C ALCULATING PC W AVEGUIDE P ROPERTIESwiki.phy.queensu.ca/hughes/images/f/fd/MPB_tutorial.pdf · MPB uses Guile and Libctl to control its simulations. EVERYTHING YOU NEED TO KNOW

CALCULATING PC WAVEGUIDE PROPERTIES

Mark Patterson – 2009 October 29

PROBLEM DEFINITION

We want to calculate the properties of an ideal PC waveguide mode:

Dispersion: omega(k),

Group velocity: d omega(k) / d k,

Bloch mode electric and magnetic field.

COMPUTATIONAL METHODS

FDTD PWE

Calculate eigen-frequencies ✓ ✓

Calculate group velocity - ✓

Obtain fields - ✓

Sweep over k-space - ✓

Fast - ✓

Radiation or other loss ✓ ✗

Finite Size ✓ ✗

GUI Interface ✓ ✗

✓ easy to do. - difficult. ✗ don’t try.

COMPARISON OF BAND STRUCTURES

0 0.1 0.2 0.3 0.4 0.51.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8x 1014

Wavevector, k [2!/a]

Freq

uenc

y [H

z]

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Wavevector [2!/a]

Freq

uenc

y [c

/a]

MPB FDTD

FDTD

Lumerical FDTD (commercial) or Radiant (see Cole).

Necessary for structures with material loss, radiation loss, or finite size.

Using Fourier techniques, frequency resolution converges linearly with simulation time. Some improvements can be made using harminv (http://ab-initio.mit.edu/wiki/index.php/Harminv).

http://ab-initio.mit.edu/wiki/index.php/Harminv




LUMERICAL TUTORIALS

http://www.lumerical.com/fdtd_online_help/fdtd_online_help_summary.php

3D slab structure with square lattice: http://www.lumerical.com/fdtd_online_help/pc_bandstructure_3d_planar.php

2D structure with triangular lattice (supercell technique): http://www.lumerical.com/fdtd_online_help/pc_bandstructure_triangular_lattice.php





http://www.lumerical.com/fdtd_online_help/pc_bandstructure_3d_planar.php






http://www.lumerical.com/fdtd_online_help/pc_bandstructure_triangular_lattice.php






MPB

MPB: MIT Photonic Bands; uses the plane wave expansion technique.

Free (GPL) software written by Stephen Johnson and John Joannopoulos et al.

They also have an FDTD, MEEP.

Mature software: started development in 1999. Last updated in 2003.

Johnson still answers questions on the mailing list.

RESOURCES

Home page and user guide: http://ab-initio.mit.edu/wiki/index.php/MIT_Photonic_Bands

My notes and installing on Fedora: http://wiki.phy.queensu.ca/hughes/index.php/MPB

Questions: Mailing list (see home page).

Theory: Steven G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Optics Express 8, no. 3, 173-190 (2001). doi:10.1364/OE.8.000173.

http://ab-initio.mit.edu/wiki/index.php/MIT_Photonic_Bands




http://wiki.phy.queensu.ca/hughes/index.php/MPB

http://wiki.phy.queensu.ca/hughes/index.php/MPB

http://www.opticsinfobase.org/abstract.cfm?URI=oe-8-3-173






http://dx.doi.org/10.1364/OE.8.000173

http://dx.doi.org/10.1364/OE.8.000173

PLANE WAVE EXPANSION

Write Maxwell wave equation as an eigenvalue problem.

Expand the mode in plane waves (spatial Fourier transform) for evaluating the curls.

Use fancy matrix methods to calculate the eigen-frequencies and -vectors (field profiles).

!"!

1!r(r)

!"H(r, ")"

=#"

c

$2H(r, ")

! = minh

h†Ah

h†h

WAVEGUIDE SUPERCELL

The plane wave expansion technique models periodic media only.

For a waveguide, we use a supercell.

You cannot obtain radiation modes with this method.

UNITS

Lengths are dimensionless, usually one normalizes to the periodicity, a.

Frequencies are in units of c/a.

Wave vectors are in units of 2π/a. The band edge is at 0.5.

OVERVIEW

CTL Filempb Console

(frequency, group

velocity)

Command line

Command Line

mpbi

mpb-mpi

mpbi-mpi

HDF5 Files (Fields) MATLAB

Input Compute Output Analyse

OVERVIEW

CTL Filempb Console

(frequency, group

velocity)

Command line

Command Line

mpbi

mpb-mpi

mpbi-mpi



OVERVIEW

CTL Filempb Console

(frequency, group

velocity)

Command line

Command Line

mpbi

mpb-mpi

mpbi-mpi



OVERVIEW

CTL Filempb Console

(frequency, group

velocity)

Command line

Command Line

mpbi

mpb-mpi

mpbi-mpi



OVERVIEW

CTL Filempb Console

(frequency, group

velocity)

Command line

Command Line

mpbi

mpb-mpi

mpbi-mpi



MPB INPUT

WHAT LANGUAGE ARE WE USING?

Lisp is a very old (50 years) programming language.

Scheme is a dialect of Lisp.

GNU Guile is intended to provide a scripting and extensibility interface for libraries and programs. It is promoted by the GNU project and uses Scheme.

Libctl is an extension to Guile intended for running scientific simulations. It is written by Steven Johnson.

MPB uses Guile and Libctl to control its simulations.

E V E RY T H I N G Y O U N E E D T O K N O W A B O U T L I S P

SCHEME PRIMER

Comments begin with a semicolon:

Everything is a function:

… even creating and setting variables:

; This is a comment

(function-name value1 value2)

(define sq32 1) ; Define n and initialize value(set! sq32 (/ (sqrt 3) 2)) ; Change value

LIBCTL INPUT FILES

Usually, provide a *.ctl file to run the simulation. Think of the ctl file as a shell script, but with more parentheses.

Alternatively, mpb can be started interactively and controlled from a command prompt. You would manually type the commands from the ctl file.

Special variables are used as inputs to the simulation.

A ctl file can just run a simulation or be a complex script.

INPUT PARAMETERS

(define-param) is an alternative to (define). It allows the default value to be overridden on the command line.

Proper use of this means you can run lots of simulations with a single shared ctl file.

(define-param a 475e-9)(define-param h (/ 180e-9 a))

DESCRIBE THE SUPERCELL LATTICE

Set the geometry variable.

For a non-rectangular lattice, use basis1, basis2, basis3:

What I use for a waveguide:

(set! geometry-lattice (make lattice (size 1 1 (* 10 thickness)) (basis1 sq32 0.5) (basis2 sq32 -0.5)))

(set! geometry-lattice (make lattice (size 1 (+ (* supercell-yi sq32) (* 2 dy)) supercell-z)))

(set! geometry-lattice (make lattice (size 1 5 no-size)))







(set! geometry-lattice (make lattice (size 1 5 no-size)))

One of the special input variables







(set! geometry-lattice (make lattice (size 1 5 no-size))) Unit cell dimensions in units of the basis vectors.

One of the special input variables








One of the special input variablesFor a 2D structure, use no-size.
















Override default Cartesian basis.









Override default Cartesian basis.


DESCRIBE THE STRUCTURE

(set! geometry (list

(make block (material (make dielectric (index ind))) (size infinity infinity h) (center 0 0 0) )

(make cylinder (material air) (center 0.5 (+ (* 1 sq32) dy) 0) (radius r1) (height infinity) ) (make cylinder (material air) (center 0.5 (- (* -1 sq32) dy) 0) (radius r1) (height infinity) ) (make cylinder (material air) (center 0 (+ (* 2 sq32) dy) 0) (radius r ) (height infinity) ) (make cylinder (material air) (center 0 (- (* -2 sq32) dy) 0) (radius r ) (height infinity) )))

The structure is described by setting geometry:

Start with a block of dielectric:

And drill holes in it:















K-POINTS

The calculation is performed once for each Bloch wavevector specified in the k-points list variable.

(define Gamma (vector3 0 0 0))

(define K (lattice->reciprocal (vector3 0.5 0 0) ) )

(set! k-points (interpolate (- numk 2) (list Gamma K)))

Create vector3 variables for the symmetry points:

Remember, this is in reciprocal space:

Create a list of evenly sampled points.

K-POINTS

The calculation is performed once for each Bloch wavevector specified in the k-points list variable.

(define Gamma (vector3 0 0 0))

(define K (lattice->reciprocal (vector3 0.5 0 0) ) )

(set! k-points (interpolate (- numk 2) (list Gamma K)))

Create vector3 variables for the symmetry points:

Remember, this is in reciprocal space:

Create a list of evenly sampled points.

This list could have more than 2 entries.

OPTIONAL PARAMETERS

Number of grid points per lattice unit length:

Number of sub-cell points for calculating average ε:

Frequency eigensolver tolerance, a relative value:

Number of bands to solve for (you should always set this):

(set! resolution (vector3 24 24 24))

(set! mesh-size 10)

(set! tolerance 1e-8)

(set! num-bands 18)

RUN THE SIMULATION

The (run*) commands start the simulation.

It is useful to use alternate run commands to enforce a desired symmetry and e.g. suppress the TM modes.

You can have multiple (run*) commands in one file.

(run display-group-velocities fix-efield-phase output-efield)

(run-zeven) (run-yodd) (run-te)

RUN THE SIMULATION





Run command


RUN THE SIMULATION





Run command Commands to execute after solving each k-point.


RUN THE SIMULATION







RUN THE SIMULATION







INTERPRETING SYMMETRIES

ADVANCED SCRIPTING

You can do full blown scripting within the ctl file. I have examples for programatically generating device geometry and fitting the band structure to experiments by tweaking the geometry.

Libctl has support for 3-vectors 3x3 matrices, complex numbers, minimization, root finding, derivatives and integrals.

MPB can compute field energy, energy in objects, and do math with fields.

CTL BEST PRACTICE

Use (define-param) liberally and make the ctl file generic.

Always display-group-velocities.

Always output-efield.

Recycle code.

Don’t use the targeted eigensolver. It is slow and inaccurate.

RUNNING SIMULATIONS

RUNNING SIMULATIONS (COMMAND LINE)

mpb r=0.3 simulation.ctl > simulation.out

mpb-split 2 r=0.3 simulation.ctl > simulation.out

Starts 2 serial jobs, each with half of the k-points.

mpbi, mpbi-splitAssumes ε(-x) = ε*(x)Twice as fast, half the memory.No error checking!






Capture the console output.






Override a (define-param) variable. Always check this.
















RUNNING SIMULATION IN PARALLEL

MPB is parallelized (MPI). Use mpb-mpi and mpbi-mpi.

Sometimes it crashes the node, but Cole said he’d fix it.

Running across nodes tends to be slow.

This transposes x and y.

OUTPUT

OUTPUT

Prints the frequencies and, if specified, group velocity to the console.

Saves dielectic data to epsilon.h5 file. If specified, creates files for E, D, and H field files.

e.g.: e.k05.b16.zeven.h5

Always check the epsilon.h5 file.

Delete the extra field files promptly.

HDF5 FILES

The fields are save in HDF5 files.

A data file for structured binary data.

Modern MAT files are actually HDF5 files in disguise.

Tools:Use h5utils and mpb-data on command line.Use hdf5* in MATLAB.Use PyTables in Python.

ANALYSIS

COMMAND LINE TOOLS

h5ls

Examine the structure of the data file.

mpb-data

Set-up derived fields such as converting to a rectangular grid.

h5topng

Visualize fields on slices through the volume.

H5TOPNG

Part of h5utils.

Useful for quick checks of the structure.

Read the man page.

h5topng -0x0 epsilon.h5

MATLAB TOOLBOX

See the wiki to get the source.

Provides the MPBBandstructure class for parsing console output and loading fields. Start by creating a new object based on an output file:

mpb = MPBBandstructure(file, 'zeven') ;

Some functions have implicit assumptions about a rectangular supercell.

If you used (run-zeven).

MPBBANDSTRUCTURE OBJECT

>> display(mpb) ;MPBBandstructure object: k: [5x1 double] kVect: [5x3 double] freqs: [5x18 double] vg: [5x18 double] vgVect: [5x 3x18 double] numK: 5 numBands: 18 pathroot: '/home/mark/Masters/Matlab Library/MPB/

test_data/waveguide' filename: 'waveguide.out' symmetry: 'zeven' isMPI: false interchange: [0x0 struct] vars: [1x1 struct]

PLOTTING BAND STRUCTURES

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Wavevector [2!/a]

Freq

uenc

y [c

/a]

clf ;plotBands(mpb) ; % Plot all the bandsplotBands(mpb, 13, 'r') ; % Highlight the waveguide band

epsilon.h5: epsilon, x=centre+0

y

z

epsilon.h5: epsilon, x=1

y

z

epsilon.h5: epsilon, y=centre+0

x

z

epsilon.h5: epsilon, z=centre+0

xy

VISUALIZING STRUCTURESplotFields3D(mpb, 'epsilon') ;

e.k42.b13.zeven.h5: e, x=centre+0

y

z

e.k42.b13.zeven.h5: e, x=1

y

z

e.k42.b13.zeven.h5: e, y=centre+0

x

z

e.k42.b13.zeven.h5: e, z=centre+0

xy

VISUALIZING FIELDS

plotFields3D(mpb, mpb.numK, 13, 'e') ;

USING THE FIELDS

Use readField() to load epsilon and field data.

Returns a 4D (x-by-y-by-z-by-component) matrix.

Takes care of transposition for MPI runs and band interchanges.

>> epsilon = readField(mpb, 'epsilon') ;>> epsilonCentre = ...

epsilon(ceil(end/2), ceil(end/2), ceil(end/2)) ;>> disp(['The membrane has a dielectric constant of ' ...

num2str(epsilonCentre) '.']) ;The membrane has a dielectric constant of 12.0826.

USING THE FIELDS 2

Use getScale() to get the coordinates.

>> [scaleX scaleY scaleZ] = getScale(mpb) ;>> e = readField(mpb, ki, band, 'e') ;>> ey = e(:,:,:,2) ;% I could also have used ey = readField(mpb, ki, band, 'y') ;>> [maxi maxi] = max(ey(:)) ;>> [xi yi zi] = ind2sub(size(ey), maxi) ;>> disp(['The maximum value of the electric field ' ...

'y-component occurs at (x,y,z) = (' ...num2str(scaleX(xi)) ', ' num2str(scaleY(yi)) ', ' ...num2str(scaleZ(zi)) ').']) ;

The maximum value of the electric field y-component occurs at (x,y,z) = (0, 0.3997, 0).

USER DATA

It can be useful to associate data (parameters, band index) with the output file.

Add (usually by hand) to the top of the output file:

These will be automatically loaded into MATLAB:

POSTPROCESS: Set vars double a to 4.14e-7POSTPROCESS: Set vars double r to 0.286POSTPROCESS: Set vars integer band to 13

>> disp(mpb.vars)

a: 4.1400e-07 r: 0.2860 band: 13

INTERCHANGE BANDS

25 30 35 40 45 500.26

0.265

0.27

0.275

0.28

0.285

0.29

0.295

0.3

0.305

Wave vector index

Freq

uenc

y [c

/a]

25 30 35 40 45 500.26

0.265

0.27

0.275

0.28

0.285

0.29

0.295

0.3

0.305

Wave vector index

Freq

uenc

y [c

/a]

POSTPROCESS: Swap bands 7 8 indices 32 to 34

Without Swap Command With Swap Command

FLAG MPI

Did you remember that you have to transpose the fields if you use mpb-mpi?

POSTPROCESS: MPI

MISC. ADVICE

Save parameters and the band index with output files.

The POSTPROCESS lines must be at the top.

Use MPBBandstructures and inputs to later calculations.

If you are interpolating fields between k-points, be careful of the phase.

Cache the fields if you load them a lot.

QUESTIONS?

C ALCULATING PC W AVEGUIDE P ROPERTIESwiki.phy.queensu.ca/hughes/images/f/fd/MPB_tutorial.pdf · MPB uses Guile and Libctl to control its simulations. EVERYTHING YOU NEED TO KNOW

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