def find_roots(wl, epsilon, alpha, beta):
"""Find roots of characteristic equation.
Given a wavelength, a 3x3 tensor epsilon and the tangential components
of the wavevector k = (alpha,beta,gamma_i), returns the 4 possible
gamma_i, i = 1,2,3,4 that satisfy the boundary conditions.
"""
omega = 2. * S.pi * c / wl
K = omega**2 * mu0 * epsilon
k0 = 2. * S.pi / wl
K /= k0**2
alpha /= k0
beta /= k0
alpha2 = alpha**2
alpha3 = alpha**3
alpha4 = alpha**4
beta2 = beta**2
beta3 = beta**3
beta4 = beta**4
coeff = [
K[2,
2], alpha * (K[0, 2] + K[2, 0]) + beta * (K[1, 2] + K[2, 1]),
alpha2 * (K[0, 0] + K[2, 2]) + alpha * beta *
(K[1, 0] + K[0, 1]) + beta2 * (K[1, 1] + K[2, 2]) +
(K[0, 2] * K[2, 0] + K[1, 2] * K[2, 1] - K[0, 0] * K[2, 2] -
K[1, 1] * K[2, 2]), alpha3 * (K[0, 2] + K[2, 0]) + beta3 *
(K[1, 2] + K[2, 1]) + alpha2 * beta * (K[1, 2] + K[2, 1]) +
alpha * beta2 * (K[0, 2] + K[2, 0]) + alpha *
(K[0, 1] * K[1, 2] + K[1, 0] * K[2, 1] - K[0, 2] * K[1, 1] -
K[2, 0] * K[1, 1]) + beta *
(K[0, 1] * K[2, 0] + K[1, 0] * K[0, 2] - K[0, 0] * K[1, 2] -
K[0, 0] * K[2, 1]), alpha4 * (K[0, 0]) + beta4 * (K[1, 1]) +
alpha3 * beta * (K[0, 1] + K[1, 0]) + alpha * beta3 *
(K[0, 1] + K[1, 0]) + alpha2 * beta2 * (K[0, 0] + K[1, 1]) +
alpha2 * (K[0, 1] * K[1, 0] + K[0, 2] * K[2, 0] -
K[0, 0] * K[2, 2] - K[0, 0] * K[1, 1]) + beta2 *
(K[0, 1] * K[1, 0] + K[1, 2] * K[2, 1] - K[0, 0] * K[1, 1] -
K[1, 1] * K[2, 2]) + alpha * beta *
(K[0, 2] * K[2, 1] + K[2, 0] * K[1, 2] - K[0, 1] * K[2, 2] -
K[1, 0] * K[2, 2]) + K[0, 0] * K[1, 1] * K[2, 2] -
K[0, 0] * K[1, 2] * K[2, 1] - K[1, 0] * K[0, 1] * K[2, 2] +
K[1, 0] * K[0, 2] * K[2, 1] + K[2, 0] * K[0, 1] * K[1, 2] -
K[2, 0] * K[0, 2] * K[1, 1]
]
gamma = S.roots(coeff)
tmp = S.sort_complex(gamma)
gamma = tmp[[3, 0, 2, 1]] # convention
k = k0 * \
S.array([alpha *
S.ones(gamma.shape), beta *
S.ones(gamma.shape), gamma]).T
v = S.zeros((4, 3), dtype=complex)
for i, g in enumerate(gamma):
H = K + [[-beta2 - g**2, alpha * beta, alpha * g],
[alpha * beta, -alpha2 - g**2, beta * g],
[alpha * g, beta * g, -alpha2 - beta2]]
v[i, :] = [
(K[1, 1] - alpha2 - g**2) *
(K[2, 2] - alpha2 - beta2) - (K[1, 2] + beta * g)**2,
(K[1, 2] + beta * g) * (K[2, 0] + alpha * g) -
(K[0, 1] + alpha * beta) * (K[2, 2] - alpha2 - beta2),
(K[0, 1] + alpha * beta) * (K[1, 2] + beta * g) -
(K[0, 2] + alpha * g) * (K[1, 1] - alpha2 - g**2)
]
p3 = v[0, :]
p3 /= norm(p3)
p4 = v[1, :]
p4 /= norm(p4)
p1 = S.cross(p3, k[0, :])
p1 /= norm(p1)
p2 = S.cross(p4, k[1, :])
p2 /= norm(p2)
p = S.array([p1, p2, p3, p4])
q = wl / (2. * S.pi * mu0 * c) * S.cross(k, p)
return k, p, q
import scipy scipy.sort_complex # OUT: <function sort_complex at 0x972cb1c> scipy.sort_complex(1,33,12) # OUT: Traceback (most recent call last): # OUT: File "<input>", line 1, in <module> # OUT: TypeError: sort_complex() takes exactly 1 argument (3 given) a=[2,33,1,12,3] scipy.sort_complex(a) # OUT: array([ 1.+0.j, 2.+0.j, 3.+0.j, 12.+0.j, 33.+0.j]) a=[2+12.j,33+1.j,1+13.j,12,3+10.j] scipy.sort_complex(a) # OUT: array([ 1.+13.j, 2.+12.j, 3.+10.j, 12. +0.j, 33. +1.j]) a=[2+12.j,33+1.j,1+13.j,12,3+10.j,3+3.j] scipy.sort_complex(a) # OUT: array([ 1.+13.j, 2.+12.j, 3. +3.j, 3.+10.j, 12. +0.j, 33. +1.j])
def find_roots(wl, epsilon, alpha, beta):
"""Find roots of characteristic equation.
Given a wavelength, a 3x3 tensor epsilon and the tangential components
of the wavevector k = (alpha,beta,gamma_i), returns the 4 possible
gamma_i, i = 1,2,3,4 that satisfy the boundary conditions.
"""
omega = 2.0 * S.pi * c / wl
K = omega ** 2 * mu0 * epsilon
k0 = 2.0 * S.pi / wl
K /= k0 ** 2
alpha /= k0
beta /= k0
alpha2 = alpha ** 2
alpha3 = alpha ** 3
alpha4 = alpha ** 4
beta2 = beta ** 2
beta3 = beta ** 3
beta4 = beta ** 4
coeff = [
K[2, 2],
alpha * (K[0, 2] + K[2, 0]) + beta * (K[1, 2] + K[2, 1]),
alpha2 * (K[0, 0] + K[2, 2])
+ alpha * beta * (K[1, 0] + K[0, 1])
+ beta2 * (K[1, 1] + K[2, 2])
+ (K[0, 2] * K[2, 0] + K[1, 2] * K[2, 1] - K[0, 0] * K[2, 2] - K[1, 1] * K[2, 2]),
alpha3 * (K[0, 2] + K[2, 0])
+ beta3 * (K[1, 2] + K[2, 1])
+ alpha2 * beta * (K[1, 2] + K[2, 1])
+ alpha * beta2 * (K[0, 2] + K[2, 0])
+ alpha * (K[0, 1] * K[1, 2] + K[1, 0] * K[2, 1] - K[0, 2] * K[1, 1] - K[2, 0] * K[1, 1])
+ beta * (K[0, 1] * K[2, 0] + K[1, 0] * K[0, 2] - K[0, 0] * K[1, 2] - K[0, 0] * K[2, 1]),
alpha4 * (K[0, 0])
+ beta4 * (K[1, 1])
+ alpha3 * beta * (K[0, 1] + K[1, 0])
+ alpha * beta3 * (K[0, 1] + K[1, 0])
+ alpha2 * beta2 * (K[0, 0] + K[1, 1])
+ alpha2 * (K[0, 1] * K[1, 0] + K[0, 2] * K[2, 0] - K[0, 0] * K[2, 2] - K[0, 0] * K[1, 1])
+ beta2 * (K[0, 1] * K[1, 0] + K[1, 2] * K[2, 1] - K[0, 0] * K[1, 1] - K[1, 1] * K[2, 2])
+ alpha * beta * (K[0, 2] * K[2, 1] + K[2, 0] * K[1, 2] - K[0, 1] * K[2, 2] - K[1, 0] * K[2, 2])
+ K[0, 0] * K[1, 1] * K[2, 2]
- K[0, 0] * K[1, 2] * K[2, 1]
- K[1, 0] * K[0, 1] * K[2, 2]
+ K[1, 0] * K[0, 2] * K[2, 1]
+ K[2, 0] * K[0, 1] * K[1, 2]
- K[2, 0] * K[0, 2] * K[1, 1],
]
gamma = S.roots(coeff)
tmp = S.sort_complex(gamma)
gamma = tmp[[3, 0, 2, 1]] # convention
k = k0 * S.array([alpha * S.ones(gamma.shape), beta * S.ones(gamma.shape), gamma]).T
v = S.zeros((4, 3), dtype=complex)
for i, g in enumerate(gamma):
H = K + [
[-beta2 - g ** 2, alpha * beta, alpha * g],
[alpha * beta, -alpha2 - g ** 2, beta * g],
[alpha * g, beta * g, -alpha2 - beta2],
]
v[i, :] = [
(K[1, 1] - alpha2 - g ** 2) * (K[2, 2] - alpha2 - beta2) - (K[1, 2] + beta * g) ** 2,
(K[1, 2] + beta * g) * (K[2, 0] + alpha * g)
- (K[0, 1] + alpha * beta) * (K[2, 2] - alpha2 - beta2),
(K[0, 1] + alpha * beta) * (K[1, 2] + beta * g)
- (K[0, 2] + alpha * g) * (K[1, 1] - alpha2 - g ** 2),
]
p3 = v[0, :]
p3 /= norm(p3)
p4 = v[1, :]
p4 /= norm(p4)
p1 = S.cross(p3, k[0, :])
p1 /= norm(p1)
p2 = S.cross(p4, k[1, :])
p2 /= norm(p2)
p = S.array([p1, p2, p3, p4])
q = wl / (2.0 * S.pi * mu0 * c) * S.cross(k, p)
return k, p, q