knv)
upart_diag = self.upart_diag(n, u, jnu, jnpu, u, jnu, jnpu)
vpart_diag = self.vpart_diag(n, v, knv, knpv, v, knv, knpv)
upart_off = self.upart_off(n, u, jnu, u, jnu)
vpart_off = self.vpart_off(n, v, knv, v, knv)
return np.sqrt(val_u * (a * (a * upart_diag + b * upart_off) + b *
(b * upart_diag + a * upart_off)) - val_v *
(a * (a * vpart_diag + b * vpart_off) + b *
(b * vpart_diag + a * vpart_off)))
def u_jnu_jnpu_pec(num_n, num_m):
us = np.empty((2, num_n, num_m))
jnus = np.empty((2, num_n, num_m))
jnpus = np.empty((2, num_n, num_m))
for n in range(num_n):
us[0, n] = jnp_zeros(n, num_m)
us[1, n] = jn_zeros(n, num_m)
jnus[0, n] = jv(n, us[0, n])
jnus[1, n] = np.zeros(num_m)
jnpus[0, n] = np.zeros(num_m)
jnpus[1, n] = jvp(n, us[1, n])
return us, jnus, jnpus
def cutoffHE2(b):
nu = 2
a = rho * b
i = ivp(2, u1 * a) / (u1 * a * iv(2, u1 * a))
X = (1 / (u1 * a)**2 + 1 / (u2 * a)**2)
P1 = jn(nu, u2 * a) * yn(nu, u2 * b) - yn(nu, u2 * a) * jn(nu, u2 * b)
P2 = (jvp(nu, u2 * a) * yn(nu, u2 * b) -
yvp(nu, u2 * a) * jn(nu, u2 * b)) / (u2 * a)
P3 = (jn(nu, u2 * a) * yvp(nu, u2 * b) -
yn(nu, u2 * a) * jvp(nu, u2 * b)) / (u2 * b)
P4 = (jvp(nu, u2 * a) * yvp(nu, u2 * b) -
yvp(nu, u2 * a) * jvp(nu, u2 * b)) / (u2 * a * u2 * b)
A = 2 * nu / (u2 * b)**2 - 1 / (nu - 1)
return (n22 / n32 * (i * P1 + P2) * (n12 * i * P3 + n22 * P4) +
(A * i * P1 + A * P2 + i * P3 + P4) * (n12 * i * P1 + n22 * P2) -
n32 * nu**2 * X**2 * P1 * (A * P1 + P3) - n22 * nu * X *
(nu * X * P1 * P3 + 2 * P1 * P4 - 2 * P2 * P3))
def effective_coated_N1(self, filling, polarization, orders=1):
'''
Calculate effective medium parameters using the
coated cylinder model
'''
R = self.radius
k = self.k_host
mu_h = 1.0
R2 = R / np.sqrt(filling)
#aa0 = self.coeff(1,polarization)
sum_ = 0.0 + 0j
if polarization in ['H', 'TE']:
if orders < 1: orders = 1
for m in np.arange(1, orders + 1):
# bessel functions
J0 = special.jv(m, k * R2)
H0 = special.hankel1(m, k * R2)
JD0 = special.jvp(m, k * R2)
HD0 = special.h1vp(m, k * R2)
aa1 = self.coeff(m, 'TE')
denom = JD0 + HD0 * aa1 + 0j
numer = (J0 + H0 * aa1) + 0j
sum_ = sum_ + m * numer / denom
term = self.host_eps / (k * R2)
return term * sum_ + 0j
else:
for m in np.arange(1, orders + 1):
J0 = special.jv(m, k * R2)
H0 = special.hankel1(m, k * R2)
JD0 = special.jvp(m, k * R2)
HD0 = special.h1vp(m, k * R2)
aa1 = self.coeff(m, 'TM')
denom = JD0 + HD0 * aa1 + 0j
numer = (J0 + H0 * aa1) + 0j
sum_ = sum_ + m * numer / denom
term = self.host_mu / (k * R2)
return term * sum_ + 0j
def cyl_modal_coefs(N, kr, arrayType):
"""
Modal coefficients for rigid or open cylindrical array
Parameters
----------
N : int
Maximum spherical harmonic expansion order.
kr: ndarray
Wavenumber-radius product. Dimension = (l).
arrayType: str
'open' or 'rigid'.
Returns
-------
b_N : ndarray
Modal coefficients. Dimension = (l, N+1)
Raises
-----
TypeError, ValueError: if method arguments mismatch in type, dimension or value.
Notes
-----
The `arrayType` options are:
- 'open' for open array of omnidirectional sensors
- 'rigid' for sensors mounted on a rigid baffle.
"""
_validate_int('N', N)
_validate_ndarray_1D('kr', kr, positive=True)
_validate_string('arrayType', arrayType, choices=['open', 'rigid'])
b_N = np.zeros((kr.size, N + 1), dtype='complex')
for n in range(N + 1):
if arrayType is 'open':
b_N[:, n] = np.power(1j, n) * jv(n, kr)
elif arrayType is 'rigid':
jn = jv(n, kr)
jnprime = jvp(n, kr, 1)
hn = hankel2(n, kr)
hnprime = h2vp(n, kr, 1)
temp = np.power(1j, n) * (jn - (jnprime / hnprime) * hn)
temp[np.where(kr == 0)] = 1 if n == 0 else 0.
b_N[:, n] = temp
return b_N
def linton_evans_coeffs(k, incAng, numCylinders, centres, radii, M):
# Four-cylinder problem
#if k<=20:
# M=40
#elif 20<k<=40:
# M=80
#elif 40<k<=60:
# M=120
#elif 60<k<=120:
# M=180
#elif k>120:
# M=300
N = numCylinders
origin = centres
a = radii
i = np.arange(0, N).reshape(-1, 1)
j = np.arange(0, N).reshape(1, -1)
R = np.sqrt((origin[j, 0] - origin[i, 0])**2 +
(origin[j, 1] - origin[i, 1])**2)
alpha = np.angle((origin[j, 0] - origin[i, 0]) + 1j *
(origin[j, 1] - origin[i, 1]))
q = np.repeat(np.arange(N), 2 * M + 1).reshape(-1, 1) # rows
p = np.repeat(np.arange(N), 2 * M + 1).reshape(1, -1) # cols
m = np.repeat([np.arange(-M, M + 1)], N, axis=0).reshape(-1, 1) # rows
n = np.repeat([np.arange(-M, M + 1)], N, axis=0).reshape(1, -1) # cols
# Right-hand vector
Iq = np.exp(
1j * k *
(origin[q, 0] * np.cos(incAng) + origin[q, 1] * np.sin(incAng)))
exponential1 = np.exp(1j * m * (np.pi / 2 - incAng))
RHS = -Iq * exponential1
# Left-hand side matrix
Znp = jvp(n, k * a[p]) / h1vp(n, k * a[p])
exponential2 = np.exp(1j * (n - m) * alpha[p, q])
Hnm = hankel1(n - m, k * R[p, q])
LHS = Znp * exponential2 * Hnm
Identity = np.identity(2 * M + 1)
for cyl in range(N):
LHS[cyl * (2 * M + 1):(cyl + 1) * (2 * M + 1),
cyl * (2 * M + 1):(cyl + 1) * (2 * M + 1)] = Identity
# Solve for values of Anq
A = np.linalg.solve(LHS, RHS)
return A
def fb_1_grad(i, k, x, y, x0=0, y0=0):
X = x - x0
Y = y - y0
r = np.sqrt(X**2 + Y**2)
angle = np.arctan2(Y, X)
idx = i + 1
j = special.jv(idx, k * r)
dj = special.jvp(idx, k * r)
c = np.cos(idx * angle)
s = np.sin(idx * angle)
dx = dj * k * X / r * s - j * c * Y / (X**2 + Y**2) * idx
dy = dj * k * Y / r * s + j * c * X / (X**2 + Y**2) * idx
return dx, dy
def _ehceq(self, neff, wl, nu):
u, w = self._uw(wl, neff)
v2 = u * u + w * w
nco = self.fiber.maxIndex(0, wl)
ncl = self.fiber.minIndex(1, wl)
delta = (1 - ncl**2 / nco**2) / 2
jnu = jn(nu, u)
knu = kn(nu, w)
kp = kvp(nu, w)
return (jvp(nu, u) * w * knu + kp * u * jnu * (1 - delta) -
jnu * sqrt((u * kp * delta)**2 + ((nu * neff * v2 * knu) /
(nco * u * w))**2))
def fb_ca_dk(i, k, x, y, nu=1, x0=0, y0=0, phi0=0, sym=None):
X = x - x0
Y = y - y0
r = np.sqrt(X**2 + Y**2)
#r[r==0] = 1e-16
angle = np.arctan2(Y, X) - phi0
if sym == None:
idx = i + 1
if sym == "odd":
idx = 2 * i + 1
if sym == "even":
idx = 2 * i + 2
s = np.sin(nu * idx * angle)
return r * special.jvp(nu * idx, k * r) * s
def eigensystem(b, m, ri, ro):
r""" Computes the function associated with the eigensystem of the
convected wave equation. The location of the zeros for this function
correspond to the eigenvalues for the convected wave equation.
The solution to the Bessel equation with boundary conditions applied
yields a system of two linear equations.
.. math::
A x =
\begin{bmatrix}
J'_m(\mu \lambda) & Y'_m(\mu \lambda) \\
J'_m(\mu) & Y'_m(\mu)
\end{bmatrix}
\begin{bmatrix}
x_1 \\ x_2
\end{bmatrix}
= 0
This procedure evaluates the function
.. math::
det(A) = f(b) = J_m(b*ri)*Y_m(b*ro) - J_m(b*ro)*Y_m(b*ri)
So, this procedure can be passed to another routine such as numpy
to find zeros.
Args:
b (float): coordinate.
m (int): circumferential wave number.
ri (float): inner radius of a circular annulus.
ro (float): outer radius of a circular annulus.
"""
f = sp.jvp(m, b * ri) * sp.yvp(m, b * ro) - sp.jvp(m, b * ro) * sp.yvp(
m, b * ri)
return f
def _ehceq(self, neff, wl, nu):
u, w = self._uw(wl, neff)
v2 = u*u + w*w
nco = self.fiber.maxIndex(0, wl)
ncl = self.fiber.minIndex(1, wl)
delta = (1 - ncl**2 / nco**2) / 2
jnu = jn(nu, u)
knu = kn(nu, w)
kp = kvp(nu, w)
return (jvp(nu, u) * w * knu +
kp * u * jnu * (1 - delta) -
jnu * sqrt((u * kp * delta)**2 +
((nu * neff * v2 * knu) /
(nco * u * w))**2))
def coeff_int(self, order, polarization, freq_index=None):
#if self.frequency == None:
# raise Exception("ERROR: Frequency range has not been specified.")
# for backward compatibility only
mu1 = self.host_mu
mu2 = self.cyl_mu
if freq_index >= 0:
z1 = self.kR_host[freq_index]
z2 = self.kR_cyl[freq_index]
k0 = self.k0[freq_index]
kR_cyl = self.kR_cyl[freq_index]
kR_host = self.kR_host[freq_index]
else:
z1 = self.kR_host
z2 = self.kR_cyl
k0 = self.k0
kR_cyl = self.kR_cyl
kR_host = self.kR_host
bn = self.coeff(order, polarization, freq_index)
### POLARIZATION
if polarization in ['TE', 'H']:
JD1 = special.jvp(order, z1)
JD2 = special.jvp(order, z2)
HD1 = special.h1vp(order, z1)
numer = HD1 * bn + JD1
denom = JD2
return numer / denom
elif polarization in ['TM', 'E']:
J2 = special.jv(order, z2)
J1 = special.jv(order, z1)
H1 = special.hankel1(order, z1)
numer = H1 * bn + J1
denom = J2
return numer / denom
def JnYn(maxn, resolution):
delta = numpy.pi / resolution
z_table = numpy.mgrid[min(minz_j(maxn), minz_y(maxn)) : max(maxz_j(maxn), maxz_y(maxn)) : delta]
J_table = []
Jdot_table = []
Y_table = []
Ydot_table = []
for n in range(maxn + 1):
J_table.append(special.jn(n, z_table))
Jdot_table.append(special.jvp(n, z_table))
Y_table.append(special.yn(n, z_table))
Ydot_table.append(special.yvp(n, z_table))
return z_table, J_table, Jdot_table, Y_table, Ydot_table
def Bescoef(k,r,a,n):
"""
The calculates the Bessel and Hankel function coef for the
incident and scattered wave solution.
spec.jv(v,z) - Bessel Func of the first kind of order v
spec.jvp(v,z) - first derivative of BF of the first kinda of order v
spec.h1vp(v,z) - first derivative of Hankel Func of order v
spec.hankel1(v,z) - Hankel func of the first kind of order v
"""
kr , ka = k*r , k*a
coef = -(spec.jvp(n,ka,n=1)/spec.h1vp(n,ka,n=1))*spec.hankel1(n,kr)
return coef
def linton_evans_coeffs(k,incAng,numCylinders,centres,radii,M):
# Four-cylinder problem
#if k<=20:
# M=40
#elif 20<k<=40:
# M=80
#elif 40<k<=60:
# M=120
#elif 60<k<=120:
# M=180
#elif k>120:
# M=300
N = numCylinders
origin = centres
a = radii
i = np.arange(0,N).reshape(-1,1)
j = np.arange(0,N).reshape(1,-1)
R = np.sqrt((origin[j,0]-origin[i,0])**2+(origin[j,1]-origin[i,1])**2)
alpha = np.angle((origin[j,0]-origin[i,0]) +1j*(origin[j,1]-origin[i,1]))
q = np.repeat(np.arange(N),2*M+1).reshape(-1,1) # rows
p = np.repeat(np.arange(N),2*M+1).reshape(1,-1) # cols
m = np.repeat([np.arange(-M,M+1)],N,axis=0).reshape(-1,1) # rows
n = np.repeat([np.arange(-M,M+1)],N,axis=0).reshape(1,-1) # cols
# Right-hand vector
Iq = np.exp(1j*k*(origin[q,0]*np.cos(incAng)+origin[q,1]*np.sin(incAng)))
exponential1 = np.exp(1j*m*(np.pi/2-incAng))
RHS = - Iq * exponential1
# Left-hand side matrix
Znp = jvp(n,k*a[p]) / h1vp(n,k*a[p])
exponential2 = np.exp(1j*(n-m)*alpha[p,q])
Hnm = hankel1(n-m,k*R[p,q])
LHS = Znp * exponential2 * Hnm
Identity = np.identity(2*M+1)
for cyl in range(N):
LHS[cyl*(2*M+1):(cyl+1)*(2*M+1),cyl*(2*M+1):(cyl+1)*(2*M+1)] = Identity
# Solve for values of Anq
A=np.linalg.solve(LHS,RHS)
return A
def _hefield(self, wl, nu, neff, r):
rho = self.fiber.outerRadius(0)
k = wl.k0
nco2 = self.fiber.maxIndex(0, wl)**2
ncl2 = self.fiber.minIndex(1, wl)**2
u = rho * k * sqrt(nco2 - neff**2)
w = rho * k * sqrt(neff**2 - ncl2)
v = rho * k * sqrt(nco2 - ncl2)
jnu = jn(nu, u)
knw = kn(nu, w)
Delta = (1 - ncl2/nco2)/2
b1 = jvp(nu, u) / (u * jnu)
b2 = kvp(nu, w) / (w * knw)
F1 = (u * w / v)**2 * (b1 + (1 - 2 * Delta)*b2) / nu
F2 = (v / (u * w))**2 * nu / (b1 + b2)
a1 = (F2 - 1) / 2
a2 = (F2 + 1) / 2
a3 = (F1 - 1) / 2
a4 = (F1 + 1) / 2
a5 = (F1 - 1 + 2 * Delta) / 2
a6 = (F1 + 1 - 2 * Delta) / 2
if r < rho:
jmur = jn(nu-1, u * r / rho)
jpur = jn(nu+1, u * r / rho)
jnur = jn(nu, u * r / rho)
er = -(a1 * jmur + a2 * jpur) / jnu
ephi = -(a1 * jmur - a2 * jpur) / jnu
ez = u / (k * neff * rho) * jnur / jnu
hr = Y0 * nco2 / neff * (a3 * jmur - a4 * jpur) / jnu
hphi = -Y0 * nco2 / neff * (a3 * jmur + a4 * jpur) / jnu
hz = Y0 * u * F2 / (k * rho) * jnur / jnu
else:
kmur = kn(nu-1, w * r / rho)
kpur = kn(nu+1, w * r / rho)
knur = kn(nu, w * r / rho)
er = -u / w * (a1 * kmur - a2 * kpur) / knw
ephi = -u / w * (a1 * kmur + a2 * kpur) / knw
ez = u / (k * neff * rho) * knur / knw
hr = Y0 * nco2 / neff * u / w * (a5 * kmur + a6 * kpur) / knw
hphi = -Y0 * nco2 / neff * u / w * (a5 * kmur - a6 * kpur) / knw
hz = Y0 * u * F2 / (k * rho) * knur / knw
return numpy.array((er, ephi, ez)), numpy.array((hr, hphi, hz))
def coef(self, h, w, alpha):
"""Return the coefficients of TE- and TM- components which compose
the hybrid mode.
Args:
h: A complex indicating the phase constant.
w: A complex indicating the angular frequency
alpha: A tuple (pol, n, m) where pol is 'M' for TM-like mode or
'E' for TE-like mode, n is the order of the mode, and m is
the number of modes in the order and the polarization.
Returns:
a: A complex indicating the coefficient of TE-component
b: A complex indicating the coefficient of TM-component
"""
e1 = self.fill(w)
e2 = self.clad(w)
pol, n, m = alpha
w = w.real + 1j * w.imag
h = h.real + 1j * h.imag
if e2.real < -1e6:
if pol == 'E':
norm = self.norm(w, h, alpha, 1.0 + 0.0j, 0.0j)
ai, bi = 1.0 / norm, 0.0
else:
norm = self.norm(w, h, alpha, 0.0j, 1.0 + 0.0j)
ai, bi = 0.0, 1.0 / norm
else:
u = self.samples.u(h**2, w, e1)
v = self.samples.v(h**2, w, e2)
knv = kv(n, v)
knpv = kvp(n, v)
jnu = jv(n, u)
jnpu = jvp(n, u)
ci = -n * (u**2 + v**2) * jnu * knv / (u * v)
if pol == 'E':
ci *= (h / w)**2
ci /= (e1 * jnpu * v * knv + e2 * knpv * u * jnu)
norm = self.norm(w, h, alpha, 1.0 + 0.0j, ci)
ai = 1.0 / norm
bi = ci / norm
else:
ci /= (jnpu * v * knv + knpv * u * jnu)
norm = self.norm(w, h, alpha, ci, 1.0 + 0.0j)
bi = 1.0 / norm
ai = ci / norm
return ai, bi
def _hefield(self, wl, nu, neff, r):
rho = self.fiber.outerRadius(0)
k = wl.k0
nco2 = self.fiber.maxIndex(0, wl)**2
ncl2 = self.fiber.minIndex(1, wl)**2
u = rho * k * sqrt(nco2 - neff**2)
w = rho * k * sqrt(neff**2 - ncl2)
v = rho * k * sqrt(nco2 - ncl2)
jnu = jn(nu, u)
knw = kn(nu, w)
Delta = (1 - ncl2 / nco2) / 2
b1 = jvp(nu, u) / (u * jnu)
b2 = kvp(nu, w) / (w * knw)
F1 = (u * w / v)**2 * (b1 + (1 - 2 * Delta) * b2) / nu
F2 = (v / (u * w))**2 * nu / (b1 + b2)
a1 = (F2 - 1) / 2
a2 = (F2 + 1) / 2
a3 = (F1 - 1) / 2
a4 = (F1 + 1) / 2
a5 = (F1 - 1 + 2 * Delta) / 2
a6 = (F1 + 1 - 2 * Delta) / 2
if r < rho:
jmur = jn(nu - 1, u * r / rho)
jpur = jn(nu + 1, u * r / rho)
jnur = jn(nu, u * r / rho)
er = -(a1 * jmur + a2 * jpur) / jnu
ephi = -(a1 * jmur - a2 * jpur) / jnu
ez = u / (k * neff * rho) * jnur / jnu
hr = Y0 * nco2 / neff * (a3 * jmur - a4 * jpur) / jnu
hphi = -Y0 * nco2 / neff * (a3 * jmur + a4 * jpur) / jnu
hz = Y0 * u * F2 / (k * rho) * jnur / jnu
else:
kmur = kn(nu - 1, w * r / rho)
kpur = kn(nu + 1, w * r / rho)
knur = kn(nu, w * r / rho)
er = -u / w * (a1 * kmur - a2 * kpur) / knw
ephi = -u / w * (a1 * kmur + a2 * kpur) / knw
ez = u / (k * neff * rho) * knur / knw
hr = Y0 * nco2 / neff * u / w * (a5 * kmur + a6 * kpur) / knw
hphi = -Y0 * nco2 / neff * u / w * (a5 * kmur - a6 * kpur) / knw
hz = Y0 * u * F2 / (k * rho) * knur / knw
return numpy.array((er, ephi, ez)), numpy.array((hr, hphi, hz))
def JnYn(maxn, resolution):
delta = numpy.pi / resolution
z_table = numpy.mgrid[min(minz_j(maxn),minz_y(maxn)):
max(maxz_j(maxn), maxz_y(maxn)):delta]
J_table = []
Jdot_table = []
Y_table = []
Ydot_table = []
for n in range(maxn+1):
J_table.append(special.jn(n, z_table))
Jdot_table.append(special.jvp(n, z_table))
Y_table.append(special.yn(n, z_table))
Ydot_table.append(special.yvp(n, z_table))
return z_table, J_table, Jdot_table, Y_table, Ydot_table
def calc_bessel(x, y, radius, gradr=False):
rho = (x**2 + y**2)**0.5
zero = special.jn_zeros(0, 1)
# See RT2 p75 for derivation
norm_fac = np.pi**0.5*special.jv(1, zero)*radius
z = rho*zero/radius
in_beam = rho <= radius
E = pyfftw.byte_align(special.jv(0, z)*in_beam/norm_fac)
if gradr:
gradrhoE = special.jvp(0, z, 1)*in_beam/norm_fac*zero/radius
gradxE = pyfftw.byte_align(mathx.divide0(gradrhoE*x, rho))
gradyE = pyfftw.byte_align(mathx.divide0(gradrhoE*y, rho))
return E, (gradxE, gradyE)
else:
return E
def jac(self, h2, args):
"""Return Jacobian of the characteristic equation
Args:
h2: A complex indicating the square of the phase constant.
args: A tuple (w, n, e1, e2), where w indicates the angular
frequency, n indicates the order of the modes, e1 indicates
the permittivity of the core, and e2 indicates the permittivity
of the clad.
Returns:
val: A complex indicating the Jacobian of the characteristic
equation.
"""
w, pol, n, e1, e2 = args
h2comp = h2.real + 1j * h2.imag
u = self.u(h2comp, w, e1)
v = self.v(h2comp, w, e2)
jus = jv(n, u)
jpus = jvp(n, u)
kvs = kv(n, v)
kpvs = kvp(n, v)
du_dh2 = -self.r / (2 * u)
dv_dh2 = self.r / (2 * v)
te = jpus / u + kpvs * jus / (v * kvs)
dte_du = (-(u * (1 - n**2 / u**2) * jus + 2 * jpus) / u**2 +
jpus * kpvs / (v * kvs))
dte_dv = jus * (n**2 * kvs**2 / v + v *
(kvs**2 - kpvs**2) - 2 * kvs * kpvs) / (v**2 * kvs**2)
tm = e1 * jpus / u + e2 * kpvs * jus / (v * kvs)
dtm_du = e1 * dte_du
dtm_dv = e2 * dte_dv
if n == 0:
if pol == 'M':
val = dtm_du * du_dh2 + dtm_dv * dv_dh2
else:
val = dte_du * du_dh2 + dte_dv * dv_dh2
else:
dre_dh2 = -(n / w)**2 * jus * (jus * (
(1 / u**2 + 1 / v**2)**2 - self.r * h2comp *
(1 / u**4 - 1 / v**4)) + jpus * 2 * h2comp *
(1 / u**2 + 1 / v**2)**2)
val = ((dte_du * du_dh2 + dte_dv * dv_dh2) * tm +
(dtm_du * du_dh2 + dtm_dv * dv_dh2) * te + dre_dh2)
return val
def func_jac(self, h2, *args):
"""Return the value and Jacobian of the characteristic equation
Args:
h2: A complex indicating the square of the phase constant.
Returns:
val: 2 complexes indicating the left-hand value and Jacobian
of the characteristic equation.
"""
w, pol, n, e1, e2 = args
h2comp = h2.real + 1j * h2.imag
u = self.u(h2comp, w, e1)
v = self.v(h2comp, w, e2)
jus = jv(n, u)
jpus = jvp(n, u)
kvs = kv(n, v)
kpvs = kvp(n, v)
du_dh2 = -self.r / (2 * u)
dv_dh2 = self.r / (2 * v)
te = jpus / u + kpvs * jus / (v * kvs)
dte_du = (-(u * (1 - n**2 / u**2) * jus + 2 * jpus) / u**2 +
jpus * kpvs / (v * kvs))
dte_dv = jus * (n**2 * kvs**2 / v + v *
(kvs**2 - kpvs**2) - 2 * kvs * kpvs) / (v**2 * kvs**2)
tm = e1 * jpus / u + e2 * kpvs * jus / (v * kvs)
dtm_du = e1 * dte_du
dtm_dv = e2 * dte_dv
if n == 0:
if pol == 'M':
f = tm
val = dtm_du * du_dh2 + dtm_dv * dv_dh2
else:
f = te
val = dte_du * du_dh2 + dte_dv * dv_dh2
else:
f = (tm * te - h2comp * (n / w)**2 *
((1 / u**2 + 1 / v**2) * jus)**2)
dre_dh2 = -(n / w)**2 * jus * (jus * (
(1 / u**2 + 1 / v**2)**2 - self.r * h2comp *
(1 / u**4 - 1 / v**4)) + jpus * 2 * h2comp *
(1 / u**2 + 1 / v**2)**2)
val = ((dte_du * du_dh2 + dte_dv * dv_dh2) * tm +
(dtm_du * du_dh2 + dtm_dv * dv_dh2) * te + dre_dh2)
return f, val
def fb_ca_grad(i, k, x, y, nu=1, x0=0, y0=0, phi0=0, sym=None):
X = x - x0
Y = y - y0
r = np.sqrt(X**2 + Y**2)
#r[r==0] = 1e-16
angle = np.arctan2(Y, X) - phi0
if sym == None:
idx = i + 1
if sym == "odd":
idx = 2 * i + 1
if sym == "even":
idx = 2 * i + 2
j = special.jv(idx * nu, k * r)
dj = special.jvp(idx * nu, k * r)
c = np.cos(nu * idx * angle)
s = np.sin(nu * idx * angle)
dx = dj * k * X / r * s - j * c * Y / (X**2 + Y**2) * nu * idx
dy = dj * k * Y / r * s + j * c * X / (X**2 + Y**2) * nu * idx
return dx, dy
def Y(self, w, h, alpha, a, b):
"""Return the effective admittance of the waveguide mode
Args:
w: A complex indicating the angular frequency
h: A complex indicating the phase constant.
alpha: A tuple (pol, n, m) where pol is 'M' for TM-like mode or
'E' for TE-like mode, n is the order of the mode, and m is
the number of modes in the order and the polarization.
a: A complex indicating the coefficient of TE-component
b: A complex indicating the coefficient of TM-component
Returns:
y: A complex indicating the effective admittance
"""
pol, n, m = alpha
e1 = self.fill(w)
e2 = self.clad(w)
en = 1 if n == 0 else 2
if e2.real < -1e6:
if pol == 'E':
val = h / w
else:
val = e1 * w / h
else:
u = self.samples.u(h**2, w, e1)
jnu = jv(n, u)
jnpu = jvp(n, u)
v = self.samples.v(h**2, w, e2)
knv = kv(n, v)
knpv = kvp(n, v)
val_u = 2 * np.pi * self.r**2 / en
val_v = val_u * ((u * jnu) / (v * knv))**2
upart_diag = self.upart_diag(n, u, jnu, jnpu, u, jnu, jnpu)
vpart_diag = self.vpart_diag(n, v, knv, knpv, v, knv, knpv)
upart_off = self.upart_off(n, u, jnu, u, jnu)
vpart_off = self.vpart_off(n, v, knv, v, knv)
val = (val_u * (h / w * a *
(a * upart_diag + b * upart_off) + e1 * w / h * b *
(b * upart_diag + a * upart_off)) - val_v *
(h / w * a *
(a * vpart_diag + b * vpart_off) + e2 * w / h * b *
(b * vpart_diag + a * vpart_off)))
return val
def vConstants(self, ri, ro, neff, wl, nu, EH):
a = numpy.zeros((4, 4))
n = self.maxIndex(wl)
u = self.u(ro, neff, wl)
urp = self.u(ri, neff, wl)
if neff < n:
B1 = jn(nu, u)
B2 = yn(nu, u)
F1 = jn(nu, urp) / B1
F2 = yn(nu, urp) / B2
F3 = jvp(nu, urp) / B1
F4 = yvp(nu, urp) / B2
c1 = wl.k0 * ro / u
else:
B1 = iv(nu, u)
B2 = kn(nu, u)
F1 = iv(nu, urp) / B1 if u else 1
F2 = kn(nu, urp) / B2
F3 = ivp(nu, urp) / B1 if u else 1
F4 = kvp(nu, urp) / B2
c1 = -wl.k0 * ro / u
c2 = neff * nu / urp * c1
c3 = constants.eta0 * c1
c4 = constants.Y0 * n * n * c1
a[0, 0] = F1
a[0, 1] = F2
a[1, 2] = F1
a[1, 3] = F2
a[2, 0] = F1 * c2
a[2, 1] = F2 * c2
a[2, 2] = -F3 * c3
a[2, 3] = -F4 * c3
a[3, 0] = F3 * c4
a[3, 1] = F4 * c4
a[3, 2] = -F1 * c2
a[3, 3] = -F2 * c2
return numpy.linalg.solve(a, EH)
def _tmfield(self, wl, nu, neff, r):
N = len(self.fiber)
C = numpy.array((1, 0))
EH = numpy.zeros(4)
ri = 0
for i in range(N-1):
ro = self.fiber.outerRadius(i)
layer = self.fiber.layers[i]
n = layer.maxIndex(wl)
u = layer.u(ro, neff, wl)
if i > 0:
C = layer.tetmConstants(ri, ro, neff, wl, EH,
constants.Y0 * n * n, (0, 3))
if r < ro:
break
if neff < n:
c1 = wl.k0 * ro / u
F3 = jvp(nu, u) / jn(nu, u)
F4 = yvp(nu, u) / yn(nu, u)
else:
c1 = -wl.k0 * ro / u
F3 = ivp(nu, u) / iv(nu, u)
F4 = kvp(nu, u) / kn(nu, u)
c4 = constants.Y0 * n * n * c1
EH[0] = C[0] + C[1]
EH[3] = c4 * (F3 * C[0] + F4 * C[1])
ri = ro
else:
layer = self.fiber.layers[-1]
u = layer.u(ro, neff, wl)
# C =
return numpy.array((0, ephi, 0)), numpy.array((hr, 0, hz))
def eval_d_u_outwards(self, arg, sid):
# Debug function
# Only for true arc. At least at the moment
k = self.k
R = self.R
a = self.a
nu = self.nu
r, th = domain_util.arg_to_polar_abs(self.boundary, a, arg, sid)
d_u = 0
for m in range(1, self.m_max+1):
ac = self.fft_a_coef[m-1]
if sid == 0:
d_u += ac * k * jvp(m*nu, k*r, 1) * np.sin(m*nu*(th-a))
elif sid == 1:
d_u += ac * jv(m*nu, k*r) * m * nu * np.cos(m*nu*(th-a))
elif sid == 2:
d_u += -ac * jv(m*nu, k*r) * m * nu * np.cos(m*nu*(th-a))
return d_u
def cylinder(wavenumber,xPlotPoints,yPlotPoints,radius=1.0):
""" Analytical potential on hard cylinder with [1,0] incident wave (Jones)"""
x = xPlotPoints
y = yPlotPoints
theta = np.arctan2(y,x)
numPoints = len(x)
nans=False,False
N = 100
while any(nans)==False:
N += 50
n = np.arange(0,N)
neumann = 2*np.ones(N); neumann[0]=1
dJ = jvp(n,wavenumber*radius,1)
H1 = hankel1(n,wavenumber*radius)
dH1 = h1vp(n,wavenumber*radius,1)
nans = np.isnan(dJ) + np.isnan(H1) + np.isnan(dH1)
dJ = dJ.compress(np.logical_not(nans))
H1 = H1.compress(np.logical_not(nans))
dH1 = dH1.compress(np.logical_not(nans))
neumann = neumann.compress(np.logical_not(nans))
n = n.compress(np.logical_not(nans))
total = (-neumann * (1j**n) * dJ * H1 / dH1).reshape(-1,1)
cosines = np.cos(n.reshape(-1,1)*theta)
total = total * cosines
incPotential = np.exp(1j*wavenumber*x).reshape(numPoints)
fullPotential = np.sum(total,axis=0) + incPotential
return fullPotential
def z(self,x):
'Z(x) equation that keeps showing up in waveguide modes'
m, n, chi = self.m, self.n, self.root
return yvp(m,chi,1)*jn(m,x)-jvp(m,chi,1)*yn(m,x)
def rJnp(r,n): return (0.5*sqrt(pi/(2*r))*jv(n+0.5,r) + sqrt(pi*r/2)*jvp(n+0.5,r))
def root_equation(self,m,c,x):
'''Radial root equation of Phi for TE mode'''
return yvp(m,x,1)*jvp(m,c*x,1)-jvp(m,x,1)*yvp(m,c*x,1)
def rfnd(self, n, x, d):
return ss.jvp(n,x,d)
def E_theta(self, r, theta, z):
return -(jvp(self.l,k1*r,1])/k1) * np.cos(self.l* theta)*np.sin(k3*z)
if __name__ == '__main__': mesh = [25, 50, 100, 200, 300,500,1000,2000] convergence = np.zeros((0,2)) for nPoints in mesh: y = homoCircle(nPoints, 1.0, 2.0, 1.0, 1.0, 1) scatMat = y.computeScatteringMatrix(y.Mmax) analScatMat = np.zeros(2*y.Mmax+1, dtype=complex) zc = y.nc*y.k zo = y.no*y.k eta = y.nc/y.no for i in range(2*y.Mmax+1): m = i-y.Mmax num = -(eta*jvp(m,zc)*hankel2(m,zo)-jn(m,zc)*h2vp(m,zo)) den = eta*jvp(m,zc)*hankel1(m,zo)-jn(m,zc)*h1vp(m,zo) analScatMat[i] = num/den err = np.amax(np.abs(np.diag(analScatMat)-scatMat)) print(err) print(analScatMat) # -- Mean areas of triangles convergence = np.insert(convergence, len(convergence), [np.mean(y.areas), err],axis=0) fig1 = plt.figure(figsize=(5,3)) ax1 = fig1.add_subplot(111) plt.plot(convergence[:,0],convergence[:,1], ls='--', marker='o', label=r"Numerical Error") p = np.polyfit(np.log(convergence[:,0]),np.log(convergence[:,1]),1) ynew = np.exp(np.polyval(p,np.log(convergence[:,0])))
def coeq(v0, nu, fiber):
r1, r2 = fiber.rho
n1, n2, n3 = fiber.n
na = sqrt(max(fiber.n)**2 - n3**2)
K0 = v0 / (na * r2)
beta = K0 * n3
if n1 > n3:
u1 = K0**2 * (n1**2 - n3**2)
s1 = 1
else:
u1 = K0**2 * (n3**2 - n1**2)
s1 = -1
if n2 > n3:
u2 = K0**2 * (n2**2 - n3**2)
s2 = 1
else:
u2 = K0**2 * (n3**2 - n2**2)
s2 = -1
s = -s1 * s2
u1r1 = u1 * r1
u2r1 = u2 * r1
u2r2 = u2 * r2
X = (u2r1**2 + s * u1r1**2) * nu * beta
if s1 == 1:
Y = jvp(nu, u1r1) / jn(nu, u1r1)
else:
Y = ivp(nu, u1r1) / iv(nu, u1r1)
ju2r1 = jn(nu, u2r1) if s2 == 1 else iv(nu, u2r1)
nu2r1 = yn(nu, u2r1) if s2 == 1 else kn(nu, u2r1)
jpu2r1 = jvp(nu, u2r1) if s2 == 1 else ivp(nu, u2r1)
npu2r1 = yvp(nu, u2r1) if s2 == 1 else kvp(nu, u2r1)
ju2r2 = jn(nu, u2r2) if s2 == 1 else iv(nu, u2r2)
nu2r2 = yn(nu, u2r2) if s2 == 1 else kn(nu, u2r2)
j1u2r2 = jn(nu+1, u2r2)
n1u2r2 = yn(nu+1, u2r2)
M = numpy.empty((4, 4))
M[0, 0] = X * ju2r1
M[0, 1] = X * nu2r1
M[0, 2] = -K0 * (jpu2r1 * u1r1 + s * Y * ju2r1 * u2r1) * u1r1 * u2r1
M[0, 3] = -K0 * (npu2r1 * u1r1 + s * Y * nu2r1 * u2r1) * u1r1 * u2r1
M[1, 0] = -K0 * (n2**2 * jpu2r1 * u1r1 +
s * n1**2 * Y * ju2r1 * u2r1) * u1r1 * u2r1
M[1, 1] = -K0 * (n2**2 * npu2r1 * u1r1 +
s * n1**2 * Y * nu2r1 * u2r1) * u1r1 * u2r1
M[1, 2] = X * ju2r1
M[1, 3] = X * nu2r1
D201 = nu * n3 / u2r2 * (j1u2r2 * nu2r2 - ju2r2 * yn(nu+1, u2r2))
D202 = -((n2**2 + n3**2) * nu * n1u2r2 * nu2r2 / u2r2 +
n3**2 * nu * nu2r2**2 / (nu - 1))
D203 = -(n2**2 * nu * ju2r2 * n1u2r2 / u2r2 +
n3**2 * nu * nu2r2 * j1u2r2 / u2r2 +
n3**2 * nu * (j1u2r2 * nu2r2 +
n1u2r2 * ju2r2) / (2 * (nu - 1)))
D212 = -(n2**2 * nu * nu2r2 * j1u2r2 / u2r2 +
n3**2 * nu * ju2r2 * n1u2r2 / u2r2 +
n3**2 * nu * (n1u2r2 * ju2r2 +
j1u2r2 * nu2r2) / (2 * (nu - 1)))
D213 = -((n2**2 + n3**2) * nu * j1u2r2 * ju2r2 / u2r2 +
n3**2 * nu * ju2r2**2 / (nu - 1))
D223 = nu * n2**2 * n3 / u2r2 * (j1u2r2 * nu2r2 - ju2r2 * n1u2r2)
D30 = M[1, 1] * D201 - M[1, 2] * D202 + M[1, 3] * D203
D31 = M[1, 0] * D201 - M[1, 2] * D212 + M[1, 3] * D213
D32 = M[1, 0] * D202 - M[1, 1] * D212 + M[1, 3] * D223
D33 = M[1, 0] * D203 - M[1, 1] * D213 + M[1, 2] * D223
return M[0, 0] * D30 - M[0, 1] * D31 + M[0, 2] * D32 - M[0, 3] * D33
def z_dash(self,x):
''' Z'(x) equation for waveguide mode '''
m, n, chi = self.m, self.n, self.root
# jvp(m,x,r) is the rth derivative of the bessel function of order m evaluated at x
return yn(m,chi)*jvp(m,x,1)-jn(m,chi)*yvp(m,x,1)
def timeDelayEigenfunctionIm(r,m,nc,k): eps2 = np.imag(nc*nc) return np.imag(-1j*k*eps2*2.0*np.pi*r*r*np.conj(jn(m,nc*k*r))*jvp(m,nc*k*r))
def sMatrixElementCoefficients(k,n1,n2,m): A = -n1*hankel2(m,n2*k)*jvp(m,n1*k)+n2*jn(m,n1*k)*h2vp(m,n2*k) B = n1*hankel1(m,n2*k)*jvp(m,n1*k)-n2*jn(m,n1*k)*h1vp(m,n2*k) Ap = -n1*n1*jvp(m,n1*k,2)*hankel2(m,n2*k)+n2*n2*jn(m,n1*k)*h2vp(m,n2*k,2) Bp = n1*n1*jvp(m,n1*k,2)*hankel1(m,n2*k)-n2*n2*jn(m,n1*k)*h1vp(m,n2*k,2) return A, B, Ap, Bp