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/* Copyright (C) 2005-2022 Massachusetts Institute of Technology.
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*/
/* Compute the transmission spectrum through a 4-layer 1d Bragg mirror,
and compare to the result from the analytical transfer matrices.
The transmission spectrum is computed via the dft_flux feature,
which dynamically updates the DFTs of the fields on the flux plane
as we go along. */
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <meep.hpp>
using namespace meep;
using std::complex;
const double nhi = 3, nlo = 1;
const double wlo = nhi / (nlo + nhi);
const int Nperiods = 4;
const double zsize = 10;
double eps_nlo(const vec &) { return nlo * nlo; }
double eps_bragg(const vec &pt) {
double z = pt.z() - zsize * 0.5;
if (fabs(z) * 2 > Nperiods)
return nlo * nlo;
else {
double zi;
double zf = modf(z, &zi);
if (zf < 0) zf += 1;
if (zf < wlo)
return (nlo * nlo);
else
return (nhi * nhi);
}
}
typedef complex<double> matrix2x2[2][2];
/* multiply m by transfer matrix from n1 to n2 */
inline void byT12(matrix2x2 m, double n1, double n2) {
complex<double> m00, m01, m10, m11;
double td, tod;
double n12 = n1 / n2;
td = 0.5 * (1 + n12);
tod = 0.5 * (1 - n12);
m00 = m[0][0];
m01 = m[0][1];
m10 = m[1][0];
m11 = m[1][1];
m[0][0] = m00 * td + m01 * tod;
m[0][1] = m00 * tod + m01 * td;
m[1][0] = m10 * td + m11 * tod;
m[1][1] = m10 * tod + m11 * td;
}
/* multiply m by propagation matrix through dz of index n, frequency w */
inline void byP(matrix2x2 m, double n, double w, double dz) {
complex<double> p, pc;
p = exp(complex<double>(0, n * w * dz));
pc = conj(p);
m[0][0] *= p;
m[0][1] *= pc;
m[1][0] *= p;
m[1][1] *= pc;
}
inline double abs2(complex<double> x) {
double ax = abs(x);
return ax * ax;
}
void bragg_transmission_analytic(double freq_min, double freq_max, int nfreq, double *T,
double *R) {
for (int i = 0; i < nfreq; ++i) {
double omega = 2 * pi * (freq_min + i * (freq_max - freq_min) / (nfreq - 1));
matrix2x2 Tm = {{1, 0}, {0, 1}};
for (int j = 0; j < Nperiods; ++j) {
byT12(Tm, nlo, nhi);
byP(Tm, nhi, omega, 1 - wlo);
byT12(Tm, nhi, nlo);
byP(Tm, nlo, omega, wlo);
}
complex<double> refl = -Tm[1][0] / Tm[1][1];
T[i] = abs2(Tm[0][0] + refl * Tm[0][1]);
R[i] = abs2(refl);
}
}
void bragg_transmission(double a, double freq_min, double freq_max, int nfreq, double *T, double *R,
bool use_hdf5) {
const grid_volume gv = volone(zsize, a);
structure *s = new structure(gv, eps_bragg, pml(0.5));
fields f(s);
f.use_real_fields();
structure s0(gv, eps_nlo, pml(0.5));
fields f0(&s0);
f0.use_real_fields();
vec srcpt(0.1), Tfluxpt(zsize - 0.1), Rfluxpt(0.1);
gaussian_src_time src((freq_min + freq_max) * 0.5, 0.5 / fabs(freq_max - freq_min), 0,
5 / fabs(freq_max - freq_min));
f.add_point_source(Ex, src, srcpt);
f0.add_point_source(Ex, src, srcpt);
dft_flux ft = f.add_dft_flux_plane(Tfluxpt, freq_min, freq_max, nfreq);
dft_flux fr = f.add_dft_flux_plane(Rfluxpt, freq_min, freq_max, nfreq);
dft_flux ft0 = f0.add_dft_flux_plane(Tfluxpt, freq_min, freq_max, nfreq);
dft_flux fr0 = f0.add_dft_flux_plane(Rfluxpt, freq_min, freq_max, nfreq);
while (f0.time() < nfreq / fabs(freq_max - freq_min) / 2)
f0.step();
/* we want to subtract the fields for the reflection... */
if (use_hdf5) {
/* simulate a case where the normalization is done
by a separate run and saved to a file */
fr0.save_hdf5(f, "flux", "reflection");
fr.load_hdf5(f, "flux", "reflection");
fr.scale_dfts(-1.0);
// clean up after ourselves: delete the file
h5file *ff = f.open_h5file("flux", h5file::READONLY);
ff->remove();
delete ff;
}
else
fr -= fr0;
while (f.time() < nfreq / fabs(freq_max - freq_min) / 2)
f.step();
double *flux = ft.flux();
double *flux0 = ft0.flux();
for (int i = 0; i < nfreq; ++i)
T[i] = flux[i] / flux0[i];
delete[] flux;
flux = fr.flux();
for (int i = 0; i < nfreq; ++i)
R[i] = -flux[i] / flux0[i];
delete[] flux;
delete[] flux0;
delete s; // tests whether okay to delete s before f
}
inline double max2(double a, double b) { return (a > b ? a : b); }
inline double min2(double a, double b) { return (a < b ? a : b); }
inline double max2a(double a, double b) { return max2(abs(a), abs(b)); }
inline double sqr(double x) { return x * x; }
/* The discretization errors tend to result in a *shift* of the spectral
features more than a change in their amplitude. Because these features
are very sharp (e.g. at the gap edges), it is more appropriate to compute
errors via the distance from a point to the curve, rather than just
the difference of the abscissae. That's what this function does. */
double distance_from_curve(int n, double dx, double ys[], double x, double y) {
double d = meep::infinity;
for (int i = 1; i < n; ++i) {
double theta = atan2(ys[i] - ys[i - 1], dx);
double L = sqrt(sqr(dx) + sqr(ys[i] - ys[i - 1]));
double x0 = x - (i - 1) * dx;
double y0 = y - ys[i - 1];
double x0p = x0 * cos(theta) + y0 * sin(theta);
double y0p = y0 * cos(theta) - x0 * sin(theta);
if (x0p < 0)
d = min2(sqrt(sqr(x0) + sqr(y0)), d);
else if (x0p > L)
d = min2(sqrt(sqr(x - i * dx) + sqr(y - ys[i])), d);
else
d = min2(abs(y0p), d);
}
return d;
}
void doit(bool use_hdf5) {
const int nfreq = 100;
const double freq_min = 0.1, freq_max = 0.5;
double *T = new double[nfreq];
double *R = new double[nfreq];
bragg_transmission(40.0, freq_min, freq_max, nfreq, T, R, use_hdf5);
double *T0 = new double[nfreq];
double *R0 = new double[nfreq];
bragg_transmission_analytic(freq_min, freq_max, nfreq, T0, R0);
double dfreq = (freq_max - freq_min) / (nfreq - 1);
if (0) { // output transmission & reflection spectra for debugging
master_printf("transmission:, freq (c/a), T, R, T0, R0\n");
for (int i = 0; i < nfreq; ++i)
master_printf("transmission:, %g, %g, %g, %g, %g\n", freq_min + i * dfreq, T[i], R[i], T0[i],
R0[i]);
}
double maxerrT = 0, maxerrR = 0;
for (int i = 0; i < nfreq; ++i) {
double errT = distance_from_curve(nfreq, dfreq, T0, i * dfreq, T[i]);
double errR = distance_from_curve(nfreq, dfreq, R0, i * dfreq, R[i]);
if (errT > maxerrT) maxerrT = errT;
if (errR > maxerrR) maxerrR = errR;
if (errT * sqr(freq_min / (freq_min + i * dfreq)) > 0.01)
meep::abort("large error %g at freq = %g: T = %g instead of %g\n", errT, freq_min + i * dfreq,
T[i], T0[i]);
if (errR * sqr(freq_min / (freq_min + i * dfreq)) > 0.01)
meep::abort("large error %g at freq = %g: R = %g instead of %g\n", errR, freq_min + i * dfreq,
R[i], R0[i]);
}
master_printf("Done (max. err in T = %e, in R = %e)\n", maxerrT, maxerrR);
delete[] R0;
delete[] T0;
delete[] R;
delete[] T;
}
int main(int argc, char **argv) {
initialize mpi(argc, argv);
verbosity = 0;
#ifdef HAVE_HDF5
doit(true);
#endif
doit(false);
return 0;
}
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