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<td><img src="tri-rods-ez1.gif" border="1" width="83" height="96"></td>
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<td><img src="tri-rods-ez6.gif" border="1" width="83" height="96"></td>
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<p>Your images should look the same as the ones above.  If we hadn't
included <code>fix-efield-phase</code> before
<code>output-efield-z</code> in the ctl file, on the other hand, yours
would have differed slightly (e.g. by a sign or a lattice shift),
because by default the phase is <em>random</em>.

<p>When we look at the real parts of the fields, we are really looking
at the fields of the modes at a particular instant in time (and the
imaginary part is half a period later).  The point in time (relative
to the periodic oscillation of the state) is determined by the phase
of the eigenstate.  The <code>fix-efield-phase</code> band function
picks a canonical phase for the eigenstate, giving us a deterministic
picture.

<p>We can see several things from these plots:

<p>First, the origin of the band gap is apparent.  The lowest band is
concentrated within the dielectric rods in order to minimize its
frequency.  The next bands, in order to be orthogonal, are forced to
have a node within the rods, imposing a large "kinetic energy" (and/or
"potential energy") cost and hence a gap.  Successive bands have more
and more complex nodal structures in order to maintain orthogonality.
(The contrasting absence of a large TE gap has to do with boundary
conditions.  The perpendicular component of the displacement field
must be continuous across the dielectric boundary, but the parallel
component need not be.)

<p>We can also see the deep impact of symmetry on the states.  The K
point has C<sub>3v</sub> symmetry (not quite the full C<sub>6v</sub>
symmetry of the dielectric structure).  This symmetry group has only
one two-dimensional representation--that is what gives rise to the
degenerate pairs of states (2/3, 4/5, and 7/8), all of which fall into
this "p-like" category (where the states transform like two orthogonal
dipole field patterns, essentially).  The other two bands, 1 and 6,
transform under the trivial "s-like" representation (with band 6 just
a higher-order version of 1).

<h2><a name="diamond">Diamond Lattice of Spheres</a></h2>

<blockquote>
"Then were the entrances of this world made narrow, full of sorrow and
travail: they are but few and evil, full of perils, and very
painful." (<i>Ezra</i> 4:7)
</blockquote>

<p>Now, let us turn to a three-dimensional structure, a diamond
lattice of dielectric spheres in air.  The basic techniques to compute
and analyze the modes of this structure are the same as in two
dimensions, but of course, everything becomes more complicated in 3d.
It's harder to find a structure with a complete gap, the modes are no
longer polarized, the computations are far bigger, and visualization
is much more difficult, for starters.

<p>(The band gap of the diamond structure was first identified in:
K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic
gap in periodic dielectric structures," <i>Phys. Rev. Lett.</i>
<b>65</b>, 3152 (1990).)

<p>The control file for this calculation, which can also be found in
<code>mpb-ctl/examples/diamond.ctl</code>, consists of:

<h3><a name="diamond-ctl">Diamond control file</a></h3>

<pre>
(set! geometry-lattice (make lattice
                         (basis-size (sqrt 0.5) (sqrt 0.5) (sqrt 0.5))
                         (basis1 0 1 1)
                         (basis2 1 0 1)
                         (basis3 1 1 0)))

; Corners of the irreducible Brillouin zone for the fcc lattice,
; in a canonical order:
(set! k-points (interpolate 4 (list
                               (vector3 0 0.5 0.5)            ; X
                               (vector3 0 0.625 0.375)        ; U
                               (vector3 0 0.5 0)              ; L
                               (vector3 0 0 0)                ; Gamma
                               (vector3 0 0.5 0.5)            ; X
                               (vector3 0.25 0.75 0.5)        ; W
                               (vector3 0.375 0.75 0.375))))  ; K

; define a couple of parameters (which we can set from the command-line)
(define-param eps 11.56) ; the dielectric constant of the spheres
(define-param r 0.25)    ; the radius of the spheres

(define diel (make dielectric (epsilon eps)))

; A diamond lattice has two "atoms" per unit cell:
(set! geometry (list (make sphere (center 0.125 0.125 0.125) (radius r)
                           (material diel))
                     (make sphere (center -0.125 -0.125 -0.125) (radius r)
                           (material diel))))

; (A simple fcc lattice would have only one sphere/object at the origin.)

(set-param! resolution 16) ; use a 16x16x16 grid
(set-param! mesh-size 5)
(set-param! num-bands 5)

; run calculation, outputting electric-field energy density at the U point:
(run (output-at-kpoint (vector3 0 0.625 0.375) output-dpwr))
</pre>

<p>As before, run the calculation, directing the output to a file.
This will take a few minutes (2 minutes on our Pentium-II); we'll put
it in the background with <code>nohup</code> so that it will finish
even if we log out:

<pre>
unix% nohup mpb diamond.ctl &gt;&amp; diamond.out &amp;
</pre>

<p>Note that, because we used <code>define-param</code> and <code>set-param!</code> to define/set some
variables (see the <a
href="http://ab-initio.mit.edu/libctl/doc/user-ref.html">libctl
manual</a>), we can change them from the command line.  For example,
to use a radius of 0.3 and a resolution of 20, we can just type <code>mpb
r=0.3 resolution=20 diamond.ctl</code>.  This is an extremely useful feature,
because it allows you to use one generic control file for many
variations on the same structure.

<h3><a name="fccunits">Important note on units for the diamond/fcc
lattice</a></h3>

<p><a href="user-tutorial.html#units">As usual,</a> all distances are
in the "dimensionless" units determined by the length of the lattice
vectors.  We refer to these units as <i>a</i>, and frequencies are
given in units of <i>c/a</i>.  By default, the lattice/basis vectors are
unit vectors, but in the case of fcc lattices this conflicts with the
convention in the literature.  In particular, the canonical <i>a</i>
for fcc is the edge-length of a cubic supercell containing the lattice.

<p>In order to follow this convention, we set the length of our basis
vectors appropriately using the <code>basis-size</code> property of
<code>geometry-lattice</code>.  (The lattice vectors default to the
same length as the basis vectors.)  If the cubic supercell edge has
unit length (<i>a</i>), then the fcc lattice vectors have length
sqrt(0.5), or <code>(sqrt 0.5)</code> in Scheme.

<h3><a name="diamond-bands">Gaps and and band diagram for the diamond lattice</a></h3>

<p>The diamond lattice has a complete band gap:

<pre>
unix% grep Gap diamond.out
Gap from band 2 (0.396348703007373) to band 3 (0.440813418580596), 10.6227251392791%
</pre>

<p>We can also plot its band diagram, much as for the tri-rods case
except that now we can't classify the bands by polarization.

<pre>
unix% grep freqs diamond.out > diamond.dat
</pre>

<p>The resulting band diagram, with the complete band gap shaded
yellow, is shown below.  Note that we only computed 5 bands, so in
reality the upper portion of the plot would contain a lot more bands
(which are of less interest than the bands adjoining the gap).

<p align="center"><img src="diamond-bands.gif" width="446"
height="346" alt="diamond band diagram">

<h3><a name="diamond-vis">Visualizing the diamond lattice structure and bands</a></h3>

<p>Visualizing fields in a useful way for general three-dimensional
structures is fairly difficult, but we'll show you what we can with
the help of the free <a
href="http://www.ssec.wisc.edu/~billh/vis5d.html">Vis5D</a>
volumetric-visualization program, and the <code>h5tov5d</code>
conversion program from <a
href="http://ab-initio.mit.edu/h5utils/">h5utils</a>.

<p>First, of course, we've got to rectangularize the unit cell using
<code>mpb-data</code>, as before.  We'll also expand it to two
periods in each direction.

<pre>
unix% mpb-data -m 2 -r -n 32 epsilon.h5 dpwr.k06.b*.h5
</pre>

<p>Then, we'll use <code>h5tov5d</code> to convert the resulting
datasets to Vis5D format, joining all the datasets into a single file
(<code>diamond.v5d</code>) so that we can view them simultaneously if
we want to:

<pre>
unix% h5tov5d -o diamond.v5d -d data-new epsilon.h5 dpwr.k06.b*.h5
</pre>

<p>Note that all of the datasets are named <code>data-new</code> (from
the original datasets called <code>data</code>) since we are looking
at scalar data (the time-averaged electric-field energy density).  No
messy field components or real and imaginary parts this time; we have
enough to deal with already.

<p>Now we can open the file with Vis5D and play around with various
plots of the data:

<pre>
unix% vis5d diamond.v5d &amp;
</pre>

<p>If you stare at the dielectric function long enough from various
angles, you can convince yourself that it is a diamond lattice:

<p align="center"><img src="diamond-eps.gif" width="394" height="394" alt="diamond epsilon plot">

<p>The lowest two bands have their fields concentrated within the
spheres as you might expect, flowing along more-or-less linear paths.
The second band differs from the first mainly by the orientation of
its field paths.  The fields for the first band at U are depicted
below, with the strongest fields (highest energy density) shown as the
most opaque, blue pixels.  Next to it is the same plot but with an
isosurface at the boundary of the dielectric superimposed, so you can
see that the energy is concentrated inside the dielectric.

<p align="center">
<img src="diamond-b1.gif" width="394" height="394" alt="diamond band 1 plot">
<img src="diamond-b1-eps.gif" width="394" height="394" alt="diamond band 1 plot, epsilon superimposed">

<p>The first band above the gap is band 3.  Its field energy densities
are depicted below in the same manner as above.  The field patterns
are considerably harder to make out than for the lower band, but they
seem to be more diffuse and "clumpy," the latter likely indicating the
expected field oscillations for orthogonality with the lower bands.

<p align="center">
<img src="diamond-b3.gif" width="394" height="394" alt="diamond band 3 plot">
<img src="diamond-b3-eps.gif" width="394" height="394" alt="diamond band 3 plot, epsilon superimposed">

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