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# -*- coding: utf-8 -*-
# polarization grating from C. Oh and M.J. Escuti, Optics Letters, Vol. 33, No. 20, pp. 2287-9, 2008
# note: reference uses z as the propagation direction and y as the out-of-plane direction; this script uses x and z, respectively
import meep as mp
import numpy as np
import math
import matplotlib.pyplot as plt
resolution = 50 # pixels/μm
dpml = 1.0 # PML thickness
dsub = 1.0 # substrate thickness
dpad = 1.0 # padding thickness
k_point = mp.Vector3(0,0,0)
pml_layers = [mp.PML(thickness=dpml,direction=mp.X)]
n_0 = 1.55
delta_n = 0.159
epsilon_diag = mp.Matrix(mp.Vector3(n_0**2,0,0),mp.Vector3(0,n_0**2,0),mp.Vector3(0,0,(n_0+delta_n)**2))
wvl = 0.54 # center wavelength
fcen = 1/wvl # center frequency
def pol_grating(d,ph,gp,nmode):
sx = dpml+dsub+d+d+dpad+dpml
sy = gp
cell_size = mp.Vector3(sx,sy,0)
# twist angle of nematic director; from equation 1b
def phi(p):
xx = p.x-(-0.5*sx+dpml+dsub)
if (xx >= 0) and (xx <= d):
return math.pi*p.y/gp + ph*xx/d
else:
return math.pi*p.y/gp - ph*xx/d + 2*ph
# return the anisotropic permittivity tensor for a uniaxial, twisted nematic liquid crystal
def lc_mat(p):
# rotation matrix for rotation around x axis
Rx = mp.Matrix(mp.Vector3(1,0,0),mp.Vector3(0,math.cos(phi(p)),math.sin(phi(p))),mp.Vector3(0,-math.sin(phi(p)),math.cos(phi(p))))
lc_epsilon = Rx * epsilon_diag * Rx.transpose()
lc_epsilon_diag = mp.Vector3(lc_epsilon[0].x,lc_epsilon[1].y,lc_epsilon[2].z)
lc_epsilon_offdiag = mp.Vector3(lc_epsilon[1].x,lc_epsilon[2].x,lc_epsilon[2].y)
return mp.Medium(epsilon_diag=lc_epsilon_diag,epsilon_offdiag=lc_epsilon_offdiag)
geometry = [mp.Block(center=mp.Vector3(-0.5*sx+0.5*(dpml+dsub)),size=mp.Vector3(dpml+dsub,mp.inf,mp.inf),material=mp.Medium(index=n_0)),
mp.Block(center=mp.Vector3(-0.5*sx+dpml+dsub+d),size=mp.Vector3(2*d,mp.inf,mp.inf),material=lc_mat)]
# linear-polarized planewave pulse source
src_pt = mp.Vector3(-0.5*sx+dpml+0.3*dsub,0,0)
sources = [mp.Source(mp.GaussianSource(fcen,fwidth=0.05*fcen), component=mp.Ez, center=src_pt, size=mp.Vector3(0,sy,0)),
mp.Source(mp.GaussianSource(fcen,fwidth=0.05*fcen), component=mp.Ey, center=src_pt, size=mp.Vector3(0,sy,0))]
sim = mp.Simulation(resolution=resolution,
cell_size=cell_size,
boundary_layers=pml_layers,
k_point=k_point,
sources=sources,
default_material=mp.Medium(index=n_0))
tran_pt = mp.Vector3(0.5*sx-dpml-0.5*dpad,0,0)
tran_flux = sim.add_flux(fcen, 0, 1, mp.FluxRegion(center=tran_pt, size=mp.Vector3(0,sy,0)))
sim.run(until_after_sources=100)
input_flux = mp.get_fluxes(tran_flux)
sim.reset_meep()
sim = mp.Simulation(resolution=resolution,
cell_size=cell_size,
boundary_layers=pml_layers,
k_point=k_point,
sources=sources,
geometry=geometry)
tran_flux = sim.add_flux(fcen, 0, 1, mp.FluxRegion(center=tran_pt, size=mp.Vector3(0,sy,0)))
sim.run(until_after_sources=300)
res1 = sim.get_eigenmode_coefficients(tran_flux, range(1,nmode+1), eig_parity=mp.ODD_Z+mp.EVEN_Y)
res2 = sim.get_eigenmode_coefficients(tran_flux, range(1,nmode+1), eig_parity=mp.EVEN_Z+mp.ODD_Y)
angles = [math.degrees(math.acos(kdom.x/fcen)) for kdom in res1.kdom]
return input_flux[0], angles, res1.alpha[:,0,0], res2.alpha[:,0,0];
ph_uniaxial = 0 # chiral layer twist angle for uniaxial grating
ph_twisted = 70 # chiral layer twist angle for bilayer grating
gp = 6.5 # grating period
nmode = 5 # number of mode coefficients to compute
dd = np.arange(0.1,3.5,0.1) # chiral layer thickness
m0_uniaxial = np.zeros(dd.size)
m1_uniaxial = np.zeros(dd.size)
ang_uniaxial = np.zeros(dd.size)
m0_twisted = np.zeros(dd.size)
m1_twisted = np.zeros(dd.size)
ang_twisted = np.zeros(dd.size)
for k in range(len(dd)):
input_flux, angles, coeffs1, coeffs2 = pol_grating(0.5*dd[k],math.radians(ph_uniaxial),gp,nmode)
tran = (abs(coeffs1)**2+abs(coeffs2)**2)/input_flux
for m in range(nmode):
print("tran (uniaxial):, {}, {:.2f}, {:.5f}".format(m,angles[m],tran[m]))
m0_uniaxial[k] = tran[0]
m1_uniaxial[k] = tran[1]
ang_uniaxial[k] = angles[1]
input_flux, angles, coeffs1, coeffs2 = pol_grating(dd[k],math.radians(ph_twisted),gp,nmode)
tran = (abs(coeffs1)**2+abs(coeffs2)**2)/input_flux
for m in range(nmode):
print("tran (twisted):, {}, {:.2f}, {:.5f}".format(m,angles[m],tran[m]))
m0_twisted[k] = tran[0]
m1_twisted[k] = tran[1]
ang_twisted[k] = angles[1]
cos_angles = [math.cos(math.radians(t)) for t in ang_uniaxial]
tran = m0_uniaxial+2*m1_uniaxial
eff_m0 = m0_uniaxial/tran
eff_m1 = (2*m1_uniaxial/tran)/cos_angles
phase = delta_n*dd/wvl
eff_m0_analytic = [math.cos(math.pi*p)**2 for p in phase]
eff_m1_analytic = [math.sin(math.pi*p)**2 for p in phase]
plt.figure(dpi=150)
plt.subplot(1,2,1)
plt.plot(phase,eff_m0,'bo-',clip_on=False,label='0th order (meep)')
plt.plot(phase,eff_m0_analytic,'b--',clip_on=False,label='0th order (analytic)')
plt.plot(phase,eff_m1,'ro-',clip_on=False,label='±1 orders (meep)')
plt.plot(phase,eff_m1_analytic,'r--',clip_on=False,label='±1 orders (analytic)')
plt.axis([0, 1.0, 0, 1])
plt.xticks([t for t in np.arange(0,1.2,0.2)])
plt.xlabel("phase delay Δnd/λ")
plt.ylabel("diffraction efficiency @ λ = 0.54 μm")
plt.legend(loc='center')
plt.title("homogeneous uniaxial grating")
cos_angles = [math.cos(math.radians(t)) for t in ang_twisted]
tran = m0_twisted+2*m1_twisted
eff_m0 = m0_twisted/tran
eff_m1 = (2*m1_twisted/tran)/cos_angles
plt.subplot(1,2,2)
plt.plot(phase,eff_m0,'bo-',clip_on=False,label='0th order (meep)')
plt.plot(phase,eff_m1,'ro-',clip_on=False,label='±1 orders (meep)')
plt.axis([0, 1.0, 0, 1])
plt.xticks([t for t in np.arange(0,1.2,0.2)])
plt.xlabel("phase delay Δnd/λ")
plt.ylabel("diffraction efficiency @ λ = 0.54 μm")
plt.legend(loc='center')
plt.title("bilayer twisted-nematic grating")
plt.show()
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