dft.cpp

/* 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
 */

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <string.h>
#include <algorithm>
#include <assert.h>
#include "meep.hpp"
#include "meep_internals.hpp"

using namespace std;

namespace meep {

std::vector<double> linspace(double freq_min, double freq_max, size_t Nfreq) {
  double dfreq = Nfreq <= 1 ? 0.0 : (freq_max - freq_min) / (Nfreq - 1);
  std::vector<double> freq(Nfreq);
  if (Nfreq <= 1)
    freq[0] = (freq_min + freq_max) * 0.5;
  else
    for (size_t i = 0; i < Nfreq; ++i)
      freq[i] = freq_min + i * dfreq;

  return freq;
}

struct dft_chunk_data { // for passing to field::loop_in_chunks as void*
  component c;
  int vc;
  std::vector<double> omega;
  complex<double> stored_weight, extra_weight;
  double dt_factor;
  bool include_dV_and_interp_weights;
  bool sqrt_dV_and_interp_weights;
  bool empty_dim[5];
  dft_chunk *dft_chunks;
  int decimation_factor;
  bool persist;
};

dft_chunk::dft_chunk(fields_chunk *fc_, ivec is_, ivec ie_, vec s0_, vec s1_, vec e0_, vec e1_,
                     double dV0_, double dV1_, component c_, bool use_centered_grid,
                     complex<double> phase_factor, ivec shift_, const symmetry &S_, int sn_,
                     const void *data_) {
  dft_chunk_data *data = (dft_chunk_data *)data_;
  if (!fc_->f[c_][0]) meep::abort("invalid fields_chunk/component combination in dft_chunk");

  fc = fc_;
  is = is_;
  ie = ie_;
  s0 = s0_;
  s1 = s1_;
  e0 = e0_;
  e1 = e1_;
  dV0 = dV0_;
  dV1 = dV1_;

  persist = data->persist;

  c = c_;

  /* for adjoint calculations, we want to pad
  (or expand) the dimensions of the dft region
  to account for boundary effects. We will pad
  by 1 pixel in each dimension, while ensuring
  we don't step outside of the chunk loop itself
  */
  if(persist){
    is_old = is_;
    ie_old = ie_;
    is = max(is-one_ivec(fc->gv.dim)*2,fc->gv.little_corner());
    ie = min(ie+one_ivec(fc->gv.dim)*2,fc->gv.big_corner());
  }

  if (use_centered_grid)
    fc->gv.yee2cent_offsets(c, avg1, avg2);
  else
    avg1 = avg2 = 0;

  stored_weight = data->stored_weight;
  extra_weight = data->extra_weight;
  scale = stored_weight * phase_factor * data->dt_factor;

  /* this is for e.g. computing E x H, where we don't want to
     multiply by the interpolation weights or the grid_volume twice. */
  include_dV_and_interp_weights = data->include_dV_and_interp_weights;

  /* an alternative way to avoid multipling by interpolation weights twice:
     multiply by square root of the weights */
  sqrt_dV_and_interp_weights = data->sqrt_dV_and_interp_weights;

  shift = shift_;
  S = S_;
  sn = sn_;
  vc = data->vc;
  decimation_factor = data->decimation_factor;

  const int Nomega = data->omega.size();
  omega = data->omega;
  dft_phase = new complex<realnum>[Nomega];

  N = 1;
  LOOP_OVER_DIRECTIONS(is.dim, d) { N *= (ie.in_direction(d) - is.in_direction(d)) / 2 + 1; }
  dft = new complex<realnum>[N * Nomega];
  for (size_t i = 0; i < N * Nomega; ++i)
    dft[i] = 0.0;
  for (int i = 0; i < 5; ++i)
    empty_dim[i] = data->empty_dim[i];

  next_in_chunk = fc->dft_chunks;
  fc->dft_chunks = this;
  next_in_dft = data->dft_chunks;
}

dft_chunk::~dft_chunk() {
  delete[] dft;
  delete[] dft_phase;

  // delete from fields_chunk list
  dft_chunk *cur = fc->dft_chunks;
  if (cur == this)
    fc->dft_chunks = next_in_chunk;
  else {
    while (cur && cur->next_in_chunk && cur->next_in_chunk != this)
      cur = cur->next_in_chunk;
    if (cur && cur->next_in_chunk == this) cur->next_in_chunk = next_in_chunk;
  }
}

void dft_flux::remove() {
  while (E) {
    dft_chunk *nxt = E->next_in_dft;
    delete E;
    E = nxt;
  }
  while (H) {
    dft_chunk *nxt = H->next_in_dft;
    delete H;
    H = nxt;
  }
}

static void add_dft_chunkloop(fields_chunk *fc, int ichunk, component cgrid, ivec is, ivec ie,
                              vec s0, vec s1, vec e0, vec e1, double dV0, double dV1, ivec shift,
                              complex<double> shift_phase, const symmetry &S, int sn,
                              void *chunkloop_data) {
  dft_chunk_data *data = (dft_chunk_data *)chunkloop_data;
  (void)ichunk; // unused

  component c = S.transform(data->c, -sn);
  if (c >= NUM_FIELD_COMPONENTS || !fc->f[c][0]) return; // this chunk doesn't have component c

  data->dft_chunks =
      new dft_chunk(fc, is, ie, s0, s1, e0, e1, dV0, dV1, c, cgrid == Centered,
                    shift_phase * S.phase_shift(c, sn), shift, S, sn, chunkloop_data);
}

dft_chunk *fields::add_dft(component c, const volume &where, const double *freq, size_t Nfreq,
                           bool include_dV_and_interp_weights, complex<double> stored_weight,
                           dft_chunk *chunk_next, bool sqrt_dV_and_interp_weights,
                           complex<double> extra_weight, bool use_centered_grid,
                           int vc, int decimation_factor, bool persist) {
  if (coordinate_mismatch(gv.dim, c)) return NULL;

  /* If you call add_dft before adding sources, it will do nothing
     since no fields will be found.   This is almost certainly not
     what the user wants. */
  if (!components_allocated)
    meep::abort("allocate field components (by adding sources) before adding dft objects");
  if (!include_dV_and_interp_weights && sqrt_dV_and_interp_weights)
    meep::abort("include_dV_and_interp_weights must be true for sqrt_dV_and_interp_weights=true in "
          "add_dft");

  dft_chunk_data data;
  data.persist = persist;
  data.c = c;
  data.vc = vc;

  if (decimation_factor == 0) {
    double src_freq_max = 0;
    for (src_time *s = sources; s; s = s->next) {
      if (s->get_fwidth() == 0)
        decimation_factor = 1;
      else
        src_freq_max = std::max(src_freq_max, std::abs(s->frequency().real())+0.5*s->get_fwidth());
    }
    double freq_max = 0;
    for (size_t i = 0; i < Nfreq; ++i)
      freq_max = std::max(freq_max, std::abs(freq[i]));
    if ((freq_max > 0) && (src_freq_max > 0))
      decimation_factor = std::max(1, int(std::floor(1/(dt*(freq_max + src_freq_max)))));
    else
      decimation_factor = 1;
  }
  data.decimation_factor = decimation_factor;

  data.omega.resize(Nfreq);
  for (size_t i = 0; i < Nfreq; ++i)
    data.omega[i] = 2 * pi * freq[i];
  data.stored_weight = stored_weight;
  data.extra_weight = extra_weight;
  data.dt_factor = dt / sqrt(2.0 * pi) * decimation_factor;
  data.include_dV_and_interp_weights = include_dV_and_interp_weights;
  data.sqrt_dV_and_interp_weights = sqrt_dV_and_interp_weights;
  data.empty_dim[0] = data.empty_dim[1] = data.empty_dim[2] = data.empty_dim[3] =
      data.empty_dim[4] = false;
  LOOP_OVER_DIRECTIONS(where.dim, d) { data.empty_dim[d] = where.in_direction(d) == 0; }
  data.dft_chunks = chunk_next;
  loop_in_chunks(add_dft_chunkloop, (void *)&data, where, use_centered_grid ? Centered : c);

  return data.dft_chunks;
}

dft_chunk *fields::add_dft(const volume_list *where, const std::vector<double> &freq,
                           bool include_dV_and_interp_weights, bool persist) {
  dft_chunk *chunks = 0;
  while (where) {
    if (is_derived(where->c)) meep::abort("derived_component invalid for dft");
    complex<double> stored_weight = where->weight;
    chunks = add_dft(component(where->c), where->v, freq, include_dV_and_interp_weights,
                     stored_weight, chunks, persist);
    where = where->next;
  }
  return chunks;
}

void fields::update_dfts() {
  am_now_working_on(FourierTransforming);
  for (int i = 0; i < num_chunks; i++)
    if (chunks[i]->is_mine()) chunks[i]->update_dfts(time(), time() - 0.5 * dt, t);
  finished_working();
}

void fields_chunk::update_dfts(double timeE, double timeH, int current_step) {
  if (doing_solve_cw) return;
  for (dft_chunk *cur = dft_chunks; cur; cur = cur->next_in_chunk) {
    if ((current_step % cur->get_decimation_factor()) == 0) {
      cur->update_dft(is_magnetic(cur->c) ? timeH : timeE);
    }
  }
}

void dft_chunk::update_dft(double time) {
  if (!fc->f[c][0]) return;

  const int Nomega = omega.size();
  for (int i = 0; i < Nomega; ++i)
    dft_phase[i] = polar(1.0, omega[i] * time) * scale;

  int numcmp = fc->f[c][1] ? 2 : 1;

  PLOOP_OVER_IVECS(fc->gv, is, ie, idx) {
    size_t idx_dft = IVEC_LOOP_COUNTER;
    double w;
    if (include_dV_and_interp_weights) {
      w = IVEC_LOOP_WEIGHT(s0, s1, e0, e1, dV0 + dV1 * loop_i2);
      if (sqrt_dV_and_interp_weights) w = sqrt(w);
    }
    else
      w = 1.0;
    realnum f[2]; // real/imag field value at epsilon point
    if (avg2)
      for (int cmp = 0; cmp < numcmp; ++cmp)
        f[cmp] = (w * 0.25) * (fc->f[c][cmp][idx] + fc->f[c][cmp][idx + avg1] +
                               fc->f[c][cmp][idx + avg2] + fc->f[c][cmp][idx + (avg1 + avg2)]);
    else if (avg1)
      for (int cmp = 0; cmp < numcmp; ++cmp)
        f[cmp] = (w * 0.5) * (fc->f[c][cmp][idx] + fc->f[c][cmp][idx + avg1]);
    else
      for (int cmp = 0; cmp < numcmp; ++cmp)
        f[cmp] = w * fc->f[c][cmp][idx];

    if (numcmp == 2) {
      complex<realnum> fc(f[0], f[1]);
      for (int i = 0; i < Nomega; ++i)
        dft[Nomega * idx_dft + i] += dft_phase[i] * fc;
    }
    else {
      realnum fr = f[0];
      for (int i = 0; i < Nomega; ++i)
        dft[Nomega * idx_dft + i] += std::complex<realnum>{fr * dft_phase[i].real(), fr * dft_phase[i].imag()};
    }
  }
}

/* Return the L2 norm of the DFTs themselves.  This is useful
   to check whether the simulation is finished (whether all relevant fields have decayed).
   (Collective operation.) */
double fields::dft_norm() {
  am_now_working_on(Other);
  double sum = 0.0;
  for (int i = 0; i < num_chunks; i++)
    if (chunks[i]->is_mine()) sum += chunks[i]->dft_norm2(gv);
  finished_working();
  return std::sqrt(sum_to_all(sum));
}

double fields_chunk::dft_norm2(grid_volume fgv) const {
  double sum = 0.0;
  for (dft_chunk *cur = dft_chunks; cur; cur = cur->next_in_chunk)
    sum += cur->norm2(fgv);
  return sum;
}

static double sqr(std::complex<realnum> x) { return (x*std::conj(x)).real(); }

double dft_chunk::norm2(grid_volume fgv) const {
  if (!fc->f[c][0]) return 0.0;
  double sum = 0.0;
  size_t idx_dft;
  const size_t Nomega = omega.size();
  /* looping over chunks that have been "expanded"
  for adjoint calculations requires some care. Namely,
  we want to make sure we don't double count the padding
  and can replicate results with different chunk combinations.
  */
  if (persist) {
    grid_volume subgv = fgv.subvolume(is,ie,c);
    LOOP_OVER_IVECS(subgv, is_old, ie_old, idx) {
      for (size_t i = 0; i < Nomega; ++i)
        sum += sqr(dft[Nomega * idx + i]);          
    } 
  } 
  /* note we place the if outside of the
  loop to avoid branching. This routine gets
  called a lot, so let's try to stay efficient
  (at the expense of uglier code).
   */
  else{
    LOOP_OVER_IVECS(fgv, is, ie, idx) {
     idx_dft = IVEC_LOOP_COUNTER;
     for (size_t i = 0; i < Nomega; ++i)
        sum += sqr(dft[Nomega * idx_dft + i]); 
    }
  }

  return sum;
}

// return the maximum decimation factor across
// all dft regions
int fields::max_decimation() const {
  int maxdec = 1;
  for (int i = 0; i < num_chunks; i++)
    if (chunks[i]->is_mine())
      maxdec = std::max(maxdec, chunks[i]->max_decimation());
  return max_to_all(maxdec);
}

int fields_chunk::max_decimation() const {
  int maxdec = std::numeric_limits<int>::min();
  for (dft_chunk *cur = dft_chunks; cur; cur = cur->next_in_chunk)
    maxdec = std::max(maxdec, cur->get_decimation_factor());
  return maxdec;
}

// return the maximum abs(freq) over all DFT chunks
double fields::dft_maxfreq() const {
  double maxfreq = 0;
  for (int i = 0; i < num_chunks; i++)
    if (chunks[i]->is_mine())
      maxfreq = std::max(maxfreq, chunks[i]->dft_maxfreq());
  return max_to_all(maxfreq);
}

double fields_chunk::dft_maxfreq() const {
  double maxomega = 0;
  for (dft_chunk *cur = dft_chunks; cur; cur = cur->next_in_chunk)
    maxomega = std::max(maxomega, cur->maxomega());
  return maxomega / (2*meep::pi);
}

double dft_chunk::maxomega() const {
  double maxomega = 0;
  for (const auto& o : omega)
    maxomega = std::max(maxomega, std::abs(o));
  return maxomega;
}

void dft_chunk::scale_dft(complex<double> scale) {
  for (size_t i = 0; i < N * omega.size(); ++i)
    dft[i] *= scale;
  if (next_in_dft) next_in_dft->scale_dft(scale);
}

void dft_chunk::operator-=(const dft_chunk &chunk) {
  if (c != chunk.c || N * omega.size() != chunk.N * chunk.omega.size())
    meep::abort("Mismatched chunks in dft_chunk::operator-=");

  for (size_t i = 0; i < N * omega.size(); ++i)
    dft[i] -= chunk.dft[i];

  if (next_in_dft) {
    if (!chunk.next_in_dft) meep::abort("Mismatched chunk lists in dft_chunk::operator-=");
    *next_in_dft -= *chunk.next_in_dft;
  }
}

size_t my_dft_chunks_Ntotal(dft_chunk *dft_chunks, size_t *my_start) {
  // When writing to a sharded file, we write out only the chunks we own.
size_t n = 0;
  for (dft_chunk *cur = dft_chunks; cur; cur = cur->next_in_dft)
    n += cur->N * cur->omega.size() * 2;

  *my_start = 0;
  return n;
}

size_t dft_chunks_Ntotal(dft_chunk *dft_chunks, size_t *my_start) {
  // If writing to a single parallel file, we are compute our chunks offset
  // into the single-parallel-file that has all the data.
  size_t n = my_dft_chunks_Ntotal(dft_chunks, my_start);
  *my_start = partial_sum_to_all(n) - n; // sum(n) for processes before this
  return sum_to_all(n);
}

size_t dft_chunks_Ntotal(dft_chunk *dft_chunks, size_t *my_start, bool single_parallel_file) {
  return single_parallel_file ? dft_chunks_Ntotal(dft_chunks, my_start) :
                                my_dft_chunks_Ntotal(dft_chunks, my_start);
}

// Note: the file must have been created in parallel mode, typically via fields::open_h5file.
void save_dft_hdf5(dft_chunk *dft_chunks, const char *name, h5file *file, const char *dprefix, bool single_parallel_file) {
  size_t istart;
  size_t n = dft_chunks_Ntotal(dft_chunks, &istart, single_parallel_file);

  char dataname[1024];
  snprintf(dataname, 1024,
           "%s%s"
           "%s_dft",
           dprefix ? dprefix : "", dprefix && dprefix[0] ? "_" : "", name);
  file->create_data(dataname, 1, &n);

  for (dft_chunk *cur = dft_chunks; cur; cur = cur->next_in_dft) {
    size_t Nchunk = cur->N * cur->omega.size() * 2;
    file->write_chunk(1, &istart, &Nchunk, (realnum *)cur->dft);
    istart += Nchunk;
  }
  file->done_writing_chunks();
}

void save_dft_hdf5(dft_chunk *dft_chunks, component c, h5file *file, const char *dprefix, bool single_parallel_file) {
  save_dft_hdf5(dft_chunks, component_name(c), file, dprefix, single_parallel_file);
}

void load_dft_hdf5(dft_chunk *dft_chunks, const char *name, h5file *file, const char *dprefix, bool single_parallel_file) {
  size_t istart;
  size_t n = dft_chunks_Ntotal(dft_chunks, &istart, single_parallel_file);

  char dataname[1024];
  snprintf(dataname, 1024,
           "%s%s"
           "%s_dft",
           dprefix ? dprefix : "", dprefix && dprefix[0] ? "_" : "", name);
  int file_rank;
  size_t file_dims;
  file->read_size(dataname, &file_rank, &file_dims, 1);
  if (file_rank != 1 || file_dims != n)
    meep::abort("incorrect dataset size (%zd vs. %zd) in load_dft_hdf5 %s:%s", file_dims, n,
          file->file_name(), dataname);

  for (dft_chunk *cur = dft_chunks; cur; cur = cur->next_in_dft) {
    size_t Nchunk = cur->N * cur->omega.size() * 2;
    file->read_chunk(1, &istart, &Nchunk, (realnum *)cur->dft);
    istart += Nchunk;
  }
}

void load_dft_hdf5(dft_chunk *dft_chunks, component c, h5file *file, const char *dprefix, bool single_parallel_file) {
  load_dft_hdf5(dft_chunks, component_name(c), file, dprefix, single_parallel_file);
}

dft_flux::dft_flux(const component cE_, const component cH_, dft_chunk *E_, dft_chunk *H_,
                   double fmin, double fmax, int Nf, const volume &where_,
                   direction normal_direction_, bool use_symmetry_)
    : E(E_), H(H_), cE(cE_), cH(cH_), where(where_), normal_direction(normal_direction_),
      use_symmetry(use_symmetry_) {
  freq = meep::linspace(fmin, fmax, Nf);
}

dft_flux::dft_flux(const component cE_, const component cH_, dft_chunk *E_, dft_chunk *H_,
                   const std::vector<double> &freq_, const volume &where_,
                   direction normal_direction_, bool use_symmetry_)
    : E(E_), H(H_), cE(cE_), cH(cH_), where(where_), normal_direction(normal_direction_),
      use_symmetry(use_symmetry_) {
  freq = freq_;
}

dft_flux::dft_flux(const component cE_, const component cH_, dft_chunk *E_, dft_chunk *H_,
                   const double *freq_, size_t Nfreq, const volume &where_,
                   direction normal_direction_, bool use_symmetry_)
    : freq(Nfreq), E(E_), H(H_), cE(cE_), cH(cH_), where(where_),
      normal_direction(normal_direction_), use_symmetry(use_symmetry_) {
  for (size_t i = 0; i < Nfreq; ++i)
    freq[i] = freq_[i];
}

dft_flux::dft_flux(const dft_flux &f) : where(f.where) {
  freq = f.freq;
  E = f.E;
  H = f.H;
  cE = f.cE;
  cH = f.cH;
  normal_direction = f.normal_direction;
  use_symmetry = f.use_symmetry;
}

double *dft_flux::flux() {
  const size_t Nfreq = freq.size();
  double *F = new double[Nfreq];
  for (size_t i = 0; i < Nfreq; ++i)
    F[i] = 0;
  for (dft_chunk *curE = E, *curH = H; curE && curH;
       curE = curE->next_in_dft, curH = curH->next_in_dft)
    for (size_t k = 0; k < curE->N; ++k)
      for (size_t i = 0; i < Nfreq; ++i)
        F[i] += real(curE->dft[k * Nfreq + i] * conj(curH->dft[k * Nfreq + i]));
  double *Fsum = new double[Nfreq];
  sum_to_all(F, Fsum, int(Nfreq));
  delete[] F;
  return Fsum;
}

void dft_flux::save_hdf5(h5file *file, const char *dprefix) {
  save_dft_hdf5(E, cE, file, dprefix);
  file->prevent_deadlock(); // hackery
  save_dft_hdf5(H, cH, file, dprefix);
}

void dft_flux::load_hdf5(h5file *file, const char *dprefix) {
  load_dft_hdf5(E, cE, file, dprefix);
  file->prevent_deadlock(); // hackery
  load_dft_hdf5(H, cH, file, dprefix);
}

void dft_flux::save_hdf5(fields &f, const char *fname, const char *dprefix, const char *prefix) {
  h5file *ff = f.open_h5file(fname, h5file::WRITE, prefix);
  save_hdf5(ff, dprefix);
  delete ff;
}

void dft_flux::load_hdf5(fields &f, const char *fname, const char *dprefix, const char *prefix) {
  h5file *ff = f.open_h5file(fname, h5file::READONLY, prefix);
  load_hdf5(ff, dprefix);
  delete ff;
}

void dft_flux::scale_dfts(complex<double> scale) {
  if (E) E->scale_dft(scale);
  if (H) H->scale_dft(scale);
}

dft_flux fields::add_dft_flux(const volume_list *where_, const double *freq, size_t Nfreq,
                              bool use_symmetry, bool centered_grid, int decimation_factor) {
  if (!where_) // handle empty list of volumes
    return dft_flux(Ex, Hy, NULL, NULL, freq, Nfreq, v, NO_DIRECTION, use_symmetry);

  dft_chunk *E = 0, *H = 0;
  component cE[2] = {Ex, Ey}, cH[2] = {Hy, Hx};

  // the dft_flux object needs to store the (unreduced) volume for
  // mode-coefficient computation in mpb.cpp, but this only works
  // when the volume_list consists of a single volume, so it suffices
  // to store the first volume in the list.
  volume firstvol(where_->v);

  volume_list *where = use_symmetry ? S.reduce(where_) : new volume_list(where_);
  volume_list *where_save = where;
  while (where) {
    derived_component c = derived_component(where->c);
    if (coordinate_mismatch(gv.dim, component_direction(c)))
      meep::abort("coordinate-type mismatch in add_dft_flux");

    switch (c) {
      case Sx: cE[0] = Ey, cE[1] = Ez, cH[0] = Hz, cH[1] = Hy; break;
      case Sy: cE[0] = Ez, cE[1] = Ex, cH[0] = Hx, cH[1] = Hz; break;
      case Sr: cE[0] = Ep, cE[1] = Ez, cH[0] = Hz, cH[1] = Hp; break;
      case Sp: cE[0] = Ez, cE[1] = Er, cH[0] = Hr, cH[1] = Hz; break;
      case Sz:
        if (gv.dim == Dcyl)
          cE[0] = Er, cE[1] = Ep, cH[0] = Hp, cH[1] = Hr;
        else
          cE[0] = Ex, cE[1] = Ey, cH[0] = Hy, cH[1] = Hx;
        break;
      default: meep::abort("invalid flux component!");
    }

    for (int i = 0; i < 2; ++i) {
      E = add_dft(cE[i], where->v, freq, Nfreq, true,
                  where->weight * double(1 - 2 * i), E, false, std::complex<double>(1.0,0),
                  centered_grid, 0, decimation_factor);
      H = add_dft(cH[i], where->v, freq, Nfreq, false, 1.0, H, false, std::complex<double>(1.0,0),
                  centered_grid, 0, decimation_factor);
    }

    where = where->next;
  }
  delete where_save;

  // if the volume list has only one entry, store its component's direction.
  // if the volume list has > 1 entry, store NO_DIRECTION.
  direction flux_dir = (where_->next ? NO_DIRECTION : component_direction(where_->c));
  return dft_flux(cE[0], cH[0], E, H, freq, Nfreq, firstvol, flux_dir, use_symmetry);
}

dft_energy::dft_energy(dft_chunk *E_, dft_chunk *H_, dft_chunk *D_, dft_chunk *B_, double fmin,
                       double fmax, int Nf, const volume &where_)
    : E(E_), H(H_), D(D_), B(B_), where(where_) {
  freq = meep::linspace(fmin, fmax, Nf);
}

dft_energy::dft_energy(dft_chunk *E_, dft_chunk *H_, dft_chunk *D_, dft_chunk *B_,
                       const std::vector<double> &freq_, const volume &where_)
    : E(E_), H(H_), D(D_), B(B_), where(where_) {
  freq = freq_;
}

dft_energy::dft_energy(dft_chunk *E_, dft_chunk *H_, dft_chunk *D_, dft_chunk *B_,
                       const double *freq_, size_t Nfreq, const volume &where_)
    : freq(Nfreq), E(E_), H(H_), D(D_), B(B_), where(where_) {
  for (size_t i = 0; i < Nfreq; ++i)
    freq[i] = freq_[i];
}

dft_energy::dft_energy(const dft_energy &f) : where(f.where) {
  freq = f.freq;
  E = f.E;
  H = f.H;
  D = f.D;
  B = f.B;
}

double *dft_energy::electric() {
  const size_t Nfreq = freq.size();
  double *F = new double[Nfreq];
  for (size_t i = 0; i < Nfreq; ++i)
    F[i] = 0;
  for (dft_chunk *curE = E, *curD = D; curE && curD;
       curE = curE->next_in_dft, curD = curD->next_in_dft)
    for (size_t k = 0; k < curE->N; ++k)
      for (size_t i = 0; i < Nfreq; ++i)
        F[i] += 0.5 * real(conj(curE->dft[k * Nfreq + i]) * curD->dft[k * Nfreq + i]);
  double *Fsum = new double[Nfreq];
  sum_to_all(F, Fsum, int(Nfreq));
  delete[] F;
  return Fsum;
}

double *dft_energy::magnetic() {
  const size_t Nfreq = freq.size();
  double *F = new double[Nfreq];
  for (size_t i = 0; i < Nfreq; ++i)
    F[i] = 0;
  for (dft_chunk *curH = H, *curB = B; curH && curB;
       curH = curH->next_in_dft, curB = curB->next_in_dft)
    for (size_t k = 0; k < curH->N; ++k)
      for (size_t i = 0; i < Nfreq; ++i)
        F[i] += 0.5 * real(conj(curH->dft[k * Nfreq + i]) * curB->dft[k * Nfreq + i]);
  double *Fsum = new double[Nfreq];
  sum_to_all(F, Fsum, int(Nfreq));
  delete[] F;
  return Fsum;
}

double *dft_energy::total() {
  const size_t Nfreq = freq.size();
  double *Fe = electric();
  double *Fm = magnetic();
  double *F = new double[Nfreq];
  for (size_t i = 0; i < Nfreq; ++i)
    F[i] = Fe[i] + Fm[i];
  delete[] Fe;
  delete[] Fm;
  return F;
}

dft_energy fields::add_dft_energy(const volume_list *where_, const double *freq, size_t Nfreq,
                                  int decimation_factor) {

  if (!where_) // handle empty list of volumes
    return dft_energy(NULL, NULL, NULL, NULL, freq, Nfreq, v);

  dft_chunk *E = 0, *D = 0, *H = 0, *B = 0;
  volume firstvol(where_->v);
  volume_list *where = new volume_list(where_);
  volume_list *where_save = where;
  while (where) {
    LOOP_OVER_FIELD_DIRECTIONS(gv.dim, d) {
      E = add_dft(direction_component(Ex, d), where->v, freq, Nfreq, true, 1.0, E,
                  false, 1.0, true, 0, decimation_factor);
      D = add_dft(direction_component(Dx, d), where->v, freq, Nfreq, false, 1.0, D,
                  false, 1.0, true, 0, decimation_factor);
      H = add_dft(direction_component(Hx, d), where->v, freq, Nfreq, true, 1.0, H,
                  false, 1.0, true, 0, decimation_factor);
      B = add_dft(direction_component(Bx, d), where->v, freq, Nfreq, false, 1.0, B,
                  false, 1.0, true, 0, decimation_factor);
    }
    where = where->next;
  }
  delete where_save;

  return dft_energy(E, H, D, B, freq, Nfreq, firstvol);
}

void dft_energy::save_hdf5(h5file *file, const char *dprefix) {
  save_dft_hdf5(E, "E", file, dprefix);
  file->prevent_deadlock(); // hackery
  save_dft_hdf5(D, "D", file, dprefix);
  file->prevent_deadlock(); // hackery
  save_dft_hdf5(H, "H", file, dprefix);
  file->prevent_deadlock(); // hackery
  save_dft_hdf5(B, "B", file, dprefix);
}

void dft_energy::load_hdf5(h5file *file, const char *dprefix) {
  load_dft_hdf5(E, "E", file, dprefix);
  file->prevent_deadlock(); // hackery
  load_dft_hdf5(D, "D", file, dprefix);
  file->prevent_deadlock(); // hackery
  load_dft_hdf5(H, "H", file, dprefix);
  file->prevent_deadlock(); // hackery
  load_dft_hdf5(B, "B", file, dprefix);
}

void dft_energy::save_hdf5(fields &f, const char *fname, const char *dprefix, const char *prefix) {
  h5file *ff = f.open_h5file(fname, h5file::WRITE, prefix);
  save_hdf5(ff, dprefix);
  delete ff;
}

void dft_energy::load_hdf5(fields &f, const char *fname, const char *dprefix, const char *prefix) {
  h5file *ff = f.open_h5file(fname, h5file::READONLY, prefix);
  load_hdf5(ff, dprefix);
  delete ff;
}

void dft_energy::scale_dfts(complex<double> scale) {
  if (E) E->scale_dft(scale);
  if (D) D->scale_dft(scale);
  if (H) H->scale_dft(scale);
  if (B) B->scale_dft(scale);
}

void dft_energy::remove() {
  while (E) {
    dft_chunk *nxt = E->next_in_dft;
    delete E;
    E = nxt;
  }
  while (D) {
    dft_chunk *nxt = D->next_in_dft;
    delete D;
    D = nxt;
  }
  while (H) {
    dft_chunk *nxt = H->next_in_dft;
    delete H;
    H = nxt;
  }
  while (B) {
    dft_chunk *nxt = B->next_in_dft;
    delete B;
    B = nxt;
  }
}

direction fields::normal_direction(const volume &where) const {
  direction d = where.normal_direction();
  if (d == NO_DIRECTION) {
    /* hack so that we still infer the normal direction correctly for
       volumes with empty dimensions */
    volume where_pad(where);
    LOOP_OVER_DIRECTIONS(where.dim, d1) {
      if (nosize_direction(d1) && where.in_direction(d1) == 0.0)
        where_pad.set_direction_max(d1, where.in_direction_min(d1) + 0.1);
    }
    d = where_pad.normal_direction();
    if (d == NO_DIRECTION && gv.dim == D2 && beta != 0 && where_pad.in_direction(X) > 0 &&
        where_pad.in_direction(Y) > 0)
      d = Z;
    if (d == NO_DIRECTION) meep::abort("Could not determine normal direction for given grid_volume.");
  }
  return d;
}

dft_flux fields::add_dft_flux(direction d, const volume &where, const double *freq, size_t Nfreq,
                              bool use_symmetry, bool centered_grid, int decimation_factor) {
  if (d == NO_DIRECTION) d = normal_direction(where);
  volume_list vl(where, direction_component(Sx, d));
  dft_flux flux = add_dft_flux(&vl, freq, Nfreq, use_symmetry, centered_grid, decimation_factor);
  flux.normal_direction = d;
  return flux;
}


dft_flux fields::add_mode_monitor(direction d, const volume &where, const double *freq,
                                  size_t Nfreq, bool centered_grid, int decimation_factor) {
  return add_dft_flux(d, where, freq, Nfreq, /*use_symmetry=*/false, centered_grid, decimation_factor);
}

dft_flux fields::add_dft_flux_box(const volume &where, double freq_min, double freq_max,
                                  int Nfreq) {
  return add_dft_flux_box(where, meep::linspace(freq_min, freq_max, Nfreq));
}

dft_flux fields::add_dft_flux_box(const volume &where, const std::vector<double> &freq) {
  volume_list *faces = 0;
  LOOP_OVER_DIRECTIONS(where.dim, d) {
    if (where.in_direction(d) > 0) {
      volume face(where);
      derived_component c = direction_component(Sx, d);
      face.set_direction_min(d, where.in_direction_max(d));
      faces = new volume_list(face, c, +1, faces);
      face.set_direction_min(d, where.in_direction_min(d));
      face.set_direction_max(d, where.in_direction_min(d));
      faces = new volume_list(face, c, -1, faces);
    }
  }

  dft_flux flux = add_dft_flux(faces, freq);
  delete faces;
  return flux;
}

dft_flux fields::add_dft_flux_plane(const volume &where, double freq_min, double freq_max,
                                    int Nfreq) {
  return add_dft_flux_plane(where, meep::linspace(freq_min, freq_max, Nfreq));
}

dft_flux fields::add_dft_flux_plane(const volume &where, const std::vector<double> &freq) {
  return add_dft_flux(NO_DIRECTION, where, freq);
}

dft_fields::dft_fields(dft_chunk *chunks_, double freq_min, double freq_max, int Nf,
                       const volume &where_)
    : where(where_) {
  chunks = chunks_;
  freq = meep::linspace(freq_min, freq_max, Nf);
}

dft_fields::dft_fields(dft_chunk *chunks_, const std::vector<double> &freq_, const volume &where_)
    : where(where_) {
  chunks = chunks_;
  freq = freq_;
}

dft_fields::dft_fields(dft_chunk *chunks_, const double *freq_, size_t Nfreq, const volume &where_)
    : freq(Nfreq), where(where_) {
  chunks = chunks_;
  for (size_t i = 0; i < Nfreq; ++i)
    freq[i] = freq_[i];
}

void dft_fields::scale_dfts(complex<double> scale) { chunks->scale_dft(scale); }

void dft_fields::remove() {
  while (chunks) {
    dft_chunk *nxt = chunks->next_in_dft;
    delete chunks;
    chunks = nxt;
  }
}

dft_fields fields::add_dft_fields(component *components, int num_components, const volume where,
                                  const double *freq, size_t Nfreq, bool use_centered_grid,
                                  int decimation_factor, bool persist) {
  bool include_dV_and_interp_weights = false;
  bool sqrt_dV_and_interp_weights = false; // default option from meep.hpp (expose to user?)
  std::complex<double> extra_weight = 1.0; // default option from meep.hpp (expose to user?)
  complex<double> stored_weight = 1.0;
  dft_chunk *chunks = NULL;
  for (int nc = 0; nc < num_components; nc++)
    chunks = add_dft(components[nc], where, freq, Nfreq, include_dV_and_interp_weights,
                     stored_weight, chunks, sqrt_dV_and_interp_weights, extra_weight,
                     use_centered_grid, 0, decimation_factor, persist);

  return dft_fields(chunks, freq, Nfreq, where);
}

/***************************************************************/
/* chunk-level processing for fields::process_dft_component.   */
/***************************************************************/
complex<double> dft_chunk::process_dft_component(int rank, direction *ds, ivec min_corner, ivec max_corner,
                                                 int num_freq, h5file *file, realnum *buffer, int reim,
                                                 complex<realnum> *field_array, void *mode1_data, void *mode2_data,
                                                 int ic_conjugate, bool retain_interp_weights,
                                                 fields *parent) {

  if ((num_freq < 0) || (num_freq > static_cast<int>(omega.size())-1))
    meep::abort("process_dft_component: frequency index %d is outside the range of the frequency array of size %lu",num_freq,omega.size());

  /*****************************************************************/
  /* compute the size of the chunk we own and its strides etc.     */
  /*****************************************************************/
  size_t start[3] = {0, 0, 0};
  size_t file_count[3] = {1, 1, 1}, array_count[3] = {1, 1, 1};
  int file_offset[3] = {0, 0, 0};
  int file_stride[3] = {1, 1, 1};
  ivec isS = S.transform(is, sn) + shift;
  ivec ieS = S.transform(ie, sn) + shift;

  ivec permute(zero_ivec(fc->gv.dim));
  for (int i = 0; i < 3; ++i)
    permute.set_direction(fc->gv.yucky_direction(i), i);
  permute = S.transform_unshifted(permute, sn);
  LOOP_OVER_DIRECTIONS(permute.dim, d) { permute.set_direction(d, abs(permute.in_direction(d))); }

  for (int i = 0; i < rank; ++i) {
    direction d = ds[i];
    int isd = isS.in_direction(d), ied = ieS.in_direction(d);
    start[i] = (std::min(isd, ied) - min_corner.in_direction(d)) / 2;
    file_count[i] = abs(ied - isd) / 2 + 1;
    if (ied < isd) file_offset[permute.in_direction(d)] = file_count[i] - 1;
    array_count[i] = (max_corner.in_direction(d) - min_corner.in_direction(d)) / 2 + 1;
  }

  for (int i = 0; i < rank; ++i) {
    direction d = ds[i];
    int j = permute.in_direction(d);
    for (int k = i + 1; k < rank; ++k)
      file_stride[j] *= file_count[k];
    file_offset[j] *= file_stride[j];
    if (file_offset[j]) file_stride[j] *= -1;
  }

  /*****************************************************************/
  /* For collapsing empty dimensions, we want to retain interpolation
     weights for empty dimensions, but not interpolation weights for
     integration of edge pixels (for retain_interp_weights == true).
     All of the weights are stored in (s0, s1, e0, e1), so we make
     a copy of these with the weights for non-empty dimensions set to 1. */
  vec s0i(s0), s1i(s1), e0i(e0), e1i(e1);
  LOOP_OVER_DIRECTIONS(fc->gv.dim, d) {
    if (!empty_dim[d]) {
      s0i.set_direction(d, 1.0);
      s1i.set_direction(d, 1.0);
      e0i.set_direction(d, 1.0);
      e1i.set_direction(d, 1.0);
    }
  }

  /***************************************************************/
  /* loop over all grid points in our piece of the volume        */
  /***************************************************************/
  vec rshift(shift * (0.5 * fc->gv.inva));
  int chunk_idx = 0;
  complex<double> integral = 0.0;
  component c_conjugate = (component)(ic_conjugate >= 0 ? ic_conjugate : -ic_conjugate);
  LOOP_OVER_IVECS(fc->gv, is, ie, idx) {
    IVEC_LOOP_LOC(fc->gv, loc);
    loc = S.transform(loc, sn) + rshift;
    double w = IVEC_LOOP_WEIGHT(s0, s1, e0, e1, dV0 + dV1 * loop_i2);
    double interp_w = retain_interp_weights ? IVEC_LOOP_WEIGHT(s0i, s1i, e0i, e1i, 1.0) : 1.0;

    complex<double> dft_val =
        (c_conjugate == NO_COMPONENT
             ? w
             : c_conjugate == Dielectric
                   ? parent->get_eps(loc)
                   : c_conjugate == Permeability
                         ? parent->get_mu(loc)
                         : complex<double>(dft[omega.size() * (chunk_idx++) + num_freq]) / stored_weight);
    if (include_dV_and_interp_weights && dft_val!=0.0) dft_val /= (sqrt_dV_and_interp_weights ? sqrt(w) : w);

    complex<double> mode1val = 0.0, mode2val = 0.0;
    if (mode1_data) mode1val = eigenmode_amplitude(mode1_data, loc, S.transform(c_conjugate, sn));
    if (mode2_data) mode2val = eigenmode_amplitude(mode2_data, loc, S.transform(c, sn));

    if (file) {
      int idx2 = ((((file_offset[0] + file_offset[1] + file_offset[2]) + loop_i1 * file_stride[0]) +
                   loop_i2 * file_stride[1]) +
                  loop_i3 * file_stride[2]);

      dft_val *= interp_w;

      complex<double> val = (mode1_data ? mode1val : dft_val);
      buffer[idx2] = reim ? imag(val) : real(val);
    }
    else if (field_array) {
      IVEC_LOOP_ILOC(fc->gv, iloc);         // iloc <-- indices of parent point in Yee grid
      iloc = S.transform(iloc, sn) + shift; // iloc <-- indices of child point in Yee grid
      iloc -= min_corner;                   // iloc <-- 2*(indices of point in DFT array)

      // the index of point n1 or (n1,n2) or (n1,n2,n3) in a 1D, 2D, or 3D array is
      // (for a 1D array) n1
      // (for a 2D array) n2 + n1*N2
      // (for a 3D array) n3 + n2*N3 + n1*N2*N3
      // where NI = number of points in Ith direction.
      int idx2 = 0;
      for (int i = rank - 1, stride = 1; i >= 0; stride *= array_count[i--])
        idx2 += stride * (iloc.in_direction(ds[i]) / 2);
      field_array[idx2] = interp_w * dft_val;
    }
    else {
      mode1val = conj(mode1val); // conjugated inner product
      if (mode2_data)
        integral += w * mode1val * mode2val;
      else
        integral += w * mode1val * dft_val;
    }

  } // LOOP_OVER_IVECS(fc->gv, is, ie, idx)

  if (file) file->write_chunk(rank, start, file_count, buffer);

  return integral;
}

// get variables that are needed by complex<double> fields::process_dft_component
void fields::get_dft_component_dims(dft_chunk **chunklists, int num_chunklists, component c, ivec &min_corner, ivec &max_corner, size_t &array_size, size_t &bufsz, int &rank, direction *ds, size_t *dims, int *array_rank, size_t *array_dims, direction *array_dirs) {
  /***************************************************************/
  /* get statistics on the volume slice **************************/
  /***************************************************************/
  volume *where = &v; // use full volume of fields
  bufsz = 0;
  min_corner = gv.round_vec(where->get_max_corner()) + one_ivec(gv.dim);
  max_corner = gv.round_vec(where->get_min_corner()) - one_ivec(gv.dim);

  for (int ncl = 0; ncl < num_chunklists; ncl++)
    for (dft_chunk *chunk = chunklists[ncl]; chunk; chunk = chunk->next_in_dft) {
      if (chunk->c != c) continue;
      ivec isS = chunk->S.transform(chunk->is, chunk->sn) + chunk->shift;
      ivec ieS = chunk->S.transform(chunk->ie, chunk->sn) + chunk->shift;
      min_corner = min(min_corner, min(isS, ieS));
      max_corner = max(max_corner, max(isS, ieS));
      size_t this_bufsz = 1;
      LOOP_OVER_DIRECTIONS(chunk->fc->gv.dim, d) {
        this_bufsz *= (chunk->ie.in_direction(d) - chunk->is.in_direction(d)) / 2 + 1;
      }
      bufsz = std::max(bufsz, this_bufsz);
    }
  am_now_working_on(MpiAllTime);
  max_corner = max_to_all(max_corner);
  min_corner = -max_to_all(-min_corner); // i.e., min_to_all
  finished_working();

  /***************************************************************/
  /***************************************************************/
  /***************************************************************/
  rank = 0;
  array_size = 1;
  LOOP_OVER_DIRECTIONS(gv.dim, d) {
    if (rank >= 3) meep::abort("too many dimensions in process_dft_component");
    size_t n = std::max(0, (max_corner.in_direction(d) - min_corner.in_direction(d)) / 2 + 1);

    if (n > 1) {
      ds[rank] = d;
      dims[rank++] = n;
      array_size *= n;
    }
  }
  if (array_rank) {
    *array_rank = rank;
    for (int d = 0; d < rank; d++) {
      if (array_dims) array_dims[d] = dims[d];
      if (array_dirs) array_dirs[d] = ds[d];
    }
  }
}

/***************************************************************/
/* low-level [actually intermediate-level, since it calls      */
/* dft_chunk::process_dft_component(), which is the true       */
/* low-level function] workhorse routine that forms the common */
/* backend for several operations involving DFT fields.        */
/*                                                             */
/* looks through the given collection of dft_chunks and        */
/* processes only those chunks that store component c, using   */
/* only data for frequency #num_freq.                          */
/*                                                             */
/* the meaning of 'processes' depends on the arguments:        */
/*                                                             */
/*  1. if HDF5FileName is non-null: write to the given HDF5    */
/*     file a new dataset describing either                    */
/*      (A) DFT field component c (if mode_data1 is null), or  */
/*      (B) mode field component c for the eigenmode described */
/*          by mode_data1 (if it is non-null)                  */
/*                                                             */
/*  2. if HDF5FileName is null but pfield_array is non-null:   */
/*     set *pfield_array equal to a newly allocated buffer     */
/*     populated on return with values of DFT field component  */
/*     c, equivalent to writing the data to HDF5 and reading   */
/*     it back into field_array.                               */
/*                                                             */
/*  3. if both HDF5FileName and field_array are null: compute  */
/*     and return an  overlap integral between                 */
/*      (A) the DFT fields and the fields of the eigenmode     */
/*          described by mode_data1 (if mode_data2 is null)    */
/*      (B) the eigenmode fields described by mode_data1       */
/*          and the eigenmode fields described by mode_data2   */
/*          (if mode_data2 is non-null).                       */
/*     more specifically, the integral computed is             */
/*      < mode1_{c_conjugate} | dft_{c} >                      */
/*     in case (A) and                                         */
/*      < mode1_{c_conjugate} | mode2_{c} >                    */
/*     in case (B).                                            */
/*                                                             */
/* if where is non-null, only field components inside *where   */
/* are processed.                                              */
/***************************************************************/
complex<double> fields::process_dft_component(dft_chunk **chunklists, int num_chunklists, int num_freq,
                                              component c, const char *HDF5FileName, complex<realnum> **pfield_array,
                                              int *array_rank, size_t *array_dims, direction *array_dirs,
                                              void *mode1_data, void *mode2_data, component c_conjugate,
                                              bool *first_component, bool retain_interp_weights) {

  /***************************************************************/
  /***************************************************************/
  /***************************************************************/
  int ic_conjugate = (int)c_conjugate;
  if (component_index(c) == -1) {
    ic_conjugate = -((int)c);
    num_chunklists = 1;
    c = chunklists[0]->c;
  }

  ivec min_corner, max_corner;
  int rank;
  direction ds[3];
  size_t array_size, bufsz, dims[3];
  get_dft_component_dims(chunklists, num_chunklists, c, min_corner, max_corner, array_size, bufsz, rank, ds, dims, array_rank, array_dims, array_dirs);

  if (rank == 0) {
    if (pfield_array) *pfield_array = 0;
    return 0.0; // no chunks with the specified component on this processor
  }

  /***************************************************************/
  /* buffer for process-local contributions to HDF5 output files,*/
  /* like h5_output_data::buf in h5fields.cpp                    */
  /***************************************************************/
  realnum *buffer = 0;
  complex<realnum> *field_array = 0;
  int reim_max = 0;
  if (HDF5FileName) {
    buffer = new realnum[bufsz];
    reim_max = 1;
  }
  else if (pfield_array)
    *pfield_array = field_array = (array_size ? new complex<realnum>[array_size] : 0);

  complex<double> overlap = 0.0;
  for (int reim = 0; reim <= reim_max; reim++) {
    h5file *file = 0;
    if (HDF5FileName) {
      file = open_h5file(HDF5FileName, (*first_component) ? h5file::WRITE : h5file::READWRITE);
      *first_component = false;
      char dataname[100];
      snprintf(dataname, 100, "%s_%i.%c", component_name(c), num_freq, reim ? 'i' : 'r');
      file->create_or_extend_data(dataname, rank, dims, false /* append_data */,
                                  sizeof(realnum) == sizeof(float) /* single_precision */);
    }

    for (int ncl = 0; ncl < num_chunklists; ncl++)
      for (dft_chunk *chunk = chunklists[ncl]; chunk; chunk = chunk->next_in_dft)
        if (chunk->c == c)
          overlap += chunk->process_dft_component(rank, ds, min_corner, max_corner, num_freq, file,
                                                  buffer, reim, field_array, mode1_data, mode2_data,
                                                  ic_conjugate, retain_interp_weights, this);

    if (HDF5FileName) {
      file->done_writing_chunks();
      file->prevent_deadlock(); // hackery
      delete file;
    }
    else if (field_array) {
/***************************************************************/
/* repeatedly call sum_to_all to consolidate full field array  */
/* on all cores                                                */
/***************************************************************/
#define BUFSIZE 1 << 20  // use 1M element (16 MB) buffer
      complex<realnum> *buf = new complex<realnum>[BUFSIZE];
      ptrdiff_t offset = 0;
      size_t remaining = array_size;
      while (remaining != 0) {
        size_t size = (remaining > BUFSIZE ? BUFSIZE : remaining);
        am_now_working_on(MpiAllTime);
        sum_to_all(field_array + offset, buf, size);
        finished_working();
        memcpy(field_array + offset, buf, size * sizeof(complex<realnum>));
        remaining -= size;
        offset += size;
      }
      delete[] buf;
    }
  } // for(int reim=0; reim<=reim_max; reim++)

  if (HDF5FileName)
    delete[] buffer;
  else {
    am_now_working_on(MpiAllTime);
    overlap = sum_to_all(overlap);
    finished_working();
  }

  return overlap;
}

/***************************************************************/
/* routines for fetching arrays of dft fields                  */
/***************************************************************/
complex<realnum> *fields::get_dft_array(dft_flux flux, component c, int num_freq, int *rank,
                                        size_t dims[3]) {
  dft_chunk *chunklists[2];
  chunklists[0] = flux.E;
  chunklists[1] = flux.H;
  complex<realnum> *array;
  direction dirs[3];
  process_dft_component(chunklists, 2, num_freq, c, 0, &array, rank, dims, dirs);
  return collapse_array(array, rank, dims, dirs, flux.where);
}

complex<realnum> *fields::get_dft_array(dft_force force, component c, int num_freq, int *rank,
                                        size_t dims[3]) {
  dft_chunk *chunklists[3];
  chunklists[0] = force.offdiag1;
  chunklists[1] = force.offdiag2;
  chunklists[2] = force.diag;
  complex<realnum> *array;
  direction dirs[3];
  process_dft_component(chunklists, 3, num_freq, c, 0, &array, rank, dims, dirs);
  return collapse_array(array, rank, dims, dirs, force.where);
}

complex<realnum> *fields::get_dft_array(dft_near2far n2f, component c, int num_freq, int *rank,
                                        size_t dims[3]) {
  dft_chunk *chunklists[1];
  chunklists[0] = n2f.F;
  complex<realnum> *array;
  direction dirs[3];
  process_dft_component(chunklists, 1, num_freq, c, 0, &array, rank, dims, dirs);
  return collapse_array(array, rank, dims, dirs, n2f.where);
}

complex<realnum> *fields::get_dft_array(dft_fields fdft, component c, int num_freq, int *rank,
                                        size_t dims[3]) {
  dft_chunk *chunklists[1];
  chunklists[0] = fdft.chunks;
  complex<realnum> *array;
  direction dirs[3];
  process_dft_component(chunklists, 1, num_freq, c, 0, &array, rank, dims, dirs);
  return collapse_array(array, rank, dims, dirs, fdft.where);
}

/***************************************************************/
/* wrapper around process_dft_component that writes HDF5       */
/* datasets for all components at all frequencies stored in    */
/* the given collection of DFT chunks                          */
/***************************************************************/
void fields::output_dft_components(dft_chunk **chunklists, int num_chunklists, volume dft_volume,
                                   const char *HDF5FileName) {
  int NumFreqs = 0;
  for (int nc = 0; nc < num_chunklists && NumFreqs == 0; nc++)
    if (chunklists[nc]) NumFreqs = chunklists[nc]->omega.size();

  // if the volume has zero thickness in one or more directions, the DFT
  // grid is two pixels thick in those directions, but we want the HDF5 output
  // to be just one pixel thick in those directions. solution: first get the
  // fields in array form (as get_dft_array), then collapse degenerate dimensions
  // and export the collapsed array to HDF5. in this case the max_to_all() below
  // is needed to make sure everybody agrees on how many frequencies there are,
  // because some processes' field chunks may have no overlap with dft_volume,
  // in which case those processes will think NumFreqs==0.
  bool have_empty_dims = false;
  LOOP_OVER_DIRECTIONS(dft_volume.dim, d)
  if (dft_volume.in_direction(d) != 0.0) have_empty_dims = true;

  h5file *file = 0;
  if (have_empty_dims && am_master()) {
    char filename[100];
    snprintf(filename, 100, "%s%s", HDF5FileName, strstr("%.h5", HDF5FileName) ? "" : ".h5");
    file = new h5file(filename, h5file::WRITE, false /*parallel*/);
  }
  am_now_working_on(MpiAllTime);
  if (have_empty_dims) NumFreqs = max_to_all(NumFreqs); // subtle!
  finished_working();

  bool first_component = true;
  for (int num_freq = 0; num_freq < NumFreqs; num_freq++)
    FOR_COMPONENTS(c) {
      if (!have_empty_dims) {
        process_dft_component(chunklists, num_chunklists, num_freq, c, HDF5FileName, 0, 0, 0, 0, 0,
                              0, Ex, &first_component);
      }
      else {
        complex<realnum> *array = 0;
        int rank;
        size_t dims[3];
        direction dirs[3];
        process_dft_component(chunklists, num_chunklists, num_freq, c, 0, &array, &rank, dims,
                              dirs);
        if (rank > 0 && am_master()) {
          array = collapse_array(array, &rank, dims, dirs, dft_volume);
          if (rank == 0) meep::abort("%s:%i: internal error", __FILE__, __LINE__);
          size_t array_size = dims[0] * (rank >= 2 ? dims[1] * (rank == 3 ? dims[2] : 1) : 1);
          double *real_array = new double[array_size];
          if (!real_array) meep::abort("%s:%i:out of memory(%lu)", __FILE__, __LINE__, array_size);
          for (int reim = 0; reim < 2; reim++) {
            for (size_t n = 0; n < array_size; n++)
              real_array[n] = (reim == 0 ? real(array[n]) : imag(array[n]));
            char dataname[100], filename[100];
            snprintf(dataname, 100, "%s_%i.%c", component_name(c), num_freq, reim ? 'i' : 'r');
            snprintf(filename, 100, "%s%s", HDF5FileName, strstr(".h5", HDF5FileName) ? "" : ".h5");
            file->write(dataname, rank, dims, real_array, false /* single_precision */);
          }
          delete[] real_array;
        }
        if (array) delete[] array;
      }
    }
  if (file) delete file;
}

void fields::output_dft(dft_flux flux, const char *HDF5FileName) {
  dft_chunk *chunklists[2];
  chunklists[0] = flux.E;
  chunklists[1] = flux.H;
  output_dft_components(chunklists, 2, flux.where, HDF5FileName);
}

void fields::output_dft(dft_force force, const char *HDF5FileName) {
  dft_chunk *chunklists[3];
  chunklists[0] = force.offdiag1;
  chunklists[1] = force.offdiag2;
  chunklists[2] = force.diag;
  output_dft_components(chunklists, 3, force.where, HDF5FileName);
}

void fields::output_dft(dft_near2far n2f, const char *HDF5FileName) {
  dft_chunk *chunklists[1];
  chunklists[0] = n2f.F;
  output_dft_components(chunklists, 1, n2f.where, HDF5FileName);
}

void fields::output_dft(dft_fields fdft, const char *HDF5FileName) {
  dft_chunk *chunklists[1];
  chunklists[0] = fdft.chunks;
  output_dft_components(chunklists, 1, fdft.where, HDF5FileName);
}

/***************************************************************/
/***************************************************************/
/***************************************************************/
void fields::get_overlap(void *mode1_data, void *mode2_data, dft_flux flux, int num_freq,
                         complex<double> overlaps[2]) {
  component cE[2], cH[2];
  switch (flux.normal_direction) {
    case X:
      cE[0] = Ey;
      cH[0] = Hz;
      cE[1] = Ez;
      cH[1] = Hy;
      break;
    case Y:
      cE[0] = Ez;
      cH[0] = Hx;
      cE[1] = Ex;
      cH[1] = Hz;
      break;
    case R:
      cE[0] = Ep;
      cH[0] = Hz;
      cE[1] = Ez;
      cH[1] = Hp;
      break;
    case P:
      cE[0] = Ez;
      cH[0] = Hr;
      cE[1] = Er;
      cH[1] = Hz;
      break;
    case Z:
      if (gv.dim == Dcyl)
        cE[0] = Er, cE[1] = Ep, cH[0] = Hp, cH[1] = Hr;
      else
        cE[0] = Ex, cE[1] = Ey, cH[0] = Hy, cH[1] = Hx;
      break;
    default: meep::abort("invalid normal_direction in get_overlap");
  };

  dft_chunk *chunklists[2];
  chunklists[0] = flux.E;
  chunklists[1] = flux.H;
  complex<double> ExHy = process_dft_component(chunklists, 2, num_freq, cE[0], 0, 0, 0, 0, 0, mode1_data,
                                               mode2_data, cH[0]);
  complex<double> EyHx = process_dft_component(chunklists, 2, num_freq, cE[1], 0, 0, 0, 0, 0, mode1_data,
                                               mode2_data, cH[1]);
  complex<double> HyEx = process_dft_component(chunklists, 2, num_freq, cH[0], 0, 0, 0, 0, 0, mode1_data,
                                               mode2_data, cE[0]);
  complex<double> HxEy = process_dft_component(chunklists, 2, num_freq, cH[1], 0, 0, 0, 0, 0, mode1_data,
                                               mode2_data, cE[1]);
  overlaps[0] = ExHy - EyHx;
  overlaps[1] = HyEx - HxEy;
}

void fields::get_mode_flux_overlap(void *mode_data, dft_flux flux, int num_freq,
                                   std::complex<double> overlaps[2]) {
  get_overlap(mode_data, 0, flux, num_freq, overlaps);
}

void fields::get_mode_mode_overlap(void *mode1_data, void *mode2_data, dft_flux flux,
                                   std::complex<double> overlaps[2]) {
  get_overlap(mode1_data, mode2_data, flux, 0, overlaps);
}

/* deregister all of the remaining dft monitors
from the fields object. Note that this does not
delete the underlying dft_chunks! (useful for 
adjoint calculations, where we want to keep
the chunk data around) */
void fields::clear_dft_monitors() {
for (int i = 0; i < num_chunks; i++)
  if (chunks[i]->is_mine() && chunks[i]->dft_chunks) chunks[i]->dft_chunks = NULL;
}

// return the size of the dft monitor
std::vector<size_t> fields::dft_monitor_size(dft_fields fdft, const volume &where, component c){
  ivec min_corner, max_corner;
  int rank, reduced_rank;
  direction dirs[3], reduced_dirs[3];
  size_t array_size, bufsz, dims[3], reduced_dims[3], reduced_stride[3], stride[3];
  dft_chunk *chunklists[1];
  chunklists[0] = fdft.chunks;

  get_dft_component_dims(chunklists, 1, c, min_corner, max_corner, array_size, bufsz, rank, dirs, dims);
  reduce_array_dimensions(where, rank, dims, dirs, stride, reduced_rank, reduced_dims, reduced_dirs, reduced_stride);
  std::vector<size_t> reduced_dims_vec = {reduced_dims[0], reduced_dims[1], reduced_dims[2]};

  return reduced_dims_vec;
}

std::vector<struct sourcedata> dft_fields::fourier_sourcedata(const volume &where, component c, fields &f, const std::complex<double>* dJ){
  const size_t Nfreq = freq.size();

  ivec min_corner, max_corner;
  int rank, reduced_rank;
  direction dirs[3], reduced_dirs[3];
  size_t array_size, bufsz, dims[3], reduced_dims[3], reduced_stride[3], stride[3];
  dft_chunk *chunklists[1];
  chunklists[0] = chunks;

  f.get_dft_component_dims(chunklists, 1, c, min_corner, max_corner, array_size, bufsz, rank, dirs, dims);
  reduce_array_dimensions(where, rank, dims, dirs, stride, reduced_rank, reduced_dims, reduced_dirs, reduced_stride);
  size_t reduced_grid_size = reduced_dims[0]*reduced_dims[1]*reduced_dims[2]; // total number of points in the monitor

  std::vector<struct sourcedata> temp;

  for (dft_chunk *f = chunks; f; f = f->next_in_dft) {
    assert(Nfreq == f->omega.size());
    vec rshift(f->shift * (0.5 * f->fc->gv.inva));

    std::vector<ptrdiff_t> idx_arr;
    std::vector<std::complex<double> > amp_arr;
    std::complex<double> EH0 = std::complex<double>(0,0);
    sourcedata temp_struct = {component(f->c), idx_arr, f->fc->chunk_idx, amp_arr};
    int position_array[3] = {0, 0, 0}; // array indicating the position of a point relative to the minimum corner of the monitor

    LOOP_OVER_IVECS(f->fc->gv, f->is, f->ie, idx) {
      IVEC_LOOP_LOC(f->fc->gv, x0);
      IVEC_LOOP_ILOC(f->fc->gv, ix0);
      x0 = f->S.transform(x0, f->sn) + rshift;
      ix0 = f->S.transform(ix0, f->sn) + f->shift;

      double dJ_weight = 1; // weight for linear interpolation
      int nd = 0;
      LOOP_OVER_DIRECTIONS(f->fc->gv.dim, d){
        if (where.in_direction(d) > 0) position_array[nd++] = int((ix0.in_direction(d)-min_corner.in_direction(d))/2);
        else dJ_weight *= (1-abs(x0.in_direction(d)-where.in_direction_min(d))/(f->fc->gv.inva)); // based on distances
      }

      // index when dJ is flattened to a one-dimenional array
      size_t idx_1d = (position_array[0]*reduced_dims[1]+position_array[1])*reduced_dims[2]+position_array[2];

      if (f->avg1==0 && f->avg2==0){ // yee_grid = true
        temp_struct.idx_arr.push_back(idx);
        for (size_t i = 0; i < Nfreq; ++i) {
          EH0 = dJ_weight*dJ[reduced_grid_size*i+idx_1d];
          if (is_E_or_D(temp_struct.near_fd_comp)) EH0 *= -1;
          EH0 /= f->S.multiplicity(ix0);
          temp_struct.amp_arr.push_back(EH0);
        }
      }
      else{ // yee_grid = false
          // four or two neighbouring points in the yee lattice are involved in calculating the value at the center of a voxel
        ptrdiff_t site_ind[4] = {idx,idx + f->avg1,idx + f->avg2,idx + f->avg1 + f->avg2};
        for (size_t j = 0; j < 4; ++j){
          temp_struct.idx_arr.push_back(site_ind[j]);
          for (size_t i = 0; i < Nfreq; ++i) {
            EH0 = dJ_weight*dJ[reduced_grid_size*i+idx_1d]*0.25; // split the amplitude of the adjoint source into four parts
            if (is_E_or_D(temp_struct.near_fd_comp)) EH0 *= -1;
            EH0 /= f->S.multiplicity(ix0);
            temp_struct.amp_arr.push_back(EH0);
          }
        }
      }
    }
    temp.push_back(temp_struct);
  }
  return temp;
}

} // namespace meep