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# Computes the radiation pattern of a dipole antenna
# positioned a given height above a perfect-electric
# conductor (PEC) ground plane and compares the result
# to analytic theory.
import math
import matplotlib
import numpy as np
import meep as mp
matplotlib.use("agg")
import matplotlib.pyplot as plt
resolution = 200 # pixels/um
n = 1.2 # refractive index of surrounding medium
h = 1.25 # height of antenna (point dipole source) above ground plane
wvl = 0.65 # vacuum wavelength
r = 1000 * wvl # radius of far-field circle
npts = 50 # number of points in [0,pi/2) range of angles
angles = 0.5 * math.pi / npts * np.arange(npts)
def radial_flux(sim, nearfield_box, r):
E = np.zeros((npts, 3), dtype=np.complex128)
H = np.zeros((npts, 3), dtype=np.complex128)
for n in range(npts):
ff = sim.get_farfield(
nearfield_box, mp.Vector3(r * math.sin(angles[n]), r * math.cos(angles[n]))
)
E[n, :] = [np.conj(ff[j]) for j in range(3)]
H[n, :] = [ff[j + 3] for j in range(3)]
Px = np.real(E[:, 1] * H[:, 2] - E[:, 2] * H[:, 1]) # Ey*Hz-Ez*Hy
Py = np.real(E[:, 2] * H[:, 0] - E[:, 0] * H[:, 2]) # Ez*Hx-Ex*Hz
return np.sqrt(np.square(Px) + np.square(Py))
def free_space_radiation(src_cmpt):
sxy = 4
dpml = 1
cell_size = mp.Vector3(sxy + 2 * dpml, sxy + 2 * dpml)
pml_layers = [mp.PML(dpml)]
fcen = 1 / wvl
sources = [
mp.Source(
src=mp.GaussianSource(fcen, fwidth=0.2 * fcen),
center=mp.Vector3(),
component=src_cmpt,
)
]
if src_cmpt == mp.Hz:
symmetries = [mp.Mirror(mp.X, phase=-1), mp.Mirror(mp.Y, phase=-1)]
elif src_cmpt == mp.Ez:
symmetries = [mp.Mirror(mp.X, phase=+1), mp.Mirror(mp.Y, phase=+1)]
else:
symmetries = []
sim = mp.Simulation(
cell_size=cell_size,
resolution=resolution,
sources=sources,
symmetries=symmetries,
boundary_layers=pml_layers,
default_material=mp.Medium(index=n),
)
nearfield_box = sim.add_near2far(
fcen,
0,
1,
mp.Near2FarRegion(center=mp.Vector3(0, +0.5 * sxy), size=mp.Vector3(sxy, 0)),
mp.Near2FarRegion(
center=mp.Vector3(0, -0.5 * sxy), size=mp.Vector3(sxy, 0), weight=-1
),
mp.Near2FarRegion(center=mp.Vector3(+0.5 * sxy, 0), size=mp.Vector3(0, sxy)),
mp.Near2FarRegion(
center=mp.Vector3(-0.5 * sxy, 0), size=mp.Vector3(0, sxy), weight=-1
),
)
sim.run(until_after_sources=mp.stop_when_dft_decayed())
return radial_flux(sim, nearfield_box, r)
def pec_ground_plane_radiation(src_cmpt=mp.Hz):
L = 8.0 # length of non-PML region
dpml = 1.0 # thickness of PML
sxy = dpml + L + dpml
cell_size = mp.Vector3(sxy, sxy, 0)
boundary_layers = [mp.PML(dpml)]
fcen = 1 / wvl
# The near-to-far field transformation feature only supports
# homogeneous media which means it cannot explicitly take into
# account the ground plane. As a workaround, we use two antennas
# of opposite sign surrounded by a single near2far box which
# encloses both antennas. We then use an odd mirror symmetry to
# divide the computational cell in half which is effectively
# equivalent to a PEC boundary condition on one side.
# Note: This setup means that the radiation pattern can only
# be measured in the top half above the dipole.
sources = [
mp.Source(
src=mp.GaussianSource(fcen, fwidth=0.2 * fcen),
component=src_cmpt,
center=mp.Vector3(0, +h),
),
mp.Source(
src=mp.GaussianSource(fcen, fwidth=0.2 * fcen),
component=src_cmpt,
center=mp.Vector3(0, -h),
amplitude=-1 if src_cmpt == mp.Ez else +1,
),
]
symmetries = [
mp.Mirror(direction=mp.X, phase=+1 if src_cmpt == mp.Ez else -1),
mp.Mirror(direction=mp.Y, phase=-1 if src_cmpt == mp.Ez else +1),
]
sim = mp.Simulation(
resolution=resolution,
cell_size=cell_size,
boundary_layers=boundary_layers,
sources=sources,
symmetries=symmetries,
default_material=mp.Medium(index=n),
)
nearfield_box = sim.add_near2far(
fcen,
0,
1,
mp.Near2FarRegion(
center=mp.Vector3(0, 2 * h), size=mp.Vector3(2 * h, 0), weight=+1
),
mp.Near2FarRegion(
center=mp.Vector3(0, -2 * h), size=mp.Vector3(2 * h, 0), weight=-1
),
mp.Near2FarRegion(
center=mp.Vector3(h, 0), size=mp.Vector3(0, 4 * h), weight=+1
),
mp.Near2FarRegion(
center=mp.Vector3(-h, 0), size=mp.Vector3(0, 4 * h), weight=-1
),
)
sim.plot2D()
plt.savefig("antenna_pec_ground_plane.png", bbox_inches="tight")
sim.run(until_after_sources=mp.stop_when_dft_decayed())
return radial_flux(sim, nearfield_box, r)
if __name__ == "__main__":
src_cmpt = mp.Ez # TM/P: Hz or TE/S: Ez
Pr_fsp = free_space_radiation(src_cmpt)
Pr_pec = pec_ground_plane_radiation(src_cmpt)
# The radiation pattern of a two-element antenna
# array is equivalent to the radiation pattern of
# a single antenna multiplied by its array factor
# as derived in Section 6.2 "Two-Element Array" of
# Antenna Theory: Analysis and Design, Fourth Edition
# (2016) by C.A. Balanis.
k = 2 * np.pi / (wvl / n) # wavevector in free space
Pr_theory = np.zeros(
npts,
)
for i, ang in enumerate(angles):
Pr_theory[i] = Pr_fsp[i] * 2 * np.sin(k * h * np.cos(ang))
Pr_pec_norm = Pr_pec / np.max(Pr_pec)
Pr_theory_norm = (Pr_theory / max(Pr_theory)) ** 2
plt.figure()
plt.plot(np.degrees(angles), Pr_pec_norm, "b-", label="Meep")
plt.plot(np.degrees(angles), Pr_theory_norm, "r-", label="theory")
plt.xlabel("angle (degrees)")
plt.ylabel("radial flux (normalized by maximum flux)")
plt.title(
f"antenna with {'E' if src_cmpt==mp.Ez else r'H'}$_z$ polarization above PEC ground plane"
)
plt.axis([0, 90, 0, 1.0])
plt.legend()
plt.savefig("radiation_pattern.png", bbox_inches="tight")
print(f"norm:, {np.linalg.norm(Pr_pec_norm - Pr_theory_norm):.6f}")
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