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[Slide 1] Hey everyone.

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Welcome back.

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So this is lecture 1.22 and this is going to
be a summary of the overall Unit 1.

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Kind of looking back at what did we learn
from all the information that I presented.

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[Slide 2] So the first really big take home
message of this unit is that

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we can formulate an Eigenproblem for
Maxwell's equations.

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And that that Eigenproblem can be
solved using the Bloch Theorem.

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So why is that so important?

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First of all, that will allow us to
predict the relationship between

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frequency and wave vector and then it will
also tell us a lot about if we have any

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kind of unusual or non-trivial behavior
such as photonic bandgaps in the system.

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And then that of course can lead to lots
of different applications down the road,

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which I'll talk in a little
more detail about later.

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But for now, we can just
basically first of all write down

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the Maxwell Eigenproblem, which again
is just a combination of a Faraday's law and

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Amphere's Law with Maxwell's correction.

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And enforcing the transversality of each
because there are no magnetic mono poles.

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And then we can use the Bloch solution
which is just a combination of

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a periodic function for each.

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Which can be also written as
a Fourier series times a plane wave.

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And then that allows us to turn what
otherwise looks like a very unwieldy 

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analytical problem into
a matrix algebra type problem.

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And then you can see that actually
the matrix formulation kind of looks this.

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And then we can actually solve
that using a tool that's already

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in existence called MIT Photonic Bands.

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[Slide 3] And you can access it at this website,
jdj.mit.edu/mpb.

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And then once we have that capability then
we can start to calculate all kinds of

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things when we have
periodic dielectric media.

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So, if we start with 2D structures,

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which we could say are kinda the first
real innovations in photonic crystals,

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given that 1D periodic structures have
been known for more than a century.

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Then we find that we get
actually potentially photonic

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bandgaps both in transverse-electric and
transverse-magnetic

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polarizations which basically
correspond to E fields or

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H fields being in the plane of
the 2D Photonic Crystal structure.

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And in particular, if we have a triangular
lattice of holes that are punched

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into high index medium, we find that
in this special region of radii,

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we can actually get fairly significant
bandgaps and we can actually tune

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the bandgap based on the radius as well,
both for TE and TM at the same time.

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[Slide 4] And then second we actually find
that we can have 1D periodic

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structures that are etched
into index-guided structures.

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And then this gives rise to a set
of bandgaps that are below what

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we call the light line.

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And then when we're below the light
line in this yellow region,

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then we can actually also
add defects on top of that.

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And then these defects create
strongly localized modes.

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And they're partially localized
because of the photonic bandgap or

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periodicity in one direction.

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But also partially localized by the index
guiding effect which creates confinement

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at one or more bandgap frequencies which
are somewhere in the yellow region.

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And they can be tuned based on the nature
of the defect that you introduce.

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[Slide 5] And then we also found something
very similar holds if we have a 2D

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structure that has a finite
thickness with periodicity in the

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x and y directions which we
call photonic crystal slab.

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And then in the photonic crystal slab
again we have a bandgap which is

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depicted in this case in blue and
then we can also again create defects.

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In this case I've created a whole
array of defects in a line and so

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this creates kind of a wave guide.

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And then this wave guide actually can
be index guided out of the board and

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you can see that here in
the far right illustration,

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that it exponentially decays up and
down in the z direction.

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And at the same time it can flow
smoothly in the lateral directions.

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And you can even adjust

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the direction of the defects to
go wherever you need it to go.

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[Slide 6] And then also we found that we can create
3D photonic crystals out of composites

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of different 2D structures that are
stacked layer-by-layer into 3D capability.

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And they have actually true 3D photonic
bandgaps in all possible directions and

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polarizations.

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And then we also found that
we can have confinement,

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not just in the lateral direction,
but also in the vertical direction.

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Even if we were to remove
the high index materials.

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So, we could actually get away
from relying on refractive index

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wave guiding in this 3D structure,
and have a true 3D confinement,

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with very high quality factor and
very low volume.

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[Slide 7] And then in terms of the solvers that
we talked about, the key methodology,

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was using MPB which essentially
reformulates the essential Eigenproblem,

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which we mentioned at the beginning
in a very efficient way.

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And it essentially rewrites the H field
as a discreet Fourier Transform problem.

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And then that can be solved using a
combination of fast Fourier transforms and

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then inverse fast Fourier transforms.

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And it's augmented by
tensor-based averaging.

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And essentially afterwards,
performs a conjugate gradient minimization

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of a block Rayleigh quotient, which is to
related to Rayleigh-Ritz minimization.

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Which is very common variational
method in quantum mechanics.

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So in short we showed that it's easy
to calculate band structures and

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we can learn a lot from them.