Dielectric patch array

Transmission spectrum.

import matplotlib.pyplot as plt
import numpy as np
from matplotlib.colors import ListedColormap
from scipy.constants import c, e, h

import nannos as nn

Results are compared to the reference [Tikhodeev2002].

eps_quartz = 2.132
eps_active = 3.97

N = 2**7
period = 0.68
l_patch = 0.8 * period

Define the lattice

lattice = nn.Lattice(([period, 0], [0, period]), discretization=(N, N))

Define the slab layer with a square patch

epsilon = lattice.ones() * eps_quartz
square = lattice.square(center=(0.5 * period, 0.5 * period), width=l_patch)
epsilon[square] = eps_active

slab = lattice.Layer("Slab", thickness=0.12)
slab.epsilon = epsilon

cmap = ListedColormap(["#dddddd", "#73a0e8"])


plt.figure(figsize=(3, 2.5))
im = slab.plot(cmap=cmap)
cbar = plt.colorbar(im[0], ticks=[eps_quartz, eps_active])
plt.xlabel(r"$x$ ($\mu$m)")
plt.ylabel(r"$y$ ($\mu$m)")
plt.title(r"$\varepsilon$")
plt.axis("scaled")
plt.tight_layout()
plt.show()

Define the simulation

sup = lattice.Layer("Superstrate", epsilon=1)
sub = lattice.Layer("Substrate", epsilon=eps_quartz)
stack = [sup, slab, sub]


def compute_transmission(fev):
    w = h * c / e / (fev * 1e-6)
    pw = nn.PlaneWave(wavelength=w, angles=(0, 0, 90))
    sim = nn.Simulation(stack, pw, 100, formulation="original")
    R, T = sim.diffraction_efficiencies()
    print(f"f = {fev}eV")
    print("T = ", T)
    return T


freqsev = np.linspace(1, 2.6, 101)
fev_adapted, transmission = nn.adaptive_sampler(
    compute_transmission, freqsev, max_bend=10, max_x_rel=0.001, max_df=0.005
)

f = 1.0eV
T =  0.875079841479671
f = 1.016eV
T =  0.8733086223140683
f = 1.032eV
T =  0.8715383631793254
f = 1.048eV
T =  0.8697632452161226
f = 1.064eV
T =  0.8679753643630375
f = 1.08eV
T =  0.8661639185691241
f = 1.096eV
T =  0.8643139580601631
f = 1.112eV
T =  0.8624043894850065
f = 1.1280000000000001eV
T =  0.8604046400838723
f = 1.1440000000000001eV
T =  0.8582687627131041
f = 1.16eV
T =  0.8559242630674468
f = 1.176eV
T =  0.853248908507815
f = 1.192eV
T =  0.8500162551486128
f = 1.208eV
T =  0.8457424321117241
f = 1.224eV
T =  0.8391067025346531
f = 1.24eV
T =  0.8237634300182944
f = 1.256eV
T =  0.8386779297361336
f = 1.272eV
T =  0.8588077634164764
f = 1.288eV
T =  0.8636751296187558
f = 1.304eV
T =  0.8657239751848143
f = 1.32eV
T =  0.8667531922358099
f = 1.336eV
T =  0.8673049000582131
f = 1.3519999999999999eV
T =  0.8676023288070067
f = 1.3679999999999999eV
T =  0.867755635682578
f = 1.384eV
T =  0.867826078859081
f = 1.4eV
T =  0.8678507527654894
f = 1.416eV
T =  0.8678536045987653
f = 1.432eV
T =  0.8678508873425411
f = 1.448eV
T =  0.8678540863103513
f = 1.464eV
T =  0.86787159396402
f = 1.48eV
T =  0.8679097202856495
f = 1.496eV
T =  0.8679733303581473
f = 1.512eV
T =  0.8680662633974666
f = 1.528eV
T =  0.8681916195246092
f = 1.544eV
T =  0.8683519654283142
f = 1.56eV
T =  0.8685494917004616
f = 1.576eV
T =  0.8687861459781006
f = 1.592eV
T =  0.8690637648599786
f = 1.608eV
T =  0.8693842359783003
f = 1.624eV
T =  0.8697497500218311
f = 1.6400000000000001eV
T =  0.8701632865790704
f = 1.6560000000000001eV
T =  0.8706297494237245
f = 1.6720000000000002eV
T =  0.8711592140740528
f = 1.6880000000000002eV
T =  0.8717790347907778
f = 1.704eV
T =  0.8726033789586823
f = 1.72eV
T =  0.8749303101689936
f = 1.736eV
T =  0.875209108674189
f = 1.752eV
T =  0.874018101264678
f = 1.768eV
T =  0.8748855705530771
f = 1.784eV
T =  0.8761956741227044
f = 1.8eV
T =  0.8772752161166499
f = 1.816eV
T =  0.8784981407096406
f = 1.832eV
T =  0.8783193204583583
f = 1.8479999999999999eV
T =  0.87913143638086
f = 1.8639999999999999eV
T =  0.8803321548944225
f = 1.88eV
T =  0.8816844652984465
f = 1.896eV
T =  0.8831262937824182
f = 1.912eV
T =  0.8846306402673378
f = 1.928eV
T =  0.8861822333849293
f = 1.944eV
T =  0.8877707163615692
f = 1.96eV
T =  0.8893880420072322
f = 1.976eV
T =  0.8910272821706133
f = 1.992eV
T =  0.8926819992532338
f = 2.008eV
T =  0.8943458647504972
f = 2.024eV
T =  0.8960123897023824
f = 2.04eV
T =  0.8976747001143647
f = 2.056eV
T =  0.899325317703472
f = 2.072eV
T =  0.9009559161569054
f = 2.088eV
T =  0.9025570236568881
f = 2.104eV
T =  0.9041176360673845
f = 2.12eV
T =  0.9056246911735147
f = 2.136eV
T =  0.9070623292829234
f = 2.152eV
T =  0.9084108220304947
f = 2.168eV
T =  0.9096449750081104
f = 2.184eV
T =  0.9107316723219699
f = 2.2eV
T =  0.9116259735256356
f = 2.216eV
T =  0.9122646683665366
f = 2.232eV
T =  0.9125551512430047
f = 2.248eV
T =  0.9123551829951925
f = 2.2640000000000002eV
T =  0.9114336793410751
f = 2.2800000000000002eV
T =  0.9093886409693751
f = 2.2960000000000003eV
T =  0.9054580357168606
f = 2.3120000000000003eV
T =  0.8980276458916097
f = 2.3280000000000003eV
T =  0.8831396572552425
f = 2.3440000000000003eV
T =  0.8491994955129943
f = 2.3600000000000003eV
T =  0.763570988661018
f = 2.3760000000000003eV
T =  0.7584567496138948
f = 2.3920000000000003eV
T =  0.9124831849670401
f = 2.408eV
T =  0.9472430348192934
f = 2.424eV
T =  0.9525365054953068
f = 2.44eV
T =  0.9465871455188564
f = 2.456eV
T =  0.7412617636861561
f = 2.472eV
T =  0.9639128776982578
f = 2.488eV
T =  0.9632136370015142
f = 2.504eV
T =  0.9625908360098736
f = 2.52eV
T =  0.96321837683802
f = 2.536eV
T =  0.9638868276185957
f = 2.552eV
T =  0.9652453162011098
f = 2.568eV
T =  0.96503451064423
f = 2.584eV
T =  0.9650926772357361
f = 2.6eV
T =  0.9652127678832587
f = 1.216eV
T =  0.8428785851695566
f = 1.232eV
T =  0.8336026432436981
f = 1.248eV
T =  0.7837331290594014
f = 1.28eV
T =  0.86183396094491
f = 1.296eV
T =  0.8648868907819696
f = 1.312eV
T =  0.8663202608944343
f = 1.712eV
T =  0.8732795543975511
f = 1.728eV
T =  0.888214618514185
f = 1.744eV
T =  0.8738705943543547
f = 1.76eV
T =  0.8743181340505738
f = 1.776eV
T =  0.8756338213489687
f = 1.824eV
T =  0.8787597228461658
f = 1.8559999999999999eV
T =  0.8797059517126271
f = 2.2720000000000002eV
T =  0.910587314753518
f = 2.2880000000000003eV
T =  0.9077303861140447
f = 2.3040000000000003eV
T =  0.90234185732466
f = 2.3200000000000003eV
T =  0.8919461925042316
f = 2.3360000000000003eV
T =  0.8699183521015544
f = 2.3520000000000003eV
T =  0.815589316681806
f = 2.3680000000000003eV
T =  0.7153337600487691
f = 2.3840000000000003eV
T =  0.8556882895073519
f = 2.416eV
T =  0.9514215369225753
f = 2.432eV
T =  0.9513811712493273
f = 2.448eV
T =  0.926387963870859
f = 2.464eV
T =  0.9575863177373318
f = 2.48eV
T =  0.9638122666273472
f = 2.544eV
T =  0.9666385977220846
f = 2.56eV
T =  0.9650219553682995
f = 1.228eV
T =  0.8366596015649678
f = 1.236eV
T =  0.8295699219383965
f = 1.244eV
T =  0.8138594679113729
f = 1.252eV
T =  0.8180195246027361
f = 1.264eV
T =  0.8530967050692031
f = 1.284eV
T =  0.8628572593658421
f = 1.292eV
T =  0.8643397263927852
f = 1.716eV
T =  0.8738552175580628
f = 1.724eV
T =  0.8777889940327103
f = 1.732eV
T =  0.8816670208448514
f = 1.74eV
T =  0.8740767096821042
f = 1.748eV
T =  0.8739035028594004
f = 2.3240000000000003eV
T =  0.8879662472807288
f = 2.3320000000000003eV
T =  0.8772302140410604
f = 2.3400000000000003eV
T =  0.8607699705810344
f = 2.3480000000000003eV
T =  0.8344419195438209
f = 2.3560000000000003eV
T =  0.7918638033140023
f = 2.3720000000000003eV
T =  0.7217260878368489
f = 2.3800000000000003eV
T =  0.8095009420161203
f = 2.3880000000000003eV
T =  0.8896614477468741
f = 2.4000000000000004eV
T =  0.9369536974663364
f = 2.42eV
T =  0.9522731948212323
f = 2.436eV
T =  0.9496810879189841
f = 2.444eV
T =  0.9405550658490183
f = 2.452eV
T =  0.8782826521076641
f = 2.46eV
T =  0.9252157493644315
f = 2.468eV
T =  0.9628398924617436
f = 2.476eV
T =  0.9640038365299946
f = 2.528eV
T =  0.9634404320358981
f = 2.548eV
T =  0.9656956601055311
f = 2.556eV
T =  0.9650776959993574
f = 1.238eV
T =  0.8269715864139365
f = 1.242eV
T =  0.8196171289469852
f = 1.246eV
T =  0.8047294942663055
f = 1.25eV
T =  0.7925021808831064
f = 1.254eV
T =  0.8308281807399327
f = 1.26eV
T =  0.8478708494845185
f = 1.268eV
T =  0.8564640721618737
f = 1.276eV
T =  0.8605261679804405
f = 1.722eV
T =  0.8759522900864198
f = 1.726eV
T =  0.8814806420706693
f = 1.73eV
T =  0.8896384699870566
f = 1.738eV
T =  0.8744263865675015
f = 2.3640000000000003eV
T =  0.7344930580830582
f = 2.3740000000000006eV
T =  0.7369171494729554
f = 2.3900000000000006eV
T =  0.9022704535971096
f = 2.3960000000000004eV
T =  0.9273378337045731
f = 2.412eV
T =  0.9498349608534649
f = 2.428eV
T =  0.952258784604358
f = 2.4379999999999997eV
T =  0.948368624452553
f = 2.442eV
T =  0.9441138079826908
f = 2.4459999999999997eV
T =  0.9351643086980231
f = 2.45eV
T =  0.910599672244609
f = 2.4539999999999997eV
T =  0.8068868013608411
f = 2.458eV
T =  0.8535677107137496
f = 2.4619999999999997eV
T =  0.9489040580393502
f = 2.466eV
T =  0.9612079442052016
f = 2.4699999999999998eV
T =  0.9635896315856931
f = 2.54eV
T =  0.9653315057983358
f = 1.245eV
T =  0.8099241274035947
f = 1.2469999999999999eV
T =  0.7972016418487191
f = 1.249eV
T =  0.7637016225848388
f = 1.251eV
T =  0.807882010726431
f = 1.2530000000000001eV
T =  0.8253062730436738
f = 1.255eV
T =  0.8351695551755751
f = 1.258eV
T =  0.8440082674293623
f = 1.262eV
T =  0.8507997891292914
f = 1.266eV
T =  0.8549453070957892
f = 1.27eV
T =  0.8577329209102207
f = 1.7229999999999999eV
T =  0.8767241639779254
f = 1.725eV
T =  0.8793007948680457
f = 1.7269999999999999eV
T =  0.884537858385385
f = 1.729eV
T =  0.8907578125360167
f = 1.7309999999999999eV
T =  0.885643912924382
f = 1.734eV
T =  0.8770604200833987
f = 2.3660000000000005eV
T =  0.7226947284435292
f = 2.37eV
T =  0.7145325028051333
f = 2.3980000000000006eV
T =  0.9326652271568051
f = 2.404eV
T =  0.9431942238265526
f = 2.4139999999999997eV
T =  0.9507336212483538
f = 2.418eV
T =  0.9519275978640516
f = 2.434eV
T =  0.9506577161568968
f = 2.441eV
T =  0.9454561432306491
f = 2.445eV
T =  0.9381582585674206
f = 2.447eV
T =  0.9313522398553993
f = 2.449eV
T =  0.9197502740591198
f = 2.453eV
T =  0.8492344731662907
f = 2.457eV
T =  0.7884725694262412
f = 2.461eV
T =  0.9401540892032634
f = 2.463eV
T =  0.9542294853247199
f = 2.465eV
T =  0.9597639596296397
f = 2.469eV
T =  0.9632880167636726
f = 2.532eV
T =  0.9635703403388872
f = 2.5460000000000003eV
T =  0.9661319241251238
f = 1.733eV
T =  0.8788550629193908
f = 1.7349999999999999eV
T =  0.8759307715741134
f = 1.737eV
T =  0.8747385613914042
f = 2.365eV
T =  0.7281765951458781
f = 2.369eV
T =  0.7139973968125348
f = 2.3710000000000004eV
T =  0.7170898239180566
f = 2.373eV
T =  0.7283892865065307
f = 2.4059999999999997eV
T =  0.9454384420187295
f = 2.41eV
T =  0.9486877073573503
f = 2.451eV
T =  0.8975471710364511
f = 2.455eV
T =  0.7579412492487018
f = 2.459eV
T =  0.8989414162983563
f = 2.467eV
T =  0.9621800536482643
f = 2.5380000000000003eV
T =  0.9643477684883892
f = 2.542eV
T =  0.9665132321638428
f = 2.362eV
T =  0.7486560100870004
f = 2.3670000000000004eV
T =  0.7183206747796664

Figure 4 from [Tikhodeev2002].

plt.figure()
plt.plot(fev_adapted * 1000, transmission, c="#be4c83")
plt.ylim(0.4, 1)
plt.xlabel("frequency (meV)")
plt.ylabel("Transmissivity")
plt.tight_layout()

Plot the fields at the resonant frequency of 2456meV

fev = 2.456
w = h * c / e / (fev * 1e-6)  # /1000

pw = nn.PlaneWave(wavelength=w, angles=(0, 0, 90))
sim = nn.Simulation(stack, pw, 151, formulation="tangent")
E, H = sim.get_field_grid("Superstrate", shape=(N, N))

Ex, Ey, Ez = E[:, :, :, 0]
Hx, Hy, Hz = H[:, :, :, 0]
nE2 = np.abs(Ex) ** 2 + np.abs(Ey) ** 2  # + np.abs(Ez)**2
nH2 = np.abs(Hx) ** 2 + np.abs(Hy) ** 2  # + np.abs(Hz)**2

Electric field

extent = [0, period, 0, period]
x, y = np.linspace(0, period, N), np.linspace(0, period, N)

plt.figure()
plt.imshow(epsilon.real, cmap="Greys", origin="lower", extent=extent)
plt.imshow(nE2, alpha=0.9, origin="lower", extent=extent)
plt.colorbar()
s = 3
plt.quiver(x[::s], y[::s], Ex[::s, ::s].real, Ey[::s, ::s].real, color="w")
plt.xlabel(r"$x$ ($\mu$m)")
plt.ylabel(r"$y$ ($\mu$m)")
plt.title("$E$")
plt.tight_layout()
plt.show()

Magnetic field

plt.figure()
plt.imshow(epsilon.real, cmap="Greys", origin="lower", extent=extent)
plt.imshow(nH2, alpha=0.9, origin="lower", extent=extent)
plt.colorbar()
plt.quiver(x[::s], y[::s], Hx[::s, ::s].real, Hy[::s, ::s].real, color="w")
plt.xlabel(r"$x$ ($\mu$m)")
plt.ylabel(r"$y$ ($\mu$m)")
plt.title("$H$")
plt.tight_layout()
plt.show()

Estimated memory usage: 546 MB