From 4f6ff70efe042925a9fdc1309341e560e23659a4 Mon Sep 17 00:00:00 2001 From: Homer Reid Date: Thu, 1 Feb 2018 12:09:45 -0500 Subject: [PATCH 1/3] new clean branch for modeExpansion code --- doc/docs/ModeExpansion.md | 794 +++++++++ doc/docs/ModeExpansionFiles/FluxVsLP.png | Bin 0 -> 198727 bytes doc/docs/ModeExpansionFiles/L0.0_p0_f85.png | Bin 0 -> 141686 bytes doc/docs/ModeExpansionFiles/L0_p0.mpg | Bin 0 -> 368640 bytes doc/docs/ModeExpansionFiles/L1.0_p0_f85.png | Bin 0 -> 155764 bytes doc/docs/ModeExpansionFiles/L1_p0.mpg | Bin 0 -> 374784 bytes doc/docs/ModeExpansionFiles/L2.0_p0_f85.png | Bin 0 -> 112371 bytes doc/docs/ModeExpansionFiles/L2_p0.mpg | Bin 0 -> 362496 bytes doc/docs/ModeExpansionFiles/L3.0_p0_f85.png | Bin 0 -> 126952 bytes doc/docs/ModeExpansionFiles/L3.0_p1_f85.png | Bin 0 -> 127534 bytes doc/docs/ModeExpansionFiles/L3.0_p2_f85.png | Bin 0 -> 128120 bytes doc/docs/ModeExpansionFiles/L3_p0.mpg | Bin 0 -> 360448 bytes doc/docs/ModeExpansionFiles/L3_p1.mpg | Bin 0 -> 360448 bytes doc/docs/ModeExpansionFiles/L3_p2.mpg | Bin 0 -> 360448 bytes .../ModeExpansionFiles/ModeProfiles2D.png | Bin 0 -> 258343 bytes doc/docs/ModeExpansionFiles/Taper2D.svg | 1453 +++++++++++++++++ doc/docs/ModeExpansionFiles/Taper2D_p0.png | Bin 0 -> 362195 bytes doc/docs/ModeExpansionFiles/Taper2D_p1.png | Bin 0 -> 386333 bytes doc/docs/ModeExpansionFiles/Taper2D_p2.png | Bin 0 -> 133910 bytes doc/docs/ModeExpansionFiles/TaperData.png | Bin 0 -> 43097 bytes doc/docs/ModeExpansionFiles/wvg-taper.cpp | 1 + doc/docs/ModeExpansionFiles/wvg-taper.py | 363 ++++ libmeepgeom/Makefile.am | 5 +- libmeepgeom/wvg-taper.cpp | 562 +++++++ src/dft.cpp | 383 +++++ src/meep.hpp | 74 +- src/mpb.cpp | 514 ++++-- 27 files changed, 4053 insertions(+), 96 deletions(-) create mode 100644 doc/docs/ModeExpansion.md create mode 100644 doc/docs/ModeExpansionFiles/FluxVsLP.png create mode 100644 doc/docs/ModeExpansionFiles/L0.0_p0_f85.png create mode 100644 doc/docs/ModeExpansionFiles/L0_p0.mpg create mode 100644 doc/docs/ModeExpansionFiles/L1.0_p0_f85.png create mode 100644 doc/docs/ModeExpansionFiles/L1_p0.mpg create mode 100644 doc/docs/ModeExpansionFiles/L2.0_p0_f85.png create mode 100644 doc/docs/ModeExpansionFiles/L2_p0.mpg create mode 100644 doc/docs/ModeExpansionFiles/L3.0_p0_f85.png create mode 100644 doc/docs/ModeExpansionFiles/L3.0_p1_f85.png create mode 100644 doc/docs/ModeExpansionFiles/L3.0_p2_f85.png create mode 100644 doc/docs/ModeExpansionFiles/L3_p0.mpg create mode 100644 doc/docs/ModeExpansionFiles/L3_p1.mpg create mode 100644 doc/docs/ModeExpansionFiles/L3_p2.mpg create mode 100644 doc/docs/ModeExpansionFiles/ModeProfiles2D.png create mode 100644 doc/docs/ModeExpansionFiles/Taper2D.svg create mode 100644 doc/docs/ModeExpansionFiles/Taper2D_p0.png create mode 100644 doc/docs/ModeExpansionFiles/Taper2D_p1.png create mode 100644 doc/docs/ModeExpansionFiles/Taper2D_p2.png create mode 100644 doc/docs/ModeExpansionFiles/TaperData.png create mode 120000 doc/docs/ModeExpansionFiles/wvg-taper.cpp create mode 100644 doc/docs/ModeExpansionFiles/wvg-taper.py create mode 100644 libmeepgeom/wvg-taper.cpp diff --git a/doc/docs/ModeExpansion.md b/doc/docs/ModeExpansion.md new file mode 100644 index 000000000..26a568494 --- /dev/null +++ b/doc/docs/ModeExpansion.md @@ -0,0 +1,794 @@ +# Eigenmode decomposition of arbitrary field configurations + +*Eigenmode decomposition* exploits Meep's interconnectivity +with the [MPB][MPB] mode solver to express an arbitrary +time-harmonic field configuration as a superposition of +the normal harmonic modes of your structure. + +[TOC] + +## Theoretical background[^1] + +Consider a waveguide structure of infinite extent in the $x$ +direction with constant cross section in the transverse +$[\vec\rho=(y,z)]$ directions. For any given +angular frequency $\omega$ we may solve the time-harmonic +Maxwell equations to obtain the *normal modes* of the +structure---an infinite set of vector-valued +functions of the transverse coordinates +$\{\mathbf{E}^\pm_n(\vec{\rho}), \mathbf{H}^\pm_n(\vec{\rho})\}$, +with associated propagation constants $\{\beta_n\}$, +that furnish a complete expansion basis for +time-harmonic electromagnetic fields at frequency $\omega$. +That is, given any arbitrary frequency-$\omega$ field +configuration of the form +$$ \mathbf{E}(\mathbf{r},t) = \mathbf{E}(\mathbf{r}) e^{-i\omega t} $$ +$$ \mathbf{H}(\mathbf{r},t) = \mathbf{H}(\mathbf{r}) e^{-i\omega t} $$ +we have the *exact* expansions + +$$ + \mathbf{E}(\mathbf{r}) = + \mathbf{E}(x,\vec{\rho}) = + \sum_{n} \left\{ \alpha^+_n \mathbf E^+_n(\vec \rho)e^{+i\beta_n x} + + \alpha^-_n \mathbf E^-_n(\vec \rho)e^{-i\beta_n x} + \right\} + \tag{1} +$$ +$$ + \mathbf{H}(\mathbf{r}) = + \mathbf{H}(x,\vec{\rho}) = + \sum_{n} \left\{ \alpha^+_n \mathbf H^+_n(\vec \rho)e^{+i\beta_n x} + + \alpha^-_n \mathbf H^-_n(\vec \rho)e^{-i\beta_n x} + \right\} + \tag{2} +$$ +where (as discussed further [below](ModeExpansion.md#UnderTheHood)) +the expansion coefficients $\{\alpha^{\pm}_n\}$ +may be extracted from knowledge of the time-harmonic +fields $\mathbf{E},\mathbf{H}$ on any cross-sectional +surface $S$ transverse to the waveguide. + +The idea of mode expansion in Meep is to compute +the $\{\alpha_n^\pm\}$ coefficients above for any +*arbitrary* time-harmonic field distribution +resulting from a Meep calculation. In calculations +of this sort, + ++ the $\{\mathbf{E},\mathbf{H}\}$ fields on the RHS + of equations (1a,b) above will be frequency-domain + fields stored in a `dft_flux` object in a Meep + run, where you will have arranged this `dft_flux` object + to live on a cross-sectional surface $S$ transverse + to the waveguide; + +- the $\{\mathbf{E}^\pm_n,\mathbf{H}^\pm_n\}$ eigenmodes + and $\{\beta_n\}$ propagation constants are computed + automatically under the hood by [MPB][MPB] as normal modes + of an infinitely extended waveguide with the same + cross-sectional material distribution that your structure + has on the transverse slice $S$, and + +- the $\alpha_n^\pm$ coefficients for as many bands + as you like are computed by calling `get_eigenmode_coefficients(),` + as discussed below. + +## Main function prototype + +The highest-level interface to the mode-expansion implementation +in Meep is the libmeep function `meep::fields::get_eigenmode_coefficients,` +callable from C++ or python. This routine makes use +of several [lower-level libmeep functions](ModeExpansion.md#OtherRoutines) +that you may also find useful; +these are documented +[below](ModeExpansion.md#OtherRoutines) +and their use is illustrated in the tutorial that follows. + +The C++ prototype for the top-level routine is + +```c++ +std::vector + fields::get_eigenmode_coefficients(dft_flux *flux, + direction d, + const volume &where, + std::vector bands, + kpoint_func k_func=0, + void *user_data=0); +``` +where + ++ `flux` is a `dft_flux` object pre-populated with frequency-domain field data resulting from a time-domain Meep calculation you have run to tabulate fields on a cross-sectional slice perpendicular to your waveguide + ++ `d` is the direction of power flow in the waveguide + ++ `where` is a `volume` describing the cross-sectional surface $S$ + ++ `bands` is an array of integers that you populate with the indices of the modes for which you want expansion coefficients + ++ `user_func` is an *optional* function you supply to provide initial estimates of the wavevector of a mode with given +frequency and band index; its prototype is + +```c++ + vec (*kpoint_func)(void user_data, double freq, int band); +``` + +which returns a `vec` giving your best guess for the +wavevector of the `band`th mode at frequency `freq`. + +The return value of `get_mode_coefficients` is an array +of type `cdouble` (short for `std::complex`), +of length `2 * num_freqs * num_bands`, where `num_freqs` +is the number of frequencies stored in your `flux` object +(equal to `flux->Nfreq`) and `num_bands` is the length +of your `bands` input array. +The expansion coefficients $\{\alpha^+,\alpha^-\}$ +for the mode with frequency `nf` and band index `nb` +are stored sequentially starting at slot +`2*nb*num_freqs + nf` of this array: + +````c++ + std::vector coeffs=f.get_eigenmode_coefficient(...) + fields::get_eigenmode_coefficients(dft_flux *flux, + direction d, + const volume &where, + std::vector bands, + kpoint_func k_func=0, + void *user_data=0); + + int num_bands = bands.size(); + int num_freqs = Flux->Nfreq; + for(int nb=0; nb +$$ w(x) = + \begin{cases} + w_A, \qquad &x < -\frac{L}{2} \\[5pt] + T_p(x), \qquad &x \in \left[ -\frac{L}{2}, +\frac{L}{2}\right] \\[5pt] + w_B, \qquad &x > +\frac{L}{2} \\ + \end{cases} + \tag{3} +$$ +where the taper function $T_p(x)$ is a $C^{p}$ function, +i.e. $p$ is the index of its first discontinuous derivative. +For the cases $p=\{0,1,2\}$, the taper functions are +$$ T_p(x)=\begin{cases} + w_0 + \Delta \left(\frac{x}{L}\right), \qquad &p=0 \\[5pt] + w_0 + \Delta \Big[ \frac{3}{2}\left(\frac{x}{L}\right) + -2\left(\frac{x}{L}\right)^3 + \Big],\qquad&p=1,\\[5pt] + w_0 + \Delta \Big[ \frac{15}{8}\left(\frac{x}{L}\right) + -5\left(\frac{x}{L}\right)^3 + +6\left(\frac{x}{L}\right)^5 + \Big],\qquad&p=2 + \end{cases} +$$ +where +$$ w_0\equiv \frac{w_A+w_B}{2}, \qquad \Delta = w_B - w_A$$ +are the average and difference of the smaller and larger waveguide +thicknesses. + +Here are pictures of the $p=0,1,2$ taper geometries for the case of a +taper of length $L=4$ between waveguides of thickness $w_A=1$ +and $w_B=3$. (See below for the python code that produces these +plots.) + +

p=0 Taper

+ +![Taper2D_p0.png](ModeExpansionFiles/Taper2D_p0.png) + +

p=1 Taper

+ +![Taper2D_p1.png](ModeExpansionFiles/Taper2D_p1.png) + +

p=2 Taper

+ +![Taper2D_p2.png](ModeExpansionFiles/Taper2D_p2.png) + +In these figures, the dashed lines at $x=x_{A,B}$ +indicate the locations of cross-sectional planes +that we will use in our calculation: +the plane at $x=x_A$ is where we will place an +eigenmode source in our Meep calculation to describe +incoming power entering from the smaller waveguide, +while the plane at $x=x_B$ is where we will +tabulate the Fourier-domain fields in our Meep +calculation and determine their overlaps with +the eigenmodes of the larger waveguide to +compute mode-expansion coefficients. + +### User-defined material function + +Because the material geometries we will be studying here +are too complicated to be described as assemblies of +the [usual geometric primitives like blocks and cylinders](Python_User_Interface.md#GeometricObject), +we will instead write our own [user-defined material function](Python_User_Interface.md#material_function), which inputs the coordinates of a point in space +and fills in a [medium structure](Python_User_Interface.md#medium) +for the material properties at that point. +Actually, since the material geometry in this case involves a +spatially-varying but otherwise simple (isotropic, linear, lossless) +dielectric function, we can get away with the slightly simpler +[user-defined epsilon function](Python_User_Interface.md#epsilon_function), +for which we need only furnish a function of position that +returns a scalar relative permittivity. This is implemented +by the `my_eps_func()` routine in `wvg-taper.py;` note that +it invokes a subroutine `w_func` that evaluates [equation (3) above](ModeExpansion.md#Equation1) +to compute the $x$-dependent waveguide width $w(x)$. + +```python +################################################## +# x-dependent width of waveguide +################################################## +def w_func(x, L, p, wA, wB): + if L==0: + return wA if x<0 else wB + x0=x/L + if (x0 < -0.5): + return wA; + elif (x0 > +0.5): + return wB; + elif p==2: + return 0.5*(wA+wB) + (wB-wA)*x0*(15.0 + x0*x0*(-40.0 + x0*x0*48.0))/8.0; + elif p==1: + return 0.5*(wA+wB) + (wB-wA)*x0*(1.5 - 2.0*x0*x0); + else: # default t p==0, simple linear taper + return 0.5*(wA+wB) + (wB-wA)*x0; + +################################################## +# user-defined function for position-dependent material properties +################################################## +def my_eps_func(loc, L, p, wA, wB, eps_out, eps_in): + + if ( abs(loc.y) > 0.5*w_func(loc.x, L, p, wA, wB) ): + return eps_out; # outside waveguide + else: + return eps_in; # inside waveguide +``` + +We can pass `my_eps_func` as the value of the +`epsilon_func` keyword argument to the +[`Simulation` class constructor](Python_User_Interface.md#SimulationClass); +however, because this expects a function of just a single +argument (the spatial point), we use a `lambda` construction +to package the remaining arguments, i.e. something like + +```python +eps_func = lambda loc: my_eps_func(loc, L, p, wA, wB, + eps_ambient, eps_waveguide) + +sim=mp.Simulation( cell_size=mp.Vector3(2*LX, 2*LY), + resolution=resolution, + boundary_layers=[mp.PML(DPML)], + epsilon_func = eps_func + ) +``` + +The [`wvg-taper.py`](ModeExpansionFiles/wvg-taper.py) code defines +a class called `wvg-taper` that accepts keyword arguments for +various geometric parameters and instantiates a `Simulation` object +as in the code snippet above. For example, here's how we +made the pictures of the structures shown above: +a couple of examples involving waveguides and tapers of +various geometries: + +```python +>>> execfile("wvg-taper.py"); +>>> wt=wvg_taper(wA=1, wB=3, L=4, p=0); wt.plot_eps(); +>>> wt=wvg_taper(wA=1, wB=3, L=4, p=1); wt.plot_eps(); +>>> wt=wvg_taper(wA=1, wB=3, L=4, p=2); wt.plot_eps(); +``` +The `plot_eps()` class method that produces these plots +just calls [`Simulation.get_array`](Python_User_Interface.md) +to get a `numpy` array of ε values at the grid +points, then plots it using the `imshow` routine in +matplotlib: + +```python + def plot_eps(self): + + eps=self.sim.get_array(center = mp.Vector3(0,0), + size = self.sim.cell_size, + component = mp.Dielectric) + plt.figure() + plt.imshow(eps.transpose()) + plt.show(block=False) +``` + +### Visualizing eigenmode profiles + +Next, before doing any timestepping let's calculate +and plot the field profiles of some waveguide modes, +for both the smaller and larger waveguides. +This calculation is done by the `plot_modes` +function in the `wvg_taper` class; you can look +at the [full Python code](ModeExpansionFiles/wvg-taper.py) +to see how it's done in full detail, but +here is a synopsys: + ++ For the lowest-frequency ($n=1$) eigenmode of the + smaller waveguide, and for the first several + eigenmodes of the larger waveguide, we call the `meep::fields::get_eigenmode` + routine in libmeep. + This routine inputs a frequency (`fcen`), an integer (`nb`), and a + `meep::volume` specifying a cross-sectional slice through our + geometry, then invokes [MPB][MPB] to determine the `nb`th + eigenmode at frequency `fcen` for an *infinitely extended* + waveguide with constant cross section matching that of + our slice. For example, to compute the $n=1$ eigenmode + for an infinite waveguide whose cross section matches + that of our structure at $x=x_A$, we say + +````python +nb = 1; # want first eigenmode +vA = mp.volume( mp.vec(xA, -YP), mp.vec(xA,+YP) ) # cross section at x_A +modeA = f.get_eigenmode(fcen, mp.X, vA, vA, nb, k0, True, 0, 0.0, 1.0e-4) +```` + +The return value of `get_eigenmode` is a data structure +containing information on the computed eigenmode; for +example, to get the group velocity or propagation vector +of the mode you could say + +````python +vgrp = get_group_velocity(modeA); +k_vector = get_k(modeA) +```` + +Alternatively, you can call the `meep::fields::output_mode_fields` +routine to export the $\mathbf{E}$ and $\mathbf{H}$ +components of the eigenmode (at grid points lying in the cross-sectional +plane) to an HDF5 file, i.e. + +````python +f.output_mode_fields(modeA, fluxA, vA, "modeA"); +```` + +where `fluxA` is a [`meep::dft_flux`][DFTFlux] structure +created for the cross section described by `vA`. This will +create a file called `modeA.h5` containing values of +field components at grid points in `vA`. + ++ Having computed eigenmodes with `get_eigenmode` and written their + field profiles to HDF5 files with `output_mode_fields`, we + can read the data back in for postprocessing, such as (for example) + plotting eigenmode field profiles. This is done by the `plot_fields` + routine in [`wvg-taper.py`](ModeExpansionFiles/wvg-taper.py); + the basic flow looks something like this: + +````python + + # open HDF5 file + file = h5py.File("modeA.h5", 'r') + + # read array of Ey values on grid points + ey = file["ey" + suffix + ".r"][()] + 1j*file["ey" + suffix + ".i"][()]; + + # plot real part + plt.plot(np.real(Ey)) +```` + +The `plot_modes` routine in `wvg-taper.py` repeats this process for the +lowest mode $x_A$ (`ModeA1`) and the first several modes at $x_B$ +(`ModeB1...B6`) and plots the results: + + +![ModeProfiles2D.png](ModeExpansionFiles/ModeProfiles2D.png) + + +### Adding an eigenmode source and timestepping + +The next step is to add an *eigenmode source* inside the +smaller waveguide---that is, a collection of Meep point +sources, lying on the cross-sectional surface at $x_A$, +whose radiated fields reproduce the fields of a +given waveguide eigenmode at a given frequency: + + +````python +sources = [ mp.EigenModeSource(src=mp.GaussianSource(fcen, fwidth=df), + center=mp.Vector3(xA,0.0), + size=mp.Vector3(0.0,LY), + eig_band=band_num + ) + ] +self.sim=mp.Simulation( cell_size=mp.Vector3(LX, LY), + resolution=resolution, + boundary_layers=[mp.PML(DPML)], + force_complex_fields=True, + epsilon_func = eps_func, + sources=sources + ) +```` + +Next, we timestep to accumulate Fourier-domain fields +on a cross-sectional plane within the larger waveguide. +This is done by the `get_flux()` method in +[`wvg_taper.py](ModeExpansionFiles/wvg_taper.py). + +The timestepping continues until the instantaneous +Poynting flux through the flux plane at $x_B$ +has decayed to 0.1% of its maximum value. +When the timestepping is finished, the Fourier-domain +fields on the plane at $x_B$ are stored +in a [`dft_flux`](DFTFlux) object called `fluxB.` +and we can call `meep::fields::output_flux_fields` +to export the fields to an HDF5 file, similar to +`output_mode_fields` which we used above: + +````python +f.output_flux_fields(fluxB, vB, 'fluxB') +```` + +This produces a file called `fluxB.h5`. One slight +difference from `output_mode_fields` is that `dft_flux` +objects typically store field data for multiple +frequencies, so the field-component datasets +in the `HDF5` file have names like `ey_0.r`, `ey_1.i`. + +### Visualizing DFT fields + +Having written Fourier-transform fields to HDF5 +files, we can read in the data and plot, as we +did previously for mode profiles. In the `wvg_taper.py` +code this is again handled by the `plot_fields` +routine. +Here are the results of this process for +a few different values of the taper length $L$ and +smoothness index $p$: + +![FluxVsLP.png](ModeExpansionFiles/FluxVsLP.png) + +Take-home messages: + ++ For $L=0$ (no taper, i.e. abrupt junction) the fields + at $x_B$ look nothing like the fields of the lowest + eigenmode for the larger structure (second row + of [this plot](ModeExpansion.md#ModeProfiles)); + clearly there is significant contamination from + higher modes. ++ As we increase the taper length and the smoothness index + the fields at $x_B$ more and more closely resemble the + lowest eigenmode fields, indicating that the taper is + working to transfer power adiabatically from lowest mode + to lowest mode. + +### Making movies + +The `get_flux()` routine in the +[`wvg_taper.py`](ModeExpansionFiles/wvg_taper.py) +supports a keyword parameter +`frame_interval` which, if nonzero, defines an +interval (in meep time) at which images of the +instantaneous Poynting flux over the entire geometry +are to be written to `.h5` files. +The default is `frame_interval=0`, in which case +these images will not be written. + +If you specify (say) `frame_interval=1` +to `get_flux()` for a geometry with (say) taper +length $L=1.2$ and smoothness index $p=1$, you +will get approximately 100 files with names like +````bash + L1.2_p1_f1.png + L1.2_p1_f2.png + L1.2_p1_f3.png + ... + L1.2_p1_f105.png + ... +```` + +To assemble all these frame files into a movie +using [FFMPEG](https://www.ffmpeg.org/), go like this: + +````bash + # ffmpeg -i 'L1.2_p1_f%d.png' L1.2_p1.mpg +```` + +(Note that the string `%d` in the input filename +is a wildcard that will match all integer values; it needs +to be in single quotes to protect it from shell expansion.) + +Here are the movies for the various cases considered above: + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + +
TaperMovie
L=0 + + +
L=1, p=0 + + +
L=2, p=0 + + +
L=3, p=0 + + +
L=3, p=1 + + +
L=3, p=2 + + +
+

+ + + +### Extracting mode-expansion coefficients + +Finally, we call `get_mode_coefficients` to compute +the inner product of the Meep DFT fields in the larger waveguide +with each of a user-specified list of eigenmodes of the larger +waveguide to compute the fraction of the power carried by +each mode. + +### Intra-modal scattering losses vs. taper length and smoothness + +Repeating this calculation for many taper lengths $L$ and +smoothness indices $p=0,1$ yields the following plots +showing the rate of decay of inter-mode scattering losses +as the taper length $L\to\infty.$ + +![TaperData.png](ModeExpansionFiles/TaperData.png) + + +## Related computational routines + +Besides `get_eigenmode_coefficients,` there are a few +computational routines in `libmeep` that you may find useful +for problems like those considered above. + +### Routine for computing MPB eigenmodes (in `mpb.cpp`) +```` + void *fields::get_eigenmode(double &omega, + direction d, const volume &where, + const volume &eig_vol, + int band_num, + const vec &kpoint, bool match_frequency, + int parity, + double resolution, + double eigensolver_tol); +```` + +Calls MPB to compute the `band_num`th eigenmode at frequency `omega` +for the portion of your geometry lying in `where` (typically +a cross-sectional slice of a waveguide). `kpoint` is an initial +starting guess for what the propagation vector of the waveguide +mode will be. + +### Routines for working with MPB eigenmodes (in `mpb.cpp`) + +The return value of `get_eigenmode` is an opaque pointer to +a data structure storing information about the computed eigenmode, +which may be passed to the following routines: + +```` +// get a single component of the eigenmode field at a given point in space +std::complex eigenmode_amplitude(const vec &p, void *vedata, component c); + +// get the group velocity of the eigenmode +double get_group_velocity(void *vedata); + +// free all memory associated with the eigenmode +void destroy_eigenmode_data(void *vedata); +```` + +### Routines for exporting frequency-domain fields (in `dft.cpp`) + +```` + void output_flux_fields(dft_flux *flux, const volume where, + const char *HDF5FileName); + + void output_mode_fields(void *mode_data, dft_flux *flux, + const volume where, + const char *HDF5FileName); +```` + +`output_flux_fields` exports the components of the (frequency-domain) fields +stored in `flux` to an HDF5 file with the given file name. `where` is the +`volume` passed to the `flux` constructor. In general, `flux` will store +data for fields at multiple frequencies, each of which will + +`output_mode_fields` is similar, but instead exports the components of the eigenmode +described by `mode_data` (which should be the return value of a call to `get_eigenmode`). + + +### Routines for computing overlap integrals (in `dft.cpp`) +```` + std::complex get_mode_flux_overlap(void *mode_data, + dft_flux *flux, + int num_freq, + const volume where); + + std::complex get_mode_mode_overlap(void *mode1_data, + void *mode2_data, + dft_flux *flux, + const volume where); +```` + +`get_mode_flux_overlap` computes the overlap integral +(defined by [equation (*) above](#OverlapEquation)) +between the eigenmode described by `mode_data` +and the fields stored in `flux` (for the `num_freq`th stored +frequency, where `num_freq` ranges from 0 to `flux->Nfreq-1`.) +`mode_data` should be the return value of a previous call to +`get_eigenmode.` + +`get_mode_mode_overlap` is similar, but computes the overlap +integral between two eigenmodes. (`mode1_data` and `mode2_data` may be +identical, in which case you get the inner product of the +mode with itself; by the normalization convention used in MPB, +this should equal the group velocity of the mode.) + + +## Under the hood: How mode expansion works + +The theoretical basis of the mode-expansion algorithm +is the orthogonality relation satisfied by the normal +modes: +$$ \left\langle \mathbf{E}_m^{\sigma} \right| + \left. \mathbf{H}^\tau_n \right\rangle + =C_{m}\delta_{mn}\delta_{\sigma\tau} + \qquad \Big( \{\sigma,\tau\}\in\{+,-\}\Big) + \tag{4} +$$ +where the inner product involves an integration over +transverse coordinates: + +$$ \left\langle \mathbf{f} \right| \left. \mathbf{g} \right\rangle + \equiv + \int_{S} + \Big[ \mathbf{f}^*(\vec \rho) \times \mathbf{g}(\vec \rho)\Big] + \cdot \hat{\mathbf{n}} \, dA + \tag{5} +$$ +where $S$ is any surface transverse to the direction of propagation +and $\hat{\mathbf{n}}$ is the unit normal vector to $S$ (i.e. +just $\hat{\mathbf{z}}$ in the case considered above). +The normalization constant $C_{m}$ is a matter of convention, +but in [MPB][MPB] it is taken to be the group velocity of the +mode, $v_m$, times the area $A_S$ of the cross-sectional surface $S$: +$$ C_m = v_m A_S. \tag{6} $$ + +Now consider a Meep calculation in which we have accumulated +frequency-domain $\mathbf E^{\text{meep}}$ and $\mathbf H^{\text{meep}}$ +fields on a `dft-flux` +object located on a cross-sectional surface $S$. Invoking the +eigenmode expansion [(1)](ModeExpansion.md#EigenmodeExpansions) +and choosing (without loss of generality) the origin +of the $x$ axis to be the position of the cross-sectional plane, +the tangential components of the frequency-domain Meep fields +take the form +$$ \mathbf E^{\text{meep}}_\parallel + = \sum_{n} (\alpha_n^+ + \alpha_n^-)\mathbf{E}_{n\parallel}^+, + \tag{7} +$$ +$$ \mathbf H^{\text{meep}}_\parallel + = \sum_{n} (\alpha_n^+ - \alpha_n^-)\mathbf{H}_{n\parallel}^+, + \tag{8} +$$ +where we used the well-known relations between the tangential +components of the forward-traveling and backward-traveling field +modes: +$$ \mathbf{E}^+_{n\parallel} =+\mathbf{E}^-_{n\parallel}, + \qquad + \mathbf{H}^+_{n\parallel} =-\mathbf{H}^-_{n\parallel}. +$$ +Taking the inner product (5) of both sides of equations (7) and (8) +with the $\mathbf{H}$ and $\mathbf{E}$ fields of each eigenmode +and using equations (4) and (6), we find +$$ \left\langle \mathbf{H}_m + \right|\left. \mathbf{E}^{\text{meep}} \right\rangle + =+(\alpha_n^+ + \alpha_n^-) v_m A_S +$$ +$$ \left\langle \mathbf{E}_m + \right|\left. \mathbf{H}^{\text{meep}} \right\rangle + =-(\alpha_n^+ - \alpha_n^+) v_m A_S +$$ +Thus, by evaluating the integrals on the LHS of these equations---numerically, +using the MPB-computed eigenmode fields $\{\mathbf{E}, \mathbf{H}\}_m$ +and the Meep-computed fields $\{\mathbf{E}, \mathbf{H}\}^{\text{meep}}\}$ +as tabulated on the computational grid---and combining the results +appropriately, we can extract the coefficients $\{\alpha^\pm_m\}$ +in the expansion (1). This calculation is carried out by the +routine [`meep::fields::get_mode_flux_overlap`](ModeExpansion.md#OverlapRoutines). +(Although simple in principle, the implementation is complicated by +the fact that, in multi-processor calculations, the Meep fields +needed to evaluate the integrals are generally not all present +on any one processor, but are instead distributed over multiple +processors, requiring some interprocess communication to evaluate +the full integral.) + +The Poynting flux carried by the Meep fields (7,8) may be expressed +in the form +$$ S_x = \frac{1}{2}\text{Re } + \left\langle \mathbf{E}^{\text{meep}}\right| + \left. \mathbf{H}^{\text{meep}}\right\rangle + = \frac{1}{2}\sum_n \left\{ |\alpha_n^+|^2 - |\alpha_n^-|^2) \right\} v_n A_S +$$ +and thus the fractional power carried by any one (forward- or backward-traveling) +eigenmode is given by +$$ \text{fractional power carried by }\pm\text{-traveling mode }n= + \frac{|\alpha_n^\pm|^2 v_n A_S}{2S_x} +$$ + +[^1]: + The theory of waveguide modes is covered in many references; + one that we have found useful is [Snyder and Love, *Optical Waveguide Theory* (Springer, 1983)](http://www.springer.com/us/book/9780412099502). + + +[MPB]: https://mpb.readthedocs.io/en/latest/ +[DFTFlux]: https://meep.readthedocs.io/en/latest/Scheme_User_Interface/#Flux_spectra.md diff --git a/doc/docs/ModeExpansionFiles/FluxVsLP.png b/doc/docs/ModeExpansionFiles/FluxVsLP.png new file mode 100644 index 0000000000000000000000000000000000000000..f3c562c26d4d4ba23314b56e510f0c0885ff5afc GIT binary patch literal 198727 zcmeFZg;$l^8aKL36bV5DX$eUI3F(qlq#IEh1!<6yRJU}8qEgZz3aF%nNHFQK=57jo$nLBc;@m~R+PoXCd0;HFu1pFN~vNn*nAiaR^pk{ z@He5k5ry!_Ne78r>Sy5Pe#XQPzCUYwQ^x^=!8b(z9m^0)H^*QuV{S>^PCoKc=(S0`<;JMF~Psz z+$3rV{`VX4Bs1%i|9(d(?LhbMH(^Zwd#iso<9{3Y&rV?e7Z(4^tG|Qcf7RmOr1)RC z{r`It86R@&xksY#n&+0+@6y%QJg49N5-+1&+C(<*)|Tc!>|zn!`uR-ak&U&2fE4MMqa^rq`;WQXCrpy3%Nst`F z@KB9x)$)zz)lmhivmj`(+?C^bUGMQsYHZ!TQWrb-jlKu28M#jIerxl6+^>1eV=N@n 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