def test_anisotropic_xi_eigenvectors(rnd_data1, rnd_data2, rnd_data3,
rnd_data4, rnd_data5):
"""
Check whether eigenvectors fulfill the QEVP.
"""
lc = LocalCoordinates("1")
myeps = rnd_data1 + complex(0, 1) * rnd_data2
# ((epsxx, epsxy, epsxz), (epsyx, epsyy, epsyz), (epszx, epszy, epszz)) = \
# tuple(myeps)
m = AnisotropicMaterial(lc, myeps)
n = rnd_data3
n = n / np.sqrt(np.sum(n * n, axis=0))
x = np.zeros((3, 5))
k = rnd_data4 + complex(0, 1) * rnd_data5
kpa = k - np.sum(n * k, axis=0) * n
((_, _), (Mmatrix, Cmatrix, Kmatrix)) \
= m.calcXiQEVMatricesNorm(x, n, kpa)
(eigenvalues, eigenvectors) = m.calcXiEigenvectorsNorm(x, n, kpa)
#print(eigenvalues)
should_be_zero = np.ones((4, 3, 5), dtype=complex)
for j in range(5):
for k in range(4):
should_be_zero[k, :, j] = np.dot(
(Mmatrix[:, :, j] * eigenvalues[k, j]**2 +
Cmatrix[:, :, j] * eigenvalues[k, j] + Kmatrix[:, :, j]),
eigenvectors[k, :, j])
assert np.allclose(should_be_zero, 0)
def test_anisotropic_xi_determinants(rnd_data1, rnd_data2, rnd_data3,
rnd_data4, rnd_data5):
"""
Check whether xi zeros from polynomial fulfill the QEVP and the
associated GLEVP. This also verifies whether A6x6 and B6x6 are
constructed correctly.
"""
lc = LocalCoordinates("1")
myeps = rnd_data1 + complex(0, 1) * rnd_data2
# ((epsxx, epsxy, epsxz), (epsyx, epsyy, epsyz), (epszx, epszy, epszz)) = \
# tuple(myeps)
m = AnisotropicMaterial(lc, myeps)
n = rnd_data3
n = n / np.sqrt(np.sum(n * n, axis=0))
x = np.zeros((3, 5))
k = rnd_data4 + complex(0, 1) * rnd_data5
kpa = k - np.sum(n * k, axis=0) * n
xiarray = m.calcXiNormZeros(x, n, kpa)
((Amatrix6x6, Bmatrix6x6), (Mmatrix, Cmatrix, Kmatrix)) \
= m.calcXiQEVMatricesNorm(x, n, kpa)
should_be_zero_1 = np.ones((4, 5), dtype=complex)
should_be_zero_2 = np.ones((4, 5), dtype=complex)
for j in range(5):
for xi_num in range(4):
should_be_zero_1[xi_num, j] = np.linalg.det(
(Mmatrix[:, :, j] * xiarray[xi_num, j]**2 +
Cmatrix[:, :, j] * xiarray[xi_num, j] + Kmatrix[:, :, j]))
should_be_zero_2[xi_num, j] = np.linalg.det(
(Amatrix6x6[:, :, j] -
Bmatrix6x6[:, :, j] * xiarray[xi_num, j]))
assert np.allclose(should_be_zero_1, 0)
assert np.allclose(should_be_zero_2, 0)
def test_anisotropic_xi_calculation_polynomial(rnd_data1, rnd_data2, rnd_data3,
rnd_data4):
"""
Random epsilon tensor, Random k vector and n unit vector in z direction.
Polynomial coefficients for eigenvalue equation from
the numerical calculations via np.einsum and the analytical expressions
given below (p4a, ..., p0a) should be identical. The test should work
for real and complex epsilon and k values.
"""
lc = LocalCoordinates("1")
myeps = rnd_data1 + complex(0, 1) * rnd_data2
((epsxx, epsxy, epsxz), (epsyx, epsyy, epsyz), (epszx, epszy, epszz)) = \
tuple(myeps)
m = AnisotropicMaterial(lc, myeps)
n = np.zeros((3, 5))
n[2, :] = 1.
x = np.zeros((3, 5))
k = rnd_data3 + complex(0, 1) * rnd_data4
kpa = k - np.sum(n * k, axis=0) * n
kx = kpa[0]
ky = kpa[1]
(p4v, p3v, p2v, p1v, p0v) = m.calcXiPolynomialNorm(x, n, kpa)
# TODO: Maybe generalize to arbitrary n vectors
p4a = epszz * np.ones_like(kx)
p3a = (epsxz + epszx) * kx + (epsyz + epszy) * ky
p2a = epsxz*epszx + epsyz*epszy - (epsxx + epsyy)*epszz \
+ (epsxx + epszz)*kx**2 + (epsxy + epsyx)*kx*ky \
+ (epsyy + epszz)*ky**2
p1a = (epsxz + epszx)*kx**3 \
+ (epsxz*epsyx + epsxy*epszx - epsxx*(epsyz + epszy))*ky \
+ (epsyz + epszy)*kx**2*ky + (epsyz + epszy)*ky**3 \
+ kx*(-(epsxz*epsyy) + epsxy*epsyz - epsyy*epszx + epsyx*epszy + (epsxz + epszx)*ky**2)
p0a = -(epsxz*epsyy*epszx) + epsxy*epsyz*epszx + epsxz*epsyx*epszy \
- epsxx*epsyz*epszy - epsxy*epsyx*epszz + epsxx*epsyy*epszz \
+ epsxx*kx**4 + (epsxy + epsyx)*kx**3*ky \
+ (epsxy*epsyx - epsxx*epsyy + epsyz*epszy - epsyy*epszz)*ky**2 \
+ epsyy*ky**4 + kx**2*(epsxy*epsyx + epsxz*epszx - epsxx*(epsyy + epszz) \
+ (epsxx + epsyy)*ky**2) + kx*(\
(epsyz*epszx + epsxz*epszy - (epsxy + epsyx)*epszz)*ky \
+ (epsxy + epsyx)*ky**3)
should_be_zero_4 = p4a - p4v
assert np.allclose(should_be_zero_4, 0)
should_be_zero_3 = p3a - p3v
assert np.allclose(should_be_zero_3, 0)
should_be_zero_2 = p2a - p2v
assert np.allclose(should_be_zero_2, 0)
should_be_zero_1 = p1a - p1v
assert np.allclose(should_be_zero_1, 0)
should_be_zero_0 = p0a - p0v
assert np.allclose(should_be_zero_0, 0)
def test_anisotropic_xi_eigenvalues(rnd_data1, rnd_data2, rnd_data3, rnd_data4,
rnd_data5, rnd_data6):
"""
Comparison of eigenvalue calculation for xi from random complex material
data. Comparing polynomial calculation from determinant, from quadratic
eigenvalue problem and analytical calculation from sympy.
"""
lc = LocalCoordinates("1")
myeps = np.zeros((3, 3), dtype=complex)
myeps[0:2, 0:2] = rnd_data1 + complex(0, 1) * rnd_data2
myeps[2, 2] = rnd_data3 + complex(0, 1) * rnd_data4
((epsxx, epsxy, _), (epsyx, epsyy, _), (_, _, epszz)) = \
tuple(myeps)
m = AnisotropicMaterial(lc, myeps)
#n = n/np.sqrt(np.sum(n*n, axis=0))
n = np.zeros((3, 1))
n[2, :] = 1
x = np.zeros((3, 1))
k = rnd_data5 + complex(0, 1) * rnd_data6
kpa = k - np.sum(n * k, axis=0) * n
(eigenvalues, _) = m.calcXiEigenvectorsNorm(x, n, kpa)
xiarray = m.calcXiNormZeros(x, n, kpa)
# sympy check with analytical solution
kx, ky, xi = sympy.symbols('k_x k_y xi')
exx, exy, _, eyx, eyy, _, _, _, ezz \
= sympy.symbols('e_xx e_xy e_xz e_yx e_yy e_yz e_zx e_zy e_zz')
#eps = Matrix([[exx, exy, exz], [eyx, eyy, eyz], [ezx, ezy, ezz]])
eps = sympy.Matrix([[exx, exy, 0], [eyx, eyy, 0], [0, 0, ezz]])
v = sympy.Matrix([[kx, ky, xi]])
m = -(v * v.T)[0] * sympy.eye(3) + v.T * v + eps
detm = m.det().collect(xi)
soldetm = sympy.solve(detm, xi)
subsdict = {
kx: kpa[0, 0],
ky: kpa[1, 0],
exx: epsxx,
exy: epsxy,
eyx: epsyx,
eyy: epsyy,
ezz: epszz,
sympy.I: complex(0, 1)
}
analytical_solution = np.sort_complex(
np.array([sol.evalf(subs=subsdict) for sol in soldetm], dtype=complex))
numerical_solution1 = np.sort_complex(xiarray[:, 0])
numerical_solution2 = np.sort_complex(eigenvalues[:, 0])
assert np.allclose(analytical_solution - numerical_solution1, 0)
assert np.allclose(analytical_solution - numerical_solution2, 0)
def test_anisotropic_xi_calculation_det(rnd_data1, rnd_data2, rnd_data3,
rnd_data4, rnd_data5, rnd_data6):
"""
Random epsilon tensor, Random k vector and n unit vector in z direction.
Determinant of the propagator from numerical calculations
via np.einsum and from analytical expression given below should coincide.
The test should work for real and complex epsilon and k values.
"""
lc = LocalCoordinates("1")
myeps = rnd_data1 + complex(0, 1) * rnd_data2
((epsxx, epsxy, epsxz), (epsyx, epsyy, epsyz), (epszx, epszy, epszz)) = \
tuple(myeps)
m = AnisotropicMaterial(lc, myeps)
n = np.zeros((3, 5))
n[2, :] = 1.
x = np.zeros((3, 5))
k = rnd_data3 + complex(0, 1) * rnd_data4
xi = rnd_data5 + complex(0, 1) * rnd_data6
kpa = k - np.sum(n * k, axis=0) * n
kx = kpa[0]
ky = kpa[1]
# TODO: Maybe generalize to arbitrary n vectors
det_analytical = xi**4*epszz \
+ xi**3*((epsxz + epszx)*kx + (epsyz + epszy)*ky)\
+ xi**2*(epsxz*epszx + epsyz*epszy - (epsxx + epsyy)*epszz\
+ (epsxx + epszz)*kx**2 + (epsxy + epsyx)*kx*ky\
+ (epsyy + epszz)*ky**2)\
+ xi*((epsxz + epszx)*kx**3\
+ (epsxz*epsyx + epsxy*epszx - epsxx*(epsyz + epszy))*ky\
+ (epsyz + epszy)*kx**2*ky + (epsyz + epszy)*ky**3\
+ kx*(-(epsxz*epsyy) + epsxy*epsyz - epsyy*epszx\
+ epsyx*epszy + (epsxz + epszx)*ky**2))\
-(epsxz*epsyy*epszx) + epsxy*epsyz*epszx\
+ epsxz*epsyx*epszy - epsxx*epsyz*epszy\
- epsxy*epsyx*epszz + epsxx*epsyy*epszz + epsxx*kx**4\
+ (epsxy + epsyx)*kx**3*ky \
+ (epsxy*epsyx - epsxx*epsyy + epsyz*epszy - epsyy*epszz)*ky**2 \
+ epsyy*ky**4 + kx**2*(epsxy*epsyx + epsxz*epszx \
- epsxx*(epsyy + epszz) + (epsxx + epsyy)*ky**2)\
+ kx*((epsyz*epszx + epsxz*epszy \
- (epsxy + epsyx)*epszz)*ky + (epsxy + epsyx)*ky**3)
should_be_zero = det_analytical - m.calcXiDetNorm(xi, x, n, kpa)
assert np.allclose(should_be_zero, 0)
def test_anisotropic_xi_calculation_polynomial_zeros(rnd_data1, rnd_data2,
rnd_data3, rnd_data4,
rnd_data5):
"""
Random epsilon tensor, random k vector and random n unit vector.
Check whether the roots calculated give really zero for the determinant.
The test should work for real and complex epsilon and k values.
"""
lc = LocalCoordinates("1")
myeps = rnd_data1 + complex(0, 1) * rnd_data2
# ((epsxx, epsxy, epsxz), (epsyx, epsyy, epsyz), (epszx, epszy, epszz)) = \
# tuple(myeps)
m = AnisotropicMaterial(lc, myeps)
n = rnd_data3
n = n / np.sqrt(np.sum(n * n, axis=0))
x = np.zeros((3, 5))
k = rnd_data4 + complex(0, 1) * rnd_data5
kpa = k - np.sum(n * k, axis=0) * n
should_be_zero = np.ones((4, 5), dtype=complex)
xiarray = m.calcXiNormZeros(x, n, kpa)
for i in range(4):
should_be_zero[i, :] = m.calcXiDetNorm(xiarray[i], x, n, kpa)
assert np.allclose(should_be_zero, 0)
frontsurf = Surface.p(lc1,
shape=Conic.p(lc1, curv=0),
aperture=CircularAperture.p(lc1, maxradius=10.0))
rearsurf = Surface.p(lc2,
shape=Conic.p(lc2, curv=0),
aperture=CircularAperture.p(lc3, maxradius=10.0))
image = Surface.p(lc3)
elem = OpticalElement.p(lc0, name="crystalelem")
no = 1.5
neo = 1.8
myeps = np.array([[no, 0, 0], [0, no, 0], [0, 0, neo]])
crystal = AnisotropicMaterial.p(lc1, myeps)
elem.addMaterial("crystal", crystal)
elem.addSurface("stop", stopsurf, (None, None))
elem.addSurface("front", frontsurf, (None, "crystal"))
elem.addSurface("rear", rearsurf, ("crystal", "crystal"))
elem.addSurface("image", image, ("crystal", None))
s.addElement("crystalelem", elem)
sysseq = [("crystalelem", [("stop", {}), ("front", {}),
("rear", {
"is_mirror": True
}), ("image", {})])]
rnd_data4 = np.random.random((3, 3)) # np.zeros((3, 3))#
# isotropic tests
# bk7 = material.ConstantIndexGlass(lc1, n=1.5168)
# sf5 = material.ConstantIndexGlass(lc2, n=1.6727)
myeps1 = 1.5168**2 * np.eye(3)
myeps2 = 1.6727**2 * np.eye(3)
# anisotropic materials
# myeps1 = rnd_data1 + complex(0, 1)*rnd_data2
# myeps2 = rnd_data3 + complex(0, 1)*rnd_data4
crystal1 = AnisotropicMaterial(lc1, myeps1, name="crystal1")
crystal2 = AnisotropicMaterial(lc2, myeps2, name="crystal2")
elem.addMaterial("crystal1", crystal1)
elem.addMaterial("crystal2", crystal2)
elem.addSurface("stop", stopsurf, (None, None))
elem.addSurface("front", frontsurf, (None, "crystal1"))
elem.addSurface("cement", cementsurf, ("crystal1", "crystal2"))
elem.addSurface("rear", rearsurf, ("crystal2", None))
elem.addSurface("image", image, (None, None))
s.addElement("AC254-100", elem)
sysseq = [("AC254-100", [("stop", {}), ("front", {}), ("cement", {}),
("rear", {}), ("image", {})])]
from pyrateoptics.raytracer.globalconstants import degree
from pyrateoptics.raytracer.analysis.optical_system_analysis import\
OpticalSystemAnalysis
logging.basicConfig(level=logging.DEBUG)
wavelength = 0.5876e-3
rnd_data1 = np.random.random((3, 3)) # np.eye(3)
rnd_data2 = np.random.random((3, 3)) # np.zeros((3, 3))#
lc = LocalCoordinates("1")
myeps = np.eye(3) + 0.1 * rnd_data1 + 0.01 * complex(0, 1) * rnd_data2
# aggressive complex choice of myeps
# myeps = np.eye(3) + 0.01*np.random.random((3, 3))
crystal = AnisotropicMaterial(lc, myeps)
# definition of optical system
s = OpticalSystem(matbackground=crystal)
lc0 = s.addLocalCoordinateSystem(LocalCoordinates(name="object", decz=0.0),
refname=s.rootcoordinatesystem.name)
lc1 = s.addLocalCoordinateSystem(LocalCoordinates(name="m1",
decz=50.0,
tiltx=-math.pi / 8),
refname=lc0.name) # objectDist
lc2 = s.addLocalCoordinateSystem(LocalCoordinates(name="m2_stop",
decz=-50.0,
decy=-20,
tiltx=math.pi / 16),
refname=lc1.name)
def test_anisotropic_xi_eigenvalues(rnd_data1, rnd_data2, rnd_data3, rnd_data4,
rnd_data5, rnd_data6):
"""
Comparison of eigenvalue calculation for xi from random complex material
data. Comparing polynomial calculation from determinant, from quadratic
eigenvalue problem and analytical calculation from sympy.
"""
lc = LocalCoordinates.p("1")
myeps = np.zeros((3, 3), dtype=complex)
myeps[0:2, 0:2] = rnd_data1 + complex(0, 1) * rnd_data2
myeps[2, 2] = rnd_data3 + complex(0, 1) * rnd_data4
((epsxx, epsxy, _), (epsyx, epsyy, _), (_, _, epszz)) = \
tuple(myeps)
m = AnisotropicMaterial.p(lc, myeps)
#n = n/np.sqrt(np.sum(n*n, axis=0))
n = np.zeros((3, 1))
n[2, :] = 1
x = np.zeros((3, 1))
k = rnd_data5 + complex(0, 1) * rnd_data6
kpa = k - np.sum(n * k, axis=0) * n
(eigenvalues, _) = m.calcXiEigenvectorsNorm(x, n, kpa)
xiarray = m.calcXiNormZeros(x, n, kpa)
# sympy check with analytical solution
kx, ky, xi = sympy.symbols('k_x k_y xi')
exx, exy, _, eyx, eyy, _, _, _, ezz \
= sympy.symbols('e_xx e_xy e_xz e_yx e_yy e_yz e_zx e_zy e_zz')
#eps = sympy.Matrix([[exx, exy, 0], [eyx, eyy, 0], [0, 0, ezz]])
#v = sympy.Matrix([[kx, ky, xi]])
#m = -(v*v.T)[0]*sympy.eye(3) + v.T*v + eps
#detm = m.det().collect(xi)
#soldetm = sympy.solve(detm, xi)
# hard wired solution to the above code to avoid deadline issues during
# testing with pytest
soldetm = [
-sqrt(2) *
sqrt(exx - exx * kx**2 / ezz - exy * kx * ky / ezz -
eyx * kx * ky / ezz + eyy - eyy * ky**2 / ezz - kx**2 - ky**2 -
sqrt(exx**2 * ezz**2 - 2 * exx**2 * ezz * kx**2 + exx**2 * kx**4 -
2 * exx * exy * ezz * kx * ky + 2 * exx * exy * kx**3 * ky -
2 * exx * eyx * ezz * kx * ky + 2 * exx * eyx * kx**3 * ky -
2 * exx * eyy * ezz**2 + 2 * exx * eyy * ezz * kx**2 +
2 * exx * eyy * ezz * ky**2 + 2 * exx * eyy * kx**2 * ky**2 +
2 * exx * ezz**2 * kx**2 - 2 * exx * ezz**2 * ky**2 -
2 * exx * ezz * kx**4 - 2 * exx * ezz * kx**2 * ky**2 +
exy**2 * kx**2 * ky**2 + 4 * exy * eyx * ezz**2 - 4 * exy *
eyx * ezz * kx**2 - 4 * exy * eyx * ezz * ky**2 + 2 * exy *
eyx * kx**2 * ky**2 - 2 * exy * eyy * ezz * kx * ky +
2 * exy * eyy * kx * ky**3 + 4 * exy * ezz**2 * kx * ky -
2 * exy * ezz * kx**3 * ky - 2 * exy * ezz * kx * ky**3 +
eyx**2 * kx**2 * ky**2 - 2 * eyx * eyy * ezz * kx * ky +
2 * eyx * eyy * kx * ky**3 + 4 * eyx * ezz**2 * kx * ky -
2 * eyx * ezz * kx**3 * ky - 2 * eyx * ezz * kx * ky**3 +
eyy**2 * ezz**2 - 2 * eyy**2 * ezz * ky**2 + eyy**2 * ky**4 -
2 * eyy * ezz**2 * kx**2 + 2 * eyy * ezz**2 * ky**2 - 2 *
eyy * ezz * kx**2 * ky**2 - 2 * eyy * ezz * ky**4 + ezz**2 *
kx**4 + 2 * ezz**2 * kx**2 * ky**2 + ezz**2 * ky**4) / ezz) /
2,
sqrt(2) *
sqrt(exx - exx * kx**2 / ezz - exy * kx * ky / ezz -
eyx * kx * ky / ezz + eyy - eyy * ky**2 / ezz - kx**2 - ky**2 -
sqrt(exx**2 * ezz**2 - 2 * exx**2 * ezz * kx**2 + exx**2 * kx**4 -
2 * exx * exy * ezz * kx * ky + 2 * exx * exy * kx**3 * ky -
2 * exx * eyx * ezz * kx * ky + 2 * exx * eyx * kx**3 * ky -
2 * exx * eyy * ezz**2 + 2 * exx * eyy * ezz * kx**2 +
2 * exx * eyy * ezz * ky**2 + 2 * exx * eyy * kx**2 * ky**2 +
2 * exx * ezz**2 * kx**2 - 2 * exx * ezz**2 * ky**2 -
2 * exx * ezz * kx**4 - 2 * exx * ezz * kx**2 * ky**2 +
exy**2 * kx**2 * ky**2 + 4 * exy * eyx * ezz**2 - 4 * exy *
eyx * ezz * kx**2 - 4 * exy * eyx * ezz * ky**2 + 2 * exy *
eyx * kx**2 * ky**2 - 2 * exy * eyy * ezz * kx * ky +
2 * exy * eyy * kx * ky**3 + 4 * exy * ezz**2 * kx * ky -
2 * exy * ezz * kx**3 * ky - 2 * exy * ezz * kx * ky**3 +
eyx**2 * kx**2 * ky**2 - 2 * eyx * eyy * ezz * kx * ky +
2 * eyx * eyy * kx * ky**3 + 4 * eyx * ezz**2 * kx * ky -
2 * eyx * ezz * kx**3 * ky - 2 * eyx * ezz * kx * ky**3 +
eyy**2 * ezz**2 - 2 * eyy**2 * ezz * ky**2 + eyy**2 * ky**4 -
2 * eyy * ezz**2 * kx**2 + 2 * eyy * ezz**2 * ky**2 - 2 *
eyy * ezz * kx**2 * ky**2 - 2 * eyy * ezz * ky**4 + ezz**2 *
kx**4 + 2 * ezz**2 * kx**2 * ky**2 + ezz**2 * ky**4) / ezz) /
2, -sqrt(2) *
sqrt(exx - exx * kx**2 / ezz - exy * kx * ky / ezz -
eyx * kx * ky / ezz + eyy - eyy * ky**2 / ezz - kx**2 - ky**2 +
sqrt(exx**2 * ezz**2 - 2 * exx**2 * ezz * kx**2 + exx**2 * kx**4 -
2 * exx * exy * ezz * kx * ky + 2 * exx * exy * kx**3 * ky -
2 * exx * eyx * ezz * kx * ky + 2 * exx * eyx * kx**3 * ky -
2 * exx * eyy * ezz**2 + 2 * exx * eyy * ezz * kx**2 +
2 * exx * eyy * ezz * ky**2 + 2 * exx * eyy * kx**2 * ky**2 +
2 * exx * ezz**2 * kx**2 - 2 * exx * ezz**2 * ky**2 -
2 * exx * ezz * kx**4 - 2 * exx * ezz * kx**2 * ky**2 +
exy**2 * kx**2 * ky**2 + 4 * exy * eyx * ezz**2 - 4 * exy *
eyx * ezz * kx**2 - 4 * exy * eyx * ezz * ky**2 + 2 * exy *
eyx * kx**2 * ky**2 - 2 * exy * eyy * ezz * kx * ky +
2 * exy * eyy * kx * ky**3 + 4 * exy * ezz**2 * kx * ky -
2 * exy * ezz * kx**3 * ky - 2 * exy * ezz * kx * ky**3 +
eyx**2 * kx**2 * ky**2 - 2 * eyx * eyy * ezz * kx * ky +
2 * eyx * eyy * kx * ky**3 + 4 * eyx * ezz**2 * kx * ky -
2 * eyx * ezz * kx**3 * ky - 2 * eyx * ezz * kx * ky**3 +
eyy**2 * ezz**2 - 2 * eyy**2 * ezz * ky**2 + eyy**2 * ky**4 -
2 * eyy * ezz**2 * kx**2 + 2 * eyy * ezz**2 * ky**2 - 2 *
eyy * ezz * kx**2 * ky**2 - 2 * eyy * ezz * ky**4 + ezz**2 *
kx**4 + 2 * ezz**2 * kx**2 * ky**2 + ezz**2 * ky**4) / ezz) /
2,
sqrt(2) *
sqrt(exx - exx * kx**2 / ezz - exy * kx * ky / ezz -
eyx * kx * ky / ezz + eyy - eyy * ky**2 / ezz - kx**2 - ky**2 +
sqrt(exx**2 * ezz**2 - 2 * exx**2 * ezz * kx**2 + exx**2 * kx**4 -
2 * exx * exy * ezz * kx * ky + 2 * exx * exy * kx**3 * ky -
2 * exx * eyx * ezz * kx * ky + 2 * exx * eyx * kx**3 * ky -
2 * exx * eyy * ezz**2 + 2 * exx * eyy * ezz * kx**2 +
2 * exx * eyy * ezz * ky**2 + 2 * exx * eyy * kx**2 * ky**2 +
2 * exx * ezz**2 * kx**2 - 2 * exx * ezz**2 * ky**2 -
2 * exx * ezz * kx**4 - 2 * exx * ezz * kx**2 * ky**2 +
exy**2 * kx**2 * ky**2 + 4 * exy * eyx * ezz**2 - 4 * exy *
eyx * ezz * kx**2 - 4 * exy * eyx * ezz * ky**2 + 2 * exy *
eyx * kx**2 * ky**2 - 2 * exy * eyy * ezz * kx * ky +
2 * exy * eyy * kx * ky**3 + 4 * exy * ezz**2 * kx * ky -
2 * exy * ezz * kx**3 * ky - 2 * exy * ezz * kx * ky**3 +
eyx**2 * kx**2 * ky**2 - 2 * eyx * eyy * ezz * kx * ky +
2 * eyx * eyy * kx * ky**3 + 4 * eyx * ezz**2 * kx * ky -
2 * eyx * ezz * kx**3 * ky - 2 * eyx * ezz * kx * ky**3 +
eyy**2 * ezz**2 - 2 * eyy**2 * ezz * ky**2 + eyy**2 * ky**4 -
2 * eyy * ezz**2 * kx**2 + 2 * eyy * ezz**2 * ky**2 - 2 *
eyy * ezz * kx**2 * ky**2 - 2 * eyy * ezz * ky**4 + ezz**2 *
kx**4 + 2 * ezz**2 * kx**2 * ky**2 + ezz**2 * ky**4) / ezz) /
2
]
subsdict = {
kx: kpa[0, 0],
ky: kpa[1, 0],
exx: epsxx,
exy: epsxy,
eyx: epsyx,
eyy: epsyy,
ezz: epszz,
sympy.I: complex(0, 1)
}
analytical_solution = np.sort_complex(
np.array([sol.evalf(subs=subsdict) for sol in soldetm], dtype=complex))
numerical_solution1 = np.sort_complex(xiarray[:, 0])
numerical_solution2 = np.sort_complex(eigenvalues[:, 0])
assert np.allclose(analytical_solution - numerical_solution1, 0)
assert np.allclose(analytical_solution - numerical_solution2, 0)